ガラーキン法による不確定曲げ問題の解法

(1)

Architectural Institute of Japan

NII-Electronic Library Service

Arohiteotural エnstitute 　of 　Japan

1

論　　文

1

UDG 　624．042．1：624．73

　　　日本建築学会構造系論文報告築第 419 号・1991 年 1月 Journal　of　Struct，　Constr．　Engng ，　AIJ，　No．419，　Jan，，1991

GALERKIN

METHOD

TO

ANAL

．

YZE

SYSTEMS

」

_WITH

STOCHAST

_．

IC

FLEXURAL

RIGIDITY

ガラーキン_法による不確定曲げ問題の _解_法

Tsd

姻Oshi

’

TAKADA

＊

　　　高田毅士 r

　Anew 　stochas しic　analysis 　method 　based　on　the　Bubnov −Galerkin　method 　is　_proposed 　herein　fof estimati ．ng　the　response 　variabihty 　of　systems 　with 　spaUally 　varyihg 　flexural　rigidity ．　Such’a

flexural　rigidity 　is　idealized　as　a　mu 旦ti−dimensiQnal，　statist 互cally　homogeneous，　continuous 　Gau’s− sian 　stochastic 　field．、ln出e　formulation　of　this　met 卜od ，　a　set　of　deterministic　trial　functions　are　in． trgduced _，_andhence　the　resulting 　response 　can ．be　expressed 　in　terms　of 　the　trial　functLons_with non ．Gaussian　randDm 　coefficien ヒs，　Two 　kinds　of　techniqlies　for　approximating 　the　respQIlse 　statis ．

bcs　are　utilized ：afirst ・order 　perturbation　technique 　and 　the　Monte　Carlo　slmulation 　technique ，

Bending　_probiems　in　which 　the．flexural　rigidity 　either ．of　elastic 　beams　or　rectangular 　_plates　has

spatial 　yariability　are　treated．　Two　numerica 里 examples ，　a　both　erld・fixed　bem　a且d　a　fQur−edge

blamped

　square 　plate，　are　present ’

ed　aldng 　with 　the　conv ’entional 　stochastic 　finite　element 　method ．

The　result 　frQm　the　proposed　method 　shows 　reasonably 　godd　agr毎ement 　with ．　those．　from　the　con − ventio 皿al　method ．　Finally，　the　_proposed　method 　is　expected 　to　make ．it　_possible　not　only ．to　

de

． velop 　a　new 　3tochastic　finite　element 　method

，　but　also　to　treat　dynamic　and ／or　nQnlinear 　stochas −

tic　problerP_，s　

by

　virtue 　Qf_．　the　Galerkin　apProxi 皿 ation ．　　　　　　

鹽

　Ke　_f’WOiitg 　

l

　Flexu厂at　rigZ−dity呶厂iation，　stochastic 　

fietd

，　Galerkin　method ，　bendingρ厂0 わ’翩，　stochastic

　　　　　　　　　differential　equation

1

．　htroduction

　．The　stochastic 　responses ・of　systems 　with 　spatially 　varying 　material 　properties，　in　_genera1，　become

statistically 　non −

homogeneous

，　non ・

Gaussian

　stochastic 　

fields

　particularly　when 　the　material 　properties

are　idealized　as　stochasticfields ．　Furthermore

，　itis　extremely 　

difficult

．to　find　the　exact 　solution 　in　most cases 幽｝_．

Th

_丘_s

，．many 　approximate 　treatments　have　

been

　introduced．　From ・the　viewpoint 　of　such

approximat ．

ions2

｝_，

Table

l

illustrates

_the_{classification} _of_the_stochastic _analyses _treating_system

stQchasticity 　issues　so　far。　As　is　evident _，　there　are　three　kinds　of　claSsifications _，　The　

first

　one 　is

associated 　with 　_the　representation

’

of　randomness 　

involved

in

　the　stochastic ．systems ・．　

The

　second

indicates

how

　the　response 　analysis 　is　implbmented ，　The　Iast　one 　is　associated 　with 　

how

　the　response

variability 　

is

　evaluated ，　

These

　classifications 　are　

described

　more 　pr_拿ci爭ely　in　the　

following

．

The

first

　classification 　

is

　the　stafting 　point　

in

　carrying 　out　the　stochastic 　analyses ，．　

The

　concept 　of　the

stochastic 　

field

is

_珊ade　use　

gf

　to、represent 　the　sp_耳tially　

fluctuating

　material 　properties．　

More

　often ，　the stochastic 　

field

　is　

discretized

　into　several 　finite　sub ・

domainS

　for　simplicity 　of　the　ensuing 　analysis _．

Furthermore

，　without 　using 　the　stochastic 　

field

　concept ，　several 　random 　yariabl6s　are　sometimes 　used ．

Which　treatment　is　most 　appropriate 　is　depehdent　not 　only 　upon 　the　statistical ．nature 　of ．the　randQmness under 　considera 亡ion，　but　also　upon 　the　response 　analysis 　to　be ．　performed

　Regarding　the　secQnd 　classification _，

．

numerous 　procedures，　e．g ．　spatially 　continuous 　methods 　and such 　spatially 　

diScrete

　methoCIS 　as　the　

finite

　element 　methQd _（

FEM

_）and ．the　

finite

difference

　method

（FDM ）are　available 　since 　

fundamental

　equations 　in　most 　prQblems　are　

difficult

　to　solve 　even 　in　a 本艱告の一部は，1990年日本建築学会大会で発表した．

＊ ShimizuCorporation_，OhsakiResearchInstitute_，．M．E

皿g，清水建設株式会社大崎研究室・工修

一

₁₀₇

一

(2)

ArchitecturalInstitute of Japan

deterministic

sense.

The

author thinks that all these _procedurescan effectively

be

used as a t/oolto

evaluate the

input-output

relationship even

in

the stochastic _problems.

The last classification might very well bethe subject most energeticaliy studied,so far regarding the stochastic _problems. Techniques forevaluating the response variability can be classified in'totwo

different

ones _:statistical and non-statistical techniques3'.

Since

the resulting response,

in

general, cannot

have

any

known

_probability

distribution,

the Monte

Carlo

simulation technique as ene of the

statistical techniques will producethe most accurate resuits when itisimplemented with a sufficiently

large

sample size. This technique isoften costly and time-censumihg, however, and t'herefore non-statistical techniques such as _perturbationand

hLerarchy

techniqueshave

been

_proposed.

The previousresearch en stochastic analyses is,summarized

below.

Spatially

continuous analytical

methods are surveyed first,Bolotin4i solved an infinite

beam

lying

on the stochastic elastic foundation, which is modelled

by

a random

function

(stochastic

field),

and assumed thattheresponse can

be

linearly expanded intothe summation of several _perturbedfunctions and obtained the explicit

form

of

the response, Baker et al.5'. extendecl thisideato a finitebeam, Recently,Bucher et al,EideTivedthe solution ina closed

form

for

statically

incleterminate

stochastic

beam

structures.

They

solved

directly

the stochastic

differential

equation under _the condition that the

bending

flexibility

isidealizedas a

Gaussian,

continuous, statistically homogeneous, stochastic

field.

