Engineering Structures 32 (2010) 3725–3733

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

The projection gradient algorithm with error control for structural reliabilityFrédéric Duprat ∗, Alain Sellier, Xuan Son Nguyen, Gérard PonsUniversité de Toulouse, Laboratoire Matériaux et Durabilité des Constructions, LMDC UPS-INSA, 135 Avenue de Rangueil, 31077 Toulouse Cedex 4, France

a r t i c l e i n f o

Article history:Received 22 February 2010Received in revised form24 June 2010Accepted 9 August 2010Available online 16 September 2010

Keywords:ReliabilityProbabilistic methodsRackwitz–Fiessler algorithmProjection gradientDesign pointError control

a b s t r a c t

Nowadays, probabilistic approaches are frequently used in the design of new civil engineering structuresand the durability analysis of existing constructions. Hasofer–Lind’s reliability index is one of the mostpopular reliabilitymeasures due to its relevance and ease of use, and is now referred to inmany structuraldesign codes. This index can be determined by several algorithms dealing with minimization underconstraint such as the well known Rackwitz–Fiessler algorithm based on the projection gradient method.The drawback of this algorithm lies in the estimation of the gradient vector of the limit-state function,which is often carried out by finite differences for non-explicit functions, resulting from the Finite ElementMethod for instance. If the perturbation chosen in the estimation of the gradient vector gives a variationof the output lower than the numerical accuracy of the limit-state function, the algorithm could lead toerroneous results or even not converge. In order to circumvent this problem, an original technique issuggested in this paper called ‘‘Projection gradient with error control’’. The principle of the proposedtechnique is to attach to Rackwitz–Fiessler’s algorithm a procedure for judiciously determining theperturbation in the calculation of the gradient vector by finite differences, accounting for the numericalprecision of the limit-state function. The efficiency and interest of the proposed procedure is emphasizedthrough various examples.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The design of civil engineering structures is based on pro-cedures and models integrating the design constraints such asloads, material properties and geometry. The uncertainties affect-ing the design parameters are of a random nature. Probabilistic ap-proaches are consequently suitable tools for assessing structuralrisk and reliability as they include a probabilistic description of therandom uncertainties. A common practical measure of the struc-tural reliability is Hasofer–Lind’s reliability index denoted β [1].This index is defined in the standardized space of reduced normaland independent variables as the minimum distance from the ori-gin to a point of the failure surface, the so-called design point P∗.The reliability index β can be determined by several algorithmsdealing with minimization under constraint such as the Rack-witz–Fiessler algorithm [2]. This algorithm, like those derived fromit to improve its robustness and efficiency, or other gradient-basedalgorithms, requires computation of the gradient vector composedof the partial derivatives of the failure function with respect to thestandardized variables. When the failure function is defined froma numerical model that only provides approximations of the ex-act solution to the mechanical problem, the partial derivatives are

∗ Corresponding author. Tel.: +33 561 559 930; fax: +33 561 559 900.E-mail address: frederic.duprat@insa-toulouse.fr (F. Duprat).

0141-0296/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2010.08.017

in turn computed approximately, which can lead to errors or evennon-convergence in some cases. Although fruitful developmentshave been achieved in the finite element field for calculating thesensitivity of model response to design variables, the finite differ-encemethod remains a practical and general tool for evaluating thepartial derivatives. The aforementioned problem may arise due topoor choice of the perturbation applied to variables in this eval-uation. In order to overcome this drawback, a simple and efficientprocedure is proposed for choosing the perturbation accounting forthe numerical precision of the model.

2. Determining the reliability index

The determination of Hasofer–Lind’s index is a constrained op-timization problem. The function to be minimized is the Euclideandistance‖u‖ in the standardized space under the constraintG(u) =

0, where G() is the failure function. If u∗ is the solution of the opti-mization with β = ‖OP∗

‖ = ‖u∗‖, then

u∗= argmin(‖u‖)G(u)=0. (1)

G(u) = 0 defines the failure surface, the frontier between thesafe side with G(u) > 0 and the unsafe side with G(u) < 0.According to design codes,G() defines a limit-state function, whichis expected to be continuous in the safe side and the unsafe side,at least in the neighbourhood of the design point P∗. This explainsthe reason why most optimization algorithms used in structural

3726 F. Duprat et al. / Engineering Structures 32 (2010) 3725–3733

reliability apply to differentiable constraint functions, althoughother non-gradient-based algorithms exist that also aim to solveEq. (1), such as the simplex algorithm [3], tailored for structuralreliability analysis [4] or, more recently, the bundle algorithm [5].

Rackwitz and Fiessler proposed an iterative method whichsolves Eq. (1) by generating a series of points converging towardsthe optimal solution [2]. Every new point u(k+1) of the series atiteration (k + 1) is determined from the previous point u(k) asfollows

u(k+1) =∇G(u(k))

‖∇G(u(k))‖2(uT

(k)∇G(u(k)) − G(u(k))) (2)

where ∇G(u(k)) is the gradient vector of G(u(k)).Iterations stopwhen the expected accuracy εp is reached on two

successive coordinates for every axis i

|ui(k+1) − ui(k)| ≤ εP . (3)

The descent or search direction vector reads:

d(k) =∇G(u(k))

‖∇G(u(k))‖2(uT

(k)∇G(u(k)) − G(u(k))) − u(k). (4)

Thus the new iteration point can be expressed as:

u(k+1) = u(k) + λ(k)d(k) (5)

where λ(k) is the step size, constant and equal to 1 in the Rack-witz–Fiessler algorithm. The main improvements made to this al-gorithm by researches have focused on how to optimize the stepsize in order to make the convergence more robust and stable. Liuet al. [6], Zhang et al. [7] suggested adding a line search scheme inwhich the step size is selected to minimize a merit function.

