Calcady™
Home / Scientific / Cantilever Beam Deflection Calculator

Cantilever Beam Deflection Calculator

Calculate the maximum deflection at the free end of a cantilever beam under point load or uniform distributed load. Verify serviceability against code limits like L/360.

Solve exact structural sagging of free-end architectural spans by deriving applied forces against material stiffness bounds.

Newtons
Meters
GPa
m⁴
Determined heavily by the 'I-Beam' shape/cross section dimensions vs a flat rectangle.

Cantilever Tip Deflection

Maximum Sag (Metric)

4.5
Millimeters Displacement (mm)

Maximum Sag (Imperial)

0.177
Inches Displacement (in)
Flexure Multiplier AlertLength Cubed (L³) Penalty = 27.0x
Email LinkText/SMSWhatsApp

What is The Vulnerability of Cantilevers?

A cantilever is a rigid structural beam anchored solidly at only one end, while the other end hangs freely in space (like a diving board). Because the free end has no vertical support, it physically bends downward when a heavy point-load force pushes on the tip. Civil Engineers demand this Deflection calculation to ensure bridges don't snap and skyscraper balconies don't sag and collapse under human weight.

Mathematical Foundation

δmax=P L33 E I\delta_{max} = \frac{P \, L^3}{3 \, E \, I}
δmax\delta_{max}= The maximum vertical deflection (sag) at the very tip of the free-hanging beam.
PP= The exact magnitude of the Point-Load crashing down on the unguarded tip of the beam (in Newtons).
L3L^3= The length of the beam sticking out, multiplied by the power of 3. This is why long bridges sag so aggressively.
EE= Young's Modulus of Elasticity. The chemical stiffness of the material. Steel fights bending brilliantly (high E). Soft pine wood bends easily (low E).
II= Area Moment of Inertia. The physical shape geometry of the cross-section. An "I-Beam" shape creates a massive moment of inertia, fighting bending incredibly well compared to a flat square.

Laws & Principles

  • The L-Cubed Danger: Because the Length variable is cubed ($L^3$) in the numerator, doubling the length of your balcony doesn't double the sag—it explodes the sag by 800% ($2^3=8$). Extending a cantilever beam outwards rapidly demands exponentially reinforced steel to remain stable.
  • The Role of I-Beams: Why are skyscraper bones shaped like the capital letter "I"? Because placing the metal as high and as low as geometrically possible away from the geometric center maximizes the Area Moment of Inertia ($I$). This makes the beam crushingly rigid against vertical bending while aggressively saving on the cost and weight of the solid metal.

Step-by-Step Example Walkthrough

" A 3.0 meter steel diving board (E = 200 GPa) is bolted to concrete at one end. The board has an Inertia profile of 0.00005 m^4. A heavy human applying 5,000 Newtons stands on the unguarded edge. "

  1. 1. Calculate Cubed Length Numerator: 5000 × (3.0 ^ 3) = 135,000.
  2. 2. Calculate Stiff Denominator: 3 × (200,000,000,000 Pa) × 0.00005 = 30,000,000.
  3. 3. Divide to find meters of sag: 135,000 / 30,000,000 = 0.0045 meters.
  4. 4. Convert to normal human measurements.
Final Result: The steel board deflects by precisely 4.5 millimeters (about 0.17 inches) under the load. It remains highly rigid and safe.

What is Cantilever Beam Deflection: Euler-Bernoulli Beam Theory?

Cantilever beam deflection is governed by Euler-Bernoulli beam theory (1750), which relates beam curvature to bending moment through the product EI (flexural rigidity). A cantilever — fixed at one end, free at the other — is statically determinate: the fixed support provides both a reaction force and a reaction moment. Because all bending is absorbed at the fixed end with no midspan support, cantilevers deflect significantly more than simply supported beams of equal span and loading. The free-end deflection formulas derive from integrating the moment-curvature relationship M(x) = EI·d²y/dx² twice along the beam length.