Finally,

they _pointedout the need

touse either the_perturbationtechnique or the

Monte

Carlo

simulation technique since the resulting response

does

not have a

linear

relationship with the.original stechastic

field.

Among

these analytical methods, the lastone _yieldsthe most accurate results

from

a methodological _pointof view.

On

theother hand, methods

based

on spatial

discretizati6n,

such as

FEM

and

FDM,

have

been

so

far

proposed. Many researchers7)･S) utilize thefiniteelement scheme and the _perturbationtechnique to

establish thestochastic

finite

element method

(SFEM)

which essentially requires the

discretization

of

theoriginal stochastic

field,

Vanmarcke

et al.9'_and

Der

Kiureghian

_et _al.'O',

givingconsideration tothe

representation of the original stochastic field,_proposedthe stochastic

FDM

or

FEM

by

using the "local

4verage" definedby the spatial average of the original stochastic

field

over the finitesub-domain.

Furthermore,very recently, thepresentauthor

has

prop'osedtheconcept of the "local

integral"which is

defined

by

the spatial

integration

of the relevant

deterministic

weighting

functions

ancl the originaLl stochastic

field

oyer the

finite

elementL". He then showed thatthe stochastic element stiffness matrices can

be

expressed

in

terms of seyeral

local

integrals,

and thateither the_perturbationtechnique or the

Monte

Carlo

simulation technique can still be used.

Regarding the statistical techniques,

Shinozuka

and

his

associates showed the _generaltechnique

for

digitally

_generatingmulti-dimensional and multi-variate stochastic

fields

by

using the trigonometric

series

for

the furtheruse of the Monte

Carlo

simulation methodi2]-i`'.

Since

the

Monte

Carlo

sirnuiation

technique requires a

large

amount of numerical effort, theresponse surface methocl'5)･'6]for reduci'ng the

sample size and the

Neumann

expansion technique'7)

for

reducing the computational time required

in

an

individual

run

have

been

_proposed,

From

theclassification mentiened above, the_presentauthor would liketoclaim thateven with linear elastic _problems, rnost of the stochastic analyses

have

not _yetmethodologically reached the _present

levelalready achieved

by

the current

deterministic

analyses.

Most

analytical stochastic methods simply

utilize the

deterministic

selution and the_perturbationtechnique, which must beverified

by

_theMonte

Carlo

simulation method. Regarding mest SFEMs and SFDMs, the discretizationof the original stochastic

field

can remarkably

facilitate

the analyses so that

deterministic

computer codes can conv'eniently

be

used.

However,

they sometirnes require

finer

discretization

thatsubsequently means an

increaseincomputational cost'''.

In

other words, the conventional

SFEMs

and

SFDMs

are

highly

dependent

upon the

fiher

discretization

for

realizing theeriginal stochastic

field.

Therefore,

regardless of whether the analytical method or the

discretized

method

is

used, _{.considerable}care

is

essential although theywill obviously produce some sort of answer,

(3)

-Architectural Institute of Japan

Inthisi

junctllre,

the analytical method such as

'

isproposed inthe

literature6]

isquiteuseful

for

providing valuable insightintoestablishing anew

approach]B' as well as intoverifying the,solution

from

any other approaghes.

Although

thismethod

is

applicable torelatively simple systerns

consist-ingof trusses or

beams,

itisnot applicable to

complex systems with

higher

static indeterminacy or

higher

dimensionality.

･Needecl,therefore, isa method that can treat such complex stochastic systerns without using spatial

di$cretization

of the original stochastic

field

and that makes it

technique,

This'is

the major motivation

behind

For

thesereasons, a stochastic analysis method

and the Bubnov-Galerkin method w

This

inethod

can treat'the con'tinuous

discrete

_Inethods inTable 1.Bending _pToblems in

spatial variation are

tion. Itwill then beshown thatone of two kin

technique or the Monte

Carlo

simulatien tech '

variability

is

evaluated.

presentedalong with the conventional

SFEM,

possiblenot only to

formulate

a new SFEM,.

but

problems

by

virtue of the

Galerkin

'

Tabie1Classificationofstochasticanalyses Representationof

inyolyedstochasdcfieldResp(mseevaluationmethodResponsestadedeseyalultiontachrig-ePTeviousworks astattSteCBmh),Shimmats) Oentinuous stochasdrfieldsSpatial]yeandnunus metthnd(analyticul)en-sta"sti.atian)Bolati-},BaLe) tatistsnc Spatianydiscrvte rnethodffM{FDM)Nen-statisticaltech.1lahedall],IS) tutsti' ' imvdiables Ctisaetini stochastieficki) Spatiallycontinunns metlwod(analytieal)on-stansncvec. SpatiaUydiserete methodCFEMFDM)Statistiealtech.Astill14),wan'slS), ytuEki17) on･statisttctoc,Nut),BabeX Vanmackesc

feasible

to _perform even the Monte

Carlo

simulation

the _presentstudy and this _paper. ' ･

will

be

_proposed

based

on thestochastic

fietd,theory

'

hichis,oneof thleapproximations even

for

deterministic

_problems.

stochastic fieldand isclassified between the analytical and the

which flexural rigidity of eithet beams or

Plates

has

syste'matically and conveniently

formulated

by

virtue of the Galerkin

dsof approximations, either the first-order_perturbation

mque, can effectively

be

utilized when the response

To

exemp]ify thevalidity of the_proposedmethod, numerical exarnples will be

Finally,the _proposeclmethod isexpected tomake it

also to treat

dynamic

and/or nonlinear ･stochastic

approximatlon.

2.

Formulatienof Bending Problems by Galerkin Method

2,1 StochasticExpreSsion of

Flexural

Rigiclify

Treated

here

will

be

such _problems inwhi ¢

h

the

flexural

rigidity of either one-dimensional

beams

or

two-dimensional _platesspatially varies.

Here,

it

is

assumed thatsuch

flexural

rigidity

denoted

by

K(x)

is

expressecl as

K(x)=Ko(1+f(x)}, :''･･･････-････'･'''''･'･''･"'''"""--'-''H-'H--'''-''''''''''･-･t-･---･･･--･"<1)

in_which K,isan expected rigidity <K{x)>of K(x), and

f(x)

isa fluctuating_partwhich

is

assumed to

constitute a multi-dimensional, homogeneous,

Gaussian

stochastic field.Without lackof generality,

f(x>

has zero expectation and the auto-correlation

function

such that

<f{x)>=O, aricl･-･---･･--･--･---T･-･--･-･--･･････････････････---･---･--･･･････--･-･---

(2)

<f(x+e)f(x)>=R.(e). ･･･････----･----･---･･･-･･･-･---･-･･････-ti･････-･--w,･i,..,(3)

In

Eq.

₍

1

),

note thatK(x)represents EI

for

an elastic straight

beam

case, Eh'!12

(1-u')

for

an elastic

'

isotropic

platecase with

E

being

Young's

modulus, Ia sectional

inoment

ofinertia, v aPoissionratio, and

haplate

thickness.

' 2,2

Derivation

of

Solution

based on Galerkin Method .

The

pro61em to

be

solved hereisfindingan appr6ximate solution w(x) tothetrue

deflection

thatcan satisfy the followingtwo equations, i.e. _, the

fundamental

_.equilibriumequation

defined

ina

domain

v

tt

'

atid the

boundary

conditions at the

boundaries

S:

' '

L(zv)-p=O

xEV, ･･--J---･･･-･･-･-･･-･･････-･･･-･････----･-L･-･--････････-･･----:･,l(4)

S(zv)-q=O

xES, ･･-･･････---･-･---･---･･･････-･---･--･････-･･･････-･--･･,･･-･･(5)

in

which

L<

)

isa stochastic differentialoperator which is

linear

with respect

both

tothe deflectionancl

tothe stochastic flexuralrigidity K(x), while S(

)

isassurned tobea deterministicoperator, _p and _q are

deterministic

_quanti'tiesthatimply respectively a

clistributed

load

'and

boundary

conditions.

(4)

-109-Architectural Institute of Japan

Considering

the

beam

case,

L{

)

is

LC

)=ddi,

(EI

_ddi,),

''''''''H''H'''''''''-'''-'''''''''''''-''H'''''`'"""''"-H--･･-･･･････--･･-･･････

₍₆₎

while

L(

)

of the_plate case can be found as

LO=

8i,

[K(

aOi,+v oOi,)]+2

a.Oa2y

[Ka-v)

oxOa2y]+

aai,

[K(

oaye,+voax2,)],･･--(7)

in

which

it

should

be

noted thatthe

Poisson

ratio v

is

consideredSo

be

deterministic

inthis_paper to

simplify the ensuing analysis.

Since

such solution satisfying bothEqs.

(

4

)

and

(

5

)

isnot always obtainable even indeterministic:

problems,many approximate _procedureswere _proposedespecially

for

deterministic

_plates,such a$ the

energy methods,

FDM

and

FEM.

The Bubnov-Galerkin method isone of the energy rnethods, which is also understood to be one of the rnethods ef weighted residual

CMWR)

and isidentifiedas the

Rayleigh-Ritz

method'g).

Returning to the stochastic _problems, Eq.(4) is,in_general,not solvable since the stochastic

differentia]operator L(

)

involvesthe stochastic

fielcl

f(x)

characterized only ina stochastic manner,

i.e.

,

its

mean and aute-correlation

function.

As

an exception tothis,

Shinozuka20)

and

Bucher

et al.6', whose work was

based

on _the_theory of

differential

equations, solved

beam

_problemsinwhich they made

thesarne

Gaussian

assumption

for

theflexural

flexibility

1/K(x) rather than _therigidity K(x),as isseen

in

Eq.

₍

1 _).

Th.e

Galerkin

method

begins

with selecting tTial

functions

which satisfy the

boundary

conditions

prespecified

by

Eq.

(

s

).

For

the stocha$tic _problems,however, itis_quite

difficult

to establish aset of

stochastic trial

functions

compatible with the

deterministic

boundary

conditions.

In

this_paper,these

trial

functions

aie _given

deterministically.

Therefore,

the

Galerkin

solution to

be

derived

in

the

following

isapproximate

both

in a

deterministic

and ina stochastic sense,

Using the

deterministic

trial

functions

_ip.{x},

the approximated

deflection

w(x) can'be expressed in

terms of the

linear

summation

N

w(x)= £ an¢n(x)=

dit(x)a

'''''""'"-'"'''''''H'"-"'''''''"'-''''''''''''''HH''''''''''H'"･････-

(

8

)

n=1

with a trial

function

vector

di(x),

each component of which must

be

lineaTly

independentand complete

but

not necessarily orthogonai :

¢ '(x)=l ¢,(x)g6!(x)-･ipN(x)L 'H''"････''･･････---･･･-･--･･････---･-･･-･--･-･････t･---･･･-････--･･-(9)

where a

is

an undetermined coefficient vector which turns out to be stochastic.

The approximation of the deflection_giveninEq,

₍

8

)

obviously states thatthe true deflectionas a non-homogeneous, non-Gaussian, continuous stochastic fieldisapproximated

by

the stochasti.c

function

series that consists of the spatially continuous

deterministic

functions

with the random coefficients,

This

formulation

may

be

an

implicit

discretization

in which the non-homogeneous, continuous stochastic

field

constituted

by

the true

deflection

is

decomposed

into

finite,

non-homogeneous,

continuous stochastic

function

spaces.

Using

Eq.

(

8

),

the residual

between

theapproximated and thetrue solutions

is

then orthogc}nalized

to the trial

fuhctions

ina

domain

V:

.L{L(w)-p}edx=o.

･･････-･････-････-･-･･･-･･･････-･-･････-････････････--･--･----･･-･･-･････-･･････-･･(io)

Substituting

Eq.

(

8

)

intotheabove and takingintoaccount that_L(

₎

isalinearoperator with respect

to zv, the above equation turns out to

be

an algebraic equation of the undetermined vector _a:

.L]eL(di')dxa==X]pdidx･･-･････････i･･･････････････-････-･･-･･････････････････････---････････--････ao

with

L(

¢t)==IL( ¢,)L(ip,)-L(ip.)I. ･---･･---･-･･-･:`--･････v･-･-･--･･-･･-･-･･-････--･--･･･-･-････-･-･･(12)

(5)

ArchitecturalInstitute ofJapan

Usihg

the

following'

naming cdnvention,

Eq.(ll)

can

be

written

ih

the simple

forin

Ka=P,

where-･････-･･････････-･･･････････････i･･･--･-･･･････････････;-･････････-･-･･･････--･----･･i･(13)

K=

Xl

¢

L(dit)dx,,

_,and･･･････-･････････:･･････････'･',""'･'''-･･+-･---･･-r････-･････t････-･･･--････-･･:

(14)

'

P=fpdidx. ''H'''""''''''''''''"''r'''-'''''''""'''''""-'''"H"'''''''''''''''''''-'"r"''r';(15)' '

The-i-j cemponent of the rnatrix K is

' '

kii--f_¢iL{ ¢j)dx. '"''''''''''''""'H''･--･･-･･---･--･･･---･-"'''''''''''''''''"-'""'''''(16)

'

For

the

beam

with the

determihistic

lengthl,vaTious

boundary

conditions cari

b6

c6nsideredl

If

the

' '

boundary

condition ishomogeneous, k.

become

the simple

form

_,

h.=KoX't{1+f(x))ip;

¢

fdx,

''''''''''''''''''HH'''''H''''''''''''HH'''""-'''''''''''''''H""''''(17)

in

which the

double

_prime means the second

derivative

with respect to the spatial coordinate x.

For

a rectangular isotropic_platewith 2

l.

and 2

l.

beingrespectively

lengths

in

both

directions,

le.

are

found

as.. .

'

.

kw=:K,fil.L:.X<i+f(x,y))[(

¢

:+v

¢

:o>ip:+2(i-v)el'e3"+(ip:-o+vdi:-)ip;"]dxdy,

.･･････････-<ls) '

where '=afax, "=O/Oy,. FTom the _above two _equations, it_can

be

_observed that h._can

be･divided

into.two_parts :a

deterministic

and a stochastic _part.The expressions of Eqs.

(17)

and

(18)

are _quite

similar to"a

_local

_integral"

which theauthor hasrecently _proposedi').Itisinterestingtonote thateven

if_¢, are orthogonal each other, K does not

become

cliagonal

due

to the presence'of

f(x).

