Structural system reliability assessment using directional simulation

ELSEVIER

Structural

Safety 16 (1994) 23-37

Structural system reliability assessment using directional simulation * R.E. Melchers Department of Civil Engineering and Surveying, The University of Newcastle, Newcastle, NSW 2308, Australia

Abstract

The theory for the estimation of the reliability of a structural system which is subject to one or more load processes requires knowledge of the surface defining the mechanical response of the system. For many systems this surface may be known only implicitly and defined through a method of structural analysis. In the formulation presented herein, the surface is obtained using an established probabilistic structural analysis technique. The reliability estimation is formulated in the load process space. In this space, directional simulation is employed to estimate the outcrossing rate and the initial (zero time) probability of failure and hence to estimate the probability of failure at any subsequent time. Two examples using discrete rectangular pulse (Poisson) processes are described. Keywords: Probability; Outcrossing; processes; Poisson; Pulses

Structural

systems; Failure;

Monte

Carlo; Sampling;

Importance;

Random

1. Introduction The estimation of the probability of failure of a structural system of some complexity and subject to more than one load process has been central to much of recent work in structural reliability theory. Efforts to solve problems of this type have adopted essentially two types of approaches; the use of stochastic process theory together with simulation [l-5], with a modification of first order type methods [6,7] with asymptotic ideas [8,9], or the use of various refinements of the “time-integrated” approach in which the processes are replaced by life-time equivalent random variables [lo-121. When allowance must be made also for time-varying structural properties and/ or non-stationary loading, recourse may be had to process simulation approaches, although for practical problems stochastic process theory is probably the only

* Discussion is open until June 1995 (please submit your discussion paper to the Editor, Ross B. Corotis). 0167-4730/94/$07.00 0 1994 Elsevier SSDI 0167-4730(94)00026-M

Science

B.V. All rights reserved

24

R.E. Melchers/Structural

Safety 16 (1994) 23-37

viable approach. Various efforts in this direction have been reported recently [13-151. A useful overview of the theory particularly for rectangular wave renewal processes with nonstationary characteristics has been given by Rackwitz [16]. In most of this work a continuing problem appears to be the dimensionality of the structural reliability problem, with each random resistance variable and each loading random variable contributing directly to the dimensions of the total space. Some attempts to work in the generally much smaller space of the loads only have been described for so-called time invariant problems [17,18]. Recently, this approach has begun to be studied [4,19] also for time variant systems since it develops rather naturally from outcrossing considerations such as described by Hasofer et al. [2]. As will be shown, the time variant reliability problem may be formulated in the (load) process space and evaluated with the use of directional simulation. This has been discussed earlier for purely continuous Gaussian processes [4], for mixed discrete rectangular (Poisson) processes having Gaussian amplitudes [20] and for structural systems subject to both types of loading [21]. Unlike most earlier techniques, in the present approach the domain boundary specifying the safe-unsafe interface for the structural system as a whole need not be known explicitly at any time. As will be seen, this aspect may be handled through the use of an appropriate structural analysis procedure. In the present paper, the essential theoretical background is outlined first. This is followed by comments on the use of existing probabilistic structural analysis programs to obtain estimates of the domain boundary. An example application will show that the computation times can be excessive and that measures need to be taken to reduce the amount of redundant computation in the part of the algorithm dealing with structural analysis. Several observations about the particular value of the directional simulation approach close the paper. The present work differs in intent from attempts to solve the “load-combination” problem as most commonly posed, in which loads (or load effects) are added algebraically. Such an approach implies validity of the superposition theorem. This assumption clearly is not admissible for nonlinear structural systems.

2. Load space formulation Let the problem be formulated in the space of the m-dimensional load (process) vector Q(t) (Fig. 1). Also, let there be an m-dimensional random vector R representing the load capacity vector of the structural system, so defined that each component of R corresponds in direction and sense to one (and only one) component of Q. Evidently R is a direct function of the components which contribute to the overall structural system strength. Let these components be described by the random vector X of dimension k (zs=-m, in general). Also, let the joint PDF of R be given by f,(). If, as will be assumed here, there are no components of X involved in describing Q(t), it follows that R = R(X) and that Q(t) is independent of R. This restriction is convenient but not necessary [l]. Also, it is possible for Q to exhibit a degree of dependence between its various components: this might occur at different locations on the structure, such as on the legs of a jacket-type offshore platform as a result of wave loading.

