Seite - 1477 - in Book of Full Papers - Symposium Hydro Engineering

in the vertical and horizontal directions. The same coupling is applied at the interface between the reservoir and foundation (Refer to Figure 1). Coupled Dam-Reservoir-Foundation System FEM Modeling In the present study, the standard finite element technique is adopted utilizing Galerkin’s method in which the structure displacement vector is discretized as (Zienkiewicz 1991). 𝑢 = 𝑁 𝑢 ?̅? ………….……………………….. (8) And the fluid is similarly discretized as 𝑝 = 𝑁 𝑝 ?̅? ………………………….………... (9) Where u and p are the nodal parameters of each field and Nu and Np are appropriate shape functions. The standard finite element technique is adopted utilizing Galerkin’s method in which the discrete equation of the concrete dam and the rock foundation dynamic responses reads. M?̈? +𝐶 ?̇? + 𝐾 ?̅? − 𝑄 ?̅? +𝑓 = 0 ……………...... (10) ∫ 𝑁 𝑢 𝑇 𝑛 𝑝 𝑑 Г = (∫ 𝑁 𝑢 𝑇 𝑛 𝑁 𝑃 𝑑 ГГ1 )?̅? = 𝑄 Г1 ?̅? …..... (11) In which [𝑀 ],[𝐶 ] 𝑎 𝑛 𝑑 [𝐾 ] are mass, damping and stiffness matrices of the structure respectively. ?̅? , ?̇? and ?̈? are displacement, velocity and acceleration vectors respectively. In the above n is the direction vector of the normal to the interface. Standard Galerkin’s discretization applied to the fluid equation and its boundary equations leads to: S?̈? + Ƈ?̇? + 𝐻 ?̅? − 𝑄 𝑇 ?̈? + 𝑞 = 0 ……..………………..…… (12) in which [S], [Ƈ] ,[H] and 𝒒 are pseudo fluid mass matrix, pseudo fluid damping matrix, pseudo fluid stiffness matrix and prescribed flux vector respectively, Q is a transformation matrix which transforms the acceleration of structure to fluid pressure and also transforms the hydrodynamic pressure into applied loads on the structure to simulate fluid structure interaction. 𝑆 = −∫ 𝑁 𝑃 𝑇 1 𝐶 2 𝑁 𝑃 𝑑 𝛺 𝛺 + ∫ 𝑁 𝑃 𝑇 1 𝑔 𝑁 𝑃 𝑑 𝛺 Г3 …………………. (13) Ƈ = ∫ 𝑁 𝑃 𝑇 1 𝐶 2 𝑁 𝑃 𝑑 𝛺 Г4 …... …………………………………... (14) H = ∫ 𝛻 𝑁 𝑇 𝛺 𝛻 𝑁 𝑑 𝛺 ……..……………...........……………… (15) The coupled equation of the dam – reservoir - foundation system subjected to earthquake ground motion can be presented as follows (Zeidan 2014): [ M O QT S ]{ ü p̈ }+[ C 0 0 C̃ ]{ u̇ ṗ } +[ K −Q 0 H ]{ u p} = [ M.I.üg(t) − ρQT. I. üg(t) ] ..… (16) Case of Study KOYNA Dam The KOYNA Dam, situated in the Maharashtra State, India, is part of the KOYNA Hydroelectric Project, which aims to supply water to western Maharashtra as well as hydropower to 1477