梁-柱問題に対する混合型境界要素法(梗概)

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[m

veitsc]

UDC:624.04:624.072.2:ee4.0ts.2

A

MIXED

JournalofStruetualand ConstTuctienHiRfi\kzavaXasYvagfi

EngineeringNo.350,Apri],1985 M 350 ny･maM 6e 4 4 n

BOUNDARY

ELEMENT

METHOD

FOR

BEAM-COLUMNS

'

by

SEIICHI

TASAKA*,

Member of A.I,

J.

'

1.Introduction

The recent

development

of the

boundary

elemeBt method

(BEM)

in various fieldsinengineering looks remarkable, ascan

be

$een inRefs,i'5) Especially,'fairly_{successful applications ofthemethod tosuch}

problems that

thefiniteelement method

(FEM)

seems tobeunsuitable･to use havebeen realized. However, therestillexistsome

technical

difficuities

inherentlyihvolved inthe

(direct)

BEM, one of which isconceTned with theexistence of the

(complet

or exact) fundamentalsolution fora _given differentialequation. In fact;we'usually' encounter the

non-existence of complete fundarnentalsolutions for_generalinhomogeneous bodiesalthough differentialoperators

involvedare relatively simple. Generalnon-self-adjoint _problems are another noticeable caseof thenon-existence of complete fundamentalsolutions. Furthermore,inthe so-called displacement-typeBEM theremay occur thelossof accuracy in resulting stresses ifapproximate

gr

incomplete･fundamental solutions are employed.

.

Recently,several studies

for

overcoming thesedifficultieshavebeenmade. Brebbia'}hasshown applications of

the BEM toinhomogeneoustwo-dimensionalelasticity and _{potentialproblems} wherein inhomogeneity isidealized

bytheassemblage ofa certain number of homogeneous regions, resulting a _globallydiscontinuousinhoniogeneous region, so thata _general continuous variation of inhomogeneitycannot be'treated.Butterfield6･T)analysed

continuously inhomogeneottsproblems inpotentialflowsand beams on elastic foundation,The fundaniental

solutions for the correspondipg homogeneous cases are used therein, and itwas shown that the effect of

inhomogeneityappears as the occurance of domainintegrals.The similar approach hasbeenemployed also by

Tanaka and TanakaS,9)forthermoelastic and elastodynamic _ptoblems. Ontheother hand,Tosaka and KakudaiO･'i･'Z)'

'

developed a mixed-type procedure inthe BEM, in which the _principaldifferential operator is canonically

'

decomposed

and the

fundamental

solution forthedecomposeddifferentialoperator isused. Theyhaveapplied'this scheme to the fourth-orderself-adjoint and non-self-adjoint elasticity and eigenvalue _problems. However,their application' islimitedtohomogeneeuscases and theinvestigationof applichbility and efficiency of the mixed scheme

to inhomogeneous-casesisstill leftopen.

Thispaperdealswiththe mixed scheme inthe BEM applied to homggeneousand inhomogeneous linerelastic

beam-dolumnstogetherwith _the sirnply-supported end condition. The formulationisbasedon the method of weighted residuals and integratienby _parts,'whichisequivalent to the methodology dueto _{Green'sformula,}

Althoughtheproblem treated

herein

isquitesimple, some fundamentalnew featuresof themixed BEM may bewell recognized. Totally three mixed schemes are _presented.The firsttwo schems are applied to homogeneous

beam-coiumnswherein theconcept of the`degree of incompleteness'of fundarnentalsolutions isillustrativelyshown

which hasnot beenconsidered

by

Tosaka and Kakuda

(see

also'Section'2), The third scheme isused for

inhomogeneous cases. This scheme can beobtained by a simple modification of thg second scheme, and ismore

easily _performableas compared with theprocedure giveq byButterfieldand others. Resultsof numerical examples

are

given,'

and the approximate solutioris are compared with the exact onesiS) forthe estimation of accuracy and convergence.

2.Summary on the Mixed Scheme

A briefcomparative study bf themixed schemes inthe FEM and BEM ismade forclarifying thecharacteristic of ' _{Research Associate,} Dr. Eng,, Department_of Regional_{Planning, ToyohastiiUniversityef}

Technology-37-NII-Electronic Library Service

the mixed BEM and itsinterrelationto the mixed FEM

The mixed FEM has beeneriginated

by

Herrmann'-for

platebendingproblems. The scheme consists of theinde- PL> eP

pendent approximation of displacementand stress

components. This corresponds tothe _appropriate decom-

rx

y

.PO.:l'/i,O,",Off,l.h.ifth.e.r,.O;geJ..di.f,fe:::,ti,atdiOffP.e,r,a.t:'.ltO.pa,:.et'Xg? Fig.1 A m6dei ef

EEarn-coiumn

The mixed BEM is

based

on the same approach. We will illustrateitina relatively _general situation for

(homogeneous)

beam-column_problems

_(Fig.

1).

2.1 Beam-Cdumn Equations

We hereprepare equations of equilibrium of beam-colums, The equation of equilibrium of a homogeneous

beam-column$ubjected to a lateral

distributed

load,_q=q(x}, and a compressive･axial load,P, is_given by

EIttf"'+Ptti'=q---･--･-･-･･-･･-･･-･-･･----･-･･･-･-･･-････--J･-･･-･････-･･---･-･･--･-･･-･･---･--･-･-(.1)

where El impliestheflexuralrigidity, w= ttKx) isthelateraldisplacementand

(

>'==d(

)ldx.

