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  1. Home/
  2. Dinesh Kumar/
  3. Understanding the concepts on Degrees of Freedom

Understanding the concepts on Degrees of Freedom

  Q] 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…

    • Dinesh Kumar

      updated on 16 Jul 2023

     

    Q] 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 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?

     

     

    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:-

    Step:-1

    Step 2:

    Step 3:

    Step 4:

    Step 5:

    Step 6:

    Now we need to obtain the base shear for each of the 5 modes of vibration, as shown below.

    Step:-7

    Combined base shear as per SRSS rule:-

    Step:-8

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

    RESULT:-

    • Calculated Ln , Mn and Γn 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 are obtained.
    • As per 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.

     

     

     

     

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