Abstract
Regarding the importance of the interaction effect in the dynamic behavior of a concrete dam and reservoir, the process of hydrodynamic wave propagation in the reservoir and its effects on the response of a concrete gravity dam are examined in this article by presenting a new probabilistic model. For this purpose, a program has been written in an APDL environment of Ansys software capable of dynamic analysis of a system and presenting probabilistic models. The Monte Carlo method with the LHS method has been applied for the probabilistic model, in which the essential parameters in the dam-reservoir interaction problem include excitation frequency, reservoir height, and reservoir length as input variables, and the effect of their changes on the hydrodynamic wave propagation process in the reservoir and the dynamic response of the dam are evaluated. For a better understanding of the dam-reservoir interaction concept and to show the capability of the presented model, two simple and isolated models of the concrete gravity dam and reservoir system have been considered. According to the results, the distribution trend of hydrodynamic pressure along the reservoir can be obviously recognized. Also, the results present the dependence of the responses and the propagation status on the excitation frequency and its relationship with the system frequency and the reservoir natural frequency.
1. Introduction
The interaction of the dam and reservoir is one of the outstanding issues in the seismic design of concrete dams. The dam-reservoir interaction generates hydrodynamic pressure during the ground motion that interacts with the dam and influences the response of the system. In dynamic analysis of the dam-reservoir system, the water compressibility and dam motion hydrodynamic pressure is induced and propagated upwards, but the distribution conditions and its dissipation process depend on the excitation frequency and the reservoir conditions.
Westergaard [1] conducted the first research on modeling the dam and reservoir interaction effect by introducing the added mass model. His model is a two-dimensional system of a rigid dam and reservoir affected by the horizontal ground motion. Westergaard revealed that the interaction force depended on the acceleration caused by the ground vibration, and its distribution on the upstream face of the dam is almost parabolic [1]. An essential disadvantage of the rigid Westergaard model is that the model considers a rigid dam and ignores the compressibility of water and it is restrict to specific vibration frequencies. The model does not consider any propagation and dissipation of compressive pressure waves.
By introducing a model similar to the Westergaard model, Kotsubo revealed that if the first specific frequency of the reservoir is significantly higher than the dominant frequency related to the seismic ground motion, the Westergaard added mass approximation would be validated [2]. Chopra also examined the effects of water compressibility on the force caused by the interaction of the dam and reservoir by introducing a model similar to the Westergaard model and determined that the waves propagate in the compressible water and the interaction forces are dependent on the acceleration that is imposed on the fluid [3].
The dam has been assumed to be rigid in most of the models presented in the field of interaction, and the response is not involved in calculating the interaction forces. Chopra and Chakrabarti[4] investigated the effects of dam flexibility on the interaction force. They explained that the natural frequencies of the coupled dam-reservoir system are varied from the noncoupled system. Chopra and Chakrabarti revealed that if the ratio of the first specific frequency of the dam to the first specific frequency of the reservoir is greater than two, the effects of water compressibility can be ignored. They concluded that if the effects of the dam and reservoir interaction are considered in the model, the dam response will be considerably increased [4].
The presented analytical solutions in the previous studies can uniquely be applied for the simple geometry and boundary conditions of the reservoir. For very complex geometries, hydrodynamic wave equations can be solved numerically with proper discretization for the reservoir amplitude. The features related to the infinity of the reservoir require to be presented by choosing a suitable numerical model at the limit distance so that analysis can be performed based on the absence of reflected waves from the far boundary.
Special methods are required to correctly solve the dam interaction with the reservoir, in which the analysis domain is much greater than the structure. The direct modeling of the reservoir with numerical methods needs much computational effort due to the infinity of the reservoir. Hence, the dam and reservoir model turns into a minor system in which proper boundary conditions have been imposed on it.
The infinity of the reservoir length is very significant in wave dynamics. The waves moving towards the infinite boundary are not reflected to the studied range in these conditions. The energy transfer from the upstream face of the dam to the far field boundary is irreversible due to wave transmission and damping. It is required that this range is truncated at a limited distance for numerical modeling, and the condition of proper propagation is applied at the truncated location according to the infinite length of the reservoir.
