Define Ld and lap splice, derive Ld

Definitions

• Development length ($L_d$): the length of bar embedded in concrete that is needed to develop the design stress in the bar through bond: $L_d=\dfrac{\phi\,0.87f_y}{4\tau_{bd}}$.
• Lap splice: a joint between two bars made by overlapping them side by side for a specified length, so that force passes from one bar to the other through the surrounding concrete. The lap length is at least $L_d$ (and not less than $30\phi$ for flexural tension, $2L_d$ for direct tension and $24\phi$ for compression).

Derivation of $L_d\le 1.3\dfrac{M_1}{V_u}+L_0$

Near a simple support the shear is nearly constant, so $M=V_ux$. Let $M_1$ be the moment of resistance of the section with all bars stressed to $0.87f_y$. The applied moment reaches $M_1$ at $x=M_1/V_u$ from the support centre, so the bars must be fully developed there. The length of bar available for development is the distance to that section plus the anchorage $L_0$ beyond the support centre:

$$L_{avail}=\frac{M_1}{V_u}+L_0$$

For the bar to be safe against pull-out, $L_d\le L_{avail}$. The compressive reaction at the support confines the bar and increases bond, so the code allows the first term to be increased by 30%:

$$L_d\le\frac{1.3\,M_1}{V_u}+L_0$$

Here $M_1$ is the moment of resistance of the section assuming all bars are stressed to $0.87f_y$, $V_u$ is the factored shear at the section, and $L_0$ is the sum of the anchorage beyond the centre of the support and the equivalent anchorage value of any hook or mechanical anchorage. (The limit of the greater of $d$ and $12\phi$ applies to $L_0$ at a point of inflection.)