three-dimensional finite element analysis on …

Architectural Institute of Japan

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ArchitecturalInstitute of Japan

(tar:iJ-I] H4alkVkmsreXitttsu rgSl7e, 115-123, 1999#3HJ. Struct. Constr. Eng., AIJ, Ne. Sl7, 115'123, Mar., 1999

THREE-DIMENSIONAL FINITE ELEMENT ANALYSIS

ON COMPRESSIVE STRENGTH OF CONCRETE PRISMS

WITH SEVERAL HEIGHT/WIDTH RATIOS '

3 }ftJiiJfi waee$ee6Ei: ik 6 ffNr."J-- S Lka) sc (it 6 - > 7 ]) - F ts aa)JIIwtSfimaee)bFft

Hisato HOTTA' and Chang-Geun CHO"

va M Jt. .A., ig g me

In order te investigate the effects of restraint conditions and stress concentrat;ons on the compressii,e

strength of con"'ete by three-dimensiona] nunlinedl' finite element ana]ysis, concrete prisms with the seve]'al

heightlwldth ratios(1i!w) under compressive }oads were analyzed. Cone;'ete muterials were mode]ed as an

elastic sb'ain-hardening p]asticity materials in the pre-fa;]ure i'egion. and as an elastic iiftd lineai'

sb'ain-softening rnaterials with t.he concept ef shem' retention factor in tension in the post'crac]cinn' reglen.

For unrestrained loading suJ'faces, the compressive ultirnate load caiTying capucities were the same regardless

of the effects uf height!width raties, however, for restrained loacling surfaees, ulie shurLer the height', 1'.he

higher the compressive ultimate load caiTying capacity is. It is clemonstrated that the compi'essive streng(h of

concrete ehanges accordlng to the restraint cenditions.

.lfeywords : constitative relationsliip of concr?ig strain-hardenitrg plasticit.v, finite element anal.vsis,

aniaxiat compressive strellgth

ii )･/a 1) -FiKhSLF.ll. ew#iJILue{t. gecffpttwut. -"thiEmsgutU

1. INTRODUCTION

rn general, the compressive strength of cencrete in stsMcturul members cliffers from that in a ey]inder test As shown

in Fig. 1, the compressive strengths in two regions of a conci'ete bcam, tbe region under bending only nt point B und

tlie regien under shear combined witli bending at point A, differ from each other. In a paper by Takiguchl, K., et al.i}, it

is demonstrated that the compressive strength of eoncrete near the end support of colLimn ur beam, subjec{ed to bending

combined svith shear, is highly increasecl compared with the uniaxial cempressive strength obtaiRed from a eylinder test.

This effeet ls rnninly due tu the restraint condition. The stress state in the region uncler sheai' comL}ined ivlth bending,

point A in Fig. 1, is a multiaxial stress state since the concrete is restrained. 'l'he

failure. condition ef eoncrete uncler

multiaxlal stress state is svell knewn, and the strength becomes veirs, high compared with the uniaxia] c()mpressive

strength for condition of hydrostatic pressure under high eemprcssion. A]though iL is presently questionable if thc

difference in concrete strength, due to the effect of the resa'aint condition. is directly related to the multiaxial stress

state, a study in terms of the triaxial constitutive reiationship of concrete gives an apprupriate understanding cencerning

the relationship between the compresshre strength and the effect of the restraint condition in the concrete.

v.

Fig. I Cotnpressive strengths in (wo .regiens of a concrete beam

' Assoc. Prof., Tekyo Institute ef Technology, Dr. Eng.

" Graduate Student, Tekyo lnstitute of Technology, M. Eng.rkttl*X\ inthnt ･ I.･waRstImetV jc#wath ･ I-S

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In genera], the uniaxia] compressive strength is determined by testing a cylinder which hus a diameter of 10cm and a

height of 20cm. But, by tests, Gonnerman, H. F., et al.2) and Murdeek, J, W,, et a].,3) have demonstrated that the

compressive strength of concrete changes with the heightidiameter ratio of specimen. The smal]er the height of specimen,

the higher the compressive strength, and lt is lcnown that the reason is the end friction bet"'een concrete surface and

steel loading plate. If the end friction is eliminated, the effect of heightldiarneter ratio on cempressive s.lrength

disappears, By Van Mier, J, G. M.,4) when conerete prisms ef varying height are ]oaded between dry i'igld steeHonding

plates as opposed to flexib]e p]ates, completely dlfferent strengths result. rn the case of d", rigid steel ]{)ading p]ates, tdie

apparent compressive strength of the concrete increases as the ptdsm height decreases. In centrast. when f]exible ]oadTng

plates are used, the effect dlsappears, and almost the same peak strength is measared independent of the prlsm height',

An investigatien of the cempressive strength of concrete prisms having several height/width ratios pi'uvides iL good

example of the effect of restraint conditions. since it is easy to understand Lhe mechanism ancl the behavior nut vnly

from experimental tests but also in terms of the triaxiat constltutivc re]atienship for concrete. Tlie effeet t}f different

heightfwidth ratios for a given friction conditiun of the loading surface ean be considered in order to investigate the

effect of restraint condition on concrete strength. The puipose of this study, using l.hree'dimensivna] finite element

analysis, is to investigate the behavior of concrete prisms having several heightfwidth ratios, and to understand tl]e

relationship between the effect of restraint condition and Lhe compressive strength of Lhe cc)nerete.

