Earthquake response of concrete arch dams: aplastic–damage approach

Omid Omidi*,† and Vahid Lotfi

Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran

SUMMARY

There are several alternatives to evaluate seismic damage-cracking behavior of concrete arch dams, amongwhich damage theory is the most popular. A more recent option introduced for this purpose is plastic–damage (PD) approach. In this study, a special finite element program coded in 3-D space is developedon the basis of a well-established PD model successfully applied to gravity dams in 2-D plane stress state.The model originally proposed by Lee and Fenves in 1998 relies on isotropic damaged elasticity in combi-nation with isotropic tensile and compressive plasticity to capture inelastic behaviors of concrete in cyclic ordynamic loadings. The present implementation is based on the rate-dependent version of the model, includ-ing large crack opening/closing possibilities. Moreover, with utilizing the Hilber–Hughes–Taylor timeintegration scheme, an incremental–iterative solution strategy is detailed for the coupled dam–reservoirequations while the damage–dependent damping stress is included. The program is initially validated, andthen, it is employed for the main analyses of the Koyna gravity dam in a 3-D modeling as well as a typicalconcrete arch dam. The former is a major verification for the further examination on the arch dam. Theapplication of the PD model to an arch dam is more challenging because the governing stress condition ismultiaxial, causing shear damage to become more important than uniaxial states dominated in gravity dams.In fact, the softening and strength loss in compression for the damaged regions under multiaxial cyclicloadings affect its seismic safety. Copyright © 2013 John Wiley & Sons, Ltd.

Received 1 December 2011; Revised 27 April 2013; Accepted 30 April 2013

KEY WORDS: plastic–damage; arch dam; Koyna dam; dam–reservoir interaction; seismic analysis

1. INTRODUCTION

Seismic safety of concrete dams as one of the main infrastructures needed for flood control or watersupply is still an engineering concern, because it is increasingly evident that the seismic designconcepts used at the time of construction of most existing dams were simplistic and inadequate.Furthermore, the population at risk located downstream of dams continues to expand. Therefore, thecollapse of dams would cause a socioeconomic tragedy. The growing concern of the seismic safetyof such critical structures has caused great interest for reevaluating existing dams by using nonlinearmodels that are able to predict crack initiation and its propagation through dam body. Damage andfailure analyses of arch dams are very complicated because they need to be considered as 3-Dsystems with semi-unbounded foundation rock and reservoir domains. The dynamic analysis of archdams should consider the following factors [1]: dam–water interaction, wave absorption at thereservoir boundary, water compressibility, dam–foundation interaction, spatially varying groundmotion around the canyon, vertical contraction and peripheral joints opening and slippage, and,finally, a possible cracking in mass concrete monoliths.

*Correspondence to: Omid Omidi, Department of Civil and Environmental Engineering, Amirkabir University ofTechnology, Tehran, Iran.†E-mail: o.omidi@gmail.com

Copyright © 2013 John Wiley & Sons, Ltd.

EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2013; 42:2129–2149Published online 22 July 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.2317

Constitutive theory of concrete materials has been one of the main themes of research for somedecades. Plasticity, continuum damage mechanics, and combined plasticity and damage mechanicsare the most common theories that have been used for the description of concrete constitutivebehavior. By the term damage-based, a class of smeared crack models such as fixed and rotatingcrack models is also included. Besides, a robust and efficient algorithm as a crucial issue ofnumerical implementation needs to be employed when a crack growth analysis in concrete dams iscarried out. Plasticity theory has been widely utilized in concrete modeling. Damage mechanics as arelatively simple approach simulates concrete fracture by a degradation of stiffness due to thedevelopment of micro-cracks. Many isotropic and anisotropic damage models have been developedfor concrete. However, because of the composite nature of concrete, its complicated behaviorespecially in cyclic loadings cannot be satisfactorily reflected in the usual constitutive theories ofmaterials such as pure plasticity and pure damage mechanics theories [2]. In fact, the plasticitytheory fails to address the degradation of the material stiffness due to micro-cracking, and thecontinuum damage mechanics also fails to describe the irreversible deformations and the inelasticvolumetric expansion in compression. Therefore, because both micro-cracking and irreversibledeformations are contributing to the nonlinear response of concrete, a constitutive model shouldaddress equally the two physically distinct modes of irreversible changes. Several effective plastic–damage (PD) models have been developed for concrete [2–13].

Replacing the isotropic elastic properties with orthotropic constitutive relations, Vargas-Loli andFenves used fixed smeared crack models to represent the fracture zone in concrete dams [14]. Theyemployed the model in seismic analysis of the Pine Flat dam in which the resulting crack profileswere diffused while numerical instability was also reported because of sudden energy release insmall fractured elements. El-Aidi and Hall developed a fixed smeared crack model and seismicallyapplied it to concrete gravity dams [15]. However, arbitrariness of the shear retention factor and abuildup of shear stresses along the fixed crack surface have been reported as disadvantages of thefixed smeared crack approach. Bhatacharjee and Leger employed a rotating smeared crack model asan improved smeared crack approach in the seismic analysis of gravity dams [16]. The modeladdresses softening by the secant stiffness in the direction of maximum principal strain. Theydefined the viscous damping on the basis of the tangent stiffness to include the energy dissipation.Espandar and Lotfi compared the application results of a non-orthogonal smeared crack model withan elastic perfectly plastic model to concrete arch dams, and it was concluded that the non-orthogonal smeared crack approach can redistribute the state of stresses and produces a morerealistic profile of stresses in the dam [17–19]. Mirzabozorg and Ghaemian proposed a smearedcrack to model static and dynamic behaviors of mass concrete in three-dimensional space [20]. Thedeveloped model examined in the seismic fracture analysis of the Koyna dam simulates the tensilefracture on mass concrete and contains pre-softening behavior, softening initiation, fracture energyconservation, and strain-rate effects under dynamic loads.

Several researchers have used continuum damagemechanics to simulate mass concrete behavior in dams.Formulating the damage evolution, loading/unloading conditions, and crack opening/closing rules on thebasis of the total strain, Ghrib and Tinawi proposed an orthotropic damage model [21, 22]. Cervera et al.developed an isotropic damage model that accounts for the different behaviors of concrete in tension andcompression, each with its own damage surface and evolution law [23, 24]. Valliappan et al. evaluatedthe seismic response of gravity and arch dams by damage mechanics approach [25, 26]. Gunn proposed a3-D damage model for static analysis of arch dams [27, 28]. Ardakanian et al. successfully employed thedamage-based model proposed by Gunn for dynamic evaluation of arch dams [29].

Although much effort has been devoted to the development of coupled damage-plasticityconstitutive laws for concrete in the recent years, there are a few studies in which a PD model isemployed to assess the earthquake-induced damage of concrete dams [30, 31]. Lee and Fenvesproposed a PD model for seismic analysis of concrete dams [32]. They applied the PD model,which had been implemented in 2-D plane stress state, to the Koyna gravity dam as a widely usedbenchmark problem. The PD model proposed by Lee and Fenves is developed herein in three-dimensional formulation and implemented in a special finite element program, SNACS [33], for theseismic application to a typical arch dam. In this respect, the present study can be considered to bethe first for this class of PD models to study the nonlinear seismic behavior of concrete arch dams.

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Dam–reservoir interaction affecting the response of concrete dams has been widely investigated inthe context of linear analyses. However, there are a few investigations carried out on the nonlinearanalysis of concrete dams, including dam–water interaction. Utilizing a staggered solution schemefor the coupled dam–reservoir equations, Ghaemian and Ghobarah evaluated the seismic cracking ofconcrete gravity dams by a smeared crack model. The smeared crack model was found to beinfluenced by the dam–water interaction solution in comparison with the added mass approachcommonly used [34, 35]. Considering the dam–water interaction, Calayir and Karaton examined theapplication of the smeared crack model proposed by Bhatacharjee and Leger to the Koyna dam [36].

The rest of the paper is outlined as follows. Section 2 describes the main features of the rate-dependent PD model in the usual cracking state along with introducing the yield surface and theflow rule employed. The theoretical issues involved in the large cracking concept to properlyaddress the cyclic behavior of concrete subjected to large tensile strains are also highlighted in thissection. By using single-element tests, some basic validations for the 3-D implementation arediscussed in Section 3. The incremental–iterative solution strategy implemented for the timeintegration of the coupled dam–reservoir equations in a nonlinear context is derived in Section 4.The Koyna gravity dam is then analyzed in a 3-D modeling, and the resulting crack pattern iscompared with the available experiment. Finally, by applying the extended model to the seismicanalysis of an arch dam, it is demonstrated that the PD model is eminently suitable for damage andfailure analyses of arch dams.

2. MAIN FEATURES OF PLASTIC–DAMAGE MODEL

The PD model implemented herein for three-dimensional analysis of concrete arch dams wasestablished by Lubliner and coauthors in 1989, which was mainly appropriate for analyses withmonotonic loadings [3]. The model was later modified by Lee and Fenves in 1998 to simulatecracking and crushing of concrete under cyclic or dynamic loadings [2]. Their proposed model isbased on isotropic damaged elasticity in combination with isotropic multi-hardening plasticityformulated in the effective (undamaged) stress space. Although the damage part of the model isisotropic, two distinct damage variables are defined, which correspond to the two main failuremechanisms as tensile cracking and compressive crushing. Furthermore, two hardening variablesconnected to these failure mechanisms control the evolution of the yield surface [2]. By followingLubliner et al. [3], these hardening variables are referred to as the PD variables herein.

2.1. Constitutive law and damage evolution equation

An overview of the rate-independent model as the backbone of the rate-dependent extension is initiallypresented. The basic constitutive equations of the PD model in tensor notation are

« ¼ «e þ «p; s ¼ 1� Dð Þs ; s ¼ E0 : «e

_«p ¼ _l rsΦ ; _k ¼ _l H s;kð Þ ; D ¼ D s;kð Þ (1)

where «,«e and «p are the total, elastic, and plastic strains, respectively; s and s are the stress andthe effective stress tensors, respectively; E0 is the initial (undamaged) elastic stiffness tensor; l isthe plastic consistency parameter; Φ is the flow potential function; D is the stiffness degradationvariable; and the vector H is the plastic modulus derived considering plastic dissipation.Moreover, k is the PD vector containing the normalized PD variables in tension andcompression (i.e., k= {kt,kc}

T), playing the role of hardening variables herein. The evolutionequations of kt and kc are initially constructed by considering uniaxial loading conditions, andsubsequently, they are extended to multiaxial states [2]. By using exponential functions, theuniaxial stress is assumed to be related to the scalar plastic strain symbolized by ep. In thiscondition, the PD variables are represented as

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kℵ ¼ 1gℵ

Z ep

0sℵ dep; gℵ ¼

Z 1

0sℵ dep ℵ 2 t; cf g (2)

where gℵ called the specific fracture energy is the fracture energy normalized by the characteristiclength as the localization zone size, lℵ (i.e., gℵ =Gℵ/lℵ). In a finite element analysis, this isemployed to ensure mesh objectivity. Different choices are available for lℵ. Here, it is includedas the side of an equivalent cube having the same volume as the tributary volume at each Gausspoint of a 3-D solid element. It is clear that 0≤ kℵ≤ 1 and _kℵ≥0.

The PD model utilizes the yield function of Lubliner et al. with the modifications proposed by Leeand Fenves to give different evolutions of strength in tension and compression [2, 3]. It takes thefollowing form in terms of effective stresses:

F s;kð Þ ¼ f s;kð Þ � cc kð Þf s;kð Þ ¼ 1

1� a

ffiffiffiffiffiffiffiffi3 J2

pþ a I1 þ b kð Þ smax

� �� g �smax� �h i (3)

in which smax is the maximum principal effective stress. The Macauley bracket h � i is defined byhxi= (x + |x|)/2. The parameters a, b and g have the following definitions:

a ¼ f b0=f c0ð Þ � 12 f b0=f c0ð Þ � 1

; b kð Þ ¼ cc kcð Þct ktð Þ 1� að Þ � 1þ að Þ; g ¼ 3 1� roctð Þ

2 roct � 1(4)

where fb0/fc0 is the ratio of the initial yield strengths under biaxial and uniaxial compression; ct and ccdenote the effective tensile and compressive cohesion (positive values utilized here), respectively; roctis the ratio of the octahedral shear effective stress (i.e., toct ¼

ffiffiffiffiffiffiffiffiffiffiffiffi2J2=3

p) on the tensile meridian to that

on the compressive meridian (i.e., roct ¼ toctð ÞTM= toctð ÞCM) at the initial yield condition for any givenI1 such that 0.5≤ roct≤ 1.0. The value of roct = 2/3 suggested by Lubliner et al. leads to g= 3 [3].Containing the parameter g, which appears only in triaxial compression, the yield function better

predicts the concrete behavior in compression under confinement. F and _l obey the loading/

unloading conditions: _l≥0; F s;kð Þ≤0; _lF s;kð Þ ¼ 0. Furthermore, a Drucker–Prager hyperbolicfunction is employed here as the plastic potential function:

Φ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2H þ 2 J2

qþ apI1; bH ¼ e0 ap f t0 (5)

where ap is the dilatancy parameter, ft0 is the maximum uniaxial tensile strength, and e0 is theeccentricity parameter. It is noted that the original 3-D formulation of Lee and Fenves utilizes thelinear function (i.e., Equation (5) when e0 = 0), which causes severe numerical difficulties in return-mapping process because of its apex’s singularity. Such a modification, which is inevitable for the3-D implementation, requires a part of the model to be reformulated and makes an extra iterationnecessary within the local iteration to compute the plastic consistency parameter as the main step ofthe stress return algorithm [37]. SNACS program utilizes the spectral return-mapping algorithmefficiently derived for this class of PD models [33, 38]. Although the damaged elastic stiffnessremains isotropic, the model is accurately capable of capturing the tensile damage and compressivedamage as the two major damage phenomena. Based on Lee-Fenves PD model [2, 32], amultidimensional degradation is defined by interpolating between the two main damage variables:

D s;k� � ¼ 1� 1� Dc kcð Þð Þ 1� s s

� �Dt ktð Þ� �

(6)

where the damage variables in tension and compression denoted by Dt and Dc, respectively, areexplicit functions of the PD variables in tension and compression [2]. They can have values rangingfrom zero, corresponding to the undamaged state, up to one, which represents the fully damagedstate (i.e., 0≤Dℵ≤ 1). Moreover, s is the recovery parameter such that 0≤ s≤ 1 and used to include

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the elastic stiffness recovery during elastic unloading process from tension to compression. Setting aminimum value, s0, for s, it may be written as [2]

s s� � ¼ s0 þ 1� s0ð Þ r s

� �; r s

� � ¼ X3i¼1

si� � !

=X3i¼1

si�� �� !

; 0≤r s� �

≤1 (7)

where si’s (i = 1, 2, 3) are the principal effective stress components and r is a multiaxial stress weightfactor ranging from zero when all principal stresses are negative to 1 when they are positive. Thestiffness is recovered upon crack closure as the load changes from tension to compression. On theother hand, it is not recovered as the load changes from compression to tension once crushingmicro-cracks have developed.

2.2. Large cracking modification

In a cyclic response, tensile behavior is close to pure damage, whereas response in compression iscloser to pure plasticity, that is, an approximately secant path in tension and an elastic path incompression are followed in an unloading state. In fact, the crack opening/closure mechanismbecomes similar to a discrete crack after a large damage is sustained in tension region, that is,kt> kcr where kcr is an empirical value usually near to unity [32]. At such a tensile damage level,the evolution of plastic strain caused by tensile damage is stopped. The modified evolution relationproposed by Lee and Fenves is utilized herein to treat large crack opening/closing and reopeningprocess in the continuum context [32]. By employing an intermediate effective stress, es , theevolution equation of plastic strain is modified as

es ¼ E0 : «� e«pð Þ 2 esj F es;kð Þ≤0f g_«p ¼ 1� rð Þ _e«p; _e«p ¼ _l r~sΦ; r ¼ r es� (8)

where e«p is referred to as the intermediate plastic strain, and r is the weight function causing the largecrack modification to be applied for the states, which have positive principal stresses. To make theeffective stress on the basis of the plastic strain admissible in the stress space, one has to employ anew damage variable denoted by Dcr at the crack damage corrector, making the evaluated effective stressreturn back onto the yield surface. It is determined by the plastic consistency condition for a continuedloading ( _Dcr > 0) such that F 1� Dcrð Þ s;kð Þ ¼ 0, from which Dcr ¼ 1� cc kð Þ=f s;kð Þ [32].The stiffness degradation variable is redefined considering large cracking as

D ¼ 1� 1� Dcð Þ 1� s Dtð Þ 1� s Dcrð Þ (9)

With utilizing the spectral return-mapping algorithm proposed in [38], the numericalimplementation including the algorithmic tangent operator and the stress update algorithm modifiedto consider large cracking possibilities has been detailed by Omidi and Lotfi in [39].

2.3. Visco-plastic regularization

Some of the convergence difficulties in material models addressing softening and degradation in behaviorare treated by using a visco-plastic regularization of constitutive equations. Besides, it will improve meshobjectivity in structural simulations [32]. Both plastic strain and degradation variable are regularizedherein by adding viscosity with the Duvaut–Lions regularization. The visco-plastic strain rate tensor,_«vp, and the viscous degradation variable, Dv, for the visco-plastic system are defined, respectively, by

_«vp ¼ 1m

«p � «vpð Þ ; _Dv ¼ 1

mD� Dvð Þ (10)

where m, which is called the viscosity parameter, shows the relaxation time of the visco-plastic model; «p

andD are the plastic strain and degradation variable computed in the inviscid backbonemodel. The stress–strain relation for the visco-plastic model is given as [32]

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s ¼ 1� Dvð Þ E0 : «� «vpð Þ (11)

2.4. Visco-elastic-damage damping stress

Several approaches are available to include material damping effects in a transient analysis, amongwhich the most common one is based on the constant damping mechanism. This kind of dampingutilizing the linear form of visco-elastic damping stress introduces artificial damping forces as cracksopen with large relative velocity, and the damping forces then restrains the crack opening bytransferring artificial stresses across the crack. This artificial resistance results in unreal diffusedcracks in the cases where cracking should be highly localized [40, 41]. Three alternatives have beenemployed to solve this problem: (i) setting the damping term to zero upon cracking of an element[15]; (ii) using the tangent stiffness matrix rather than the elastic one [16]; and (iii) utilizing thedegraded stiffness in computation of the visco-elastic damping stress [32]. By following Lee andFenves, a nonlinear form of visco-elastic damping, which includes the degradation damage variable,D, computed within the rate-independent backbone model, is employed here as

x «; _«ð Þ ¼ bR 1� Dð ÞE0 : _« (12)

where bR is the coefficient of viscous damping and calibrated to provide a damping ratio at thefundamental period. This kind of damping stresses introducing damage-dependent damping mechanismrepresents stiffness proportional term for damping of damaged material.

3. BASIC VALIDATIONS OF PLASTIC–DAMAGE MODEL

By using SNACS program [33], several single-element tests initially validating the 3-D extendedmodel are discussed herein. In all cases, a = 0.12 and ap = 0.2 are used as the model parameters.

3.1. Cyclic loading example under large tensile strain

To examine how the large cracking phenomenon is being captured in the model, one subjects a singleelement to full cyclic uniaxial loadings under a large tensile strain. The following material propertiesare utilized: E0 = 31.7 GPa, n = 0.18, f 0t = 3.48 MPa, f 0c = 27.6 MPa, Gt = 12.3 N/m, Gc = 1750 N/mand lt = lc = 25.5 mm.

