Paper

Materials and Design 31 (2010) 278–286

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Materials and Design

journal homepage: www.elsevier .com/locate /matdes

Plasticity models for concrete material based on different criteria includingBresler–Pister

Tayfun Dede *, Yusuf AyvazDepartment of Civil Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 March 2009Accepted 12 June 2009Available online 17 June 2009

Keywords:Failure analysisConcretePlastic behaviorBresler–Pister criterion

0261-3069/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.matdes.2009.06.018

* Corresponding author. Tel.: +90 462 3772638; faxE-mail address: [email protected] (T. Dede).

The purpose of this study is to investigate nonlinear behavior of reinforced concrete (RC) structures withthe plasticity modeling. For this aim, a nonlinear finite element analysis program is coded in MATLAB.This program contains several yield criteria and stress–strain relationship for compression and tensionbehavior of concrete. In this paper, the well-known criteria, Drucker–Prager, von Mises, and Mohr Cou-lomb, and a new criterion-Bresler–Pister are taken into account. The elastic–perfectly plastic and Saenzstress–strain relationships in compression and tension stiffening in tension behavior of concrete are usedwith four different yield criteria mentioned above. The proposed models are in good agreement with theexperimental and analytical results taken from the literature. It is concluded that the coded program, theproposed models, and Bresler–Pister criterion can be effectively used in nonlinear analysis of reinforcedconcrete beams.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Considerable constitutive models have been proposed to definenonlinear behavior and stress–strain relationship of reinforcedconcrete (RC) material. These models can be classified into ortho-tropic models, nonlinear elastic models, plasticity models, endo-chronic models, fracture mechanics models and micromodels [1].Using these models, several studies have been made in the fieldof nonlinear analysis of RC structure to predict the behavior ofreinforced concrete structures more reliable. Arslan [2] investi-gated the sensitive of the Drucker–Prager modeling parametersand the use of it in plasticity theory for shear design of RC beams.Park and Klingner [3] presented a nonlinear analysis study of RCmembers by using plasticity multiple failure criteria. Wang andHsu [4] applied the nonlinear finite element analysis to varioustypes of RC structures using a new set of constitutive models. Bra-tina et al. [5] presented a study on materially and geometricallynonlinear analysis of RC planar frames by dealing with the fiber-based constitutive equations of concrete and steel. Zhao et al. [6]studied the load-deflection and failure characteristics of deep RCcoupling beams. Pankaj and Lin [7] used two similar continuumplasticity material models to examine the influence of the materialmodeling on the seismic response of RC frame structures. Belmou-den and Lestuzzi [8] investigated post peak modeling and nonlin-ear performance of RC structural walls. Bischoff [9,10],

ll rights reserved.

: +90 462 3772606.

Stramandinoli and Rovere [11] and Dede and Ayvaz [12] studiedon RC structures by considering tension stiffening effect.

Among the models given above, plasticity models need a yieldfunction, a hardening rule, a flow rule and a stress–strain relation-ship to construct the plastic material matrix for the plastic behav-ior of concrete. A review of the literature indicates that there arenot any studies based on the Bresler–Pister criterion for plasticbehavior of concrete. This yield function can be found in the booksconcerning with the plasticity theory. But, its plasticity materialmatrix or any application of this function to the RC structures isnot found.

In this paper, derivation of plastic material matrix based onBresler–Pister yield function and two applications of this functionto the RC beams are presented. For this aim, a nonlinear finite ele-ment analysis program is coded in MATLAB. This program containsseveral yield criteria and stress–strain relationship for compressiveand tensile behavior of concrete. In the nonlinear analysis, thewell-known criteria, Drucker–Prager, von Mises, and Mohr Cou-lomb and as a new criterion, Bresler–Pister, are taken into account.The elastic–perfectly plastic and Saenz stress–strain relationship incompressive and tension stiffening in tensile behavior of concreteare used with four different yield criteria mentioned above.

2. Yield criteria for concrete

The concrete is assumed to be elastic until it reaches the yieldlimit. Beyond yielding, plastic deformations take place. So, residualplastic deformations remain after removing the loading. A consid-erable amount of formulations have been proposed for concrete as

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Nomenclature

f 0c uniaxial compressive cylinder strengthf 0t uniaxial tensile strengthh angle of similaritys effective von Mises stress/ internal friction angled kronecker deltaa, k material parametersecr cracking strain of concretercr cracking stress of concreterf, ef control point coordinates on stress–strain curveroct octahedral normal stresssoct octahedral shear stressep concrete strain corresponding to rp

rp peak concrete compressive stressqx reinforcement ratio in global direction of the X axisqy reinforcement ratio in global direction of the Y axisc cohesion

D elastic material-stiffness tensorDc material matrix of concreteDep elastic–plastic material-stiffness tensorDp plastic material-stiffness tensorDs material matrix of equivalent reinforcing bar elementsE Young’s modulusf yield functionHp plastic hardening modulusI1 first invariant of stress tensorJ2 second invariant of stress deviator tensorJ3 third invariant of deviatoric stress tensorK initial tangent moduluss deviatoric stresse strainee elastic strainep plastic strainr stress

Table 1Test data for Bresler–Pister Criterion.

Test roct=f 0c soct=f 0c

r1 ¼ f 0t13

�f 0tffiffi2p

3�f 0t

r3 ¼ �f 0c � 13

ffiffi2p

3

r2 ¼ r3 ¼ �f 0bc � 23

�f 0bc

ffiffi2p

3�f 0bc

T. Dede, Y. Ayvaz / Materials and Design 31 (2010) 278–286 279

a yield function such as Drucker–Prager, von Mises, Mohr Cou-lomb, Tresca, Rankine, William Warnke, Ottosen, Hsieh Ting Chen,and Bresler–Pister [13]. The well-known yield function for Druc-ker–Prager, von Mises, and Mohr Coulomb are given by the follow-ing equations, respectively [14].

f ¼ aI1 þffiffiffiffiJ2

p� k ð1Þ

f ¼ffiffiffiffiJ2

p� k ð2Þ

f ¼ I1 sin /þ 12

3ð1� sin /Þ sin hþffiffiffi3pð3þ sin /Þ cos h

h i ffiffiffiffiJ2

p� 3c cos / ð3Þ

cosð3hÞ ¼ 3ffiffiffi3p

2J3

J3=22

: ð4Þ

where f is yield function, a and k are the material parameters, c iscohesion, / is internal friction angle, I1 is the first invariant of stresstensor, J2 is the second invariant of deviator stress tensor, J3 is thethird invariant of deviator stress tensor, and h is angle of similarity.

The Bresler–Pister criterion is the extension of Drucker–Pragercriteria. This yield function in terms of octahedral stresses is givenby

soct

f 0c¼ a� b

roct

f 0c

� �þ c

roct

f 0c

� �2

: ð5Þ

where a, b, and c are the material parameters of this yield function.These parameters can be established by using available experimen-tal test data given in Table 1 [14]. In this table, �f 0t and �f 0bc are the nor-malized strengths, f 0t is uniaxial tensile strength, f 0c is uniaxialcompressive cylinder strength, f 0bc is equal biaxial compressivestrength, roct is octahedral normal stress and soct is octahedral shearstress.

The octahedral normal and shear stresses are given by the fol-lowing equations, respectively.

soct ¼ffiffiffiffiffiffiffiffi23

J2

r

roct ¼I1

3

ð6Þ

The normalized strengths are given by the following equations.

�f 0t ¼f 0tf 0c; �f 0bc ¼

f 0bc

f 0c: ð7Þ

When these experimental test data are substituted into Eq. (5), theparameters a, b and c can be obtained by solving a system of threelinear equations given below.