To model the concrete prisms under the uniaxiai compressive ]oad, an isepai'ameb'ic sulid e]ement with eJghELn{,des

was used, Concrete matei'ia] models are formu]ated as incrementu] strain-hardening p]asticity mode]s "'ith a

Drucker-Prager type surface in compression. In tensile regic]ns after cracking', concrete materials are Tnodeled us [imear

strain-softening materials, and the shear transfer mechanism is reduced linearly in accerdance -'itli the concept c)f shen]'

retention factor. The prisms were 10cmX10cm square in cross section and had heights of 5cm, 10cm, ]5em, and 20cm.

To investigate the development of cempressive strength in relation {o heightfwidth ratios, the loading sttrfaces 'of

prisms

were assumed te be both restrained and uni'estrained in the lateral dlrection. The non]inear lm]blem was solved by an

incremental and iterative method. The tangentia) stiffness matrix was updated at each loading step or when t,he cbncrete

newly cracked･,

2. STRAIN-HARDENING PLASnCITY MODEL OF CONCRETE

In current stucly, for analysis of conerete prisms, conerete material is considered as an elastic st]'ain-hardening

plasticity model in the pre-failure region. In the strain-hardening regime, the formulation follews a classicaHsotrepic

hardening plasticity with the Drucker-Prager type yield and loading function5i.

2.1 DRUCKER-PRAGER TYPE SURFACE

In the initial stage of joading, concrete is treated as a homogeneotss and isotropie eiastic matei'iu]. Fu]' lhe stag'e of

stress between the initial yield surface and the failure surface, the non]inear behnvior of concrete is assumed to fol]owthe cenventional theorlJ of strainuhaJ'dening p]asticity. Te detei'mine the current stlite of yielding ancl the failure surfaee,the Di'ucker-Prager type of loading function can be expressed as:

f= VJ2 +ali-k' =O, fi.VJ:taift- k, =O (IL{2}

t twhere a and k are material constants, li is the first invariant of the stress tensor, and Y,, t's the secund ini,ariitnt of

the stress cleviator tensor. Ttie constants a, and k, can be determined in te]ms of the LmiaNi"i coinpressii'e sti'ength f.'

and the equa]-biaxia] compr ¢ ssive strength fb. as

VS f,.-f. t,E f,. f,. -,"r.

'' 32f,.T/L' 7.',ICf=3 2f,.･. f.･ , (s'),(4)

Tfie initial yield surface fe and the plastic potential function Q, respectii,ely, also have the Dnickei"Prager type,

surface.

2.2 tNCREMENTAL STRAIN-HARDENING RELATtONSHIP

Based on the theery of plasticity, the total strain lncrement ;s taken t{} be the sum t)f the e]as.tic nncl plasticlncrements

dEii=dsff -- dE{j (5}

in which the elastic strain increment, dE:, is re]ated to the stress increment, claii, by the genei'a]ized Ilook's ]aw as

dO ,･,･=Pik, dE:.s (C'})

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whei'e Diiki is the tensor of elastic modulus.

For plastic strain increment dEe･i, the necessary cennection between the loading function and the stress-strain relation

is mude by the flow rule. "rhen tlie cuiTent stress state reaehes the yield sur £ace. the materlal is ln a state of piastic

flour upon further loading. In particu]ar it, is assumed that the 1}]astic sts'ain ;nci'einent is propoi:tional tc) the sta'ess

gradient ef a quantlq, teiined the plastic potential Q, known as the nonasst/)ciatecl nosv i'ule, so (hat

dE{･, r- dX /rt,,

(7)

where dX is a pesitive proportienality ct/)nstant termed the plastic multiplier and the grudient ef the potential surface

OQ/ v' c,i defines the clirection of the plast{c strain increment vector. The, dadvatives of the ]oading and p]astic

I)otential funcLions can generajly be expressed by, respectively:

-S･ZI,,, -

-;-iiE-,,6･i--b9i, J, s-,i･ ,n,1.

t,･, ,9,5i,, -

g2. s,+-/r- ,Q,. s,,+ tu,!ttlt,,,

(sL(g}

where, 8ii is the Kronecker delta, sii is the deviatoric su'ess tensor, and t,, = s,,,,s,･,i

' (.2f3)J:8,i.