This test exemplifies the ability of the model to simulate stiffness recovery when the status changesfrom tension to compression and vice versa. The result with the loading paths A to G is illustrated inFigure 1(a). During the tensile unloading and subsequent compressive loading (path A–B or D–E–F),

Figure 1. Crack opening/closing simulation: (a) stress–strain curve and (b) visco-plastic effects.

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the stiffness is properly recovered. Large crack opening/closing is simulated during path C–D–E. Thecrack closes at point E, and the elastic stiffness is recovered. It yields in compression again to reach pointF. Subsequent reopening of the crack is noticed along path F–G during unloading. In Figure 1(b), the fullcyclic loading response of the rate-dependent model is compared with the rate-independent result. Thestrain rate of 0.0002 sec-1 is imposed through the full cycles. The differences in the responses emergeonly at the softening and hardening regions. The stiffness recovery is well represented during theunloading from tension to compression in the analysis including rate dependency.

3.2. Concrete under 3-D compression

The g term in the yield function is adjusted to have better prediction in 3-D applications loaded incompression under confinement. The results of the model are compared for this type of loadingbetween the cases of g= 0 and g= 3. The material properties are the same as those utilized for thecyclic loading test. The numerical results under three sets of confining stresses are presented inFigure 2. The confining stresses are increased proportional to the imposed strain in the thirddirection for all these cases. It is clearly observed that the enhancements of strength and ductilitydue to confinement are satisfactorily captured.

3.3. Shear panel test

With imposing a special in-plane boundary condition as illustrated in Figure 3(a), a 3-D single-elementmesh with unit dimensions is used to study multiaxial cyclic behavior of a concrete panel. Thisexample was adopted from [38]. Such a boundary condition imposed to the element does not provide apure shear state. The out-of-the-plane boundary condition is considered as plane stress. The materialproperties are E0 = 30.0 GPa, n=0.2, f 0t = 3.3 MPa, f 0c = 30.0 MPa, Gt = 250 N/m, Gc = 25000 N/m andlt = lc = 0.5 m. The results are compared in Figure 3(a) for three different sets of displacements as 20,40, and 80 steps per one cycle of loading to assess the accuracy of the solution. Excellent agreementexists between the three analyses through all stages. This indicates good accuracy of the solution over alarge range of strain increments. Figure 3(b) illustrates the effects of rate dependency on the results byconsidering different viscosity ratios. The imposed shear strain rate is 0.0004 sec-1.

To examine how well the stiffness degrades during the cycles, one can subject the panel to several cyclicloadings. The hysteresis curves computed by the rate-independent model are given in Figure 4(a). Thesoftening and strength loss of the panel are precisely observed under multiaxial cyclic loading.Furthermore, Figure 4(b) in which the time represents loading sequence illustrates the evolutions oftensile and compressive damages. This example clearly shows that multiaxial stress states may causetensile and compressive damages to occur simultaneously, which is usually interpreted as shear damagealthough compressive stresses do not reach the compressive strength limit [38, 42].

Figure 2. s1� e1 curves under three different confinements when (a) g = 0 and (b) g = 3.

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4. TIME INTEGRATION SCHEME WITH DAM–RESERVOIR INTERACTION

The Lagrangian–Eulerian formulation, in which displacement and pressure are unknown variables for damand reservoir, respectively, is utilized to construct the governing coupled equations for dam–reservoirsystem [43]. Herein, the fundamental equations of equilibrium are briefly outlined first for the dam andthe reservoir separately. Afterward, the handling of the coupled governing equations is presented by theHilber–Hughes–Taylor (HHT) time integration scheme in an incremental–iterative solution strategy.

4.1. Coupled dam–reservoir equations in a nonlinear context

The equations of motion may be written in vector-matrix notation as follows for the discretized damunder specified free-field ground acceleration:

M €U þ P u; _uð Þ ¼ Rst þQT p ; €U ¼ €u þ J ag (13)

where M is the mass matrix of dam body; u, _u and €u represent the nodal vectors of the displacement,velocity, and acceleration relative to the ground motion, respectively; €U is the vector of absolute nodal

Figure 3. Shear stress versus shear strain for the shear panel under cyclic loading: (a) with different size steps(20, 40, and 80 steps) and (b) effects of rate dependency.

Figure 4. Shear panel test under several loading loops: (a) shear stress vs. shear strain (b) evolutions oftensile and compressive damage variables.

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acceleration; P denotes the restoring force vector; Q is often referred to as interaction or couplingmatrix; p represents the vector of nodal hydrodynamic pressures and Rst is the static loads vectordue to weight of the dam and hydrostatic pressure applied to upstream face of the dam; agdenotes the vector of ground accelerations; and J is a matrix with each of the three rows equalto a 3� 3 identity matrix.

It is noted that in the current implementation, the element’s restoring force vector, Pe, is given by theintegral of the total stress including damping stress over the element volume as

Pe ¼Z

VeBT sþ xð Þ dVe (14)

where B is the strain–displacement transformation matrix. In this condition, the stiffness and dampingmatrices of an element (i.e., Ke= @ Pe/@ ue and Ce ¼ @Pe=@ _ue) are derived as

Ke ¼Z

VeBT Eats B dVe ; Ce ¼ bR

ZVeBT 1� Dð ÞE0 B dVe (15)

where Eats = @ (s+x)/@ « denotes the modified algorithmic tangent stiffness with damping. The visco-elastic damping stress is included in Eats to keep the convergence rate rather unaffected by rapidchanges of damping forces [32]. The tangent operator, which is consistent with the stress updatingalgorithm, is derived from the linearized constitutive relations around the given state. Consideringlarge cracking possibilities and visco-plastic regularization, the derivation details of the algorithmictangent stiffness have been given in [39]. Furthermore, according to El-Aidi and Hall [15], themass-proportional term for the damping matrix has been omitted, because it would provide someartificial numerical stability during time-marching process.

