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 Tn and their corresponding mode shapes, φn 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 (T5 = 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 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?
• 

Solution:

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 SRSS rule.

PROCEDURE: -

Result:

• Calculated Ln , Mn and Γn for each of the five modes from the given equations.

The Ln values for the five modes are:

4.121M, -1.298M, 0.687M, 0.381M, 0.175M respectively.

The Mn values for the five modes are:

3.861M, 3.861M, 3.851M, 3.861M, 3.861M.

The Γn values for the five modes are:

1.067, -0.33, 0.178, -0.09, 0.04 respectively.

• Calculated the effective modal mass Mn* and the modal mass participating ratio for the 5 modes.

The values of Mn* for the five modes are:

4.39M, 0.43M, 0.12M, 0.03M, 0.007M respectively.

And the values of modal mass participating ratios of the 5 modes are:

89%, 8.6%, 2.4%, 0.6%, 0.1% respectively.

• The base shear for each of the 5 modes of vibration are obtained.

The base shear for the 5 modes are:

2.619 kips, 0.719 kips, 0.272 kips, 0.067 kips, 0.011 kips

• As per SRSS rule the combined base shear is obtained.

And as per the SRSS rule the combined base shear is 3.20 kips.

• The two modes are to be considered such that a modal mass participating ratio of 90% is obtained and according to this the two modes are M₁^* & M₂^*