Cantilever Beam Load at Free End Calculator

Free End Load Cantilever Analysis

Cantilever Beam Load at Free End Formula

Formula

\[ \theta = \frac{P \, L^2}{2 E I} \qquad \delta_x = \frac{P \, x^2 (3L - x)}{6 E I} \]

Where:

\(P\) — applied load at the free end (lbs or N)
\(L\) — length of the beam from the fixed support to the free end (in or m)
\(E\) — elastic modulus of the material (psi or Pa)
\(I\) — moment of inertia of the beam cross-section (in⁴ or m⁴)
\(x\) — distance from the fixed end to the point where deflection is evaluated
\(\theta\) — slope (rotation) at the free end, in radians
\(\delta_x\) — vertical deflection at position \(x\)

The first formula gives the rotation of the beam at its tip; the second gives the deflection at any distance \(x\) from the fixed support. Setting \(x = L\) in the second formula yields the familiar maximum tip deflection \(\delta_{max} = \frac{P L^3}{3 E I}\).

Cantilever Beam Load at Free End – Calculation Example

Example: a steel cantilever beam, 120 in long, carries a 500 lb load at its free end. The steel has an elastic modulus of 29,000,000 psi and the cross-section has a moment of inertia of 150 in⁴. Find the slope at the free end and the deflection 100 in from the fixed support.

\[ \theta = \frac{P L^2}{2 E I} = \frac{500 \times 120^2}{2 \times 29{,}000{,}000 \times 150} = \frac{7{,}200{,}000}{8{,}700{,}000{,}000} = 0.008276 \text{ rad} \]

\[ \delta = \frac{P x^2 (3L - x)}{6 E I} = \frac{500 \times 100^2 \times (3 \times 120 - 100)}{6 \times 29{,}000{,}000 \times 150} = \frac{1{,}300{,}000{,}000}{26{,}100{,}000{,}000} = 0.0498 \text{ in} \]

Result: the slope at the free end is 0.0083 rad (about 0.47°), and the beam deflects 0.0498 in at 100 in from the wall. At the very tip (\(x = L\)) the deflection reaches its maximum of 0.0662 in.

How to Use the Calculator

Enter the Applied Load \(P\) in lbs — the force acting at the free end of the beam.
Enter the Elastic Modulus \(E\) in psi. For steel use about 29,000,000 psi; for aluminum about 10,000,000 psi; for wood roughly 1,000,000–1,600,000 psi depending on the species.
Enter the Moment of Inertia \(I\) in in⁴ for the beam cross-section. For a rectangular section, \(I = \frac{b h^3}{12}\), where \(h\) is the dimension in the direction of bending.
Enter the Beam Length \(L\) in inches, measured from the fixed support to the point where the load is applied.
Enter the Deflection Point \(x\) in inches — the distance from the fixed end where you want to know the deflection. Use \(x = L\) for the maximum deflection at the tip.
Read the Slope at Free End (in radians) and the Deflection at your chosen point (in inches).

What Is a Cantilever Beam with a Tip Load?

A cantilever beam is fixed rigidly at one end — built into a wall, column, or foundation — and completely free at the other. When a single concentrated load \(P\) acts at the free end, the beam bends downward, with zero deflection and zero slope at the fixed end and the largest movement at the tip. This is one of the most fundamental cases in structural mechanics, and it appears everywhere in practice: balcony slabs, aircraft wings, diving boards, shelf brackets, sign posts, and robotic arms all behave essentially as cantilevers.

The calculator is based on the standard Euler–Bernoulli beam theory solution for this loading case. The deflected shape is a smooth cubic curve: nothing moves at the wall, and the deflection grows rapidly toward the free end. Because the deflection formula contains \(x^2\) and \(x^3\)-type terms, doubling the load or the length has a dramatic effect — deflection scales linearly with \(P\) but with the cube of the length. A beam twice as long deflects eight times as much under the same tip load.

The slope result tells you how much the beam rotates at the tip, in radians. Small values are normal in stiff structures; if you need degrees, multiply the radian value by \(\frac{180}{\pi} \approx 57.3\). The deflection result tells you how far the beam sags at your chosen point. Most design codes limit deflection to a fraction of the span — for example \(L/240\) or \(L/360\) for live load in building design. For the example beam above, \(L/240 = 0.5\) in, so a tip deflection of 0.066 in would easily pass such a limit.

