3連および一方向均等無限連続耐震壁の各基本節点剛性マトリックス

(1)

Knt

gN

UDC:69.022:69Y.B41

JouTnal

of

Structural

and

ConstTuction

Enginee[ing

(T[ansactions

of

AIJ)

No,374,

April,

1987

aseR#kasurxnNtae.m

M

374 g･uetu

62 _44n

FIUNDAMENTAL

NODAL

STIFFNESS

MATRICES

OF

3-CONTINI[JED

AND

INFINITELY

CONTINIUED

FRAMED

SIHIEAR

WALLS

by

MASAHIDE

TOMII"

and

TETSUO

YAMAKAWA"*,

Members

of

A.LJ.

1. Introduction

A

very effective way

to

increase

the

earthquake resisting capacity of reinforced concrete

frame

structures

is

to

properly

arrange monolithically

infilled

wall

_panels

in

the

frames.

Hereafter.

the

frame

stiffened with a

monotithicaity

infilled

wali

_panet

is

referred

to

as "shear wall".

The

Architectural

Institute

of

Japan

(AIJ)

recommends

that

building

structures should respond

to

within

the

nearly elastic range, witheut

damage

to

be

repaired,

during

moderate earthquake motions which would occur several

times

during

the

use of

the

buildings'].

On

the

other

hand,

as

the

lateral

stiffness of ashear wall

is

much

larger

than

that

ofa celumn,

the

elastic

behaviour

of non･damaged

frame

structures with shear walls under seismic excitations

tends

to

be

remarkably

_governecl

by

the

elastic

behaviour

of

the

shear walis.

Therefore,

in

order

to

analyze

the

elasticbehavieur of

fiame

structures

by

matrix method of structutal analysis, an accuTate evaluation of

the

nodal stiffness matrices with respect

to

_the

_boundary

nodes

of

arbitrary

continued

shear walls

is

required.

This

_paper

is

lirnited

to

the

discussion

of

the

arbitrary continued shear walls whose aspect of each one-bay one-story

_part

is

the

sarne and symmetric with respect

to

the

longitudinal

and

transversal

center

lines

of

the

infilled

wall

_panel.

In

the

Transactions

of

AIJ!)F3),

the

authors

have

reported

the

exact nodal stiffness matrices of single and

2-continued

shear

walls

by

using

the

analytical

solutiens

(hereafter

referred

te

as

"known

solutions")

derived

by

Temii

et

al.4)"6)

These

nodal

stiffness

matrices

are

full

matrices which express

preciseLy

the

complete relations

between

all

the

force

components

and all

the

displacement

components at

each

boundary

node

of

the

shear

wall,

The

nodal

force

components are

the

hoTizontal

foTce

X,

vertical

force

Y

and moJnent

Mlh

(the

momefit

M

is

divided

by

the

story

height

h

so

that

its

dimension

would

be

the

same

as

those

of

the

other

components

X

and

Y),

while

the

nodal

displacement

cornponents

aie

the

horizontal

displacement

a,

vertical

disp]acementI

and angle of rotation

he

(the

angle

of

rotation

_e

is

multiplied

by

the

story

height

h

se

that

its

dimension

would

be

_the

same

as

those

of

the

other components

of

and

T).

In

this

_paper,

the

fundamental

nodal stiffness

matrix

of

a

one-bay

or

one-story

3-continued

shear watt

(hereafter

referred

to

as "3-continued

shear wall")

is

formulated.

Also

a

_procedure

which

gives

accurately

the

fundamental

nodal

stiffness

matrix

of

a

one-bay

or

one-story

infinitely

continued

sheai wall

(hereafter

referred

to

as "infinitely continued shear wail")

by

using

the

known

solutions

is

_proposed.

This

_procedute

and

the

formulated

fundamental

nodal stiffness matrix of

the

_3-continued

shear wall,

together

with

the

fundamental

noda} stiffness matrices

of

single

and

2-continued

shear

walls already reported, will

be

used

to

derive

accurately

the

nodal stiffness matrix of an

arbitrary

continued shear wall,

This

derivation

will

be

reported

in

another

_paper.

The

use

of

the

nodal

stiffness

matrix

makes

it

possible

to

analyze

the

exact

elastic

behaviour

of

the

frame

structures with

shear

walls

under

seismic

excitations,

By

comparing

this

exact

elastic

behaviour

with

the

approximate one

given

by

using

the

equivalent

bracing

model or

the

equivalent colurnn model, an adequate and

_practical

method can

be

developed

for

the

calculation

of

the

elastic

stiffness

of

the

shear wall,

"

_D.Eug.,

_Professor,

_Kyushu

_University.

e"

_M.Eng.,

_Research

_Associate,

_Kyushu

_University.

(Manvseript

recelved

October

2B,

1986)

(2)

2. The

Fundamental

Nodal

Stiffness

Matrix

of

3-Continued

$hear

Wall

In

oTdeT

to

formulate

the

nodal

stiff-ness matrix

K

of an arbitiary continued shear wall, we use

the

nodal stiffness rnatrix _sK,. of

the

3-continued

load

term

model').

In

order

to

obtain _3KL, we clarify

the

fundamental

nodal stiffness matTix _sK* of

the

_3-continued

shear wall as shown

in

Fig,1.

Fundamental

components of nodal

forces

and

nodal

displacements

of

the

3-continued

shear walls as weli as

those

of

the

single

or

2-continued

shear

walls can

be

classified

into

4 basic

types

as

follows.

Fundamental

components

Type

I=

and

transversal

center

Fundamental

components

Type

ll

=

transversal

center

Fundamental

components

Type

M=

center

line

and

Fundamental

components

Type

rv!

Table1

ferces

on one-bay

three-story

2

1 Ili

Db

Vi

rh

h

t

a bHD:'l]b.Hbb

"'

Fz-Fig.1

Definitiens

of

the

nodal

peint

numbers, nodal externar

forces,

nedal

dispLacements

and aspect of

the

ene-bay

three-story

fTamed

shear wall

antisymmetric with respect

to

both

longitudinal

of

the

shear wall

the

shear

wall

components

which are antisymmetric with respect

te

_the

longitudinal

with respect

to

the

transversal

center

line

of

the

shear wall

components which are symmetric with respect

to

the

longitudinal

enodal extemal wal1

3 i

i

'

5 ,'

tz2-A2

Hu

-...･i'-7-I2?-Y2-M"n.

