地震荷重を受ける耐震壁の力学挙動に及ぼす床スラブの影響に関する解析的研究(梗概)

(1)

Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan

[su

N] . ･, .･ JeurnaLef Strueturaland ConstTuctipnEngineering H"ftev\ftMuskdescvazaff UDc:6g. o22 ':bgg.

lj41

:6g. o2s. 22:sso. 344 (Transaetions ef AIJ)No.377,Julv,lgB7 ･ rg377 g-wa" 62 'ny7fi

'

'_. _.

1 /.

tt/ '

_tt

EFFECTS

_ttt

OF

MONOLITHIC

_ttt

FLOOR

_ttt

SLABS

ON

_tt

THE

_t/t/

MECHANICAL

BEHAVIOUR

OF,FRAMED

SHEAR

WALLS

t t

SUBJECTED

TO

EARTHQUAKE

LATERAL

LOADS'''

'

' '

. . . .･

by

Masahide

TOMII',.

Tetsuo

YAMAKAWA"!

.and

Toshiharu.

NINOMIYA+,",

Members of A.I.J.,

/, , . ,,/ .

1.Introduction,.- .., . . , ,. ,

Itisconsidered thatone of the necessary stttdies foraseismic designisto discussthe effects of floorslabs.on the

mechanical

behaviour

of framedshear wall

(hereafter

refeFred to as."shear wall") subjected toearthquake late.ral

loads,since the shear walls arranged inthe reinforced concrete framestructures are monolithic with floorslabs._,The monolithic fleorslabs

(hereafter

referred toas "floor-slabs") _transfer _the

eqrthquake lateralloadsasdistiibutedloads on the

beanis

adjaceqt towall _panel

(hereafter

referred toas "beams").

Also

theaxial,

flexura.1

and shearing stiffnessesof beamsare enhanced bythe floQrslabs. Inthis_paperthe effects of floorslab on the_,angular_distortionof thecolumns adjacent touncracked wall _panel

(hereafter

referred to as･"columns"} are discussedusing analytical solutions

(concerning

shear walls) expressed intermsof Fourierseriesi).Oneaspect ofthediscussionconcerns with

themechanism forapplying the lateral

_load

and the other with thestiffness evalqation of beams. As aconsequence the nodal stiffness matrixZ) of sheqr wall, considering the effect oftfloor slabs, iscalculated

(as

_presentedin numerical examples) inorder to analyze frame structures stiffened with monolithic wall _panels using matrix

displacementmethod,

On the other hand, when the diagonalshear cracks occur inthe vvall panel, the wall panel becomesas an anisotropic _platehavingand behavesthediagonalcompression fieldbyshear, and

beams

and columns, togetherwith

thefloerslab, restrain thedilatationofthecracked wall _panel.Thus the cracked shear wall exhibits complicated

mechanical behavioui.The elastic analyses of thesingle-bay single-story and thesingle-bay two-storyor two-bay single-story cracked shear walls

(hereafter

referred to as "single shear wall" and "2-continued shear wall" respectiyely) havebeenreported by Torniiet al.3)'5),assuming the walls to be 45-degreeorthotropic _plates. Inthispapertheelastic analyses of uncracked and cracked reinforced concrete shear walls infinitely continuous in

one or two directions

(hereafter

referred teas "infinitely

continued shear wall") as weH as thoseof single and 2-continuedshear walls are _performedusing Airy'sstress functionof 45-degreeorthotropic elastic _plates.And the effects of monolithic floor slabs on stress resultants and deformations of the beams are clarified,

2.The Effectsof MonolithicFloorSIabson Angular Distortionof UncrackedShear Watl

Inorder toexamine theeffects of floorslabs, the angular distortionsof _single-baysingle-story and single-story infinitelybontinueduncracked shear walls subjected te'earthquake lateralloads

(as

illustratedinFig.1) are

discussedinthefollowingfourcases.

I:Lateralloadsare applied as uniform distributedIoadstothe axis of thebeams, where flexural,shearing and

axial stiffnesses of the beams are real.

ii:Lateralloadsare applied as cencentrated loadstothenodal _pointsat fourcorners of theshear wall, where flexural,shearing and axial stiffnesses of the beams are real.

iii:Axialstiffness of thebeams isassumed toberigidi, while flexuraland shearing stiffnesses of _thebeamsare

real.

* _D. _Eng., _Professor,_Kyushu _University.

# _M. _Eng.. _Research_Associate._Kyushu_University. """ _M. _Eng., _Structural _Engineer, NikkenSekkeiLtd. {Manuseiipt received becember 10,1986)

(2)

-102-Architectural Institute of Japan

ArchitecturalInstitute ofJapan

h

Concentrated loads Di$tributed loads

Single-bay single-story shear walLs

Fig.1 Uncrackedshear

Table1 The aspect ratios of single-bay

numerieal examples Q e Q - - -> ' e - <-Q Q Q

e

' ･ e'Q ' t/

Concentratedloads 'Distributed _loads

Single-storyinfiniteLy continued shear waltg

walls subjected to earthquake lateralloads

single-story shear wall or unit shear wall adopted infiye

Aspectratios'x ab

Bb

_%

Bc

No.11.714 _O.171 _2.5 _O.171 _3.333

No.2O.5 _O.3 _4' _O.2' ₄

ericalmples'No.31

O.3 3 O.2 4

No.42 _O.2 ₃ _O.2 ₄

'

Ne.53 _O.1 ₂ _O.2 ₄

Note: The _proportion of the shear wall adopted as example No. 1

(see

Fig.

fordefinitions of

_espect

ratios _of shear wall} is equal to that _of the

shear wall adopted as the designexample shown inthe AIJ Standard

StructuralCalculqtionof Reinfo,rcedConcrete Structures

(revised

in

1982).

iv

:Axial stiffness of thebearnsisassumed to

be

rigid, while

flexural

anclshearing

stiffnesses of the beams are increased

together _twiceland fivetimes:

Thenumerical calculations of angular

distor-tions forfive_proportionsof shear wall

(see

Table1) are carried out''for each cdse

men-tioned above.

