Calculation of Stiffness in Structural elements
Challange: 2 Calculation of Stiffness in Structural elements
Question 1:
Compute lateral stiffness of the one story frame with an intermediate realistic stiffness of the beam. The system has 3 DOFs as shown. Assume L = 2h and Elb = Elc
Ans:
We use Definition of stiffness influence coefficients to solve the problem.
The system has the 3 DOF as shown in fig-a
The elements ki1 in the first column of the stiffness matrix we impose unit displacement in DOF u1, with u2 = u3 = 0, as shown in fig-b
The elements ki2 in the second column of the stiffness matrix are determined by imposing u2 = 1 with u1 = u3 = 0, as shown in fig-c
Similarly, the elements ki3 in the third column of the stiffness matrix can be determined by imposing displacements u3 = 1 with u1 = u2 = 0
Thus the 3×3 stiffness matrix of the structure is known and the equilibrium equations can be written. For a frame with Ib = Ic subjected to lateral force fS, they are
------------(a)
From the second and third equations, the joint rotations can be expressed in terms of lateral
displacement as follows
-------------(b)
Substituting Eq. (b) into the first of three equations in Eq. (a) gives
Thus the lateral stiffness of the frame is
Question 2:
For the following structures:
- Determine the number of degrees-of-freedom for dynamic analysis
- Establish the equation of motion
- Calculate their natural frequencies
a)
ANS:
The number of degrees-of-freedom for dynamic analysis for above system is 1,
The equation of motion is given by,
mu+ku=0 -------(1)
hence, we consider the stiffness of frame is given by,
k=3EI/h3
Now we have to substitute the value of k in eqn-1
Therefore,
mu+ku=0
mu+(3EI/h3)(u)=0
mu+(3EI/h3)(1)=0,
Now Natural frequency (fn)
--------(1)
Now,
--------(2)
Where m=mass =M
now substitute the value of k in (2)
--------(3)
Now,
--------(4)
Substitute the above vale in equation (4),
------------(5)
Substitute above value in equation (1),
b)
ANS:
The number of degrees-of-freedom for dynamic analysis for above system is 1,
The equation of motion is given by,
mu+ku=0 -------(1)
hence, we consider the stiffness of frame is given by,
Stiffness of 1st column = K1= 12EI/h3
Stiffness of 2nd column = K2= 12EI/h3
Stiffness of 3rd column = K3= 3EI/h3
K= 12EI/h3 + 12EI/h3 + 3EI/h3
= 27EI/h3
Now we have to substitute the value of k in eqn-1
Therefore,
mu+(27EI/h3)(u)=0
Now Natural frequency (fn)
--------(1)
Now,
--------(2)
Where m=mass =M
now substitute the value of k in (2)
--------(3)
Now,
--------(4)
Substitute the above vale in equation (4),
------------(5)
Substitute above value in equation (1),
Question 3:
Consider the propped cantilever shown in the figure below. The beams are made from the same steel section and have lengths as shown on the diagram. Determine the natural period of this system if a large mass, M, is placed at the intersection of the beams at point A. The weight of the beams in comparison with the mass M is very small.
ANS:
The natural period of this system if a large mass, M, is placed at the intersection of the beams at point A.
w=ku, There fore k=w/u
Now, we know that w= Mg therefore , M=w/g
and the natural period,
Question 4
Determine an expression for the effective stiffness of the following systems:
ANS:
