Structural reliability assessment based on directional vector approximation method

Computers ind. Engng Vol. 33, Nos 3-4, pp. 749-752, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-8352/97 $17.00 + 0.00

Pergamon PII: S0360-8352(97)00238-6

STRUCTURAL RELIABILITY ASSESSMENT BASED ON DIRECTIONAL VECTOR APPROXIMATION METHOD Masaaki Yonezawa Dept. of Industrial Engineering Kinki University Higashi-Osaka, Osaka, Japan

Shoya Okuda Dept. of Mechanical Engineering Kumano Technical College Kumano, Mie, Japan

ABSTRACT A new numerical integration method for calculating structural failure probabilities based on a directional vector approximation is proposed. The approach is based on a division method of the surface of the unit hypersphere, in which the directional vectors are determined by relating to the finite element meshes allocated on the surface of the unit hypersphere. The failure probability is calculated by using the values of area of the finite element meshes and the upper probabilities of the chi square distribution. Numerical examples are provided to show the validity of the proposed method, which gives efficiently good approximate values of structural failure probabilities. © 1997 Elsevier Science Ltd KEY WORDS: Structural reliability, Failure probability, Probability integration, Directional simulation, Directional vector, Chi square distribution,

1. INTRODUCTION

Consider a structural reliability problem including n basic random variables expressed as 17 dimensional

It is usually difficult to obtain the probability integration of the structural failure in reliability problems, then various analytical methods and simulation techniques such as first order second moment reliability method, Ditlevsen's lower and upper bounds of failure probabilities, importance sampling using design points and directional simulation methods[I]-[7] have been studied to evaluate the structural failure probability efficiently.

standard normal vector

g(X)

X = ( X ] , X 2 , . . . , Xn) T. Let

denote a limit state function

function), Le., g ( X ) > 0

(performance

corresponds to a safety state

and g ( X ) <_0 represents a failure state of the structural system, the structural failure probability /~f is given by, Pf = p r o b [ g ( x ) < O] =

The directional simulation method is one of excellent tools for the structural reliability assessment. In a regular directional simulation, random direction vectors are generated according to a uniform density distributed on the unit hypersphere and the distance from the origin to the limit state surface is searched in each random direction to estimate a conditional failure probability, which is evaluated by an upper probability of the chi square distribution.

where f ( x )

~f ( x ) d x g( x)<_O

(1)

is a joint p.d.f, of X .

Introducing a length R and a direction A of vector X =RA, Pf is given by an integration of the conditional failure probability in the direction A = a as shown in Eq.(2).[2]

P f = P r o b [ g ( X ) < O] = P r o b [ g ( R A ) < O]

A new numerical integration method is proposed here to evaluate the structural failure probability based on a directional vector approximation method, in which the surface of the unit hypersphere is divided analytically into a specified number of finite element meshes and the failure probability is approximated by using the centerline vectors normal to each finite element meshes and evaluated by the sum of the product of the area of each mesh and the upper probability of the chi square distribution.

Since R is a random normal variable, its square R 2 is chi-square distributed, a is a realization of a random unit vector, which is uniformly distributed according to a uniform distribution fA(a) on the n dimensional unit

2. STRUCTURAL FAILURE PROBABILITY

the distance from the origin to the limit state surface in

= [ Prob[g(Ra) < 01 A =.].fA (.)d.

(2)

all a = I {l-z2.

("a2)}fA(a) da

all a

hypersphere ~ .

749

centered around the origin, t;, denotes

750

Proceedings o f 1996 1CC&IC

the direction A = a ,

and Z 2 denotes a cumulative

distribution function of the chi square distribution with n degrees of freedom. A conceptual relation of a conditional failure probability in the direction A = a and other factors is illustrated in Fig.l. The uniform density function fA(a) is given by[8]

1- X,2(r.2) .. ~ r(./2)

' fA(y

l

f A ( a ) = 2~rn/2 = ~

(3)

where F(.) is the gamma function and S is a surface area of ~2n , i.e., a constant. In a regular directional simulation process[2], a random direction vector a is generated according to the uniform distribution fA(a) and the distance from the origin to

surface Fig.1

Conditional failure probability in the direction

A=a

the limit state surface is searched in each random direction a to estimate the conditional failure probability,

J

which is evaluated by the upper probability of Z~

'

J

Limit

distribution. Instead of using random directional vectors to calculate conditional failure probabilities, direction vectors are determined, in this paper, by relating to the finite element meshes allocated regularly on the surface of ~ . ,

~ Unitsp-"~ere0 3

/

/

~

/

~/~

~ ~ .

