Understanding the concepts on Degrees of Freedom
Question 1: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.
Solution:
(a) Modified Mass Matrix(M):
(i) For the first mode:
Where n=1
The fundamental period of vibration (T1 = 2 seconds) and its associated mode shape:
Modified Mass Matrix(`Mundersetn`) = `phin^Txxmxxphiundersetn`
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 (n) = 1
`Munderset1`= [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]]`
`Munderset1`= `0.111+0.410+0.801+1.162+1.375`
`Munderset1`= 3.860m.
(ii) For the second mode:
Where n = 2
The second mode of vibration (T2 = 0.6852 seconds) and its associated mode shape:
Modified Mass Matrix(`Mundersetn`) = `phin^Txxmxxphiundersetn`
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 (n) = 2
`Munderset2` = [-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]]`
`Munderset2` = 0.801+1.375+0.410+0.111+1.16`
`Munderset2` = 3.860`.
(iii) For the third mode:
Where n = 3
The third mode of vibration (T3 = 0.4346 seconds) and its associated mode shape:
Modified Mass Matrix(`Mundersetn`) = `phin^Txxmxxphiundersetn`
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 (n) = 3
`Munderset3` = [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]]`
`Munderset3` = `1.375+0.111+1.162+0.410+0.801`
`Munderset3` = 3.860m.
(iv) For the fourth mode:
Where n = 4
The fourth mode of vibration (T4 = 0.3383 seconds) and its associated mode shape:
Modified Mass Matrix(`Mundersetn`) = `phin^Txxmxxphiundersetn`
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 (n) = 4
`Munderset4` = [-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]]`
`Munderset4` = `1.162+0.801+0.111+1.375+0.410`
`Munderset4` = 3.860m.
(v) For the fifth mode:
Where n = 5
The fifth mode of vibration (T4 = 0.2966 seconds) and its associated mode shape:
Modified Mass Matrix(`Mundersetn`) = `phin^Txxmxxphiundersetn`
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 (n) = 5
`Munderset5` = [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]]`
`Munderset5` = `0.410+1.162+1.375+0.801+0.111`
`Munderset5` = 3.860m.
(b) Modal Property (`Lundersetn`):
(i) For the first mode:
Where n = 1
`therefore`Modal Property (`Lundersetn`) = `phin^Txxmxxiota`
where `iota` = Influnce factor = `[[1],[1],[1],[1],[1]]`
For the first mode (n) = 1
`Lunderset1` = [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]]`
`Lunderset1` = `0.334+0.641+0.895+1.078+1.173`
`Lunderset1` = 4.121m.
(ii) For the second mode:
Where n = 2
`therefore`Modal Property (`Lundersetn`) = `phin^Txxmxxiota`
where `iota` = Influnce factor = `[[1],[1],[1],[1],[1]]`
For the second mode (n) = 2
`Lunderset2` = [-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]]`
`Lunderset2` = [0.895]+[-1.173]+[-0.641]+[0.334]+[1.078]
`Lunderset2` = -1.297m.
(iii) For the third mode:
Where n = 3
`therefore`Modal Property (`Lundersetn`) = `phin^Txxmxxiota`
where `iota` = Influnce factor = `[[1],[1],[1],[1],[1]]`
For the third mode (n) = 3
`Lunderset3` = [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]]`
`Lunderset3` = [1.173]+[0.334]+[-1.078]+[-0.641]+[0.895]
`Lunderset3` = 0.683m.
(iv) For the fourth mode:
Where n = 4
`therefore`Modal Property (`Lundersetn`) = `phin^Txxmxxiota`
where `iota` = Influnce factor = `[[1],[1],[1],[1],[1]]`
For the fourth mode (n) = 4
`Lunderset4` = [-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]]`
`Lunderset4` = [-1.078]+[0.895]+[0.334]+[-1.173]+[0.641]
`Lunderset4` = -0.381m.
(v) For the fifth mode:
Where n = 5
`therefore`Modal Property (`Lundersetn`) = `phin^Txxmxxiota`
where `iota` = Influnce factor = `[[1],[1],[1],[1],[1]]`
For the fifth mode (n) = 5
`Lunderset5` = [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]]`
`Lunderset5` = [0.641]+[-1.078]+[1.173]+[-0.895]+[0.334]
`Lunderset5` = 0.175m.
(c) Modal Participation Factor ( `Gammaundersetn`):
(i) For the first mode:
Where n = 1.
