Understanding the concepts on Degrees of Freedom

AIM:- to calculate the base shear force and combined base shear for the given modes and to find the number of modes required to achieve 90% of the model mass participating ratio

GIVEN DATA:-

• the structure has 5 degrees of freedom
• 5 mode shape corresponding to the period of vibrations
• pseudo-spectral acceleration to calculate the base shear

CALCULATION OF M, L, M* and BASE SHEAR

• AT MODE SHAPE 1

=4.121/3.86

=1.067

=(4.121)^2 /3.86

=4.39m

• model mass participation ratio

M*/m

=4.399m / 5m

=0.879

• base shear at mode

=4.39 x 0.27g (at fundamental period of vibration T=2 sec psuedo spectralacceleration is 0.27g)

=1.1853

• AT MODE SHAPE 2

=(-1.297) / 3.863

=-0.335

=(-1.297)^2 / 3.863

=0.435m

• model mass participation ratio

M*/m

=0.435m / 5m

=0.09

=0.435 X 0.78g (at fundamental period of vibration T=0.6852sec psuedo spectralacceleration is 0.78g)

=0.339g

• AT MODE SHAPE 3

=0.683/ 3.861

=0.177

=(0.683)^2/ 3.861

=0.1208m

• model mass participation ratio

M*/m

=0.1208m / 5m

=0.02

=0.1208 X 1.03g (at fundamental period of vibration T=0.4346sec psuedo spectralacceleration is1.03g)

=0.1244g

• AT MODE SHAPE 4

=(-0.381)/ 3.861

=(-0.098)

=(0.381)^2/ 3.861

=0.0375m

• model mass participation ratio

M*/m

=0.038m / 5m

=0.0076

=0.0375 X 1.03g (at fundamental period of vibration T=0.3383sec psuedo spectralacceleration is 1.03g)

=0.0386g

• AT MODE SHAPE 5

=0.175/ 3.861

=0.045

=(0.175)^2/ 3.861

=0.008m

• model mass participation ratio

M*/m

=0.008m / 5m

=0.0016

=0.008 X 1.03g (at fundamental period of vibration T=0.3383sec psuedo spectralacceleration is 1.03g)

=0.00824g

RESULTS:-

1. base shear at first mode shape is 1.185g
2. base shear at second mode shape is 0.339g
3. base shear at third mode shape is 0.124g
4. base shear at fourth mode shape is 0.0386g
5. base shear at fifth mode shape is 0.008g

• COMBINED BASE SHEAR AS PER SRSS RULE IS 1.239g

1. modal mass participation ratio of the first mode is 0.88
2. modal mass participation ratio of the first mode is 0.09
3. modal mass participation ratio of the first mode is 0.02
4. modal mass participation ratio of the first mode is 0.075
5. modal mass participation ratio of the first mode is 0.0016

• The first two modes are required to achieve 90% of the modal mass participation ratio