Truss Solver (Method of Joints)

How it works

A pin-jointed truss carries load only as tension or compression in its straight members. If the truss is statically determinate, the number of members plus three support reactions equals twice the number of joints, and the forces follow from equilibrium at every joint.

ΣFx = 0, ΣFy = 0 at every joint

The page builds all joint equations together and solves them by Gaussian elimination, then checks that every joint balances (the equilibrium check at the bottom is the largest leftover force, which should be a tiny fraction of a newton). This is the same result as working joint by joint by hand. The supports are a pin at the left and a roller at the right, so the horizontal reaction is zero for vertical loads. For beams and frames see Beam Calculator and Structures and Strength of Materials.

Worked example

A Warren truss of 4 panels, span 8 m and height 2 m, with 10 kN on each of the four top joints. Total load 40 kN, so each reaction is 20 kN. For the middle top chord, take moments about the bottom joint at x = 4 m: C × 2 = 20 × 4 − 10 × 3 − 10 × 1 = 40, so C = 20 kN compression. For the bottom chord next to the middle, take moments about the top joint at x = 3 m: T × 2 = 20 × 3 − 10 × 2 = 40, so T = 20 kN tension. The end diagonal carries the whole reaction vertically: its force is 20 × √(1² + 2²) ÷ 2 = 22.36 kN compression, the largest in the truss. The solver gives the same values and shows that the diagonals next to the middle carry no force under this symmetric load.

Things to know

The model is an ideal pin-jointed truss with loads at the joints only. Real trusses have bending from loads between joints and joint stiffness; check chords with a beam model if loads act along a member.

A compression member can buckle at a force far below its yield strength: check slenderness with the column formulas on Columns and Buckling.

Pratt trusses carry the long diagonals in tension (good for steel), Howe trusses in compression (the traditional timber design). The chord forces are the same for the same loads, the web members differ.

Zero-force members still matter: they brace long compression members and carry load if the loading changes.

Self weight is not included: add it as extra load on the joints. Use factored loads for design, and combine with the safety factors your code requires.

Common questions

What does statically determinate mean?

The member forces can be found from equilibrium alone: members + 3 reactions = 2 × joints. If there are fewer members the truss is a mechanism; if there are more it needs deformation (stiffness) analysis.

Why is the top chord in compression?

Under downward loads the truss bends like a beam: the top is squeezed and the bottom is stretched. Chord force is approximately bending moment divided by truss height, so a deeper truss has smaller chord forces.

Can I load the bottom joints instead?

Not in this version: loads are on the top joints. Put the load on the top and add the equivalent loads yourself for a bridge-style bottom-loaded truss.

Sources

• Hibbeler, R. C., Structural Analysis (method of joints, statically determinate trusses).
• Beer, Johnston, Mechanics of Materials and Vector Mechanics for Engineers: Statics (analysis of trusses).

Last reviewed: November 2, 2026