_'

Lm-tt3

I:,e-1,,e3x

I4-.-.

6..""5

ILil3leVI

Y4M6X6-YGMeife-8

.xf.

l-kUsJ-e7I-?,T

H

The

components

which are

lines

The

components

which are symmetric with respect

te

both

longitudinal

and

lines

of

The

symmetnc

The

representative components

Pi

and

Pe

of the

batancing

and unbalancing

fundamental

components of

th

framed

shear

I･-1

I-2

I-]

1-4

I-5

I-6

TypeIg-･ci`lliS'".,"i'

t+U,PFFZ-.,,.5-lim:-)

t>i.ii.idixl'

e,,',･<=U"''v

dYlas-izyv

Y?

a...SStyr)

eliieylt

G.,.i:,fie'v

ag)Mi

ev

{-Mlb

c(rt,;･,･th,i･

'e.ii･sDG'''U

ll-1

]-2

]-]

]-g

ll-5

]-6

Type11

--T-.Xl k

--:ttttte,'.i',=>xit e.'',',dittttttt

ttYft-ii

ttttttO'･'i!','ffyr

sii:.g

tGa"i

ey

±

t

K:･:'･l'iDMll

Ciii･il-bttt

IU-l

Ill-2

I]-5

Ill-4

M-5

Ill-6

TypeII[

4

--..xm

--(-xin")

Ql:,.:e++ =>.'ii,.4Xm C.;.il'.Ottttcpt'e=

;tYti

tl

tttS;IiiiCyti.

e!.･i･.,'Jttt

)rtG-m

go

Cl･l･;･:.･)Nti.-:-.-ec,,i,.IDtt

IV-1

IV-2

IV-]

IV-4

IV-5

IV-6

TypeIV--..-xrv

--<=/1'li',rpx,.

±

=>'.'ic=/.::

ttYi,

tt

ee(-y"-)

O;'iil,;iffrrc

o,'ite'eG

GoN"rv

uo

(II･il'il-.pMft-C,':.l.I.,D

Note

:D2)3)4)

_Nodalnot

inSymbolSymbolSymbol

forces

in

parentheses

are

dependent

fundamental

compQnents and those

parentheses

are

independent

fundamental

components.

'

_denotes

ba[ancing

fundamental

_components _on _edge _nodal

_points.

"+

denotes

balancing

fundernentaL

cornponents on

inner

nodal

peints.

o

denotes

_unbalancing

fundamental

_components.

(3)

center

line

and antisymmetric with respeet

to

the

transversal

center

line

of

the

shear wall

Also,

the

above

classifications are applied

to

the

fundatnental

components of nodal

forces

and nodal

displacements

of

the

unit

shear

wall of

4-continued

type

(defined

in

Section

3),

Table

1

shows

the

fundamental

components of nedal

forces

acting on

the

3-continued

shear wall,

The

nodal

forces

P

can

be

_given

as

the

sum

of

their

fundamental

components

P*

and

1>e,

where

P#

is

self-balancing

forces

and

PS

is

unbalancing

forcese)･3).

Forces

PS

and

_P"

are

expressed

by

the

sum

of

the

absolute values of

their

representative

components

P"

and

P'

as

_given

in

Eq.(1

),

where _zP"

is

the

sum of absolute values of

P"

and _.Pe

the

sum of absolute values

of

f'O

2).

_Here,

_[,

_Tl,,

_i

,7},]

is

the

matrix which

transforms

.P" and .ff)'

into

P,

and

the

eiements of

this

matrix are

_presented

in

Eq.

(

1')

which shows

Eq,

(

1 )

in

detail.

p-p"+p"-[,

T},

i

,z,]

i

i-P-

-"-

)

･･-･･･････-･･･････-･･･････-････････-･･-･･････････････････-･･････--･･･-･-･･････････-,(

1 >

t

.Po

I

Xl

E,

Es':t'

is-ig

X7-ib-Yl

Vz

ffsY,

Y5

Y6

E,

iaMilhfi2xnkofk'MXE!h-maxofb-pm

1o-4

11 -x..1

t

1al-1

-1.S .O.S

-1-S

eC)e5

-1.S

-O.S -1.s -o.s

1eX

E

-X

z

oZx-XO.sk

o.

sk

-z

O.wt

O.sA

-x

1 z

O.scL

O-SA

el

O.Sk

o.sA

-z

1-1

1-z

x-1

:gl

zx

oZ-l

1 1-1.1

z-1

.,x

g

1-1-11

-1-1

1

1 xz-s-1l-1

dl1

1-1-X1

zl

-z-1

ka-1-!

z-1

-z

z

1-1-1

z

-z-1

z

g

z-1-zz-X

zoZ

:

,

l

I

l

I

!

l

;

l

!

l

:

:1

:x

g:-z

i

gi-1

:

"

:1

:x

xs

1lzz

1x

4xg

4XIO

4V?

4Tfte4uaiam

4xk

4zfib

4Taza

4vk'4agra4c

if7Z;

4Xfut

CYfu

eyee4C

fu/h4?itSa!h

4Xfo

4Xge

4Vge6anIYfa6anMfoWttt)...-iA"-6nfu

4Vge

..(1')

Note

that

blanks

in

the

matrix

indicate

zero elements and

that

A

is

the

aspect ratio which

is

the

span-height ratio

of

the

one-bay

one-story

_part

of

3-continued

shear wall,

The

nodal

displacements

_e

are

given

as

the

sum of

their

fundamental

components

fi*

and

_a",

where

aO

are

displacements

due

to

strains and

fiO

are

displacements

due

to

rigid-body motions2)･3).

