Renée El

Melhem Bat

Blaise Pascal

R Gourdon

M Jean

Marc Lancelin

M Gérard Scorletti
email: gerard.scorletti@ec-lyon.fr

Isabelle Mme

Von Bueltzingloewen

Keywords: RC structures, RC slabs, impact at low velocity, dynamic, material nonlinear behaviors, deterministic approaches, reliability approaches, failure probability Structures en béton armé, dalles en béton armé, impact à faible vitesse, dynamique, comportement nonlinéaire des matériaux, approches déterministes, approches abilistes, probabilité de défaillance output variable . . . . . . . . . . . . . . . . . . . . . . 6.5.4.2 Dispersion

Reinforced concrete (RC) structures are subjected to several sources of uncertainties that highly aect their response. These uncertainties are related to the structure geometry, material properties and the loads applied. The lack of knowledge on the potential load, as well as the uncertainties related to the structure features shows that the design of RC structures could be made in a reliability framework. This latter allows propagating uncertainties in the deterministic analysis. However, in order to compute failure probability according to one or several failure criteria mechanical and stochastic models have to be coupled, which can be very time consuming and in some cases impossible. Indeed, either the complexity of the deterministic model considered implies important computing time (from minutes to hours) or reliability methods evaluating failure probability require a too large number of simulations of the deterministic model.

The platform OpenTURNS is used to perform the reliability analysis of three dierent structures considered in the present study and propagate uncertainties in their physical models. OpenTURNS can be linked to any external software using the generic wrapper. Therefore, OpenTURNS is coupled to CASTEM to study the reliability of a RC multiber cantilever beam subjected to a concentrated load at the free end, to Abaqus to study the reliability of RC slabs which are subjected to accidental impacts, and to ASTER to study the reliability of a prestressed concrete containment building.

Among structures considered in this study, only the physical problem of reinforced concrete (RC) slabs subjected to impact is investigated in detail. The design of such type of structures is generally carried out under static or pseudo-static loading. In the case of dynamic loads such as impacts, the force applied on the structural member is often assessed from energetic approaches and then, for the sake of simplicity, an equivalent pseudo-static loading is considered for its design. However, in some particular cases, the dynamic response of the structure cannot be simplied to a quasi-static response. The transient dynamic analysis has to be performed accounting for the main physical processes involved. Thus accurate models are needed to describe and to predict the capacity of the RC member and in particular the impact force.

This study focuses on RC slabs impacted by adropped object impact during handling operations within nuclear plant buildings. When a dropped object impacts a RC slab, damage can arise at the impact zone depending on the impact energy and the relative masses of the colliding bodies. An optimal design requires taking into account the potential development of nonlinearities due to the material damage (concrete cracking, steel yielding, etc.).

The aim of this study is to address the issue of reliability computational eort. Two strategies are proposed for the application of impacted RC slabs. The rst one consists in using deterministic analytical models which predict accurately the response of the slab. In general, the analytical model assumptions are constraining and only few slab congurations can be studied. The objective is to reduce the computational cost to the minimum and to explore as far as possible how to use these types of analytical models. In the opposite case, when nite element models are needed, the second strategy consists in reducing the number of simulations needed to assess failure probability or substituting the nite element model by a meta-model less expensive in computational time.

The rst part of this study describes the behavior of reinforced concrete slabs subjected to impact and determines the failure modes of slabs according to their characteristics and impact conditions.

this study only low velocities are considered. Then two deterministic models are used and evaluated: a 3D nite element model simulated with the commercial code Abaqus/Explicit and an analytical mass-spring model. For the simulation based on nite element method (FEM), the RC slab and the impactor are fully modeled. The steel is modeled as an elasto-plastic material with hardening and, the concrete behavior is described by the damaged plasticity model. The model is validated with experiments carried out on several RC slabs under drop-weight loads at Heriot-Watt University by Chen and May [START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF]. As an alternative modeling, a simplied analytical model of the slab is used.

The model consists in a two degrees of freedom mass-spring system which accounts for potential viscous damping. The rst (resp. the second) spring represents the stiness of the slab (resp. the contact). A frequency decrease approach is used to describe the degradation of RC slabs stiness.

The natural frequency of the slab is updated as a function of the maximal displacement reached during the loading.

Next, a reliability analysis of reinforced concrete slabs under low velocity impact is performed.

Firstly, the choice of random variable inputs and their distributions, failure criteria and probabilistic methods are discussed. In the case where the nonlinear behavior of materials is considered in the model, the failure criterion is dened according to the maximum displacement of the slab at the impact point. Finally, the probabilistic method dedicated to assess failure probability is chosen. This latter has to be adapted in order to reduce the computational cost and keeping a good accuracy.

The Monte Carlo and importance sampling methods are coupled with the mass-spring model, while FORM is used with the nite element model. The two strategies used to reduce the computational cost of a reliability analysis are compared in order to verify their eciency to calculate the probability of failure. Finally, a parametric study is performed to identify the inuence of deterministic model parameters on the calculation of failure probability (dimensions of slabs, impact velocity and mass, boundary conditions, impact point, reinforcement density). This study helps to locate the most critical impact points for the slab design, and illustrate how to optimize the design of reinforced concrete slabs under impact in terms of material choice, concrete properties and ratio of reinforcement.

Résumé

Dans le domaine du génie civil, le dimensionnement des structures en béton armé est essentiellement basé sur des démarches déterministes. Cependant, les informations fournies par des analyses déterministes sont insusantes pour étudier la variabilité de la réponse de la dalle. En outre, la réponse des structures en béton armé sont fortement inuencés par de nombreuses sources d'incertitudes. Le manque de connaissance des charges appliquées ainsi que les incertitudes liées à la géométrie de la dalle et les caractéristiques des matériaux nécessitent donc l'utilisation d'une approche abiliste qui permet la propagation de ces incertitudes dans les analyses déterministes. De nombreuses méthodes basées sur la théorie des probabilités ont été développées, elles permettent d'une part de calculer la probabilité de défaillance des structures et, d'autre part, d'étudier l'inuence de la variabilité des variables de conception sur le comportement de la structure étudiée. L'approche abiliste est basée sur le principe de couplage mécano-abiliste qui consiste à coupler un modèle stochastique, contenant les caractéristiques probabilistes des variables aléatoires d'entrée, et un modèle déterministe, permettant d'obtenir une estimation quantitative des variables de sortie ainsi que d'évaluer les fonctions de performance qui décrivent les états défaillants de la structure étudiée. Cependant un couplage mécano-abiliste peut être très exigeant en temps de calcul, soit à cause des modèles déterministes dont un seul lancement peut être de l'ordre de quelques minutes à quelques heures en fonction de son degré de complexité, soit à cause des méthodes d'évaluation de la probabilité de défaillance qui exigent parfois un nombre d'appels très important au modèle déterministe.

Dans cette étude, il est proposé de s'appuyer sur la plateforme généraliste de traitements des incertitudes OpenTURNS, développé par EDF R&D, EADS et la société PHIMECA pour eectuer les analyses abilistes. Cette plateforme open source, permet de nombreuses méthodes de traitement des incertitudes, Monte Carlo, Cumul Quadratique, méthodes de abilité FORM/SORM, tirage d'importance, ..., qui méritent d'être testées dans le cadre d'applications du génie civil. Cette plateforme possède une interface qui facilite le couplage avec n'importe quel logiciel externe. Dans le cadre de cette thèse, la méthodologie propre aux problématiques des ouvrages du génie civil est développée et validée tout d'abord sur un cas simple de structures en béton armé. Le cas d'une poutre encastrée en béton armée est proposé. Le système est modélisé sous CASTEM par une approche aux éléments nis de type multibre. Puis la abilité d'une dalle en béton armé impactée par une masse rigide à faible vitesse est étudiée en couplant OpenTURNS à Abaqus. Enn, une enceinte de connement en béton précontrainte modélisée sous ASTER est étudiée d'un point de vue probabiliste.

Parmi les structures considérées dans cette étude, seul le problème physique des dalles en béton armé soumises à impact est examiné en détail. Le problème d'impact fait partie des problèmes dynamiques nonlinéaires les plus diciles parce qu'il pose le problème du contact accompagné par une variation temporelle des vitesses. Le comportement d'une dalle en béton armé soumise à impact est très complexe. Sa réponse peut s'exprimer en deux termes: la réponse globale et la réponse locale.

Le cas où l'impacteur a une vitesse initiale importante, la dalle a une réponse locale caractérisée par un endommagement qui se localise près de la zone d'impact où la pénétration, la perforation et l'écaillage ont lieu. Avec une vitesse initiale faible, la réponse globale se caractérise par la ssuration progressive des faces comprimée et tendue, l'endommagement est lié à une réponse en mode de v défaillance est aussi essentiel dans une étude abiliste, un seul critère de défaillance est étudié dans le cas où le comportement nonlinéaire des matériaux est considéré dans le modèle. Le critère est lié au déplacement maximal que peut subir la dalle au point d'impact. Le seuil de déplacement est choisi de façon à ne pas avoir de plastication dans les armatures ou au moment de l'apparition des ssures dans le béton. La dernière étape à considérer dans une étude abiliste nécessite à bien choisir la méthode probabiliste à utiliser pour calculer la probabilité de défaillance, elle doit être adaptée au type du modèle déterministe et pour des temps de calcul optimisés. Dans cette étude, nous avons couplé les méthodes Monte Carlo et simulation d'importance avec le modèle de type masse-ressort.

FORM est utilisée avec le modèle par éléments nis, ce qui permet un nombre d'appels réduit au modèle par éléments nis.

Les deux stratégies utilisées pour réduire le temps de calcul d'une étude abiliste sont comparées an de vérier l'ecacité de chacune pour calculer la probabilité de défaillance. Enn, une étude paramétrique est réalisée an d'étudier l'eet des paramètres d'entrées des modèles déterministes sur le calcul de la probabilité de défaillance. Nous avons choisi dans cette étude de considérer les paramètres liés aux dimensions de la dalle, la vitesse d'impact, les conditions aux limites de la dalle, la position du point d'impact et le taux de ferraillage. Cette étude permet de localiser les points d'impact les plus défavorables pour la réponse de la structure, et d'illustrer les moyens d'optimiser le dimensionnement des dalles en béton armé sous impact en termes de choix des matériaux, carac-

2.2

Accidental dropped object impact during handling operations within nuclear plant buildings

2.3

Response types during impact on plates according to [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF] . . . . . . . . . . . . . . . .

2.4

Structure and impactor deformations according to the impact type [START_REF] Daudeville | Concrete structures under impact[END_REF] : a) Initial conditions (t=0), b) Hard impact (t>0), c) Soft impact (t>0) . . . . . . . . . . . . . . . . .

2.5

Mass-spring model proposed by Eibl [START_REF] Eibl | Soft and hard impact[END_REF] to simulate an impact [START_REF] Daudeville | Concrete structures under impact[END_REF] . . . . . . . . . . .

2.6

Classication of low velocity impacts according to Koechlin and Potapov [START_REF] Koechlin | Classication of soft and hard impacts -application to aircraft crash[END_REF] . . . . . .

2.7

Overall response failure modes of impacted RC slabs [START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF]

2.9

Energy transformation process during an impact event [START_REF] Miyamoto | Design concept for reinforced concrete slab structures under soft impact loads[END_REF] . . . . . . . . . . . . . . .

2.24

Mass-spring models of impact on composite structures according to [START_REF] Abrate | Modeling of impacts on composite structures[END_REF] . . . . . . . . .

2.25

The two mass-spring-damper models used by [START_REF] Delhomme | Simulation of a block impacting a reinforced concrete slab with a nite element model and a mass-spring system[END_REF] to analyze SDR response: (a) contact: 10 Principle of importance sampling method, illustration in the standard space [START_REF] Deheeger | Couplage mécano-abiliste: SMART -méthodologie d'apprentissage stochastique en abilité[END_REF] . . . .

4.11

Principle of directional simulation method, illustration in the standard space [START_REF] Deheeger | Couplage mécano-abiliste: SMART -méthodologie d'apprentissage stochastique en abilité[END_REF] . . . . 4.12 Principle of Latin hypercube sampling method, illustration in the standard space [START_REF] Deheeger | Couplage mécano-abiliste: SMART -méthodologie d'apprentissage stochastique en abilité[END_REF] . 4.13 Linear stepped model for tension stiening [35] . . . . . . . . . . . . . . . . . . . . . . .

6.2

Discretization of multiber beam into elements, nodes and degrees of freedom [START_REF] Guedes | A bre timoshenko beam element in castem[END_REF] . . .

6.3

Geometrical properties of the multiber beam . . . . . . . . . . . . . . . . . . . . . . .

6.4

Beam displacement distribution estimated with Kernel Smoothing method . . . . . . .

6.5

Comparison of the beam displacement sample to several parametric distributions . . . . In civil engineering eld, the main problem consists in nding an optimal design that ensures to maximum the continuous safety of structures throughout its service life. The design should also ensure a satisfactory structural performance with an economical cost. Reinforced concrete structures are subjected to several sources of uncertainties that highly aect their response. These uncertainties are related to the structure geometry, material properties and the loads applied. Engineers have always used traditional and deterministic approaches to simplify the problem and account for these uncertainties through the application of safety factors according to limit states. However, these factors are based on engineering judgment, they are calibrated to a certain level of security and presented as xed values. In addition, they do not provide sucient information on the eect of dierent uncertain variables on the structural safety. Consequently, for an optimal and robust design that guarantees a real prediction of the behavior of structures, uncertainties should be taken into account and propagated in the structural deterministic analysis. To address this issue, probabilistic approaches were developed to correctly model the variation of input variables and to ensure a higher reliability of representative mechanical models used for the design of new structures or the monitoring of performance of existing structures. These approaches allow assessing failure probability of structures according to one or several criteria, but also studying the inuence of uncertain variables on the structure response. Sensitivity analysis can be performed to analyze the variation of structure resistance during the design phase or the surveillance testing phase with respect to conventional parameters, such as loads and material properties, but also with respect to nonlinear inaccurately known parameters such as the creep in concrete or its damage due to cracking. Thus, structural reliability analysis based on the principle of combining a stochastic model with a deterministic model enables to deduce a more realistic estimate of the safety margin in terms of the variation of dierent variables. Stochastic models should include the probabilistic characteristics of random input variables including a suitable probability distribution and appropriate values of their mean and coecient of variation. Deterministic models allow predicting structural response and evaluating the variables of interest, they can be based on analytical, empirical or numerical deterministic approaches.

Therefore, the physical problem and mechanical phenomena applied should be evaluated before performing reliability analysis. Among structures considered in this study, only the physical problem of reinforced concrete (RC) slabs subjected to impact is investigated in detail. Such types of problem is classied as low velocity impact event and considered as one of the most dicult nonlinear dynamic problems in civil engineering because it involves several factors, including the nonlinear behavior of concrete, the interaction eects between the concrete and reinforcement, and contact mechanics between the slab and the impactor. The complexity of impact problems also lies in the dynamic response of RC slabs and the time-dependent evolution of impact velocity. Design of RC slabs to resist impact loads represents an area of research that is increasingly gaining importance. Their design is generally based on static method which is often not suciently accurate to describe an impact phenomenon. An optimal design requires taking into account the main physical processes involved and the potential development of nonlinearities due to material damage. Numerous studies were carried out on the transient behavior of RC slabs under low velocity impact and three approaches were usually used. Many researchers studied the actual response of impacted slab with the development of crack patterns and resulting failure modes through full-scale impact tests [START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF][START_REF] Zineddin | Dynamic response and behavior of reinforced concrete slabs under impact loading[END_REF]. Advanced numerical approaches, which take into account materials constitutive models and contact algorithms, have proven to be reliable to analyze the failure modes of RC slabs subjected to impact [START_REF] Berthetrambaud | Analytical advantages and numerical modeling of impact testing on a structurallydissipating rock-shed slab (sdr)[END_REF][START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF][START_REF] Mokhatar | Computational analysis of reinfoced concrete slabs subjected to impact loads[END_REF]. Simplied analytical methods were also developed and used as an initial approximation of the impact behavior of slabs [START_REF] Abrate | Modeling of impacts on composite structures[END_REF][START_REF] Ceb | Concrete structures under impact and impulsive loading[END_REF][START_REF] Delhomme | Simulation of a block impacting a reinforced concrete slab with a nite element model and a mass-spring system[END_REF], they are generally based on simplifying assumptions and account for the complexity of the studied phenomenon. However, these approaches are deterministic and do not consider the randomness of the basic variables of an impact problem.

These variables aect the response of the slab which has also a random character, thus reliability analysis seems to be the more appropriate to study the behavior of slabs under impact. Based on this approach, a number of studies were carried out to estimate the reliability of dierent types of RC structures. Low and Hao [START_REF] Low | Reliability analysis of direct shear and exural failure modes of rc slabs under explosive loading[END_REF] performed a reliability analysis of RC slabs subjected to blast loading using two loosely coupled single degree of freedom system and Val et al. [START_REF] Val | Reliability evaluation in nonlinear analysis of reinforced concrete structures[END_REF] evaluated the reliability of plane frame structures with ultimate limit state functions. However, reliability analyses that consider the eect of uncertainties in a low velocity impact phenomenon are scarce.

Research motivation

The lack of knowledge on the potential load applied to a RC structure, as well as the uncertainties related to its features (geometry, mechanical properties) shows that the design of RC structures could be made in a reliability framework. Structural reliability provides the tools necessary to account for these uncertainties and evaluate an appropriate degree of safety. This research discusses the use of reliability analysis for three dierent civil engineering applications of dierent degrees of complexity.

The platform OpenTURNS is used to perform the reliability analysis of the RC structures considered in the present study and propagate uncertainties in their physical models. OpenTURNS is used because it oers a large set of methods that enable the quantication of the uncertainty sources, the uncertainty propagation and the ranking of the sources of uncertainty. However, OpenTURNS should be linked to the software used to perform deterministic analysis. The link of any external software to OpenTURNS can be made using the generic wrapper and is called wrapping. Wrapping allows an easy connection to the deterministic simulation as well as an easy evaluation of its response due to several features provided in OpenTURNS. A wrapper is a Python based programming and necessitates the description of input and output variables using regular expressions that enable the substitution of the values of these variables with random values in the initial deterministic model.

Thus, the input le of the deterministic code and the output le corresponding to output variable values should be specied, since the rst will be modied according to the random values generated and the second will be read by OpenTURNS. The command to submit the deterministic model should be also provided in the wrapper, it allows running the external software using the specic random values generated for each call of the wrapper. A wrapper works as follows:

Values of random variables are generated in OpenTURNS.

The wrapper uses the input le of the deterministic code in order to create a new le corresponding to the random values generated. The new le will be placed in a temporary directory.

Once the submission command of the deterministic model is accomplished, the wrapper reads the results le and send the mechanical response obtained to OpenTURNS in order to perform a reliability analysis for the structure considered.

The capacity of OpenTURNS to be linked to dierent nite element software is tested through the three applications considered in this study. Therefore, OpenTURNS is linked to CASTEM to study the reliability of a RC multiber cantilever beam subjected to a concentrated load at the free end. It is also coupled to Abaqus in order to study the reliability of RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings.

Furthermore, a combination of ASTER and OpenTURNS is considered in the aim of studying the reliability of a prestressed concrete containment building. In the case where a structure is presented with an analytical model, OpenTURNS is combined with Matlab.

The combination of mechanical and stochastic models is necessary in reliability analysis in order to compute failure probability for one or several failure modes, which can be very time consuming and in some cases impossible. Indeed, either the complexity of the deterministic model considered implies important computing time (from minutes to hours) or reliability methods evaluating failure probability require a too large number of simulations of the deterministic model.

Study scope and objectives

There are many objectives behind this research. One of these objectives is to discuss the principles of structural reliability analysis, as well as the types of uncertainty related to the structure and the deterministic model used. This study describes the basic statistical concepts required to perform a reliability analysis in structural engineering, as well as the probabilistic methods available in OpenTURNS to assess failure probability of existing or new structures according to a specic limit state function.

Another main objective of the current research is to address the issue of computational eort of reliability analysis. Thus, two strategies are proposed to accurately assess failure probability with the minimum computational cost:

The rst strategy consists in using deterministic analytical models that are as simple as possible, but that are able to predict the structural response appropriately. In general, the constraining assumptions of analytical models allow studying only a few congurations of the structure. Thus, the objective is to reduce the computational cost of the deterministic analysis to the minimum, while also explore as far as possible how to use this type of simplied models. Due to their low computational time, these models can be coupled with simulation probabilistic methods that need to carry out several deterministic analyses.

The second strategy is used when the structure is simulated with a nite element model.

It consists in choosing an appropriate probabilistic method where failure probability can be assessed from a small number of simulations of the numerical model, or substituting the nite element model by a meta-model less expensive in computational time.

The reliability of three dierent structures is examined in this study:

A simple application to a RC beam is rstly used as a preliminary example in the aim of mastering the basics of OpenTURNS and examining the eciency of several probabilistic methods available in OpenTURNS. This application enables to compare probabilistic methods with an example necessitating low computational eort in order to initially select an appropriate method to use in the case of more complex structures depending on the deterministic model considered and the computational eort constraints required.

The second application focuses on RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings. In this case, dropped objects are characterized by small impact velocities and damage can arise at the impact zone depending on the impact energy and the relative masses of the colliding bodies. The following example is used to compare the strategies proposed to reduce the computational eort of reliability analysis, as well as to perform parametric studies in order to locate the most critical impact points for the slab design and illustrate how to optimize the design of reinforced concrete slabs under impact in terms of dimensions, material properties and reinforcement ratio. The reliability of slabs subjected to impact assuming an elastic linear behavior of materials is examined rst followed by the reliability of slabs when nonlinear behaviors of concrete and steel are considered in deterministic models.

The third application consists of a prestressed containment building in order to study the stress evolution with time in the containment during periodic surveillance testing carried out 20 years after its implementation. The problem is examined under aging phenomena, including relaxation of reinforcement, creep and shrinkage of concrete. The aim of this application is to investigate the eciency of meta-modeling strategy in reliability analysis of very complicated structures. In this case, the deterministic model is approximated by a polynomial function and the reliability analysis is performed for this approximation. The computational eort of this strategy depends on the number of simulations needed to obtain an accurate approximated model.

As previously mentioned, only the physical problem of reinforced concrete (RC) slabs subjected to unplanned impact events due to dropped objects is investigated in detail. The aim of this application is to provide an ecient procedure to predict RC slabs response under low velocity impact. Numerical and analytical approaches are used and models are validated using physical experimental tests from the literature which provide high quality input data and a wide range of output results. As a part of the fundamental objectives of this research, the following investigations are carried out in order to be able to examine the structure performance and carry out designs:

Study the complex phenomenon of impact and the structural behavior associated to impact by: classifying the impact response types of RC slabs according to the impactor mass and velocity, dening the impact event types according to the impactor deformability, material properties and impact velocity, describing the failure modes of RC slabs under impact according to their characteristics and impact conditions, presenting the energy transformation process of the impact energy to slabs as dierent forms of energy.

Characterize concrete and steel materials properties by: describing their mechanical behaviors according to several stress-strain relationships proposed in the literature, choosing suitable material constitutive models to use in nite element analysis and identifying properly their parameters, providing an accurate statistical description of material variables intervening in the reliability analysis of impacted RC slabs.

Choose an appropriate nite element software for impact problems by: presenting the dierent steps necessary to create a nite element model using the software selected, describing the principles of nite element components implemented in the software and adopted for the model.

Develop a three-dimensional nite element model of an impacted slab in Abaqus using an explicit conguration, which allows taking into account the nonlinear behavior of concrete and modeling the contact between the impactor and the slab. The model should also allow a better representation of the actual structure geometry and boundary conditions. The model is validated with experiments carried out on several RC slabs under drop-weight loads at Heriot-Watt University by Chen and May [START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF]. Then, the model adopted is used to model RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings.

Develop a simplied analytical model that consists of two degrees of freedom mass-spring system which accounts for potential viscous damping, and describes the degradation of slab stiness using a frequency decrease approach

The modeling procedure adopted in this study simulates the impacted slabs as close as possible to the actual structure without any geometrical or material simplications. The full geometry of slabs is considered in the model and no symmetry boundary conditions are modeled in order to be able to study several congurations of impact position. The model is parametrized through a code developed using a Python script in Abaqus Python edition. It should be noted that strain rate eect is not taken into account in this research as only low velocity impacts are studied.

Thesis contents

Chapter 2 This chapter provides a classication of impact response types of RC slabs according to the impactor mass and velocity. Then a denition of impact event types is introduced and the behavior of RC slabs subjected to a transient dynamic load is described by determining their failure modes according to their characteristics and impact conditions. An overview of approaches used to determine slabs response is also given and several studies from the literature performed on their transient behavior under low velocity impact are detailed. Next, the nonlinear static and dynamic mechanical material behaviors are presented and many uniaxial stress-strain relationships for steel and concrete in compression and tension proposed in the literature are provided.

Chapter 3 This chapter discusses the choice of deterministic software used in the present study to simulate the problem of RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings. First, the basic phases of nite element analysis are described and the dierent steps which are necessary to create a nite element model in Abaqus are presented. Furthermore, plasticity constitutive models used to represent concrete and steel in numerical simulations are discussed by describing and identifying their fundamental parameters, and contact algorithms used for modeling the interaction between two bodies are presented. Other numerical features which are necessary for simulating impact analysis are also discussed. Chapter 4 In this chapter, the principles of structural reliability analysis, as well as the types of uncertainty related to the structure and the deterministic model used, are presented. Next, the steps of a reliability analysis are detailed and the methods used in OpenTURNS for uncertainty quantication, uncertainty propagation and sensitivity analysis are described. Finally, statistical descriptions of random variables intervening in RC structures are examined according to several studies in the literature. Chapter 5 In this chapter, two deterministic models are used and evaluated for the problem of impacted slabs. The rst model consists of a 3D nite element model simulated with Abaqus/Explicit.

The steel is modeled as an elasto-plastic material with hardening and, the behavior of concrete is described by the damage plasticity model. A detailed step-by-step procedure for creating FE models of impacted slabs with Abaqus is described. This model is used for Chen and May RC slabs [START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF] and validated with experimental results. The model adopted is used to simulate RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings. Then, a simplied analytical model is also used for these slabs. It consists of a two degrees of freedom mass-spring system which accounts for potential viscous damping. A frequency decrease approach is used to describe the degradation of the slab.

Chapter 6 In this chapter, 3 case studies are used in order to address the issue of solving reliability problems in the domain of civil engineering. The rst application consists of a RC multiber cantilever beam and is used in the aim of mastering the basics of OpenTURNS and examining the probabilistic methods proposed in OpenTURNS to estimate failure probability in structural reliability analysis in terms of their accuracy, precision and computational eort. The second application is the problem of RC slabs subjected to impact in the aim of addressing the issue of computational cost of reliability analysis and proposing computational strategies allowing the accurate assessment of the failure probability for minimum computational time. Firstly, the problem of impacted slab is studied assuming a exural mode of failure and an elastic behavior for steel and concrete. Following this, the problem of impacted slabs is studied using deterministic models that take into account the nonlinear material properties. In both cases, elastic and nonlinear behaviors, a parametric study is performed to identify the inuence of deterministic model parameters on the reliability of RC slabs under low velocity impact. The third application consists of the prestressed concrete containment building of the Flamanville nuclear power plant and is considered in the aim of presenting a procedure to be followed to study the response of very complicated structures in a reliability framework.

Polynomial chaos expansion is used to simplify the physical model and study the reliability of the containment.

Chapter 2

Reinforced concrete slabs behavior under impact: Dynamic response and literature review

Introduction

In civil engineering eld, reinforced concrete (RC) structures are often subjected to some extreme dynamic loadings due to accidental impacts of rigid bodies that may occur during their service life with a very low probability of occurrence. The failure resulting consequences of structures subjected to such extremely severe loadings might be extremely high, which makes their analysis and design very complex especially when working with nonelastic materials. Impact loadings are characterized by a force of considerable magnitude applied within a short duration, they may be caused by falling rock impact on protection galleries, missile and aircraft impact on nuclear containments, vehicles collision with buildings or bridges, ships or ice crash impact with marine and oshore structures, ying objects due to natural forces such as tornados and volcanos, fragments generated due to military or accidental explosions on civil structures, or by dropped objects impact on industrial or nuclear plant oors as a consequence of handling operations. Figure 2.1 shows some examples of impact loading cases in civil engineering eld. For all these applications, the problem of impact has to be studied with convenient considerations since structures undergo, depending on their stiness, dierent failure modes such as exure and punching shear failure, in addition to dierent types of local damage such as crushing, cracking, scabbing, spalling and perforation. Furthermore, the velocity, angle and position of impact, as well as the shape, mass and rigidity of impactors vary broadly from one application to another.

Design of RC slabs to resist impact loads represents an area of research that is increasingly gaining importance and numerous studies were carried out by many researchers on the behavior of RC slabs subjected to dynamic impact loading, especially under high velocity regimes [START_REF] Belov | Stress analysis of concrete and reinforced concrete slab structures under a high velocity impact[END_REF][START_REF] Duranovic | Impulsive loading on reinforced concrete slabs -local failur propagation[END_REF][START_REF] Herve | Simulation numérique de l'endommagement de dalles en béton armé impactées par un réacteur d'avion idéalisé[END_REF][START_REF] Jahidulislam | Numerical study on concrete penetration/perforation under high velocity impact by ogive-nose steel projectile[END_REF][START_REF] Jovall | Airplane crash simulations: comparison of analyses results with test data[END_REF].

However, when slabs are exposed to unplanned impact events such as accidental collisions with dropped objects, the problem is classied as low velocity impact event. Several studies that will be detailed in section 2.3 were carried out on the transient behavior of RC slabs under low velocity impact, but there is a growing need to more investigate the behavior of slabs for low velocity situations.

In the available design codes of civil engineering, the design of reinforced concrete (RC) structures subjected to impact is generally based on approximate static method. The dynamic forces during impact are usually considered by applying an equivalent dynamic impact factor and converted into a static force of equal magnitude. Such static design is often not suciently accurate to describe an impact phenomenon, as not only the maximum value of the impact force is crucial to predict the The use of approximate static methods results in either an under-or over-dimensioned design for structures and, in some particular cases, the dynamic response of structures cannot be simplied to a quasi-static response. Moreover, an equivalent dynamic force can be just used in the case when structures withstand exural mode. Nevertheless, during impacts, structures do not only fail under exural mode but they may fail under punching shear or local damage, and higher modes of vibration can be expected. Thus the transient dynamic analysis has to be performed accounting for the main physical processes involved and accurate models are needed to describe and predict the structural behavior of RC members when subjected to impact loading. An optimal design that guarantees the structure resistance with an economical cost requires taking into account the potential development of nonlinearities due to the material damage (concrete cracking, steel yielding, etc.).

Abrate [START_REF] Abrate | Impact on composite structures[END_REF] indicated that the study of impact on composite structures such as RC slabs involves many dierent topics (contact mechanics, structural dynamics, strength, stability, fatigue, damage mechanics, micromechanics) and that the prediction of dynamic response of composite structures to impact can be made by using analytical or nite element models which must appropriately account for the motion of the impactor, the overall motion of the structure, and the local deformation in the contact zone and in the area surrounding the impact point. Abrate also indicated that the parameters which aect the impact resistance of composite structures are the properties of the matrix, the reinforcing bers, the ber-matrix interfaces, the size, the boundary conditions, and the shape, mass, and velocity of the impactor.

The present study focuses on RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings (Figure 2.2). In this case, dropped objects are characterized by small impact velocities and damage can arise at the impact zone depending on the impact energy and the relative masses of the colliding bodies. The current chapter rst provides a classication of impact response types of RC slabs according to the impactor mass and velocity. Then a denition of impact event types is introduced and the behavior of RC slabs subjected to a transient dynamic load is described by determining their failure modes according to their characteristics and impact conditions. Afterwards, an overview of approaches used to determine slabs response is given and several studies from the literature performed on their transient behavior under low velocity impact are detailed. Next, the nonlinear static and dynamic mechanical material behaviors are presented and many uniaxial stress-strain relationships for steel and concrete in compression and tension proposed in the literature are provided. During handling and maintenance operations within nuclear plant buildings, impacts on RC slabs by foreign objects as tools and nuclear material packages can be expected to occur. Impacts create internal damage that depends on the load (velocity and mass of the impactor) and may causes severe reductions in strength [START_REF] Abrate | Impact on composite structures[END_REF]. The internal damage is often undetectable by visual inspection, hence the eects of dropped object impacts on RC slabs must be understood and taken into account in the design process. According to Abrate [START_REF] Abrate | Modeling of impacts on composite structures[END_REF], composite structures subjected to impact have dierent types of behavior which must be properly evaluated in order to interpret experimental results and select convenient analytical or numerical models. Abrate studied and classied the currently available models used to predict the dynamic of impacts between a foreign object and a composite structure in the aim of presenting an approach to select an appropriate model for each particular case study.

He showed that developing a model to analyze the overall response of the structure and the contact phenomenon requires understanding the dierent types of structural behavior, as well as the eects of several parameters on the impact dynamics.

According to [START_REF] Abrate | Impact on composite structures[END_REF], when the impactor enters in contact with the structure, shear and exural waves propagate away from the impact point and reect back when they reach the back face. These waves propagate at dierent speeds and decline progressively in response to materials damping, energy diusion and several geometrical eects [START_REF] Olsson | Experimental validation of delamination criterion for small mass impact[END_REF]. Thus, the impact response can be controlled by wave propagation and strongly inuenced by impact duration [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF]. The impact duration inuence is illustrated in Figure 2.3, which shows three dierent types of response for very short, short and long impact times.

The response dened for very short impact times is dominated by three-dimensional wave propagation through the thickness direction and the impact time is in the order of waves transition time.

This response is unaected of structure size and boundary conditions and is generally associated with ballistic impacts (Figure 2.3.a). A denition of ballistic impacts was given by [START_REF] Abrate | Impact on composite structures[END_REF] as impacts Figure 2.3: Response types during impact on plates according to [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF] resulting in complete perforation of the structure with a ballistic limit equal to the lowest initial velocity of the impactor causing this perforation. The ballistic limit increases with the structure thickness and is aected by material densities and mechanical properties. For short impact times, the response is governed by exural waves and shear waves and is typical for impact by hail and runway debris (Figure 2.3.b). In this case, the response remains independent of boundary conditions as long as main waves have not reached a boundary. For long impact times, the response is quasi-static inuenced by structure dimensions and boundary conditions. A typical example of long impact times is dropping of heavy tools. In this case, the time needed by waves to reach structure boundaries is much less than impact duration and the quasi-static response is in the sense that deection and load have the same relation as in a static case (Figure 2.3.c). The two latter response types can be associated with the case of RC slabs subjected to foreign object impact.

A criterion related to the three-dimensional stress distribution under the impactor was proposed by Abrate [START_REF] Abrate | Impact on composite structures[END_REF] to classify impact dynamics by distinguishing between high-velocity and low-velocity impacts. His criterion consisted of dening low-velocity impacts in cases where stress wave propagation through the thickness has no signicant role. In this case, damage is not introduced in the early stages of impact, but it is initiated after the overall structure bending behavior is established. For low-velocity impacts, stress levels remain low due to several wave reections through the thickness and the structure dynamic properties are usually not aected by the presence of damage.

Conversely, Abrate indicated that damage for high-velocity impacts is introduced during the rst few compressive wave reections through the thickness, while overall structure behavior is not yet initiated. Abrate's criterion dened high-velocity impacts as cases related to the ratio of impact velocity versus the speed of propagation of compressive waves through the thickness direction, so that this ratio must be larger than the maximum failure strain. During high velocity impacts, failure near the structure back face may occur due to tensile stresses generated by the compressive wave after reection. A velocity of 20 m/s was suggested by Olsson [START_REF] Olsson | Experimental validation of delamination criterion for small mass impact[END_REF] as the upper limit of low velocity impacts, while the suggested limit of medium velocity ranges from 20 to 100 m/s. Very high velocities were considered to be greater than 2 km/s [START_REF] Abrate | Impact on composite structures[END_REF]. Classifying impact events by using velocity is the most common, however this classication is highly relative. For example in the case of impactors dropped at low velocity (a few m/s) on large plates, the response was demonstrated to be associated with high-velocity impact since it is governed by wave propagation [START_REF] Olsson | Experimental verication of a theory for the impact response of composite plates[END_REF].

Further, the initial kinetic energy of the impactor is an important parameter that aects the structure response [START_REF] Abrate | Impact on composite structures[END_REF][START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF]. In consequence, although the kinetic energy is the same, a structure subjected to an impact of a large mass with a low initial velocity behaves dierently than in case of a smaller mass with higher velocity. For the rst problem, the indentation eect is negligible and the impact may involve an overall response of the structure. In the other case, damage initiates at earlier stages and the response can be introduced by deformations in a small zone surrounding the impact point. Impactors with small mass cause more localized deection and higher impact loads with a signicant eect of the indentation on absorbing the impact energy that can be restituted to the impactor or transferred to the structure. The inuence of impactor mass on the response behavior can be associated to Figure 2.3 as mentioned in [START_REF] Olsson | Analytical prediction of large mass impact damage in composite laminates[END_REF]. A ballistic response occurs for very small masses, moderately small masses cause a response dominated by shear and exural waves and large masses cause a quasi-static response.

However, Olsson [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF] showed that response and damage are not inuenced only by impact velocity, duration or initial kinetic energy, but also by the impactor vs. structure mass ratio. Olsson derived a simple criterion for impact plates response controlled by exural waves and distinguished between small-mass, intermediate-mass and large-mass impacts. A sucient condition for smallmass impact response is that dominating exural waves does not reach the boundaries during the impact duration. For central impact on a quasi-isotropic plate, small-mass criterion corresponds to impactor masses less than one fth of the mass of the plate area aected by the impact:

M/M p ≤ 1/5 (2.1)
The response with intermediate-mass impactors is more complex and the mass criterion becomes:

1/5 < M/M p < 2 (2.2)
For large-mass impactors, the exural wave reaches boundary conditions and a quasi-static impact response occurs for:

M/M p > 2 (2.3)
where M is the impactor mass and M p is the mass of the impacted plate. For non-central impacts, M p is calculated as the mass of a square zone centered at the impact with a side that coincides with the closest plate edge [START_REF] Olsson | Experimental validation of delamination criterion for small mass impact[END_REF]. For square plates, the condition of small-mass impact response corresponds to a mass ratio less than 1/4 (M/M p ≤ 1/4). It should be noted that this mass criterion becomes irrelevant for impact velocity resulting in penetration [START_REF] Olsson | Experimental validation of delamination criterion for small mass impact[END_REF].

Types of impact

An existing denition allows normally classifying impact events into either hard or soft according to the impactor deformability. For hard impact, impactor is generally considered as rigid and its deformation is negligible compared to the structure deformation (Figure 2.4.b). In this case, the impact kinetic energy is to a great extent absorbed by the structure deformation and the impactor remains undamaged or barely deformed during impact. Hard impact results in both local and overall dynamic response of the structure and failure may occur due to complicated stress waves. According to [START_REF] Miyamoto | Nonlinear dynamic analysis of reinforced concrete slabs under impulsive loads[END_REF], the shape and dimensions of impactor, as well as impact velocity are the essential parameters that may aect the classication of an impact as hard.

However, in the case of soft impact, the structure deformation is very small compared to the impactor. In other words, the structure that resists to impact is assumed to remain undeformed, while the impactor is strongly damaged and the impact kinetic energy is fully transferred into deformation energy of the impactor (Figure 2.4.c). The propagation of stress waves is considered as negligible and the structure failure response can be compared to that under static loading with no rebound of the impactor. An impact can be considered as soft owing to the failure region and the ratio between the impacted structure and the impactor masses [START_REF] Miyamoto | Nonlinear dynamic analysis of reinforced concrete slabs under impulsive loads[END_REF]. The case of soft impact can be illustrated by a crash of a vehicle against a rigid wall or a crash of an airplane against a thick concrete containment wall of a nuclear power plant [START_REF] Ceb | Concrete structures under impact and impulsive loading[END_REF]. This basic and well-known denition of soft and hard impacts was proposed by Eibl [START_REF] Eibl | Soft and hard impact[END_REF] in order to consider the dissymmetry of an impact event along with the dierence between structure and impactor responses. The dissymmetry of an impact event is due to the fact that the structure is stationary before impact, while the impactor moves with an initial velocity V (0). As mentioned earlier, one of the two bodies in collision shows small deformations compared to the other one during soft and hard impact, which involves a dissymmetry in the behaviors of the structure and the impactor.

The classication of Eibl is based on a mass-spring model (Figure 2.5) that consists of the structure (m 2 ) and the impactor (m 1 ) masses, a contact spring associated with a stiness k 1 and a second spring with a stiness k 2 . The contact spring simulates the force that arises when the two bodies are in contact, while the second represents the structure behavior and its resisting force to impact. The mass-spring model is governed by two dierential equations that describe the motion of the two bodies and can be expressed as follows:

m 1 ẍ1 (t) + k 1 [x 1 (t) -x 2 (t)] = 0 m 2 ẍ2 (t) -k 1 [x 1 (t) -x 2 (t)] + k 2 x 2 (t) = 0 (2.4)
In the case of a soft impact (x 1 (t)

x 2 (t)), the contact force F (t) can always be deduced only from the impactor deformation by solving the rst independent equation of the following system:

m 1 ẍ1 (t) + k 1 x 1 (t) = 0 m 2 ẍ2 (t) + k 2 x 2 (t) = F (t) (2.5) with F (t) = k 1 x 1 (t).
The second equation allows determining the structure response under an independently acting force F (t).

Hard impacts require to consider the local as well as the overall deformations of the structure, thus the two equations of system (2.4) cannot be independent and spring stinesses must be determined. Section 2.3.2.2 shows how to calculate k 1 and k 2 for hard impacts as indicated by the Euro-International Concrete Comity (CEB) [START_REF] Ceb | Concrete structures under impact and impulsive loading[END_REF] that assumes the same characterization of soft and hard impact as Eibl.

Nevertheless, Koechlin and Potapov [START_REF] Koechlin | Classication of soft and hard impacts -application to aircraft crash[END_REF] found that the denition of Eibl is qualitative and not precise enough to make a distinction between soft and hard impacts for low velocity impact events. Alternatively, they proposed a more precise quantitative classication which is based on the material properties and the impactor velocity, and permits to predict the type of impact that is expected even before the observation of the structural response during impact. They indicated that Figure 2.5: Mass-spring model proposed by Eibl [START_REF] Eibl | Soft and hard impact[END_REF] to simulate an impact [START_REF] Daudeville | Concrete structures under impact[END_REF] considering the impactor deformation as described previously as the essential property of the impact type specication is not completely true since the terms hardness and softness are relative and the designation of a rigid impactor or a rigid structure is not identical for all studies that examine the eect of impact on RC structures. Consequently, they showed that the characterization of soft and hard impacts is related to the structure failure mode and to the fact that the impactor penetrates or not the structure. They indicated that in the case of soft impacts, failure occurs due to a shear cone failure mode, the structure is not damaged and the impactor is crushed. On the contrary, perforation is a consequence of a local failure and the impactor penetrates the structure for hard impacts.

In order to dene an impact type, the new criterion of Koechlin and Potapov consists in comparing the structure compressive strength σ s with the stress applied by the impactor during impact. This latter stress represents the sum of the impactor limit strength σ i and a second component that depends on the impactor density ρ i and velocity v i . This criterion allows verifying if the structure withstands the impact and is expressed by:

σ s = σ i + ρ i v 2 i (2.6)
Equation (2.6) can be written in terms of non-dimensional parameters, which permits to compare impacts at dierent scales and gives the boundary between hard and soft impacts (Figure 2.6):

σ i σ s + ρ i v 2 i σ s = 1 (2.7)
This boundary provides the type of failure that could occur during impact and species whether the structure fails due to a direct perforation or a punching shear. In other words, it represents the boundary between the potential global and local failure modes.

Numerous experimental studies were carried out to assess the eect of soft and hard impact on the concrete behavior and the response of RC structures. Koechlin and Potapov presented and classied these tests using their new criterion that is based on an explicit graphic representation as shown above. Few experimental tests that were considered by their authors as soft impact are available in the literature, namely EDF tests [START_REF] Dulac | Impact testing of reinforced concrete slabs[END_REF] and Meppen tests [START_REF] Jonas | Experimental investigations to determine the kinetic ultimate bearing capacity of reinforced concrete slabs subjected to deformable missiles[END_REF][START_REF] Nachtsheim | Impact of deformable missiles on reinforced concrete plates -comparisonal calculations of meppen tests[END_REF][START_REF] Nachtsheim | Interpretation of results of meppen slab tests -comparision with parametric investigations[END_REF][START_REF] Rudiger | Experimental and theoretical investigations on the impact of deformable missiles onto reinforced concrete slabs[END_REF]. Other experiments with soft missiles [START_REF] Kojima | An experimental study on local behaviour of reinforced concrete slabs to missile impact[END_REF][START_REF] Ohno | Local damage of reinforced concrete slabs by impact of deformable projectiles[END_REF][START_REF] Sugano | Local damage to reinforced concrete concrete structures caused by impact of aircraft engine missiles -part 2. evaluation of test results[END_REF][START_REF] Sugano | Local damage to reinforced concrete concrete structures caused by impact of aircraft engine missiles -part 1. test program, method and results[END_REF] were incorrectly referred as soft impact due to the reason that the impactor is deformable. However, Koechlin and Potapov found that, according to their criterion, these tests must be classied as hard impact since the failure mode of slabs is similar to the hard impact one. Kojima tests [START_REF] Kojima | An experimental study on local behaviour of reinforced concrete slabs to missile impact[END_REF] were performed in the aim of investigating the local behavior of RC slabs subjected to missile impact with varying the missile and slabs properties. Soft-nosed and hard-nosed missiles were used to study the degree of slab damages in both cases, while critical reinforcement ratio and slab thickness were evaluated in order to analyze slabs resistance to impact and prevent perforation. Ohno et al. [START_REF] Ohno | Local damage of reinforced concrete slabs by impact of deformable projectiles[END_REF] investigated experimentally the eect of missile nose shape on the local damage of RC slabs when impacted by deformable missiles.

Five shapes of missiles were used for the impact tests and empirical formulas for perforation and scabbing were developed. The investigations carried out by Sugano et al. [START_REF] Sugano | Local damage to reinforced concrete concrete structures caused by impact of aircraft engine missiles -part 2. evaluation of test results[END_REF][START_REF] Sugano | Local damage to reinforced concrete concrete structures caused by impact of aircraft engine missiles -part 1. test program, method and results[END_REF] Moreover, it should be noted that tests performed by Kojima [START_REF] Kojima | An experimental study on local behaviour of reinforced concrete slabs to missile impact[END_REF] and Sugano et al. [START_REF] Sugano | Local damage to reinforced concrete concrete structures caused by impact of aircraft engine missiles -part 2. evaluation of test results[END_REF][START_REF] Sugano | Local damage to reinforced concrete concrete structures caused by impact of aircraft engine missiles -part 1. test program, method and results[END_REF] with rigid missiles were also classied as hard impact by Koechlin and Potapov, which is in accordance with Eibl classication. CEA-EDF experiments [START_REF] Fiquet | Study of the perforation of reinforced concrete slabs by rigid missiles -experimental study, part ii[END_REF][START_REF] Goldstein | Study of the perforation of reinforced concrete slabs by rigid missiles -experimental study, part iii[END_REF][START_REF] Gueraud | Study of the perforation of reinforced concrete slabs by missiles -general introduction and experimental study, part i[END_REF] were also considered as hard impact for both criteria. They were performed by CEA (Commissariat à l'énergie atomique et aux énergies alternatives/ French Alternative Energies and Atomic Energy Commission ) and EDF to predict the perforation limit of RC slabs subjected to rigid missile impacts with varying slabs thickness, concrete strength, reinforcement ratio and impact velocity. Likewise, Koechlin and Potapov described impacts with high velocity as hard impacts, which is in agreement with the usual observations of Bischo and Perry [START_REF] Bischo | Compressive behaviour of concrete at high strain rates[END_REF].

Failure modes

Structures subjected to impact loading show dierent behavior compared to that under static load, as a result of the transient and localized properties of impact loading. Furthermore, dynamic properties of materials are dierent than those under static loading and may be aected by the strain rate or load rate. Therefore, a solid understanding of this type of structures behavior under an impact event is essential to develop a satisfactory design and prevent their collapse. The prediction of the behavior of RC slabs when they are subjected to impact loading is of considerable complexity since impact event involves several phenomena and slabs have been observed to react both locally and globally. Failure modes of slabs during impact change with time and are inuenced by stress waves and inertial forces. The impact behavior of concrete is also considered as quite complicated due to heterogeneity of material. Several parameters may signicantly inuence the impact response of RC slabs:

Type of impact plays a role in dissipating energy.

Slab dimensions and boundary conditions control the stiness of the slab.

Material properties have a signicant inuence on the slab transient response by aecting the contact and overall slab stinesses.

Impactor characteristics including impact velocity, shape, position, mass and rigidity inuence the impact dynamics.

Changes in slab stiness arise from cracking or crushing of concrete, yielding of reinforcement, or other mechanisms such as local unloading behaviors and post-cracking dilation [START_REF] Hrynyk | Behaviour and modelling of reinfoced concrete slabs and shells under static and dynamic loads[END_REF].

A RC slab subjected to a dropped object will be exposed to a transient dynamic load. As a result, both local and global dynamic response of the slab can be induced. The identication of the various modes of failure that occur during impact is important to correctly evaluate the dynamic response.

The elastic-plastic response of RC slabs may cause two main overall response failure modes: exural failure or punching shear failure [START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF] (Figure 2.7). At exural failure mode, the slab bends strongly and fails due to excessive tension stresses which lead to the formation of cracks through the RC slab thickness and yielding in tension reinforcement. At the punching shear failure mode, a shear cone that engenders tensile stresses within the transverse steel is created and the slab fails due to excessive shear stresses. The kinematic mechanism of punching was described by [START_REF] Theodorakopoulos | Ultimate punching shear strength analysis of slab-column connections[END_REF] for a static load: the application of an increasing load to a RC slab presents a roughly circular crack that appears around the loading zone on the slab tension surface and propagates subsequently into the compressed zone of concrete. Then new exural cracks arise and inclined shear cracks are observed. The further increase in load develops curved shear cracks which aect steel in tension and compression zone.

Punching shear mode is characterized by brittle response with minimal tension cracking damage and little or no yielding of reinforcement [START_REF] Miyamoto | Design concept for reinforced concrete slab structures under soft impact loads[END_REF]. However, an intermediate failure mode between exural and punching shear failure modes can also be expected [START_REF] Abbasi | Nonlinear nite element modelling of failure modes in rc slabs[END_REF]. A combined exure-shear failure mode is characterized by a exural failure at the early stages of loading followed by a transition into the punching shear failure mode.

The local damage of impacted slabs can be divided into four local failure modes: surface crushing, spalling of concrete and formation of a crater on the impacted surface, scabbing of the rear face and perforation [START_REF] Murthy | Impact analysis of concrete structural components[END_REF] (Figure 2.8). The severity of the local damage is primarily dependent upon the velocity of the impactor [START_REF] Kennedy | A review of procedures for the analysis and design of concrete structures to resist missile impact eects[END_REF]. According to [START_REF] Jovall | Airplane crash simulations: comparison of analyses results with test data[END_REF][START_REF] Murthy | Impact analysis of concrete structural components[END_REF] when the impactor strikes the slab for low impact velocities, a failure surface mode occurs. At failure surface, concrete is fallen o the impacted face of the slab at and around the impact zone. The impactor penetrates the slab to low depth and then bounces back. The further increase in velocity causes ejection of concrete from the upper surface and spalling occurs producing a spall crater in the surrounding zone of impact. The impactor penetrates The evaluation of each of the above failure modes is completely dicult either experimentally or numerically. For example, complete perforation usually takes place under high-velocity impact when slab deformations are localized in the impact zone, but it can also be achieved under lowvelocity impacts after the slab reaches its overall deections [START_REF] Abrate | Impact on composite structures[END_REF]. Spalling and scabbing generally occur during bending failure, while they cannot be observed during a full penetration [START_REF] Zineddin | Dynamic response and behavior of reinforced concrete slabs under impact loading[END_REF]. These failure modes must be taken into account when designing RC slabs under impact: it is obvious that exural failure mode represents the critical parameter to design longitudinal reinforcement, punching shear is important to determine shear reinforcement, while spalling and scabbing helps to evaluate the appropriate concrete properties [START_REF] Berthetrambaud | Analytical advantages and numerical modeling of impact testing on a structurallydissipating rock-shed slab (sdr)[END_REF].

The onset and growth of damage in slabs represent another aspect that should be treated as well to a better understanding of slabs dynamic behavior under impact, since the main orientation of major cracks can be helpful to distinguish the corresponding failure mode during impact event.

According to Abrate [START_REF] Abrate | Impact on composite structures[END_REF] that studied the impact on composite structures, during a low-velocity impact event, two types of cracks are observed, namely tensile cracks and shear cracks. Tensile cracks are created by tensile exural stresses and appear when in plane normal stresses exceed the concrete tensile strength. Shear cracks are inclined relative to the normal to the midplane and indicate a signicant role of shear stresses. Cracks initiate the damage process and follow a complicated and irregular path in concrete. For thick slabs, cracks are rst induced on the upper surface subjected to impact as a result of high and localized contact stresses, then they progress from the top downwards. For thin slabs, cracks are induced on the bottom surface due to bending stresses, which leads to a reversed shear cracks pattern. Therefore, a detailed prediction of nal cracks pattern is very dicult, but it is not necessary since its shape has no important contribution on the reduction in slab residual properties. dominate the slab behavior with a likelihood of punching mode. In case of soft impact, a direct shear failure could take place at the boundary conditions due to high stresses. These results are in agreement with Miyamoto and King [START_REF] Miyamoto | Design concept for reinforced concrete slab structures under soft impact loads[END_REF] study that also considered the eect of loading rate on slab failure mode. Miyamoto and King proposed a dynamic design procedure for RC slabs subjected to soft impact loads and found that failure modes are aected by loading rates, slabs fail under exural mode for low loading rates while the punching shear failure is dominant for higher loading rates. The experiments of Zineddin and Krauthammer showed also that the response of a slab is aected by the amount of steel reinforcement and the drop height (i.e. the impact velocity), RC slabs may change their failure mode from a exural failure at low drops to punching one under higher drops. As well, Martin [START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF] indicated that the overall response failure of a RC slab depends upon the impactor velocity and reinforcement strength: for lower impact velocities exural failure is more likely while for high impact velocities punching shear failure occurs; for a strong reinforcement exural failure is more likely and for a weaker one punching shear failure takes place. However, reinforcement ratio shows an adverse eect on RC slabs behavior under impact, as indicated by Abbasi et al. [START_REF] Abbasi | Nonlinear nite element modelling of failure modes in rc slabs[END_REF]. Abbasi et al. mentioned that for weakly reinforced slabs failure is identied by a global and pure exural mode with a measure of ductility due to reinforcement plasticity, whereas increasing reinforcement ratio enhances the punching capacity and slabs may fail under combined exural-shear mode for medium levels of reinforcement and under punching shear mode for high reinforcement ratio.

