Question 18.1: The uniform five storey shear frame with rigid beams shown i......
The uniform five storey shear frame with rigid beams shown in Fig. 18.4 is subjected to ground acceleration. All the floor masses are m and all stories have same height and stiffness k. Assume the displacement to increase linearly with height above base; formulate the equation of motion for the system and determine natural frequency.
1. Determine general properties
\tilde{k}=\Sigma ~k_j~\left(\psi_j-\psi_{j-1}\right)^2
=k\left(\frac{1}{25}+\frac{1}{25}+\frac{1}{25}+\frac{1}{25}+\frac{1}{25}\right)
=\frac{k}{5}
m=\sum ~m_j ~\psi_j^2
= m ~\frac{(1~+~4~+~9~+~16~+~25)}{25}=\frac{11}{5} ~m
\bar{L}=\Sigma~ m_i \psi_i=\frac{m}{5}(1+2+3+4+5)=3~ m
2. Formulate equation of motion
\frac{11}{5}~ m \ddot{z}+\frac{k}{5}~ z=-3 ~m \ddot{u}_g
\ddot{Z}+\frac{k}{11 ~m} ~Z=\frac{-15}{11} ~\ddot{u}_g
\omega_n=0.302 ~\sqrt{\frac{k}{m}}
The above is 6% higher than actual = 0.28 \sqrt{\frac{k}{m}}.
