Understanding the concepts on Degrees of Freedom

Aim:

To compute the values of the seismic parameters based on the given structural data.

Formation of 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]]`

Influence vector:

Since the degrees of freedom move along the direction of the lateral force, the influence vector will be a unit.

`iota`= `[[1],[1],[1],[1],[1]]`

`L_n = phi_n^T*m*iota`

`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 = 4.121m`

Similary for other modes

`L_2 = -1.297m`

`L_3 = 0.683m`

`L_4 = 0.381m`

`L_5 = 0.175m`

`M_n = phi_n^T.m.phi_n`

`M_1 = [[0.334m,0.641m,0.895m,1.078m,1.173m]] [[0.334],[0.641],[0.895],[1.078],[1.173]]`

`M_1 = 3.861m`

`M_2 = [[-0.895m,-1.173m,-0.641m,0.334m,1.078m]]*[[-0.895],[-1.173],[-0.641],[0.334],[1.078]]`

`M_2 = 3.861m`

`M_3 = [[1.173m,0.334m,-1.078m,-0.641m,0.895m]]*[[1.173],[0.334],[-1.078],[-0.641],[0.895]]`

`M_3 = 3.861m`

`M_4 = [[-1.078m,0.895m,0.334m,-1.173m,0.641m]]*[[-1.078],[0.895],[0.334],[-1.173],[0.641]]`

`M_4 = 3.861m`

`M_5 = [[0.641m,-1.078m,1.173m,-0.895m,0.334m]]*[[0.641],[-1.078],[1.173],[-0.895],[0.334]]`

`M_5 = 3.861m`

`Gamma_n = L_n/M_n`

`Gamma_1 = (4.121m)/(3.861m) = 1.067`

`Gamma_2 = (-1.297m)/(3.861m) = -0.336`

`Gamma_3 = (0.683m)/(3.861m) = 0.177`

`Gamma_4 = (0.381m)/(3.861m) = 0.099`

`Gamma_5 = (0.175m)/(3.861m) = 0.045`

effective modal mass `M_1^star`= `L_1^2/M_1`

`M_1^star =`4.398m

Participation ratio = 0.8796

Similarly,

`M_2^star =`0.435m

Participation ratio = 0.087

`M_3^star =`0.121m

Participation ratio = 0.024

`M_4^star =`0.037m

Participation ratio = 0.0074

`M_5^star =`0.008m

Participation ratio = 0.0016

We can verify the total mass of the building m+m+m+m+m=5m

Adding all effective modal mass = `M_1^star + M_2^star + M_3^star + M_4^star + M_5^star = 5`

Hence OK.

From the above participation ratios of modal mass, we can understand that sum of Mode-1 and Mode-2 are enough to obtain minimum 90% of the participation.

0.8796 + 0.087 = 0.9666 = 96.66%

From the given pseudo-acceleration design spectrum graph, we will note the design acceleration for each modes as follow.

For `T_1, A_1 = 0.279`

`T_2, A_2 = 0.769`

`T_3, A_3 = 1.039`

`T_4, A_4 = 1.039`

`T_5, A_5 = 1.039`

Base shear for Each mode,

`V_(bn) = M_n^star * A_n`

`V_(b1) =`1.227m

`V_(b2) =`0.334m

`V_(b3) =`0.126m

`V_(b4) =`0.038m

`V_(b5) =`0.008m

Combined base shear by SRSS method = `root()((1.227^2 + 0.334^2 + 0.126^2 + 0.038^2 + 0.008^2))`

`V_b =` 1.278m