純曲げを受ける鉄筋コンクリートばりの破壊集中性

(1)

[$

11

UDC:624. 072. 7.012:624. 04 :539.

42Journal

of

Structural

and

Coostructien

Engineering

{Transactions

of

AIJ)

No.

371,

January,

1987

HSMas\ftMtaXas1vafifi

ig

371

e･main

62

ff1fi

CURVATURE

LOCALIZATION

IN

PLASTIC

DEFORMATION

'

RANGE

OF

RC

BEAMS

UNDER

FLEXURE#

by

SHIGEMITSU'HATANAKA*,

YOSHIO

KOSAKA**,

YASUO

TANIGAWA'*"

and

RYUJI

MIWA'"",

Members

of

A.

I.

J. '

gl.

Introduction

For

the

application

of

_plastic

design

method

to

reinforced concrete

<RC)

frarnes,

it

is

the

first

requisite

that

constitutive

RC

members are

ductile

eneugh

for

the

redistribution of

moment

and

for

the

formation

of

_plastic

collapse mechanism,

To

discuss

analytically

the

ductility

of

RC

members,

the

effects

of

various

factors

shoulcl

be

fully

examined on

the

rotational

capacity

and

the

length

of

th'e

failure

zone

(or

idealistically

the

_plastic

hinge)

of

RC

members3}'6).

The

toughness

of

RC

beams

under

flexure

is

often

evaluated

in

the

test

of

the

beams

subjected

to

two

same

concentrated

loads

acting

at

the

trisectien

_points,

which

offers

the

fundamental

information

of

the

mechanical

properties

of

the

beams.

Provided

that

the

mechanical characteristics of eonstitutive materials

in

arbitrary sections are uniform, moment

(M)-curvature

{ip)

relationships

would

be

constant

over

the

region

between

the

two

loaded

points

(flexural

span)

i.

_e.

failure

in

the

flexural

span

where

moment

is

constant

would

occttre

uniformly

except

for

the

region adjacent

to

tbe

loaded

points,

According

to

the

observation of

the

failure

_patterns

of

RC

beams

in

experiments,

however,

failure

zone

is

concentrated

in

a

finite

length,

which

is

usually about

h

to

2 h

(where,

h

is

the

height

ef

Rq

beams)`)･')-9),

Such

failure

or

curvature

localization

i$

considered

to

be

resulted

from

the

unavoidable

nonuniformities of

the

internal

stresses especially adjacent

to

the

loaded

_points,

material characteristics of concrete and steel,'bond

characteristics

between

concrete and steel, etc.

Various

M-

¢

relations,

therefore,

may

inevitably

be

measured

in

experiments

depending

on

the

measurement

region of

the

curvature.

This

fact

is

considered one of

the

reasons why

the

same results

have

not

been

reported on

the

applicability

of

the

stress

(a)-strain

(e)

behavior

ef

concrete

te

the

_plastic

deformation

analysis of

RC

beam

'

tionsiO)-]3).

For

the

concrete specimen even under

the

uniform stress state, on

the

oth,er

hand,

the

failure

zone

is

localized

in

a

limited

region

of

the

specimen')･S),i`)･i5),

and

then

the

measured strain

in

the

stress

descending

range

is

_quite

different

depending

on

the

length

and

_position

of

the

strain measurement region.

Therefore,

in

order

_to

attain

agood

accuracy

in

the

deformation

analysis

of

RC

members under

flexure,

the

a-e relation of concrete corresponding

to

the

compressive

zone

of

RC

member

should

be

applied

to

an appropriate certain region,

instead

of

applying

the

relation

of a conventional

cylindrical

specimen

to

any region.

There

aTe

two

main

purposes

in

the

present

study.

One

is

to

exarnine

the

cttrvature

localization

in

the

constant

flexural

moment

region

of

RC

beams

in

plastic

deformation

range,

The

other

is

to

discuss

the

applicability of

the

uniaxial stress-strain relati6nship

of

concrete

to

the

bending

deformation

analysis of

the

RC

beams,

g2.

Experimental

procedures

The

following

two

series of

experiments

were

carried

out.

One

is

a

test

on

RC

beams

ttnder

flexure.'Outline

of

the

experiment

is

shown

in

Table

_1.

Testing

variables

include

the

water-cement ratio

(WIC),

spacing

of stirrups

(S).

#

This

paper

is

based

on

the

earlier works

_presented

in

Refs.I)

and

2).

*

Fellowship,

the

Japan'Society

fo[

the

Promotion

of

Science,

Dr.

of

Eng.

**

_Professor,

_Department

_of

_{Architecture,}

_Nagoya

_University,

_Dr.

_of

_Eng.

*"

_Professor,

_Departrnent

_of

_{Architecture,}

_Mie

_University,

_Dr.

_of

_Eng.

*#*

_Structural

_Engineer,

_Shimizu

_Corp.

_Ltd.,

_M.

_of

_Eng.

(Manuscript

received

May

17,

19g6)

(2)

-27-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan

'

volume

fraction

of steel

fiber

(VCr},

tensile

reinforcement

ratio

(Pn,

value

of

P.IPt

(P,:compressive

reinforcement ratio), and

length

of

flexural

span

{tb}.

Details

of

the

RC

beams

are

shown

in

Fig.1.

The

other

is

a

test

on concrete

prisms

under

uniaxial

compression.

Outline

of

the

experiment

is

shown

in

Table

2. The

experimental

fac-tors

include

the

heightlwidth

ratio

(H/D)

of specimen, existence

of

frictiori

between

specimen and

load-ing

_platens,

and

the

others

corres-ponding

to

the

factors

of'

the

com-pressive

zones of

the

RC

beams.

For

all

the

_prisms,

the

section

size

was set

to

9.7

×

9.7cm,

and

con-crete was cast

horizontally,

whieh corresponds

to

the

casting

direction

'

of

the

RC

beams.

Ordinary

Portland

cement,

river

sand'(maximum

size

:

5 mm),

river

'

gravel

(size

range:5-15

mm), and

steel

fiber

(size:O.5XO.5

×

30

mm,

tensile

strength:7000kgfl

cmZ)

were

prepared

for

the

fabrica-tion

of

concrete.

