RD-RI52 164 A SIMULATION STUDY OF MODELS FOR COMBINATIONS OF RADM 1/1LOADS(U) NAVAL POSTGRADUATE SCHOOL MONTEREY CA J K NONSEP 84

UNCLASSIFIED F/G 9/2 N

1.8.

IIIJI25 1111_!-

MICRCOP RESLUTON TST HAR

NAIKN~i i~f~l "

NAVAL POSTGRADUATE SCHOOLMonterey, California

co

DTiC&y ELECTE

APR 5 W56

THESISA SIMULATION STUDY OF MODELS

FORCOMBINATIONS OF RANDOM LOADS

Amhor -or

"_'.._ 7h ui' Advisor: . .. "- -':.. .. -A:

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A Simulation Study of Models for Master's Thesis;Combinations of Random Loads September 1984

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Noh, Jang Kab

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Naval Postgraduate SchoolMonterey, California 93943

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Constant LoadShock Load

First-passaae Time

': ,,S~L Cotl~l o r- -'eso v do It Iec~ r, ,i dentify by block numbarlThis thesis describes a model for a combnation of random loads acting

upon a physical structure, such as a building or ship. The various loadsrepresented in a model might be winds, tidal effects, or even earthquakes or

*• snow loading. Asumptotic results are given for the first-passage time for theload combination process to exceed a given stress level exceeding structuralstrength. The accuracy of using the asymptotic results to approximate the

"first-passage time, or time to structural failure, distribution is assessedb simulation, o.

" DD F o43 1473 E:'1ON OF NO, 65 ICOSOLETE1 UNCLASSIFIED*. 1 " ,SECURITY CLASSIFICATION OF THIS P^G9 (When Dore Entered)

&BSTACT T

This thesis descriles a model for a comtination of

randcm icads acting upon a physical structure, such as a

building or ship. The various loads represented in the

rodel right be winds, tidal effects, or even earthquakes or

snow loading. Asymptotic results are given for the first-

passage time for the load combination process to exceed a

given stress level Exceeding structural strength. The accu-

racy of usin~g the asymptotic results to approximate the

first-passage time, cr time tc structural failure, distribu-

t_;on is assessed by simulation.

3S

• S .

S

S -.

-S

S. ..

Approved for public release; distribution unlimited.

A Simulation Study of Modelsfor

Combinations of Random Loads

by

Noh, Jang KabMajor Korea Air Force

B.S., Korea ir Force Academy, 1974

Suhmitted ir partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN OPERATION RESEARCH

from the

NAVAI POSTGRADUATE SCHOOLSeptember 1984

Author: A~/~ "

Approved by:bbbbbbbbbbb

"--.---'/a

repartment of Operation Fsearch"

Dean of Information and Policy Fciences

2.-

TABLE OF CONTENTS 0

I. DESCRIPTION OF A STOCHLSTIC MODEL FOR

CCMBINATIONS OF RANDOM LOADS ... ........... 8

II. ASYMPTOTIC RESULTS FOR THE DISTRIEUTION OF THE

.IRST-PASSAGE TIME Tx ...... ............... 11

A. THE DISTRIBUTION OF TX FOR LARGE X ....... .11

B. THE TAIL OF THE DISTRIBUTION OF Tx FOR

FINITE X ........ ................... 12

III. CG1PUTER SIMULATION ............... 16

IV. ANALYSIS OF THE SIMULATION RESULTS .. ........ 21

A. THE DISTRIBUTION OF Tx FOR INCHEASING X . . . 21

1. Constant and Shock Load Magnitudes

Have Identical Exponential

Distributions ...... .............. 21

2. Constant and Shock Load Magnitudes

Have Different Exponential

Distributions .... .............. 23 S3. Arrival Rate of Shock Loads Depends

on the Constant Load lagnitude ..... 24

4. Load Magnitudes Have a Pareto

Distribution ..... ............... 25

3. -,!E EYPONENTIAL TAIL OF THE DISTRIBUTION

CF T% FOR FINITE X ..... .............. 26

C. CONCLUSIONS ...... ................. 27

V. SIMULATION ALGCRITHM ....... ............... 45

A. VARIAELE DEFINITION 4............. 45

S. ALGORITHM ....... .................. 5"

4-S •.

