Cantilever Beam Deflection Calculator

Calculate maximum deflection and slope for cantilever beams under point, uniform, or moment loads.

Steel=200, Al=70, Concrete=30
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Cantilever Deflection Is Highly Sensitive to Length and Section Depth

A cantilever is fixed at one end and free at the other, so bending moment and deflection build toward the fixed support under transverse loading. For an end point load P, the free-end deflection is δ=PL³/(3EI) and slope is θ=PL²/(2EI). For a uniform load w over the entire beam, free-end deflection is wL⁴/(8EI).

Flexural rigidity EI combines material stiffness E with geometric stiffness I. The strong L³ or L⁴ dependence means a modest increase in span can cause a large deflection increase. The formulas assume small deflection, linear elasticity, and Euler-Bernoulli beam behavior.

End load: δtip=PL3/(3EI);   UDL: δtip=wL4/(8EI)
SymbolMeaningWhy it appears / units
PEnd point loadN.
wUniform load intensityN/m.
LCantilever lengthm.
EIFlexural rigidityN·m².

The fixed end usually carries the largest bending moment, while the free end has the largest deflection in these simple load cases. Stress and deflection should both be checked because serviceability can govern before material yield.

Use scaling to test a cantilever result. Increasing E or I must reduce deflection, while increasing the applied load must increase it. For an end-load case, doubling L should multiply δ by eight. If the calculator trend violates one of those relationships, recheck the selected load case, support condition, and units.

Worked Examples

Example 1: Steel cantilever: P=1000N, L=2m, E=200GPa, I=1×10⁶mm⁴
δ=1000×8/(3×200e9×1e-6)
Result: 13.3 mm deflection
Check if acceptable for application
Example 2: Concrete balcony: w=5000N/m, L=3m, E=30GPa, I=50×10⁶mm⁴
δ=5000×81/(8×30e9×50e-6)
Result: 33.75 mm — may be excessive
Typical limit: L/250=12mm for concrete
Example 3: End load
P=500N, L=1m, E=200GPa, I=1×10−6m⁴
Result: δtip≈0.833mm
The cubic length dependence makes span a powerful design variable.
Example 4: Doubling cantilever length
same end load, E, I; L doubles
Result: tip deflection increases 8×
For an end point load, δ is proportional to L³.

Common Mistakes

⚠️
Using only supported formulas for a cantilever

Support conditions change moment distribution and deflection coefficients dramatically.

⚠️
Mixing section units in I

Because I contains length to the fourth power, mm⁴-to-m⁴ conversion errors are severe.

⚠️
Ignoring shear deformation in short deep beams

Euler-Bernoulli theory can underestimate deflection when shear deformation is important.

Frequently Asked Questions

Why L³ (or L⁴) matters so much?
Deflection ∝ L³ for point loads. Doubling beam length → 8× more deflection. This is why long spans require disproportionately larger beams — not just for strength but for stiffness.
Deflection limits in practice?
Structural codes specify limits (L/240 to L/500). L/360 for live loads in floors. L/240 for total loads. Plastered ceilings below: L/360. These prevent cracking and feel of 'springiness'.
Where is maximum bending moment in an end-loaded cantilever?
At the fixed support. For a point load P at the free end, the maximum moment magnitude is PL.
Why is tip deflection so sensitive to length?
The end-load formula contains L³ and the uniform-load formula contains L⁴. Length increases bending leverage along the entire beam.
How can section shape reduce cantilever deflection?
Increase the second moment of area I by placing material farther from the neutral axis, often by increasing section depth.
Does a cantilever have zero slope at the fixed end?
Yes in the ideal fixed-support model. Both deflection and rotation are constrained to zero at the built-in end.
Why does cantilever deflection grow so strongly with length?
For a common end-loaded cantilever, tip deflection scales as δ=FL3/(3EI). The cubic dependence means doubling length increases ideal deflection eightfold if load, E, and I stay unchanged. This strong length sensitivity is why span errors and unit mistakes can dominate beam-deflection calculations.

Formula Explorer connections

Interpretation: This engineering-physics relationship connects load, material property, geometry, deformation or system response. Assumption: Confirm material linearity, geometry, support conditions and safety convention. Small deformation, elastic behavior and ideal loading are common assumptions.

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