Formulate the equations of motion for the system of Problem 9.14 subjected to ground displacements…

Formulate the equations of motion for the system of Problem 9.14 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, h, and the lateral stiffness k = 12/h3 of the smaller column. How do these governing equations differ from the case of identical ground motion ug(t) at all column supports?

Problem 9.14

Problem 9.14

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Problem 9.14

Figure P9.14 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 ux , uy , and uθ 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 = 12 /h3 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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