Keywords: earth/rockfill dam, crest settlement, simplified methods, dynamic analysis. Introduction .... behaviour of a potential sliding block of an embankment.
Reliability of simplified methods for evaluation of earthquake-induced displacement in earth and rockfill dams Mojtaba E. Kan and Hossein A. Taiebat School of Civil and Environmental Engineering, University of New South Wales, NSW Abstract: The simplified procedures for evaluation of earthquake induced displacement in earth and rockfill dams are widely used in practice. These methods are simple, inexpensive, and substantially less time consuming as compared to complicated numerical approaches. They are especially recommended to be used as a screening tool, to identify embankments with marginal factor of safety, assuming that these methods always give conservative estimates of deformation. However recent studies show that application of these methods may not be conservative in some cases, especially when the tuning ratio of a dam is within a certain range. In this paper the fundamental theory behind the simplified methods is critically reviewed and practical guidelines are presented that can be used to identify cases where the simplified procedures may not be conservative. Keywords: earth/rockfill dam, crest settlement, simplified methods, dynamic analysis.
Introduction Evaluation of the effects of earthquakes on embankment dams and the safety of the dams during and after earthquakes is an important issue in the design of earth and rockfill dams. Although a low percentage of failures or accidents of earth dams have been due to earthquake loading (Fell et al., 2005), the few reported cases are all among the important engineering problems. This is due partially to the unexpected behaviour of the dams and partially to unknown factors affecting the response of dams under earthquake loading. In some cases, the failure has been due to liquefaction of the dams or their foundations, among the classical cases are the failure of San Fernando Lower Dam in 1971 (Seed et al., 1973) and El Cobre Dam in Chile in 1965 (Dobry and Alvarez, 1967). However, there are a large number of cases where earthquakes resulted in sliding and lateral spreading of the embankment and settlement of the dam crest. The deformation patterns of embankment dams under earthquake loading can be visualised as shown in Figure 1 (Ambraseys, 1958). The magnitude of the crest settlement should be less than the free board of the dam to prevent overtopping and breach. In the design of dams, or in evaluating the earthquake response of existing dams, it is important to evaluate the crest settlement due to earthquake loading. As a general rule, the magnitude of the crest settlement should be less than the freeboard of the dam to prevent overtopping and breach. Different approaches have been proposed for the analysis of earth dams under earthquake loading, ranging from pseudo static stability analysis to simplified dynamic procedures, and to complex numerical analyses. A brief review of these methods is given in the following section with more emphasis on the simplified dynamic procedures. A critical assessment of the simplified procedures, which are widely used in practice, will be presented. Finally the conditions beyond which the simplified procedures cannot be relied upon will be introduced.
Figure 1: Major deformation patterns in earth dams (After Ambraseys, 1958, taken from Yan, 1991)
Methods of estimating deformations of dams The pseudo static stability analysis is probably the oldest subjected to earthquake loading. In this method the effect of an earthquake is considered by application of a horizontal acceleration to a potential sliding wedge in a stability analysis. The acceleration is normally taken as a portion of the peak ground acceleration (Fell et al., 2005). While some design guidelines (such as US Army Corps of Engineers, 1984; ANCOLD, 1998; USBR, 2000) have recommended this method as a screening tool for preliminary studies, others (for instance Fell et al., 2005) are of the opinion that the simplified assumptions used in this method make it unsuitable for use as a design procedure. One limitation in this method is that the
earthquake acceleration is applied as a constant value over the whole height of the embankment, regardless of the location of the potential sliding wedge. Another important deficiency of this method is that it does not take into account the cyclic nature of the earthquake loading, and therefore cannot consider loading reversal. Therefore, a factor of safety of less than one obtained by this method does not necessarily mean the embankment will fail. Also, since this method is based on conventional limit equilibrium analyses, it cannot give any indication of the probable deformation of an embankment. Several empirical relationships have been proposed to predict the crest settlement of earth and rockfill dams subjected to earthquake loading. These relationships have been formulated based on the responses of existing dams after earthquakes. Amongst the first contributions in this area is the relationship proposed by Jansen (1990) as: ππ
β= οΏ½48.26 (10 )8 (ππππ β πππ¦π¦ )οΏ½οΏ½οΏ½πππ¦π¦
[1]
where β is the deformation of the dam crest under an earthquake loading of magnitude M, km is the maximum induced acceleration at the crest, and ky is the yield acceleration of a potential sliding mass; that is the horizontal acceleration which results in a factor of safety of unity for the sliding mass and can be obtained by appropriate methods such as limit equilibrium analyses. Swaisgood (1995 and 1998) proposed relationships for evaluation of crest settlement induced by earthquake loading based on the responses of 54 dams. These relationships were subsequently modified (Swaisgood, 2003) using a larger data base of 69 dams and presented as: S(%) = ππ (6.07 ππππππ+0.57 ππβ8.00)
[2]
where S is the relative settlement of dam crest, as a percentage of the total height of dam and its alluvium foundation, and PGA is the peak ground acceleration. Figure 2 shows the variation of crest settlement, in logarithmic scale, with PGA. The scattered data in this figure indicate the inherent errors in the relationship could be an order of magnitude.