APPEN'DIX A: COMPUTEP PROGRAI .............. 47

ilIST C! rEFFRENCFS ................... 59

INITIAL DISTRIBUTION lIST ............... 60

i ,; 2 T. -: I-

i A -.!.jj,'t ,t CO ews

j-,1. t specieal II --

-..jII

.-.i l l- - .- -- ..' -.i .' .".. -...- i-. .- --i i > .> . -' -. . '.. i. --i i - -i L i'. -.i .- ...:, . >ii >i > -ii .21..i .l- 1 -12 .i."i '. - ..2 1 .-'.i.. " -. .. .i

LIST OF TABLES

A.1. Identical Exponential( case A-i ) ........... 28

A.2. Identical Exponential( case A-2 ) ........... .29

A.3. Identical Exponential( case A-3 ) ........... .. 30

B.1. Different Exponential( case B-1 ) ........... .. 31

B.2. Different Expcnential( case B-2 ) ........... .. 32

B.3. Different Exponential( case B-3 ) . ........ 33

C.1. Varying Arrival Rate( case Z-1 ) .... ......... 34

C.2. Varying Arrival Rate( case C-2 ) ... ......... .. 35

C.3. Varying Arrival Pate( case C-3 ) ... ........ 36

D.1. Pareto Distritution( case D-1) .... ......... .. 37

r.2. ?areto Distritution( case D-2 ). ......... 38

D.3. Pareto Distrihution( case D-3) .......... 39

E.1. X(x) and r(x) ( case E ) .... ............. .. 40

2.2. Quantiles ( case E ) ...... ............... 41

.I. 1((x) and r*(x) ( case F ) ...... ............. 42

F.2. Quantiles ( case F ) ...... ............... 43

G.A. Confidence Interval in case E ..... ......... 44

-

-

• - • "

|C

LIST OF FIGURES

1.1 Vhe 'Load Comiration Process.............10

13.1 Generating T....................17

I7

I. DESCRIPTION OF A STOCHASTIC MODEL

FOR COMBINATIONS OF RANDOM LOADS

many physical structures are threatened by cgmtinations S

of loads of varying ragnitudes from various sources. For

instance, bridges, piers, dams, and buildings cin experience

loads frcm wind, snow, ice, tides, earthquakes, and so

forth. In many instances the total load or stress experi-

erice 1-v a structure varies in time in an apparently random

fash icn. Certain components of the loads vary ratAer

slowly; others occur more nearly as impulses, such as those

associated with winds or earthquakes. The problem is to

de-sign i structure to withstand the superposition of raindom

loals from many sotrces with at least an approximately

understocd Irobability.

The purpose of this thesis is to describe and investi-

gate certain simple but somewhat realistic probabilistic

loal models for use in design, and perhaps safety assessment

of structures. In this thesis we confine attention tc the

superposition of just two load types; shock loads, and

constant loads. For example, winds gusts, flash flools, ind

earthquakes have varying magnitudes and have relatively

short durations in ccmparison to the times between their

cccnrrences. These will be modelled ds instantaneous shock

loads. Cn the other hand, snow, ice, or water accumulation,

or even the presence of slowly moving vehicles present loadsthat remain nearly ccnstant in time, occasionally changing 4

tr) new levels. These will be modeled as constant loads that

change infrequently. Throughout this invEstigation it will

i1 assured that the effective strauss exprtc by several

types o loads acting simultaneously can he expressed as a

linear combination cf the component loals, the load

comporents heing treated as stochastic processes.

2

The times between changes in magnitude of the constant

load Frocess are independently and identically distributed

with distribution function H. The successive magnitudes of

the constant load are independently and identically distrib-

uted with distributicn function F. Given the constant load

jprocess, the shock load process is a compound Poisson

process. The successive shock load magnitudes are indepen-

dently and identically distributed with distribution func-

tion G. The conditicnal rate of arrival of shocks, given

that tle magnitude cf the constant load process is x, is

4(x)let X(t) be the magnitude of constant load process at

time t; and Y(t) be the magnitude of shock load process at

tiwe t; Z(t)=X (t) +Y(t) is the superposition of the two loads

at time t, and M (t) =SUP Z(s), the maximum load combination

in :C,t , see figure 1.1.

Let Tx=[inf tO: Z(t)>x ) ie, the first-passage time

for the load combination to exceed a stress level x.

will represent the time to failure of a structure whose

strength is x, and is subjected to a stress history

[Z (t) ; t O].

Gaver and Jacobs (1981) studied the above inolel for the

case in which the distribution H is exponential and the

shock load rate m is a constant independent of the shock

load process. Related load coibination models that have

been studied include Peir and Cornell (1973) [Ref. 1] wen

(1I-) [Pef. 21 Pearce and Wen (1983) [Ref. 3].