Ifthebeam-column

isinhomogeneousowing tothecontinuously variable flexuralrigidity which may byexpressed as El=EI(x), then

the equation of equilibrium becomes

(EIto")"+Pw"=q･･･-･･-･-･･--･---･･-･･･-･-･･-･･----.-..･"...H"...-",,",.H,.."...m.".,k..H..(2)

Suppose that the beam-columnissimply-supported ;itsboundarycondition is_given by

tvCx)=O and tV'

(x)=O,

x=O, l,･･--･--･-･J-･･--･--･---･---･･-･･---･･--･--･･-･･---･･-･･-･･-･･-･-<3

)

where x=O and x=l implythe leftand right ends of the beam-column.

We will consider thenormal

(non-critical)

behaviorof thebeam-column,so thatthe bucklifigisassumed _tonot

take _place.

2.2 Mixed

Schemes

inthe FEM and BEM

LetHi bea Hilbertspace, which isa space of functions,say

f=f(x),

such that the followingcondition is satisfied :

f`tf2+Cfi)2Id.<..,.H,,H.,H.,".H,.H"".,..m-,.".,",,-.,.",.",,",,H,,...,-.,.k..k..:..･-･･･-･･･--･--.･,-･(4)

LetH:be a space of functions,

f.

inHi'and eyery

f

iszero at

the

boundary.Hence, Hlis a subspace of H'.In our

problem thedeflection,w, and bendingmoment, M, are zero attheboundary,so thatbothwand Afare inH:.This

' impliesthatHS isthe admissible space.

We firsttreatthemixed FEM. From theReissnervariational _principleitfollowsthatthecondition of the zero first variation forany independentadmissible variations iw,and 6M of the foliowingfunctional

(see,

e._g., Washizui5))

n=SX'`12 ttfM'+M'+p( tef

)i+2auidx,

wandM in HS---･･-･･-･･-･･-･---･-･･････--･--･･--･･-･-･-･( s

)

'

gives rise to the equation of equilibrium ol

(homogeneous)

beam-columns

(cf.(1

))

:

:t-+"pi'

of=a

1

'H--''"'"''"'"''"'-"'H--H''-'"''"'H'''"'''H'"'''"''"''""H-""''"'"''"'''-H"'(6) where

P==PIEI

anda= _qIEI. LetH" bean approximate solution space, which isa finitedimensionals-bspace of

H:, and tvhand _Mh beapproximate solutions of w and M, respectively, foundfromHh. The functionalin

(

_s

)

is now replaced by the approximate functienal:

nh=SJC`l2{u;h)'(M"r+(Mh)'tp((u,"r)'+2atvhldx,

tohandM"inHh･--･･-･･---･J･･･--･･-･･--･--････--･--･--･--･--････････--･･-･･･-･･-･･･--･-･･･----･･-(7)

Sincebothtvh and Mh are inH", the zere firstvariation of nh in

(

7

)

with respect to w" and _M" resttltsinthe

symmetric mixed finiteelement equation

(i.e.,

the stiffness equation>. Hence,wn and Mh can beapproximate solutions ina weak sense of

(6).

Comparing

(

5

)

and

(

6

),

we can see thatthesoltttions wand M obtained by.theindependentextremization of the

functionalin

(5>

are identicalwith the solutions of the followingsimultaneous variational equatiens:

-38-q(x)

'

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Xi(M-

ztf

)e

dx=o

.)('i(M"+pof-a)odx=o forany admissible weighting

(or

Jf`(Me+

ttr _{dio dx=o}

X"l(M'+Pw')

di'+aeldx=o

.-,.".,-",.-.H,,".,".,.-,,-.,-.,".,-.,.",.",.",,k...""""".,",,H.<8)

test) function_e; i,e,,_O isin_H}. Integratingby _parts,

(8)

becomes

",,",,",,".,".,.",..,.,.,,,.,.",.,.""""".,"...,.i.,--..H..aH.-H..-:,.-,,-''-(9)

Sub$titutingttrh, M and di,respectiyely, in

(9)

with ehaset of basisfunctionsinHh, we again

arrive atthemixed finiteelernent equation, Note that inthe mixed FEM the weighting functionO isadmissible and

itsapproximation dihhasthe ldcalbasis

functions

inHlton each

finite

element.

We now suppose thatdiin

(9)

isnot admissible, Then, boundaryterms appear in

(9>

so,that

.ft(Mdi+

ttio'

)dx=[ed

e]

Xi{(M'+pw'

)o'+aditdx=[M'

e]+pttcfdi]

inwhich

[･]

denotesthe boundaryterm,

certain boundarymethod will beemployed. sense with an appropriate choice of the wei

The usual mixed BEM doesnot beginwith

=[ed e]-[wdi']

.",.".,".,.",."..",.",.""".,.",.",,".,".,".,"."",.",,H,･(10)

Inthiscase the fermalismof the mixed FEM isvielated, and insteada

The mixed BEM isan alternative _procedureforsolving

(

6

)

ina weak

ghting function.

(10)

butwith the followingadjoint variational form:

.('`(Mdi-we")dx

Xi{-(M+bw)

e"+a ¢

}dx=[M'

di]+ptw'o]-[M ¢']-ptwdi']I

-t-t-T-t-t-tt-ttt--tt-1;t-t---+-tl---t---t----(11)

which can beobtained byintegrationby_partsof

(lo).