Alternative algorithms to those derived from the gradient pro-jection method exist, such as Polak–He’s algorithm [8] tailored forreliability purposes [9] or algorithms based on the Lagrangian ex-pression associated with Eq. (1):

L(u, γ ) = f (u) + γG(u) (6)

where f (u) is the objective function. The solution that minimizesL(u) can be found by using a second order Taylor expansion of thegradient of L(u) together with the optimality conditions L(u) = 0and G(u) = 0. Finding this root sequentially is the principle of theSequential Quadratic Programming method in which an updatingscheme for the Hessian matrix of G(u) is needed. An iterative andefficient schemewas proposed by Liu andDer Kiureghian [10]withthe objective function f (u) = ‖u‖2/2. Abdo and Rackwitz [11] sug-gested using the augmented Lagrangian

L(u, γ ) = f (u) + γG(u) + cG(u)2/2 (7)

as a merit function for optimizing the step size in the Rackwitz–Fiessler algorithm, with the objective function f (u) = ‖u‖2. Thischoice considerably simplifies the updating scheme for the Hes-sian matrix.

The algorithms shortly described here are globally conver-gent and are expected to be more robust than the original Rack-witz–Fiessler algorithm.However, a common feature among all thealgorithms is that they use the gradient vector ∇G(u) of the limit-state function to define the search direction, and their efficiencydepends on the quality of the estimate of ∇G(u). As the limit-statefunction is defined in the physical space, it is necessary to calculatepartial derivatives in the standardized space from partial deriva-tives in the physical space as follows:

∂G(u)∂ui

= ∇G(u)i = ∇G(x)T∂x∂ui

= ∇G(x)T∂T−1(u)

∂ui(8)

where ∂G(u)∂ui

is the partial derivative of the limit-state functionwithrespect to the standardized variable ui,∇G(x) is the gradient vector

Fig. 1. Problem posed by the minimal perturbation value.

with respect to physical randomvariables andu = T (x) is theNataftransformation.

In practical problems where G(x) is computed from the finiteelement method, analytical expressions of the partial derivativesare rarely available. The direct differentiation method involves de-velopment of analytical derivatives of the finite element responseand its implementation as a part of the finite element code. Someimplementations have been carried out for static and dynamic in-elastic problems, where response sensitivities were obtained withrespect to material parameters, nodal coordinates, cross-sectionalgeometry and nodal loads for a variety of material models and el-ement formulations [7,9,12,13]. However, for complex mechanicalmodels combining several types of element and problem (ageingchemical–mechanical behaviour of concrete for example), theseimplementations become somewhat cumbersome. For the sake ofsimplicity, general purpose reliability programs often resort to thefinite difference method for computing the gradient vector [4,14–17], which remains attractive and easily practicable:∂G(x)∂xi

≈G(x + 1xiei) − G(x)

1xi(9)

where1xi is the perturbation and ei is the unit vector for the axis xi.The truncation error in Eq. (9) results from the neglected terms

in the Taylor series expansion of the perturbed function G(x), andcan be reduced by using a small perturbation1xi. The perturbationsize should nevertheless be chosen in compliance with themecha-nical model precision. The finite element response of a non-linear mechanical model is twofold: one part is deterministic andone part is random, resulting from the computational inaccuraciesaccepted in the solving procedures (convergence threshold) andthe inevitable numerical round-off errors (machine code preci-sion). It can be reasonably stated that the random part, denoted1G(x) in Fig. 1, is proportional to the value of the response itselfwith aworking computational precision imposed by the engineer’sknowledge as the proportionality factor. Hence, if the perturbation1xi is too small, say 1xi < 1xmin in Fig. 1, it may lead to a dif-ference G(x + 1xiei) − G(x) that is smaller than the random part1G(x) of the response, and thereafter lead to poor efficiency, evennon-convergence of the search algorithmwhatever the underlyingmethod used for finding the design point.

In order to get a best estimate of a finite difference, it is neces-sary to impose a judicious value of the perturbation: sufficientlylarge to make the random part of the mechanical response asnegligible as possible and small enough to avoid significant trun-cation error. Furthermore, as iterations of the search algorithmproceed, the partial derivative for the axis xi should be estimatedusing Eq. (9) only if the coordinate of the new iteration point is farenough from the previous one.

To avoid the pitfalls of calculating the gradient vector response,the response surfacemethods can alternatively be employed. By it-eratively constructing an explicit approximation of the limit-statefunction and combining it with a classical search algorithm forfinding the design point, these methods can give satisfactory re-sults: the approximating function generally behaves very muchlike the true limit-state function in the neighbourhood of the de-sign point. Nevertheless, thesemethods often becomedramaticallycostly in computation resources for high dimension problems [18].

F. Duprat et al. / Engineering Structures 32 (2010) 3725–3733 3727

3. Projection gradient method with error control

The above mentioned considerations regarding the problemposed by theminimal perturbation value have been taken into con-sideration in the improvement to the Rackwitz–Fiessler algorithmproposed in this paper. The integrated procedure combines thegradient vector calculated at the previous iteration and the work-ing computational precision of the mechanical model in order tostate the new perturbation value. It is assumed that the limit-statefunction of the structure is defined as the difference between a re-source variable R(x) and a demand variable S(x):

G(x) = R(x) − S(x). (10)

Alternative expressions are sometimes used, for instance G(x)= R(x)/S(x) − 1. In most engineering fields however it is easier tohandle with limit-state functions or safety margins having a phys-ical meaning and unity.

