On the calculation of the joint probability density of slightly non-linear stochastic processes

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Sonderdruck aus Sonderheft

GAMM-Tagung Lyngby 19

Band 58 1978 TECH!USCHE UNIVERSITEIT Laboratorium voor Scheepshydromechanica Archlef Mekeiweg Z 2628 CD Deift Tot.: 015.788873-Faz 015-781838

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Optiiiiization, Stochastic ProcêsBes and Mathenatica1 Methods in Economics T 481 ZAMM 58, T 481 T 482 (1978)

J. JUNCBER JENSEN / P. TERNDRtTP PEDEBSEN

On the Calculation o the Joint Probability Density

oI Slightly Non-Linear Stochastic Processes

Introduction

In order to obtain more realistic design cöncepts of structural elements a great deal of attention has been paid

in the recent years to the problem of replacing the empirical safety factors by more welldefmed critena based upon

a prescribed probability of failure The failure criteria can often be formulated as quadratic relations involving

stochastic variables with known distribution fùnctions. As eíamples, we can mention y. MisEs' yield criterion, plate buckling criteria for combmed loading and hunt load criteria for beams and frames Then, to calculate the piobabiIity of failure either the probability density fôr quadÑtic variables has to be calculated or simulation tech-niques have to be used. The first possibility is the topic of the present paper.

Formulation

Consider two normalized stochastic variables , i given by the quadratic relations

=

_±

_{n = ßi + rßiji,}

(1)

where the mutually independent variables ; i = 1, 2, ... , N, are standard normally distributed. The summation convention is used, implying that each product is suhimed over all repeated: suffixes, from i to N. Without loss of generality we take

₌

= L Furthermore, it will be assumed that oß = O.

It has been shown by LONGUET HIGUENS [1] and VrNJE [2] that in the cases where e is a small parameter, the

joint probability density p( n) can be suitably evaluated by a perturbational method using the cumulants of the distribution, yielding

=

_{Ji + E ¿}

'

HqHp()' He()

. (2)

I Sr

I r'=i p+q=0

I.

p,q0

Here He( ) denotes the HERMITE polynomial of order p. The coefficieíit H;q can be found, see ref. [2], from the following identity in the dummy variables 8 and t

Sr

lf

_i

v'Cr E

' (E

kgi __(82 +t9)

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r=1 p+q=O n=i \=0 ¿=0 2 _/

p,qO

where 'the coefficient ,j are the cumulants of the distributiOn. In the following, a very, effective method for the

calculation of H is described.

Using the results given, in ref. [1] and [2] it is found that the cumulants XA can be written

= kkt»1'2 ±hw

. (4)

where the coefficients kkj and h .aEe independent Of e and given by

=

2k+1_2k!l.!{-_PZ2.1

_{+ PZ-i,_i ±}

1 Pßß}

h =

left! where iterms V jterms i, i O

Pii = E oe

.... "hkß1m ...ß,g

i or j <O,

i terms j terms

The summation convention is applied in each matrix product.

V The summations in formulae (6) are to be. taken over all distinct, irreducible matrii products. Thus any

suffix must appear twice and only twice in each product, with the restriction that no matrix o andß may have

two equal sii,ffixes and that the suffixes on and ß, in formula (6 a) be different Thereby each sum contains (z + ) U (z j I) terms It should be noticed that the cyclic nature of P is not taken into account since it yields no computa

tional advantages. V

Substitution of equations (4) and (5) into the right-hand side of the identity (3) and use of a well known for mula from the theory of multinominal coefficieñts yield a new identity, .which holds for each value of r

Sr r (g(i))a (g(2))ai (g(r))ai. ,V

E

2'

_{" E}

.