(See

Fig.2fordefinitionsand aspect ratios of each unit shear wall.) The results are _presented i.nTables2and.3, Ifi

Table2,the_values _shown inthe.columns _with

headingsRilR'm and RiilRtiiare ratios of

angular

distortions

inducedincases

i

and

ii,

coefficient x'.of the angular distortionsof th xtsQl(Gtl). Noticethatthe angular

dependenton the _preportionsof shear 'walls

No.3).Thissu

case iiare used. Alsothe

shear wall incase ･iimay become remarkable smaller.' However the angulat distortionsin are nearly equal to those incase iii

(see

Inorder todiscusstheeffect of floorsla case ivtothoseincase

iii

, where flexuralan

rh

L

: '

Qc,H

respectively,

e shear wall incase

distortions

(compare

ggeststhatthe actual angulhr distortionsof

discrep.anciesbetweenangular distertionsof as

case

Table 2)

bsen angular distortiensof shear dshearing stiffnesses

e

'

Fig.'2.C6nfiguratio4and aspebt '

'

tothoseincase

iji

.

iii'

of qolumns inease

ii

are

rbuch

numerical 6xamplesNo._1,

cloumns will beoverestimat6d

columns an

thespan lengthbecomeslongerand

j

are almost uni

wall,

of the'beamsare' '

abI

{

u

Ibc

2 for x =ft ab"?

%.

b,b Dc(c=

T'

bcBc " t

ratios of each unit shear wall

Inthistable,R,,,represents the where the angular distortionsare computed as

largerthan thoseincase iand

No.4and No.5with No.2and iftheangular

distortioRs

in dthose definedat midspan of the

theaxial stiffness of beam

formthroughouttheshear walL and these values

' ' '

the ratios of angular distortionsin

increased byenlarging the

beam

(3)

-Architectural Institute of Japan

Table2 Ratios RilRm and RiilRmof angutar distortiensof theshear walls

'

'Single-'baysingle-stQryshearwalLS,Single-storyinfinitelycontinued

shearwalls--' Locations Column Midspafi Ri"ag Column

,Midspan R.H111(2 Ratios t. Ri/R;-[,Ri･,/Rili 'Ri/Ri" 'Ru/R;;;(n.tZ)Ri/RiilRii/R.iii,Ri/R"iRil"/Riii,(GtZ) iNo.1O.991.361.00O.75O.811.001.38O.99O.7SO.79 No.2o.9g1.,.1.001.00.O.9,9..O.46O.99tt1.001.00' _,P.9,9 'O.48 opo-aEasx'o-es.ytsE=z No.3O.99il.OS'-l,OOO.94O.61O.99'･･1;051.00O.94O.61 Nb.4O.991.39O.99O.73O.761,.OO1.40O,99O.73O.76 No.51.002.94-1.00-O,12O.871.02 '3.04 '1,Ol-O.13O.86 '

Note: 1)Riisthe angular distortions of the $hear wall subjected to _lateral

uniform distributed load e･incase i.

'

2)Riiisthe. angular distortionsof.the shear

ivall

subjected to Iateral

concentrated loafd

ap

in case ii. . ･

3)Riilisthe angular distortionsof the shear wall subjected to lateral load

Q

in case iii.

4>T,h,Y..",ai:?S,hS.hP.W.".fif",,l,O.i:,M",.W,`t.hf?:.adL",gg.IR,,i"idi,?8?,iC.O.r,re.SfPO,"h/`n,gh.t.O,the

wall incase iii,inwhich the values of Ri;; ate computed as KdiQ/(GtZ)

, where G is modutus of elasticity inshear, 1.tisthickness of wall

and

Z

iscenter-to-cenrer distanceef columns'.of the shear walL

t ttt t t t t ' t ttt tt t t ' t tt tt t t t t '

Table3' T.heef.fectof'the width of bearns'ontheraliog _RtJZRattof,ang'ular diFtoltions

/t t ttt tt t t t t t tt tt '

Note :1) Rii'iisthe angular distontionsof shear,walls iny case

ili,

where the'beain width tb ]vallthickness rati6 'Bb` is2;'5.' ''

2)lRi, _isthe angular distortiops of shear walls in case iv,'- , './

t t.

Y2h.es're

'thebeam width to wal'! thicknp,ss rat,ios.

Bbs

are 5..Oand.

･･ '3) _¢ ･ 'is''therate of･ enlargement for.thewidth of beam$. ･'''･

.t .t/. . .t. /

/tttt t .t /t . t. t/ // / /./ltl.

width,

qteprovided,,as shown ,inTable3..Thistable shows,,th4t ,the ,effectisgenerally･small,''

Con.sequgntly,theaxial stiffness of the,beams can be,assumedto berigid, in

distributedthrough thefloorslabs are tre4ted_.asconcentrated loaason the･noda

F,hearw411s. Thi.sresult is._.usefulin,,thefprmula,tionof the _Qedal stiffness-matrix of the shear

3.

The results discussedinSection2,suggestthatthe_axial _stiffness _of thebeams_can-be _assumed _tQbe

formulationof the fundamental nodal stiffness matrix Ki of u.ncracked shea,r wall,

(see

Tables

4 and s) is

qerived

,asthe analytical selutiops, a _genera! stiffness mqtrix

k

of shear walls is easily by making use of transformationmatrix2).

Note

_,thatTypes

I,

ll,

--

₁₀₄

-'Numerical .tSingle-baYshearwallsingle-storyt/_Single-story

continuediri'finit'elY'shearwalls examples/

th-.2 ¢ -5

.O-2, ¢ =5

No.1 _o.97 _O.95 '

O.97 'O.9.5.

'',No.2'o.93 'O.88

O.95 O,.92tt

No.3 _o.93 _O.88 _oLgsL 'o'.9i

No.4 _o.96 _O.94 tt.0.97 'O.95,/ttt

No.5 _o.98.

'O.97' O.98 o..,9''7'

'

t//t t t.t

'

case theeatthquake lateralloads

1

pointsatfourcorners of ungracked

wall.

Numerical Examples on Fundamenytal'Nodal

_$tiffne$s

_Matrix.

ctSingte Shear,Watl.Restrained byFloorSlab

rigid inthe

Ifeach K" .of

fundamental

type formulated

M

and

rv

to b,ementioned･below･.are,the

(4)

basictypesof the

fundamental

compenents of nodal forces

'and

nodal displacementsof theshear wallsZ}.

