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?

Aim :-

• To compute L_n , M_n and Γ_n for each of the five modes from given Question.
• To calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes.
• To obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule.

Procedure :-

Step :- 1

To Calculate M_n.

Step :- 2

To Calculate L_n.

Step :- 3

To Calculate Γ_n.

Step :- 4

To Calculate M_n*_.

Step :- 5

To Calculate M_n*/m_j.

Step :- 6

To Calculate V_bn.

Step :- 8

Modal Mass participation ratio

Result :-

• Computed L_n , M_n and Γ_n for each of the five modes from given Question.
• Calculated the effective modal mass M_n* and the modal mass participating ratio for the 5 modes.
• Obtained the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule.