Seismic Deformation of Earth Dams: A State-of-the-art Review

Abstract

Earthquakes are natural phenomena that cause terrible disasters for human society. Many of the disasters caused by earthquakes around the world serve as warnings to us about the fragility of life and the immaturity of science - especially construction science. For earth dams, which are made up mostly of compacted earth, the most common type of construction in the world, an earthquake can have enormous consequences if it causes a dam failure. Thus, proper seismic safety evaluations are of the utmost importance in terms of assuring the safety of earth dams. Along with the advancements in science and technology, knowledge in the field of earthquake engineering has been expanding. Many studies have been done on the effects of earthquakes on earth dams and many methods have been developed to estimate the deformation of these dams during earthquakes. This paper firstly presents the observed seismic performance of earth dams in past earthquakes which can help provide insight into the general effects of earthquakes on earth dams as well as the seismic response of dams during an earthquake. The paper also summarizes the methods presently available for estimating the seismic deformation of earth dams. These methods can provide design engineers with confidence when dealing with one of the most difficult geotechnical engineering challenges they will ever face.

1. Introduction

An earthquake is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth’s crust that creates seismic waves.  According to the National Earthquake Information Center, there are around 55 earthquakes per day worldwide, or 20,000 per year [1], and the overall trend shows an increase in their frequency [2]. Earthquakes can occur at any time of the year; the world can expect 16 earthquakes in any given year that include 15 of magnitude 7.0 or greater and one of magnitude 8.0 or greater [3]. Seismic activity of sufficient strength may affect the environment or the integrity of structures constructed on the crust of the Earth.

By observing the performance of earth dams in past earthquakes, it is possible to investigate the cause of dam failures and to analyze the seismic response of the dams. According to statistics, about 2% of dam failures are caused by seismic activity [4]. Most of these failures occur to small, homogeneous earth dams, which are the most common type of dams found because of their cost-effectiveness. As is known, a dam is a water barrier constructed across a stream or river to hold water back and form a reservoir that serves many purposes. These purposes include providing water for human consumption, irrigation for industrial use, and aquaculture, as well as protecting an area from flooding. However, with the storage of a very large volume of water, a dam is no different from a “water bomb” when it is damaged. When a dam is damaged, the huge volume of water in the reservoir will flow downstream with extremely high velocity, which can cause a catastrophic disaster [5, 6]. It is clear that such a disaster has the potential to create a high risk to life and property in the downstream area for dams and reservoirs built in areas of high seismic activity. Thus, in order to ensure the safety of earth dams, it is of utmost importance to conduct proper seismic safety evaluations. In the 1950s–1960s, earth dams were designed and constructed based on static loading when knowledge and capabilities of the seismic design were not available. The seismic safety of an earth dam was evaluated by the factor of safety based on the pseudo-static method with a seismic coefficient, which represents the seismic effect of the earthquake. Since advancements in earthquake engineering have been made, displacements are now seen to provide a better criterion than the factor of safety in the design of earth dams under earthquake loading conditions, particularly after a consideration of the magnitude of deformations or change in configuration [7, 8].

Recently, there has been a great deal of research whose results provide methods for evaluating the seismic deformation of earth dams. Therefore, it is vital to choose an appropriate and reasonable method for application in each particular situation, which will give confidence to the design engineers. By synthesizing information from various sources, this study presents a review of the effects of earthquakes on earth dams through the observed seismic performance of these types of dams during past earthquakes and the state-of-the-art methods adopted for estimating the seismic deformation of earth dams.

2. Observed seismic performance of earth dams during past earthquakes

Observations of the behavior of earth dams in past earthquakes provide a brief overview of the seismic performance and the damage to all types of dams as well as insight into the effects of earthquakes on earth dams, or the seismic response of dams during an earthquake. This work highlights the need for seismic stability evaluations when assessing the overall performance of an earth dam in a seismic area. The study of earthquakes has been going on for many centuries. The earliest recorded evidence of an earthquake dates back 3000 years and occurred in China. Records from Japan and the eastern Mediterranean go back nearly 1600 years [9].

Historically, numerous dams have suffered little or no deformation; just a few dams have sustained significant damage or collapsed entirely as a result of an earthquake. These dams were mostly hydraulic fill dams, or relatively old, small, earthfill dams of a possibly poor design. Meanwhile, well-built dams on stable foundations are unlikely to have been damaged by moderate earthquakes [10, 11, 12, 13].

