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

ANSWER 1

The element Ki1 in first column, we determine stiffness matrix by imposing

u1=1, with u2=u3=0

The element Ki2 in first column, we determine stiffness matrix by imposing

u2=1, with u1=u3=0

The element Ki3 in first column, we determine stiffness matrix by imposing

u3=1, with u1=u2=0

QUESTION 2(a)

For the following structures:

• Determine the number of degrees-of-freedom for dynamic analysis
• Establish the equation of motion
• Calculate their natural frequencies

ANSWER 2(a)

QUESTION 2(b)

For the following structures:

• Determine the number of degrees-of-freedom for dynamic analysis
• Establish the equation of motion
• Calculate their natural frequencies

ANSWER 2(b)

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.

ANSWER 3

Two beams contribute to overall stiffness of the system in a parallel manner

This implies that the system stiffness can be found from the direct addition of the stiffness of the individual beams

QUESTION 4

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

ANSWER 4

1. An expression for the effective stiffness of figure(a)

• K=3EI/L3

2. An expression for the effective stiffness of figure(b)

• K=(3EI/L3) + (3EI/L3) = 6EI/L3

3. An expression for the effective stiffness of figure(c)

• K=(12EI/L3) + (12EI/L3) = (24EI/L3)