do not necessarily reflect the views of UKDiss.com.
THE INFLUENCE OF DAM-FOUNDATION INTERACTION EFFECTS ON THE SEISMIC RESPONSE OF EARTH DAMS
Earth dams play a major role in the management of water resources, including water supplies, flood controls and hydropower, therefore the failure of such structures is likely to cause large socioeconomic damage. With the aftereffects of catastrophic earthquake events, it is of public and industrial interest to have a better understanding of the seismic response of earth dams. This dissertation explores the effects of dam-foundation interaction on the modal response of the La Villita earth-filled dam.
The research uses finite element capabilities and an analytical method based on the shear beam model. The finite element methods include a modal analysis using ANSYS’s Modal Analysis System, and a linear dynamic time-domain analysis. Two parametric studies are investigated; the effects of varying of thickness and shear modulus of the foundation layer.
The results of the different modal frequencies of the dam and response spectra of the various scenarios are found. The findings are conclusive on the significant damping effects of the foundation layer, however the shear modulus study showed little impact on the overall response.
Table of Contents
List of Figures
Figure 3 Comparison of a typical seismic code design spectrum (bold, solid line) for soft soils to actual spectra observed from catastrophic earthquakes presented for spectral amplification (Gazetas, G. and Mylonakis, G., 2001)
List of Tables
Earth dams play a major role in the management of water resources, including water supplies, irrigation, flood controls and hydropower. They are a type of embankment made from compacted earth with the core consisting of an impermeable material surrounded by pervious material.
Earthquakes are known for their catastrophic aftereffects. They occur when there is a release of energy in the earth’s crust. The amplitude of an earthquake is commonly measured using the Richter scale which quantifies the magnitude logarithmically. Each magnitude relates to a quantity of energy released.
The seismic response of a structure is concerned with how it reacts to seismic loads or ground accelerations caused by earthquakes. Unlike most structural loads, seismic loads are transient, usually acting over a short period of time (between a few seconds to a few minutes). The proximity to the earthquake source will determine the magnitude of shaking felt by the structure (the amplitudes of the seismic waves).
The seismic studies of earth dams are of significance, as failures of such structures are likely to result in massive socioeconomic damage; the failure of Teton Earth Dam in 1976 resulted in the death of 14 persons and an estimated property damage of up to $1 billion (Wallace, C.L., 1976). Subsequently, it is of public and industrial interest to have an increased understanding of how earth dams respond to seismic loads.
Due to the significance of seismic design and the increased requirements for dams to resist seismic effects, in addition with the technological advancements allowing for more in-depth analyses, there have been various studies dedicated to this topic in the past 40 years, for example:
- Use of finite element in predicting displacements;
- Dam-valley interaction effects;
- Dam-reservoir interaction effects.
As well as valley and reservoir interaction effects, dam-foundation interaction effects, or more commonly known as soil-structure interaction, need to be considered. It assesses the influence of the soil layer beneath a structure on its response to loading, and vice versa. The properties of the dam and foundation material play an important role in this as they govern the stiffness’s of the material(s), which will alter the response of the dam through phenomena such as damping and resonance.
This dissertation uses a review of literature to establish the current knowledge regarding the seismic response of earth dams with a focus on dam-foundation interaction effects, and the phenomena which accompany it. Failure modes of earth dams are explained to give understanding of how seismic loading may affect the structure. The importance of dam-foundation interaction is explained and shown through soil-structure interaction case studies. Different parameters are also discussed with their effects on seismic response. Finally, current methods of seismic analysis are described which lead onto the methods used in this study.
The study conducts two parametric experiments into the dam-foundation interaction effects on an existing earthen dam, the La Villita dam in Mexico. They investigate the influence of the foundation layer on the dam’s modal response by varying 1) the thickness of the foundation layer and 2) the foundation material’s shear modulus. The study uses an analytical method based on the shear beam model, and numerical methods using finite element capabilities to find the modal frequencies and spectra of the dam under the different scenarios, which are then compared to find any trends.
This dissertation seeks to accomplish the following:
- To review current knowledge on the seismic response of earth dams with a focus on dam-foundation interaction;
- To evaluate current methods of seismic analysis of earth dams;
- To investigate dam-foundation interaction effects on the La Villita dam through modal analyses, by undertaking parametric studies on;
- The thickness of the foundation layer;
- The varying stiffness’s of dam and foundation material.
There are a range of dams consisting of different designs and materials, each with unique failure modes. Despite variations in earth dams, their common failure modes can be largely contributed to the following three problems: hydraulic, seepage or structural issues.
- Hydraulic – Water can cause erosion on the surface of the dam. If this is not maintained, gullies may eventually form which can induce localised slip failures. Erosion of the toe can also lead to failure as the dam relies on friction at the base to prevent sliding. If the spillway capacity is exceeded, water will overflow the dam causing erosion at the crest, this can lead to overtopping; earth dams are especially susceptible to this kind of failure. Statistics for large embankment dams up to 1986 show that failure due to overtopping contributed to 35.9% of all known failure cases, whilst sliding contributed 5.5% (Foster et al, 2000).
- Seepage – Small concentrated or uncontrolled channels may form within the dam, transporting material, this is called seepage. This process will gradually increase the size of the channel and could lead to wash out of the dam. This can also occur in the foundation. Concentrated seepage may eventually cause piping, resulting in foundation material being washed downstream. This loss of material can result in the dam settling or sinking. Statistics also show that failure due to piping through the embankment and foundation contributed to 30.5% and 14.8% respectively to all known failure cases (Foster et al, 2000).
- Structural – Common signs of structural failure are the presence of cracks or movement of the dam such as settlement or sliding. For example, shearing failure in the dam can cause sliding in the embankment or foundation.
Earthquakes can cause a combination of failure modes to occur, for example:
- Overtopping due to breaking waves or settlement of the dam from seismic motion.
- Cracks may form due to displacements in the ground, leading to leakages and piping failure.
- Seismic motion is may cause structural failure, such as a crack, this may shorten a seepage path resulting in piping failure (Ohio Department of Natural Resources Division of Water, 1994).
- The loss of freeboard can also occur due to movement from slope failures or differential tectonic movements.