Table 1 lists the performance of some earth dams and embankments in past earthquakes that have case history value for the dam engineering field. The list includes the principal dam parameters (type of dam, dam height (H), information on the earthquake to which they were subjected (earthquake magnitude (M) and epicentral distance (D)), and a damage assessment.

Table 1: Observed performance of earth dams during past earthquakes

Dam information			Earthquake data			Damage assessment	Reference
Name (Country)	Type	H (m)	Name, Date	M	D (km)
O. San Andreas (USA)	E	8.5	San Francisco, 19 Apr 1906	8.3	0	Minor damage, cracking on crest	[14, 15]
Ono (Japan)	E	49	Kanto, 01 Sep 1923	8.2	51	Serious damage, cracking on crest, local slides on downstream face	[16, 17]
L. Van Norman (USA)	HF	42.7	San Fernando, 09 Feb 1971	6.5	11.2	Major damage, widespread liquefaction and major slope failures, overtopping of crest	[10, 18]
		38.1	Northridge, 17 Jan 1994	6.7	9.4	Noticeable damage, crest settlement and movement, cracking on crest, sand boils, sinkhole on upstream face	[18, 19]
La Villita (Mexico)	ECRD	60	Michoacan, 19 Sep 1985	8.1	44	Minor damage, cracking on crest, upstream and downstream face	[20, 21]
Austrian (USA)	E	56.4	Loma Prieta, 17 Oct 1989	7.1	11.5	Serious damage, crest cracking and settlement	[22, 23, 24, 25]
Masiway (Philippines)	E	25	Luzon, 16 Jul 1990	7.7	5	Serious damage, upstream face slumped, cracking on crest and upstream face	[26, 27]
Los Angeles (USA)	E	39.6	Northridge, 17 Jan 1994	6.7	9.5	Minor damage, crest cracking, settlement	[19, 28]
Koyoen (Japan)	E	9.1	Kobe, 17 Jan 1995	6.9		Collapse, massive sliding failures	[29, 30]
Shui-Chih (Taiwan)	E	30	Chi-Chi, 21 Sep 1999	7.6		Minor damage, cracking on crest and upstream face	[31]
Chang (India)	E	15.5	Bhuj, 26 Jan 2001	7.7	13	Serious damage, cracking on crest and upstream face, slope failure on upstream	[32, 33, 34, 35, 36]
Tapar (India)	E	15.5	Bhuj, 26 Jan 2001	7.7	43	Moderate damage, cracking on crest and upstream face, crest settlement	[34, 35, 37]
Zipingpu (China)	CFRD	156	Wenchuan, 12 May 2008	7.9	7.0	Minor damage, Crest settlement, damaging small parts of the face slab	[38, 39, 40, 41, 42]
Fujinuma (Japan)	E	18.3	Tohoku, 11 Mar 2011	9.0	8.0	Collapse	[43, 44, 45, 46]

Note: E: Earth fill, HF: Hydraulic fill, ECRD: Earth core rock fill, CFRD: Concrete face rock fill

3. Effects of earthquakes on earth dams

From the observed seismic performance of earth dams during past earthquakes, it is clear that the effects of earthquakes on dams are mainly related to excessive displacements due to either instability without liquefaction or failure due to liquefaction [47].

Excessive displacements are the result of dams being subjected to greater loads during earthquakes than those experienced under static conditions. The earthquake loading is of short duration, cyclic, and involves motion in horizontal and vertical directions. This earthquake loading may cause settlement as well as longitudinal and transverse cracking, particularly near the dam crest. Crest settlements are important as they may reduce the freeboard between the dam crest and the reservoir level, which could result in overtopping in the worst-case scenario. Cracking is also important because it can lead to leakage through the dam, internal erosion, and piping failure.

Earthquakes can also cause dams to liquefy. Flow liquefaction can occur when the strength of the soil drops below the level needed to maintain stability under static conditions as a result of the earthquake-induced increase in pore pressure. Therefore, static gravitational forces generate flow failures, which can result in large movements. Flow failures have caused the collapse of earth dams and other slopes, as well as the failure of foundations [9].

Seed [48] and Fell et al. [49] summarized the following possible ways in which an earthquake may affect earth dams:

• - Instability of the upstream and downstream slopes of the dam and large deformations

  - Differential settlements and cracking due to active faults passing through the dam foundation

  - Transverse cracks occurring as a result of differential movements among the embankment, abutments, and spillway structures

  - Development of open cracks, or the opening of previously closed joints in the foundation, close to the core-foundation contact

  - Damage to outlet works passing through the embankment and differential settlements causing leakage and potential piping erosion

  - Overtopping of the dam due to massive tectonic activity in the reservoir basin

  - Overtopping of the dam by waves caused by earthquake-induced landslides into the reservoir from the valley sides

The potential risks of an earthquake to cause the aforementioned effects can range from insignificant to catastrophic depending on the seismic activity, the foundation materials, the topography of the area where the dam is located, the type and detailed construction of the dam, and the water level in the reservoir at the time of the earthquake [49, 50, 51].

Therefore, to avoid most of the possible types of dam failures, the earthquake-resistant design requires many precautions. For this purpose, experience and good judgment are needed. Seed [52] provided the following measures:

• - Placement of the dam in an area with no active faults

  - Provision of enough freeboard to allow for slope slumping or dam subsidence

  - Use of wide transition sections for filter materials that are not prone to cracking

  - Use of wide cores for self-healing materials in the event that cracking occurs

  - Thorough assessment of the stability of slopes adjacent to the reservoir

  - Measures to mitigate the risk of overtopping caused by seiches or slope failures into the reservoir

4. Seismic performance criteria for earth dams:

Engomoen et al. [53] suggested that the following criteria be satisfied for the seismic performance of earthen dams:

• - The dam and foundation are comprised of materials that do not easily liquefy and do not include sensitive clays.

  - The dam is well built and compacted to at least 95% of the laboratory maximum dry density, or to a relative density greater than 75%.

  - The slopes of the dam are 2.5:1 (horizontal:vertical) or flatter, and/or the phreatic line is well below the downstream face of the embankment.

  - The peak horizontal acceleration at the base of the dam is no greater than 0.35 g.

  - The static factor of safety for all the potential failure surfaces involving the loss of crest elevation is greater than 1.5, with levels of pore-water pressure that could reasonably be expected immediately prior to the earthquake.

  - The minimal freeboard with active or joint-use storage is 3 to 5% of the embankment height, but never less than 0.9 m.

  - There are no appurtenant features that would be affected by minor embankment movements or that could result in internal erosion or other modes of failure.

5. Methods available for estimating seismic deformation

Since the 1950s and 1960s, many researchers have been interested in seismic analysis. Along with advancements in science and technology, seismic analysis for earth dams has evolved significantly. There are numerous methods available for estimating the deformations of earth dams in response to seismic loading. These methods range from comparing case histories of dams subjected to similar loading, through simplified analysis, to advanced numerical analysis. As a result, the level of complexity and the time it takes to do the analysis have also increased with these methods.

However, none of the methods, not even one of the advanced methods, has been proven to accurately predict the actual deformation shape and magnitudes. It is common practice to recommend estimating seismic deformations in stages. In view of this point, the database and simplified methods should not be assumed to be conservative. When the estimated deformations are close to acceptable deformations, more advanced methods should be used or deformation-reducing measures should be applied. In general, the analysis should start from the simplest method of analysis, and proceed to more rigorous methods of analysis, until acceptable and justifiable results are reached.

The present study only focuses on methods of estimating the deformations which may occur in earth dams and their foundations where liquefaction and significant strain weakening do not occur. These methods are briefly summarized here and then presented in more detail in the following sections:

• - Empirical database methods: based on the observed performance of existing dams, these methods are useful for rapidly predicting how a dam will react to a deterministically derived earthquake by empirical equations or charts.

  - Simplified methods: commonly based on the principles of Newmark [7], these methods assume that the post-earthquake deformations are negligible and that the deformations occurring during an earthquake are caused by the inertia forces induced by the earthquake.

  - Advanced methods: these methods cover a wide range of sophistication and complexity. They include dynamic analyses using total and effective stress methods and linear and nonlinear models that vary in their degrees of sophistication in terms of how the levels of pore pressure develop and are coupled to deformations.

5.1 Empirical methods

Several empirical relationships have been proposed to estimate the deformation of earth dams subjected to earthquake loading. These relationships have been formulated based on the responses of existing dams after earthquakes.

