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 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

SOLUTION:

By using Moment distribution method:

Given the Degree of freedom is '3'

Now,

• Lets consider: Ki1 in first column
  • to determine the stifness by imposing u1=1 and u2 = u3 = 0
• consider Ki2 in second column
  • to determine the stifness by imposing u2=1 and u1 = u3 = 0
    
    
• consider Ki3 in third column
  • to determine the stifness by imposing u3=1 and u1 = u2 = 0

therefore:

From the 2nd and 3rd equations, the jointrotations can be expressed interms of lateral displacements as follows:

Now By susstituting the above equation in the first equation, as shown below:

• `fs = ( {24EIc}/h^3 - {EIc}/h^3 6/{7h} (6h. 6h) (1/1)) u1 = 96/7 {EIc]/h^3 u1`

finally lateral stiffness of frame is `{96/7} (EIc}/h^3`

• K = `{96/7} (EIc}/h^3`

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

AIM: To Determine the following

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

SOLUTION:

a]

1] The no. of degree of freedom for dynamic analysis is 1

2] the equation of motion is,

mu + ku = 0 -- (1)

here, we consider the stiffness of frame : K = 3EI/h^3

Now, we have to substitute the K in the eq(1)

:. Equation of Motion = mu + ku =0

mu + [3EI/h^3].u = 0

mu + [3EI/h^3].(1) = 0

3] For natural frequency : (fn)= 1/ Tn

now wn = root {k/m}

here, m = mass(M)

by substituting the value of m and k in Wn

thus, the Natural of Frequency of

Wn = root .(3EI/h^3/m)

b]

1] The no. of degree of freedom for dynamic analysis is 1

2] the equation of motion is,

mu + ku = 0 ---(1)

we need to find the stiffness of three column

• stiffness of 1st column: k1 = 12EI/h^3
• stiffness of 2nd column: k2 = 12EI/h^3
• stiffness of 3rd column: k3 = 3EI/h^3

K = 12EI/h^3 + 12EI/h^3 + 3EI/h^3 = 27EI/h^3

Now, we need to substitute the K in the eq(1)

:. Equation of Motion = mu + ku =0

mu + [27EI/h^3].u = 0

3] For natural frequency : (fn)= 1/ Tn --(1)

now Wn = root {k/m} --(2)

here, m = mass(M)

by substituting the value of k in eq(2)

thus, the Natural of Frequency of

Wn = root (27EI/h^3/M) --(3)

Now, Tn = `{2pi} / {Wn}`---(4)`

substitute eq (3) in eq)(4)

Tn = `{2pi}/ root[(27EI/h^3)/{EM}]` = `[2pi root .Mh^3]/[root .27EI]`

Now, sybstitute in eq (1)

fn = 1/[{2pi root. Mh^3}/root .27EI] = {root. 27EI}/{2pi root .Mh^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.

AIM: To Determine the natural period of this system if a large mass, M, is placed at the intersection of the beams at point A

SOLUTION:

A) The two beams contribute to the overall stiffness of the system in a parallel manner.

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

Lateral stiffness of cantilever beam (Kc) = `{3EI}/{Lc^3}`

Lateral stiffness of simply supported beam (Kb) = `{48EI}/{Lb^3}`

Now we need to add the lateral stiffness of both cantilever beam and simply supported beam.

• K=Kc + Kb

• K= `{3EI}/{Lc^3}`+ `{48EI}/{Lb^3}`

• K = `3EI{1/{Lc^3} + 16/{Lb^3} }`

Natural Frequency (fn) = `1/{Tn}`

• Wn =

Now substitute the k value in the above equation.

• Wn =

The natural period of the system can be found from this natural frequency using the expression, below.

• Tn = `{2pi}/{Wn}`
• Tn =

Question 4

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

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

SOLUTION:

a. For fig.(a) the expression for the effective stiffness is,

• K = `{3EI}/L^3`

b. For fig.(a) the expression for the effective stiffness is,

• K = `{3EI}/L^3` + `{3EI}/L^3`= `{6EI}/L^3`

c. For fig.(a) the expression for the effective stiffness is,

• K = `{12EI}/L^3`+ `{12EI}/L^3`= `{24EI}/L^3`