Fast failure mode identification for ductile structural system reliability

Structural

safety

ELSEVIER

Structural Safety 13 (1994) 207-226

Fast failure mode identification for ductile structural system reliability ** Q. X i a o and S. M a h a d e v a n * Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA

Abstract

This paper presents a method for fast plastic hinge imposition and system collapse mode identification in large ductile framed structures. The proposed method follows the general scheme of the truncated enumeration method, but makes use of the statistical correlations between different member limit states to impose groups of hinges at each selection level instead of one plastic hinge at a time as used in the previous studies, thus making the search for significant failure modes very rapid. This results in an efficient and accurate method for system reliability estimation. The proposed method is illustrated with the help of numerical examples.

Key words: Failure probability; Limit states; Enumeration; Plastic hinges; Random variables; Statistical correlation; System reliability; Failure sequences; Failure modes; Mechanisms

I. Introduction

The estimation of failure probability of structural systems has been an active area of research for more than two decades. During this period, efficient procedures have been developed for reliability evaluation of individual limit states in large structures. However, the computation of system reliability which is affected by many interacting limit states presents considerable difficulty and expense. The search for computationally efficient procedures to estimate system reliability has resulted in several approaches. These include enumeration-type techniques [1-3], and other methods such as the stable configuration approach (e.g. [4]), and mathematical programming techniques (e.g. [5]). Among the enumeration techniques, the truncated enumeration method [2] provided a systematic and rational derivation and generalized earlier methods in this category, such as the branch and bound method [6], incremental load method [1,7-9], etc. This paper proposes a technique which follows the general scheme of the truncated * Corresponding author. ** Discussion in open until October 1994 (please submit your discussion paper to the Editor, Ross B. Corotis). 0167-4730/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0167-4730(93)E0030-F

208

~), Xia~J, ?~. Mahadci an

Structur./ Suh'(v 13 (19041 207 226

enumeration method, but makes use of statistical correlations among component failures t due to the existence of common loads, cross sections, and material properties among different members) to speed up the enumeration. This is found to result in substantial savings in computation, thus making system reliability analysis practically feasible for large structt~e~.

2. Ductile structural system reliability Several commonly used assumptions are applied here: (1) Elasto-plastic framed structures are considered. The failure of a section means the imposition of a plastic hinge and an artificial m o m e n t at this section. (2) The structural uncertainties are represented by considering only the moment capacities as random variables. (3) The order of loads and loading paths are not considered. (4) Geometrical second-order and shear effects are neglected. The effect of axial forces on the reduction of m o m e n t capacities are also neglected. Based on the upper-bound theorem of plasticity [10], failure of a ductile frame is defined as the formation of a kinematically admissible mechanism due to the formation of plastic hinges at a certain number of sections. The mechanisms can be identified from the structural analysis when the total stiffness matrix becomes singular. The technique of automatic generation of safety margins [11] is used in this paper to generate the performance functions of the failure events in the failure sequences. The maximum number of potential failure paths which result in the collapse modes for a structure can be expressed as [3]: N!

Nmax- (r-- l)!

(1)

where N is the number of the potential plastic hinges and r is the degree of redundancy. It can be seen from this formula that there is an astronomically large number of potential failure paths in large-scale structures. For examples, Nmax is 6.095 × 101° for the two-story two-bay building shown in Fig. 9. However, in most cases only a small fraction of Nm~, contribute significantly to the overall failure probability. Therefore, many types of systematic component failure generation procedures have been developed to identify this set of failure paths with relatively higher failure probability; these may be referred to as significant failure sequences, and the estimate of system failure probability based on these sequences is expected to be close to the true answer. Generally, a complete failure sequence (from intact to partial or total collapse) constitutes a failure mode, i.e., S i = E~ i) n E~2i) N . . .

n E~ i) n " . .

