Understanding the concepts on Degrees of Freedom

UNDERSTANDING THE CONCEPTSN 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 (T4 = 0.2966 seconds) and its associated mode shape:

• Ln, Mn and Tn for each of the five models are to be computed from equation 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?

AIM:

To compute the values of seismic parameters based on the given structural data.

STEP 1:

STEP 2:

STEP 3:

STEP 4:

STEP 5:

STEP 6:

• Ln, Mn, Tn values are calculated for each of the 5 modes.
• The base shear force for each of the 5 modes is obtained.
• Effective modal mass and the modal mass participating ratio for 5 modes are obtained.