Using the flow rule ancl carr},ing out the mathematical derivation, the constitut'ive equation in the elastic

strain-hardening plastic model can be exprcssed as

dai,･=(D,/ii,i-D'ljky)dEkt {10}

ln which the plastic stiffness tensor DZ･ki has the form

i)e,ki= tDiiJ,ti, oOuS,,, -:iCltSl],,'

Ppqlti al.a)

whei'e /

-

h' "'a.., D""'p" sig., -E.illllE{::

emp,.,,'st' it 'SEi"',/

CJIii "' of,., -t/ilifl,.:' {]i'b}

c= op,y oOo9,, fv:[l.l'J?,, t6a9,

"e "

a-L･c)

and o. is ulie effective stress, tiEi, is the effective p]tistic strain increinent. Once the louding {ind plastic potential

function ai'e defined for n isotropic hai'dening plastic muterial, the constitutive relation ean be directly de]'is'ecl frotn Eqns.

10-11 in n rat.her sts'aiglitf()rsvarcl manner.

3. CONCRETE CRACKING AND STRAIN-SOFTENING MODEL IN TENSION

3.1 CRACKS IN THREE DIMENSION

Concrete in tension is modeled as a linem' clastic and strain softenlng material by the smeared erack itpproach. 1'he

principnl sti'esses and their directions ai'e computed initially in uncrac]{ed concrete. If the maximum prlnclpal st,ress for

some reasen exceeds a limiting valvte, a crack is assumed to form in a plane orLhogonal to this st)'ess. After this, the

behav{ur of that zone of euncrete becomes oi'thotrepic us shown ln Fig. 2. The locul matci'ia] axes in the given zone

coincide with the principa] stress directien. After eracking, new sets of cracks can be fermed which are peipenclicular tt)

the previous cracks. In this model, the limiting values to deflne the onset of craclcing are given as fo]]ows, by Al-Noury,

et ai.6):(a)

for triaxial tension

o,,=f, i=J,2,3 ".2.a)

(b) for tension-tension-compression, linearly decreasing tensile strength expressions

o,i=ft (i. -7-.1'･

l) t'or fi3go

"2.b)

(c} for lensionmcoinpression-coinpressioii, linenn' decreasing tensilo streng(h expressions

o,i. f, (1+ ,;T,i',)(l+';i) for o,,fi3 i-O (l2.c}

whet'e f, is the tensi}e strength of concrete,

In the cnse of a triaxial stress state both {n the strain-hnrdening plastic rcg'ion and in the craek region, stresses ar'e

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calcu]ated as followgng initially the tensi}e strain-softening state, and subsequent]y the triuxia] st'ress

strainuhardening plastic region.

S3

i zz i,

2

,, ,,,, :Iilr, =l

q

+

q

i

-m- EE,

!

/

(a) One craclc (b}'Twocraclcs ' (e}Three cracks

e

a)o

Fig. 2. 0rthogonal crack model in three-climension Fig. 3. Linear tensi]e. strain-soCten;ng

3.2 STRAIN-SOFTENING IN TENSILE CRACK

After concrete cracks in tension, It is 1{nown as demonstrated by Peterson, P. E.7) that the

immediately released to zero but is gradually releaspd by strainLsoftening 1)ehaviDr, In.t.hls paper, in

Chen, W. F.,8) after concrete eracl(s in tension, the stress-strain relation is taken 'as

a ]lnear

shown in Fig. 3, The total strain increment[AE} is decomposed into tsvo parts, the concrete st]'ain'

and the erack strain increment {AEC"], and the strain-softening modulus Et can be derived as fo]]esvs,

{As} r･ {AEc"}+{AE"'} , l, = /i; ' iL/:. ･ Clm = "e"

"f//,9i

'

'

'where

Gf is defined as the fracture energy of conciete required to create one unit of area of a contineus

i$ the,crack band width.

3.3 SHEAR TRANSFER MECHANISM FOR CRACKED SURFACE ' ' Experimental results indicate that a eonsicieiab]e

tGc sti'ess can be transferecl aeioss the reugh surfaces ef

l,,,,. '

]/k'

i?ti,11,:'

:.l,ILi'ffk..ii gii

e

:f'

1":,l.3./'.

'1'

,.I

'lell"i,,t

gr

i::.'Llliit,,2t`

;,. x

Xx to the craclced shear modu]us Gc asafunetion of tlie -

Fig.4 ,Craclced sheai- modu]us Gg=a. G(l- -Eel;,;,),

for Ei <E<E,,,

' ' tt '

In the present worlc, from Kolmar, W. et a].,9] the cracked ,shear medulus is assume.d. to ,be

function of ti)e cun'ent tensile strain after ci'agking as shown in Fig. 4. , tt t ' '4. FINITE ELEMENT MODELS FOR CONCRETE PRISMS ttt

Based on the detailed desciiptions in the previuus sections, a three-dimensional finite element

the NFERC was developed, where the 8-noded hexahedrai e]ements as shown in Fig. 5 svere used to

solids. Concrete prisms had 10cmX10cm square cross sections. Finite elemenl mesh generations for

height of 20cm, 15cm, 10cm, nnd 5cm, respectively, are ii]ustrated in Fig. 6. With these mesh

'

modeled as the symmeb'ic moclel of the 118 part, und is compai'ed svith the fu]l size m.nc]e] t,o '

e]ement size,

'['he beunclaiy conclitions fur the top and the bottem sui'faces in ct/)ntnet "'ith the

a'ssutined to be either complcte restraint in the lateral direction, tvith no s]il) between the steel lvading

surfaee of prism, or complete unrestraint in the lateral direction.

