A fink truss with gusset-plate joints

Look at the joints in Figure 1 before touching a single equation. Two kinds are drawn. At the apex, where both rafters end and the center web arrives, and at both supports, where rafter and bottom chord end together and the reaction lands as one more force, three or more members converge on a single point — and the drawing shows a gusset plate: one flat plate, every member lapped onto it and bolted or welded through it. Everywhere else — the two rafter mid-points and the middle of the bottom chord — a plain pin disc is drawn: there the chord simply passes through, continuous, and the webs hang onto it. Where members must end and be joined, a plate; where they run through, a pin.

Now the honesty ruling, and it is part of the engineering, not a shortcut: the drawing shows gussets, but the analysis idealizes every joint as a frictionless pin. A real gusset does add some moment stiffness — the members can no longer rotate freely against each other — but ignoring it is conservative for the member forces: the axial forces we compute are the full load path, and any moment the plate attracts only shares the work. Pretending the plate is a pin spares us a frame analysis and costs us nothing we care about. That one idealisation is the method of joints.

The truss is symmetric and symmetrically loaded — 12 kN of purlin load at each of the three rafter nodes, 3 × 12 = 36 kN in all — so the supports share it equally:

With every joint declared a pin in Step 1, the method of joints is open for business: walk from pin to pin, always picking a joint with at most two unknown members, and let each joint’s two equilibrium equations settle everything on the spot. The rafter slope is (sin θ = 0.4472, cos θ = 0.8944) — hold on to those two numbers; they run the whole solution.

Start at joint 1 — a gusset in the drawing, a pin in the analysis — where only two members meet the reaction. Vertical equilibrium finds the lower rafter:

Compression — the rafter pushes the reaction back up the slope. Horizontal equilibrium then hands over the bottom-chord tie:

Climb to joint 4, the rafter mid-point with its 12 kN purlin load — a plain pin, the rafter passing straight through. Resolving along and across the rafter:

The rafter’s compression eases from 40.25 kN below the purlin to 26.83 kN above it, while the inclined web props the rafter from below and carries that joint’s share of the load down toward the bottom chord.

At the apex, joint 5 — the biggest gusset on the drawing, still just a pin to the analysis — symmetry makes the two upper rafters equal at −26.83 kN, and their horizontal thrusts cancel. Vertically, their upward push exceeds the 12 kN sitting on the ridge, and the center web takes up the difference in tension:

The center web hangs the apex load down to the bottom chord — a tension member doing a hanger’s job. That leaves joint 2, the bottom-chord middle, which we never used: every force arriving there was found elsewhere, so its equilibrium is a built-in audit. It passes: the two outer webs’ verticals () balance the +12 kN hanger, their horizontals cancel, and the two 36.0 kN chord panels run straight through. Symmetry fills in the mirror half — members 5-6 (−26.83), 6-3 (−40.25), 6-2 (−13.42) and 2-3 (+36.0) match their twins — and all nine members are on the books.

Put this truss next to the 12 m Fink roof truss example: the span grew from 12 to 16 m and the rise from 3 to 4 m — the whole geometry scaled by 4/3, so the slope is identical — and the purlin loads grew from 10 to 12 kN, a factor of 1.2. Now compare the answers, member by member:

Every force is exactly 1.2× its smaller twin — the load factor, and only the load factor. The 4/3 geometric scale changed nothing, because every joint equation in Steps 2–4 was written in terms of sin θ and cos θ, and scaling every coordinate leaves every angle alone.