Structural design of a fabricated I-beam composed of spliced web plate, flanges and L-stiffeners 1

[anedelcu2002]

Hello everyone! I have been tasked with designing an I-beam out of a web plate, horizontal flanges and L-shaped corner stiffeners (angles?) as individual parts, with bolted connections being the only ones allowed. The corner stiffeners transmit the loads and hold the web and the flanges together. The objective of the beam design is to resist a nominal bending load (expressed as a distributed force load) while also minimising weight.

While this should be simple in practice by approximating the beam as an integral structure and using bending stress/moment of inertia calculations, the crux of the design problem is that the individual, physical parts I have to use are all approximately two thirds of the expected length of the beam. This means that the web and the flanges will present a splice (or a "cut") at a certain point along the beam - obviously leading to a weakness in the load path transmission. Moreover, the maximum allowable weight constraint leads to an impossibility of placing stiffeners all along the length of the beam in all four corners. As such, a configuration has to be selected so that this fabricated/compound beam can be approximated as an integral I-beam. I have attached two pen sketches of the I-beam cross-section and of the beam loading respectively:

Having played with the idea, I've come up with the following design considerations. Firstly, the cuts in each member will have to be positioned away from each other - if two cuts overlap, a clear weak point occurs with only the stiffener carrying the load across. Secondly, the cuts/splices will have to be placed as far away from the base (clamping) of the beam (where the moment diagram shows the moment is maximum) as possible, so the weak spots are under as little stress as possible. Thirdly, the failure area will generally be the point of a cut, as the weakness resulting from a non-uniform load path will lead to earlier failure than column buckling of the web plate or pure shearing failure of the material. Finally, a stiffener segment will always need to be placed to cover a cut and streamline load transmission.

I have identified two possible failure points: the splice of the lower flange, where the tension from the bending loading could cause the bolts in the stiffener connection to shear off and cause a chain reaction of failure in the cuts; then, the splice in the web plate, where the shear loading could simply displace the web plate segments from one another, leading again to failure. Based on these failure conditions, I have considered two possible configurations: the first one with the web plate cuts at closer to the base of the beam, with the flange cuts further away from the base (lowering the load on the flange splices and thus the vulnerability to the first failure case); the second has the web plate cuts as far away from the base as possible, with the web cuts closer to the base. Both configurations would obviously have stiffeners covering and connecting the cuts and transmitting loads. A possible issue would be the length of the stiffeners, which due to weight constraints can't cover the entirety of the beam and as such might only run from the base to the first cut and then also cover a small segment surrounding the second cut.

This is a fringe design case and I did not find a lot of academic data regarding such constructions. Most of the above comes from intuition, so I would need a sanity check from more experienced engineers. As such, (finally) I have a number of questions:

1. Are my failure predictions for the beam correct? Is there another failure condition I have not considered which is more probable? Moreover, would the solutions I presented be useful in dealing with failure? (obviously, if you can find a better configuration, please say so)
2. Would it be more effective to run the stiffeners from the base to the first cut and then a small segment over the second cut or to cover both cuts with medium-length segments, allowing for some unstiffened web + flange portions (thinking of Saint-Venant's principle for the load distribution here)?
3. How could I estimate a configuration like the two I presented to be an integral beam? I am only a first year aerospace student, and my knowledge of translating such very real problems into theoretical models is very very limited.

If you're still here, thank you so much for reading (and hopefully for answering)!

 

[Settingsun]

That's a long post but I don't blame you. I sometimes forget the nonsense that students are put through in order to learn the basics.

First, I assume you mean 'supports' when you say 'base' of the beam. 'Base' is really only used for columns. Let me know if I've misunderstood.

I haven't put much thought into this. My instinct is to splice the web near the zero-shear point to minimise stress on that connection; and to splice the flanges as close as you can to zero-moment within the problem's constraints. There's no problem with splicing both flanges at the same location as that's quite normal - it always happens in practice when you have a proper beam (web + 2 flanges as a single hot-rolled piece cut to length). This is all quite normal in long beams, especially steel bridges.

One complication is that you can't run the corner angles full length. Therefore you want them near the supports where the shear force is largest in order to achieve composite action. In between, the axial force in your flange plates will be constant because you can't transmit shear flow between web and flanges [actually not necessairly true - see the edit at the end of this post], so the web needs to handle all the change in bending moment over that length. My guess is you need to splice the web at a location close enough to the supports that you also have the corner angles continuous from the support to beyond the web splice, but this is something I'm less confident about. I'll leave you to run the detailed calculations to prove me right or wrong.

Edit: a corner angle on only one side of the web probably does the job if you provide enough bolts. That might be the trick at the heart of the problem.

You could also have a pair of angles back-to-back vertically at the web splice to transfer shear force. A simple plate is used in practice but you have artificial constraints.

 

[anedelcu2002]

steveh49, thank you so much for your answer! Indeed, when I mean 'base' I mean the clamp supports (which might themselves impose some extra stress concentrations - horrible).

Indeed, intuitively it makes sense to splice the components further away from the supports, but the loading case (in both 'real' and test system) leads to the moment being larger as one approaches the supports, and with the shear as well. As such, the zero-moment and zero-shear points are both at the (non-clamped, free) end of the beam. The point you raised about the webs being spliced in the same point is interesting, but it makes me wonder whether the heightened stress in the angle stiffeners on the top and bottom would interfere and thus lead to earlier failure.

As for the corner angles, the testing system does not allow stiffener placement within a small distance from the base (probably to allow for clamping of the web), which leads me to believe that there is no point (beyond keeping the web and the flanges acting together with short one-bolt stiffeners) in covering the whole length until and including the web splice. I will probably use the solution I mentioned above (short angles with a single bolt keeping the members together) in conjunction with longer, multiple-bolt stiffeners covering the cuts and the areas around them generously.

I have run some preliminary calculations on the strength of the bolts with regards to shearing and tension, and the results show that failure in the bolts should not happen before the material of the stiffeners fails (stainless steel for the bolts versus aluminum 2024 T-3). As such, I expect the material of the set of double stiffeners (one on each side of the web) to fail before the bolts themselves do. To my (untrained) mind, this should mean that the stiffeners fail by shearing out of the bearing, which means I also need to consider doubling the set of bolts for each stiffener, placing them in parallel, and adding a generous margin of material in order to normalise the stress.

About reinforcing the web splice, I was thinking to maybe use pre-cut portions of the flange material as shear splices on each side. However, this would only be necessary if I was certain that the first area of failure would be the web cut, something I'm definitely not certain of. However, it would be much easier to consider the failure cases as stiffener-based and not member based - after all, the design problem, as you remarked, is the design of effective stiffeners. I think I need to do some plastic deformation and bearing stress calculations involving the local failure for a bolt: either the bearing material shearing off or just 'popping off' of the bolt.

Thank you for your input! I will try to split the design problem into several smaller problems I can solve with my limited knowledge, then iterate for weight reduction.

 

[Settingsun]

Hi, I hadn't spotted that it was a cantilever. I assumed simply supported span, so disregard comments relating to different position of peak moment and shear. I thought the moment would be maximum at midspan.

A one-side web splice plate is better than relying on the angle stiffeners alone for shear. Not necessarily impossible with just the stiffeners, but that arrangement is really a 'student' solution. We wouldn't do it with a 'real' structure without good reason - or more likely a bad reason that just can't be avoided.

If you're allowed to cut pieces out to save weight (and that gives you a better score), look at cutting holes in the web near the unsupported end.