Thefundamentalnodal stiffness matrices K:of Type [ and Kreof Type

rv,

whose components are symmetric with

re$pect tothelongitudinalcenter line of the wali _panel, can be_given approximately bypin-jointing rigid bars at the nodal points of the shear wall along the beams,Thesestiffnesk matrices are treatedas approximate solutions

pertainingto the stiffness matrix of the shear wa!ls with rigid floorslabs

<see

Tables4 and 5),

'

Table4

'

-.

The representative components P* and PO of the balancing and unbalancing fundamental components of thenodal external forceson singLe-bay single-story shear wall

I-1 I-2 I-3

TypeIy.tSExSi".xt,

"u43S'

fl

'

,gk

*

i)Ca

p

Mi ll-1 ll--2 ll--3

TypeII."zzzE

fti

LZ"

t"Yk.

EZZ,

""

M".

CEZ)

M-1 M-2 M-3

TypeIIIgm

;

xin

"

va'tL

tYk.

CZZZ)

Mfu

IV-1 IV-2 IV-3

TypeIV;vatY

t-Yft

m""

Cva3

ytrv

Note:1} Nodal forcesin_parenthesesare dependentfttndamentalcomponents parenthesesare independentfundarnentalcomponents,

2) Syrnbol * denotesbalaricingfundamentalcornponents. 3) Symbol o denotesunbalancing fundamental components,

and those not in

Table5The numericalexamples of thefundamental stiffness rnatrixK* of single-bay single-stoiy shear wall

Fundamental type Type I" Type II

Type

III Type IV Neglecting rnonolithic floorslab 4Xt4Yf.Et

[3.oes

(

sym.g';l:jl:i･

4Xft4Yft4Millh=Et3.854O.4S3 -O.197

sym.7.626 O.362O.170 ui vtiheft

{

4Y*m4Mli,1,ll.,,[ _'.g;2:,3g,s

]I,glll

{

4Xft4M"ivlhl-Et[3.:.2i '

gl:l:]I,:lrl

Correction term for

considering ,

monolithic floor slab

(i:til-E,[

O.761sym.:'ill)I:lll 4Xil4Y *I4Mti/h=Etkbn o o sym.ooo uft Vdeihei

{i,X

il,,l

-Et[ kdnsym.

:

]Ih:l'li

Note: kbll'kdn= oo

(5)

-Architectural Institute of Japan

Sincethereis no late:alloadstobalancethe system inType

M

(see

Table4),the effect of, floerslabs,on KM may

be neglected. The lengths of the bearpsdo not change bythe nodal displacements of Type,I.Thereforeitis rpeaninglesstopin-jointing rigid bars at the nodal points ef the sheqr wall along thebeamsfor considering the effect

of floorslabs. Asa result, only theanalytical solutions of Kr are required, where theaxial stiffness of thebeamsis

assurnecl to berigid. Inthis_paperthe_analytical _solutions _ofKr_ofthe_shear .walls_with floor_{slabs are obtained,} _and

after that,thecorrection term to beadded to KI of the shear walls without flooislab isderived.

Numerical examples･ for_K" of thesingle shear wall are _presentedinTable_5.The _proportionof the shear wail is eqtial to that shown inthedesign example inthe AIJStandard

for

Struct'uralCalculation ef ReinforcedConcrete

Structures

(revised

in1982)

_(see

Fig.2and shear wall No.₁inTable _1).Poissonratio isassumed to be1!6.

4. ElasticAnalysis ot Cracked Shear Watl lnfinitetyContinuous in･One or Two Directions

The elastic analyses, bywhich mechanical behaviourof single and 2-continuedcracked shear walls was clarified assuming shear cracked wall _panelsas 45-degreeorthotropic _plates,havebeenreported

by

Tomii et al.3)'5).Inthis

paper,the analysis isapplied toanalyze cracked shear wall infinitelycontinued in one or two directionsand

subjected to external forces_thatare _polarsymmetric with respect to the centers of the wall and theintermediate member

(see

Fig.3).Sincetheassumptions ahd methods intheelastic analysis of this_paperarethesame as those of single and _2-eontinuedshear walls3]-5), except foreqllilibrium equations on horizentaland vertical cross sections of unit shear walls

(see

Fig.4),detaileddiscussionson the analytical methods 'and_thederivationsof equations are omitted. The two typical characteristics aTe as follows.'

'

i

)

The deflectionof intermediatemembers isexpressed byodd functionsand a,constant. The odd functions

'

represent flexural

'and

shearing deformations of the intermediatemembers. The constant represents the rigid-body displacementof intermecliatemembers duetodilatationof theshear cracked wall panel. Therefore,if

the wall _panel isah isotropicelastic _plate,thedeflectienof each intermediatemember isexpressed byodd

functionsonly. _.

'

ii

)

Theaxial deformationof each intermediateraember isalso expressed byodd functionsand aconstant. Inthis

case, theodd functionsrepresent theelongation of each intermediatemember duetothedilatationof thecracked wall _panel and the constant represent therigid-body displacement.The odd functionsare not necessary to express the axial deformationof each intermediatemember whose adjacent wall _panelsare isotropiceiastic plates.

[SHEAR

WALL 11

[SHEAR

WALL 2]

---.-1'

M.-"

･, ---.1/e-g. .-.--f/

f-

g.

--."T-""e.

2Lh

eh z

_Tl"-

T

---1 h

ieh

`lli

i

±

l-T-

-L-:

ojtL

z

Qh7

"･N

-

zg

{a)

Single-bay infinitely

gontinued

shear wall

Note:1)

2)

Fig.3 Infinitelycontinued shear

-106-t

,

t

!

,

-,

t

eh-' z ah"' z eh-T /

,,te1Y,`1,Y1,1,1"fe---v･:-J.M--`e----J/-.---

-

z- .

(b)

Infinitelycontinued shear walls Theunit partsmentioned bythe thick brokenlines are analyzed.

s/

Externalforces_persingle-bay are illustrated.

walls havingshear cracked wallpanels and their external forces

fNt?h

iTTA

hi QftiT-L

(6)

tN

Mb ---L-,>O Mb

e".""-

t.

