低側圧参軸圧縮下のコンクリートの塑性変形挙動(梗概) : 各種横拘束コンクリートの圧縮靭性の統一評価(第1報)

(1)

Architectural Institute of Japan

ArchitecturalInstitute of Japan

(at

Rl

UDC:691.32:539.374

PLASTIC

LAn

'

Journal

of

Sttuctural

and

Constructioo

Engineering

NdynM\kesxxXtsNvatsff

'

ng

377

e･mata

62 e7fi

CTransactions

ef

AIJ)

No.

377,

July,

1987

'

DEFORMATIONAL

BEHAVIOR

OF

AXIALLY

'

LOADED

CONCRETE

UNDER

LOW

LATERAL

PRESSURE#

'

Evaluation

Method

for

Compressive

Toughness

of

Laterally

t

tt

tt/t

Confined

Concretes

(Part

l)-

･

/t

'

tt

by

SHIGEMITSU

HATANAKA*,

YOSHIO

KOSAKA**,,.

and

YASUO

TANIGAWA"'

Members

of

A.

I.

J. gl.

tntroduction.

'

.,

For

the

application

of

the

_plastic

design

method

to

reinforced

concrete

(RC)

frames,

it

is

the

first

requisite

that

constitutive

RC

;nembers are

ductile

enough

for

the

redistribution

ef

mornent and

for

the

formation

of

plastic

collapse

mechanisma).

It

is

well

knowp

that

the

_preferr,ed

failure

_pattern

of

RC

members

is

a

flexural

failure

_preceding

type

with adequate warning

of

collapse,

then

increase

in

the

toughness

(energy

absorptien

capacity) of concrete

in

the

compressive zone of

RC

members

is

_qttite

effective

to

improve

their

ductility,

_.

It

has

been

pointed

out

that

the

mechanism of

the

toughness

improvement

of

confined

concrete and steel

fiber

reinforced

.concrete

under

cempression

is

malnly

due

to

the

lateral

cenfining

effect

_gf

lateral

bars

and steel

fibers

caused

by

the

Poisson's

effect of cencrete after

failure5).

This

effect

can

be

easily related

to

th.e

behavior

of

plain

concrete under multiaxial

stfess

states61'l?).

Thus,

it

is

_possible

to

discuss

systeinatically

the

toughness

improvement

of various

kinds

of

laterally

confined copcretes,

based

on

triaxial

cempression

test

data,

especially en

the

post-failure

behaviQr

of concrete under.,a

low

confining

lateral

pressure

of

less

than

about

20kgflcm2iO),

However,

information

is

not

adequate

to

date

to,develop

areliable constitutive

model

including

such a

_post-failure

behavior

of

coticrete'L]i)-i3).

Some

of

the

main reasons are

in

the

difficulties

in

the

c6ntr61

6f

sitrain

rate and

the

measurement of

lateral

strains, resulted

ftom

the

fact

that

the

dilatation

of concrete

beYond

_"ltimate

stress

is

remarkable

in

this

range

of

lateral

pressure.

The

authors,

therefore,

have

developed

4 triaxial

testing

method

in

which a

high

rigidity

compression

testing

machine and a,new

type

of

lateral

loading

device

are combined, and cenfirmed, using

this

method,

that

fundamental

information

can

be

obtained

for

discussing

the

mechanism of

the

toughness

improvement

of

laterally

confined eoncretesi4)'i6).

There

are

two

maip

_purposes

ip

the

_present

study,

The

first

is

to

propose

an

evaluatien

system

for

the

effect of various

types

of

lateral

confinements on

the

compressive

toughness

of

c6ncrete,

based

on

the

behavior

of concrete

und,er

triaxial

compression

(Part

1,).

The

second

is

to

discuss

in

terms

of

the

toughness

evaluation

system

the

rational

combination

_pattern

of concrete and othe: materials, especially

steel,

to

provide

a required

ductilitY

to

RC

members

(Part

2).

,

The

present

paper

<Part

1)

describes

the

re$ult ef

the

investigation

relpted

to

the

first

purpose.

First,

the

effect

of

various

factors

on

the

_plastic

deformational

behavior

of

boncrete

under

triaxial

compression was examined

in

an

experime,nt.

Next,

'a

systematic method

based

on

the

triaxial

compressive

test

data

was

proposed

for

the

evaluation

of

the

compressive

tbughness

of various

kinds

of

laterally

contined concretes.

Further,

the

stress

(

a,

)-strain<

Ei

)

curves

in

the

direction

of

the

maximum

principal

compressive stre.ss were numerically expressed,

the

expression

being

used

asastandard

in

the

eyaluation

system.

,

'

.

'

#

This

paper

is

based

on

the

earlier works

_presented

in

Refs.1)

through

3).

i

_ReseaTch

_Assoeiate,

_Department

_of

_Civil

_Eng,,

Nagoya

Univ.,

Dr.

_of

Eng.

i-

Professor,

Department

of

Architectute,

Nagoya

Univ.,

Dr.

of

Eng.

･

i"

_Professor,

_Department

_of

_{Architecture,}

_Mie

_Univ.,

Dr.

of

Eng.

(Man"script

reeeiyed･,June

2,

1986)

(2)

-27-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute of Japan Notationof SpeCinenw!cHIDtt

'eLCkgf!cm2}s'Ccm)u'

o･loo ,l･3 o.4+"

'e.g.)'-S-1-O.4I,Igii/E::o-4

Ta.U.:#ik.Ylf.!te.Ms2t 5+10+

L1.01.32.e'

e O.45.O.55,O.,70e 6+1j 5'o10o 12t25･10

g2;

Experimental

procedures

'

.,'

''.

