Maximum-entropy meshfree method for nonlinear static analysis of planar reinforced concrete structures

Transcription

1 Maximum-entropy meshfree method for nonlinear stati analysis of planar reinfored onrete strutures Giuseppe Quaranta 1, Sashi K. Kunnath, Natarajan Sukumar Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, United States Abstrat A meshfree method for nonlinear analysis of two-dimensional reinfored onrete strutures subjeted to monotoni stati loading is presented. The onrete model is implemented in the ontext of the smeared rotating rak approah. The stress-strain relationship for steel bars aounts for the surrounding onrete bonded to the bar (tension stiffening effet). The priniple of virtual work (variational form) is used to setup the nonlinear system of equations. Maximum-entropy basis funtions are used to disretize the two-dimensional domain and bakground ells are adopted to failitate the numerial integration. The generalized displaement ontrol method is implemented to solve the nonlinear system of equations and to obtain the softened strutural response beyond the maximum load apaity. The proposed meshfree methodology is used to study the nonlinear behavior of reinfored onrete shear walls. Comparisons with experimental data and finite element analysis indiate that the proposed maximum-entropy meshfree method is a viable approah for nonlinear simulations of planar RC strutures. Keywords: generalized displaement ontrol method, meshfree method, maximum-entropy basis funtions, nonlinear stati analysis, reinfored onrete, shear wall. 1. Introdution The response of reinfored onrete (RC) strutures is typially solved within the framework of the finite element (FE) method, thus requiring a priori disretization of the domain through the definition of a mesh. Beause of the intrinsi (geometrial and/or material) nonlinearities in most problems of pratial interest dealing with RC strutures, signifiant effort has gone into improving existing models for simulating RC behavior as well as enhaning the auray and reliability of solutions obtained by FE methods. On the ontrary, very little effort has gone into exploring new numerial methods for the analysis of RC strutures that may have the potential to establish new and effetive paradigms in modern strutural analysis. 1 Corresponding author. gquaranta@udavis.edu. Tel.: +1 (530)

2 To this end, meshfree (or meshless or element-free) methods are gaining popularity as effetive tools for advaned numerial analyses. To date, meshfree methods are routinely used for very speialized appliations in omputational and solid mehanis. In ontrast, less effort has been devoted toward applying meshfree methods in ommon strutural engineering problems. In order to explore the viability of meshfree-based omputational tools in the field of strutural engineering, Yaw and o-workers [1] presented a blended FE and meshfree Galerkin approximation sheme to study the inelasti response of plane steel frames with J 2 plastiity. The same authors also proposed a meshfree o-rotational formulation for two-dimensional ontinua in the presene of small strains with elasti and elasto-plasti material behavior [2]. Current trends in meshfree appliations on the analysis of RC strutures essentially exploit their natural ability in modeling stati or dynami frature phenomena in brittle materials. For instane, Rabzuk and o-workers [3] presented a geometrially non-linear three-dimensional rak method in whih the element free Galerkin (EFG) method was used for onrete modeling, a ohesive zone model was adopted after rak initiation and the reinforement (modeled by truss or beam elements) is onneted by a bond model to the onrete. A oupled FE-meshfree method was also presented by Rabzuk and Eibl [4] for the analysis of pre-stressed onrete beams. In this paper, a meshfree method is developed for nonlinear analysis of two-dimensional RC strutures subjeted to stati monotoni loading. To this end, onrete modeling issues are first addressed (Setion 2), and then the adopted model for unraked, partially raked and fully raked onrete under plane stress ondition is presented. This is followed by the desription of the steel onstitutive model (Setion 3). In Setion 4, the meshfree-based nonlinear system of equations is derived from the variational weak form, and maximum-entropy basis funtions are used to disretize the domain. Subsequently, the adopted numerial strategy to solve for the resulting nonlinear system of equations is disussed. In Setion 5, the validity of the method is demonstrated through omparisons with experimental data and FE analysis of shear walls subjeted to monotoni stati loading. Finally, some losing remarks are made in Setion 6 on the future development of the present work. 2. Conrete modeling 2.1. Modeling of raked onrete The FE analysis of RC strutures needs an appropriate approah for modeling raked onrete, e.g. smeared rak [5] and disrete rak approahes [6]. Smeared rak based models tend to be more popular than the disrete rak models due to their improved ability to aount for multi-axial stress 2

3 states in onrete, and also beause they are more onvenient from a programming point-of-view. A smeared rak approah is intended to preserve the ontinuity of the displaement field over the domain of influene of the raked node by representing many (fititious) finely spaed raks parallel to the dominant disrete (real) rak. With this ontinuum approah, the loal displaement disontinuities at raks are artifiially smeared over the tributary area, and the behavior of raked RC is represented by average stress-strain relations for both onrete and steel. Although meshfree methods provide an effetive theoretial bakground to deal with disrete representation of raks, the use of a smeared rak approah aims at supporting future blended FEmeshfree approahes for nonlinear analysis of RC strutures. In fat, as long as standard FE methods have strong limitations in disrete rak modeling, a smeared approah an be used in both FE and meshfree domains, thus providing a unique framework for onrete rak modeling. A pure smeared rotating rak model is hene adopted in this study. Several studies demonstrate that smeared rak models for RC modeling are physially onsistent and show good agreement with experimental data. On the other hand, it is well known that FE analyses using work-softening brittle materials within smeared rak models show signifiant mesh-sensitivity problems. Our preliminary analyses of unreinfored onrete strutures with smeared rak models revealed that a similar issue also ours in meshfree methods. In partiular, it was found that the load apaity dereases moderately as the nodal spaing redues whereas the slope of the softening branh inreases onsiderably, whih is onsistent with FE analyses [7]. To alleviate the above issues, averaging onept in tension stiffening [8] and the rak band model [9,10] are implemented. The width h of the rak band is assumed to be equal to the square root of the area of the domain of influene for retangular shapes or to its diameter in the ase of irular shapes. The stress-loking problem in the viinity of dominant raks that is attributed to the smeared rak model an be redued by employing the smeared rotating rak model, whih is more popular than the smeared fixed rak model [5,8,9] Constitutive matrix for onrete The x-y oordinate system represents the geometrial oordinates at the integration point. On the other hand, the 1-2 oordinate system represents the mehanial (loal) oordinates at the integration point and oinides with the prinipal stress diretions. Plane stress ondition is assumed and the angle between the x-axis and the 1-axis is denoted as θ. It has been experimentally observed that diretions of prinipal strains in the onrete deviate somewhat from the diretions of prinipal stress, Vehio and Collins [11] onluded that it is a reasonable simplifiation to assume that the prinipal strain axes and the prinipal stress axes for the onrete oinide. Let ε = {ε x ε y 3

4 γ xy } T be the strain in the geometrial system of referene. Therefore, the following transformation holds ε Τ ε (1) ( ) = 12 ( θ ) in whih ε (12) = {ε 1 ε 2 γ 12 } T is the strain vetor in the loal system of referene and T (θ) is the strain transformation matrix. For the implementation proedure in this study, a tangent material stiffness matrix is onsidered [8,12]. The relation between inremental stresses dσ (12) ={dσ 1 dσ 2 dτ 12 } T and inremental strains dε (12) ={dε 1 dε 2 dγ 12 } T in the 1-2 oordinate system of referene is dσ = D dε (2) ( 12) ( 12) Referring to the prinipal axis 1-2, it is assumed that the tangent onstitutive matrix D for onrete before raking takes the form [13] E 1 ν E 1E D = 2 ν E 1E2 E2 0 (3) 1 ν ( 1 ν ) G in whih E 1 and E 2 are the onrete tangent moduli, G is the shear modulus and ν is the Poisson ratio. One onrete raking is initiated either in 1-diretion or 2-diretion, the tangent onstitutive matrix given by Eq. (3) is replaed with the following E D = 0 E (4) ν 0 0 G as in Ref. [12]. As the loading progresses, a set of smeared raks fully opens into the onrete. In this ase, Eq. (4) is replaed with the following [14] D (5) = 0 E µ G where E 2 is the tangent modulus of onrete parallel to the rak diretion and µ is the shear retention fator introdued to provide for shear frition aross fully opened raks. Various forms of this shear fator have been proposed until now. However, numerous analytial results have demonstrated that the partiular value hosen for µ (between 0 and 1) does not appear to be ritial, but values greater than zero are suggested to prevent numerial instabilities [9,14]. 4

5 Upon further loading, a seond set of smeared raks an form in the diretion normal to the first set of smeared raks. In this ase, the onstitutive matrix is: D = (6) 0 0 µ G Moreover, when the onrete strain exeeds the ultimate onrete strain in ompression in one or two diretions, onrete failure by rushing ours. In this ase, the element is assumed to lose its strength ompletely and is not able to arry any more stress. Beause of material nonlinearities, all quantities in the above onstitutive matries usually vary as funtions of the stress and strain fields Conrete strength due to onfinement effet Let f be the uniaxial onrete ompressive strength and ε the onrete strain at f.. For unonfined onrete, f = f and ε = ε, in whih f and ε are the uniaxial ompressive strength and the orresponding strain for onrete without (or with modest or ineffiient) onfinement reinforement. It is well known that the uniaxial onrete ompressive strength and the orresponding strain inrease if appropriate onfinement reinforement is provided. To aount for this phenomenon, the model proposed by Hoshikuma and o-workers [15] is onsidered f = f ρ f 1 z y, z ε = ρ z f y, z f (7) where f y,z is the yielding stress of the onfinement reinforement and ρ z the onfinement reinforement ratio. Parameters 1 and 2 depend on the shape of the onfined setion. For square setions, Hoshikuma and o-workers [15] suggested 1 = 0.2 and 2 = 0.4. It is assumed that onfinement has no effet on onrete tensile strength Conrete strength under biaxial stress It is well known that the onrete strength signifiantly depends on the urrent stress-strain state. It is assumed that both the uniaxial onrete ompressive strength and the orresponding strain depend on the urrent stress- and/or strain-state at the onsidered integration point, i.e., f p = α f (8) ε = αε (9) p where α is the saling fator depending on stresses or strains in the assumed prinipal diretions. 5

6 The strength enhanement fator for onrete subjeted to ompression in the 2-diretion, arising from the ompressive stress σ 1 ating in the 1-diretion, is given by [16]: 2 σ 1 σ 1 α = (10) f f Similarly, the strength enhanement fator for onrete subjeted to ompression in the 1-diretion, arising from the ompressive stress σ 2 ating in the 2-diretion, is given by σ 2 σ 2 α = f f 2 (11) In the tension-ompression state, the major prinipal stress is tensile and it redues the ompressive strength in the minor prinipal diretion. In this ase, the saling fator adopted in this study takes the form 1 α = ε ε + 1 (12) where ε + is the prinipal tensile strain. It is assumed that the onrete tensile strength does not depend on the stress-strain state Relationship between biaxial strains and uniaxial strains The uniaxial strains ( ) = { ε ε γ } ( 12) ( 12) T ε are obtained from the biaxial strains as follows [5,8]: ε = Vε (13) The adopted projetion matrix V is 1 ν ν12ν21 1 ν12ν 21 ν21 1 V = (14) ν ν ν12ν A lak of agreement exists on the numerial values to be used for ν 12 and ν 21. He and o-workers [5] onsidered ν 12 = ν 21 = ν, as is ommon in most of the existing literature. On the ontrary, Zhu and Hsu [17] experimentally observed that the Poisson effet is haraterized by the Poisson ratio for onrete before raking whereas ν 12 and ν 21 are different from ν when onrete raks, and 6

