About Truss Analysis Calculator (Method of Joints)
The truss analysis calculator solves a planar (2D) statically determinate truss by the method of joints. You define the joints with their coordinates and support conditions, the members connecting them, and the loads applied at the joints. The solver assembles the two equilibrium equations at each joint and solves the resulting linear system for every member axial force and support reaction.
Member forces are reported as tension (positive) or compression (negative). Before solving, the tool checks static determinacy using m + r = 2j, where m is the number of members, r the number of reaction components, and j the number of joints, and warns when the truss is unstable or statically indeterminate so you only solve cases the method of joints can handle.
How It Works
- Define the joints: x and y coordinates and a support type (pin, roller, or free).
- Define the members by selecting the two joints each member connects.
- Apply joint loads as horizontal (Fx) and vertical (Fy) components.
- The solver checks m + r = 2j for determinacy, then assembles and solves the 2j joint equilibrium equations for the member forces and reactions.
Worked Example
A triangle truss has a pin at A (0,0), a roller at B (4,0), and a free apex C (2,3) carrying a 10 kN downward load. Taking moments about A gives the roller reaction R_By = 10*2/4 = 5 kN, and vertical equilibrium gives R_Ay = 5 kN. Resolving at the apex, the two diagonals AC and BC each carry -10*sqrt(13)/6 = -6.01 kN (compression), and the bottom chord AB carries +10/3 = +3.33 kN (tension). The truss is determinate because m + r = 3 + 3 = 6 = 2j.
Formulas
- Static determinacy (planar truss)
m + r = 2 j (determinate); m + r < 2j unstable; m + r > 2j indeterminate- Joint equilibrium (method of joints)
at each joint: sum Fx = 0 and sum Fy = 0- Member force projection
F_x = S * (dx / L), F_y = S * (dy / L)- Reaction components by support
pin -> Rx and Ry (2); roller -> Ry (1); free -> none (0)
Standards & References
- Static equilibrium (statics)
- Hibbeler, Structural Analysis (method of joints)
- Hibbeler, Engineering Mechanics: Statics
Frequently Asked Questions
What does the truss analysis calculator solve?
It solves a planar statically determinate truss by the method of joints, returning the axial force in every member (tension or compression) and the reaction at every support, given the joint geometry, supports, members, and joint loads.
How is static determinacy checked?
The tool compares m + r with 2j, where m is members, r is reaction components, and j is joints. Equality means determinate; m + r less than 2j is unstable, and greater than 2j is statically indeterminate, which the method of joints cannot solve uniquely.
What is the sign convention for member forces?
Tension is positive and compression is negative. A member in tension pulls its end joints toward each other; a member in compression pushes them apart. Loads and reactions use +x to the right and +y upward.
Why might the solver report a geometrically unstable truss?
Even when m + r = 2j, a truss can be unstable if its geometry is degenerate, for example with collinear members or a mechanism. The solver detects this as a singular equilibrium matrix and reports it instead of returning meaningless forces.