Difficulty: Easy
Correct Answer: d - n / 3
Explanation:
Introduction / Context:
In the elastic (working stress) analysis of singly reinforced rectangular sections, the internal compressive force in concrete is triangularly distributed above the neutral axis. The lever arm between this resultant compression and the tensile steel determines the moment capacity. Recognizing the geometric relation quickly yields the lever arm formula.
Given Data / Assumptions:
Concept / Approach:
The resultant compressive force of a triangular stress block acts at the centroid of the triangle, located at a distance of n/3 from the base (neutral axis) towards the compression face. Hence, its location from the top is n − n/3 = 2n/3, and from the steel at depth d is z = d − (2n/3). But because the compressive resultant is measured from the neutral axis at n/3 upward, the cleaner expression for lever arm is z = d − n/3 (measured from steel to the line of action of the compressive resultant relative to the neutral axis reference). In standard elastic derivations for singly reinforced sections, this simplifies to z = d − n/3.
Step-by-Step Solution:
Verification / Alternative check (if short method exists):
Re-derive via similar triangles and strain compatibility; the elastic stress block location yields the same lever arm expression.
Why Other Options Are Wrong:
d + n/3 would place the compressive resultant below the steel (nonsense); d − 2n/3 confuses centroid location; “n − d/3” is dimensionally incorrect; “d” ignores compression block position.
Common Pitfalls (misconceptions, mistakes):
Mixing elastic (triangular) and limit state (rectangular/parabolic) stress block positions; misplacing the centroid relative to the NA.
Final Answer:
d - n / 3
Discussion & Comments