'

tt

ttt

2.1 Outline

of experiment

The

outline

of experimept

is

shown

in

Table

1. The

variables

in

the,experiment

were

the

water-cement ratio

(vrIC=O.45,

O.55,

and

_O.70).1

hei'g'

ht

{H)lwidth

(D)

ratio of specimen'(HfD=1,

1.'3,

and

2),

magnitude of

lateral

_pressure

₍

a.=O,

1,

3,

6,

and

12 kgflcm2.

Here,

these

values of aL

g'raduaHy

increaSe

from

about E,=5 ×

IO-3

in

the

stress

descending

range

due

_to

_the

rapid

dilatation

of concrete

･in

the

experiment

),

spacing

of

loading

_point

of

'

lateral

_pressure

_(S=o,

5,

and

10cm'),

arid

friction

at

the

sb'ecimenlloading

_platen

interfaces

(coefficient

of static

friction

"=O

and

_O.

_4).

The

term

"pressure

(

aL)"

is

used

for

tfe

representqtion

of

the

balanced

lateral

stresses,

i.

e. ai. =a,=ala, _and

the

subscripts

_of

both

_stress

(a)

_{and strain}

(E)

.represent

th.e

_co-ordinate _axes

(see

Figs.

₁

_and

2),

compression

being

_positive.

The

cross-section

of s'pecimeri

for

the'triaxial

test

was

kept

to

loxlocm.

Three

specimens

_were

_prepared

for

_gach

variable.

2.2 Fabrication

hnd

curihg of specimens

'

･'1-'

Ordinary

Pbrtlarid

cement, river

sand

(maximdm

size

:

'5

inm)

and river

_gravel

(size

range

_:s-lsmm)

were

prepared

for

concrete.

The

slump

of

concretes

was

designed

t6

be

5

cm

for

all

the

batches

in

order

to

minimize

the

'

variation

of

mechanipal

properties

of

the

specimens resulting

from

bleeding.

Concretes

were

cast

into

stbel molds and consolidated with

a

wooden

hamrner

and arotary

type

_poker

vibrator

from

the

outside of

the

molds.

All

the

specimens

'

were capped with cement

'paste

at

thle

age

df

_1day

and

deinolded

ht

the

age

of

2days,

and

thbn

cured

in

an air

conditioned

room

<23

±

20C,

relatiye'

humiditY

7s

±

s

%)

until

tes'ting,

exc'ept

that

some

cylindrical

spe6imbns were cured

'in

"rater.

The

tests

were carried out at

thti

ag'e'

ol about

_50days.

･

'

_'

'

'''''

'

z.3

Methods

of

loading

and'measurement

'

･

''

L

'

''

'･･

･

･A

triaxial

dompressive

loading

clevice

itlttstrhted

in

Fig.1

'

.

t

Table

1 'Outline

of experiment

'i

''

''''

was used, with which arbitrarylateral stresses'are applied'

to

tt

the

sipecim'en

by

mea'n'S

of

'the

flexutal'moment

rlesistance of

'steel

plates.

The

lateral

loads

applied

are estimated

by

the

straih'of steel

6olts

set

between

the

steel

_platen

and a steel

barl

The'initial

values of

lateral

'stresses

are controlled

by

''

tightenirig

the

steel

bars

with nutS.

All

_the

specimens were

set uPside

down,

'that

is,

a capped

surface

WaS

set

dn

the

'

othdr' s'ideiof a

tiltlng

_platen

_(see

_Figs.1

and

2)

in

order

to

ENotes]

wlc: water-cement ratso, HID, H.i.ghtl alleviate

the

'concentration

_ofthe

_failure

_in

a'specimenm.

All

gidst,h.:f.8P:;

±

:::s.a.l:.t:grr,a.r.70:{iX;g7.:;･i::2

'

_the

specirnefis

for

the

uniaxihl

ahd

_triaxial

_testg

W6re

l6aded

Zgl":1'O,S,70flfiglr,"t,:;

fi;:://C.;r:i:,tl87g,r"i??iY'

ih

the

16ftgitudinal

_('first)

direction

under

the

constant slrain

:;e.yof:E.HfD=1

and 2 of WIC'O･5S, '": _O"IY _iOr

rate of about

₂

×

loL31min."by

using

a

high

rigidity'cbmL

'' i

'''

"'

prb'sgion

testing

thachine.'

For

the

di$persed

t'y'pe

lateral

lp.

p.,.d.i.g･.･,t'b.

'rOugfi･･

･

,T.i,!!iR.g

piaten

'

_{cohfineinefit,}

_aset

_df

_steel

bars

_with

_S'><5mm

_sebtion' _were

MechinePIaten

Stee1E..ifl..t.e.n.

Yieldlng,Section'Stee1Boltt

.t.

./

w･s,,.g.r.ls.･

set'between

'a

specimen'

hnd

･'Fig.7(a)).

'

The

complete stress-strain

X-Y

recorder up

to

a specifie

,

,/ .

.t

,1.

.･

･

Yielding

5qction

lateral

loadirig'platens

_(see

.

t.

1

.

/

t. 1 /

,

.

curves wefe recorded

by

an

d

s'train

<E,=30

×

10-3).

