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 El_b = El_c

ANS

• u1=1
• stiffness coefficient of column translation = 2x12EI/h^3
• Stiffness coefficient of beaam rotation = EI/h^2
  

• u2=1
• stiffness coefficient of column rotation at right end = 6EI/h^2
• stiffness coefficient of column + beam rotation at left end = 6EI/h

• u3=1
• stiffness coefficient of column rotation at left end = 6EI/h^2
• stiffness coefficient of column + beam rotation at right end = 6EI/h

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

ANS 2 (A)

• The system has SINGLE DEGREE OF FREEDOM structure.
• So stiffness coefficient is k = 3EI/h^3
• The equation of motion shall be Mu+ku=0

ANS 2 (B)

• The system has SINGLE DEGREE OF FREEDOM structure.
• Two column with both end fixed & one column has pinned-fixed ends.
• So stiffness coefficient will be as follow.

k1=12EI/h^3
k2=12EI/h^3
k3=3EI/h^3

• For parallel stiffness elements
  ke = (k1+k2+k3) = 27EI/h^3

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

• Both Elements have same material and section. it means they have same modulus of elasticity.
• When mass is applied at the point, both elements will respond simultaneously.

Question 4

Determine an expression for the effective stiffness of the following systems:

ANS 4 (A)

ANS 4 (B)

ANS 4 (C)

• The stiffness coefficient of the pinned beam is k1 = 48EI/L^3
• The effective stiffness coefficient for the system has to be formed by adding the relative axial stiffness provided.