Understanding the concepts of Degrees of Freedom

Q 1] A five-storey structure with 5 translational DOFs is shown here. Each storey has mass ‘m’. The eigenvalue problem has been solved and the 5 periods of vibration Tn and their corresponding mode shapes, φn have 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:

• Ln, Mn and Γn for each of the five modes are to be computed from Equations below.
• Also, calculate the effective modal mass Mn* 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 the 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 Ln, Mn and Γn for each of the five modes from the given equations.

To calculate the effective modal mass Mn* 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 the SRSS rule.

PROCEDURE:-

RESULT:-

• Calculated Ln , Mn and Γn for each of the five modes from the given equations.
• Calculated the effective modal mass Mn* and the modal mass participating ratio for the 5 modes.
• The base shear for each of the 5 modes of vibration is obtained.
• As per the SRSS rule the combined base shear is obtained.
• The two modes are to be considered such that a modal mass participating ratio of 90% is obtained.