7 introdued the Zhu/Hsu ratios to replae it. All the above ited studies, however, laim that the final omparison with experimental data is satisfatory. Therefore, it seems that more experimental investigations are needed to resolve this issue. Following a onventional approah, in this study it is assumed that ν 12 = ν 21 = ν. As the urrent strength of onrete is alulated aording to the biaxial stress state, the uniaxial stress-strain relationship reflets the biaxial stress state. Therefore, one the uniaxial strains are obtained and the onrete ompressive strength is modified aounting for the biaxial stress state, the tangent modulus E 1 or E 2 an now be obtained by differentiating the uniaxial onstitutive stress-strain relationship for onrete ( ) σ ε with respet to 1 ε = ε or ε = ε 2, respetively Stress-strain relationship for onrete in tension The response of onrete in tension is usually regarded as linear until the tension strength f t has been reahed, and is desribed by a Rankine-type priniple stress riterion. At strain ε t, rak initiation and loalization of a narrow proess zone ours. Subsequently, a softening behavior starts and it is usually modeled as a straight line, a pieewise linear branh, or a desendingexponential urve. In this study, the stress-strain relation for onrete in tension is based on the model proposed in Ref. [18] E t 0ε if 0 ε εt σ ( ε ) = εt ε ϕ fte if ε > εt (15) whih is shown in Fig. 1. In Eq. (15), φ is a (positive) shape parameter and E t0 is the initial Young s modulus for onrete in tension. To redue the lak of objetivity in the smeared rak model, the rak band theory by Bažant and Oh [10] is often adopted for the purpose of modeling tension in onrete [9,18,19]. Given G f, the mode-i frature energy of onrete, if the following ondition is satisfied G E h <, (16) f t 0 2 ft then the shape parameter φ is G f 1 ϕ = ε t (17) hf 2 t If Eq. (16) is not fulfilled, then Bažant and Oh [10] proposed to modify the onrete strength in tension as follows 7

8 f t 2G f E t0 = (18) h along with a vertial post-peak stress drop of the stress-strain relationship for onrete in tension. Sine a tangent-based implementation is used in this study, a vertial post-peak stress drop may lead to numerial problems. Therefore, an exponential post-peak softening response is still adopted as in Eq. (15) and the onrete strength in tension given by Eq. (18). In doing so, the shape parameter φ and the ultimate strain of the onrete in tension ε tu are now pure material parameters, and their numerial values should be quite small in order to approximate a vertial stress drop without ausing numerial instabilities. Finally, the mode-i frature energy of onrete G f is alulated as speified in the CEB-FIP model ode [20] G f [ N mm] [ MPa] 0.7 f = α f 10 (19) 3 with f ( 1.25d max [ mm] 10) α =10 +, where d max is the maximum aggregate size in onrete. Beause of the lak of reliable experimentally-alibrated models, it is assumed that the frature energy of onrete does not depend on onfinement effet or biaxial stress states Stress-strain relationship for onrete in ompression The uniaxial strain-stress relationship proposed by Popovis [21] is used in this study for onrete β σ ( ε ) = ε β 1+ εp β p εp ε f (20) where β > 1 is a shape fator (see Fig. 1). The softening branh for onfined onrete is modeled as proposed in Ref. [15]. In doing so, the post-peak Popovis model is replaed with a straight line having a slope: E ( f ) 2 = 11.2 ρ f z y, z. (21) The model for onfined onrete is illustrated in Fig. 1. Conrete failure by rushing ours one the ultimate strain of onrete in ompression is ahieved.. 8

9 2.8. Shear modulus for onrete Darwin and Peknold [13] provided a rational tangent shear modulus to be used in Eq. (3) in suh a way that the results are independent with respet to axis rotation. The pre-raking shear modulus for onrete in this ase is 1 ( ) ( ) 2 ν G = E + E 2ν E E 4 1 (22) Several analytial, semi-analytial and empirial formulations have been proposed for the postraking tangent shear modulus of onrete. In this study, the post-raking value of onrete G is alulated as proposed in Ref. [22] G σ σ = ( ε ε ) 1 2 (23) 2.9. Poisson ratio The Poisson ratio ν for onrete typially ranges between 0.15 and 0.22, but it has been experimentally observed that the Poisson ratio also depends on the urrent strain and stress states. A value for the Poisson ratio about 0.20 has been shown to be quite satisfatory for monotoni loading in tension-tension and ompression-ompression [13]. Beause of the lak of well-settled experimental evidene, a onstant Poisson ratio is assumed in this study Unloading-reloading sheme for onrete Unloading-reloading an loally our in onrete even under monotoni loading. It may be due to numerial issues (during iterations to ahieve equilibrium) whih depend on the smeared rotating rak model. Furthermore, a realisti RC struture has the ability to redistribute stress after a rak opens, whih often leads to unloading in the onrete. To this end, the simplified unloadingreloading sheme adopted by He and o-workers [23] is assumed in this study to deal with RC strutures subjeted to monotoni loading (see Fig. 1). This simple unloading-reloading sheme provides satisfatory results for ylially loaded RC strutures with typial steel reinforing ratios [5]. On the ontrary, it is observed a signifiant differene for low steel reinforing ratios. Therefore, this unloading-reloading sheme is a reasonable simplifiation for monotoni loadings only, beause the auray at the strutural level is not affeted. 9

