Dalia S Youssef

Abdel Massih

Abdul-Hamid Soubra

Bak Kong Low

Reliability-Based Analysis and Design of Strip Footings against Bearing Capacity Failure

Keywords: Bearing capacity, Earthquake loads, Limit analysis, Probability, Reliability, Shallow foundations, Punching, Failure loads

This paper presents a reliability-based approach for the analysis and design of a shallow strip footing subjected to a vertical load with or without pseudostatic seismic loading. Only the punching failure mode of the ultimate limit state is studied. The deterministic models are based on the upper-bound method of the limit analysis theory. The random variables used are the soil shear strength parameters and the horizontal seismic coefficient. The Hasofer-Lind reliability index and the failure probability are determined. A sensitivity analysis is also performed. The influence of the applied footing load on the reliability index and the corresponding design point is presented and discussed. It was shown that the negative correlation between the soil shear strength parameters highly increases the reliability of the foundation and that the failure probability is highly influenced by the coefficient of variation of the angle of internal friction of the soil and the horizontal seismic coefficient. For design, an iterative procedure is performed to determine the breadth of the footing for a target failure probability.

Introduction

In the analysis and design of geotechnical structures, all the input data have some degree of uncertainty and, thus, may be considered as random variables or stochastic processes. Geotechnical engineers have always recognized the presence of uncertainty in their analysis and design. However, traditional deterministic models simplify the problem by considering the uncertain parameters to be deterministic and by accounting for the uncertainties through the use of a global safety factor, which is essentially a "factor of ignorance." This factor is derived based on past experience and does not reflect the inherent uncertainty of each parameter. A reliability-based analysis or design is more rational, since it takes into account the inherent uncertainty of each input variable. Nowadays, this is possible because of the improvement of our knowledge on the statistical properties of the soil ͑Phoon and Kulhawy 1999͒.

In the last decade, the reliability-based analysis has been extensively applied to the slope stability problem ͓see, for instance, Christian et among others͔. However, very few authors have investigated the reliability-based analysis and design of foundations. Some ͑Griffiths [START_REF] Griffiths | Bearing capacity of spatially random soil: The undrained clay Prandtl problem revisited[END_REF][START_REF] Fenton | Probabilistic foundation settlement on spatially random soil[END_REF], 2003, 2005;[START_REF] Griffiths | Bearing capacity of rough rigid strip footing on cohesive soil: Probabilistic study[END_REF][START_REF] Przewlocki | A stochastic approach to the problem of bearing capacity by the method of characteristics[END_REF][START_REF] Popescu | Effect of random heterogeneity of soil properties on bearing capacity[END_REF] modeled the uncertain parameters as random processes and examined the effect of the spatial variability of these parameters on the settlement or on the bearing capacity of foundations. Most of these studies considered the uncertainty of one single parameter ͑such as the soil elastic modulus, the cohesion, or the angle of internal friction of the soil͒. Others modeled the uncertainties of the different parameters as random variables and conducted reliability-based analysis using formulas of the bearing capacity factors ͑Cherubini 2000; Low and Phoon 2002͒. These approaches have the merit of simplicity, but also some shortcomings because they are based on approximate theoretical formulations.

In this study, a reliability-based analysis and design of a strip foundation resting on a c -soil is presented. Two performance functions may characterize the footing behavior: The serviceability limit state and the ultimate limit state. Only the punching failure mode of the ultimate limit state is analyzed herein. Two loading cases are considered. The first examines the case of a strip footing subjected to a vertical load only; the second considers the case of a vertically loaded strip footing situated in a seismic area and subjected to a pseudostatic seismic loading. Two rigorous deterministic limit analysis models are used. The uncertainties of the soil shear strength parameters and the seismic coefficient are modeled as random variables. The basic concepts of the theory of reliability are described next, followed by the two deterministic models, and discussions of the probabilistic numerical results based on these models.

Basic Reliability Concepts

Two different measures are commonly used in the literature to describe the reliability of a structure: The reliability index and the failure probability.

Reliability Index

The reliability index of a geotechnical structure is a measure of the safety that takes into account the inherent uncertainties of the input variables. A widely used reliability index is the Hasofer and Lind ͑1974͒ index defined as the shortest distance from the mean value point of the random variables to the limit state surface in units of directional standard deviations, namely ␤ HL = min͓R͑͒ / r͔͑͒ ͑Fig. 1͒. Its matrix formulation is ͑Ditlevsen 1981͒

␤ HL = min xF ͱ ͑x -͒ T C -1 ͑x -͒ ͑ 1͒
in which x = vector representing the n random variables; = vector of their mean values; C = covariance matrix; and F = failure region. The minimization of Eq. ͑1͒ is performed subject to the constraint G͑x͒ ഛ 0 where the limit state surface G͑x͒ = 0, separates the n dimensional domain of random variables into two regions: A failure region F represented by G͑x͒ ഛ 0 and a safe region given by G͑x͒ Ͼ 0.

The classical approach for computing ␤ HL by Eq. ͑1͒ is based on the transformation of the limit state surface into the rotated space of standard normal uncorrelated variates. The shortest distance from the transformed failure surface to the origin of the reduced variates is the reliability index ␤ HL .

An intuitive interpretation of the reliability index was suggested in Low and Tang ͑1997a, 2004͒, where the concept of an expanding ellipse ͑Fig. 1͒ led to a simple method of computing the Hasofer-Lind reliability index in the original space of the random variables using an optimization tool available in most spreadsheet software packages. When there are only two uncorrelated nonnormal random variables x 1 and x 2 , these variables span a two-dimensional random space, with an equivalent onesigma dispersion ellipse ͓corresponding to ␤ HL = 1 in Eq. ͑1͒ without the min.͔, centered at the mean values ͑ 1 N , 2 N ͒ and whose axes are parallel to the coordinate axes of the original space. For correlated variables, a tilted ellipse is obtained. Low and Tang ͑1997a, 2004͒ reported that the Hasofer-Lind reliability index ␤ HL may be regarded as the codirectional axis ratio of the smallest ellipse ͑which is either an expansion or a contraction of the 1-ellipse͒ that just touches the limit state surface to the 1dispersion ellipse. They also stated that finding the smallest ellipsoid that is tangent to the limit state surface is equivalent to finding the most probable failure point.

Failure Probability

Given a vector of n random variables X and a performance function defined by G͑x͒, the failure probability P f is defined by

P f = ͵ G͑x͒ഛ0 f͑x͒dx ͑2͒
where f͑x͒ = joint probability density function of the random variables X. By introducing the indicator function I͑x͒ defined as

I͑x͒ = ͭ 1 if G͑x͒ ഛ 0 0 elsewhere ͮ ͑3͒
The integral that gives the failure probability ͓Eq. ͑2͔͒ can be written as

P f = ͵ ⍀ I͑x͒f͑x͒dx ͑4͒
where ⍀ = total domain. A brief description of the methods used in this paper for the evaluation of this integral ͑i.e., FORM, MC, and IS͒ follows.

First-Order Reliability Method

From the first-order reliability method ͑FORM͒ and the Hasofer-Lind reliability index ␤ HL , one can approximate the failure probability as follows:

P f Ϸ ⌽͑-␤ HL ͒ ͑ 5͒
where ⌽͑•͒ = cumulative distribution function of a standard normal variable. In this method, the limit state function is approximated by a hyperplane tangent to the limit state surface at the design point.