As one of the earliest contributions in this field, Jansen [54] proposed the following relationship among the earthquake magnitude (M), maximum acceleration at the crest (a_c), yield acceleration of a potential sliding mass (a_y), that is, the horizontal acceleration that results in a safety factor of unity for the sliding mass and can be determined using appropriate techniques, such as a limit equilibrium analysis, and total settlement at the crest (∆) as Equation 1:

  

$$\Delta =\left[48.26{\left(\frac{M}{10}\right)}^8(a_c-a_y)\right]/\sqrt{a_y}$$(1)

Figure 1 provides the correlation between the value of the amplification of a_c and the peak ground acceleration (a_max).

Figure 1: Amplification of maximum acceleration at crest of dams during an earthquake (Jansen, 1990)

Bureau [55] presented an empirical relationship that relates the relative crest settlement (∆/H) (%), the friction angle of the soil material of a dam, and the Earthquake Severity Index (ESI), that is defined as Equation 2, as shown in Figure 2.

  

$$ESI=a_{max}{(M-4.5)}^3$$(2)

Figure 2: Relative crest settlement versus earthquake severity index (Bureau, 1997)

Swaisgood [56, 57, 58] developed an empirical relationship that relates the normalized crest settlement (∆/(DH+AT)) (%), the earthquake moment magnitude (M), and the peak ground acceleration (a_max), based on a database of the seismic response of case histories of dams versus earthquake loading, as shown in Figure 3.

Figure 3: Relative normalized crest settlement versus peak ground acceleration (Swaisgood, 2014)

Pells and Fell [59] classified dams by damage due to earthquakes based on the records of 305 dams, with the classification based on the earthquake magnitude (M), peak ground acceleration (a_max), and the type of material used for the dam. According to Table 2, the damage class reflects six classes or groups of types of damage to dams with crest settlement percentages ranging from less than 0.03% to more than 5%. Based on the plots of damage contours versus earthquake magnitude and peak ground acceleration for earthfill dams from Pells and Fell [59], this method can be used to estimate the magnitude of crest settlement for each damage class given in Table 2.

Table 2: Crest settlement based on damage classification of dams under earthquake loading (Pells and Fell [59])

Damage class		Maximum relative crest settlement* (%)
Number	Description
0	No or slight	< 0.03
1	Minor	0.03 – 0.2
2	Moderate	0.2 – 0.5
3	Major	0.5 – 1.5
4	Severe	1.5 - 5
5	Collapse	> 5

* Maximum relative crest settlement is expressed as a percentage of the maximum dam height (from the foundation to the dam crest)

Singh and Debasis [60] proposed a strong correlation between the ratio of the peak horizontal ground acceleration and the yield acceleration (a_y /a_max)for estimating earthquake-induced crest settlements based on 152 case histories of the performance of dams during earthquakes, as shown in Figure 4. The correlation displayed in this figure should not be regarded as an exact estimator of the crest settlement; rather, it is a simple tool that gives a reasonably conservative estimate of the crest settlements for embankment dams under earthquake loading.

Figure 4: Crest settlement versus a_y/a_max (Singh and Debasis [60])

5.2 Simplified methods

5.2.1 General principles

The analytical concepts of these methods generally follow the original ideas proposed by Newmark [7]. He provided a method or a set of tools for the evaluation of the permanent or irreversible seismic displacement of a slope. In his research, a potential sliding mass is modeled as a rigid block on a potential sliding plane, and is subjected to a constant gravitational force and transient, time-dependent, earthquake acceleration. During the earthquake duration, the earthquake acceleration may or may not exceed the yield or critical acceleration which reflects the characteristics of the slope, including geometrical and geotechnical parameters. The yield acceleration of the block can be determined from a conventional limit equilibrium analysis by calculating the inertial forces required to lower the factor of safety against sliding to 1.0. The block will only move downslope if the earthquake acceleration becomes larger than the yield acceleration of the block. Assuming that the record of the earthquake-induced acceleration for a block is known, the displacement of the block can be derived by the double integration of those portions of the earthquake acceleration exceeding the yield acceleration of the block. The procedure is illustrated in Figure 5.

Figure 5: Concept of Newmark approach (Newmark [7])

The approach should not be applied where embankments or their foundations are susceptible to liquefaction or strain weakening because it will significantly underestimate the displacements [49].

