How BeamSolve Works
A transparent look at the engineering methods, formulas, and standards behind every calculation. BeamSolve uses the Finite Element Method (FEM) with Euler-Bernoulli beam theory to deliver accurate structural analysis.
1. Finite Element Method (FEM) Solver
BeamSolve discretizes your beam into 80 Euler-Bernoulli beam elements, each with 3 degrees of freedom per node (vertical displacement, rotation, axial displacement). This gives a system of 243 DOFs for high accuracy.
Element Stiffness Matrix
Each element uses the standard 6×6 Euler-Bernoulli stiffness matrix combining bending and axial stiffness:
Axial terms: k = EA/L × [1, -1; -1, 1]
The global stiffness matrix K is assembled by summing element contributions, then boundary conditions are applied by eliminating constrained DOFs. The system K·u = F is solved using LU decomposition.
Internal Forces
After solving for displacements u, internal forces are recovered element-by-element:
V(x) = shear force from element end forces
M(x) = bending moment from element end moments
N(x) = axial force from element end forces
w(x) = deflection from nodal displacements
2. Biaxial Bending (Strong + Weak Axis)
BeamSolve runs two independent FEM solvers simultaneously:
Strong Axis (Y-axis)
Uses Ix (major moment of inertia). Handles vertical loads, distributed loads, moments. Produces V, M, N, and w diagrams.
Weak Axis (Z-axis)
Uses Iy (minor moment of inertia). Handles lateral loads. Produces Vz, Mz, and wz diagrams.
The weak-axis solver uses a simplified 2-DOF/node model (lateral displacement + rotation) with the same 80-element mesh.
3. Load Types
| Load Type | Description | How It Works |
|---|---|---|
| Point Load | Concentrated force at a point | Applied as equivalent nodal forces using shape functions. Positive = downward. |
| Angled Load | Point load at angle θ | Decomposed into Fy = F·cos(θ) and Fx = F·sin(θ). |
| Distributed Load | Uniform load over a span | Converted to consistent nodal loads using work-equivalent formulation. |
| Trapezoid Load | Linearly varying (q1 to q2) | Each element gets interpolated load intensity, converted to equivalent nodal forces. |
| Moment | Concentrated moment | Applied directly to the rotational DOF of the nearest node. |
| Axial Load | Force along beam axis | Applied to the axial DOF. Positive = tension, negative = compression. |
| Lateral Load | Force perpendicular to web | Solved by the weak-axis FEM solver using Iy. |
4. Eurocode 3 (EN 1993-1-1) Code Checks
When a standard profile and material are selected, BeamSolve performs the following checks:
§6.2.5 — Bending Resistance
UC = MEd / Mc,Rd
Uses elastic section modulus Wel (conservative Class 3 assumption). γM0 = 1.0.
§6.2.6 — Shear Resistance
UC = VEd / Vpl,Rd
§6.2.4 — Axial Resistance
UC = NEd / Npl,Rd
§6.2.9 — Combined Biaxial Bending + Axial
5. AISC 360-22 Code Checks
Chapter F — Flexure
φb = 0.90, Mn = Wel × Fy
Chapter G — Shear
φv = 1.00, Vn = 0.6 × Fy × Aw
Chapter D/E — Axial
φc = 0.90, Pn = A × Fy
Chapter H — Combined
H1-1: biaxial interaction with N + My + Mz
6. Stability Checks (Buckling & LTB)
§6.3.1 — Flexural Buckling (EC3)
Nb,Rd = χ × A × fy / γM1
| Curve | α | Use |
|---|---|---|
| a | 0.21 | Hot-rolled I, h/b > 1.2 |
| b | 0.34 | Hot-rolled I, h/b ≤ 1.2 |
| c | 0.49 | Welded I, hollow sections |
| d | 0.76 | U-channels, angles |
§6.3.2 — Lateral Torsional Buckling (EC3)
χLT = 1/[ΦLT + √(ΦLT² - λ̄LT²)]
Mb,Rd = χLT × Wel × fy / γM1
7. Profile Library
European (EN 10365)
IPE, HEA, HEB, HEM, UNP, UPE
Hollow Sections
SHS, RHS, CHS per EN 10210/10219
American (AISC)
W-shapes, S-shapes, C-channels, HSS
Custom Mode
Enter your own E, I, A, Wx
8. Units & Precision
Metric (SI)
mm, N/kN, kNm, MPa
Imperial (US)
in, lbs/kips, kip·in, ksi
All internal calculations in N and mm. 64-bit floating point. Verified against analytical solutions with errors < 0.1%.