Understanding the concepts on Degrees of Freedom

Aim:To calucate the base shear of the 5 storey building of five transational dof using modal response spectrum analysis method.

step by step procedure

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:

1.Natural circular frequency of vibration(Wn).

W1=2π/T1

=2π/2

=3.14 rad/sec

w2=2π/T2

=2π/0.6852

=9.17rad/sec

w3=2π/T3

= 2π/0.4346

=14.45rad/sec

w4=2π/T4

= 2π/0.3383

=18.572rad/sec

w5=2π/T5

=2π/0.2966

=21.18rad/sec

modified mass matrix

M1=[0.334 0.641 0.895 1.078 1.173 ]
=3.861m

M2=[ -0.895 -01.173 -0.641 0.334 1.078 ]

=3.861m
M3=[ 1.173 0.334 -1.078 -0.641 0.895 ]

=3.861 m

M4=[ -1.078 0.895 0.334 -1.173 0.641 ]

=3.861 m

M5=[ 0.641 -1.078 1.173 -0.895 0.334 ]

=3.861 m

modal property

L1=[0.334 0.641 0.895 1.078 1.173 ]

=4.121m

L2=[ -0.895 -01.173 -0.641 0.334 1.078 ]

=-1.297 m

L3=[ 1.173 0.334 -1.078 -0.641 0.895 ]

=0.683 m

L4=[ -1.078 0.895 0.334 -1.173 0.641 ]

=-0.381 m

L5=[ 0.641 -1.078 1.173 -0.895 0.334 ]

=0.175 m

modal pariticipation factor:

Γ1=L1/M1

=4.121/3.861

=1.067

Γ2=-1.297/3.861

=-0.335

Γ3=O.683/3.861

=0.177

Γ4=-0.381/3.861

=-0.0986

Γ5=0.175/3.861

=0.045

effective mass model:

M*1=(4.121 x 4.121)/3.861

=4.398 m

M*2=(-1.297 x -1.297)/3.861

=0.435 m

M*3=0.121 m

M*4 =0.037 m

M*5=0.008 m

modal mass participation ratio = M*n/Mj

total mass of the building=Mj=m+m+m+m+m

=5m

mode1=0.88

mode2=0.09

mode3=0.024

mode4=0.007

mode5=0.001

To achieve minimum 90% of modal participating ratio first two modes is enough .

Base shear

Vb1=Mn*An

for knowing the An we need to use is 1893 code book.

A1=0.27 g

A2=0.76 g

A3=1.02 g

A4=1.02 g

A5=1.02 g

Vb1=4.398m x 0.27g

=1.187 mg

vb2=0.435m x 0.76 g

=0.33 mg

vb3=0.123 mg

Vb4=0.0037 mg

Vb5=0.008 x 1.02

=0.008 mg

we need to combined base shear for all five modes can be obtained from srss method(square root of sum of all squares)

F=

Vb=1.23 mg

base shear for the building is 1.23 mg.

we can calcuate the seismic forces at the nodes make them combined to get the total seismic force at the base of the structure.

Result

I calucated the base shear of 5 storey building of five transational dof using modal response spectrum analysis method.