Problem $3$

A uniform slab is supported by three identical frames as shown in Figure $3.$ Derive the stiffness matrix of an individual frame with reference to the three degrees of freedom indicated in the figure. Assume that the beam as well as the columns are axially rigid. Obtain the lateral stiffness of a frame by static condensation of the stiffness matrix to eliminate degrees of freedom $u_2$ and $u_2.$ Then obtain the equations of motion for free vibrations of the slab$-$frame system in terms of the three degrees$-$of$-$freedom defined about the mass center.

The welght of the slab including the superimposed dead load is $100l\frac{b}{f}{t}^2.$

Figure $3$