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Strength of Beams

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SillyOldDuffer10/09/2020 10:53:37
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In a recent thread on Boring Bars Steviegtr asked which was stronger, square or round section bars. Good question, that I got wrong by saying for the same weight of metal round bar is stronger than square. Duncan Webster corrects my mistake in Steve's thread.

Looking it up, I'd misremembered. I was thinking of tube versus solid rod. However none of my books have a simple comparison of the effect of cross-sectional shape on beams. Instead they all have a Chapter full of hard sums showing how to work it out. Duncan applied the maths to prove me wrong in Steve's thread: not easy.

Can the team provide a simple answer to the following. Assuming the same weight of metal is used to make a beam of the same length, which shape bends least under load? ( Assume the beam is rigidly fixed to a wall at one end, and weights are added at the other.) Some examples:

shapes.jpg

Just for starters, the first two rectangles are identical except one bar is mounted horizontally, while the other is vertical. Experimentally bending a steel-rule shows immediately that the same lump of metal behaves very differently. Held horizontally, the rule bends easily - a child can do it. Held vertically, it's a job for superman. As the same shape is both strong and weak depending on load direction, the engineer has to understand and choose cross-sections appropriately.

Michael mentioned resistance to torque (twisting force) in the Boring Bar thread as being something a round section would resist better than square. I don't know, but suspect an ⌶ girder is stronger right/left and up/down than a tube, but the tube wins if the beam is twisted. Several other shapes might be useful depending on which forces a beam has to resist. How about ⌴, ⎔, △, and ☆?

Dave

ega10/09/2020 11:09:25
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The I girder can be modified to become a castellated beam yielding greater depth and stiffness for the same or slightly less weight.

David Davies 810/09/2020 11:28:13
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''The I girder can be modified to become a castellated beam yielding greater depth and stiffness for the same or slightly less weight.''

 

As often seen on frames for HGV semi-trailers.

 

Dave

Edited By David Davies 8 on 10/09/2020 11:29:29

Martin Kyte10/09/2020 11:39:03
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Re castellated beam

Interesting way they make those. The webs are cnc cut (Water jet or Plasma) so that the two half profiles stack inside each other to fit onto a blank less than the final width of the finished web. The 'points are then butt welded to form a web with a series of circular hole down the middle. Add top and Bottom and you have your castellated beam.

regards Martin

Speedy Builder510/09/2020 11:58:56
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You don't say in which direction the load will be applied, and how long the beam should be. I seem to remember something about "crippling strengths" back in my HNC strength of material days back in the '60s. Also don't forget elliptical tubes as used on aircraft struts.

Clive Brown 110/09/2020 12:00:42
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The Stress Engineer's "Bible" used to be Roark's "Formulae for Stress and Strain". Nowadays a snip on Amazon @ £90 I see.

It'll probably have all the shapes that SOD illustrates, and more.No need for a computer!

Michael Gilligan10/09/2020 12:27:10
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This should get you started, Dave : **LINK**

http://www.kmp.tul.cz/system/files/hearn_krouc_nekruh_tenkost_profilu.pdf

But I must mention a couple of points:

  1. I presume we are discussing stiffness, not strength
  2. My comment about the round section for a boring bar used the word ‘optimum’ ... this was intentional; because all the likely load-cases need to be considered when deciding what’s best for a particular application.

MichaelG.

.

P.S. ___ Important caveat

I only do the ‘arm-waving’ version of this stuff ... The sums are much to hard for me.

Edited By Michael Gilligan on 10/09/2020 12:30:15

Ron Colvin10/09/2020 12:30:05
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Posted by Martin Kyte on 10/09/2020 11:39:03:

Re castellated beam

Interesting way they make those. The webs are cnc cut (Water jet or Plasma) so that the two half profiles stack inside each other to fit onto a blank less than the final width of the finished web. The 'points are then butt welded to form a web with a series of circular hole down the middle. Add top and Bottom and you have your castellated beam.

regards Martin

 

I had question regarding how cellular beams are formed. Posted, then immediately realised what the answer was.

Edited By Ron Colvin on 10/09/2020 12:41:10

Michael Gilligan10/09/2020 12:33:00
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Posted by Ron Colvin on 10/09/2020 12:30:05:

It is the advantage of producing Cellular beams that have got me confused. I can not see how they can be produced, without every circular cutout in the final beam requiring half a circle of waste material.

.

The weight reduces more than the stiffness ...

MichaelG.

Ron Colvin10/09/2020 13:13:07
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Posted by Michael Gilligan on 10/09/2020 12:33:00:

Posted by Ron Colvin on 10/09/2020 12:30:05:

It is the advantage of producing Cellular beams that have got me confused. I can not see how they can be produced, without every circular cutout in the final beam requiring half a circle of waste material.

 

.

The weight reduces more than the stiffness ...

 

 

I apologise, I had hoped that no one would notice before I deleted my original post. When I first visualised the cutting of a cellular beam I mistakenly foresaw that waste material would be involved. Directly after posting I realised that this would not be so.

Ron

 

Edited By Ron Colvin on 10/09/2020 13:14:59

Michael Gilligan10/09/2020 13:26:49
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Posted by Ron Colvin on 10/09/2020 13:13:07:

I apologise, I had hoped that no one would notice before I deleted my original post.

.

No problem, Ron ... and certainly no need for apology

MichaelG.

Clive Foster10/09/2020 14:39:25
2373 forum posts
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Pending discovery of a concise and simple listing there are several online calculators.

This calculator looks as if it might help with getting ther right anwers for any sensible shape **LINK** . It allows you do design custom sections, within reason, too. Only given it a quick look, not proper use.

This one looks easy but only does standard type beams defined by cross sectional area **LINK** .

