Question $1$

A $2-$story $3$D shear building is given below. It is composed of infinitely rigid rectangular slabs

with mass $m_{\text{s}\text{l}\text{a}\text{b}1}$ and $m_{\text{s}\text{l}\text{a}\text{b}22}$ on four flexible columns attached by rigid connections to the slabs

and clamped at the base. The degrees of freedom $u_x,u_y$ and $u_{}$ are defined at the centroid of

each slab.

a$)$ Construct mass and stiffness matrices of the building and write the differential equations of

motion for ground motion exciations in $x$ and $y$ directions $($i$.$e$.g_{gx}^{}$ and $u_{gy}^{}).$

b$)$ Comment on the resulting equations of motion in terms of coupled modes. Assume no

damping.

Story $1$: $m_{\text{s}\text{l}\text{a}\text{b}\text{l}}=90$ tons, Column A & B $=0\mathrm{.}3m**0\mathrm{.}3m,$ Column C & D$=0\mathrm{.}4m**0\mathrm{.}4m.$

Story $2$: $m_{\text{s}\text{l}\text{a}\text{b}2}=60$ tons, Column A & B $=0\mathrm{.}15m**0\mathrm{.}15m,$ Column C & D $=0\mathrm{.}2m**0\mathrm{.}2m.$