水平荷重をうけた多層鉄筋コンクリートフレーム柱の鉛直耐力(梗概)

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Vl

Jo-inat

of

StT"ctural

and

Construction

Engineering

Hptgees\ftnemekasYNtsfi

UDC:624.o7s.2.ol2:624.o42.7:62o.1:624.o2

{TranSaCtionsofAIJ)No.360,

February,

lgs6

ng

36o

e･wan

61

ijzA

THE

VERTICAL

LOAD

CARRYING

CAPACITY

OF

THE

COLUMNS

OF

MULTI-STORY

REINFORCED

'

CONCRETE

FRAMES

WITH

THE

EXPERIENCE

OF

HORIZONTAL

LOADING

'

by

TAKAYUKI

SHIMAZU'

and

S.M.

PARVEZ

MOHIT*',

Members

of

A.

I.

J.

'

1.

Introduction

It

has

been

recognized

from

vario'us ehrthquake experiences as we31 as

the

results of stiuctuial analysis

that

weak-beam, strong-column

type

Qf

frames

is

suitable

for

aseismic

design

of

buildings

because

in

this

type

of

frames,

total

deformation

at

the

top

of

buildings

induced

by

earthquakes can

be

distributed

to

aconsiderable extent uniformly

along

the

height

of

buildings.

Owing

to

this

desirable

trend

of

uniform

deformation,

mqximum

interstory

dTift

of

this

type

of

frames

will

be

much

smaller

even

for

major

earthquakes

than

_that

of

strong-beam,

weak-column

_type

of

frames,

Furthermore

residual

interst.ory

drift

after earthquakes may also

be

very small,

However,

there

wM

be

much

_possibility

of

danger

in

the

continuation of

the

use of

the

reinforced concrete

buildings

consisting o{ such

type

of

frames

after

earthquakes

because

the

beams

of

frames

have

already much softened regions at

the

ends

inspite

of

little

evidence of

damage

caused

by

earthquakes

in

appearance,

due

to

the

restoring

force

characteristics of reinforced

'concrete.

Questions

arise on

the

structural safety of such

type

of

frames

against vertical

loads

after earthquakes.

How

is

the

column stability?

How

much

is

the

vertical

load

carrying

capacity

of

beams?

And

what about

the

tirne-dependent

deformation

of

beams?

etc･･

Howevei,

the

philosophy

of structural

design

of weak-beam, strong-column

frames

has

not

_yet

been

established

until very recently.

So

there

seems

to

be

no

liteiatuie

on

the

studies

done

to

answer

these

_questions

up

to

date,

'

though

the

studiesi)-3)･has

been

reported on

the

stabi]ity of single story

frame

coiumns.

Experimental

stucly

has

been

initiated

in

our

laboratory

with an aim

to

answer

these

_questions

and

to

_propose

appropriate

design

methods of weak-beam, strong-column

type

of

frames,

taking

into

account

the

vertical resistance

'

after'earthquakes.

This

paper

presents

the

one of a series of studies on

the

vertical

resistance

of reinforced concrete

frames

with

the

s

experience of

hoTizontal

Ioading,

In

thi$

_paper

focus

is

_placecl

on

the

_problem

of column stability.

The

philosophy

of

limit

design

method

is

also

intToduced

into

structural

design

of

frames

to

replace

the

currently

-used

elastic

design

methods.

In

the

next

Section,

the

equations are

derived

for

estimating

the

vertical

load

cauying capacity of

the

columns

of

multi-story reinforced concrete

frames

with

the

experience

of

high

level

of

horizontal

loading

upto

_{pest-yielding}

range.

Following

this

derivation

of

the

equations,

the

experimental works are

_presented

to

verify

the

validity of

them.

'

The

abstract

of

this

paper

was already reported

in

Ref.

_4.

2.

Derivation

of

the

Equations

It

seemS

that

it

i･nvolves

much

difficulty

to

find

out

the

solution

for

the

stability

_problem

of

the

columns

of

'

_Professor,

University

_of

Hiroshima,

Dr.

_of

Eng.

*i

_Graduate

_Student,

_UniveTslty

_o{

_Hiroshlrna,

Mr.

_of

Eng,

Manuscript

received

March

11, 19S5

-119-Architectural Institute of Japan

ArchitecturalInstitute of Japan

multi.story

frame

with

the

experience of

horizontal

loading

because

the

behaviour

of reinforced concrete

frame

become

already much complicated when subjected

to

horizontal

load

upto

the

ine]astic

range.

However,

in

this

_paper

the

energy

method

on

elastic

stability5)

is

usect

for

the

solution of

this

problem

by

paying

attention

to

the

restoring

force

characteristics

of

reinforced concrete members, which

can

approximately

be

assumed

to

be

of origin oriented

type

with

degrading

stiffness.

It

betcome

easier

to

use

this

method

for

weak-beam,

strong-column

fiame

already subjected

to

ho[izontal

load

upto

_{post-yie}ding}

range as explained

below.

The

expression

for

the

strain

energy

of

bending

for

the

system

as

shown

in

Fig.1

becomes

u-gJg"(Eiin)･[

di.y,t

]'･d.---･･-･-･---･･--･-･･･---････-･･----･--･･-･･---･･(

i

)

The

decrease

in

the

potential

energy of

the

vertical

load

due

to

the

lowering

of

the

_point

of apptication of

the

load

and

the

work

done

by

the

bending

moment at each

bearn

end

becomes

u-e.(":

]q,･[

g:l

]'･

dx +S

_£

.

M,･a･･･----･--･-･-･-･--････--･--･-･････-････-･･･-･･･････-･-･･-(

2

)

The

cTitical va!ue of

the

compTessive

force

is

obtained

from

the

cofldition

U=U,---･･---･-･---･･･-･･-･-･---･----･･---･---･･---･---･･･--･･---(3)

Beams

of

the

frames

may

be

assumecl

to

be

subjected

to

anti-syrnmetrical

bending

at

the

starting of

instability

under vertical

load,

With

the

axes

taken

as

indicated

in

Fig.1,

the

bending

moment at any cross section

becomes

n n

Z]

q,.(at-y)-Z

M.,

and

the

differential

equation

of

the

deflection

curve

is

t i

Eu･dd.2g=#.

_{q,(s,-y)-:;.]M,･-･･-･･-･-･---･-･-･---･･---･-･-･･･--･---･(4)}

The

following

assumptions were made

for

the

weak-beam, strong-column

frame

with

the

experience of

horizontal

loading

upto

_{post-yielding}

range.

1)

All

the

beam

ends

_yield

and so

the

rotational stiffness

is

Kei=Myi1anaxi'H-"m'm''''''m'm'-'-''''mH-'-'-"'-""'"'''''''-'''"''"''"''m-"--'m'''m---(5)

in

which

M.,

is

the

_yield

strength

and

can

be

calculated

with

_O.

_9.A,.

a.-d

accorcling

to

R,

C,

Cocle

of

AIJ6)

and

da..,

is

the

value ef

the

maximum rotation angle of each

beam,

uncler

horizontal

load

as shown

in

Fig.

2

(a)

and

(b).

2)

The

cross section of column

is

assumed uniform along

the

overall

height.

The

equivalent

flexural

rigidity

{Ef).,

of columns

is

evaluated

based

on

the

restoring

force

characteristics of columns as will

be

explained

in

the

Section

5.

The

next step

is

how

to

assume

the

deflection

cuTve of

the

column when under

the

action of acompressive

load.

The

_general

expression

for

the

deflection

curve of a column with

fixed

end can

be

represented

by

the

series

y= t!,[1-cos 2nli

]+zs,[1-cos

32

EHX

]+･-･--+A,[1-cos

(2

h2-Hl)nX

]..---.,.-.-,.-"-..-"m..(

6

)

However,

it

usually

involves

much

difficulty

to

_get

the

solution

by

usiRg

this

series

for

the

highly

statical

indeterminate

st[ucture.

In

this

_paper

the

following

deflected

shape was assumed,

taking

into

account

that

horizontal

loading

made

first-orcler

mode

for

the

fiames.

Fig.1

Frame

mode{

foT

veitical

load

anaiysis

-120-Fe'--

6H _uH i 1 i i i i 1

-:

erruxi

l:

l

-

emax1

:

d

{a)Fig.2

Assumed

ma,in _p,ttttt 'STitttttt

.KeT''''

i.tl't/'te1/lemaxi

fo)

NII-Electronic Library Service

y..a.[1

.]"....-,."..".-...--.m..,.-.h.--.-h-...m..-.---.,..,･-･･･-.."･-･-･-･･･-･･---(7)

The

solution can

be

obtained

by

selecting

the

value

of

m so as

to

make

the

critical

load

a minimum.

In

general

vertical

loads

are

transmitted

from

beams

to

coLumns at

each

floor.

The

following

equation can

be

dbtained

for

vertical

load

carrying capacity of

n.q,.

by

assuming

the

deflected

shape of

Eq,

7

when equally

distributed

IDads

are applied at each

flooT

for

uniform story

height

frame.

_.

a'q2r+b'qcr+c=O-'HrHH-'''m'H'-'"HH-'''"-""-''h''-''--'-'H''"'"''-''"-""-HHHH"ll"'(8) where

,

t

t

t

t

a-,z"=,[t(te.,[t]"l2.2.:,[i]th..i.,[[t]m+i-[Jii]m+il 2M+1

+2mi+'

_{i([t]!M'i-[i;li}

]

ll･-･･-･･-･-･･･--･-･･--･-･-･-･･･--･---･--(g)

t

t

b--2.M

･tn.

[ltt,

[t)"-.i.,

([i]""-[jii

]m+i)}

'

'

_il.li,

_Ker

[S]M'i]-(2M.2(E{))eH'

,･

S.,

[t]'"Ti

･-･･---･････-･-･-..----.m

mm-..--",",ao)

c=

Z:i

･[,Z"..,

:I

£

.,

K.,･[-i}]"-']t-(ES}ee

te.,

K,,.[

;i]'"T2]

.-.-.-.m."",-.-"..,-".""".-.-.,-"oo

The

vertical

load

carrying capacity

VL.

can also

be

obtained as

follows

when

concentrated

loads

instead

of

distributecl

loads

are applied at

the･

top

for

uniform

story

height

frame

as

explained

in

the

next

Section.

a'W:r+b'PVIr+c=O'"H-H''h"''"H"'-"'""'''-''--H"'""''-''"'H'H"""mH':v'H'H"H'H'HH'H'H-(12}

'

where

'

'

'

2

1

.".H""H.-".,-..""-･-･-･-･----･･-･"'--H'"''--"H"HH'"'H'"-H-"'Hm"(13)

+

m+1

2m+1

b==

'2HM

',i.,'[il.ll,

Ke:'

[t]""(-ll'

Tn}1

['ll]"'i+

ml+

1

[

j'

-n

l

]""

1]-

(2"mt!(EI

IPi;,

･･-･･･--･--.･･･a4)

c='.MHZ2't?.,(S.,Ke"['S]re']]t-MZ{HE,l)eq･Si.]K,,･[-A/]t""----.--.-...m....m-"h.-.-.-m..h"...os).