A phenomenon can occur where due to the shaking of the ground, substantial loss of strength in a saturated or partially saturated soil is experienced, this is called liquefaction. This can take place in the embankment dam or foundation soil. It was not until the mid-1960’s that liquefaction was considered important when carrying out seismic design. This was due to two major earthquake events in 1964; the Good Friday earthquake in Alaska (West M.E., 2014) and Niigata in Japan (Japan National Committee on Earthquake Engineering, n.d.), where wide spread liquefaction and subsidence occurred, causing severe damage to buildings. It was also during this period that the importance of instrumentation to record strong ground motions was recognised to monitor dam response as a consequence of liquefaction in the lower dam of Upper San Fernando dam, where soil softening was induced (Marcuson, W., Hynes, M. and Franklin, A., n.d.). Liquefaction was believed to only occur in sands until the 1983 Borah Peak earthquake where major causes of damage was due to the liquefaction of saturated granular deposits (Stokoe et al, 1988), and so it was recognised that liquefaction could be exhibited in gravelly soils too.
When an external force drives at a frequency equal to another system’s modal frequency (also known as resonant frequency), large amplitudes of oscillations occur within the system, this phenomenon is called resonance. At these frequencies, the system is more able to store and transfer energy. This means even small, periodic driving forces have the ability to produce large displacements in structures. Figure 1 presents a typical idealised spectrum showing the first three modal frequencies of a system. Prominent peaks correspond to the distinct modal frequencies, and it can be seen that a small change in frequency near a modal peak causes a large change to the amplitude. The first modal frequency is called the fundamental frequency, and the other modal frequencies can be referred to as the harmonic frequencies. Note the amplitude always tends towards 1 at low frequencies due to the motions being applied at a rate which will not cause resonance effects.
Another phenomenon which occurs during oscillations is damping. Damping is considered beneficial to the structure under seismic analysis as it contributes to the energy dissipation of the energy transfer. A free vibration without damping will continue to oscillate with the same amplitude as there are no energy losses within the system. In reality, there is always damping in a system as there are frictional interactions, as a result, with a single period of oscillation, the amplitudes will decrease over time until the motion stops.
With increased damping of a system, the amplification at a modal frequency decreases and the curve shifts to the left (Figure 2), meaning the frequencies at which the modal frequencies occur are smaller and the fundamental periods of vibration are elongated, this is called a softening response of a system.
Soil Structure Interaction (SSI) is the process of assessing the influences a soil layer has on a structure’s structural response. In the past, it was assumed that SSI was always beneficial to a structure as it was shown to reduce its overall seismic response, and therefore could be neglected; this is prevalent in many American design codes such as in the Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (2009). The belief behind this is due to oversimplifications, including the over prediction of damping of structures due to foundation impedances derived from homogeneous half-space conditions of the soil (Gazetas, G. and Mylonakis, G., 2001) leading to smaller accelerations and stresses in the structure and foundation than in reality. However, recent studies have shown that SSI does not always provide a beneficial effect to structures regarding seismic response, and in some case, may be largely detrimental.
Figure 3 Comparison of a typical seismic code design spectrum (bold, solid line) for soft soils to actual spectra observed from catastrophic earthquakes presented for spectral amplification (Gazetas, G. and Mylonakis, G., 2001)
Seismic code spectra in Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (1997) assumes a decrease in amplification after a T period of 1.0 second, however it can be observed that higher spectral ordinates have occurred in some cases, especially prevalent in the Mexico City event of 1985. In addition, two other earthquakes in Figure 3, Bucharest 1977 and Kobe Takatori 1995, had damage associated with SSI effects (Mylonakis, G. and Gazetas, G., 2000). In the case of Mexico City, buildings were “selectively” more damaged. Storey buildings of six or more floors experienced greater damage due to their increased fundamental periods. Slender structures are more flexible, and therefore less stiff, resulting in higher fundamental periods, this in addition with the soil layer beneath the storey buildings resulted in a resonance frequency similar to the vibrations in the ground (Lockridge, P., 1994). This contradicts the assumption adopted by design codes as in some case, an increase in fundamental period due to SSI effects can lead to an increased response of the structure despite increased damping effects.
An advantage of earth dams is that they can be built directly on soft soils (a weak foundation material) as they can accept some settlement, whilst other types of dams, i.e. concrete dams, require better quality foundation material such as bedrock. The presence of a soft foundation layer adds flexibility to a structure, increasing its fundamental period of vibration.
As well as affecting the fundamental period of a structure, soft soils affect the amplification of seismic waves as it can elongate the wave’s period. This can lead to resonance with the increased fundamental period of the structure. A contributor to site amplification is the velocity at which soils transmit shear waves (S-waves). S-waves cause materials to oscillate and are considered to be the primary cause of earthquake damage. The shear wave velocity, Vs, of a soil is a material property and can be defined by Equation (1):
G is the shear modulus, and ρ is the density.
As a result, waves travel slower in softer soils as the shear modulus is smaller. The shear velocity is also affected by clay content, porosity, water content and overburden, to a lesser extent. It has been correlated that stronger shaking is experienced where the shear wave velocity is lower (U.S. Geological Survey, n.d.). It should be noted that the shear wave velocity is largely dominated by the shear modulus as variations in the density of soils is small.
A number of analysis methods have been developed in the past century. They can be categorised into linear or non-linear, and static or dynamic analyses, and can be solved either using analytical or numerical methods. Numerical methods are commonly used in seismic design as analytical methods are not always possible – they are sometimes impossible, or too time-consuming to solve. As earth dams are made from compacted soils, slope stability plays a major role in the analysis of such structures. Slope stability problems can be solved through simplified hand calculations and compatibility equations, but more currently, Finite Element Modelling (FEM) is used for more complicated analyses, as a computer is much more suited to solving the required number of operations.
Linear analysis deals with linear-elastic behaviour. The Young’s Modulus of the structure does not change (the stress-strain relationship remains linear) and the strains are assumed to remain in the elastic region. However, materials also exhibit plastic behaviour, therefore there is a limitation on the effectiveness of this method. The critical situations with dams deal with permanent deformations, this would indicate the material has deformed plastically. During strong earthquake shaking, it is likely the dam would have exhausted its yield strength and deform beyond its yield point. This means a linear-elastic approach would not be appropriate for soft soils and high magnitude earthquakes for analysis looking at failure.
Nonlinear analysis can incorporate nonlinear elasticity, as well as plasticity. There are various models with different levels of complexity which nonlinear analysis can incorporate. For example, elastic perfectly-plastic assumptions can be made (Figure 4a), or linear elastic-plastic (Figure 4b) which separates the elastic and plastic regions by two gradients. However more complicated models may include nonlinear elastic or plastic regions. This means softening or hardening properties of the material may also be modelled. Figure 4c is an example of a material which exhibits kinematic hardening; it has multiple plastic regions which change over the ranges of strain.
