High Order Limit State Functions in the Response Surface Method...

High Order Limit State Functions in the

Response Surface Method for Structural

Reliability Analysis

Henri P. Gavin ∗, Siu Chung Yau

Department of Civil and Environmental Engineering,

Duke University, Durham NC 27708-0287

Abstract

The stochastic response surface method (SRSM) is a technique for the reliabilityanalysis of complex systems with low failure probabilities, for which Monte Carlosimulation (MCS) is too computationally intensive and for which approximate meth-ods are inaccurate. Typically, the SRSM approximates a limit state function witha multidimensional quadratic polynomial by fitting the polynomial to a number ofsampling points from the limit state function. This method can give biased approx-imations of the failure probability for cases in which the quadratic response surfacecan not conform to the true limit state function’s nonlinearities. In contrast to re-cently proposed algorithms which focus on the positions of sample points to improvethe accuracy of the quadratic SRSM, this paper describes the use of higher orderpolynomials in order to approximate the true limit state more accurately. The useof higher order polynomials has received relatively little attention to date becauseof problems associated with ill-conditioned systems of equations and an approxi-mated limit state which is very inaccurate outside the domain of the sample points.To address these problems, an algorithm using orthogonal polynomials is proposedto determine the necessary polynomial orders. Four numerical examples comparethe proposed algorithm with the conventional quadratic polynomial SRSM and adetailed MCS.

Key words: Chebyshev polynomial, Failure probability, Monte Carlo, Statisticaltest, Structural reliability, Response surface

∗ Corresponding author. Tel.: +1 919 660 5201; Fax.: +1 919 660 5219.Email address: [email protected] (Henri P. Gavin).URL: http://www.duke.edu/~hpgavin/ (Henri P. Gavin).

Preprint submitted to Elsevier Science 21 February 2007

1 Introduction

In system reliability analysis, Monte Carlo Simulation (MCS) [17] is the onlyknown technique to accurately estimate the probability of failure, Pf , regard-less of the complexity of the system or the limit state. MCS becomes com-putationally intensive for the reliability analysis of complex systems with lowfailure probabilities and MCS can be infeasible when the analysis requires alarge number of computationally intensive simulations. In such cases, the firstorder reliability method (FORM) [22] and the second order reliability method(SORM) are also difficult to apply since the true limit state usually cannot beeasily expressed explicitly.

The stochastic response surface method (SRSM) [10] is a recently-developedtechnique which can provide an efficient and accurate estimation of struc-tural reliability regardless of the complexity of the failure process. The SRSMapproximates the true limit state function using simple and explicit mathe-matical functions (typically quadratic polynomials) of the random variablesinvolved in the limit state function. By fitting the response surface to a num-ber of designated sample points of the true limit state, an approximated limitstate function is constructed. As the approximated limit state function is ex-plicit, FORM or SORM can be applied to estimate the probability of failuredirectly. Alternatively, MCS can be used efficiently since the evaluation of theresponse surface function requires very little computational effort.

Recent studies have investigated the limitations of the SRSM and have shownthat the method fails to estimate the probability of failure accurately in someproblems with highly nonlinear limit state functions and in some problemswith low probabilities of failure [4,14,21]. Most of the proposed improvementsto the SRSM have been concerned with the relocation of the sample pointsto positions close to the true limit state or design point, so that a quadraticpolynomial can better approximate the true limit state function in that region[2,4,12,16,21]. These studies have shown that if a second order polynomial isused, methods of experiment design (simply working on the locations of samplepoints) do not neccessarily improve results in cases where the a quadratic formcan not accurately fit the limit state function over a broad domain. If the truelimit state function is highly nonlinear, the accuracy of the approximationdepends very much upon the location and distribution of the sample points[19]. Therefore, for highly nonlinear limit states, experimental design alonecannot solve the problem.

In this study, a different approach is proposed, suggesting the use of higherorder polynomials, in order to approximate the true limit state over a broadregion of the space of random parameters. The use of higher order polyno-mials has received little attention because doing so can require an excessively

2

large number of sample points. It can also result in ill-conditioned systemsof equations, and huge differences between the approximated and true limitstate functions outside the domain of the sample points [8,19,21]. This studybriefly discusses how to determine polynomial orders so as to avoid unneces-sary higher order terms and associated ill-conditioning. The method involvesthe use of Chebyshev polynomials, a statistical analysis of the Chebyshevpolynomial coefficients, and a statistical analysis of the high-order responsesurface.

2 Background, Review and Motivation

2.1 Introduction to SRSM

In a large and complex structural system, the true limit state is usually implicitand expressed in terms of a set of n random variables, such as material proper-ties, dimensions, and load densities. Let X denote the 1×n vector containingthese variables and let g(X) be the true limit state function. The functiong(X) = 0 defines the true limit state, and g(X) < 0 indicates a failure condi-tion of the system. The failure of probability is given by Pf = Prob[g(X) < 0].

In the stochastic response surface method (SRSM), the true limit state func-tion, g(X), is approximated by a simple and explicit mathematical expressiong(X), which is typically a k-th order polynomial, with undetermined coeffi-cients. The value of the true limit state function is evaluated at a number ofsamples of X, to determine the unknown coefficients such that the error ofapproximation at the samples of X is minimized.

2.2 The Second Order SRSM

The selection of the form of the approximated limit state function, g(X), i.e.the response surface, ideally should be based on the shape and the nonlinearityof the true limit state function, g(X). Since g(X) is usually unknown, therehas been a tendency to develop a generic form for a response surface whichcan be applied across a wide range of strutucal reliability problems. The mostcommon form is the quadratic polynomial [10],

g(X) = a +n∑

i=1

biXi +n∑

i=1

ciX2i (1)

3

Higher order polynomials are usually not used because if g(X) is of a muchhigher degree than g(X), ill-conditioned systems of equations may be encoun-tered [19,17,21] and there may be huge differences between g(X) and g(X)outside the domain of sample points [8]. Comparatively, though the use of thequadratic polynomial has no physical justification, it provides a simple andreadily-available smoothing surface which allows easy identification of a singledesign point [19]. This is particularly useful for the application of FORM andSORM on the approximated limit state and the iterative scheme of responsesurface approximation which locates the design point on the approximatedlimit state [2].

