側柱または側ばりのせん断破壊によって支配される1スパン1層鉄筋コンクリート造耐震壁の水平耐力算定式(梗概)

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LATERAL

SHEAR

CAPACITY

OF

ONE-BAY

ONE--STORY

REINFORCED

CONCRETE

FRAMED

SHEAR

WALLS

WHOSE

EDGE

COLUMNS

OR

EDGE

BEAMS

FAIL

IN

SHEAR

'

by

Masahide

TOMII'

and

Fumiya

ESAKI'",

Members

of

A.I.J.

from

the

horizontal

direction

in

the

wall

prop-agate

to

the

ends

_{of eqge columns}

and

_edge

beams

with

the

same

angle

e,

When

either

the

ends of

edge columns

or

those

of

edge

beams

fail

in

shear

the

wall

fails

in

shear at

the

same

time

'because

it

cannot

bear

th6

increment

of

the

lateral

force

due

to

the

re-distribution.

Consequently,

the

lateral

shear

capacity

of

shear

walls

is

the

lateral

force

carried

by

the

shear wall when

either

the

ends of edge columns or

those

of edge

beams

1,

lntroduction

The

shear

failure

modes of asymmetric one-bay one-stoTy

framed

shear wall

{hereafter

referred

to

as "shear wall")

can

be

classified

into

two

typical

types

ksee

Fig.

D.

If

the

boundary

frame

can

bear

_the

reaction

due

to

_the

dilation

of

the

shear cracked

infilled

wall

_panel

(hereafter

referred

to

as "wall")

_forming

_diagonal

compressien

field,

the

type

is

the

slip

failure

of

the

wall shown

in

Fig.

1

a.

On

the

other

hand,

if

the

boundary

frame

cannot

bear

the

reaction,

the

type

is

the

shear

failure

of edge columns or edge

beams

(see

Fig.1b).

If

the

edge columns

fail

in

shear,

bearing

'

capacity _of

the

_{shear wall}

decreases

_and

the

_{upper stories supported}

by

the

_fihear

_{wall are}

in

danger

_of

falling.

To

prevent

a such

dangerous

brittle

failure,

it

is

necessary

to

estimate

the

lateral

shear capacities of shear walls

corresponding'

t6

the

shear

failure

modes.

Although

some estimating methods of

the

lateral

shear capacity of

shear

viTalls

failing

in

shear

have

been

pro-posedi}-4},

they

are not

based

on

the

shear

failure

modes except

the

Mochizuki's

_{proposition`).}

Dr.

Mochizuki

has

shown

the

expressions

for

each shear

failure

mode.

However,

his

analysis makes

it

difficult

to

_present

the

rational

expressions,

because

the

member stresses necessary

to

calculate

the

lateral

shear capacity are

determined

by

using

the

assumption

different

from

the

analytical results of

the

cracked

shear

wallsS).

The

objective

of

this

paper

is

to

_present

the

expresgion

for

estimating adequately

the

lateral

shear capaeity of shear

walls

dominated

by

the

shear

failure

of

their

edge columns or edge

beams.

The

expression

is

derived

by

considering

the

_{mechanism of shear tesistance couespondipg}

to

the

failure

mode of a shear wall.

In

this

derivation,

the

_rational

assumptions

based

_on

the

_{orthotropic elastic}

plate

analysis of

the

cracked shear walls are used and

the

multiple

regression analysis of

the

experimental results

is

made.

'

2.

Assumptions

for

Analysis

To

derive

the

expression,

the

following

assumptions are llsed

(see

Fig.1

b).

1)

The

shear

cracks

inclining

at

_the

angle

_e

'

g}+trtiipZfraiiurq

_t8

tlshearga

_±iureg'et

r;,h

LE･

+iz

.g2g33u

bb

]be

g

2

g

2-e.2 sPeareraok

_.'s'

L.

'a

_･.,

.'

'

.pl,,'.,i･:･

_'

.l.SII..4.･Ig-'

Q,l

_I

(a)

The of

d.

La,

t,-a, pattern of a sltp a walZ

Fig.1

Typical

_patterns

ef

+g.

2t

ttt

fa±lure

kzleenter

'sectton,b-.tg.h

2 .-gh2V

-eni2-4T

(b)

"a pattern of a shear failnre

of a boundary frame

shear

failuie

of shear walls

i

_Professor

_of

_Structural

_Engineering,

_DepaTtment

_of

_{ATchitecture,}

_Faculty

_of

_Engineering,

_Kyusyu

_Univ.,

_Dr.

_Eng,

#

_Research

_Assistant

_of

_Structural

_Engineering,

_Department

_of

_{Architecture,}

_Faculty

_of

_Engineering,

_Kyusyu

_Univ.

,

Mr,

Eng.

(M,an"s:[ipt

Teccived

October

13,

1986)

-81-fail

in

shear.

The

lateral

force

is

_given

by

summing

the

forces

acting on

the

a-b-c-･d

section

in

the

wall and

the

ultirnate shear strength of

the

ends

of

edge members.

2)

At

ultimate,

the

shear

cTacks

in

the

wall

which

cause

the

shear

failure

of

the

ends of edge columns or edge

bearns

propagate

across

the

horizontal

line

or

vertical one at

the

center of

the

wall,

The

wall reinforcements crossing

these

cracks at

the

a-b

and

c-d

s.ections

yield

in

tension,

3>

At

ultimate,

the

shear

force

_per

unit

length,

_.Q.1

t,

acti"g on

the

b-c

section

{hereafter

refeTred

to

as

"centeT

section")

is

equal

to

_Q.atcs)1l

or

Qaubstll

(Q.o[..],

Q.otbsi=lateral

shear

capacities

dominated

by

sheai

failure

of edge

columns and

that

of edge

beams,

respectively),

The

resttlts of

the

elastic

analysis

of

shear walls

by

assuming

their

cracked wall

to

be

_45-degree

orthotropic

_plate

_prove

that

this

assumption

is

rational

irrespective

of

the

extent

of

the

diagonal

shear

cracks

of

the

wal16).

4)

All

shear

cracks

crossing

the

center

section

incline

at

the

angle

e

from

the

horizontal

line.

At

ultimate,

the

wall teinforcements crossing

these

shear

cracks

near

the

center section

_yield

in

tension

at

the

cracks.

3.