Mathematical Foundation

Cantilever — Point Load at Free End
δmax=PL33EI\delta_{max} = \frac{P L^3}{3 E I}
δmax\delta_{max}= Maximum deflection at the free end of the cantilever (in or mm). This is the point that deflects most.
PP= Concentrated point load applied at the free end (lbf or N).
LL= Total length of the cantilever from fixed support to free end (in or mm). Note: L is cubed — span has enormous effect.
EE= Modulus of elasticity (Young's modulus) of the beam material: Steel = 29,000 ksi; Aluminum = 10,000 ksi; Douglas Fir = ~1,700 ksi. Governs material stiffness.
II= Area moment of inertia of the cross-section about the bending axis (in⁴ or mm⁴). Tabulated in AISC/NDS tables for standard sections.
Cantilever — Uniform Distributed Load (UDL)
δmax=wL48EI\delta_{max} = \frac{w L^4}{8 E I}
ww= Uniform distributed load intensity (lb/in or N/mm). Convert from lb/ft: divide by 12. Note: L is raised to the 4th power — span sensitivity is extreme.
L4L^4= When comparing to a simply supported beam under UDL (δ = 5wL⁴/384EI), the cantilever under UDL (1/8 coefficient) deflects approximately 24× more for equal span. The fixed-end moment in a cantilever carries all the bending with no relief from an opposite reaction.

Laws & Principles

  • Flexural Rigidity EI: The product E × I is called flexural rigidity. It is the fundamental measure of a beam's resistance to bending deformation. Increasing E (stiffer material) or I (deeper section) proportionally reduces deflection. A W12×26 (I = 204 in⁴) at E = 29,000 ksi → EI = 5,916,000 kip-in². A 2×12 Douglas Fir joist (I ≈ 178 in⁴) at E = 1,700 ksi → EI = 302,600 kip-in² — about 20× less rigid. This explains why wood deflects far more than steel for equivalent spans.
  • L³ vs. L⁴ Sensitivity: Under a point load, deflection scales with L³. Under UDL, it scales with L⁴. Doubling the cantilever span: point load deflection increases 8× (2³); UDL deflection increases 16× (2⁴). This is why long cantilevers are extremely deflection-sensitive — small span increases cause disproportionate serviceability problems. A cantilever that passes the deflection check at 6 ft may be 4× over the limit if extended to 9.5 ft (same load, same section).
  • Serviceability vs. Strength: Building codes distinguish two limit states. Strength limit state: beam must not yield (σ ≤ Fb). Serviceability limit state: beam must not sag excessively. Most building codes (IBC, ASCE 7) limit cantilever live-load deflection to L/180 and total deflection to L/120. Floors over plaster or masonry use L/360 and L/240. A 6-ft cantilever (L = 72 in) with L/180 limit: δ_max ≤ 0.40 in. With L/360: δ_max ≤ 0.20 in. These are independent checks — a beam can pass strength and fail serviceability, or vice versa.

Step-by-Step Example Walkthrough

" A balcony cantilever: W10×33 A36 steel, 6-ft cantilever (free end), 3,000 lb point load at free end. Confirm serviceability (L/180 limit for exterior cantilever). "

  1. Inputs: P = 3,000 lb = 3 kips; L = 6 ft = 72 in; from AISC: W10×33 → I = 170 in⁴; E = 29,000 ksi.
  2. EI = 29,000 × 170 = 4,930,000 kip-in².
  3. δ = PL³ / (3EI) = 3 × 72³ / (3 × 4,930,000) = 3 × 373,248 / 14,790,000 = 1,119,744 / 14,790,000.
  4. δ = 0.0757 in.
  5. Serviceability limit: L/180 = 72/180 = 0.400 in.
  6. Check: 0.0757 in < 0.400 in — PASSES with 81% reserve.
Final Result: Maximum free-end deflection = 0.076 in. DCR_service = 0.076/0.400 = 0.19 — significantly over-capacity for serviceability. The W10×33 is well-suited for this application; if using a W8×24 (I = 82.7 in⁴), deflection would be 0.156 in — still passing at DCR = 0.39.

Quick Answer: How do you calculate cantilever beam deflection?