On

the

other

hand,

the

vertical

lpad

carrying

capacity

for

the

frames

subjected

to

veTtical

load

only can

be

calcultited using

the

following

cummulative

strength

equation

on

axial compression strength7L

_.

wrt=Ae'Fc+As'ay''''''"'''''"w-"''-'-''-"'"''-'-'-"'"'''"''"'-''''''''-'"'"'-"---'"-'"-'(16)

3.

Test

Programme

The

specimens used

_,for

this

study were _{single-bay, multi-story reinforced concrete}

lrames.

A

typical

_m,odel

'

L..

#.: Aema

...um,

l,t!t

r-.

w

Fig.3

A

s'ingle･bay, mu!ti-story

reinforced concfete

bllilding

plan

with the

location

of the

interloi

fTqrne,

used as the

testspecimen

7

]2G

toiizenua1

1pading

Verc±cal toading

'

'

Fig.4

Assumptions

for

the

_point

of

application of

loadings

Fig,5

u

:

um:

!l

pm

The

specimen with reinforcement

(C63-42H}

Architectural Institute of Japan

ArchitecturalInstitute of Japan

specimen

(C

63-42

H)

was sllpposed

to

represent asix-sto[ied

frame,

one of

the

single-bay

frames

in

rows as shown

in

Fig,

3.

However,

the

nurnber of stories of

the

specimens was reduced

to

four

for

the

simplicity oi

the

application of

loads.

Fig.

4

explains why

this

reduction of

the

numbers of

the

steries was made,

The

effects of

this

reduction,

that

are

the

relations

between

the

models

and

the

prototypes

will

be

discussed

quantitatively

later

in

this

Section.

Fig.

s

shows

the

overall

dimension

and

the

reinforcement of

specimen

(C

63-42H).

The

calculated

maximum

shear

coefficient at collapse mechanism under

horizontal

Ioading

was

O.

07

for

this

weak-beam, strong-column specimen

with axial

force

level

of

O.2

F.bD

for

columns.

The

value of

O,

07

is

increased

to

O.

09

when

the

upper

two

more

bearns

are consiclered.

Usually

shear walls are allotted

to

resist about

two-third

of

total

shear.

Thus

the

building

including

this

frame

can

be

assumed

to

resist

}ateral

force

of about

30

%

of

_gravity

load,

Stirrups

and

hoops

were

_providecl

to

ensure sufficient

ducti]ity,

according

to

the

R.

C.

Code

of

AIJ.

The

number of specimens

totaled

eight.

Out

of which, six were

four-storied

and

the

other

two

were

two-storied

frames.

Table

1

shows

the

dimensions

and

the

cross section

_properties

of all

the

eight specimens.

The

first

two

specimens were

designed

basically

in

accordance with

the

currently used methods

in

such a way

that

the

strength

of

beams

was

allotted smaller

foT

upper stories and

_greater

for

lower

stoiies.

Particularly

the

difference

of strength

between

the

upper and

lower

stories

forthis

fTame

was made

_greater

than

that

of

_prototype

to

see

the

effects

distinctly

as shown

in

Table2.

0n

the

other

hand

the

other six specimens were

designed

according

to

the

iimit

design

method

in

the

way

that

the

strength of

bearns

was arranged uniforrnly along

the

height

with

total

value of stTength

being

the

same as

in

the

above

first

type

of

frames

for

four-storied

ones.

Table

3

shows

the

calculated

results

of

vertical

load

carrying

capacity

obtained

by

using

Eq.

<

8

)

and

Eq.

(12)

for

both

the

models

(specimens)

and

the

prototypes

vith

the

changing value of each variable.

The

effects

of

the

equivalent

flexurai

rigidity

(EI)..=a..El

of columns,

beam

end rotational stiffness

Kei

and

the

deflected

shape

(for

_6F..,)

are shown with

two

values

for

each

case.

Tablel

Cross

section

_properties

of members of specimens

Table2

Comparison

of

beam

strength ratios

between

prototypes

and models

Specimer1--e.H(tm)C63-42HC6]-41Ue6s-q-C83-41Hcs-C63-21HC-c6]-a2v170017oe17oeuoegoeHoops&stirr-s

2(mm)BOO800soe820soe daIum tHlltIi.[o-3--op

_{;Fst,-+I,--iultce-s-#e}

7L-,F

,[I]-r4t3--di

-t7ei.14-Ioeco-s-4dil"elermclc

se

±

Fle-11i･Dii,

NA

Beem6tsHk8at

32hio-11i[]rfi･

//i"

i

"

l･T2-S,2-s

e-e-,ilD:,-1-e10mncte

GigiF!!itll

tiFmkT.2-3.1di

I,git'-11i,

mptmbeeoHeetpt en"mno=

Based cricurrently ttsed elastie degign method

O.34-O.66ooO.99

O.34oO.99

Based

on

lim

±t de$ign nethod

U

.66

g

,.