Energy considerations

An impact event can be dened as an interaction of two bodies where mechanical energies are transformed. Miyamoto and King [START_REF] Miyamoto | Design concept for reinforced concrete slab structures under soft impact loads[END_REF] mentioned that the energy criterion would be the most ecient method of designing concrete structures under impact loads, especially for a exural failure mode. They show that structural failure of a concrete structure subjected to impact likely occurs if the structure is not capable of absorbing all of the energy transmitted during impact. Therefore it is necessary to examine the energy balance, especially in case of evaluating the results of numerical impact analyses. As illustrated in Figure 2.9, before the impact there is only the impactor kinetic energy of which the main part is then transmitted to the RC slab when the impactor hits the slab.

After impact a small part of the impactor kinetic energy is transferred to slab as kinetic energy due to slab vibrations that occur as a result of impact, while a considerable part of this energy is converted into the energy absorbed by the structure. The energy not transmitted into the structure is progressively converted into elastic and plastic strain energy in the impactor as a result of its elastic and plastic deformation. The energy absorbed by the RC slab involves irrecoverable (plastic) energy Figure 2.9: Energy transformation process during an impact event [START_REF] Miyamoto | Design concept for reinforced concrete slab structures under soft impact loads[END_REF] and recoverable strain (elastic) energy. These energies are absorbed by the mechanisms of concrete and reinforcement [START_REF] Jovall | Airplane crash simulations: comparison of analyses results with test data[END_REF], as well as elastic and plastic parts of strain energy in slab are associated with its deformation after impact [START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF]. A part of the recoverable strain energy is subsequently re-transmitted to slab as kinetic energy in response to slab vibrations, as indicated in Figure 2.9 by a dashed line. In addition to plastic deformation, the irrecoverable energy in RC slabs results from the formation of cracks, friction and damping. Thus, a part of the initial kinetic energy of the impactor is dissipated as damage energy due to concrete damage and as viscous energy due to viscous damping inside the RC slab.

Martin [START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF] indicated that an energy balance of an impact on a RC slab can be written as follows:

E I kin0 = E I kin1 + E I str1 + E S kin1 + E S str1 + E S dam + E vis (2.8)
with E I kin0 = kinetic energy of impactor before impact E I kin1 = kinetic energy of impactor after impact E I str1 = strain energy of impactor after impact E S kin1 = kinetic energy of concrete slab after impact E S str1 = strain energy of concrete slab after impact E S dam = damage energy due to damage of concrete E vis = viscous energy due to viscous damping Miyamoto et al. [START_REF] Miyamoto | Integrated analytical procedure for concrete slabs under impact loads[END_REF] found that the kinetic energy transmitted to a RC slab structure by an impacting body as well as the energy absorbed by the structure are aected by the nal structural failure mode and RC slabs may respond in several ways as indicated by [START_REF] Hrynyk | Behaviour and modelling of reinfoced concrete slabs and shells under static and dynamic loads[END_REF]. For overall response failure modes the energy absorbed is due to global structural bending or shear deformations of the slab, while the main part of impact energy is dissipated through local damage mechanisms for local response failure modes. In case where both local and global dynamic response of the slab are involved, the initial kinetic energy of impactor is dissipated through a combination of structure deformations and local damage mechanisms. Moreover, Miyamoto et al. indicated that the energy absorbed by slabs under exural failure mode is higher than in case of punching shear failure mode which results in less energy being dissipated. The process of dissipating energy is also aected by mechanical properties of the RC structure and the impactor, it can vary signicantly depending upon the type of impact event (hard or soft) [START_REF] Hrynyk | Behaviour and modelling of reinfoced concrete slabs and shells under static and dynamic loads[END_REF][START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF]. Therefore, the energy transformation process of the impactor kinetic energy to dierent forms of energy mentioned in equation 2.8 is inuenced by the impactor velocity, the slab rigidity and the reinforcement of the concrete slab; e.g. in case of a strongly reinforced slab subjected to a soft impact with a high initial velocity, the majority of the impact energy is converted into strain energy of the impactor [START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF]. The impactor shape and size have also a considerable eect on the energy absorption mechanism since failure modes can be quite dierent, consequently impactors with nose shape dissipate more energy during penetration than at impactors and the energy absorbed during a perforation failure mode increases with impactor diameter [START_REF] Abrate | Impact on composite structures[END_REF].

Methods to determine the response of RC slabs under impact

This section presents the three approaches that are usually used to determine the global and local dynamic responses of RC slabs subjected to impact loading. Several studies from the literature that are considered as important investigations in the context of impact phenomenon eect on RC slabs are also presented for each approach. These three approaches can be classied as follows:

Experimental methods based on either laboratory or eld studies and leading to empirical formula;

Simplied analytical solutions such as spring-mass system;

Advanced numerical approaches such as nite element method.

Experimental approaches

This approach, based on either laboratory or eld studies, is considered to be fundamental for extending the understanding of the dynamic behavior of RC slabs under the action of impact loading.

The main aim of this approach is to study the actual response of the slab including the development of crack patterns and the resulting failure modes. Various data can be gathered from tests such as the variation of the impact load with respect to the slab deection, the impactor acceleration, the reactions at supports and the reinforcing steel strains. Experimental data are important for validating analytical and numerical models. The accuracy of the measurement of physical parameters depends on observations and the equipment used to store data from the test. Empirical expressions which are derived from experimental data are especially important due to their simplicity to represent the complex impact phenomena and can be useful to get a rst order approximation of impact eects.

However, experimental approach is not a cost-eective practicable solution since it requires a series of full-scale tests in order to perform a reliable estimation of RC structures response. This approach can be an excessively time-consuming and costly procedure, especially in terms of providing the necessary test materials and equipment.

Zineddin and Krauthammer tests

Zineddin and Krauthammer [START_REF] Zineddin | Dynamic response and behavior of reinforced concrete slabs under impact loading[END_REF] performed an experimental study in the aim of understanding the dynamic behavior of structural concrete slabs under impact loading, and investigating the eect of reinforcement type and the applied load on the dynamic response of RC slabs. 9 slabs of 3.353x1.524 m span and 90 mm thickness were tested and reinforced with three dierent types of reinforcement placed along the slabs width and length at spacing of 152 mm, as follows: Slabs were impacted with a cylindrical drop hammer with a 2608 kg mass and 250 mm diameter.

The impact mass was dropped from dierent heights of 152 mm, 305 mm or 610 mm at the center of slabs in order to examine the variation of slab failure modes under low and high drop heights.

Slabs were supported on a sti steel frame of 305 mm width and bolted on all four sides of the top face with steel channels of the same width (Figure 2.10). Zineddin and Krauthammer mentioned that boundary conditions could be described as somewhere between simply supported and xed.

Output that were gathered from these impact tests included impact force-time history, slab deection, reinforcement strains, as well as accelerations that were measured for the slab, the steel frame and the impact mass. Cracks propagation was also examined by means of high-speed videos.

Zineddin and Krauthammer studied the transition of slab behavior modes in term of force-time history curves and indicated that, according to these curves, four stages could be distinguished:

1. Slab resistance to the impact load as an elastic plate, followed by concrete failure at the impact zone.

2. Reinforcement resistance with severe cracking of concrete around the perimeter of the impacted zone.

3. Steel yielding that started in bars under the impact mass and propagated radially in the non-impacted zone.

4. Increase in the slab deection at the center, which may lead to a concrete ejection or slab perforation. Figure 2.11 shows these four stages for 3 slabs with 3 dierent types of reinforcement and impacted with a drop height of 152 mm. These curves indicate that slab response is aected by reinforcement type and, particularly by reinforcement ratio. Although all curves had a similar shape, the peak load for slab with lower reinforcement ratio (Reinf. 1) was not signicant as the peak for other slabs and was reached at earlier time. Reinforcement ratio had also an important inuence on slab failure modes, as can been seen in Figure 2.12 which presents the failure modes of two slabs impacted with a drop height of 610 mm but with dierent reinforcement ratio. In fact, slabs with more steel reinforcement showed a localized punching shear failure (Figure 2.12.a), whereas slabs with less steel reinforcement failed by brittle failure of concrete with a considerable amount of debris ejected from the bottom surface (Figure 2.12.b). Reinforcement bars were yielded for both slabs, but one longitudinal steel bar was cut in the case of slab with lower reinforcement ratio (Reinf. 1) due to the excessive shear failure that occurs at the impact zone and to its location at the mid-thickness of the slab (Figure 2.12.c).

Zineddin and Krauthammer found that slabs response was also aected by drop height of the impact mass, hence RC slabs may change their failure mode from a exural at low drops to punching one under higher drops. Slabs impacted with the lower drop height of 152 mm were designed in the objective to fail by exural failure mode regardless of their reinforcements. Nevertheless, only slab reinforced with Reinf. 3 type underwent exural failure with large cracks on the bottom surface, and the two slabs reinforced with Reinf. 1 and Reinf. 2 types failed in a brittle manner since the change in the failure mode was compensated by the strength increase due to the loading rate. For University by Chen and May [START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF]. The aim of these impact tests was to investigate the behavior of RC slabs subjected to high-mass, low-velocity impact and to provide sucient input data with high quality for the validation of numerical procedures.

Six square slabs were tested under impact loads using a drop-weight impact system. Four 760 mm square slabs with a depth of 76 mm (Slabs 1-4) were reinforced with 6 or 8 mm diameter high The striker that struck the slabs was released with a velocity of up to 8.7 m/s and consisted of a mass, a load cell and an impactor. The tests were performed with a striker mass of up to 380 kg and a stainless steel hemispherical impactor of a 90 mm diameter dropped at the center of slabs (Figure 2.15a), except for Slab 3 which was impacted by a mild steel cylindrical impactor of a 100 mm diameter (Figure 2.15b). Although the vertical movement of the impactor was not restrained and a second impact might occur after a rebound, only the rst impact of duration of 20-30 ms was considered to study the slabs behavior. Table 2.1 summarizes the details of impact test for each slab. These experimental studies, considered as well monitored, allowed gaining a better understanding of the impact behavior of RC slabs. The transient impact force-time histories were determined using a load cell placed between the mass and the impactor. Strain gauges placed inside bars and accelerometers were used to measure transient reinforcement strains and accelerations, respectively.

Experimental output were sampled at a rate of 500 kHz and ltered with a Butterworth lter using a cuto frequency of 2 kHz. The local impact response of slabs was recorded with a high-speed camera to show the development of failure modes in terms of cracking, scabbing and spalling.

Impact force-time histories show a plateau for slabs 2, 3, 4 and 6 after reaching the peak load (Figure 2.16). The presence of this plateau is related to the fact that local failures due to penetration and scabbing, accompanied by reinforcement yielding, were observed. As can be seen in Figures 2.17.d and 2.17.e, scabbing was not fully developed for slab 5 while more scabbing was observed on slab 6 which have the same geometric and material properties as slab 5, but impacted by a higher mass. Moreover, these two slabs of 150 mm thick show a less amount of concrete debris ejected due to scabbing than the 76 mm thick slabs. For slab 4 which is similar to slab 2 but impacted with a higher velocity, a signicant amount of penetration of the impactor in the slab occurred, indicating that the thickness used was not sucient to prevent penetration for velocities higher than 6.5 m/s (Figure 2.17.c). As illustrated in Figures 2.17.a and 2.17.b, the size and shape of scabbing zone could be aected by the impactor shape since the stress wave caused by the progressive contact between the impactor and the slab could propagate more uniformly in the case of a hemispherical impactor.

Thus, the scabbing zone on slab 2 bottom face caused by the hemispherical impactor had a more circular shape than that created on slab 3 with a at impactor.

From these experimental investigation results, it can be seen that several parameters such as the slab thickness, the impactor mass, velocity and shape have a signicant eect on RC slabs behavior under impact.

Hrynyk tests

The research program of Hrynyk [START_REF] Hrynyk | Behaviour and modelling of reinfoced concrete slabs and shells under static and dynamic loads[END_REF] layers of a slab with additional links at the center that were used in the aim of stiening the impact region. In order to reduce axial connement eects with ensuring an overall stability boundary conditions during tests, the four corner supports were restrained to translate vertically and free to rotate in all directions, while lateral restraint conditions varied with a simple pinned support, two supports restrained to translate laterally in only one direction, and a support free to translate laterally in all directions (Figure 2.18.a). Slabs were impacted sequentially with a drop-weight of a constant velocity of 8.0 m/s but an increasing mass ranging from 120 kg to 300 kg. The contact surface of the impactor with slabs had a at square shape of 300x300 mm and the impact was dened as hard. For all except one of the experiments, slabs were impacted with an initial mass of 150 kg which was increased for the subsequent impacts as indicated in Table 2.2. Hrynyk considered that the impact testing of slabs could be terminated in case of a completion of the impact loading protocol presented in Table 2.2, a signicant decrease of the measured support reaction forces between two successive impacts, or a potential damage of the instrumentation under an additional impact.

Experimental results were obtained using detailed instrumentation and data measurement techniques such as load cells to measure reaction forces at the four corner supports, accelerometers to estimate the transverse acceleration behavior of slabs and the applied impact force, potentiometers to determine transverse and lateral slab displacements, and strain gauges to evaluate the magnitude and rate of strain in reinforcing bars. Furthermore, a high-speed video camera was used to observe cracking patterns, mass penetration and concrete scabbing. observed during three consecutive impacts with masses ranging from 150 kg to 210 kg. Cracking patterns were marked after each impact event. As can be seen, under the rst impact, cracks developed circumferentially along the reinforcing bars conguration and more densely in the impact region, while no scabbing or penetration was apparent. Then a punching shear failure occurred at the location of circumferential cracks as a result of subsequent impacts, associated with extensive concrete scabbing and mass penetration.

Hrynyk varied the reinforcement ratio parameter in order to study its inuence on the response of slabs. It was found that slabs stiness increased with the reinforcement ratio while it had a limited inuence on slabs capacity to absorb impact energy. The nal cracking patterns of three RC slabs with dierent reinforcement ratio presented, for all cases, an extensive damage at the impact region with a concrete scabbing occurring on the bottom surface (Figure 2.20). The bottom reinforcement layer bars were more exposed in the case where the slab was less reinforced. It should be noted that the experimental program of Hrynyk provided all data necessary for a numerical analysis since it also included a series of tensile coupon tests to evaluate the stress-strain curve of reinforcing steel, as well as a series of concrete cylinder and bending prism tests which were performed to characterize the properties of concrete and investigate its compressive and tensile behaviors. Furthermore, in comparison to the RC slabs response, SFRC slabs showed a dierent behavior as the addition of steel bers resulted in an important increase in slabs strength and postcracking stiness. Slabs with higher ber volume fraction led to reduce the development of the localized punching behavior and prevented a fully scabbing of concrete from the bottom surfaces of slabs. The ber volume fraction had a signicant inuence on the midpoint displacement responses, the support reaction magnitudes and the increasing of impact energy capacities of slabs that tended to fail more toward exural mode.

Analytical approaches

In order to be able to choose an appropriate and accurate simplied model, the eect of impact on composites has been extensively studied by several researchers [START_REF] Abrate | Modeling of impacts on composite structures[END_REF][START_REF] Christoforou | Impact dynamics and damage in composite structures[END_REF][START_REF] Christoforou | Analysis of impact response in composite plates[END_REF][START_REF] Christoforou | Scaling of low-velocity impact response in composite structures[END_REF][START_REF] Christoforou | Eect of exibility on low velocity impact response[END_REF][START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF]. They indicated that understanding the dierent types of response and the knowledge of parameters governing the response type are necessary for the selection of an appropriate analytical model. Abrate [START_REF] Abrate | Impact on composite structures[END_REF] specied that several analytical models may be used to predict the impact dynamics, namely spring-mass models, energy-balance models, innite plate models and models based on plate theory.

In this section, analytical mass-spring models developed in various research studies to predict the dynamic response of RC slabs subjected to low impact velocity are presented.

The analytical approach, which is commonly regarded to be the most representative simplied method to simulate an impact event between a slab and an impactor, consists in a one or more degrees of freedom mass-spring system. The slab and the impactor are modeled as two colliding masses and springs are used to represent the local and global behaviors of the slab. In the case of hard impact (the impactor is considered to be rigid), the analytical model is based on the assumption that the deformation and failure of the impactor are negligible. The local slab response is normally neglected and the approach focuses on the global response which plays an important role for low impact velocities. Analytical approaches have limited applicability since they are based on assumptions which separate the local and global response of the slab, therefore they can only be used as an initial approximation of the slab impact behavior.

Tonello model

Tonello IC design oce used a mass-spring model with one degree of freedom to represent impact on RC slabs [START_REF] Tonello | Généralité et approche de modèles simples[END_REF]. Their approach consisted in estimating the maximum dynamic load reached during an impact on the structure and took into account the elasto-plastic behavior of materials. Several assumptions were adopted for this model: the impact is considered as soft, the impactor stays in contact with the slab after impact (no rebound or penetration are allowed), and the slab equivalent mass is calculated based on a static bending deformation (Figure 2.21).

CEB model

The Bulletin of Information initiated by CEB [START_REF] Ceb | Concrete structures under impact and impulsive loading[END_REF] provides simplied models in the aim of helping and guiding civil engineers in designing RC structures subjected to impact. This bulletin shows that, in the case of a hard impact where the kinetic energy of the impactor is absorbed by the structure deformation, an impact problem on RC slabs can be reduced to a mass-spring model with two degrees of freedom as shown in Figure 2.22. In this case, the slab local behavior must be considered as well as its overall deformation and the two dierential equations of equilibrium for the two masses can be written as:

m 2 ü2 + R 2 (u 2 -u 1 ) = 0 m 1 ü1 + R 1 u 1 -R 2 (u 2 -u 1 ) = 0 (2.9)
m 1 and m 2 represent the two colliding masses. The spring R 1 represents the overall behavior of the slab and has a nonlinear force-displacement relationship which can be considered similar to a static behavior (Figure 2.23.a), while R 2 represents the local behavior and simulates the contact force between the impactor and the structure. The nonlinear relationship of R 2 is more dicult to determine, but can be obtained by using a nonlinear nite element code. CEB indicated that the general force-displacement relationship of R 2 in the contact zone of a solid, e.g. concrete, shows an elastic compression phase in the range 0 < u < u 1 , followed by an elastic-plastic phase for u 1 < u < u 2 where irreversible internal damage occurs (Figure 2.23.b).

Abrate model

Abrate [START_REF] Abrate | Modeling of impacts on composite structures[END_REF] has presented an approach for selecting an appropriate model for analyzing the impact dynamics for each particular case. He classied models according to how the structure is modeled: spring-mass models, energy balance models, complete models, and a model for impact on innite plates.

The complete model consists of the impactor mass M 1 , the eective mass of the structure M 2 , the nonlinear contact stiness K, the shear stiness K s , the bending stiness K b and the nonlinear membrane stiness K m (Figure 2.24.a). This model presents two situations for which a nonlinear SDOF can provide accurate predictions of the contact force history. The rst situation is when the overall deection of the structure is negligible compared to the local indentation. In this case, the response of the structure can be modeled by the spring in Figure 2.24.b and the equation of motion is:

M 1 ẍ1 + kx 3/2 1 = 0 (2.10)
The second situation is when the membrane stiening is signicant and the deections of the structure are large. In this case, the local indentation is negligible and the equation of motion of the nonlinear SDOF model is: The principle of the model consisted in separating the contact phase between the block and the slab from the slab free vibration phase after impact. Thus, two mass-spring-damper models were generated: the contact model with two degrees of freedom that simulates the contact phase during the rst milliseconds (Figure 2.25.a) and the post-impact model with one degree of freedom that simulates the free vibration phase for several seconds (Figure 2.25.b).

M ẍ + k b x + k m x 3 = 0 (2.11)
Delhomme et al. [START_REF] Delhomme | Simulation of a block impacting a reinforced concrete slab with a nite element model and a mass-spring system[END_REF] indicated that M * and k s which represent the equivalent mass and the static stiness of the slab, respectively, are not constant during the impact phenomenon since they depend on the slab surface in bending. Thus, M * 1 is lower than M * 2 and k s1 is higher than k s2 , since in the contact phase only a reduced surface is considered in bending, while the equivalent mass and the stiness are calculated for the total slab surface for the post-impact model.

The contact was considered as an elasto-plastic in order to take into account the cracking of concrete and the Hertz contact law was used to estimate the contact force. Therefore, the equations of the contact model can be written as:

m b (ü b + g) + k c (u b -u s ) 3/2 = 0 M * 1 üs -k c (u b -u s ) 3/2 + k s1 (u s ) + c 1 us = 0 (2.12)
where m b , u b are respectively the mass and the displacement of the block, u s is the slab displacement at the impact point. The contact stiness k c and the impedances c were derived from experimental results.

Yigit and Christoforou model

The model shown in Figure 2.26.a was proposed by [START_REF] Yigit | Limits of asymptotic solutions in low-velocity impact[END_REF] to model low velocity impact response of composite plates. In this model, m s represents the modal mass and the impactor has a mass of m i and an initial velocity of v 0 . The plate stiness K st and the linear contact stiness K y account for the structural contribution and the local contact behavior, respectively. The amount of energy transferred to the structure during impact is presented by the impedance c. This full model is used for impacts that do not correspond to small or large mass impacts.

According to [START_REF] Christoforou | Scaling of low-velocity impact response in composite structures[END_REF], if the size of the structure is very large and the impactor mass is very small, the impact is dened as an innite structure impact which can be modeled as a SDOF (Single Degree Of Freedom) system, where the mass of the impactor is supported by the local contact stiness and [START_REF] Yigit | Limits of asymptotic solutions in low-velocity impact[END_REF] the structure acts as a damping mechanism (Figure 2.26.b). During this kind of impacts, the waves are not reected back from the boundaries.

On the other hand, if the impactor mass is very large, the mass of the structure can be neglected and the impact response can be modeled with a SDOF system, where the mass of the impactor is supported by the local contact stiness and the static stiness of the structure in series (Figure 2.26.c). This type of impact is dened as a quasi-static impact and its duration is relatively long.

Finite element approaches

In order to solve a complete impact problem and to gain knowledge of the physical behavior of slabs, numerical approaches such as nite element (FE) method can be considered to be more appropriate to determine the behavior of RC slabs subjected to impact loading. A three-dimensional nite element model allows modeling, as close as possible to the real case, the various aspects of any complex problem including the boundary conditions and the loads applied to the structure. In the case of impacted slabs, the RC slab and the impactor can be fully modeled and nonlinearities due to material behaviors can be taken into account. The numerical simulation of RC slabs subjected to impact loads is a complex phenomenon. Therefore an accurate simulation requires using an appropriate model for concrete and a contact algorithm which can be used in nonlinear explicit nite element analysis. Once the FE model eectiveness is veried with precision, numerical simulations can replace costly full-scale tests by providing more details of the dynamic response and an expansion of experimental measurements which would be inaccessible during tests. They also provide a large output data including the stress and deformation elds and make possible to perform sensitivity analyses and parametric studies to determine the inuence of one parameter or another. Despite all these advantages that can oer FE approaches, the main diculty consists in choosing appropriately the set of numerical tools in order to accurately simulate the impact phenomenon and to identify the induced failure modes due to transient dynamic loading.

Berthet-Rambaud et al. model

Berthet-Rambaud et al. [START_REF] Berthetrambaud | Analytical advantages and numerical modeling of impact testing on a structurallydissipating rock-shed slab (sdr)[END_REF] investigated numerically the response of SDR protection galleries in order to improve their design and reduce costs in response to the increasing request of using such type of structures to provide protection in mountainous regions against rockfalls risk. These galleries are composed of a RC slab which is directly subjected to falling rock impacts, and steel fuse supports that allow transmitting slab reactions to the sub-structures and dissipating energy. Berthet-Rambaud et al. used ABAQUS to propose a 3D model of the slab and supports (Figure 2.27) and compared their numerical results to experimental tests that were conducted by Delhomme et al. [START_REF] Delhomme | Simulation of a block impacting a reinforced concrete slab with a nite element model and a mass-spring system[END_REF] on a 1/3 reduced scale system model. The slab of 4.8 m wide and 12 m length was modeled and meshed with C3D8R solid elements with four layers in the thickness of 0.28 m. The slab was impacted, at various positions, by a cubic RC block of 450 kg mass and a velocity varying from 17.2 to 24.2 m/s.

The block was introduced in the FE model respecting the actual block geometry of the experiments and meshed with C3D4 tetrahedral elements. The slab is supported by two lines, each with 11 steel fuse supports spaced at a distance of 1.14 m. The supports were considered with all their components in the model, but their simulation will not be detailed in the present study (for more details about supports simulation, see [START_REF] Berthetrambaud | Analytical advantages and numerical modeling of impact testing on a structurallydissipating rock-shed slab (sdr)[END_REF]). The slab and block reinforcements were represented by bar elements which were embedded within the concrete mesh in order to create a perfect bond between steel and concrete. Steel was modeled as an elasto-plastic material with hardening, while the PRM (Pontiroli-Rouquand-Mazars) damage model that uses one scalar damage variable [START_REF] Pontiroli | Comportement au soue des structures en béton armé, analyse expérimentale et modélisation[END_REF] was used to represent concrete in the numerical simulation and was implemented in the FE code using a Fortran subroutine. The kinematic contact algorithm was adapted to model the contact between the slab and the block using hard contact for the normal interaction and a Coulomb-type friction law for the tangential reaction. The explicit time integration method was used to solve the problem and the Hillerborg regularization method [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and nite elements[END_REF] was integrated to reduce mesh dependence eects. An analysis of two successive impacts at the same point and under the same conditions was performed. After the rst impact, the damaged zone was repaired and the damage of elements corresponding to this zone was reduced to zero. However, the potential damage on the remaining part of the slab around the repaired part is taken into account by introducing three damaged zones in the model. The damage distribution was supposed to have a cone shape that starts at the impact point and goes through the slab thickness (Figure 2.28). The eect of gravity was included in the model through a step of duration of 0.15 sec that allows a numerical gravity application before the impact step.

A parametric study on the tangential friction at the slab and the impactor interface was carried out since this parameter has a considerable inuence on the sliding of the impactor corner in contact with the slab. An optimal value of 0.15 was found to be in agreement with experimental results and permitted to correctly identify the experimental impact phase (Figure 2.29.a). Following the calibration on the friction parameter, a comparison between experimental and numerical results of the vertical displacement evolution at several points of the slab proves the accuracy of Berthet-Rambaud et al. model in predicting the overall structure behavior. Figure 2.29.b depicts the vertical displacement of a point close to the impact zone and shows that slab oscillations were very well simulated. However, numerical amplitude values were found to be high after the rst oscillation, which may be related to a general damping problem or to the complexity of boundary conditions.

Mokhatar and Abdullah model

The aim of the study performed by Mokhatar and Abdullah [START_REF] Mokhatar | Computational analysis of reinfoced concrete slabs subjected to impact loads[END_REF] was to verify the accuracy of numerical modeling technique in assessing the response of RC slabs and the steel reinforcement failure mechanism when subjected to impact loading. A FE model was carried out using ABAQUS in order to gain a better understanding of impacted slabs behavior and was validated with Slab 3 experimental results of Chen and May tests [START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF] described in section 2.3.1.2. The model was divided into four main parts: the concrete region of slab, the steel reinforcement, the steel support and the steel projectile (Figure 2.30). The concrete region was modeled using eight-node continuum solid elements (C3D8R) while the steel support was considered as rigid and modeled with undeformable discrete rigid elements (R3D4). These two parts were connected to each other using the tie contact technique after assembling all elements. A proper bond action using the embedded technique was created between the solid elements of the concrete region and the two-node beam elements of steel reinforcement. The interaction between the steel impactor and the concrete region was dened as surface-to-surface contact with the kinematic contact method for mechanical constraint formulation and a friction coecient of 0.2. Material properties, as well as stress-strain and damage curves were chosen based on several experimental works [START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF][START_REF] Jankowiak | Identication of parameters of concrete damage plasticity constitutive model[END_REF][START_REF] Tahmasebinia | Numerical modelling of reinforced concrete slabs suject to impact loading[END_REF]. The increasing of concrete compressive and tensile strengths due to impact load was considered with a rate of 1.5.

The FE model was validated with experimental results using impact force-time history and the nal crack pattern. First, a mesh sensitivity was performed in order to nd the suciently rened mesh which provides a reasonably accurate result (Figure 2.31.a). Then, the transient impact load was estimated for three constitutive models of concrete, the Concrete Damage Plasticity (CDP) considered having a brittle-cracking behavior, and the Drucker-Prager and the Cap-Plasticity models characterized by their ductile behaviors. The results show that ductile models can evaluate the RC slabs behavior under dynamic impact loading and provide a good agreement with experimental results rather than brittle-cracking model (Figure 2.31.b). However, it can be seen from (Figure 2.32) that the nal crack pattern obtained using the CDP model shows a good correlation with experimental results, but no eect of spallation could be observed which does not correspond to the actual slab failure mode.

Trivedi and Singh model

The numerical analysis performed by Trivedi and Singh [START_REF] Trivedi | Prediction of impact induced failure modes in reinforced concrete slabs through nonlinear transient dynamic nite element simulation[END_REF] is among other numerical studies that focus on identifying the global and local failure modes of RC slabs subjected to impact using ABAQUS. Due to symmetry, Trivedi and Singh simulated one fourth of RC slabs of Zineddin and Krauthammer experimental tests [START_REF] Zineddin | Dynamic response and behavior of reinforced concrete slabs under impact loading[END_REF] [START_REF] Mokhatar | Computational analysis of reinfoced concrete slabs subjected to impact loads[END_REF] As mentioned in section 2.3.1.1, slabs were impacted by a cylindrical drop hammer which was not introduced in Trivedi and Singh study, hence no contact modeling was presented. Alternatively, a time dependent pressure loading applied at the slab center was considered to model the impact loading and was distributed over a circular area with the same diameter as the hammer. The pressure loading values were calculated by distributing the peak load over the impacted zone area, while their amplitude curves in term of time were obtained using load-time histories available in Zineddin and Krauthammer study (Figure 2.33.b). The amplitude values represent the ratio of the load value at a certain time to the peak load.

Slabs were supposed to be fully constrained at the edges (no displacement is allowed in the three directions), while displacement was restrained vertically on the x-plane of symmetry and horizontally on the y-plane of symmetry. The concrete region of slabs was modeled as a 3D deformable part and meshed with 3D continuum elements, whereas reinforcement was considered as a deformable wire part and meshed with linear truss elements. No slip between concrete and steel was assumed, hence reinforcement was embedded in the concrete part dened as the host component. Material behaviors were incorporated in the FE model using the CDP model for concrete and a plasticity model for steel.

Two approaches for post-cracking softening model were used to dene tension stiening for CDP model, namely the limiting strain and fracture energy based approaches. Results showed that the fracture energy cracking criterion gave mesh independent results which were in agreement with [START_REF] Trivedi | Prediction of impact induced failure modes in reinforced concrete slabs through nonlinear transient dynamic nite element simulation[END_REF] those obtained in experiments, but results using the strain criteria were inconsistent due to a mesh sensitivity problem. Therefore, Trivedi and Singh limited their discussions to the fracture energy approach and recommended its use as a robust method to study the behavior of RC structures under impact. Besides the tension stiening, the eect of strain rate was investigated in the aim of developing an adequate nonlinear dynamic FE model. Based on Tedesco et al. study [START_REF] Tedesco | A strain-rate-dependent concrete material model for adina[END_REF], Trivedi and Singh used a rate of 1.5 for concrete in compression and a rate in range of 1.7 to 1.75 for concrete in tension. These rates are functions of the strain rate and allow taking into account the increase in compressive and tensile strengths due to a dynamic loading. However, no signicant change was observed in numerical results due to the use of strain rate, since the problem was associated with low strain rates of the order of 10 -4 to 10 -2 s -1 .

Trivedi and Singh studied several criteria in order to predict the failure modes of slabs, such as inection point, limiting strain based failure, bi-axial failure, shear failure, strain in steel and tensile damage. Only some results using the rst two criteria are presented here: Miyamoto et al. identication by inection point using the slab deformed shape [START_REF] Miyamoto | Analysis of failure modes for reinforced concrete slabs under impulsive loads[END_REF]. As can be seen in gure 2.34.a, for a RC slab with one mesh of longitudinal and transversal steel bars impacted with a height of 610 mm, the formation of an inection point in the deected prole indicates the transition from exure to punching shear failure which occurs at the slab center under the impact region. A exural mode can be identied by a smooth deection curve with parabolic shape (Figure 2.34.b).

Limiting strain based failure criterion by evaluating strain variations in XZ and YZ planes at various elements of concrete selected at the top and the bottom of slabs, and localized at and distant from the impact zone. Based on this criterion, a slab has a mixed mode failure of exure and shear when strain in top elements localized at the impact zone exceeds the compressive limiting strain and the top distant elements show tensile behavior. In addition, bottom elements near the impact zone and distant elements have tensile and compressive strain, respectively. When a slab fails in exure, elements at the impact zone show a crushing failure at the top surface and cracking failure at the bottom surface (Figure 2.35).

Mechanical behavior of concrete

As a strongly heterogeneous material, concrete shows a complex nonlinear mechanical behavior characterized by softening response in tension and low conned compression. Softening is dened as decreasing stress with increasing deformation beyond the concrete compressive strength, and results in a reduction of the unloading stiness of concrete and irrecoverable deformations localized at cracks. However, the connement of concrete with reinforcement results in a signicant increase in the strength of compressed concrete and in a ductile hardening response described by increasing stress with increasing deformations. In the current study, uniaxial compressive and tensile stressstrain curves based on experimental observations are adopted and dened in terms of concrete properties in compression and tension, respectively, as follows:

σ c = f (E c , f c , f cu , ε c , ε c1 , ε cu1 ) (2.13)
Where σ c represents the concrete compressive stress values calculated, E c is the Young's modulus of concrete, f c is the maximum compressive strength of concrete, f cu is the compressive stress of concrete corresponding to the ultimate strain, ε c represents the concrete compressive strain value corresponding to the stress σ c , ε c1 is the compressive strain of concrete at the compressive strength f c , and ε cu is the ultimate strain of concrete in compression.

σ t = f (E c , f t , ε t , ε cr ) (2.14)
Where σ t represents the concrete tensile stress values calculated, f t is the tensile strength of concrete, ε t represents the concrete compressive strain value corresponding to the stress σ t , and ε cr is the tensile strain at concrete cracking at the tensile strength f t .

Uniaxial compression

The properties of concrete in uniaxial compression are obtained from cylinder tests or cube tests. The nonlinear stress-strain behavior under uniaxial compressive stress, shown in Figure 2.36, is divided into three regions: linear elastic, non-linear plastic (hardening region) and post-peak stress (softening region). The rst region, in which the concrete behaves almost linearly, is observed during 30-60% of the maximum uniaxial compressive strength f c . At this level of strain, the specimen deformation is recoverable and localized cracks are initiated but they do not propagate. This could be linked to the balance of energy present in the concrete specimen which is in this case less than the energy required to create new cracks.

In the plastic regime the response is typically characterized by stress hardening followed by strain softening beyond the peak stress. Beyond the limit of elasticity, the deformation is no longer recoverable and the stress-strain curve begins to deviate from a straight line. The concrete behaves in a non-linear manner up to the peak stress and the crack system continues to grow slowly with the increasing of the applied load. During this phase, cracks cause an increase in volume and a small lateral expansion associated with Poisson's ratio eect is observed.

Immediately after the peak stress, the concrete undergoes strain softening and the lateral expansion increases dramatically with the cracks propagation. The energy released by the propagation of a crack is greater than the energy needed for propagation. The load carrying capacity of the test specimen decreases and the stress-strain curve starts to decrease until a crushing failure occurs at the ultimate strain. The shape of the stress-strain curve depends on the concrete strength which continues to behave in a linear manner up to a higher stress level than that for normal strength concrete. Also higher strength concrete must have a steeper descending curve and exhibits a brittle failure mode due to the fact that the specic fracture energy of concrete in compression does not increase much with the concrete strength.

When the concrete specimen is unloaded from any point of the strain softening branch of the stress-strain curve, the nonlinear behavior can be easily noted from the residual deformation present in the stress-strain curve. The concrete stiness is reduced because of the increase of the internal damage which causes irrecoverable deformation in loading.

Compression stress-strain curve without detailed laboratory test results The way in which the concrete behaves can be accurately determined on the basis of uniaxial compression test results. However, problems arise when no such test results are available and when the only available quantities for the analysis are the compressive strength and the modulus of elasticity of concrete.

Many uniaxial stress-strain relationships for concrete in compression were proposed in the literature, such as the expressions developed by [3,4,[START_REF] Cervera | Nonlinear analysis of reinforced concrete plates and shells using a three dimensional model[END_REF][START_REF] Desayi | Equation for the stress-strain curve for concrete[END_REF][START_REF] Kratzig | An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[END_REF][START_REF] Majewski | The mechanics of structural concrete in terms of elasto-plasticity[END_REF][START_REF] Pavlovic | Bolted shear connectors vs. headed studs behaviour in push-out tests[END_REF][START_REF] Wahalathantri | A material model for exural crack simulation in reinforced concrete elements using abaqus[END_REF][START_REF] Wang | Nonlinear nite element analysis of concrete structures using new constitutive models[END_REF].

Desayi & Kirshnan formula [START_REF] Desayi | Equation for the stress-strain curve for concrete[END_REF] Among many formulas proposed in the literature to represent nonlinear stress-strain characteristics of concrete, Desayi and Krishnan suggested a simplied equation that describes the ascending and descending parts of concrete compressive stress-strain curve in a single expression as follows:

σ c = E c ε c 1 + ( εc ε c1 ) 2 (2.15)
Madrid parabola [START_REF] Cervera | Nonlinear analysis of reinforced concrete plates and shells using a three dimensional model[END_REF][START_REF] Cervera | Nonlinear analysis of reinforced concrete plate and shell structures using 20-noded isoparametric brick elements[END_REF] Considered as a good relation by CEB, the conventional Madrid parabola is a parabolic equation that allows dening the uniaxial stress-strain curve of concrete in compression as given below:

σ c = E c ε c [1 - 1 2 ( ε c ε c1 )]
(2.16)

However, Kmiecik and Kaminski [START_REF] Kmiecik | Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration[END_REF] mentioned that this function is not accurate enough to correctly describe the behavior of concrete. They also suggested, in case of parabolic relations, using a lower value of initial modulus E c so as to ensure that the parabola curve passes through the correct value of f c .

Eurocode 2 curve (EN 1992-1-1) [4,[START_REF] Ecp | Commentary eurocode 2[END_REF] According to Eurocode 2 (EC2), concrete acts elastically up to 0.4f c . Then, the compressive stress σ c for hardening and softening parts is dened as a function of uniaxial strain ε c by using the concrete compressive strength f c and expressed by:

σ c = f c kη -η 2 1 + (k -2)η (2.17) with k = 1.1 E c ε c1 /f c and η = ε c /ε c1 .
EC2 provides a formulation of the strain at peak stress in term of the compressive strength (f c in MPa), which can be used for normal and high strength concrete:

ε c1 = 0.7f 0.31 c (2.18)
A constant value ε cu = 0.0035 can be used for concrete with a characteristic value f ck less than 55 MPa, however for f ck ≥ 55 MPa the ultimate strain value should be replaced by:

ε cu = 2.8 + 27 [(98 -f c ) /100] 4 (2.19)
σ c = f c / 1 + {(η 1 -1) / (η 2 -1)} 2 (2.20)
where

η 1 = ε c /ε c1 and η 2 = (ε c1 + ε c0 ) /ε c1 . ε c0
σ c = f c 1 β - sin(µ αt 1 .α t 2 π/2) β. sin(α t 2 π/2) + µ α , ε cu 1 < ε c < ε cu 2 (2.21) where µ = (ε c -ε cu 1 )/(ε cu 2 -ε cu 1 ) and β = f c /f cu 1 .
At the end of the sinusoidal descending part at strain ε cu 2 , concrete strength was reduced to f cu 2 by a factor α = f c /f cu 2 . Pavlovic et al. adopted a value of 20 to the reduction factor α and 0.03 to the ultimate strain ε cu 2 . Factors α t 1 and α t 2 , which represent the governing tangents angles at the starting and end points of the sinusoidal curve respectively, were chosen in order to obtain a stress-strain curve with a smooth overall shape, hence α t 1 = 0.5 and α t 2 = 1.0.

Chinese code curve (GB50010-2002) [3] Chinese code for design of concrete structures is considered as the only code among many design codes that allows dening uniaxial concrete strength at high strains. The ascending and descending parts of stress-strain curve are described by two dierent softening parameters related to the material strength and their values generally range from 0.4 to 2.0. According to the design Chinese code, the uniaxial compressive stress-strain curve of concrete can be estimated using the following equations:

σ c = f c α a η + (3 -2α a )η 2 + (α a -2)η 3 , η ≤ 1 σ c = f c η/ α d (η -1) 2 + η , η > 1 (2.22)
where η = ε c /ε c1 , while α a and α d represent the ascending and descending softening parameters and they are expressed in term of the concrete strength (f c in MPa) as follows:

α a = 0.0475f c (2.23) α d = 0.0485f c (2.24)
Wang and Hsu curve [START_REF] Wang | Nonlinear nite element analysis of concrete structures using new constitutive models[END_REF] Wang and Hsu proposed two equations to describe the ascending and descending parts of concrete stress-strain curve in compression. These equations are expressed in terms of concrete compressive strength f c , concrete strain at maximum compressive stress ε c1 , which is considered having a value of 0.002; and a softened coecient ζ that is a function of reinforcement ratios in x and y directions:

       σ c = ζf c 2 εc ζε c1 -εc ζε c1 2 , if εc ζε c1 ≤ 1 σ c = ζf c 1 -εc/ζε c1 -1 4/ζ-1 2 , if εc ζε c1 > 1 (2.25)
ζ represents the reduction in compressive stress resulting from locating reinforcing bars in the compressed zone, hence the value ζ = 1.0 is considered in case of plain concrete behavior when no reinforcement is taken into account. It should be noted that, in order to avoid any potential numerical problem when using the following concrete stress-strain curve in a FE analysis, Wang and Hsu suggested a minimal value of 0.2ζf c of the compressive stress in the descending part.

Majewski curve [START_REF] Majewski | The mechanics of structural concrete in terms of elasto-plasticity[END_REF] Although Eurocode 2 assumes a limit of 0.4f c for the linear elastic part of concrete stress-strain curve, Majewski found that a linear elasticity limit should increase with the uniaxial concrete compressive strength. Elasticity limit e lim is considered as a scale factor and calculated as a percentage of stress to concrete strength according to this formula (f c in MPa):

e lim = 1 -exp (-f c /80) (2.26)
Consequently, Majewski adopted a linear stress-strain relation for the initial stresses (σ c ≤ e lim f c ). Beyond this limit, hardening and softening parts were described by a nonlinear relation that depends on several parameters, including the elasticity limit:

σ c = E c ε c , if σ c ≤ e lim f c σ c = f c (e lim -2) 2 4(e lim -1) ( εc ε c1 ) 2 -f c (e lim -2) 2
2(e lim -1)

( εc ε c1 ) + f c e 2 lim 4(e lim -1) , if σ c > e lim f c (2.27)
On the basis of experimental results, Majewski [START_REF] Majewski | The mechanics of structural concrete in terms of elasto-plasticity[END_REF] proposed the following formulas to approximate the values of ε c1 and ε cu :

ε c1 = 0.0014[2 -exp(-0.024f c ) -exp(-0.140f c )] ε cu = 0.004 -0.0011[1 -exp(-0.0215f c )] (2.28)
Kratzig and Polling curve [START_REF] Kratzig | An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[END_REF] Based on the recommendations of the Model code 1990 [2],

the concrete compressive stress-strain relation suggested by Kratzig and Polling was divided into three part to describe the phases of elasticity, hardening and softening of concrete behavior. Thus, three formulations were derived and expressed as follows:

         σ c = E c ε c , if σ c ≤ f c /3 σ c = E ci εc fc -(εc/ε c1 ) 2 1+(E ci ε c1 fc -2) εc ε c1 f c , if f c/3 < σ c ≤ f c σ c = 2+γcfcε c1 2fc -γ c ε c + γcε 2 c 2ε c1 -1 , if σ c > f c (2.29)
The rst two equations describe the ascending part up to the maximum compressive strength f c at ε c1 by assuming an elasticity limit equal to one third of f c . In the hardening equation, the modulus E ci was used to guarantee the stress-strain curve to pass through the point (ε ce , f c /3) where ε ce denotes the elastic strain of concrete corresponding to the end of linear elastic phase. The modulus E ci can be dened in terms of initial Young's modulus E c and compressive strength f c :

E ci = 1 2E c f c ε c1 2 - f c ε c1 + 3 2 E c (2.30)
The third equation represents the descending part of concrete stress-strain curve and includes the descent function γ c that permits to take account for the descending part dependency on the specimen geometry, which ensures mesh independent results in numerical analysis. γ c controls the area under the stressstrain curve, it depends on the constant material parameter G cl that represents the localized crushing energy and on the characteristic length l c derived from the respective element mesh. Consequently, the descent function can be expressed by:

γ c = π 2 f c ε c1 2 G cl lc -1 2 f c ε c1 (1 -b c ) + b c fc Ec (2.31)
where b c is a constant factor assumed equal to the ratio of the plastic strain to the inelastic strain (ε

pl c = b c ε in c ) with 0 < b c ≤ 1.
Values b c = 0.7 and G cl = 15KN/m were found by [START_REF] Birtel | Parameterised nite element modelling of rc beam shear failure[END_REF][START_REF] Mark | Computational modelling of failure mechanisms in reinforced concrete structures[END_REF] to t well with experimental data from literature. For solid elements, the characteristic length is

calculated by l c = V 1/3 e
where V e represents the element volume.

Wahalathantri curve [START_REF] Wahalathantri | A material model for exural crack simulation in reinforced concrete elements using abaqus[END_REF] Wahalathantri et al. developed a complete stress-strain curve for concrete under uniaxial compression by only using compressive strength, they also suggested slight modications from the original version of Hsu and Hsu [START_REF] Hsu | Complete stress-strain behaviour of high-strength concrete under compression[END_REF] in order to be used and comparable with the damaged plasticity model in Abaqus. Modications were made only for concrete with maximum compressive strength up to 62 MPa. In the ascending part, concrete acts elastically up to 0.5f c .

Beyond this limit, compressive stress values can be calculated in terms of the compressive strength f c , the strain at peak stress ε c1 and a parameter β which depends on the stress-strain curve shape:

σ c = β (ε c /ε c1 ) β -1 + (ε c /ε c1 ) β f c (2.32) with β = 1 1 -[f c / (ε c1 E c )] (2.33) ε c1 = 8.9 × 10 -5 f c + 2.114 × 10 -3 (2.34)
and the initial modulus E c is given by:

E c = 1.2431 × 10 2 f c + 3.28312 × 10 3 (2.35)
Wahalathantri et al. formula can be used only for stresses between 0.5f c at elastic strain ε ce in the ascending part and 0.3f c at ultimate strain ε cu in the descending part. Wahalathantri et al. suggested to iteratively calculate ε cu through the equation 2.32 for σ c = 0.8f c . It should be noted that in the above equations, σ c and E c are expressed in kip/in 2 ( 1M P a = 0.145037743kip/in 2 ).

Uniaxial tension 2.4.2.1 Uniaxial tensile stress-strain curve

Indirect tensile tests as cylinder splitting and bending prisms are often preferred over direct tension test and are used as an indirect way to determine concrete properties in tension (Figure 2.37). The mechanical behaviors of concrete under the action of tensile and compressive loadings have many Figure 2.37: Uniaxial tensile stress-strain curve for concrete [START_REF] Peterson | Crack growth and development of fracture zones in plain concrete and similar materials[END_REF] similarities. In general the limit of elasticity is observed to be about 60-80% of the uniaxial tensile strength which corresponds to the onset of micro-cracking in the concrete material. Above this level the response of concrete exhibits highly nonlinear behavior and the formation of microcracks is represented macroscopically with a softening stress-strain response. The tensile stress drops gradually with increasing deformations until a complete crack is formed. The reduction in strength of the test specimen resulting from the opening of initial cracks in concrete leads to increase the stress-strain curve nonlinearity. When the concrete specimen is unloaded from any point of the strain softening branch of the stress-strain curve, the unloading response is weakened and the material elastic stiness appears to be damaged.

Tensile stress-strain curve without detailed laboratory test results In the present section, several uniaxial stress-strain relationships for concrete in tension proposed in the literature are detailed, such as the expressions developed by [3,[START_REF] Hordijk | Tensile and tensile fatigue behaviour of concrete; experiments, modelling and analyses[END_REF][START_REF] Kratzig | An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[END_REF][START_REF] Wahalathantri | A material model for exural crack simulation in reinforced concrete elements using abaqus[END_REF][START_REF] Wang | Nonlinear nite element analysis of concrete structures using new constitutive models[END_REF].

Wang & Hsu curve [START_REF] Wang | Nonlinear nite element analysis of concrete structures using new constitutive models[END_REF] Wang and Hsu divided concrete behavior in tension into two ascending and descending parts, the rst describes the elastic phase while the second indicates the softening stress-strain response and is given in terms of cracking strength of concrete f t , strain at concrete cracking ε cr and the rate of weakening n:

σ t = E c ε t , if ε t ≤ ε cr (2.36) σ t = f t ( ε cr ε t ) n , if ε t > ε cr (2.37)
Wang and Hsu adopted a value of n = 0.4. However, this parameter can be used to calibrate the relation for a given simulation since tension stiening may considerably aect numerical results, hence Kmiecik and Kaminski [START_REF] Kmiecik | Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration[END_REF] proposed a range of values from 0.4 to 1.5. This tensile stressstrain curve is characterized by a sharp change at the cracking strain, which may result in some numerical diculties during a FE analysis. For similar cases, Wang and Hsu suggested to dene a short plateau at the peak point.

Chinese code curve (GB50010-2002) [3,[START_REF] Zhang | Nonlinear dynamic analysis of the three gorge project powerhouse excited by pressure uctuation[END_REF] According to the Chinese code for design of concrete structures, the uniaxial tensile stress-strain curve of concrete includes ascending stiening and descending softening parts. When concrete is subjected to tension, it is assumed to exhibit initially a linear elastic response up to the uniaxial tensile strength f t . After the tensile stress reaches f t and as stresses increase, concrete response becomes nonlinear and the softening stress-strain curve recommended by the Chinese code is given in term of the peak tension strain ε cr corresponding to f t :

σ t = f t (ε/ε cr ) α t ( ε εcr -1) 1.7 + ε εcr (2.38)
α t is the coecient at the descent stage of the uniaxial tension stress-strain curve of concrete and determined by:

α t = 0.312f t (2.39)
Wahalathantri curve [START_REF] Wahalathantri | A material model for exural crack simulation in reinforced concrete elements using abaqus[END_REF] The proposed modied part where stress is gradually reduced from maximum tensile stress f t to 0.77f t between strain of values ε cr and 1.25ε cr ,

The primary cracking part where stress decreases steadily to 0.45f t between strain values 1.25ε cr and 4ε cr ,

The secondary cracking part that ends at point (8.7ε cr , 0.10f t ).

Kratzig and Polling curve [START_REF] Kratzig | An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[END_REF] Kratzig and Polling considered a linear and elastic response of concrete in tension until the maximum strength f t is reached. They assumed that at strain ε cr a crack is initiated and occurs as tensile stress in concrete exceeds f t . Thereafter, a descending exponential curve was adopted to represent softening behavior of concrete in tension after crack initiation. Their tensile stress-strain relation after cracking can be expressed as:

σ t = f t e (εcr-εt)/γt (2.40)
In the above equation, γ t is a shape parameter that controls the area under the stress-strain curve and depends on the fracture energy G f and the characteristic length l t :

γ t = ε cr -ε cr /2 (2.41)
in which ε cr is the strain after which crack expands and is estimated by the following expression:

ε cr = G f l t f t (2.42)
The estimation of fracture energy and characteristic length values is detailed in the next section.

Uniaxial stress-crack opening curve

Another approach could be also used to dene concrete behavior in tension, namely the energy balance approach that showed a reasonable agreement with results from a tensile test according to Hillerborg et al. study [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and nite elements[END_REF]. Hillerborg et al. assumed that the response of concrete under tension is linear until the fracture surface is reached and a linear softening branch beyond cracking was adopted (Figure 2.38.a). Consequently, concrete post-cracking behavior is governed by the equivalent fracture energy criterion that is based on the amount of energy absorbed by the formation of a unit area of crack surface. Cracks arise in planes perpendicular to the direction of maximum principle tensile stress when the maximum tensile strength of concrete is reached, they present a highly irregular shape by following the weakest path in the material. In fracture mechanics experiments, the fracture energy is determined as the ratio of the total energy that is supplied to fracture a specimen (area under the load-deformation relation) to the fractured cross-sectional area [START_REF] Hordijk | Tensile and tensile fatigue behaviour of concrete; experiments, modelling and analyses[END_REF]. In Hillerborg et al.

frictitious crack model, the fracture energy G f is assumed as a material property and presented by the area under the stress-crack opening relation:

G f = ˆσdw (2.43)
Nowadays, this model is presented in most nite element codes and used to ensure mesh independence of results since the amount of absorbed energy is not very sensitive to the mesh size [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and nite elements[END_REF].

This enables to perform nite element analysis with a rather coarse mesh and treat complicated problems with a limited computational cost.

Descending branch after cracking can be also presented by a bi-linear or tri-linear tension softening curve. Haidong [START_REF] Haidong | Strengthening of RC slabs with openings using CFRP systems[END_REF] adopted in his analysis a bi-linear softening branch with a maximum crack width of 4.6G f /f t at which no stress is transferred by the opened crack ( The simplied concrete equation for uniaxial tensile loading of Hordijk [START_REF] Hordijk | Tensile and tensile fatigue behaviour of concrete; experiments, modelling and analyses[END_REF] is based on the principle of the frictitious crack model proposed by Hillerborg et al. [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and nite elements[END_REF] and depends on the specimen geometry. The corresponding tensile stress-strain relation consists of a loading linear part up to the strength f t and a nonlinear descending part which is derived from the following stress-crack opening relation:

σ t = f t 1 + c 1 w w c 3 e -c 2 w wc - w w c (1 + c 3 1 )e -c 2 (2.44)
The relation between crack opening and stress was found to be the most important input parameter for the nonlinear fracture mechanics analyses of concrete [START_REF] Peterson | Crack growth and development of fracture zones in plain concrete and similar materials[END_REF]. In the previous equation, the crack opening is expressed as a product of inelastic strain ε in t and characteristic length l t :

w = l t ε in t = l t ε t -σ t E -1 c (2.45)
Tajima et al. [START_REF] Tajima | 3d nite element cyclic analysis of beam/column joint using special bond model[END_REF] indicated that the denition of the characteristic length of concrete in compression and tension are similar, hence w is smeared over the average element length l t which can be calculated for 3D problems in term of the element volume by

l t = V 1/3 e
according to [START_REF] Mark | Computational modelling of failure mechanisms in reinforced concrete structures[END_REF] or l t = 2 3 3Ve 4π according to [START_REF] Tajima | 3d nite element cyclic analysis of beam/column joint using special bond model[END_REF]. Hillerborg et al. [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and nite elements[END_REF] dened the characteristic length in terms of concrete Young's modulus, fracture energy and tensile strength with

l t = G f Ec f 2 t .
Hordijk used a test series on normal-weight concrete from the literature in order to determine the unknown variables in the stress-crack opening relation. The best t was obtained for c 1 = 3, c 2 = 6.93 and w c = 160µm where w c represents the critical crack opening.