2 tXY=

-#2

:Z

r

_i,

h

gLh

zL

,Q.eer.t,rff..tXg Mtr

e<--T--

Alt, .LN･

t

x

`

1Cin'-t

glhlrubeLe"iT..,.t+

V Tti .-, N 1 --!-)e

eva

ic+

NCI.

k

-# y x

'"k7txy'

e <--rp-

J,N

ah

t-

T-qcf

÷

Fi"ZebNbi

FTtc,

,

iiis"

1Ih,,1

t ' t bc

"

-:'

liil3.I[be

r etpo

{a)

SHEAR WALL1

_{Q=1,

ru=O)

{b}

SHEAR WALL2

_(Q;1,

N=O)

SHEAR WALL 1'

(Q=O,

N=!} SHEAR WALL 2'

(e=O,

_ru=1)

Fig.4 ThebeundaTystresses and external forcesof the unit paTtef infinitelycontinued shear walls

Intbenumerical exampl.es, the elastic analyses are carried out forthe infinitelycontinued shear wall whose single-bay single-story unit _partshave the same _{preportions･as}those forthesingle shear wali adoPted inSection3.

The elastic constants of the cfacked wall _panels are _given byEqs.

(

1

>-(

4

>5].

Ei=("t+n(Pd+2(:,Cill;l,)Psl]cE "'''''''''"''"''"L''"''"''"''''''-''-''H''''''H''-H･･-･･H･"･･･J･･-･･-･-( 1

)

'

Ei ==[ptc+nlPd+2("".`iPn;.) Ps])cE '''--'-'"''-''''''-'''"'''-'"'''''''''''''"'''''''H''H'･-･-･･-･･-･･-･･･(2

)

G.= "il+2nPS .E .H.,-,.,,",,.,.H,,.,.,.,...H...H..k.,H,,m,.,.,.-,.",.".,..,...."..H...k...,...,,H,,"

(,

3'}

h=

ftt

M={i ,."･,--･--･･･-･･-･････",,-,,.,t,.,,".,･-.,--･`.-,,-,,.,,-".,.,･･--･･--･･･-･･･-･･･-･･-･･---,-Hk.･･･.,.",.,.,(4

>

where

E,,E2:Young's modulus forthe _principaldirection1of elasticity of the wall _paneland that forthe _principal

direction2 of elastieity of the Wall _panel

Gi!:shearmodulus in the rectangular coordinates 1and 2of the wall _panel

n :modular ratio

(n=

_.ElcE=10)

.E :initialYoung's modulus of concrete .E :Young]smodulus of thereinforcement

p. :ratio of diagonalshear reinforcement area to the _gross concrete'area of a diagonalsection

(pd=O}

p.:ratio of horizontalor vertical shegr reinforcement area te the_grossconclete area of avertieal or horizontal section

(p.=O.

O025)

ut : rediuction coefficient of the Young's modulus bf concrete inthedirectionperpendicular tothe cracks inthe_, wall panel

(i.

e. intheprincipal

directien

l of the wall _panel)

pt.:redution coeffcient of the Young's modulus of concrete inthe

airection

of the cracks inthe wall _panel

(i.

e. in the _principaldirection₂of the wall _panel)

(".==1)

gn :reductiofi coefficient of the shear modulus of concrete inthe rectangular coordinates 1and 2of the wall panel

(#i!==pte}

-

107

(7)

Thebeam-to-columnconnections are _given by Eq.(5). ''

E=E,, G=::;E･･･････-･･:･･･

'

are assumed tobea rigid

Fig.5The deformattonsat the bounda[ybetweenwall and frame[SHEAR WALL ]]

,.

t/. - /

Fig.7

ttThe fiheqring

stress _tc.ipthe_.wallan.d.

t4eshgaring

forces_Qband Q.on the cross sectien of

{he

frame

[SHEAR

WALL 1]

'

zone,Theelastic constants of the beamsand columns

'

","."."HL"H-:Li."H"."""hH"H-{s)

Fig.6The deformationsat the frame

_[SHEAR

WALLbound4fy_2]

betweenwalland

Fig,8The she.aring stress lt. in forces_Q,and Q.on the

[SHEAR WALL 2] thecrosswal]and sectlenthe shearing of theframe Nb(tenston} _Nb(tenston> ' t''"tt't

-r.:..･,.iii'･,T;,ti',f:.･:::･{.;･,'i'1'i･f,'.'･`.i-,'S':･,,tL.:'tT.it..t:,'t･t.･.l,.f.,L..:;itV:':i.:f:"t.i./.itt.F;.l:t" ,,.fi,i.",i-l.,",.r･va&'-i#.L･=.i't-1･i-g'"...S"'･･,-";."ti:-1.i.:..,.･.--,--:;r--･f･/":.:::･,i.t,.X･;,･lt･i,,tltk--:#-ig,.c･trt.,l,,,.r.;g,F.i"･i,t. ..

' ' l..L..t= i'l, Scele:ox,wO.ttiiS(

£

tT OO.S1 Nb.Nc- _(q) .-tttt'///L･ii"-I,{:s,:S,Ii･l ut=o ----utgO.1

';'"itli':'rFt･'.e=:'t'":･l:_{lt''tJttt.t.:.ttt:lil,li:j,･.4･1.Ltt-""'i.t..:tt:'::L'1},,.,L.[:ti.:.:.tt./.:.:.1.:.t:1:::i:r'!T,:.'tttttttt:;tttttttl"tt.ttttttttt_Scale:/':ili

....veOttt11'5(ilhTlilj'i mb,NcOttti(e)i'i',,a..L:. "t-o -'--･Ut=O.1'r'i"'.'t::':f:.r.xt:../;.:.;:t"..:."=.:.tl;r.;:/-" tt:3,S".:fr･･''t'is,,tt.1: ---'"Ut=1 _l ny--Ut=1

i

' :LL-ms.,'-:ri..L-,･t:'･t:-!:.,･;;;:.t",ti,zl;k-1:,t･IiJ:'i:it,:..,･t･:tl-r-tt･,:･tlli.e..:.i'I=/Lt.x,.･tii,:,.t.ijL'l:-/,lt/Vglt-.i,l-.tll.-Tg.-':.U.".e-･: :"t':':'iICriNt.;':t':tt.ttt'lz･L!t:t.:::t/":":f.:.:t.t:t-:'l,.Fh..tjtt::-r-t.-=-A-- ' L"iV..i.imah,':..'l;..ime.',-t.:'.l..il'it J/t,--1;.ft;i',,s,,r..:. ve(eompresston)tt1:1/-.-1;;,Flt'.::li.//1;i"'1' ''a::(campression)Nc(tens ±on> 't :f4,'ii"Yl::'l'],･.l･･r･1.'E!t""t"fEs-r l.,.t':':.::::s:':"i"gi;1,:,1..."'t -)-{･./.,::.=!c.L::..:'.:"'t":'i';ttttt/Z,Z･･L,1,･J-･f,･:-,t-･{ %(compression)E',,}i.Ij･tttUx(eompTesston)'Nc(tensten)'.L,;:Ltttttt'.]stLtt".::::F!.,rt;':i:t:.z:i't-tt:ttt;:tt:t ' ].ixT'lt;tn:;:='t'I\'=Jrt"'l-ttrt-xtt･-'･-,'t･.,･,,,;:i',':.･=s'fitl･:.i:]-',･"t-':.l:!i･:,.,.'.',t.1-t't:.'.･i･l-t::}';･91S' ' ...-..=.tf4Sl-'.."ttr:'tt,T-.=,.,..[..,T,7M,t,.-