magnitude, PGA and embankment material type. In this contribution the dams are categorised into six groups with different percentage of crest settlement in the range of less than 0.03% to more than 5%, measured with respect to the maximum dam height, as shown in Table 1 (from Fell et al. 2005). Table 1: Crest settlement based on damage classification of embankment dams Damage class Maximum crest settlement (%) Number Description 0 No or slight 0.03 1 Minor 0.03β0.2 2 Moderate 0.2β0.5 3 Major 0.5β1.5 4 Severe 1.5β5 5 Collapse >5
Sing and Roy (2009) evaluated the main parameters affecting crest deformation of dams under earthquake loading using the records of the performances of 152 dams of different types. The results of their studies show there is a strong correlation between ky/PGA and crest settlement. Also, crest settlements appear to depend on To/TP (the ratio of the natural period of the dam in the fundamental mode, To, to the predominant period of the earthquake motion, Tp) and marginally on the earthquake magnitude. No quantitative equation for estimation of crest settlements was proposed. Numerical modelling of the dynamic performance of dams is the most sophisticated approach for estimation of the crest settlement. In a numerical analysis, the nonlinear dynamic behaviour of a dam is simulated by different techniques such as finite element or finite difference methods. Such a procedure is expensive, time consuming and relatively complicated, and therefore may be used only in the final steps of dam design or when preliminary screening shows a marginal factor of safety for a dam. Different numerical methods utilise simplified assumptions, mainly for the behaviour of earthfill and rockfill materials. Although these methods are much more sophisticated than other approaches, the accuracy of their prediction of crest settlement of dams is yet to be established. One of the widely used methods in prediction of the crest settlement is the simplified method proposed by Makdisi and Seed (1978). This method is based on the analytical approach proposed by Newmark (1965) and numerical dynamic analyses of some typical dams. Since it is routinely used in practice, an overview of this method is given in the next section in more detail, followed by a critical evaluation of the method. Simplified semi-empirical methods
Figure 2: Crest settlement of dams under earthquake loading (Swaisgood, 2003)
Pells and Fell (2002 and 2003) classified dams for damage due to earthquakes based on the records of 305 dams, with the classification based on earthquake
Newmark (1965) in the fifth Rankin Lecture proposed a method for evaluation of deformation of slopes and embankments under earthquake dynamic loading. This method, which became the basis of what are known as the simplified methods, is based on the assumption that the behaviour of a potential sliding block of an embankment under earthquake loading is similar to a sliding mass on an inclined surface. An earthquake loading may cause the
block to slide if its acceleration becomes larger than the yield acceleration of the block, ky. The yield acceleration of a potential sliding block is a horizontal acceleration which results in yielding (or failure) of the block with irrecoverable deformation. Only if the acceleration induced by an earthquake becomes larger than the yield acceleration, can permanent displacement of the block occur. Assuming that the record of the earthquake induced acceleration on a block is known, the displacement of the block can be derived by double integration of the earthquake acceleration record exceeding the yield acceleration of the block. Figure 3 shows the basic concepts of the Newmark method.
Figure 4: Variation of seismic acceleration along the height of embankments (Makdisi and Seed, 1978)
Figure 3: Concepts of Newmark approach (Newmark, 1965)
Application of Newmark method of deformation analysis of embankment dams was investigated further by Goodman and Seed (1966) and Sarma (1975). It was concluded that this method can estimate the deformation of dams constructed of dry or compacted cohesionless material, provided the variation of the yield acceleration along the dam height could be included in the analysis... Seed and Martin (1966) and Ambraseys and Sarma (1967) presented a method to evaluate the variation of earthquake acceleration along the dam height using the shear beam theory for a triangular dam cross section. Makdisi and Seed (1978) also modified and improved the original Newmark method by including the effects of dam deformability during earthquakes. They evaluated the variation of the induced acceleration along the dam height approximately as a function of the crest acceleration. This contribution was made based on a series of twodimensional finite element analyses of real and hypothetical dams with heights ranging from 30m to 60m, constructed of compacted cohesive or stiff cohesionless materials, and subjected to earthquakes with magnitudes ranging from 6.5 to 8.25. The stiffness of the embankment material was assumed to be dependent on the strain level. The results of their work is presented in Figure 4, where kmax is the maximum earthquake induced acceleration at depth y, measured from the crest of a dam of height h, and ΓΌmax is the maximum crest acceleration.