(2)

N E mi (i)

where m i is the number of events in the ith sequence in which Then the system probability of failure is expressed as P~ys -- P~ys = P ( S 1 U $2 U -.- U SN,)

E (i)

is the j t h failure event. (3)

Q. Xiao, S. Mahadevan / Structural Safety 13 (1994) 207-226

209

where N s is the number of the significant failure sequences identified. The subscript s denotes the use of significant failure sequences to calculate system failure probability. Equation (3) is the general expression for different kinds of the member behavior properties. However, Eq. (3) can be simplified for elasto-plastic member behavior. The collapse of a ductile structure is the formation of a mechanism, defined as a failure mode. Each complete failure sequence leads to a collapse mechanism. A collapse mechanism can be reached through many failure sequences; the limit state function of the last failure event in any failure sequence leading to this mechanism is the same regardless of the failure sequence and is the failure mode equation for this mechanism [3]. Therefore, the computation of system failure probability using the union of failure modes is much simpler than the use of Eqs. (2) and (3). It is also apparent that the use of failure modes is more accurate, because the limit state equation of any failure mode defines its failure domain more accurately than the significant failure sequences leading to this mode. This statement can be mathematically explained as follows. The failure domain of the ith failure mode can be accurately expressed as nt

F / = U (E~ k)n "'" AE(mk])

(4)

k=l

where n i is the number of total potential failure sequences leading to the ith failure mode. If n~ is the number of the significant failure sequences leading to this failure mode, then the failure domain in Eq. (4) becomes smaller, i.e.,

P(e,)

e

U

ne

.

(5)

kk=l

Under the assumptions mentioned earlier, the failure domain of the failure mode in Eq. (4) can simply be expressed as nr

np

E a i j Mp j + EbijPj <-
(6)

J

where Mpj and Pj are the plastic moment capacity and external load respectively, aij and bij are constants related to the geometrical properties of the structure; n r is the number of plastic hinges already imposed, and np the number of external loads. Assume that N m failure modes are identified by N S significant failure sequences. Then Eq. (3) in term of the failure mode is expressed as

P~ys "" Pffs

=

P( F1 U F 2 U

"'"

U FN.,).

(7)

The subscript m denotes the use of failure modes to calculate system failure probability. It is obvious that Psys is a better estimate of P~y~ than P~s due to the approximation involved in Eq. (5). Therefore, using failure modes to evaluate the system failure probability is more accurate and simpler than using the significant failure sequences for ductile structures. Since the failure event E (g) in Eq. (2) is influenced by the plastic as well as elastic properties of the ductile structure while the failure event F i is only expressed in terms of the plastic properties, a question arises whether the significant failure sequences always correspond to the significant failure modes. However, most example applications of the enumeration approach

21 ()

Q. Xiao. 5. Maha~h'Fan / Structural Sa]~'O' 13 (l q941 207 220

have found that the significant failure sequences do correspond to the significant failure modes [12]. The reason is as follows: A significant failure sequence is obtained by the following truncation criterion:

P( E. N .." N Em~ , N E,,,,) >~TP,,,

!8i

where m~ is the number of failure events in this failure sequence. Pref is a reference value (see also Section 3.2) and 3' may be referred to as the truncation or bounding parameter, whose value is based on the desired degree of accuracy. In Eq. (8), the last e v e n t Em~ is the failure event F of the corresponding failure mechanism. Therefore,

P( F) = P ( E m , ) >~P( E, N "'" N E,,,~ , N E,,,~)>~ YPre,"

(9)

It is obvious from Eq. (9) that if the probability of a failure sequence is greater than 7Pr~f, the probability of the corresponding failure mode is also greater than yP~f. Therefore, the significant failure sequences can identify the significant failure modes. Therefore, the computational strategy of enumeration techniques for ductile structures is to identify significant failure sequences in order to identify significant failure modes. System reliability is then estimated by considering the union of significant failure modes. Earlier enumeration techniques impose only one plastic hinge at each selection level and perform reanalysis of the modified structure during the search for significant failure sequences. As mentioned earlier, this results in a large number of failure paths and structural reanalyses, which makes the computation very expensive even for small-size structures. Further, much work is wasted because many of the identified significant failure sequences lead to the same significant failure mode. Therefore, an efficient procedure is proposed here to identify the significant failure modes for large elasto-plastic structures.

3. Proposed method The enumeration approach typically has two operations: the branching operation, and the bounding operation. A third operation, referred to here as the grouping operation, is proposed here to impose groups of plastic hinges instead of only one hinge at each selection level, based on the use of statistical correlations among m e m b e r failures.

3.1. Grouping operation The limit states for many member failures have several common random variables related to loads and resistances. For example, several members may have the same cross section due to architectural and economic considerations. All members are influenced by the entire set of loads on the structure, with varying degrees of influence. Also, many loads could be identical or highly correlated; for example, the dead loads due to the weight of a floor may be the same on different members, and the wind loads on different floors are all functions of the same intensity variable. Therefore, there could be significant statistical correlations among the limit states of c o m p o n e n t failures, which means that the joint failure probability of several sections alongwith one selected section could be significant.

