Probabilistic analysis of ultimate seismic bearing capacity of strip foundations

Adm Hmrouni,Bdreddine Sbrti,Dniel Dis

aLaboratory InfraRes,Mohammed Cherif Messaadia University,Souk-Ahras,Algeria

bDepartment of Civil Engineering,Badji Mokhtar-Annaba University,Annaba,Algeria

cLaboratory LMGHU,Skikda University,Skikda,Algeria

dSchool of Automotive and Transportation Engineering,Hefei University of Technology,Hefei,230009,China

eGrenoble Alpes University,Laboratory 3SR,Grenoble,France

Keywords:Strip foundations Seismic bearing capacity First-order reliability method(FORM)Response surface methodology(RSM)Reliability Genetic algorithm(GA)

ABSTRACT This paper presents a reliability analysis of the pseudo-static seismic bearing capacity of a strip foundation using the limit equilibrium theory.The first-order reliability method(FORM)is employed to calculate the reliability index.The response surface methodology(RSM)is used to assess the Hasofer-Lind reliability index and then it is optimized using a genetic algorithm(GA).The random variables used are the soil shear strength parameters and the seismic coefficients(khand kv).Two assumptions(normal and non-normal distribution)are used for the random variables.The assumption of uncorrelated variables was found to be conservative in comparison to that of negatively correlated soil shear strength parameters.The assumption of non-normal distribution for the random variables can induce a negative effect on the reliability index of the practical range of the seismic bearing capacity.

1.Introduction

Uncertainty is an important issue in engineering design as geotechnical engineers can basically introduce uncertainty in the design when using a global safety factor.Reliability methods have therefore become promising when assessing the effect of uncertainty on geotechnical structure design.The designs using reliability assessment were applied to many geotechnical engineering projects(e.g.Mollon et al.,2009a,b,2011,2013;Grif fiths and Fenton,2001;Grif fiths et al.,2002;Kulhawy and Phoon,2002).

Many theories have also been used to study the seismic bearing capacity of a strip foundation(e.g.Budhu and AlKarni,1993;Dormieux and Pecker,1995;Soubra,1997).Their results indicated that the value of the bearing capacity decreased with the increase of the seismic acceleration coefficient.Inertia forces in the soil mass decrease the bearing capacity of the soil and,as a result,the bearing capacity of the foundation decreases.In recent years,some researchers such as Zeng and Steedman(1998),Garnier and Pecker(1999),Askari and Farzaneh(2003),Gajan et al.(2005),Knappett et al.(2006),and Merlos and Romo(2006)have drawn the same conclusions by using the dynamic centrifuge tests.Using the characteristics method,Cascone and Casablanca(2016)evaluated the static and seismic bearing capacity factors for a shallow strip foundation by the pseudo-static approach.Other researchers such as Pane et al.(2016)numerically obtained the bearing capacity of soils under dynamic conditions.Sha fiee and Jahanandish(2010)employed the finite element method to determine the seismic bearing capacity of strip foundations with various seismic coefficients and friction angles.They also presented curves relating the seismic bearing capacity factors to the seismic acceleration coefficient.

In this context,the homogeneous soils and seismic properties are used to analyze the seismic bearing capacity of strip foundations.The bearing capacity is calculated using a single deterministic set of parameters.Reliability analysis is then used to assess the combined effects of uncertainties and provide a logical framework for selecting the bearing capacity that is appropriate for a degree of uncertainty and the failure consequences.Thus,the reliability assessment useful for providing better engineering decisions is performed as an alternative to the deterministic assessment.

Over the last fifteen years,the reliability analysis of shallow foundations subjected to a centered static vertical load has been studied by Fenton and Grif fiths(2002,2003),Sivakumar Babu et al.(2006),and Youssef Abdel Massih et al.(2008).However,the reliability analyses of shallow foundations subjected to inclined,eccentric or complex loads are rarely investigated(Ahmed and Soubra,2014).Probabilistic approaches for seismic bearing capacity of shallow foundation are seldom elaborated in the literature(Youssef Abdel Massih et al.,2008;Baroth et al.,2011).Johari et al.(2017)used the slip lines method coupled with the random field theory to estimate the seismic bearing capacity of strip foundations.The bearing capacity factors Ni(Nc,Nqand Nγ)are assessed stochastically,with the values depending on friction angle.

