PSI - Issue 42

H. Mazighi et al. / Procedia Structural Integrity 42 (2022) 1714–1720 H.Mazighi and M.K.Mihoubi / Structural Integrity Procedia 00 (2019) 000 – 000

1716

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2. Finite element modelling and equations The finite element (FE) discretization of the differential equation defines the displacement of the dam structure is descripted as below: ̈ + ̇ + = + (1) The gravity and hydrodynamic forces acting on the upstream face of the dam are defined as: = − ̈ ( ) (2) = (3) F p is a function of parameters of the nodal pressures vector of the reservoir water pressure, P, through the transformation matrix, Q, which is determined as: = ∫ (4) Also, the energy dissipation inside the dam is characterized by Rayleigh damping matrix and is calculated by the following equation: = + (5) Which α and β are constant of damping (as 5%) of the highest and lowest modes in relation to the dyn amic response. The natural frequency ω1 is found to be 18.61 rad/sec according to the first and last modes of vibrations is calculated. This gives the values of α = 0.806 and β = 0.00164 Chopra, (2001). 2.1. Concrete Damaged Plasticity (CDP) In plasticity modelling, the tensor is divided in two parts, the elastic strain and plastic strain et Ghaedi al. (2017): = + (6) The stress tensor is defined as: = (1 − ) ̅ = (1 − ) 0 ( − ) (7) Where d can be in the range of 0 (undamaged) to 1 (fully damaged). However, the effective stress can be calculated as: ̅ = ( ⁄1 − ) = 0 ( − ) (8) 3. Material properties and loadings 3.1. Material properties The non-linear material properties of the Koyna dam are illustrated in table Bhattacharjee and Léger, (1993):

Table 1. Material properties Properties

Dam body

Young modulus E Poisson’s ratio ν

3.1e+10

0.2

Mass density ρ

2643 36.31

Dilatation angle ψ

Tensile failure stress f t

2.9

3.2. Seismic loading Two transverses’ accelerations of Koyna and Saguenay earthquakes are applied to the Koyna dam as shown in Fig.1.