連続耐震壁の節点剛性マトリックス(梗概)

(1)

Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan

Eiigc,6g,.]o,,,,,,.,,,,,,,,,

B:".a,i.,Otfi.S.tg".C"A'f})""Nd..C,o,",rlTM'c,`i,ah",E,n,zkneering

_{sm,if,Nzirarzx,ifAffwr.ff}

'

NODAL

STIFFNESS

MATRIX

OF

CONTINUED

FRAMED

SHEAR

WALL

'

by

MASAHIDE

TOMII"

and

TETSUO

YAMAKAWA"

'

Members

of

A.

I.

J.

1. Introduction

The

authors

have

already

clarified

the

nodal

stiffness

matrices _,fi of single-bay single-story

framed

shear walli)

(hereafter

referred

to

as "single shear wall") and ,k of single-bay

two-story

or

two-bay

single-story shear wal12)

{hereafter

referred

to

as

"2Jcontinued shear

wall"),

using analytical

solutions3)-5)

expressed

in

terms

of

Fourier

series

(hereafter

referred

tQ

as "analytical solutions"),

These

_nodal

_stiffness _matrices _express

_the

_relationship

between

nodal external

fofces

P{I)i,

7,

Mlh)'

_ana _nodal

displacement$

5(a,

_T,

_hb).

_The

_horizontal

and vertical cross _sections _of

_the

_{said shear} _{walls are}

_symmetric

_with _respect

_to

_their

_centers.

In

this

_paper,

the

authors

_propose

a method

to

estimate with

high

accuracy

the

nodal

stiffness

matrix

k

of

arbitrary-bay

arbitrary-story

framed

shear walls

(hereafter

referred

_to

as "continued shear wall") using

the

fundamental

nodal

stiffness

matrix

.K*

'of

single-bay

･or

single-story

infinitely

continued shear wallSL7)

(hereafter

referred

to

_as

"infinitely

_centinued

_shear _wall") _and

_the

_nodal _stiffness

_matrices

,I and ,R.

The

continued

shear wall

is

composed

_of

the

_{sarne units of a single}

_shear

_{wall whose} _aspects _Df

_horizontal

_{and vertical}

_{cross-sections}

_are symmetrlc,

'

2. The

nodal

stiifness

matrix

of

continued

shear

watl

composed

ot

the

units

ot

2-continued

shea[

walls

The

nodal stiffness matrix

k

of continued

shear

wall cannot

be

correctly

obtained, even

if

_,ff

of

sillgle shear wall

is

superposed

by

direct

stiffness method,

This

can

be

_proven

_{quantitatively}

using

2-continued

shear wall whose _,k

is

given

by

the

analytical

solutions2)･4)･5).

'

The

reasons

for

the

inability

_to

_obtain

k

f

t,

"hb

t

.

fttbb

ij:f.;M,i:ag`22i:'[Z,:8".ILtkO,:i,,:,:,O.V.e,;'

es,-":be

,

bb

Fig.1

Distributed

loads

to obtain ,k of

2-continued

shear wall

by

at nodal

_points.

assembling

the

,K' of single shear wall whose cross sectional aspect

In

order

to

eliminate

the'se

tWo

_prob-

_.

IsunsyrnmetTic

lems,

the

distributed

loads

whose

abso-

'

,

lute

values are

same

but

the

signs

are

different,

must

be

applied

_fespectively

.

_-

iK'

-aiong

the

_{axis of each}

_overlapping

_member whose width

is

haly6d

as

indi6ated

in

Fig.1,

The

reaL

load

_term

due

_to

these

distributed

loads

can

be

obtained

by

2-tracting

_£

_,Kl

from

the

nodai stiffness

t=1

-2K

2K

2;continued

shear wall

Fig.2

Method

for

2-continued

lk,.

1ft'

1ftt

Assemblage

of

2-continued

load

single

shear

walls

term

model

calculating the stiffness matrix _,Int of

load

term

model i

_D.

_Eng.,

_Professor,

_Kyushu

_Uniyersity.

#

_M.

_Eng.,

_Rese4rch

_Asseciate,

_Kyushu

_Uniyersity,

(Manuscript

[eceived

July

31,

1987)

'

(2)

Architectural Institute of Japan

NII-Electronic Library Service

t-

r

matrix _,R of

2-continue,d

shear wall

as

shown

m

Fig,

2,

where

ll.l,

iKl

is

assembled using

the

nodal stiffness matnx ,K'

of single

shear

wall whose aspect

of

cross section

is

unsymmetric.

Consequently,

the

real

load

term

defined

by

nodal stiffness matrix ,k,,

is

expressed

by

Eq,

(

1 ).

,k,,=,k-i,rl･････････-･･t･････････--･･･････････････････-･--･--･････.･",,H...･.H...,,,.,,,.,,...,..,.,...,.,,,o)

tdi1

.

Here,

,ft1 which

is

made complicated

by

the

unsymmetric cross section of

the

single

shear wall must

be

derived

to

obtain

tKnL.

If

the

shear walls are

con-tinued

in

horizontally

and vertically,

the

calculation of _,k

becomes

more cornpli-

'f-

-

+

=

cated

because

the

overlapping members

sm2iSeSesMh2,LargewaOifis

D.p)il/IIEI.at2.K-.D.TFI,E(ectio,Rte.?L,iOad;ftrnMv

2-centinued

whose width

must

be

halved

are

doubled

shear wall

Hence,

the

nodal stiffness matrix _,kL of

2it

ik

_2-continued

_lQad

_term

mode! zRL

2-continued

load

term

model

serves

as a

correction

term

_for

A.,,fi,.

'

NO'e:giwWldd,t:

gl

:.O".:g.a,ryY

_bb,ea,.M,S

ll,S

rhe.ail

:?e,',,i

_..,.

