Failure Theories at Elastic Limit: The maximum principal stress theory is known as

Introduction:
Failure theories predict yielding or fracture under multiaxial stress states. Among classical criteria, each theory emphasizes a different stress or energy measure. This question asks you to correctly name the theory corresponding to the maximum principal stress criterion at the elastic limit.

Given Data / Assumptions:

• Material is ductile or brittle within elastic range for theory comparison.
• Principal stresses are well-defined at the point of interest.
• We are identifying the name attached to the maximum principal stress theory.

Concept / Approach:
The maximum principal stress theory states failure occurs when the largest principal stress equals the material's uniaxial failure stress. Historically, this is called Rankine's theory and is especially used for brittle materials. Other named theories correspond to different criteria (shear stress, strain energy, etc.).

Step-by-Step Solution:
Step 1: Recall mapping: maximum principal stress → Rankine.Step 2: Distinguish from Tresca (maximum shear stress) and Von Mises (distortion energy).Step 3: Confirm St. Venant and Haig relate to other strain/energy concepts, not maximum principal stress naming.Step 4: Select Rankine's theory as the correct naming.

Verification / Alternative check:
Design handbooks list Rankine as the principal stress criterion, suitable for brittle materials where tensile strength governs crack initiation.

Why Other Options Are Wrong:
Guest's/Tresca's — Maximum shear stress criterion, not principal stress.St. Venant's — Associated with principal strain or energy-based ideas, not the maximum principal stress naming.Haig's — Related to strain energy approaches (e.g., octahedral shear), not Rankine.Von Mises's — Distortion energy criterion, not principal stress criterion.

Common Pitfalls:
Confusing principal stress (Rankine) with shear-based (Tresca) or energy-based (Von Mises) criteria; assuming all names are interchangeable.

Final Answer:
Rankine's theory.