Presentation on
Constitutive Models for Concrete In Plasticity
Presented By Ansari Abu Usama
DEPARTMENT OF APPLIED MECHANICS DEPARTMENT GOVERNMENT COLLEGE OF ENGINEERING AURANGABAD 2015-16
Guided By Dr. M. G. Shaikh
C!"#!" •
Introduction
•
b$ectives
•
%iterature survey
•
System Develo&ment
•
Performance Analysis
•
'eferences
C!"#!" •
Introduction
•
b$ectives
•
%iterature survey
•
System Develo&ment
•
Performance Analysis
•
'eferences
Introduction •
Constitutive modelin( of concrete.
•
"heory of Plasticity )ield )ield
criteria
*ardenin( +lo,
rule
rule
b$ective "o study the elasto&lastic behavior of concrete under different loadin( conditions and to simulate the same by com&utational model.
%iterature survey Han and Chen 1!"5# •
In
this ,ork- the five&arameter model of /illam/arnke- and the four&arameter model of *sieh"in(Chen ,as ado&ted. +or the /illam/arnke five&arameter model
In ,hich r c and r t are the com&ressive and tensile deviatoric len(ths at the meridians 01 and 201 res&ectively and are related to by
•
,here- are material constants- and the constants satisfy the condition +or the *sieh"in(Chen four&arameter model
,here a- b- c and d are material constants.
+i(ure 3.4. A non uniform hardenin( &lasticity model
+i(ure 3.3. )ield Surface and +ailure Surface in the Model
•
*ere a DruckerPra(er ty&e of &lastic &otential function is assumed ,here 5 constant- 6 re&resents the &lastic dilatation factor
"he incremental elastic&lastic constitutive relation is (iven by ,here the &lastic stiffness tensor has the form in ,hich
+i(ure 3.7. Com&arison for com&ressioncom&ression loadin(s to 8u&fer9s data :8u&fer et al.- 4;2;< :a< Com&arison for unia=ial and bia=ial com&ressive loadin(s to 8u&fer9s data. :b< Com&arison for bia=ial com&ressive loadin( to 8u&fer9s data
+i(ure 3.>. Com&arison for com&ressiontension loadin(s to 8u&fer9s data :8u&fer et al.- 4;2;<
+i(ure 3.?$Com&arison for tensiontension loadin(s to 8u&fer9s data :8u&fer et al.- 4;2;<.
"he
author concluded that the model has been sho,n to &roduce results in (ood a(reement ,ith e=&erimental data for a ,ide ran(e of stress states.
"he
model has also re&resentin( the im&ortant characteristics of concrete behavior- includin( brittle failure in tension- ductile behavior in com&ression- &ressure sensitivity- and volumetric dilatation under com&ressive loadin(.
"he
model is fle=ible and can fit a ,ide ran(e of e=&erimental data.
Han and Chen 1!"%#& •
In this &a&er constitutive model include /illam/arnke five &arameter or *sieh"in(Chen four&arameter failure surfacethe non uniform hardenin( rule- the nonassociated flo, rule.
"he failure surface can be e=&ressed in a (eneral form as
+or the /illam/arnke five&arameter model
+or the *sieh"in(Chen four&arameter model
A nonassociated flo, rule ,ith a DruckerPra(er ty&e of &lastic •
&otential function is ado&ted
"he incremental elastic&lastic constitutive relation is (iven by
,here 5 isotro&ic elastic tensor h5 a scalar ,hich is e=&ressed as
+i(ure 3.2.Com&arison for
+i(ure
[email protected]&arison for "ria=ial
Bia=ial Com&ressive %oadin( to
Com&ressive %oadin( to A+/% Data
A+/% Data
S'(a)n S*+'en)n, •
"he constitutive euation is (iven by
+i(ure 3. Schematic Descri&tion of StrainS&ace +ormulation for Material ,ith #lasto Plastic Cou&lin( #ffect :a< Stress and Strain Increments (b) Plastic+racturin( /ork
"he &ro&osed constitutive model ,as a&&lied in !onlinear +inite #lement Pro(ram !+AP to obtain the deformational res&onse of a concrete s&litcylinder test.