2IL=2k-

S.,

ikt=

2.ll,

(iff{'ikt)

Fig'3

PoUaPdlitCea,t;i

C;l:deeClt]

;}lerm

,Kpand real

load

term zKAL

in

z-continued

-t-

+(2K-ZtKl)=xKn+2KRL

t.i

e

"-･---･-･-"('2)

Here,

2kL contains 2KD and 2KRL, whe're tkD

is

the

correction

term

for

the

duplica-tion

of stiffness

due

to

overlapping

of

members and ,IftL

is

the

real

load

terrr}

given

by

Eq.{1).

The

relationship

among

these

load

terms

is

illustrated

in

Fig.3.

To

show

clearly

this

relationship,

a

schematic

diagram

is

given

as

illust.rated

in

Fig.

4. In

this

example,

the

propottions

of sihgle-bay

single-story

part

of

2-conti-nued

shear walls

are

'identical

to

those

of

the

single shear wall used

in

the

design

:-

-:

t:

--.:.1:.:

+

EEEE

EccE

EEEE

Fig.4

+

.t-1

_-..l

,

-le----i',---l---..

---..-i---}--r"-"

+

term2Kp

erm2k.

-

o

@

'

,kD and ,IRL

in

zR. of

2-continued

shear wall subjected

to

a

pair

of

symmetric

horizontal

forced

displacernent$

at

the

nodal

peints'

of the

intermediate

bearn

:.--+.---:--+.---:-:`.'tt---..'---

e'-1 = =

+

]+,+,--t4---,,,'.-.t...--l.i----,--.---t=

-

Assemblage

of

Assemblage

of

2-continued

Continued

shear

wall

single

shear

walls

load

term

models

'

Fig.5

Examples

of

the

nodal stiffness matrix of continued shear wall

in

whlch no single shear walls _{are connected}

in

succession of more than twe

in

one

di[ection

･

--

₈₀

(3)

Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute ofJapan

,

example

shown

in

the

AIJ

Standard

for

Structural

Calculatiop

of

Reinforced

Concrete

Structures

(revised

in

1982).

Also,

the

eleinents of

the

fundamental

nodal stiffness matix ,KT

of

2-continued

load

term

model whose

preportions

are

equal

to

those

of

the

above-mentioned

2-continued

_shear

_wall _are

indicated

in

Table1.

Considering

these,

_,K,

defined

by

Eq.

(

2 _>

is

used.

Therefore,

_the

above-mentioned

_problems

in

i

)

and

_ji

)

will

Table1

Numerical

examples concerning

fundamental

nodal stiffness matrix _,Kt of

2-continued

load

term

modet

-ovoEE'ts-vdio-xLo-m

1o)"xcaD 1o-oo=.-m -vvoEEts-v･No-tsbo"co so-tn=.asxcaD

to)-Type

I

Type

_'II

Type

III

Type

IV

Type

I

Type

Ii

Type

III

Type

IV

Neglectingmonolithic

floor

siab

4X!4vr2rlh

2e.A

4xli2Xrrle

4Yfi4agA

Et

O.IS6

sum.

o.11･g

o.loo

o.o4e

o.oss

e.l34

o.osl

-O.OSI

-O.025

-O.O16

O.321 -O.3eS

O.461 -O.821

-O.449

O.694 .eym,

O.022-O.022

O.032 O.OOI

ufv{vYhelt ufi uit vtiheft

2Xti.

4rti4Mfi,th.EtO.067

BYM.

O.O08

-O.O05

o,lss

o.eo7

-o.ooo

utmtvtihelir

4Xfi2Yrvt4)(ril"2Mirlh

-Et

-O,Oll

O.027

-O.OOO

--O.061

O.OOO

O.OOD

sym.

O.O16-O.033

o.ooo-･O,022

uNvMhetrvheirv" ' -/h =Et

O.Oll

o,o3s

o.oe7

-O.138

O.O07

O.O02

sym.

-O.Oll

O.064LO.O02-O.030

uful''vf1'

4Xtr4Yfi2Y

i'

4"{li!h

Et

O.12S

sym.

O.O16

-O.135

-O.O05

O.O02

-O.OIS

-O.OOO

-2.613

O.O06

o.ooo

uftvivffhe'.

lhlh

=E

-O.026

sym.

O.O03

-O.OOI

O.021 O.OOO

O.OOO

-O.O02,

O.OOO

O.OOI

-O.024

Utllt: vtihelirheff

,xNrr

rvtqMttYlh-Et-O.O04

O.OOS

-O.OOI

sym.

-o.ooe

o.ooo

e.ooo

uft vNk

he".

C6rrection

term

monolithic

floorforslabconsidering

4xr4Vf2ylt

2M!klh

.tEt

-O.178

-O.129

-O.084

eym.

-O.122

-O.e4

-O.096

-O.03

-O.079

-D.03

-O.O0

u!vfv{telt

2Xti-4Vti4Mli,za-EtO.093

--O.O16

-O.O05

sym.

b.oo6-o.oe3

O.O09

utit vtihefu

･h

4Et -lh

O.081-O.106

O.230

sym.

O,O12

-O.173

O.Dll

O.160 O.O04

-O.O04

-O.OOI

ut ui vlhel

!h.EtO.380

sym.

O.267-O.188-O.039

O.027.-O.O04

u+rv v"+het.

(4)

-81-Architectural Institute of Japan

NII-Electronic Library Service

be

eliminated simvltangously.

This

will make

the

lo4d,telm,paodel.e,asy

te

deal

with.

As

a consequence

the

ngdal. stiffness

matrix

K

of continued

_shear

walls,

in

which no. more

than

two

single shear walls are connected

in

succes-sien

in

one

direction,

can

be

obtained easily and with

high

'accuracy.

