Keywords: Thermal break system, thermal dilatation, thermos-hydro-mechanical analysis, cracking, fatigue strength

This work is

Résumé

Pour les bâtiments conçus avec une isolation interne, les ponts thermiques à la jonction entre le plancher et les voiles sont un problème commun, car ils créent une perte de chaleur de l'intérieur du bâtiment. Les systèmes de rupteur thermique (TBS), composés d'éléments structuraux (barres d'armature) et de matériaux isolants, sont utilisés pour réduire cette perte de chaleur et transférer les forces de cisaillement du plancher aux voiles. Le système d'isolation du TBS génère un gap de température entre le plancher et le voile. En conséquence, alors que les voiles sont exposées aux actions climatiques et sont dilatées et contractées à plusieurs reprises, alors que le plancher se rapporte à une température constante et ne présente aucune variation volumétrique, ainsi les TBS sont soumis aux grands déplacements imposés. Cette thèse illustre l'effet de la dilatation et de la contraction thermiques des voiles, qui créent des forces supplémentaires dans les TBS, et par conséquent la fissuration dans les voiles. Un modèle numérique d'un quart d'étage d'un bâtiment est soumis aux actions climatiques à Embrun, en France. Un côté du voile en forme de L est faire face au sud, et l'autre à l'ouest. L'analyse thermique, hydrique et mécanique est réalisée avec le logiciel Castem. Dans analyse thermique et hydrique, la température de l'air, le flux de rayonnement solaire et l'humidité ambiante sont définis à partir des bases de données de Météo France, et sont appliqués sur la surface extérieure des voiles. Les résultats du premier calcul, par analyse thermo-hydro-mécanique (THM) avec la loi comportement élastique pour le béton, confirment l'important de contraint en traction se produit dans le coin des voiles et les éléments du béton à proximité du TBS, au niveau de la connexion 'voile-plancher'. De plus, les TBS qui se trouvent près du coin des voiles présentent le cisaillement horizontal et la force axiale significative et risquent d'être rompus en raison de la résistance à la fatigue. Baser sur ces résultats, un deuxième calcul, qui tient compte de la fissuration dans les voiles par la loi comportement: élastique endommagement pour le béton, est fait pour indiquer des efforts plus réalistes dans les TBSs, et analyser les fissurations dans les voiles.

Mots clés : Rupteur thermique, dilatation thermique, analyser thermo-hydro-mécanique, fissuration, résistance à la fatigue.

Table of contents

Table of contents

Introduction General introduction

In the context of the sustainable development, reduction of the energy consumption for the residential building has been recasted through the European Directive 2010/31/EU [1], and the concept of 'nearly zero-energy building' is raised in this directive. A study on energy performance of residential buildings [2] indicate that the annual energy consumption increases by 38-42% because of the thermal bridges. The thermal bridge is a common issue, which is known as a path for creating heat loss at the connections of the slab-wall and the slab-balcony. In the past decades of the building practice, two insulation systems exist : the external and internal insulation systems. Internal insulation of a buildings is interrupted at the interface of the slab-wall due to the reason of transferring mechanical loads from the slab to the wall system. The discontinuity of the internal insulation is the origin of the thermal bridge, and eventually, lead to increase the energy consumption in the building. Even though the external insulation ignores this configuration, still it cannot avoid the discontinuity at the slab-balcony connection.

Thermal break systems (TBS) are developed for the purpose of solving the thermal bridge, and are composed of insulation material and structural elements (most of the time : rebar). The TBSs are either placed at the slab-wall or slab-balcony connection, and are meant to increase the thermal resistance and to assure the load transferring. The presence of the TBSs generate a temperature gap, which separates the temperature field from the walls to the slab for the case of internal insulation, and from the balcony to the slab for the case of external insulation. The thermal expansion and contraction of the walls are constrained by the insulated part of the building. The displacement restrains then create supplemental forces in the TBSs, and also in the nearby concrete elements. Those forces may be large enough to provoke damage in the TBSs, and the cracking of the surrounding concrete, see in [Fig. 1]. Together with the damage of concrete in tension and compression, the TBSs are imposed with large displacements, and may be submitted to repeated yielding during the life of the building. This repeated yielding is a concern with the consideration of fatigue strength of the TBSs.

Introduction

Aims and scopes

Previous researches related to the TBSs focus on di↵erent scopes, and are varied from the thermal performance to the mechanical behavior under the static and seismic loads. The e↵ect of the climatic actions on the building with internal insulation and equipped with TBSs has never been presented before. The only similar previous studies were di↵erently observed in the thesis of G. Le Bloa [3] and in the paper of P.Heng et al. [4], which focused on the seismic behavior of the TBSs and fatigue resistance of TBS at the slab-balcony junction in buildings with external insulation, respectively. The research of this thesis stands as the first milestone to discover and to post the questions for this innovating building structure.

The objective of this PhD thesis is thus to investigate a building, which is internally insulated and equipped with the TBSs, under the climatic actions. The investigation highlights on the stresses evolution and damage distribution on the walls, as well as the supplemental forces pertained by the TBSs.

Introduction

di↵erent scales, which are : the three dimensional numerical modeling at building scale and the scale of material constitutive law modeling. In numerical modeling, the scale of numerical model corresponding to the TBS is considered as the intermediate scale. The TBSs are not completely modeled by the macro elements (spring) nor the refined model. But the simplification by distributing the shell elements is made due to the similarity of geometry of the steel profiles in the TBS. Moreover, the walls are modeled by the conventional solid elements, but the simplified modeling with solid-shell is made for the slab, and hypothesis of Bernoulli is introduced at the junction of solidshell. Globally, the scale of finite element modeling in this thesis can be considered as the intermediate scale, which balances between the capacity of the calculation and the appropriate degree of the finite elements modeling, which are necessary to provide the reasonable results. Regarding to the scale of phenomenon modelings, two principal phenomenons are the multiphysics, which include the shrinkage and creep models, then the non linearity behavior of the materials such as : concrete and TBS. It is worth to be mentioned that the couplings among thermo-hydro-mechanical analysis is a weak coupling. There is no interaction from the structural e↵ect (crack) to thermal and hydric analysis nor in the opposite way.

On the other hand, the climatic actions exposed on the building are extracted from the database of Méteo France [START_REF] France | [END_REF], and are applied to the exterior surface of the walls with the boundary conditions as air convection and flux radiation. The coe cient of emissivity related to the color of the building is also taken into account.

The decision of 3D modeling at building scale is imposed by the geometrical complexity of the system under the following considerations. Indeed, the movements of the walls interact with the slab through the TBS, and walls interact with each other at every corner of the building cannot be described simply by any 2D models. Besides that, TBSs are submitted both to horizontal shear by elongation and contraction of the walls, as well as, to axial forces in the zone of the corner. The simulation at the building scale is proposed, and is also a challenge to the capacity of the central calculation system. The super computer (72 core, 1.2 To of RAM), which is based at the laboratory of civil and mechanical engineering and in the group of GEOSAX (LGCGM), gives the possibility for performing such calculation. Anyway, the full modeling with solid elements is too expensive in calculation time. A combination of solid and shell elements are applied. Considering the complexity of the problems in this thesis, the modeling is limited to a building with rectangular geometry, and only a quarter of a building's storey is modeled (L-shape wall).

A refined material modeling is mandatory in order to include the delayed strains and hydric variations that have already been underlined. Moreover, a large cracking of the concrete walls is expected and will play evidently an important role in the distribution of forces and displacements in the structure. This refined material modeling again make the simulation become heavy. In order to obtain the desired balance between the accuracy and e ciency, the simplified models of shrinkage and creep are developed.

Even more important, the nonlinear behaviors of the concrete and TBSs are relied on the damage elasticity of µ model of Mazars et al. [START_REF] Mazars | A new 3d damage model for concrete under monotonic, cyclic and dynamic loadings[END_REF] and the classical elasto-plasticity model for stainless steel [START_REF] Ryś | Constitutive modelling and identification of parameters of 316l stainless steel at cryogenic temperatures[END_REF], respectively. A more detailed discussion is given in the outline of the thesis.

A previous study of T.T.H. Nguyen [START_REF] Nguyen | Macroscopic modeling of reinforced concrete joints : Application to thermal break elements subject to earthquake loadings[END_REF] has introduced the TBSs into the numerical model by the spring elements that is already implanted in Castem [START_REF]Code de calcule aux eléments finis cast3m[END_REF]. The macro elements contribute to reduce the calculation load, and also provide adequate degrees of freedom (3 rotations, 3 displacements) for analyzing the TBSs. As mentioned in the multi-scale approach, the TBSs are not completely modeled in the scale of macro elements, a choice of intermediate scale is already explained previously. In addition, here the spring elements are used to connect between the shell on the wall and the shell on the slab.

Introduction

creep models are reconstructed, and the parameters are identified from the experimental results as described in the literature review.

To calculate the shrinkage strains in the numerical simulation, solving hydric analysis is necessary, in which a simplified drying model is applied, then linear shrinkage model is adopted. The creep strain (" creep ) is the stress dependent strain, which is influenced by the concrete's age when the load are applied and the environmental conditions. To include the creep strain into the calculation, the rate type creep law (RTCL), which is a popular numerical method to calculate the incremental creep strain ( " creep ) based on the stress rate, is used.

With the RTCL, two chains of Kelvin Voigt model are considered for representing the basic creep (" bc ) and the drying creep (" dc ), respectively. The parameters of the creep compliance are initially identified from the creep compliance curve of EC2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF] by the fitting method, then are modified by the relative humidity of concrete. Besides that, identifying the parameters from the creep compliance curve of EC2 also give a possibility to obtain a variation of the parameter with time evolution. This could help to introduce aging e↵ect of concrete into the creep calculation. An attention should be payed, the creep is mentioned for many times in this thesis, but in the reality of the calculation, the concrete is not a visco-elastic behavior. A module is implanted and only to calculate the creep strain, but the total sti↵ness of the concrete is not considered with the aging e↵ect.

For the TBS, the classical elasto-plasticity model for the stainless steel is used. The parameters corresponding to the horizontal and vertical behavior of the TBSs are identified from the force-displacements curves of the experiments, which were conducted at the LGCGM. There is no experiment has been performed for observing the axial behavior of the TBSs at the slab-wall connection. Then the axial behavior are calibrated from the stress-strain curve proposed by Eurocode 3 [START_REF]Eurocode 3 : Design of steel structures -part 1-4 : General rules -supplementary rules for stainless steels[END_REF] for the stainless steel.

The final numerical simulation is the combination of the nonlinear constitutive law of the materials to the numerical modeling at building scale. These simulations are detailed in the following chapters, and the grade of complexity in the simulations are commenced from the elastic analysis for Chapter 3 to the elastic damage analysis in Chapter 4. These simulations allow to observe for the results of stress, strain and damage distribution in the walls, as well as the plastification of the TBSs. Moreover, some cases of parametric studies are proposed for understanding the influence of the initial conditions.

-Chapter 3 :

In Chapter 3, the THM analysis with an elastic short term behavior for the concrete and elasto-plasticity for the TBS stands for the first benchmark of the simulation in this thesis. The numerical simulation is based on the combination of the selected mo-dels including shrinkage and creep phenomenons, which are described in chapter 2. Furthermore, those e↵ects are only considered for the walls structure. The developed finite elements modeling at the intermediate scale is also applied. The motivation to start this simulation is also for indicating the conservation condition of the method of analysis.

The steps of the simulation initially start from thermal and hydric analyses. The climatic actions considered in thermal analysis are the air temperature and the solar radiation, and the humidity signal for the hydric analysis. The results of the temperature and relative humidity distribution are used to generate the thermal strain and the shrinkage strain. Both strains are introduced into the mechanical calculation as the input data. The objectives of this analysis are to confirm the correctness of the implanted codes and the response of the simplified finite elements modeling at the building scale under the climatic actions in terms of the stress in the walls and internal forces in the TBSs.

Exposing to 1 year of the climatic actions costs 20 days of calculation time, which is 3 times faster than the full simulation with the µ model presented in Chapter4. Due to this benefit, cases studies with di↵erent initial conditions, in term of the temperature and relative humidity, are performed under the configuration of the THM analysis with elastic behavior for the concrete. The results of the numerical simulation shown that a significant stress level in tension occurs in the concrete, especially at the corner of the walls and the concrete elements nearby the TBSs. Those tensile stresses even exceed the limit strength of concrete in tension, then crackings must take place in the walls. In addition, it is observed that the initial conditions of temperature have influence on the order of magnitude of the stresses in the walls. On the other hand, the horizontal dilatation e↵ect of the walls has no influenced to the stresses in the slab. Moreover, the TBSs that are close to the corner of the walls pertain the maximum internal forces, and exceed the limit strength. Those repeated dilatation and contraction in the walls seem as the cyclic imposed displacement to the TBSs, and should be also concerned by the fatigue strength of the TBSs. The damage accumulation in the TBSs due to the fatigue analysis is performed.

-Chapter 4 :

As summarized in the elastic analysis, the tensile stresses in the walls are important, then a simulation that takes into account crackings in the wall is proposed in Chapter 4. The simulations of THM analysis with elastic damage behavior for the concrete, elasto-plastic behavior for the TBSs and elastic behavior for the reinforcement in the walls, are performed. It should be noticed that the reinforcement distribution is corresponding to a very low reinforcement ratio that does not allow to control the cracking. The minimum reinforcement directly comes from the real reinforcement plan with the configuration of wall-slab connected by the TBSs.

Introduction

In the full simulations, two configurations are selected, which are the roof floor and the intermediate floor. Those configurations are di↵erentiated by imposing the vertical loading, which is corresponding to the self weigh of the upper floors structure to the walls in the simulation, 3 MPa for the intermediate floor and 0.7 MPa for the roof floor. Besides that, the color e↵ect of the building surface is also considered in the simulations, and this e↵ect is included by modifying the radiation signals. The calculation time for the full simulation could go to 2 months for an exposure to the thermal loading of 1 year, and each calculation uses 12 core of the CPU and 150Go of the RAM.

The results corresponding to the intermediate floor case are interesting. Those results presented a uniform damage distribution on the surface of the walls (skin e↵ect), cracks at the corner, and several local damages (vertical cracks) on the interior surface, especially the concrete elements that are surrounding the TBSs, and are close to the corner. The displacements of the full simulations are larger than the one depicted in Chapter 3. Several displacement jumps are observed and those jumps are corresponding to the cracks in the walls. The supplemental displacements and displacement jumps are concerned by horizontal shear and axial force in the TBSs, especially at the corner of the walls. The increasing the horizontal shear leads to increase the damage accumulation in the TBSs. In the worst case of the simulation, the numbers of TBSs, with damage accumulation for 50 years larger than 1, are 3 to 4 times of the results calculated from the elastic analysis in Chapter 3. This observation gives an important information that the elastic analysis is not conservation to estimate the damage accumulation in the TBSs. This should be worried by the design engineer who mostly used the industry engineering software with the elastic behavior for the material for estimating the internal forces in the TBSs.

At the end, general conclusions are provided. Critical remarks for buildings internally isolated and equipped with TBSs are raised. The limitations of the proposed simplified finite element modeling and numerical simulation are also described, and the perspectives for improving the current simulations are proposed. The site measurement of a real building is necessary to confirm with the results of the simulation.

Chapter 1

Literature review 1.1 Introduction

The literature review is mainly composed of 3 parts, which are the analysis of thermo-mechanical behavior, thermo-hydro-mechanical behavior and the studies related to the TBSs. The existent thermo-mechanical studies on the large scale concrete structures (bridge, dam), which are sensitive to the thermal loading, are discussed. The thermal loading cause the total length of the structures expand and contract. Because of the restraint of the structural elements, the thermal e↵ects create stresses that result in damage in the elements, and in the case of concrete, cracking. The climatic actions that are considered for those structures are also explained. A simplified study of a building equipped with the TBSs, and modeled by shell elements with elastic behavior, is presented. However the study is based on an industry engineering software, and only focus on the internal forces in the TBSs. From a scientific point of view, this study is not su cient to describe the response of the building. The nonlinear behaviors of the materials, cracking in the concrete, long term e↵ects (shrinkage & creep) and repeated yielding of the TBS are indeed important factors that have not been taken into account.

In order to include the shrinkage and creep e↵ects, the previous studies of the THM behavior of the concrete are presented. The experiment of CEOS.fr [START_REF]Large beam specimens with restrained shrinkage (rg8)[END_REF] is a good example to explain the e↵ect of the climatic actions on the massive concrete elements. Besides that, experiments of shrinkage and creep are cited, and are used for identifying the parameters of the simplified THM model in the next chapter. The existent numerical models for drying behavior, shrinkage and creep, are presented. Furthermore, the original version of the damage elasticity model of Mazars is explained, then the modified version of µ model for adapting to the cyclic loading is added.

Eventually, the studies related to the TBSs, which cover from the experiments to the numerical model of nonlinear behavior, are described. Furthermore, there exist some 16 Chapter 1 : Literature review studies, which try to describe the influence of the horizontal e↵ect to the TBS, are also introduced.

Analysis of thermo-mechanical behavior 1.2.1 Thermal behavior

The thermal analysis are mandatory for some special structures such : bridges, dams for hydroelectric station, nuclear power plants and concrete pavements. Those structures are exposed to critical environmental conditions [START_REF] Gottsater | Validation of temperature simulations in a portal frame bridge[END_REF][START_REF] Abid | Experimental analysis of temperature gradients in concrete box-girders[END_REF][START_REF] Liu | Temperature response in concrete under natural environment[END_REF][START_REF] Etzion | The thermal behavior of a concrete 'finned' wall in a hot-arid zone[END_REF][START_REF] Nam | E↵ect of daily temperature variations on the continuous deflection profiles of airfield jointed concrete pavements[END_REF]. The nonlinear thermal loading and thermal restraint are able to generate fatal tensile stresses in the structures. Whereas, often the thermal analysis in the residential buildings mostly aims to observe the thermal performance and the energy consumption rather than the thermo-mechanical behavior of the structures [2,[START_REF] Wakili | Experimental and numerical thermal analysis of a balcony board with integrated glass fibre reinforced polymer gfrp elements[END_REF][START_REF] Dumitrescu | The influence of thermal bridges in the process of buildings thermal rehabilitation[END_REF][START_REF] Theodosiou | Thermal bridging analysis on cladding systems for building facades[END_REF]. As a consequence, they are not interested for this work, and only some of the special structures will be focused.

Erik et al. [START_REF] Gottsater | Validation of temperature simulations in a portal frame bridge[END_REF] developed a model for determine the temperature di↵erences between deck and abutments in portal frame bridges, see [Fig. 1.1]. The thermal e↵ects include the air temperature, wind speed, solar radiation and also soil beside the bridge. The author indicated that the temperature distribution on the bridge's deck is greatly influenced by the solar radiation. In this thermal simulation, the solar radiation is presented by the heat flux (q s ), and the coe cient of emissivity is suggested. Besides that, the air temperature is applied at the boundary conditions as convection. Anyways the data of the solar radiation and air temperature are not clearly presented.

slab surrounded by air. The model was later also verified for a hollow concrete box cross-section [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF], and used for temperature simulation in a portal frame bridge by Gottsäter et al. [3]. In the later study, the model was used for both comparison of simulated temperature with thermal load cases in Eurocode, and investigation of material and geometry parameter influence on the thermal load magnitude. Other models for temperature simulation have been described by e.g. Elbadry and Ghali [START_REF]Eurocode 3 : Design of steel structures -part 1-4 : General rules -supplementary rules for stainless steels[END_REF], Mirambell and Aguado [START_REF] Gottsater | Validation of temperature simulations in a portal frame bridge[END_REF], Xia et al. [START_REF] Abid | Experimental analysis of temperature gradients in concrete box-girders[END_REF], Westgate et al. [START_REF] Liu | Temperature response in concrete under natural environment[END_REF] and Zhu and Meng [START_REF] Etzion | The thermal behavior of a concrete 'finned' wall in a hot-arid zone[END_REF].

However, no studies have, to the authors' knowledge, investigated the temperature distribution in a portal frame bridge where the presence of soil next to the structure affects the temperature in the structure. Therefore, no model for temperature simulation has been verified for use in a portal frame bridge. Due to this effect on the temperature distribution by the soil, the already existing models cannot be applied directly and new measurements on the specific bridge type are required.

In this study, the temperature in a portal frame bridge is both measured and simulated. The simulation model developed by Larsson [START_REF]Large beam specimens with restrained shrinkage (rg8)[END_REF] is used, and is validated for use in portal frame bridges with adjacent soil and fill. The comparison between measured and simulated temperatures was conducted for a 12-month period, thereby covering both summer and winter conditions, and temperature measurements were conducted both in the bridge deck and in one of the abutments. The simulation model included a cross section of the concrete structure and asphalt paving, and also the adjacent soil, which affects the temperature in the structure.

Temperature measurements

Temperature was measured in a portal frame bridge located about 1 km west of Lund, Sweden, its exact position being 55°41′58″ N, 13°8′12″ E, from the 6th of December 2016 to the 6th of December 2017. The location is approximately 15 m above sea level and the surroundings are flat fields, except to the east where there is a copse with low trees, see Fig. 1. The bridge deck is at the same level as the ambient ground level, and the road under the bridge is thereby lower than the ambient ground level. The geometry of the bridge section is shown in Fig. 2. The information shown in the figure was obtained from the Bridge and Tunnel Management database (BaTMan) of the Swedish Transport Administration [START_REF] Nam | E↵ect of daily temperature variations on the continuous deflection profiles of airfield jointed concrete pavements[END_REF].

The locations for temperature measurement are shown in Fig. 3. Holes were drilled in which thermocouples were inserted, and thereafter the holes were injected with mortar. Thermocouples of type "K" were used, which consist of one nickel-chrome wire and one nickelaluminum wire. The properties and functionality of thermocouples is described more extensively in e.g. American Society for Testing and Materials [START_REF] Wakili | Experimental and numerical thermal analysis of a balcony board with integrated glass fibre reinforced polymer gfrp elements[END_REF] and Eckert and Goldstein [START_REF] Dumitrescu | The influence of thermal bridges in the process of buildings thermal rehabilitation[END_REF]. The thermocouples were placed in the longitudinal mid-section of the bridge and connected to data loggers of model Microedge Site-Log LPTM-1, using a sample period of 1 min. Measurements were performed in the southern half of the bridge due to the low amount of direct sunshine reaching the southern abutment; this was preferable since sunshine was only included for the top surface of the bridge deck in the simulation model. To completely avoid sunshine reaching the abutments was however difficult at the latitude of the bridge, 56°north, since the positions of sunrise and sunset vary greatly over the year.

Thermal simulations

The model for thermal simulations used in this study was developed by Larsson [START_REF]Large beam specimens with restrained shrinkage (rg8)[END_REF]. It describes heat transfer through conduction, as well as heat flux along surfaces facing air. The latter is described in Eq. (1) where q is the total heat flux in W/m 2 , qs is heat flux from solar radiation, qc is heat flux due to convection and qr is heat flux due to long wave radiation.

(a)

minus the mean temperature of the nodes in the abutment. As the transition from bridge temperature to abutment temperature occurs in the frame corner, as shown in [3], nodes < 1 m from the frame corner were excluded from the comparison in order not to underestimate the temperature difference.

Although the geometry and given weather data do not correspond to a worst case scenario to be considered in a design situation, the results shown in Fig. 8 can be seen as yet another indication that the temperature difference given in Eurocode to be considered in design for crack width limitation [1] may be exaggerated for the specific bridge type. A quasi-permanent load shall be used in crack width limitation, which corresponds to a load exceeded during "a large fraction" of the reference period, and is in some cases seen as the mean or median load value [START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF]. The given value in this load case, 15 °C, is significantly higher than a median value obtained from the data in Fig. 8. This holds true even if the given load value of 15 °C is assumed to be a characteristic value, thus multiplied with a reduction factor, equaling 0.5 in thermal load cases for bridges, to obtain the quasi-permanent value. However, more data are needed in order to determine new realistic load values and corresponding reduction factors, along with considerations of the influence of both geometry and climate data.

When comparing the temperature difference according to the simulation shown in Fig. 8 and the estimated deviation between temperature difference according to measurement and simulation shown in Fig. 7, no clear correlation can be seen between large temperature differences and large deviations. In fact, at the time for the largest positive temperature difference at the end of May, Fig. 7 shows the value 0.3 °C, indicating a larger temperature difference in the simulation than in the measurements. When investigating the corresponding relation at the times for the 5 largest positive and negative temperature differences, only one coincides with a large negative deviation between simulation and measurement, and the mean difference between measured and simulated temperature difference between the parts for these 10 extreme value occasions becomes 0.13 °C. The mean value for the entire period is -0.02 °C. In order to avoid the risk of underestimating the temperature difference, extreme values obtained in future simulations should be increased by a value ΔT. For the bridge investigated in this study, the value can be approximated to ΔT = 1.5 °C, as larger deviations were found to be almost nonexistent in this study. For mean and median values, a smaller increase is sufficient, as large deviations are unlikely to persist during longer periods of time.

Temperature distribution within the model

In Fig. 9, the temperature distributions at the times for the largest negative (bridge deck colder than abutment) and positive (bridge deck warmer than abutment) differences are given. Fig. 9 clearly shows that the temperature change from the deck to the abutment is gradual and occurs in the frame corner area, and that gradients over the bridge deck and abutment can be present simultaneously. The largest positive and negative temperature differences over the abutment cross sections during the simulation period were 7 °C and -6 °C respectively, calculated as the temperature at the side facing air minus the temperature at the side facing the fill. Since previous work has shown that the way the temperature varies within the structural parts has a large impact on the resulting stresses [3], a thermal load case describing the difference in temperature between structural parts could also describe the transition between temperature in the deck and abutment, as well as gradient over the cross sections. The determination of these load values and distributions are however outside the scope of this paper.

It can also be noted in Fig. 9 that the temperature changes rapidly in the lower part of the abutment, just below the level of the road surface under the bridge. This causes a difference between temperature in the abutment and the foundation, which could also be considered in design according to Eurocode EN 1991-1-5, Section 6.1.6 [1]. The difference is caused by the foundation temperature being determined by the surrounding soil to a very large extent, while the abutment is also affected by the air temperature and convection. A more thorough investigation of this temperature difference is however outside the scope of this study, but the results show the need to consider the soil in future simulations, investigating temperature differences in this bridge type.

Conclusions and future work

In this study, a 2D FE-model for simulating temperature in a portal frame bridge with adjacent soil has been evaluated by comparing simulation results with measurements in a bridge during a 12-month period. The focus of the evaluation has been on comparing the temperature difference between measurement locations in the bridge deck and in the abutment.

The results show that the simulation model was capable of predicting the temperature distribution adequately, especially when the temperature distribution within the model is concerned. The temperature variations were larger in the bridge deck than in the abutment, which showed a slower response to weather changes and thus did not reach as high temperatures in summer and low temperatures during winter. This demonstrates the influence of both the radiation and the soil next to the abutment, and also verifies that there can be a temperature difference between the two structural parts.

For the temperature difference between abutment and bridge deck, Peng et al. [START_REF] Liu | Temperature response in concrete under natural environment[END_REF] conducted a study to observe the temperature response of concrete under the natural environment by measuring temperature in concrete samples. The concrete specimens dimensions were 150mm ⇥ 150mm ⇥ 150mm with the geometry of a cylinder, and the thermocouples were placed at di↵erent positions, see in [Fig. 1.2]. Correspondingly, the concrete specimen for temperature response under natural environment without sheltered condition was directly exposed to the sun, and others was under the shelter. The temperature response of the concrete is a↵ected by the solar radiation, the di↵erence between temperatures recorded with shelter and without shelter is 10 o C, see [Fig. 1.3]. The author concluded that the solar radiation cannot be neglected in the concrete structures that are exposed to the natural environment. The thermal investigations as presented above always include the air temperature and solar radiation signals, but the data of those signals are not fully described. However, previous studies describing the solar radiation signal are found in [START_REF] Maxwell | Measuring and modeling solar irradiance on vertical surfaces[END_REF][START_REF] Etzion | The thermal behavior of a concrete 'finned' wall in a hot-arid zone[END_REF][START_REF] Rigollier | On the clear sky model of the ESRA -European Solar Radiation Atlas with respect to the Heliosat method[END_REF]. The residential building and the bridge structures undergone by the temperature signal are not influenced by the orientations. In contrast, the solar radiation absorbed by the building envelops is a↵ected by the orientations (south, west, etc), and also depend on either the vertical or horizontal elements of the building. E.L. Maxwell et al. [START_REF] Maxwell | Measuring and modeling solar irradiance on vertical surfaces[END_REF] measured the global solar radiation during a clear day on a vertical plan for di↵erent orientations, see in [Fig. 1.5]. Those di↵erences are originally caused by the movement of the earth. To describe this movement, the sun positions are presented as the solar azimuth and solar altitude angles, see in [Fig. 1.6]. Rigollier [START_REF] Rigollier | On the clear sky model of the ESRA -European Solar Radiation Atlas with respect to the Heliosat method[END_REF] proposed the equations, Eqs. (1.1) and (1.2), for calculating those angles. The global solar radiation is projected to the di↵erent orientations by multiplying with the corresponding angles.

-Solar altitude :

s = sin 1 (sin sin + cos cos cos!) (1.1)
where : is the altitude corresponding to the geography (degree). is the solar declination angle (degree).

! is the solar hour angle.

sin↵ s = cos sin!/cos s (1.3) if sin↵ s < 0 : ↵ s = cos 1 (cos↵ s ) if sin↵ s > 0 : ↵ s = cos 1 (cos↵ s )
Figure 6. Solar radiation

Geometry of solar movements as seen from the earth

Base on the documentation of the European Solar Radiation Atlas (ESRA) [4], the knowledge of the geometrical parameters describing the position of the sun as seen from the earth is essential when information on solar radiation for a specific location and time is required. Three fundamental parameters are needed to determine the position of the sun as seen from any point of the earth. The latitude of the site of observation.

The Julian day number.

The time of day expressed as an hour angle from solar noon.

Figure 7. Geometry of Earth-Sun system (cited from ESRA)

There are three angles for locating the position of the sun : Declination angle (δ); the solar altitude angle (γs); Solar azimuth angle (αs).

THE SOLAR RADIATION ATLAS -3 Basics of solar radiation 32

Solar azimuth angle

The direction of the sun is given by its azimuth angle. The solar azimuth angle, a s , is the angle between the vertical plane containing the direction of the sun, and the vertical plane running true North South. It is measured from due South in the Northern Hemisphere and from due North in the Southern Hemisphere. The azimuth angle has a positive value when the sun is to the West of the South-North meridian, i.e. during afternoon in solar time. These angles may be converted into bearings from true north, but it is important to adopt the correct definition of azimuth angle in using the algorithms that follow. Refer

c. Solar azimuth angle

The solar azimuth angle, αs is the angle between the vertical plane containing t direction of the sun, and the vertical plane running true North South. It measured from due south in the northern hemisphere and from due north in t southern hemisphere. By ESRA, the azimuth angle calculated as:

𝑐𝑜𝑠𝛼 𝑠 = (𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝛾 𝑠 -𝑠𝑖𝑛𝛿)/𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝛾 𝑠 𝑠𝑖𝑛𝛼 𝑠 = 𝑐𝑜𝑠𝛿𝑠𝑖𝑛𝜔/𝑐𝑜𝑠𝛾 𝑠 if 𝑠𝑖𝑛𝛼 𝑠 < 0𝑡ℎ𝑒𝑛𝛼 𝑠 = -cos -1 (𝑐𝑜𝑠𝛼 𝑠 ) if 𝑠𝑖𝑛𝛼 𝑠 > 0𝑡ℎ𝑒𝑛𝛼 𝑠 = cos -1 (𝑐𝑜𝑠𝛼 𝑠 )
The position angle of the sun described above was presented in the [figure 7 8].

Thermo-mechanical simulation

Thermo-mechanical analysis is addressed by many researchers for large scale concrete structures [START_REF] Abid | Experimental analysis of temperature gradients in concrete box-girders[END_REF][START_REF] Flaga | Early age thermal shrinkage crack formation in bridge abutments[END_REF][START_REF] El-Tayeb | Thermal analysis of reinforced concrete beams and frames[END_REF][START_REF] Mirzazadeh | E↵ects of low temperature on the static behaviour of reinforced concrete beams with temperature di↵erentials[END_REF][START_REF] Jendele | Multiscale hydro-thermo-mechanical model for early-age and mature concrete structures[END_REF]. The daily and seasonal temperature variations can generate displacements and rotations in the same magnitude as those due to the dead and live loads. The thermal restraint and thermal gradient may generate tensile stresses, and create cracking to the structures. In Eurocode, the temperature distribution in concrete structure is decomposed into uniform, linear and non linear temperature profiles [Fig. 1.7], and the first two profiles provoke normal force and bending moment. In the design procedure for the residential building, thermal e↵ects are taken into account explicitly only through the distance between dilatation joint. For example, 30 m to 35 m for the region of Aples and massif center is given by the national annex of France.

Section 4 Representation of actions

(1) Daily and seasonal changes in shade air temperature, solar radiation, reradiation, etc., will result in variations of the tenlperature distribution within individual elenlents of a structure.

(2) The magnitude of the thermal effects will be dependent on local climatic conditions, together with the orientation of the structure, its overall mass, finishes (e.g. cladding in buildings), and in the case of building structures, heating and ventilation regimes and thermal insulation.

(3) The temperature distribution within an individual structural element may be split into the following four essential constituent components, as illustrated in Figure 4.1 : a) A uniform temperature component, /). Tu ; b) A linearly varying tenlperature difference component about the z-z axis, /). TMy ; c) A linearly varying temperature difference component about the y-y axis, /). T MZ ; d) A non-linear temperature difference component, /). T E • This results in a system of self-equilibrated stresses which produce no net load effect on the element. (5) For the purpose of deriving thermal effects, the coefficient of linear expansion for a material should be used. A simplified numerical investigation on a building (15m⇥32m) equipped with TBSs is explained by Le Bloa through an internal technical report of the engineering o ce, INGENOVA [START_REF] Bloa | E↵et de la dilatation thermique des bâtiments munis de boitier isolant structurel slabe[END_REF]. The numerical modeling is made of shell elements for the walls, and springs for the TBSs. The author conducted the numerical simulation with an engineering software, Graitec. The elastic behavior is used, and only the uniform temperature profile as described in [Fig. 1.7] is considered. The simulation indicated that no significant stresses are observed in the concrete at the connection wall-slab, see [Fig. 1.9], and the first TBS on the corner absorbed the maximum horizontal shear. Even though, the investigation is only corresponding to the industry engineering level, and cannot fully describe the stresses and strains distribution in the wall. But this study gave a noticeable remark to a large deformation at the corner of the building.

Figure 1.8 -view of the building and connection wall-slab [START_REF] Bloa | E↵et de la dilatation thermique des bâtiments munis de boitier isolant structurel slabe[END_REF] Essam et al. [START_REF] El-Tayeb | Thermal analysis of reinforced concrete beams and frames[END_REF] made the investigation of thermal e↵ects on reinforced concrete beams and frames. Di↵erent temperature gradients, uniform, linear and nonlinear, are Figure 1.9 -internal horizontal force on the wall and internal force in the TBS [START_REF] Bloa | E↵et de la dilatation thermique des bâtiments munis de boitier isolant structurel slabe[END_REF] applied to observe the response of the beams and frames. The stress profiles for the frames analysis exceed the limit strength of concrete in tension. The expansion due to the temperature gradient and combination with the thermal restraint are the origin for tensile stresses. The author emphasized that cracking contributed to the release of significant portion of temperature restraint or even eliminated it.

Figure 1.10 -stress profiles at di↵erent sections of the frame [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF] The study of Babara et al. [START_REF] Flaga | Early age thermal shrinkage crack formation in bridge abutments[END_REF] is interested on thermal-shrinkage strains and stresses in RC wall-on-slab structure. Cracks are found commonly above the joint between the wall and the foundation at the construction phase. A typical crack of a bridge abutment is presented in [Fig. 1.11]. This kind of cracking is caused by the restraint between the concrete foundation and the latter concrete wall of the frame system. The construction phases originally cause di↵erent thermal strain in those concrete elements. When the concrete wall of a frame system is constructed, the concrete foundation has cooled down and also a part of shrinkage strain is occurred. Whereas, during the heating phase of the concrete walls, the displacement at the bottom of concrete wall is limited by the foundation element, and results in compression stress for lower fiber of the walls. While in the cooling phase of the concrete wall, the bottom part of the walls experience tensile stresses due to a huge resistance from the previously cast foundation. Those tensile Chapter 1 : Literature review stresses are significant and might exceed the limit strength of concrete in tension. This phenomenon is similar to what happens in a building equipped with internal insulation, particularly when TBSs are presence. The temperature distributions in the walls vary with the daily air temperature and solar radiation signals, whereas in the slab of the building, the temperature distribution is almost constant. The dilatation and contraction of the walls are thus restrained by the slab system due to the presence of the TBSs at the interface. Considering the building walls as the wall system of the bridge, and the cap of the foundations as the concrete slab, then the thermal strain on the concrete wall of the building is restrained due to the TBSs placed at the slab-wall connection. Under the climatic actions, the stress generated by the thermal restraint on the concrete wall can lead to a significant level of stress in tension, and cracking may occur on either the interior or the exterior surface of the walls.

Drying process of concrete

Concrete structure exposed to the ambient humidity experiences with the pore moisture transfer. The imbalance of relative humidity entails a di↵usion of water vapor contained in the concrete samples through the air. An evaporation of liquid water occurs in order to maintain the liquid vapor equilibrium [START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF]. The evolution of relative humidity of the concrete a↵ects directly the evolution of shrinkage and creep strain. Furthermore, the drying of concrete is also the origin of micro crack formation of concrete at the early age. Concrete surface can lose its relative humidity only a few days after being exposed to the environment, whereas, the concrete core of massive members take more than 10 years to dry to the environmental humidity [START_REF] Jirasek | Inelastic analysis of structures[END_REF].