+i(ure 3.;+inite #lement Mesh for Analysis of S&littin( "ension of Concrete Cylinder
+i(ure 3.40%oadDeflection Curve
Author
has verified in an e=tensive com&arison ,ith a ,ide ran(e of available e=&erimental results- the &ro&osed ,orkhardenin( model.
/orks ,ell in modelin( the im&ortant features of concrete behavior- e.(. the brittle failure in tensionthe ductile behavior in com&ression- the hydrostatic sensitivities- and the volumetric dilation under com&ressive loadin(s.
"he
&ro&osed strainsoftenin( model ,orks ,ell in bia=ial and tria=ial com&ressive loadin(s ,ith a relatively lo, hydrostatic com&ressive stress.
•
H.a/-Teh H. e' a$ 1!"!# &
In this &a&er an elastic strainhardenin( &lastic model is &ro&osed.
)ield functions is define as "he failure surface for bia=ial tension in this investi(ation is defined
as
,here 5 the ma=imum com&ressive stren(th of concrete. 5 mean normal stress. 5 octahedral shear stress.
•
and 5 &rinci&al stresses- ,ith
/hen concrete is sub$ected to a combined tensioncom&ression stress state- the yield function is defined as
/here
+or bia=ial com&ression- the yield function is defined as
,here •
+i(ure 3.44. )ield surface of concrete in t,odimensional &rinci&al stress &lane
•
"he sim&lest von mises yield function used as &lastic &otential function
+inally the incremental stress strain constitutive euation for concrete can be e=&ressed as
where
+i(ure 3.43. Com&arison of model ,ith bia=ial +i(ure 3.47 Com&arison of model ,ith combined tenstion test :a< 54E4 tensioncom&ression test 4E0.0?3
+i(ure 3.4>. Com&arison of model ,ith bia=ial com&ression test :a< 5 4E0.?3
Iand) I/(an e' a$2001#& •
In this model dama(e is uantified by the volumetric e=&ansion .
"he four&arameter *sieh"in(Chen :*"C< criterion ,as used as the functional sha&e for the loadin(- failure .
DruckerPra(er ty&e criterion ,as selected as the &lastic &otential function for nonassociated &lastic flo, in the model
A constitutive euation is (iven as
+i(ure3.4? Com&arison of "ria=ial
+i(ure 3.42. Ma=imum and 'esidual
'es&onses bet,een Analytical and
Stren(th #nvelo&e of A=isymmetric
#=&erimental 'esults of Fie et al.
"ria=ial Com&ression Com&arison of
:4;;?<
model ,ith "est 'esults of Fie et al. :4;;?< and Ansari and %i :4;;<
"he &ro&osed constitutive model is used to analye the mechanical res&onse of reinforcedconcrete columns ,ith a circular cross section usin( +# &acka(e.
+i(ure 3.4@ :a< +# Mesh of "est Columns :b< Analytical versus #=&erimental 'esults of Columns
Author concluded that A model In
sho, (ood a(reement ,ith e=&erimental result.
simulatin( the e=&erimental res&onse of reinforced
concrete columns ,ith circular cross section- the model &roduced stren(th and deformation estimates consistent ,ith observed values. "he
by
rate of &ost &eak softenin( ,as some,hat affected such
limitations
as
ne(lectin(
lon(itudinal steel and hoo& fracture.
bucklin(
of
Pe'e( G(a e' a$ 2002#& •
"he aim of authours is to model the load resistance and the deformation ca&acity in unia=ial- bia=ial and tria=ial com&ression.
