Figure P9.23 shows a uniform slab supported on four identical 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 stiffness about any diametrical axis is as noted. With the DOFs selected as ux , uy , and uθ , formulate the equations of motion for the system subjected to ground displacements uga(t), ugb(t), ugc(t), and ugd (t) in the x-direction at the supports of columns a, b, c, and d, respectively. These equations governing the dynamic component (= total displacement − quasi-static component) of the displacements ux , uy , and uθ should be expressed in terms of m, b, and k = 12/h3 of the columns. How do these governing equations differ from the case of identical ground motion ug(t) at all column supports?
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