In the design of a built-up beam, the area A_p of cover plates required in one flange is expressed in terms of the section modulus deficiency and the lever arm. Which of the following formulas is correct?

Introduction / Context:Cover plates are provided in flanges of built-up beams to increase section modulus when the rolled section alone is insufficient. The required area depends on the moment demand and geometry of the section.

Given Data / Assumptions:

• Z_required = required section modulus
• Z_beam = section modulus of rolled section
• h = distance between flange centroids

Concept / Approach:The additional section modulus is supplied by cover plates. The formula links the deficiency ΔZ = Z_required − Z_beam to the cover plate area by ΔZ = A_p * h.

Step-by-Step Solution:Determine Z_required from design bending momentCalculate deficit: ΔZ = Z_required − Z_beamRelate ΔZ to cover plate: ΔZ = A_p * hSolve: A_p = (Z_required − Z_beam) / h

Verification / Alternative check:This aligns with plastic centroid approximation and common textbook derivations.

Why Other Options Are Wrong:

• Using + instead of − exaggerates section modulus
• Multiplication Z_required × Z_beam is dimensionally invalid
• (Z_beam − Z_required) / h would be negative if reinforcement is required

Common Pitfalls:Neglecting bolt/rivet hole deductions, using wrong h, or misplacing plates asymmetrically.

Final Answer:A_p = (Z_required − Z_beam) / h