igt'.':'tL'S,T,f:.ii:tr:pt1-ti･,,';;'l,ltt'･IE:-',4:･;'l･tt,','.-I.i,tf;:.i,:'tt,t':'tL;L:/.t:?･ig･`g,･itL;.,,'t"･i･:

' ' ' l

Fig.9 The normal stresses ay and a, inthe wall and the axial forcesNb and N. on the cross section of.the frame

[SHEAR

WALL 1]

108

-Fig.10ttThe normal stresses ele and eb in the wall and the axial forces'N,.andM'on the cross section ef the

frame _[SHEARWALL2] '

(8)

ArchitecturalInstitute of Japan ' ' ' tt li･ ÷ -t･'/-'･.trt.:L,:--../,:.･1;:,.-.tEl/-r･i'-/,,･",/''

I-N

- vt;o v ,r 1

i:lfiifi.,,tig!--".th･･;:--･tkt-il.g･:"f-,Ls･l.siii

---- ' ut=o ' Ut=O,1Ptn12(fFt)' ---Alv .:ii.'t'-･',tt,i"f.v･:-･i･:-;･:;･'-･::':;''"･ -- Scele:O1--.-N.ix--'LN tv.tt:t-Lt-sttt-t''

i;.{;t'tttl/{'g,2tii,i::;"t-':'is'u'.･--"･utnO.1 -Ut=1") Scale:O12,N, - sTt' 4.L4,;:;･･;l-'S:,iL"Tt,...,. ' :r=,-,-ttttt ' '

Flg.11The deformationsat the boundary betweenwall and frarne[SHEAR WALL 1']

/.1'1/･1･Il..ttit ･

l

'1'Scale:oe,s ･lax,veL-ff-J(tt/) ' _oo.s rtNb,Ne-(N) utino ---Ut=O.1 'T-uttl Illlsls/l'tttt

ilLi

i'1'i:'ttt

f

l

llli

/t

ll

1'f--""--" ---F/.'t -l-'kojZ.o:p-.es-s-ien}-ax"c(COmpressten) ili.'l.'L･',iidl'l['j'--:

Flg.12 The deformations

_et

the boundary_betweenwall and frame

(SHEAR

WALL 2']

tt.,,.."ttttttttt/ttl111i"ii!l-1.l//･I'1tttt't-ttttt..d-...l;.,:.'1,'L't./,:;lt'/･-..;-lt:,i/'s/1":./t:':''1:'1't.tt:tttt li:cale:'Ox.'epttS(lh)i "b,-NcttS{m! ve.e -"'"+-Ut-O.1 -Ut=1 t-t'';i"i'''tttttt'i:tt't::.t.-.l:-tlve-･･'li,/･1:/, ±

;l:.tl,i,･;1

ttilixtvllwl't'1.)･f'i: ttttttttttt

l

t-tttttttt±:':.r,r.a:.---, ....":,.-,,1:,dt.tl....;_{ttt-:l::::1-:-d::.:']1.[I:"r''l:t:1.,}--tby,

---:--9T(E.lotp.-ress-ton)

Mc(cempresston) ':,ttL''ttltlt.'r..t,.:",hiL,l'r.i./:1::1-,:.:'l:f:/e---""''t:-t･::tI11.i-ttt-t-s.J" '

FIg.13 The normal stresses at and ab inthe wall and the Fig.14 The normal stresses tc.and obeinthe wall and the

axial forcesNb and N, on the cross section of the axial fercesN.and _M on thecross section of the

frame[SHEAR WALL 1'] frame[SHEARWALL 2']

'

' 'Here

itisshown

how

thestress

distribution

and

defermation

of the frameand warl _panelchange acc.ording tothe

propagation of thediagonal shear cracks inthe wall _panel, This _prepagation makes the diagonal tension of the

concrete unreliable i.e. _",=O

(see

Figs,5-14),When _",=1, itmeans thatthe'uncracked infilledwall panelbehaves

as an isotropicelastic _plate. ,

'

From thenumerical results

(shown

inFigs.s-10) concerning infinitely continued framed shear wall subjected to

lateralload

_Q,

the followingobservations are obtained. ,

i

}

As"ldecreases,thecompressive normal stresses q± and av on the cross sections at the midspan and midhight of the shear wall remarkably increase

(see

Figs.9and 10),However the shearing stresses _lt,on the sections are

'

hardly affected by _p,

(see

Figs.7 and _8).

ii

)

AxialforcesNband N.of beamsand columns becomelargetensionsand the wall _paneldilatesas thediagonal

shear cracks inthe wall _paneldevelop

{see

Figs.9 and le).This factisalso observed inFigs,5 and ₆which

show the defermationsof cracked wall _panel.