Asyrtotic results which appear in Gaver and Jacobs

(19R4) and Jacobs (1984) will be described in chapter 2 for

the distrilution of Ix as x oe and for the tail of the

0 stribution of Tx for f-nite x.

in chapter 3, a simulation program for the load

combination model will be described.

X t}

Y '-

I - - - - - -t - -lo- - - -

z t

I, a s

0 ---- I

I---'"

I I

asymtotic results of chater 2 to approximate the

istribution, of Tx will be described.

.7

4--!

o b D m S____-____________ n _______

"'" " " "" " " " '__'__"_... __"__" __"_"" __..... _______ ...... "____--__ --__ -

010

n-S

10w

II. ASYMPTOTIC RESULTS FOR THE DISTRIBUTION OF

THE FIRST-PASSAGE TIME TK

A. THE DISTRIBUTION CF TX FOR LARGE X

--his section suroarizes some of the results found in

GdVer and Jacobs (1981) [Ref. 4] for the model in which the

distribution H is exEcnential(x) and the shock load arrival

r.ite Ai is constant independent of the constant load Frocess.

virst, we consider the distribution function of the

raximum load combinaticn to occur luring 'O,t], M(t).

H (t) =P {;(t) _< X)=P('I(t) <x,T, >t) +[M (t)_5x,T, <t} ---------- (2-1)

where T, is the arrival time of the first shock.

A renewal argument yields;

F (M (t) _<x,T, >t) = e- o'eXPC-j.j - )) c1) j

M f (t) ~x ,r T<t] =f\)e'Ayd~JeP4&io.~(t, A~

where ?(x)=1-r(x).

Next take Laplace transforms with respect to t of (2-1).

=x ( )+ A i 4 -- )(-)---------------- (2-2)

h here

~ )=Jj~4 ~ 4& -~ ~ (y)--------------------------3

FEnce

--------------------- (2-4)

:t seems to be difficult tc invert H,(g) for any inter-

estin choice of the distributions F(x) and G(x). However,

useful information can still he gleaned from equation ('-L).

Fir-t, from the lefirition of T,

I {' (t) <x} =P (T, >t) --------------------------- (2-5)

Thus, ( = ( >t) t-------------------- (2-6)

et -+o, in equation (2-6) to find that

01

m W E (" ) HX()0) /-A-- --(-) .. ---------------- (2-7)

The next result is the limiting property for the first

time the load combination process exceeds a given level x.

The limiting distribution of TX is exponential in the sense

thatlin P fm (x) 4 TX>t) =e - t - - - - - - - (2-8)X-4 X.

where x,=inf (t;F*G(t)=1]

The Erccf of this result for a more general model that

includes that of chapter 2 can be found in Jacobs (1984)

[Ref. 5].

B. 7RE TAIL OF THE DISTRIBUTION OF TX FOR FINITE X

In this section we will summarize results for the asymp-

totic behavior of P ( x>t) for large t and fixed x as tle

for the model in which the distribution H is exponential and

the shock load arrival rate is constant. More letails and

results fcr a more ceneral model that includes that of

chapter 2 can Le ?unl in Jacobs (1984)

Frcm the equations (2-1) and (2-5)

S(T t) = e, Fcd*) Pe-A 4(- _t

+ Al.I e'-s t L'Fav e'A!&:F-0)S PC ( > -S) ----- (2-9)

let

L (ds)= f jC(df)Ae A[+4?W(j*. ))5 ds (2-10)

1 (t) = f gd ) A ) (,_ e-[ L-----------(2-11)

1(0= ) F(d-) (2-12)

The renewal equation (2-9) gives;

P(T)>t)=g(t)+ f4L(dS) HX(t-s) (2- 131

wherE g (t) =f' F CdV9) e-[A+AX&(X-)t (- 14)

*ollowing the development in Feller (1971; page 376)

[Ref. 61 it will be assumed that there -xists a number -

such that

J e L )--------------------(2-15)

12

"4'- /-C.

This roct is unique and since the distribution of I is

defective, is positive.

