Cantilever Beam Deflection Calculator
Calculate the maximum deflection, bending moment, and reaction forces in a cantilever beam with a point load at the free end.
Supports rectangular sections or custom moment of inertia.
What Is a Cantilever Beam? A cantilever beam is a structural member that is fixed (rigidly supported) at one end and free at the other. The fixed end resists both vertical forces and bending moments. Common examples include diving boards, balconies, aircraft wings, crane jibs, and overhanging roof structures.
Maximum Deflection For a cantilever with a point load P applied at the free end, the maximum deflection occurs at the free end:
delta_max = PL³ / (3EI)
Where:
- P = applied point load (N)
- L = beam length (m)
- E = Young’s modulus of the material (Pa)
- I = second moment of area (moment of inertia) of the cross-section (m⁴)
Bending Moment The maximum bending moment occurs at the fixed support:
M_max = P × L
This is the moment the fixed support must resist. It increases linearly with both load and beam length.
Reaction Forces At the fixed end (the wall or support), two reactions exist:
- Vertical reaction force: R = P (equal and opposite to the applied load)
- Fixed-end moment: M = P × L (the support must provide this moment to prevent rotation)
Moment of Inertia for a Rectangular Section For a rectangular cross-section with width b and height h:
I = b × h³ / 12
The moment of inertia measures the cross-section’s resistance to bending. Doubling the height (h) increases I by a factor of 8 — which is why tall, narrow beams (like I-beams) are far more efficient than short, wide ones.
Design Considerations In structural design, both stress and deflection must be checked:
- Bending stress: sigma = M × y / I, where y = h/2 for a rectangular section. This must be below yield strength.
- Deflection limit: Building codes typically limit deflections to L/360 (for floor beams) to L/180 for various structural applications.
Material and Section Selection A stiffer material (higher E) or a deeper cross-section (much higher I) dramatically reduces deflection. The term EI is called the flexural rigidity — it is the product of material stiffness and section geometry that governs beam behavior.
Superposition for Multiple Loads If multiple loads act simultaneously, deflections from each load can be calculated separately and added together (superposition), provided the material behaves elastically and deformations remain small.