R.E. Melchers/Stnxtural

Safety 16 (1994) 23-37

25

failure domain

Fig. 1. Realisations c = 0.

of a directional

ray and of the domain boundary

S,, showing also the surface element

AS, for

Because radial symmetry is implicit in the polar coordinate formulation to be described below, it is advantageous to have f,() in such a form that the variances in each qi direction are of similar magnitude. This can be achieved by simple linear transformation. In the load space the probability of failure of a structural system may be given in terms of polar coordinates as [4]

sphere

where the conditional probability of failure pf(s 1a) is a function of the scalar structural strength S = s > 0 as defined along the radial direction specified by the direction cosines A = a. Evidently, S and A are related to the m-dimensional real space R" defined in R through R - c = S .A,where s = 0 at point C = c, the centre of the polar co-ordinate system. In Eq. (l), the conditional PDFfs ,.(> is defined on s for a given direction A = a; evidently it is a function of R and it might depend also, in general, on A. The unit vector of direction cosines has the PDF given by f,O. The location of c is of interest; no obvious rules for its selection appear to exist apart from noting that the point should be chosen to lie in the “safe” domain D (as defined by the mechanics of the system-see below) and to expose as much as possible of the surface S, defining D, when viewed from C = c. In the special case to be pursued here in which Q represents the space of load processes only, the point C = c = 0 usually is convenient (Fig. 1). It is important to understand that fs, J) arises from the fact that the problem has been formulated in the space defined by all the load processes and that therefore the mechanical response of a structural system, described by the domain boundary S, will need to reflect the probabilistic nature of the structural system itself. This means that S, is not known with certainty and may be seen as a probabilistically described “boundary” in the space of the load processes. In the above formulation, for the direction a, the random variable S represent the probabilistic nature of the structural system. [As should readily be evident, this formulation may be generalised to include time dependent resistance or member properties in the load space [4] but this detail will be ignored herein for simplicity of exposition.]

R.E. Melchers /Structural

26

Safety 16 (1994) 23-37

Monte Carlo and numerical integration schemes to employ (1) have been suggested in a variety of settings [1,22,23]. In most of these difficulties arose because of the large space of integration. Recently, Engelund and Rackwitz [24] have also noted this in their comparative study of various Monte Carlo importance sampling techniques. The load space formulation has obvious advantages. The inner, conditional, integral in [l] is one dimensional, corresponding to the most elementary structural reliability problem but weighted to account for polar coordinates. It is likely, therefore that it can be evaluated by simple means, such as numerical integration. Also, the required amount of multiple integration, as defined by m will be manageable, since in most realistic structural reliability problems, m is relatively small. To employ expression (1) it is necessary that both p&s I a) and f,,.() can be derived. The first of these can be obtained from (i) time invariant probability theory and (ii) using the outcrossing rate of a vector process.

3. Conditional failure probability pf( s / a) Let failure of the structural of the safe domain, sometime let it be assumed that for outcrossings are independent systems). Then the conditional established result

system be defined as the first time violation of S,, the boundary during the period [O, tJ, where t, is a known design life. Also, a given direction A = a, and for the point S = s, individual (Poisson) events (as is reasonable for highly reliable structural failure probability p&s 1a> can be bounded from above using the

P~(s~~)~p,(o,~l~)+[i-p~(O,sl~)]~

1-w

i

---iS,f(t~Wt [ 0

(2)

Ii

where ~~(0, s I a) is the failure probability at time t = 0 and v,‘(t) is the outcrossing rate of the vector process Q(t) out of the safe domain D. As is known, expression (2) may be simplified in various ways. If v;() is independent of time (a common assumption) and if p,(O, s I a) is small relative to unity, then P& I a) =a+(O, s I a) + {I - ew[-vi+

I a)t,]]

(3)

while for systems of high reliability Pf(S I a) ‘t,.