Static analysis is carried out when the inertia forces (Newton’s First Law of Motion) can be neglected and when the response of the system does not vary with time. It is commonly used in structural design as many of the loading combinations are static (they do not change over a short period).
Dynamic analysis is used when the loading is applied at a frequency greater than the damping effects of the structure. The dynamic load causes the structure to vibrate rather than a single direction deformation, thus dynamic analyses are time dependent and are assessed over a period of time. Dynamic analyses have higher capabilities compared to their static counterpart as they can model accelerations, as well as other dynamic effects such as damping.
Combination of Analyses
There can be a combination of linear/nonlinear and static/dynamic analyses. For example, nonlinear dynamic analysis applies a time history record to a model and is a widely-accepted method of analysis as capabilities are higher compared with linear-static analysis. The time histories can be displacement, velocity or acceleration with respect to time, which can be derived from one another. However nonlinear analysis will usually be more time and computational demanding.
In FEM, it is common to carry out a combination of analyses, for example a static analysis is used to model gravity and hydrostatic pressures in a dam as these loads are assumed to not vary over time, then a dynamic analysis can be done to model the acceleration time history of an earthquake.
The pseudo-static approach defines a series of forces acting on a body multiplied by a seismic coefficient, and uses slope stability techniques to find a minimum factor of safety against sliding. These seismic coefficients are derived from judgement and slope behaviour during past earthquake events (Marcuson, W., Hynes, M. and Franklin, A., n.d.), however recent studies have looked into new theoretical methods of estimating the seismic coefficient, such as relating the value of the coefficient to a function of the horizontal resultant acceleration time history of the sliding mass (Papadimitriou, A., Bouckovalas, G. and Andrianopoulos, K., 2013). The pseudo-static method assumes the earth dam behaves as a rigid body and ignores the fluctuation of forces in both direction and magnitude. According to Terzaghi (1950), the method is based on oversimplified assumptions as the “horizontal acceleration acts permanently on the slope material and in one direction only, therefore conveying an inaccurate concept”. A further study by Seed (1981) found that earth dams with sandy soils showed a significant loss of strength due to shaking, therefore this type of analysis would be inappropriate for sandy soils as the modulus of the material is assumed not to change in this analysis method. However, for soils which show no significant loss of strength due to earthquakes (clayey soils), pseudo-static analysis is found appropriate (Seed, 1981). Another consideration with this method is that it considers slope failure as the sole problem, however as described in Section 2.1, there are other modes of failure which can occur.
Finite Element Modelling
Finite Element Analysis (FEA) is not a direct method of analysis but a useful tool which can be used to carry out analysis. FEA is widely used because of its powerful capabilities. One such example is its ability to closer model real site conditions, e.g. different bodies of a model can be defined with different materials, therefore different parameters. FE can also be used to model complex behaviour, e.g. soil is a difficult material to analyse, however FE software can include yield and failure models such as the Mohr-Coulomb or Drucker-Prager failure criterion.
There are limitations to FE methods; it requires the user to have knowledge of both the software and the analyses, as the results cannot be taken confidently without validation. This is usually carried out by checking the results with calculations using first principles and other analytical or empirical solutions. FE packages are also usually costly and some computational power is required, therefore there is an element of dependence on resources.
Past research includes investigations on the various parameters which affect the seismic response of earth dams. Some considerations are:
- Dam geometry – Plane-strain conditions only accurately hold for infinitely long dams. Dams in narrow valleys (frequent in mountainous regions), exhibit a stiffer response due to the presence of rigid abutments creating 3-D stiffening effects. For dams in narrow canyons, such as the La Villita dam, results can be calibrated to take into account the stiffening effects for static and dynamic analyses; this has been done in Pelecanos et al (2015).
- Soil stiffness – Soil stiffness depends on soil shear modulus and the confining pressure (which varies point to point in an earth dam). With increasing distance from the crest and inclined surfaces, the average shear modulus across dams increase as G/Gmax is affected by the effective stress to an extent (Towhata, I., 2008).
- Permeability of soil – Han et al (2016) undertook a parametric study of two critical factors; the permeability of dam material and vertical ground motion. They found that when the permeability of the soil was lowered, the dynamic response in both directions of the dam was affected, in particular greater peak accelerations were found in the vertical direction.
- Dam-reservoir interaction – Pelecanos, L., (2013) did a study on dam-reservoir interaction and found that the effects were insignificant for earth dams. This is likely due to the sloped upstream face and the large volume of earth dams, with the inertia effects of the earth dam being significant compared to that of the reservoir.
A number of researchers have used FEA capabilities to predict and reproduce the seismic response of dams. A common approach of these studies is to analyse a well-documented case history, ideally one with site parameters and seismic records readily available. One such study by Han et al (2016), looks at the Yele rockfill dam during the 2008 Wenchuan earthquake using the dynamic hydro-mechanically coupled finite element method. Validation of the model is initially obtained through static and dynamic analysis against monitoring data recorded before and during the Wenchaun earthquake. Then the predicted seismic response is analysed and compared to the observed data, in particular the deformed shape, crest settlements and acceleration distribution pattern.
The results obtained from Han et al’s static model (Figure 5) are in general agreement with the monitored data which ensures the correct initial stress state for the subsequent dynamic analysis.
The results obtained from the dynamic analysis (Figure 6) compared with the monitored data in terms of acceleration response spectra and acceleration time histories also exhibit good agreement.
The failure modes of earth dams have been reviewed, with the most likely causes of failure and the failure modes related to seismic motions identified. The effects of resonance and damping has been described in relation to SSI effects, and the importance of considering these effects are shown. It has also been discussed that the foundation material plays a crucial role in determining the seismic waves which propagate from the subsoil into the dam structure. It should be apparent that the assumption of SSI effects being negligible is dangerous and should be considered in analysis and design for large structures. Some methods of seismic analysis has been mentioned with their benefits and limitations stated. Examples of parametric studies has been given to reflect the complexity of seismic models of earth dams and the range of variables which can contribute to the seismic response. Lastly, the benefits of using FEM has been shown through the existing work of researchers by its capability of predicting seismic response. This dissertation now attempts to further explore the effects of SSI between the dam and foundation through FEM capabilities.