Steel

bars

whose

mechanical

_properties

are

shown

in

Table3

were

used

as

longitudinal

reinforcing

bats

and

stirrups,

All

the

specimens were

demolded

at

the

age

of

3days,

and

then

cured

in

the

atmosphere

ef

laboratory,

The

numbers

of specimen

_prepared

for

each

factor

were one

(two

or

three

for

main

factors)

for

the

RC

beams

and

three

for

the

_prisrns.

'

Methods

of

loading

and

measure-ments

for

the

RC

beams

are

illus-trated

in

Fig.2.

The

RC

beams,

whose supported span was

9 h

(

=9

×

19.36#174.2cm),

were

sub-Table1Outlineef

experimentof

RCbeamsunderflexure

Notattenof speeitaenvfCl}wlCptct)YAbShapespitch ofstirrup O.45 _2:l2hAms 3h A A

!.4O.04

10

p..ss-2sl2.e3h,St7S,o,S,10

ss-A+gf42T[.o.3s selOenA-tvpeettrrdp vlcnyo.ss SS-O-2hTpts-[,kb:ethnc. o 10 h, le 2.1O.19 As,Ale o.le O.70 _O.04

3h

10

1.5O.S5

5,IO

[Notes]

Vf: Volume

fraction

of steel fiber. WIC: Water-cement ratle･t Pt: Tefisile

rein-forcetnent

ratSej

Y:

%IPtt

Pe: Compressive reSnforeement retio,

ib!

Length of flexural

span, h: Height of RC beam.

oA-type

Table2Outline

ofexperiment ofprlsmsunderuniaxial

B-type comp{esslon Netatienof'Epeclmenvf(")wl,CHIDuShape&pitch ofhoop

'

O.4S2O.4Aav-rA

i-Qt4-o.4S=5em

oA..A As

oO.552

_{.Ao,,Ale.A,As,Ble,Bs}

Ao".Ao.As A-typeheop

w!c=o.ss

O.70

_2O.4

Am,Aeml"

L5O.S5 Atu+.Ale,As

tNoteslVf:

Volume fraction of steel fiber. W!C: Watet-cement ratio, HIDt Slenderness

'ratiot

v: Appreximate value of ceefficient

ef static

frictien

between specimen and

loading

platen,

';

_dnly

fo: specimen without belts.

s[

NArtypeanMype

"mplo.oem

03e4@10cm"3.scm

b t

1.5lt

]h th

th

Lth

･12h

Flg.1Detailsof

RCbeam(55-ALo)

Fig.2Methods

of

Tebte3

bloading

Mechanical

andmeasurement

propertles

,,,[]l

ptb=9.7

cmd=17.4 em h=19.4 cm

Rubber

e(iZisheet

for

RC

bEalll'

of steelbars Kindfy(kgflem2)fu{kgflem2)p(") D13 4010 6oeo24-1 DIO

4050

566e25.B

e4-H

sleo

542021.3

"4-L

3310 3s7e29.9

"3

3SEO 422014.2

tNetes}g{;.:igrig,streSSr

fu: Elengatien)taximum

jected

to

two

same concentrated

loads

acting

at

the

trisection

_points,

and

the

central region of

the

length

of

2 h

in

the

'

flexural

span

(tb=3h)

was

divided

into

four

regions, with screw

bolts

being

set

between

the

regions

for

the

'

measuTement

of curvature.

in

each region,

The

beams

were

loaded

until

the

ayeraged'value of

dip

(where

d

is

the

effective

depth

of

the

RC

bearns)

in

the

measurement region

becomes

_O.

_1.

To

_prevent

shear

failure,

steel

frames

were set and

tightened

by

nuts along

the

shear span of

the

beams.

as shown

in

Fig.2.

The

_prisms

were

loaded

under

the

(3)

-28-constant

strain

rate of about

2

×

10-31min.

up

to

E=15

×

10-3

by

using a

high

rigidity compression

testing

machinei`).

Longitudinal

strains of

the

_prisms

were measured

by

a couple of

deformation

transducers

(measurement

length

:

17.8Cm)

attached

to

the

specimen

by

means of steel

frames.

53. Test

results

and

discus$ions

3.1 Moment-curvature

eurves

of

each

region

Fig.

3(a)

and

(b)

show

typical

examples

of moment

{M/bd')-curvature

{dip)

curves

'(

hereinafter

referred

to

as

M-di

eurves) measurea

in

each

region.

The

failure

_patterns

of specimens, ultimately

being

in

compressive

failure

mode, are

illustrated

together

with

them.

It

is

observed

that

the

measured

_M-ip

curves are significantly

different

from

each other,

depending

on

the

relation

between'･the

'locations

of compressiye

failure

zone

(4)

and

curvature

measured regiofis

(indicated

by

the

numbers

of

1 to

4 in

the･figures).

That

is,

there

exist

two

types

of

regions.

One

is

Ihe

region where eurvature

increases

_gradually,

which corresponds

to

the

failure

zone.

The

other

is

the

region where

curvature

decreases

or

the

･increment

of

curvature

rapidly

decreases,

which

corresponds

to

the

non-failure zone,

Such

correspondence

of

the

curvature

localization

in

the

flexural

span with

the

failure･pattern

of

an

RC

beam

is

observed

for

all

the

specimens

tested.

Moreover,

since

_the

relationships

of

both

curvature-compressive and-tensile

fiber

strains

in

each region are

found

to

be

almost unique, respectively,

as

seen

in

Fig.

4,

mechanical characteristics of

each

section

are not considered

to

be

significantly

different.

Such

diverging

_phenomenon

of

the

M-ip

curves

is

considered mainly

due

to

the

instability

of

the

mechanical

behavior

of

concrete

in

the

moment

descending

range,

i.

e.

in

the

strain

softening range of concrete

in

the

compressive zone

of

RC

beamsi},S)ii4),i5}.

Fig.

5

shows an example

ef

the

diverging

_phenomena

of a-e curves

from

the

uniaxiat compressive

test

of

the

cylindrical

specimen of

HID=Zi'].

Note

_that

_the

averaged curvatute

of

the

diverging

_point

tends

to

become

larger

with

increase

in

the

ductility

of

the

RC

bearns.

Fig.

