The following figure shows a simple $2$D frame structure. C$1,$ C$2$ and C$3$ indicate the columns, B$1$ and B$2$ the beams and M the total mass of the frame. The total mass of the frame is assumed to be concentrated at a single point as shown. The dimensions of column and beam member are assigned differently to each student in the following table. The cross$-$sectional views are shown under each column member.

$$ Beams are assumed to be axially rigid, i$.$e$.$

$$ Young$$s Modulus

Beams are assumed to be axially rigid, i$.$e$.EA=$

Young's Modulus $E=3{10}^{7}k\frac{N}{m^2}$

Mass of the structure $M=60t(=60000kg)$ Again, the total mass of the frame is assumed to be concentrated at floor level and beams are assumed

to be axially rigid i$.$e$.EA=.$ Such a model is referred to as shear building and lateral stiffness of the

building is given by

$k={\displaystyle \underset{i}{\overset{\text{\#}of\text{c}\text{o}\text{l}\text{u}\text{m}\text{n}\text{s}}{}}}\frac{12E_i{I}_i}{h_i^3}$

where $E$ is the Young's Modulus, I the moment of inertia of column cross section and $h$ the height of

column. Based on the formula and following data, determine the natural period of the building $T_n.$

C$1$ Bx$=60$cm By $=40$cm

C$2$ Bx$=80$cm By$=30$cm

C$3$ Bx$=70$cm By$=30$cm

Beams bw$=30$cm h$=50$cm