Outcrossing rates of marked poisson cluster processes in structural reliability

Outcrossing rates of marked Poisson cluster processes in structural reliability Karl Schrupp and Ruediger Rackwitz Technische Universitiit Miinchen, Munich, Federal Republic of Germany (Received October 1987)

A marked Poisson cluster process (PCP) is defined as a model for live loads in buildings. The outcrossing rate for this PCP and the superposition of such processes are derived for the determination of structural failure probabilities. For the equilibrium process a Poisson limit theorem for the failure probability and an asymptotic approximation for the outcrossing rate are given. Numerical results for normally distributed marks are presented, and comparisons with two other similar load models are made. Keywords: cluster processes, outcrossing asymptotic analysis

Introduction Various attempts have been made to realistically describe structural loads by appropriate stochastic models and to solve the corresponding combination problem when calculating structural reliability. The class of marked jump processes has been found useful when describing certain loading phenomena (e.g., live loads in buildings). Most of these models rest on the assumption that occurrence, intensity, and duration of the load pulses are independent random variables. Only under this assumption and, in part, in special forms of the load pulses has the combination problem found practical solutions.~-8 However, real loading phenomena often exhibit pronounced dependencies of various kinds. There may be amplitude dependencies (Rackwitzg), dependencies between pulse durations, or dependencies between pulse amplitude and pulse duration (Madsen~°). One of the more important dependencies is occurrence clustering. Wen 6 and Wen and Pearce ~ (the latter reference should also be consulted for a general discussion of the matter of dependence for structural loads) give examples where such clustering can be observed in reality. Wen and Pearce used a kind of Bartlett-Lewis Iz cluster process to handle the problem. By the method of load coincidence they constructed an approximation for the distribution of the lifetime maximum value for the sum of such load processes.

482

App.I. Math. Modelling, 1988, Vol. 12, October

rate, structural

reliability,

In this paper a more general clustering phenomenon is described by a marked point process. It may be used when modelling loads due to vehicle traffic, but other applications in nonstructural areas can also be visualized. The maximum lifetime distribution of sums or more complicated functions of such processes is approximated by the well-known outcrossing rate method. Denote by PF(t) = P(T <~ t), t /> 0, the distribution function of the time T to first failure (i.e., entrance of the marked point process X(t) into the failure domain F C_ R~). We obtain an upper bound to the failure probability ~3 t

Pv(t) <~PF(0) + E[M(t)] = PF(0) + I K(O')do" (1) 0

where PF(0) is the initial failure probability, E[M(t)] is the mean value of the point process of exits of X(t) into F, and K(~) is the instantaneous crossing rate of M(t) into F. A lower bound to PF(t) involving higher moments of M(t) can also be given, but the well-known asymptotic approximation ~3.~4 P F ( t ) - I -- exp[-- ElM(t)]]

(2)

valid under certain conditions, generally is of more practical interest. In the following, a formally strict description and several properties of the unmarked Poisson cluster point

© 1988 Butterworth

Publishers

Poisson cluster processes in structural reliability: K. Schrupp and R. Rackwitz process are presented. Then a special marked Poisson cluster process (PCP) is defined. Its crossing rate out of safe domains of structural states is given. The results are extended to sums of such processes. The Poisson convergence theorem leading to equation (2) is given together with an asymptotic formula for the crossing rate out of arbitrary domains. The results are illustrated by some examples.

Its expected value is given by ~z E[N(t)] = f E[Np(~;t)]E[dN,.(cr)] 0 I

;,fl4(t

=

-

(4)

o-)do-

0

with

Unmarked cluster point process

H(o') = ~ , Fi*(o')P(Z >1 i) = E[Np(O;tr)]

In the context of point process theory, clustering means that points which occur along a time scale can be separated into main points, the cluster centers, and subsidiary points, the points within a cluster. For our purposes the following point process is introduced: N(t) =

f

Np(~r;t) dN~(o')

