Bending and Shear Force Distribution in Cantilever Beams

Cantilever beams are widely used in mining and heavy industrial equipment, particularly where support is only possible on one end. This design allows the free end to project outward, supporting loads such as conveyor extensions, crane jibs, and loading platforms.

Understanding how bending moments and shear forces distribute along a cantilever is critical for safe, efficient, and long-lasting designs.

1. What Is a Cantilever Beam?

A cantilever beam is fixed rigidly at one end and free at the other.

• The fixed end resists bending moments and shear

• The free end experiences maximum deflection

• Common in overhanging structures where support under the load is not possible

2. Bending Moment Distribution

For a point load $P$ at the free end:

Mmax=P⋅LM_{max} = P \cdot LMmax​=P⋅L

• Moment is maximum at the fixed support and zero at the free end

• The moment diagram is triangular, peaking at the support

For a uniform load $w$:

Mmax=wL22M_{max} = \frac{wL^2}{2}Mmax​=2wL2​

• Again, maximum at the fixed end, zero at free end

3. Shear Force Distribution

For a point load $P$:

$$V=P$$

Shear force is constant along the entire length.

For a uniform load $w$:

Vsupport=wLV_{support} = wLVsupport​=wL

Shear force is maximum at the fixed end and decreases linearly toward the free end.

4. Mining Application Example: Loader Arm Extension

A cantilevered loader arm in a coke plant must support a bucket weighing 5000 N at 2.5 m from the pivot.

Mmax=5000×2.5=12500 NmM_{max} = 5000 \times 2.5 = 12500 \, \text{Nm}Mmax​=5000×2.5=12500Nm $$V=5000\text{N}$$

This calculation shows that the fixed joint must withstand the full bending moment and shear simultaneously.

5. Design Considerations for Cantilever Beams in Mining

✔️ Reinforce fixed ends to handle peak bending moments
✔️ Use stiff, high-strength materials to reduce deflection
✔️ Minimize length-to-depth ratio for better stiffness
✔️ Regularly inspect fixed supports for fatigue cracks

Cantilever beams concentrate the highest stresses at their fixed ends, making precise calculation and robust design essential in mining environments. By understanding bending and shear distribution, engineers can optimize material use, increase safety, and extend service life.