Understanding the concepts on Degrees of Freedom

Answer

The Mass Matrix is Given as below

1. Calculation of Mass Matrix (M):

2. Calculation of Modal Property (L):

3. Calculation of Modal Participation Factor (Γ):

4. Calculation of Effective Modal Mass (M*):

5. Calculation of Modal Participation Ratio

To obtain a Modal Mass Participation Ratio of 90%, the First Two nodes are sufficient

6. Calculation of Psuedo Spectral Accelaration (Ah):

• We need to Find the respective Psuedo Spectral Accelaration corresponding the respective Time Period of different nodal shapes.
• T1 = 0.27 g
• T2 = 0.75 g
• T3 = 1.0 g
• T4 = 1.0 g
• T5 = 1.0 g

7. Calculation of Base Shear (Vb):

8. Average Base Shear :

• The average base shear is calculated by the method of Square Root of Sum of Square.

Result:

The Average Base Shear = 1.238 mg