Experiments on drying process of concrete

Many researchers conducted experiments on drying of concrete not only to observe the relative humidity di↵usion in the concrete, but also to investigate the drying shrinkage strain. The authors [START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF][START_REF] Oshita | Modeling of water migration phenomenon in concrete as homogeneous material[END_REF][START_REF] Xi | Moisture di↵usion in cementitious materials moisture capacity and di↵usivity[END_REF][START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF] performed experiments on the moisture transferring of ordinary and high performance cement pastes and concrete. The relative humidity profiles and the evolution of relative weight loss are measured. Those results are then used for identifying the parameters of the simplified drying model in the next chapter.

[Fig. 1.12] presents moisture content distribution and its evolution on ordinary (BO) and high performance (BH) concrete, respectively [START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF]. Those experiments are conducted on cylinder concrete samples with the dimension of Ø16 ⇥ 10cm, and are submitted for drying at RH=50%. The moisture content distribution of BO varies corresponding to the drying time, whereas there is no obvious variation of moisture content for the BH in function of the drying time. Besides that, the moisture content reduces significantly at the surface compared to the core for the BO. Figure 1.13 -weight loss evolution [START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF][START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF] (Baroghel, [START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF] [Fig. 1.13] introduces the evolution of weight loss for ordinary and high performance concrete (BO & BH) of Baroghel [START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF]. In addition, the experiments of Granger [START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF],

Chapter 1 : Literature review which are performed with the concrete corresponding to those of the nuclear plants, are also illustrated. However, the concrete used by granger are still considered as ordinary concrete since their compression strength are only around 50 MPa. Both experiments of Baroghel and granger have a similar dimension for the samples. At the beginning, a significant weight loss is observed, except for the BH. The ordinary concrete presents higher weigh loss than the one of the BH, and becomes stable when t tend to infinity.

Conclusion

The experiments on the drying concrete present a moisture distribution of concrete samples after exposing to a low humidity environment. The moisture distribution of concrete is nonuniform, and the moisture content reduce quickly on the surface for the ordinary concrete. Whereas for the high performance concrete, the moisture content is less variable. The drying process for the concrete building significantly depends on the surrounding environment, ambient humidity.

Cracking of concrete

The cracking state of the concrete is usually determined by comparing stresses or strains to a limit defined for a uni-axial state of stress. In a building, the concrete is subjected to a multi-axial loading, and stress in one direction can either reduce or increase the limit strength in the other directions. In this work, it is necessary to consider many di↵erent sources of cracking, which cover from the stresses due to permanent or live load, thermal expansion or contraction and the phenomenons at the early age. The early age cracking is believed to be caused by the e↵ects of two kinds of stresses : the auto equilibrated stress system due to the thermal gradient between surface and core of the concrete during heating phase, and the stress provoked by external restraints of the structure that limit the global displacements during the contraction of concrete (cooling phase).

Bi-axial loading

Many di↵erent authors [START_REF] Kupfer | Behavior of concrete under biaxial stresses[END_REF][START_REF] Shang | Experimental study of strength and deformation of plain concrete under biaxial compression after freezing and thawing cycles[END_REF]36] have tried to determine the resistance of concrete for a bi-axial state of stress. The study performed by Kupfer et al. is often considered as a reference. They made experiments of bi-axial compression, bi-axial tension and compression-tension for di↵erent stress ratios. The results allow to construct the limit strength surface of the concrete, see [Fig. 1.14]. For the case of bi-axial compression, the limit strength of concrete reaches up to 1.3 times of the uni-axial limit stress. Regarding to the compression-tension stress, the ultimate strength in compression reduces gradually while the tensile stress increases. For bi-tensile stress, there is no significant di↵erence from the uni-axial limit stress in tension.

cracking at early age

The majority of concrete structures in European countries are designed in accordance with the Eurocode 2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF], which uses a performance based approach to check the durability of concrete at the serviceability limit state (SLS). For massive and more general special structures, the EC2 does not fully reflect the complete behavior of massive elements, for which thermo-hydro-mechanical e↵ects are important [START_REF] Mazars | Control of cracking, a major issue for reinforced concrete structures[END_REF] (J. [START_REF] Mazars | Control of cracking, a major issue for reinforced concrete structures[END_REF]. In this context, French Civil Engineering Community launched a joint national research program [START_REF]Large beam specimens with restrained shrinkage (rg8)[END_REF] (CEOS.fr) on a massive restrained shrinkage beam (RG8) in order to understand the early age cracking pattern of those structures.

A massive concrete beam is cast and exposed to the ambient environment for the purpose of thermal and hydric transferring, and eventually to understand the THM behavior of the structure, see in [Fig. 1.15]. Two steel struts are placed parallel to the beam in order to restrain the displacement of the beam. The cracking patterns are presented in [Fig. 1.16]. The first cracking appeared in the central part of the beam, and was generated due to the blocking of the contraction of the beam by the steel struts. The detection of cracking evolution is made by optical fiber sensors for measuring the velocity of displacements and strains. They are measured in three di↵erent zones, see in [Figs. 1.17a and 1.17b], which are the top, middle and bottom of the section. The first peak of the velocity of the displacement occurs at around 70 hours, and the other cracking takes place at around 240 hours based on the evolution of the velocity of the strain.

A simplified modeling of the massive reinforced concrete beam (RG8) at an early age is proposed by Mazars [START_REF] Mazars | Simplified modeling strategy for the thermomechanical analysis of massive reinforced concrete structures at an early age[END_REF]. The simplified model is based on a multi-fiber beam discretization that includes a thermo-mechanical model coupled with a concrete damage From 0 to 50h, the force evolution of the steel struts is in tension, whereas concrete and reinforcement are in compression. During the cooling phase, the damage appears at the middle span of the beam at 76h based on the results of the simulation. Whereas, the force evolution in the concrete and the reinforcement changes from compression to tension due to the restraint by the steel struts. At 103h, the damage distribution continues on both sides of the previous one, and in the experiment, the second cracking is observed at 167h.

Conclusion

Cracking is prevalent in the concrete structures. For some special structures and massive concrete elements, the minimum reinforcement prescribed by EC2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF] or other dedicated rules is not su cient for validating the service limit state. The experiment Rapport d'avancement-Exploitation des résultats expérimentaux du bloc RG8bis 36 Sur les résultats obtenus à court terme, on arrive à détecter des sauts de déformation indiquant la date d'apparition de la première fissure. On remarque que dans une zone d'exploitation, les cordes vibrantes disposées dans la même direction (longitudinale ou transversale), mesurent des déformations d'amplitude comparable même si elles ne sont pas dans la même section.

Cependant on remarque systématiquement que les déformations mesurées dans la direction transversale (indiquée par (*) sur les légendes) sont supérieures aux déformations mesurées dans la direction longitudinale. Cela s'explique par la combinaison de deux effets. En effet le blocage de la déformation longitudinale du bloc RG8bis par les butons entraine un effet poisson dans la direction transversale. A cela s'ajoute le fait que la déformation thermique du bloc dans la direction transversale est libre. Les sauts de vitesse de déformation mécanique des capteurs à corde vibrante (Figure 42) permettent de distinguer nettement l'apparition de la première fissure aux environs de 70 heures et une autre fissure aux environs de 240 heures. Les enregistrements acoustiques réalisés par ADVITAM ont permis de détecter un pic important d'énergie le 10 avril 2010 entre 9 h et 14 h (Figure 43), confirmant ainsi l'apparition de la première fissure à cette échéance. De même des pics d'énergie observés par ADVITAM (Figure 44) le 17 avril 2010 confirment l'apparition d'une fissure aux environs de 240 heures. Selon les analyses acoustiques, une fissure serait apparue entre le 12 avril et le 14 avril (Figure 45). Les vitesses de déformation mécanique des cordes vibrantes montrent un léger saut de déformation (Figure 42) à 167 h qui pourrait éventuellement être attribué à cette fissure. 1 m from the central section. The following results pertain to these sections, which were instrumented with strain gauges on the rebar, vibrating wires close to the upper and lower surfaces, and 3 optical fibers distributed on a median plane of the beam covering 2.5 m of its central zone. Relative forces developed in the central section-Within a given cross-section, three main structural elements were considered: concrete, rebar, and struts. From the strain gauges located on the rebar, and assuming that they remain elastic, it is straightforward to deduce the stress acting at a given time on each element, from which the force Frb absorbed by the set of reinforcements within a given section can be deduced. Strain gauges were also placed on the struts. With knowledge of the strut surface area, the force Fst being exerted on them can also be deduced. At a given time, each cross-section is in equilibrium. From this balance, the force Fc supporting the concrete section can be deduced as: [START_REF] Nam | E↵ect of daily temperature variations on the continuous deflection profiles of airfield jointed concrete pavements[END_REF] in which Es is the Young's modulus for steel, Ai and εi are, respectively, the area and strain on rebar i I, and Ast and εst are, respectively, the cross-sectional area of each strut and corresponding strain.

Analyse de l'apparition des fissures

st st s i i s st rb c A t E A t E t F t F t F ) ( 2 ) ) ( ( ) ( ) ( ) ( (
The evolution in these three global forces is compared in Figure 4b. It should be emphasized that the comparison between calculation (solid line) and experiment (dashed line) is in good agreement.

After a period during which the struts are in tension and both the concrete and rebar in compression, the sign of the forces changes at around 50 h, with an initial crack appearing on the concrete (hour 71 measured and hour 76 calculated). Due to bending caused by the gravity load, the crack is located in the central beam section. This phenomenon leads to a concrete force peak followed by a sudden decrease of this force, with a concurrent increase in the rebar-related force. At 103 h, the calculation forecasts a second series of cracking, which, due to the diffuse description of the damage, appears to be an extension on both sides of the previous cracking area (Figure 4c). This second cracking process was observed experimentally at approx. 167 h.

Local stress evolution in the rebar. During this restrained shrinkage process, the rebar remained elastic. Figure 5b provides the stress evolution over time that corresponds to both the upper rebar (blue line) and lower rebar (red line) in the central section for the first 100 h. The jump caused by rebar loading due to the initial cracking is clearly visible. For the lower rebar, this jump has been correctly predicted, and the difference obtained for the upper rebar is the counterpart due to use of the Timoshenko beam approach (a plane section remains plane, which is not the case in practice).

From 0 to 120 h: Maturation and Cracking Period

Relative forces developed in the central section-Within a given cross-section, three main structural elements were considered: concrete, rebar, and struts. From the strain gauges located on the rebar, and assuming that they remain elastic, it is straightforward to deduce the stress acting at a given time on each element, from which the force F rb absorbed by the set of reinforcements within a given section can be deduced. Strain gauges were also placed on the struts. With knowledge of the strut surface area, the force F st being exerted on them can also be deduced. At a given time, each cross-section is in equilibrium. From this balance, the force F c supporting the concrete section can be deduced as:

F c (t) = [F rb (t) + F st (t)] = [ (E s i (t)A i ) + 2E s st (t)A st ] (17)
in which E s is the Young's modulus for steel, A i and i are, respectively, the area and strain on rebar i I, and A st and st are, respectively, the cross-sectional area of each strut and corresponding strain. The evolution in these three global forces is compared in Figure 4b.

It should be emphasized that the comparison between calculation (solid line) and experiment (dashed line) is in good agreement. on a massive concrete beam of CEOS.fr aims at predicting the cracking pattern of the concrete at an early age, where the high percentage of reinforcement is included. In this experiment, crackings are generated by the restraint of shrinkage strain during the cooling phase. The good estimation of the simplified model, which is based on the damage model of Mazars under the benchmark of CESO.fr, is interesting for this thesis that confronts to the similar numerical simulation, THM analysis.

Drying shrinkage

The drying shrinkage is a common problem in concrete material, and is known to create crackings on the surface of the concrete elements. Those crackings mean the Chapter 1 : Literature review existence of tensile stresses, which go beyond the limit strength. The tensile stresses are originally caused by the nonuniform relative humidity distribution in concrete.

Structural e↵ect of drying shrinkage

Drying of the concrete material is a long term process, which can take more than 10 years for reaching to the core of concrete. The drying shrinkage strain is more important on the skin of the concrete, where the relative pore moisture decreases quickly [Fig. 1.19 (a)]. The drying shrinkage creates those tensile stresses on the surface of concrete, and compression stresses in the core of concrete [Fig. 1.19 (c)]. [START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF][START_REF] Iding | Prediction of Shrinkage Stresses and Deformations in Concrete[END_REF][START_REF] Meftah | An elasto-plastic damage approach for the modeling of concrete submitted to the mechanical induced e↵ects of drying shrinkage[END_REF] Some researchers performed elastic calculations, and shown that the stress distribution on the surface of concrete exceed the limit strength of concrete in tension. As observed on a bridge pile, cracks propagation on the surface are presented. At the final stage of the drying process, the relative humidity of the core of concrete is reduced, and the drying process on the surface of concrete finished. A contraction zone takes place in the core of concrete, and the tensile stress in the core increases, whereas compression stress happens on the surface of concrete.

Experiments on drying shrinkage

The drying process creates thus a structural e↵ect to the concrete (cracking on the surface). The measure of drying shrinkage strain from experiment, which includes this structural e↵ect, creates di culties for proposing a drying shrinkage model. It is necessary to separate the drying shrinkage strain (including the structural e↵ect) from those of the material behavior [START_REF] Benboudjema | Structural e↵ects of drying shrinkage[END_REF]. To reduce the structural e↵ect, drying shrinkage experiment for the samples with small dimension in the thickness are made, and are compared with the thick samples.

-Thin dimensional sample :

The experimental results observed in [Fig. 1.20] are obtained separately by Baroghel and Obeid [START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF][START_REF] Obeid | Experimental identification of biot's hydro-mechanical coupling coe cient for cement mortar[END_REF]. For the cement paste material, the drying shrinkage strain varies in proportion to the variation of the relative humidity. The shrinkage strain is nearly constant for the relative humidity between 0 and 55%, and then decreases linearly from 55% to 90%.

The drying shrinkage strains corresponding to di↵erent drying rates are also presented in [Fig. 1.21]. R. Day et al. [START_REF] Day | The e↵ect of rate of drying on the drying creep of hardened cement paste[END_REF] observed that for a slow drying rate, the shrinkage strain is proportional to the variation of the relative humidity, whereas at a fast drying rate, the drying shrinkage strains are separated into two segments corresponding to [50% to 65%] and [65% to 100%]. -Thick dimensional sample :

The [Fig. 1.22] presents the drying shrinkage strain in function of the moisture loss for di↵erent concrete compositions [START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF] (Grange 1996). Those results are obtained from tests with cylinder samples (16cm of diameter and 100cm of height), and the results are recorded at middle height of the samples. Three zones are observed from the shrinkage curves. Initially, the slope of the drying shrinkage strain is small, when the samples lose their weight, but the shrinkage strain is not yet developed. Granger gives a reason due to the cracking on the surface of the samples. Besides that, a linear variation of the drying shrinkage strain is observed on the middle part, where the slope increases. For the long term drying process, the relative humidity gradient reduces, and the previous cracking partially closes, then the slope of shrinkage strain decreases.

Conclusion

Di↵erent drying shrinkage experiments for the concrete and cement paste are described. However, the experiments corresponding to the concrete are interested in this thesis. The drying shrinkage strain presents a linear part when plotting with the variation of relative humidity and moisture loss. Anyways, those linear parts are also limited to 55%-90% for the relative humidity. Because of those linear variations, there are many existing simplified drying shrinkage models based on this physical phenomenon, see in Section 1.4.3.1. The linear part of the shrinkage strain can be used for identifying the parameters of the simplified drying model.

Creep strain

Creep strain is considered as one of the long term e↵ects of the concrete, and should be taken into account for the long term design of the building structure. Creep strains are generally decomposed into the basic creep and the drying creep strains. The parametric study related to the creep strains indicated that the age of concrete while loading and the relative humidity made change to the creep strains.

Basic creep

Jirasek and Bažant [START_REF] Jirasek | Inelastic analysis of structures[END_REF] gave the definition of basic creep as the creep occurring at constant moisture content (without exchange with environment). The age of concrete, and concrete properties such as mechanical strength and elastic modulus are considered as the factors that a↵ect the basic creep [START_REF] Gaucher | Fluage du béton : tentative de caractérisation du comportement rhéologique non linéaire dans la représentation par intégrales multiples / Christophe Gaucher[END_REF][START_REF] Bazant | Mathematical modeling of creep and shrinkage of concrete[END_REF][START_REF] Wittmann | Einflub des feuchtigkeitsgehaltes auf das kriechen des zementsteines[END_REF][START_REF] Frech-Baronet | New evidences on the e↵ect of the internal relative humidity on the creep and relaxation behaviour of a cement paste by micro-indentation techniques[END_REF]. The influences of the age of concrete when load is applied, and of the relative humidity in concrete on the basic creep strain are illustrated below :

-Influence of the age of concrete when apply load on the basic creep :

Niyogi et al. [START_REF] Niyogi | The influence of age at time of loading on basic and drying creep[END_REF] performed the basic creep experiments, in which the concretes are loaded at di↵erent ages. The results are presented as the the basic creep coe cient in function with the ages of concrete when load is applied. The creep strains are recorded at di↵erent durations after loading (7 days, 21 days and 47 days) [Fig. 1.23]. The basic creep strain increases while the young concrete is loaded, and decreases when loading to older concrete, and this is due to the fact that the concrete sti↵ens. In [Fig. 1.24], the evolution of the basic creep strains are presented with the di↵erent values of relative humidity. The basic creep strain decreases while the relative humidity decreases. Under a constant load, the less evaporable water concrete contains the less basic creep occurs in the concrete [START_REF] Acker | Creep and shrinkage of concrete : physical origins and practical measurements[END_REF] The concrete samples loaded and dried simultaneously develop more important creep strains than those of the samples loaded with pre-drying. This phenomenon is firstly known as 'Pickett e↵ect', in which the total creep strain is the sum of the basic creep strain and the additional strain that is called as the drying creep strain. [START_REF] Bažant | E↵ect of cracking in drying and shrinkage specimens[END_REF] Bažant and Raftshol suggest that the creep rate depends not only on the water potential, but also on the rate of variation of this potential. In [Fig. 1.25], the drying creep strain can increase up to five times of the basic creep strain. The greater the drying concrete experiences, the more drying creep strain increases. The drying process of concrete elevates the local stress peaks within the microstructure of calcium silicate hydrates, and thus increase the rate of bond breakages [START_REF] Bažant | Microprestresssolidification theory for concrete creep.2003 ;i : Aging and drying e↵ects[END_REF].

In [Fig. 1.26], the drying creep strains are presented in function of the variation of moisture loss in cement paste samples for the fast and slow drying rates. The drying creep strain is proportional to the moisture loss for the slow drying rate, whereas, a bi-linear segment is observed for the fast drying rate. The wetting creep strains, which occur after first drying, follow a linear relationship with the moisture loss.

Interaction of creep and cracking

The creep strain is assumed to give time deformation with no significant material damage for stress levels under 40% of the limit strength [START_REF] Mazzotti | Nonlinear creep, poisson's ratio, and creep-damage interaction of concrete in compression[END_REF]. Furthermore, the creep law is generally assumed to be linear with respect to the stress history, and this hypothesis agree very well with the test results in the case of basic creep (no moisture content ) is applied in the first and third cycle, in which a linear viscoelasticity is observed in the stress-strain curve. Besides that, high stresses level varied from 60% to 80% of the limit strength is applied to the second and forth cycle, and the non linear stress-strain relation is presented in [Fig. 1.27a]. Furthermore, the creep strain evolutions corresponding to di↵erent stresses level are plotted in [Fig. 1.27b]. The creep strain produced by the low stress level (1 st and 3 rd cycle) is less than those of the high stress level (2 nd and 4 th cycle). The higher stress is applied, the higher creep strain is obtained. To include this phenomenon in the coupled problem between creep model and damage model, the e↵ective stresses are used for predicting the creep strain in the numerical simulation.

Conclusion

The previous researches on the creep e↵ect have separated the total creep stain as : basic and drying creep strain. The basic and drying creep are depending di↵erently on the relative humidity, and should be calculated separately. The parameters, which influence the creep strain, are summarized. The stress level presented a great e↵ect on the creep strain. A linear viscoelasticity is observed for the low stress level, and nonlinear for the high stress level. The complexity of the creep e↵ect is still concerned by many researchers, and especially for numerical modeling. The evolution of creep strain for a unit of stress is used to propose the creep compliance function in the model. The relation between the creep compliance with the incremental creep strain in numerical simulation is discussed in the next section. The drying process of concrete is caused by the imbalance of the relative humidity between the concrete and the environment, and the experiments of the drying concrete are already introduced in Section 1.3.1.1. The transfers of water and air from inside of the concrete to the exterior environment are described the following modes [START_REF] Oshita | Modeling of water migration phenomenon in concrete as homogeneous material[END_REF][START_REF] Coussy | Evaluation de la perméabilité à l'eau liquide des bétons à partir de leur perte de masse durant le séchage[END_REF] :

-The transport of the liquid water caused by the pressure gradient. To describe this movement, the law of Darcy is usually adopted :

V l = K ⌘ l k rl rp l (1.4)
where V l [m/s 1 ] is the velocity of liquid water of filtration, K [m 2 ] is the intrinsic permeability of the material, ⌘ l [kg.m 1 .s 1 ] is the viscosity of the liquid, k rl is the relative permeability of liquid water and p l [Pa] is the pressure of saturating liquid.

-Transportation of humidity caused by the concentration gradient (govern by water vapor di↵usivity). It can be described by the first law of Fick :

di↵ = D dif f rC vl (1.5)
where C vl is the concentration of water vapor [Kg.m 3 ], D dif f [m 2 .s 1 ] is the difussivity of water vapor and di↵ [Kg.m 2 .s 1 ] is the flow of water vapor.

Chapter 1 : Literature review -Transfer of the dried air caused by the pressure gradient. It can also be described by the law of Darcy :

V a = K ⌘ a k ra rP a (1.6)
where V a [m.s 1 ] is the velocity of dried air, ⌘ a [Kg.m 1 .s 1 ] is the viscosity of dried air, k ra is the relative permeability of dried air and P a [Pa] is the pressure of dried air.

-the di↵usion of water under the phase of vapor depends on the convection. The vapor of water is leaded by the migration of dried air :

conv = C vl V a (1.7)
where conv [Kg.m 2 .s 1 ] is the flow water vapor by convection.

The mass balance equations corresponding to l (liquid) ; v (vapour) and a (air) [START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF] (Baroghel et al. 1999) are :

@( ⇢ l S l ) @t = 5 ( S l ⇢ l V l ) µ l!⌫ (1.8) @ [ ⇢ v (1 S l )] @t = 5 ( (1 S l )⇢ v V v ) + µ l!⌫ (1.9) @ [ ⇢ a (1 S l )] @t = 5 ( (1 S l )⇢ a V a ) (1.10)
where l and g are the porosity of liquid and gas, whereas ⇢ is the density of the fluids (liquid ; vapour and air). S l is the degree of saturation, and V i is the velocity of fluids. Furthermore, µ l!v is the rate of mass of liquid water changing into vapour per unit of overall material volume.

Simplified drying model

The di↵usion process of the relative humidity of concrete with the environment is started after the formworks of structure are removed. Within a thin surface layer, water is quickly evaporated until the hygro equilibrium between the concrete and the surrounding atmosphere is reached. The moisture migration can be calculated by a linear di↵usion equation based on the second law of Fick [START_REF] Wittmann | Total deformation of loaded drying concrete[END_REF][START_REF] Xi | Moisture di↵usion in cementitious materials moisture capacity and di↵usivity[END_REF][START_REF] Zhao | Computational model for internal relative humidity distributions in concrete[END_REF][START_REF] Abbasnia | Prediction of free shrinkage strain related to internal moisture loss[END_REF]. This suggests that the moisture flux, J, is proportional to the gradient of the pore relative humidity, and is expressed by [START_REF] Abbasnia | Prediction of free shrinkage strain related to internal moisture loss[END_REF] :

J = k grad h (1.11) @h @t = C r [krh] = r [D eq (h)rh] (1.12)
where D eq [m 2

.s 1 ] is the coe cient of di↵usion, and depends on the variation of humidity. The expression of D eq (h) is proposed by [START_REF] Xi | Moisture di↵usion in cementitious materials moisture capacity and di↵usivity[END_REF] [START_REF] Xi | Moisture di↵usion in cementitious materials moisture capacity and di↵usivity[END_REF]) :

D(h) = D o h 1 + a ⇣ 1 2 10 b(h 1) ⌘i (1.13)
where D o , a and b are the material parameters and depend on the value of e/c. Furthermore, the Eq. (1.13) does not include the damage variable of concrete, so it considers that the cracking does not influence significantly the drying process of concrete. [START_REF] Bazant | E↵ect of cracking on drying permeability and di↵usivity of concrete[END_REF][START_REF] Sicard | Influence of creep and shrinkage on cracking in high strength concrete[END_REF] Bažant and Sicard shown that the opening of drying induced cracks ranges from 25 µm to 50 µm, and bažant proposed that the cracks have no influence while the opening of cracking is less than 100 µm.

Conclusion

The physical drying model presents the drying process of concrete under di↵erent form of di↵usion. A number of parameters are needed for the complete model of drying process, in which those parameters are not constant, and depend on the variation of the relative humidity. In this research framework, the buildings simply experience with the ambient humidity, then a simplified drying model is chosen, and this also contributes to reduce the calculation time of the simulations in chapter 3 and 4.

Damage model of concrete

To describe the mechanical behavior of concrete structures, many constitutive laws are proposed. The elasto-plasticity ; damage elasticity and damage elasto-plasticity framework are used in literature for representing the cracking pattern caused by tension stress and post peak behavior of the concrete in tension and compression.

Damage elasticity model

The damage theory firstly appears at the beginning of the 60's, and the objective is to predict the creep e↵ect in the metallic materials. The damage model is then used to represent the cracking e↵ect for the mechanical behavior of concrete.

On the scale of micro-cracking, the damage model is based on the e↵ective stresses concept. Material after cracking is decomposed into an e↵ective area ( S) and cracking area (S -S) [Fig. 1.28]. Under the e↵ective stresses (˜ ), the apparent stresses ( ) are only taken by the e↵ective area :

(n) = S(n) S(n) ˜ (n) (1.14)
whereas, the damage variable (D) is defined by :

D = S(n) S(n) S(n) (1.15)
Chapter 1 : Literature review The reduction of e↵ective cross section is due to the crack formation, the e↵ective stress applied on the non cracked part of the material is related to the apparent stress by :

= (1 D)˜ (1.16)
The e↵ective stress is thus given by : ˜ = E : " : "

: " e (1.17)

The constitutive equation is then :

= (1 D)E : " : "

: " e (1.18) whereas, the elastic strain is calculated by :

" e = " " th " creep " sh (1.19)
where, " ; " th ; " creep and " sh are the total strain, thermal strain, creep strain and shrinkage strain, respectively. D is the damage variable, which indicates the damage degree of concrete during loading, and D varies from 0 to 1 (0 for non damage and 1 is full damage). Mazars [START_REF] Mazars | Application of Continuous Damage Mechanics to Strain and Fracture Behavior of Concrete[END_REF] proposed the expression of equivalent strain (" eq ) to control the damage variables, where " eq can be defined from the principal strain as :

" eq = q h" 1 i 2 + + h" 2 i 2 + + h" 3 i 2 + (1.20)
The equivalent strain is an indicator of the state of tension in the material which generates the damage, and the loading function of the damage is :

f = " eq K(D) (1.21)
Observation of modeling of THM behavior of concrete 39 Initially, K(D = 0) = " d0 , where " d0 is the limit strain of tensile stress.

In order to capture the di↵erences of mechanical responses of the material in tension and compression, the damage variable is split into two parts :

D = ↵ t D t + ↵ c D c (1.22)
The weighting coe cients ↵ t and ↵ c are defined as functions of the principal values of the strain :

↵ t = 3 X i=1 ✓ h" t i ih" i i + " 2 eq ◆ , ↵ c = 3 X i=1 ✓ h" c i ih" i i + " 2 eq ◆ (1. 23
)
" t i = (1 D)C t , " c i = (1 D)C c (1.24)
The evolution of damage is provided in an integrated form, and in function of the equivalent strain [START_REF] Mazars | Application of Continuous Damage Mechanics to Strain and Fracture Behavior of Concrete[END_REF] (Mazars and Lemaitre, 1985) :

D t = 1 " d0 (1 A t ) " eq A t exp [B t (" eq " d0 )] (1.25) D c = 1 " d0 (1 A c ) " eq A c exp [B c (" eq " d0 )] (1.26)
where, A t and B t are the parameters corresponding to tensile strength of the model.

A c and B c are for compression.

It is required in order to avoid spurious mesh sensitivity, the energy dissipation due to fracture per unit length is a constant [START_REF] Matallah | A practical method to estimate crack openings in concrete structures[END_REF][START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF]. Based on the energetic regularization, the crack band theory is proposed. Bažant and Oh [START_REF] Bažant | Microplane model for progressive fracture of concrete and rock[END_REF] consider the fracture zone to have a certain width h over which micro-cracks are uniformly distributed.

G f = Z 1 0 d (1.27)
where, is the crack opening displacement, and the crack opening displacement is taken as the fracture strain accumulated over the width h e of the finite element :

= " uco h (1.28)
then the fracture energy per unit width (h e ) is calculated as the area under the complete stress-strain curve :

G f = h e Z 1 0 d" (1.29)
The fracture energy is the area under uni-axial tensile stress-strain diagram, and the damage evolution here is given by those of [START_REF] Matallah | A practical method to estimate crack openings in concrete structures[END_REF] :

D = 1 " d0 " exp [B t (" d0 ")] (1.30)
Chapter 1 : Literature review

G f h e = Z 1 0 ((1 D)E" " ")d" (1.31)
Then :

G f h e = f t " d0 2 + f t B t (1.32)
where h e is the dimension of the finite element, and f t is the limit strength in tension.

In the simplified modeling of Mazars as discussed in Section 1.3.2.2, the author gave a reference for calculating the fracture energy through the crack band approach [START_REF] Bazant | Concrete fracture models : Testing and practice[END_REF][START_REF] Planas | Measurement of the fracture energy using three-point bend tests : Part 2-influence of bulk energy dissipation[END_REF]. The crack band width is expressed in Eq. (1.33), and E t is the post-peak tangent sti↵ness :

h e = 2G f f 2 t ✓ 1 E 1 E t ◆ 1 (1.33) 1.4.2.

Damage µ model of Mazars

The original damage model of Mazars is only used for the monotonic loading, and is able to describe the dissymmetry of tension and compression stress-strain curve. However, the unilateral e↵ect (reclosing of crack) is not taken into account by this model. A new version of the damage model [START_REF] Mazars | A new 3d damage model for concrete under monotonic, cyclic and dynamic loadings[END_REF] [START_REF] Mazars | A new 3d damage model for concrete under monotonic, cyclic and dynamic loadings[END_REF] has been proposed for the purpose of 3D cyclic and dynamic loadings. Unilateral behavior (crack opening and closure) is introduced by the use of two internal variables, and the threshold surface is then associated with each of those variables (compression, tension).

-Damage evolution :

The equivalent strain, see Eq. (1.20), in the damage model of Mazars is defined by " t and " c for cracking and crushing of concrete, respectively :

" t = I " 2(1 2⌫) + p J " 2(1 + ⌫) (1.34) " c = I " 5(1 2⌫) + 6 p J " 5(1 + ⌫) (1.35)
where :

I " = " 1 + " 2 + " 3 (1.36) J " = 0.5 ⇥ (" 1 " 2 ) 2 + (" 2 " 3 ) 2 + (" 3 " 1 ) 2 ⇤ (1.37)
The two loading surfaces are :

f t = " t Y t ; f c = " c Y c (1.38)
Y t and Y c are the maximum values reached during the loading path :

Y t = sup [✏ 0t , max(" t )] ; Y c = sup [✏ 0c , max(" c )] (1.39)
" 0t and " 0c are the limit strains in tension and compression, respectively.

The damage D is directly correlated with the thermodynamic variables Y t and Y c through the driving variable Y :

Y = rY t + (1 r)Y c
(1.40)

r = P h i i + P | i | (1.41)
r is the triaxial factor, and varies from 0 for compression to 1 for tension.

The expression of damage D is similar to the equation given by the original damage model of mazars :

D = 1 (1 A)Y 0 Y Aexp( B(Y Y 0 )) (1.42)
where :

Y 0 = r" 0t + (1 r)" 0c (1.43) A = A t (2r 2 (1 2k) r(1 4k)) + A c (2r 2 3r + 1) (1.44) B = r (r 2 2r+2) B t + (1 r (r 2 2r+2) )B c (1.45)
A c ; B c ; A t and B t are the material parameters, which are directly defined from compression and tension tests.

-Model response to various loading paths :

The damage µ model is used to follow a cyclic uni-axial load path as illustrated in [Fig. 1.29]. For the entire loading path, the internal variables (Y t and Y c ) are continuously increasing. Furthermore, the e↵ective damage evolution is initially activated due to tensile stress, and decreases while the loading switches from the tension to the compression (closure of cracking). The trend of the e↵ective damage variable depends on 'r', which interprets as the part of damage activated by local stress. The stress-strain curve corresponding to the loading path is presented in [Fig. 1.30], and the unilateral e↵ect is clearly exhibited from tension stress (t = t 3 ) to the compression stress(t = t 5 ). The sti↵ness of the concrete is recovered in the compression part when the loading is in the direction of crack-closing, and damage caused by crashing of concrete is activated once the loading surface of the compression is exceeded. The damage due to distributed cracking has been treated in structural analysis as strain softening. When the stress-strain curve exhibits a negative slope or the matrix of tangential moduli ceases to be positive definite, the strain softening damage tends to localize in a zone of vanishing volume. The energy dissipation per unite volume is constant, but the vanish of damage zone volume cause the indicated structure to fail at zero energy dissipation [START_REF] Pijaudier-Cabot | Nonlocal damage theory[END_REF][START_REF] Peerlings | Gradient enhanced damage for quasi-brittle materials[END_REF][START_REF] Pijaudier-Cabot | Continuous Damage Models for Fracture of Concrete[END_REF]. Mesh refinement leads to a vanishing energy dissipation and strain localizes in an area with decreasing volume [START_REF] De Vree | Comparison of nonlocal approaches in continuum damage mechanics[END_REF] (Vree et al. 1995).

The principal of non local continuum model is to replace the equivalence strain (" eq ) of the local model with its average :

" = 1 V r (x) Z ⌦ (s)" eq (s + x)ds (1.46)
with :

V r (x) = Z ⌦ (s)ds (1.47)
where ⌦ is the volume of the structure, V r (x) represents the volume at point x, and (s) is the weight function :

(s) = exp ✓ 4ksk 2 l 2 c ◆ (1.48)
l c is the internal length of non local damage model.

Conclusion

In order to introduce the post peak behavior of concrete material, the damage model is developed. To capture the non linear and post peak behavior of concrete, strain softening is considered by reducing the concrete sti↵ness with the damage variable (D). The evolution of the damage variable strongly depends on the amplitude of the principal strain compared to the limit strain in either compression or tension. Moreover, the damage distribution zone is generally considered as the cracking zone. Besides that, the unilateral e↵ect is also presented in the literature, in which the µ damage model of [START_REF] Mazars | A new 3d damage model for concrete under monotonic, cyclic and dynamic loadings[END_REF] aims to take into account the closure of crack opening. The sti↵ness matrix recovers while the loading path switches from tension to compression, and to achieve this objective, the damage variable is not only depended on the strain evolution but also on the local stress value.

The problem of localized strain in a zone of vanishing volume is also known as a numerical problem in the damage model, in which the non local damage model means to solve this type of numerical issue. The equivalent strain in the local damage model is averaged with the weigh function ( ), which is related to the internal characteristic length.

Drying shrinkage model

Simplified drying shrinkage strain

The observation of the evolution of the drying shrinkage strain from previous experimental results confirmed that the drying shrinkage strain depends greatly on the moisture distribution in concrete structures. Many authors [START_REF] Wittmann | Total deformation of loaded drying concrete[END_REF][START_REF] Bažant | Drying creep of concrete : constitutive model and new experiments separating its mechanisms[END_REF][START_REF]Shrinkage as influenced by strain softening and crack formation, dans 'creep and shrinkage of concrete[END_REF] suggest a linear drying shrinkage model :

" " " sh = K h sh ḣ1 (1.49)
Chapter 1 : Literature review where, " " " sh is the drying shrinkage strain rate vector, K h sh hydrous compressibility factor, ḣ is the di↵erence relative humidity and 1 is unit vector.

The drying shrinkage strain can be also evaluated by a linear function of water content. Many experiments [START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF][START_REF] Torrenti | Modélisation du retrait du béton en ambiance variable[END_REF][START_REF] Benboudjema | Interaction between drying, shrinkage, creep and cracking phenomena in concrete[END_REF] of drying shrinkage presented in the previous section also confirmed this phenomenon :

" " " sh = K c sh Ċ1 (1.50)
where, K c sh [m 3 .l 1 ] is proportional coe cient, and is identified from a linear part of the curve of drying shrinkage strain-moisture loss.

Both Eqs. (1.49) and (1.50) are proposed directly through the observation of drying shrinkage experiments. The simplified drying shrinkage model limit to the drying of concrete structure with the relative humidity varies from 50% to 100%. When relative humidity decreases below 50%, drying shrinkage of concrete passes to the nonlinear part. It should be noticed that both equations are based on the phenomenology of drying shrinkage experiments and does not deeply discuss on the origin of drying shrinkage.

