Finite Element Analysis of Concrete Structures Using PlasticDamage Model in 3-D Implementation

Transcription

1 Finite Element Analysis of Concrete Structures Using PlasticDamage Model in 3-D Imlementation O. Omidi1 and V. Lotfi,* Received: October 009 Acceted: August 010 Abstract: Neither damage mechanics model nor elastolastic constitutive law can solely describe the behavior of concrete satisfactorily. In fact, they both fail to reresent roer unloading sloes during cyclic loading. To overcome the disadvantages of ure lastic models and ure damage aroaches, the combined effects need to be considered. In this regard, various classes of lastic-damage models have been recently roosed. Here, the theoretical basics of the lastic-damage model originally roosed by Lubliner et al. and later on modified by Lee and Fenves is initially resented and its numerical asects in three-dimensional sace are subsequently emhasized. It should be mentioned that a art of the imlementation in 3-D sace needs to be reformulated due to emloying a hyerbolic otential function to treat the singularity of the original linear form of lastic flow roosed by Lee and Fenves. The consistent algorithmic tangent stiffness, which is utilized to accelerate the convergence rate in solving the nonlinear global equations, is also derived. The validation and evaluation of the model to cature the desired behavior under monotonic and cyclic loadings are shown with several simle one-element tests. These basic simulations confirm the robustness, accuracy, and efficiency of the algorithm at the local and global levels. At the end, a four-oint bending test is examined to demonstrate the caabilities of the model in real 3-D alications. Keywords: lastic-damage; cyclic loading; consistent algorithmic tangent stiffness; numerical imlementation; concrete model; lasticity theory 1. Introduction There have been significant efforts in comutational mechanics to describe the behavior of concrete using various roosed models. As the softening zone is known to result from the formation of microcracks, the strain softening is one of the most imortant asects of concrete behavior. As well as the hardening behavior which occurs under comressive loading, a constitutive model for concrete materials should reresent the softening behavior correctly. The stiffness degradation is another imortant issue to simulate the crack oening and closing under cyclic loading. The cracking rocess in concrete can be categorized into several aroaches including discrete crack, smeared crack, damage mechanics * Corresonding Author: vahlotfi@aut.ac.ir 1 PhD Candidate of Structural Eng., Civil and Environmental Eng. Deartment, Amirkabir University of Technology, Tehran, Iran. Professor of Structural Eng., Civil and Environmental Eng. Deartment, Amirkabir University of Technology, Tehran, Iran. and lasticity-based models. Among different versions of lasticity-based models, the lasticdamage constitutive law is a suitable aroach to simulate concrete behavior due to its caabilities of including not only inelastic strain but also stiffness degradation. It is well recognized that cracking reduces the stiffness of concrete structural comonents. Therefore, in order to accurately model the degradation in the mechanical roerties of concrete, the use of continuum damage mechanics is necessary. However, the concrete material also exeriences some irreversible deformations during unloading such that the continuum damage theories cannot be used alone. Therefore, the nonlinear material behavior of concrete can be recisely simulated by two searate material mechanical rocesses: damage and lasticity. Plasticity theory has been widely used alone to describe the concrete behavior [1-4]. The main characteristic of these models is a lasticity yield surface that includes ressure sensitivity, ath sensitivity, non-associative flow rule, and work or strain hardening. However, such models failed to address the degradation of the material stiffness due to micro-cracking. On the other International Journal of Civil Engineerng. Vol. 8, No. 3, Setember

2 hand, the continuum damage theory has been also emloyed alone to model the material nonlinear behavior such that the mechanical effects of the rogressive micro-cracking and strain softening are reresented by a set of internal variables which act on the elastic behavior (i.e. decrease of the stiffness) at the macroscoic level [5-7]. Furthermore, concrete behavior has several asects such as irreversible deformations, inelastic volumetric exansion in comression, and crack oening/closure effects that cannot be reresented by the two above-mentioned methods alone. Since both micro-cracking and irreversible deformations are contributing to the nonlinear resonse of concrete, a constitutive model should address equally the two hysically distinct modes of irreversible changes. Combinations of lasticity and damage are usually based on isotroic hardening combined with either isotroic (scalar) damage or anisotroic (tensor) damage. Isotroic damage is widely used due to its simlicity such that different tyes of combinations with lasticity models have been roosed in the literature. There are two aroaches for this combination. One relies on stress-based lasticity formulated in the effective (undamaged) sace [8-11] and another alternative is founded on stress-based lasticity in the nominal (damaged) stress sace [1,13]. However, the couled lastic-damage models formulated in the effective sace are numerically more stable and attractive [9,14]. The urose of this aer is to discuss comutational issues involved in the modeling of lain concrete in view of an aroach which takes advantages of both lasticity and damage mechanics alternatives. The model referred to as lastic-damage in literature emloys two internal variables as the damage arameters in tension and comression searately. This constitutive model was originally roosed by Lubliner et al. [1] and later modified by Lee and Fenves [15]. It is also worthwhile to mention that in this latter study, comutational asects of the 3-D formulation are also briefly resented along with the more comlete -D examles with lane stress behavior from numerical imlementation oint of view. In 3-D sace, the rocess of returnmaing needs to be reformulated due to 188 emloying a hyerbolic otential function to treat the singularity of the original linear form of lastic flow roosed by Lee and Fenves. In the resent study, the details of 3-D imlementation of the model are discussed. Moreover, a secial finite element rogram is develoed in that context which is generally ideal for three dimensional nonlinear analyses of concrete structures under cyclic loading. It should be added that in the resent work, the rateindeendent lasticity and the isotroic continuum damage mechanics theories are alied to the elastic and inelastic constitutive laws. Additionally, the infinitesimal deformation theory, which is reasonable and adequate for concrete, is assumed as well. An outline of the remainder of this aer is as follows: In Section, the framework of the lastic-damage model is summarized. It includes the basic concets emhasizing on uniaxial state in tension and comression and then multiaxial behavior. This section is followed by introducing the stiffness degradation which lays an imortant role in cyclic loading states. The yield and otential functions utilized in the study are introduced at the end of this section. In the next section, the comutational issues involved in the numerical integration of the model are discussed. Section 3 also resents the rocedure used for the stress calculation of each Gauss oint within global iterations. Moreover, the algorithmic tangent stiffness which accelerates the convergence rate in the nonlinear global solution is formulated in that art. Finally in Section 4, some numerical simulations including several one-element validation tests and two real alications are resented to demonstrate different features of the model and to establish comarisons with some available exerimental data.. Theoretical Basics of Plastic-Damage Model Based on the incremental theory of lasticity, the strain tensor, İ, is decomosed into the elastic art, İ e, and the lastic art, İ : İ İe İ (1) International Journal of Civil Engineerng. Vol. 8, No. 3, Setember 010

