Monthly Mechanics: Columns

Just as the vast majority of bridges are beam bridges, the vast majority of buildings are post-and-beam buildings. Horizontal and vertical elements are easy to build and maximize use of space, making them the only rational choice for all but deliberately fancy buildings like stadiums and opera houses. The horizontal elements are the beams, and they work mainly in bending. (We’ve explored beams a few times already.) The vertical elements are the posts, or columns.

(Smaller buildings, including most houses, have loadbearing walls as their vertical elements. You can think of a loadbearing wall as a spread-out column.)

columns2

If you peel back the finishes of a typical building, you’ll see a grid of columns like this one.

A well-designed column is subject to dead loads and live loads from directly overhead, based on the tributary area of the floor above. With no horizontal or eccentric (off-center) loads, a column acts in pure compression. In this way, columns are sort of the opposite of cables.

But unlike cables, which fail only by yielding, columns can fail in two ways. One is crushing, the compression equivalent of yielding. The other is buckling. Even without horizontal forces to induce bending, long unbraced columns will bend anyway.

columns_crushingbuckling

Crushing | Buckling

How do you know if a column will fail by crushing or by buckling? The answer brings together many topics from previous Monthly Mechanics articles. Crushing is governed by the equation Pc=fc*A, where fc is the material’s compressive strength and A is the column area. Buckling is governed by the equation Pb2*E*I / L2, where E is the elastic modulus, I is the moment of inertia, and L is the unbraced length.

columns_example

Let’s investigate the above concrete column. (Concrete columns are almost always reinforced, but for compression the rebar is negligible.) Crushing load is Pc = 4000 psi * 144 in2 = 576 kip. Buckling load is Pb = π2 * 360000 psi *1728 in4 / (120 in)2 = 426 kip. So this column fails first by buckling. A shorter or stockier concrete column would fail first by crushing. Steel columns almost always fail by buckling.

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