Understanding the concepts on Degrees of Freedom

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

M_n`= 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

M_n`= 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)= 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`= 0.8801+1.375+0.410+0.111+1.16

`Munderset2`= 3.860m`

(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

M_n`= 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)= 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

M_n`= 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)= 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`= 1.375+0.111+1.162+0.410+0.801

`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

M_n`= 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)= 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`= 1.375+0.111+1.162+0.410+0.801

`Munderset5`= 3.860m`

(b) Modal Property (`Lundersetn`):

(i) For the first mode:

Where n=1

∴Modal Property

(`Lundersetn`)`= ϕ n^ T × m × ι`

where ι = 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

∴Modal Property

(`Lundersetn`)`= ϕ n^ T × m × ι`

where ι = Influnce factor =`[[1],[1],[1],[1],[1]]`

For the first 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

∴Modal Property

(`Lundersetn`)`= ϕ n^ T × m × ι`

where ι = Influnce factor =`[[1],[1],[1],[1],[1]]`

For the first 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

∴Modal Property

(`Lundersetn`)`= ϕ n^ T × m × ι`

where ι = Influnce factor =`[[1],[1],[1],[1],[1]]`

For the first 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

∴Modal Property

(`Lundersetn`)`= ϕ n^ T × m × ι`

where ι = Influnce factor =`[[1],[1],[1],[1],[1]]`

For the first 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(`Γundersetn`) :

(i) For the first mode:

Where n = 1.

∴Modal Participation Factor (`Γundersetn`) = `(L n)/(Mn)`

For the first mode (n)=1

`Γunderset1`= `(L1)/(M1)`

`Γunderset1`= `(4.121)/(3.860)`

`Γunderset1`= 1.067

(ii) For the second mode:

Where n = 2.

∴Modal Participation Factor (`Γundersetn`) = `(L n)/(Mn)`

For the first mode (n)=2

`Γunderset2`= `(L2)/(M2)`

`Γunderset2`= `(-1.297)/(3.860)`

`Γunderset2`= -0.335`

(iii) For the third mode:

Where n = 3.

∴Modal Participation Factor (`Γundersetn`) = `(L n)/(Mn)`

For the first mode (n)=3

`Γunderset3`= `(L3)/(M3)`

`Γunderset3`= `(0.683)/(3.860)`

`Γunderset3`= 0.177`

(iv) For the fourth mode:

Where n = 4.

∴Modal Participation Factor (`Γundersetn`) = `(L n)/(Mn)`

For the first mode (n)=4

`Γunderset4`= `(L4)/(M4)`

`Γunderset4`= `(-0.381)/(3.860)`

`Γunderset4`= -0.098`

(iv) For the fifth mode:

Where n = 5.

∴Modal Participation Factor (`Γundersetn`) = `(L n)/(Mn)`

For the first mode (n)=5

`Γunderset5`= `(L5)/(M5)`

`Γunderset5`= `(-0.175)/(3.860)`

`Γunderset5`= 0.045`

(d) Effective Modal Mass (`M^**undersetn`):

(i) For the first mode:

Where n = 1

∴Effective Modal Mass (`M^**undersetn`) = `(L n^2)/(Mn)`

`M^**underset1` = `(L 1^2)/(M1)`

`M^**underset1` = `((4.121)^2)/(3.860)`

`M^**underset1`= 4.398m.

(ii) For the second mode:

Where n =2

∴Effective Modal Mass (`M^**undersetn`) = `(L n^2)/(Mn)`

`M^**underset2` = `(L2^2)/(M2)`

`M^**underset2` = `((-1.297)^2)/(3.860)`

`M^**underset2`= 0.435m.

(iii) For the third mode:

Where n =3

∴Effective Modal Mass (`M^**undersetn`) = `(L n^2)/(Mn)`

`M^**underset3` = `(L3^2)/(M3)`

`M^**underset3` = `((0.683)^2)/(3.860)`

`M^**underset3`= 0.121m.

(iv) For the fourth mode:

Where n =4

∴Effective Modal Mass (`M^**undersetn`) = `(L n^2)/(Mn)`

`M^**underset4` = `(L4^2)/(M4)`

`M^**underset4` = `((-0.381)^2)/(3.860)`

`M^**underset4`= 0.038m.