/'o

(5)

qk(tr) = P(D,, = k)

for Nc(t) = 0

N~(t) / Z No(~',;t) ~'i=

the renewal function of N,(O;o') (F ~* denoting the ith convolution of F). We also shall need the distribution of the number of subsidiary processes D~, which are active at time instant ~r, o" C (0,~). Using the fact that N,.(o') is a homogeneous Poisson process and some standard results in probability theory, it can be shown ~2 that the distribution V(D,,) is a Poisson distribution; i.e.,

(3)

0

=

i=l

= exp ( - ,~(o')) ('~)Y'" for N¢(t) E N

k • No

with parameter

I

o-

where (i) No(or) is a homogeneous Poisson counting process with intensity parameter A > 0 and jump times ~'i, i E N (~o = 0), and

(ii)

A(o') = A l R ( u )

du

and

0

R(u) = ~ pi(l - Fi*(u)) i=1

Z(o')

Np(o';t) = ~ l({p7 E (¢r,t]}) j=l

is a process counting the number of events of a renewal process in the interval (o-,t] with renewal times PT, J E No (p~ = or) and continuous waiting time distribution function F(u), u >>- O,

R(o-) = P(3',, ~< =) is the probability that the subsidiary process is still active at time or. The distribution of the duration L = L(o-) = y o=%Ai' A0 --- 0, of the subsidiary process starting in o- is easily derived by conditioning on the event {Z(cr) = i}: G(l) = P(L(o') <~ l) = ~ piF"*(l)

p(A ~< u) = P(Aj <~ u) =P(PI+I-PT<~u)=F(u)

jENo

which has expectation E[A] = p.. The renewal process is assumed to be finite; i.e., there exists a probability distribution with e(Z(o-) = k) = pk

p k = 1 and

(6)

i=O

k E No

E[Z]<

For the subsidiary process we further determine the distribution of the forward recurrence time 3'~ which is the time to the next event in Nu(r;;=) from an arbitrary time instant o', o" E (ri,=). This distribution is again derived by using results in Lewis ~2 and by remembering that the jump times z,. under the condition {D, = k} have the following distribution: ~5 1-

kENo

Process (3) is similar to the Bartlett-Lewis point process.~2 Cluster centers are the starting points T,- of the subsidiary processes Np(~-i;~) = Z0"~). The clusters are the points in each subsidiary process. For the derivation of the important stochastic properties of the subsidiary process Np(tr;t) and the cluster process N(t) it is assumed that the random variables ,7-,.,PT, and Z(r~) are mutually independent for i,j E N. By some well-known results of point process theory ~5 it can be shown that the cluster point process (3) is nonstationary, regular, and evolves with aftereffects.

P('ri <- I") = I I A(u) du

r~
(7)

0

with ~ = fg A(u) du and A(u) as in equation (5). Thus, we obtain for the forward recurrence time P(T~ ~< hlZ. = k) o-

O'--T

f (F(.+ h)-

=; 0

(8)

0

where H*(u) = E?=o P ( Z > i)P*(u)

Appl. Math. Modelling, 1988, Vol. 12, October

483

Poisson cluster processes in structural reliability: K. Schrupp and R. Rackwitz The inner integral in equation (8) represents the probability of a renewal in the interval (u,u + h] given that there are more than i renewals and the lifetime up to the ith renewal is not greater than u. As mentioned before, the cluster process (3) is nonstationary and evolves with aftereffects. Stationarity for the cluster process can be regained by using a limit operation common in renewal theory. Instead of using the process (3) which starts at time instant zero, another point process is considered which starts in the infinite past. The resulting point process which is observed at time zero, the so-called equilibrium point process, is stationary. Further information and details about this limit operation can be found in Westcott. ~6 Further derivations are mainly restricted to this equilibrium point process. Equation (4), for example, reads ~7

E[N(t)] = AtE[Z]

(9)

Some other characteristics of this equilibrium point process which, to a large extent, can be found in Lewis '7 are repeated here for easy reference. The number of subsidiary processes D which are active at time instant tr is given by a Poisson distribution with parameter A~ = A/zE[Z] = AE[A]E[Z]; i.e.,