Nonlinear drying shrinkage model

Di↵erence from the previous model, nonlinear drying shrinkage model is described by the theory of nonlinear poroelasticity [START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF][START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF][START_REF] Coussy | Mechanics of porous continua[END_REF][START_REF] Schrefler | A general model for the mechanics of saturated-unsaturated porous materials[END_REF]. In this method, the concept of e↵ective stress is used. The e↵ective stress is described as the sum of the stresses caused by the external force and pore pressure. A general expression of e↵ective stress is given :

d 0 0 0 = d + dP sol 1 (1.51)
where, d is the total increment stress, dP sol is the pressure applied on the solid by surrounding fluid of environment (moisture, air). Moreover, the total pressure dP sol represents the sum of fluid in the pore, and is a↵ected by a weight coe cient :

dP sol = b i dP i (1.52)
where, i=w and i=g are corresponding to water liquid and gas, respectively. P g , the pressure gas is equal to the atmosphere pressure (P g =P atm =constant). The coe cient b w is replaced by liquid water saturation (S l ) in order to fit more easily with the experimental data (Baroghel-Bouny et al. 1999).

dP sol = S l dP w (1.53)
The expression P w is defined from the Kelvin's equation for expressing the liquid-vapour equilibrium in the atmosphere condition :

P w = ⇢ w RT M v ln(h) (1.54)
In the case that the apparent stress is zero, by replacing Equ. 1.54 into Equ. 1.51, drying shrinkage strain is :

" " " sh = 1 K Ṗsol 1 (1.55) " " " sh = 1 K ⇢ w RT M v S l h ḣ 1 (1.56)
where, R is the gas constant, M v is the molar mass of water, h is relative humidity and ⇢ w is the water density. The degree of liquid water saturation is given by expression of [START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF] (Baroghel-Bouny et al. 1999) :

S l = w⇢ s ⇢ w (1.57) 1.4.3.

Comparison of the drying shrinkage models

Based on the drying shrinkage experiments conducted by Granger in Section 1.3.3.2, Benboudjema [START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF] simulated the drying shrinkage, and compared between the linear shrinkage and nonlinear shrinkage models to the experiments, see in [Fig. 1.31]. Both drying shrinkage models are simulated with the nonlinear behavior of concrete. This comparison presented a good prediction of the nonlinear shrinkage model corresponding to all phases of the shrinkage strain.

Conclusion

The simplified drying model of concrete is described by a non linear di↵usion equation Eq. (1.12). Moisture distribution of concrete structure exposed to surrounding environment is obained by imposing a bounding condition under the form of convection. The two equations of drying shrinkage strain Eqs. (1.49) and (1.50) are corresponding to a phenomenology model. The proposed equations of linear drying shrinkage model are based on a linear part of the experimental curve of the drying shrinkage strain in function of the relative humidity. Obeid et al. [START_REF] Obeid | Experimental identification of biot's hydro-mechanical coupling coe cient for cement mortar[END_REF] also proposed a bilinear Chapter 1 : Literature review variable for drying shrinkage strain, which are : K h sh = 398µm • m 1 for 0%<h<66%, and K h sh = 568µm • m 1 for 66%<h<95%.

An alternative drying shrinkage model is the nonlinear based on poroelasticity theory. This approach give a possibility for predicting the drying shrinkage strain by imposing the liquid water pressure to the solid skeleton. The drying shrinkage simulation tests of Benboudjema [START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF] indicated a good agreement of shrinkage strain of the nonlinear shrinkage model to those of the experiments. However, for a large scale of the simulation, the complexities of the nonlinear model make the simulation become heavy. In the study of this thesis, the buildings are mostly exposed to a medium humidity (50% -60%), then the relative humidity of concrete should vary from 60% to 100%. The estimation of shrinkage strain as linear is thus consistent, and the linear shrinkage model is also possible to predict the shrinkage strain at this part.

Creep model

Basic creep strain

The basic creep strain is defined as " bc , under a constant loading, the basic creep is computed by :

" " " bc (t) = J(t, t 0 ) (1. 58
)
where t is the time corresponding to the calculation of strain, t' is the age of concrete when applying load, and J is the compliance function of the basic creep.

For a variable stress evolution, the principle of superposition is usually used. The stress history is decomposed into infinitesimal step, where d (t 0 ) is considered as constant for the various time steps of t'. By summing all the infinitesimal strain :

" " " bc = (0)J(t) + 1 X i=1 d i J(t t 0 ) (1.59)
This approach can only used for a small scale structure. For a large finite element structure, storing the stress history is the weakness of this method. There is not only the superposition method, the Rate type creep law (RTCL) is developed to overcome that weakness. The RTCL is govern by the direct integration, and the detail of the RTCL is described in Appendix B.

" " " bc (t) = Z t 0 J(t, t 0 )d (t 0 ) (1.60)
The expression of the creep compliance function is generally based on the theory of non aging linear viscoelasticity [START_REF] Bazant | Solidification theory for concrete creep. i : Formulation[END_REF][START_REF] Bazant | Concrete creep at variable humidity : constitutive law and mechanism[END_REF][START_REF] Bazant | Creep and shrinkage prediction model for analysis and design of concrete structures-model b3[END_REF]. The compliance function of creep is approximated to the results of experiment by using either a combination of multiple chains of Maxwell or Kelvin-Voigt [Fig. 1.32] :

CHAPITRE I

-67 -L'avantage important de ce choix pour la complaisance de fluage est qu'il n'est pas nécessaire de stocker en mémoire l'histoire des contraintes. En effet, l'utilisation d'une telle complaisance permet de réécrire l'équation I-58 sous la forme d'équations différentielles, qui peuvent alors être résolues de façon incrémentale [START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF], Benboudjema et al. 2000b).

Cette formulation peut être étendue au cas vieillissant en faisant varier les paramètres de la chaîne de Maxwell ou de Kelvin-Voigt en fonction de l'âge du matériau (Bažant et Chern 1985, de Schutter 1999). Toutefois, certaines restrictions doivent être vérifiées afin de garantir le caractère strictement monotone de la complaisance de fluage (Bažant 1998). La décomposition en série de Dirichlet adoptée dans l'équation I-59 garantit que la déformation de fluage propre sera bornée. Un débat sur le comportement à l'infini du fluage propre a lieu au sein de la communauté scientifique. Certains chercheurs pensent que cette déformation est bornée [START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF], d'autres non (Bažant et Chern 1985). Or, les résultats expérimentaux ne permettent pas de confirmer l'une ou l'autre des deux hypothèses, étant donné que les essais ne dépassent pas souvent trois ans. Ainsi, d'autres fonctions de complaisance de fluage sont proposées dans la littérature afin d'obtenir une complaisance de fluage non asymptotique, telles que des fonctions puissances ou logarithmiques.

( a ) ( b ) σ σ σ σ 1 k 2 k n k 1 k 2 k n k 1 η 2 η n η 1 η 2 η n η σ σ σ σ 1 k 2 k n k 1 k 2 k n k 1 η 2 η n η
( ) ( ) ( ) ( ) 0 1 t t fp t t Jt t d t t ε σ υ ′= ′= ′ ′ = - ∫ ( I -6 0 ) Figure 1.32 -(a) chain of Maxwell, (b) chain of Kelvin-Voigt [76] (Benboudjema, 2002 )
J(t) = i=n X i=1 1 k i 0 B @1 e k i ⌘ i t 1 C A (1.61)
⌧ i = ⌘ i k i (1.62)
where, k i [MPa] are the elastic parameters of each element, ⌘ i [MPa.s] are the viscosity parameters of each element of Kelvin-Voigt. The creep compliance function curve plays an important role in the creep strain calculation, and also varies with di↵erent loading ages of concrete. In [Fig. 1.33], for loading of young concrete, the creep compliance function curve moves vertically for di↵erent loading ages. When loading of old concrete, the creep compliance function shifts horizontally from the left to the right.

The advantage of using either Maxwell or Kelvin-Voigt to compute the compliance function is to avoid of storing the stress history during the calculation. The Eq. (1.60) can be rewritten under the form of a di↵erential equation by using the RTCL [START_REF] Jirasek | Inelastic analysis of structures[END_REF] and there are some internal variables that are necessary to be updated for each loading step. Furthermore, this approach can also include the aging e↵ect of concrete, while the second order of di↵erential equation is needed to be solved. The elastic parameters seen in Eq. (1.61) are not constant, and should be in function of t'.

The aging e↵ect of concrete is due to the hydration process of concrete after the formwork being removed. The influence of the relative humidity on the basic creep strain has been pointed out previously, see in [Fig. 1.24]. This is the reason that the creep compliance function should also include the environmental conditions and vary through the age of concrete. To include the hydration e↵ect of concrete, Benboudjema [START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF] proposed to divide the elastic parameters of the Kelvin-Voigt by the relative humidity :

k bc i = k i h (1.63) ⌘ bc i = ⌘ i h (1.64)
Once the relative humidity decreases, the elastic parameters of the basic creep compliance function increase, and results in increasing sti↵ness of the concrete, and finally, the basic creep strain reduces. Besides the creep compliance function that is based on the Kelvin-Voigt model, the short form of B3 model is considered as the most empirical part and yet not too unrealistic [START_REF] Jirasek | Inelastic analysis of structures[END_REF][START_REF] Bazant | Concrete creep at variable humidity : constitutive law and mechanism[END_REF] :

J(t, t 0 ) = 1 E 0 + q s ln ⇥ 1 + ((t 0 ) m + ↵)(t t 0 ) n ⇤ + C d (t, t 0 , t 0 ) (1.65)
where C d is the additional creep compliance function (drying creep), the typical value of the model parameters are m=0.5, n=0.1. Whereas, ↵ = 0.001, and = 0.3, in which E 0 = E 28 /0.6 and q s = 11.4/E 28 , furthermore, E 28 is the conventional elastic modulus of concrete at the age of 28 days.

Drying creep strain

As described in [Section 1.3.4.2], the drying creep takes place while the loading and drying process of concrete are conducted simultaneously. The drying creep strain strongly depends on the relative humidity of concrete, Bažant and oh [START_REF] Bažant | Microplane model for progressive fracture of concrete and rock[END_REF] proposed a relation between the viscoelastic parameters of the Maxwell chain model and the relative humidity : 1

⌘(t) = ' h ⌧ E(t) (1.66) ' h = ↵ h + (1 ↵ h )h 2 (1.67)
in which ' h is the empirical function of pore humidity, and ↵ h is found to vary from 0.1 to 0.5, whereas E and ⌧ denote as the elastic parameters and the relaxation time of the Maxwell chain model (⌘ = ⌧ E).

Besides that, Benboudjema [START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF] replaced the Maxwell model by the Kelvin-Voigt model [Fig. 1.34], in which the viscoelastic parameters depend on the velocity of the relative humidity of concrete. The constitutive law of the drying creep strain is given by :

-108 -des liaisons n'est plus immédiate, comme c'est le cas du modèle de Bažant et Chern (1985). En effet, la rigidité des liaisons entre les particules de C-S-H retarde ce processus. Ainsi, la loi constitutive de la déformation de fluage de dessiccation intrinsèque s'écrit : ⌘ fd rupt " " " dc + K fd " " " dc = (1.68)

Ciment non hydraté

Hydrates

⌘ fd rupt = ⌘ fd ✓ fd h ḣi (1.69)
where, h ḣi = ḣ if ḣ < 0, and h ḣi = 0 if ḣ > 0. The presence of h ḣi aims to control the domination of the drying creep strain in the total creep strain. Once the structure is rewetting, then the drying creep strain becomes zero, and there is only the basic creep strain :

⌘ fd " " " dc K fd " " " dc h ḣi = h ḣi (1.70)
in the case that the velocity of relative humidity and stress are constant in function of time, the drying creep strain expresses as :

" " " dc (t) = J dc (t) (1.71) J dc = 1 E dc  1 exp ✓ t ⌧ dc ◆ (1.72)
Benboudjema [START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF] concluded that the advantage of the current drying creep model as that the kinetic and amplitude of the drying creep strain is controlled independently by the elasticity and viscosity parameters. Thus E dc is for controlling asymptotic of the drying creep strain, and the characteristic time (⌧ dc ) is for the drying creep strain :

⌧ dc = ⌘ fd E dc h ḣi (1.73)
Chapter 1 : Literature review this approach gives the possibility to adjust horizontally the compliance function of drying creep strain. However, the drying creep model of Benboudjema [START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF] does not consider the aging e↵ect of concrete as the elastic parameters of the compliance function (J dc ) do not vary in function of time.

A creep model, which used the Burger chain and implemented in a poromechanical framework, is developed by [START_REF] Sellier | Towards a simple and unified modelling of basic creep, shrinkage and drying creep of concrete[END_REF][START_REF] Sellier | Concrete creep modelling for structural applications : non-linearity, multi-axiality, hydration, temperature and drying e↵ects[END_REF] depends on the creep strain, corresponding to a consolidation phenomenon [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF].

The model is formulated to be usable in finite element codes so as to handle the modelling of large structures. Thus, the concrete microstructure is not explicitly modelled, but the constitutive equations reflect several important underlying phenomena induced by concrete heterogeneity. First, the viscous phenomena are assumed to take place in the C-S-H inter-layer (idealized in stage 1 of Fig. 2), while other hydrates and aggregates are taken to be non-viscous.

When a stress is applied to the material, as symbolized by red arrows in stage (2) of Fig. 2, an instantaneous elastic response occurs. It is modelled by the elastic level in Fig. 1. However, if the stress lasts for a long time, a viscous sliding of C-S-H sheets occurs (stage (3) in Fig. 2). This sliding includes a reversible creep corresponding to the reversible arrangement of inter-layer water molecules [START_REF]Eurocode 3 : Design of steel structures -part 1-4 : General rules -supplementary rules for stainless steels[END_REF]; corresponding to the Kelvin module in Fig. 1. It is also assumed that some interlayer bonds can break and repair to form a new configuration (as in plasticity theory), leading to irreversible basic creep. Thi responds to the non-linear Maxwell module in Fig. 1. W loaded, the stress in the solid skeleton transits in C-S-H sions (non-viscous phase in Fig. 2), but, during creep, stress supported by C-S-H inter-layer is relaxed due to haviour, leading to a stress concentration in non-visc the stress increases on the non-viscous phases while the same magnitude in C-S-H. As the creep velocity is the micro-stress applied to the C-S-H, this phenome creep velocity reduction assimilable to a consolidatio phenomenon can be modelled by the Kelvin module as distribution is reversible, However, as the C-S-H matri the other non-viscous phases, a purely viscous creep also possible, it is why a Maxwell module must be used t versible creep strain. Moreover, during the C-S-H matrix rial rearrangement occurs, and a configuration that blo flow can be reached (schematized by stage (4) in Fig qualified as an interlocking phenomenon. Once the blocked, the stresses concentrate on the non-viscous in the loading level is sufficient, induce a micro structural (5) in Fig. 2). This damage has been revealed experiment emission [START_REF] Ryś | Constitutive modelling and identification of parameters of 316l stainless steel at cryogenic temperatures[END_REF] but seems to have only negligible effects on ties as illustrated in the experimental results provided damage induced by creep takes place at a micro-level, creep strain without changing the elastic modulus. Wh occurs, as schematized in stage ( 6 The permanent creep corresponding to the Maxwell strain derived directly from the elastic strain component. The proportionality between the permanent strain and the elastic strain can be assumed as :

@" M I @t = " E I ⌧ M ref C c I (1.74)
where ⌧ M ref is a reference characteristic time, " M I refers to the Maxwell strain, C c I is a standard function for a consolidation state of material, and " E I is the elastic strain. The consolidation function bases on the expression of exponential, in which k is creep coe cient for the reference temperature. The objective of the consolidation function aims to consider the decreasing of creep strain velocity in function of time as observed in the creep experimental test. The creep model given by EC2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF] is based on the mean cross section approach for predicting the creep strain corresponding to a constant stress. Furthermore, in this creep model, concrete is considered as an aging linear visco-elastic material. Creep strain after a given duration of loading can be predicted from a creep coe cient ('), which depends the on age of concrete at loading and a function for describing the development of creep with time. For a constant stress applied at t 0 :

C c I = 1 k exp " 1 k ✓ " M I " E I ◆ + # (1.75) k = k ref C T C H C M (1.
" cc (t, t 0 ) = (t 0 ) E c,28 '(t, t 0 ) (1.77)
where, '(t, t 0 ) is the creep coe cient, and the stress dependent strain (" c ) is expressed as :

" c = (t 0 )  1 E c (t 0 ) + '(t, t 0 ) E c,28 (1.78)
" c = (t 0 )J(t, t 0 ) (1.79)
in which J(t, t 0 ) is the creep compliance function that represents the total stress dependent strain per unit stress, and the creep coe cient is given by :

'(t, t 0 ) = ' 0 c (t, t 0 ) (1.80)
the notional creep coe cient may be estimated from :

' 0 = ' RH • (f cm ) • (t 0 ) (1.81)
and

' RH =  1 + 1 RH/100 0.1 3 p h • ↵ 1 • ↵ 2 (1.82) (f cm ) = 16.8 p f cm (1.83) (t 0 ) = 1 0.1 + (t 0 ) 0.2 (1.84)
whereas the development of the creep with time is expressed by :

c (t, t 0 ) =  (t t 0 ) H + (t t 0 ) 0.3 (1.85)
and

H = 1.5 • h • ⇥ 1 + (1.2 • RH/100) 18 ⇤ + 250↵ 3  1500↵ 3 (1.86) ↵ 3 =  35
f cm 0.5 (1.87)
Chapter 1 : Literature review and RH is the ambient humidity.

In this model, due to the actual non-linear behavior of concrete some prediction errors are inevitable when linear superposition is applied to the creep of concrete under the variable stress. The disadvantages of this creep model are that it is impossible to independently calculate the basic creep and the drying creep, which are strongly linked to di↵erent configuration of the relative humidity of concrete. Besides that, the creep compliance function of this model is only usable for the superposition approach, but not for rate type creep law approach, see in Appendix B.4. However, for a design engineer, the empirical formulation gives the facility for constructing the creep compliance function and leads to include the creep strain in structural design corresponding to di↵erent humidity conditions.

Conclusion

Di↵erent creep models are presented in the above literature review. Many authors proposed to separate creep strains as the basic and the drying creep strain. Those creep strains depend di↵erently on the relative humidity configurations. Two methods for calculating the incremental creep strain have been presented, superposition and RTCL. The RTCL gives an advantage of avoiding of storing the stress history, and this is a good option in the large finite elements simulations. The creep compliance function of RTCL relies on the Kelvin Voigt model, and the identification of the parameters is discussed in the next chapter.

Thermal break system

The previous studies [4,[START_REF] Nguyen | Macroscopic modeling of reinforced concrete joints : Application to thermal break elements subject to earthquake loadings[END_REF][START_REF] Lebloa | Comportement sismique d'une connexion hybride acier-béton utilisée comme rupteur de ponts thermiques[END_REF][START_REF] Keo | Experimental study of the behavior of a steel-concrete hybrid thermal break system under vertical actions[END_REF][START_REF] Larbi | Thermal and mechanical analysis of thermal break with end-plate for attached steel structures[END_REF][START_REF] Dikarev | Experimental and numerical thermal analysis of joint connection 'floor slab and balcony slabe' with integrated thermal break[END_REF][START_REF] Goulouti | Aramid/glass fiber-reinforced thermal break thermal and structural performance[END_REF][START_REF] Wakili | Experimental and numerical thermal analysis of a balcony board with integrated glass fibre reinforced polymer gfrp elements[END_REF][START_REF] Goulouti | Thermal performance evaluation of fiber-reinforced polymer thermal breaks for balcony connections[END_REF][START_REF] Goulouti | Aramid/glass fiber-reinforced thermal break structural system performance[END_REF][START_REF] Keo | Low-cycle fatigue life of a thermal break system under climatic actions[END_REF] of the TBSs di↵erently focus on the thermal performance and mechanical behavior. The researchers aim to find a balance between the thermal resistance and the mechanical strength of the TBSs. However, forces happening in the TBSs and in their nearby concrete elements, when buildings are submitted to climatic actions, have never been investigated by a refined scientific study.

Investigation of thermal performance of TBSs

All the studies of the TBS related to the thermal performance have presented the advantage of the TBSs in term of solving the thermal bridge issue. However, those studies only focused on the thermal resistance of the TBSs, and do not really concern to the research framework of this thesis. At here, the author presents a representative study of Kostiantyn et al. [START_REF] Dikarev | Experimental and numerical thermal analysis of joint connection 'floor slab and balcony slabe' with integrated thermal break[END_REF]. They conducted the thermal simulation on a conventional TBS that is connected at the slab-balcony and with the configuration of internal insulation. The conventional TBS is composed of the insulation material, which is penetrated by the rebars. The boundary conditions in thermal analysis are simply described as hot side ( 45 sed for the measurement in the guarded hot box were built in several stages. At a first stage beam was constructed. After that the insulating element was assembled within a wooden rmal break). Then the thermocouples were fixed with thermal transfer ribbon. Fig. 2 shows inforcement steel rods and some of the thermocouples. The latter were fixed on critical sites warm side of the structure.

Characterized mechanical behavior of TBS by experiments

Several experimental tests on a typical TBS (SLABE ZN r ) have been conducted at the laboratory of civil and mechanical engineering (LGCGM) during the thesis of G. Le Bloa [START_REF] Bloa | Caractérisation du comportement sismique d'une connexion hybride plancher-voile utilisée comme rupteur de ponts thermiques[END_REF]. This typical TBS is only used for the building internally insulated. The experiments focused on the vertical and horizontal behavior of the TBS by applying the load in horizontal and vertical directions. The results of the experiments are used to validate the numerical modeling, and also for analyzing the seismic behavior of the TBS. the hydraulic jacks, whereas, the displacements are measured from four displacements sensors that are placed at four corners of the slab. In [Fig. 1.38], the maximum force reaches to 120 KN for each module of the TBS, and the displacement is 13mm. To estimate the plastic load and displacement, the method of Aribert and A. Lachal [START_REF] Lachal | Formulation de la rupture par fatigue de connecteurs acier-béton pour des solicitations de type sismique-applications aux assemblages[END_REF] is used. The secant modulus is define from a linear part of the force-displacement curve, which is between the original point and (2/3)F y . The abscissa value corresponding to F y is 2S y . In Table 1 The geometry of the specimen is already discussed in the previous section. Di↵erence from the vertical load test, the hydraulic jacks are placed in horizontal direction, and are supported by an additional steel structure. The forces are applied to the lateral surfaces of the slab, see [Fig. 1.40]. Furthermore, a permanent load is considered on the slab by placing a number of concrete blocks. The force-displacement curve of the horizontal load test is plotted in [Fig. 1.41], and the maximum force can reach up to 190 KN. A large ductility of the TBS for the horizontal displacement is obtained by comparing to the results of vertical load test. Besides that, the plastic load and displacement are also obtained by the same method as presented in the vertical load test, see

Experiment on vertical loading

The studies of horizontal e↵ects to the TBSs

The studies related to the horizontal e↵ect on the TBSs are concerned by many researchers [4,[START_REF] Lebloa | Comportement sismique d'une connexion hybride acier-béton utilisée comme rupteur de ponts thermiques[END_REF][START_REF] Keo | Low-cycle fatigue life of a thermal break system under climatic actions[END_REF]. Those horizontal e↵ects include seismic loading and thermal expansion from the balcony.

G. Le Bloa [3,[START_REF] Lebloa | Comportement sismique d'une connexion hybride acier-béton utilisée comme rupteur de ponts thermiques[END_REF] described the mechanical behavior of the TBS (SLABE ZN r ) Thermal break system Extending from the study of Keo [START_REF] Keo | Low-cycle fatigue life of a thermal break system under climatic actions[END_REF], Heng [4] performed the parametric studies including the uncertainties related to the strength, the loads, and the accumulated damage. The author presented a reliability-based analysis for the calibration of the safety factor that taken into account these uncertainties. Both, Keo and Heng, indicated that the e↵ect of the accumulated damage is significant for the case simulated with the climatic actions of the Embrun city, which is located in mountainous region and is subjected to a continental climate. This information contributed to reduce the case studies in this thesis since the most unfavorable climatic conditions are defined.

Thermal break system 59 T. Nguyen et al. [START_REF] Nguyen | Macroscopic modeling of reinforced concrete joints : Application to thermal break elements subject to earthquake loadings[END_REF] developed the constitutive law corresponding to a typical TBS, which is composed of high performance fiber reinforced concrete (Ductal) and reinforcement steel, see [Fig. 1.45]. This constitutive law is then applied to the spring elements that are already developed internally by Castem [START_REF]Code de calcule aux eléments finis cast3m[END_REF]. The author proposed to model the TBS by spring elements (JOI1) for the simplified numerical model in her thesis, and this spring element gives a good agreement in term of force-displacement curve between the simplified model and the experiment.

Constitutive law modeling of stainless steel

Maciej RYS [START_REF] Ryś | Constitutive modelling and identification of parameters of 316l stainless steel at cryogenic temperatures[END_REF] proposed a constitutive law for the description and identification of the parameters of the model of the stainless steel. The theory relies on the notion of local state, and involves one state potential for the state laws, and a dissipation potential for the description of the irreversible phenomena. The classical laws of kinematic and isotropic hardening are considered in this model.

The Helmholtz free energy of the material is described as the sum of elastic (E) ; inelastic (I) and chemical (CH) terms :

= ⇢ E + ⇢ I + ⇢ CH (1.88) ⇢ E = 1 2 " e ij E ijkl " e kl
(1.89)

⇢ I = 1 3 C p ↵ p ij ↵ p ij + R p 1  r p + 1 b p exp( b p r p ) (1.90) ⇢ CH = (1 n)⇢ CH + n⇢ CH ↵ 0 (1.91)
Since that this thesis is not interested with the chemical energy, then this part is not described in detail in this section. The Clausius-Duhem inequality for isothermal case is :

ij "ij ⇢ ˙ 0 (1.92)
The time derivative of Helmholtz free energy as a function of internal state variables is :

˙ = @ @" e ij "e ij + @ @↵ p ij ↵p ij + @ @r p ṙp + @ @⇠ e ij ⇠ (1.93)
Substituting the rate of the Helmholtz free energy into Clausius-Duhem inequality :

ij ⇢ @ @" e ij ! "e ij + ij "p ij ⇢ @ @↵ p ij ↵p ij ⇢ @ @r p ṙp ⇢ @ @⇠ ⇠ 0 (1.94)
the thermodynamic forces are obtained from the above equations :

ij = ⇢ @ @" e ij = E ijkl " e kl = E ijkl ⇣ " kl " p kl ⇠" bs kl ⌘ (1.95) X p ij = ⇢ @ @↵ p ij = 2 3 C p ↵ p ij (1.96) R p = ⇢ @ @r p = R p 1 [1 exp( b p r p )] (1.97) Z = ⇢ @ @⇠ = ⇢ @ I @⇠ (⇢ CH ↵ 0 CH ) (1.98)
The steel plasticity is described by a classical threshold of kinematic and isotropic hardening :

F p = f p = J 2 ( ij X p ij ) y R p (1.99)
the equations involving the dissipation potentials take the form :

"p ij = ˙ p @F p @ ij (1.100) ↵p ij = ˙ p @F p @ ij (1.101) ṙp = ˙ p @F p @R p = ˙ p (1.102)

Conclusion and prospects

61

the multiplier ˙ p is obtained from the consistency condition :

ḟ p = @f p @ ij ⇣ ˙ ij Ẋp ij ⌘ + @f p @R p Ṙp = 0 (1.103)
the force rate presented in Eq. (1.103) are given by the following formula :

˙ ij = E ijkl ⇣ "kl "p kl ⇠" bs kl ⌘ (1.104) Ẋp ij = 2 3 C p "p ij + 1 C p @C p @⇠ X p ij ⇠ (1.105) Ṙp = b p (R p 1 R p ) ṙp +  1 R p 1 @R p 1 @⇠ R p + (R p 1 R p ) r p @b p @⇠ ⇠ (1.106)
C p ; b p and R p 1 are the parameters of the model.

Conclusion

The studies of the TBSs in thermal performance and mechanical behavior are described, and the researchers desired to optimize the TBS for those two objectives. However, the mechanical studies of the TBSs mostly remain with a small scale of simulations and experiments. The experiments of SLABE ZN r conducted at LGCGM have defined the vertical and horizontal behaviors of this TBS. But unfortunately, there is no experiment are conducted for describing the low fatigue strength of this TBS. Anyways, the studies of Heng and Keo related to the fatigue strength of the TBS SUNE are a good reference for estimating the damage accumulation of the TBSs in the simulation of this thesis. The simplified numerical model of T.T. Nguyen proposed to represent the TBSs by the macro element (spring), and this approach gave well agreement between the simulation and experiment. Furthermore, Maciej developed an elasto-plasticity model for describing the behavior of the stainless steel, which is based on the kinematic and isotropic hardening.

Conclusion and prospects

The studies related to thermo-mechanical behavior, thermo-hydro-mechanical behavior and TBSs are presented. The previous thermo-mechanical simulation indicated the e↵ect of thermal loading on the structure. The thermal restraints could generate the tensile stress, and even exceed the limit strength of concrete. The thermal loading resulted by the complex climatic actions is suspected as a risk for the building equipped with the TBSs. Even worse, this kind of study has not been fully described through a scientific approach.

The THM behavior of concrete is discussed by the observation from experiments and numerical models. The THM analysis included thermal ; shrinkage and creep effects. Those e↵ects are presented as thermal strain, shrinkage strain and creep strain, Chapter 1 : Literature review and are based on the temperature and relative humidity distribution in concrete. The buildings exposed to the complex climatic actions also experience with those e↵ects. In addition, the nonlinear behavior of concrete with elastic damage models are indicated. The damage variable leads to reduce the concrete sti↵ness, then increase the total displacement of the structure.

The TBSs are studied by many researchers for di↵erent purposes. Some interested in thermal resistance of the TBS, and aimed to decrease the energy consumption in the buildings. Others preferred to investigate the mechanical behavior of the TBSs by doing experiments and simulations. The important knowledge that the author obtained from those studies were related to the macroscopic modeling of the TBSs and the nonlinear behavior. The results of the experiments for the vertical and horizontal loadings are essential for identifying the parameters of the classical elasto-plasticity model.

Based on the described models in this chapter, a certain number of the existence simplified models are selected. A simplified THM model, which covers from the physical models such as : drying behavior, shrinkage and creep models to the nonlinear models of the material are developed, and aims to balance between the accuracy and e ciency. Those selected models are reconstructed. The validation of the reconstructed models, as well as the identification of the parameters are presented in Chapter 2.

Chapter 2

Simplified modeling of the THM behavior of the concrete and the mechanical nonlinear behavior of the TBS

Introduction

As explained in the introduction of the thesis, the simplified modeling of concrete must include a nonlinear mechanical model in order to take into account the damage variable corresponding to tension and compression. The drying and drying shrinkage model mean to include the auto-equilibrium stresses, which are caused by the nonuniform relative humidity of the concrete. Furthermore, a RTCL is used to consider the creep e↵ect, which leads to the relaxation of the stresses in the building in function of time. In the literature review, Section 1.4, the di↵erent existence models are presented, and in current chapter, the appropriate choices have been made. This chapter presents the implementation of those selected models, and the identification of the parameters with the experimental results. In the same way, the classical elasto-plasticity model of the TBS is reconstructed, and the parameters are identified from the experiments presented in Section 1.5.2 of Chapter 1.

Mechanical model of concrete

The elastic damage model of Mazars (µ model), which is presented in Section 1.4.2.2, is selected for describing the nonlinear behavior of concrete in this thesis. This model is developed in order fulfill the original damage model of Mazars that cannot be used in cyclic load. The µ model is improved while the threshold surface is with the damage variable corresponding to tensile and compression stresses. In Section 1.3.2.2, this model is used for the THM analysis under the benchmark of CEOS.fr, and the crackings 64 Chapter 2 : Simplified THM behavior of concrete and Nonlinear behavior of TBS evolution are compared. The model is reconstructed and implanted into the calculation code of Castem, and the simulation tests in uni-axial and bi-axial are performed.

Uni-axial test of µ model

The uni-axial simulation tests in tension and compression are performed for the implanted µ model. The results are compared with the experimental test of Gopalaratnam [START_REF] Gopalaratnam | Softening response of plain concrete in direct tension[END_REF] for the tension and with Eurocode for the compression, see in [Fig. 2.1]. The compositions of concrete, which are used by Gopalaratnam, are w/c=0.45, s/c=g/c=2, and this is similar to the ordinary concrete. Supposing the ordinary concrete is 3 MPa in tension and 30 MPa in compression, the parameters for tensile and compression behavior of the model are respectively identified from experiment and EC2, see in Table 2.1. Those parameters are served for the full simulation presented in Chapter 4. An attention should be payed, the B t that is corresponding to the post peak behavior of concrete in tension is modified by the size of finite element, see Eq. (1.32).

µ model E (Pa) G f (N/m) ⌫ " 0t " 0c A t B t A c B c ↵(1/ o C) 31E+9 70

Bi-axial test of µ model

The bi-axial tests of Kupfer et al. [START_REF] Kupfer | Behavior of concrete under biaxial stresses[END_REF] are described in Section 1.3.2.1, the di↵erent ratios of z / x are tested, and the surface of concrete strength under bi-axial loading is depicted in [Fig. 1.14] of Chapter 1. In this section, simulation tests with the reconstructed µ model are performed in order to ensure the correctness of the implanted model. The dimensions of the sample are 20cm ⇥ 20cm ⇥ 5cm, and the applied stresses ratios, concrete strength and identified parameters are all the same as those presented in the paper of the µ model of Mazars [START_REF] Mazars | A new 3d damage model for concrete under monotonic, cyclic and dynamic loadings[END_REF], see in Table 2.2.

In [Fig. 2.3], the stress ratio, z /f ck , is plotted with the corresponding strains (" z , " y ). The results of the compression-compression and compression-tension are compared between the simulations and experiments of Kupfer. The model provides a good pre-

µ model f ck (MPa) f t (MPa) E (GPa) ⌫ " 0t " 0c A t B t A c B c 32.7 3.2 30 0.2 1e-4 3e-4 1 1e+4 1.25 517
Table 2.2 -parameters and concrete properties [START_REF] Mazars | A new 3d damage model for concrete under monotonic, cyclic and dynamic loadings[END_REF] diction of the stresses and strains, especially for the stress corresponding to the loading direction (Z), whereas there are some over and under estimations for the ultimate stress in the perpendicular direction (Y).

! " ! # = 0.52! " ! " ! # = 0.052! "

Di↵usion equation for drying model

As mentioned in [Section 1.4.1.2], since the drying process of residential building is mostly caused by the exchange of relative humidity between the concrete and the environment. In this thesis, the simplified drying model is used, and is governed by a di↵usion equation :

@h @t = r [D eq (h)rh] (2.1)
where D eq [m 2

.s 1 ] is the coe cient of di↵usion, and depends on the variation of humidity. The expression of D eq (h) is proposed by [START_REF] Xi | Moisture di↵usion in cementitious materials moisture capacity and di↵usivity[END_REF] [START_REF] Xi | Moisture di↵usion in cementitious materials moisture capacity and di↵usivity[END_REF]) :

D(h) = D o h 1 + a ⇣ 1 2 10 b(h 1) ⌘i (2.2)
where D o , a and b are material parameters and depend on the value of e/c.

Boundary condition

The exchange of moisture from inside of concrete to exterior environment happen at the surface of the concrete structure [Fig. 2.4]. Many authors give the suggestion of selecting boundary condition as convection. The convective flux at the boundary conditions for presenting the humidity transfer between the concrete and environment is written as [START_REF] Torrenti | Modélisation du retrait du béton en ambiance variable[END_REF] [START_REF] Torrenti | Modélisation du retrait du béton en ambiance variable[END_REF]) :

q f = H f (c)(h h eq ) (2.3)
where H f is the exchange coe cient for convection, h is the relative humidity of concrete (unknown), h eq is the environmental humidity and c is moisture content. The coe cient H f is obtained by the expression of Torrenti et al. [START_REF] Torrenti | Modélisation du retrait du béton en ambiance variable[END_REF] H

f (c) = f ((2C 0 C eq ) C) (2.4)
whereas, f is the parameter of the material (m/s), C 0 is the initial moisture content, C eq is moisture content corresponding to the ambient humidity, and C is moisture content in concrete. Besides, the relation between the relative humidity and moisture content is described by the model BSB, which is enhanced from the BET model [START_REF] Xi | Moisture di↵usion in cementitious materials moisture capacity and di↵usivity[END_REF] (Xi et al. 1994) :

C(h) = AkV m h (1 k h) [1 + (A 1)k h] (2.5)
where, A, k and V m are the constant of the model BSB. The reconstructed drying model is simulated in the same configuration of the experiments of Granger and Baroghel [START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF][START_REF] Baroghel-Bouny | Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials[END_REF]. This is to validate the correctness of the implanted code and also to identify the parameters of the model. The experiments are conducted on cylinder samples with the diameter of 16 cm, and 10 cm high, in which the samples are at the age of 28 days when starting experiment. The experiments are performed in a room temperature, and lateral surface of the samples are exposed to 50% of humidity. The top and bottom surface of the samples are shielded from drying, and the moisture loss are recorded for di↵erent types of concrete.

The BSB model described in Eq. (2.5) is used for presenting the relation between water content and the relative humidity of concrete. The parameters of Civaux B11 & Penly are already identified in the thesis of Benboudjema [START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF]. Whereas those of BO are calibrated from the water vapor desorption of Baroghel for the concrete of w/c=0. [START_REF]Code de calcule aux eléments finis cast3m[END_REF] In [Fig. 2.8], the evolution of weigh loss are compared between the simulations and the experiments. The drying model and the identified parameters give a good prediction for the weigh loss of the concrete. Furthermore, the results correspond to Civaux and Penly of the simulation are also confront to the simulation of Benboudjema (dashed blue curves). Besides that, the evolution of relative humidity profiles for the Civaux B11 are introduced in [Fig. 2.9]. The humidity distribution is not uniform in the concrete, at the skin, the concrete loss its relative humidity faster than those of the core. It takes years to dry the core of the concrete to the ambient humidity level. The drying shrinkage, which is applied in this thesis, is based on the linear shrinkage model, see in [Section 1.4.3.1]. Even though the linear shrinkage model cannot fully describe all phase of shrinkage strain evolution, but for the variation of relative humidity between 50% to 100%, the model still provide an acceptable results. The incremental shrinkage strain, see in Eq. (1.49), are very similar to the one of the thermal strain.