Inthesirnilar manner tothemixed FEM, wh and Mh, bothin

Hh, are substituted into

(11)

with e an appropriate fundamentalsolutien which

generally

hasnot'local' basis

functions.Sincediisnot admissible, _we can selectit _so thatthe symbol of domaini.ntegrationin

(11)

is eliminated,

However,thecomplete elimination isusually unattainable, siothatwe reach a set of discreteequations involving

bothinteriorand boundarytermsintheformof discreteintegralequations. Thismay besolved by,forexarnple,

iteration_procedures.Thus,we can see thatthemain differencebetweenthemixed FEM and BEM isinthe selection of available classes of weighting functions.Remarkthatthecondition thatbothtvt`and Mh aTe inHhisrequired also

forthe mixed BEM becausethe integralterms in

(10

should befinite.

Itshould benoted thatinthe application of the mixed BEM to

(ll

),

thechoice of diisnot ttnique. Furthermore, we may assume differentweighting functionsforthefirstand second equations in

(11).

Sincetheusual

direct

BEM uses fundarnentalsolutions as weighting functions,thisversatility inthe selection of fundamentalsolutions will give rise tothe concept of the `degree

of incompleteness'of fundamentalsolutions. This concept willbeillustratedlater

on, and itseffect on the accuracy of approximate solutions is

discussed

_thereinas well.

ny

Finally,itmay beworth while to mention that the mixed BEM always ttses `incomplete' lundamental

solutions

beeauseef thefactthattheorder ef differentialoperatois isreduced byaeertain appropriate decomposition of the

original equation ;the completeness can besatisfied'only forthe original equation. The appearance of domain

' integralsisa natural and directeonsequence of this fact.

3.Applications

The mixed BEM summarized inthe_previoussection isnow applied tothe displacementand stress analyses of

beam-columns.Totallythree'mixedschemes are _presenSed,which are refered toas Schemes1,2and 3,Notethat

each scheme isnot cenfined to the simply-suppoted case butappiicable to other boundaryconditions, However, since differentboundgry _{conditions generally}leadto differentadmiss{ble _spaces, the_previousdiscussion_givenfor

the simply-sdpported case may no longerholdforother boundaryconditions.

Schemes1and 2are concerned with homogeneousbeam-colttmnsthe equation of equilibrium of which is_givenby

"

).

Thetwo schemes willillustratethe`degree

of incompleteness'of fundamentalsolutions _;Scheme 1employs the

fundamental solution of Helmholtz'sdifferentialoperater and Scheme2doesthat of theLaplacian.Scheme₃deals

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with inhomogeneous

beam-columns

_governed by

(

2

).

Thisscheme isobtained

by

asimple modification ofScheme

'

2. Hence,, the exteption.for inhomogeneous cases.i$,straightforward. , 1

'' ''' ' ' ' ' 3.1 Scheme1 ･･'/. '

We iFtroduce a new dependent variable,. M=M(x}, fgr,the decolnposi･tion of

(

1,)to

Fhe,fgllowing

simultaneouF equatlons _:

It'+UAS',M=a,

x,Lp

l

''`"H"'H''''''''''''''"-'-:-'-H''H''H''H''"i'""'`"''"''''''''''''''"'''"''r''"r"''"':(12) '

Define thetwo functiens

_e,(x)

and

a,{x)

satisfying the,following equations, respectiyely, ../.. . _,

:-/i[:l:fiIXale,L,=',,.,,,

I

'"""''"'"''"''-''''`'''"''"`'"''H''-''m'-"'";;"''"'"''I"'""''"'`"''"''"''k,"(i3)

t tt

'

where i( ･,･

)implies

the Dirac

delta

functiop.T)hesolu,tiens to

(13)

qregiven as, .

ipt(x)=lx-gl12andiie(x)=sin(Alx-el)/2A-･･--･･-･･-･----･-･--･-･･-･･-･･･-･･･-･･･--･･-･･-･･････-･-･･-(14)

'

We use thesetwo functionsas weighting functionsforthe firstand second equations, respectively, ef

(12).

Applyingthe methed of weighted regiduals in the formof

(11)

to

(12)

yields

.' '

w,=(w,+w,+etk-a-e)a+X;iix-elMdx)i2 '

l.H,.",.,".,(is)

Me=IMt(ocosAg+NidlcosA(t-e)+Q,sinAe-Q,sinA(l-e)+.L'`apinA.,1x-e1dx]12A1 _{. .} Furthermore,the discreteboundaryequation associated with

(15)

can be ob.tained by'putting

e=

Oafid･･e=l as

follows:

'

'

wo- wi+ _{la=X'ixM dx}

,,.. , ' ' '

ttts.w,+la=-.J[-i(t-g)Md,x. . ･ ....:.,.:,..:..,,:..,...:.:...i...`.･i･･････････t････-･(16)

Nifo-XMtcosAl+QisinAl=Jf'asinxxdx '-' ' ' '

AMeeosAl-Mli+QosinXl=-.Jf`asin,X(l-x)dl.p., ,,, '',' ･.

Equation

₍₁₆₎

includes e{ght

boundary

terms, four of which are _preseribedby theboundary condition, so that we can

'

write down

(16)

as the following-matrix-vectorequation: ,

4X4Ku+4X4Lv=f･.".H,.".,",.":.".:.-,."..H.."..:,.;.,,-.1".,..L,.,.,...th.H.,-.,.-,...,,",,'".,...."...,H(17)

where the vectors u and v consist of, respectiyelY,' the unknown and known boundaryternis,and ' '

f==lll',G,Jl;Ji

l',

T･transposeJit-･"--･-･-･･････-･･-･･--i--･--･-･-･･-･-･-･･-･-･･･-1･--･-･･-･--･--･-･--･･-(18) ' inwhich ･' ' ' '' ･' ' ･･' --' ,',

"-JC`xM

dF= AxS.xs

PI,i.

