In plastic design and collapse analysis, if Q denotes the load factor, S is the plastic shape factor, and F is the factor of safety used in elastic design (based on permissible stress), identify the correct relationship among these quantities.

Introduction / Context:
Plastic design compares ultimate (collapse) capacity to service (working) levels via a load factor. The reserve strength from plastic redistribution is captured by the shape factor S = Mp / My, while the elastic factor of safety F = σ_y / σ_perm links yield stress to permissible working stress.

Given Data / Assumptions:

• Shape factor S = Mp / My (plastic to yield moment).
• Elastic factor of safety F = σ_y / σ_perm.
• Load factor Q = Collapse load / Working load.

Concept / Approach:
At collapse, Mp governs. At first yield, capacity is My; at working level, allowable stress is σ_perm, so the working moment is limited to My / F. Therefore, Q = (Mp) / (My / F) = (Mp / My) * F = S * F.

Step-by-Step Solution:
1) Write S = Mp / My.2) Working-limit capacity = My / F (elastic permissible level).3) Load factor Q = Collapse / Working = Mp / (My / F) = S * F.

Verification / Alternative check:
Dimensional check is trivial (dimensionless ratios). The relation is quoted in plastic design texts for sections without premature local buckling.

Why Other Options Are Wrong:

• Additive or subtractive relations (S ± F) have no theoretical basis in collapse design.
• Q = F − S also lacks physical meaning.

Common Pitfalls:

• Confusing factor of safety F with load factor Q.
• Applying S of one shape (e.g., rectangle = 1.5) to another (I-beams ≈ 1.1–1.2) without checking.

Final Answer:
Q = S x F.