Th.e.se

single

s.hear

walls

have

symmetrical

horizoptql

and

yertical

cro.ss sections

and

are

identical

in

size

as,

illustratea

in

Fig,

5. If

the

effect

of

floor

slabs on

the

n6dal

sitiffness

matrix of shear

wall

is

to

be

considiered,

it

is

necessary

to,

add

the

correction,termB)

to

2IL of

2-continued

load

･

term

model without

floor

slab

(see

Table

1).

3. The

nodal

stiffness

matrix

ot

continued

3hear

wall

in

whiqh

more

than

two

single

shear

walls

are

connected

in

succession

in

one

direction

In

order

to

obtain

the

nodal stiffness matrix'

k

of

the

contiilued

shear

wall

in

which

more

than

two

single

shear

wgllfi are connected

in

t

/

'

_t:t

-.:

''

il

:-

'

Jr

U:

,

hE:

pa

'

e

'

g

e

V

eo

･i

.Tab!e2,Loadtermmodels

ofcontinuedshearwall sing]eAssemblagesheafofwallsLoadtermrnedelsSlnglstory''e.-bay'shearmulti-waU

'

_Single-bay

two-storySingte-baythree-storySingie-bay

four-stery

Single-bay

twq-story

frarnedshearwall

'

12+'tt.,2' 'f'UT'L-Ht.'

Singte-bay1

''

::ttt/t'

three-story1+1.t/

tt''1]･:ttt

'''m-framedshearwall1

s.:.,,,ttttt/t' '

''

'

Single-bay1

tt'tt

':tttlttt'tt.tt'ttttt

four:st'ory'1

2,'ttt/t'tt'3/''ttt./tttt-4

framed･

shearwail3+'t/ttttt,1,ttttt/t

'tttttt''4-ttttt'-4 /a//

,lt/tttttt'

't/tt'/tttttttt

'

'hE;

,S?

,1

,.

Note

:

1)

Numbeis

in

circles

inditiate

'orders

to assemblb

ithe

1'.

''

'

_single shear waLls and

the

load

term

inedels so as

to

obtain

the

K

ef continued

'

shear watl,

'

2)

Single-bay

four-story

loae

term

_m'i

del

is

disregarged

asahigh erder

load

te;m model,

''

,

'

tir-'

ig

ir

i:

der,

hg:

'

gg

g

e

E]

)

eE)

B7

,.

System

with a peir of syrnmetrie foreed

dispiscements

---Xl,l;lJ･

X7･･'//t"`

i:yg.gr`.-MjlthM'gihM'sth.'rkNote

-et

System

with

ttt

s

_pair

of antisyrnhnetriC

The

elements

equations.

stiffhess matrix

k21ek211k212

of

-k211

-k!11I

510

d

,

-k2ie

no,ksn

i

-(ili5)

klg

1･ks12

sKr encircled

by

:

assumed zero sipcg

the

nkt

k61ok6n

@

k'6l1k61o

(EES)

k511

g

::::d

kglok911･･k912

kllo

21o

-kill

Iilll)).

::i:

lk6n

ksn

klol

klOlkgn

forced

displacernents

'

@k

2･11

'

'k21

1kvo:ksllkeukGloksio'i

kl12k212

gks12

61kEgi

ksg

':t.ul't･us'tU5-*U7'tVl-t,V3

t'V'5'tV7r':iheihe3.

't'hes1

-t･he7.

:

.

0:The

elements assumed zere order

to

rnake the number.of unknown elements equal the nurnber of conditienl

t

'

O:

soltitions can

'not

be

obtaineq

from

.K'G)･7)L

'

FIg.

₆

3-continued

iead

term model sllbjected

to

ipairof

symil,Letnc or antisymrnetTic

forced

displaceFients

gnd

its

nodal

(5)

Architectural Institute of Japan

NII-Electronic Library Service

successiQn

in

one

direction,

a

single

shear

wall and continued

load

term

models

must

be

cembined as

illustrated

in

Table

_2.

'In

this

way,

if

the

load

term

rnodels

can

be'

defined

from

low

to

high

order

load

_term

models,

it

will enable us

to

derive

the

nedal stiffness matrix of continued shear

'

walls,

Here,

the

smaller

the'influence

coefficients

become

at on'e nodal

_point

of

the

shear wall,

the

farther

this

nodal

_point

is

from

the

nodal

_point

subjected

to

forced

displacement

This

means

that

the

influ-ence

coefficients will converge

to

zeTo

as

the

load

term

models

becOme

continued,

i.e.,

higher

order

load

term

models,

Also,

if

the

3=continued

load

term

rnodel

is

adopted as

_,the

highest

oicler

load

_term

model,

the

fundamental

elastic

behaviour

of

internal

force

systeme)

can

be

express-ed.

This

force

_syste.m

does

not

cause

restraint reactions even

if

idetitical

forcea

displacements

are applied'to each unit shear wall

<a

single-bay single-story

_part

of

continued

shear walls)7).

Co,nsequent-ly,

2-

and

3-cohtinued

load

_term

'models

are

adopted

'in

the

case of more

than

3-continued

shear walls;ana continued

load

term

models

higher

than

3-continued

ones are

ignored.

The

elements of nodal stiffness matrix ,Kr are

defined

by

the

influlence

coeffi-cients

of

3-contifiued

load

term

model subjected

to

a couple of unit

forced

dis-placements

which are symmetric or anti-symmetric _with _respect

to

the

_c.enter

line

of

continued shear walls.

Therefore,

_,kr

is

distinguished

from

the

nodql stiffness

ma-trix

,K, whose elements are

defined

by

influence

coef!icients

at each nodal

_point

'

of

the

3-continued

load

term'

model.

