--- title: "Derive Ld and explain its use" subject: "Design of RCC Structure" chapter: 6 key: "semester_7/design_of_rcc_structure:derive-ld-and-explain-its-use" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/derive-ld-and-explain-its-use site: BCT Engineering --- # Derive Ld and explain its use Design of RCC Structure · Chapter 6: Design for bond and development length - Asked in: 2082 Bhadra, 2+2 marks - Video: [Derivation for development length ! RCC !](https://www.youtube.com/watch?v=vYC_vbV7fYo) — RS EDUCATION - 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/derive-ld-and-explain-its-use - Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-6 ## Question Derive formula for development length ($L_d$) for rebar in RCC member. Explain the use of development length in RCC construction works. ## Short answer L_d = 0.87 f_y φ/(4 τ_bd) is derived from force equilibrium: bar tension 0.87 f_y πφ²/4 equals bond resistance τ_bd πφ L_d. It is used for anchorage of bars beyond critical sections, lap splices, curtailment, support anchorage, and in beam-column joints to ensure full stress development. ## Answer #### Derivation **Development length** $L_d$ is the length of bar needed on either side of a section so that the stress $\sigma_s$ in the bar can be developed by bond with the concrete (IS 456 cl. 26.2.1). Consider a bar of diameter $\phi$ stressed to $\sigma_s=0.87f_y$ at a section and anchored over a length $L_d$. ```text concrete +---------------------------+ | T = σs.(πφ²/4) <==[bar]=====>| +---------------------------+ |<------- Ld -------->| bond stress τbd on πφ.Ld ``` Tensile force in the bar: $T=\sigma_s\times\dfrac{\pi\phi^2}{4}$. Resisting bond force over the embedded length: $F=\tau_{bd}\times\pi\phi\,L_d$. For equilibrium, $T=F$: $$ \sigma_s\frac{\pi\phi^2}{4}=\tau_{bd}\,\pi\phi\,L_d\ \Rightarrow\ L_d=\frac{\phi\,\sigma_s}{4\,\tau_{bd}}=\frac{0.87f_y\,\phi}{4\,\tau_{bd}} $$ Here $\tau_{bd}$ is the design bond stress (IS 456 cl. 26.2.1.1): 1.2, 1.4, 1.5, 1.7 and 1.9 N/mm$^2$ for plain bars in tension in M20, M25, M30, M35 and M40. These values are increased by 60% for deformed bars and by 25% for bars in compression. For Fe415 deformed bars in M20: $\tau_{bd}=1.2\times1.6=1.92$ N/mm$^2$, so $L_d=\dfrac{0.87\times415\,\phi}{4\times1.92}=47\phi$. #### Use of development length in RCC construction 1. **Anchorage of bars:** bars are extended by $L_d$ beyond the point where full stress is required (beam bars beyond the column face, bars into supports and into footings). 2. **Lap splices:** the lap length is based on $L_d$ ($\ge L_d$ and $\ge 30\phi$ in flexural tension, $\ge 2L_d$ and $\ge 30\phi$ in direct tension). 3. **Curtailment of bars:** bars are extended by $d$ or $12\phi$ beyond the theoretical cut-off point, and the development length from the section of maximum stress is checked. 4. **Support anchorage:** at simple supports, $L_d\le 1.3M_1/V+L_0$ is checked. 5. **Column-footing and beam-column joints:** dowels and column bars must be anchored by $L_d$ (in seismic work, measured from the critical section). 6. **Hooks and bends:** the anchorage value of a standard hook is $16\phi$ and of a 90$^\circ$ bend $8\phi$, which reduces the straight length required. Development length depends on the bar diameter, steel grade, concrete grade, bar surface (plain or deformed), and whether the bar is in tension or compression. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In deriving Ld, which forces are equated? A. Shear force and bond stress B. Concrete compression and steel tension C. Bar weight and bond force D. Bar tension σs πφ²/4 and bond resistance τbd πφ Ld Answer: D. For equilibrium T = F, giving Ld = 0.87 fy φ/(4τbd). (Key point: Derivation of Ld) 2. How is Ld used in lap splices? A. Lap length is half of Ld B. Lap length is unrelated to Ld C. Lap length is always 12φ D. Lap length is taken not less than Ld (and not less than 30φ in flexure, 2Ld in direct tension) Answer: D. The lap must transfer the bar force through bond, so it is based on Ld. (Key point: Use in lap splices) 3. What is done at the theoretical cut-off point of a bar? A. The bar is stopped exactly at that point B. The bar is extended by the full span C. The bar is extended by 4φ only D. The bar is extended by d or 12φ (greater), and Ld from the maximum-stress section is checked Answer: D. Bars are extended beyond the theoretical cut-off by the greater of d and 12φ, with Ld also checked. (Key point: Use at curtailment) 4. How do standard hooks and bends affect the straight length required? A. They increase the length needed B. They have no anchorage value C. They replace the need for any bond D. Anchorage values of 16φ (hook) and 8φ (90° bend) reduce the straight length needed Answer: D. A standard hook has an anchorage value of 16φ and a 90° bend 8φ, reducing the straight Ld required. 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