Derive Ld and explain its use

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 $L_d$.

  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.