For the first iteration, as the sensitivity of themechanicalmodelto variables is not known yet, a large perturbation is applied:

1xi = c1(εR + εS)xi (11)

where c1 is a magnifying factor (c1 > 1), εR and εS are workingcomputational precisions of bothmodels. For the ongoing iteration(k), 1xi is given by:

(1xi)(k) = c2

1G(x(k−1))

∇G(x(k−1))i

= c2

εRR(x(k−1)) + εSS(x(k−1))

∇G(x(k−1))i

(12)

where c2 is a magnifying factor (c2 > 1), 1G(x(k−1)) = εRR(x(k−1)) + εSS(x(k−1)) is the maximal numerical inaccuracy of thelimit-state function and ∇G(x(k−1))i =

∂G(x(k−1))

∂xiis the ith compo-

nent of the gradient vector at the previous iteration (k − 1).The working computational precisions are stated according to

the features of the limit-state function. If G() is analytical a usualchoice is εR = εS = 10−8 (for double precision floating numbers).IfG() results from an iterative numerical scheme, εR and εS must bechosen according to the employed convergence criteria. From theexperience of the authors, relevant values of themagnifying factorsc1 and c2 lie between 5 and 10. The influence of these factors on theresults obtained is emphasized hereafter.

As previously mentioned, the distance between two consecu-tive points given by the iterative search procedure for finding thedesign point should be a criterion for updating the partial deriva-tive for the axis under consideration. Consequently the followingcondition applies for updating the component ∇G(x(k))i: k−m=k0+1

((xi)(m) − (xi)(m−1))

≥ (1xi)(k0) (13)

where ((xi)(m) − (xi)(m−1)) is the algebraic distance between twosuccessive points along axis i, for which no update of the compo-nent ∇G(x(k))i has been achieved yet, (k0) denotes the iterationnumber corresponding to the last update of the partial deriva-tive and (k) is the ongoing iteration (see Fig. 1). (1xi)(k0) is theminimum required perturbation stated by Eq. (12). The condi-tion in Eq. (13) allows the overall computational cost to be re-duced by avoiding useless computation of the partial derivativesfor axes where the optimum has already been reached. This issimilar to the use of omission sensitivity factors. The omissionsensitivity factor for a mean value is defined as the ratio of the re-liability index determined, with the variable in question taken atits mean value, to the original reliability index. An omission sen-sitivity factor very close to one exhibits that the variable couldbe omitted in determining the reliability index, say fixed to its

Table 1Input variables — example #1.

Variable Distribution Mean Stand. dev.

x1 Lognormal 10 5x2 Normal 25 0.8x3 Normal 5 0.2x4 Lognormal 0.0625 0.0625

mean value. Based on approximate omission sensitivity factors,an implementation of these factors in the search algorithm wasproposed by Madsen [19], requiring the number of variables tobe modified as iterations proceed. In PGEC, the sensitivity crite-rion given by Eq. (13) is equivalent to the use of omission sensi-tivity factors. It is finally found that the proposed improvementof determining the perturbation 1xi as a function of the workingcomputational precision without systematically updating the par-tial derivatives in the course of iterations leads to the convergenceof the Rackwitz–Fiessler algorithm being facilitated.

4. Examples

Five examples are presented below in order to illustrate the effi-ciency of the proposedmethod and the computational cost benefitbrought by not systematically updating the partial derivatives.

The two first examples deal with explicit limit-state functionsand detailed comparison is made with other frequently employedalgorithms. The results of interest are the value of the reliabilityindex β , the coordinates of design point P∗, the value of the limit-state function at this point with respect to its value at the medianpoint (space origin) |G(u∗)/G(u0)|, which is expected to be zero ifP∗ lies exactly on the failure surface, and the number of runs of thelimit-state function Nr . Nr reflects the efficiency of the methodswith respect to the computational cost. The proposed technique issubsequently referred to as the PGEC method (projection gradientwith error control).

The three last examples concern implicit failure functions:some comparisons are made with results from the literature whenavailable.

4.1. Example #1

The limit-state function is a polynomial used as a response sur-face for reliability analysis of a pipeline [6]:

G(x) = 1.1 − 0.00534x1 − 0.0705x2 − 0.226x3 + 0.998x4− 0.00115x1x2 − 0.0149x1x3 + 0.0717x1x4+ 0.0135x2x3 − 0.0611x2x4 − 0.558x3x4

+ 0.00117x21 + 0.00157x22 + 0.0333x23 − 1.339x24. (14)

The following algorithms are under consideration: the Liu–DerKiureghian algorithm (SQP) and the Zhang–Der Kiureghian al-gorithm (iHLRF), implemented in OpenSees [10,11], the Abdo–Rackwitz algorithm (RFLS), theRackwitz–Fiessler algorithm (HLRF),and the Nelder–Mead algorithm (SPLX), implemented in Com-rel [13].

