Understanding the concepts on Degrees of Freedom

1.Ans

AIM

• A five storey structure with 5 translational DOF is given 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.
• 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.

PROCEDURE

Step-1

For model-1

• We know that, Modified mass matrix, `Mn=Phin^T*m*Phin`
• Where, `n=1`
• And 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]]`
• For the first mode, `M1=[[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]]`
• `M1=3.861m`

For model-2

• We know that, Modified mass matrix, `Mn=Phin^T*m*Phin`
• Where, `n=2`
• And 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]]`
• For the second mode, `M2=[[-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]]`
• `M2=3.861m`

For model-3

• We know that, Modified mass matrix, `Mn=Phin^T*m*Phin`
• Where,`n=3`
• And 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]]`
• For the third mode,`M3=[[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]]`

• `M3=3.861m`

For model-4

• We know that, Modified mass matrix, `Mn=Phin^T*m*Phin`
• Where, `n=4`
• And 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]]`
• For the fourth mode,`M4=[[-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]]`
• `M4=3.861m`

For model-5

• We know that, Modified mass matrix, `Mn=Phin^T*m*Phin`
• Where, `n=5`
• And 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]]`
• For the fifth mode,`M5=[[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]]`
• `M5=3.861m`

Step-2

For model-1

• We know that, Model property, `L=Phin^T*m*iota`

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

• Where, n=1

• And 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]]`

• For the first mode, `L1=[[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]]`

• `L1=4.121m`

For model-2

• We know that, Model property, `L=Phin^T*m*iota`

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

• And 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]]`

• For the second mode

  `L2=[[-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]]`
• `L2=-1.297m`

For model-3

• We know that, Model property, `L=Phin^T*m*iota`

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

• Where n=3
• And 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]]`
• For third mode, `L3=[[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]]`
• `L3=0.683m`

For model-4

• We know that, Model property, `L=Phin^T*m*iota`

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

• Where n=4
• For Fourth mode, `L4=[[-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]]`
• `L4=-0.381`

For model-5

• We know that, Model property, `L=Phin^T*m*iota`

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

• Where n=5
• For fifth mode, `L5=[[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]]`
• `L5=0.175m`

Step-3

for model-1

• Modal participation factor, `Gamma=L/M`
• For the first mode, `Gamma1=frac{L1}{M1}`
• Substitute the value of L1 and M1 in above
• `Gamma1=frac{4.121}{3.861}=1.067`
• `Gamma1=1.067`

for model-2

• Modal participation factor, `Gamma=L/M`
• For the second mode, `Gamma2=frac{L2}{M2}`
• Substitute the value of L2 and M2 in above
• `Gamma2=frac{-1.297}{3.861}=-0.335`
• `Gamma2=-0.335`

for model-3

• Modal participation factor, `Gamma=L/M`
• For the third mode, `Gamma3=frac{L3}{M3}`
• Substitute the value of L3 and M3 in above
• `Gamma3=frac{0.683}{3.861}=0.176`
• `Gamma3=0.176`

for model-4

• Modal participation factor, `Gamma=L/M`
• For the fourth mode, `Gamma4=frac{L4}{M4}`
• Substitute the value of L4 and M4 in above
• `Gamma4=frac{-0.381}{3.861}=-0.098`
• `Gamma4=-0.098`

for model-5

• Modal participation factor, `Gamma=L/M`
• For the fourth mode, `Gamma5=frac{L5}{M5}`
• Substitute the value of L5 and M5 in above
• `Gamma5=frac{0.175}{3.861}=0.045`
• `Gamma4=0.045`

Step-4

For model-1

• Effective model mass, `M^**=frac{L n^2}{Mn}`
• Where, n=1
• For the first mode, `M1^**=frac{L1^2}{M1}=frac{4.121^2}{3.861}=4.398m`
• `M1^**=4.398m`

For model-2

• Effective model mass, `M^**=frac{L n^2}{Mn}`
• Where, n=2
• For the second mode, `M2^**=frac{L2^2}{M2}=frac{-1.297^2}{3.861}=0.436m`
• `M2^**=0.436m`

For model-3

• Effective model mass, `M^**=frac{L n^2}{Mn}`
• Where, n=3
• For the Third mode, `M3^**=frac{L3^2}{M3}=frac{0.683^2}{3.861}=0.121m`
• `M3^**=0.121m`

For model-4

• Effective model mass, `M^**=frac{L n^2}{Mn}`
• Where, n=4
• For the fourth mode, `M4^**=frac{L4^2}{M4}=frac{-0.381^2}{3.861}=0.038m`
• `M4^**=0.038m`

For model-5

• Effective model mass, `M^**=frac{L n^2}{Mn}`
• Where, n=5
• For the fifth mode, `M5^**=frac{L4^2}{M5}=frac{0.175^2}{3.861}=0.008m`
• `M5^**=0.008m`

Step-5

For model-1

• Modal mass participation ratio = `frac{Mn^**}{mj}`
• Where, `mj` = Total mass of the building = `m+m+m+m+m=5m`
• Modal mass participation ratio of First mode (n=1)= `frac{4.398m}{5m}=0.88`

For model-2

• Modal mass participation ratio = `frac{Mn^**}{mj}`
• Where, `mj` = Total mass of the building = `m+m+m+m+m=5m`
• Modal mass participation ratio of second mode (n=2)= `frac{0.436m}{5m}=0.09`

For model-3

• Modal mass participation ratio = `frac{Mn^**}{mj}`
• Where, `mj` = Total mass of the building = `m+m+m+m+m=5m`
• Modal mass participation ratio of third mode (n=3)= `frac{0.121m}{5m}=0.02`

For model-4

• Modal mass participation ratio = `frac{Mn^**}{mj}`
• Where, `mj` = Total mass of the building = `m+m+m+m+m=5m`
• Modal mass participation ratio of forth mode (n=4)= `frac{0.038m}{5m}=0.01`

For model-5

• Modal mass participation ratio = `frac{Mn^**}{mj}`
• Where, `mj` = Total mass of the building = `m+m+m+m+m=5m`
• Modal mass participation ratio of fifth mode (n=5)= `frac{0.008m}{5m}=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

step-6

For modal-1

• Base shear, `Vb=Mn^** *An`
• For the first mode, `Vb1=4.398m*0.27g=1.187mg`
• `Vb1=1.187mg`

For modal-2

• Base shear, `Vb=Mn^** *An`
• For the second mode, `Vb2=0.436m*0.72g=0.314mg`
• `Vb2=0.314mg`

For modal-3

• Base shear, `Vb=Mn^** *An`
• For the third mode, `Vb3=0.121m*1.03g=0.125mg`
• `Vb3=0.125mg`

For modal-4

• Base shear, `Vb=Mn^** *An`
• For the fourth mode, `Vb4=0.038m*1.03g=0.039mg`
• `Vb4=0.039mg`

For modal-5

• Base shear, `Vb=Mn^** *An`
• For the fifth mode, `Vb5=0.008m*1.03g=0.008mg`
• `Vb5=0.008mg`

Step-7

• Average base shear is calculated by the method of Square root of sum of square (SRSS) =
• `Vb=sqrt(Vb1^2+Vb2^2+Vb3^2+Vb4^2+Vb5^2)=sqrt(1.187^2+0.314^2+0.125^2+0.039^2+0.008^2)=1.23mg`
• `Vb=1.23mg`