Deflection control, sufficient stiffness method

Idea of the method

Calculating the actual deflection takes a long time, so IS 456 cl. 23.2.1 allows deflection to be controlled indirectly. If the ratio of span to effective depth of a beam, slab or cantilever does not exceed a permitted value, the deflection will be within the limits of cl. 23.2 (final deflection including creep, shrinkage and temperature $\le \text{span}/250$, and the part occurring after finishes are applied $\le \text{span}/350$ or 20 mm, whichever is less). This is the empirical, or sufficient stiffness (span/depth ratio), method.

Procedure

1. Take the basic span/effective depth ratio:

2. Modify it by factors:
   • $k_t$: modification factor for tension steel from Fig. 4 of IS 456, which depends on $p_t$ and the service stress $f_s = 0.58 f_y \dfrac{A_{st,req}}{A_{st,prov}}$. Less steel and a lower stress give a larger factor (up to 2.0).
   • $k_c$: modification factor for compression steel from Fig. 5, based on $p_c = 100A_{sc}/bd$ (maximum 1.5).
   • $k_f$: reduction factor for flanged beams from Fig. 6, based on the ratio $b_w/b_f$ only (0.8 when $b_w/b_f \le 0.3$). For flanged beams, $p_t$ and $p_c$ for Figs. 4 and 5 are worked out on the area $b_f d$.
3. For spans above 10 m (other than cantilevers), the basic value is multiplied by $10/\text{span (m)}$; for cantilevers longer than 10 m, deflection must be calculated.
4. The permissible ratio is

$$\left(\frac{l}{d}\right)_{perm} = \left(\frac{l}{d}\right)_{basic} \times k_t \times k_c \times k_f$$

5. Compare it with the actual $l/d$. If actual $\le$ permissible, deflection is satisfied; if not, increase $d$ (or the steel) and check again.

Notes

• For two-way slabs, the check is made on the shorter span.
• This method is a conservative approximation of Annex C (the detailed method, which adds short-term, creep and shrinkage deflections).