K h sh is suggested to identify from the linear part of shrinkage strain-relative humidity curve, moreover, [START_REF] Bažant | Drying creep of concrete : constitutive model and new experiments separating its mechanisms[END_REF] [START_REF] Xi | Moisture di↵usion in cementitious materials moisture capacity and di↵usivity[END_REF]) also give the another express to K h sh :

K h sh = " 0 s g s (t) (2.6)
where, " 0 s is the ultimate shrinkage strain (table 3.2 EC2), and g s (t)=E(t 0 )/E(t).

validation of the implanted linear shrinkage model

Granger performed the drying shrinkage and creep tests on the same concrete composition, but the geometry of the samples are di↵erence. For the drying shrinkage, the samples share the same geometry as those of the drying test, see in Section 2.3.1.3. Whereas, the creep test is performed with the samples of 16 cm for diameter and 100 cm height, and 12 MPa is applied on the top surface of the sample, see [Fig. The mechanical properties of the concrete samples are presented in The drying shrinkage simulation are compared with the experiments and also with the drying shrinkage simulation of Benboudjema (dashed blue line), see [Fig. 2.11]. Only the results of Civaux and Penly are presented, and the same phenomenon is observed for the BO. The simulations provide a good prediction of shrinkage strains to those of the experiments on the linear part. In [Fig. 2.12], the deformed configurations of the samples after drying for 3 days, 28 days and 3 years are presented. The contraction zone is observed at the surface of concrete, where the relative humidity greatly reduces.

Conclusion

The linear shrinkage model is implanted, comparison between the simulations and the experiments indicate the limitation of the model. This model cannot describe the asymptotic zone of the shrinkage strain evolution. As described in Section 1.4.3.3, using the nonlinear shrinkage model, Benboudjema presented a good estimation of shrinkage strains to the experiments. Even thought, there is limitation in the linear shrinkage model, but this limitation can capture almost 2/3 of shrinkage strain, and because in the reality, the building is not exposed to complete a drying environment. The environmental humidity is variable, and this gives enough argument for the reason to use linear shrinkage model for the latter simulations in Chapter 3 and 4.

Creep e↵ect of concrete

Creep compliance function of RTCL

The creep model in this thesis is based on the RTCL that is described in [Appendix B.4]. The advantage of this approach is to avoid of storing the stresses history, and this is important for the large finite elements structure. The creep compliance functions of the RTCL are presented by the Kelvin-Voigt model, and are written as :

" creep = Z t 0 J(t, t 0 )d (t 0 ) (2.7)
where the total creep compliance function is the sum of the basic and drying creep compliance :

J(t, t 0 ) = J bc (t, t 0 ) + J dc (t, t 0 ) (2.8)
in which, J bc and J dc are the creep compliance functions corresponding to the basic and drying creep, respectively.

J bc (t, t 0 ) = 1 E 0 + n X i=1 1 E i,bc (t 0 )  1 exp ✓ (t t 0 ) ⌧ i ◆ (2.9) J dc (t, t 0 ) = n X i=1 1 E i,dc (t 0 )  1 exp ✓ (t t 0 ) ⌧ i ◆ (2.10)

Identification of the parameters of creep compliance

Many creep calculations are limited by the aging e↵ect of concrete. Because there is no enough experiments for identifying the parameters of the creep model in function with time. Facing to this di culty, we consider to identify the parameters of the creep compliance function from the creep compliance curve of Eurocode 2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF]. The creep compliance of EC2 is varied with the age of concrete, the relative humidity and the strength. Thus, the elastic parameters of Kelvin-Voigt can be identified at di↵erent age of concrete by using the fitting method.

The creep compliance function proposed by EC2 is the total creep compliance curve, and cannot be judged as the basic or drying creep. An assumption is proposed, by considering that at RH=95%, the creep compliance is dominated by the basic creep, and the drying creep strain is ignored for the high relative humidity of concrete. The elastic parameters of the basic creep are initially identified from J EC2 at RH=95%, see [Fig. 2.13], and then are calibrated for the di↵erent relative humidity by the expression : It has to be noticed that the relative humidity presented in Eq. (2.11) is not obtained from the drying shrinkage model, the relative humidity of the creep model is calculated by a proposed equation, see Eq. (2.12). The objective of the proposed equation is to calibrate the total creep compliance of the Kelvin-Voigt model in accordance with the creep compliance of EC2, and the constants in Eq. (2.13) are obtained from the calibration with the di↵erent relative humiditys. This approach also gives advantage for calibrating the creep model to the results of the experiments, in which the parameters in Eq. (2.13) are changeable. To ensure the correctness of the proposed assumptions, the total creep compliance (J Kelvin ), see in Eq. (2.8), is calculated for di↵erent relative humidity, for example at RH=70%, based on the identified parameters. J Kelvin is compared with the creep compliance of EC2 (J EC2 ) calculated at RH=70%, J Kelvin superimposes on J EC2 , see in [Fig. 2.15].

E bc (t 0 ) = E bc,95% h (2.
h(t) = h i +  1 exp ✓ t ⌧ dry ◆ (h ex h i ) (2.12) ⌧ dry = ✓ D K s ◆ a ; a = 2
The identified parameters of the basic and drying creep compliance are used by the RTCL, see [Appendix B.4], for calculating the incremental creep strain. To minimize the memory used by the calculation, the implanted code is written for storing only the total creep strain, but does not store both of the basic and drying creep strain. This procedure might not satisfy for people who deeply study on the creep e↵ect, but in term of the large finite element structure with di↵erent types of finite elements and complexities in non linear material behaviors, this simplification is necessary.

Validation of the implanted creep model

The total creep tests of Granger are used for the validation of the creep model. The geometry and mechanical properties of the concrete are the same as those of the shrinkage tests, see in Section 2.3.2.2. Exposing to 50% of the ambient humidity and loading 12 MPa simultaneously, the total strains are measured at the middle of the samples.

The creep compliance curves of EC2 are constructed based on the mechanical properties of the corresponding concretes. Then the parameters of basic and drying creep compliance are identified from the creep compliance curves of EC2 through the procedure discussed in Section 2. 3 The total strain of the experiment is compared with those of the simulation, see in [Fig. 2.16]. The total strain is the sum of mechanical strain, drying shrinkage strain and creep strain. The identified parameters lead the results of the simulation to have a good agreement with the experiments. This good agreement is not only depend on the identified elastic parameters of creep compliance function, but it is also serve to calibrate the parameter (K s ) as described in Eq. (2.13). The evolution of stress profiles are illustrated in [Fig. 2.17]. In the core of concrete, the stresses are uniform, whereas on the surface of concrete, the stresses is initially in tension because of the shrinkage e↵ect. Once the relative humidity of concrete becomes stable, the shrinkage e↵ect is reduced, and the stresses return to the compression due to the pre-loading of 12 MPa at the beginning.

Algorithm of THM analysis of concrete

The summary of the algorithm for solving the thermo-hydro-mechanical problem in this thesis is presented in [Fig. 2.18]. As mentioned in the part of introduction, the implanted algorithms of this thesis can be grouped into two tool boxs. First, in the box of thermal and hydric analyses, the generations of temperature, radiation and humidity signals obtained from database of Météo France is developed. Those signals are introduced into the simulations with di↵erent types of boundary conditions. It should be noticed that due to the similarity of the equation of di↵usion between thermal and hydric analyses, even thought in castem, there is no specific tool box to conduct hydric analysis, but in this thesis, the tool box of thermal analysis is used and the parameters of the models are replaced by those of hydric analysis. The second tool box is corresponding to mechanical part. The results of thermal and hydric analyses are converted to thermal and shrinkage strains, and are projected into mechanical simulation as imposed strains. The modules corresponding to creep and nonlinear models of the materials are integrated in the mechanical part. The module of creep model is used to compute the Simplified THM behavior of concrete 79 supplemental free deformation to the structure, but there is no aging e↵ect in the global sti↵ness. The sti↵ness of the concrete is modified by the damage variable, which is activated from the limit strains. Again, in this thesis, it is a weak coupling simulation, there is no interaction from thermal and hydric analyses to the mechanical analysis.

Identification of parameters of elasto-plasticity of the TBS

The TBSs are mostly made of the stainless steel, then the classical elasto-plasticity model for the stainless steel described in Section 1.5.4 is used. However, this model originally used for describing the pure stainless steel behavior, whereas the TBSs are embedded into the concrete walls and slab. Then the model presents the favorable behavior of the TBS, and it cannot be considered that the TBS introduced in the simulations is corresponding to the SLABE ZN r . The numerical tests are performed, and the force-displacement curves are compared between the experiments as described in Section 1.5.2 and the simulations, see in [Fig. 2.19]. The parameters of the model correspond to horizontal and vertical behaviors are identified. For the axial behavior of the TBS, no experiment is performed. Without any good option, the stress-strain curve recommended by the expressions in the annex of Eurocode 3 for the stainless steel [START_REF]Eurocode 3 : Design of steel structures -part 1-4 : General rules -supplementary rules for stainless steels[END_REF] is selected, and the identified parameters are presented in

Conclusion

The simplified modeling of thermo-hydro-mechanical behavior of concrete is developed. The simplified model includes nonlinear behavior of concrete based on damage elasticity model, the drying model, shrinkage and creep model and elasto-plasticity model for the TBS. The parameters of the simplified models are identified from the results of experiments as presented in the literature review. The simplified drying model allows to calculate the relative humidity distribution in concrete, then the drying shrinkage strain is generated from those distributions through the linear shrinkage model. In the calculation of creep strain, the creep compliance function identified from EC2 is proposed, and the elastic parameters of the basic and drying creep are modified by the relative humidity of concrete. Moreover, the elasto-plasticity model of the stainless steel is chosen, and the parameters of the model are identified from the force-displacement curves of the TBS of the experiments.

Anyway, the simplified model still has its limitation. As mentioned for the linear shrinkage model, the drying shrinkage strains do not fully correct at the asymptotic Besides that, the elasto-plasticity model of the TBS purely describes the nonlinear behavior of stainless steel, and does not indicate the behavior of the SLABE at the slabwall connection. Furthermore, there exist some imperfections in the model, but limited to the capacity of the calculation, the scale of the finite element structure and the complexities in the nonlinear behaviors, the errors caused by those simplifications and assumptions should be discussed in terms of the results in chapter 3 and 4, especially while a real site measurement of a building is possible.

Chapter 3

Thermo-Hydro-Mechanical analysis (elastic behavior for the concrete)

Introduction

In this chapter, THM analysis with the elastic behavior for the concrete stands for the reference numerical simulation. The simulation is based on the simplified THM modeling, which has already been presented in Section 2.3. The description of the thermal and hydric analysis begins with the explanation of material properties and boundary conditions. The results of thermal and hydric analysis are discussed. In THM analysis, the numerical modeling at building scale including the concrete elements and the TBSs is introduced. Di↵erent cases studies, which are derived from di↵erent initial conditions of thermal and hydric loading, are defined, and the simulations are performed with 1 year of those loadings. The results are compared between the cases studies, in which the observations take into account the displacement evolution of the walls, stresses and strains distribution in the walls and slab, and the internal forces in the TBSs. Finally, a simplified calculation based on Robot Structural Analysis (RSA) is performed, and this is to strengthen the results obtained from the full simulation of the THM model.

Thermal and hydric analysis 3.2.1 Climatic actions

The climatic actions in this thesis are daily air temperature, solar radiation signals and ambient humidity signal. These signals are obtained from the database of Météo France [START_REF] France | [END_REF], and are corresponding to the Embrun city of France. The selection of this city refers to the previous studies of Heng and Keo [4,[START_REF] Keo | Low-cycle fatigue life of a thermal break system under climatic actions[END_REF], see in Section 1.5.3, which indicated high damage accumulation in the TBSs with the climatic actions of this city.

Chapter 3 : THM analysis (elastic behavior for the concrete)

Temperature signal generation

To construct a complete temperature signal including daily and seasonal variations, two items are concerned. A cosine equation Eq. (3.1) and the daily maximum and minimum temperatures. The proposed equation is based on the study of Peng, see in Section 1.2.1, which presented the measured daily air temperature evolution. In this equation, the T max and T min are corresponding to maximum and minimum temperatures of each day. Furthermore, t max and t min are respectively time corresponding to maximum and minimum temperatures. Those input data are obtained from the database of Météo France [START_REF] France | [END_REF]. In [Fig. 3.1], the daily and seasonal temperature variation are constructed, and are ready for introduce into thermal analysis. The maximum and minimum temperatures of the year obtained from Météo France are then compared to those proposed by EN 1991-1-5 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF]. The national annex of France EN 1991-1-5/NA presents the maximum and minimum shade air temperature with an annual probability of being exceed of 0.02, and for the Embrun city, they are :

T (t) = T min + T max T min 2  1 cos ✓ ⇡ t t min t max t min ◆ ( 3

Temperature

Météo France NF EN 1991-1-5

T max 34 o C 40 o C T min -10.3 o C -25 o C Table 3.

-maximum and minimum temperature

The T max and T min corresponding to the Météo France are lower than those of the EN 1991-1-5 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF]. However, the temperature distribution of the building's walls is not only influenced by the shade air temperature, but also the solar radiation.

Thermal and hydric analysis 85

Solar radiation signal

The solar radiation has an important e↵ect on the results of thermal analysis, especially in term of the dilatation. To construct the radiation signals, two items are necessary. They are the global radiation, and the evolution of sun's angles in order to project this radiation on a vertical plan and to di↵erent orientations (east, south, west, north), seen in [Fig. 3.2]. It should be noticed that the radiation obtained from Météo France [START_REF] France | [END_REF] is a global horizontal radiation, seen in [Fig. 3.2]. Whereas, to compute the sun's angles, a description is given in the literature review, seen in Section 1.2.1. An attention should be payed here, the radiation signal is introduced directly in the boundary condition as flux (q) in thermal analysis, and is not calculated through an equation with the consideration of constant of Stefan ( r ), seen in Appendix A. In [Fig. 3.3], due to space limitation, only some selected projected signals are presented in di↵erent season, and the full variation of the signals are found in the appendix. People might get confuse while see that the radiation in summer is even smaller than the one in winter. Again, the radiation exposed to a building walls is a projected signal on the vertical plan. As seen in [Fig. 3.2], the sun's angle in winter is lower than in summer, this is the reason for which we have more important projected radiation in winter than in summer. Moreover, To include the color e↵ect of the building, the coecient emissivity (" s ) is considered. In the current studies, the simulations are based on the building with the worst situation, then " s equals to 1. However, in the table 5.2 of EN 1991-1-5 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF] proposed " s =0.9 for the dark colored surface. The modified radiation due to the color e↵ect is mostly influence the dilatation e↵ect during the daylight.

Ambient humidity signal

The ambient humidity signal is also obtained from the database of Météo France [START_REF] France | [END_REF], the data are monthly averaged humidity, see [Fig. 3.4]. This signal is used in hydric analysis, and is applied on the exterior surface of the walls, while the interior environment, a constant humidity of 60% is considered. The ambient humidity varies between 50% to 75%, and the lowest humidity takes place in summer. The variation humidity again proves that the envelope of the building is in the extreme drying condition. The limitation of the linear shrinkage model discussed in Section 1.4.3.3 of Chapter 1 is already large enough to estimate the shrinkage strain in the first 2/3 of strain evolution. Besides that, due to the reason of monthly average, several jumps are observed in the signal. But the type of boundary condition in hydric analysis is convection, and those jumps are reduce by the fact of using convection.

Numerical modeling of thermal and hydric analyses

The thermal and hydric analyses are performed separately by Castem [START_REF]Code de calcule aux eléments finis cast3m[END_REF], which are based on the finite elements method, see in Appendix A. Described in the introduction of the thesis, the numerical modeling refers to a quarter of a building's storey (Lshape wall), see in [Fig. 3.5]. A preliminary study of a wall internally insulated and connected to the slab do not present any significant temperature variation in the slab, see in Appendix C. As a result, the thermal and hydric analyses are only performed with the walls. The walls are 16m and 7.5m long for the south and west orientations, respectively, and the choice of those orientations is already explained in Section 3.2.1.2. Furthermore, the walls are 3m height, and 20cm for the thickness of concrete, and are modeled by 8 nodes solid elements. The insulation layers are discussed in the following section.

Thermal properties and boundary conditions

The total thickness of the walls includes the concrete part and the insulation system. In [Fig. 3.6], the external and internal faces of the walls are covered by 1cm of plaster, and 10cm of expanded polystyrene is placed beside the interior face of concrete wall. The thermal properties of those materials are presented in Table 3.2 :

To avoid complicating the modeling, instead of modeling the insulation layers, in appendix D of EN 1991-1-5 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF], the total thermal resistance of the corresponding insulation layers can be calculated through the expression :

R tot = R in + X i d i i + R out (3.2)
and R in =0.13 ; R out =0.04.

h int = 0.27W/(m 2 K) ; T int =20 o C Thermal convection (ext) h ext = 22.5W/(m 2 K) ; T ext =T air Initial temperature, T ini =10 o C Table 3.

-thermal properties and parameters of the model

The air temperature is introduced into the simulation through the boundary condition of convection, and the flux of convection is :

q c = h(T f T ) (3.3)
and the coe cient of thermal transmittance (h) is calculated through :

h = 1 R tot (3.4) h ext = 1 R out + d plaster plaster ; h int = 1 R in + d plaster plaster + d pol pol (3.5)
then this coe cient is calculated for the exterior and interior surface of the wall by taking into account the thermal resistance of insulation materials, see in Table 3.2. In addition, the initial temperature of the walls (T ini ) equals to 10 o C, which is the annual average temperature, and is recommended by EC1 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF]. The air temperature signal is applied to the exterior surface (T ext ) of the wall, while on the interior surface (T int ), a constant temperature of 20 o C described in the table 5.1 of EN 1991-1-5 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF] is used, see in Table 3.2.

Hydric properties and boundary conditions

In Section 2.3.1, the parameters of the simplified drying model corresponding to the BO (ordinary concrete) are used in hydric analysis. The ambient humidity then applies to the exterior surface of the walls by the boundary condition of convection. In Castem, there is no module for conducting the simulation of humidity transfer. Nevertheless, the equations of thermal transfer are similar to the equations of the simplified drying model, see Section 1.4.1.2. Both models are governed by the expressions :

8 > > > < > > > : ⇢C p @T @t r( rT ) = 0 ; Q T = k(T T e ) @h @t r (D(h)rh) = 0 ; Q h = H f (h h e ) (3.6)
then, to perform the hydric analysis, the module of thermal transfer is used with the following substitution :

8 > < > : T , h and , D(h) ⇢ , 1 and Q T , Q h C p , 1 and k , H f (3.7)
The coe cient of convection (H f ) is discussed in Section 2.3.1.2 of Chapter 2. But that expression can only use for the exterior surface, but for the interior surface, while including the insulation layer, then it is not applicable. In the thesis of Tran Le [START_REF] Tran Le | Etude des transferts hygrothermiques dans le béton de chanvre et leur application au bâtiment (sous titre : simulation numérique et approche expérimentale[END_REF], the relation between the coe cient of convection used in thermal analysis and the one used in hydric analysis is described by the number of Lewis (L e ), seen in Eq. (3.8) :

L e = h T H f ⇢ a C pa (3.8)
where, L e = 1 (recommended value), h T coe cient of convection in thermal analysis, seen in Table 3.2 as h ext and h int , H f coe cient of convection in hydric analysis, ⇢ a is the air density (1.2Kg/m 3 ) and C pa is the air specific heat (1005JK 1 Kg 1 ). Due to this reason, since h int used for interior surface already take into account the insulation layer, then H f calculate from Eq. (3.8) is also include those insulation layer. The parameters H f for exterior and interior surface are presented in Table 3.3 : C, respectively. In table 5.2 indicated by EN 1995-1-5 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF] has introduced the expression for estimating T out for building above the ground level :

H f (Ext) H f (Int) 2.32 ⇥10 3 2.15 ⇥10 4
T out = T max + T 5 (3.9)
and based on the national annex, T max corresponding to Embrun city equals 40 o C, and

T 5 =10 o
C for the dark colored surface, then :

T out = 40 + 10 = 50 o C and T max,simu = 49 o C (3.10)
The maximum temperature (T max ) obtained from the simulation is very close to the T out calculated through EN 1995-1-5 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF] corresponding to the dark colored surface. Furthermore, the evolution of temperature profiles in the thickness of south and west walls at t=272 days are plotted, see [Fig. 3.7]. The nonlinear temperature distribution in the thickness of the walls is noticed, and are highlighted for those of the south wall.

As described in Section 1.2.2, EC1 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF] has separated the temperature profile as the uniform temperature ( T U ), linear temperature ( T M ) and nonlinear temperature ( T E ). The temperature profiles obtained from the simulation are decomposed into

M th = N th Y G (3.12)
and Y G is the center of gravity of the surface below the the T profiles.

Chapter 3 : THM analysis (elastic behavior for the concrete)

The T U and T M are illustrated in [Fig. 3.9]. On the south wall, the maximum T U is 28 o C, whereas the T M is not much vary, and is around 10 o C. The seasonal variation is noticed for T U , and becomes important in summer. In addition, thermal strain (" th ) is calculated by multiplying T with ↵ th =1.10 5 . The mechanical stresses exist only when the displacements at the extremities of the walls are blocked, and the stresses are defined as : = " th E c . In [Fig. 3.10], the simplified estimated stresses are calculated from the average " th in the thickness. The maximum tensile stresses on the south and west go above 4 MPa. Anyways, those estimated stresses are rough for describing stresses distribution in the walls. The corner e↵ect and Thermal and hydric analysis 93 TBSs are also the factors that will influence the stresses.

Results of simplified drying model (hydric analysis)

Since the applied ambient humidity signal on the exterior surface of the south and west walls is the same, then the relative humidity distribution for the walls in both orientations is identical. The relative humidity evolution are presented in [Fig. 3.11] for the exterior and interior surfaces. The variation of relative humidity on the exterior surface is in accordance with the applied humidity signal. On the exterior surface, the concrete loses its relative humidity quickly because the walls are exposed to the environment directly. Whereas on the interior surface, because of the insulation layers, the relative humidity drops progressively. The variation of relative humidity in concrete indicate that the concrete wall is not corresponding to an extreme drying case, and this argument supports the assumption of using the linear shrinkage model in the simulation of this thesis. The relative humidity profiles, which are presented in [Fig. 3.12], indicate the skin e↵ect for the shrinkage phenomenon. This e↵ect is known to create the crackings on the surface of concrete, because of the nonuniform relative humidity distribution between the surface and the core. As presented in Section 1.3.2.2 for the experiment performed by CEOS.fr [START_REF]Large beam specimens with restrained shrinkage (rg8)[END_REF], the early age cracking is observed for the restrained shrinkage beam. Comparing to the drying simulation test described in Section 2.3.1.3, the drying process of the building is complex. The uncertainties are caused by dissymmetry of external and internal applied humidity and also the coe cient of convection, which is considered in the boundary condition. This coe cient depends on many factors such as : the velocity of the flux, surface condition, properties of the materials [START_REF] Tran Le | Etude des transferts hygrothermiques dans le béton de chanvre et leur application au bâtiment (sous titre : simulation numérique et approche expérimentale[END_REF].

Conclusion

The objectives of thermal and hydric analysis are to present the climatic actions that are taken into account, and to explain the approach for obtaining and verifying those signals. After that, the boundary conditions corresponding to the signals for thermal and hydric analysis are introduced. Furthermore, the material properties and the technique for performing hydric analysis based on the simplified drying model in Castem are indicated. The initial conditions of thermal and hydric analysis are presented. Anyways, there is no good solution for the choice of the initial conditions. The whole storeys of the building do not experience with the same initial conditions during and after the construction. As explained, the recommendation of Eurocode for choosing the initial condition is adopted.

In thermal analysis, the maximum and minimum of shade air temperatures obtained from Météo France [START_REF] France | [END_REF] are lower than those of the Eurocode 1 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF]. Moreover, the coe cient emissivity is considered as 1 rather than 0.9 of the Eurocode for the dark colored surface. However, the result of the T max = 49 o C in the simulation almost equals to the T out = 50 o C calculated through the national annex of France. Besides that, the estimated stresses indicate the significant tensile stresses in the wall, which go beyond the limit strength of concrete.

In hydric analysis, an important relative humidity variation on the skin is noticed. This observation is considered to generate an extra tensile stresses to the surface of the walls. However, there exists some uncertainties parameters in the hydric analysis, which are explained in the previous section, and the simulation is needed to be verified with the experiments.

Mechanical analysis

Numerical modeling at building scale

The numerical simulation in this chapter is based on a numerical modeling at the building scale, see in [Fig. 3.13]. The dimensions of the walls remain the same as those of the thermal analysis, see in [Fig. 3.5]. In addition, the slab of the building is modeled and is placed at the middle height of the walls. The thermal mechanical analysis with the same degree of shape function for the finite elements are known to generate the detrimental stresses. An example for the bar element is presented in Appendix B in order to illustrate this problem. To avoid those stresses, the degree of shape function for the finite elements are increased when performing the simulation in mechanical part.

In current simulation, the walls are modeled by cubic solid elements with 20 nodes, where in thermal and hydric simulations, only the solid elements with 8 nodes are used. A fine mesh is applied to the corner of the wall, because in the previous study of Bloa [START_REF] Bloa | E↵et de la dilatation thermique des bâtiments munis de boitier isolant structurel slabe[END_REF], a large deformation of the walls is observed at the corner. Besides that, at the middle height of the wall, where the TBSs are distributed for the slab-wall connection, the mesh of the finite elements is also refined. The smallest dimension of the finite element is 3cm⇥3cm⇥5cm, and there are 6 elements in the thickness of the walls.

A band of the slab with 1m width is modeled by cubic solid element with 8 nodes, and the rest is modeled by 4 nodes shell elements. Since a constant temperature is considered for the slab, then no deformation is generated from the slab. As a consequence, the degree of shape function of the finite elements for the slab is lower than the one of the walls. At the connection of solid-shell elements, the hypothesis of Bernoulli is applied, in which the section plan remains plan.

As explained in the introduction of the thesis, finite element modeling of the TBS is considered as the intermediate scale. The combination of shell-spring-shell elements is selected, but there exist also the evolution of di↵erent finite elements combination before the final option is adopted, and they are presented in Appendix C. In [Fig. 3.13], the construction plan and the numerical modeling of the TBS are introduced. The TBS, SLABE ZN r used for internal insulation, is composed of steel profiles (Z & N) and rebar. At the construction site, 4 horizontal rebars are placed in the walls at the same level as the TBSs. 10cm of steel profiles and rebar are embedded into the walls, and the other side is embedded into the slab by lefting a gap of 6cm from the wall to the interface of the slab for installing the insulation layer.

In the numerical modeling, the shell elements are modeled, and are distributed through the length of the walls for representing the steel profiles and the rebars of the TBSs and also in the walls. After that, the thickness of the shell elements is equivalent to the described structural elements. Furthermore, the steel-concrete contact is considered as perfectly rigid. A node of the shell element of the walls is connected to another node of the shell elements of the slab by the spring elements, see [Fig. 3.13]. Then the elasto-plasticity model for the stainless steel, as explained in Section 1.5.4, is applied to the springs. Even though, the numerical modeling of the TBSs is not exactly corresponding to the real situation, but to balance between the accuracy and e ciency, the equivalence thickness is the adequate choice for the modeling at building scale. Moreover, the minimum reinforcement proposed by EC2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF] of the European standard

Boundary conditions

The boundary conditions of the structure are presented in [Fig. 3.14]. The top and bottom surfaces of the walls are applied separately with the rigid displacement in vertical direction (Z), where all the nodes in the lines of the top or bottom surface are displaced vertically the same. The choice of this boundary condition refers to the deformation shape of the wall. In [Fig. 3.14], the cross section of the wall at the middle height of each floor is deformed without rotation. Furthermore, the lateral surfaces of the walls and slab are considered as symmetrical conditions, and the borders of the shell elements are blocked with the rotations. In addition, the lateral surfaces of the slab (red color) are blocked with the vertical displacement (Z) in order to give the stability to the structure.

Thermal and hydric loading

To avoid the confusion of the initial temperature between thermal and mechanical analysis, those initial temperatures are symbolized as T ini for thermal analysis and T ref for the mechanical analysis. The same explanation is also applied to the hydric analysis. The thermo-hydro-elasticity analysis is performed with 1 year of thermal and hydric loading. Even through continuing the simulation over 1 year, the results of displacements and stresses evolution repeat similarly to the one of the first year cycle. Besides that, the thermal and hydric loadings are delayed to 6.5 and 10.5 days in order to avoid the temperature shock at the beginning of the signal due to the di↵erence between T ini and T air , see in [Fig. 3.15]. The selections of T ref at 6.5 and 10.5 days are also to observe the influence of the initial conditions. It should be noticed that T ref in Table 3.4 is the averaged temperature in the walls instead of a uniform temperature. The cases studies are summarized in 3.

Material properties for THM analysis

As presented at the beginning of the chapter, the concrete wall, slab, reinforcement are modeled by elastic behavior, whereas the TBSs is elasto-plastic behavior.

The parameters for the nonlinear behavior of the TBS are presented in Table 2.10 of Section 2.4. Besides that, the steel-concrete contact is considered as perfectly rigid, and the elastic parameters of the concrete and reinforcement are described in Table 3.5 :

materials E (MPa) ⌫ ↵ (1/ o C) reinforcement percentage Concrete 31000 0.2 1E-5 - Reinforcement 210000 0.3 1E-5 A s,v =0.0027A c , A s,h =0.0012A c
Table 3.5 -materials properties Moreover, since the shrinkage and creep are considered in the simulation, then the parameters of drying and drying shrinkage model, which are explained in Sections 2.3.1.3 and 2.3.2.2, respectively for the ordinary concrete (BO), are used. However, the elastic parameters for the creep model are changed, the concrete used in the residential building is supposed to have 30MPa in compression and 3MPa in tension. Then the parameters are identified through the procedure discussed in Section 2.3.3.2. The identified parameters are presented in [Fig. 3.16], and in function of the concrete's age. The elastic parameters increase as the concrete's age increase, and the identified parameters corresponding to the age of 28 days are described in Table 3

Results of thermo-hydro-elasticity analysis

The THM analysis is performed for 1 year of climatic actions, and the calculation with elastic behavior for concrete consumes at least 20 days of calculation time. The observation of the results focus on 3 structural elements, which are the walls, the slab and the TBSs. The stress, strain and displacement evolution are extracted for the walls and slab, whereas the internal forces and relative displacement are for the TBS. Besides that, the fatigue analysis based on the low fatigue strength of TBS SUNE is proposed.

Displacement evolution of the walls

The displacement distribution and deformed configuration, which are depicted in [Fig. 3.17], is corresponding to the maximum U y (t=251.6 days). The U max in X and Y direction happen at the extremity of the walls (corner). For example, on the south wall,

E dc [START_REF] France | [END_REF] E dc [START_REF] Mazars | A new 3d damage model for concrete under monotonic, cyclic and dynamic loadings[END_REF] E dc (8) To clarify the bending e↵ect in the walls, the displacement profiles are plotted through the length of the walls, and are corresponding to the middle height and middle plan in the wall, see in [Fig. 3.18]. The bending e↵ect on the west wall is more important than the one on the south as presented in axial displacement profiles, U y and U x for the west and south, respectively. Furthermore, those bending zones only extend to 2m from the corner. Regarding to the horizontal displacements, they increase proportional linearity from the middle to the corner of the walls. The displacement evolution depicted in [Fig. 3.19] are extracted at the black dot as saw in [Fig. 3.17]. The displacement in X and Y directions are respectively corresponding to the west and south's wall. The displacement in Y direction is larger than the one in X direction, and this is because of the total dimension of the south's wall. A simple estimation of the free displacements based on the averaged " th + " sh in the thickness of the walls is made, and is compared to those of the maximum displacements, see Table 3.7. The estimated displacements are higher than those of the simulation about 25% for U x , and 42% for U y . Those over estimations prove that the displacements of the walls are restrained by the insulated parts of the building.

The comparison of the displacements between case 1 and 2 are described in [Fig. 3.20], in which the displacement di↵erence ( U x = U x,1 U x,2 ) is divided by the U max of case 1. There are 20% and 15% di↵erence between case 1 and 2 for U y and U x , respectively. With lower initial temperature (T ref =5 o C), the displacements are larger

Mechanical analysis 103

From the observation of the displacements, a large horizontal displacement (U y ) is noticed on the south wall. This kind of displacement is restrained by the TBSs, and lead to increase the internal force in the TBSs at the connection south's wall-slab. The estimated displacements also present the e↵ect of displacement restraint in the walls. Those displacement restraints contribute to generate the tensile and compression stresses to the walls system. The displacement profiles give a significant information for the axial displacements. The axial displacement present the bending e↵ect to the walls, especially the west wall with an influence zone of 2m from the corner.

Stress and strain in the wall

As mentioned in the previous section, displacement restraints between the walls and slab are observed. The thermal and shrinkage e↵ects generate the imposed strains, in which the displacement restraints create either compression and tensile stresses in the walls. The stresses distribution and evolution are illustrated in order to understand the harmfulness of those e↵ects in the walls.

-Stresses distribution analysis :

In order to be consistent with the presentation of the displacement, stress distribution ( y ) in the walls is shown in [Fig. 3.21] at t=251.6 days, and is expressed in the global axis. Under the dilatation phase, a uniform compression stress is observed at the middle part of the wall. However, a large tensile stresses concentrate at the corner of the walls, and even while the walls are under the dilatation phase of a day. The origin of those stresses might be caused by the bending e↵ect at the corner. Furthermore, at the corner, the stresses are obviously influenced by the zone with TBSs and without TBSs. A detail stresses analysis is presented in the following section, and the stresses evolution are extracted at di↵erent locations of the walls. C, the maximum tensile stresses are 3.7 MPa and 4.48 MPa for the west and south's walls, respectively. Consistency to those of the maximum displacements (U x & U y ), the evolution of stress profiles one on the west and other on the south wall are presented corresponding to t=202 days and t=251 days, respectively. A large stress variation is observed on the exterior surface, and tensile stresses mostly happen at the night when the temperature drop or the concrete is contraction. In Table 3.8, the estimated stresses calculated from the displacements restraint ( U ) are compared with the stresses obtained from the simulation. It should be noticed that the stresses of the simulation are averaged in the the thickness of the walls. The displacement restraints are obtained by subtracting between the displacement of simulation (U simu ) and the free displacements, see in Table 3.7 of the previous section. The estimated stresses are close to those of the simulation, this proves that the bending caused by the dilatation e↵ect of the south wall as described in Section 3.3.5.1 is limited to the zone of the corner. The stresses in the middle of the walls can be computed from the restrained part of the displacements. The comparisons of the stresses evolution for case 1 & 2, which are presented in [Fig. 3.22], are described by / max , see in [Fig. 3.23]. The stresses di↵erence ( ) is divided by the maximum stress ( max ) of case 1. Those comparisons are made for both, the south and west walls, and they present a very similar evolution. At the beginning, a constant of 10% di↵erence between case 1 and 2 is observed, then a trend is noticed at 100 days < t < 130 days. After that, the evolution becomes constant again. The stresses di↵erence presented in [Fig. 3.23] might be influenced by either the relaxation of creep model or the nonlinear behavior of the TBSs. Normally, with a constant imposed strain, the relaxation stresses should increase as an exponential curve, and become stable as t tends to infinity. As a results, even staring with di↵erent initial conditions, but the stresses di↵erence should become constant while t is large. The creep Chapter 3 : THM analysis (elastic behavior for the concrete) strains evolution corresponding to those stresses evolution in [Fig. 3.22] are illustrated, see in [Fig. 3.24].

estimation ( U , MPa) simulation (middle, MPa) Percentage West x = E c (U x,simu U x,esti ) L x = -1.9 x = -1.62 17% South y = E c (U y,simu U y,esti ) L y = -2.74 y = -2.655 3%
The creep strains evolution presented in [Fig. 3.24] also indicates the trend at 100 days<t<130 days as those of the stresses. The " creep are computed from the elastic strain (" e ) with the corresponding elastic parameters of Kelvin-voigt model. Whereas the " e are obtained from the subtraction of the total strains (" total ), and the " total are depended on the total displacements (U). The total displacements of the walls are computed with the total sti↵ness of the structure including those of the TBSs (spring elements). Then, the plastified of spring elements should be contributed into this e↵ect, and this explanation is keeped to continue to the results of the TBSs in Section 3.3.5.4. As described in the stress distribution, the significant tensile stresses are observed at the corner of the walls. The stresses evolution are extracted for the west and south walls at the corner, see [Fig. 3.25]. Due to the corner e↵ect, the maximum tensile stress at the corner is about 2 times of those at the middle. Regarding to the stress profiles, the exterior surface is always in tension, whereas the interior surface is in compression.

Besides that, the similar stresses di↵erence between case 1 and 2 as the one presented in [Fig. On the interior surface, the stresses are influenced by the zone with TBS and non TBS. To observe those e↵ects, the stresses evolution are extracted for di↵erent locations of interior surface of the wall, see [Fig. 3.26]. At the middle of the wall, the zones with and without TBS are not so di↵erent. Whereas, at the corner, the zone with TBSs presents the significant stresses ( x and z ), and the dispersion is observed when comparing those stresses as the data points, see [Fig. 3.26b]. The zone influenced by the TBSs extends to 2m from the corner, which is coherent to the zone with bending e↵ect as depicted in [Fig. 3.18]. All the observations here also apply to the results of case 2 (T ref = 10 o C). The stresses generated on the walls are complex, and change follow by the locations in the walls and the initial conditions. From the stresses distribution, two remarkable points are noticed, which are : concentrated stresses at the corner due to the bending and the influence zone of the TBSs on the interior surface. The stresses evolution on the exterior and interior surface, at the middle and corner of the walls, are extracted.