The propagation condition and location of the truncated boundary for the infinite reservoir are essential to calculate the propagated pressure waves in a numerical model. Various studies have been conducted to examine and define the propagation behavior of hydrodynamic pressure waves caused by dam and reservoir interactions. The boundary condition available to calculate an infinite domain with a truncated boundary has been the subject of multiple studies in continuous mechanics [5–7].
Zienkiewicz and Bettess [8] introduced a model for the condition of the infinity of the reservoir to solve the pressure wave equation in the reservoir. They explained that the Sommerfeld boundary condition is proper for a very long reservoir model and could be included in the discretization of the finite element model of the fluid domain. According to studies conducted by Zienkiewicz and Bettess, applying the Sommerfeld boundary condition can inhibit the reflection of pressure waves at the truncated boundary in the vertical direction [8].
Studies reveal that reservoir condition is a significant factor influencing the pressure wave propagation process and the dam response, particularly in numerical models. In numerical models, the infinite length of the reservoir requires to be truncated in a finite distance, and a proper boundary condition must be applied at the truncated boundary. The reservoir truncated location and applying the appropriate boundary condition had been significant challenges in preceding research. One of the most proper boundaries for the appropriate truncated boundary is the Sommerfeld boundary condition, which operates as a damper at the boundary [9, 10].
Bayraktar et al. studied the effect of the reservoir length on the seismic response of gravity dams to near- and far-fault ground motions using the finite element method. They considered the linear and nonlinear behavior for body concrete. Obtained results illustrated that the length of the reservoir affects the seismic responses, considerably. Also, the induced stress on the dam body is more due to near-fault ground motions in comparison with far-fault ground motions [11].
Consequently, it can be stated that it is essential to choose the length of the reservoir rationally in addition to defining the appropriate far field boundary so that sufficient accuracy is provided for the responses and the computational effort not to be required excessively. Also, the reservoir water level plays an essential role in developing the hydrodynamic force generated in the reservoir.
Nowadays, energy dissipation used in the structures with application of seismic isolation and protection systems. One of the factors in the economic project of new concrete dams and safety valuation of available dams in seismic areas is the control and dissipation of the induced hydrodynamic pressure induced by the dam-reservoir interaction. To reduce the dam-reservoir-foundation interaction effect on seismic responses, hydrodynamic dampers was used on the upper face of a dam body connected to the reservoir. The effects of the soft material on reducing the seismic response of the dam were investigated by Hall et al. They reported promising results in the reductions of the hydrodynamic force exerted on the dam due to the effect of the isolation layer [12, 13]. Hatami studied the reduction of hydrodynamic pressure using an isolation layer as an absorbing boundary at the contact site of the dam and reservoir using analytical solutions in the frequency domain [14].
Khiavi et al. studied the effect of isolation layer on seismic optimization of concrete gravity dams using rubber damper in the upstream face of the dam. They applied the probabilistic method using Monte Carlo simulation for geometry optimization of a rubber damper. The details of rubber damper characteristics and their effect on the seismic performance of the system have been presented in the research by Khiavi et al. [15].
The research shows that one of the important discussions in the field of the effect of the dam-reservoir interaction on the dynamic response is the problem of the effect of the truncated length of the infinite reservoir in the numerical model and also the effect of the vibration frequency on the dam response.
Such arguments have been considered in previous studies, considering the simple geometry and boundary conditions for the dam-reservoir system using analytical solution.
It is obvious from the studies that an analytical solution was not possible for complex geometry and boundary conditions and it needs a numerical solution. Due to the high computational effort in numerical methods, it is possible to perform parametric dynamic analysis for special cases.
This article presents the Monte Carlo method as an effective tool for optimizing and designing in the uncertainty space by FE-ANSYS. ANSYS is a software program based on the finite element method and is used to give more accurate results for complicated geometries. In this method, the geometry is divided into small parts and the boundary conditions are applied. Finally the results are obtained for each node or element [16].