In equation (1), a, bi and ci, are the 2n + 1 unknown coefficients. The valuesof the coefficients can be determined via singular value decomposition using aset of sample points from the true limit state function, g(X). The number ofsample points must be larger or at least equal to the number of coefficients.Among various sampling methods, a common experiment-design approach isto evaluate g(X) at 2n+1 combinations of µi and µi±hσi, as shown in Figure1a, where µi and σi are the mean and standard deviation of Xi, and h is anarbitrary factor. In this case, the number of sample points is just sufficientfor the determination of the coefficients, thereby minimizing the number ofsample points and the number of evaluations of the limit state function.

In order to capture the nonlinearity of the true limit state more precisely,mixed terms are sometimes included [15] into the quadratic polynomial g(X):

g(X) = a +n∑

i=1

biXi +n∑

i=1

ciX2i +

n−1∑

i=1

n∑

j=i+1

dijXiXj (2)

where the number of coefficients is now 1+2n+ n(n−1)2

. In this case, 3k factorialdesign is a common sampling approach. In 3k factorial design, sample pointsare chosen at all possible combinations of the mean values µi and µi ± hσi, asshown in Figure 1b. In 3k factorial sampling, the number of sample points canbe excessively large if the number of random variables, n, is large. An alternatesampling strategy, ‘spherical’ 3k factorial design, can also be applied, whereevery sample point except the center point has the same distance to the origin,as shown in Figure 1c.

The two basic forms of the SRSM mentioned above can approximate the truelimit state function, g(X), accurately for rougly linear and quadratic limitstates. But when the shape of the true limit state is not close to linear orquadratic, the parameter h, which controls the size of tbe sampling domain,plays a significant role in the accuracy of the 2nd order SRSM approximation[11]. For example, consider the approximation of

g(X) = −0.16(X1 − 1)3 − X2 + 4 − 0.04 cos(X1X2) (3)

4

−2 0 2−2

−1

0

1

2

X1

X2

(a) 2n+1 Combinations

−2 0 2−2

−1

0

1

2

X1

X2

(b) 3k Factorial design

−2 0 2−2

−1

0

1

2

X1

X2

(c) Spherical 3k Factorial

Fig. 1. Different sample methods (µ1 = µ2 = 0, σ1 = σ2 = 1, h = 1)

−5 0 5−5

0

5

x1

x 2

h =1

−5 0 5−5

0

5

x1

x 2

h =2

−5 0 5−5

0

5

x1

x 2

h =3

−5 0 5−5

0

5

x1

x 2

h =4

True limit stateSampling pointApprox limit state

Fig. 2. Influence of parameter h in the quadratic approximation of the limit state

using equation (2). As shown in Figure 2, as long as the quadratic polynomialcannot conform to the true limit state, the 2nd order response surfaces dependcritically on the value selected for h. Since the form of the true limit statefunction and the failure probability are usually unknown before the approxi-mation and vary from problem to problem, there is no typical value of h whichcan be applied across a wide range of reliability problems.

2.3 Improvements on the Second Order SRSM

For structures with high reliability and highly nonlinear limit states, the ap-proximation using the sampling methods in Figure 1 may be inaccurate be-cause the sample points, which are located around the mean values, can be farfrom the true limit state, g(X) = 0. Some recent developments suggest algo-ritms of repeated response surface approximations to shift the sample pointscloser to the true limit state, or the design point [4,16,21].

5

Bucher and Bourgund [4] proposed an algorithm to locate a new center ofsample points xm, which is closer to the true limit state than the mean valuevector µ, by adaptive linear interpolation:

xm = µ − g(µ)µ − xD

g(µ) − g(xD)(4)

where xD is the approximated design point, defined as a point lying on theapproximated limit state that is closest to µ. For standardized random vari-ables, µ=0 After an approximation of the true limit state function, equation(4) is applied. A new set of sample points, which are combinations of xmi andxmi ± hσi, are used to construct a second approximated limit state function,and the reliability is estimated based on this new approximated limit state. Adrawback of the method is a doubling of the computational effort. The proce-dure also assumes a single design point and fails to incoporate mulitple designpoints.

To approximate the true limit state function more accurately around the sin-gle design point, a sequence of repeated response surface approximations canbe persued [3]. Rajashekhar and Ellingwood [21] suggested that an iterationof the procedure proposed by Bucher and Bourgund and that a reduction ofh in the later iterations can improve the approximation. Liu and Moses [16]proposed the iteration of the procedure until a convergence criterion is satis-fied. Gupta and Manohar [12] recognized the difficulty in the approximationof limit states with multiple design points. They proposed a method whichinvolves the use of a higher order polynomial and Bucher and Bourgund’s al-gorithm to incorporate the multiple design points in the approximation. Theirsample strategy is complicated and requires considerably many sample points.

Most of the recent improvements of the second order SRSM require a signifi-cant increase in the computational effort, but they may not produce accurateapproximations in highly nonlinear limit state functions or limit states withmulitple design point. For example, consider the application of iterative Bucherand Bourgund procedure at constant h on the limit state funciton in equation3, as illustrated in Figure 3. The process starts with sample points centeredat the origin. In each figure a 2nd order response surface approximation (solidline) is determined from a set of sample points (circles) to locate a designpoint, xD (star), which is the point on the approximation closest to the ori-gin. The previous response surface(s) are shown as dashed lines, and the truelimit state function is shown as a dotted line. The next response surface isdetermined from a set of sample points centered at xm. The upper row of thefigure corresponds to h=1 and the lower row of the figure corresponds to h=2.As shown in the figure, though the quadratic polynomial could accurately fitthe local nonlinearity around the region of the sample points, the approxima-tion outside the domain can be so inaccurate that it affects the estimate the

6

−10 0 10

−5

0

5

x1

x 21st Reponse Surface (h=1)

−5 0 5

−5

0

5

x1

x 2

2nd Reponse Surface (h=1)

−5 0 5

−5

0

5

x1

x 2

3rd Response Surface (h=1)

−5 0 5

−5

0

5

x1

x 2

1st Response Surface (h=2)

−5 0 5

−5

0

5

x1

x 22nd Response Surface (h=2)

−5 0 5

−5

0

5

x1

x 2

3rd Response Surface (h=2)

Sampling Point

Design Point

True Limit State

Previous Approximated RS

Approximated RS

Fig. 3. Iterative Response Surface Approximation

probability of failure significantly.