The

displacements

Da

and

Od

are

expressed

in

terms

of

their

_a"

and

_eO

as shown

in

Eq.(2),

where

[iZr,l,ron]

is

the

matrix which

tfansforms

a*

and

aO

into

a.

ff-5v+S･=[,IT.i,171.]iI-1･l---･

-･

･

-･

-

･･-

･

･-

-･･

-

･-

-････

₍₂₎

By

taking

the

inverse

of

this

matrix, or

by

applying

the

contragredience

theorem,

e"

and

aO

are

derived

as shown

in

Eq.(3>.

i

Pal'1

l

±

[t

Zn

l

_iTLso]"5=[i

T})i

i

_iTlo]`

5'''''''''"'"''''''-''''''''''''''''''''''''''''''''''''''''''''''''･･････-･････(

3)

Using

the

fundamental

nodal stiffness matrix _,K" which

defines

the

relation

between

_,P#

and

ae,

,P*

can

be

expressed

in

terms

of

a'

as shown

in

Eq.(4).

(4)

.P*=,K*a*･-･･･-･･･-･･･-･･･-･{4)

Table2

shows ,K'

in

detail

for

every

basic

type.

Using

Eqs.(1>,

₍₃₎

and

<4

),

Eq.(5)

which

_gives

the

_general

nodal stiffness matrix _,K of

the

3-con-tinued

shear wail, where

the

size

bf

this

matrix

is

equal

to

the

total

number of

degrees

of

freedorn

for

the

3-continued

shear

wall

can

be

deTived

:-

rft::

l;

::l

:;t,;

o-sK

a

}

----･--･----･(5)

3. The

Fundamental

Nodal

Stiffness

Matrix

of

Unit

Shear

Wall

ot

4-Continued

Type

A

unit shear wall

is

the

part

of

the

infinitely

continued shear wall whose

elastic

behaviour

repeats

itself

at every equal number of

bays

or stories.

If

the

unit shear wall

is

a

_4-continued

shear wall, we refer

to

it

a$ "unit shear wall of

4-continued

type".

In

this

Section

these

descriptions

are restricted

to

the

unit

shear wall of one-bay'iour-story

type.

However,

they

can also

be

applied

to

the

unit shear wal! of

four-bay

The

deformations

of

intermediate

components whic

In

this

_paper

antisymmetric components

termed

fundamental

components

Type

B

in

the

intermediate

beam

of a one-bay

components

Type

I,

Type

IV

and

fundamental

components

Type

ll

_,

Type

M

(see

Section

2>,

The

fundamental

elastic

behaviour

of

the

unit

shear

wall of

2,-continued

type

is

_given

by

analyzing

one-bay

two-story

shear walls whose stiffnesses of edge

beams

are replaced

by

the

equivalent

stiffness as shown

in

Table

3. The

unit shear wall of

4-continued

type

can

be

obtained

by

connecting

two

unit shear walls

of

2-continued

type.

Since

the

unit shear wall of

4-con-tinued

type

is

ashear wall

in

which

the

same elastie

behaviottr

is

repeated

at

intervals

of

four

shear walls

in

an

in-finitely

continued shear wall,

its

nodal

Table2The

funclamental

sheaT wallstiffnessmatrlx,K*ef one-baythree-steryframed

Type

I

Type

II

Type

III

Type

IV

xPk

= ]K*6*

4xl4xit4y!4ylt4Ml,Yh

=Et Fall3ta12aKa22 sym.

3ta133ka233Ka333kalg3ka243Ka343kaU4sua153Ka2SSka353ka"53kaS5

4xtt4xfsc4Ytr4Ytlt4Mlil4MlrY

=Et

exbn

sym. 3kbtzesb"3Kbl3stb233tcb333th14exb2"axb

y,abg"

3rbisstbzaaxbzz3rb4ssubssaxbraorb2sstb3saxb46axbs6aub66

4xfuk4Yrtr4y

fut4MFtir/h4Mlilt

tEt

4xN4xN4Y

X'4Mti,/h4MfeXh

Et

3rCll

sym.

3KC123KC223te133SC233ke33

3KC143ter+3Key+3tC-1-

3rC153ke2S3tC3S3KCgs3kC55

3kdll

6YM.

3sdi23kd223kdl33Kd23skd333itd143td2"3Kdsu3xd443rdis3(d2s3Kd3s3fd-s3kdss ul u:-vl vYhel" uf uit vf vft+hefthefium vti vtithefuhe"m'ui uft+

vxthervhekrvt

Note

:

E

and

tindicate

Young's

rnodulus

and

thickness

of

wall

respectively.

one-story

type.

beams

in

the

continued shear wall are

decomposed

into

two

fundamental

h

are

either

antisymmetric or

symmetric

with respect

_to

_the

center

of

the

beam,

as shown

in

Fig.

2.

are

termed

fundamental

components

Type

A

and

the

symmetric

on,es are

.

In

the

paper3),

the

fundamental

components

Type

A

and

Type

B

two-story

framed

shear wall are respectively classified

into

fundamental

b

S!SSiS;..4t

£

lkkllg!t

Type

AI:

EIIEI

ZIE!!]

I]'

Ti

imMiti

it

iflMItr

Type

B

Fig.2

b

-.--...

ic2Ci･zt1Z

---.--.

ti+-i---L---.

'

ttt

tlt

ttr

Two

fundamental

elastic

behavior

infinitely

comtinued

framed

shear

2U

L:

l""

of

inteTmediate

watl

2 bisT

"･e

beam

,

in

li

--1]･

IT

one-bay

(5)

-IOI-elastic

behaviour

is

different

from

that

Tabte3

Equivalent

stiifness of edge members

in

single and one-bay

of

the

4-continued

shear walls.

The

two-story

fiamed

shear walls

representative

components

P#

and

Pe

with respect

to

nodal

forces

of

the

unit

shear wall

of

4--eontinued

type

are

shown

in

Table

4. The

number

of

those

representative components are equal

to

the

total

_number _of

degrees

_of

freedom

_Note

_:

_EI,

_GAs

_and

_EA

_are

_elastic

_stiffness

_of

_intermediate

for

the

unit

shear

wall

of

4-continued

members.

type.

In

the

unit shear wall of

_4-continued

type

as

well

as

in

the

4-continued

shear wall,

there

are

30

representative components with respect

to

nodal

external

forces

(see

Table

4)

and also

30

as

for

nodal

displacements.

