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Re: Calculation of flange stresses for underhung trolleys

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> From: Ahltomjo(--nospam--at)aol.com

> (page 9), had a question concerning the design calculation of the lower
> flange loading capacity of a steel beam to be used to support an underhung
> crane.  It further asked if there are any published ASD or LRFD design

I have been through this same analysis a couple times. I was surprised,
the first time, how high my stresses seemed to be. However, I find that
the results of this "simple" analysis and those based on the example in
Roark's (infinite continuous cantilever plate rigidly supported on one
edge) are very similar.

I also believe that you need to consider bi-directional stresses due to
the bending moment in the member resulting from the crane load. Also the
intersection of the web and flange of a hot rolled memeber are areas of
residual tensile stresses. All this tension and no mention of fatigue
cracking!

> He further investigates the capacity based on flange bending using the
> section modulus of an  "equivalent cantilever" beam whose effective width is
> 2xe (e is the eccentricity of the load measured from the k1 point as 
> presented in the AISC Manual) and whose depth is equal to the thickness of
> the flange.  The allowable moment is equal to the section modulus times the

Remember, it's just a model, reality is not so clean about the
rectangular beam that you describe. I prefer to think in terms of a
triangular cantilever with the base at the k1 radius. Net result is the
same section in bending at the web but different transition issues in
the flange.

> calculation.  The "equivalent cantilever" beam is considered to have free
> unsupported longitudinal edges.  The moment is equal to the local load times

An easy assumption. The actual bending will occur along (yield) lines
much closer to the point of wheel support. 

> the eccentricity.  The stress calculation results in e cancelling out in the
> numerator and denominator and is thus independent of the eccentricity.  In

This is very convenient because you don't need to know anything about
the trolley wheels across the flange.

> decreases.  The result is a very low capacity and seems to be unrealistic and
> very conservative.  It would seem that taking into account the continuity of
> the longitudinal edge of the beam (instead of assuming it to be a cantilever
> slice) would dramatically influence the capacity calculation. It would also
> seem to be more realistic to use a greater effective width of the "equivalent
> cantilever" member.  Possibly, something more like the 3.5 x k value used in
> determining the tensile effective area.

Superposition of the restraint by bending of the flanges at the end of
the assumed cantilever will be small thus rendering the simple analysis
somewhat conservative. One could go so far as to include the tensile
(catenary) stress in the flange and the general tendency of the outer
edges of the tension flange to curl toward the NA. Lots of work for very
little return.

Don't forget, there is probably another pair of wheels, not far away,
that is doing nasty things to the flange on one side of the cantilever
model. Add this to the fact that everything changes as the trolley moves
along the beam and especially near each end ...

> Is there a more realistic assessment of the flange bending capacity which
> accounts for the continuity of the "equivalent cantilever" beam's edge
> restraint?  More importantly, is there a prescribed ASD or LRFD procedure?

The best thing is for you to do the math on your proposals and determine
the influence on the outcome. If you think that you can get significant
returns, I would be glad to review it.

-- 
Paul Ransom, P. Eng.
Civil/Structural/Project
Burlington, Ontario, Canada
<mailto:ad026(--nospam--at)hwcn.org> <http://www.hwcn.org/~ad026/civil.html>