Due to its simplicity, this approach is widely used in engineering practice to estimate the permanent displacement of earth dams. Much research has shown the basic principles of Newmark’s approach to be valid [14, 61, 62, 63, 64, 65, 66].

5.2.2 Types of simplified methods

At present, the analytical procedures for estimating earthquake-induced displacements can be grouped into three types:

• - Rigid block-type procedures: they ignore the dynamic response of the potential sliding mass

  - Decoupled procedures: they account for the dynamic response, but “decouple” this response from the sliding response of the potential sliding mass

  - Coupled procedures: they “couple” the dynamic and sliding responses of the potential sliding mass

It is found that, if the slope geometry, soil properties, and earthquake ground motions are known and the yield acceleration is correctly evaluated, this approach can predict the displacement of a potential sliding mass reasonably well. In other words, in order to apply Newmark’s approach to predict earthquake-induced displacements, both an input acceleration time series, corresponding to the earthquake ground motion, and a yield acceleration representative of the seismic response of the sliding mass are required. It is common to choose sites that have been shaken by an earthquake of the same magnitude, are close to the source of the earthquake, and have the same ground conditions in order to figure out the site-specific acceleration history. However, the selection of ground motions and appropriate scaling factors that achieve the desired level of shaking for a specific site can be very difficult, and usually requires a certain level of expertise and judgment [67].

Therefore, several variations to Newmark’s approach have been proposed. They require structural parameters (yield acceleration (a_y) and first fundamental period of the dam (T_D)) and ground motion parameters (peak ground acceleration a_max, peak ground velocity v_max, earthquake magnitude (M), Arias intensity (I_a) (a measure of the strength of a ground motion and determines the intensity of shaking by measuring the acceleration of transient seismic waves), predominant period of the acceleration spectrum (T_p), spectral acceleration at a degraded period (S_a (1.5T₁)), etc.

The calculated displacement values are correlated to the yield acceleration and to one or more of the characteristic ground motion input parameters in the database of the ground motion. A summary of the regression equations of the common simplified methods is provided in Table 3.

