Understanding the concepts on Degrees of Freedom

Q.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 T_n and their corresponding mode shapes, φ_n 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 (Mn) = ϕnT×m×ϕnϕnT×m×ϕn

Mass Matrix (m) =

For the first mode (n) = 1

M1= [0.334 0.641 0.895 1.078 1.173]

M1= 0.111+0.410+0.801+1.162+1.375

M1= 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 (Mn) = ϕnT×m×ϕnϕnT×m×ϕn

Mass Matrix (m) =

For the second mode (n) = 2

M2 = [-0.895 -1.173 -0.641 0.334 1.078]

M2 = 0.801+1.375+0.410+0.111+1.16

M2 = 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 (Mn) = ϕnT×m×ϕnϕnT×m×ϕn

Mass Matrix (m) =

For the third mode (n) = 3

M3 = [1.173 0.334 -1.078 -0.641 0.895]

M3 = 1.375+0.111+1.162+0.410+0.801

M3 = 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 (Mn) = ϕnT×m×ϕnϕnT×m×ϕn

Mass Matrix (m) =

For the fourth mode (n) = 4

M4 = [-1.078 0.895 0.334 -1.173 0.641]

M4 = 1.162+0.801+0.111+1.375+0.410

M4 = 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 (Mn) = ϕnT×m×ϕnϕnT×m×ϕn

Mass Matrix (m) =

For the fifth mode (n) = 5

M5 = [0.641 -1.078 1.173 -0.895 0.334]

M5 = 0.410+1.162+1.375+0.801+0.111

M5 = 3.860m

(b) Modal Property (Ln):

(i) For the first mode:

Where n = 1

∴Modal Property (Ln) = ϕnT×m×ιϕnT×m×ι

where ι = Influence factor =

For the first mode (n) = 1

L1 = [0.334 0.641 0.895 1.078 1.173]

L1 = 0.334+0.641+0.895+1.078+1.173

L1 = 4.121m

(ii) For the second mode:

Where n = 2

∴Modal Property (Ln) = ϕnT×m×ιϕnT×m×ι

where ι = Influence factor =

For the second mode (n) = 2

L2 = [-0.895 -1.173 -0.641 0.334 1.078]

L2 = [0.895]+[-1.173]+[-0.641]+[0.334]+[1.078]

L2 = -1.297m

(iii) For the third mode:

Where n = 3

∴Modal Property (Ln) = ϕnT×m×ιϕnT×m×ι

where ι = Influence factor =

For the third mode (n) = 3

L3 = [1.173 0.334 -1.078 -0.641 0.895]

L3 = [1.173]+[0.334]+[-1.078]+[-0.641]+[0.895]

L3 = 0.683m

(iv) For the fourth mode:

Where n = 4

∴Modal Property (Ln) = ϕnT×m×ιϕnT×m×ι

where ι = Influence factor =

For the fourth mode (n) = 4

L4 = [-1.078 0.895 0.334 -1.173 0.641]

L4 = [-1.078]+[0.895]+[0.334]+[-1.173]+[0.641]

L4 = -0.381m.

(v) For the fifth mode:

Where n = 5

∴Modal Property (Ln) = ϕnT×m×ιϕnT×m×ι

where ι = Influence factor =

For the fifth mode (n) = 5

L5 = [0.641 -1.078 1.173 -0.895 0.334]

L5 = [0.641]+[-1.078]+[1.173]+[-0.895]+[0.334]

L5 = 0.175m

(c) Modal Participation Factor (Γn):

(i) For the first mode:

Where n = 1

∴Modal Participation Factor (Γn) = Ln/Mn

For the first mode (n) = 1

Γ1 = L1/M1

Γ1 = 4.121m/3.860m

Γ1 = 1.067

(ii) For the second mode:

Where n = 2

∴Modal Participation Factor( Γn ) = Ln/Mn

For the first mode (n) = 2

Γ2 = L2/M2

Γ2 = −1.297m/3.860m

Γ2 = -0.335

(iii) For the third mode:

Where n = 3

∴Modal Participation Factor (Γn) = Ln/Mn

For the first mode (n) = 3

Γ3 = L3/M3

Γ3 = 0.683m/3.860m

Γ3 = 0.177

(iv) For the fourth mode:

Where n = 4

∴Modal Participation Factor (Γn) = Ln/Mn

For the first mode (n) = 4

Γ4 = L4/M4

Γ4 = −0.381m/3.860m

Γ4 = -0.098

(iv) For the fifth mode:

Where n = 5

∴Modal Participation Factor(Γn) = Ln/Mn

For the first mode (n) = 5

Γ5 = L5/M5

Γ5 = −0.175m/3.860m

Γ5 = 0.045

(d) Effective Modal Mass (M*):

(i) For the first mode:

Where n = 1

∴Effective Modal Mass (M*) = Ln2/Mn

M1* = L12/M1

M1* = (4.121m)2/3.860m

M1* = 4.398m

(ii) For the second mode:

Where n = 2

∴Effective Modal Mass (M*) = Ln2/Mn

M2* = L22/M2

M2* = (−1.297m)2/3.860m

M2* = 0.435m

(iii) For the third mode:

Where n = 3

∴Effective Modal Mass (M*) = Ln2/Mn

M3* = L32/M3

M3* = (0.683m)2/3.860m

M3* = 0.121m

(iv) For the fourth mode:

Where n = 4

∴Effective Modal Mass (M*) = Ln2/Mn

M4* = L42/M4

M4* = (−0.381m)2/3.860m

M4* = 0.038m

(v) For the fifth mode:

Where n = 5

∴Effective Modal Mass (M*) = Ln2/Mn

M5* = L52/M5

M5* = (0.175m)2/3.860m

M5* = 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(n) = Mn*/mj

= M1*/mj

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

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

For the second mode (n) = 2

Modal Mass Participation Ratio (n) = Mn*/mj

= M2*/mj

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

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

For the third mode (n) = 3

Modal Mass Participation Ratio (n) = Mn*/mj

= M3*/mj

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

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

For the fourth mode (n) = 4

Modal Mass Participation Ratio (n) = Mn*/mj

= M4*/mj

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

Modal Mass Participation Ratio (4) = 0.007

(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

For the fifth mode (n) = 5

Modal Mass Participation Ratio (n) = Mn*/mj

= M5*/mj

Modal Mass Participation Ratio (5) = 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):

Base Shear (VB) = Mn*. An

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 = 0.76

For T3:

A3 = 1.03

For T4:

A4 = 1.03`

For T5:

A5 = 1.03

(i) For the first mode:

Where n = 1

Base Shear (VB) = Mn*. An

VB1 = M1*.A1

VB1 = 4.398m×0.27g

VB1 = 1.187mg

(ii) For the second mode:

Where n = 2

Base Shear (VB) = Mn*. An

VB2 = M2*.A2

VB2 = 0.435m×0.76g

VB2 = 0.330mg

(iii) For the third mode:

Where n = 3

Base Shear (VB) = Mn*. An

VB3 = M3*.A3

VB3 = 0.121m×1.03g

VB3 = 0.124mg

(iv) For the fourth mode:

Where n = 4

Base Shear (VB) = Mn*. An

VB4 = M4*.A4

VB4 = 0.038m×1.03g

VB4 = 0.039mg

(v) For the fifth mode:

Where n = 5

Base Shear (VB) = Mn*. An

VB5 = M5*.A5

VB5 = 0.008m×1.03g

VB5 = 0.008mg

(f) Average Base Shear (VB):

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

∴ Average Base Shear (VB) = √VB1²+VB2²+VB3²+VB4²+VB5²

= √1.187²+0.330²+0.124²+0.039²+0.008²

Average Base Shear (VB) = 1.23mg