6 shows

the

effect

of spacing of stirrups

(S)

on

_M-ip

curves

avereged

ever

the

whole measurement

length

<2

h).

According

to

the

figure,

the

ductility

of

the

averaged

M-ip

curye

does

not neccessarily

correspond

to

that

of

the

_compressive

touhgness

of

the

concrete

in'the

compressive

zone.

The

ductility

of

55-A,,,

RC

beam

specimen

is

considered

_to

be

underestimated,

because

failure.extends

to

the

outside

of

the

curvatttre measurement region.

NvAx= 4fi"Eq"b5:!s

g

:

g

:

o SS-Ato{1) ¢1

-Ctr,:!L!L7byh.F

==..-.--

L-

s"

:

,

;

,

:

t::;"2

. --.--"rxli.:

N

x. × . tPt../.-.t',n.,lt/1'l'lt'.tpt..t'

'ttttttttt'Oppositesid

'

4321 x"3 side(----)

2 e

2 4

G

S CURvAruRE(xlo-2)

dth

f

{a)

55-Aio

specimen

10

nfits o-wo5:efi vxiR:o 55-Alo(1)

"1

`.

.-L-

"1

/t

-'H

"3

/

-"-

"4

_/'

::

1

'

l･:

'il"l'lill.

` .is

b"':i{i

ile"kfii･il. n ',ll'iii. `

"i/tf'br'

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ss-As-o.2 tptt.tdi.tt.t{･ptts't''rlj"iif,1''PPosi '4321

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×

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th2

i'de4

side(---"

Hg.

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2

4

6

s le

euRvATvREtxlo-2) .a"

Curvature-fiber

straias relations of each region

e

#N-Eg)e5tsee:::e.

oO

o

₂

4

fi e

CURVATVRE

{xlo-.2)

_,de

(b)

55-As-O,2

specimen

Flg.3

M-

¢ curves

in

each region

10

2

Fig5

a-e

4

6

STRAnT

(xlo-3)

curves

in

each region s

(4)

-29-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute ofJapan eNvm<E

'on

"7NeRe"ovS:Rsge lp lp lp

s

_'

.Ll

-...

/"

S-SCMS---s-'T-

'-

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'-s.

WIC=O.A-type

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s=1,Scm "s x

N s,

s sN

X

.

Xhs.

-

x

'

Nx.i!l'stb

Xtsrco7-F.

'

Saco cm

opt anvAxE

Lanee'Hq."o"s::y9

xEy2

Fig.7

x

× 5Sstirrup

"'

-E";g9n

-s

S"2eem

e

Fig.6

M-

¢

2

curves 4 6 S CURVATVRE

C

xlo-2

)

d"

'

averaged over whole measured

'

IO

reglon o'

CVRVAT.URE,

"

Id6alized

failure

zene model

r"--.--'-'"-"'.-'

_::;"be'<K'sx

-NNx.

'N･sillNx

×

SM

'N

X

l[fC,//;LS.5,'i...p

''NK`XS"tb

s_NSs

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s-

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H S-10eni o

Fig.8

2 Predieted

4

6 e

lo CuRvATuRE

(xre-2)

.d"

M-e

curves

for

measured region

3.2 Idealized

failure

zone model

.

To

evaluate

_{quantitatively}

the

effect of various

factors

on

the

M-

_¢

relations

in

the

flexural

span of

the

RC

beams,

the

flexural

span

is

idealized

into

two

zenes according

to

the

degree

of

failttre,

'as

shown

in

Fig.

_7.

In

the

two-zones

moclel,

the

diyerging

point

of

M-ip

curve may

be

defined,

for

example,

as

the

point

where

the

stress

block

index

k,ki,

or

the

hveraged

stress of

the

compressive zone

of

an,RC

beam,

is

maximumi')･iS).

The

stiffness

<El)

of

the

'

unloading

curve

is

assumed

to

be

equal

to

the

initial

value.

･

'

Idealization

of

the

flexural

span of

the

RC

beams

was carried

out

based

on

the

measured

M-ip

curve

in

each region.

For

the

calculation

of

the

curvature

in

each region, curvature

distribution

was assurned

to

be

symmetric with re$pect

to

the

center of

the

idealized

failure

zone when

failure

zone extended

to

the

outside of

the

measured region, and

to

be

isosceles

triangular

when

the

ldealized

failure

and

the

non-failure zones coexisted

in

a

divided

region.

For

the

determination

of

the

idealized

failure

zone,

cornpressive

fiber

strain at

the

initiation

of

the

compressive

failure

of

the

RC

beams

was set

to

range

from

about

4 to

6

×

10-3.

referring

to

the

failure

_pattern

and

the

M-e

curves of each

specimen.

Narnely,

the

regions where

the

compressive

fiber

strain exceeds

(4-6)

×

10-'

were

treated

as

the

idealized

failure

zones, p "aajp N eO,45 0.55 O.70 w/cWater-cement ratio p "Aajn pto

. o,' -A-type stlrlup e B-type stirrup

5

7.5

10 S{cm)

Spaeing

of co stiuups p "ptel P pt e e As eAiO

(a}

D "pmetp N o o o

"

'

oAs -AIO

{b)

s.pp

pt o, O O.2 O.4

T

(d]

P.!P,

ratio

<e)

FIg.9

Effects

ef e w

O

1.5

vf

(z}

(c)

Volume

fraction

of steel

fiber

I,4 2,1 2.8 pt

<z)

Tensile

reinforcement

varlous

factors

on

length

p t-et p N o

a

s

e -AIO eAm

2h

3h

4h

tb

Length

of

ftexural

sPan zene

(l.)

'

ratio

(f)

of

failure

(5)

Fig,8

shows

M-ip

curves calculated

on

the

assumption

that

the

center

of

the

idealized

failure

zone

locates

at

the

'

center

of

the

measurement region

(2h).

The

inconsistency

of

the

ductility

factor

of

the

_M-ip

curves with

the

toughness

of

the

concrete

in

the

compressive zone,

seen

in

Fig.6,

is

not observed

here.

_,

3,3

Relation

between

length

and

ductili,ty

of

idealized

failure

zone

'

Figs.9(a>

through

(f)

_show

the

_{effect of each} _{experirnental}

_factor

_on

_the

_length

_of

_idealized

_failure

_zone

_<l.,

hereinafter

simply

_referrecl

to

_as

failure

zone).

