Introduction:
When a cantilever beam is loaded at its free end, it experiences various internal stresses due to the applied load. These stresses are distributed across the cross-section of the beam. The maximum compressive stress, also known as the principal stress, is an important parameter to determine the beam's strength and stability.

Explanation:
The maximum compressive stress in a cantilever beam loaded at its free end develops at the bottom fiber of the beam. This can be understood by considering the behavior of the beam under loading.

1. Bending Stress Distribution:
When a cantilever beam is subjected to a load at its free end, it undergoes bending deformation. The top fibers of the beam experience tensile stresses, while the bottom fibers experience compressive stresses.

2. Neutral Axis:
The neutral axis is an imaginary line within the beam cross-section that experiences no stress during bending. It lies at the centroid of the cross-section and divides the beam into top and bottom fibers.

3. Maximum Compressive Stress:
The maximum compressive stress occurs at the bottom fiber of the beam. This is because the bottom fibers are located farthest away from the neutral axis and experience the highest compressive forces during bending.

4. Stress Distribution:
As we move away from the neutral axis towards the bottom fiber, the compressive stress increases. At the bottom fiber, the stress reaches its maximum value. Conversely, as we move away from the neutral axis towards the top fiber, the stress decreases until it becomes zero at the neutral axis.

5. Importance of Maximum Compressive Stress:
The maximum compressive stress is crucial in determining the beam's strength and stability. If the compressive stress exceeds the material's compressive strength, it may lead to failure, such as buckling or crushing of the beam.

Conclusion:
In a cantilever beam loaded at its free end, the maximum compressive stress develops at the bottom fiber of the beam. This occurs due to the bending deformation and stress distribution across the beam's cross-section. Understanding the distribution of stresses is essential for designing beams that can withstand the applied loads without failure.