Derive an equation which could be used to calculate the bending moment

[oxon88]

Homework Statement

The calculation for the maximum bending moment is to be verified
experimentally using a strain gauge bonded to the outer surface of the
beam, at the point where the maximum bending moment occurs.
Derive an equation which could be used to calculate the bending
moment from the measured strain value. State the meaning of all
symbols used in your equation.

Homework Equations

M / I = σ / y = E / R


Stress = Force / Area = F/A

strain = Change in Length / Original Length

1 / R = M / EI

can anyone provide some guidance?

 

[PhanthomJay]

Homework Statement

View attachment 64121

The calculation for the maximum bending moment is to be verified
experimentally using a strain gauge bonded to the outer surface of the
beam, at the point where the maximum bending moment occurs.
Derive an equation which could be used to calculate the bending
moment from the measured strain value. State the meaning of all
symbols used in your equation.

Homework Equations

M / I = σ / y = E / R


Stress = Force / Area = F/A

strain = Change in Length / Original Length

1 / R = M / EI

can anyone provide some guidance?

You are missing in your relevant equations the relationship between stress and strain. Hint: check out the stress strain graph for an ideal elastic material.

 

[AlephZero]

You are missing in your relevant equations the relationship between stress and strain.

You already have the stress-strain relation in your equation 1 / R = M / EI.

You need an equation to get the strain from the radius of curvature.

You also need to find where the maximum bending moment occurs, so draw a shear force and bending moment diagram.

 

[PhanthomJay]

The problem states that the maximum bending moment and its location along the beam has already been determined, by calculation, and you wish to verify that value experimentally by applying a strain gauge on the outer surface at that point determined by calculation. The strain gauge records the strain. Now the question is knowing that strain, what is the bending moment at that location? You can use the stress-strain relation or the curvature-strain relationship, your choice.

 

[oxon88]

Thanks for the replies. I have calculated the bending moment in a previous question.

The maximum bending moment is 138.4 kN-m, which occurs at 5.6m from the left end.

 

[oxon88]

You already have the stress-strain relation in your equation 1 / R = M / EI.

You need an equation to get the strain from the radius of curvature.

You also need to find where the maximum bending moment occurs, so draw a shear force and bending moment diagram.

can you provide any help with the equation to get the strain from the radius of curvature?

 

[oxon88]

anyone?

would the equation be M = (σ * I) / y

 

[PhanthomJay]

can you provide any help with the equation to get the strain from the radius of curvature?

would the equation be M = (σ * I) / y

no, you already have that equation that relates moment to I and stress at a distance y from the neutral axis. You also have correctly written the equation that relates moment to E and I and the radius of curvature, R. But your strain gauge is recording strain , at the outer fibers of the beam. So you must now consider the equation which relates stress to strain, which is stress =(strain)(E), to then solve for stress and then moment, OR, the equation that relates radius of curvature to strain, which you should look up because it is not often memorized or calculated, to solve for curvature and then moment. Note that the value of y must be at where strain is recorded at the outer fibers.

 

[oxon88]

ok i see.

ε_max = Z * M / EI

M = bending moment
Z = distance from the neutral layer to the outer tensile layer
K = curvature of the beam
E = Young's Modulus
I = second moment of area

 

[PhanthomJay]

That equation is correct. But I thought you wanted to find another one using the radius of curvature, K?

 

[oxon88]

could i use ε_max = k*z

 

[PhanthomJay]

You didn't show how you arrived at your equation for strain equals ZM/EI, which is correct . Did you use strain = stress/E?
If instead you used strain = Z/K, you would get the same result. Incidentally, the max distance to the neutral axis is usually denoted by 'c', not 'Z'.

 

[oxon88]

is it not, strain = Z*K ?

 

[PhanthomJay]

is it not, strain = Z*K ?

Why no, strain is a dimensionless quantity, it has no units. If strain = Z*K, the units would be in (length)^2, since both Z and K have length units.