10 3. Modeling of steel reinforement 3.1. Constitutive matrix for reinforement Longitudinal reinforing steel bars are treated as an equivalent uniaxial layered material plaed at the depth of the enterline of the bars and smeared over the region of bar effet [8,14,24,25]. As many layers are used as there are layers of bars in the ross setion, eah with its uniaxial properties oriented along the axis of the bars. The tangent onstitutive matrix for the kth steel reinforement layer is ρsk Esk 0 0 D sk = (24) in whih ρ sk is the reinforement ratio and E sk the tangent modulus for the kth steel reinforement layer, both varying as funtion of the urrent strain state. A perfet bond between the onrete and the reinforement is assumed. Steel bars were also modeled as smeared reinforement with perfet bond by Bao and Kunnath [25] for post-peak FE-based analysis of two-dimensional RC strutures under monotoni loadings. Sine the bond degradation of reinforing bars is one of the most important issues in assessing the seismi performane of RC strutures, the bond-slip between the onrete matrix and the reinforement needs to be modeled appropriately when onsidering yli loadings. For eah reinforement layer, the uniaxial strains of steel ε sk are alulated from the uniaxial strains of onrete ε ( 12) in Eq. (13) as follows: (, ) ( 12) ε = Τ θ θ ε (25) sk s sk where T s (θ,θ sk ) the strain transformation matrix and θ sk is the angle between the x-axis and the enterline of the bars belonging to the kth reinforement layer. Therefore, the tangent modulus E sk an be now obtained by differentiating the uniaxial onstitutive stress-strain relation for steel ( ) σs ε s with respet to s sk in Eq. (25). ε = ε. The strain ε sk will be the first element of the vetor εsk alulated 3.2. Stress-strain relationship for reinforing steel A stress-strain relationship for steel bars embedded in onrete is somewhat different from that of a bare steel bar beause the surrounding onrete bonded to the bar auses tension-stiffening effet. As a onsequene, the smeared stress-strain relationship of steel bars embedded in onrete must be 10

11 obtained by averaging the stresses and strains between two raks. In this study, the envelope stress-strain relation adopted by Mansour and o-workers [26] is assumed E ε if ε ε σs ( ε s ) = ε s f ( B) B if ε > ε ε y s0 s s n y s n (26) ( ) ε = ε (27) n y B 1.5 ( t y ) max { s } B = f f ρ,0.25% (28) where E s0 is the initial (elasti) modulus of steel, ρ s the reinforement ratio, f y the yielding stress and ε y the yielding strain. The model is illustrated in Fig. 2. All these quantities are referred to the orresponding kth reinforement layer. The ultimate strength of reinforing steel is denoted as f su. A simple bi-linear elasto-plasti hystereti model with kinemati hardening is adopted to model the unloading-reloading behavior of reinforing steel bars, see Fig. 2. This simplified model is routinely used in modeling reinforing bars in RC strutures, suh as in Kabeyasawa and Milev [12] for modeling RC shear walls subjeted to yli loadings. Although more aurate models were reently proposed to desribe the unloading-reloading behavior of reinforing steel bars, this simplified sheme provides satisfatory results at the strutural level under the assumption of monotoni loading. 4. Variational formulation and disrete equations 4.1. Equilibrium equation The variational weak form is formulated in the global x-y oordinate system. The displaement approximation u h is of the form h T u = Φ d (29) in whih Φ is the matrix that onsists of meshfree basis funtions and d is the vetor of nodal parameters. The strain-displaement relation is ε = Bd (30) where B is the strain displaement matrix. The weak form (priniple of virtual work) for problems in strutural mehanis leads to the equilibrium equations: ext int f f = 0 (31) 11

12 where ext f is the vetor of external loads f = Φ b dω + Φ t dγ (32) ext T T Ω Γt and int f the vetor of internal fores int T f = B σ dω (33) Ω In Eq. (32) and Eq. (33), b is the body fore vetor ating on Ω, t the presribed tration on the natural boundary Γ t and σ = {σ x σ y σ xy } T the stress vetor. Generally Eq. (31) is a nonlinear system of equations, beause the unknown stress field is a nonlinear funtion of the strain field. As a onsequene, an appropriate inremental-iterative numerial proedure has to be adopted to solve for the unknown displaement field. The inremental-iterative form of the equilibrium equation is i i i ext ˆ i K d = λ f + r (34) ( j 1) j j ( j 1) in whih i denotes the urrent load inrement step, j is the urrent iteration number, d i j is the inrement of the vetor of nodal parameters, ext f ˆ is the referene external load vetor, λ i j is the load inrement parameter (proportional stati loading), r i (j 1) is the unbalaned fore vetor = i ext i int i r f f (35) ( j 1) ( j 1) ( j 1) In Eq. (34), K i (j 1) is the tangent stiffness matrix formed at the beginning of the jth iteration and it is based on the known information arried out at the (j 1)th iteration. It is obtained as an assembly proess of the 2 2 nodal tangent stiffness matries K IJ (d i (j 1)) n ( ) ( ( ) ) n i i T i K( 1) = K IJ d, ( 1) = I d j j, j 1 J Ω I J I J B C d B Ω Α Α (36) where I, J = 1,,n denote two nodes within Ω and C(d i (j 1)) is the tangent onstitutive in the global system of referene x-y. The integral in Eq. (36) vanishes if I and J do not belong to the same loal support domain. One D (d i (j 1)) and D sk (d i (j 1)) are alulated as in Eqs. (3)-(6) and in Eq. (24), they are rotated into the x-y oordinate system, thus obtaining C (d i (j 1)) and C sk (d i (j 1)), respetively. Having done so, the tangent onstitutive matrix C(d i (j 1)) in Eq. (36) is ( ( ) ) ( ) n k ( ) sk ( ( ) ) C d C d C d (37) = + i i i j 1 j 1 j 1 k where n k is the total number of reinforing layers. 12

13 Numerial integration based on bakground ells is used to evaluate the integrals in Eq. (32), Eq. (33) and Eq. (36). For example, the 2 2 nodal tangent stiffness matries K IJ (d i (j 1)) in Eq. (36) is evaluated as follows n n g i T i ( ( 1) ) = j ω ( ) ; ( j 1) ( ) ( ) K d B x C x d B x J (38) IJ g I Qg Qg J Qg g g where ω g is the Gauss weighting fators for the gth Gauss point (g = 1,,n g ) at x Qg and J g is the Jaobian matrix for the area integration of the bakground ell ( = 1,,n ), at whih the Gauss point x Qg is loated On the solution of the nonlinear system of equations The main numerial issue lies in the solution of a set of equations whih are nonlinear beause of the nonlinear behavior of, both, onrete and reinforing steel. Moreover, it should be kept in mind that limit points an our in the final load-displaement urves due to material nonlinearities suh as work-softening or geometri nonlinearities. Although the maximum load apaity of a struture is the main interest in strutural analysis and design, it may be important to investigate the entire response even beyond the ourrene of a limit point in order to identify failure modes as well as post-peak behavior. Yang and Shieh [27] proposed the so-alled Generalized Displaement Control Method (GDCM) with Generalized Stiffness Parameter (GSP). Cardoso and Fonsea [28] reently demonstrated that the GDCM is an ar-length method with orthogonal onstraints. The method was originally applied to geometrially non-linear analyses and the results were found to be superior to most typial solvers. Reent results regarding the FE analysis of RC strutures [29] and in-filled steel-onrete omposite olumns [30] demonstrated the validity of the GDCM in handling geometrially nonlinearities and softening-working materials suh as onrete. In this work, the GDCM with GSP is used for nonlinear meshfree analyses, and the most important steps are desribed below Generalized displaement ontrol method A onvenient deomposition for Eq. (34) is the following K i d ˆ i = ext f ˆ (39) j j ( 1) i i i K d ɶ = r (40) ( j 1) j ( j 1) where i i ˆ i i d = λ d + d ɶ (41) j j j j 13

