--- title: "Neutral axis ratio in WSM beam" subject: "Design of RCC Structure" chapter: 2 key: "semester_7/design_of_rcc_structure:neutral-axis-ratio-in-wsm-beam" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/neutral-axis-ratio-in-wsm-beam site: BCT Engineering --- # Neutral axis ratio in WSM beam Design of RCC Structure · Chapter 2: Working Stress Method of Design - Asked in: 2067 Asar (old course), 12 marks - Video: [RCC | Lecture 10| Working Stress Method |Depth of Neutral Axis| Stress in concrete and steel | GATE](https://www.youtube.com/watch?v=ExoL8sUG7Qg) — Civil Masonry - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/neutral-axis-ratio-in-wsm-beam - Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-2 ## Question The moment of resistance of a rectangular reinforced concrete beam section having width b mm and overall depth D mm is $0.85bd^2$. The stresses in the extreme fiber of the concrete and in the steel are not to exceed 7 N/mm$^2$ and 140 N/mm$^2$ respectively and the modular ratio equals 18.33. Determine the ratio between the depth of neutral axis from the compression fiber and the effective depth of the beam. The beam is reinforced for tension side only. ## Short answer Neutral axis depth ratio x/d = 0.394. Section is under-reinforced, steel stress governs; solving equilibrium gives this ratio. ## Answer **Given:** $M_r=0.85\,b\,d^2$ (N·mm), $\sigma_{cbc}=7$, $\sigma_{st}=140$ N/mm$^2$, $m=18.33$, singly reinforced. Let $n=x/d$. **Step 1: Balanced (critical) neutral axis** $$ k=\frac{m\sigma_{cbc}}{m\sigma_{cbc}+\sigma_{st}}=\frac{18.33\times7}{18.33\times7+140}=0.4782 $$ $$ j=1-\frac{k}{3}=0.8406,\qquad Q=\tfrac12\sigma_{cbc}kj=1.407 $$ Balanced moment $=1.407\,bd^2$. The given moment $0.85\,bd^2$ is smaller than this, so the section is **under-reinforced** and steel stress governs ($\sigma_{st}=140$ N/mm$^2$), with the concrete stress below 7 N/mm$^2$. **Step 2: Equations for an under-reinforced section** Equilibrium of the transformed section: $\dfrac{b x^2}{2}=mA_{st}(d-x)$, so $$ A_{st}=\frac{b\,d\,n^2}{2m(1-n)} $$ Moment about the compression centroid: $$ M_r=\sigma_{st}A_{st}\,d\left(1-\frac{n}{3}\right)=\frac{\sigma_{st}\,b\,d^2\,n^2\left(1-\frac{n}{3}\right)}{2m(1-n)} $$ **Step 3: Solve for $n$** Setting $M_r=0.85\,bd^2$: $$ \frac{140\,n^2\left(1-\frac{n}{3}\right)}{2\times18.33\,(1-n)}=0.85 $$ Solving numerically (trial and error): $n=0.394$. **Check:** concrete stress $\sigma_c=\dfrac{\sigma_{st}\,n}{m(1-n)}=4.97$ N/mm$^2$ $<7$, consistent with an under-reinforced section; $n