K truss bridge: short members for tall panels, and indeterminacy

A single diagonal across a panel 3 m wide and 3 m high is 4.24 m long, and a 60 mm by 3 mm steel tube that long buckles at 24.0 kN when pushed. A K truss bridge avoids that member. It meets the diagonals halfway up a split vertical, so the longest diagonal is 3.35 m and the buckling load rises to 38.4 kN. The cost is a more complicated frame, and statics alone no longer finds every force.

Twenty-four members around split verticals

The K truss preset spans 12 m, stands 3 m high and has four panels of 3 m. Each interior vertical is cut at mid-height, 1.5 m up, and the two halves are tied to the chords by pairs of diagonals that meet at that midpoint, one aimed at the top chord and one at the bottom. The members are the four segments of each chord, two end posts, six half-verticals and eight diagonals: 24 in all, over 13 joints.

The load is 20 kN in three equal loads of 6.67 kN on the interior bottom joints, and each support takes 10 kN.

One member too many: indeterminate by one degree

The other three presets are determinate: their members plus reactions equal twice the joints. The K truss is not. Here 24 members plus 3 reaction components make 27, while two equations per joint over 13 joints give 26. One extra member means one extra unknown.

In plain words: equilibrium says the forces must balance at every joint, and for the K truss that leaves a family of answers instead of one. A member that is a little stiffer takes a little more load, because it stretches less for the same force. So the actual forces depend on the member sizes, not only on the shape and the loads. A determinate truss gives the same answer whatever the tubes are; the K truss does not.

The solver in the designer handles this directly. It uses the stiffness of each member, so the figures below belong to the 60 mm by 3 mm tube in every member. Change the diagonals from that tube to 90 mm by 4 mm and the forces move, though only slightly here: the bottom chord drops from 13.0 to 12.9 kN, one inner diagonal rises from 4.10 to 4.24 kN. The frame mass jumps from 278 kg to 392 kg for a gain in failure load of about one kilonewton. Sizes, in short, belong in the question when you ask what a K truss carries.

Hangers, a zero member and the 3.35 m diagonal

The chords behave as in any truss: the bottom chord pulls with 13.0 kN in the middle, the top chord pushes with 13.7 kN, and both fall to 10.0 kN at the ends. The half-verticals beside the loaded joints hang them: B to K and D to M pull with 8.2 kN. At the centre, L to H carries 0.0 kN under this symmetric load, a member that would only matter if the load moved.

The outer diagonals do the heavy lifting. K to A pushes with 11.18 kN and K to F pulls with the same. The reaction of 10 kN at support A is split between the end post and the diagonal K to A, 5.0 kN of vertical force each, which is why the end post carries only 5.0 kN.

The governing member is that diagonal, K to A: 3.354 m long, 11.18 kN in compression, 29.1 percent of its Euler load of 38.4 kN at the applied load. The bridge fails at 20 × 3.433 = 68.7 kN. Tension members sit below 10 percent.

Where a K truss bridge earns its extra members

A K truss bridge is for panels that are tall compared with their width, or spans where a single long diagonal would buckle early. It earns its extra members by making every compression member short. In the full comparison of truss bridge designs, the six-panel K fails at 165.8 kN, three times the Pratt or Howe load and twice the Warren, for 316 kg against 239 and 253.

It is a poor first project. Joint by joint, the method of joints stalls where the extra member sits, and a model of 24 glued members has 13 joints to get right; the Warren truss bridge is the better start. Stiffening only the two support diagonals, K to A and M to E, buys little at four panels: the failure load moves from 68.7 to 70.2 kN, because the top chord at the middle takes over.