The La Villita dam is a 60m earth and rockfill embankment dam located in Guerroro, Mexico, 350km southwest of Mexico City and sits on an average foundation layer of 70m. It has a capacity of 510 million cubic metres (IndustryAbout, 2015) and houses a hydroelectric power plant. The dam was completed in 1973 and is currently still in use, however since construction, it has experienced six major earthquakes, five of which are believed to have caused significant localised deformation of the dam (Elgamal A-W. M. et al, 1990). There is strong-motion instrument installed near the crest of the dam which recorded the dynamic response due to these earthquake events, this includes records of the permanent displacements.
This dam is chosen as the basis of study in this dissertation because of the readily available data in literature.
Figure 7 shows the known material properties of the La Villita dam from Pelecanos, L., (2013).
Modal frequencies, previously discussed in Section 2.3, are the frequencies at which resonance will occur between the system and the driving force, in this case; the earth dam and the earthquake.
In this study, analysis is conducted to find the modal frequencies of the La Villita dam. This is achieved using three analysis methods:
- Linear solutions for the inhomogeneous shear beam method by Dakoulas, P. and Gazetas, G., (1987);
- Modal analysis in the frequency-domain using the ANSYS built-in Modal analysis system;
- Dynamic, time-domain analysis with harmonic motions using ANSYS’s Transient Structural analysis system.
The analysis methods are used to compare three scenarios with varying foundation thicknesses; no foundation layer, 40m foundation layer and 70m foundation layer. The objective of this study is to analyse how the modal frequencies of the La Villita dam changes with the foundation depth, therefore how the soil foundation layer beneath the dam influences its modal response.
The three scenarios in this investigation use the original geometry of La Villita dam but with the thickness of the foundation layer are altered. Scenario 1 asseses the dam alone with no foundation layer influence (Figure 8). This scenario is used to compare the effects of not considering a foundation layer in the analysis. Scenario 2 models the dam with a reduced foundation layer depth of 40m (Figure 9). This scenario is to analyse the effects of varying foundation depths. Scenario 3 assesses the La Villita dam in its original geometry; the dam on a 70m foundation layer (Figure 10).
Scenario 1 – No Foundation Layer
Scenario 2 – 40m Foundation Layer
Scenario 3 – 70m Foundation Layer
For this study, the dam and foundation is modelled as a single homogeneous material, consisting of one zone. Table 1shows the simplified properties used in the study based off average values in Figure 7.
|Shear Modulus||200 MPa|
Using Equation (1), the shear wave velocity can be found;
The following assumptions have been made for this analysis:
- The FE models are 2-dimensional so plane-strain conditions are used.
- The dam and foundation consist of the same material, thus have the same properties.
- Changes of the shear modulus due to confining pressures are ignored and an average value is taken to represent the material.
- The reservoir does not affect the seismic response as found in a previous study (Pelecanos, L., 2013) therefore it is not modelled.
- This dissertation does not look at failure of the dam, the analyses conducted assume small deformations therefore the dam remains elastic and any nonlinearity of the dam is not considered.
This section uses the simple analytical closed-form solutions derived by Dakoulas, P and Gazetas, G in Seismic Lateral Vibration of Embankment Dams in Semi-Cylindrical Valleys (Gazetas, G., 1987). The solutions are based on the shear beam model and offers exact solutions to the modal frequencies of earth dams.
The fundamental period for a dam can be found using from Equation (2) (Gazetas, G., 1987);
Hd is the height of the dam and Vs is the shear wave velocity.
When considering the presence of a foundation layer on a dam, the fundamental period of the whole system needs to be accounted for. The effects of this can be estimated analytically using Figure 11.
h is the depth of the foundation layer
H is the height of the dam
L is the width of the dam cross section
T1 is the fundamental period of the dam itself
T̃1is the fundamental period of the system
Other modal periods of the dam can also be found analytically (Gazetas, G., 1987) using the nth natural circular frequency;
βn is the nth root of J0 (Bessel’s function). β1 = 2.4, β2 = 5.52 and β3 = 8.65.
H is the height of the dam
V is the shear wave velocity
The natural circular frequency, ω, is also referred to as the angular frequency:
fis the frequency.
Therefore, for the nth modal period;
Figure 11 is then used to find the second and third modal periods. To calculate the modal frequencies, the inverse of the modal period is taken;
The tabulated results can be found in Appendix A – Results for Foundation Thickness Effects.
The results in Figure 12 show that the modal frequencies at each mode are smaller where the foundation is thicker. The difference between scenario 1 and 2 results are much greater than between scenario 2 and 3. Despite an almost 70% increase in foundation depth from scenario 2 to 3, the modal frequencies are relatively close together.
The smaller modal frequencies at higher foundation depths show the response of the system is softer. The large difference between scenario 1 and 2 results of around 50% emphasises the difference between considering the structure as an isolated object as opposed to a system with the foundation. However, the smaller differences between scenario 2 and 3 indicate it is likely that as the depth of the soil foundation increases, the additional influence on the seismic response decreases. This means there eventually would be a limit where further increase in foundation depth would have no additional, or very little, effect on a dam’s seismic response.
The trend for each scenario is linear due to the nature of the analytical equations, however this would be unexpected in reality due to viscous damping effects. Equations (2) and (3) only consider the height of the dam and shear wave velocity of the material to affect the modal period. Other considerations such as the width of the dam is not accounted, however a slenderer dam would be expected to have different modal frequencies compared to a wider dam. Another limitation is that Figure 11 does not consider the type of material in the foundation; for softer foundation materials, the modal frequencies of the system may be smaller than calculated. Therefore, this method should be used as only an estimate of modal frequencies and results should be used in conjunction with other analysis methods.
The investigation in this section uses a frequency-domain modal analysis in ANSYS’s analysis system.
The models are created as 2D shapes, as per the geometries in Section 4.1.1.
The diagram below (Figure 13) show the boundary conditions (BC) applied to the models in ANSYS.
Double hatched line represents fixities in x and y axes.
Single hatched line represents fixity in the x axis.
The models are completely restrained at the base of the dam for scenario 1 and at the base of the foundation in scenarios 2 and 3. In scenario 2 and 3, the sides of the foundation layer are restrained horizontally, but are free to move vertically. This is to represent the real foundation conditions as the rigidity of the soil embedded in the ground will act as horizontal restraints. However, as the surface of the soil layer is free, it is able to deform vertically.
Eight-noded quadrilaterals are used to form the mesh with an average size of 5m x 5m. Figure 14 is an example of the mesh layout used for the model in scenario 3.