A three&arameter yield surface is used ,hich is (iven by
,here m is defined as and the elli&tic function as
•
"he
&lastic &otential function used here is
"he
novel hardenin( &arameter is define as
+i(ure 3.4 "he yield surface in
+i(ure 3.4; "he sha&e of the yield
the &lane.
surface in the deviatoric &lane for different hydrostatic stresses
+i(ure 3.30 Unia=ial com&ression tests : < re&orted by 8u&fer et al. :4;2;< com&ared to the model &rediction
+i(ure3..34 Bia=ial com&ression test : < re&orted by 8u&fer et al. :4;2;< com&ared to the model &rediction
+i(ure 3.33 "ria=ial com&ression tests from Imran :4;;>< com&ared to the constitutive model
"he author concluded that the model &redicts the load resistance and the deformation ca&acity of &lain concrete in unia=ial- bia=ial and tria=ial com&ression.
#=&erimental results for stren(th and deformation behaviour ,ere found to be in (ood a(reement ,ith the model &rediction.
And(e3 L)'e4a e' a$2002# & •
"he (eneralied stress strain relations for anisotro&ic elastic solids ,here is the strain tensor- is the stress tensor and is the material constants of orthotro&ically dama(ed solid
substituting the value of in the above stress strain relation the following tensor function was obtained
Deterioration •
of the brittle rocklike materials due to load a&&lied can be described by the dama(e evolution euation e=&ressed in the form of the tensor function
,here as the third one is eual to ero and A- B- F and H are constant "herefore
Pe'e( G(a and M)an )(ae4 2005# •
"he
(eneral stressHstrain relation for this model is
,here 5stress 5dama(e variable 5isotro&ic elastic stiffness 5 effective stress "he
dama(e variable , is a function of the internal variable i.e.
,here the dama(e function one
monotonically (ro,s from ero to
+i(ure 3.37Model res&onse under unia=ial
+i(ure 3.3> Model res&onse under
tension ,ith unloadin( com&ared to
unia=ial com&ression ,ith unloadin(
e=&erimental results re&orted by
com&ared to e=&erimental results re&orted
Go&alaratnam and Shah
by 8arsan and irsa.
Th(ee-*)n' 7end)n, 'e' •
"he material &arameters are #530 GPa J5 03 f t 53.> MPa fc 5 3> MPa 50.00403? and As 5 3 and ' 53? mm and m54
+i(ure 3.3?-3.32 Geometry- loadin( setu& and finite element mesh for the three&oint bendin( test. Com&arison of the analysis of the test on the fine mesh ,ith the e=&erimental bounds
+i(ure 3.3@ Com&arison of the analyses of the three&oint bendin( test on three different meshes
Author sho,s that the nonlocal dama(e&lastic model for concrete can
&rovide
a
mesh
inde&endent
descri&tion
of
various
combinations of tensile and com&ressive failure.
"o kee& the number of &arameters limited- only one scalar dama(e variable ,as considered.
Va)) 8$ Paan)4*a*. e' a$200%# & •
Aim to describe the stren(th and deformational behaviour of both normal and hi(hstren(th concrete under multia=ial com&ression.
A three&arameter hydrostatic&ressure sensitive loadin( surface ,as selected
"he hardenin( &arameter is define as
•
A softenin( function :c< is assumed to have follo,in( form
Plastic &otential function taken here is
+i(ure 3.3 Com&arison bet,een
+i(ure 3.3; Com&arison bet,een
analytical and e=&erimental results
analytical and e=&erimental results
:8u&fer et al.- 4;2;< for concrete under
:8u&fer et al.- 4;2;< for normal concrete
unia=ial com&ression in both a=ial and
under euibia=ial com&ression
lateral directions
+i(ure 3.70 Com&arison bet,een analytical and e=&erimental results :Imran- 4;;>< for normal concrete under tria=ial com&ression and various confinement levels
Author
concluded that the model &erformance ,as evaluated a(ainst
e=&erimental results and it ,as verified that the ultimate stren(thdeformation ca&acity and residual stren(th of confined concrete ,ere &ro&erly ca&tured. Model
follo,s an o&en structure- allo,in( easy recalibration usin( selected
9)an, :. e' a$ 201;# & •
In this study- a four&arameter yield function &ro&osed by *sieh et al. :4;7< is ado&ted.
"he elasto&lastic dama(e constitutive euation can be ,ritten as
"he material &arameter used for numerical simulation are f Kc543. MPa- E 532.@ GPa- λ50.3?. "he &arameters a- b- c- and d, are 3.0002- 0.;4>- ;.4743- and 0.33430- res&ectively- and w54L402.
+i(ure 3.74 Stressstrain curves of unia=ial tension :a< and com&ression :b< :unit MPa
+i(ure 3.77 Ma=imum first &rinci&le +i(ure 3.73 Dam system and monitorin( &oints stress distribution of the slice of dam
+i(ure 3.7> Dama(e factor contour ma& of the slice of dam usin( the dama(e model &ro&osed in soft,are Abaus
+i(ure 3.7? Com&arison of horiontal dis&lacement :a< and vertical dis&lacement :b< of the slice of dam head
"he author concluded that the model reduce the limitation and lacuna of the traditional dama(e constitutive models for concrete.
"he model reflect different stren(th characteristics of concrete in tension and com&ression.
model can also be a&&lied in concrete (ravity dam.
System Develo&ment •
E<e()/en' P(*=ed.(e& !ineteen
standard cubical s&ecimens of the ordinary concrete C30 s&lit into three (rou& as Grou& 4:four s&ecimens<- Grou& 3:seven s&ecimens< and Grou& 7:ei(ht s&ecimens<- ,ere unia=ially tested.
S&ecimen
of Grou& 4-,ere sub$ected to com&ressive stress in main confi(uration as sho,n in +i(ure 7.4.
"he
loadin( &ro(ramme for Grou& 3 and Grou& 7 s&ecimens consisted of four cycles.
cycle
4- the com&ressive load in the direction of the x3 a=is.
cycle
3 -the com&ressive load in the direction of the x7 a=is above the &oint of initial crackin( of the concrete.
cycle Cycle
7- the com&ressive load in the direction of the x3 a=is. >-com&ressive load a&&lied in the direction of the x7 a=is
+i(ure 7.4 Cycles of the loadin( of the s&ecimens of (rou& 3 and 7 tested in main confi(uration
+i(ure 7.3 Cycles of the loadin( of the s&ecimens of (rou& 3 and 7 tested in au=iliary confi(uration
•
The*(e')=a De=()')*n *+ M*de&-
stress strain relations for anisotropic elastic solids where is the strain tensor, is the stress tensor and is the material constants of orthotropically damaged solid Substituting the value of in the above stress strain relation the following tensor function was obtained
•
"he follo,in( nonlinear stress strain relations ,ere obtained for unia=ial com&ression.
,here is a lon(itudinal &rinci&al strain and are transvers &rinci&al strain. The
following non-linear stress strain relations were obtained for biaxial compression.
•
where
S)/.a')*n *+ 'he M*de )n MATLAB&
%ite,ka et al.:3003< have conducted analytical and e=&erimental study of dama(e induced Anisotro&y of concrete as e=&lain above. In this dissertation ,ork the same model is simulated in MA"%AB .
"able 7.4. Material &ro&erties and constants for concrete C30 N%ite,ka et al.:3003
Unit
Group 1
Group 2
Group 3
E0
MPa
!!!
"#$!!
"#$!!
V0
-
!."
!.!