In the finite element model of the dam-reservoir system, it is necessary to cut the reservoir to a suitable length until the pressure waves to the upstream of the reservoir are dissipated correctly. Also, the effect of excitation frequency is due to its relationship with the normal frequency of the reservoir and the possibility of the occurrence of a resonance that can significantly increase the response of the system.
The reason for considering the reservoir water level as one of the variables is that the effect of the dam and reservoir interaction is important for large volume of water and considering the compressibility of water, which is directly related to reservoir height. So, in the probabilistic model of the system, the reservoir length and height and the vibration frequency are considered as input variables and their variation on dynamic performance of the dam and reservoir are investigated.
In recent years, various tools and methods have been presented to predict the behavior of models and optimization, among them are the machine learning method and the use of probabilistic models. The use of such methods reduces the computational effort and analyzes and makes it possible to generalize the results for wider ranges. In recent years, the method of machine learning to optimize engineering models and uncertainty analysis with predicting has received attention, among which we can mention Fu et al. research in the field of improving the performance of energy systems and distribution networks [17–19]. They used the statistical machine learning model to optimize the distribution network performance and capacitor planning considering uncertainties in photovoltaic power. They also used the second-order reliability method to estimate the failure probability of an energy system and proposed a suitable method with high accuracy and low computational effort. The performance of the probabilistic model is similar to machine learning. With the difference that in machine learning, data training is needed, but in probabilistic methods, probability distribution models are used for prediction and analysis.
1.1. Probabilistic Analysis
One of the new techniques in design is based on the possibilities and considering a variable range for design parameters. Similarly, one important aspect of performance-based earthquake engineering is an accurate estimate of seismic response and structural capacity. There are two major sources of uncertainty in seismic performance of the structure: the uncertainties associated with the randomness and physical uncertainties caused by modeling assumptions, deletions, or existing errors. Adequate understanding of responses expected from the structure can be influential on the probabilistic analysis regarding the structure of the uncertainty. Calculating different types of seismic performance given the physical uncertainty commonly involved the use of safety factors or standard dispersion criteria. A very robust method given both sources of uncertainty in earthquake engineering is probabilistic analysis using Monte Carlo simulation [20].
Several studies have been conducted on this type of analysis. Altarejos et al. applied a Monte Carlo simulation to study the reliability of a concrete gravity dam against slip. They considered ten parameters of variables in their study and determined the results as a graph of the probability of failure against slip for each variable [21]. Mirzabozorg et al. used the Monte Carlo method to create nonuniform three-dimensional ground motion to compare the effect of uniform and nonuniform ground motion on the performance of the Dez arc dam. Their research revealed that the responses obtained from considering nonuniform ground motion are different from considering uniform and can increase the responses of system structures [22]. Alembagheri and Seyedkazemi used Monte Carlo probabilistic analysis and the Latin hypercube sampling method and determined that the modulus of elasticity and tensile strength of concrete play a more critical role in the ultimate strain ratio against earthquakes [23]. Pasbani Khiavi considered the effect of hydrodynamic pressure absorption in the reservoir bed using the probabilistic analysis in assessing the seismic response of the dam. His research showed the trend of the effect of variation of the reservoir bottom absorption coefficient on seismic responses [24].
Studies revealed that the Monte Carlo probability analysis, assuming uncertainty due to the random nature of materials and geometry, has been extensively applied in structural uncertainty analysis. Researchers mainly attempt to provide solutions to use algorithms properly in software to optimize structure behavior.
In this article, a probabilistic analysis of Monte Carlo based on the LHS method is provided with the writing code in the ANSYS APDL to understand the effects of dam-reservoir interaction and the process of wave propagation in the reservoir by studying the seismic sensitivity of the concrete gravity dam to the excitation frequency and the height and length of the reservoir for models without and with the isolation layer. It should be remarked that the details of the Monte Carlo method have been stated in the Ansys software manual [16].
According to the discussed contents, in this article the wave propagation process and the reservoir effects for different ranges of excitation frequency are examined by introducing a new probabilistic model utilizing the Monte Carlo and LHS methods using Ansys software. For this means, a program has been written in the APDL environment in the Ansys software.
