## Introduction

The cracking behaviour of concrete dams has been the subject of extensive research during the past decade because few dams suffered severe cracking during earthquakes. Rescher (1990) indicated that most concrete gravity dams will experience cracking even under operational loading conditions and moderate earthquake ground motions. Therefore, the assumption of linear behavior may not be appropriate in the analysis of the seismic response of concrete gravity dams.

Concrete dams are distinguished from other structures because of their size and their interactions with the reservoir and foundation. The results obtained from the nonlinear analysis of concrete dams are strongly dependent on the approach to modelling of these interactions. It is a difficult task to develop a comprehensive analytical model to include both nonlinearity and interaction effects. The size effect can also influence the properties of the dam concrete. The fracture properties of normal concrete can be determined using laboratory tests. However, the dam concrete differs from the normal weight concrete because of aggregate size and its poor strength. Little information is available on the fracture properties of the dam concrete. The fracture surface of the dam concrete specimens is characterized by mainly aggregate failure.

Saouma et al. (1991) attempted to measure fracture toughness of a concrete specimen in the laboratory which was considered to be similar to dam concrete. They concluded that a definitive decision cannot be made concerning the results and their accuracy. Bruhwiler and Wittmann (1990) carried out a dynamic test to determine the material properties of the dam concrete under high rate of loading and an initially applied compression load. They found that the fracture energy of dam concrete is 2 to 3 times higher than that of ordinary concrete. The reason is related to the tensile strength characteristic of dam concrete. The tensile behavior of concrete can be divided into two stages. In the first stage, the behavior is linear until the tensile strength is reached. In the second stage, strain softening behavior is observed. Fracture energy is sensitive to the tensile stress. In addition, increasing the preloading decreases the fracture energy.

To understand the nonlinear behaviour of concrete dams, modelling of the cracking and damage process is needed. Bazant and Oh (1983) proposed a fracture mechanics approach as a blunt smeared crack band. The proposed approach represented a significant advance in comparison to the linear fracture theory. The strain softening of the material was considered based on the fracture parameters, fracture energy, uniaxial tensile strength and crack band width. Fracture energy can be determined from the complete stress-strain curve. Formulas were derived to give the fracture parameters.

Two classes of solutions can be found in the nonlinear study of concrete gravity dams. Discrete crack approach is the first class of solutions which is based on the variable mesh approach. Two methods of linear elastic fracture mechanics LEFM, and nonlinear fracture mechanics NLFM, can be used in this approach. The second class of solutions is the continuum model in which a fixed finite element mesh is used. Smeared crack model and damage mechanics are the two methods of solution in this class. These two families of fracture model have been investigated in parallel. They also called global approach and local approach. In the global approach, the cracks are simulated by discontinuity in the continuum and the stresses are obtained using linear elastic fracture mechanics(LEFM). This method is coupled to a numerical technique such as finite elements or boundary elements. On the other hand, the local approach of fracture is based on changes in the constitutive law governing the behaviour of concrete. The integrity of the structure is not required during the solution process.

Bhattacharjee and Léger (1994) applied NLFM to predict the response of concrete gravity dams. The experimental work done on a model of a concrete gravity dam and a small beam specimen confirmed the applicability of the proposed NLFM approach. The coaxial rotation crack model gives a better response than the fixed crack model. Léger and Leclerc (1996) studied the nonlinear response of concrete gravity dams subjected to different earthquake ground motions. They found that the response is sensitive to time variation of the input motion. Most of the time, cracking response showed that the crack starts from the downstream side and moves toward the upstream side. This form of cracking does not promote dam instability. The cracks are either horizontal or they sloped downward. They found that the vertical ground motion acceleration component is not critical in seismic cracking response of dams.

The nonlinear response of a concrete gravity dam with an initial distribution of temperature gradient when subjected to the earthquake was studied by Léger and Bhattacharjee (1995). They used frequency-independent added mass matrix as a representative of dam-reservoir-foundation interaction. The reservoir and foundation were modelled as a series of dampers and springs such that the same response can be obtained for the linear response of the crest when compared with the case of actual interaction. Under earthquake excitation, when a rigid foundation is assumed with no reservoir bottom absorption, no crack was observed at the top part of the dam. A crack was formed at the foundation level.

Bhattacharjee and Léger (1993); Léger and Bhattacharjee (1994) studied the energy response of concrete gravity dams. They used a stiffness proportional damping with -method of integration. Newton-Raphson iteration technique was used to remove the unbalanced load. An energy balance error approach is used as a measure of damage. The seismic analysis of Koyna dam under both horizontal and vertical components of the earthquake was conducted. Without introducing the numerical damping, the analysis stopped after the first few seconds because of energy balance error due to spurious deformation of some elements. No discrepancies were found in the results of the analysis before the occurrence of instability, when compared with the case of = -0.2 in which the analysis was successfully completed. Dissipated fracture energy is negligible in comparison to other sources of energy dissipation. The reservoir effect was represented by added mass technique.