In previous researches,different types of simulation approaches were used to assess the reliability of geotechnical systems,inwhich the response surface methodology(RSM)is basically used.Monte Carlo simulation(MCS)(Wang et al.,2010)and importance sampling(IS)(Mollon et al.,2009a)offered the implied estimates of the system failure probability(Pf).However,they are rather timeconsuming (e.g. finiteelementmethod or finitedifference method).Different types of RSMs such as classic RSM,artificial neural network(ANN)based RSM(Cho,2009)and Kriging-based RSM(Zhang et al.,2013)have been proposed to overcome this disadvantage.However,they are all approximate methods which cannot provide precise estimates.

This paper presents a reliability analysis of the seismic bearing capacity of a strip foundation under pseudo-static seismic loading.The uncertain parameters are modeled by random variables.These variables are the soil shear strength parameters and the seismic coefficients(khand kv).Only the punching failure mode of the ultimate limit states is studied.The deterministic model is based on the limit equilibrium theory(Budhu and Al-Karni,1993).The Hasofer-Lind reliability index(βHL)was adopted to calculate the reliability of the seismic bearing capacity.The RSM optimized by the genetic algorithm(GA)have been used to find the approximate performance function and deriveβHL.The RSM optimized by GA saves computation time compared with the conventional RSM methods(Hamrouni et al.,2017a,b,2018).The in fluence of normal and non-normal parameters distribution as well as the correlation between soil shear strength parameters on the failure probability is studied.

2.Ellipsoid approach in reliability theory

The safety of geotechnical structures can be represented by itsβHLvalue which takes the inherent uncertainties as input parameters.TheβHL(Hasofer and Lind,1974)is the most widely used indicator in the literature.Its matrix formulation is(Ditlevsen,1981)

whereμis a vector of mean values,x is a vector representing the n random variables and C is a matrix covariance.

The minimization of Eq.(1)is performed using the constraint G(x)≤0 where the n-dimensional domain of the random variables is separated by the limit state performance(G(x)=0)into two regions:an unsafe region F represented by G(x)≤0 and a safe region given by G(x)＞0.Eq.(1)is used in a form of the classical method to calculateβHL,which is based on the transformation of the performance limit state initially defined in the space of the physical variables.This state must be shown in the space of the normal random variables,centered,reduced and uncorrelated,which is also called standard space.TheβHLis the shortest distance between the origin of the space and the state boundary surface.

Low and Tang(2004)proposed an interpretation ofβHL.The concept of iso-probability ellipsoid leads to a simpler calculation method forβHLin the original physical variables(see Fig.1).Low and Tang(2004),Mollon et al.(2009b),Lü et al.(2011),Low(2014)and Hamrouni et al.(2017a,b,2018)demonstrated that the ellipticity(ratio between the axes)of the critical dispersion ellipsoid corresponds to the value ofβHL,which is the smallest ellipsoid dispersion that just touches the limit state surface to the unit dispersion ellipsoid,i.e.the one obtained forβHL=1 in Eq.(1)without minimization.

They also stated that the intersection point between the critical dispersion ellipsoid and the equivalent performance limit state surface is called the design point(see Fig.1).In the case of nonnormal random variables,the Hasofer-Lind method can be extended.A transformation of each non-normal random variable into an equivalent normal random variable with an averageμNand a standard deviationσNwas proposed by Rackwitz and Flessler(1978).Using the above-mentioned procedure,the transformation makes it possible to estimate a solution in a reduced space.The equivalent parameters evaluated at the design pointare given by

whereΦandφare the cumulative density function(CDF)and the probability density function(PDF)of the standard variables,respectively;FXiand fXiare the CDF and PDF of the original nonnormal random variables,respectively.The CDFs and PDFs of the real variables and the equivalent normal variables identified at the design point on the performance state surface are assimilated after derivation of Eqs.(2)and(3).