For a better understanding of the dam-reservoir interaction concept and to show the capability of the presented model, two models of the concrete dam have been considered: The simple model and the model of dam-reservoir-foundation with the isolation layer in the upstream face.
The adjusted program has dynamic analysis capability with applying the interaction effects of structure and fluid, and different probabilistic models can be defined in it with different input parameters.
The model is first analyzed considering two cases of static analysis and dynamic analysis for a rigid dam to validate the accuracy of the presented program. Then, the effect of reservoir length and height and the effects of the loading frequency on critical responses are examined.
In this research, the Monte Carlo method is used to study of the changes in important parameters on the concrete gravity dam and reservoir system considering interaction effects. A key feature of the Monte Carlo simulation is that sampling points are located in the space of random input variables. The Monte Carlo simulation produces a random sample of N points for each input variable and specifies the corresponding result for each of the N values. Then, it processes and displays the results as standard deviations, sensitivity, and probability density curves. To describe the scatter of the data, the uniform distribution has been used.
For probabilistic simulation, the excitation frequency and reservoir height and length have been introduced as input variables of the uniform distribution. Additionally, the maximum displacement of the dam crest, the principal tensile and compressive stresses of the dam heel and toe, and the hydrodynamic pressure at the heel of the dam have been selected as output variables.
In probabilistic analysis, the number of required simulations should be such that the average value of the output variable reaches the appropriate convergence according to the number of simulations. So, in this study the Ansys software settings are selected for the Monte Carlo probabilistic method according to Table 1:
2. Dynamic Equations
If the reservoir water is assumed to be a compressible, nonviscous fluid, the hydrodynamic pressure P caused by the ground motion in the upstream face of the dam is represented by the Helmholtz equation as follows [4, 9, 10]:
In the above equation, is the sound velocity, is the Laplacian operator in the two-dimensional state, and t is also the time variable. The boundary conditions of the reservoir are defined as follows:(i)At the interface of dam-reservoir and reservoir-foundation, the boundary condition is defined as follows: In the above equation, is the outgoing normal vector at the solid and fluid interface and the normal component of the acceleration at the solid and fluid interface and ρ is the fluid density.(ii)At the far end of the reservoir, the Sommerfeld boundary condition is used as(iii)If the effect of surface waves is neglected at the free surface of the reservoir, the boundary condition will be defined as
Equations (1)–(6) can be written in the following matrix form using the finite element method [9]:where , , and are called the mass, damping, and stiffness matrices of the fluid, respectively. , ,and are the node pressure vector, node acceleration vector, and ground motion acceleration, respectively.
The dynamic equations of the finite elements of the structure, which involves the dam and the foundation, when exposed to ground motion, are expressed by the finite element method as follows:, , and are the mass, damping, and stiffness matrices of the structure, respectively, and is the hydrodynamic pressure generated by the solid and fluid interaction.
Details of the finite element model of the governing equations of the system have been provided in the Ansys software manual.
3. Case Study
The Pine Flat dam model with a height of 122 m, reservoir with a height of 116 m, and flexible foundation has been selected as a case study. Figure 1 shows the geometric properties of the considered model, where all dimensions are in meters. The characteristic of material parameters of the system have been shown in Table 2. The speed of sound waves in water is 1438.66 m/s and water density is 1000 kg/m3.
The massless foundation model has been used. The mass-less foundation model is the best model for expressing the foundation behavior in interaction issues according to previous studies the flexibility of the foundation is considered in the simple massless model, while the effects of inertia and damping are eliminated. The size of the massless foundation model does not have to be very large and should only provide an acceptable estimate for the flexibility of the foundation. Therefore, in order not to consider the mass of the foundation, its density is considered zero.
The Newmark method was used for numerical integration in which its parameters were selected as β = 0.25 and γ = 0.5 and a time step of = 0.05 sec has been selected for analysis. Latin hypercube sampling (LHS) method has been used in Monte Carlo probabilistic analysis. Figure 2 shows the discretization of the finite element model of the case study:
For dynamic analysis, a horizontal component of harmonic excitation is applied to the system defined as follows:where is the applied acceleration, is the acceleration of gravity, ω is the excitation frequency, and t is time for dynamic analysis.