Table 3: Simplified methods

Type	Method	Equation		Reference
Rigid	Sarma (1975)	$$\mathrm{\text{log}}\left(\frac{D}{C{a}_{max}{T}_p^2}\right)=0.85-3.91\left(\frac{a_y}{a_{max}}\right)$$	(3)	[68]
	Franklin and Chang (1977)	$$D=35\frac{v_{max}^2}{a_{max}}exp(-6.91\frac{a_y}{a_{max}}){\left(\frac{a_y}{a_{max}}\right)}^{-0.38}$$	(4)	[68]
	Ambraseys and Menu (1988)	$$\mathrm{\text{log}}\left(D\right)=0.90+\mathrm{\text{log}}\left[{(1-\frac{a_y}{a_{max}})}^{2.53}{\left(\frac{a_y}{a_{max}}\right)}^{-1.09}\right]$$	(5)	[69]
	Yegian et al. (1991)	$$\mathrm{\text{log}}\left(\frac{D}{N_{\mathrm{\text{eq}}}{a}_{max}{T}_D^2}\right)=0.22-10.12\left(\frac{a_y}{a_{max}}\right)+16.38{\left(\frac{a_y}{a_{max}}\right)}^2-11.48{\left(\frac{a_y}{a_{max}}\right)}^3$$	(6)	[70]
	Watson-Lamprey and Abrahamson (2006)	$$\mathrm{\text{ln}}\left(D\left(cm\right)\right)=5.470+0.451(\mathrm{\text{ln}}\left(S_a(T=1s)\right)-0.45)+0.0186{(\mathrm{\text{ln}}\left(S_a(T=1s)\right)-0.45)}^2+0.596(\mathrm{\text{ln}}\left(A_{\mathrm{\text{RMS}}}\right)-1.0)+0.656\left(\mathrm{\text{ln}}\left(S_a(T=1s)/a_{max}\right)\right)-0.0716{\left(\mathrm{\text{ln}}\left(S_a(T=1s)/a_{max}\right)\right)}^2+0.802(\mathrm{\text{ln}}\left(\mathrm{\text{Du}}{r}_{\mathrm{\text{ac}}}\right)-0.74)+0.0763{(\mathrm{\text{ln}}\left(\mathrm{\text{Du}}{r}_{\mathrm{\text{ac}}}\right)-0.74)}^2-\frac{1}{0.581(\mathrm{\text{ln}}\left(a_{max}/a_y\right)+0.193)}$$	(7)	[67]
	Jibson (2007)	$$\mathrm{\text{log}}\left(D\left(cm\right)\right)=0.215+\mathrm{\text{log}}\left[{(1-\frac{a_y}{a_{max}})}^{2.341}{\left(\frac{a_y}{a_{max}}\right)}^{-1.438}\right]$$	(8)	[71]
		$$\mathrm{\text{log}}\left(D\left(cm\right)\right)=-2.710+\mathrm{\text{log}}\left[{(1-\frac{a_y}{a_{max}})}^{2.335}{\left(\frac{a_y}{a_{max}}\right)}^{-1.478}\right]+0.424M$$	(9)
		$$\mathrm{\text{log}}\left(D\left(cm\right)\right)=2.401\mathrm{\text{log}}{I}_a-3.481\mathrm{\text{log}}{a}_c-3.230$$	(10)
		$$\mathrm{\text{log}}\left(D\left(cm\right)\right)=0.561\mathrm{\text{log}}{I}_a-3.833\mathrm{\text{log}}\left(\frac{a_c}{a_{max}}\right)-1.474$$	(11)
	Saygili and Rathje (2008)	$$\mathrm{\text{ln}}\left(D\left(cm\right)\right)=5.52-4.43\left(\frac{a_y}{a_{max}}\right)-20.93{\left(\frac{a_y}{a_{max}}\right)}^2+42.91{\left(\frac{a_y}{a_{max}}\right)}^3-28.74{\left(\frac{a_y}{a_{max}}\right)}^4+0.72\mathrm{\text{ln}}\left(a_{max}\right)$$	(12)	[72]
		$$\mathrm{\text{ln}}\left(D\left(cm\right)\right)=-1.56-4.58\left(\frac{a_y}{a_{max}}\right)-20.84{\left(\frac{a_y}{a_{max}}\right)}^2+44.75{\left(\frac{a_y}{a_{max}}\right)}^3-30.5{\left(\frac{a_y}{a_{max}}\right)}^4-0.64\mathrm{\text{ln}}\left(a_{max}\right)+1.55\mathrm{\text{ln}}\left(v_{max}\right)$$	(13)
	Rathje and Antonakos (2011)	$$\mathrm{\text{ln}}\left(D\right)=\mathrm{\text{ln}}\left(D_{PGA,M}\right)+3.69T_D-1.22{\left(T_D\right)}^2$$	(14)	[73]
Decoupled	Makdisi and Seed (1978)	$$\frac{D}{a_{max}{T}_D}=f\left(\frac{a_y}{a_{max}}\right)[chart-basedmethod]$$	(15)	[14]
	Hynes-Griffin and Franklin (1984)	$$\mathrm{\text{log}}\left(D\left(cm\right)\right)=-0.116{\left(\frac{a_y}{a_{max}}\right)}^4-0.702{\left(\frac{a_y}{a_{max}}\right)}^3-1.733{\left(\frac{a_y}{a_{max}}\right)}^2-2.854\left(\frac{a_y}{a_{max}}\right)-0.287$$	(16)	[74]
	Bray and Rahtje (1998)	$$\mathrm{\text{log}}\left(\frac{D}{k_{max}{D}_{5-95}}\right)=1.87-3.477\left(\frac{a_y}{a_{max}}\right)$$	(17)	[75]
Coupled	Bray and Travasarou (2007)	$$\mathrm{\text{ln}}\left(D\left(cm\right)\right)=-0.22-2.83\mathrm{\text{ln}}\left(k_y\right)-0.333\mathrm{\text{ln}}{\left(k_y\right)}^2+0.566\mathrm{\text{ln}}\left(k_y\right)\mathrm{\text{ln}}\left(a_{max}\right)+3.04\mathrm{\text{ln}}\left(a_{max}\right)-0.244{\left(\mathrm{\text{ln}}\left(a_{max}\right)\right)}^2+0.278(M-7)$$	(18)	[76]

5.3 Advanced methods

Advanced methods have been developed to address some of the key issues that are not adequately considered in the simplified methods. This development has yielded a family of analytical methods, including finite-element, finite-difference, distinct-element, and discrete-element modeling. In practice, these methods may be divided into two main categories: total stress methods and effective stress methods.

