Application of Drucker-Prager Plasticity Model for Stress- Strain Modeling of FRP Confined Concrete Columns

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1 Available online at Proedia Engineering 14 (211) The Twelfth East Asia-Paifi Conferene on Strutural Engineering and Constrution Aliation of Druker-Prager Plastiity Model for Stress- Strain Modeling of FRP Confined Conrete Columns Jiafei Jiang 1a, Yufei Wu 2b, and Xuemei Zhao 3 1,2,3 Deartment of Building and Constrution, City University of Hong Kong, China Abstrat Existing researh works have identified that Druker-Prager (DP) lastiity model is aable of modeling the stressstrain behavior of onfined onrete. However, the auray of the model largely deends on the adequate evaluation of its arameters that determine the yield riterion, hardening/softening rule and flow rule. U to date, most researh works mainly fous on the first two riteria. The lasti dilation angle is the major arameter that governs the DP flow rule. This aer addresses the lasti dilation roerties of onrete for FRP onfined irular onrete olumns under the theoretial framework of DP model in the ommerial software ABAQUS. Through areful analyses of test results for FRP onfined onrete olumns, it is found that the lasti dilation angle is a funtion of axial lasti strain and the lateral stiffness ratio. A simle model for the lasti dilation angle is subsequently develoed. With the imlementation of this model, the finite element analysis results fit well with the exerimental stress-strain urves for olumns with both low and high onfinement. 211 Published by Elsevier Ltd. Oen aess under CC BY-NC-ND liense. Keywords: Druker-Prager, Plastiity model, FRP onfined onrete, Plasti dilation, ABAQUS. 1 INTRODUCTION Nowadays, researhers have been exloring suitable models for onfined onrete inluding FRP onfined onrete to study the behavior of onfined onrete strutures and develo design guidelines for engineering aliations. As emirial/semi-emirial models are limited to availability of exerimental data, the omutational onstitutive models are attemted more extensively in reent years as it an a Presenter: jjiang4@student. ityu.edu. hk b Corresonding author: yfwu@ityu.edu.hk Published by Elsevier Ltd. Oen aess under CC BY-NC-ND liense. doi:1.116/j.roeng

2 688 Jiafei Jiang et al. / Proedia Engineering 14 (211) rovide a more general framework for the nonlinear behavior of onrete. Although the onrete mehanis is highly omlex, some of the rinile features an be atured by the onstitutive models based on the theory of lastiity (Pekau et al.1992). A lastiity model suitable for onfined onrete should inlude the following features: ressure deendene; ath deendene; non-assoiative flow rule; work or strain hardening; and limited tensile strength. It has been demonstrated by Karabinis et al. (22) that the behavior of simle onrete strutural members an be aurately estimated by Druker-Prager (DP) tye lastiity model whih is also adoted in urrent study. In a lastiity model, the arameters related to frition angle and ohesion govern the yielding and hardening riteria, while the arameter related to lasti dilation determines the flow rule. Extensive researhes have been done on the lasti dilation rate for steel onfined onrete (Karabinis and Kiousis 1994; Rousakis et al. 2; Karabinis and Rousakis 22; Oh 23). For FRPonfined onrete, relatively few works have been undertaken. Mirmiran et al. (2) and Karabinis et al. (28) adoted a onstant lasti dilation rate in some studies. However, Yu et al. (21) showed that the lasti dilatation rate varies with the lasti strains and the lateral stiffness. In this study, the lasti dilation for FRP onfined ylinder is extensively investigated, leading to a model for finite element analyses of FRP onfined onrete. 2 DRUCKER-PRAGER PLASTICITY MODEL The DP model was roosed by Druker and Prager in1952. It an well desribe ressure-sensitive materials suh as rok, soil and onrete. Similar to other lastiity models, there are three riteria ontrolling the framework of DP tye model and hene the auray of reditions by the model. The numerial studies in this aer are based on the linear extended DP model that is built in the ommerial software ABAQUS. Details of the model are briefed in the following setions. 2.1 Yielding riterion and Hardening/softening riterion The yielding funtion for the DP model is in the form of I1 J2 f( K) k( ), (1) J, (2) ( ) [1 (1 )( ) ] 2 K 2 K J2 f K where I 2, J l / 3, (3) 3 l 1 2 in whih the subsrit denotes axial diretion and l reresents lateral diretion; is the frition arameter, whih is the sloe of the yield surfae in the stress sae; k is the hardening/softening funtion whih governs the develoment of subsequent yielding surfae. The funtion f(k) is an indiret exression of Lode s angle ombining seond and third invariant of deviatori stress, J 2 and J 3. K is a material arameter that aounts for stress-ath with the variation of shear strength under a given hydrostati ressure and determines the shae of yielding funtion in deviatoti lane, ranging from.778 to 1. In the ase of uniform onfinement, f(k) equals to 3 irrelevant to K.