Crude Monte Carlo Simulation

Monte Carlo is the most robust simulation method in which samples are generated with respect to the probability density of each variable. For each sample, the response of the system is calculated. An unbiased estimator of the failure probability is given by

P ˜f = 1 N ͚ i=1 N I͑x i ͒ ͑ 6͒
where N = number of samples and I͑x͒ is as defined in Eq. ͑3͒. The coefficient of variation of the estimator is given by

COV͑P ˜f͒ = ͱ ͑1 -P f ͒ P f N ͑7͒

Importance Sampling Simulation

Generally, for a given target of the coefficient of variation, the crude Monte Carlo simulation requires a large number of samples, i.e., a large computation time. This is especially the case for small values of the failure probability P f . The importance sampling ͑IS͒ simulation method is a more efficient approach; it requires fewer sample points than the crude Monte Carlo method. In this approach, the initial sampling density is shifted to the design point in order to concentrate the samples in the region of greatest probability density within the zone defined by G͑x͒ ഛ 0. The design point may be determined by using any of the classical methods such as Rackwitz-Fiessler algorithm ͑Rackwitz-Fiessler 1978͒, Low and Tang's ellipsoid approach ͑Low and Tang 1997a, 2004͒, etc. An estimator of the failure probability P f is obtained as follows ͑Melchers 1999͒:

P ˜f = 1 N ͚ i=1 N I͑v i ͒ f͑v i ͒ h͑v i ͒ ͑8͒
where h͑•͒ = new sampling density centred at the design point, and v = vector of sample values with probability density function h͑•͒.

The coefficient of variation of the estimator is given by ͑Melchers 1999͒

COV͑P ˜f͒ = 1

P f ͱ 1 N ͫ 1 N ͚ i=1 N ͩI͑v i ͒ f͑v i ͒ h͑v i ͒ ͪ 2 -͑P f ͒ 2 ͬ ͑9͒

Sensitivity Factors

The sensitivity factors convey the relative importance of the random variables in affecting reliability. The "omission sensitivity factor" is used in this paper. It gives the relative error in the reliability index when a random variable is replaced by its deterministic mean value. It is given by

␥ i ͑ i ͒ = ␤ HL ͑x i = i ͒ ␤ HL ͑10͒
where ␤ HL = value of the reliability index calculated when all parameters are considered as random variables, and ␤ HL ͑x i = i ͒ = reliability index determined when x i is replaced by its deterministic mean value.

Reliability Analysis of Shallow Strip Foundations

The aim of this paper is to perform a reliability analysis of a strip footing resting on a c -soil and subjected to a vertical load. The case of a vertically loaded strip footing situated in a seismic area is also studied. Quasi-static representation of earthquake effects using the seismic coefficient concept is adopted. The earthquake acceleration for both the soil and the structure is assumed to be the same: Only the horizontal seismic coefficient K h is considered, the vertical seismic coefficient often being disregarded. The deterministic models are based on the upper-bound method of the limit analysis theory ͑Soubra 1999͒. Due to uncertainties in soil shear strength parameters and horizontal seismic coefficient, the cohesion c, the angle of internal friction , and the seismic coefficient K h are considered as random variables. The performance function used in the reliability analysis is defined with respect to the bearing failure of the soil. It is given as follows:

G = P u P S -1 ͑11͒
where P u = ultimate foundation load, and P S = applied load. One may use another performance function as G = P u -P S . However, this leads exactly to the same value of the reliability index since the Hasofer-Lind reliability index does not vary with alternative but equivalent definitions of the performance function. Notice, however, that the first definition has an advantage over the second one since it makes it possible to introduce the concept of the safety factor as follows: G = F -1.

Limit Analysis Models

Two deterministic models are used in this paper. In these models, the upper-bound theorem of limit analysis is applied to the bearing capacity problem of a strip footing using two kinematically admissible failure mechanisms referred to in the following as M1 and M2. These mechanisms were presented by Soubra ͑1999͒.

The approach is simple and self-consistent and it obtains rigorous upper-bound solutions in the framework of limit analysis theory.

Although the results given by this approach are upper-bound solutions, they are the smallest upper-bounds against the available ones given by rigid block mechanisms. In some cases ͑i.e., for weightless soil͒, they are the exact solutions, since they are equal to the results given by the lower-bound method.

M1 is a translational symmetrical multiblock failure mechanism ͑Fig. 2͒ and is used for the computation of the bearing capacity of a vertically loaded footing. M2 is a translational nonsymmetrical multiblock failure mechanism and is suitable for the calculation of the bearing capacity of vertically loaded foundations situated in seismic areas by a pseudostatic approach ͑Fig. 3͒. The calculation of the bearing capacity is performed by equating the total rate at which work is done by the foundation load, the soil weight in motion, the horizontal seismic loads ͑in the case of seismic loading͒, and the ground surface surcharge to the total rate of energy dissipation along the lines of velocity discontinuities. For both M1 and M2, it is found that an upper-bound on the bearing capacity of the soil is given as follows:

P u = 1 2 ␥B 2 N ␥ + qBN q + cBN c ͑12͒
in which the bearing capacity factors N ␥ , N q , and N c can be expressed in terms of the geometrical parameters of each mechanism. For the M1 mechanism, these factors are given as follows:

N ␥ = -͑f 1 + f 2 ͒ ͑ 13͒ N q = -f 3 ͑14͒ N c = 2͑f 4 + f 5 + f 6 ͒ ͑ 15͒
where the expressions of f i ͑i =1, ... ,6͒ are given in Appendix I.

For the M2 mechanism, the bearing capacity factors are given by

N ␥ = - 1 sin͑␤ 1 -͒ + K h cos͑␤ 1 -͒ ͑g 1 + K h g 2 ͒ ͑16͒ N q = - 1 sin͑␤ 1 -͒ + K h cos͑␤ 1 -͒ ͑g 3 + K h g 4 ͒ ͑17͒ N c = 1 sin͑␤ 1 -͒ + K h cos͑␤ 1 -͒ ͑g 5 + g 6 ͒ ͑ 18͒
where the expressions of g i ͑i =1, ... ,6͒ are given in Appendix II.

The ultimate load of the foundation is obtained by minimization of Eq. ͑12͒ with regard to the mechanism's geometrical parameters. For further details on the failure mechanisms, the reader can refer to Soubra ͑1999͒.

Ellipsoid Approach via Spreadsheet

Low and Tang ͑1997a, 2004͒ showed that the minimization of the Hasofer-Lind reliability index can be efficiently carried out in the Excel spreadsheet environment. The spreadsheet approach is simple and easy to understand because it works in the original space of random variables and does not require the additional step of transforming x to u where u is a transformed vector of the random variables in the uncorrelated Gaussian space. However, the optimization in original space is not preferred from a computational perspective ͑STRUREL 1991͒. This is because optimization in standardized space is mathematically more desirable in nonlinear optimization. For example, when minimizing the quadratic form of Eq. ͑1͒ in the original space, in some cases the correct solution is obtained only when the solver option "use automatic scaling" is activated. As an alternative, Cholesky factorization of the convariance matrix can be used. The robustness of the latter approach is investigated in Phoon ͑2004͒.

When the random variables are nonnormal, the Rackwitz-Fiessler equivalent normal transformation was used to compute the equivalent normal mean x N and the equivalent normal standard deviation x N . The iterative computations of x N and x N for each trial design point are automatic during the constrained optimization search.

In the present paper, by the Low and Tang method, one literally sets up a tilted ellipsoid in the Excel spreadsheet and minimizes the dispersion ellipsoid subject to the constraint that it be tangent to the limit state surface using the Excel Solver with the automatic scaling option. Eq. ͑1͒ may be rewritten as ͑Low and Tang 1997c, 2004͒

␤ HL = min xF ͱ ͫ x -x N x N ͬ T ͓R͔ -1 ͫ x -x N x N ͬ ͑19͒
in which ͓R͔ -1 = inverse of the correlation matrix. This equation will be used ͓instead of Eq. ͑1͔͒ since the correlation matrix ͓R͔ displays the correlation structure more explicitly than the covariance matrix ͓C͔.