P(D = k) = qk = exp ( - Ae) ~.l

kENo

(10)

Equation (10) reveals some insight into the structure of the cluster process N(t). The expected length of a subsidiary process Z(tr) = Np(o-;oo) is E

(11)

Aj = E[ZIE[A] k./=0

and the distribution function of the forward recurrence time in the equilibrium process can be shown to be h

p(7<~h ) = _ txlof(l _ F ( u ) ) d u

h~>0

(14)

Marked Poisson cluster process The previous point process model can be generalized by assigning a mark to each renewal. The marks are independent, identically distributed (i.i.d.) random vectors which assume values in a mark space which is a part of R I. Formally, we have the following definition for the PCP: No(t)

X(t) = ~ ~ Ajl({Np(r,;t) =j})I({L(~',) >t t}) (15) i=lj=l

with Aj i.i.d, random vectors with distribution V(A) = V(Aj). It is further assumed that Aj, r,-, p~, and Z(o-) are mutually independent. In structural reliability the interpretation of equation (15) is as follows: Load changes occur according to the point process equation (3). To each change there is associated a random vector which characterizes the load, for example, by attributes such as amplitude, oscillator frequency, and pulse shape. If a failure domain F C_ R I is assumed, the structure fails if the process (15) enters this domain. A typical realization of the PCP is shown in Figure 1. In order to bound the probability of failure according to equation (1), let MF(t) denote the counting process which counts the number of exits of X(tr) from F into F during the time interval (0,t]. As shown in Schrupp ~8 the expected value of Mr(t) is I

The expected length of a renewal interval of N~(t) is E[~',-+ I

--

Ti] =

E[MF(t)] = / K(o') dtr

(16)

0

1 A

(12)

--

Now, if E[A]E[Z] < l/A, the expected length of a cluster does not exceed the expected waiting time for the next cluster and, thus, the clusters usually are separated. The probability that clusters overlap is small. For E[A]E[Z] = l/A, the expected number of Z~ is 1; i.e., there is always one subsidiary process active at lime instant o- and the clusters change with the same rate as the clusfer centers. This resembles very much the weft-known case o f a Poisson square wave process. For E[A]E[Z] > I/A the clusters overlap, but in the equilibrium case the expected number of Z,, is finite, provided that E[A]E[Z] < ~. This is exactly the condition ensuring the existence of the stationary process. The distribution function of the lifetime of a cluster is given by

with outcrossing rate

K(cr) = limlp(MF(o-,tr + h) = 1) hloh

and where {MF(tr, tr + h) = 1} denotes the event that the counting process MF has an event in (tr,tr + h]. For the calculation of the instantaneous crossing rate K(o-) in equation (16), we observe that the PCP (15) can have three different types of jumps in a small interval (tr, tr + h] (compare Figure I): A jump occurs if (i) a new subsidiary process is generated in (o-,o- + h], denoted by {N,(tr,o- + h) = 1}, (ii) an active subsidiary process dies out in (o',tr + h], denoted by {N2(o-,~r + h) = 1}, (iii) an active subsidiary process has a renewal in (o-,tr + h], denoted by {N3(o',tr + h) = 1}.

I

P(L <~l)

fR(u)du

I p.E[Z] o

t>~o

(13)

484 Appl. Math. Modelling, 1988, Vol. 12, October

For small h the regularity of the cluster point process ensures that these three cases exhaust all possibilities for a jump in the PCP.

Poisson cluster processes in structural reliability: K. Schrupp and R. Rackwitz L("[ i.I }

Xltl

A.

L('til A

.....

I.........

I r

I I I

- - - ~

. . . . .

