Truss Analysis Calculator
Solve planar pin-jointed trusses with the method of joints: support reactions, member forces in tension or compression, zero-force members, and determinacy checks.
∑Fx = 0, ∑Fy = 0 (per joint) — assume tension positive
Truss Master analyzes any 2D pin-jointed truss — draw it on the grid, apply loads and supports, and get member forces, reactions, and displacements with worked joint-by-joint derivations.
Zero-Force Member Rules
Spotting zero-force members first collapses the truss to a much smaller system — every rule below is applied automatically by the solver before any equations are written.
Unloaded two-member joint
A joint with only two members and no external load has both members zero-force.
Unloaded three-member joint
A joint with three members where two are collinear: the non-collinear third member is zero-force.
Load collinear with two members
A joint whose external load acts along the line of two collinear members: the third member is zero-force.
Simple Triangular Truss · A(0,0) pin · B(100,0) roller · C(50,87) · 10 kN ↓ at C
// Joint-by-joint equilibrium (member length √(50² + 87²) = 100.6):
Joint C · ΣFy: F_AC·(87/100.6) + F_BC·(87/100.6) = 10 ⟹ F = −5.77 kN
Joint A · ΣFx: F_AB + F_AC·(50/100.6) = 0 ⟹ F_AB = +2.88 kN
// Negative sign = compression; positive = tension, exactly as assumed.
Symmetry gives R_Ay = R_By = 5 kN immediately. At the apex the two inclined legs share the 10 kN load through their vertical components; the horizontal components cancel, and the bottom chord carries the resulting tie force in tension — the classic triangular-roof result.
One cut, three unknowns
Cut the truss through no more than three members. Take moments about the intersection point of two of the cut members — their forces vanish and the third force is solved directly, bypassing intermediate joints.
k = EA / L
In indeterminate trusses, member forces split in proportion to axial stiffness k = EA/L — stiffer members attract more load. This is the principle behind the matrix stiffness solution the app uses as its ground truth.
Engineering Formulas & Governing Equations
Joint Equilibrium (Method of Joints)
Equation 01Each pin joint is a particle in equilibrium; solving joints in sequence propagates forces member by member.
Static Determinacy
Equation 02m = members, r = reaction components, j = joints. Equality means the 2j equilibrium equations close the problem.
Zero-Force Member Rules
Equation 03Detectable by inspection before any arithmetic — the fastest simplification of any truss.
Method of Sections
Equation 04The moment equation isolates one unknown member force directly, without stepping joint by joint.
Member Stiffness
Equation 05Stiffness governs load distribution in indeterminate trusses: stiffer members attract more force.
Calculation Details & Clarifications
Solve in Truss Master
Need indeterminate trusses, per-member materials, or an A4 structural report? Open the complete web application.
Open Truss Master FreeTruss Master · m + r = 2j determinacy • No install needed