Vibration tests and analysis of a model arch dam

Vibration tests were conducted on a 1/24-scale model of the North Fork Dam, a double-curvature arch dam, to determine natural frequencies, mode shapes and hydrodynamic pressures. The mode shapes and natural frequencies were determined from tests using two vibrators mounted on the crest of the dam. Hydrodynamic pressures at the dam/reservoir interface were determined from tests in which the vibrator was attached to the downstream foundation of the dam. The hydrodynamic pressures calculated using Westergaard's theory and a theory for arch dams developed by Perumalswami and Kar accurately predicted the measured pressure at frequencies below the first mode frequency of the dam. The differences in the two theories were insignificant. The Structural Analysis Program (SAP), a linear three-dimensional (3-D) finite element code, was used to compute mode shapes and frequencies for the dam with its base fixed and with a foundation. Numerical solution schemes used in the finite element analysis consisted of a Ritz analysis and a subspace iteration method. Calculations were conducted for both full and empty reservoir conditions. The accuracy of the Ritz analysis improved considerably as more nodes in flexible regions of the dam were loaded. However, the lowest eigenvalues were computed using the subspace iteration method. For the full reservoir, the natural frequencies decreased by 20-30 per cent when the foundation was included in the finite element model. The difference was less when the reservoir was empty. The calculations using the subspace iteration scheme and including the foundation agreed closely with experimental mode shapes and corresponding natural frequencies.     

Effective ambient vibration testing for validating numerical models of concrete dams

Ambient vibration tests were conducted on a 56 metre high concrete gravity dam to measure its modal properties for validating a finite element model of the dam–reservoir–foundation system. Excitation was provided by wind, by reservoir water cascading down the spillweir, and by the force of water released through outlet-pipes. Vibrations of the dam were measured using accelerometers, and 3-hour data records were acquired from each location. Data were processed by testing for stationarity and rejecting non-stationary portions before Fourier analysis. Power spectra with low variance were generated from which natural frequencies of the dam were identified clearly and modal damping factors estimated. Modal analysis of the frequency response spectra yielded mode shapes for the six lowest lateral modes of vibration of the dam. The finite element model for the dam was analysed using EACD-3D, and the computed mode shapes and natural frequencies compared well with the measured results. The study demonstrates that ambient vibration testing can offer a viable alternative to forced vibration testing when only the modal properties of a dam are required. Copyright © 1999 John Wiley & Sons, Ltd.     

Coupled hydrodynamic response of concrete gravity dams using finite and infinite elements

This paper presents the application of the finite element method for analysing the two-dimensional response of reservoir-dam systems subjected to horizontal ground motion. The interaction between the dam and the reservoir as well as the compressibility of water has been taken into account. The complete system has been considered to be composed of two substructures, namely the reservoir and the dam. To take into account the large extent of the reservoir, it has been idealized using specially developed infinite elements coupled with standard finite elements while the dam is represented using finite elements alone. Structural damping of the dam and radiation damping in the fluid phase have been accounted for in the analysis. It is concluded that the effect of radiation damping is considerable at high frequencies of excitation. The coupled response of the system is significantly large at and near the fundamental natural frequency of the system in comparison to the uncoupled responses. The method is computationally quite economical, capable of taking into account the arbitrary geometry of the system and is recommended for practical application. Further applications and extensions of the approach to three dimensional analyses are possible.     

Full-scale dynamic testing and analysis of a reservoir intake tower

Dynamic tests were conducted on a 50 m high intake tower at Wimbleball dam in the U.K. The results were compared against predictions from a corresponding numerical model. The aim of this work was to validate the assumption that the compressibility of the reservoir water is not a significant factor in the seismic analysis of intake towers. Three sets of tests were conducted on different occasions with different water levels in the reservoir. In the first two tests, modal characteristics of the tower were determined from the measured responses under ambient, hammer and human excitation. These results were used in planning the final set of tests where rotating eccentric mass exciters were used to vibrate the tower. Structural accelerations and hydrodynamic pressures were measured over the height of the tower for three important bending modes of vibration. The finite element method was used to develop a numerical model for Wimbleball tower. The tower was discretized with traditional solid elements and the reservoir with incompressible fluid elements. This model was analysed to predict the modal characteristics and harmonic responses of the tower and reservoir under the various conditions imposed during the dynamic tests. Theoretical predictions of the tower's accelerations and hydrodynamic pressures in the reservoir were compared against the test results. Excellent agreement was found for the natural frequencies and mode shapes while predictions of the harmonic responses were only fair. The observed responses of the tower and reservoir support the assumption that reservoir compressibility is not a significant factor in the seismic analysis of towers of this size.     

