Structural Analysis · 7 min read
How to solve truss member forces (method of joints, fast)
A practical workflow for solving 2D pin-jointed trusses by hand: reactions first, then joint-by-joint with sign conventions that don't trip you up.
Solving a 2D truss looks intimidating until you've done a couple. The method of joints is just F = ma applied to each pin in turn. Get the workflow right and you can solve a typical roof truss in a few minutes.
Assumptions you're making
- All joints are perfect pins (no moment transfer)
- All loads applied only at joints
- Member self-weight ignored (or lumped at joints)
- Members carry pure axial force — tension or compression only
Step 1 — Solve the reactions first
Treat the whole truss as a free body. Apply ΣM = 0 about one support to find the other reaction, then ΣFy = 0 for the remaining one. Don't try to start at a joint until you've got the reactions — it's much harder.
Step 2 — Pick a starting joint
Pick a joint with no more than 2 unknown member forces. Usually that's a support joint — you've got the reaction and only 2 members meeting there. From there you can solve those 2 forces using ΣFx = 0 and ΣFy = 0 at the joint.
Step 3 — Sign convention that works
Step 4 — Walk through the truss
Once you've solved one joint, the forces in those members are known. Move to a neighbouring joint where only 2 unknowns remain. Continue until every member is solved. Always check by going to the last joint — both equilibrium equations must be satisfied automatically (otherwise you have an arithmetic error somewhere).
Step 5 — Identify zero-force members
Two rules to spot them quickly: (1) if a joint has only 2 non-collinear members and no external load, both members are zero-force. (2) if a joint has 3 members where 2 are collinear, the third is zero-force. Spotting these saves time and gives you a sanity check.
When to use the method of sections instead
If you only need the force in one specific member (typically a chord member at mid-span), the method of sections gets you there in one step: cut the truss through the member of interest, take moments about a convenient point, and solve directly. Faster than walking joint-by-joint.
Quick worked example
Simple Pratt truss, 4 m span, 2 m height, vertical load 10 kN at apex. Reactions by symmetry: RA = RB = 5 kN. Joint A: members AC (diagonal) and AB (bottom chord). Geometry: AC at 45°. ΣFy: 5 + FAC·sin45° = 0 → FAC = −7.07 kN (compression). ΣFx: FAB + FAC·cos45° = 0 → FAB = +5 kN (tension). Symmetric joint B gives the same. Apex joint C: vertical equilibrium gives the apex web force; horizontal gives the top chord. Done in 3 joints.
Frequently asked questions
How do I check if a truss is statically determinate?
For a 2D truss: m + r = 2j, where m = members, r = reactions, j = joints. If m + r < 2j the truss is a mechanism (unstable). If m + r > 2j it's statically indeterminate and the method of joints alone can't solve it — you need stiffness methods or matrix analysis.
What if a load isn't at a joint?
Strictly speaking, a load between joints means the member carries bending as well as axial force — the pin-joint assumption breaks down. For roof purlins, the standard fix is to check the rafter member separately as a beam under the purlin load.
Are zero-force members useless?
No — they're often there to brace longer members against buckling, or to handle alternative load cases (e.g. unbalanced snow load) where they aren't zero. Don't remove them just because they're zero in one load case.