3 The elastic art is defined as the recoverable ortion in the total strain, which for linear elasticity, is given by: İe E 1 : ı () where the elastic stiffness, E, is a rank four tensor, and ı is the stress tensor. Based on equations (1) and (), the stress and strain relation can be written as: ı E : (İ İ ) (3) If scalar damage in stiffness degradation is assumed, the elastic stiffness is defined as: E (1 D) E0 (4) where D is the degradation variable and E0 is the initial stiffness tensor. Substituting equation (4) into equation (3) leads to the following relation: ı (1 D) E0 : (İ İ ) (5) This equation may also be written as: ı (1 D) ı (6) as the effective stress, where ı is denoted given as: (7) ı E0 : (İ İ ) According to equation (6), the constitutive relation for elastolastic resonse is decouled from the degradation damage resonse, which rovides advantages in the numerical imlementation. Therefore, the strength function for the effective stress is used in this model to control the evolution of the yield surface, such that calibration with exerimental results becomes convenient. The lastic strain rate, which is evaluated by a flow rule, is assumed to be related to a scalar otential function ). For a lastic otential defined in the effective-stress sace, it is given by: İ O ı) O. Omidi and V. Lotfi (8) where ı) w) w ı and O is a nonnegative function referred to as the lastic consistency arameter. In addition to the lastic strain, another internal variable set is required to reresent the damage states. Consequently, it is assumed that the lastic-damage variable k is the only necessary state variable and its evolution is exressed as: ț O H (ı, ț ) (9) As shown later, the function H can be derived considering lastic dissiation. At this stage, uniaxial behavior of concrete is initially considered for tension and comression in order to describe the basic concets of the model. Subsequently, this is extended to multiaxial states under comlex loading utilized for a general 3-D imlementation..1. Definition of Damage Variables The model utilizes two damage variables to reresent tensile and comressive damage indeendently, which is required to simulate concrete behavior under cyclic loading. These damage variables would have values within the range of zero to one. Considering a uniaxial state of stress, the state variable ^t, c` is introduced. In that resect, tension and comression states are denoted by t and c, resectively. Accordingly, each uniaxial strength function is exressed in terms of its corresonding lasticdamage arameter, k. In the Barcelona model (i.e., initial form of the lastic-damage model) which was originally roosed by Lubliner and co-authors [1], the uniaxial stress is assumed to be related to the scalar lastic strain variable symbolized by (İ : ı f (İ ) f 0 ª (1 a )ex( b İ ) a ex( b İ ) (10) where a, b are constants, and f 0 is the initial yield stress, defined as the maximum stress without damage. The above strength function is assumed to be factorized as: ı (1 D ) ı (11) 189

4 In this relation, D and ı are defined as the degradation damage variable and the effective stress resonse for the uniaxial state, resectively. It should be mentioned that the degradation damage variables are also defined as increasing functions of lastic-damage variables. They can take values from zero, corresonding to the undamaged concrete, u to one, which reresents fully damaged concrete. An exonential form is normally assumed for the degradation, D, as below: 1 D ex( c İ ) (1) where c is a constant. Thus, the effective stress can be written as follows based on equations (10), (11) and (1): ı f ( İ ) ª f 0 «(1 a ) ex( b İ ) «1 c b a ex( b İ ) c b º»»¼ (13) The uniaxial stress curves are illustrated in Fig. 1. It is noted in these diagrams that total strain is decomosed into two arts, the elastic strain İ e and the lastic art (İ. Furthermore, the lastic-damage variables in tension and comression are defined based on the following relations [1,15]: ț 1 g ³ İ ı di ; g 0 ³ f ı di (14) 0 where g is the secific fracture energy, defined as the fracture energy normalized by the localization size zone, also referred to as the characteristic length, l (i.e., g = G / I ). It is noted that 0 d ț d 1. Substituting Eq. (10) into Eq. (14) yields the relation between k and İ, which can be substituted back into equation (10) to exress the uniaxial stress in terms of k, i.e., V f (N ). Similarly, the effective stress and degradation damage variables can be stated in terms of k as ı f (ț ) and D D (ț ), resectively. It is also worthwhile to mention that the arameters utilized in equation (13) (i.e., a, b and c ) could be determined by emloying the usual arameters defined in Fig. 1, as well as some additional enforced values of tensile degradation at V t 0.5 f t0 (i.e., D t ) and comressive degradation at V c f cm (i.e., D c ) [9,15]... Damage Evolution in Multiaxial Behavior By utilizing Eqs (10) and (14), the damage evolution equation for the uniaxial state can be written as [9]: N 1 f (N ) H g (15) To extend the uniaxial version of damage evolution equation to the multi-dimensional case, the scalar lastic strain rate H in Eq. (15) is assumed to be evaluated in the general threedimensional case by the following relation: H Fig. 1. Uniaxial local curves : (a) in tension, (b) in comression 190 įc (1 r (ıˆ )) H ˆmin į t r (ıˆ ) H ˆmax (16) where į is Kronecker delta and Hˆ max, are, Hˆ min the algebraically maximum and minimum eigenvalues of rate of the lastic strain tensor, resectively. The scalar quantity r (ıˆ ) is a weight International Journal of Civil Engineerng. Vol. 8, No. 3, Setember 010

