Semi Empirical Slope Stability Analyses of Mohmand Dam using Limit Equilibrium and Finite Element Methods

This study presents a framework for semi-empirical slope stability analysis of Mohmand dam, an important ongoing mega concrete faced rockll dam hydropower project in Pakistan. The project comprises of 213 m high hybrid dam that will produce 800 megawatt of clean hydropower energy in addition to an effective ood mitigation. Also, it will supply water for both irrigation and drinking to the provincial capital city, Peshawar. In this study, nite element and limit equilibrium methods have been used for slope stability analysis and factors of safety have been computed for all anticipated loading conditions including earthquake loading. The rockll samples of main dam were obtained from the construction material site of Mohmand dam and the input parameters for slope stability analysis were obtained both empirically and through laboratory testing. Results of both limit equilibrium and nite element analyses have been compared and it was observed that the latter is more conservative than the former except for earthquake loading. The implications of current ndings have been demonstrated using an important case study of an independent dam site that would boost the condence of practitioners.


Introduction
An accurate method for analysis of slope stability has a major impact on the assessment of safety of a project, whereas an appropriate method of stability analyses would result in both safety and economy.
For instance, a small factor of safety (FOS) would result in a signi cant difference in construction cost and time (Bromhead 1992), Slope stability analyses is generally conducted using classical limit equilibrium methods (LEM) and nite element methods (FEM) given their obvious limitations are overshadowed by their nancial and computational advantages (Aryal 2006). For instance, the limit equilibrium approaches rely on geometrical and spatial characteristics of anticipated failure surfaces, whereby the shear strength of the material is insu cient to resist the applied shear stresses (Gri ths and Lane 1999). Similarly, recent research on computational geomechanics have signi cantly enhanced rigor and robustness of FEM approaches (Hammouri et al. 2008), hence making its use for slope stability analyses more reliable (Aryal 2008). Contrary to LEM, no prior knowledge about shape and location of the failure surface is required. In essence, Athania et al. (2015) concluded that FEM provides more reliable and conservative estimates of factors of safety than the LEM.
For brevity, the LEM satis es equilibrium of forces or moments at the limit state based on the principle that as long as resisting shear strength of material remains higher than the destabilizing shear stresses on a pre-determined failure plane, the slope remains stable. Mathematically, this is expressed as the ratio of resisting to destabilizing forces in the slope, termed as factor of safety (FOS), which quanti es the stability of a slope. In contrast, the FEM is regarded as a more detailed and accurate approach that analyses the slope stability problem based on stress-strain constitutive relationship to capture material behaviour and determines the failure surface by looking at the developing shear stresses (Gri ths and Lane 1999). These stresses, if greater than the material strength, can lead to slope failure. A slope stability problem is typically analysed by FEM through the use of two possible approaches, namely; elastoplastic stress analyses and strength reduction method. In former, the model is loaded incrementally Loading [MathJax]/jax/output/CommonHTML/fonts/TeX/fontdata.js to failure and the number of times the failure load is greater than the eld load gives the FOS, while in latter, the shear strength of the material is decreased to cause slope failure. Mohmand and Bajaur Districts. The surrounding districts will bene t from the water supplies of the Mohmand Dam. This study purports to compare the outcomes of both LEM and FEM approaches to analyse slope stability of Mohmand Dam using post-earthquake stability by pseudo static and dynamic analysis. Results for various anticipated loading conditions such as end of construction, full supply level and seasonal variation are also evaluated.

Project Description And Geology
The dam will be founded on Schist, left and right banks will be underlain by hard Quartz Talcosic Schist and Chlorite Mica Schist respectively. The spillway will be on Quartz Talcosic Schist. Powerhouse will be constructed on Graphitic Schist. Geological map of the dam site is shown in the Figure 1. The bedrock is in hard, fresh, and intact condition (WAPDA 2017). The under-construction dam is a concrete face rock ll dam (CFRD). The main dam body will comprise of rock ll Material of different grades/qualities. A concrete slab will span the upstream face acting as a barrier to retain the reservoir due to the highly permeability of rock ll. To model the dam by both LEM and FEM, evaluation of rock ll material properties is a pre-requisite and a critical part of this study.
The size of rock ll material used in this project is large enough to necessitate the use of large scale triaxial test apparatus. However, due to non-availability of such equipment with the authors, shear strength properties of the rock ll were determined using cores obtained from the rock ll and tested using traditional triaxial apparatus. Later, the rock ll parameters were obtained by correlating the results of core-testing with the literature references. Obtained parameters serve as input for LEM and FEM Models.
Rock ll samples to be tested were collected from the actual quarry site to be used for dam construction, The sample collection site is shown in Figure 2. Laboratory testing carried out for the study is discussed in the following section.

Laboratory Investigations
For the current study, Petrographic Analysis is performed on rock ll sample collected at site to determine its mineral composition. Petrographic Analysis results are used to assess the suitability of the material to be used as concrete aggregate and to con rm the rock classi cation carried out in the eld. Rebound hammer tests were also performed in accordance with ASTM D5873 -00. For brevity, hammer spring was pressed by depressing the plunger until the hammer was triggered and the impact occurred.
Height of hammer rebound was measured to the nearest whole number and UCS was calculated using speci c equation for the material under consideration.

Numerical Model And Parameters
Two commercially available software packages, each LEM and FEM based, are used to evaluate slope stability. FOS is computed using input parameters derived using both empirical relations and laboratory test results. The Finite Element Model was meshed with 15 nodal elements with suitable mesh re nement. Mohr-Coulomb model was used as the material model for both LEM and FEM analysis, ( Figure 3).