Low and Tang(1997,2004)implemented an inclined ellipsoid and an optimization algorithm tominimize the dispersion ellipsoid.Eq.(1)can then be rewritten as

where R-1is the inverse of the correlation matrix R.The configuration of the ellipsoid can be presented by this equation.

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

The probability of failure is approximated using the first-order reliability method(FORM)as follows:

2.1.RSM optimized by GA

If the objective function has a known analytical form,βHLmaybe easily calculated.When using numerical calculations,it is impossible to obtain an explicit analytical form of the objective function.The RSM can then be used to approach this function by successive iterations to calculateβHLand the design point.An algorithm based on the RSM proposed by Tandjiria et al.(2000)was used in this work consequently.This method approximates the function of performance by an explicit function of the random variables using an iterative process.The quadratic form(see Eq.(6))of the approximate performance function is the most widely used form in the literature(second-order polynomial with squared terms):

where aiand biare the coefficients to be determined,and xirepresents the random variable.

For a higher accuracy purpose,a more complex performance function(Eq.(7))can be used which contains quadratic and crossed terms:

The parameters aiand bijin Eq.(7)can be determined using the iterative method,but it seems to be rather time-consuming(Youssef Abdel Massih and Soubra,2008;Mollon et al.,2009b).This method is used for a specific point of the limit state,thus it has to repeat this calculation for determination of other reliability index values.In this paper,the parameters aiand bijwill be calculated by an optimization using GA(Bouacha et al.,2014;Hamrouni et al.,2017a,b,2018).The coefficients aiand bijare determined from a number of deterministic calculations using values of the variables xi,by the least-squares regression analysis.

In this study,the parameters of the optimization problem for parameters aiand biare translated into chromosomes with a data string.To begin with the procedure of GA,an initial population is needed.The size of the initial population depends on the nature of the problem,and it usually contains several hundreds and thousands of possible solutions(in our study,a number of 50 was chosen).This population is generated randomly,covering the whole range of possible solutions,i.e.the research space(Tang et al.,1996).

The minimum square error(MSE)is represented by the fitness function in the GA approach to compare the results obtained with Eq.(7)and the deterministic results.This permits to determine the values of aiand bij,on which no constraints occur.

Several possible solutions are obtained from the variables space and the physical conditions of these solutions are compared.If no solution is reached,a new population is created from the original(parent)chromosomes using “crossover”and “mutation”operations.From two random solutions(parents),the crossover forms a child(new solution)by the exchange of genes.Mutation is used to maintain population diversity by randomly switching a single variable into a chromosome,as the process converges towards a solution.The operation of the GA process is detailed in the flowchart shown in Fig.2.The key advantages of GA are described as follows:

(1)It is a population-based approach and thus considers a wide range of possible solutions;and

(2)The mutation process restricts the solution to local minima that can occur in alternative solution techniques.

ThestopcriteriaforGAsearchoperationarecrucialandprobably difficult.Since GA is a stochastic global optimization technique,it is rather difficult to know when the algorithm has reached its optimum.Withthe measures of convergence anddiversity,itis possible to compare the performances of the GA during operation state.

3.Deterministic model

Budhu and Al-Karni(1993)assumed the logarithmic failure surfaces as shown in Fig.3 for determination of the seismic bearing capacity of soils.They modified the commonly used static bearing capacity equations of Meyerhof(1963)to obtain the dynamic bearing capacity as follows:

The bearing capacity factors in Eq.(8)are computed by the following equations:

where B is the width of the foundation,Dfis the depth of the foundation,φ is the friction angle,c is the cohesion of soil,γ is the unit weight,s is the shape factor,d is the depth factors,i is the inclination factor,andHisthedepthof thefailure zonefromthegroundsurface.

