ACTA Mechanica-2012

8/2/2019 ACTA Mechanica-2012

1/18

13

Acta Mechanica

ISSN 0001-5970

Volume 223

Number 1

Acta Mech (2012) 223:109-124

DOI 10.1007/s00707-011-0546-3

System reliability analysis of spatialvariance frames based on random field

and stochastic elastic modulus reductionmethod

Lu Feng Yang, Bo Yu & J. Woody Ju


2/18

13

Your article is protected by copyright and

all rights are held exclusively by Springer-

Verlag. This e-offprint is for personal use only

and shall not be self-archived in electronic

repositories. If you wish to self-archive yourwork, please use the accepted authors

version for posting to your own website or

your institutions repository. You may further

deposit the accepted authors version on a

funders repository at a funders request,

provided it is not made publicly available until

12 months after publication.


3/18

Acta Mech 223, 109124 (2012)DOI 10.1007/s00707-011-0546-3

Lu Feng Yang Bo Yu J. Woody Ju

System reliability analysis of spatial variance framesbased on random field and stochastic elastic modulusreduction method

Received: 5 April 2011 / Published online: 4 October 2011 Springer-Verlag 2011

Abstract This paper presents the stochastic elastic modulus reduction method for system reliability analysisof spatial variance frames based on the perturbation stochastic finite element method (PSFEM) and the localaverage of a random field. The stochastic responses and reliability index of each element of a structural frameare characterized by the PSFEM and the first-order second-moment method, to properly handle the correlationstructures and scale of fluctuation of random fields. A strategy of elastic modulus adjustment for the estimationof system reliability is developed to determine the range and magnitude of elastic modulus reduction, by takingthe element reliability index as a governing parameter. The collapse mechanism and system reliability indexof a stochastic framed structure are determined through iterative computations of the PSFEM. Compared withthe failure mode approaches in traditional system reliability analysis, the proposed method avoids two majordifficulties, namely the identification of significant failure modes and estimation of the joint probability offailure modes. The influences of the correlation structure and scale of fluctuation of the random field uponsystem reliability are investigated to demonstrate the accuracy and computational efficiency of the proposed

methodology in system reliability analysis of spatial variance frames.

1 Introduction

It has long been recognized that a fully satisfactory estimation of the reliability of a structure must be basedon a system approach. However, the system reliability analysis of the spatial variance structure presents twomajor difficulties [1]. One is the identification of significant failure modes of a redundant structure; the otheris the estimation of joint failure probability contributed from the significant failure modes. It is very difficultto enumerate all collapse modes of a statically indeterminate structure that has many possible modes or pathsto cause failure. However, among these possible failure modes or paths, only some contribute considerablyto the structural system failure probability while others have low probability of occurrence. To identify theseprobabilistically dominant failure modes or paths, many methods have been developed, such as the incremental

L. F. Yang supported by the Guangxi Lab Center of Science and Technology (LGZX201002).

L. F. Yang (B) B. YuKey Laboratory of Disaster Prevention and Structural Safety of Ministry of Education,School of Civil Engineering and Architecture, Guangxi University, Nanning 530004, ChinaE-mail: [email protected].: +86-771-3236827

J. W. Ju (B)Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095, USAE-mail: [email protected]

J. W. JuChang-Jiang Scholar Chair Professor, Guangxi University, Nanning, China


4/18

110 L. F. Yang et al.

load approach [2], the beta-unzipping method [3], the truncated enumeration method [4], the mathematicalprogramming technique together with the Monte Carlo simulation [5], etc. On the other hand, the evaluation ofjoint failure probability is often intractable even if the potential failure modes can be identified. The numericalintegration methods [6,7] and several approximate methods, such as the point estimation methods [8] and

various bounding techniques [9,10], have been developed in the last three decades.To date, the system reliability analysis has not been widely used in the safety evaluation of complex engi-neering structures, as there still remain many pending problems to be solved. For example, nearly all failuremode approaches in existing system reliability analysis describe the structural parameters and external loadsas random variables, ignoring the spatial variances of these parameters, thus introducing additional modelingerrors. Therefore, a random field should be employed to model and describe random quantities embedded inthe stochastic structures.

Over the past three decades, several methods for the discretization of a random field have been developed[1113]. Among these, the local average method (LAM), which is easy to implement and insensitive to thetypes of correlation structures, has been widely adopted in various stochastic structural analyses [13,14].Typically, the second-order perturbation technique and stochastic variational principle-based perturbation sto-chastic finite element method (PSFEM) [15,16] combined with the LAM provide a powerful tool for stochasticanalysis and reliability evaluation of stochastic structures with small variations.

The plastic limit analysis approach is one of the important bases for structural system reliability analysis.In recent years, the plastic limit analysis method based on various elastic modulus adjustment procedures hasmade considerable advance [1722]. In particular, the elastic modulus reduction method (EMRM) for limitanalysis and safety evaluation of complex structures was developed in recent years [ 2022]. In the EMRM,the element bearing ratio (EBR) and its degree of uniformity are defined to describe the loading state of eachelement as well as the redistribution of internal forces in a structure. We also refer to the elastic-damagestiffness reduction mechanisms and damage mechanics developed by Ju [23,24], Ju and Lee [25] as well as Juet al. [26]. On the basis of linear elastic finite element analysis, numerous sets of statically admissible stressdistributions can be generated. This method was applied to stochastic structures for system reliability analysisof frame structures while random quantities are still described as random variables [27].