In

formulating

_K

of continued shear wall,. ,Kr must

be

trarrsformed

into

3KL.

In

Fig,6,

,Kt

is

a

symmetric matrix whose

upper

triangular

elements represent

the

symmetric

forced

displacement

system

and

the

lower

ones

represent

the

antisyrpmetric one.,It

is

impossible

to

define

the

ele-ments of _sKr as uniquely as

those

ol tRL of

+

tt

'''tt.'''

tt.t'tt'tt''''/.

''

''''''''/t'

t.

+''tt.tt'tt'-1/t

'''''''''tttt'.ttt

t.ttt.''''.1''ttttt'''''tt

'tt/..i''''''Itt'tttt'tt'''

ttt-

_･''''''''

Assemblage

of

Assernblage

of

.

Assembtage

of

Single-bay

sjngle shear walls

2-continued

load

3-continued

10ed

_continued _sheaF _wall

,

term

.models

term models

Load

terrn rnodel of single-bay

'

･

jnfinitety

eontinued shear watl

Note:Symbol

O

indicates

edge nodal

points

ef single shear

walls and

load

terrn

models,

Fig.7

Single-bay

infinitely

continued shear wall composed of assemblage

of,single shear walls _and

load

_termmodels

'

,..

Table3

Inde-tformagnitudgofabsolutevaluesoftheelements

ef sK:

-2:

Apair-offorced

43

_{nodaldisplacements}

65 _--+

r-t

"t

e7

Ul Vl

hei

'tXl1,1,1+1l,1,2+11,1,O.1

-tXl2,1,1+22,1,2

÷

22,1,Ct+2

-+Exs3,1,1.33,1,2.33,1,O--3

g-tX72,1,1"22,1,2.22.1,O+2

comx-trt1,2,1+11,2,2;11,2,os1

o--tr3･2,2',1+22,2,2.22.,2,O..2

Las-ders1,2,ltl1,2,2+11,2,O-ul

fi-+

'mEr72,2,1+22,2,2.22,2.0.2

tt

¢ hm-Mllh1,O,1.11,O,2+11,O,os1

,-

't

.9

Milh2,O,1+22,O,2.22,O,os2

-

'

L.-

't

･8ttMsAt-t2,O,1+22,O,2+22,O,O+2

o

_{M71h2,O,1+22,O,2.22,O,O--2}

･o

-t

,o='Xl1,i,1.11,1,1+11,1,O+1

s=

£

Eek･op

.xl-tXs---X7,

2,1,1+21,1,1.12,1',1.2

_{2,1,1.21,1,1.12,1,1+2}

2,1,O+22,1,os22,1,O+2

-t,2rl1,1,1+!1,1,1+11,1,O-,1

k.as--r]2,1.1.22,1,1+22,1,os2

'EE-krs3,1,1+32,1,1+22,1,O"2

},--t172,1,1+22,1,1.22,1,O+2

･"'trt<Mllh-+･1,O,1+11,O,1+1'1,O,O+1

M]llt2,O,1+22,O,1+22,OiO,･2

-deMs/h3.0,1+33,O,1+32,O,or2

-tMTIh2.0,1+22,O,1.22,O,O-.2

-ii

Index

fo[

theLmagnittide. ef absoLute values ofthe

etements of _iKr

Ihdex

fer

the magnitude

due

te the components of

ferced

nodat

'dispLecements

Index

for

themagnitude

due

tethe cernponents of restraint

reattions

1ndex

fot

the magnitude

due

to the'locatiens of noda) points

-

83

(6)

Architectural Institute of Japan

NII-Electronic Library Service

the

2-continued

load

term

model,

because

the

analytical solutions

of

3-continued

shear wall cannot

be

obtained

yet.

As

a result,

the

elements

of _,kr are approximated using

the

mechanical

properties

of ,kr and

the

fundamental

nodal

stiffness matrix _.,K* of

infinitely

continued shear wall.

Using

,It

(shown

in

Fig.

6),

,r of single shear wall,

5nd

the

already

derived

,k.of

2-continued

load

term

model, ..k

of

infinitely

continued shear wall can

be

formulated

as

shown

in

Fig.7.

The

unknown elements of ,kT

can

be

obtained

by

sttbstituting

the

elements of ..K'

(which

are

6btaified

analytically6)･'))

to

the

influence

coefficients of

the

'

infinitely

continued

shear wall,

This

sheal wall

is

subjected

to

a

_pair

of symmetric

or

antisymmetric

unit

forced

displacements:

However,

the

total

ngmber

of

equations

resulting

frorn

these

substitutions

(plus

equilibrium

equations}

is

56. 0n

the

other

hand,

the

total

number of unknown elements of ,Kr

is

84. Consequently

these

unknown elements of skr cannot

be

determined

uniquely,

because

the

total

number

of

unknown

elerpents

is

2s

more

than

the

number of

the

equations,

Therefore,

28 unknown

elements of smaller value should

be

assumed

zero

so

as

to

･

/

E*.1

,

.

JkLl'

riide.l'

-

r

-"

voOLo-o.:6EExen

Eo-cotaco"=ofiooas-aco.-v

--kx

f

e

(

"

1232

12l2

1222

)

>

)

"2

s

(

>

2

3 )+

"

1232

1212

1222

"

/IN

'IL2

1 +

-g

2

1

2 -kevh

3

'tLg

g

1232

1212

l222

)

2

1 "･

g2

2 ･g2

)

2

2 2g2'

2

/

voo-o-v-L-oEEta-:c<

_{E9a.eco-=oEooto-nop.-v}

-tx

.ky

-.*erh

)

'

)

1>212

1232

₎

>

)

1)2

)

32 2.'

