local bending stress calculation in long beams 1

[Guideon]

Hi everyone,

Recently I faced a problem in calculating bending stress in a long UPN profile "flange" due to concentrated force.
It seems that the regular/familiar formula for bending stress in a finite/short element does not applicable in local bending of long/infinite beam. See sketch attached for clarification.
An example to such calculation I found in elevator's std for "T" rails calculation which express the stress in the "flange" root:

\sigma=\frac{1.85F}{c^2}

My question is how to calculate such local bending stress in a "flange" of std profiles such UPN/IPN...and sources/books in the subject.

Thank you,
Guideon

 

[TERIO]

I haven't used it for this, but maybe the AISC Specification for Structural Steel Buildings, Section J10 "Flanges and Webs with Concentrated Forces" and the associated commentary could give you something to work with.

 

[ESPcomposites]

Generally, we use a concept of "effective width". If you are looking at a top view, the effective width is 45 degrees from the horizontal, in both directions, for a total of 90 degrees. In other words, b = 2x. 45 degree is typical, but other angles may be considered.

Brian

http://www.espcomposites.com

 

[Ron]

In simple bending, there is no limitation on "L" except the resulting stress. An "infinite" "L" would have to be computed based on an effective "L", which could be related to beam stiffness relative to the concentrated load; however, I don't see a practical application where the beam would be considered infinite in length for computing bending stress.

A sketch of your premise might be helpful.

 

[Guideon]

Hi everyone,

Thanks for the answers,

Brian,
You are right, I found something like your answer in my notes, with one little correction 2b=x, and the only problem with this assumption is, that the bending stress equation you get is ?=3P/t^2, which is x independable (means that no matter what is the distance of the force P from the "flange" root, the stress has the same value).

Ron,
Upon your request, hereby another sketch which I hope that
explains to you the local bending in a flange of long beam.

Guideon

 

[Ron]

Thanks, Guideon. I would treat this as a plate bending issue, considering the flange to be a plate restrained on 3 sides. I would consider the length of the plate to be the length from no deformation-to-no deformation (deformation centered) and the width of the plate to be 1/2 the flange width, unless there is upward deformation of the opposite side of the flange.

The could be modeled easily in FEA, but obviously can be approximated by hand.

As for the beam itself, you should check torsion, particularly if the concentrated load is near an end where the top flange is restrained.

 

[ESPcomposites]

Guideon, are you sure the b=x/2 is correct? This is not a very generous effective width, but would be conservative.

As far as being x independent, that is the way it works out. Its a bit conservative for small x, and is mostly used at root (corner of your beam).

The aircraft industry has been using this approach for decades, with a total subtended angle usually 90 degrees (45 to both sides of the horizontal).

Brian

http://www.espcomposites.com

 

[Guideon]

Thank you Brian,

I found (simple geometry) that you are right, it should be b=2x!
I also lie on your experiance in the aircraft industry and accept the conservative approach you discribed.

Guideon

 

[ESPcomposites]

You could also try an FEM to satisfy your curiosity. I think I did it while back and it worked out reasonably well. The problem with the FEM is that it may peak out some in the middle. In reality, that would locally yield and redistribute (for ductile materials). So then you are forced to do a nonlinear FEM to get a better idea, and it starts to become impractical for such a simple problem. If you are working with a brittle material or doing fatigue analysis, the effective width approach may not be good enough.

The 45 degrees is just a guideline. Some use more, some use less, based on test data, FEM, etc. You can think of it as a method and not necessarily something that captures the exact state of stress (which varies along the width b).

Brian

http://www.espcomposites.com

 

[Guideon]

Thank you again Brian,

Guideon

 

[msquared48]

That's why we use bearing/web stiffeners at point load locations, to eliminate the possible problem.

Mike McCann
MMC Engineering

 

[ESPcomposites]

Msquared48. Good point, if the load is large, you definitely want a web, fitting, etc.

In aircraft at least, you will have wires and systems installations that run along the length of beams. In these cases, the flange is sufficient to support the load. It wouldn't be economical to add webs everywhere you want to hang a light load.

Forgot to mention, I believe Niu's analysis book has a curve that is based on test data for these types of problems as well. It better addresses the effect of x (i.e. the apparent dilemma Guideon observed). If the load is close to the corner, you get an improved benefit due to the fastener head, etc. The 45 degree rule works better when x is relatively large.

Brian

http://www.espcomposites.com