Understanding the concepts on Degrees of Freedom

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.

aim afive storey structure with 5 translational dof is shown herre. each storey has massm the eigenvalue problem hass been solved nd the 5 periods of vibration Tn and their corresponding mode shapes sigh 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 shap

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

answer

effective model space(M+) =Ln2/Mn

for the firstnode M*1=(L1)2/M1

M1* =(4.121m)2/3.861m

m1*=4.398m

similarly m2=0.436

m3* =0.121

m4* =0.038m

m5* =0.008m

modal mass participitation ratio mn*/mj

where mj= total mass of the building=m+m+m+m+m=5m

similarly second mode n=2=0.09

n=3 =0.02

n=4 =0.01

n=5= 0.002

adding first two mass participitation ratio wee get 0.88+0.09=0.97

to achieve minimum 90% of modal mass participitation ratio first two are enough

base shear vb=Mb*.An

for the first node vb1= 4.398mx0.27g

for n=1

vb1=1.187mg

similarly

vb2=0.327mg

vb3=0.126mg

vb4 =0.039mg

vb5 =0.008mg

average base shear is calculated by the method

finally obtain the base shear vb=1.15mg