A steel frame $(E=29,000$ksi$)$ has two columns that are fixed at the base and pinned at the connection with the roof girder. The columns are bending about the strong axis. The girder is rigid. The damping ratio of the frame is $2\%.$ There is a rotating machine on the roof inducing a force $F(t)=200lbsin(5t).$ The girder plus the machine weigh $5\mathrm{.}5$kips, and each column weighs $600lb.$ Calculate:

a$.$ The equivalent mass at roof level $($consider one half of column mass concentrated at roof level$)$

b$.$ The horizontal stiffness of the frame

c$.$ The steady state amplitude of vibrations

d$.$ The maximum bending stress induced in the columns due to dynamic excitation.