Loading Conditions
Analyses are performed for all anticipated loading conditions including end of construction, full supply level, minimum water level and earthquake loading. Furthermore, given that the project area falls in a tectonically active zone due to collision between the north moving Indian plate and the Eurasian plate i.e. Zone 2B (BCP 2007), post-earthquake stability was also examined using both pseudo static and dynamic analysis. The recommended ground motion value corresponding to maximum credible earthquake is 0.68 g, peak ground acceleration of 0.16g was xed for the operation basis earthquake (OBE). Equivalent horizontal force component for pseudo static analysis (OBE) (= 2/3 of PGA i.e. 0.107 and for MCE = 1/2 of PGA i.e. 0.34) was used, while the vertical component in pseudo static analysis was kept zero. To perform dynamic time history analysis, design response spectra was developed with a PGA of 0.68 (Figure 4(a)) and the following Idriss Equation was used to develop the design spectra.
where, Y is ground motion parameter, M is earthquake magnitude, R is the closest distance to the source in Km, F is style of faulting factor (i.e. F =0 for a strike slip fault; F =1 for a reverse fault and F =0.5 for an oblique source), ε is standard error term (natural logarithm) and α o , α1, α2, β 0 , βl, β2 are Constants.
The design response spectrum was matched with sample time history of Loma Prieta Earthquake recorded at Lexington Dam California on 18th October 1989 (Figure 4 For brevity, it suggests that the Friction angle for rock ll varies from 45 0 (lowest density, poorly graded weak particles) to as high as 60 0 for high density well graded particles. For Slope Stability Analysis using critical conditions, a conservative estimate of friction angle was taken as 45°.

Results And Discussion
As    Table 3, where the slopes are stable under all loading conditions except for MCE condition in pseudo static analysis. The probable reason for this discrepancy would be that the dynamic analysis uses actual time history of the earthquake as input parameter that may or may not be represented truly by the assumed pseudo static condition. Hence, the pseudo static approach is conservative than the governing dynamic analysis based on actual time history. In case a structure fails in pseudo static analysis, dynamic analysis should be performed to reassure the structure's safety. As Figure 7(a) shows, the horizontal displacements of the embankment soil at full supply level are larger than those at the bottom. Although phreatic pressure is minimum at the top, soil mass at the bottom is more con ned therefore displacement is maximum at the top. Similarly, Figure 7( at the top due to lesser con nement than the bottom that may result in overall settlement of the dam body. Figure 8 presents the variation of factor of safety (FOS) against the elevation of water level in the reservoir. Figures 8(a) and 8(b) show the results of both LEM and FEM analyses using empirical factors, respectively, while Figures 8(c) and 8(d) present the plots of both LEM and FEM analyses using laboratory parameters, respectively. For end of construction (EOC) stage, water level is zero so the stabilizing force on the dam is minimum which provides minimum value of factor of safety. Factor of safety is maximum when reservoir is full (555 m) in all cases. During seasonal variations, water level is reduced to 430 m, which reduces the FOS than that at full supply in the reservoir. Notably, the factor of safety markedly drops when the reservoir at full supply would be hit by an earthquake. For instance, the FOS is lowest for the case of maximum credible earthquake (MCE) shown in Figs. 8(a) and 8(b).
Overall, the FOS value for LEM was 11% -19% higher compared to the FOS obtained through FEM. The reason being that FEM uses stress-strain constitutive relationship and incorporates material stiffness which results in a more accurate and representative behaviour of material, that is shown by moreconservative values compared to the LEM. Such behaviour is also con rmed by Aryal et al. (2005) wherein FEM based calculations resulted in a smaller value of FOS than LEM based method. LEM based models are based on simplistic approach and these methods ignore some crucial complex mechanisms of soil and rock ll failure. Many parameters: for instance, the dilation angle (ψ) and the horizontal to vertical stress ratio (k 0 ) are not included in an LEM analysis as compared to FEM. Results of this study have shown the effect of these parameters on slope stability and on the FOS. Although the location of critical slip surface may occasionally be the same for both methods, the FEM provides results based on stresses and deformations without providing prior slip surface. Hence, FEM can be considered as a more reliable technique to model slopes of the dam under consideration compared to LEM.

Case Study
Mangla dam in Mirpur district is a multipurpose project constructed in 1967 to conserve the water of river Jhelum, irrigation releases and generation of power up to1000 MW. Main features of the project include three earth dams with a maximum height of 380 ft. and a total length of 8 miles. clearly suggested that the LEM based methods are often proved non-conservative because they overestimate the factor of safety that is fully consistent with the ndings of this study. Figures 9(a) and 9(b) show the images of slopes before and after the failure, respectively.

Conclusions
In this paper, results have been reported from the comparison between LEM and FEM based slope stability analyses for a CFRD dam based on input parameters calculated using both laboratory tests and empirical methods. Factor of safety was computed and compared for various loading conditions of the dam. Stability of the dam was also checked for pseudo static and dynamic earthquake loading. Following main conclusions can be drawn from the presented work: The uncon ned compression test is a more conservative approach which gives smaller values of UCS than both the point load strength index test and the rebound hammer test. Similarly, the shear strength parameters calculated through empirical methods and equation often over-estimate than those obtained through laboratory testing of rock sample samples collected from the site.
In practice, the design engineers prefer limit equilibrium method over nite element method believing it is conservative and the design will have an additional factor of safety (FOS   Typical layout for analysis using; (a) limit equilibrium method, and (b) nite element method.  Plot between shear stress and normal stress for friction angle.    (a) Slope during construction before failure, and (b) slope along Mangla Reservoir Rim just after occurrence of failure.