Several parameter calculators here **LINK**

Clive

SillyOldDuffer10/09/2020 14:59:00
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Posted by Speedy Builder5 on 10/09/2020 11:58:56:

You don't say in which direction the load will be applied, and how long the beam should be. Also don't forget elliptical tubes as used on aircraft struts.

I meant a beam sticking out from a wall with a weight on the end like so:

beam.jpg

I didn't think the actual length matters, because all the beams are the same weight. Now Speedy points it out, could be I've over-simplified. For example, metal cut out of an I girder to castellate it could be reused to make the beam longer...

Why is nothing ever easy?

Duty calls; I shouldn't be playing with a computer.

Dave

Andrew Johnston10/09/2020 16:04:30
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Despite calling it strength I think SoD means stiffness. If all beams are the same length and weigh the same then they must have the same cross sectional area. The deflection of the beam shown by SoD is inversely proportional to the moment of inertia. So it's simply a case of comparing moments of inertia for constant area to find the stiffest shape. All the information needed is in Machinery's Handbook.

Andrew

Michael Gilligan10/09/2020 16:56:03
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Posted by Andrew Johnston on 10/09/2020 16:04:30:

Despite calling it strength I think SoD means stiffness.

.

I’m glad we agree on that, Andrew

MichaelG.

stephen goodbody10/09/2020 16:59:05
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Hi Dave,

Your questions are good ones. In fact they're so important that they're one of the first things covered in mechanical and civil engineering university degree courses - or at least that used to be the case.The specific situation you raise is known as a "cantilevered beam", there are many other potential scenarios of course.

It's far too much to cover in any detail here, and the math can be a bit intimidating, but here are a couple of "food for thought" items that may be of interest.

You can probably imagine that there are two factors in play here; (i) bending loads (bending moment) about the fixed point on the wall, and, (ii) shear (breakage) of the beam. In both cases the loads are imposed by both the applied weight and by the mass of the beam itself. Both the maximum bending moment and the maximum shear force occur at the point where the beam is attached to the wall and decrease as you move towards the end.

This leads to the first food-for-thought item - cantilevered beams have to be strongest at their attachment point. This is where the bending moment and shear stress are greatest, and hence cantilevers are generally designed with decreasing mass-per-unit-length as you move away from that attachment point. Hence this is why cantilevered beams often taper (or otherwise decrease in area) from their fixed ends towards their unsupported end - they need to be less strong as you move away from the fixed point and you want to have less mass as you move away from the fixed point.

The second item is more complex, and I won't go into the detail, but suffice to say that an I beam is essentially the optimum strength-versus-weight compromise (and hence strength-versus cost compromise). A plain rectangle (with the long length vertical) is generally strongest of all, but contains a lot of heavy material which is doing very little for the beam's bending strength. With that insight, If you look at an I beam end-on you can see it's really just a rectangle with the least-beneficial bits removed. This is why I beams are so ubiquitous in the world - they're cheapest for the strength they provide.

Best regards

Steve

SillyOldDuffer10/09/2020 17:01:28
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Posted by Michael Gilligan on 10/09/2020 16:56:03:
Posted by Andrew Johnston on 10/09/2020 16:04:30:

Despite calling it strength I think SoD means stiffness.

.

I’m glad we agree on that, Andrew

MichaelG.

We all agree! I do mean stiffness...

Martin Kyte10/09/2020 21:10:32
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Posted by Michael Gilligan on 10/09/2020 12:33:00:

Posted by Ron Colvin on 10/09/2020 12:30:05:

It is the advantage of producing Cellular beams that have got me confused. I can not see how they can be produced, without every circular cutout in the final beam requiring half a circle of waste material.

 

.

The weight reduces more than the stiffness ...

MichaelG.

If you take the profile of the top half and the profile of the bottom half. Shift the top half so the points in the top half sit in the middle of the semicircles in the bottom half. Now you can move the two profiles closer together so the protrusions of the top half fit into the gaps of the bottom and vice versa. The cutouts are then quarter circles joined at the corners with the corners at 45 degrees. By my reconing that saves about half the diameter of the final cutout for the full length of the beam less the eyeshaped cutouts and trimming the ends.

I wish I had some drawing software on this computer, it would be a lot easier to show you instead of explain. It also saves an awfull lot of weld on the middle joint which is possibly the big attraction.

regards Martin

 

Edited By Martin Kyte on 10/09/2020 21:12:10

Michael Gilligan10/09/2020 21:29:06
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Posted by Michael Gilligan on 10/09/2020 12:33:00:

Posted by Ron Colvin on 10/09/2020 12:30:05:

It is the advantage of producing Cellular beams that have got me confused. I can not see how they can be produced, without every circular cutout in the final beam requiring half a circle of waste material.

 

.

The weight reduces more than the stiffness ...

MichaelG.

.

Just for clarity:

When responding to Ron, I should have only quoted his opening sentence

“It is the advantage of producing Cellular beams that have got me confused.“

[ I was summoned to lunch, and missed the opportunity to edit my post ]

MichaelG.

.

Edit: That said ... Here’s an interesting page:

https://www.kloecknermetalsuk.com/westok/products/westok-cellular-beam/

Edited By Michael Gilligan on 10/09/2020 21:34:21

John Olsen10/09/2020 22:17:15
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It gets worse, because for the same weight of material, by making the webs or the wall thickness thinner, you can make the section larger, eg a larger diameter tube with a thinner wall, or an I beam that is taller, like the castellated beam achieves. But if you go too far with this process, the compression side will become very prone to buckling. It is quite hard to predict at what load buckling will occur, since it depends on initial straightness and whether the load is applied truly along the axis.There are ways to reduce the chance of that, for instance by corrugating the walls of the tube along the length. Of course none of this is very applicable to a boring bar, which has to fit inside a given starting hole.

John

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