An efficient approach for frequency-domain and time-domain hydrodynamic analysis of dam–reservoir systems

An efficient procedure is developed for the hydrodynamic analysis of dam–reservoir systems. The governing equations of hydrodynamic pressure in the frequency as well as time domain are derived in the framework of the scaled boundary finite element method. The water compressibility and absorption of reservoir sediments can be conveniently taken into consideration. By extending the reservoir to infinity with uniform cross-section, only the dam–reservoir interface needs to be discretized to model the fluid domain, and the hydrodynamic pressure in the stream direction is solved analytically. Several numerical examples including a gravity dam with an inclined upstream face and an arch dam with a reservoir of arbitrary cross-section are provided to demonstrate the computational efficiency and accuracy of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

A solution procedure for the earthquake analysis of arch dam-reservoir systems with compressible water

Two processes using the Newmark implicit integration scheme are presented for the analysis of the earthquake response of a three-dimensional model for arch dam-reservoir systems including the effect of compressibility of the water. The solid structure and fluid regions are modelled separately, and the forcing functions at the interface are due to the hydrodynamic pressures from the reservoir acting on the upstream face of the dam wall, and the accelerations from the dam wall acting in turn on the reservoir. For the purposes of an initial investigation, elastic properties are assumed for the material of the dam, whilst in the reservoir radiation damping at the upstream boundary has been included, but bottom absorption has not. The excitation is provided by means of a combisweep which is fashioned so that its continuously varying frequencies pass through the fundamental frequencies of both the arch dam-reservoir system and the reservoir alone. Consequently the response is highly resonant, thus providing a severe test for the numerical procedures. From the numerical results obtained for an example problem it is concluded that both schemes provide an acceptable solution to the problem posed, and the possibility of enhancement to cater for more complex situations is discussed.     

Effects of reservoir bottom absorption on earthquake response of concrete gravity dams

A procedure is presented to analyse the response of concrete gravity dams due to horizontal and vertical earthquake ground motion components considering dam-water interaction and partial absorption of hydrodynamic pressure waves at the reservoir bottom into the foundation medium. The effects of reservoir bottom absorption on the hydrodynamic force on a rigid dam are examined first. The harmonic response of an idealized dam cross-section is presented for a wide range of parameters characterizing the properties of the dam, the impounded water and the foundation medium. Based on these frequency response functions the effects of dam-water interaction and of reservoir bottom absorption in the response of dams due to horizontal and vertical components of ground motion are investigated.     

A non-reflecting boundary condition for the finite element modeling of infinite reservoir with layered sediment

The design of seismic resistant concrete gravity dam necessitates accurate determination of hydrodynamic pressure developed in the adjacent reservoir. The hydrodynamic pressure developed on structure is dependent on the physical characteristics of the boundaries surrounding the reservoir including reservoir bottom. The sedimentary material in the reservoir bottom absorbs energy at the bottom, which will affect the hydrodynamic pressure at the upstream face of the dam. The fundamental parameter characterizing the effect of absorption of hydrodynamic pressure waves at the reservoir bottom due to sediment is the reflection coefficient. The wave reflection coefficient is determined from parameters based on sediment layer thickness, its material properties and excitation frequencies. An analytical or a closed-form solution cannot account for the arbitrary geometry of the dam or reservoir bed profile. This problem can be efficiently tackled with finite element technique. The need for an accurate truncation boundary is felt to reduce the computational domain of the unbounded reservoir system. An efficient truncation boundary condition (TBC) which accounts for the reservoir bottom effect is proposed for the finite element analysis of infinite reservoir. The results show the efficiency of the proposed truncation boundary condition.     

Effect of reservoir bottom on earthquake response of concrete dams

The absorption of hydrodynamic pressure waves at the reservoir bottom has dominant effects on the structural response of the dam when subjected to ground motion. In the present study, a model is proposed for the absorption effects of the reservoir bottom in the earthquake analysis of dams. The model utilizes the wave reflection coefficient approach and is based on the solution of the wave equation in a sediment layer of viscoelastic material with a constant thickness overlying an elastic, semi-infinite foundation. Numerical studies were conducted to evaluate the effect of the sediment layer thickness and material properties as well as the effect of reflection of waves from the underlying rock. It is shown that the current approach of assuming the wave reflection coefficient at the reservoir bottom based on the characteristics of the sediment material and excluding the effect of the reflected waves from the underlying rock, may significantly underestimate the seismic response of the dam.     