1 )

'

2

1 >

2

2 b

･

,lt'

1212

1222

1232

slt2

3

2 (4,.,

itN

2 2jlL

e

a

2 1222

1222

)e

)a

)e

2'

3

2

3 2'

2

2 z

)･

'

Note

:l)

Numbers

indicate

thesmalLorder

ef magnitude of the

in

coefficients.'

'

z)

_e:Influence

coefficients assumed

to

be

zero

･

O

:

Unknown

inftuence

_coefficients

O

:

lnfluence

coefficients assumed

to

be

zero since

the

solutions can not

b'e

obtained

from

..K'

6)･').

Fig.8

Srnall

order of the magnitude of

the

influence

coefficients and classifications of

_given

6T

unknown

influence

coefficients at'the nodal

_p6ints

of the

3-continued

leacl

term

model subjected･to a

_pair

of symmetrical or

antisymmetrical

forced

disptacements

84

(7)

Architectural Institute of Japan

NII-Electronic Library Service

Table4Numericalexamplesconcemlngfundamental

_{nodalstiffness} _matrix

,Ktof

3-continued

load

term

models

-ov oEE L o-v aso-ts"o"m ,oo"=-xcon ,o-tu=.-co-o*"\ -o v oeE L o -v as o-tu L o-op 1o-tn=･asMcaD 'vv-=""Hok\ M

Type

I

Type

il

TypeIII

Type

IV

Type

I

Type

II

Type

III

Type

IV

Neglectingmonolithicf!eor

slabCorrection

rnonolithictermfloorforslabconsidering

4q4X{'4vl4Tit4Ht.A,

axin'4Yin4r

h'4Hthlh4Hfu.17,

.E -E

.E

.OOY

-.OOS

O

.OIS

eym,

-.OOS

O

-.O14

-,ool

_.elt

.eol-.OOI.eo3

,O04･..OOI O

-.OS4

.002

_,O02

_O

.ne

o

evm,

.177

-.O06

O eym, .eoo

.elG

o

.,162

e

-,

.150

0

-.e

s-. ulul'vtvl'hel'

.OIS

O

.OO

.oo7

-,eog

o-.oo

,O04

O-.OOo

,oo-,oo

uft uiA vi vretei' uttvfuvMetieit

4xl4xl-4vf4Vlk4wt,1

4xil4Tti4Vit.4Hfu!h4Hrr!lt .Et

et

-.eog

.oos

evm.

O

-.OIS

-,OOI

,O09 0

.017

.001

.oos

o evm.

-.OOI

,eol-.eo6..O02

-.Oll

,OOI

-.els

.oee

o

-,oel

,O07 O

.OOI

-..Ow

O-.OOIe

.eoo-,ov

uf ul" vlvl'-el' ufutvti vitt ethheh'

4xN4xftt4vN'4Mrvlh4HNIh

-E

･o

_.os4

_,-.31S

_.oe2

o o

gum. oe

-.ao

-.v7

o-.eoo

,oo-,OO

uN uNvrv-ervett -t+lh

.Et

.Et･ -Et

.Et

.ooo

eum.oel

ym.

-.ooo

o-,O02

,ooo.ooo

-,ooo-.ooo

-.O02-.OOI

-.

-.Q02

-.O02

eum,

.oeo

evm, o

-,ooo

o

-.oeo

.ooo

,ooo

-,Doo

_.oeo

..ooo

..ooo.oeo

e

o

.ool

oo6o

.oeo

O

-.OOOO

D

-,ooe

o

-,eol

o

-.eoo.oooo

.ooo

e

-.elo

o.oe6

-,eoo-.eeo

o

.oooo

.eoo o

-.ooo

o

-.o.o

.o-.o trltcrvlvl"elh ufiptft-vftvfttefteft-utmt vfi vfulheitheti' uftuN-vrceNese+ 4xr4xl'yr-rr'4Hl+lh 4x"4xrvt4rrvlqMkza4MNIh -Et Et

-.e72

eum,

,183

O-.4y3.oes.e23

-,olo-,eo2

sum,

e

-,los

-.e17

,10S

-.071

-.D02

,2il

,O07

.ooe

o

-.ooe-.oe2

o-.oeo

oo

.eo

-.,

"tul'vSvl'

.ef'

urvttift-vNeleft"

(8)

-85-Architectural Institute of Japan

NII-Electronic Library Service

obtain

a

highly

accurate stiffngss matrix _,fi of

3-continued

shear wall.

In

dolng

this,

it

is

necessary

to

decide

the

order of nodal

_p6ints

in

which

the

influence

coefficients of

3-continued

load

term

rnodel

(subjected

to

a

pair

of symmetric or antisymmetng

forced

displacements)

are assumed zero.

_...

'･,

i

Since

it

is

considered

that

the

characteristics

of

both

,RT'

4nd

,kr

are

similar,

the

results cpncerning

the

elements

'

of

2-continued

load

t6rm

thoddl

given

6y

the

analyt{cal

golutions

can

be

applied

to

,RT,

Hence,

the

foliowing'

'

assumptlons are

_glven.

.

,

.

1 )

The

index

which expresses

the

magnitudes

of

absolute

values of

the

elements

of

,Rt

is

given

by

considering

synthetically

the

indices

expressing

the

magnitude

due

to

the

locations

of nodal

points,

the

magnitude

due

to

'

the

comp6tients

of

influence

coefficients

,and

the

magpitude.

due

to

the

components of

forced

nodal

displacements.

These

indices

must

be

decided

so

as

to

'Satisfy

the

reciprocal

'theorern.

