# Truss & Frame Solver (Stiffness Method)

> Forces, moments, reactions and deflections of 2-D/3-D trusses, beams and frames. Describe a truss, a continuous beam or a 2-D frame line by line — nodes, a steel or timber section, members with optional hinges, pin, roller and fixed supports, springs, point loads and distributed loads — and get the node displacements, the support reactions, the axial force, shear and bending moment of every member, the stress and a utilisation check against the yield strength, an Euler buckling check of the compression members and an equilibrium check. Diagrams show the member forces in colour, the deformed shape and the bending moments. Solved with the direct stiffness method and a sparse Cholesky solver; part of the Proofline math notebook. A checking and teaching model, not certified design.

Part of usefulweb.site (https://usefulweb.site/). Part of [Proofline](https://proofline.usefulweb.site/) (ownerApp: proofline). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://prooflinetrussframesolver.usefulweb.site/ (interactive tool at https://prooflinetrussframesolver.usefulweb.site/tool)
- JSON API: POST https://prooflinetrussframesolver.usefulweb.site/api/v1/calculate with a JSON object of inputs (schema: https://prooflinetrussframesolver.usefulweb.site/api/v1/tool)
- OpenAPI: https://prooflinetrussframesolver.usefulweb.site/openapi.json
- MCP (Streamable HTTP): https://prooflinetrussframesolver.usefulweb.site/mcp - tool `proofline_truss_frame_solver`

## Inputs
- type (select): Structure type
- model (textarea, required): Structure (one statement per line) - One statement per line, # starts a comment. node A 0 0 (x y in m; 3-D: x y z) · material steel E=210 fy=235 (GPa, MPa) · section S1 A=28.5 I=1943 W=194 (cm², cm⁴, cm³; I needed for beams and buckling checks, W for bending stress) · member M1 A B [section=S1] [material=steel] [hinge=start|end|both] [K=1] [truss] · support A pin | fixed | roller | roller-x | ux uy rz · spring B ky=5000 (kN/m; kr in kN·m/rad) · load B Fx=10 Fy=-20 M=5 (kN, kN·m) · udl M1 qy=-5 (kN/m per metre of member; qx, qn = normal to the member).
- deflectionLimit (number): Deflection limit L/… - Beam spans that sag more than span/limit are flagged (250 is a common serviceability figure; check the code that applies).

## Method
- Direct stiffness method: bar elements (axial stiffness EA/L along the member direction) for trusses in 2-D and 3-D; Euler–Bernoulli beam-column elements (EA/L, 12EI/L³, 6EI/L², 4EI/L, 2EI/L) for 2-D frames, transformed to global axes and assembled into a sparse matrix.
- Uniform member loads enter as consistent fixed-end forces (wL/2, wL²/12). Hinges release the end moment by static condensation of the element; the released rotation is recovered afterwards. A node where only hinged or pin-ended members meet has an undefined rotation, which is held fixed.
- The system is reordered with reverse Cuthill–McKee and solved by an envelope Cholesky factorisation; a failing pivot means the structure is a mechanism and names the free direction.
- Member forces from the element end forces; shear and moment along each member from equilibrium, deflections from Hermite shape functions plus the exact fixed-end solution of the uniform load — exact for Euler–Bernoulli beams.
- Checks (not design-code verifications): stress |N|/A + |M|/W against fy, Euler load Pcr = π²EI/(KL)² for compression members, span deflection against L/limit, and the sum of reactions against the applied loads.
- Verified against Hibbeler Statics Example 6.1 (forces 500 N T, 707.1 N C, 500 N T), PL³/3EI and 5wL⁴/384EI, the propped cantilever (3wL/8, wL²/8), the symmetric three-bar truss P/(1 + 2cos³θ), a tripod and mechanism detection.

## Limits
- Not certified engineering. A linear-elastic, first-order model for checking and teaching: no safety factors, second-order (P-Δ) effects, imperfections, lateral-torsional or local buckling, shear deformation, connection design or code combinations. Have structures that matter designed by a qualified engineer.