The non-correlated random variables used in this example aredefined in Table 1. Convergence criteria in PGEC were εp = 5 ×

10−3 and εR = εS = 10−8. Convergence criteria are also stated inComrel. In OpenSees the convergence criteria are:

|G(u∗)/G(u0)| ≤ ε1 (15)

checking for the proximity of P∗ to the failure surface, and

‖u∗− α∗Tu∗α∗

‖ ≤ ε2 (16)

checking that the gradient vector at P∗ points towards the space

3728 F. Duprat et al. / Engineering Structures 32 (2010) 3725–3733

Table 2Reliability index and convergence criteria — example #1.

Technique β Nr |G(u∗)/G(u0)| ‖u∗− α∗Tu∗α∗

‖ max |u∗

i −ui(k∗−1)|

SQP 1.5296 57 4.24 × 10−6 8.71 × 10−4 n.a.iHLRF 1.5296 49 2.40 × 10−5 6.95 × 10−4 n.a.RFLS 1.5296 25 2.10 × 10−7 n.a. <5 × 10−3

HLRF 1.5296 25 2.10 × 10−7 n.a. <5 × 10−3

SPLX 1.5294 55 1.14 × 10−6 n.a. <5 × 10−3

PGEC (c1 = c2 = 1) 1.5296 25 2.10 × 10−7 6.40 × 10−7 1.70 × 10−3

PGEC (c1 = c2 = 5) 1.5296 23 1.84 × 10−7 6.51 × 10−7 1.71 × 10−3

PGEC (c1 = c2 = 104) 1.5298 20 1.44 × 10−6 1.67 × 10−6 7.37 × 10−4

Table 3Design point — example #1.

Technique u∗

1 u∗

2 u∗

3 u∗

4

SQP 1.499 −1.152 × 10−1−8.040 × 10−2 2.675×10−1

iHLRF 1.499 −1.142 × 10−1−7.775 × 10−2 2.706×10−1

RFLS 1.496 −1.147 × 10−1−7.957 × 10−2 2.845×10−1

HLRF 1.496 −1.147 × 10−1−7.957 × 10−2 2.845×10−1

SPLX 1.496 −1.187 × 10−1−7.601 × 10−2 2.849×10−1

PGEC (c1 = c2 = 5) 1.496 −1.148 × 10−1−7.976 × 10−2 2.845×10−1

Table 4Input variables — example #2.

Variable Description Distribution Mean value Stand. dev.

x1 Bending moment (MN m) Normal 0.015 0.00525x2 Effective depth (m) Normal 0.30 0.015x3 Steel yielding strength (MPa) Lognormal 550 55x4 Reinforcing steel area (m2) Normal 308 × 10−6 30.8 × 10−6

x5 Block factor Normal 0.5 0.05x6 Width (m) Normal 0.20 0.01x7 Concrete strength (MPa) Lognormal 38 5

Table 5Reliability index and convergence criteria — example #2.

Technique β Nr |G(u∗)/G(u0)| ‖u∗− α∗Tu∗α∗

‖ max |u∗

i −ui(k∗−1)|

SQP 4.2829 55 1.31 × 10−5 2.15 × 10−4 n.a.iHLRF 4.2829 50 1.60 × 10−5 5.53 × 10−4 n.a.RFLS 4.2173 40 7.79 × 10−7 n.a. <5 × 10−3

HLRF 4.2174 40 7.57 × 10−7 n.a. <5 × 10−3

PGEC (c1 = c2 = 1) 4.2176 40 1.60 × 10−8 1.60 × 10−7 1.62 × 10−3

PGEC (c1 = c2 = 10) 4.2176 34 1.66 × 10−8 1.60 × 10−7 1.62 × 10−3

PGEC (c1 = c2 = 5000) 4.2177 33 1.08 × 10−7 1.53 × 10−7 5.00 × 10−4

origin. Default values for these criteria are ε1 = ε2 = 10−2. Theresults are reported in Tables 2 and 3.

As it can be seen in Table 2, the influence of the magnifyingfactors c1 and c2 remains limited in the case of an analytical limit-state function. Since the assessment of the gradient vector is fairlyaccurate in such case, the sole expected effect of the magnifyingfactors is globally to save runs of the limit-state function, that isobtained for c1 = c2 ≥ 5.

The reliability index results are almost the same for all thetechniques employed. As far as computational cost is concerned,the algorithms that appear themost relevant are those inwhich theoptimization of the step size in the descent direction is made withminimum effort, or even not made at all, say RFLS, HLRF and PGEC.Five iterations are needed in these algorithms for convergence toa design point very close to the failure surface to be achieved. Bysaving two calculations of partial derivatives (with c1 = c2 =

5), PGEC is the least costly algorithm. It can be noted that theconvergence criterion used in PGEC (Eq. (3)) is as, or more, severethan those in iHLRF and SQP (Eqs. (15) and (16)). For the latter,requiring supplementary runs of gradient evaluation to optimizethe step size of the search direction, it is worth noting that thecomputational cost is highly dependent on the perturbation valueused in finite difference for estimating the partial derivatives. In

OpenSees, the perturbation value is proportional to the standarddeviation of random variables: the results presented above wereobtained with proportionality factors of 1/1000 for SQP and1/1500 for iHLRF. With the default value 1/1000 for iHLRF, thenumber of computations jumps to Nr = 70 for identical reliabilityresults.

4.2. Example #2

The limit-state function is the safety margin of a reinforcedconcrete section subjected to a bending moment:

G(x) = x2x3x4 −x5x23x

24

x6x7− x1. (17)

The random variables are listed in Table 4. The same conver-gence criteria and magnifying factors were imposed here as in theprevious example. The results are reported in Tables 5 and 6, ex-hibiting again the influence of the magnifying factors c1 and c2. Asalready noticed this effect is the reduction of the computationalcost while maintaining the same accuracy of results.

