Derive an equation for the maximum bending moment

[Big Jock]

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


M/I = σ/y = E/R



my attempt so far is

M= E/R x I
Totally unsure if that is correct and really be doing with a hand from someone more clued up than I am.

Many thanks in advance

 

[Simon Bridge]

Please explain your reasoning.

What is "strain" as measured by the gauge?
What is "bending moment"?
Do you see that these concepts will be related? How does bending moment give rise to strain?

 

[Chestermiller]

Have you learned the relationship between the tensile strain, the radius of curvature, and the distance from the neutral axis?

Chet

 

[Big Jock]

Eventually got there I believe with a final answer of

M= EIk would that be correct?

Have all the workings etc to prove it...

 

[Chestermiller]

See this thread: https://www.physicsforums.com/showthread.php?t=715041

 

[Big Jock]

So what I have is wrong your telling me?... I seen that thread a long time ago and thought it was irrelevant as he was using the max strain plus it was hardly a derived equation. If you can shed anymore light please do...

 

[Chestermiller]

So what I have is wrong your telling me?... I seen that thread a long time ago and thought it was irrelevant as he was using the max strain plus it was hardly a derived equation. If you can shed anymore light please do...

You are supposed to express the answer in terms of the maximum strain. The maximum strain ε is equal to the distance t of the outer surface from the neutral axis divided by the radius of curvature:

ε = t/R

So R = t/ε

You are supposed to substitute this for R into your equation for the bending moment to get the relationship between the bending moment and the maximum strain (at the outside of the bend).

Chet

 

[Big Jock]

I started by taking k=1/R= change in radians/ change in x
when I calculated the strain for my equation I had strain= change in length/original length
then strain = (radius of curvature - distance from the neutral axis)x change in radians-radius of curvature x change in radians.
Which got me to strain = -y/r
stress = Youngs modulus x strain
stress= -youngs modulus x distance from neutral axis/ radius of curvature.
Which I integrated down to M= EI/r
then I got to EIk

also there is no mention what so ever of the beams thickness of width that is why I went down this route

 

[Chestermiller]

I started by taking k=1/R= change in radians/ change in x
when I calculated the strain for my equation I had strain= change in length/original length
then strain = (radius of curvature - distance from the neutral axis)x change in radians-radius of curvature x change in radians.
Which got me to strain = -y/r
stress = Youngs modulus x strain
stress= -youngs modulus x distance from neutral axis/ radius of curvature.
Which I integrated down to M= EI/r
then I got to EIk

also there is no mention what so ever of the beams thickness of width that is why I went down this route

I think the answer they were looking for was M=EIε_max/y_max, where ε_max is the measured tensile strain on the outside of the bend.

 

[Big Jock]

so what I have derived in the strain section is wrong. Can you send me a link to show how this works or list the workings please, to tidy it up as it would seem Iam not to far away...

 

[Chestermiller]

so what I have derived in the strain section is wrong. Can you send me a link to show how this works or list the workings please, to tidy it up as it would seem Iam not to far away...

You have the correct equation for the strain as a function of y, but the strain is being measured only at the surface where y = ymax. The idea is to measure the strain at the surface, and, from that, determine the bending moment. You know that the strain at the surface is equal to the distance of the surface from the neutral axis divided by the radius of curvature. So, by measuring the strain at the surface and the distance of the surface from the neutral axis, you know the radius of curvature. Once the radius of curvature is known, the bending moment is known from EI/R. What you were being asked to do was to eliminate R from the two equations, and calculate M as a function of ε_max. I hope this makes sense.

Chet

 

[Big Jock]

Got lost a bit now...How do I do this? I only ask as I have never been taught about this.

strain is being measured only at the surface where y = ymax. The idea is to measure the strain at the surface, and, from that, determine the bending moment. You know that the strain at the surface is equal to the distance of the surface from the neutral axis divided by the radius of curvature. So, by measuring the strain at the surface and the distance of the surface from the neutral axis, you know the radius of curvature

 

[Chestermiller]

Got lost a bit now...How do I do this?

strain is being measured only at the surface where y = ymax. The idea is to measure the strain at the surface, and, from that, determine the bending moment. You know that the strain at the surface is equal to the distance of the surface from the neutral axis divided by the radius of curvature. So, by measuring the strain at the surface and the distance of the surface from the neutral axis, you know the radius of curvature

You measure the strain ε_max at the outer surface of the beam. This strain is related to the radius of curvature R and the distance of the surface from the neutral axis y_max by

ε_max=y_max/R

The bending moment M is related to the radius of curvature R by:

M=EI/R

If we combine these two equations, we get

M = EIε_max/y_max

Using this equation, if you measure the strain on the outer surface of the beam, you can calculate the bending moment. So this is an indirect method of experimentally determining the bending moment.

 

[Simon Bridge]

@Big Jock - you got lucky. We don't normally do your homework for you.
It looks like you didn't understand the experiment you were doing.
The relationship between stuff in equations and stuff you measure is very important to understanding physics.
Best practice is to review a lab class a couple of days beforehand to make sure you understand it.
The derivation is one of the things you could have done beforehand.
(Assuming, of course, that you get the lab notes/instructions before the lab...)

 

[Big Jock]

Simon our right I don't follow this procedure as I am learning online doing this all myself. I don't have access to lab classes so if you can point me in the right direction I would be grateful...

 

[Simon Bridge]

Oh I see - for future point out that it is self-study, not homework.

The problem you described was for a practical physics or engineering exercise - typically called a "lab class".
A student would go to a special room that includes the bar already set up with a strain-gauge and everything and they'd have to carry out the experiment.

To understand what Chestermiller is telling you, you need to go back and make sure you understand the concepts you need. Their definitions and descriptions will be in the resources you are using.
You need to know what "bending moment" is etc.