LetL~(d 'v eAS Ld-------------------------- (2-16)L* (ds)= e""x) L(ds) -(-6

g (t) Y ) 8Lt) (2- 17)

- > (xt) M ------------ (2-18)

Then( >t)=g (t) + 0 PO (T,> (t-s) L* (ds) --------- (2-19)

it is assumed that the assumption for the key renewal

theorem holds. Arplication of the key renewal theorem

yields

1 (T >t) 0 (t) It (2-20)

0

=J'F(dy) / (2-21)

Rewriting the equation (2-15), the equation determining

is1= f X' 5 A 4 ACV(X )S AS

r- fX cdq) ----------- (2-22)

and

=-- ----)------- (2-23)

The key renewal theorem implies;

Lim e%(Ie 3 (Tx >t) . . ... ' . (2-24)t- Cc Aj ,rF(dt ) -

Examles: CX A(tK- ) -J)

It appears to be difficult to solve for X analytically

in general. We will discuss the numerical computation of '

for two examples. In both examples, A= =1 and the

distxiibutior. of the shock and constant load magnitudes F and

G are of the form;F(x)= c: x:-

..............................................

W "

(X) r--

M

In example 1, a=h=1 and in example 2, a/b=2.

To compute the : for F(x)= e- " ,(x) = e-4

let xf (7) = fcdx ) r 1 d vC'-i) -- - - - - (2-25)where A= A =1.

Fcr model 1,

if n)(,-a~e'.e - -- (2-26)for X<<+e - x andxtl.

TjrX=1, f(x) has a singular point, so by using the

L'IICPITA1 3RI [Ref. 7]

im f(x)= Lim g'(x)/h'(x) 0

where gb,) is the numerator and h(x) is the denominator

Cf the function )For X =I ,

f (1) (ex - e x ) ---- (2-27)

Fcr model 2,

f (W) = z Y ' -

+ e-ab 2-28)

for %<1+e-' and -Y1. ui teaYor k=l f(:) has also singular point, so by using the same . -

.ethcd as ured in model 1;

forj( =1 a(I) = (,ebx

Lifferen tiat ion of the ecuation (2-25) gives

+ -- ------------ (2-30)

E.cuation (2-30) is greater than 0 for 0_<X<1 e and V1. So

the function f (x) is a monotone increasing function in the

interval 0 X<1+e - . Using this result, we compute the Ix

from the ecjuation (2-22) by applying the bisection method

[Ref. 83.

U.sin the computed * we compute the constant I such that;

is the ccnstant of equation (2-24).

14

° -.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..o °" .•• ° i°° . ' °o o .°'-°°°.°'° .io.-. i°

°.°.. -. % .•°° 'o°-°'j ° . ' ° °-.° ' °° °'% , ,

For case a=b=l,

(~~q3~ {(- e)-eiv(xe -') Z. 4 J- (2-31)

For case a/b=2,

(I Z (IYQ* e4)

+3 e-2b .f( 6X ~ -% 2-32)-o~~e4 )3

0

III. COMPUTEP SIMULATION

This section describes tLe computer simulation moel.

'Ae denote the constant load arrival rate by A, the shock

load arrival rate byA, the distribution of constant load

magnitude by F (x), and the distribution of shock load magni-

tude by G(x). The simulation model will be used to study

the quality of the approximations of the distribution of TX

by the asymptotic results described in chapter 2.

The following describes the simulation algorithm. The

stress levels are x, <x<x3 . . .. .. . . <x.

1, Set T=0, NO =0.

2, Set V=0, generate a constant load magnitude,C, and

duration interval Tc. Find the largest index i>N,

such that x- :<C.

Put N, equal to that index and

set

-A; =T for N, <i-<N,

set N, =N,

3, Generate a shock load magnitude,S, and a time until

the shock occurs,Ts.

4, If the shock interval does not exceed the excess life

of the constant load duration,Tc, find the largest

index i>N, such that x _<C+S.

Let N, be that index.

Set T =T+V+Ts for No, <iNI.

Set N =N,,V=V+s, Tc=Tc-Ts. Go to step 3.

5, I:f the shock interval exceeds the excess life tirE

of the constant load duration,

set T=T+Tc, go to step 2.

16

........................ ..

-'" "". """""":"""":". :"" """. '"""""""""" '".. "" " ".: --: -,_.. - .-. .-.-"- - - - -:-_:

One realization cf the simulation is completed when one

TX is comiuted for each level x; ,15i!5n. 3ach siimulatior.

consisted of 5000 replications. Further letails are

Z(t) - - Tx4-

I x4

x3 JxTx2

xl- Tx 1 . ...

Figure 3.1 Generating T.

iI~clided in chapter 5.

-he key subprograrps are following. Other programs will

bc exilained by reading from the output results.

FAKDC ; generate the Tx.

CC.L-UT ; compute the P(Tx=3), true P(Tx=0), sample nean,

true mean, standard deviation, coefficient of

variance.

