Understanding the concepts on Degrees of Freedom

 

  Given :-

• 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 :-

                                        `Phi`₁ = `{(0.334),(0.641),(0.895),(1.078),(1.173):}`

• The second mode of vibration (T₂ = 0.6852 seconds) and its associated mode shape :-

                                        `Phi`₂ = `{(-0.895),(-1.173),(-0.641),(0.334),(1.078):}`

• The third mode of vibration (T₃ = 0.4346 seconds) and its associated mode shape :-

                                        `Phi`₃ = `{(1.173),(0.334),(-1.078),(-0.641),(0.895):}`

• The fourth mode of vibration (T₄ = 0.3383 seconds) and its associated mode shape :-

                                        `Phi`₄ = `{(-1.078),(0.895),(0.334),(-1.173),(0.641):}`

• The fifth mode of vibration (T₄ = 0.2966 seconds) and its associated mode shape :-

                                        `Phi`₅ = `{(0.641),(-1.078),(1.173),(-0.895),(0.334):}`

 

  AIM :-

• To find L_n , M_n and Γ_n for each of the five modes which are to be computed from the Equations below.
• Also, calculate the effective modal mass (M_n*) and the modal mass participating ratio for the 5 modes.

                                     

• Also, obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule.

                                    

• Also, describe how many modes are to be considered such that a modal mass participating ratio of 90% is obtained?
• Pseudo Acceleration Design Spectrum :-

                      

 

 

  ANSWER :-

• Mass Matrix, m = `[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]`
• Since the degree of freedom moves along the direction of the lateral force, the Infuence Vector will be unity.
• Infuence Vector, `iota` = `[[1],[1],[1],[1],[1]]`

 

  1. Calculation of L_n :-

• L_n = `Phi^T`_n m`iota`

 

  For mode 1,

• L₁ = `Phi^T`₁ m`iota`

        `=>` L₁ =  `[[0.334,0.641,0.895,1.078,1.173]]` `[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]` `[[1],[1],[1],[1],[1]]`

        `=>` L₁ = 4.121m

 

  For mode 2,

• L₂ = `Phi^T`₂ m`iota`

        `=>` L₂ =  `[[-0.895,-1.173,-0.641,0.334,1.078]]` `[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]` `[[1],[1],[1],[1],[1]]`

        `=>` L₂ = -1.297m

 

  For mode 3,

• L₃ = `Phi^T`₃ m`iota`

        `=>` L₃ =  `[[1.173,0.334,-1.078,-0.641,0.895]]` `[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]` `[[1],[1],[1],[1],[1]]`

        `=>` L₃ = 0.683m

 

  For mode 4,

• L₄ = `Phi^T`₄ m`iota`

        `=>` L₄ =  `[[-1.078,0.895,0.334,-1.173,0.641]]` `[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]` `[[1],[1],[1],[1],[1]]`

        `=>` L₄ = -0.381m

 

  For mode 5,

• L₅ = `Phi^T`₅ m`iota`

        `=>` L₅ =  `[[0.641,-1.078,1.173,-0.895,0.334]]` `[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]` `[[1],[1],[1],[1],[1]]`

        `=>` L₅ = 0.175m

 

  2. Calculation of Modified Mass Matrix, M_n :-

• M_n = `Phi^T`_n m`Phi`_n

 

   For mode 1,

• M₁ = `Phi^T`₁ m`Phi`₁

        `=>` M₁ =  `[[0.334,0.641,0.895,1.078,1.173]]``[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]``[[0.334],[0.641],[0.895],[1.078],[1.173]]`

        `=>` M₁ = 3.861m

 

   For mode 2,

• M₂ = `Phi^T`₂ m`Phi`₂

        `=>` M₂ = `[[-0.895,-1.173,-0.641,0.334,1.078]]``[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]``[[-0.895],[-1.173],[-0.641],[0.334],[1.078]]`

        `=>` M₂ = 3.861m 

 

   For mode 3,

• M₃ = `Phi^T`₃ m`Phi`₃

        `=>` M₃ = `[[1.173,0.334,-1.078,-0.641,0.895]]``[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]``[[1.173],[0.334],[-1.078],[-0.641],[0.895]]`

        `=>` M₃ = 3.861m 

 

   For mode 4,

• M₄ = `Phi^T`₄ m`Phi`₄

        `=>` M₄ = `[[-1.078,0.895,0.334,-1.173,0.641]]``[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]``[[-1.078],[0.895],[0.334],[-1.173],[0.641]]`

        `=>` M₄ = 3.861m 

 

   For mode 5,

• M₅ = `Phi^T`₅ m`Phi`₅

        `=>` M₅ = `[[0.641,-1.078,1.173,-0.895,0.334]]``[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]``[[0.641],[-1.078],[1.173],[-0.895],[0.334]]`

        `=>` M₅ = 3.861m

 