14 Based on Eq. (41), the displaement inrement vetor u i j is determined by making use of the meshfree basis funtions and enforing the essential (displaement) boundary onditions. The total displaement vetor of the struture at the end of the jth iteration u i j is omputed as u = u + u (42) i i i j j ( j 1) The total applied load vetor at the jth iteration of the ith inremental step ext f i j relates to the referene load vetor ext f ˆ as follows f = f + f = f + λ f ˆ (43) ext i ext i ext i ext i i ext j j j ( j 1) ( j 1) It is understood that the following initial onditions hold ( i 1) ( i 1) ( i 1) K = K, f = f, u = u (44) i ext i ext i 0 l 0 l 0 l where l is the last iteration of the last inremental step. The load inrement parameter λ i j is an unknown and is determined by imposing a onstraint ondition. For the first iterative step j = 1, the GSP is introdued. For any load inrement i at j = 1, GSP 1 i is defined as GSP i 1 1 if i = 1 1 T 1 = ( uˆ 1 ) ( uˆ 1 ) if i 2 T ( i 1) i ( uˆ ) ( uˆ ) 1 1 (45) Therefore, for any load inrement i at the first iterative step j = 1, λ i 1 is determined based on Eq. (45) λ = ± λ GSP (46) i 1 i where λ 1 1 is an initial value of the load inrement parameter. The sign in Eq. (46) depends on the sign of the GSP. In fat, the GSP is negative only for the load step "immediately after" a limit point whereas it will always be positive for the other load steps. This is beause the numerator is always positive, but the denominator ould be negative if the two vetors have different diretions. The GSP by itself is a useful indiator for hanging the loading diretions. Therefore, initially let λ i 1 be of the same sign as λ (i 1) 1 : if GSP i 1 is negative, then λ i 1 is multiplied by 1 to reverse the diretion of the loading. For the iterative step j 2, the load inrement parameter λ i j is alulated as 14

15 λ = i j T ( i 1) i ( uˆ 1 ) ( uɶ ) j T ( i 1) i ( uˆ 1 ) ( uˆ ) j (47) Here, ( 1) ˆ i 1 u is the displaement inrements generated by the referene load ˆp at the first iteration (j = 1) of the (i 1)th (previous) inremental step, and u ˆ i j and 15 i uɶ j denote the displaement inrements generated by the referene loads and unbalaned fores, respetively, at the jth iteration of the ith inremental step. Note that meshfree basis funtions are required to alulate the displaement inrements on solving the linear system of equations in Eq. (39) and Eq. (40). For the first inrement i = 1, 0 u ˆ 1 will be taken equal to 1 u ˆ1. A onvergene riterion based on, both, the inremental displaements and the unbalaned fore vetor is used. The flowhart of the numerial method is shown in Fig Maximum-entropy basis funtions On denoting x a (a = 1, m) as the nodal oordinates, the displaement field in Eq. (29) an be rewritten as m h u ( x) = φa ( x) d a (48) a There are many hoies to define meshfree basis funtions, for example moving least squares (MLS) approximants, radial basis funtions (RBF) and maximum-entropy (max-ent) approximants, to name a few. An overview about the onstrution of meshfree basis funtions is presented in Ref. [31]. The priniple of maximum entropy postulated by Jaynes [32] on the basis of the Shannonentropy was reently exploited in order to onstrut meshfree basis funtions. Shannon-entropy based onstrution of max-ent basis funtions on polygons was proposed by Sukumar [33], and a modified entropy funtional was exploited by Arroyo and Ortiz [34]. Introduing the notion of prior weight funtion, Sukumar and Wright [31] obtained the max-ent basis funtions whih generalizes the entropy funtional onsidered in Ref. [34]. The first appliation of the maximum entropy meshfree (MEM) method in the field of strutural engineering is presented in Ref. [2]. Max-ent basis funtions are promising beause they are a onvex ombination and possess a variation diminishing property as well as a weak Kroneker-delta property on the boundary [34]. An impliation of the weak Kroneker-delta property is that essential boundary onditions an be imposed as in FE methods. This is a noteworthy advantage with respet to most meshfree approximants (i.e., MLS), whih require speial tehniques to enfore essential boundary onditions. Max-ent approximants also allow to blend FE and meshfree basis funtions in a

16 seamless fashion. For ompleteness, a short review about the derivation of max-ent basis funtions follows. The max-ent basis funtions φ a (a = 1, m) are obtained by solving the following optimization problem [31] ( x) ( x) m φ a max φ ( ) ln m a x φ R + a w a (49) subjeted to m φ a ( x ) = 1 (50) a m φ a ( x) ξa ( x) = 0 (51) a where ξ a =x a x are shifted nodal oordinates and w a (x) is the prior weight funtion (initial guess for φ a ). The first onstraint in Eq. (50) imposes the partition of unity property in order to represent rigid body translations (zeroth-order reproduibility). Basis funtions that satisfy the set of onstrains in Eq. (51) an reprodue a onstant strain field exatly (first-order reproduibility). One the variational problem in Eqs. (49)-(51) is solved by means of the Lagrange multipliers tehnique, the following result for the max-ent basis funtions is obtained [31]: φ a ( x) where Z and Z Za = Z ( xη ; ) ( xη ; ) ( ; ) = w ( ) exp{ } xη x η ξ (53) a a a m ( ; ) = Zb ( ; ) xη xη (54) b is the so-alled partition funtion. The Lagrange multipliers vetor η* is obtained by solving the dual optimization problem η 2 η R { ( xη) } = arg min ln Z ; that leads to the following system of nonlinear equations: 16 (52) (55)