L ......... I

i

t Zl-1

Figure I

I

i

i

I

c:>l

I

I T'i

"tl. 1

T h e P o i s s o n c l u s t e r p r o c e s s (PCP)

The conditional intensity rates

tribution with p a r a m e t e r / 3 :~ O. W e obtain b -- A(I - po)f(A(l - Po))

1

ri(cr, K) = lim ~P(Ni(~,tr + h) = IIZ,, = k) h+on i= i,2,3

kENo

17) = A(I - Po) f e-a(l-P°)"/3e-t3"du 0

and their limits for the steady state o - ~ = are given below. The proofs can be found in Schrupp.~S

/3,~(1 - po) = A(I

Lemma 1

1

Let k E No and F ' = f . T h e n 1

b = b(o-,k) = l i m E P ( N t ( o - , o +/7) =

I

Po) + / 3 = 1//3 + I/A(1 - Po)

-

= E[A,] + I/E[A/¢(0,1)]

fiZz, =

k)

hlon

= A(I - po)f(A(I - Po))

(18)

with f(¢(o')) = hlira I i 0~P(Nl(o',o" + h) = l l/~/,(0,o'), Nl(0,cr))

being the conditional intensity of N~ given the entire history of ]V, and Nt up to time instant ¢, N, a Poisson process with rate A(I - Po),

(19)

Observing that E[A,] > 0, we obtain b < A(I - Po) and equation (19) is plausible. The quantity A(I - P0) is the rate of the Poisson process N,. The process N, starts with a delay measured from the last event of A/c. Thus, the intensity has to be smaller and depends on the first waiting time distribution.

Lemma 2 Let k E N. T h e n 1

d(o-,k) = lim ~P(N2(o',o" + h) = IIZ,, = k) h ~0

f(s)

/.

= I" e - ' " f ( u ) du 0

the L a p l a c e t r a n s f o r m o f f , and ~-(o-) = o- - min {u ~< otAl,.(O,u) = A/,.(0,o')} the b a c k w a r d r e c u r r e n c e time o f A/,. at instant o-. To make this result more transparent, let us c o n s i d e r the following e x a m p l e with V(A~) an exponential dis-

J

0

k=O

with ff f r o m equation (7). lim d(cr, k) -

o~=

k

E[Z]E[A]

(21)

Result (21) for the stationary case is again plausible.

Appl. Math. Modelling, 1988, Vol. 12, October

485

Poisson cluster processes in structural reliability: K. Schrupp a n d R. Rackwitz In the long run the expected cluster length simply is E[Z]E[A]. Clusters die out with a rate equal to the reciprocal value of this cluster length.

into F during (0,t], we obtain with equation (16) and equation (28),

Lemma 3 Let k E No. Then

which can be used to bound the failure probability in the equilibrium case as in equation (1). If the mark space is strictly positive, then there cannot be an outcrossing due to the dying out of a subsidiary process. In this case, the last summation term in equation (28) disappears. Also, for A,, -> A, the second summation term is dominant.

1

c(o,k) = lim 7P(N3(o-,tr + h) = IIZ~ -- k) h~0r/ Or

z¢

O'--'r

0

E[M'i:(t)] = Kt

(29)

0

(22) k lim c(mk) = c(=,k) = o.--.= E[A]

(23)

Again the stationary result (23) is evident and coincides with that of a usual renewal process. We now state the main result of this section.

Theorem The outcrossing rate K(tr) of the PCP (15) is given by K(~) = b ~ q~(o')B(F,k) + ~, d(tr,k)qk(¢r)D(F,k) k=0

k=l

Superposition

The combination of load processes results in summations over the different components in accordance with, or in good approximation to, the physical context. First of all, we investigate the superposition of s E N i.i.d. copies of a PCP which yields a simple generalization of the results in the foregoing section. Only the equilibrium results are given. Let s

X(t) = ~, X,(t) ~,=1

+ ~ c(o',k)qk(o')C(F,k)

(24)

k=l

s

= ~ I,=l

with b, d(tr,k), and c(mk) given in equations (18), (20), and (22), respectively, and qk(tr) given in equation (5):

B(F,k) = f P(A + x E F ) f k * ( x ) d x .I

kEN (25)

B(F,O) = P(A E F)I({0 E F})

D(F,k) = f (A + x E T')f ~k- i~, (x) dx ,.I r

k >I 2 (26)

D(F, 1) = P(A E F)I({0 E F}) d RI

x (x) dx k i> 2 C(F, I) = P(A ~ F)P(A ~ Y').