Cantilever beam deflection uses δ = PL³/(3EI) for a point load at the free end, or δ = wL&sup4;/(8EI) for a uniform distributed load. Example: W10×33 steel, 6-ft cantilever, 3,000 lb point load → δ = 3,000 × 72³ / (3 × 29,000,000 × 170) = 0.076 in. L/180 limit = 0.40 in → passes with 81% reserve. The dominant variable is span L, which is cubed (point load) or raised to the fourth power (UDL) — doubling the cantilever span increases deflection 8× or 16× respectively. This is why cantilevers are much more deflection-sensitive than simply supported beams: a simply supported beam under UDL deflects δ = 5wL&sup4;/384EI — the cantilever formula (1/8 coefficient) vs. the simply supported formula (5/384 coefficient) means cantilevers deflect approximately 24× more for equal span, load, and section.

Beam Deflection Formulas by Loading and Support Condition

Beam Type Loading Max Deflection Formula Location of Max δ Relative Stiffness
Simply Supported Uniform Load (w) 5wL&sup4; / 384EI Midspan Baseline (1.0×)
Simply Supported Point Load at Center (P) PL³ / 48EI Midspan 2nd stiffest (same span)
Fixed-Fixed (both ends fixed) Uniform Load (w) wL&sup4; / 384EI Midspan Stiffest (5× stiffer than SS)
Propped Cantilever (1 end fixed) Uniform Load (w) wL&sup4; / 185EI ~0.58L from fixed end Between SS and fixed-fixed
Cantilever (free end) Point Load at Free End (P) PL³ / 3EI Free end 16× more than SS center PL
Cantilever (free end) Uniform Load (w) wL&sup4; / 8EI Free end 24× more than SS UDL

IBC / ASCE 7 Deflection Limits by Application

Application Live Load Limit Total Load Limit Reason
Roof — Non-plastered ceiling L/180 L/120 Ponding risk; visual sag
Roof — Plastered ceiling below L/360 L/240 Plaster cracking prevention
Floor — No plaster L/360 L/240 Serviceability; occupant comfort
Floor — Plaster or brittle finishes L/480 L/360 Tile / plaster cracking prevention
Balcony / exterior cantilever L/180 L/120 Visual perception; waterproofing drain
Lintel / header over masonry L/600 L/480 Masonry veneer cracking; stiff requirement

Pro Tips & Cantilever Design Errors

Do This

  • ✓Use consistent units throughout — moment of inertia I must match the force and length units used for E and L. In Imperial: if L is in inches, P is in pounds, and E is in psi (lb/in²), then I must be in in&sup4; to get δ in inches. E for steel = 29,000,000 psi = 29,000 ksi. Mixing E in ksi with P in lbs gives a result 1,000× too large. The safest practice: convert everything to inches and pounds before calculating, then convert the result to more convenient units. A W12×26 steel cantilever, 8 ft (= 96 in), with P = 2,000 lb and E = 29,000,000 psi: δ = 2,000 × 96³ / (3 × 29,000,000 × 204) = 2,000 × 884,736 / (17,748,000,000) = 0.0997 in. Checking L/180: 96/180 = 0.533 in — beam passes comfortably.
  • ✓Check deflection against the appropriate code limit for your specific application — the limit varies from L/120 (lenient) to L/600 (extremely strict). The IBC deflection table distinguishes between live-load deflection (which fluctuates and causes cyclic stress) and total-load deflection (which includes dead load and can cause permanent sag). For balcony cantilevers: live-load L/180 applies to people and furniture weight. For lintels supporting masonry veneer: L/600 is common for the total load limit because masonry cracks at very small differential settlements. Always verify which limit applies to your construction type before using L/360 as a default.

Avoid This

  • ✗Don't extend a cantilever by adding a few inches without recalculating — the L³ relationship means small span increases cause large deflection increases. A cantilever with δ = 0.38 in at L = 72 in (right under L/180 = 0.40 in limit) requires only a 1.7% span increase (72 to 73.2 in) to push deflection over the limit. Because δ ∝ L³: δ_new = δ_old × (L_new/L_old)³ = 0.38 × (73.2/72)³ = 0.38 × 1.050 = 0.40 in — now failing. In practice, field cuts, connection adjustments, and as-built conditions regularly add 1–3 inches to designed cantilever lengths. Always include a DCR cushion ≤ 0.85 for cantilevers to absorb construction tolerance.
  • ✗Don't use the simply supported beam deflection formulas for a cantilever — they give results up to 24× too small, creating a dangerous false sense of compliance. The two most common beam formulas are δ = 5wL&sup4;/384EI (simply supported, UDL) and δ = wL&sup4;/8EI (cantilever, UDL). The ratio: (1/8) ÷ (5/384) = 384/(8×5) = 9.6. Wait — that's 9.6× more deflection in the cantilever, not 24×. The 24× comparison is for equal total load W = wL distributed: cantilever W×L³/(8EI) vs SS 5W×L³/(384EI) → (1/8)/(5/384) = 48/5 = 9.6. Both formulas apply to different span references — the key point is that cantilever deflection is dramatically larger than simply supported. Never copy a simply supported formula for a cantilever application.

