Understanding the concepts on Degrees of Freedom

Modal Shape (ɸ₁) =

Modal Shape (ɸ₂) =

Modal Shape (ɸ₃) =

Modal Shape (ɸ₄) =

Modal Shape (ɸ₅) =

Mass Matrix (m) =

Step 1:-

To find the Modified mass matrix (M).

Modified mass matrix (M) = ɸ_n^T .m. ɸ_n

For Mode 1:-

n = 1

M₁ = ɸ₁^T .m. ɸ₁

M₁ = (0.334 0.641 0.895 1.078 1.173) x x

M₁ = 3.861 m

Similarly,

For,

Mode 2, M₂ = 3.861 m

Mode 3, M₃ = 3.861 m

Mode 4, M₄ = 3.861 m

Mode 5, M₅ = 3.861 m

Step 2:-

To find the Modal property (L).

Modal property (L) = ɸ_n^T .m. Ɩ

Where Ɩ is Influence vector (Assuming 100% utilization of Seismic force)

Ɩ =

For Mode 1:-

n = 1

Modal property (L₁) = ɸ₁^T .m. Ɩ

L₁ = (0.334 0.641 0.895 1.078 1.173) x x

L₁ = 4.121 m

Similarly,

For,

Mode 2, L₂ = -1.297 m

Mode 3, L₃ = 0.683 m

Mode 4, L₄ = -0.381 m

Mode 5, L₅ = 0.175 m

Step 3:-

To find the Modal property (Γ).

Modal participation factor (Γ) = L/M

For Mode 1:-

n = 1

Modal participation factor (Γ ₁) = L₁/M₁

Γ ₁ = 4.121 m / 3.861 m

Γ ₁ = 1.067

Similarly,

For,

Mode 2, Γ ₂ = -0.335

Mode 3, Γ ₃ = 0.177

Mode 4, Γ ₄ = -0.098

Mode 5, Γ ₅ = 0.045

Step 4:-

To find the Effective modal mass (M_n^*).

Effective modal mass (M_n^*) = L_n²/M_n

For Mode 1:-

n = 1

Modal participation factor (M₁^*) = L₁²/M₁

M₁^* = (4.121 m)² / 3.861 m

M₁^* = 4.398 m

Similarly,

For,

Mode 2, M₂^* = 0.436 m

Mode 3, M₃^* = 0.121 m

Mode 4, M₄^* = 0.038 m

Mode 5, M₅^* = 0.008 m

Step 5:-

To find the Modal mass participation ratio.

Modal mass participation ratio = M_n^*/m_j

Where m_j is Total mass of the building

m_j = m + m + m + m + m

m_j = 5 m

For Mode 1:-

n = 1

Modal mass participation ratio = M₁^*/m_j

= 4.398 m / 5m

= 0.879

Similarly,

For,

Mode 2, MMPR = 0.087

Mode 3, MMPR = 0.024

Mode 4, MMPR = 0.007

Mode 5, MMPR = 0.002

Adding first two modal participation ratios.

We get 0.879 + 0.087 = 0.966

To achieve minimum 90% of modal mass participation ratio, First Two modes are enough.

Step 6:-

To find the Base Shear (V_B).

Base Shear = M_n^* . A_n

Where A_n is Pseudo Acceleration.

For Mode 1:-

n = 1

Base Shear = M₁^* . A₁

V_B1 = 4.398 m x 0.27 g

V_B1 = 1.187 mg

Similarly,

For,

Mode 2, V_B2 = 0.327 mg

Mode 3, V_B3 = 0.126 mg

Mode 4, V_B4 = 0.039 mg

Mode 5, V_B5 = 0.008 mg

Average Base Shear is calculated by the method of Square-Root-of-Sum-of-Squares.

V_B = (V_B1)² + (V_B2)² + (V_B3)² + (V_B4)² + (V_B5)²

V_B = 1.238 mg