As far as reliability outputs are concerned, values of the reliabil-ity index are slightly different when the SQP and iHLRF algorithmsare compared with the RFLS, HLRF and PGEC algorithms. For the

F. Duprat et al. / Engineering Structures 32 (2010) 3725–3733 3729

Table 6Design point — example #2.

Technique u∗

1 u∗

2 u∗

3 u∗

4 u∗

5 u∗

6 u∗

7

SQP 3.074 −9.854 × 10−1−1.610 −2.307 4.939 × 10−2

−2.484 × 10−2−6.501×10−2

iHLRF 3.076 −9.855 × 10−1−1.609 −2.305 4.937 × 10−2

−2.484 × 10−2−6.498×10−2

RFLS 3.016 −9.560 × 10−1−1.718 −2.194 4.774 × 10−2

−2.391 × 10−2−6.256×10−2

HLRF 3.013 −9.535 × 10−1−1.716 −2.201 4.710 × 10−2

−2.368 × 10−2−6.201×10−2

PGEC (c1 = c2 = 10) 3.011 −9.526 × 10−1−1.715 −2.206 4.665 × 10−2

−2.356 × 10−2−6.198×10−2

Table 7Computation of partial derivatives — example #2.

Table 8Omission sensitivity factors — example #2.

Variable x1 x2 x3 x4 x5 x6 x7

β 6.5159 4.3241 4.6407 4.8411 4.2179 4.2177 4.2190γi(µi) 1.545 1.025 1.100 1.148 1.000 1.000 1.000

Fig. 2. Structural system — example #3 (Unit: m).

latter, the closeness of the design point to the failure surface andthe computational cost aremore satisfactory. It is finally found thatthe PGEC algorithm is the least costly (with c1 = c2 = 10): bysaving six computations of the partial derivatives with respect tovariables x5– x7 at iterations 4 and 5 (see Table 7) according to thecriterion given by Eq. (13), PGEC allows useless runs of the limit-state function to be avoided. The omission sensitivity factors havebeen computed for this example. Results for these are reported inTable 8. It can be seen that variables x5–x7 are the least influential.

4.3. Example #3

The case of a multi-storey multi-span metallic portal frame isdealt with (see Fig. 2). The limit-state function is the margin withrespect to a prescribed value δmax of the displacement δ at the topof the structure:

G(x1, . . . , x21) = δmax − δ(x1, . . . , x21). (18)

The x-vector contains 21 basic variables translating differentproperties of structural components. The structural data and thestatistical parameters are reported in Tables 9 and 10. Somevariables are assumed to be correlated. All loads are correlated bya coefficient of correlation ρF = 0.95. All cross section propertiesare correlated by ρAiAj = ρIiIj = ρAiIj = 0.13. The two differentmodulus of elasticity E1 and E2 are correlated byρE = 0.9. All other

Table 9Frame element properties — example #3.

Element Modulus of elasticity Moment of inertia Cross section

1 E1 I5 A52 E1 I6 A63 E1 I7 A74 E1 I8 A85 E2 I1 A16 E2 I2 A27 E2 I3 A38 E2 I4 A4

variables are assumed to be uncorrelated. The internal languageof the CAST3M free finite element software [20] was used in thisstudy, as for the following examples, for both finite element andprobabilistic calculations. Convergence criteria in PGEC were εp =

5 × 10−2, εR = 10−8 and εS = 10−4. The results are reported inTable 11 for several values of δmax and c1 = c2 = 10.

It is worth noting that the convergence criteria are fulfilled sat-isfactorily, whatever the value of the reliability index is, which isincreased with larger value of δmax. In Table 11 Ns is the numberof saved runs of the finite element model resulting from the cri-terion given by Eq. (13). As seen in Table 12, this criterion allowsthe gradient to be updated only when required as the computa-tion goes along. The absence of update of a partial derivative ishence not definitive in the course of the search procedure. This in-teresting feature gives the PGEC algorithm some adaptability in thesearch procedure for finding the design point. The study of the ef-fect of the magnifying factors c1 and c2 reveals in Table 13 that therange [5–10] for these ones leads to a fairly good compromise be-tween the precision and the saving of runs.

4.4. Example #4

In this example, the risk of depassivation of the reinforcementsembedded in concrete is addressed. Reinforcing bars are depas-sivated when pH decreases due to the ingress of pollutants suchas chlorides or carbon dioxide. Once depassivated, steel is proneto corrosion if sufficient moisture and oxygen are available in theneighbourhood of reinforcing bars. The case of a concrete beam ex-posed to carbonation is considered here. As cracks due to the ap-plied load facilitate the ingress of carbon dioxide, the mechanicalbehaviour is taken into account togetherwith the diffusion processand the chemical aspects. The overall modelling is detailed in [21].The limit-state function for the durability limit-state is defined asfollows:

G(x) = x1 − d(x2, x3, x4, x5). (19)

The carbonation depth is given by the function d() which iscomputed from a finite element model. The steep slope profileof the calcium content in concrete leads to strong non-linearityin the function d(). In this model, a fine mesh in both spaceand time was used in order to ensure satisfactory convergence ofcomputation procedures. The carbonation depth profiles obtainedfor several exposure times can be seen in Fig. 3. In function d(), thecarbonation depth is given along the line A–B.

The random variables are listed in Table 14. Convergencecriteria in PGEC were εp = 5 × 10−2, εR = 10−7 and εS = 10−4.