(v) For the fifth mode:

Where n =5

∴Effective Modal Mass (`M^**undersetn`) = `(L n^2)/(Mn)`

`M^**underset5` = `(L5^2)/(M5)`

`M^**underset5` = `((0.175)^2)/(3.860)`

`M^**underset5`= 0.008m.

(e) Modal Mass Participation Ratio:

(i) For the first mode:

Where n = 1;

Modal Mass Participation Ratio = `(Mn^**)/(mj)`

Whereas mj = Total mass of the building.

mj = m+m+m+m+m

mj = 5m.

Modal Mass Participation Ratio = `(Mn^**)/(mj)`

Modal Mass Participation Ratio(1) = `(4.398)/(5)`

Modal Mass Participation Ratio(1) = 0.879.

(ii) For the second mode:

Where n = 2;

Modal Mass Participation Ratio = `(Mn^**)/(mj)`

Whereas mj = Total mass of the building.

mj = m+m+m+m+m

mj = 5m.

Modal Mass Participation Ratio = `(Mn^**)/(mj)`

Modal Mass Participation Ratio(2) = `(0.435)/(5)`

Modal Mass Participation Ratio(2) = 0.087.

(iii) For the third mode:

Where n = 3;

Modal Mass Participation Ratio = `(Mn^**)/(mj)`

Whereas mj = Total mass of the building.

mj = m+m+m+m+m

mj = 5m.

Modal Mass Participation Ratio = `(Mn^**)/(mj)`

Modal Mass Participation Ratio(3) = `(0.121)/(5)`

Modal Mass Participation Ratio(3) = 0.024.

(iv) For the fourth mode:

Where n = 4;

Modal Mass Participation Ratio = `(Mn^**)/(mj)`

Whereas mj = Total mass of the building.

mj = m+m+m+m+m

mj = 5m.

Modal Mass Participation Ratio = `(Mn^**)/(mj)``

Modal Mass Participation Ratio(4) = `(0.038)/(5)`

Modal Mass Participation Ratio(4) = 0.024.

(v) For the fifth mode:

Where n = 5;

Modal Mass Participation Ratio = `(Mn^**)/(mj)`

Whereas mj = Total mass of the building.

mj = m+m+m+m+m

mj = 5m.

Modal Mass Participation Ratio = `(Mn^**)/(mj)`

Modal Mass Participation Ratio(5) = `(0.008)/(5)`

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(`VBundersetn`):

Base Shear(`VBundersetn`) = `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:

A₁ = 0.27g

For T2:

A₂ = 0.76g

So for T3:

A₃ = 1.03g

For T4:

A₄ = 1.03g

For T5:

A₅ = 1.03g

(i) For the first mode:

Where n = 1;

Base shear (`Vbundersetn`) = `Mn^** xx An`

`VBunderset1` = `M1^** xx A1`

`VBunderset1` = `4.398 xx 0.27`

`VBunderset1` = 1.187mg.

(ii) For the second mode:

Where n =2;

Base shear (`Vbundersetn`) = `Mn^** xx An`

`VBunderset2` = `M2^** xx A2`

`VBunderset2` = `0.435 xx 0.76`

`VBunderset2` = 0.330mg`

(iii) For the third mode:

Where n =3;

Base shear (`Vbundersetn`) = `Mn^** xx An`

`VBunderset3` = `M3^** xx A3`

`VBunderset3` = `0.121 xx 1.03`

`VBunderset3` = 0.124mg`

(iv) For the fourth mode:

Where n =4;

Base shear (`Vbundersetn`) = `Mn^** xx An`

`VBunderset4` = `M4^** xx A4`

`VBunderset4` = `0.038 xx 1.03`

`VBunderset3` = 0.039mg`

(v) For the fifth mode:

Where n =5;

Base shear (`Vbundersetn`) = `Mn^** xx An`

`VBunderset5` = `M5^** xx A5`

`VBunderset5` = `0.008 xx 1.03`

`VBunderset5` = 0.008mg`

(f) Average Base Shear (`VundersetB`):

We can calculate the average base shear by the method of square root of sum of square.

∴ Average Base Shear (`VundersetB`) = `sqrt (V B 1^ 2 + V B 2^ 2 + V B 3^ 2 + V B 4 ^2 + V B 5 ^2`

=`sqrt (1.187^ 2 + 0.330^ 2 + 0.124^ 2 +0.039 ^2 + 0.008 ^2`

Average Base Shear (`VundersetB`) = 1.23mg.