5.3.1 Total stress methods

The total stress methods are based on the total stress concept, and the pore pressure development during an earthquake cannot be calculated with them. Therefore, these methods are used in situations where the seismically induced pore pressure is negligible. There are basically two analyses available:

• - Equivalent linear analysis: this is simply a linear response analysis and was developed for approximating the nonlinear behavior of soil under cyclic loading. An equivalent-linear analysis cannot predict the permanent deformation directly, but the output acceleration time history can be used in a simplified deformation analysis by the Newmark method and for developing acceleration time histories for a deformation analysis. The various equivalent linear response programs commonly used include QUAD4M [77] and QUAKE/W [78].

  - Nonlinear analysis: this analysis analyzes the actual nonlinear inelastic stress-strain behavior of the soil using direct numerical integration in the time domain to compute the development of permanent strain throughout an earthquake. This analysis was firstly modelled with a finite element by Prevost [79] using DYNAFLOW. The accuracy of a nonlinear finite element analysis is mostly determined by the stress-strain or constitutive models used. The seismic performance of slopes has been analyzed with two- and three- dimensional finite element analyses using both cyclic stress-strain models [80] and advanced constitutive models [81, 82, 83, 84].

5.3.2 Effective stress methods

The effective stress methods comprise the most comprehensive approach for estimating dam deformations due to earthquake loading. This approach allows for the modeling of the pore pressure development during earthquake shaking. In addition, the approach is able to account for the loss of strength and stiffness due to the increase in pore pressure during shaking, and it is also able to incorporate the residual strength after liquefaction. The effective stress methods can be divided into three catergories: fully coupled, semi-coupled, and uncoupled.

• - Fully coupled effective stress analysis: the soil is treated as a two-phase medium in this analysis, consisting of the soil phase and the water phase. This analysis utilizes elasto-plastic models based on the kinematic hardening theory of plasticity (utilizing multi-yield surfaces) or the boundary surface theory with a hardening law. In general, the fully coupled prediction of the pore pressure under cyclic stress is extremely complicated and challenging. The accuracy of the results is completely dependent on the quality of the input data. The various fully coupled programs commonly used include DNAFLOW [84, 85], PLAXIS [86], ABAQUS [87], and FLAC [88].

  - Semi-coupled effective stress analysis: this analysis is more robust and less susceptible to numerical difficulties. In this analysis, empirical relationships are used to relate cyclic shear strain/stress to pore pressure. In general, a semi-coupled analysis is less complex and often requires input data obtained in the laboratory or in the field. Some semi-coupled programs commonly used include DESRA-2 [89], DSAGE [90], TARA-3 [80], and FLAC [88].

  - Uncoupled effective stress analysis: with this analysis, the pore pressure responses due to an earthquake are estimated separately from laboratory testing or an empirical relationship. Then they are incorporated into a nonlinear elasto-plastic model in order to obtain permanent deformations. The uncoupled analysis is widely used in practice, and has been implemented in the most recent version of the dynamic FLAC software [88].

6. Conclusion

This paper has given an overview of the effects of earthquakes on earth dams through their observed seismic performance during past earthquakes. One or more factors may have a significant effect on how a dam responds during an earthquake. As a result, the appropriate methods for estimating the seismic deformations of earth dams should be used. The three families of methods have been presented in an attempt to make these methods more accessible for use by practicing engineers. The field of earthquake engineering is evolving, from the simplest methods, as empirical database methods, through simplified methods, to complex advanced methods.

The methods discussed in this paper can provide design engineers with some confidence when dealing with one of the most difficult geotechnical engineering problems that they will ever have to face. The more sophisticated methods require experts to help with the analysis and the selection of properties, and thus, should only be used by people who know exactly what they are doing. As a result, a well-designed dam is certainly within the capabilities of the profession of dam engineering. However, the methods for estimating seismic deformations are often approximate and even the most complex methods have a range of probable outcomes due to their own limitations as well as the natural uncertainties of earthquakes. Therefore, the methods for estimating the seismic deformation of earth dams still need to be expanded and improved in order to capture the failure mechanisms more accurately as well as to reduce these uncertainties and ensure the safety of the dams.

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