Fracture energy without detailed laboratory test results Several relations were proposed in the literature to estimate the fracture energy in case of lack of experimental data [2,[START_REF] Bazant | Crack band theory for fracture of concrete[END_REF][START_REF] Hordijk | Tensile and tensile fatigue behaviour of concrete; experiments, modelling and analyses[END_REF][START_REF] Oh-Oka | Inuence of short cut ber mixing and curing time on the fracture parameter of concrete[END_REF].

Hordijk energy [START_REF] Hordijk | Tensile and tensile fatigue behaviour of concrete; experiments, modelling and analyses[END_REF] The fracture energy of Hordjik can be calculated in terms of the tensile strength f t and the critical crack opening w c by the integration of equation 2.44:

G f = f t w c 1 c 2 1 + 6 c 1 c 2 3 - 1 c 2 + c 3 1 1 c 2 + 3 c 2 2 + 6 c 3 2 + 6 c 4 2 + 1 2 1 + c 3 1 exp(-c 2 ) (2.46)
By replacing c 1 and c 2 by their values as mentioned previously, the fracture energy can be estimated by:

G f = 0.195w c f t (2.47)
CEB-FIP energy [2,[START_REF] Quaranta | Maximum-entropy meshfree method for nonlinear static analysis of planar reinforced concrete structures[END_REF] CEB-FIP model code proposed a relation between the fracture energy and compressive strength of concrete. Fracture energy G f is calculated for mode I in N/mm in term of coecient α f that depends on the maximum aggregate size in concrete d max :

G f = α f f c 10 0.7 (2.48)
with f c representing the compressive strength of concrete in MPa and α f = (1.25d max + 10) .10 -3 , where d max is introduced in mm and may vary between 2 and 32 mm. As can be seen, G f has generally a tendency to increase with f c and d max . In the present study, a value of 16 mm is adopted for d max .

Oh-Oka et al. energy [START_REF] Oh-Oka | Inuence of short cut ber mixing and curing time on the fracture parameter of concrete[END_REF][START_REF] Tajima | 3d nite element cyclic analysis of beam/column joint using special bond model[END_REF] Oh-Oka et al. also determined the fracture energy of concrete in N/mm in term of the compressive strength according to the following formula:

G f = 0.23f c + 136 1000 (2.49)
Bazant and Oh energy [START_REF] Bazant | Crack band theory for fracture of concrete[END_REF][START_REF] Miyamoto | Impact-resistant design of rc slabs using immune algorithms[END_REF] Based on numerous experimental results, Bazant and Oh proposed an equation of the fracture energy of concrete in terms of Young's modulus E c , tensile strength f t and the maximum size of coarse aggregate d max :

G f = (0.3044f t + 2.72) f 2 t d max E c (2.50)
G f is calculated in kgf /cm, E c and f t are given in kgf /cm 2 (kgf /cm 2 = 0.0980665 M P a) and d max is in cm.

Mechanical behavior of steel

Considered as a composite material, reinforced concrete consists of steel reinforcement embedded in plain concrete. Steel is used in RC structures to carry tensile forces and to recover the low tensile strength of concrete. Inclusion of steel reinforcement is very eective to control the development of cracks by distributing cracks uniformly over the cracked regions, but it does not prevent the cracking of concrete under tension. The behavior of reinforcing steel may control the response of RC structures subjected to impact, thus the prediction of fundamental characteristics and behavior of steel is necessary. In the current study, uniaxial steel stress-strain curves based on experimental observations are adopted and dened in terms of steel properties as follows:

σ s = f (E s , E s1 , f y , f u , ε s , ε y , ε sh , ε u ) (2.51)
Where σ s represents the steel stress values calculated, E s is the elastic modulus of steel, E s1 is the deformation modulus of steel at the hardening phase, f y is the yield strength of steel, f u is the ultimate stress of steel corresponding to the ultimate strain, ε s represents the steel strain value corresponding to the stress σ s , ε y is the yield strain of steel at the yield strength f y , ε sh is the strain of steel at which strain hardening initiates, and ε u is the ultimate strain of steel.

Uniaxial tension

Steel behavior can be determined from coupon tests of bars loaded monotonically in tension. These tests allow obtaining a typical stress-strain curve for reinforcing steel in tension to failure (Figure 2.39). Steel response exhibits an initial linear elastic region, a yield plateau, a strain hardening region and a post-ultimate stress region. The compression stress-strain curve of reinforcing steel is assumed to be equal and opposite of its curve in tension.

The linear elastic region is up to the steel yield strength f y and steel stiness is governed by the elastic modulus E s at low strains. Stress in this region is expressed as:

σ s = E s ε s (2.52)
After the strain ε y corresponding to the yield strength, steel response is referred as the yield plateau or Lüder's plateau. At the beginning of this region, steel behavior shows a slight drop in strength below the initial yield strength. Then, stress remains steady at this lower yield strength and steel behaves nonlinearly and plastically. Steel stress-strain curve is generally idealized and assumed to be horizontal in the yield plateau region with an average strength equal to the material yield strength. Thus, the idealized stress-strain relationship in this region is:

σ s = f y (2.53)
The yield plateau size depends on the steel tensile strength, e.g. the yield plateau is considerably shorter for high-strength, high-carbon steels than for low-strength, low-carbon steels [START_REF] Wang | Reinforced concrete design[END_REF]. The strain-hardening region starts at the idealized point of coordinates (ε sh , f y ) and ends at the ultimate point of coordinates (ε u , f u ). At this phase, stress increases again with strain and exceeds the yield strength by 30 to 60 % at the ultimate point, and stiness decreases with permanent plastic deformation. The ratio of ultimate strength to yield strength depends on the steel specication and grade [START_REF] Lowes | Finite element modeling of reinforced concrete beam-columns bridge connections[END_REF]. At the ultimate point, the maximum tensile load is resisted and the tensile stress-strain curve of steel has zero slope.

Finally, in the post-ultimate region, stress reduces with strain until steel reinforcement fracture occurs and load capacity is lost. However, steel stress-strain curve is assumed to end at the ultimate point since its shape at the post-ultimate region is related to the gauge location and the length over which the data are collected [START_REF] Yu | Inelastic modeling of reinforcing bars and blind analysis of the benchmark tests on beam-column jjoint under cyclic loading[END_REF].

Idealizations

This section discusses idealized stress-strain curves proposed in the literature to describe steel behavior. For practical application, typical stress-strain curve of steel can be idealized in two dierent ways depending on the desired level of accuracy. For the rst idealization, steel is considered as a linear elastic, perfectly plastic material with a bilinear stress-strain curve. The rst branch is linear elastic up to yield strength f y and the second branch represents the yield plateau with a stress equal to f y . In this idealization, the stress increase due to strain hardening is neglected. This idealization was found to be sucient to analyze RC structures under service load conditions [START_REF] Chong | Numerical modelling of time-dependent cracking and defromation of reinforced concrete structures[END_REF] and very satisfactory in the case of low-strength, low-carbon steels characterized by a strain hardening much greater than the yield strain [START_REF] Taqieddin | Elasto-plastic and damage modeling of reinforced concrete[END_REF]. The design equations of the ACI code are based on this assumption [START_REF] Taqieddin | Elasto-plastic and damage modeling of reinforced concrete[END_REF]. However, this approximation underestimates steel stresses at high strains in cases where the onset of yielding is directly followed by steel hardening [START_REF] Taqieddin | Elasto-plastic and damage modeling of reinforced concrete[END_REF]. According to Kwak and Filippou [START_REF] Kwak | Nonlinear fe analysis of rc structures under monotonic loads[END_REF], an elastic-perfectly plastic curve may lead to numerical convergence problems when used in a FE analysis in cases where a RC structure is highly aected by reinforcing steel hardening.

On the contrary, this idealization was adopted by Kratzig and Polling [START_REF] Kratzig | An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[END_REF] to derive an elasto-plastic damage model for reinforced concrete and was found to be successfully used to model RC structures.

The second idealization assumes a linear elastic, linear plastic steel behavior with hardening.

The stress-strain curve consists of two linear branches, the elastic branch is up to yield strength f y while stress varies in the plastic branch between f y and f u for strain varying between ε y and ε u , respectively. This approximation is generally more used to estimate steel stresses at strains higher than yield strain and considered as more accurate since it takes into account the strain hardening eect. According to [START_REF] Taqieddin | Elasto-plastic and damage modeling of reinforced concrete[END_REF], this approximation would be more appropriate than the rst idealization to model the steel behavior and allows assessing the corresponding ductility of a structure investigated under high strains. Several researchers adopted the second idealization to represent steel behavior [START_REF] Birtel | Parameterised nite element modelling of rc beam shear failure[END_REF][START_REF] Cao | Experimental study on the seismic performance of recycled concrete brick walls embedded with vertical reinforcement[END_REF][START_REF] Kwak | Nonlinear fe analysis of rc structures under monotonic loads[END_REF]. Kwak and Fillippou [START_REF] Kwak | Nonlinear fe analysis of rc structures under monotonic loads[END_REF] indicated that assuming a gradually increasing plastic hardening branch immediately after yielding for the steel stress-strain curve does not aect the accuracy of results. Britel and Mark [START_REF] Birtel | Parameterised nite element modelling of rc beam shear failure[END_REF] used a value of E s1 = 1111 M P a for the deformation modulus of plastic-hardening branch, while Cao et al. [START_REF] Cao | Experimental study on the seismic performance of recycled concrete brick walls embedded with vertical reinforcement[END_REF] estimated E s1 in term of steel Young's modulus E s as:

E s1 = 0.01E s (2.54)
In order to predict the ultimate punching shear strength of slab-column connections, the second idealization was also used by Theodorakopoulos and Swamy [START_REF] Theodorakopoulos | Ultimate punching shear strength analysis of slab-column connections[END_REF] by assuming a value of E s1 = 5 GP a to express the strain hardening eect up to a stress equal to 1.2 f y . However, the elastic part before yielding was divided into two linear branches, a branch up to 0.8 f y with a modulus of elasticity E s = 200 GP a and a branch between stresses of 0.8f y and f y with a modulus equal to:

E s = 0.2f y 0.002 + 0.2f y /E (2.55)
The yield strain can be calculated using the following expression:

ε y = 0.002 + f y /E (2.56)

Steel-concrete bond

Stress-strain curve of reinforcing steel bars embedded in concrete is slightly dierent from that of a bare steel bar due to tension-stiening eect caused by the surrounding concrete bonded to steel. During a tension-stiening phenomenon, the bond action between concrete and reinforcement develops a number of cracks in the structure and results in the redistribution of tensile loads from concrete to steel. The response of a RC structure with tension-stiening is more complicated and stier than the response with a brittle failure [START_REF] Taqieddin | Elasto-plastic and damage modeling of reinforced concrete[END_REF]. The development of bond between concrete and steel is represented by three stress transfer mechanisms, namely mechanical interaction, chemical adhesion and surface friction [START_REF] Chong | Numerical modelling of time-dependent cracking and defromation of reinforced concrete structures[END_REF]. The mechanical interaction is the main mechanism that maintains the composite interaction between the two materials and for which load transfer can be idealized as a continuous stress eld developed at the deformed surface of reinforcement and concrete [START_REF] Taqieddin | Elasto-plastic and damage modeling of reinforced concrete[END_REF]. Bond stresses located along the surface between reinforcement and the surrounding concrete eectively transfer tensile stresses between reinforcing bars and concrete. Steel and the surrounding concrete are considered as full bonded and behave as a unit as long as the loading applied to the RC structure does not exceed the bond stress capacity. Stresses at the interface of concrete and steel increase with increasing the load applied, which results in a localized bond stress exceeding the bond capacity and a localized damage that propagates gradually to the surrounding concrete. Consequently, steel starts to yield at cracked sections and a signicant movement between the reinforcing steel and the surrounding concrete may occur. The quality of bond between reinforcing steel and concrete has a considerable inuence on cracks distribution and aects their width and spacing. For this reason, the interface properties between concrete and steel are very important to analyze RC structures and hence steel can be included in numerical analyses with an equivalent stress-strain relation that considers bond eects by assuming a perfect bond between concrete and steel elements.

The main dierence between a bare steel bar behavior and that of a steel bar embedded in concrete is related to the stress value at which steel yields. For an embedded steel bar, stresses Figure 2.40: Average tensile stress-strain curve for reinforcing steel embedded in concrete [START_REF] Belarbi | Constitutive laws of concrete in tension and reinforcing bars stiened by concrete[END_REF] in reinforcing steel between cracks is less than the yield stress at cracks since tensile forces can still partially be resisted by the concrete matrix located between cracks. Therefore, yielding of an embedded bar traversing several cracks starts at an average value f n which is less than the yield strength of a bare steel bar f y (Figure 2.40). In addition, stress-strain relationship of reinforcing steel bars embedded in concrete can be obtained by averaging stresses and strains between cracks. A bilinear average stress-strain relation was proposed by Belarbi and Hsu [START_REF] Belarbi | Constitutive laws of concrete in tension and reinforcing bars stiened by concrete[END_REF] based on experimental results:

σ s = E s ε s , if |ε s | ≤ ε n σ s = f y (0.91 -2B) + 0.02 + 0.25B εs εy , if |ε s | > ε n (2.57)
As illustrated in Figure 2.40, the average behavior of steel embedded in concrete still maintains an initial elastic phase up to a stress value f n at which yielding of steel bars starts. The second branch is expressed in term of a parameter B that determines the dierence between f n and f y . The parameter B is a function of the ratio f t /f y and the reinforcement steel ratio ρ s limited to a minimum value of 0.25%:

B = (f t /f y ) 1.5 /max{ρ s , 0.25%) (2.58)
with f t is the tensile strength of concrete. In Figure 2.40, ε n is the limiting boundary strain dened as follows:

ε n = ε y (0.93 -2B) (2.59)
The average stress-strain curve proposed by Belarbi and Hsu that incorporates the eect of tension stiening was adopted by several researchers [START_REF] Mansour | Cyclic stress-strain curves of concrete and steel bars in membrane elements[END_REF][START_REF] Quaranta | Maximum-entropy meshfree method for nonlinear static analysis of planar reinforced concrete structures[END_REF][START_REF] Taqieddin | Elasto-plastic and damage modeling of reinforced concrete[END_REF] to dene the behavior of reinforcing steel embedded in concrete.

Conclusion

In civil engineering eld, RC structures are often subjected to some extreme dynamic loadings due to accidental impacts of rigid bodies that may occur during their service life with a very low probability of occurrence. In the available design codes of civil engineering, the design of RC structures subjected to impact is generally based on approximate static method. However, the transient dynamic analysis has to be performed accounting for the main physical processes involved and accurate models are needed to describe and predict the structural behavior of RC members subjected to impact loading.

This study focuses on RC slabs subjected to accidental dropped object impact during handling operations within nuclear plant buildings. The objective of this chapter is to provide a general overview of research programs which have been carried out in regards to analyzing structures under general loading conditions, evaluating global response of RC slabs under drop-weight impact loading conditions. A classication of impact response types of RC slabs according to the impactor mass and velocity was provided. Then types of impact event and the transient behavior of RC slabs subjected to impact are described and identied. In order to properly predict the behavior of RC slabs under impact, impact dynamics, type of impact, failure modes and energy consideration should be examined and identied. The impact response of RC slabs depends on several parameters, including slab dimensions, material properties and impact conditions. In this chapter, uniaxial compressive and tensile stress-strain curves of concrete and steel based on experimental observations in the literature are adopted.

Chapter 3

Deterministic analysis: Software and steps

Introduction

This chapter discusses the choice of deterministic software used in the present study to simulate the problem of RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings. First, the basic phases of nite element analysis are described and the dierent steps which are necessary to create a nite element model in Abaqus are presented. Furthermore, plasticity constitutive models used to represent concrete and steel in numerical simulations are discussed by describing and identifying their fundamental parameters, and contact algorithms used for modeling the interaction between two bodies are presented. Other numerical features which are necessary for simulating impact analysis are also discussed.

Finite element analysis

As an alternative of experimental approach, nite element analysis (FEA) represents an advanced engineering tool to design structural and civil engineering applications. Based on the nite element method, FEA is a numerical technique for obtaining approximate solutions of partial dierential equations as well as of integral equations. These equations may be generated from complex structures with complicated features, for which it is dicult to nd an analytical solution. Thus, the simulation procedure should be performed with an intensive attention in order to improve the quality of FEA studies and obtain models that comply with the physical properties of the actual structure as far as possible. Computational tools based on the nite element method are increasingly improved with the development of computer technology, thus FEA is widely used as a powerful and helpful technique to assess complex engineering problems. Once the numerical model is validated with experimental results, FEA enables an accurate calculation of stress distributions and an evaluation of the inuence of model parameter variations.

A nite element model is based on dividing the complex structure into several small elements of nite dimensions that are connected to associated nodal points. Each element is assigned with a specic geometric shape and appropriate material properties and has its own functions of dependent variables. These functions can be interpolated using shape functions and describe the element response in terms of the value of dependent variables at a set of nodal points. These functions and the actual element geometry are used to determine the equilibrium equations that express the displacement occurring at each node in term of the external forces acting on the element. The unknown displacements are determined by solving a system of nite element equations that are generated, with the element stiness matrices, through the equilibrium equations and stress-strain relationships.

Generally, FEA procedure involves the following three basic phases:

The pre-processing phase represents the most consuming in time among the three phases, it is the phase where the numerical model of the physical problem should be dened and ready to be submitted to the solution or analysis phase. It consists in dividing the model into a number of discrete interconnected elements with common discrete nodes and applying certain appropriate boundary conditions with reasonable assumptions. Suitable nite element types should be selected correctly and a suciently rened mesh should be created, especially in regions where high stresses are expected. The following process of creating the mesh, choosing suitable elements connected with their respective nodes, and dening boundary conditions is referred to as discretization of the physical problem.

The solution/Analysis phase represents the phase of dening the analysis type and submitting the nite element model to nite element code solver. The analysis type can be either static or dynamic depending on the structure response to applied loads. A numerical output database will be generated by solving a series of linear or nonlinear equations. In this phase, numerical output can be controlled in the nite element code solver and quantities such as displacements, stresses, reactions, or other variables can be determined.

Post-processing phase represents the nal phase that provides a visualization environment of results by using a post-processing software. Results can be displayed in contour plots or other approaches which assist to get a better evaluation and interpretation of the simulation results. However, at this stage, the task of interpreting results should be performed carefully in order to verify the accuracy of FEA and assess whether the assumptions adopted during simulation are satised and accurate. These assumptions related to the structure geometry, boundary and loading conditions, material properties, and interaction between dierent parts of the model will potentially aect the results, as well as mesh sizes that may have important eects on predicted solutions.

The three basic phases of FEA procedure also can be used to describe the main framework of any available FE software. Hence, almost all FE software are based on the same fundamental principle of FE method and have the same general purpose that includes three components, namely, the pre-processor, the processor and the post-processor. A pre-processor establishes the FE model and the input data for the processor that solves the dierent equations generated during analysis. A post-processor helps in reviewing and interpreting results, as well as checking the validity of the solution.

Choice of commercial FE software

Several commercial software packages are developed to assist in reducing the complexities of FEA application. Due to advancement of computer technology, FE computer software have evolved to become so powerful to conduct sophisticated analyses and assist in design process stages. However, every commercial computer software is developed with its own capabilities, features and algorithms that are dierent from others. Most FE code are computer-aided design (CAD) software developed with a graphical user interface (GUI) that signicantly reduces the actual application of FEA and provides a powerful tool for engineers to perform analyses with minimal eort and without knowing the governing equations or the limitations of the analysis process. Recent computer-aided engineering (CAE) software oer a robust and reliable technology to automate critical intensive tasks and integrate CAD. They permit to graphically generate complex 3D geometries and to automatically create rened mesh by merely indicating the desired mesh density. Most FE code are written using programming languages such as Fortran, C and C++ that provide, with very less additional code, the advantage of writing subroutines for purposes of adding new elements to the element library and dening unique material models.

Choosing among the currently available commercial software is a very important task that is based on requirement of applications and involves several criteria such as analysis type, ease of use, eciency, software limits, computational cost, technical support and training. Furthermore, the accuracy of numerical results is very dependent on the prociency and the fundamental theoretical knowledge of the engineer in FEA. Most of the time, engineers will interact only with the pre-and post-processor components of the software, which may lead to the danger of using its tools as a black box in case of the engineer does not understand what happens in this essential part between the two processors. Mistakes like accepting results with an inappropriate mesh, not considering reasonable assumptions for an analysis and misrepresenting model properties can still be easily made despite all the modern and sophisticated improvements in FE software. For example, the automatic mesh generators that are provided in most commercial software oer a robust tool for meshing complex geometries, but if misused, they would result in a loss of accuracy and a dramatic increase in computational time that may even exceed available hardware and software resources.

Therefore and before starting an analysis, a FE model must be extracted from the real problem in order to solve the problem using the most ecient modeling method with the less computational time and a high degree of reliability. Likewise, the selection of the right FE software is absolutely crucial, as well as understanding its limits and the computational resources.

The present study focuses on RC slabs subjected to low impact velocity, thus the FE software chosen should include the ability of creating 3D geometries, applying impact conditions and dening nonlinear material behaviors. In addition, it should be capable of simulating contact and interaction between dierent parts of the structure, which is highly required for an impact problem. Given the many available software, the well-known commercial FE software, Abaqus, is used to illustrate how to simulate the geometric and material properties of the slab and impactor, as well as to dene contact, boundary and initial conditions. Data input for FEA with Abaqus can be done through:

Abaqus interactive edition by using the graphical user interface Abaqus/CAE, Abaqus Keyword edition by using an input le written with a text editor (.inp), or Abaqus Python edition by using a script written with Python (.py)

Components of an Abaqus analysis model

Abaqus is a series of powerful engineering programs that are based on the FE method with a high quality of pre-and postprocessing capabilities. The user-friendly nature of Abaqus oers the possibility of modeling and solving even the most complicated engineering problems. Simulations ranging from relatively simple and linear to highly dicult and nonlinear can be performed easily with Abaqus by only providing the engineering data such as the structure geometry, material behaviors, loads, boundary and initial conditions. Abaqus provides complete numerical solutions and controls their eciency by automatically choosing appropriate values for load increments and convergence tolerances and continually adjusting them during the analysis in order to obtain accurate results.

Abaqus is divided into modules that permit to build a FE model by performing a successive passage through modules. Each module denes a fundamental step in the modeling process. In the present section, the dierent steps which are necessary to create a FE model in Abaqus and the data that must be included are presented:

Part module: The structure geometry can be divided into multiple parts, each part represents a main component of the model and is created out of a two-dimensional sketch.

Property module: Mechanical behaviors and properties of materials can be specied by using the Abaqus material library that contains an extensive list of linear and nonlinear material models, or by creating new material models through several subroutines that are provided in Abaqus and programmed in Fortran language. The behavior of most typical engineering materials can be simulated with Abaqus, which includes metals, concrete, composites, polymers, and so on. Section properties of the physical structure are also dened in this module. Each section is associated with a material and assigned to a correspondent part of the geometry.

Assembly module: Geometry of the model is dened by creating instances of each part which is independent of each other. Instances are positioned relative to one another in a global coordinate system in order to assemble the model. Mesh module: Generating nite element meshes can be actually done either on the part or the assembly module. Meshing involves choosing appropriate element types and mesh density.

Various types of geometrical elements that can model virtually any geometry are available in Abaqus element library. All elements use numerical integration and must refer to a section property denition.

Conguring analysis module: The analysis procedure type must be dened in order to solve the numerical problem. Abaqus can be used to solve general static and dynamic response problems and consists of two main analysis products, Abaqus/Standard and Abaqus/Explicit. Abaqus/Standard allows carrying out static analyses where the structure has a long-term response to the applied load and solves a coupled system of equations implicitly at each increment. Abaqus/Explicit is particularly used to simulate transient dynamic events and uses an explicit formulation to perform a large number of small time increments without the need of solving a system of equations at each increment. The whole analysis procedure may comprise one or multiple steps. Thus, as many steps as needed must be created for the case load by providing the required parameters and controlling the output data requested for each step.

Load module: Specifying boundary and initial conditions and applying loads to the structure are step-dependent. Loads can be applied in Abaqus with many dierent ways which include concentrated or distributed loads, pressures and loads per unit volume such as due to gravity or acceleration. Zero-valued, including symmetry conditions, or non-zero boundary conditions can be imposed to model regions where the displacements and/or rotations are known. Most loads or boundary conditions can follow an amplitude curve varying with time. Non-zero initial conditions can be also specied as predened elds, these include initial stresses, velocities or temperatures.

Interaction and constraints module: Interaction and constraints between dierent parts or surfaces of the model can be dened. Abaqus provides two algorithms for simulating contact, general contact algorithm that allows dening contact between all model regions and contact pairs algorithm that describes contact between two surfaces. Interaction properties are used to enhance contact modeling, normal or tangential behavior can be dened with friction or contact damping. Abaqus includes also several constraint types such as rigid bodies, coupling constraints and surface-based ties. Constraints are used to couple a group of nodes motion to the motion of other nodes.

Output control module: A large amount of output variables can be generated with an Abaqus simulation, they depend on the analysis and element types used. Output are specied with Field and History output managers that generate a default output request, but also enable to control and manage the analysis requested output in order to only produce results that are required to be interpreted and avoid excessive disk space.

Job module: A job is associated with the model and submitted to the solver for analysis. Submission of input les can be performed through Abaqus/CAE or manually from the command prompt. Abaqus oers the possibility of checking incorrect or missing data through a data check analysis that can be used before running a simulation and helps in minimizing the probability of errors in the model.

Visualization module: This module is only available in Abaqus/CAE which writes the analysis results to an output database le with the extension (.odb). The evaluation of results are performed interactively using the program Abaqus/viewer that allows viewing and displaying results related to the model through a variety of options including color contour, X-Y and deformed shape plots. Note that three other types of output are available in Abaqus and results can be written to the Abaqus data le (.data) in tabular form, to the Abaqus restart le (.res) that uses restart data to continue the analysis, and to Abaqus results le (.l) that can be used with third-party software for subsequent postprocessing. Moreover, a Python script can be used to read data from a (.odb) le and gather numerical results as tables, plots or animations.

Conguring analysis

Selecting an appropriate approach to solve the equation governing the response of the nite element analysis of a specic problem is very important since an incorrect solution and a highly computational analysis may result from choosing the wrong simulation method. Two methods are available in Abaqus to solve a nite element problem, namely the implicit and explicit methods. Both methods consist of a numerical direct time integration scheme to evaluate the unknown displacement at the end of the time step. Implicit method is generally used for static, harmonic or modal analyses in which the time dependency of the solution is not a signicant factor, while explicit method is used for blast and impact problems involving high deformation and time dependent solution. However, Abaqus allows using the implicit method to perform dynamic analysis of problems in which inertia effects are considered, and using the explicit method quasi-static problems that experience convergence diculties in implicit analysis methods. Implicit direct integration is provided in Abaqus/Standard, while explicit direct integration is provided in Abaqus/Explicit [5].

The direct-integration dynamic procedure provided in Abaqus/Standard oers a choice of implicit operators using the implicit Hilber-Hughes-Taylor operator for integration of the equations of motion, while Abaqus/Explicit uses the central-dierence operator. In an implicit dynamic analysis the integration operator matrix must be inverted and a set of nonlinear equilibrium equations must be solved at each time increment. In an explicit dynamic analysis displacements and velocities are calculated in terms of quantities that are known at the beginning of an increment. Therefore, the global mass and stiness matrices need not be formed and inverted, which means that each increment is relatively inexpensive compared to the increments in an implicit integration scheme.

However, the size of the time increment in an explicit dynamic analysis is limited because the centraldierence operator is only conditionally stable, whereas the implicit operator options available in Abaqus/Standard are unconditionally stable. Thus, there is no such limit on the size of the time increment that can be used for most analyses in Abaqus/Standard (accuracy governs the time increment in Abaqus/Standard).

The explicit scheme is only stable if the size of the time step is smaller than the critical time step size for the structure being simulated. This very small time step size requirement for stability thereby makes explicit solutions recommendable for short transient situations. But, even though the number of time steps in an explicit solution may be orders of magnitude greater than that of an implicit solution, it may be more ecient than an implicit solution since no matrix inversion is required. Very Large deformation problems such as impact analysis can result in millions of degrees of freedom eectively increasing the size of stiness matrix. The implicit scheme requires considerable computation time, disk space, and memory because of the iterative procedure involving the inversion of a large global stiness matrix (determined by the number of degres of freedom).

The computational cost for problems based on the implicit scheme is roughly proportional to the square of the number of degrees of freedom, while the analysis cost for problems based on the explicit scheme rises only linearly with problem size. Hence there is a need for an explicit method which would prevent the inversion of stiness matrix. It is possible to solve complicated, very general, three-dimensional contact problems with deformable bodies in Abaqus/Explicit. Problems involving stress wave propagation can be more ecient computationally in Abaqus/Explicit than in Abaqus/Standard.

Material properties 3.6.1 Concrete constitutive model

Modeling concrete within nite element packages is a very complicated task due to the complexity of its behavior. For an accurate simulation of RC structures, both elastic and plastic behavior of concrete in compression and tension should be taken into account using a proper material model in nite element analysis. Simulation of compressive behavior should include elastic behavior of concrete, as well as strain hardening and softening regimes. Simulation of concrete under tension should include tension softening, tension stiening and interaction between concrete and steel elements.

There are dierent theories available in the literature to model the constitutive behavior of concrete, namely elasticity theory, viscoelasticity theory, viscoplasticity theory, classical plasticity theory, endochronic theory, fracture mechanics, continuum damage mechanics and stochastic approach [2,[START_REF] Koh | A plastic-damage model for lightweight concrete and normal weight concrete[END_REF]. Among these, the most commonly used frameworks are based on plasticity, continuum damage mechanics and combinations of plasticity and damage mechanics.

Plasticity theory was widely used to describe concrete behavior [START_REF] Grassl | Concrete in compression: a plasticity theory with a novel hardehard law[END_REF][START_REF] Menetrey | A triaxial failure criterion for concrete and its generalization[END_REF]. Stress-based plasticity models obviously have advantages over elastic theory since they allow modelling concrete behavior when subjected to triaxial stress states, as well as representing hardening and softening regimes.

The main characteristic of these models is a plastic yield surface that corresponds at a certain stage of hardening to the strength envelope of concrete and includes a nonassociative ow rule [START_REF] Grassl | Concrete in compression: a plasticity theory with a novel hardehard law[END_REF][START_REF] Menetrey | A triaxial failure criterion for concrete and its generalization[END_REF].

The expression of the yield surface includes a hardening-softening function and inelastic strains are considered in order to represent realistically the deformations in concrete [START_REF] Grassl | Cdpm2: A damage-plastcity approach to modelling the failure of concrete[END_REF][START_REF] Koh | A plastic-damage model for lightweight concrete and normal weight concrete[END_REF][START_REF] Menetrey | A triaxial failure criterion for concrete and its generalization[END_REF]. Nevertheless, such models do not incorporate a damage process to capture the variations of the elastic unloading stiness upon mechanical loading accurately. As a result, they are not able to describe unilateral eects and the reduction of material stiness due to microcracks [START_REF] Jason | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[END_REF][START_REF] Koh | A plastic-damage model for lightweight concrete and normal weight concrete[END_REF]

(Figure 3.1.a).
Conversely, continuum damage mechanics models are based on the concept of a decrease of the elastic stiness [START_REF] Grassl | Cdpm2: A damage-plastcity approach to modelling the failure of concrete[END_REF][START_REF] Jason | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[END_REF]. The continuum damage theory was also widely used to model the nonlinear behavior of concrete [33,[START_REF] Mazars | Continuum damage theory: Application to concrete[END_REF]. For these models, phenomena such as strain softening, stiness decrease and unilateral eects due to progressive microcrackings and microvoids are taken into account [START_REF] Jason | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[END_REF][START_REF] Koh | A plastic-damage model for lightweight concrete and normal weight concrete[END_REF]. The mechanical eect of these phenomena is represented by a set of internal state variables that describes the decrease of the elastic stiness at the macroscopic level [START_REF] Jason | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[END_REF][START_REF] Mazars | Continuum damage theory: Application to concrete[END_REF].

Nevertheless, such models do not consider the plastic strain (Figure 3.1.b), hence they are unable to describe irreversible deformations nor predict dilatancy behavior under multiaxial loadings [START_REF] Grassl | Cdpm2: A damage-plastcity approach to modelling the failure of concrete[END_REF][START_REF] Jason | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[END_REF].

On the other hand, several models were proposed in literature as a combination of isotropic damage and plasticity in order to model both tensile and compressive behaviors of concrete [START_REF] Jason | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[END_REF][START_REF] Lee | Plastic-damage model for cyclic loading of concrete structures[END_REF] since neither pure elastic damage models nor pure elastoplastic constitutive models can satisfactorily describe the concrete behavior. Both models fail in capturing the evolution of unloading stinesses accurately [START_REF] Jason | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[END_REF][START_REF] Omidi | Finite element analysis of concrete structures using plastic-damage model in 3-d implementation[END_REF]. Thus, damage-plastic models are used to overcome this problem since both irreversible deformations and microcrackings contribute to the nonlinear behavior of concrete (Figure 3.1.c). Damage plasticity models are formulated in the eective (undamaged) stress space combined with a strain based damage model, which is numerically more stable and more appropriate to represent tension and compression behaviors of concrete [START_REF] Grassl | Cdpm2: A damage-plastcity approach to modelling the failure of concrete[END_REF][START_REF] Omidi | Finite element analysis of concrete structures using plastic-damage model in 3-d implementation[END_REF]. Plastic yield function is no longer expressed in term of the applied stress, but it is written in term of the eective stress that represents the stress in undamaged material in between the microcracks [START_REF] Jason | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[END_REF].

Several plasticity models which have been implemented in ABAQUS were widely used to represent concrete in numerical simulations, namely the Drucker-Prager/Cap model, the brittle cracking model and the concrete damage plasticity model. These models require multiple and complex parameters, which are generally obtained from material tests. Plasticity models available in ABAQUS are decomposed into an elastic part and a plastic part.

They are usually formulated in terms of a yield surface, a ow rule and evolution laws that dene the hardening. The yield function represents a surface in eective stress space, for which a failure or damage state can be determined. For a plastic-damage model the yield function arrives at:

F (σ, εpl ) ≤ 0 (3.1)
The states of stress corresponding to material failure are on this surface while the states of safe behavior are inside.

The plastic ow is governed by a ow potential function G(σ) which is also dened in the eective stress space according to nonassociative ow rule:

εpl = λ ∂G(σ) ∂ σ (3.2)

Concrete damage plasticity model (CDP)

For an accurate simulation of the response of RC slabs under impact, a realistic representation of concrete behavior under dynamic loads is necessary. The concrete model adopted for numerical analysis should include the hardening behavior that occurs under compressive loading, the softening behavior that results from the formation of microcracks, the damage that describes the decrease in material stiness, stiness recovery related to crack opening and closing under cycling loading, and rate dependence. Among dierent models available in Abaqus, CDP model represents a convenient model to simulate concrete behavior due to its capabilities to represent irreversible deformations but also stiness degradation [6]. A comparison to experimental results has proved the accuracy of this model to capture the complete behavior of concrete up to failure, and to be the most stable regime for modeling concrete nonlinear behavior [START_REF] Mohamed | Prediction of the behavior of reinforced concrete deep beams with web openings using the nite element method[END_REF]. Therefore, the CDP model is used for concrete to study the response of RC slabs under impact in the present study CDP model is a continuum, plasticity-based, damage model for concrete that uses concepts of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity to represent the inelastic behavior of concrete. It is based on the models proposed by Lubliner et al.

[113] and Lee and Fenves [START_REF] Lee | Plastic-damage model for cyclic loading of concrete structures[END_REF], and designed for applications in which the concrete is subjected to arbitrary loading conditions, including cyclic loading. The model assumes that the main two failure mechanisms are tensile cracking and compressive crushing of the concrete material. Two hardening variable related to concrete failure mechanisms under tension and compression, respectively, are used in the CDP model in the aim of controlling the evolution of the yield surface. The model takes into account the degradation of the elastic stiness induced by plastic straining both in tension and compression by introducing two independent scalar damage variables for tension and compression, respectively. The elastic behavior of concrete in CDP model is assumed to be isotropic and linear.

Plastic ow and yield surface

The plastic ow potential function and the yield surface in CDP model make use of two stress invariants of the eective stress tensor, namely the hydrostatic pressure stress (or the eective hydrostatic stress), p = -1 3 trace ( σ) and the Mises equivalent eective stress, q = 3 2 ( S : S) where S is the eective stress deviator (or deviatoric part of the eective stress tensor), dened as S = σ + pI.

The CDP model assumes a nonassociated potential plastic ow that allows a realistic modeling of the volumetric expansion under compression for concrete [START_REF] Omidi | Finite element analysis of concrete structures using plastic-damage model in 3-d implementation[END_REF]. The ow potential G used for this model is the Drucker-Prager hyperbolic function:

G = ( f t tan ψ) 2 + q2 -p tan ψ (3.3)
where f t is the uniaxial tensile strength of concrete. The parameters ψ and determine the shape of the ow potential surface: ψ is the dilation angle measured in the p -q plane at high conning pressure, while is referred to as the eccentricity of the plastic ow potential surface. The dilation angle represents the angle of inclination of the failure surface towards the hydrostatic axis in the meridian plan [START_REF] Kmiecik | Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration[END_REF]. The ow potential eccentricity aects the exponential deviation of G from the linear Drucker-Prager ow potential [START_REF] Mark | Computational modelling of failure mechanisms in reinforced concrete structures[END_REF] (Figure 3.2) and denes the rate at which the function approaches the asymptote (the ow potential tends to a straight line as the eccentricity tends to zero) [START_REF] Zhang | Nonlinear dynamic analysis of the three gorge project powerhouse excited by pressure uctuation[END_REF].

The nonassociative ow rule, which is used in the CDP model requires a yield surface that determines states of failure or damage. The model uses a yield condition based on the loading function proposed by Lubliner et al. [START_REF] Lubliner | A plastic-damage model for concrete[END_REF], with the modications proposed by Lee and Fenves [START_REF] Lee | Numerical implementation of plastic damage model for concrete under cyclic loading : Application to concrete dam[END_REF] to account for dierent evolutions of strength under tension and compression. The yield function used in CDP model is expressed in the eective stress space and depends on two hardening variables ε pl t and ε pl c :

F = 1 1 -α (q -3αp + β(ε pl ) < σmax > -γ < -σ max >) -σc (ε pl c ) = 0 (3.4)
with ε pl t and ε pl c are the plastic strain of concrete in tension and compression, respectively. In this expression,σ max represents the maximum principal eective stress and < . > is the symbol of the Macaulay bracket which is dened by < x >= 1 2 (|x| + x). Moreover, α is a dimensionless material constant and depends on the ratio of concrete strengths under biaxial and uniaxial compression as follows:

α = (f b /f c ) -1 2(f b /f c ) -1 , 0 ≤ α ≤ 0.5 (3.5)
A biaxial laboratory test is necessary to dene the value of α [100] (Figure 3.3.a). Typical experimental values of α are within 0.08 and 0.12 [START_REF] Lubliner | A plastic-damage model for concrete[END_REF], while Abaqus allows a range of values between 0 and 0.5 [6]. f b represents the biaxial compressive strength of concrete and f c is its uniaxial compressive strength.

β is a material function of the two hardening variables ε pl t and ε pl c and includes the ratio of the biaxial to uniaxial compressive strength. it is expressed in terms of the eective tensile stress σt and eective compressive stress σt as follows:

β = σc (ε pl c ) σt (ε pl t ) (1 -α) -(1 + α) (3.6)
The coecient γ represents a dimensionless material constant included in the CDP model only for the stress states of triaxial compression when σmax < 0 [START_REF] Zhang | Nonlinear dynamic analysis of the three gorge project powerhouse excited by pressure uctuation[END_REF] to better predict the concrete behavior in compression under connement [START_REF] Omidi | Finite element analysis of concrete structures using plastic-damage model in 3-d implementation[END_REF]. γ is calculated in term of the coecient K c that controls the failure surface in the deviatoric cross section:

γ = 3(1 -K c ) 2K c -1 (3.7)
K c is the ratio of the second invariant on the tensile meridian to that on the compressive meridian at any given value of the pressure invariant p [6]. Physically, it represents the ratio of the distances between the hydrostatic axis and respectively the compression meridian and the tension meridian in the deviatoric cross section [START_REF] Kmiecik | Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration[END_REF] (Figure 3.3.b). In Abaqus, K c must satisfy the condition 0.5 ≤ K c ≤ 1.0. For K c = 1, the deviatoric cross section of the failure surface becomes a circle.

Stress-strain curve for uniaxial compression

The constitutive equation under uniaxial compression for the CDP model takes the following form [6,[START_REF] Lubliner | A plastic-damage model for concrete[END_REF]:

σ c = (1 -d c )E c (ε c -ε pl c ) = E(ε c -ε pl c ) (3.8)
where d c is the damage variable in the compression zone, it can take value from zero, representing the undamaged material, to one, representing the total loss of strength. E c is the initial (undamaged) elastic stiness of the material, while E = (1 -d c )E c is the degraded elastic stiness in compression. The eective compressive stress is dened as:

σc = σ c (1 -d c ) = E c (ε c -ε pl c ) (3.9)
where ε pl c is the equivalent plastic strain in compression.

Inelastic strains ε in c are used in the CDP model to describe the hardening rule. As illustrated in Figure 3.4, the compressive inelastic strain is dened as the total strain minus the elastic strain ε ce corresponding to the undamaged material,

ε in c = ε c -ε ce (3.10)
where

ε ce = σ c E 0 (3.11)

Stress-strain curve for uniaxial tension

The stress-strain relation under uniaxial tension is similar to that in compression and takes the following form [6,[START_REF] Lubliner | A plastic-damage model for concrete[END_REF]:

σ t = (1 -d t )E t (ε t -ε pl t ) = E(ε t -ε pl t ) (3.12)
where d t is the damage variable in the tension zone and E = (1 -d t )E c is the degraded elastic stiness in tension.

The eective tensile stress is dened as:

σt = σ t (1 -d t ) = E c (ε t -ε pl t ) (3.13)
where ε pl t is the equivalent plastic strain in tension. The values of d t range from 0 to 1 representing the undamaged material and the total loss of strength, respectively. The strain-softening behavior for cracked concrete is dened by the phenomenon called tension stiening which is required in the CDP model. The tension stiening eect depends on several factors such as the density of reinforcement, the quality of the bond between the rebar and the concrete, the relative size of the concrete aggregate compared to the rebar diameter, and the mesh ( [7]).

The tension stiening can be specied in two ways:

A postfailure stress-strain relation in term of the cracking strain ε ck t which is used in the CDP model for numerical analyses. The strain after cracking is dened as the dierence between the total tensile strain and the elastic strain ε te for the undamaged material (Figure 3.5):

ε ck t = ε t -ε te (3.14)
where

ε te = σ t E c (3.15)
The specication of a postfailure stress-strain relation introduces mesh sensitivity in the results, especially in cases with little or no reinforcement.

A fracture energy cracking criterion which is adequate to reduce the concern of the mesh sensitivity. Hillerborg [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and nite elements[END_REF] denes the energy required to open a unit area of crack (G f ) as a material parameter, using brittle fracture concepts. CDP model assumes a linear loss of strength after cracking (Figure 2.38.a). The cracking displacement at which complete loss of strength takes place is, therefore, w c = 2G f /f t . Descending branch after cracking can be also presented by a bi-linear or tri-linear tension softening curve. In Abaqus, the fracture energy cracking model is introduced by specifying the postfailure stress as a tabular function of cracking displacement.

Damages

The variation of the elastic modulus of concrete can be due to two damage mechanisms of concrete in tension and compression, respectively [START_REF] Burlion | Compaction and tensile damage in concrete: constitutive modelling and application to dynamics[END_REF]. In uniaxial tension, plastic strains are very small compared to those in compression, and concrete nonlinear response induces a lot of damage due to mircrocraking. Positive strains controls the tension damage by producing microcrak opening and To account for this dierence, the CDP model considers two independent damage variables, namely the tensile damage d t and compressive damage d c . As previously mentioned, these damages can take values ranging from zero, for the undamaged concrete, to one, for the fully damaged concrete. Experiments to determine the damage variable in terms of the equivalent plastic strains are very dicult to perform, thus many researchers studied the decrease of elastic stiness and proposed expressions to tensile and compressive damages [START_REF] Burlion | Compaction and tensile damage in concrete: constitutive modelling and application to dynamics[END_REF][START_REF] Deng | Experimental research on damage parameter of concrete in non-uniform stress eld[END_REF][START_REF] Grassl | Cdpm2: A damage-plastcity approach to modelling the failure of concrete[END_REF][START_REF] Koh | A plastic-damage model for lightweight concrete and normal weight concrete[END_REF][START_REF] Lee | Plastic-damage model for cyclic loading of concrete structures[END_REF][START_REF] Lubliner | A plastic-damage model for concrete[END_REF][START_REF] Mark | Computational modelling of failure mechanisms in reinforced concrete structures[END_REF][START_REF] Mazars | Continuum damage theory: Application to concrete[END_REF]. Damage evolution of concrete in tension and compression can be derived from uniaxial stress strain curves and formulated as increasing functions of the inelastic strains. Pavlovic et al. [START_REF] Pavlovic | Bolted shear connectors vs. headed studs behaviour in push-out tests[END_REF] as well as Jankowiak and Lodygowski [START_REF] Jankowiak | Identication of parameters of concrete damage plasticity constitutive model[END_REF] compared undamaged and damaged concrete responses in order to determine the expressions of tensile and compressive damages, respectively, as follows:

d t = 1 - σ t f t (3.16) d c = 1 - σ c f c (3.17)
However, Korotkov et al. [START_REF] Korotkov | Determination of dynamic reaction in concrete oors of civil structures of nuclear power plant in accidental drops of heavy objects[END_REF] and Wahalathantri et al. [START_REF] Wahalathantri | A material model for exural crack simulation in reinforced concrete elements using abaqus[END_REF] assumed the tensile damage variable to be equal to the ratio of the cracking strain to the total tensile strain, and similarly dened the compressive damage variable as the ratio of the crushing strain to the total compressive strain.

In Abaqus, the accuracy of damage evolution should be evaluated by ensuring that no negative and/or decreasing plastic strain are produced with the increase of stress [7]. Tensile and compressive plastic strains can be expressed in Abaqus for the CDP model in terms of the cracking and crushing strains, respectively, as follows:

ε pl t = ε ck t - d t (1 -d t ) σ t E c (3.18) ε pl c = ε in c - d c (1 -d c ) σ c E c (3.19)
Negative and/or decreasing plastic strains are indicative of incorrect damage evolutions, which leads to generate error message in Abaqus before the analysis is performed [7].

Constitutive model of reinforcing steel

The uniaxial behavior of steel shows that steel can be modeled as an elasto-plastic material with isotropic strain hardening characterized by an increase in stress with the increase of inelastic strain.

Thus, plasticity models are found to be the simplest and most computationally ecient models for predicting steel behavior.

In Abaqus, the von Mises yield criterion, associative ow rule and isotropic hardening are typically used for modeling structural steel. The von Mises yield criterion can be written as:

F (σ, R) = q -f y -R(p) ≤ 0 (3.20)
where f y is the yield stress, R(p) is the isotropic hardening stress and q is the von Mises equivalent stress (see 3.6.2.1).

The evolution of the plastic strain is governed by the following plastic ow rule:

εp ij = λ ∂f ∂σ ij (3.21)
with εp ij is the plastic strain rate and λ is the plastic multiplier.

Finite elements

Abaqus provide a wide variety of element types available, it is important to select the correct element for a particular application. Choosing an element for a particular analysis can be simplied by considering specic element characteristics: rst-or second-order, full or reduced integration, hexahedra/quadrilaterals or tetrahedra/triangles. By considering each of these aspects carefully, the best element for a given analysis can be selected.

In this study, hexahedral solid elements with 8 nodes with reduced integration (C3D8R) are used during the simulations of the problem involving contact-impact of reinforced concrete (Figure 3.6). These solid (or continuum) elements in Abaqus can be used for complex nonlinear analyses involving contact, plasticity, and large deformations. These elements use linear interpolation in each direction and are often called linear elements or rst order elements. These elements have only three displacement degrees of freedom and are stress/displacement elements. Hourglassing can be a problem with rst-order, reduced-integration elements (such as C3D8R) in stress/displacement analyses. Since the elements have only one integration point, it is possible for them to distort in such a way that the strains calculated at the integration point are all zero, which, in turn, leads to uncontrolled distortion of the mesh. First-order, reduced-integration elements in Abaqus include hourglass control, but they should be used with reasonably ne meshes. There are advantages involved with the use of the hexahedral elements since they provide solution of equivalent accuracy at less computational cost and allow a better convergence rate and a more uniform mesh compared to tetrahedral elements. The computational eciency of reduced-integration element is very high compared to the fully integrated element, especially in the problems involving contact-impacts. Truss elements have no initial stiness to resist loading perpendicular to their axis. The most common applications of trusses at large strains involve yielding metal behavior in which cases the material is eectively incompressible.

4-node R3D4 rigid elements are used to dene the surfaces of rigid bodies presented in the FEM (Figure 3.6). R3D4 elements can be used in three-dimensional analysis to dene master surfaces for contact applications. Rigid elements must always be part of a rigid body. By default in Abaqus/Explicit, rigid elements do not contribute mass to the rigid body to which they are assigned.

To dene the mass distribution, you can specify the density of all rigid elements in a rigid body.

When a nonzero density and thickness are specied, mass and rotary inertia contributions to the rigid body from rigid elements will be computed in an analogous manner to structural elements.

Contact modeling between slab and impactor

Abaqus/Explicit provides two algorithms for modeling contact and interaction problems: the general contact algorithm and the contact pair algorithm. Both algorithms can be used with threedimensional surfaces. The general contact algorithm allows very simple denitions of contact with very few restrictions on the types of surfaces involved. The contact pair algorithm has more restrictions on the types of surfaces involved and often requires more careful denition of contact.

The general contact and contact pairs algorithms in Abaqus/Explicit dier by more than the user interface; in general they use completely separate implementations with many key dierences in the designs of the numerical algorithms. Contact denitions are greatly simplied with the general contact algorithm which allows generating contact forces to resist node-into-face. To dene a contact pair, pairs of surfaces that will interact with each other must be indicated.

Abaqus/Explicit uses two dierent methods to enforce contact constraints with the contact pair algorithm, namely the kinematic contact algorithm that uses a kinematic predictor/corrector contact algorithm to strictly enforce contact constraints and the penalty contact algorithm that has a weaker enforcement of contact constraints but allows for treatment of more general types of contact. There are dierences in the way in which the kinematic and penalty contact algorithms enforce contact constraints (Figure 3.7). Using Abaqus analysis guide [5], the dierences can be described as follows: The kinematic algorithm is a predictor/corrector method. When kinematic contact is considered in the analysis, Abaqus/explicit carries out a predictor phase and a corrector phase in each time increment. In the predictor phase the kinematic state of the model is advanced by ignoring any contact conditions. This can result in overclosure or penetration. In the corrector phase of the time increment, an acceleration correction is applied to the slave and master nodes to correct for this predicted penetration, while conserving momentum. This correction results in a nal conguration for increment in which the slave node is exactly in compliance with the master surface.

The penalty algorithm uses an explicit approach to enforce contact constraints. In contrast to the kinematic algorithm, a corrector phase is not processed for an increment. Rather, an interface spring is inserted automatically between the slave node and the master face in increment to minimize the contact penetration. The force associated with the interface spring is equal to the spring stiness multiplied by the penetration distance. Therefore, a small residual penetration will exist since contact forces are not generated unless there is some amount of penetration at the beginning of the increment.

In a mechanical contact simulation the interaction between contacting bodies is dened by assigning a contact property model to a contact interaction, including a constitutive model for the contact pressure-overclosure relationship that governs the motion of the surfaces and a friction model that denes the force resisting the relative tangential motion of the surfaces. The contact pressureoverclosure relationship used in this study is referred to as the hard contact model (Figure 3.8.a).

Hard contact implies that the surfaces transmit no contact pressure unless the nodes of the slave surface contact the master surface. No penetration is allowed at each constraint location and no limit to the magnitude of contact pressure can be transmitted when the surfaces are in contact. 3.9 Constraints 3.9.1 Approaches to represent steel in RC numerical analysis Typically, three distinct approaches are available in nite element analysis to represent steel reinforcement in a 3D nite element model of a RC structure, namely smeared approach, embedded approach and discrete approach. These approaches allow the interactions between the concrete elements and the reinforcement elements by specifying the constraints on the degrees of freedom of the relevant nodes.

The classical isotropic

In the smeared or distributed approach (Figure 3.9.a), the distribution of reinforcing steel is assumed to be uniform over the concrete elements in the appropriate direction. As a result, the total stiness of a smeared RC nite element depends on the stiness of concrete and that assigned to steel reinforcement. In addition, the material properties in a smeared nite element are estimated according to composite theory based on individual properties of concrete and reinforcement. In this approach, reinforcing bars are considered in the model using a ratio that depends on the size of the concrete structure and the amount of steel used. Therefore, it is not necessary to discretely simulate the bars as they are distributed in the nite model assuming a complete compatibility between steel and concrete. Abaqus allows modeling this approach using the rebar option to dene layers of uniaxial reinforcement in concrete elements as smeared layers with a constant thickness.

The thickness of layers is equal to the ratio of each reinforcing bar area to the spacing between bars [5]. This approach is usually applied for large structural models when modeling details of the reinforcement is not essential to evaluate the overall response of the structure [START_REF] Murthy | Impact analysis of concrete structural components[END_REF][START_REF] Tahmasebinia | Numerical modelling of reinforced concrete slabs suject to impact loading[END_REF], or when the structure is heavily reinforced with a huge number of reinforcing bars and an exact denition of every single reinforcing bar is dicult [START_REF] Chong | Numerical modelling of time-dependent cracking and defromation of reinforced concrete structures[END_REF].

The embedded approach allows an independent choice of concrete mesh, it is generally used to create a bond and arbitrarily dene the reinforcing steel regardless of the mesh shape and size of the concrete element (Figure 3.9.b). Each reinforcing bar is considered as an axial member and its stiness is evaluated independently from the concrete elements. The displacements of the reinforcement elements are compatible with those of the surrounding concrete elements. In this approach, the identication of the intersection points of concrete elements with reinforcement bars leads to additional nodes created in the model and located at the intersection points. This results in increasing the number of degrees of freedom in the analysis, and hence the computational time becomes more important. Abaqus allows the embedded element technique to model reinforcing bars as a group of elements embedded in 'host' solid elements that represent the concrete elements in a RC structure.

Abaqus searches for the intersection nodes of the host and embedded elements and constrains their translational degrees of freedom to the interpolated values of the corresponding degrees of freedom of the host element [5]. This approach is generally found to be very advantageous in cases where modeling the reinforcement of a structure is complex [START_REF] Murthy | Impact analysis of concrete structural components[END_REF], and to be more ecient than the discrete approach in term of the computational eort and improving the simulated interaction [START_REF] Tahmasebinia | Numerical modelling of reinforced concrete slabs suject to impact loading[END_REF][START_REF] Yamagushi | Accurate and ecient method for analysis of reinforced concrete structures[END_REF].

The discrete approach is based on the use of separate elements to represent the reinforcing steel.