The parameters ec,eqand eγare the seismic factors that are estimated using the following equations:

where khand kvare the horizontal and vertical seismic coefficients,respectively;and D is given by the following equation:

4.Reliability analysis of seismic bearing capacity

For the failure mechanisms of Budhu and Al-Karni(1993),in this paper,the deterministic results presented consider the case of a shallow strip foundation with a width of 2.5 m and a depth of 1 m.The unit weight of the soil is 18 kN/m3.The values of the internal friction angle and cohesion are 30°and 20 kPa,respectively.By using Eq.(8),the ultimate seismic bearing capacity reaches only 729.51 kPa with khand kvvalues of 0.2 and 0.06,respectively.

Fig.2.Principle of optimization with GA.

Fig.3.Soil failure mechanism under static and seismic conditions assumed in the theory of Budhu and Al-Karni(1993).

The Algerian seismic regulation(RPA,2003)recommended that,in Algeria,kvequals±0.3khfor several types of structures such as foundations,retaining walls and slopes.Our study takes this linear relationship between the two seismic coefficients,and will be discussed only with the parameter kh.

4.1.Performance function

Three random variables used in this study are the soil shear strength parameters(c and φ)and the seismic coefficient(kh).The values of the mean and the variation coefficient are chosen and presented in Table 1.Two cases are studied:random variables with normal and non-normal distribution,respectively.The parameters c and khare assumed to follow a lognormal distribution and φ is considered to follow a Beta distribution to better represent the friction angle(Fenton and Grif fiths,2003).The parameters of the Beta distribution are determined fromthe meanvalue and standard deviations of φ.A negative correlation between the variables c and φ (ρc,φ)are considered equal to-0.5.

The performance function used in this study is given by

(a)In this text,100 sample points taken from the direct Monte Carlo method were used to calculate qudusing the random variables.

(b)Coefficients(ai)of the performance equation G(φ,c,kh)are optimized by GA using 100 sample points.

Table 1 Probabilistic model.

(c)The Matlab optimization tool(fmincon)is utilized to determine the minimum value ofβHLand the corresponding design point(φ*,c*,k*h)using the condition G(x)≤ 0(G(x)is presented in step b).

The advantage of GA is that it moves in the search space with more possible solutions.The successful use of GA depends on how accurately and quickly it converges to the optimal solution,avoiding local minima to reduce the computation time.However,the major disadvantage of GA in case of a large number of variables is that it requires a significant computation time before the optimal solution is finally reached.In our case study,the number of variables is 11 and the computation time is less than 1 min,which makes it reasonable to choose this optimization method.

4.2.Numerical results

Eq.(11)is proposed to approximate the performance limit state.The case with normal uncorrelated variables is used to present the efficiency of GA approach.After 100 running of the GA,the best results are selected to illustrate the best combination that satisfies Eq.(11).The MSE is 0.19 with a1=710.48405,a2=-12.1999,a3=0.37112,a4=-18.54146,a5=-0.1532,a6=184.02496,a7=1575.78991,a8=3.78538,a9=-103.37221,a10=132.45428 and a11=-14.90744.

For the case with normal uncorrelated variables,theβHLvalues obtained after convergence are 3.981,2.691 and 1.795 for qu,minvalues of 200 kPa,300 kPa and 400 kPa,respectively.These indices correspond to the failure probabilities of 0.003%,0.357%and 3.629%,respectively,using the FORM.

A good way to validate the convergence of the GA optimization approach is to consider the value provided by the model at the design point.In case of normal uncorrelated variables,qudvalues provided by the deterministic model and the quadratic polynomial are 200.02 kPa and 198.31 kPa,301.5 kPa and 298.42 kPa,and 399.2 kPa and 399.9 kPa,respectively,which can be compared with the acceptable maximum efficacy of 200 kPa,300 kPa and 400 kPa.In this paper,the use of a GA is very effective to optimize the unknown parametersoftheperformancefunction.Thus,a quadratic polynomial with crossed terms between parameters is used as the function of response surface that permits to obtain a good approximation of the performance limit state in previous analyses.

5.Reliability index,design point and partial safety factors

Table 2 shows the results ofβHL,design points(φ*,c*and k*h),and partial factors for different bearing capacity limit values.The calculations were performed for several cases:normal variables,non-normal variables,variables correlated or not.Note that the values ofβHLand partial factors increase with the decrease of the bearing capacity limit value.The results ofβHLare also shown in Fig.4,with a negative correlation between the shear strength variables.It is the same as a lower extent when considering nonnormal variables.

