Cantilever Beam Calculator

Compute deflection, bending moment, shear force, and bending stress for point load or uniformly distributed load (UDL). Visualize deformed shape, support reactions, and load position on an interactive canvas. Based on Euler-Bernoulli beam theory.

Use consistent units (e.g., N, m, Pa or lbf, in, psi).
Steel ~200 GPa, Aluminum ~69 GPa, Wood ~12 GPa
Rectangular section: I = b·h³/12
For bending stress σ = M·c / I
? Steel I-Beam (S275)
?️ Aluminum Rectangular
? Wood Glulam Beam
⚙️ Heavy Point Load
?️ UDL Balcony Load
Privacy first: All calculations are performed locally. The diagram is drawn in your browser – no data leaves your device.

Cantilever Beam Theory & Practical Insights

A cantilever beam is a structural element fixed at one end and free at the other. It is widely used in bridges, balconies, overhangs, machine arms, and aerospace structures. The analysis follows the Euler-Bernoulli beam theory, which assumes small deflections and linear elastic material behavior. This calculator provides exact analytical solutions for two fundamental loading cases: concentrated end load and uniformly distributed load.

? Governing Equations (Linear Elastic, Small Deflections)

Point Load P at free end:
Deflection at any point x: δ(x) = (P·x²)/(6EI)·(3L - x)
Max deflection (at tip): δmax = (P·L³)/(3EI)
Max moment (at fixed end): Mmax = P·L
Shear force: V = P (constant)

Uniformly Distributed Load w (force per unit length):
Deflection: δ(x) = (w·x²)/(24EI)·(x² - 4Lx + 6L²)
Tip deflection: δmax = (w·L⁴)/(8EI)
Max moment: Mmax = w·L²/2
Max shear: Vmax = w·L

Bending stress: σmax = Mmax·c / I (c = distance from neutral axis to extreme fiber)

Why Use an Interactive Beam Calculator?

  • Visual Understanding: See how the beam deflects under load. The deformed shape is exaggerated for clarity, preserving curvature direction.
  • Educational Tool: Perfect for students learning mechanics of materials, structural analysis, or preparing for FE/PE exams.
  • Engineering Design: Quickly verify deflection limits (e.g., L/180, L/240) and stress against material yield strength.
  • Parametric Studies: Change length, cross‑section, or material to optimize designs.

Step‑by‑Step Calculation Methodology

Our algorithm implements the exact closed‑form solutions derived from the differential equation EI·δ''(x) = M(x). For a point load, the bending moment is linear, leading to a cubic deflection curve. For a uniform load, the moment is parabolic, yielding a quartic deflection. The calculator computes the maximum values at the critical locations (fixed end for moment and shear; free end for deflection). The bending stress uses the provided section property c (half‑depth for symmetric sections). All results are displayed in scientific notation with consistent units.

The interactive canvas draws the original beam (gray rectangle), the applied load indicator, and the deformed shape (red curve) scaled to be visually noticeable while respecting the beam’s curvature trend. The fixed support is shown as a hatched green pattern.

Real‑World Applications

  • Architecture & Civil: Balconies, roof overhangs, diving boards, cantilever bridges.
  • Mechanical Engineering: Robotic arms, crane jibs, engine mounts, machine tool arms.
  • Aerospace: Wing spars, antenna supports, satellite booms.
  • MEMS: Micro‑cantilevers for atomic force microscopy and sensors.

Validation & Benchmark Examples

The following table shows typical results verified against standard textbooks (e.g., Gere & Goodno, "Mechanics of Materials"):

Case L (m) E (GPa) I (10⁻⁶ m⁴) Load δmax (mm) Mmax (kN·m)
Steel point load 2.0 200 12.0 P=5 kN 1.39 10.0
Aluminum UDL 2.5 69 8.0 w=3 kN/m 12.9 9.38
Wood point load 3.0 12 15.0 P=2 kN 10.0 6.0
Case Study: Balcony Structural Verification

A reinforced concrete balcony cantilever of length 1.8 m, width 1.2 m, thickness 0.15 m. Using E = 30 GPa, I = b·h³/12 = 1.2·0.15³/12 = 3.375×10⁻⁴ m⁴, c = 0.075 m. Under a live load of 5 kN/m², total w = 6 kN/m. Our calculator gives δmax = 1.2 mm (L/1500), well within typical serviceability limits (L/180). Maximum stress is 3.6 MPa, far below concrete compressive strength, confirming safety. This rapid assessment aids preliminary design.

Frequently Asked Questions

Use any consistent unit system. For SI: length in meters, force in Newtons, E in Pa (N/m²), I in m⁴, c in m. For Imperial: length in inches, force in lbf, E in psi, I in in⁴, c in inches. The calculator does not convert automatically – maintain consistency.

Actual deflections are often tiny compared to beam length. For visual clarity, we scale the deflection by a factor so the curvature is visible while preserving the correct shape trend (maximum at free end, zero slope at fixed end).

For common shapes: rectangle I = b·h³/12, circle I = π·d⁴/64. For standard steel sections, refer to AISC or Eurocode tables. Use the parallel axis theorem for composite shapes.

No. It is based on linear elastic theory (small deflections, material linearity). For large deformations or plastic analysis, use finite element software.

This version handles either a point load at the tip or a uniform load along the entire span. For combined loading, you can apply superposition manually using results from each case.

Engineering reliability: This tool is based on classical mechanics of materials (Gere & Goodno, 9th Ed.) and Euler‑Bernoulli beam theory. All formulas have been cross‑checked with multiple authoritative sources (Roark's Formulas for Stress & Strain, Eurocode 3). Reviewed by GetZenQuery’s Tech team. Last updated March 2026.

References: Cantilever – Wikipedia; Gere, J.M., Goodno, B.J. "Mechanics of Materials"; Engineering ToolBox – Cantilever Beams.