A simplified estimation of the stresses is confronted to those of the simulation. This shows that the stresses in the middle part of the wall are caused by the displacements restraint. Whereas, at the corner, the dilatation e↵ect of the other wall creates a bending that generates tensile stresses. However, the stresses either at the corner or the middle of the walls still exceed the limit strength of concrete in tension. Regarding to the interior surface, local stresses created by the TBSs are only observed at the zone of corner, and the maximum of this zone is 2m from the corner. Due to the space limitation, the stresses evolution in the walls for di↵erent locations are not fully presented, but are placed in Appendix D.

Stress results on the slab

The stresses distribution ( y ) on the slab are presented in [Fig. 3.27] at t=202.7 days. The stresses concentrate at the corner, and di↵use to the center of the slab. The stresses evolution are extracted at 2 points in the corner of the slab, see [Fig. 3.28]. The stresses evolution are not seasonal variation, and vary between -2 MPa to 2 MPa. The horizontal deformation of the wall thus has no e↵ect on the slab, and the same conclusion also applies for other cases with di↵erent initial conditions.

T ref = 5 o C)

Results of internal forces of the TBSs

The internal forces in the TBSs are generated by the displacement restraints between the walls and the slab. Since the slab does not present any volumetric change, then the yielding of the TBSs relates directly to the dilatation and contraction from the walls. In the simulation, the TBSs are modeled by the spring elements as described in [Fig. 3.13] of Section 3.3.1, and each TBS (1m width) is represented by 10 springs elements. The force profiles and force-displacement curves are extracted. Besides that, the estimated damage accumulation based on low cycle fatigue strength of the TBS SUNE is performed.

-Analysis the internal forces of the TBS :

In [Fig. 3.29], the force profiles, one on the south and the other on the west. On the left of [Fig. 3.29] are the force profiles corresponding to t=251.6 days, and on the right are envelop curves of the force profiles. The forces profiles illustrate that the important horizontal shear happen to the TBSs on the south, the axial force is for the TBSs on the west. In the envelop curves, we observe that the numbers of spring elements to be plasticized are not the same between case 1 and case 2. For example, in the envelop curve of horizontal shear, there are 8m of spring elements from the corner are plasticized for case 1 and only 6m for case 2.

Furthermore, the influence zone of the axial force and horizontal shear always commence from the corner of the wall, in which the influence zone of axial force is within 2m from the corner. Among the internal forces, it is the horizontal shear that is harmful to a large numbers of the springs rather than the axial force. Moreover, the number of the plasticized spring elements also made evidence to the stress di↵erence between case 1 and case 2 that we observed previously in [Fig. As explained in the force profiles, the important horizontal shear and axial force are corresponding to the TBSs on the south and west, respectively. In [Fig. 3.31], the horizontal shear evolution and shear-displacement curve of the 1 st spring element at the corner are extracted for case 1 and 2. The results of case 2 are only to shift downward the shear evolution of case 1. But if looking into the envelop curve of the displacement profiles, the order of magnitude of plastic displacement and numbers of spring corresponding to case 1 (T ref = 5 o C) are all larger than case 2 (T ref = 10 o C). Besides that, on the right side of [Fig. 3.30c], the evolution of number of the plastified springs are depicted. Again, we notice that the numbers of plastified springs is more important for case 1, and especially while in summer season.

-Axial force of the TBS on the west wall :

The axial force evolution and force-displacement curve of the 1 st spring element on the west wall are plotted in [Figs. 3.31a and 3.31b]. For better understanding, the axial force evolution are presented as the data points. The maximum and minimum axial forces are about the elastic limit strength. But looking into the plastified displacement evolution, see in [Fig. 3.31c], the spring element is almost plastified at every moment compared to those of the horizontal shear. However, the axial and horizontal behavior of the TBS are really di↵erent. Explained in Section 2.4, the axial behavior is corresponding to the perfectly stainless steel given by Eurocode 3 [START_REF]Eurocode 3 : Design of steel structures -part 1-4 : General rules -supplementary rules for stainless steels[END_REF]. Whereas the experiment of TBS for the horizontal loading, see in [Fig. 2.19] of Section 2.4, presents a large reservation of ductility.

Similar to the explanation of the horizontal plastic displacement profiles, we also observe that the envelop curve of the axial plastic displacement of case 1 is larger than the one of case 2. At around t=100 days, the number of the plastified springs for case 1 increase, while the one of case 2 remains constant. In summarized, due to the di↵erent initial conditions, the dilatation e↵ect of case 1 is more important than case 2. This important dilatation creates more displacement to the walls and more displacement restraint to the springs. The plastified in the springs also contribute to influence the stresses in the walls.

-Accumulated damage of the TBSs :

In the literature review, see in Section 1.5.3, Heng and Keo [4,[START_REF] Keo | Low-cycle fatigue life of a thermal break system under climatic actions[END_REF] constructed the fatigue strength of the TBS SUNE based on the experiments for low cycle fatigue loading. Unfortunately, we do not have this results for the SLABE ZN r . However, the strength of the TBS SUNE is known to be lower than the one of SLABE ZN r , because its steel profiles are welding and are sensible to cyclic loading. Purpose of using fatigue strength of SUNE is only to estimate the order of magnitude of the damage in the TBSs based on the horizontal shear obtained from the simulations. The damage accumulation of the TBSs at di↵erent position of the south's and west's wall-slab connection are plotted in [Figs. 3.34a and 3.34b], respectively. The damage of TBSs on the south are the most important, and the damages decrease from the TBS at corner to the middle as an exponential evolution. This kind of damage evolution is also observed in the previous study of Keo [START_REF] Keo | Low-cycle fatigue life of a thermal break system under climatic actions[END_REF] with di↵erent length of balcony. With damage estimation for 50 years, there are at lease 3 TBSs that have D (50y) > 1. Regarding to the TBSs implanted on the west wall, the accumulated damages are small, and they are less than 1 even for D (50y) . Besides that, there are no significant influences from the initial conditions to the damage distribution in the TBSs are observed.

-Conclusion :

The displacement restraints between walls and slab have produced important internal forces in the TBSs that cannot be neglected. Even though the using of designed fatigue strength of TBS SUNE is not corresponding to the real one, still the damage accumulations of the TBSs are large, and need for a better estimation. Anyways, the current studies with elastic behavior for concrete may be considered as a favorable situation for the building. The damages in the wall structure might aggravate the damage accumulation in the TBSs.

Comparison with simplified model of Robot Structural Analysis

Description of the simplified model of RSA

In this section, a simplified numerical model with elastic behavior for the walls, slab and TBSs is developed by using an engineering software, Robot Structural Analysis (RSA). The walls and slab are modeled by the shell elements, whereas the TBSs are represented by the spring elements. It should be noticed that each spring is connected between a node of the wall to a node of the slab. Besides that, the boundary conditions remain the same as discussed in Section 3.3.2. Regarding to the loading, in order to be consistent to the results presented in Section 3.3.5, the " th and " sh corresponding to the maximum displacement (U y ) at t=251.6 days, see in [Fig. 3.19], are used.

Comparison of the results

-Displacement :

The deformed configuration and displacement distribution (U y ) are compared between Castem and RSA. The bending at the corner is also detected by the simplified analysis of RSA, and horizontal displacement of the south wall (U y ) increases linearly from the middle to the corner. The displacements at the corner corresponding to U x and U y are summarized in Table 3 The axial force and horizontal shear profiles are presented together for Castem and RSA, see in [Fig. 3.37]. A similar phenomenon for the axial force and horizontal shear is observed for both simulations. As explained in Section 3.3.5.4, the horizontal shear increases progressively from the middle to the corner, and the axial forces are only significant at the zone of corner. Since the elastic behavior is used for the springs in the analysis of RSA, then there is no interruption from the plastic displacements to the forces in the springs.

Conclusion

The analysis of RSA has helped to assure the results obtained from the simplified THM model at the building scale by simulating with Castem [START_REF]Code de calcule aux eléments finis cast3m[END_REF]. Even though, there exist some errors between the results of Castem and RSA. We consider that those errors are caused by the type of element (solid for castem ; shell for RSA), the distribution of " th and " sh for the loading in shell element of RSA, elastic behavior of the spring and no creep e↵ect is considered for RSA. However, in term of the order of magnitude for

Conclusion

The THM simulation with elastic behavior for concrete commences with thermal and hydric analysis, then the thermal and shrinkage strains are calculated and are imposed into the mechanical simulation. In thermal analysis, the maximum temperature obtained from the simulation is very close to the one calculated from Eurocode 1 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF]. The hydric analysis present the important variation of the relative humidity of concrete on the skin. The nonuniform distribution of temperature and relative humidity are observed in the thickness of concrete.

The THM numerical modeling at building scale is fully described in terms of the finite elements mesh, the choice of boundary conditions, materials behaviors and the parameters of drying, drying shrinkage and creep model. Di↵erent cases studies are proposed in order to understand the influence of the initial conditions, thermal and hydric loading. Case 1 and 2 are corresponding to

T ref = 5 o C and T ref = 10 o C, respectively.

Conclusion 121

The results of the current studies focus on the walls and the TBSs. The stresses on the exterior surface of the walls go beyond the limit strength of concrete in tension, and are highlighted for those at the corner. With the RSA analysis, we also observed a similar stresses distribution in the walls. At the middle, the estimated stresses calculated by displacement restraints are very close to the one obtained from the simulation. Moreover, at the corner, bending caused by the dilatation e↵ect of the south's wall is noticed. This e↵ect contribute to increase the tensile stresses at the zone of the corner. Besides that, the stresses on the interior surface are varied significantly between the zone with TBS and far from TBS. Close to the corner, the concrete elements nearby the TBSs present an important local stress rather than the zone far from TBS. Furthermore, this influenced zone by the TBSs extends to maximum of 2m from the corner. Regarding to the influence of the initial conditions, there are 15% to 25% of stress di↵erence for

5 o C of initial temperature di↵erence ( T ref ).
Based on the fatigue analysis, the significant accumulated damage are observed for the TBSs implanted in the zone of the corner of the walls, especially the south wall. This is not only because of the total dimension of the south wall, but it also due to the important dilatation e↵ect, which is corresponding to the flux of solar radiation. In the current simulation, we observed several TBSs with D (50y) > 1 corresponding to the TBSs implanted on the south wall. Even though, performing the simulation with RSA, the dilatation e↵ect of the walls to the TBSs still indicated important internal forces in the TBSs at the corner.

However, the elastic behavior of concrete structure contributes to limit the global displacements, and results as less plastified in the springs. Since the tensile stresses in the walls are noticed as larger than the concrete strength, it is necessary to pursue a simulation by taking into account crackings in the concrete. A deeper study is proposed, and the THM analysis with the damage elastic behavior for the concrete is presented in the next chapter.

Chapter 4 THM analysis (elastic damage behavior for the concrete)

Introduction

The simulations in this chapter are initiated from the results of the Chapter 3. With the elastic behavior for the concrete, a large tensile stress distribution in the walls, especially at the corner, is observed, see in Section 3.3.5.2. Furthermore, the accumulated damage in the TBS due to the repeated yielding of the horizontal shear is an issue, see Section 3.3.5.4. It is necessary to perform a new simulation by considering the nonlinear behavior of the concrete. The elastic behavior for concrete is changed to elastic damage behavior (µ model). The numerical modeling at building scale, the boundary conditions and the hydric loading remain the same as the one already presented in Section 3.3.1 of Chapter 3. In this Chapter, the objective is also to approximate the simulations to be close to a real building configuration. After switching to the elastic damage behavior, two principal building configurations are considered. They are the configuration without openings and with openings in the walls, at here openings means windows. Moreover, we consider to include the color e↵ect of the building surface by applying " s corresponding to a light colored surface. Besides that, the additional vertical dead load corresponding to the self weigh of the upper floor structures is included. The analysis of the results focuses on the damage distribution in the walls, and internal forces in the TBSs. Those results are crossly compared with di↵erent types of method of analyses and di↵erent types of building configurations.

Thermal analysis for the light colored surface 4.2.1 Generating the solar radiation for light colored surface

The color of the building's surface influences the temperature distribution in the walls. NF EN 1991-1-2 [START_REF]Eurocode 1 : Actions on structures -part 1-2 : Actions sur les structures[END_REF] proposed the expression to calculate the flux of solar radia-Chapter 4 : THM analysis (elastic damage behavior for the concrete) tion by including the coe cient of emissivity (" s ), see in Eq. (4.1).

q s = " s ' ⇥ (T r + 273) 4 (T m + 273) 4 ⇤ (4.1)
where, ' is the shape factor (recommended value by N.A : 1), is the constant of Boltzmann, T r is the e↵ective temperature of solar radiation, T m is the temperature of the element at the surface, and " s is the coe cient of emissivity (table 5.2 EN-1991-1-5) [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF].

In this thesis, the flux of solar radiation is obtained directly from Météo France [START_REF] France | [END_REF]. The flux that is applied to the light colored building surface is calculated as : In table 5.2 EN-1991-1-5, " s for light colored surface is 0.7, and the modified solar radiation with the corresponding " s is illustrated in [Fig. 4.1]. As presented in Section 3.2.1.2, the solar radiation has an important e↵ect for heating the building, changing " s would influence only the variation of the maximum temperature (T max ), but not really for the minimum temperature (T min ). The results of thermal analysis in Section 3.2.5 corresponding to " s = 1 are compared with the one of " s = 0.7.

q s = " s q meteo (4.2)
Q (W/m 2 ) Q (east) Q (north) Q (south) Q (west)

Results of thermal analysis

The temperature evolutions on the exterior surface of the walls are presented in [Fig. The estimation of T out through the table 5.2 indicated by EN 1991-1-5 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF] for the light colored surface building is :

T out = T max + T 4 (4.3)
and based on the national annex, T max corresponding to Embrun city equals 40 o C, and

T 4 =0 o
C for the light colored surface, then :

T out = T max = 40 o C (4.4)
Again, the maximum temperature obtained from the simulation are closed to T out . In Section 3.2.5, only " th is used for estimating the stresses. In this Chapter, the relative humidity of concrete remains the same as presented in Section 3.2.6, the sum of " thm =" sh +" th corresponding to the cases (" s = 1 and " s = 0.7) is used for estimating the stresses. The estimated stresses are calculated through the average of " thm E c , see [Fig. 4.3]. The tensile stresses, which are caused by the contraction e↵ect, still remain at a noticeable value. Whereas the compression stresses calculated by " s = 1 are more important than those of " s = 0.7, and this is because of the reduction in the dilatation e↵ect.

Conclusion

Even changing the " s corresponding to the light colored surface, the maximum temperature calculated through the simulation is still close to the one calculated by EN 1991-1-5 [START_REF]Eurocode 1 : Actions on structures -part 1-5 : Gernal actions -thermal actions[END_REF]. However, the comparison of temperature evolution between " s = 1 and " s = 0.7 shows that the dilatation e↵ect reduces for " s = 0.7, but the contraction e↵ect does not change. Less dilatation may lead to reduce bending e↵ects in the west wall, which have been described in the deformed configuration of Chapter 3. Even though, the estimated stresses are considered as the case with fully displacements restraint and are not really corresponding to the real one, but those averaged stresses still give a sight to tensile stresses in the walls.

Mechanical analysis 4.3.1 Materials properties for THM analysis

The material properties of the TBSs and reinforcement have been already presented in Section 3.3.4, which are elasto-plasticity for the TBS and elastic behavior for the reinforcement. The parameters for shrinkage and creep models remain the same as those have been introduced in Chapter 3. As discussed in Section 2.2, the damage elasticity of µ model is used for describing the nonlinear behavior of the concrete. Besides that, the parameters of the model are described in the uni-axial test of Chapter 2, see in Table 2.1.

Reinforcement in the walls

The reinforcement in the walls are modeled by bar elements, and perfectly rigid of steel-concrete contact is applied, see in [Fig. 4.4]. In the design of reinforced concrete structure, the vertical reinforcement in the walls must be selected between the minimum percentage reinforcement (A v,mini ) and A v calculated from the applied load. Anyways, since the concrete is good with compression, then the concrete strength is enough to resist to the compression load, and A v,mini is applied. Eurocode 2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF] recommends the A v,mini = 0.002A c , whereas the horizontal reinforcement is even less, which is 25% of A v . In the current simulations, we adopt the minimum percentage of reinforcement in the concrete walls, see in Table 4.1. In other word, we do not control the cracking in the walls, and the damage would appear when it passes the limit strength. It must be pointed out that often, in France, a single layer of reinforcement is used on the exterior surface of the walls. There is no reinforcement at all on the interior surface. This case of the reinforcement distribution has not been considered in the present work. In fact, considering only one layer of rebar would have provoked a very di cult convergence in the simulation. Even though with two layers of reinforcement as being used in this work, the numerical simulations were already di cult to realize.

%A

Mechanical, thermal, hydric loadings

As explained in the introduction of the Chapter, the simulations in this Chapter mean to approximate to a real building configuration. Based on the first benchmark of the simulation in Chapter 3, for instant, the constitutive law of concrete is changed to elastic damage behavior. In additional, if considering the floor level as intermediate floor, see in [Fig. Fig. 4.5], then the vertical load corresponding to the self weigh of the upper floors structure is applied to top and bottom surface of the walls as pressure. The dead loads include the self weight of the walls, 3m band width of slab and the permanent load of 3KN/m 2 corresponding to a residential building. Each floor is corresponding to 0.35 MPa of stress in the walls. Considering that there are 9 floors structure on the top the current walls, then it is 3 MPa, that is applied to the top and bottom surface of the walls. Describing in Section 3.2.7, the initial conditions are adopted with the recommendation of Eurocode [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF], in which T ref is the averaged temperatures that the building experiences with. The averaged shade air temperature at Embrun is 10 o C, and the mechanical simulation is delayed to t=10.6 days in order to avoid the temperature shock at the beginning of temperature signal, and also to have an average temperature distribution in walls about 10 o C. The principal case study is summarized in Table 4.2. Taking advantage from this principal case study, two configurations of the building, without and with openings (windows), are performed for the simulations. Besides the principal case, the other case studies with di↵erent initial conditions and coe cient of emissivity are summarized in Table 4.3. Since the simulations are time consuming, those studies from case 2 to 4 is only performed for the configuration without openings. They are served to explore the worst situation that the building may experience with, in term of the cracking in the concrete walls and the damage in the TBSs based on the fatigue verification.

T ref = 5 o C h ref = 82% 3 MPa (intermediate) 1 4
t=6.5 days

T ref = 5 o C h ref = 82% 0.7 MPa (roof) 1
Table 4.3 -summary of the cases studies (without openings only)

Results of thermo-hydro-damage elasticity analysis (principal case + without openings)

In this thesis, we present mostly the results of the principal cases with both configurations. In this section, the results simulation without openings are discussed. The discussion of the results focus on two parts, which are the walls and the TBSs. For the walls, the results presented as the displacements, strains, damage and stresses distribution. Whereas, for the TBS, still the force-displacement curves and the damage accumulation are presented. Furthermore, the comparison of the results with di↵erent types of method of analysis (elastic Vs damage analysis) is performed.

Displacement evolution of the walls

The displacement distribution presented in [Fig. 4.6] is corresponding to the maximum U y at t=246.6 days. With elastic damage behavior for the concrete, the maximum displacements are almost 2 times larger than the one presented in Chapter 3. The deformed configuration also indicates bending on the west and south walls. the bending on the west wall is larger. Besides that, those bending e↵ects are observed within 2 m from the corner. Whereas, the horizontal displacement profiles remain increasing linearly from the middle of the walls.

In [Fig. 4.8], two displacements evolution are extracted for U x and U y at the black dot as saw in [Fig. 4.6]. Two remarkable points are noticed, first the displacements obtained from the simulation with elastic damage behavior for the concrete are larger than those presented in Chapter 3 with the elastic behavior for the concrete. Second, the significant displacement jumps are observed at t=165 days for U x , and at t=126 and 367 days for U y . The existence of the displacement jumps refer to important cracking in the walls. To confirm with this information, the corresponding damages evolution in the walls are extracted in the next section.

Damage distribution on the walls

To give a first look to the damage distribution in the walls, the evolution of the damage distribution are presented for di↵erent instants, which begin from the first 40 days to 415 days, see in Table 4.4. The detail analysis of the vertical cracks and the uniform damage distribution are provided in the following section.

After simulating for 40 days, the damages begin at the corner and go through the thickness to the exterior surface, as well as the local damages in the concrete elements, For 70 days, the vertical cracks on the interior surface of the walls with the total high of the storey are noticed, and those cracks are close to the corner. From 70 to 310 days, the number of vertical cracks on the interior surface increase. Furthermore, the uniform damage distributions on the skin of the walls are developed on the exterior and interior surfaces, except the south wall, only the uniform damage distribution on the interior surface is observed. After 365 days, the vertical cracks on the interior surface of south wall also distribute to the exterior surface through the thickness of the wall. Moreover, the uniform damage distribution on the exterior surface of the south wall, especially at the middle of the wall appears.

0
-Analysis the damage distributions and the displacement jumps :

As mentioned in Section 4.3.4.1, the displacement jumps are corresponding to the cracks in the walls. To better detect the displacement jumps and the cracks, in the post processing of the experiment of CEOS.fr [START_REF]Large beam specimens with restrained shrinkage (rg8)[END_REF], the illustration of displacement velocity ( U/ t) have been proposed. At here, the displacement velocity ( U/ t) and the corresponding damage evolution are extracted from the simulation, see in [Fig. 4.9]. tension and compression occur, whereas there is only the positive principal strain (" 1 ) that present an important increment. This increment of " 1 explains that the vertical crack is caused by the tensile stresses.

Furthermore, one step of damage distribution (t=51.47 days) before the vertical crack (t=51.52 days) has shown that, the damage in the concrete elements surrounding the TBSs already existed, see in [Fig. 4.10]. The development of the vertical crack distributes from the zone with TBS to the zone far from TBS. Besides that, the damage of the concrete elements at the zone with TBS is provoked by the horizontal shear. Because the observation of the horizontal strain (" y ) have the same evolution as " 1 . After the first vertical crack appeared on the south wall, the second vertical crack takes place immediately at the bending zone of the west wall, see in [Fig. 4.11]. To investigate whether this local damage is caused by the bending e↵ect due to the dilatation from the south. The imposed strains (" th +" sh ) corresponding to the south and west are illustrated in the left side of [Fig. 4.11], and the damage evolution is on the right. The maximum damage happens at the same instant while the imposed strain on the south wall comes to the maximum value of the day. Besides that, this local damage is located in the bending zone of the west wall.

Besides the vertical cracks, on the interior surface, the uniform damage distributions on the exterior and interior surfaces only appear on the skin of the walls. As discussed in Section 3.2.6 of Chapter 3, the relative humidity profiles in the thickness of concrete Chapter 4 : THM analysis (elastic damage behavior for the concrete)

-Conclusion :

The damages distribution on the walls are considered as the damage at the corner, damage distribution in vertical lines and uniform damage distribution on the skin. The damage at the corner is noticed for all simulations in this Chapter, and it goes through the thickness of the walls. Besides that, it always occurs before the other damage distributions. The vertical cracks on the south and west walls do not come from the same reason. The vertical cracks on the south wall are due to the horizontal displacement restraints at the connection wall-slab by the TBSs. Whereas the vertical cracks on the west wall locate in the bending zone, which is caused by the dilatation e↵ect from the south wall. Furthermore, the uniform damages distribution on the skin are explained as the contraction of concrete on the surface due to the loss of relative humidity, which is known as the shrinkage e↵ect. As indicated in Section 4.3.2 of reinforcement distribution in the simulations, we do not control the propagation of the crack, the reinforcement is only respect to the standard of the minimum percentage.

Stress distribution on the walls

The maximum and minimum principal stresses distribution corresponding to the first vertical crack on the interior surface at t=51.52 days is illustrated in [Fig. 4.13]. The minimum principal stress is more important than the maximum principal stress. Once the vertical cracks on the interior surface appear, the minimum principal stress on the exterior surface increases, see in [Fig. 4.13 (b)].

The displacement profiles are extracted one for the south and other for the west in order to understand the deformed shape of the walls after the vertical crack taken place, see in [Fig. 4.13 (c)]. The bending e↵ect of the south and west walls are completely di↵erent. On the west, the wall is bent by the dilatation of the south wall. On the south, the wall is bent at di↵erent portions, especially between each vertical cracks.

The stresses evolution at di↵erent locations on the exterior surface of the walls are extracted with the global axis system, see in [Fig. 4.14]. At the first 40 days, the damage distribution is not important, and the stresses obtained from the elastic damage analysis are close to those calculated from the elastic analysis of Chapter 3. But after the vertical cracks and the uniform damage distribution appeared, the stresses evolution on the south and west walls present a very di↵erent evolution.

On the west, even though the vertical cracks takes place (50<t<170 days), the evolution of the stress in compress part is still similar to those of the elastic analysis, and only the tensile stresses are reduced due to the damage distribution on the exterior surface. After t>175 days, the damage on the exterior surface of the west wall is about 0.9, and the stresses tend to zero.

On the south wall, after the vertical cracks on the interior surface appeared (t=51 and 123 days), the stresses change to the compression, and is even larger than those in the elastic analysis. But after t>165 days, while the second vertical crack on the west wall occurred, then y in compression is released.

Those vertical cracks made change to the stresses distribution in the walls, especially the south wall. Since the stresses on the south wall change to compression, that is the reason for which we do not observe the uniform damage on the exterior surface of the south wall until t>310 days, see in [Fig. 4.4]. The mechanical strains, " y and " x corresponding to y and x , are extracted, see in [Fig. 4.15]. This is to clarify the post peak behavior of concrete, because in the stress evolution, after the damage goes to 1, and the stresses reduce to almost zero. After the uniform damage appeared, both " x and " y are all in positive. Those positive strains activate the damage in tension and results to reduce the stress. A general explanation of the forces in the spring element is given in Section 3.3.5.4 while the results of internal force in the TBSs are discussed in Chapter 3. The TBSs absorb more internal force when simulating with the elastic damage behavior for the concrete wall. This come from the observation of the displacement evolution in Section 4.3.4.1. However, the relevant force profiles and evolution of the spring are discussed in the following parts.

-Analysis the internal forces of the TBS :

To be consistent with the maximum displacement (U y ) presented in [Fig. 4.6], horizontal shear and axial force profiles are extracted at t=246.6 days for the spring elements located on the south wall, see on the right of [Fig. 4.16]. Furthermore, the envelop curves of the forces profiles are compared between elastic and damage analysis, see on the left of [Fig. 4.16]. For the horizontal shear, at t=246.6 days, the number of spring elements, which exceed the elastic limit strength of the TBS, is almost 2 times of the number of spring elements simulated with elastic behavior for the concrete. Comparison for the envelop curves, still the damage analysis increases the internal force in the springs. The axial force profiles are also extracted, and are interested. Two remarkable points are noticed from the axial force profiles. First, no matter with elastic or damage analysis, the important axial force always focus at the corner, and is tension force. Second, with damage analysis, while the vertical crack appear, then the axial force in the springs increases, see in [Fig. 4.16b].

The observation in the horizontal shear profiles renews the influence zone, which is a↵ected by the dilatation of the walls. This also gives information that cracking in the walls contributes to increase the internal forces in the TBSs, as well as, the number of TBSs to be plasticized. Furthermore, the axial force is always important for the elastic analysis and damage analysis. Depicted in the force profiles, the 1 st spring absorbs the maximum horizontal shear and axial force, in the following part, the force evolutions corresponding to the 1 st spring at the corner are extracted. The horizontal shear corresponding to the 1 st spring, one on the south and one on the west wall, are illustrated in [Figs. 4.17a and 4.17b] as horizontal shear evolution and shear-displacement curve. At the beginning, while the vertical cracks in the walls are not taken place, the shear evolution for elastic analysis and damage analysis is the same. But once the vertical cracks appear, the horizontal shear increases. As explained in the displacement evolution of the walls, the vertical cracks create several displacement jumps, and those jumps are imposed to the springs as displacement (c-right) number of plastified spring control, and result to increase the horizontal shear. Besides the shear evolution, the shear-displacement curve is interested, the horizontal shear in the spring is plasticized significantly in one direction, and is corresponding to the expansion phase in the walls. If observing the shear-displacement curves between the south and the west, it is noticed that the springs on the south are more concerned than the springs on the west.

The envelop curve of plastic displacement profiles of the springs on the south and west walls are presented on the left of [Fig. 4.17c], and two observations are noticed. The cracks in the walls increase the number of TBSs to be plasticized and also the magnitude of the plastic displacement. First, there are 12 TBSs implanted on the south wall (10 springs = 1 TBS = 1 m) are plasticized and 4 TBSs for the west. The number of plastified springs increase at the same instant while cracks in the wall occurred, see on the right of [Fig. 4.17c]. Seconde, in the evolution of plastic displacement of the 1 st spring, see in [Fig. 4.17c], the increments of plastic displacement are observed at the same instant as those of the vertical crack in the walls.

-Axial force of the TBS :

In [Fig. 4.8], the illustration of the axial force are presented in the same way as those of the horizontal shear. In the axial force evolution, there is not much variation, and the maximum axial forces of the springs at the south and west close to 20 KN, see in [Figs. 4.18a and 4.18b]. But in the axial force-displacement curves, the relative displacement of the spring on the west is more important than the one on the south. Again, those large relative displacement, about 5 mm for the west directly relates to the displacement jumps. To be more precise about the axial behavior between the springs on the south and west, the plastic displacement profiles and evolution are illustrated in [Fig. 4.18c]. First, only the spring elements within 2 m from the corner are a↵ected. The maximum plastic displacement of the spring on the west is almost 2 times of the one on the south. As a result, the axial force should be concerned by the 1 st TBS on the west wall. Observing with the plastic displacement evolution, the increasing of the plastic displacement is corresponding while the vertical cracks take place. The accumulated damage of the TBSs are recalculated based on the shear displacement obtained from the simulation. As described in Chapter 3, the rainflow approach is used for counting the numbers of occurrences corresponding to the maximum forces and displacements. The occurrence numbers are compared with the design fatigue strength of the TBS SUNE, see in [Fig. 4.19]. A point with the highest energy (F max X a ) is observed for a single occurrence, we simply call this point as instantaneous point. The point is corresponding to the displacement jump at t=367.6 days. The evolutions of the relative displacement, horizontal shear and shear-displacement curve are illustrated in [Fig. 4.20] for the first TBS on the south. Those results are presented in order to understand the instantaneous point described in the [Fig. 4.19]. At t=367.6 days, a significant relative displacement and shear are noticed, after that several cycles with small amplitude of displacement and shear are observed, see in [Figs. 4.20a and 4.20b]. The maximum forces corresponding to the instantaneous point is at t=380.6 days. In fact, the rainflow approach counts from t=367 to 381 days as a half cycle, and this half cycle produce the highest energy presented as the instantaneous point.

The illustration of shear-displacement curves, see in [Fig. 4.20c], between the simulation and the one corresponding to experiment of horizontal shear of the SLABE ZN r is to clarify that in a single occurrence, the energy of the instantaneous point remains in the limitation of the SLABE ZN r .

The damage evolutions are plotted for the 1 st TBS at the corner, one for the south and one for the west, see in [Fig. 4.21]. The TBS on the south is much more a↵ected by the horizontal shear than the one on the west. The damage increase significantly while the vertical cracks occurred, especially at t=367 days. The simulation is continued further than 1 year by repeating the same thermal and hydric loadings, the damage accumulation in the TBSs still increases while the cracks happened in the walls. In [Fig. 4.22], the damage accumulation of the TBSs in 1 year is plotted for the TBSs at di↵erent positions. The damage in the TBS declines from the maximum value at the corner to those implanted at the middle of the walls. The same observation is also applied to the TBS on the west. But however, the magnitude of damage of TBSs on the south is 7 times larger than the one on the west. In order to estimate the damage in the TBSs for 50 years (D 50 ) of building's life, the damage accumulation in one year is multiplied with 50. However, this is not a really good way to estimate the damage accumulation for such long duration. It should be noticed that the increment of the damage accumulation relates with the crackings in the walls, and once those crackings reach to their final form, then the damage accumulation might latter become stable.

Even though, due to the fact that the damage accumulation of the TBSs is calculated from the energy based approach. This method is consistent with the Miners rule that does not take into account the e↵ect of previous loadings on fatigue resistance. With this assumption that each annual damage is independent, and the random variable D 50 follow a normal distribution [4].

The D [START_REF] Niyogi | The influence of age at time of loading on basic and drying creep[END_REF] of the TBSs on the south and west, see in [Fig. 4.23], are calculated by multiplying with 50. On the west, there is only 1 TBS that has D 50 > 1 with the damage analysis. Whereas, the elastic analysis does not detect any damage in the TBS. Looking into the damage accumulation profiles of the TBSs on the south, there are 5

TBSs have D 50 > 1 for damage analysis and 2 TBSs for elastic analysis. The TBSs on the south are more concerned to the e↵ect of horizontal deformation from the walls. Table 4.5 -number of TBS with D 50 > 1

In Table 4.5, the results of damage accumulation in the TBS are summarized and are compared between di↵erent method of analyses. An important information is pointed out, the elastic analysis underestimates the damage accumulation in the TBSs, and it is not the conservation method. This information should be concerned by the design engineers whom only use the industry engineering software in order to obtain the internal forces in the TBSs, and finally to estimate the damage accumulation.

-Conclusion :

The results including damage evolution and damage profiles of the TBSs indicate that the cracking in the concrete walls contribute to aggravate the damage accumulation in the TBSs. Furthermore, comparing with di↵erent method of analyses shown that the elastic analysis is not conservation for estimating the damage in the TBSs due to the cyclic fatigue analysis. Besides that, based on the damage accumulation profiles of the TBSs, the TBSs on the south are concerned by horizontal shear rather than those on the west. The damage in the concrete shortens the duration for D T BS = 1, and increases the number of TBS that has D 50 >1. Regarding to the axial force, it is significant, no matter with elastic analysis or damage analysis, the important axial force in tension is always noticed, especially for the 1 st TBS on the west.

Even though, in this fatigue verification, the fatigue strength is not corresponding to the SLABE ZN r , but in term of magnitude of the maximum damage and the number of TBSs, the horizontal e↵ect to the TBSs is at risk and should be taken into account in the structural design of the building.

Results of thermo-hydro-damage elasticity analysis (principal case + with openings

An extended study based on the principal case in the previous section, see in Table 4.2, is proposed at the final stage of this thesis, in which the openings (windows) in the walls are considered, see in [Fig. 4.24]. The motivation to launch the simulation with this configuration means to approximate to a real building configuration. The previous simulation is still considered as a very fundamental simulation, there is rare to have a building without openings in the reality. Other configurations used for the previous simulation including the thermal and hydric loadings, material properties, parameters of shrinkage and creep models, boundary conditions and minimum reinforcement remain the same. It should be noticed that the reinforcement is cut out at the openings of the walls and there are no supplemental reinforcement at the corner of the opening, see in [Fig. 4.24 (b)].

Damage distribution in the walls

To give a more realistic view to the cracks propagation in the walls, supposing that one floor on the top and bottom of the intermediate floor are not influence by the floor level, and they might present a similar cracks propagation, then the damage distribution of the intermediate floor is copied to the upper and lower floors, see in Table 4.6.

Interior Exterior

Table 4.6 -evolution of damages distribution on the walls, 40 -415 days

In [Table 4.6], the evolution of damage distribution on the exterior and interior surfaces of the walls are illustrated at the first 40 days and after simulating for 1 year. For the first 40 days, the vertical cracks at the corner, at the bending zones and at the openings are noticed. The vertical cracks at the openings even get through the thickness of the walls. An attention should be payed to the crack propagation after simulating for 1 year. If observing on the exterior surface, the damage distribution seem not to Chapter 4 : THM analysis (elastic damage behavior for the concrete) be degradation, since, there are only the common cracks at the openings and at the corner. If only looking to the damage distribution on the exterior surface, it is hard to imagine that there exists such degradation on the interior surface.

The simulation estimates a large crack propagation for the interior surface. If those kinds of crackings do exist in a real building configuration and are hidden behind the insulation layers, then it should be worried. Even worse, those damage distributions are not only limited to the building equipped with the TBSs, but also to a building constructed with a conventional wall-slab connection. An supplemental simulation with the conventional wall-slab connection is confirmed with those damage distributions. Those vertical cracks are caused by the displacement restraint between the walls and slab. Observing on the real buildings surfaces, the similar cracks at the openings are noticed, see in [ Even though the simulation has been switched to a real building configuration with the openings, the vertical cracks are inevitable. Those damage distributions obtained from the simulations with or without openings should become an argument to question to the minimum reinforcement, and also the consideration of thermal loading in the design of the buildings. In the national appendix of France in Eurocode 2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF], it even proposes no reinforcement in the walls while the walls is under the compression load. However, the simulations in this thesis are conducted with the extreme climatic condition, and it remains a long way to go through in order to verify if the minimum reinforcement is enough to prevent the walls from such crackings.

Internal forces in the TBSs

As observed for the internal forces with the previous simulation, see in Section 4.3.4.4, the crackings in the concrete walls create the displacement jumps, and those jumps are concerned to the springs on the south and west as horizontal shear and axial force, respectively. In [Fig. 4.26], the horizontal shear and axial force corresponding to the springs on the south and west are depicted. Furthermore, the comparisons with the elastic analysis are provided. For the horizontal shear, still the result with damage analysis is larger than the one with elastic analysis. The shear-displacement curve presents that the spring is plas-Chapter 4 : THM analysis (elastic damage behavior for the concrete) ticized in one direction, which is the expansion phase, and the relative displacement could go to 5 mm. For the axial force of the spring on the west, with damage analysis, the springs are significantly plasticized. Moreover, no matter with elastic or damage analysis, the amplitude of the axial force is not so di↵erent.