a ¼ffiffiffi2p

3�f 0t

�f 0bcð8�f 0bc þ �f 0t � 3Þ=D

b ¼ffiffiffi2pð4�f 02bc � �f 0bc � �f 0bc

�f 0t þ �f 0tÞð1� �f 0tÞ=Dc ¼ 3

ffiffiffi2pð3�f 0t

�f 0bc � �f 0bc � 2�f 0tÞ=D

ð8Þ

where

D ¼ ð2�f 0bc � 1Þð2�f 0bc þ �f 0tÞð1þ �f 0tÞ ð9Þ

Substituting Eq. (6) into Eq. (5) and rewriting Eq. (5), the Bresler–Pister yield function in terms of stress invariant can be obtained,and it is given as

f ¼ c9f 02c

� �I21 �

b3f 0c

� �I1 �

ffiffiffiffiJ2

pffiffiffi3p

f 0c

! ffiffiffiffiJ2

p� a: ð10Þ

3. Plastic material matrix for concrete based on Bresler–Pistercriterion

In the plasticity theory, total strain can be assumed to be thesum of the elastic strain and plastic strain as given in Eq. (11),and stress increment, drij, for strain increment, deij, is given inEq. (12) [15].

deij ¼ deeij þ dep

ij ð11Þdrij ¼ Dep

ijkldeij ð12Þ

where Depijkl is elastic–plastic material matrix. In the case of associ-

ated flow rule the general form of this matrix is given as,

Depijkl ¼ Dijkl þ Dp

ijkl ð13Þ

280 T. Dede, Y. Ayvaz / Materials and Design 31 (2010) 278–286

where Dijkl is the elastic material matrix of the element (see Section5), and

Dpijkl ¼ �

Dijkl@f@rij

@f@rpq

Dpqkl

hþ @f@rij

Dijkl@f@rkl

ð14Þ

where

h ¼ UHp @f@s

ð15Þ

where

U ¼ 1s@f@rij

rij ð16Þ

where s is von Mises effective stress s ¼ffiffiffiffiffiffiffi3J2

p� �and Hp is slope of

uniaxial stress–strain curve.The gradient @f

@rijcan be written as

@f@rij¼ @f@I1

@I1

@rijþ @f@J2

@J2

@rijþ @f@J3

@J3

@rij: ð17Þ

Taking the derivatives of Eq. (10) with respect to I1, J2 and J3, the fol-lowing equations can be obtained.

@f@I1¼ 2c

9f 02cI1 �

b3f 0c

@f@J2¼ � 1ffiffiffiffiffiffiffi

6J2

pf 0c

@f@J3¼ 0

ð18Þ

The gradient @I1@rij

and @J2@rij

are called to be the kronecker delta anddeviatoric stress tensor, respectively, and they are given in Eqs.(19) and (20), respectively.

@I1

@rij¼ dij ¼

1 0 00 1 00 0 1

264

375 ð19Þ

@J2

@rij¼ sij ¼ rij �

I1

3dij: ð20Þ

By substituting Eqs. (18)–(20) into Eq. (17), the following equationcan be obtained.

@f@rij¼ 2c

9f 02cI1 �

b3f 0c

� �dij �

1ffiffiffiffiffiffiffi6J2

pf 0c

!sij ð21Þ

The gradient @f@s based on Bresler–Pister criterion can be obtained,

and it is given below.

@f@s¼ �

ffiffiffi2p

3f 0c: ð22Þ

Strain, ε

Stre

ss, σ

K=1.0

K=1.2

K=1.4

K=1.6

K=1.8

K=2.0

σp

a

Fig. 1. Stress–strain curve of (a) Saenz and (b) elastic–per

4. Stress–strain curves for concrete

In order to define stress–strain relationship for concrete severalstress–strain curves are proposed by researchers [4,16–25]. TheSaenz [19] and elastic–perfectly plastic stress–strain relationshipused for the behavior of concrete in compression are given inEqs. (23) and (24), respectively, and they are given in Fig. 1.

rc ¼ rp

K ecep

� �� ec

ep

� �2

1þ A ecep

� �þ B ec

ep

� �2þ C ec

ep

� �3 ð23Þ

rc ¼ �ec

ep

� �rp if ep < ec < 0

rc ¼ �rp if ec < ep < 0ð24Þ

where

A ¼ C þ K � 2; B ¼ 1� 2C; C ¼ KðKr � 1ÞðKe � 1Þ2

� 1Ke

ð25Þ

K ¼ Eep

rp; Ke ¼

ef

ep; Kr ¼

rp

rfð26Þ

where ef and rf are the control point coordinates on descendingbranch of stress–strain curve, rc is concrete compressive stress, ec

is concrete compressive strain, rp is peak concrete compressivestress, ep is concrete compressive strain corresponding to rp, andE is modulus of elasticity of concrete.

