HPC Tier Module
Finite Element Beam / Truss Solver
2D beam and truss finite-element solver, checked against closed-form deflections and equilibrium.
This tool runs inside GDBS HPC. What it computes and the authorities it applies are below.
The problem
Structural prototyping needs quick displacement and member-force checks, but full FEM packages are heavy to set up and a hand calculation is slow and error-prone for anything past a couple of members.
The local-first solution
This solver builds a 2D model of nodes and truss or Euler-Bernoulli beam elements, assembles the global stiffness matrix, and solves K u = F by Gaussian elimination in your browser - recovering displacements, reactions, and member forces, validated against closed-form cantilever and simply-supported deflections. Deterministic f64 math, nothing uploaded.
What it does
Truss (axial EA/L) and beam/frame (Euler-Bernoulli Hermite, 3 DOF per node) elements
Global stiffness assembly with nodal point loads (Fx, Fy, M) and fixed-DOF boundary conditions
K u = F solve by Gaussian elimination with partial pivoting
Nodal displacements, support reactions, and per-element internal actions (axial, shear, moment)
Validation against closed-form cantilever, simply-supported, and method-of-joints references
GeoNum drift probe of the elimination pivots to flag ill-conditioning
Honest scope
EXACT: the element stiffness formulas, global assembly, Gaussian-elimination solve, and force recovery are exact linear algebra in f64; E, A, and I are dated spec inputs the result echoes back. NOT modeled: this is linear-elastic, small-displacement, static analysis only - no nonlinearity, buckling, dynamics, shear deformation, distributed loads, thermal/prestress, self-weight, or 3D. Not engineering advice - verify any load-bearing design against the governing code and a licensed PE.
Authorities cited
- Logan, D. L. (2017). A First Course in the Finite Element Method, 6th ed. Cengage. Ch.2-3 (spring/bar/plane-truss elements, EA/L stiffness), Ch.4 (plane-frame / beam element, the 6x6 Euler-Bernoulli + axial stiffness and the T^T k T rotation).
- Hughes, T. J. R. (1987). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall (reprint Dover 2000). Assembly of the global stiffness matrix and imposition of essential boundary conditions.
- Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002). Concepts and Applications of Finite Element Analysis, 4th ed. Wiley. The beam/frame element stiffness and the direct stiffness method.
- Gere, J. M. & Goodno, B. J. (2018). Mechanics of Materials, 9th ed. Cengage. Beam deflection formulas: cantilever tip = P L^3/(3EI), tip slope = P L^2/(2EI); simply-supported central = P L^3/(48EI).
- Bathe, K.-J. (2014). Finite Element Procedures, 2nd ed. The Euler-Bernoulli (Hermite cubic) beam element and Gaussian-elimination solution of K u = F.
- AISC 360-16, Specification for Structural Steel Buildings. Material modulus E = 29,000 ksi (~ 199.95 GPa) used as the steel default (confirm for your grade).
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Size a member with a built-in check
Run the analysis in your browser with nothing uploaded, then route displacements, reactions, and the conditioning verdict into a Sandbox workspace or attach them to a Worklog case.