The nonlinear prob]em for numerical analysis is solved by the incrementa] ftnd itei'ative method

Newton-Raphson approach. The tangehtial' stiffness matrix is recalculated for each ]oad in6rement or

newly eraclcs. Convergence criteria in terrns of incrementa1 noda] displacements are adopted in order

iterative cycje when the solution is consiciered to be sufficlent]y accurate.

rvlaterial properties of concrete prisms were assumed te be as follDws; unittxia} comprcssive

uniaxia] tensile strength of 3.22 MPa, Young's modu]us of :l1700 MPa, Poisson's ratio of O.22; (lf of

20mm, and the uniaxial stress-straln cui've ls adoptecl･as that presented by Labbane, M.. et al.iQ} The

S3 S3crz zsz

szcr

Slcr Sl1,- li

!- -

1 /1

slate

dr

,for the

mode]

fe

tensi]e stress is not

acc[}rdance svitl)

strain-softening moclel us

Mcrement { AE "" }'

(13),U4),(15)

cracl{ ;inct tvl

ameunt of shem/

cracked coticrete.

mechanism i$ the cruck

ancl bar slze also

aeuunt for agg'reg'atel

appropiiat.e value

uncracked slieai'

(16}

'

redueed as, a ]ingai'

prograrn fur cencrete,

inodel the concrete

concrcre prisms. witl)

generatpons, each prisin is

investigal'e the effect uf,

loading plnles wei'e

p]ute und thc end

of the inodiflecl

when t}ie toncreie' tu teJ'm;nute t/he

strenglli r,f :i2.2 "IP",

124Nlm, u]f Uf･

palllmeters ol' tile

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Di'uckeruPrager surfaces are assumed te be O.07 for a and crg, 2.36 for K and )gf in the associated f}ow ni]e. Up Lo 25%

of the uniaxial compressive strength, the stress-strain curve ls assumed to be linear elastlc. The analysis is performed

by displacement control}ed incremental loading appliecl to the top of specimen. The triaxia] stress-strain re]at,ionships

obtained from the foregoing models are compared with the test resu]ts by Kotsovos, M. I}., et al,.ii} in Fig. 7. The

maximum va]ues in this model are slightly lower than in the test results, since the triaxia] st:'ength is underestimated in

the DruclcerLPrager failure sgrface formulated by tlie biuxial stress strengths in Eqns. :S and 4. The fuilure sm'face of

the Drucker-Pragei' type is circular on the deviatoric p]ane, but that of the aetual concrete is near]y triangulai' fo]' tensi}e

and small compressive stresses, and becomes increasingly convex for higher compressive su'esscs. C)nly for the case

where tl)e angle ef similai'ity is neai'ly 60 O, the Dnicker-Pi'ager surface can be satisfied as a concrete

failLve

surface,

sinee

at a eorner

of

convex triangulm' surface of the concrete, the angle of similarity is 60 ".

Here, the angle of

slmi]arities is exprcssed as follows;

e- cos'i[ 3!2.t-3-

jf.Y,] (].7)

and Jp and J3 are the second and third invariants of the stress deviator tensor. From the analysis of prisms, the angles

of

siml]arities at ai'bitrary

points

from the maximum principai stress axls in the deviatoric section were eaicu]ated and

are shown

in Table 1. The values are nearly 60 t.

This result clernonstrates that this ･yield and loading surface, which

has the triaxia] stress-strain rclatlon shown in Fig. 7, can mode] accurntely the 1}ehuvior of Lhe concreLe prisms tinder

compressive loads.

8

7

l i

5

1

b

,

i

i

i

4d -- t

.1 yiJ-

Hexahedral

Angles of

6

z., 2

lsepm'ametrlc

similarlties at

3

Fig. 5.Table

].

(a) H=LtO [,m

element Fig. 6.

inner sultace of prisms

{b)

Finite

H=15 cm

element

-2oe

a .l $ E

g -lse

8

? -loe cr E

: : X -50 va ff co

o

Fig. 7.

and

;ll'e

tllehelght

of

J. "f.,

in

angle ef similarity

(clegree)

{c> H=10 cin (d) T{=5 eni

mesh for concrete specimens

loeationsanglc of similarity

(degree)locations

speclmen

top at outer

top at mner

middle at outer

middle at inner

l/

t)f H=20cm

52.78

55.28

59.96

58.93

speclmen

top at outer

top at lnner

middle at outer

middle at inner

:

ef I{=10cm

50.86

59.54

59.16

59.42

speclm.enef H=15cm tt

specimen of H=5cm t- '

.

e-

･

)tt,:1 - s･ i,

`i

.

"･/,t･--x{･i'gg

--･!.-".･f.

.

.

