how to determine if a member react an applied moment by bending or an axial load times a moment arm

[ajroc]

Hello,

I have a quick question. I have a I beam whose cross section which is built up using rectangular members. The section has a moment applied to it. When I do an analysis of the section using FEA, it is obvious that a portion of the applied moment is reacted by bending in the web, and the rest by axial loads in the top and bottom caps. I can determine what portion of the applied moment is reacted by the webs and caps by comparing EI's, however I do not have a good explanation as to why the web bends and the caps are in extension other than intuition.

As I have many sections to size, I do not want to build FEA models for all, but would like to create a tool to do so quickly by hand.

Looking at a rectangular member located at the section CG , I know the area moment of inertia is 1/2 bh^3. If that member moves away from the section CG (say the upper cap), I= 1/12*b*h^3+b*h*d^2.

If I take the ratio of the two I's (with the member away from the section CG in the numerator) the result is 1+12d^2/h^2. the term 12d^2+h^2 is the additional moment of inertia (and thus additional stiffness) that member has because it is away from the section CG.

Using this second term in the ratio (12d^2/h^2), I can predict for any member in my I beam (say the cap), how much bending will occur and how much of the applied moment will be reacted out using a axial load times moment arm to section CG.

I am struggling with how rigorous this is - does this method make sense? Am I overlooking something? I have attached a simple example to help explain my method.

thanks in advance for your help

 

[SWComposites]

why don't you just calculate and compare the stress on the top and bottom surface of each element of the cross section? If the stresses are close (with say 5-10%) then assume the element is axially loaded.

 

[ajroc]

Thanks for the tip SW. I originally started down that path (calculating bending and axial stresses for each member) however I found that it was not easy to implement in a program. That I why I started down the path I described above as it is much simpler to execute.

 

[rb1957]

you've got a beam section made up of rectangles ... then simply make a section properties table ...
x y A=x*y yc A*yc A*yc^2 Ic
where yc is the centroid of the rectangle (x*y) and Ic is the I of the rectangle about it's centroid
then yNA = sum(A*yc)/sum(A)
and INA = sum(A*yc^2+Ic)-sum(A)*yNA^2

then calc bending stress at whatever y you like (My/INA)
of course, y is the distance from the NA ...

if this doesn't help, pls post a pic of your cross-section

 

[SWComposites]

how hard can it be to calculate P/A and Mc/I in a spreadsheet???

 

[ajroc]

I think there might be a little bit of misunderstanding as to what I want to do. I am not looking for Mc/I and P/A, but looking to evaluate the load sharing characteristics and specifically if a member reacts out a moment using axial load times moment arm or reacts out the moment by bending.

First off, I have several configurations to evaluate, different ways to build up the section using rectangular areas, and thousands of load cases. I prefer to code up a program to do that rather than build a huge spreadsheet.

rb, I attached an excel spreadsheet with a simple I beam to illustrate what I am trying to do here. I'll attach a screen shot if that helps. I am using the spreadsheet as a "proof of concept" before I start coding.

Yesterday I found a method in Bruhn (Sec A13.10) that may also work. The method solves for both the axial and bending stress in a non-homogenous beam, but I am not done investigating if it will work.

I am still curious why the ratio of 12d^2/h^2 predicts so well the ratio of bending to axial stress and load in a member, and was hoping someone had some insight to share with me.

 

[prex]

Any part of a section in bending reacts a moment by an axial load (the resultant of the stress distribution) times a moment arm (distance of NA to the centroid of the part) and by a bending distribution (a linearly varying stress with zero resultant).
So what you are trying to do is unclear to me: you seemingly need to decide whether the axial distribution prevails on the bending one by some unknown criterion, but to which purpose I can't understand.

prex
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[paddingtongreen]

I suspect that you would get more help if you posted a picture of the the member and it's end conditions and loading. It is difficult, trying to reverse engineer the problem statement from your spreadsheet. I don't understand the differences in your result options, it sounds to me like a distinction without a difference. A diagram please.

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[dhengr]

Ajroc:
Structures don’t act the way you would like to code them to act. Structures act in a way forced on them by the way they have been detailed and fabricated and loaded, and you had better do your analysis and coding accordingly, to account for that. Until you learn this, you should probably steer clear of designing airplanes. What you think you are seeing from your FEA, may have more to do with how you modeled your beams and their parts, and the types of elements you used in that modeling, than it does to the way the beams actually act. You have to do a better job of explaining your problem, how it works, what it does, and what you are trying to do. Show a sketch with dimensions, loads, beam cross section, etc., so we can see what you are talking about. What are the sizes of your caps/flanges, and the size and thickness of your webs? Are the flanges acting as frame or truss members and the webs just acting as a shear panel or tension field to keep the frame from racking?

