Cantilever beam with a uniform load

A beam is fixed at one end, free at the other, and carries a load w spread evenly along its length, in newtons per metre, for example its own weight.

Typical examples:

a cantilevered shelf or tray loaded evenly
a sign arm or a light pole arm under its own weight
a robot or machine arm sagging under its own weight

Calculator

Enter the span and the load, choose a cross-section and a material. Results, verdicts and diagrams update as you type. Lengths are in millimetres; decimal comma or point both work.

Case

Cross-section

Span and load

Span L

Distributed load w

N/m

Solid rectangle dimensions

Width b

Height h

Material

Required safety factor on yield

Results

Maximum deflection	f	13.23	mm
Span-to-deflection ratio	L/f	151.2
Maximum bending moment	M_max	2000	N·m
Maximum bending stress	σ_max	83.33	MPa
Fixed end reaction	R	2000	N
Fixed end moment	M_A	2000	N·m
Maximum shear stress	τ_max	1.25	MPa
Maximum combined stress (von Mises)	σ_eq,max	83.33	MPa

OK — Bending stress vs yield strength: σ_max = 83.33 MPa, fy = 235 MPa, safety factor 2.82 ≥ required 1.5.

OK — Combined stress (von Mises) vs yield strength: σ_eq,max = 83.33 MPa, fy = 235 MPa, safety factor 2.82 ≥ 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

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.cantilever.uniform_load v1, beam.combined_stress v1.

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

Formulas

Bending moment, stress and shear are largest at the fixed end; the deflection is largest at the free end.

Maximum deflection, at the free end: f_{max} = \frac{w L^4}{8 E I}
Maximum bending moment, at the fixed end: M_{max} = \frac{w L^2}{2}
Fixed-end reactions: R = w L, \qquad M_A = \frac{w L^2}{2}
Deflected shape (x from the fixed end): y(x) = -\frac{w x^2 \left(6L^2 - 4 L x + x^2\right)}{24 E I}
Maximum bending stress (c: distance from the neutral axis to the extreme fiber): \sigma_{max} = \frac{M_{max}\, c}{I}
Shear stress over the section (Jourawski): \tau(y) = \frac{V\, Q(y)}{I\, b(y)}
Combined stress (von Mises): \sigma_{eq} = \sqrt{\sigma^2 + 3\,\tau^2}

Symbols: L span, F point load, w load per unit length, E Young's modulus, I second moment of area of the section, c distance from the neutral axis to the extreme fiber, V shear force, M bending moment, Q first moment of area, b section width at height y.

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.

Other beam cases

Reference: Classical beam theory (Euler–Bernoulli) — Double integration of the elastic curve equation E·I·y″ = M(x), with the support boundary conditions. Module beam.cantilever.uniform_load, version 1.