Equation

(13)

yieldsthe solution fora:

a=KriP. '-･･''''''""'''''"H'HHH''''''''''''''''"''''m''''''''''HH'''H"'"''''''''''''･""''''''''''(l9>

Finally,the approximate

deflection

can

be

,evaluated as

in

/' w(x)==e'(x)a= ¢t(x)K'iP. ---･･･-･･･-･････--･･-･-･-･-･････---･---････-:-･･---･･･････-･v<20) ' ' '_'Carefully

exarnining

Eq,

(2o),

the statis'tics of the

deflection

can

be

evaluated

after

those of a are evaluated. From Eq.

(14

),

the components ofthetriatrix'K become Gaussianrandom variables since the

stochastic differentialoperator L linearlycontains the original

Gaussian

field

f(x).

Note,

however,

thatthe vector a

is

no longer

Gaussian

since itrequires the matrix

ipversi6n

of the

Gaussian

K.

_,The

expectation and auso-correlation functionof the

deflection

can

formally

be

written as :

<w<x)>=dit(x)<a>,

･--･w"'""""-'"HH-'HHH'''''''''''H'"""'"H'''''''''''''''''H:''''H''''H(21)

Rww(x,g)=<w(x)w(u>>=dit(t)<aa`>

di(g).

--･･･J-･･･････----･-･･････････:･--･･--･･････-････L-(22)

Next, assuming that the deflectioncan beobtained, the moment

force

denoted

by

a(x) isestimated

employing the

differentiation

cr(x)!K(x)M(w(x)), ･･･････"H"-"''-'''"""''H'HHH'''''H-"'-"'''""''''''''''H--H'HH'''(23)

in

which _M{

)

is

a

linear,

deterministic

differentialoperator with respect to the

deflection.

For

the

beam

case,

it

is

expressed as

Mo=-

dZ

"...,,,,.,.,.,.,...H.."...,.,.,.,."".,.,,,････････････-･---･････-･-･･---(24)

For the _plate operator

MO==

where the

first

respectively.

dx2'

case,.

it

becomes

thevector

ff( aOi,+v

oOiz)

-(aOy22,+v

aOx22)

-""-HHH'

q=

,) ,5.ai, ,

second and third rows are '

m,".,H-･･･---･---H"Hr'--"-""'(25) '

associated with the moment

forces

M.., M., and _Ml.,

(6)

ArchitecturalInstitute ofJapan

Similarly

to

Eqs.

(21

)

and

(22),

the expectation and theauto-correlation functionof thernoment

force

can

be

obtainecl :

<a(x)>=<K(x)M(zv(x))>=iMt(O(x))l<K(x)a>,

'-''"'''･･--'''''''''''''""''･''''''''''''''''･･'"(26)

'

R..{x,u)=:<K(x)M(w(x))M'(w(g))K(g)>=:1lft(

¢

(x))<K{x)aatK(g)>M(O(y)),

････-･････-･････(27) .ith MO=IMOMO--MOI'. ･････････････････････････-･･･････････-･･-･-･-･･--････････---･･-････-･･J-･･(28) 2.3 First-erderApproximation

As seen in

Eqs.(21),

<22),

(26)

and

(27),

it

is

impossible

to rigorously evaluatg the statistical moments of the undetermined random vector a since thisvector

is

not

Gaussian,

To

overcorne this

difficulty,

the first-orderapproxirnation can be efficiently used as has been done inmost stochastic analyses. Employing the Taylorexpansion of Eq.

<19)

with respect tothe basicrandom variables

h.

at

theirexpected values and takingup tothe

first

term,

Eq.(19)

turnsout to

be

NN

a=aO+ZZaljAh"''-'''''''''''''''''''''''"''''''''''''"''''''''"''''''''""'"'`-'`''''･･'･････--･･(29)

ij

.ith

Ah.=h,,-<k,,>, ･--･---･･･----･-･･･････-･･-･････--･････-･･････-･･-･･-･-･-･･･--････---･-･･････-･･(30)

where aO and _a:･,are

aO=<K>-iP, ･･･-･----･"''H'''''''''H'"H"'''''''''''''''''''''-'''''''''''H"'"''''''''''''''''''''"i(31)

,- aa

OK

w in,.<ta>=-<K>-i

ehw

<K>-iP'

k"'''"'''''''''-'''''''''''''''''H''''''--''''''''''''E(32)

a"-

ah

Using Eq,

_(26),

expectations appearing inEqs,

(21)

and

(22)

are approximated as

<a>= aO, -･･--･･･-･-･････t-''･･････''''"''''''''''''"""'''''"'""""'-'''''''''''"'''''''''''''''''''(33)

N rv NN

<aa5=aO(aO)t+ £

ZXZ]aL(alt)t<AhijAicht>･

m"''''"'''''--''''''''''''''''''''''''''''''''''''(34)

ijtt

Similarly,

the expectations appearing inEqs.(26) and

(27)

become

<K(x)a>

:KoaO, ''･-''-''･･''-'''･''''''･'-･･'･･''-'''''''''''''-''''''-'''--''-'''''''''''''''"'''''･---(35)

<K(x)aa'K(g)>=:KS[aO(aO)t+<f(x)f(g>>aO(aO)'+*.

#.

aO(aL)'l<f(x)Ah">+<f(g)Ah">l

+#>l])]#al･j(aLDt<Z!hijZ!hict>].'"'-'""-"'H''''"''''''-'--'--''''･･----･･･(36)

As

seen

in

theabove, theresponse statistics can

finally

be

expressed

in

termsof thecharacteristics of

the original stochastic field

f(x),

Using Eqs.

(17)'and

(18),

<f(x)Ah.> and <AkijAhht>inthe aboye can be evaluated as in:

<f(x)Akw>=X]Vii(r)R"<x,r)dr. ･･L-･･･--･･････-･--･･-･････････････････････-.-･･････-･---･-･･･-･･-･･(37)

<Ah"Ahnt>=fX]

V,,(ri)Vki(r:)R"(rb

ri)dridr!'･･-･-･-････-･-･･･"･･････'･-･-･-･---･････-･-･-･･････-････-･

(38)

.ith

V"=

l

(ipf+,ip:o)

¢

f+2(1

m ¢ ."b)ip_¢ ";,'o ¢

;o+(ip:.+,di7)e;.

ffO.r,

bpr,at:,S.

2,4

Monte

Carlo

Solution

The

Monte

Carlo

simulation technique

is

an effective alternative when the

first-order

approximation

is

not appropriate _primarily

due

tothestrong nonlinear relationship

between

theoriginal stochastic

field

and theresponse field.A

digital

_generationtechnique of multi-dimensional,

Gaus$ian

stochastic field

f(x)

is adepted firstly'2'.The samples of the

basic

random variables

h,,

are secondly realized through the numerical integrationof Eq,(16). Using these samples, Eq.(19) issolved and the response variability of the deflectionand the bending rnoment are subsequently evaluated. Inthe simulation methbd, only the sample size should

be

carefully

determined

although no approximations are

inyolved

as

far

as the evaluation of the statistical response isconcerned.