Reinforcement can be modeled using truss or beam elements connected to the concrete mesh nodes in Figure 3.9: Approaches to represent steel in RC numerical analysis: a) smeared approach, b) embedded approach, c) discrete approach [START_REF] Murthy | Impact analysis of concrete structural components[END_REF] Figure 3.10: Tie constraint to tie two surfaces together in Abaqus [5] the way that the mesh discretization boundary of concrete elements must overlap the direction and location of the steel reinforcement elements (Figure 3.9.c). As a result, concrete and reinforcement elements share common nodes and they are superimposed at some specied same regions. Perfect bond is usually assumed between concrete and steel. However, in cases where slip of reinforcing bars with respect to the surrounding concrete is of concern, ctitious spring elements of no physical dimension can be used to model the bond-slip between reinforcing steel and concrete. These elements are dened as linkage elements to connect concrete nodes with reinforcement nodes having the same coordinates, and are determined by their mechanical properties and a specic bond-slip relationship [START_REF] Chong | Numerical modelling of time-dependent cracking and defromation of reinforced concrete structures[END_REF][START_REF] Murthy | Impact analysis of concrete structural components[END_REF].

Tie constraint

A surface-based tie constraint in Abaqus is used to dene a fully constrained contact behavior by providing a simple way to tie two surfaces together for the duration of a simulation. It uses a masterslave formulation to prevent slave nodes from separating or sliding relative to the master surface. As a result, an assignment of slave and master surfaces should be carried out to dene the contacting surfaces. Typically, a rigid surface is always considered as master surface, and slave surface mesh must be suciently rened to interact with the master surface. In case where two deformable surfaces need to be tied, master surface should be that of the material with higher Young's modulus or larger cross-sectional area [7]. This type of constraints enables to tie translational and rotational motions between two surfaces, hence each of the nodes of the slave surface is constrained to have the same motion as the node of the master surface to which it is closest. Abaqus eliminates the degrees of freedom of the slave surfaces nodes that are constrained and allows the tie constraints only in cases where the two surfaces to tie are close to one another (Figure 3.10) [5].

Output

For the analysis of RC slabs subjected to impact, a large amount of output can be requested and evaluated in Abaqus:

Energies: kinetic energies for whole model and slab (ALLKE), internal energy (ALLIE), energy dissipated by plastic deformation (ALLPD), recoverable strain energy (ALLSE), articial strain energy (ALLAE) and energy dissipated by viscous eects (ALLVD). When concrete damage is taken into account in simulations, the damage dissipation energy (ALLDMD) is also evaluated.

Distributions of stresses (S11, S22, S33) on the upper and bottom surfaces of the RC slab at dierent given times of the simulation.

Distributions of accelerations (A11, A22, A33) on the upper and bottom surfaces of the RC slab at dierent given times of the simulation.

Distributions of velocities (V11, V22, V33) on the upper and bottom surfaces of the RC slab at dierent given times of the simulation.

Distributions of displacements (U11, U22, U33) on the upper and bottom surfaces of the RC slab at dierent given times of the simulation.

Distributions of stresses (S11) in the upper and lower reinforcement layers at dierent given times of the simulation.

Graphical visualization of the cracking patterns in the slab obtained by assuming that cracking initiates at points where the tensile equivalent plastic strain is greater than zero and the maximum principal plastic strain is positive.

Displacements, velocities and accelerations for the impactor, as well as for dierent nodes of the slab.

Evolution of contact load (CFN3) between the impactor and the slab in terms of time and slab displacement at the point of impact.

Compressive damage variable (DAMAGEC), tensile damage variable (DAMAGET) and the equivalent plastic strain (PEEQ) at dierent elements positioned at the center and the corner of the impacted face of the slab.

Chapter 4

Reliability analysis: principles and methods

Introduction to reliability

The concept of security is very old and inseparable from the state of knowledge of nature's laws and the concept of lifetime. It is simultaneous to construction practices and related to uncertain events as well as to the notion of codes and regulations. The early builders built for their personal use and were therefore interested in what the structure fullls the functions that had been assigned.

By referring to the history of constructions, many structures are no longer in use because of a lack of knowledge about nature's laws, in particular materials and loadings, but mostly because of the time elapsed since their application. It was only in the 18th century that numerical approaches have replaced by empirical approaches. Numerical approaches allowed reducing the concept of security to a deterministic notion since they are based on the analysis of stresses: stress is obtained by a "safety factor" that comprises a certain number of unknowns and limits them to a single factor.

In the fties, Freudenthal became the head of the safety probabilistic approach that has made a signicant progress in the domain of structures [START_REF]The statistical aspect of fatigue of materials[END_REF][START_REF]The safety of structures[END_REF][START_REF]Safety and the probability of structural failure[END_REF][START_REF] Freudenthal | On the statistical interpretation of fatigue tests[END_REF]. This approach is based on the uncertain aspect of loadings on structures, material properties and mechanical models used for design. Thus, any probabilistic approach requires the acquaintance of a risk that is not identied and does not exist in deterministic approaches. The control of risk would rst result from expert knowledge and algorithmic knowledge (rules and techniques). Both knowledge increase exponentially with human development. The reliability theory and its tools were initiated in the modern era based on probability theory. Recently, probabilistic approaches were introduced in the domain of civil engineering to control risks related to design choices. This control can be included in design process by combining a stochastic data model to a mechanical data model for a mechanical behavior [START_REF] Lemaire | Fiabilité des structures: couplage mécano-abiliste statique[END_REF] (couplage mécano-abiliste).

Reliability is dened as the probability so that an equipment could be used without failure for a given period of time, under specied operational conditions (Grand Larousse). Considered as the ability of a structure to achieve a required function under certain given conditions and within a given timeframe, reliability can be applied to any system that is expected to correctly carry out its functions, e.g. a civil engineering structure. Reliability theory is based on methods using probabilities to assess failure probability that is mathematically considered as the complement of the reliability of the unit. A reliability approach of a mechanical system allows examining the statistical dispersion of quantities, rejecting values out of tolerance and verifying their variability.

Many methods based on probability theory were developed, they permit to calculate the probability of failure of structures on the one hand, and to study the inuence of variability of design parameters on the structural behavior on the other hand. Uncertainties are modeled in terms of the mean, the variance and the probability density of input variables. Reliability approach was widely used in order to ameliorate the design of RC structures. Arafah [START_REF] Arafah | Reliability of reinforced concrete beam section as aected by their reinforcement ratio[END_REF] studied the reliability of RC beam sections by considering a limit state in exion and using Monte Carlo method. Araujo [START_REF] Araujo | Probabilistic analysis of reinforced concrete columns[END_REF] presented a reliability analysis of RC columns by combining a FEM with MC in order to calculate the mean and standard deviation of the ultimate load applied to column. Val et al. [START_REF] Val | Reliability evaluation in nonlinear analysis of reinforced concrete structures[END_REF] proposed a probabilistic method to evaluate the reliability of RC frames with respect to an ultimate limit state. The method consisted in coupling a nonlinear FEM with FORM by considering all the possible sources of uncertainty. Low et al. [START_REF] Low | Reliability analysis of reinforced concrete slabs under explosive loading[END_REF] performed a reliability study on RC slabs subjected to explosion. Two deterministic models were considered: a FEM and an analytical model of one degree of freedom. The comparison of these two models was performed in a probabilistic framework by estimating the failure probability with MC in the case of FEM, and with FORM/SORM in the other case. Two failure criteria related to the slab displacement and the deformation in reinforcement were considered. The simplied model was used in a parametric study to evaluate the eect of various parameters on the slab reliability.

In for reliability analysis and can be easily connected to any external computational code. These features allow using OpenTURNS for dierent engineering and industrial problems. In addition, a complete documentation is provided for OpenTURNS and can be found on its dedicated website www.openturns.org.

In this chapter, the principles of structural reliability analysis, as well as the types of uncertainty related to the structure and the deterministic model used, are presented. Next, the steps of a reliability analysis are detailed and the methods used in OpenTURNS for uncertainty quantication, uncertainty propagation and sensitivity analysis are described. Finally, statistical descriptions of random variables intervening in RC structures are examined according to several studies in the literature.

Principles of structural reliability analysis

A civil engineering structure is a complex system composed of more or less simple elements behaving in series or in parallel. This structure should provide a number of functions (mechanical strength, durability, ...) with a certain level of reliability. In general, structural analysis and design are based on deterministic methods and attempt to assure reliability by the application of safety factors.

However, due to many sources of uncertainties associated with loads, material properties, geometry and modeling approach of structures, probabilistic concepts have to be used. Thus, structural reliability analysis should be considered and functions are performed by not exceeding a certain threshold level.

The aim of structural reliability analysis is to determine the probability that the structure does not perform its functional requirements. Structural reliability theory is based on the probabilistic modeling of the uncertainties and provides methods to estimate the probability of failure. The failure of the structure is dened through the concept of limit state which represents a boundary between satisfactory and unsatisfactory performance of the structure.

Sorensen [START_REF] Sorensen | Notes in structural reliability theory and risk analysis[END_REF] has provided the main steps in a reliability analysis:

1. Selection of an accepted or specied target reliability level.

2. Dening failure modes of the structure, i.e. exure, shear, torsion, fatigue, local damage.

3. Decomposition of failure modes into series systems or parallel systems of single components, in the case where failure modes consist of more than one component. A series system leads to the failure of the entire system if one of the system components fails, while for a parallel system no failure is induced as long as not all the system components fail.

4. Modeling the uncertainty of deterministic parameters (loads, dimensions, materials properties) by random variables and specifying the probability distribution types and statistical parameters for these variables and the correlation between them.

5. Denition of limit state functions corresponding to each component in failure modes (ultimate and serviceability limit states). Limit states are formulated in terms of the basic random variables.

6. Estimating the probability of failure for each failure mode (simulation methods, approximations methods, response surface methods).

7. Performing sensitivity analyses in order to evaluate the sensitivity of the probability of failure with respect to stochastic variables.

According to [START_REF] Pandher | Reliability analysis of strength characteristics of bre-reinforced y ash concrete[END_REF], the acceptable probabilities for structural failure are in order of 10 -3 for serviceability limit states and 10 -6 for ultimate limit states.

Sorensen [START_REF] Sorensen | Notes in structural reliability theory and risk analysis[END_REF] showed that the estimated probability of failure should be considered as a nominal measure of reliability since the information used in reliability analyses are generally not complete.

Uncertainty sources

Reliability analysis requires the knowledge of uncertainties related to the structure or system studied.

Nevertheless, before proceeding to reliability analysis and collecting information about uncertainties, an engineer needs to know that engineering systems are exposed to several types of uncertainty and each type requires a dierent process to be used in the evaluation of reliability. The uncertainties in structures or systems can be classied into ve groups [START_REF] Sorensen | Notes in structural reliability theory and risk analysis[END_REF][START_REF] Faber | Failure consequences and reliability acceptance criteria for exceptional building structures[END_REF], namely physical uncertainties, measurement uncertainties, statistical uncertainties, model uncertainties and others.

Physical uncertainties

Physical uncertainties are typically associated with loads, geometry of the structure and mechanical properties of materials. A perfect knowledge of geometric dimensions is impossible since their variability highly depends on errors in manufacturing and construction process. The uncertain aspect of material properties is due to heterogeneity that describes the variability of material characteristics from one point to another even in the same structural element. Loadings that can be classied as permanent, unstable or accidental actions represent also an uncertain aspect in time since their time history cannot be predicted in advance with certainty. Consequently, reliability analysis allows representing these uncertainties by random variables and/or stochastic processes and elds of which statistical parameters must be evaluated properly. In most cases, random variables dened by probability distributions are sucient to model the uncertain quantities. A stochastic process can be dened as a random function of time in a way that the value of the stochastic process is considered as random variable for any given point in time. Stochastic elds are dened as a random function of space and the value of the stochastic eld is a random variable for any given point in the structure.

Measurement uncertainties

Measurement uncertainties are caused by inaccurate measurements of variable values and are attributed to uncontrolled factors and imperfections in test conditions. Because of several uctuations in the environment, test procedures, tools and so on, repeated measurements of the same physical variable do not give the same value. In addition, resistance of actual physical structures includes more uncertainties than structures tested under restrained laboratory conditions as they are subjected to more uncertain environments that may be sometimes very severe. Thus, an engineer should respond to this type of uncertainty by collecting a large number of observations, which may be limited to the availability of resources such as money and time, as well as to the scale of the structure considered. The procedure of collecting observations can provide sucient information about the measured quantity variability and may lead to a certain level of condence in the value used for design.

Statistical uncertainties

Uncertainties in reliability analysis can also result from statistical parameters that are signicantly dependent on sampling methods, variance estimation and choice of probability distribution type for random variables. These parameters should be properly assessed and a condence interval should be estimated and associated for each parameter. Statistical uncertainties are due to incomplete statistical information owing to limited sample sizes of observed quantities. Statistical parameters can be considered as random variables and are used to t an input data sample to a mathematical function [START_REF] Saporta | Probabilités, analyse des données et statistiques[END_REF].

Model uncertainties

Model uncertainties are associated with structural idealizations of deterministic models used to predict the physical behavior of structures and their response. Mechanical models are based on several assumptions and can be performed using an empirical, analytical or numerical approach.

However, these assumptions considered in deterministic models can lead to inaccuracy results and simple models do not take into account all possible factors that inuence the structural behavior.

Therefore, mechanical models introduce another type of uncertainty related to modeling error. They are considered as an approximate prediction of the structural behavior since they always show some dierence compared to actual response. The dierence between calculated and actual responses can be used to develop a statistical description of the modeling error that can be introduced as an additional random variable in reliability analysis.

Other uncertainties

Other sources of uncertainty that cannot be covered by reliability analysis can arise from the denition of certain parameters, the performance of structures, human factors, i.e. the competence and experience of builders and engineers.

Reliability approach

The aim of deterministic models is to analyze structural behavior and predict the physical phenomena observed by evaluating mechanical loads or stresses (S) and resistances (R). Nevertheless, a validation of these models is necessary in order to reduce the dierence between the response of the actual structure and that of the model. Input data of a calculation model are generally known with only some signicant digits in spite of their uncertain aspect. By neglecting uncertainties the calculated results may deviate signicantly from reality, hence they should be taken into account during analysis. This can be accomplished by coupling the deterministic model with a probabilistic model that allows modeling uncertainties and performing structural reliability analysis. Design requirements become more complex by introducing uncertainties as the structure need to satisfy different criteria of failure, security and durability. The design should involve all relevant components so that the structure does not reach its limit state. For example, a structure must be designed in such a manner that its resistance is greater than the applied loads. In this case, the structure limit state can be expressed with a single random variable as Z = R -S . Figure 4.1 illustrates a volume integral of a two dimensional joint probability density function, where the two random variables are the load and the resistance, and failure is dened as the event when the load exceeds the resistance.

Reliability approaches have the advantage of providing a realistic examination of uncertainties and controlling their eect on structures response. However, their application shows some limitations as they replace the reassuring notion of safety factor and conrm the acceptance of risks.

In addition, limitations also arise as a result of lack of probabilistic knowledge and reasoning in engineering, lack of sucient data, the diculty of considering human errors and the novelty of development of tools and software to analyze reliability problems. A reliability approach is based on a statistical knowledge of basic random variables such as geometric properties, material characteristics and actions. Each variable requires a coecient of variation (COV) and a probability distribution dened by statistical parameters such as mean and standard deviation. Then by using probabilistic methods, such as simulation or approximation methods, failure probability can be assessed by studying all possible failure modes in order to make better decisions for an economical design. For each mode, the equilibrium between loads and resistances is dened by a specic failure criterion represented by a limit state function (g(X)) which is expressed in term of input random variables, but also depends on the deterministic model and the threshold of the output variable considered. A limit state function represents the state beyond which a structure can no longer perform the function for which it was designed and separates safe domain D s from failure domain D f , the latter corresponds to the domain for which the limit state function in negative (g(X) ≤ 0 ) (Figure 4.1). Let's rst consider the simplest case of Z = R -S , safe domain corresponds to values that are reasonable and physically possible and acceptable, while failure domain corresponds to inadmissible values of the variable Z.

Modeling of uncertainty

In practice, probabilities are rarely known and need to be estimated, which represent the most dicult part of reliability analysis. But before addressing this issue, uncertainties in structures or systems need to be quantied. This section describes how to model and quantify uncertainties in OpenTURNS, as well as the essential parameters and characteristics to include in a reliability analysis. Uncertainties are generally introduced in a probabilistic model as input variables assigned with a random aspect. Thus, the mathematical modeling of random variables is an essential task in any probabilistic formulation. A random variable and a particular realization of this variable are denoted in this dissertation with an uppercase letter X and a lowercase letter x , respectively. Only continuous random variables are considered in this study.

The probability density function (PDF), f X (x) , allows modeling the uncertainty of a random variable. The cumulative distribution function (CDF), F X (x) , allow estimating directly the probability that a random variable is found to have a value less or equal to a specic value. For continuous distributions, CDF represents the area under the PDF from minus innity to the value desired and is given by:

P (X ≤ x) = F X (x) = ˆx -∞ f X (x)dx (4.1)
PDF is hence expressed as: To dene PDF, some parameters are needed to be identied, such as mean, variance, coecient of variation, etc. The number of parameters depends upon the nature of uncertainty and the type of distribution. Therefore, the mean µ X of a continuous random variable X with PDF f X (x), also known as the rst central moment, can be calculated as:

f X (x) = dF X (x) dx (4.2) Figures 4.2
µ X = ˆ+∞ -∞ x f X (x)dx (4.3)
The corresponding variance of X, also known as the second central moment and denoted as Var (x ) , indicates how data are distributed about the mean and can be estimated as: The standard deviation, denoted as σ X , is expressed in the same units as the mean value and is calculated as the square root of the variance:

V ar(X) = ˆ+∞ -∞ (x -µ X ) 2 f X (x)dx
σ X = V ar(X) (4.5)
The coecient of variation of a random variable, denoted as COV (X ) or COV is introduced as the ratio of the standard deviation to the mean. It is a nondimensional parameter that allows indicating the degree of dispersion in the random variable. A high COV indicates a signicant uncertainty and an important aspect of randomness in the variable. For a deterministic variable, COV (X ) is equal to zero.

COV (X) = σ X µ X (4.6)

Joint probability density function

From a practical point of view and in order to correctly formulate a reliability problem, it is necessary to model uncertainties jointly and to not limit the analysis to random variables modeled separately.

Therefore, a reliability analysis should involve the modeling of multiple random variables. This can be accomplished by considering the joint probability density function f X (x) , which allows more information to be provided. In case of multiple random variable, X represents the random vector of input variables and x is a particular observation of this vector. Each variable in X is a random variable that can be related to structure parameters such as geometry, material properties or loads.

f X (x) permits to calculate the probability that each of random variables exists in any particular range of specic values of that variable and satises the following condition:

ˆRn f X (x 1 , x 2 , ..., x n )dx 1 dx 2 ...dx n = 1 (4.7)
In case of independent variables, the joint density is the product of the probability distributions of variables. For two random variables, the joint probability density function can be described by a three-dimensional plot as show in Figure 4.1.

Covariance and correlation

Covariance and correlation analyses are generally used to study the dependence or independence between two random variables in the aim of extracting as much information as possible for a reliability analysis. The covariance of two random variables X 1 and X 2 , COV (X 1 , X 2 ) , represents the second moment about their respective means µ X 1 and µ X 2 , and indicates the degree of linear relationship between X 1 and X 2 . COV (X 1 , X 2 ) can be calculated as:

Cov(X 1 , X 2 ) = µ X 1 X 2 -µ X 1 µ X 2 (4.8) with µ X 1 X 2 = ´+∞ -∞ ´+∞ -∞ x 1 x 2 f X 1 X 2 (x 1 , x 2 )dx 1 dx 2 .
Statistically independent variables have a covariance equal to zero.

The correlation coecient, ρ X 1 X 2 , also represents the degree of linear dependence between two random variables and its values range between -1 and +1. ρ X 1 X 2 can be expressed as a nondimen- sional parameter by: the two variables, with a correlation coecient value that is expected to range between 0 and 1. In this case, the relationship between X 1 and X 2 is positive since X 2 increases as X 1 . However, in the case of a nonlinear relationship between two random variables (Figure 4.4.c), ρ X 1 X 2 is equal to zero.

ρ X 1 X 2 = Cov(X 1 , X 2 ) σ X 1 σ X 2 (4.9)
In the present study, the correlation between random variables is not taken into consideration due to the lack of information provided on the relationship between the dierent parameters of the structures considered.

Variable of interest

The aim of a reliability analysis is to assess the probability of a variable of interest exceeding a certain threshold. Variables of interest represent the output variables of the deterministic model considered in the study. The propagation of uncertainties of input variables is performed through the deterministic model towards output variables, hence these variables are aected by the randomness of the basic input variables and have also a random character. Therefore, a description of the randomness of a variable of interest is essential to provide the necessary information on its uncertainty. OpenTURNS allows evaluating the statistical characteristics of an output variable through a dispersion analysis and a distribution analysis. In engineering applications concerning structural reliability, a variable of interest can be related to forces (resistance of structure), deections, stresses, strains, energies, etc. An output random variable is denoted in this dissertation as Y .

Dispersion analysis

The purpose of a dispersion analysis is to characterize statistical parameters of one or more quantities of interest. This involves determining the statistical moments (e.g. mean and standard deviation) of the output variable, which depends on the accuracy desired and the method used. The quality of this estimation depends on the output sample size, the amount of uncertainty present in the corresponding variable and the condence of level required for the prediction. A condence interval corresponds to a range of values in which the mean of a variable are located. Two dierent methods can be used in OpenTURNS in order to estimate statistical moments of a variable of interest:

Monte Carlo method which is a numerical integration method using sampling. This method consists in performing a sucient number of random simulations of the input vector X and evaluating the value of Y for each simulation. Then, the mean and standard deviation of the output sample can be estimated as:

µ Y = 1 N N i=1 y i (4.10) σ Y = 1 N N i=1 (y i -µ Y ) 2 1/2 (4.11)
The condence interval that represents the mean estimation uncertainty permits to control the dierence between the estimated and exact values. Its size decreases with the increase of the number of simulations and the convergence of the estimators to exact values can be assured if the sample size tends to innity. However, the only limitation of this method resides in cases where the number of simulations is not suciently high, for example in case of a deterministic model with a signicant computational eort. This can result in an important uncertainty for the estimations of µ Y and σ Y , hence it is necessary to choose another propagation method, such as the quadratic combination method, to estimate the central uncertainty of Y .

Quadratic combination method which enables to assess the central dispersion of an output variable by using a Taylor decomposition of Y in the vicinity of the mean point (µ X 1 , µ X 2 , ..., µ Xn ) of random vector X. This method depends on the order of the Taylor decomposition. Open-TURNS allows obtaining the estimations of µ Y and σ Y with a Taylor approximation at rst or second order, which may lead to inaccurate results in highly nonlinear deterministic models.

However, quadratic combination method does not require any assumptions on the PDF type of input variables, it only depends on their mean value and their dispersion.

Distribution analysis

The aim of a distribution analysis is to determine the PDF of output variables. The distribution of a variable of interest can be determined using its values calculated by the physical model through a sucient number of simulations. The determination of the distribution that ts an output sample

D n = max |F Y (y) -F S Y (y)| (4.12)
where F Y (y) is the theoretical CDF of the assumed distribution and F S Y (y) is the sample CDF. D n is a random variable and its CDF is related to a risk of error α as:

P (D n ≤ D α n ) = 1 -α (4.13) D α
n is the critical value of Kolmogorov-Smirnov test which is identied by default in OpenTURNS.

The assumed distribution is considered as accepted is the maximum dierence D n is less or equal to D α n . OpenTURNS introduces the notion of p-value which is equal to the limit error probability under which the assumed distribution is rejected (p-value= P (D n ≤ D α n ) and compares it to the p-value threshold α. Thus, the assumed distribution is accepted only if p-value is greater than the value α (a value of 0.05 is used in this study).

QQ-plot is based on estimates of the quantiles. The principle is to estimate the α-quantiles that correspond to the output sample q S Y (α) and the probability distribution chosen q Y (α). Then, the points (q S Y (α), q Y (α)) are plotted and their positions according to the diagonal are veried.

Consequently, the assumed distribution is considered as acceptable if these points are close to the diagonal.

In case where no parametric distribution t the output sample, OpenTURNS provides a nonparametric statistical technique that allows representing the probability distribution of a random variable graphically and without referring to any statistical model, namely the Kernel Smoothing method. This method depends on the size of the sample.

Probability of failure

The aim of structural reliability is to determine the probability that the structure does not perform its functional requirements. This probability or failure probability P f may be determined by the integral (4.14) in term of the limit sate function:

P f = P [g(X) < 0] = ˆg(x)≤0 f X (x)dx
P f = ˆg(x)≤0 f X (x)dx = ˆH(u)≤0 f U (u)du (4.15)
where f U (u) is the density function of the distribution in the standard space, this distribution is spherical and invariant by rotation. The Rosenblatt transformation [START_REF] Lebrun | Do rosenblatt and nataf isoprobabilistic transformations really dier?[END_REF] is the isoprobabilistic transformation used by default by OpenTURNS when random variables are independent and can only be used if the density function f X (x) of all random variables is well known. The second step consists calculating the reliability index of Hasofer-Lind β [START_REF] Lemaire | Fiabilité des structures: couplage mécano-abiliste statique[END_REF] by solving the following constrained optimization problem:

β = min( n i=1 u 2 i ) H(u i )≤0 (4.16)
Geometrically, β is the distance of the design point from the origin of the standard space (Figure 4.6).

OpenTURNS uses the algorithm of Rackwitz-Fiessler [START_REF] Rackwitz | Structural reliability under combined random load sequences[END_REF] to solve this type of problem. This algorithm is the most widely used in structural reliability as it can be simply formulated and gives satisfactory results in practice. Resolving this problem allows assessing the value of β, as well as the coordinates of the design point P * in the standard space. This point represents the most likely point of failure, or in other words the point on the limit state surface nearest to the origin. Afterwards, the limit state function is approximated by a Taylor expansion of rst order for FORM and of second order for SORM around the design point. FORM and SORM require a reduced number of calls of the deterministic code and their computational eort is independent of the magnitude of failure probability. They ensure a reasonable accurate estimate of failure probability [START_REF] Dolinski | First-order second-order moment approximation in reliability of structural systems: critical review and alternative approach[END_REF], while SORM improves the accuracy of result and gives a very good estimate for large values of β [START_REF] Shetty | Probabilistic fatigue assessment of welded joints[END_REF]. FORM and SORM lose their accuracy when the limit state surface is highly nonlinear in the vicinity of the design point (main curvatures at the failure surface at the design point are large) [START_REF] Shetty | Probabilistic fatigue assessment of welded joints[END_REF], or when there may exist other important local optimum points (the contribution of the global optimum point P * is not taken into account) [START_REF][END_REF]. In this case, failure probability is only a function of the geometric properties of the limit state surface (the design point position and the main curvatures). It is recommended not to use FORM and SORM in the case of several failure criteria.

FORM FORM approximates the failure domain to the half-space dened by means of the hyperplane which is tangent to the limit state surface at the design point (Figure 4.7). This approximation of rst order permits to calculate failure probability using equation (4.17), where Φ is the standard normal cumulative distribution function (CDF).

P f ≈ Φ(-β) (4.17)
SORM SORM consists in approximating the failure surface to a quadratic surface at the design point, with the same main curvatures as the true failure surface at the design point (Figure 4.8). To calculate the main curvature κ i , the Hessian matrix

(H ij (f ) = ∂ 2 f ∂x i ∂x j
) should be determined since the terms of second orders are considered. Failure probability can be calculated by the Breitung formula (Equation 4.18) [START_REF] Breitung | Asymptotic approximations for multivariate integrals with an application to multinormal probabilities[END_REF] which estimates the probability with an asymptotic analysis, in the sense β → ∞ with βκ i xed.

P f ≈ Φ(-β) n-1 i=1 (1 + βκ i ) -1/2 (4.18)
This expression seems as the product of the approximation of FORM by a correction factor expressed in terms of the principal curvatures at the most likely point of failure.

P f ≈ 1 N N i=1 I(x i ) (4.19)
In this equation, N is the number of generated samples of the random vector X, I(x i ) describes the indicator function so that I(x i ) = 1 if g(x i ) ≤ 0 and I(x i ) = 0 otherwise. The accuracy of the estimate certainly depends on the number of simulations and the magnitude of failure probability.

For a small value of N the estimation of P f may be subjected to a considerable uncertainty, thus a large number of simulations may be required to reduce this uncertainty to an acceptable level. A large number of samples is also required in case of low probabilities. Therefore the eciency of Monte The concept of Monte Carlo appears to be simple, however, its application in structural reliability analysis where very low probabilities usually appear may be impossible for complex structures that necessitate a huge computational eort during deterministic analyses. Therefore, and in order to achieve eciency of simulation methods, the number of simulations needs to be greatly reduced [START_REF] Haldar | Probability, reliability and statistical methods in engineering design[END_REF]. For this purpose, several variance-reduction techniques were developed to increase eciency by reducing the variance or the error of the estimate of probability of exceeding a threshold. The expected or mean value are not inuenced by these alternative techniques and the number of simulations required to assess failure probability decreases signicantly. The most commonly used variance-reduction methods are importance sampling, directional simulation and Latin hypercube sampling.

Importance sampling (IS) Importance sampling consists in generating samples that lead more frequently to failure instead of extending them uniformly over all possible values of random variables.

The distribution of sampling points is concentrated in the region of most importance that mainly contributes to failure probability (Figure 4.10). In order to obtain results in the desired region, samples are simulated from a new probability density function h U (u) instead of the original joint probability density function f X (x) and the probability of failure is given by:

P f ≈ 1 N N i=1 I(u i ) f X (u i ) h U (u i ) (4.20)
In this expression, h U (u) is known as the importance sampling density function used to generate samples and centered at the design point, U is the random vector of samples based on the new density function such that h U (u) > 0 in the failure domain [START_REF][END_REF].

Although the eciency of this method depends on the selection of the density h U (u) (the optimum importance sampling density function gives the minimum variance, i.e. the variance of P f is equal to zero), importance sampling is generally recognized as the most ecient variance reduction technique. It is very eective for estimating low failure probabilities and allows obtaining more Figure 4.10: Principle of importance sampling method, illustration in the standard space [START_REF] Deheeger | Couplage mécano-abiliste: SMART -méthodologie d'apprentissage stochastique en abilité[END_REF] information on failure domain since samples are concentrated in the region of high density of failure probability around the design point. Importance sampling produces accurate and satisfactory results if the design point is correctly identied and reduces signicantly the number of simulations for an appropriate choice of h U (u) (OpenTURNS uses a normal importance distribution, centered on the standard design point). In addition, the number of samples aects the eciency of the sampling method and sucient simulations in the failure region are needed. Indeed, the number of simulations required to assess an accurate estimation of failure probability increases with the dimension of input random vector. As the estimation of failure probability by importance sampling method depends on the identication of the design point, this method shows the same disadvantages of FORM and SORM related the nonlinearity of the limit state surface and the number of optimum points.

Directional simulation (DS) Directional simulation is also frequently used for the structural reliability assessment. This method is more economical than Monte Carlo in term of computational eort and requires no additional information. Directional simulation is based on the concept of conditional probability and implicates the rotational symmetry of multi-normal probability density in the standard space. The simulations are performed radially, hence random directions are generated in the space instead of generating random simulation points and determining whether these simulations are in failure or safe domain (Figure 4.11). Random directional vectors are generated according to a uniform density distributed on the unit hypersphere and are expressed in polar coordinates. For each random direction, the intersection of the direction and the limit state surface is determined iteratively and the distance ρ i from the origin to the limit state surface is searched. Failure probability is evaluated in the conditional directional by a Chi-Square probability function with n degrees of freedom (χ 2 n ) as follows:

P f ≈ 1 N N i=1 1 -χ 2 n (ρ i ) 2 (4.21)
OpenTURNS requires the use of a numerical solver at every step of directional simulation method, this solver necessitates a certain number of calls of the external deterministic code in addition to the number of simulations required by directional simulation to assess failure probability. As a simulation probabilistic method, directional simulation has the characteristic that failure probability converges to an accurate value if the number of simulations increases. However, in the case of large number of Figure 4.11: Principle of directional simulation method, illustration in the standard space [START_REF] Deheeger | Couplage mécano-abiliste: SMART -méthodologie d'apprentissage stochastique en abilité[END_REF] random variables, Monte Carlo requires less simulations than directional simulation to converge to a solution of similar accuracy [START_REF] Grooteman | An adaptive directional importance sampling method for structural reliability[END_REF].

Latin hypercube sampling (LHS) Latin hypercube sampling is also another simulation based technique to evaluate reliability. This method is based on a stratied sampling strategy, hence it allows a better cover of the domain of input variables variation. The sampling procedure consists in dividing the domain of each random variable into several intervals and assigning the same probability to all intervals. The number of intervals depends on the sample size generated for each variable.

From each interval, a value is selected randomly with respect to the probability density in the interval. A combination of random variable intervals permit to form the hypercube. In this method, all the areas of the space are represented by input values, which results in a smoother sampling of the probability distributions. The estimator of failure probability with Latin hypercube sampling method is similar to that followed in Monte Carlo method and the governing is given in Eq. (4.19).

Latin hypercube sampling can also be combined with importance sampling method in the aim of reducing the failure probability variance. In this case, simulations are generated uniformly in an hypercube centered at the design point (Figure 4.12) and failure probability can be estimated by the following equation:

P f ≈ V (p) N N i=1 I(x i ) f X (u i ) (4.22)
where V (p) is the volume of the hypercube.

OpenTURNS does not allow assessing failure probability by combining Latin hypercube sampling with importance sampling, but also it requires input variables to be independent so that this method could be applicable. Latin hypercube sampling methods can give comparable results to Monte Carlo, but with a reduced number of calls of the deterministic model. It also improves the manner of generating samples and enables a better exploration of the domain variations of input variables [START_REF][END_REF].

As all simulation probabilistic methods, a small number of simulations may not give accurate results, while large sample size may not be convenient for deterministic models that are time consuming. Taylor response surface is very useful to study central tendencies and determine the PDF of the variable of interest. In this case, the expansion should be performed around the point of mean values of input variables. However, if the aim is to assess the probability of exceedance of a threshold, the Taylor expansion should be performed around the design point and its accuracy must be justied.

The Least Square method enables to replace the initial model limit state function by a linear response surface of type:

g(X) = AX + B (4.23)
where A is the matrix of the coecients to be determined, and B is the constant term. The coecients can be determined from a sample of output values obtained with the initial model and calculated for dierent values of the input variables. The Least Square method allows determining these coecients by minimizing the quadratic error between the output values of the initial model and the approximate output values obtained by the response surface for the same input values.

The polynomial chaos expansion allows approximating the initial deterministic model by a response surface considered as a projection of the physical model in a truncated Hilbertian orthonormal basis:

g(X) = P -1 i=0 α i ψ i • T (X) (4.24)
where T is an isoprobabilistic transformation which transforms the input random vector into a reduced vector of independent components. In OpenTURNS, the Nataf transformation is used if the input random vector has a Normal copula, while the Rosenblatt transformation is used for all other cases [START_REF][END_REF]. ψ i represents a multivariate polynomial basis in the Hilbertian space associated with a particular polynomial family. OpenTURNS provides 4 polynomial family types, namely Hermite, Legendre, Jacobi and Generalized Laguerre orthogonal polynomials. The polynomial chaos is described by a degree P and a nite number of coecients α i that represent the unknown quantities of the problem. These coecients are evaluated in OpenTURNS using least squares strategy through a nite number of realizations of the input physical vector. The principle consists in selecting the realizations according to a convenient experimental design and minimizing the square residual between the initial model and meta-model responses.

In order to construct a basis of the meta-model with polynomial chaos expansion, three strategies are implemented in OpenTURNS:

The Fixed strategy corresponds to a polynomial chaos basis with a specic number of terms and depends on the degree of the polynomial chaos and the number of input random variables.

The Sequential strategy does not take into consideration the number of terms nor the degree of the polynomial chaos. First, an initial basis is proposed and then updated iteratively depending on a convergence criterion based on the dierence between the physical model response and the polynomial chaos prediction.

The Cleaning strategy consists in generating an initial basis with a nite number of terms and eliminating the non-ecient terms (i.e. the terms associated to coecients less than a given value). Then, a new term is generated and the polynomial basis is updated according to the new coecient. The procedure is repeated iteratively until only the most signicant terms are considered.

Comparison of methods according to some reviews in the literature

Reliability results are sensitive to the used reliability analysis method [START_REF] Ghoulbzouri | Evaluating some reliability analysis methodomethod in seismic design[END_REF][START_REF] Ghoulbzouri | Reliability analysis of reinforced concrete buildings: comparison between form and ism[END_REF][START_REF] Ghoulbzouri | Reliability analysis in performance-based earthquake engineering[END_REF]. Thus, probabilistic method should be properly chosen for a particular problem. Two methodologies were proposed by [START_REF] Ghoulbzouri | Evaluating some reliability analysis methodomethod in seismic design[END_REF][START_REF] Ghoulbzouri | Reliability analysis of reinforced concrete buildings: comparison between form and ism[END_REF][START_REF] Ghoulbzouri | Reliability analysis in performance-based earthquake engineering[END_REF] in order to compare approximation and simulation probabilistic methods in term of their eciency in assessing the same results of failure probability. The rst consisted in coupling the probabilistic methods directly to the nite element model, while response surface method was used to simplify the model. The approximated model was also coupled to the same probabilistic methods.

The inuence of random variables PDF was also investigated.

The physical problem examined in [START_REF] Ghoulbzouri | Evaluating some reliability analysis methodomethod in seismic design[END_REF][START_REF] Ghoulbzouri | Reliability analysis of reinforced concrete buildings: comparison between form and ism[END_REF][START_REF] Ghoulbzouri | Reliability analysis in performance-based earthquake engineering[END_REF] is performed within the context of performance based seismic engineering design. The nite element model is based on a nonlinear static pushover analysis in [START_REF] Ghoulbzouri | Evaluating some reliability analysis methodomethod in seismic design[END_REF][START_REF] Ghoulbzouri | Reliability analysis of reinforced concrete buildings: comparison between form and ism[END_REF], while the problem is studied assuming a linear elastic behavior in [START_REF] Ghoulbzouri | Reliability analysis in performance-based earthquake engineering[END_REF]. Ghoulbzouri et al. [START_REF] Ghoulbzouri | Evaluating some reliability analysis methodomethod in seismic design[END_REF] studied the reliability of two-story RC building structure taking the uncertainties of three input variables into account. OpenSees was used to estimate failure probability using three dierent types of probability distribution for random variables (lognormal, normal and gamma).

Goulbzouri et al. [START_REF] Ghoulbzouri | Reliability analysis of reinforced concrete buildings: comparison between form and ism[END_REF] developed their reliability analysis to the case of four-story RC building structure. OpenSees was also used to assess failure probability and the same random variables were considered and modeled using a lognormal distribution. Another reliability analysis four-story RC building structure was also performed by Goulbzouri et al. [START_REF] Ghoulbzouri | Reliability analysis in performance-based earthquake engineering[END_REF] using FERUM and considering 15 random lognormal variables related to the structure geometry, material properties and loading conditions. The limit state function examined in the three references was related to the horizontal displacement at the roof of the structure in order to fulll the requirements of the Moroccan seismic code RPS 2000.

Goulbzouri et al. found that the direct coupling of probabilistic methods with the nite element model does not predict the same results as the approximated response surface coupled to the same probabilistic methods. This latter underestimates in general failure probability, but it allows performing the reliability analysis with less computational eort compared to the direct coupling with the nite element model. The computational eort of response surface method depends only on the time needed to identify the approximated function.

Sensitivity analysis

The objective of reliability sensitivity analysis is to identify input variables that mostly contribute to the variability of the variable of interest and that of failure probability. Sensitivity analysis is important to gain more information about the deterministic model and its probabilistic behavior, especially for models with large number of input variables and those that represent a high nonlinear response and several failure modes. Results from sensitivity analysis can be used to ameliorate structural design in engineering, for examples the dimension of a design problem can be reduced by determining the variables that have a negligible probabilistic eect, the probabilistic response can be also improved by reducing the uncertainty in random variables. Several methods permit to estimate the sensitivity of input random variables:

Quadratic combination method that is based on a Taylor expansion of the output variable and evaluates the importance factors (IF) of input variables on the estimator of the output variable mean:

IF x i = ∂g ∂x i (µ X ) (4.25)
FORM permit to calculate direction cosines that describes the sensitivity of each standard variable U i on the reliability index around the design point.

IF x i = ∂β ∂u i (u * ) (4.26)
However, it is impossible to compare one factor to another since variables have dierent physical units. Therefore, importance factors are multiplied by normalized factors in OpenTURNS, which enables the comparison of these factors independently of the original units of the deterministic model variables.

Statistical descriptions of random variables

In order to study the reliability of RC structures, a statistical description of intervening random variables must be provided. The basic information required is the probability distribution (PDF) of each random variable and its statistical parameters, such as the mean and the coecient of variation (COV). According to [START_REF] Jonsson | Reliability analysis of structural concrete elements[END_REF], the mean and COV of variables should be representative of values that would be expected in actual structures in-situ. A sucient data are required in order to obtain a reasonable estimate of the probability distribution. However, in other cases where the data do not exist it is dicult to develop statistical description for variables. Hence, a literature review is conducted in this section in order to examine statistical descriptions of random variables intervening in RC structures according to several studies in the literature.

Concrete properties

In practice, the strength of concrete in a structure may dier from its specied design strength [START_REF] Mirza | Statistical description of the strength of concrete[END_REF] and may not be uniform throughout the structure [START_REF] Jonsson | Reliability analysis of structural concrete elements[END_REF]. According to [START_REF] Jonsson | Reliability analysis of structural concrete elements[END_REF], the major sources of variations in concrete strength are due to variations in material properties and proportions of concrete mix, variations in mixing, transporting, placing and curing methods, variations in testing procedures and the rate of loading, and nally the size eects.

Compressive strength of concrete

Mirza et al. [START_REF] Mirza | Statistical description of the strength of concrete[END_REF] have derived a relationship for the COV in-situ compressive strength as:

V c = (V 2 ccyl + 0.0084) 1/2 (4.27)
where V ccyl is the COV of the cylinder tests. For average control V ccyl is about 0.15 and 0.12 for 20.68 MPA and 34.47 MPa concretes, respectively, and V c can be taken as 0.18 and 0.15. The concrete strength f c is assumed to follow a normal distribution and the COV of the concrete insitu compressive strength can be considered to vary between 0.15 and 0.18. These values were also suggested by [START_REF] Ellingwood | Development of a probability based load criterion for american standard a58[END_REF][START_REF] Macgregor | Safety and limit states design for reinforced concrete[END_REF][START_REF] Macgregor | Statistical analysis of resistance of reinforced and prestressed concrete members[END_REF] A reliability analysis was performed by Jonsson [START_REF] Jonsson | Reliability analysis of structural concrete elements[END_REF] on RC beams in bending and concrete oshore structures which are exposed to extreme environmental load such as icebergs and waves.

The mean value of the compression strength considered in this study and presented in table 4.2 is adjusted for the rate of loading. In the paper of Val et al. [START_REF] Val | Reliability evaluation in nonlinear analysis of reinforced concrete structures[END_REF], a probabilistic method for reliability evaluation of plane frame structures with respect to ultimate limit states was proposed. The mean value of the compressive strength was estimated by:

f cm = f c + 8 (4.28)
Val et al. assumed that the compressive strength has a normal distribution, thus the COV may be obtained by:

COV fc = 8 1.645f cm (4.29)
Sun [START_REF] Sun | Probabilistic analysis of reinforced concrete bridge girders under corrosion[END_REF] has studied the eect of corrosion on the reliability of bridge girders. In this study and according to [START_REF] Macgregor | Statistical analysis of resistance of reinforced and prestressed concrete members[END_REF], the concrete strength f c is assumed to follow a normal distribution with COV of 0.16. [35] have used Monte Carlo simulation in order to assess the variability of deections of RC beams and one-way slabs with known statistical data and probability. Their results showed that the variability of deections can be high due to the random nature of the parameters related to the concrete and steel strength, as well as the structural dimensions. The parameters were considered to be independent except for the modulus of elasticity and the tensile strength of concrete which were considered correlated with the compressive strength of concrete. The values of variables in this study were given in 'psi' for strengths and in 'in' for dimensions. The mean of the concrete compressive strength was given by: 0.675f ck + 1100 ≤ 1.15f ck (4.30) with a COV equal to 0.176.

Choi and Kwon

Braverman et al. [START_REF] Braverman | Structural performance of degraded reinforced concrete members[END_REF] have performed a study to evaluate, in probabilistic terms, the eects of agerelated degradation on the structural performance of reinforced concrete members at nuclear power plants. The paper focuses on degradation of reinforced concrete exural members (beams and slabs) and shear walls due to the loss of steel reinforcing area and loss of concrete area (cracking/spalling).

As shown in Table 4.2, the mean value of the in-situ concrete strength and its COV considered in this paper for RC slabs were based on [START_REF] Ellingwood | Probabilistic descriptions of resistance of safety-related structures in nuclear plants[END_REF][START_REF] Macgregor | Statistical analysis of resistance of reinforced and prestressed concrete members[END_REF].

In order to combine the dynamic analysis of the structural slab subjected to blast loading with its various aspects of uncertainties, a reliability study has been performed via the setting up of displacement and strain performance functions [START_REF] Low | Reliability analysis of reinforced concrete slabs under explosive loading[END_REF]. This paper presents also results from a parametric investigation of the reliability of RC slabs under blast loading. The results showed that the variation of Young's modulus of concrete has a largest inuence on the failure probability of the slab, followed by the yield strength of steel bar and the crushing strength of concrete. The mean values of random variables adopted in this study present the dynamically enhanced values with an enhancement factor of 1.4 for the concrete compressive strength.

Low and Hao [START_REF] Low | Reliability analysis of direct shear and exural failure modes of rc slabs under explosive loading[END_REF] have used two loosely coupled SDOF systems to model the exural and direct shear responses of one-way RC slabs subjected to explosive loading. Incorporating the effects of random variables of the structural and blast loading properties, as well as the strain rate eect caused by rapid load application, failure probabilities of the two failure modes were analyzed.

Failure probabilities of the two failure modes were evaluated by considering statistical variations of material strengths, structural dimensions, and blast loading properties. Strain rate eect on material strengths were also taken into account in the analysis. Therefore, the mean values of random variables considered in this study present the dynamically enhanced values with an enhancement factor of 2.3 for the concrete compressive strength.

The aim of the study of Firat and Yucemen [START_REF] Firat | Uncertainty analysis for reinforced concrete members constructed in turkey[END_REF] was to analyze the dierent sources of uncertainties involved in the mechanical properties of materials and geometrical quantities of RC beams, columns and shear walls in Turkey. They have showed that the in-situ compressive strength of concrete (f ck ) can be expressed in term of the compressive strength of the cylindrical specimens (f c ) (Eq. 4.31):

f ck = N fc .f c (4.31)
where N fc is the overall bias in f c and is equal to 0.72. In this study, the COV of concrete strength is found to be equal to 0.18.

Wisnieski et al. [START_REF] Wisniewski | Probability-based assessment of existing concrete bridges -stochastic resistance models and applications[END_REF] have presented probabilistic models of ultimate shear and bending resistance of RC and prestressed concrete bridge cross-sections. Statistical material and geometrical parameters were dened according to [START_REF] Nowak | Reliability of structures[END_REF][START_REF] Nowak | Calibration of design code for buildings (aci 318): part 1 -statistical model for resistance[END_REF][START_REF] Wisniewski | Safety formats for all assessment of concrete bridges -with special focus on the precast concrete[END_REF]. All random variables were considered normally distributed. Remaining parameters were considered as deterministic and their representative values were taken from [4]. Wisnieski et al. have indicated that the value of COV of concrete compressive strength for plant cast concrete (COV = 0.09) is smaller than the one for concrete cast in-situ (COV = 0.12). In addition, they showed that the concrete compressive strength is correlated with the concrete elasticity modulus and its tensile strength, with coecients of correlation C = 0.9 and C = 0.7, respectively. According to [START_REF] Mirza | Statistical description of the strength of concrete[END_REF], the COV of the in-situ tensile strength of concrete can be taken equal to 0.18 which is the value assumed for the compressive strength, and it was assumed that the tensile strength follow a normal distribution.

Several equations exist in literature in order to calculate the tensile strength of concrete in term of its compressive strength (Table 4.3).

According to [START_REF] Ceb | Ceb-p model code 1990 -ceb bulletin d'information[END_REF], the concrete tensile strength may be estimated on the basis of the concrete compressive strength using the following relationship:

0.95( f c f co ) 2/3 ≤ f t ≤ 1.85(( f c f co ) 2/3 (4.32)
where f t is the concrete tensile strength in MPa, f c is the concrete compressive strength in MPa and f co is 10 MPa.

According to [START_REF] Val | Reliability evaluation in nonlinear analysis of reinforced concrete structures[END_REF], the value of the tensile strength may be estimated from its relationship with f c :

f t = α ft f c 2/3 (4.33)
where α ft was taken as an independent normal random variable and the COV of f t was obtained from:

COV 2 ft = COV 2 α f + (2COV fc /3) 2 (4.34)
Choi and Kwon [35] have provided an equation of the tensile strength and the in-situ compressive strength of concrete (the strengths are given in psi):

f t = 8.3 f ck (4.35)
In addition, they have considered the tension stiening eect of concrete in their study in order to include the eect of cracking in the analysis. The tension stiening eect of concrete can be Figure 4.13: Linear stepped model for tension stiening [35] estimated by a tension stiening parameter, β t dened as β t = ε tu /ε ti . This parameter was used to characterize the post-peak of the tensile stress-strain relationship and was also considered as a random variable in order to model the variability of post-tensile response of concrete, which was represented as a linear reduction in the tensile stress as shown in Figure 4.13. This parameter was also considered as a random variable in the study of [START_REF] Val | Reliability evaluation in nonlinear analysis of reinforced concrete structures[END_REF] and it was assumed that it follows a normal distribution with a mean equal to its nominal value and a COV equal to 0.15 since no available data are available for this parameter.

A study on the cover failure in concrete structure following concrete deterioration has been carried out by Choo et al. [START_REF] Choo | A study on the cover failure in concrete structure following concrete deterioration -8155[END_REF]. The purpose of this study was to analyze the relationship between the range of the steel expansion and the crack creation and propagation. The tensile strength was directly calculated from the compressive strength of concrete using the following equation:

f t = 0.18f 0.7 c (4.36)
f t and f c are given in MPa.

Pandher [START_REF] Pandher | Reliability analysis of strength characteristics of bre-reinforced y ash concrete[END_REF] has studied the reliability-based Partial safety factors for compressive strength of concrete with partial replacement of cement by Fly ash having bers. Pandher has assumed that the tensile strength of concrete is approximately one tenth of its compressive strength.

Young's Modulus

Although several equations are available in the literature to estimate the static modulus of elasticity of concrete (Table 4.4).

In [START_REF] Aci | Prediction of creep, shrinkage and temperature eects in concrete structures[END_REF], the concrete elastic modulus was also estimated on the basis of the compression strength:

E c = 33ρ 1.5 c f 0.5 c (4.37)
where f c is the compressive strength (psi), E c is the elastic modulus (psi), ρ c is the weight density of the concrete (lb/f t 3 ).

Jonsson [START_REF] Jonsson | Reliability analysis of structural concrete elements[END_REF] has proposed to use the following approximate expression of the Young's modulus in the cases where only the compressive strength of the concrete is known:

E c = 5500 f c (4.38)
In the paper of Val et al. [START_REF] Val | Reliability evaluation in nonlinear analysis of reinforced concrete structures[END_REF], the statistical variation of the modulus of elasticity of concrete was expressed through the variability of the compressive strength of concrete and a coecient α E :

E c = α E (0.1f c ) 1/3 (4.39)
f c and α E were considered to be independent random variables and the COV of E c was obtained through the following relation:

COV 2 Ec ≈ COV 2 α E + (COV fc /3) 2 (4.40)
In [START_REF] Choo | A study on the cover failure in concrete structure following concrete deterioration -8155[END_REF], the Young's modulus of concrete was calculated as follows:

E c = 4.73 f c (4.41)
where E c is estimated in GPa.

Steel properties

In order to understand the eects of their variability on the reliability of RC slabs subjected to impact, the variability of the mechanical properties of reinforcing steel needs to be studied. According to [START_REF] Mirza | Variability of mechanical properties of reinforcing bars[END_REF], the variations of yield strength and modulus of elasticity of steel may be caused by varying rolling practices and quality control measures used by dierent manufacturers, as well as possible variations in cross-sectional area, steel strength, and rate of loading.

Yield strength of steel

Dierent researches have been carried out to evaluate the statistical parameters for yield strength of reinforcement (Table 4.5). In [START_REF] Macgregor | Safety and limit states design for reinforced concrete[END_REF], the probability distribution for the yield strength of steel bars and stirrups was assumed to be lognormal. The study of Mirza and MacGregor [START_REF] Mirza | Variability of mechanical properties of reinforcing bars[END_REF] was based on a sample that included 3947 bars taken from 13 sources. They have found that the probability distribution of yield strength of steel could be represented by either a beta distribution or normal distribution.

In [START_REF] Mirza | Strength and ductility of concrete slabs reinforced with welded wire fabric[END_REF], the stress-strain curve for reinforcing bars was assumed to be composed of three regions: 3. Region III, for which 0.01 < ε ≤ ε u , consists of a parabola in term of the ultimate stress

f = f y + (f u -f y ) ε -0.01 ε u -0.01 1/2 (4.44)
where f u , ε u , E s and f y are the ultimate strength, ultimate strain, modulus of elasticity, and yield strength of reinforcement.

The ultimate strain of reinforcing bars was assumed to be a deterministic variable and the ultimate strength was assumed to be 1.55 times the yield strength of reinforcing bars. The probability distribution of the yield strength reinforcing bars was assumed to follow a beta distribution with a mean value of 460 MPa and a COV of 0.09.

The aim of the study of Bournonville et al. [START_REF] Bournonville | Statistical analysis of the mechanical properties and weight of reinforcing bars[END_REF] was to evaluate the variability of the physical and mechanical properties of reinforcing steel produced throughout the United States and Canada and to develop expressions to represent the probability distribution functions for yield strength of reinforcing bars. The representation of the statistical distribution of the yield strength was expressed using a beta distribution which was found to provide a reasonably accurate description of the yield strength distribution. The general form of the beta probability density function is: Nowak and Szeszen [START_REF] Nowak | Calibration of design code for buildings (aci 318): part 1 -statistical model for resistance[END_REF] have focused on the yield strength distribution of steel reinforcing bars through 416 samples of Grade 60 reinforcement. It was found that, regardless of the bar size, a normal distribution provides a good representation for the yield strength of steel.

P DF = Cx( f -LB D ) α x( U B -f D ) β
Wisnieski et al. [START_REF] Wisniewski | Probability-based assessment of existing concrete bridges -stochastic resistance models and applications[END_REF] showed that the yield strength of steel is correlated with its ultimate strength, with a coecient of correlation C = 0.85.

