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 the values of seismic parameters based on the given structural data

Step - 01 :-

* 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 :-

• For each of the five modes, Ln, Mn, and n must be calculated using the equations below.
• Calculate the modal mass participating ratio and effective modal mass Mn* for each of the five modes. It will be necessary to evaluate these parameters using the mass matrix.

• Get the combined base shear according to the SRSS rule as well as the base shear for each of the five vibrational modes. Below is provided the pseudo acceleration design spectrum to determine the base shear for each mode.

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

Answer's :-

Step - 02 :-

Step - 03 :-

Step - 04 :-

Step - 05 :-

*

* Result :-

• For each of the 5 modes, the values of Ln, Mn, and Tn are calculated.
• For each of the five modes, the base shear force is determined.
• For five modes, the effective modal mass and the modal mass participating ratio are determined.