Introduction:
Local shear failure refers to the failure of soil beneath a footing, where the failure surface does not reach the ground surface. It is important to understand the factors that contribute to local shear failure in order to design and construct safe and stable foundations.

Explanation:
The correct answer to the given question is option 'A': Compression of soil under the footing. Let's understand why this is the correct answer by considering the other options:

Option B: Ultimate bearing capacity is not well defined
- Ultimate bearing capacity is a measure of the maximum load that a foundation can support without causing shear failure.
- While it is true that ultimate bearing capacity is not well defined in local shear failure cases, it is not the reason why the failure surface does not reach the ground surface.

Option C: Failure is defined by large settlements
- Large settlements can indicate a failure in the foundation, but they are not the reason why the failure surface does not reach the ground surface in local shear failure.
- Settlements occur when the soil beneath the footing undergoes excessive deformation, but they do not necessarily cause local shear failure.

Option D: All of the mentioned
- This option includes all the options mentioned above, but none of them provide a correct explanation for why the failure surface does not reach the ground surface.

Option A: Compression of soil under the footing
- The correct answer is option 'A' because local shear failure occurs due to the compression of the soil beneath the footing.
- When a load is applied to a footing, the soil directly beneath it undergoes compression.
- The compression of the soil causes an increase in contact stress and shear stresses within the soil.
- If the shear strength of the soil is exceeded, shear failure occurs in the form of a shear surface.
- However, in the case of local shear failure, the shear surface does not reach the ground surface and remains confined within the soil beneath the footing.
- This is because the bearing pressure on the soil decreases with depth, and the shear strength of the soil increases with depth.
- Therefore, the shear failure surface does not extend to the ground surface and remains localized beneath the footing.

Conclusion:
In local shear failure, the failure surface does not reach the ground surface because of the compression of the soil under the footing. This occurs due to the increase in contact stress and shear stresses within the soil, which can lead to shear failure. Understanding the factors contributing to local shear failure is crucial for designing safe and stable foundations.