**5.1.1 Vertical dam**

For a rigid dam-reservoir system with a vertical upstream face as shown in Fig.3, the whole reservoir was flat so that the whole reservoir was modeled by the far field alone. This example's aim was only to test the correctness and efficiency of the SBFEM in Eqs.(7, 8, 16) of the far field. The whole reservoir was discretized by the SBFEM alone using 10 and 20 3 noded SBFEM elements, respectively. The hydrodynamic pressure acting on the damreservoir interface from a reflection coefficient *<sup>r</sup>* 0.95 and these two mesh densities was plotted in Fig.4. The coefficient *Cp* was defined as *p aH* and <sup>1</sup> *c H* 2 , where *p* denoted the amplitude of hydrodynamic pressure acting on the dam-reservoir interface.

Fig. 3. Vertical dam-reservoir system

Fig. 4. Hydrodynamic pressure on vertical dam-reservoir interface from different meshes

Two-dimensional dam-reservoir systems subjected to horizontal harmonic ground

For a rigid dam-reservoir system with a vertical upstream face as shown in Fig.3, the whole reservoir was flat so that the whole reservoir was modeled by the far field alone. This example's aim was only to test the correctness and efficiency of the SBFEM in Eqs.(7, 8, 16) of the far field. The whole reservoir was discretized by the SBFEM alone using 10 and 20 3 noded SBFEM elements, respectively. The hydrodynamic pressure acting on the dam-

denoted the amplitude of hydrodynamic pressure acting on the dam-reservoir interface.

Cantilevered dam

<sup>H</sup>

Fig. 4. Hydrodynamic pressure on vertical dam-reservoir interface from different meshes

2

*y* 95.0

4

1 

> 1

0 2 4 6

*Cp*

in the upstream direction were studied. For simplicity, here the dam

 

10 3-noded elements

*r*

20 3-noded elements

*aH* and

0.95 and these two mesh densities was

 

1

1 

<sup>1</sup> *c H* 2 , where *p*

**5. Numerical examples** 

accelerations *i t a ae*

**5.1.1 Vertical dam** 

was assumed to be rigid.

**5.1 Harmonic response of reservoir** 

reservoir interface from a reflection coefficient *<sup>r</sup>*

Fig. 3. Vertical dam-reservoir system

0.5

*H*

1

plotted in Fig.4. The coefficient *Cp* was defined as *p*

Results from different mesh densities were the same. The hydrodynamic pressure obtained by using 10 3-noded SBFEM elements and the corresponding analytical solutions (Weber, 1994) corresponding to different *<sup>r</sup>* were plotted in Fig.5. The SBFEM solutions were the exact same to the analytical solutions. Furthermore, a *Cp* figure of a point located at *y H* 0.6 corresponding to *<sup>r</sup>* 0.8 was shown in Fig.6. The SBFEM solution and the analytical solution (Weber, 1994) were the same.

Fig. 5. Hydrodynamic pressures on vertical dam-reservoir interface caused by different *r*

Hydrodynamic Pressure Evaluation of

agreement with Sharan's results.

0.5

*y H* 1

Fig. 8. Hydrodynamic pressure acting on gravity dam

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

Acceleration Ramped

0.02 Time (sec)

 and 0.5 

*<sup>r</sup>* 0.75

*r* 

adopted to obtain the response of the dam-reservoir interaction problems.

integration parameters 0.25

a

Fig. 9. Horizontal acceleration excitations

Reservoir Subjected to Ground Excitation Based on SBFEM 101

isoparametric acoustic fluid finite elements, while the far field was still modeled by 10 3 noded SBFEM elements. Their meshes were shown in Fig.7. Solutions from Eq.(29) and the literature (Sharan, 1992) were plotted in Fig.8. Results obtained by Eq.(29) were in excellent

SBFEM(*L*=0.001*H*)

*r*

0.95

Sharan's solution

1 

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

0 0.5 1 1.5

*Cp*

0.5 <sup>1</sup>

. An iteration scheme (Fan et al., 2005) was

El Centro

Fig. 6. *Cy H <sup>p</sup>* 0.6 for different 
