Understanding the concepts on Degrees of Freedom

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.

The fundamental period of vibration (T₁ = 2 seconds) and its associated mode shape:

The second mode of vibration (T₂ = 0.6852 seconds) and its associated mode shape:

The third mode of vibration (T₃ = 0.4346 seconds) and its associated mode shape:

The fourth mode of vibration (T₄ = 0.3383 seconds) and its associated mode shape:

The fifth mode of vibration (T₄ = 0.2966 seconds) and its associated mode shape:

Solution:

A five storey building building with their respective mode shapes and time period of vibration has been given and their repective base shear needs to be calculated.

• L_n , M_n and Γ_n for each of the five modes are to be computed.
• Also, calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes. The mass matrix will be required to evaluate these parameters.

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]]`

Modal mass matrix (M) = `Phi`_n^T * m * `Phi`_n

For the first mode, Mode 1 (n=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]]``[[0.334],[0.641],[0.895],[1.078],[1.173]]`

By doing matrix multiplication

= [0.112m+0.410m+0.801m+1.162m+1.376m]

M1=3.861m

Similarly for other modes the modal mass matrix is calculated as '3.861m' as the mode shapes for all the storeys are similar to each other and as per the manual claculation the values are more or less close to 3.861m. So,

M2=3.861m, M3=3.861m, M4=3.861m, M5= 3.861m.

Modal Property Calculation:

Ln = `Phi`_n^T * m * `iota`

where `iota`= influence vector and assuming 100% uitilization of seismic force it is `[[1],[1],[1],[1],[1]]`

Mode 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]]`

By doing matrix multiplication

=[0.334m+0.641m+0.895m+1.078m+1.173m]

L1=4.121M

Mode 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]]`

By doing matrix multiplication

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

L2=-1.297M

Mode 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]]`

By doing matrix multiplication

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

L3=0.683M

Mode 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]]`

By doing matrix multiplication

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

L4=-0.381M

Mode 5:

=[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]]`

By doing matrix multiplication

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

L5= 0.175M

Modal participation factor calculation:

Γn = Ln/Mn

Mode 1:

=4.121/3.861

Γ1=1.067

Mode 2:

=-1.297/3.861

Γ2=-0.335

Mode 3:

=0.683/3.861

Γ3=0.176

Mode 4:

=-0.381/3.861

Γ4=-0.09

Mode 5:

=0.175/3.861

Γ5=0.04

Effective Modal mass (Mn*) calculation:

Mn*=(Ln)2/Mn

Mode 1:

=(4.121)2/3.861

M1*=4.39M

Mode 2:

=(-1.297)2/3.861

M2*=0.43M

Mode 3:

=(0.683)2/3.861

M3*=0.12M

Mode 4:

=(-0.381)2/3.861

M4*=0.03M

Mode 5:

=(0.175)2/3.861

M5*=0.007M

Modal mass participating ratio calculation:

=Mn*/Mj

where Mj=Total mass=M+M+M+M+M=5M

Mode 1:

=4.39M/5M=0.88=88%

Mode 2:

=0.43M/5M=0.086=8.6%

Mode 3:

=0.12M/5M=0.024=2.4%

Mode 4:

=0.03M/5M=0.006=0.6%

Mode 5:

=0.007M/5M=0.001=0.1%

Base shear (Vb) calculation:

Vb=Mn* x An

where An is the pseudo-acceleration which is obtained using the time acceleration graph and obtained using the time given

T1 = 2 seconds, A1= 0.27g

T2 = 0.6852 seconds,A2= 0.76g

T3= 0.4346 seconds,A3= 1.03g

T4 = 0.3383 seconds,A4= 1.03g

T5= 0.2966 seconds,A5= 0.76g

Mode 1:

=4.39m x 0.27g

V1=1.18 mg

Mode 2:

=0.43 x 0.76

V2=0.32 mg

Mode 3:

=0.12 x 1.03

V3= 0.12 mg

Mode 4:

=0.03 x 1.03

V4=0.03 mg

Mode 5:

=0.007 x 0.76

V5=0.005 mg

Total base shear calculation using SRSS rule

V= √(V1)2+(V2)2+(V3)2+(V4)2+(V5)2

= √(1.18)2+(0.32)2+(0.12)2+(0.03)2+(0.005)2

Total shear = 1.29 mg

Results:

• L_n , M_n and Γ_n for each of the five modes are computed.
• Effective modal mass Mn* is also computed for all the five modes.
• The modal mass participating ratio is also computed for all the modes and to achieve 90% of modal mass participating ratio the first two modes are enough.
• Base shear for each modes and total base shear using SRSS rule are computed.