.,

1'

.'

"

.,' ', ,' .' ･' '--1'

b=-X'(l-x)Mdx=-Ax･>](l-x,)M,

-･ - , ' - _i

ifl'

.L"asihin dx=fAt'll]atsiriint "

1.

1･

.'.･ . ,

lllei-JC'iisinX(l-'x)dic==Ax]>l]a,sinx(IL'x,)i

'i ,.'' . .'''''

'- '･

t t tt

Here,theintegralterms are discretizedbytherectangular rule with N thetotalnurnber of dividedsubdomains of

[O,

l],Ax=xt+i-x`=llN ･andxt thg coordinate value,of the.i-th.node,,i=1,2,･･･,N+1. The similar nume.rical

i4tegrationfQrmulaisapplied to."5),as welL. ... , .. , , ., ,. , .

/ t

･Equation

<17)

'togetherwith

(15･)'are

solv,ed 'byiteratiQq･Qnthebasisof suceessi･ye approximations intheform

Kz"""'+Lv=fbl', n=e.1,2,--･p"t･--･-"･-.---･･`-･--･･-･･:`･t--･･･-`･--･･･-･-･･-･-･･-･･･--･--･･--･-･'H-'-･-(19),

fromwhich theunknown boundary termS, tveand,ua are obtained.

qand

Qe

are calculated bythe central difference

formula.Note that insteadof'the iterationprocedrire we can solye

(15)

and

(17.)

as simultaneous argebraic

equations. 'However,wheri the number of divided･subdomains increases,a large.scale 'non:symmetricand aspally

densecoefficient matrix should be'treatedinthiscase. On the･otherhand, the size of the eoefficientmatiix inthe

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iteration_proceduredependson the number of unknowns on theboundaryand not on thetotalnttmber of unknowns in

the domainwith boundary.Hence, thechange ofdiseretization intheinteriorof thedomaincanbeeasily _performed.

3.2

Scheme2 '

'

Thefurthersimplification ofScheme 1leadstothe_{presentprocedure;}we use thesolutien tothe firstequation of

(13)

<see

(14))

also forthe second equation of

(12)

as theweighting function.Equation

<12)

isnow rewritten as

M= tv' . ,

..."...･.,,-.,...-..H..H..H....i.,",,H,,-,,".,.",.-,,",,",,H..-･･･--･.-･･-J･･--･-･･･-･･･--･--･<20)

M"+A' ttS'=ff ,

The similar calculation _gives rise to '

l

we=(ttfo+w,+ca-(l-e)a+X`lx-elMdx]!2 M,=[1lfi+M,+eQ,-(l-e)Q,-xSXt1x-eiMdx+X`1x-e1adt]12

XixM

dx

-X"(l-x>M dx

Xixa

dx

-Xt{l-x)a dx ･････-･･-･･･-･-････-･･-･･-･･--･-<21) and we-ws+la= wo-wi+la== ･-･･-･･････-･･--･･--･･････-･･-･･-･･--･--･･-･･--･-･･･-･･-(22) A2(wo-wi+le)+ll4b-Mi+lQi= Xt(wo-wi+l&)+Mh-Mi+tQo= ' '

The matrix-vector equation of

(22>

can now be written as

' 4X`Gu+4X4Hb=f.,T,,H.,.-."..".･-･･･-･･･-･･･-･･--･--･･･-･-,-･･-･-･-･-･･-･--･･-･･-･･･-････････-･-･･-･･･････････-･-･･(23) ' where. _. ' f=III,GsJl,Jil"''''''-''-'''"''"'''''''''''''"''"""''''"''"''"-'H''-HT'H''H'''''''''''''''""'':'''''''''''''"'''(24) and

A=JC'xM dx=Ax

S.x,

M,

G=-Xi(l-ic)M

d.= -A.

\(t-x,

)M,

,

A=.Ctxa

dx :Ax\x,a, '

A==-.Jf'(l-x,),adx :TAx

>

](l-

_x,

)a,

The similar numerical integratiQnformulaisapplied to

(21)

as well.

' ,

Equations

<21)

and

(23)

are solved byiteration,This case

differs

fromScheme 1insuch asen$e that thesecond

equation of

(21

)

appears as an integralequation with respect _toM. Hence, _thedouble iterationisrequired here.We

write the iterationscheme for

(23)

as .

GtiM+i)+ev=tin',n=o,1,2,+-･- -･･･--･･--･･･-･･･-･･-････････-･--･･-･･-･･･-･･･-/･--･--･･-･･･-･･･-･･･････-･--･-･(25)

Furthermore, the iterationscheme forthe second equation of

(21)

isdefined by '

M[,M+i]!I(M{M)+AM+i,+J-･･-･･-:･･-･･-･;･--･･-･･･-･･･-･･･-･,･････-･･-･･-T･-･･-･･-････････-････････････････---･(26) ' ' where , ' .. I(Mlmb=-x2drZ1x,-e1M"]12 t j Ain+i)=IMtn+U+Mr+i)+Qtn+i)-(l-e)Qr+i)I12 ' J=AxZlxrela,12 ' '' t t t and

e=xt

.

Noticethat the square bracketsand _parentheses implythe innerand outer iterations,respectively. ,

'

Compared with Scheme 1, itcan beseen thatthe differenceof the degreeof incompletenessof fundamental

solutions indttcesherethatof-the iterationform,Itcan _generallybesaid thatthe higherthe degreeof incompleteness

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is,themore thecomputationai effort is,so thatthe limitationof simplification inthe use of fundamentalsolutions

may. exist in_practicalview _points.