Modulus of elasticity

The modulus of elasticity of steel has been found to have a small dispersion and to be more or less insensitive to the rate of loading or the bar size. But it was assumed to be a random variables in some researches (Table 4.6). According to [START_REF] Mirza | Flexural strength reduction factor for bonded prestressed concrete beams[END_REF], the probabilistic distribution of the modulus of elasticity for reinforcing bars can be considered as normal with a mean value equal to the specied value and a coecient of variation of 0.033. In [START_REF] Mirza | Strength and ductility of concrete slabs reinforced with welded wire fabric[END_REF], the modulus of elasticity of reinforcing bars was taken to be normally distributed with a mean value of 200000 MPa and a standard deviation of 6600 MPa.

Dimensions

The variability of geometrical properties of RC slabs is generally a consequence of inaccuracies in the construction process and the curing operation of concrete [START_REF] Firat | Uncertainty analysis for reinforced concrete members constructed in turkey[END_REF][START_REF] Jonsson | Reliability analysis of structural concrete elements[END_REF]. Therefore, the dimensional characteristics of RC members constructed at site are dierent from the values specied in the f y (psi) E s (ksi)

Choi and Kwon (2000) [35] -29200 0.024 Normal [START_REF] Mirza | Variability of mechanical properties of reinforcing bars[END_REF][START_REF] Naaman | Reliability of partially prestressed beams at serviceability limit states[END_REF] design of the structure. Mirza and MacGregor [START_REF] Mirza | Variations in dimensions of reinforced concrete members[END_REF] demonstrated that variations in dimensions can signicantly aect the size and the strength of concrete members, and thus dimensions must be considered as random variables. They also indicated that all suggestions for geometric properties should be considered as preliminary since they are based on interpretation of available data. Most researchers have indicated that the probability distributions for the geometric properties should be taken as normal [START_REF] Mirza | Flexural strength reduction factor for bonded prestressed concrete beams[END_REF][START_REF] Macgregor | Statistical analysis of resistance of reinforced and prestressed concrete members[END_REF] (Table 4.7). According to Hong et al. [START_REF] Hong | Calibration of material resistance factors in csa s474-m1989[END_REF] and Sun [START_REF] Sun | Probabilistic analysis of reinforced concrete bridge girders under corrosion[END_REF], the COV of dimensions decreases as the RC member size increases and for massive elements the eect of uncertainty in dimensions is negligible. The paper of Udoeyo and Ugbem [START_REF] Udoeyo | Dimensional variations in reinforced concrete members[END_REF] focuses on a study of dimensional variations of structural members measurements which were carried out on beams, columns and slabs. Based on this study, normal distributions are recommended to represent the probability distributions of all dimensions and slab thickness presents nominal values ranging from 150 to 180 mm and a coecient of variation between 0.0183 and 0.0689.

According to [START_REF] Jonsson | Reliability analysis of structural concrete elements[END_REF], the diameter of reinforcement can be considered one of the dimensional random variables involved in evaluating reliability of RC structures and it was modeled with a lognormal distribution with a mean equal to 1.0048 times the nominal diameter and a COV of 0.02. Mirza

and MacGregor [START_REF] Mirza | Probabilistic study of strength of reinforced concrete members[END_REF] also considered a lognormal distribution for the diameter of bars with a mean equal to 1.01 times the nominal diameter and a COV of 0.04.

Another dimensional variable which may be considered as a random variable is the cover of concrete. In [START_REF] Braverman | Structural performance of degraded reinforced concrete members[END_REF] , the mean of the cover was taken equal to 4.45 cm with a COV of 0.36.

Conclusion

The aim of structural reliability analysis is to determine the probability that the structure do not perform its functional requirements. A reliability approach is based on a statistical knowledge of basic random variables such as geometric properties, material characteristics and actions. Failure probability can be assessed using probabilistic methods such as simulation and approximation methods, or response surface methods allowing the approximation of the deterministic model with a polynomial function. The rst order (FORM) and second order (SORM) approximation of limit state function methods consist rst in transforming random variables of the physical space into standardized and independent random variables that dene the basic vectors of the standard normal space. Next, the most likely point of failure is identied which represents the point on the limit state surface nearest to the origin in the standard space and also called the design point. Afterwards, the limit state function is approximated by a Taylor expansion of rst order for FORM and of second order for SORM around the design point. FORM approximates the failure domain to the half-space dened by means of the tangent hyperplane to the limit state surface at the design point, and provides accurate results in case of linear limit state functions in the standard space. SORM approximates the failure surface to a quadratic surface at the design point and gives a very good estimate for large values of β. Probabilistic simulation methods are commonly used to estimate failure probability. Random A 3D nonlinear explicit nite element (FE) analysis was proven to be eective for transient and dynamic impact analysis [START_REF] Murthy | Impact analysis of concrete structural components[END_REF]. 3D models allow simulating the behavior of 3D structures more realistically than 2D models, especially that nowadays, computational resources are not an issue anymore. However, an engineer should keep in mind that complicated models are not necessarily more trustworthy than simpler models since they may contain large numbers of errors. The aim should not be to develop a model with an exact representation of the physical problem or the most accurate model possible, but rather to develop the simplest model that provides an idealized representation of the problem and though enables to represent the structural components and behaviors with accuracy. The assumptions adopted during a FE model construction should be addressed correctly in order to verify and evaluate FE results quality. For this reason, valid answers should be provided to this type of questions: Is the analysis static or dynamic? Do the material behaviors include nonlinearities and plasticity? What are the main physical loadings applied? How to dene boundary conditions? Which are the suitable FE elements to use? How to create a regular mesh?, etc.

In general, the analysis of the response of a RC slab subjected to impact loading is complex due to the many non-linearities involved. Dierent factors contribute to the nonlinear behavior of the reinforced concrete such as the nonlinear stress-strain response, the damage due to crushing and tensile failure, the eect of the loading rate, the interaction between the concrete and the reinforcement. In order to accurately simulate the structure and obtain detailed information from nite element approach, all these factors and contact algorithms must be correctly incorporated into the nite element model. Section 5.2 is included as an illustrative example to show how to develop a FEM of an impacted slab in Abaqus by performing a successive passage through its modules. This section describes in detail the 3D FE model of Chen and May RC slabs [START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF] subjected to impact using Abaqus v6.11. The analysis is performed with an explicit conguration that allows a better representation of impact problems. In order to improve the modeling process and optimizing the postprocessing of results, the model is parametrized through a code developed using the Python programming language. This code can easily be read using Abaqus/CAE and allows obtaining a better consistency between the geometry and the quality of mesh. Python script enables to systematically prioritize the accuracy of results in the regions of interest. The model developed is then validated with the experimental results of Chen and May tests. The model adopted is used to model RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings. Then, a simplied analytical model is also used for these slabs. It consists of two degrees of freedom mass-spring system which accounts for potential viscous damping.

A frequency decrease approach is used to describe the degradation of slab stiness.

FEM of Chen and May slabs 5.2.1 Creating model

All input data should be specied in consistent units in Abaqus. In addition, selecting a coordinate system to use during the FE analysis is of high importance in order to create the FE model and interpret results. In this study, all variables used to simulate the FEM of impacted slabs are given in the SI system of units and results are automatically displayed with SI units. Model assembly and results are displayed according to the default global coordinate system in Abaqus. This coordinate system consists in a right-handed Cartesian system with z-axis perpendicular to xy-plane. Positive values of z-axis are oriented upward in the vertical direction. In this dissertation, the 1-axis, 2-axis and 3-axis denote the x-axis, y-axis and z-axis, respectively. The origin (x 1 , x 2 , x 3 ) of the global coordinate system is located at the center of the lower surface of slabs.

Creating parts

Numerical simulations of RC structures require an accurate representation of the geometry of all the structural components that constitute the feature of the structure. Every component may aect the structural response and excessive simplications in geometry will obviously lead to inaccurate results. Therefore, experience and engineering judgement are necessary to develop an appropriate model for a specic type of problem. FE analysis allows considering complex geometries and a physical representation of the FEM close to the actual structure. The complex mechanical behavior of RC slabs under impact necessitates an adequate 3D simulation in terms of slab and impactor characteristics.

In

Part of slab

The rst part created for Chen and May impacted slabs problem represents the slabs concrete region surrounded by a steel frame. A three-dimensional, deformable solid part is created by sketching the 2D square geometry of dimension equal to the total width of slab and frame. The sketch is placed on a square sheet of size equal to the approximate size specied to create the part. The sketcher is dened by a certain number of units square depending on the approximate size. The sketch created for slab is then extruded in the z-axis direction with a depth equal to the slab thickness. The geometries of slab concrete region and steel frame are continuous, hence they are merged in the same part and not separated in two dierent parts in order to reduce the problem. The merging process permits to facilitate the desired meshing technique and to avoid creating constraints to tie the two parts together, which would results in reducing the computational cost of numerical analysis.

Therefore, the full geometry of slab is partitioned with many partitions to create the steel frame around the concrete region, which allows the simplication of the meshing process leading to an easier discretization and a higher quality of structured mesh.

The full geometry of slab represents a solid single cell. Partitions are performed by cutting the cell with a plane that passes completely through the cell. The method used to dene the cutting plane in the aim of creating the frame geometry is the PartitionCellByPlanePointNormal method.

In this method, partition is created by selecting a point on the cutting plane then selecting an edge or datum axis that denes the normal to this plane.

Part of reinforcement

The geometry of reinforcement is considered identical to that of the experiments of Chen and May

Part of impactor

Impactors used in Chen and May experiments have a complex shape. They are composed of two sections, the rst consists of the mass with which slabs are impacted while the second represents the section of impactor that enters in interaction with RC slabs. Slabs were tested with two dierent shapes of the interaction impactor section, a steel hemispherical section of 90 mm diameter was used to test Slabs 1,2,4,5 and 6, while a steel cylindrical section of 100 mm diameter was used to test Slab 3. Details of this section shape are presented in Figure 2.15.

In the FEM, impactors geometry is considered identical to that of the experiments of Chen and May. A three-dimensional, deformable solid part is created by sketching the 2D impactor prole and then revolving it by an angle of 360°about the y-axis of the impactor part coordinate system. The mass and interaction sections of the impactor are considered in one part in order to avoid creating constraints to tie the two sections together. However, the impactor is partitioned with a cutting plane in order to distinguish both sections and facilitate the generation of mesh. The cutting plane is dened with a point and a normal vector, and the partition scheme adopted for hemispherical impactors is illustrated in Figure 5.3. The upper part represents the mass section is highlighted in dark gray, while the bottom part represents the interaction section highlighted in light gray.

Part of support

In Chen and May experiments, slabs were supported on 4 cylindrical supports (Figure 2.14.c).

Thus, in order to have a better representation of the actual structure and develop realistic structural supports, these latter are developed in the FEM as rigid bodies having a cylindrical shape with a diameter equal to the steel frame width and a thickness of 10 mm. The shape of rigid body does not

Creating materials

Material properties greatly inuence the structural response of RC slabs and play an essential role in dissipating impact energy. Thus, proper material models capable of representing both elastic and plastic behaviors of materials are required for the nite element analysis. In Abaqus, Property module is used to dene materials and sections, as well as to assign sections to each element of mesh of the relevant part. Material properties can be modeled in Abaqus as isotropic, orthotropic, and anisotropic. In the current study, materials are assumed to be homogeneous and isotropic having the same properties in all directions. This results in only two independent quantities to identify for each material, namely the Young's modulus and Poisson's ratio. In addition, density must be dened for all materials used in any dynamic analysis where inertia eects are important.

Abaqus includes an extensively material library that can be used to model most engineering materials, including metals, polymers, concrete, fabrics and composites [7]. It also allows creating new material models through subroutines. For concrete, Abaqus provides three dierent constitutive models, namely the smeared crack concrete model, the brittle cracking model and the concrete damaged plasticity model. Among these crack models, the concrete damaged plasticity model is adopted in the present study as it is the only model that can be used in both Abaqus/Standard and Abaqus/Explicit, which make it very useful for the analysis of RC structures under both static and dynamic loadings. Furthermore, CDP model is the only model that allows representing the complete inelastic behavior of concrete in both tension and compression including damage characteristics. The classical metal plasticity model available in Abaqus is used to dene the elastic-plastic behavior of steel, which permit to express the steel stress as a tabular function of plastic strain.

Table 5.1 shows the linear mechanical properties of concrete and steel materials used in the FE model for all Chen and May slabs, while Table 5.2 shows their nonlinear mechanical properties that are used to dene the stress-strain curves for each slab materials.

Identication of constitutive parameters for CDP model

The fundamental constitutive parameters for CDP model consists of four parameters which identify the shape of the ow potential surface and the yield surface in the three-dimensional space of stresses, namely ψ, , K c and the ratio f b /f c . A biaxial failure in plane state of stress and a triaxial test of concrete are necessary to identify these four parameters, while a uniaxial compression and uniaxial tension tests are needed to be carried out to describe the evolution of the stress-strain curves of concrete (the hardening and the softening rule in tension and compression).

Flow potential parameters Physically, ψ is interpreted as a concrete internal friction angle, it represents the angle of inclination of the failure surface towards the hydrostatic axis, measured in the meridional plane (Figure 3.2). Jankowiak and Lodygowski [START_REF] Jankowiak | Identication of parameters of concrete damage plasticity constitutive model[END_REF] supposed that ψ ranges from 34°to 42°, while Kmiecik and Kaminski [START_REF] Kmiecik | Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration[END_REF] indicated that a value of 40°is usually assumed in simulations.

In the CDP model the plastic potential surface in the meridional plane assumes the form of a hyperbola (Figure 3.2). The ow potential eccentricity is a small positive value which expresses the rate of approach of the plastic potential hyperbola to its asymptote. In ABAQUS user's manual [7],

it is recommended to assume = 0.1, which implies that the material has almost the same dilation angle over a wide range of conning pressure stress values. Increasing the value of provides more curvature to the ow potential, implying that the dilation angle increases more rapidly as the conning pressure decreases. For = 0, the surface in the meridional plane becomes a straight line.

Yield surface parameters Experimental results for biaxial loading on concrete reported by Kupfer et al. [START_REF] Kupfer | Behavior of concrete under biaxial stresses[END_REF] are the most reliable to determine the ratio of biaxial strength to compression strength (f b /f c ) which represents the point in which the concrete undergoes failure under biaxial compression. Kupfer et al. [START_REF] Kupfer | Behavior of concrete under biaxial stresses[END_REF] found that this ratio ranges from 1.10 to 1.20, while Lubliner et al. [START_REF] Lubliner | A plastic-damage model for concrete[END_REF] reported a range of 1.10 to 1.16. Jankowiak and Lodygowski [START_REF] Jankowiak | Identication of parameters of concrete damage plasticity constitutive model[END_REF] indicated that f b /f c ratio is sensitive to the change of parameters ψ and . ABAQUS user's manual [7] species a default value of 1.16.

Physically, K c is interpreted as a ratio of the distances between the hydrostatic axis and respectively the compression meridian and the tension meridian in the deviatoric cross section (Figure 3.3). Typical values range for K c is between 0.64 and 0.8 [START_REF] Lubliner | A plastic-damage model for concrete[END_REF]. According to experimental results, Majewski [START_REF] Majewski | The mechanics of structural concrete in terms of elasto-plasticity[END_REF] indicates that the value of K c increases slowly with the decrease of mean stress and that it is equal to 0.6 for mean normal stress equal to zero. ABAQUS user's manual [7] recommends to assume K c = 2/3. This ratio must always higher than 0.5 and when the value of 1 is adopted, the deviatoric cross section of the failure surface becomes a circle (as in the classic DruckerPrager strength hypothesis).

Table 5.3 summarizes the parameters of yield surface and ow potential adopted in this study according to several research previously mentioned.

C9

Wahalathantri curve [START_REF] Wahalathantri | A material model for exural crack simulation in reinforced concrete elements using abaqus[END_REF] Compression behavior The CDP model has the potential to develop complete stress-strain curves of concrete for compression and tension separately based on experiment results. In this study, no experimental results are available to perform an analysis of the stress-strain curve for the concrete. Therefore, the expressions considered to describe the stress-strain curve are based on several studies in literature (see section 2.4.1) with some modications in order to t the curve with the behavior of CDP model.

The stress-strain relations proposed by [3,[START_REF] Cervera | Nonlinear analysis of reinforced concrete plates and shells using a three dimensional model[END_REF][START_REF] Desayi | Equation for the stress-strain curve for concrete[END_REF][START_REF] Pavlovic | Bolted shear connectors vs. headed studs behaviour in push-out tests[END_REF][START_REF] Wahalathantri | A material model for exural crack simulation in reinforced concrete elements using abaqus[END_REF][START_REF] Wang | Nonlinear nite element analysis of concrete structures using new constitutive models[END_REF] represent parabolic stress-strain relations for both the ascending and descending curve (see 2.4.1). They show that the nonlinear elastic behavior of concrete occurs almost from the beginning of the compression process. The CDP model requires that the elastic behavior of the material to be linear, therefore the nonlinear elastic behavior in the initial stage of these curves was neglected and replaced in this study by a linear branch. A linear elasticity limit can be assumed as 0.4f c according to Eurocode 2 [4]. Beyond this limit, the strain-hardening and strain-softening regimes keep their parabolic shapes according to the relations proposed by [3,[START_REF] Cervera | Nonlinear analysis of reinforced concrete plates and shells using a three dimensional model[END_REF][START_REF] Desayi | Equation for the stress-strain curve for concrete[END_REF][START_REF] Pavlovic | Bolted shear connectors vs. headed studs behaviour in push-out tests[END_REF][START_REF] Wahalathantri | A material model for exural crack simulation in reinforced concrete elements using abaqus[END_REF][START_REF] Wang | Nonlinear nite element analysis of concrete structures using new constitutive models[END_REF]. The stress-strain relations proposed by [4,[START_REF] Kratzig | An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[END_REF][START_REF] Majewski | The mechanics of structural concrete in terms of elasto-plasticity[END_REF] can directly be used in the CDP model since their elastic behavior is linear.

The evolution of the scalar damage variable for compression is determined using equation 3.17.

Compressive stress and damage of concrete is included in Abaqus as tabular functions of the crushing strain. Table 5.4 summarizes the compressive concrete behaviors considered for the CDP model in this study. The rst column contains the symbol attributed to each compressive stress-strain curve.

Tensile behavior The choice of tension stiening parameters is important since too little tension will introduce unstable behavior in the overall response of the model and it will be dicult to obtain numerical solutions. In order to model the behavior of concrete in tension in the FEM, several expressions for stress-strain curve, stress-crack opening curve and fracture energy are adopted from the literature (see section 2.4.2.1). Expressions of tensile stress-strain curves developed by [3,[START_REF] Kratzig | An elasto-plastic damage model for reinforced concrete with minimum number of material parameters[END_REF][START_REF] Wang | Nonlinear nite element analysis of concrete structures using new constitutive models[END_REF][START_REF] Wahalathantri | A material model for exural crack simulation in reinforced concrete elements using abaqus[END_REF] can be directly used in the CDP model since their initial elastic behavior is linear.

Tensile stress is included in Abaqus as tabular functions of the cracking strain.

As previously discussed in section 3.6.2.3, the tension stiening can be specied by a fracture energy cracking criterion by directly specifying the fracture energy G f as a material property. Thus, tensile behavior of concrete is also modeled with stress-crack opening curves. Three types of concrete stress-crack opening curve are used with linear, bilinear and trilinear descending softening branches,

G4

Bazant and Oh energy [START_REF] Bazant | Crack band theory for fracture of concrete[END_REF] respectively (Figure 2.38). Finally, the tensile behavior of concrete is modeled by considering only the fracture energy as material property. According to ABAQUS user's manual [7], typical values of G f range from 40 N/m for a typical construction concrete (with a compressive strength of approximately 20 MPa) to 120 N/m for a high-strength concrete (with a compressive strength of approximately 40 MPa). The relations proposed by [2,[START_REF] Bazant | Crack band theory for fracture of concrete[END_REF][START_REF] Hordijk | Tensile and tensile fatigue behaviour of concrete; experiments, modelling and analyses[END_REF][START_REF] Oh-Oka | Inuence of short cut ber mixing and curing time on the fracture parameter of concrete[END_REF] are adopted in this study to estimate the fracture energy.

The evolution of the scalar damage variable for tension is determined using equation 3.16. Damage of concrete are included in Abaqus as tabular functions of the cracking strain. Table 5.5 summarizes the tensile concrete behaviors considered for the CDP model in this study. The rst column contains the symbol attributed to each tensile description of concrete.

Identication of parameters for reinforcing steel stress-strain curve

Idealizations are usually used to model the reinforcing steel for a numerical analysis. In this study, ve stress-strain curves are proposed. The rst assumes a linear elastic, linear plastic steel behavior with hardening and is based on the idealization proposed [START_REF] Theodorakopoulos | Ultimate punching shear strength analysis of slab-column connections[END_REF]. The parameters of this curves related to ultimate strength, as well as yield and ultimate strains are presented in section 2.5.2. For other curves, the values of yield strain and ultimate strain of steel are considered from the literature with ε y = 0.25 [START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF] and ε u = 0.4 [START_REF] Taqieddin | Elasto-plastic and damage modeling of reinforced concrete[END_REF]. The idealization proposed by [START_REF] Taqieddin | Elasto-plastic and damage modeling of reinforced concrete[END_REF] models a linear elastic, perfectly plastic material with a yield plateau of stress equal to the yield strength of steel. The stress-strain curve proposed by [START_REF] Belarbi | Constitutive laws of concrete in tension and reinforcing bars stiened by concrete[END_REF] represents the average behavior of steel embedded in concrete with a bilinear stress-strain curve taking hardening of steel into considerations. For more details, see section 2.5.2.

In Abaqus, stress-strain curve of steel is included as tabular function of the plastic strain. Table 5.6 summarizes the stress-strain curves considered for the reinforcing steel in this study. The rst column contains the symbol attributed to each stress-strain curve of steel.

Dening and assigning section properties

Dening sections is an essential step to create nite element models in Abaqus. A denition of section requires information about properties of a part or a region of a part, such as the cross-sectional area and the material type to be assigned to this region. Each section is created independently and must refer to a material name. However, a single section can be assigned to several regions or parts as necessary and a single material can be referred to as many sections as necessary. Once sections are created, section properties can be then assigned to the homogeneous set that contains the region or part in question. As a result, the properties of the material referring to a particular section are also assigned to the region or part considered, as well as to all relevant instances in the assembly.

Assigning sections to instances is automatically performed in Abaqus and elements related to those instances have the same section properties specied. Property module in Abaqus allows several types of sections, including homogeneous solid sections, shell sections, beam sections and truss sections.

A homogeneous solid section is the simplest section type that can be dened since the only information required in this case are a material reference and the section thickness for 2D regions. 3 other sections are created to model Chen and May slabs under impact. A homogeneous solid section that refers to concrete material is assigned to the concrete part of slabs highlighted with light gray in Figure 5.1. A second homogeneous solid section is also created and assigned to the steel frame surrounding the concrete region of slabs (section in dark gray in Figure 5.1). This section refers to steel material and is also used to describe properties of the bottom section of the impactor part (section in light gray in Figure 5.3). Due to a lack of information, impactor and frame materials are assumed to have the same mechanical properties and behavior as steel reinforcement. The third homogeneous solid section created refers also to steel material and is assigned to the upper section of the impactor part (section in dark gray in Figure 5.3), but the density used for steel material of this section is magnied for each slab case in order to obtain the value of mass with which the impactor hits the corresponding slab (Table 2.1).

In the case of rigid bodies, no section can be dened. They are simulated with point mass and rotary inertia property features available in Abaqus. A mass must be assigned to a discrete rigid part that refers to a reference point and whose motion is governed by the motion applied to this point.

Dening the assembly

Assembly module is used in order to create the nal geometry of the FEM by dening instances of parts which are independent of each other. Each part created in Abaqus is oriented in its own local coordinate system independently of other parts of the model. When an instance of a part is created, The nal model of Slab 2 is displayed in a three-dimensional view (Figure 5.6). The triad in the lower-left corner of the gure indicates the orientation of the global coordinate system, the origin of this system is located at the center of the lower surface of slabs.

Meshing the model

Generating meshes in Abaqus is performed in the Mesh module that provides a variety of tools and meshing techniques to simulate structures of dierent geometries. Meshing techniques are used to control mesh characteristics and automatically generate a mesh that meets the need of a particular structure analysis. Free, structured and swept techniques are available in Abaqus with the possibility of using the medial axis or advancing front meshing algorithm. The automated process of these techniques results in a mesh that exactly conforms to the original geometry of the structure.

Partition toolset can be used in the Mesh module to create partition for structures of complex edges. In Mesh module, the element type to be assigned to the mesh must be chosen by specifying the element family, geometric order and shape. It should be noted that if an instance part is created as independent, the mesh must be performed in the Assembly module. However, the mesh must be operated in Part module for any dependent instance part.

In the current study of impacted RC slabs, A 3D solid element, the eight-node continuum element (C3D8R) is used to develop the mesh of the concrete slab and the steel frame, as well as to create the mesh of the impactor. The reinforcement is modeled with two nodes linear 3D truss elements (T3D2), while the rigid parts of supports are meshed with four-node tetrahedral elements (R3D4) using free meshing technique with the medial axis algorithm (Figure 5.7). Finite elements size of reinforcement SizeElReinf is considered equal to 0.01 m, and cylindrical supports are modeled with a seed equal to 0.0025 m.

Slab

4 dierent types of mesh are used to model the slab part (Mesh1, Mesh2, Mesh3, Mesh4) and then compared in the aim of choosing the appropriate mesh that gives the most similar results to those of experiments. For all mesh types, the steel frame is modeled with 3 elements in width.

Dierent partition schemes are considered in order to emphasize on the impact zone and allow a better transformation of energy and force from the impactor to the slab.

Mesh1 Thus, the rst mesh consists of several partitions that permit to mesh the concrete region of slab into three dierent densities of mesh (Figure 5.8). Partitions are performed in 3D by cutting the solid slab cell with a plane that passes completely through the cell. The sweep meshing technique with the medial axis algorithm are used to generate the mesh of the slab. The region at the center Mesh2 Mesh2 is the same as Mesh1 with only one dierence related to the size of elements in the thickness direction. For Mesh2, elements along the thickness of the slab have dierent size, but a bias command is used in order to have ner mesh at the upper surface of the slab. This surface is where the impact loading is applied, hence it is preferred to be modeled with ne mesh allowing a better representation of the impact zone in the FEM and more accurate results. Consequently, the thickness of the slab is modeled with ne elements and a bias ratio of 2 (Figure 5.9). is to concentrate the ne mesh density at the impact point since the region that is highly aected during an impact analysis represents the contact zone of the impactor and the slab. For Mesh3, the thickness is also meshed with ne elements and a bias ratio of 2 (Figure 5.11) and the mesh is generated using the structured meshing.

Mesh4

The concrete region of slab and the steel frame are partitioned in 3D with 2 cutting planes in x-direction and 2 cutting planes in y-direction in order to create an square that refers to the impact zone. Next, this square is partitioned in both diagonal directions to be able to create ner mesh at the impact point (Figure 5.12). The diagonals of the impact zone are meshed using the bias command with a maximum element size SizeElImpactZoneM ax of 10 mm at the square corner and a minimum element size SizeElImpactZoneM in of 1 mm at the center of the slab. The concrete this mesh allows having ne mesh density only in the impact zone where the loading is applied. For Mesh4, the thickness is also meshed with ne elements and a bias ratio of 2 (Figure 5.13) and the mesh is generated using the sweep meshing technique with the medial axis algorithm.

The Python script written for Mesh4 allows changing the size of the impact square by only specifying the desired dimension d impact , then the mesh can be automatically generated according to the seeds provided for slab and impact square edges by running the script in Abaqus/CAE (Figure 5.14). Another important advantage of Python script written for Mesh4 is that it allows studying the eect of impact position by only specifying the coordinates of impact point in x-direction x impact and in y-direction y impact . The mesh can be then automatically generated according to the impact square by running the script in Abaqus/CAE, and the impactor will be always located above the center of the impact square. (Figure 5.15). The script enables to perform the impact analysis of RC slabs at any point on the slab. A second mesh type is considered for the impactor and used in the model with Mesh4 of the slab.

In this case and in order to allow a better contact between the slab and the impactor, more partitions are needed. First, the impactor must be partitioned with a cutting plane passing through the region of intersection of its cylindrical and hemispherical parts. Next, the hemispherical part is partitioned to create a ner mesh at the center of the impactor surface that enters in contact with the slab.

The mesh of the hemispherical part is generated using the structured meshing technique, while the cylindrical part is meshed using the sweep meshing technique with the medial axis algorithm. The size of elements in the cylindrical part is considered the same as that in the previous mesh, while the hemispherical part is meshed using the bias command with a maximum element size of 10 mm and a minimum element size of 1 mm.

Conguring analysis

Abaqus can be used to solve dierent types of problems such as static, dynamic, quasi-static, seismic, etc. Analyses are congured in Abaqus through one or multiple steps to describe the various loading conditions of the problem. Steps are created using the Step module and two types of analysis steps can be dened in Abaqus, namely general response analysis steps and linear perturbation steps.

General analysis steps can be used to analyze the overall structural response that can be linear or nonlinear. Linear perturbation steps are used to calculate the linear perturbation response of the problem such as extraction of natural frequency, transient modal analysis and response spectrum analysis.

In order to determine the dynamic response of RC slabs under low velocity impact of dropped objects, only one dynamic/explicit step is created for the simulation as the problem consists of a single event. In this step, initial conditions are applied to the impactor and the time period is set to 0.03 s with a maximum time increment of 1e-6. The current analysis consists of two steps, the impact step and the initial step generated automatically in Abaqus and in which the boundary conditions are applied.

Applying loads, boundary and initial conditions to the model

As previously mentioned in section 3.4, Abaqus provides several ways to introduce loads, boundary conditions and initial conditions in nite element analysis. Specifying boundary and initial conditions and applying load conditions to the structure is step-dependent, which means that the step or steps at which these conditions become active should specied in the FEM.

To simulate the motion of the dropped object, the impactor is dropped at the center of Slab 2

and Slab 3 with a velocity of 6.5 m/s, while the impactor hits Slab 4 with an initial velocity of 8.0 m/s. Slab 5 and Slab 6 are subjected to an impact velocity of 8.7 m/s and 8.3 m/s, respectively.

Velocity values are considered as in experimental tests of Chen and May (Table 2.1). In the nite element model of each slab, the impactor hits the slab with a constant velocity that is applied to each of the nodes of the impactor in the direction perpendicular to the slabs and opposite to z-axis direction. In this study, the eect of impact loading of dropped objects on the behavior of RC slabs using a nite element analysis is examined in the presence of gravity load. Therefore, the analysis of Chen and May slabs is carried out by applying a gravity load of 9.81 m/s 2 on the whole model.

It should be noted that gravity load does not act on rigid bodies included in the FEM, namely the four cylindrical supports on which the slab is supported. Gravity load is not prescribed as a function of time nor dened according to an amplitude curve. In this case, Abaqus assumes that the loading is applied instantaneously at the beginning of the step. Gravity load is created at the impact step and acts in the negative z-axis direction.

The motion of a rigid body is governed by that of its reference point. Therefore, boundary conditions used to constrain the cylindrical support instances are applied to their reference points that are located at the center of the bottom circular surface of supports, respectively. The same boundary conditions are applied to each of the cylindrical supports on which the slab is supported, thus it is more convenient to group the reference points of these rigid bodies into a single set of geometry. The ENCASTRE boundary condition is used to fully constrain the movement of the relevant reference points and set their degrees of freedom to zero. Translational displacements of the reference points in the x-, y-and z-directions, as well as their rotations about the x-, yand z-axes, remain restrained during the whole analysis. An additional boundary condition is applied to the model in order to ensure the motion of the impactor in the vertical direction only.

Thus, a set of geometry that includes the upper circular surface of the impactor is created and the displacement/rotation boundary condition is used to prescribe its displacement and rotation degrees of freedom. A nonzero displacement in the z-axis direction is prescribed to the impactor upper surface set, while the remaining degrees of freedom are constrained to zero. Boundary conditions of supports and impactor are applied at the initial step.

Creating interaction and constraints

In the Assembly module, the various parts of the model are positioned relative to each other in order to create the nal geometry of the model but they are still unconnected. Therefore, interaction and constraints between parts or surfaces need to be dened to connect instances properly to each other.

As previously mentioned in section 3.4, Abaqus provides two algorithm to simulate contact between dierent regions in the model and several constraints types to couple the motion of a group of nodes to that of other nodes. In the current study, the embedded and tie constraints are used to represent the reinforcing steel bars in the model and to tie the four cylindrical supports to the slab bottom surface, respectively. Both algorithms for simulating contact are used and compared, namely general contact algorithm and contact pairs algorithm.

The embedded approach is used to create bond between the two instances of steel reinforcement and the slab instance and overcome the mesh dependency of the discrete approach. In this case, it is not necessary to adapt the concrete mesh to overlap the common reinforcement nodes. Concrete elements of slabs are grouped in a set of geometry and considered as the host elements. The set of truss reinforcement elements are embedded in the set of three-solid elements of concrete that constrain the translational degrees of freedom of the embedded reinforcement nodes. The embedded constraint available in Abaqus couples the nodal degree of freedom automatically assuming a full bond action between the reinforcement and concrete elements with no relative slip.

Four surface-to-surface tie constraints are used to create proper interaction between the discrete rigid elements of cylindrical supports and the solid elements of the steel frame and the concrete region of slab. Each tie constraint connects the upper surface of a cylindrical support to the corresponding quarter of the slab bottom surface. The upper surfaces of supports are designated to be the master surface as they represent the rigid surfaces, while the bottom surfaces of slab are dened as slave surface. As cylindrical supports are positioned at the intersection points of the slab concrete region and the steel frame, the slave surfaces of slab must include the bottom surfaces related to the slab concrete region as well as those related to the steel frame. A tie constraint is mesh-independent, the master surfaces are modeled with a coarser mesh. No relative motion between master and slave surfaces is allowed throughout the whole analysis, thus shear interaction is avoided between the elements of the selected surfaces.

To analyze the impact problem of RC slabs, the contact algorithms available in Abaqus are used and compared in order to model the contact between the slab and the impactor. The rst algorithm is the general contact algorithm which allows very simple denitions of contact with very few restrictions on the types of surfaces. The second algorithm is the contact pair algorithm that represents more restrictions on the types of surfaces involved. The contact pair algorithm necessitates the denition of the contact surfaces. Thus the surface-to-surface contact is used to describe the interaction between the upper surface of the slab and the impactor hemispherical surface. Both algorithm need to dene the mechanical interaction property model in Abaqus/Explicit, while the contact pair algorithm requires more denition such as the sliding formulation and the contact constraint enforcement method.

Concerning the interaction properties, the hard contact pressure-overclosure relationship is used to dene the mechanical normal interaction of the contact model. The hard contact implies that any contact pressure can be transmitted between surfaces when they are in contact, while no contact pressure is transmitted when surfaces separate. The frictional behavior between the slab and the impactor is dened using the penalty friction formulation to dene the mechanical tangential interaction. These formulation allows some relative motion between the surfaces in contact. A uniform coecient of friction of 0.45 is assumed for contact surfaces.

For contact pair algorithm, the nite sliding formulation which allows any arbitrary motion of surfaces is used to account for the relative motion of the upper surface of the slab and the impactor surface. The position of surfaces in contact changes during the analysis, then the element faces and nodes that are in contact change. In order to enforce contact constraints, both the pure master-slave kinematic and penalty contact methods are used and compared in the aim of nding the best method to model the contact in the FEM. The impactor surface is considered as the master surface since the surfaces attached to rigid bodies must always be designated as master surfaces in Abaqus/Explicit.

The slave surface in this case must be the upper surface of the slab since this latter represents the deformable body and must be with the ner mesh in order to avoid signicant penetration for hard contact.

Output requests

In order to compare numerical results to those obtained from the experiments of Chen and May, several Field and History output are dened in the Step module, but only the contact force versus time is presented in this study. The aim is to compare the force-time histories of dierent slabs with experimental results in order to nd the nite element features that allow a good agreement with experiments. Therefore, the contact force on the master surface of the impactor in the global z-direction (CFN3) is evaluated. It should be noted that a Butterworth lter of order 6 is used in the model as Chen and May used a Butterworth lter with a cut-o frequency of 2000 Hz to lter results and reduce noise. In Abaqus, ltering of contact force-time history can be applied only in the case of general contact. For kinematic and penalty contact, results are presented with no ltering as it is not allowed for contact pairs in Abaqus/Explicit.

Creating an analysis job

The nal step in a nite element analysis in Abaqus is to create a job that permits to analyze the model. Therefore, the Job module is used to create a job and submit it for analysis.

Discussion of results of Chen and May slabs FEM

First, the FEM of Chen and May slabs is validated for Slab2 using the C6T1S1 simulation. (C refers to the compression stress-strain curve, T refers to the tension stress-strain curve and S refers to the steel stress-strain curve).

Comparison of mesh types

The problem of impact is a very complex problem that involves several nonlinearities. Thus, it is important to create a mesh that allows an accurate transfer of impact energy to the slab. The region of concern in this type of problem lies at the impact where the impactor hits the slab and damage under compression caused by impact is concentrated in this zone. This region needs more attention and more rened mesh is necessary as the main physical mechanisms of RC slabs under impact occur at this region. 4 types of meshes are used in this study as indicated in section 5.2.6.

Mesh1 allows a more rened mesh at the impact zone compared to the remaining regions of the slab.

Mesh2 is the same as Mesh1 with the only dierence that elements in thickness direction do not have the same size, more rened elements are used at the upper surface where the impact phenomenon occurs. Mesh3 is the mesh allowing the most rened mesh at the impact point among the meshes considered for Chen and May slabs, elements size varies linearly in the edges direction as well as in the thickness direction. However, an excessive renement could lead to distortion problem due to the aspect ratio of very small elements at the impact point. For this reason, Mesh4 is developed in order to provide a rened mesh at the impact zone with an acceptable aspect ratio of elements.

Mesh4 allows a rened mesh only at the selected zone of impact, while elements at the slab part surrounding the impact square have the same size.

For all mesh types, the three-dimensional model of Slab2 with 5 elements in the thickness consists of:

3168 truss elements of type T3D2 with 2970 nodes to represent reinforcement, 460 solid elements of type C3D8R with 648 nodes to represent the impactor (except for Mesh4), 828 rigid elements of type R3D4 with 836 nodes to represent the cylindrical supports. The impactor is modeled with 736 solid elements with 953 nodes when Mesh4 is used, as a dierent mesh is considered for the impactor in this case. Although the mesh of impactor used with Mesh1, 2 and 3 presents a better aspect ratio for elements, the mesh used with Mesh4 is more convenient to represent the concentrated forces acting on hemispherical impactor geometry. The concrete region of Slab2 as well as the steel support consist of: 67280 solid elements of type C3D8R with 82134 nodes for Mesh1 and Mesh2, 69620 solid elements of type C3D8R with 84966 nodes for Mesh3, 33020 solid elements of type C3D8R with 40566 nodes for Mesh4.

As can be seen, Mesh4 allows reducing the number of elements and nodes, and hence the number of degrees of freedom in the numerical analysis, in comparison to other proposed meshes. Consequently, the computational eort is reduced. The comparison of experimental force-time curve of Slab2 with those obtained with the dierent mesh types in this study shows the eciency and accuracy of developing Mesh4 in predicting the behavior of the slab (Figure 5.18). The peak impact force is accurately estimated and is obtained at the same time as experimental results, while the curve shape in the after peak phase shows some dierence in comparison to experimental results. The accurate value of peak force obtained can be attributed to a good estimation of contact stiness using general contact algorithm. Impact duration does match with the tests. In numerical analysis, the slab and the impactor remain in contact for about 12 ms, while the contact duration is estimated by 14 ms in experiments.

Comparison of contact algorithms

The inuence of the contact algorithm used to model the contact phenomenon between the slab and the impactor is also studied. Figure 5.19 shows the contact force-time curve obtained using general contact algorithm and contact pair algorithm. The kinematic and penalty methods used to enforce constraints in contact pair algorithm are also compared. As can be seen, both algorithms provide good results which are in agreement with experimental results. The general contact slightly presents more accurate results, especially for the initial linear phase of the curve and the peak value of contact force. The reason is that Abaqus allows using lters to lter the force-time output curve only when the contact is dened as general algorithm. For contact pair algorithm, force-time curves are presented with no ltering. The dierence between general contact and contact pair algorithms is that the rst permits to specify automatically the contact surfaces in the nite element model, which can be very useful for complex contact problems. The contact pair algorithm necessitates specifying each of the surfaces that are in interaction.

Mesh sensitivity

A mesh sensitivity analysis is conducted in order to choose a convenient mesh that is suciently rened to converge to an accurate solution with the minimum computational eort. Therefore, several sensitivity analyses are performed:

The number of elements in the thickness direction is investigated, and it is found that 5 elements are sucient to obtain numerical results that are in good agreement with experiments.

Using 7 or 9 elements in the thickness direction does provide a more improvement in results (Figure 5.20).

The maximum and minimum size of elements in the impact zone is studied. Elements are generated in this zone using the bias seeding option. It can be seen that there is no need to use very small elements at this region and that a maximum size of 0.01 m and a minimum size of 0.001 are sucient to converge to a unique solution (Figure 5.20).

The size of elements surrounding the impact region is varied using a ner, intermediate and coarser mesh. Numerical results obtained with elements of seed of 0.01m are in good agreement with experimental results, while smaller elements do not give more accurate results (Figure 5.21).

The size of elements in reinforcement is studied. It can be seen that the size of these elements has no eect on results (Figure 5.21).

Choice of materials behavior

In this study, the dynamic problem of impact is investigated in order to predict the structural behavior of RC slabs. Although the impactor is modeled as a deformable body in the FEM of Chen and May slabs, the response of impactor is not of concern. The reason for which the impactor is modeled as deformable body is related to the aim of allowing the best quality of transmission of impact energy between the impactor and the slab. The response of slab is more important in terms of deformations and damages. The upper surface of the slab is the rst element that enters in collision with the impactor, this surface is under compression due the compressive wave generated by the impactor at the moment of impact. Then, the compressive wave progresses in the slab thickness and reects from the bottom surface as a tensile wave. Thus, it is important to rst specify the behavior of concrete in compression, as it is the rst physical mechanism involved during an impact event.

Thereafter, its behavior in tension can be studied.

The most dicult parameters to identify for the CDP model are those related to the behavior of concrete in tension and compression. In order to accurately dene the concrete behavior in a numerical analysis of an impact problem, several relationships that describe the compressive stresses in concrete in term of compressive strain are considered. Those stress-strain curves are based on several experimental results from the literature (see Chapter 2). Force-time curves obtained using these curves in the model are depicted in Figures 5.22-5.24. It can be seen that compressive stressstrain curves with a higher resistance to crushing allow a better estimation of the peak impact force (C6 and C9), while compressive stress-strain curve with more softening allow a linear plateau as experiments and a better estimation of the shape of the force-time curve after the rst peak.

Consequently, the curve C6 is adopted since it provides the best t with experimental results.

Subsequently, the behavior of concrete in tension is studied. CDP model permits to describe the tensile behavior of concrete using a stress-strain curve, a stress-cracking displacement curve and a fracture energy value. Several cases from the literature are considered and compared (Figures 5. 25-5.28). It can be seen that for stress-strain curves T1 gives results which are in good agreement with experiments. However, the trilinear stress-cracking displacement curve D3 seems to be the most accurate way to describe concrete in tension in the CDP model. It provides a good estimation of the peak impact force, it also predicts correctly the plateau in the force-time curve. Curves based on the fracture energy cracking criterion seem to be more ecient as they permit a better assessing of deformation and cracks opening. These curves are more adequate to reduce the concern of the mesh sensitivity, however they results in more computational eort compared to tensile stress-strain curves.

Figures 5. 29-5.30 show that the steel behavior has no signicant eect on the prediction of force-time curves.

Validation of model for other slabs

C6T1S1 and C6D3S1 simulations with the relevant constitutive behaviors of materials are found to give the most accurate in the case of Slab2. Thus, both simulations are used to validate the model for the other slabs of Chen and May experiments (Figures 5. 31-5.32). As can be seen, using stress-cracking displacement curve to describe the tensile behavior of concrete in numerical analysis provides more accurate results for all slabs then using tensile stress-strain curve.

In the case of Slab3, a at impactor is used in experiments as well as in the FEM. The peak impact force is overestimated in this case indicating the diculty of general contact algorithm in accurately modeling the interaction between the slab and the impactor when a at surface is considered for the impactor. In the case of hemispherical impactor, the general contact algorithm allows a better interaction between the two bodies in collision (Slab2, Slab4, Slab5). Slab4 is impacted with higher velocity than in the case of Slab2 and Slab3, the FEM allows a good prediction of the peak value for higher velocity values. For Slab3, 4 and 5 the times at which the impact force reaches its peak are in The following section discusses the FEM of a RC slab which is subjected to accidental dropped object impact during handling operations within nuclear plant buildings. From the previous section where the FEM proposed for impact problems is validated with experiments from the literature, an accurate FEM of impacted slabs should introduce dierent factors that involve several nonlinearities in the response of impacted RC slabs. The factors that should be correctly incorporated into in the model and that contribute to the nonlinear behavior of the reinforced concrete are the nonlinear stress-strain response, the damage due to crushing and tensile failure, the interaction between the concrete and the reinforcement, and the contact algorithm. In order to accurately simulate the structure and obtain detailed information from nite element approach, all these factors and contact algorithms must be correctly incorporated into the nite element model. However, since the FEM of slab in nuclear plant is used for reliability analysis, the aim is to choose nite element features that allow the minimum computational cost, but also provide an accurate representation of the impact phenomenon. The slab is studied in a reliability framework for 2 cases:

Assuming an elastic behavior of steel and concrete materials.

Assuming a nonlinear behavior of steel and concrete materials. The choice of stress-strain curves adopted is based on those used in the FEM of Chen and May slabs by considering the relations that give a good agreement with experimental results with the minimum computational eort. In order to evaluate the transmission of impact energy to the slab during an impact phenomenon and consider the interaction between the slab and the impactor, the impactor is incorporated into the model as a sphere rigid body with 3600 kg mass and 30 cm diameter, dropped at the center of the slab with a velocity of 7.7 m/s. Table 5.7 shows the linear mechanical properties of concrete and steel materials used in the FE model for the slab in nuclear plant, while Table 5.8 shows its nonlinear mechanical properties that are used to dene the stress-strain curves for each slab materials.

Finite element features used for elastic case are detailed in Chapter 6. Only the FEM of nonlinear case is detailed in this section. Mesh 4 is adopted from the study of Chen and May slabs as it is the most suitable to use in the case of an impact problem. This mesh allows a better simulation of the contact between the slab and the impactor and hence a better estimation of impact force. A 3D solid element, the eight-node continuum element (C3D8R) is used to develop the mesh of the concrete slab. Abaqus/Explicit provides three alternative kinematic formulations for the C3D8R solid

f c (MPa) f t (MPa) f y (MPa)
in nuclear plant 40 3.5 500 element, namely the average strain formulation, the orthogonal formulation and centroid formulation. This latter is selected for the nonlinear case in the aim of reducing the nite element analysis computational eort since it is the most economical among the three formulations. As illustrated in Figure 5.33.a, elements along the slab thickness have dierent size with ner elements located at the upper surface of the slab. The thickness is modeled with ve elements in the thickness direction and a bias ratio of 2. The slab is modeled with two dierent densities of mesh. The impact zone is modeled with ner mesh, it consists of a square of 1.0 m with a maximum element size of 10 cm and a minimum element size of 1 cm. The mesh around the impact zone is considered uniform in the xy-plane with a size of 10 cm. The reinforcement is modeled with truss elements (T3D2) of 10 cm of size and the embedded approach is used to create the bond between the steel reinforcement and concrete. This approach allows independent choice of concrete mesh and arbitrarily denes the reinforcing steel regardless of the mesh shape and size of the concrete element. The impactor is meshed with four-node tetrahedral elements (R3D4) having a constant seed of 2 cm.

The contact pair algorithm is selected in order to model the contact and interaction problems between the slab and the impactor since the model involves only two contact surfaces, namely the upper surface of the slab and the impactor surface. In addition, it is more ecient in term of computational cost when used with kinematic contact constraints. The nite sliding formulation which allows any arbitrary motion of surfaces is used to account for the relative motion of the upper surface of the slab and the impactor surface. The contact constraints are enforced with a pure master-slave kinematic contact algorithm which does not allow the penetration of slave nodes into the master surface (surfaces attached to rigid bodies must always be dened as master surface in Abaqus/Explicit). Two interaction properties are presented in the model, a hard contact normal interaction dened as hard contact and a penalty tangential interaction with a friction coecient of 0.45.

Stress-strain curves of concrete in tension and compression considered are those proposed by Wang and Hsu [START_REF] Wang | Nonlinear nite element analysis of concrete structures using new constitutive models[END_REF], while steel is modeled according to Theodorakopoulos and Swamy curve [START_REF] Theodorakopoulos | Ultimate punching shear strength analysis of slab-column connections[END_REF].

The FEM of the slab consists of 32072 nodes and more than 27400 elements (21800 elements of type C3D8R, 4448 elements of type T3D2, 1152 elements of type R3D4), and a simulation takes a total time of 43 minutes for an impact step of 0.1 s. The Python script written for this enables to easily change the impact position, as well as add stirrups (Figure 5.33.b), in the FEM (the main FEM of this slab does not include shear reinforcement).

Mass-spring models (MSM)

In order to simulate the impact of a dropped object on RC slab, two simplied analytical models of the slab are used. The two models consist in a two degrees of freedom mass-spring system which accounts for nonlinearity of RC by a stiness degradation approach. The second model also accounts for potential viscous damping. The input parameters that involve in these models are slab geometry, reinforcement ratio, concrete and steel densities, impactor mass and velocity. Outputs of these models are the time-history curves of slab displacement at impact point, slab velocity at impact point, impactor displacement and impactor velocity. From the mechanical equilibrium of the two masses, two coupled ordinary dierential equations of second-order are derived and solved in time by an approximated method, namely the Runge-Kutta fourth order method:

d 2 x i dt 2 + kc m i x i -kc m i x s = 0 d 2 xs dt 2 -kc ms x i + kc+ks ms x s = 0 (5.1)
with the initial conditions

ẋi (0) = v, ẋs (0) = 0, x i (0) = x s (0) = 0 (5.2) Assuming that X = x i x s and M = kc m i -kc m i -kc ms (kc+ks) ms
, the system of equations for the mass-spring model without damping can be written in a matrix form as:

d 2 X dt 2 + M X = 0 (5.3)

Mass-spring model with damping

In this model, viscous damping is taken into consideration in addition to the slab and impactor stinesses and masses (Figure 5.35). It is assumed that the structural damping is to be of the viscous type according to [START_REF] Biggs | Introduction to structural dynamics[END_REF], i.e., the damping force is opposite but proportional to the velocity.

The impedance c c accounts for the viscosity of the contact, while c s accounts for the viscosity of the slab. In this case, the two dierential equations can be expressed by:

d 2 x i dt 2 + kc m i x i -kc m i x s + cc m i ẋs -cc m i ẋs = 0 d 2 xs dt 2 -kc ms x i + kc+ks ms x s + -cc ms ẋi + (cs+cc) ms ẋs = 0 (5.4)
with the same initial conditions considered for the previous model and the equations are also solved in time by the Runge-Kutta fourth order method.

Assuming that X =

x i x s , C = cc m i -cc m i -cc ms (cs+cc) ms , and M = kc m i -kc m i
kc ms (kc+ks) ms

, the system of equations for the mass-spring model with damping can be written in a matrix form as:

d 2 X dt 2 + C dX dt + M X = 0 (5.5)

Identication of MSM parameters

In order to dene the equivalent two-degrees of freedom systems presented in the previous section, it is necessary to evaluate the parameters of these models. The equivalent system is selected so that the deection of the concentrated mass is equal to the deection of the slab at the impact point.

Eective mass of slab In order to obtain the slab mass of the equivalent system, the mass of the real slab is multiplied by a transformation factor α M [START_REF] Biggs | Introduction to structural dynamics[END_REF].

m s equi = α M × m s (5.6)
According to [START_REF] Christoforou | Scaling of low-velocity impact response in composite structures[END_REF][START_REF] El-Dakhakhni | Validity of sdof models for analyzing two-way rc panels under blast loading[END_REF], the equivalent mass of the slab can be obtained by equating the kinetic energy for the equivalent and real system:

1 2 ( ¨αM ρdA) × ( du imp dt ) 2 = 1 2 ¨A ρ du(x, y) dt 2 dA (5.7) α M = ˜A ρφ 2 (x, y)dA ˜ρdA (5.8)
In the above equations, u imp (t) is the deection of the slab at the impact point and u(x, y, t) represents the deection at any point on the slab of x-and y-coordinates; φ(x, y, t) = u(x,y,t) u imp(t) is the deected shape function; A is the slab area and ρ represents the mass per unit area of the slab.

According to [START_REF] Biggs | Introduction to structural dynamics[END_REF], the mass transformation factor, for two-way slabs with simple and xed supports, is evaluated on the basis of an assumed deected shape of the actual structure which is taken to be the same as that resulting from the static application of the dynamic loads. The approximation of the deected shape is based on the classical plate theory and is considered to satisfy the boundary conditions of simply supported slabs:

u(x, y, t) = ∞ m=1 ∞ n=1 W mn (t) sin mπx a sin nπy b
(5.9)

In the following, only the rst mode will be considered, that is, m = n = 1. Thus, the deected shape will be:

u(x, y, t) = W 1 (t) sin πx a sin πy b

(5.10)

The maximum deection at the impact point of the slab (x = a/2, y = b/2) is therefore u imp = W 1 (t).

After substituting the u imp (t) and u(x, y, t) in Equation 5.7, the equivalent mass of the slab is found to be equal the fourth of its real mass.

Slab stiness A frequency decrease approach is used to describe the RC's degradation which is due to cracks in concrete and yielding in steel. This approach allows considering the nonlinearities presented in RC. The slab stiness decrease is identied by quasi-static pushover tests in term of the displacement of the slab at the center. The initial value of slab stiness is estimated to be equal to 1.8e8 N/m for the main case of study with a thickness of 0.5 m. Figure 5.36 shows the variation of the initial value of slab stiness in term of the slab thickness, as well as the stiness decrease in term of displacement for several values of thickness.

Contact stiness In impact dynamics analysis, a detailed description of the contact between the impactor and the structure during impact is not required and statically determined contact laws can be used [START_REF] Abrate | Impact on composite structures[END_REF]. Thus, a contact law relating the contact force to the indentation is needed to dene the contact between slab and impactor. The indentation is dened as the dierence between the displacement of the impactor and the displacement of the slab at the impact point.

The local indentation α = x b -x d is represented by the spring (k c ) which can be linear or nonlinear. Thus during the loading phase of the impact, the impact force can be obtained by using a linearized contact law (q = 1) or an Hertzian contact law (q = 3/2).

F c = k c α q (5.11)
For Hertzian contact between a plate and a hemispherical indentor, the contact stiness is given by [START_REF] Abrate | Modeling of impacts on composite structures[END_REF]:

k c = 4 3 ER 1/2
(5.12)

where R represents the impactor radius [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF] and E is given by

1 E = 1 -ν 2 s E s + 1 -ν 2 i E i (5.13)
In the present study, we assumed that the impactor is rigid so E is in this case equal to:

1 E = 1 -ν 2 s E s (5.14)
By substituting the parameters by their values in the above equations, the contact stiness is found to be equal to 7.29e 9 N/m for a linearized contact law and 1.88e 10 N/m 3/2 for a Hertzian contact law.