Ultimate

Shear

Strength

ot

the

Ends

of

Edge

Columns

or

Edge

beams

In

the

case when ashear wall

is

subjected

to

the

polar

symmetric

forces

with respect

to

the

center of

the

wall

(see

Fig.2),

the

shear

forces

(the

ultimate shear strengths),

Q..,

Q.,

and

the

axial

compressive

ferces,

N..,

N.b,

acting

on

the

shear

failure

section

of

the

edge

columns and edge

beams

at ultimate can

be

obtained easily

by

the

equilibrium

condition

based

on

the

assumption mentioned

in

section

_2.

(1)

the

shear

force,

Q..,

and

the

axial compressive

force,

IV.,,

acting on

the

shear

failure

section

of

the

edge

columns

at

ultimate

:

Q.,:=S(Q.,,..-.Q.-.Q.)=i

I(i-

i+DC-ih'

COt

e)

Q.,,..,-(h'-2

D. tan e)tp.o,.)

･････J･･････-･････

o

a

)

iv.,=

iS

(t}

diQ..,et+.N.-.ivL,+N)=S

I(!}

di+

h'L(t+?C)

tan

e)

Q.,,,.,+

rtp.a,.+N]

･･･-･･-･･･-･･-

(2

a

)

(2)

the

shear

force,

Q.,,

and

_the

axial

compressive

force,

M,,

acting on

the

shear

failure

section of

the

edge

bearns

at ultimate:

Q.,

=;

(tl,

Q.,,,.r.N.+.N.)=i:

((3-

h+De-ti'

tan

e)

Q..,.-(r-2

D,

_cot

e)

tp..y.l

･-･--･-･･･

(i

b

)

IVLtb=S'(g-Quorbs)+rQw+eQ.)==S((ip+l'"(h+?b)COte)Q.oce.)+h'tp.a,.)････-･･･-･･-･--･･････-･--･--(2b)

where

'

.Q., .IV.=horizontal component and vertical

one

of

the

forces

<kg

)

acting

on

the

a-b

and

c-d

section at

ultimate

(see

Fig.2)

{tension

is

taken

as

positive)

Case

(

1

)

.IVLe =pvay.t(l'-L)

Case

<

2

)

rN.=p.a..t(l'-2DbcoteTL)

.Q., .IV.=shear

force

and compressive

one

(kg)

acting

on

the

center

section

at ultimate, ,IVI,,

is

obtained

by

aR

given

by

the

equilibrium of

the

forces

acting

on

the

triangular

element

(see

Fig.2,

At

ultimate,

T=(Quo[cs]t

Quo(bs))ltl,

rav=ayv)"O{e')h

i'S2

,

ptshear

failure

sectton ef edge member

In

the case

that

edge columns

In

the

case

that

edge

bearns

fail

in

shear

faiL

in

shear

Fig.2

The

ultirnate

forces

and

_polar

symmetric sections

foT

-82-itt

tZne

Foices

acting on the

b-c

section

NII-Electronic Library Service

ptcase

(D''''d''tt''t'j'''''t

''''

''''

t--''''''t-'

dr

xease

(ti)''"'tt"i''-.t'''''

''tttt

''tt''''''j

sk case

(M)a)

' t'''''' "''''''''.

''''''''''','tt1'

-

eafie

(M)b)

case

Civ)a)

case

(iv)b)

Tht

loading

conditions feT shear

CASE(l) Push a diagonal and pull other d±agenal ef the boundary frai]e tn the same fetce,.

CASE(iO Pushadtegenal of the boundary

frame.

,

:fft:[:l#l

::lg

:hg2:gOsntl:s:fa:::gb:::d:;l{sfg:M:atiure

edge rnembers, betveen thetr

fa

±

iure

seetions respeativeiy assunied

cAsE(i.).)

i,i,gi.li:'xii'S'iwh!liiE:ill"iilil':i.il[,i::,illl.l,l,i

a

,l

i

:,ii.,l.11.illi,IEi,,n

h

il[,,:ll::[il,,l

i ,11:,:ll

:l

±

CASE(tv)b)

Gtve

untferrn shear stTesses aleng the axts of

fa

±

lure

edge beams, between the centers ef their- edge

tions, and g±ve shear stiesses elong the ax ±s ef the edge columns wh ±ch de not fa±

J.

Fig.3

The

values of

the

factof

ip,

related

to

the

shear conditions of

typical

experiments

Case

₍

1

)

.N.'== aRtL=(

9"tOiCS}

tan

e-p.ob.)

tL

'

case

(

2

)

_.N.=aktL=(

Q"tOibS]

tan

e-p,abo)

tL

P.,ph=vertical

wall reinforcement ratio and

horizontal

one

ai,.,a.h=yield strengths of vertical wall reinforcement and

horizontal

one

(kg!cm2)

t=thickness

of

the

wall

(cm)

,

L=length

ef

the

center section

(cm)

l=distance

from

center

to

center

of

edge

colurnns

(cm)

h==distance

from

.center

to

center of edge

beams

(cm)

l'=clear

span of

the

boundary

fTame

{cm)

'

,

h'=clear

height

of

the

boundaTy

frame

(cm)

6,,

bb=widths

of

the

edge

column

and

edge

beam

.(crn)

･D,,

Db=depths

of

the

edge column and edge

bearn

(cm)

N=vertical

load

applied

to

'the

shear

wall

(kg)

The

factor

ip

is

determined

by

the

loading

condition

of

the

external

load

except

the

_vertical

load

N

(see

Figs.

2

and

3).

'

In

order

'to

determine

the

ultimate shear strength of edge colurnns or edge

beams,

Q.t

(Q.,

for

edge columns and

'

Q.,

for

edge

beams),

the

multiple regression analysis of

Q..

is

made with regard

to

7

data

for

Q..

(data

of

the

specimens whose edge

columns

fail

in

shear) and

14

data

for

Q.b

Cdata

of

the

specimens whose edge

beams

fail

in

shear) of

18

specimens'

(see

Table1)

which

satisfies

the

foitowing

conditions.

'

1>

The

specimens are shear walls subjected

to

the

_polar

symmetric

forces

with respect

to

the

center

of

the

wall.

2>

The

angle,

e,

of shear cracks

in

the

wall

is

known.

.

3>

The

lateral

load,

Q,

applied

to

shear walls

increases

after

the

occurrence of

the

shear

cracks

along

_to

_the

a-b

and

c-d

in

the

wall.

'

The

forces,

Q.c,

N.,,

Q.b,

and

N.b,

employed

for

the

multiple regression analysis are obtained

by

substituting

the

'expeFimental

lateral

shear capacity

for

Q.ot.st

or

Q.qb.)

in

Eqs.(1a)-(2b).

