Understanding the concepts on Degrees of Freedom

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?

AIM:

To compute L_n , M_n and Γ_n for each of the five modes from the given equations.

To calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes.

To obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule.

SOLUTION:

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_(n)^T.m.phi_n`

`[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_1`=3.861

Model property(L)=``phi_(n)^T.m.L

Where L =influence vector (Assuming 100% utilization of seismic force)=`[[1],[1],[1],[1],[1]]`

For the first mode (n=1)=`[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=`4.121m

Modal participation factor(`(Gamma)=L/M`

For the first mode ,`Gamma=L_1/M_1`

`nbsp; `Gamma_1=4.121/3.381m`

`Gamma_1=1.067`

Similarly,`Gamma_2=0,335`

`nbsp; `Gamma_3=0.177`

`Gamma_4=-0.098`

`Gamma_5=0.045`

Effective modal mass(`(M^(*))=L_n^2/M_n`

`nbsp; For the first mode,`M_1^*=L_1^2/M_1`

(n=1)

`M_1^*=4.398`

Similarly,`M_2^*=0.436m`

`M_3^*=0.121m`

`M_4^*=0.038m`

`M_5^*=0.08m`

Modal mass participation ratio=`M_n^*/m_j`

Where ,`M_j=`Total mass of the building

=m+m+m+m+m

=5m

Modal masss participation of first mode(n=1)=4.398m/5m=0.88

Similarly,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 achieve minimum 90% of modal 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=1.187mg`

Similarly , `V_B2=0.327mg`

`V_B3=0.126mg`

`V_B4=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=1.15mg`

RESULT:

Adding first two mode mass participation ratios

We get 0.88+0.09=0.97

Base shear

`V_B=1.15mg`