Space truss calculator
A three-dimensional pin-jointed truss solved with three degrees of freedom per node: a square pyramidal roof truss under an apex load, with every member force, the support reactions and the apex deflection. The same solver a tower or a space frame needs.
| Member | Group | Nodes | Length (m) | Force (kN) |
|---|---|---|---|---|
| 1 | Base | 0–1 | 6.00 | 8.4 |
| 2 | Base | 1–2 | 6.00 | 8.4 |
| 3 | Base | 2–3 | 6.00 | 8.4 |
| 4 | Base | 3–0 | 6.00 | 8.4 |
| 5 | Diagonal | 0–2 | 8.49 | 4.4 |
| 6 | Sloping | 0–4 | 5.20 | -20.0 |
| 7 | Sloping | 1–4 | 5.20 | -14.6 |
| 8 | Sloping | 2–4 | 5.20 | -20.0 |
| 9 | Sloping | 3–4 | 5.20 | -14.6 |
Positive is tension. The diagonal is redundant and takes the asymmetric share of the support reactions.
K·u = F, 3 DOF per node · Assembled from direction cosines in x, y and z.
N = EA/L · (Δu·cx + Δv·cy + Δw·cz) · Axial only, as a pin-jointed truss requires.
Σ N·(Δz/L) = W · The check that the solve is right.
A clear path from inputs to results.
A roof truss in two dimensions is a plane frame; a tower, a hip roof or a space frame is not. This solves a three-dimensional pin-jointed truss with three degrees of freedom per node — a square pyramidal roof truss under an apex load, with every member force and the reactions.
How to use
- Set the base width, apex height and the load at the apex.
- Enter the member area and the modulus.
- Read the peak compression in the sloping members, the tension in the base and the apex deflection.
The formula
Reference: Direct stiffness method, three degrees of freedom per node, pin-jointed members.
Worked example
6 m square pyramid, 3 m high, 40 kN apex load
- 1,500 mm² members, 200 GPa.
- 1
Five nodes, nine members. Assemble the 15×15 global matrix, restrain the four base nodes in z plus a pin and a roller in plan, solve the eight free degrees of freedom.
Sloping members in compression, base in tension, equilibrium satisfied at 40 kN.
Interpreting the result
The reactions are not equal even though the geometry is symmetric, because the supports are not: a pin plus three rollers is the minimum restraint that keeps a space truss stable in plan.
Assumptions & limitations
- Pins at every joint.
- Linear elastic, small displacement.
- Self-weight and buckling not included.
- One fixed topology: square pyramid.
Frequently asked questions
Why are the reactions not equal?
Because a space truss needs a pin and two rollers in plan, and that restraint set is asymmetric even on a symmetric geometry.
Can I model a tower?
Not in this form; the topology is fixed to a square pyramid. The solver underneath handles any node and member list.
What does the diagonal do?
It is redundant — the pyramid is stable without it — but it takes the asymmetric share of the load path.
Is the apex deflection realistic?
It is the elastic axial deflection; joint slip and connection flexibility add to it.
Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.