Beam calculator

Test version: four basic cases, four cross-sections, results update as you type.

Choose a case, enter the span and the load, pick a cross-section and a material. Lengths are in millimetres. Decimal comma or point both work.

Case
Cross-section
Span and load
mm
N
Solid rectangle dimensions
mm
mm
Material

Results

Maximum deflectionf1.102mm
Span-to-deflection ratioL/f1814
Maximum bending momentM_max500N·m
Maximum bending stressσ_max20.83MPa
Left support reactionR_A500N
Right support reactionR_B500N
Maximum shear stressτ_max0.3125MPa
Maximum combined stress (von Mises) σ_eq,max 20.83 MPa
OK — Bending stress vs yield strength: σ_max = 20.83 MPa, fy = 235 MPa, safety factor 11.28 ≥ required 1.5.
OK — Combined stress (von Mises) vs yield strength: σ_eq,max = 20.83 MPa, fy = 235 MPa, safety factor 11.28 ≥ required 1.5.

Deflection is given without a verdict: the admissible deflection depends on what the beam is for. Compare f with your own requirement.

Diagrams

F L
Shear force V [N] -600 -400 -200 0 200 400 600 0 500 1000 1500 2000 x [mm] peak 500 N
Bending moment M [N·m] 0 100 200 300 400 500 0 500 1000 1500 2000 x [mm] peak 500 N·m
Combined stress σ_eq (von Mises) [MPa] 0 5 10 15 20 25 0 500 1000 1500 2000 x [mm] peak 20.83 MPa
Deflection y [mm] -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0 500 1000 1500 2000 x [mm] peak -1.102 mm

Sign conventions: dM/dx = V; sagging moment positive (cantilevers hog, so M < 0); deflection negative downward. σ_eq is the largest value over the section height at each position.

Section and material

  • Solid rectangle: I = 720000 mm⁴, c = 30 mm, A = 2400 mm².
  • Structural steel S235: E = 210 GPa, fy = 235 MPa (guaranteed minimum). Source: fy: EN 10025-2, minimum ReH for nominal thickness ≤ 16 mm. E: EN 1993-1-1, 3.2.6.

Assumptions

  • Euler–Bernoulli beam: plane sections stay plane, small deflections and rotations.
  • Linear elastic, homogeneous and isotropic material.
  • Constant cross-section along the span.
  • Self-weight neglected.
  • Shear deformation neglected (valid for slender beams, span much larger than section height).
  • Not checked: lateral-torsional buckling, local buckling of thin walls, fatigue, stress concentrations.
  • Shear stress by the Jourawski formula: mean value across the width (approximate for round sections).
  • Equivalent stress by the von Mises criterion, scanned over the section height at each position.
  • Local stresses under concentrated loads and at the supports are not included.

References: Classical beam theory (Euler–Bernoulli) — Double integration of the elastic curve equation E·I·y″ = M(x), with the support boundary conditions. Classical strength of materials, Jourawski (Zhuravskii) shear formula — τ(y) = V·Q(y) / (I·b(y)), mean shear stress across the width b at height y. Von Mises yield criterion — σ_eq = √(σ² + 3τ²) for a normal stress σ combined with a shear stress τ. Modules beam.simply_supported.center_load v1, beam.combined_stress v1.

Engineering aid only. Results are indicative and must be checked by the user before any design decision.