Understanding the concepts on Degrees of Freedom
1.AIM:
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 Tn and their corresponding mode shapes, φn has been derived.
Step:1
For mode 1:
`pai`=[0.334
0.641
0.895
1.078
1.173]
we kow that,
M1=(0.334 0.641 0.895 1.078 1.173)x
[0.334
0.641
0.895
1.078
1.173]
=(154459/40000)m
=3.861m
Similarly,
For
Mode 2,M2=3.885
Mode 3,M3=3.8613
Mode 4,M4=3.8611
Mode 5,M5=3.8612
Now,
We need to calculate Ln
Ln=`pai`n^TmL
therefore,
L1=(0.334 0.641 0.895 1.078 1.173)xm(1
1
1
1)
=4121/1000
L1=4.121
Similarly,
L2=-1.29
L3=0.68
L4=0.381
L5=0.175
We need to calculated the Mn
M*n=(Ln^2/Mn)
M1=4.121^2/3.8614
M*1=4.398m
For M*2=0.432m
M*3=0.119m
M*4=0.0357m
M*5=0.0079m
Now,we need to calculate rn=Ln/Mn
we know that,
r1=L1/m1
=4.121/3.8614
r1=1.067
Similarly,
r2=-0.336
r3=0.1761
r4=0.0986
r5=0.045
Now we need to claculated shear base,Vbn
therfore,
Vbn=Ln^2/mn An=Mn*An
we know that,
pseudo acceleration,An Value from graph
For Mode1,Force Vector
f1=(m1)(`pai`)(r1)(a1)
f1=M=(0.334
0.641
0.895
1.075
1.173)x1.067x0.27
Vb1=1.121
Similarly,
Vb2=0.31
Vb3=0.121
Vb4=0.035
Vb5=0.001