5 factor, which is within the range zero to one. Denoting the ram function as x² ( x x ) /, r (ıˆ ) is defined by: r (ıˆ ) 3 i 1 ıˆi ² 3 i 1 ıˆi (17) As two lastic-damage variables are indeendently used in tension and comression, the lastic-damage vector is defined as ț (ț t țc )T. Substitution of eq. (16) into eq. (15) results in an evolution equation for the general case in the form of: ț h(ıˆ, ț ) : ݈ (18) D 1 (1 Dc (țc )) (1 Dt ( ț t )) () Where both Dt and Dc are defined in equation (1). Eq. () does not cature the correct crack oening/closing behavior, which can be imlemented by elastic stiffness recovery during elastic unloading rocess from tensile state to comressive state. Thus, D is modified by multilying the tensile degradation in eq. () by a arameter which is a function of the stress state, s (ıˆ ) : D 1 (1 Dc ) (1 s (ıˆ ) Dt ) (3) where 0 d s d 1 and is referred to as the stiffness recovery factor. A ossible form of s is: where h in matrix notation is written as: r (ıˆ ) f t (ț t ) 0 gt ˆ h (ı, ț ) 0 0 (1 r (ıˆ )) f c ( țc ) gc s (ıˆ ) 0 (19) Furthermore, the flow rule in eq. (8) can be modified into one for İ : ݈ O ıˆ)ˆ (0) By substituting Eq. (0) into Eq. (18), the damage evolution equation can be written as: ț O H (ıˆ, ț ) ; H h : ıˆ)ˆ (1).3. Stiffness Degradation The degradation of stiffness, which is caused by microcracking, occurs in both tension and comression and becomes more significant as the strain increases. Under cyclic loading, the mechanism of stiffness degradation gets more comlicated due to oening and closing of the microcracks. Since the scalar degradation, which is assumed to deend on only k, is considered in the resent study, exact simulation of the comlicated degradation henomena is very difficult. As the model is accurately caable of caturing two major damage henomena, the uniaxial tensile and comressive ones, degradation in multi-dimensional behavior can be ossibly evaluated by interolating between these two main degradation damages such as: O. Omidi and V. Lotfi s0 (1 s0 ) r (ıˆ ) (4) in which r is defined in Eq. (17) and s0 is a constant to set the minimum value of s [9]. It is also aarent that r (ıˆ ) r (ıˆ ) due to scalar degradation assumtion..4. Yield Surface Since concrete behaves differently in tension and comression, the lasticity yield criterion cannot be assumed to be alike. Considering the same yield surface for both tension and comression in concrete materials can lead to over/under estimate of lastic deformation [1,15]. The Mohr-Coulomb criterion is the most well-known yield function for frictional materials such as concrete. In this model, the J3 term is included to make the yield surface more realistic. In Lubliner's model, the algebraically largest rincial stress is used instead of J3 [1]. This is also adoted in the resent study. The yield surface is exressed here in terms of effective stress tensor and the lastic-damage variables as: F (ı, ț ) 1 ª Į I1 3 J ȕ (ț ) ıˆ max ² 1 Į J ıˆ max ² º cc (ț ) ¼ (5) where arameters I1t and J are the first and second invariants of the effective stress tensor. 191