Additional information on Solver's options and algorithms can be found in the Microsoft Excel Solver's help file and at www.solver.com. The FORM implementation procedure in the spreadsheet is described in Low ͑2005͒ and Low and Tang ͑1997a,b,c, 2004͒. Some Excel files are available at http:// alum.mit.edu/www/bklow.

Probabilistic Numerical Results

For both M1 and M2 failure mechanisms, the numerical results presented in this paper consider the case of a shallow strip foundation with breadth B =2m. The soil has a unit weight of 18 kN/ m 3 . No surcharge loading ͑q =0͒ is considered in the analysis. Different values of the coefficients of variation of the angle of internal friction and cohesion are presented in the literature. For most soils, the mean value of the effective angle of internal friction is typically between 20 and 40°. Within this range, the corresponding coefficient of variation as proposed by Phoon and Kulhawy ͑1999͒ is essentially between 5 and 15%. For the effective cohesion, the coefficient of variation varies between 10 and 70% ͑Cherubini 2000͒. For the coefficient of correlation, Harr ͑1987͒ has shown that a correlation exists between the effective cohesion c and the effective angle of internal friction . The results of Wolff ͑1985͒ ͑ c, = -0.47͒, Yuceman et al. ͑1973͒ ͑-0.49ഛ c, ഛ -0.24͒, Lumb ͑1970͒ ͑-0.7ഛ c, ഛ -0.37͒, and Cherubini ͑2000͒ ͑ c, = -0.61͒ are among the ones cited in the literature. In this paper, the illustrative values used for the statistical moments of the shear strength parameters and their coefficient of correlation c, are as follows: c = 20 kPa, = 30°, COV c = 20%, COV = 10%, and c, = -0.5. These values are within the range of the values cited above.

Mechanism M1

For the configuration presented above, the ultimate footing load determined for the mean values of the soil shear strength parameters is P u = 2,136.72 kN/ m. For the probability distribution of the random variables, two cases are studied. In the first case, referred to as normal variables, c and are considered as normal variables. In the second case, referred to as nonnormal variables, c is assumed to be lognormally distributed while is assumed to be bounded and a beta distribution is used ͑Fenton and Griffiths 2003͒. The parameters of the beta distribution are determined from the mean value and standard deviation of ͑Haldar and Mahadevan 2000͒. For both cases, correlated or uncorrelated variables are considered.

Probabilistic Failure Surface

The conventional deterministic approach for the calculation of a safety factor or an ultimate load on a soil mass is based on the minimization of these functions over a range of trial failure surfaces. The surface of minimum factor of safety or ultimate load is referred to as the critical deterministic surface. A common approach to determine the reliability of a stressed soil mass is based on the calculation of the reliability index ␤ HL corresponding to this surface ͑Christian et al. 1994͒. In this paper, a more rigorous approach is used. It consists of the determination of the reliability index by minimizing the quadratic form of Eq. ͑19͒ not only with respect to the random variables, but also with respect to the geometrical parameters of the failure mechanism ͑ , ␣ i , ␤ i ͒. Twelve rigid blocks ͑i.e., n = 12 in Fig. 2͒ are considered for each side of the footing. Therefore, the minimization is performed with respect to 27 parameters ͑ , ␣ i , ␤ i , c , ͒. The surface obtained corresponding to the minimum reliability index is referred to here as the critical probabilistic surface. The reliability index calculated with respect to the critical probabilistic surface is smaller ͑i.e., more critical͒ than the one calculated by using the critical deterministic surface. Had one considered the deterministic critical surface, a slightly higher reliability index value of 3.49 would have been obtained instead of 3.27 for the case with normal uncorrelated soil shear strength parameters. In all subsequent results, for both the M1 and M2 mechanisms, the reliability index is calculated using the probabilistic failure surface.

Reliability Index and Sensitivity Factors

Table 1 presents the Hasofer-Lind reliability index and the corresponding design point for different values of the vertical applied load P S varying from small values up to the deterministic ultimate load. This ultimate load is the one for which the design point is equal to the mean point for normal variables and equivalent normal mean point for nonnormal variables. The table also gives the omission sensitivity factors of the angle of internal friction and the cohesion. All these results are presented for normal and nonnormal, correlated and uncorrelated shear strength parameters. For all cases, the reliability index decreases with the increase of the applied load P S ͑i.e., with the decrease of the safety factor F = P u / P S ͒ until it vanishes for an applied load equal to the deterministic ultimate load. This case corresponds to a deterministic state of failure for which F = 1 using the mean values of the random variables and the failure probability is equal to 50%.

The comparison of the results of correlated variables with those of uncorrelated variables ͑Fig. 4͒ shows that the reliability indices corresponding to uncorrelated variables are smaller than those of negatively correlated variables for both normal and nonnormal variables. One can conclude that assuming uncorrelated shear strength parameters is conservative in comparison to assuming negatively correlated shear strength parameters. For instance, when the safety factor is equal to 4.27 ͑i.e., P S = 500 kN/ m͒, the reliability index increases by 40% if the variables c and are considered to be negatively correlated. Fig. 4 also shows that for small values of the safety factor, the results of normal and nonnormal variables are nearly identical. A difference appears for large values of the safety factor.

From Table 1, when P S = 700 kN/ m, the most probable failure point for uncorrelated and correlated normal variables is found to be at ͑c* = 12.45 kPa, * = 22°͒ and ͑c* = 16.55 kPa, * = 19.88°͒, respectively. These are the points of tangency of the ␤ HL -ellipses with the limit state surface. Notice that the limit state surface divides the combinations of ͑c , ͒ that would lead to failure from the combinations that would not. The ͑c , ͒ values defining the limit state surface are obtained by searching c ͑or ͒ for a prescribed ͑or c͒ that achieve both the conditions ͑i͒ a minimum ultimate load P u and ͑ii͒ a safety factor F = P u / P S = 1. For this purpose, a numerical procedure was coded in Microsoft Excel Visual Basic. It calls the Excel Solver iteratively in order to simultaneously satisfy the two conditions above. Fig. 5 provides graphical representation of the reliability analysis for both corre-lated and uncorrelated shear strength parameters in the physical space of the random variables. One can easily see that negative correlation between shear strength parameters rotates the major axis of the ellipse from the horizontal direction.

The values ͑c* and *͒ of the design points corresponding to different values of the vertical applied load can give an idea about the partial safety factors of each of the strength parameters c and tan as follows:

F c = c c * ͑20͒ F = tan͑ ͒ tan * ͑21͒
For uncorrelated shear strength parameters, the values of c* and * at the design point are smaller than their respective mean values and increase with the increase of the applied load. For negatively correlated shear strength parameters, c* slightly exceeds the mean for some values of the applied load.

The values of the omission sensitivity factors have shown that the effect of the randomness of the angle of internal friction on the reliability index is much more pronounced than that of the cohesion, especially for the case of nonnormal random variables.