In this paper, key random quantities are modeled as random fields which are discretized by the local averagemethod (LAM); the PSFEM is employed to compute stochastic responses while the element reliability indexof the frame structures is produced by the first-order second-moment (FOSM) method. A strategy of elasticmodulus reduction is developed for stochastic structures by defining the degree of uniformity of reliabilityindex (DURI) as well as the reference reliability index (RRI), such that the stochastic elastic modulus reductionmethod (SEMRM) is deployed for the estimation of system reliability of spatial variance structures throughiterative analysis. Compared with traditional failure mode approaches in existing system reliability analysis,the proposed methodology avoids the difficulties in the identification of significant failure modes and theevaluation of joint probability of failure of those significant failure modes. Several numerical examples arepresented to demonstrate the accuracy and computational efficiency of the proposed framework.

2 The local average method for discretization of a random field

Let us start by considering a continuous wide-sense homogeneous isotropic random field {X(x, )} withmultiple components, where is the outcome of the random parameter. Here, x denotes the coordinate vector.The values of the mean and variance of

{X(x, )

}are specified as

X = X1, X2, . . . , XMT ; 2X = 2X1 , 2X2 , . . . , 2XMT , (1)where Xi , Xi and 2Xi denote the mean, standard deviation and variance of the i th component of the randomfield {X(x, )}. There are M components in total. For a given spatial point x0, {X(x0, )} is the random vectorcontaining M random variables. Conversely, for a given outcome 0, {X(x, 0)} represents a realization ofthe random field.

A frame component forms an angle of with the x-axis, as illustrated in Fig. 1. The random parametersconcerning the material, geometry and load can be modeled as a multicomponent random field with two dimen-sions along the planar straight bar component. According to the relationship between the polar coordinate rand the Cartesian coordinates x and y, the random field {X(x, )} canbe denoted as {X(r, )} along the straightcomponent.


5/18

System reliability analysis of spatial variance frames 111

o

Le

y=x tan

y

x

x=y/ tan

rr

Fig. 1 A planar element of a frame structure

We assume that the random field is discretized into N elements; the local average of {X(r, )} over theelement e with the length Le yields a set of random variables:

{X}e =Xe1, X

e2, . . . , X

eM

T, (2)

where Xe

i denotes the local average of the i th component of{X(r, )}, Xi (r, ), over the element e, whichcan be defined as:

Xei =1

Le

Le

Xi (r, ) dr; e = 1, 2, . . . , N, i = 1, 2, . . . , M. (3)

The mean and variance of the local average Xei are:

EXei

= Xi , (4)Var

Xei

= EXei Xi2 = ii (Le) 2Xi , (5)where ii (Le) represents the variance function of Xi (r, ) which can be defined as [14]

ii (Le) =2

Le

Le

1 r

Le

ii (r)dr; (6)

here, ii (r) is the autocorrelation function of Xi (r, ).For random variables Xei and X

ej evaluated by local averaging over the element e, the covariance of the

two variables is [14]

Cov

Xei , Xej

= Xi Xj i j (Le), (7)

where i j (Le) is the variance function of Xi (r, ) and Xj (r, ):

i j (Le) =2

Le

Le

1 r

Le

i j (r)dr, (8)

where i j (r) represents the correlation function between Xi (r, ) and Xj (r, ).

For random variables Xi and X

j achieved by local averaging of the ith and j th components Xi (r, ) and

Xj (r, ) over two arbitrarily situated planar linear elements and , respectively, as exhibited in Fig. 2, the

covariance of Xi and X

j can be defined as follows [14]:

Cov

Xi , X

j

= Xi Xj

LL

L

L

i j (r)dl ds. (9)


6/18


y

x

yk

l

s 0

0

o

p

yp

yk

xpxk xp xk

A

L

L

B

r

Fig. 2 Two arbitrarily situated planar elements and

Here, L and L are the lengths of elements and ; r represents the distance from point A with the coor-

dinates (x, y) located in the element to point B with the coordinatesx , y

located in the element .

If the elements and are not parallel to the y-axis, one has

y = yp + x xp tan 0; y = yp + x xp tan 0, (10a)where

tan 0 =yk ypxk xp

; tan 0 =yk ypxk xp

. (10b)

Here, the coordinatesxp, yp

, (xk , yk),

xp , yp

and

xk , yk

denote the ends of the elements

and , respectively. Accordingly, we arrive at

ds = dx /cos 0; dl = dx /cos 0. (11)The distance r between points A and B is

r = x xp tan 0 x xp tan 0 + yp yp2 + x x2. (12)Substitution of Eq. (12) into Eq. (9) leads to

Cov

Xi , X

j

= Xi Xj

L L cos 0 cos 0

L

L

i j (r) dx dx. (13)

If both the elements and are parallel to the y-axis, the distance r between points A and B is

r =

y y2 + xp xp2. (14)

Then, the covariance of Xi and X

j gives

Cov

Xi , Xj

= Xi Xj

L L

L

L

i j (r)dy dy. (15)

Further, if the element is parallel to the y-axis, but the element is not, one expresses the distance r as

r =

yp +x xp

tan 0 y

2 + x xp2. (16)The covariance of Xi and X

j then takes the form

Cov

Xi , X

j

= Xi Xj

LL cos 0

L

L

i j (r) dx dy. (17)


7/18


Similarly, if the element is parallel to the y-axis, while the element is not, the distance r becomes

r =y

yp +

x xp

tan 0

2 +

xp x

2

. (18)

Accordingly, the covariance of Xi and Xj reads

Cov

Xi , X

j

= Xi Xj

LL cos 0

L

L

i j (r) dy dx. (19)

All the random variables Xei (e = 1, 2, . . . , N; i = 1, 2, . . . , M), as displayed in Eq. (2), can be assembledinto a vector:

{X} = X1, X2, . . . , XnT , (20)where n = M N denotes the number of random variables resulting from discretization of the random fields.

