List of Tables

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Table of Symbols

A_i	Tributary area of a liquid free surface node projected in the horizontal plane.
${\bf B}$ or ${\bf B}_o$	Linear strain-displacement matrix.
${\bf B}_l$	Nonlinear strain-displacement matrix.
${\bf C}_s$	Solid domain damping matrix.
${\bf C}_{fs}$	Liquid-solid interaction matrix.
${\bf C}_{sf}$	Solid-liquid interaction matrix.
${\bf C}_{\phi h}$	Liquid-free surface interaction matrix.
${\bf E}$	Constitutive matrix.
${\bf F}_{ext}$	External force/discharge vector.
${\bf F}_{int}$	Internal force/discharge vector.
${\bf F}_{I}$ and ${\bf F}_{E}$	Interface and external body forces, respectively.
F_n and F_t	Normal and tangential contact forces, respectively.
g	Acceleration of gravity.
${\bf G}$	Ground motion vector.
${\bf h}$	Vector of liquid free surface heights.
H	Tank height.
${\bf J}$	Element Jacobian matrix.
${\bf K}_{s}$ , ${\bf K}_{f}$ and ${\bf K}_{h}$	Solid, liquid and free surface element stiffness matrices, respectively.
${\bf K}_{\lambda}$	Contact/constraint matrix.
l_x	Projected width of liquid element.
L₁, L₂ and L₃	Nondimensional area coordinates.
${\bf n}$	Unit vector normal to the element face.
N_i	Shape function at node i.
${\bf M}^j_f$	Liquid element face mass matrix.
${\bf M}_s$	Structural mass matrix.
P	Liquid total pressure.
${\bf P}$	Contact penetration vector.
${\bf q}$	Lamina transformation matrix.
${\bf r}$	Nodal position vector.

R	Radius of a cylindrical tank or base width of a rectangular tank.
${\bf R}_d$ and ${\bf R}_k$	Residual vectors of the dynamic and kinematic conditions, respectively.
${\bf R}_f$	Residual vector of liquid domain nodes.
T_a	Tributary surface area of a solid domain node.
${\bf u}$ , ${\bf\dot{u}}$ and ${\bf\ddot{u}}$	Nodal displacement, velocity and acceleration vectors, respectively.
${\bf u'}$	Transformed displacement vector.
${\bf v}_3$	Fiber direction.
v_n and $\dot{u}_n$	Liquid and solid normal velocity at a solid-liquid interface, respectively.
${\bf V}$	Velocity of a fluid particle.
${\bf V}_{r}$	Velocity of a fluid particle relative the fluid mesh.
${\bf x}_i^t$ and ${\bf x}_i^b$	Top and bottom nodal coordinate vectors, respectively.
x', y' and z'	Transformed lamina coordinate system.
$\Gamma_f$	Liquid domain boundary.
$\Gamma_s$	Free surface boundary.
$\Gamma_w$	Liquid flexible boundary.
$\theta$	Nodal rotational vector.
$\lambda$	Lagrange multipliers vector.
$\nu_x$ and $\mu$	Free surface damping parameters.
$\Pi$	Energy functional.
$\Omega_s$ and $\Omega_f$	Solid and liquid domains, respectively.
$\phi$ and $\Phi$	Velocity potential function and vector, respectively.
$\bigtriangledown^2$	Laplacian differential operator.
$\bigtriangledown$	Gradient differential operator.
$\xi$ , $\eta$ and $\zeta$	Element local non-dimensional coordinates.
$\rho_s$ and $\rho_f$	Solid and liquid mass densities, respectively.
$\sigma$	Stress and strain vectors, respectively.

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A. Zeiny
2000-09-06