17 m ( ) φ ( ) ( ) η ln Z η = x ξ x = 0 (56) a a a Standard Newton-based solvers an be used for Eq. (56), and the onvergene is typially very fast. One the onverged Lagrange multipliers η* are alulated, the gradient of the max-ent basis funtions are evaluated as follows: 1 1 w w φ = φ ( ) + φ m a b a a ξa H H A b (57) wa b wb where w m b A = φbξ b (58) b wb and H is the Hessian matrix ( ) m H = ln Z η = φξ ξ (59) η η b b b b As the prior weight funtion, quarti polynomials are used in this study: w a q + 8q 3q 0 q 1 x = (60) 0 q > 1 ( ) where q = xa x ra and r a is the radius of the basis funtion support at node a. 5. Validation 5.1. Analysis of RC shear walls The nonlinear response of a variety of two-dimensional RC strutural elements under plane stress e.g., beams and shear walls may be analyzed by means of the MEM method. Current approahes in modeling RC shear walls an be lassified as follows [25]: beam-olumn type models in whih flexure is the dominant mode of response, multi-spring based maro-models and FE based mirosopi models. In this study, the proposed MEM method is adopted as an alternative tehnique for the analysis of RC shear walls Comparison with experimental data To verify the reliability of the presented meshfree method, the large sale RC shear walls tested by Lefas and o-workers [35] are investigated and numerial simulations are ompared to experimental 17

18 results. In their work, Lefas and o-workers onsidered two types of RC shear walls having onstant thikness. Type I walls were 750 mm high, 750 mm wide and 70 mm thik. Type II walls were 1300 mm high, 650 mm wide and 65 mm thik. Speimens onsisted of two lateral ribs, a top slab and a bottom base blok. The vertial and horizontal reinforement omprised high-tensile deformed steel bars of 8 and 6.25 mm diameter, respetively. Additional horizontal reinforement in the form of stirrups onfined the wall edges. Mild steel bars of 4 mm diameter were used for this purpose. Lateral ribs with onfinement reinforement provide two onealed olumns with retangular shape at eah edge of the shear wall. The RC shear wall speimens were subjeted to the ombined ation of a distributed vertial load and a horizontal load at the upper beam (loadontrolled testing). In this study, speimens SW14 and SW15 are investigated for Type I RC shear walls whereas speimens SW21 and SW22 are onsidered for Type II RC shear walls. Input data on the imposed load, onrete and reinforing steel are listed in Table 1 and 2. The top slab served to distribute both the horizontal and the vertial loads. The base blok, lamped to the laboratory floor, simulated a rigid foundation. Having this in mind, the base blok is replaed with a set of deformation onstraints and the vertial and horizontal load are uniformly distributed over the wall width. Starting from the input data listed in Table 1 and 2, the remaining mehanial data are determined as follows. Values for the onrete ompressive strength 18 f are speified as 85% of the ube ompressive strength values. For onrete in tension, E t0 is alulated on assuming ε t = [26] and f t = 0.33(f ) 0.5 [16]. The mode-i frature energy of onrete is alulated based on the maximum aggregate size d max of 10 mm. The initial (elasti) modulus for steel is 200,000 MPa. A small number of preliminary runs were performed to investigate β, ν and µ values in the ranges [2.00,5.00], [0.15,0.20] and [0.10,1.00], respetively. The GDCM is performed by assuming λ 1 1 = and the tolerane for onvergene was set to A regular nodal arrangement is used at first. The adopted nodal spaing is 75 mm 75 mm for speimens SW14 and SW15 and 65 mm 65 mm for speimens SW21 and SW22. Assuming this set of input data, the final load-displaement urves in Fig. 4 are obtained. Numerial results appear to be in satisfatory agreement with the experimental observations. Obviously, some of the disrepanies may be attributed to the unavoidable unertainties and approximations in, both, models and parameters, whih are not insignifiant in RC modeling. However, another fator may also be the soure of disagreement between numerial and experimental results. It is observed that numerial results for the initial lateral stiffness are higher than those observed experimentally. This may be due to the foundation flexibility of the RC shear wall whih has not been onsidered in this analysis. In this simulation the base blok was replaed with a set of rigid onstraints, thus

19 overestimating the shear wall foundation stiffness. The overestimation is less severe for walls with imposed axial loads. The presene of axial loading more losely approximates the rigid onstraints assumed at the base blok. Therefore, under zero or very low axial loads, replaing the base blok with a set of rigid onstraints is less onsistent with the experiment. Following this argument, one may justify why the agreement with experimental data is better for SW22 than for SW15. In fat, the shear wall flexibility inreases onsiderably with respet to the base blok as the shear wall height doubles, and then replaing the base blok with rigid onstraints is more appropriate in SW22 than in SW15. However, the foundation flexibility should have no signifiant effets on the load apaity as well as on its post-peak behavior beause the failure mehanism essentially involves the behavior of the shear wall. About the numerial proedure, it is found that 2 iterations are needed for eah inrement when onsidering SW14 and SW21. A larger number of iterations (3 or 4) are needed to ahieve the equilibrium at eah inrement for speimens SW15 and SW Comparison with FE methods Experimental data from Ref. [35] were also used by many other authors to validate their FE based analyses [16,24,36,37]. Likewise, the main features of the load-displaement urves obtained by the meshfree method presented in this paper are ompared to FE simulations. A omparison between the obtained load-displaement urves and those presented in Ref. [36] is shown in Fig. 4. Kwan and He [36] adopted a smeared rak approah in onrete modeling as well as onsidering raked onrete to be an orthotropi material, but there is no information on the reinforing steel model in their work. They also onsidered the onfinement effet in modeling the ompressive onrete strength at the lateral ribs and a standard displaement ontrol method was used to trak the postpeak behavior of the shear walls. Aounting for the differenes in RC modeling, the presented MEM method and the FE analysis by Kwan and He [36] are in satisfatory agreement. The meshfree results are ompared to that in Ref. [37]. Park and Kim [37] presented a miroplane based FE analysis of planar RC strutures, and validated their method using load-displaement urves also onsidered in the present study, that is the experimental responses for SW21 and SW22 speimens. A qualitative omparison between the obtained results in Fig. 4 and those presented in Ref. [37] onfirms the validity of the presented MEM method. Speifially, the illustrated meshfree method provides a better estimation of the load apaity for the SW21 speimen, along with a more onsistent predition of the orresponding displaement. The meshfree-based estimation of the ultimate displaement for this shear wall is in very good agreement with the FE results of Park and Kim [37]. A general good agreement is observed in the simulation of the SW22 speimen, even if 19

