Q$1$: A two$-$storey frame is shown in Figure Q$1.$ The horizontal beams are infinitely stiff

relative to the vertical columns.

Figure Q$1$

Design data:

Bending stiffness of columns: El$=81\mathrm{.}35$ MNm$^2.$

Mass per unit length of the horizontal beams: $m^{\text{'}}=10,000k\frac{g}{m}.$

Stiffness of the horizontal spring restraining the top floor: $k=50M\frac{N}{m}.$

It can be assumed that damping ratio is $2\%$ for all relevant modes of horizontal

vibration.

For the structural frame shown in Figure Q$1$:

b$)$ Calculate the natural frequencies of the system.