3 Jiafei Jiang et al. / Proedia Engineering 14 (211) Flow rule In the DP model, the lasti otential funtion G governs the flow rule. The inrements of lasti strain an be found by, G I dij G J tan 1 ; 3 2 onstant ij 3, (4) Where is a non-negative salar arameter;, the lasti dilation angle, is diretly related to the sloe of lasti volumetri strain, v, and shear strains, s, whih is of great signifiane in the mathematial modeling of ressure deendent material (Rousakis et al. 28). In the ase of uniformly onfined ylinder, 3( d 2 d ) 3d 3 di 3 tan, (5) 2( d d ) 2d 2 dj 2 ' l v 1 ' l s ( 2 v l), l l [(1 v) l v ] E E, (6) in whih is termed as the lasti dilation rate; E and v are the elasti Young s modulus and Poisson s ratio, resetively. A negative value indiates a volumetri omation tendeny while a ositive value indiates a volumetri exansion tendeny. A transition oint arises when is zero in volumetri deformation. Here a omression is onsidered as a negative value and tension ositive. 3 PLASTICITY DILATION In a assive FRP onfining system, the lateral dilation of onrete indues linear elasti onfinement from FRP. Figure 1a illustrates tyial volumetri deformations. The volumetri deformation rate will vary during loading and be influened by the lateral stiffness, 2E f t f /D. As the elasti volumetri deformation is always in omation and develoed in a relatively onstant rate, the lasti volumetri strain ontributes more to the hange in total volumetri deformation (see Fig. 1b). Therefore, aurate redition on lasti deformation ounts in the DP tye lasti model. Volumetri strain C37.7CF C38CF4.4 C38CF6 C38CF8 Axial lasti strain (a) Volumetri strain Elasti volumetri deformation Plasti volumetri deformation Axial lasti strain (b) Total volumetri deformation Figure 1: Volumetri deformation for FRP onfined onrete

4 69 Jiafei Jiang et al. / Proedia Engineering 14 (211) Previous studies Previous studies on the lasti dilation started from steel onfined ases. Karabinis and Kiousis (1996) modeled the dilatation rate in an asymtoti relationshi with lasti strains. Oh (23) roosed a monotoni funtion regressed from the data generated from emirial models for ative onfinement. In the ase of FRP onfinement, Karabinis and Rousakis (22) initially adoted their revious asymtoti funtion. The lasti dilation rate dereased from -.6 to 3, whih indiates lasti volumetri omation only and ends in zero lateral-to-axial lasti strain ratio. Rousakis et al. (28) reently suggested a onstant dilation rate for different onrete strengths while the onstant varies with the modulus of onfinement. Mirmiran et al. (2) found a zero lasti dilation rate ould give reasonably lose redition for C29.6 onrete with 6 lies of FRP through a trial-and-error roedure but ointed out that the onstant rate ould not reresent the true dilation tendenies. Yu et al. (21) demonstrated that the flow rule reflets the effet of lasti deformation and the rate of onfinement inrement an lead to reasonably lose redition of the behavior of FRP-onfined onrete. They resented the roedures for obtaining the variation of the otential funtion arameters. However, a systemati and feasible method was not rovided. 3.2 Test observation The test data olleted in this work inlude 6 FRP onfined seimens (diameter D = 152 mm and height H = 35 mm) tested by Teng et al. (27), 23 FRP onfined seimens (diameter D = 152 mm and height H = 35 mm) tested by Jiang et al. (27) and 5 ontrol seimens (D = 15 mm and H = 3 mm) tested by Wang et al. (28). The unonfined onrete strength ranges from 3 MPa to 5 MPa. From the test data, is alulated from Eqs. 5-6, the orresonding E alulated through ACI 318 formula and v set as.2. A similar trend is observed in all urves that relate with the axial lasti strain in absolute value as illustrated in Fig. 2a. begins from a negative value,, and inreases to the maximum oint, ( r, m ). Afterwards, it dereases to an asymtoti value, u. Figure 2b shows a series of suh urves with variation in lateral stiffness of FRP and unonfined onrete strength. The omarison shows that the desending urve moves downward with the inrease in lateral stiffness relative to unonfined onrete strength (2E f t f /Df ) ( r, m ) C39.6GF1 C38CF1 C39.6GF u 2-2 C39.6GF3.1.2 C38CF C38CF8-4 Axial lasti strain (a) -6 Axial lasti strain (b) Figure 2: Plasti dilation urves obtained from tests

