Calculation of Stiffness in Structural elements
Q.1) Answer -
ki1 in the first column,we determine the stiffness matrix by imposing u1=1 , u2=u3=0
Ki2 in the second column,we determine the stiffness matrix by imposing u2=1,u1=u3=0
Ki3 in the third column,we determine the stiffness matrix by imposing u3=1,u1=u2=0
EIc/h^3 [24 6h 6h ] {u1} {fs}
[6h 6h^2 h^2 ] {u2} = {0}
[ 6h 6h^2 6h^2 ] {u3} {0}
From the second and third equation,the joint rotation can be expressed in terms of lateral displacement
{u2} = - [6h^2 h^2 ]^-1 [6h]u1 = -6/7h [1]u1
{u3) [h^2 6h^2] [6h] [1]
Substitute the equation,
fs = (24EIc/h^3 - EIc/h^3 * 6/7h (6h 6h)[1] u1 = 96EIc/7h^3 *u1
[1]
Lateral stiffness of frame = K = 96EIc/7h^3
Q.2) Answer -
a)
The number of degree of freedom for dynamic analysis is 1
The equation of motion is -
mu + ku = 0
mu = mass and ku = stiffness of frame
Stiffness of frame = K = 3EI/h^3
Now,put the value of k in equation,
mu + ku = 0
mu + 3EI/h^3 (1) = 0
natural frequency (fn) = 1/Tn
now,Wn = √k/M
m= mass
Wn = √3EI/h^3 / M
now, Th = 2π /Wn
Th = 2π / √3EI/h^3 / M
Fn = 1 / Tn
= 1 / 2π/√3EI/h^3 / M
= √3EI / 2π√Mh^3
b)
The number of degree of freedom for dynamic analysis is 1
The equation of motion = mu + ku =0
Stiffness of first column = K1 = 12EI/h^3
Stiffness of second column = K2 = 12EI/h^3
Stiffness of third column = K3 = 3EI/h^3
total stiffnss K =K1+K2+K3
= 12EI/h^3 + 12EI/h^3 + 3EI/h^3
= 27 EI/h^3
Now,put the value of k in equation,
mu + 27EI/h^3 = 0
Natural frequency (fn) = 1/Tn
Now, Wn= √k/M
= √27EI/h^3 / M
Now, Tn = 2π / Wn
= 2π / √27EI/h^3 /M
= 2π√Mh^3 / √27EI
put the value of Tn in natural frequency equation ,
Fn = 1 / 2π√Mh^3/√27EI
= √27EI / 2π√Mh^3
Q.3)Answer -
Lateral stiffness of cantilever beam (Kc)= 3EI/Lc^3
Lateral stiffness of simply supported beam (Kb) = 48EI/Lb^3
now,total stiffness = K = Kc + Kb
K = 3EI/Lc^3 + 48EI/Lb^3
Taking 3EI common,
K = 3EI ( 1/Lc^3 + 16/Lb^3)
Now,Natural frequency (fn) = 1/Tn
Wn = √k/M
Substitute the value of k in equation,
Wn = √3EI((1/Lc^3 + 16/Lb^3)) / √M
Now,Tn = 2π / Wn
Tn = 2π x √M / √3EI((1/Lc^3 + 16/Lb^3))
Q.4) Answer -
A) An expression for the effective stiffness of fig (a)
K = K1+k2
k1 = stiffness of beam
k2 = stiffness of spring
K = 3EI/L^3 + K2
B) An expression for the effective stiffness of fig (b)
K = K1+K2+K3
K3 = stiffness of spring
K = 3EI/L^3 + 3EI/L^3
= 6EI/L^3 + K3
C) An expression for the effective stiffness of fig (c)
K = K1 + K2
K = 12EI/L^3 + 12EI/L^3
= 24EI/L^3