The stress–strain curve of concrete in tension proposed byWang and Hsu [4] is shown in Fig. 2a. The ascending and descend-ing braches of this curve are given by the following equation.

rt ¼ Eet if et � ecr

rt ¼ rcrecr

et

� �0:4

if et > ecr

ð27Þ

where rt is concrete tensile stress, et is concrete tensile strain, rcr isconcrete cracking stress, and ecr is concrete cracking strain.

The other stress–strain curve of concrete in tension used in thispaper is Vecchio 1982 curve [26]. This curve is shown in Fig. 2b andits stress–strain relationship is given by the following equation.

rt ¼rcr

1þffiffiffiffiffiffiffiffiffiffiffiffiffi200etp if 0 < ecr < et ð28Þ

5. Material matrix of finite element

The material matrix of a finite element is constructed to be thesum of the material matrices of the concrete and reinforcement. Inthis calculation, the reinforcement embedded in the concrete ele-ments is represented by an equivalent element. The materialmatrices of concrete and reinforcement are given, respectively, as:

Strain, ε

Stre

ss, σ

σp

εp

b

fectly plastic model for concrete under compression.

Strain, ε

Stre

ss, σ

σcr

εcrStrain, ε

Stre

ss, σ

σcr

εcr

a b

Fig. 2. Stress–strain curve of: (a) Wang and Hsu [4] and (b) Vecchio 1982 model for concrete under tension.


½D� ¼ ½Dc� þ ½Ds� ¼E

1� v2

1 v 0v 1 00 0 ð1� vÞ=2

264

375þ

qxEs 0 00 qyEs 00 0 0

264

375

ð29Þ

where Dc and Ds are the material matrices of concrete and equiva-lent reinforcement elements, respectively, Es is the modulus of elas-ticity of reinforcement, qx and qy are the reinforcement ratios inglobal directions of the x and y axes, respectively.

6. Applications

The applicability and verification of the developed program aredemonstrated by comparing the results obtained in this study withthe experimental and analytical results of two different RC beams,Bresler–Scordelis beam and J4 beam, given below.

6.1. Bresler–Scordelis beam

The first RC member used to validate the program coded is Bres-ler–Scordelis beam. It is simply supported RC beam [4,27] and isshown in Fig. 3. The longitudinal reinforcement consists of foursteel bars with total area of 2580 mm2. The concrete has a com-pressive strength of 24.5 MPa and elastic modulus of 21,300 MPa.The elastic modulus of steel bars is 191,400 MPa.

In the finite element modeling, 4-noded rectangular plane-stress element is used. This element has two displacement degreesof freedom at a point and eight displacement degrees of freedom inan element. Perfect bond between concrete and reinforcement isassumed.

Since the method used herein is a numerical method, the finiteelement method, there is always some error in the results, depend-

Fig. 3. Bresler–Sco

ing on the mesh size used to solve the problem. Therefore, for thesake of accuracy in the results, rather than starting with a finiteelement mesh size, the mesh size to produce the desired accuracyis determined. To find out the required mesh size, convergence ofthe maximum displacement is checked for different mesh sizes.In conclusion, the results have an acceptable error when usingapproximately 70 elements. Therefore, 70 elements which is alsothe number of the elements used in the literature are used in thisstudy in order to compare the results obtained in this study withthe experimental and theoretical results given in the literature. Fi-nite element modeling of this beam is given in Fig. 4.

The results of the nonlinear analysis of this beam by using yieldfunction of Drucker–Prager criterion with two different tensionstress–strain curves ([4], Vecchio 1982) for the tension behaviorof concrete and with two different compression stress–straincurves (elastic–perfectly plastic and Saenz) for the compressionbehavior of concrete are given in Fig. 5. These results are comparedwith each other and with the experimental result taken from theliterature [4,27]. As seen from this figure, the load–displacementcurves obtained in this study are in good agreement with theexperimental result.