Q. Xiao, S. Mahadevan / Structural Safety 13 (1994) 207-226

211

A sufficient number of hinges are n e e d e d to obtain a collapse mechanism. The problem is how to identifs' these sufficient hinges to reach the mechanism. Earlier studies [2,6] impose one hinge at each step, progressively leading to a mechanism. This results in a large number of reanalyses of the structure requiring considerable computational time for a large-scale structure. As mentioned earlier, the generation of the significant failure sequences is only to produce the significant failure modes. The last events of the significant failure sequences are of importance but the history of plastic hinge formation is of no interest. Therefore, to accelerate the whole search process, a group of plastic hinges may be imposed on the structure at each selection level until the structure becomes a collapse mechanism. The procedure of imposing a group of plastic hinges in the truncated enumeration method is developed as follows. Assume that section k is selected for the next new damaged section after sections 1 , . . . , k - 1 are damaged. To speed up the search process, a section i whose conditional failure probability satisfies the criterion.

P(Ei/Ek)>~A, i=l,...,N-k

(10)

will be assumed to fail together with section k, where A is a chosen grouping parameter (0 < A < 1), and N is the number of potential hinges. It is reasonable to let section i to fail alongwith section k if its conditional failure probability given the failure of section k is relatively high. Assume that m sections are selected to fail alongwith the k t h section; then the path probability for the failure sequence that includes these sections is expressed as p(k) • path = p

Ei nE kn L\i=l

Ekj

(11)

"

j

p~k) path is smaller than the path probability obtained by imposing the same m hinges one by one. First consider m = 1 to explain the above statement. Assume that the k th section needs to be explored after imposing plastic hinges at k - 1 sections. Let a section k I be selected to fail alongwith section k using Eq. (10) (the subscript 1 indicates that m is equal to 1). The limit

X2

-.......

~

I~-.....-.,,P(E~NEk~) g k l =0

P(E~NEkdk) I gkl/k =0 \ gk=0 X1 Fig. 1. Progressive vs. simulations imposition of plastic hinges.

2ii!

Q. Xia~), 5. Mahade~'un , Mru~ tural ,~afi'ty 13 (1994) 207--220

states of sections k and k~ are g~ = 0 and gk, = 0 for two random variables as shown in Fig. i The path probability of the simultaneous failure of sections k and k~ is E~ A ( E k nEk,)

r,(k) = p • path

(J2)

i

If, instead of the above, section k I is explored after section k has been explored (its limit state is gk,/k = 0 in Fig. 1) the path probability of the failure of section k~ is •P(k')=Ppath

Eif3(EkOEk,/~) •

(13)

The three limit states gk = 0, gk, = 0, gk,/k = 0 intersect at one point due to the physics of the problem. That is, the intersection point represents the change in the limit state of section k I due to the failure of section k. Before the failure of section k, the line gk~ = 0 represents the limit state for section k~; after the failure of section k, the line gkl/~ = 0 represents the limit state for section k~, while section k continues to carry a m o m e n t equal to its plastic m o m e n t capacity. It is clear from Fig. 1 that P ( E k • Ek,/k) is larger than P ( E , N Ek,). Therefore, Eq. (12) is smaller than Eq. (13). The above illustration may be generalized for m > 1 to indicate that the path probability of progressive failure of the sections is larger than that of the simultaneous failure of the same sections. This result is used later during the truncation operation.

3.2. Truncation or bounding operation This operation is to discard the insignificant failures sequences by comparing their failure probabilities to the system failure probability Pf, i.e., the ith failure sequence is ignored if its failure probability satisfies the following criterion [2]:

Pi < 6Pf

(14)

where 6 is a constant, with a chosen value based on the required degree of accuracy. Since Pf is unknown, it is replaced by the maximum path probability Pf* among the significant failure sequences [2,6]. Then the truncation criterion becomes

