The examples of analyzing the structure of a building as you investigate that I discussed over the last two days were fairly straightforward in terms of calculations and were fairly forgiving in terms of accuracy. Today’s example is not, because it contains everyone’s favorite way to have numbers not work: non-linearity.

If you look at the beam formulas I was talking about before, they are linear with respect to most of the variables. They are linear with respect to load: double the load and you double the shear, moment, shear stress, and bending stress. The deflection calcs are linear with respect to the beam moment of inertia; the bending stress is linear with respect to the beam section modulus. The big place that these formulas are non-linear is the beam span: doubling the beam span increases the bending stress by a factor of four and the deflection by a factor of eight.

Some calculations are inherently non-linear. The interaction between compression and bending is a good example: columns (or other compression members) can fail in buckling and the likelihood of that kind of failure increases if the column is not straight. Bending moments make members not straight. So a beam-column (a column that is subject to some degree of bending in addition to the compression along its length) is weaker in compression than a column with no bending. Early steel codes did not really take this into account, but simply added the maximum compression stress from overall compression to the maximum compression stress from bending and compared that to the allowable compression for the column. More and better research into how columns act led to changes in the codes, starting in the mid-1900s, and we have now have interaction formulas. (See equation H1-3 in the AISC Specification for Structural Steel Buildings to see how ugly this can get.) Playing around with those equations, it becomes clear that they are badly non-linear, so that a small increase moment will greatly increase the combined stress when the compression is near the allowable limit. This is taken into account in design all the time, so it’s not a big problem with new buildings.

What happens with existing buildings? Everything’s fine as long as the steel is in good condition and you’re not changing anything. Of course, if that’s true, why did the owner even hire an engineer? A lot of low- and mid-rise steel-frame buildings (in the past and now) have moment frames, where the columns are designed to carry both moment and compression; but the old buildings were designed, by our standards, incorrectly, without taking interaction into account. So if we have to re-analyze the columns we may get a nasty surprise. (Or maybe not, since we tend to allow higher stress in old steel, under our current codes, than the old codes did.) What if a column is at the facade and has rusted a bit? The asymmetry created by the rust, or the asymmetry created by a lot of possible repairs, causes a new bending moment in the column. Or take the example in the picture above: we have a cantilever transfer girder to allow a column to offset. (This is in a factory that made planes during World War II, and the alteration that created the offset was to allow for the production of larger planes.) If the girder is free to rotate over the top of the lower column, treating the column as a pin support for that beam, then no moment is imparted to the column. But if the girder-to-column connection is not free to rotate, the column is going to see bending moments. That connection could have been given some weld later on by someone trying to strength it, or the rivets could have rusted a bit and their expansion jammed the column cap plate tight to the girder flange.

In short, when dealing with structure that tends towards non-linear response, like columns, we need to be more careful and more accurate than with ordinary beam.