Combined shear and bending stress

[JSPILLER3]

I am designing a retrofit in a power plant and I wish to add a single point load to an existing beam. It is a simple span or at least being solved as such so it should be fairly straight forward. However I am having some doubts about the shear and bending allowable not being additive. The interaction equation is section H is straight forward about axial stress combined with bending, but I have yet to find something for shear and bending. ( I believe this could be because typically shear and bending stresses rarely share their maximums at the same point.) The beam is nearly at its maximum for both shear and bending, so the interaction equation if applicable would be >1. Is that applicable or is there a better code or equation to follow. Thanks for any help you can offer.

John Spiller

 

[jayrod12]

If you think about where the shear and bending stresses occur on the cross section then that should answer your question. I would assume for bending you are looking at the elastic section modulus S and not the plastic section modulus Z and for the shear capacity you are only accounting for the material in the web and not including what is in the flange. Using those values when designing instead of the whole section resisting each stress than you don't need an interaction diagram.

 

[JoshPlumSE]

Some foreign codes have combined force equations for bending + shear. Eurocode is the one that comes to mind. I haven't looked at it recently, but I think it comes into play more for deep girders than for a typical rolled section.

 

[JSPILLER3]

Yes, that is correct, bending will be for an elastic section and shear strictly through the web. My concern arose when looking at the stress diagram across the cross section. Shear is practically nothing at the flanges, which is the highest stress in bending. Once you get just below the flange though, do you not have an area of additive stress concentrations. The top of the web (just below the flange) is at a maximum and constant stress along the web for shear, and still relatively high for bending.


I have been reading maguire and other text, but I may be over thinking things now or lack the depth of understanding at the moment.

 

[rb1957]

you're right, below the flagne there can be high bending stress (< max) and high shear stress (< max).

personally, i'd combine them as a principal stress.

you beam is pretty dodgy if this is your concern

Quando Omni Flunkus Moritati

 

[JSPILLER3]

I am determining the required reinforcement(if any) for an existing beam I am adding load to.

 

[rb1957]

nest angles into the I flanges ?
for something like 1/2 the span ??

Quando Omni Flunkus Moritati

 

[JSPILLER3]

For those interested, this was the answer from AISC on the matter.

Interaction between shear and normal stresses need not be considered in the design of main members (or connection elements). This has been discussed in committee a number of times and we have always decided to neglect this interaction. If you decide to incorporate some type of combined shear and bending check, you will have to decide what approach is best based on your own judgment.

You could look to the following for some guidance. Part 10 in the 14th Edition AISC Manual include a combined shear and moment check on Page 10-104. The form used is (Vr/Vc)^2 + (Mr/Mc)^2 <=1.0. You could also refer to Section H3.2 in the AISC Specification (a free download at

http://www.aisc.org/2010spec)

as there is an equation provided that combines axial, bending, shear and torsion. For just axial and moment, this would look like (Vr/Vc)^2 + (Mr/Mc) <= 1.0.

The commentary for Section H3.2 provides additional guidance stating “Several interaction equation forms have been proposed in the literature for load combinations that produce both normal and shear stresses. In one common form, the normal and shear stresses are combined elliptically with the sum of the squares (Felton and Dobbs, 1967). In a second form, the first power of the ratio of the normal stresses is used. The latter form is somewhat more conservative, but not overly so (Schilling,1965), and this is the form used in this Specification.” Equations for each of these are also provide in the Commentary and match with what is shown above.

 

[jayrod12]

Thanks for that JSPILLER3, I guess if you are concerned you could limit each stress to 70% of max which when using that eq would work out to just less than 1.0

 

[CELinOttawa]

We were just discussing this in another thread, however I think you've gotten your answer...

Should it be of interest, there are cases where the failure to include a combined actions check has resulted in failure. It is rare since you have to also eat through all of the load factors, resistance factors, inherent additional capacity (ie: Most sections have Fy greater than the specified minimum without excessive residual strains limiting the effectiveness), etc.

I routinely apply combined actions formulae for situations where I have high moment or high shear, and believe that the "old standard" of neglecting this is a high-visibility failure away from change.