!.!
f c
MPa
-%.#
-%.#
-%.#
A
MPa-
.&&'"!(
.$$'"!(
."$!'"!(-%
-%
-%
)."*&'"!(
)."#$'"!(
-&
-&
-
-
-".#%'"!(-
&.+**'"!(
%.$!+'"!(
$
-) ".%$'"!(
-) !.+)*'"!(
".**+'"!(-$
ant
B C
D
MPa- MPa-"
MPa-"
$."!"'"!(-&
Stress vector for the unia=ial com&ression case ,ere taken from the e=&eriment for lon(itudinal strain calculation is 5N0 3.>?4 3.@;33 ?.@72> .7?2 44.?@44 4>.7?7@
[email protected]@@ 4;.2;2; 34.2774 37.2272 O. %on(itudinal strain vector obtained as an out&ut of the MA"%AB &ro(ram is 5 N 0 0.0004 0.0003 0.0007 0.000? 0.0002 0.000 0.0040 0.0043 0.004> 0.004@ O. Similarly stress vector for the unia=ial com&ression case ,ere taken from the e=&eriment for lateral strain calculation is 5N0 3.00 2.0@>? ;.?3@ 43.7?@3 4?.477 4;.7>0 33.4@?? 37.@2@?O. %ateral strain vector obtained as an out&ut of the &ro(ram is 5 N 0 0.0704 0.02> 0.40@0 0.4>23 0.4;2@ 0.7>>> 0.?74> 0.@;7O. •
+i(ure 7.7 !ormalied stress vEs lon(itudinal strain curve for unia=ial com&ression
+i(ure 7.> !ormalied stress vEs lateral strain curve for unia=ial com&ression
"able 7. 3. #=&erimental data and constants for Concrete A and B tested by • %i(ea. N%ite,ka et al.:3003
Unit
Concrete A
Concrete B
MPa
*#!!
%!+!!
-
!."#
!."#
f c
MPa
-"&.#
-+."&
A
MPa-
&.&%'"!(-%
".+&$'"!(-%
B
MPa-
%.%%'"!(-&
.#*#"'"!(-&
C
MPa-"
-%.)&$'"!(-)
-".&$*$'"!(-)
D
MPa-"
#.%%+'"!(-)
).!$&'"!(-)
Stress vector for the Bia=ial com&ression case ,ere taken from the e=&eriment Nconcrete AO- for lon(itudinal strain calculation is 5N 03.000 ?.4230 @.;370 44.00>3 43.20> 4>.3?? 4?.2>20
[email protected]>2? 4.30@0O.
%on(itudinal strain vector obtained as an out&ut of the MA"%AB &ro(ram is 5N 0 0.0004 0.0003 0.0007 0.000? 0.0002 0.000@ 0.004 0.0042 0.0033O.
•
Stress vector for the Bia=ial com&ression case ,ere taken from the e=&eriment Nconcrete AO- for lateral strain calculation are 5N 0 3.4?2@ 7.@4>7 ?.4;30 @.7@ 40.>3>0 43.230@ 4>.>4@; 42.0??>
[email protected]?72 4.4@34O. 5N 0 4.447 7.4;?4 ?.4;30 @.@4 40.40>? 43.?00; 4>.4@7 42.4>3 4@.@@3@ 4.3;4;O. %ateral strain vectors obtained as an out&ut of the MA"%AB &ro(ram are 5N 0 0.03>4 0.0>33 0.0207 0.0;2 0.4>32 0.4;77 0.3>@; 0.7>;> 0.>@7? 0.24?3O. 5N 0 0.044? 0.077> 0.0?2; 0.0;4 0.47@> 0.3002 0.3;02 0.?>>@ 0.@2?? 0.272O.
+i(ure 7.? !ormalied stress vEs lateral strain curve for bia=ial com&ression
+i(ure 7.2 !ormalied stress vEs lon(itudinal strain curve for bia=ial com&ression.
Performance Analysis "o
validate the &erformance of the model em&loyed in the MA"%AB- simulated results ,ere com&ared to the e=&erimental and theoretical results.