/f

/ ftf

./

Sl=S2=S3

- 44MPa

-]- 24MPa

---- 19MPe

5. RESULTS OF ANALYSIS

For the ana]ysis of two moclels, the syrnrnetric models of Y8 size

str ¢ ss-strain curvcs obtained from the reactions on the loading surfaces

unrestrained loacling surface, the compressive stress-strain cui's'es m'e

however, in the case of a restrained loading surfaee, the shorter the

ean'ying capacity. Resu]ts agree with the experimental results of Murdock,

heights are sbown in Fig. 9. For a 1ieighL of 5cm, the physicul tests i'esulted

unlaxial compressive strengtii for conerete strengths of 52, 26, ancl 14 MPa,

increase of 1.64 times for a uniaxial compressive strength ef 32 MPa.

o.oo2 e.ooe -o.ee2 4.oo4 o.oo6

Tensllet2,3} STRAINS Cempresslve{1)

1'riaxiul sta'ess-strain re]atlonship

the fu]] size mode]s, the compressive

shown in Fig. 8. In the case of an

same, regardless of heightlwidth ratios,

prism, the higher the u]timate load

et a].Z) Results for cy]inders of varying

strengths of 1,4, 1.8, and 2.0 times the

respectively. The analyses gave a strenglh

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60 6o

'

STRAINS STRAINS

(a) By l/8 symmetric models (b) By fu]] models

Heights(cm} extends througliout the centm1 region of the specimen.

Fig. 9. Influence of the different height en strength

' '

Ili :: II li H=

t5

CiiL･lff

･IT ･ITi ･p '

:ll l: ll ll lll ll ll ll lII lll"I lII :ll,i l! lIT tl

it, si :l Il Ill Il li :! II: [ll! lll lll !

,'F,

'fi

'fi n

II"l ll ll H=scm

lII l: Il [!･ ;':TR'ri ':i

tti

st

Hy l/ ]: ::

ll :l .

'

(e] !5.S3 MPa (0 35.75MPa la) 2s.6sMpa fo) 41.54MPa

Fig. 10. Crac]c distributions on uuter surface as function of calculatecl uniaxial eompressive stress

The maximum and the minimum principal stress distJlbut,iups en end surfaees, for resb'ai'ned ]oading surfaees, are

shown in Fig, ll. The tensile state is positive ancl the compressive state ls negative. For prisms of 20cm and 15cm'

height, large stress concentration exists near the coina' of the loading surface. With decreasing height, huwever, ]ai'gg

stress coricentrations alse developed near the center of the loading sLirface, ,especia]ly ln the case of H!5cm.

In Fig, l2, the dlstributions of the maximum principal stresses for inner su]'faces of prisms are shewn fer rest"'ainEd

loqding' surfaces. For the restrained }oading surfaces, stress s,tates varied according to the height!width rntio uf the

specimen. For the case of a height of 20cm, as shown in Fig. 12 (a). there ai'e twu expliflE zones, a. biaxial c"Tnpressive

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zenc near

deereases,

the triaxia]

the ]oadlng surfacc and a uniaxial

the triaxia! stress state increases

stiess state is extensive not unly

compressive zofie at the iniddlc uf

neai' the middle of the prlsms. For

near the cnd surface of the prism

the specimen. As the height of the prism

a height of 5cm, ag shown in Fig. 12 (d),

1)ut alsu near the middle uf the prism.

in

Ilre,

D

),git

]:il.7:.,iL'-.:.,L.t･.IF,'･l.i:ilP,''Ed']

'/oc

I:DS/O/5,O. 3

lt ltl PANLIPp. STF[V, /J. t/F/IAtlALr,TREt,t, t/,ttHIP/

ta'

''n[rvT[.

,JOtl.t,Ml FltEP

Cft) H - ,.'n /in

:;[E

et]l

stt/ts:liv]c]Elt

'''//r['

/'

,/pt "v///./O./tTr7.t//,/Pt//1/1.Al.L,,/Ptt.. tS)1,/PO

/"INPANtlPALSTrESr,/IAF.1//N/tv/JLSTR[r,S.ttl.WPn

'Ezo[D5o,].Ol.:OCT-.1t//1H-

16 cm

iaeEes:;lt:neo/o)

i.]tt/a2EE]e

,," pp//'lp. .'p[t:/vp ,,NIEIttV:ESE.ttt.'t/,/pts'1]

//'-

IODS/O S.Z..

M.tP-///IP,ttt ///-x./AtStnEEt'q']''2//c'

]o]. oPE:SlvPt/.].lt

/P.

ZZ5S/O S.: .303

(t} 1' 10 cm

I:::[:1::1]/"//t

'eo]o-]'o].o]:o,Ioa7P/'oa.[tE[

!:,:,..".PJL.,::'..:'!.:.t,",L"p':

'"''

'o-

[oo./o/s t [.E g Eo

inHt PRt/ IP.1 .ltFSS//-Ps, u/-"1,./ tl,ESt.]7 1E/,/pt

ittt/gewli,llllgg oooL/o/' ,ott.t, tEqOtEEr "ltt'H // Cd)H.Z,m

itti/[s

5'sS5ts

g'I;S::l:::i;r]te

Flg.11. Principal stresson loadings"rface of prism as function ofca]culateduniaxial compressive stress

2e.5eMPa10

1.1.1"4.di-7le.OlMPa

o.2""4"e.12 ISTSMP-

e1234S

fa)H=20cm {blH=IScm (t]HtlOcm

Fig. t2. Maximum principal stress on inner surface of prism as funclion ef

Strain'hardening ratios for each loacl step of the calculated compi'essive stress

surfaces are shown in Fig. 13. For a given lecation, those ratiog are percentages

restrained conditions. Due to the development of the niaxia] cDmpressive stress

case uf H=2ecm, as shown in Fig. 13 (a}, largc i'atios are confined to the miclclle

p]'ogressive]y extended from the middle to the ends of tbe prigmg as the prlsm

(c), ancl (d).

Frum the development of cracking and the ]ocation of critica] zone, it {s apparent

fuilLn'e shape wiLli spa]]ing due to tensile ci'aeks c'aused by jateral expansion

speclmen and the cc)mp]'essive failure in the inicldle uf the specinicn. This result

s]iape dernonstrated l)y the experiments t)S Hdnsen. H.. et dl.,E) and 1)y Newman

little different frorn the result nf Che two'dimensional axisvmmetrie finite element

where tensile cracks do not des,e]op. A dec]'ement iJi prigm height vauseg the reRion

the region ef triaxial stress states also incrcasc ultimate]y, the extension uf

which the concrete is restruined inf]uences direct]y the i"ci'eage in thc, peak strength

iinma'.'MM,.

i, 1: fd)H-Scm

caleulated uniaxial eompressive sb'ess

of the prisms for regti'ttinecl ]oading

of the failiu'e st]'ess for the gnverning

states near the loacling surfaees, in the

of the specimen. Huwes'cr, lai'ge ratios

height decreuses as shuwn iTi F]g. 1:l (b),

that the concrete prlsm has u conica]

at the outer surface in the rniddle "f tlie

gives a good ttgrcenicnt witli the failm'e

, K et a].B) Moreever, tliis resLtlt is a

mode] in the paper by Otcogen. N. s.,H)

' c}f regtrained coTici'ete Lo increase, su

the triaxiul cempressive sti'css state by

lll COtnPl'e5"t)Il.

NII-Electronic

121-Library




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6. CONCLUStON

From three dlmensiona] finite e]ement analyses of concrete prlsms under compressive ]oads and svith the severa}

heightfwidth ratios, it is concluded that:

1) For unrestrained loading surfaces, the compressive stress-strain curves are the same regardless of the heightlwidth

ratio, For restrained loading siirfaces, however, the shorter the height ef the prism, the greater the peak cumpressive

strength, and the more extensive the triaxial stress state, The compressive strength of concrete varies direct]y with the

restraint conditions.

2) For heightXwidth ratios of 2.0, 1.5, and 1.0, there are twn stress zones, a triaxia] compressive zone witli the 1arge

stress concentrations near the end surface a.nd a uniaxial cDmpressive zone neai' the middle of the, prism.

3) The failure rnechanism of prisms with heightiwidth ratios of O.5 to 2.0 can be predicted to have a cenica] fai]w'e

shape resulting frem tensile spalling cracks caused by iatera] expansioR of the outei' surface in the micldle of the prismand ceinpressive failure in the middle of the prlsm.

REFERENCES1. Takiguchi, K., hnai, 1<., and Mizobuchi, T., "Ceinpressive Stretigth oi ConcretE Around Critical Sectio" of R/C Cohi[n" under Ce:npressie[i-LSendiixg-S]ieai'," J. Slmict. CeTisn'. Eng., A", No. ･496, S3-90, Jun., 1997.Z, Gonnei'tnan, H, F.,

"Effeet of Size an{l Shape of Test Specilnen en Cempressive Strength of Concrere'. Proc. ASI'M, 25, PaiT il, L925, pp. 21i7-5U.

S. Murdock, J. W. [tnd Kesler, C/. E., "Effect

of Length te "la]neter Ratio ef Speci]nen on the Aplmrc]!t Cg[npressive Streiigt]] of CeT ¢ rete, A5TM Bu].. Apri].

1957, ppr 6S-73.

4. Van Mier, J. G. M., "Scaling

in Tensile aiid Cotnpressive Fractiwe of Conciete", Applicataons ef Fraeliue Mechanics te Rei[]fo]vect Co[]e]'ete, Edite[l by

Carpinteri, A., Elsevier App]ied Science, 1992, pp. 95-l35.5. Drucker, D. C. and Prager, SY,,

"Soil Mechariics and Plastic Analysis or limit Design," Q"arterly of Applled MathemHtics, i'e]. 16, 1952, pp. I57' 165.

6. Al-Noury, S, I, and Baiigash, Y., "?restressed

Concrete Contairunent Structures - Ctrc"inferential Hoop Tendon Calculation," 9th trxternational Cenferetice on

Stnictural Mechanics ln Reactor Technolegy CSMiRT), Lausanne, l9B7, l'ol. 4, pp. 395-401.

7. Petersen, P. E., "CTack

Grovvth and Deve]opment o Fractvre Zones iT) Plai" Cencrete andi Similm' Materkds," Report TVBM-100G, Viv, of Bui]ding Mnterials,

LuTicl Ii]stitute ef Technology, Lund, Svvedeit, l9Sl.

S. Chen, W. F., "Constitutive

Equations For Engineering Materials - Vol. 2: P]asticity And Modeling," Elsevier, 1994, pp. SSI-RS7.

9. Kolillai', W. and MehJhom, G., "Comparison of Shear Fefinulations fo:' Cracked Reinfoixred Cenerete Elements", in Procee{ling ef the IriLeriia[ioiLHI Cenfere])ce on Computer Aided Analysls and Design of Concrete Stnictures, Part I, (Eds. Daiijanic, F., et al.) Pineridge Press, Swanses, U. K., vpl33-l47, l9Saj.

Ie. Labbane, M., Saha, N. K. and Ting, E, C., "Yield Criterion And Loading FuT]ctior] Fer Co"crete Plasticity," Inteniatienal Jounial of Se]ic]s Rucl Stractures, Ve]. 30, No. ,9, pp, 1269-12S8, 1993.Il. Kotsovos, M. D. and Nevvman, J. B.,

"A mathematical DescriptioT] of the Defonnational Behavior of Ce"a'ete uTider Complex Loatii[]g," Maguzine of Coibci'ele

Research, IS79, 31, pp. 77'SO.

12, Hftnsen, H., Kiellancl, A,, Nielsen, K. E, C., and Thaa]ow, S., ''Comptzssive

Stiength ot Concrete - Ciibe or Cy]inder,'' RILEM Bulletin, No. I7, I)ee,, l962, :)p.

23-SO. ･

13, ,Ne,wtman, K. an{i Sivaldason, O. T., ''Testing

Machine and Specimen Chftracteristics ancL their Efteet on tlic "lo(]e of 1/)eformarioii, Failure, an[S Strength of

Materials," I'receedings ef the Institution of "Ieclianical ET)gineers, Vel. ISO, Part 3A, 1965-66, pp. 399-4]O.I4. 0ttosen,

'N. S.,

"Evu]vatien of Coricrete Cy]inder Tests Using Finite Elements," Jovmal et the Engimeering Mee]ianics l)ivisioii, ASCE, Vol. 1LO, No. 3, MaJvh,

･l9S4,

pp. ,165-4Sl.

