Understanding the concepts on Degrees of Freedom

AIM: Understanding the concepts on Degrees of Freedom

Questions:

• 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.
• 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?

Answer:

Step 1: Calculation of Modified Mass Matrix (M_n):

M_n = ɸ^T_n*m* ɸ_n

1. Mode 1

M₁ = ɸ^T₁*m* ɸ₁

M₁ = 3.861M

2. Mode 2

M₂ = ɸ^T₂*m* ɸ₂

M₂ = 3.861M

3. Mode 3

M₃ = ɸ^T₃*m* ɸ₃

M₃ = 3.861M

4. Mode 4

M₄ = ɸ^T₄*m* ɸ₄

M₄ = 3.861M

5. Mode 5

M₅ = ɸ^T₅*m* ɸ₅

M₅ = 3.861M

Step 2: Calculation of Modal Property (L_n):

L_n = ɸ_n*m*ɩ

1. Mode 1

L₁ = ɸ₁*m*ɩ

L₁ = 4.121M

2. Mode 2

L₂ = ɸ₂*m*ɩ

L₂ = -1.965M

3. Mode 3

L₃ = ɸ₃*m*ɩ

L₃ = 5.086M

4. Mode 4

L₄ = ɸ₄*m*ɩ

L₄ = 0.381M

5. Mode 5

L₅ = ɸ₅*m*ɩ

L₅ = 0.175M

Step 3: Calculation of (Γ_n):

Γ_n = L_n/M_n

• Γ₁ = L₁/M₁ = 4.121M/ 3.861M = 1.067
• Γ₂ = L₂/M₂ = -1.965M/ 3.861M = -0.509
• Γ₃ = L₃/M₃ = 5.086M/ 3.861M = 1.317
• Γ₄ = L₄/M₄ = 0.381M/3.861M = 0.099
• Γ₅ = L₅/M₅ = 0.175M/ 3.861M =0.045

Step 4: Calculation of Effective Modal Mass (M_n^*):

M_n^* = L_n²/M_n

• M₁^* = L₁²/M₁ = (4.121M)²/ 3.861M = 4.39M
• M₂^* = L₂²/M₂ = (-1.965M)²/ 3.861M =0.98M
• M₃^* = L₃²/M₃ = (5.086M)²/ 3.861M = 6.70M
• M₄^* = L₄²/M₄ = (0.381M)²/3.861M = 0.04M
• M₅^* = L₅²/M₅ = (0.175M)²/ 3.861M = 0.01M

Step 5: Calculation of Modal Mass Participating ratio:

M_j = Total Mass = M+M+M+M+M = 5M

• For Mode 1 = M₁^*/ M_j =0.88
• For Mode 2 = M₂^*/ M_j =0.086
• For Mode 3 = M₃^*/ M_j =0.024
• For Mode 4 = M₄^*/ M_j =0.008
• For Mode 5 = M₅^*/ M_j =0.002

Step 6: Calculation of Base Shear (V_bn):

V_bn = M_n^*x A_n

T₁ = 2 sec ; A₁ = 0.27g

T₂ = 0.6852 sec ; A₂ = 0.76g

T₃ = 0.4346 sec ; A₃ = 1.03g

T₄ = 0.3383 sec ; A₄ = 1.03g

T₅ = 0.2966 sec ; A₅ = 0.76g

a. Mode 1

V_b1 = M₁^*x A₁ = 4.40M * 0.27g = 1.188Mg

b. Mode 2

V_b2 = M₂^*x A₂ = 1.00M * 0.77g = 0.77Mg

c. Mode 3

V_b3 = M₃^*x A₃ = 6.70M * 1.03g = 6.901Mg

d. Mode 4

V_b4 = M₄^*x A₄ = 0.04M * 1.03g = 0.041Mg

e. Mode 5

V_b5 = M₅^*x A₅ = 0.01M * 0.77g = 0.77Mg

Step 7: Calculation of combined base shear as per SRSS Rule:

(V_b)² = V_b1² + V_b2² + V_b3² + V_b4² + V_b5²

V_b = 1.188² + 0.77² + 6.901² + 0.041² + 0.77²

V_b = 7.086

Step 8: Calculation of Total modes to be considered

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

M₁^*/ M_j + M₂^*/ M_j = 88% + 8.6% = 96.6%