Model & response
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. Lateral loads act perpendicular to the beam’s web, causing bending about the weak axis.
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) (vertical) and Fx = F·sin(θ) (axial). Both components are solved independently. |
| Distributed Load (UDL) | Uniform load over a span | Converted to consistent nodal loads using work-equivalent formulation for each element within the loaded region. |
| Trapezoid Load | Linearly varying load (q1 to q2) | Each element gets a linearly interpolated load intensity, converted to equivalent nodal forces. |
| Moment | Concentrated moment at a point | 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. Produces N-diagram. |
| Lateral Load | Force perpendicular to web | Solved by the weak-axis FEM solver using Iy. Produces Mz, Vz, wz diagrams. |
4. Eurocode 3 (EN 1993-1-1) Code Checks
When a standard profile and material are selected, BeamSolve performs the following checks per EN 1993-1-1:
§6.2.5 — Bending Resistance
UC = MEd / Mc,Rd
Uses elastic section modulus Wel (conservative Class 3 assumption). γM0 = 1.0.
For weak-axis bending: Wz = Iy / (b/2), then Mz,Rd = Wz × fy / γM0.
§6.2.6 — Shear Resistance
UC = VEd / Vpl,Rd
Shear area Av = max(h × tw, 0.5 × A) for I-sections (simplified per §6.2.6(2)).
§6.2.4 — Axial Resistance
UC = NEd / Npl,Rd
§6.2.9 — Combined Biaxial Bending + Axial
Simplified linear interaction formula (conservative). If no axial load is present, the N-term drops out and only biaxial bending is checked.
5. AISC 360-22 Code Checks
When AISC 360 is selected as the design code, BeamSolve applies LRFD resistance factors:
Chapter F — Flexure
φb = 0.90
Mn = Wel × Fy
UC = Mu / (φb × Mn)
Chapter G — Shear
φv = 1.00
Vn = 0.6 × Fy × Aw
UC = Vu / (φv × Vn)
Chapter D/E — Axial
φc = 0.90
Pn = A × Fy
UC = Pu / (φc × Pn)
Chapter H — Combined
H1-1a: Pr/Pc ≥ 0.2 → Pr/Pc + 8/9(Mry/Mcy + Mrz/Mcz)
H1-1b: Pr/Pc < 0.2 → Pr/(2Pc) + (Mry/Mcy + Mrz/Mcz)
6. Stability Checks (Buckling & LTB)
§6.3.1 — Flexural Buckling (EC3)
When you enter a buckling length Lcr and an axial load is present:
Φ = 0.5 × [1 + α(λ̄ - 0.2) + λ̄²]
χ = 1 / [Φ + √(Φ² - λ̄²)]
Nb,Rd = χ × A × fy / γM1
| Buckling Curve | α (imperfection factor) | Typical Use |
|---|---|---|
| a | 0.21 | Hot-rolled I-sections, h/b > 1.2, tf ≤ 40mm |
| b | 0.34 | Hot-rolled I-sections, h/b ≤ 1.2 (default) |
| c | 0.49 | Welded I-sections, hollow sections |
| d | 0.76 | U-channels, angles, T-sections |
§6.3.2 — Lateral Torsional Buckling (EC3)
When you enter an unbraced length LLTB:
λ̄LT = √(Wel × fy / Mcr)
χLT = 1 / [ΦLT + √(ΦLT² - λ̄LT²)]
Mb,Rd = χLT × Wel × fy / γM1
Where It ≈ (2b·tf³ + (h-2tf)·tw³) / 3 and Iw = Iy·(h-tf)² / 4.
AISC Buckling (Chapter E)
If KL/r ≤ 4.71√(E/Fy): Fcr = 0.658^(Fy/Fe) × Fy
If KL/r > 4.71√(E/Fy): Fcr = 0.877 × Fe
φcPn = 0.90 × Fcr × A
7. Profile Library
BeamSolve includes a comprehensive library of standard steel profiles with verified section properties:
European (EN 10365)
IPE 80–600, HEA 100–1000, HEB 100–1000, HEM 100–1000, UNP 80–400, UPE 80–400
Hollow Sections
SHS (square), RHS (rectangular), CHS (circular) per EN 10210/10219
American (AISC)
W-shapes (W8–W44), S-shapes, C-channels, HSS sections
Custom Mode
Enter your own E, I, A, Wx values for non-standard sections or other materials.
Each profile includes: h, b, tw, tf, Ix, Iy, Wx, A, and mass. All values are sourced from official standards and manufacturer data.
8. Units & Precision
Metric (SI)
Length: mm • Force: N/kN • Moment: kNm • Stress: MPa • Deflection: mm • Inertia: mm&sup4;
Imperial (US)
Length: in • Force: lbs/kips • Moment: kip·in • Stress: ksi • Deflection: in • Inertia: in&sup4;
All internal calculations are performed in N and mm for maximum numerical precision. Unit conversion happens only at the display layer. The solver uses 64-bit floating point arithmetic throughout.