Cantilever Beam Load at Any Point Calculator

Analyze Cantilever Deflection with Point Load

Cantilever Beam with Load at Any Point Formula

Formula

\[ \theta_{free} = \frac{P a^2}{2 E I} \] \[ \delta(x) = \begin{cases} \dfrac{P x^2 (3a - x)}{6 E I}, & x \leq a \\[6pt] \dfrac{P a^2 (3x - a)}{6 E I}, & x \geq a \end{cases} \]

Where:

\(P\) — applied point load (lbs)
\(E\) — elastic modulus of the beam material (psi)
\(I\) — area moment of inertia of the cross-section (in⁴)
\(L\) — total length of the beam (in)
\(a\) — distance from the fixed support to the load (in)
\(x\) — distance from the fixed support to the point where deflection is evaluated (in)

The slope formula gives the rotation at the free end of the beam. The deflection formula is piecewise: the first expression applies between the support and the load, and the second applies from the load to the free end, where the beam bends in a straight (rigid-body) line.

Cantilever Beam with Load at Any Point – Calculation Example

Example: a steel cantilever beam is 120 in long with \(E = 29{,}000{,}000\) psi and \(I = 150\) in⁴. A 500 lb load acts 80 in from the fixed support. Find the slope at the free end and the deflection 90 in from the support.

\[ \theta_{free} = \frac{P a^2}{2 E I} = \frac{500 \times 80^2}{2 \times 29{,}000{,}000 \times 150} = \frac{3{,}200{,}000}{8{,}700{,}000{,}000} = 0.000368 \text{ rad} \]

Since the deflection point \(x = 90\) in lies beyond the load \(a = 80\) in, use the second expression:

\[ \delta = \frac{P a^2 (3x - a)}{6 E I} = \frac{500 \times 80^2 \times (3 \times 90 - 80)}{6 \times 29{,}000{,}000 \times 150} = \frac{608{,}000{,}000}{26{,}100{,}000{,}000} = 0.0233 \text{ in} \]

Result: the slope at the free end is approximately 0.000368 rad (about 0.021°) and the deflection at 90 in from the support is 0.0233 in downward.

How to Use the Calculator

Enter the Load \(P\) in pounds — the point force applied to the beam.
Enter the Elastic Modulus \(E\) in psi. For steel this is typically 29,000,000 psi; for aluminum about 10,000,000 psi.
Enter the Moment of Inertia \(I\) in in⁴ for the beam cross-section (available from standard beam tables).
Enter the Beam Length \(L\) in inches, measured from the fixed support to the free end.
Enter the Load Position \(a\) in inches — the distance from the fixed support to where the load is applied. It must be between 0 and \(L\).
Enter the Deflection Point \(x\) in inches — the distance from the fixed support where you want the deflection.
Read the Slope at Free End (in radians) and the Deflection (in inches) below the inputs.

What Is a Cantilever Beam with a Load at Any Point?

A cantilever beam is fixed rigidly at one end and free at the other. In many practical situations the load does not act at the free tip — for example, a pipe or duct supported partway along a bracket, a sign mounted mid-span on a pole arm, or a machine bolted to a platform extension. In these cases the load sits at a distance \(a\) from the fixed support, and the beam beyond the load carries no bending moment; it simply rotates as a rigid body.

This changes the deflected shape compared to a tip-loaded cantilever. Between the support and the load, the beam follows a curved elastic shape described by the first formula. Beyond the load, the beam remains straight but sloped, so any point in that region deflects according to the second formula. The maximum deflection always occurs at the free end, and its value is found by substituting \(x = L\) into the second expression: \(\delta_{max} = \frac{P a^2 (3L - a)}{6EI}\).

The slope at the free end depends only on the load magnitude and its distance from the support, not on where along the beam you measure it afterward — the portion beyond the load rotates uniformly. A small slope in radians means a nearly horizontal beam; multiply by \(57.3\) to convert to degrees if needed.

