The Influence of Initial Cracks on the Crack Propagation Process of Concrete Gravity Dam-Reservoir-Foundation Systems

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1 See discussions, stats, and author profiles for this publication at: The Influence of Initial Cracks on the Crack Propagation Process of Concrete Gravity Dam-Reservoir-Foundation Systems Article in Journal of Earthquake Engineering August 2015 DOI: / CITATION 1 READS authors: Gaohui Wang Wuhan University 44 PUBLICATIONS 318 CITATIONS SEE PROFILE Wenbo Lu Wuhan University 173 PUBLICATIONS 1,049 CITATIONS SEE PROFILE Chuangbing Zhou Nanchang University 304 PUBLICATIONS 3,320 CITATIONS SEE PROFILE Wei Zhou Wuhan University 188 PUBLICATIONS 1,034 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Micromechanics of granular materials and Simulation View project Fluid coupled-dem View project All content following this page was uploaded by Gaohui Wang on 20 November The user has requested enhancement of the downloaded file.

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4 Journal of Earthquake Engineering, 19: , 2015 Copyright A. S. Elnashai ISSN: print / X online DOI: / The Influence of Initial Cracks on the Crack Propagation Process of Concrete Gravity Dam-Reservoir-Foundation Systems GAOHUI WANG 1, WENBO LU 1, CHUANGBING ZHOU 2, and WEI ZHOU 1 1 State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, China 2 Nanchang University, Nanchang, China A scaled-down 1:40 model of a gravity dam with the initial notch in the upstream wall is analyzed using the XFEM. Furthermore, the cracking process of Koyna dam including dam-reservoir-foundation interaction under the 1967 Koyna earthquake is also simulated numerically by employing the XFEM. The computed distribution of cracking damage is consistent with other numerical methods and the model experiment. Subsequently, a typical concrete gravity dam with different crack cases is employed as a numerical example. The effects of the initial crack position and length on the crack propagation and seismic response of dam-reservoir-foundation systems are studied. Keywords Concrete Gravity Dams; Initial Cracks; XFEM (Extended Finite Element Method); Failure Mode; Dynamic Crack Propagation Process; Dam-Reservoir-Foundation Interaction 1. Introduction Due to the low tensile resistance of concrete, mass concrete dams, which are distinguished from other concrete structures because of their size and their interactions with the reservoir and foundation, are likely to experience cracking at their base or at the upstream and downstream faces caused by internal and external temperature variations, shrinkage of the concrete, differential foundation settlement, previous earthquakes or other reasons [Ingraffea, 1990; Feng et al., 1996]. Duringastrongseismicevent, theseinitialcracks with limited depth will actually propagate sufficiently deep into the body of the dam, which may considerably alter the seismic performance of the structure, and thereby endanger the safety of the installation. To predict the possible crack profiles and the corresponding seismic capacity of concrete gravity dams with initial cracks is of outmost necessity to ensure safe operations of the structures. Significant research efforts on studying the cracking behavior of concrete gravity dams have been made over the last decade [Ayari and Saouma, 1990; Bhattacharjee and Léger, 1994; Lee and Fenves, 1998; Calayir and Karaton, 2005; Zhong et al., 2011; Panet al., 2011; Omidi et al., 2013; Zhang et al., 2013a]. However, few attempts have been done to analyze the crack propagation process of concrete gravity dams with initial cracks in their base or at the upstream and downstream faces. The crack propagation analysis of two scaled-down 1:40 models of a gravity dam with an initial notch on the upstream surface Received 25 November 2014; accepted 16 February Address correspondence to Gaohui Wang, State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan , China. wanggaohui@whu.edu.cn Color versions of one or more of the figures in the article can be found online at 991