5 Jiafei Jiang et al. / Proedia Engineering 14 (211) Proosed model The tyial lasti dilation urves in Fig. 2 an be well reresented by the following equation: 2 a b( ) 2 1 d( ), (7) where a, b,, d and are oeffiients to be determined. Careful study of the ritial values m, r, and u in Fig. 2a reveals that these values are losely related to the relative lateral stiffness ratio 2Et f f ' Dfo, (8) with the relationshis as shown in Fig. 3. The initial value and sloe of are not affeted by and an be reresented by the following onstants d ( ) 37, 157. (9) M d 8 m = R 2 = m = u= u = R 2 = (a) r r= R 2 = (b) Figure 3: m, r, and u From the relationshis for m, r, and u in Fig. 3, the oeffiients in Eq. 7 an be derived to give M ( ) (.25e6 2.52e6 4.27e7)( ), 35 (1) ( ) ( e5)( ) M ( ) (.25e66.73e6)( ), ( ) ( e5)( ) (11) 3.4 Disussions Subjeted to uniaxial omressive loading, both onrete ore and FRP exand laterally due to the Poisson s effet. The onrete ore will exhibit larger lateral dilation after the damage indued by

6 692 Jiafei Jiang et al. / Proedia Engineering 14 (211) miroraking whih auses faster lateral exansion and a larger Poisson s ratio of the onrete. The onfinement will not take effet until the lateral-to-axial strain ratio of the onrete larger than the Poisson s ratio of the jaket. Therefore, will take an initial negative (volumetri ontration) that is irrelevant to the lateral stiffness ratio, similar to unonfined onrete. When the axial load inreases, the miroraking in onrete develos quikly and hene inreases quikly. One the onfinement takes effet, the lateral onstraint from jaket will ounterat onrete lateral exansion, leading to a redued rate of inrease of. At a artiular lateral onfinement, reahes its eak m and starts to redue, whih indiates the beginning of derease in the exansion rate. When the interation between the jaket and the onrete is stabilized, dereases to an asymtoti value, u. It is obvious that the higher the lateral stiffness is the larger onstraint FRP an exert. As a result, an inrease in the lateral stiffness ratio will derease m and u, as well as r indiating an earlier onset of the turning oint, whih an be learly seen in Fig. 3. However, the damage annot be revented before the lateral-to-axial strain ratio of onrete reahes the Poisson s ratio of FRP. Clearly, a stiffer jaket or a larger will ause an earlier onset of the eak oint in Fig. 2b whih leads to the redution of r with an inrease in, as shown in Fig. 3b. However, this redution slows down and stabilized to a value around.1. Similarly, the inability to further revent damage in onrete, when further inreases after a suffiiently large value, slows down the sloe of the m vs. urve. Meanwhile, u will aroah to a lower bound at large values as a further inrease in the stiffness of the FRP jaket annot further restrain the lateral exansion to a smaller rate than the Poisson s ratio of FRP. 4 VERIFICATION OF PROPOSED MODEL 4.1 Imlementation of finite element analysis in ABAQUS The onrete olumn is modeled as 1/8 of a ylinder in the modeling with aroriate boundary onditions. The FRP sheet is modeled as elasti laminar with orthotroi elastiity in lane stress without bending stiffness. The elasti modulus of FRP is only designated in the fiber diretion. Its orresonding Poisson s ratio is set as.3. There is no relative sli between FRP and the onrete. The loading is under axial dislaement ontrol alied on the to of the onrete, without alying diretly on the FRP. The onrete is onsidered to be isotroi elasti body. The elasti modulus, E, is alulated in aordane with ACI 318. The Poisson s ratio, v, is set to be.2. With the assumtion of a onrete frition angle of 54 (Yu et al. 21), the hardening arameter k for eah seimen an be alulated through the yielding surfae funtion (Eq. 1) from test data under ertain axial lasti strain. For a artiular axial lasti strain, the hardening funtion k() an be obtained. For a artiular seimen, there is a ertain loading history or ath in the stress sae. Therefore, it is irrelevant to the yielding and hardening rule on the FEA results. The roosed lasti dilation urve (Eqs. 1-11) is inutted into ABAQUS as tabular data through SDFV otion. 4.2 Sensitivity study As onstant value was adoted in the revious studies by many researhers, there is a ossibility that the stress-strain resonse is insensitive to the lasti dilation urve. Therefore sensitivity study is undertaken to find out how sensitive of the arameter, M, m and u are to the stress-strain urve. The referene seimen is a C39.6 ylinder with 2 layers of glass FRP (Teng et al. 27). The base values for the four arameters are -5.86, , 52.6 and 35.4, resetively. With a 5% deviation from the base value of and M, the differene is 2.3% and 2.4% for the axial strain, resetively. However, a 2% deviation for m will indue over 3% differene in the axial strain; and a 3% deviation for u will

