Understanding the concepts on Degrees of Freedom

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.

The fundamental period of vibration (T₁ = 2 seconds) and its associated mode shape:

The second mode of vibration (T₂ = 0.6852 seconds) and its associated mode shape:

The third mode of vibration (T₃ = 0.4346 seconds) and its associated mode shape:

The fourth mode of vibration (T₄ = 0.3383 seconds) and its associated mode shape:

The fifth mode of vibration (T₄ = 0.2966 seconds) and its associated mode shape:

• L_n , M_n and Γ_n for each of the five modes are to be computed from Equations below.
• Also, calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes. The mass matrix will be required to evaluate these parameters.

• Finally, obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule. The pseudo acceleration design spectrum to obtain the base shear for each mode is provided below.

• How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?

ANS:

1. 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]]`

modifed mass matrix (m)=`phi underset(n)()overset(T)() . m . phi underset(n)`

`or the 1st 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),(0.895),(1.078),(1.173))`

`M underset(1)()`=3.861m

2.model property (L)=`phi underset(n)() overset(T)() . m.i`

where `i`= influence vector (assumeig 100% uilization of seismic forces) =`((1),(1),(1),(1),(1))`

for the 1st model= (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 underset(1)`=4.121m

3.model participation factor`tau=(l/m)`

`tau=4.121/3.861m`

`tauunderset(1)`=1.067

`tauunderset(2)`=-0.335

`tauunderset(3)`=0.177

`tauunderset(4)`=0.098

`tauunderset(5)`=0.045

4.Effective model mass `M^*=``(Lunderset(n)overset(2)() ) / Munderset(n)()`

for 1st mode `M^*underset(1)`=`(4.121overset(2)() ) /3.861m`=4.398m

`M^*underset(1)`=4.398

`M^*underset(2)`=0.436

`M^*underset(3)`=0.121

`M^*underset(4)`=0.038

`M^*underset(5)`=0.008

5.model mass participation ratio =`M^*underset(n)()`/`m underset(j)()`

where `m underset(j)`=total mass of the building =`m+m+m+m+m`

model mass participation ratio of the 1st model (n=1)=`(4.398 cancel(m) )/(5 cancel(m))`=0.88

2nd mode =(n=2)= 0.09

3rd mode (n=3)=0.02

4th mode (n=4)=0.01

sth mode (n=5)=0.002

adding 1st two models mass participation ratios we get 0.88+0.09=0.974

To achive minimum 90% of model mass participation ratio 1st 2 models are enough.

6.Base shear `Vunderset(B)`= `Munderset(n)`x `Aunderset (n)`

for the 1st mode `Vunderset (B1)`=`4.398 times 0.27mg`=1.187m`

`Vunderset(B2)`=0.327g

`Vunderset(B3)`=0.126g

`Vunderset(B4)`=0.039g

`Vunderset(B5)`=0.008g

Average BAse shear is calculated by the method of Square root of sum of square(SRSS)

`Vunderset(B)() =sqrt( Vunderset(B1)()overset(2)()+Vunderset(B2)()overset(2)()+Vunderset(B3)()overset(2)()+Vunderset(B4)()overset(2)()+Vunderset(B5)()overset(2)()`

=1.15mg.