2.4 Higher Order Approximated Limit State Functions

A polynomial of a fixed degree, e.g. a quadratic polynomial, is unable to ap-proximate the limit states accurately for a wide range of structural reliabilityproblems, since the shape of the limit state varies for each problem. Thereforeit is preferable to use a more flexible polynomial which has a form suited tothe nonlinearity of the true limit state. A flexible polynomial not only pro-duces more accurate results, but is also more robust to the variations in thepositions of sample points. This paper proposes the use of statistical analysesof trial response surfaces, in order to determine the appropriate order of anapproximated limit state at small expense of computational effort. This avoidsany excessively high order terms, which can involve ill-conditioned systems ofequations. Compared to the recursive response surface approximation, whichinvolves a small domain of sample points in each iteration, the objective ofthis paper is to provide an accurate estimation of the failure probability bysampling a boader domain size only once. After the approximation, the esti-mated probability of failure can serve as an indication of the domain size of

7

sample points, given that the approximated limit state can conform well tothe true limit state.

3 The High Order Stochastic Response Surface Method, HO-SRSM

The method proposed in this study approximates the true limit state functionwith an approximate polynomial response surface of arbitrary order,

g(X) = a +n∑

i=1

ki∑

j=1

bijXji +

m∑

q=1

cq

n∏

i=1

Xpiq

i , (5)

where the coefficients bij correspond to terms involving only one random vari-able, and the coefficients cq correspond to mixed terms, involving the productof two or more random variables. The polynomial order, ki, the total numberof mixed terms, m, and the order of a random variable in a mixed term, piq,are determined in the algorithm below.

The algorithm of the proposed method, called the High Order Stochastic Re-sponse Surface Method (HO-SRSM), has four stages. First, orders for theresponse surface are identified. Second, the number and types of mixed termsare determined. These first two stages result in the formulation of the higherorder polynomial to be used for the response surface. After the formulation ofthe higher order polynomial is completed, the coefficients of the higher orderresponse surface polynomial are estimated in the third stage, using singularvalue decomposition. Fourth, MCS is carried out on the response surface todetermine the probability of failure, Pf .

3.1 Polynomial Orders

In the first stage of the method, the mixed terms are neglected and the poly-nomial orders, ki, are determined by statistically and numerically testing thesignificance of polynomial coefficients. The statistical test for these coefficientsrequires that the coefficients be statistically uncorrelated. The numerical testfor the coefficients requires that all local maxima and minima of all poly-nomial basis functions have the same magnitude. These two conditions aresatisfied by fitting the limit state function in a basis of Chebyshev polynomi-als. The orthogonality of the Chebyshev polynomials guarantees uncorrelatedcoefficients, and Chebyshev polynomials evaluated over the domain [−1, 1] arebounded to within the range of [−1, 1], as shown in Figure 4. A Chebyshev

8

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

λ

TM

(λ)

M=1M=2M=3M=4

Fig. 4. Chebyshev polynomials of different orders

polynomial [20] of degree M in λ is given by

TM(λ) = cos(M arccos λ) , (6)

where min(TM(λ)) = −1, and max(TM(λ)) = 1, for all λ such that −1 ≤ λ ≤1 .

The polynomial TM(λ) has M roots in the interval [−1, 1] at

λ = cos

(

π(m − 12)

M

)

where m = 1, . . . ,M (7)

The discrete orthogonality relation for Chebyshev polynomials is given by:

M∑

m=1

Ti(λm)Tj(λm) =

0 : i 6= j

M/2 : i = j 6= 0

M : i = j = 0

(8)

where λm(m = 1, . . . ,M) are the M roots of TM(λ) given by equation (7)and M must be greater than or equal to the larger of i and j. If the range ofthe desired sample points is other than the interval [−1, 1], the sample pointsmust be interpolated to the interval [−1, 1].

9

The orders of the variables ki in equation (5) are estimated one-by-one alongdimension Xi using one-dimensional Chebyshev polynomials,

gi(Xi) = d0T0(λi) + d1T1(λi) + d2T2(λi) + . . . + dkiTki

(λi), (9)

where λi is the interpolated values of Xi from interval [µi −hordσi, µi +hordσi]to [−1, 1], i.e.

Xi = µi + hordλσi (10)

Parameter hord is the domain of the sampling points used to determine thepolynomial degree of the approximation. Ideally the zeros of the Chebyshevpolynomial should be interpolated to the interval about the design point ofthe true limit state, instead of µi, but the location of the true design pointis unknown at this stage of the algorithm. In the estimation of the order ki

for a random variable Xi, all other variables, X1,. . .,Xi−1,Xi+1,. . .,Xn, are setto their mean values. (µ1, ..., µi−1, µi+1, ...µn). This approximated limit statefunction serves only to determine the appropriate polynomial order ki in equa-tion (5), and is not used to estimate the structural reliability. One-dimensionalapproximation, instead of a multi-dimensional approximation, is used becauseit is much more computationally efficient, especially in cases involving a largenumber of random variables.

The Chebyshev polynomial coefficients, dj, are determined by the least squaresmethod.

d = [TTT]−1TTgi(xi) (11)

where the Tjk = Tj(λk), λk is the k-th root of TK(λ), and gi(xi) is a vector ofthe values of true limit state function evaluated with discrete values of randomvariable Xi set to

xik = µi + hordλkσi where k = 1, . . . , K , (12)

and with all other elements of X set to their mean values.

The coefficient covariance matrix,

Vd = [TTT]−1K∑

k=1

(gi(xik) − gi(xik))2/(K − ki) , (13)

is diagonal, due to the discrete orthogonality relationship of the Chebyshevpolynomials, given in equation (8). In fact, every diagonal term of Vd, except

10

the first, is

σ2d =

K

2

K∑

k=1

(gi(xik) − gi(xik))2/(K − ki) . (14)

Because all Chebyshev polynomials are bounded to within [−1, 1] the contri-bution of Tj to gi is related only to dj.

The variance of the coefficients, σ2d, is used to reveal the statistical significance

of each of the terms in equation (9), indicating which terms could be neglectedwithout significant influence. The test of statistical significance of an individualterm dj in equation (9) involves the test of the null hypothesis, H0: the truecoefficient of the term is 0. The test is performed by calculating values of thet-statistics [6],

tj =dj

σd

. (15)

Using a two-sided test and 90% confidence intervals, if the absolute value oftj is smaller than the value of t0.05 = 3.499, the null hypothesis can not berejected and the Tj(λi) term is determined to be statistically insignificant.Also if dj is less than one percent of the other coefficients, in magnitude, thenTj(λi) is determined to be numerically insignificant, because Tj(λi) is boundedwithin [-1,1] for all i and j.

The algorithm starts with ki = 3, the approximation first determines whethera second order is sufficient for Xi. If T3(λi) is determined to be statistically ornumerically insignificant, it is concluded that the response surface should besecond order in Xi, i.e. ki=2. On the other hand, if T3(λi) is significant, anotherapproximation including a T4(λi) term is carried out and the significance ofthe forth order term is tested. These iterations continue until the highest orderterm is determined to be insignificant or the order is estimated to be at mostfive. It is presumed that it is very rare to exceed the fifth order. By removingthe insignificant high order terms, the orders k1, . . . , kn can be kept small yetsufficient to provide an accurate approximation of the limit state.

3.2 Mixed Terms

After the orders, ki, of the polynomials in Xi, i = 1, · · · , n, are determined,the formulation of the mixed terms can be determined. As a rule, the numberof mixed terms should be kept as small as possible. In general, a mixed termcan be expressed as Xp1

1 Xp2

2 . . . Xpnn . There are two criteria for a valid mixed

term: (1) the power of a variable in a mixed term should not be larger than

11

Table 1Valid mixed term powers in a limit state function where k1 = 2, k2 = 3, k3 = 3

q p1q p2q p3q

1 0 1 1

2 1 0 1

3 1 1 0

4 0 1 2

5 0 2 1

6 1 0 2

7 2 0 1

8 1 2 0

9 2 1 0

the estimated order of the variable alone, i.e., pi ≤ ki and (2) the total orderof the mixed term,

∑

i pi, should not be larger than the highest order term,i.e.,

∑

i pi ≤ max(ki). For example, in a limit state function of three variables,suppose it is found in stage 1 that k1=2, k2=3, k3=3, then the nine validmixed terms, cq

∏ni=1 X

piq

i , (and q=1, . . . , 9), have the powers listed in Table1, where q is the mixed term with coefficient cq.

For large and complex problems, it may be worthwhile to examine and com-pare the results of several reasonable values of hord after the number of mixedterms are calculated, since the computational cost for the determination ofthe order is relatively small compared to the approximation of the true limitstate. It may be desirable to use a value for hord resulting in a small numberof coefficients in order to reduce the computational cost, as there is no generalguideline to decide which hord should be used before Pf is estimated.

3.3 Response Surface Approximation

Once the response surface has been formulated, the coefficients are estimatedvia singular value decomposition using sample points from the true limit statefunction. In this study, the number of sample points is chosen to be twice thenumber of coefficients. This number is more than sufficient and was found tobe adequate in the examples considered here. A “full factorial design” has Psample points where

P =n∏

i=1

(ki + 1) . (16)

12

In problems with n > 3, the value of P , even 3n, is much larger than thenumber of coefficients. Thus uniformly distributed random sample points aretaken within the domain [µ + hregσ,µ − hregσ], where µ is a vector of themean values of X and σ is a vector containing the standard deviation of X.Analogous to the parameter h in regular 2nd order SRSM, the parameter hreg

here indicates the size of the domain of the sample points used in the regressionfor the polynomial coefficients. The true limit state function is evaluated at theuniformly-distributed sample points within the domain [µ+hregσ,µ−hregσ].The use of randomly distributed sample points may increase the randomness ofthe results, but the number of sample points are believed to be large enough todiminish the effect of their randomness, while still being much smaller than Por a 3n design. If the true limit state function is more accurately approximatedby a higher-order limit state function, then the regression coefficients of thehigher order aproximation will be less sensitive to hreg than those of a 2nd

order approximation, because the 2nd order approximation will be accurateonly with smaller domains.

3.4 Monte Carlo Simulation

In the fourth stage, a full scale MCS on the approximated limit state is carriedout to determine the probability of failure, Pf .

Recall that in the proposed method, there are two input parameters hord andhreg, while the regular quadratic SRSM has one domain size parameter. Pa-rameter hord is the size of the domain of the sampling points used to determinethe polynomial degree of the response surface, and hreg is the size of domain ofthe sampling points used to determine the coefficients of the response surface.

4 Numerical Examples

The performance of the proposed method is illustrated by four numerical ex-amples. The first two examples have explicit hypothetical limit state functionsand the other two are realistic structural reliability problems. In each example,the probability of failure is estimated by the following three methods:

(1) Direct full scale Monte Carlo simulation (MCS).(2) Stochastic response surface method which approximates the true limit

state with a quadratic polynomial including the mixed terms (regular 2nd

order SRSM). The spherical 3k factorial design is used to locate the posi-tions of sample points. The effect of the parameter hreg on the estimationresult is investigated for a set of values of hreg. In the first example, the

13

procedure proposed by Bucher and Bourgund will be applied for furthercomparison.

(3) High order stochastic response surface method (HO-SRSM). In each ex-ample, a range of values of parameter hord are used to identify appropriatepolynomial orders. Then the limit state is approximated for that polyno-mial order. Again, different values of hreg will be used to investigate itsinfluence on the estimated failure probabilities.

In addition, the values of adjusted R2, indicating the accuracy of the approx-imation, are calculated from the statistical data of each approximated limitstate [6]. The definition of adjusted R2 is given by

AdjustedR2 =(P − 1)R2 − N

P − (N + 1)(17)

where P is the total number of sample points, N is the total number of coef-ficients, and

R2 = 1 −

∑Pi=1 (g(xi) − g(xi))

2

∑Pi=1 (g(xi) − g)2 (18)

The mean value of the limit state function, g, is given by

g =1

P

P∑

i=1

g(xi) (19)

The adjusted R2 is bounded by the interval [0,1]. If its value is large, theapproximated limit state is close to the true limit state at the sample points.

The general-purpose computer code for the HO-SRSM, and examples are avail-able at http://www.duke.edu/˜hpgavin/HOSRSM/.