Furthermore, one can apply Galerkin method to the wave equation and impose related boundaryconditions to obtain the following relation for the discretized reservoir domain [43]:

G €p þ L _pþH p ¼ �r Q €U (16)

where G,L and H are called generalized mass, damping, and stiffness matrices, respectively. Toconstruct the coupled equations set of the system consistently, the governing equations for the damand reservoir are restated as the following, respectively:

M €U þ F u; _u; pð Þ ¼ Rst ; F u; _u; pð Þ ¼ P u; _uð Þ �QT p (17)

G €p þ F0p; _pð Þ ¼ �r Q €U ; F

0p; _pð Þ ¼ L _pþH p (18)

Now, one may set up the following coupled equations:

M 0

r Q G

�€u

€p

�þ F

F0

�¼ R

R0

�(19)

where R =Rst�M J ag and R0 =�rQ J ag. With consideringu, _u and €u as total displacement, velocityand acceleration vectors with the following definitions:

u ¼ u

p

�; _u ¼ _u

_p

�; €u ¼ €u

€p

�(20)

and denoting M as the total mass matrix, Equation (19) is written in a more compact form:

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M €uþ F ¼ R; M ¼ M 0

r Q G

�; F ¼ F

F0

�; R ¼ R

R0

�(21)

4.2. Iterative solution strategy with the Hilber–Hughes–Taylor time integration scheme

Numerical damping associated with the time-stepping schemes used for integrating second-ordersystems of equations over time stabilizes the numerical integration by damping out the unwantedhigh-frequency modes. For the Newmark method, numerical damping also affects the lower modesand reduces the accuracy of integration scheme from second order to first order. For the HHTmethod, numerical damping affects only the higher modes and always maintains second-orderaccuracy. Larger numerical damping values are usually necessary for problems involving rigid bodyrotational motion and dynamic contact/impact [32].

Utilizing the HHT integration scheme, the discrete form of the governing equilibrium equation,Equation (21), is written at the time step n+ 1 as

M €unþ1 þ F unþ’; _unþ’

� �� Rnþ’ ¼ 0 (22)

where ’= 1 + a and a are the free parameter controlling the amount of the numerical dissipation,which is increased by decreasing a. The recommended range for a is [�1/3, 0] to achieveunconditional stability, second-order accuracy, and good high-frequency dissipativecharacteristics. Furthermore, on the basis of the generalized midpoint scheme, the interpolatedquantity is defined as

�ð Þnþ’ ¼ 1� ’ð Þ �ð Þn þ ’ �ð Þnþ1 (23)

It is noted that this interpolation scheme is only used for the equations of motion (i.e.,unþ’ and _unþ’)and the applied force, Rnþ’ . Because employing the generalized midpoint scheme in the stresscomputation does not have robust numerical performance [44], the stresses needed for the restoringforce, Pnþ’, are integrated on the basis of the shifted backward-Euler scheme [38], in which theinternal variables are directly computed at tnþ’ rather than tnþ1. The shifted scheme becomes the

original method in [tnþ’� 1,tnþ’], for which the given variable set «nþ’; «pnþ’�1;knþ’�1

n ois

utilized to find the updated variable set snþ’; «pnþ’;knþ’

� :

Because Equation (22) is nonlinear with respect to u , a global iteration scheme is employed toupdate it by defining the residual of Ψ at the time step n + 1 as

Ψnþ1 ¼ Μ €unþ1 þ Fnþ’ � 1� ’ð Þ Rn � ’ Rnþ1 (24)

By applying the Taylor’s expansion to this residual and eliminating the higher-order terms, the

equation @Ψ=@u� � ið Þ

nþ1du ¼ �Ψið Þnþ1 is established for the ith iteration in which @Ψ=@u is the

Jacobian of the residual Ψ with respect to u . The vector of du is subsequently utilized to update

unþ1 (i.e., uiþ1ð Þnþ1 ¼ u ið Þ

nþ1 þ du). From Equation (24), the Jacobian is derived as

@Ψ@u

�nþ1

¼ M@€unþ1

@unþ1þ @Fnþ’

@ _unþ’

@ _unþ’

@ _unþ1

@ _unþ1

@unþ1þ @Fnþ’

@unþ’

@unþ’

@unþ1

! (25)

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Furthermore, according to the Newmark formulation, €u and _u are defined as

€unþ1 ¼ a0 unþ1 � unð Þ � a2 _un � a3€un; _unþ1 ¼ a1 unþ1 � unð Þ � a4 _un � a5€un (26)

in which, a0 through a5 are the coefficients of the Newmark approach. By using the above relationsalong with Equation (23), the Jacobian matrix yields

@Ψ=@u� �

nþ1 ¼ a0Mþ ’ a1Cnþ’ þKnþ’

� �(27)

where C and K are called total damping and stiffness matrices, respectively as

C ¼ C 0

0 L

�; K ¼ K �QT

0 H

!(28)

After obtaining du, €u and _u are updated at the end of the iteration by

€uiþ1ð Þnþ1 ¼ €u

ið Þnþ1 þ a0du ; _u

iþ1ð Þnþ1 ¼ _u

ið Þnþ1 þ a1du (29)

Moreover, the initial values for these vectors at the beginning of a time step would be as

€u0ð Þnþ1 ¼ �a2 _un � a3€un; _u

0ð Þnþ1 ¼ �a4 _un � a5€un; u 0ð Þ

nþ1 ¼ un (30)

Finally, the HHT time integration algorithm is summarized in Table I. It is worthwhile to mentionthat under certain circumstances, it is possible to utilize the pseudo-symmetric technique relying onsymmetric matrices for efficiency purposes [33, 43, 45].

Table I. The HHT time integration algorithm with dam–reservoir interaction.

0. i ¼ 0; agnþ1;€u

0ð Þnþ1 ¼ �a2 _un � a3€un; _u

0ð Þnþ1 ¼ �a4 _un � a5€un and u

0ð Þnþ1 ¼ un:

1. Rnþ1 ¼ Rst �M J agnþ1, R0nþ1 ¼ �r Q J agnþ1 and Rnþ1 ¼

Rnþ1

R0nþ1

�.

2. u ið Þnþ’ ¼ 1� ’ð Þ un þ ’ u ið Þ

nþ1.

3. «nþ’ ¼ B ueð Þ ið Þnþ’, then obtain snþ’ and Dnþ’ on the basis of the plastic–damage model.

4. _uið Þnþ’ ¼ 1� ’ð Þ _un þ ’ _u

ið Þnþ1.

5. _«nþ’ ¼ B _ueð Þ ið Þnþ’, then compute xnþ’ ¼ bR 1� Dnþ’

� �E0 _«nþ’.

6. P ið Þnþ’ ¼

Xnel

ZVeBT snþ’ þ xnþ’

� dVe .

7. F ið Þnþ’ ¼ P ið Þ

nþ’ �QT p ið Þnþ’, F

0 ið Þnþ’ ¼ L _p ið Þ

nþ’ þH p ið Þnþ’ and F

ið Þnþ’ ¼

F ið Þnþ’

F0 ið Þnþ’

0@ 1A.

