Inputs
Formula used
This calculator uses Euler–Bernoulli beam theory with constant E and I. End deflection δ and end slope θ for common cantilever load cases are:
- Point load at free end: δ = P L³ / (3EI), θ = P L² / (2EI)
- Point load at distance a: δ = P a² (3L − a) / (6EI), θ = P a² / (2EI)
- Uniform load (full length): δ = w L⁴ / (8EI), θ = w L³ / (6EI)
- End moment: δ = M L² / (2EI), θ = M L / (EI)
- Triangular, max at free end: δ = 11 w0 L⁴ / (120EI), θ = w0 L³ / (8EI)
- Triangular, max at fixed end: δ = w0 L⁴ / (30EI), θ = w0 L³ / (24EI)
Also reported: EI, tip stiffness 3EI/L³, maximum shear, and maximum fixed-end moment for the chosen case.
How to use this calculator
- Enter the beam length L and select its unit.
- Enter material modulus E and section inertia I.
- Select a load case, then enter the needed load value(s).
- Pick output units for deflection and slope.
- Click Calculate to view results above the form.
- Use Download CSV or Download PDF for reporting.
Example data table
| Case | L (m) | E (GPa) | I (cm⁴) | Load | Estimated tip deflection |
|---|---|---|---|---|---|
| Point load at free end | 1.5 | 200 | 8.33 | P = 1000 N | ≈ 67.5 mm |
| Uniform load (full length) | 1.5 | 200 | 8.33 | w = 500 N/m | ≈ 28.1 mm |
| End moment | 1.5 | 200 | 8.33 | M = 200 N·m | ≈ 13.5 mm |
Professional notes on cantilever end deflection
Why tip deflection matters
Cantilever beams show up in brackets, machine arms, crane jibs, sensor mounts, and canopies. Tip deflection is a key serviceability check because too much movement causes misalignment, vibration, rubbing, and poor fit-up. Many designs target limits such as L/250 to L/500, depending on precision and comfort requirements.
Inputs that drive results
Length dominates bending compliance. For a tip point load, δ ∝ L³, so doubling span increases deflection about 8×. For a full-span uniform load, δ ∝ L⁴, so doubling span increases deflection about 16×. Stiffness depends on EI: raising either modulus E or inertia I reduces deflection linearly.
Because the model is linear, you can compare scenarios quickly by changing one input at a time. Keep units consistent; the calculator converts everything to SI internally before reporting your selected output units.
Load cases supported
This tool covers six common cases: point load at the free end, point load at distance a, uniform load across the span, a pure end moment, and two triangular loads (maximum at the free end or at the fixed end). These approximate many real load distributions when you can represent the effect with an equivalent load.
Typical material data
Use realistic modulus values. Structural steel is typically about 200 GPa, aluminum alloys near 69 GPa, and many titanium alloys around 110 GPa. Polymers can be 1–5 GPa, and wood varies widely with grain and moisture. A 10% change in E produces a 10% change in predicted deflection.
Section inertia and shape effects
The second moment of area I is geometric and can change dramatically with shape. For a rectangle, I = b h³ / 12, so increasing depth is very effective. For a solid round section, I = π d⁴ / 64. Tubes often provide higher I per mass, improving stiffness without large weight penalties.
Practical accuracy limits
The formulas assume small deflection, constant cross-section, linear elasticity, and Euler–Bernoulli bending (shear deformation neglected). For short, deep beams, shear can increase deflection. Flexible supports, joints, cracks, or large rotations also reduce accuracy. For critical parts, validate using detailed analysis or testing.
Interpreting the reported outputs
Along with tip deflection and slope, the calculator reports flexural rigidity EI, estimated tip stiffness 3EI/L³, and the maximum fixed-end shear and moment for the selected case. Deflection and slope govern alignment and usability; fixed-end moment often governs stress and fatigue checks.
FAQs
1) Which theory does this calculator use?
It uses Euler–Bernoulli beam theory with constant E and I. The approach assumes linear elasticity and small deflection, so superposition applies across the supported load cases.
2) What is the second moment of area I?
I measures how strongly a cross-section resists bending about a chosen axis. Larger I means higher stiffness and lower deflection. It depends on shape and orientation, not material.
3) What happens if the point load location a is greater than L?
The calculator flags this as invalid because the load must lie on the span. Set a ≤ L, measured from the fixed end, to model a bracket-mounted component correctly.
4) How do I keep units consistent?
Pick length units for L and a, choose E in Pa-family units, and enter I in a matching length-to-the-fourth unit. The tool converts internally to SI and reports outputs in your selected units.
5) When should I worry about shear deformation?
When the beam is short and deep, or when the material is very compliant, shear can add noticeable deflection. In those cases, consider a Timoshenko beam approach or finite-element modeling.
6) Can I use this for composite or tapered beams?
Only as an approximation. If E or I changes along the length, the constant-EI formulas are not exact. Use an equivalent EI estimate or a more detailed method for better accuracy.
7) What are quick ways to reduce deflection?
Shorten the span, increase section depth to raise I, choose a higher-modulus material, reduce the applied load, or add a brace/support to change the boundary condition from cantilever to a stiffer configuration.