Truss bridge designs compared: Warren, Pratt, Howe and K trusses

Load a 12 m truss that stands 3 m high with 20 kN spread over its bottom joints, and four layouts that all look sensible on paper turn into four quite different bridges. One of them goes at 55 kN. Another carries 166 kN before a single member fails. Truss bridge designs differ less in outline than in where they put their long members, and a long member in compression is the most expensive thing in a truss.

Three of the four layouts carry the names of the people who patented them. William Howe patented his in 1840, with timber diagonals and wrought-iron vertical rods, and early railroads built many of them. Thomas and Caleb Pratt patented the reverse in 1844. James Warren’s triangulated layout, with no verticals at all in its basic form, was patented in England in 1848. The fourth, the K truss, is defined by its shape: verticals split at mid-height, with pairs of diagonals meeting there.

Truss bridge designs side by side on one load

For the comparison every layout got the same 12 m span, the same 3 m height, six panels, and 20 kN divided into five equal 4 kN loads on the interior bottom joints. Every member is the same steel tube, 60 mm across with a 3 mm wall. Six panels is the count of the Pratt and Howe presets, and the K presets use an even count so that a split vertical stands exactly at mid-span; the Warren and K rows are the six-panel presets, warren6 and k6.

Layout	Members	Joints	Largest tension, kN	Largest compression, kN	Failure load, kN	Frame mass, kg
Warren	23	13	11.3	12.0	81.9	253
Pratt	21	12	10.7	12.0	55.3	239
Howe	21	12	12.0	12.0	55.3	239
K	36	19	11.7	12.3	165.8	316

Look first at the forces. They are nearly the same: no layout asks any member for more than about 12 kN in either direction, because the load, the span and the height fix the moment the chords have to resist. The layouts differ in failure load by a factor of three, and the forces cannot explain that. Length does.

Why compression members decide the result

A steel tube in tension fails by yielding, and the 60 mm tube yields at 134 kN whatever its length. In compression it buckles first, and the Euler load, P = π²EI/L², falls steeply with length (the tension and compression page gives the rule): 69 kN at 2.5 m, 43 kN at 3.16 m, 33 kN at 3.6 m. Each layout in the table broke at its longest heavily pushed member. The Pratt and Howe end posts are 3.6 m long and carry 12 kN, so they buckle at 55 kN of applied load. The Warren’s end diagonal is 3.16 m. The K truss ends in a 2.5 m diagonal, which is why it holds out to 166 kN.

The K truss pays for that with 36 members and 316 kg. Per kilogram of frame it still comes out ahead: 0.53 kN of failure load per kg, against 0.32 for the Warren and 0.23 for the Pratt and the Howe.

The Pratt and the Howe give identical numbers in the table, and that deserves a straight explanation. With one tube size everywhere, the end post governs both. Upgrade only the two end posts to the next tube, 90 mm by 4 mm, and the picture splits: the Pratt then fails at 180 kN, the Howe at 92 kN, both at 270 kg. The Howe’s diagonals are in compression and are 3.6 m long, so once the end post is no longer the weak point a diagonal is. The Pratt’s long diagonals pull, and a pulled member does not buckle.

Which layout to build first

Of these four truss bridge designs, build the Warren truss bridge. It has few members for its span, every diagonal is the same length, and every force in it can be found by hand with the method of joints in a few minutes. The table also shows it is a good bargain: 0.32 kN of failure load per kg, better than the Pratt or the Howe.

Move to the Pratt when the aim is to learn why the diagonals are thin, and to the Howe to see the same bridge with its tension and compression swapped. The K truss comes last: it is statically indeterminate by one degree, so equilibrium alone cannot find its forces.

Where each layout fits

Among practical truss bridge designs the Warren suits short spans and model bridges, where few members mean few glued or bolted joints. The Pratt suits steel, because its long diagonals are the members that pull. The Howe suits timber, which is strong in compression and awkward in tension joints, with iron rods doing the pulling. The K truss is a layout for tall panels, where a single diagonal across a panel would be too long to resist buckling.

Every preset is in the truss bridge designer, and the truss analysis pages show how the forces behind the table are worked out by hand. The four-panel K preset, for one, goes at 68.7 kN, at the 3.35 m diagonal beside the support.

How these figures were computed

Each row is one of the designer’s presets: Warren and K at six panels, Pratt and Howe at six, all at 12 m span and 3 m height, with 20 kN split equally over the five interior bottom joints. The designer also offers four-panel Warren (48.0 kN) and K (68.7 kN). Forces come from the direct-stiffness solver, which was checked against hand calculations. Failure load scales the load until the first member reaches its capacity: yield in tension, the lesser of yield and Euler buckling in compression, with E = 200 GPa and 250 MPa yield. Self-weight is left out and joints are ideal pins, so real bridges fail earlier. The K truss forces depend on member sizes because it is indeterminate; the table uses equal tubes throughout.