Accumulated damage of the TBSs

As already explained in Sections 3.3.5.4 and 4.3.4.4 with the method to calculate damage accumulation in the TBSs, in this section, the damage profiles in the TBSs (D 50 ), which are estimated for 50 years of building's life, are illustrated in [Fig. 4.27]. Again, those damage profiles are obtained by multiplying the annual damage accumulation in 1 year with 50. For the TBSs on the west, considering with the openings, result from both elastic and damage analysis do not have the TBSs with D 50 > 1. But, for the TBSs on the south, there are two remarkable points should be pointed out. First, changing the configuration to the openings, the numbers of TBSs with D 50 > 1 reduce. Second, the elastic analysis still underestimates the damage accumulation in the TBSs, and the damage analysis presents two TBSs with D 50 > 1, see in Table 4.7. In this section, the summarized of the results corresponding to cases 2, 3 and 4 are presented. The observations focus on the damage distribution in the walls and the accumulated damage in the TBSs. As stated in Section 4.3.3 that the proposed simulations mean to explore the worst conditions that the buildings might be experience with, and are only performed for the configuration without openings.

Damage distribution in the walls

The damages distribution in the walls for di↵erent cases are presented in [Figs.

T ref = 5 o C h ref = 82% 0.7 MPa (roof) 1
In the case of roof floor [Fig. 4.30], the damages distribution change. Instead of having a specific local damage as a vertical line, at here, there are only the local damage zones at the corner of the interior surface. Those damage zones locate at the concrete elements that are surrounding the TBSs. On the exterior surface, the damage distribution is uniform. But di↵erence from the intermediate floor, the uniform damage does not only happen to the skin, there is almost half of the walls thickness are damaged.

The mechanical loading has much influence to the damage distribution in the walls rather than the initial conditions or the colored e↵ect from the building's surface. The imposed mechanical loading is the opposite e↵ect to the climatic actions (" th + " sh ). The compression load has compensated with the tensile stresses caused by the shrinkage As described in Chapter 3, the calculation of the accumulated damage in the TBSs is based on the designed fatigue strength of the TBS SUNE. In this section, the da-mage evolutions of the 1 st TBSs at the corner of the south and west are illustrated in [Fig. 4.31]. For the TBS on the south, again in case 2 and 3, which are corresponding to the intermediate floor, present an important damage increment while the vertical cracks in concrete occurred. Whereas for the roof floor (case 4), the damage evolution is far lower than those 2 cases. Regarding to the TBS on the west, the damage in the TBS remains small, and there are no significant di↵erence between the cases studies.

-TBSs on the south :

In [Fig. 4.32], the accumulated damage for 1 year and 50 years are calculated for the TBSs implanted on the south wall. As described in the previous section, D 50 is obtained by multiplying the annual damage accumulation of 1 year with 50. Each figure, the damage of the TBSs are plotted in function with the positions of the TBS, which begin from the corner. The damages reduce from the corner to the TBSs located at the middle. For damage accumulation of 50 years, there are 8 TBSs with D 50 > 1 for the intermediate floor cases, and only 3 TBSs for the roof floor. This results give a very interested information, in which the colored e↵ect seems not much influence to the damage in the TBSs rather than the floor level. The accumulated damages of the TBSs implanted on the west wall are presented in [Fig. 4.33]. No matter in which conditions, there is only the first TBSs at the corner indicates the highest damage degree, and even for the intermediate and roof floor level. The TBSs on the west are not really concerned by the horizontal shear e↵ect.

-Conclusion :

As presented in the accumulated damage for the TBSs at di↵erent positions, the TBSs on the west are not a↵ected by the horizontal shear, but they may be concerned by the axial force in the TBSs. The summary of the numbers of TBSs with D 50 > 1 are described in Table 4.8. For the west, there is no TBS with D 50 > 1. But, for the south, cases 2 and 3 present a large number of TBSs with D 50 > 1. It gives an important information that lower the reference temperature (T ref ), the damage accumulation in the TBSs increases. Furthermore, the same observation for the elastic analysis is that this method underestimate the internal forces in the TBSs and results as a non conservation to evaluate the damage in the TBSs.

D

Conclusion

The proposed simulations with the nonlinear behavior for the concrete have been performed in order to understand the contribution of the crackings in the concrete elements to the internal forces in TBSs. Case 1, which is considered as the principal case study, is corresponding to the intermediate floor with 3 MPa loading on the top and bottom surfaces of the walls. Besides that, the simulations of principal case (case 1) are performed with two configurations of the building, which are without openings and with openings. Others cases studies, see in Table 4.3, are simulated only with the configuration without openings, and mean to find the worst situation that the buildings might confront.

For the results of the damage distribution in the walls while simulating without openings and with openings, there are two remarkable points are noticed. First, both configurations present a similar crack propagation, which include the vertical cracks at the corner and at the bending zones, crack of concrete elements at the interface of the TBSs. But with the openings, there exist supplemental vertical cracks at the openings of the walls, and this is because of the reduction of e↵ective cross section.

Second remark is corresponding to the damage distribution between the exterior and interior surfaces. The cracks happened on the interior surface is more important than the one on the exterior surface. Most of the vertical cracks and cracks at the interface of wall-slab connection are observed on the interior surface. Furthermore, those crackings are not obviously noticed since they are hidden behind the insulation layers, and are not monitored.

The vertical cracks corresponding to the south and west walls do not come from the same reason. On the south, the damage initially take place at the concrete elements that is surrounding the TBSs, especially for these close to the corner. The damage in these elements are caused by the shear restraint of the slab through the TBSs. The vertical cracks are then developed from those damage concrete elements at the interface of TBSs. Whereas, on the west, the vertical cracks take place in the bending zone. Initially, the concrete elements at the interface of the TBS are damage due to the dilatation e↵ect from the south, and the displacements are restrained by the TBSs in axial direction. Once the dilatation e↵ect becomes more and more important, then the west wall is bent and the vertical cracks occurred. It should be pointed out that while the vertical cracks took place then the displacement jumps are noticed, especially for the extremity with free displacement. Those jumps are applied to the TBSs at the corner as the large imposed displacements, and should be worried.

For the TBSs, the damage accumulation is obtained from the fatigue analysis. It should be mentioned again that the choice of fatigue strength of the TBS for estimating the damage accumulation is not the one corresponding to the TBS used for wall-slab connection (internal insulation), but for balcony-slab connection (external insulation). There are also two focal points regarding to the results of internal forces and damage accumulation in the TBSs. First, as concluded in the previous paragraph, a large imposed displacement caused by the jumps concerned to the TBSs at the corner under two directions. For TBS on the south, it is horizontal shear, and for TBS on the west, it is axial force. The horizontal shear and axial force are plasticized in one direction, which is during the expansion phase in the walls, and for the TBS on the west, the tensile force dominate.

Second, comparing the damage accumulation in the TBSs with di↵erent method of analyses, elastic analysis and elastic damage analysis, the elastic analysis is not conservation to estimate the damage accumulation in the TBSs. Both configurations, without and with openings, present that the crackings in the walls create the jumps, and the jumps increase the damage in the TBSs. As a result, there are several TBSs have D 50 > 1, see in Tables 4.5, 4.7 and 4.8. An attention should be payed also for the TBS on the west, which is a↵ected by the axial force, there is no fatigue analysis for the TBSs corresponding to the axial force. But from the results, the axial force remains important in tension, and does not relate to the methods of analyses or any configurations of the building (without or with openings).

Conclusion 163

In the case of roof floor, there is more damage in the walls, and almost half of the walls thickness from the exterior surface are damaged. Besides that, the local damage zones take place at the corner of the interior surface, and the concrete elements nearby the TBSs, as well as, no vertical cracks are observed. The floor level has significantly influenced the damage distribution in the walls. However, those damage distributions seem not contribute to aggravate the damage accumulation in the TBSs. The order of magnitude of the damage and numbers of TBSs for D (50) > 1 are not so di↵erent from the elastic analysis. A quick loss of the structure sti↵ness before the important dilatation e↵ect arrive might be the reason for the observation in the TBS.

Between the parameters : color e↵ect of the building surface and floor level, it seems like the floor level or the imposed mechanical load in the walls has played an important role to influence the results in terms of damage in the walls and in the TBSs. The dark and light colored surface of the buildings are not much changing in the results, as seen in Section 4.3.6. Even with the light colored surface or less dilatation e↵ect, the damage in the walls and TBSs remain an inevitable phenomenon.

In summary, the dilatation and contraction e↵ects of the walls to the TBSs are concerned by the intermediate floors of the buildings. In the worst case of the simulations, while considering a building is constructed during the cooling season or with low reference temperature, the numbers of TBSs with D (50) > 1 are 3 to 4 times larger than the one obtained from elastic analysis. In other word, there are 8 to 9 TBSs with D (50) > 1, then the cracks in the walls have a↵ected to the damage in the TBSs. Even though, in the fatigue analysis, the fatigue strength is not corresponding to the TBS used for internal insulation, but the order of magnitude of the damage in the TBSs and the numbers of TBS with D 50 > 1 still present an aggressive value that cannot be neglected. The large damage distribution in the walls, especially for the interior surface should be worried. This observation also questions to the minimum reinforcement in walls and the distance of the dilatation joint. 166 Conclusions and perspectives the numerical point of view, it still existed the complexity in the nonlinear behavior of the concrete. Therefore, the modelings of concrete walls and slab were made of mixing volumetric elements and shell elements, where the partial refined meshes were used at the corner of the walls and the concrete elements surrounding the TBSs. An attention has been addressed to the modeling at the interface between those two kinds of finite elements. In addition, representation TBS in numerical model again gave another complexity, a simplification by using shell-spring-shell elements was proposed.

Once the simplified THM model was implanted and the numerical model was developed, first computation was performed with an elastic behavior for the concrete and elasto-plasticity for the TBS, in order to limit computation time. The objectives of this computation were to validate the model, to understand the behavior of the building and to analysis the e↵ects of the di↵erent assumptions that were made. As could be expected, these elastic computations presented large stresses and strains in the walls, especially at the corner. As a consequence, a new computation by taking into account of crackings in the concrete was mandatory. Moreover, significant repeated yielding in the TBS was noticed, and a prospective verification of fatigue strength of the TBS was carried out.

Afterward, a full nonlinear simulation was conducted with a limited number of case studies due to the heavy of computation time. In these simulations, they confirmed the formation of several cracking patterns in the walls, which are : a uniform crack propagation on the exterior surface, vertical cracks through the thickness of the walls from the interior surface and approximated to the corner, and the last is the damage in concrete surrounding the TBS, mainly at the slab level. Those crackings are not surprise since the walls are not well reinforced. The minimum reinforcement percentage is not enough to be able to control the cracking. A similar conclusions have been obtained while a case of usual building without TBS is simulated. In addition, the nonlinear simulation indicated more important repeated yielding in the TBS, and risk of failure in fatigue for the TBSs near the corner of the buildings. Simulating only 1 year of the life of a building, the results shown that a major damage in TBS increases while the cracking forms in the walls. It is likely that, once the cracking in the walls reaches to its final form, the damage growth in the TBS will be limited.

The severity of those conclusions must be attached with the following considerations. The simulations are performed with the extreme climatic conditions, and the cross walls are not considered in the current configuration. The presence of the cross walls might generate the supplemental sti↵ness and limit the relative displacement between the wall and slab. Even if the TBS seem at risk of failure with the current simulations, but the resistant in fatigue is not corresponding to the real one, as this resistance has not been determined experimentally for any TBS made for internal insulation. At here, it is adopted from the design resistance of a TBS used for external insulation, which is more critical in fatigue. In this thesis, the estimation of the damage in TBS for 50 years is only based on the damage accumulation in 1 year, and this is thus largely unfavorable. However, the results in this thesis cannot give further comment on this topic. Moreover, numerous simplification and hypothesis had been done in order to be able to perform the computation.

As a general conclusion, this thesis has set in evidence that the rebar density in the walls in France is not su cient, and this results as large cracking pattern that can be observed on any real concrete buildings. On the other hand, the TBSs are at risk of fatigue failure. Therefore, the last conclusion calls for the determination of fatigue resistance of the existing TBSs, and for an experimental investigation of forces in TBS in a real building. This thesis presented its limitation in the full numerical analysis, the model is not powerful enough to conduct a numerical simulation of a realistic building with a significant duration. The experimental observation is urgent for pursuing a step further, in terms of modifying a better THM model, and also producing an ecient model for representing the behavior of TBS in building with internal insulation. In the following, a more detail contribution of this work, and the perspectives are given.

Detailed research contributions

-Implanted simplified THM model in finite element code :

Due to the complexity of the e↵ects to be included and the scale of the finite element structure in this work, based on the analyses in the literature review, we selected and reconstructed the models by balancing between the complexity, accuracy and e ciency. Our capacity of calculation machine cannot support a simulation with consideration of all parameters. A simplified modeling of THM behavior for the concrete, which includes a simplified drying model, linear drying shrinkage model, Rate type creep law, elastic damage behavior for the concrete and the elasto-plasticity model of stainless steel for the TBS, is developed, and is integrated into the calculation code of CASTEM.

To ensure the correctness of those implanted models, the simulation tests are performed, and the parameters are identified from the results of the experiments as described in the literature review. For creep model in this thesis, we introduced the creep compliance function of Eurocode for calculating the incremental creep strain. Before this, the creep including aging e↵ect faces to the di culty of identifying the elastic parameters in function of time. All of those complexities are also the reason to limit our numerical model at building scale to only a quarter of one storey.

-THM simulation at building scale with elastic behavior :

After the simplified THM model is constructed, an analysis at building scale with elastic behavior for the concrete and elasto-plastic behavior for the TBSs is made.

Conclusions and perspectives

Reason for starting with the elastic analysis is to economize calculation time. Besides that, this analysis is also used for understanding the response of the proposed numerical model at building scale, the implanted nonlinear model and the selected boundary conditions.

Afterwards, there exist the double complexity in numerical modeling for the mechanical analysis such as solid-shell combination and representation of the TBS by the macro elements, spring. Those simplified numerical modelings are important and have been tested, but they are not all depicted in this thesis due to the space limitation. The solid-shell has been used for the slab modeling, furthermore, the shell elements with equivalent thickness were used for representing the rebars in the walls, and in the zone of TBSs. In this analysis, the temperature and relative humidity distributions in the concrete are defined with time evolution. A significant non linear temperature gradient is observed in the south wall, and this non constant temperature generates important bending e↵ect in the walls. Investigating the relative humidity profiles, the skin e↵ect, which is related to a quick loss of relative humidity on the exterior surface of the wall, was observed in the hydric analysis. Those nonuniform relative humidity distribution between the core and surface of concrete is a common issue that is known to generate tensile stresses, and result in cracks on the surface.

In the analysis with the elastic behavior for the concrete, significant tensile stresses were observed at the corner, and are almost two times larger than those on the middle of the walls. On the other hand, no important stresses were observed for the slab. Besides that, the deformed configuration of the walls illustrated an important bending e↵ect to the wall on the west. This bending e↵ect is caused by dilatation from the south wall, which pushes the west wall to bend. Due to the double e↵ects, large dimension and significant solar radiation signal, the horizontal displacement on the south wall is essential. Large horizontal displacement means more imposed displacement to the TBSs. As a result, the horizontal shear has a↵ected the TBSs on the south, especially for those close to the corner, whereas it has less influence on the TBSs of the west. Regarding to the axial force in the TBSs, as noticed for the bending e↵ect in the walls, the TBSs at the corner of the west wall pertained the maximum axial force due to displacement restraint of the dilatation e↵ect from the south wall. A fatigue damage estimation for the TBSs, which is caused by repeated horizontal shear, is made. Considering the building's life as 50 years, many TBSs exceed the damage value of 1.

Due to the fact that the tensile stresses in the elastic analysis exceeded the limit strength of concrete in tension, it is necessary to upgrade the current simulation to nonlinear analysis with an elastic damage behavior for the concrete. This attempt is to investigate the cracking patterns in the walls, and whether the damage in concrete would influence the results of damage accumulation in the TBSs. If there is no significant change in the results between the elastic and elastic damage analysis, then we might be able to limit the studies in the future to only the elastic analysis.

Conclusions and perspectives

169

-THM simulation at building scale with elastic damage behavior :

As stated previously, the analysis with elastic damage behavior for concrete is proposed. In this simulations, the conclusions should be grouped into two groups, which are the methods of analyses and the configurations of the finite elements modeling. In addition, there exists a limit of parametric study, which includes the color of the building surface, reference temperature and floor level.

Writing in the conclusion, only the conclusions corresponding to the simulation with elastic damage analysis and configuration with openings are focused. First, the displacements in the walls for the simulation with nonlinear behavior in concrete are larger than those of the elastic analysis, and several displacement jumps, which are corresponding to the vertical cracks in the walls, are observed. Besides that, several types of cracks propagation in the walls are underlined. They are the cracks at the corner of the wall, uniform crack on the surfaces, and vertical cracks through the height of the storey that even get through the thickness of the walls.

The crack at the corner is caused by the contraction of the walls from the south and west, because it occurred while the temperature in the walls dropped. The uniform crack on the surface is provoked by shrinkage e↵ect, because it happened only on the surface of the walls, and especially on the exterior surfaces, which are exposed to the environment directly. The local damages, which are considered as the vertical cracks do not come from the same reason for those on the south and west walls. Those on the south are caused by the shear restraint of the slab through the TBSs, and the damages are initially distributed from the concrete elements at the interface of the TBSs. Whereas, for the vertical cracks on the west wall, they are located at corner or in the bending zone of the wall, and again they are initiated from the damage of concrete elements at the slab-wall interface, which is caused by the axial force in the TBSs. Those crackings are not surprise in the concrete building, especially while the reinforcement is not calculated for crack control, and they are not only limited to the building equipped with TBSs.

Regarding to the internal forces in the TBSs, for the axial force, it remains the TBSs on the west that absorbed the maximum axial force. The plastic displacement obtained from the elastic damage analysis is larger than the one calculated through the elastic analysis. Furthermore, the significant increment of the plastic displacement is corresponding with the displacement jumps. Moreover, the TBSs concerned by axial forces are only located in a zone within 2m from the corner, and those axial forces are in tension, and do not influence by the methods of analyses or the configurations. The horizontal shear mainly focuses on the TBSs implanted on the south wall. Comparing to the elastic analysis, the results obtained from the elastic damage analysis are double in terms of the order of magnitude of the damage accumulation in the TBSs and the number of TBSs with D 50 > 1. Whereas, the D 50 of TBSs on the west still stay below 1. The current simulation made evidence that cracks in the walls, especially the Conclusions and perspectives vertical cracks, gave a significant influence to the damage in the TBSs. This observation also indicates that the elastic analysis is not conservation to estimate the internal forces and the damage accumulation in the TBSs. The attention should be payed by the design engineers whom only estimate the force in the TBSs by the industry software.

As mentioned, three studies over the surface's color, di↵erent reference temperatures and floor level are investigated by the elastic damage analysis. For the surface color, there are light and dark colored surfaces are taken into account. The reference temperatures while the buildings are constructed, at here, 5 o C and 10 o C are chosen. For the floor level, the intermediate floor and roof floor are carried out by imposing mechanical load in the walls. The surface's color does not much influence to the damage patterns that have been described in previous paragraph. Indeed, for the dark colored surface, it presented more damage in the walls than the light colored surface. Concerning to the TBSs, there is no significant change to the damage accumulation in the TBS.

Regarding to the reference temperature, lower reference temperature has increased more dilatation e↵ect and generated more damage to the walls, but sill remain a similar damage patterns as illustrated before. Besides that, the damage accumulation in the TBSs become more important than those simulated with higher reference temperature. Surprisingly, for the roof floor level, the damage distribution in the walls are aggravated. The uniform damage does not only happen to the surface, but it distributes into the thickness of the walls, and there are some local damages on the interior surface, especially at the slab level where the TBS is connected between walls and slab. But, no vertical cracks are noticed in the walls. In contrast, when observing the damage accumulation in the TBSs, we did not find out any important di↵erence between the elastic and elastic damage analysis. Quickly losing a large part of the sti↵ness of the structure, and not even until the important dilatation e↵ect taken place might be the reason for this observation.

The results and conclusion of the simulations in this thesis are the benchmark for the future researches in this domain. They still need a deeper study even if the complex studies are time consuming. There are some remarks that can be extended from the current simulation. First, would the damage distributions in the walls remain similar after simulating for several years of loading signals ? As we have seen that the cracks in the walls have a↵ected to the damage accumulation in the TBSs, then the number and duration, for which the TBSs are fully damaged, remain uncertainty part. Besides that , the consideration of 1/4 of a building scale and with openings, and the symmetry conditions are the remaining works that have not been fully clarified. Nevertheless, this thesis presented an essential damage in the TBSs due to dilatation and contraction e↵ect from the walls that cannot be neglected, and it is important to go further for the studies in this domain.

Perspectives

The numerical investigations of this thesis try to include all e↵ects that the building might experience with. The observations and questions raised in this framework are to draw an attention to the problem with the building equipped with TBSs under the climatic actions, that have never been noticed. Nevertheless, the current simulation must be compared to a real constructed building in order to improve or validate the selected simplified THM model, and the simplifications that have been made in the numerical modeling.

-Experiment :

From the experimental point of view, there are several experiments relate to the concrete and TBS should be performed. First, for the concrete, the measurement of temperature and relative humidity in concrete with the configuration of wall + internal insulation under the climatic condition should be established. Those results would help to validate and to improve the parameters in thermal and hydric analyses.

Regarding to the experiment in the TBS, the investigation of the axial behavior in the TBS and the construction of fatigue strength of the TBS used for wall-slab connection should be described experimentally. Those experiments are necessary in order to identify the parameter of the elasto-plasticity model corresponding to the axial behavior of the TBS. Besides that, the real fatigue strength curve would help to estimate accurately the damage accumulation in the TBSs and also the duration for reaching D T BS to 1.

Last but not least, the building constructed with the technique of reinforced concrete walls, equipped with or without TBSs and with internal insulation should be monitored. The simulations estimate a large crackings and disorder of the concrete elements on the interior surface. But in reality, those information are not possible to be noticed due to the insulation layers. Again, this seems like the history of the corrosion of steel roof structure hidden by the ceiling. The measurements of the displacements on the wall and slab, the strain, crack opening and forces in the TBSs should be monitored in the protocol of the experiment.

-Simulation :

From simulation point of view, there are also several improvements should be considered. The nonlinear behavior of steel-concrete contact for the TBSs remains the missing work in this thesis. This local e↵ect might a↵ect the internal forces in the TBSs. The plasticity in the reinforcement should be included. As seen in the simulation of Chapter 4, once the concrete cracks, the force in the concrete drops and it is absorbed by the reinforcement. Those forces are large enough to plasticize the reinforcement in

Source de chaleur ponctuelle

Une source de chaleur ponctuelle Q est définie par la puissance thermique reçue par le système. Elle s'exprime en W.

Source de chaleur volumique

Une source de chaleur volumique q est définie par la puissance thermique générée par unité de volume. Elle s'exprime en W/m 3 .

Figure A.1 -solide under the di↵erents thermal e↵ects

The heat equation in a solide V can be written as :

8 ! x 2 V : ⇢C p Ṫ div( • ! gradT ) q v = 0 (A.1)
with the law of Fourier :

8 ! x 2 V : ! q ( ! x ) = • ! gradT (A.2)
and boundary condition : W/(m 2 .K 4 ). ' s imposed flux (W/m 2 ).

8 ! x 2 S T : T = T p (A.3) 8 ! x 2 S ' : ! n • ( • ! gradT ) = ' s + h(T f T ) + " r r (T 4 1 T 4 ) (A.4)
To solve the equation Eq. (A.1), we are using the formulation of Galerkin by multiplying equation Eq. (A.1) with a test function T ⇤ and do intergration in the domaine of V

Z V T ⇤ ⇢C p Ṫ dV + Z V ! gradT ⇤ • • ! gradT dV Z S' T ⇤ (' s +h(T f T )+" r r (T 4 1 T 4 ))dS Z V T ⇤ qdV = 0
(A.5) for x 2 S T , T ⇤ = 0, T = T p and the initial condition was T (x, y, t 0 ) = T 0

A.2 Discretize to finite elements

The solide V can be discretized into many small elements V e and those elements composed by the nodes. In [Fig. A.2], a solide V was discretized into many elements and the mesh should contain the number of nodes and number of elements :

6

Méthode des éléments finis -Dans l'équation (5.1) la fonction T doit être dérivable deux fois et une fois dans l'équation (5.5).

Ces équations sont dites respectivement forme intégrale forte et forme intégrale faible de l'équation di↵érentielle (4.1). -Sous certaines conditions de régularité, les formulations (4.1) et (5.5) sont équivalentes.

Forme discrétisée : éléments finis

La solution analytique de l'équation (5.5) est en général inaccessible. On est donc conduit à chercher une solution approchée par une méthode numérique : la méthode des éléments finis. Cette méthode est un cas particulier de la méthode de Galerkin : le champ de températures et les fonctions test appartiennent au même espace de dimension finie.

Discrétisation du domaine : maillage

Le domaine V est décomposé en sous-domaines V e de forme géométrique simple (les éléments) reliés entre eux en des points appelés noeuds. Cette opération s'appelle maillage (figure 3). Le maillage est défini par la table des noeuds et la table des éléments.

Représentation élémentaire (ou locale) du champ de températures

Le champ de températures T e (x, y, z; t) dans l'élément (e) a pour expression : Finite elements method for thermal elasticity analysis

B.1 General

The analysis of thermo-mechanical mean to determind the stress and deplacement evolution in the domaine (⌦) under the distribution of temperature, surface load ! f ( ! x , t) and imposed displacement ! U ( ! x , t). In this chapter, only the distribution of temperature T ( ! x , t) which was calculated from the previous chapter will be applied.

Figure B.1 -loading type on a solid element

In the problem static, stress and displacements are the solutions for the equation below :

8 ! x 2 ⌦ : div + ! f v = ! 0 (B.1)
and the boundary condition :

8 ! x 2 ⌃ : • ! n = ! f (B.2) 8 ! x 2 ⌃ u : ! u = ! u (B.3) 8 ! x 2 ⌦ : " = 1 2 ⇣ ! u ⌦ ! r + ! r ⌦ ! u ⌘ (B.4)
Finite elements method for thermal elasticity analysis the law of behavior :

8 ! x 2 ⌦ : = E : (" " th ) (B.5) " = C : + " th (B. 6
)

" th = ↵ th (T T 0 )I (B.7)
By integration the equation Eq. (B.1), we have :

Z ⌦ u ⇤ i ✓ @ ij @x i + f v i ◆ d⌦ = 0 (B.8) with : 8 ! x 2 ⌃ u , u ⇤ i = 0 B.

Discretize to finite elements

The solide will be discretized into many small elements and each element was composed by the nodes as shown in the [Fig. B.2], in which q is the vector of displacement at node. The displacement at a poin of (x,y) can be define by the interpolation function N(x,y) :

 u x (x, y) u y (x, y) =  N A (x, y) 0 N B (x, y) 0 N C (x, y) 0 0 N A (x, y) 0 N B (x, y) 0 N C (x, y) 2 6 6 6 6 6 6 4 q A 1 q A 2 q B 1 q B 2 q C 1 q C 2 3 7 7 7 7 7 7 5
(B.9) [U (x, y)] = [N (x, y)][q] (B.10)
and the strain was :

["(x, y)] = 2 4 N A,x 0 N B,x 0 N C,x 0 0 N A,y 0 N B,y 0 N C,y N A,x N A,y N B,x N B,y N C,x N C,y 3 5 [q] (B.11)
Problem of detrimental stress in thermal elasticity analysis 179

["(x, y)] = [B(x, y)][q] (B.12)
principle of virtual work :

Z Ve : " ⇤ dV = Z Ve f v u ⇤ dV + Z ⌃ e f u ⇤ dS (B.13) Z Ve [ " ⇤ ] T [ ]dV = Z Ve [ u ⇤ ] T [f v ]dV + Z ⌃ e [ u ⇤ ] T [f ]dS (B.14) [ q ⇤ ] T Z Ve [B] T [ ]dV = [ q ⇤ ] T Z Ve [N ] T [f v ]dV + Z ⌃ e [N ] T [f ]dS ! (B.15) Z Ve [B] T [E](" " th )dV = Z Ve [N ] T [f v ]dV + Z ⌃ e [N ] T [f ]dS (B.16) Z Ve [B] T [E][B]dV [q e ] = Z Ve [N ] T [f v ]dV + Z ⌃ e [N ] T [f ]dS + Z Ve [B] T [E][" th ]dV (B.17) [K e ][q e ] = [F e ] + [F th ] (B. 18
)
where :

[K e ] = Z Ve [B] T [E][B]dV (B.19) [F e ] = Z Ve [N ] T [f v ]dV + Z ⌃ e [N ] T [f ]dS (B.20) [F th ] = Z Ve [B] T [E][" th ]dV (B.21)
Assembly the matrix sti↵ness of all elements, we will need to solve the system equations Eq. (B.23)

[K] = X e [K e ] (B.22) [K][Q] = [F e ] + [F th ] (B.23)

B.3 Problem of detrimental stress in thermal elasticity analysis

The problem of detrimental stress is commonly happened in thermal elasticity analysis, in which this problem is mainly caused by the inequality of degree of shape function between thermal analysis and thermal elasticity analysis. To clearly presented this problem, an example of one dimensional barre is given : Finite elements method for thermal elasticity analysis an element of barre is composed by two nodes, and node a1 is blocked with displacement. The thermal and mechanical analysis share the same degree of interpolation function, and only thermal loading is considered in the example.

⇥ K ⇤ U = F th (B.24)  1 1 1 1  a 1 = 0 a 2 = EA↵ th (✓ 1 + ✓ 2 ) 2  1 1 (B.25) a = ↵ th (✓ 1 + ✓ 2 )L 2  0 1 (B.26) = E ✓ ↵ 2 L ↵ th ✓ ✓ 1 + ✓ 2 2 + ✓ 2 ✓ 1 2 x ◆◆ (B.27) = ✓ E↵ th ✓ 2 ✓ 1 2 x ◆ 6 = 0 (B.28)
The result of this problem should a free stress result, but in Eq. (B.27), the compensation of stress does not equal to 0. This is due to the mechanical strain ", which is calculated by B*U, and B is the derivative of shape function, whereas, the calculation of thermal strain " th is not needed to do the derivative on shape function. For this reason the compensation of strain is not equal to zero.

 a 2 a 3 = L↵ th 2  4(✓ 1 ✓ 3 ) ✓ 1 + 5✓ 3 (B.30)  a 2 a 3 = L↵ th 8  3✓ 1 + ✓ 3 4(✓ 1 + ✓ 3 ) (B.31)
The stress result in barre element :

= E ⇣ B d a ↵ th N T ✓ (✓ ✓ 0 ) ⌘ (B.32) = E↵ th ✓ 1 8 ⇥ 4x 2x + 1 ⇤  3✓ 1 + ✓ 3 4(✓ 1 + ✓ 3 ) 1 2 ⇥ 1 x 1 + x ⇤  ✓ 1 ✓ 3 ◆ = 0 (B.33)
The stress equal to zero is the result of free stress in thermal elasticity analysis.

B.4 Rate type creep law (RTCL)

In a large structural analysis, the storage of stress history at each integration point of each finite element costs expensively or is even impossible for the current computer capacities. To avoid those di culties, the integral of stress-strain relation in Eq. (B.34) is approximated by a rate type stress-strain relation based on either the Kelvin chain or the Maxwell chain model of aging viscoelasticity [START_REF] Jirasek | Inelastic analysis of structures[END_REF] [START_REF] Jirasek | Inelastic analysis of structures[END_REF]. The previous stress history is characterized by the current value of several internal variables, since the number of internal variables is constant, then the computational cost per step is also constant.

"(t) = C v Z t 0 J(t, t 0 )d (t 0 ) + " 0 (t) (B.
J(t, t 0 ) = 1 E 0 + n X i=1 1 E i (t 0 )  1 exp ✓ (t t 0 ) ⌧ i ◆ (B.35)
and the total strain is given by : where :

"(t) = Z t 0 J(t,
" 0 (t) = Z t 0 1 E 0 d (t 0 ) = 1 E 0 (t) (B.38) " i (t) = Z t 0 1 E i (t 0 )  1 exp ✓ t t 0 ⌧ i ◆ d (t 0 ), i= 1
i (t) = Z t 0 1 ⌧ E i (t 0 ) exp ✓ t t 0 ⌧ i ◆ d (t 0 ) (B.40)
combining Eq. (B.39) and Eq. (B.40), we obtain :

" i (t) + ⌧ "(t) = Z t 0 d (t 0 ) E i (t 0 ) (B.41)
For the case of aging material, the parameters E i is not constant, to eliminate the integral form, the Eq. (B.41) is di↵erentiated again :

"i (t) + ⌧ i "i (t) = ˙ (t) E i (t) (B.42)
to generalize the previous equations into the multiaxial stress, the Eq. (B.42) becomes :

E i (t) "(t) + ⌧ i E i (t)" i (t) = C v ˙ (t) (B.43)
In the di↵erential formulation, the e↵ect of stress history is described by a finite number of internal variables that characterize individual units of the corresponding rheological chain. The internal variables are written as :

i (t) = ⌧ i "i (t) (B.44)
the Eq. (B.43) is replaced :

˙ i (t) + 1 ⌧ i (t) = C v ˙ (t) E i (t) (B.45)
To simplify the problem, we first describe the problem in uniaxial case and then extend to multiple dimensions. Denoting :

t = t n+1 t n (B.46) = (t n+1 ) (t n ) (B.47) E (n+1/2) i = E i (t n + t/2) (B.48)
the Eq. (B.45) becomes :

˙ i (t) + 1 ⌧ i i (t) = tE (n+1/2) i (B.49)
solving the first order di↵erential equation :

i (t) = ⌧ i tE (n+1/2) i  1 exp ✓ t t n ⌧ i ◆ + (n) i exp ✓ t t n ⌧ i ◆ (B.50)
the value of i at the end of the step is given by :

(n+1) i = i (t n+1 ) = i E (n+1/2) i + i (n) i (B.51) where i = exp ✓ t ⌧ i ◆ , i = ⌧ i t (1 i ) (B.52)
the strain increment over the time step is evaluated by integrating Eq. (B.44) :

" i = 1 ⌧ i Z t n+1 tn i (t)dt (B. 53
) " i = 1 i E (n+1/2) i + (1 i ) (n) i (B.54)
the total strain increment is :

" = M X i=0 " i = 1 E 0 + M X i=1 1 i E (n+1/2) i ! + M X i=1 (1 i ) (n) i (B.55)
The first term on the right hand side can be evaluated as the strain increment due to a change of the applied stress, whereas the second term is the strain increment due Finite elements method for thermal elasticity analysis to the creep :

1 E = 1 E 0 + M X i=1 1 i E (n+1/2) i (B.56) " creep = M X i=1 (1 i ) (n) i (B.57)
Finally, the creep strain is evaluated by the updated internal variable ( i ), and the current internal variable is calculated by the results of the internal variable of the previous step. For the initial condition of ( i ), we set i = 0. Besides that, to coupling between the creep model and mechanical model, the stress ( ) in Eq. (B.51) is replaced by the e↵ective stress (˜ ). In fact, only the non cracking part is assumed for the creep e↵ect, and [Eqs. (B.51) and (B.57)] are written as :

" creep = M X i=1 (1 i ) (n) i (B.58) (n+1) i = i ˜ E (n+1/2) i + i (n) i (B.59)
Results of the preliminary studies 4.2. Between the west and south walls, the damage in the south wall is more important, as observed in the temperature profiles, the nonlinear temperature profiles on the south wall are also significant. Always at the corner, the damage distribution become the most vulnerable.

207

Introduction général

Dans le contexte du développement durable, la réduction de la consommation d'énergie pour le bâtiment résidentiel a été refondu par la directive européenne 2010/31/UE [1], et le concept de "bâtiment à presque zéro énergie" est évoqué dans cette directive. Une étude sur la performance énergétique des bâtiments résidentiels [2] indique que consommation d'énergie augmente de 38-42% en raison des ponts thermiques. La pont thermique est un problème commun, qui est connu comme une voie pour créer une perte de chaleur aux connexions de la plancher-voile et de la plancher-balcon. Au cours des dernières décennies de la pratique du bâtiment, il existe deux systèmes d'isolation : les systèmes d'isolation externe et interne d'un bâtiment est interrompu à l'interface de la dalle-mur en raison de transfert des charges mécaniques de la dalle au système mural. La discontinuité de l'isolation interne est à l'origine du pont thermique, et finalement, conduire à l'augmentation la consommation d'énergie dans le bâtiment. Même si l'isolation extérieure ignore cette configuration, toujours elle ne peut pas éviter la discontinuité à la connexion plancher-balcon.

Les systèmes de rupture thermique (TBS) sont mis au point dans le but de sont composées de matériaux isolants et d'éléments structuraux (en général : barres d'armature). Les TBS sont placés au niveau de la connexion plancher-voile ou plancher-balcon, et sont destinés à augmenter la résistance thermique et à assurer le transfert de la charge. La présence des TBS génère un écart de température qui sépare la température de la voile à la plancher pour le cas d'isolation interne, et du balcon à la dalle pour le cas de l'isolation externe. L'expansion thermique et la contraction des murs sont contraints par la partie isolée du bâtiment. Le déplacement entrave ensuite créer des e↵orts supplémentaires dans les TBS, ainsi que dans le béton à proximité éléments. Ces e↵orts peuvent être su samment importantes pour provoquer des dommages dans les TBS, et fissuration dans le béton, voir [Fig. D.11]. Avec les dommages de béton en tension et compression, les TBS sont imposés avec de grands déplacements, et peut être soumis à des rendements répétés pendant la durée de vie du bâtiment. Cette la résistance à la fatigue des TBS est préoccupante.

