Calculation of Stiffness in Structural elements
Lateral stiffness of the one-story frame:
This structure can be analyzed by any of the standard methods, including moment distribution. Here we use the definition of stiffness influence coefficients to solve the problem.
The system has three Degree of freedom. To obtain the first column of the 3×3 stiffness matrix, we impose unit displacement in DOF ul, with u2 = u3 = 0.
From the second and third equations, the joint rotations can be expressed in terms of lateral displacement as follows:
Substituting Eq. (2) into the first of three equations in Eq. (1) gives
Therefore the lateral equation of frame is
2)
For the following structures:
Determine the number of degrees of-freedom for dynamic analysis
Establish the equation of motion
Calculate their natural frequencies
Answer:
2a)
Degree of freedom = 1
Equation of motion = mu + Ku = 0
Stiffness of this frame, k = 3El/h^3
Now, Substitute the K value in the equation of motion.
Therefore, Equation of Motion = mu + 3El/h^3 = 0
For Natural frequency, wn = k/m
In this frame m = Mass (M)
Substitute the value of m and K in wn formula
Therefore, Natural frequency of wn = 3EI/h^3/M
2b)
Degree of freedom = 1
Equation of motion = mu + Ku = 0
Stiffness of this frame, k = (12El/h^3) + (12EI/h^3)+(3EI/h^3)
= 27EI/h^3
Substitute the value of k in natural frequency formula
Here f or e, Naturalequencyofwn= root
(K/M)
√ (27EI/h^3/M)
3a) Natural Period of the System:
In this question two beams connected with one another. One is Cantilever beam and other one is Simply supported beam.
So,
For lateral stiffness we need to add both beam stiffness
We know that,
4) Effective Stiffness:
a)
b)
An effective stiffness for the Simply supported beam is K = 3EI/L^3
- c) An effective stiffness for the Simply supported beam is
K = (3EI/L^3) + (3EI/L^3)
= 6 (EI/L^3)
