Cantilever Beam Deflection Calculator
Calculate deflection, bending moment, and reaction forces in a cantilever beam with a point load.
Supports rectangular sections or custom moment of inertia.
What Is a Cantilever Beam? A cantilever beam is a structural member that is fixed (rigidly supported) at one end and free at the other. The fixed end resists both vertical forces and bending moments. Common examples include diving boards, balconies, aircraft wings, crane jibs, and overhanging roof structures.
Maximum Deflection For a cantilever with a point load P applied at the free end, the maximum deflection occurs at the free end:
delta_max = PL³ / (3EI)
Where:
- P = applied point load (N)
- L = beam length (m)
- E = Young’s modulus of the material (Pa)
- I = second moment of area (moment of inertia) of the cross-section (m⁴)
Bending Moment The maximum bending moment occurs at the fixed support:
M_max = P × L
This is the moment the fixed support must resist. It increases linearly with both load and beam length.
Reaction Forces At the fixed end (the wall or support), two reactions exist:
- Vertical reaction force: R = P (equal and opposite to the applied load)
- Fixed-end moment: M = P × L (the support must provide this moment to prevent rotation)
Moment of Inertia for a Rectangular Section For a rectangular cross-section with width b and height h:
I = b × h³ / 12
The moment of inertia measures the cross-section’s resistance to bending. Doubling the height h increases I by a factor of eight, which is why tall narrow beams and I-beams are far more efficient than short wide ones.
Design Considerations In structural design, both stress and deflection have to be checked, and they are separate failures with separate limits.
- Bending stress: sigma = M × y / I, where y = h/2 for a rectangular section. This must stay below the material’s strength. The calculator checks it against a representative figure for whichever material you pick, and you can type your own if you know it.
- Deflection limit: codes typically cap deflection at L/360 for floor beams and allow up to L/180 for many other cases.
Which one governs depends on the span. A short deep beam nearly always fails the stress check first; a long slender one hits the deflection limit long before it gets anywhere near yielding, because deflection grows with the cube of length while stress only grows linearly. That is why long cantilevers are almost always sized by stiffness rather than strength.
Representative bending strengths, for the check the calculator runs:
| Material | Figure used | What it actually is |
|---|---|---|
| Steel | 250 MPa | A36 structural yield |
| Aluminum | 276 MPa | 6061-T6 yield |
| Copper | 70 MPa | Annealed yield, hard-drawn is far higher |
| Wood along grain | 40 MPa | Softwood modulus of rupture, and allowable design stress is roughly a quarter of it |
| Concrete | none | Plain concrete has almost no tensile strength and cracks in bending, so it must be reinforced, and reinforced design is beyond this calculator |
Material and Section Selection A stiffer material (higher E) or a deeper cross-section (much higher I) dramatically reduces deflection. The product EI is called the flexural rigidity, and it is what actually governs how a beam behaves.
Superposition for Multiple Loads If multiple loads act simultaneously, deflections from each load can be calculated separately and added together (superposition), provided the material behaves elastically and deformations remain small.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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