computational procedures in structural reliability

7/29/2019 computational procedures in structural reliability

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Computational Procedures in Structural ReliabilityG. I. SchuEller and V. Bayer

Instituteof Engineering MechanicsUniversity of InnsbruckA - 6020 InnsbruckAustriaAbstract

Mainly due lo lack of respective available data andpmcednres, traditional methods of analysis and designof engineering structures, generally, neglect the phys-ical, i .e. intr insic uncertainties of the various para-meters involved. I n the present paper it i s shown howthe treatment of structural systems becomes feasibleby interrelating mechanical and probabilistic modelingprocedures, e.g. by efi cient structural analyses as wellas accurate stochastic procedures such as Monte Carlotype simulations. Numeri cal examples are presented.

1 IntroductionThe sophistication in mechanical modeling of struc-tures has progressed considerably, particularly withinthe last three decades. This was mainly due to thedevelopment of the Finite Element method (FEM),which, by utilizing high-speed computers, proved tobe computationally most efficient. This goes along

with an equally dramatic development of material sci-ence which provided significant additional insight inmaterial properties. The latter contains informationon statistical variation of strength as wel as its spa-tial correlation, etc. In addition, great efforts havealso been put into load observation programs. Theresults of these programs] again, reflect the statisticalvariation of loads in space and time.It is interesting to observe that, despite of thesedevelopments, analysis and design procedures, by andlarge, still followed the traditional lines, i.e. select-ing from the spectra of values for each parameter aparticular value, e.g. maximum values for loads,minimum or mean values for material properties,and mainly empirically based safety factors. In otherwords, the degreeof accuracy between structural, i.e.stress analysis on one hand and load, material andsafety analyses on the other hand still diverge signifi-

cantly, as it was already observed in 1964by Freuden-thaZ[16].The theory of structural reliability however prevides a tool to treat the uncertainties involved in theanalysis and design process by rational means. Asadditional data become available] the results of theanalysis certainly improve. This is due to the resultingimprovement of the probabilistic modeling which al-lows a better extrapolation of the distributions to thetails of interest. Hence, the credibility of the resultsof the structural reliability analysis depends stronglyon both the mechanical and the probabilistic model-ing. Additionally the accuracy of the computationalprocedures to determine the reliability estimates is ofequal importance. For reasons of practicability theyalso have to meet the requirement of efficiency. Partic-ularly for practical problems] the limit state function

g(x), which refers to the respective failure condition- e.g. collapse, serviceability] etc. - n general is notknown explicitely. In fact, quite frequently i t is knownonly pointwise from structural (FE - ) analyses.Forthose cases the so-called Response Surface Method( RSM) roves to be most instrumental.The present paper concentrates on the computa-tional aspects and hence the assessment of variousmethods to calculate the failure probabilities in viewof accuracy] efficiency and the possibility of practicalapplication.

2 Methods to determine failure prob-abilities2.1 General remarks

The basic problems in structural reliability are onone hand concerned with the determination of thelimit state function g(x), and on the other hand with

0.8186-3850.8193 $3.00 8 1993 I HBB 552


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the solutionof a generallymulti-dimensional integralPI = J fx(x)d x

d x ) 50where f x (x ) s the joint density function of the ran-dom variables involved. While the limit state functioncan be determined by utilizing the well-known pro-cedurea of structural analysis - preferably the FEM-, the solution of the multi-dimensional integral basbeen, and still is, subject to detailed investigation,i.e. by utiliring either approximate, accurate and ex-act procedures, respectively. Due to the nature of theproblem, which liesin the properties of the limit statefunction, i.e. the boundaries of the integration do-main, as well as in the high dimensionality, the lastmentioned possibility is not tractable. Also numer-ical integration is not practicable because it is tootime consuming. For these reasons the major develop-ments concentrated on approximate (first and secondorder) as well as accurate (efficient simulation) relia-bility methods. For an early review of both lines ofdevelopments it isreferred to [12], [ 24.The following discussion concentrates mainly on themethods capability in context with practical applica-tion. In this context it is important to stress, that anystructural reliability analysis must reflect the samefeaturesof the mechanical modelingas used in deter-ministic analysis. In other words, nonlinear mechani-cal effects with respect to geometrical (e.g. PA-effect,etc.) and material (e.g. plasticity effects, etc.) non-linearities have to be included in the model.2.2 Approximate methods