'

The

f.actor

of

the

bending

moment, which can not

be

obtained'by

the

equilibrium condition even

if

under

the

specific

loading

condition

(mentioned

at

the

beginning

of

this

_paragraph),

is

neglected since

the

tiltimate

shear

,strength,

Q..,

is

more affected

by

the

axial

force,

N..,

than

by

the

bending

moment

acting

on

the

!ailure,section

of

the

edge members.

･

The

equation

<3)

for

the

ultimate

_shear

strength

of

edge members,

Qof,

is

obtainecl

froqt

_this

analysis.

'

_The

_data

_of

₃

_{shear walls}

in

_{which whether}

_the

_shear

failure

_occurred

in

_the

_{edge celumns or}

distinguished

'are

included

in

both

data

of

Q..

and

qub,

in

the edge

beams

ifi

not

83

Table1Data

of

18

specimens applied

to

the

rnultiple regression analysis ancl

symmetric

forces

(Greup

A,)

another specimen which are subjected

to

the

_polar

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_-1

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±

speetmem not applied tD the multiple regressien anfilysts

Qu,=cQut+NQzar+rQuf=(cip+Ne

illX+Tipp"ayg)bD=(8.58+O.262

{llf+O.374pgayg)bD

(kg>'''''''･(

3)

where

N.,=axial

compressive

force

(kg)

acting on

the

shear

failure

section

of

edge members

<IV..

for

edge

columns

and

N.,

for

edge

beams)

bD=total

sectional area

(cmZ)

of edge members

(b.D,

for

edge columns and

bbDb

for

edge

bearns)

p.!ratio

of

total

sectional area

of

longitudinal

reinforcing

baTs

to

sectional area of concrete

in

edge members

ake.!=yield

strength

of

longitudinal

reinforcing

bars

<kglcmZ)

Although

the

range of

the

shear reinforcement ratio

in

edge members,

_p.,

(=O-O.O123.

see

Table

1)

and

the

cornpressive

strength

of

concrete,

E,,

(=170--295

kglcme,

see

Table

1)

corresponding

to

the

data

applied

to

this

analysis are very wide,

the

ultimate shear strength ofedge members,

Q.,,

is

scarcely affected

by

these

factors,

This

fact

suggests

that

the

variation

to

distinguish

the

contribution of confining reinforcement

to

the

ultimate shear

strength

of

edge

members

from

the

contribution

of concrete cannot

be

obtained

by

this

analysis

because

the

ratio of

the

othei

contribution,

,Q.r, except

the

eontribution of

the

axial

force

and

longitudinal

reinforcing

bars

to

the

lateral

shear capacity of shear walJs, _..Q.,,x.,

is

less

than

10

%

and

because

of

fewdata.

This

_problem

is

to

be

inyestigated

hereafter

more

in

detail

by

using

rnany

data

having

large

ratio

of

cQtcr

to

exQuevsi.

The

comparison

between

the

experimental ultimate shear

so

strength of

the

edge members, _..Q.!lbD, and

the

calculated

one,

Q.flbD,

by

Eq,(3)

is

shown

in

Fig.4.

Although

Dr.

Mechizuki

has

conducted

the

experiments

of

shear walls subjected

to

the

polar

symmetric

forces

with

respect

_to

the

center

of

the

wall,

_the

Mochizuki's

₁₂

shear

walls

(see

Table

2)

are not applied

to

the

multiple regresslon

analysis.

The

reason

for

_this

is

_that

it

is

difficult

_to

estimate

the

contribution

of

the

longitudinal

reinforci,ng

bars

to

the

ultimate shear strength,

Q.f,

since steel

_plates

are

embed-ded

in

the

center

of

the

section of

their

boundary

frame

in

order

that

the

compressive

and

ten$ile

loads

applied

to

the

shear walls

in

the

direction

of

the

two

diagonal

lines

may

be

distributed

along

the

edge members.

However,

the

lateral

shear capacity,

Q.,tx.),

calculated

by

assuming

that

_p.a..

is

the

sum of

the

amount of

longitudinal

reinforcing

bars,

_.pg.ay., and

the

amount

of

steel

_plates,

84

=

{--c'sht."iR

su

1

60

40

20

Fig.4

2!dataappliedtothemultiple eregress ±o"analysis xMochtzukt's12data . multiplecerielatioRcoefficiemt

(wttheutMoehizuki'sdata)O.980.

etse

't.

''

xlxx1:

'd

/e

/

'Quf

Nuf

bD=8'58+O,262+obD'374Pgcryg

1

O

20

40

60

Quf

bD

(kglcrn2}

-The

comparison

between

the experimental

strellgth of the edge members, _..Q.f!bD,

theoretical

one,

QzarlbD

80

shearto

NII-Electronic Library Service

Table2

Data

of

the

Mechizuki's

12

spectmens subjected to the

_polar

･symrnetric

forces

(Group

A,}

OOLV)DlBEt4H REFE-ReNCESPECIHENzCcm)h(cm)DeCcm)bc(cm)DbCcm}bb{tm)tCcm}Pscz)vetscm)pgc:)%g(fSt2)Pg{:)Ckeg(-ckm)Pcc"k.z,N(ten)e(degree}･e=imorfs(Ack.)IzaorceJcdikm2)ttTuo(bsJ(:eEfir.)PAILUREHe)E 6AO,]S-R"-152.037,o1.D4,S1.04.S1.So,]s2148S.ISt200oS,ISt2000]21oaoo61.363,761.1=olumn O.]S.RW.2S2.031.e7.04.S7.04.51.SO.3S214SS.ISt2000S.ISt2ooe327oaeo6S.761.;61,1besm 1,OS-RW-2Sl.O17.o1,a4,5T.O4.SLS1.05214S5.IS+2000S.ISt2000321o40o19.e77.1S4.0[olumn 7A1.0JRW-1'17.017.o7.a4,51.04.S1.SO,7o2]144,7Shle374.IS+la31251o4So19.079,S79.S･zolumn 1.0-R".1]7.011,O7.04.5T.e4.S1.SO.70233aq.7s-le314,7Stle31251 4Se81.S79,S79.5celumn SAO,3S-NR.1s2,O17,O7.04,ST.e4.S1.So.3s22Tls.se+2128S.S6.212S3!S1,es4oo61.973.861.Sbeam O,3S-NR-2s2.0]7,o7,O4,Szo4.51.SO.ls12TS5.56+212S5.S6i:12S11S1.SS40o67.973.B6?.8beam 1.0S-NR-2S2,O37,O1.o4,51,O4.S1.SLDS22735.S6.212SS,S6,212S31S1.eB4Do76.3S7.98S.4beam 9AO.3S---t67.031,O1.o4.5?.o4.S1.So.ls20126.ISt217Sfi,ISA211S26Soaoe46.J5a,4SS.4column O.35-W.!67.0]7,O7.04.5T.O4.S1.SO.]52e126.IS-217B6.IS+217S26So]sos2.6S2,S53,S･beam O,].W-1e7.o]1.o7.04.5T.e4.S1,So.ro20126.15+tlT86.ISt217e174o]so61.3SS,672.Scoltimn O.7---267.017.07,O-,s7.e4.SLSO.7020126.IS+217e6.IS.!pe274olso61.6SS.672,B:oluinn "ote:

