Flexural bond stress and Ld proof

Definitions

• Anchorage bond: the bond developed over the embedded length of a bar that lets it carry its full design stress without slipping (developed at the ends of bars, at supports and at cut-offs).
• Flexural bond: the bond stress caused by the change in bar force along the span as the bending moment changes (it is high near the supports, where shear is high).

Flexural bond stress is a function of shear force

Derivation. For a beam section, the tension in the steel is $T=\dfrac{M}{jd}$, where $jd$ is the lever arm. Over a small length $dx$, the change in tension is

$$dT=\frac{dM}{jd}=\frac{V\,dx}{jd}$$

This change is resisted by bond over the perimeter of the bars, $\Sigma o$, along $dx$:

$$\tau_{bf}\,\Sigma o\,dx=\frac{V\,dx}{jd}\ \Rightarrow\ \boxed{\tau_{bf}=\frac{V}{\Sigma o\ jd}}$$

Here $\Sigma o$ is the sum of the perimeters of the tension bars ($\Sigma o=n\pi\phi$) and $jd\approx0.9d$. Flexural bond stress is therefore directly proportional to the shear force $V$. It is highest near the supports, where $V$ is large, so it is checked there.

Proof of the condition at simple support

Condition at a simple support (IS 456 cl. 26.2.3.3). At a simply supported end the moment is small, but the positive-moment bars must still be able to develop their full design stress at the sections where they are needed. To ensure this, the bar diameter is limited so that

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

Proof.

Near the support the shear $V$ is almost constant, and the bending moment grows linearly from zero at the support centre:

$$M=V\,x$$

Let $M_1$ be the moment of resistance of the section when all the tension bars at it are stressed to $0.87f_y$. The applied moment reaches $M_1$ at $x=M_1/V$ from the support centre, so the bars must be fully developed by this point.

  bar end   support centre             section where M = M1
     |<- L0 ->|<-------- M1/V -------->|
     ==========================================  bar
     |<--- available length = L0 + M1/V --->|

Available length of bar $=$ (distance from the support centre to that section) $+$ (anchorage beyond the support centre, $L_0$):

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

The bar must satisfy $L_d\le L_{avail}$. The compressive reaction at the support confines the concrete around the bar and so increases bond. The code allows for this by increasing the first term by 30%:

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

where $M_1$ = moment of resistance of the section assuming all the steel at the section is stressed to $0.87f_y$; $V$ ($=V_u$) = factored shear at the section; $L_0$ = sum of the anchorage beyond the centre of the support and the equivalent anchorage value of any hook or mechanical anchorage. (At a point of inflection, $L_0$ is limited to the greater of the effective depth $d$ and $12\phi$.) The factor 1.3 is used when the bar ends are confined by a compressive reaction; otherwise $M_1/V$ is used.