Table3

Cornparison

of calculated verticat

having

the

experience of reversed

deTensile _{reinforcimt ratio}

load

carrying capacities

between

models and

horizontal

loadings

(x}prototypes

a.=lt3 a,i=2/3

Y-6.(:]i

y=6･ft yt6,cg)' y=6･}

KefellrrbliLvexiKei"zaiKeie-!.talLiKei=eZlrtZS/iKeie-!mxt!LiKet=deiKei=eMllva),iKsi"i}lli-C63-42H10,8118,22IS.4727,6S13,S221.6318.IO30.92 Ce3-41HIS.8926.S324.4345.9521.3137.7727.194S.86 utAnNkUC6S-41H18.8334.7724.3845.8921,2237.6727.0948,76

SECS3-41H22.0838.712S.1049,S427.e644.1733.9656,19

C63-21H13.4919.2216.3024.3819.8426.9723.8132.59

063-42Hla,6o33.10'

l9.1933.2122.I836.oe22.533B,38

C63-blH20.3S33.9123.6942.8q24.114e.7127.01'47,39

mzaAeyo.Hage-C65.41H20.2S33.6S23.6242,7423.9940.S326.9047.2a C8]-41H24.1442.2527.9748,6830.93SO.!933,52SS:94

C63-21H12.9D19.7714,9023,32IS.032S.812e.28r29.79

Ceuuxi

is

at themxinnm total

height-drift

of

2Z)

{mit:

tan)

--122-NII-Electronic Library Service

'

Table4

Me'chanical

properties

'of

materiats

ReinfereemEnt

a.oip].2eZ.3di1.0diCancretet

o4670.0as6o.e336S,O31se.oFc=3Sl,6

_{'Es(KID)L9L92,12,1Ee=2.4xlOS}

UU5990.067ZO.06710.0SS20.0-"J

-Average _{of eight specirnensLmitt kgtan2}

Table5

Programme

foi

herizentaMoading

CycleNe,imtotalhe ±

_ght-drift

q,IH(percgnt)

1,2 +O.2S

3,4

io.se S,6 tl.00

7,8

t2,ao

x

.

tt

illk41

t'di

ttttttttttttttt.'tt'

tttttt

lt.stgr ±

ttttttttttttttD.T.

'

'tttt

ttde'tttt

t''

'

.

tttt/ttttttttt

Ian:alh:aqing

tt

tt''tttt'1tnsism

"t1////: Ja;.c. D.T.DlsppmtT=aiisinoers

w.s,g. wnxe straul

_gege

L.c. Inad

Cell'

Fig.6

Setup

fo[

loading

and measuring

instrurnents

According

to

the

Eql

{

8,)

and

Eq.

(12)I

the

vertical

load

carrying

capacity

of

the

models

can

be

_estimated

to

be

nearly

the

same as

that

of

_prototypes

except

the

C

63-42

H

type

frame

whose stTength

distributions

were considerabLy

changed

between

both

the

ones.

It

can also

be

seen

from

Table

3

that

the

effects of

Ke[

value on

the

n･q,. or

WL.

Is

much

greater

than

those

of

the

a. values,

The

effects of

the

assumed

deflected

shape are not so

_great.

The

mechanical

_properties

of

the

materials used

for

the

construction of

these

specimens are

_given

in

Table

4.

Five

specimens with

its

numbers ending with

the

alphabet "H"

<e,

g,

C

63-41

H>

were

tested

under

displacement

controlled reversed

horizontal

load,

during

which

there

was a

_previously

applied constant vertical

load

(lVill(2

E,bD)l

=o.

2)

on

the

columns.

At

the

end of

last

cycle of reversed

horizontal

loading,

the

vertical

load

on

the

columns was

increased

monotonically until

the

frame

failed.

The

other

three

specimens

with

the

alphabet 'V'

_ih

_its

numbers were

tested

only under monotonically

increasing

vertical

load

upto

failure.

Both

the

positive

and negative

horizonta!

Ioading

were applied on

the

outer

surface

of

the

columns

at

the

uppetmost

beam

center

level,

while

_the

_two

_point

vertical

load

was applied on

the

top

of

the

two

columns,

Fig.

6

shows

the

setup

for

thb

test

as well as

the

loading

and

measuring

apparatus.

The

ve'rti'cal

loading

apparatus was

designed

in

such a way

that

the

top

of

the

specimens could move

freely

in

its

_plane,

In

aseismic

design

of

buildings,

story

deflection

is

usually

limited.

The

_permissibte

interstory

drift

is

_prescribed

to

be

O.5

_percent

for

moderate earthquakes

in

the

Building

Standard

Laws

of

Japan8).

However,

it

has

been

recognized

that

several

times

more

interstory

drift

than

_O.

₅

_percent

should

be

allowed

for

major earthquakes.

In

this

programme,

an ultimate

total

height

drift

_(6H!H)

inducecl

by

inajor

earthquakes was assumed

to

be

2.o

peicent.

Table5

shows

the

_programme

for

the

reversed

horizontal

loading.

4.

Test

Results

The

load-deflection

curves

for

reversed

horizontal

loading

test

of

all

the

specimens are nearly

similar.

Upto

the

total

height-drift

of

_O,

₅

_percent,

the

hysteresis

loops

are spindle shaped.

As

the

drift

increases,

the

shape

_gradually

changes

into

inverted

S

shaped

ones,

while

the

equivalent stiffness

decreases.

At

the

end of

the

last

cycle'of

horizontal

loading,

the

residual

total

height-drift

was about

_1.0

_percent

for

all

the

specimens.

Fig.7

shows

the

P-a.

curves

for

two

typical

specimens.

W-aH

curves

for

four

different

specimens are

_given

in

Fig,8.

0ne

can readiLy compare

the

top

displacement,

ak,

against

the

increasing

vertical

load,

W,

for

the

same

type

of specimen with or without

the

_{experience of}

_{eversed

horizontal

Ioading.