(4)

Both p,(O, s I a) and v,+
4. Local outcrossing rate For loads modelled as continuous processes, the (conditional) local outcrossing rate through an elementary domain boundary AS at point S = s (or, equivalently, R = r) on S, is given by

R.E. Melchers/Structural

Safety 16 (1994) 23-37

27

Fig. 2. Local outcrossing through AS.

the generalised Rice formula. Details of this have been discussed previously [4]. For loads modelled as rectangular Poisson pulse processes, the basic theory for the system outcrossing rate has been provided by Breitung and Rackwitz [25]. If the ith process has a mean rate of arrival v,,,~ and a mean duration pmi such that pumivmi= 1, the conditional local rate of the vector process Q(t) crossing out of the safe domain D through the elemental surface AS, is given by (for n, > 0)

4

A=a = S=s

2[

vmi{4(rli>

-

Fi(r2i)){1

-Fi(‘i)j

* 1ni 1‘fe,,,(‘(i)>

* “Ll

i=l

where feCi, is the probability that the vector load process lies along rli, rZi in Fig. 2 with

Here n is the outward unit normal given that q is at AS, on the limit state surface, so that or qi = ri for all i. Fi(rli) denotes the probability that Ti< rli for the ith process, and F(rii) - Fi(rZi) denotes the probability that qi lies in D given that q is at AS (see Fig. 2). For independent Q,(t) it follows that f i(qci,) =fe(q)/fei(qi) =fe(‘)/fei(‘i) and for systems having high reliability Fi(rli> - Fi(rzi> = &?rli) = e(ri). If the pulses always return to zero at the end of each pulse duration, the term F;:(rii) - ~i(r,i) in (5) reduces to unity for most practical problems. If the Poisson pulses are of the “mixed” type and “sparse”, such that vmi - pmi +c 1, q = r,

Fig. 3. Schematic “mixed” pulse process.

28

R.E. Melchers/Structural

which means that most of the time in Fig. 3. In principle, expression However, in a practical simulation Fig. 3) are unlikely to be detected better, therefore, to separate out evaluate each of these separately. A=a

womb

m

C

C

j=l

i=l

=

[ vi((

Safety 16 (1994) 23-37

each process is “off”, their PDFs may be described as shown (5) may be used to obtain the conditional outcrossing rate. technique, the probability contributions for zero load (see unless a zero load component is simulated precisely. It is the various combinations of “off’ and “on” states, and to With this approach, expression (5) becomes

Pi + qiF,(ri))qi[l

-Fi(ri)])

. I ni

I *.fQ,i,(rci,)]

* AS,

S=s

where F,()is the improper CDF (i.e., that part of the CDF for which q > 0) for the ith process, with corresponding improper PDF fi; vi is the pulse rate conditional on the ith process being “on”; pdi is the mean pulse duration for the ith process (i.e., when “on”), so that yipdi = 1; and vmi is the mean pulse rate (unconditional) as before (for the whole process). Note that vmi = viqi, where qi = probability of the ith process being “on”. Also, the number of combinations of “on”/“off” states is given by ncomb = t

2U-‘1.

(8)

e=l

By integrating over all s and a, expression (8) may be made unconditional

4 =/ unit

fA(")/;-,"~b[

sphere

E[ (( Vi

J

Pi+qiE(ri))qi[l

-fi(‘i)])

i=l

-8

fe,,,(‘ci))] .fSIA(‘I u)s(nj-l)(a ‘n

XI ni 1

’ ds da 1-J i

where nj is the number of “active” components of the load process vector Q(t) for the jth combination. Note that the yi depend directly on s. Also f,() reflects the probability of the various processes being in “on” and “off’ states. As for continuous Gaussian processes [4], the term s@-‘)(u en)-’ corrects the elemental surface area AS, at radius s to the equivalent surface area on the unit sphere. 5. Initial failure probability p,(O) At t = 0, the pulse load process vector Q(t) becomes a vector of random variables, and the failure probability p,(O) is the volume under f,() in the failure domain D’. This may be evaluated using the directional simulation approach, as has been discussed extensively in the literature [1,22,23,26]. An alternative exposition follows. To estimate the probability content, use may be made of sectors I,, I, of arbitrary span, as shown in Fig. 1. These may be approximated by the sectors of finite size E,, E,. Clearly the approximation improves with increased sampling of A. Further, it follows that for polar co-ordinates in R”, the volume element at radius Y is r (m-*)*dY*du so that at radius S=s for given A = a