20 the meshfree-based predition of the ultimate displaement seems to be lower than that in Ref. [37] Effets of different nodal arrangements The effets of different nodal arrangements on the load-displaement response are shown in Fig. 5 for speimen SW22. The irregular nodal arrangement was obtained by perturbing the regular one, and the magnitude of the perturbation was 3%. Further strategies may be implemented to improve the numerial method as well as the reliability of the final results. For instane, the use of nodal integration tehniques along with appropriate stabilization methods an be exploited to ahieve a ompletely meshfree method. Non-loal approahes an also be used to improve the objetivity of smeared rak models based results Stress-strain predition Experimental data of the shear walls A2 and A3 tested by Pang and Hsu [38] are finally taken into aount. The panels were 55 in. high, 55 in. wide and 7 in. thik. The maximum ompressive ylinder onrete strength was 5.98 ksi and 6.04 ksi for A2 and A3, respetively. The orresponding strain values were and Reinforement was arranged at 45 deg with respet to the x axis. The steel reinforing ratio is for A2 and for A3, with equal amounts in the two perpendiular diretions. The orresponding yield strengths were ksi and ksi, respetively. The prinipal ompressive and tensile stresses of equal magnitude were applied in the vertial and horizontal diretion, respetively. Suh a proportional stati loading leads to a shearbased stress ondition in the 45 diretion. The applied shear stress and the shear strain were monitored along the system referene of the reinforing bars. The results in Fig. 6 show a good agreement between experimental data and numerial simulations. The yielding point of the reinforing steel is evident and well predited. The maximum applied shear stress was reahed when the onrete began rushing. From this point on, the urves went into the desending branhes that are satisfatorily simulated. The meshfree-based simulation of the A3 speimen is in good agreement with the FE analysis of Wang and Hsu [38]. 6. Conlusions This study presented a maximum-entropy meshfree method for material nonlinear analysis of twodimensional RC strutures subjeted to proportional monotoni stati loading. Conrete and reinforing steel modeling were first disussed, and the generalized displaement ontrol method was implemented in order to solve the final nonlinear system of equations. As a validation study, RC shear walls were onsidered, and omparison of numerial simulations with experimental data 20

21 as well as finite element (FE) analysis demonstrated the suitability of the presented maximumentropy meshfree method. However, meshfree methods are not exempt from drawbaks, i.e. a larger omputational effort than FE approahes. A ompetitive omputational framework an be formulated in suh a way that FE methods are used where meshless approahes are not an effiient option whereas meshfree methods are adopted where FE based tehniques are known to be problemati. Moving from this onsideration, next studies will address a blended FE-meshfree analysis of RC strutures subjeted to yli loadings. In doing so, the use of max-ent approximants is partiularly suitable beause it provides a seamless bridge between FE and meshfree basis funtions. 21

22 Referenes [1] Yaw L.L., Kunnath S.K., Sukumar N. Meshfree method for inelasti frame analysis. Journal of Strutural Engineering, 135(6), , 2009 [2] Yaw L.L., Sukumar N., Kunnath S.K. Meshfree o-rotational formulation for twodimensional ontinua. International Journal for Numerial Methods in Engineering, 79(8), , 2009 [3] Rabzuk T., Zi G., Bordas S., Nguyen-Xuan H. A geometrially non-linear threedimensional ohesive rak method for reinfored onrete strutures. Engineering Frature Mehanis, 75(16), , 2008 [4] Rabzuk T., Eibl J. Numerial analysis of prestressed onrete beams using a oupled element free Galerkin/finite element approah. International Journal of Solids and Strutures, 41(3-4), , 2004 [5] He W., Wu Y.F., Liew K.M. A frature energy based onstitutive model for the analysis of reinfored onrete strutures under yli loading. Computer Methods in Applied Mehanis and Engineering, 197(51-52), , 2008 [6] Yang Z.J., Chen J. Finite element modelling of multiple ohesive disrete rak propagation in reinfored onrete beams. Engineering Frature Mehanis, 72(14), , 2005 [7] Pietruszzak S., Stolle D.F.E. Deformation of strain softening materials. Part I: Objetivity of finite element solutions based on onventional strain softening formulations. Computers and Geotehnis, 1(2), , 1985 [8] Mo Y.L., Zhong J., Hsu T.T.C. Seismi simulation of RC wall-type strutures. Engineering Strutures, 30(11), , 2008 [9] Kwak H.G., Filippou F.C. Finite element analysis of reinfored onrete strutures under monotoni loads. Tehnial report no. UCB/SEMM-90/14, University of California Barkeley, 1990 [10] Bažant Z.P. Oh B.H. Crak band theory for frature of onrete. Materiaux et Construtions, 16(93), , 1983 [11] Vehio F.J., Collins M.P. Modified ompression-field theory for reinfored onrete elements subjeted to shear. ACI Journal, 83(2), , 1986 [12] Kabeyasawa T., Milev J.I. Modeling of reinfored onrete shear walls under varying axial load. Transations of the 14 th International Conferene on Strutural Mehanis in Reator Tehnology (SMiRT 14), Lyon, Frane, August 17-22,