In

these

₃₀

representative cemponents

there

are

three

representative

components

pertaining

to

unbalancing

forces

and rigid-body

displacements

tespectively

(see

I

-8,

M-8,

IV-8

for

unbalancing

fundamental

components

in

Table

_4),

Either

_the

I-5,

I-6,

I-7

or

M-5,

M-6,

M-7

shown

in

Table4

is

independent.

This

independence

can also

be

applied

to

either

ll-5,

U-6,

ll-7

or

IV-5,

IV-6,

rv-7,

shown

also

in

Table4.

The

elements

of

the

fundamental

nodal stiffness matrix _4K4 of

the

unit shear wall of

_4-continued

type

are

shown

in

Table

s

by

using

the

elements of _,K" of unit sheaT wall of single

type

and

the

elements of _,K"

of

unit

shear

wall of

2-continued

type.

The

encircled

elements

sb;3s, Kczas

and

rcdii,A

in

Table

5

are

the

infinite

influence

coefficients

corresponding

to

the

ll

-4,

M-3

and

IV-1

(which

are

shown

in

Table4).

As

a result, we can

observe

Table4

The

represeittative compeitents

_Pt

and

P"

of the

balancing

and unbalancing

fundamental

components of

the

nodal externat

forces

on unit

framed

shear wall ef

4-eontinued

type

flexural

stiffness

shearing

stiffnessaxial.stiffness

TypeA

1-EI2

1liCAs

oo

TypeB

_oo

co

17EA

Type

I

Type

Il

Type

III

Type

IV

I-1I

n-1-eoee

BB

xftri.OQ･Ill-1

IV-1"xn.

=

I-2

ll-2eoeots-XA,,-p

o-5,Ill-2:-t

i-:,JLb

"

_th

IV-2"-"

ltrv

I-3-"yr'

e"U-3.A,.."

'v

-Yft

g'

u

M-5-m

IV-]c

2"t"rv

c

2 I-･"e

)

tvi

ll-4::,.i

,':'V"J

:i,

Mlt

.:})

IV-4c

Q

pt,rv･

Q

1-5--po

mpce-A11

/Bl,

/･//tAl.'･J;'-B',ri.:･.'A'.1,i

--ppm

aj.iL-n-s-s.

@-p

B1'

':A'::J'.･,.･

B.:1･.:.1･A';･/i.:Bl･.･

de q÷

-.Aal

Il-5`t-op

deIV-5

A

l-6

s

?･

"･

pft

U-6

d YnA [

M-6ot,

1

Yfu'

rv-6,･

t

ftt

I-7g

c

b

rfa

ll-7

Ill-71a

"

Mftk

IV-1e

)

1-8e

e

)Mo

?

Ill-8Xo

IV-8""Ye

Note

:

1)

Influence

coefficients

due

to

the

fundamental

components

II-4,

III-3

and

IV-1

are

infinite.

2)

Dotted

lines

indicate

internal

forces

of one-bay

infinitely

Centinued

fraTTied

shear walL

3)

Symbols

k

and

",

indicate

balancing

components of unit

framed

shear wall of single type･

4)

Symbol

kt

indicates

balancjng

components

of unit

framed

shear

wail

of

one-bay two-storY

tYPe･

5)

Symbol

o

indicates

fundamental

unbalancing

components.

(6)

-102-that

18 independent

nodaL repre-sentative components which satisfy

tfie

balancing

force

system

exist

in

the

unit shear wall of

4-continued

type.

In

these

₁₈

components,

the

ll

-3

ancl

M-2

of

the

internal

force

systems

(see

Table

4)

yield

elongations and

flexural

deformations

without

exter-nal

forces

at nodal

_points.

4. Discu$sion

on

the

Efastic

haviour

at

Nodal

Points

of

lnfinitely

Continued

Shear

Wall

If

the

same unit shear walls of

4-continued

type

are connected

in

succession, an

infinitely

continued shear wall can

be

obtained,

in

which an

identical

elastic

behaviour

is

repeated at every

4-continued

shear wall.

Therefore,

the

influ-ence

coefficients

due

to

unit

forced

displacements

at

intervals

of

three

nodal

_points

on

the

infinitely

con-tinued

shear wall can

be

_given

by

using

the

fundamental

nodal

stiff-ness matrix ,K* of

the

unit shear

wall

of

4-continued

type.

In

this

paper,

these

influence

coefficients

are

assumed

to

be

the

ones

resulting

from

a

unit

forced

displacement

at

a

nodal

_point

on

the

infinitely

con-tinued

shear wall.

Through

numer-ical

calculations we could clarify whether

this

assumption

is

suitable or not.

The

shape of one-bay ene-story

part

of

inlinitely

continued shear wall

(see

Fig.3)

in

the

numerical

examples

is

i

Standard

for

Structural

Calculation

common.

Taon

the

infinitely

continued

shear

nodal

_points

of

the

intervals

are

displacements.

points

far

fiom

the

nodal

_point

Consequently,

Table5The

fundamental

one-bay

four-storystiffness

typematrix`K*

ofunitframedshearwall of

Type

I

Type

II

Type

II{

Type

IV

zp-=･-Kdeet

16Xf16Yfl6Yt'16Ml/hZ"x"1't--,4Yit

4MrS,h

Et

4an,A4pn,-

O

_:

,

4s2zAO

o:

,

4tc

uA

4,c

1ec

,

---....-...

kcLgit

i

sym.

:

I

2mCllA

o

2zC

_{lz"4P22,A22r,1ZzF2Z2ti,3} u: vi v!th.e-f.al" vf*he!"

16xt[16Xff16Ytr16Mi/h

4Xff 4Y

fi

±

4MtrTh

st

Et Et 4tb-B

'O

_kbn,e

4ptllB

e

4kbns

.r--.--.-..

