Understanding the concepts on Degrees of Freedom

A five storey structure with 5 translational DOF :

i ) GIVEN :

• Each storey has mass ‘m’.

• The eigenvalue problem has been solved.

• The 5 periods of vibration T_n and their corresponding mode shapes, φ_n has been derived.

ii ) TO FIND :

A. ) 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.

B. ) 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.

C. ) How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?

A . )

1. The fundamental period of vibration (T₁ = 2 seconds) and its associated mode shape:

• Mass matric (m) = `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]`

• Modified mass matrix (M) = `phi_n^T` . m . `phi_n`

For the first mode, (n=1) = [0.334 0.641 0.895 1.078 1.173] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[0.334],[0.641],[0.895],[1.078],[1.173]]`

`M_1` = `[[0.334m,0.334,0.334,0.334,0.334],[0.641,0.641m,0.641,0.641,0.641],[0.895,0.895,0.895m,0.895,0.895],[1.078,1.078,1.078,1.078m,1.078],[1.173,1.173,1.173,1.173,1.173m]]` `[[0.334],[0.641],[0.895],[1.078],[1.173]]`

`M_1` = `[[0.111556m,0.214094,0.29893,0.360052,0.391782],[0.214094,0.410881m,0.573695,0.690998,0.751893],[0.29893,0.573695,0.801025m,0.96481,1.049835],[0.360052,0.690998,0.96481,1.162084m,1.264498],[0.391782,0.751893,1.049835,1.264498,1.375929m]]`

`M_1` = 0.111556m + 0.410881m + 0.801025m + 1.162084m + 1.375929m

`M_1` = 3.861 m

• Model property (L) = `phi_n^T` . m . `phi_n`

Where `iota` = Influencer vector (Assuming 100% utilization of seismic force) = `[[1],[1],[1],[1],[1]]`

For the first mode, (n=1) = [0.334 0.641 0.895 1.078 1.173] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[1],[1],[1],[1],[1]]`

`L_1` = `[[0.334m,0.334,0.334,0.334,0.334],[0.641,0.641m,0.641,0.641,0.641],[0.895,0.895,0.895m,0.895,0.895],[1.078,1.078,1.078,1.078m,1.078],[1.173,1.173,1.173,1.173,1.173m]]` `[[1],[1],[1],[1],[1]]`

`L_1` = 0.334m + 0.641m + 0.895m + 1.078m + 1.173m

`L_1` = 4.121 m

• Model Participation factor (`Gamma`) = `L/M`

For the first mode = (n=1) `Gamma` = `L_1/M_1`

`Gamma_1` = `4.121/3.861`

`Gamma_1` = 1.067

• Effective model mass (M*) = `L_n^2/M_n`

For the first mode, (n=1) M* = `L_1^2/M_1`

M* = `(4.121 m)^2/3.861 m`

M* = 4.398 m.

• Model mass Participation ratio = `M_n^*/m_j`

Where `m_j` = Total mass of the building = m + m + m + m + m = 5m

Model mass participation ratio of first mode (n=1) = 4.398 m / 5m = 0.88 m

• Base shear (`V_B`) = `M_n^* . A_n`

For the first mode, (n=1) `V_B__1` = 4.398 m x 0.27 g

`V_B__1` = 1.187 mg

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

• Mass matric (m) = `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]`

• Modified mass matrix (M) = `phi_n^T` . m . `phi_n`

For the second mode, (n=2) = [-0.895 -1.173 -0.641 0.334 1.078] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[-0.895],[-1.173],[-0.641],[0.334],[1.073]]`

`M_2` = `[[-0.895m,-0.895,-0.895,-0.895,-0.895],[-1.173,-1.173m,-1.173,-1.173,-1.173],[-0.641,-0.641,-0.641m,-0.641,-0.641],[0.334,0.334,0.334,0.334m,0.334],[1.078,1.078,1.078,1.078,1.078m]]` `[[-0.895],[-1.173],[-0.641],[0.334],[1.073]]`

`M_2` = 0.801025m + 1.375929m + 0.410881m + 0.111556m + 1.162084m

`M_2` = 3.861 m

• Model property (L) = `phi_n^T` . m . `phi_n`

Where `iota` = Influencer vector (Assuming 100% utilization of seismic force) = `[[1],[1],[1],[1],[1]]`

For the second mode, (n=2) = [-0.895 -1.173 -0.641 0.334 1.078] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[1],[1],[1],[1],[1]]`