According

to

Figs.9(a)

through

(d),

_there

seems

to

be

a

_positive

correlation

between

_the

_value _of

h

_and

_the

_toughness

_{of concrete}

_in

_the

_compressive _zone

_of

_the

_RC

_beams,

_including

the

effect of compressive reinforcement,

According

to

Fig.9(e),

the

tensile

reinforcement ratie

<P,=1.4-2.8

%)

has

little

influence

on

the

value of

l.

as

far

as

the

_present

experimental

results are concerned.

Generally,

failure

zone

length(h)

is

censidered

to

become

larger

as

the

value of

P,

increases,

due

to

the

increase

in

the

depth

of neutral

axis

at

the

compressive

failure.

Further

examination

is

required

for

the

wider range of

P,,

especially when

stirrups

are used.

As

_seen

in

Fig.

9(f),

the

effect

of

the

flexural

span

(

l,=2

_h-4

_h)

on

the

value of

l.

is

scarcely notable so

far

as

the

_present

_experimental _result

_goes.

_However,

_the

_failure

_zone

_may _extend

_t6

_the

_whole

_flexura!

span,

it

the

section

of

beam

is

designed

to

be

_considerably

_ductile.

_In

_such _acase,

_the

_behavior

_in

_the

_flexural

_{span may}

_be

_more

_or

_less

affected

by

_the

constraint of

loading

_points,

so

that

_the

_failure

zene

is

restricted within

the

span.

The

relation _of

thg

toughness

_of

_concrete

in

_the

_compressive _zone _with

_the

_value

_of

_l.

_is

_{investigated,}

_using

a-e

curves

obtained

from

the

uniaxial compression

test

of

the

concrete

specimens.

Fig.

10 shows

the

_plots

of

the

values of

l.

versus

the

toughness

₍

T,,

area under a-E curve ttp

until

e=15

×

10-3).

Numbers

added

to

the

_plots

indicate

those

of

the

_beam

_specimens

(see

_Table

_4).

_Rather

_strong _correlation

_(CR

_:

_coefficient

_of

correlation) exists

between

the

toughness

_{of compressive}

_cqncrete

_and

_the

_value _of

_h.

'

_･

'

Fig.11

shows

experimental

M-

_¢

curves

averaged

over

the

failure

zones which are

defined

based

on

the

aforementioned

idealization

_method.

The

_curves _are

_very

_ductile

_for

_all

_the

_specimens,

_the

_experimental

_factors

having

less

influence

on

them

than

on

the

M-ip

curves averaged over

the

whole measured

length.

Fig.

12

shows

the

relation

between

_the

ductility

factor

_of

_the

_M-ip

_curves

_in

_the

_failure

_zone

_and

_the

_toughness

of etn

a

e

g

"

Tabte4

Specimen

numbet

Notation Notation

No.of

_'speclmen

No.ot

_'speelmen

14S-AIO 145S-O-2h 27D-Alo ISS5-Alo-2h 155-OCI)IES5-e-4h 4S5-O[2)17SS-AID-4h

sSS-As(1}ISS5--Alo-1.4Pt

E5S-AsC2}19S5-Alo-2.S?t

755-A7.S 20S5-As-O.2y

e5S-Alofyt2155-Alo-O.2T

95S-AloCl.}22S5-As-O.4T

10SS-Alo.{2)2]S5-Alo-O.4T

11S5-AloC3)!455-As-1.SVf 12SS-Bs !555-Alo-1.5Vf 13S5-BIO

[Netes]

-;"4-L _is _used

for

_stirrup Ce4-H

for

other spectrnens)

gp

M

lsr-t-ilseoa]

';'i'/:i _i

/i

i

Lp=L30+O.66Tl CR=O.S1 ,.,

s g'io

/T

b

;1

ii

,wal.'.fi''g.

･.･eq-3

"

orvv rpqx.

8e-,s)sMtn-:Rg=oe

Nextu

,Qe

t=ISxlO O 1

2

3 4 Tl

(kgflcm2)

Fig.

10 Effect

of compressive

toughlless

of

on

length

of

failuTe

zone

(t.]

5 conerete

(TO

2

Fig.

₁₁

4

6

S

CURVATV RE

C

xlo-2

)

d"

'

M-e

curves

in

failure

zone

3

10

O

1

2

3

4

s

Tl

[kgf!crn2)

Fig.12

Effect

of eompressive

toughness

of concrete

{T,)

en

ductility

factor

{A.Vpt)

of

RC

beam

(6)

-31-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan elza n t e N p=smo

"y"o.s

S,

,.,k/ 1.1 2"

rt･

b .pt-ect.D.:za: eev aeM eoN ad' With belt

--"With

no bolt rr

E}IELt

s NNN NNs

Xxli

li

li.x.

NN

>lls.lls

-` s

-.

N s4sem

X

Ns"`

..

s;-7.sem

× NN:KN w!cte.SS NNs SNtis sljlocm A-type hooP Ss..

s:co'em

e

o s, le IS 20 2S

_e24sB

10 12

14

16

!B 2e

"o.sl"v

sTRAiN

(

x le-3

)

,e

Fig.13

Relation

between

length

((.}

and

ductility

factor

Fig.14

Measured

a-e cttrves

(pt.slip.>

of

failuTe

zone

the

concrete

in

the

compressive zone,

Here,

the

toughness

up until e=15 ×

10-3

is

used

as

the

compressive

toughness

of

concrete again, and

the

ductility

factor

in

the

failure

zone of

the

RC

beams

is

defined

as

the

ratie

of

the

curvature

(A.)

at

O.8

M. (M.

:

maximum

moment)

in

moment

descending

range

to

the

curvature

'(

¢.)

at

yielding.

The

considerably

strong correlation

is

recognized

between

the

compressive

toughness

of

concrete

and

the

ductility

factor

'

of

the

M-e

curves

in

the

failure

zone.

･

Fig.13

shows

the

_plots

of

the

values of

l.

versus

the

ductility

factor

pt./e.

in

the

failure

zone.