I+(% s 10)= /“f,(p -a + c)p”-*dp. s

(10)

RX. Melchers/Structural

The unconditional

Safety 16 (1994) 23-37

29

initial failure probability becomes

(11)

P,(O)= / unit sphere

In (11) the inner integral represents the volume under f,() in the failure domain for a given location s along the radial direction a. The middle integral obtains the expected value of the inner integrals, using fS ,A as the PDF. As noted previously [20] computationally it is more efficient to consider elemental parts of the inner integrand summed after weighting by fslA (i.e., changing the order of integration), thus (see also Fig. 1) p,(o) = / unit sphere

fA(u)/m[/pfs,~(s I a) +p(p *a +c)p’m-l)dp da. 0

0

Since the inner integral is the cumulative distribution P~(O)=[-~~

f,ca)[~m~~,a(~~~).~~(~.~+~)~‘m-l)dp]

(12)

function (CDF) for S 1A, it follows that da.

(13)

sphere

6. Determination of f, ,A The second term which needs to be known in order to solve Eq. (1) is the PDF for the system strength in any given radial direction, fS, A. In the load space the system strength is described by a probabilistically defined safe domain D and for most structural systems, D is defined through one or more limit state functions GJq, X) = 0, i = 1,. . . . If these are known explicitly, the conditional probability density function f,,,&) can be obtained in general by multiple integration along the radial direction s (i.e., for given A = a). The technique described in [4] may be employed. For complex structures, the system limit state function(s) (as distinct from those for the elements or components) are unlikely to be known explicitly. This situation is of considerable practical interest. FSIA(s I a) may be obtained for any given direction a and at point s by considering the structure subject to a deterministic loading (i.e., at point 4 = T(X)). It then follows that (i) since the load space must be bounded in all directions by structural strength limitations, pf ,J s ) + s as s + ~0,(ii) since, by definition, structural failure under zero loading cannot occur; pf,Js) at s = 0, and hence (iii) the cumulative probability function along the ray is given by [4] (see Fig. 4) p&h

= Pr(S
=Fsl&).

(14)

From this Fs, JS I a) may be approximated as a piecewise function through evaluating (14) along a for different values of S. Also, differentiation produces fs, JS I a). For framed structural systems, available methods [27,28] to determine structural system failure probability under one parameter random variable loading (i.e., all loading governed by one random variable) may be employed but using, instead, a deterministic load vector (4) which is a function of s. Herein the value of fs, ,&s I a) at any radius S = s will be obtained using the procedure of Melchers and Tang [28].

R.E. Melchers /Structural

I fstA(), Fsd

Fig. 4. Realisation

)

Safety 16 (1994) 23-37

cdf of G( )la

of a directional ray and definition of F,, .O and Fs, .O in load process space.

In principle, the integration shown in Eq. (13) for pf can be carried out directly by evaluation of f,,,
7. Some observations In the formulation used above, it has been assumed that the structure has the property that its probability of failure is independent of the path followed by the load process vector in reaching the domain boundary. This is a major assumption and one which is not always justified. However, it is common to most techniques currently available; because there are innumerable load realisations possible, accounting for load path dependence significantly complicates structural mechanics reliability assessment-if it is possible at all. In most cases the only realistic approach is to ignore load path dependency-alternatively some particular cases can be identified a priori as being critical for the analysis [27,29]. Through formulating the problem in the load (process) space and with the existence of a completely enveloping domain boundary, f,O is unlikely to need to be biased in a particular way through importance sampling techniques and might be assumed uniformly distributed unless contrary information exists. One such exception is for load processes having one or more significant zero component(s) such as the “mixed” Poisson pulse processes considered herein. Another is when there is correlation between the load processes. Further, algorithmic rules might be applied to ensure that directional samples are not taken too close to each other, to avoid repeating (almost) identical calculations along (slightly) different A = a directions. As noted previously [21], in integrating along a realisation ray A = a to account for the uncertain domain boundary, the critical components contributing to the structural system strength are automatically considered in the present procedure. This follows from the way F,,,(s Ia) is obtained using a structural system analysis which allows for structural system strength uncertainty. [This is unlike the usual importance sampling approaches which require