SYM. o

o

4P31s ,,:::,l-:l,l

2asbizK

o

22cb23,4op33Aljb2v2rtb3vlyb4it4 uft UilT vi.h.e.t.. ui' v±

Ikhetlls

4kanp

O

4slis

sym.

ol

@'

i

'

-22-gc'

fi.A

?22ge,ln'

-

-2tt'

c-

lb

"

i

4pn-

2zs2en

:

l

Z!e3y

dri

o

k!,nA4pbaA O

4sbss,A

O

4td22A

sym. ,1,,::l:::i

lab2nyA

o

bjb23A2zb3yZ2rb2kAzab3,"q"b"of

ulir VfiIheth'+i'fi' vtithe'm' u Vheh.e u vhe

Note

:

1)

The

encircled values of elements are

infinite

(see

Table

4}.

2}

The

E

and

t

indicate

Yovng's

modulus

and

thickness

of

wall

respectively.

3)

Notations

and

subscripts

for

elements :

kall,A

LE'I･1.Ega,Si8i,Cib,e:,a,,Vf'}O.r.,O,f.e.d,g,ei.be.afM.;.it

f,.rned

shear waii of

N.tS.i"tgiZe.

Of8,25C.O".tiS".".e.dt t.YfPrYpe

i

(Notations

for

eiements

of

TYpe

II,

III

and

IVI

are

b,

c

and

d

respectively};

Elements

in

_IKk

of unit

_framed

shear wall of single

type

{2k

indicates

the

elements

in

_2Kft

of

unit

framed

shear wall of

2-continued

type).

design

example

in

the

AIJ

Reinforced

Concrete

Structures

(revised

in

1982),

where

the

shape

is

orced

displacements

at

intervals

of

four

nodal

points

From

this

table,

it

is

observed

that

the

influence

coefficients

at

the

central

small cornpared with

those

at nodal

_points

subjected

to

unit

_.forced

,

since

the

infilled

wall

panel

which

transfers

stresses

to

adjacent unit shear

the

values of

the

influence

coefficients at nodal

,

to

a

unit

forced

dSsplacement

tends

to

converge

rapidly

to

zero.

fluence

coefficients

due

to

aunit

forced

displaceinent

at anodal

_point

on

the

dentical

to

that

of

the

one-bay

one-story

shear

wall adopted as

the

of

ble

6 _presents

the

major

influence

coefficients

due

to

unit

f

wall. very

The

reason

for

this

is

that

walls can absorb strain energy

together

with

the

boundary

frames

subjected

(7)

rh

L

2,

'4 :::J.LI;;..11.:.:t.rJ;tr'J.J

tttttttt

_tt

_t

_tt

_-.

ttt-tttttt:t

It..'.s.-.-,.:,,..

_ttt

_t

tttt

_{ttttt.t:-.ttttt}

_tt

t-t

'

t-ttt-..ttttt.'

1.

Jg

t-t.;..-..::....-'.:.'.t.'tL,,.

'･'.L-'･:..･t::,De

t

N

zzbT

t

F-

zratios

of.one-bay

bb

-Ng.3

Configuration

and aspect

Xbe

ene-story

A

aS=1.714

Ph

Qg)

a

h

GO.171

%s

llSt

u

2.s

op

(xt?

s

-ffm

0.

171 be

Se

87"

3.333 part

of

infinitely

contintied

framed

shear watl

Table6

Principal

influence

coefficients

due

to

unit

forced

displacernents

at

intervals

of

four

nodal

points

on

infinitely

continued

framed

shear walls

One-bay

infinitely

continued

framed

shear wall

One-story

infinitely

¢

ontinued

framed

shear

wall

I-

il

Y-

'v

c)ttti

Ty

.---...

m.-e

r･,,,,i1LLl1e

"ii

_'

t

9-';ttl,

'1

,''t''Jll

jUh

-

h-e

il

--r-v

1 tw

11 infinitely

continued

shear

wall

by

using

the

fundarnental

nodal

stiffness

matrix _,KS of

the

unit shear wall of

4-･continued

type.

5. The

Fundamental

Nodal

Stiffness

Matrix

of

lnginitely

Continued

Shear

YVall

There

are

two

fundamental

types

of

elastic

behaviour

in

the

infinitely

continued shear wall.

One

is

due

to

nodal

external

force

system which

_yields

restraint reaction at each nodal

point

due

to

a

pair

of

symmetric

or

antisymmetric

nodal

forced

displacements.

The

other

is

due

to

internal

force

system which

does

not

_yield

restraint reactions even

if

symrnetric or antisymmetric

identical

forced

deforrnations

are applied

to

each unit shear wall

{see

Section

3

and

Table

4).

In

order

to

formulate

the

nodal stiffness matrix _.K of

the

arbitrary continued shear walls'}, we

define

in

this

paper

the

influence

coefficients

due

to

a

_pair

of symmetric or antisymmetric nodal

forced

displacements

on

the

infinitely

continued shear wall.

In

Fig.4,

we

assume

the

influence

coefficients

to

be

zero

at

all

nodal

_points

farther

than

the

third

_poiflt

from

the

nodal

_points

(1

and

2)

subjected

to

forced

displacements.

(8)

-104-As

a result, we

can

define

the

influence

coefficients

due

to

a

_pair

of symmetric or antisymmetric unit nodal

forced

displacements

as

the

elements of

the

fundamental

nodal stiffness matrix

.K'

of

the

infinitely

continued shear wall, as

shown

in

Fig.

4. Each

element of _.K'

(shown

in

Fig.

4)

is

_given

by

considering

the

nodal external and

internal

force

systems,

presented

respectively

in

Tables

7

and

8,

and

the

equilibrium conditions

indicated

in

Table

9. Howeverthe

influence

coefficients

as2

and

ass

due

to

apair of symmetric

forced

displacements

cannot

be

obtained.

Hence

they

are

approximated

by

the

elements of

the

_2-continued

shear wall.