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For example,for a bearing capacity limit value equal to 300 kPa,the comparison of the results between the correlated or uncorrelated variables shows that the value ofβHLcorresponding to the negatively correlated variables is greater than that of the uncorrelated variables case.For the same bearing capacity limit values,the value ofβHLdecreases by about 10%if one considers non-normal variables.Wecan concludethatthesimplifiedassumption considering the uncorrelated normal variables is safer compared tomore complex probabilistic models.This can therefore lead to uneconomic designs.

Table 2 Indices of reliability,design points,and partial safety factors.

Fig.4.Reliability index related to the performance limit.

Engineers are interested in reliability index values of 2＜βHL＜4,suggesting that taking into account the non-normal variables has an in fluence onβHL.This observation is related to the fact that the distribution functions of normal and non-normal variables differ in the zones of different design points obtained.The randomvariables(φ,c)and khhave reverse effects on the behavior of the model.For example,a low bearing capacity is induced by the reduction of(φ,c)and the increase of(kh).

The coordinates(φ*,c*and k*h)of the design points obtained for bearing capacity limit values can be used to calculate the partial factors Fφ,Fcand Fkhas follows:

where μφ,μcand μkhrepresent the mean values of friction angle,cohesion of soil and horizontal seismic coefficient,respectively.

Factors for each bearing capacity limit value are also provided in Table 2.For uncorrelated variables,with the increase of the bearing capacity limit values,the partial factors are even lower,and almost are equal to 1 for the limiting case.In the case of negatively correlated variables,it is sometimes observed that the value of the design point c*slightly exceeds the average value of c,which gives Fcless than 1 and large values of Fφand Fkh.This is due to the negative correlation between c and φ.This correlation implies that if the value of c is smaller than its mean,then the values of φ and khwill probably be high.For this reason,a case with Fcless than 1 does not necessarily indicate failure,as long as the value of Fφis high.This conclusion is similar to those of Youssef Abdel Massih and Soubra(2008),Mollon et al.(2009b),and Hamrouni et al.(2017a,b,2018).

Fig.5.Failure probability in relation to the performance limit.

From theβHLvalues obtained by RSM optimized by GA,the failure probability values are provided directly by the FORM approximation,as shown in Fig.5.Taking into account a negative correlation between c and φ and the use of bounded laws instead of the normal laws can significantly reduce the failure probability,when other parameters remain unchanged.The assumptions of normal and uncorrelated laws are quite acceptable.It is also observed that the failure probability is much more sensitive to the variations of φ and khthan c.

6.Conclusions

An analysisbased on the reliability of seismic bearing capacityof strip foundation subjected to a vertical load with pseudo-static seismic loading is presented in this paper.The main conclusions are drawn as follows:

(1)The use of GA is very effective in optimizing the unknown parameters of the performance limit function.A quadratic polynomial function with crossed terms between the parameters makes it possible to obtain a satisfactory approximation of the limit performance state in the previous analyses.

(2)Assumption of negatively correlated shear strength parameters(c,φ)was found conservative with respect to uncorrelated variables.For uncorrelated shear strength parameters values,the design point value c*slightly exceeds the average value of c,which gives partial safetycoefficients Fcless than 1 and large values of Fφand Fkh.For this reason,a case with Fcless than 1 does not necessarily indicatefailure,as long as the value of Fφis high.

(3)For the higher values of the minimum seismic bearing capacity,the reliability indexβHLis lowand induces a very high failure probability value,indicating the vulnerability of this structure.

(4)The simplified assumption considering a normal variable is safer compared to more complex probabilistic models(nonnormal variables).This can lead to uneconomical designs.

Conflicts of interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have in fluenced its outcome.

Acknowledgments

The authors are grateful to the Ministry of Higher Education and Scientific Research of Algeria for supporting this work by offering an 11-month scholarship to the first author at the 3SR laboratory of Grenoble Alpes University,France.