Understanding the concepts on Degrees of Freedom

Objectives:

• We have to understand the shapes and modes of the multi storey building.
• After that we have to manually calculate the base shear for a multi storey building in the model analysis.
• At finally make a view on the terminologies that are involved in the model analysis.

Given values in the question:

The 5 modes are given in the questions and their modes are also given in the question.

The elevation is also given in the question i.e. Φ₁, Φ₂, Φ₃, Φ₄, Φ₅.

• The fundamental period of vibration (T₁=2 sec) and its associate mode shape.
• The second mode of vibration (T₂=0.6852sec) and its associated mode shape.
• The 3^rd mode of vibration (T₃=0.4346 sec) and its associated mode shape.
• The 4^th mode of vibration (T₄=0.3383 sec) and its associated mode shape.
• The 5^th mode of vibration (T₅=0.2966 sec) and its associated modes shape.

The values we have to be calculate it from the question.

• We have to calculate L_n, M_n, T_n for each of the 5 modes.
• Second thing we have to calculate the effective model mass M_n and the model mass participating ratio for the all 5 modes. Since the (mass matrix will be required).
• Now obtain the base share for each 5 modes of vibration and the combined best share as per SRSS rules (Square root of sum of squares).
• udo acceleration given in terms of g (for vertical loading we have gravitational acceleration), this graph is drawn against the time period
• How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?

Calculations:

First considered the mass at every floor is m.

Now,

Mass matrix (m) =

Modified mass matrix (M). = Φ_n^T .m. Φ_n

For the first mode, = [0.334 0.641 0.895 1.078 1.173]

n=1,

M₁ =3.681

Calculation of Model Property:

Model Property (L) = Φ_n^T .m. l

where, l = Influence vector =

for the first mode, = [0.334 0.641 0.895 1.078 1.173]

n=1,

L₁ = 4.121 m

Calculation of Model participation factor:

Model participation factor (T) = L/M

For the first mode, T = L₁/M₁

n =1

T₁= 4.121/3.861

T₁ = 1.067 m

Similarly,

T₂= -0.335

T₃= 0.177

T₄= -0.098

T₅= 0.045

Calculation of Effective Model mass

Effective Model mass (M*) = L_n²/M_n

For the first mode, (M₁*) = L₁²/M₁

M₁*= (4.121)/3.861

= 4.398 m

Similarly,

M₂*= 0.436 m

M₃*= 0.121 m

M₄*= 0.038 m

M₅*= 0.008 m

Calculation of Model mass participation ratio:

model mass participation ratio = M_n*/m_j

where, mj = total mass of the building = m+m+m+m+m = 5m

Model mass participation ratio of the first mode (n=1) = 4.398 m /5 m = 0.88

Similarly,

Second mode (n=2) = 0.09

Third mode. (n=3) = 0.02

Fourth mode. (n=4) = 0.01

Fifth mode. (n=5) = 0.002

To achieve the. Minimum 90% of model mass participation ratio. First 2 modes are enough because

0.88+0.09 = 0.97

Calculating Base shear.

Base shear (Vb) = Mn* . An

for the first mode, Vb1 = 4.398 m x 0.27, (n=1)

Vb1 = 1.187 mg

Similarly,

Vb2 = 0.327 mg

Vb3 = 0.126 mg

Vb4 = 0.039 mg

Vb5 = 0.008 mg

Now the Average base shear is calculated by the method of SRSS (Square root of sum of square).

√(Vb1)² +(Vb1)² + (Vb1)² + (Vb1)² + (Vb1)²

Vb = 1.15 mg