Since

the

influence

coefficients

at

nodal

_points

7

and

8 (see

Fig.4)

are estimated

to

be

small,

they

are assumed

to

be

zero except

for

the

ones

determined

by

considering

the

internal

force

systems,

The

nurnerical results

illustrated

in

Tables

₁₀

and

_ll

pertain

to

the

fundamental

the

infinitely

continued shear wall whose one-bay one-story

_part

Fig.4

Symmetric

forced

displacement

systems

e

<----･+Xl-tXl"txs-tX7

-tYl'tY3-+Ys-tr7

'tMil}z-kM]lh-+Mslh-t

iM71h

21436587

=Et

==Et

.Et

't

"->

al:a21a31

@

"l'

2468

-(i!)

o

lt

57,!,

o

g

-.fV

21436s87

a2 _2-tl':-;,a2 3@

@ @

la'I-'//t

@

b12

b13

''H

tF.N -1

bn='b-2:

b23

r

t

bn

ba3

:1fi"2,'

(Ei)

Note

1?

-(iEI)

O

bi3

cn

--an@

b2e-<[il]})

C230

@

b33

lo33

0 @

0 :

Elements

O:

Elements

t'1:

Elements

O:

Elements

@:

Elements

shear

walls

except

for

the

one-bay

infinitely

+--a-tUl'tVl

-+hel

-tul-+Vl

-'thel

HtUl-+Vl

+he

Antisymmetric

forced

di$placement

system

'

.

"kXl-tX3-tXs"hI7

-tYl-krs-krs-hY7

'tMllh-ikM3/h-+MsPz-kM71h

=et =et

=Et

.

l..>

alla2!-<{!l) a31

@

o-a22@

@g

o

T

A

(2

4

6 87t--o

aZ2@ a2 30

@

bu

b2

2='b[

[412):

bs2

･lbl'ttl

13s7

O

b13

cls

"a2!-(I!)

_b23-([ii)

c2s

@

be3

css

@

@)

@

-({III})

encircled

by

:

assumed

to

be

zero

obtained

by

internal

force

systems

obtained

by

equilibrium

equations

replaced

by

those

of

one-bay

two-story

ones

mentioned

_above

are

_given

as

displacements

at

interyals

of

four

continued

framed

shear

wall.

)

--;`-+Ul-kVlhek1

'tUl"tVl

-thel

-.kUl--Vl

kh6

framed

Elements

the

influence

coefficients

nodal

pomts

on

Influence

coefficients of ene-bay

infinite]y

continued

framed

shear wall subjected

to

a

_pair

of symmetric or antisymrnetric

forced

nodaL

displacements

(9)

-105-Table7Influence

coefficients

due

to

unit

force

framed

shear watl

d

displacements

at

intervals

of

four

nodal

_points

on one-bay

infinitely

continued

x

y

Ftrh

Symmetricforceddisplacementsystems

il

e

÷

all-> a21->2a31b

-.

D

a22a32-a32tsO

"

h-e

e->

a2aa33-Cl31

mOgt--'

,

r Ol 1

-a22

1 1a32va321 uO ₁ 1 1 L

l

bn

,

b22

₁

2b32

I

1

---1

tt1t11L

LL"1

bl]

t

b23l"

2b,3

_,1

1 r

r

--n-l

g

e

o2

-a23-a31vassS

-e

gS{IKsg

n

b,,2

b,,D2b,,2)

(gssg

?

ell} e23ilC33))

AntisymrnetricforceddispLacementsystems

u

]all

1a21

t

1"2ael

L

1

1 F

J't,F-)'tlL-,t1

il---'1

,J

O il

->

_a22ii

lq

qlL32g9e!2[

l

ll

he6+1a23

:3-a3][=o

1

l

o

-a22a32ra32

eO

1111J1l11

'

i

t[

t

b,,ik

-

b221Ie

"

2b32

1I

e

-

lj"

"

IIe

t)

b13

b232b13

11I',,11d

e

g

'

ot

I

R-a23

i

-a33g33i

S

-o:

l-eeeee

12

b,,lIe

2 b,,i!e

)

_thS3)

iie

][e

2

tr(1

----TL.

2

c13? C23? 2ee3?) ,1,''1'i

Table8

Note

Relations

:

1)

Equitibrium

condjtions must

be

satisfied ameng

influence

enelosed

_by

breken

lines

_(see

_Table

_9).

2)

Symbol

o

indicates

the nodal

_points

subjected te a unit

between

internal

forces

and

forced

nodal

displacements

of one-bay

in

coefficients

forced

dispLacement.

finitely

continued

fTamed

Forced

nodal

displacements

lnternal

force$

-=o'sodiok"=.-tuL-mvat-x

Y

shearwall

Symmetric

_forced

dLsplacernent

systems

(

Vi{o) "Vice)

o-2n+1

(t>o)

G-2n-1

(iCO)

::i,li,

D."li:.l.1

.0

_o)

ia2n-x-x-x-rt"2n

H

iz

2n-1

.e-

÷

t-3"-2-1)

(21)

. ÷

43H

iv

2n+1

n "-2n#-1n-1n"2

"y

9-x

Antisymmetric

disp]acement

forcedsystems

Ui(O) "Ui(e) e

n

Vt{e) "

2n-1,

Vi(o)

o

2n+1,

v

.Z!N(

x

2n(n+1)

x

Vi

_(e)

"-2n+1

i{e)"-C2n+1)

hei(o)

=

het(.)

=

(-4n+2)IA

hei(o)#het(.)n-(4n+2)IX

{i>o)(i<o)

(i>o)

ci<o)

(i>o)

(i<o)

M

rtsiM"2

-12143VM

nv-2n ts-1n-ln=2

Note

:Subscriptso and edenote odd and evennodal

_{pointnumbersrespectively.}

(10)

-106-Table9Eqttilibrium

equations satisfied among

the

influence

coefficients

due

to

unit

forced

displacements

atintervals offeur nodal

peints

on one-bay

infinitely

continued

framed

shear wall

Syrnmetric

forced

displacement

systemsr

$12+b22+b32=O

}13+b23+b3s=O

Antisymrnetric

forced

x

fllta21+a31'O

displacement

systemsM

+llLt3+b23+b33-}(?t;2+b

-l}lt13-c23+c33T1-2(l}!tl3+b

'

22+b32)-(a22+2a32)mO

23+b33)-Ca23+ha33)=O

Note

:

)L

=

Zfl]

iSthespan-heightratio

of

unitframedshearwall

ofsingletype.