Fluid–foundation interaction in the seismic response of gravity dams

The seismic response of a dam is strongly influenced by its interaction with the water reservoir and the foundation. The hydrodynamic forces in the reservoir are in turn affected by radiation of waves towards infinity, wave absorption at the reservoir bottom, and cross-coupling between the foundation below the dam and the reservoir bottom. The fluid–foundation interaction effect, i.e. the wave absorption along the reservoir bottom, can be accounted for by using either an approximate one-dimensional (1D) wave propagation model or a rigorous analysis of interaction between the flexible soil along the base and the water. The rigorous approach requires enormous computational effort because of (a) cross-coupling between the foundation of the dam and the soil below the reservoir and (b) frequency dependence of the boundary condition along the fluid-foundation interface. The analysis can be simplified by ignoring the cross-coupling and by using the approximate 1D wave propagation model. The effects of each of these two simplifications on the accuracy and computational efficiency of the procedure used for the seismic response analysis of a dam are examined. Analytical results are presented for the complex frequency-response functions as well as the time histories of the response of Pine Flat dam to Taft and E1 Centro ground motions.     

Effects of reservoir bottom absorption and dam-water-foundation rock interaction on frequency response functions for concrete gravity dams

The linear response of an idealized concrete gravity dam monolith to harmonic horizontal or vertical ground motion is presented for a range of the important system parameters that characterize the properties of the dam, foundation rock, impounded water and reservoir bottom materials. Based on these frequency response functions, the effects of alluvium and sediments at the reservoir bottom on the response of the dam, including its interaction with the impounded water and foundation rock, are investigated. It is shown that the partial absorption of hydrodynamic pressure waves by the reservoir bottom materials has an important effect on the dynamic response of concrete gravity dams.     

Earthquake analysis of concrete gravity dams including reservoir bottom absorption and dam-water-foundation rock interaction

The available substructure method for the earthquake analysis of concrete gravity dams, including the dynamic effects of the impounded water and the flexible foundation rock, is extended to include the effects of alluvium and sediments invariably present at the bottom of actual reservoirs. Modelled approximately by a reservoir bottom that partially absorbs incident hydrodynamic pressure waves, these effects are incorporated into the continuum solution for the hydrodynamic pressure. The dam-water-foundation rock system is idealized as a two-dimensional system and analysed under the assumption of linear behaviour. An example earthquake analysis is presented to demonstrate the results obtained from the analytical procedure. Computation times for several cases illustrate the efficiency of the analytical procedure. In particular, the additional computation time required to include reservoir bottom absorption is shown to be very small.     

Two-dimensional dynamic analysis of concrete gravity and embankment dams including hydrodynamic effects

An analysis procedure in the frequency domain is developed for determining the earthquake response of two-dimensional concrete gravity and embankment dams including hydrodynamic effects; responses of the elastic dams and compressible water are assumed linear. The dam and fluid domain are treated as substructures and modelled with finite elements. The only geometric restriction is that an infinite fluid domain must maintain a constant depth beyond some point in the upstream direction. For such an infinite uniform region, a finite element discretization over the depth is combined with a continuum representation in the upstream direction. The fluid domain model approximately accounts for interaction between the fluid and underlying foundation medium through a damping boundary condition applied along the reservoir bottom, while the dam foundation is assumed rigid. Several examples are presented to demonstrate the accuracy of the fluid domain model and to illustrate dam responses obtained from the analysis procedure.     

Hydrodynamic pressures and response of gravity dams to vertical earthquake component

Hydrodynamic pressures and structural response of concrete gravity dams, including dam-reservoir interaction, due to the vertical component of earthquake ground motions are investigated. The response of the dam is approximated by the deformations in the fundamental mode of vibration, and the effects of deformability of bed rock on hydrodynamic pressures are recognized in the analysis. Expressions for the complex frequency response functions for the dam displacement, dam acceleration and lateral hydrodynamic force are derived. These results along with the Fast Fourier Transform algorithm are utilized to compute the time-history of responses of dams of 100, 300 and 600 ft height, with full reservoir, for different values of elastic modulus of mass concrete: 3.0, 3.5, 4.0, 4.5 and 5.0 million psi, to the vertical component of El Centro, 1940, and Taft, 1952, ground motions. It is concluded that the hydrodynamic forces caused by vertical ground motion are affected substantially by damreservoir interaction and depend strongly on the modulus of elasticity of the dam. The dam response to the vertical component of ground motion is compared with that due to the horizontal component. It is concluded that because the vertical component of ground motion causes significant hydrodynamic forces in the horizontal direction on a vertical upstream face, responses to the vertical component of ground motion are of special importance in analysis of concrete gravity dams subjected to earthquakes.     