`therefore`Modal Participation Factor( `Gammaundersetn` ) = `frac{L n}{M n}`
For the first mode (n) = 1
`Gammaunderset1` = `frac{L 1}{M1}`
`Gammaunderset1` = `frac{4.121 m}{3.860 m}`
`Gammaunderset1` = `1.067`
(ii) For the second mode:
Where n = 2.
`therefore`Modal Participation Factor( `Gammaundersetn` ) = `frac{L n}{M n}`
For the first mode (n) = 2
`Gammaunderset2` = `frac{L 2}{M2}`
`Gammaunderset2` = `frac{-1.297 m}{3.860 m}`
`Gammaunderset2` = -0.335.
(iii) For the third mode:
Where n = 3.
`therefore`Modal Participation Factor( `Gammaundersetn` ) = `frac{L n}{M n}`
For the first mode (n) = 3
`Gammaunderset3` = `frac{L 3}{M3}`
`Gammaunderset3` = `frac{0.683 m}{3.860 m}`
`Gammaunderset3` = 0.177`.
(iv) For the fourth mode:
Where n = 4.
`therefore`Modal Participation Factor( `Gammaundersetn` ) = `frac{L n}{M n}`
For the first mode (n) = 4
`Gammaunderset4` = `frac{L 4}{M4}`
`Gammaunderset4` = `frac{-0.381 m}{3.860 m}`
`Gammaunderset4` = -0.098`.
(iv) For the fifth mode:
Where n = 5.
`therefore`Modal Participation Factor( `Gammaundersetn` ) = `frac{L n}{M n}`
For the first mode (n) = 5
`Gammaunderset5` = `frac{L5}{M5}`
`Gammaunderset5` = `frac{-0.175 m}{3.860 m}`
`Gammaunderset5` = 0.045`.
(d) Effective Modal Mass (`M^*undersetn`):
(i) For the first mode:
Where n = 1
`therefore`Effective Modal Mass (`M^*undersetn`) = `frac{L n^2}{Mn}`
`M^*underset1` = `frac{L1^2}{M1}`
`M^*underset1` = `frac{(4.121m)^2}{3.860m}`
`M^*underset1` = 4.398m.
(ii) For the second mode:
Where n = 2
`therefore`Effective Modal Mass (`M^*undersetn`) = `frac{L n^2}{Mn}`
`M^*underset2` = `frac{L2^2}{M2}`
`M^*underset2` = `frac{(-1.297m)^2}{3.860m}`
`M^*underset2` = 0.435m`
(iii) For the third mode:
Where n = 3
`therefore`Effective Modal Mass (`M^*undersetn`) = `frac{L n^2}{Mn}`
`M^*underset3` = `frac{L3^2}{M3}`
`M^*underset3` = `frac{(0.683m)^2}{3.860m}`
`M^*underset3` = 0.121m`
(iv) For the fourth mode:
Where n = 4
`therefore`Effective Modal Mass (`M^*undersetn`) = `frac{L n^2}{Mn}`
`M^*underset4` = `frac{L4^2}{M4}`
`M^*underset4` = `frac{(-0.381m)^2}{3.860m}`
`M^*underset4` = 0.038m`
(v) For the fifth mode:
Where n = 5
`therefore`Effective Modal Mass (`M^*undersetn`) = `frac{L n^2}{Mn}`
`M^*underset5` = `frac{L5^2}{M5}`
`M^*underset5` = `frac{(0.175m)^2}{3.860m}`
`M^*underset5` = 0.008m`
(e) Modal Mass Participation Ratio:
(i) For the first mode:
Where n = 1;
Modal Mass Participation Ratio = `frac{Mn^*}{mj}`
Whereas mj = Total mass of the building.
mj = m+m+m+m+m
mj = 5m.
Modal Mass Participation Ratio = `frac{Mn^*}{mj}`
Modal Mass Participation Ratio(n) = `frac{Mn^*}{mj}`
= `frac{M1^*}{mj}`
Modal Mass Participation Ratio(1) = `frac{4.398m}{5m}`
Modal Mass Participation Ratio(1) = 0.879.
(ii) For the second mode:
Where n = 2;
Modal Mass Participation Ratio = `frac{Mn^**}{mj}`
Whereas mj = Total mass of the building.
mj = m+m+m+m+m
mj = 5m.
Modal Mass Participation Ratio = `frac{Mn^**}{mj}`
For the second mode (n) = 2
Modal Mass Participation Ratio (n) = `frac{Mn^**}{mj}`
= `frac{M2^**}{mj}`
Modal Mass Participation Ratio (2) = `frac{0.435m}{5m}`
Modal Mass Participation Ratio (2) = 0.087.
