Mechanical design — In strength-based design, the “factor of safety” (FoS) is defined as the ratio of the material's limiting strength to the allowable service (working) stress used for design. Which statement correctly gives this ratio?

Introduction:
The factor of safety (FoS) is a cornerstone of mechanical and pressure equipment design. It provides a margin between the material’s limiting capacity (ultimate or yield strength per code) and the allowable service stress that we permit in operation. Understanding the correct definition prevents under- or over-design.

Given Data / Assumptions:

• Design is based on allowable stress concepts.
• Ultimate stress refers to ultimate tensile strength (or a code-defined limit, sometimes yield).
• Working stress is the maximum expected stress in normal service.

Concept / Approach:
FoS is a ratio greater than 1 that ensures the working stress is safely below the limiting stress. In simple form: FoS = ultimate stress / working stress. Codes may define allowable stress as ultimate/yield divided by a code factor (e.g., 3.5, 1.5), but the fundamental definition remains a ratio that provides design margin.

Step-by-Step Solution:
Identify limiting strength: ultimate tensile strength (or code limit).Define working (allowable) stress: the stress used in calculations for service.Compute FoS = ultimate stress / working stress; hence working stress = ultimate / FoS.

Verification / Alternative check:
Check dimensional sense: a ratio of two stresses is dimensionless. For a given FoS, higher ultimate strength permits higher working stress in direct proportion.

Why Other Options Are Wrong:

• Ultimate to breaking stress: ambiguous and not a standard definition (breaking is usually the same specimen’s ultimate event).
• Working to ultimate: this is the reciprocal of FoS, sometimes called “utilization factor,” not FoS.
• None of these: incorrect because the standard ratio is well established.

Common Pitfalls:
Confusing FoS with a code allowable stress factor; mixing yield vs. ultimate basis—always follow the applicable design code (e.g., ASME, IS, EN) for which limit is used.

Final Answer:
ultimate stress to working stress