Pi < 7Pf*

(15)

where y is a truncation parameter, defined in Eq. (8). Pf* may not always be close to Pf for ductile structures (refer to Eqs. (4) and (7)). This is specially true of the path probability in the proposed method (/)path is calculated by simultaneously imposing a group of hinges (Eq. (11)); this is smaller than that obtained by imposing the hinges one by one). Thus the approximate nature of Pf* prevents the precise evaluation of the degree of accuracy achieved by using the parameter 7. It is desirable to obtain Pf* as early as possible in order to discard more insignificant failure sequences. A branching operation is used in previous methods [2,6] for this purpose; at every selection level, each section is examined whether it can maximize its corresponding path probability. This operation is implemented all the way to the end because

Q. Xiao, S. Mahadevan ~Structural Safety 13 (1994) 207-226

213

one does not known where the most significant failure sequence occurs. Therefore, more computational time is spent on finding Pf*. Based on the above analysis, the path probability of the first complete failure sequence obtained by branching, grouping and bounding operations is used as a reference eref" Once eref is calculated, only the truncation and grouping operations are performed. Pref is obtained as follows. Starting from the intact structure, a section k is chosen and examined whether its path probability is maximized, i.e.,

maxe[(kN1Ei) ~Ek] k

(16)

[~i=1

where k refers to any of the remaining potential hinge locations in the structure. Therefore, the path probability of the first complete significant failure path is

Pref = maxP s

I(sal ) 1 Ei A Es

(17)

[\i=1

where s refers to any of the remaining potential hinge locations in the structure. The structure becomes a mechanism after the imposition of a plastic hinge on section s. Other details to obtain Pra are shown in the flow chart in Fig. 2. Finally, section k and m other sections are chosen to be explored after k - 1 sections have failed if ei

Pkath = e

CIE~n

i

Ej

>~ TPref.

(18)

j

It was mentioned earlier than the grouping operation results in a lower value for Pkath compared to imposing the plastic hinges one by one. This could create the possibility of neglecting some important failure sequences. However, such a possibility is counteracted by the fact that Pref is also reduced by the grouping operation as shown in Fig. 2. It should also be kept in mind that the goal is to identify the significant failure modes, not sequences; generally, many failure sequences lead to the same failure modes. Furthermore, the parameter 7 controls the number of sequences; if more sequences are desired, a lower value of 3' may be chosen.

3.3. Calculation of the path probability Equation (11) is the joint probability of many failure events states. Its exact or numerical evaluation is difficult when k >~ 3. Numerical methods have been developed to evaluate the approximation for a multinormal integral [13-15]. Alternatively, several approximate bounds for Eq. (11) have also been developed for the sake of ease in computation [2,3,6]. For example, Ditlevsen's upper bound [16] for the path probability is

P

E i ~< 1 i

P(EI) +

max 0, i=2

P(ff'i)- Y'~ P j=l

C~

(19)

Q. Xiao, S. Mahadet,an /' Stru('tural 5afi'tv 13 (1094) 207-220

214

Intact structure

)

Choose section k that ~ n ~

Select m other sections by P ( E i / E ~ ) >_ ,\

,t

. .i.

Impose hinges at k and m other sec~ons

No Go to me

next level Impose a Image --i only at k section

L

'l

Yes

Fig. 2. Flow chart of steps to obtain

eref"

where Ei is the complimentary event of failure e v e n t E i. The sequence of n events should be arranged in an increasing order in term of their failure probabilities to ensure narrow bounds. This bound has also been expressed in different forms [3,12]. An alternative second-order upper bound was suggested by Murotsu [18] as

P

<~min[P(Ei~E,)], i , j = 1 , 2 i=

i¢:j

n.

(20)

215

Q. Xiao, s. Mahadevan / Structural Safety 13 (1994) 207-226

A weakened version ot the above upper bound may be obtained by first selecting the with the minimum failure probability among n events to compute

g i <~m i n P(Emi n f~

P

,

j = 2,...,n.

event

Emi n

(21)

i In the above equation, E 1 = E m i n. This bound is used in the following numerical examples. Caution should be exercised when bounds are used for the multiple intersection probability of large systems because the accuracy of the bounds may not be sufficient.

3.4. Criteria for the reduction of computational time The following criteria further improve the efficiency of the proposed method: (1) Efficiency in the bounding operation: If the failure probability of a section at a selection level is less than YPref, then this section is rejected as a candidate for the bounding operation since its path probability will definitely be less than YPref" Referring to Eq. (18), it is seen that if P(E k) is less than ")/Pref, then the path probability e;ath is also less than ~/eref. Using this criterion helps avoid unnecessary computation of path probabilities. (2) Efficiency in the grouping operation: Equation (10) may also be written as

P ( E i A E k ) >1AP(Ek).