:::,g.g;:,m:;::d.:,i2,gh;,:･:S"::.::,ez

g:":g.eE,eg,:..:fag.rC,kf.z,.:,gZg,s.flYg

.'f`

gy.g,b.D.;',;::g".f.g.::S.::,tr,g

:::

_,t.h,e

:O.Sai

sveg are the seetionel area and the yield strength of steel plate embedded in the beundary frame re6pecttvely,

'

.p..obg, and

pgayg;100

kglcm2

when

poayg>100

kglcm2

agrees well with

the

experimental

one with regard

to

the

Mochizuki's

12

shear walls as well as

the

ana!yzed shear

walls.

This

fact

indicates

that

in

the

case

of

the

shear walls

having

the

edge mernbers

as

well

as

thoSe

of,

the

shear walls

analyzed

in

this

_paper,

the

restraint

effect

of

the,

boudarY

frame

is

dominated

by

the

"Tension

_Ring

action" rather

than

the

flexural

resistance against

the

dilation

of

the

c;acked wall and

_proves

that

Q.,

is

more

affected

by

the

axial

force

considered

in

the

multiple regression analysis

than

by

the

bending

moment

neglected

i'n

the

analysis as mentioned

above,

-4.

_Lateral

_Shear

_Capacities

_of

_Shear

_Walls

From

the

assumptions

for

this

analysis and

Eq.

(

3

),

the

Lateral

shear capacities,

Q.oc..],

_qnd

Q.otb.,,

of shear walls

subjected

to

the

_polar

symmetric

forces

with respect

to

the

center of

the

wall are

_given

by

the

following

equations.

Ip

the

_case

that

the

_{ends of edge}

columns

fail

in

shear:

phayh

(!l(ll-

£

_iD.

c

tan

e)+,e

£

:ic.Dc+.di

£

:f.ahrg

+.ip

(p.a..+

i\/,)

tl-･-･-･-･L---･-･-'･-<4a)

i+

2I

?.c-a-.di

tan e)

(i+

cot

e)-.di

3/

(i+

;I

£

.')di

In

the

case

that

the

encls of eclge

l'

£ Db

£

agakeg

.,Pvalrv(h.

_h.

_th'

+Nipphayh

ue[bs]==

tl'-'''''''-''''-'-"'''-'''-'''<4b)

Q

beams

fail

in

shear:

cot

e)+.di

£

tbhbPb+.di

i+?2,b

-a-.ip

cot

e)

(i+

Xh9b-X/l,

tan e)-.ip

-Eir

(i+e9.c)ip

where

a.=total

sectienai area

(cm2)

of

the

IDngitudinal

reinfercing

bars

of

the

edge column

in

Eq.

{4a

),

a,=

p.b,D,,

and

the

edge

beam

in

Eq.<4b),

a.=p.bbDb.

In

the

tests

of

simply

supported

coupled

shear

walls and

cantilever

shear

walls,

the

shear wall

is

subjected

to

the

polaT

asymmetric

forces,

The

components of

the

forces

are

decomposed

into

_polar

symmetry and antimetry with

respect

to

the

center of

the

wall.

The

inflection

_point

of

these

sheaT walls

is

apart

from,

the

center

point

of

the

wall

(see

Fig,5).

The

lateral

shear capacity,

Q.,(..,,

of such shear walls can

be

also

_given

by

Eq.

(4

a

)

if

the

following

assumptions

for

the

effects of

the

_polar

antimetric components

on

the

stiesses and

}ateral

shear capacity of shear walls are used

:

'

(

1

)

The

axial

forces

(compression

is

_positive),

N..,

acting on

the

failure

section of

left

and right edge columns

'

are

in

¢reased and

decreased

by

a(htlt)Q.Dc,., where

ht

{s

the

distance

between

the

inflection

_point

and

the

center

point

of

the

wali and a

is

statically

indeterminate

_positive

value

less

than

1.

The

a(h,ll)Q...st

is

statically

indeterminate

forces

which

keep

_gquilibrium

of rnoment about

the

center

point,

O,

of

the

wall

together

with

the

'

inctease

of

bending

moment,

AMc,.

and

AMcn

(the

AMcL

and

AM,,

are

_generti11y

not equal), acting on

the

failuJe

'

'

section of

left

and right edge columns and

the

lateral

force,

Q.,[..h

actin'g on

the

horizontal

section at

the

inflection

point

of shear wells

(see

Fig.s).

The

shear

fgrce,

Q.,,

is

increase

and

decrease

by

NeAIV}=Nda(hill}Q.of,et,

but

the

sum _of

Q..

_of

left

and _{right edge columns,}

ZQ..,

does

_{not change since}

the

_{ultimate shea{ strength of}

the

_{end of edge}

columns

is

_given

by

Eq.(3).

(2)

The

wall reinforcement which

crosses

shear

cracks at

the

a-b

and

c-d

section

_yields

in

tension.

The

sum of each

lateral

steel

force,

.Q., and

that

of

each

vertical steel

force,

_.N., aeting

on

the

a-b

and

c-d

sections, and

lateral

concrete

force,

_,Q., and vertical

concrete

force,

_.IVI,, acting on

the

centersection,

b-c,

in

the

wall

do

not

change.

Therefore,

the

lateral

force

and

vertical

force

acting on

the

a-

b-

c-d

section

in

the

wall

de

not change,

5.

Comparison

Between

Experimental

Value,

e=Q.ovs,

and

Calculated

Value,

Q.o[t.)