  3. Calculation of Γ_n :-

• Γ_n = L_n/M_n

 

   For mode 1,

• Γ₁ = L₁/M₁  

        `=>` Γ₁ = (4.121m/3.861m)

        `=>` Γ₁ = 1.067

   For mode 2,

• Γ₂ = L₂/M₂  

        `=>` Γ₂ = (-1.297m/3.861m)

        `=>` Γ₂ = -0.336

   For mode 3,

• Γ₃ = L₃/M₃  

        `=>` Γ₃ = (0.683m/3.861m)

        `=>` Γ₃ = 0.177

   For mode 4,

• Γ₄ = L₄/M₄  

        `=>` Γ₄ = (-0.381m/3.861m)

        `=>` Γ₄ = -0.098

   For mode 5,

• Γ₅ = L₅/M₅  

        `=>` Γ₅ = (0.175m/3.861m)

        `=>` Γ₅ = 0.045

 

  4. Calculation of Effective Modified Mass, M_n* :-

• M_n* = `L^2`_n/M_n

 

   For mode 1,

• M₁* = `L^2`₁/M₁

        `=>` M₁* = (`(4.121m)^2`/3.861m)

        `=>` M₁* = 4.399m

   For mode 2,

• M₂* = `L^2`₂/M₂

        `=>` M₂* = (`(-1.297m)^2`/3.861m)

        `=>` M₂* = 0.435m

   For mode 3,

• M₃* = `L^2`₃/M₃

        `=>` M₃* = (`(0.683m)^2`/3.861m)

        `=>` M₃* = 0.121m

   For mode 4,

• M₄* = `L^2`₄/M₄

        `=>` M₄* = (`(-0.381m)^2`/3.861m)

        `=>` M₄* = 0.037m

   For mode 5,

• M₅* = `L^2`₅/M₅

        `=>` M₅* = (`(0.175m)^2`/3.861m)

        `=>` M₅* = 0.008m

 

  5. Calculation of Modal Mass Participating Ratio :-

• Participating Ratio = M_n* / M_j

          where, M_j = M₁* + M₂* + M₃* + M₄* + M₅* = 5m (sum of mass of each floor)

 

    For mode 1,

• Participating Ratio = M₁* / M_j = 0.8798

    For mode 2,

• Participating Ratio = M₂* / M_j = 0.087

    For mode 3,

• Participating Ratio = M₃* / M_j = 0.0242

    For mode 4,

• Participating Ratio = M₄* / M_j = 0.0074

    For mode 5,

• Participating Ratio = M₅* / M_j = 0.0016

  Verifying,

  The total mass of the buiding = m + m + m + m + m = 5m

  Adding all effective modal mass = M₁* + M₂* + M₃* + M₄* + M₅* = 5m

   Hence OK

• From the above participation ratios of modal mass, we can say that sum of Mode 1 and Mode 2 are enough to obtain a minimum of 90% of the participation.

• 0.8796 + 0.087 = 0.9666 = 96.66%

 

  6. Calculation of Base Shear (V_bn) :-

• V_bn = A_{n *} M_n*

   From the Pseudo Acceleration design Spectrum graph, Design Acceleration for each modes are :-

• For T₁ = 2 seconds, A₁ = 0.279
• For T₂ = 0.6852 second, A₂ = 0.769
• For T₃ = 0.4346 second, A₃ = 1.039
• For T₄ = 0.3383 second, A₄ = 1.039
• For T₅ = 0.2966 second, A₅ = 1.039

    For mode 1,

• V_b1 = A_{1 *} M₁*

        `=>` V_b1 = (0.279) * (4.399m)

        `=>` V_b1 = 1.227m

    For mode 2,

• V_b2 = A_{2 *} M₂*

        `=>` V_b2 = (0.769) * (0.435m)

        `=>` V_b2 = 0.334m

    For mode 3,

• V_b3 = A_{3 *} M₃*

        `=>` V_b3 = (1.039) * (0.121m)

        `=>` V_b3 = 0.126m

    For mode 4,

• V_b4 = A_{4 *} M₄*

        `=>` V_b4 = (1.039) * (0.037m)

        `=>` V_b4 = 0.038m

    For mode 5,

• V_b5 = A_{5 *} M₅*

        For T₅ = 2 seconds, A₅ = 0.27g  

        `=>` V_b5 = (1.039) * (0.008m)

        `=>` V_b5 = 0.008m

 

  7. Calculation of Combined Base Shear as per SRSS rule :-

• `V^2`_B = `V^2`_b1 + `V^2`_b2 + `V^2`_b3 + `V^2`_b4 + `V^2`_b5 

        `=>``V^2`_B  = `(1.227m)^2+(0.334m)^2+(0.126m)^2+(0.038m)^2+(0.008m)^2`

        `=>` V_B = `sqrt (1.634)` m

        `=>` V_B = 1.278m