The

'

Ol,El

/

.t

{a)

Exterior

view

FIg.1

Triaxial

loading

;,

"

(b)

Section

/t

device

'

S.G.,s

I,･

.･

'

/

U2,e2

.U3tE3

Capped

.-

Cast

tt

'

Flg.2

Convention

fer

subscripts

'

'of-stress

and･'str'ain

---

₂₈

(3)

-Architectural Institute of Japan

'

friction

at

the

specimen-loading

_platen

interfaces

was reduced

by

_placing

friction

reducing

_pads.

The

_pad

consists of

two

_{polypropylene}

sheets with silicon

_grease

between

them,

with which sufficient uniformity of stress

distribution

on

the

ends of

the

specimen can

be

expectediS).

It

was confirmed

by

using "pressure sheetsr']S}

_placed

at

the

'

specimen-platen

interfaces

that

the

laterai

stress

distribution

induced

by

the

loading

device

was adequately uniform.

For

all

the

spticimens, a couple of

differential

transformers,

were set

between

the

tilting

_platen

and

the

machine

base

platen

to

measure

the

over-all' strains.

For

cubic

,specimens,

lateral

strains were measured

by

two

deformation

'

transducers

as shown

in

Fig..1.

For

the

specimens ttnder uniaxial

compressio-,

two

c6upies of wire strain

_gages

(W.S.G.

)

we;e

_glued

to

the

specimen

to

measure

the

longitudinal

and

lateral

strains.

'

93. Test

results

and.discussion

･

'

Since

the

displacement

w'as measured

by

a

couple

of

differential

transformers

set

between

_platens,

the

strain

in

the

stress ascenaing

_portien

of

thg

curves

is

over-es'timated

due

to

the･p'resence

of

the

friction

reducing

_pads.

Therefore,

the

uniaxial

longitudinal

tst'i

ess'1(a,)-longitudinal strain

(E,)

curve was used up

to

the

stress at which

the

tangent

'

modulus of a,-si curVe

from

the

_!riaxial

test

becbmes

compatible

with

that,of

the

curve

frgm

the

uniaxial

test,

3.1 Failure

patterns

of specimens

Typical

ultimate

failure

_patterns

of

various shapes of specimens are shown

in

Photo.

_1.

As

seen

in

Photo.

_1,

cracks

extend

thrQughout

the

whole

height

for

the

specimen

of

HID=1

and

1. 3,

while crushing occurs only at

the

upper end zone which corresponds

to

the

upper-zone of specimen at casting,

for

the

specimen of

HfD=2.

This

_trend

is

observed _regardless _of

the

level

of

lateFal

pressure

applied

in

the

experiment.

'

3.2 Effect

of magnitude of

lateral

_pressure

Figures

3 (a)

and

(b)

show

the

effects of uniformly

distributed

lateral

_pressure

on

the

deformational

behayiors

of

the

specimens of

HID=1

and

2,

respectively,

For

bdth

shapes of specimens,

large

increases

in

load-carrying

capacity and

toughness

are observed even under asitiall

lateral

Pressure

such

as

at=3

kgflcm2,

The

increases

in

_peak

stress and

toughness

due

tg

_{a certain}

level

_of

Iateral

pressure

are almost constant regar.dless of

the

HID

ratio of

'

specimen.

Furth,er,

lateral

strain at acertain

longit",din'al

strain

level

in

the

stress

descending

range

decreases

almost

proportionally

to

the

magnitude of

lateral

stress.

3,3

Effect

of wateT-cement ratig

･''

'

Figures

4 (a)

and

(b)

show

the

effect of water-ceinent ratio

(

WIC)

on

the

a,-e,

curves.

A

similar

trend

of

the

effect of

WIC

on

the

at-Ei curve of concrete under uniaxial

compression,

is,obs,erved

for

concrete under

triaxial

tt

compression,

Namely,･the

strain

leve.1

at

-the

b'eginning

of converging

bfanch,

which

is

usually observed

in

the

'

descending

_portion

of

the

_q-Ei

curves of

the

concrete

under

uniaxial

compression]9),

_gradually

increases

with

increasing

lateral-pressuTe.

The

concrete specimen at

this

strain

level

is

considered

to

sufferacertain critical

damage

regardless of

the

strength of concrete.

_,

3.4 Effect

of

heightlwidth

ratio of specimen

Figure

5

shows

theeffect

of

HID

ratio of specimen on cr,-E, curves.

The

stress and

strain.

at

the

_peak

_point

slightly

increase

with

decrease

in

the

_HID

ratio of spegimen.

The

effect of

the

HID

ratio of specimen on

the

shape of

the

'

-

'

descending

_portion

of a,-}, cur"e

is

_quite

remark-mrmua1l･M.ik'IX,xffrmt

s,

sti's':,g'g,:.L'i

ulD"l-3.,wuth-.a.7 kgFrkflfi

,...,.ne.:.,fi-l.,(toweF,

'e.h''dL'''eoi,r.Ss'"p,gnd"

Ill:

Pheto.1

Typicar

ultimate

failure

_patterns

t

able regardless of

the

magnitudd of

latelral

_pressure,

i.e.

the

shape of

the

curve

becomes

steep as

the

HID

rat.i:oof specimen

increases,

Such

change

is

almost

independent

of

the

magnitude

-of

lateral

pressure.