(22)

The checking of Eq. (22) involves the numerical evaluation of the bivariate normal integral to compute the joint probability. To do this for every combination of E i and E k (i = 1 , . . . , N - k) involves many integrations, especially if N is large. To reduce the computational time, only the sections that satisfy P ( e i ) >~~ e ( E k )

(23)

are selected first. These are then further checked with the criterion of Eq. (22). When the value of Pik, the correlation coefficient between the ith and kth limit states, is unity (perfect correlation), Eq. (22) becomes Eq. (23). For cases where Pik is less than 1, Eq. (23) selects a larger failure domain than Eq. (22), since P(E i) is always greater than P(E i N Ek). In other words, the member limit states that satisfy Eq. (22) are a fraction of those that satisfy Eq. (23). The filtering through Eq. (23) helps to avoid the estimation of joint probability integral for many unnecessary combinations of E i and E k. (3) Reduction of the number of significant failure sequences: Assume that m + 1 hinges are imposed at the kth selection level (see Fig. 3). Later, if one of these m + 1 hinges such as k~ is explored and the same locations of other m hinges are identified to fail by the grouping operation at the same selection level, the failure paths identified after this selection level will be the same as before. Such a sequence is not pursued any further, resulting in the savings of a large amount of computational effort.

3. 5. Procedure of the proposed method The above criteria and techniques are incorporated into a systematic failure sequence generation strategy [2] to ensure that all the significant failure sequences within the defined

21(~

Q. Xiao. S. Mahadevan / Structural Safety 13 (1094; 207-226

',,

\y..

x

,

()

kin© Fig. 3. Reduction of the number of significant failure sequences.

bounding criteria are selected. The main steps of the proposed method are shown in the flow chart in Fig. 4, assisted by Fig. 5. In Fig. 5, the smaller circles stands for the sections identified by the grouping operation. One point needs to be mentioned here: If the path probability is less than YPref or the modified structure becomes a mechanism after a group of hinges is imposed, then only one hinge is imposed at the selected section to avoid missing the significant failure sequence or to obtain the failure mode equation (see Fig. 4).

3.6. Effect of the parameters ~ and ,~ The values of the parameters y and h control the efficiency and accuracy of the proposed method. The effect of ~ is simple: the smaller the value of 3/, the better the accuracy of the system reliability estimate, but the number of failure sequences is large, resulting in high computational expense. The effect of the parameter A is more complicated. As the value of A is decreased, more hinges can be imposed alongwith a selected hinge. But when A is very small, too many hinges are imposed at each step, leading to the formation of a collapse mechanism. This results in imposing failure only at the selected section and more partitions for the failure sequences (see Section 3.5). The effects of y and A are investigated in the second numerical example. The result indicates that there is an optimum value for A which can make the search process most efficient. But this value may be different for different structures. It is noted that the proposed method will in fact become the truncated enumeration method if the value of A is taken as 1.0.

217

Q. Xiao, s. Mahadevan/ Structural Safety 13 (1994) 207-226

Selecta sectionk that saasfies: i, [(n,~,) n E.] >_~e,.,

Selectm other sectio~ by P(E,/Ek) > A

I

Discardthis patrol failure sequence

1N°

'l Yes

Imposehinges at k and 1 ._~ Imposea hingeonly m other sections I at k secdon

'r

No Yes

. . ' . . ~ / G o to the xes same level

30tothe

nextlevel Out'puta SFM

]

I Go to the 'Llower level

No'<" Go to ~. L

~

wer leveI

Fig. 4. Flow chart of the proposed method. After the significant failure m o d e s are obtained by the p r o p o s e d m e t h o d , Ditlevsen's second-order bounds for the union of the multiple events are used to evaluate the system reliability [16]. 3. 7. Application to other types o f structures

T h e p r o p o s e d m e t h o d may also be extended to structures with semi-brittle behavior, i.e., the m e m b e r has some residual strength after reaching its capacity. Therefore, during structural reanalysis, a plastic hinge is imposed with residual m o m e n t capacity. In that case, even

l ,~

~). Xia~). S. M a h a d ( , t a l t ,, htr~ctural Sq[(~tv 13 (1~)94) ~

~gt)

The first failure sequence is obtained "l,.._j/

•'"''~"i

(~

-.