Sample prcbability

F __1: : __ is obtained from

the sample.

rue probability-(,-x=0)=1-P(TX>0)=1-P (M(0)_<x} =l- J,'F(dv)=F (t) i s

obtained from the equation (2-5)

The sample mean

and

the true mean E (Tx) is obtained in special

cases from the equation (2-7).

See details in Gaver and Jacobs (1981).

Sample standard deviation

= , (-ry- ) is obtained from the sample,

anI

Sample coefficient of variance

CC'V)- is obtained from the sami!e

mean and sample standard deviation.

CCMP ; compute tIe sample guantiles, and the quantiles

cf an exponential distribution with mean E(Tx).

A sample quantile is

=(oxsanple size)th order statistic of sarple and

e.( =-E(T.)I.( -. 4) is the yuantile of exponential

distribution with mean E (Tx)

CCM ; compute the quantile of the sample conditional

distribution given T >0 and the exponential distrib-

ution with mean -(Tx)/P(Tx>O).

7he sample guantile of the conditional distribut-

ion

x (number of positive TY) ]th order statistic

is obtained for sample cf positive Tx.

The conditional quantile of the exponential dist-

ribution with mean E(Tx)/P(T)>0) is-E (-r")

pL v>o) n (1-dL).

SK ; corpute the I such that;A___

1(= ( 'K(x) FL) -I by applying the

bisection method.

INT ; compute the constant [ such thatA

b' substituting the Ik found in SK.

18

"' -°~~ ~ ~ ~~ ~ ~ ~~~~~~~ ~~~~ " - " - " "" " --" . , . . " - ., - ' . . . -- - .- -. -- , - . . - . . .- - . ... . . . ..' - . . . . - . .

The following cases were simulated.

"istribution case A A(c) F(x) G(x)

a,ex-cnential 1 1 1 1-e-e

(a=b=l) 2, 1 2

3, 2 1

b,expcnential 1, 1 1 1-e-2 1-e~z

(a=2,b=1) 2, 1 2

3, 2 1

c,expcnential 1, 1 1/c 1-e-" 1-e-l

with varying 2, 1 e- "

arrival rate 3, 1 e C"

d,Fareto case 1, 1 2 1 -,b 1-e-X

2, 1 2 I-(,) 1

3, 1 2 1- 1- -

TLese different settings will be referred to as case (A-i) ,(A- 2) , (1%-3) , (B-l), (B- 2) , (B-3) , (C- 1) , (C- 2) , (C- 3) ,

(-1) , (D-2) , (D-3) respectively throughout this paper. F'or

each above case, we compare the asymptotic result (2-8) and

the experimental results.

The next setting is to examine the exponential approxi-

mation of the tail cf the distribution of Tx for finite x.

The cases of identical exponential distributions,

Y(x)=G(x)= e-Y ani lifferent exponential distributiors,

.(x)=e "Ax G(x)=e -bx where a/b=2

were considered.

The following conditions are considered.C-, A XI=' (x) =e -X

X =A1 F(x) =e-7' ,(x) =e-'

19

These different settings will he referred to as case (E) and

case (F) respectively. For each case, compute the X., and

constant V" and compare the .]uantiles of the asymrtotic

results (2-24) to the simulated data for each level of x.

2I0

I S

, "I

I I

I I

, I

. . .. I. . . . . . . .

IV. ANALYSIS OF THE SIMULATION RESULTS

As indicated in chapter 2 the exact distribution of Tx

is difficult to obtain in general. In this chapter sirula-

tion will be used to study the accuracy of the approxima-

tions of the distribution of T by the exponential

distributions suggested by the asymptotic results of chapter

A. THE DISTRIBUTION CF TX FOR INCREASING X

:he liriting result (2-8) indicated that the distribu-

tion of Tx approaches the exponential as level x increases.

T1,is result suggests that the distribution of T% can be

dpproximated by an exponential distribution at least for

large x. In this section this exponential approximation will

be studied via a simulation.

1. Constant and Shock Load Ma _nitudes Have 7Tlentica.

_Z2nential Distributions

In the first collection of three molels to be

considered f (x) =7 (x) = e-x. The times between constant load

changes are exlonential with parameterA and the shock load

arrival rate .4A is a constant independent of the constant

l1al process.

Tor these cases it is possible to derive an analyt-

ical expression for E(Tk).

-+.a)e=f -- (4-I),7 't A.

In the Tables, P (Ty =)) refers to tLe simulated

samFIe Frobability that TY =0; X-BAR 3ives the si mlated .

