I’ve used the phrase “boundary conditions” in a number of blog posts, and it occurred to me recently that (1) its meaning is probably not clear to non-engineers, (2) its meaning to most engineers doesn’t help them understand the way it applies to our somewhat oddball work, and (3) it plays a vital role in how we investigate existing buildings. So here’s a belated discussion.

As I’ve said many times, any form of beam analysis relies on simplifying assumptions. Fortunately, those assumptions are close to reality most of the time and are typically conservative when they are not. To keep from getting bogged down in minutia, I’m going to use the standard assumptions below: the beams are made of a perfectly elastic material (steel comes close), the beams have a doubly-symmetric shape so they are not subject to warping when loaded concentrically (a solid rectangle, rectangular tube or wide-flange shape loaded by continuous floors resting on top come close), plane sections remain plane, etc. These assumptions all matter, and if they are not true the results can be drastically different. So an obvious corollary to the next paragraph is that field investigation has to check to see if there is visible behavior or other reason to believe these assumptions are not true. Finally, let’s assume that all of the beams being discussed for the models are the same size, which would be true if, say, I had measured a beam’s geometry and concluded it was an 12-inch-deep American Standard I-beam weighing 32 pounds per foot. That beam has been around in exactly that form since 1896 and is still in use, just barely, today. There were similar beams in use since the 1870s.

Say I’ve found that 12-inch beam in an area where the floor framing layout is not exposed, and I’m wondering what it means. Here are the analysis diagrams for two possibilities. First, a simply-supported beam

Second, a cantilever:

In these diagrams, the beam is the same and the floor loading is the same, so E and I are the same in both cases. If I know the length of the span, I know L (unlike whomever drew these versions of the diagrams, I’m going to make it easy to read by using the upper-case L); if I know the spacing of this beam from its neighbors, I know W. If I know all that, I can make a good case for it being either a simply-supported beam or a cantilever, since the simply-supported beam has one-quarter the maximum moment if everything else is equal. Run the numbers on both cases, see which it agrees with. If I know the length but not the spacing, or the spacing but not the length, I can come up with the scenarios where the two different sets of end conditions work.

These line diagrams don’t look much like real beams to an engineer and they don’t look anything like real beams to non-engineers. The simply-supported beam has the boundary conditions that no moment can be transferred at its ends, but vertical shear can be. The cantilever beam has the boundary condition on its left end that neither moment not shear can be transferred, and on its right that both can be. That’s it: the previous two sentences are, in terms of analysis, the entire difference between the two different beam types. But the actual conditions are not necessarily as simple as those three idealized end conditions (in order: simple support, free, and restrained.)

Free ends can and often do resemble the idealized model. The edge of a cantilever balcony or marquee is unsupported (in fact, not in contact with any other structure) and can transfer neither moment nor shear. That can get complicated quickly if there is something there that is not meant to be structural but has some structural capacity, like posts running from balcony to balcony on a high-rise to carry handrails.

Simple supports almost never look like the model. You can create a true simple support easily enough – lay a beam across a couple of sawhorses – but in real structure beams are typically connected at the ends and those connections are never perfect shear-only moment-free transfers of load. Similarly, the idealized restrained support has zero rotation under load but real-life versions usually have some rotation. A beam embedded in brick walls at each end was almost certainly meant to be simply supported, but the embedded ends may be able to transfer a substantial moment to the walls. Such a beam will over-perform the simply-supported model, at least until the end moments damage the brick. This matters because trying to determine the load on the beam by reverse-analyzing it (starting with the maximum defection – D in the formulas above – and working backwards to find the moment M and load W) will give a wrong answer if the deflection has been kept low because the beam is getting some help from sort-of cantilevering in addition to the regular beam action.

More to follow…

The picture at the top, in case you were wondering, is marker number 123 along the border between Arizona and Mexico, as photographed in the early 1890s.

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