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

Answer:

• Unit displacement u1= 1
• Stiffness co efficient of column translation(2 similar)

2X12EI/h³

• Stiffness co efficient of beam rotation(2 ends)

EI/h²

• Unit displacement u2= 1
• Stiffness co efficient of column rotation at right end

6EI/h²

• Stiffness co efficient of column + beam rotation at left end

4EI/h +4EI/L

4EI/h + 4EI/2h = 6EI/h

• Unit displacement u3= 1
• Stiffness co efficient of column rotation at left end

6EI/h²

• Stiffness co efficient of column + beam rotation at right end

4EI/h +4EI/L

4EI/h + 4EI/2h = 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)

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.

Consider the propped cantilever . 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.

so,

The lateral stiffness we need to add the both stiffness of beam

therefore we know,

For simple supported beam lateral stiffness is=48EI/L^3

For cantilever supported beam lateral stiffness is=3EI/L^3

Add both beam stiffness,

K=(48EI/L^3+3EI/L^3)

K=51(EI/L^3)

The natural frequency, wn=root(k/m)

wn =root(51(EI/L^3)/m

Question 4

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

Expression for the effective stiffness :

• The effective stiffness of cantilever beam is K=48EI/L^3
• The effective stiffness of simply supported beam is K=3EI/L^3
• The effective stiffness of 2 simply supported beam is K=3EI/L^3+3EI/L^3

=6EI/L^3