Com&arison of the stress strain curves obtained by simulation in MA"%AB for the s&ecimens of Grou& 4Grou& 3 and Grou& 7 ,ith the theoretical as ,ell as e=&erimental results for concrete under unia=ial com&ression is sho,n in +i(ure >.4- >.3 and >.7.
+i(ure >.4. %on(itudinal and transverse strain for Grou& 4 of the s&ecimen vEs lon(itudinal com&ressive stress
+i(ure >.3. %on(itudinal and transverse strain for Grou& 3 of the s&ecimen vEs lon(itudinal com&ressive stress
+i(ure >.7. %on(itudinal and transverse strain for Grou& 7 of the s&ecimen vEs lon(itudinal com&ressive stress
"he
above curves sho,s (ood a(reement ,ith e=&erimental and theoretical &rediction for the s&ecimens under unia=ial com&ression
Com&arison of the stressstrain curves obtained by simulation in MA"%AB for bia=ial com&ression ,ith e=&erimental data for Concrete A and Concrete B tested by %i(a and the relevant theoretical results is sho,n in +i(ure >.> and >.?.
:a<
:b<
+i(ure >.>. Stressstrain curves for Concrete A sub$ected to bia=ial com&ression a< k 5 Q3 E Q7 5 0.?- b< k 5 4.0
:a<
b
+i(ure >.?. Stressstrain curves for Concrete A sub$ected to bia=ial com&ression a< k 5 Q3 E Q7 5 0- b< k 5 4.0
'eferences 4. *elmut ku&fer. *ubert k. *ilsdorf and hubert rusch-:4;2;< “Behavior of Concrete Under Biaxial Stree! ACI $ournal au(ust . 3. /illam- 8. .- and /arnke- #. P",:4;@>< #Contit$tive %odel for the &riaxial Behavior of Concrete,# IABS# Seminar on Concrete Structure Sub$ected to "ria=ial Stresses Pa&er IIIl- Ber(amo- Italy- May- 4;@> 7. D.. *an and /.+. Chen-:4;?< Ra non$nifor' hardenin laticit* 'odel for concrete 'aterial! Mechanics of Materials > :4;?< 37703. >. D.. *an and /.+. Chen-:4;@< “contit$tive 'odelin in anal*i +f concrete tr$ct$re ournal of #n(ineerin( Mechanics- Tol. 447- !o. >- A&ril4;@. ASC#. ?. *suam"eh *u and /illiam C. Schnobrich-:4;;< “contit$tive 'odelin of concrete b* $in nonaociated laticit*! ournal of Materials in Civil #n(ineerin(- Tol. 4- !o. >- !ovember- 4;;. ASC#2. Sheikh- S. A.- and "oklucu- M. ". :4;;7<. VV'einforced concrete columns confined by circular s&irals and hoo&s.WW ACI Struct. .- ;0:?<- ?>3H??7 @. By Is,andi Imran and S. . Pantao&oulou-:3004< “laticit* 'odel for concrete $nder triaxial co'reion!ournal of #n(ineerin( Mechanics- Tol. 43@- !o. 7- March- 3004. ASC#.
'eferencesX . Peter Grassl - 8arin %und(ren- 8ent Gylltoft-:3003< “Concrete in co'reion a laticit* theor* with a novel hardenin law! International ournal of Solids and Structures 7; :3003< ?30?H?337. ;. '. 'aveendra Babua- Gurmail S. Beni&ala and Arbind 8. Sin(hb:300?
Pa(es 3443>2 40.Peter Grassl and Milan iraW sek ,:3002< “-latic 'odel with non.local da'ae alied to concrete! international $ournal for numerical and analytical methods in (eomechanics Int. . !umer. Anal. Meth. Geomech.- 3002 70@4H;0. 44.Tassilis 8. Pa&anikolaou - Andreas . 8a&&os-:300@< “Confine'ent. enitive laticit* contit$tive 'odel for concrete in triaxial co'reion! International ournal of Solids and Structures >> :300@< @034H@0>