3 The stochastic finite element method

The functional of potential energy of a linear structural system of continuum reads [15]:

= 1

2

V

{}T[D]{}dV S

{u}T{F}dS, (21)

where {} is the strain vector, [D] is the elasticity matrix, {u} is the displacement vector, {F} denotes the bound-ary traction vector, and V denotes the domain of interest. In addition, S signifies the boundary of the domain V,which can be simply divided into two parts: the traction boundary S and the prescribed displacement boundary

Su . Based on the second-order perturbation techniques [15], one can obtain the zeroth-, first- and second-orderexpanded expressions, respectively, of the functional of potential energy:

= 12

V

{}T D {}dV S

{u}T F dS, (22a)i =

1

2

V

{}T[D]i {} + 2{}T

D {}i dV S

{u}T{F}i {u}Ti

F

dS, (22b)

i j =1

2

V

{}T[D]i j {} + 2{}Ti

D {}j + 2{}T[D]i {}j + 2{}T[D]j {}i+ 2{}T D {}i j dV

S

{u}T{F}i j + {u}Ti j F+ {u}Ti {F}j + {u}Tj {F}i dS, (22c)where i, j = 1, 2, . . . , n. Moreover, () signifies the value of () evaluated at the mean of the random field;()i denotes the first-order partial derivatives of () with respect to the i th component of the random fieldXi , evaluated at the mean of the random field; ()i j signifies the second-order partial derivatives of () withrespect to Xi and X

j , evaluated at the mean of the random field.

According to the stochastic variational principle [16], only the highest order expansion of the functional,that is i j herein, is necessary and sufficient to derive all governing equations of the perturbation stochasticfinite element method (PSFEM):

K {a} = {P}; K {a}i = {P}i [K]i {a}; K {a}II = {P}II, (23)


8/18


where

{a}II =1

2

ni=1

nj=1

{a}i j Cov

Xi , Xj

, (24a)

{P}II =1

2

ni=1

nj=1

{P}i j 2[K]i{a}j [K]i j {a} CovXi , Xj . (24b)Here, [K], {a} and {P} denote the structural stiffness matrix, nodal displacement vector and the equivalentnodal force vector, respectively.

Once the nodal displacement vectors {a}, {a}i and {a}II are obtained from Eqs. (23) and (24), the meanand variance of the internal force vector {P}e of the element e can be expressed as:

E{P}e = Ke {a}e + {a}eII Pde {P}eII + 12

ni=1

nj=1

{P}ei j Cov(Xi , Xj ), (25)

Var {P}e =n

i=1

n

j=1

CovXi

, Xj K

e

{a}

e

j + [K

]e

j{ a}

e

{Pd

}e

jT K

e

{a}

e

i + [K

]e

i { a}

e

{Pd

}e

i ,

(26)

where

Pde

and {Pd}ei are the equivalent nodal force vector and its first-order partial derivatives with respectto Xi , evaluated at the mean { X}.

4 The SEMRM for system reliability analysis

In this section, the stochastic elastic modulus reduction method (SEMRM) is adopted for system reliabilityevaluation of structural frames. According to probabilistic structural analysis [1,3,27], it is not necessarilytrue that only the element with the lowest reliability index would fail at each stage during the failure processfor a stochastic structural system. However, it is inevitable that the elements with reliability indices lower

than a certain threshold will be damaged, and one or more of those elements would fail. Consequently, thedistribution of internal forces keeps changing such that the redistribution of internal forces often occurs in theprocess from damage to failure for a ductile structural system.

Here, the elastic moduli of elements with reliability index lower than the reference value will be reduced inorder to simulate the structural damage and the redistribution of internal forces in a stochastic structural systemthrough an iterative analysis. In each iterative step, the reliability index of each element can be approximatelyestimated by the first-order second-moment (FOSM) method as follows:

i =Ri Si

2Ri+ 2Si

, (27)

where Si and 2Si

are the mean and variance of load effects of the element i achieved by the PSFEM and the

LAM. Moreover, Ri and

2

Ri are the mean and variance of resistance of the element i , which can be evaluatedaccording to the material and geometric parameters of the element i .The reference reliability index (RRI), 0k, for the elastic modulus reduction in the kth iteration can be

defined as

0k = mink +

maxk mink dk, (28)

where dk signifies the degree of uniformity of reliability index (DURI) in the kth iteration:

dk =k + minkk + maxk

(29)

in which maxk , mink and k denote the maximum, minimum and the average of the elements reliability

indices in the kth iteration, respectively.


