Understanding the concepts on Degrees of Freedom

Aim: To calculate the base shear of 5 storey building

Procedure:

It is mentioned in the given data ,that each storey has mass 'm' and 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 can be calculated by using the Equations given below.


Calculation of Mn,





For mode 1,

[0.334 0.691 0.895 1.078 1.173].`[m]*((0.334),(0.641),(0.895),(1.078),(1.173))`

By matrix multiplication,


Mn=[0.112m+0.412m+0.801m+1.162m+1.376m]


Mn=3.861m


similarly, for mode 2,Mn=3.866

for mode 3,Mn=0.683

for mode 4, Mn=3.861

for mode 5, mn=3.860




To calculate L_n ,






If=1,where degree of freedom is in same direction of siesmic force

For mode 1,

`[0.334,0.691,0.895,1.078,1.173]*((1),(1),(1),(1),(1))*[m]`

Ln=4.171m

similarly, for mode 2,Ln= -1.297m

for mode 3,Ln=0.683m

for mode 4, Ln=-0.381m

for mode 5, Ln=0.175m



Calculation of `Mn^**`



For mode 1,Ln=4.171m and Mn=3.861,substituiting in the formula ,we get,


`Mn^**`=`4.171^2/3.861`=4.51


For mode 2,Ln=-1.297m and Mn=3.860,substituiting in the formula ,we get,


`Mn^**`=0.436

For mode 3,Ln=0.683m and Mn=3.861,substituiting in the formula ,we get,


`Mn^**`=0.121

For mode 4 ,Ln=-0.381m and Mn=3.861,substituiting in the formula ,we get,


`Mn^**`=0.036

For mode 5,Ln=0.175m and Mn=3.860,substituiting in the formula ,we get,


`Mn^**`= 0.008




Calculation of Γ_n ,




For mode 1,Ln=4.171m and Mn=3.861,substituiting in the formula ,we get,


Γ_n=`4.171/3.861`=1.080


For mode 2,Ln=-1.297m and Mn=3.860,substituiting in the formula ,we get,


Γ_n=-0.3360

For mode 3,Ln=0.683m and Mn=3.861,substituiting in the formula ,we get,


Γ_n=0.171

For mode 4 ,Ln=-0.381m and Mn=3.861,substituiting in the formula ,we get,


Γ_n=-0.099



For mode 5,Ln=0.175m and Mn=3.860,substituiting in the formula ,we get,


Γ_n= 0.008


Calculation of base shear:

Calculation of pseudo spectrum acceleration:-

Property or Quantity	Mode 1	Mode 2	Mode 3	Mode 4	Mode 5
T	2	0.6852	0.4346	0.3383	0.2966
An	0.279	0.769	1.039	1.039	1.039

- referred from the graph of pseudo spectrum acceleration versus period.



Base shear calculation:






We can find out all Vbn values for all the modes by simply substituting the values of Lnand Mn in the above equation.

For mode 1,Vbn=`4.171^2/3.861*0.279-(3.861*0.279)`

Vb1=0.180

similarly,for mode 2,Vb2=-2.632


for mode 3,Vb3=-3.886


for mode 4,Vb4=-3.972

for mode 5, Vb5=-4.00

	Mode 1	Mode 2	Mode 3	Mode 4	Mode 5
Tn in seconds	2	0.6852	0.4346	0.3383	0.2966
Ln	4.171	-1.297	0.683	-0.381	0.175
Mn	3.861	3.860	3.861	3.861	3.860
Γn	1.080	-0.3360	0.177	-0.099	0.0453
An	0.279	0.769	1.039	1.039	1.039
Vbn	0.180	-2.632	-3.886	-3.972	-4.00

We should consider 3 modes such that, a modal mass participating ratio of 90% is obtained.