.. 1

* .. . . ... .. *.~.:. .. : . . . V . . . .-. ..-.-. . . . . .. _ . . . . , .-.. , . .. . .. . :

sa m-le mear of T, ; ST-DEV is the simulated standard

deviation of TI ; VAS(X-B) is the variance of the sample

mean; COEF-V is the sample standard deviation divided by the

sample muean. T P (-x=O) is the theoretical probability that

T.=0, W(x) ; T x-bar is the theoretical mean of T× computed

from the equation (4-1).

The simulated coefficient of variation in all three

cases decreases as the level x increases becoming quite

close to the theoretical exponential value of 1 when x=5.

7o compare the simulated distribution of Tx to the

approximating exponential distribution, guantiles fror the

simulated ata were computed and compared to the corre-

sponding puantiles cf an exponential distribution having

theoretical mean (4-1) . The simulated ,juantiles were

computed as described in chaeter 3. The exponential

4-iuantile is given hyq =a.ln (I-S.

wh.ere a is tie mean cf the exnonential.for each level of x, the first row in the tables

(A.), (A.2) , (A.3) for the quantiles of the distributicn of

:x jives the juantiles of the simulated data and the second

row -jivts the corresponding exponential gaantiles for an

tx-oren-tidi listrIbution having theoretical mean (4-1).

Cne way the distribution of Tx differs from an expo-

n.ritiak is that it has an atom at 0. In particular,

S=0) =F (x) =e

for the case considered here.

Cuantiles were usel to compare the simulated condi-tional Cistribution of T, given T, >0 to an exponential

distrilution having mean B(T, )/P(T,>O). The simulate] quan-

tiles fcr ti.e conditional distribution were compute! as

dcscribed ir chapter 3. These .uantiles were comparel to

those of an exponential distribution having theoretical meanE- (T ) /P (7r>0).

22

. . . . . . . .. . . . . .

In the tables (A. 1), (A. 2), and (A.3) for the quan-

tiles of the conditicnal distribution for each level x, the

first row slows the simulated quantile and the seccnd row

the corresponing approximating exponential yuartile.

The guantiles of the simulated conditional distribi-

tion are ruch closer to their exponential approximation than

the guantiles for the unconditional distribution to their

exponential approximation. However the quantiles for the

uiiconditional distrilution get closer to those of their

approximating exponential as x increases.

Comparing tables (A. 1) , (A. 2) , and (A. 3) , it appears

tLat increasing the arrival rate of shock or the rate of

change of the constant load magnitude decreases the quan-

tiles. The change of arrival rate of shocks appears to have

the greater effect.

2. 2onstant and Shock Load :aanitudes Have Diferent

Exponential Distributions

In the next three cases, the distribution of the

sLocl, load magnitudes is exponential with parameter 1 and

the distribution of the constant load magnit-ides is exponen-

tial with mean 0.'). The other model assumptions in tables

(.1), (P.2) , (P.3) correspond to those in tables (A. 1),(A ) a ! (A. 3).

As before, for each level x, the first row ir the

guantile table gives the simulated 3uantile and the second

row gives the approximating exponential i uantile using the

cyxponential distribution with theoretical mean. Comparing

the guantiles of T( in table (B. 1) with (A.1) (respectively

(P.2) with (A.2) and (P.3) with (A.3)) , it is seen that

ecreasing the wean constant load magnitudes has increased

the cuantiles. rurtler, it arnears that the convergence of . -

the distribution of T- to ar. exponential is faster in the

case of smailer mean constant load magnitule.

23

0

7he exponential approximation to the conditional

distribution of Ty given TA>0 has theoretical mean

E(Tx ) /P (Tx >0). Once again for small levels x,it appears

that tIe exponential approximation to the conditional

distributior of 7 is better than that for the unconditional

one.3. Arrival Fate of Shock Loads Depends on the Constant

load Maqnitude

In the next 3 cases, constant load magnitudes and

shock 23ad magnitudes are exponential with mean 1. Constant

load magnitudes change according to a Poisson process with

rate 1. Given the ccnstant load magnitude at time t is C,

the probahility a shock load will arrive in the time

interval [t,t+h] is AA(C)h+o(h)

In case (C-1) the ccnditional shock load arrival

rate is l(r)=C. Comrarison with table (A.1) indicates that

conditional shock arrival rate AA(C)-I =C has the effect of

decreasing the quantiles of Tx. Further the exponential

approximation to the quantiles of Tx is not as good as in

case (A-i).The approximating exponential distribution to the

conditional distribution of Tx given T,>0 has a mean of

T(x )/P (-x>0). The quantiles of the exponential apiroxima-tio1 to the guantiles of the conditional distributicn appear

to Ic closer than those for the unconlitional distribution.