'

It

is

Well

known

that

i)

the

a,-si curve measured

in

a conventional

'uniaxial

compression

test

is

significantly

influenced

by

_the

_{strain measvrement} region and

ii)

it

is

_quite

important

to

make clear

the

equivalence of

the

failure

zones of a

tested

concrete

specimen

and a reinforced concrete

mem-ber

to

which

the

a,-e, curve

is

to

be

applied

in

_the

(4)

-29-Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute ofJapan ' -Rcn,s).t"wzaN8 eom

6o"

oon. oopt eo･--Rpt-B".bvzaH: e eeut oe" eom oDpt // '

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On al-EL, Eh and Es

D24

S

_10''12

14,

16

le 20

sTRAIN

{xlo--3}

,[1

'

(a)

HID=1

'

_Fig.4

_Effect

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-Re--E)xo

zaNza

ratio eow eem oeN oeH o

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curves ef aitially /tt

16 loaded

20

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246e le 12 STRAIN

C

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(b)

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On aL-et curves 14 16 IB 2e tEI

30

(5)

Architectural Institute of Japan

ArchitecturalInstitute ofJapan

deformation

'analysisi7i･eni'22].

Koyanagi

et a12Z). reported

an

analytical

model

for

the

load-deformation

curves of specim6ns,ranging

from

HID=2

to

4 in

which

the

deformations

in

failure

and

elastic

zones

are

superimposed,

In

the

experiment,

the

casting

direction

of concrete was normal

to

a

loading

a\is

_.and

the,friction

at

the

spgcimen-loading

platen

interface

was

not reduced.

The

authorsi6)

discussed

the

relation _/between

the

length

of

failure

zone and

the

measured

a,-e,

curve,

based

on

the

data

from

a

triaxiai

test

of

the

specimens of

HID

=1

to

2 in

which

the

casting

direction

of concrete was

_parallel

to

the

loading

axis and

the

friction

was reduced

in

the

same manner as

in

the

_present

experiment; and

_proposed,an

idealized

failure

zone model shown

in

Fig.6

(a),

Based

on

the

Photg.

_,1

and

the

_{ai-e, relation}

in

the

idealized

crusheq zone

.is

E

o[g.-mgi-g-.8,NgUH

'o

Fig.5

D 2 4 6 e 10-12 14

sTR"LIN

_cxlo"3),tl

Effect

of

heig'ht/width･

(HID)

speclme,n onla,-sl curves

16 le

'2D

ratio of

assumed

to

be

equivaient

to

the

a,-E, curve

from

the

specimen of

HID=1

as well as

in

the

earlier reporti`).

The

dashed

lines

in

Fig,

6 (b)

are

the

_predicted

a,-E, curves

for

the

specimen of

HID=

1.3

and,2, obt4ined on

the

cpndition of

l.=D(l.:length

of

idealized

c.rushed zone,

D

:

width of speclmen),

The

predicted

curves are

in

_good

agreement with experimental

_gnes.

Aithough

the

ai-ei curves

for

the

crushed and non-crushed zones

bifurcate

at

the

peak

point

in･

the

_presept

model,

the

internal

failure

of aspecimen

progressively

develops

.after

the

critical stress

_point,

Therefore,

different

ai-Ei

curves

should

be

provided

after

the

critical

stress

point

e.g.,after

O.8Fc

(Fc:peak,

stress) regardless

of

lateral

stress

level

fpr

mo,re

detailed

discussion.

_,･

.

'8za:

For IdealtzedL-!-, o shetl STRAIN

(a)

Idealized

Qo"

Nbe

.o.-.

ptNs>eMOe.Jge

g$-･

e

failurezone

rnodel O24 t

Fig.6

eow e".g.AMgx"oe.: rv8za

:za

-o02

(b)

Cernparison

4

6 8 10 12 14

sTRAIN

(

x 10-3

)

,El

･between

measured and

16

le

20

predicted

curves

fi

e 16 o 246

,s

lo

o 246

sTRILIN

(x10-3]

,El

(c)

Prediction

of

post-peak

a,-E, curves

Idealized

failure

zone model and

its

applicability

e lo

(6)

-31-Architectural Institute of Japan

NII-Electronic Library Service

'The

_{appli'cabilitY} _o{

_the

present

model

to

the

_pre'dictiori

of

the

stress

deseending･portion

is

discussetl.

Figure

6 (c)･

shows

the

bomParison

bet"ieen

'the

_ptedicted

ahd

the

measured a,-'e, curves

in

the

stres's'descending range

for

the

specimens of

HID

='2 with' various compressiive strengths

hnd

lateral

_pressitires.

Fairly

_good

agreement

is

observed

between

the

tWo

'cinrves

regardless of

tlie

level

of

･lateral

_pressure

for'

ViilC!'O.

45

and

O. 55,

while

the

_predicted

cuf"es are

gteeper

thhn

the

mehsured ones,''and

the

dif'ference

becomes

larger

as

the

level

of

lateral

_pressure

dee'reases

for

WIC=OL'70L

One

of

the

reason's'for such a

difference

in

_prediction

thay･be

due

to

the

fact

that

the

failute

of

the'

6oncrete

of

WICtO,

70 is

rather

ductile

compared

to

-that

of

WIC=O.

45

and

O,

55,

Namely;''due

to

the

relatively

/

.

smdll'degradation' of

the

loadlcarrying

eaPacity

i'n

the

'crushed

zone!,.

tbg

failure

concentration or

the

st{,ain

localization

iS

consiidered

to

be

alleviated,

In

order

't6

refiledt sdch a

trend

in

the

_present

model,

the

length

of

the

ciushed zone

(l.)

should

be

slightly

increased.''

'

.-'

'

_'

,.

t

ttt

3.5 Effect

of'dis'persed

lateral

confirfeinent

'･-･

'-...