4. Modeling and Analysis
A program has been written in Ansys software in APDL environment for modeling and seismic analysis of the studied system and presenting a probabilistic model. In the model, the interaction among the dam, reservoir, and foundation has been considered by using the proper components that determine the fluid compressibility behavior. According to the conditions of the concrete gravity dam behavior and the geometric shape of the reservoir, the dam model has been recognized as two-dimensional with plane stress behavior. SOLID182 elements have been applied to discretize the solid parts of concrete, and FLUID29 elements have been applied for the fluid part, and for adjacent and nonadjacent fluid of the structure, which is a suitable element to display fluid compressibility.
4.1. Probabilistic Analysis
The considered model has been influenced by the horizontal acceleration to better understand the sensitivity of the dam-reservoir-foundation system to different loading frequencies and reservoir conditions.
In the first step, modal analysis of the system is performed and the model frequencies are extracted to compare the system frequency and the natural frequency of the reservoir with the excitation frequency and their effect on the system response and pressure propagation process should be examined. Table 3 exhibits the first frequencies of the model. It should be remarked that the natural frequency of the reservoir is equal to 19.49 and has been obtained from , where C = 1440 m/s shows the sound wave velocity and H = 116 m is the maximum height of the reservoir.
4.1.1. Model Validation
In the first step to validate the prepared program, the model without the isolation layer is analyzed for both static and dynamic cases, assuming the rigid dam and compressible fluid and the results are compared with the analytical solution and the Westergaard model to assess the accuracy and capability of the presented numerical model in dealing with the problem of the dam and reservoir interaction. Figures 3 and 4 reveal a comparison of the time history response of the static and dynamic pressure with the analytical solution for the considered model.
According to Figures 3 and 4 and comparing the curves, it is possible to understand the reasonable accuracy of the presented model for problems in which the interaction of solid and fluid is considered.
4.1.2. The Effect of Excitation Frequency
In this section, the proposed model is reviewed and analyzed, assuming that the fluid inside the reservoir is compressible, which is the natural state. The effect of the excitation frequency on the dynamic response is estimated using the proposed probabilistic model in the first step of sensitivity analysis. For this purpose, the height and length of the reservoir have been considered constant and equal to 116 m and 488 m, respectively. The loading frequency has been selected as the input variable, and the sensitivity curve of the responses concerning the frequency changes is extracted. Figures 5–8 provide the sensitivity of critical responses to the excitation frequency for models without and with upstream isolation.
The critical case in Figures 5 and 8 is that the curves have a maximum value and a comparison of frequency values at these maximum points shows that these frequencies are approximately close to the first frequency of the system, and in this case, resonance occurs and the responses reach to critical values.
4.1.3. Reservoir Height Effects
The reservoir height is selected as the input variable in the next step to show the effect of water level on the dynamic behavior of the model of the simple and isolated concrete gravity dam. The sensitivity of responses is investigated for frequencies smaller and more significant than the reservoir natural frequency. Figures 9–16 show the sensitivity curves of the responses to changes in the reservoir level for different excitation frequencies ( and ).
According to the curves, it is possible to see the effect of the height of the reservoir on the response of the model for excitation frequencies smaller and larger than the natural frequency of the reservoir. For a frequency greater than the natural frequency of the reservoir, the model shows a different and irregular behavior.
4.1.4. Reservoir Length Effects
In the next step, the reservoir length is selected as the input variable to better present the propagation process in the proposed probabilistic model, and the sensitivity of responses to length changes is investigated for different frequencies, including frequencies smaller and larger than the reservoir natural frequency to identify the wave propagation trend and assess the effect of the truncated length of the reservoir on the dynamic representation that was a challenge in the past research.
Figures 17–24 show the sensitivity curves of the responses to changes in reservoir length for two cases of excitation frequencies.
