Understanding the concepts on Degrees of Freedom

Answer to the Question No. 1

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.

Answer:

Aim: Aim to the question is to given 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.

Objective & Introducton: Objective of the question:

• To understand modes of multi-storey building.
• Manually calculated the base shear for a story building in model analysis.
• Understand the terminologies involved in mdel analysis

Procedure:

Pseudo acceleration in terms of for vertical loading we have yet gravitational acceleration, we are analysis Horizontal load. Horizontal deformation Horizontal acceleration is required.

To convert the horizontal force into relative base shear for a particular high rise building we neec to convert Horizontal mass into force with the relative acceleration only it cannot be it suppose to be Downward direction for the horizontal mass we have to multiple acceleration factor that is called Pseudo Spectrum acceleration which will be shown of this graph show time period in the x direction nd Pseudo acceleration in the y direction.

Calculation:

Mass matrix (m) =`[[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]`

Modifiedcmass matrix (M) = `Φnt . m. Φn`

For the First mode=

(n=1)

`[[0.334,0.641,0.895,1.078,1.173]][[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]] [[0.334],[0.641],[0.895],[1.078],[1.173]]`

M1 = 3.861m

Model Property (L) = `Φnt . m.l`

where `l` = influencer vector (Assuming 100% utilization of seismic force)= `[[1],[1],[1],[1],[1]]`

For the first mode=

(n=1)

`[[0.334,0.641,0.895,1.078,1.173]][[m,0,0,0,0],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]] [[1],[1],[1],[1],[1]]`

L1= 4.121m

Model participation factor = L/M

For the first mode, Model participation factor = `(L1)/(M1)`

(n=1)

Model participation factor-1 = 4.121/3.861m

= 1.067

similarly

Model participation factor-2 = -0.335

Model participation factor-3 = 0.177

Model participation factor-4 = -0.098

Model participation factor-5 = 0.045

Effective Model mass (M*) = `(Ln^2)/(Mn)`

For the first mode, M1 = `(L1 )/(M1)`

(n=1)

M1* = `(4.121m)^2/(3.861m)`

M1* = 4.398m

Similrly,

M1*= 0.436m

M1*= 0.121m

M1*= 0.038m

M1*= 0.008m

Model Mass participation Ratio = `(Mn)/(mj)`

Where , mj = Total mass of building = m+m+m+m+m= 5m

model mass participation ratio of first mode (n=1) = 4.398/5m= 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

Adding Firstt wo mode mass participation ratio we get 0.88+ 0.09 = 0.97 to achieve minimum 90% of model mass participation ration, First two modes are enough.

Base share can be calculated by multiplying effective model mass with pseudo acceleration. A

Base shear (VB) = `Mn**.An`

For the first mode , VB1 = 4.398m x 0.27g

(n=1)

VB1 = 1.187mg

similarly, VB2= 0.327 mg

VB3 = 0.126mg

VB4 = 0.039mg

VB5 = 0.008mg

Average Base shear is calculated by the method of Square roott sum of square

= `√ (Vb1) + (Vb2^2) + (Vb3^2) + Vb4^2 + Vb5^2`

VB = 1.15g