Refer to Table 1 for the properties of the soil material used in the model.
The analysis setup in ANSYS only requires the model with the correct boundary conditions and properties. The system undergoes undamped vibration purely due to oscillations with no external loads applied, this is because the natural circular frequency of a system is dependent on the stiffness and mass, and is not a function of the input load, this is further explained:
The equation of motion for a multi-degree of freedom system (Kramer, S.L., 1996) can be written as:
m is the mass
üis the acceleration with respect to time
c is the damping coefficient
u̇is the velocity with respect to time
k is the stiffness
uis the displacement with respect to time
Qis the external loading
The first term,
mü, represents the inertial force, the second term,
cu̇, is for viscous damping and the final term,
ku, is the elastic spring force.
For a system with undamped free vibrations,
Q=0. Substituting this into Equation (7)gives;
By definition, if a mass is under simple harmonic motion, it’s acceleration is directly proportional to its displacement;
Equation (11) gives the undamped, natural circular frequency of a system. It can be seen it is dependent on the stiffness and mass of a system, and is not a function of the input load, as mentioned earlier.
The tabulated results can be found in Appendix A – Results for Foundation Thickness Effects.
The results in Figure 15 show similar trends to the results obtained in the shear beam analysis; the modal frequencies at each mode are smaller where the foundation is thicker. The difference between the first two scenarios are also greater than between the last two. The changes between each mode (the gradients) are not constant. The increase between the first and second modes for all scenarios are much greater than for the second and third modes. However, at the change between the fourth and fifth modes, the gradient steepens again. This means the second, third and fourth modes occur at closer frequencies than with the first and fourth modes.
As the overall trends in Figure 15 are similar to the shear beam results, the conclusions drawn are the same; the modal frequencies occur at smaller frequencies with higher foundation depths showing the response of the system is becoming softer with increasing depth of soil foundation. Likewise, the difference between scenarios 1 and 2 results are much greater than between scenarios 2 and 3, with a difference between scenario 1 and 2 results of around 40%. However, the smaller differences between scenario 2 and 3 agree with the previous results; as the depth of the soil foundation increases, the additional influence on the seismic response decreases. The changes in gradient between each mode are similar for all the scenarios which suggests the foundation layer affects the overall response of the dam but not the modes individually, i.e. there’s no greater effect on the second or third mode on a softer or stiffer system.
The limitations of this analysis can largely be contributed to the model used; it is a 2D simplification of the La Villita dam, therefore additional effects such as narrow canyon stiffening effects are not accounted for. The modal analysis in the ANSYS family group is a linear analysis, therefore it assumes no nonlinearity of the model.
The time-domain analysis method uses a linear-elastic setup to find the modal response of the dam using the Transient Structural analysis system in ANSYS by applying dynamic motions to the model.
Geometry and Boundary Conditions
The geometry setup is the same as in Section 4.3. The same vertical boundary conditions are used, however there are no fixities in the horizontal direction, this is because the applied motions are in that axis.
The properties used in this analysis are as per Table 1. Additional coefficients are required for this analysis as the applied motions must be damped so that unrealistic deformations are not achieved.
The damping of the system is modelled using Rayleigh damping. Rayleigh damping is calculated from the mass and stiffness matrix of the system (Woodward, P.K & Griffiths, D.V, 1995);
Where [M] and [K] are the mass and stiffness matrices, respectively.
The matrix is solved using ANSYS, however the coefficients α and β need to be inputted, they are the mass-proportional and stiffness-proportional damping coefficients respectively. They can be found from Equations (13) and (14) (Pelecanos, L. 2013);
ξt is the target damping ratio (5%)
ω1 the first modal circular frequency of the system
ω3 the third modal circular frequency of the system
|Rayleigh Damping α||0.993||0.565||0.418|
|Rayleigh Damping β||0.00172||0.00301||0.00395|
The dynamic analysis applies harmonic displacements to the base of the dam or foundation layer. This causes the dam body to sway back and forth (Figure 16), deforming periodically. The accelerations at the crest of the dam are measured to obtain an acceleration time-history. The models are ran until a steady state value of acceleration is obtained, usually between 5 to 15 seconds, dependent on the input frequency of the harmonic motion. Figure 17 is an example acceleration time-history output; the amplitudes of the first three seconds are irregular, but after the amplitudes are seen to reach a steady state.
The harmonic displacements are generated from accelerations. The accelerations are found using the sinusoidal expression from Equation (15), the displacements are then found through double integration Equation (16);
A is the amplification (taken as 1)
ω is the input circular frequency of the harmonic motion
t is the time
It should be noted that the magnitude of the input displacements are arbitrary as this only affects the amplitude of the peaks of the response spectra and not their relative positions.
Each period of motion is divided by 24 to obtain the points used to plot the input harmonic displacement. This also gives the timesteps needed to run each analysis. This analysis is repeated for a number of input circular frequencies.
The plots below show the steady state accelerations found from the dynamic analysis plotted against the ratio between the input circular frequency of the harmonic displacement and the angular frequency of the system (ω/ωdam or ω/ωsystem) found using the inhomogeneous shear beam equations in Section 4.2.1. Resonance happens when the frequency of the driving force matches the modal frequency of the system, where in this case the driving force is the input harmonic motion and the system is the dam or dam and foundation. Therefore, it is expected that when the ratio is equal to 1 (where the input frequency matches the harmonic frequency of the system), maximum amplification should occur and therefore a modal peak is expected. The accelerations are plotted on the y axis and are denoted as amplitude, this is because the actual magnitude of the accelerations is arbitrary.
This scenario looks at the dam alone. The first peak occurs before it is expected (ω/ωd = 1). This means the modal frequency of the dam is underestimated using the shear beam equations. Two smaller but prominent peaks can also be observed at a ratio of 1.6 and 2.25. This is expected as the amplification decreases over the range of frequencies (see Figure 1). As the peaks correspond to where resonance occurs, the modal frequencies of the dam can be found (Table 3).
|Input Frequency (Hz)||1.62||3.24||4.56|
This scenario looks at the dam with a 40m foundation layer.
Similar to scenario 1, the first peak occurs slightly before 1, meaning the modal frequency of the dam and system has been underestimated. The second peaks occurs at 2.34, however the third peak is not as defined and also has a similar amplitude to the second. Table 4 shows the modal frequencies of the system at which the peaks occur.
