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

  1. Nov 20, 2013 #1
    1. The problem statement, all variables and given/known data

    Capture.PNG

    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.

    2. Relevant 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?
     
    Last edited: Nov 20, 2013
  2. jcsd
  3. Nov 20, 2013 #2

    PhanthomJay

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    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.
     
  4. Nov 20, 2013 #3

    AlephZero

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    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.
     
  5. Nov 20, 2013 #4

    PhanthomJay

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    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.
     
  6. Nov 21, 2013 #5
    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.
     
  7. Nov 27, 2013 #6
    can you provide any help with the equation to get the strain from the radius of curvature?
     
  8. Dec 12, 2013 #7
    anyone?

    would the equation be M = (σ * I) / y
     
  9. Dec 14, 2013 #8

    PhanthomJay

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    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.
     
    Last edited: Dec 14, 2013
  10. Dec 15, 2013 #9
    ok i see.

    εmax = 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
     
  11. Dec 15, 2013 #10

    PhanthomJay

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    That equation is correct. But I thought you wanted to find another one using the radius of curvature, K?
     
  12. Dec 15, 2013 #11
    could i use εmax = k*z
     
  13. Dec 15, 2013 #12

    PhanthomJay

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    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'.
     
  14. Dec 15, 2013 #13
    is it not, strain = Z*K ?
     
  15. Dec 15, 2013 #14

    PhanthomJay

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    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.
     
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