Understanding the concepts on Degrees of Freedom
Aim -
1) To calculate Ln, Mnand Γn for each of the five modes.
2) To calculate the effective modal mass Mn* and the modal mass participating ratio for the 5 modes.
3) To find the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule.
1) To find out the Modal mass (`M_n`) of a given storey .
A) For 1st Node - T= 2 sec
i) We know that ,
Modal Mass (`M_n`) = `phi_n^T m phi_n`----------------------------------(A)
Wherre ,
m = Mass Matrix
`phi_n` = Mode shape @nth node
`phi_n^T` = Transpose of Mode shape .
ii) So , For 1st node = `T_1 = 2 sec`
`phi_n = ((0.334) , (0.641) , (0.895) , (1.078) , (1.173))` ------------------------------------( given)
`:.phi_n^T = [ 0.334 , .641 , 0.895 , 1.078 , 1.173 ]`
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]]` --------------( Each Storey has same mass 'm')
iii) put all this value in`eq^n (A)`
`:.Modal Mass = M_n` = `[ 0.334 , .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))`
`:.`Solving above matrix , we get
`M_1 = 3.861m`
iv) To Calculate `L_n`
We know that ,
`L_n =phi_n^T m i`-----------------------------------------(B)
Wherre ,
m = Mass Matrix
`phi_n^T` = Transpose of Mode shape .
i = Influence vector = ( Take influence factor as Unitary )
`= ((1) ,(1) ,(1) ,(1) ,(1))`
`:.L_n=[ 0.334 , .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))`
`:.` Solving above Matrix , we get
`L_1 = 4.121 m`
v)To find the Modal Participation factor - `Γn`
We know that ,
`Γn = (L_n)/M_n`
So put the value of `L_1 = 4.121 m` and `M_1 = 3.861m`
we get ,
`:.Γ_1 = (4.121m)/(3.861m)`
`:.Γ_1 = 1.067`
vi) To calculate the Effective Modal Mass
we know that ,
`M_n^ ** = (L_n^2)/(M_n)`
put the value of `L_1 = 4.121 m` and `M_1 = 3.861m`
`:.M_1^ ** = (4.121m)^2/(3.861m)`
`:.M_1^ ** = 4.398 m`
vii) To calculate the modal mass participating ratio -
we know that ,
Modal mass participating ration is given by = It is ratio of Effective Modal mass (`M_n^**`) to the Total Mass (mj)
= `(M_n^**) / (m_j)`
So Total mass of 5 storey = m+m+m+m+m
= 5m
`:.M_1^** / m_j`
= `(4.398m) / (5m)`
= `0.88 = 88%`
viii) To find out the base shear -
Now we have to calculate the Base Shear (`V_b`)
We know that ,
`V_b = M_1^** *A_1`
To calculate A1 value refer the following graph
i.e Pseudo spectral acceleration vs Time period
we get value of A1 in terms of Pseudo acceleration
So , For T1 = 2 sec
A1 = 0.27 g ( From graph )
`:.V_b = M_1^** *A_1`
`:.V_b = 4.398m^** *0.27g`
`:.V_(b_1) = 1.187 mg`
B) For 2nd Node - T =( 0.6852 sec)
i) We know that ,
Modal Mass (`M_n`) = `phi_n^T m phi_n`----------------------------------(A)
Wherre ,
m = Mass Matrix
`phi_n` = Mode shape @nth node
`phi_n^T` = Transpose of Mode shape .
ii) So , For 2nd node = `T_2 = 0.6852 sec`
`phi_2 = ((-0.895) , (-1.173) , (-0.641) , (0.334) , (1.078))` ------------------------------------( given)`
`:.phi_n^T = [ -0.895 , -1.173 , -0.641 , 0.334 , 1.078 ]` `
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]]` --------------( Each Storey has same mass 'm')
iii) put all this value in`eq^n (A)`
`:.Modal Mass = M_n` = `[- 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))`
`:.`Solving above matrix , we get
`M_2 = 3.861m`
iv) To Calculate `L_n`
We know that ,
`L_n =phi_n^T m i`-----------------------------------------(B)
Wherre ,
m = Mass Matrix
`phi_n^T` = Transpose of Mode shape .
