Analysis of Pine Flat Dam Considering Fluid-Soil-Structure Interaction and a Linear-Equivalent Model

Abstract

A 2D plane strain model is adopted. After some sensitivity studies an average mesh size of 15 m is selected, resulting in about only 700 nodes. Two methods are considered for fluid-structure interaction modelling: full finite element and Westergaard’s added mass approach, leading to non-significant differences in the results, at least for relatively low input motion. The dam nonlinear response is analyzed through an equivalent linearization technique, based on conventional damage model. Damage is tensile strain controlled and results in a lowered effective concrete Young Modulus. Implementation of the method requires an iterative procedure, which converges in a few iterations. Damage development in dam can be measured by the calculated effective frequency. An output is that the Taft input motion does not generate damage either in the dam body or at the dam-foundation interface. Under ETAF, with the adopted definitions of damage, the dam fails at about 9 s.

Appendix

The following appendix describes the procedure used to find an incoming wave at a depth z parting from the deconvolved signal at the same depth. The foundation damping is neglected throughout the whole procedure. Let:

• $\gamma \left( {z,t} \right)$ be the deconvolved signal at a depth z and time t;

• $\gamma \left( {0,t} \right)$ be the signal at the free-surface at time t;

• $\gamma_{i} \left( {z,t} \right)$ be the incoming signal at a depth z and time t;

• $\gamma_{r} \left( {z,t} \right)$ be the reflected signal at a depth z and time t;

• T be the wave-propagation time in the foundation.

We can write, by definition:

$$ \gamma \left( {z,t} \right) = \gamma_{i} \left( {z,t} \right) + \gamma_{r} \left( {z,t} \right) $$

(2)

Since there is no damping in the foundation:

$$ \left\{ {\begin{array}{*{20}c} {\gamma_{i} \left( {z,t} \right) = \frac{1}{2}\gamma \left( {0,t + T} \right)} \\ {\gamma_{r} \left( {z,t} \right) = \frac{1}{2}\gamma \left( {0,t - T} \right)} \\ \end{array} \Rightarrow } \right.\gamma_{r} \left( {z,t} \right) = \gamma_{i} \left( {z,t - 2T} \right) $$

(3)

Inserting (3) into (2) gives:

$$ \gamma \left( {z,t} \right) = \gamma_{i} \left( {z,t} \right) + \gamma_{i} \left( {z,t - 2T} \right) $$

(4)

Since, for $t < 0$, $\gamma_{i} \left( {z,t} \right) = 0$ we can write:

$$ \gamma_{i} \left( {z,t} \right) = \gamma \left( {z,t} \right) for t < 2T $$

(5)

For a time $t \ge 2T$, the incoming motion $\gamma_{i} \left( {z,t - 2T} \right)$ has already been determined by means of (5), hence, we can find $\gamma_{i} \left( {z,t} \right)$ using:

$$ \gamma_{i} \left( {z,t} \right) = \gamma \left( {z,t} \right) - \gamma_{i} \left( {z,t - 2T} \right) for t \ge 2T $$

(6)

The above procedure was tested for Taft’s deconvolved acceleration and the incoming wave found was the same as half the free-surface acceleration provided by the Contributors. ETAF’s incoming signal is illustrated on Fig. 13.

Fig. 13

ETAF’s incoming signal