5 992 G. Wang et al. subjected to equivalent hydraulic and weight loading was first studied by Carpinteri et al. [1992]. Assuming multiple initial notches of different sizes at different locations along the upstream face of the model dams, Shi et al. [2003] used a well-quoted scale model of a concrete gravity dam to analyze multiple discrete cracks in concrete and obtain various kinds of cracking behaviors. Barpi and Valente [2000] employed the cohesive crack model to investigate the behavior of a concrete gravity dam of 103 m high with an initial crack in the upstream face. Their results showed that the initial notch in the upstream face served as the starting point of a crack that propagated toward the foundation during the loading process. Bolzon [2004] compared the merits of Linear Elastic Fracture Mechanics (LEFM) and cohesive crack approach on the evaluation of safety against ultimate failure of large concrete gravity dams with an initial notch at the base. Batta and Pekau [1996] extended the two-dimensional boundary element procedure for analyzing the propagation of a single discrete crack to simultaneous multiple cracking in concrete gravity dams. Bhattacharjee et al. [1994] considered both the coaxial rotating crack model (CRCM) and the fixed crack model with a variable shear resistance factor (FCM-VSRF) to study the nonlinear response of a model concrete gravity dam with an initial notch on the upstream surface and a full-scale concrete gravity dam with the pre-assigned imperfection located on the upstream side at the elevation of the downstream slope change. Their results showed that the ultimate response of the full-scale dam was not sensitive to the depth of initial imperfection placed on the upstream side. Oliver et al. [2002] presented the strong discontinuity approach to observe the fracture process of a reduced model of a concrete gravity dam with an initial notch on the upstream surface. Tinawi et al. [2000] conducted the shake table experiment and numerical simulation on four 3.4-m-high plain concrete gravity dam models with initial notch to study their dynamic cracking and evaluate the seismic safety. Zhang and his co-workers [2013b] presented the extended finite element method (XFEM) to analyze the seismic crack propagation of concrete gravity dams with initial cracks at the upstream and downstream faces. Pekau and Yuzhu [2004] used the distinct element method (DEM) to study the seismic behavior of the fractured Koyna dam during earthquakes. Their results showed that the safety of the dam was ensured if the crack shape was horizontal or upstream-sloped, and it was very dangerous if the crack slopes downstream. Javanmardi et al. [2005] combined the discrete crack model with a theoretical model for uplift pressure variations along a cracked dam to study the seismic stability of concrete gravity dams. Wang et al. [2006] studied the stability of a gravity dam on jointed rock foundation and the seismic stability of the fractured Konya gravity dam using 3D-discontinuous deformation analysis (3D-DDA). Pekau et al. [2006] and Zhu et al. [2007] proposed a rigid model and a flexible FE model to study the seismic behavior of cracked concrete gravity dams. Mirzayee et al. [2011] proposed a hybrid distinct element-boundary element (DE-BE) approach for modeling the nonlinear seismic behavior of fractured concrete gravity dams considering dam-reservoir interaction effects. Shi et al. [2013] used the scaled boundary polygons coupled with an interface element to investigate the crack propagation in concrete gravity dams with a preset notch on the upstream face. Jiang et al. [2013] developed a dynamic contact model and a simplified reinforcing steel constitutive model to analyze the failure process of the cracked gravity dam with and without reinforcement. Cracking plays an important role in the concrete structural behavior, and the modeling of crack growth is a problem of great importance in the simulation of failure. In this paper, the extended finite element method (XFEM), which is based on the cohesive segments method in conjunction with the phantom node technique, is used to study the crack propagation and nonlinear fracture behavior of concrete gravity dams with initial cracks under earthquake conditions. The XFEM can be used to simulate crack initiation and propagation along an arbitrary path, since the crack propagation is not tied to the element boundaries in

6 Initial Cracks on the Crack Propagation Process 993 a mesh. A scaled-down 1:40 model of a gravity dam with an initial notch on the upstream wall is analyzed for accuracy verification. Furthermore, the cracking process of Koyna dam with considering the effects of dam-reservoir-foundation interaction under the 1967 Koyna earthquake is also simulated numerically by employing the XFEM. Lagrangian approach is used for the finite element modeling of dam-reservoir-foundation interaction problem. Subsequently, the seismic cracking analyses of a typical concrete gravity dam with initial cracks at different locations along the upstream and downstream faces are performed. The influence of the initial crack length on the crack propagation process of concrete gravity dams subjected to design earthquake is also discussed. The crack propagation process and failure modes of the dam with cracks are obtained. 2. Extended Finite Element Method for Dynamic Crack Analysis In the traditional formulation of the FEM, the existence of a crack is modelled by requiring the crack to follow element edges. In contrast, the crack geometry in the extended finite element method (XFEM), which was first introduced by Belytschko and Black [1999], need not be aligned with the element edges. The method provides flexibility and versatility in the numerical modeling of crack propagation, without requiring the finite element mesh to conform to the existence cracks and the remeshing for crack growth due to the approximation for a displacement vector function added to model the presence of a crack. In the XFEM, a crack is modeled by enriching the classical displacement-based finite element approximation with the framework of the partition of unity method (PUM), previously proposed by Melenk and Babuška [1996], which allows local enrichment functions to be easily incorporated into a finite element approximation. This enrichment functions typically consist of the near-tip asymptotic functions that capture the singularity around the crack tip and a discontinuous function that represents the jump in displacement across the crack surfaces. By implementing the generalized Heaviside function [Moës et al., 1999], the method was further enhanced, avoiding taking into account the complicated mapping for arbitrary curved cracks. The XFEM technique used to model crack initiation and propagation in concrete gravity dams is described in the following section The XFEM Approximation The XFEM enriches a standard displacement based finite element approximation with discontinuous functions. The approximation for a displacement vector function u with the partition of unity enrichment (Fig. 1) in the XFEM takes the form [Moës et al., 1999; ABAQUS, 2011]: u xfem (x) = u i N i (x) + b j N j (x) H (x) + i I j J }{{} only Heaviside nodes ( 4 ) N k (x) c l1 k F l1 (x) + ( 4 ), (1) N k (x) c l2 k F l2 (x) k K 1 l=1 k K 2 l=1 }{{} only crack tip nodes where x ={x, y} is the two-dimensional coordinate system, I is the set of all nodes in the mesh, N i (x) is the shape function associated with node i, u i are the classical degrees