9/18


The elastic modulus of the element with reliability index lower than the RRI should be reduced in the kthiteration according to the following expression:

Ee

k+1 = Eek

ek

0

k

q

for ek < 0k

Eek for ek

0k(30)

where ek denotes the reliability index of the element e in the kth iteration; Eek and E

ek+1 signify the elastic

moduli of the element e in the kth and the next iteration, respectively. Further, q represents the adjustmentfactor for the reduction of elastic modulus and may take an arbitrary value in the closed interval [0.1, 3.0], asdemonstrated later in our numerical examples. The iterative procedure will stop when dk meets the followingcriterion for convergence:

|dk dk1|dk

; k 2, (31)

where represents the prescribed admissible error. In what follows, we adopt 1.0 106 for this value.If the criterion for numerical convergence described in Eq. (31) is satisfied after Nf iterations, it indicates

that the entire structural system is at the plastic limit state and may fail as soon as a single element fails. Sincethe potential failure element is limited to the elements with reliability indices lower than the RRI, the failureinteraction of the frame with n elements can be modeled by a series system with M1 elements, where M1denotes the numbers of the potential failure elements. Therefore, the probability of failure of the structuralsystem can be defined as:

Pf s = P

U1 U2 UM1

, (32)

where Ui (i = 1, 2, . . . , M1) defines the event that the element i would fail. The system reliability index cannow be derived from Pf s as follows:

S = 1(Pf s ), (33)

where 1(

) is the inverse standard normal cumulative distribution function.

Many methods were developed for estimating the joint probability of failure in Eq. (32), including thefirst-order bounding method (FOBM) [28], the second-order bounding method (SOBM) [29], the third-orderbounding method (TOBM) [30], the equivalent correlation coefficient method (ECCM) [8] and the probabilis-tic network evaluation technique (PNET) [31]. Each one of these methods can be employed to characterize thejoint probability of failure of the structural frame; then, the system reliability index can be derived by Eq. (33)accordingly.

5 Numerical examples

5.1 Portal frame

The geometry and loading conditions of a portal frame are illustrated in Fig. 3, with the elastic modulus of

2.0 105

MPa and carrying the stochastic external loads F1 and F2. The black dots denote the locations ofpotential plastic hinges, and the numbers in the circles signify the serial numbers of elements. The cross-sectional area and moment of inertia of beam and column elements are modeled as random variables andrepresented by A and I, respectively. The plastic resistances of elements are described as random variablesand denoted by Mb and Mc, respectively. The statistical characteristics of all the random variables are listedin Table 1.

The proposed SEMRM is employed to calculate the system reliability of the stochastic frame structures.The serial numbers of the potential failure elements and their probabilities of failure as well as the elementreliability indices are listed in Table 2, in which the adjustment factor q takes the value of 0.6.

In the process of the iterative computation of the SEMRM, the redistribution of reliability indices is sim-ulated by adjusting the elastic modulus of each element, as exhibited in Fig. 4. It is apparent that the initialreliability indices of element 1 and element 2 are higher than those of the others and keep decreasing duringthe iterations. By contrast, the initial reliability indices of element 3 and element 4 are lower than those of the


10/18


1

2 3

4

F1

F2

3000mm

3000mm3000mm

x

y

o

Fig. 3 The geometry and loading conditions of a portal frame

Table 1 Statistical characteristics of random variables

Variable Mean value Coefficient Distributionof variation type

A 1.3500

105 mm2 0.05 Normal

I 2.2781 109 mm4 0.05 NormalMb 3.7969 109 MPa 0.05 NormalMc 4.3664 109 MPa 0.05 NormalF1 3.3000 106 N 0.10 NormalF2 2.5000 106 N 0.10 Normal

Table 2 Reliability index and failure probability of potential failure element

Element Reliability Failure probabilityindex

1 3.091851 9.945633 1042 3.091984 9.941178 1043 3.091835 9.946169 104

4 3.091892 9.944260 104

0 10 20 30 401

2

3

4

5

6

7

Iteration step

Elementreliabilityindex

element 1

element 2

element 3

element 4

Fig. 4 The iterative process of element reliability index

others but keep increasing during the iterations. The reliability indices of all elements are adjusted to the samelevel in about 20 steps of iterations; this behavior shows that the system reliability of the frame depends on thereliability level of all elements. Moreover, it implies that the collapse mechanism of the frame is an integralfailure mode.

In the process of iterations, the DURI increases gradually as displayed in Fig. 5 and converges to approx-imately 1.0 in about 20 steps of iterations. This implies that the reliability indices of all elements in the frame


11/18


0 10 20 30 400.45

0.65

0.85

1.05

Iteration step

DURI

Fig. 5 The iterative process of the DURI

0 10 20 30 403.0

3.2

3.4

3.6

3.8

Iteration step

RRI

Fig. 6 The iterative process of the RRI

Table 3 System reliability index and failure probability of the portal frame by the SEMRM

Method Reliability index Failure probability

FOBM [2.654456, 3.091835] [9.946169 104, 3.971815 103]SOBM [3.091835, 3.091851] [9.945633 104, 9.946169 104]TOBM [3.091835, 3.091851] [9.945633 104, 9.946169 104]ECCM 3.091851 9.945631 104

are adjusted to the same level when the structural failure occurs, as demonstrated in Fig. 4. The RRI decreasesalong with the iterations, as shown in Fig. 6, and converges to approximately 3.09 in about 20 steps of iterations.

Based on Eq. (32) and the reliability of the potential failure elements listed in Table 2, the system reliabilityindex as well as the joint probability of failure of the frame system can be estimated by means of the FOBM,the SOBM, the TOBM and the ECCM. The corresponding results are listed in Table 3. We observe that thesystem reliability index predicted by the SEMRM is approximately 3.09.