3.3 Scheme3

,The last scheme _presentedhereisforthe analysis of inhomogeneous beam-columns.This scheme is essentially equivalent toScheme Z except forthe mixed form of theoriginal eqttation to which the scheme is applied _;the

introductionof a dependent variable, _N=N(x>, into

<

₂

)

_yields

w"=NIEI, EI=EI(x)

N"+Ptd'=q

'"''"''"''"'''"''"''k''"''"''"''"'''H-'H''H''H''-''''i''H''--･--･--･---･."...(27)

The use of the sarne ftindamentalsolution as thatinScheme 2and the _performance of the similar calculation

eventually result in

w,=( tu,+ w,+

ea-(i-e)

_{a+JC`lx2elNfEi dxl12}

.,.H,,",.H,.,."..-･･"-･-･･-(28)

M=[IVh+Nt+ene-(l-e)Rt-PJC`lx-elNIEIdx+.)('ilx-Hqdxl12

and

an- wt+ la=.('`xN/EI _dx

tet,-w,+ _{ta=-JC`(t-x)NIEI dx}

L--.,...".,"..".,.H.."...",.H..."""-.-..,-･･--(29)

P(wo- wi+ la

)+M-lv}+

IR,=.L"xqdx

P(wo-w,+la)+ivi-ivi+IR,=-.t'`a-x)qdx

where R=N'. Remark thattheequivalency of Schemes2and 3indicatesthatthe effect of inhomogeneity can be expressed more _generallybythe incompletenessof fund'amental

solutions

(cf,

Butterfield6,')), qo

4.NumerjcalExampleS

P-->U

EI eP

The followingthreeexamples are

(See

Fig･2)･

(a)

Examplel

L

treated

Example1:Homogeneousbeam-columnsubjected to a

uniform-lydistributedlateralload,_{q= qo,}and a compressive axial load,

P.

Example Z:Homogeneous

beam-column

subjected to end moments, MA and _Ms, and a compressive axial load.P. M,isin

the positive

(clockwise)

direction

and M. isnegative. The

subscripts _A and B imply the leftand right ends of the

beam-column,respectively.

Example 3:Inhomogeneous beam-columnsubjeeted to the

sarne loadsas Example 2.

InExample 3the variable flexural rigidity EICX)

isdefinedby

EI(x)==EL+Eh(xll)"･････-････-･-(3o>

where EL=EI(O) and EL+Eh=EI(l)(see

Fig.3).We will take ttpvalues of a as follows:

a=lo-, lo-3,lo-i,lo-i,1, lo, 1cr,

103,lo4Note

thata=O and a=co can be considered to represent homogeneousbeam-columnshaving

EI=EL+Eh and EI :EL, respectively, although the discontinuityoccurs at x=O and

x= l.Hence,thesecases may leadtotheexact

-42-P{ MA

_ng

P{

n

Ei (b} Example 2' MA MB

M

EICx) (c) Example3

Fig.2 Numericalexamples

y,

y,

ct=oe<oc<1 ct-1 EIB 1<ct<copt=co _EIA k x

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Table 1 Boundary yalues

{a)Example

1Scheme 1 Scheme2 Div 10 30 so lo 30 50 Iter 2 2 2n=5m=4 s4 54 [xect eeen%Qt O.04581LO.045al O.5499.e.q499 O.046Z5.O.04625 O.5166-O.4833 e.04628-O,04628 O.5100-o.4geo e.04581tO.04587 O.5500-D.4500 o,-o, o,-o, 04624D46Z4516J4834D.-o. D.-o. 04S260462651004900o.-o. o,-o. an630046305000'5000 OCIO-S}= D.O (b}ExampteZScheme 1 Scheme 2 Div 10 30 so 10 30 50 Iter z 2 2n=7m=5 76 ]6 Exact

eoeLQe% O.595a-o,4gse-O.0495-O.0505O.5629-O.5296-O.O15E-O.O]61e.ss63-D.5363.O.OIOO-o.DloeO.5959-O,4959

o.o o.e o.-o. o. o, S6295296ooO.5563-O.5363 o.o o.o o-ooo,5463,5453.o.o oo o-5> = o.o

Table2 Mid-point values

(a)Example1Scheme

1 Scheme 2

Div ID 3e 50 10 30 so

Iter 2 2 2n=7m=5 16 16

Exact

wemq O.]396o.o1.139o,oO.1395o.e1.139o.oO.1395o.o1,I39o.oO.139So.o1.140o.oool'o.1395.o.139oO,1395o,o1,13gD.Ooo1o.13g5,o.T39.o

ooo-5} =o.o

(b)Example

2Scheme 1 Scheme2 Div le 30 50 10 3e 5D ]ter 2 2 2n=5m=4 s4 54 Exect

wemQ o,e146oo,oo.l]gqo.oO.O1451o.oO.1395o.oO.O1450o.oO.1395o.oO.e1462o.oO.1396o.ooooo.O1450,e.139S.oo.o.o.o.O1449e139Soe,e,o,o.e1449o1395o

ooo-5) = o,o

5

8:eegOetuMa=<

-5

Fig.4 Beundaryvalttes of theangte ofrotatlon lnExampLe 3

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solutions, so thatwe can observe the asympto-tic behaviorof the approximate sglutions as

a---O and a-od. . ･

'

Only

numerical values are of interestwhen

comparillg the approximate solutions with the

exact ones. So we assume the fol16wing

'

numerical

date

:

qo=1, P=1, MA=Mh=1 EI=1, Eza=Eh=1, l=1

inwhich the absolute values a'reshown foTP,

Ml and Me, The real bendingmoment and shearing forceare obtained, respectively, by

m=-EIM=-N and

Q=ml-Pe

=-R-Pe

Table 1shows the boundaryvalues of the

angle of rotation and shearing forcefor

Exam-ples1and 2,each of which hasbeensolved by'

bothSchemes 1 and 2, The notation Div

impliesthenumber of dividedsubdomains and

lter the number of iterationswherein nt

stands forthe maximum'numbet of inner

iterationsduringn outer

iteTations.