The

evident

correlation

is

recongnized

between

them,

although

the

coefficient of cerrelation

is

slightly small cornpared with

those

ef

Fig,

lo

and

12. That

is,

the

increase

in

the

ductility

in

a

failure

zone also accelerates

the

extension

of

the

failure

zone,

and

as a

result,

the

_plastic

deformation

capacity

df

the

whole

RC

member

increases,

3.4 Applicability

of

stress-strain relationship of concrete

for

_prediction

of

deformational

behavior

df

RC

beam

The

applicability

of

the

a-E

curves

measured

from

the

test

of

concrete

prism

specimens

of

HID=2

to

the

deformation

analysis

of

the

RC

beams,

is

di'scussed

here.

The

shape

of

HID=2'is

considered

the

mos.t

general,

and

therefore

much

experimental

data

are

available.

As

illustrated

in

Fig.

14,

screw

bolts

were

set

in

the

spgcimens as

in

the

flexural

spans

of

RC

beams.

According

to

the

measured a-E

curves,

the

stress

descending

portions

become

ductile

due

to

the

existence

of

the

screw

bolts,

regardless of

the

spacing

of

hoops.

In

the

_present

study,

therefore,

a-E curves which reflect

the

effect

of

the

s

¢rew

bolts

were represented

by

the

follewing

formulas,

5tress

ascending

portion

(Esl)

:

-

naE

_{.."...",,",,...",,,:H.."..:.,,-,.-,,".,"....".,".,".,".,-,,.".,".."..",,-HH-.,".,",,--(1)}

s

-

na'1+Eto

'

where,

S=alE,,

Fl

:

compressive

strength,

E=elE.,

e.

:

strain

at

compressivestrength, n.

:

empirical constant.

Stress

descending

po.rtio.n.

{E.>1)

:

S=

n.-nld+X

x.

････-:-:･-･･-･･･--････-･･-･･-･･--･･-･--･･-･･-･･-･･--･--･--･･-･････,"...･･,"...,...,.,".,H.,(

Z

)

where,

X=a(E-1)"+1;a,

ln,

n.:empirical

constants,

Eqs.

(

1 )

and

(

2 )

were

_proposed

by

Popovics

and

the

authors, respectivelyi3).

It

was

already

confirmed

tbat

most

of

the

descending

_portions

of

the

normalized

a-E

curves

(S-E

curves) can

be

well represented

by

using

Eq,

(

2 ),

independently

of

the

kind

of

concrete.

Figs.

IS(a)

and

(b>

show

the

comparisons

between

measured and calculated

{a=O.5,

m=1.3)

S-E

curves

and

a-E curves, respectively.

The

M-

¢ relations

for

RC

beams

obtained

by

an

analysis and

the

_present

experiment

were

compared,

The

M-e

curves

were

calculated

under

the

following

fundamental

assumptions

:

D

A

cross section which was

plane

before

loading

remains

plane

under

load,

ii)

The

a-e relation

of

steel

bar

may

be

expressed

by

a

bi-linear

equation,

"i)

The

a-E relation of compressive concrete

is

_givenby

the

expressions

for

the

prisrnatic

specimens of

HID=2,

Figs.

16<a)

and

(b)

show

the

coTnparison

between

the

analytical

M-

¢

curve

and experimental curves

in

various

(7)

o

HA9m e:za m$ege v:es.za e boO b

.rv-e)M-8Has

oov eom

123

4, S6

RELATIVE STRArN

{E}

(a)

S-E

curves

oept oeH

7

1.2 1.S

!.4].eB

cNvq=

-N,RigtssEeR

em ee ony opt e

ss

s. AIOsst;xsx---kZlSsx

X

NN SsN NNx

ls N2h N

3h

Cal.

-.."

ExP 3h5.ill Nlp=1.-25h

e O 246 8 10

12

14 STRAIN

C

x 10-3

}

,E

(b)

a-E curves

FigT5

Comparisons

between

measured

curves blvflxs .N-eu.bvEreg epm ee ov opt e 2

4

6 B

CUHVATuRE

(xlO'2)

,

d"

(a)

55-A,,

specirnen

5S-A.

_3h

rN:lsh.'i.---

"tw

-

Ns

X

hNN

tt

X

･'"NN

X L

hNs Xx

_Xs..s-t

XX

N..x

--"::S

Nx XNs

2h

NKxs2!:h N3b

ttt.z.it'''

10

15 IS 20 and calculated

regions

(failure

zone

(l.),

2h

region,

and･3h

region;where,

h:height

of

beam,

the

center

of

the

failure

zone

is

assumed

to

be

located

at

the

cen'ter

of

the

regions)

estimated

from

measured

curves

by

using

the

idealized

failure

zone model.

According

to

the

figures,

the

experimental

M-ip

curves

in

different

regions

are

quite

different

from

each

other,

with

the

analytical

curves

being

located

between

them.

Therefore,

in

applying such analytical

M-

_¢

relations

to

_the

deforma-tion

analysis of

RC

members,

it

is

necessary

to

clarify

the

region

(corresponding

measurement region)

to

which

the

analytical

M-ip

relation should

be

applied.

Comparisons

such as

Fig.I6

were carried out

fer

all

the

specimens, and

the

lengths(t.)

of

the

corresponding measurement regions,

in

which

the

ductility

factor

(-de.sley)

of averaged

M-to

the

figure,

the

value

of

t,,

decreases

with

the

'

was obtained

between

t.

'and

T,.

la=

The

illustration

in

the

figure

shows

the

prism

specimen of

HID=2,

from

which

axis

locates

at

the

middle

height

of a

beam

section

oo

Ffig.

16 {

fi

de .. =

eo

Fig.17

2

4

CVRVATURE

(b)

55'Am

Comparison

between

experimental curves

length

6Cxle-2) , speclmenanalytical

in

varieus

S

10 dfp

M-e

curve and rneasurement

ip

curves

is

almost

the

sarne as

that

of

analytical

curves,

plots

of

the

values of

l.

versus

the

tottghness

(TJ

of concretes

in

the

compressive

increase

in

the

compressive

touhgness

{Z)

4.0-O.68T,--･･-･･--･--･--･･-･･-･･-･-･---･--･--･･-･-･･---･---･-･･････-･･---･---･･--･･-･･-(3)

dimensional

ralation

between

the

flexural

span

of

the

RC

beams

and

the

a-E cttrves

for

the

analysis were

measured.