(7)

Z

l

Z

Figure1 A both end-fixed stochastic bearn

.3 .2 .1. F v. _O s -.1 -.2 -.3 O :2 .4 .6 .8 1 Spatialceerdinate=!t

Figure2 Shape of trialfti'nctions_di.(x)

When

one wants toevaluate only the variability of the

deflection,

o]ly the term <aat>isneeded,

as

Seen

in Eqs.(21) and

(22).

Itis.ofgreat

interesttonote that

it

is

not necessary to_generate

the otiginal stochastic

fielcl

f(x)

in

mul'ti-dimensional

space, rather only thesamples of the

random vector a _qre neecled. The samples of a can

be

realized, basecl on thecovariance matrix of

h,,6)I

This confirms that the problem 'ofthe stochastic fieldistransformed

into

thatof the

finite

number of random variqbles

by

means of _,the

decompositionof the

deflection

field.

out the Monte

Carlo

sirnulation.

AAHvpv O.O04 O.O03 O.O02 O.OOI oe Figure3 -.1 -O.05 =Hvotsv O.05 .1 .2 .4 .6 .8 Spatialcoordinate=/l

S.patialdistributionof mean

deflection<w(x)> o Figtire4 .2 .4 '.6 Spatialcoerdinatexll

Spatialdistributionof mean

moment <M(x}>

.8

bending

1 1

This leads.totremendous savings in numerical effort in carrying

3. Numerical

Examples and Discussions

3..I Both

End-fixed

Stochastic

Beam

. . ,

A

both

end-fixed beam is analyzed as

in

Figure 1.The uniformly

distributed

unit loadis statically and

deterministically

acting _along the beam axis,

Unit

mean

flexural

rigidity

(K,=1)

isassumed.

The

following

trial

functions

_¢.(x) are intuitivelyselected: ,

ip.(x)==x(l-x)sin

"i x:(n=:1,2,･･]N), -･･････---･･･････-････----････････-'･･･････-･･-`'-"-'･-････:

(39)

t t

where

l

isthe beam length.The

boundary

conditions at

both

ends can

be

easily found to

be

satisified

by

' 'the

trial

func.tions.

These trialfunctionstake intoaccount that anti-sy-mmetric

de.flection

modes with

respect tothe midspan may exist since the

flexural

ngidity spatially varies despitethe symmetry of the

load_patterfiand the

boundarY

conditions.

Figure2

shows the shape of

ip.(x).

The auto-correlation

function

R,v(6)isnow taken as

R.(e)=aSe-["!b)!,･･････････････-･---･-･-････････---････････-･････--･････--･---･････････-･----･--･･･-･(40)

where _qris a standard

tieviation

associated with

f{x)

and

b

is

"a correlation

distance"

that

irnplies

how

fast

thecorrelation

decays

along the

beam

axis. _qr is set equal to

O.

1.

The

bZlvalue isadopted intwo

ways:

b!l=1.0

and

bll==Q.1.

The

former

case represents the more smoothly fluctuatinRstochastic

(8)

ArchitecturalInstitute of Japan O.OO03 " O.OO02 r"sep-sctts O.OOOI o O .2 .4 .6 .e ! Spatialcoordinate xlt

Figure5 Spatialdistributionof standard deviation

of deflectionVar[w(x) O.O06 -T_o.O04

gg,

O.O02 o O .2 .4 .6 .8 1 Spatialcoordinatexlt

Figure6 Spatialdistributionof standard deviation

of bendingmoment Var M(x)

O.OO03 O.O06 EM7) EM" , e

{lli

O･OO02 ."-.No･Oo4 -C]. i

ts

g d s t'E o.oeol

So.oo2

tsti:kg=ta o o O 1 2 3 4 5 O 1 2 3 4 S Non-dimensionalcorrelationdistanceblt Nen-dimensionalcorTelatiendistancebXt

Figure7 var w(U2) vs. correlation distancebll Figure8 Var M(o) vs. correlation distancebl,l

field.

Adopting the

first-order

_perturbationtechnique

described

in

the _previoussection, the cerrelation

terms, i.e._,

<f<x)Ah.>

and

<Ak.Ah,,>,

must

be

evaluated.

These

integrations

in

Eqs.

(37)

and

(3s)

are carried out numerically since itis

difficult

toevaluate theseterms inan explicit

form,

The number of expansien N ischanged to 2, 4 and 6 inorder to see the solution convergence,

Figures3and 4show thespatial

distribution

of the mean deflectionand the mean bending rnoment

along with the results from theconventional first-order_{perturbation-based}SFEM". Itcan beobserved

that the mean solution

frorn

the _proposed method converges as

IV

increases.Such convergence is slightly slower in the

bending

moment response.

Here,

50 sub-elements

for

bll=O.1

and 10 sub-elements for bll=1.0 are used inthe conventional SFEM.

Figures5 and 6 show the same _plotsof the standard deviationof the deflectionand the

bending

moments foTthe above two cases. The convergence of the standard deviationof the

deflection

can

be

observed to

be

excellent while that of the

bencling

moment is.sloweras _IVincreases

for

_bll=O. _1,

Figures7 and 8 are the standard

deviation

of the midspan

deflection

and the end moment,

respectively, when the

bll

value changes.

In

most of the

bll

range, the _proposed method _produces a result close to that

from

the conventional method.

'

From these results, the followingstatement can bemade, Only a few triaHunctions are needed to

produceaccurate results

in

the _proposedmethod, while theconventional

SFEM

requires a

large

number

of

finite

elements.

However,

itshould

be

noted that this agreement

is

meaningful only

from

the

first-order

approximation _perspective, and the appropriateness of thisapproxirnation can

be

examined

(9)

Table2 Comparison of results

' ' / t ' Galerkinmethod Conventional ,･SFEM7)''Deterministic_Exact Splution22)

N.1N=4N=9N=16N=25

×410'4 ×10･ Cen'berdeflection

<w(o,o)>

O.0243O.0243' O.0229O.0229'O.0232. O.0225O.0207 O.0202

Var[w(O,O)] o.oe213O,O0213O.O0201e,oo2olO.O0203O.OO175O.OO158

-Centermoment

(M..(o,o)> O.1410O.1410O.0997O,0997O.1143O,1107O,0946 O.0924

Var[M.xCO,O)]'O.O063O.O063O.O036O.O036O.O037O.O032O.O025 -Edgemoment <Mx:(-t,O)) -O.153-O.153rO.176･.O,176-O.190-OJ92.O.203-O.2052 Var[M..(-l,P)]O,Ol16O.O098O,O097O.O090O.O091O.O080e,oo7g -' Sumo _£SquareErrer ewC%) 6.2 4.9 1.0 O.8 .. " F ' SumofSquareError eMxx(%) -- 41.2 22.0 13.2 5.8 - h h -SumofSquareError sMxyC%) 34.2 12.0 4.6'4,2 - - -

e_arethesum _ofthe square _errorto

Fhe

_specifiecl _case(No= 25).

'

only through the

Monte

Carlo

sirnulation technique.

3,2

Four

Eclge-clamped

Stochastic

Plate

･' ･

A

square

plate.witha stochastic

flexural

rigidity

is

now analyzed.-

The

_plate issubjected to a

deterministicuniform unit load.Like the

bearn

ekarnple, the trial functionsin

the

two-dimensional

space are selected as ･ '' ･ '

ip.(x;y)=

¢..,(x,

ke)=(l2-x!){l2-y2)sin

n2Xtn

(x+Dsin

n2Ylrr

(y+b'

:

in='1,2,L･･N>.

