Understanding the concepts on Degrees of Freedom

Aim:

• To understand the mode shapes of multi storey buildings
• Manually calculate the base shear for a multi storey building in model analysis
• To understand the technologies involved in the modal analysis.

Calculation:

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.

• 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?

Given data:

The fundamental period of vibration (T₁ = 2 seconds) and its associated mode shapes.

The second mode of vibration (T₂= 0.6852 seconds) and its associated mode shapes.

The third mode of vibration (T₃= 0.6852 seconds) and its associated mode shapes.

The fourth mode of vibration (T₄= 0.6852 seconds) and its associated mode shapes.

The fifth mode of vibration (T₅= 0.6852 seconds) and its associated mode shapes.

Modal participation factor t= L/M

As by the above value,

For the first mode, t= 4.121m/ 3.861m= 1.067

For the second mode, t= -1.297m/ 3.861m= -0.335

For the third mode, t= 0.683m/3.861m= 0.176

For the fourth mode, t=0.381m/3.861m= -0.098

For the fifth mode, t= 0.175m/ 3.861m= 0.045

Effective model mass(M*) = L_n²/M_n

Where, n = respective modes,

By the above formula and value,

M₁* = (4.121m)^2/ 3.861m= 4.398m

M₂* = (-1.297m)^2/ 3.861m=0.436m

M₃* = (0.683m)^2/3.861m= 0.121m

M₄* = (0.381m)^2/3.861m= 0.038m

M₅* = (0.175m)^2/ 3.861m= 0.008m

Modal mass participation ratio= M_n*/m_j

Where m_j = Total mass of the building = m+m+m+m+m = 5m

For first mode, 4.398m/5m= 0.88

For second mode, 0.436m/5m= 0.09

For third mode, 0.121m/5m= 0.02

For fourth mode, 0.038m/5m= 0.01

For fifth mode, 0.008m/5m= 0.002

Adding first two modes participation ratios, we get 0.88+ 0.09 = 0.97

To achieve minimum 90%of modal participation ratio, First two modes are enough