The con-yergence tolerance was assumed tobe'10-`in

every case. Itcan beseen thata relatively

largeportion of the approximate solutions reveals unsymmetrical results. Althoughthis

.result

is_quiteunsatisfactory, the

unsymmetric-itygradually disappearsas the nllmber of

dividedsubdomains increases, so that thel

convergence isexpectable, We can see also

thatSchemes 1 and 2 have.the_41mp.stsame,

accuracy forExample 1,while a certain

diffe-rent degreeof accuracy isseen forExample2,

This may imply that thedegree of

inc6mplete-ness of fundarnentalsolutionsi isnot

necessari-ly concerned directlywith the accuracy of

numerical results, Rather,the symmetrical

usage of weighting functions

{Scheme

2) seems to be adequite foTa certain kind of

problems. InTable2 we present the results

obtained at the middle _pointof the domain.

Fairly_good accuracy and convergence are attained

Theboundaryvalues and mid-point values of zero inthiscase. The number of iterationsan reveals again unsymmertrical results, butthe

deflection,angle of rotation and ･b

-44-'xlo-2･. 1 .le･82ts -n5 o /t ct o xlo-5 2 .1eu,ELg-1oerv-a[< 5 o lo-410-]io-2 io:i i io (a)Deflectien ' ' ' ' t/ lo2lo3lo4 ct oo 2 :28elE5m olo-qio-]io-2 io-i i

(b)

Angleof lo lo2 rotattop1010oo 1/ _ct o

o io-4 io-] io-2 io'i i io io2 ioi io4 co

(c)Bending.moment

Fig.5 Mid-point values inExample 3 (Div=10)

' d '

'

several displacementand stress cemponents forEtiample 3are _giveninFigs.4and 5.Theshearing forcewas always

dcornputation time w6re alrnost equal' to thoseof Scheme 2,Fig.4

convergence isfairlysatisfactory. For the mid-point values of the

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mid-pqint values converge somewhat fasterthan the _boundaryvalues. From thisresult we may conclude thatthe

present mixed BEM igwell applicable to also inhomogeneous_problems. '

5.

Conclusions

'

We haveconsidered the mixed scheme inthe

(direct)

BEM applied tothe displacement and stress analyses of

homogenousand inhomogeousbearn-columns.Forthe_purpQse of

_glarifying

thecharacteristic of the mixedBEM and

itsinterrelationto themixed FEM, abriefcomparative study of the mixed schemes intheFEM and BEM hasfirst

been made for a simply-supported homogeneousbeam-columnin a somewhat _general situation involving _the

fundamental methodologY of the mixed procedure.In applications, totallythree mixed boundary element schernes, refered to as Schemes _1,2 and 3, havebeen_given wherein Schemes 1and 2are concernecl with homogeneous

beam-columnsand Scheme 3 isforinhoinogeneous_ones.

The

difference

betweenSchemes 1 and 2' represents, on the ene hand, the conception of the `degTee

of

incompleteness'of

fundamental

solutions;the degreeishigherinScheme₂

(or

lower_inScheme _{1).On theother}

hand,S¢heme2uses the same fundamentalsolution forthetwo decomposedequations, while Scheme₁is

different

_;

thesymmetrical usage of weighting functionsisrealized inScheme _2.Scheme₃hastheessentially same structure as

Scheme 2, so that _{the extensionforinhomogeneous}cases isstraightforward･.

The resulting discreteinterior and boundaryequations havebeen solved by iteration.Iteration_procedurei$ convenient fer_thealternation of discretization'inthe interiorof thedomain since the coefficient matrices dependonly

on the boundarydiscretization.'

Some numerical examples havebeen analysed, and effects of the above mentioned contrastive _propertiesin

Schemes

1and 2 and inhomogeneity concerning

Scheme

3havebeeninvestigated.The numerical results haveshown as awhole fairly_good accutacy'and convergence. This may demonstratethevalidity of the_presentmixea BEM for

homogeneousand inhomogeneousbeam-columns. '

Acknowledgement The autherwould liketo express histhanks tothereferees fortheir valuable comments and'

cntlclsm. Reterences

1)

_Brebbia,

C.A., :The Boundary Element Methed forEngineers,PentechPress,L6ndon, lg7s.

2) Brebbia,C,A,

_{ed,)

:Recent Advancesin Beundary _{ElementMethods, PentechPress,}Lendon, 1978.

3) Brebbia,C.A. (ed.):New DevelopmentsinBou4dary Element Methods,Proceedingsof the SecendInter._Seminoron

RecentAdvances inBounaryEIement Methods,,March,1980.. ''

4),Brebbia,C.A.

{ed,

)

:BoundaryElement

_"t[ethgds,

Proceedingfiof _the ThirdInter.Seminor,Irvine,Califernia,July,

198L･ ･ - ･

-･

s) Banerjee,P.K. and Butterfield,R.

