Calculate strain from bending moment?

[LDC1972]

Homework Statement

Simply supported beam, all values known now I believe as this is the last question on it. I have spent a long time trying to work this out:

The bending moment is to be verified using a strain gauge bonded to the outer surface of the beam at the point of max bending moment.
Derive an equation which could be used to calculate the bending moment from the measured strain value.

Homework Equations

k = M/EI
ε(max) = kz,

The Attempt at a Solution

I know the position and value of the max bending moment, so I "think" this may be correct. But I've spent so long I have confused myself a lot. Please be gentle :-)

I have come up with this so far:

k = M/EI
ε(max) = kz,

and therefore ε(max)= (z) M/(EI)

I think I've typed the above equation correctly. I mean it as follows: strain(max) = z multiplied by the result of M divided by EI.

Z is the distance from the neutral layer to the outer layer in a tensile condition.
M is bending moment
EI is flexural stiffness (E is youngs and I is 2nd moment of area)
K is curvature of the beam or could be replaced with 1/R where R is the radius of curvature in radians.

Can anyone confirm the above or point me correctly?

 

[Chestermiller]

Homework Statement

Simply supported beam, all values known now I believe as this is the last question on it. I have spent a long time trying to work this out:

The bending moment is to be verified using a strain gauge bonded to the outer surface of the beam at the point of max bending moment.
Derive an equation which could be used to calculate the bending moment from the measured strain value.

Homework Equations

k = M/EI
ε(max) = kz,

The Attempt at a Solution

I know the position and value of the max bending moment, so I "think" this may be correct. But I've spent so long I have confused myself a lot. Please be gentle :-)

I have come up with this so far:

k = M/EI
ε(max) = kz,

and therefore ε(max)= (z) M/(EI)

I think I've typed the above equation correctly. I mean it as follows: strain(max) = z multiplied by the result of M divided by EI.

Z is the distance from the neutral layer to the outer layer in a tensile condition.
M is bending moment
EI is flexural stiffness (E is youngs and I is 2nd moment of area)
K is curvature of the beam or could be replaced with 1/R where R is the radius of curvature in radians.

Can anyone confirm the above or point me correctly?

Looks good.

 

[LDC1972]

Thank you! So much time spent and to have an expert say "looks good" makes it so rewarding!

Have clicked your thanks button ;-)

Thanks again!

 

[oxon88]

how would this be re-arranged for M?

M = I * ε_max * E / Z

 

[rezeew]

I can confirm that your equations and approach are correct. To calculate the strain from the bending moment, you can use the formula ε = kz, where k is the curvature of the beam and z is the distance from the neutral layer to the outer layer. This equation assumes that the beam has a linear stress-strain relationship and is within the elastic limit.

To verify the bending moment, you can use a strain gauge bonded to the outer surface of the beam at the point of maximum bending moment. The strain gauge will measure the strain at that point, and using the equation above, you can calculate the bending moment.

To further verify your results, you can compare the calculated bending moment to the expected bending moment based on the known values of the beam's properties (E, I, and z). If they are within a reasonable margin of error, then your calculations are likely correct.

I hope this helps. Keep up the good work!