31

Safety 16 (1994) 23-37

R.E. Melchers /Structural

I

Ql(O

cm

t

I-III x2

Xl

Xl

15

,

20

I-

t

Fig. 5. Example 1.

the critical structural system limit state function to be identified, usually employing the critical point of local maximum likelihood.] In the present technique, multi-parameter optimisation to identify candidate points of maximum likelihood has been replaced by repeated structural analysis along a directional realisation (i.e., A = a> of the possible load process vector. However, the present technique does require evaluations of the structural system capacity for given, deterministic, load vector outcomes biased to the lower tail, together with repeated estimation of n, the local outward normal vector.

8. Example 1 Consider a very simple structural frame (Fig. 5). The frame is composed of ideal elastic-plastic material having the strength properties X, = N(360, 54) and X, = N(400, 72). Two independent sparse rectangular Poisson processes model the loadings Q, and Q2. They have Gaussian pulse heights Q, = N(100, 10) and Q2 = N(50, 15) and occur with intensities vmi and

Table 1 Typical results for outcrossing rates Poisson process Loading parameters

Outcrossing rate Melchers

urn1

urn2

Ild

PELm2

/year

/year

yr

yr

5 5 20 20

0.2 5 20 20

0.01 0.01 0.01 0.025

0.01 0.01 0.01 0.025

1201

Present method*

CPU (s)

Present method2

CPU (s)

Error %

0.0093 0.077 0.365 0.44

0.0087 0.072 0.331 0.403

860 876 869 884

0.0088 0.062 0.35 0.44

55 72 61 67

6 8 7 8

Notes: 1. Call to analysis algorithm every sampling. 2. Call to analysis algorithm 3 times only (see text). 3. All computation times for VAX 8550 computer.

32

RE. Melchers/Structural

Safety 16 (1994) 23-37

durations ,uUmi(see Table 1). Results for this problem have been reported earlier [20] using directly the known limit state functions for the problem [12]. For testing the algorithm proposed herein, these limit state functions were ignored and recourse was had to the “analysis algorithm” of Melchers and Tang [28] (i) to obtain the critical limit state functions and (ii) to aid in calculation of the probability of failure. This technique, like other similar methods, relies on the use of repeated linear elastic analyses of the structure to search for the critical failure (or collapse) modes. It is known that the results of such a procedure do not always correspond precisely to the collapse modes of rigid plastic theory. In the first approach to the problem, the “analysis algorithm” of Melchers and Tang [28] was called for every point S = s along A = a to perform the integration required by Eq. (9). It was found that this produced identical critical collapse modes for virtually every realisation s for a given a. As this represents a considerable waste of computation, an approximate approach was adopted in which the critical collapse mode was determined only three times-at points spread some way apart along A = a. It was then assumed that the critical collapse mode remained at the previously determined mode as s was increased. Hence the evaluation carried out by the “analysis algorithm” was largely reduced to evaluation of the structural failure probability for a given s and a. Table 1 shows results for the outcrossing rate obtained by the present approaches compared to those obtained more directly [20]. It is seen that the results from the present approaches somewhat underestimate, rather consistently, those obtained earlier. The present results were all obtained with 80 radial direction integration points. As is evident from Table 1, the strategy of evaluating only a few times the analysis algorithm for collapse modes substantially reduced the overall computation times. It might be noted, moreover, that the analysis algorithm [28] is not optimised or very efficient, and could be improved in terms of its capability and speed for system probability estimation. Within the given computational system, the estimate for the outcrossing rate may be improved, in principle, by increasing the number of radial integration points. It was found, however, that the coefficient of variation of the outcrossing rate estimate does not show significant improvement beyond about 80 radial points, the value adopted here. 9. Example 2 The frame shown in Fig. 6 was considered earlier for estimation of outcrossing rate. For the properties shown in Table 2 and using the limit state functions shown in Table 3, the mean outcrossing rate was estimated at about 0.07 per year [12]. A considerably lower value was estimated by the procedure outlined earlier in the present paper. This led to a re-examination of the proposed approach and a separate evaluation of the main contributors to the outcrossing rate, using directly the limit state functions given in Table 3. Given the sparseness of all the loading processes except Q,, as shown in the last column of Table 2, it is clear that load combinations involving Q2, Q3 and/or Q4 are of negligible occurrence; those involving Q,, are much more likely. Also, Q, acting alone is the most likely occurrence. It follows that of the collapse modes shown in Table 3, modes 8, 5, 6 and 2 and those involving Q, (i.e., the remaining modes) are likely to be of importance, in that order. (Note that mode 8 was shown incorrectly in [12] as being for Q2.>