An experimental assessment of the added mass of some plates vibrating in water

A reservoir of water is contained by a concrete valley block, a ferrocement wall and a steel plate. Both wall and plate contain an array of pressure transducer sockets (Figures 1 and 2). Using the M.A.M.A.¹ equipment pure modes of vibration are excited. Frequency and mode shape are measured with the reservoir empty. When the reservoir is full hydrodynamic pressure is also measured. These hydrodynamic pressures are compared with Chopra's² two-dimensional, series solution, which includes compressibility of water, and with two- and three-dimensional finite element solutions of Laplace's equation, which do not include compressibility. Chopra's solution is unsatisfactory for modes which contain a vertical node line. The best agreement between experimental and theoretical hydrodynamic pressure is obtained when the latter is obtained from three-dimensional solutions of Laplace's equations, indicating that compressibility does not play a significant rǒle. This conclusion is supported by agreement between experimental frequencies (reservoir full) and those calculated using added mass obtained from the Laplace solution. Similar conclusions were reached from tests on a floating steel plate, suspended in the surface of the reservoir by a soft spring. Here, dynamic pressure measurements were not made, reliance being placed on agreement between calculated and measured frequencies and mode shapes.     

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.     

Modal analysis of a DAM–Foundation system based on an FE–BE method in the time domain

A modal analysis procedure based on an FE–BE method in the time domain is first formulated and then applied to a dam–foundation system. In the application, horizontal and vertical impulsive responses are calculated for the system having six different impedance ratios. Modal characteristics such as natural frequencies, damping ratios and mode shapes are evaluated from the Fourier spectra of the responses. The proposed procedure allows analysis of not only the underdamped but also the overdamped modes. According to the analysis, the radiation damping pertinent to the vertical vibration is half of that pertinent to the horizontal vibration and the interaction effect on the modes is not negligibly small when the impedance ratio exceeds 0·3.     

Hydrodynamic effects in dynamic response of simple arch dams

The dynamic responses of simple arch dams, with different radius to height ratios are analysed for three conditions: the dam alone without water, and the dam with full reservoir, considering water to be compressible in one case and neglecting water compressibility in the other case. The complex frequency response functions for accelerations at the dam crest due to the three components of ground motion—upstream-downstream component, cross-stream component and vertical component–are presented. Based on these results, the effects of dam-water interaction, of water compressibility, and of bank motions on dam response are investigated.     

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.     

Numerical simulation of earthquake excited dam-reservoirs with irregular geometries using an immersed boundary method

In this work, a ghost-cell immersed boundary method is proposed for the hydrodynamic response of earthquake excited dam-reservoirs. The numerical method employs a second order accurate two-step projection algorithm including compressibility effects in pressure field due to earthquake. The effects of reservoir bottom absorption are treated by introducing damping terms into the momentum equations. Hydrodynamic response of earthquake excited dam with a sloping face is simulated to demonstrate the accuracy of the present numerical method. Numerical results compared with previous numerical and analytical solutions show that the present immersed boundary method can accurately compute the hydrodynamic forces on inclined and curved dam faces including the effects of water compressibility and reservoir bottom absorption for the possibility of resonance. The proposed numerical method was shown to have significant advantages in computational time and memory usage for the hydrodynamic simulation of large dam-reservoirs with arbitrary geometries. Hydrodynamic forces on a double curvature arch dam subjected to real earthquake induced ground motion are also simulated to demonstrate the capability of the method.     

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.     

Nonlinear dynamic response of concrete gravity dams: cavitation effect

A finite element method for the dynamic analysis of concrete gravity dams is presented. Displacement based formulation is used for both fluid and structural domains. During severe ground motion, the impounding fluid in the reservoir may separate from the dam and cause forming of micro bubbles. As a result, the compressibility of water is reduced. This nonlinear phenomenon of the reservoir is termed cavitation. When the direction of the ground motion is changed, the micro bubble's region of fluid collapses, and an impact will occur. By using different damping ratios in the fluid and solid domains the spurious oscillations which were caused by the impact are removed. The cavitation is confined to the upper part of the reservoir, where it has an effect of paramount importance on the tensile stresses. To illustrate the cavitation effect, the response of the non-overflow monolith of the Pine Flat dam subjected to the first 6.5 s of the May 1940 El-Centro, California earthquake, is considered. In order that the cavitation phenomenon take place more widely, maximum acceleration was scaled to give an amplitude of 1 g.     

Experimental and finite element studies of the forced vibration response of morrow point dam

Measured accelerations and water pressures obtained during a recent forced vibration test on a large thin arch dam at high water are compared to predictions from a finite element model for which water compressibility is both included and neglected. The numerical model is calibrated using the antisymmetric response data because they are only slightly affected by water compressibility; good agreement is obtained. In the effort to reproduce the symmetric response data, for which water compressibility plays a strong role, the calibrated model shows better correlation when water compressibility is included, but the agreement is still inadequate. A successful isolation of the fundamental water resonance from the experimental data shows significantly different features from those of the numerical water model, indicating possible inaccuracy in the assumed geometry and/or boundary conditions for the reservoir. Some other results at low water level are also included.     