In order to validate the precision and computational efficiency of the SEMRM, the traditional failure modeapproach (FMA) is employed to evaluate the system reliability of the frame. Eight significant failure modes ofthe frame structure are identified according to the FMA as well as the limit state functions, reliability indicesand corresponding failure probabilities, as listed in Table 4. The system reliability index of the frame canbe obtained from the joint probability of the eight failure modes, which can be estimated by the FOBM, theSOBM, the TOBM, the ECCM and the PNET. For comparison, the Monte Carlo simulation (MCS) techniqueis also employed with 5 million cycles. These system reliability results are summarized in Table 5, which shows


12/18


Table 4 Significant failure modes of the portal frame

Limit state function Reliability Failure probabilityindex

Z1

=4Mb

+2Mc

3000F1

3000F2

=0 3.088169 1.006970

103

Z2 = 2Mb + 4Mc 3000F1 3000F2 = 0 3.572171 1.770171 104Z3 = 2Mb + 2Mc 3000F1 = 0 5.008424 2.743877 107Z4 = Mb + 3Mc 3000F1 = 0 5.375582 3.816785 108Z5 = 4Mc 3000F1 = 0 5.730992 4.992249 109Z6 = 4Mb 3000F2 = 0 7.202758 2.950332 1013Z7 = 3Mb + Mc 3000F2 = 0 7.590942 1.587940 1014Z8 = 2Mb + 2Mc 3000F2 = 0 7.962255 8.446594 1016

Table 5 System reliability index and failure probability of the portal frame by the FMA

Method Reliability index Failure probability

FOBM [3.039676, 3.088169] [1.006969 103, 1.184165 103]SOBM [3.074761, 3.082958] [1.024771 103, 1.053355 103]TOBM [3.077451, 3.077467]

[1.043839

10

3, 1.043897

10

3

]ECCM 3.051625 1.138031 103PNET 3.088084 1.007259 103MCS 3.089402 1.002800 103

L2

L1

L1

F1

F2

F3

2F3

1

2

3 4

5

6

7 8

xo

Fig. 7 The geometry and load conditions of a one-bay two-story frame

that the system reliability indices rendered by the TOBM and the ECCM are somewhat lower than those bythe SEMRM, while the PNET and the MCS achieve results agreeing well with those by the SEMRM. As theMCS can be regarded as a benchmark, the accuracy of the proposed SEMRM is demonstrated by this portalframe example.

5.2 One-bay two-story frame

Example 2 considers a one-bay two-storyframe exhibited in Fig. 7, loaded by four stochastic external loads withamplitudes F1, F2, F3 and 2F3; their mean values and coefficients of variation are tabulated in Table 6. Theelastic modulus E, story height L1 and span length L2 are all deterministic: E = 2.1105 MPa, L1 =3,000mmand L2 =4,800mm. The beams cross-sectional area, moment of inertia and plastic resistance are all randomvariables and represented by Ab, Ib and Mb, respectively. Similarly, the columns cross-sectional area, momentof inertia and plastic resistance are also random variables and represented by Ac, Ic and Mc, respectively. Theirstatistical characteristics are displayed in Table 6. The black dots in Fig. 7 denote the locations of potentialplastic hinges, and the numbers in the circles signify the serial numbers of elements.

In order to provide a benchmark to verify the computational effectiveness of the proposed SEMRM,the traditional failure mode approach (FMA) is employed to estimate the system reliability of the frame.Eight significant failure modes of the frame structure are identified according to the FMA. The limit state


13/18


Table 6 Statistical characteristics of random variables

Variable Mean value Coefficient Distributionof variation type

Ab 1.2000

105 mm2 0.10 Normal

Ac 1.2250 105 mm2 0.10 NormalIb 1.6000 109 mm4 0.10 NormalIc 1.2505 109 mm4 0.10 NormalMb 3.6000 109 MPa 0.10 NormalMc 3.2156 109 MPa 0.10 NormalF1 1.6000 106 N 0.15 NormalF2 3.2000 106 N 0.15 NormalF3 1.3000 105 N 0.15 Normal

Table 7 Significant failure modes of the one-bay two-story frame

Limit state function Reliability Failure probabilityindex

Z1=

4Mb

0.5F2L2=

0 3.644054 1.341890

104

Z2 = 4Mc + 2Mb 0.5F2L2 = 0 5.352326 4.341540 108Z3 = 4Mc + 6Mb 0.5F1L2 0.5F2L2 4F3L1 = 0 5.485902 2.056830 108Z4 = 4Mc + 3Mb 0.5F2L2 3F3L1 = 0 5.497299 1.928260 108Z5 = 6Mc + 2Mb 0.5F2L2 3F3L1 = 0 5.988532 1.058720 109Z6 = 8Mc + 2Mb 0.5F2L2 4F3L1 = 0 6.665395 1.319770 1011Z7 = 4Mc + 4Mb 0.5F1L2 4F3L1 = 0 7.719356 5.845940 1015Z8 = 4Mc 3F3L1 = 0 9.006916 1.059660 1019

Table 8 System reliability index and failure probability of the one-bay two-story frame by the FMA

Method System reliability index System failure probability

FOBM [3.643975, 3.644054] [1.341886 104, 1.342297 104]SOBM [3.644005, 3.644025] [1.342038 104, 1.342140 104]