i = Influence vector = ( Take influence factor as Unitary )
`= ((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))`
`:.` Solving above Matrix , we get
`L_2 = -1.297 m`
v)To find the Modal Participation factor - `Γn`
We know that ,
`Γn = (L_n)/M_n`
So put the value of `L_2 = -1.297 m` and `M_2 = 3.861m`
we get ,
`:.Γ_2 = (-1.297m)/(3.861m)`
`:.Γ_2 = -0.336`
vi) To calculate the Effective Modal Mass
we know that ,
`M_n^ ** = (L_n^2)/(M_n)`
put the value of `L_2 = -1.297 m` and `M_2 = 3.861m`
`:.M_2^ ** = (-1.297m)^2/(3.861m)`
`:.M_2^ ** = 0.436 m`
vii) To calculate the modal mass participating ratio -
we know that ,
Modal mass participating ration is given by = It is ratio of Effective Modal mass (`M_n^**`) to the Total Mass (mj)
= `(M_n^**) / (m_j)`
So Total mass of 5 storey = m+m+m+m+m
= 5m
`:.M_2^** / m_j`
= `(0.436m) / (5m)`
= `0.0872 = 8.72%`
viii) To find out the base shear -
Now we have to calculate the Base Shear (`V_b`)
We know that ,
`V_b = M_2^** *A_2`
To calculate A2 value refer the following graph
i.e Pseudo spectral acceleration vs Time period
we get value of A2 in terms of Pseudo acceleration
So , For T1 = 0.6852 sec
A2 = 0.76 g ( From graph )
`:.V_b = M_2^** *A_2`
`:.V_b = 0.436m^** *0.76g`
`:.V_(b_2) = 0.33 mg`
C) For 3rd Node - T =( 0.4346 sec)
i) We know that ,
Modal Mass (`M_n`) = `phi_n^T m phi_n`----------------------------------(A)
Wherre ,
m = Mass Matrix
`phi_n` = Mode shape @nth node
`phi_n^T` = Transpose of Mode shape .
ii) So , For 3rd node = `T_3 = 0.4346 sec`
`phi_3 = ((1.173) , (0.334) , (-1.078) , (-0.641) , (0.895))` ------------------------------------( given)`
`:.phi_n^T = [ 1.173 ,0.334 , -1.178 , -0.641 , 0.895 ]``
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]]` --------------( Each Storey has same mass 'm')
iii) put all this value in`eq^n (A)`
`:.Modal Mass = M_n` = `[1.173 , 0.334 , -1.178 , -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) , (-1.178) , (-0.641) , (0.895))`
`:.`Solving above matrix , we get
`M_3 = 3.85m`
iv) To Calculate `L_n`
We know that ,
`L_n =phi_n^T m i`-----------------------------------------(B)
Wherre ,
m = Mass Matrix
`phi_n^T` = Transpose of Mode shape .
i = Influence vector = ( Take influence factor as Unitary )
`= ((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))`
`:.` Solving above Matrix , we get
`L_3 = 0.687m`
v)To find the Modal Participation factor - `Γn`
We know that ,
`Γn = (L_n)/M_n`
So put the value of `L_3 = 0.687m` and `M_3 = 3.85m`
we get ,
`:.Γ_3 = (0.687m)/(3.85m)`
`:.Γ_3 = 0.178`
vi) To calculate the Effective Modal Mass
we know that ,
`M_n^ ** = (L_n^2)/(M_n)`
put the value of `L_3 = 0.687 m` and `M_3 = 3.85m`
`:.M_3^ ** = (0.687m)^2/(3.85m)`
`:.M_3^ ** = 0.122 m`
vii) To calculate the modal mass participating ratio -
we know that ,
Modal mass participating ration is given by = It is ratio of Effective Modal mass (`M_n^**`) to the Total Mass (mj)
= `(M_n^**) / (m_j)`
So Total mass of 5 storey = m+m+m+m+m
= 5m
`:.M_3^** / m_j`
= `(0.122m) / (5m)`
= `0.0244 = 2.44%`
viii) To find out the base shear -
Now we have to calculate the Base Shear (`V_b`)
We know that ,
`V_b = M_3^** *A_3`
To calculate A3 value refer the following graph
i.e Pseudo spectral acceleration vs Time period
we get value of A3 in terms of Pseudo acceleration
So , For T3 = 0.4346 sec
A3 = 1.03 g ( From graph )
`:.V_b = M_3^** *A_3`
`:.V_b = 0.122m^** *1.03g`
`:.V_(b_3) = 0.126 mg`
D) For 4th Node - T =( 0.3383 sec)`
i) We know that ,
Modal Mass (`M_n`) = `phi_n^T m phi_n`----------------------------------(A)
Wherre ,
m = Mass Matrix
`phi_n` = Mode shape @nth node
`phi_n^T` = Transpose of Mode shape .