R.E. Melchers /Structural

M3

33

Safety 16 (1994) 23-37

Ql

2Q4

i

+

M2 Ml

L2=20

L2 = 20

I I

I

Fig. 6. Example 2.

Note also that the modes shown in Table 3 were selected from earlier work on the basis of their contribution to time-independent failure probability estimates. These do not necessarily correspond in importance to outcrossing rate values.

Table 2 Properties of random variables and processes Variable

IJ,

u

Xl

70 150 70 90 120

10.5 22.5 10.5 13.5 18

x2

X X‘I X5 ::

Q3 Q4

28 16 21 7

8.4 4.48 5.88 1.75

V

pd

0.5 0.2 0.2 3.0

1.0 0.1 0.2 0.006

Table 3 Failure mode equations Mode

Failure mode equation G1=5X,+3X,+3X3+2X,-lOQ,-lOQ,-48Q, G, = 6X, - 36Q, G3 = 5X, +4X, +2(Y, + X, + X5>- 10(Q1 + Qz + QJ-48Q4 G4 = 5X, +3X, + X,-10Q1 -36Q, G,=2X,+2X,-lOQ, G, = XI +3X, - lO,Q, G, = 5X, +3X, + X,+4X, - lOQ, - lOQ, -48Q2, G, = 4X, - lOQ,

q = “on”

0.5 0.02 0.04 0.068

34

Table 4 Outcrossing

R.E. Melchers/Structural

Safety 16 (1994) 23-37

rates (see appendix)

ad(s) acting

Mode(s)

q = “on”

121

0.5

2

0.02 0.04 0.018

Q4 Total for individual loads

QI+Qz+Q~ Q,+Q,+Q, Q,+Q,

3 1, 7 4

Vi+

0.22 x 0.15 x 0.20x 0.21 x 0.43 x

10-3 10-5 10-7

10-3 10-3

0.0004 0.00018 0.009

Table 4 shows the contributions to the total outcrossing rate of modes 8,5, 6 and 2, assuming the pulses return to zero automatically. The total of these results is a “lower” bound estimate of the total outcrossing rate. In view of the rates shown and occurrence probabilities, it is evident that the joint occurrence loads and associated failure probabilities are likely to have little impact and hence the “lower” bound represents at least the order of magnitude of the correct result. Interestingly, in this type of problem the directional simulation approach has a degree of affinity with the “load coincidence” method of Wen [12], in that the major contributors to the result are the individual loads, acting alone, and for which the outcrossing rate contribution is easily calculated since the outcrossing rate calculation reduces to a single probability calculation [Appendix]. For rectangular Poisson processes this was noted by Breitung [9] and is evident also in Wen and Chen’s [12] sample calculations. This correspondence between Wen’s approach and the directional simulation approach given here is due directly to the selection of c = 0, the origin of the load space and the realisation in expression (7) that the simulation needed to be modified in order not to miss important contributions to the outcrossing rate result. For estimating the contributions due to joint occurrences of loads, however, the directional simulation approach as outlined herein has the capacity to estimate the contributions rather more accurately than is possible by the approximations inherent in the corresponding part of the “load coincidence” procedure.

