Q$1.$ a$)$ Please examine Reference Document related to the "static condensation technique".

b$)$ Consider the $3-$story frame building model shown in the figure. It is assumed that the columns are massless and their flexural rigidities are as provided in the figure. There exists a lumped mass at each floor as shown. Neglect axial deformations in all elements. Obtain the differential equation of motion of this idealized structure for the lateral DOFs $($i$.$e$.u_i)$ only, subjected to a horizontal earthquake ground motion $u_g^{}(t)$ by considering the following two cases separately:

beams are flexurally rigid,

beams on each story have flexural stiffness EI$.$

Consider $EI=3600kN{m}^2.$ Use Rayleigh damping with $5\%$ damping ratio for modes $1$ and $2.$ Employ "static condensation technique" whenever necessary. Compare and discuss the results briefly.