(eds.)

_{:Developmentsin,BoundaryElgmentMethods 1and 2,AppliedScience}

Publishers, London, 1979 and ]982.

6) Butterfield,R,_, :An application of the boundaryelement method to_potentialflowproblems in_generallyinhomegeneous

bedies,in2), ]23-135. '

'

7) Butterfield,R.,:New concepts illustratedby old _problems,in5)-1, l-20. ･

'

8} Tallaka,M. and Tanaka,K._, :Onnumerical solution scheme forthermoelastic_{problems ininhernogeneous}media bymeans

of boundary-volumeelement, Z.A. M,P., _{60,1980,719-723.}

9) Tanaka, M. and Tanaka, K.,:On boundary-volumeelement discretizationof inhemogeneous elastodynamic _problems,

Appl.Math. Modelling,5,1981.194-198.

Io) Tosaka,N, and Kakuda,K._, :New integralequation method for approximate selutiens of bolindaryvalue _problems Partsl

and _2,Trans.ArehitecturalInst._{Japan,No. 321,l982,49-55and No.329,1983,43-51(inJapanese).}

11) Tosaka,N, and Kakuda,K., :New integralequation method forapproximate solutions of eigenyalue _problems,Trans.

AtchitecturatInst.Japan,No. 328,1983,36-43(i]Japanese), '

12) Tosaka,N.and Kakuda,K., :Anintegralequation method fornen･self-adjoint eigenvalue problems and its applicatiens to

non-conservative stability _problgms,In.ter.J.Num. Meth. Eng., 20, l984, 131-141.

13) Timoshenko, S.P. and Geie,J.M,,:Theoryof ElasticStability,,McGraw-Hill,_1961.

14) Herrmann,L.R.,:Finite-elementbendinganalysis for_plates,J.Eng. Mech, Div,,Proc.ASCE, 94, 1967, 13-25.

15) Washizu, _K., :VariationalMethods inElasticityand Plasticity,PergarnonP:ess,New York, 1975.

-45-NII-Electronic Library Service 【研 究 論 刻 凵DC ：624 ．　04 ：624．072．2 ：624．075．2 日本建築 学 会 構 造系 諭 文 報 告 集・ 第 350号・昭和 6e 年4月

梁

一

柱 問題

に

対 す

る

混 合

型

境 界 要素

法

†

（

梗 概 ）

正 会 員　 田 坂 誠 　 　一＊ 　

L

」廴 論　境 界 積 分 方 程 式法に有 限 要素離 散 化の_概 念 を導 入して得 られ る境 界 要 素 法 （BEM ）は近 年 様々 な分 野で用い ら れてお り，特に有限 要素法 （FEM ）の 適 用が種々 の意 味で制 約 的で あ る よ う な諸 問 題へ の応 用 に おいて有用な成 果 を見 ることが できる。 一方， BEM の一般 的 適 用にお ける本 質 的な制 約の ひとつ _{と し}て基 本 解の_存在の_問題が_挙げら れ_， こ れ は現 在なおク リ』テ ィカ ル な問題 とし て残さ れ たま まである。 　 与え ら れ た方 程 式 を 直 接 法 的BEM に より離 散 化し て_得られ る_{境 界}方 程 式が領 域 内 部の未 知 量を含まな いと き，重み関 数 とし て使 用さ れる基 本 解は完 全 （complete ） であるといわ れ る。こ の条 件 を満たさない基 本 解は不 完 全 （incomplete）で あり， その場 合に は 一_{般に未 知 量に} 関す る領域積分 項 が 境 界 方 程 式の 中に表れる。 工学 的 諸 問題に おいて完 全 基 本 解が存 在 する例はま れであり，線 形の場 合でも非均質体の_{平 衡}方程式や， よ り 一_{般に は非} 自己随伴作用素を有す る支配方程式に は通常完 全 基本解 は存在し ない。 　完 全基 本解の_{存 在}しな い問 題を対 象と し たBEM で は近 似スキーム の構 成に多様 性が見 られ る。 特に非均質 体に _対す る BEM の応 用に おい て，　 Butterfield6・’）は対 応 する均質体の基本解を利 用し た BEM につ いて論じ， 非 均質性の効果が領 域 積 分 項の発 生と なっ て表れ ること

