Understanding the concepts on Degrees of Freedom
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 Tn and their corresponding mode shapes, φn has been derived.
Given:
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:
- Ln , Mn and Γn for each of the five modes are to be computed from Equations below.
- Also, calculate the effective modal mass Mn* 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?
ANSWER:
First we calculating the mass matrix(m).
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]]`
Now we calculate the transformed mass matrix for all modes-`M=Phi^T*m*Phi`
1)Modified mass matrixs:(M)
For mode-1
M_1=Phi_1^T*m*Phi_1
i.e, `M_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]]*[[0.334],[0.641],[0.895],[1.078],[1.173]]`
`M_1=[[0.334m,0.641m,0.895m,1.078m,1.173m]]*[[0.334],[0.641],[0.895],[1.078],[1.173]]`
`M_1=3.86m`
For mode-2
`M_2=Phi_2^T*m*Phi_2`
i.e, `M_2=[[-0.895,-1.173,-0.641,0.334,1.078]]*[[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]]*[[-0.895],[-1.173],[-0.641],[0.334],[1.078]]`
`M_2=[[overset(-0.895)(m),overset(-1.173)(m),overset(-0.641)(m),0.334m,1.078m]]*[[-0.895],[-1.173],[-0.641],[0.334],[1.078]]`
`M_2=3.86m`
For mode-3
`M_3=Phi_3^T*m*Phi_3`
i.e,`M_3=[[1.173,0.334,-1.078,-0.641,0.895]]*[[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]]*[[1.173],[0.334],[-1.078],[-0.641],[0.895]]`
`M_3=[[1.173m,0.334m,overset(-1.078)(m),overset(-0.641)(m),0.895m]]*[[1.173],[0.334],[-1.078],[-0.641],[0.895]]`
`M_3=3.86m`
For mode-4
`M_4=Phi_4^T*m*Phi_4`
i.e,`M_4=[[-1.078,0.895,0.334,-1.173,0.641]]*[[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]]*[[-1.078],[0.895],[0.334],[-1.173],[0.641]]`
`M_4=[[overset(-1.078)(m),0.895m,0.334m,overset(-1.173)(m),0.641m]]*[[-1.078],[0.895],[0.334],[-1.173],[0.641]]`
`M_4=3.86m`
For mode-5
`M_5=Phi_5^T*m*Phi_5`
`M_5=[[0.641,-1.078,1.173,-0.895,0.334]]*[[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]]*[[0.641],[-1.078],[1.173],[-0.895],[0.334]]`
`M_5=[[0.641m,overset(-1.078)(m),1.173m,overset(-0.895)(m),0.334m]]*[[0.641],[-1.078],[1.173],[-0.895],[0.334]]`
`M_5=3.86m`
Now we have to determine modal proprety.(L)
2)Modal property(`L_n`):`phi_n^T*m*l`
Where,`l=` infulence matrix is aunit matrix,`[[1],[1],[1],[1],[1]]`
For mode-1:
(`L_1`):`phi_1^T*m*l`
`L_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=[[0.334m,0.641m,0.895m,1.078m,1.173m]]*[[1],[1],[1],[1],[1]]`
`L_1=4.12m`
For mode-2:
(`L_2`):`phi_2^T*m*l`
`L_2=[[-0.895,-1.173,-0.641,0.334,1.078]]*[[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_2=[[overset(-0.895)(m),overset(-1.173)(m),overset(-0.641)(m),0.334m,1.078m]]*[[1],[1],[1],[1],[1]]`
`L_2=-1.297m`
For mode-3:
(`L_3`):`phi_3^T*m*l`
`L_3=[[1.173,0.334,-1.078,-0.641,0.895]]*[[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_3=[[1.173m,0.334m,overset(-1.078)(m),overset(-0.641)(m),0.895m]]*[[1],[1],[1],[1],[1]]`
`L_3=0.683m`
For mode-4
(`L_4`):`phi_4^T*m*l`
`L_4=[[-1.078,0.895,0.334,-1.173,0.641]]*[[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_4=[[overset(-1.078)(m),0.895m,0.334m,overset(-1.173)(m),0.641m]]*[[1],[1],[1],[1],[1]]`
`L_4=-0.381m`
For mode-5
(`L_5`):`phi_5^T*m*l`
`L_5=[[0.641,-1.078,1.173,-0.895,0.334]]*[[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_5=[[0.641m,overset(-1.078)(m),1.173m,overset(-0.895)(m),0.334m]]*[[1],[1],[1],[1],[1]]`
`L_5=0.175m`
Now, we have to calculate the 3)Modal participation factor;`Gamma_n`
It is defined as the ratio of modal property (`L_n`) to modified mass matrix(M).