In

regard

to

the

specimens

(see

Tabte

1-4)

which

experimental values and

the

calculated ones

by

Eqs.

(1)

The

speeimens

are

shear

walls

whose

lateral

edge columns

or

edge

beams.

r.li.2LeLua

Fig.5

leveltton-

hi-t-of tn[Zec-pe±nt

k

,

"

.

:,eet,r..,..,

The

antimetric

forces

with respect

to

the

longitudina

center

line

of a shear wall acting en theends of edge columns

due

to

the

_polar

asymmetric cemponent ef the external

forces

satisfy

the

following

cenditions,

the

comparison

(4a)

and

(4b)

is

shown

in

Table5

and

F

shear

capacity

is

dominated

by

the

shear

failure

1

between

the

ig.9.of

the

end

of

Table3Data

of

54

specimens which satisfy

the

condition

asymmetric

forces

(Group

B,)

(Quo[cot!tl))O.1FcandpslO.

25

_%

and which aresubjectedto

the

_polar

REFE-RENCESPECIMENt(em)h(cm)Dc(cm)be(am)nyCDM)bb'(em)t{em)Ps(z)%("k.')Pg(x)Oyg(fFl},,Fe{gek."N(ton)e(degTee)･exiuorcsc"k.:'-Tuofes){iikF[a) IB13 Sl.OSLO6.06.06.04.0].oL2229DO4.I23700350o41oS9.7s6.e 15 Sl,OSLO6,O6,D6.04.04.0O,9229004.72370031Se42o49,O46.3

!B42

Sl.OSl.O6.06.06.04.02.01.S330004.722900438e41o68.673.9 49 51.0Sl.O6,O6,O6.04,O3,O1,223eoo4,722900470o4!oSl,O56.S

S4

Sl.OSLO6.06.06.04,O4.0O,P2leoo4.7229004S4o43o3S.74E.6

3BA-2

aL371.1,10.210.22S,4!O.24.43,IS351S4.9131743SBo4Soro4.o104,4 A-4 SL3ILIIO.210.22S.410.24.41.SS351S4.91.31SB302o40o9e,77S,2 A-S S!,371.110,210,22S.410.24.4O.79351S4.9132]7366o40o7B,O67.S B-2

Sl,37Ll10.210.225.410.24.43.IS3SIS2.]636243S9o38o9S,3S9.8

B-4 81.37Ll10.210.22S,410,24,41.58351S2.7614933S2o38o81,270,O B-e 8L37Ll10.210,22S.41,O,24,4O.793SIS2.761493344o38o64.7S9.1 C-2

S6,471.lS.110.22S.410,24,43,153SISS.S2376S394o40oSO.68S,O

C-4 B6.4]LlS.110,22S,410,24,41.5B3S15S.523744302o39oSS.661,4 c-g S6,47LlS.110.225.4le.24.4O.793SIS5.S237SS324o38o51.94S.O

4BlbU-2al60.097.S12.719,112.719,1S,1o.so33362.09]IB4226o40o･57.S46.9

lbl-2b160.097.812.7r9.112,719,1S,1e.so33362.0931eA244o40o4fi.746.9

31r-1160.09],S12,19,S12.79,SS,1o.so33364,19318421So3So23.43S,1

3el-3160.097.e12,719,112,730,S5,lo.so131fi1.]131S4219oaso16.]55.7 R-1160.09],S12.719.112.719,ZS,1O.2S36532.133184197o41o19.S40,1 R-S160.D97,S12.719,1l2.719.1S.1O.2S36532.133102232o31o12.832.2 VR-3160,P97,812.719.112.719.1S.1O.5029912,1331S4218ofiso37,849.3 3A2-1Sl.361.a10.212.710.212.74.4O,5033363.3031S42Slo16oS8,3S2,4

3A2-2BL361.010,212.710.212.74.4O.2S33363,3011S4!ooo40o39.3SO.2

3A2-3Sl.361.0le.212.710.212.74.4o.so]1362,2031a4219o42o44,4S3,2 4Bll-4161.66LOID,212.710.212.74.4D.SOS1362.2031S4269e3So40.641.0 At-A16],661.D10,212.710,212.74,41.0029511,OS3184221oS3o43.142,e Al.B167,66LO10.212.710,2Z2.74.41.oe29Sl1,OS31S4231o3So50.S43.1 A2"B167.66LO10.212,710.Z12.74,4L5e1951L05333620Bo32o45.547.3 Nv-115Z482,612.712.712.712.7S,1o.so29911.773184276o]8o]9.S34.4 VRR-116e.o97.B12.717.Sl2.7l7.SS.1o.se29912.2S318422So4]D41,147.4 MS-1141.3ao,oIX712.712,712,7S,10.2729914,963184220o42D31.144.2 SB6 161.6les.712.]19.112.719,15.1O.2527612,10133o4222o40o43.035.S IO 16],6105.112.]19.112.719,15.1O.2S27614,723114236o40o54.14S.4 13 1fi1.610S.71!.7!9.112.]19,15.1o.so400S2,10]0231,8Bo42o49.448.7 25 167,610S.712.719,112,119,15,1O,5033722,102S12420e4So4S.846.0 32 167,6105.7i2.719.112,119,15,1o.so3S132,103SIS274o4]D53,1SO,4 ls 16],6105,712,719,112,719,15.1o,so3S132,103SIS260'o4So48.3SLS

37

167,610S.712.719,112.719,15,1o.so3S132,103SIS2e8o45o43,OSLS

41 167.6105.712.719,112.719,15.1O.5032974.7234as232o40o56.3S6.4 4S 167.610S,712.719.1IZ.719.11.Ee.25]19D2.103016207o36o32.823.S

so

167.6la5.712.719.112.719.11.6O.5031192.1032SS167o43e32,S36.S Sl l67,510S,7IZ,719,112.719.1].6O.50]4992.103248174o4So40.239.4