･1-

(4o

Here, 2l isa _platelengthinboth

directions,

Tihe

origin is selected inthe center of the _plate,

The

numerical analysis is

done

under the conditions

K,=1

and y=O.3.

The

two-dimensional auto-correlation

function

R,t,<g,_rp)

is

now taken as ',･

Rtt(g,n)=o;e-"e'+iniVb, ･･--･･････････････････････-･･････････-･････-･-････････････････-･･････････････････････

(42)

where _q,

is

a standard

deviation

of

f(x,

_y)and

is

set equai toO,1,and b

is

atwo-dimensionalcorrelation'

distance

and isassumed to

be

2

l.

Since

the above correlation,function isseparable intotwo

directions,

numerical integrationso'f<f(x,y)Ahi,->and <Ah.Ah.> are not curnbersome

jobs.

'

The first-order_perturbation technique isutilized again

in

the _proposed method.

Similar

to_the

previous example, the･conventional SFEM" isalso implemented by using a 10× 10 mesh

division.

He.re, an ACM non-conforming rectangular

bending

plate elemen･t2`' is adopted since it can yield good

results in

deterministic

p'roblems.･The mesh

divis'ion

_was

determined

not only from the solution accuracy ina

deterministic

sense,

but

also from the characteristics of the stochastic

field

f(x,.y).

Table

2 lists

the response statistics at specific locations

from

beth

methods.

The

mean response 'from

both

rnethods can

be

compared with the

deterministic

exact･solutionl2) since thetwo methods adopt the

first-order

approximatipn., ,Thereissome $light

difference

in

magnitude

between

the results from

both

methods, These differencesinthe standard

deviation

may come from those of the mean values.

Here, in order to see the solution convergence

in

the _proposed method, thefollowingsum of the

'square_error _to _the _specified _case

_(N6)

_is

_intreduced

_:

f(

Var[AKx,y)]-

Var[A,,(x,y)])2dV

Eis=""

xJ

var[A.,(x,y)]dv

, '''''''''''''"''''''-'""-''''''''`''""'""''

₍₄₃₎

-115-'

(10)

ArchitecturalInstitute of Japan o O.oo91 O.oo37

geictz

ar w :,y ]fromGalerkinmethed(N =2S) O,oo16

ar_[Ml._(:,y frDmGalerkinmethod _(N= 2S) Vhr

[M.,

(s,y)]ffomGalerkin method (IV= 2S)

O.oo79 A o4 t o.ooes

y

ar w :,y) frernSFEMI}(10x 10) ar[M.. =,v ]fromSFEM7)(IOx lO) Nigr

[M..(=,y)]

fromSFEMV(10 x lO)

Figure9

Spatial

distributionof stanclard deviationof various responses

where Var[

]denotes

the yariance of the argument. _A.(v,_y) represents the response, e.g. w,

Mle.,

M.. and Mk.,when the totalnumber of superposition Arisused. Itcan beobserved fromthe tablethat

the sum of the square error _graduallyapproaches zero as

N

increases although themean and variance values at specific

locations

do

not converge well,

This,

behavior

of theresponse at specific locationscan

be

considered to result

from

the selected trialfunctions, which includeanti-symmetric mode shapes

havinga zero value at the center of the _plate,Therefore,thesolution conveigence cannot

be

improved

even

if

N

increases

from

1 to 4 and

from

9 to 16,

Figureg compares the results

from

both

methods regarding the spatia]

distributions

of the standard

deviationof theresponse. As isevident, the _proposedmethod agrees well with theoyerall tendency of

the results from theconventional

SFEM.

4. Conclusions

A

new analysis method

for

systems with spatially fluctuating

flexural

rigiclity was _proposed,

This

method,

based

on the

Galerkin

approximation in which the _trial

functions

are assumed to

be

deterministic,

suggests that a new stochastic finiteelement method can

be

deyeloped,

Through

numerical exarnples, the results

fiom

this method were confirrned to

be

close to those from the

conventienal stochastic

finite

element method. Finally,by virtue of the

Galerkin

approximation, this methocl isexpected to

be

applicable to

dynamic

andlor nonlinear stochastic _problems.

5.

Acknowledgment

Some

of the ideas_presentedhereinarose during_1987,at which _time _the'author stayed at

Columbia

Universityas avisiting scholar of ProfessorM. Shinozuka.

The

author

deeply

acknowledges hisuseful

advice. '

Reterences

1) Takada, T.:ResponseVariability_and_Reltabilityof Beams with StochasticMechanical'Property,Transactjon_ofAIJ,

No.409, pp.67-74,l990(lnJapanese)

2) Takada,T. :on Analytica]Approximations inStochasticFiniteElementMethod, Pfoc, ofthe9thSympesium on ReliabMty

EngineeringinDesign,Societyof Malerial Science,Yokohama, 1989 <inJapanese)

(11)

3）Liu，　W．　K．

，　and　Belytschko，　T，　and 　Mani，　A．：Random　Field　Fmite　Elements，　Internatienal　JQurnal　

for　Numerical

　　Methods．in　Engineering，　VQI，23，　pp．1831−1845，1986

4）　Bo監otin ，　V．　V．：Prohabi’istic　MethedandReliability　Probtems　in　Stru‘tu厂ai 　Design，　Translated　by　S．　Kobayashi，　et　al．，Baifu．kan

　　Publishing　Company，1981 _〔in　Japanese）

5｝　Baker，　R．，ZeiLoun，　D．　G，　and　Uzan，_亅．：Ana且ysisQf　a　Beam　on　Random　ElasIic　Support，　Soils　and　Found≧tions，　Japanese

　　Society　o「Soil　Mec卜anics 　and 　Foundation　Engineerlng，　Vol，29，　NQ．2，　_pp．24−36，　June，1989，

6）Bucher，　 C，　G．　 and　Shinozuka，　 M ．；Sしructural 　RespQnse 　Variab互hヒ_y _旺，　 Journal　of　EM ，　 ASCE ，　 Vol．114，　 No．12，、

　　 pp．2035−2854，　1988

7）　Nakagifi，　S．　and 　Hisada，　T．；JntredtcctionげStechasti‘Finite　Etentent　Method，　Baifu−kan　Publishing　Company，1986_（in

　　 Japanese）

8）Baecトer，　G．　B．　and 　Ingra，　T，　S．；S10chastic　FEM 　in　Settlement　Predictions，　Journa【of 　GT，　ASCE ，　Vol．107，　No．4，

　　 pp，449−463，　1981

g_｝Vanmarcke _，　E．　and　Grigqriu，　M．　l　Stochastic　Fini亡e　E【ement 　Analysis　of　Simp［e　Beams，　Jourロa ［of　EM ，　ASCE ，　Vol．10g，

　　 No，5_，　pp．1203−1214，1983　　　　　　　・

10）Der　Kiureghian_，　A_．：Finite　Elemeht　Methods　in　Strucしロral　Safety　Stud｛es，　Stru砌 r認S_願_{吻跏} _めedlted 　by　J，　T．−P，　Yao，　et

　　al．，ASCE ，　New　York，　NY，1985

11＞Takada，　T．_；Stochastic　Finlte　Element 　Method 　with 　Concept　or 　Local　Integra且_，　Tra皿saction 　of　AIJ，　No．399，　PP，49−57，