7 994 G. Wang et al. FIGURE 1 Enriched nodes in the XFEM. of freedom for node i. J I is the set of nodes whose shape function support is cut by a crack, b j is the vector of corresponding additional degrees of freedom for modeling crack faces (not crack-tips). If the crack is aligned with the mesh, b j represent the opening of the crack, H(x) is the Heaviside function. K 1 I and K 2 I are the set of nodes whose shape function support contains the first and second crack tips in their influence domain, respectively. c l1 k and cl2 k are the vector of corresponding additional degrees of freedom which are related to the modeling of crack-tips, as the near-tip regions are enriched with four different crack functions. F l1 (x) and F l2 (x) are crack-tip enrichment function. If there is no enrichment, then the above equation reduces to the classical finite element approximation u fem (x) = i u in i (x). The first term on the right-hand side of the above equation (Eq. (1)) is applied to all the nodes while the second term is valid for nodes whose shape function support is cut by the crack interior, and the third (fourth) term is used only for nodes in which shape function support is cut by the crack tip Enrichment Functions To model the discontinuity in displacement field, the enrichment function H(x) which we refer to as a generalized Heaviside enrichment function is implemented in simulation of powder-die contact surface, in which the function H(x) takes on the value of +1 above the crack, and 1 below the crack. The function H(x)isgivenby H(x) = { 1 1 if(x x ).n 0, (2) otherwise where x is a sample gauss point, x (lies on the crack) is the closest point to x (Fig. 2), and n is the unit outward normal to the crack at x. Figure 2 illustrates the discontinuous jump function across the crack surface. In order to model the crack-tip and also to improve the representation of crack-tip fields, crack-tip enrichment functions are used in the element which contains the crack tip. For an isotropic material, the crack-tip enrichment functions F l (r, θ) which are also shown in Fig. 2 are given as

8 Initial Cracks on the Crack Propagation Process 995 FIGURE 2 Representation of normal and tangential coordinates for a smooth crack. [ r {F l (r,θ)} 4 l=1 = θ sin 2, r cos θ 2, r sin θ sin θ 2, r sin θ cos θ ], (3) 2 where (r, θ) is the local polar coordinate system with its origin at the crack tip, and θ=0is tangent to the crack at the tip. Note that the first function in (3) r sin θ / 2, is discontinuous across the crack plane whereas the last three functions are continuous. It bears emphasis that the near-tip discontinuity can be represented with other sets of functions, or even a single function which is discontinuous across the crack tip geometry. Multiple cracks can be treated in the above framework, by incorporating additional discontinuous and near-tip enrichment. The cohesive segments method [Remmers et al. 2008] in conjunction with phantom nodes proposed by Song [2006], has been used to model the damage onset and progression and simulate crack propagation. The XFEM based cohesive segments method in conjunction with phantom nodes can be used to simulate crack initiation and propagation along an arbitrary, solution-dependent path in the bulk materials for brittle or ductile fracture, since the segments are not restricted to being located along element boundaries, but can be located at arbitrary locations and in arbitrary directions, allowing for the resolution of complex crack patterns. In this case the near-tip asymptotic singularity is not needed, and only the displacement jump across a cracked element, which is described using the phantom node method, is considered. Therefore, the crack has to propagate across an entire element at a time to avoid the need to model the stress singularity. Phantom nodes which are superposed on the original real nodes, are utilized to represent the discontinuity of the cracked elements, as illustrated in Fig. 3 [Song et al. 2006]. Propagation of a crack along an arbitrary path is made possible by the use of phantom nodes that initially have exactly the same coordinates than the real nodes and that are completely constrained to the real nodes up to damage initiation. In an uncrack element, each phantom node is completely constrained to their corresponding real nodes (n 1 to n 4 ). But when crossed by a crack at Ɣ c, the element is partitioned into two subdomains, A and B. The discontinuity in the displacement is made possible by adding phantom nodes (n 1 to n 4 ) superimposed to the original nodes. The existing element is replaced by two sub-elements, referred to as element A and element B. Each sub-element is formed by a combination of some real nodes (the ones corresponding to the cracked part) and phantom nodes (the ones corresponding to the respective part of the original element). The two sub-elements are constituted by the nodes n 1, n 2, n 3, and n 4 (n A ) and n 1, n 2,n 3, and n 4 (n B ). Each phantom node and its corresponding real node are no longer tied together and can move apart. Both elements are only partially active, the active part of element A is A and the active part of element B is B. This is represented numerically in the definition of the displacement field: the displacement of a point with coordinates x is computed by

9 996 G. Wang et al. FIGURE 3 Representation of cracked elements by implementation of phantom node method. { u A (x, t) = u A j (t) N j (x), x A u B (x, t) = u B j (t) N j (x), x B. (4) The approximation of the displacement field is then given by: u (x, t) = u A j (t) N j (x) H ( f (x)) + u B j (t) N j (x) H (f (x)), (5) j n A }{{} j n B }{{} u A (x,t) u B (x,t) where n A and n B are the index sets of the nodes of superposed element A and element B, respectively; f (x) is the signed distance measured from the crack. The crack normal opening δ n and the tangential sliding δ t are shown in the following equation: { δn = n[u] δ t = [u] nδ n, (6) where [u] is the jump in the displacement given as [u] = j n A N j u A j j n B N j u B j. (7) In order to control the magnitude of separation, the cohesive law is defined. A separation occurs when the cohesive strength of the cracked element is zero, after which the phantom and the real nodes move independently. This method which provides an effective and attractive engineering approach has been used to simulate the initiation and growth of multiple cracks in solids by Remmers et al. [2008] and Song et al. [2006]. The detail of the flowchart for crack propagation simulation can be found in Shi et al. [2008] and Ye et al. [2012].