`L_2` = `[[-0.895m,-0.895,-0.895,-0.895,-0.895],[-1.173,-1.173m,-1.173,-1.173,-1.173],[-0.641,-0.641,-0.641m,-0.641,-0.641],[0.334,0.334,0.334,0.334m,0.334],[1.078,1.078,1.078,1.078,1.078m]]` `[[1],[1],[1],[1],[1]]`

`L_2` = -0.895 - 1.173 - 0.641 + 0.334 + 1.078

`L_2` = -1.297 m

• Model Participation factor (`Gamma`) = `L/M`

For the second mode = (n=2) `Gamma` = `L_2/M_2`

`Gamma_2` = `-1.297/3.861`

`Gamma_2` = -0.335

• Effective model mass (M*) = `L_n^2/M_n`

For the second mode, (n=2) M* = `L_2^2/M_2`

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

M* = 0.436 m.

• Model mass Participation ratio = `M_n^*/m_j`

Where `m_j` = Total mass of the building = m + m + m + m + m = 5m

Model mass participation ratio of second mode (n=2) = 0.436 m / 5m = 0.09 m

• Base shear (`V_B`) = `M_n^* . A_n`

For the second mode, (n=2) `V_B__2` = 0.436 m x 0.75 g

`V_B__2` = 0.327 mg

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

• Mass matric (m) = `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]`

• Modified mass matrix (M) = `phi_n^T` . m . `phi_n`

For the third mode, (n=3) = [1.173 0.334 -1.078 -0.641 0.895 ] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[1.173],[0.334],[-1.078],[-0.641],[0.895]]`

`M_3` = `[[1.173m,1.173,1.173,1.173,1.173],[0.334,0.334m,0.334,0.334,0.334],[-1.078,-1.078,-1.078m,-1.078,-1.078],[-0.641,-0.641,-0.641,-0.641m,-0.641],[0.895,0.895,0.895,0.895,0.895m]]` `[[1.173],[0.334],[-1.078],[-0.641],[0.895]]`

`M_3` = 1.375929m + 0.111556m + 1.162084m + 0.410881m + 0.801025m

`M_3` = 3.861 m

• Model property (L) = `phi_n^T` . m . `phi_n`

Where `iota` = Influencer vector (Assuming 100% utilization of seismic force) = `[[1],[1],[1],[1],[1]]`

For the third mode, (n=3) = [1.173 0.334 -1.078 -0.641 0.895 ] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[1],[1],[1],[1],[1]]`

`L_3` = `[[1.173m,1.173,1.173,1.173,1.173],[0.334,0.334m,0.334,0.334,0.334],[-1.078,-1.078,-1.078m,-1.078,-1.078],[-0.641,-0.641,-0.641,-0.641m,-0.641],[0.895,0.895,0.895,0.895,0.895m]]` `[[1],[1],[1],[1],[1]]`

`L_3` = 1.173 m + 0.334 m - 1.078 m - 0.641 m + 0.895 m

`L_3` = 0.683 m

• Model Participation factor (`Gamma`) = `L/M`

For the third mode = (n=3) `Gamma` = `L_3/M_3`

`Gamma_3` = `0.683/3.861`

`Gamma_3` = 0.177

• Effective model mass (M*) = `L_n^2/M_n`

For the third mode, (n=3) M* = `L_3^2/M_3`

M* = `(0.683 m)^3/3.861 m`

M* = 0.121 m.

• Model mass Participation ratio = `M_n^*/m_j`

Where `m_j` = Total mass of the building = m + m + m + m + m = 5m

Model mass participation ratio of third mode (n=3) = 0.121 m / 5m = 0.02 m

• Base shear (`V_B`) = `M_n^* . A_n`

For the third mode, (n=3) `V_B__3` = 0.121 m x 1.04 g

`V_B__3` = 0.126 mg

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

• Mass matric (m) = `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]`

• Modified mass matrix (M) = `phi_n^T` . m . `phi_n`

For the fourth mode, (n=4) = [-1.078 0.895 0.334 -1.173 0.641] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[-1.078],[0.895],[0.334],[-1.173],[0.641]]`

`M_4` = `[[-1.078m,-1.078,-1.078,-1.078,-1.078],[0.895,0.895m,0.895,0.895,0.895],[0.334,0.334,0.334m,0.334,0.334],[-1.173,-1.173,-1.173,-1.173m,-1.173],[0.641,0.641,0.641,0.641,0.641m]]` `[[-1.078],[0.895],[0.334],[-1.173],[0.641]]`

`M_4` = 1.162084m + 0.801025m + 0.111556m + 1.375929m + 0.410881m

`M_4` = 3.861 m

• Model property (L) = `phi_n^T` . m . `phi_n`

Where `iota` = Influencer vector (Assuming 100% utilization of seismic force) = `[[1],[1],[1],[1],[1]]`