Résumé en Français

Objective et contribution

Les recherches antérieures liées aux TBS se concentrent sur di↵érentes portées et sont variées de la performance thermique au comportement mécanique sous le statique et sismique l'e↵et des actions climatiques sur le bâtiment avec isolation interne et équipé de TBS n'a jamais été présenté auparavant. Les seules études similaires précédentes ont été observés di↵éremment dans la thèse de G. Le Bloa [3] et dans le document de P.Heng et al. [4], qui se concentraient sur le comportement sismique des TBS et la résistance à la fatigue des TBS à la jonction plancher-balcon dans les bâtiments avec isolation extérieure, respectivement. La recherche de cette thèse est la première étape à découvrir et à a cher le questions pour cette structure de bâtiment innovante.

L'objectif de cette thèse de doctorat est donc d'étudier un bâtiment, qui est en interne isolé et équipé des TBS, sous les actions climatiques. L'enquête faits saillants sur l'évolution des contraintes et la répartition des dommages sur les voiles, ainsi que les e↵orts supplémentaires visées par les TBS.

Pour atteindre l'objectif de cette thèse, une simulation numérique basée sur des éléments finis méthode est développée, et de nombreux phénomènes devront être prises en compte. Tout d'abord, les actions climatiques sont complexes, dans lequel le rayonnement solaire et les signaux de température de l'air sont des variations quotidiennes et saisonnières. Cette double complexité génère une charge thermique non linéaire dans l'épaisseur des voiles en béton. De plus, ces variations périodiques intérêt avec le fluage et le retrait du béton. Ces deux les déformations retardées sont fortement a↵ectées par les variations hydriques dans le béton. En conséquence, la simulation de cette thèse est une analyse thermo-hydro-mécanique (THM). Les simulations numériques sont effectuées par un logiciel de recherche bien connu, Castem [START_REF]Code de calcule aux eléments finis cast3m[END_REF]. Ce choix se rapporte également aux di↵cultés dans les simulations. Castem [START_REF]Code de calcule aux eléments finis cast3m[END_REF] est un système de code source. Il est alors possible d'intégrer l'algorithme de retrait, de fluage, les comportements non linéaires du béton et non linéaire du TBS dans le code calcul des éléments finis. Les codes implantés peuvent être séparés en deux domaines. Tout d'abord, l'analyse hydrique, le rétrécissement et les modèles de fluage sont développés dans di↵érents modules et sont combiné avec l'algorithme principal de Castem [START_REF]Code de calcule aux eléments finis cast3m[END_REF] dans l'environnement 'UNPAS' de Gibiane. Deuxièmement, le comportement non linéaire des matériaux sont implantés dans les codes sources de Castem [START_REF]Code de calcule aux eléments finis cast3m[END_REF], qui est connu dans l'environnement de l'Esop. Il convient de souligner que la mise en oeuvre des codes et des algorithmes n'est pas la seule contribution à la point de vue informatique. Mais la technique pour e↵ectuer la simulation avec le grand échelle et de nombreux types des éléments finis, et plus important encore, la technique de post-traitement les résultats sont tous la deuxième partie principale dans le calcul environnement de cette thèse.

Dans la simulation numérique, l'approche multi-échelles, qui est largement utilisé pour résoudre numérique complexe, est adoptée. Cette approche vise également à simplifier les problèmes de modélisation numérique et de modélisation du comportement 220 Résumé en Français des matériaux du grand fini structure des éléments. L'approche multi-échelles dans cette thèse est décomposée en deux échelles di↵érentes, qui sont : la modélisation numérique tridimensionnelle à l'échelle du bâtiment et l'échelle de la modélisation de la loi constitutive matérielle. Dans la modélisation numérique, l'échelle du modèle numérique correspondant au TBS est considérée comme l'échelle intermédiaire. Les TBS ne sont pas entièrement modélisés par les macro-éléments (ressort) ni par le modèle ra né. Mais la simplification par la distribution des éléments shell est faite en raison de la similité de la géométrie des profilés en acier dans le TBS. De plus, les voiles sont modélisés par les éléments solides classiques, mais la modélisation simplifiée avec coque solide est faite pour la dalle, et l'hypothèse de Bernoulli est introduite à la jonction de la coque solide. Globalement, l'échelle de la modélisation d'élément finis dans cette thèse peut être considérée comme le échelle intermédiaire, qui équilibre la capacité de calcul et degré de la modélisation d'élément finie, qui sont nécessaires pour fournir le raisonnable résultats. En ce qui concerne l'échelle des modélisations de phénomènes, deux phénomènes principaux sont les multiphysiques, qui comprennent les modèles de rétrécissement et de fluage, puis la non linéarité comportement des matériaux tels que : le béton et le TBS. Il vaut la peine d'être mentionné que les couplages entre l'analyse thermo-hydro-mécanique est un couplage faible. Il n'est pas une interaction de l'e↵et structural (fissure) à l'analyse thermique et hydrique, ni de la manière opposée. D'autre part, les actions climatiques exposées sur le bâtiment sont extraites de la base de données de Météo France [START_REF] France | [END_REF], et sont appliquées à la surface extérieure des voiles avec les conditions limites comme la convection de l'air et flux du rayonnement. Le coe cient de l'émissivité liée à la couleur du bâtiment est également prise en compte.

La décision de la modélisation 3D à l'échelle du bâtiment est imposée par la complexité géométrique du système sous les considérations suivantes. En e↵et, les mouvements de les voiles interagissent avec la dalle à travers le TBS, et les voiles interagissent entre eux à chaque coin du bâtiment ne peut pas être décrit simplement par des modèles 2D. En outre que les TBS sont soumis à la fois au cisaillement horizontal par allongement et contraction de la les voiles, ainsi que, aux e↵orts axiales dans la zone du coin. La simulation au bâtiment Il s'agit également d'un défi pour la capacité du système de calcul central. Le super ordinateur (72 core, 1.2 To de RAM), qui est basé au laboratoire de génie civil et mécanique et dans le groupe de GEOSAX (LGCGM), donne la possibilité d'e↵ectuer un tel calcul. En tous cas, la modélisation complète avec éléments est trop cher dans le temps de calcul. Une combinaison d'éléments solides et de coque sont appliquées. Compte tenu de la complexité des problèmes dans cette thèse, la modélisation est limité à un bâtiment à géométrie rectangulaire, et seulement un quart d'un bâtiment étage modélisé (voile en L).

Une modélisation des matériaux ra nés est obligatoire pour inclure les déformations di↵érées et les variations hydriques qui ont déjà été soulignés. En outre, une grande fissuration des voiles en béton est attendue et jouera évidemment un rôle important dans la distribution des e↵orts et des déplacements dans la structure. Cette modélisation de matériaux ra nés à nouveau faire la simulation devenir lourde. Afin d'obtenir l'équilibre souhaité entre le précision et e cacité, les modèles simplifiés de retrait et de fluage sont développés. Plus important encore, les comportements non linéaires du béton et des TBS sont l'élasticitique endommagement du µ modèle de Mazars et al. [START_REF] Mazars | A new 3d damage model for concrete under monotonic, cyclic and dynamic loadings[END_REF] et l'élasticitéplasticité classique modèle pour l'acier inoxydable [START_REF] Ryś | Constitutive modelling and identification of parameters of 316l stainless steel at cryogenic temperatures[END_REF], respectivement. Une discussion plus détaillée est donnée dans le aperçu de la thèse.

Une étude précédente de T.H. Nguyen [START_REF] Nguyen | Macroscopic modeling of reinforced concrete joints : Application to thermal break elements subject to earthquake loadings[END_REF] a introduit les TBS dans le numérique modèle par les éléments à ressort déjà implantés dans Castem [START_REF]Code de calcule aux eléments finis cast3m[END_REF]. Les macro éléments contribuent à réduire la charge de calcul et fournissent également des degrés adéquats de la liberté (3 rotations, 3 déplacements) pour l'analyse des TBS. Comme mentionné dans l'approche multi-échelle, les TBS ne sont pas entièrement modélisés dans l'échelle des macro éléments, un choix d'échelle intermédiaire est déjà expliqué précédemment. En outre, ici les éléments de ressort sont utilisés pour relier la plaque dans le voile et la plaque dans la plancher.

Aperçu de la thèse

Cette thèse de doctorat se compose principalement de 4 chapitres. Au début, un présentation de la problématique de cette thèse de doctorat. En outre, le processus et les méthodes de recherche sont également résumés.

-Chapitre 1 :

Le chapitre 1 présente principalement les recherches des études précédentes. La littérature couvre à la fois les expériences et les simulations numériques. Ces études sont regroupés en trois niveaux, qui sont l'analyse thermo-mécanique, thermo-hydromécanique l'analyse et les études des TBS. Les structures en béton telles que : pont, barrage et éléments massifs en béton sensibles à la charge thermique, sont décrits. L'essai expérimental de CEOS.fr [START_REF]Large beam specimens with restrained shrinkage (rg8)[END_REF] est une bonne référence pour la compréhension le comportement THM des structures en béton. En outre, les modèles existants concernant le comportement de séchage, le retrait et les e↵ets de fluage du béton sont mis en évidence. Le modèle modifié de dommages élastiques de Mazars pour la charge cyclique est également discuté. Finalement, l'état de l'art, qui se rapportait au TBS, met l'accent sur la loi constitutive et les expériences de chargement vertical et horizontal à une configuration du plancher-voile connectée par les TBSs.

-Chapitre 2 :

Les modèles pour les e↵ets à long terme et les comportements non linéaires, qui sont inclus dans l'analyse THM pour le béton et le TBS, sont présentées au chapitre 2. Pour le béton, le µ modèle du Mazars [START_REF] Mazars | A new 3d damage model for concrete under monotonic, cyclic and dynamic loadings[END_REF], élastique endommagement, est sélectionné et reconstruit. Les paramètres de la traction et de la compression sont identifiés respectivement à partir de l'expérience et du modèle proposé par EC2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF], et correspondent à un béton ordinaire. La sélection de ce modèle est en raison de la complexité du chemin de chargement, qui est considéré comme la variation quotidienne et périodique. D'autre part, les modèles de séchage, de retrait et de fluage sélectionnés sont reconstruits, et les paramètres sont identifies de l'expérimental résultats décrits dans la recension des écrits.

Pour calculer les déformations de rétrécissement dans la simulation numérique, résoudre l'analyse hydrique est nécessaire, dans lequel un modèle de séchage simplifié est appliqué, puis rétrécissement linéaire modèle est adopté. La déformation par fluage (" creep ) est la déformation dépendante des contraintes, qui est influencée par l'âge du béton lorsque la charge est appliquée et les conditions environnementales. Pour inclure la contrainte de fluage dans le calcul, le 'rate type creep law' (RTCL), qui est une méthode numérique populaire pour calculer la déformation de fluage incrémentale (" creep ) sur la base de l'état de contrainte, est utilisé. Avec la RTCL, deux chaînes du modèle Kelvin Voigt sont considérées pour représenter le fluage de base (" bc ) et le fluage de séchage (" dc ), respectivement. Les paramètres du la conformité au fluage est initialement identifiée à partir de la courbe de conformité au fluage d'EC2 [START_REF]Eurocode 2 : Calcule des structures en béton -partie 1-1 : Règles générales et règles pour les bâtiments[END_REF] par la méthode de montage, puis sont modifiés par l'humidité relative du béton. En outre que, en identifiant les paramètres à partir de la courbe de fluage d'EC2 donnent également une possibilité d'obtenir une variation du paramètre avec l'évolution du temps. Cela pourrait aider pour introduire l'e↵et de vieillissement du béton dans le calcul du fluage. Une attention devrait être payé, le fluage est mentionné pour de nombreuses fois dans cette thèse, mais dans la réalité de la calcul, le béton n'est pas un comportement visco-élastique. Un module est implanté et seulement pour calculer la déformation du fluage, mais la rigidité totale du béton n'est pas prise en compte avec l'e↵et de vieillissement.

Pour le TBS, le modèle classique élastoplasticité pour l'acier inoxydable est utilisé. Les paramètres correspondant au comportement horizontal et vertical des TBS sont identifiés à partir des courbes e↵ort-déplacement des expériences, qui ont été menées à la LGCGM. Aucune expérience n'a été réalisée pour observer le comportement axial du TBS au niveau de la connexion plancher-voile. Ensuite, le comportement axial est étalonné de la courbe contrainte-déformation proposée par l'Eurocode 3 [START_REF]Eurocode 3 : Design of steel structures -part 1-4 : General rules -supplementary rules for stainless steels[END_REF] pour l'acier inoxydable.

La simulation numérique finale est la combinaison de la loi constitutive non linéaire des matériaux à la modélisation numérique à l'échelle du bâtiment. Ces simulations sont détaillé dans les chapitres suivants, et le degré de complexité dans les simulations sont de l'analyse élastique du chapitre 3 à l'analyse élastique endommagement dans le Chapitre 4. Ces simulations permettent d'observer les résultats des contraintes, des déformations et des dommages distribution dans les voiles, ainsi que la plastification des TBS. En outre, certains des études paramétriques sont proposées pour comprendre l'influence de la valeur initiale conditions.

-Chapitre 3 :

Au chapitre 3, l'analyse THM avec un comportement élastique à court terme pour le béton et l'élasto-plasticité pour le TBS est la première référence de la simulation dans cette thèse. La simulation numérique est basée sur la combinaison des modèles sélectionnés, y compris les phénomènes de retrait et de fluage, qui sont décrits au chapitre 2. De plus, ces e↵ets ne sont pris en compte que pour la structure des parois. La modélisation des éléments finis développés à l'échelle intermédiaire est également appliquée. La motivation pour lancer cette simulation est également d'indiquer les conditions de conservation de la méthode d'analyse.

Les étapes de la simulation commencent par des analyses thermiques et hydriques. les actions climatiques prises en compte dans l'analyse thermique sont la température de l'air, le rayonnement solaire et le signal d'humidité pour l'analyse hydrique. Les résultats de la distribution de la température et de l'humidité relative sont utilisés pour générer la déformation thermique et les déformation di↵érées. Les déformations sont introduites dans le calcul mécanique en tant que données d'entrée. Les objectifs de cette analyse sont de confirmer la justesse des codes implantés et la réponse de la modélisation simplifiée des éléments finis à l'échelle du bâtiment sous les actions climatiques en termes de contraintes dans les voiles et des e↵ort internes dans les TBSs. L'exposition à 1 an des actions climatiques coûte 20 jours de temps de calcul, qui est 3 fois plus rapide que la simulation complète avec le µ modèle présenté au cha-pitre4. Grâce à ce bénéfice, études de cas avec di↵érentes conditions initiales, en termes de température et l'humidité relative, sont e↵ectuées sous la configuration de l'analyse THM avec comportement élastique pour le béton. Les résultats de la simulation numérique ont montré qu'un niveau de traction significatif se produit dans le béton, en particulier au coin de la voile et les éléments de béton à proximité des TBS. Ces contraintes de traction dépassent même la résistance limite du béton en tension, puis les fissures doivent avoir lieu dans les voiles. En outre, il est observé que les conditions initiales de température ont une influence sur l'ordre de grandeur des contraintes dans les voiles. D'autre part, l'horizontale e↵et de dilatation des voiles n'a aucune influence sur les contraintes dans la dalle. En outre, les TBSs qui sont proches du coin des voiles portent sur les e↵orts internes maximales, et dépasser la limite de résistance. Ces dilatation et la contraction répétées dans les voile le déplacement cyclique imposé aux TBS, et devrait également être concerné par la résistance à la fatigue des TBS. L'accumulation de l'analyse de la fatigue est e↵ectuée.

-Chapitre 4 :

Comme le résume l'analyse élastique, les contraintes de traction dans les voiles sont importantes, puis une simulation qui prend en compte les fissures dans la voile est proposée dans le chapitre 4. Les simulations de l'analyse THM avec le comportement élastique endommagement pour le béton, comportement elasto-plastique pour les TBS et comportement élastique pour les armatures dans les voiles, sont e↵ectués. Il convient de noter que la distribution de renfort est à un taux d'armature très faible qui ne permet pas de contrôler la fissuration. L'armature minimum provient directement du plan de ferraillage réel avec le configuration de voile-plancher connectée par les TBS.

Dans les simulations complètes, deux configurations sont sélectionnées, soit l'étage du toit et l'étage intermédiaire. Ces configurations sont di↵érenciées en imposant la charge verticale, qui correspond à l'auto-pesée de la structure des étages supérieurs aux murs dans la simulation, 3 MPa pour le plancher intermédiaire et 0,7 MPa pour le plancher de toit. En outre, l'e↵et de couleur de la surface du bâtiment est également pris en compte dans les simulations, et cet e↵et est inclus en modifiant les signaux de rayonnement. Le temps de calcul pour la simulation complète pourrait aller à 2 mois pour une exposition à la charge thermique de 1 an, et chaque calcul utilise 12 coeur de CPU et 150Go de la RAM.

Les résultats correspondant au cas de plancher intermédiaire sont intéressants. Ces résultats présente une répartition uniforme des dommages à la surface des voiles (effet peau), les fissures au coin et plusieurs dommages locaux (fissures verticales) sur la surface intérieure, en particulier les éléments de béton qui entourent les TBS et sont proches du coin. Les déplacements des simulations complètes sont plus importants que celui décrit au chapitre 3. Plusieurs sauts de déplacement sont observés et ces sauts correspondent aux fissures dans les voiles. Les déplacements et les sauts de déplacement supplémentaires sont concernés par le cisaillement horizontal et l'e↵ort axiale dans les TBS, en particulier au coin des voiles. L'augmentation du cisaillement horizontal entraîne une augmentation de l'accumulation de dommages dans les TBS. Dans le cas le plus défavorable de la simulation, le nombre de TBS, avec une accumulation de dommages pendant 50 ans supérieure à 1, sont 3 à 4 fois les résultats calculés à partir de l'analyse élastique du chapitre 3. Cette observation donne une information importante que l'analyse élastique n'est pas de conservation pour estimer l'accumulation de dommages dans les TBS. Cela devrait inquiéter l'ingénieur concepteur qui a surtout utilisé le logiciel d'ingénierie industrielle avec le comportement élastique du matériau pour estimer les e↵orts internes dans les TBSs.

Conclusion

Dans cette thèse, nous avons essayé de développer une modélisation numérique, qui est capable d'analyser un bâtiment intérieurement isolé et équipé de systèmes de rupture thermique (TBS) sous les actions climatiques. Pour réaliser la simulation proposée, le modèle mécanique réaliste pour le béton en tenant compte de la fissuration, e↵et des déformations di↵érées et les échanges hydriques ont été adoptés. Pour le TBS, le comportement non linéaire a été inclus. De plus, l'analyse thermique a été e↵ectuée en tenant compte de deux types de conditions limites : la convection de l'air et le flux de rayonnement solaire.

Par la suite, une simulation non linéaire complète a été e↵ectuée avec un nombre limité d'études de cas en raison du temps de calcul élevé. Dans ces simulations, ils ont confirmé la formation de plusieurs profils de fissuration dans les voiles, qui sont : une propagation uniforme des fissures sur la surface extérieure, des fissures verticales à travers l'épaisseur des voiles depuis la surface intérieure et approchées jusqu'au coin, et le dernier est les dommages dans le béton entourant le TBS, principalement au niveau du plancher. Ces fissures ne sont pas surprenantes puisque les voiles ne sont pas bien renforcés. Le pourcentage d'armature minimal n'est pas su sant pour pouvoir contrôler la fissuration. Des conclusions semblables ont été obtenues alors qu'un cas de bâtiment habituel sans TBS est simulé. De plus, la simulation non linéaire a indiqué des rendements répétés plus importants dans le TBS, et le risque de défaillance en fatigue pour les TBS près de l'angle des bâtiments. Simulant seulement un an de la vie d'un bâtiment, les résultats ont montré qu'un dommage majeur au TBS augmente alors que les fissures se forment dans les voiles. Il est probable qu'une fois que la fissure dans les voiles atteindra sa forme finale, la croissance des dommages au TBS sera limitée.

La gravité de ces conclusions doit être jointe avec les considérations suivantes. Les simulations sont e↵ectuées dans des conditions climatiques extrêmes, et les murs refend ne sont pas prises en compte dans la configuration actuelle. La présence des murs refend pourrait générer la rigidité supplémentaire et limiter le déplacement relatif entre la voile et le plancher. Même si le TBS semble à risque d'échec avec les simulations actuelles, mais la résistance en fatigue ne correspond pas à la vraie, car cette résistance a n'a pas été déterminé expérimentalement pour tout TBS conçu pour l'isolation interne. Ici, il est adopté à partir de la résistance de conception d'un TBS utilisé pour l'isolation externe, qui est plus critique en fatigue. Dans cette thèse, l'estimation des dommages au TBS pour 50 ans n'est basée que sur l'accumulation des dommages en 1 an, ce qui est donc largement défavorable. Cependant, les résultats de cette thèse ne peuvent donner d'autres commentaires sur ce sujet. De plus, de nombreuses simplifications et hypothèses avaient été faites afin de pouvoir e↵ectuer le calcul.

En conclusion générale, cette thèse a mis en évidence que la densité des barres d'armature dans les voiles en France n'est pas su sante, ce qui donne lieu à des fissures importantes qui peuvent être observées sur n'importe quel bâtiment en béton réel. D'autre part, les TBS risquent de subir une défaillance par fatigue. Par conséquent, la dernière conclusion exige la détermination de la résistance à la fatigue des TBS existantes et une étude expérimentale des e↵orts au TBS dans un immeuble réel. Cette thèse a présenté ses limites dans l'analyse numérique complète, le modèle n'est pas assez puissant pour mener une simulation numérique d'un bâtiment réaliste avec une durée significative. L'observation expérimentale est urgente pour aller plus loin, en termes de modification d'un meilleur modèle de THM, et aussi pour produire un modèle e cace pour représenter le comportement du TBS dans les bâtiments avec isolation interne. Dans ce qui suit, une contribution plus détaillée de ce travail, et les perspectives sont données.

Perspectives

Les études numériques de cette thèse tentent d'inclure tous les e↵ets que le bâtiment pourrait connaître. Les observations et les questions soulevées dans ce cadre sont d'attirer l'attention sur le problème avec le bâtiment équipé de TBS sous les actions climatiques, qui n'ont jamais été remarquées. Néanmoins, la simulation actuelle doit être comparée à un bâtiment construit réel afin d'améliorer ou de valider le modèle THM simplifié sélectionné, et les simplifications qui ont été faites dans la modélisation numérique.

-Expérimentation :

Du point de vue expérimental, plusieurs expériences se rapportent au béton et le TBS. Premièrement, pour le béton, la mesure du température et humidité relative dans le béton avec la configuration du voile + isolation interne dans les conditions climatiques devrait être établie. Ces résultats valider et améliorer les paramètres des analyses thermiques et hydriques.

En ce qui concerne l'expérience du TBS, l'étude du comportement axial dans le TBS et la construction de la résistance à la fatigue du TBS utilisé pour la connexion voileplancher devraient être décrites expérimentalement. Ces expériences sont nécessaires pour identifier le paramètre du modèle élasto-plasticité correspondant au comportement axial du TBS. En outre, la courbe de résistance à la fatigue réelle aiderait à estimer avec précision l'accumulation de dommages dans les TBS et aussi la durée pour atteindre D T BS à 1.

Enfin et surtout, le bâtiment construit avec la technique des voiles en béton armé, équipé ou non de TBS et avec isolation interne doit être surveillé. Les simulations estiment une grande fissuration et un désordre des éléments de béton sur la surface intérieure. Mais en réalité, ces informations ne peuvent pas être remarquées en raison des couches isolantes. Encore une fois, cela ressemble à l'histoire de la corrosion de la structure de toiture en métallique cachée par le plafond. Les mesures des déplacements sur la voile et la plancher, la déformation, l'ouverture des fissures et les e↵orts dans les TBSs doivent être surveillées dans le protocole de l'expérience.

-Simulation :

Du point de vue de la simulation, il y a également plusieurs améliorations à envisager. Le comportement non linéaire du contact acier-béton pour les TBS reste le travail manquant dans cette thèse. Cet e↵et local pourrait a↵ecter les e↵orts internes dans les TBS. La plasticité dans le renfort devrait être incluse. Comme on l'a vu dans la simulation du chapitre 4, une fois que le béton fissure, l'e↵ort dans le béton chute et elle est absorbée par l'armature. Ces e↵orts sont assez grandes pour plastifier les armatures dans le béton.

La configuration de la modélisation des éléments finis devrait être plus proche d'un configuration réelle du bâtiment en incluant les murs refend, les étages multiples et les di↵érents rapports des ouvertures. Comme expliqué dans la conclusion du chapitre 4, les murs refend pourraient contribuer à limiter les déplacements relatifs entre les voiles et le plancher en raison de la rigidité supplémentaire.

Ensuite, les études paramétriques doivent être e↵ectuées. D'ici, les principaux paramètres à étudier sont les signaux de rayonnement ayant des orientations et des conditions climatiques avec les di↵érentes géographie. La simulation avec des orientations di↵érentes aiderait à comprendre l'e↵et de la condition limite symétrique, parce qu'en réalité, la charge thermique n'est pas symétrique. L'étude des di↵érentes géographies des conditions climatiques permet de confirmer concrètement les fissures dans la voile conçue avec le minimum d'armature. Dans le code de conception, la charge thermique est considérée comme la chargement secondaire ou même négligée en raison du joint de dilatation (massif central : 35 m). Cependant, les bâtiments construits sont observés avec les fissures sur la surface extérieure, et les réponses simplifiées pointent toujours à la retrait ou les ouvertures dans les voiles sans aucune preuve scientifique pour soutenir. Title : Effect of climatic actions on buildings with internal insulation and thermal break: Multi-scale approach and thermo-hydro-mechanical modeling.

Keywords : Thermal break system, thermal dilatation, THM analysis, cracking, fatigue strength Abstract : For buildings designed with internal insulation, thermal bridges at the junction between slabs and walls are a common issue, since they create heat loss from inside of the building. Thermal break systems (TBS), which are composed of structural elements (rebar) and insulation material, are used to reduce this heat loss and to transfer shear forces from the slab to the walls. Insulation system of TBS generates a temperature gap from wall to slab. As a consequence, while the walls are exposed to climatic actions and is repeatedly dilated and contracted, whereas the slab pertains a constant temperature and does not present any volumetric variation, thus the TBSs are submitted to large imposed displacements. This thesis illustrates the effect of the thermal dilatation and contraction of the walls, which create a supplemental force in the TBSs, and consequently cracking of the walls. A numerical model of a quarter of a building's storey is submitted to the climatic actions computed at Embrun city, in France.

One side of the L-shape wall is supposed to face to the south, and the other one to the west. The thermal, hydric and mechanical analysis are performed with the software Castem. In thermal and hydric analysis, air temperature, flux of solar radiation and ambient humidity are defined from databases of Météo France, and are applied on the exterior surface of the walls. The results of the first calculation, by thermo-hydromechanical analysis (THM) with elastic behavior, confirm that a significant stress level in tension occurs in the concrete at the corner of the walls and the concrete elements nearby the TBS at the slab level. Furthermore, the TBSs that are close to the corner of the walls pertain the highest horizontal shear and axial force, and are at risk to be failure by the fatigue strength. Based on those results, a second calculation, which is taken into account cracking in the walls by the elastic damage behavior for the concrete, is made for indicating more realistic forces in the TBSs, and analyzing the cracking pattern of the walls.

4 . 3 61 2

 4361 Creep strain by EC2 . . . . . . . . . . . . . . . . . . . . 51 1.4.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.5 Thermal break system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.5.1 Investigation of thermal performance of TBSs . . . . . . . . . . . 52 1.5.2 Characterized mechanical behavior of TBS by experiments . . . 53 1.5.2.1 Experiment on vertical loading . . . . . . . . . . . . . . 53 1.5.2.2 Experiment for horizontal loading . . . . . . . . . . . . 55 1.5.3 The studies of horizontal e↵ects to the TBSs . . . . . . . . . . . 56 1.5.4 Constitutive law modeling of stainless steel . . . . . . . . . . . . 59 1.5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 1.6 Conclusion and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . Simplified modeling of the THM behavior of the concrete and the mechanical nonlinear behavior of the TBS 63 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 1 -

 1 Figure 1 -description of building equipped with TBS exposed to climatic actions

Fig. 1 .Fig. 2 .

 12 Fig. 1. Bridge seen from west. Photo taken in July 2017.

Fig. 9 .Figure 1 . 1 -

 911 Fig. 9. Temperature distribution in the simulation model at the time for the largest negative (left) and positive (right) temperature differences.

Figure 1 . 2 -

 12 Figure 1.2 -(left) concrete specimen ; (right) position of sensors[START_REF] Liu | Temperature response in concrete under natural environment[END_REF]

Figure 1 . 3 -

 13 Figure 1.3 -temperature evolution : (left) under shelter ; (right) without shelter[START_REF] Liu | Temperature response in concrete under natural environment[END_REF]

Chapter 1 :

 1 Figure1.4 -overall view of the case study building[2]

Figure 1 . 5 -

 15 Figure1.5 -Averaged measurement of solar irradiance on a vertical plan[START_REF] Maxwell | Measuring and modeling solar irradiance on vertical surfaces[END_REF]

Figure 3 . 6 . 2 .Fig. 3 . 6 . 3

 362363 Fig. 3.6.3 Geometry of the earth-sun system from the viewpoint of an observer at the position P on earth's surface.

Figure 8 .Figure 1 . 6 -

 816 Figure 8. Solar azimuth and solar altitude seen from observation point P (ESRA) Figure 1.6 -sun position [22]

Figure 4 . 1 :( 4 )

 414 Figure 4.1: Diagrammatic representation of constituent components of a temperature profile (4) The strains and therefore any resulting stresses, are dependent on the geometry and boundary conditions of the element being considered and on the physical properties of the material used. When materials with different coefficients of linear expansion are used compositely the thermal effect should be taken into account.

15 Figure 1 . 7 -

 1517 Figure 1.7 -decomposition of temperature profiles based on EN 1991-1-5 [27]

Figure 1 .

 1 Figure 1.11 -A typical cracking pattern observed in early ages of concrete curing in a bridge abutment [23]

Figure 1 .

 1 Figure 1.12 -moisture content distribution [29] (V. Baroghel-Bouny et al. 1999)

Figure 1 .

 1 Figure 1.14 -concrete strength under bi-axial loading [34] (Kupfer et al. 1973)

Chapter 1 :Figure 1 . 15 -

 1115 Figure 1.15 -Experiment of restrained shrinkage beam (RG8) [10] (CEOS.fr, 2010) Rapport d'avancement-Exploitation des résultats expérimentaux du bloc RG8bis

Figure 52 :Figure 1 .

 521 Figure 52 : Relevé de fissure réalisé in situ

  Figure 42 les vitesses de déplacement des capteurs à fibre optique et les vitesses de déformation mécanique des capteurs à corde vibrante.

Figure 41 :

 41 Figure 41 : Sauts de déplacement mesurés par les fibres optiques

Figure 42 :Figure 1 .

 421 Figure 42 : Sauts de déformation mesurés par les cordes vibrantes

  Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 15 instrumented with strain gauges on the rebar, vibrating wires close to the upper and lower surfaces, and 3 optical fibers distributed on a median plane of the beam covering 2.5 m of its central zone. 4.2.1. From 0 to 120 h: Maturation and Cracking Period

Figure 4 .

 4 Figure 4. Evolution during the 0-120 h time frame: (a) the 3 section elements: struts (red), rebar (blue), concrete (green); (b) global forces supported by the 3 systems in the central beam section (dashed line: experiment-solid line: calculation); and (c) damage contour at 76 h (time of initial cracking) and 103 h (time of extended cracking).

Figure 4 .

 4 Figure 4. Evolution during the 0-120 h time frame: (a) the 3 section elements: struts (red), rebar (blue), concrete (green); (b) global forces supported by the 3 systems in the central beam section (dashed line: experiment-solid line: calculation); and (c) damage contour at 76 h (time of initial cracking) and 103 h (time of extended cracking).

Figure 1 .

 1 Figure 1.18 -(a) 3 sections elements : struts (red), rebar (blue), concrete (green) ; (b) force evolution by 3 systems in the central beam section (dashed line : experiment, solid line : calculation) ; (c) damage distribution [39] (J. Mazars et al. 2018)

Figure 1 .Figure 1 .Figure 1 . 21 -

 11121 Figure 1.19 -(a) pore humidity distribution during drying, (b) free shrinkage of slices imagined to be cut our, (c) shrinkage stresses and (d) crack propagation on surface of a bridge pile (Métro ligne B, Rennes) [42]

Figure 1 .

 1 Figure1.22 -drying shrinkage strain in function of moisture loss[START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF] (Grange, 1996)

Figure 1 . 23 -

 123 Figure 1.23 -basic creep coe cient in function of loading age (7 days, 21 days and 47 days : during of loading) [50] (Niyogi et al. 1973)

Figure 1 . 25 -

 125 Figure 1.25 -relative creep strain in function of relative humidity [51] (Acker et al. 2001)

Figure 1 .

 1 Figure 1.26 -drying and wetting creep strain of cement paste [45] (Day et al. 1984)

Figure 1 .

 1 Figure 1.27 -(a) sress-strain curve (7 days age at loading) ; (b) creep strain evolution (7 days age at loading) [54] (Mazzotti and Savoia 2001)

1. 4

 4 Observation of modeling of THM behavior of concrete 1.4.1 Drying model 1.4.1.1 Physical model of drying process

Figure 1 .

 1 Figure 1.28 -definition of e↵ective area [61] (Benboudjema et al. 2014)

Chapter 1 :

 1 Literature review

Figure 1 .Figure 1 .

 11 Figure 1.29 -evolution of loading path and e↵ective damage [7] (Mazars et al. 2015)

Figure 1 . 31 -

 131 Figure1.31 -drying shrinkage strain[START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF]

Figure I- 33

 33 Figure I-33 Chaînes de Maxwell (a) et chaînes de Kelvin-Voigt (b).

  Afin de proposer une description plus physique de l'aspect vieillissant,Bažant et Prasannan (1989) et Bažant et al. (1997) suggèrent de considérer que les éléments constituants la pâte de ciment ne sont pas dépendants de l'âge. Ainsi, la dépendance du fluage à l'âge est alors attribuée : Soit à la déposition graduelle du gel de C-S-H du fait de la réaction d'hydratation (paragraphe § I-3.4.1.1.2, Bažant et Prasannan 1989). Le gel de C-S-H est supposé nonvieillissant et viscoélastique (ou visqueux). Le C-S-H qui se dépose est alors susceptible de contribuer à la reprise du chargement macroscopique. Lorsque la réaction d'hydratation s'arrête, le vieillissement est dû à d'autres réactions chimiques (par exemple la polymérisation des C 3 S), prises en compte à travers la même expression. Ainsi, la déformation de fluage propre visqueuse fp ε s'écrit en fonction de la fraction d'hydrates formées ( ) t υ et de la complaisance de fluage (non vieillissante) J :

Chapter 1 :

 1 Figure 1.33 -creep compliance function by unite creep curves for di↵erent age of loading [30] (M. Jirasek and Bažant 2001)

Figure II- 19 Figure II- 20

 1920 Figure II-19 Mécanisme du fluage de dessiccation intrinsèque.

Figure 1 .

 1 Figure1.34 -rheology model for drying creep[START_REF] Benboudjema | Modélisation des déformation di↵erées du béton sous sollicitation biaxiles. Application aux enceintes de confinement de bâtiments reacteurs des centrales nucléaires[END_REF] (Benboudjema, 2002)

  (A. Sellier and L. Lacarriere 2009, A.Sellier et al. 2016). The rheological scheme presented in [Fig. 1.35] composed of two branches : the left part represents the solid behavior, with an elastic spring to model the instantaneous behavior, a Kelvin-Voigt module is for the reversible creep, and the Maxwell module is for the permanent strain. The right branch represents the e↵ect of the hydric forces.

  ) of Fig. 2, it allows flow until the next interlocking occurs on another si viscous phases are not yet damaged. The relative impor ible creep and permanent creep depend on the proporti clusions per unit volume of material as shown in homog

Fig. 1 .Figure 1 .

 11 Fig. 1. Idealized rheological scheme for poro-mechanical creep model.

  76) in which C T , C H , and C M are independent in function with the temperature, relative humidity and non linear e↵ect of mechanical loading. The expressions of C T , C H and C M are fully described by Alain Sellier et al [83]. 1.4.4.3 Creep strain by EC2

  o C) and cold side (11 o C), and the temperature distribution is presented in [Fig. 1.36]. The presence of the TBS improved the thermal resistance, and reduced the thermal bridge. But as the number of rebars increased, the thermal transfer caused by the rebars are also considerable. Kostiantyn Dikarev et al. / Energy Procedia 85 (2016) 184 -192 ral assumptions rete connection «floor slab -balcony slab» was selected for the analysis. Two types of this ere examined during the experiment. In the first case we investigated a model of a ard connection with floor slab (a continuous steel reinforce concrete slab), as shown in Fig. we studied a model of the same concrete connection interrupted by a thermal break, as samples were put into guarded boxes made of cellular polystyrene. b) ing a model of the conventional balcony board connection with floor slab; (b) specimen including a model of the concrete connection interrupted by a thermal break

Fig. 2 .Section 1 - 1 Section 2 - 2 Fig. 5 (

 211225 Fig.2. Thermal break, the reinforcement steel rods and some of the thermocouples hermocouple positions is given in Fig.3(a) and Fig.3(b). Then, cement was poured into the ick wall was built up. The dimensions are also indicated in Fig.3. A layer of 10 mm of nded polystyrene) was applied to the cold side of the brick wall. Other thermocouples were of the sample and between the brick wall and the insulation layer. In the case with onnection thermal break is absent but the thermocouples are placed in the same points as in ting element (Fig.3 (b)).

Figure 1 .

 1 Figure 1.36 -(left) conventional TBS ; (right) temperature distribution[START_REF] Dikarev | Experimental and numerical thermal analysis of joint connection 'floor slab and balcony slabe' with integrated thermal break[END_REF]

-

  Figure 1.37 -geometry of specimens [93] (G. Le Bloa et al. 2013)

Figure 1 .Figure 1 .

