Define the factor of safety (FoS) used in machine design and structural engineering: choose the correct ratio.

Introduction / Context:
The factor of safety (FoS) provides a margin between the limiting material strength and the expected working (allowable) stress during service. It accounts for uncertainties in loading, material variability, and analysis assumptions.

Given Data / Assumptions:

• Allowable (working) stress is used for design checks.
• Limiting strength may be ultimate or yield, depending on design philosophy.
• Linear elastic behavior assumed up to the allowable.

Concept / Approach:
By definition, FoS = limiting stress / working stress. For ductile design against yielding, some codes use yield stress; for ultimate strength design, ultimate stress is used. The classical general definition uses ultimate stress in the numerator.

Step-by-Step Solution:

Verification / Alternative check:
Confirm with design codes: allowable stress = limiting stress / FoS ⇒ rearranging yields FoS as above.

Why Other Options Are Wrong:
Working/ultimate is the reciprocal.Breaking/ultimate or ultimate/breaking are ill-defined for ductile design and do not match standard FoS definition given.

Common Pitfalls:
Mixing yield-based and ultimate-based FoS; ignoring different FoS for different failure modes (buckling, fatigue).

Final Answer:

ultimate stress to working stress