For the fourth mode, (n=4) = [-1.078 0.895 0.334 -1.173 0.641] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[1],[1],[1],[1],[1]]`

`L_4` = `[[-1.078m,-1.078,-1.078,-1.078,-1.078],[0.895,0.895m,0.895,0.895,0.895],[0.334,0.334,0.334m,0.334,0.334],[-1.173,-1.173,-1.173,-1.173m,-1.173],[0.641,0.641,0.641,0.641,0.641m]]` `[[1],[1],[1],[1],[1]]`

`L_4` = -1.078 + 0.895 + 0.334 - 1.173 + 0.641

`L_4` = -0.381 m

• Model Participation factor (`Gamma`) = `L/M`

For the fourth mode = (n=4) `Gamma` = `L_4/M_4`

`Gamma_4` = `-0.381/3.861`

`Gamma_4` = -0.098

• Effective model mass (M*) = `L_n^2/M_n`

For the fourth mode, (n=4) M* = `L_4^2/M_4`

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

M* = 0.038 m.

• Model mass Participation ratio = `M_n^*/m_j`

Where `m_j` = Total mass of the building = m + m + m + m + m = 5m

Model mass participation ratio of fourth mode (n=4) = 0.038 m / 5m = 0.01 m

• Base shear (`V_B`) = `M_n^* . A_n`

For the fourth mode, (n=4) `V_B__4` = 0.038 m x 1.03 g

`V_B__4` = 0.039 mg

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

• Mass matric (m) = `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]`

• Modified mass matrix (M) = `phi_n^T` . m . `phi_n`

For the fifth mode, (n=5) = [0.641 -1.078 1.173 -0.895 0.334] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[0.641],[-1.078],[1.173],[-0.895],[0.334]]`

`M_5` = `[[0.641m,0.641,0.641,0.641,0.641],[-1.078,-1.078m,-1.078,-1.078,-1.078],[1.173,1.173,1.173m,1.173,1.173],[-0.895,-0.895,-0.895,-0.895m,-0.895],[0.334,0.334,0.334,0.334,0.334m]]` `[[0.641],[-1.078],[1.173],[-0.895],[0.334]]`

`M_5` = 0.410881m + 1.162084m + 1.375929m + + 0.801025m + 0.111556m

`M_5` = 3.861 m

• Model property (L) = `phi_n^T` . m . `phi_n`

Where `iota` = Influencer vector (Assuming 100% utilization of seismic force) = `[[1],[1],[1],[1],[1]]`

For the fifth mode, (n=5) = [0.641 -1.078 1.173 -0.895 0.334] `[[m,1,1,1,1],[1,m,1,1,1],[1,1,m,1,1],[1,1,1,m,1],[1,1,1,1,m]]` `[[1],[1],[1],[1],[1]]`

`L_5` = `[[0.641m,0.641,0.641,0.641,0.641],[-1.078,-1.078m,-1.078,-1.078,-1.078],[1.173,1.173,1.173m,1.173,1.173],[-0.895,-0.895,-0.895,-0.895m,-0.895],[0.334,0.334,0.334,0.334,0.334m]]` `[[1],[1],[1],[1],[1]]`

`L_5` = 0.641m - 1.078m + 1.173m - 0.895m + 0.334m

`L_5` = 0.175 m

• Model Participation factor (`Gamma`) = `L/M`

For the fifth mode = (n=5) `Gamma` = `L_5/M_5`

`Gamma_5` = `0.175/3.861`

`Gamma_5` = 0.045

• Effective model mass (M*) = `L_n^2/M_n`

For the fifth mode, (n=5) M* = `L_5^2/M_5`

M* = `(0.175 m)^2/3.861 m`

M* = 0.008 m.

• Model mass Participation ratio = `M_n^*/m_j`

Where `m_j` = Total mass of the building = m + m + m + m + m = 5m

Model mass participation ratio of fifth mode (n=5) = 0.008 m / 5m = 0.002 m

• Base shear (`V_B`) = `M_n^* . A_n`

For the fifth mode, (n=1) `V_B__5` = 0.008 m x 1 g

`V_B__5` = 0.008 mg

B. ) Average base shear is calculated by the method of square root of sum of square :

`sqrt(Vb1^2+Vb2^2+Vb3^2+Vb4^2+Vb5^2)`

`V_B` = 1.15 mg

C. ) How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?

• Adding first two mode participation ratios, we get 0.88 + 0.09 = 0.97

Note : To Achieve minimum 90% of modal mass participation ratio, first two modes are enough.