ii) So , For 4th node = `T_4 = 0.3383 sec`
`phi_4 = ((-1.078) , (0.895) , (0.334) , (-1.173) , (0.641))` ------------------------------------( given)`
`:.phi_n^T = [ -1.078 ,0.895 , 0.334 , -1.173 , 0.641 ]`
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]]` --------------( Each Storey has same mass 'm')
iii) put all this value in`eq^n (A)`
`:.Modal Mass = M_n` = `[-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)`
`:.`Solving above matrix , we get
`M_4 = 3.861m`
iv) To Calculate `L_n`
We know that ,
`L_n =phi_n^T m i`-----------------------------------------(B)
Wherre ,
m = Mass Matrix
`phi_n^T` = Transpose of Mode shape .
i = Influence vector = ( Take influence factor as Unitary )
`= ((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))`
`:.` Solving above Matrix , we get
`L_4 = -0.381m`
v)To find the Modal Participation factor - `Γn`
We know that ,
`Γn = (L_n)/M_n`
So put the value of `L_4 = -0.381m` and `M_4 = 3.861m`
we get ,
`:.Γ_4 = (-0.381m)/(3.861m)`
`:.Γ_4 = -0.098`
vi) To calculate the Effective Modal Mass
we know that ,
`M_n^ ** = (L_n^2)/(M_n)`
put the value of `L_4 = -0.381 m` and `M_4 = 3.861m`
`:.M_4^ ** = (-0.381m)^2/(3.861m)`
`:.M_4^ ** = 0.030 m`
vii) To calculate the modal mass participating ratio -
we know that ,
Modal mass participating ration is given by = It is ratio of Effective Modal mass (`M_n^**`) to the Total Mass (mj)
= `(M_n^**) / (m_j)`
So Total mass of 5 storey = m+m+m+m+m
= 5m
`nbsp; `:.M_4^** / m_j`
= `(0.030m) / (5m)`
= `0.006 = 0.6%`
viii) To find out the base shear -
Now we have to calculate the Base Shear (`V_b`)
We know that ,
`V_b = M_4^** *A_4`
To calculate A4 value refer the following graph
i.e Pseudo spectral acceleration vs Time period
we get value of A4 in terms of Pseudo acceleration
So , For T3 = 0.3383 sec
A4 = 1.03 g ( From graph )
`:.V_b = M_4^** *A_4`
`:.V_b = 0.030m^** *1.03g`
`:.V_(b_4) = 0.031 mg`
E) For 5th Node - T =( 0.2966 sec)`
i) We know that ,
Modal Mass (`M_n`) = `phi_n^T m phi_n`----------------------------------(A)
Wherre ,
m = Mass Matrix
`phi_n` = Mode shape @nth node
`phi_n^T` = Transpose of Mode shape .