10.Conclusion When the structural reliability problem is formulated in the space of the load processes, it is possible to use a directional simulation approach to estimate (an upper bound of) the mean outcrossing rate and the zero time probability of failure. Existing truncated enumeration methods for the evaluation of conditional structural failure probabilities may be used to develop the,probabilistic description of the probabilistic system. The feasibility of this approach was demonstrated using a relatively simple example. It was argued also that the solution strategy obviates the need for explicit limit state functions for the system (although local member limit state functions are still required). It was noted that computational times are very much a function of the efficiency of the structural analysis method employed.

R.E. Melchers/Structural

35

Safety 16 (1994) 23-37

Notation

= vector co-ordinates in Q space for origin of hyper polar co-ordinate = domain = expectation operator EO = joint probability density function in X fx() = limit state function (ith) GiO ho = sampling probability density function k = dimension of X = dimension of Q m n = direction cosines of unit outward normal = probability of failure (unconditional) Pf P,(X) = conditional probability of failure = vector of load processes Q(t) R(x) = vector of structural resistance parameters corresponding to Q S = radial direction (from point c) = surface area s.4 = surface of domain D SD t = time = structure life time t, X = vector of basic random variables a = vector of direction cosines of hyper polar co-ordinate system AS = elemental surface area = vector of circumferential co-ordinates in hyper polar space 4 = vector of mean values for X m, = outcrossing rate for domain D 4

Fl

system

Acknowledgment

The assistance of Dr Chun Qing Li in carrying out the numerical work for the examples is acknowledged. This research is supported by the Australian Research Council.

Appendix:

approximate

individual

load outcrossing

rate contributions

When only one load (e.g., QJ is active, and the others are all “off’, expression (9) reduces to

Vofi=[Iovi{ (Pi +9iF(s))9i[1

-

F(s)])fp(i)(s)

*fS(‘)

* ds

where the s(‘Q-‘)(a *n>-’ term in (9) becomes unity, references to A direction is fixed along Qi and Ti=s, in the direction along Qi. approximated by noting that for pulses returning automatically to zero, equal to unity; in addition the term feci,(S) now represents largely the

(94 may be deleted as the The integral may be the term (pi + qiF()) is probability of all loads

R.E. Melchers/Structural Safety 16 (1994) 23-37

36

other than Qi being in the “off’ state (see definition of feCi, for Eq. (5)). It follows that (9a) becomes

But since feCi, is not dependent on S, it may be taken out of the integral. The remaining integral represents the probability of the load Qi being greater than the structural strength S = S. Rewriting feCi,() as poff, it follows that Eli= Viq,~( Qi > S) ‘poff.

(9c)

The term p(Qi > S) can be evaluated using the FOSM method since both the pulse height and the strengths are Gaussian. The results are readily verified to be as shown in Table 5. Table 5 Component outcrossing rates Load

yi

qi

P(Qi > 8

POff

1

0.5

0.5

0.967x 1O-3

2

0.1

0.02

0.154 x 10-2

3

0.2

0.04

0.54 x10-5

4

3

0.018

0.85 x~O-~

(1-0.02)x (1 - 0.04) x (1 - 0.18) = 0.924 (1- 0.5) x (1 - 0.04) x (1 - 0.018) = 0.471 (1-0.5)x(1 - 0.02) x (1 - 0.018) = 0.481 (1 - 0.5) x(1 - 0.02) x(1 - 0.04) = 0.470

0.22 x10-3 0.146x 1O-5 0.2

x10-7

0.21 x10-3

References 111Ditlevsen, O., Hasofer, A.M. Bjerager, P. and Olesen, R., Directional simulation in Gaussian processes,

Prob.

Engrg. Mech., 3(4) (1988) 207-217.

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