(

Dividing the surface of ~2n analytically into a

specified number N¢ of finite element meshes, the area of which are sj's, ( j = 1 , 2 , . . . ,

N c ) as illustrated in

Fig.2, then the structural failure probability defined by Eq.(2) is rewritten as the sum of the integration of the conditional failure probability in the direction A = aj over the domain of each finite element mesh as follows:

N¢

(4)

j=l all aj.

Fig.2

Finite element meshes on the unit sphere f'23 .

where a j is a set of directional vectors within the limit of the region of area s], and r j

denotes the distance from

the origin to the limit state surface in the direction of A =aj In place of a j in the integral of Eq.(4), the centedine

S j=I where rhj is the distance from the origin to the limit state surface in the direction bj as shown in Fig.3.

vector bj is used as a representing directional vector related to

sj.

Since an infinitesimal increment of

3.DIVISION OF SURFACE OF UNIT HYPERSPHERE

direction vector da corresponds to that of hypersurface element ds on ~ n , then by using the relation da = ds and f A ( a j ) = ] / S ,

Eq.(4)is approximated as follows:

Nc

In the proposed method, the evaluation of structural failure probability defined by the Eq.(2) is reduced to a problem how to divide the surface of the unit hypersphere into finite element meshes and how to calculate the distance from the origin to the limit state surface. A procedure divide the surface of ~ .

j= 1

sj

is as follows:

Changing the angles of arguments ~l,~2,'",~n-I

in

polar coordinates at constants steps a ] , a 2 , ' " , G n - 1

Proceedings of 1996 ICC&IC

751

respectively, the area of each hypersurface element s2 , corresponding to coordinates of arguments is given by[8] ^

(ji+l)a!

s./ =

f

COS n-z

(/'l+l)a2

OldO 1

JlCt I (Jn-2+l)an-2

""

f

cos n-3

02dO 2

J2(Z2

J

(6)

(j._l+l)an_l

COS{gn-2d0.-2

Jn-2Ctn-2

I

J

don-1

jn-lan_l

Nc

(7)

S=y~ sj j=l

The ranges of arguments are defined in Eq(8)

_X_
,(k = 1,2,...,n-2)

(8)

{ 1, ql +1) ell, J2a2} {1, ~1 +1) a1,(]2 +1) a2}

O<_On_1 S 2 z Division

numbers

-...,

mk,(k = 1 , 2 , . . . , n - l )

of

each

( r , 6"1' 02~ =

argument, step angles a k , (k = 1 , 2 , . . . , n - l ) and the total division number N c of the surface of the unit

/

" ~ ~ S j

{1, Jl a l , J 2 a 2 } / '

{1,Jl a l , (/2 +1) a 2 }

Fig.3 Centerline vector bj and coordinates of edges of

hypersphere are related as follows:

the j th finite.element mesh /g

,(k = 1 , 2 , . . - , n - 2 )

ctk = - mk 2z t~n_ 1 =

(9)

-

ran-1

n-I

N~ = l-I mt

(lO)

k=l

The surface of £'2, is finally divided into N c finite

processing time compiled in FORTRAN 77 on SPARC LT2, AS1000/20 computer are listed in Tables. [Case I ] Consider a simple portal frame structure subjected to a total gravity load w , an equivalent static earthquake load KW, where K is a seismic load coefficient. The limit state functions of dominant failure modes are given in Eq.(11),[5]

element meshes sj (j=1,2,..., N c) as shown in Fig.3, and each mesh is numbered as 1, 2 ..... N c. The

gl (M, K) = 4 M 1 - KWh

centerline vector bj is also determined analytically from

g2 (M, K) = 4 M l + 2 M 2 - KWh - W(I / 2)

the polar coordinates of edges of the .] th finite element

g3(M,K) = 2M 1 +2M 2 -W(l/2)

mesh and r/,j, the distance from the origin to the limit

g4 (M, K) = 2 M 1 + 4 M 2 - KWh - W(l / 2)

state function in the direction b ] , can be calculated by

(11)

an appropriate method such as the Newton's method.

where M 1 , M 2 are plastic bending capacities of the columns and the beam, h is the frame height and 1 is the beam length.

