Understanding the concepts on Degrees of Freedom

AIM: To calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes.

PROCEDURE:

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.