6 The maximum rincial stress is also denoted by ıˆ n 1. Moreover, a is a dimensionless constant and it deends on the ratio of yield strengths under biaxial and uniaxial comression, ( i.e., fb0 / fc0): Į ( f b0 f c0 ) 1 ( f b0 f c0 ) 1 (6) Originally, Lublinear et al. resented ȕ and J as constant arameters [1]. However, Lee and Fenves [9] later modified ȕ as a dimensionless variable which is written as a function of the two lastic-damage arameters, kt and kc : ȕ (ț ) cc (țc ) (1 Į ) (1 Į ) ; J ct (ț t ) 3 (1 U c ) Uc 1 (7) where ct and cc,which are equal to f t (ț t ) and f c (țc ), denote the effective tensile and comressive cohesions (ositive values utilized here), resectively. Moreover, Uc is the ratio of corresonding values of J for tensile and comressive axes at any given value of the hydrostatic ressure, I1t. It is usually assumed as a constant, with a tyical value of /3 for concrete [1], which results in the value of J =3. Since the influence of coefficient J disaears in stress states other than triaxial comression (i.e., ı max < 0), the corresonding term in Eq. (5) can be viewed as a minor modification to imrove the redictive caability of the model under stress states other than the lane stress conditions. Moreover, the yield surface can be reresented alternatively as F (ı, ț ) Fˆ (ıˆ, ț ) [16].. 5. Plastic Potential Function The non-associative lasticity flow rule, which is imortant for realistic modeling of the volumetric exansion under comression for frictional materials such as concrete, is emloyed here. The Drucker-Prager linear function was utilized by Lee and Fenves in their -D studies [9,16] as the otential flow: ) (ı ) J Į I1 (8) where the arameter a should be calibrated to give roer dilatancy [9]. It is well-known that this tye of surface has a singularity at the aex. Although this will not cause any difficulty for the lane stress -D condition considered by Lee and Fenves [9], it would be roblematic for the 3-D imlementation of the resent study and it requires secial attention. The strategy chosen herein is to emloy the Drucker-Prager hyerbolic function as equation (9) which has been deicted in Fig. as well. ) (ı ) E H J Į I1 ; E H İ1Į f t0 (9) where İ 1 is a arameter which adjusts how the flow otential aroaches its corresonding linear function. Since the hyerbolic function is continuous and smooth, the flow direction is always uniquely defined. Fig.. Drucker-Prager hyerbolic function along with its asymtote in the meridian lane 19 International Journal of Civil Engineerng. Vol. 8, No. 3, Setember 010

7 Table 1. Return-maing rocess (local iteration) 3. Numerical Imlementation 3.1. Linearization of Damage Evolution Equations The damage evolution equation (i.e., Eq. (1)) can be exressed in discrete form based on the backward-euler method: 'ț 'O H (ıˆ n 1, ț n 1 ) (30) 0. j = 0; σˆ n +1 = σˆ ntr+1; κ (0) n +1 = κ n 1. Obtain λ. Comute σˆ n+1 3. Q(n +j )1 = κ (n +j )1 + κ n + λ H (σˆ n +1, κ (n +j )1 ) Q (n +j )1 TolL THEN 4. IF Exit. 5. Form (dq dκ )(n +j )1 ( j) dq ( j) 6. Solve δκ = Q n +1 for δκ dκ n +1 This may also be rewritten as: ț n 1 ț n O H (ıˆ n 1, ț n 1 ) (31) where O is used instead of 'O for simlicity. Since the above equation is a nonlinear relation, an iterative solution scheme for the effective stress, k and O is required to be carried out (this is referred to as the local iteration). For this urose, the Newton-Rahson aroach is utilized here [9]. The residual for Eq. (31) denoted by Q is: Q(ıˆ n 1, ț n 1, O ) ț n 1 ț n O H (ıˆ n 1, ț n 1 ) 7. κ (n +j +11) = κ (n +j )1 + δκ 8. j = j + 1 and GOTO Ste 1. model are integrated over a time interval (tn, tn 1 tn 't ) using an imlicit backwardeuler scheme, which is unconditionally stable [16,17]. The stress and the lastic strain at ste n+1 are exressed as: ı n 1 ı n 'ı ; İ n 1 İ n 'İ (35) (3) This residual equation, which needs to be satisfied (i.e.,q=0), is iterated using the NewtonRahson scheme as: From Eq. (6), the stress can be written in the following form: ı n 1 (1 Dn 1 ) ı n 1 (36) ( j) dq dț įț n 1 Q (n j )1 (33) in which dq / dk is the Jacobian matrix and įț is used to udate the lastic-damage vector, kn+1 : ț (n j 11) ț (n j )1 įț (34) The local iteration scheme, which actually lays the role of the return-maing rocess in this model, is summarized in Table 1. The required formulas to comute O and ıˆ n 1 that are emloyed in this rocedure will be derived in the following sections. 3.. Stress Comutation From a numerical oint of view, the nonlinear behavior of concrete model may be treated as an imlicit time discrete strain-driven roblem. Accordingly, the time discrete equations of the O. Omidi and V. Lotfi Moreover, based on Eq. (7), the effective stress becomes: ı n 1 E0 : (İ n 1 İ n 1 ) (37) ı ntr 1 E0 : 'İ ı ntr 1 ( G0 'e. 0 'ș I ) tr where the trial effective stress, denoted by ı n 1, is defined as: ı ntr 1 E0 : (İ n 1 İ n ) (38) The above rocedure for comutation of stress tensor can be interreted as the three-ste redictor-corrector method: tr 1) Elastic redictor: ı n 1 ) Plastic corrector: -E0: 'İ 3) Damage corrector: -Dn+1 ıˆ n 1 193

