Understanding the concepts on Degrees of Freedom

Question 1:

A five storey structure with 5 translational DOF is shown here. Each storey has mass ‘m’. The eigenvalue problem has been solved and the 5 periods of vibration T_n and their corresponding mode shapes, φ_n has been derived.

The fundamental period of vibration (T₁ = 2 seconds) and its associated mode shape:

The second mode of vibration (T₂ = 0.6852 seconds) and its associated mode shape:

The third mode of vibration (T₃ = 0.4346 seconds) and its associated mode shape:

The fourth mode of vibration (T₄ = 0.3383 seconds) and its associated mode shape:

The fifth mode of vibration (T₄ = 0.2966 seconds) and its associated mode shape:

• L_n , M_n and Γ_n for each of the five modes are to be computed from Equations below.
• Also, calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes. The mass matrix will be required to evaluate these parameters.

• Finally, obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule. The pseudo acceleration design spectrum to obtain the base shear for each mode is provided below.

• How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?

Answers:-

AIM: To compute Ln, Mn, Tn, for of the five modes from the given equations.

1. To calculate the effective modal mass M*_n, and the modal mass participating ratio for the 5 modes.
   
2. To obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule.

PROCEDURE:

a.CALCULATION OF MODIFIED MASS MATRIX, Mn:-

b.Caluclation of modal property(Ln):-

C.Caluculation of Γn:-

d.Calculation of effective modal mass(Mn*):-

e.Calculation of Base shear (V_bn):-

6.Combined base shear

7.The two modes are considered such that a modal mass participating ratio of 90% is obtained.

ie, (M₁*/M₁) + (M₂*/M₁) = 87% + 8.7% = 95.7%

Result:

1. Ln, Mn, Γn values are calculated for each of the 5 modes.
   
2. The base shear force for each of the 5 modes are obtained.
   
3. Effective modal mass and the modal mass participating ratio for 5 modes are obtained.