Columns

A column carries load along its length, from the end it is pressed on to the end it stands on. It is the simplest job in a structure, and short, thick columns do it simply: they get squeezed, and if the load is large enough the material itself gives way. Long, slender ones fail differently and much sooner. They bow out sideways and fold, while the stress in the material is still well below what it could take. The frame above tests four identical struts side by side, so you can watch that happen.

Push on a short block of wood and the only thing resisting you is the material. Its fibres shorten, and at a stress around 4,000 psi (28 MPa) for structural Douglas fir they fold over in a band across the grain: it crushes. The load it takes is simply that stress times its area, however the ends are held.

A long stick behaves differently. No stick is perfectly straight and no load is perfectly central, so the load always pushes slightly off the line of the wood. That bends it a little, which moves the load further off line, which bends it more. Below a certain load the stick’s stiffness wins and it settles at a small bow. At that load, the buckling load, stiffness no longer wins: the bow runs away, and the stick folds. Nothing in the material has failed yet. Buckling is a failure of shape, not of strength, and it gives little warning.

Raise the load in the figure and watch the struts. Each one bows a little from the start, and the bow grows faster as its own buckling load gets near: at half that load it is twice its starting crook, at nine-tenths ten times. Then the strut goes over, and its load cell stops climbing, because a buckled strut can carry no more.

Leonhard Euler worked out the load in 1744. For a straight, elastic column pinned at both ends:

Nothing in it says how strong the material is. A weak material and a strong one of the same stiffness buckle at the same load. What matters is stiffness, the shape of the section, and above all the length, which enters squared: double the length and the column buckles at a quarter of the load.

The section matters through I, and a column buckles about the axis where I is smallest. A 2×4 is 3½ in deep one way and only 1½ in the other, so it bows across its thin side, as every strut in the frame does. Engineers fold area and inertia into one number, the radius of gyration r = √(I/A), and measure a column by its slenderness ratio, length over r. For a rectangle r is the thickness divided by √12, so timber codes use the simpler length over least thickness, L/d, and cap it at 50.

Euler’s formula is for pinned ends, free to turn. Hold the ends differently and the buckled shape changes, and with it the load. The trick is to find the stretch of the column that bends like a pinned one: the length between the two points where its curve changes direction and the bending moment is zero. That stretch is the effective length, kL, and it goes into the formula in place of L. Light Effective length in the legend to see it marked beside each strut.

Pinned

k = 1.0. Both ends turn freely. The strut bows into a single arc, and the whole length counts.

Fixed–pinned

k = 0.7. The socket at the foot keeps it square, so the curve has to turn back near the bottom. Only the upper seven tenths bends like a pinned column.

Fixed–fixed

k = 0.5. Held square at both ends, it bows in an S at each end with a single arc in the middle half. Half the length, four times the load.

Fixed–free

k = 2.0. A flagpole: fixed at the foot and free to sway at the top. It is half of a pinned column twice its height, the other half being its mirror image below the base. Its dimension line runs on, dashed, into the floor.

For the four 2×4 struts in the frame, each 8 ft (2.44 m) between the points where its ends are held:

Same wood, same length: the ends alone make a sixteen-fold difference. None of the four comes close to crushing, which would take 93.5 kN (21,000 lb). The k values are the textbook ones for perfect fixings. Real sockets and base plates give a little, so design guides use more cautious figures: 0.65 for fixed–fixed, 0.8 for fixed–pinned and 2.1 for the flagpole.

Switch the member to a 4×4 and then a 6×6. Thickness raises the buckling load much faster than the crushing load, because I grows with the cube of the thickness and the area only in proportion to it. In the 4×4 the flagpole, the pinned strut and the fixed–pinned strut still buckle, at 25, 101 and 207 kN, but the fixed–fixed one would need 404 kN to buckle and crushes first, at 218 kN (49,000 lb). In the 6×6 only the flagpole buckles, at 154 kN; the other three all crush at the same 539 kN (121,000 lb), because crushing does not care how the ends are held.

The figure uses the classic picture: a column fails at its buckling load or its crushing load, whichever is lower. Real columns in the middle range, like the 4×4 fixed–pinned strut whose two loads nearly meet, fail below both, because the bending from the bow adds to the direct squeeze. A real one would give way at roughly two-thirds of either figure. Design codes blend the two limits into one curve that runs smoothly from short columns, governed by strength, to long ones, governed by Euler.

A load that misses the centre of the column adds bending to the squeeze: one face is compressed more than the other. If the load stays inside the central zone of the section, the kern, every part of the section is still in compression. For a rectangle the kern is a diamond reaching a sixth of the width each way from the centre, so a load within the middle third of each side is safe from tension. Push it further out and the far face starts to pull apart. Wood and steel can take that; unreinforced masonry cannot, which is why piers and walls of stone and brick are proportioned to keep their loads in the middle third.

Nobody builds with a free-standing 8 ft 2×4 post. Its slenderness of 64 is past the code limit, and even with pinned ends it would buckle at under 2,000 lb. Yet the same 2×4 stands in every wall of a wood-framed house, carrying the floors and roof above. The difference is the sheathing and drywall nailed to both faces. They stop the stud bowing across its thin side, so it can only buckle the other way, across its 3½ in depth, where its slenderness is 27.

Bracing is the cheapest way to strengthen a column. A single brace at mid-height halves the effective length and multiplies the buckling load by four. It is why tall steel columns are tied into the floors at every storey, why built-up wood columns are nailed together along their length, and why a stair or a balcony post that looks too thin is often fine once it is fixed at both ends.

The struts are Douglas fir-larch No. 1 at the dressed sizes of each member, with a stiffness of 1.7 × 10⁶ psi (11.7 GPa) and a crushing strength taken as 4,000 psi (27.6 MPa), typical for a short test on structural lumber; the design value the code allows for this grade is 1,500 psi. Each strut starts with a crook of a thousandth of its length, about 2.5 mm, magnified by 1 / (1 − P/P_E) as the load rises. The readout gives the true bow. The drawing shows it five times larger, eased off toward a cap so that a buckled strut stays on the page; a real one would keep bending until it snapped.

Every strut in the frame carries the same load from its own screw jack, measured by its own load cell. The steel frame closes the loop: the jacks push down on the struts and up on the crosshead, the struts push down on the base beam, and the two columns hold crosshead and base beam together in tension. The free top of the flagpole sits on a rocker under a plate on rollers, so it can lean and slide while the load still pushes straight down.