**5.2.2 Gravity dam**

102 Hydrodynamics – Natural Water Bodies

As the cross section of the vertical dam-system as shown in Fig.3 was uniform, a near-field fluid domain was not necessary and the whole reservoir was modeled by a far-field domain

weight per unit length of the cantilevered dam was 36000*kg/m*. The height of the cantilevered dam *H* was 180*m*. The dam was modeled by 20 numbers of simple 2-noded beam elements with rigidity *EI* (=9.646826×1013*Nm*2), while the whole fluid domain was modeled by 10 numbers of 3-noded SBFEM elements, whose nodes matched side by side with nodes of the dam. In this problem, the shear deformation effects were not included in the 2-noded beam elements. Time step increment was 0.005*sec*. The pressure at the heel of dam subjected to the ramped horizontal acceleration shown in Fig.9 was plotted in Fig.10 and Fig.11. Analytical solutions of deformable and rigid dams were from the literature (Tsai et al., 1990) and the literature (Weber, 1994), respectively. In Fig.11, analytical solutions (Weber, 1994), solutions from the SBFEM in the full matrix form (Wolf & Song, 1996b) and solutions from the SBFEM in the diagonal matrix form (Li, 2009) were plotted with circles, rectangles and solid line, respectively. Solutions from the SBFEM and analytical solutions were the same. In the literature (Li, 2009), it was found that diagonal SBFEM formulations

Fig. 10. Pressure at the heel of deformable dam subjected to ramped horizontal acceleration

is 1000*kg/m*3. The

alone. Sound speed in the reservoir is 1438.656*m/s* and the fluid density

need much less computational costs than those in the full matrix.

**5.2.1 Vertical dam** 

This example was analyzed to verify the accuracy and efficiency of the FEM-SBFEM coupling formulation for a dam-reservoir system having arbitrary slopes at the damreservoir interface. The density, Poisson's ratio and Young's modulus of the deformable dam are 2400*kg/m*3, 0.2 and 2.5×1010*N/m*2, respectively. The fluid density is 1000*kg/m*3 and wave speed in the fluid is 1438.656*m/s*. The height of the dam *H* is 120*m*. A typical gravitydam-reservoir system and its FEM and SBFEM meshes were shown in Fig.12. The dam and the near-field fluid were discretized by FEM, while the far-field fluid was discretized by the SBFEM. 40 numbers and 20 numbers of 8-noded elements were used to model the dam and the near-field fluid domain, respectively, while 10 numbers of 3-noded SBFEM elements were employed to model the whole far-field fluid domain. Note that the size of the nearfield fluid domain can be very small compared to those used in other methods. In this example, the distance between the heel of the dam and the near-far-field interface was 6*m*

Fig. 12. Gravity dam-reservoir system and its FEM-SBFEM mesh

Hydrodynamic Pressure Evaluation of

Reservoir Subjected to Ground Excitation Based on SBFEM 105

(a) Vertical deformable dam

(b) Gravity dam

Fig. 14. Displacement at top of dam subjected to ramped horizontal acceleration

(=0.05*H*). The pressure at the heel of the gravity dam caused by the horizontal ground acceleration shown in Fig.9 was plotted in Fig.13. The time increment was 0.002*sec*. Results from SBFEM were very close to solutions from the sub-structures method (Tsai & Li, 1991). The displacements at the top of vertical and gravity dams subjected to a ramped horizontal acceleration were plotted in Fig.14. The displacement solutions of vertical dam from the SBFEM were the same with analytical solutions (Tsai et al., 1990). Fig.15 showed the displacement at the top of gravity dam subjected to the El Centro horizontal acceleration. At early time, the displacements obtained by the present method agreed well with substructure method's results (Tsai et al., 1990), especially at early time.

Fig. 13. Pressure at the heel of gravity dam subjected to horizontal acceleration

(=0.05*H*). The pressure at the heel of the gravity dam caused by the horizontal ground acceleration shown in Fig.9 was plotted in Fig.13. The time increment was 0.002*sec*. Results from SBFEM were very close to solutions from the sub-structures method (Tsai & Li, 1991). The displacements at the top of vertical and gravity dams subjected to a ramped horizontal acceleration were plotted in Fig.14. The displacement solutions of vertical dam from the SBFEM were the same with analytical solutions (Tsai et al., 1990). Fig.15 showed the displacement at the top of gravity dam subjected to the El Centro horizontal acceleration. At early time, the displacements obtained by the present method agreed well with sub-

(a) Ramped acceleration

(b) El Centro acceleration

Fig. 13. Pressure at the heel of gravity dam subjected to horizontal acceleration

structure method's results (Tsai et al., 1990), especially at early time.

Fig. 14. Displacement at top of dam subjected to ramped horizontal acceleration

Hydrodynamic Pressure Evaluation of

pp.302-311

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Fig. 15. Displacement at top of gravity dam subjected to El Centro horizontal acceleration
