Civil
Cantilever Beam
Beam fixed at one end, free at the other — bending under load
A cantilever beam is rigidly fixed at one end and free at the other, projecting outward without intermediate support. Loads at the free end create maximum bending moment at the fixed support. Stress distribution: tension on top fiber (or bottom, depending on load direction), compression on the opposite face. Tip deflection for a tip load: δ = PL³/(3EI). Used in balconies, aircraft wings, diving boards, signposts, MEMS accelerometers. Failure modes: bending stress at root, deflection limits, fatigue, lateral-torsional buckling.
- BoundaryFixed at one end, free at other
- Tip deflectionδ = PL³/(3EI) for tip load
- Max momentAt fixed support
- Max stressOuter fibers at root
- UseWings, balconies, signs, MEMS
- FailureBending, deflection, fatigue
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Why cantilevers matter
- Aerospace. Wings, tail surfaces, control horns.
- Buildings. Balconies, canopies, awnings.
- Bridges. Cantilever bridges, overhanging spans.
- MEMS. Accelerometer and AFM probe tips.
- Signs and lights. Highway gantries, lamp posts.
- Diving boards. Classic teaching example.
- Robotics. Manipulator arms.
Common misconceptions
- Stress is uniform. Highest at root, zero at tip.
- Deflection scales linearly. Cubic with length.
- More material is stiffer. Distribution matters more than mass.
- Fixed means rigid. Real connections have some flexibility.
- Tip load is worst. Distributed load can produce higher root moment.
- Static loads only. Fatigue at root is often the limit.
Frequently asked questions
How does a cantilever support load?
The fixed support resists both vertical force and bending moment. As load is applied at the free end, the beam bends — top fibers stretch (tension), bottom fibers compress, and the moment increases linearly toward the support. Maximum bending stress occurs at the outer fibers at the fixed end. The support reactions: a vertical force equal to the load, and a moment equal to the load times the lever arm.
What's the deflection formula?
For a tip load P on a cantilever of length L, modulus E, second moment of area I: tip deflection δ = PL³/(3EI). For uniformly distributed load w: δ = wL⁴/(8EI). Cubic and quartic dependence on length means doubling the length increases deflection by 8× or 16×. Engineers often deflection-limit cantilevers before stress limits become critical.
What's the maximum bending stress?
σ = Mc/I, where M is bending moment, c is distance from neutral axis to outer fiber, I is second moment of area. For a cantilever with tip load P: M_max = PL at the fixed support, so σ_max = PLc/I. The combination M/I × c — section modulus S = I/c — captures the cross-section's bending efficiency. I-beams put material at large c for high S.
How is deflection minimized?
Increase EI (stiffness). For a rectangular cross-section, I = bh³/12 — depth h has cubic effect. Doubling depth gives 8× the stiffness. Choosing high-modulus materials (steel E=200 GPa, CFRP composites E=140+ GPa) helps. Truss-like structures (I-beams, hollow tubes) put material where it's needed. Reducing length is the most powerful — but often impossible — option.
What's an aircraft wing as a cantilever?
Wings are cantilevers attached to the fuselage, loaded by lift along their span. Lift distribution increases bending moment and shear toward the root, so spar caps thicken near the fuselage. Modern wings use I-beam-like spars, ribs, and skin together as a torsion box. Wing flex during flight is normal — Boeing 787 wingtips deflect over 7 m at limit load.
Why fatigue critical?
Cantilever roots see the highest stress range under repeated loading — a crack propagation hot spot. Aircraft wings cycle through flight loads thousands of times; bending stresses at the root drive fatigue cracks. Designers use generous fillet radii at fixed ends, reducing stress concentration. Inspection focuses on root regions.
When use cantilever vs simple support?
Cantilever where one end must be free — balconies, wings, diving boards, lab probes. Simply supported beams (pinned at both ends) are 4× stiffer for the same length and material. Continuous beams over multiple supports outperform both. Cantilevers are used when geometry demands them, accepting deflection penalties for design freedom.