The First Order Reliability Method (FORM) isbased on a linearization of the limit state function.The basic random variablesX are at first transformedto standard normal variablesY, .g. by Nataf trans-formation, see [22]. Uncorrelated variables are thenobtained by multiplyingY with the triangularized co -variance matrix. The limit state function is linearizedat the point of maximum likelihood. The geometricalinterpretation of this so-called design point in stan-dard normal space, U*, s the point on the limit sur-faceg(u) with shortest distance to the origin, see Fig.1. This distance is called reliability index p, see e.g.[19], [28]. In this context it should be noted that thebasic notion of this formulation goes already back toan early work of F reudenthal [15], who denoted thisdistance by p . The search for the design point can becarried out by any nonlinear optimization procedure.

For this purpoee the NLPQGAlgorithm [271, whichutilizes adifferentiable augmented Lagrange functionproved to be most useful.From linearization the failure probability yieldsP/ =1- @(P )=@(-P ) (1)

wherea(.) s the standard normal distribution.

failure

Fig. 1. - Design po int U* and reliability index p inuncorrelated standard no rmal space.

Shortcomings of FORM are that the design pointmay not be found, or that several local minima of thedistance to the origin exist. This can be the caseforhighly nonlinear or not differentiable limit state func-tions. Results obtained with the FORM procedure,however, reveal errors of several orders of magnitudewhen compared to exact or accurate methods such assimulation, as the numerical examples will show.

The Second Order Reliability Method (SORM) uti-lizesaquadratic approximation of the limit state func-tion at the design pointn- 1

g(x) =2 , - ni .zi2i =lwith K the curvaturesof the limit state function eval-uated at the design point. If the design point can notbe uniquely determined, i.e. if 1 local minimaof /3 ex-ist, the failure probability can be expressedas (after[41, [ m

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However, difficulties with the determination of the de-sign points, as mentioned above, may al so apply hereas well.TheSORMhas been proved to approach the exactsolution for /34 0 in [4], [6], though realistic valuesare in the range of 3to 5.

Since for problems with non-normal variables, evenfor a linear limit state function, the function revealsnonlinear properties when transformed into the nor-mal space- which is then linearized again at the de-sign point when applying FORM, t is suggested in[5], [15],and [28] to perform the approximation in theoriginal space, i.e. to use the maximum likelihoodpoint a~the design point, also for non-normal randomvariables, wherein the first work the method is basedon asymptotic analysis.2. 3 Accurate methods

The Monte Carlo simulation procedure is a pow-erful tool to solve the multi-dimensional integral overthe failure domain. It givesan unbiased estimator forthe failure probability.An indicator function is defined as follows:

1 ifg(x) 500 ifg(x) > OThe failure probability then yields

(4)

which is interpreted as the expected value

An unbiased estimator for the expectation is. N

(7)with xi the ith realization of a simulated sample ofsizeN . The variance of the estimation is

U;, = E m EEf1)21= ( Pf - P ? ) l N (8 )

The estimation obtained by Monte Carlo simulationapproaches the exact failure probability for N -+ 00.Thus, the computational effort is very high and themethod is not tractable for practical applications

wherepj w lo-', because the number of simulationshas to be significantly higher than the inverse value ofp j .This led to the developmentof several variance re-duction techniques for Monte Carlo simulation. Someof them are presented in the following.The idea of importance sampling is to concentratethe samples in the region which contributes most tothe total failure probability, see [17, 26, 291. For thispurpose a weighted simulation is carried out. Theweighting function hx(x) is introduced as follows,without any lossof accuracy:

pf =J I(g(x)) * hd9 'hx(x) dx (9)hX(4-mwhich is the expectation with respect to the weightingfunction hx(x)

The unbiased estimation for p j is now carried outby simulating samples xi following the distributionhx(x) and then dividing the estimation by h~(x) :

The variance of pf can be estimated as

and becomes zero for an ideal weighting function, de-fined only in the failure domain Xlg(x)


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design point than FORM, because the simulation isspread withinan area around the point which is iden-tified by the optimization procedure. However, simu-lation procedures require higher computational effortthan FORM. t should be noted, that the main taskin a practical application of reliability analysisisgen-erally the mechanical modeling and determination ofthe limitstatefunction by structural analysis. Com-pared to this, the computational efforts for simulationare of lesssignificance.

