Calculation of Stiffness in Structural elements
Q1. 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.
The system has the three DOFs shown in Figure.
To obtain the first column of the 3x3 stiff ness matrix.
we impose unit displacement in DOF u1,with u2 = u3 = 0.
These are determined using the stiffness coefficients for a uniform flexural element.
Motion Equation is this case.
The elements ki2 in the second column of the stiffness matrix are determined by imposing u2 = 1with u1= u3 = 0
Motion equation in this case
Similar y , the elements ki3 in the third column of the stiffness matrix can be determined by imposing displacements u3 = 1with u1= u2 =0.
Thus the 3x3 stiffness matrix of the structure is known and the equilibrium equations can be written. For a frame with lb = lc subjected to lateral force fS, they are
The joint rotations can be expressed in terms of lateral displacement as follows:
On solving we get
Substituting the above equation into the first of three equations gives
Q2. For the following structures:
- Determine the number of degrees-of-freedom for dynamic analysis
- Establish the equation of motion
- Calculate their natural frequencies
Ans.
No of Degree of freedom
Only 1 DOF
Equation of motion:
The standard equation of motion is given by m+ku=O Stiffness (k) of the frame is given by,
3EI/h
By substituting the above value of stiffness in equation of motion we get the equation as follows:
Natural frequency:
Stiffness (K) value is substituted in the equation
Rearranging the equation we get the equation as follows now
we get
we have
(B)
No of Degree of freedom
Only 1 DOF
Equation of motion:
The standard equation of motion is given by m+ku=O Stiffness (k) of the columns is given by,
Natural frequency:
Stiffness (K) value is substituted in the equation
Q3. 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.
Q4. Determine an expression for the effective stiffness of the following systems:
Ans.
(A)
(B)
(C)