4.1 Example 1 - Hypothetical limit state with 2 variables

A hypothetical limit state with two independent standard normal variables isconsidered.

g(X) = −0.16(X1 − 1)3 − X2 + 4 − 0.04 cos(X1X2) (20)

The limit state, g(X) = 0, has the shape shown in top-left figure in Figure5. The limit state function is roughly cubic in X1 and quadratic in X2. Thecosine term represents the small effects of higher-order terms. The true value

14

−5 0 5

−5

0

5

x1

x 2

True Limit State

−5 0 5

−5

0

5

x1

x 2

HO−SRSM (hord

=2, hreg

=1)

−5 0 5

−5

0

5

x1

x 2

HO−SRSM (hord

=2, hreg

=3)

−5 0 5

−5

0

5

x1

x 2

HO−SRSM (hord

=2, hreg

=5)

True Limit StateSampling pointApprox limit state

Fig. 5. True limit state and approximated higher-order limit states in Example 1

Table 2Determination of polynomial orders for Example 1

hord k1 k2

1.0 2 2

2.0 3 2

3.0 3 2

4.0 3 2

5.0 3 2

6.0 3 2

7.0 3 2

of Pf = 0.000095 is obtained by a direct full scale MCS of 10,000,000 samplesize.

Table 2 suggests that the parameter hord in the HO-SRSM has almost no in-fluence in the estimation of the polynomial orders in this problem, since allvalues of hord larger than 1.0 generate exactly the same results. The failure todetect the third order at hord = 1 is reasonable because over a small samplearea, the limit state does not demonstrate sufficient nonlinearity. The poly-nomial orders given by hord from 2 to 7, i.e., k1 = 3 and k2 = 2, are used toapproximate the true limit state function via the HO-SRSM.

15

1 2 3 4 5 6 710

10

10

10

10

−5

−3

−2

−1

−4

100

hreg

Pro

babi

lity

of F

ailu

re, p

f HO−SRSM2nd order SRSMMCS

1 2 3 4 5 6 70.75

0.8

0.85

0.9

0.95

1

hreg

Adj

uste

d R

2

Fig. 6. Estimation results of Example 1

The shapes of the approximated limit states at different hreg in the regular 2nd

order SRSM and HO-SRSM are shown in Figures 2 and 5 respectively. Theestimation of Pf by the HO-SRSM is consistent over a large range of hreg, ascompared to the regular 2nd order SRSM, as shown in Figure 6. The Pf ofthe HO-SRSM ranges from 0.000067 (-29%) (at hreg=2) to 0.00011 (+16%)(at hreg=5), while that of the 2nd order SRSM ranges from 0.000015 (-84%)(at hreg=1) to 0.25 (+2,600,000%) (at hreg=6). The estimation results of the2nd order SRSM are very heavily dependent on hreg. Comparatively, there isno obvious trend over values of hreg in the HO-SRSM. All the values of Pf

mentioned above are obtained by running a MCS of 1,000,000 sample size onthe response surface.

The lower plot of Figure 6 suggests the reasons for the large errors in theregular 2nd order SRSM. For the 2nd order SRSM, at hreg=1 and at hreg=2, thevalues of R2 are large, implying the approximation is fairly accurate withinthe domain and the inaccurate Pf estimation is due to a lack-of-fit of theapproximated limit state outside the small domain. While hreg increases, thevalue of the adjusted R2 decreases because as the sample domain becomeslarger, the quadratic polynomial increasingly fails to conform accurately tothe shape of the true limit state within the domain. On the other hand, theHO-SRSM method is able to capture the high nonlinearity regardless of thesize of the domain.

16

Table 3Iterative Bucher and Bourgund procedure for Example 1

hreg Iteration Adjusted R2 Pf

1.0 0 0.9970 0.000015

1.0 1 0.9894 0.000015

1.0 2 0.9906 0.000014

2.0 0 0.9689 0.000026

2.0 1 0.9510 0.000018

2.0 2 0.9578 0.003095

An iterative scheme of the procedure proposed by Bucher and Bougund is car-ried out at hreg=1 and 2. Small sampling areas are chosen since it is expectedthat the samples are taken near the design point. The procedure proposed byBucher and Bourgund does not improve the accuracy of the estimation of Pf inthis example. As shown in Table 3, the approximated limit state labeled ‘2ndResponse Surface’ and ‘3rd Response Surface’ do not approximate the truelimit state better than the one labeled ‘1st Response Surface’ as illustrated inFigure 3. At the second iteration at hreg=2, the estimated value of Pf becomesexceptionally large. The values of adjusted R2 are consistent and close to 1throughout the iteration process, implying the procedure may have led to anaccurately approximated limit state around an inaccurate design point in thiscase.

4.2 Example 2 - Hypothetical limit state with 3 variables

A second hypothetical example, with three independent standard normal vari-ables, is considered:

g(X) = −0.32(X1 − 1)2X22 − X2 + X3

3 − 0.2 sin(X1X3) (21)

A sine function is included to prevent the hypothetical limit state from be-ing perfectly fit by a third order polynomial. The value of Pf = 0.037216 isobtained by a direct full scale MCS of 10,000,000 sample size.

As in Example 1, parameter hord has no influence on the estimation of theorders of the polynomials, as shown in Table 4. All values of hord give exactlythe same results.

The failure probability estimation results of the HO-SRSM again are consistentover different values of parameter hreg, while the estimated Pf in the regular2nd order SRSM is heavily dependent on hreg. In the proposed method, Pf

17

Table 4Determination of polynomial orders for Example 2

hord k1 k2 k3

1.0 2 2 3

2.0 2 2 3

3.0 2 2 3

4.0 2 2 3

5.0 2 2 3

6.0 2 2 3

7.0 2 2 3

1 2 3 4 5 6 710

−4

10−2

100

hreg

Pro

babi

lity

of F

ailu

re, p

f

1 2 3 4 5 6 70.5

0.6

0.7

0.8

0.9

1

hreg

Adj

uste

d R

2

HO−SRSM2nd order SRSMMCS


ranges from 0.023212(-37.6%) (at hreg=6) to 0.046269 (+24.3%) (at hreg=3),and in the regular 2nd order SRSM, Pf increases from 0.000255(-99%) to0.446921(+1100%) from hreg=1 to hreg=6. All of the estimated Pf values areobtained by running a MCS of 1,000,000 sample size on the approximatedlimit state function. The lower plot in Figure 7 suggests that the accuracy ofthe regular 2nd order SRSM decreases much more significantly than does theHO-SRSM as hreg increases.