8 The damage art is imlemented searately, because it is decouled from the lastic corrector art. The backward-euler scheme is used to integrate the lastic strain: 'İ O ı) (39) Using the mentioned flow rule, the increment of lastic strain can be rewritten in the slit form as: 'e 'İ 'e 'ș I 3 sn 1 O E H sn 1 (40) ; 'ș 3 O Į (41) in which, s denotes the deviatoric art of the J. Moreover, by effective stress and s substituting Eq. (41) into Eq. (37), the effective stress is reformulated as: ı n 1 G0 sn 1 ı ntr 1 O E H sn Į I (4) If the deviatoric and volumetric arts in Eq. (4) are searated and slightly maniulated, the following relations can be concluded: sn 1 sn 1 sn 1 ( I1 ) n 1 sntr 1 'İ ( I1tr ) n 1 9 K 0 O Į O Pn 1 '݈ PnT 1 ; '݈ ı ntr 1 O ıˆ)ˆ Pn 1 ıˆ ntr 1 PnT 1 (47) (48) By using the Drucker-Prager hyerbolic lastic function as the flow rule, one can show that by emloying the sectral return-maing briefly discussed above, Eqs. (39) and (4) are written in the context of their eigenvalues as the following, resectively: sˆ n 1 O E H sˆ n 1 (44) ıˆ n 1 (45) (46) As roved in [16], any eigenvector of matrix tr ı n+1 is also an eigenvector of matrix ı n 1, which tr leads to the sectral decomosition form of ı n 1: sn 1 E H sn 1 Pn 1 ıˆ n 1 PnT 1 where Pn+1 is the non-singular matrix whose columns are the orthonormal eigenvectors of ı n+1 and ıˆ n 1 is the diagonal matrix of eigenvalues of ı n+1. Moreover, since an isotroic material behavior is assumed, there exists a function )ˆ such that )ˆ (ıˆ ) ) (ı ). Thus, the lastic strain increment, 'İ, can also be written in the sectral decomosition form [17]: '݈ where s tr and I1tr denote the deviatoric art and the first invariant of the trial effective stress, resectively. When rincial stress terms exist in a yield function, the return-maing algorithm based on a sectral decomosition of the stress is more efficient. This concet is reviewed briefly below and the reader is referred to other references for details [16,17]. It is well known that any symmetric matrix such as the effective stress 194 ı n 1 (43) sntr 1 sntr 1 G0 O matrix can be factorized by the sectral decomosition: ıˆ ntr 1 Į I G0 sˆ n 1 O E sˆ H n 1 (49) 3. 0 Į I (50) Moreover, since scalar degradation is assumed, the stress is also decomosed by the same eigenvectors as those of the trial effective stress (i.e., P in Eq. (46)). This means that it can be calculated as: ı n 1 Pn 1 (1 Dn 1 ) ıˆ n 1PnT 1 (51) It is also worthwhile to mention that by substituting Eqs. (43)-(45) and (50) into the discrete version of lasticity consistency condition (i.e., Fˆ (ıˆ n 1, ț n 1 ) =0), the following International Journal of Civil Engineerng. Vol. 8, No. 3, Setember 010

9 relation is obtained for the consistency arameter, O : O 3 ˆ tr sn 1 Kn 1 (ıˆ1tr ) n 1 (1 Į ) cnc 1 6 G0 sˆ n 1 G0 (sˆ1 ) n 1 Kn Į 9. 0 Į Į E H sˆ n 1 E H sˆ n 1 Į ( I1tr ) n 1 (5) where K is a new function defined as:, IF (ıˆ1tr ) n 1! 0, Otherwise ȕn 1 Kn 1 J (53) Since O in equation (5) is deendent on sˆ n 1, it cannot be exlicitly comuted. Furthermore, O needs to satisfy Eq. (44). Therefore the consistency arameter is obtained from an iterative strategy using these two relations Consistent Algorithmic Tangent Stiffness When the Newton-Rahson algorithm is emloyed in global iterations, the rate of convergence is strongly deendent on the elastolastic tangent stiffness. Using the algorithmic tangent stiffness, which is consistent with the local iteration as well, guarantees the quadratic convergence in global iterations [16]. The consistent algorithmic tangent stiffness is defined by combining the linearized equations having been used in the local iteration algorithm derived to comute stresses. After the converged effective stresses and damage variables are comuted for the given strain, all the residual equations, Eqs. (54)-(56), are assumed to be satisfied Stress Calculation Procedure Comutation of stress is based on the classic ste by ste iterative method which is comosed of elastic rediction and lastic-damage correction. During this rocess, as well as stress, ΣM, other variables such as İ, kt, kc, Dt and Dc are udated. From an imlementation oint of view, it is more convenient that Dt and Dc are considered as a vector similar to k (i.e., D=(Dt Dc)T). The comutational rocedure for the stress at ste n+1 can be summarized as Table. Q(ıˆ n 1, ț n 1, O ) 0 (54) Fˆ (ıˆ n 1, ț n 1 ) 0 (55) ı n 1 E0 : (İ n 1 İ n 1 ) (56) Having equation (54), the total differential of Q becomes: dq wq wq ˆ wq : dı do dț wț wo wıˆ (57) Table. Stress calculation rocedure for an integration oint ε n +1, ε n, κ n, Dn σ ntr+1 = E0 : (ε n +1 ε n ) σ ntr+1 = Pn +1 σˆ ntr+1 PnT+1 Fˆ (σˆ ntr+1, κ n ) IF 0 THEN (Elastic loading/unloading state) tr ˆ ˆ 3.1. σ n +1 = σ n +1 ; ε n +1 = ε n ; κ n +1 = κ n ; Dn +1 = Dn (Plastic loading state) 4.1. Return-maing rocess (local iteration to comute σˆ n +1, κ n +1, λ ) 4.. εˆ = λ ˆΦˆ 4. ELSE σ 4.3. ε n +1 = ε n + Pn +1 εˆ PnT Dn +1 = D(κ n +1 ) ˆ 5. sn +1 = s (σ n +1 ) 6. Dn +1 = 1 (1 Dnc+1 )(1 sn +1Dnt +1 ) 7. σ = P σˆ P T n +1 n +1 n +1 n σ n +1 = (1 Dn +1 ) σ n +1 O. Omidi and V. Lotfi 195

