Understanding the concepts on Degrees of Freedom

Question:

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:

`Phi underset(1)() = {(0.334),(0.641),(0.895),(1.078),(1.173):}}`

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

`Phi underset(2)()={(-0.895),(-1.173),(-0.641),(0.334),(1.078):}}`

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

`Phi underset(3)() = {(1.173),(0.334),(-1.078),(-0.641),(0.895):}}`

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

`Phi underset(4)() = {(-1.078),(0.895),(0.334),(-1.174),(0.641):}}`

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

`Phi underset(5)() = {(0.641),(-1.078),(1.173),(-0.895),(0.334):}}`

• 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.

`Munderset(n)()=Phiunderset(n)()^TmPhiunderset(n)` , `Lunderset(n)()=Phiunderset(n)()^Tml` , `Γunderset(n)()=(Lunderset(n)())/(Munderset(n)())` , `Munderset(n)()^**=(Lunderset(n)()^2)/(Munderset(n)())`

• 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.

`Vunderset(bn)()=(Lunderset(n)()^2)/(Munderset(n)())Aunderset(n)()=Aunderset(n)()Munderset(n)()^**`

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

AIM.:

• 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.

`Munderset(n)()=Phiunderset(n)()^TmPhiunderset(n),Lunderset(n)()=Phiunderset(n)()^Tmiota,Γunderset(n)()=(Lunderset(n)())/(Munderset(n)()) ,Munderset(n)()^**=(Lunderset(n)()^2)/(Munderset(n)())`

• Then 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.

`Vunderset(bn)()=(Lunderset(n)()^2)/(Munderset(n)())Aunderset(n)()=Aunderset(n)()Munderset(n)()^**`

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

PROCEDURE.:

• Step-1.:- To calculate the Modified Mass Matrix, `Munderset(n)()`.

`:. Munderset(n)()=Phiunderset(n)()^TmPhiunderset(n)()`

• For mode 1,

`Munderset(1)()=Phiunderset(1)()^TmPhiunderset(1)()`

`:.Munderset(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]]`

`:. Munderset(1)()=3.861`

• For mode 2,

`Munderset(2)()=Phiunderset(2)()^TmPhiunderset(2)()`

`:.Munderset(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]]`

`:. Munderset(2)()=3.861`

• For mode 3,

`Munderset(3)()=Phiunderset(3)()^TmPhiunderset(3)()`

`:.Munderset(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]]`

`:. Munderset(3)()=3.861`

• For mode 4,

`Munderset(4)()=Phiunderset(4)()^TmPhiunderset(4)()`

`:.Munderset(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]]`

`:. Munderset(4)()=3.861`

• For mode 5,

`Munderset(5)()=Phiunderset(5)()^TmPhiunderset(5)()`

`:.Munderset(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]]`

`:. Munderset(5)()=3.861`

• Step-2.:- To calculate `Lunderset(n)()`.

`Lunderset(n)()=Phiunderset(n)()^Tmiota`

• For Mode 1,

`Lunderset(1)()=Phiunderset(1)()^Tmiota`

`:.Munderset(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]]`

`:.Lunderset(1)()=4.121m`

• For Mode 2,

`Lunderset(2)()=Phiunderset(2)()^Tmiota`

`:.Munderset(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]]`

`:.Lunderset(2)()=-1.297m`

• For Mode 3,

`Lunderset(3)()=Phiunderset(3)()^Tmiota`

`:.Munderset(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]]`

`:.Lunderset(3)()=0.683m`

• For Mode 4,

`Lunderset(4)()=Phiunderset(4)()^Tmiota`

`:.Munderset(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]]`

`:.Lunderset(4)()=0.381m`

• For Mode 5,

`Lunderset(5)()=Phiunderset(5)()^Tmiota`

`:.Munderset(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]]`

`:.Lunderset(5)()=0.175m`

• Step-3.:- To calculate `Γunderset(n)()`.

`Γunderset(n)()=(Lunderset(n)())/(Munderset(n)())`

• For Mode 1,

`Γunderset(1)()=(Lunderset(1)())/(Munderset(1)())`

`:.Γunderset(1)()=(4.121m)/(3.861m)`

`:.Γunderset(1)()=1.067`

• For Mode 2,

`Γunderset(2)()=(Lunderset(2)())/(Munderset(2)())`

`:.Γunderset(2)()=(-1.297m)/(3.861m)`

`:.Γunderset(2)()=-0.336`

• For Mode 3,

`Γunderset(3)()=(Lunderset(3)())/(Munderset(3)())`

`:.Γunderset(3)()=(0.683m)/(3.861m)`

`:.Γunderset(3)()=0.177`

• For Mode 4,

`Γunderset(4)()=(Lunderset(4)())/(Munderset(4)())`

`:.Γunderset(4)()=(-0.381m)/(3.861m)`

`:.Γunderset(4)()=-0.098`

• For Mode 5,

`Γunderset(5)()=(Lunderset(5)())/(Munderset(5)())`

`:.Γunderset(5)()=(0.175m)/(3.861m)`

`:.Γunderset(5)()=0.045`

• Step-4.:- To calculate Effective Modified Mass, `Munderset(n)()^**`.

