Trusses

A beam carries its load by bending. Its top is squeezed, its bottom is stretched, and the material near the middle does little. A truss keeps only the parts that work hard: a top chord to take the squeeze and a bottom chord to take the stretch, set far apart. Between them runs a web of lighter members arranged in triangles. The result spans further than a beam of the same weight, and every member in it is simply pushed or pulled.

Three bars pinned at their ends make a shape that cannot change unless one of the bars changes length. Four bars make a frame that folds flat without any bar stretching at all, as the racking bay in lateral stability shows. A truss is a chain of triangles, each one stiff, sharing sides with its neighbours. Each rectangular panel in a flat truss is held square by its diagonal. Take the first diagonal out in the figure and see: the end panel folds without any member stretching, and the whole truss drops onto the cribbing set under it.

Because the triangles do the holding, the joints do not have to. Treat every joint as a pin that lets the members turn freely, put the loads on the joints, and each member can only be pushed or pulled along its length: compression or tension, never bending. That is what makes a truss efficient. A member in pure tension or compression uses its whole cross section evenly; a beam uses mostly its top and bottom edges.

Chords

The members that run the full length of the truss, top and bottom. Under gravity the top chord is squeezed and the bottom chord is stretched, exactly like the top and bottom of a beam. Cut through any panel and the two chords form a couple that resists the bending moment there: chord force is close to the moment divided by the depth between the chords.

Web

The verticals and diagonals between the chords. They carry the shear, passing the load from panel to panel toward the supports. They work hardest near the ends, where the shear is largest, and least in the middle.

Panel point

A joint where web members meet a chord. The stacks of blocks in the figure sit on them.

Heel

The supported end of a pitched truss, where top and bottom chords meet over the bearing.

A truss is only free of bending if its loads arrive at the joints. A load set down halfway along a chord makes that piece of chord a small beam between two panel points, and it has to be sized for bending as well as for its axial force. That is why roof trusses carry purlins at their panel points, and why the blocks on the bench stand over the joints. For the same reason the centre lines of the members at a joint should meet at one point; when they miss, the joint twists them and adds bending they were not designed for.

Each stack carries the roof that its panel point gathers: on the parallel-chord trusses, 4 ft of span per stack; on the Fink, 6 ft, so its stacks are half again as tall. The two loads that would sit over the bearings go straight into the supports and are left off.

The truss on the bench is built as timber-and-iron trusses were in the nineteenth century: every member that is pushed is a timber, and every web member that is pulled is a steel rod. A rod can be thin because tension only straightens it. A strut must be stocky, because a slender member in compression bows sideways and buckles long before the material itself is crushed. So the drawing is its own force diagram: the thin lines are in tension, the thick members in compression. Switch between the two first trusses to see it flip.

Pratt

Diagonals slope down toward the middle of the span. Under gravity they hang the load from one panel to the next and are stretched; the verticals are squeezed. Its long members, the diagonals, are the ones in tension, which is the economical way round, so it is the usual pattern for steel trusses.

Howe

The mirror image: diagonals slope up toward the middle, so they are squeezed and the verticals are stretched. It suited timber with iron rods for verticals: the timbers bear on each other end to end, and nuts on the rods could be tightened to take up any slack in the joints.

Warren

No verticals, only diagonals forming a row of triangles, alternately pushed and pulled. Its diagonals are all the same length and its joints all alike, which makes it simple to fabricate. Verticals are sometimes added to shorten the panels of the top chord, which is in compression.

Some members drawn faint in the figure carry no force at all under these loads. In the Pratt, the end panels of the bottom chord: the reaction goes straight up the end post instead. In the Howe, both end posts, the end panels of the top chord and the middle vertical. In the Warren, the two diagonals at midspan, where the shear is zero. They are not wasted. They hold the joints in place, shorten the free length of chords that would otherwise buckle, and start working the moment the load is uneven: snow drifted to one side, wind, a hoist hung from one panel point.

The deeper the truss, the further apart the chords and the smaller the forces they need to resist the same moment. Halve the depth and the chord forces double. The web gains less from depth: the verticals of a Pratt carry the shear whatever the depth, and the diagonals fall toward it as they steepen. The figures below are for the Pratt on the bench, 24 ft between bearings, with four blocks on each stack (1.61 kN/m, 110 lb/ft of roof).

Past a point, more depth buys little and costs a great deal: longer webs, a taller building, more surface to clad. Flat trusses are commonly made between a fifteenth and a tenth of their span deep; pitched roof trusses take their depth from the roof slope, often a quarter to a fifth of the span, and bowstrings a sixth to an eighth.

Fink

The shape of a gable roof, with its web folded into a W. The short members are struts that prop the top chord at its midpoints; the long ones are ties running from the ridge down to the bottom chord. Here the chords work hardest at the heels, where the truss is shallowest. Under the same roof load, the top chord carries 14.0 kN at 4 in 12 and 8.0 kN at 8 in 12.

Bowstring

A curved top chord rising from a straight bottom chord. Here the curve is a parabola, the same shape as the bending moment under a load spread evenly along the span, and the result is striking: every web member carries exactly nothing. The top chord is an arch, the bottom chord is the tie that holds its feet together, and the force in the tie is the same in every panel. The webs are there for the day the load is uneven. Take one out and the bowstring still stands, because it carried nothing; but the open panel is now a mechanism held only by the balance of the loads, and the first uneven one would fold it.

Other shapes include Belgian trusses with only inclined webs, scissors trusses that raise the ceiling at the cost of much larger chord forces, crescents with both chords curved. They all work the same way. Choose the pattern so the long members pull and the short ones push, and give the truss as much depth where the moment is largest as the building allows.

On the bench every joint is a pair of ¼ in steel plates, one on each face, bolted through the timbers with ¾ in bolts; the rods are pinned between the plates. Heavy timber trusses are still made this way, or with split rings and shear plates set into the wood. Steel trusses are welded or bolted through gusset plates. The light trusses that frame most houses are made of 2× lumber joined by toothed metal plates pressed into both faces, as in trussed rafters.

A truss is stiff only in its own plane. Seen end on it is a thin sheet, and its top chord, squeezed along its length, will buckle sideways unless something holds it. In a building the purlins, decking and bracing do, and while trusses are being set, temporary bracing does.

The chords are 4×8 Douglas fir-larch, the timber struts 4×6, all 3½ in thick so the plates lie flat. The rods are A36 steel, sized for the heaviest load the bench applies; none needs to be thicker than ¾ in, the smallest size used. The forces come from a pin-jointed model of the truss solved by the stiffness method, with timber at 11.7 GPa and steel at 200 GPa. They depend only on the geometry and the loads, so they are exact for the ideal truss; the sag also depends on the stiffnesses, and a real bolted truss sags more, because each bolt slips a little in its hole before it bears. Two crib stacks of 4×4s stand under the bottom chord to catch the truss if it fails, as they do in any test lab. The Pratt weighs about 200 kg (450 lb), and that weight is left out, as if the gauge were zeroed with the truss in place. In the drawing the sag is shown thirty to sixty times larger than life, less as the load grows; the readout gives the true value.