S4

167,6IDS,712.719,112.719.!7.6O.503S212.1011SS147o40o34,236,3 ss 320.010S,712.719,l12.719,15.1O,5036752.le]2692]2o43o30.94LB

se

32e.D105,712.719.112.719.15.1e,so35eB2,103424204o40o30.641.4

60 320.010S.712.719.112,719.1S.1O,5035692,103248200o3So37,SIS.4 SBWC-16240,O152.032.032.012.032.016.0O.4634e2L7437003oeo3So41,339.6 9Bs-o-s100.0110.0IS.OIS,O20,OIS,O4,4O,S921801,273110237o47e43,94J.S S-O-10100,O110.0IS.OIS.O20.0IS.O4,3O.S92180L273110202o41e4S,I42,3 R-O-5100,O110,O15,OIS,O20.0IS.O4.4O.S721801.273710259o4So36,645.4 R-O-10100,O110,O15.015.020.0IS.O4,5O.5621SOL27371021So42o]5,841.4 S-30-10100.0110.015.0IS.O20.015.04.7O.S421SO1.273]ro2292].]S3o66,26].7 10BNo.2gs,e9S.O12.012.030.012.04.0O.27S6602.003660Z7S20,O40oS4.268,3 No.495.09S.O12,O12.030,O12,O4,OO,27S6602,oe36602]520.04So77.47S,9

86

NII-Electronic Library Service

'

Table4Data

ef

13

specimens which satisfy

the

condition

(Q

asymmetric

forces

(Group

B,)

.or.et1tl))O.

1

Fc

and

p.<O.

Z5

%

and which are subjected

to

the

pelai

REFE-RENCESPECIMENI(cm)h(cm)De(em)bc(cm)Db(cm)bb(cm)t(cm)Pe(z)%(:ciifkm>Pg{mee9(:cEfifk.)Fc{SIIir.)N{ton)e(degree)dee=imoCcs{S.2)

-Tuo{cel(;.iftrk.)

1]5 sLeSl.O6.06.06.04.02.0o

-4.123100321oSlo31.447.T

3BA-O

SL371,110,210.22S.410.24.4o

-4.913416330o40o60.243,6

B-O

SL371,1le.210,22S,410.24.4o

-2.163515]Slo4SO41,Z4L9

4B'lbl-1so,o･4e.96.49.S6.4'9,S2,So

.2.0031e4243o44o40.932,6

lbl-2160.097,812.719.112,719,15.1o

.2.0031Ba212e36o27,226.e

C-1 Bl.361.010.212.710.212.74,4o

-3.323184359osoo5],S51,1

c-s

SL361,O10.212.710.212,74.4o

-3.3231e4227o41o40,O43,7

4nl-2Sl.361.0ID.212.7ID,212.74.4o

'2.2031S4231o35o32.131.1

4Blt3111.861.0IO.212.7ID.212.74,4o

-2.2031S4229o46o2S,S34.9

4Bl.4167.661.010.212.710.212.74.4o

-2.2031e4246o40o29,S26.9

6BA-120S.O145.02S,O2S,O2S,Ols.e7.SO.IS34202.S74770237o40o6L6S6.2 A-220S.O145.0!s.o.2S.O25.0IS.O7.SO.1934202.S74770336o40o61,8S6,5

7BwA:120S,O145.02S,O2S,O2S,OIS.O7,4O.19S6002,S54270246o33oS6.050,7

120

100

seTEx..

60=

g:

4o

1

2o

tl9spectmensefCTeUPAliS4'specllnensefGrovpBl O12spectmemsofCrovPA2A13spee ±rmensofCreupB2 i Note:thedefin ±tto"ofeEch Crouptsmenttenedin _i Table5 i

tt

tai .-ei ii

'

.e-en

g.NS

pdio-"'-.it.k..iKi. ahian4L 41,A",/a";" ".x"7

'ahA

llfilsFc 1

o

loo

2oo

3oe

4oo

soo･

-FtCkglorn')

Fig.6

The

relatienship of

the

shea[ strengtb,

Qtiove)1,tt,

and compfessive strength of concrete,

FZ,

of shear walls

(2)

The

angle,

e,

between

the

shear

crackg

in

the

vyall

and

the

horizontal

direction

is

known,

(

3

)

The

shear walls satisfy

the

condition,

Q.o[cs)

or

Quo(bs))O.

1

F}

tl#Qcn

because

the

lateral

shear

capac-ity

of

the

shear wall whose

Q.,[..}

or

Q.o[b.]

is

les.s

than

the

lateral

force,

_Q..

at

first

shear cracking

is

domin-ated

by

Q,.

(see

Fig,6).

'

Group

A

consists of

the

specimens subjected

to

the

polar

symmetric

forces

and

Group

B

consists of

the

specimens

subjected

to

the

polar

asymmetric

forces.

'

When

the

amount of

the

wall reinforcement and

that

of

the

longitudinal

reinforcement of

the

beundary

frame

are very

large,

the

calculated values are

larger

than

the

experimeptal ones.

Therefore,

7060=sx..

40ib..

1le

detE

Fig.7

-19spe[tmensofGroupALiS4spec ±mensofCreuPBl O12spectmensofCroupA2a11specdmensefGreupB2 NotE:Thedef ±ntt±enefeachCroupismenttomed ±mTable5

o

'1/ MSfdiit

:

slb"1Q>i-6".it-t".dv,

-ijSl1dg-p,11

i /i 4tx-i.e`X.0t1 esezorfs) psa=-als?t7ETfiJPstry -1

o

_bl1 vhenPscry>30kgfomt

ts"t":b1

p.a=O.4p.av+ISiglcm' lO4data20304060soleo12 - p,f,(kgforn')

The

contributien of the wall reinforcement

(Note

:

The

p.a,

denotes

the

p.abe.

ofvertical wall

leinforcement

andthephop.of

horizontal

wal] reinforcement)

to

the

lateral

shear capacity of shear walls

160 120=stae

sobn4

1

4o

Fig.8

e19spectmensefGroupA!i54speetmensef'GteupBl O12spectmensefGrDupAlA1]specimensefCroupBl NotetThedeflntttenofeschCrouptsmeottened

-tnTableS

/1 'x'aTr..4...i

']

"".61"i-!6tg"i-e-PgOVg..1J,stskslCn' iiLAdiPg' ttf.".3"-L .A･tt'li.

:'elil

pgomeRuarfs)PgOyg

st/i''

vhenpgC"g>BOkgfem'auerfsJ

lka=O,3ppavo+56kgfan'

g

1

o40

- PsOvg

(kg!em')

The

contribution of

th

on

the

failure

bearns

to

the

lateral

the

contribution of

the

waLl reinforcement and

that

of

the

longitud

of

the

boundary

frame

to

the

lateral

shear capacity

of

shear

walLs

are

modified as

Eq.

(

specimens which

include

the

specimens

applied

to

the

multiple regression analy

the

edge members

(see

Figs,7

and

8).

