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Earthquake Dam-Reservoir Interaction Analysis of Arch Dams

R. Taherzadeh1, G. Potin2

1Tractebel Engie, Gennevilliers, France. reza.taherzadeh@tractebel.engie.com2 Tractebel Engie, Gennevilliers, France. gildas.potin@tractebel.engie.com

ABSTRACT: The dam-reservoir interaction analysis of arch dams under seismic loadings is an essential topic in design process. Most of the arch dams are placed in seismic areas, sometimes with high seismic hazard and level, and in consequence the risks induced are of great importance. This paper aims to study the seismic Fluid-Structure Interaction (FSI) in arch dams. For this purpose, the coupled Boundary Elements and Finite Elements (BE-FE) analysis is used to take into account the FSI. The results obtained by the BE-FE modelling of a case study are compared with the classical Westergaard simplied approach. The inuence of considering the uid as compressible or incompressible is also examined with the numerical BE-FE model.

Keywords: Earthquake, Dam-Reservoir Interaction, Numerical Analysis

1. INTRODUCTION

In dynamic analysis of the arch dams, the dam body is generally modeled throughout the Finite Element Method. To address uid-dam body interactions, it is necessary to take into account the reservoir in the model. The modeling of dam–reservoir interaction can be carried out by different methods. In order to avoid the great difculties from such a complex interaction, the overall problem is solved by applying additional masses to the upstream face of the dam. These masses represent the hydrodynamic pressure and are traditionally determined by the conservative Westergaard approach (Westergaard, 1933). In the case of a signicant dam/reservoir interaction, this method may not be realistic enough to assess the dam response. A simple but accurate alternative solution is proposed in the present work. The BE-FE analysis can take into account the dam body exibility as well as the radiation damping of the semi-unbounded reservoir (Clouteau D. and Aubry D., 2004). In this solution, the dam body is modeled by a FE method and the reservoir and the rock foundation by a BE method. There is a coupling between both methods in a subspace with dimensions smaller than the ones of the original FEM approximation subspace. Consequently, this technique is useful to govern a very high computational demand when modeling a 3D complex geometry of the reservoir.

2. SEISMIC FLUIDS-STRUCTURE INTERACTION

In the following, a description of the uid-structure interaction of an arch dam with a simple physical model is presented. As shown in Figure 1, a spring-mass-damper system is employed to represent the coupled FSI system. In this model, the dam body motion is represented with a single degree of freedom. During the earthquake, some of the uid in the reservoir follows the movement of the dam body. This phenomenon introduces an additional mass to the dam, generating the inertial effect. The semi-unbounded reservoir is a way to generate the radiation damping for the new system. The additional mass linked to the dam body throughout a simple spring and damper in parallel. The added spring associated to the additional mass representing the resonance of the reservoir. Furthermore, the added damper may represent the wave radiation into the reservoir. It can be clearly seen that the equivalent natural frequency of the uid-structure system is smaller than the one of the dam body due to the inertial effect of the uid. In addition, the spectral response of the uid-structure system can be modied because of the damping effect of the semi-unbounded reservoir.

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(a) (b)

Fig. 1: Seismic Fluid-Structure Interaction of an arch dam (a) Simple physical model and (b) Spectral response with and without FSI

3. COUPLING BE-FE METHOD

In the coupling between FE and BE methods, the dam body is modeled by FE method and the semi-nite reservoir as well as the innite ground are modeled by the boundary method. The two methods are then coupled by using the sub-structure technique (Tardieu B., Zghal H., Aubry D., Ozanam O., 1993). In this modeling, the dam body is supposed to be linear and elastic and the uid is either compressible or incompressible, assuming that the ow is irrotational. In the uid domain, the pressure eld satises the wave equation and on the free surface is zero. As far as the interface between the uid and the dam body is concerned, the displacement and pressure elds satisfy the local equilibrium and the normal continuity. Sub-structure technique is commonly used for the problem of the interaction between the different sub-domains. Herein, this technique is applied for the problem of the uid-structure interaction. In order to reduce the size of the model, the nite element method based on modal analysis is used to solve the global problem in a reduced system. Thus, the displacement eld in the dam body can be decomposed into two modal bases: the three translation modes due to displacement of the basis of the dam and for example the 20 rst eigenmodes with xed base conditions. Thanks to this decomposition, the global problem size is reduced to “3+20” instead of the degrees of freedom. The coupled FSI model is then solved by introducing the impedance of the uid computed by the BE method in the dynamic reduced structure system.

4. CASE STUDY

To better study the FSI, the case of an arch dam under construction has been chosen. It is located in a seismic area, in Turkey with the PGA of 3m/s2. The cantilever crown presents a variation in its thickness from 56m at the base to 8m at the crest. The height of the dam is 181m and the length at its crest is around 220m. This dam will produce 332 megawatts provided by a hydraulic head of 166 m. The consideration of a linear elastic behavior of the concrete in the dam body is made. The concrete in the dam body is assumed to be linearly elastic, with the following properties: density=2.35t/m3; Poisson’s ratio = 0.2; Young’s modulus = 20GPa; and a hysteretic damping factor of 5%. The foundation rock is assumed to be massless with the Young’s modulus=10GPa and Poisson’s ratio=0.2. The density of water is 1t/m3 and the wave velocity in compressible uid is assumed to be 1440m/s. The gure below shows the FE model of the dam and the foundation. Solid elements are employed in the modeling of the dam body and the rock foundation. The BE method is used for modeling the semi-unbounded reservoir domain by using quadrilateral elements which are introduced on the upstream water-dam interface and on the reservoir-rock interface. At the location of about one time of water depth, the BE model is truncated.

