Understanding the concepts on Degrees of Freedom

General:

Forming the mass matrix (m) as m_ij = 0 where i≠j & the story masses at the main diagonal where m₁₁ for the first story & m₅₅ for the 5^th story.

Mass Matrix (m) =

Applying the following equations of the modal analysis:

Studying the 5 mode shapes of the structure brings the following results:

For Mode 1

* The modal transformed mass matrix (M₁) =

= 3.861 m

* The modal property (L₁) =

= 4.121 m

* The modal participation factor (ᴦ₁) = L₁/M₁ = 3.861m/4.121m = 1.067

* The modal effective mass (M^*₁) = (L₁)²/M₁ = (3.861m)²/4.121m = 4.398 m

* The modal mass participation ratio = M^*₁/∑m = 4.398m/5m = 0.88

* Enter the response spectrum curve with T₁ = 2 sec. to get the pseudo acceleration (A_n) = 0.27g where g is the gravitational acceleration.

* The base shear (V_b1) = M^*₁*A_n = 4.398m*0.27g = 1.187 mg

Using the same manner to calculate the same parameters for the rest of the mode shapes:

For Mode 2

* The modal transformed mass matrix (M₂) =

= 3.861 m

* The modal property (L₂) =

= -1.297 m

* The modal participation factor (ᴦ₂) = L₂/M₂ = -0.335

* The modal effective mass (M^*₂) = (L₂)²/M₂ = 0.436 m

* The modal mass participation ratio = M^*₂/∑m = 0.09

* Enter the response spectrum curve with T₂ = 0.6852 sec. to get pseudo acceleration (A_n) = 0.76g

* The base shear (V_b2) = 0.327 mg

For Mode 3

* The modal transformed mass matrix (M₃) =

= 3.861 m

* The modal property (L₃) =

= 0.683 m

* The modal participation factor (ᴦ₃) = L₃/M₃ =0.177

* The modal effective mass (M^*₃) = (L₃)²/M₃ = 0.121 m

* The modal mass participation ratio = M^*₃/∑m = 0.02

* Enter the response spectrum curve with T₃ = 0.4346 sec. to get pseudo acceleration (A_n) = 1.03g

* The base shear (V_b3) = 0.126 mg

For Mode 4

* The modal transformed mass matrix (M₄) =

= 3.861 m

* The modal property (L₄) =

= -0.381 m

* The modal participation factor (ᴦ₄) = L₄/M₄ =-0.098

* The modal effective mass (M^*₄) = (L₄)²/M₄ = 0.038 m

* The modal mass participation ratio = M^*₄/∑m = 0.01

* Enter the response spectrum curve with T₄ = 0.3383 sec. to get pseudo acceleration (A_n) = 1.03g

* The base shear (V_b4) = 0.039 mg

For Mode 5

* The modal transformed mass matrix (M₅) =

= 3.861 m

* The modal property (L₅) =

= 0.175 m

* The modal participation factor (ᴦ₅) = L₅/M₅ =0.045

* The modal effective mass (M^*₅) = (L₅)²/M₅ = 0.008 m

* The modal mass participation ratio = M^*₅/∑m = 0.002

* Enter the response spectrum curve with T₅ = 0.2966 sec. to get pseudo acceleration (A_n) = 1.03g

* The base shear (V_b5) = 0.008 mg

Final Results:

* Only the first 2 modes of shapes achieve 0.88 + 0.09 = 0.97 mass participation ratio which is greater than the minimum of 90% as per the code.

* The combined base with the SRSS method (Square Root of the Sum of Squares) =

sqrt [(1.187mg)²+(0.327 mg)²+(0.126 mg)²+(0.039 mg)²+(0.008 mg)²] = 1.15 mg

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