According to the curves, the effect of reservoir length on the responses can be recognized. Figure 17 shows that for , i.e., the excitation frequency smaller than the natural frequency, no pressure distribution occurs, and the behavior of the system is similar to that of an incompressible fluid. In this case, the pressure is nearly completely absorbed in a short distance from the dam. Therefore, if the truncated boundary is selected outside the dissipation zone, the hydrodynamic behavior of the reservoir will not be affected, even if there is no absorb boundary.
While the conditions are different for excitation frequency values larger than the natural frequency () and the pressure is propagated along the reservoir to upstream. Hence, the model presents a different behavior for this case and the hydrodynamic pressure in the most distant parts is also significant.
As shown in Figures, the curves of case fluctuate in the short length of the reservoir, which can be due to the reflection of hydrodynamic pressure waves from the truncated boundary. In this case, the pressure is not entirely dissipated to the upstream.
4.2. Hydrodynamic Pressure Distribution
Since the hydrodynamic pressure distribution in the reservoir is essential in interaction problems, the hydrodynamic pressure distribution contour (in Pa unit) for the incompressible water and compressible water under excitation with different frequencies states has been established for models and displayed in Figures 25–28 for a better presentation of the pressure propagation.
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Figures 25 and 26 illustrated that for an incompressible fluid, no pressure distribution occurs for both cases of excitation frequencies.
According to contour 27, it can be observed that the hydrodynamic pressure wave does not propagate upwards for the excitation frequency lower than the reservoir natural frequency and is absorbed at a short distance from the dam. In this case, the behavior of the system is similar to that of an incompressible fluid, while Figure 28 shows that the conditions are varied for the excitation frequency higher than the reservoir natural frequency, and the reservoir is influenced by hydrodynamic pressure waves even at a distance far from the dam.
Also, according to the contours, it can be seen that in the incompressible fluid model, the isolation layer does not have much effect on reducing the response.
The reason for this is that in the incompressible fluid model, there is no special interaction between the dam and the reservoir. While in the compressible fluid model, the isolation layer significantly reduces the hydrodynamic pressure applied from the reservoir to the dam due to the interaction.
5. Discussion
In this study, the effects of excitation frequency and reservoir height and length on the behavior of the dam-reservoir-foundation system were examined by presenting a probabilistic model with programming in the APDL environment of Ansys software. The model was analyzed by considering the range of changes for excitation frequency and reservoir height and length, and sensitivity curves were extracted. The model results provided a clear and proper understanding of the dam-reservoir interaction and the process of propagation of hydrodynamic pressure waves in the reservoir, along with the effect of compressibility and excitation frequency. According to the results, it is probable to explain the relationship between the excitation frequency, system frequency, and reservoir natural frequency in the dynamic behavior of the system, as well as the pressure wave propagation process for different frequencies. The results show a different representation of the hydrodynamic wave propagation caused by the interaction for the excitation frequency lower and higher than the reservoir natural frequency. While wave propagates over a significant length of the reservoir for the frequency higher than the reservoir natural frequency, it is required to select the proper length for the reservoir for dynamic analysis of the system in the numerical model. The next point is that the sensitivity curve shows that a resonance happens, and the response reaches its maximum value when the excitation frequency is close to the main frequency of the system.
The proposed model has many capabilities, including the following:(1)According to the wave propagation process and the responses obtained, the truncated location of the reservoir can be determined in numerical models for any vibrational frequency. This has been an essential challenge in the past research in discussing the interaction of the dam and reservoir in numerical models.(2)The advantage of using the upstream isolation layer in the concrete dam and reservoir system for all types of vibrations and any height and length of the reservoir can be observed.(3)The effects of the reservoir level on the dynamic response of a concrete dam for various vibrational frequencies can be clearly seen.(4)The effects of the loading frequency on the dynamic performance of the system can be seen and investigated the critical situation at the time of resonance where the vibration frequency approaches the system frequency.
Finally, it can be concluded that the prepared model is capable of considering all cases related to the dynamic analysis of the concrete gravity dam and the uncertainties in the analysis.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.