Consider the three$-$story shear frame $($beams are infinitely rigid$)$ given above. EI$,$ h $($story height, same for each story$),$ and the m values are $7500$Kn$/$m$2,5$m and $17$ton respectively. Neglect axial deformations of the members.

a$)$

Construct the stiffness matrix.

b$)$

Construct the mass matrix and write the equation of motion.

c$)$

Find the natural frequencies and the mode shapes by using eigenvalue analysis. Normalize each mode so that the roof displacement is unity.

d$)$

Sketch the mode shapes.

e$)$

Check the orthogonality properties w$.$r$.$t$.$ mass and stiffness.

f$)$

Let$$s assume $90\%$ model mass participation ratio is enough for the analysis. How many modes should you consider then? Show your calculations.