,,

, . (7a).

p+q=0 n=i a a1.a2. ... ar. .

where

j+2 i

g(i) =

_{'g,i,)ei+2_iti + E g? 8iii:}

_{(7 b)}

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(6a)'

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T 482 Optimierung, Stochastik und mathematische Methoden der Wirtschaátswissenschaften with

(i)

P

- P

+ -

g(i) =

-

P

2

ji+1,i-1

2

ji+2,i-2 '

2i _2j 3,%

The summation is to be taken over all integer values of a1, a2, ... , a, satisfying

a

r r

' ma,,, = r and

.' am

=

m=i m=i

From equation (7b) it follows that the product q(i)g(k) can be written

j+1'+4 j-f-L

=

_{+ .'}

_{Gi-'-1t' +}

_L

i-0 i=0 1=0

ii, w4 rn,

G'

=

Y (i),(k) G

= L

ri(i) V) G2

=

11 ,1(k)

+

> (1(J) (J(k)

i

1mi,im'

i 2,ni'2,ini' i 2,ma1,fn4

m-rn1 mm, rn=m, 1l4=1!

where

with

(S a)

(Sb)

=

max (0, i - k - 2) ,

in2 = min

(j +

2, i)

=

max (O, i - k) , in4 = min (j,i) In general, it will be seen that the product

n r+2m

(y(i))a1 ((2))a4 (g(r))ar _.L' _Eirnrn8'2" , (9)

ì,=0 i0

where the coefficients are found using the scheine (8) repeatedly. Finally, we obtain from equations (7a) and (9)

ii;,,

=

2T

j.

y-. Equr,, (IO)

n=mnx(i,u) ,, a1!a2. a. where u

= --

(p + q -

r).

A special application of equation (1)(2) has been considered in detail in reference [2, where 7) is taken as the time-derivative of the stationary process . Then by assuming that the processes ¿ have narrowbanded spectra, it follows that the probability > ] that the individual maxima of exceed some level is given by

PRm,, > ]

f17)I p( 77) di)!!¡Ij p(O, 1) d77 (11) and, as shown in [2], the integration can be carried out analytically.

Numerical example

The theory has been programmed for an IBM 370/165 computer. In the numerical calculations the infinite limit on the summation in equation (2) is replaced by a finite number im. In order to check the convergence of the method the probability

_{> ] for}

= ¿ + e was calculated according to equations (2) and (1 1 ) , for variousvalues

of , and m. This problem can be solved analytically, see Li [3]. It was found that : (i) the series should be

trun-cated after the e'°-term, i.e. m

=

10, since the range of convergence only increases slightly for higher values of m, whereas the computer time increases approximately as rn45 with a CPU-time of approximately 5 sec. for rn

=

10, (ii) in the range .04 .2, should not be chosen higher than about. 5/e in order to obtain convergence with m

=

10 or, alternatively, the desired probability levels should not be chosen smaller than about e

Solutions have been reached for other values of matrices a, _ß, andß than those used above, and also for

other values of the number N of the variables . These solutionsindicate that the above-mentioned conclusion (ii) approximately holds good in general.

Finally, consider a piece of material witha yield stress c, and loaded by a normal stress o and a shear stress r. The y. MISES yield criterion states that when Oef

=

o + 3v2 o yielding will occur. If cr and r are taken as sta-tionary GAussian, stochastic, narrow-banded processes, the probability P

=

max > a] that the individual peak reference stresses aref,,,x exceed the yield stress o, will be a useful measure of the probabifity of failure. If, as an example, we take o.,

=

.6325cr + a, and r

=

r,, where a, and r, are stochastic processes with zero means and variances equal to (.l5o)2 and (.05a,,)2, respectively, we find P

=

5.4. 1O-. It should be mentioned that if the pro-bability P is calculated according to the yield criteria, linearized in the stochastic variables a, and r,' wefind P

=

= 4. 10-6, thus underestimating the probabilityof failure by a factor 102.

References

1 LONGUET-HIGGENS, M. S., The effect of non-linearities on statistical distributions in the theory of sea waves, J. Fluid Mechanics,

17, 459-480 (1963).

2 VINJE, T.; SKJORDAL, S. O., On the calculation of the statistical distribution of maxima of slightly non-linear, quadratic, stationary

stochastic variables, International Shipbuilding Progress, 22, 265-274 (1975).

3 Lru, Y. K., Probability distribution of stress peak in linear and non-linear structures, AIAA 1, 1133-1138 (1963).

Anschrift: Dr. J. JUNCRER JENSEN; Dr. P. TEENDEUP PEnxasEN, Instituttet for Skibs- og Havteknik, Danmarks Tekniske Hojskole,