Understanding the concepts on Degrees of Freedom
Understanding the concepts on Degrees of Freedom
TASK:
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.
The fundamental period of frequency, `T_1=2 ` seconds and its associated mode shape: `Phi_1=[[0.334],[0.641],[0.895],[1.078],[1.173]]`
The second mode of vibration, `T_2=0.6852 ` seconds and its associated mode shape: `Phi_2=[[-0.895],[-1.173],[-0.641],[0.334],[1.078]]`
The third mode of vibration, `T_3=0.4346 ` seconds and its associated mode shape: `Phi_3=[[1.173],[0.334],[-1.078],[-0.641],[0.895]]`
The forth mode of vibration, `T_4=0.3383 ` seconds and its associated mode shape: `Phi_4= [[-1.078],[0.895],[0.334],[-1.173],[0.641]]`
The fifth mode of vibration, `T_5=0.2966 ` seconds and its associated mode shape: `Phi_5=[[0.641],[-1.078],[1.173],[-0.895],[0.334]]`
- Ln, Mn, and Γn for each of the five modes are to be computed from the 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 the 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 calculate `L_n, M_n` and `Gamma_n` for each of the five modes
- To calculate the effective modal mass `M_n`* and the modal mass participating ratio for the 5 modes.
- To find the base shear for each of the 5 modes of vibration and the combined base shear as per the SRSS rule.
Calculations:
Calculating the entire parameters for mode 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]]`
Now, we have to obtain the transformed mass matrix 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`
Now, we need to find the modal property, `L_n`.
`L_n=Phi_n^Tm iota`
Where, `iota=` Influence matrix which is a unit matrix, `[[1],[1],[1],[1],[1]]`
`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`
The modal participation factor, `Gamma_n=L_n/M_n`
`Gamma_1=(4.12m)/(3.86m)=1.067`
Seismic force for mode 1:`f_n=mPhi_nGamma_nA_n`
`f_1=mPhi_1Gamma_1A_1`
Where `A_1=0.27g` for a time period of 2 seconds.
`f_1=[[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]] *1.067*0.27g`
`f_1=[[0.334m],[0.641m],[0.895m],[1.078m],[1.173m]] *1.067*0.27g`
`f_1=[[0.096mg],[0.185mg],[0.258mg],[0.310mg],[0.338mg]]`
We know that Weight(newton), W=mg, and this can be substituted in the above equation.
`f_1=[[0.096W],[0.185W],[0.258W],[0.310W],[0.338W]]`
Now, to find out the base shear for mode1, all the values of the seismic forces can be added.
I.e., `V_(b1)= 0.096W+0.185W+0.258W+0.310W+0.338W=1.187W`
Similarly, we need to find out the base shear for all the modes.
`V_(b1)` can also be calculated by finding out the effective modal mass, `M_n`*
`M_1`*=`(L_n)^2/M_n=(4.12m)^2/(3.86m)=4.397m`
Hence, modal mass participating ratio, `(M_n**)/m` for mode 1 = `(4.397m)/(5m)=0.88=88%`
`V_(b1)=M_n`*`*A_n=4.397mxx0.27g=1.187mg=1.187W`
For mode 2:
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 have to obtain the transformed mass matrix 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= [[-0.895m,-1.173m,-0.641m,0.334m,1.078m]] * [[-0.895],[-1.173],[-0.641],[0.334],[1.078]]`
`M_2= 3.86m`
Now, we need to find the modal property, `L_n`.
`L_n=Phi_n^Tm iota`
Where, `iota=` Influence matrix which is a unit matrix, `[[1],[1],[1],[1],[1]]`
`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= [[-0.895m,-1.173m,-0.641m,0.334m,1.078m]] * [[1],[1],[1],[1],[1]]`
`L_2=-1.297m`
The modal participation factor, `Gamma_n=L_n/M_n`
`Gamma_2=(-1.297m)/(3.86m)=-0.336`
Seismic force for mode 2: `f_n=mPhi_nGamma_nA_n`
`f_2=mPhi_2Gamma_2A_2`
`A_2=0.76g` for a time period of 0.6852 seconds.
`f_2=[[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]] *-0.336*0.76g`
`f_2=[[-0.895m],[-1.173m],[-0.641m],[0.334m],[1.078m]]*-0.336*0.76g`
`f_2=[[0.228mg],[0.299mg],[0.164mg],[-0.085mg],[-0.275mg]]`
We know that Weight (newton), W=mg, and this can be substituted in the above equation.
`f_2=[[0.228W],[0.299W],[0.164W],[-0.085W],[-0.275W]]`
Now, to find out the base shear for mode 2, all the values of the seismic forces can be added.
I.e., `V_(b2)= 0.228W+0.299W+0.164W-0.085W-0.275W=0.331W`
`V_(b2)` can also be calculated by finding out the effective modal mass, `M_n`*
`M_2`*=`(L_n)^2/M_n=(-1.297m)^2/(3.86m)=0.436m`
Hence, modal mass participating ratio, `(M_n**)/m` for mode 2 = `(0.436m)/(5m)=0.087=8.7%`
`V_(b2)=M_n`*`*A_n=0.436m xx 0.76g=0.331mg=0.331W`
For mode 3:
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 have to obtain the transformed mass matrix 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,-1.078m,-0.641m,0.895m]] * [[1.173],[0.334],[-1.078],[-0.641],[0.895]]`
`M_3= 3.86m`
Now, we need to find the modal property, `L_n`.
`L_n=Phi_n^Tm iota`
Where, `iota=` Influence matrix which is a unit matrix, `[[1],[1],[1],[1],[1]]`
`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,-1.078m,-0.641m,0.895m]] * [[1],[1],[1],[1],[1]]`
`L_3=0.683m`
The modal participation factor, `Gamma_n=L_n/M_n`
`Gamma_3=(0.683m)/(3.86m)=0.177`
Seismic force for mode 3: `f_n=mPhi_nGamma_nA_n`
`f_3=mPhi_3Gamma_3A_3`
`A_3=1.03g` for a time period of 0.4346 seconds.
`f_3=[[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]] *0.177*1.03g`
`f_3=[[1.173m],[0.334m],[-1.078m],[-0.641m],[0.895m]]*0.177*1.03g`
`f_3=[[0.214mg],[0.061mg],[-0.196mg],[-0.177mg],[0.163mg]]`
We know that Weight (newton), W=mg, and this can be substituted in the above equation.
`f_3=[[0.214W],[0.061W],[-0.196W],[-0.177W],[0.163W]]`
Now, to find out the base shear for mode 3, all the values of the seismic forces can be added.
I.e., `V_(b3)= 0.214W+0.061W-0.196W-0.177W+0.163W=0.125W`
`V_(b3)` can also be calculated by finding out the effective modal mass, `M_n`*
`M_3`*=`(L_n)^2/M_n=(0.683m)^2/(3.86m)=0.121m`
Hence, modal mass participating ratio, `(M_n**)/m` for mode 3 = `(0.121m)/(5m)=0.024=2.4%`
`V_(b3)=M_n`*`*A_n=0.121m xx 1.03g=0.125mg=0.125W`
For mode 4:
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 have to obtain the transformed mass matrix 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= [[-1.078m,0.895m,0.334m,-1.173m,0.641m]] * [[-1.078],[0.895],[0.334],[-1.173],[0.641]]`
`M_4= 3.86m`
Now, we need to find the modal property, `L_n`.
`L_n=Phi_n^Tm iota`
Where, `iota=` Influence matrix which is a unit matrix, `[[1],[1],[1],[1],[1]]`
`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= [[-1.078m,0.895m,0.334m,-1.173m,0.641m]] * [[1],[1],[1],[1],[1]]`
`L_4=-0.381m`
The modal participation factor, `Gamma_n=L_n/M_n`
`Gamma_4=(-0.381m)/(3.86m)=-0.099`
Seismic force for mode 4: `f_n=mPhi_nGamma_nA_n`
`f_4=mPhi_4Gamma_4A_4`
`A_4=1.03g` for a time period of 0.3383 seconds.
`f_4=[[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]] *-0.099*1.03g`
`f_4=[[-1.078m],[0.895m],[0.334m],[-1.173m],[0.641m]]*-0.099*1.03g`
`f_4=[[0.109mg],[-0.091mg],[-0.034mg],[0.119mg],[-0.065mg]]`
We know that Weight (newton), W=mg, and this can be substituted in the above equation.
`f_4=[[0.109W],[-0.091W],[-0.034W],[0.119W],[-0.065W]]`
Now, to find out the base shear for mode 4, all the values of the seismic forces can be added.
I.e., `V_(b4)= 0.109W-0.091W-0.034W+0.119W-0.065W=0.038W`
`V_(b4)` can also be calculated by finding out the effective modal mass, `M_n`*
`M_4`*=`(L_n)^2/M_n=(-0.381m)^2/(3.86m)=0.037m`
Hence, modal mass participating ratio, `(M_n**)/m` for mode 4 = `(0.037m)/(5m)=0.0074=0.74%`
`V_(b4)=M_n`*`*A_n=0.037m xx 1.03g=0.038mg=0.038W`
For mode 5:
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 have to obtain the transformed mass matrix for mode 5: `M_5=Phi_5^T m Phi_5`
I.e., `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,-1.078m,1.173m,-0.895m,0.334m]] * [[0.641],[-1.078],[1.173],[-0.895],[0.334]]`
`M_5= 3.86m`
Now, we need to find the modal property, `L_n`.
`L_n=Phi_n^Tm iota`
Where, `iota=` Influence matrix which is a unit matrix, `[[1],[1],[1],[1],[1]]`
`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,-1.078m,1.173m,-0.895m,0.334m]] * [[1],[1],[1],[1],[1]]`
`L_5=0.175m`
The modal participation factor, `Gamma_n=L_n/M_n`
`Gamma_5=(0.175m)/(3.86m)=0.045`
Seismic force for mode 5: `f_n=mPhi_nGamma_nA_n`
`f_5=mPhi_5Gamma_5A_5`
`A_5=1.03g` for a time period of 0.2966 seconds.
`f_5=[[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]] *0.045*1.03g`
`f_5=[[0.641m],[-1.078m],[1.173m],[-0.895m],[0.334m]]*0.045*1.03g`
`f_5=[[0.029mg],[-0.049mg],[0.054mg],[-0.041mg],[0.015mg]]`
We know that Weight (newton), W=mg, and this can be substituted in the above equation.
`f_5=[[0.029W],[-0.049W],[0.054W],[-0.041W],[0.015W]]`
Now, to find out the base shear for mode 5, all the values of the seismic forces can be added.
I.e., `V_(b5)= 0.029W-0.049W+0.054W-0.041W+0.015W=0.008W`
`V_(b5)` can also be calculated by finding out the effective modal mass, `M_n`*
`M_5`*=`(L_n)^2/M_n=(0.175m)^2/(3.86m)=0.0079m`
Hence, modal mass participating ratio, `(M_n**)/m` for mode 5 = `(0.0079m)/(5m)=0.0016=0.16%`
`V_(b5)=M_n`*`*A_n=0.0079m xx 1.03g=0.008mg=0.008W`
Hence, Base shear has been obtained for 5 modes of vibration.
Now, Combined base shear by SRSS Rule.
`V_b=sqrt((V_(b1))^2+(V_(b2))^2+(V_(b3))^2+(V_(b4))^2+(V_(b5))^2)=sqrt((1.187W)^2+(0.331W)^2+(0.125W)^2+(0.038W)^2+(0.008W)^2)=1.239W`
`V_b=1.239W`
Total mass of building= 5m
Total effective modal mass= 4.397+0.436+0.121+0.037+0.0079= 4.9989m `~~`5m
Hence Total mass of the building is verified.
Now, The Modal mass participation ratios obtained is:
Mode 1- 88%
Mode 2- 8.7%
Mode 3- 2.4%
Mode 4- 0.74%
Mode 5- 0.16%
From the above values, it is clear that the sum of mode 1 & mode 2 will exceed 90% .
I.e., 88% + 8.7% = 96.7%