7 Jiafei Jiang et al. / Proedia Engineering 14 (211) ause 12% differene in the axial strain. Therefore the stress-strain resonse is sensitive to m and u but insensitive to the variation of and M. 4.3 Verifiation The lasti dilation angle alulated with Eqs has an average orrelation oeffiient of.874 to the test results (Table 1). The test seimens in Table 1 have 14 different tyes. The simulation results an well redit the omression strength and its orresonding strain at FRP ruture (Fig.4). The average errors are 1.48% and 5.79% for strength and axial strain, resetively (Table 1). Considering the test sattering of 5.9% in strength and 16.6% in strain between different seimens in the same ategory, the FEM results are onsidered as exellent. The orrelation oeffiients are as high as.9996 and.982 for strength and its axial strain, resetively. Therefore roosed model erforms well for these olumns. Stress (MPa) Stress(MPa) C44.2CF2 C44.2CF1 C47.6CF3 Jiang et al.(27) Simulation result Strain C45.9GF1 C45.9GF3 C45.9GF2 Jiang et al.(27) Simulation result Strain Stress(MPa) Stress(MPa) C33.1GF1 C37.7CF C38CF8 C38CF6 C38CF4 Jiang et al. (27) Simulation result Strain C39.6FG3 C39.6GF1 C39.6GF2 Teng et al.(27) Simulation result Strain Figure 4: Simulation and test results omarison 5 CONCLUSION The lasti dilation model is essential for FEM analyses of FRP-onfined onrete using DP lastiity model. Different from atively-onfined onrete, the lasti dilation angle varies with the axial lasti strain and the lateral stiffness ratio. An exliit model for the lasti dilation angle is develoed in this work through analytial study of test results and FEM simulations by ABAQUS. The good agreement with test results in both the dilation angle itself and the redited stress-strain resonse urves demonstrates the auray and effetiveness of the roosed model.

8 694 Jiafei Jiang et al. / Proedia Engineering 14 (211) Table 1: Comarison of simulation and test results Soure Teng et al. (27) Jiang et al. (27) Seimen ID Seimen No f MPa Fiber tye Ply No R 2 (FEA) Error (%) f' f' (FEA) C39.6GF glass C39.6GF glass C39.6GF glass C33.1GF glass C45.9GF glass C45.9GF glass C45.9GF glass C38CF arbon C38CF arbon C38CF arbon C37.7CF arbon C44.2CF arbon C44.2CF arbon C47.6CF arbon Error (%) Aknowledgements The work desribed in this aer was fully suorted by a grant from the City University of Hong Kong (Projet No ). The authors are also grateful to Professor JG Teng s grou for roviding the test data and Dr GM Chen for his hel on the use of ABAQUS. Referenes [1] A. Mirmiran1 and M. Shahawy (1997). Dilation harateristis of onfined onrete. Int J Meh Cohesive-Fritional Materials. 2(3), [2] B. Oh (23). A lastiity model for onfined onrete under uniaxial loading. Ph.D Thesis. Deartment of ivil engineering, Lehigh University. [3] J.G. Teng, T. Yu, Y.L. Wong, and S.L. Dong (27). Hybrid FRP-onrete steel tubular olumns: Conet and behaviour. Constrution and Building Materials. 21(4): [4] T. Jiang and J.G. Teng (27). Analysis-oriented stress-strain models for FRP-onfined onrete. Engineering Strutures. 29(11), [5] L. M. Wang and Y.F. Wu (28). Effet of orner radius on the erformane of CFRP-onfined square onrete olumns: Test. Engineering Strutures. 3(2), [6] T. Yu, J.G. Teng, Y.L. Wong, and S.L. Dong (21). Finite Element Modeling of Confined Conrete-I:Druker-Prager Tye Plastiity Model. Engineering Strutures. 32(3), [7] T. C. Rousakis, A. I. Karabinis, P. D. Kiousis and R. Tefers (28). Analytial modelling of lasti behaviour of uniformly FRP. Comosites Part B:Engineering. 39(7-8),