Assumptions and Limitations

Small deflections: the formulas assume the slope is small (typically well under 0.1 rad) and that the load is applied vertically. For large rotations the results become inaccurate.
Linear elastic material: the material must stay below its yield stress, and \(E\) must be constant along the beam.
Constant cross-section: \(I\) is assumed uniform over the full length. A tapered or stepped beam needs a more detailed analysis.
Rigid fixed support: the wall or support is assumed perfectly rigid. In reality, connection flexibility or rotation at the base increases the actual deflection.
Load at the very tip only: if the load is distributed, applied at an intermediate point, or combined with other loads, use superposition carefully or a different loading case. Self-weight of the beam is not included unless you add it as an equivalent load.
Deflection only: this tool does not compute bending moment or shear. For a tip load the maximum bending moment is \(M = P L\) at the fixed end — always check stresses separately before finalizing a design.

When sizing real members, deflection is only one check among several. You may also want to verify load capacity limits, for example with the Maximum Floor Live Load Capacity Calculator, or check a supported slab with the Concrete Slab Maximum Wall Load Calculator. If your cantilever is part of a roof structure, the Rafter Length Calculator can help you lay out the supporting members.

Frequently Asked Questions

How do I calculate cantilever deflection by hand?

Use the formula \( heta = rac{P L^2}{2 E I} \\) for the tip slope and \\( heta = \frac{P L^2}{2 E I} \) for slope and \\( \\delta_x = \frac{P x^2 (3L - x)}{6 E I} \) for deflection. Work through it step by step: square the length, multiply by the load, then divide by \\( 2 E I \) or \\( 6 E I \). Keep all units consistent — if \\( E \) is in psi, then \\( P \), \\( L \) and \\( x \) must be in lbs, inches, and inches. The deflection comes out in inches directly.

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

Yes, as long as you stay consistent. If you enter \\( P \) in newtons, \\( E \) in pascals (or MPa with lengths in mm and \\( I \) in mm⁴), and \\( L \) and \\( x \) in meters, the results come out in meters. A common metric combination is \\( P \) in N, \\( E \) in MPa (N/mm²), \\( I \) in mm⁴, and lengths in mm — this yields deflection in mm and slope in radians. Never mix psi with metric lengths.

What is the maximum deflection of a cantilever with an end load?

The largest deflection occurs at the free end, \\( x = L \\): \\( \\delta_{max} = \frac{P L^3}{3 E I} \). Note the cubic dependence on length — a beam twice as long sags eight times as much. If you need this tip value in the calculator, simply enter the deflection point \\( x \) equal to the beam length \\( L \).

How do I find the moment of inertia I for my beam?

It depends on the cross-section. For a solid rectangle, \\( I = \frac{b h^3}{12} \), where \\( h \\) is the dimension aligned with the bending direction — orienting the deeper dimension vertically dramatically increases stiffness. For a circular section, \\( I = \frac{\pi d^4}{64} \). For standard steel shapes (W-beams, channels, tubes), look up the tabulated value in a properties table, and for built-up sections combine the parts with the parallel axis theorem.

What is an acceptable amount of deflection?

Most building codes limit deflection to a fraction of the span. For cantilevers, the limit is commonly applied to twice the projection length, so an \\( L/240 \) check means \\( \\frac{2L}{240} \) where \\( L \) is the cantilever length. Typical live-load limits are \\( L/360 \) for finishes that crack easily and \\( L/240 \) for more tolerant situations. Always verify the applicable code, and remember that deflection alone does not confirm a safe design — check bending stress and shear too.

Why is my calculated slope in radians, and how do I convert it to degrees?

Radian measure is standard in beam theory because the formulas derive directly from calculus. To convert, multiply by \\( \frac{180}{\pi} \approx 57.296 \): a slope of 0.05 rad equals about 2.86 degrees. If your slope exceeds roughly 0.1 rad (about 5.7°), the small-deflection assumption behind the formulas starts to break down, and the results become increasingly conservative in error.

Does this work for a distributed load or a load applied partway along the beam?

No — these formulas are only for a single concentrated load at the very tip. For a uniformly distributed load, the tip deflection is \\( \frac{w L^4}{8 E I} \) and the tip slope is \\( \frac{w L^3}{6 E I} \). For a load at an intermediate point \\( a \) from the support, different case formulas apply. You can combine independent load cases by superposition (add deflections and slopes), since the underlying theory is linear.