The

crack

_patterns

of

four

different

specimens under

different

loading

are

_given

in

Fig,

9,

The

crack

_patterns

for

horizontal

_loading

of

_all

the

four-storied

_Epecimens

are nearly similar,

i,

e. ,

the

cracks appearecl mainly on

the

beams

-123-Architectural Institute of Japan

ArchitecturalInstitute ofJapan

Fig.7Hysteresis

Ioops

under

-so

-60Fig.8-40

-20

O

20

Load-deflection40

osrm)curves

herizontal

Loading

-6o

-4a

T2o

o 2o 4o under veTtlcat

loading

･

l

- e At:theloading C63-42Hend of

horizontal

'

-At

theleariing

C83-41Hend

of

botuontal

Fig.9

Craek

･

,

C63-42H C63-42V At theend af vertieal loading

･･

･

,

CS3-41H

C83-41V

At the end o[ vertical leading

patternsunder

differentloadings

t de

-7.-5-P,1

t,/1

ttl

_tt/t/

tt/t/

_ttt1/

tttttt[

_t//

'"t'

tt//ttttlt/',,',,1!i,･:11'.,vllt/t:ttttttt/1ttt/t//t///:

C63-42HVI･Vll:'L':'

13SZ-1

:ittttttt :///tt/ttt 1:ttttt

///t

l･:',S,/tttiS:r'/t///tii//t/tttttitt/tt!ittttttt//t//tt/t//th'///

'-P-#-,1//t tt//

't/1

txt//

ttt/

_t///

_.,/t/,t/11L

"ttlttt/t/t///tt//ttl･,･"lt/I//t/t/t:lt//1t///t/11///tlkJ/,1'

}3S?

_///t /1/tt 1/tt i://tt

lfis1

/ltt

//J

t:/t

dt/ttt///t/ittl/ttt/tttt/t/ttii;'"11tt:l///t/t//tt',-･CS]-4M

i/ttt"

'' 1' k,tatthepeakefcyaletw).1'i

4e302010O102030403o2eloO10203040Cum)

Fig10

Deflected

shapeunderhori2ontal

leacling

with avery

few

cracks at

the

bottom

of

the

columns, while

the

two-storied

one

had

considerable

cracks

at

the

bottom

of

the

columns as well as on

the

beams,

The

specimens with

the

experience of reversed

horizontal

loading,

'failed

finally

against

the

stability under monotonically

increasing

vertical

load,

while

those

without

the

experience of

horizontal

loading,

failed

under

direct

compression.

"

_All

_the

_five

_{specirnens show}

_the

_similar

_deflected

_{shape under reversed}

_horizontal

_load.

_Each

_of

_the

_four-storied

frames

failed

finally

on

the

3rd

floor

bearn-column

_joint

under

its

ultimate

level

of vertical

load,

while

the

-124---NII-Electronic Library Service

Tabie6

Experirnentedresults

of all

the

specimens FarHai ±zantaLLx)ading ForVerttcallaadi]g

SpectmenNo.

Pu(kg)

6H.(mn)

Post.Nega.Aver.Pos

±

.Nega.'peqcrm)va(tdn)6va(mm)k6H(nm)

C6]-42H.302.S2SO,S276.33D.e]4.0.25.011.0S-79.S-34.6 c6]-q2v''

-J--32.27e.sLS

c6]-qA32S.O275.0300.03],3]o.o-13.32D.20･"41.2-4S,O C6S-41H]oo.o2SO.O290.032.02S.O-20.0IB,47-3S.7-3e.8

C83-41H450,O162,･S406.]30.DI10.D.17.127.36.12.0-SS,8

CS3-41V-J

---.16.80S.96.7

C6]-21H･q32.sloo.o266.118.0IS.O-6.S20.8S'i-17.9'

-17.9

C63-21V'-

.r･..26.5S[LOL3

Table7Correlationofcalculatedandexperimented

values of

horizontal

load

and

the

calculated

values of the coefflcients, a. and

fi

B

p.drg) SPecimm1eq Cal.DCP･E]cp./Cal.

C63-42HO,40O.93244.0276.3L13

C63-42HO.331.0D2sq.o'

300.0].06

C6S-41HO,5]O.80304.0290.0O,9S

C83-41HO,34O.99336.0406,31,21

C63-21HO,331.00334.0366.31.IO

two-storied

one

failed

at

the

2

nd

floor

level.

Fig.

10

shows

the

deflected

shape of

typical

specimens under reversed

horizontal

loading,

The

overall results of all

the

specimens are

listed

in

Table6,

5.

Discussions

i)

Maximum

strength under

horizontal

loading:

Table

7

shows

the

comparison

between

test

results and calculatecl values

for

the

maximum strength of

five

specimens under

horizontal

loads.

This

ta.ble

also shows

the

values of

the

calculated equivalent

flexural

rigidity,ratio

aH

for

columns of specimens.

These

calculated values were obtained

by

assuming

the

moment-curvatu,re

diagrarn

for

cross section of a column as shown

in

Fig.

11

and using

the

followlng

two

equations,

by

neglecting

the

variation of

the

axial

forces

incluced

in

columns

during

horizontal

loading.

n

Z

M.,+ev.-

VeG&,.

Pu=i='

H

'HH"H"""HH"HHH''H''H''"''HH"H''H''H'H"HH"'H"HHHH"'i''HHHHHH"H<17)

where

BM.

is

the

resisting moment at

the

bottom

of

the

lowest

column and

llib.

aH.

is

the

effect

due

to

eccentric vertical

'

loading.

o,

:"E"i-te.,"st,"`･[gt+H-H,]

.

aH=a..=

'

6u.

HH"H''HHH"HHHH''H''m"HH"H'H''HHHH''H"H'H'HH-HH<i8)

li

where

a.

is

the

elastic

deflection

of

the

cantilever when

_yield

moment of

beam

is

applied at each

floor

level

in

addition

to

the

horizontal

load.