Impedances Biggs [START_REF] Biggs | Introduction to structural dynamics[END_REF] indicated that the inclusion of damping in multidegree analysis involves some rather troublesome problems since there is little theoretical means for determining the nature of the damping. The damping was assumed to be viscous, thus the damping force applied to a mass is directly proportional to the velocity of the structure at the impact point and may be expressed by:

F v = -c ẋ (5.15)
where c is the impendence and ẋ is the velocity of the mass. Biggs also indicated that the magnitude of c is extremely dicult to determine. Thus, in order to be sure that the damping coecients are reasonable, it is convenient to introduce the concept of critical damping which represents the amount of damping that would completely eliminate vibration.

For a SDOF, the critical damping is given by:

c cr = 2 √ km (5.16)
while for multiple degree of freedom, the critical damping matrix is dened by [START_REF] Zabaras | Viscous damping approximation of laminated anisotropic composite plates using the nite element method[END_REF]:

C cr = 2M -1/2 K M -1/2 = 0
(5.17)

According to [START_REF] Yigit | Limits of asymptotic solutions in low-velocity impact[END_REF], the contact impedance c c is only valid during contact and damping mechanism is due to the energy transfer from the impactor to the structure. Denitions of the impedances of impactor and plates were given by [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF] for small-mass impact models for composite plates. The impedance of the plate is expressed by:

c s = 8 ρD * (5.18)
where ρ is the mass per unit area of the slab and D * the equivalent bending rigidity which is dened as:

D * ≈ D 11 D 22 (A + 1)/2 where A = (D 12 + 2D 66 )/ D 11 D 22 (5.19)
By substituting the parameters by their values in the above equations, the slab impedance is found to be equal to 5.52e 6 N/(m/s).

The impactor impedance is dened for an Hertzian contact (q=3/2) by:

c c = k 2/5 c ( 5 4 m i )v 1/5 0 (5.20)
and for a linear contact (q=1) by:

c c = k c m i (5.21)
By substituting the parameters by their values in the above equations, the contact impedance is found to be equal to 1.26e 7 N/(m/s) for a linearized contact law and 3.01e 6 N/(m/s) for a Hertzian contact law.

5.5 Discussion of results of deterministic models of the slab in nuclear plant

Comparison of FEM and MSM results

The aim of using a mass-spring model is to develop a simplied model economical in computational cost, but also allows an accurate and precise estimation of the structural response. Thus, both MSM models proposed for the slab in nuclear plant assuming a nonlinear behavior of material are compared with its FEM. The values of MSM parameters identied previously are used as preliminary values to initially present the model, then they are calibrated in order to nd the best t with FEM results. As a result, the values adopted for the MSM are as follows:

The equivalent mass of the slab: m s equi = 0.015 × m s .

The initial value of slab stiness varies according to the case studied in terms of the input variable values.

The contact stiness:

k c = 2e9 N/m.
The impedance of contact: c c = 1e5 N/(m/s).

The impedance of slab: c s = 1.7e5 N/(m/s). The following comparison permits to show the importance of taking the potential viscous damping in the analytical analysis of an impact problem. As can be seen, the mass-spring model without damping overestimates the slab and impactor displacements, as well as their velocities. The reason is that this type of models does not accurately model the dissipation of energy related to the plastic deformation of materials, formation of cracks, friction and damping. Considering only the stiness degradation approach to describe the reinforced concrete degradation due to cracks in concrete and yielding in steel seems then insucient, hence viscous dampers with a constant impedance should be used to provide a better dissipation of energy in the system. Thus, the mass-spring model with damping gives more accurate results.

Accordingly, several cases are studied to compare the MSM accounting for viscous damping and the FEM by varying several input variables involved in the MSM, including the impactor mass and velocity for the same kinetic energy, the impactor velocity, the impactor mass, the slab thickness, the steel diameter and the concrete density. 38 cases are investigated and detailed in Annex B.

The cases studied enable to determine the limitations of the MSM proposed, as well as the range of values of its variables for which the simplied model is veried. As a result, the two degrees of freedom mass-spring model which accounts for potential viscous damping and for nonlinearity of RC by a stiness degradation approach can be used for the following conditions:

Low-velocity high-mass impacts when the same impact energy is considered (v = 2.5 m/s, m = 34151 kg, v = 5.0 m/s, m = 8537 kg, v = 7.7 m/s, m = 3600 kg, v = 10.5 m/s, m = 1936 kg).

For these cases, the maximum values of slab displacement, impactor displacement and slab velocity, as well as the shape of the corresponding curves, are in good agreement with numerical results. The simplied model shows less eciency in predicting the shape of the impactor velocity in term of time for low-velocity high-mass impacts, although it allows a good estimation of the impactor velocity maximum value. In the cases of high-velocity low-mass impacts, the MSM allowed assessing the maximum values of slab and impactor displacements. However, the shape of their curves in term of time shows a great dierence with respect to those obtained with the FEM for the after contact phase. For these cases, the MSM estimates accurately the maximum value and the curve shape of the impactor velocity. The maximum values of slab velocity is not properly assessed with the MSM for high-velocity low-mass impacts, but the shape of slab velocity-time curve is predicted with precision for the after contact phase.

Low-velocity impacts when the same mass is considered (v = 2.5 m/s, v = 5.0 m/s, v = 7.7 m/s, v = 10.5 m/s). In these cases, the maximum values and the curves shape of the slab and impactor displacements show a good agreement with numerical results, respectively. The MSM is very ecient in estimating the slab and impactor velocities (maximum values and curves shape) for all cases studied by varying the initial velocity value, including high-velocity impacts.

High-mass impacts when the same velocity is considered (m = 2000 kg, m = 3000 kg, m = 3600 kg, m = 5000 kg, m = 6000 kg. For these cases, the MSM results are in good agreement with numerical results for all the output variables examined. The maximum values and the curves shape are properly assessed. However, for low-mass impacts (m ≤ 1000 kg) the MSM loses its accuracy and cannot be used to estimate impacted RC slab response as it is governed by more localized deection.

Thick slabs (e ≥ 0.4 m). For these cases, the MSM results are in good agreement with numerical results for all the output variables examined. This may be related to the fact that thin slabs are more subjected to penetration and perforation failure modes than thick slabs for the same impact conditions, and the MSM does not allow predicting these types of failure modes.

All values of steel reinforcing bars considered. In these cases, the maximum values and the curves shape of all output variables show a good agreement with numerical results.

All values of concrete density considered. In these cases, the maximum values and the curves shape of all output variables show a good agreement with numerical results. It should be noted that the MSM presents a higher precision and accuracy for high concrete density (ρ c ≥ 3500 kg/m 3 ).

In the present section, only two cases are presented. Figure 5.38 represents the slab displacement at the impact point, slab velocity at the impact point, impactor displacement and impactor velocity in term of time for dierent values of masses and velocities with the same impact energy. The gure shows that the response of the slab varies with the mass and the velocity of the impactor. As can be seen, the MSM provides results which are in good agreement with numerical results, especially for the phase where the slab and the impactor are in contact. When the impactor is separated from the slab, the MSM does not permit to accurately predict the numerical results. For large masses and low velocities the time of contact between the slab and the impactor, as well as the displacement at the impact point, is higher than the cases of small masses and higher velocities. For these latter cases, when the impactor strikes the slab the damage initiates and the impactor rebound earlier.

These results, which agree with [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF], demonstrate that the slab response to impact is inuenced by impactor mass and impact duration and not only by impact velocity.

Chapter 6

Structural reliability: application to RC structures 6.1 Introduction

The determination of limit state functions is necessary to perform structural reliability analysis. But it is typically impossible to dene an explicit expression of a limit state function, and to assess the probability of failure with analytical integration, especially in cases of complex structures modeled with the nite element method. To address this problem, numerous probabilistic methods can be used to estimate failure probability as previously mentioned in Chapter 4. Selection of the most suitable method for a particular type of problem is not apparent since the accuracy and computational eort of probabilistic methods depend on several factors, such as degree of nonlinearity of the limit state function, type of random variable distributions, number of random variables and their variance [START_REF] Eamon | Evaluation of accuracy and eciency of some simulation and sampling methods in structural reliability analysis[END_REF].

First, a simple application to a RC beam is performed in the aim of mastering the basics of OpenTURNS and examining the probabilistic methods proposed in OpenTURNS to estimate failure probability in structural reliability analysis in terms of their accuracy, precision and computational eort. This application is used to address the issue of solving reliability problems in the domain of civil engineering and to propose solutions based on the case studied.

Next, the problem of RC slabs subjected to low velocity impact is addressed. The aim is to address the issue of computational cost of reliability analysis and to propose computational strategies allowing the accurate assessment of the failure probability for minimum computational time. The rst strategy consists in using deterministic analytical models involving low computational time. The second one consists in choosing an appropriate probabilistic method where the failure probability is assessed from a small number of simulations. Probabilistic methods, such as Monte Carlo, FORM, SORM and importance sampling, are adopted from the beam application and used to calculate the probability of failure. The choice of random variables and their distributions, as well as failure criteria are discussed according to several studies in the literature. Firstly, the problem of impacted slab is studied assuming a exural mode of failure and an elastic behavior for steel and concrete.

Several deterministic models are used and combined with a suitable probabilistic method. Studying slabs under elastic behavior is important to examine the limitations of deterministic models and to adopt the most convenient procedure to use in case of nonlinear behavior. Following this, the problem of impacted slab is studied using deterministic models that take into account the nonlinear material properties. In both case, elastic and nonlinear behaviors, a parametric study is performed to identify the inuence of deterministic model parameters on the reliability of RC slabs under low velocity impact.

Finally, a third application is also considered in the aim of presenting a procedure to be followed to study the response of very complicated structures in a reliability framework. For this purpose, the prestressed concrete containment building of the Flamanville nuclear power plant is used. The deterministic model of the containment building consists of a numerical model in which the full process of aging is taken into account, including relaxation of the reinforcement, creep and shrinkage of concrete. This application allows examining stress evolution with time in the containment building during periodic surveillance testing carried out 20 years after its implementation. Polynomial chaos expansion is used to simplify the physical model and study the height at which the containment is under tension.

Application to RC beams

In order to illustrate the principle of each step of a structural reliability analysis presented in Chapter 4 and compare the available probabilistic methods in terms of their accuracy and computational eort, the case of a RC beam modeled with CASTEM is studied. Combining CASTEM with OpenTURNS is possible using the wrapper interface that allows linking any external code to the OpenTURNS library and using it through the library functionalities.

Deterministic model

The rst step in reliability analysis consists in providing a deterministic model (analytical or numerical) that correctly represents the structure studied in order to predict the structural response.

The case of a RC cantilever beam is proposed from experiments in the framework of the LESSLOSS project [108] (Figure 6.1). The beam under investigation is xed at only one end and subjected to a concentrated load at the free end. A numerical model based on a multiber FE approach is proposed to examine the response of the cantilever beam. Multiber nite elements are used to develop the mesh of the beam (Figure 6.2), they permit to describe the behaviors of concrete and reinforcement with a one-dimensional way and are assumed to follow the kinematics of the Timoshenko beam theory which allows taking into account shear deformation eects in the cross section. Materials are supposed to have an elastic behavior.

Probabilistic model

The variables which may be subjected to statistical variations in RC cantilever beams under an end load condition can be classied into ve categories:

1. Dimensional variables related to the geometric properties of the beam: length and section width 2. Dimensional variables related to the position of reinforcement and its size: reinforcement section width and its eccentricity 3. Variables related to the concrete properties: Young's modulus and Poisson's ratio of concrete.

4. Variables related to the steel properties: modulus of elasticity and Poisson's ratio of steel.

5. Variables related to loading: concentrated force at the free end.

For this application, the probabilistic characterization of random variables is not based on the literature review detailed in Chapter 4. The aim of this example is to study the eect of type of random variable distributions, number of random variables and their variance on the estimation of failure probability. Thus, rst, all random variables are assumed to have a lognormal distribution with a mean value equal to the nominal value and a COV of 0.1. Table 6.1 represents the random

Loading variables

F orce1

Maximum force applied 1000 N 0.1 Lognormal variables adopted in the present study in the case of multiber beam, their probability distribution, their mean and COV.

Failure criteria

Failure of structure is investigated according to a displacement failure criterion. Failure probability is estimated for the following limit state function:

g = f max -f (6.1)
where f max is the threshold of the displacement of beam at the free end and f is the displacement at the free end calculated by CASTEM. f max is considered as an additional random variable to the variables presented in Table 6.1 and also modeled with a lognormal distribution and a COV of 0.1.

Discussion of results of beam reliability 6.2.4.1 Sensitivity analysis

A detailed sensitivity analysis is performed in this section in order to identify the random variables that mostly contribute to the variability of the displacement of cantilever beams. 10 random variables are considered, including the threshold of displacement. The aim is to select the most signicant variables which control the beam response. Importance factors are calculated using Quadratic combination method and FORM and detailed in Table 6.2.

In order to verify the accuracy of these estimations, several sensitivity studies are performed.

First, a study of mechanical sensitivity that enables to estimate the variation of the limit state function when variable change is conducted. This study is purely deterministic and measures the sensitivities with respect to the design variables [START_REF] Lemaire | Structural reliability[END_REF]. In order to be able to make comparisons, a non-dimensional quantity, namely the mechanical elasticity Si , is calculated as:

Si = ∂g(X) ∂x i (x = x r ) x r g(x r ) (6.2)
where x r are particular representative values of dierent input variables, such as the mean.

Mechanical sensitivity allows distinguishing stress variables from resistance variables and is very useful to select the random variables to take into consideration. The sign of Si values that are calculated with CASTEM determines the type of variables. If Si is a positive value then the variable is a resistance variable, while a negative value indicates a stress variable. The mechanical elasticity multiplied by the COV of the variable permit to study the variability of the corresponding variable. This can be achieved by comparing the product Si × COV i to the variable mean. The variability of a variable is considered high in the case where | Si × COV i | is higher than its mean (mean of this product is equal to 0.0746). Table 6.3 presents the stress and resistance variables for the cantilever beam and shows if their variability aects highly the beam response.

Then a statistical sensitivity analysis is performed by calculating omission factors ζ i that express the relative error of the reliability index when a random variable is replaced by a deterministic value [START_REF] Lemaire | Structural reliability[END_REF]:

ζ i = β X i β (6.3)
β X i is the reliability index calculated for the case where all variables are considered as random, except X i which is considered as deterministic. β is the reliability index in case where all variables are random. An omission factor value close to unity indicates that the corresponding variable has no eect on the failure probability estimation. Table 6.4 shows omission factors of random variables of the cantilever beam, as well as their inuence on the estimation of failure probability for a displacement criterion.

Next, the variability of variables in terms of their mean and standard deviation are evaluated.

The reliability index is calculated for the case where all random variables, except for one variable, have the same mean and standard deviation as mentioned in Table 6.1. The variation of this variable is assessed for four cases by multiplying its mean or standard deviation by 2 or by 4. The variable sensitivity is considered high if the calculated reliability index varies by more than 50% compared to β [START_REF] Croston | Etude expérimentale du comportement d'une poutre en béton armé en exion 3 points reparée par matériaux composites (approche probabiliste)[END_REF] (β = 2.1221). Table 6.5 shows the variability of each random variable of the cantilever beam according to their mean and standard deviation values.

As mentioned earlier, the mechanical and statistical sensitivity studies are performed in order to verify the accuracy of results provided by Quadratic combination and FORM with respect to importance factors. Table 6.6 summarizes the results of these studies and shows the inuence of each variable on the estimation of failure probability for a displacement criterion in case of a cantilever beam. As can be seen, the most inuential variables are related to the beam dimensions (hB1 and Lb), which is in good agreement with results found by Quadratic combination and FORM (Table 6.2). In addition, these results show that mean and standard deviation values have an important eect on the estimation of failure probability. Thus, they must be properly identied in structural reliability analysis in order to assess accurate estimation. After examining the most inuential input variables, the sensitivity analysis performed in the previous section allowed reducing the number of random variables to two (beam width and length).

µ 1i = 2 × µ i µ 2i = 4 × µ i Inuence σ 1i = 2 × σ i σ 2i = 4 × σ i
The next step is to analyze the dispersion and the distribution of the output variable, which is in this case the displacement at the free end of the beam. For this purpose, 100000 random simulations of the input vector are generated using the numerical integration Monte Carlo method (see section 4.4.2.1). Displacement values are determined for each simulation, they are then arranged in increasing order and subdivided into several equal intervals. Their percentage frequency values are calculated for each interval of displacement in order to plot the histogram that enables to determine the distribution of the variable of interest (Figure 6.4). The displacement probability distribution is rst represented graphically using the Kernel Smoothing method. As can be seen, the number of simulations used is sucient to obtain a very smooth Kernel density estimate and displacement values are distributed around a mean value of 0.614 mm with a standard deviation of 0.254 mm.

Note that the estimated density captures the peak that characterizes the mode, and represents a nonparametric alternative to the tting of a parametric probability density function.

Another parametric way to determine the distribution that ts the output sample of the beam displacement can be also used. It consists in selecting a parametric distribution to model the randomness behavior of the variable of interest and verify if it is in good agreement with the output 2 subset of cases where random variables taken into consideration are presented with identical distribution type (normal, lognormal), and 1 subset that contains all cases with mixed distributions (i.e. cases where random variables are modeled with dierent distribution types).

5 subset of cases where random variables taken into consideration are presented with the same COV (0.05, 0.1, 0.2, 0.3, 0.5), and 1 subset that contains all cases with mixed COV (i.e. cases where random variables have dierent COV values).

For each subset, the mean value and COV of P f (method) P f (M C) ratio are calculated for all methods. MC is assumed to be the method that gives the most accurate results, hence other methods are compared to MC and examined in terms of their accuracy and precision. Accuracy is estimated in term of the mean value of P f (method) P f (M C) ratio and refers to closeness to MC results. Precision is estimated in term of the COV of P f (method) P f (M C) ratio and refers to the degree of variation of results. A particular method is supposed to give the same results as MC in the case where the corresponding failure probabilities ratio has a mean value of 1.0 and a COV of 0.

It should be noted that, although a sucient number of cases is necessary to make better conclusions on probabilistic methods eectiveness, the number of cases studied is limited to 27 and not all possible cases are considered because of the computational eort needed by dierent probabilistic methods to estimate failure probability for each case. However, this procedure can be useful to have a general understanding of each method limitations and how to select an appropriate method for a specic type of problem. Computational eort The computational eort required for a reliability analysis depends on the computational time needed by the physical model to evaluate the response of the structure considered, as well as on the number of calls of the deterministic model needed to assess failure probability. In the current study, minimizing the computational eort of a reliability analysis is of a great concern, especially for deterministic models, such as nite element models, that are very time consuming to evaluate the output variable values. Therefore, the computational eort of each probabilistic method is examined in term of the number of calls of the deterministic model in order to analyze their eciency and help to initially select an appropriate method that provides accurate results and reduces considerably the reliability analysis computational cost for a specic type of problem. The number of calls of the deterministic model is highly dependent on the magnitude of failure probability in the case of simulation methods, and on the convergence criterion of AbdoRackwitz algorithm to nd the design point in the case of approximation methods.

By comparing the number of calls of the deterministic model required by the various methods for the beam problem, it is clear that approximation methods can reduce this number tremendously and independently of the failure probability magnitude and ensures a reasonable accurate estimate of failure probability for all cases studied in comparison with MC (Tables 6.8-6.10). However, a degree of verication for a specic type of problem is necessary to ensure that FORM and SORM provides convenient results, notably in case of highly nonlinear limit state functions. In the computations performed by means of simulation probabilistic methods for cantilever beam, the COV that represents the desired degree of precision on failure probability estimate is xed at 10% with a condence level of 95% and the number of simulations required to assess failure probability is limited to a maximum allowed value of 100000 simulations. It should be noted that the number of calls of deterministic method for simulation methods depends on the convergence criterion in term of this COV and a more signicant number of simulations is needed for less values of this COV. MC is well-known to be the most eective and robust tool to estimate failure probability, but this method becomes very time consuming when computing small failure probabilities due to the large number of calls of the deterministic model required in such a case. As can be seen in Tables 6.8-6.10, for cases 1, 10 and 19, failure probability is of the order of 1E-6 and values provided by MC are not accurate since the maximum allowed value of 100000 simulations is not sucient to reduce the uncertainty on the estimate of failure probability to an accepted level. Thus, the eciency of MC is not veried for these cases and a larger number of simulations is necessary to assess an accurate estimate. Alternatively, simulation methods based on a variance-reduction principle are more ecient in term of accuracy and provide similar results as FORM and SORM for cases with low failure probabilities. In addition, the number of simulations is signicantly reduced by 97.5% for IS (number of simulations < 2500) and 69% for DS (number of simulations < 31000). LHS is as much time consuming as MC (number of simulations > 100000) but leads to an accurate estimate of probability exceeding a displacement threshold.

For other cases studied, the number of calls of deterministic model is presented in term of failure probability magnitude for dierent probabilistic methods (Figure 6.7). As can be seen, FORM and SORM are the less time consuming methods and the corresponding number of calls does not depend on the failure probability magnitude. For DS, the number of calls of the deterministic model varies according to the random radial directions generated. For failure probabilities higher than 0.01 and in the cases of large number of random variables, MC requires less simulations than DS to converge to results of similar accuracy. LHS gives comparable results to MC for all cases studied, but no reduction in computational eort is observed, as expected, for any magnitude of failure probability. Considering the duration of computations associated to IS, it is clear that, among simulation probabilistic methods, IS is the less time consuming since it requires only a limited number of evaluations of the deterministic model, and the number of simulations decreases with the increase of failure probability. But, for low failure probabilities, IS is more time consuming in comparison with approximation methods. IS loses its accuracy in the cases of random variables with A mean value close to 1.0 indicates that the relevant method provides similar results to MC and a COV close to 0 indicates an estimation of failure probability with high precision.

As can be seen in Figure 6.8, simulation and approximation methods give good results for most cases and failure probabilities are estimated with high accuracy in comparison to MC, the mean of probabilities ratio for each subset of random variables number is identical. For approximation methods, accuracy decreases as the number of random variables increases. This may be due to the fact that, as the number of random variables increases, the limit state surface becomes more complex with potentially other local optimum points and AbdoRackwitz algorithm may converge to a local but not a global optimum point. For both methods (FORM and SORM) there is a signicant dierence in precision for the subset of 2 and 7 random variables. For a higher number of random variables, overall precision of SORM is observed to be low with a high degree of variation

(C OV P f (SORM ) P f (M C) = 0.175).
For IS, precision decreases with the number of random variables. Low precision as well as low accuracy results are particularly found for random variables with high and mixed COV (cases 14, 15, 23 and 24). It should be noted that the case 23 with 7 random variables of a COV of 0.5 is not considered for IS to represent the mean and COV of probabilities ratio.

Otherwise, the ratio P f (IS) P f (M C) is found to have a COV of 0.439. When LHS and DS are considered, precision increases signicantly with the number of random variables and a the estimate of failure probability is achieved with a COV less than the desired degree of variation of 0.1.

According to the sensitivity analysis performed to identify the random variables that mostly contribute to the variability of the displacement of cantilever beams, the variation of failure probability in terms of the force applied at the free end of the beam and the number of random variables The rst corresponds to the case where all variables are considered as random variables (10 variables).

The second corresponds to the case of the 7 variables that aect the most the estimate of failure probability with respect to their omission factor (Table 6.4).

The third represents the case of the most signicant variables identied by Quadratic combination method (3 variables, Table 6.2).

The fourth represents the case with the 2 most inuential variables related to the beam dimensions (Table 6.6 ).

Statistical description of these variables is detailed in Table 6.1. As can be seen in Figure 6.9, a good agreement is found between the estimated values of failure probabilities in the case where all random variables are taken into account and the case where the number of random variables is reduced to two after performing a sensitivity analysis. The variation of failure probability is presented in term of the force applied at the free end of the beam, which indicates that the dierence between failure probabilities estimated for each case is not signicant. To emphasize the importance of considering all the most inuential parameters in a reliability analysis, the case of one random variables is studied. In this case, only the beam width hB1 is represented as a random variable and its length Lb is considered as a deterministic variable. Results with one random variable shows a substantial dierence in comparison to cases where the 2 most inuential variables are modeled as random variables. Figure 6.9 illustrates the importance of performing a sensitivity analysis since a bad choice of random variables may lead to erroneous values of P f . In addition, sensitivity analysis helps Another study of the eect of the type of random variable distributions is performed. For this purpose, the main case of study is considered with two random variables, namely hB1 and Lb.

Both random variables are modeled with the same type of distribution by considering all parametric The values of distribution statistical parameters are estimated in terms of the statistical moments for each random variable. First, the mean and COV values of the beam displacement are evaluated in order to examine which distributions provide the same results for the variable of interest. Table 6.11 shows that the displacement mean value is not aected by the type of probability distribution, but its variance is highly dependent on the distribution type and the displacement COV may reach a value of 12.22 in the case where random variables are modeled with Logistic distribution. Among the probability distributions considered, only 8 distributions allow the same statistical characterization of the beam displacement that represents the output variable of the problem. Afterwards, the inuence of these 8 probability distributions on the beam reliability is studied. Figure 6.11 indicates that reliability depends signicantly on the probability distribution type used for random variables.

Results obtained in the cases of Beta, Gamma, Lognormal and Normal distributions show a good agreement regardless of the probabilistic method used and the dierences between the estimated values of failure probability are inconsiderable. The results obtained in the present section illustrate the necessity of selecting the appropriate probability distribution type to model variable uncertainties. Thus, it is not sucient to provide the mean and COV of random variables, but an appropriate PDF must also be specied for each variable.

Eect of COV In order to compare the accuracy and precision of probabilistic methods in assessing similar results to MC and examine the eect of the COV of random variables, all cases studied are divided into 6 subset of 0.05, 0. In order to have a better understanding on the eect of COV of random variables, the case of 7 random variables modeled with Lognormal distribution is considered assuming that all random variables have the same COV value. The variation of failure probability is rst presented in term of the mean of the force applied at the free end of the beam for dierent values of COV (Figure 6.13), then presented in term of the COV for dierent values of the mean of the force (Figure 6.14). MC is used to assess failure probability with a maximum allowed value of 1E6 simulations. Figure 6.13 shows that the inection point of curves corresponds to the case where the limit state function is equal to zero (G = f max -f = 0). At this point, the value of the displacement threshold is equal to the mean value of the calculated displacement and failure probability is equal to 0.5.

The corresponding force value is approximately of the order of 2800 N. Furthermore, approximate models obtained from several response surface types are combined with dierent probabilistic methods in order to study the eciency of response surface methods in assessing failure probability. For each type, failure probability is compared with that obtained in the case of the beam FEM combined with the dierent probabilistic methods (Figure 6.18). Least Square method, Taylor expansion and polynomial chaos expansion with sequential strategy are found to signicantly underestimate failure probability, while results provided by polynomial chaos FEM combined with a particular probabilistic method. It should be noted that the computational eort needed to assess failure probability in the case of approximate models combined with any probabilistic method is negligible (time of order of several seconds).

Conclusion remarks

The problem of the cantilever beam is used as a simple example in the aim of mastering the basics of OpenTURNS and examining the probabilistic methods proposed in OpenTURNS to estimate failure probability in structural reliability analysis. The principle of each step of a reliability analysis is illustrated and the available probabilistic methods are compared in terms of their accuracy, precision and computational eort. The FEM of the RC beam is modeled with CASTEM and combined with OpenTURNS using the wrapper interface.

This application allows drawing some preliminary conclusions which highly served the purposes of other types of applications in order to initially select an appropriate probabilistic methods, given computational eort constraints. Although Monte Carlo method is a robust method for several structural reliability analysis, it is often impractical to use due to the large number of simulations required to reach the desired level of accuracy. Comparing other probabilistic methods to MC, IS is the method that gives best results in terms of accuracy and precision for dierent subset examined, except for the subset of random variables with high COV (COV=0.5). For this subset, IS loses its accuracy in estimating MC failure probabilities. DS is signicantly aected by the number of random variables and is less precise for high random variables cases (7 random variables). DS is the method that gives the highest values of the COV of probabilities ratio leading to less condence in failure probabilities estimated with this method. LHS gives comparable results to MC, but no reduction in computational eort is observed. For FORM and SORM, failure probabilities are estimated with high accuracy in comparison to MC, except in cases of high COV (COV>0.2). In term of reliability analysis computational eort, FORM and IS are the less consuming methods in comparison to other methods. However, these conclusions could not be generalized for every problem in reliability analysis and some degree of verication should be conducted to ensure the eciency of methods used.

6.3 Application to RC slab subjected to impact -Elastic behavior

In this section, a reliability analysis of RC slabs under low velocity impact is presented assuming a exural mode of failure and an elastic behavior for steel and concrete. The behavior of RC slabs in bending has two major phases before the failure, the rst is related to the increase of deection, the second is related to steel yielding [START_REF] Favre | Dimensionnement des structures en béton -Aptitude au service et éléments de structures[END_REF]. Probabilistic methods used in this section to study the reliability of impacted slabs are the most known methods in the reliability domain, namely FORM, SORM, MC and IS. Several criteria of failure are considered, they are related to the displacement of slab at the impact point, the peak force of impact, stresses in reinforcement, and the impact energy absorbed by the slab. To study the dynamic eect of impact applied to RC slabs, 3 deterministic models are used and evaluated: a 3D FEM simulated with Abaqus/Explicit, a simplied model based on the plate theory [START_REF] Reddy | Mechanics of laminated composites plates and shells: theory and analysis[END_REF] and a two degrees of freedom mass-spring system [START_REF] Ceb | Concrete structures under impact and impulsive loading[END_REF]. The aim is to select the most appropriate model to use in RC slabs reliability analysis, while accurately representing the mechanical behavior of the slab studied and reducing the computational cost to the minimum. The model chosen is then used in a parametric study in order to investigate the eect of several input parameters of the deterministic model on the calculation of failure probability. This section describes the 3D FE model of the RC slab subjected to impact within nuclear plant buildings using ABAQUS/Explicit. Geometry of slab and impactor, reinforcement and material properties, as well as initial and boundary conditions are present in section 5.4. A 3D solid element, the eight-node continuum element (C3D8R) is used to develop the mesh of the concrete slab. As illustrated in Figure 6.19.a, the mesh of the concrete slab contains ve elements in the thickness direction with three dierent densities of mesh. The elements at the center of the slab have dimensions of 5x5x10 cm, while the elements at the corners have dimensions of 5 cm in three directions. Thus, the elements of regions between the center and corner regions are of 5x10x107 cm dimensions. The reinforcement is modeled with truss elements (T3D2) (Figure 6.19.b) and the embedded approach is used to create the bond between the steel reinforcement and concrete. This approach allows independent choice of concrete mesh and arbitrarily denes the reinforcing steel regardless of the mesh shape and size of the concrete element.

The impactor is meshed with four-node tetrahedral elements (R3D4). The denition of contact properties and interaction types is very important for impact problems, but in this case of study this denition is limited to a normal interaction and a hard contact with nite sliding. The contact pair algorithm is used in order to model contact between the upper surface of the slab and the impactor surface. The contact constraints are enforced with a pure master-slave kinematic contact algorithm which does not allow the penetration of slave nodes into the master surface (surfaces attached to rigid bodies must always be dened as master surface in Abaqus/Explicit).

At rst, nonlinearities due to concrete and steel behaviors are not considered, both materials are assumed to have an elastic behavior. The only nonlinearity present in the FEM is due to the contact algorithm between the impactor and the slab. The FEM of the slab consists of 53000 nodes The Classical Laminated Plate Theory is used to develop an analytical solution for RC slabs. This theory is based on the kinematic assumptions of Kirchho-Love for thin plates with small deections.

The approximate Navier solution is used to estimate the deection of the mid-surface of the slab (w 0 ). Thus, the deection of slab can be expressed as:

w 0 (x, y, t) = ∞ m=1 ∞ n=1
W mn (t) sin(αx) sin(βy) (6.4) The impactor in this model is replaced by a concentrated force q(x, y, t) applied at the impact point and expressed as following:

q(x, y, t) = ∞ m=1 ∞ n=1
Q mn (t) sin(αx) sin(βy) (6.5) where Q mn (t) = 4Q 0 (t) ab sin(αx 0 ) sin(βy 0 ) for impacted slab cases (x 0 , y 0 ) are the coordinates of the impact point.

The aim is to determine the coecients W mn (t). This can be achieved by integrating the following constitutive relation over the slab surface:

-D 11 ∂ 4 w 0 ∂x 4 + 2 (D 12 + 2D 66 ) ∂ 4 w 0 ∂x 2 ∂y 2 + D 22 ∂ 4 w 0 ∂y 4 +q(x, y, t) = I 0 ẅ0 -I 2 ∂ 2 ẅ0 ∂x 2 + ∂ 2 ẅ0 ∂y 2 (6.6)
For this case of study, the coecients W mn (t) can be estimated as:

W mn (t) = 4 sin(αx 0 ) sin(βy 0 ) abw mn I 0 t 0 Q 0 (τ ) sin(w mn (t -τ ))dτ

(6.7)
where a and b are the slab length and width, respectively. Values of W mn (t) are determined analytically assuming a sinusoidal evolution in time of impact loading [START_REF] Lee | On the dynamic response of laminate circular cylindrical shells under impulse loads[END_REF].

This model is developed using Matlab, it consists of the same input variables as the FEM except those related to the impactor. Impactor velocity, mass and radius are replaced in this model by an impact force which is estimated by means of the FEM. The impact time is taken equal to that of the FEM impact step.

Mass-spring model

A simplied model of two degrees of freedom mass-spring system is developed using Matlab. This model was proposed by CEB [START_REF] Ceb | Concrete structures under impact and impulsive loading[END_REF] to study the behavior of slabs subjected to hard impact. In this case, the kinetic energy of the impactor is absorbed by the structure deformation, and the slab local behavior as well as its overall deformation must be considered. The model consists of two masses, that represent the impactor and slab masses, respectively, and of two springs that describe the stiness of slab and the contact force, respectively. The dierential equations of equilibrium of the two masses are previously detailed in Section 1.3. the same as those used for the FEM. It should be noted that the transformation factor is the only parameter calibrated for the MSM in order to nd the best t with the FEM.

Probabilistic model

Assuming an elastic behavior for steel and concrete, the variables which may be subjected to statistical variations in RC slabs under impact can be classied into four categories:

1. Dimensional variables related to geometric properties of the slab: length, width and thickness of the slab.

2. Variables related to the impactor and its kinetic energy: radius, mass and velocity of the impactor.

3. Variables related to concrete properties: Young's modulus and density of concrete.

4. Variables related to steel properties: diameter of bars.

It should be noted that the length and width of slab are not introduced into deterministic models to model their relevant uncertainties, but they are considered to cover a wide range of slab geometries.

The probabilistic characterization of random variables is based on some literature reviews. Table 6.12 represents the random variables adopted for this application, their probability distribution, their mean and COV. Mean values of random variables are taken equal to their nominal values, and their COV is determined according to references cited in the fourth column. Probability distributions are chosen according to the reliability based code calibration of JCSS (Joint Committee on Structural Safety) [START_REF]Probabilistic model code[END_REF]. JCSS code contains several rules and basics that are essential for the design of new structures or evaluating the response evolution with time of already existing structures in a reliability framework. The modulus of elasticity of steel is considered as deterministic variable due to its small dispersion [START_REF] Mirza | Variability of mechanical properties of reinforcing bars[END_REF]. It should be noted that the following probabilistic characterization was adopted before performing the literature review detailed in section 4.6.

Failure criteria

Gay and Gambelin [START_REF] Gay | Dimensionnement des structures[END_REF] described the problem that an engineer could expect during the analysis and design of RC structures under certain loading conditions. This problem involves two fundamental components: Stiness design for which the structure must not endure displacements higher than a particular threshold dened in design specications (serviceability limit states in Eurocode).

Load-carrying capacity design for which the structure must continue to perform its functional requirements under the applied loadings (ultimate limit states in Eurocode).

The rst aspect to consider for the application of impacted RC slabs assuming an elastic behavior of materials is the displacement of slab at the impact point. Thus, the rst criterion is dened in term of this displacement:

g 1 = u max -u (6.8)
In this equation, u max is the displacement threshold and u represents the maximum displacement of slab at the impact point. u max is considered according to Eurocode specications [START_REF] Narayanan | How to design concrete structures using Eurocode 2: Introduction to Eurocodes[END_REF] equal to a 250 , with a is the length of slab. The values of u are calculated according to the deterministic model used and the values of input random variables generated in OpenTURNS. The RC slab is modeled assuming an elastic behavior of materials, however this assumption does is not representative of the actual response and the maximum displacement of slab is underestimated of approximately 60% in this case [START_REF] Berthetrambaud | Structures rigides soumises aux avalances et chutes de blocs: modélisation du comportement mécanique et caractérisation de l'interaction "phénomène-ouvrage[END_REF]. Therefore, a coecient of 1.5 is introduced in the expression of the displacement failure criterion in order to take into account plastic deformations related to actual materials behavior. The displacement criterion is then expressed as:

g 1 = u max -1.5u (6.9)
A second aspect related to stresses in reinforcement can be also examined. Berthet-Rambaud [START_REF] Berthetrambaud | Structures rigides soumises aux avalances et chutes de blocs: modélisation du comportement mécanique et caractérisation de l'interaction "phénomène-ouvrage[END_REF] supposed that the failure of SDR protective slabs initiates when yielding of main reinforcement occurs. Thus, a second failure criterion can be expressed in term of steel stresses as following:

g 2 = f y -σ s (6.10)
where f y is the yield strength of reinforcement and σ s is the stress in reinforcement. For this ultimate limit state function, failure probability can be assessed only through the combination of slab FEM with OpenTURNS, stresses in reinforcement cannot be evaluated using the mass-spring model nor that based on the plate theory. This criterion can be dened in such a way that stresses in any nite element of reinforcement should not reach the yield strength of steel, but exceeding the threshold of stresses in one steel element does not necessarily lead to slab failure. Therefore the variable of interest of steel stress is replaced by the variable related to the number of reinforcement FE elements n A in which yielding occurs, and a threshold n Amax equal to the number of FE elements in 3 longitudinal bars and 3 transversal bars is assigned.

g 2 = n Amax -n A (6.11)
An energy criterion for RC slabs subjected to impact was dened by Miyamoto and King [START_REF] Miyamoto | Design concept for reinforced concrete slab structures under soft impact loads[END_REF].

They indicated that this criterion would be the most ecient method of designing RC structures under impact loads, especially for a exural failure mode. The criterion consists of that structural failure is likely to occur if the slab is not capable of absorbing all of the energy transmitted during impact collision. A part of the impactor kinetic energy E I kin0 is transferred to slab as kinetic energy and irrecoverable strain energy E S plas1 that results from the formation of cracks, friction and damping. Therefore, a third ultimate limit state function corresponding to this criterion is considered assuming that the slab should absorb 80% of the impact energy:

g 3 = 0.8E I kin0 -E S plas1 (6.12)
Another important criterion related to the impact force should also be examined. The evolution in time of impact force represents one of the main problems of impacts on RC structures. It depends on two colliding bodies geometry, their stinesses and the initial conditions of the impactor. Thus, a fourth ultimate limit state function is considered:

g 4 = F max -F (6.13)
where F max represents the ultimate load-carrying capacity of slab and from is determined a limit load analysis, F is the peak of impact force calculated with the slab FEM. 6.3.4 Discussion of results of slab reliability (elastic behavior)

Comparison of deterministic models

In order to be able to use simplied analytical model in reliability analysis, their eectiveness in predicting the response of slab should be rst veried. Thus, mass-spring model and that based on plate theory are compared to the nite element model. Table 6.13 shows the displacement of slab at the impact point obtained with dierent models developed to evaluate the behavior of impacted RC slabs assuming an elastic behavior of materials. It can be seen that the mass-spring model allows a better estimation of displacement at the center of slab for dierent values of velocity with a maximum error of only 1.5%. However, the analytical model based on plate theory is less precise and estimate the displacement with a maximum error of 5%.

Next, deterministic models are compared in a reliability framework. Failure probability is assessed using FORM method by combining the dierent deterministic models to OpenTURNS for several values of velocity. Figure 6.20 shows that failure probability increases with the mean of velocity for the displacement criterion. Results obtained with MSM are in good agreement with failure probabilities estimated by coupling FORM with the FEM. In addition, the MSM is found to be more eective in term of computational cost. Thus, it is selected to study the eect of input variables on the estimate of failure probability in the case of impacted slabs with elastic behavior of materials. to failure probabilities obtained with the FEM combined with FORM in order to choose the most suitable probabilistic method to use in the parametric study. The MSM is used due to its reduced computational time which allows combining it with simulation probabilistic methods, and because it permits to accurately predict the physical response of slab as proven in the previous section. A comparison of dierent failure criteria dened in section 6.3.3 is important for a better understanding of the behavior of RC slabs subjected to impact with an elastic behavior of materials.

Figure 6.21 shows failure probability in term of the mean of impact velocity for the dierent failure criteria considered for this application. It can be seen that the slab is more resistant to displacement and steel yielding, while it is less resistant to the contact force for the same values of impact velocity mean. In this case and for an optimal design, it is necessary to evaluate the maximum contact force value that the slab can sustain, as well as the impactor velocity for which this maximum value of force is reached. Once this velocity is determined, the performance of slab under displacement and steel yielding criteria is subsequently veried.

In this gure, the performance of slab regarding its capacity of dissipating impact energy is also examined. Unlike other criteria, failure probability decreases with the impact energy for this criterion as a result of the energy dissipated by steel and slab deformations that increase with velocity. The performance of slab in dissipating energy can be improved by considering the nonlinear behavior of concrete and steel, which allows modeling concrete damage and the onset of plastic deformations.

It should be noted that this criterion does not indicate a failure mode of slab, it is used to study the energy transmitted to RC slabs during an impact phenomenon.

Parametric study

The following parametric study is performed in the aim of investigating the eect of dierent input variables on the estimation of failure probability for the application of impacted RC slabs. For this purpose, the MSM is selected because, among the deterministic models proposed for this problem, it is the most ecient in term of computational eort for a reliability analysis. In addition, the MSM model is veried to accurately predict the displacement of slab at the impact point. Thus, the parametric study is carried out by combining the MSM with FORM for the displacement criterion and the eect of several input variables on the reliability of impacted RC is examined. Variables such as slab dimensions, concrete density, steel bars diameter, impactor mass and rigidity of slab are considered and failure probability is estimated for dierent values of the mean of the variable studied.

Slab length Failure probabilities are calculated for several mean values of the slab length that varies between 2.35 and 12 m. Figure 6.22 shows that, for a simply supported slab, reducing the slab length leads to an increase in failure probability for the displacement criterion and that this probability depends on the impactor velocity. Thus, an optimal design of RC slabs requires to properly determine the structure dimensions and choose the appropriate values depending on impact velocity in order to improve the performance of slab. It can be seen that a low variation in length

Concrete density

The eect of the mean of concrete density is small in comparison to that of the slab length and thickness, especially in the cases where v=7.7 m/s and v=24.4 m/s (Figure 6.24). Therefore, this variable can be considered as deterministic variable in the case of low failure probabilities.

Steel bars diameter In order to study the eect of steel bars diameter on the reliability of impacted RC slabs, dierent diameters of reinforcement are considered while retaining the same number of longitudinal and transversal bars. Figure 6.25 shows that the diameter of bars has an inconsiderable inuence on failure probability for all the impact velocity values considered. However, this inuence should be examined in the case where the plastic behavior of steel is introduced in the analysis of RC slabs.

Impactor mass Figure 6.26 illustrates the variation of failure probability in term of the mean value of the impactor mass. It can be seen that the mass has the same inuence as the impactor velocity and failure probability increases with the mean value of mass. For v=7.7 m/s, failure probability remain in the domain of low probabilities for a mass less than 6000 kg. However, for velocities higher than 7.7 m/s, the eect of slab becomes more signicant and failure probability varies greatly in term of the mean value of mass. Increasing the impactor mass and velocity lead to an increase in impact energy, so a slab with the same dimensions and material properties lose its performance to sustain the impact force applied. Slab rigidity One of the most important parameters in a mass-spring model is the rigidity of the spring that represents the stiness of slab. Figure 6.27 shows that increasing the slab stiness results in a decrease of failure probability and improve the slab performance for a displacement criterion. This can be due to the fact that a more rigid slab is subjected to smaller displacement, hence its failure probability for a displacement criterion is lower. RC slabs stiness can be increased depending on the slab dimensions and material properties.

Conclusion remarks

In this section, an application of reliability analysis to RC slabs under low velocity impact assuming a exural mode of failure and an elastic behavior for steel and concrete is presented. Several criteria of failure related to the displacement of slab at the impact point, the peak force of impact, stresses in reinforcement, and the impact energy absorbed by the slab are considered. Probabilistic methods such as FORM, SORM, MC and IS are used to assess failure probability. In order to study the dynamic eect of impact applied to RC slabs, 3 deterministic models are used: a 3D FEM simulated with Abaqus/Explicit, a simplied model based on the plate theory and a two degrees of freedom mass-spring system. The aim is to select the most appropriate model to use in RC slabs reliability analysis, while accurately representing the mechanical behavior of the slab studied and reducing the computational cost to the minimum. The analytical MSM is seen to be very eective in predicting the same values of slab displacement at the impact point as the numerical model, as well as in estimating failure probabilities very similar to those obtained with the FEM for a displacement criterion. The MSM is more convenient than numerical models to use in reliability analysis in term of computational eort. Thus, it is used in a parametric study in order to investigate the eect of several input parameters of the deterministic model on the estimation of failure probability of RC slabs. The impact velocity is the most inuential variable, followed by the slab length, thickness and stiness. Results obtained for this type of problem can be useful for the design of RC slabs subjected to impact since they permit to identify the values of input variables for which failure probability is not in the range of low probabilities.

Studying the response of impacted slabs assuming an elastic behavior of materials represents a preliminary step in the procedure of evaluating the reliability of RC slabs under low velocity impact.

This application is initially considered to simplify the problem and to avoid complicating structural reliability analysis of slabs as a rst step. However, it is clear that nonlinearities due to materials behavior have a signicant eect on RC slabs response, in particular that cracking in concrete under tension and plastic deformations in concrete at the impact zone and in reinforcement reduce signicantly the peak of contact force obtained with an elastic model. The eect on nonlinearity due to material properties is discussed in the next section.

Probabilistic model

In the case when the nonlinear behavior of materials is considered in the FEM, the variables which may be subjected to statistical variations in RC slabs under impact can be classied into ve categories:

1. Dimensional variables related to the geometric properties of the slab: length, width and thickness of the slab.

2. Dimensional variables related to the position of reinforcement and its size: concrete cover and diameter of bars.

3. Variables related to the concrete properties: Young's modulus, density, compressive strength and tensile strength of concrete.

4. Variables related to the steel properties: modulus of elasticity and yield strength of steel.

5. Variables related to the impactor and its kinetic energy: radius, mass and velocity of the impactor.

In this study, the probabilistic characterization of random variables is based on some literature reviews mentioned in previous section 4.6 and in which the statistical descriptions of material properties and geometry variables for reinforced concrete structures are relatively well documented. For variables related to the impactor, statistical descriptions that represent the variation of accidental dropped objects impact during handling operations within nuclear plant buildings are not available in the literature. Table 6.15 represents the random variables adopted in the present study, their probability distribution, their mean and COV, as well as the variables to be considered as deterministic.

Concrete properties

As previously mentioned, stress-strain curves of concrete in tension and compression considered to study the reliability of RC slabs under impact assuming a nonlinear behavior of materials are described as stress-strain relations. These relations vary in terms of the tensile strength, f t , and the compressive strength, f c , respectively. Thus, the variation of these curves depends on the statistical description of f t and f c .

Compressive strength of concrete Mirza et al. [START_REF] Mirza | Statistical description of the strength of concrete[END_REF] estimated the mean value of the compressive strength to be equal to 0.8f c assuming that f c follows a normal distribution and the COV can be considered to vary between 0.15 and 0.18.

Tensile strength of concrete Tensile strength mean value is chosen according to Pandher [START_REF] Pandher | Reliability analysis of strength characteristics of bre-reinforced y ash concrete[END_REF] assuming that it is approximately one tenth of the concrete compressive strength. The COV is taken equal to 0.18 according to Mirza et al. [START_REF] Mirza | Statistical description of the strength of concrete[END_REF] that studied the variation of concrete tensile strength in-situ. Mirza et al. found that f t follows a normal distribution.

Young's modulus The elastic modulus of concrete is generally considered as a normally distributed variable with a COV of 0.1 [START_REF] Mirza | Statistical description of the strength of concrete[END_REF] . Collins and Mitchell [START_REF] Collins | Prestressed concrete basics[END_REF] approximated the Young's modulus in MPa by E c = 5500 √ f c .

Steel properties

As previously mentioned, stress-strain curve of steel considered to study the reliability of RC slabs under impact assuming a nonlinear behavior of materials are described as stress-strain relations.

These relations vary in term of the yield strength, f y . Thus, the variation of this curve depends on the statistical description of f y .

Yield strength of steel The distribution of steel yield strength is chosen according to Mirza and MacGregor [START_REF] Mirza | Strength and ductility of concrete slabs reinforced with welded wire fabric[END_REF] assuming that it follows a normal distribution. The mean of yield strength is generally considered equal to the nominal value with a COV of 0.08-0.1.

Modulus of elasticity

The modulus of elasticity of steel is considered as deterministic variable due to its small dispersion with respect to other variables. Only the thickness of slab is taken as random variable in the case of impacted slabs assuming a nonlinear behavior of materials. It is considered as normally distributed with a mean value equal to its nominal value and a COV of 0.05 [START_REF] Mirza | Variations in dimensions of reinforced concrete members[END_REF]. The length and width are described as deterministic variables since slab dimensions are important, thus the eect of their uncertainty is negligible [START_REF] Hong | Calibration of material resistance factors in csa s474-m1989[END_REF].

Steel bars diameter is also considered as a dimensional random variable and is modeled as a normal distribution with a mean equal to the nominal diameter and a COV of 0.024. The variation of concrete cover is not taken into account in this study.

Failure criterion

Several output variables may be obtained as result from a nite element model. These variables allow dening limit state functions corresponding to the failure criteria studied. However, the only variable of interest considered for impacted slabs assuming a nonlinear behavior of materials is the displacement of the impact point on the front face of the RC slab, u. Thus, a serviceability limit state function can be written as:

g = u max -u (6.14)
where u max is the displacement threshold and u represents the maximum displacement of the slab at the impact point. Since no information available in the literature to choose the value of u max , a deterministic parametric study is performed in order to estimate the displacement for which yielding in tensile steel occurs. The displacement of steel yielding is evaluated for dierent impact energies, and it is found that the stresses in reinforcement exceed the yield strength for a displacement of 50 mm as mean value. In order to nd the best t with the FEM and estimate the eect of the slab and impactor properties on the mass-spring model parameters, a 2 k factorial experiment is designed in the aim of expressing the slab stiness in terms of slab thickness, steel diameter, mass and velocity of the impactor. Table 6.16 represents the low and high level for the specied factors, but also the values of the mass-spring model parameters which are found to give similar results to the FEM.

As discussed earlier, the two strategies proposed in this study to reduce the computational eort of reliability analyses consist in using simulation probabilistic methods with the mass-spring model and FORM with the nite element model. A comparison of these two strategies leads roughly to the same value of failure probability, except in the case of an impact velocity of 13.2 m/s and a mass of 1225 kg (Table 6.17. This may be related to the samples generated by IS and MC and which do not correspond to the mass and velocity selected range for the design of experiments. However, the second strategy is more time consuming since at least 36 simulations of the FEM are needed in order to assess the failure probability with an average time of 45 minutes per simulation. As a result, the rst strategy is more eective in term of computational cost but cannot be used if the simplied model is not proven to be able to predict the behavior of the RC slab and give the same results as the FEM.

Parametric study

Impactor velocity Figure 6.28 shows failure probabilities for dierent values of velocity mean for the displacement criterion. As can be seen, failure probability increases with the impact velocity for Figure 6.28: Eect of the mean of impactor velocity for the displacement criterion (m=3600 kg)

the displacement criterion and impact duration increases with the impact energy. The displacement at impact point can be related to the overall deection of slab, to the motion of shear cone during impact, or to spalling that occurs at the upper surface of slab. RC slabs are observed to undergo exural failure under low impact velocities [START_REF] Martin | Comparision of dierent constitutive models for concrete in abaqus/explicit for missile impact analyses[END_REF] along with scabbing. The local response of scabbing can dominate the slab behavior for impacts with low velocity as they are classied as short impact time according to [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF][START_REF] Olsson | Analytical prediction of large mass impact damage in composite laminates[END_REF]. Furthermore, shorter impact duration leads to less critical yielding in steel [START_REF] Zhang | Analyse et dimensionnement d'ouvrages de protection contre les chutes de blocs[END_REF], hence less displacement values are observed. However, under high impact velocities shear type failure occurs, but the impactor could perforate the slab if its velocity is suciently high. Therefore, higher displacements are observed at the impact point leading to an increase in failure probability with impact velocity. According to [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF] (see Figure 2.3), high-velocity impacts can be associated with short impact times or long impact times depending on the slab stiness and material properties. Consequently, for long impact times the displacement at impact point can be involved due to an overall structural response that is aected by the boundary conditions and slab size, and governed by several waves reection through the thickness reaching slab boundaries. For short impact times, the local damage failure modes that govern the slab response are caused by exural and shear waves with no signicant propagation through the thickness. As can be seen, it is very dicult to predict the slab behavior in term of the mean of impactor velocity as the response of RC slabs under impact is signicantly inuenced by several parameters, including slab and impactor characteristics. Displacement at the impact point in case of perforation are more important than in case of exural failure, thus it is important to design RC slabs to withstand smaller displacement values under impact to prevent perforation and scabbing.

Impactor mass As previously discussed, slabs subjected to large-mass low-velocity impacts have dierent behavior than slabs subjected to small-mass high-velocity impacts. This is can be seen in Figure 6.29 that shows how the reliability of RC slabs subjected to impact is inuenced by the mean value of the impactor mass. The mass has the same inuence as the impactor velocity and failure probability increases with the mean value of mass. Values of maximum displacement and impact duration are more signicant for higher masses and yielding in steel occurs leading to more Figure 6.29: Eect of the mean of impactor mass for the displacement criterion damage in the slab for higher impact energy. Impacts of large mass and low initial velocity involve an overall response of the RC slab as the dominating exural waves reach the boundaries during the impact duration. This type of impact is classied as long impact times according to Olsson [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF] and local damage has small eect on slab response. Conversely, impacts of small mass and high velocity cause deformation at a small zone surrounding the impact point leading to more localized deection. Thus, the slab undergoes smaller displacements for smaller mass and failure probabilities for the displacement criterion increases with the mean of the impactor mass. Accordingly, impacts with v=5 m/s and m > 3600 kg can be classied as large-mass low velocity impacts using Abrate classication [START_REF] Abrate | Impact on composite structures[END_REF]. However, this classication does not satisfy the criterion of Olsson [149] that indicated that small-mass impacts correspond to impactor masses less than one fth of the mass of the slab. The ratio of the mean of the impactor mass to the mean of the slab mass is always less than 1/5 for all the mean values of the impactor mass considered. According to Olsson criterion all cases studied must be classied as small-mass impacts. This is in contrast with Abrate classication, hence it is necessary to take into account all parameters present in an impact problem in order to be able to properly estimate the behavior of impacted RC slabs. It should be noted that small masses are more critical for a given impact energy because they result in higher impact loads and less energy absorbed by the slab.

