RCC slab design (working stress approach): If the maximum bending moment for a simply supported slab (per metre width) is M kg·cm and the material strength factor (moment of resistance factor) is Q, what is the required effective depth d of the slab?

Introduction / Context:This question checks your understanding of the working-stress design relation for slabs, where the ultimate (allowable) moment capacity per metre width is expressed using a material constant Q. Slabs are normally designed per 1 m width, so breadth b = 100 cm when using kg·cm units.

Given Data / Assumptions:

• Simply supported slab designed by working-stress method.
• Maximum bending moment per metre width = M kg·cm.
• Moment of resistance factor (material factor) = Q.
• Per metre width: b = 100 cm.

Concept / Approach:For slabs in WSM, the limiting (safe) moment of resistance is M = Q * b * d^2. Here Q groups material and section constants. Rearranging gives the effective depth d needed for the demand moment M.

Step-by-Step Solution:Start with M = Q * b * d^2.Use b = 100 cm for one-metre strip of slab.Rearrange: d^2 = M / (Q * b) = M / (Q * 100).Therefore d = sqrt(M / (Q * 100)).

Verification / Alternative check:Dimensional check: M (kg·cm) divided by Q (kg/cm^2) and by b (cm) leaves cm^2; the square root gives cm, as required for d.

Why Other Options Are Wrong:d = sqrt(M / Q): Ignores b = 100 cm and overestimates depth.d = M / (Q * 100): Linear instead of square-root dependence.d = sqrt(Q / M) or d = M / Q: Dimensionally inconsistent.

Common Pitfalls:

• Forgetting to use b = 100 cm for a 1 m strip.
• Mixing SI (N·mm) with kg·cm units; be consistent.

Final Answer:d = sqrt(M / (Q * 100))