`Munderset(n)()^**=(Lunderset(n)()^2)/(Munderset(n)())`

• For Mode 1,

`Munderset(1)()^**=(Lunderset(1)()^2)/(Munderset(1)())`

`:.Munderset(1)()^**=((4.121m)^2)/(3.861)`

`:.Munderset(1)()^**=4.399m`

• For Mode 2,

`Munderset(2)()^**=(Lunderset(2)()^2)/(Munderset(2)())`

`:.Munderset(2)()^**=((-1.297m)^2)/(3.861)`

`:.Munderset(2)()^**=0.435m`

• For Mode 3,

`Munderset(3)()^**=(Lunderset(3)()^2)/(Munderset(3)())`

`:.Munderset(3)()^**=((0.683m)^2)/(3.861)`

`:.Munderset(3)()^**=0.121m`

• For Mode 4,

`Munderset(4)()^**=(Lunderset(4)()^2)/(Munderset(4)())`

`:.Munderset(4)()^**=((-0.381m)^2)/(3.861)`

`:.Munderset(4)()^**=0.037m`

• For Mode 5,

`Munderset(5)()^**=(Lunderset(5)()^2)/(Munderset(5)())`

`:.Munderset(5)()^**=((0.175m)^2)/(3.861)`

`:.Munderset(5)()^**=0.008m`

• Step-5.:- To calculate Modal Mass Participation Ratio.

Participating Ratio `=(Munderset(n)()^**)/(Munderset(j)())`

`Rightarrow Where,Munderset(j)()=Munderset(1)()^**+Munderset(2)()^**+Munderset(3)()^**+Munderset(4)()^**+Munderset(5)()^**=5m`(`:'`sum of mass of each floor)

• For Mode 1,

Participating Ratio `=(Munderset(1)()^**)/(Munderset(j)())=4.399/5=0.8798`

• For Mode 2,

Participating Ratio `=(Munderset(2)()^**)/(Munderset(j)())=0.435/5=0.087`

• For Mode 3,

Participating Ratio `=(Munderset(3)()^**)/(Munderset(j)())=0.121/5=0.0242`

• For Mode 4,

Participating Ratio `=(Munderset(4)()^**)/(Munderset(j)())=0.037/5=0.0074`

• For Mode 5,

Participating Ratio `=(Munderset(5)()^**)/(Munderset(j)())=0.008/5=0.0016`

`Rightarrow :.Munderset(1)()^**+Munderset(2)()^**+Munderset(3)()^**+Munderset(4)()^**+Munderset(5)()^**=5m`(`:'`sum of mass of each floor)`,So O.K.

From the received participation ratio we can say that the sum of the participation ratio of the two model mass will be greater than 90%.

• Step-6.:- To calculate Base Shear,`Vunderset(bn)()`.

`Vunderset(bn)()=Aunderset(n)()Munderset(n)()^2`

From the Pseudo Acceleration design Spectrum Graph,

`rightarrow Tunderset(1)()=2sec, Aunderset(1)()=0.279`

`rightarrow Tunderset(2)()=0.6852sec, Aunderset(2)()=0.769`

`rightarrow Tunderset(3)()=0.4346sec, Aunderset(3)()=1.039`

`rightarrow Tunderset(4)()=0.3383sec, Aunderset(4)()=1.039`

`rightarrow Tunderset(5)()=0.2966sec, Aunderset(5)()=1.039`

• For Mode 1,

`Vunderset(b1)()=Aunderset(1)()Munderset(1)()^**=(0.279)**(4.399)=1.227m`

• For Mode 2,

`Vunderset(b2)()=Aunderset(2)()Munderset(2)()^**=(0.769)**(0.435)=0.334m`

• For Mode 3,

`Vunderset(b3)()=Aunderset(3)()Munderset(3)()^**=(1.039)**(0.121)=0.126m`

• For Mode 4,

`Vunderset(b4)()=Aunderset(4)()Munderset(4)()^**=(1.039)**(0.037)=0.038m`

• For Mode 5,

`Vunderset(b5)()=Aunderset(5)()Munderset(5)()^**=(1.039)**(0.008)=0.008m`

• Step-7.:- To calculate Combined Base Shear.

`Rightarrow Vunderset(B)()^2=Vunderset(b1)()^2+Vunderset(b2)()^2+Vunderset(b3)()^2+Vunderset(b4)()^2+Vunderset(b5)()^2`

`Rightarrow Vunderset(B)()^2=(1.227)^2+(0.334)^2+(0.126)^2+(0.038)^2+(0.008)^2`

`RightarrowVunderset(B)()=sqrt(1.634)`

`:. Vunderset(B)()=1.278`