Unlike the previous scenarios, the first peak occurs very close to the expected value of 1. This means the calculated natural frequency of the system is close to the actual value used from the analysis. However, this response spectrum can be seen to be less defined than the ones previous ones as after ω/ωsystem = 3, the spectrum does not appear to have another prominent peak or have a significant decrease in amplitude. It is unclear why this occurs and may be due to a number of reasons, one being; not all input frequencies are ran, therefore there may be a missing peak or drop at an untried frequency. Overall the reliability of this set of results is in question. Table 5 shows the modal frequencies of the system at which the peaks occur.
As with the shear beam and frequency-domain results, the same trends occur in the time-domain analysis (Figure 21); the modal frequencies at each mode are smaller for higher foundation depths. However, one noticeable difference between these results and the previous sets, are the third mode results. In Figure 12 and Figure 15, the scenario 2 and 3 modal frequencies are significantly lower than scenario 1. But in Figure 21, the third modal frequencies for scenarios 2 and 3 are higher.
The response spectra for the three scenarios are compared on a plot of ω/ωd to show the effects of a foundation layer on the modal response of the system, Figure 22.
The phenomena of damping have been previously discussed in Section 2.3. Figure 2 shows the effects of damping on the modal peaks in a response spectrum. The first two modal peaks of Figure 22 show exceptional similarities with Figure 2; a reduction of around 40% and 70% in the amplitudes of scenarios 2 and 3, respectively, can be seen occurring at smaller frequencies than for the dam alone (scenario 1). This means the response spectra are translated to the left of the graph and the amplitude of peaks are reduced. This is evidence that the foundation layer plays a major role in the damping effects of a system’s dynamic response. Whilst this decrease in amplification is beneficial to the structure’s dynamic response, the reduced harmonic frequencies of the system should also be considered as resonance can still occur to a lesser extent.
It should also be noted that at low frequencies in Figure 22, the amplitude tends towards 1 irregardless of the scenario, this is as expected because at low frequencies, oscillations move too slowly for resonance effects to occur.
This analysis method requires a considerable amount of run time and computational effort to analyse the models. A contribution of this is the large number of required input data. As multiple analyses are required to obtain enough data to plot the response spectrum, the total run time can be substantial. For the scope of this dissertation, a limited number of analyses were run to obtain a good representation of the frequency spectrum, however it should still be noted that some peaks may have not been found, and for a better defined response spectrum, more input frequencies will need to be tried.
The analysis is linear; no nonlinearity of the dam is assumed. However in reality, permanent deformations may be observed if the dam was under harmonic motions. Considering this is only a 2D dimensional analysis, 3D analysis is likely to return more conclusive results, however the time required to create the model and run times will be exponentially greater. Nonlinearity will also contribute significantly to model run time.
The correlation between the foundation thickness and the modal response of the dam show that a thicker foundation layer leads to a softer response, likely due to damping effects. Another apparent trend was the decreasing influence of a thicker foundation layer on the dam; the change in the modal frequencies were much greater from scenario 1 to 2, than 2 to 3 indicating there is likely to be a point where further increase in foundation depth does not influence of dam’s dynamic response.
This study uses two of the analysis methods in the previous study to analyse how the modal frequencies of the La Villita dam changes with varying shear moduli of the foundation; 160MPa, 200MPa and 240MPa.
The two analysis methods are:
- Modal analysis in the frequency-domain using the ANSYS built-in analysis system;
- Dynamic, time-domain analysis with harmonic motions using ANSYS’s Transient Structural analysis system.
The inhomogeneous shear beam method is not used for this study as it does not consider the stiffness of the foundation.
For this analysis, the original dimensions of the La Villita dam is used; a 60m high dam on a 70m foundation layer, see Figure 10.
The same properties in Table 1 are used for the dam. However, for the foundation, the shear modulus varies. Table 6 shows the shear moduli corresponding to each scenario. Scenario 1 is a softer material, scenario 2 is the original shear modulus and scenario 3 is a stiffer material. The values of shear modulus are based on Figure 7b, where the spatial variations of the dam shear modulus are known.
|Scenario||Shear Modulus, G (MPa)|
The same boundary conditions and mesh settings are used from Section 4.3, the methodology also remains the same.
The tabulated results can be found in Appendix B – Results for Foundation Moduli Effects.
The variations in Figure 23 between the scenarios remain relatively constant throughout, with the modal frequencies decreasing by only 8% at mode 1 and 15% at mode 5 from a shear modulus change of 240MPa to 160MPa. Like with the previous frequency-domain analysis in the foundation thickness investigation, the gradients between the first and second modes, and fourth and fifth modes are steeper than the second and third gradients.
Despite a 50% increase in shear modulus on a 70m foundation layer (from scenario 1 to 3), the modal frequencies only experience an 8%-15% change. This suggests that the stiffness of the soil foundation layer does not play a prominent role in the modal response of the earth dam.
The same boundary conditions and mesh settings are used from Section 4.4, but there is a small difference to the methodology; the dam and foundation in this study need to be defined as separate regions to assign the different material properties. The boundary where the two materials meet, mesh merge is used to link the nodes together. Finer meshes are required for more connection points, however this increases the run time. To maintain a reasonable run time, a limited number of contact points were used with the 5m x 5m mesh.
The response spectra for the three scenarios are compared on a plot of ω/ωdam to show the effects of a varying shear moduli on the modal response of the system, Figure 22. The solid black line is a reference response spectrum of the dam with no foundation layer influence.
The first modal peak for scenario 2 and 3 are identical, with the second peak of scenario 2 being softer than scenario 3. The first peak for scenario 1 occurs at a lower frequency, with a reduced amplitude of 23% of scenarios 2 and 3.
As found in the results of Section 4.5, the response of the system is softer with the foundation layer than with the dam itself. However, these results agree with the frequency-domain trends in Section 5.2.1, where the varying shear modulus of the foundation layer does not appear to have a significant effect on the modal response of the La Villita dam. Likewise, some softening of the system can also be seen, in particular with scenario 1, and to a lesser extent in scenario 2.
There is some uncertainty with the reliability of the results for scenario 2 and 3 as it is expected to have some variation in the shape of the first modal peak. A major limitation of this analysis method is the use of merged nodes. Shorter analysis times were preferred over a finer mesh in this analysis, therefore only a limited number of merged nodes were possible with a 5mx5m mesh. If the analysis was repeated with a more refined mesh, it is likely a better response spectrum for scenario 1 or 2 could be achieved.
Some softening of the system can be seen when the shear modulus of the foundation layer is lower, however the overall extent of this is small, especially in comparison to the effects found in the previous study of varying foundation thickness.