18

0 0.5 1 1.5 2−400

−300

−200

−100

0

100

200

300

400

t (sec)F ex

t (N

)m

Fext

(t)

fnl

k

r(t)

c

Fig. 8. Example 3 - A simple dynamic problem with an hysteretic force

4.3 Example 3 - Dynamic Problem of One Degree of Freedom

This example examines a simple dynamic problem of a hysteretic [23] one de-gree of freedom system excited by a full sine-wave force impulse. The equationof motion is given by:

m r(t) + cr(t) + kr(t) + fnl(t) = fext(t) (22)

z =(

1 −1

2z2(sign(rz) + 1)

)

r

ry

fnl = fyz

The description, mean and coefficient of variation of each random variable islisted in Table 5. All variables are lognormal and are assumed to be indepen-dent. The limit state function is defined as:

g(X) = 0.04 − maxt

|r(t)| (23)

where r(t) is the acceleration of the mass. This limit state means that thesystem fails if the maximum acceleration over time exceeds 0.04 m/s2. Thevalue of Pf generated by the direct MCS of 100,000 sample size is 0.00529.

Table 6 shows the polynomial orders for different values of hord. Only theorders in the force magnitude Fp and force period Tp differ at different valuesof hord, while the other five variables have orders of two. Fp increases fromorder 2 to 5 and Tp fluctuates between 2 and 5 at as hord increases. Thissuggests an important finding, that the excitation force and period in problemswith a hysteretic restoring force have a significant nonlinear effect on thepeak acceleration of the mass. The fluctuating order of Tp may be because

19

Table 5Statistical properties of random variables for Example 3

Variable Description Units Mean value C.O.V.

m Mass kg 1,000 0.05

k Stiffness N/m 4,000 0.10

c Damping Coefficient N/m/s 250 0.20

fy Yield Force N 400 0.10

ry Yield Displacement m 0.01 0.15

Fp Force Amplitude N 400 0.20

Tp Force Period s 0.5 0.50

its effect on the limit state cannot simply be described by integer powers. Inthe estimation of polynomial orders, the required number of evaluations ofthe true limit state varies for different values of hord. At hord=1, there areonly 24 evaluations and at hord=5 there are 39. Compared to the hundredsof evaluations in the process of limit state approximation, the computationaleffort spent in the order estimation is modest.

The estimations of the failure probability are shown Figure 9. In the regu-lar 2nd order SRSM, the estimated result, which is based on 36 coefficientsand 2187 sample points, is positively correlated with the parameter hreg. Itincreases from 0.00199 (-62%) to 0.00962 (82%) from hreg=1 to hreg=6. Inthe HO-SRSM, in order to show the effect of parameter hord, two approxi-mate limit states are constructed using the values of hord=3.0 and hord=5.0.The estimated Pf for hord=3.0 ranges from 0.00366(-31%) (hreg=2) to 0.00562(+6%) (hreg=6). For hord=5.0, Pf has estimated values from 0.00332 (-37%)(hreg=6) to 0.00600 (+13%) (hreg=2). Both cases of hord=3.0 and 5.0 involve288 coefficients and 576 sample points. The two estimation results do not showany correlation with the parameter hreg and are more accurate than the 2nd

order SRSM, in general. The lower plot shows that the values of adjusted R2

are about the same for both hord, and they are significantly higher than thoseof the 2nd order SRSM.

To roughly indicate if the value of hord is appropriate and at what value of hreg

the estimation result is accurate, the standard normal cumulative distributionfunction (cdf) may be used. After Pf is estimated, it can be compared withthe normal cdf at the negative value of hord, Φ(−hord). If the estimated Pf

is larger than Φ(−hord) by one or two order of magnitude, the value of hord

may be too large. An excessively-large value of hord may result in inefficientcomputation because large hord values give higher estimated orders and hencea larger number of coefficients and sampling points. For this problem, theestimated Pf is about 0.005. The standard normal cdf at −hord =-3.0 has

20

Table 6Determination of order of variables for Example 3

hord m k c fy ry Fp Tp

1.0 2 2 2 2 2 2 2

2.0 2 2 2 2 2 2 3

3.0 2 2 2 2 2 3 4

4.0 2 2 2 2 2 3 5

5.0 2 2 2 2 2 4 3

6.0 2 2 2 2 2 4 5

1 2 3 4 5 60

0.002

0.004

0.006

0.008

0.01

hreg

Pro

babi

lity

of F

ailu

re, p

f

1 2 3 4 5 60.85

0.9

0.95

1

hreg

Adj

uste

d R

2

HO−SRSM (hord

=3)

HO−SRSM (hord

=5)

2nd order SRSMMCS


a value of 0.001350 and has a value of 2.867 × 10−7 at −hord=-5.0. Since2.867×10−7 is largely different from the actual Pf of 0.005, it can be suggestedthat at hord=5, the sampling domain is too large. In this specific problem,using hord=3 does not save any computational time compared to hord=5. Butin general a smaller value of hord would result in smaller estimated order andless required computation. Nonetheless, compared to the regular 2nd orderSRSM, both values of hreg provide an accurate estimation of Pf . Similarly, areasonable value of hreg could be obtained by this comparison. In fact, if bothhord and hreg are equal to 3.0, the result is 0.00497 (-6%), which is sufficientlyaccurate. This approach may be inaccurate for the determination of hreg for

21

the regular 2nd order SRSM because the shape of the approximated limit statemay be different from the true limit state.

4.4 Example 4 - Dynamic Problem with Six Degrees of Freedom

Figure 10 shows the problem of a four-story building excited by a single pe-riod sinusoidal impulse of ground acceleration. The building contains isolatedequipment on the second floor. The motion of the lowest floor is resisted bya nonlinear hysteresis force [23] due to the building’s base isolation bearingsand an additional stiffness force, if its displacement exceeds dc. Each floorhas a mass of mf and between floors the stiffness and damping coefficientare kf and cf respectively [13]. For simplicity, and to decrease the number ofvariables, mf , kf and cf are assumed to be the same for each floor. On the sec-ond floor there are two isolated masses, representing isolated, shock-sensitiveequipment. The larger mass is connected to the floor by a relatively flexiblespring, k1, and a damper, c1, representing the isolation system. The smallermass is connected to the larger mass by a relatively stiff spring, k2 and damper,c2, representing the equipment itself. There are six degrees of freedom, four atthe floors and two at the equipment mass blocks. The statistical parametersof the basic random variables are listed in Table 7. All variables are assumedto be lognormal and independent.