10 Initial Cracks on the Crack Propagation Process 997 In the XFEM, the mesh is not required to conform to the geometric discontinuities. Two signed distance functions per node are generally required to describe the crack location, including the location of crack tips, in a cracked geometry. The first signed distance function describes the crack surface, while the second is used to construct an orthogonal surface so that the intersection of the two surfaces gives the crack front. The first signed distance function is assigned only to nodes of elements intersected by the crack, while the second is assigned only to nodes of elements containing the crack tips. 3. Crack Propagation Analysis of a Scale Gravity Dam Model with Initial Notch based on XFEM A scaled model of a concrete gravity dam tested by Carpintieri et al. [1992] is analyzed using the XFEM. The model, which contains a horizontal notch of 15 cm on the upstream face located at a quarter of the dam height, is loaded with an equivalent hydraulic load in order to induce a curved crack that propagates from the tip of the notch towards the downstream face. Numerical simulations of this test were reported in several public actions using the FEM in combination with CCMs [Shi et al., 2003; Barpi and Valente, 2000]. Finite element model for the dam with the setup is shown in Fig. 4.The mesh of the dam is adequately refined at the lower portion of the dam, in which crack propagations are expected. The material properties of the model dam are: elasticity modulus E=35700 MPa, Poisson s ratio v=0.1, tensile strength σ u =3.6 MPa, and fracture energy G f =184 N/m. The density of the material is assumed to be 2400 kg/m 3. Following an unsuccessful experimental attempt to simulate the self-weight condition, in which an unstable failure occurred along the base of the model, the repaired model as the second model has been tested without any adjustment of the self-weight condition. In the present study, only the second model is analyzed, and the predicted response of the single crack is compared with the documented experimental and discrete crack analysis results of Carpinteri et al. [1992]. The crack mouth opening displacement (CMOD) at a rate of 1.2 μm/s is applied as the control parameter in the present nonlinear analyses performed using the XFEM. The hydraulic thrust was generated by means of a servo-controlled actuator with a 2000 kn capacity and applied to FIGURE 4 Dam model, Scale 1:40, notch depth of 15 cm (dimensions in cm).

11 998 G. Wang et al. Applied force (kn) Experimental result (Carpinteri, 1992) Cohesive crack model XFEM CMOD (mm) FIGURE 5 Total upstream face load vs. CMOD displacement. (a) Numerical results from the XFEM procedure (b) Previous research results [Carpinteri et al., 1992; Barpi and Valente, 2000] FIGURE 6 Numerical and experimental crack trajectories for concrete gravity dam. the upstream side. This force was distributed in four concentrated loads whose intensity is indicated in Fig. 5. This force is gradually increased until the failure of the dam. The relations between the total upstream face load and CMOD obtained from the XFEM, cohesive crack model [Barpi and Valente, 2000] and experimental results [Carpinteri et al. 1992] are shown in Fig. 5. As can be seen from Fig. 5 that these curves are close to each other. The predicted peak load of the numerical simulation based on the XFEM agrees well with the experiment results. Figure 6 shows the computed and experimental crack trajectories. It can be noted that the occurring crack profile shows a good agreement with the experimental results from the model test. By comparing the current results with previous research results [Carpinteri et al., 1992; Barpi and Valente, 2000], notice that the curved character of the crack trajectory is correctly captured by the XFEM, indicating that the XFEM procedure can predict effectively the crack propagation process in concrete gravity dams with initial cracks. 4. Seismic Crack Propagation Analysis of Koyna Gravity Dam-Dam-Reservoir-Foundation System The Koyna concrete gravity dam, which is one of a few concrete dams that have experienced a damaging earthquake, is selected for numerical application. This problem has been extensively analyzed by a number of investigators. In this article, the dynamic response analyses of Koyna dam are performed by using the XFEM. The time histories of the Koyna earthquake are shown in Fig. 7. The finite element model of the dam-reservoir-foundation

12 Initial Cracks on the Crack Propagation Process Horizontal component 0.4 Vertical component Acceleration (g) Acceleration (g) Time (s) (a) Horizontal component Time (s) (b) Vertical component FIGURE 7 Koyna earthquake on December 11, 1967: (a) horizontal component and (b) vertical component. (a) Dam-reservoir-foundation system (b) Concrete gravity dam FIGURE 8 Finite element discretization for the dam- reservoir-foundation system of Koyna dam: (a) dam-reservoir-foundation system and (b) concrete gravity dam. interaction system is given in Fig. 8a, and the dimensions of the dam are given in Fig. 8b. The material parameters for the Koyna dam concrete are as follows: the elasticity modulus E= MPa, the Poisson s ratio ν=0.2, the mass density ρ=2643 kg/m 3, the fracture energy is 250 N/m. The tensile and compressive strength of the dam are 2.9 and 24.1 MPa, respectively. The foundation rock is assumed to be linearly elastic. In order to avoid reflection of the outgoing waves, the foundation rock is assumed to be massless. The elasticity modulus and Poisson s ratio of the foundation rock are taken as 21.6 GPa and 0.20, respectively. The fluid is assumed to be linearly elastic, inviscid, and irrotational. The bulk modulus and mass density of the fluid are taken as 2.07 GPa and 1000 kg/m 3. A dynamic magnification factor of 1.2 is considered for the tensile strength to account for strain rate effects. The energy dissipation of the dam and foundation is considered by the Rayleigh damping method with 5% damping ratio. The maximum reservoir water level of 96.5 m is considered. Applied loads include self-weight of the dam, hydrostatic, uplift, hydrodynamic, and earthquake forces. The static solutions of the dam due to its gravity loads and hydrostatic loads are taken as initial conditions in the dynamic analyses of the system. It should be noted that the seismic water pressure effects inside the cracks are not considered in the analysis. Further studies of the effect of seismic water pressure on the crack propagation and dynamic response of the dam are deemed necessary. The formulation of the fluid system based on Lagrangian approach is given according to Wilson and Khalvati [1983] and Calayir and Dumanoğlu [1993]. In this approach,

