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

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Discussion Overview

The discussion revolves around deriving an equation to calculate the bending moment from measured strain values in a beam, specifically focusing on the relationship between stress, strain, and curvature. Participants explore various equations and concepts related to bending moments, strain gauges, and the stress-strain relationship.

Discussion Character

  • Homework-related, Technical explanation, Exploratory

Main Points Raised

  • Some participants emphasize the need for an equation that relates strain to the radius of curvature and the bending moment.
  • Others mention the importance of the stress-strain relationship and suggest using the equation 1 / R = M / EI as a starting point.
  • A participant points out that the maximum bending moment has been previously calculated and provides its value and location along the beam.
  • There is a discussion about the appropriate equations to use for calculating strain from the radius of curvature, with suggestions to consider both stress-strain relations and curvature-strain relationships.
  • Some participants propose various forms of equations, such as M = (σ * I) / y and εmax = Z * M / EI, while others challenge or refine these proposals.
  • There is a clarification regarding the notation used for the distance from the neutral axis, with a suggestion that it is typically denoted by 'c' rather than 'Z'.
  • Participants express uncertainty about the correct formulation of strain in relation to curvature and distance from the neutral axis.

Areas of Agreement / Disagreement

Participants generally agree on the need to derive an equation for bending moment based on strain measurements, but multiple competing views and approaches remain regarding the specific equations and relationships to use.

Contextual Notes

Some participants note that the relationship between stress and strain is crucial, and there is mention of the need to clarify the definitions and units involved in the equations discussed. The discussion also highlights the importance of identifying the location of maximum bending moment and the assumptions made in deriving the equations.

oxon88
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Homework Statement



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.

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?
 
Last edited:
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oxon88 said:

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.
 
PhanthomJay said:
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.
 
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.
 
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.
 
AlephZero said:
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?
 
anyone?

would the equation be M = (σ * I) / y
 
oxon88 said:
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.
 
Last edited:
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
 
  • #10
That equation is correct. But I thought you wanted to find another one using the radius of curvature, K?
 
  • #11
could i use εmax = k*z
 
  • #12
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'.
 
  • #13
is it not, strain = Z*K ?
 
  • #14
oxon88 said:
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.
 

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