Problem $1$

The structure is subjected to an initial displacement delta$\_(0)$ at each story and then released from rest.

$($a$)$ Derive the equations for the displacement of the structure using modal analysis.

$($b$)$ Determine the story shears.

IMPORTANT INFO:

stiffness $($k$)=12$EI$/($h$)^3$

mass matrix $[$ m $0/0$ m$/2]$

stiffness matrix $=[4$k $-2$k $/-2$k $2$k $]$

natural frequency $1=1\mathrm{.}08($k$/$m$)^1/2$

natural frequency $2=2\mathrm{.}61($k$/$m$)^1/2$

mode shape $1=\{0\mathrm{.}711\}$

mode shape $2=\{-0\mathrm{.}711\}$