Failure Probability

Figs. 6 and 7 present, respectively, the failure probability and the corresponding coefficient of variation as a function of the number of samples for both MC and IS simulations when the soil shear strength parameters are normal and uncorrelated and the footing applied load is equal to 700 kN/ m. In this paper, MC and IS simulations were performed in the standardized space of uncorrelated variables. Hence, only uncorrelated normal random variables have been generated. For nonnormal and correlated variables, the limit state surface, which is determined point by point as explained in the previous section, is transformed to the standardized space of uncorrelated normal variables using the equivalent normal transformation ͑i.e., the Rackwitz-Fiessler equations͒ for each couple of ͑c , ͒. The two equations used for the transformation of each ͑c , ͒ of the limit state surface from the physical space to the standarized normal uncorrelated space ͑u 1 , u 2 ͒ are ͑Lemaire 2005͒

u 1 = ͩ c -c N c N ͪ ͑22͒ u 2 = 1 ͱ 1 -2 ͫͩ -N N ͪ -ͩ c -c N c N ͪͬ ͑23͒
where = coefficient of correlation of c and ; and c N , N , c N , and N = respectively, the equivalent normal means and standard deviations of the random variables c and . They are determined from the translation approach using the following equations:

c -c N c N = ⌽ -1 ͓F c ͑c͔͒ ͑24͒ - N N = ⌽ -1 ͓F ͔͑͒ ͑25͒
where F c and F = non-Gaussian cumulative distribution functions of c and ; and ⌽ -1 ͑•͒ = inverse of the standard normal cumulative distribution. If desired, the original correlation matrix ͑ ij ͒ of the nonnormals can be modified to ij Ј in line with the equivalent normal transformation, as suggested in Der Kiureghian and Liu ͑1986͒. Some tables of the ratio ij Ј / ij are given in Appendix B2

of Melchers ͑1999͒, including closed form solution for the special case of lognormals. For the cases illustrated herein, the correlation matrix, thus modified, differs only slightly from the original correlation matrix. Hence, for simplicity, the examples in this study retain their original unmodified correlation matrices.

The importance sampling density function used is given in the standard uncorrelated space as follows ͑Lemaire 2005͒:

f͑u͒ = 1 ͱ 2 e -1/2͑u -u*͒ 2 ͑26͒
where u* = transformed value of the design point in the standard uncorrelated space of the random variables. The convergence of the failure probability calculated by IS is achieved for a sample size of 20,000. The corresponding coefficient of variation is about 2%, which is smaller than the commonly adopted value used in the literature ͑i.e., 10%͒. For the Monte Carlo simulation, a sample size of 5,000,000 is needed to achieve a constant value of the failure probability and a corresponding coefficient of variation of 2%. In order to have a clear visualization of the convergence of the IS method, the maximal number of samples represented on the x-axis of Figs. 6 and 7 was limited to 200,000. In the following, only IS simulation method will be used since it gives accurate results with reasonable sample size. All the subsequent calculations are performed for a coefficient of variation of the estimator of 2%. By varying the target of the ultimate load, the reliability index is calculated and the CDF is plotted in Figs. 8 and9 using FORM approximation and importance sampling ͑IS͒ simulation. From these figures, it is observed that the CDFs obtained from FORM approximation are in good agreement with those obtained from IS simulation for normal, nonnormal, uncorrelated, and correlated variables for the commonly used values of the coefficients of variation of the soil shear strength parameters ͑i.e., COV c = 20%, COV = 10%͒. This means that FORM approximation is an acceptable approach for estimating the failure probability for the commonly used values of the soil variability. In order to explain the good agreement between the two approaches, the limit state surface is plotted. Fig. 10 shows the limit state surface corresponding to an applied load P S = 1,300 kN/ m for nonnormal correlated random variables in the standard space of normal uncorrelated variables. This figure also shows the linear FORM approximation, which is tangent to the limit state surface at the design point. From this figure, it can be shown that the linear FORM approximation is very close to the exact limit state surface within the circle centred at the origin of the rotated and transformed space and having a radius equal to 3. This explains why a good agreement between the two approaches is obtained. The difference between the two failure probabilities shown in Fig. 10 is 1.25%. For larger values of the coefficients of variation ͑i.e., COV c = 50%, COV = 20%͒, the difference between the two approaches increases and attains 3.32% as can be shown from the values given in Fig. 11. This can be explained by the curvature of the limit state surface near the design point. Since the limit state surface is convex with respect to the origin of the rotated and transformed space, the FORM approximation overestimates the failure probability. 12 presents the CDFs of the ultimate load for normal, nonnormal, correlated, and uncorrelated variables as given by FORM. When no correlation between shear strength parameters is considered, one can notice a more spread out CDF of the ultimate load ͑i.e., a higher coefficient of variation of the ultimate load͒. The chosen probability distribution ͑i.e., normal, lognormal, and beta distribution͒ does not significantly affect the values of the failure probability.

Fig. 13 presents the effect of the coefficient of variation of the shear strength parameters on the CDF of the ultimate load. It can be seen that a small change in the coefficient of variation of highly affects the CDF curve. On the other hand, the CDF curve is not sensitive to changes in the uncertainty of the cohesion. Thus, the failure probability is highly influenced by the coefficient of variation of . The greater the scatter in , the higher the failure probability. This means that accurate determination of the uncertainties of the angle of internal friction is very important in obtaining reliable probabilistic results. In contrast, the coefficient of variation of c does not significantly affect the failure probability.

Reliability-Based Design

The conventional approach used in the design of a shallow foundation is to prescribe a target safety factor ͑generally F =3͒ and to determine the corresponding breadth B of the footing. Recently, a reliability-based design approach ͑RBD͒ has been used by several authors ͓e.g., Low ͑2005͒ and Phoon et al. ͑2003͒, among others͔. This approach was used in this section. It consists of the calculation of B for a target reliability index of 3.8 as suggested by Eurocode 7 for the ultimate limit states ͑Calgaro 1996͒. This foundation breadth is called hereafter "probabilistic foundation breadth." Fig. 14 presents the probabilistic foundation breadth for different values of the coefficients of variation of the shear strength parameters and their coefficient of correlation. This figure also presents the deterministic breadth corresponding to a safety factor of 3. The probabilistic foundation breadth decreases with the increase of the negative correlation between the shear strength parameters and the decrease of their coefficients of variation. It can become smaller than the deterministic breadth for the common values of the soil variability ͑i.e., COV = 10%, COV c = 20%, -0.7ഛ c, ഛ -0.3͒. For high values of the coefficients of variation and small correlation coefficient, the design breadth for normal variables is higher than that of nonnormal variables. This means that, for the case in hand, assuming normal distributions for the random variables is more conservative than assuming nonnormal distributions. As a conclusion, the deterministic footing breadth may be higher or lower than the reliability-based footing width, depending on the uncertainties.

Contrary to Eurocode 7, which prescribes constant values of the partial safety factors F c and F , the present RBD has the advantage of providing different values of these factors depending on the soil variability. These factors are the optimal ones and are determined rigorously by a maximization of the failure probability for a given soil variability. Hence, a reliability-based design has the merit of explicitly reflecting the correlation structure, the standard deviations, and the probability distributions of the underlying random variables, and of automatically seeking the design point without relying on prescribed values of partial safety factors.

Mechanism M2

The number of rigid blocks used in the M2 mechanism ͑Fig. 3͒ is equal to 12 since further increase in the number of blocks improves the optimal solution by less than 0.5%. For the seismic coefficient, an exponential distribution ͑Exp D͒ and an extreme value type II distribution ͑EVD͒ are used ͑Haldar and Mahadevan 2000͒. The mean value of the horizontal seismic coefficient is assumed to be 0.15. The angle of internal friction is considered to follow the beta distribution.