To avoid the search for the design point and to ob-tain an improved weighting function for simulation,the Adaptive Sampling Procedure (ADSAP) has beendeveloped, see [8], [lo]. The idea is to obtain informa-tion about the failure domain by an initial simulationwith relatively high sampling variance and to improvethe weighting function after each simulation run. Thefailure domain is described by the first and secondsta-tistical momentWI = E[Xlg(x)501 (14)E[YYT]= EIXXT Ig(x )501 (15)

which are estimated by initial simulation. A normaldistribution can be uniquely defined by two statisti-cal moments, hence it is utilized as weighting functionhy(y) or simulation. For the second and succeedingsimulation runs the weighting function is adapted byutilizing the statistical moments estimated from theprevious run. Usually the variance of the estimatedpf decreases with each adaption of the weighting func-tion.Experience with the ADSAP procedure shows thatat least four simulation runs are necessary to obtaina reasonable variance for Pf . Although this causeshigher computational effort than for ISPUD, ADSAPis much more robust towards highly nonlinear or notdifferentiable limit state functions.

Directional Sampling also does not require the de-termination of the design point. I t has the furtheradvantage of higher accuracy for cases where the dis-tance between the failure domain and the respectivemean values of the variables is large. The procedure isperformed in the standard normal space by simulatingunit direction vectors AT =[cos0; in01,where0 isuniformly distributed. For each realization of a theconditional failure probabilitypf a can be accuratelyevaluated by integration. The limit state function hasto be expressed by polar coordinatesR, A with Rthe distance to the origin. The boundaries for inte-gration are the pointsr* aon the surfaceg(r,a)=0.

Note that several points lying in one directiona mayexist. If there exists only one such point, then theconditional failure probability isdefined as:P f b = P[ '2 r'(41

= 7 fRIA(rb) dr.*(a)= 1- X: (r *(a)) (16)The total failure probability is the expected value ofthe conditional failure probabilities, estimated fromsimulation by

Directional Simulation is especially suitable for limitstate surfaces similar to a hypersphere around the ori-gin. It can be improved by importance sampling ofthe vector a (see e.g. [1],[13])and adaptive strategies(P11)3 Response SurfaceMethod

For the computationof the failure probability of astructure by any of the previously described methods,the determination of the limit stateof the structure isoneof the main tasks to be carried out. Thelimit state- as stated above- can be definedfor example by totalcollapse of the system, fatigue or the loss of service-ability, etc. The evaluationof the limit state functionof the basic variables is carried out by mechanical,i.e. structural analysis. Explicit formulations of thelimit state function are rarely available for structuresused in practice. Usually they are only available forsmall systems and with simplifications in the struc-tural analysis. To reach the limit state, in particularif it is definedas total collapse of a structure, nonlineareffects such as plasticity or PA effects, etc., have tobe considered. As already mentioned above, requiredstructural analysis can be quite accurately carried outby FEM, yielding single points of the structural re-sponse for aset of input variables.The Response Surface Method (RSM) utilizes thesepoints to approximate the exact limit state functionby interpolation. Again it should be noted that, de-spite this approximation, all effects in the mechanicalmodel can be taken into account to calculate the var-ious points. In this regard, RSM is an interface be-tween the mechanical and probabilistic analysisof astructure.

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Earlier approaches, e.g. 1141, [25], propose to cal-culate the structural responseg(x) at the mean val -ues of the variables Zi and at Z i f n e oxi (withn e.g. =1.5 - 3). As these points are generally lo-cated still in the safe domain, the limit stateg(x)=0can only be reached by extrapolation, which may causeadditional inaccuracy.In applying simulation techniques as described insection 2.3, it is decided, by utilizing the indicatorfunction I (g(x)),whether or not a sample s in the safeor the failuredomain (seeeqns. (4), (5)). Thereforethe limit state g(x) =0 is of main interest. It hasbeen suggested in [ll] o calculate points on g(x)=0,which can be carried out by incrementally increasingan initial set of variables until failure occurs. Thisleads to a more accurate approximation of the limitstate than the above approach.