3730 F. Duprat et al. / Engineering Structures 32 (2010) 3725–3733

Table 10Input variables — example #3.

Variable Unit Distribution Mean value Standard deviation

F1 kN Gumbel max 133.454 40.04F2 kN Gumbel max 88.97 35.59F3 kN Gumbel max 71.175 28.47E1 kN/m2 Normal 2.173752 × 107 1.9152 × 106

E2 kN/m2 Normal 2.379636 × 107 1.9152 × 106

I1 m4 Positive normal 0.813443 × 10−2 1.08344 × 10−3

I2 m4 Positive normal 1.150936 × 10−2 1.298048 × 10−3

I3 m4 Positive normal 2.137452 × 10−2 2.59609 × 10−3

I4 m4 Positive normal 2.596095 × 10−2 3.028778 × 10−3

I5 m4 Positive normal 1.081076 × 10−2 2.596095 × 10−3

I6 m4 Positive normal 1.410545 × 10−2 3.46146 × 10−3

I7 m4 Positive normal 2.327853 × 10−2 5.624873 × 10−3

I8 m4 Positive normal 2.596095 × 10−2 6.490238 × 10−3

A1 m2 Positive normal 0.312564 0.055815A2 m2 Positive normal 0.3721 0.07442A3 m2 Positive normal 0.50606 0.093025A4 m2 Positive normal 0.55815 0.11163A5 m2 Positive normal 0.253028 0.093025A6 m2 Positive normal 0.29116825 0.10232275A7 m2 Positive normal 0.37303 0.1209325A8 m2 Positive normal 0.4186 0.195375

Table 11Reliability index and convergence criteria — example #3.

δmax (cm) β Nr Ns |G(u∗)/G(u0)| ‖u∗− α∗Tu∗α∗

‖ max |u∗

i −ui(k∗−1)|

3 1.342 52 14 4.29 × 10−4 1.69 × 10−3 3.63 × 10−2

6 3.398 91 63 1.72 × 10−5 2.00 × 10−5 2.58 × 10−2

9 4.573 119 79 1.74 × 10−4 1.55 × 10−4 4.93 × 10−2

Table 12Computation of partial derivatives — example #3 (δmax = 6 cm).

Table 13Influence of c1 and c2 factors — example #3 (δmax = 9 cm).

c1, c2 β Nr Ns |G(u∗)/G(u0)| ‖u∗− α∗Tu∗α∗

‖ max |u∗

i −ui(k∗−1)|

1 4.573 178 42 1.91 × 10−6 3.59 × 10−5 2.81 × 10−2

2 4.573 156 64 6.95 × 10−6 3.96 × 10−5 2.73 × 10−2

4 4.573 139 81 2.89 × 10−5 2.79 × 10−5 2.24 × 10−2

6 4.573 135 85 2.80 × 10−5 2.93 × 10−5 2.33 × 10−2

8 4.573 123 75 3.27 × 10−5 1.24 × 10−4 4.87 × 10−2

10 4.573 119 79 1.74 × 10−4 1.55 × 10−4 4.93 × 10−2

12 4.573 121 99 7.43 × 10−4 5.07 × 10−4 3.92 × 10−2

50 4.558 140 168 9.17 × 10−4 6.59 × 10−4 1.54 × 10−2

Table 14Input variables — example #4.

Variable Description Distribution Mean value Stand. dev.

x1 Concrete cover (mm) Lognormal 20 4x2 Coefficient of diffusion (m2/s) Lognormal 1 × 10−8 0.8× 10−8

x3 Tortuousity Uniform (0.1–0.9) 0.5 0.23x4 Concrete strength (MPa) Lognormal 35 5x5 Live load (kN/m2) Gumbel max 1.04 0.4

The results for an exposure time of 5 years are reported in Tables 15and 16.

The reliability indices and coordinates of the point P∗ suppliedby PGEC and by the response surface method used in [20] are

F. Duprat et al. / Engineering Structures 32 (2010) 3725–3733 3731

(a) 1 month. (b) 5 years. (c) 20 years. (d) 35 years.

Fig. 3. Carbonation fields at various times (half cross-section at mid-span).

Table 15Reliability index and convergence criteria — example #4.

Technique β Nr |G(u∗)/G(u0)| ‖u∗− α∗Tu∗α∗

‖ max |u∗

i −ui(k∗−1)|

RSM [20] 3.998 43 3.17 × 10−4 8.70 × 10−6 9.62 × 10−3

PGEC 3.923 26 4.48 × 10−4 5.53 × 10−4 1.54 × 10−2

HLRFa

a Not converged.

Table 16Design point — example #4.

Technique u∗

1 u∗

2 u∗

3 u∗

4 u∗

5

RSM [20] 3.601 1.312 × 10−1−9.212 × 10−4

−1.731 1.601×10−2

PGEC 3.076 −7.814 × 10−2−2.547 × 10−2

−2.026 4.837×10−2

similar even though some differences exist. These latter probablyresult from the approximation of the true failure function givenby the response surface, which might not be sufficiently accuratein the vicinity of point P∗. In contrast, it is not surprising thateven satisfactory convergence results with PGEC are less tight thanwith RSM. This is due to the fact that the approximate failurefunction handled by RSM is explicit as are the gradients as well.Nevertheless, PGEC exhibits an advantage regarding the numericalcost: 10 runs of the mechanical model are saved as shown inTable 17.