In

using

these

equations

the

relationship of a+fi =!413 _{was used as shown}

in

Fig.

I1

by

_assuming

that

M,IM.=113

and

_¢

_,ldi.=1!9.

The

rnoment･curvature

diagram

in

Fig.11

were

determined

mainly

for

the

simplicity

of

the

relationship

between

aH and

fl

but

the

curve

in

Fig.

11

is

considered

to

represent

basically

the

fLexural

_properties

of

g

1

weror

la

T

l/3

!

'

-T...---

_V-

""""'

"""-'

'

/

'

e

1

"'7'----

)ly

'

1

t st.

1ine

1

1

'

x

'

_P"e

'"'i

Th

l

st,

1ine

:

/

ptlll ..

lic/r.

g,

1

"buLl-:3-ltXL" 1

'

:::::::::/::/:::////:/

lierla

T

113

h

te9 ot'

119 -"ley. 1

The

relatienship

between

mornent

level

force

level

and

the

curvatu[e of column

section or axiaLcross O

063.42HO

C63-41Hac6s-41H

･

e CS3-41H-C63-21H

:/:::d::1

:

:....ttL-:-...

:

!

l

l

1

'

Fig.

11

s.:---/a

Fig,12

lt3

1 -

eq

Determination

of the calculated vallleS of

VVI.

frorn

the

intersection

between

the

curve

by

Eq.

(12)

and

VVZ.1

W}u-av

curve

-125-Architectural Institute of Japan

ArchitecturalInstitute of Japan

the

cross section of reinforced concrete columns.

The

caleulated values

of

a. were

_O.

_33-O.

_53.

This

means

that

all

the

specimens

except

C

65-41

H

almost reach

the

strength at collapse mechanism

(P==

1)

of

frames

in

calculations.

It'can

be

seen

from

Table

7

that

the

test

values of maximum

strength

are

a

little

higher

than

but

nearly

the

same

as calculated

ones.

iO

Veutical

load

carrying capacity:

The

calculated values of vertical

load

carrying capacity were obtained

by

using

Eq.(12)based

on

the

following

assumptlons,

1}

The

top

deflection

is

assumed

the

same with

the

maximum

deflection

under

horizontal

loading

te

determine

the

value of

6L,..,.

On

the

otherhand

the

deflected

Shape

along

frame

height

is

obtained

by

trial

and errors so

that

it

can

be

nearly

the

same as

that

selected as

to

make

the

critical

load

a minimum.

Z>

The

equivalent

flexural

rigidity of

the

cross section of columns varies with

the

axial

force

level

acting on

the

cross section,

based

on

the

relationship as shown

in

Fig,

_11.

It

seems very

difficult

to

obtain

the

exact values of equi,valent

flexural

rigidity of

the

section at any

level

of axial

force

appliedS),

Thus

more

_practical

method

is

adepted.

Fig,12

shows

the

method of

how

to

determine

the

vertical

load

carrying capacity, which

is

obtained as

the

intersection

of

the

cuive

due

to

Eq.

(12)

and

the

assumed relationship

between

axiai

force

level

versus equivalent

flexural

rigidity ratio a..

It

can

be

seen

from

Table9

that

test

results are nearly

the

same as

the

calculated ones

except

the

specimen

C

63-42

H,

C

65-41

H

and

V

specimens.

The

maximum

top

drift

of

C

63-42

H

specimen

was

by

far

greater

under

vertical

loading

with

deflected

shape of

high

ordercurve

than

that

under

horizontal

loading.

That

means

the

values

of

rotation of upper

beams

_go

beyond

the

_point.A

as shown

in

Fig.2(b)

and

the

equivalent stiffness

becomes

smaller

than

the

assumed

constant value upto

_point

A.

It

seems

due

to

the

variation

of

beam

strength along

the

height

in

this

$pecimen.

Thus,

there

_is

also

a

possiblity

that

the

vertical

load

carrying

capacity

of

the

colurnns

of

actual

frames

in

accordance with currently used methods

is

considerably

lower

than

_predicted

by

Eq.

_(8).

Table

9

also

shows

the

va]ues

of

a. ranging

from

_O,

₆₆

_to

_O.

_88,

which

are

_quite

different

from

those

of a..

Fig.

13

Table9

Correlation

of experimented and calculated

ultimate vertical

loacl

caTrying capacity and

Table8

Assumed

yalues of

beam

end rotation at total

the

calculated yalues of the coefflcient, a.

height-dr{ft

of

2

%

for

calcutation ntx(om)xtH?::fff'}l2,zf.1

4!701.00o.osoe･O,0400

3130'O.77O,035Se,e3o6 42'90O.S3O.Dl19O,0212

!50O.29o.eo2oO,Ol18

2901.00--O,0400 21soo.ssJ-D,0222 tema.i-46-E:

_{pe...i-26･g2}

VertiealInad

SPectlnmeqEbcP-(ten)Cal.(tor1)Exp.tCal(rat

±o)

C6]-42HO.S811.0513.S3o.so

C63-42vJ-3Z.2731.IS],03

C63-4!HO.662o.2e2e.87O.97

C6S-41HO.7318,4721.7SO,8S

C83-41HO.6627.3627.331.00

C83-4IVtt36.8e40.SSO.9!

C63-2!HO.6920.SS19.941.04

C63-21V--26.SS31,a5o.ss

63-42H

_{--..pF4--...h'L-.-NvsLx}

_{5C63-41He6S-4s3-as5CFIeor)}

2.rrF2sNN q

X

2

N

'li4

hh _N s 1 n N s,L3 N

.3N3xC63-213

N x 1 h 1 h N 1rtFl.7 s v x 1 1 Lt z

,h

1 _T

-

,N L Ebcp. x:,

.----caLy]`'cftpuyJ

2 't',12Nt1112LL1,72Lt1NTt2 G G G G

so6e4o2o O4020 e2o OCurn}

SO

60

40

20 O

20

O

Fjg.