(27)

The outcrossing rate K of the stationary equilibrium PCP simplifies to. K = lim K(o') = I " = b ~ qkB(F,k) + ~ kqkD(F,k) k=0 E[Z]E[A] k= i 1

=

k~=,kqkC(F,k)

(28)

with qk = exp ( - A~)A,~/k!, b given in equation (18) and A, = AE[A]E[Z] given in equation (9). Denoting by M~(t) the equilibrium number of crossings of X(o-) from

488

N,-(I)

=

~. ~A~'"I({N~(Ti;t)})I({L"(~'i) >i t}) i=l

j=l

(30) be the superposition of s i.i.d. PCPs X,(t). As will be seen, the calculation of the crossing rate of equation (30) is rather straightforward. The only difference to the univariate case is that the number of active processes at instant ~r needs to be considered. Observing that

D=~D" v=l

C(F,k) = f P(A + x E F)P(A + x E T~) f~k - ~,

+~

of Poisson cluster processes

Appl. Math. M o d e l l i n g , 1988, Vol. 12, October

with D v the number of active clusters in the component u, and using the i.i.d, condition, it follows that V(D) is Poissonian with parameter sAe with Ae from equation (5). Thus, the results of the last section can be used with this slight modification. More generally, the process X(t) is defined as in equation (3), all variables are independent, but now the cluster processes NT, have different distributions. This can be described by different waiting time distributions V(A v) or different occurrence distributions V(Z ~) in the component processes u = 1. . . . . s. Both results in a modification of V(D) = V(Z~= ~D~) and a change in the conditional intensity functions given in Lemmas I-3. The first type of modification is simple. V(D) is still Poissonian, but now with parameter A = Z~= t)t~ and ;t, = E[Z ~] as in equation (5) with the obvious modifications. More subtle is the change of the intensity functions when calculating K(tr). The crossing rate K of the superposition process (30)

Poisson cluster processes in structural reliability: K. Schrupp and R. Rackwitz for different distributions of N~, can be shown to be

K = ~ r,.b ~ ~ qkB(F,k) t,= I

k=O

generates in the limit. By scaling, the distribution of MF, is blown up to obtain a nondegenerate limit distribution. Under conditions (32) and (33) one obtains the following limit result for the point process (34):~8

+ ~, r~d ~ ~'~ kqkD(F,k) v=l

t)tk ,lim - = P(I(/1F.(t) = k) = exp ( - k-~,

k=l

k E No

(35)

s

+ ~. r~c" ~. kqkC(F,k) .v=l

(31)

k=l

with

P~(t) = 1 - P(MF(t) = 0 ) = 1 - exp ( - K t )

i~ 1,

r.=--

h

and

b'=h(l

-p~)ff(h(1 -P~))

The last case to be considered here is when V(A~ .v) # V(A~.~'); i.e., the distribution of the marks in the processes are different. In this case we obtain different compositions of the convolution factors in equations (25)-(27). These factors can be given explicitly but are rather complicated.~8

Poisson convergence for the equilibrium process and an asymptotic approximation for the outcrossing rate Using equation (!) and the results of the last two sections, we can give an upper bound for the failure probability. We now discuss another approach by using the fact that under some specific conditions the equilibrium point process of crossings MF(t) has an asymptotic Poisson distribution. This limit distribution can be used to approximate the failure probability PF(t), a procedure which is common in deriving limit distributions for the extreme values of stochastic processes. 14 The first condition for the process MF(t) tO approach the Poisson process is the mixing condition; i.e., sup sup IP(A N B) - P(A)P(B)] <- g(t) A

(32)