`Gamma_n=L_n/M_n`
For mode-1:
`Gamma_1=L_1/M_1`
`Gamma_1=(4.121m)/(3.861m)=1.067`
`Gamma_1=1.067`
For mode-2:
`Gamma_2=L_2/M_2`
`Gamma_2=-(1.297m)/(3.861m)=-0.335`
`Gamma_2=-0.335`
For mode-3:
`Gamma_3=L_3/M_3`
`Gamma_3=(0.683m)/(3.861m)=0.177`
`Gamma_3=0.177`
For mode-4
`Gamma_4=L_4/M_4`
`Gamma_4=(-0.381m)/(3.861m)=-0.099`
`Gamma_4=-0.099`
For mode-5
`Gamma_5=L_5/M_5`
`Gamma_5=(0.175m)/(3.861m)=0.045`
`Gamma_5=-0.045`
After finding the modal participation factor now we have to calculate the effective modal mass(`M^**=L_n^2/M_n`)
4)effective modal mass(`M^**=L_n^2/M_n`):
For mode-1:
`M_1^**=L_1^2/M_1`
`M_1^**=(4.121m)^2/(3.861m)=4.398m`
`M_1^**=4.398m`
For mode-2:
`M_2^**=L_2^2/M_2`
`M_2^**=(-1.297m)^2/(3.861m)=-0.335m`
`M_2^**=0.436m`
For mode-3:
`M_3^**=L_3^2/M_3`
`M_3^**=(0.683m)^2/(3.861m)=0.121m`
`M_3^**=0.121m`
For mode-4
`M_4^**=L_4^2/M_4`
`M_4^**=(-0.381m)^2/(3.861m)=0.038m`
`M_4^**=0.038m`
For mode-5
`M_5^**=L_5^2/M_5`
`M_5^**=(0.175m)^2/(3.861m)=0.008m`
`M_5^**=0.008m`
Now we have to calculate the modal mass participation ratio.
5)Modal mass particiopation ratio=`M_n^**/m_j`
Where `m_j`is total mass of the buuilding=m+m+m+m+m= 5m.
For mode-1:
`M_1^**/m_j= (4.938m)/(5m)=0.88`
For mode-2:
`M_2^**/m_j= (0.436m)/(5m)=0.09`
For mode-3:
`M_3^**/m_j= (0.121m)/(5m)=0.02`
For mode-4:
`M_4^**/m_j= (0.038m)/(5m)=0.01`
For mode-5:
`M_5^**/m_j= (0.008m)/(5m)=0.002`
As per the question, there are sum of first two modes are enough to considered such as to obtain the MODAL MASS PARTICIPATING RATIO as 90%. i.e, 0.88+0.09=0.97.
Now we have to caluclate the base shear for each mode(`V_B`)
Base shear(`V_B=M_n^**. A_n`)
For mode-1:
`V_(B1)=4.398m *0.27g=1.187mg`
For mode-2:
`V_(B2)=0.436m *0.76g=0.327mg`
For mode-3:
`V_(B3)=0.121m *1.03g=0.126mg`
For mode-4:
`V_(B4)=0.038m *1.03g=0.039mg`
For mode-5:
`V_(B5)=0.008m *1.03g=0.008mg`
Now we have to caluculate the average base shear by the method of Square root of sum of square:
i.e,`V_B=sqrt(((V_(B1))^2+(V_(B2))^2+(V_(B3))^2+(V_(B4))^2+(V_(B5))^2)`
`V_B=sqrt(((1.187)^2+(0.327)^2+(0.126)^2+(0.039)^2+(0.008)^2)`
`V_B=1.15mg`
Finally the percentages of Modal mass participation ratio are obtained as:
mode 1- 88%
mode 2-8.7%
mode 3-2.4%
mode 4-0.74%
mode 5-0.16%
Total effective modal mass= 4.397m+0.436m+0.121m+0.037m+0.008m=4.9989m`~~`5m
Hence, total mass of the building is verified.
the average base shear value is 1.15mg is calculated by SRSS Rule.