**5.2 Transient response of dam-reservoir system**

Consider transient responses of dam-reservoir systems where dams were subjected to horizontal ground acceleration excitations shown in Fig.9. In the transient analysis, only the linear behavior was considered, the free surface wave effects and the reservoir bottom absorption were ignored, and the damping of dams was excluded. Dams were discretized by the FEM, while the response of the reservoir was solved by Eq.(30). The FE equation of dam and Eq.(30) was solved by Newmark's time-integration scheme with Newmark integration parameters 0.25 and 0.5 . An iteration scheme (Fan et al., 2005) was adopted to obtain the response of the dam-reservoir interaction problems.

Fig. 9. Horizontal acceleration excitations

Hydrodynamic Pressure Evaluation of

**5.2.2 Gravity dam** 

Reservoir Subjected to Ground Excitation Based on SBFEM 103

Fig. 11. Pressure at the heel of rigid dam subjected to ramped horizontal acceleration

dam are 2400*kg/m*3, 0.2 and 2.5×1010*N/m*2, respectively. The fluid density

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

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

is 1000*kg/m*3 and

#### **5.2.1 Vertical dam**

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 alone. Sound speed in the reservoir is 1438.656*m/s* and the fluid density is 1000*kg/m*3. The 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 need much less computational costs than those in the full matrix.

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

Fig. 11. Pressure at the heel of rigid dam subjected to ramped horizontal acceleration