2)

Among

the

magnitudes

of

the

influence

coefficients, which

depend

on

the

locations

of

nodal

points,

the

ones

at nodal

_points

1,

2 (which

are

the

farthest

nodal

p6ints

from

nodai

points

s,

6

subjected

to

distributed

load

)

are

the

smallest and

their

indices

are

1. _,

,

i

We

indicate

the

index

expressing

the

magnitudes' of abso}ute values,

y.bich

are estirpated using

the

above

'

assumptions,

of

the

elements of ,kT

by

the

increasing

order of

these

rnagnitudes as shotvn

in

Table

3. Since

irt.is

difficult

to

estimate

the

orders ef magnitudes of absolu

£

e,

Values

of

the

influenCe

coefficients

at

nodal

points

3,

4 and

'

r

'

s,

6 "n

the

casd of

forced

displacements

actihg

on

the

intermediate

nodal

_points

3,

4),

the

elements of ,Kt

corresponding

't6'tfi6

influence

coefficients at nodal

points

3,

4

and

5,

6

are obtaihed

bY

solving

the

56

equations.

Fig.8

shows

not'6nly

the

orders

of

magnitudes of absolute values of

the

elements

of _,kt

but

also

the

_given

ancl

unknown

elements.

Table4

gives

the

values of

the

elerhents of

the

fundamenLal

nodal stiffness matrix ,Kt of

3-continued

load

term

model whose

sing}e-bay

single-story

part

is

identical

with

that

of

.the

2-continqed

load

term

/

･

.1

model

in

Section

2.'

.'.'

･.

･,

'

4. Numerical

examples'

'

'-.･

,

_.

In

order

to

discuss

the

roles of

the

duplicate

correction'terrns !RD and sXD and of

the

real

load

terms

±

itRL

and 3KRL, numerical calculations coneeTning

internal

force

syst'ernS

'ofinfinitely

continued shear walls arp carried out.

The

load

terms

_:fiD, !kaL

and

slD, 3rRL are contained respectively

in

tkL and 3kL

of

the

2-

and

3-continued

load

term

models.

The

analytical

solutions

of

these

intethal

force'systems

have

been

obtained

in

the

papers3)･`)･'}.

In

these

calculations,

the

aspects of cross

igection

and sizes of

unit

shear walls are

identical

with

those

of

the

unit shear walls

of

standard

type

in

Section

2. These

numerichl

calc,ulations

are carned out

for

each

internal

force

system of

Types

I,

ll,

M,and

IV

as

illustrated

in

Fig.9.

Note

that

Types

I,

ll,

M

and

IV

are

the

basic

types

of

the

fundamental

components of nodal

forces

and nodal

displacements

of shear walls as

discussed

in

the

paperi),

,

,t./.

.

'

AL-s,:QITY:I

_AMul

'

U-QI

J-yilV-Mni,

''

TypeI

Type

II

Type

II]

-e,e;LM

DQi

ttt

hl'-IIQrxt

zH-elx

T

{I{?

ix

･

e=elx

･

Type

I

(Q

rk

per

single

bay}

Type

I

.

(-Xrr)

Xii)

.-X."M

'Xu

･

'

Type

ll

･

1

ttt

'

-Mrvgp

t

)Miv

,

Type

lV

;･･

i

,Type

II

(XIper

single story,

Yll

_per

single

bay)

'''

Fig.g

InteTnal

forc'es

acE/1'hgon the

infinite]y

continued sfie'ar"'an

TYu

)

t.

(.

-

/

S=Yn

(9)

-86-Architectural Institute of Japan

NII-Electronic Library Service

Since

the

elementary

beam

theory

is

applied

to

the

boundary

frame

of wali

_panel,

the

nermal

strain

in

the

directien

perpendicular

to

the

_{axis of}

the

member

does

not

yield.

Consequently,

this

fact

is

considered

in

calculating

the

deformation

of

the

shear wall

using

I

-beam

theory.

The

clear

height

of shear wall

is

regarded as

I-beam

and

the

intermediate

_members _are

treated

_{as rigid}

body.

_On

_the

_basis

_of

_these

_{considerations,}

_the

_shear,

_flexural

_{and axial} stiffness of_an

infinitely

_continued

.shear

_{wall are}

defined.

Each

_of

_these

_stiffne'ss

_is

_assumecl

_te

_be

_the

_standard

stiffness

K,,

and each stiffness of

uhit

shear wall obtained

by

the

analytical solutiOns

is

expressed

in

terms

of

fi,.

The

shear

stiffness of

infinitely

centintted shear wall of

Type

I

is

defined

at

two

fiodal

_points

of

the

unit shear wall

in

the

vertical

direction.

On

the

other

hand,

the

axial stiffness

of

infinitely

continued shear wall

of

Type

_ll

is

defined

at

two

_noclal

_points

_of

the

unit shear wall

in

the

direction

associated with

that

of

internal

forces,

As

mentioned

in

Section

3,

the

elements of ,Kr are

decided

in

erder

to

express

the

fundamental

elastic

behaviour

of

infinitely

continued _shear _wall.

Therefore,

_the

_values _of

_the

_stiffness

_of

_infinitely

_continued _shear _wall

_calculated

_using

2-

and

3-continued

load

term

models agree well with

those

resulted

from

using

the

analytical solutions

;

and

their

residual errers are zero

as

_shown

in

Table

5. If

the

_3-continued

_Ioad

_term

_model

_is

_used,

_the

_stiffness

_of

_{shear wall}

_infinitely

continued

in

two

_directions

_is

computed with

high

accuracy as

indicated

in

Table

_5.

_Alse,

it

_can

be

_observed

from

this

table

that

_slp

is

_generally

_smaller

_as _compared _with

,fi,

concerning

all

kinds

of

internal

force

system.

