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Beam Theory - Billski can you elaborate here?

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Fenix

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I'm reading through Beam Theory and finding it an excellent description an ultra novice can follow.
But then I came to this and it has me held up:

*******

Bending – if we take a little section of the beam and put a bending moment on each end of the section, the little section of beam becomes curved (a tinyamount) and stresses build up across the section in relation to how far we are from the neutral axis. And because plane cross sections remain plane, even in the deflected beam, the stresses are linear with distance from the neutral axis. And so the resistance to bending goes with E (we know that strain and stress are related linearly byYoung’s Modulus, E) and with I, the Second Area Moment of Inertia of the section. So now we have: Bending stiffness at any point along a beam is EI.

What else? Well, if we know the moment bending the beam, we can also calculate the axial stresses any place up and down along the cross section. Axial stress sigma is M*z/I where the moment is M, z is how far up or down we are from the neutral plane, and I is the Second Area Moment of Inertia ofthe section. Highest stress is at the farthest point on the cross section from neutral axis.


*******

I'm good up to "because plane sections remain plane" Can someone maybe state that in a different phrase or further explain please?
I think I follow "stresses are linear with distance from neutral" - it simply means the increase in stress is a linear function as you move outward (from neutral axis) and not an exponential function??

Next I will be asking what are "E" and "I" and "second moment of inertia". I found second moment of inertia on Google but the description seems to be written to people who already know what it is.

Perhaps I'm too inexperienced for your thread? Sorry.
 

wsimpso1

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I'm good up to "because plane sections remain plane" Can someone maybe state that in a different phrase or further explain please?
I think I follow "stresses are linear with distance from neutral" - it simply means the increase in stress is a linear function as you move outward (from neutral axis) and not an exponential function??
You have more of the theory than you think.

Those surfaces we "cut" on each side of our little segment of beam, starts as a parallel planar surfaces and they stay planar when we bend the beam, although they do become non-parallel when the section is bent. Imagine holding a block of rubber and bending it. The end faces remain planar. This results in the 'stresses are linear with distance' comment. Strain and stress are related by a constant called Young's modulus, abbreviated by E. You can look this up. Pitch a couple bucks to Wikipedia if you go there too.

Next I will be asking what are "E" and "I" and "second moment of inertia". I found second moment of inertia on Google but the description seems to be written to people who already know what it is.
E is "Young's Modulus", see above. I is second area moment of inertia. You can look it up too. It is the integral of dA*y^2 of the cross section, where dA is a tiny chunk of the section area and y is the distance from neutral axis to the tiny chunk of area. For common shapes, there are established formulae and then the Parallel Axis theorem for combining things. I represents how much resistance to bending a shape has and is essential to figuring out stress in a structure under bending. EI then becomes how much resistance to bending a shape made of something has, and is essential to figuring out angular and out-of-plane deflection, resonant frequencies, etc. E is material while I is shape. Any beam with a certain EI has the same bending stiffness as any other beam with the same EI.

Perhaps I'm too inexperienced for your thread? Sorry.
Naw. Welcome to the Monkey House.

Billski
 

Fenix

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I think I understand the "planar". Like if you open a book a couple of inches. The pages remain "in plane" as in each one is still a flat sheet of paper, but there are gaps between them as you move away from the binding, as in no longer parallel. Of course a board or aluminum bar does not experience these "gaps" but I think the point is that we analyze them as if any given piece of the material "remained planar". I don't really connect the significance of "staying in plane" dictating that the function remain linear or become something exponential though.

OK I need to digest this. Stress vs Strain, Hooke's Law, Poissons Ratio and all this a couple hundred years ago!! Now I DO feel stupid.
Why wasn't all this in my science class? I want my money back - LOL. But yes, I do contribute to Wikipedia - what a cool idea it was to connect the intellect of your peers without the injection of an agenda, at least usually.

Thanks Billski for the additional direction. I need to resolve all of the above (stress, strain, and grasp why: The volume of materials that have Poisson’s ratios less than 0.50 increase under longitudinal tension and decrease under longitudinal compression. Really? They get "bigger when stretched?)
I'll be back.
 

wsimpso1

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I think I understand the "planar". Like if you open a book a couple of inches. The pages remain "in plane" as in each one is still a flat sheet of paper, but there are gaps between them as you move away from the binding, as in no longer parallel. Of course a board or aluminum bar does not experience these "gaps" but I think the point is that we analyze them as if any given piece of the material "remained planar". I don't really connect the significance of "staying in plane" dictating that the function remain linear or become something exponential though.
What is with this exponential stuff? We are working in the linear region. In metals, it is below yield. In composites, it is below first fiber failure. IIRC "With load comes extension" was Hooke's expression and it is linear. Find strain (dL/L) multiply by Young's Modulus (E) and you have stress. It really is that simple below yield. The theory has been known for centuries and is well demonstrated.

OK I need to digest this. Stress vs Strain, Hooke's Law, Poissons Ratio and all this a couple hundred years ago!!
...
Thanks Billski for the additional direction. I need to resolve all of the above (stress, strain, and grasp why: The volume of materials that have Poisson’s ratios less than 0.50 increase under longitudinal tension and decrease under longitudinal compression. Really? They get "bigger when stretched? I'll be back.
Hmm. Poisson's Ratio of 0.5 indicates zero volume change. 0.3 is typical of elastic deformation in metals. Get into wood or plastic foam and you have a lot of air in the mix, behaviour can seem strange indeed. air is pretty darned compliant stuff. Go with rods of stuff 30-40 times as stiff as the glue it is swimming in and you again get wonky behaviour - try out the thought experiment of pulling across the fibers. The fibers are stiff and the resin is soft, so much of the motion is in the resin, change in axial dimension is darned close to zero, and yeah, the definition of Poisson's ratio gets stretched too, with values greater than 0.5... I am just a mechanical engineer who uses the stuff that means understanding its limitations and working within it.

