Analysis of Short Cantilever Beams

[TJAvery]

I'm looking for a simple method (i.e. hand calculation, not FEA) for finding the shear and bending stresses in a short cantilever beam. In particular, I'm interested in span/depth ratios of 1.0 or less.

Roark's (6th edt.) has some info in article 7.10. However, it is not helpful for my simple cantilever.

Is there perhaps a simple SIF that I can apply to the traditional equations:

bending stress = M·c/I
shear stress (ave) = F/A

Many thanks in advance.

-TJ Avery

 

[EnglishMuffin]

The bending stress that you quote is at the extreme fibre. However, the shear stress is zero at that location and the maximum is likely to occur close to the neutral axis (see Roark 7.1). To determine that maximum, there is certainly a factor which needs to be applied to the average shear stress, which depends on the shape of the section (see Roark 7.1 again, or for a derivation see Timoshenko's strength of materials for example). Just how fancy are you trying to get with this ? Do you have FEA ?

 

[EnglishMuffin]

Sorry - my first sentence should have read "The bending stress that you quote is a maximum at the extreme fibre".

 

[EnglishMuffin]

Also, by the way, there is another thread currently open which appears to be addressing a somewhat related topic :
thread507-78355

 

[TJAvery]

Thanks for your replies.

I'm not trying to get fancy with this, or else I'd use FEA I'm trying to do a simple hand-calc to check the work of others.

I'm just concerned that the very low ratio of span/depth (1.0 or less) is outside the validity of the equations I stated in my original question.

Roark's states this is true, and that actual stresses are higher than predicted with those equations. However, Roark's does not give equations that cover my particular problem with a short cantilever.

Roark's gives a table for a simply supported beam and also an example of a gear tooth. Neither of these apply to a simple, short catenary beam. (well, the gear tooth does, but the equation is very complicated and the geometry is complex as compared to a rectangular beam)

So, can I use the equations I stated and expect +/- XX% of accuracy? Or should I seek out other guidance to provide some sort of factor that increases the calculated stresses to a more realistic value?

many thanks,
TJ Avery

 

[JWB46]

Hi TJAvery

Is this a steel beam and are you concerned about strength or do you need stresses for fracture mechanics or fatigue assessment or something like that?

 

[ishvaaag]

You may resource to equations used for the design of Tee Seats in steel. You may look for orientation in the Mathcad Collaboratory site, Civil engineering folder for a

Steel welded corbel

worksheet.

Most likely the more critical item is the strut in compression in the web. You may look for every resource on such kind of inclined struts for K and width, since there is ample variation of opinion. In one of such resources the width of the inclined strut in the web is 1/8 of the total projected available width in such direction. And 2/3 of the length at the axis of such strut is taken as buckling length. This may still be conservative on both accounts.

 

[TJAvery]

Hi JWB46,

Yes, this is a steel beam. I'm just performing a quick check to assess strength. I'd like to calculate stresses (e.g. bending, shear) accurately for the short cantilever.

thanks,
TJ Avery

 

[ishvaaag]

Form my viewpoint for length 75% of the depth a moment concept is inappropriate. Things of these dimensions use to be recommended lately be looked at in capacity design viewpoint. Just as corbels, "true" deep beams and cantilevers, rigid footings etc in reinforced concrete. The St. Venant, Bernoulli, Beam of Young etc hypotheses do not hold for these short members, and these are quoted at the start of the theories introducing moment and shear for linear elements.

Look at

http://www.engin.brown.edu/courses/en175/airy/airy.htm

For rectangular sections tin heory of elasticity there are closed form solutions for stresses for a point load at the tip and uniform load in the cantilever. I have not seen one for double tees.

So except some set closed formulae holds the meaningful checks, one would have to go FEM to talk in terms of stresses, and one would have to think beyond that about stability of the thing. For a small number of these items, and if of moderate proportions, one and the owner himself may fare better by going conservative when in uncharted territory.

 

[haynewp]

My first thought was to design it like a stiffened seat like Ishvaaag said, that's if it is a steel wide flange or tee section you are using. Salmon and Johnson's "Steel Structures Design and Behavior" has examples on these.

 

[JWB46]

Hi,
Given that it is only strength you are interested in, you don't really need exact stresses. Normal beam theory is not strictly exact, as a few of the guys have said.

I would suggest you invoke the 'lower bound theorem of plasticity'.
The gist of this is as follows : if you can set up an equilibrium system (ie stress distribution that is in equilibrium with the applied loads) and the yield stress is not exceeded in this system then the assumed system is safe, in the sense that the true failure load will be greater. The only assumption in this is that displacements are small (i.e., it doesn't apply if buckling is an issue.)
If you use 'allowable stresses' rather than the yield stress then generally you can take it that the buckling issue is ruled out (e.g., d/t ratios taken into account etc).
An example of this is the very common practice of allowing beam flanges to take all the bending and the web to take all the shear either as a shear panel or as an inclined strut, again as one of the guys has implied earlier.

This lower bound theorem is totally legit and provable and is the basis for what used to be Cl 4.3.5 in API RP2a and similar clauses in other codes.
Hope this helps. Good luck anyway