13 1000 G. Wang et al. (a) Numerical results from the XFEM procedure (b) Experimental results from the model test [Mridha and Maity, 2014] FIGURE 9 The final failure mode of Koyna dam under the 1967 earthquake. (a) numerical results and (b) experimental results (Mridha and Maity 2014). displacements are selected as the variables in both fluid and structure domains. Fluid is assumed to be linearly elastic, inviscid and irrotational. Some details of Lagrangian formulation for dynamic interaction of dam-reservoir-foundation systems can be found in Zhang and Wang [2013c]. Dynamic crack propagation analysis of Koyna gravity dam under the 1967 earthquake is conducted employing the XFEM based cohesive segments method made in ABAQUS program. The integration time step used in the analysis is 0.01 s. The response of cohesive behavior in the enriched elements in the model is specified. The maximum principal stress failure criterion is selected for damage initiation. Figure 9 shows the final crack profile of Koyna dam under the 1967 Koyna earthquake obtained from the numerical simulation and the model test [Mridha and Maity, 2014], it can be noted that the occurring crack profile match reasonably well with the experimental results from the model test. By comparing the current results with the Koyna dam prototype observation [Chopra and Chakrabarti, 1973], the model test [Mridha and Maity, 2014], and previous research results [Calayir and Karaton, 2005; Pan et al., 2011; Das and Cleary, 2013; Omidi et al., 2013], it may be concluded that the XFEM procedure can predict effectively the crack propagation process in concrete gravity dams under seismic conditions. 5. Seismic Crack Propagation Analysis of Concrete Gravity Dam-Reservoir-Foundation Systems with Initial Cracks 5.1. FEM Model and Material Properties The Guandi gravity dam is currently under construction in Southwest China. The dam is located in a strong earthquake region with design peak ground acceleration (PGA) of 0.34 g. A typical non overflow monolith of the dam, which is 142 m high with a 138 m deep reservoir, is employed to model its seismic damage process. Four-node plane strain quadrilateral isoparametric elements with 2 2 Gauss integration are utilized to discretize

14 Initial Cracks on the Crack Propagation Process 1001 (a) Dam-reservoir-foundation system (b) Dam FIGURE 10 Finite element meshes of the dam-reservoir-foundation system. TABLE 1 Material properties of the dam Concrete Modulus (GPa) Poisson s ratio Density (kg/m 3 ) Compressive strength (MPa) Tensile strength (MPa) Fracture energy (N/m) C C C the rock foundation and dam structure as shown in Fig. 10. Foundation corresponding to twice the dam height from the dam heel to upstream, from the dam toe to downstream and from the dam bottom downward is modeled with massless foundation. Since the presented research is aimed at the seismic damage process of concrete gravity dams, the nonlinearity of the foundation rock is not considered. In addition, Lagrangian approach is used for the finite element modeling of dam-reservoir-foundation interaction problem. The material parameters of the fluid are assumed as aforementioned. The energy dissipation of the monolith is considered by the Rayleigh damping method with 5% damping ratio. The traditional massless foundation approach is utilized herein to account for the dam-foundation interaction. Applied loads include self-weight of the dam, hydrostatic, uplift, hydrodynamic, and earthquake forces. The static solutions of the dam due to its gravity loads and hydrostatic loads are taken as initial conditions in the dynamic analyses of the system. Three indices of concrete are employed, i.e., C15, C20, and C25. The static material properties are listed in Table 1. The tensile strength is taken as 10% of its compressive counterpart. The elasticity modulus and Poisson s ratio of the foundation rock are taken as 21.6 GPa and 0.20, respectively. To account for the effect of strain rate, modulus and strength for the dam and modulus for the foundation rock are increased by 30% according to the Code for Seismic Design of Hydraulic Structures in China. Both stream and vertical directions are subjected to earthquake excitation.

15 1002 G. Wang et al The Influence of the Initial Crack Position In order to predict the locations of potential cracking, stress analysis of the intact Guandi gravity dam subjected to the recorded Koyna earthquake with the design peak ground acceleration (PGA) of 0.34 g as well as the static loads (concrete dead weight plus reservoir hydrostatic pressure) is first performed. Envelope of the resulting principal tensile stresses for the dam is presented in Fig. 11. It can be seen from that the zones of high tensile stress exceed the tensile strength of concrete are mainly obtained on the downstream surface (near the change in downstream slope) and the upstream surface (near the slope change at two heights of 8 m and 48 m above the base, the dam heel and the upper part of the upstream surface). On the slope change and the dam heel, due to the stress concentration, the stress singularity results in a computed high tensile stress almost three times the tensile strength of concrete. Based on the above, the Guandi concrete gravity dam with six different sets of small initial cracks from the pre-assigned imperfection, located on the upstream and downstream faces, is assumed, as shown in Fig. 12. The initial crack at the dam heel is also considered, but an initial crack modeled at this location did not propagate during seismic analysis. The seismic performance and crack propagation process of the cracked concrete gravity dams are investigated for each of these six models of cracking. In order to examine the influence of the initial crack position on the crack propagation trajectory, depths of the initial cracks are all assumed to be 0.4 m, which is introduced separately on the downstream and the upstream faces of the dam. Their corresponding XFEM discretizations are depicted in Fig. 10, and titled as Case I for an intact profile with no initial crack, Case II an upstreamsloped crack at the downstream face (Crack C1), Case III for the profile with a horizontal crack at the upstream face near the change in the slope of the face at a height of 8 m above the base (Crack C2), Case IV for a horizontal crack at the upstream face near the slope change at a height of 48 m above the base (Crack C3), and Cases V, VI, and VII for a horizontal crack on the upstream face at the a height of m, 110 m, and 95 m above the base (Crack C4, Crack C5, and Crack 6), respectively. A coefficient of friction of 0.7 is assumed for all the cases to consider the effective interlock in the cracks, and the cohesion coefficient of the crack is set at zero. FIGURE 11 Envelope of principal tensile stresses for seismic analysis without cracks.

16 Initial Cracks on the Crack Propagation Process 1003 FIGURE 12 Details of Koyna dam with different initial cracks. It is evident that the model for the actual cracking of the dam is more likely to be associated with crack C1, crack C2, and crack C3. This is because the changes in the slope of the downstream and the upstream faces provide a singular point for the first crack formation. However, the initial cracks are not solely stress-induced but may also arise due to a variety of other causes including shrinkage, temperature effects, etc., the additional three fracture models are included in the following examination of the fracture process of the dam. In order to investigate the influence of initial cracks on the seismic performance and crack propagation of concrete gravity dams, dynamic crack process and response analysis of the intact Guandi gravity dam under the 1967 earthquake with the design peak ground acceleration (PGA) of 0.34 g are first conducted employing the XFEM-based cohesive segments method. Figure 13 shows the crack propagation process of the intact dam (Case I) as predicted using the XFEM procedures. As shown, no penetrating crack appears in this case, and the failure mechanism is formed of two main damage zones, one near the dam heel and one at (a) Time A (t = 4.18 s) (b) Time B (t = 4.24 s) (c) Time C (t = 4.55 s) (d) Time D (t = 5.56 s) FIGURE 13 Crack propagation process of the dam with no initial crack (Case I) at four selected times. (a) Time A (t=4.18 s), (b) Time B (t=4.24 s), (c) Time C (t=4.55 s), and (d) Time D (t=5.56 s).

17 1004 G. Wang et al. the change in downstream slope. An initial crack in the dam is predicted to initially occur near the changes in the slope of the downstream face at 4.18 s. At this location, stresses are concentrated and the tensile stresses take large values. As the vibration characteristics, the crack extends deeper inside of the dam at approximately a 45 angle to the vertical. The crack trajectory curves down due to the compressive stresses resulting from rocking of the top block. Furthermore, an initial crack in the dam heel is observed at 4.24 s, which is probably due to stress concentration. It can be noted that the crack at the slope change stops expanding after 5.56 s, and the crack extends into the dam about 10 m. Seismic crack propagation analyses of the dam with initial cracks C1 C6 considered separately are presented in Figs , respectively. It can be seen from that the initial crack positions have significant influence on the seismic performance and crack propagation process of concrete gravity dams, which will cause more severe damage to the dam body than the intact dam profile. In some cases, cracks extend completely to the opposite face, penetrating the whole section of the dam and separating the crest from the upper part of the dam. Figure 14 shows the crack propagation process of the Guandi gravity dam with initial crack C1. As shown, the crack propagation process for the Case II is more or less similar to the Case I that the dam with no initial crack, as shown in Fig. 13. But smooth curvature discrete cracks penetrating the elements are obtained in the Case II. At t=3.42 s, the initial crack C1 near the changes in the slope of the downstream face is beginning to propagate almost perpendicular to the downstream surface. With the on-going acceleration excitation, the crack extends deeper inside of the dam. The crack trajectory curves down due to the (a) Time A (t = 3.42 s) (b) Time B (t = 4.24 s) (c) Time C (t = 4.60 s) (d) Time D (t = 4.63 s) FIGURE 14 Crack propagation process of the dam for single initial crack C1 (Case II) at four selected times. (a) Time A (t=3.42 s), (b) Time B (t=4.24 s), (c) Time C (t=4.60 s), and (d) Time D (t=4.63 s). (a) Time A (t = 4.26 s) (b) Time B (t = 4.61 s) (c) Time C (t = 4.67 s) (d) Time D (t = 5.00 s) FIGURE 15 Crack propagation four selected times. (a) Time A (t=4.26 s), (b) Time B (t=4.61 s), (c) Time C (t=4.67 s), and (d) Time D (t=5.00 s).