Like PGEC, the original HLRF algorithm was implemented inthe CAST3M software. The perturbation value for evaluating thegradient vector in the HLRF algorithm was set at 1xi = σi/1000,where σi is the standard deviation of the variable xi. It can beseen in Table 18 that convergence is not reached after 6 iterations(not even beyond); the value of reliability index is not stabilizedand the value of the limit state function is far from zero. Thisarises because the perturbation value does not suit the numericalinaccuracy of limit state function adequately, and is below theoptimum value. The non-convergence is not caused by an irregularor discontinuous profile of the failure surface.

4.5. Example #5

The case of a wide prestressed footbridge is considered in thisexample. The longitudinal cross-section and cross-section over thesupport can be seen in Fig. 4. The structure was designed accord-ing to Eurocode 2 [22]. The prestressing force of 23 MN was de-termined in such a way that only the minimum area of reinforcingsteel was required: under the characteristic combination of loadsand for the upper and lower values of the prestressing force, thetensile stress in concrete is limited to the average tensile strength.

Table 17Computation of partial derivatives — example #4.

Table 18Comparison between PGEC and HLRF — example #4.

Iteration 1 2 3 4 5 6

PGEC β 3.298 5.210 3.951 3.984 3.926 3.923|G(u∗)/G(u0)| 0.375 0.220 0.026 0.016 0.0006 0.0004

HLRF β 3.741 3.852 1.893 1.608 3.687 4.901|G(u∗)/G(u0)| 1.032 0.316 0.126 1.011 0.695 0.548

The structural durability is affected by the concrete cracking andthe requirement for durability is therefore that the effective stressdoes not exceed the effective tensile strength, taking into accountthe concrete damage. The limit state function is hence

G(x1, . . . , x9) =x1

1 − dtp(x2, . . . , x9)−

σM(x2, . . . , x9)1 − dM(x2, . . . , x9)

(20)

where dtp() is the concrete damage on reaching the maximumstrength (variable x1), dM() is the effective concrete damage, andσM() is the normal stress, both computed at pointM in Fig. 4.

3732 F. Duprat et al. / Engineering Structures 32 (2010) 3725–3733

Table 19Input variables — example #5.

Variable Description Distribution Mean value Stand. dev.

x1 Tensile strength (MPa) Lognormal 3 0.6x2 Modulus of elasticity (MPa) Lognormal 37600 9400x3 Modulus of compressibility (MPa) Lognormal 208890 41778x4 Characteristic strain for the consolidation of spherical viscosity Lognormal 10−4 3.2 × 10−5

x5 Viscosity for the reversible creep (MPa/d) Lognormal 62668 12534x6 Moisture pressure (MPa) Lognormal 58 14.21x7 Drying creep stress (MPa) Lognormal 11.7 3.1x8 Chemical factor Lognormal 3.1 × 10−4 9.748×10−4

x9 Pedestrian live load (kN/m2) Gumbel max 2.437 0.9748

Table 20Reliability index and convergence criteria — example #5.

Exposure time (y) β Nr Ns |G(u∗)/G(u0)| ‖u∗− α∗Tu∗α∗

‖ max |u∗

i −ui(k∗−1)|

10 3.261 55 15 1.99 × 10−5 4.22 × 10−4 8.60 × 10−3

50 2.582 38 2 6.90 × 10−4 6.45 × 10−3 3.48 × 10−2

90 1.897 40 10 9.01 × 10−4 1.50 × 10−3 3.01 × 10−2

Fig. 4. Prestressed footbridge studied (units: cm). Main design inputs: the concrete strength class is C40/50, the tensile strength of strands is 1860 MPa, and the pedestrianlive load is 3.71 kN/m2 .

Creep and shrinkage of the concrete are factors that reducethe prestressing force, and have been included in the functionsσM(), dtp() and dM() which are actually outputs of a global finiteelementmodel, geometrically reduced to a quarter of the structure(see Fig. 5). This global model comprises several comprehensivecoupled sub-models for creep, shrinkage and mechanical damageof the concrete, accounting for the main parameters that influencecreep and shrinkage: the ambient humidity, the dimensions ofthe element, the composition of the concrete, the maturity ofthe concrete when the load is first applied and the duration andmagnitude of the loading. The mathematical formulation of thesub-models derives from the damage theory for a visco-elasticmaterial submitted to consolidation. The modelling is detailedin [23].

The random variables are listed in Table 18. Very few statisticaldata exist in the literature for the creep and shrinkage of concrete:most of the surveys are incomplete. Data are hence missing forthe parameters of the model, which are expected to be uncertainin a broad range. From the assumption that creep and shrinkagefunctions given in EC2 present an uncertainty of 20%–30% [22],and from available collected experimental data, propositions havestill been derived in Table 19. From a physical point of view, andowing to possible numerical issues, lognormal distributions weresuggested.

Convergence criteria in PGEC were εp = 5 × 10−2, εR = 10−5

and εS = 10−4. The results obtained for several exposure times arereported in Table 20.

Fig. 5. Stresses just after tensioning.

It can be seen that the number of runs of the limit state functionremains reasonably low. The convergence criterion, Eq. (3), onthe distance between two successive search points, which is theonly one active in the procedure, allows the other convergencecriteria, Eqs. (15) and (16), to be satisfactorily fulfilled. The updatefor estimating the coordinates of the gradient vector, as stated byEq. (13), again leads to a saving of runs of the limit state functionfor this example, as shown in Table 21.