10 the relation for do : which is exected to be zero (i.e.,dq=0). Taking this into consideration, the following relation is obtained by rearranging this latter equation: dț Tțı : dı TțȜ d O do 1 wq wq wıˆ : wț wıˆ wı TțȜ wq wq wț wo (59) dı 1 ıˆ Fˆ : dıˆ ț Fˆ dț TFˆı TFȜ ˆ TF ı and TF Ȝ 0 (61) 0 (6) dd TDı : dı TDȜ d O ıˆ Fˆ : wıˆ wı ț Fˆ Tțı ț Fˆ TțȜ where the following definitions for TDı and TDȜ are being emloyed: (63) E0 : (di di ) ı) d O O w ) wı : dı TDı wd wıˆ wd wd Tțı : wıˆ wı wd wț (73) TDȜ wd wd TțȜ wd wț (74) (64) Eq. (70) could lead to the following form by utilizing relation (7): (65) wı wi >(1 Dn 1 ) I ı n 1 wı wi n 1 >(1 Dn 1 ) I ı n 1 dı S : di S : ı) d O (67) Substituting Eq. (67) into Eq. (6), one obtains : wı TDȜ ı n 1 wi wo wi (75) Emloying Eqs. (68) and (69) in Eq. (75), the consistent algorithmic tangent stiffness is concluded: (66) By emloying equation (66) and defining a 1 1 modified stiffness S E0 O w ) wı, Eq. (65) may be rewritten as: 196 (7) are defined as below: in which the total differential of the lastic strain is: di (70) Substituting Eq. (58) into Eq. (71) gives the differential of the degradation as: Moreover, based on Eq. (56), the total differential of the stress becomes: dı ı n 1 dd (1 Dn 1 ) dı Since the total degradation, D, is a function of k and ı (i.e., Dn 1 D (D(ț n 1 ), ıˆ n 1 ) ), the following is concluded: wd wd wd wıˆ dd dț : : dı (71) wd wț wıˆ wı Substituting Eq. (58) into Eq. (61), it yields: where (68) (60) Similarly, from Eq. (55) the total differential of function F gives: TFˆı : dı TFȜ ˆ do : di which is substituted back into equation (67) to obtain: S : ı) TFˆı : S dı S : di (69) TFˆı : S : ı) TFȜ ˆ Furthermore, based on equation (36), the total differential of the stress tensor is written as: (58) where Tțı and TțȜ would have the following form, resectively: Tțı TFˆı : S T ˆ : S : ı) T ˆ FȜ Fı : S TFˆı : S TFˆı : S : ı) TFȜ ˆ S : ı) ı n 1 TFˆı : S TDȜ ˆ TFˆı : S : ı) TFȜ (76) International Journal of Civil Engineerng. Vol. 8, No. 3, Setember 010

11 It is noted that the derived algorithmic tangent stiffness, Eq. (76), is not symmetric. This is as the result of non-associative flow rule and existence of the degradation comonent. 4. Numerical Examles This section describes numerical simulations used to validate the imlemented lastic-damage model in its 3-D develoment. The numerical algorithm discussed above has been imlemented in a secial finite element rogram (i.e. SNACS [18]) using a series of FORTRAN subroutines. This art is divided into two subsections including single-element tests and structural alication. In all executions, the model arameters of Į, J, Į, s0 and İ 1 are equal to 0.1, 3.0, 0., 0.0 and 0.1, resectively. Furthermore, in all cases D t =0.51 and D c=0.4. It should be also mentioned that the dislacement control aroach is used to aly the loadings Single-element Tests The imlementation is initially validated by alying the model to evaluate the numerical resonse of several examles for which exerimental results are available. Thereafter, the results of some additional numerical tests are resented. The first two verifications discussed here confirm the basic caabilities of the resent lastic-damage model in reresenting tensile and comressive behavior of concrete under monotonic loading. For this urose, one 8-node isoarametric solid element with x x Gauss integration with lch=5.4 mm is used. For all cases, unless otherwise secified, the following material roerties are utilized: Table 3. Material roerties for the one-element tests E0 ν f t f c Gt Gc (GPa) - (MPa) (MPa) (N/m) (N/m) Monotonic Uniaxial Loadings Performance of the model for the fundamental loadings such as uniaxial tensile and comressive loading tests is evaluated and comared with the corresonding exeriments and results from Lee and Fenves by -D imlementation. It should be mentioned that in the uniaxial comressive case, elastic modulus, E0 of 31.7 GPa is utilized instead of 31.0 GPa. Fig. 3 deicts the simulated tensile and comressive stress-strain curves, resectively. It is observed that the numerical results agree well with the exerimental data and also with the solutions of Lee and Fenves Monotonic Biaxial Loadings To test numerical behavior of the model under biaxial state of stresses, the test results for biaxial Fig. 3. Monotonic uniaxial loadings comared with exeriment [19,0], and also with Lee and Fenves study [9]; (a) tensile test, (b) comressive test O. Omidi and V. Lotfi 197

