Understanding the concepts on Degrees of Freedom

• 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.
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• 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?

  

• ANSWER :

RESULT:

• Calculated Ln, Mn & 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.