(a) (b) (c)Fig. 2: (a) Upstream view of the reservoir-dam body; (b) Numerical model of the arch dam with its massless foundation; (c) Ground acceleration and

5%-damped response spectra

The rst studied parameter is the impedance of the uid. Figure 3 shows the real and the imaginary parts of the impedance for both compressible and incompressible uids. The dependency of the impedance on the frequency is heavy. There is a parabolic increase of the real part with frequency for an incompressible uid, resulting from the inertial effect of the uid (-ω2m).

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(a) (b)Fig. 3: Impedance of the uid computed by the couple BE-FE model for (a) the horizontal model and c) the rst eigenmode

The imaginary part as well as the real part increase with frequency. As seen in Figure 3, some kind of the uctuation behavior due to the uid resonance in the reservoir is exhibited. The radiation damping is represented throughout the imaginary part of the impedance. The radiation damping is caused by the wave propagation into the semi-unbounded reservoir.

Figure 4 shows the harmonic frequency response of the dam at the crest as a result of the FSI analysis. It can be seen that the new resonances of the dam body in the coupled FSI system take place at lower frequencies than the ones obtained with no interaction. This is due to the additional mass of the uid. We can also observe that there is no difference in the fundamental frequencies obtained with the coupled FSI system considering either compressible or incompressible uid. However, the dam response is strongly reduced with the compressible uid thanks to the energy absorption by the additional damping produced by the wave radiation into the semi-unbounded reservoir.

Fig. 4: Harmonic frequency response of dam at crest with taking into account the FSI

Similar conclusions are observed in terms of the relative time history displacement between top and bottom of the dam body. The compressibility of the uids reduces signicantly the dam response. The same remark can be made regarding the damping effect of the compressible uid in dam body motion.

Fig. 5: Comparison between compressible and incompressible uid based on the BE-FE model; Relative displacement time history response between top and bottom of the dam at the crown including uid-structure interaction

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The comparison of the BE-FE model with the Westergaard approach is studied below. It is clearly shown that the Westergaard approach heavily overestimates the dam response.

Fig. 6: Comparison between Westergaard approach and the BE-FE model; Relative displacement time history response between top and bottom of the

dam at the crown including uid-structure interaction

5. RISK MANAGEMENT ANALYSIS

The complex structures located in the seismic area can be excited with the heavy interaction in contact with soil/water. In this case, taking into account this phenomena and using the sophisticated method is strongly recommended. The case study of this paper shows that the seismic response of the arch dam depends on the uid-structure interaction as well as the method of computation. Thus, the latters can be introduced as the topics to study in risk management analysis categorized as the preventive measures against the earthquake risk in detailed design phase of a project. In Artvin dam seismic analysis, it was seen that using the sophisticated BE-FE model provides the opportunity to save on project cost. In some cases in which the rst natural period of the structure, without the interaction, is located on the ascending branch of the response spectrum, the uid structure interaction phenomena increases the natural period of the couple system leading to amplify the response of the structure.

6. CONCLUSION

In the present work, a simple but powerful numerical tool based on the coupled BE-FE model is presented. It aims to study the behavior of arch dams under seismic loading. This method can efciently determine the dam response, particularly for a complex 3D geometry of the reservoir. Based on the results obtained by the BE-FE model, it was seen that the uid compressibility heavily reduces the dynamic response of the dam due to the wave propagation into the semi-unbounded reservoir. The comparison between the Westergaard approach and the coupled BE-FE model shows that the Westergaard approach is very conservative in earthquake design of the cases study high arch dams. On other words, using the FSI analysis by the coupled BE-FE model can be a mean to optimise an arch dam project located in a seismic area.

7. REFERENCESClouteau, D. and Aubry, D. (2004). Computation soil-structure interaction, Boundary elements methods for soil-structure interaction,

chapter 2, Kluwer Academic Publishers.

Fluid Structure Interaction, Arch Dam, Proceedings of the 12th International Benchmark Workshop, ICOLD, Graz, Austria, October (2013).

Taherzadeh, R., Eid, N. and Daux, C. (2013). An efcient numerical model based on the couple nite elements-boundary elements method for seismic analysis of arch dam-reservoir interaction, Hydropower international conference and exhibition, Bilbao, Spain.

Tardieu, B., Zghal, H., Aubry, D. and Ozanam, O. (1993). Méthode simpliée de prédimensionnement des barrages poids en zone sismique, 3ème Colloque National AFPS Génie parasismique et aspects vibratoires dans le génie civil, Saint-Rémy-lès-Chevreuse, France.

West ergaard, H.M. (1933). Water pressures on dams during Earthquakes, Transactions of the American Society of Civil Engineers, ASCE, n° 1835, vol. 98.