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NII-Electronic Mbrary



Arohiteotural エnstitute 　 of 　 Japan

和文要約

1．はじめ

　一

般に構造物のコンクリート強度は、シリンダー実験のそれと

は異なっている。曲げとせん断を受ける梁支点のコンクリート強

度は、シリンダー強度に比べ、大幅に上回る場合がある事が報告

されている。この事は、拘束状態が異なっているからだと一般に

説明されている。拘束問題を多軸応力状態での構成関係で把握さ

れるかは疑問であるが、コンクリー

トの多軸応力での構成関係を

基にして、圧縮強度が拘束条件にどのような影響を受けるかを調

べることでその影響を把握する必要がある。特に、圧縮を受ける

径高さ比の異なる角柱は、拘束条件が圧縮強度に及ぼす影響を表

わすことで、その問題を簡単に把握される一例と考えられる。

　本砥究は、径高さ比の異なるコンクリー

ト角柱の 3 次元非線形

有限要素モデルにより、実際のコンクリート構造部材の中で拘束

条件および応力集中の変化がコンクリー

トの圧縮強度に及ぼす影

響に関して検討することを目的とする。高さ20cm 、15cm 、10cm 、

5em の 10em × 10cm の正方形断面を有するコンクリー

ト角柱に

対して解析による検討を行う。

2．コンクリートの弾性尸歪み硬化塑性モデル

　破壊までのコンクリート材料は弾性一

歪み硬化塑性材料と仮定

し、降伏面および載荷関数を Drucker−Prager基準に基づき定式

化している。Drucker−Prager基準による破壊面、初期降伏面、塑

性ポテンシャル関数はそれぞれ式（2）〜（6）のように与えられる。塑

性論に従って歪み増分は、弾性歪み増分と式（7）の流れ則で定義

される塑性歪み増分との和で定式化される。最終的に、弾性一

歪

み硬化塑性材料の応カー歪み関係は式（IO＞で求められ、塑性剛性

テンソルは式（11、a）となり、硬化パラメータは式（11，b）と式（11．c）

により求められる。

3．引っ張り側のひび割れと歪み軟化モデル

　引っ張り側コンクリート材料は分散ひび割れモデルと考え、線

形弾性一歪み軟化材料と考えている。3次元でのひび割れ発生は、

図 2 のように主応力に対して直交方向の平面に生じると仮定する

直交ひび割れモデルと考え、3 軸応力状態でのひび割れの限界値

は式（12）と仮定する。

　ひび割れ後、引っ張り側のコンクリートは、図 3 のように総歪

み増分はコンクリー

ト歪み増分とひび割れ歪み増分との和で、式

（13）〜式（15）と仮定する。

　ひび割れ面でのせん断伝達メカニズムは、式（16＞のように線形

的にせん断係数が低．．ドすると仮定する。

4．コンクリート角柱の有限要素モデル

　図 3 に示すような 8節点八面立体要素を用いて、第 2 および第

3 章で述べた仮定を基にコンクリート材料の 3 次元非線形有限要

素解析プログラム NFERC を開発した。解析対象は 10cmXIOcm

の正方形断面を有する高さ 20cm 、15　cm 、10　 cm 、5cm の角柱

で、荷重と構造体の対称性を考え、供試体の 1／8 の対称部分だ

け解析を行っている。供試体の要素分割はそれぞれ図 4 に示すと

おりである。コンクリー

ト載荷面と載荷釧板は完全フリーと完全

付着の二つで、載荷面の付着条件を考慮している。

　コンクリート材料値は、圧縮強度を 32．2MPa 、引っ張り強度を

3．22MPa 、ヤング係数を 31700MPa 、ボアソン比を 0．22 と考え、

応カー歪み曲線は文献 10 で示されているものを使用し、圧縮強

度の 25％までは線形弾性と仮定する。

5．結果考察

　図 7 には、本研究で定式化されたコンクリー

ト材料のモデルに

より、3 軸応力での応カー歪み関係を示しているが、試験に比べ

解析により得られた応力の最大値が少し小さくなっている。これ

は式（3＞および式（4）に用いられた 2 軸応力強度による降伏面が過少

評価されたからである。表 1 に示すように、各供試体の代表的な

節点で計算された相似角が 60°に近づくことは、円形三角形であ

るコンクリー

トの破壊面が Druckel・−Prage ど基準と相似角が 60 °で

一致することを考えると、1 軸圧縮強度解析のモデルとして 3 軸

応力下での Drucker ・Prager 基準は妥当性を示していることを証

明する。

　図 8 によると、載荷面が完全フリーの場合、高さが異なること

とは関係なく圧縮強度が一

定であるが、載荷面が完全固定の場合、

高さが短くなると圧縮強度が高くなっている。図 9 では、その結

巣が実験と良く一致していることを表わしている t，

　載荷面が完全フリー

の場合、ひび割れは発生していないが、図

10 は、載荷面が完全固定の場合での外側面でのひび割れ分布を表

わしている。中央断面の外側面からひび割れが発生し、外側面に

沿って徐々に載荷面側に向け伸びている。

　図 ll は、載荷面が完全固定の場合の載荷面での主応力分布を

表わしている。高さ 20から 10cm の場合、外側面で応力が集中し

ているが、高さ 5cm の場合、外側面より中央部分で応力が集中し

ている。

　載荷面が完全フリー

の場合、全てが一

軸応力状態を表わしたが、

図 12 のように、載荷面が完全固定の場合、応力状態は、径高さ

比の影響を受け異なっているし、高さ 20cm の場合、載荷面側で

は三軸応力状態となっているが、中央面側ではほぼ一

軸応力状態

となっており、二つの応力状態で明らかに区分することができる。

しかし、高さが短くなると段々三軸応力状態の領域が増え、高さ

5cm の場合、載荷面から中央面までほぼ三軸応力状態となってい

る。

　図 13 は、載荷面が完全固定の場合、内部面での塑性硬化率を

表わしている。高さが 2C｝cm の場合、載荷面側では高応力状態で

あまり塑性硬化が進んでいないが、中央面側では中心から塑性歪

み硬化が進んで、圧縮破壊の限界領域になっている，，載荷面側と

中央面側でのその差は大きくなっているが、高さが短い場合、そ

の差は小さくなっている。

　ひぴ割れの発生、主応力分布、塑性歪み一

硬化分布から考える

と、角柱の破壊は中央断面の外から横方向の膨張によるひび割れ

が進展すると共に中央面からの塑性歪み硬化が進み、文献 12 と 13

の試験に示されているような円錐型に破壊することが判断される。

6．まとめ

　径高さ比が異なるコンクリー

ト角柱の 3 次元有限要素解析を行

い以下の知見が得られたtt

゜丿1

2）

3）

載荷面が付着されていない場合、ほぼ一

軸応力状態で、径高

さ比は圧縮強度変化に影響を及ばないが、付着されている場

合、高さが短くなることにより、角柱の圧縮強度が高くなり、

三軸応力状態が広げられている。その事は、拘束条件が圧縮

強度に影響．を及ぼすことが確かであるのを言正明する。

高さ 20Cm 、15cm 、1Dcm の場合、供試体の応力状態は載荷

面での 3軸応力状態と中央面での 1軸応力状態の二っの領域

に区分される。角柱は、中央断面の外側からの膨張によるひび割れと共に中

央面側での塑性歪み硬化の進展により、試験に示されている

ように、円錐型の破壊になりことが予測される。

（1998年 3 月10日原稿受理，1998年 10月 15 日採用決定）

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