(iii) For the third mode:
Where n = 3;
Modal Mass Participation Ratio = `frac{Mn^**}{mj}`
Whereas mj = Total mass of the building.
mj = m+m+m+m+m
mj = 5m.
Modal Mass Participation Ratio = `frac{Mn^**}{mj}`
For the third mode (n) = 3
Modal Mass Participation Ratio (n) = `frac{Mn^**}{mj}`
= `frac{M3^**}{mj}`
Modal Mass Participation Ratio (3) = `frac{0.121m}{5m}`
Modal Mass Participation Ratio (3) = 0.024.
(iv) For the fourth mode:
Where n = 4;
Modal Mass Participation Ratio = `frac{Mn^**}{mj}`
Whereas mj = Total mass of the building.
mj = m+m+m+m+m
mj = 5m.
Modal Mass Participation Ratio = `frac{Mn^**}{mj}`
For the fourth mode (n) = 4
Modal Mass Participation Ratio (n) = `frac{Mn^**}{mj}`
= `frac{M4^**}{mj}`
Modal Mass Participation Ratio (4) = `frac{0.038m}{5m}`
Modal Mass Participation Ratio (4) = 0.007.
(v) For the fifth mode:
Where n = 5;
Modal Mass Participation Ratio = `frac{Mn^**}{mj}`
Whereas mj = Total mass of the building.
mj = m+m+m+m+m
mj = 5m.
Modal Mass Participation Ratio = `frac{Mn^**}{mj}`
For the fifth mode (n) = 5
Modal Mass Participation Ratio (n) = `frac{Mn^**}{mj}`
= `frac{M5^**}{mj}`
Modal Mass Participation Ratio (5) = `frac{0.008m}{5m}`
Modal Mass Participation Ratio (5) = 0.001.
Now we can verify the total mass of the building by adding all the effective modal mass:
Total mass of the building = m+m+m+m+m
= 5m
Whereas the total of the effective total mass is:
= `4.398+0.435+0.121+0.038+0.008`
= `5`
Hence the total of the effective modal mass = 5
whereas the total mass of the building = 5
Hence O.K.
Now we will be adding the first two mode of the mass participation ratio i.e; (0.879,0.087)
= `0.879+0.087`
= `0.966`
= 96.66 %
So in order to achieve the minimum 90% of the modal mass participation ratio,first two modes are enough.
(f) Base Shear (`VB undersetn`):
Base Shear (`VBn`) = `Mn^**xxAn`
So from the given pseudo-acceleration design spectrum graph,we will be able to find the value of `An`. The pseudo-acceleration design spectrum graph is given below:
So for T1:
`A underset1` = 0.27g
For T2:
`A underset2` = 0.76`
For T3:
`A underset 3`= 1.03`
For T4:
`A underset4`= 1.03`
For T5:
`A underset5`= 1.03`
(i) For the first mode:
Where n = 1;
Base Shear (`Vb undersetn`) = `Mn^**xxAn`
`VB underset1` = `M1^**xxA1`
`VB underset1`= `4.398mxx0.27g`
`VB underset1`= `1.187 mg`.
(ii) For the second mode:
Where n = 2;
Base Shear (`Vb undersetn`) = `Mn^**xxAn`
`VB underset2` = `M2^**xxA2`
`VB underset2`= `0.435mxx0.76g`
`VB underset2`= `0.330 mg`.
(iii) For the third mode:
Where n = 3;
Base Shear (`Vb undersetn`) = `Mn^**xxAn`
`VB underset3` = `M3^**xxA3`
`VB underset3`= `0.121mxx1.03g`
`VB underset3`= `0.124 mg`
(iv) For the fourth mode:
Where n = 4;
Base Shear (`Vb undersetn`) = `Mn^**xxAn`
`VB underset4` = `M4^**xxA4`
`VB underset4`= `0.038mxx1.03g`
`VB underset4`= `0.039 mg`
(v) For the fifth mode:
Where n = 5;
Base Shear (`Vb undersetn`) = `Mn^**xxAn`
`VB underset5` = `M5^**xxA5`
`VB underset5`= `0.008mxx1.03g`
`VB underset5`= `0.008 mg`
(f) Average Base Shear (`VundersetB`):
We can calculate the average base shear by the method of square root of sum of square.
`therefore` Average Base Shear (`V undersetB`) = `sqrt(VB1^2+VB2^2+VB3^2+VB4^2+VB5^2)`
= `sqrt(1.187^2+0.330^2+0.124^2+0.039^2+0.008^2)`
Average Base Shear (`VundersetB`) = 1.23mg.