The results of the nonlinear analysis of this beam by using yieldfunction of von Mises criterion with two different tension stress–strain curves ([4], Vecchio 1982) for the tension behavior of con-crete and with two different compression stress–strain curves(elastic–perfectly plastic and Saenz) for the compression behaviorof concrete are given in Fig. 6. These results are compared witheach other and with the experimental result taken from the litera-ture [4,27]. As seen from this figure, the load–displacement curvesobtained in this study are in good agreement with the experimen-tal result.

The results of the nonlinear analysis of this beam by using yieldfunction of Mohr Coulomb criterion with two different tension

rdelis beam.

1,2, …, 88 : Node numbers

1 2 10

61 70

11 1 11 12

78 88

P

1828.8 mm

127

425

.5 m

m

1 2 70

2 3

, , …, : Element numbers

Fig. 4. Finite element modeling of Bresler–Scordelis beam.

0 2 4 6 8Displacement (mm)

0

100

200

300

Loa

d (k

N)

This study Tension model Compression model

Wang and Hsu (2001) Elastic perfectly plastic

Wand and Hsu (2001) Saenz

Vecchio 1982 Elastic perfectly plastic

Vecchio 1982 SaenzExperimental

Fig. 5. Load–displacement curves of Bresler–Scordelis beam for Drucker–Prager criterion.

Displacement (mm)

0

100

200

300

Loa

d (k

N)






0 2 4 6 8 10

Fig. 6. Load–displacement curves of Bresler–Scordelis beam for von Mises criterion.


stress–strain curves ([4], Vecchio 1982) for the tension behavior ofconcrete and with two different compression stress–strain curves(elastic–perfectly plastic and Saenz) for the compression behaviorof concrete are given in Fig. 7. These results are compared witheach other and with the experimental result taken from the litera-ture [4,27]. As seen from this figure, the load–displacement curvesobtained in this study are in good agreement with the experimen-tal result.

The results of the nonlinear analysis of this beam by using yieldfunction of Bresler–Pister criterion with two different tension

stress–strain curves ([4], Vecchio 1982) for the tension behaviorof concrete and with two different compression stress–straincurves (elastic–perfectly plastic and Saenz) for the compressionbehavior of concrete are given in Fig. 8. These results are comparedwith each other and with the experimental result taken from theliterature [4,27]. As seen from this figure, the load–displacementcurves obtained in this study are in good agreement with theexperimental result.

In general, the results obtained using the compression models,tension models and the yield criteria considered in this study are

0 2 4 6 8 10Displacement (mm)

0

100

200

300

Loa

d (k

N)






Fig. 7. Load–displacement curves of Bresler–Scordelis beam for Mohr Coulomb criterion.

Displacement (mm)

0

100

200

300

Loa

d (k

N)






0 2 4 6 8

Fig. 8. Load–displacement curves of Bresler–Scordelis beam for Bresler–Pister criterion.


in good agreement with the experimental result. Especially, the re-sults of new criterion, Bresler–Pister, show excellent agreementwith the results obtained by using the other criteria and with theexperimental result.

6.2. Simply supported J4 beam

The second RC member used to validate the program coded is J4beam. It is simply supported [28,29] and is shown in Fig. 9. The lon-gitudinal reinforcement consists of two steel bars with total area of1021 mm2. The concrete has a compressive strength of 33 MPa andelastic modulus of 26,200 MPa. The elastic modulus of steel bars is203,000 MPa.

Fig. 9. J4 b

The finite element mesh convergence of this beam is also stud-ied. It is concluded that the results have an acceptable error whenusing approximately 45 elements. This element number is also thenumber of the elements used in the literature. Therefore, using thiselement number makes possible compare the results obtained inthis study with the experimental and theoretical results given inthe literature. Finite element modeling of this beam is given inFig. 10.

The results of the nonlinear analysis of this beam by using yieldfunction of Drucker–Prager criterion with two different tensionstress–strain curves ([4], Vecchio 1982) for the tension behaviorof concrete and with two different compression stress–straincurves (elastic–perfectly plastic and Saenz) for the compression

eam.

1 1 2 9

10

37 45

2 3 9 10 11

51 60

P

1850 mm

51

457

mm

Fig. 10. Finite element modeling of J4 beam.