Makdisi and Seed (1978) also evaluated the deformation of potential sliding blocks as a function of the dynamic properties of the dam and the earthquake, as shown in Figure 5. The displacement of the sliding block, U, is assumed to be in the horizontal direction. Further adjustment is required to evaluate the displacement in the direction of the slide and the magnitude of the crest settlement. In this procedure, the dynamic properties of the dam, in terms of the maximum crest acceleration, ΓΌmax, and the fundamental period, To, should be evaluated by proper procedures (e.g. using shear beam theory or numerical methods). Makdisi and Seed (1979) proposed a simplified method to calculate the dynamic properties of embankments, i.e., ΓΌmax and To which are required for calculation of crest deformations. This method is mainly based on the shear beam theory with some simplifications, using a trial and error procedure on the mobilised shear strain level and the dynamic response of the dam body, although in current practice it is more common to perform such a calculation by commercial codes (e.g. Ziaie Moayed et al., 2008 and Ghahreman Nejad et al., 2010). Despite recent evidence (e.g. Swaisgood, 2003) which show that the deformation of earth and rockfill dams under earthquake loading is mainly in the form of lateral spreading rather than block slipping, the Makdisi and Seed (1978) method is still widely used in practice as an acceptable design tool to evaluate the range of crest settlements. Most of the references and industry guidelines (e.g. ANCOLD, 1998) recommend this method be used as a screening tool for evaluation of deformation of dams under earthquake loading before performing complicated numerical analyses.
displacements, i.e. lower displacements. Also it was shown that at high values of tuning ratio, To/Tm (ratio of the fundamental period of the soil system to the mean period of the earthquake motion), the displacements are generally underestimated if the decoupled approximation is employed. Note that Tm here is referred to the mean period of the earthquake motion which is defined by Rathje and Bray (1998) as: ππππ =
1 ππ ππ β πΆπΆππ 2
β πΆπΆππ 2 .οΏ½ οΏ½
[3]
where Ci is the Fourier amplitude of the entire accelerogram of the ground motion and fi is the discrete Fourier transformed frequencies between 0.25 and 20Hz.
Figure 5: Variation of normalised displacement with normalised yield acceleration (Makdisi and Seed, 1978)
Is the simplified method conservative? The procedure originally proposed by Newmark (1965) and later modified by Makdisi and Seed (1978) has long been believed to be a conservative method for assessing the deformation of embankment dams, and therefore has been regarded as an appropriate tool for screening of dams with marginal factors of safety. The procedure is that the simplified method is utilised first to evaluate the crest settlement of the dam under seismic loading. Only if the settlement obtained by this method is larger than (or close to) the freeboard of the dam is a more detailed method, such as numerical analysis, employed to evaluate the deformation more accurately. Nevertheless, this method was proposed in the 1970βs when the use of computer-based analytical techniques for engineering purposes was very limited. Makdisi and Seed (1978) highlighted the limitation of their method, stating that βit is a procedure based on few analyses in limited range of applicability and should be improved in future investigationsβ. The results of some recent studies show that the method may not be conservative for some cases. Rathje and Bray (1999) investigated application of the simplified methods for evaluation of the deformations of landfills under earthquake loading and concluded that application of the sliding block method is not always conservative. They used a coupled model in which analysis of dynamic response is performed concurrently with calculation of the movements of the sliding blocks. This relatively simple model was analysed under 19 recorded ground motions as well as a sinusoidal input acceleration. It was shown that for high values of ky/kmax (ratio of the yield acceleration for a given block to the maximum earthquake induced acceleration) the decoupled approach (i.e., Makdisi and Seedβs method) provides less conservative estimates of
For the majority of the cases considered by Rathje and Bray (1999), when the decoupled displacements were non-conservative, the calculated displacement tends to be small. The reason is that high tuning ratios normally give a low value for kmax and a high ky/kmax, which result in small displacements (refer to Figure 5). Very high tuning ratios (e.g. 2 to 4) and very low ky/ kmax ratios could also result in large displacements. For these cases, the displacement obtained by the decoupled method can differ significantly from that of the coupled displacement, providing a significantly non-conservative estimate of the crest displacement. Rathje and Bray (1999) also concluded that the rigid sliding block analysis (referred to as the Newmark approach here), is not generally conservative when the frequency content of the input motion is close to the fundamental period of the embankment, that is when 0.2
