Understanding the concepts on Degrees of Freedom

1)

• we have the 5 story building,

• also, we have the 5 period of vibrations and their mode shapes which are listed below

1. first fundamental period of vibration is T1= 2 seconds and mode shape is ϕ1 =
2. second period of vibration is T2= 0.6852 seconds and mode shape is ϕ2=
3. third period of vibration is T3= 0.4346 seconds and mode shape is ϕ3=
4. fourth period of vibration is T4= 0.3383 seconds and mode shape is ϕ4=
5. fifth period of vibration is T5=0.2966 seconds and mode shape is ϕ5=

modified mass matrix:

mass matrix (m) =

modified mass matrix (M) =

for,

• M1=

the first mode (n=1) =

M1=3.861m = M2= M3= M4=M5

modal property(L):

modal property (L) =

where =influence vector (assuming 100% utilization of seismic force)=

for,

the first node (n=1)=

L1= 4.121m

the second node(n=2)=

L2=-1.297

the third node(n=3)=

L3=0.683

the fourth node(n=4)=

L4=-0.381

the fifth node(n=5)=

L5=0.175

modal participation factor( ):

modal participation factor ()= Ln/Mn

for the first node (n=1), =L1/M1= 4.121m/3.861m=1.067

for the second mode(n=2),=L2/M2=-1.297m/3.861m= -0.335

for the third mode(n=3),= L3/M3= 0.683m/3.861m=0.177

for the fourth mode(n=4),=L4/M4= -0.381m/3.861m= -0.098

for the fifth mode(n=5),=L5/M5=0.175m/3.861m=0.045

effective modal mass (Mn*):

effective modal mass (Mn*)= Ln^2/Mn

for the first mode,M1*=L1^2/M1=4.121m^2/3.861m=4.398m

for the second mode,M2*=L2^2/M2=(-1.297m)^2/3.861m=0.436m

for the third mode, M3*=L3^2/M3=0.683m^2/3.861=0.121m

for the fourth mode,M4*=L4^2/M4=-0.381m^2/3.861m=0.038m

for the fifth mode,M5*=L5^2/M5= 0.175m^2/3.861m=0.008m

modal mass participation ratio=Mn*/mj:

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

modal mass participation ratio of first mode(n=1)=M1*/mj= 4.398m/5m= 0.88

for second mode(n=2)= M2*/mj= 0.436m/5m= 0.09

for third mode(n=3)= M3*/mj= 0.121m/5m= 0.02

for fourth mode(n=4)= M4*/mj= 0.038m/5m= 0.01

for fifth mode(n=5)= M5*/mj= 0.008m/5m= 0.002

adding first two mode mass participation ratio, 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):

where, pseudo-spectral acceleration is obtained from graph between period, T(seconds) Vs Pseudo-spectral acceleration

for the first mode, Vb1= M1*.A1= 4.398m*0.27g= 1.187 mg

for the second mode, Vb2=M2*.A2=0.436m*0.75g=0.327 mg

for the third mode,Vb3=M3*.A3=0.121m*1.04g= 0.126 mg

for the fourth mode,Vb4=M4*.A4=0.038m*1g=0.038 mg

for the fifth mode, Vb5=M5*.A5=0.008m*1g =0.008 mg

average base shear is calculated by the method of square root of sum of square

'squaroot' of (Vb1^2)+(Vb2^2)+(Vb3^2)+(Vb4^2)+(Vb5^2)

='squaroot' of (1.187mg^2)+(0.327mg^2)+(0.126mg^2)+(0.038mg^2)+(0.008mg^2)

=1.15 mg

average base shear (Vb)=1.15 mg