8. Ψið Þnþ1 ¼ Μ €u

ið Þnþ1 þ F

ið Þnþ’ � 1� ’ð Þ Rn � ’ Rnþ1.

9. If Ψið Þnþ1

��� ���≤TolG, then exit.

10. @Ψ=@u� � ið Þ

nþ1¼ a0Mþ ’ a1Cið Þnþ’ þK

ið Þnþ’

� .

11. Solve @Ψ=@u� � ið Þ

nþ1du ¼ �Ψið Þnþ1 for du.

12. €uiþ1ð Þnþ1 ¼ €u

ið Þnþ1 þ a0du; _u

iþ1ð Þnþ1 ¼ _u

ið Þnþ1 þ a1du and u iþ1ð Þ

nþ1 ¼ u ið Þnþ1 þ du.

13. i= i+ 1, then go to Step 2.

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4.3. Verification of the dam–water interaction modeling

By utilizing the derived time integration algorithm implemented in SNACS program, a linear dynamicanalysis of the Pine Flat dam is carried out, and the response is compared with the correspondingresults reported by Chopra et al. [46]. It is clear that in a linear analysis, a nonlinear iterativeprocedure can be utilized. However, the system’s equations converge in the first iteration of eachstep. The 3-D analysis is performed in a state of plane stress over the tallest monolith of the dam.The idealization of 20-node isoparametric elements utilized to discretize solid and fluid domains isdepicted in Figure 5(a). The truncation length of the reservoir was checked to be adequate for theconsidered horizontal excitation depicted in Figure 5(b).

Rayleigh damping coefficients are determined such that damping would be equivalent to 5% ofcritical damping for frequencies close to the first and third vibration modes. The concrete propertiesare E0 = 22.75 GPa, n = 0.2 and unit weight of 24.8 kN/m3. Moreover, the bottom of the reservoir isassumed to be completely reflective, and pressure wave velocity of water is 1440m/s. Prior to theseismic loading, the dam is subjected to gravity and hydrostatic loads.

The crest stream displacement history is compared with the corresponding result of Chopra et al.[46] in Figure 6(a). It should be mentioned that the employed approach in the work of Chopra et al. isbased on infinite elements for the reservoir discretization in a frequency domain analysis. Theenvelopes of maximum tensile stresses are also compared in Figure 6(b) between the twomentioned analyses. The comparison reveals that the accuracy of the present implementation iswell enough for the solution of the coupled governing equations although it is being carried out intime domain.

Figure 5. Pine Flat dam: (a) mesh and (b) the S69E component of Taft earthquake.

Figure 6. SNACS’s results compared with the responses obtained by Chopra et al. [46]: (a) time history ofcrest stream displacement and (b) envelope of maximum tensile stresses.

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5. FRACTURE RESPONSE OF THE KOYNA DAM IN 3-D MODELING

This section focuses on the seismic fracture response of the Koyna dam, which has beenbroadly examined by many investigators as the main validation of their proposed concretemodels [16, 20, 32, 36, 47]. The Koyna dam is a 103-m high concrete gravity dam located in Indiaand constructed in 1963. It was subjected to the 1967 earthquake of magnitude 6.5 and was severelydamaged [48]. Its scaled model was also experimentally tested in 1988 [49].

5.1. Model description and loading

The simulation is based on the 3-D modeling of a slice of the tallest monolith depicted in Figure 7(a).The cracked configuration experimentally observed by a shaking table test is also shown in thisfigure [49]. The eight-node isoparametric solid and fluid elements having two integration points ineach direction are utilized to discretize the dam and reservoir domains, respectively. As illustrated inFigure 7(b), two meshes are employed in the simulation to examine mesh objectivity. The materialproperties for the PD model are E0 = 30.0 GPa, n=0.2, ft0 =2.9 MPa, fc0 =� 24.1 MPa, Gt = 200 N/m,Gc = 20000 N/m and m=0.15 Δt. Moreover, Δt=0.005 sec and a=� 0.3 are selected as the analysisparameters. The thickness of the slice is 1m, and the dam body is assumed to be in a state of planestress. The length of the reservoir simulated in the model is 2.5 times of the reservoir height, whereasthe reservoir bottom is assumed to be partially reflective with the wave reflection coefficient equal to0.85. The unit weights of concrete and water are assumed as 25.8 and 9.81 kN/m3, respectively. Thestiffness proportional damping factor is computed such that 3% damping is being captured for thefundamental vibration period.

At the first step of loading, the dam is subjected to gravity load, and then, the hydrostatic pressure isapplied to the upstream face. The static loads are applied at negative range of time in 10 incrementseach before the earthquake excitation is employed. Afterward, 10 sec of the horizontal and verticalcomponents of the 1967 earthquake records [32] are applied to the base of the dam and reservoirstarting at time 0. The time integration is performed for 15 sec of which the last 5 sec is free vibration.

5.2. Plastic–damage analysis

The analysis commences with applying static loads while the dam remains elastic with no damage atthe end of this step. Damage to the dam initiates during the earthquake loading. Figure 8(a) showsthe stream and vertical displacements at the left corner of the crest relative to the ground motion,whereas positive horizontal displacement is in the downstream direction. Furthermore, the contourplots of tensile damage, compressive damage, large-crack damage, and degradation variables at theend of the analysis are illustrated in Figure 8(b). These are denoted as Dt, Dc, Dcr and D,respectively. The resulting crack profile is divided into three major parts. The rigid foundationcauses the stress concentration to occur near the base, and consequently, a crack propagates into thedam along the lowest layer of the elements. As observed in Figure 8(b), a crucial damage has

Figure 7. Koyna gravity dam: (a) geometry (unit, meter) and (b) coarse and fine meshes.

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occurred near the downstream change of slope where the localized band of damaged elements isformed. Curving down due to the rocking motion of the upper block, this downstream crack spreadstoward the upstream direction. Furthermore, a horizontal crack propagating toward the curved part ofthe downstream crack is formed during a major deformation. As noticed, the damage has very welllocalized on just one layer of the elements along the upstream face. The compressive damage variable,Dc, depicted in Figure 8(b) shows that a few number of the damaged elements experience shear damage.

Contour of the large-crack damage variable, Dcr, shown in Figure 8(b) reveals that most of thedamaged elements also undergo large cracking phenomenon. It is worthwhile to mention that allcracks have occurred in the body by the time 4.410 sec, which corresponds to the second majorexcursions of the crest in the upstream direction. Oscillating back and forth during the remainder ofthe analysis including the free vibration, the upper portion of the dam remains stable while theupstream and downstream cracks alternatively close and open because of rigid body rotation ofthe upper block. It is simply observed from the presented displacement histories. The distribution ofthe stiffness degradation variable, D, in Figure 8(b) indicates that most of the cracks close undercompressive stresses at the end of the analysis. The well-localized damage pattern predicted bySNACS is consistent with those numerically captured by other investigators [20, 47] as well as thecracked zone obtained in the scaled experimental test [49].