を示し た。Tanaka　and 　Tanaka8・9）等_{も 同}様の _観点か ら

非 均質体の_熱弾性， 動弾性 問題の解 析を行っ た。しか し な が ら， 高階の問 題では近 似 解に対す る高次の可微分性 が要 求さ れ る た め， 一_{般に変 位お よ び応力 共に精度 良}い 解 を求め るこ と は 困 難で あ る。一方，Tosaka　and Kakudai°・n・i2》 等は混 合 型ス _キーム を用い た BEM にっ い て論じ， 自 己 随伴お よ び非自己随伴 作用 素 を 含む弾 性， 固有値 問 題へ の応 用を示 し た。し か し なが ら，そ の適 用 例は均 質 体の_場合に_限られ て お り_， また使 用さ れ る 基本 解の多様性につ い て も議 論は な さ れて いない。 　本論では， 均 質お よ び非 均 質線形弾性梁 一_柱問題に対 †_{本 論は文 献}己_{嗹 修正，加 筆し た も ので あ る}。 零 _豊 橋技 術 科 学大 学　助 手 ・工博 　（昭和59年 5 月 15日原 稿受 理 日，昭 和59年11月26日改 訂 原 稿受 理 　日，討 論 期限 昭和 60 年 7 月 末日〕 ，す る 混合型 BEM の適用に つ い て考 察を加え る。混合 型スキーム は変位と応 力を独立に近 似す る手法であ り， これ は支配 方程式の高 階微分作用素を低階作用素の連 立 形 式に変換す るこ と に対 応し てい る。こ こで は合 計 3つ の 混 合 型BEM を示し， 各ス キーム の内容お よ び特 徴 につ い て比 較 検 討す る。ま た，い くつ かの数 値 計 算 例 を 通じ て近 似 解の精 度，収 束 等を 調べ ，_各スキーム の適 用 性につ い て検 討 を 行 う。 　 2．・△ 型ス　ーム の　 ・均 質お よび 非 均 質 梁一柱 の平 衡 方 程式 を準 備し，単 純 支持を有する均 質 梁一柱に 適 用さ れ る混 合型 FEM とBEM の比 較にっ い て述べ て い る。 Reissnerの 変分原理か ら導か れ る混 合 型 FEM では重み関 数が許 容 関数 と 同 じ ク ラス に属 して い るが， 混 合 型BEM で は使 用さ れ る重み関数の ク ラスが異な る。これ が両 者の最も顕著な相違と考え ら れ るが，近 似 解の クラス は両 者 共に同等で あ る。 す な わ ち，混合型の 長 所 として_， 変 位 （w ）と応 力 （M ）は共に_{区 分的連続} であれ ばよい。（これ は （11 ）式に おいて Φ” ＝ _{δ と}い う条件を暗に仮 定し て いることにな る が，この条件はこ こに述べ _{る混 合 型} BEM に対 し，お そ ら く最も自然な 形 式 を与え るもの と思わ れ る．。）注 意すべ _きこと は， 混 合 型 BEM で は常に不 完全 基本解 が 用 い られ る の で，領 域 積 分 項の発 生は避け難い とい うこと で ある。したがっ て_， 完 全 基本解の存在す る問題にこ の_手法 を適 用 するこ との実 質 的 有 効 性とい う もの は 期待で き ないで あろう。 　 豊 一」里　均 質お よび 非 均 質 梁一柱問題に対して， 合 計3っ の混合型BEM （スキーム 1− 3_）の_定式化を 行う。ス キーム 1と2は均 質体， ス キーム 3は非 均 質 体 を対 象と してい る。各ス キーム は種々 の境界条 件に対し て適 用し得る。スキーム 1と2に は異 なる基 本 解 が 使 用 され ，そ れに より基 本 解の ‘不 完 全 度 の相 違お よ び重 み関数の_対称，非対称性の_相違等が考 慮さ れてい る。領 域積 分 項を含む境界 方 程 式お よび 内点で の未知量に関す る内 部 方 程 式は反 復 手 法で解か れる。ス キーム 3は ス キーム 2と本 質 的に同 等であ り，これ より領 域 積分項の 存在が 非 均質性の故 とい う よ り は，む し ろ よ り一般に基 本 解の不 完 全 性に よ るこ と がわ か る。 　

9L

！

Egdil

：

EWPt

以下の三例につ い て解 析結果を示す。 ・一　

46

一 N工 工一Eleotronio _Library

NII-Electronic Library Service 　例1：等分布荷重と軸 圧 縮 力 を受 ける均 質体。 　例2：端モ ーメ ン トと軸 圧縮力を受け る均 質 体。 　例 3：端モーメ ン トと軸圧縮 力を受ける非 均質体。 　 例 1と 2で は各々 に対し てスキーム 1と2の 双方が適 用さ れ，例 3はスキーム 3を 用い て解 析し ている。結果 の概 略は以 下の通り で ある。近 似解は_幾分の_非対称性を 示 し，精 度は領 域 分 割 数に依 存 する ；領 域 分割数の増 加 と共に_精度は向上し，対称 的 な 分 布に収 束す る。スキー ム 1と2にお ける基本解の_不完全度の_相違は，必ずしも 近 似 解の精 度の有 意 差に は結 び付か ない。む し ろ， そ の 対 称 的 使 用に より精 度の向 上が実 現され るこ と『も あり_得 る。 反 復 回 数か ら予 想され る計 算 時 間はス キーム 1が最 も短く，ス_キーム 2と3では大 差 がない。す な わ ち， 基 本解の不完全 度はスキーム の複 雑さに関 連し て お り．同 じ不 完全度を有する スキーム 2と3は均質体， 非 均質体 に か か わ らず実 質 的に同 等の結 果 （精 度，計算時 間 等 ） を与え るもの と推 定される。 　昼」墮」 彊　均 質， 非均 質梁一柱 問 題に対 する混合型 BEM の定 式 化 を示し，若干の数 値 計 算 例 を 通じ て本手 法の_妥当 性 を検 討し た。混合型ス キーム の主た る_有効性 は変 位 と応 力 をほ ぼ等しい_{精 度}で求 め得るという点に置 か れて いる。本 論に述べ たBEM に適 用さ れ る 混合 型 スキーム は，_{完 全 基 本 解}が利 用で きな い広い クラスの _問 題に対し て も ひ とつの有効な手 法にな り得る も の と思わ れ る。 文 　 献 a _{） 田 坂 誠}一；梁一柱 問題 に_対す る混 合 型 境 界 要 素スキー 　 　ム，_第8_回マ ト リッ ク ス解 析 法に関す る シンポ ジ ウム， 　 　日本鋼構造飽 会，7月，1983． 一

₄₇

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