It was shown for the M1 mechanism ͑Fig. 13͒ that the failure probability of a vertically loaded footing is more sensitive to a variation of the angle of internal friction than the cohesion. For the seismically loaded footing in hand, Fig. 15 presents the effect of the randomness of the soil cohesion on the reliability index for both exponential and extreme value distributions of the seismic coefficient. In this figure, the cohesion c is assumed to be either a lognormally distributed random variable with the commonly used Reliability Index, Failure Probability, and Sensitivity Factors Tables 2 and3 present the reliability results ͑i.e., Hasofer-Lind reliability index, the corresponding design point, the omission sensitivity factors, and the failure probability͒ for different values of the applied load P S varying from small values up to the deterministic ultimate load defined earlier for both exponential and extreme value distributions on K h . The reliability index decreases with the increase of the vertical applied load P S . The exponential distribution on K h gives more conservative results than the extreme value distribution on K h . The values of the omission sensitivity factors suggest that, when the applied load is relatively small compared to the deterministic ultimate load, whether is random or deterministic has only a minor effect on the reliability index since ␥͑ ͒Ϸ1 for both types of the probability distributions on K h . On the other hand, the randomness of the seismic coefficient has a significant effect on the reliability index. Hence, for small values of the applied load, one can neglect the uncertainty in . In contrast, for higher values of the applied load, the uncertainties of both the angle of internal friction and the seismic coefficient should be considered in the reliability analysis.

Probability Distribution of the Punching Safety Factor Fig. 16 shows the CDF of the punching safety factor obtained from FORM approximation for the exponential distribution and for different values of the coefficient of variation of the extreme value distribution. It can be seen that the choice of the probability distribution on K h significantly affects the probability distribution of the safety factor. The dispersion of F is significantly larger when an exponential distribution is assumed for the seismic coefficient.

Probabilistic Design

For the design of a vertically loaded footing situated in seismic area, a reasonable target failure probability of 1% is used. Fig. 17 presents the probabilistic foundation breadth for the extreme value distribution on K h and for different values of the coefficient of variation of K h . The deterministic breadth corresponding to a safety factor of 3 is also shown. The applied load is 700 kN/ m. The probabilistic foundation breadth increases with the increase of the coefficient of variation of K h . It can be greater than the deterministic breadth for large values of the coefficient of variation.

Conclusion

A reliability-based analysis and design of a shallow strip foundation subjected to a vertical and a pseudostatic horizontal seismic loading was performed. Only the punching failure mode of the ultimate limit state was studied. Two rigorous deterministic models based on limit analysis failure mechanisms were used.

For the vertically loaded footing, the reliability index calculated based on the critical probabilistic surface is more critical than that determined using the critical deterministic surface. The reliability index decreases with the increase of the applied load P S . The assumption of uncorrelated shear strength parameters is conservative in comparison to that of negatively correlated parameters. The reliability index is much more sensitive to than to c, especially for the case of nonnormal variables. The greater the scatter in , the higher the failure probability. This means that the accurate determination of the uncertainties of the angle of internal friction is important in obtaining reliable probabilistic results. FORM approximation is an acceptable approach for estimating the failure probability for the commonly used values of the soil variability. When no correlation between shear strength parameters is considered, a more spread out CDF of the ultimate load was obtained. For the case in hand, the chosen probability distribution does not significantly affect the values of the failure probability. The probabilistically-designed foundation breadth decreases with the increase of the negative correlation between the shear strength parameters and the decrease of their coefficients of variation. Contrary to Eurocode 7, which prescribes constant values of the partial safety factors F c and F , a reliability-based design has the advantage of providing different values of these factors depending on the soil variability and correlation structure. These factors are by-products of a reliabilitybased design and reflect parametric sensitivities from case to case in a way rigid partial safety factors cannot.

For the seismically loaded footing, the extreme value distribution was used to model the uncertainties of K h . For values of the vertical load much below the deterministic ultimate load, one can neglect the uncertainty in . In contrast, for higher values of the vertical load, the uncertainties of both and K h should be considered in the reliability analysis.

sin ␤ i sin ͩ ␤ i --͚ j=1 i-1 ␣ jͪ ϫ ͟ j=2 i sin 2 ␤ j sin 2 ͑␣ j + ␤ j ͒ ͟ j=1 i-1 sin͑␣ j + ␤ j -2͒ sin͑␤ j+1 -2͒ ͬ g 2 = sin 2 ␤ 1 sin 2 ͑␣ 1 + ␤ 1 ͒ ͚ i=1 n ͫ sin ␣ i sin͑␣ i + ␤ i ͒ sin ␤ i cos ͩ ␤ i --͚ j=1 i-1 ␣ jͪ ϫ ͟ j=2 i sin 2 ␤ j sin 2 ͑␣ j + ␤ j ͒ ͟ j=1 i-1 sin͑␣ j + ␤ j -2͒ sin͑␤ j+1 -2͒ ͬ g 3 = sin ␤ 1 sin͑␣ 1 + ␤ 1 ͒ sin ͩ ␤ n --͚ j=1 n-1 ␣ jͪ ϫ ͟ j=2 n sin ␤ j sin͑␣ j + ␤ j ͒ ͟ j=1 n-1 sin͑␣ j + ␤ j -2͒ sin͑␤ j+1 -2͒ g 4 = sin ␤ 1 sin͑␣ 1 + ␤ 1 ͒ cos ͩ ␤ n --͚ j=1 n-1 ␣ jͪ ϫ ͟ j=2 n sin ␤ j sin͑␣ j + ␤ j ͒ ͟ j=1 n-1 sin͑␣ j + ␤ j -2͒ sin͑␤ j+1 -2͒ g 5 = sin ␤ 1 cos sin͑␣ 1 + ␤ 1 ͒ ͚ i=1 n ͫ sin ␣ i sin ␤ i ͟ j=2 i sin ␤ j sin͑␣ j + ␤ j ͒ ͟ j=1 i-1 sin͑␣ j + ␤ j -2͒ sin͑␤ j+1 -2͒ ͬ g 6 = sin ␤ 1 cos sin͑␣ 1 + ␤ 1 ͒ ͚ i=1 n-1 ͫ sin͑␤ i -␤ i+1 + ␣ i ͒ sin͑␤ i+1 -2͒ ϫ ͟ j=2 i sin ␤ j sin͑␣ j + ␤ j ͒ ͟ j=1 i-1 sin͑␣ j + ␤ j -2͒ sin͑␤ j+1 -2͒ ͬ

Fig. 1 .

 1 Fig. 1. Design point and equivalent normal dispersion ellipses in the space of two random variables

Fig. 2 .Fig. 3 .

 23 Fig. 2. Failure mechanism M1 for static bearing capacity analysis

Fig. 4 .Fig. 6 .

 46 Fig. 4. Reliability index versus safety factor for COV c = 20%, COV = 10%

Fig.

  Fig.12presents the CDFs of the ultimate load for normal, nonnormal, correlated, and uncorrelated variables as given by FORM. When no correlation between shear strength parameters is considered, one can notice a more spread out CDF of the ultimate load ͑i.e., a higher coefficient of variation of the ultimate load͒. The chosen probability distribution ͑i.e., normal, lognormal, and beta distribution͒ does not significantly affect the values of the failure probability.Fig.13presents the effect of the coefficient of variation of the shear strength parameters on the CDF of the ultimate load. It can be seen that a small change in the coefficient of variation of highly affects the CDF curve. On the other hand, the CDF curve is not sensitive to changes in the uncertainty of the cohesion. Thus, the failure probability is highly influenced by the coefficient of variation of . The greater the scatter in , the higher the failure probability. This means that accurate determination of the uncertainties of the angle of internal friction is very important

Fig. 8 .

 8 Fig. 8. Comparison of the CDFs of the ultimate load given by FORM and IS for normal variables

Fig. 10 .

 10 Fig. 10. Limit state surface and FORM approximation for nonnormal correlated variables, for COV c = 20%, COV = 10%

Fig. 12 .