12 Fig. 4. Biaxial loading tests along with uniaxial resonses: (a) tension, (b) comression tensile case and comressive case are comared with their uniaxial cases in Fig. 4a and 4b, resectively. The same material data considered for uniaxial loading are emloyed for these cases. It is noted that as exected, the strength of concrete is redicted to slightly decrease for the biaxial loading in tension, while it shows a significant increase for the biaxial loading in comression Triaxial and Constrained Biaxial Tension Tests In order to evaluate the imlemented rocedure for the singularity treatment of the aex, two examles including triaxial test and constrained biaxial test are examined in this subsection. Fig. 5a comares the numerical redictions for triaxial case along with the uniaxial and biaxial results obtained reviously. The third direction is restrained in the constrained biaxial test. It is noted that tensile stress develos in that direction as equal tensile strains are exerted in the other two directions. Furthermore, the three tensile stresses become equal at certain stage and remain equal afterwards. From that oint forward, states of stresses corresond to the oint on the aex of the lastic otential surface. The result of this test (Fig. 5b) demonstrates the caability of the strategy adoted here to overcome this singularity. Fig. 5. (a) Uniaxial, biaxial, triaxial tensile loading results; (b) Constrained biaxial test 198 International Journal of Civil Engineerng. Vol. 8, No. 3, Setember 010

13 Triaxial Comression Test As mentioned above, Lubliner's yield function contains a term, γ, which can better redict the concrete behavior in comression under confinement. The numerical results of the resent model for this tye of loading are comared between the case of γ =0 and γ =3. The material roerties adoted in the simulations are the same as used for the uniaxial comression test. The numerical redictions of secimens under three sets of confining stresses, namely, σ = σ 3=0.0 MPa, σ = σ 3=-3.75 MPa and σ = σ 3=-7.50 MPa, are reroduced in Fig. 6. It should be mentioned that the confining stresses are increased roortional to the imosed strain in the third direction for all these cases. It can be clearly seen that the enhancement of strength and ductility due to the comressive confinement are satisfactorily catured by the resent model Cyclic Uniaxial Loadings Herein, cyclic loading alications are resented. The main objective is to examine the caability of the model in caturing stiffness degradation in both tensile and comressive loadings. The material roerties are the same as those of Table 3 excet E0=31.7 GPa and D c =0.38. Figs. 7a and 7b illustrate the numerical results from two uniaxial cyclic loading cases which are comared against the exerimental data in each case. For both cases, the stiffness degradation is seen to be roerly simulated at each unloading/reloading cycle. However, it is noted that hysteresis on reloading cannot be catured by the model because of the rate-indeendent elastic loading/unloading assumtion. It is noted that in some structures the dominant failure mechanism is shear damage [1] which in this lastic-damage model, both damages in tension and comression are exected in such examles Full Cyclic Loading In the last case, the single-element is subjected to cyclic tensile-comressive-tensile load (Fig. 8). This erfectly illustrates the ability of the model to simulate stiffness recovery when the status changes from tension to comression and vice versa. The tensile strength after comressive loading is degraded and it means that the tensile damage and the elastic stiffness are considerably affected by the comressive damage. 4.. Structural Alication The analysis of a notched concrete beam Fig. 6. V 1 H1curves under three different confinements: (a) γ O. Omidi and V. Lotfi =0 and (b) γ =3 199

14 Fig. 7. Cyclic uniaxial loading results in comarison with exerimental result [19,0]: (a) tensile test and (b) comressive test Fig. 8. Numerical solution for full cyclic loading case monotonically loaded under four-oint bending is investigated herein to examine the imlemented algorithm in a real 3-D alication. This roblem has been extensively studied by researchers in both exeriments and numerical modeling [,3]. The secifications of the test are illustrated in Fig. 9a. A 3D 8-node finite element mesh for the symmetric left art of the secimen is also shown in Fig 9b. The material roerties used in this alication are defined in Table 4. It should be mentioned that the size of the elements in the exected localization zone is used here as the characteristic length,. To calibrate the material data for the resent model, slightly lower values than in the exerimental test investigated by Hordijk [] are assumed for the tensile strength and fracture energy in tension. To evaluate the Fig. 9. Four-oint bending test: (a) geometry of the secimen and boundary conditions (b) 3D 8-node finite element mesh 00 International Journal of Civil Engineerng. Vol. 8, No. 3, Setember 010

15 Table 4. Material roerties for the notched beam E0 ν 5. Conclusions f t f c Gt Gc (GPa) - (MPa) (MPa) (N/m) (N/m) accuracy of the solution, three analyses are carried out with different number of increments; these are 15, 30 and 60 equal stes which are considered within the alied maximum dislacement. Fig. 10a shows the load P versus the load oint deflection for the numerical simulations and the corresonding exerimental test. The three analyses excellently agree with one another, confirming robust convergence of the algorithm and good accuracy of the solution over a large range of dislacement increments. The damage develoment at the end of the analysis is also illustrated in Fig. 10b which is clearly showing the localization in the model. Constitutive relations of the lastic-damage model roosed by Lee and Fenves are reviewed and its comutational asects in 3-D imlementation mainly related to the singularity of the original lastic otential function are discussed in this aer. It should be noted that the imlementation and examles carried out by Lee and Fenves were limited to -D lane stress roblems. In 3-D develoment, a art of the imlementation of the return-maing rocess needs to be reformulated due to emloying the Drucker-Prager hyerbolic otential function as a treatment for the aex s singularity of the linear one. The effective stress concet is used to simulate stiffness degradation. The numerical integration is erformed by the backward-euler scheme, which is known to give an unconditionally stable method. The sectral return-maing which has more advantages with resect to the general returnmaing is utilized. To model dilatancy of concrete accurately, a non-associative flow rule which causes unsymmetrical algorithmic tangent stiffness is used. Through the numerical examles by several one-element tests, the imlemented lasticdamage model is known to give the results that agree well with exerimental data for concrete under cyclic loading as well as monotonic loading. It is also demonstrated that stiffness degradation and crack oening/closing are considered roerly. In site of the assumtion of isotroy in damage art of the model, this lasticdamage model ermits to obtain reliable results in view of the rediction of the failure behavior of concrete. The damage rocess in a notched beam under four-oint bending is investigated numerically, where a comarison is also made with the available exerimental result. This real alication shows robustness and accuracy of the imlemented algorithm in caturing the global softening behavior in concrete structures while it is well reresenting the exected localization. References Fig. 10. Four-oint bending test: (a) Load-deflection curves for different number of stes comared with the exeriment result [], (b) Tensile damage at the end of the analysis O. Omidi and V. Lotfi [1] William, K.J. and Warnke, E.P. (1975), Constitutive model for the triaxial behavior of concrete, International Association of Bridge 01

