Deflection, crack width control, crack formula

Methods of controlling deflection

• Keep the span/effective-depth ratio within the permitted value $= (l/d)_{basic} \times k_t \times k_c \times k_f$ (basic 7, 20 and 26 for cantilever, simply supported and continuous members; IS 456 cl. 23.2.1).
• Increase the depth or the amount of tension steel, add compression steel, and use good-quality concrete with proper curing.
• Limit the total deflection to span/250 and the part after finishes to span/350 or 20 mm.

Methods of controlling crack width

• Limit the spacing of bars (Table 15 and cl. 26.3.3: in slabs, main bars at not more than $3d$ or 300 mm, whichever is less) and use small-diameter bars well distributed over the tension zone.
• Provide adequate cover and side-face reinforcement of 0.1% of the web area in beams whose web depth exceeds 750 mm.
• Keep the steel stress low, provide the minimum steel, and cure properly.
• The surface crack width should not exceed 0.3 mm (0.2 mm where the member is exposed to moisture or an aggressive environment), cl. 35.3.2.

Design surface crack width (IS 456 Annex F)

$$w_{cr} = \frac{3\, a_{cr}\, \varepsilon_m}{1 + \dfrac{2\,(a_{cr} - c_{min})}{h - x}}$$

where

• $a_{cr}$ = distance from the point considered (on the surface) to the surface of the nearest longitudinal bar,
• $c_{min}$ = minimum clear cover to the longitudinal bar,
• $h$ = overall depth, $x$ = depth of the neutral axis,
• $\varepsilon_m$ = average strain at the level considered:

$$\varepsilon_m = \varepsilon_1 - \frac{b_t\,(h - x)(a' - x)}{3\,E_s A_s\,(d - x)}$$

$\varepsilon_1$ is the strain at the level considered (calculated ignoring the stiffening of the concrete in the tension zone), $b_t$ is the width of the section at the centroid of the tension steel, $a'$ is the distance from the compression face to the point considered, and $E_s$ and $A_s$ are the modulus and area of the tension steel.