Fig. 5. Systematiccomponent failure generation of the proposed method. identical-looking mechanisms will have different limit state equations for failure sequences with different last plastic hinge locations. This increases the number of significant failure modes. All the other steps are identical to those for a ductile structure. Generally, the proposed method can be readily applied to structural systems whose collapse limit states are independent of the history of component failures.

4. Numerical examples Two steel frames are used to demonstrate the efficiency of the proposed method, compared with the method of truncated enumeration. All the random variables are assumed to be normal. The values of the parameters A and ~, are chosen to be 0.5 and 0.6 for the two examples.

4.1. One-story two-bay frame (Fig. 6) The following AISC [19] sections are used: W16X57 for the beams, W14X53 for the columns. The statistics for the loads and resistances are shown in Table 1. To demonstrate the efficiency of the proposed method, a complete significant failure path from the proposed method and truncated enumeration method are compared in Fig. 7. The numbers in the circles are the plastic hinge locations. The sections in the lower circle in Fig. 7(a) are chosen by the grouping criterion of Eq. (10). It is seen that three selection levels are n e e d e d in the proposed method instead of seven selection levels of the same hinge locations in

219

Q. Xiao, s. Mahadeuan / Structural Safety 13 (1994) 207-226

P1

I1

P2

P2

21~13

4L 152

I '° 9

61~

11

.

.1_13

,xs0

w

-1

Fig. 6. One-story two-bay frame.

Table l Statistics of variables for the one-story two-bay frame example

Mean COV

Mb (kip in)

Mc (kip in)

PI (kip)

P2 (kip)

3969.0 0.15

3292.4 0.15

60.0 0.30

100.0 0.15

@

:G Ca)

(b) Fig. 7. Failure sequence imposition: (a) proposed method; (b) truncated enumeration [2].

2211

Q. Xiao. S. Mahadel,an / Structural Sa[~'ty 13 (19q4) 207-220

I

Mode

I

/31 = 2.954

7 ii !

Mode

2 '~2= 3.102

/ <9

/3a = 3.!56

/

~

Mode a,

/34 = 3.261 Fig. 8. Significant failure modes for the one-story two-bay frame example.

a complete failure path. Thus much time is saved in computation, especially in structural analysis, which is expensive for large structures. Four significant failure modes are identified for the structure; these are shown in Fig. 8. The corresponding results in Table 2 show that the number of significant failure sequences are reduced from 463 to 93 compared to the truncated enumeration method. Second-order bounds Table 2 Number of significant failure sequences for the one-story two-bay frame example Mode

Truncated enumeration

Proposed

1 2 3 4 Sum Time (s)

259 80 52 72 463 370

47 20 14 12 93 77

Q. Xiao, S. Mahadevan / Structural Safety 13 (1994) 207-226

221

P2

P1

P1

2V2

1.,

4X108"

_f -I

Fig. 9. Two-story two-bay frame.

on the probability of union of these four significant failure modes are 0.00200 and 0.00389. A system reliability index has been used in the literature [17,20] as /3sys = - ~ - l ( P s y s ) . Using this definition, the corresponding bounds on the system reliability index are obtained as 2.662 and 2.878.

4.2. Two-story two-bay frame (Fig. 9) The following AISC [19] sections are used: W14X34 for the beams, W12X30 for the columns. The statistics for the loads and resistances are shown in Table 3. Three significant failure modes are identified by the truncated enumeration method as well as by the proposed method, and are shown in Fig. 10. The corresponding results in Table 4 show that the total number of the significant failure sequences in reduced from 60 536 to 204. The bounds on the system reliability index are 2.342 and 2.527 for those three significant failure modes. From these two examples, it can be seen that the proposed method results in great savings in computational effort compared to the truncated enumeration method. The proposed method appears to become increasingly more economical as the structure becomes larger.

Table 3 Statistics of variables for the two-story two-bay frame example

Mean COY

Mb (kip in)

Mc (kip in)

P1 (kip)

P2 (kip)

2270.0 0.15

1792.0 0.15

40.0 0.15

10.0 0.37

227.

