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.

• 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?

Ans:

• To calculate Ln,Mn">Ln,Mn��,�� and Γn">ΓnΓ� for each of the five modes
• To calculate the effective modal mass Mn">Mn��* and the modal mass participating ratio for the 5 modes.
• To find the base shear for each of the 5 modes of vibration and the combined base shear as per the SRSS rule.

Procedure:

Step-1:

Given Data:

Step-2: 

• To calculate Ln,Mn">Ln,Mn��,�� and Γn">ΓnΓ� using above given Formulae and given date for Mode-1Γn">
  
• using those calculate Seismic Force for mode-1 and later you calculate base shear for mode-1
• contiue this procedure for remaining 4 modes also 
• Finally add all base shear which are calculated individually we got a total base shear 
• calculate effective modal mass  individually hence calculate modal mass participating ratio 
• Due to this ratio we got exceed 90 percent modal ratio 
• and also total effective modal mass is equal to given total mass of frame hence it is verified.