Hydrodynamic effects in a reservoir with semicircular cross-section and absorptive bottom

A three-dimensional dam-reservoir system under seismic load is analysed. The dam is assumed to be rigid. The reservoir is an infinite channel with semi-circular cross-section. The exact analytical solution, based on the assumption of potential fluid motion is presented, as well as numerical results for selected parameters.The most significant parameters are: the direction and frequency content of the seismic input; the radiation damping at the reservoir bottom; and the compressibility of the fluid. The response of the system depends strongly on the direction of the input ground motion. This is shown by the transfer functions as well as by the pressure time histories due to two earthquakes with different frequency content. The energy absorption at the reservoir bottom is important. A simple plane-wave model shows, that even for a rock foundation, the amount of transmitted energy can reach up to 80%. For comparison the case without bottom absorption is also shown. Compressbility has to be included to capture the resonance effects. The exact analytical solution is also used to verify numerical results obtained by a new method that combines a finite element model with a rigorous radiation boundary for the infinite channel in the time domain.     

A FE–BE method for dynamic analysis of dam–foundation–reservoir systems in the time domain

By coupling FEM and BEM, a numerical method was developed for dynamic response analyses of dam–foundation–reservoir systems in the time domain. During formulation, the weighted residual procedure was applied to the coupling of several equations of motion for solid and fluid in the FE and BE regions, and an algorithm similar to the Newmark beta procedure was finally obtained. The algorithm is advantageous in that it takes into account all the effects of dam–foundation, dam–reservoir and reservoir–foundation interactions, as well as of the absorption of both elastodynamic and hydrodynamic waves at the boundaries of the foundation and the reservoir. To demonstrate the validity of the present method, the impulsive response of a dam–foundation–reservoir system was calculated using the algorithm, and showed a good agreement with the existing results obtained by other researchers.     

Nonlinear earthquake analysis of high arch dam–water–foundation rock systems

A nonlinear finite element model for earthquake response analysis of arch dam–water–foundation rock systems is proposed in this paper. The model includes dynamic dam–water and dam–foundation rock interactions, the opening of contraction joints, the radiation damping of semi‐unbounded foundation rock, the compressibility of impounded water, and the upstream energy propagating along the semi‐unbounded reservoir. Meanwhile, a new equivalent force scheme is suggested to achieve free‐field input in the model. The effects of the earthquake input mechanism, joint opening, water compressibility, and radiation damping on the earthquake response of the Ertan arch dam (240 m high) in China are investigated using the proposed model. The results show that these factors significantly affect the earthquake response of the Ertan arch dam. Such factors should therefore be considered in the earthquake response analysis and earthquake safety evaluation of high arch dams. Copyright © 2011 John Wiley & Sons, Ltd.     

Earthquake analysis of gravity dams including hydrodynamic interaction

A general procedure for analysis of the response of gravity dams, including hydrodynamic interaction and compressibility of water, to the transverse horizontal and vertical components of earthquake ground motion is presented. The problem is reduced to one in two dimensions considering the transverse vibration of a monolith of a dam, and the material behaviour is assumed to be linearly elastic The complete system is considered as composed of two substructures—the dam, represented as a finite element system, and the reservoir, as a continuum of infinite length in the upstream direction governed by the wave equation. The structural displacements of the dam (including effects of water) are expressed as a linear combination of the modes of vibration of the dam with the reservoir empty. The effectiveness of this analytical formulation lies in its being able to produce excellent results by considering only the first few modes. The complex frequency response for the modal displacements are obtained first. The responses to arbitrary ground motion are subsequently obtained with the aid of the Fast Fourier Transform algorithm An example analysis is presented to illustrate results obtained from this method. It is concluded that the method is very effective and efficient and is capable of producing results to any desired degree of accuracy by including the necessary number of modes of vibration of the dam.     

Two‐dimensional modelling of ice cover effects for the dynamic analysis of concrete gravity dams

A two‐dimensional numerical model for determining the effects of the presence of an ice cover on the dynamic behaviour of large gravity dams is presented. Analytical predictions are compared to results obtained during a series of extensive dynamic tests on a large gravity dam. Data were obtained during summer and severe winter conditions to investigate the dynamic interactions between the dam, foundation, reservoir and the ice cover. The analysis includes ice‐reservoir interaction as well as the effects of water compressibility, flexible foundation and reservoir bottom absorption. Good agreement with the experimental findings is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

Analysis of absorption effects on the dynamic response of dam reservoir systems by boundary element methods