 12 Fig. 12. Comparison of the CDFs of the ultimate load for normal, nonnormal, correlated, and uncorrelated variables

Fig. 14 .

 14 Fig. 14. Comparison between probabilistic and deterministic design

Fig. 16 .Fig. 17 .

 1617 Fig. 16. CDF of the punching safety factor for P S = 450 kN/ m

  al. ͑1994͒; Low and Tang ͑1997b,c͒; Low et al.

͑1998͒; Hassan and Wolff ͑1999͒; El-Ramly et al. ͑2002͒, ͑2003͒;

Table 1 .

 1 Reliability Index and Sensitivity Factors

	P S ͑kN/m͒	c* ͑kPa͒	* ͑deg͒	␤ HL	␥͑ c ͒	␥͑ ͒	F c	F
				͑a͒ Uncorrelated normal variables
	500	8.87	20.69	4.17	1.30	1.39	2.25	1.53
	700	12.45	22.00	3.27	1.21	1.60	1.61	1.43
	900	14.76	23.47	2.54	1.16	1.81	1.35	1.33
	1,100	16.35	24.84	1.95	1.13	2.01	1.22	1.25
	1,300	17.49	26.08	1.45	1.11	2.20	1.14	1.18
	1,500	18.35	27.19	1.02	1.09	2.38	1.09	1.12
	1,700	19.00	28.17	0.66	1.08	2.56	1.05	1.08
	1,900	19.52	29.06	0.34	1.08	2.74	1.02	1.04
	2,100	19.93	29.86	0.05	1.02	2.91	1.00	1.01
	2,136.72	20.00	30.00	0.00	-	-	1.00	1.00
				͑b͒ Correlated normal variables
	500	10.80	19.41	5.87	0.92	0.99	1.85	1.64
	700	16.55	19.88	4.48	0.88	1.17	1.21	1.60
	900	18.89	21.66	3.38	0.87	1.36	1.06	1.45
	1,100	19.89	23.46	2.53	0.87	1.54	1.01	1.33
	1,300	20.29	25.09	1.85	0.87	1.72	0.99	1.23
	1,500	20.40	26.51	1.29	0.87	1.90	0.98	1.16
	1,700	20.36	27.76	0.82	0.87	2.07	0.98	1.10
	1,900	20.22	28.86	0.41	0.88	2.23	0.99	1.05
	2,100	20.04	29.83	0.06	0.93	2.40	1.00	1.01
	2,136.72	20.00	30.00	0.00	-	-	1.00	1.00
				͑c͒ Uncorrelated nonnormal variables
	500	12.46	18.35	4.59	1.20	5.44	1.61	1.74
	700	14.39	20.98	3.41	1.18	5.27	1.39	1.51
	900	15.76	23.02	2.58	1.15	4.87	1.27	1.36
	1,100	16.80	24.67	1.94	1.13	3.89	1.19	1.26
	1,300	17.60	26.04	1.42	1.12	3.51	1.14	1.18
	1,500	18.25	27.22	0.99	1.12	3.29	1.10	1.12
	1,700	18.77	28.25	0.62	1.13	3.16	1.07	1.07
	1,900	19.21	29.15	0.30	1.19	3.03	1.04	1.04
	2,100	19.59	29.96	0.02	1.37	2.17	1.02	1.00
	2,114.15	19.61	30.00	0.00	-	-	1.02	1.00
				͑d͒ Correlated nonnormal variables
	500	14.99	16.77	6.20	0.89	4.03	1.33	1.92
	700	17.41	19.45	4.51	0.89	3.99	1.15	1.63
	900	18.82	21.68	3.34	0.89	3.76	1.06	1.45
	1,100	19.57	23.59	2.47	0.89	3.05	1.02	1.32
	1,300	19.92	25.22	1.79	0.89	2.80	1.00	1.23
	1,500	20.01	26.64	1.23	0.90	2.66	1.00	1.15
	1,700	19.96	27.88	0.77	0.92	2.57	1.00	1.09
	1,900	19.82	28.97	0.37	0.97	2.49	1.01	1.04
	2,100	19.63	29.95	0.02	1.13	1.80	1.02	1.00
	2,114.15	19.61	30.00	0.00	-	-	1.02	1.00

Table 2 .

 2 Probabilistic Results When K h Follows the Exponential Distribution

	P S kN/m	* ͑deg͒	K h *	␤ HL	␥͑ ͒	␥͑ K h ͒	P f ͑%͒
	200	28.58	0.70	2.44	1.02	2.03	3.87
	450	28.77	0.45	1.71	1.03	2.62	6.52
	700	28.84	0.32	1.25	1.06	1.95	11.45
	1,000	29.01	0.22	0.80	1.09	1.34	21.41
	1,300	29.40	0.15	0.37	1.20	0.47	35.33
	1,600	30.01	0.10	0.00	-	-	50.00

Table 3 .

 3 Probabilistic Results When K h Follows the Extreme Value Distribution and COV Kh = 40%

	P S kN/m	* ͑deg͒	K h *	␤ HL	␥͑ ͒	␥͑ K h ͒	P f ͑%͒
	200	28.86	0.71	3.22	1.02	1.54	0.06
	450	28.08	0.44	2.65	1.02	1.69	0.41
	700	26.90	0.27	2.05	1.10	1.18	2.00
	1,000	27.08	0.16	1.11	1.45	0.97	13.25
	1,300	29.15	0.14	0.31	2.18	0.54	37.96
	1,450	30.00	0.13	0.00	-	-	50.00

Acknowledgments

The first two writers would like to thank the Lebanese National Council for Scientific Research ͑CNRSL͒ and the French organization EGIDE for providing the financial support for this research.

Appendix I

The expressions of the functions f i ͑i =1, ... ,6͒ are given as follows:

The expressions of the functions g i ͑i =1, ... ,6͒ are given as follows:


Jérôme Boero
email: jerome.boero@oxand.com

Franck Schoefs
email: franck.schoefs@univ-nantes.fr

Bruno Capra
email: bruno.capra@oxand.com

Nicolas Rouxel
email: nicolas.rouxel@equipement.gouv.fr

Technical management of French harbour structures Part 2: Current practices, needs -Experience feedback of owners

The French harbour system has an essential role: on the one hand for French business, tourism and industry; and on the other hand for European defence. Consequently, maintenance of harbour assets is a stake of major importance. This paper focuses on the analysis of actual maintenance practices, associated constraints, but also capacities to fulfil maintenance programs of the built assets of French harbour owners. The needs in risk management and control are also highlighted. The results are presented by focusing the analysis on observed levels of heterogeneity.

Introduction

The challenge for the owners consists in guaranteeing the operation of ageing structures (wharves, dry docks, dikes, etc.) with necessary levels of safety, reasonable costs and acceptable availability conditions. Moreover, the opportunities for harbour development are more and more limited in the face of increasing environmental constraints and decreasing space available for development with low costs for building. To deal with this problem, the GEROM project (Risk Management of maritime and river structures in harbours) was created within the GIS MRGenCi (Grouping of Scientific Interest "Risk management in Civil Engineering") and lies on a partnership between Oxand and the Research Institute in Civil and Mechanical engineering of Nantes. The GEROM project was undergone from 2006 to 2009. In the first phase, with the support of the CETE de l'Ouest (Ministry of ecology, energy, sustainable development and town and country planning), a national inquiry was carried out with owners about the technical management of their harbour structures. This paper focuses on maritime harbours only. The national inquiry was prepared with the aim of a risk analysis and the results are presented by centring the analysis on the observed levels of heterogeneity. Due to the multiple topics addressed during the national investigation, results are presented in two parts. This paper deals with the second part and is focused especially on: a) Current practice of maintenance (policies and strategies). b) Organization, constraints. c) Capacities to fulfil operations (maintenance programs).