The effect of hydrodynamic pressure inside the crack in the seismic analysis of concrete gravity dams was investigated by Tinawi and Guizani (1994). The pressure inside the crack does not change the response of the dam significantly. It was found that under high frequency content earthquakes, the hydrodynamic pressure inside the crack may increase when higher modes are significant. At the base of the dam, the hydrodynamic pressure may be 50% higher than the hydrostatic pressure.

The nonlinear response of the Pine Flat dam was studied using the discrete approach (Wepf et al. 1993). A fictitious crack approach was used to model the crack tip. Reservoir interaction was modelled using a boundary element. Linear response of the dam was compared with EAGD-84 code analysis and good agreement was found. The nonlinear response of the dam including reservoir interaction was strongly affected in comparison with the added mass approach. The slope of the reservoir bottom strongly influenced the nonlinear response. The aggregate interlock effect was found to be important in the final cracking configuration of the dam.

The cracking response of a concrete gravity dam when subjected to earthquake loading can be different if nonuniform damping or uniform damping including the damping due to cracking is considered (Barrett et al., 1991). In the analysis, the dam was represented by a small number of elements. When the bottom few elements were cracked, a noticeable change in the response was observed.

Using different computer codes, Singhal (1991) found that the Wester-gaard's added mass approach yields higher values for crest displacement and stress than that obtained using other approaches. The reservoir bottom ab-sorbtion and water compressibility did not change the response significantly.

Pekau et al. (1991, 1995) and Pekau and Batta (1991) presented a method to study the cracking of concrete gravity dams using the principle of Linear Elastic Fracture Mechanics (LEFM) and boundary element mode superposition analysis. The model was checked by a shake table test of cantilever beam made of gypsum. The impact of cracking surface was modeled as a load pulse.

Ayari and Saouma (1990) proposed a model for simulation of discrete crack closure. The model was applied in the dynamic analysis of the Koyna dam (India) under both horizontal and vertical components of the earthquake. The results were obtained for 5 seconds only of the earthquake in which the numerical damping was less than 10%.

Nonlinear seismic response of concrete gravity dams was studied by Skrikerud and Bachmann (1986). Fracture mechanics analysis using discrete crack approach was applied. The model was capable of initiation, opening, closing and reopening of discrete cracks. Special treatment was used to model aggregate interlock effect. The model was applied to a dam of rigid foundation with empty reservoir. The crack pattern was found to be very sensitive to the parameters chosen for the analysis. The first four seconds of an artificially generated time history was used for the purpose of analysis. The analysis stopped due to excessive damage. Nonlinear response of concrete gravity dams was also studied by Feltrin et al. (1990). A rigid foundation was assumed for Pine Flat dam and the reservoir interaction was included. The nonlinearity in concrete behaviour included the strain softening and aggregate interlock. Response of the linear model with and without the reservoir interaction was determined. Nonlinear response of the empty reservoir was studied by scaling the ground motion until cracking occurred. The cracks started at the top part from down stream face of the dam near the slope discontinuity and moved horizontally. A different response was observed under the effect of reservoir interaction. The first crack started at the foundation level and then it followed by a crack at the top part of the dam at the same location of the crack of empty reservoir case. The crest displacement was found to be higher than that of the empty reservoir. They concluded that the effect of dam-reservoir interaction must be included in the nonlinear analysis.

El-Aidi and Hall (1989 a,b) investigated the nonlinear response of concrete gravity dams. The water cavitation in addition to cracking of concrete was considered. Despite the difficulties involved, the nonlinear model was applied for the case of preformed base crack, top crack and homogeneous dam without any cracks. In the case of homogeneous dam, the top crack initiated at t = 1.95 sec. Soon after initiation of the top crack, it went through the dam body and almost separated the top part from the rest of the dam. During the rest of the analysis, no other cracks were observed and only rocking and opening and closing of the crack were observed.

Fenves and Vargas-Loli (1988) proposed a method for dam-reservoir interaction which resulted in a symmetric matrix representation of the total equation of the system. The nonlinearity of the reservoir was introduced into the proposed method to investigate the reservoir interaction effect. They found that the effect of cavitation is not significant in the response of the structure.

Mlakar (1987) studied the nonlinear dynamic behaviour of concrete gravity dams using the ADINA code. It was found that the crack first started at the base. Then cracking initiated at the top part near slope discontinuity. The cracks near the slope discontinuity propagated instantaneously and passed through the cross section.

In this chapter the nonlinear fracture response of concrete gravity dams due to seismic loading is investigated. The dam-reservoir interaction is included in the time domain analysis using the method of staggered displacement. Smeard crack approach based on a nonlinear fracture mechanics crack propagation criterion is used to study the cracking and response of concrete gravity dams.

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