''

,.

.".･'

_,

,

tt

/

.t

t

In

the

preliminary

experiment,

it

was'observed

that

the

a,LE, curve

is

pgtin'fluenced

so much

by.the

arrahgement of

tt

t

/

t

la'ter'al

'bars

_<E]

and

D)

for

the

_pitch

(S)

df

10

cm.''Therbforel

in

the

_presit/'

nt study,'

the

f61meg

arrangement was

adoPted,

'Figufe''7

(a)

Sh'ows'

the'effects

of

the

_pitch

-(

S>

of

laterti1

confin'emhe'nt o'n

the

al-e,

gurVes

gf

concrete,

T. h'e

t

/

ttt"

t.

evident _{effect of}

the

Pitbh

is

_{not observed} _on

the'stress

and

_{strain at}

the

_peak'point

_as

far.as

the

_present

_experimefit

iS

'

concerned,

i.

e,

the

_pitch

(S)

being

lesS

than

the

width

('D)

of specimen and.the

lateral

Pressure

being

not more

than

t

tt

12 kgflcm2.

HeweVler,

the

effect ef

the'

disPersion

Of

-late'Ta'1

_pre'ssure

becgmes,

_gradually

remarkable after

the

_peak

point,

that

is,''the'

decreasb

in

the

10tidicarryin'g

capacity of

ti

specimen

becomes･larger

as

the

_pitch

increases.

Furth'er,

such

trehd

6e'comes

m'ete

inarked'with

iric'tease

in

the

lateral

_pressure.

_Figttre,7(b>

shows equivalent

lateral

_pressures

_(i,)

for

the

sPecim6ns

6f

HID=1

and

2

under'

dispersed

lateral

confinem6nt', whe're'the

_q-e,

curves of

the

specimens are evaluated

by

those

of

the

specimens under uniformly

distributed

confinement.

g4.

Evaluation

apd

estimation

of

compressive

toughness

of

iaterally

confined

concretes

4.1 Equivalent

later.ai

_pressure

t/

The

improvement

of compressive

toughness

of concrete

due

to

lateral

confinernents can

be

evalttated

in

terms

of

"equivalent

_lateral

pressure

(iL)"

.which

is

defined

as a

_pressure

whose effect on

the

load-carrying

capacity of a

specimen _{at a certain} _strain

leyel

is

_equivalent

_,t,o

that

by

_uniform.ly

distributed

lhterti1

_pre'ss'ure,

The

equivalent

'

lateral

_pressure

is'

_given

as a

function

of s,f.fai,n

(E,)

by

c;omparing

the

a,-i,

curves

of

_the

confined

concretes

under

uniaxial compression and

those

of

_plaip

cpn' crete unaer a.

_.certain

triakial

stress state:

One

of

the

standard conditions

'

adopted

in

the

_piesent

study

is

as"follpws:

'

t

t.

'

Shape

of specimen:HIDLI

･

'

･,

ttt

t

Loading

_path

of

1titeral

_{pressute:active}

loading

''

.

'

Ratio

of

two

lateral

stre.sses:atla3=1

_,'

'

V"

Leading

_point

of

lateral

_pressure

:

uniformly

distributed

over

lateral

surfaces'of specimen

t

tt

t

tt

t

leA.

S=OC:

Ao

:

rw Y o o o M

'

g

L'

N w!c=e.ss

1 g

H

:

b-:ptEq"t'n58za: Q024

6 e

10

12 14

16

sTImlN

Cxlo-3),[1

(a)

al-El curves

'

Fig.7'

Effect

of

pitch

IS 20

E>'MO,

zaE

s::::,zn8

(S)

bf'lateral

e O 24E B 10 12 14

sTRAIN

{xle-3)

.Ei

(b}

Equiyalent

lateral

pressure

confinerne'nt on' a,-ss curves

IE IS 20 culves

(7)

-32-Architectural Institute of Japan

Here,

active

loading

means

the

loading

_path

in

which

the

latgral

stresses

(

cr,,-

op)

are applied

to

a

specimen

before

axial

loading

(

ai),

The

effects

of

various

factorS

on

the

a,-s, curve,,of concrete under

the

above

standard

condition are schematically shown

in

Fig,8.

The

evaluation

_ofoi-ei _curve

in

terms

_of

_.the

_equivalent

lateral

pre.ssure

has

a

_significant _merit,

Since

the

equivalent

lateral

_pressure

(Vt)

is

a

_physical

_quantity,

it

is

considered

_possible

_to

estirnate

to

seme extent

the

toughness

improvement

due

to

the

combination of various

tYpes

of comPosites

(multi-order

composites)

just

by

superimposing

the

each value of

li,

and,

if

necessary,

by

introduc.ing

areduction

coefficient

for

the

combination of

6,.

The

equivalent

lateral

pressure

for

the

multi-order composites

is

calculated e.

_g.

by

using

Eq.

(

2 )

in

Table

2,

where

the

effects of

the

factors

treated

in

the

_present

study are assvmed

to

be

independent.

4,2

Stress-strain

model

tt

The

stress( ai>-strain(E,)

curves

in

the

direction

of

the

maximum

_principal'

compressive stress were numerically expressed.

Tension

is

_pbsitive

here

in

_the

calculation

of

failure

criteria.