16 and Structural Engineers, Seminar on Concrete Structure Subjected to Triaxial Stresses, Paer III-1, Bergamo, Italy. [] Menetrey, Ph. and Willam, K.J. (1995), Triaxial failure criterion for concrete and its generalization, ACI Structural Journal 9, Journal of Numerical and Analytical Methods in Geomechanics, 6: 1 4. [11] Wu, J.U., Li, J. and Faria, R. (006), An energy release rate-based lastic-damage model for concrete, Int. Journal of Solids and Structures, 43: [1] Lubliner, J., Oliver, J., Oller, S. and Onate, E. (1989), A lastic-damage model for concrete, Int. Journal of Solids and Structures, 5: [3] Grassl, P., Lundgren, K. and Gylltoft, K. (00), Concrete in comression: a lasticity theory with a novel hardening law, Int. Journal of Solids and Structures 39: [4] Sadrnejad, S.A. and Ghoreishian Amiri, S.A. (010), A simle unconventional lasticity model within the multilaminate framework, Int. Journal of Civil Engineering, 8(): [13] Menzel, A., Ekh, M., Runesson, K. and Steinmann, P. (005), A framework for multilicative elastolasticity with kinematic hardening couled to anisotroic damage, Int. Journal of Plasticity, 1(3): [5] Loland, K.E. (1980), Continuous damage model for load-resonse estimation of concrete, Cement & Concrete Research, 10: [14] Abu Al-Rub, R.K. and Voyiadjis, G.Z. (003), On the couling of anisotroic damage and lasticity models for ductile materials, Int. Journal of Solids and Structures, 40: [6] Mazars, J. and Pijaudier-Cabot, G. (1989), Continuum damage theory: Alication to concrete, Journal of Engineering Mechanics, 115: [7] Lubarda, V.A., Kracjinvovic, D. and Mastilovic, S. (1994), Damage model for brittle elastic solids with unequal tensile and comressive strength, Engineering Fracture Mechanics 49, [15] Lee, J. (1996), Theory and imlementation of lastic-damage model for concrete structures under cyclic and dynamic loading, PhD dissertation, Deartment of Civil and Environmental Eng., University of California, Berkeley, California. [8] [9] Yazdani, S. and Schreyer, H.L. (1990), Combined lasticity and damage mechanics model for lain concrete, Journal of Engineering Mechanics Division, ASCE, 116: Lee, J. and Fenves, G.L. (1998), A lasticdamage model for cyclic loading of concrete structures, Journal of Engineering Mechanics, ASCE, 14: [10] Gatuingt, F. and Pijaudier-Cabot, G. (00), Couled damage and lasticity modeling in transient dynamic analysis of concrete, Int. 0 [16] Lee, J. and Fenves, G.L. (001), A returnmaing algorithm for lastic-damage models: 3-D and lane stress formulation, Int. Journal for Numerical Methods in Engineering, 50: [17] Simo, J.C. (199), Algorithm for static and dynamics multilicative lasticity that reserve the classical return-maing schemes of the infinitesimal theory, Comuter Methods in Alied Mechanics and Engineering, 99: [18] Omidi, O. (010), SNACS: A rogram for Seismic Nonlinear Analysis of Concrete Structures, Deartment of Civil and Environmental Eng., Amirkabir University of Technology, Tehran, Iran. International Journal of Civil Engineerng. Vol. 8, No. 3, Setember 010

17 [19] Goalaratnam, V.S. and Shah, S.P. (1985), Softening resonse of lain concrete in direct tension ACI Journal, 3: [0] Karsan, I.D. and Jirsa, J.O. (1969), Behavior of concrete under comressive loading, Journal of structural division, ASCE, 95(1): [1] Arabzadeh, A., Rahaie, A.R. and Aghayari, R. (009), A simle strut-and-tie model for rediction of ultimate shear strength of RC dee O. Omidi and V. Lotfi beams, Int. Journal of Civil Engineering, 7(3): [] Hordijk, D.A. (1991), Local aroach to fatigue of concrete, PhD dissertation, Delft university of technology, Delft, the Netherlands. [3] Nguyen G.D. and Korsunsky A.M. (008), Develoment of an aroach to constitutive modeling of concrete: isotroic damage couled with lasticity, Int. Journal of Solids and Structures, 45: