Understanding the concepts on Degrees of Freedom
ANSWER :
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
Modified mass matrix (M) = φn T * m * φn
for the first mode, = [0.334 0.641 0.895 1.078 1.173 ] * m 0 0 0 0 * 0.334
(n=1) 0 m 0 0 0 0.641
0 0 m 0 0 0.895
0 0 0 m 0 1.078
0 0 0 0 m 1.173
M1 = 3.861 m
Modal property (L) = φn T . m ,l
l = influence vector ( Assuming 100% utilization of seismic force) = 1
1
1
1
1
for the first mode, = [0.334 0.641 0.895 1.078 1.173 ] * m 0 0 0 0 * 1
(n=1) 0 m 0 0 0 1
0 0 m 0 0 1
0 0 0 m 0 1
0 0 0 0 m 1
L1 = 4.121 m
Modal participation (Γ ) = L/M
for the first mode, Γ = L1/M1
Γ1 = 4.121/ 3.861
= 1.0673
Effective modal mass (M*) = L^2 / Mn
for the first mode, M1* = L1^2 / M1
= 4.121 m ^2 / 3.861
= 4.398508 m
Modal mass participation ratio= Mn*/Mj
= 4.398508 m / 5m
= 0.8797
Base shear (Vb) = Mn* An
for the first mode,Vb1 = 4.398 m * 0.27
Vb1 = 1.187 mg
similarly ,,
for the second mode = [-0.895 -1.173 -0.641 0.334 1.078 ]* m 0 0 0 0 * -0.895
(n=2) 0 m 0 0 0 -1.173
0 0 m 0 0 -0.641
0 0 0 m 0 0.334
0 0 0 0 m 1.078
M2 = 3.861 m
L2 = -1.297
Γ3 = 3359
M2* = 0.43233 m
Modal mass participation ratio= M3*/ 5m
= 0.43233 m / 5m
= 0.086
for the second mode,Vb2 = 0.086 m * 0.27
Vb2 = 0.02322 mg
for the third mode, = [1.173 0.334 -1.078 -0.641 0.895 ] * m 0 0 0 0 * 1.173
(n=3) 0 m 0 0 0 0.334
0 0 m 0 0 -1.078
0 0 0 m 0 -0.641
0 0 0 0 m 0.895
M3 = 3.861m
L2 = 0.683
Γ4 = 0.176
M3* = 0.120 m
Modal mass participation ratio= M4*/ 5m
= 0.120 m / 5m
= 0.024
for the third mode,Vb3 = 0.024 m * 0.27
Vb3 = 0.00648 mg
for the Fourth mode, = [-1.078 0.895 0.334 -1.173 0.641 ] * m 0 0 0 0 * -1.078
(n=4) 0 m 0 0 0 0.895
0 0 m 0 0 0.334
0 0 0 m 0 -1.173
0 0 0 0 m 0.641
M4 = 3.861m
L4 = - 0.381
Γ5 = -0.0986
M 4* = 0.0378
Modal mass participation ratio= M4*/ 5m
= 0.0378 m / 5m
= 0.0075
for the fourth mode,Vb4 = 0.0075 m * 0.27
Vb4 = 0.002025 mg
for the Fifth mode, = [ 0.641 -1.078 1.173 -0.895 0.334 ] * m 0 0 0 0 * 0.641
(n=4) 0 m 0 0 0 -1.078
0 0 m 0 0 1.173
0 0 0 m 0 -0.895
0 0 0 0 m 0.334
M5 = 3.861
L5 = 0.175
Γ6 = 0.045
M5* = 0.00793
Modal mass participation ratio= M5*/ 5m
= 0.07930 m / 5m
= 0.0150
for the fifth mode,Vb4 = 0.0150 m * 0.27
Vb5 = 0.00405 mg
Adding first 2 modal mass participation ratio we get 0.97 so get above 90%
VB = 1.01 mg