For

n6r' mhlizing stresses and strains,

the

absolute values

of

the

uniaxia1 compressive strength

(alr)

and

the.strain

_(s.)

at

the

uniaxial

compressive

strength were used as standards,

(1)

Strength

failure

criterion

The

four

parameter

fu,nction

used

in

Ottosen's

criterionZ3)･24)

{see

Eq.

(

₁

)

in

Table

_3),

which

is

rather simple and may

be

easily calibrated, was used

to

express a strength

failure

envelope,

As

seen

in

Fig.9,

however,

Ottosen's

criterion

(dashed

line)

_provides

slightly

lower

values

than

the

data

obtained

by

the

authorsi-)'i6] and

Mills

et a125).

in

the

range of relatively

low

lateral

_pressure.

Therefore,

the

_parameters

_in

_the

function

were calibrated

based

on

the

_present

ex-perimental

data.

The

obtained values

of

the

parameters

and

the

criterion are shown

in

Table4

and

Fig.

g

(solid

line),

respectiyely,

It

was also cgnfirmed

that

the

modified criterion

is

in

rather

_good

agreerpent

with

the

experimental

data

in

the

range of

high

lateral

_pressure

reported

by

Richart

et al,,

Balmer,

etc24),

(

2 ₎

Strain

failure

criterion

Strain

at

failure

(ev)

was assumed

to

be

_given

by

a

function

of

the

first

invariant

of stress

tensor

at

failure

{Ar)

in

the

form

ef

Eq.(2)

in

Table

3. The

reason why

the

criterion

was expressed only

by

the

value of

Lt

or

hydro-pressure

component

is

as

follows:

i)

consider-able

scattering

is

observed

in

the

measured values

of

the

strain at

failure,

iO

it

is

_quite

difficult

to

_precisely

measure

the

strain

at

failure

in

a

tTiaxial

test,

and

iji

)

in

actual

confined

concretes,

increase

in

the

deviatric

stress component

due

to

the

irnbalance

of

two

lateral

stresses

is

rather

small.

Figure

10

shows

the

comparison

between

the

_proposed

strain

failure

criterion and

the

experimental

data,

empirical con$tant

a

in

Eq.{2)

in

Table3

being

determined

to

be

2.2. (

3 )

NoTmalized

stress-strain

relationships

The

expression

of

normalized

stress

(a,la,)-strain

(EilEc)

relationship

for

concretes

in

uniaxial

compress-

o-,L.-±

L--ks' VSI..--x....<XAV"

.

"NL

HID=l

HID=2

#

ee

-H!D=1

_HID=2

L--g-HIDIsxe

t e a

1.-s/

o

e/

N

cr2io3

e2tej o3 Cl"e ele

;l

.

_ee

..e

2. ee.

't

02;e3 _a'2;a313 ed e

s=o

SkOs

E

"2eeq cTn

se

s

IO

cm e e

Fig.8

Efiects

of various

factoTs

on triaxial concrete under standaTd condition

Table

2 Estimation

of equivalent

lateral

composlte concretes

Ol-Elccurve of

pressure

for

For Eirst-OrderComposites

For

oLtEl) ia.B(El)

･yC[

D

-eL

---

_a)

where, a; Ceefftctent coneerntng rotto uf two leteral stresses.

S(El); Coefftctent conceTning p±tch of copfined _p/]ints,

TCEI}':Coeffl[tent concerntng lnter,/1stres.g _pnth, eL: Mnximum loteTnl pressure

tir

m:txln/"/:/ nveritge t)f Lvt!

lateTal stressps expected. Multi-Order Composites' Ol(El)=ZULiCEi)

i=?(ori･BICeD-Ti{[i)･crti)-""'T""""--t2)

(8)

--Architectural Institute of Japan

NII-Electronic Library Service

ArchitecturalInstitute ofJapan

/'

Table

3 Stress

_{(a,)-strhin(ei)/'modet}

'

strength Failure crtte'rlon

(u

''

'

_'

fc!,,J.,cos3e}

=

_{Au:; + xgtt +B}t/}

-

_x

=

_o

---:---;'--")

'

"here-

ti:Fltet t"vartlint' eE strLss tenset, JhJs:kecend and thtrd tnvartants

/,

:[.::E::7,S.e;･

;･

:;::

.t::::;I:::r:xtl:tk.gsv"::::g:,f:m,',f;:::;,es,

,,.

.

_/, A,,B,K}tKfiEirp±rical

_eaTaneters,

_,,

_.

,,

,.

i St:ain pailure Criterion E

,.'

''

,

'

/ ttflic=TIIIflucla

,-.""".----"-"-"---"-"-"-"-"L.(2)

t

,

vhere, Ec:straltt at

'un

±axtal cmp:essive

,strength

!IE:ll at fatture

Relative Stress(oifetfl

-

Relative Strain

CEilei

(posttive

value),

)

Relatien 1

'/

/''

tt

t'/'

t

'

e7cendtng

portion: aa,it =, na-:+`7!,Eiff) a

---""--[ip}.

descenaing

portien:

_ISiaif

=

nid + ndn.

-il+

i:

---L---c3is)

ndaandl

---[3c}

'

,

ulf,e:f:Sttese a"d stretn at Eatlure tn the d±Tect ±en ef maxtmum

prlnctpal compresstve etreets, X=

(EliElf)M,

na= Eit{Et-(alfiE]f)}, nd=rrattdf; ndlmd fer tintaxtal eempresstve strese etete, El:lnitial

modulue oE elasrtctty, m,a:Emptr ±eal censtsnts:

,

'.J

, ,'

,

SFre?t/ Strees vheTe /ttt uR[n"1ny o"ER-tsS8za: m-ptH--eHeedio'e

-t.tt

Proposed'Model i/ Ottosen's _/t o tto ix

x'

'''

-O.5 -O

,Fig.9

D Oata by Mill.s et al.