Duptication

of members

due

to

_{superpositiQn} _{of single} _shear

_walls

_is

_mostly _compensated

_by

,I,.

On

the

other

hand,

Tables

Roles

of

2-continued

load

terms

,I,=,r,+,k., and

3-continued

load

terms 3kL= alD+sknL as corfection

terms

for

stiffness

K

of shear walls subjected to'

internal

forces

(Numerical

examples)

Correction

values

2-continued

loadterm-continued

loadtermResidualerrors

shearInternal

wallforcesR,h-.Ko

Q/{2utlh+2vlkz]

t t

xl(2un.>,r/C2vlh)

･MlC2ul(Zh)},

Ml{2vf!(Zge)}

knth!?ts

K;t-!K-t-K-,,h1XM.

2K-olK-eh

2K'Rb1ffth

2R,/K',,

3K'e'1K--th

aKLRE･1Kth

3Rs1Rth

{K-o,k-aK-p+3K-'p)}ffi,A

{itEtth-etit,e+s'tCn:)}!R,h

{Rstn-aK-L+3k`)]1itth

KHtnyfaio2':C･f"fd,".r,d,Io

Qr(MDO.97C.t,1:+,T}k';)"-II･,-O,13100)'

o.37(loe)

e.24.(loo)

-O.14(;e6)

O.36(98)

O.2294O.Ol{-6)

O.Ol(2)

O.026)

o

o(o)(o)

Single-bay

infinitely

continued

shearwall

Yr'I1.00

EAil7r-O.03100

O.45(100)

O.42(100)

-O.03O,24O.21100(54)CSI)o.oeo

O.21(46)

O.21(4)

o

o(o)(o)

Mru1.00E"r,

-O.05(100)

O.14(100)

o.o(le)

-O.OS102

O.13(,95)

o.oeC2)O.OO-･2

O.Ol(S)

O.Ol<B)

o

o(o)(o)

Q,(Mr)1.01tT,S+as'2oif,-O.02(IOO)

o.esQoo)

O.2(100)

-O.03119

O.OS(109)

O.02(100)O.OO-19

--O.OO(-9)

o.oo(o)

o

o(o)

'

Single-story

infinitely

continued

shearwallX･I:1.00

tEAT,-O.03100)

_O.16(100)

O.13(100)-O.03100

O.15(96)

O.i2(9S)o.ooo

O.O'1(4)

O.Ol(5)

o

o<o)

o{o)

Mry1.00

tEtT,-O.Ol100

O.02CIOO)

O.Ol<1)

-O,Ol100

O.02(100)

O.Ol{100)o.ooo

o.ooCo)

ooo()

o

e

oCe)()

erx(Q;f)O.99

_{haAdi-if,-o.16{100)}

_O,42<100)

o.26aeo)

-O.17106)

O.39(94>

O.22(B6)O.Ol-6

O.Ol{2)

O.02{7)

o.oo(o

O.02(4)

O.02(7)

Infinitely

continued

shearwall'

X:r1.00

zenTr-O.39100)

O.22(100)

-O.17

-O.39100

O.21(95)

-O.18(7)o:ooo

O.Ol{4)(-)

o.ooo

O.OO(l)(-)

Yrr1.01EAII-T,-O.31(iOO)'

_O.57(100)

O,26(100)

-e.31(loo>

O.32(S7)

O.Ol(6)o.oo(o}

O.26(a6)

O.26(99)

o.ooo

-O.Ol<-3)

-e.ol(-s)

Single-bayinfin.itely.

continuedshearwall

ingle-storyinfinitely

continuedshearwallInfinitelycontinued shearwall

ShearstiffnessGAw

_ctt

_aht

att

FlexuralstiffnessErE{bc(t+Pe)1-(be-t)(Z-De)3}1!2e{bb(h+tn))S-(bb-t)(h-Db)S}ln

AxialstiffnessEA

_{e(2beDal-Zrt)}

_EC2bbDb+h,t)

Horizontal:e(bbPb+h't)

Vertical:E(beDa+Z't)

Shapefactor

forshearing

deformationitwr

O.98

O.92 Note:Numericalvaluesin(

_{)indicatetheratiosinpercentfor}

the anaiytical solutions.

(10)

--Architectural Institute of Japan

NII-Electronic Library Service

Table6Influence

coefficients of shear wall subjected toa

paiT

displacements

at edge nodal

points

(Unit

:

Et)of

symmetricor anlisymmetricunitforced nodal

ep-=o.-oL-".atoooo=o=-"-=.-r

?

fiLXh

Symmetricunit

forced

nodaldisplacernents

a

-cr-Ee" )::,

1L--"

t'r'

O.1p

o.esF..--D

ttt

oE

E]

,LL,'1,'E ,,,1

O.1-1L...i

.05-he

¢

_macE]Ne

[]

ct

::,1ii'

o.os-lda''

O.05-

e.os-Antisymmetricunit

forced

nodaldisplacements

il

-:::

ug

ll

1L..1

O.1-

O.05-Jti

tst

g

:::

O.1-

'1-''

o.os-h-eejrvrvRrv

es

vil

e.os-''

O.05-Table7Influencecoe

displacementsfficients

ofshear wall subjected

to

a

pair

of at edge nodal

points

(Unit:Et)

symmetrlcor antisyrnmetric umtforced nodal

Symmetricunitforcednodaldisplacements

,

Antisymmetric.unitforeed]odaSdisplacements

i

-v

h-e

il

i

he

...Tza;

----･m':D]:

---.-,"-r[]I=:'fi'---a;

,n.m,m･ml[mz

tttttt.tt･-･---･utl-'['[sii

r1H

O.1A

o,osH

1- O.1-

_mp

n:'6L-s88s2tt-r

-t-O/1- 1-

O.05-

O.1- 1H

o.os-

Blho.os-O.05H e.os-

Rtgs

D.OS-'

(11)

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NII-Electronic Library Service

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---

Assemblage

ef single shear walls on}y

---

Assemblage

of single shear walls and

2-centinued

load

terrn models

Assernblage

of single shear walls,

2-

and

3-continued

loati

term

models

Fig.