Spatiotemporal perimeter of the national inquiry on the harbour assets management

This inquiry was performed from January 24 th to May 12 th 2006, and was completed in the first quarter 2007. The geographical perimeter covers a total amount of 7 Autonomous Harbours (AH), 16 National Interest Harbours (NIH) and 3 Military Harbours (MH) located along the French metropolitan coast. The status of AH have changed since the law of July 4th, 2008 and since its implementing decrees, in particular those of October 9th, 2008. The 532 harbours, that have been decentralized since 1984, and managed by the General Councils of departments or Municipalities, were not integrated in this study because of their large number. However, some of them were interviewed in order to have an overall view. Further to the recent decentralization of the NIH since January 01 st 2007, the status of NIH no longer exists. In this article, this status was preserved to make the distinction with harbours already decentralized in 1984. The referees are technical managers of harbour structures.

A very good level of return was obtained. In fact, of the 23 contacted harbours (AH, NIH, MH), 19 were interviewed, representing approximately 85% of the targeted owners.

Maintenance practices

The stages which allow owners to ensure risk management of civil engineering infrastructure are presented in figure 1. According to this pyramid, the specific objectives concerning the harbour system are to expose: a) The framework which governs the technical management of the harbour structures.

b) The methods used by owners to build the previous pyramid. c) The actions needed at every stage.

Framework for the technical management of harbour structures

The framework which governs the risk management of the harbour structures differs according to the status (MH, AH, NIH) and is more or less formalized (explicit or implicit).

MH have a framework for the levels 1 (Knowledge) and 2 (Inspection) of the pyramid. Inspection (various actions and rules of implementation) is required as defined in the Instruction n°125/DEF/TM/T relative to the inspection of the structures of civil engineering of July 11th, 1996 established by the ministry of the Defence (DCTIM, 1996). Contrary to military harbour structures, but also road assets, the French State does not impose a framework for the management of civil harbour structures (AH, NIH). However, more and more owners are inspired by the technical instruction of October 19th, 1979 (ITSEOA 79 modified 95) for the inspection, maintenance and repair of the infrastructures along the national road network (DIRECTION DES ROUTES, 1995) or recommendations which result from specific working groups (AIPCN, 2001 and2007).

Methodologies of asset management

During the national inquiry, various methodologies of asset management were identified. Technical managers can be classified in three different degrees of advancement: "Reflection"; "Implementation"; "Application" (see figure 2a).

A quarter of the respondents (MH, AH, NIH) are working on the conception of best methodology adapted to the harbour structures, while another quarter is implementing it.

In both cases, they are dealing with formalized methodologies which are based on a risks approach or on notation systems, based on the structure's state (estimated after inspections) and sometimes coupling with their strategic importance in the studied system.

The remaining half already applies a methodology of management: 50% use formalized methodologies and 50% use non formalized methodologies because they consider that the number of structures and the linear kilometres are not sufficient to look for more advanced decision-making tools.

For the three different degrees of advancement in terms of asset management methodology, the percentage of built assets concerned is appreciably the same (see figure 2b).

Inspection of harbour structures

Most owners continuously monitor their structures by the means of the harbour office staff, equipment staff or statutory undertaker (Chambers of Commerce and Industry, etc.). This type of inspection allows to identify the accidental events (ship collision), the failing fender systems or safety equipments (ladders), but also to detect the appearance of defects on the open area (misalignment of the rails and settlements, etc.). These defects can have heavy consequences ion the short-medium term for the security of the people, as well as the safety and the availability of the installations.

To prevent such events from occurring, approximately two thirds of the respondents began a periodic inspection of their harbour structures, completed if necessary by a particular inspection. The implementation of this type of inspection is relatively recent (in the past 10 years). Since 1996 and the technical instruction n°125/DEF/TM/T, the MH started elementary periodic visits and detailed inspections of the structures. The elementary visits are carried out internally by engineers. They aim to detect visually the defects without specific means others than simple optical instruments, light means of control and common measuring instruments. Their frequency is annual, however according to the characteristics, the age and the state of the structure, a more important interval can be fixed, without exceeding 3 years. The detailed inspections are more complete than the elementary visits. They consist of a diagnosis which concerns all of the structure's components. Specific means of measurements and tests can be implemented if necessary. For example, the immersed components are inspected by divers. This specific task is usually performed by private companies. The frequency of detailed inspections is fixed at 5 years, but it can be modulated between 4 and 7 years or less in case of necessity. For AH and NIH, the periodic inspection of all the structures is similar to the elementary periodic visits, and the frequency varies between 6 months and 1 year. Results of inquiry show that the number of structures really inspected within the periodic inspection framework is a function of the budget and\or the available staff resources and periodicity is more and more spaced out in time. For this reason, half of the owners prioritize the periodic inspection according to the strategic interests associated with the structures. If necessary, a localised inspection completes the inspection process. These actions are similar to the periodic inspection most of the time, but they are implemented in the following cases: a) Accidental events (storms, collisions, etc.). b) Appearance of defects. c) Diagnosis before repairs. d) New needs (replacement of the harbour equipment) and exceptional events. e) Strategic stakes. f) Recent constructions (before official acceptance of work, ten-year guarantee). To complete their knowledge on the state of the structures, owners appeal to additional investigations. The most frequent investigations are: a) Steel residual thickness measurements by ultrasound. b) Chloride ion penetration profiles in reinforced or prestressed concrete components. c) Bathymetric measurements. d) Ground investigation with sampling, laboratory and in situ testing (by geologic radar and/or pressuremeter tests) and groundwater observations. e) Topographic survey. f) Components instrumentation (tie-rods for example) during the first years after construction to follow the global behaviour of structures (vibrating wire sensors, electrical strain gauges, sensors measuring the water level (piezometers)). In the case of MH, these actions are often included in the contents of the detailed inspections.

Common maintenance, repairs of harbour structures

The common maintenance and repairs are defined by owners according to the following criteria: a) Accidental events (ship collision). In this case, the repair or the reconstruction of the damaged part is paid for by the ship owner's insurance. b) First urgency actions to avoid a global collapse of the structure. c) Detection of important defects. d) Preventive maintenance to avoid the irreversibility of the phenomena on the mechanical performances of a structure. e) For the mobile structures, the respect of conditional maintenance criteria (service rate) established by the owners' experience feedback. f) Strategic stakes associated with the structures. g) Efficiency of the actions. h) Cost of the actions. i) Available workforce to ensure the success of repair projects. The different actions can be classified according to the impact measured on the performance of a structure. During the national inquiry, two intervention levels were chosen: common maintenance and repairs. Maintenance is the recurrent day-to-day, periodic, or scheduled work that is required for a facility to be used for its designed purpose. It includes routine work undertaken to prevent damage or deterioration of a facility that otherwise would be more costly to restore. Repair is the restoration of a facility to such a condition that it can be used for its designed purpose. The repair is done by overhaul, reconstruction, or replacement of deteriorated constituent parts or materials that have not been accomplished through maintenance (HQUSACE, 2001). The more common concerns in common maintenance waterfront facilities are: a) Routine painting, replacement of fender components and safety equipment. b) Filling of settlements on the open area. c) Localized rejointing of masonry structures. d) Routine patching of small concrete spalls and cracks. e) Painting and protective coating of some piers. When the actions of common maintenance present a generalized character, they are considered as actions of repair. Common maintenance organization is presented in figure 3. Common maintenance of structures in the MH and AH is assigned to private companies through contracts with order forms. On the contrary, common maintenance is performed by internal workforce in the NIH. Only a single NIH breaks this rule. In this case, the statutory undertaker is in charge of common maintenance. The organization of repairs is presented in figure 4. Repairs are generally made by private companies because they require specific skills and means, but also important human resources during a long period. However, some NIH carry out specific works and they have at their disposal internal skills and technical means (materials) for carrying out operations such as: a) Replacement of dolphins. b) Careening of the lock's gate except their painting.