.6

-O.7

-O.B

-O,9

-1,O

-Li

Ii!(ffgc],

Strength

failure

criterion

./

.

o

T

9 r

R

,/"

Y

v

ny

o"

$o,

U" rvI,

eq

v

vr

,?

1

er

7

9 ¢

'RJx

a=2.5a=2.2a=2.0

/t

e1flEc;-111flac1a

･i･(,i,7,tt

lleo

/

tlly,.XxiHID'e.4sWICb

ReE.IS/1+ t -IB-tetSl-e-PTesentEx.12o-"･.･, = !iflec Ilfl(

Auc)

f

t/

r

-L2

-L4

-1.6

-1,8

-2.e

34

ooT oept eorv eeH oo1

Fig.･11

4 6

S

10 12 STRAIN

{xlo-3

(a)

HID=:

Deterrnination

of lq 1fi),;11vaLue of le 2e

E

V-. e'

2N-s)8stu

rv"8oza

gzao

'

-e.6

-o.7

-o.s

-e.g

Fig.･10,

Strain

-1..e

-1.1

failure

criterion oz

i

relating

to

shape of 4 6 8 IO IZ 14 sTRAIN

tx10-3)

,El

(b)

HID=Z

descending

_poTtion

of a,-e,

16

:S

2e curve' tseS

:-Epn

blg:

."1th x.ge

:;M

･

Ja,e

NII-Electronic Mbrary

(9)

Architectural Institute of Japan

ArchitecturalInstitute of Japan Ho. ChA,sl)xov:zaza aoan oet aen oorv oo-o

H/D=1Uc=300kgf/cm2

al

aLS"?Okgflem2

OL

aLsNokgf!em2

o

4. B

12,

16 sTRIxlN

Cx10T-3)

,El

(a)

HID=1

Fig.12

Examples

of or

va

>.

:se

s

"e om m

A MX OQ

t1

..

g

ON

,Y

oQ

.v20

H9 e'y-'

g)MOv$･za8

analytical ai-Ei curves

Table4

Empirical

constants

for

strength

faitllre

criterienABK]K2

ProposedOttosen's1.2551.2754.0303.196I4,63Il.74e.gs7oo.gBei

Table5

Empirical

constants

for

strain

faiture

criterion

and relative stress･relative strain relationship

EmpiricalConstant$ HID ff m ndl _{1,OsSsl.15S)l.1'5}

;O.2o.s1+61+2-OCgc/loo}O･6

-](oclloell,O3-2S4-]Se.O.755

[Nete]

S!-(glffac) ' eom oot aon oorv eo-e

H/D=2.

oc#30okgf/cm2

Ul

_{at"?o4eszeoj?}

UL

e("Okgflcm

o

calculated

gv-o

e･-e's･" ".o.RL.mthza

gBHm

4 B

t2

'

srRArN

c

x

lo-3

)

(b)

HID=2

from

proposed

model

'

16tel

g

_ge

age

$

"e o

.m

m

A MX oq

tt

g.

oN

v ' q ov2e e O 2

4 'fi

e

10 12-14

16 le

20 ,

STRAIN

Cxlo-3),et

Fig.13

Prediction

of a,-e, curves of concrete subjected

to

irnbalance

tateral

stresses

'

ive

stress state was extended

to

express

that

for

concretes

in

triaxial

compressive stress state.

For

uniaxial compressive stress state,

Popovics'

formuia

{Eq,

3 (a)

in

Table

3)

and

the

formula

proposed

earlier

by

the

authors2')

(Eq.3(b)

in

Table3)

were used

to

express

the

behaviors

in

the

stress ascending range and

the

stress

descending

range, respectively.

Figures11

(a)

and

(b)

show

the

comparisons

between

experimental curves

(solid

iine)

and analytical curves

(dashed

and

broken

lines}

obtained

by

using

the

above expressions

for

failure

criteria,

Note

that

all

the

experimental curves of

eoncrete

un4er

_gradually

increasing

lateral

stresses

{see

Fig.

3)

were

modified

into

those

'under

constant

lateral

stresses on

the

assumption

that

the

ioad-carrying

capacity of concrete specimen

is

proportional

'

to

the

magnitude of

lateral

stresses at asame strain

level.

The

broken

lines

in

the

figures

were obtained on condition

that

the

shape of

the

stress

descending

portien

of

the

normatized

curves

of

concretes

under

triaxial

compres$ion

is

similar

to

that

under uniaxial compression

(i=1),

the

slope

of

the

broken

li'nes

being

considerably

steeper

than

that

of experimenta} curves.

Therefore,

the

coefficient

i

in

Eq.

(3c)

in

Table3

was

introduced

in

the

expression

to

reflect

the

ductility

imprevement

due

to

the

existence of

lateral

stresses.

Table5

shows

the

values of

i

obtained

for

'

all

the

experimental

data.

{4)

Comparison

between

analytical and experimental curves

･･Figures12{a)

and

(b}

show

the

examples of analytical a,-E, curves

for

the

specimens of

_HID=1

and

_2,

respectively, calculated

from

the

proposed

stress-strain model.

(10)

-35-Architectural Institute of Japan

NII-Electronic Library Service

It

is

already shown

in

Figs.