B

with lim,_= g(t) = 0 for all events A and B, where A depends only on the behavior of the process MF until time instant it, and B depends only on the behavior of MF(') after time instant t2 > q, with t2 - tt > t. Condition (32) states that the probabilistic behavior of events which are separated in time becomes asymptotically independent. The equilibrium process MF satisfies condition (32) as shown in Schrupp.~8 The second condition is achieved by a scaling of the time axis. Let (F,,),~N be a sequence of failure domains with lim m,, = lim E[MF.(I)] = 0

i.e., the Poisson convergence for the scaled point process. For practical applications equation (35) yields the approximation

(33)

(36)

where u = E[MF(I)] is given in the last two sections. Practical application of equation (36) requires the calculation of K. The most difficult part is the determination of the factors B, C, and D in equations (25)-(27). A special case will be derived later. However, for standard normally distributed marks and failure domains which have twice differentiable boundaries, asymptotic approximations for these factors can also be derived. The following result generalizes a result of Breitung, 19where an asymptotic approximation for the outcrossing rate of the classical renewal square wave process is given. Since, for independent marks, standard normality can always be achieved by an appropriate probability distribution transformation, the result is of quite general nature. 2° Let the failure domain F for k active processes be given by Fk = {x E Rk I gk(X) < 0}

(37)

and the safe domain S by Sk = {x E Rk [ gk(x) > 0}

(38)

with gk: R ~ ~ R a twice differentiable function. The factors in equations (25)-(27) can also be written as

B(F,k) = P({X(t - ) E Sk} n {x(t) E Fk + i})

(39)

B(F,0) = 1({0 E F})P(X(t) E F~)

C(F,k) = P({X(t - ) E Sk} n {x(t) @ Fk})

(40)

D(F,k) = P ( { X ( t - ) E S k } A { X ( t ) E F k _ , } )

(41)

D(F, l) = P(X(t) E T-')I({0E F}) where X ( t - ) and X(t) are the left and right limits at time instant t of the PCP, respectively. Asymptotic approximations for equations (39)-(41) can then be developed by closely following the arguments in Breitung. 19Making use ofP(A f'l B) = P(B) P(A n B) and neglecting the last summand asymptotically, one obtains k

B(F,k) = ~P(-flk+ ~) H (1 - ~ + ~)_7/.

We consider the sequence of scaled processes

](/IF.(t)= MF,(--~n )

kEN

j=l

(42)

(34) B(F,O)

The meaning of (34) is as follows: Under condition (33) the point process MF.(t) counts rare events and de-

--~ ~o(-

[3,)

k-I C(F,k) = ~p(-/3k) n (l - ~ ) - ' / 2

kE N

(43)

j=l

Appl. Math. Modelling,

1988, Vol. 12, O c t o b e r

487

Poisson cluster processes in structural reliability: K. Schrupp and R. Rackwitz k-2

D ( F , k ) ~ - f l k _ , ) l--I (1

- ~-')-~/-" k > 2

(44)

the error of a finite summation, up to K, say, can be bounded by

j=I

r(K)<~bel + E[Z

D(F, 1) ~- 0 Here, flk is the minimal distance of gk(x) = 0 to the origin: /3~ = Ilxgll = min{llxll ] X E R k, g~(x) = 0}

k=K+

I

and

E kqk< E kEN

,z - = ,

Thus, for g i v e n , > 0 we can find a K = K(,,)t,,) such that the rest of the series is smaller than ,.

(45)

and ~ are the main curvatures of gk(x) = 0 at x = xl'. F o r large values of/3k these approximations can be used in equation (28). As shown by Breitung, 19the limit result~ (35) and (36) remain valid when these approximations are used. A final remark appears suitable. An obvious difficulty in practical computations is the infinite summation required for the calculation of exact, or even asymptotic, outcrossing rates. Observing that the factors B, C, and D are not greater than unity and using the bounds

k~No

[A] +

Example We now discuss a special but practical important example in which K can be given explicitly. Let NAt) be a homogeneous Poisson process with parameter )t = I generating the cluster centers ~'i, i E N. The equilibrium crossing rate for the superposition of s E N i.i.d. PCPs is given by Ae = sE[A]E[Z], according to the fourth section. The mark space for the point process is assumed to be the real line. The distribution of the marks is V(A) = N(/z,o -2) and the failure domain is F = (a,~), a > 0. Then, according to equations (25)-(28), we have the exact formulae by using a table for normal integrals given by Owen: 2~

kqk=A~ E qk =-'2

k=K+l

k=K

i0(x+ ~ o),~ , ~ ) a

B(F,k) = _=

dx kEN

=~[((k+l)~-~)/~-k~V~+l ' V~ ' ~ ]

oo

D(F'k+l)=fdp(a-x-lx)