Slab thickness Figure 6.30 shows how the reliability of RC slabs subjected to impact is inuenced by the mean value of the slab thickness. Other values of random variables remain the same as dened in Table 6.15, including the length and width of the slab and the reinforcement ratio. As illustrated in this gure, increasing the thickness leads to lower failure probabilities for the dierent mean values of impact velocity considered as the slab deection becomes less signicant for larger thicknesses.

Increasing the thickness results in increasing the slab stiness as well as the compression concrete area, which leads to higher impact force and shorter impact times. In this case, a few compressive wave reections through the thickness are observed and the damage is initiated at the early stages of impact. According to Abrate's criterion [START_REF] Abrate | Impact on composite structures[END_REF] that classify impact events according to the ratio of impact velocity versus the speed of propagation of compressive waves through the thickness direction, the impact problem with higher thickness can be classied as high-velocity impact depending on the Figure 6.30: Eect of the mean of slab thickness for the displacement criterion impact velocity. For v=5.0 m/s, the problem can be classied as low-velocity impact for thickness value less than 0.5 m as the stress wave propagation through the thickness has no signicant role and the slab would likely fail due to its overall bending behavior. For these thickness values and a velocity equal to 5.0 m/s, the response of slab is inuenced by its dimensions and boundary conditions and the impact duration is described as long impact times according to Olsson [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF] (see Figure 2.3). The reinforcement has more inuence on the slab resistance during exural failure mode, and the slab may undergo higher displacements at the impact point. For v=5.0 m/s and thickness values higher than 0.5 m, the impact can be classied as high-velocity impact according to Abrate's criterion [START_REF] Abrate | Impact on composite structures[END_REF] and the impact duration can be described as short to very short impact times according to Olsson [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF] (see Figure 2.3). For v=7.7 m/s, the problem can be considered as high-velocity and short impact time for thickness value higher than 0.6 m. For v=10.5 m/s, the problem can be considered as high-velocity and short impact time for thickness value higher than 0.8 m. It should be noted that increasing the slab thickness improves the slab performance to perforation and results in increasing the ballistic limit dened as the lowest initial velocity of the impactor causing complete perforation [START_REF] Abrate | Impact on composite structures[END_REF].

Compressive strength of concrete In a reliability analysis, it is important to investigate the effect of material parameters on failure probability estimation and how they contribute to the strength of RC structures. Only the mean value of the compressive strength is considered since those of the tensile strength and the modulus of elasticity are estimated in term of the compressive strength.

Thus, an increase in the mean value of the compressive strength leads directly an increase in tensile strength and modulus of elasticity mean values. As can be seen in Figure 6.31, failure probability falls o with the mean value of the compressive strength. The compressive strength contributes signicantly to the slab rigidity, so that increasing the compressive strength allows lower displacement at the impact point of the slab and yielding in steel is reduced. Nevertheless, increasing slab rigidity leads to higher peak value for the impact force and punching shear failure mode could occur. A high strength concrete improves the slab performance in term of perforation compared to normal strength concrete since higher impactor velocities are needed to perforate the slab in this case. The slab resistance to penetration can be enhanced by increasing the compressive strength of concrete. ing to lower failure probabilities. However, it can be seen that for v=5.0 m/s and v=7.7 m/s, using concrete with compressive strength greater than 50 MPa does not seem to give a signicant improvement in slab reliability for the displacement criterion. Thus, using the appropriate concrete properties is important for an optimal design of impacted RC slabs to resist local failure modes, including spalling, scabbing, penetration and perforation. As in this reliability analysis increasing the concrete compressive strength means an increase in its tensile strength, scabbing takes place in a later phase of impact for the same impact conditions. This is due to the fact that scabbing occurs when the tensile stresses generated by the tensile reected wave produced during impact become equal or higher than the concrete tensile strength, and higher value of tensile strength delays the scabbing local failure mode.

Yield strength of steel The inuence of yield strength of steel on failure probability is investigated for the displacement criterion proposed for this application. As can be seen in gure 6.32, the yield strength of steel has no inuence for v=5.0 m/s and v=7.7 m/s because the displacement criterion proposed in this study is chosen in order to not allow yielding in steel. The displacement threshold is estimated for slabs with reinforcement of yield strength of 500 MPa. Thus, for higher yield strength mean value, failure probability decreases for v=10.5 m/s since the problem is classied in this case as long impact times and the overall response of slab dominates.

Reinforcement ratio In order to study the eect of reinforcement ratio on the impacted RC slabs reliability, failure probability is calculated by considering several mean values of the steel bars diameter (Figure 6.33). Other values of random variables remain the same as dened in Table 6.15, including the slab geometry and impact conditions input variables. In addition, the eect of numbers under bending mode. This may be also the reason why steel bars diameter has no signicant eect on the slab reliability, hence it will be interesting to examine its eect in case where the slab is reinforced with more than 20 longitudinal bars per layer. Although reinforcement ameliorates the ductility of RC slabs due to its plasticity, the steel ratio should be chosen with attention since a strongly reinforced slab may fail under punching shear failure mode when subjected to high velocity impacts. Consequently, it is important to design RC slabs under impact in a way which reduces the reinforcement ratio, but also ensures that no failure or large deection would occur during impact.

Stirrups As previously indicated, the Python script written for the slab subjected to dropped object within nuclear power plant enables to add stirrups to the FEM. Therefore, the eect of reinforcing the slab with shear reinforcement on failure probability is also examined (Figure 6.35).

It is found that for low impact velocities, stirrups do not improve the performance of slab for the displacement criterion proposed in the case where nonlinearities due to material behaviors are considered in the FEM. This may be due to the thickness of slab that can be described as thick slab. In the case of thick slabs subjected to low velocity impacts, cracks are rst induced at the impact zone and then progress through the thickness without leading to a reversed shear cracks pattern. However, higher velocities result in shear type failure mode for thick and stier slabs. This can be illustrated in Figure 6.35 and, as a consequence, failure probability decreases for values of mean velocity greater or equal to 10.5 m/s if shear reinforcement is considered in the FEM. It can be also expected that stirrups have more inuence on the slab displacement in case of thin slabs The bottom four edges of the slab are considered as simply supported (pinned edges).

The four faces of the slab in the xz and yz planes are considered as simply supported (pinned faces).

The bottom four edges of the slab are considered as clamped (xed edges).

The four faces of the slab in the xz and yz planes are considered as clamped (xed faces).

As can be seen in Figure 6.36, boundary conditions has a negligible eect on slab response for low velocities (v=5.0 m/s and v=7.7 m/s). This is due to the fact that in the case of impactors dropped at low velocity of a few m/s on large slabs, main exural and shear waves do not reach the boundary, which can be associated with high-velocity impacts according to [START_REF] Olsson | Experimental verication of a theory for the impact response of composite plates[END_REF] and very short impact time according to [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF]. As a result, failure probability increases with the mean of velocity as the slab undergoes higher displacements and tends to fail under exural failure mode. For v=10.5 m/s, the displacement of slab at the impact point is more important for the pinned edges boundary condition as the slab shows considerable rise in its rigidity for pinned faces and xed boundary conditions.

In this case, the problem is found to be dependent of boundary conditions and the time that main waves need to reach structure boundaries is less than the impact duration. Thus, it can be classied as low-velocity impact according to Abrate's criterion [START_REF] Abrate | Impact on composite structures[END_REF] and long impact time according to [START_REF] Olsson | Mass criterion for wave controlled impact response of composite plates[END_REF]. Therefore, failure probability corresponding to displacement criterion is estimated for ten dierent impact point positions in the directions of the width, the length and the diagonal (Figure 6.37). The results show that the most critical point for the response of slabs subjected to impact loading is at the center of the slab. The more the impact point is close to supports, the more the slab is rigid and consequently, the displacement is lower.

Conclusion remarks

The present case of study proposes strategies in order to reduce the computational time of reliability analyses. The rst strategy consists in using analytical models instead of nite element models. Due to the low computational time of the simplied model proposed, probabilistic methods like Monte Carlo and importance sampling which require too large number of simulations of the deterministic model are coupled with this model. However, this strategy does not allow taking into account all random variables that aect the slab response to impact loading and can be only used when the simplied model is veried with the nite element model. Alternatively, the second strategy consists in assessing failure probability from a small number of simulations using FORM with the nite element model. Despite its computational cost, this strategy allows a better understanding of the slab behavior when subjected to low velocity impact. The results of failure probability show high values which are not acceptable for structural failure, this conrms that this type of loading must be considered in the design of RC slabs. Therefore the slab dimensions, material parameters and reinforcement ratio must be modied and evaluated in order to get better reliability of RC slabs.

Application to containment building of nuclear power plant

The long term behavior of a prestressed concrete structure like nuclear power plants, is aected by the delayed deformations that could initiate cracks, induce prestressing losses and result in a redistribution of stresses in the structure. They could also be the cause that the structure does

Deterministic model

The evolution of containment building is modeled for t ] t c ; t V D1 [, where t c is the date at which the internal containment is prestressed and t V D1 is the date of the rst periodic surveillance testing (20 years after its implementation). Time increment is estimated in term of years, which makes a total In order to reduce the computational cost of the deterministic analysis, the geometry considered in the model is simplied as illustrated in Figure 6.40. The model is limited to an angle of 22.22 gr (symmetry with respect to vertical plans) and the containment building is modeled up to the height h = +30.0 m which is approximately equal to half of its total height. Thus, the selected zone is located between the levels of -0.6 m and +30.0 m, and the azimuth of 377.778 and 400.0 gr. In this case, the length of sector at the neutral ber is equal to 8 m. The origin O is located at the center of the containment, at z = 0 corresponding to the level 0.0 m.

Two types of prestressed cables are used, namely the vertical cables of the vessel that are vertical straight lines in the common zone and the horizontal cable of the vessel. Taking into account the above simplication of geometry of the selected zone, ten vertical cables are modeled and located The curve of sorption-desorption that gives the humidity in term of the drying of concrete

The activation energy obtained from the time-temperature superposition.

Material properties and behaviors

Concrete is modeled as a visco-elastic material with a Young's modulus of 37000 MPa, a density of 2500 kg/m 3 and a Poisson's ratio of 0.2. The prestress is adopted in such type of structures in order to avoid tension zones and thereby reduce cracking in concrete. Therefore, it is designed so that concrete in the containment building can be fully compressed when fullling its functional requirements. For this case of study, the prestress is achieved with vertical and horizontal cables of 37T15and a cross-section of 5143 mm 2 . Cross-sections of horizontal cables modeled as groups of 2, 3 or 4 cables are multiplied by 2, 3 or 4 respectively. The failure tensile stress is equal to 1814

MPa while the initial tensile stress at the anchorage is taken equal to 1429 MPa. Steel material of cables is modeled as a linear elastic material with a Young's modulus of 190000 MPa and a Poisson's ratio of 0.3. The prestress is evaluated taking into account the loss in tension along cables due to rectilinear and curvilinear friction, as well as to the relaxation in prestressing steel (2.5 % at 1000 hours, 3.0% at 3000 hours). Losses due to the creep and the shrinking of the concrete, as well as that resulting from the retreat of anchorage are also considered in the FEM. The relaxation of cables is estimated to be equal to 2.5 % at 1000 hours and 3.0% at 3000 hours, while the retreat of anchorage is supposed to be equal to zero for horizontal cables and 0.008 m for vertical cables. The model of Granger is used to represent the basic creep of concrete in the FEM.

Boundary conditions and loading

The selected zone of study necessitates choosing appropriate boundary conditions in order to take into account the symmetry (Figure 6.41):

For the surface Surf1 located at the azimuth of 400 gr, displacement in x-direction is restrained so that d x = 0.

For the surface Surf2 located at the azimuth of 377.78 gr, displacement in x-direction is restrained so that d x = -d y tan α.

For the upper surface Surf3 located at the level z = +30.0 m, rotations about x-,y-and z-axes are zero and displacement d z in z-direction is constrained to an identical value for all the surface nodes. The eect of the weight of the upper part of the containment as well as the dome part is considered in the model by applying a compression stress upon this surface equal to σ zz = 1.51 MPa.

For the surface Surf4 located at the level z = -0.6 m, the selected zone is clamped at the bottom. In addition, the foundation slab shrinkage is dened as a boundary condition of displacement in the plate base. The FE mesh of the internal containment includes the concrete part of the structure, as well as the prestressed cables that signicantly contribute to rigidity and the nonlinear behavior of the containment. The concrete part is modeled using solid FE with 8 nodes and linear integration, while prestressed cables are modeled with two nodes linear truss elements. Passive reinforcement are not taken into account in the FEM since their inuence on the structural rigidity is relatively small. As previously mentioned, the dome and the foundation slab are replaced in the model by equivalent boundary conditions.

The mesh is performed in the way that concrete nodes overlap with those of horizontal cables, but vertical cables are meshed independently. Therefore, the discretization in space of horizontal cables and the concrete part is the same and a node is located every 0.8 m. Vertical cables, as well as the concrete part in the z-direction, are discretized in space with a node every 0.4 m. Consequently, the FEM of the selected zone of the internal containment consists of 9200 nodes and 7900 elements, including 6800 solid elements assigned to concrete and 1100 truss elements assigned to steel cables. Figure 6.41 shows the mesh adopted for the selected zone study.

Failure criteria

The aim of periodic surveillance testing is to measure the global leakage rate in the internal containment, which represents the percentage of decrease in dry air masses in the containment per day. A containment should ensure a leakage rate less than 1.5% of the total mass of dry air per day. During periodic surveillance testing, tensile stresses may appear in some particular zone of the containment.

Furthermore, in very long term, beyond the expected period of time for which the containment is designed, such tensile stresses may also occur in the common zone of the containment in cases where delayed strains were underestimated during the design process.

The eect of dierential shrinkage is observed in the internal containment for height less than 5.8 m, which leads to tensile orthoradial stresses that reduce the compressive stress in concrete generated from prestressing forces. For this application, the limit state function is dened using the criterion of maximum height, that is failure occurs when the height of zone under tension is greater than the threshold h max (h max = 5.8 m).

g = h max -h The rst consists in coupling OpenTURNS and the initial deterministic code directly. The initial deterministic code is an ASTER FEM which is called by OpenTURNS for each assessment of the limit state function denoted in this case g model .

The second is based on an approximation of the function g model using the polynomial chaos expansion which enables to substitute the containment initial deterministic model with an analytical expression. The approximated limit state function is denoted in this case g M etaM odel .

The aim is to analyze the eciency of using polynomial chaos expansion for problems with an important number of random variables and a limit state which is not easy to evaluate. Thereafter,

x-axis corresponds to the number of calls to the initial deterministic model in the case a direct combination between OpenTURNS and ASTER, while it indicates the number of sample points in the design of experiments in the case of an approximate limit state function.

Deterministic output variable

This case of study presents a deterministic FE model for the selected zone of the containment that allows estimating the height at which the structure is under tension. Figure 6.43 shows the variation of the average tensile stress induced in concrete in term of the height. The average stress is estimated over the containment thickness for a given value of height. It can be seen that tensile stresses occur in the zone of height 4.906 m, which corresponds to a case belonging to the safe domain according to the failure criterion expressed in equation 6.16. This criterion is related to the height of zone under tension with a height threshold of 5.8 m.

Dispersion analysis

Figure 6.44 shows the variation of the mean of the tension zone height in term of the number of simulations of the initial deterministic model that represents also the number of sample points used to evaluate the approximate limit state function. The height mean is calculated with the initial limit state function (g model ) using the numerical integration method Monte Carlo, and with the approximate limit state function (g M etaM odel ) using Monte Carlo and the quadratic combination The COV of variation of the tension zone height is calculated as the ratio of the height standard deviation to its mean value. The higher the COV is, the more the dispersion about the mean is important. In other words, the estimation of mean value is more precise for lower values of COV.

Thus, the number of simulations used should be suciently high to reach the precision desired.

COV is expressed in percentage so the dispersion of variables of dierent units can be comparable.

However, this does not represent a problem in this case of study since the same variable is compared using dierent methods to assess the output of models. Figure 6.44 shows that the most accurate estimation of the mean of tension zone height is obtained using the quadratic combination method with the meta-model as it allows reducing the COV to minimum. This is due to the fact the quadratic combination method depends only on the limit state function, but does not depend on the number of simulations or calls of the model as in the case of Monte Carlo numerical integration method. This latter necessitates a sucient number of simulations in order to converge to the exact value of mean.

Distribution analysis

In order to verify the eciency of polynomial chaos expansion in approximating accurately the The probability distribution functions of output samples are estimated using Kernel Smoothing method that enables to represent those distributions graphically. As can be seen in Figure 6.47, the distribution function resulting from the meta-model is smoother than that resulting from the initial model, especially around the mean value of the height, or in other words, in the zones of high probability density. This problem can be solved by increasing the number of sample points in the design of experiments or by using a higher order for the polynomial chaos expansion [START_REF] Perrin | Prise en compte des données expérimentales dans les modèles probabilistes pour la prévision de la durée de vie des structures[END_REF]. A higher order of polynomial chaos permits a better representation of the third and fourth-order statistical moments that measure the skewness and the kurtosis of a distribution, respectively.

The graphical statistical test QQ-plot is used with quantiles estimated at 95% to compare the distributions of the height sample obtained with the initial model and that obtained with the approximated model. Figure 6.48 shows that quantile points are very close to the diagonal, which means that the distributions of the two samples are almost identical. It should be noted that the probability distribution obtained with the initial FEM model is not properly estimated since only 1000 random simulations are used to create the output sample. Thus, a higher number of random simulations is needed for a better estimation.

Failure probability

The aim of this application is to study the likelihood of failure of the containment under tension stresses at a certain height. Subsequently, only the limit state function approximated by the polynomial chaos method is considered and failure probability is estimated by combining this function to dierent probabilistic simulation and approximation methods. The eect of the number of sample points in the design of experiments is examined in the aim of illustrating the importance of using a sucient number of points to accurately estimate failure probability with a limit state function approximated by polynomial chaos expansion. Figure 6.49 shows failure probability and reliability index estimated with g M etaM odel that is coupled to dierent probabilistic methods for the failure to exact values is achieved. The convergence criterion of Mohammadkhani [START_REF] Mohammadkhani | Contribution à l'étude de la redondance dans les ponts: Analyses des mécanismes de défaillance par surfaces de réponse[END_REF] is used in this study, it consists in calculating the dierence of reliability indices of two successive iterations (2 dierent number of sample points). Mohammadkhani indicated that the convergence is reached when the dierence of reliability indices is less than 10 -3 or 10 -6 depending on the probabilistic model and numerical model considered. According to this criterion, the convergence between 600 points and 1000 points is achieved. In the case of 1000 points, only IS provides the same failure probability compared to Monte Carlo. Approximation methods do not give satisfactory results, which may due to the presence of large curvatures at the failure surface approximated with polynomial chaos expansion. As it is known, FORM and SORM do not allow a good estimate of failure probability in the case of highly nonlinear limit state functions. Convergence of simulation methods is evaluated for a condence interval at level 0.95, which indicates that the true value of failure probability is, with a probability close to 1, within the range of condence interval. x-axis in Figure 6.50 corresponds to the number of calls to the meta-model by a simulation method (number of simulations), while y-axis indicates the estimate of failure probability assessed using this method. Simulation methods can be classied according to their condence interval and the number of simulations needed to assess failure probability. It can be seen that MC shows a high precision as it provides the tighter condence bounds for failure probability, but it necessitate a signicant number of simulations (more than 60000 simulations). However, IS is more Figure 6.52 shows the eect of the number of points in the design of experiments on the shrinkage coecient importance factor. It can be seen that, for higher number of points, the importance factor calculated with the quadratic combination method and which indicates the eect of shrinkage coecient on the estimator of height mean, trends to converge towards a steady value. However, the importance factor calculated using FORM and which describes the sensitivity of shrinkage coecient In order to verify the eciency of the meta-model approximated with polynomial chaos expansion, the output sample of height obtained with the approximated model is compared to that obtained with the initial FEM. The points (h model , h M etaM odel ) are plotted on the same graph and their position with respect to the diagonal is examined. A point on the diagonal indicates that h model = h M etaM odel .

A good approximation of the initial model can be deduced in the case where the points are close to the diagonal. In the other case, the number of points in the design of experiments should be increased. Figure 6.53 shows that the number of points (h model , h M etaM odel ) that are close to the diagonal increases with the number of points in the design of experiments. However, widely spaced points from the diagonal indicate that the meta-model does not estimate accurately the height value obtained with the FEM for the relevant values of input variables. Therefore, a higher number of point in the design of experiments (>1000) is needed in order to reach the desired convergence.

Conclusion remarks

The application of reliability analysis to a containment building permits to investigate the eectiveness of the polynomial chaos method in estimating the initial physical model with an approximated model. The comparison is performed on the estimation of statistical moments of the output variable (dispersion analysis), as well as on the determination of its probability distribution (distribution analysis). The reliability analysis results are obtained by combining the meta-model with various approximation and simulation methods. Simulation methods are more accurate than approximation methods and allow more precise failure probability values. Monte Carlo and importance simulation provide the same value of failure probability and reliability index. However, the precision obtained should be veried since no reference results are available for comparison.

In conclusion, the eciency of the polynomial chaos method is related to several factors, including the limit state function to be estimated, the degree of the polynomial chaos, the number of points in the design of experiments and the variation of random variables. Dierent methods are used to estimate the reliability of the containment. The eciency of the response surface method used in this study necessitate the convergence of results towards a steady value for all types of conditions.

These conditions can be related to the physical complexity and the precision of the mechanical model, geometric properties of limit state function, number of random variables considered, ... Conclusions and perspectives

Conclusions

For an optimal and robust design that guarantees a real prediction of the behavior of structures, uncertainties should be taken into account and be propagated in the structural deterministic analysis.

Probabilistic approaches were developed to correctly model the variation of input variables. The aim of these approaches is to assess failure probability of structures according to one or several criteria and to study the inuence of uncertain variables on the structure response. Structural reliability analysis is based on the principle of combining a stochastic model with a deterministic model. Stochastic models should include the probabilistic characteristics of random input variables including a suitable probability distribution and appropriate values of their mean and coecient of variation. Deterministic models allow predicting structural response and evaluating the variables of interest, they can be based on analytical, empirical or numerical deterministic approaches. The combination of mechanical and stochastic models can be very time consuming. Thus, the issue of computational eort of reliability analysis is addressed and strategies are proposed to accurately assess failure probability with the minimum computational cost.

This research discusses the use of reliability analysis for three dierent civil engineering applications of dierent degrees of complexity. The platform OpenTURNS is used to perform the reliability analysis of the RC structures considered in the present study and propagate uncertainties in their physical models. Thus, the reliability of three applications are examined: a RC multiber cantilever beam subjected to a concentrated load at the free end, RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings, a prestressed concrete containment building.

The physical problem of reinforced concrete (RC) slabs subjected to impact is classied as low velocity impact event and considered as one of the most dicult nonlinear dynamic problems in civil engineering because it involves several factors, including the nonlinear behavior of concrete, the interaction eects between the concrete and reinforcement, and contact mechanics between the slab and the impactor. The complexity of impact problems also lies in the dynamic response of RC slabs and the time-dependent evolution of impact velocity. Several parameters may signicantly inuence the impact response of RC slabs:

Type of impact play role in dissipating energy.

Slab dimensions and boundary conditions control the stiness of the slab.

Material properties have a signicant inuence on the slab transient response by aecting the contact and overall slab stinesses.

Impactor characteristics including impact velocity, shape, position, mass and rigidity inuence the impact dynamics.

A detailed step-by-step procedure for creating FE models of impacted slabs with Abaqus is described in this study. the FEM models correctely dierent factors that contribute to the nonlinear behavior of the reinforced concrete such as the nonlinear stress-strain response, the damage due to crushing and tensile failure, the eect of the loading rate, the interaction between the concrete and the reinforcement. The analysis is performed with an explicit conguration that allows a better representation of impact problems. The model developed is validated with the experimental results of Chen and May tests and used to model RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings. Numerical results obtained are in good agreement for several slabs tested.

The reliability analysis of beam allows concluding that selection of the most suitable method for a particular type of problem is not apparent since the accuracy and computational eort of probabilistic methods depend on several factors, such as degree of nonlinearity of the limit state function, type of random variable distributions, number of random variables and their variance. In addition, it is found that FORM and IS are the less consuming methods in comparison to other methods.

The reliability analysis of impacted RC slabs shows that analytical MSM can be very eective in predicting the same values of slab displacement at the impact point as the numerical model, as well as in estimating failure probabilities very similar to those obtained with the FEM for a displacement criterion. Studying the response of impacted slabs assuming an elastic behavior of materials represents a preliminary step in the procedure of evaluating the reliability of RC slabs under low velocity impact. This application is initially considered to simplify the problem and to avoid complicating structural reliability analysis of slabs as a rst step. However, it is clear that nonlinearities due to materials behavior have a signicant eect on RC slabs response, in particular that cracking in concrete under tension and plastic deformations in concrete at the impact zone and in reinforcement reduce signicantly the peak of contact force obtained with an elastic model.

The application of reliability analysis to a containment building permits to investigate the eectiveness of the polynomial chaos method in estimating the initial physical model with an approximated model. It is found that the eciency of the polynomial chaos method is related to several factors, including the limit state function to be estimated, the degree of the polynomial chaos, the number of points in the design of experiments and the variation of random variables.

Perspectives

For Chen and May experiments [START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF], only the transient impact force-time history obtained with numerical model is compared to experimental results. However, Chen and May provide more results such as accelerations, transient reinforcement strains, and damages on the upper and bottom surfaces of slabs. Thus, it is better to study also the eciency of the FEM proposed for Chen and May slabs in predicting other experimental results.

Other studies exists in the literature that studied experimentally the behavior of RC slabs under impact. Zineddin and Krauthammer tests [START_REF] Zineddin | Dynamic response and behavior of reinforced concrete slabs under impact loading[END_REF], as well as Hrynyk tests [START_REF] Hrynyk | Behaviour and modelling of reinfoced concrete slabs and shells under static and dynamic loads[END_REF], are very important in term of gaining a better understanding of the impact behavior of RC slabs. Therefore, validating the FEM for dierent slab geometries and reinforcement ratio under dierent impact conditions seems very important to study the limitation of the proposed FEM, especially that Hrynyk tests include successive impacts on the same slab. They also tested the impact behavior of steel ber reinforced concrete, which represents an important problem to investigate numerically.

The behaviors of concrete and steel are considered in the FEM similar to their static behavior.

However, an impact problem is a dynamic problem and concrete has dierent behavior under impact than under static load. Thus, using subroutines in Abaqus to implement a new concrete model that takes into account its behavior under impact would provide more accurate prediction of the structural response.

In the current study, the evolution in time and in space of uncertainties related to loading and material properties, respectively, are not taken into account. These types of variation necessitate modeling the random functions as stochastic eld if the evolution in space is studied, and as stochastic process if the evolution in time is considered. The latest version of OpenTURNS allows more probabilistic methods such as kriging and subset methods, it also allows the denition of randomness as stochastic eld or stochastic process. Thus, it would be interesting to compare the new available methods with those used in this study in order to examine their eciency and precision in assessing failure probability. In addition, the description of the evolution in time of uncertainties related to loading and that in space related to material properties permits a better presentation of the physical problem in reliability analysis if data are available.

In this study, dierent failure criteria are examined, but independently. However a system analysis is necessary since output variables considered in structural reliability analysis does not reach their maximum values simultaneously. For example, the contact force reaches its maximum value at the rst phases of contact while that of displacement occurs at the end of contact. Yielding of steel and energy dissipation may occur at any time during the impact phenomenon. Thus, it is necessary to study the reliability of slabs in the nonlinear case for several failure criteria, separately or using a system analysis.

Several relationships are presented in this study to describe the behavior of concrete in tension and compression based on research from the literature. It is found that these curves aect highly the response of the slab, thus a reliability analysis can be performed in order to investigate the eect of the tensile and compressive stress-strain curves of concrete on the probability of failure for one or several failure criteria.

Triangular distribution Spécialité : Génie Civil RESUME: An optimal design of reinforced concrete structures requires to take into account the potential development of nonlinearities due to materials damage (concrete cracking, steel yielding, etc.). In addition, the lack of knowledge on the potential load, as well as the uncertainties related to the features of the structure (geometry, mechanical properties) shows that the design of RC structures could be made in a reliability framework. This latter allows propagating uncertainties in the deterministic analysis. However, in order to compute failure probability for one or several criteria, mechanical and stochastic models have to be coupled which can be very time consuming and in some cases impossible. Indeed, either the complexity of the deterministic model considered implies important computing times (from minutes to hours) or reliability methods evaluating the failure probability require a too large number of simulations of the deterministic model. The platform OpenTURNS is used to study the reliability of a RC multifiber cantilever beam, a RC slabs subjected to low impact velocities, and a prestressed concrete containment building. Only the physical problem of RC slabs subjected to impact is investigated in detail in this study. This study focuses on RC slabs which are subjected to accidental dropped object impact during handling operations within nuclear plant buildings. In this case, dropped objects are characterized by small impact velocities and damage can arise at the impact zone depending on the impact energy and the relative masses of the colliding bodies. The aim of this study is to address the issue of computational cost of reliability analysis and to propose computational strategies allowing the accurate assessment of the failure probability for minimum computational time. The first strategy consists in using deterministic analytical models involving low computational time. The second one consists in choosing an appropriate probabilistic method where the failure probability is assessed from a small number of simulations. The first part of this study describes the behavior and the failure modes of RC slabs subjected to impact. Then two deterministic models are used and evaluated. The first model consists of a 3D finite element model simulated with ABAQUS/Explicit. The steel is modeled as an elasto-plastic material with hardening and, the behavior of concrete is described by the damaged plasticity model. The model is validated with experiments from the literature. A simplified analytical model is also used, it consists in a two degrees of freedom mass-spring system which accounts for potential viscous damping. A frequency decrease approach is used to describe the degradation of the slab. This study includes a reliability analysis of RC slabs under low velocity impact. Firstly, the choices of random variables inputs and their distributions, failure criteria and probabilistic methods are discussed. In the case where materials have nonlinear behavior, the failure criterion is defined according to the maximum displacement of the slab at the impact point. The two strategies used to reduce the computational cost of a reliability analysis are compared in order to verify their efficiency to calculate the probability of failure. Finally, a parametric study is performed to identify the influence of deterministic model parameters on the calculation of failure probability.

       f X (x) = 2(x-

σt

  téristiques du béton et pourcentage d'armatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eective stress of concrete in tension Si Mechanical elasticity of variable X i β Reliability index of Hasofer-Lind Flow potential eccentricity in the CDP model X Random vector of input variables µ X Mean of a random variable ψ Dilation angle of concrete in the CDP model ρ X 1 X 2 Correlation coecient of two random variables X 1 and X 2 Standard deviation of a random variable ε c1 Compressive strain of concrete at the compressive strength f c ε ce Compressive elastic strain of concrete corresponding to the end of linear elastic phase ε cr Tensile strain of concrete at the tensile strength f t ε cu Ultimate strain of concrete in compression ε c Concrete compressive strain corresponding to the stress σ c Plastic strain of concrete in compression ε sh Strain of steel at which strain hardening initiates ε s Steel strain corresponding to the stress σ s xiii ε te Tensile elastic strain of concrete corresponding to the end of linear elastic phase ε t Concrete tensile strain corresponding to the stress σ t Yield strain of steel at the yield strength f y ζ i Omission factor of variable X i c c Impedance that accounts for the contact viscosity in mass-spring models c s Impedance that accounts for the slab viscosity in mass-spring models COV (X), COV Coecient of variation of a random variable COV (X 1 , X 2 ) Covariance of two random variables X 1 and X 2 Deformation modulus of steel at the hardening phase E s Elastic modulus of steel F X (x) Cumulative distribution function of a random variable f X (x) Probability density function of a random variable f X (x) Joint probability density function of random variables f b Biaxial compressive strength of concrete f ck Characteristic value of concrete f cu Compressive stress of concrete corresponding to the ultimate strain f c Uniaxial compressive strength of concrete f t Uniaxial tensile strength of concrete f u Ultimate stress of steel corresponding to the ultimate strain f y Yield strength of steel g(X) Ratio of the second invariant on the tensile meridian to that on the compressive meridian at any given value of the pressure invariant (parameter of CDP model) k c Contact stiness in mass-spring models k s Slab stiness in mass-spring models m i Impactor mass in mass-spring models m s Slab mass in mass-spring models N Number of generated samples or simulations of the random vector X ne Number of elements in slab thickness direction P f Failure probability SizeElImpactZoneM ax Maximum nite element size of slab in the impact zone SizeElImpactZoneM in Minimum nite element size of slab in the impact zone SizeElReinf Finite elements size of reinforcement SizeElSlab Finite elements size of slab edges loading cases in civil engineering eld: a) Rockfall on protection galleries, b) Aircraft impact on nuclear containments, c) Dropped objects impact during handling operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  : a) Flexural failure, b) Punching shear failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Local damage failure modes of impacted RC slabs [119] : a) Surface failure, b) spalling, c) scabbing, d) perforation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. 10

 10 Impact test system used by Zineddin and Krauthammer [189] . . . . . . . . . . . . . . . 2.11 Load-time history for slabs impacted with a drop height of 152 mm [189] . . . . . . . . . 2.12 Failure mode of slabs impacted with a drop height of 610 mm and reinforced with reinforcement type: (a) Reinf. 3, (b) Reinf. 1, (c) Reinf. 1-Failure of one steel bar [189] . . . 2.13 Cracks patterns on top surface of slabs reinforced with reinforcement type Reinf. 3 and impacted with a drop height of : (a) 305 mm, (b) 610 mm[189] . . . . . . . . . . . . . . 2.14 Details of slabs (dimensions in mm): a) 760 mm square slabs, b) 2320 mm square slabs, c) Boundary conditions [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Details of impactors (dimensions in mm): a) Hemispherical impactor, b) Cylindrical impactor [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Transient impact load of slabs 2-6 [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Local damage of slabs bottom faces after impact: (a) Slab 2, (b) Slab 3, (c) Slab 4, (d) Slab 5, (e) Slab 6 [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Details of slabs (dimensions in mm): a) Slab with additional links at the impact region, b) Boundary conditions [84] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 Cracking patterns on the bottom surface of a slab with a longitudinal reinforcement ratio of 0.42%: a) Impact #1, b) Impact #2, c) Impact #3 [84] . . . . . . . . . . . . . . . . . 2.20 Final cracking patterns on the bottom surface of slabs with a longitudinal reinforcement ratio of: a) 0.273%, b) 0.42%, c) 0.592% [84] . . . . . . . . . . . . . . . . . . . . . . . . . 2.21 One degree of freedom mass-spring model used in Tonello IC design oce [174] . . . . . 2.22 Simplied model of CEB for a hard impact [29] . . . . . . . . . . . . . . . . . . . . . . 2.23 Nonlinear force-displacement relationship for spring: (a) R 1 , (b) R 2 [29] . . . . . . . . .

  model, (b) post-impact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 4.

5 . 1

 51 Part of slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Part of reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Part of impactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Part of support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Reinforcement in the model assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.6 Assembly of the model after creating instances . . . . . . . . . . . . . . . . . . . . . . . 5.7 Mesh of cylindrical support instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Partitions used to create Mesh1 and Mesh2 . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Mesh1 and Mesh2 of slab in xy-plane and in the direction of thickness . . . . . . . . . . 5.10 Partitions used to create Mesh3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Mesh3 of slab in xy-plane and in the direction of thickness . . . . . . . . . . . . . . . . 5.12 Partitions used to create Mesh4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Mesh4 of slab in xy-plane and in the direction of thickness . . . . . . . . . . . . . . . . 5.14 Mesh4 with dierent sizes of impact region . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Mesh4 for dierent impact positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16 The impactor mesh used with Mesh1, Mesh2 and Mesh3 of slab . . . . . . . . . . . . . 5.17 The impactor mesh used with Mesh4 of slab . . . . . . . . . . . . . . . . . . . . . . . . 5.18 Comparison of dierent mesh types proposed for Slab2 . . . . . . . . . . . . . . . . . . 5.19 Comparison of contact algorithms available in Abaqus for Slab2 meshed with Mesh4 . .5.20 Sensitivity of mesh in term of the number of elements in the thickness direction and the size of elements in the impact zone for Slab2 meshed with Mesh4 . . . . . . . . . . . . . 5.21 Sensitivity of mesh in term of the size of elements in the slab and the size of elements in the reinforcement for Slab2 meshed with Mesh4 . . . . . . . . . . . . . . . . . . . . . . 5.22 Comparison of concrete compressive stress-strain curves for Slab2 . . . . . . . . . . . . 5.23 Comparison of concrete compressive stress-strain curves for Slab2 . . . . . . . . . . . . 5.24 Comparison of concrete compressive stress-strain curves for Slab2 . . . . . . . . . . . . 5.25 Comparison of concrete tensile stress-strain curves for Slab2 . . . . . . . . . . . . . . . 5.26 Comparison of concrete tensile stress-strain curves for Slab2 . . . . . . . . . . . . . . . 5.27 Comparison of concrete tensile stress-cracking displacement curves for Slab2 . . . . . . 5.28 Comparison of fracture energy values for Slab2 . . . . . . . . . . . . . . . . . . . . . . . 5.29 Comparison of steel stress-strain curves for Slab2 . . . . . . . . . . . . . . . . . . . . . . 5.30 Comparison of steel stress-strain curves for Slab2 . . . . . . . . . . . . . . . . . . . . . . 5.31 Comparison of C6T1S1 and C6D3S1 simulations for Slab3 and Slab4 . . . . . . . . . . 5.32 Comparison of C6T1S1 and C6D3S1 simulations for Slab5 and Slab6 . . . . . . . . . . 5.33 Finite element model of slab: a) meshing, b) reinforcement . . . . . . . . . . . . . . . . 5.34 Mass-spring model without damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.35 Mass-spring model with damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.36 Slab stiness decrease in term of displacement for several values of thickness and the variation of the stiness initial value in term of the thickness . . . . . . . . . . . . . . . 5.37 Comparison of mass-spring models with and without damping . . . . . . . . . . . . . . 5.38 Comparison of FEM and MSM for the same kinetic energy . . . . . . . . . . . . . . . . 6.1 RC beam tested experimentally in the framework of the LESSLOSS project [108] . . . .

6 . 6

 66 QQ-plot test to graphically compare the beam displacement sample to several parametric distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Number of calls of beam FEM in term of failure probability magnitude for dierent probabilistic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Eect of the number of random variables on the accuracy and precision of dierent probabilistic methods in comparison to MC . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Variation of failure probability in terms of the force applied at the free end of the beam and the number of random variables considered . . . . . . . . . . . . . . . . . . . . . . 6.10 Eect of the distribution type of random variables on the accuracy and precision of dierent probabilistic methods in comparison to MC . . . . . . . . . . . . . . . . . . . . 6.11 Eect of the type of random variables distribution on the estimation of failure probability for dierent probabilistic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Eect of the COV of random variables on the accuracy and precision of dierent probabilistic methods in comparison to MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Variation of failure probability in term of the mean of the force for dierent values of COV161 6.14 Variation of failure probability in term of the COV for dierent values of the mean of the force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Comparison of the beam displacement distributions estimated with the initial model and Taylor expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 Comparison of the beam displacement distributions estimated with the initial model and Least Square method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Comparison of the beam displacement distributions estimated with the initial model and polynomial chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.18 Comparison of failure probability estimated with the initial model and several response surface types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.19 Finite element model of slab: a) meshing, b) reinforcement . . . . . . . . . . . . . . . . 6.20 Failure probability with dierent deterministic models and displacement criterion . . . 6.21 Comparison of dierent failure criteria in term of impact velocity mean . . . . . . . . . 6.22 Eect of the mean of slab length for the displacement criterion . . . . . . . . . . . . . . 6.23 Eect of the mean of slab length for the displacement criterion . . . . . . . . . . . . . . 6.24 Eect of the mean of concrete density for the displacement criterion . . . . . . . . . . . 6.25 Eect of the mean of steel bars diameter for the displacement criterion . . . . . . . . . 6.26 Eect of the mean of impactor mass for the displacement criterion . . . . . . . . . . . . 6.27 Eect of the mean of slab stiness for the displacement criterion . . . . . . . .

Figure 2 . 1 :

 21 Figure 2.1: Some accidental impact loading cases in civil engineering eld: a) Rockfall on protection galleries, b) Aircraft impact on nuclear containments, c) Dropped objects impact during handling operations

Figure 2 . 2 : 2 . 2

 2222 Figure 2.2: Accidental dropped object impact during handling operations within nuclear plant buildings

Figure 2 . 4 :

 24 Figure 2.4: Structure and impactor deformations according to the impact type [45] : a) Initial conditions (t=0), b) Hard impact (t>0), c) Soft impact (t>0)

Figure 2 . 6 :

 26 Figure 2.6: Classication of low velocity impacts according to Koechlin and Potapov [95]

Figure 2 . 7 :

 27 Figure 2.7: Overall response failure modes of impacted RC slabs [119] : a) Flexural failure, b) Punching shear failure

Figure 2 . 8 :

 28 Figure 2.8: Local damage failure modes of impacted RC slabs [119] : a) Surface failure, b) spalling, c) scabbing, d) perforation

Figure 2 . 10 :

 210 Figure 2.10: Impact test system used by Zineddin and Krauthammer [189]

Figure 2 . 11 :Figure 2 . 12 :

 211212 Figure 2.11: Load-time history for slabs impacted with a drop height of 152 mm [189]

Figure 2 . 13 :

 213 Figure 2.13: Cracks patterns on top surface of slabs reinforced with reinforcement type Reinf. 3 and impacted with a drop height of : (a) 305 mm, (b) 610 mm[START_REF] Zineddin | Dynamic response and behavior of reinforced concrete slabs under impact loading[END_REF]

  yield steel bars at the bottom and the top with a concrete cover of 12 mm. Slab 5 and Slab 6 were 2320 mm square and 150 mm thick and reinforced with 12 mm diameter high yield steel bars at the bottom and the top with a concrete cover of 15 mm. The slabs, shown in Figure 2.14, consisted of two parts: a reinforced concrete region surrounded by a steel support of 17.5 mm width, where the support was clamped at the four corners to restrain horizontal and vertical displacements (Figure 2.14.c).

Figure 2 . 14 :Figure 2 . 15 :

 214215 Figure 2.14: Details of slabs (dimensions in mm): a) 760 mm square slabs, b) 2320 mm square slabs, c) Boundary conditions[START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF]

Figure 2 . 16 :

 216 Figure 2.16: Transient impact load of slabs 2-6 [34]

Figure 2 . 17 :

 217 Figure 2.17: Local damage of slabs bottom faces after impact: (a) Slab 2, (b) Slab 3, (c) Slab 4, (d) Slab 5, (e) Slab 6[START_REF] Chen | Reinforced concrete members under drop-weight impacts[END_REF]

Figure 2 .

 2 Figure 2.19 illustrates the typical damage behavior and cracking patterns of RC slabs that were

Figure 2 . 18 :Figure 2 . 19 :

 218219 Figure 2.18: Details of slabs (dimensions in mm): a) Slab with additional links at the impact region, b) Boundary conditions[START_REF] Hrynyk | Behaviour and modelling of reinfoced concrete slabs and shells under static and dynamic loads[END_REF]

Figure 2 . 20 :

 220 Figure 2.20: Final cracking patterns on the bottom surface of slabs with a longitudinal reinforcement ratio of: a) 0.273%, b) 0.42%, c) 0.592%[START_REF] Hrynyk | Behaviour and modelling of reinfoced concrete slabs and shells under static and dynamic loads[END_REF]

Figure 2 . 21 :

 221 Figure 2.21: One degree of freedom mass-spring model used in Tonello IC design oce[START_REF] Tonello | Généralité et approche de modèles simples[END_REF]

Figure 2 . 22 :Figure 2 . 23 :

 222223 Figure2.22: Simplied model of CEB for a hard impact[START_REF] Ceb | Concrete structures under impact and impulsive loading[END_REF]

Figure 2 .

 2 Figure2.24: Mass-spring models of impact on composite structures according to[START_REF] Abrate | Modeling of impacts on composite structures[END_REF]

Figure 2 . 25 :Figure 2 . 26 :

 225226 Figure 2.25: The two mass-spring-damper models used by [47] to analyze SDR response: (a) contact: model, (b) post-impact model

Figure 2 .

 2 Figure 2.27: 3D FE model of SDR protection galleries proposed by Berthet-Rambaud et al. [19]

Figure 2 . 28 :

 228 Figure 2.28: Damage distribution in and around the repaired zone[START_REF] Berthetrambaud | Analytical advantages and numerical modeling of impact testing on a structurallydissipating rock-shed slab (sdr)[END_REF]

Figure 2 . 29 :

 229 Figure 2.29: Comparison of numerical and experimental results: (a) Calibration on friction parameter, (b) Vertical displacement at a point near the impact zone[START_REF] Berthetrambaud | Analytical advantages and numerical modeling of impact testing on a structurallydissipating rock-shed slab (sdr)[END_REF]

Figure 2 . 30 :

 230 Figure 2.30: Details of nite element model of Mokhatar and Abdullah[START_REF] Mokhatar | Computational analysis of reinfoced concrete slabs subjected to impact loads[END_REF]

  using a 3D inelastic nite element model (Figure 2.33.a).

Figure 2 . 31 :

 231 Figure 2.31: Comparison of numerical and experimental transient impact force with dierent: (a)Mesh densities, (b) Constitutive concrete models[START_REF] Mokhatar | Computational analysis of reinfoced concrete slabs subjected to impact loads[END_REF]

Figure 2 . 32 :

 232 Figure 2.32: Final crack pattern of Slab3 bottom face: (a) Experimental result, (b) Numerical result

Figure 2 . 33 :

 233 Figure 2.33: Details of Trivedi and Singh FE model: (a) One fourth of a slab with 2 meshes of longitudinal and transversal steel bars, (b) Time-amplitude curve in case of a hammer dropped with a height of 610 mm[START_REF] Trivedi | Prediction of impact induced failure modes in reinforced concrete slabs through nonlinear transient dynamic nite element simulation[END_REF]

Figure 2 . 34 :

 234 Figure 2.34: Identication of slabs failure mode by inection point criterion: (a) Transition from exure to punching shear, (b) Flexural mode[START_REF] Trivedi | Prediction of impact induced failure modes in reinforced concrete slabs through nonlinear transient dynamic nite element simulation[END_REF]

Figure 2 . 35 :

 235 Figure 2.35: Identication of a slab exural failure mode by strain based failure criterion: (a) Top elements, (b) Bottom elements[START_REF] Trivedi | Prediction of impact induced failure modes in reinforced concrete slabs through nonlinear transient dynamic nite element simulation[END_REF]

Figure 2 . 36 :

 236 Figure 2.36: Uniaxial compressive stress-strain curve for concrete[4]

Figure 2 . 38 :

 238 Figure 2.38: Concrete stress-crack opening curve with: (a) Linear softening branch [7], (b) Bi-linearsoftening branch[START_REF] Haidong | Strengthening of RC slabs with openings using CFRP systems[END_REF], (c) Tri-linear softening branch[START_REF] Tajima | 3d nite element cyclic analysis of beam/column joint using special bond model[END_REF]

Figure 2 .

 2 38.b). Tajima et al.[START_REF] Tajima | 3d nite element cyclic analysis of beam/column joint using special bond model[END_REF] used a tri-linear softening branch with fracture energy as depicted in Figure2.38.c.

Figure 2 . 39 :

 239 Figure2.39: Experimental tensile stress-strain curve for reinforcing steel[START_REF] Lowes | Finite element modeling of reinforced concrete beam-columns bridge connections[END_REF]

Figure 3 . 1 :

 31 Figure 3.1: Unloading response of: (a) elastic plastic, (b) elastic damage, (c) elastic plastic damage models[START_REF] Jason | An elastic plastic damage formulation for concrete: Application to elementary tests and comparison with an isotropic damage model[END_REF]

Figure 3 . 2 :

 32 Figure 3.2: Drucker-Prager hyperbolic function of CDP ow potential and its asymptote in the meridian plane[START_REF] Kmiecik | Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration[END_REF]

Figure 3 . 3 :

 33 Figure 3.3: Yield surface for the CDP model in: (a) plane stress, (b) the deviatoric plane corresponding to dierent values of K c [6]

Figure 3 . 4 :

 34 Figure 3.4: Response of concrete to uniaxial loading in compression for CDP model[6]

Figure 3 . 5 :

 35 Figure 3.5: Response of concrete to uniaxial loading in tension for CDP model[6]

  Three dimensional truss elements having two degrees of freedom (T3D2) are used to represent the reinforcement in FEM (Figure 3.6). Truss elements are used in two and three dimensions to model slender structures that support loading only along the axis or the centerline of the element. No

Figure 3 . 6 :

 36 Figure3.6: Finite elements used for the problem involving contact-impact of reinforced concrete[5]

Figure 3 . 7 :

 37 Figure 3.7: Constraints with the contact pair algorithm: a) Kinematic contact, b) Penalty contact[5]

  Coulomb friction model is used to model the friction between the contacting bodies in terms of the stresses at the interface of the bodies (Figure 3.8.b). The basic concept of the Coulomb friction model is to relate the maximum allowable frictional (shear) stress across an interface to the contact pressure between the contacting bodies. In the basic form of the Coulomb friction model, two contacting surfaces can carry shear stresses up to a certain magnitude across their interface before they start sliding relative to one another. The friction coecient needs to be dened for the mechanical tangential interaction.

Figure 3 . 8 :

 38 Figure 3.8: Contact interaction properties: a) Hard contact model, b) Coulomb friction model[5]

Figure 4 . 1 :

 41 Figure 4.1: Limit state function[START_REF] Faber | Failure consequences and reliability acceptance criteria for exceptional building structures[END_REF]

Figure 4 . 2 :

 42 Figure 4.2: PDF of Gauss, Lognormal, Uniform and Weibull distributions with same mean and standard deviation [107]

  Figures 4.2and 4.3, respectively, show the probability density and cumulative distribution functions for four types of probability distributions of a random variable (Gaussian, Lognormal, Uniform and Weibull).

Figure 4 . 3 :

 43 Figure 4.3: CDF of Gauss, Lognormal, Uniform and Weibull distributions with same mean and standard deviation [107]

Figure 4 . 4 :

 44 Figure 4.4: Correlation of two random variables

Figure 4 .

 4 Figure 4.4 depicts the physical properties of the correlation coecient. In Figure 4.4.a, the two random variables are uncorrelated, the correlation coecient is expected to be zero and no linear relationship exists between them.Figure 4.4.b indicates a non perfect linear relationship between

  can be established graphically by plotting a histogram (Figure 4.5). A histogram can be developed by arranging the values of Y in increasing order and subdividing them into several equal intervals in each of which the number of observations is drawn. Then a distribution that ts the histogram needs to be selected to model the behavior of the randomness. OpenTURNS proposes 17 parametric distribution types to represent the uncertainty of a continuous variable, including normal, lognormal, Gamma, Weibull, etc. The values of the distribution parameters are estimated in terms of the output sample statistical moments. The probability density functions available in OpenTURNS are listed in Appendix A with their formulas and their statistical parameters. Once the type of distribution is selected and its parameters are evaluated, a statistical test of goodness-of-t must

Figure 4 . 5 :

 45 Figure 4.5: Representation of a sample of an output variable by a histogram[START_REF] Lemaire | Structural reliability[END_REF]

Figure 4 . 6 :

 46 Figure 4.6: Reliability index in standard space[START_REF] Croston | Etude expérimentale du comportement d'une poutre en béton armé en exion 3 points reparée par matériaux composites (approche probabiliste)[END_REF]

Figure 4 . 7 :

 47 Figure 4.7: Approximation of limit state surface by FORM [75]

Figure 4 . 8 :

 48 Figure 4.8: Approximation of limit state surface by SORM [75]

Figure 4 . 9 :

 49 Figure 4.9: Principle of Monte Carlo method, illustration in the standard space[START_REF] Deheeger | Couplage mécano-abiliste: SMART -méthodologie d'apprentissage stochastique en abilité[END_REF]

Figure 4 . 12 :

 412 Figure 4.12: Principle of Latin hypercube sampling method, illustration in the standard space[START_REF] Deheeger | Couplage mécano-abiliste: SMART -méthodologie d'apprentissage stochastique en abilité[END_REF]

1 .( 4 . 42 ) 2 .

 14422 Region I, for which 0 < ε ≤ f y /E s , consists of a straight line from zero stress to the yield stress f = E s ε Region II, for which f y /E s < ε ≤ 0.01, consists of a straight horizontal line and a stress equal to the yield stress f = f y (4.43)

(4. 45 )

 45 The variable f represents the yield strength value, and the variable D represents the dierence between the values for the upper bound U B and the lower bound LB. The values of the parameters presented in equation 4.45 were evaluated for dierent grades and sizes of reinforcement bars. For A 615 Grade 75 reinforcement, the average of the probability distribution of yield strength was estimated for all bar sizes, with: C = 37335, α = 2.59, β = 1882.86, LB = 75000, U B = 4000000, D = 3925000.

  Abaqus, Part module is used to divide the structure geometry into multiple parts. Each part denes a main component of the model and needed to be meshed separately. The Part module permits to create deformable, discrete rigid, analytical rigid, or Eulerian parts. The simulation of Chen and May slabs has 4 main components, namely the slab, the reinforcement, the impactor and the supports (for more details about these components geometry, see section 2.3.1.2). Slab part represents the slab concrete region surrounded by a steel frame and reinforcement part includes longitudinal and transversal bars of one layer of reinforcement. Impactor part describes the striker geometry used in experiments and support part represents the shape of cylinder components on which the slab is supported.

Figure 5 . 1 :

 51 Figure 5.1: Part of slab

Figure 5 .

 5 [START_REF][END_REF] shows a three-dimensional view of the part which is used to develop the typical geometry of Chen and May slabs and highlighted the partition scheme adopted. The section in dark gray represents the steel frame, while the light gray represents the concrete part of slabs. Details of the geometry and dimensions of Chen and May slabs are illustrated in Figure2.14. Slabs have two dierent geometries: 760 mm square slabs with a thickness of 76 mm and 2320 mm square with a thickness of 150 mm. Steel frame that surround the concrete region is considered with a width of 17.5 mm.

(Figure 2 .

 2 14). Reinforcement part includes longitudinal and transversal bars of one layer of reinforcement. It consists of 11 longitudinal and 11 transversal bars in the case of 760 mm square slabs, and of 15 longitudinal and 15 transversal bars in the case of 2320 mm square slabs. The reinforcement is modeled as a 3D deformable wire part using the wire option in Abaqus. The 2D sketch is created by modeling a single transversal bar and a single longitudinal bar, then using the linear pattern option available in Abaqus. The spacing of reinforcement bars is equal to 60 mm for 760 mm slabs and 142 for 2320 mm slabs.

Figure 5 .

 5 Figure 5.2 shows a three-dimensional view of the modeling of longitudinal and transversal reinforcement in 760 mm square slabs. It should be noted that no shear reinforcement is present in Chen and May slabs, otherwise an additional part should be considered to incorporate stirrups in the FEM.