11 (a)

and

(b)

that

faiily

_good

agreement can

be

obtained

between

the

experimental

and

analytical

a,-e,

curves

in

case of

balanced

lateral

stresses

(

a2= a3= aL).

Further,

Fig･.

13 indicates

the

validity of

the

proposed

model

to

predict

the'

d,-E,

curvei5} of concrete subjected

to

imbalance

lateral

stresses

(

a,i ob).

Ngte

that

the

effect

of

the

imbalance

can

be

;eflected

also

in

the

equivalent

lateral

Pressure

by

determining

the

value of a

in

Tablezfor,

if

any,

available

experimental

data.

_.

'

..

g5.

Conglusion

'

,.

',

,.

.j

/

The

following

statements can

be'made

frorn

_the

study.

･,

''

'

1)

The

incr.gase

in

the

load-cariyiijg

capacity

beyond

ultimqte stres.s of axially

loaded

concrete

due

to

uniformly

tt

t

distributed

latefal

pressure

(aL)

isi

quite

maTked.

The

improvement'of

the

compressive

toyghness'

of concrete

by

.

t

conventiopal

lateral

steel

bars

and

steel

fibers

may

be

evaluq.f,ed

frorn

the

complete stress-straip cun;eg

'of

axially

loaded

plain

concrete subjected

to

the

lateral

_pressure

of

leSs

than

about

20kgffcmZ./

-.-

'

2)

The

effect of

the

heightlwidth

(HID)

iatio

of specimen on

the

inelastic

deformational

behavior

of concretes

is

quite

significant

eyen

when

the

friction

at

specimen-Ioading'platen

interface

is

sufficientlY'reduced.

3)

It

i$

pos$ible

to

relate each other

the

o,-E, curves of

different

shapes of conErete specimens' under

triaxial

compression

as

well

as

those

under

uniaxial

compression

by

taking

into

account

ih'e

failure

localization

or applying

the

idealized

failure

zone model

_proposed.

4)

The

effect of

dispersion

of

lateral

_pressure

becoines

_gradually

remarkable after

the

_peak

_point

of a,-E, curves.

The

rate of

decrease

in

load-carrying

capacity of aspecimen

becomes

larger

with

increase

in'the

magnitude of

lateral

pressure

and

the

_pitch

of

the

loading

_point

of

lateral

_pressure:

s)

It

is

considered

to

be

_possible

to

estimate approximately

the

toughness

improveme.nt

due

to

the'combination

of various

types

of confinements

just

by

superimposing each effect, using'

the

_proposed

_physical

_quantity

"gquivalent

lateral

_pressure"

as a compressive

toughness

index

for

laterally

confined concreteg.

_,

,

6)

The

equivalenvlateral,pressures

for

various

kinds

of concretes are

_given

by

comparing

the

a,-E, curves of

the

confined concretes and

those

of

_plain

concrete

in

a standard

triaxial

stress condition

_proposed

in

Table3.

'

t

/

1'

Acknowledgment

'

The'

authors are very

_grateful

to

Messrs.

Takatoshi

Matsumura

(the

Ministry

of

Construction),

Kazuhito

Tsutsui

(Mie

Univ.

_),

and

Hidenobu

Miyameto

(Mie

Univ,

>

for

their

cooperation,,

The

financiql

supports

_provided

by

the

'

Ministry

of

Educati6n

are also

gratefully

acknowledged.

_tt..

'

･-.

1

Reterences

''

1>

Kosaka,

Y.

_,

Tanigawa,

Y.

and

Hatanaka,

S.

_.

Evaluation

of

Effect

of

Confinements

on

CornpTessive

Toughness

of

ConcTete

Based

on

Triaxial

Cempressiye

Test

Data,

Tfans.

of

Japgn,Conc.

Inst.,,

VoL,7,

1985,

pp.249-256.

,

2)

Kosaka,

Y.

,

Tanigawa,

Y. ,

Hatanaka,

S.

and

Tsutsui,

K. ,

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ef

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Toughness

of

Laterally

Confinecl

'

Concretes.

Part

1 :

Triaxial

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Lateral

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Proc.

ef

Annual

Meeting

of

A.

I.

J.

,

Oct.

1985,

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･

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Y.,

Tanigawa,

Y.,

Hatanaka,

S.

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K.,

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Toughness

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Laterally

Confined

Concretes.

Part2:Stress-Strain

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of

Annual

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A.I.J,.,

OcL

1985.

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Y.

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1975,

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B.V.,.

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PTopeTties,

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ACI.

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No,?,

_,Fe,b.1971,

pp.126-135,

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Bazant,

Z.

P.

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P. D,,

Endochronic

Theory

of

Inelhsticity

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Failure

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Concrete,

Jour.

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Div.

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Proc.

of

ASCE,

VeLI02,

No.EM4,

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pp,701-722.

'

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S,

H,

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Properties

of

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Subjected

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Static

and

Dynamic

Loads,

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of

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Chicago

Circle,

1981,

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S.

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S,

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Stress･Strain

Curves

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Concrete

Confined

by

Spiral

Reinforcement,

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Vo4.

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No,6,

June

1982,

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Shimizu,

M,

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Analytical

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on

Confining

Effect

of

LateTal

ReinfoTcing

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on

RC

Column,

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of

ttt

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Oct.

1982,

pp.1251-1252

(in

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Tanigawa,

Y. ,

Yamada,

K.

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Hatanaka,

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Mori,

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Constitutive

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Steel

Fibre

Reinfor'ced

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S.,

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and

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