5. Conclusion

In this paper, an improvement of the Rackwitz–Fiessler algo-rithm has been proposed for cases where the model associated

F. Duprat et al. / Engineering Structures 32 (2010) 3725–3733 3733

Table 21Computation of partial derivatives — example #5 (10 year exposure).

with the limit-state functionpresents some randomcomputationalinaccuracies. These are part of the outputs supplied by the FiniteElement Method when resorting to non-linear solving procedures.If the computational inaccuracies are not accounted for in estimat-ing the coordinates of the gradient vector by finite differences, theiterative search procedure for finding the design point may con-verge erroneously or not at all. In order to circumvent this draw-back, a dedicated procedure for calculating the perturbation to beused in finite differences was achieved and attached to the Rack-witz–Fiessler algorithm. Simultaneously, the procedure takes intoaccount the working computational precision assigned by the en-gineer’s knowledge and depending on the finite element mod-elling, and allows the number of runs to be reduced thanks acriterion for the updating of the partial derivatives with respectto the input variables. The overall algorithm, called the PGECalgorithm, was compared to enhanced algorithms commonly im-plemented in dedicated software. From examples considered withexplicit failure functions, PGEC exhibits satisfactory behaviour: forsimilar convergence constraints, the results obtained are very closeto those supplied by enhanced algorithms in terms of the relia-bility index and the coordinates of the design point. Furthermore,PGEC is the least costly algorithm numerically speaking. Some ex-amples with strongly non-linear finite element modelling werealso addressed, dealing with durability issues of concrete struc-tures exposed to carbonation, or subjected to creep and shrinkageeffects. Beyond efficiently fulfilling the convergence criteria, thePGEC algorithm allows a significant number of runs to be saved.The proposed technique could also give rise to improvements inthe other search algorithms derived from the projection gradientor gradient-based and resorting to finite differences for estimatingthe gradient vector.

Acknowledgements

The CEA (french Atomic Energy Commission) is thanked for theprovision of the Cast3M software to the LMDC in its developmentversion for education and research.

References

[1] Hasofer AM, Lind NC. An exact and invariant second moment code format.J Eng Mech 1974;100:111–21.

[2] Rackwitz R, Fiessler B. Structural reliability under combined random loadsequences. Comput Struct 1979;9:489–94.

[3] Nelder JA, Mead R. A simplex method for function minimization. Comput J1965;7:308–13.

[4] COMREL & SYSREL users manual. RCP Consult. Munich; 1987–2004.[5] Bonnans JF, Gilbert JC, Lemarechal C, Sagastizabal C. Optimisation numérique:

aspects théoriques et pratiques. In: CollectionMathématiques et Applications.Springer; 1999.

[6] Liu P-L, Der Kiureghian A. Optimization algorithms for structural reliability.Struct Saf 1991;9:161–78.

[7] Zhang Y, Der Kiureghian A. Finite element reliability methods for inelasticstructures. Report No. UCB/SEMM-97/05. Berkeley (CA): Department of Civiland Environmental Engineering, University of California; 1997.

[8] Polak E. Optimization — algorithms and consistent approximations. Appliedmathematical sciences, vol. 124. Springer Verlag; 1997.

[9] Haukaas T, Der Kiureghian A. Strategies for finding the design point in non-linear finite element. Struct Saf 1991;9:161–78; Probab Eng Mech 2006;21:133–47.

[10] Liu PL, Der Kiureghian A. Finite element reliability of geometrically non-linearuncertain structures. J Eng Mech 1991;17:1806–25.

[11] Abdo T, Rackwitz R. A new β-point algorithm for large time-invariant andtime-variant reliability problems. In: Der Kiureghian A, Thoft-Christensen P,editors. Proc. of the 3rd IFIP WG 7.5; 1991.

[12] Haukaas T, Der Kiureghian A, Fujimura K. Structural reliability software at theUniversity of California, Berkeley. Struct Saf 2006;28:44–67.

[13] Reh S, Beley JD, Mukherjee S, Khor EH. Probabilistic finite element analysisusing ANSYS. Struct Saf 2006;28:17–43.

[14] Schuëller GI, Pradlwarter HJ. Computational stochastic structural analysis(COSSAN) — a software tool. Struct Saf 2006;28:68–82.

[15] Gollwitzer S, Kirchgäßner B, Fischer R, Rackwitz R. PERMAS-RA/STRURELsystem of programs for probabilistic reliability analysis. Struct Saf 2006;28:108–29.

[16] Lemaire M, Pendola M. PHIMECA-SOFT. Struct Saf 2006;28:130–49.[17] Tvedt L. Proban — probabilistic analysis. Struct Saf 2006;28:150–63.[18] Nguyen XS, Sellier A, Duprat F, Pons G. Adaptive response surface method

based on a double weighted regression technique. Probab EngMech, availableonline 27 April 2008.

[19] Madsen HO. Omission sensitivity factors. Struct Saf 1988;5:35–45.[20] CAST3M. Finite element software. Commissariat à l’Energie Atomique.

http://www-cast3m.cea.fr.[21] Duprat F, Sellier A. Probabilistic approach to corrosion risk due to carbonation

via an adaptive response surface method. Probab Eng Mech 2006;21:207–16.[22] Eurocode 2: EN 1992-2. ICS: 91.080.40; 93.040. Design of concrete structures,

part 2: concrete bridges.[23] Sellier A, Buffo-Lacarrière L. Vers une modélisation simple et unifiée du fluage

propre, du retrait et du fluage en dessiccation du béton. Eur J Environ Civil Eng2009;13:1161–82.