Using reciprocal theorems for dynamic and static boundary value problems, boundary integral equations are presented for wave propagation in elastic, isotropic media and compressible, inviscid fluids in the time domain as well as in the frequency domain. For the analysis of fluid–soil and fluid–structure systems, suitable coupling conditions are prescribed along the interfaces. The numerical treatment of the boundary integral equations consists of a point collocation and of a discretization of the boundary, in which constant and linear approximation functions are assumed. Step-by-step integration is applied to the time-dependent equations, where again the states are taken to be linear and constant over each time interval. These boundary element procedures are used to analyse the response of dams due to horizontal and vertical ground motions considering dam–water interaction and absorption of hydrodynamic pressure waves at the reservoir bottom or at the far end into the soil medium. Both the frequency response and the impulse generated transient response are investigated.     

Natural frequencies of arch dam reservoir system—by a mapping finite element method

A simple mapping finite element method is used to calculate the coupled natural frequencies and mode shapes of realistic arch dam reservoir systems in which the dam is circular cylindrical with non-uniform cross-section. This method, in which both the dam and the reservoir domains are mapped into geometrically simpler shapes using cylindrical-polar transformations, is found to give accurate results, achieved simply and economically. Results of analysis show that hydrodynamic interaction has a substantial effect on the coupled natural frequencies and mode shapes; also that the effect of water compressibility in the type of dams considered can be ignored without significant loss of accuracy. A simple method is also presented for predicting the water compressibility effect before undertaking detailed response analysis.     

地震SV波斜入射下沉管隧道的地震响应分析

在ABAQUS黏弹性人工边界时域波动方法的基础上,首先运用等效应力输入方法实现地震SV波倾斜入射,半空间算例验证该方法具有较好的计算精度,进而基于所建立的斜入射方法研究地震波斜入射对海河沉管隧道地震响应的影响。计算结果表明:SV波斜入射情况下,沉管隧道的地震响应规律与垂直入射时具有明显差异;随入射角增加,沉管隧道结构应力增大,应力较大点出现在沉管隧道的四个角点及隔墙与底板、顶板的连接处,其中中隔墙为最薄弱点;随入射角增加,侧墙和隔墙的相对最大水平位移增大,其中中隔墙位移最大;随入射角增加,沉管隧道结构竖向加速度峰值明显增大。因此在沉管隧道结构抗震设计中应考虑地震波斜入射的影响。     

Influence of seismic input mechanisms and radiation damping on arch dam response

In this study, two different earthquake input models are introduced, i.e. massless foundation model and viscous-spring boundary input model considering radiation damping. Linear elastic and nonlinear contraction joint opening analyses of the 210 m high Dagangshan arch dam under construction in China are performed using the two different earthquake input models. First, the responses of the three-dimensional (3-D) canyon without the dam are analyzed, respectively, with massless-truncated foundation and with viscous-spring boundary; second, linear and nonlinear analyses of the dam–foundation system are performed and compared by using the two input models. Hydrodynamic effects are considered using finite element discretization for incompressible reservoir fluid. It is concluded that stresses and displacements and contraction joint opening in the dam are significantly reduced both in linear and nonlinear analyses when using viscous-spring boundary model. Interestingly, in the case of linear analysis of the Dagangshan, the massless foundation input model with a relatively higher damping ratio of 10% leads to a comparable response of the dam to that using viscous-spring boundary model. In addition, the maximum tensile stresses from nonlinear analysis are 10–25% larger than that of the corresponding linear cases due to a partial release of the arch action.     

Three‐dimensional models of reservoir sediment and effects on the seismic response of arch dams

The important effects of bottom sediments on the seismic response of arch dams are studied in this paper. To do so, a three‐dimensional boundary element model is used. It includes the water reservoir as a compressible fluid, the dam and unbounded foundation rock as viscoelastic solids, and the bottom sediment as a two‐phase poroelastic domain with dynamic behaviour described by Biot's equations. Dynamic interaction among all those regions, local topography and travelling wave effects are taken into account. The results obtained show the important influence of sediment compressibility and permeability on the seismic response. The former is associated with a general change of the system response whereas the permeability has a significant influence on damping at resonance peaks. The analysis is carried out in the frequency domain considering time harmonic excitation due to P and S plane waves. The time‐domain results obtained by using the Fourier transform for a given earthquake accelerogram are also shown. The possibility of using simplified models to represent the bottom sediment effects is discussed in the paper. Two alternative models for porous sediment are tested. Simplified models are shown to be able to reproduce the effects of porous sediments except for very high permeability values. Copyright © 2004 John Wiley & Sons, Ltd.