Cantilever beam with a load at the free end
A cantilever is a beam that is rigidly fixed at one end and free at the other. It is one of the most common structures you run into in practice: a shelf bracket, a cantilevered balcony, a jib, a sign post or a machine arm all behave like this.
This page solves the textbook case: a single concentrated force acting downward at the free end of the beam. The calculator below is already loaded with this exact setup, so you only need to change the length, the load, the cross section or the material to get your own numbers.
Calculator
This calculator is already set up for the problem above. Change the length, the load, the profile or the material and it recalculates immediately; the address bar keeps a shareable link to your exact configuration.
Input data:
Beam:
Material: Construction steel (Generic)
Profile:
| Ax: 78.54mm2 | Jxx: 981.75mm4 |
| Asy: 78.54mm2 | Iyy 490.87mm4 |
| Asz: 78.54mm2 | Izz: 490.87mm4 |
Simulation results:
Displacements (mm):
| # | x | y | zz |
|---|
Reactions (newtons):
| # | x | y | zz |
|---|
Direct link to this setup: https://beamcalc.net/en/solve/cantilever-end-load#v1-AQEAAHpEAACgQZQBAQEAAAAAAADIQgAAAAA
How the beam is modelled
- The left end is rigidly fixed: it can neither move nor rotate.
- The right end is free: it carries no support at all.
- A vertical point load acts at the free end, pointing down.
- The beam is a solid steel rod, 20 mm in diameter and 1000 mm long.
How to read the results
The largest deflection always appears at the free end, and the slope is largest there as well. Deflection grows with the cube of the length, so doubling the beam length makes it eight times more flexible — this is the single most important thing to keep in mind when sizing a cantilever.
The reactions reported by the calculator are the shear force and the bending moment that the fixed support has to provide. The moment at the fixed end is the design quantity: it is where the beam is most likely to break or to yield.
Formulas
| Formula | What it gives |
|---|---|
delta_max = F * L^3 / (3 * E * I) | Deflection at the free end. F is the load, L the beam length, E the elastic modulus and I the bending moment of inertia of the cross section. |
theta_max = F * L^2 / (2 * E * I) | Slope at the free end, in radians. |
M_max = F * L | Bending moment at the fixed end, the most stressed section of the beam. |
FAQ
Where does a cantilever beam break first?
At the fixed end. The bending moment grows linearly along the beam and reaches its maximum F * L at the support, so that is where the stresses are highest.
Why does a small extension of a cantilever cause such a large deflection?
Because deflection is proportional to the cube of the length. Doubling the free length multiplies the deflection by eight, while the load-carrying capacity only grows linearly.
Does the calculator account for shear deformation?
Yes. The underlying Frame3DD kernel uses a stiffness formulation that includes shear, so short and stocky beams are reported more accurately than with the pure Euler-Bernoulli formula above.
Can I use this page for a shelf bracket or a balcony arm?
Yes, as long as the structure really is a single prismatic beam fixed at one end and loaded at the other. Real brackets with gussets or tapered sections are stiffer, so treat the result as a conservative estimate.
More beam examples
- Cantilever beam with a load in the middle A beam fixed at one end and loaded at midspan: where the load sits changes the deflection, and this page shows by how much.
- Steel pipe as a cantilever A clamped steel pipe loaded at its free end: deflection and reactions for a hollow circular section.
- Aluminium cantilever arm An aluminium rod fixed at one end and loaded at the other: why aluminium arms sag, and how to size one that does not.