Cantilever Beam Calculator
This calculator computes deflection, end slope, maximum bending moment, shear and bending stress for a cantilever beam under a point load, a uniform load, or an end moment. For a tip load P, the result is deflection equals PL³ divided by 3EI, where L is span, E modulus, I the section's second moment of area.
Three load cases, one set of formulas
Every cantilever problem on this page reduces to the same three load cases, and each one has a closed-form answer you can check by hand in about thirty seconds. Point load at the tip, uniform load along the span, or a pure moment applied at the free end — pick one, and the deflection, slope, bending moment and shear all fall out of the same beam theory: Euler–Bernoulli, small deflections, a linear-elastic material, with I taken from our second moment of area calculator if you ever need it for a shape other than a plain rectangle.
| Load case | Max deflection | Slope at free end | Max bending moment | Max shear |
|---|---|---|---|---|
| Point load P at the tip | PL³ / 3EI | PL² / 2EI | PL (at the wall) | P |
| Uniform load w over L | wL⁴ / 8EI | wL³ / 6EI | wL² / 2 (at the wall) | wL |
| End moment M₀ | M₀L² / 2EI | M₀L / EI | M₀ (constant) | 0 |
Notice the pattern: the point load and the UDL both put the maximum bending moment and the maximum shear at the fixed support, not at the tip — that's where the beam actually fails, a detail students get wrong on exams more often than you'd think. The end-moment case is the odd one out: shear is zero everywhere, because nothing pushes sideways on the beam; a couple applied at the free end simply bends it into an arc of constant curvature.
Worked example: a 500 mm steel bracket under a tip load
Take a 10 mm thick, 25 mm wide steel flat bar, 500 mm long, clamped at one end like a shelf bracket, with a 150 N load (about 15 kg) hung at the free tip. Structural and plain carbon steels run close to 200 GPa, so that's the modulus we'll use.
Start with the section: I = bh³/12 = 25 × 10³ / 12 = 2083 mm⁴. Plug that into δmax = PL³/(3EI): 150 × 500³ / (3 × 200,000 × 2083) = 18,750,000,000 / 1,250,000,000 = 15.0 mm of tip deflection. The slope there, θ = PL²/(2EI) = 150 × 500² / (2 × 200,000 × 2083), works out to 0.045 rad, close to 2.58°.
Bending moment is largest at the wall, Mmax = PL = 150 × 0.5 = 75 N·m, and the shear force stays constant along the whole length at Vmax = P = 150 N. The bending stress at that wall section is σ = 6M/(bh²) = 6 × 75,000 / (25 × 100) = 180 MPa, comfortably under the roughly 250 MPa yield of mild steel — though "comfortably" is relative, and it's worth running that number through a proper factor of safety calculator rather than eyeballing it.
Why the height matters more than the width
Look at the rectangle formulas again and one thing jumps out: width b sits to the first power, height h to the third. Double the width and stiffness doubles. Double the height instead and stiffness climbs eightfold, while bending strength, which goes with bh², quadruples. That's the entire reason structural sections — I-beams, joists, even a plain flat bar on edge rather than flat — are always deeper than they are wide when bending resistance is the goal. If you're comparing a few candidate cross-sections, our second moment of area calculator and section modulus calculator will get you I and Z = I/c for each one faster than redoing the algebra by hand.
When these formulas stop telling the truth
These formulas assume small deflections, a linear-elastic material that hasn't yielded, a perfectly rigid wall, and a beam slender enough that shear deformation near the support is negligible. Push any one of those and the tidy PL³/3EI family starts lying to you. A beam that's short and deep relative to its span deflects more than Euler–Bernoulli predicts, because shear strain — which this calculator ignores — starts to contribute meaningfully once the beam stops being slender. A support that flexes even slightly, a bolted bracket rather than a welded one, say, adds deflection the formula never accounts for. And a beam stressed past yield simply stops obeying Hooke's law, full stop.
Before trusting a result for anything load-bearing, compare the bending stress against the material's yield strength; our factor of safety calculator turns that comparison into one clean number. If you need the full moment and shear distribution along the span rather than just the two peak values, the shear force diagram calculator plots both end to end.
Units: GPa, mm⁴, and the usual mistakes
Unit mistakes, not the formulas themselves, are where most hand calculations actually go wrong. Young's modulus is usually quoted in GPa, but mixing it directly with millimetres and newtons needs E in N/mm² — numerically identical to MPa, so 200 GPa becomes 200,000 N/mm², not 200. The second moment of area is even easier to fumble: I from a rectangle in millimetres comes out in mm⁴, and forgetting to convert before comparing it with a value quoted in m⁴ is the single most common reason a deflection answer comes back wrong by a factor of a trillion. This calculator takes b and h in millimetres and E in GPa and does the conversion for you, but it's worth knowing where the trap sits before you reach for a spreadsheet instead.
For other support conditions — simply supported, propped, fixed at both ends — our beam deflection calculator covers those cases, and the rest of our engineering calculators, including the bending stress calculator, pick up where this page leaves off.
Frequently Asked Questions
What is the deflection formula for a cantilever beam with a point load at the free end?
Maximum deflection is delta = PL cubed divided by 3EI, occurring at the free end where the load is applied. P is the load, L the span from wall to tip, E the Young's modulus of the material, and I the second moment of area of the cross-section. Doubling the span multiplies deflection by eight.
Where is the maximum bending moment in a cantilever beam?
For both a point load and a uniform load, the maximum bending moment occurs at the fixed support, not at the free end, because that's where the internal resistance to the applied load accumulates over the full length. Moment is zero at the free tip and grows steadily toward the wall.
How do I calculate deflection for a cantilever beam under a uniformly distributed load?
Use delta_max = wL to the fourth divided by 8EI, where w is the load per unit length, L the span, E the modulus and I the second moment of area. This gives the tip deflection; the shape along the span follows a fourth-order curve, flat at the wall and steepest at the tip.
What is the maximum allowable deflection for a cantilever beam?
There's no universal number — allowable deflection depends on the application, the governing design code, and whether appearance, vibration or clearance is the limiting concern. A lab shelf bracket and an aircraft wing tolerate very different amounts of tip movement, so check the relevant structural code rather than a rule of thumb.
Does changing the cross-section shape change which formula I use?
No — the deflection, slope, moment and shear formulas on this page depend only on load, span and EI, not on the shape of the section. What changes is how you compute I itself; a rectangle uses bh³/12, while an I-beam or circular section needs its own second moment of area formula instead.
What's the difference between applying a force and applying a moment at the free end?
A point load bends the beam into a curve whose curvature is largest at the wall and zero at the tip, with deflection equal to PL³/3EI. An applied moment produces constant curvature along the whole length, a constant internal bending moment equal to M0, and zero shear force anywhere in the beam.
Why does the shear force stay constant along a cantilever with a point load?
Because no other load intercepts the beam between the tip and the wall, the internal shear force required for vertical equilibrium is the same at every cross-section and equals the applied load, P. It only changes where an additional point load or distributed load enters the picture.
Can I use these formulas for an I-beam or a circular cross-section?
The deflection, slope, moment and shear formulas apply to any prismatic cantilever regardless of shape; only the second moment of area I changes. This calculator assumes a solid rectangle with I = bh³/12. For other profiles, compute I separately and substitute it into the same equations.