Here,

providing

that

a

_4etttral

and

that

the

effect of strain

_gradient

can

be

negligible,

the

'

1 2

3

4

5

Tl

Ckgflan2)

Effeet

ef

toughne$s

{Ti)

of compressive concrete en corresponding measurement

length

(a.)

were

obtained.

Fig.

17

shows

the

zones of

the

RC

beams.

According

(8)

Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan

HID=2Pli'smspeclmen -tpt

(a}

Effect

of

distance

M.

o x"!e m6

ZI

-Zi

ratio of neutral vHx rda axis

(Xi)

pt

ol 2 3 4 S TI

(kgflcm2}

(b}

Effect

ef

toughness

of compressive zone of

RC

beam

(T,)

on the value of

_X,

at

kik:

being

maximum

Fig.18

Correspondence

between

compressive zone of

RC

beam

and

prism

specimen

corresponding

measurement

length

(l.)

should

be

equal

to

1 h.

Some

of

the

reasons

for

the

result

in

Fig.17,

showing

the

variation of

the

value of

t.

instead

of

the

constant value of

1 h,

may

be

considiered

as

follows

:

D

The

less

the

compressive

toughness,

_the

more

remarkably

is

influenced

the

a-e relation of

crete

in

the

compressive zone

by

_the

strain

dient,

the

behavior

of concrete after

failure

ing

more

ductile,

iD

The

lo.c.

ation

of

the

neutral axis

(see

Fig.

18(a))

at

the

settlement of

failure

zone,

_generally

after

compressive

failure,

varies

depending

on

the

toughness

in

the

compressive

zone.

That

is,

_the

depth

of

the

compressive zone

is

dependent

on

its

toughness,

axis

<Xi)

when

the

stress

block

in

increase

with

the

decrease

in

the

toughness

(

T,

},

w

.

Figs.19(,a)

and

(b)

show

the

comparisons

between

corresponding measurement

lengths

(

l.)

given

by

Eq.

(

3 Fig.

20 shows

the

applicability

of

Eq.

(

3 .)

the

test,

curvatures were measured with

two

couples surfaces of a

beam

by

means of steel

frames,

are agaiq obtained

betwcen

the

curves.

S4.

Conclusiens,

'

The

.cuTvature

localization

in

the

flexura

-34-Nvnxx

-e-'EqgtiEyg

ptvAx= LNnE)tssEeeR ptvnxs

-r'"'Eq"o"tiEeeR

oca owo otr etw o eco ee ew opt e

o

(a}

2 4 6 CURVATURE

(-lo-2)

Effect

of wateT-cement s , ddiratio

(WIC)

10 o

Fig.19

8 6e

est ept o 2 4

6･

S

10

cuRvATuRE

rxlo-2)

,

ddi

(b)

Effect

of spacing of stirrup

(S}

Cornparisons

between

analytical

M-

¢ cllrve and

,experimental

curve

in

correspondins measurement

length

_(l.)

Fig,

18(b)

shows

the

rela,tion

between

the

toughness

in

the

compressive zone

(T,)

dex

hi

k3

becomes

maximum,

hich

consists with

the

te4dency

of

the

value of

l.

shown

in

Fig

analytical

)･

to

the

prediction

of

the

test

results

obt,ained

_,earl

of

deformatien

and

the

length

of measurement region was

2 h.

.O･

2･･

4

E U 10

CURvATuRE

(rlo-2)

,

de

Fig.2e

Appiicability

of expression

Of

correspending

measuremeFtTlenglh.(tm)

and

the

distance

}atio

of

a

neutral

or at compressive

failureM,iM.

The

valtte of

_X,

tends

to

,17.

M-di

curves and experimental curves

in

t,he

Fairly

good

agreements

are obtained

between

the

curves.

ier

by

the

authorsi3].

In

transducers

attached

tq

the

top

and

bottom

Fairlygood.agreements

(9)

then

the

applicability of

the

llniaxial stress-strain relationship

for

concrete

to

the

deformation

analysis of

the

RC

beams

were

discussed.

The

following

statements can

be

drawn

from

the

study.

'

1)

The

curvature

loealization

in

the

flexural

span

of

RC

beams

is

remarkable, and

_the

span

can

be

divided

into

the

failure

and

th.e

non-fail.ure zoneg.

_,

_.

2)

There

exist

evident

correlations

between

the

toughneSs

of

concrete

in

the

compreksive

zofi'e,

the

length

of

the

failure

zone, and

the

ductility

factor

in

the

failure

zone of

RC

beams.

3)

The

increase

in

the

ductility

in

the

failure

zone also accelerates

the

extension of

the

failure

zone,

and

as a

result,

the

plastic

deformation

capacity

of

the

whole

RC

member

increases.

4)

Experimental

moment-curvature curves are

_quite

different

according

to

the

curvature measured region

even

in

the

flexural

span of

RC

beams

due

to

the

failure

or

the

curvature

localization,

s)

By

using

Eq.

<

3'

),

prediction

of

the

region

l.

is

_possible

_te

which

cross

sectional

moment-curvatllre

relation

'

analyzed with

the

stress-strain curve of

the

concrete specimen of

HID=2

should

be

applied.

Reterences

O

Kosaka,

Y.

,

Tanigawa,

Y.,

Hatanaka,

S.,

and

Miwa,

R. :

Curyatuie

Localizatioq

in

Plastic

Deformation

Range

of

RC

Beams

under

Ftexure.

Trans.

of

Japan

Conc.

Inst,,

Vol,7,

1985,

_pp,519-526.

.

2)

Kosaka,

Y,

,

Tanigawa,

Y.

,

Hatanaka,

.S.

, and

Miwa,

R. :

Stress-Strain

Relations

for

C6ncretes

in

Plastic

Hing.e

of'RC

Beam,

Proc.

of

Annual

Meeting

of

Tokai

Branch

of

A.I.J.,

N6,24,

_1986,

_pp.113-116

(in

_Japanese).

'

3)

Kosaka,

Y.

and

Morita,

S. :

Reinforced

Conctete

StTuctuTe,

Maruzen,

1975,

pp.385

(in

Japanese).