TOBM [3.643996, 3.644011] [1.342111 10

4,1

.342191 10

4

]ECCM 3.644089 1.341701 104PNET 3.643975 1.342297 104

Table 9 Element reliability index with different adjustment factors

q Number Reliability Failure probabilityindex

0.5 7 3.644055 1.341880 1048 3.644041 1.341953 104

1.5 7 3.644049 1.341911 1048 3.644050 1.341906 104

2.5 7 3.644048 1.341917 1048 3.644051 1.341901 104

function, reliability index and the corresponding probability of failure for each failure mode are summarizedin Table 7. Accordingly, the system reliability index of the frame and the probability of system failure canbe estimated by the FOBM, the SOBM, the TOBM, the ECCM and the PNET; these results are presented inTable 8. It is observed that the system reliability index predicted by the traditional system reliability methodsis approximately 3.644.

It should be pointed out that the random parameters of the frame must be modeled as random variables inthe FMA. To facilitate the comparison of the system reliability results between the FMA and the SEMRM, theassumption of random variables for all the random parameters remains effective here when adopting the SEM-RM to evaluate the system reliability of the stochastic two-story frame structure example. The serial numbersof potential failure elements and the probabilities of failure as well as the corresponding reliability indices arepresented in Table 9. Here, the adjustment factor q in Eq. (30) takes the values of 0.5, 1.5 and 2.5, respectively.


14/18


0 5 10 15 202.5

4.5

6.5

8.5

Iteration step

Elementreliab

ilityindex

element 1

element 2

element 3

element 4

element 5

element 6

element 7

element 8

Fig. 8 The iterative process of element reliability index

0 5 10 15 200.60

0.65

0.70

0.75

0.80

Iteration step

DURI

Fig. 9 The iterative process for the DURI

We observe that the potential failure elements are always element 7 and element 8 in three different numericalcases; the respective reliability indices of the corresponding elements are approximately 3.644. It is clear thatthe value of the adjustment factor q has little effect on the accuracy of the SEMRM.

In the process of numerical iterations for the SEMRM, the redistribution of element reliability indices ofthe frame is simulated by adjusting the elastic modulus of each element, as illustrated in Fig. 8. Here, theadjustment factor q takes the value of 2.0 without loss of generality. We observe that the initial reliabilityindices of element 5 and element 6 are higher than those of other elements and continue decreasing duringsubsequent iterations. By contrast, the initial reliability indices of element 7 and element 8 are less than thoseof other elements and continue increasing during subsequent iterations though they remain significantly lowerthan others during the entire process of iterations. Therefore, the system reliability of the two-story frame

is largely dependent upon the reliability level of element 7 and element 8; the collapse mechanism of framesystem features a local failure mode. In the process of iterations, the DURI gradually increases and convergesto approximately 0.746 in about 17 steps of iterations using q = 2.0; see Fig. 9. Moreover, the RRI decreasesduringthe iterations as exhibited in Fig. 10 and converges to approximately 5.994 in about 17 steps of iterations.

Based on Eq. (32) and the reliability of the potential failure elements listed in Table 9, the system reliabilityindex as well as the joint probability of failure of the frame can be estimated by means of the FOBM, theSOBM, the TOBM and the ECCM. The corresponding numerical results are rendered in Table 10, whichshows that the system reliability index produced by the SEMRM is approximately 3.644 in all three cases ofq = 0.5, q = 1.5, and q = 2.5. Comparison of the numerical results by the SEMRM in Table 10 with those bythe FMA in Table 8 demonstrates the accuracy and computational effectiveness of the proposed methodology.

In order to investigate the influence of the adjustment factor q on the numerical results by the SEMRM, thesystem reliability index s and corresponding iterative steps are estimated with q varying from 0.1 to 3.0. Thecorresponding numerical results are summarized in Table 11. It is apparent that s remains almost constant,


15/18


0 5 10 15 205.9

6.0

6.1

6.2

6.3

Iteration step

RRI

Fig. 10 The iterative process for the RRI

Table 10 System reliability index and failure probability of the one-bay two-story frame by the SEMRM

q Method System reliability index System failure probability

0.5 FOBM [3.461748, 3.644041] [1.341953 104, 2.683401 104]SOBM [3.538277, 3.642846] [1.348199 104, 2.013736 104]TOBM [3.642585, 3.642585] [1.349568 104, 1.349568 104]ECCM 3.644089 1.341701 104

1.5 FOBM [3.461748, 3.644049] [1.341912 104, 2.683401 104]SOBM [3.537899, 3.642564] [1.349677 104, 2.016621 104]TOBM [3.442242, 3.642242] [1.351368 104, 1.351368 104]ECCM 3.644089 1.341701 104

2.5 FOBM [3.461748, 3.644048] [1.341917 104, 2.683401 104]SOBM [3.537937, 3.642565] [1.349676 104, 2.016334 104]TOBM [3.442242, 3.642242] [1.351367 104, 1.351367 104]ECCM 3.644089 1.341701

104

while the number of iterative steps decreases as q varies from 0.1 to 3.0. Therefore, q has negligible effecton the accuracy of the SEMRM, but much on its computational efficiency. The higher the adjustment factorq is, the fewer the number of iterations is for the SEMRM. Furthermore, the numerical results converge in13 iterative steps (or less) when q 2.5, thus demonstrating the computational efficiency of the proposedSEMRM methodology.

