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

_{AIM: To compute lateral stiffness of one storey building with the given parameters}

_PROCEDURE:

Step 1: The system has 3 DOF's, so it makes three equations one for each DOF

Kij =Force/ moment required/ developed at 'i'th coordinate due to unit displacement (Deflection/ rotation) at 'j'th coordinate

Step 2: The stiffness matrix of the above problem can be written as

Step 3: To get the stiffness matrix values

Equation 1:

Consider unit force at U1 = 1 and U2 = U3 = 0, we get

Equation 2:

Consider unit force at U2 = 1 and U1 = U3 = 0, we get

Equation 3:

Consider unit force at U3 = 1 and U1 = U2 = 0, we get

Step 4:

• From the question EIc = EIb and L = 2h, so the stiffness matrix from the above equations

• From the second and third equations, the joint rotations can be expressed in terms of lateral displacement as follows:

• Substituting this in the above equation and on solving we get,

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)

Answer:

• The degree of freedom is 1
• The equation of motion => mu + fd + fs = p(t)
• mu + ku = p(t)

• Natural frequancy

B) ANSWER:

• The degree of freedom is 1
• As the beam is rigid (i.e., flexural rigidity M.I = infinity)
• 

• The equation of motion => mu + fd + fs = p(t)

  • mu + ku = p(t)
• 

• Natural frequency

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:

• In the above diagram, there is two beams are connected to each other
• One is a cantilever beam and the other is simply a supported beam.

• To calculate the lateral stiffness of cantilever beam:

• To calculate the lateral stiffness of simply supported beam:

Total stiffness => k = Kb + Kc

• Natural frequency,

• Thus,

• The time period is given as,
• 

Question 4

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

AIM: To determine the expression for the effective stiffness for the given structures

System (a):

The effective stiffness of the system with components in series

System (b)

As the beam is rigid i.e., EI = infinity

System (c):