Free vibration of asymmetric shear wall-frame buildings

The Galerkin method of weighted residuals is used to determine the frequencies and associated mode shapes of asymmetric shear wall-frame structures. The governing equations are formulated using the continuum approach by idealizing the structure as a shear-flexure beam. Varying properties along the height of the building are considered. The effect of translational, rocking and torsional flexibilities of the foundation on the natural frequencies is also investigated. The method presented herein utilizes polynomial and transcendental displacement functions, and is found to be simple, versatile and efficient.     

Seismic safety analysis of tall concrete dams,investigation and insights on critical challenges

This paper discusses critical and potentially controversial issues related to the seismic safety of tall concrete dams. These include the seismic input at a dam site, the effective treatment of the damage-rupture process, and the consideration of compressibility of reservoir water for hydrodynamic pressure. Major challenges to currently popular but questionable treatments of these critical problems are presented. Insights and additional research on these critical challenges are emphasized and explained based on prior published works of the author. More reasonable alternatives to dealing with these potentially controversial problems are provided in light of engineering practice in China. First, the design seismic input at depth as deconvoluted from an arbitrarily selected recorded accelerogram at a control point of an artificially developed free-field surface with the elevation of the dam crest is difficult for engineering projects to accept as appropriate. It may be more reasonable to use the design seismic incident motions as half of the ground surface motions from seismic safety analyses obtained from deterministic or probability approaches conducted by seismologists according to approved standards or guidelines. Second, since seismic damage to the dam must be estimated separately following uniaxial tensile and compressive experimental damage evolution rules, a simplified and realistic nonlinear elastic model is proposed as an alternative to the plastic-damage coupling model, which is very complex and includes assumptions based on a number of uncertainties. Finally, the effect of the reflection coefficient for compressibility of reservoir water on hydrodynamic pressures is very sensitive. The notion that the applied unified reflection coefficient at the reservoir bottom could be frequency-dependent and exhibit a significant variability in space as confirmed by field tests is questionable. To neglect the compressibility of reservoir water it may be closer to engineering practice at present.     

Dynamic response of a staggered wall-beam structure

A method for the dynamic analysis of staggered wall-beam frames is developed using consistent mass terms which are derived and given in simple terms. The method uses effective stiffnesses for wall-beam elements developed in an earlier paper. Experiments using a nine storey 1 : 15 scale perspex model are described. The first three natural frequencies of the model were obtained using two methods: sinusoidal external excitation of the structure with the base fixed and white noise excitation employing a single degree-of-freedom shake table, in the latter method with and without the addition of mass throughout the model. Agreement between analytical predictions of the first three natural frequencies and mode shapes and experimentally determined values is considered satisfactory, particularly for the first two modes. The lumped mass assumption gave reasonable results for these two cases, whereas the consistent mass theory gave reasonable results for the first three natural frequencies.     

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.     

CFRP composite retrofitting effect on the dynamic characteristics of arch dams

The purpose of this study is to investigate the effect of retrofitting dynamic characteristics of a damaged laboratory arch dam model, subsequently repaired with high-strength structural mortar and strengthened with composite carbon fiber reinforced polymer. This study constructed in laboratory conditions is a prototype arch dam–reservoir–foundation model. Five test cases of ambient vibration on the arch dam model illustrate the changes in dynamic characteristics: natural frequency, mode shape, and damping ratio, before and after retrofitting. The ambient vibration tests collected data from the dam body during vibrations by natural excitations which provided small impacts and response signals from sensitivity accelerometers placed at crest points. Enhanced Frequency Domain Decomposition Method in the frequency domain extracts the experimental dynamic characteristics. At the end of the study, experimentally identified dynamic characteristics obtained from all test cases have been compared with each other. Apparently, after the retrofitting, the natural frequencies of the dam body increased considerably, demonstrating that the retrofitting, including repairing and strengthening is very effective on the flashback of initial dynamic characteristics.     

A semi-analytical method for predicting the outflow hydrograph due to dam-break in natural valleys

The paper presents a semi-analytical method for predicting the flow rate hydrograph due to a hypothetical sudden and total dam failure in a natural valley. The method generalizes the approach proposed by Hunt for the dam-break problem in a rectangular frictionless sloping channel to a valley with a cross-section area expressed by a power-law function of water depth, in order to take into account the most common shapes of natural valleys. The parameters of the deriving model can be set by exploiting data usually available concerning the dam section geometry and the reservoir storage-depth curve. The application of the technique to three different reservoirs is discussed. The results show that the flow rate hydrographs obtained at the dam site agree with the ones calculated by means of a finite volume numerical code based on two-dimensional shallow water equations. The method requires moderate computational and data collecting effort, so it can be regarded as a useful alternative to other procedures commonly adopted in the practice.