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 T_n and their corresponding mode shapes, φ_n 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)`