Understanding the concepts on Degrees of Freedom
AIM -
Question 1:
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.
ANS - 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) = `phi_nt.m.phi_n`
for the first mode = `[0.334 ,0.641, 0.895, 1.078, 1.173]``[ [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]]```[[0.334],[0.641],[895],[1.073],[1.178]]`
(n=1)
`M_1=3.861m`
MODEL PROPERTY (L) = `phi_nT..m.t`
Where(t) = infiluence 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],[0,m,0,0,0],[0,0,m,0,0],[0,0,0,m,0],[0,0,0,0,m]]```[[1],[1],[1],[1],[1]]``
`L_1 = 4.121`M
MODEL PARTICIPATION FACTOR (`(Gamma)=L/M`
FOR THE FIRST MODE `Gamma=L_1/M_1`
(n=1)
`Gamma =4.121/3.861`
simmilarly, `Gamma_2=-0.335`
`Gamma_3=0.177`
`Gamma_4=-0.098`
`Gamma_5=0.045`
Effective Model Mass (M*) =`L_n^ 2 /M_n`
For the first mode `M_1` = `L_ 1^ 2/M_n`
(`n=1`)
`M_1`*=`(4.121m)^2/(3.861)`
M = 4.398m
`M_2=0.436 m`
`M_3=0.121m`
`M_4=0.038m`
`M_5=0.008M`
MODEL MASS PARTICIPATION RATIO=`M_n/M_j`
where `m_j =`total mass of the building =m+m+m+m+m =5m
Modal mass participation ratio of first mode (n=1)=4.398m/5m=.88
simillarly, Second Mode (n=2)=0.09
Third Mode (n=3) = 0.02
Fourth Mode (n=4) = 0.01
Fifth Mode (n=5) = 0.002
Adding First two Mode Mass Participation Ratios, We get 0.88+0.09=0.97
TO Achive Minimum 90% of model Mass Participation Ratio First Two Modes Are Enough.
BASE SHEAR (`V_B)= M_n.A_n`
For The first mode `V_B1=4.398m`*0.27g
(n=1)
`V_B1=`1.187 mg
simmilarly, `V_B2`=0.327mg
`V_B3`=0.126mg
`VB_4`=0.039mg
`V_B5`=0.008mg
Average Base Shear is Calculated By the method of square root of Sum of Square
`sqrt((V_b1^2)+(V_b2^2)+(V_b3^2)+(V_b4^2)+(V_b5^2))`
`V_b=1.15 mg`
`Gamma =4.121/3.861`
simmilarly, `Gamma_2=-0.335`
`Gamma_3=0.177`
`Gamma_4=-0.098`
`Gamma_5=0.045`
Effective Model Mass (M*) =`L_n^ 2 /M_n`
For the first mode `M_1` = `L_ 1^ 2/M_n`
(`n=1`)
`M_1`*=`(4.121m)^2/(3.861)`