The results show that there is a significant difference in the modal response of the dam when comparing analysis of only the dam to a system consisting of the dam and foundation. The response is significantly softer, with a reduction of about 70% in the amplification, in the analysis with a 70m foundation layer compared to one without a soil foundation. The thickness of the foundation layer also had an influence on the response, with the change between a 40m and 70m foundation, a decrease in amplification of just under 60% was found.
Interestingly, on the other hand, lower shear moduli of the foundation layer showed to cause some softening of the response, but overall had no significant impact on the modal response.
The tabulated summary of the results found for the first three modes from the investigation of foundation thicknesses can be found in Appendix A – Results for Foundation Thickness Effects.
Figure 25 compares the modal frequencies found for the first three modes of each scenario using the different analysis methods. It is clear that the first modal frequency of the systems for all methods exhibit good agreement with one another. However, this good agreement does not hold for the second and third modes. For scenario 1 and 2, the shear beam method show the highest frequencies, but interestingly, for scenario 3, the highest frequency is found by the time-domain method. In all cases, the frequency-domain analysis gave the lowest modal frequencies. It is also interesting to note that the shear beam and time-domain results for scenario 3 in Figure 25c are similar. As this only occurs one, it is difficult to say whether it is due to the methods validating one another, or if it is an anomaly. Without further investigation, it is also difficult to state which method is the most accurate, however it should be noted that the shear beam method uses equations which are only approximations, therefore only estimated results are expected. The results for the frequency and time-domain analysis are relatively dependent on the complexity of the ANSYS model, for example it can be improved by using a 3D model to closer model the site boundary or by incorporating nonlinear effects, however this would greatly increase the software run time and require more complex analysis of the system.
Figure 26 compares the modal frequencies found for the first two modes of each scenario using the different analysis methods. Similar to the results in Figure 25, the first modal frequency of the systems for the methods exhibit good agreement with one another, but this agreement does not extend to the second mode. Also like the previous results, in all cases, the frequency-domain analysis gave the lowest modal frequencies.
The results here show a high level of uncertainty for the values of the second and third modal frequencies as there is quite a disparity between them. Despite this uncertainty, the first modal frequency is the commonly the most significant mode as the largest amplification occurs at that peak, therefore these methods are still viable in modal response analysis. Further investigation into the accuracy of estimating second and third modes should be explored.
The studies in this dissertation has looked at the effect of the foundation layer influence on the dam’s modal response through varying foundation layer thickness and foundation layer shear modulus. The results are quite conclusive with the foundation layer thickness playing a dominant role in the dampening effects of the dam. Conversely, the shear modulus of the foundation was found to have little impact on the modal response, however the results from this analysis may not be fully conclusive due to the limitations in the model used. It would be recommended for a more complex FE model to be utilised in this study.
The three analysis methods validate one another as they showed good agreement in finding the first modal frequency of the dam and foundation. However, there is some uncertainty with the methods if subsequent modal frequencies are required.
Overall, the influence of the dam-foundation interaction effects on the seismic response of the La Villita dam has been explored using relatively quick methods of analysis. Much more in-depth FE models should be used in the analysis of earth dams for design, but the methods show their adequacy in determining an approximate value.
This study has only covered two parametric studies, and the methods used had several limitations. There are still many further areas of studies which can be explored further to the results found in this dissertation. They are summarised below:
- Further to the results of this investigation:
- As only two foundation depths were investigated, the incorporation of more depths can be done to achieve a more defined trend. It was found that an additional increasing foundation depth had a decreasing influence on the response, following this trend, a limit is likely where further increase in foundation thickness provides no additional contribution to the modal response. Further analysis can be done to find this limit, and may even be able to develop analytical equations to estimate this.
- In the shear modulus investigation, only the shear modulus for soil foundation and thus weak material was tried. Stiffer foundation material such as bedrock with a much greater shear modulus may show different effects as it was shown that higher foundation shear moduli showed a stiffer response.
- Damping of the system was found to be an influence of the foundation, however this also means modal periods are elongated. This can be further investigated with real earthquake data to find the common earthquake modal periods to see if resonance effects are any more or less likely to occur. This can also be looked at in conjunction with the elongation of seismic waves in soft soils.
- Other parametric studies include:
- Multiple foundation layers consisting of different soil material.
- Case studies of earth dams with different dam, foundation and site properties.
- Based on the limitations discussed:
- Further investigation into the accuracies of the methods used, in particular agreements in the results of second and third modal frequencies.
- As the analysis methods used in this dissertation were all linear, nonlinear effects are of interest as failures in dams are related to plastic deformations.
- Development of more complex FE models will lead to higher capabilities and more would be obtained from results, therefore further to this investigation, a better model can be developed.
- Based on the literature discussed:
- It was mentioned in Section 2.5 that the presence of soft soils can amplify seismic waves. An investigation can be done to incorporate the effects of wave amplification based on the shear modulus of soils. This can also be related to the speed of the seismic waves travelling in different shear moduli as waves travel slower in softer material and it has been correlated that shaking is experienced stronger when waves are slow.
- Soil softs were found more susceptible to liquefaction, resonance effects can be looked at in conjunction with this – liquefaction suggests a highly viscous material, it is likely that damping effects would be greater in this. The balance between the beneficial damping effects and detrimental effects of liquefaction can be explored.
This dissertation only touches some dam-found interaction effects, however it is evident the problem cannot be easily answered due to the various complexities with many factors playing a role in the beneficial and detrimental contributions. There are still gaps in the knowledge, and much scope for further investigation.
Results for the first three modal frequencies calculated from the Shear Beam Method.
|Modal Period (s)||0.496||0.216||0.138||0.868||0.378||0.242||1.19||0.49||0.313|
|Modal Frequency (Hz)||2.02||4.63||7.25||1.15||2.65||4.13||0.84||2.04||3.19|
First five modal frequency results from ANSYS modal analysis in the frequency-domain.
|Modal Frequency (Hz)|
|Foundation Layer Depth (m)||1||2||3||4||5|
Summary of the first three modal frequencies of the three scenarios found from the different analysis methods.
|Analysis Method||Modal Frequency (Hz)|
|No Foundation Layer||40m Foundation Layer||70m Foundation Layer|
|Shear Beam Method||2.02||4.63||7.25||1.15||2.65||4.13||0.84||2.04||3.19|
|Frequency Domain Analysis||1.63||2.37||2.69||1.00||1.45||1.67||0.79||1.13||1.30|
|Time Domain Analysis||1.62||3.24||4.56||1.01||2.73||4.05||0.81||2.23||3.44|
First five modal frequency results from ANSYS modal analysis.
|Modal Frequency (Hz)|
|Foundation Shear Modulus, G (MPa)||1||2||3||4||5|
Chadwick, W., Casagrande, A., Coombs, H., Dowd, M., Fucik, E., Higginson, R., Leps, T., Peck, R., Seed, H. and Jansen, R. (1976) Report to U.S. Department of the Interior and State of Idaho on Failure of Teton Dam. Idaho: U.S. Department of the Interior, pp. V.
