A cantilever is a structure that is fixed at one end and free at the other, such as a balcony, diving board, crane arm, or bridge segment during construction. Because the free end is unsupported, loads create bending that must be carried back to the fixed support. Engineers study cantilevers to predict internal forces, maximum stress, and deflection before a structure is built.
This makes cantilever analysis essential for safe, efficient designs that use material where it is needed most.
In a simple cantilever beam with a downward point load at the tip, the bending moment is largest at the wall and decreases to zero at the free end. The fixed support must provide a vertical reaction force and a resisting moment to keep the beam in equilibrium. The beam bends into a curve, with the greatest deflection occurring at the free end.
Real cantilever structures often use deeper sections, trusses, reinforcement, or counterweights to reduce stress and limit movement.
Key Facts
- For a cantilever with a tip load P and length L, the support reaction force is R = P.
- For a cantilever with a tip load P, the maximum bending moment at the fixed support is Mmax = P L.
- For a cantilever with a tip load P, the tip deflection is δmax = P L^3 / (3 E I).
- Bending stress in a beam is σ = M y / I, so stress increases when moment M or distance y from the neutral axis increases.
- For a cantilever with a uniformly distributed load w, the maximum support moment is Mmax = w L^2 / 2.
- Increasing the second moment of area I greatly reduces bending stress and deflection.
Vocabulary
- Cantilever
- A cantilever is a beam or structure fixed at one end and unsupported at the other.
- Fixed support
- A fixed support prevents translation and rotation, allowing it to provide reaction forces and a resisting moment.
- Bending moment
- Bending moment is the internal turning effect in a beam caused by external loads.
- Deflection
- Deflection is the displacement of a structural member from its original shape under load.
- Second moment of area
- The second moment of area describes how a cross section resists bending based on how its material is distributed around the neutral axis.
Common Mistakes to Avoid
- Putting the maximum bending moment at the free end is wrong because the moment is zero where there is no lever arm beyond the load and largest at the fixed support.
- Ignoring the fixed support moment is wrong because a cantilever needs both a reaction force and a resisting moment to remain in rotational equilibrium.
- Using beam length linearly for deflection is wrong because tip deflection for a point load follows δmax = P L^3 / (3 E I), so length has a cubic effect.
- Treating all beam shapes with the same area as equally stiff is wrong because bending stiffness depends strongly on I, not just cross-sectional area.
Practice Questions
- 1 A 2.0 m cantilever beam carries a 500 N downward point load at its free end. Find the vertical reaction force and maximum bending moment at the fixed support.
- 2 A cantilever has L = 3.0 m, P = 800 N, E = 200 GPa, and I = 6.0 x 10^-6 m^4. Calculate the tip deflection using δmax = P L^3 / (3 E I).
- 3 A balcony designer can choose either a shallow solid rectangular beam or a deeper beam with the same material and weight. Explain which is usually better for reducing deflection and why.