Understanding the concepts on Degrees of Freedom

Aim: To calulate the effective modal mass 'Mn' and the modal mass participating ratio for the 5 modes shown from the figure.

Also to calculate base shear for each of 5 modes of vibration and the combined base shear as per SRSS rule.

Answer:

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)=`underset(n)(phi)^T . m . underset(n)(phi)`

For the first mode,(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]]`

`underset(1)(M)`=3.861 m

Modal Property(L)=`underset(n)(phi)^T .m . iota`

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

For the first mode,= `[[0.334,0.641,0.895,1.078,1.73]]``[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,m,0]]``[[1],[1],[1],[1],[1]]`

`underset(1)(L)`=4.121 m

Modal participation factor(`lceiling`)=`frac(L)(M)`

For the first mode,(n=1) `lceiling`=`frac{underset(1)(L)}underset(1)(M)`

`lceilingunderset(1)`=`frac(4.121 m)(3.861 m)`

`lceilingunderset(1)`= 1.067

Similarly, `lceilingunderset(2)`=-0.335

`lceilingunderset(3)`=0.177

`lceilingunderset(4)`=-0.098

`lceilingunderset(5)`=0.045

Effective Modal Mass (`M^*`)=`frac{underset(n^2)(L)}{underset(n)(M)}`

For the first mode,(n=1) `underset(1)(M^*)=frac{underset(1^2)(L)}{underset(1)(M)}`

`underset(1)(M^*)=frac{(4.121m)^2}{3.861m`

`underset(1)(M^*)=4.398m`

Similarly, `underset(2)(M^*)=0.436m`

`underset(3)(M^*)=0.121m`

`underset(4)(M^*)=0.038m`

`underset(5)(M^*)=0.008m`

Modal mass participation ratio= `frac{underset(n)(M)^*}{underset(j)(m)}`

where, `underset(j)(m)`=Total mass of the building=m+m+m+m+m=5m

Modal mass participation ratio of First mode (n=1)= `frac(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)=0.002

Adding first two mode mass participation ratios,we get 0.8+0.09=0.97

To achieve minimum 90% of modal mass participation ratio, First two modes are enough.

Base Shear(`underset(B)(V)`)=`underset(n)(M)^* . underset(n)(A)`

For the first mode,(n=1) `underset(B1)(V)`=4.398m x 0.27 g

`underset(B1)(V)=1.187 mg`

Similarly, `underset(B2)(V)=0.327 mg`

`underset(B3)(V)=0.126 mg`

`underset(B4)(V)=0.039mg`

`underset(B5)(V)=0.008mg`

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

=`sqrt((Vb1^2)+(Vb2^2)+(Vb3^2)+(Vb4^2)+(Vb5^2)`

`underset(B)(V)=1.15mg`