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.

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:

• Ln , Mn and Γn for each of the five modes are to be computed from Equations below.
• Also, calculate the effective modal mass Mn* and the modal mass participating ratio for the 5 modes. The mass matrix will be required to evaluate these parameters.

• Finally, obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule. The pseudo acceleration design spectrum to obtain the base shear for each mode is provided below.

• How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?

Answer

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

Modified mass matrix (M) = ФT. m. Qn

For the first mode, = (0.334 0.641 0.895 1.078 1.173]

(n=1)
M, = 3.861 m

Modal property (L) = ФnT.m. l
Where l = Influence vector (Assuming 100% utilization of seismic force) =1,1,1,11

For the first mode,

0.334 0.641 0.895 1.078 1.173
(n=1)

L4 = 4.121 m

Modal participation factor (r) = L / M
For the first mode, Г = L, / My

(n=1)
Г1 = 4.121 m / 3.861 m
Г, = 1.067
Similarly,

Г2 = -0.335
Гз = 0.177
Гд= -0.098
Г5 = 0.045

Effective Modal mass (M*) = L,? / M.
For the first mode, M,* = L,2 / M,
(n=1)
M1 * = (4.121 m)2 / 3.861 m
M1* = 4.398 m

Similarly,

M2* = 0.436 m
M3* = 0.121 m
M4* = 0.038 m
M5* = 0.008 m


Modal mass participation ratio - M,* / m1
Where, mj = Total mass of the building = m+m+m+m+m = 5m
Modal mass participation ratio of First mode (n=1) = 4.398 m / 5 m = 0.88
Similarly,

Second mode (n=2) = 0.09
Third mode (n=3) = 0.02
Fourth mode (n=4) = 0.01
Fifth mode (n=5) = 0.002


Adding First two mode mass participation ratios, we get 0.88 + 0.09 = 0.97
To achieve minimum 90% of modal mass participation ratio, First two modes are enough

Base shear (Vb) = Mn* An
For the first mode, Vb1 = 4.398 m x 0.27 g
(n=1)
Vb1 = 1.187 mg
Similarly,

VB2 = 0.327 mg
VB3 = 0.126 mg
VBa = 0.039 mg
Vss = 0.008 mg

Average Base shear is calculated by the method of Square root of sum of square
(Vb1^2) + (Vb2^2) + (Vb3^2) + (Vb4^2) + (Vb5^2)
Vb = 1.15 mg