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 COMPUT Ln, Mn AND Tn FOR EACH OF THE FIVE MODES FROM THE GIVEN EQUATION
• TO CALCULATE THE EFFECTIVE MODAL MASS Mn* AND THE MODAL MASS PARTICIPATING RATIO FOR THE 5 MODES.
• TO OBTAION THE BASE SHEAR FOR EACH OF THE 5 MODES OF VIBRATION AND THE COMBINED BASE SHEAR AS PER SRSS RULE.

PROCEDURE: -

STEP:-1

STEP: -8

The two modes are to be considered such that a model mass participating ratio of 90% is obtained

M₁^* / M_j + M₂^*/M_j

= 88% + 8.6%

= 96.6%

RESULT:

HENCE,

• COMPUTED Ln, Mn AND Tn FOR EACH OF THE FIVE MODES FROM THE GIVEN EQUATION
• CALCULATED THE EFFECTIVE MODAL MASS Mn* AND THE MODAL MASS PARTICIPATING RATIO FOR THE 5 MODES.
• OBTAION THE BASE SHEAR FOR EACH OF THE 5 MODES OF VIBRATION AND THE COMBINED BASE SHEAR AS PER SRSS RULE.