13

Comparisolt

of experimented and calculated

horizontal

deflections

llnder maximum vertieal

load

after

horizontal

toading

-126-NII-Electronic Library Service

shows

the

cornparison of

deflected

shape at rnaximum vertical

load

between

test

results.and

calculated

ones.

On

the

_otheT

hand

the

_vertlcal

Load

_{carrying capacity of columns of specimens}

C

83-41

V

and

C

63-21

V

without

the

expe[ience of

horizontal

loading

is

a

little

smaller

than

those

calculated

by

Eq,

(16).

The

vertical

load

6arrying

capacity

decreased

due

to

the

local

failure

at

the

_point

of apptication of

the

load,

Longitudinal

reinforcement can also

be

_judged

to

be

ineffective

for

the

values of

VVZ.

and also

for

those

of

ilJL.

as shown

for

C

65-41

H

in

Table

9.

61

Conclusions

'

･

Based

en

the

study reported

herein,

the

following

conclusions may

be

made.

1)

Simplified

analytical

method

has

been

developed

to

evaluate

the

vertical

load

carrying

capacity of

the

columns of

the

frames

after subjected

to

reversecl

horizontal

loads,

This

method was

derived

from

the

energy method

based

on

the

assumption

that

all

the

bearn

ends

_yield

with rotational stiffness

being

the

equivalent elastic one, while

that

the

resistance

ef

columns

are

determined

from

the

assumed,

relationship

between

axial

force

level

and

the

equivalent

flexural

rigidity ratio of

the

cross section.

2)

Experimental

works

have

beeri

conducted

to

verify

the

yalidity of

the

above method.

The

numbeT of specimen

dealt

with,

totaled

eight.

Six

were

four-storied

and

the

_qther

two

were

two-storied

frames,

These

model

frames

were

subjected

to

concentrated vertical

loads

at

the

top

of

the

columns, with or without

the

experience of

horizontal

loadings.

This

concentrated

loading

methocl

was

adopted

from

the

simplicity

of

the

application of

loads,

after

establishing

the

relation

between

the

models

and

the

_prototypes

having

distributed

loads

at

every

floor

level,

on

the

vertical

loacl

carrying capacity

by

using

the

above methods.

3)

It

has

been

found

that

the

_proposed

method

_gives

the

_good

_predictions

of

the

vertical

load

carrying caPacity

Pli}.

of

the

columns of

rpulti-story

weak-beam, strong-column

frames

with

the

experience of reversed

horizontal

loading

upto

_{pest-yielding}

range.

The

values of

IIi}.

were about

two-third

of axial compression strength

W}.

as

predicted

{or

the

frames

in

which strength of

beams

was arranged uniformly, along

the

height,

while was about one-third of

-Cl.,

which was considerably smaller

than

_predicted,

for

the

f[ames

in

which strength of

beams

was allotted smaller

for

upper stories and

greater

for

lower

stories.

Test

results suggests

that.

Iimit

design

method

is

superlor

to

currently used method with respect

to

the

safty against

_gravity

load

after major earthquakes.

4)

Further

study

is

needed

to

_get

_general

conclusions

by

aclding experiFiental studies on statically

determinate

structure

frames

to

the

abpve

experiments

as well as

by

developing

the

more

precise

calculation

methods.

7.

Acknowledgemerit

Thls

study

has

been

conducted at

the

Structural

Engineering

Department

of

the

University

of

Hiroshima.

The

authors would

like

to

thank

the

assistance of

tAe

staffs,

_particularly

Mr.

H.

Araki

and

Mr.

E.

Najima

of

the

Earthquake

Enginee'ring

Laboratory,

The

authors

acknowledge

the

cooperation

ef

S.

Hayashi

and

H.

Kawasaki,

senior

_year

_students.

The

_{encouragements of}

Profs.

Y.Mukudai

_and

M.Hanai

_{are also}

_greatly

_appreciated,

APPENDIX

Reduction

6f'Equatiens

₍

_s

_)-ol]

'

Assumptions

:

1)

qt=qi==･･L･･･=qi='""'=qn=qcr

2)

H,=Hln,

Ht=2Hln"･････,H`;iH!n"･････,H.=nH/n=H

3)

y=o(#)M;

4)

y,-fi(ilS'/)M

s)

e==gxy=l7!

(#)m'];

6)

a=

aHm

({;,

)m-]

'

'

7)

Mt=Kete;

8)

CEI)eq=avEl

u

=t;

11"{EI)e.

(

d,t,Z,g

)'

dx

=

2(

EII

)..

X"

(M

)'d

t

･

-127-Architectural Institute of Japan

ArchitecturalInstitute of Japan

=2(il).,1"(S)qt(yt'y)-#Ketet]tdx

=2(iD.,

X"

[(#

q`(

Yt-

y)]'-2#

qt(

yt-y);l]

Keia+(# K,,e,)t]

=2(EII

)..

f"

[qZr(#

(yt-

y)l'-2

ger#(yi-

y)

#

Keien+(#

Keia)']

=a'q;.+b'qtt+c'

u

==S

X"

Sl]

_qi

(

:\

)2dx

+t

,i.,

M,

e

=={}q..X"(gg)'dx+Si,Mte

==b"qcr+c'

':U=Ul,

hence

a=aL････････Eo.{9)

b

==

b'-

b"''-"mEq･

(]O)

c==c'-c"･----･Ee.(11)

An

exampte of

the

aboye

integrations

as

below:

x"#

y,d.=x"'

£

.

y,dx+L"i

s.,

y,dx+･･････+x]Ir,

, £ "=.

y,dx

Notations

A,:

area of,concrete

A.:total

area

of reinforcement

At:area

of

tensile

reinforcement

b:width

of

cross

section

D:depth

of cross section

d

:

depth

of

center

of

tensile

reinforcement

from

maximum compressed

fibre

E,:Young's

rnodulus of concrete

EI:elastic

flexural

rigidity of cross section of column

EL,:varying

flexural

rigidity of cross section of coiumn

(EI).,

:

equivalent

flexural

rigidity of cross section of column

E.:Young's

modulus of steel

F:c

:

maximum compression strength

in

concrete

H:height

of

frame

from

top

of

the

foundation

to

the

uppermost

beam

level

H}:height

of

frame

from

_top

of

the

foundation

to

the

ith

beam

level

i

:

story number,

1-4

J':story

number,

1-4

Kei:equivalent

retational stiffness at

beam

end

l:length

of

beam

from

axis

to

axis

of

columns

Me:cracking

moment

of

coiumn

M,:bending

moment of

beam

M,:yeild

moment of column

M.t:yeild

moment of

beam

m:constant, usecl

in

shape

function

of column

n:total

number of stories

P:horizontal

load

P.:ultimate

horizontal

load

q..

:

catculated critical vertical

load

at every

beam-column

joints

q,

:

vertical

load

on column

from

beams

U:strain

energy of

bending

of colurnn

Ul:decrease

in

potential

energy

due

to

lowering

of

the

_point

of application

of

-128-'

NII-Electronic Library Service

ptt

:vertical

load

VII

:

working' vertical

load

on columns

during

horizontal

loading

IeC,.

:

calculated critical vertical

load

.JVI.:calculated

cumulative vertical

load

Wl,

:

measured ultimate vertical

load

x:an

aTbitrary

distance

along

the

height

of

frame

from

the

_top

of

the

foundation

y:horizontal

displacement

of

frame

at

the

height

of x

Ai-Ak

:

constants, used

in

'the

Fourer

Series

of shape

functien

a:flexural rigidity reduction ratio of cross

$ection

of

column

aH

:

flexural

rigidity reduction ratio of cross section of column

du[ing

horizontal

loading

a.

:

flexural

rigidity reduction ratio _{of cross section of column}

during

_vertical

loading.

fi

:

moment reduction Tatio of column

a

:

honzontal

displacement

at

the

uppermost

beam

level

for

anaiysis under vertical

load

a.

:

horizontal

displacement

at

the

uppermost

beam

level

for

elastic

analysis under

horizonthl

loading

iH:measured

horizontal

displacement

at

the

uppermost

beam

level

ROH

:

residual

horizontal

displacement

at

the

uppermost

beam

level

at

the

end of

horizontal

or vertical

loading

'

'

a".

:

horizontal

displacement

at

the

uppermost

beam

level

at

the

ultimate stage of

horizontal

or vertical

loading

6.

:

horizontal

displacement

at

the

ith

beam

level

for

analysis under vertical

load

di,

:

curvatur'e at cracking of column section

to.

:

curvature at

_yielding

of column section

a.

:

maximum strength of reinforcement

a,:yield strength of reinforcement

'a:rotation

at

beam

end

'

emaxi

:

maximum rotation at

beam

end under

horizontal

load

'

'

REFERENCES

'.

'

1)

American

Cocrete

lnstitute

:

"Symposium

on

Reinforced

Concrete

Columns",

ACI

Publication

SP-13,

_pp.55-156,

₁₉₆₆

2)

Ferd,

J,S.,

Chang,

D.C.

and

Breen,

J.E.

;"Behavior

of

Conciete

Colurnns

Unde[

Controlied

Lateral

Defermation",

Journal

of

thE

ACI,

Proceedings

V.78,

January-February

19811No.1,

pp.3-20

'

3)

Ford,

J.

S.

,

Chang,

D,

C.

and

Breen,

J.

E.

:

"Behavior

of

UnbTaced

Multipanel

Cencrete

Frames",

Journal

of

the

ACI,

'

PToceedings

V,78,

March-April

1981!No.2,

pp.99-l15

'

4)

Mohit,

S.

M.P,

and

Shimazu,

T.

:

"The

Vertical

Load

Carrying

Capacity

of the

Columng

of

Multi-Story

Frames

with the

Experience

of

Horizontal

Loading",

Reprint

of

Chugoku

Meeting

of

Architectural

lnstitute

of

Japan,

March,

1985,

Vol,

12,

pp.113-116

s)

Timoshenko,

S.P.

and

Gere,

J.M.

:

"Theo[y

of

Elastic

Stability",

McGTaw-Hill

International

Book

Company,

Secend

Edition,

1982,

pp,82-162,

163--211

6)

ArchitecturaHnstitute

ofJapan

:

"AIJ

_Standard

_{forStructural}

_CalcuLation

_of

_Reinforced

6oncrete

_Stfuctures",

_l982,

pp.213

(in

Japanese

edition)

･

7)

Muto,

K.

:

"Pla$tic

_Design

of

Reinforced

Concrete

$tructures",

Part

2

in

EarthquakeResistant

Design

SerLes,

Maruzen

Co,

Ltd.,

pp.

84-92,

'

8)

Ministry

of

ConstTuction.

Japan

:

`"Building

_Standard

_Law

_Enforcernent

_Order",

1981,

Chapter

_M,

Sectiofi

8,

Article

82-2