23 [13] Darwin D., Peknold D.A. Inelasti model for yli biaxial loading of reinfored onrete. Tehnial report, Strutural Researh Series 409, College of Engineering. University of Illinois at Urbana-Champaign, 1974 [14] Hu H.T., Shnobrih W.C. Nonlinear finite element analysis of reinfored onrete plates and shells under monotoni loading. Computers and Strutures, 38(516), , 1991 [15] Hoshikuma J., Kawashima K., Nagaya K., Taylor A.W. Stress-strain model for onfined reinfored onrete in bridge piers. Journal of Strutural Engineering, 123(5), , 1997 [16] Vehio F.J. Finite element modeling of onrete expansion and onfinement. Journal of Strutural Engineering, 118(9), , 1992 [17] Zhu R.R.H., Hsu T.T.C. Poisson effet of reinfored onrete membrane elements. ACI Strutural Journal, 99(5), , 2002 [18] Noh S.Y., Krätzig W.B., Meskouris K. Numerial simulation of servieability, damage evolution and failure of reinfored onrete shells. Computers and Strutures, 81(8-11), , 2003 [19] Krätzig W., Pölling R. An elasto-plasti damage model for reinfored onrete with minimum number of material parameters. Computers and Strutures, 82(15-16), , 2004 [20] CEB-FIP model ode (design ode), Comité Euro-International du Béton. Thomas Telford Ltd, 1993 [21] Popovis S. A numerial approah to the omplete stress-strain urve of onrete. Cement and Conrete Researh, 3(5), , 1973 [22] Zhu R. H., Hsu T. T. C., Lee J. Y. Rational shear modulus for smeared rak analysis of reinfored onrete. ACI Strutural Journal, 98(4), , 2001 [23] He W., Wu Y.F., Liew K.M., Wu Z. A 2D total strain based onstitutive model for prediting the behaviors of onrete strutures. International Journal of Engineering Siene, 44(18-19), , 2006 [24] Kwak H.G., Kim D.Y. Material nonlinear analysis of RC shear walls subjet to monotoni loadings. Engineering Strutures, 26(11), , 2004 [25] Bao Y., Kunnath S.K. Simplified progressive ollapse simulation of RC frame-wall strutures. Engineering Strutures, 32(10), , 2010 [26] Mansour M., Lee J.-Y., Hsu T.T.C. Cyli stress-strain urves of onrete and steel bars in membrane elements. Journal of Strutural Engineering, 127(12), , 2001 [27] Yang Y.-B., Shieh M.-S. Solution method for nonlinear problems with multiple ritial points. AIAA Journal, 28(2), ,

24 [28] Cardoso E.L., Fonsea J.S.O The GDC method as an orthogonal ar-length method. Communiations in Numerial Methods in Engineering, 23(4), , 2007 [29] Gomes H.M.M., Awruh A.M. Some aspets on three-dimensional numerial modeling of reinfored onrete strutures using the finite element method. Advanes in Engineering Software, 32(1), , 2001 [30] Lakshmi B., Shanmugam N.E. Nonlinear analysis of in-filled steel-onrete omposite olumns. Journal of Strutural Engineering, 128(7), , 2002 [31] Sukumar N., Wright R.W. Overview and onstrution of meshfree basis funtions: From moving least squares to entropy approximants. International Journal for Numerial Methods in Engineering, 70(2), , 2007 [32] Jaynes E.T. Information theory and statistial mehanis. Physial Review. 106(4), , 1957 [33] Sukumar N. Constrution of polygonal interpolants: a maximum entropy approah, International Journal for Numerial Methods in Engineering, 61(12), , 2004 [34] Arroyo M., Ortiz M. Loal maximum-entropy approximation shemes: a seamless bridge between finite elements and meshfree methods. International Journal for Numerial Methods in Engineering, 65(13), , 2006 [35] Lefas I.D., Kotsovos M.D., Ambraseys N.N. Behavior of reinfored onrete strutural walls: strength, deformation harateristis, and failure mehanism. ACI Strutural Journal, 87(1), 23-31, 1990 [36] Kwan A.K.H., He X.G. Finite element analysis of effet of onrete onfinement on behavior of shear walls. Computers and Strutures 79(19), , 2001 [37] Park H., Kim H. Miroplane model for reinfored-onrete planar members in tensionompression. Journal of Strutural Engineering, 129(3), , 2003 [38] Pang X.B., Hsu T.T.C. Behavior of reinfored onrete membrane elements in shear. ACI Strutural Journal, 92(6), , 1995 [39] Wang T., Hsu T.T.C. Nonlinear finite element analysis of onrete strutures using new onstitutive models, Computers and Strutures, 79(32), ,

25 List of figures Fig. 1. Conrete modeling (not to sale) Fig. 2. Modeling of steel reinforement (not to sale) Fig. 3. Generalized displaement ontrol method for meshfree nonlinear analyses Fig. 4. Comparison between experimental data, finite element analysis and maximum-entropy meshfree method Fig. 5. Effets of different nodal arrangements Fig. 6. Comparison between shear stress-shear strain experimental urves and maximum-entropy meshfree method List of tables Table 1. Load and onrete data from Ref. [35] Table 2. Reinforing steel data from Ref. [35] 25

26 Shear wall identifiation Type Speimen Vertial load [kn] Cubi uniaxial onrete strength [MPa] I II SW SW SW SW Table 1 Type Yield strength [MPa] Ultimate strength [MPa] 8 mm high-tensile bar mm high-tensile bar mm mild-steel bar Table 2 26

27 ó f t unonfined å u å` u å å ` f ù å t å tu å f u onfined E f ` f Fig. 1 27

28 ó s f su f n E s0 E s0 å n å su å s E s0 Fig. 2 28

29 Fig. 3 29