On the other hand,the followingobservations are obtained fromthe numerical results concerning infinitely

,continuedframedshear wall subjected tovertical loadIV.Sinceshearing stresses inwall _paneland columns areso

small thatthey can'be disregarded,the stress diagrams are omitted inthis _paper, .

i)

Beforetheshear cracks occur inthewall _panel,_thestory

drift

ofthe shear wall iszero. When the

diagonal

shear cracks occur inthe wall _panelthe story drift_yieldsinthedirectionopposite tethat of the-lateraltead

(see

Figs.ll and 12). .' ･

'

iD Aspt decreases,the compressive norrnal stres,s alr on thehorizontal cross section at themidhight of thewall

-

109

(9)

M

pa'neldecreases.As a resnlt the compression N} ori the cross section of the column increases

(see

Figs.13 and

'

14). ･･

5. Effect$ot MonolithicFloorSIab on the Mechanieal Behaviour otBeams ctSingle-baylnfinitelyContinued

Cracked Shear'Wall ,

t tt t

When the

diagonal

shear cracks occur inthe wall pan'el,the wall panelbecomesas an anisotropic _plate,behaves the diagonalcompression fieldand dilates.,Themonolithic floorslabs togethet with theboundaryfrarneof the wall panel restrain the dilatationof thecrAcked'wall _panel.Su6sequently, _the role of the monolithic floorslab in restraining _thedilatatien of the cracked shear wall isdiscussedbasedon axial force and deformation of intermediate

beam ofsingle-bay infinitely.continuedfrained sheqr wall.. The floor slab is assumed to be an infinitelysbreading

plate.Thisplateis a homogeneous one with thickness t.and is assumed tohaveplain stress. Thedilatingforces

due

toelongation of thebeams are assumed to be a pair of concentrated loadsP as illustratedinFig.15.Inthisfigure, therelative

displacemepts

dueto_a_pair_{of symmetri.c} loadsP cenfgrm tothe_elongatien _of

bgams

duetodilatatiQn_of thecracked shear wall subjected toconstant'shear forces

Q

and restraint forcesX simultaneousiy. The restraint

forcesX areequal tothe_pairof concentrated loadsP

(see

Figs.15 and 16).The horizontaldisplacementsu at _points

subjected tothe _pairof loads_P

(see

Fig.15>afe _given by Eq,(6).

U=EsPl(Ets)m''-"'''"'H''H''H'':''''''"''"''""'H''H''"''H''''''H'''''''"''"'''''''''''"''"''-''"''H''H-''<6)

where .

'

Jz,=(ii.")(3-u)(i-in2(llCD.)]-(i+") _･

/ .)

.'' - ' On the other

hand

the horiiontal

-

Q

'' ' nodal displacem'ent ''o'f tAe cracked

i/

ho,im:,

:

ii

W. '#":,t' ,i,:

g,;

g

.iy

ll/li,il/f,r

l

'

i

d

/,c.:

i

//.i

p

l,.i

e

l

".ii

m _.due to the elongation of the beam.

, Thesedisplacements,u, and uE are

pressed by the flexibilitymatrix in

terms of shear forces

_Q

and restraint

t t

Q-

, , ,' 'fercesx _as Eq,

(,7).

. Fig'i5 SQi

:.gdie

bi.a.Y.ifi.

[:,

ltefiiY..C,O,nit2:"::bCJI.a.Ctk.eddt.Sh.epa.ri:'gifi

S#yb.je;t.etlt,eS,h.e.a.l,f.O.rCepS

1 l

uU,i

l

=::

St[

s

Sli'.

ll]l

lll

due toeiongation of the,encased intermedia!e beam

. .' '

The restraint f6rcesx alL"1'L(ta7iled /t ttt / /t tt ---- ･-)

e

T$hJ

tSh

h=Qth

",

,l`･Slih

e,hl'

1･Slfh

Q<--･'

,f_./y _{.. ','} _Vt'.

L

z:-

.. Fig.16 Single-bayinfinitelycontinued

of numerical eitamples -110-'' '''' x･-xe

x,

.e' '' ' _' ' .i, 'Ttl

i},

l.tt･IS

'

shear walls and the external forces

bymaking u 'and P inEq.

{

6

)

lentto'u.' and X in'Eq.

(7)

' ' ' tively as Eq,

(

8

),

' /t /t

' X!.{ '

li:;th

_9"-'""--(8)

In,Fig.'17,･the relat'ion 'betweenu,

and _", is _given by'the ratio,of the

thicknes$ of wall, _t, to the thickness

'ofslab, t.,thatis,tlt..The

ordin-i

ate in Fig.17 is 'ufilunl.iots=e,in

whicfi uu1..e,teLe'repTesents the

tion of thebeam of cracked shear wall

(10)

ArchitecturalInstitute of Japan un"i

i?e::

1.0 e.go,u O.7 O.6o.se.4 e.3 e.2O.1 o .---L.>Q

zahthlz

QhTLltt t"ts

ft-1-ll

±Thlcknessefwell ts:Th[cknessofslab -10,vgt,Ps-e.ISI ttsm 3 O.5o.1 f

o 'o.z os o.s o.e 1.o - ve

'

UI

Fig.17 Elongationratios _..lv..o of each interrnediate

' b'eam te'O

Fig.17suggests thatthefloorslab can not _perfectly

restrain the di14tationof the shear cracked wall _panel.

Ontheother hand,the relation betweenaxial foreeN,at theend of the beamof cracked shear wall and _", is

' . e o,2 os o.6 o.s 1.o

shown by_parameter tlt.inFig.18.Itis observed that _- i,t

the axial tension at theend of thebeamisdecreaSedbY _Fig.

Is The ratio of axial force

lvh

to shearing forceQat the the restraint dueto _thefloorslab. Consideringthese _end _of _each _intermediate_beam

facts,itcan bededucedthatthe sheai failurecan hardly

occur at the end of the beam of shear wall with floor.slab.

6.Conclusions

1) Theearthquake lateralloadsdistributedthTough the floorslabs are treated as concentrated loadson thenodal pointsatfourcorners of uncracked shear walls. Inthiscase, iftheaxi'al stiffness of thebeams isassumed tQberigid, the approximate solutions _pertainingtothe nodal displacementsof the shear walls subjected toearthquake lateral

leadscan beobtained with accuracy. Thisresuit isuseful intheformulationof the nodal stiffness matrix of the shear

wall, .

'

2) When the diagonalshear cracks occu: inthewall panel,thewall _paneldilatesand theaxial tensions _yieldsin

theboundaryfrarneof the wall _panel,Howevertheaxial tensionatthe end of beam doesnot become largeifthe

restraint action of monolithic floorslabs isexpected. References

1} Tomii,M., Hiraishi,H.,

"Elastic _Analysis

of Framed Shear Walls byConsideringShearingDeformationof t,heBeamsand

Columnsof Their Bonndary Frames, Part I, ll, M", Trans.of AIJNo.273,Nov. 1978,pp.25-31,No.274,Dec. 1978, pp,75-85, No.275, Jan. 1979,pp.45-53,

2} Tomii,M..Yamakawa, T.,"Relations _between_the_Nodal_Forces

and theNodatDisplacementen theBounda[yFrarnes of

RectangularElastic Framed ShearWalls,PaTtV", TTans,of AIJ, No.241,Mar. 1976, pp.79-89.