Figure 5 . 2 :Figure 5 . 3 :

 5253 Figure 5.2: Part of reinforcement

Figure 5 . 4 : 3 Young

 543 Figure 5.4: Part of support

  Truss sections are used to model slender 2D or 3D structures that provide axial strength but no bending stiness. Thus, a truss section is specied and assigned to reinforcement part. It consists of steel material and a cross-sectional area equal to one steel bar area. Slabs 2 and 3 are reinforced with steel bars of 6 mm diameter, Slab 4 is reinforced with steel bars of 8 mm diameter, while Slabs 5 and 6 are reinforced with steel bars of 12 mm diameter.

Figure 5 . 5 :

 55 Figure 5.5: Reinforcement in the model assembly

Figure 5 . 6 :Figure 5 . 7 :

 5657 Figure 5.6: Assembly of the model after creating instances

Figure 5 . 8 :Figure 5 . 9 :

 5859 Figure 5.8: Partitions used to create Mesh1 and Mesh2

Figure 5 . 10 :Figure 5 . 11 :

 510511 Figure 5.10: Partitions used to create Mesh3

Figure 5 . 12 :Figure 5 . 13 :

 512513 Figure 5.12: Partitions used to create Mesh4

Figure 5 . 14 :

 514 Figure 5.14: Mesh4 with dierent sizes of impact region

Figure 5 . 16 :Figure 5 . 17 :

 516517 Figure 5.16: The impactor mesh used with Mesh1, Mesh2 and Mesh3 of slab

Figure 5 . 18 :

 518 Figure 5.18: Comparison of dierent mesh types proposed for Slab2

Figure 5 . 19 :

 519 Figure 5.19: Comparison of contact algorithms available in Abaqus for Slab2 meshed with Mesh4

Figure 5 . 20 :Figure 5 . 21 :

 520521 Figure 5.20: Sensitivity of mesh in term of the number of elements in the thickness direction and the size of elements in the impact zone for Slab2 meshed with Mesh4

Figure 5 . 22 :

 522 Figure 5.22: Comparison of concrete compressive stress-strain curves for Slab2

Figure 5 . 23 :Figure 5 . 27 :Figure 5 . 28 :

 523527528 Figure 5.23: Comparison of concrete compressive stress-strain curves for Slab2

Figure 5 . 29 :Figure 5 . 30 :

 529530 Figure 5.29: Comparison of steel stress-strain curves for Slab2

Figure 5 . 31 :Figure 5 . 32 :

 531532 Figure 5.31: Comparison of C6T1S1 and C6D3S1 simulations for Slab3 and Slab4

Figure 5 . 33 : 2 . 3 . 4 .

 533234 Figure 5.33: Finite element model of slab: a) meshing, b) reinforcement

Figure 5 . 34 :

 534 Figure 5.34: Mass-spring model without damping

Figure 5 . 36 :

 536 Figure 5.36: Slab stiness decrease in term of displacement for several values of thickness and the variation of the stiness initial value in term of the thickness

Figure 5 . 37 :

 537 Figure 5.37: Comparison of mass-spring models with and without damping

Figure 5 . 38 :

 538 Figure 5.38: Comparison of FEM and MSM for the same kinetic energy

Figure 6 . 1 :Figure 6 . 2 :Figure 6 . 3 :

 616263 Figure 6.1: RC beam tested experimentally in the framework of the LESSLOSS project [108]

Figure 6 . 4 :Figure 6 . 5 :

 6465 Figure 6.4: Beam displacement distribution estimated with Kernel Smoothing method

Figure 6 . 7 :

 67 Figure 6.7: Number of calls of beam FEM in term of failure probability magnitude for dierent probabilistic methods

Figure 6 . 8 :

 68 Figure 6.8: Eect of the number of random variables on the accuracy and precision of dierent probabilistic methods in comparison to MC

Figure 6 . 9 :

 69 Figure 6.9: Variation of failure probability in terms of the force applied at the free end of the beam and the number of random variables considered

Figure 6 . 10 :

 610 Figure 6.10: Eect of the distribution type of random variables on the accuracy and precision of dierent probabilistic methods in comparison to MC

Figure 6 . 11 :

 611 Figure 6.11: Eect of the type of random variables distribution on the estimation of failure probability for dierent probabilistic methods

Figure 6 . 12 :

 612 Figure 6.12: Eect of the COV of random variables on the accuracy and precision of dierent probabilistic methods in comparison to MC

Figure 6 .

 6 [START_REF] Baudin | An industrial software for uncertainty quantication in simulation[END_REF] shows failure probability values as a function of the COV. The linear equation of P f = 0.5 describes the limit state function that corresponds to the point of intersection of the curves presented in the previous gure. This gure permits to evaluate the range of low failure probabilities in term of the mean value of the force applied at the free end of the beam. It can be seen that, for a force less than 200 N and a COV less than 0.4, failure probability represents values less than 0.005.

Figure 6 . 13 :Figure 6 . 14 :Figure 6 . 15 :

 613614615 Figure 6.13: Variation of failure probability in term of the mean of the force for dierent values of COV

Figure 6 . 16 :Figure 6 . 18 :

 616618 Figure 6.16: Comparison of the beam displacement distributions estimated with the initial model and Least Square method

6. 3 . 1

 31 Deterministic models 6.3.1.1 Finite element model

Figure 6 . 19 :

 619 Figure 6.19: Finite element model of slab: a) meshing, b) reinforcement

2 . 2 .

 22 The rigidity of slab and the contact force-time are determined by means of the FEM. The equivalent mass of slab in the MSM is obtained by multiplying the mass of the real slab with a transformation factor. The slab mass, as well as the transformation factor, highly depend on slab dimensions, material densities, bars diameter, and number of longitudinal and transversal reinforcement bars. Impactor velocity, mass and radius are

Figure 6 . 21 :Figure 6 . 22 :

 621622 Figure 6.21: Comparison of dierent failure criteria in term of impact velocity mean

Figure 6 . 24 :

 624 Figure 6.24: Eect of the mean of concrete density for the displacement criterion

Figure 6 . 26 :Figure 6 . 27 :

 626627 Figure 6.26: Eect of the mean of impactor mass for the displacement criterion

6. 4

 4 Application to RC slab subjected to impact -Nonlinear behavior 6.4.1 Deterministic models Deterministic FEM model used for the slab subjected to accidental dropped object impact within nuclear plant building assuming a nonlinear behavior of materials is detailed in section 5.4, while the simplied model used consists of a two degrees mass-spring model which is detailed in section 5.4.2.

Figure 6 . 31 :

 631 Figure 6.31: Eect of the mean of concrete compressive strength for the displacement criterion

  Figure 6.32: Eect of the mean of steel yield strength for the displacement criterion

Figure 6 . 33 :

 633 Figure 6.33: Eect of the mean of steel diameter for the displacement criterion

Figure 6 . 35 :

 635 Figure 6.35: Eect of stirrups for the displacement criterion

Figure 6 . 36 :

 636 Figure 6.36: Eect of boundary conditions for the displacement criterion

Figure 6 . 37 :Figure 6 . 38 :

 637638 Figure 6.37: Eect of impact position for the displacement criterion

  of 15 time increments between t c and t V D1 . The connement building of Flamanville nuclear power plant consists of a prestressed cylindrical concrete vessel of 45 m of diameter and 65.05 m of height (Figure 6.39). The vessel is capped with a dome and is supported on a at foundation slab. The plate (gousset) linking the bottom of the internal containment to the foundation slab has height of 3.6 m and an extra thickness of 1.10 m.

Figure 6 . 39 :Figure 6 . 40 :

 639640 Figure 6.39: Geometry of the containment building of the Flamanville nuclear power plant

Figure 6 . 41 :

 641 Figure 6.41: Mesh of the selected zone of study of the containment building

6. 5 . 1 . 5

 515 Delayed strains in concreteThe internal containment structure shows three dierent thicknesses related to the slab foundation, the plate and the common part of the containment. The dierence of thickness along the structure necessitates the prediction of drying shrinkage and creep of concrete in terms of the geometry and boundary conditions of each zone of dierent thickness. Delayed strains in concrete may aect the sustainability of concrete structures for several reasons. Shrinkage strains ε r may induce cracking in concrete until causing steel reinforcement corrosion, while creep strains ε rf may result in a signicant reduction in the prestressing force.

Figure 6 . 42 :

 642 Figure 6.42: Horizontal and vertical cables considered in the reliability analysis of containment building

(6. 16 ) 6 . 5 . 4

 16654 Discussion of results of containment building reliabilityPolynomial chaos method with cleaning strategy is used to study the reliability of the containment building of Flamanville nuclear power plant. As previously demonstrated in the reliability study of cantilever beam, the cleaning strategy is found to be the most ecient strategy, among those available in OpenTURNS for polynomial chaos, to estimate failure probability and analyze the dispersion of variables of interest. Two steps are necessary to perform the study:

Figure 6 . 43 :

 643 Figure 6.43: Variation of the average tensile stress in term of height

  limit state function of the initial deterministic model, the sample of height obtained with the initial model function g model and that obtained with the approximated function g M etaM odel for the same realizations of the input physical vector are compared. The aim is to compare two statistical samples having the same population and the same size of 1000.A histogram is considered as a graphical tool with which the identication of the distribution of a variable can be established. It can be used to verify if the meta-model provides a good estimation of the height output sample obtained with the initial FEM. In order to facilitate the comparison, both histograms obtained through the initial model and the approximate model, respectively, are plotted on the same graph (Figure6.45). x-axis corresponds to the height values in m, while y-axis indicates the frequency values in %. Examining the histograms of the output samples allows some preliminary

Figure 6 . 44 :Figure 6 . 45 :

 644645 Figure 6.44: Mean and COV of height in term of the number of simulations

Figure 6 .

 6 Figure 6.46: CDF of tension zone height samples obtained with initial model and meta-model for 1000 simulations

Figure 6 . 47 :Figure 6 . 48 :Figure 6 . 49 :

 647648649 Figure 6.47: PDF of tension zone height samples obtained with initial model and meta-model for 1000 simulations

Figure 6 .

 6 Figure 6.50 illustrates the convergence of failure probability assessed using dierent simulation methods coupled the limit function approximated with 1000 points in the design of experiments.

Figure 6 . 50 :Figure 6 .

 6506 Figure 6.50: Convergence at level 0.95 of the estimate of failure probability for dierent probabilistic simulation methods

Figure 6 . 51 :Figure 6 . 52 :

 651652 Figure 6.51: Importance factors of containment random variables using quadratic combination method and FORM

Figure 6 . 53 :

 653 Figure 6.53: Comparison of initial model and meta-model for dierent number of points in the design of experiments

  Figure B.8: Displacement and velocity of slab and impactor: v = 2.5 m/s

  MOTS-CLÉS: RC structures, RC slabs, impact at low velocity, dynamic, material nonlinear behaviors, deterministic approaches, reliability approaches, failure probability Laboratoire de recherche : LGCIE -Laboratoire de Génie Civil et Ingénierie Environnementale Directeur de thèse: Ali LIMAM Président de jury : Composition du jury : Laurence CURTIL, Fabrice GATUINT, Abdellatif KHAMLICHI, Elias BOU-SAID, David BERTRAND, Ali LIMAM

  . . . . . 6.28 Eect of the mean of impactor velocity for the displacement criterion (m=3600 kg) . .6.29Eect of the mean of impactor mass for the displacement criterion . . . . . . . . . . . . 6.30 Eect of the mean of slab thickness for the displacement criterion . . . . . . . . . . . . 6.31 Eect of the mean of concrete compressive strength for the displacement criterion . . . 6.32 Eect of the mean of steel yield strength for the displacement criterion . . . . . . . . . 6.33 Eect of the mean of steel diameter for the displacement criterion . . . . . . . . . . . . 6.34 Eect of numbers of transversal and longitudinal bars for the displacement criterion . . 6.35 Eect of stirrups for the displacement criterion . . . . . . . . . . . . . . . . . . . . . . . 6.36 Eect of boundary conditions for the displacement criterion . . . . . . . . . . . . . . . . Horizontal and vertical cables considered in the reliability analysis of containment building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.43 Variation of the average tensile stress in term of height . . . . . . . . . . . . . . . . . . 6.44 Mean and COV of height in term of the number of simulations . . . . . . . . . . . . . . 6.45 Histogram of tension zone height samples obtained with initial model and meta-model for 1000 simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.46 CDF of tension zone height samples obtained with initial model and meta-model for 1000 simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.47 PDF of tension zone height samples obtained with initial model and meta-model for 1000 simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.48 Comparison of initial model and meta-model output samples using QQ-plot . . . . . . . 6.49 Failure probability and reliability index estimated with dierent probabilistic methods depending on the number of points of the design of experiments . . . . . . . . . . . . . 6.50 Convergence at level 0.95 of the estimate of failure probability for dierent probabilistic simulation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Chapter 1
	Introduction
	1.1 Background

6.37 Eect of impact position for the displacement criterion . . . . . . . . . . . . . . . . . . 6.38 Containment building [161] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.39 Geometry of the containment building of the Flamanville nuclear power plant . . . . . 6.40 The selected zone of the containment building to be modeled . . . . . . . . . . . . . . . 6.41 Mesh of the selected zone of study of the containment building . . . . . . . . . . . . . . xix 6.42 6.51 Importance factors of containment random variables using quadratic combination method and FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.52 Importance factor of concrete shrinkage coecient in term of number of points in the design of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.53 Comparison of initial model and meta-model for dierent number of points in the design of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 2 . 2

 22

			: Impact loading protocol [84]
	Impact #	1	2	3	4	5	6	7	8	9	10
	Drop-weight mass (kg)	150	180	210	240	240	270	270	300	300	300
	Amongst the eight slabs, only impact tests of the four RC square slabs of 1.80 m width and
	130 mm thickness are presented in the following section. The slabs were reinforced with top and
	bottom layers with equal amount of reinforcement in x and y directions (ρ x = ρ y ). Upper and lower
	reinforcing layers were joined with steel links in the corners and the center regions of slabs to prevent
	reinforcement movement during casting. Figure 2.18.a presents the geometry and reinforcement

Table 2 .

 2 3: Values of ε c0 parameter for high strength concrete[START_REF] Ecp | Commentary eurocode 2[END_REF]

	f ck (MPa)	50	60	70	80	90
	ε c0 (10 -3 ) 0.807 0.579 0.338 0.221 0.070

and the descending branch of stress-strain curve for high strength concrete (f ck ≥ 50 MPa) should be formulated by:

  is the parameter for high strength concrete and its value can be taken from Table2.3. Concrete compressive strength can be estimated from its characteristic value by f c = f ck + 8.

Pavlovic et al. curve

[START_REF] Pavlovic | Bolted shear connectors vs. headed studs behaviour in push-out tests[END_REF]

Pavlovic et al. found that the EC2 compressive stress-strain curve is not accurate to estimate the crushing strength of concrete when high crushing strains are expected, as in the case of slabs subjected to impact. Plasticity curve in EC2 considers concrete compression behavior only up to the ultimate strain ε cu 1 limited to a maximum value of 0.0035, which may lead to unrealistic overestimation of concrete strength. Consequently, Pavlovic et al.

suggested an extension of the compressive stress-strain curve beyond the EC2 ultimate strain and proposed a new value. The extension was performed with a sinusoidal descending curve between two points, the rst point corresponds to EC2 ultimate strain (ε cu 1 , f cu 1 ) while the second is the end of the sinusoidal part at Pavlovic et al. ultimate strain (ε cu 2 , f cu 2 ). The sinusoidal part is dened by the following equation:

  Wahalathantri et al. selected the experimentally validated tensile stiening model of concrete proposed by Nayal and Rasheed[START_REF] Nayal | Tension stiening model for concrete beams reinforced with steel and frp bars[END_REF] in order to dene the stress-strain

curve of concrete in tension. Nayal and Rasheed model consists of two linear descending parts and has the ability to accurately simulate the response during primary and secondary cracking stages. However, Wahalathantri et al. indicated that using Nayal and Rasheed curve for damaged plasticity model under tension in Abaqus leads to runtime errors due to a sudden vertical stress drop at the cracking tensile strain ε cr . To avoid this problem, Wahalathantri et al. introduced a tensile stress-strain curve with three linear descending parts characterized by a gradual reduction in stress:

  the current study, the platform OpenTURNS (version 13.2) is used under Linux environment to perform reliability approach for several types of RC structures. The name OpenTURNS is for open source Treatment of Uncertainty, Risk 'N Statistics. OpenTURNS was developed as a C++ library and a Python TUI by EDF R&D, Airbus Group and Phimeca Engineering at the beginning of 2005, with the collaboration of EMACS in 2014 [14]. OpenTURNS is an open source software dedicated to uncertainty propagation by probabilistic methods, it includes a set of ecient mathematical methods

Table 4 .

 4 1: Typical limit states for structures[START_REF] Melchers | Structural reliability analysis and prediction[END_REF]

	Limit state type	Description	Examples
	Ultimate (safety)	Collapse of all	Tipping or sliding, rupture, progressive
		or part of structure	collapse, plastic mechanism, instability,
			corrosion, deterioration, re, fatigue
	Damage		Excessive or premature cracking, deformation
	(often included in above)		or permanent inelastic deformation
	Serviceability	Disruption of normal use	Excessive deections, vibrations, local
			damage, etc.

  In addition, FORM does not give the same results with respect to IS for both methodologies proposed. In general, FORM overestimates failure probability when coupled with the nite element model, but shows good accuracy when coupled to the approximated model. Goulbzouri et al. recommended the use of IS in the case of nite element model, although it is hugely time consuming in comparison to FORM. Goulbzouri et al. indicated also the importance of specifying appropriate probability density functions to model uncertainties and that is not sucient to describe random variables only by mean values and COV. However, the choice of PDF has a reduced eect when surface response based reliability analysis is performed. It should be noted that these conclusions could not be generalized and should be used with attention for other problems when assessing failure probability.

Table 4 .

 4 2: Summary on compressive strength of concrete according to various references

	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.
		f c (M P a)	f c (M P a)
	Mirza et al. (1979) [123]	20.68	19.03	0.18	Normal	-
		27.58	23.37	0.18	Normal	-
		34.47	27.77	0.15	Normal	-
	Jonsson (1992) [91]	-	27.77	0.18	Normal	[123]
	Val et al. (1997) [177]	30	38	0.13	Normal	-
		-	f c + 8	8 1.645fcm	-	[30]
	Sun (1999) [168]	40	-	0.16	Normal	[115]
	Braverman et al. (2001) [25]	27.58	24.49	0.16	Normal	[58, 115]
	Low and Hao (2001) [110]	-	51.2	0.11	Normal	[123]
	Low and Hao (2002) [111]	-	84.08	0.11	Normal	[123]
	Firat and Yucement (2008) [62]	C14 to C30*	29.87	0.105	-	-
	Wisnieski et al. (2009) [182]	30	30	0.12	Normal	[144, 145, 181]
	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.
		f c (psi)	f c (psi)
	Choi and Kwon (2000) [35]	-	0.675f ck + 1100 0.176	Normal	[123, 126, 139]

* TS Turkish standards: Requirements for design and construction of RC structures 4.6.1.2 Tensile strength of concrete

Table 4 .

 4 3: Summary on tensile strength of concrete according to various references

	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.
		f c (M P a)	f t (M P a)
	Mirza et al. (1979) [123]	20.68	2.11		0.18	Normal	-
		27.58	2.34		0.18	Normal	-
		34.47	2.52		0.18	Normal	-
	Val et al. (1997) [177]	-	α ft f c	2/3	Eq.(4.34)	Normal	[4]
		30	α ft = 0.3 COV α f = 0.15 Normal -
	Braverman et al. (2001) [25] 27.58	2.46		0.18	Normal	[58, 115]
	Choo et al. (2008) [37]	-	0.18f 0.7 c	-	-	-
	Pandher (2008) [153]	-	f c /10	-	-	-
	Wisnieski et al. (2009) [182] 30 (C = 0.7) 2.0		0.2	Normal	[144, 145, 181]
		50 (C = 0.7) 2.85		0.2	Normal
	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.
	Choi and Kwon (2000) [35]	f c (psi) -	f t (psi) 8.3 √ f ck	0.218	Normal	[123, 139]
	Ref.	Tension	Mean of	COV	PDF	Main Ref.
		stiening	β t
	Choi and Kwon (2000) [35]		3.0		0.11	Normal	[139]
	Val et al. (1997) [177]		0.6		0.15	Normal	[30]

Table 4 .

 4 4: Summary on Young's modulus of concrete according to various references

	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.
		f c (M P a)	E c (M P a)
	Jonsson (1992) [91]	27.77 -	24530 5500 √	f c	0.1 -	Normal -	[123] [43]
	Val et al. (1997) [177]	30	-			0.08-0.1	Normal	[123]
		-	α E (0.1f c ) 1/3	Eq.(4.40)	-	[30]
		30	α E = 2.15 × 10 4 COV α E = 0.05 Normal -
	Braverman et al. (2001) [25] 27.58	26200			0.18	-	[58, 115]
	Low and Hao (2001) [110]	51.2	31200			0.1	Normal	[123]
	Low and Hao (2002) [111] Choo et al. (2008) [37]	84.08 -	38500 4.73 √	0.1 f c (GP a) -	Normal -	[123] -
	Wisnieski et al. (2009) [182] 30	33000			0.08	Normal	[144, 145, 181]
		50	37000			0.08	Normal
	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.
		f c (psi)	E c (psi)
	ACI (1992) [11] Choi and Kwon (2000) [35]	--	33ρ 1.5 c f 0.5 c 60400 √ f ck	-0.119	-Normal	-[123, 139]

Table 4 .

 4 5: Summary on yield and ultimate strengths of steel according to various references

	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.
	Mirza and MacGregor (1979) [125]	415	460	0.09	Beta	[126]
		27.58	23.37	0.18	Normal	-
		34.47	27.77	0.15	Normal	-
	Jonsson (1992) [91]	-	445.34	0.093	LogNormal	[129]
	Braverman et al. (2001) [25]	415	455	0.1	LogNormal	-
	Bournonville et al. (2001) [23]	-	-	-	Beta	-
	Low and Hao (2001) [110]	-	556	0.08	Normal	[126]
	Low and Hao (2002) [111]	415	-	-	Normal	[126]
	Nowak and Szeszen (2003) [145]	-	595	0.08	Normal	-
	Firat and Yucement (2008) [62]	-	0.9f y	0.09	-	-
	Wisnieski et al. (2009) [182]	500	500	0.05	Normal	[144, 145, 181]
	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.
	Mirza and MacGregor (1979) [125]	-	1.55f y	-	-	[126]
	Wisnieski et al. (2009) [182]	500	575 (C = 0.85) 0.05	Normal	[144, 145, 181]

f y (M P a) f y (M P a) f y (M P a) f u (M P a)

Table 4 .

 4 6: Summary on elasticity modulus of steel according to various references

	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.
		f y (M P a)	E s (M P a)
	Mirza and MacGregor (1979) [125]	-	200000	0.033	Normal	[126]
	Mirza et et al. (1980) [124]	-	specied value	0.033	Normal	-
	Jonsson (1992) [91]	445.34	200000	0.033	Normal	[129]
	Ref.	Nominal of	Mean of	COV	PDF	Main Ref.

Table 4 .

 4 7: Summary on dimensions of RC members according to various references

	Ref.	Mean of	COV	PDF	Main Ref.
	Mirza et al. (1980) [124]	nominal	-	Normal	-
	MacGregor et al. (1983) [115]	nominal	-	Normal	-
	Jonsson (1992) [91]	beam width	0.013	Normal	[129]
		= 50 cm
		beam width	0.005	Normal
		= 13 cm
	Udoeyo and Ugbem (1995) [176]	150 to 180 mm	0.0183 to 0.0689	-	-
	Low and Hao (2001) [110]	1x3x0.2 m	0.05	Normal	[127]
	Low and Hao (2002) [111]	as designed	0.03	Normal	[127]
	Wisnieski et al. (2009) [182]	Slab thickness	0.04	Normal	[144, 145, 181]
		= 25 cm
	Ref.	Mean of	COV	PDF	Main Ref.
		d A
	Mirza and MacGregor (1982) [128]	1.01d A	0.04	LogNormal	-
	Jonsson (1992) [91]	1.0048d A	0.02	LogNormal	-
		35.34 mm	0.024	Normal	[126]
	Ref.	Mean of	COV	PDF	Main Ref.
		c
	Braverman et al. (2001) [25]	4.45 cm	0.36	-	-
	Jonsson (1992) [91]	4.85	0.087	Normal	[129]
	simulations are generated and the structure response is evaluated for each simulation. Monte Carlo
	is the most simple simulation technique. Special methods, such as importance sampling, directional
	simulation and Latin hypercube sampling are developed to optimize the strategies of sampling in the
	aim of reducing the computational eort related to the number of calls of the external deterministic
	model.

A detailed step-by-step procedure for creating FE models of impacted slabs with Abaqus is described in this chapter. Abaqus/CAE is used to visualize the FE model geometry and properties, as well as to view results. However, Abaqus Python edition is used to create the model since a parameterbased model is denitely required to perform reliability analyses that necessitate a written code of the deterministic model to be coupled with probabilistic methods. The model Python script includes variation of geometric and mesh sizes, material properties, number of steel bars and initial conditions. Python script permit to easily take into consideration variations of the model geometric parameters and to generate mesh in term of these variations which is not possible with an input le (.inp).

Table 5 .

 5 2: Nonlinear mechanical properties of concrete and steel materials for Chen and May slabs

	Slab	Concrete

Table 5 .

 5

		3: Parameters of the damaged plasticity model
	Parameter	Dilation angle ψ	Eccentricity	f b /f c	K c	Viscocity µ
	Value	38	0.1	1.12	0.0666	0.0
		Table 5.4: Compressive concrete behaviors for CDP model
	Symbol		Curve name			Ref.
	C1		Desayi and Kirshnan curve		[49]
	C2		Madrid parabola			[31]
	C3		Eurocode 2 curve			[4]
	C4		Pavlovic et al. curve			[154]
	C5		Chinese code curve			[3]
	C6		Wang and Hsu curve			[180]
	C7		Majewski curve			[116]
	C8		Kratzig and Polling curve		[99]

Table 5 .

 5

	5: Tensile concrete behaviors for CDP model
	Symbol	Curve name	Ref.
	Stress-strain curve
	T1	Wang and Hsu curve (n=1)	[180]
	T2	Wang and Hsu curve (n=0.5)	[180]
	T3	Wang and Hsu curve (n=1.5)	[180]
	T4	Chinese code curve	[3]
	T5	Wahalathantri curve	[178]
	T6	Kratzig and Polling curve	[99]
	Stress-crack opening curve
	D1	Hillerborg et al. curve	[81]
	D2	Haidong curve	[78]
	D3	Tajima curve	[170]
	Fracture energy
	G1	Hordijk energy	[83]
	G2	CEB-FIP energy	[2]
	G3	Oh-Oka et al. energy	[146]

Table 5 .

 5

		6: Idealized stress-strain curves for reinforcing steel
	Symbol	Curve name	Ref.
	S1	Theodorakopoulos and Swamy curve	[173]
	S2	Taqieddin curve	[171]
	S3	Kratzig and Polling curve	[28]
	S4	Cao et al. curve	[99]
	S5	Belarbi and Hsu curve	[16]

Table 5 . 7

 57

	Concrete	2500	35000	0.2
	Steel	7850	200000	0.3

: Linear mechanical properties of concrete and steel materials for the slab in nuclear plant Material Density kg/m 3 Young's modulus (MPa) Poisson's ratio

Table 5 .

 5 8: Nonlinear mechanical properties of concrete and steel materials for the slab in nuclear

	plant
	Slab	Concrete

Table 6 .

 6 2: Importance factors of beam random variables

	Lb	hB1	dbar	ExcA E c	ν c	E s ν s F orce1 f max
	Quadratic	0.3529	0.3228	0.0128	0.0124	0.0203	0.0	0.0032	0.0	0.0396	0.2358
	combination
	FORM	0.3753	0.4015	0.0243	0.0236	0.0223	0.0	0.0061	0.0	0.047	0.0997

Table 6 .

 6 3: Stress and resistance variables of the cantilvever beam and their variability inuence

	Random Variable	µ i	Unit	Si	Type	| Si × COV i |	Inuence
	Lb	2.75	m	-2.07	Stress	0.207	high
	hB1	0.25	m	2.01	Resistance	0.201	high
	dbar	0.02	m	0.0069	Resistance	0.0007	low
	ExcA	0.1	m	0.293	Resistance	0.0293	low
	E c	25	GPa	0.499	Resistance	0.0499	low
	ν c	0.2	-	-0.0006	Stress	0.0000	low
	E s	200	GPa	0.195	Resistance	0.0195	low
	ν s	0.3	-	-0.0001	Stress	0.0000	low
	F orce1	1000	N	-0.693	Stress	0.0693	low
	f max	0.0015	m	1.69	Resistance	0.169	high

Table 6 .

 6 4: Omission factors of the cantilvever beam variables and their inuence on P f

	Random Variable	β X i	γ i	Inuence
	Lb	2.7785	1.309	high
	hB1	2.8962	1.365	high
	dbar	2.1564	1.016	medium
	ExcA	2.1559	1.016	medium
	E c	2.1467	1.012	medium
	ν c	2.1221	1.000	low
	E s	2.8735	1.354	high
	ν s	2.1221	1.000	low
	F orce1	2.1811	1.028	medium
	f max	2.1220	1.000	low

Table 6 .

 6 5: Variability of the cantilever beam variables according to their mean and standard deviation

	values
	Random Variable	Inuence of the mean value	Inuence of the standard deviation value

  i inuence µ i inuence σ i inuence inuence Lb

							Inuence
	Lb	5.7891	9.9750	high	1.4167	1.1869	high
	hB1	1.5688	4.0158	high	1.9994	1.1435	high
	dbar	4.0489	6.2843	high	2.0344	1.0806	medium
	ExcA	4.0182	6.3017	high	2.0358	1.7097	medium
	E c	3.1490	4.3862	medium	2.0475	1.7433	medium
	ν c	2.1204	2.1182	low	2.1221	2.1221	low
	E s	2.8766	4.0282	low	2.0987	2.0092	low
	ν s	2.1219	2.1211	low	2.1221	2.1221	low
	F orce1	0.4860	0.8769	high	1.9885	1.6975	medium
	f max	3.8401	5.4505	high	1.9531	1.3468	high
	Table 6.6: Summary of sensitivity studies for the cantilever beam
	Random Variable	Mechanical sensitivity	Statistical sensitivity		Total
		Variable type	Inuence
		Stress	high	high	high	high	high
	hB1	Resistance	high	high	high	high	high
	dbar	Resistance	low	medium	high	medium	medium
	ExcA	Resistance	low	medium	high	medium	medium
	E c	Resistance	low	medium	medium	medium	medium
	ν c	Stress	low	low	low	low	low
	E s	Resistance	low	high	low	low	low
	ν s	Stress	low	low	low	low	low
	F orce1	Stress	low	medium	high	medium	medium
	f max	Resistance	low	low	high	high	medium
	6.2.4.2 Distribution analysis

ζ

Table 6 .

 6 7: Results of Kolmogorov-Smirnov test to verify beam displacement distribution

Table 6 .

 6 8: Failure probability estimated with dierent methods for all the cases studied of the cantilever beam (number of random variables=2)

	Case	Random variables	Distribution	COV	FORM	SORM	MC	IS	DS	LHS
	1	hB1, Lb	Lognormal	0.05	3.26E-06	3.03E-06	6.00E-06	2.86E-06	3.29E-06	2.00E-06
	2	hB1, Lb	Lognormal	0.1	0.01176	0.01098	0.01004	0.00967	0.01237	0.011486
	3	hB1, Lb	Lognormal	0.2	0.12427	0.11593	0.12273	0.12656	0.24328	0.11294
	4	hB1, Lb	Lognormal	0.3	0.21348	0.19846	0.18263	0.18459	0.21168	0.19014
	5	hB1, Lb	Lognormal	0.5	0.29791	0.27239	0.24675	0.25535	0.21762	0.21467
	6	hB1	Lognormal	0.1	0.13678	0.13414	0.12865	0.13254	0.15701	0.13162
		Lb	Lognormal	0.3
	7	hB1, Lb	Normal	0.1	0.01147	0.01072	0.00974	0.01036	0.00982	0.01115
	8	hB1	Normal	0.1	0.01308	0.01265	0.01199	0.01274	0.01249	0.01334
		Lb	Lognormal	0.1
	9	hB1	Lognormal	0.1	0.00975	0.00885	0.00828	0.00809	0.01081	0.00852
		Lb	Normal	0.1

Table 6 .

 6 9: Failure probability estimated with dierent methods for all the cases studied of the cantilever beam (number of random variables=3)

	Case	Random variables	Distribution	COV	FORM	SORM	MC	IS	DS	LHS
	10	hB1, Lb	Lognormal	0.05	3.45E-06	3.26E-06	6.00E-06	3.28E-06	3.72E-06	2.32E-06
		f max	Lognormal	0.05
	11	hB1, Lb	Lognormal	0.1	0.01470	0.01378	0.01291	0.01717	0.01316	0.01318
		f max	Lognormal	0.1
	12	hB1, Lb	Lognormal	0.2	0.13781	0.12905	0.11029	0.10655	0.11422	0.13766
		f max	Lognormal	0.2
	13	hB1, Lb	Lognormal	0.3	0.23292	0.21762	0.22649	0.21090	0.18678	0.18043
		f max	Lognormal	0.3
	14	hB1, Lb	Lognormal	0.5	0.32725	0.30371	0.28458	0.27807	0.38822	0.34040
		f max	Lognormal	0.5
	15	hB1	Lognormal	0.1	0.13937	0.13684	0.11772	0.25863	0.14539	0.20151
		Lb	Lognormal	0.3
		f max	Lognormal	0.1
	16	hB1, Lb	Normal	0.1	0.01421	0.01369	0.01446	0.01466	0.01322	0.01313
		f max	Normal	0.1
	17	hB1	Normal	0.1	0.01561	0.01554	0.01519	0.01637	0.01997	0.01704
		Lb	Lognormal	0.1
		f max	Normal	0.1
	18	hB1, Lb	Normal	0.1	0.01449	0.01356	0.01298	0.01349	0.01311	0.01328
		f max	Lognormal	0.1

Table 6 .

 6 10: Failure probability estimated with dierent methods for all the cases studied of the cantilever beam (number of random variables=7)

	Case	Random variables	Distribution	COV	FORM	SORM	MC	IS	DS	LHS
	19	hB1, E c	Lognormal	0.05	3.76E-06	3.61E-06	6.00E-06	3.58E-06	3.94E-06	2.59E-06
		dbar, ExcA, E s	Lognormal	0.05
		Lb	Lognormal	0.05
		F orce1	Lognormal	0.05
	20	hB1, E c	Lognormal	0.1	0.01798	0.01557	0.01648	0.01899	0.01643	0.01632
		dbar, ExcA, E s	Lognormal	0.1
		Lb	Lognormal	0.1
		F orce1	Lognormal	0.1
	21	hB1, E c	Lognormal	0.2	0.14949	0.13680	0.11741	0.11175	0.13177	0.14121
		dbar, ExcA, E s	Lognormal	0.2
		Lb	Lognormal	0.2
		F orce1	Lognormal	0.2
	22	hB1, E c	Lognormal	0.3	0.24779	0.22652	0.18385	0.21789	0.19206	0.21644
		dbar, ExcA, E s	Lognormal	0.3
		Lb	Lognormal	0.3
		F orce1	Lognormal	0.3
	23	hB1, E c	Lognormal	0.5	0.34869	0.33052	0.26618	0.71120	0.24515	0.28015
		dbar, ExcA, E s	Lognormal	0.5
		Lb	Lognormal	0.5
		F orce1	Lognormal	0.5
	24	hB1, E c	Lognormal	0.1	0.04633	0.04056	0.03791	0.03459	0.04606	0.03702
		dbar, ExcA, E s	Lognormal	0.3
		Lb	Lognormal	0.1
		F orce1	Lognormal	0.2
	25	hB1, E c	Normal	0.1	0.01798	0.01634	0.01640	0.01867	0.01755	0.01659
		dbar, ExcA, E s	Normal	0.1
		Lb	Normal	0.1
		F orce1	Normal	0.1
	26	hB1, E c	Lognormal	0.1	0.01716	0.01692	0.01648	0.01567	0.01796	0.01745
		dbar, ExcA, E s	Lognormal	0.1
		Lb	Normal	0.1
		F orce1	Normal	0.1
	27	hB1, E c	Normal	0.1	0.01836	0.01709	0.01651	0.01766	0.01727	0.01621
		dbar, ExcA, E s	Normal	0.1
		Lb, F orce1	Lognormal	0.1

Table 6 .

 6 1, 0.2, 0.3, 0.5 and mixed distributions (mixed term refers to cases where random variables are modeled with dierent COV values). The mean values and COV of probabilities ratio are also compared to MC results in order to estimate the accuracy and the precision of probabilistic methods with respect to the COV of random variables.Cases 1, 10 and 19 are considered in this step, they represents the cases of a COV equal to 11: Failure probability estimated with dierent methods for dierent types of probability

	distribution (number of random variables=2)
	Type	Displacement		P f
		µ (mm) σ (mm) COV FORM	SORM	MC	IS	DS	LHS
	Beta	0.614	0.254	0.42	0.01154	0.01060	0.01069	0.01133	0.01187	0.00968
	ChiSquare	0.614	5.206	8.48	0.44222	0.44215	0.39102	0.35001	0.49413	0.34021
	Exponential	0.614	0.254	0.42	0.02135	0.01763	0.01594	0.01842	0.01702	0.02245
	Gamma	0.614	0.254	0.42	0.01157	0.01079	0.00941	0.00955	0.01061	0.01185
	Gumbel	0.614	0.254	0.42	0.01533	0.01408	0.01192	0.01502	0.03935	0.01499
	Histogram	0.614	0.415	0.72	0.04571	-	0.02605	0.03206	-	0.027282
	Laplace	0.614	0.254	0.42	0.01054	0.01438	0.01551	0.01286	0.01585	0.01249
	Logistic	0.614	12.79	20.84	0.47817	0.47311	0.07802	0.06679	0.04044	0.08630
	Lognormal	0.614	0.254	0.42	0.01176	0.01098	0.01004	0.01001	0.01237	0.00892
	NonCentral	0.614	7.92	12.89	0.46761	0.46361	0.12517	0.10236	-	0.10101
	Student
	Normal	0.614	0.254	0.42	0.01147	0.01071	0.00974	0.01216	0.00982	0.00838
	Rayleigh	0.614	1.33	2.16	0.26306	0.22506	0.25424	0.29613	0.26058	0.27137
	Student	0.614	12.22	19.91	0.46794	0.46302	0.11042	0.09995	0.06498	0.09391
	Triangular	0.614	0.809	1.32	0.24727	0.23364	0.21294	0.19674	0.21188	0.21763
	Truncated	0.614	0.254	0.42	0.011471	0.10691	0.00974	0.01215	0.00982	0.00838
	normal
	Uniform	0.614	1.14	1.86	0.35463	0.32210	0.26335	0.22915	0.24028	0.24516
	Weibull	0.614	0.254	0.42	0.01621	0.01694	0.01567	0.01494	0.01888	0.01735

Table 6 .

 6 12: Random variables of the RC slab and their statistical descriptions (elastic behavior)

	Variables	PDF	Mean	COV	Ref.
	Geometric properties of the slab
	Length, a	Normal	8.1 m	0.05	[127]
	Width, b	Normal	4.85 m	0.05	[127]
	Thickness, e	Normal	0.5 m	0.05	[127]
	Impactor variables
	Radius, r	Normal	15 cm	0.05	[38]
	Mass, m	Lognormal	3600 kg	0.05	[38]
	Velocity, v	Lognormal	7.7 m/s	0.1	[38]
	Concrete properties
	Young's modulus, E c	Lognormal	35 GPa	0.1	[123]
	Density, ρ c	Lognormal	2500 kg/m 3	0.1	lack of information
	Reinforcement properties
	Bars diameter, d A	Lognormal	20 mm	0.1	lack of information

Table 6 .

 6 13: Comparison of displacement of slab at the impact point for dierent deterministic models and dierent values of velocity (elastic behavior) As previously discussed in section 4.4.3, approximation methods such as FORM and SORM are based on an approximation of the limit sate function, while simulation methods such as MC and IS are based on a number of samples of random variables to calculate failure probability. These methods are combined with the MSM and results are compared

	Velocity	FEM	TP		MSM
	(m/s)	u 3 (mm)	u 3 (mm)	Error (%)	u 3 (mm)	Error (%)
	7.7	14.85	14.11	4.95	14.62	1.54
	10.5	18.29	17.97	1.74	18.24	0.27
	13.2	21.51	21.23	1.32	21.57	0.27
	15.5	24.26	24.09	0.67	24.23	0.14
	19.7	29.24	28.88	1.24	29.19	0.17
	24.4	34.32	33.36	1.94	34.13	0.55

Figure 6.20: Failure probability with dierent deterministic models and displacement criterion 6.3.4.2 Comparison of probabilistic methods For this application, two types of probabilistic methods are used and compared, namely approximation methods and simulation methods.

Table 6 .

 6 [START_REF] Baudin | An industrial software for uncertainty quantication in simulation[END_REF] shows that FORM coupled with the MSM gives the closest values of failure probabilities

to those estimated with the numerical model. MC overestimates failure probabilities for velocities equal to 10.5 and 13.2 m/s, this may be related to the fact that MC necessitates a huge number of simulation for low probabilities range and that the transformation factor is considered as constant parameter for all samples generated by MC and is not calibrated in terms of slabs variables, such as slab dimensions, material densities and bars diameter.

Table 6 .

 6 14: Comparison of probabilistic methods combined with the MSM for the displacement

	criterion
	Velocity	FEM	MSM
	(m/s)	FORM	FORM	SORM	MC	IS
	7.7	0.00016	0.00021	0.00001	-	0.00043
	10.5	0.03469	0.06715	0.06542	0.023611	0.06924
	13.2	0.44949	0.48965	0.49107	0.75781	0.47835
	15.5	0.86815	0.83744	0.99508	0.84211	0.98103
	19.7	0.99906	0.99149	0.99999	0.96552	-
	24.4	1.0	1.0	1.0	1.0	1.0
	6.3.4.3 Comparison of failure criteria

Table 6 .

 6 [START_REF] Bazant | Crack band theory for fracture of concrete[END_REF]: Deterministic variables, random variables and their statistical descriptions

	Variables	Nominal	Mean		COV	PDF	Ref.
	Geometric properties of the slab
	Length, a	8.1 m	Deterministic		[82, 168]
	Width, b	4.85 m	Deterministic		[82, 168]
	Thickness, e	0.5 m	0.5		0.05	Normal	[127]
	Position and size of reinforcement
	Concrete cover, c	3.5 cm	Deterministic		-
	Bars diameter, d A	20 mm	20		0.024	Normal	[126]
	Concrete properties
	Compressive strength, f c	40 MPa	0.8f c		0.15	Normal	[123]
	Density, ρ c	2500 kg/m 3	Deterministic		-
	Tensile strength, f t Young's modulus, E c	3.5 MPa 35 GPa	f c /10 5500 √	f c	0.2 0.08	Normal Normal	[153] [91, 123]
	Reinforcement properties
	Yield strength, f y	500 MPa	500		0.05	Normal	[126]
	Modulus of elasticity, E s	200 GPa	Deterministic		[110]
	Impactor variables
	Radius, r	15 cm	Deterministic		lack of information
	Mass, m	3600 kg	3600		0.1	Lognormal	lack of information
	Velocity, v	7.7 m/s	7.7		0.1	Lognormal	lack of information
	6.4.2.3 Dimensions

Table 6 .

 6 [START_REF] Belarbi | Constitutive laws of concrete in tension and reinforcing bars stiened by concrete[END_REF]: Factorial experiment factors and mass-spring model parameters

	Factors	e	d A	v	m
	Low level	0.3 m	6 mm	2.5 m/s	2000 kg
	High level	0.7 m	30 mm	13.2 m/s	6000 kg
	m s	k s	c s	k c	c c
	kg	N/m	N.s/m	N/m	N.s/m
	0.015M slab	f (e, d A , v, m)	1.7e 5	2e 9	1e 5

Table 6 .

 6

	17: Failure probability with FE and mass-spring models and displacement criterion
			FE model	Mass-spring model
	v (m/s)	m (kg)	FORM	IS	MC
	Constant impact energy
	7.7	3600	0.6824	0.6731	0.6557
	10.5	1936	0.5475	0.5167	0.5311
	13.2	1225	0.4150	0.7539	0.3895
	Constant mass
	5.0	3600	0.1861	0.2152	0.1853
	10.5	3600	0.7073	0.7288	0.7226
	Constant velocity
	7.7	2000	0.2992	0.2976	0.2646
	7.7	5000	0.7247	0.6883	0.6927
	6.4.4 Discussion of results of slab reliability (Nonlinear behavior)
	6.4.4.1 Comparison of strategies

Table 6 .

 6 [START_REF] Berthetrambaud | Structures rigides soumises aux avalances et chutes de blocs: modélisation du comportement mécanique et caractérisation de l'interaction "phénomène-ouvrage[END_REF]: Random variables of the containment building and their descriptions

	Variables	Description	Mean	COV	PDF
	Concrete properties
	E c	Young's modulus of concrete	37 GPa	0.1	Lognormal
	ν c	Poisson's ratio of concrete	0.2	0.1	Lognormal
	Reinforcement properties

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservés

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure 5.24: Comparison of concrete compressive stress-strain curves for Slab2

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure 5.25: Comparison of concrete tensile stress-strain curves for Slab2

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure 5.26: Comparison of concrete tensile stress-strain curves for Slab2

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.2: Displacement and velocity of slab and impactor: v = 5 m/s, m = 8537 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.3: Displacement and velocity of slab and impactor: v = 7.7 m/s, m = 3600 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.4: Displacement and velocity of slab and impactor: v = 10.5 m/s, m = 1936 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservés Figure B.5: Displacement and velocity of slab and impactor: v = 13.2 m/s, m = 1225 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.6: Displacement and velocity of slab and impactor: v = 15.5 m/s, m = 888 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.7: Displacement and velocity of slab and impactor: v = 19.7 m/s, m = 550 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.9: Displacement and velocity of slab and impactor: v = 5 m/s

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.10: Displacement and velocity of slab and impactor: v = 7.7 m/s

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.11: Displacement and velocity of slab and impactor: v = 10.5 m/s

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.12: Displacement and velocity of slab and impactor: v = 13.2 m/s

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.14: Displacement and velocity of slab and impactor: v = 19.7 m/s

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.16: Displacement and velocity of slab and impactor: m = 1000 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.17: Displacement and velocity of slab and impactor: m = 2000 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.18: Displacement and velocity of slab and impactor: m = 3000 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.19: Displacement and velocity of slab and impactor: m = 3600 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.20: Displacement and velocity of slab and impactor: m = 5000 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.21: Displacement and velocity of slab and impactor: m = 6000 kg

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.23: Displacement and velocity of slab and impactor: d A = 0.008 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.24: Displacement and velocity of slab and impactor: d A = 0.01 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.25: Displacement and velocity of slab and impactor: d A = 0.012 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.26: Displacement and velocity of slab and impactor: d A = 0.014 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.27: Displacement and velocity of slab and impactor: d A = 0.016 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.28: Displacement and velocity of slab and impactor: d A = 0.02 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.29: Displacement and velocity of slab and impactor: d A = 0.03 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.31: Displacement and velocity of slab and impactor: ρ c = 1600 kg/m 3

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.32: Displacement and velocity of slab and impactor: ρ c = 2000 kg/m 3

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.33: Displacement and velocity of slab and impactor: ρ c = 2300 kg/m 3

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.34: Displacement and velocity of slab and impactor: ρ c = 2500 kg/m 3

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.35: Displacement and velocity of slab and impactor: ρ c = 3000 kg/m 3

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.36: Displacement and velocity of slab and impactor: ρ c = 3500 kg/m 3

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.37: Displacement and velocity of slab and impactor: ρ c = 4000 kg/m 3

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.38: Displacement and velocity of slab and impactor: ρ c = 4500 kg/m 3

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.40: Displacement and velocity of slab and impactor: e = 0.4 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.41: Displacement and velocity of slab and impactor: e = 0.5 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.42: Displacement and velocity of slab and impactor: e = 0.6 m

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0096/these.pdf © [F. Kassem], [2015], INSA Lyon, tous droits réservésFigure B.43: Displacement and velocity of slab and impactor: e = 0.7 m

In this equation f X (x) is the joint probability density function of the random variables X which represent the uncertainties included in the reliability analysis and g(X) is the limit state function which is related to a specic failure mode. The integration is performed over the failure domain, for which g(X) is smaller or equal to zero. Integral 4.14 is considered as the fundamental equation for reliability analysis, however it is not easy to calculate because of the complexity of the failure domain D f and the joint probability density function f X (x), which may involve a large number of variables. In addition, the estimation of failure probability becomes even more dicult when complex deterministic models such as nite element models are considered. In order to assess failure probability, rst, the limit state function should be dened for a special failure mode in term of input random variables, then several probabilistic methods can be used to solve the integral 4.14:

Direct integration method allows estimating failure probability only in some particular cases where joint density functions have simple shapes.

Numerical integration method may lead to signicant errors for very low probabilities and the integration domain is not always bounded as well.

Probabilistic simulation methods, such as Monte Carlo (MC), Latin Hypercube Sampling (LHS), Directional Simulation (DS) and Importance Sampling (IS), are based on a number of samples of random variables corresponding to failure Approximation methods are based on an approximation of the limit state function and an identication of the most probable point that permits to estimate failure probability. The limit state function is linearized in the First Order Reliability Method (FORM), while it is approximated by a quadratic surface for the Second Order Reliability Method (SORM)

Response surface methods replace the initial deterministic model by an approximation response surface expressed by a function whose values can be computed more easily.

Limit states

The response and failure modes of an engineering structure depend on the type and the magnitude of the applied load, as well as on the strength and stiness of the structure. Melchers [START_REF] Melchers | Structural reliability analysis and prediction[END_REF] indicates that limit states which describe the structural failure modes are generally divided in (Table 4.1):

1. Ultimate limit states which are mostly related to the loss of load-carrying capacity, e.g. excessive plasticity, rupture due to fatigue, deterioration, etc. 2. Damage limit states which are related to loss of strength under repeated loads and can be caused by excessive cracking or permanent inelastic deformation.

3. Serviceability limit states which are related to normal use of the structure and can be expressed in terms of deections or vibrations.

Approximation methods

The application of FORM and SORM rst requires the application of an isoprobabilistic transformation that aims to simplify the joint probability density function while failure probability retains its estimated value. This transformation associates random variables in the physical space with standardized and independent random variables that dene the basic vectors of the standard normal space.

The limit state function should also be expressed in the standard space and turns into H(U).

Then the integral of failure probability in equation 4. The output sample distribution is compared with Beta, Normal, Lognormal, Logistic, Gamma, and Gumbel distributions. The parameters of these distributions are estimated in terms of the output sample statistical moments and veried using Kolmogorov-Smirnov test (see section 4.4.2.2). A pvalue threshold of 0.05 is used and compared to the p-value calculated. The assumed distribution is considered as accepted, and thus the Kolmogorov-Smirnov test succeeds to nd a distribution that ts the output sample, only if the p-value calculated is greater than the threshold. Table 6.7

shows the distributions selected to verify if they t the beam displacement sample, as well as their statistical parameters estimated in terms of the mean and standard deviation of the displacement.

As shown, Kolmogorov-Smirnov test fails for all the distribution types selected, hence none of them is appropriate to describe the beam displacement density. Figure 6.5 provides a graphical comparison of the output sample distribution with the parametric distributions selected and shows that the Gumbel distribution gives the best t. These results are also found using the QQ-plot with quantiles estimated at 95%. The quantile points estimated with Gumbel distribution of parameters α = 4674.8 and β = 0.000532686 are the closest to the diagonal (Figure 6.6).

Comparison of probabilistic methods

The aim of this section is to discuss and compare probabilistic methods proposed in OpenTURNS (see section 4.4.3) in order to examine their accuracy, precision and computational eort. These methods are used to evaluate failure probability of the beam displacement criterion under the inuence of several parameters, such as the number of random variables, their coecient of variation and the type of their probability distribution. Parametric cases considered are detailed in Tables 6.8-6.10 with the corresponding values of failure probability estimated with dierent methods (FORM, SORM, MC, IS, DS and LHS). All cases studied are divided into several subset as following:

3 subset of cases with the same random variables number: 2 random variables (hB1, Lb), 3 random variables (hB1, Lb, f max ), and 7 random variables (hB1, Lb, E c , dbar, ExcA, E s , F orce1).

Concrete may be subjected to several types of delayed strains during its lifetime. In the case where there is no water exchange between the concrete structure and its surrounding environment, 3 types of delayed strains are observed:

Early-age autogenous shrinkage ε re is an important phenomenon in young concrete and is evaluated in term of water/cement ratio. Autogenous strains develops during hardening of concrete in the early days of casting.

Early-age thermal shrinkage ε th occurs due to excessive dierences within a concrete structure or its surroundings.

Basic creep ε f p is dened as the creep that occurs under loading conditions and increases with the increase of stress and duration of loading.

In the case where water exchange occurs, 2 other types of delayed strains of concrete are observed:

Drying shrinkage strain ε rd is a function of the migration of the water through the hardened concrete and develops slowly.

Drying creep strain ε f d represents the extra creep component at drying over the sum of shrinkage and basic creep.

Consequently, the total strain that describes the time dependent behavior of concrete can be expressed as follows:

with ε el is the instantaneous strain component.

Probabilistic model

The variables which may be subjected to statistical variations in containment building can be classied into four categories:

1. Variables related to the concrete properties: Young's modulus and Poisson's ratio of concrete.

2. Variables related to shrinkage in concrete: shrinkage coecient. COV equal to 0.1. This choice is based on the fact that all input variables taken into account in the probabilistic model should have positive random values. In addition, the COV is chosen the same for all variables in order to study the inuence of each variable mean on the estimation of the height corresponding to the tension zone without considering their eect of variation. It has be shown in the reliability analysis of RC cantilever beam that the COV value highly aects the probabilistic response and a variable with a higher COV has a signicant eect on the dispersion of output variables as well as on the estimate of failure probability. Table 6.18 represents the random variables adopted Appendix A Probability distributions in OpenTRUNS 13.2

Beta distribution

with ν is the shape parameters and Γ is the gamma function.

ChiSquare distribution

with r and t are the shape parameters, a and b are the lower and upper bounds, respectively, of the distribution, and B(r, t -r) is the beta function.

Exponential distribution

with λ is the scale parameter and γ is the location parameter.

Gamma distribution

with λ is the scale parameter, γ is the location parameter and k is the scale parameter. Γ is the gamma function.

Gumbel distribution

with α is the scale parameter, β is the location parameter.

Histogram distribution

with H i are the normalized heights of the histogram.

Laplace distribution

with λ is the scale parameter and µ is the location parameter.

Logistic distribution

with β is the scale parameter and α is the location parameter.

Lognormal distribution

with σ l is the shape parameter or the standard deviation of the log of the distribution, γ is the location parameter and µ l is the scale parameter or the median of the distribution.

Normal distribution

x -µ σ 2 (A.10) with µ and σ are the mean and the standard deviation, respectively, of the variable X.

Rayleigh distribution

with σ is the scale parameter and γ is the location parameter.

Student distribution

with B is the beta function, µ describes the most likely value and ν is a measure of dispersion.