Q. Xiao, s. Mahadutan /,Structural ,sa¢ety 13 (1994t 207 220

1 l

E

/

Mode I

3t = 2.333

Mode '~ 3~. = 2.$93 ,'

/

/

Mode 3 33 = 2.959

Fig. 10. Significant failure modes for the two-story two-bay frame example. 4.2.1. E f f e c t s o f p a r a m e t e r s A a n d y

T h e t w o - s t o r y t w o - b a y f r a m e e x a m p l e is r e p e a t e d for v a r i o u s values o f A, to investigate the effect o f t h e g r o u p i n g p a r a m e t e r . T h e results are s h o w n in Fig. 11. It is seen t h a t as the value o f Table 4 Number of significant failure sequences for the two-story two-bay frame example Mode

Truncated enumeration

Proposed

1 2 3 Sum Time (s)

824 52413 7299 60536 61098

34 109 61 204 393

223

Q. Xiao, S. Mahadeuan / Structural Safety 13 (1994) 207-226 500

400 7=0.8

300

200

Z

100

0 I

0.2

I

1

I

I

I

0.3

0.4

0.5

0.6

0.7

A

Fig. 11. Two-story two-bay frame example: Effect of grouping parameter A. I

I

I

I

10

A=0.4

E 8

8 o

~6 "6

z

2

0.1

I

I

I

I

0.2

0.3

0.4

0.5

Truncation pur~rneter

0.6

7

Fig. 12. Two-story two-bay frame example 2: Effect of 7 on the number of significant failure modes.

~_~4

Q. Xiao, S. Mahaderan / Structural Sa[etv 13 (1994) 207-226

121

T

11 f \ \ ..,., a

~10

I

°

\

~ Second order bounds * Simulation

i

\ h=0.4

"~ 9 o

8

\

\ \

7

\

i

\ i

5f 4-

0.1

t J

m

,I'_. . . . i

I

I"

0.2

0.3

0.4

""-""~

I

I

0.5

0.6

Truncation parameter 7 Fig. 13. T w o - s t o r y t w o - b a y f r a m e e x a m p l e 2: E f f e c t o f 3' o n s y s t e m failure probability.

A decreases, the number of significant failure sequences decreases up to a value of 0.4 and starts to increase once again for lower values of A. It is easy to understand the trend from A = 0.7 to A = 0.4; as A is reduced, more and more sections qualify for grouping. However, as A is reduced below 0.4, too many hinges are imposed together, leading to quick formation of a plastic mechanism. In that case, as explained in Section 3.5, only one section is imposed at the corresponding selection level. This results in the enumeration of many more failure sequences. Thus the choice of 0.4 as the value for A is the most efficient. With A equal to 0.4, the effect of the truncation parameter y is investigated, and the results are shown in Fig. 12 and 13, which shows the expected trend of increasing number of significant failure modes with decreasing values of 3', thus increasing the accuracy of estimation of the system reliability. However, the additional failure modes identified by lowering the value of ~/are observed to have high values of the reliability index, i.e., low failure probability. Thus the proposed method with the grouping operation has not missed the significant failure modes corresponding to a particular level of 3'. This results from the bounding operation (Eq. (18)) and the systematic failure sequence generation strategy (Fig. 5). The individual reliability indices for ten significant failure modes for y = 1 are: 2.833, 2.893, 2.959, 3.067, 3.095, 3.156, 3.275, 3.331, 3.375 and 3.412. The second-order bounds [16] for the system reliability index as in the previous example are 2.342 and 2.636 using eight significant failure modes. However, as shown in Fig. 13, as the number of significant failure modes increases, the second-order bounds computed by Eq. (19) become wider. Methods such as conditioning or third-order bounds may be used to obtain narrower

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bounds in such cases. The Monte Carlo simulation (100000 samples) results are also shown in Fig. 13 for comparison.