Amongst other possible functionals, here the ap-proximation by asecond-order polynomial is described(seealso [ll], 23]). The polynomial reads

1/6 I 1.4.1/7 I 1.4 I 1 . 1 . lo-I 6.1. o-

n n n

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where n is the number of basic variables. The thirdterm is symmetric, i.e. cij =cji. Hence,n(n +1)l+T l+- 2

points x have to be evaluated to form a determinedequation system for the unknown coefficientsb andC. If the equation system is overdetermined, e.g. aleast square fitting procedure can be applied.

The accuracy of the RSM, i.e. capability of theresponse surface to model the structural behaviour hasbeen investigated in [23], see Appendix.

4 Numerical examples4.1 General remarks

The first example is of theoretical type and is in-tended to show the capabilities of FORM, ISPUD.The second example describes a practical application,where the plastic failure loadof asteel cross section ofa crane type structure had to be determined. This wasaccomplished by application of the Response SurfaceMethod.

4.2 ExamplesExample 1: Suppose the limit etate function is aparabola, defined as follows:

n

whereP =3, n =10, Q =i , 4, 3. The randomvari-able Xi are independent and standard normally dis-tributed. The results obtained by FORM and ISPUDare shown in Table 1, where V denotes the so-calledstandard error of estimate, in this case defined as

For Q = 1/6 the exact solution is given in [26] as1.41.10-. It can be seen that FORM does not dependon the curvature of the limit state surface, which isinfluenced by a. FORM considerably underestimatesthe failure probability. ISPUD results are based on500 simulations.I I FORM I ISPUD I

Table 1. - Failure probabilities fo r parabolic limitstate surfaces.

Example 2: This example reflects a more practi-cal application of reliability analysis. The consideredstructure is the jib of a crane. As failure criterionthe formation of a plastic hinge in a cross section isassumed. For the sake of simplicity only one cross sec-tion of the jib is considered here. The sectional forceand moment repectively are calculated by static anal-ysis. The momentM / axial forceN - interaction forfull plastification of the cross section is evaluated forfive discrete ratios M N. Through these five pointsa quadratic polynomial is interpolated, as describedin section 3. The randomness of the load and theradius of the crane, respectively, are taken into ac-count. During the simulation procedure, M and N ,in the considerded cross section, are calculated fromthe sample sets. Therefore the dependence of the limitstate on the basic variables, load and radius, is not ofquadratic type anymore and hencecan not be formu-lated explicitely.

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Field measurements formed the basis for the fol-lowing assumptions: the load is aseumed to be lognor-mally distributed with mean 2.0.106Nand coefficientof variation 10%and the radius uniformly distributedwithin the rangeof [2.4m;28.Om.The result calculated with ISPUD is@ =0.6-10 8,with FORM i jf = 0.5 - lo. In this case, FORMoverestimates the failure probability.

6. Given t * the exact valueg(x) can be calculated.7. An indicator is defined as

With this indicator the proposed error estimatorreads . N

5 Concluding remarksIt is shown, that accurate methods, i.e. advancedsimulation procedures such as importance and adap-tive sampling- n context with the Response SurfaceMethod - meet the requirements of the utilization ofsophisticated mechanical modeling as well aa compu-tational efficiency.Approximate methods, suchas first order reliabilitymethods (FORM) may, for some cases, provide inac-curate results. Unfortunately they still lack mathe-matical based guidelines concerning their range of ap-plicability. In other words, for all new types of prob-lems the results still have to be verified by simulationtechniques. This may be considered as a consider-able drawback, particularly for practical application.Hence an immediate application of advanced simu-lation techniques is advisable. Future developmentsshould be guided towards a further improvementof theefficiency of advanced simulation procedures as well asto the development of user friendly structural reliabil-ity software.

AppendixFor an explicitely given limit state function the ac-curacy of RSM can be estimated as suggested in [ll].For this purpose a conditional sampling s carried out:

1. A weighting distribution h ~ ( x )or simulation ofsamples x is obtained by Adaptive Sampling asdescribed in section 2.3.2. The most important variableX is identified bysensitivity analysis.3. A set of random variablesx excluding I * is sim-ulated utilizing the weighting functionh ~ ( x ) .4. Values of the response surfaceg(x), which is itselfa random variable, are simulated near j ( x ) =0.5. For each realization g(xi ) and x,! the value xr iscalculated.

AcknowledgementThis research is partially supported by the Aus-trian I ndustrial Research Promotion Fund ( FFF) un-der contract no. 6/636, which is gratefully acknowl-edged by the authors.

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computational procedures in structural reliability

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