Understanding the concepts on Degrees of Freedom

Question 1:

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: To compute L_n, M_n, T_n for of the five modes from the given equations.

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

To obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule.

PROCEDURE:

Step-1:

M_n=_n^Tm_n

Mode-1:

The first mode of vibration(T₁)=2 seconds

By scientific calculator, we got M₁ as 3.861M

M₁=3.861M

Mode-2:

The second mode of vibration, T₂=0.6852 seconds

By scientific calculator, we got M₂ as 3.86M

M₂=3.861M

Mode-3:

The third mode of vibration, T₃=0.4346 seconds

By scientific calculator, we got M₃ as 3.85M

M₃=3.85M

Mode-4:

The fourth mode of vibration, T₄=0.3383 seconds

By scientific calculator, we got M₄ as 3.861M

M₄=3.861M

Mode-5:

The fifth mode of vibration, T₅=0.2966 seconds

By scientific calculator, we got M₅=3.861M

M₅=3.861M

Step-2:

L_n=_n^Tm

Mode-1:

The first mode of vibration, T₁=2 seconds

By scientific calculator, we got L₁ as 4.121M

L₁=4.121M

Mode-2:

The second mode of vibration, T₂=0.6852 seconds

By scientific calculator, we got L₂ as -1.298M

L₂=-1.298M

Mode-3:

The third mode of vibration, T₃=0.4346 seconds

By scientific calculator, we got L_3­ as 0.687M

L₃=0.687M

Mode-4:

The fourth mode of vibration, T₄=0.3383 seconds

By scientific calculator, we got L₄ as -0.38M

L₄=-0.38M

Mode-5:

The fifth mode of vibration, T₅=0.2966 seconds

By scientific calculator, we got L₅ as 0.175M

L₅=0.175M

Step-3:

Γ_n=L_n/M_n

Mode-1:

Γ₁=L₁/M₁

=4.121M/3.861M

=1.067

Mode-2:

Γ₂=L₂/M₂

=-1.298M/3.861M

=0.178

Mode-3:

Γ₃=L₃/M₃

=0.687M/3.85M

=0.178

Mode-4:

Γ₄=L₄/M₄

=-0.381M/3.861M

=-0.09

Mode-5:

Γ₅=L₅/M₅

=0.175M/3.861M

=0.04

Step-4:

M_n=L_n²/M_n

Mode-1:

M₁^*=L₁²/M₁

=4.39M

Mode-2:

M₂^*=L₂²/M₂

=0.43M

Mode-3:

M₂^*=L₂²/M₂

=0.12M

Mode-4:

M₃^*=L₃²/M₃

=0.03M

Mode-5:

M₅^*=L₅²/M₅

=0.007M

Step-5:

Modal mass participating ratio=M_n^*/M_j

M_n^*= Effective modal mass

M_j= Total mass= M+M+M+M+M=5M

Mode-1:

M₁^*/M_j=0.88=88%

Mode-2:

M₂^*/M_j=0.086=8.6%

Mode-3:

M₃^*/M_j=0.024=2.4%

Mode-4:

M₄^*/M_j=0.006=0.6%

Mode-5:

M₅^*/M_j=0.001=0.1%

Step-6:

To obtain shear force for each of the 5 modes os vibration.

V_bn=A_nxL_n²/M_n

=A_nxM_n^*

From the graph,

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=2.9666 sec, A₅=0.76g

Mode-1:

V_b1=A₁xM₁^*=2.619 kips

Mode-2:

V_b2=A₂xM₂^*=0.719 kips

Mode-3:

V_b3=A₂xM₂^*=0.272 kips

Mode-4:

V_b4=A₄xM₄^*=0.067 kips

Mode-5:

V_b5=A₅xM₅^*=0.011 kips

Step-7:

Combined base shear,

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

=root((2.613)²+(0.719)²+(0.272)²+(0.067)²+(0.011)²)

=3.20 kips

Step-8:

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%

Result:

• L_n, M_n, Γ_n values are calculated for each of the 5 modes.
• The base shear force for each of the 5 modes are obtained.
• Effective modal mass and the modal mass participating ratio for 5 modes are obtained.