The building shown has a large reciprocating engine located at level $1.$ Using the following data,

compute the maximum shearing forces that columns must resist on each floor. Using El Centro NS

components

W$3=193$ k

k$3=100$ k$/$in$.$

W$2=386$ k

k$2=100$ k$/$in$.$

W$1=386$ k

k$1=100$ k$/$in$.$

Fig. $5$

A$)$ Determine the natural frequency, natural periods, and the corresponding mode shapes.

B$)$ Verify that the modes are orthogonal as expected.

C$)$ Normalize the first mode with respect to the mass matrix $\backslash$phi $1$Tm$\backslash$phi $1=1\mathrm{.}0$

D$)$ Use the normalized first mode $($from above$)$ to verify that $\backslash$phi $1$Tk$\backslash$phi $1=\backslash$omega $2$

E$)$ Use the El Centro Response Spectrum and a damping ratio of $5\%$ to estimate the maximum base

shear and moment.

Engine

Weight of engine

$100$ Kips