Figure P$9\mathrm{.}14$ shows a uniform slab supported on four columns rigidly attached to the slab and clamped at the base. The slab has a total mass $m$ and is rigid in plane and out of plane. Each column is of circular cross section, and its second moment of cross$-$sectional area about any diametrical axis is as noted. With the DOFs selected as $u_x,u_y,$ and $u_{}$ at the center of the slab, and using influence coefficients:

$($a$)$ Formulate the mass and stiffness matrices in terms of $m$ and the lateral stiffness $k=12E\frac{\text{I}}{h^3}$ of the smaller column; $h$ is the height.

$($b$)$ Formulate the equations of motion for ground motion in $($i$)$ the $x-$direction, $($ii$)$ the $y-$direction, and $($iii$)$ the direction $d-b.$

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