Generate element stiffness matrices for spring, truss, and beam elements in FEA. Visualizes the matrix with DOF labels and material properties.
The stiffness matrix is the cornerstone of the finite element method (FEM). It relates forces (or moments) to displacements (or rotations) at element nodes through [K]{u} = {F}. Each element type has a characteristic stiffness matrix derived from its governing equations and displacement interpolation functions.
For a spring element, the 2×2 matrix is simply k and −k — force is proportional to elongation. Truss elements extend this to 2D with a 4×4 matrix that includes coordinate transformation (angle θ). Beam elements add rotational degrees of freedom, producing a 4×4 matrix from the Euler-Bernoulli beam theory with 12EI/L³ and related coupling terms.
Understanding these matrices is fundamental for structural engineering students and practicing engineers using FEA software. This calculator generates the stiffness matrix for spring, truss, and beam elements with arbitrary material and geometric properties, complete with labeled rows/columns and property verification.
Use the preset examples to load common values instantly, or type in custom inputs to see results in real time. The output updates as you type, making it practical to compare different scenarios without resetting the page.
FEA software generates these matrices automatically, but understanding them is essential for debugging models, hand-checking results, and developing intuition for structural behavior. This calculator is invaluable for engineering students and anyone learning the finite element method.
This tool is designed for quick, accurate results without manual computation. Whether you are a student working through coursework, a professional verifying a result, or an educator preparing examples, accurate answers are always just a few keystrokes away.
Spring: [K] = [[k, -k], [-k, k]]. Truss: [K] = (EA/L)[[c²,cs,-c²,-cs],[cs,s²,-cs,-s²],...]. Beam (Euler-Bernoulli): [K] = [[12EI/L³, 6EI/L², -12EI/L³, 6EI/L²], [6EI/L², 4EI/L, -6EI/L², 2EI/L], ...].
Result: 4×4 beam stiffness matrix
For a 200 GPa steel beam with I = 8.33×10⁻⁸ m⁴ (roughly a 50×20 mm rectangle) and L = 2 m: EI = 16,660 N·m², K₁₁ = 12EI/L³ = 24,990 N/m.
Use consistent units throughout your calculation and verify all assumptions before treating the output as final. For professional or academic work, document your input values and any conversion standards used so results can be reproduced. Apply this calculator as part of a broader workflow, especially when the result feeds into a larger model or report.
Most mistakes come from mixed units, rounding too early, or misread labels. Recheck each final value before use. Pay close attention to sign conventions — positive and negative inputs often produce very different results. When working with multiple related calculations, keep intermediate values available so you can trace discrepancies back to their source.
Enter the most precise values available. Use the worked example or presets to confirm the calculator behaves as expected before entering your real data. If a result seems unexpected, compare it against a manual estimate or a known reference case to catch input errors early.
Maxwell's reciprocal theorem guarantees symmetry: the force at node i due to unit displacement at j equals the force at j due to unit displacement at i. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
Degrees of freedom. Springs: 1 per node (axial). Trusses: 2 per node (u, v). Beams: 2 per node (transverse v, rotation θ). 3D beam: 6 per node.
Map each element's local DOFs to global DOFs, then add corresponding entries. This is the assembly process in FEA. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
Plane sections remain plane and perpendicular to the neutral axis (no shear deformation). Valid for slender beams (L/d > 10). For thick beams, use Timoshenko beam theory.
The global stiffness matrix transforms local (axial) stiffness into global (x, y) coordinates. The transformation uses cos and sin of the element angle.
Be consistent: if E is in GPa and A in m², then K is in N/m. Mixing units is the most common source of errors in FEA.