Billski
 

Fenix

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Hmmm Perhaps I do see the "linear function" of this stress (or strain? - I still need to read that). Is it as simple as the "amount of stretch" as you move away from the neutral axis is linear because, imagining the book again, if you open it, say two inches, there is no gap at the binding, two inches at the "edges" and 1" at the halfway point across the page, 1/4" at one quarter across the page, etc. If it were a paperback book and you opened it AND bent the covers into a "non planar" shape then the gap between the pages would not be linear as you progressed from the binding out to the edges of the open book, to the extent you "bent the covers" of the book, potentially into a parabolic shape. Sort of like that? Or do I need to go back to driving rivets and leave design to the others?
 

Fenix

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Very helpful video. Thank you. That resolves stress vs strain for one thing.
 

wsimpso1

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Hmmm Perhaps I do see the "linear function" of this stress (or strain? - I still need to read that). Is it as simple as the "amount of stretch" as you move away from the neutral axis is linear because, imagining the book again, if you open it, say two inches, there is no gap at the binding, two inches at the "edges" and 1" at the halfway point across the page, 1/4" at one quarter across the page, etc. If it were a paperback book and you opened it AND bent the covers into a "non planar" shape then the gap between the pages would not be linear as you progressed from the binding out to the edges of the open book, to the extent you "bent the covers" of the book, potentially into a parabolic shape. Sort of like that? Or do I need to go back to driving rivets and leave design to the others?
I like Mechanics of Materials by Timoshenko and Gere, there are others. Chapters on stress-strain through beam theory are VERY useful. Then most of our composites within our FOS range are essentially linear.

The classic drawing of stress vs strain with a linear region, then a long plastic region is nice and all, shows the theory. Trouble is real structural materials have elastic regions with virtually vertical lines - the softest of them in the millions of psi. Then ductile materials, when plotted with the same scales used to show a slope in the elastic range go horizontal once plastic. You can effectively model post yield behaviour for many purposes by simply using a horizontal line. In composites, the behaviour is more like very high hardness heat treated metals, with only a few percent strain after yield or first fiber failure.

So, you design to stay in the elastic region. The plastic region is mostly useful in fabrication (from complex forgings to simple bent parts) and in crash energy absorption.

Billski
 

jedi

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I'm reading through Beam Theory and finding it an excellent description an ultra novice can follow.
But then I came to this and it has me held up:
*******
Bending – if we take a little section of the beam and put a bending moment on each end of the section, the little section of beam becomes curved (a tinyamount) and stresses build up across the section in relation to how far we are from the neutral axis. And because plane cross sections remain plane, even in the deflected beam, the stresses are linear with distance from the neutral axis. And so the resistance to bending goes with E (we know that strain and stress are related linearly by Young’s Modulus, E) and with I, the Second Area Moment of Inertia of the section. So now we have: Bending stiffness at any point along a beam is EI.
.......... *******
I'm good up to "because plane sections remain plane" Can someone maybe state that in a different phrase or further explain please?
I think I follow "stresses are linear with distance from neutral" - it simply means the increase in stress is a linear function as you move outward (from neutral axis) and not an exponential function??

Next I will be asking what are "E" and "I" and "second moment of inertia". I found second moment of inertia on Google but the description seems to be written to people who already know what it is.

Perhaps I'm too inexperienced for your thread? Sorry.
The analogy of the book being fanned open is a good one. The spacing (or gap) between the pages is linear from the binding to the outer edge of the pages. The gap is equivalent to the strain (material stretch) and therefore directly related to the stress by Young's modulus which is a material dependent constant.

If the pages were to bend or curve in the radial direction, they would no longer define (or be contained within a plane) a "plane" and the distance between the pages (stress) would not be a linear function of the distance from the binding.

Since the material is uniform the relation between stress and strain is constant (Young's modulus). Therefore, the force increases in a linear (straight line) fashion with the distance to the book binding. In the case of the beam the book binding is replaced by the neutral axis (the part that does not stretch).

That is a lot of words to express a fairly simple concept. Sometimes it is necessary to stand back and look at the overall picture. It reminds me of stepping back to look at the forest and ignoring the trees.

Now if you will return to the quote reference above and click to area to expand the quote you will find the next bold print that reads "Next I will be asking what are "E" and "I" and "second moment of inertia". I found second moment of inertia on Google but the description seems to be written to people who already know what it is. "

The part here that I have found most confusing is where the "second moment of inertia" is mentioned.

OK, here we need to go back into the forest and touch a few of the trees.

We have just determined that the resisting force within the beam increases linearly with the distance from the neutral axis. We also know that a moment or torque (twisting force) is the product of a force times a distance.* Therefore, the resisting moment to bending of the beam has the distance to the neutral axis (x) included twice or what is commonly known as x squared (x*x).

It just so happens that these equations for force and bending are the exact same formula for the moment of inertia (and second moment of inertia) about the neutral axis. These are common engineering values that are frequently needed in calculations. These terms are used as a "shorthand" for the equations. The moment of inertia of common shapes are frequently used and are relatively easy to calculate or measure so that term is applied in general. The values (or formulas) for various sections are frequently contained in tables with those terms (first** moment of inertia and second moment of inertia) are used as the column headings in the table.

* If the nut is too tight to loosen with the wrench you are using, you need more torque. Get a longer wrench and you can apply more torque without having to (pull harder) apply more force.

** The first moment is implied if "first" is not mentioned.

I hope that helps.
 
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