Cantilever Beam Deflection Calculator (δ, σ)

Calcola freccia massima δ, tensione massima σ e momento d'inerzia I di una trave a mensola con carico all'estremità. Formule esatte da resistenza dei materiali, in 5 lingue.

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What a cantilever beam is and why the calculation matters

A cantilever beam is fixed at one end and free at the other: balcony overhangs, roof overhangs, brackets, cantilever shafts. Under a load at the free tip it deflects and the fibre at the fixed end is the most stressed in bending. Correct sizing means guaranteeing two things: the section resists (σ < the material limit) and the deflection stays within serviceability (typically δ < L/250).

The three formulas this calculator uses

For a cantilever with a concentrated load F at the free tip:

  • Moment of inertia (rectangular section, neutral axis parallel to the base): I = b·h³/12. It is the geometric parameter measuring how much the section "resists" bending — it depends on the cube of the height.
  • Maximum deflection at the free tip: δmax = F·L³/(3·E·I). It scales with the cube of length and inversely with the moment of inertia.
  • Maximum stress (extreme fibre at the fixed end): σmax = Mmax·c/I with Mmax=F·L and c=h/2, which for a rectangular section gives σmax = 6·F·L/(b·h²).

What the moment of inertia means

The moment of inertia measures how the material is distributed about the neutral axis. In the rectangular formula I=b·h³/12 doubling the height increases I eightfold, while doubling the base only doubles it. That is why bending members have tall, narrow sections (IPE and HEA profiles, joists on edge): for the same area you get much more I, hence less deflection and less stress.

Young modulus E: some typical values

The Young modulus is the intrinsic stiffness of a material (stress-to-strain ratio in the elastic range). Typical values:

  • Structural steel: E ≈ 200 GPa.
  • Aluminium (alloys): E ≈ 70 GPa.
  • Copper: E ≈ 110 GPa.
  • Concrete: E ≈ 25–35 GPa (by class).
  • Structural timber: E ≈ 8–13 GPa (parallel to grain).
  • Glass: E ≈ 70 GPa.

Higher E = less deflection for the same section. Stress, on the other hand, does not depend on E: it only depends on F, L and geometry.

Serviceability check: why the L/δ ratio

Beyond strength (σ < σallow), codes require limiting deflection for functional and visual reasons. Typical building limits (SLS — serviceability limit state):

  • L/250 for ordinary floors and cantilevers.
  • L/300 for visible elements or those with a ceiling below.
  • L/500 for elements supporting glazing or fragile partitions.

The calculator shows L/δ: well above the limit means a stiff cantilever; close to or below means it must be stiffened.

How to act if deflection is too high

Deflection scales with L³/(E·I). To reduce it you can:

  • Increase the height h of the section (it acts as a cube: +26% h → −50% δ, all else equal).
  • Reduce length L (huge impact: −20% L → −50% δ).
  • Change to a higher-E material (steel instead of aluminium triples the stiffness).
  • Add an intermediate support (turns the cantilever into a simply-supported beam → much smaller deflection).

Model limits

The calculator assumes: concentrated load at the tip, linear-elastic material (no yielding), constant prismatic section, small displacements (Euler-Bernoulli theory), negligible shear (valid for slender beams L/h > 10). It ignores self-weight, torsion, lateral-torsional buckling, real joints. For definitive structural checks a code-based calculation (Eurocodes, national codes) is always required.

How to use the calculator

Enter the force at the tip (in newtons — a 100 kg load ≈ 981 N), the cantilever length in metres, the Young modulus of the material in GPa, and the base and height of the rectangular section in millimetres. You get moment of inertia, maximum deflection, maximum stress at the fixed-end fibre, the L/δ ratio (useful for the serviceability check) and a qualitative assessment.

Updated July 2026

Frequently Asked Questions

What changes if the section is not rectangular?

The deflection δ=FL³/(3EI) and the moment M=FL remain valid; only the moment of inertia I changes. For a tube, an IPE or a round bar you must use the tabulated I for that profile (for a solid round bar of diameter D: I=πD⁴/64). Maximum stress is computed as M·c/I where c is the distance from the extreme fibre to the neutral axis.

Why does deflection depend on L³?

The beam's curvature at each point is proportional to the bending moment M(x), which for a cantilever with a tip load grows linearly with the distance from the fixed end. Integrating curvature twice to get displacement (elastic-line equation) raises the power of L to 3. That is why doubling the span multiplies deflection by 8.

What deflection limit must I meet?

It depends on code and use: for ordinary floors and cantilevers L/250 is common (Eurocode, quasi-permanent SLS). For elements supporting glazing, fragile partitions or ceiling plaster it drops to L/500. For non-structural or purely aesthetic elements L/200 suffices. The calculator shows L/δ for an immediate comparison.

Does the calculation include self-weight?

No: it only considers the concentrated force F at the tip. Self-weight is a distributed load producing an additional deflection δsw=q·L⁴/(8·E·I) that must be added. For light sections or high F it is negligible; for large steel or concrete cantilevers it must be accounted for separately.

Does stress depend on the Young modulus E?

No, stress σmax=6FL/(bh²) depends only on load and geometry, not on the material. E only affects deflection. Two cantilevers with identical geometry and load, one in steel and one in aluminium, have the same stress but the aluminium one deflects almost 3 times more (E ≈ 70 GPa vs 200).

Does it apply to large displacements?

No, the formula is based on Euler-Bernoulli theory and small-displacement assumption (geometric linearity). For deflections large compared to L (over about 10%) the formula overestimates δ because the heavily bent beam "shortens" and its geometry changes. Non-linear cases require FEA with large deformations.

How do I compare two materials?

The key parameter for stiffness is E, for strength the ratio σallow/density (specific strength). Steel has high E and σallow ≈ 235 MPa; aluminium has ~1/3 the E but also ~1/3 the density, so per unit mass it can be competitive. For deflection maximise E·I; for strength, W = I/c (section modulus).

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