　　1989 ｛i皿 Japanese_）or 　Takada ，　T．　and　Shinozuka，　M ．：Locanntegration　Method　in　Stochastic　Finite　E【ement 　Analy＄is，

　　 Proc．　Qf　the　5［h　ICOSSAR ，　Vol．皿，　pp．1073−1080、　San　Fiancisco，1989

12） Shinozuka，　M，：Stochastlc　Ficlds　and　Their　Digital　SimulatiDn ドStoghastic　Methads　in　Structural　Lhaamics，MartinusNijhoff

　　Publishers，1987

13） Yamazaki，　F．融ndShinozuka ，　M．：Digital　Generation　of　Non・Gaussian　Stochastic　Fie［ds，　

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14） Asill，　C．J、，Nosseir，　B．　and　Shinozuka，　M．：lmpact　Loading　on　S恤 ctures　wi 出Ra皿dQm　Properties，　Journal　ofStructuraL 　　 Mechanlcs ，　Vol．1，　No，1，　_pp．63−77，1972

15） Wong ，’F，S．：Slope　ReLiability　and 　Response　Surface　MethQd，　Journal　Qf　EM ，　ASCE ，　Vo1．111，　No．1，　pp、32−53，1985 16） Faravelli，　 L．：Response・Surface　ApProach　for　Re］iabitity　Analysis，　

Journal

　of　EM ，　 ASCE ，　 Vol．　ll5，　 No．12，

　　 pp．2763−2781，　1989

17_） Yamazaki_，　F．，ShLnozuka　，M．　and　Dasgupta，　G．：Neumann　Expansion　for　Stochastic　Finite　Element　Analysis，　Journa［o［

　　 EM ，　ASCE ，　VQI．114，　No，8，　_pp．】335−1354，1988

18） Takada ，　T．：Weigh ヒed 　Integral　Method 　in　StQchastic　Finite　E［e血ent 　Ana 且_ysis_，　to　appear 　in　ProbabiListic　Engineering

　　Mechanics，　September，1990

19） Reddy，　J．　M．：AtiPlied　Functional　Analysis　and 財囲磁 o加 ’MethOdS　in　Eπ9 飢η9_」McGraw −Hill，1986

20） Shinozuka，　M ．_；Structural　Response 　Variability，　

Journal

　of 　EM ，　ASCE ，　Vol．　ll3，　NQ．6，　_pp．825−842，1987

21） WashLzu，　K，，et 　al．：Hantibook　Of　tゐe　Finite　Etement　Method　1，　Baifu．kan　Publishing　CQmpa口y，1981，　_PP．267・_｛in　Japanese_）

22_＞ Timos卜enko ，　S，　P．　and　Krleger，　S．W ，_：Theory_げ Ptates　and 　Shetls，2nd 　edition ，　McGraw −HiU，　pp．197−205，1959

〔Manuscript　received 　May　lO，1990：Paper　Acc_叩ted　November　6，1990_）

和文要約 1，序　力学特性が空間的に_変動するような不確定構造の応答は空間的に変動し，一般に，統計的非均質，非正規型の確率場（Stochastic　field_）となり，その厳密解を得ることは容易ではない L）。したがって，t数多くの近似的取り扱いが提案されている2〕_。Table_l_は，既往の不確定解析方法を近似的取り扱いの観点から分類したものである。まず，空間的変動を有する力学特性の理想化にかかわるもので，空間的連続確率場！離散化確率場， ’ あるいは，単なる確率変数として扱ったものに分類できる．次に，構造物の応答解析手法による分類があり，空間_的に連続として扱う解析的手法，有限要素法や有限差分法のような空間的離散化手法がある。最後に，得られた応答の統計量の評価方法にかかわるもので，モンテカルロ法（以下，MCS 法と呼ぶ）に代表される統計的手法，摂動法，等を利用する非統計的手法に大別できる3 ）。　以下に不確定構造に関した既往研究について概観してみる。解法的手法として，Bolotin4 ）_は，一次元均質正規型の連続確率過程で理想化された不確定弾性支承上の無限長の梁を，確定解と一次摂動解を用いて陽な形で求・_{− 117一} N工工一Eleotronio _Library

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め，Baker らSlei ，この方法を境界を有する梁に拡張している。最近，Bucher ら 6）_は_，_梁_の _{曲げ柔性}_が_{正規型} の連続均質確率場で理想化される不確定な不静定梁構造物の応答を近似を用いずに解を誘導し，応答ば基本確率場とは_線形_関_係にないことを指摘した後，応答の_統_計_的特性を評価する際にはMCS 法や摂勤法を用いる必要があることを報告している。　一方，空間的離散化手法として，有限要素法や有限差分法のような手法が提案されている。中桐ら η_， Baecherらs｝_は_，_{有限要素法}_{と摂動法}_を_用_{いて}_，_{確率有} 限要素法（以下，SFEM と呼ぶ）を確立している。また， Vanmarcke ら9 ）は，有限要素内での基本確率場の空間平均を用いて，摂動法に基づく確率有限差分法を提案している。これらの方法はすべて力学特性が形成する基本確率場の離散化に基づいている。最近，著者 11 ）_は，要素内で規定される確定関数を重み関数とする基本確率場の要素内積分として定義される「局所積分（Local　integral_｝_」の概念を用いて新しい SFEM を提案し，摂動法や MCS 法を用いることが可能であることを示した。ここでは基本確率場は連続として厳密に扱われている。　統計的な方法では， Shinozukaら ⊥z｝−1‘）は，　 MCS 法の使用を前提として，三角級数を利用した多次元多変数の確率場を数値的に発生させる一般的方法を示し，MCS 法による不確定構造の解析例を示している。一般的に， MCS 法の精度を向上させるには計算時間がかかることから，サンプルサイズを小さくする方法］5）・tfi）や一サンプルの解析に要する計算時間を低減させる手法が提案されている17 ）。　ところで，既往不確定解析手法をこのように分類してみると，著者は，不確定解析がいまだ確定解析ほど十分に理解されておらず，さらに基礎的な研究が必要であると考えている。なぜなら，前に述べ _た_{解析的手法}_{では} ，確定解と摂動法を利用しているにとどまり，SFEM においても分割された有限要素を用いて基本確率場を表現することにより以後の解析を容易ならしめているに過ぎない。したがって，解析的手法における摂動法の精度，あるいは，空間的離散化手法における基本確率場の離散化による精度（メッシュサイズ）について，十分な検討が必要である。この点において，Bucher ら 6）の方法は，基本確率場の離散化を行わず，かつ，変分原理を利用して厳密解を求めており，他の方法で得られた解の_{検証}には有効である。しかしながら，この方法は，不静定次数の低いトラス，梁構造物には適用可能であるが_，_{不静定} 次数の_高い場合や連続体の問題に拡張するには_，不静定構造のグリーン_{関数が}_{容易}に求められず，困難である。　そこで，本報告では，力学特性が空間的変動をもつ構造物の解析に対し，確率場理論と Bubnov −Galerkin法（以下，単に Galerkin法と呼ぶ）を用いた方法を新し一 118一く提案する。本手法は，連続体の問題に拡張することは容易で，Table　lの分類では解析的方法と離散化手法の間に位置するものと考えられる。本報告では，確定静的荷重を受ける，曲げ剛性が空間的に_{変動す}る線形な梁，あるいは平板の問題が， GalerkLn近似により有効に定式化できることが示される。また，確率応答の_評価は，摂動法あるいは MCS 法が_用いられる_。 _{数値計算例}として_，両端固定梁，四辺固定板を対象に既往SFEM7 ）との_結果の比較がなされている。 2．Galerkin法による曲げ問題の定式化　曲げ剛性 K（x ）が空間的に変動する一次元梁あるいは平板の曲げ問題を考える。ただし，荷重は確定，静的に作用するものとし，境界条件は確定的に与えられているものとする。ここで_，このような曲げ剛性は式（1 ）に示すように_， _多次元均質，正規型の確率場ノ（：t）で理想化できると仮定する。なお，