4)

Corley,

W.G,

:Rotational

Capacity

of

Reinforced

Concrete

Beams,

Jour.

of

ST-Div.,

Proc.

of

ASCE,

USA,

Vel,92,

No.ST5,

Oct,

.1966,

pp.121-146,

5)

Darval,

P,

L. :

Critical

Softening

ef

Hingesin

Portal

Frames,

Jour.

ef

ST-Div.,

Proc.

ofASCE,

USA,

VoL110,

No,STI,

Jan.

1984,

pp,157-162.

6)

Darval,

P.L.

:

Load-Deflection

Curve

fer

Elastic

Softening

Beams,

Jour,

ef

ST-Div.,

Proc.

ofASCE,

USA,

Vol,llO.

No.STIO,

Oct.

1984,

pp.2536-2541.

'

7)

Suzuki,

K,,

Nakatsuka,

T.,

Suzuki,

K,,

and

Yokegi,

K. :

High

Rigidity

EecentTic

Cempression

Test

Method

and

Mechanical

Properties

of

Concrete

under

Eccentric

Compression,

Proc.

of

Annual

Meeting

of

A.

I.

J.

_,

1978,

_pp,

1717-1720

(in

Japanese).

,

8)

Phipps,

M.

E. :

The

Strain

Capacity

of

Compression-Zone

Concrete

Subjected

toShort-Term

Loading,

Magazine

of

Conc.

Res.,

UK,

Vol,28,

Ne.95,

June

l976,

pp.85-100,

9}

Sakai.

Y. ,

Iwase,

H.

,

Rokugo,

K.

, andKoyanagi,

W. :

Failure

BehaviorofTwe-Span

Continuous

Beams

underBending,

Trans.

of

Japan

Cenc.

Instl,

V61.6,

1984,

_pp.711-716.

'

10}

Mugururna,

H.

,

Watanabe,

F.

, and

Tanaka,

H,

:

Study

on

Improving

the

Flexural

Deformation

Capacity

ef

Concrete

by

Using

High

Yield

Strength

Heep,

Trans,

of

Japan

Conc.

InsL,

Vol.],

1979,

_pp,365-368

(ip

Japanese),

11>

Oiaka,

Y.

,

Suzuki,

M.

,and

Kondo,

T. :

Relationbetween

Moment

and

Curvature

forReinforced

Concrete

Member,

Trans.

of

Japan

Conc,

Inst.,

Vol,1,

_1979,

_pp.349-352

(in

Japanese).

12}

Suzuki,

K.

,

Nakatsuka,

T.

,

Enornoto,

H.

,

,and

Sumi,

K. :

On

Properties

of

Ultirnate

State

of

PC

and

RC

Beams,

Trans,

ef

Cement

Assoc.

of

Japan,

Vol.34,

1980,

pp.433-437

(in

Japanese}.

'

13)

Kosaka,

Y.

,

Tanigawa,

Y.,

and

Hatanaka,

S. :

Experimental

Study

on

Inelastic

Stress-Strain

Behavior

of

Steel

Fiber

Reinforced

Concrete

under

Compression,

Trans.

ef

A.I.J.,

No.337,

March

1984,

_pp.15-26.

14)

Kosaka,

Y.,

Tauigawa,

Y.,

Yamada.

K.,

and

Hatanaka,

S,:Stress-Strain

Relations

of

Conctete

under

Unlaxial

Compression,

Trans,

ef

Cement

Assoc.

of

Japan,

Vol.37,

1983,

pp.279-282

(in

Japanese).

/

t

ls)

Koyanagi,

W,,

Rokllgo,

K.,

and

Uchida,

Y,

:Compressive

Toughness

of

Concrete,

Trans.

of

Cernent

Assoc,

o,fJapan,

VoL37,

1983,

pp.268-271

(in

Japanese}.

IG)

Tanigawa,

Y.

,

Nishikawa,

K.

, and

Kosaka,

Y.

1 A

New

Type

of

Stiff

Testing

Machine

and

Complete

StTess-Strain

Curve

of

'

Concrete.

Trans.

of

A.I,J.,

No.260,

Oct.

1977,

_pp.9-19

(in

Japanese).

･

l7)

Morita,

S.

and

Adachi,

N. :

Properties

of

Concrete

ip

the

Compression

Zone

of

Flexural

Members,

Jour.

of

the

Society

of

Materials

Science,

VoL20,

No.208,

Jan.

1971,

pp.59-66

(in

Japanese).

18)

Muguruma,

H,

:

On

the

Compre$sive

Fiber

_StTaip

of

Concrete

at

the

Flexural

Failure

of

Reinforced

Concrete

Beam

Section,

Jour.

ef

the

Society

ef

MateTials

Science,

Vol.24,

No.26.

May

1975,

pp.441-446

"n

Japanese}.

(10)

-35-Architectural Institute of Japan

NII-Electronic Library Service

Arohiteotural エnstitute 　of 　Japan

【

論

　

文

】

UDC ：624

．

072

．

7

．

012 ：624

．

04 ：539

．

42

畢

響

覊

軅

轡弩

純

曲

げ

を受け

る

鉄筋

コ

ン

ク

1 丿

一

ト

ば

り

の

破壊集

中

性

（

梗

概

）

正会員正会員正

会

員正会員

畑

小

谷

中

阪

川

輪

重

義

恭

隆

光

＊

夫

＊＊

雄

＊＊＊

治

＊＊_＊＊

　§

1．

はじめに

　

RC

架構

の

塑

性

設計

の

際

には，

構成部材が十分延性的

で

あ

り

，

かつ

構造物が崩壊機構（

rnechanism

）を形成す

るに

足

るモ

ー

メントの

_再

分配

が

_行

われることが

前提

となる

。RC

部材

の

塑性変形挙動

，

と

くにその

延性性質

（

ductility

）を解析的

に

論ず

るには

，

’

塑性

ヒンジ

領域に

おける

変形性能

と

塑

性

ヒン

ジ

領

域

の

広

がりに

及

ぼす

各種

要因

の

影響

を把握

してお

く必

要

があ

る

。

　

