Understanding the concepts on Degrees of Freedom

Objectives

To understand modes shapes of multi storey building

Manually calculate the base shear for a multi storey building in modal analysis

Understand the terminolgies involved in modal analysis

Mass matrix (m)=

Modified mass matrix (M)= ϕn^T * m*ϕn

For the first mode =

M1=3.861 m

Modal property(L)=ϕn^T *m*l

Where l = Influence vector (Assuming 100% utilization of seismic force )=

For the first mode =

L1=4.121 m

Modal participation factor = L/M

For the first mode = L1/M1

γ1 = 4.121/ 3.861

γ1 = 1.067

Similarly γ2=-0.335

γ3=0.177

γ4=-0.098

γ5=0.045

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

For the first mode M1*=L1^2/M1

(n=1)

M1*= 4.121^2/3.861

= 4.398

Similarly M2*=0.436 m

M3*=0.121 m

M4*=0.038 m

M5*= 0.008 m

Modal mass participation ratio = Mn*/m1

Where m1= Total mass of the building = m+m+m+m+m =5 m

Modal mass participation ratio of first mode (n=1) =4.398/5 =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)=0.02

Adding first two mode mass participation ratios we get 0.08+ 0.09 =0.97

To achieve minimum 90% of modal mass participation ratio first two modes are enough

Base shear (VB) =Mn* *An

For the first mode VB1 =4.398*0.27

(n=1)

VB1=1.187 mg

Similarly VB2 = 0.327 mg

VB3 = 0.126 mg

VB4 = 0.039

VB5=0.008 mg

Average base shear is calculated by the method of square root of sum of square

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

VB = 1.15 mg