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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:
Approved -or ::;iblci releise; distribution unlimit .
.
UNCLASSIFIED
SECURITY C.ASSIFICATION " , TKtS PAGE ewen Dta Entered)
REPORT DOCUMENTATION PAGE READ INSTRUCTIONS1. REPORT NUMBER 2 GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER
TITLE (mid Subtitle) . YPE OF REPORT & PERIOD COVERED
A Simulation Study of Models for Master's Thesis;Combinations of Random Loads September 1984
6. PERFORMING ORG. REPORT NUMBER
7. AUTHORes) 8. CONTRACT OR GRANT NUMBER(&)
Noh, Jang Kab
9. PERFORMING ORGANZATION NAME AND ADDRESS 10. PROGRAM ELEMENT PROJECT, TASKAREA & WORK UNIT NUMBERS
Naval Postgraduate SchoolMonterey, California 93943
1 COTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE
Naval Postgraduate School September 1984Monterey, California 93943 13. NUMBER OF PAGES
6014 MONtTORING AGENCY NA "E & ADORESS(II different from Controlling Office) 15. SECURITY CLASS. (of thle report)
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!.. SUPOLEMEN-ARY NTC1S
q7 F '. -, , , ,F sere ode if n,ee ery mid Identify by block number)
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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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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