O,4p.a,.+18

(kglcmZ)

when

_{p.alr.>30kglcm2}

)

O.4pho,h+18

(kglcm2)

when

_phabh>30

kglcmZ

i

･･････････････････････････-･･･････････-････

O.3agayo+56bD

(kg)

when

a.a,.>80bDkg

i

80 I20 160 200 240

e

longitLdinal

reinforcing

bars

section of

the

edge celumns or edge

shear capacity of shear walls

inal

reinforcernent

s

)

by

the

investigation

on all

sis of

the

ultimate shear strength of

･---(5)

The

Sugano's

equation'}

and

the

Arakawa'$

equationZ) Tnodified

by

Dr.

Hirosawa

are

_generally

used

to

estimate

the

lateral

shear

eapacity

due

to

shear

failure

of shear walls.

The

correlation

between

the

experimental values and

the

values

calculated

by

these

equations

is

summarized

in

Table

s.

The

comparison

between

the

experimental value,

..Q.qrs,ltl, and each calculated

one,

Q.!tl,

is

shown

in

Figs.9-11,

respectively.

Althollgh

the

lateral

shear

capacities,

Q.,

are

not

classified

by

the

shear

failure

modes,

the

Suganois

equation and

the

modified

Arakawa's

equation

can

estimate

adequately

the

experimental

capacity

with

regard

to

Group

B

since

the

both

equations are

the

empirical expressions which are

obtained

by

analyzing

the

test

results

of

the

specimens of simply supported coupled

shear walls

and

cantilever shear walls subjected

to

the

_polar

asymmetric

forces,

However,

the

both

equations seem

not

to

be

suitable

for

the

estimation

of

the

experimentat capacity with regard

to

Group

A

(see

Tabie

_5).

The

exptessions

_proposecl

in

this

_paper

can

estimate

more

adequately

the

experimental capacities

of

Group

A

than

those

of

Group

B.

However,

this

_proposed

expressions are more suitable

for

the

estimation of

the

experimental

capacities

of

Tables

The

means and the ceefficlents of variation of exQuedrotIQuovsF

and _{..Q.otib)IQ.,} and tltecor[eiation coefficients of

i.o,t.=

Quua!tl

and

_E.;QVtl

=

gX,Di)･

:N<y

_':

T

120

100

so

GO

40

20

e 19 spectmens of Croup Al Ol2 speeituens ef Croup A2

iS4 speeimens of Croup B]

A

l.:..lllil

"

::

e:,s , e

i,:

re

/

p

utp..

",:pt;'.1

eli l --i

(;l

i'

1"' .i<.tLl -11'f -1

gt.t

sl iVa lopl.1E.ptf(e or

ll

Nete:

Gtorfo)[equattens

4a and 4b posed in thts peper

the deftnttton of eaeh

Creup ±s mentiened in

Table S

e

2o

4o

6o

so

loo

12o

- Qevt )(kg!rmt)

The

comparison

between

the

experimental value,

..Q.qxstltl, and thetheoreticalone,

Q.,vmltl,

by

theproposedexpression Classifirationrouofsectmens Eguationsof

euo(fs),e.

ex(?uarrsl-i51/IEr.)AlCl9speci-mens)A2(12speei-mensBl(S4speei-menS)B2(13'SPecl-mens)

euorfs)Eqs.4aand4b

mean ceeffitientef variatienZ1.0279.91,0269.7O.99215.31.0391].2 correlatienceef-fic±entofiuotfs)O.929O,S20O.919O.820

eumodtfigdArakawa'sequation

mean ceeffiCientef var ±atianZ)(O.155)<37.4){O,748)(12.9)O.99121.1O.9S7IS.3 cerrelationcoef-ficientefrmorfs)(-O.051)(O.6S6)O.S06O.843

e"Sugano'seqtsatton

mean coeffialentof vartatienCZ)(1.l9S){35.7)(1.43S){8.4)O.9632S.4O,94S21.8

coirelatiencoef-fic

±entofiuo(fs){O.6]6)(O.943)O,SSIO.709 Notes:

Fig.9

iA

,k,l=

1

I20

100

80

60

40

20

1)2)ThE values Creup Al

=

Cretsp A2 = Group Bl

=

Group

B2

=

120

100

80

60

4e

20

o

in parenLheseE denote referem:e data.

IS specimens applted to the multiple TegTesston artalysis and enother spec ±men whieh are subjected

to the polar symmetrte forces

Meehizukt's specimens subjected to the polaT $ym metTic ferces

specimens whtch satisfy thg eendition.rquorcs)ltZ)

g O.IFc and Ps

_l

O.2SZ, and which are svbjeeted te

the polar asymrnetrtc forces

speciutens whieh sattsfy the eemdttion,

(auo(es)ltl)

l O.IEc and

Ps

< O.2SZ.and whieh are subjected to

the _polar asymmetrie

forces

A54'SpeelMensof GroupBl A135epeirmemSef GroupB2 . cerrelation _A coefHciento.Sl6iiAi

itt"4Atsi-za"..hQ.=modtftedeguat

±enArakzwa's Nete:Thedeftnttinnofeach Groupismentienedin TableS

=

g･

s,l,=

1

A 54 spectme"s ef Crovp

A13 _{speetmens ef Croup} cerrelatton aoefftcient O.836 i BlU2A/ A

f

A

l

:.

A

;5.

i

l-;--"-AA A'b-a/"

'1

i a"iipa d!----L

ETTTTT'i

1

O

20

40

60

80

100

120

Q.

-tl

(kg!cm')

Fig

lo

The

cemparison

between

the

experimental value,

e=Q.o/col1tl, and

the

theoretical

one,

Q.ltl,

by

the

rnodified

Arakawa's

equation

-

88

Note:Qu"

sugano's equatten

The dEftmitien oi each

Creup is menttened in Table S

20

F]g.11

40

60

80

100

120

l40

160

Q.

{kgfem')

tl

The

comparison

between

the

experimental value, e=Q.qe.b!tl, and the theoretical one,

Q.ltl,

by

the

Sugano's

equation

NII-Electronic Library Service

Group

B

than

the

other empirical expressions.

6.

Conclusion

By

using

the

experimental

data

of

the

98

specimens,

it

is

proved

that

the

semi-theoTetical expressions

derived

in

this

_paper

can estimate

niore

adequately

the

experimental

lateral

shear

capacities

of shear walls whose

lateral

shear

capacity

is

dominated

by

the

shear

failure

of

the

end of edge column$ or edge

beams

than

the

Sugano's

empirical,

expression and

the

modified

A:akawa's

one

regardless

of

leading

condition.