I can’t see your attachments, post them in jpg or pdf format, and more people will be able to see them. This sounds like a built-up beam or beam/column problem to me. And, you should dig out your text books and review elementary beam theory, Strength of Materials, shear flow, Saint- Venant’s principle as relates to stresses at a distance from load or reaction input points, etc. It sounds like you are varying flanges and/or web sections and loads and moments and wanting to check stresses. The various parts of a built-up section do not act independently, they act in unison, as a function of how they are interconnected. But, the way you are explaining it sounds bass-acwards to me. Calculating the section properties and stresses are really pretty elementary problems and you seem to be going out of your way to complicate them due to some misunderstanding on your part. The frame/truss and tension field panel do require a different tack on the problem, but I don’t understand what you are doing, and you’re not doing a good job of explaining it.

 

[PMR06]

Your ratio of the I's is not simply 1+12d^2/h^2. You have factored out 1/12bh^3, but it's still there. 12d^2+h^2 alone has no significance. bhd^2 is significant however. It is the entire reason wide flange beams have their shape. It places the material where it will provide the most I for A.

Other than that, I'm with the others, unsure of exactly what you are after. The member doesnot decide to have its flange handle the moment as an axial load. Maybe you are trying to think of it like a truss where the chords would be axial? Like SWComposites said, it all boils down to P/A +- Mc/I.

 

[BAretired]

I can determine what portion of the applied moment is reacted by the webs and caps by comparing EI's, however I do not have a good explanation as to why the web bends and the caps are in extension other than intuition.

They all bend, but the web shows greater variation in flexural stress because it is deeper.

In the case of an 'I' shaped beam the flanges carry most of the applied moment by axial forces (tension in one and compression in the other).

Some engineers approximate the axial force in each flange as M/d where M is the applied moment and d is the distance c/c of flanges. This approximation neglects bending in the web and both flanges.

BA

 

[TLHS]

You're seeing bending stress in the web elements because they're oriented such that one of their longer axes is in the direction through which your moment resisting axial forces varies. Your plates elements have a modelled height, so when your section is resisting moment there's some variation of axial force across that distance due to the nodes being different distances away from the neutral axis. To make that work, the element will report a stress along the axis of the member equal to the averages of the stresses at the top and bottom of the element and a moment that produces a stress distribution with a peak equal to half the difference between the axial stresses at the top and bottom. This will show up as a bending stress in that plate element instead of a pure axial force along the length of the member. As you approach zero height on your plate elements, I would expect that this bending stress would approach zero as well.

Bending stresses *are* axial couples. Finite element models are not necessarily telling you more about a system. Sometimes you're just trading one set of assumptions for another. In this case, your member axial stress is likely the average at the centre of your plate and if you wanted to you could use the bending stresses to help understand what's happening between the centre and the extremes of your element.

So yeah, the bending stress is likely just an expression of the variation in axial stress across the height of the plate element.

 

[RFreund]

It seems like bending stress and axial stress are being misused or misunderstood in your posts (maybe not). What you are determining is the normal stress. There is normal stress due to bending which is assumed to vary linearly or the normal stress due to axial which is assumed constant. Not sure if this is helping but just to make sure I'm understanding what your after?

EIT

http://www.HowToEngineer.com

 

[rb1957]

you can surely replace the My/I calc which produces a linearly varying stress with a couple at the centroid of the caps.

you can determine the amount of amount absorbed by the web by reversing the s = My/I ... ie you know the stress distribution in the web (assuming it is fully effective) so you know how much moment is there.

if you use the cap couple approach you'll get a different average stress in the cap (compared with the My/I stresses, as the web is no longer absorbing it's moment).

 

[paddingtongreen]

Assuming that the flanges are continuously connected to the web:
If you were looking at purely axial compression, you would see the same stress across all parts except as noted below.
If you were looking at pure bending, you would see a gradient from a negative value at one extreme fiber, to the same positive value at the opposite extreme fiber except as noted below. This is P/A+-Mc/I. You cannot escape from it if the flanges are continuously connected.

Depending on how you model the member and the way the loads and support are applied, you might see a variation due to shear lag.

Michael.
Timing has a lot to do with the outcome of a rain dance.