The limit state function is defined by

g(X) = 12.5 (0.04 − maxt

|rfi(t) − rfi−1

(t)|)i=2,3,4

+ (0.5 − maxt

|zg(t) + rm2(t)|)

+ 2(0.25 − maxt

|rf2(t) − rm1

(t)|) (24)

where rfirefers to the displacement of i-th floor and rfi

(t) − rfi−1(t) is the

inter-story displacement of two consecutive floors. The accelerations zg andrm2

are of the ground and the smaller mass block respectively. The displace-ment rm1

is of the larger mass block, and represents the displacement of theequipment isolation system. The limit state function in equation (24) is thesum of three expressions of failure modes. The first term describes the damageto the structural system due to excessive deformation. The second term rep-resents the damage to equipment caused by excessive acceleration. The lastterm represents the damage of the isolation system. They are multiplied byweighing factors, which emphasize the three faulure modes equally. Equation(24) means that it is desirable that (1) none of the inter-story displacementsexceeds 0.04 m, (2) the peak acceleration of the smaller mass block (the equip-ment) is less than 0.5 m/s2, and (3) the displacement across the equipmentisolation system is less than 0.25 m. Failing one or two of the conditions does

22

Floor mass: mf

Story stiffness: kf

Story dampling coeff.: cf

Isolation yield force: Fy

Isolation yield displ.: Dy

kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc

Floor mass: mf

Story stiffness: kf




kc

kc

m1

m2

k1

c1

k2

c2

..z

g(t)

Dc

Dc 0 1 2 3 4

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t (sec)

z g (m

/s2 )

Fig. 10. Example 4 - Base isolated structure with an equipment isolation system on the2nd floor, and including the effects of isolation displacement limits

not necessarily lead to a failure in the limit state function, e.g. g(X < 0),but will decrease the value of the limit state function. In these simulations,the system fails mainly because of the large acceleration of the smaller mass.The estimation result of Pf by direct full scale MCS of 100,000 sample size is0.19599.

Table 8 shows the polynomial orders of the random variables. The polynomialorders for mf , kf , cf and kc remain at 2 for all values of hord. The other fourvariables dy, fy, T and A remain at orders of 2 or 3 from hord=1.0 to 3.0and then fluctuate about 4 and 5 at higher values of hord. This result showsthat the parameters of the hysteresis force as well as the excitation periodand the amplitude of an earthquake have a complicated effect on the limitstate function. To illustrate the effect of hord, results corresponding to hord=2and 6 are used to approximate the true limit state. To determine which hord

estimates the shape of the limit state more accurately, the estimated valuesof Pf are compared to the normal cdf after the approximation.

The upper plot of Figure 11 shows that Pf is about 20 percent in this prob-lem. The HO-SRSM has better estimations of Pf at hreg=1 and 2. At hreg = 1,the estimated values of Pf for the HO-SRSM are extremely accurate, being0.19325(-1.4%) and 0.19512(-0.4%) for hord=2.0 and hord=6.0 respectively. Forthe 2nd order SRSM it is 0.20374(+4.0%). At hreg=2, the estimated values ofPf are almost the same, being equal to 0.19355(-1.2%) and 0.19643(-0.2%) forhord=2.0 and hord=6.0 in the HO-SRSM, and 0.20051(+2.3%) for the 2nd orderSRSM. From hreg=3 to 6, the estimated Pf in the 2nd order SRSM increasesgradually from 0.19999(-0.2%) to 0.32196(+64%). The proposed method un-derestimates the value of Pf at values of hreg larger than 2. For hord=2.0,the smallest estimated value is 0.10035(-49%)(at hreg=6); it is 0.08554(-56%)(at hreg=4) for hord=6.0. Regarding the compuatational effort, the number ofsample points is 6561 in the 2nd order SRSM, 316 in the HO-SRSM of hord =2,

23

Table 7Statistical properties of random variables for Example 4

Variable Description Units Mean value C.O.V.

mf Floor Mass kg 6,000 0.10

kf Floor Stiffness N/m 30,000,000 0.10

cf Floor Damping Coefficient N/m/s 60,000 0.20

dy Isolation Yield Displacement m 0.05 0.20

fy Isolation Yield Force N 20,000 0.20

dc Isolation Contact Displacement m 0.5 0

kc Isolation Contact Stiffness N/m 30,000,000 0.30

m1 Mass of Block 1 kg 500 0

m2 Mass of Block 2 kg 100 0

k1 Stiffness of Spring 1 N/m 2,500 0

k2 Stiffness of Spring 2 N/m 100,000 0

c1 Damping Coefficient of Damper 1 N/m/s 350 0

c2 Damping Coefficient of Damper 2 N/m/s 200 0

T Pulse Excitation Period s 1.0 0.20

A Pulse Amplitude m/s/s 1.0 0.50

Table 8Determination of polynomial orders, ki, for variables in Example 4

hord mf kf cf dy fy kc T A

1.0 2 2 2 2 2 2 2 2

2.0 2 2 2 2 2 2 2 3

3.0 2 2 2 2 2 2 2 3

4.0 2 2 2 5 4 2 2 5

5.0 2 2 2 4 5 2 5 5

6.0 2 2 2 2 5 2 5 3

and 2106 in the case of hord=6. The number of coefficients are 45, 158 and1053, respectively.

The results of the HO-SRSM are disappointing at larger hreg values. After Pf

is estimated, it should nevertheless be noted that, in this case, the values ofboth hreg and hord should be small. Checking against the values of Φ(−hord),as mentioned in Example 3, reveals that the estimation result should be suf-ficiently accurate for hord=1 and hreg=1. Referring to Table 8, at hreg=1, all

24

1 2 3 4 5 60

0.1

0.2

0.3

0.4

hreg

Pro

babi

lity

of F

ailu

re, p

f

1 2 3 4 5 60.7

0.8

0.9

1

hreg

Adj

uste

d R

2

HO−SRSM (hord

=2)

HO−SRSM (hord

=6)

2nd order SRSMMCS


variables have 2nd order. This explains why the 2nd order SRSM performs sat-isfactorily in this method. This example shows that in problems where the 2nd

order SRSM is sufficient, the proposed method is still useful in the confirma-tion of the accuracy of the results. It is suggested that several values of hord

should be investigated before the approximation of the true limit state. Afterthe estimation of the order of variables and the number of coefficients, theresult of hord which yields the smallest number of coefficients is used to ap-proximate the limit state function in order to save computational effort. Thenthe estimated value of Pf is compared to Φ(−hord) to determine the sufficientvalue of hord. Implementation of this approach in this problem would provideas accurate an estimation of Pf as the 2nd order SRSM, but in a more effi-cient way, since the 3k factorial design involves a very large number of samplepoints.

