The failure condition for a soil can be expressed in terms of limiting shear stress, called Principal Stresses and Shear Strength.

Principal Stresses:
- Principal stresses are the maximum and minimum normal stresses that act on a soil element.
- The principal stresses determine the failure condition of the soil.
- The maximum principal stress is denoted as σ₁, and the minimum principal stress is denoted as σ₃.
- The principal stresses can be determined using various laboratory tests or by analyzing the stress distribution in a soil mass.

Shear Strength:
- Shear strength is a measure of the resistance of soil to shear deformation.
- It represents the maximum shear stress that the soil can sustain without failing.
- The shear strength of soil depends on various factors such as soil type, density, moisture content, and stress history.
- It can be determined through laboratory tests such as the direct shear test, triaxial shear test, or consolidated undrained (CU) test.

Failure Condition:
- The failure condition of soil is reached when the applied shear stress exceeds the shear strength of the soil.
- In terms of principal stresses, the failure condition can be expressed as:
τ = σ₁ - σ₃
where τ is the shear stress, σ₁ is the maximum principal stress, and σ₃ is the minimum principal stress.
- When the shear stress exceeds the difference between the maximum and minimum principal stresses, the soil fails and undergoes shear deformation.

Importance:
- Understanding the failure condition of soil is crucial in geotechnical engineering.
- It helps in designing safe foundations, slopes, retaining walls, and other structures.
- By analyzing the principal stresses and shear strength, engineers can ensure that the soil can withstand the applied loads and avoid potential failures.

In conclusion, the failure condition for a soil can be expressed in terms of limiting shear stress, which is determined by the principal stresses and shear strength of the soil. The principal stresses represent the maximum and minimum normal stresses acting on the soil, while shear strength represents the resistance of soil to shear deformation. By comparing the applied shear stress with the shear strength, engineers can determine whether the soil will fail or sustain the applied loads.