5. Conclusion

Currently used enumeration-type system reliability methods b e c o m e uneconomical when applied to large elasto-plastic structures, due to the possibility of a large n u m b e r of possible failure sequences. This paper presented an efficient computational technique to overcome this difficulty. The proposed m e t h o d is based on the automatic generation of safety margins [11] and systematic c o m p o n e n t failure generation [2,3]. In addition, it also uses the statistical correlation among the c o m p o n e n t limit states to impose a group of plastic hinges (referred to as the grouping operation) instead of one hinge at each level of exploration as used in the past studies. For overall system reliability computation, this results in the following improvements over previously used search methods: (i) the n u m b e r of selection levels for the complete failure sequences is reduced; and (ii) the total n u m b e r of the significant failure sequences is greatly r e d u c e d due to the avoidance of repetition of the failure sequences. The efficiency of the proposed m e t h o d is further increased during both the grouping and bounding operations, by preliminary elimination of potential hinge locations using simpler computations before the use of the actual criteria in these operations. Thus, the search process for the significant failure modes is greatly accelerated as d e m o n s t r a t e d in the numerical examples. The speed and accuracy of the m e t h o d are seen to be governed by the choice of the truncation or bounding p a r a m e t e r y and the grouping p a r a m e t e r A. The optimum values of these parameters represent a cost vs. benefit trade-off, and may be structure-dependent; an adaptive strategy may be required for the estimation and use of these values.

6. References

[1] F. Moses and B. Stahl, Reliability analysis format for offshore structure, in: Proc. lOth Offshore Technology Conf., Houston, 1978, paper OTC 3046, pp. 29-38. [2] R.E. Melchers and L.K. Tang, Dominant failure modes in stochastic structural systems, Struct. Saf., 2 (1984) 127-143. [3] P. Thoft-Christensen and Y. Murotsu, Application of Structural Systems Reliability Theory, Springer Berlin, 1986. [4] R.M. Bennett and A.H.-S. Ang, Formulation of structural system reliability, J. Eng. Mech., ASCE, 112(11) (1987) 1135-1151. [5] J.J. Zimmerman, J.H. Ellis and R.B. Corotis, Stochastic optimization models for structural reliability analysis, J. Struct. Eng., ASCE, 119(1) (1993) 223-239. [6] Y. Murotsu, H. Okada, M. Yonezawa, and M. Kishi, Identification of stochastically dominant failure modes, in: Proc. 4th Int. Conf. on Applications of Statistics and Probability in Soil and Structural Engineering, Florence, Italy, 1983, pp. 1325-1338. [7] M.R. Gorman, Reliability of structural systems, Report 79-2, Dept. of Civil Engineering, Case Western Reserve Univ., Cleveland, 1979. [8] F. Moses, System reliability developments in structural engineering, Struct. Saf., 1 (1982) 3-13.

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[9] F. Moses and M.R. Rashedi, The application of system reliability to structural safety, in: Proc. 4th Int. ~,or~ll on Applications of Statistics and Probability in Soil and Structural Engineering, Florence, Italy, 1983, pp. 573-584~ [10] B.G. Neal, The Plastic Methods o f Structural Analysis, Chapman and Hall, London, 3rd cd., 1977. [11] Y. Murotsu, H. Okada, M. Yonezawa and M. Grimmelt, Automatic generation of stochastically dominant modes of structural failure in frame structure, Struct. Saf.. 2 (1984) 17-25. [12] A. Karamchandani, Structural system reliability analysis methods. Report No. 83, Dept. of Civil Engineering, Stanford Univ., 1987. [13] M. Hohenbichler and R. Rackwitz, First-order concepts in system reliability, Struct. Saf, 1 (1983) 177--i88. [14] M. Hohenbichler and R. Rackwitz, A bound and an approximation to the multivariate normal distribution function, Math. Jpn., 30 (5) (1985) 821-828. [15] S. Gollwitzer and R. Rackwitz, An efficient numerical solution to the multinormal integral, Prob. Eng. Mech., 3 (2) (1988) 98-101. [16] O. Ditlevsen, Narrow reliability bounds for structural Systems, J. Struct. Mech., 7 (4) (1979) 453-472. [17] R.E. Melchers and L.K. Tang, Failure modes in complex stochastic systems, in: Proc. 4th Int. Conf on Structural Safety and Reliability, ICOSSAR '85, Kobe, Japan, 1985, pp. 97-106. [18] Y. Murotsu, M. Okada, M. Yonezawa and K. Taguchi, Reliability assessment of redundant structures, in: Proc. 3rd Int. Conf. on Structural Safety and Reliability, 1COSSAR '81, Trondheim, Norway, 1981, pp, 315-329. [19] American Institute of Steel Construction, Manual of steel construction: load and resistance factor design, Chicago, Ill., 1986. [20] G. Fu, L. Yiengwei and F. Moses, Management of structural system reliability, in: Proc. 3rd IFIP WG 7.5 Conf on Reliability and Optimization o f Structural Systems, Berkeley, Calif., 1990, pp. 113-128,