f

（x ）の平均値，自己相関関数は既知とする。　ここで，式（4），（5）で示される釣合方程式と境界条件を同時に_満足するような解に対し，Galerkin_法に基づいて，近似解を求める手法を提案する。

Galerkin

法では，確定的境界条件を満足する試験関数を導入することから始まる。このような関数を境界条件を満足するような不均質な確率場として与えることは非常に難しいことから，ここでは，確定な試験関数を用いて．いる。式（8 ）に示すように_，たわみは_{確定試験関数}ベクトルと未定係数ベクトル α の内積により近似される。この式の_{意味}する所は，真のたわみ場は，本来，非均質正規型の_{確率}_場となるが，それを確率変数を係数にも．つ _{連続}_な確定関数列の_線形和により近似したものであり，真のたわみ場を確定試験関数による分解によって一種の離散化を行ったことに他ならない。　次に，近似解と真の解との残差が試験関数と直交化され，結果として，式（4）の確率微分方程式が未定係数 α に関する確率代数方程式となる。この代数方程式の係数行列K は正規型の確率変数となり，未定係数ベ _ク_トルも確率量となる（式（17），（亅8）参照）。したがって，未定係数ベクトルの統計的特性が評価できれば，式（21），（22）に示すように，近似したたわみの統計的特性が評価できる。さらに，曲げ応力のそれについては，式（23）に示すように基本確率場が係数として再度現れ_，式（26 ），（27）のように表せる。　ここで，応_答の統計的特性を_{評価}する方法として，二種類の方法（一次近似摂動法と MCS 法）が適用可能であることを示す。未知の確率分布を持つ _{応答}の統計的特性（平均値と自己相関関数）を評価するために，摂動法が用いられる。仮に_，式（29）のような一次近似を考クると，式（21），（22），（26），（27 ）中の平均値演算が可 N工工一Eleotronio _Library

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能で，式（33 ）から式（36）のように表せる。なお，式（34 ＞，（36 ）に現れる集合平均の項，＜

f

（x）Ahw ＞，〈Ak．Ak，，〉は，式（3ア），（38）の多重積分で記述でき，結果として，応答の_{統計的特性}は基本確率場の_特性により表現できる。　一方，基本確率変数あるいは確率場と応答の間に強い非線形性が存在する場合，一次摂動法の適用が危ぶまれる。そこで，MCS 法の適用が考えられる。多次元確率場の発生には，文献 12）の方法がある。MCS 法では，

f

（x｝を発生させて，式（16）の積分を数値的に行い，式（19）をサンプルごとに解けばよい。もし，たわみの統計的特性のみを評価する場合には，式（38）で示される確率行列K の各成分間の共分散行列よりK のサンプルが発生でき，式（19＞をサンプルごとに解けばよく，わざわざ

f

｛x ）のサンプル場を発生させる必要がない。このことは，たわみ場が確定試験関数列により分解されて，確率場（無限個の確率変数から成る）の問題が有限個の確率変数の問題に置換されたことにょる利点であるu）。 3、数値計算結果と考察　1）確定一様荷重を受ける不確定曲げ剛性を有する両　　　端固定梁　両端固定条件を満足する試験関数は式（39）に示すものを，基本確率場

f

（x｝の自己相関関数は式（40＞に示すものを用いた。パラメータとして，相関長さ b，試験関数の重ね合わせ総数N を選んだ。本提案手法では，一_{次摂動法}_に_基_づ_い_て_{応答}_の_{統計的特性を} 評価するものとし，同時に一次摂動法に基づく既往 SFEM ’）も解析される。したがって，この比較は_，基本確率場（入力）から応答が形成する確率場（出力）を導く手法の比較となる。ただし，既往手法では，離散化確率場を用いているために_{相関長}さに_応じた要素分割が必要となる。ここでは，わ〃の．値で， 1，0， 0．1のケースを考え，それらのケースについて，各々，10要素，50要素の分割とした。　確定な_{問題}では_， Ga1erkin法の_特徴として_，　N を大きくすれば厳密解に近づくことが証明されており，この性質は，Fig．5やFig．6に示すたわみ_，曲げ応力の標準偏差の空間分布特性にも現れている。また， b値の小さい範囲で低次の重ね合わせ総数を用いても既往 SFEM の_{結果}と良い一致を示す。なお，ハ1に_対する_解の収束については_，応力の収束がたわみのそれよりも遅いことも観察される。　2）確定一様分布荷重を受ける不確定曲げ剛性を有す　　　る四辺固定平板　四辺固定条件を満足する試験関数は式（41）に示すものを，基本確率場

f

（x，_y）の自己相関関数は式，（42）に示すx，y方向に分離可能な指数関数型を採用した。本計算例においても，梁の問題と同様の比較を行った。 Fig．9に_，たわみ_，曲げ応力の標準偏差の空間分布を示す。両手法の結果の問に少し差が見られるものの，分布の傾向は良く一致している。また， Table　2には，　 N ＝ 25のケースに対する各ケースの応答の標準偏差の全体での二乗和誤差を式（43）で定義して，解の収束を見たもので，ハ1＝Lから16と増すにつれ，N ；25 のケー_スに対する各ケースの誤差は全体的に次第に小さくなってゆくことがわかる。ただし，板の特殊な位置（例えば，板の_{中央点）}の応答の_{平均値}・_{標準偏}差の_{収束}については必ずしもそうはならない。これは，仮定した試験関数に_依存するものであり（nx_，　ny＝．2，4，6，…の_{試験関数} では，板の_中_{央点}の_値は_常に_零_であり，これらの関数を考慮しても応答への_寄与はない _）_，つまり，板の_{中央点} に対し非対称な試験関数を考慮した結果である。 4．結　論　空間的変動を有する_{力学特性}を持つ構造物の新しい解析手法を提案した。本手法の特徴は，確定関数を試験関数とする

Galerkin

法に_基づいており，新たな SFEM の定式化を示唆しているものである。既往SFEM との比較を行い _，良好な一致が_見られた。最後に_， Galerkin 法は，動的，非線形問題，汎関数の存在しないような問題にも適用可能であることから，今後の幅広い応用が期待できるものと考えられる。一

₁₁₉

一 N工工一Eleotronio _Library