18 Initial Cracks on the Crack Propagation Process 1005 (a) Time A (t = 4.21s) (b) Time B (t = 4.62 s) (c) Time C (t = 4.66 s) (d) Time D (t = 4.87 s) FIGURE 16 Crack propagation process of the dam for single initial crack C3 (Case VI) at four selected times. (a) Time A (t=4.21 s), (b) Time B (t=4.62 s), (c) Time C (t=4.66 s), and (d) Time D (t=4.87 s). (a) Time A (t = 4.24 s) (b) Time B (t = 4.26 s) (c) Time C (t = 4.40 s) (d) Time D (t = 4.83 s) FIGURE 17 Crack propagation process of the dam for single initial crack C4 (Case V) at four selected times. (a) Time A (t=4.24 s), (b) Time B (t=4.26 s), (c) Time C (t=4.40 s), and (d) Time D (t=4.83 s). (a) Time A (t = 4.24 s) (b) Time B (t = 4.26 s) (c) Time C (t = 4.30 s) (d) Time D (t = 4.81 s) FIGURE 18 Crack propagation process of the dam for single initial crack C5 (Case VI) at four selected times. (a) Time A (t=4.24 s), (b) Time B (t=4.26 s), (c) Time C (t=4.30 s), and (d) Time D (t=4.81 s). compressive stresses resulting from rocking of the top block (Figs. 14a c). After a period of 1.18 s, i.e., at t=4.60 s, the downstream crack propagates about four-fifth through the width of the dam section (Fig. 14c). After the time instant t= 4.60 s, the crack propagates horizontally toward to upstream face. At t=4.48 s, the downstream crack extends completely to the upstream face at a height of m above the base, penetrating the whole section of the dam (Fig. 14d).

19 1006 G. Wang et al. (a) Time A (t = 4.24 s) (b) Time B (t = 4.26 s) (c) Time C (t = 4.30 s) (d) Time D (t = 4.68 s) FIGURE 19 Crack propagation process of the dam for single initial crack C6 (Case VII) at four selected times. (a) Time A (t=4.24 s), (b) Time B (t=4.26 s), (c) Time C (t=4.30 s), and (d) Time D (t=4.68 s). Dynamic crack propagation processes of the dam with initial cracks C2 and C3 are shown in Figs. 15 and 16, respectively. The crack propagation processes and the final cracking profiles for the initial cracks C2 and C3 models are more or less similar, in which a crack is observed to initially occur near the change in the slope of the downstream face and then extends into the dam, and three damage zones (the change in downstream and upstream slopes and near the dam heel) are clearly identified. For the initial crack C2 model (Case III), the initial crack C2 breaks through to the base of the dam to cause complete rupture (Fig. 15c). In addition, one crack develops near the change in downstream slope and another crack extends near the change in upstream slope (Fig. 15). For the initial crack C3 model (Case IV), the initial crack C3 at the change in the slope of the upstream face continues to grow downwards toward downstream in a slightly inclined direction (about 10 to the horizontal) with the continuous vibration of the dam. The crack propagates up to about 27.8 m and then stops propagation (Fig. 16d). The crack propagation depth at this location for Case IV is longer than the Case III. The results of the seismic crack propagation process of the dam with initial cracks C4, C5, and C6 are given in Figs As shown, it is known that the final cracking profiles with initial cracks C4, C5, and C6 are quite different from those with the initial cracks C1, C2, and C3. The final cracking trajectories for the initial cracks C4, C5, and C6 models are more or less similar, with the initial cracks breaking through to the opposite face of the dam to cause serious damage. For the initial crack C4 model (Case V), a crack in the upper of the dam upstream face propagates downwards toward downstream in a slightly inclined direction and finally reach the downstream face, penetrating the whole section of the dam (Fig. 17). In Case VI (the initial crack C5 model), the initial crack C5 approximately horizontal toward the downstream face, separating the crest from the upper part of the dam (Fig. 18). For the initial crack C5 model (Case VII), the initial crack propagation in the dam is firstly observed at 4.26 s near the initial crack C5 due to the vibration characteristics. After a stretch of horizontal propagation, the crack profile gradually curves downward due to the increasing compressive stresses on the downstream side. At t=4.68 s, the crack extends about five-sixth through the width of the dam section, and then stops propagation. The final crack profile is presented in Fig. 19d, and no penetrating crack appears in this case.

20 Initial Cracks on the Crack Propagation Process The Influence of the Initial Crack Length The finite element model (Fig. 10) with initial cracks at the downstream face (Crack C1) and the upstream face near the slope change at a height of 48 m above the base (Crack C3) is analyzed with four values of initial crack lengths, 0.2 m, 0.4 m, 1.0 m, and 2.0 m, to study the sensitivity of the predicted response to this initial parameter. The finial failure patterns of the dam with different initial crack lengths are shown in Figs A comparison of horizontal displacement time history of the dam crest with that of no initial crack is shown in Fig. 22, with positive displacement in the downstream direction. The influence of initial crack lengths on the crack propagation trajectory can be addressed by comparing Fig. 13d and Figs It can be found that the initial crack length has some influence on the crack propagation depths. When the initial crack length at the downstream face is 0.2 m, there is no penetrating crack. With the increase of initial crack length to 0.4 m, the downstream crack extends completely to the upstream face, penetrating the whole section of the dam. The ultimate failure profiles are very similar for the initial crack length of 0.4 m, 1.0 m, and 2.0 m. However, there is a significant impact on the horizontal displacement response of the dam with different initial crack lengths at the downstream face as shown in Fig. 22a. As observed by comparing Fig. 13d and Fig. 21, it can be found that the position of initial crack has significant influence on the crack propagation process of concrete gravity dams. While the crack trajectories are very similar for different initial crack lengths located in the upstream face near the slope change at a height of 48 m above the base, the reason (a) Length = 0.2 m (b) Length = 0.4 m (c) Length = 1.0 m (d) Length = 2.0 m FIGURE 20 The final crack profile of the dam with different initial crack lengths at the downstream face: (a) length=0.2 m, (b) length=0.4 m, (c) length=1.0 m, (d) length=2.0 m. (a) Length = 0.2 m (b) Length = 0.4 m (c) Length = 1.0 m (d) Length = 2.0 m FIGURE 21 The final crack profile of the dam with different initial crack lengths at the upstream face near the slope change at a height of 48 m above the base: (a) length=0.2 m, (b) length=0.4 m, (c) length=1.0 m, and (d) length=2.0 m.