1O

Fundamental

cempenents efnoda]

displacements

of single-bay nine-sto[y shear walls subjected

to

nodal external

forces

most of

the

non-conforming members

(except

at nodal

_points)

in

antisymmetric

internal

force

system of

Types

I

_,

M,

and

rV

are corrected

by

_,K,,.

However,

if

the

internal

force

of

Type

ll

is

applied

in

the

short

direction,

the

role

of iKRL

as

real

load

term

becomes

important.

The

nodal

displacements

of single-bay nine-story shear wall subjected

to

external

forces

similar

to

_the

internal

force

systems can

be

decomposed

into

representative components

a*

iTwhich aTe

illustrated

in

Fig.

10. This

figure

indicate$

that

the

Tole of

3-continued

load

term

model

is

very

important

for

_the

vertical

compenent

of

the

nodal

displacement

of

Type

ll

as

_pointed

out above.

However,

its

role

is

relatively small

for

the

other components of

the

nodal

displacements.

'

Influence

coefficients of

the

continued shear wall subjected

to

a couple of symmetric or antisymmetric

forced

displacements

at

nodal

_points

on

the

edge

members

are

illustrated

in

Tables

6

and

7. As

can

be

observed

from

these

tables,

the

effect of

3-continued

load

term

model

is

observed

to

be

very small

in

the

case of exteTnal

force

systems.

5. Conclusions

In

order

to

obtain

the

ef

continued

shear wall with

high

accuracy,

the

use of

load

term

models which are regarded as new concept

in

the

analysis of

frame

structures stiffened with monolithic wall

_panels

is

pTo.posed.

The

nodal stiffness matrices of

these

models are

formulated.

Moreovei

the

physical

meanings of

the

load

term

models are clarified and

their

validity

is

verified

through

numerical examples.

Reterences

1)

Tomii,

M.

and

Yamakawa,

T.

_, "Relations

between

the

Nodal

Forces

and the

Nodal

Displacement

on the

Beundary

_FTames

of

(12)

-89-Architectural Institute of Japan

NII-Electronic Library Service

Rectangular

Elastic

FTamed

Shear

Walls,

Part

I

,

ll

,

M,

IV,

V",

Trans.

of

AIJ,

No,

237,

Nov.

1975,

pp.

45-57,

No.

238,

Dec.

1975,

pp.37-46,

No,239,

Jan.

1976,

pp.35-42,

No.240,

Feb.

1976,

_pp.63-70,

No.241,

Maf.

1976.

pp.79-89.

2}

]l?JM,

iiii(il.lig:l

doll:Mlg#;1,a'ppT.'ii:sSot,iffneSS

Matrix

ol the

Two'Bay

or

Two･Story

Duplex

Frtirp.ed

shear

wansi',

Trans.

ef

3)

Tomii,

M.

and

Hiraishi,

H,

_,

"Elastic

Analysis

of

FIamed

Shear

Walls

by

Considering

Shearing

Defermatioh

ef

the

Bearns

and

Columns

of

Their

Boundary

Frames,

Part

_I,

_ll,

_M",

Trans.

_of

_eLIJ,

No.273,

_Nov.

I978,

pp.25-31,

No,274,

Dec.

1978,

pp.75-85,

No:Z75;

Jan.

1979,

pp.45-53.

_.

･

4)

Tomii,

M,

,

Sato,

N,

and

Inoue,

M,

,

"Elastic

_Analysis

of

Twe-Steiy

or

Two-Bay

Duplex

Framed

Shear

Walls

Subjected

to

Antisymmetrical

Loads

with respect

to

the

Axes

of

the

Interrnediate

Members

of

Their

Frames".

Trans.

of

AIJ,

No.

297,

Nov.

1980,

pp.35748.

5)

Tomii,

M.,

Inoue,

_.M.

and

Kuriyama,

K.,

"Elasti,c

Analysis

of

Two-Story

or

Two-Bay

Dllplex

Frhmed

Shear

Walls

Subjected

to

Symmetrical

Lead$

with respect _te

the

Axbs

_ef

the

Interin'ediate

M,einbeis

of

Their

Frames",

Trans.

of

AIJ,

No,Z99,

Jan,

1981,

pp.69-82..

'

''-6)

Tomii,

M.

ana

Yamakawa,

T. ,

`'Nodal

_Stiffness

_Matrix

_forArbitrary

_C6ntin"gd

_FrainEd

_Shehr

_Wall

{c6ntinued)",

Proc.

of

?aYprnaPn:::r

On

COrnPUtatiOnal

Methods

'in

Structrral.

,E.

nFi.neffi.E..g and

Fields,

Volume

lo,

Iuly

lgs6,

pp,

l4o-14s

on

7)

Tomii,

M.

and

Yamakawa.

T. .

'"Fundarnental

Nodal

Stiffness

Matrices

of

3-Continued

and

Infinitely

Continued

Framecl

Shear

Walls"`

_T}ans.

of.AIJi

No.374,

Api.

1987,

PP,98-lli･

..

_..

'

8)

Tornii.

M.

,

Yamakawa,

T.

and

Ninomiya,

T.

, "`Effects

of

Monolithic

Floor

Slabs

on the

Meghanical

Behaviour

Frarned

Shear

Walls

Subjected

to

Earthquake

Lateral

It///,fds",

Trans.

of

AIJ.

No.377,

July

I987,

pp,102-113.

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