5.3 Spatial variance of random field

As random parameters always vary with respect to time or space, they should be modeled as stochastic pro-cesses or random fields rather than random variables. In order to investigate the influence of spatial variances of

structural random parameters in Example 2 upon the system reliability, all cross-sectional areas and momentsof inertia listed in Table 6 are described as independent Gaussian random fields with wide-sense homogeneousand isotropic properties. According to the principle of stochastic process, both the scale of fluctuation andcorrelation structure of the random fields might affect the outcomes of stochastic analysis. Therefore, fourtypes of correlation structures of random fields are considered in the following:

1. The triangular correlation function (TCF) i j () and its corresponding variance function i j () take theform:

i j (r) =

1 |r|i j

, |r| i j0, |r| > i j

; i j (Le) =

1 Le3i j , Le i ji jLe

1 i j3Le

, Le > i j

(34)

in which i j represents the scale of fluctuation.


16/18


Table 11 System reliability index with different adjustment factors

q s Iterations

0.1 3.651855 2740.2 3.644089 145

0.3 3.644089 1010.4 3.644089 780.5 3.644089 640.6 3.644089 530.7 3.644089 460.8 3.644089 410.9 3.644089 371.0 3.644089 331.1 3.644089 301.2 3.644089 281.3 3.644089 251.4 3.644089 241.5 3.644089 221.6 3.644089 211.7 3.644089 201.8 3.644089 18

1.9 3.644089 172.0 3.644089 172.1 3.644089 162.2 3.644089 152.3 3.644089 142.4 3.644089 142.5 3.644089 132.6 3.644089 132.7 3.644089 122.8 3.644089 122.9 3.644089 123.0 3.644089 11

2. The exponential correlation function (ECF) and its corresponding variance function are:

i j (r) = exp

2|r|i j

; i j (Le) =

2i j

2L2e

2Le

i j 1 + exp

2Le

i j

. (35)

3. The second-order autoregressive correlation function (SACF) and its corresponding variance function read:

i j (r) =

1 + 4|r|i j

exp

4|r|

i j

, (36a)

i j (Le) =i j

2Le

2 + exp

4Le

i j

3i j

4Le

1 exp

4Le

i j

. (36b)

4. The Gaussian correlation function (GCF) and its corresponding variance function are:

i j (r) = exp

r2

2i j

; i j (Le) =

2i j

L2e

Le

i jer f

Le

i j

+ exp

L

2e

2i j

1

, (37)

where erf() denotes error function and can be expressed as

erf(x) = 2

x0

exp(u2)du. (38)

For simplicity, we assume that all cross-sectional areas and moments of inertia have the same types of corre-lation structure and the same scale of fluctuation . The results produced by the SEMRM corresponding toabove four types of correlation structures, including the TCF, ECF, SACF and GCF, are presented in Table 12,


17/18


Table 12 System reliability index via the scale of fluctuation and correlation structures

( mm) System reliability index

TCF ECF GCF SACF

1,500 3.581844 3.592637 3.580751 3.5847803,000 3.584596 3.595593 3.578759 3.5851494,500 3.607544 3.605251 3.597614 3.5988726,000 3.624785 3.614230 3.617531 3.6131047,500 3.635650 3.622135 3.633223 3.6253189,000 3.643291 3.628000 3.644891 3.635096

with varying from 1,500 to 9,000 mm. It is apparent that the system reliability indices in accordance withfour different types of correlation structures agree very well with one another. That is, the system reliabilityachieved by the SEMRM based on the PSFEM and the LAM is essentially insensitive to the type of correlationstructure of the random field. Moreover, it can be readily seen that the system reliability index is smaller whenthe structural parameters are modeled as random fields than that as random variables. Evidently, the systemreliability index increases along with the scale of fluctuation of the random fields and approaches gradually tothe value when the structural parameters are modeled as random variables. This implies that the assumption

of random variables for all random parameters of the frame structure overestimates the system reliability ofthe frame and might lead to unsafe design of structures.

6 Conclusions

Based on the local averaging of random fields, perturbation stochastic finite element method and the elasticmodulus reduction method for limit analysis of structures, this paper presents the stochastic elastic modu-lus reduction method (SEMRM) for system reliability analysis of spatial variance frame. Several numericalexamples demonstrate the computational effectiveness and accuracy of the proposed SEMRM methodology.

When the random field approach is employed to describe the random parameters of the stochastic structure,the system reliability index predicted by the SEMRM is smaller than that when the random variable approachis utilized. The system reliability index increases along with the scale of fluctuation of the random fields and

approaches gradually to the value when the random variable model is used.Comparing the SEMRM with the FMA and other traditional system reliability approaches, the proposed

methodology manifests considerable advantages as it does not require the identification of significant failuremodes. Moreover, the estimation of joint probability of failure elements in the SEMRM is much simpler thanthe estimation of joint probability of failure modes in the traditional system reliability methods. Therefore,the SEMRM is superior in computational efficiency and simplicity in comparison with the existing systemreliability methods.

Acknowledgments This research was supported by the National Natural Science Foundation of China (50768001), the ChinaScholarship Council Postgraduate Scholarship Program (2008666001) and the Guangxi Natural Science Foundation (0991020Z,0992028-7).