21 1008 G. Wang et al. Displacement (cm) No initial carck Length = 0.4 m Length = 2.0 m Time (s) (a) The downstream face Displacement (cm) Time (s) No initial crack Length = 0.4 m Length = 2.0 m (b) The upstream face near the slope change at a height of 48m above the base FIGURE 22 Time history graphs of the horizontal displacements of the dam crest with initial cracks at (a) the downstream face and (b) the upstream face near the slope change at a height of 48 m above the base. for this is because the initial crack directions are similar. The crack propagation depths at the upstream face for the initial crack length of 0.2 m, 0.4 m, 1.0 m, and 2.0 m are m, m, m, and m, respectively. Although the initial cracks at the upstream face near the slope change cause different cracking depths, there is only a slight difference in the horizontal displacement obtained with different initial crack lengths (Fig. 22b). 6. Conclusions The objective of this study is to evaluate the influence of the initial crack position and length on the seismic performance and crack propagation of concrete gravity dams with considering the effects of dam-reservoir-foundation interaction. The reservoir water is modeled using two-dimensional fluid finite elements by the Lagrangian approach. The extended finite element method (XFEM), which is based on the cohesive segments method in conjunction with the phantom node technique, is used to model the cracked concrete gravity dam and predict the crack propagation process. The performance of the XFEM procedure for the analysis of crack propagation in concrete gravity dams with initial cracks has been demonstrated in this work. For this purpose, the cracking processes of a scaled-down

22 Initial Cracks on the Crack Propagation Process :40 model of a gravity dam with an initial notch on the upstream wall loaded with an equivalent hydraulic load and the Koyna gravity dam during the 1967 Koyna earthquake are analyzed for accuracy verification. Crack propagation analysis of a scale gravity dam model with initial notch shows that the crack profile match well with the experimental results from the model test, indicating that the XFEM procedure can effectively capture the crack propagation process and the crack trajectory in concrete gravity dams with initial cracks. The crack profile of Koyna dam including dam-reservoir-foundation interaction under Koyna earthquake shows a good agreement with the results of the Koyna dam prototype observation and the model test, which suggests that the XFEM procedure can effectively predict the crack propagation process in concrete gravity dams under seismic conditions. The seismic cracking analyses of a typical concrete gravity dam with six different sets of small initial cracks at different locations along the upstream and downstream faces are performed. The results with and without initial cracks are compared, and significant differences in terms of the crack profile are observed, which indicate that the initial crack position has significant influence on the crack propagation process of concrete gravity dams. The cracked dam will cause more severe damage to the dam body than the intact dam profile. The critical cracking is found to be associated with fracture originating at the point of downstream slope change and penetrating the dam almost instantaneously to separate the crest from the upper part of the dam. Other possible patterns of initial cracks on the upstream face of the upper of the dam also result in complete rupture. The influence of the initial cracks with different lengths at the downstream and upstream faces are also discussed. The results show that the initial crack length has some impact on the crack propagation depths and displacement response. The crack trajectories are very similar for cases with different initial crack lengths. Funding The authors gratefully appreciate the supports from the Chinese National Programs for Fundamental Research and Development (973Program) (No. 2011CB013501), the Chinese National Science Fund for Distinguished Young Scholars (No ), and the Fundamental Research Funds for the Central Universities ( kf0001). References Ayari, M. L. and Saouma, V. E. [1990] A fracture mechanics based seismic analysis of concrete gravity dams using discrete cracks, Engineering Fracture Mechanics 35(1 3), ABAQUS [2011] ABAQUS User s Manual V Bhattacharjee, S. S. and Léger, P. [1994] Application of NLFM models to predict cracking in concrete gravity dams, Journal of Structural Engineering 120(4), Batta, V. and Pekau, O. A. [1996] Application of boundary element analysis for multiple seismic cracking in concrete gravity dams, Earthquake Engineering & Structural Dynamics 25(1), Belytschko, T. and Black, T. [1999] Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45(5), Barpi, F. and Valente, S. [2000] Numerical simulation of prenotched gravity dam models, Journal of Engineering Mechanics 126(6), Bolzon, G. [2004] LEFM and cohesive-crack approaches to safety evaluation of concrete gravity dams, Computational Mechanics WCCM VI in Conjunction with APCOM 04, Beijing, pp Chopra, A. K. and Chakrabarti, P. [1973] The Koyna earthquake and the damage to Koyna dam, Bulletin of the Seismological Society of America 63(2),