RC

ば

りの

曲げ靱性（

toughness

）

は_，

2 点集中載荷

実験

によって

評価

される

場合

が

多

い

。

この

場

合

，

等

曲げ

モ

ー

メント

区

間断面

の

素

材

の

力学特性

およ

び付着特性が

均

一

であれ

ば

，

同

区間

における

任意断面

のモ

ー

メン

ト

（

M

）

一

曲率

（

φ）挙

動

は

_同

一

となる_。すなわち，

載荷板付

近

を

除

け

ば同

区

間

における

破壊

は

均

等

に

生ず

るは

ず

で

あ

る

。

しかし

，

通常

，

実験

の

際

に

観察され

る

破壊

は

，

ある

領

域

に

_{集中}

し，その

領域（

本文中

では

failure

　zone と

呼

ぶ

）

は，

通

常

の

RC

ば

りではおよそ

（

1 −

2 ）

・h

（

ここに，

h

：はりせい

_｝

となる

。

等曲げ

モ

ー

メ

_{ント区間}

にこの

よ

うな

破壊域

が

集中

する理

由

として，

載

荷

点近

傍

の

応力

の

不均等性

，コンクリ

ー

トおよ

び鉄筋

の

材

質

の

不

均質性お

よ

び

コンクリ

ー

ト

ー

鉄筋間

の

付着

の

不均

一

性

な

どが考

えられるが

，

この

_{破壊集中性}

のために_，

_実

験

によって

得

られる

塑

性域

の

M 一

φ

関

係

は，

曲率

の

測定区間

によって

異

なることになる

。RC

ば

りの

曲げ靱性

について_，

平面保

持

を

仮定

した

通

常

の

_断

_面解析

による

推定

精

度

が

研究者

によって必

ず

しも

一

致

しない

理由

の

1

つは，この

曲率

の

測

定区間

の

不

一

致

に

あ

る

とも考

えられる

。

　

一

方

，コンクリ

ー

_トは

_{均等応力状}

_態

_下

_に_あ_っ _て _も

_，

_破

壊

は

供試体内

の

限

られた

領域

に

集

中

する

傾

向

があること

，

ならびに

応

力下降

域

のひ

ず

みの

実

測値

は

，

ひ

ず

みの

測定長

および測

定位置

によってかなり異なることも，

著

者

らの

_別

の

_{実験}

により

確

かめている

（

図

一

5 参照）

。そ　辱日本学術振興会　特別研究員

・

工博　＃名古屋大学　教授

・

工博＃一三重大学　教授

・

工博騨輯 _{清水建設}

・

_工_修　　（昭和 61 年 5 月 17 日原稿受理

1

のため，

RC

部材

の

曲

げ

解析

の

際

，より

精度

の

良

い

結

果

を

得

るには，

単

に

常用

の

標準供

試

体

の

所定

区

間

で

実

測

した

一

軸圧縮下

の

応力（

σ

）

一

ひ

ず

み

（

ε

）関係を用

いるのではなく，

RC

ばりの

当該部

に

相当

するコンクリ

ー

ト

供試体

の

応カ

ー

ひ

ず

み

関係を適用

するの

が適当

で

あ

る

。

　

』

本

報

では

，

構成素材

の

一

軸下

の σ

一

・

関係を用

いる

RC

部材

の

塑性変形解析

の

精度を向上

させるための

基

礎資料

を得

るこ

とを目的とし

て_，

まず

，

静的

荷

重を

受

ける

RC

ばりの

等曲

げモ

ー

メン

ト区間

にお

け

る

塑性域

の

曲率分布

を実験的

に

調

べ，

次

いで，

RC

ばりの

曲げ圧

縮部

を

模

擬

した

コンク

リ

ー

ト角柱体よ

り

得

ら

れ

る σ

一

ε

関

係

の

RC

ばりの

曲げ解析

への

適用性

について

論

じた

。

　§

2 ．

実験方法

　

RC

ばりの

曲

げ

載荷実験

では

，

表

一1

に

示

すように_，

実験要

因

として

，

コンク

リ

ー

トの

水

セメン

ト比（

W

／

C

）

，

あ

ば

ら筋

の

ピ

ッチ

（

S

）

，

鋼繊維

の

体積

混

入

率（

Vr）

，

引張

鉄筋比〔

P

，

）

，

複筋

比

（

γ

〉

，およ

び載荷

スパン

（

1

，

）を取

り

上げ

た

。RC

ば

りの

配筋図を図

一

1

に

示す

。

また

，

RC

ばりの圧

縮部

の

_{材料特性}

を

調

べるために_，

表

一

2

に

示す

各要因

を

パラメ

ー

タ

ー

とす

るコンク

リ

ー

ト角柱体

の

高

ひ

ず

み

領域

に

至

る

ま

での

一

軸圧縮実験を行

った

。

角柱体

の

断面

寸法

はすべ _て

9 ，

7x9

．

7cm

_と

_し，　

RC

ばりと

同様

に

横

打

ちによって

製作

した

。

コンク

リ

ー

ト

の

製

作

には

，

普

通

ボ

ルトラン

ド

セメン

ト

，

川砂（

5mm

未満

）

，

川

砂利（

5 〜

_15mm

_）

，および

鋼繊維（

フック

付

せん

断

ファイバ

ー，

形

状

：

O．5

×

O．

5

×

30mm

_，

引張強度

：

7000

kgf

_／

cm2

_）

を

使用

し

，

調合強度を

200

一

300

kg

’

f

／

cm2 とした

。

試験

体

の

個数

は

，

はり試験

体

については原

則

として

各要

因ごとに

1 体（

ただ

．

し

，

主

要

な

要

因

については

2 〜

3 体〉

，

角柱

試

験体

は

各要

因ごと

に

3 体と

した。

　

はり

試験体

の

載荷

および

曲率測定要領を図

一

2

に

示

す

。

すなわち

支持

スパン

9h

（

＝

174 ．

2cm

）

の三

等

分点載荷とし，

曲

げスパン

（

1

，

＝3h

）

の中央部

2h

の領

域

を

4 分割

した

各区間

の

曲率を測定

した。

終局状態

に

至

るまでの

曲率を

はりの

上

下

端面

に

取

り

付

けた

計

16 個

の

変

位計

によって

測定

した

。

また

，

同時

に

，

荷重

上

昇域

のは

一 36 一

N工工

一

Eleotronio _Library