 11 Figure 1.38 -force-displacement curve of vertical load test [93] (G. Le Bloa et al. 2013)

Chapter 1 :

 1 Figure 1.40 -loading position [93] (G. Le Bloa et al. 2013)

Figure 1 .

 1 Figure 1.41 -force-displacement curve of horizontal load test [93] (G. Le Bloa 2013)

Figure 1 .

 1 Figure 1.42 -force-displacement curve : (a) horizontal monotone, (b) cyclic load[3]

Figure 1 .Figure 1 .

 11 Figure 1.43 -(left) : shear-displacement curve, (right) : low fatigue strength of the TBS SUNE [91]

Figure 1 .

 1 Figure 1.45 -TBS of Rutherma (Schöck) [9] (T. Nguyen et al. 2014)

  0.2 9.67E-5 3E-4 0.95 1.5E+4 1.25 517 1E-5

Figure 2 .Figure 2 . 2 -

 222 Figure 2.1 -stress-strain curves : (top) tension ; (bottom) compression

Figure 2 . 3 -

 23 Figure 2.3 -(top) compression-compression, (bottom) compression-tension

Figure 2 . 4 -

 24 Figure 2.4 -schematic representation of water transportation and phase changes in cement paste [76] (F. Benboudjema, 2002)

  2.10]. The shrinkage strains of the experiments are compared with those of the shrinkage simulation, and are used to identify the K h sh at the linear part. The drying shrinkage simulations are conducted by using the relative humidity from the previous section 2.3.1.3, and is coupled with the damage elasticity model of Mazars for the concrete.

Figure 2 .

 2 Figure 2.10 -samples for drying shrinkage and total creep test[START_REF] Granger | Comportement di↵éré du béton dans les enceintes de centrales nucléaires. Analyse et modélisation[END_REF]

Figure 2 .

 2 Figure 2.11 -comparison of drying shrinkage strain (linear shrinkage model)

Figure 2 .

 2 Figure 2.13 -fitting curve between EC2 and Kelvin-Voigt for basic creep (RH=95%)

Figure 2 .

 2 Figure 2.16 -comparison of total strain between experiment and simulation

Figure 2 .

 2 Figure 2.17 -stress profiles of Civaux B11

Figure 2 .

 2 Figure 2.19 -(a) horizontal force-displacement curve, (b) vertical force-displacement curve and (c) normal force-displacement curve

Figure 3 . 1 -

 31 Figure3.1 -air temperature evolution (Embrun, France, Jan-Dec 2016)[START_REF] France | [END_REF]

Figure 3 . 2 -

 32 Figure 3.2 -(left) : global horizontal radiation, (right) : sun's angles

Figure 3 . 4 -

 34 Figure 3.4 -ambient humidity evolution, monthly average (Embrun, France,Jan-Dec 2016)[START_REF] France | [END_REF]

Figure 3 . 5 -

 35 Figure 3.5 -geometry of thermal and hydric numerical analysis

Chapter 3 :Figure 3 . 6 -

 336 Figure 3.6 -total thickness of the wall

3. 2 . 5 Figure 3 . 7 -

 2537 Figure 3.7 -temperature evolution : (b) south ; (c) west

TFigure 3 . 8 -

 38 Figure 3.8 -cross section with temperature profiles

Figure 3 . 9 -

 39 Figure 3.9 -di↵erent temperature ( T ) : (top) south ; (bottom) west

Figure 3 .

 3 Figure 3.10 -estimated stress evolution

Figure 3 .Figure 3 .

 33 Figure 3.11 -relative humidity evolution on the exterior and interior surface of the walls

Chapter 3 :

 3 THM analysis (elastic behavior for the concrete) is considered in concrete walls.

Figure 3 .

 3 Figure 3.13 -numerical modeling at building scale and TBS by spring element

Figure 3 .

 3 Figure 3.14 -boundary conditions of the structure

7 Figure 3 .

 73 Figure 3.15 -(top) thermal loading ; (b) hydric loading

Figure 3 .

 3 Figure 3.16 -elastic parameters : (top) basic creep ; (bottom) drying creep

Figure 3 .

 3 Figure 3.17 -displacement distribution and deformation configuration (dilatation e↵ect, at t=251.6 days, case T ref = 5 o C)

Figure 3 .

 3 Figure 3.18 -displacement profiles : (top) south wall ; (bottom) west wall, (at t=251.6 days, case T ref = 5 o C)

Figure 3 .

 3 Figure 3.19 -(left) : displacement evolution at the red dot in X and Y directions, (right) : averaged (" th + " sh ) estimation of free displacements simulation West U x = (" th + " sh )L x = 3.10 4 ⇥ 7.5 = 2.276mm U x = 1.815mm South U y = (" th + " sh )L y = 2.95.10 4 ⇥ 16 = 4.72mm U y = 3.304mm

Figure 3 .

 3 Figure 3.20 -di↵erence displacement between case 1 and 2

Figure 3 .

 3 Figure 3.21 -stress distribution ( y ) on the wall at t=251.6 days, (case 1, unit :MPa)

Figure 3 .

 3 Figure 3.22 -stresses evolution (exterior surface) : (a) west ; (b) south

Figure 3 . 23 -

 323 Figure 3.23 -di↵erence stress between case 1 & 2

Figure 3 . 24 -

 324 Figure 3.24 -creep strains evolution (exterior surface) : (a) west ; (b) south

Figure 3 . 25 -

 325 Figure 3.25 -stresses evolution (exterior surface) : (a) west ; (b) south

Figure 3 .

 3 Figure 3.26 -stress evolution (interior surface, case T ref = 5 o C) : (a) middle ; (b) corner

Figure 3 .Figure 3 . 28 -

 3328 Figure 3.27 -stress distribution on the slab at t=202.7 days ( y , unit : MPa)

Figure 3 .

 3 Figure 3.29 -(left) : force profiles (at t=251.6 days, case T ref = 5 o C), (right) : envelop curve of the force profiles, (a) : south, (b) : west

Figure 3 .Figure 3 .Figure 3 . 32 -Figure 3 .

 333323 Figure 3.30 -horizontal shear evolution and shear-displacement curves (1 st spring element on the south) : (a) case 1 (T ref = 5 o C) ; (b) case 2 (T ref = 10 o C) ; (c-left) envelop of plastic displacement profile ; (c-right) number of plastified spring

Figure 3 .

 3 Figure 3.34 -accumulated damage distribution of the TBS at di↵erent positions from the corner : (a) TBSs on the south ; (b) TBSs on the west

Figure 3 .

 3 Figure 3.35 -displacement distribution (U y ) : (top) Castem ; (bottom) RSA

Mechanical analysis 119 Figure 3 .

 1193 Figure 3.36 -stress distribution on exterior surface : (top) west x ; (bottom) southy

Figure 3 .

 3 Figure 3.37 -force profiles : (a) south ; (b) west

Figure 4 . 1 -

 41 Figure 4.1 -solar radiation evolution for di↵erent orientations and seasons (" s =0.7)

  4.2] for the south and west orientations. The maximum and minimum temperatures on the south wall are 39.74 o C and -5.57 o C, respectively, whereas, for the west, they are Thermal analysis for the light colored surface 125 42 o C and -6.7 o C. Besides that, the temperature corresponding to " s =1 and " s =0.7 are compared in the format of data points, see [Fig. 4.2 (b & c)]. As being expected, forT > 0, the data points are dispersal, and T (" s = 0.7) is lower than T (" s = 1). Whereas, for T < 0, the data points are very close to the reference line.

Figure 4 . 2 -

 42 Figure 4.2 -temperature evolution : (b) south ; (c) west

Figure 4 . 3 -

 43 Figure 4.3 -estimated stress : (a) south ; (b) west

Figure 4 . 4 -

 44 Figure 4.4 -reinforcement in the walls

Figure 4 . 5 -

 45 Figure 4.5 -floors corresponding to case studies

  T ref = 10 o C h ref = 78% 3 MPa (intermediate) 0.7

Chapter 4 :

 4 THM analysis (elastic damage behavior for the concrete)

Figure 4 . 6 -Figure 4 . 7 -

 4647 Figure 4.6 -displacement distribution and deformed configuration (at t=246.6 days)

Figure 4 . 8 -

 48 Figure 4.8 -displacement evolutions : (top) U x , (bottom) U y

Figure 4 .

 4 Figure 4.11 -left : average of " th +" sh , right : local damage evolution

Figure 4 .

 4 Figure 4.13 -(a & b) principal stress distribution (t=51.52 days) ; (c) displacement profiles of the walls at dash line (t=51.52 days)

Chapter 4 :

 4 THM analysis (elastic damage behavior for the concrete)

Figure 4 .

 4 Figure 4.14 -stresses evolution (exterior surface) : (top) west ; (bottom) south

Figure 4 .

 4 Figure 4.15 -strain evolution : (top) west, (bottom) south

  elastic limit (horizontal shear) horizontal shear (elastic damage, t=246.6days) horizontal shear (elasticity, t=246.6days) elastic limit (horizontal shear) envelop curve (elastic analysis) envelop curve (elastic damage analysis) elastic limit (axial force) axial force (elastic damage, t=246.6days) axial force (elasticity, t=246.6days) elastic limit (axial force) envelop curve (elastic damage analysis) envelop curve (elastic analysis) (b)

Figure 4 .

 4 Figure 4.16 -force profiles of spring on the south (left : t=246.6 days, right : envelop curves) : (a) horizontal shear ; (b) axial force

Figure 4 .

 4 Figure 4.17 -horizontal shear evolution and shear-displacement curves (1 st spring element) : (a) south ; (b) west ; (c-left) envelop of plastic displacement profiles ; (c-right) number of plastified spring

Figure 4 .

 4 Figure 4.18 -axial force evolution and force-displacement curves (1 st spring element) : (a) south ; (b) west ; (c-left) envelop of plastic displacement profiles ; (c-right) number of plastified spring

Figure 4 .Figure 4 .

 44 Figure 4.19 -number of occurrence of cycles for horizontal shear compared to fatigue strength (1 st TBS from corner) : (left) south wall ; (right) west wall

Figure 4 .

 4 Figure 4.21 -damage evolution for the 1 st TBS on the south and west

Figure 4 .

 4 Figure 4.22 -accumulated damage distribution of the TBS at di↵erent positions from the corner : (left) TBSs on the south ; (right) TBSs on the west

Figure 4 .

 4 Figure 4.23 -accumulated damage distribution of the TBS at di↵erent positions from the corner : (left) TBSs on the south ; (right) TBSs on the west

Figure 4 .

 4 Figure 4.24 -(a) : numerical modeling of walls with openings , (b) : reinforcement distribution

Fig. 4 .

 4 25].

Figure 4 .

 4 Figure 4.25 -crackings at the openings : (a) Rennes ; (b) Marseille (05/2019) ; (c) Marseille (07/2020)

Figure 4 .

 4 Figure 4.26 -forces evolution and force-displacement of the 1 st spring : (a) horizontal shear of the south ; (b) axial force of the west

Figure 4 .Table 4 . 7 - 1 4. 3 . 6

 447136 Figure 4.27 -accumulated damage distribution of the TBS at di↵erent positions from the corner : (left) TBSs on the south ; (right) TBSs on the west

  4.28 to 4.30]. Each case, the damages distribution are illustrated as the first 40 days, and after one year cycle. Cases 2 and 3 are corresponding to the intermediate floor, whereas case 4 is for roof floor. T ref = 5 o C h ref = 82% 3 MPa (intermediate) 0.7 (a) (b)

Figure 4 .

 4 Figure 4.28 -damage distribution in the wall : (a) 40 days ; (b) 415 days

Figure 4 .

 4 Figure 4.29 -damage distribution in the wall : (a) 40 days ; (b) 415 days

Figure 4 .Figure 4 .

 44 Figure 4.30 -damage distribution in the wall : (a) 40 days ; (b) 415 days

Figure 4 . 4 Chapter 4 :Figure 4 .

 4444 Figure 4.32 -accumulated damage distribution of the TBS on the south wall (left : 1 year, right : 50 years) : (a) case 2 ; (b) case 3 ; (c) case 4

Figure 1 -

 1 Figure 1 -Matériau anisotrope -Si le matériau est isotrope, la matrice de conductivité se réduit à :

Figure 2 -

 2 Figure 2 -Matériau isotrope

Finite elements method for thermal analysis where : ⇢ mass density (Kg/m 3 )

 3 C p heat capacity (J/(Kg.K)) thermal conductivity (W/(m.K)) h coe cient exchange by convection (W/(m 2 .K)). " r coe cient of emissivity. r constant of Stefan r = 5.67 10 8

Figure 3 -

 3 Figure 3 -Domaine plan discrétisé en 12 éléments reliés entre eux par 15 noeuds

Figure A. 2 -

 2 Figure A.2 -discretation into finite elements

Figure B. 2 -

 2 Figure B.2 -an element of TRI3

Figure B. 3 -

 3 Figure B.3 -example 1 of one dimensional barre

3 (

 3 Figure B.4 -example 2 of one dimensional barre

34 )

 34 Figure B.5 -(a) non aging Kelvin chain, (b) aging Kelvin chain[START_REF] Jirasek | Inelastic analysis of structures[END_REF] [START_REF] Jirasek | Inelastic analysis of structures[END_REF]

Figure•

  Figure C.2 -temperature profile

Figure C. 3 -

 3 Figure C.3 -modeling TBS by beam-spring-beam element

Figure C. 4 -Figure D. 2 -Figure D. 3 -Figure D. 4 - 4 D. 2 . 1

 4234421 Figure C.4 -damage distribution of a preliminary case study

Figure D. 5 - 4 )

 54 Figure D.5 -damage profiles at di↵erent locations of the wall : (left) west, (right) south

Figure D. 6 -Figure D. 9 -Figure

 69 Figure D.6 -damage profiles at di↵erent locations of the wall : (left) west, (right) south ; (a, b, c) : case 2, 3, 4

Figure D. 11

 11 Figure D.11 -description du bâtiment equippé avec les TBSs

Titre:

  Effet des actions climatiques sur les bâtiments avec isolation interne et rupteur thermique : Approche multi-échelles et thermo-hydro-mécanique intégrée. Mots clés : Rupteur thermique, dilatation thermique, analyse THM, fissuration, résistance à la fatigue Résumé : Pour les bâtiments conçus avec une isolation interne, les ponts thermiques à la jonction entre le plancher et les voiles sont un problème commun, car ils créent une perte de chaleur de l'intérieur du bâtiment. Les systèmes de rupteur thermique (TBS), composés d'éléments structuraux (barres d'armature) et de matériaux isolants, sont utilisés pour réduire cette perte de chaleur et transférer les forces de cisaillement du plancher aux voiles. Le système d'isolation du TBS génère un gap de température entre le plancher et le voile. En conséquence, alors que les voiles sont exposées aux actions climatiques et sont dilatées et contractées à plusieurs reprises, alors que le plancher se rapporte à une température constante et ne présente aucune variation volumétrique, ainsi les TBS sont soumis aux grands déplacements imposés. Cette thèse illustre l'effet de la dilatation et de la contraction thermiques des voiles, qui créent des forces supplémentaires dans les TBS, et par conséquent la fissuration dans les voiles. Un modèle numérique d'un quart d'étage d'un bâtiment est soumis aux actions climatiques à Embrun, en France. Un côté du voile en forme de L est faire face au sud, et l'autre à l'ouest. L'analyse thermique, hydrique et mécanique est réalisée avec le logiciel Castem. Dans analyse thermique et hydrique, la température de l'air, le flux de rayonnement solaire et l'humidité ambiante sont définis à partir des bases de données de Météo France, et sont appliqués sur la surface extérieure des voiles. Les résultats du premier calcul, par analyse thermo-hydro-mécanique (THM) avec la loi comportement élastique pour le béton, confirment l'important de contraint en traction se produit dans le coin des voiles et les éléments du béton à proximité du TBS, au niveau de la connexion 'voile-plancher'. De plus, les TBS qui se trouvent près du coin des voiles présentent le cisaillement horizontal et la force axiale significative et risquent d'être rompus en raison de la résistance à la fatigue. Baser sur ces résultats, un deuxième calcul, qui tient compte de la fissuration dans les voiles par la loi comportement: élastique endommagement pour le béton, est fait pour indiquer des efforts plus réalistes dans les TBSs, et analyser les fissurations dans les voiles.

Table of contents

 of Cracking of concrete . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.1 Bi-axial loading . . . . . . . . . . . . . . . . . . . . . . 1.3.2.2 cracking at early age . . . . . . . . . . . . . . . . . . . . 1.3.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Drying shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.1 Structural e↵ect of drying shrinkage . . . . . . . . . . . 1.3.3.2 Experiments on drying shrinkage . . . . . . . . . . . . . 1.3.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Creep strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4.1 Basic creep . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4.2 Drying creep . . . . . . . . . . . . . . . . . . . . . . . . Table of contents 1.4.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.4.2 Damage model of concrete . . . . . . . . . . . . . . . . . . . . . . 37 1.4.2.1 Damage elasticity model . . . . . . . . . . . . . . . . . 37 1.4.2.2 Damage µ model of Mazars . . . . . . . . . . . . . . . . 40 1.4.2.3 Non local damage model . . . . . . . . . . . . . . . . . 42 1.4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.4.3 Drying shrinkage model . . . . . . . . . . . . . . . . . . . . . . . 43 1.4.3.1 Simplified drying shrinkage strain . . . . . . . . . . . . 43 1.4.3.2 Nonlinear drying shrinkage model . . . . . . . . . . . . 44 1.4.3.3 Comparison of the drying shrinkage models . . . . . . . 45 1.4.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.4.4 Creep model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Introduction General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aims and scopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Literature review 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Analysis of thermo-mechanical behavior . . . . . . . . . . . . . . . . . . 1.2.1 Thermal behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Thermo-mechanical simulation . . . . . . . . . . . . . . . . . . . 1.3 Observation of experiments on THM behavior of concrete . . . . . . . . 1.3.1 Drying process of concrete . . . . . . . . . . . . . . . . . . . . . . 1.3.1.1 Experiments on drying process of concrete . . . . . . . 1.3.1.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 1.3.4.3 Interaction of creep and cracking . . . . . . . . . . . . . 1.3.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Observation of modeling of THM behavior of concrete . . . . . . . . . . 1.4.1 Drying model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1.1 Physical model of drying process . . . . . . . . . . . . . 1.4.1.2 Simplified drying model . . . . . . . . . . . . . . . . . . 2 1.4.4.1 Basic creep strain . . . . . . . . . . . . . . . . . . . . . 46 1.4.4.2 Drying creep strain . . . . . . . . . . . . . . . . . . . . 48 1.4.

  [START_REF] Acker | Creep and shrinkage of concrete : physical origins and practical measurements[END_REF]).

	1.3.4.2 Drying creep
		600
		500
	basic creep (µm/m)	300 400 200					RH=98% RH=90% RH=80.5% RH=70%
							RH=53%
							RH=42.5%
		100					RH=22.5%
		0
		0	5	10	15	20	25	30
					time (day)
	Figure 1.24 -evolution of basic creep strain with di↵erence relative humidity [48]
				(Wittmann 1970)

Table 1 .

 1 .1, the plastic load and displacement are obtained and the estimated sti↵ness is calculated.

	plastic force (KN) displacement (mm) sti↵ness (KN/m)
	103	2.86	36013

1 -calculation of vertical sti↵ness Thermal break system 55 displacement (mm)

Table 1 . 2

 12

	:

Table 1 . 2

 12

-calculation of horizontal sti↵ness

Table 1 .

 1

	Chapter 1 : Literature review

Removing this part, the model becomes a classical elasto-plasticity model, the state variables and corresponding thermodynamic forces are presented in Table

1

.3 : 3 -state variables and thermodynamic forces

Table 2 .

 2

					1 -parameters of the model
		1
		0.8				Mazars ( model)
	lim	0.6				Experiment (Gopalaratnam et Shah 1985)
	/	0.4
		0.2
		0
		0	0.5	1	1.5	2	2.5	3	3.5	4	4.5
							/ lim
		1
		0.8
	lim	0.6
	/	0.4
		0.2				Eurocode 2 Mazars ( model)
		0
		0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8	2
							/ lim

Table 2 .

 2 Three types of concrete composition, in which Civaux B11 & Penly are conducted by Granger, whereas BO (ordinary concrete) is for Baroghel, see in Table 2.3. The identified parameters of BO are then used for the drying simulation in chapter 3 and 4.

	concrete	water	cement	gravel	sand	additive
		(Kg.m 3 )	(Kg.m 3 )	(Kg.m 3 )	(Kg.m 3 )	(Kg.m 3 )
	Civaux B11	195	350	1100	637	1.225
	Penly	202	350	1012	591	1.15
	BO	w/c = 0.48	-	-	s/g = 0.62	-

3 -composition of concrete

Table 2 .

 2 , see in [Fig. 2.5]. These parameters are presented in Table 2.4 : 4 -parameters for the model BSB The coe cient of di↵usion for the concrete are plotted in function of the relative humidity through Eq. (2.2), see [Fig. 2.6]. Again the parameters corresponding to the concrete of Civaux B11 and Penly are identified by Benboudjema. For those of the BO are calibrated from the drying simulation as presented in [Fig. 2.8]. For the relative humidity varied between 70% to 100%, there are 20% to 30% di↵erence between each

	concrete	A	k	V m
	Civaux B11	1.3	0.61	74.1
	Penly	1.3	0.61	76.3
	BO	1.2	0.53	75

Figure 2.5 -comparison of water vapour desorption isotherm [29] (Baroghel et al. 1999)

di↵usion coe cient, while the relative humidity is less than 70%, the di↵usion coecients are almost the same.

Table 2 .

 2 in function of relative humidity is described. The H f increase from high relative humidity to the low relative humidity. The concretes of Civaux and Penly are corresponding to the concrete used for nuclear power station, then the H f of both concrete are very close.

					5 -parameters for coe cient of di↵usion
		1.6	×10 -9
		1.4			Civaux B11 Penly
	diffusion (m 2 .s -1 )	0.6 0.8 1 1.2			BO
		0.4
		0.2
		0
			0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
								humidity

Figure 2.6 -coe cient of di↵usion D(h)

The parameters of the material in order to compute the coe cient of convection as seen in Eq. (2.4) are presented in Table

2

.

[START_REF] France | [END_REF]

. In [Fig.

2

.7], the variation of H f

Table 2 .

 2 6 -parameters for coe cient of convection (H f )

				2	10 -3
							civaux
							penly
			f (m/s)	1.5			BO
			H	1
				0.5
				0	20	40	60	80	100
						relative humidity (%)
				Figure 2.7 -coe cient of convection (H f )
		4
		3					Civaux B11 (Exp) (Granger)
	weigh loss (%)	1 2					Penly (Exp) (Granger) BO (Exp) (Baroghel) Civaux B11 (simulation) Penly (simulation) BO (simulation) Civaux B11 (Benboudjema) Penly (Benboudjema)
		0
		0	100 200 300 400 500 600 700 800 900
					time (day)

Figure 2.8 -weigh loss evolution

Table 2 . 7

 27

	:

Table 2 .

 2 7 -mechanical properties of concreteThe tension parameters of concrete are identified by Benboudjema. The drying shrinkage simulated by Benboudjema is based on water content distribution, then K c sh is used. In this work, the drying shrinkage is based on the relative humidity distribution, and K h sh is applied. The identified parameters for the drying shrinkage simulation in this work are presented in Table2.8 :

	concrete Civaux B11 Penly BO	A t 0.95	B t 21000	K h sh 2 ⇥ 10 3 1.5 ⇥ 10 3 1 ⇥ 10 3

Table 2 .

 2

8 -tension parameter (Mazars) and shrinkage parameters

  Chapter 2 : Simplified THM behavior of concrete and Nonlinear behavior of TBS parameters of the basic creep are known, then to derive the drying creep compliance J dc,EC2 from the total creep compliance of EC2, J bc,30% is subtracted from J EC2 , see Eq.(2.14).J dc,EC2 = J EC2 J bc,Kelvin(2.14)the elastic parameters of the drying creep compliance based on Kelvin-Voigt (J dc,Kelvin ) are identified from J dc,EC2 by the fitting method, see [Fig.2.14]. As described in the identification of the basic creep, the elastic parameters of the drying creep are also identified for di↵erent concrete's ages. To consider the variation of the relative humidity, the elastic parameters are modified by the velocity of the relative humidity, see Eq. (2.15). The velocity of the relative humidity is calculated by di↵erentiating Eq. (2.12) :

							E dc (t 0 ) =	E dc,30% ḣ		(2.15)
		4	×10 -5
						Eurocode 2
	dc (1/MPa)	2 3				Kelvin-Voigt
	J	1
		10 -3 0			10 -2	10 -1	10 0	10 1	10 2	10 3	10 4
								log(t-t')
	Figure 2.14 -fitting curve between EC2 and Kelvin-Voigt for drying creep
								(RH=30%)
			8	×10 -5
		J (1/MPa)	7 4 5 6	with :	Eurocode 2 Kelvin-Voigt (bc+dc)			.751	✓	h ex 100	◆ 37.8	+ 1.104	(2.13)
			3
			10 -3 2	h i	initial relative humidity. (100%) 10 -2 10 -1 10 0 10 1	10 2	10 3	10 4
								log(t-t')

h ex ambient humidity. ⌧ dry characteristic time for drying. D e↵ective thickness. K s di↵usivity of pore water in saturated concrete. Regarding to the drying creep compliance function, the elastic parameters are initially identified from the creep compliance curve of EC2 at RH=30%. Since the elastic 76 Figure 2.15 -comparison J EC2 and J Kelvin for total creep compliance (RH=70%)

Table 2 .

 2 .3.2. The parameters are presented in Table2.9 : 9 -elastic parameters of creep compliance function of Kelvin-Voigt

	concrete		1	2	3	4	5	6	7	8
		⌧	0.001 0.01	0.1	1	10	100	1000 10000
	Civaux B11	E bc	1.457	1.347	1.3	1.5.6	2.02	1.26	7.21	2.83
		(MPa) E dc (MPa)	⇥10 6 8.77	⇥10 6 4.38	⇥10 6 3	⇥10 5 1.46	⇥10 5 0.93	⇥10 5 0.35	⇥10 4 1.45	⇥10 5 0.6

Table 2 .

 2 [START_REF]Large beam specimens with restrained shrinkage (rg8)[END_REF]. The identified sti↵ness in Table2.10 are very close to those presented in Tables 1.1 and 1.2 of Chapter 1.

	horizontal shear	K a (KN/m) C p (KN) 83000 154	R p 1 (KN) 101	b p 200	F y (KN) 87
	vertical shear	36000	118	52	205	70
	normal force	1254000	3015	113	250	193

Table 2 .

 2

10 -parameters of elasto-plasticity model of TBS

Table 3 .

 3

3 -coe cient of convection for hydric analysis

Table 3 .

 3 6 -elastic parameters of creep compliance of Kelvin-Voigt (at 28 days)

	.6 :

Table 3 .

 3 7 -estimation of the displacements (case T ref =5 o C, at t=202.7 for U x and 251.6 days for U y ) than those belong to the higher initial temperature (T ref =10 o C).

		0.3
		0.25
	max	0.2
	U/U	0.15
		0.1											U x
		0.05											U y
		0
		0	60	90	120	150	180	210	240	270	300	330	360	390
							time (day)

Table 3 . 8

 38

-estimation of the stress (case T ref =5 o C, at 202.7 and 251.6 days)

Table 3 .

 3

		.9 :
		Castem (mm)	RSA (mm)	Percentage
	U x	-1.223	-1.4	14%
	U y	3.304	3.39	6%

[START_REF] Nguyen | Macroscopic modeling of reinforced concrete joints : Application to thermal break elements subject to earthquake loadings[END_REF]

-displacements comparison at the corner between Castem and RSA at t=251.6 days

Table 3 .

 3 [START_REF]Large beam specimens with restrained shrinkage (rg8)[END_REF]. However, the maximum local stress calculated by RSA reaches to 29 MPa, which is far away from Castem.

		Castem	RSA	%	Castem	RSA	%
		(corner)	(corner)		(middle)	(middle)
	x	3.29 MPa	3.97 MPa	20% -2.51 MPa	-3.41 MPa	35%
	y	4.648 MPa	5.45 MPa	17%	-4.2 MPa	-4.62 MPa	10%

Table 3 .

 3

10 -stresses comparison between Castem and RSA at t=251.6 days

Table 4 .

 4

1 -reinforcement percentage in the walls

Table 4 .

 4 2 -the principal case study (without & with opening)

Table 4 .

 4 8 -number of TBSs with D 50 > 1

	50 > 1	south	west
	case 2	7	0
	case 3	8	0
	case 4	2	0

Chapter 3 : THM analysis (elastic behavior for the concrete)

Table 4.4 -evolution of damages distribution on the walls, 40 -415 days

• Force-displacement curve

Chapter 4 : THM analysis (elastic damage behavior for the concrete)

Finite elements method for thermal analysis

• Force-displacement curve

References

supported by the financial aid from the Allocations Régionales and the laboratory of civil and mechanical engineering (LGCGM).

I would like to take this opportunity to express my deep gratitude to Prof. Hugues

To calculate the damage accumulation, the number occurrence of the simulation (N simu ) are counted from the cycles corresponding to F max and amplitude of relative displacement (X a ) through the rainflow approach [START_REF] Ryś | Constitutive modelling and identification of parameters of 316l stainless steel at cryogenic temperatures[END_REF]. It should be noticed that, the forces and displacement here are corresponding to a complete TBS (1 TBS = 1m width = 10 spring elements). After that, the N simu and the corresponding F max X a are plotted with the designed fatigue strength of TBS SUNE. An example of accumulated damage calculation corresponding to the 1 st TBS is depicted in [Fig. 3.32]. The designed fatigue strength of SUNE is constructed from the mean curve by including a coe cient of security ( s =2) as described by Heng [4]. Eventually, the damage accumulation (D T BS ) is the sum of N simu /N SU N E .

24

• Force-displacement curve Chapter 4 : THM analysis (elastic damage behavior for the concrete)

For U x corresponding to the displacement on the west wall, an important displacement velocity ( U x / t) takes place at t=165.7 days, see in [Fig. 4.9]. On the right hand of [Fig. 4.9], the corresponding damage evolution at P1 is depicted. The vertical crack at P1 occurs at the same time as the displacement jump. Regrading to U y , several important peaks of displacement velocity are noticed, for example at t=51 days and 123. Those instants are also corresponding to the vertical cracks at P2, P3, respectively. At t=367.6 days, we saw the displacement jumps in X and Y directions, the damage concrete elements at this instant are illustrated in [Fig. 4.9]. There are plenty of concrete elements have damaged, especially at the corner, whereas on the middle there is only the damage on the surface. For instance, a damage evolution of a concrete element at the corner (P4) is extracted. Those concrete elements at the corner are fully Mechanical analysis 135 damage at same instant as the one observed in the displacement jump.

The damage variable is activated from the equivalent strains, and the equivalent strains are calculated directly from the principal strains. In order to understand those damages distributions, the equivalent strains corresponding to tension (" t ) and compression (" c ) are extracted along with the principal strains. For example, the strains corresponding to the first vertical crack at P2 are illustrated in [Fig. 4.10]. The strains are extracted at 2 points (back dots), in which one is at middle height of the wall (zone with TBS) and others is at 1m above the slab level (far from TBS). The equivalent strains are depicted on the left, and the principal strains are on the right of [Fig. 4.10]. At t=51.52 days, the significant equivalent strains in walls also present a significant skin e↵ect due to the loss of the relative humidity. Furthermore, the crack propagation on the surface of a bridge pile is also depicted in [Fig. 1.19] of Chapter 1 (literature review). For example, the damage and equivalent strains evolution are extracted on two points, one on west and others on the south, see in [Fig. 4.12]. Moreover, the corresponding equivalent strains in tension are illustrated on the right side. The damages developed on the exterior surface appear during the night when the concrete are contracted. Those damages distribution are generated by the tensile stresses, because the equivalent strains in tension exceed the limit strain, see in [Fig. 4.12].

However, the damage distribution on the surface of the south wall comes late than the one on the west. Normally, we should have those kind of uniform damage on the surface in cooling season when the concrete is contraction on the surface. In this case, it might be because of the vertical cracks in walls that are occurred at the early phase of the loading signal. As we can see that, when coming to the end of the signal (cooling phase=winter season), the uniform damage on the surface appeared. To understand whether this phenomenon is caused by the vertical cracks, a deeper analysis of the local stresses and strains are suggested for the next section.

Conclusions and perspectives

Summary and general conclusions

In this thesis, we have tried to develop a numerical modeling, which is capable to analysis a building internally insulated and equipped with thermal break systems (TBS) under the climatic actions. To achieve the proposed simulation, the realistic mechanical model for concrete by taking into account cracking, e↵ect of the delayed strains and hydric exchange have been adopted. For the TBS, the nonlinear behavior has been included. Moreover, thermal analysis has been made by considering two kinds of boundary conditions : air convection and flux of solar radiation.

No similar study, which covered all the described e↵ects, and investigated a building internally insulated and equipped with the TBSs, has been found in the literature review. This is one of the di culties of this thesis, choosing and reconstructing the models in order to take into account all the e↵ects are necessary. Besides that, the simplification of the models are mandatory for the purpose of lightening the calculation. The frame work of this thesis is described in [Fig. 4.34]. The work started from the selection and construction of the simplified THM model for the concrete, the nonlinear mechanical model of the TBS and the surrounding zone of concrete. In parallel, the simplified numerical model of the building, which was limited to 1/4 of a building storey, was developed. Indeed, even if the simplified THM model was applied, but from Conclusions and perspectives the concrete.

The configuration of the finite elements modeling should be more approximated to a real building configuration by including the cross walls, multi floors and di↵erent ratio of the openings. As explained in the conclusion of Chapter 4, the cross walls might contribute to limit the relative displacements between the walls and slab due to the supplemental sti↵ness. Afterwards, the parametric studies should be performed. At here, radiation signals with di↵erent orientations and climatic conditions for di↵erent geography are the principal parameters to investigate. Simulation with di↵erent di↵erent orientations would help to understand the e↵ect of the symmetrical boundary condition, because in reality, the thermal loading is not symmetry. The investigation with di↵erent geography of climatic conditions mean to confirm concretely the crackings in the wall designed with the minimum reinforcement. In the design code, thermal loading is considered as the secondary loading or even being neglected by the reason of dilatation joint (central massif : 35 m). However, the constructed buildings are observed with the cracks on the exterior surface, and the simplified answers always point to the shrinkage restraint or the openings in the walls without any scientific evidences to support.

A.2.1 Representation temperature in the element

The temperature T e (x, y, z, t) in an element (e) can be expressed as : where : n is number of node of the element.

N e i is the shap function correspond to the node. T e i is the temperatures at a node of the element. with the same equation we have :

A.2.2 Calculation and assembly the matrix

From the equation Eq. (A.5), we have :

where :

After assembly all matrix, we need to solve the system di↵ertial equations and the vector T is the solution of equation Eq. (A.11)

Appendix C

Results of the preliminary studies C.1 Preliminary study of thermal analysis

A preliminary study of a full thermal simulation is proposed before performing the simplified thermal analysis at the building scale. In the full thermal simulation, a concrete wall with 20cm of thickness is internally insulated by 10 cm of insulation layer (expanded polystyrene), and is connected to a slab, see in [ " sh : shrinkage strain.

List of tables

" creep : creep strain.

" th : thermal strain.

" e : elastic strain.

" 0c : limit strain of concrete in compression.

" 0t : limit strain of concrete in tension.

" s : coe cient of emissivety.

⇢ a : air density.

⇢ b : concrete density. : stress.

1

: maximum principal stress. T max : maximum temperature.

T min : minimum temperature.

T U : uniform temperature profile.

T M : linear temperature profile.

Aucune étude semblable portant sur tous les e↵ets décrits et portant sur un bâtiment isolé à l'intérieur et équipé des TBS n'a été trouvée dans la recension des écrits. C'est l'une des di cultés de cette thèse, choisir et reconstruire les modèles afin de prendre en compte tous les e↵ets sont nécessaires. En outre, la simplification des modèles est obligatoire pour alléger le calcul. . Les travaux ont commencé à partir de la sélection et de la construction du modèle THM simplifié pour le béton, du modèle mécanique non linéaire du TBS et de la zone environnante du béton. En parallèle, le modèle numérique simplifié du bâtiment, qui était limité à 1/4 d'un étage de bâtiment, a été développé. En e↵et, même si le modèle THM simplifié a été appliqué, mais du point de vue numérique, il existait encore la complexité du comportement non linéaire du béton. Par conséquent, les modélisations des voiles en béton et des dalles ont été faites en mélangeant des éléments volumétriques et des éléments de coque, où les mailles ra nées partielles ont été utilisées au coin des voiles et les éléments en béton entourant les TBS. Une attention a été portée à la modélisation à l'interface entre ces deux types d'éléments finis. De plus, la représentation du TBS dans le modèle numérique a encore une fois donné lieu à une autre complexité, une simplification par l'utilisation d'éléments shell-spring-shell a été proposée.

Une fois le modèle THM simplifié implanté et le modèle numérique développé, le premier calcul a été e↵ectué avec un comportement élastique pour le béton et l'élastoplasticité pour le TBS, afin de limiter le temps de calcul. Les objectifs de ce calcul étaient de valider le modèle, de comprendre le comportement du bâtiment et d'analyser les e↵ets des di↵érentes hypothèses qui ont été faites. Comme on pouvait s'y attendre, ces calculs élastiques ont présenté des contraintes et des déformations importantes dans les voiles, en particulier au coin. En conséquence, un nouveau calcul en tenant compte des fissures dans le béton était obligatoire. De plus, des rendements répétés importants au TBS ont été constatés, et une vérification prospective de la résistance à la fatigue du TBS a été e↵ectuée.