Application of Stochastic Response Surface Method in the Structural Reliability

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Procedia Engineering 00 (2011) Procedia Engineering 28 000–000 (2012) 661 – 664

Procedia Engineering www.elsevier.com/locate/procedia

2012 International Conference on Modern Hydraulic Engineering

Application of Stochastic Response Surface Method in the Structural Reliability Liu Ying, a* College of civil and Architectural Engineering, Nanchang Institute Technology, Nanchang 330099, China

Abstract Based on Karhunen-Loève expansion of random field, the paper utilizes Hermite polynomial chaos expansion to build the stochastic response surface function, and the unknown coefficients of the function can be calculated by probabilistic collocation approach. Then, the geometric method can be used to calculate the structural reliability. A comparison with Monte Carlo simulation shows that the proposed method can achieve high accuracy and efficiency while analyzing the property of the second-order and the third-order stochastic response surface.

© 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for Resources, © 2011 Published by Elsevier Ltd. Environment and Engineering Keywords: stochastic response surface method; structural reliability; random field; Monte Carlo simulation

1. Introduction The reliability analysis has been used more and more widely in structural engineering. For large and complex engineering structures, the reliability represents the capacity of supportment for load and environment. Therefore, how to calculate the reliability effectively has important practical significance. Various reliability analysis methods such as Monte Carlo simulation methods and Perturbation stochastic finite element method have been proposed in the literature. In general, the first step is establishing the limit state equation of structure. Response surface method is one of the efficient ways. Faravelli [1] created the certainty response surface method based on experiment, which is fussy for so many testing times. Bucher [2] set up interpolated iterative technology to simulate the limit state surface, though the computation is smaller, the checking points are approximate. Kaymaz [3] provided a weighted regression method, when the experimental points are selected in the most probability areas. Stochastic * Corresponding author. Tel.: 13970900951. E-mail address: [email protected].

1877-7058 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for Resources, Environment and Engineering doi:10.1016/j.proeng.2012.01.787

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Ying/ Procedia / Procedia Engineering (2012) 661 – 664 AuthorLiu name Engineering 0028 (2011) 000–000

response surface method (SRSM) is the extension of the traditional response surface method. The signal difference between two methods is the way of establishing the limit-state function. In this paper, stochastic response surface method is utilized to simulate the complicated relation between the input uncertainties and the system response, and transform the outputs into standard random variables through the Hermite polynomial chaos expansions. 2. Spectral stochastic finite element method 2.1 Random field discretization Generally, random field S ( x,  ) not only stands for the function of the space coordinates x , but also the function of the random variable ξ . Then, zero-mean random field is given by: (1) S ( x,  )  S ( x,  )  S ( x) where S ( x) and CSS ( x1, x2 ) are mean and covariance function of the random field. Usually, CSS ( x1, x2 ) is bounded, symmetric and positive definite, and not limited to homogenous fields, so it can be decomposed as follow 

(2)

CSS  x1 , x2    i fi  x1  fi  x2  i 0

The eigenvalues i and the associated digenfunctions fi  x  can be obtained from the solutions of the integral equation (3)  CSS ( x1, x2 ) fi  x2  dx2  i fi  x1  So the random field S ( x,  ) can be approximatively discretized into random variables as M

(4)

S  x,    S  x    i i fi ( x) i 0

where  i is a set of uncorrelated Gaussian random variables, with zero-mean, and when the expansion is approximated to the third term, it can be closed to a fine result. The random variables  i is also satisfied the follow equation (5). E[lk ]   kl (5) E[12 n ]   E[r r r ]E[k j ] k

1

n2

2

2.2 expansion of the stochastic stiffness equation For the stochastic structure, its stiffness equation can be expressed as  K u  F in which  K  are the stiffness matrix  K 

(6) (7)

  B  D B d 



Considered the randomicity of material properties, the elastic stiffness can be given by (8)  D  x,    S  x,    D0  0 where [ D ] is deterministic matrix, S  x,   express random field of material properties, with mean and correct structure are known.Therefore, combined with the equation (4), equation (9) can be obtained {ξ}  [1 , 2 , , M ]T (9) Taking formula(4), (8) into formula (7), expansion of the stiffness matrix can be received as follow

 K   B

T

M

dx [ K ]   i [ K ]i  D  x,    B  i 1

(10)

in which [ K ] is the mean of element stiffness matrix, and [ K ]i is the subtle fluctuation of the element stiffness matrix which varied with  i .Combined with the discretization of the random field, stochastic

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Liu Yingname / Procedia Engineering 28 (2012) 661000–000 – 664 Author / Procedia Engineering 00 (2011)

structural finite element equations can be expressed as: M

    K    i  K i  u F  i 1  

(11)

3. Calculation of structural reliability based on SRSM Before using SRSM, the random field of the control point P is approximated by a set of uncorrelated stochastic variables { } according to formula (4). M