To examine capability of the implemented model on mesh objectivity concern, the authors alsopresent results of the fine mesh introduced in Figure 7(b) herein. The stream displacements at thedam crest for the fine mesh are compared with the corresponding results of the coarse mesh inFigure 9(a). This demonstrates reasonably objective results for the two meshes. Although 3-Dmodels could have relatively higher level of mesh sensitivity because of the effect of possiblecracking in the out-of-plane direction, the characteristic length along with the visco-plasticregularization provides a good mesh objective result in this model. Besides, the tensile damagepattern depicted in Figure 9(b) for the fine mesh at the end of the analysis is fairly similar to thedamaged zone of the coarse mesh, although as expected, the crack bandwidth is smaller. Theinclined part of the cracking has mostly two element widths, which is inevitable when the alignmentof the elements is not coincident with the crack path.

Figure 8. Seismic results of Koyna dam with the coarse mesh: (a) time histories of the crest stream andvertical displacements and (b) damage and degradation variables at the end of the analysis.

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6. APPLICATION ON CONCRETE ARCH DAMS

In this section, the seismic nonlinear behavior of Shahid Rajaee arch dam analyzed by SNACS is discussed.The dam is 130-m high, with the crest length of 420m, and it is located in the north of Iran in the seismicallyactive foothills of Alborz Mountains, near the city of Sari [17–19]. Two analysis cases are presented herein:a linear one mainly used for comparative purposes and a nonlinear case by the PD model.

6.1. Mesh, model parameters and loading

The dam body and its reservoir are discretized by the isoparametric 20-node elements having threeintegration points in each direction. A relatively fine mesh employed in the analyses and two keyGauss points monitored are depicted in Figure 10. The reservoir length considered in the simulationis 2.75 times of the reservoir height. The finite element mesh consists of 480 solid elements and 714fluid elements for the dam body and the reservoir domain, respectively.

As it is clear in this case, with knowing that cross-canyon excitation is neglected, simulation mightbe performed on one half of the dam–reservoir system by applying symmetry condition in themidplane. However, this was disregarded, and the whole domain is discretized. Therefore, byneglecting the cross-canyon excitation, symmetric results are expected throughout the analyses, andthis fact is used as an additional check for the accuracy of the obtained results.

Dam–reservoir interaction is included in the analyses based on the technique discussed in Section 4.Meanwhile, the foundation is taken as rigid. This latter idealization was decided to reduce computationalefforts. Of course, it is evident that the assumption of rigid foundation would cause large tensile stressesnear the boundaries for the linear analysis. However, for the PD analysis, these high tensile stresses areexpected to release, and this can be considered as a challenging test for the nonlinear model.

Material properties for the PD model are E0 = 30.0GPa, n = 0.18, ft0 = 3.0MPa, fc0 =� 25.0MPa,Gt = 300N/m, Gc = 45000N/m, m= 0.15 Δt and unit weight of 24.0 kN/m3. Moreover, the analysisparameters are selected as Δt= 0.005 sec and a =� 0.2. In the analyses carried out, the Rayleighstiffness proportional damping is applied and the damping coefficient, bR, is determined such thatequivalent damping for the fundamental frequency of vibration would be 8% of the critical

Figure 10. Finite element mesh of the Shahid Rajaee arch dam and the two Gauss points monitored.

Figure 9. Seismic response of Koyna dam with the fine mesh: (a) crest displacement history of the fine meshcompared with the coarse mesh’s result and (b) tensile damage, Dt, at the end of the analysis.

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damping. The reservoir bottom is partially reflective with the wave reflection coefficient equal to 0.9.The dead weight and hydrostatic pressures at upstream face are applied as the static loading at negativetime. Maximum water depth is 122m. The earthquake excitations applied at time 0 include twocomponents of the Friuli–Tolmezzo earthquake (as mentioned, the cross canyon is neglected to havea symmetric condition) whose records are normalized on the basis of the frequency content formaximum design earthquake condition with the peak ground acceleration of 0.56 g [18].

6.2. Analysis results

For the linear case, it is observed that maximum tensile stresses for the spillway and abutment regionsreach to maximum values of 8.06 and 24.84MPa, respectively (Figure 11). These high tensile stressesare predicted to be limited to tensile strength of concrete in the PD analysis. The maximum tensilestresses of these zones occur in the arch direction and perpendicular to the abutments as expected.The maximum compressive stresses of this case are �14.43 and �11.65MPa for the spillway andabutment regions, respectively (Figure 12). Although they are less than the compressive strength, itis shown in the nonlinear analysis that compressive damages would occur because of shear damages.

In Figure 13(a) and (b), displacement histories of the mid-crest point for the PD case are compared withthose of the linear case in the stream and vertical directions, respectively. The analyses did not expose largedisplacements in the dam under the severe earthquake excitations considered, giving an initial sign of a safedesign of the dam on the basis of the adopted theories and assumptions. Comparison between the results ofthe PD model and those of the linear case shows a drift in the crest displacements in upward and upstreamdirections. This is the main feature of displacement histories in the PD approach, which is mostly due toextensive cracking of downstream face of the dam beneath the spillway block. When most of thesecracks are open, they create a cumulative plastic strain as the major source of the drift.

The envelope of maximum tensile stresses throughout the PD analysis is depicted in Figure 14. Inthe linear analysis, it was noticed that very high tensile stresses were computed at the base of thedam because of the rigid foundation assumption. In addition, high tensile stresses occurred in thespillway region in arch direction, which in reality are released with the opening of contraction joints. Itis observed that the PD model has completely bounded the amount of tensile stresses to a value closeto the tensile strength of concrete. In this regard, slight overstressing of tensile stresses (about 1.7%) isstill noticed in the dam body for this case, because of three-dimensional stress states’ effects.Furthermore, the tensile damages obtained for different regions of the dam body at the end of the

Figure 11. Envelope of maximum tensile stresses (MPa) for the linear analysis.

Figure 12. Envelope of maximum compressive stresses (MPa) for the linear analysis.

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analysis are illustrated in Figure 15. The regions undergoing tensile damage are well coincident with thehigh tensile zones depicted in Figure 14 as the envelope of maximum tensile stresses.

