Understanding the concepts on Degrees of Freedom

Ans1.

Mass matrix (m)=`[[[m,0,0,0,0]],[[0,m,0,0,0]],[[0,0,m,0,0]],[[0,0,0,m,0]],[[0,0,0,0,m]]]`

Modified mass matrix (M)=`phi_n^T.m.Phi`

For the frist model, (n=1)=`[[0.334,0.641,0.895,1.078,1.173]]*[[[m,0,0,0,0]],[[0,m,0,0,0]],[[0,0,m,0,0]],[[0,0,0,m,0]], [[0,0,0,0,m]]]*[[0.334],[0.641],[0.895],[1.078],[1.173]]`

M =`[[0.334m,0.641m,0.895m,1.078m,1.173m]]*[[0.334],[0.641],[0.895],[1.078],[1.173]]`

M=3.861m

Ans2.

Model property (L)=`phi_n^T.m.l`

where l=Influence vector (Assuming 100% utilization of seismic force)=`[[1],[1],[1],[1],[1]]`

for the frist mode,= `[[0.334,0.641,0.895,1.078,1.1173]]*[[[m,0,0,0,0]],[[0,m,0,0,0]],[[0,0,m,0,0]],[[0,0,0,m,0]], [[0,0,0,0,m]]]*[[1],[1],[1],[1],[1]]`

`[[0.334m,0.641m,0.895m,1.078m,1.173m]]*[[1],[1],[1],[1],[1]]`

`L_1` =4.121m

Ans3.

Model participation factor `(Gamma)=L/M`

For the frist model,(n=1) `Gamma=L_1/M_1`

`Gamma_1=(4.121m)/(3.861m)`

`Gamma_1=1.067`

Similarly,

`Gamma_2=`-0.335

`Gamma_3=`0.177

`Gamma_4=`-0.098

`Gamma_5=`0.045

`ffective Model mass (M^*)=`L_n^2/M_n`

for the frist mode(n=1), `M_1`*=`L_1^2/M_1`

`M_1`*=`(4.121m)^2/(3.861)`

`M_1`*=4.398m

Similarly,

`M_2^*=0.436`

`M_3^*=0.121`

`M_4^*=0.038`

`M_5^*=0.008`

Ans4.

Model mass participation ratio=`M_n^*/m_j`

Where, `m_j=`total mass of the building = m+m+m+m+m=5m

model mass participation ratio of frist mode (n=1)=4.398m/5m=0.88

similarly,

Second mode (n=2)=0.09

Third mode (n=3)=0.02

fourth mode (n=4)=0.01

fifth mode (n=5)=0.09

Adding First two mode mass participation ratios, we get 0.88+0.09=0.97

To achieve minimum 90% of model mass participation ratio, first two modes are enough

Ans5.

Base shear `(V_B)=M_n^*.A_n`

For the first mode(n=1), `V_(B1)=`4.398x0.27g

`V_(B1)=`1.187mg

Similarly,

`V_(B2)=`0.327mg

`V_(B3)=`0.126mg

`V_(B4)=`0.039mg

`V_(B5)=`0.008mg

Average Base shear is calculated by the method of Square root of sum of square

`sqrt[(Vb_1^2)+(Nb_2^2)+(Vb_3^2)+(V_4^2)+(Vb_5^2)`

`V_B=`1.15mg