'

References

1)

S.

Sugano:

Sunimaries

of

Technical

Papers

of

Annual

Meeting

of

Afchitectural

Institute

of

Japan

(A.I.J,

),

ecL

1973,

pp.1305-1306

(in

Japanese).

2)

M.

Hirosawa,

T.

Akiyamaand

M.

Shiraishi

:

Summaries

of

Technical

Papers

of

Annual

Meeting

ofA.I.J..

Ogt.

_1975,

pp.

1173-1174

(in'

Japanese).

3)

M,

Yamada/

Gihod6

Publishing

Co.

LTD.,

Aug.1976;

pp.113-]l4

(in

Japanese).

4)

S.

Mochizuki

/

On

Ultimate

Shear

Strength

of

Reinforced

Cencrete

Shear

Walls-Bearing

Strength

Contrelled

by

Shear

Failure

of

Surrounding

Frame-,

Trans.

of

A.I.J.,

No.306,

Aug.

1981,

_pp,40-50

{in

Japanese).

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M,.

Tomii,

T.

Sueoka

and

H.

Hiralshi

:

Elastic

Analysis

ofFramed

Sltear

Warts

by

Assuming

theirInfilled

Panel

Walls

to

be

5)

45-Degree

OrthetTopic

Plates

Part1

ancl

2.

Trans.

of

A.I.J.,

No.

28e,

June

1979,

pp.

101-109,

No.

284,

OcL

1979,

pp.

51-60

"n

English).

6)

M,

Tomii

and

F.

Esaki/

Surltmaries

of

Technical

Papers

of

Annual

Meeting

of

A.LJ.,

Sep.

i980,

_pp.1575-1576

(in

Japanese).

.References

df

the

Shear

Walls

Subjected

to

Polar

Symrnetric

Loads

:

_(All

in

Japanese}

IA)

M.

Tomii

and

Y.

Osaki/Trans.

of

A.LJ.,

No.51.

Sep.

1955,

pp.96-105.

No,52,

March

19.56,

pp.68-78,

2A)

M.

Tornii:Trans.

of

A.I.J.,

No.60,

Oct.

1958,

pp.389-392.

,

3A)

M.

Tomii,Trans.

o,f

A.LJ.,

No.89.

Sep.

1963,

pp.164.

M.

Tomii,

T.

Kei,

T.

Yarnaguchi

and

H.

Yamamoto:

Reports

of

Chugoku-Kyushu-Chapter

of

A.I.J.,

Feb.

I97.8,

4A)

'

pp.179-182.

5A)

M.

Yamada,

H.

Kawamura

and

A.

Inada:

Reports

of

Kinki-Chapter

ef

A.I.J.,

May

1978,

pp.125-128,

6A)

S.

Mochizuki

and

S,

Matsuo

/

Summa[ies

of

Technica]

Papers

of

Annual

Meeting

ef

A.

I.J,

_,

Sep.

1978,

pp.1637-l638.

7A)

S.

Mochizuki

and

S.

Kawabe

,

Summaries

of

Technicat

Papers

of

Annual

Meeting

of

A.

I.J.

,

Sep.

1979,

pp.1459-1460.

sA)

S.

Mochlzuki

and

Y.

Hosaka/

Summaries

of

Technical

Papers

of

Annual

Meeting

of

A.

I.J.

,

Sep.

1979,

pp.1473-1474.

gA)

S,

Mochizuki/Trans,

of

A,I.J..

No.291.

May

1980.

pp.1-10.'

10A)

F.

Esaki,

M.

Tomii

an'd

T.

Nagai/

Reports

ef

Chugeku-Kyushu-Chapter

of

A.'I.J,,

March

1981,

_pp.209-212.

References

of

the

Shear

Walls

Subjectecl

to

the

Polar

Asymmetric

leads

:

_(Al]

in

Japanese

except

3B,

4B

and

5B)

IB)

H.

Tanabe,

C.

Katsuta

and

T.

Azuma/T[ans.

of

A.I.J.,

Apr,

1934,

pp,3e6-319,

2B}

H.

Tanabe,

C.

Katsuta

and

T.

Azuma/

Trans.

oi

A,I.J.,

Apr.

I935,

pp.326-339.

3B

₎

Gerard

D.

Galletly

aad

Robert

J.

Hansen

/

Behavior

of

Reinforced

Concrete

Shear

Walts

UnderStatic

Loacl,

Massachusetts

Institute

of

Technology,

Departrnent

of

Ciyil

and

Sanltary

EnginEering,

Aug.

Igs2.

,

4B)

Jack

R.

Benjamin

and

Harry

A,

Williams:

Investigation

of shear

WaLls,

Department

of

Civil

Engineering,

Stanferd

University

Apr,

1952-Dec.

1956,

/

The

Behavier

of

One-Story

Reinforced

Concrete

Shear

Walls,

Journal

of

the

Structural

Division

ef the

American

Society

of

Civil

Engineering,

V61.83,

No.ST3,

May]957,

pp.1254-1-1254-49.

5B)

Joseph

Antebi,

Senel

Utku

and

R6beTt

J.

Han$en:

The

Response

of

Shear

Walls

to

Dynarnic

Loads,

Massachusettes

Institute

of

Technelogy,

Department

of

Civil

and

Sanitary

Engineering,

Aug.

1960.

6B)

T.

Naka

and

K.

Ryo:

Trans.

of

A.I.'J.,

No.69,

Oct.

1961,

_pp.477-480.

7B)

S.

Sugago:

Summa[ies

of

Technical

Papers

of

AnnuaL

Meeting

of

A.I.J.,

Sep.

1970,

pp.749-750,

TL

Aoyagi,

S.

Furui

and

F.

Esaki/

Sumrnaries

of

Technical

Papers

of

Annua]

Meeting

_pf

A.I,J,,

Oct.

1974,

8B)

'

pp.1381-1384.

'

gB}

T.

Achiyoshi,

Y.

Ueda,

N.

_Qgawa,

Y.

Takashima

and

H.

Takeda

:

Reports

of

Hekkaido-Chapter

ofA.I.

J.,

Match

1977,

'

pp.29-32.

]oB)

K.

Baba:

Surnrnaries

of

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Papers'

of

Annual

Meeting

of

A.LJ.,

Sep.

1978,

pp.1635-1636,