Foster, M., Fell, R. and Spannagle, M. (2000). The Statistics of Embankment Dam Failures and Accidents. 2000 NRC Canada, pp. 1005.
Ohio Department of Natural Resources Division of Water Fact Sheet. (1994). Dam Safety: Earth Dam Failures. Ohio: Ohio Department of Natural Resources.
West, M., Haeussler, P., Ruppert, N. and Freymueller, J. (2014). Why the 1964 Great Alaska Earthquake Matters 50 Years Later. Seismological Research Letters,[ONLINE] Volume 85(2), p. 245. Available at: http://srl.geoscienceworld.org/content/85/2/245. [Accessed 2 Dec. 2016]
Japan National Committee on Earthquake Engineering. (No Date). Niigata Earthquake of 1964, [ONLINE] Volume 3, p. S-79. Available at: http://www.iitk.ac.in/nicee/wcee/article/vol3_S-78.pdf. [Accessed 2 Dec. 2016]
Stokoe, K., Rix, G., Sanchez-Salinero, I., Andrus, R. and Mok, Y. (1988). Liquefaction of Gravelly Soils During the 1983 Borah Peak, Idaho Earthquake. Proceedings of Ninth World Conference on Earthquake Engineering, Volume 3, pp. III-187.
Building Seismic Safety Council of the National Institute of Building Sciences. (2009). Recommended Seismic Provisions for New Buildings and Other Structures (FEMA P-750). Washington, D.C.: Federal Emergency Managements Agency of the U.S. Department of Homeland Security, pp. 225-226.
Building Seismic Safety Council of the National Institute of Building Sciences. (1997). Recommended Seismic Provisions for New Buildings and Other Structures (FEMA P-750). Washington, D.C.: Federal Emergency Managements Agency of the U.S. Department of Homeland Security.
Gazetas, G. and Mylonakis, G. (2001). Soil-Structure Interaction Effects on Elastic and Inelastic Structures. 2001 Fourth International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics.[ONLINE] pp. 1, 3. Available at: http://scholarsmine.mst.edu/icrageesd/04icrageesd/?utm_source=scholarsmine.mst.edu%2Ficrageesd%2F04icrageesd%2Fsession13%2F2&utm_medium=PDF&utm_campaign=PDFCoverPages. [Accessed 4 Dec. 2016]
Mylonakis, G. and Gazetas, G. (2000), Seismic Soil-Structure Interaction: Beneficial or Detrimental? Journal of Earthquake Engineering, Volume 4(3). Imperial College Press, pp. 281-282.
Lockridge, P. (1994). Earthquake Damage in Mexico City, Mexico, September 19, 1985. National Geophysical Data Center, NESDIS, NOAA, U.S. Department of Commerce. [ONLINE] Available at: https://www.ngdc.noaa.gov/metaview/page?xml=NOAA/NESDIS/NGDC/MGG/Hazard_Photos/iso/xml/G01196.xml&view=getDataView&header=none. [Accessed 4 Dec. 2016]
U.S. Geological Survey. (n.d.). Soil Type and Shaking Hazard in the San Francisco Bay Area. [ONLINE] Available at: https://earthquake.usgs.gov/hazards/urban/sfbay/soiltype/. [Accessed 10 Dec. 2016]
Terzaghi, K. (1950). Mechanism of Landslides In: Application of Geology to Engineering Practice. Geological of Society of America.
Seed, B. (1981). Earthquake-Resistant Design of Earth Dams. First International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics. St. Louis: N/A, pp. 1161.
Pelecanos, L., Kontoe, S. and Zdavkovic, L. (2015). A Case Study on the Seismic Performance of Earth Dams. Volume 65(11). ICE Publishing, pp. 927-928.
Towhata, Ikuo. (2008). Geotechnical Earthquake Engineering. 2008 Springer-Verlag Berlin Heidelberg.
Han, B., Zdravkovic, L., Kontoe, S. and Taborda, D. (2016). Numerical Investigation on the Response of the Yele Rockfill Dam During the 2008 Wenchuan Earthquake. Department of Civil & Environmental Engineering, Imperial College London: 2016 Elsevier Ltd.
IndustryAbout. (2015). La Villita Dam. [ONLINE] Available at: http://www.industryabout.com/country-territories-3/1679-mexico/hydro-energy/24961-la-villita-dam. [Accessed 17 Mar. 2017]
Elgamal, A-W. M., Scott, R.F., Succarieh, M.F. and Yan, L. (1990). La Villita Dam Response during Five Earthquakes including Permanent Deformation. ASCE Library: Journal of Geotechnical Engineering, volume 116(10). [ONLINE] Available at: http://ascelibrary.org/doi/pdf/10.1061/(ASCE)0733-9410(1990)116:10(1443). [Accessed 17 Mar. 2017]
Pelecanos, L. (2013). Seismic Response and Analysis of Earth Dams. London: Department of Civil & Environmental Engineering, Imperial College of Science, Technology & Medicine, pp. 38, 108, 197-201.
Dakoulas, P. and Gazetas, G. (1985). A Class of Inhomogeneous Shear Models for Seismic Response of Earth Dams and Embankments. Soil Dynamic and Earthquake Engineering, volume 4(4).
Gazetas, G. (1987). Seismic Response of Earth Dams: Some Recent Developments. Rensselaer Polytechnic Institute, New York and National Technical University, Athens, pp. 18.
Kramer, S.L. (1996). Geotechnical Earthquake Engineering. Upper Saddle River, New Jersey: Prentice Hall, pp. 576-577.
Woodward, P.K. and Griffiths, D. V. (1995). Influence of Viscous Damping in the Dynamic Analysis of an Earth Dam using Simple Constitutive Models. Great Britain: 1996 Elsevier Science, Computers and Geotechnics, volume 19(3), pp. 250.