4. NUMERICAL EXAMPLE

The basic random variables K , M 1 , M 2 are assumed

As numerical examples, the approximate structural failure probabilities Pf are evaluated for two types of structures by the proposed method of the directional vector approximation (denoted as D.V.A)with various number N c of division of the surface of the unit hypersphere, ill" estimated by the simple directional simulation (denoted as D.S.) with the same sample size of N c is also compared. The results of P f , /3f and

to

be

independently

N(0.3, 0.]2),

normally

N(45, 6.752 )

and

distributed

with

N(673, 6.752 )

respectively and other variables are deterministic constants of W=50[N], h =4.5[m] and / =6Ira]. The results are listed in Table 1, where the value of ~6f= 4.85x 10-3 denoted as "Exact" is estimated by the Monte Carlo simulation with 8x108 samples and Ditlevsen's upper and lower bounds of Pf is also given.

Proceedings of 1996 lCC&lC

752

As shown in Table 1, P f evaluated by D.V.A with the total division number of N c =216, which is very small, falls within Ditlevsen's upper and lower bounds and gives a close value to the exact one and its processing time is shorter than that of D.S. [Case 2] Consider a structure with a nonlinear limit state function as follows:[6] 3

g(X) = -0.125(Y. A ' ~ ) - X 4 +4

(12)

i=1

Table 1

Estimates of failure probabilities and processin 9 time for case 1 Total Step angles iFailure probabilityi Time number of arguments x 10-3 (Sec) of division i (Radian) D.V.A. D.S.

Nc 900 360 216

Oil re/30 ~r / 30 i *r/18 i

a2

PI

fell5 ~r / 6 u/6

4.85 4.86 4.85

I'/` 0.4 0.22 0.17 5.47 3.46 3.51

where basic random variables X i, (i = 1,2,3,4) are standardized normal variates.

180

The results are listed in Table 2, where the value of

) f - 4.0 x 10 -4

obtained

by

the

~r/6

......... 7 2 ............. ~ / 6

tr/15 ,,r./6

4.76

4.35 0.17

......... 4 r 6 ................................. 0 = ! 5 .....

Ditlevsen's bi-modal bounds:

Gaussian

4.775x 10 -3 < P/` < 5.01 x 10 -3

integration[6] is denoted as "Exact". P/` evaluated with Exact: 4.851x 10 - 3

Nc=3456 gives a close value to the exact one. Compared with t31_ estimated by D.S., ~ f

is more ^

accurate and requires shorter processing time than Pf It can be said that AFOSM does not give good approximate value to the case of highly nonlinear convex limit state surface as Eq(12).

5. CONCLUSION This paper provides a new numerical integration method based on a directional vector approximation to evaluate structural failure probabilities. The proposed method to divide the surface of unit hypersphere is shown to be easy to implement for structural reliability problem and it can be applied to solve efficiently structural reliability problems with nonlinear and multiple limit state functions.

Table 2 Total number of division

Estimates of failure probabilities and processing time for case 2 Step angles iFailure probability I Time of arguments (Sec) x 10-4 (Radian) D.V.A. D.S.

Nc

a l' a 2, a 3

11664

zr / 18 :r / 12

4.014 4.040 4.102 4.307 4.289

3456 432 54

3.03 0.90

zr / 6 ~r / 3

4.179 5.273

1.55 0.17 0.15

AFOSM: 0.3 x ]0 - 4 Exact: P f = 4.0 x 10 - 4

REFERENCES

1. Ditlevsen, O., 1979, Narrow reliability bounds for structural systems, Journal of Structural Mechanics, ASCE, 7, 4:453-472. 2. Bjerager, P., 1988, Probability integration by directional simulation, ASCE, 114: 1286-1302. 3. Schueller, G. I. and R. Stix, 1987, A critical appraisal of methods to determine failure probabilities, Structural Safety, 4:239-309. 4. Shao, S, and Y. Murotsu, 1994, Reliability of complex structural systems using an efficient directional simulation. Structural Safety and Reliability (ed G. I. Schueller et aL), 3:1501-1510, Balkema, Rotterdam. 5. A. H-S. Ang and W. H Tang, 1984, Probability Concepts in Engineering Planning and Design, Vol.2, Decision Risk and Reliability, John Wiley and Sons, 361.

[NOTE] D.V.A.: Directional vector approximation method D.S. • Directional simulation

6. Harbitz, A., 1986, An Efficient Sampling Method for Probability of Failure Calculation, Structural Safety, 3:109-115. 7. Yonezawa, M, and Okuda, S., 1995, A directional importance simulation for structural reliability assessment, Proceedings of Asian-Pacific Symposium in Structural Reliability and its Applications (APSSRA95), Tokyo. 8. Kendall, M. G., 1961, A course in the geometry of ndimensions, Charles Griffin, London. 9. Katsuki, S and Frangopol, D. M, 1994, Hyperspace division method for structural reliability, Journal of Engineering Mechanics, ASCE, 120, 11:2405-2426.