Purely cohesive (φ = 0) slope — critical circle: In a purely cohesive soil (φ = 0) slope, the critical center of the circular slip surface lies at the intersection of:

Introduction / Context:
For homogeneous slopes, Swedish circle (method of slices) and stability charts provide critical slip surfaces. In the special case of purely cohesive soil (φ = 0), classical results locate the critical circle center using simple geometrical rules that aid preliminary design checks.

Given Data / Assumptions:

• Homogeneous soil with φ = 0 (purely cohesive).
• Planar ground surface forming a slope with a defined toe and crest.
• Classical circular slip surface assumption.

Concept / Approach:
For φ = 0 slopes, the most critical (minimum factor of safety) circle is found at the intersection of the locus of potential centers and a perpendicular erected at one-third of the slope length measured from the toe. This geometric rule stems from analyses that balance driving and resisting moments for fully cohesive materials.

Step-by-Step Solution:

Verification / Alternative check:
Stability charts (Taylor’s stability numbers for φ = 0) and trial slip-circle analyses confirm this location for the critical circle in cohesive slopes.

Why Other Options Are Wrong:

• Perpendicular bisector or two-thirds position: do not match the established φ = 0 rule.
• Directional angles with slope normal: not the classical geometric construction for cohesive slopes.

Common Pitfalls:
Applying this geometric rule to c–φ soils with φ > 0 where the critical center location differs; ignoring groundwater effects that alter stability.

Final Answer:
A perpendicular drawn at one-third of the slope length from the toe and the locus of centers