~q~\{x-klz'~/"ko" ]dx

Q

kEN D(F,I) = 0

~

C(F,k + 1)

~ \

v ~ )dx

R

= 4~ [(a - (k + l)t~)/cr ((k + l)t~ - a)/cr, V~ + l

'

v~ + i

where d~[x,y;p] = ~x)¢k(y) + fg q~(x,y;p) dp is the standardized bivariate normal distribution function with correlation coefficient p, and q~(x,y;p) is its density.

488 Appl. Math. Modelling, 1988, Vol. 12, October

k v~-45

] kEN

The factors B, C, and D could also be given analytically for marks having nonvanishing auto- and crosscorrelations, following the approach in Rackwitz. 9

Poisson cluster processes in structural reliability: K. Schrupp and R. Rackwitz 0.6-

~

EII~I=2.1 2.5 3 -L

0.5-

O. 8. " ~ --

$= | 3 t.

0.7.

°

0.4

•

0.5-

0.3

8.dl. 0.3-

8.2,

0.2-

8,

7"

5

0.1

~ ~ . ~ - ~

10

5

_-=-~_ ..... 1'O

Figure 2 Variation of E(A)

Figure 4 Variation of S

H o w e v e r , it appears difficult to assess a proper correlation matrix among the marks of different subsidiary processes, so this possibility is not pursued further. In Figures 2-4 the outcrossing rates for different values of the parameters E[A], E[Z], and s are plotted against the threshold a ~ [0,10]. In each figure one p a r a m e t e r varies while standard values are kept for the two other parameters. The standard values are E[A] = 1, E[Z] = 1, and s = I. The parameters of the normal distribution are /x = 0 and cr = 1. The summation procedure in equation (30) has been truncated at K = 30, implying an error smaller than 10 - 9 in all cases. As expected, the outcrossing rate in equation (28) is dominated by the last term, which refers to t h e l o a d changes. In Figure 5 lower and upper bounds for the failure probability are given for the standard values of the parameters. The upper bound is given in equation (1); the lower bound can be found in Bolotin, '3 page 377. As can be seen, the estimation of the failure probability by equation (I) becomes fairly accurate even for moderate values of the threshold a. In Figure 6 the outcrossing rate equation (28) is

plotted against the outcrossing rate of a Poisson square wave process with the same parameters for )t and E[z~]. This rate can be found in Breitung and Rackwitz. g The figure shows that clustering increases the outcrossing rate and, thus, cannot be ignored when modelling stochastic load processes for clustered phenomena. In Figure 7 the crossing rate equation (28) was calculated with the parameters as in Wen and Pearce. t' They used a normal distribution w i t h / z = 1 and cr = 0,3. Their expected clustei- length is E[A]E[Z] = E[L] = 0.001, the Poisson p a r a m e t e r is ,~ = 6, and the number of superposed processes is s = 2. For different values of E[A], keeping the product E[z~]E[Z] at the constant value 0.001, the outcrossing rate equation (28) is plotted against the threshold a E [1,4]. For smaller values of E[A] the outcrossing rate increases in accordance with equation (28). The change rate of the loads I/E[A] increases and, simultaneously, with it the contribution of the last s u m m a n d in equation (28). In their model Wen and Pearce obtained smaller values for K, which is due to the strong filtering effect; i.e., the probability that clusters in a single process overlap is small.

i: 13.6

8.6

8.5

,

0.5

8.4

13.4. 0.3

13.3

13.2

0.2

8.