0 2 4 6 8 10 12 14 16Displacement (mm)

0

40

80

120

160

200

Loa

d (k

N)


Wang and Hsu (2001) Elastic Perfectly Plastic

Wang and Hsu (2001) Saenz

Vecchio 1982 Elastic Perfectly Plastic

Vecchio 1982 Saenz

Demir (1998) Barzegar and Schnobrich (1986) Experimental

Fig. 11. Load–displacement curves of J4 beam for Drucker–Prager criterion.


behavior of concrete are given in Fig. 11. These results are com-pared with each other and with the experimental [29] and analyt-ical results [28,30]. As seen from this figure, the load–displacementcurves obtained in this study are in good agreement with theexperimental and analytical results.

The results of the nonlinear analysis of this beam by using yieldfunction of von Mises criterion with two different tension stress–strain curves ([4], Vecchio 1982) for the tension behavior of con-crete and with two different compression stress–strain curves(elastic–perfectly plastic and Saenz) for the compression behaviorof concrete are given in Fig. 12. These results are compared witheach other and with the experimental [29] and analytical results[28,30]. As seen from this figure, the load–displacement curves ob-tained in this study are in good agreement with the experimentaland analytical results.

Displace

0

40

80

120

160

200

Loa

d (k

N)

0 2 4 6

Fig. 12. Load–displacement curves of

The results of the nonlinear analysis of this beam by using yieldfunction of Mohr Coulomb criterion with two different tensionstress–strain curves ([4], Vecchio 1982) for the tension behaviorof concrete and with two different compression stress–straincurves (elastic–perfectly plastic and Saenz) for the compressionbehavior of concrete are given in Fig. 13. These results are com-pared with each other and with the experimental [29] and analyt-ical results [28,30]. As seen from this figure, the load–displacementcurves obtained in this study are in good agreement with theexperimental and analytical results.

The results of the nonlinear analysis of this beam by using yieldfunction of Bresler–Pister criterion with two different tensionstress–strain curves ([4], Vecchio 1982) for the tension behaviorof concrete and with two different compression stress–straincurves (elastic–perfectly plastic and Saenz) for the compression

ment (mm)





Vecchio 1982 Saenz


8 10 12 14 16

J4 beam for von Mises criterion.

0 2 4 6 8 10 12 14 16Displacement (mm)

0

40

80

120

160

200

Loa

d (k

N)





Vecchio 1982 Saenz


Fig. 13. Load–displacement curves of J4 beam for Mohr Coulomb criterion.

Displacement (mm)

0

40

80

120

160

200

Loa

d (k

N)





Vecchio 1982 Saenz


0 2 4 6 8 10 12 14 16

Fig. 14. Load–displacement curves of J4 beam for Bresler–Pister criterion.


behavior of concrete are given in Fig. 14. These results are com-pared with each other and with the experimental [29] and analyt-ical results [28,30]. As seen from this figure, the load–displacementcurves obtained in this study are in good agreement with theexperimental and analytical results.

In generally, the results obtained using the compression mod-els, tension models and the yield criteria considered in this studyare in good agreement with the experimental and analytical re-sults. Especially, the results of new criterion, Bresler–Pister, showexcellent agreement with the results obtained by using the othercriteria and with the experimental and analytical results.

7. Conclusions

Analytical models are presented for the nonlinear finite elementanalysis of reinforced concrete structures. Based on the Bresler–Pister yield function, a plastic material matrix for concrete materialis constructed. Also, different stress–strain curves of concrete fortension and compression behavior are taken into account and thewell-known criteria, Drucker–Prager, von Mises and Mohr Cou-lomb are also used for the plastic behavior of concrete.

The computer program coded in this study is useful for predict-ing the behavior of reinforced concrete structures. This programcontains the well-known criteria (Drucker–Prager, von Mises, andMohr Coulomb), a new criterion (Bresler–Pister), stress–straincurves for the compression behavior of concrete (elastic–perfectlyplastic and Saenz model), and tension stiffening model ([4], Vec-chio 1982 model).

The proposed models and Bresler–Pister criterion can be effec-tively used in nonlinear analysis of reinforced concrete beams.

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