This calculator estimates the free-end deflection and maximum fixed-end bending stress of a solid round cantilever beam carrying a point load at its free end. It uses inches, pounds-force and pounds per square inch throughout.
The model is an elastic beam estimate with a uniform circular section, fixed support and small deflection. It is useful for an early calculation, but it does not check yielding, buckling, fatigue, connection flexibility, dynamic loading or a complete structural design.
All entered values must be positive. Confirm that the actual support and load match an end-loaded cantilever. A simply supported beam, distributed load, hollow tube or load placed away from the free end requires different equations.
I = πd⁴/64
tip deflection δ = PL³/(3EI)
fixed-end bending stress σ = Mc/I = PL(d/2)/I
Here the maximum bending moment is PL and the outer-fiber distance is d/2. With P in pounds-force, L and d in inches, E in psi and I in in⁴, deflection is in inches and stress is psi. The calculator uses π≈3.1416 and rounds I to six decimals and the two displayed results to four decimals.
| Symbol | Meaning | Unit |
|---|---|---|
| L | Length from fixed support to end load | in |
| d | Diameter of the solid circular section | in |
| P | Point load at the free end | lbf |
| E | Selected elastic modulus | psi |
| I | Second moment of area about the neutral axis | in⁴ |
| δ / σ | Tip deflection / maximum bending stress | in / psi |
Take a steel cantilever with L=12 in, d=1 in, P=100 lbf and E=30,000,000 psi. First, I=π(1)⁴/64=0.0490875 in⁴. Deflection is:
δ = 100×12³ ÷ (3×30,000,000×0.0490875) = 0.03911 in
The fixed-end stress is:
σ = 100×12×0.5 ÷ 0.0490875 = 12,223.071 psi
The calculator displays about 0.0391 in and 12,223.071 psi. If diameter doubles while other inputs stay fixed, deflection falls by a factor of 16 because I depends on d⁴, while bending stress falls by a factor of 8.
Deflection describes the predicted vertical displacement at the loaded free end. Stress describes the model’s maximum elastic bending stress at the outer surface beside the fixed support. Neither value alone declares the beam safe.
Compare stress only with an appropriate allowable value that accounts for material grade, direction, temperature, fatigue, uncertainty and a suitable safety factor. Compare deflection with a serviceability limit appropriate to the assembly.
The material selector supplies nominal values. Real modulus varies with alloy, product form, temperature and fiber direction. Carbon-fiber composites in particular cannot be represented reliably by one universal isotropic modulus. Use verified material data and professional engineering review for consequential work.
Length is cubed in the deflection equation. A modest increase in span can therefore produce a large displacement increase even though stress grows only in direct proportion to length. Diameter is even more influential for stiffness because it is raised to the fourth power in I. Recheck measurements before relying on a result; confusing radius with diameter changes both outputs dramatically.
The equation also assumes the fixed support does not rotate. A flexible clamp or joint adds movement that this number cannot predict. Self-weight is omitted unless separately represented, but it is a distributed load and should not simply be entered as a point force without justification.
The circular section’s second moment of area contains d⁴. Doubling diameter makes I sixteen times larger and, under this model, reduces deflection to one-sixteenth.
Not directly. This page expects inches, pounds-force and psi. Convert every quantity consistently or use a fully SI beam calculation.
No. It omits many limit states and real connection effects. A suitable design must check verified material properties, allowable stress, serviceability, stability, fatigue and applicable codes.
No. A hollow circular section uses I=π(D⁴−d⁴)/64. Entering only the outside diameter would overstate stiffness.
For this end-loaded cantilever model, bending moment is greatest at the fixed support, and maximum tensile and compressive bending stresses occur at the outer fibers there.
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