In fact, in the case of painting, specific equipment and processes are required to ensure health and safety standards, but also to respect the environment: specialised private contractors are thus generally selected for this task.

With regards to the strategic importance of mobile structures (bridges, locks) for sea and urban transport, NIH benefits from additional means (workforce and budgets) to ensure the whole of common maintenance and repairs of mobile works. Types of actions (common maintenance, repairs or construction) most frequently performed by owners are detailed in figure 5. The MH have very few new constructions. They therefore limited their strategy to common maintenance and repair on important built assets. Unlike MH, even if they schedule common maintenance, AH follows a repair and construction logic, with projects such as Port 2000 (Le Havre) or Fos 2XL (Marseille) for adapting facilities to the new needs of sea transport. Concerning NIH, repairs or constructions are possible if the statutory undertaker contributes to the financing of the projects and depends on the perimeter of the concession. However, because of limited budgets, only common maintenance is carried out depending on the workforce available internally. Note that for risk purposes, this analysis should be completed by an economical analysis on the cost of building-as-new of the several studied harbours. This study is not in the perimeter of the present paper. The maintenance strategies are coherent with the harbours' strategies (see figure 6). MH are in an optimization process for management of the existing facilities, considering the policy of rationalization of needs defined by the ministry of defence. A part of the NIH also follows this logic. It is mainly the case of harbours with low goods traffic or with geographic constraints as the Saint-Malo harbour, which is enclosed in the city. AH and the other part of NIH combine at the same time optimization and development logics. Note that this figure illustrates the result of a combination of several complex factors (economic, environmental, political, etc.) at several scales (local, national, european, international).

Maintenance constraints

The quantitative patrimonial analysis displays heterogeneous and ageing harbour assets.

The most recent part of the latter is essential for the local, national and European economic development because it is better adapted for the recent evolution of traffic.

To maintain a required level of performance, the owners have to program and to carry out various maintenance actions. The respect of the established program depends on operation constraints, technical and financial constraints associated with the harbour structures and with their technical management.

The operation constraints mainly concern the structures which can't be temporarily stopped, even if an activity transfer is possible on another structure. It is generally the case of locks, mobile bridges, specialized berthing and mooring structures (oil, containers, etc.). In these conditions, the definition of a preventive intervention must be established through dialogue with all users. Given the difficulties which it implies (partial interventions, between operations of load and\or unloading) the maintenance of these structures is frequently postponed, generating an increase of risks.

In marine environment (swell, tides, climatic hazards, physical and chemical actions of sea water), maintenance of structures requires specific means (underwater works, works in the tide, materials in the sea). These technical constraints, coupled with the length of the structures and operation requirements, increase the maintenance costs. At the same time, the maintenance strategies drawn up by the owners are largely driven by the financial constraints: budgets and limited workforce.

The AH maintenance budget is more important than that of the NIH: this is due to a larger number of built assets and a higher turnover. It can be quantified by the ratio "million tons of raw goods by linear kilometre (Mt / km) ". On the contrary, workforce dedicated to the harbour maintenance is less important in the AH (3.5 persons fulltime equivalent) than in the NIH (8.5 persons on average: except staff allocated to the mobile structures). Note that for NIH, the maintenance personnel comprise of staff from both the statutory undertaker (Chambers of Commerce and Industry, etc.) and the owners.

The staff of the AH organize the surveillance (detailed inspections) and the project management during the reparation projects whereas surveillance and maintenance actions are given to private enterprises. The NIH have a tendency to carry out a maximum number of actions under state control: surveillance of the visible parts, common maintenance and in certain cases major repairs on locks and mobile bridges for example. It explains the existence of differences between NIH which do not contain mobile works and NIH which do. Budgets, limited workforce and high costs of maintenance actions, consolidate the fact that owners need simple and successful patrimonial management tools to optimize the maintenance. They also need global indicators to offer a vision of risks control for the decision-makers. This vision can be used to adjust or not the budget.

Expectations and needs in risk management and control

After describing the common practices of harbour structure asset management in the previous section, the reasons which motivate the owners to develop their own practices are highlighted. This evolution is translated by the expression of needs in terms of means (workforce and budgets), but also in terms of asset management methodologies.

During the national inquiry, the technical managers had to classify several impacts which lead to evolution of maintenance practices. A total amount of 11 impacts, classified in four categories were used: a) Two Security impacts IS: safety of the people, environment. b) Six direct and indirect economic impacts IC1: port dues, operating loss, built assets valuation, competitiveness preservation, brand image. c) One Budgetary impacts IC2: maintenance costs. d) Two organizational impacts IO: crisis situation, know-how transmission.

Only the first four impacts selected by each owner are taken into account. A lessening weight is applied on each: 25 for the first, 20 for the second, 15 for the third and 10 for the fourth. Then, for each of the 11 impacts, all the weights are added. Finally, for each of the four categories above (a, b, c, d), only the most important factor (i.e. with the maximum sum of weights) is kept. Then a percentage is computed and is presented on figure 7 left. Note that some results are very close (IS and IC1) and the discrepancy is sensitive to the selection of lessening weights. For use in a risk-based decision process, it is recommended to consider as equal index whose range can be inversed by modifying the assumption. It is the case for IS and IC1 if we select the following lessening weight with a power relationship (figure 7 right): 16 for the first, 8 for the second, 4 for the third and 2 for the fourth. As a perspective, such results can be implemented into a multicriteria evaluation, such as the one developed within the French National research Project RERAU (for Renewal of Non Man Entry Sewer System -in French -Réhabilitation des Réseaux d'Assainissement Urbains), in order to define the built assets performance with regard to the owner's expectations (BREYSSE et al., 2007).

Conclusions

This article presents the second part of the results of the national inquiry of maritime harbour structures technical management led in 2006-2007. Results . Results show great heterogeneities between the current practices of maintenance (policy and strategies), organization, constraints and capacities to fulfil operations (maintenance programs). These differences depend mainly on the harbour status (MH, AH, NIH). Another important information is that a quarter of the owners are beginning to use formalized asset management methods but all show interest in better organizing this process with the aim of guaranteeing the operation of ageing structures with good safety, availability conditions and reasonable maintenance costs. The ranking of stakes (impacts of maintenance) available in this paper is suitable for a use in a multicriteria risk analysis.

Acknowledgements

This study is led within GEROM (contact: franck.schoefs@univ-nantes.fr), project financed by the company OXAND and led within the Grouping of Scientific Interest MRGenCi (www.mrgenci.org). The authors thank the CETE de l'Ouest for its contribution to the inquiry phase and all the owners who were kind enough to grant time to this project.

Figure 1 .

 1 Figure 1. Simplified vision of the civil engineering infrastructure Risk Management pyramid.

  of built assets Figure2. Progress state of owners in terms of asset management methodology.

Figure 3 .

 3 Figure 3. Organization of common maintenance.

Figure 4 .

 4 Figure 4. Organization of repairs.

Figure 5 .

 5 Figure 5. Most frequent maintenance actions. Legend: CM = Common Maintenance, R = Repair, C = Construction.

Figure 6 .

 6 Figure 6. Harbour strategies: optimization and/or development of the existing spaces.

Figure 7 .

 7 Figure 7. Weights computed (percentage) for the various impacts ranked technical managers during national inquiry with two weighting assumptions.

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