In case (C-2) shock luads arrive with conditional

arrival rate A4C)- " =exp(C). Con'aring the table (C. 1) an,3

(C.2), it is seen that the quantiles of the case .4(C)1 =C are

less tha:. those for the case AA(C)"=exp(C). This is tecause

interarrival times tends to be larjer for the case

2(C) =exp (C). Comparing tables (A. 1) and (C. 2) indicatest .at the exponential approximation to the quantiles cf -x is

not as good as in the case ,=l.

24"

In case (C-3) the conditional shock arrival rate is

Q(C) =exr (-C) . Comparing tables (C. 2) and (C. 3) indicates

that the quantiles fcr the case

U(C)" =exf (-C) are less than those for the case 4t(C)- =exp (C)

ard furthermore the exponential approximation of Tx appears

to le clcse for the case (C-3) than that of for the case

(C-?). In all of the cases the simulated mean was used as

the parameter in the exponential approximations.

4. load Magnitudes Have a Pareto Distribution

In the next three cases, the times between constant

loal changes of magnitude are exponential with mean 1. Fhock",ads arrive according to a Poisson process with rate 2. In

case (D-1) the constant load magnitudes have a Paretodistribution with narameter =I and the shock load magni-

tudes are exponentially distributed with mean 1. Comparison

with tab.e (1.2) indicates that the Pareto constant loal

magnitude has the effect of decreasing the juantiles of TX.

Further the exponential approximation to the quantiles of TK

ij not jood in the Pareto case; (the approximating expcnen-

tial has a mean of te simulated sample mean).

-he approximating exponential distribution of T

given - >0 to the conditional distribution has a mean of-(-X )/P(17 >0). Once again the quantiles of exponential

ai~roximation to the guantiles or the conditional distritu-

tior appear to be clcser than those for the unconditional

diztritutior.

In case (D-2) the constant load magitides have a

?arcto listribution with narameter o(=2. Comparing tahles

(D.I) and (D .2) it is seen that the ]uantiles of the simu-

1at distribution cf Tx for the case o=2 are larger than

t~.ze for t!e cIse 0 =1. Comparing taLles (D.2) and (A.2)

indicate that the exponential approximation to the uantiles

of 7x is not as gcod for the Pareto case as for the

exponential case.

25

In case (D-3) the constant load magnitudes have

Pareto distribution with parameter o =1 and the shock load

magnitudes have Pareto distribution with c=1. In all three

cases the simulated mean was used in the approximating guan-

tiles.

B. TPE EXPONENTIAL TAIL OF THE DISTRIBUTION OF TX FOR

FINIIE X

In this section simulaticn will be used to study the

exporential approximation

suggested by the asymptotic resultJim P(T>t =+-4oo e Y

r

This is an approximation for P(Tx>t) for fixed finite x; it

sLould become more accurate as t becomes large.

:wo cases are considered. In b3th cases shock loads

arrive accorling to a Poisson process with rate 1 and

constant load magnitudes change at the times of arrival of a

Poisser process with rate 1. The shock load magnitudes have

exponential distribution with mean 1. In case (M), the

constant loal magnitude distribution is exponential with

mean 1; In case (F) it is exponential with mean 0.5.

iescription of the computation of -X and T* for these two

cases can he found in chapters 2,3. It can be seen from the

computel value of X(x) and e(x) appearing in tables, (r.1)

and (F.1) tLat X(x) is approaching the value I/E(Tx) (where

T(TY) is the theoretical mean) as x becomes larger and for

te cases computed x(x) < . Further , e(x) is approaching

1 as x becomes larger.

To study the exponential approximation to the distribu-

t*'_o_ of 7 , -uantiles from the sim ilated lata were computed.

These can bc fourd in tables ( .2) a.d (7.2). For each Ievel.

x, hc fir st row presents the simulated 1,iantile; the second

row gives the approxilratiig y-aritile computel by- - i n f 1- -') -- -- -- - -- -- - -- -- -

26

. . . . .

- .-- . . ..' .. " . "...2 i . " - '. '"--. . -. . . ......*..... . 2... ...... i .......... ..2 ... .. ... -.. .......

The third row shows the approximating a-ciuantile corputed byq ,(=- E ( y)In (1 - o) ---- -- -- -- - -- -- -- -

where the theoretical mean is used for E(T,)-

Comparing the guantiles in tables (E.2) and (F.2) it can

le seen that the approximating quantiles (X) are close to

the simulated ones even for .40. The approximating quan-

tiles computed in (I) do less well but improve as x becomes

larger. The two approximating 3uantiles approach one ancther

as x beccmes larger, as expected. Approximating quantile (I)

has the advantage,however, of being easier to compute.