References

1. Chen, J.B., Li, J.: Development process of nonlinearity based reliability evaluation of structures. Probab. Eng.Mech. 22(3), 267275 (2007)

2. Moses, F.: System reliability developments in structural engineering. Struct. Saf. 1(1), 313 (1982)3. Thoft-Christensen, P.: Consequence modified -unzipping of plastic structures. Struct. Saf. 7(24), 191198 (1990)4. Melchers, R.E., Tang, L.K.: Dominant failure modes in stochastic structural systems. Struct. Saf. 2(2), 127143 (1984)5. Corotis, R.B., Nafday, A.M.: Application of mathematical programming to system reliability. Struct. Saf. 7(24),

149154 (1990)6. Yuan, X.X., Pandey, M.D.: Analysis of approximations for multinormal integration in system reliability computation. Struct.

Saf. 28(4), 361377 (2006)7. Nadarajah, S.: On the approximations for multinormal integration. Comput. Ind. Eng. 54(3), 705708 (2008)8. Thoft-Christensen, P., Dalsgrd Srensen, J.: Reliability of structural systems with correlated elements. Appl. Math.

Model 6(3), 171178 (1982)9. Zhang, Y.C.: High order reliability bounds for series systems and application to structural systems. Comput.

Struct. 46(2), 381386 (1993)


18/18


10. Greig, G.L.: An assessment of high-order bounds for structural reliability. Struct. Saf. 11(34), 213225 (1992)11. Stefanou, G.: The stochastic finite element method: past, present and future. Comput. Method. Appl. M. 198(912),

10311051 (2009)12. Spanos, P.D., Ghanem, R.: Stochastic finite element expansion for random media. J. Eng. Mech. ASCE 115(5),

10351053 (1989)

13. Vanmarcke, E., Shinozuka, M., Nakagiri, S.: Random fields and stochastic finite elements. Struct. Saf. 3, 143166 (1986)14. Knabe, W., Przewlcki, J., Rzynski, G.: Spatial averages for linear elements for two-parameter random field. Probab. Eng.Mech. 13(3), 147167 (1998)

15. Yang, L.F., Li, Q.S., Leung, A.Y.T., Zhao, Y.L., Li, Q.G.: Fuzzy variational principle and its applications. Eur. J. Mech. ASolid 21, 9991018 (2002)

16. Yang, L.F., Leung, A.Y.T., Yan, L.B., Wong, C.W.Y.: Stochastic spline Ritz method based on stochastic variational princi-ple. Eng. Struct. 27, 455462 (2005)

17. Hamilton, R., Boyle, J.T.: Simplified lower bound limit analysis of transversely loaded thin plates using generalised yieldcriteria. Thin Wall. Struct. 40(6), 503522 (2002)

18. Adibi-Asl, R., Fanous, I.F.Z., Seshadri, R.: Elastic modulus adjustment proceduresimproved convergence schemes. Int.J. Pres. Ves. Pip. 83(2), 154160 (2006)

19. Chen, L.J., Liu, Y.H., Yang, P., Cen, Z.Z.: Limit analysis of structures containing flaws based on a modified elastic compen-sation method. Eur. J. Mech. A Solid 27(2), 195209 (2008)

20. Yang, L.F., Yu, B., Qiao, Y.P.: Elastic modulus reduction method for limit load evaluation of frame structures. Acta Mech.Solida Sin. 22(2), 109115 (2009)

21. Yu, B., Yang, L.F.: Elastic modulus reduction method for limit analysis of thin plate and shell structures. Thin Wall.Struct. 48, 291298 (2010)

22. Yu, B., Yang, L.F.: Elastic modulus reduction method for limit analysis considering initial constant and proportionalloadings. Finite Elem. Anal. Des. 46(12), 10861092 (2010)

23. Ju, J.W.: On energy-based coupled elastoplastic damage theories: constitutive modeling and computational aspects. Int.J. Solids Struct. 25(7), 803833 (1989)

24. Ju, J.W.: Isotropic and anisotropic damage variables in continuum damage mechanics. J. Eng. Mech., ASCE 116(12),27642770 (1990)

25. Ju, J.W., Lee, H.K.: A micromechanical damage model for effective elastoplastic behavior of partially debonded ductilematrix composites. Int. J. Solids Struct. 38(3637), 63076332 (2001)

26. Ju, J.W., Ko, Y.F., Ruan, H.N.: Effective elastoplastic damage mechanics for fiber reinforced composites with evolutionarypartial fiber debonding. Int. J. Damage Mech. 17(6), 493537 (2008)

27. Yang, L.F., Yu, B., Mo, Y.C.:Structural system reliability analysis based on the elastic modulus reduction method. J. GuangxiUniv. 33(3), 350353 (2008) (in Chinese)

28. Cornell, C.A.: Bounds on the reliability of structural systems. J. Struct. Div., ASCE 93(1), 171200 (1967)29. Ditlevsen, O.: Narrow reliability bounds for structural systems. J. Struct. Mech. 7(4), 453472 (1979)30. Feng, Y.: A method for computing structural system reliability with high accuracy. Comput. Struct. 33(1), 15 (1989)31. Ang, A.H.S., Ma, H.F.: On the reliability analysis of framed structures. In: Proceedings of 4th ASCE Speciality Conference

on Probabilistic Mechanics and Structural Reliability, Tucson, pp. 106111 (1979)

Date post:	06-Apr-2018
Category:	Documents
Upload:	wzhang8
View:	218 times
Download:	0 times

ACTA Mechanica-2012

Documents