Interpreting the result is usually a matter of comparing deflection against a limit. Many design codes restrict live-load deflection of beams to \(L/360\) or \(L/240\). For the example beam above, \(L/360 = 120/360 = 0.333\) in, so a 0.0233 in deflection is comfortably within that limit. When checking a floor or platform more broadly, you may also want to verify the overall load capacity with the Maximum Floor Live Load Capacity calculator.

Assumptions and Limitations

Small deflection theory: the formulas assume deflections are small relative to the beam length and that the material remains linearly elastic (below yield).
Rigid, perfectly fixed support: any rotation or flexibility at the anchor increases real deflections beyond the calculated values.
Point load only: the beam's own weight and any distributed loads are ignored. For heavy sections, add the self-weight as a separate check.
Constant cross-section: \(E\) and \(I\) are uniform along the entire length. A tapered or stepped beam requires a more detailed analysis.
Units must be consistent: this calculator works in pounds, psi, inches and in⁴. If your data is in metric units, convert first or interpret the outputs accordingly (the formulas themselves are unit-agnostic as long as inputs are consistent).
Sign convention: the deflection result is a magnitude; downward deflection under a downward load is implied.

If your design involves loads applied directly to a slab rather than a cantilever member, see the Concrete Slab Maximum Wall Load calculator, or check the required quantities for the supporting structure with the Concrete Yardage tool.

Frequently Asked Questions

How do I calculate the deflection by hand without the calculator?

First check where your evaluation point sits relative to the load. If \(x \leq a\), compute \(\delta = \frac{P x^2 (3a - x)}{6EI}\). If \(x \geq a\), compute \(\delta = \frac{P a^2 (3x - a)}{6EI}\). Multiply P by the distance terms first, then divide by \(6EI\). Keeping units in lbs, psi, and inches gives the deflection directly in inches.

Can I use this calculator with metric units (N, mm, MPa)?

Yes, but every input must use the same unit system. With newtons, megapascals (N/mm²), and millimeters, the deflection comes out in millimeters and the slope in radians. The formulas are unit-agnostic as long as you stay consistent — mixing pounds with millimeters will give meaningless results. Convert all inputs before entering them.

What happens if the load is applied exactly at the free end (a = L)?

The formulas reduce to the classic tip-loaded cantilever case: \(\delta_{tip} = \frac{P L^3}{3EI}\) and \(\theta = \frac{P L^2}{2EI}\). This is a good sanity check. If your results don't match these well-known values when a = L, your units or inputs likely contain an error.

What counts as an acceptable deflection for a cantilever beam?

Common limits are \(L/180\), \(L/240\), or \(L/360\) of the span depending on the application and applicable building code. For a plastered ceiling or brittle finishes, \(L/360\) is typical; for a simple platform, \(L/240\) may suffice. Always verify against the specific code governing your project, and remember cantilever deflections grow with the cube of length.

Why is the beam straight beyond the load point?

Past the load there is no bending moment because no force acts on that segment. It carries only constant shear transferred from the load, so it stays straight and simply tilts at the slope established at the load position. That's why the second formula is linear in \(x\) — every point beyond the load deflects as if it were attached to a rigid, rotated extension.

What are the most common mistakes when using these formulas?

The biggest ones are mixing unit systems (psi with metric lengths), measuring \(x\) or \(a\) from the free end instead of the fixed support, and using the wrong branch of the piecewise formula when \(x\) and \(a\) are close together. Also check that \(E\) is in psi not ksi — forgetting the factor of 1000 understates deflection by three orders of magnitude.

Does the calculator include the beam's own weight?

No. The formulas cover only the single point load \(P\). For a beam with significant self-weight, estimate the uniform load \(w\) (weight per inch) and add the extra tip deflection \(\frac{w L^4}{8EI}\) from self-weight alone. For light brackets and short spans this correction is usually negligible; for long, heavy sections it can matter.

Cantilever Beam with Load at Any Point Calculator