TablelOInfluence

eoefficients

due

to

a

_pair

of symrnetric or antisymmetric unit

ferced

continued

framed

shear wall

(Unit

:

Et)

nodal

displacernents

on one-bay

infinitely

vdiv-oL ditv=oEootu-n.Ev

ep-=v,-vL-g-oooatv=o2t-I

y

dyh

Symmetricforceddisp]acement

systems

il

""-b

1-4

O.1 -

o.os-"

-b

.

O.1p

1-O.05"--i

he

g)

b

-

O.05-Antisyrnmetricferceddisplacementsystems

E

.･-t

1LL.--1

-

O.1--s-

O.05-V

t

--A-

O.1-

O.05-he

e)i

Lb-oM

O.05-O.05LT4

(11)

-107-Tablell

th-=eEovasnutv

x

Influence

coefficients

due

to

a

_pair

of symmetric or antisymmetric unit

ferced

nodal

displacements

on one-story

infinitely

centinued

ffamed

shear wall

(Unit

:

Et)

-eokoL

n=diU--oevov=o==E

?

ith

Symmetric

forced

displacement

systerms

E

b

tuim---T-74=

1-n

d

e1i

O.1H

o.osH

1 I

n=fftr[[i

i

Neq

i

e

n

d

OH.05

he

b..{imii

p

Aw

j

l

i

m

9:･gos

f

o,esH

Antisymmetrtc

forced

displacernent

systems

ilbe

-Pi

e wh,

O.05H

I

v

t

mg

%

el

n

fL

e

i

r

to.osF--,

he

h

tu

tui

e

Ou.05

?

i

pt-g,･gos

i

O.05-is

identical

with

the

shear wall of standard

type

shown

in

Fig.3.

6. Conclusions

The

fundarnental

the

3-continued

shear wall

is

formulated.

The

influence

¢

oefficients

of

the

infinitely

continued shear wall subjected

to

a

_pair

of symmetric or antisymmetric unit

for

¢ed nodal

displacements

are clarified

by

using

the

fundamental

the

unit shear wall of

4-continued

type.

References

1}

"Seismic

_Loads

and

Earthquake

Resisting

Capacltles

ofBuiLdings

(1976)",

Aichitectural

lnstitute

ofJapan

(AIJ>,

Jan,

1977,

pp.5-8

(in

Japanese).

2)

Tomii,

M.

,

Yamakawa,

T. :

Relations

between

the

Nodal

Forces

and

the

Nodal

Displacement

on the

Boundary

Frames

of

Rectangular

Elastic

Framed

Shear

Walls,

Trans,

ofAIJ,

No,237,

Noy.

1975,

_pp,45-57,

No.Z38,

Dec,

1975,

pp,37-46,

Ne,239,

Jan.

1976,

pp.35-42,

Ne.24e,

Feb.

1976,

pp.63-70,

Ne.241,

Mar.

I976,

pp.79-89.

3)

Tomii,

M. ,

Yamakawa,

T,

:

Stiffness

Matrix

of the

Two･Bay

or

Two-Story

Duplex

Framed

Shear

Walls,

Trans.

of

AIJ,

No.284,

Oct.

1979,

pp,41-50,

4)

Tomii.

M.

,

Hiraishi.

H. :

Elastic

Analysis

of

Framed

Shear

Walls

by

Considering

Shearing

Deformation

of

the

Beams

and

CetumnsofTheirBoundary

Frames,

Part

I,

ll,

M,

Trans.

ofAIJ,

No.Z73,

Nov.

1978,

pp.25-31,

No.274,

Dec.

1978,

pp.75-83,

No.275,

Jan.

1979,

pp.45-53.

s)

Tomii,

M.

,

Sato,

N.

,

Ineue,

M. :

Elastic

Analysi$

of

Twe-Story

or

Two-Bay

Duplex

Framed

Shear

Walls

Subjected

to

Antisymrnetrical

Loads

with respect

to

the

Axes

of

the

Intermediate

Mernbers

of

Their

Frames,

Trans.

ef

AIJ,

No.

297,

Nov.

1980,

pp.35-48.

6}

Tomii,

M.

,

Inoue,

M. ,

Kttriyama,

K. :

ELasticAnalysis

of

Two-Story

er

Two-Bay

Duplex

FramedShear

Walls

Subjected

to

Symmetrica}

Loads

with respect tothe

Axesef

the

Intermediate

Member$

of

TheirFrames,

Trans.

of

AIJ.

No.

299,

Jan.

1981,

pp.69-82.

7)

Tomii,

M.

,

Yamakawa,

T. :

Nodal

Stiffness

Matrix

for

Arbitrary

Centinued

Framecl

Shear

Wall

(continued),

Proceedings

of

Syrnposium

on

Cbmputatinal

Methods

in

StTucturaL

Engineering

and

Fields.

Volume10,

July.

1986,

pp.

140-145

(in

Japanese}.

(12)

【

論

　

文】 UDG ；

69 ．

022 ；

699 ，

B41

日本建築学会構造系論文報告集第

374

号

・

昭和 6Z 年 4 月

3 連

およ

び

一

方向均

等

無

限

連続耐

震

壁

の

各

基

本節

点剛性

マ

ト

リ

ッ

ク

ス

（

梗

概

）

正会員正会員

富

山

苅

政

哲

英

＊

雄

＊＊

　

1．

序

　

鉄筋

コンクリ

ー

トラ

ー

メン

構造

の

耐震性能を向上さ

せるためには

，

耐震

壁を適切に配置することが

極

めて

効

果

的

で

あ

る。

建物

の

耐震設

計

にあたっては，その

建物

の

耐

用年限中

に

数回

くるような

地震を

受

けて

も補修を要す

る

よう

な

損傷

が生じないように

_，

_中

地震時の挙

動

をほぼ

弾

性範囲

に

収

めることを

，

日本建

築

学会

は

推奨

している］｝

_。

一

_方

_，

_耐

_{震壁}

_の

_水

_平

_{剛性}

_が

_柱

のそれに

比較

して

き

わ

め

て

大

きいので

，

地震

荷重を受

ける

有壁

ラ

ー

メンの

弾性挙動

は

，

耐震壁

の

_弾性挙

動

に

顕著

に

支配

される

傾向

が

あ

る。したがって

，

マト

リ

ックス

構造解析

の

手法

を

使

って

，

損

傷

を

受

けていない

有壁

ラ

ー

メンの

弾性挙動

を正しく

評価

するためには_，

任意

の

多運耐震壁

の

_{節点剛性}

マトリックス

_を

正確

に

評価す

る

必要

がある。

　