Time histories of the major stress components at the two key points being monitored are comparedbetween the LN and PD cases in Figure 16. As presented in Figure 16(a), the linear response showsvery high values for sz at the base of the central cantilever, whereas it is released during the staticloading in the PD case. The arch stress component as illustrated in Figure 16(b) is limited to thetensile strength during the seismic loading while linear analysis estimates a value of 8.06MPa.

The envelope of maximum compressive stresses is captured in Figure 17. In comparison with thelinear case, the maximum compression has increased about 26.1% and reached to �18.19MPa inthe upstream face near the spillway region. Because the high compressive stress values are stillbelow the considered compressive strength, it is not expected to have compressive damages due tocrushing. Although the maximum value of the stress components is less than that of the compressivestrength, it cannot be concluded that no compressive damage is occurred in the dam body. This ismainly due to the governing equation for damage evolution in the implemented PD model. In fact,because all three principal components of stress contribute in the damage evolution, it is expected

Figure 14. Envelope of maximum tensile stresses (MPa) for the plastic–damage analysis.

Figure 15. Tensile damage, Dt, at the end of the plastic–damage analysis.

Figure 13. Comparison of displacements at mid-crest point between the linear (LN) and plastic–damage(PD) models in (a) stream direction and (b) vertical direction.

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that the compressive damage evolves when the tensile damage is occurring and at least one of theprincipal stresses is negative.

As indicated in Section 3.3, the state of damage when both tensile and compressive damages occursimultaneously is commonly interpreted as shear damage. This phenomenon was demonstrated in theshear panel test previously discussed. The distribution of compressive damage variable at the end ofthe analysis is depicted in Figure 18. This figure together with the tensile damage pattern, whichactually shows the regions that both damage variables are arising simultaneously, illustrates theextent of shear damage occurring in these zones of the dam body because of multiaxial stressstates governed in such mass concrete structures. The compressive damage occurred at the zonesin which tensile damage is also observed highlights that most part of the damaged area in archdams sustains shear damages, which affects the safety margin with respect to the cases in whichthe tensile damage occurs alone.

Besides, the large-crack damage variable, Dcr, computed for the points undergoing large cracking ispresented in Figure 19. As observed, an extensive part of the damaged regions sustain large cracking.This emphasizes the necessity of such a modification in simulation of cyclic loaded concrete subjectedto large tensile strains while this class of PD models is utilized.

Figure 16. Comparison of stresses in the index points at the upstream face of the dam:(a) sz at the base(cantilever stress) and (b) sx at the crest (arch stress). LN, linear; PD, plastic–damage.

Figure 17. Envelope of maximum compressive stresses (MPa) for the plastic–damage analysis.

Figure 18. Compressive damage, Dc, at the end of the plastic–damage analysis.

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7. CONCLUSIONS

A special finite element program called SNACS is developed in three-dimensional space on the basisof a well-established PD model, which was successfully applied to gravity dams in 2-D plane stressstate, to investigate seismic nonlinear behavior of concrete arch dams. The model was initiallyproposed by Lubliner and coauthors in 1989 and later on modified by Lee and Fenves in 1998 tomake it applicable in cyclic loadings such as earthquake. The model accounts for the differentbehaviors in tension and compression. Tensile and compressive damages are independentlyrepresented by the concept of the fracture-energy-based multi-hardening variables. Effects ofdamage on elastic stiffness and its recovery during crack opening/closing are well simulated by asimple scalar degradation model. Although the damaged elastic stiffness is isotropic, the modeloffers the different evolutions of the tensile and compressive strengths and induced directionaldamage by plastic strains.

In this paper, an overview of the rate-dependent PD model was presented first, and then, the coupleddam–reservoir equations were integrated by the HHT scheme within an incremental–iterative solutionprocedure. By utilizing the damage-dependent damping mechanism, the varying restoring force vectoris computed in which the damping stress is consistently included. The developed program was initiallyexamined by several single-element tests. A full uniaxial cyclic test was carried out to emphasize thenecessity of large crack modification to properly simulate the cyclic behavior of plain concretesubject to large tensile strains. Furthermore, to show the performance of the model in reproducingthe softening and strength loss of concrete under multiaxial cyclic loading and the correspondingstiffness recovery during unloading and reloading, the shear panel example was studied. This testwell demonstrates that the model is capable of capturing shear damage represented by the twodistinct damages in tension and compression occurring simultaneously. Afterward, SNACS wasverified by investigating the seismic fracture response of the Koyna gravity dam as a well-knownbenchmark problem and by comparing the resulting crack pattern with the correspondingexperimental data. By employing two meshes for the Koyna dam, mesh objectivity of the currentimplementation of the model was well examined.

Finally, by utilizing the 3-D extended PD model, the Shahid Rajaee concrete arch dam wasanalyzed. The dynamic instability was not observed despite extensive damages occurring in the dambody. The conducted analysis included the dynamic effects of the reservoir and also the waveabsorption at the reservoir boundaries. It is confirmed that the PD model gives reasonable crackprofiles, which is expected from the contour of the maximum tensile stresses obtained from lineardynamic analysis. The results also reveal that the concrete arch dams can suffer significant damageduring a strong earthquake and still remain stable. The results are compared with the solutions fromtheir corresponding linear analysis, and it is shown that the structural responses of dams varyconsiderably because of damage evolution being captured by the PD model. The prominentcharacteristic of the crest displacement histories in the PD analysis is a drift in upward and upstreamdirections, which is mostly due to a cumulative plastic strain caused by extensive cracking ofdownstream face of the dam beneath the spillway block when most of these cracks are open.

Because multiaxial stress state leads to shear damages represented with two distinct damages occurringsimultaneously, it is important to carefully evaluate the damaged regions observed in an earthquakedamage analysis of an arch dam because the concrete may in certain cases partially lose its strength incompression while it has fully damaged in tension. Although it is less important for gravity dams, shear

Figure 19. Large-crack damage variable, Dcr, at the end of the plastic–damage analysis.

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damages causing the softening and strength loss in compression for the damaged regions under multiaxialloadings affect the seismic safety of an arch dam and need to be carefully captured.

Future work includes seismic damage-cracking analysis of a concrete arch dam while preexistingjoints are also simulated using discrete crack elements. Under those circumstances, it will bepossible to evaluate the seismic safety of concrete arch dams in a more realistic manner. Thecomprehensive simulation including joint nonlinearity may need a mesh refinement to predict anysort of collapse mechanism.

ACKNOWLEDGEMENTS

The supports of Iran’s National Elites Foundation and Ministry of Science, Research, and Technology of Iranare greatly appreciated. The writers also thank Professor Somasundaram Valliappan for insightful discussionsduring the first author’s visit to the University of New South Wales, Australia.

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