 

i1

M



i1

M



i2





uP  a0   ai1 1 i1   ai1i2 2 i1 , i2   ai1i2i3 3 i1 , i2 , i3  i1 1 

i1 1  i2 1 

i1 1  i2 1 i3 1 

(12)

in which, ai i i are the unknown coefficients, and the multidimensional P orders Hermite polynomial can be expressed as: 123



 p i1 ,



1

p

p

   

, i p   1 e 2

i1 ,

, i p

e



1      2

(13)

Therefore, the formula (12) can be represented as  uP

L1

a     j 0

j

j

(14)

[ ]{a}

where a j are the undetermined coefficients. a j and  are different from ai i i and  . In the sample space of { } , each { } corresponds to a point, named collocation point. Substituting each collocation point into Eqs (11) could receive the nodal displacement matrix {u} . So, the key step of SRSM is how to select collocation points reasonably. For higher precision, the collocation points are selected from the roots of the next higher order Hermite polynomial. Meanwhile, the collocation points should be close to the origin, and symmetric with the origin [4]. In general, regression-based methods could obtain fine accuracy by selecting a number of points twice of the coefficients’ quantity [5]. Substituting { } and u P into Eqs (14), a linear algebraic equation is obtained as (15) Ga   f  where {a} are the undetermined coefficients, and could be gained by linear least squares method . Take a into formula (12) can gain the limit state equation f the specific point P as follow: (16) G({ }) u0  u({}) in which u0 stands for structural failure of the threshold value, and u({ }) is the stochastic response expression of the control point. Based on the result of { } , structural reliability  can be calculated according to the geometric method. 4. Numerical study A continuous beam subjected to multiple loads (Fig.1) is considered. It is assumed that the modulus of elasticity E is a Gaussian random process with mean E  100.0kN / m2 , the correlation length x2 )  E2 exp( x1  x2 c) . Let the reliability function as c  1.0 , the covariance function Css ( x1 ,  G 1 , 2  1.45u  u 1 , 2  . 123

120kN/m

120kN

2.5m

2.5m

3.0m

5

4

3

2

1

200kN.m

3.0m

4m

Fig.1 Continuous beam

Based on SRSM, the second and the third order response surface of the midpoint flexibility can be given by

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Ying/ Procedia / Procedia Engineering (2012) 661 – 664 AuthorLiu name Engineering 0028 (2011) 000–000 u 1 , 2   -.194078 10-1  .201674 10-21  .237279 10 -22  -.202964 10 -3 12 1 -.281443 10-312 -.241072 10-3 2 2 1

(17)

Fig.2 shows the reliability  versus deviation of the modulus of elasticity  E .Results from the higher order response surface could get higher accuracy. Fig.3 shows the convergence analysis by Monte Carlo simulation with a huge number. In comparison, this paper proposed a high-efficiency method based on SRSM, with nine and seventeen points respectively in the second and third order response surface. u 1 ,  2   -.194132 10-1  .198306 10-21  .233624 10 -2 2  -.201848 10 -3 12 1 -.267879 10-31 2   -.240718 10-3  2 2  1  ..200137 10 -4 13  31  

(18)

.256174 10    .244640 10 1 2  .235244 10  2  3 2  -4

2.6

2 1 2

-4

2

2.2

Monte-carlo

2.23

2

2.225

1.8 1.6

β

β

3

2.235

SRSM(2) SRSM(3) Monte-carlo

2.4

-4

2.22

1.4 2.215

1.2 1

2.21

0.8 0.1

0.15

0.2 Modulus of elasticity,σE

0.25

0.3

2.205

0

50

100

150 200 250 Number of samples(×1000)

300

350

Fig.2  versus  E (left) Fig.3 The convergence analysis of MCFEM (right)

5. Conclusion Based on stochastic response surface method, the paper carries on the structural reliability study and makes comparison with Monte-Carlo simulation .The key advantage of SRSM is that the collocation points in the proposed method are selected for minimizing the mean square error, and from high probability regions, thus leading to fewer function evaluations for high accuracy. Comparison studies with the Monte-Carlo simulation suggest that the proposed method gives results of comparable accuracy at a lower computational cost. Acknowledgements The research reported in this paper was supported by youth funds from Nanchang Institute Technology (Contract: No.2010KJ007) and the national natural science funds (Contract: No. 51108227). References [1] Faravelli L. Response-surface approach for reliability analysis. Journal of Engineering Mechanics, 1989, 115(12): 2763-2781. [2] Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 1990, 7: 57-66. [3] Kaymaz I, McMahon C. A response surface method based on weighted regression for structural reliability analysis. Probab. Eng. Mech., 2005, 20: 11-17. [4] Isukapalli SS. Uncertainty analysis of transport-transformation models. PhD thesis, 1999, The State university of New Jersey. [5] Shuping Huang, Sankaran Mahadevan, Ramesh Rebba. Collocation-based stochastic finite element analysis for random field problems. Probabilistic Engineering Mechanics, 2007, 22: 194-205.