本論

では

，

任意

の

多連耐震

壁における

1

スパン

1 層部

分の

形状が同

じで，しか

も壁板

の

中心を通

る

縦

横軸

に

関

し

対称

の

も

のに

限

って

論を進め

る。

　

著者

らは

，

_付帯

ラ

ー

メンの

外

節

点

における

水平力

X

，

鉛直力

y

，

モ

ー

メント

M

／

h

（

モ

ー

メント

M

を階高

h

で

除

し

，

次元

を

X ，Y

にそろえて

あ

る

）

と

，

水平変位

−

，

鉛直

変位 1

，

回

転角

h

θ

（

階高

h

を回転角

θ に

乗

じ

，

次

元

を瓦ーにそろえてある

_）

の

関係

が

完全

に

表示

された

節

点剛性

マトリックスの

精密解を

，

富井

などが

明

らかにした

単独

および

2 連耐震壁

の

弾性解析解

ト6

_獄

_{以下既知}

の

_解

析解

と

呼

ぶ

｝

を

使

って

求

める方法を

，

単独

・

2 連

の

各耐震壁

については

，

日本建築学会

論文報

告集

に

発表

したZ｝

・

3）

。

別論

で

発

表

する

任意

の

多

連

耐

震壁

の

外節点

に

関

する

節点剛性

マトリックスを

，

すでに発

表

した

単独

およ

び

2 連耐震壁

の

基本節点剛性

マトリックス

を用

いて

精度

良く求め

るために

，

本論

では

，3

連耐震壁

の

基本節点剛

性

マト

リ

ックスを

定式化

するとともに

，

既

知

の

解析解を

使

って

一

方向均等無限連続耐震壁

の

基本節

点剛性

マト

リ

ックスを

，

精度良

く

求

める

方法

を

提案

する

。

　

この

節点剛性

マトリックスを用いれば

，

地震時

におけ　本研究を日本建築学会九州支部研究報告昭和

58

年

3

月

，

および昭和

60

年

3

月に発表した

。

　 ’_九_{州大学　教授}

・

_{工博}

　

牌 _九

州大学　

助

手

・

工修　　（昭和

61

年

10

月

28

日原稿受理

1

る

有壁

ラ

ー

メンの

正確

な

弾性挙動

を

究明

することが

可能

と

なる。この

正確

な

弾性挙動

を

，

ブ

レ

ー

ス

置

換

法

や

柱

置

換法を用

いて

解析

し

た

有壁

ラ

ー

メンの

弾性挙

動

と

比

較

するこ

と

により，

適切

でしかも

実

用

的

な

耐

震壁

の

弾性

剛

性

評価法

の

確立が期待

できる

。

　

2 ．3

連耐震壁

の

基本節点

剛性マトリックス

　多連耐震壁

の

節点剛性

マトリックス

K

を

定式

化するた

め

に

，

3 連荷重項

モ

デ

ルの

節点剛性

マ

トリ

ックス 3KL を

用

いる7》。そのためには

，

図

一

1

に

示

す

3 連耐震壁

の

基本節点剛性

マトリックス _3K ＊

を明

らかにする

必要

がある

。3

連

耐

震壁

の

節点外

力

と

節

点変

位

に

関す

る

基本成

分は，

単独

や

2 連耐震壁

のそれらと

同

様

に

，

4 個

の

基

本

型に

次

のように分けられる_。

　

基

本

成分

1 型

：

耐震壁

の

中心を通

る

縦横軸

に

関

しとも

　　　　

に

逆対称

な

基本成分

ll

型

：

_耐

震壁

の

中心

を

通

る

縦横軸

に

関

しとも

　　　　

に

対称

な

基本成分

　

基

本線

分型

：

耐震壁

の

中心を通

る

縦軸

に

関

し

逆対称

　　　　　　　　

で，

耐震壁

の

中心

を

通

る

横軸

に

関

し

対

　　　称

な

基本成分

　

基本成分

W

型

：

耐震壁

の

中

心を

通

る

縦

軸

に

関

し

対称

　　　　　　　　

で

，

耐震壁

の

中

心を通る

横軸

に関し

逆

　　　　　　　　

対称

な

基

本成

分

　

上記

の分

類

は

，4

連

単位

耐震壁

の

_節

点外力

と

節

点変位

の基

本成

分にも

適

用される

〔

3 項参

照

）

。

　

3 連耐震壁

の

節点外力

に

関

する

基本成分を表

一

1

に

示

す

。

節点外力

P

はつり

合

い

力

で

あ

る

基本成分

P

＊ _と

_，

不つり

合

い

力

である

P

°

の

和

で

与

えられる2）

・

3）

。

P

＊と

P

°

はそれぞれ

代表成分

P

＊ _と

_P9

_の

絶対値

の

和

で

表現

され

，

（

1 ）

式

で

与

えられる2｝

_。

EP ＊は

Pl

の

絶対値

の

和

であり

，

五

P °

は

P °

の

絶対値

の

和

である

。

［

iTPi 　

i

，

T

．

］

は。

P

＊とtpe を

P

に

変換

するための

変換

マトリックスであり，その

要素

は

（

1 ｝式を詳細

に

示

した

（

1 ’

）式

で

与

えられる

。

節点変位

δは

，

ひ

ず

みによる

基本成分

δ＊と

剛体変位

による δ

゜

の

和

で

与

えられ

，

δ＊ _と_δ

゜

_{はそれ} ぞれ

代表成分

δ噌と δ

゜

の

線形結合

で

_（2

）式

のように

表

される2）

・

3］。

［

，　

T

．

ii

Te

。

］

は δ 寧と δ

゜

を

δに

変換

する

変

換マトリックスである

。

このマトリックスの

逆

マト