Table (G.1) presents 90 confidence intervals for

/-.7jartiles with k_>0.9 for case (E). These confidence inter-

vals were computed using a large sample approximation, see

Conover (19S0; page 111-114) [Bef. 9].

Compariscn with table (E.2) suggests that the approximating

quantiles approximates the simulated ones well for these

values.

C. CCNCIUSIONS

Approximating guantile I_ is always better than approxi-

xating guantile . Approximation i approximates q( well for

.as small as 0.49. However 1 is more difficult to compute

than I .

The performance of approximating quantile . can be

irproved by changing it to

PC'Tx >o)

and using it to approximate the quantiles of the conditional

distribution of -) given T%>0; for the models considered

P (x =0) is Easy to ccrpute being equal to F (x)

27

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00

43

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00 U' 159 P. '99 U' - O V ' 0

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09 NJ' f40 N4 4' m4 O 0

rN .

V. SIMULATION ALGORITHM

A. VARIABLE DEFINITICN

T); =[inf tO ; Z(t)>Xi 3

where Z(t)=superpcsition of constant load and shock load

Nrow=raximum number of level X; ; X,<X,<X< ........... <

Ncol=waximum number of repetition of generating TK

Lc=constant load magnitude

Ls=shcck load magritude

Tc=constant load interval

Ts=shcck load interval

0pc=ccunting the number of repetition

1=counting the maximum level of X;

E. AIGOBITHM

Inut;parameters (A,,, a, b, seed numbers

Initialize the stress level X; ; X, <X,<Xl< ......... <X.,i~zc=0

Fepeat

set To =314=0

TTs=O

Pepeat

generate ccnstant load;Lc

generate constant interval;Tc

7ind L such that L=[sup i;Lcx; 3

Set T-A = TO for 1_<i!5L

M=L

Ci=Tc

A, generate the shock load;Ls

generate the shock interval;Ts

-, Ts>Ci, then

4... . -. -. ...-. ' .... . ...........-..... .. ....-. "

7TOTO+Ic

Ts=Ts-Ci

TTs=O

generate constant load;Lc

generate constant interval;Tc

Find I such that !.=[sup i; Lc aXi)

If L>!M, then

Set TA, for 11 <i!5L

M= L

Else ccntinue

Ci=Tc

Go to P

Else, Find I such that i=[sup i;Ls+Lc?*X' 3

T-ss=T'Is+Ts

Set Tx; =T.+TTs for M~<i!5L

CCiic-T-.s

Go to A

U.ntil (L=Nrow)

IFc=Tpc+ 1

Jrti. (Ipc=Ncol)

E:n d Algorithm

10

.

APPENDIX A

COMPUTER PROGRAM~

IX

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9

58

lIST OF REFERENCES

1. J.C. Peir and C.A. Cornell; Spatial and TemporalVariatilitv of live Loads, Jour. Structural Div. ASCE99 ST5 (1973) Fp 903-922.-

- Y.K. Wen; Statistical Combination of Extreme Loads,Jour Structural Div, ASCE, !03, ST5, (1977) pp

3. H.T. Pearce and Y.K. Wen; A Method for The Combinationof Stochastic Time Variin5 - 7-ft ec-9 -9H5U-ET5I

--------------------eris- No-5U7, --- V-3- 2910, civilEngineering Studies University of Tllinois,Ur una,I llinois,June 1983.

4. D.P. Gaver and P.A. Jacobs On Combinations of RandomLoads, SIAM Jour Aplied Math vol 40,No.3, June 1981.

P.A. Jacobs First-Passage Times for Comtinaticn ofRandom Loads, In preparation.

6. W. s eller An !rtroduction to Probability Theory andAoplicat!Son, eoIie-scoTe1--e ot-, -i-e

5H 7n-Y -- Y York 1971.

-7. . Rudin Principles of Mathematical Analysis, Thirdedition, mETH-7BTTI, Ne-Y .7T7 .

8. M.S. 1;azaraa and C. M. Shetty Nonlinear Proqramin_-John Filey and Son Inc,New York 7379

9. V. J. Conover Practical NonDarametric Statistics, JohnUiley and Son ,-YoEE,-- - - ,

59

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3. .9AJ Noh, Jan? Fat 3repartaiert at Operation ResearchCehter of Air ForceSeoul Korea

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