Frequently Asked Questions

What is flexural rigidity EI and how do I increase beam stiffness?

Flexural rigidity EI is the product of the material's elastic modulus (E) and the cross-section's moment of inertia (I). It appears in every deflection formula's denominator — higher EI means less deflection. To increase stiffness: (1) Use a stiffer material: Steel (E = 29,000 ksi) > Aluminum (10,000 ksi) > Douglas Fir timber (1,700 ksi). For equal I, steel is 17× stiffer than wood. (2) Use a deeper section: I scales with d³/12 for a rectangular section, so doubling the depth multiplies I by 8. A 6″ deep beam and a 12″ deep beam of equal width — the deeper beam is 8× stiffer. This is why engineered LVL beams (deep, consistent grain) outperform sawn lumber cantilevers. (3) Reduce span: Most efficient — add an intermediate support to change a cantilever to a propped cantilever, dramatically reducing both deflection and bending moment.

Why does cantilever deflection scale with L³ while uniform load cantilevers scale with L&sup4;?

The difference comes from the moment distribution. For a point load P at the free end: the bending moment at position x from the free end is M(x) = P×x. Integrating the moment-curvature equation EI·d²y/dx² = M(x) twice produces a cubic relationship in L, giving δ ∝ L³. For a uniform distributed load w: the moment at position x is M(x) = w×x²/2 — a quadratic function of x. Integrating this quadratic moment twice produces a quartic relationship in L, giving δ ∝ L&sup4;. Physically: with UDL, each additional foot of span both adds more load and increases the moment arm of all previous load. The extra power of L reflects this double accumulation effect. This is why UDL deflection grows with the 4th power — and why extending a uniformly loaded cantilever even slightly has an outsized effect on sag.

What is the difference between a cantilever and a propped cantilever, and how does it affect deflection?

A cantilever (fixed-free) has a fixed support at one end and a completely free unsupported end — statically determinate. Maximum deflection occurs at the free end. A propped cantilever (fixed-pinned) has a fixed support at one end and a pin support at the other. This makes it statically indeterminate (one extra reaction). The propped reaction dramatically reduces deflection and bending moment: for UDL w, maximum deflection of a propped cantilever is δmax = wL&sup4;/(185EI) — about 23× less than the pure cantilever (wL&sup4;/8EI). The maximum moment for a pure cantilever under UDL is wL²/2 at the fixed end. For a propped cantilever under UDL, the maximum moment is only 9wL²/128 — about 7× less. Adding a prop (column, post, or wall support) at the free end of a cantilever is structurally transformative — it converts a highly-deflecting single-support condition to a far stiffer two-support system.

How does wood cantilever deflection compare to steel, and why does timber cantilever fail more often in serviceability?

Timber has a modulus of elasticity roughly 14–17× lower than steel. A Douglas Fir No. 2 2×10 has E = 1,600,000 psi; steel is E = 29,000,000 psi. For equal cross-section I, wood deflects ~18× more than steel. In practice, wood members are sized to compensate with depth — but cantilevers are particularly punishing for wood because: (1) Creep: wood under sustained load creeps over time — long-term deflection is commonly estimated as 1.5× the elastic deflection (multiply calculated δ by 1.5 for dead loads). Steel does not creep significantly at room temperature. (2) Moisture: wood swells and shrinks with humidity; improperly dried framing lumber in a wet climate can deflect an additional 10–15% beyond design. (3) Notching: NDS prohibits notching timber beams in the tension zone on the bottom of a cantilever — field notches in the wrong location have caused cantilever balcony collapses. For wood cantilevers longer than 4–5 feet, use LVL, PSL, or steel and always check serviceability, creep, and notching compliance.

Related Calculators