Fink roof truss under gravity loads
The truss over your head. A Fink (“W”) roof truss does one job elegantly: the rafters push, the bottom chord ties their feet together so the walls feel no thrust, and the W of webs breaks the rafters into short, buckle-resistant lengths. Here we walk it joint by joint — and find the tie carrying a pure +30 kN of tension while the walls feel nothing but their share of the weight.
- Span
- 12 mrise 3 m
- Pitch
- 1:2 (26,57°)
- Geometry
- 6 nodes · 9 members
- Supports
- Pin (1) · Roller (3)
- Loads
- 10 kN ↓ at 4, 5, 6gravity at top nodes
- Rafter
- 0.8944, 0.4472
Step 1 — Reactions by symmetry
Symmetry splits the 30 kN of roof load between the two walls.
The whole truss is symmetric and symmetrically loaded, so the two supports share the total gravity load of 3 × 10 = 30 kN equally:
No moment equation needed — symmetry hands the reactions over for free, and vertically that 15 kN each is all the walls will ever feel. The interesting question is horizontal: a pitched rafter pushes outward as well as down, and the next three steps trace where that thrust goes. Our tool is the method of joints — walk from pin to pin, always choosing a joint with at most two unknown members, so that its two equilibrium equations settle everything on the spot.
Step 2 — The eaves joint, where the tie is born
Two members, two equations — compression in the rafter, tension in the tie.
Start at the eaves — joint 1, where the pin, the sloping lower rafter, and the bottom chord all meet. Only two members join here, so its two equilibrium equations pin them both down. Vertical first (the rafter cosines are cos = 0.8944, sin = 0.4472):
The lower rafter is in compression — it pushes the reaction back up. Horizontal equilibrium then reveals the star of the show, the bottom-chord tie:
That +30 kN tension is what saves the walls: the rafter’s horizontal thrust is swallowed by the tie instead of shoving the wall outward.
Step 3 — Up the rafter and across the webs
The W of webs shares the work.
Climb to joint 4, the mid-rafter node carrying a 10 kN load. Resolving along and across the rafter gives the upper rafter and the first inclined web:
Both compressive: the upper rafter keeps pushing up the slope — notice the compression easing from 33.54 kN below the load to 22.36 kN above it — while the inclined web props the rafter’s midpoint from below, leaning that joint’s share of the load down toward the tie. At joint 5, the ridge (another 10 kN), symmetry makes the two upper rafters equal and the vertical web hangs the ridge load straight down onto the tie:
Symmetry now fills in the mirror half for free — members 5-6, 6-3 and 6-2 match their twins, and the far bottom-chord panel 2-3 matches 1-2. All nine member forces are on the books.
Step 4 — Joint 2 closes the books
The joint we never used is a built-in error check.
One joint remains untouched: joint 2, the middle of the bottom chord, where the hanger, both inclined webs and both chord panels meet. Because every force arriving there was found elsewhere, its equilibrium is a free audit of the whole solution — and it passes: the two inclined webs’ verticals () balance the hanger, their horizontal components cancel, and the bottom chord holds a steady +30 kN throughout. Every joint checks.
The proof
Hand calculation vs the solver, all 9 members.
| Quantity | By hand | StructureCalcs | |
|---|---|---|---|
| Reactions | 15 / 15 kN | 15 / 15 kN | |
| Bottom chord (tie) 1-2, 2-3 | +30 kN (T) | +30 kN | |
| Lower rafters 1-4, 6-3 | −33.54 kN (C) | −33.541 kN | |
| Upper rafters 4-5, 5-6 | −22.36 kN (C) | −22.361 kN | |
| Inclined webs 4-2, 6-2 | −11.18 kN (C) | −11.180 kN | |
| Ridge hanger 5-2 | +10 kN (T) | +10 kN |
Every value was worked by hand with the classical method, then checked against this site’s solver — the same engine the Try it button opens. This agreement is re-run automatically on every build.
Now make it yours
Open this exact model in the calculator — then change a load, drag a support, and watch every diagram update in real time. The best way to build intuition is to break it and see what happens.
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