Page - 1200 - in Book of Full Papers - Symposium Hydro Engineering

where, F.S. factor of safety, τf failure shear strength of the soil, and τ shear stress of the soil. The stability of earth fill dam depends on its geometry, components, materials, properties of each component and the forces to which it is subjected. There are some analytical tools for assessing the stability of a slope by using simple failure models. These techniques are limited to slopes of homogenous material, soils, fractured rocks which are behaving like soils and for plain strain (2-D) problems only. Some of these tools are listed by [13]. For steady state seepage condition the factor of safety using Fellenius method is [5, 14]: F.S. = ∑ (𝐜 𝐋 𝐢 +(𝐖 𝐢 𝐜 𝐨 𝐬 𝛂 𝐢 −𝐮 𝐢 𝐋 𝐢 )𝐭 𝐚 𝐧 𝛗 ) 𝐢 =𝐧 𝐢 =𝟏 ∑ 𝐖 𝐢 𝐬 𝐢 𝐧 𝛂 𝐢 𝐢 =𝐧 𝐢 =𝟏 …………………………….. (8) For steady state seepage condition the factor of safety using Bishop Method is: FS = ∑ (𝐜 𝐛 𝐢 +(𝐖 𝐢 −𝐮 𝐢 𝐛 𝒊 )𝐭 𝐚 𝐧 𝛗 ) 𝟏 𝒎 𝜶 𝒊 𝐢 =𝐧 𝐢 =𝟏 ∑ 𝐖 𝐢 𝐬 𝐢 𝐧 𝛂 𝐢 𝐢 =𝐧 𝐢 =𝟏 ………………….……………………….. (9) where, 𝒎 𝜶 𝒊 = 𝒄 𝒐 𝒔 𝜶 𝒊 + 𝒕 𝒂 𝒏 𝝋 𝐬 𝐢 𝐧 𝜶 𝒊 𝑭 𝑺 c = effective soil cohesion, L = length of the bottom of the slice, b = width of the slice and equal to (L cos𝜶 ), u = pore water pressure, W= weight of the slice, 𝛂 =inclination of the bottom of the slice and 𝛗 = effective internal friction angle. 5. Finite Element Formulation The basic concept of the finite element method is to divide the problem region into subdomains (finite elements) connected at their common nodal points and that the unknown function of the field variable is defined approximately within each element. The approximate solution of each element expressed by continuous function is as follows [3]: 𝐇 = ∑ [Ni enoe e=1 ] {𝐇 𝐢 𝐞 } …………………………………………… (11) where, 𝐇 𝐢 𝐞 = nodal value of (H) for ith node in element (e), noe = total number of elements and Ni e = shape function of element (e). There are more different approaches to formulate the approximate solution of the problem. In the present study, the standard weighted residual method with Galerkin’s criterion [6] is used to approximate the solution of the unknown variable (H). Thus, equation (2) can be written in a matrix form as [3]: [𝐊 ].{𝐇 } = {𝐅 } ……………………….………………………… (12) in which {𝐇 } is the unknown nodal potential head vector, {𝐅 } is the nodal external flux vector and [𝐊 ] is the conductivity matrix given by: 𝐊 ij = ∫ ( ∂Nj ∂x 𝐊 x ∂Ni ∂xVe + ∂Nj ∂y 𝐊 x ∂Ni ∂y ) dx dy………………….. (13) where, Ve = the domain of element (e) and Ni = interpolation or shape function. Derivations of the above functions are given in detail by [3]. The solution of seepage problem with phreatic surface require a successive adjustment for the location of the phreatic surface and the finite element mesh size till the desired degree of convergence for the head H is achieved. In all iterative methods, the solution is started by using initial guess for the unknown values and the solution is obtained by repeating the solution of the system of equations successively through recurrence relations to update the old values until the solution converges closely enough to the true values within some prescribed tolerance of error [3]. 6. Safety Criteria and Verification of Model Design and safety evaluation of embankment dams should satisfy the recommended criterion of the experienced agencies in the design of embankment dams. Among the numerous dam safety regulation [13, 15, 16, 17], both USACE (2003) [18] and BDS (1994) [19] criterion are considered as the benchmark for their broad area of validation in the present study. Geostudio computer code [20] is employed in the present study to simulate seepage and stability analysis of atypical earth dam. The model is verified with reference [21] for a solved example with a zoned embankment dam for slope stability analysis. The verification is done for three cases of operation using Bishop's method. Figures (2- 4) and Table (1) show the factor of safety (F.S.) simulation of upstream and downstream slope for all cases of operations. For upstream slope, the absolute mean error (AME) is about 0.16 1200