13"1 t 5

Figure 3 Variation of E(z)

I 10

,a

110

•

Figure 5 Upper and lower bound for the failure probability

Appl. Math. Modelling, 1988, Vol. 12, October 489

Poisson cluster processes in structural reliability: K. Schrupp and Ft. Rackwitz 3 I~P PSP

e.5"

4

8.4"

5 8.3-

6

8.2

7

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Figure 6 Comparison PeP and Poisson

square wave process

(PSP)

11 12

8.9" 8. fl-

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13 14

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15 16 17 18

Figure 70utcrossing rate of PCP for the parameters in Wen

19

and Pearce

Acknowledgment This study was supported by a research grant from the Deutsche Forschungsgemeinschaft. References 1 2

Hasofer, A. M. Time dependent maximum of floor live loads. J. Eng. Mech. Div., ASCE 1974, 100, 1086-1091 Wen, Y. K. Statistical combination of extreme loads. J. Struct. Div., ASCE 1977, 103, 1079-1093

490 Appl. Math. Modelling, 1988, Vol. 12, October

20 21

Rackwitz, R. and Fiessler, B. Structural reliability under combined random load sequences. Comp. and Struct. 1978, 9, 489-494 Larrabee, R. D. and Cornell, C. A. Approximate stochastic analysis of combined loading, Research Report, MIT, Cambridge, Massachusetts, 1978 Gaver, D. P. and Jacobs, P. A. On combination of random loads. SIA'M J. App/. Math. 1981, 40, 454-466 Wen, Y. K. A clustering model for correlated load processes. J. Struct. Div., ASCE 1981, 107, 965-983 Shinozuka, M. Stochastic Characterization o f Loads and Load Combinations, Structural Safety and Reliability. Elsevier, Amsterdam, 1981 Breitung, K. and Rackwitz, R. Nonlinear combination of load processes. J. Struct. Mech. 1982, 10, 145-166 Rackwitz, R. Reliability of systems under renewal pulse loading. J. Eng. Mech. ASCE 1985, 111, 1175-1184 Madsen, H. O. Load models and load combinations. Thesis, Structural Research Laboratory, Technical University of Denmark, 1979 Wen, Y. K. and Pearce, H. T. Stochastic models for dependent load processes. University of Illinois, Urbana, Illinois, 1981 Lewis, P. A. W. A branching Poisson process model for the analysis of computer failure patterns. J. R. Statist. Soc. B 1964, 26 Bolotin, V. V. Wahrscheinlichkeitsmethoden zur Berechnung yon Konstruktionen, VEB Verlag fuer Bauwesen, Berlin, 1981 Leadbetter, M. R., Lindgren, G., and Rootzen, H. Extremes and Related Properties o f Random Sequences and Processes. New York, 1983 Snyder, D. L. Random Point Processes, Wiley, New York, 1975 Westcott, W. Results in the asymptotic and equilibrium theory of Poisson cluster processes. J. Appl. Prob. 1973, 10, 807-823 Lewis, P. A. W. Asymptotic properties and equilibrium conditions for branching Poisson processes. J. Appl. Prob. 1969, 6, 355-361 Schrupp, K. Austrittsraten yon markierten Poissonschen Clusterprozessen und ihre Anwendung in der Zuverlaessigkeitstheorie. Berichte zur Zuverlaessigkeitstheorie der Bauwerke, 77, Technische Universitaet Muenchen, Muenchen, 1986 Breitung, K. Asymptotic approximations for the maximum of the sum of Poisson square wave processes. Berichte zur Zuverlaessigkeitstheorie der Bauwerke, 69, Technische Universitaet Muenchen, Muenchen, 1984 Hohenbichler, M. and Rackwitz, R. Non-normal dependent vectors in structural safety. J. Eng. Mech. Div. ASCE 1981, EM6, 1227-1238 Owen, D. B. A table of normal integrals. Communications in Statistics, Simulation and Computation, B9, 1980