Understanding the concepts on Degrees of Freedom

Understanding the concepts on Degrees of Freedom

Q] 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:

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

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

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

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

• 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?

AIM:

           1.To compute L_n , M_n and Γ_n for each of the five modes from the given equations.

           2.To calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes.

           3.To obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule by using pseudo acceleration design spectrum provided.

PROCEDURE:

Step1 : To compute L_n , M_n and Γ_n for each of the five modes from the given equations.

Modified Mass Matrix  (M_n) :

Mode:-1

The first mode of vibration(T₁ ) = 2sec

Associated mode shape :

Mass matrix (m) : 

Modified Mass Matrix  (M1) : 

 M1 =3.861m

Similarly 

M3=3.85m

M4 =3.861m

M5 = 3.861m

2.Model property : (L_n )

 

MODE 1:

The first mode of vibration (T1) = 2 seconds

 3.Model property factor (Γ_n) :

For first mode :

(Γ₁) = L₁/M₁

_{Where ,}

L₁ =4.121m And M₁=3.861m

Γ₁ = L₁/M₁ = 4.121m/3.861m

Γ₁ =1.067

MOde 2 :

(Γ₂) = L₂/M₂

L₂ =-1.298m And M₁=3.861m

Γ₂ = L₂/M₂ = -1.298m/3.861m

Γ₂ =-0.33

Mode 3 :

(Γ₃) = L₃/M₃

L₃ =-0.687m And M₃=3.861m

Γ₃ = L₃/M₃ = -0.687m/3.861m

Γ₃ =-0.178

Mode 4 :

(Γ₄) = L₄/M₄

L₄ =0.175m And M₄=3.861m

Γ₄ = L₄/M₄ = 0.175m/3.861m

Γ₄ =-0.178

Mode 5 :

(Γ₅) = L₅/M₅

L₅ =-0.381m And M₄=3.861m

Γ₅ = L₅/M₅ = 0.175m/3.861m

Γ₄ =-0.178

4.Effective model mass (M^*)

M^* = L²/M

For Mode 1:

M^*₁ = (L₁)²/M

M^*₁ = (4.121m)^2­ /3.861m

M^*₁ =4.39m

For Mode 2:

M^*₂ = (L₂)²/M

M^*₂ = (-0.298)^2­ /3.861m

M^*₂ = 0.43m

For mode 3:

M^*₃ = (L₃)²/M

M^*₃ = (0.687m)^2­ /3.861m

M^*₃ = 0.12m

For mode 4:

M^*₄ = (L₄)²/M

M^*₄ = (-0.381m)^2­ /3.861m

M^*₄ = 0.03m

For mode 5:

M^*₅ = (L₅)²/M

M^*₅ = (0.175m)^2­ /3.861m

M^*₅ = 0.007m

5. Model Mass Participation Ratio (M_n/M_j) :

Where, 

M_n= Effective model mass

M_j = Total mass (5 story m+m+m+m+m=5m)

For Mode 1:

M₁/M_j = 4.39m/5m

M₁/M_j =0.88 = 88%

For Mode 2:

M₂/M_j = 0.43m/5m

M₂/M_j =0.086 = 8.6%

For mode 3:

M₃/M_j = 0.12m/5m

M₃/M_j =0.024 = 2.4%

For mode 4:

M₄/M_j = 0.03m/5m

M₄/M_j =0.006 = 0.6%

For mode 5:

M₅/M_j = 0.007m/5m

M₅/M_j =0.001 = 0.1%

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%

6.Base Shear (V_bn)

V_bn = M^* Aⁿ

^Where,

^{M*=Effective model mass}

^{An =The pseudo acceleration} 

We will obtain this Aⁿ in the graph provided below

Where ,

T₁ = 2sec Then A₁ =0.27g

T₂ = 0.685sec Then A₂ =0.76g

T₃ = 0.434sec Then A₃ =1.039g

T₄ = 0.338sec Then A₃ =1.03g

T₅ = 0.296sec Then A₄ =0.76g

Now for mode 1  The base shear :

V_b1 = M₁^* x A₁

V_b1 =4.39m x0.27g

V_b1 =2.619mg

Mode 2  The base shear :

V_b2 = M₂^* x A₂

V_b2 =0.43m x0.76g

V_b2 =0.719mg

Mode 3  The base shear :

V_b3 = M₃^* x A₃

V_b3 =0.12m x1.03g

V_b3 =0.272mg

Mode 4  The base shear :

V_b4 = M₄^* x A₄

V_b4 =0.03m x1.039g

V_b4 =0.067mg

Mode 5  The base shear :

V_b5 = M₅^* x A₅

V_b5 =0.07m x0.76g

V_b5 =0.011mg

7.Avarage base shear as per SRSS rule :

V_b = 3.2g

Result : 

           1. L_n , M_n and Γ_n for each of the five modes from the given equations is calculated.

           2.The effective modal mass M_n* and the modal mass participating ratio for the 5 modes is calculated successfully.

           3.The base shear for each of the 5 modes of vibration and the Avarage base shear as per SRSS rule calculated successfully by using pseudo acceleration design spectrum provided.