Effect of lateral restraint on moment capacity

The bending moment capacity of a beam depends on whether its compression flange is held against sideways movement.

Laterally restrained compression flange

The flange cannot buckle sideways, so the beam fails only when the section yields. The full plastic (or elastic) section strength is available (IS 800:2007 cl. 8.2.1):

$$M_d = \frac{\beta_bZ_pf_y}{\gamma_{m0}} \le \frac{1.2Z_ef_y}{\gamma_{m0}}$$

where $\beta_b = 1$ for plastic and compact sections, and $Z_e/Z_p$ for semi-compact ones.

Laterally unrestrained compression flange

The compression flange behaves like a column and tends to buckle sideways with a twist (lateral-torsional buckling, cl. 8.2.2). The capacity is reduced:

$$M_d = \beta_bZ_pf_{bd},\quad f_{bd} = \chi_{LT}\frac{f_y}{\gamma_{m0}},\quad \chi_{LT} = \frac{1}{\phi_{LT} + \sqrt{\phi_{LT}^2 - \lambda_{LT}^2}} \le 1$$

with $\lambda_{LT} = \sqrt{\beta_bZ_pf_y/M_{cr}}$. The longer the unrestrained length and the smaller $I_y$, $I_t$, $I_w$, the smaller $M_{cr}$ and the lower the capacity.

Example: ISMB 400, Fe 410 (computed with the IS 800 formulas)

Plastic modulus $Z_p = 1161$ cm³, restrained capacity $M_d = 264$ kN·m.

Practical effect

• Short unrestrained length (about $\lambda_{LT} \le 0.4$): no loss of capacity.
• A long unbraced length can cut the capacity to less than half; the beam then needs a larger section or lateral bracing.
• Restraint is obtained from a connected slab, cross beams, bracing or end torsional restraint, and a larger $I_y$ (wider flange) improves the unrestrained strength.