5 Conclusions

This study proposes the use of higher order polynomials in the stochasticresponse surface method, and an algorithm to determine the polynomial ordersusing a statistical analysis of polynomial coefficients. The numerical exampleshave shown that the proposed method not only provides more accurate results

25

of the probability of failure than the 2nd order SRSM in highly nonlinearproblems, but also allows an indication of the accuracy of the estimated failureprobability. Moreover, unlike the 2nd order SRSM, the failure probabilitiescomputed using the proposed method do not show any significant correlationwith the size of the domain of the sample points. The method proposed inthis paper checks the accuracy of the response surface using a goodness-of-fitcriteria and checks the failure probability by a comparison to the size of thedomain of sample points.

Future work could implement better experimental design into the method,since the method currently uses random sampling points. For example, theproposed method could be improved by shifting the location of the samplepoints closer to the limit state, as is achieved by Bucher and Bourgund, inimproving the 2nd order SRSM. Implementation of such a concept in the pro-posed method is, however, more complicated, since it may not be easy tofind a single design point if the limit state function has a highly nonlinearshape. Also, estimation results may become more inaccurate if sample pointsare concentrated at one of the multiple design points. To further improve theproposed method, relocation of the sample points to incorporate the effect ofmultiple design points, at small expense of computational time, should be in-vestigated. The fact that random sampling gives good results in our examples,however, shows that the location of sample points may not critically impor-tant, as long as appropiate polynomial orders and appropriate cross-terms areutilized and as long as the sample points lie in the domain of interest. Thecriteria in the determination of the significance of polynomial coefficients alsorequires further investigation.

6 Acknowledgments

The research described in this publication was made possible in part by PrattEngineering Undergraduate Fellow Program of Duke University. The authorsthank the reviewers for their thoughtful and insightful suggestions.

References

[1] Ayyub, B.M., and McCuen, R.H. (1997). Probability, Statistics, & Reliability

for Engineers. CRC Press.

[2] Breitung, K. and Faravelli, L. (1994). Log-likelihood Maximization andResponse Surface Reliability Assessment. Nonlinear Dynamics. 4 273-286.

26

[3] Breitung, K. and Faravelli, L. (1996). Response Surface Methods andAsymptotic Approximations. Mathematical Models for Structural Reliability

Analysis. Edited by Casciati, F. and Roberts, B. CRC Press.

[4] Bucher, C.G., and Bourgund, U. (1990). A Fast and Efficient Response SurfaceApproach for Structural Reliability Problems. Structural Safety. 7 57-66.

[5] Cox, D.C., and Baybutt, P. (1981). Methods for Uncertainty Analysis: AComparative Survey. Risk Analysis. 1(4) 251-258.

[6] Devore, J.L. (2004). Probability and Statistics for Engineering and the Sciences.

Sixth Edition. Brooks/Cole.

[7] Draper, N.R., and Smith, H. (1998). Applied Regression Analysis. ThirdEdition. John Wiley & Sons.

[8] Engelund, S., and Rackwitz, R. (1992). Experiences with Experimental DesignSchemes for Failure Surface Estimation and Reliability. Proceedings of the

Sixth Speciality Conference on Probabilistic Mechanics and Structural and

Geotechnical Reliability. ASCE 1992. 252-255.

[9] Der Kiureghian, A. (1996). Structural Reliability Methods for Seismic SafetyAssessment: A Review. Engineering Structures. 18(6) 412-424.

[10] Faravelli, L. (1989). Response Surface Approach for Reliability Analysis.Journal of Engineering Mechanics, ASCE 1989. 115(12) 2763-81.

[11] Guan, X.L., and Melchers, R.E. (2001). Effect of Response Surface ParameterVariation on Structural Reliability Estimates. Structural Safety. 23 429-444.

[12] Gupta, S., and Manohar, C.S. (2004). An Improved Response Surface Methodfor the Determination of Failure Probability and Importance Measures.Structural Safety. 26 123-139.

[13] Johnson, E.A., Ramallo, J.C., Spencer, B.F., and Sain, M.K. (1998). IntelligentBase Isolation Systems. Proceedings of the Second World Conference on

Structural Control Kyoto Japan. June 21-July 1. 367-376.

[14] Kim, S.-H., and Na, S.-W. (1997). Response Surface Method using VectorProjected Sampling Points. Structural Safety. 19 3-19.

[15] Khuri, A.I., and Cornell, J.A. (1996). Response Surfaces Designs and Analyses.

Second Edition. Marcel Dekker.

[16] Liu, V.W., and Moses, F.A. (1994). A Sequential Response Surface Methodand Its Application in the Reliability Analysis of Aircraft Structural Systems.Structural Safety. 16 39-46.

[17] Melchers, R.E. (1999). Structural Reliability Analysis and Prediction. SecondEdition. John Wiley & Sons.

[18] Myers, R.H., and Montgomery, D.C. (1995). Response Surface Methodology:

Process and Product Optimization Using Designed Experiments. John Wiley &Sons.

27

[19] Olivi, L. (1980). Response Surface Methodology in Risk Analysis. Synthesis and

Analysis Methods for Safety and Reliability Studies. Edited by Apostolakis, G.,Garribba, S., and Volta G. Pleum Press.

[20] Press W.H., Teukolsky S.A., Vetterling W.T., and Flannery B.P. (1992).Numerical Recipes in C: The Art of Scientific Computing. Second Edition.Cambridge University Press.

[21] Rajashekhar M.R., and Ellingwood B.R. (1993). A New Look at the ResponseSurface Approach for Reliability Analysis. Structural Safety. 12 205-220.

[22] Thoft-Chiristensen P., and Baker M.J. (1982). Structural Reliability Theory and

Its Applications. Springer-Verlag.

[23] Wen Y.-K. (1976). Method for Random Vibration of Hysteretc Systems. Jornal

of the Engineering Mechanics Division, ASCE. 102 EM2 249-263.

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