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The concrete gravity dams are designed to perform satisfactorily during an earthquake since the consequence of failure is catastrophic to the downstream communities. The foundation in a dam is usually modeled by a substructuring approach for the purpose of seismic response analysis. However, the substructuring cannot be used for solving nonlinear dynamic problems that may be encountered in dam-reservoir-foundation systems. For that reason, the time domain approach is preferred for such systems. The deconvolved earthquake input model is preferred as it can remove the seismic scattering effects due to artificial boundaries of the semi-infinite foundation domain. Deconvolution is a mathematical process that allows the adjustment of the amplitude and frequency contents of a seismic ground motion applied at the base of the foundation in order to get the desired output at the dam-foundation interface. It is observed that the existing procedures of deconvolution are not effective for all types of earthquake records. A modified procedure has been proposed here for efficient deconvolution of all types of earthquake records including high-frequency and low-frequency ground motions.

The number and size of hydroelectric dams increased significantly across Canada since 1910 [

Clough et al. [

Computer program SHAKE developed by Schnabel et al. [

To evaluate the response of a dam during a seismic event, the ground motion acceleration is applied at the base of the foundation, which propagates vertically through elastic wave propagation mechanism until it reaches the top of the foundation. The size of the foundation in a numerical model is finite compared to the semi-infinite foundation in the physical model. Hence, the seismic waves reflect from the artificial boundaries due to the finite size of the numerical model, which may alter the frequency contents and amplitudes of a ground motion time history signal as the wave propagates through the deformable foundation rock. To account for such wave scattering effect, it is recommended to use transmitting boundaries or deconvolved ground motion records [

In this method, first, a deconvolution analysis is performed to determine the acceleration time history that can be applied to the base of the foundation to reproduce the specified free-field acceleration time history at the base of a dam (Figure

Representation of deconvolution procedure.

As mentioned earlier, the free-field acceleration or any arbitrary signal is initially applied at the base of the foundation, and, by solving the wave propagation problem, the acceleration signal at a selected point at the top of the foundation is obtained. The synthesized and free-field acceleration signals at the top of the foundation are then compared in the frequency domain, and a correction factor for each frequency is computed using the ratio of the Fourier amplitudes of the synthesized and free-field ground acceleration signals in a given iteration. The acceleration signal applied at the base of the foundation is modified using the correction factor for each frequency. The modified acceleration history is then transformed back into time domain acceleration signal by employing IFFT, and the analysis of the wave propagation analysis for the foundation system is repeated with the modified ground acceleration applied at the base of the foundation. The procedure is iterated until the original free-field ground motion at the top of the foundation closely matches the reproduced ground motion record generated by using the modified ground motion applied at the base of the foundation. The resulting ground motion at the foundation-base would be called the deconvolved ground motion that should be used in the dynamic analysis of the dam-foundation system.

The existing iterative procedure for deconvolution as discussed in the previous section does not produce appropriate results for high-frequency ground motion records as will be shown later. However, it works quite well for the low-frequency ground motion records in some cases. To overcome such limitation, a modified procedure has been proposed in this section. Figure

Proposed deconvolution procedure.

The response spectrum produces the plots of the maximum response acceleration for all possible linear single degree of freedom systems to a given ground motion for a given level of damping (assumed 5% in this analysis). The correction factors calculated through an iterative process with the existing deconvolution procedure (Section

To determine the closeness of the response spectrum of reproduced ground motion to the free-field ground motion, the coefficient of determination (

Two geometrically different monoliths of concrete gravity dams have been considered here to study the seismic wave scattering in dam-foundation systems. Figure

Material properties.

Material | Concrete | Rock |

Elastic modulus (MPa) | 3.45 × 10^{4} |
2.76 × 10^{4} |

Poisson’s ratio | 0.2 | 0.33 |

Unit weight (kN/m^{3}) |
23.5 | 25.9 |

Dam-foundation system: (a) Geometry G-1 and (b) Geometry G-2.

Representation of constraints.

Two different suites of ground motion records containing high-frequency and low-frequency contents have been considered here. They contain both simulated and actual ground motion records. The simulated records have been chosen based on those developed in Tremblay et al. [

Response spectra for the ground motion records: (a) Montreal-horizontal components, (b) Montreal-vertical components, (c) Vancouver-horizontal components, and (d) Vancouver-vertical components.

Figures

Response spectra of the original and deconvolved ground motions for G-1 in Montreal: (a) M number 3(H); (b) M number 3(V); (c) V number 2(H); (d) V number 2(V); (e) V number 3(H); and (f) V number 3(V).

Coefficient of determination (

Deconvolved ground motions with MDP for dam-foundation, G2: (a) M number 1(H); (b) M number 1(V); (c) V number 1(H); and (d) V number 1(V).

Figure

Figure

From the above results, it can be concluded that the performance of EDP in the cases of low-frequency ground motions is better than that in the cases of high-frequency ground motions. However, in some cases, even for low-frequency ground motions, such as V number 3, the performance of EDP is not acceptable. MDP shows a satisfactory performance for both low- and high-frequency ground motions. Figure

The study presents a modified deconvolution procedure for the deconvolution of input ground motions for the use in the seismic response analysis of dam-foundation systems. While the performance of the existing deconvolution procedure is generally good for low-frequency ground motions, it may not work in all such cases; the performance of the procedure is found to be quite poor when a ground motion has high-frequency contents (e.g., for Montreal). The modified deconvolution procedure is found to perform well for both high-frequency and low-frequency ground motions. It is also observed that the deconvolution by EDP requires more iterations and the convergence is poorer compared to MDP. It is important here to note that while only two-dimensional models are considered here, the modified deconvolution procedure proposed in this study is expected to be more effective for three-dimensional dam-foundation models. Further study is required in that direction.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The financial assistance provided by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.