3.)Tomii,M., Sueoka,T., Hiraishi,H.,

"Elastic _Analysis

of Framed ShearWalls byAssuming Their InfMed Panet Watls to be 45-Degree Orthotropic PLates, PartI", Trans.of AIJ,No.280,Jun.1979, pp.101-109.

4) Tomii,M.,Hiiaishi,H.,"Elastic _Analysis

of FramedShearWalls byAssuming Their InfilledPaneL WaLls tobe4s-Degree

OrthotropicPlates,Partil",Trans.of AIJ. No.284,Oct.1979,_pp.51-60.

5) Toinli.M,. Esaki,F,, Funamoto,K., "Mechanical _Behavior

of Two-Story or Two-Bay Duplex Framed ShearWalls

Having Shear Cracked Panel Walls",TTans.of JCI,Vol.4,1982, pp.313-324.

-111--"-"--Q

S"rhgh=t

e.3 O.25 O.2 O.IS O.1 glt gfg

im

ieJ-i'-t:Thlcknessefwe]t ts:Thieknessofstab

kS.lo,v.t.ps=o.2s:

3 05 1 (Temsien)o es o, Cemptession)-o. -e.1 C.5 e.1' . NII-Electronic Mbrary

(11)

Arohiteotural エnstitute 　of 　Japan

【論　文1 UDC ：69．022 ：699．841 ：69．025．22 ：550．344 日本建築学会構造系論文報告集第 377号・昭和 62 年 7 月

地

震荷

重

を

受

ける

_耐

震

壁の

力学挙動

に

及ぼ

す

床

スラ

ブ

の

　　　　　　　　

影響

に

関す

る

解析的研究

（梗概

）

正会員正会員正会員富山井川宮政　・英哲．　雄＊＊利　　治＊整　 1．序　鉄筋コンクリートラーメン_構造に多用されている_耐_震壁は，床スラブと一体になってい，るので_，床スラブが地震荷重を受ける耐震壁の力学挙動に及ぼす影響を検討することは，耐震設計上必要な研究課題の一つと考えられる。床スラブは地震時の水平荷重を耐震壁の側ばりに分布荷重として作用させ，かつ _床スラブと一体になった側ばりの軸剛性や曲げ剛性な_どを増大させる。そこで，壁板にせん断ひび割れが発生する前の耐震壁の層閲変形角に及ぼす床スラブの影響に関して，水平荷重の作用形式　　　　　　と側ばりの剛性評価の観点から耐震壁のフ．一リエ _{級数}_解 1）_{を利用} _{して}_{検討}_{する} 。この検討結果をふまえて，マトリックス変位法による有壁ラーメンの弾性解析に用いる耐震壁の節点剛性マトリックス2〕に_，床スラブの_影響を考慮した数値計算例を示す。　一方，せん断ひび割れが壁板に発生すると，異方性化した壁板は圧力場を形成し，床スラブは付帯ラーメンとともにその壁板の広がりを拘束する。このように，せん断ひび割れ後の耐震壁は複雑な力学的挙動をする。せん断ひび割れ後の単独や 2連耐震壁の壁板を_， 4S°_{直交異} 方性弾性板と仮定したこれらの弾性解析が富井らによって発表された3ト5 》_。本論では，富井らが発表した 45°直交異方性弾性板のエアリーの応力関数3 ）を用いて_，せん断ひび割れが生ずる均等無限連続耐震壁の弾性解析を行う。次いで，この耐震壁に床スラブが取り付いた場合の付帯ラーメンの応力や変形に関して床スラブが及ぼす力学的影響を明らかにする。　2．ひび割れが発生していない耐震壁の層間変形角に　，及ぼす床スラブの影響　床スラブが耐震壁の_{層間変形角}に及ぼす影響を明らかにするために_，ひび割れが発生していない単独および無　本研究を日本建築学会大会学術講演梗概集昭和57年10月，および日本建築学会中国・九州支部研究報告第6号昭和59年3 月に発表した。　摩 _九州大学　教授・工博　韓九州大学　助手・工修　tl＊（株〉日建設計・工修　　（昭和 61 年 12月10 目原稿受理｝限連スパン 1層耐震壁の層間変形角を，次の 4’つの場合について比較検討する。　 ’ 1 _．　

i

：耐震壁の側ばりの材軸上に水平力を等分布荷重と　　　して作用させる。　

ii

：耐震壁の_節点に_{水平力}を集中荷重として作用させ　　　る。　

iii

：耐震壁の側ばりの曲げおよびせん断剛性をそのま　　　まにし，側ばりの軸剛性のみを無限大にする。　

iv

：耐震壁の側ばりの_曲げおよびせん断剛性を2お「よ　　　び5倍にし，側ばりの軸剛性を無限大にする。　層間変形角の数値計算は耐震壁の ₅種類の形状について行う。表一2より

ii

の_場合における柱の _{部材角}は，耐震壁の形状によっては iの場合のそれよりもかなり大きくなる。したがって_，柱の実際の部材角はiiの場合の解を使うと過大評価される。iiの_{場合}における柱の部材角とスパン中央位置における層間変形角の相異は，耐震壁のスパンが大きく，側ばりの軸剛性が小さくなるほど顕著になる。しかしながら， 1の場合の層間変形角は耐震壁全体にわたってほとんど一様であり，かつこれらの値は

1ii

の場合の層間変形角1にほぼ等しい。．床スラブによる側ばりの曲げおよびせん_{断剛性}の_増_大が耐震壁の層間変形角に及ぼす影響をみるために， iiiの場合の層間変形角に対する

iv

の場合の層間変形角の比を表一3に示す。表一3によれば_， _側ばりの_曲げおよびせん断剛性の増大が，耐震壁の層間変形角に及ぼす影響は一_般_に_小_さ_い_。_{したが}_っ _て_，_{壁板}_{にせ}_ん_断_ひ_び割_{れが}_発生していない耐震壁に床スラブから分布して作用する地震時の_{水平荷重}を，節点水平荷重として取り扱う場合には_，側ばりの_軸剛性を無限大と仮定することができることを示している。このときは，耐震壁の節点外力と節点変位の関係である節点剛性マトリックスを定式化する上で有用である。　　　　「　　　 ’ 　3．床スラブの影響を考慮した単独耐震壁の基本節点　　剛性マ_．トリ_、ック_スに関する数値計算例　2項での検討結果，耐震壁の基本節点剛性マトリックス K＊を定式化する場合には，側ばりの軸剛性を無限大と仮定することができる。各基本型ごとの KI が解析解一一

₁₁₂

一 N工工一Eleotronio _Library

(12)

Arohiteotural エnstitute 　of 　Japan

として _求められると_，一般的な耐震壁の節点剛性マ _トリックス K が，変換マトリックスを利用して容易に誘導されるm。　壁板の_中心を通る縦軸に_関し，対称な基本成分からなる且型の Kt およびW 型の K_髭は，剛棒を側ばり両端の節点間にかけわたすことによって_，剛床が取り付いた場合の近似解が求められる。

m

型には力のつり合いを満足する水平外力が存在しないので， Kfiに対しては床スラブの影響を考慮しない。耐震壁が

1

型の挙動をする場合には，側ばりは材長変化を起こさないので_，剛棒を付与することによっても，床スラブが取り付く場合の影響を考慮することができない。したがって，側ばりの軸剛性を無限大と仮定した

1

型の解析解 K評を求める。_本論では，その解を求めた後，すでに発表している「床スラブが取り付かない場合の毋」に累加すべ _{き「}_床スラブが取り付いた場合の補正項」を次に求めた。_単独耐震壁に関する K ＊の数値計算例を表一5に示す。　4．せん断ひび割れが壁板に発生した均等無限連続耐　　震壁の弾性解析　壁板にせん断ひび割れが発生した後の _{単独}および2連耐震壁について，それぞれ壁板を 45° 直交異方性弾性板と仮定した解析方法3 」−5 ）_と_，それらの力学挙動がすでに文献 4 ＞， 5）で明らかにされている。これらの解析方法を応用発展させて，一方向および二方向均等無限連続耐震壁が，壁板および中間部材の各中心に関して_，それぞれ点対称な外力を受ける場合を解析する。本解析の特徴として次の 2点があげられるe 　

D

中間部材のたわみは，奇関数と定数で表される。奇関数は中間部材の _曲げ変形とせん断変形を表す。一方，定数はせん断ひび割れにともなう壁板のはらみだしによる中間部材の剛体変位を意味する。したがって，壁板が等方性弾性板であれば均等無限連続耐震壁の_{中間部材}のたわみは，奇関数のみで_表される。　iD　中間部材の _{軸変位}も i）と同様に奇関数と定数で表される_。 _奇_関_数はせん断ひび割れにともなう壁板の面積膨張による伸び変位を表し，定数は剛体変位を表す。壁板が等方性弾性板であれば奇関数を必要としない。　付帯ラーメンおよび壁板の _{各応力分布}や各変形がせん断ひび割れの進展とともに，コンクリートの斜め引張応力が低下し，すなわち，Pttが零に近づくにつれて，どのように変化するかを検討する。_Pt，＝1は壁板が等方性弾性板，すなわちせん断ひび割れが生じていない耐震壁を意味する。水平外力 Q を受ける均等無限連続耐震壁の数値計算結果から，次の諸点が明らかになった。　i）μ、が低下するにつれて，階高中央における水平断面上の圧縮応力度 σx および ay は顕著に増大するが，同じ断面．．ヒのせん断応力度 τxv はμεにほとんど影響されないe 　の付帯ラーメンの_軸力 N，，N ，は異方性化が進展するにつれて，大きな引張応力となっている。これは_，せん断ひび割れの進展にともない壁板が膨張していることを示している。このことは，図一5，6の壁板の変形図からも明らかである。　一方，鉛直外力 N を受ける均等無限連続耐震壁の数値計算結果から，次の諸点が明らかになった。　

D

壁板にせん断ひび割れが発生する前は耐震壁の_層間変位が零であったが，壁板にせん断ひび割れが発生すると，層間変位が水平外力め向きと反対方向に生じる。　ii）μt の低下にともなって壁板の中央水平断面上の圧縮応力度ay は小さくなる。その_結_果_，_柱の軸圧縮応力 Ncが大きくなる。　5．せん断ひび割れが壁板に発生した 1スパン無限連　　層耐震壁におけるはりの力学挙動に及ぼす床スラ　　ブの影響　せん断ひび割れが壁板に発生すると，壁板が異方性化し圧力場を形成して膨張しようとする。壁板の面積膨張に対して，耐震壁の付帯ラーメンとともに床スラブの拘束作用が生じることになる_。そこで，床スラブの_{拘束作} 用を1スパン無限連層耐震壁の_中_{間ば}りの伸び量と軸力の変化に注目して検討する。床スラブは厚さtsを有する無限に広がる一_様な平面板として取り扱う。はりの伸びによる膨張力を図一15に示したような 1組の集中荷重 P と考える。P による相対変位を，せん断ひび割れが生じた耐震壁に一定なせん断力

Q

と拘束力X が，作用することによって生じたはりの伸び量を適合させる。　図一17に _μ_tとUn の _{関係}を，壁板の厚さ tと床スラブの_厚さ tsの比 t／tsのパラメーターで示す。図一17 より床スラブは壁板のせん断ひび割れにともなう耐震壁の面積膨張を，完全に拘束できないことを示している。一_方，図一18に μtと耐震壁の _{中間}ばり材端の軸力 N，の関係をt／tsのパラメーターで示す。図一18によれば，はり材端の_引張応力が床スラブの拘束作用によりかなり減少していることがわかる。このことは，床スラブがはりに取り付いた耐震壁では，はりのせん断破壊が起こりにくいことを示唆している　6．結　論　 1）壁板にせん断ひび割れが発生していない耐震壁に床スラブから分布して作用する地震時の_{水平荷重}を，節点水平荷重として取り扱う場合は，側ばりの軸剛性を無限大と仮定すると，節点変位に関し，良い近似解が得られる。このことは，耐震壁の節点外力と節点変位の関係である節点剛性マ _トリックスを定式化する上で有用である。　2）せん断ひび割れが壁板に発生すると壁板が膨張し，付帯ラーメンに_引張応力が生じるが，床スラブの拘束作用が期待できる場合には，はり材端の引張応力はそれほど大きくならない。一

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