ii) So , For 5th node = `T_5 = 0.2966 sec`
`phi_5 = ((0.641) , (-1.078) , (1.173) , (-0.895) , (0.334))` ------------------------------------( given)`
`:.phi_n^T = [ 0.641 ,-1.078 , 1.173 , -0.895 , 0.334 ]`
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]]` --------------( Each Storey has same mass 'm')
iii) put all this value in`eq^n (A)`
`:.Modal Mass = M_n` = `[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)`
`:.`Solving above matrix , we get
`M_5 = 3.861m`
iv) To Calculate `L_n`
We know that ,
`L_n =phi_n^T m i`-----------------------------------------(B)
Wherre ,
m = Mass Matrix
`phi_n^T` = Transpose of Mode shape .
i = Influence vector = ( Take influence factor as Unitary )
`= ((1) ,(1) ,(1) ,(1) ,(1))`
`:.L_4=[ 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))`
`:.` Solving above Matrix , we get
`L_5 = 0.175m`
v)To find the Modal Participation factor - `Γn`
We know that ,
`Γn = (L_n)/M_n`
So put the value of `L_5 = 0.175m` and `M_5 = 3.861m`
we get ,
`:.Γ_5 = (0.175m)/(3.861m)`
`:.Γ_5 = 0.045`
vi) To calculate the Effective Modal Mass
we know that ,
`M_n^ ** = (L_n^2)/(M_n)`
put the value of `L_5 = 0.175 m` and `M_5 = 3.861m`
`:.M_5^ ** = (0.175m)^2/(3.861m)`
`:.M_5^ ** = 0.007 m`
vii) To calculate the modal mass participating ratio -
we know that ,
Modal mass participating ratio is given by = It is ratio of Effective Modal mass (`M_n^**`) to the Total Mass (mj)
= `(M_n^**) / (m_j)`
So Total mass of 5 storey = m+m+m+m+m
= 5m
`:.M_5^** / m_j`
= `(0.007m) / (5m)`
= `0.0014 = 0.14%`
viii) To find out the base shear -
Now we have to calculate the Base Shear (`V_b`)
We know that ,
`V_b = M_5^** *A_5`
To calculate A5 value refer the following graph
i.e Pseudo spectral acceleration vs Time period
we get value of A5 in terms of Pseudo acceleration
So , For T5 = 0.2966 sec
A4 = 1.03 g ( From graph )
`:.V_b = M_5^** *A_5`
`:.V_b = 0.007m^** *1.03g`
`:.V_(b_5) = 0.0072 mg`
3) Now we have to calculate the Combine base shear by using SRSS method
`V_b = sqrt [V_(b_1)^2 + V_(b_2)^2 + V_(b_3)^2 + V_(b_4)^2 + V_(b_5)^2`
put the Values of
`V_(b_1) = 1.187`
`V_(b_2) = 0.33`
`V_(b_3) = 0.126`
`V_(b_4) = 0.031`
`V_(b_5) = 0.0072`
Substitute the base shear values in above eqn
`V_b = sqrt [1.187^2 + 0.33^2 + 0.126^2 + 0.031^2 +0.0072^2`
`:.V_b = 1.22mg`
4) How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?
From above results we can say that
Only 2 modes of Modal participation is required to obtained above 90%
`:.`Modal Participation of node 1 + Modal Participation of node 2 = 88% + 8.72%
= 96.72%
Results -
i) The combined Base Shear is calculated by using SRSS method is = 1.22mg
ii) Two modes are to be considered such that a modal mass participating ratio of 90% is obtained .
iii)
| Parameters | Mode 1 | Mode 2 | Mode 3 | Mode 4 | Mode 5 |
| (`M_n`) = `phi_n^T m phi_n` | 3.861 | 3.861 | 3.85 | 3.861 | 3.861 |
| `L_n =phi_n^T m i` | 4.121 | -1.297 | 0.687 | -0.381 | 0.175 |
| `Γn = (L_n)/M_n` | 1.067 | -0.336 | 0.178 | -0.098 | 0.045 |
| `M_n^ ** = (L_n^2)/(M_n)` | 4.397 | 0.436 | 0.122 | 0.030 | 0.007 |
| Modal mass participating ratio | 88% | 8.72% | 2.44% | 0.6% | 0.14% |
| `V_b = M_n^** *A_n` | 1.187 | 0.33 | 0.126 | 0.031 | 0.0072 |