Formulas: Lateral-Torsional Buckling

Formula 1

(a) When $L_{b} \leq L_{p}$ , the limit state of lateral-torsional buckling does not apply.

L_{b} \leq L_{p}

(b) When $L_{p}<L_{b} \leq L_{r}$

L_{p}<L_{b} \leq L_{r}

M_{n}=C_{b}\left|M_{p}-\left(M_{p}-0.7 F_{y} S_{x}\right)\left(\frac{L_{b}-L_{p}}{L_{r}-L_{p}}\right)\right| \leq M_{p}

L_{b}>L_{r}

M_{n}=F_{c r} S_{x} \leq M_{p}

where

$L_{b}=$ length between points that are either braced against lateral displacement of the compression flange or braced against twist of the cross section, in. (mm)

$F_{c r}=\text { critical stress, ksi (MPa) }$

M_{c r}=C_{b} \frac{\pi}{L_{b}} \sqrt{E I_{y} G J+\left(\frac{\pi E}{L_{b}}\right)^{2} I_{y} C_{w}}

L_{p}=1.76 r_{y} \sqrt{\frac{E}{F_{y}}}

L_{r}=1.95 r_{t s} \frac{E}{0.7 F_{y}} \sqrt{\frac{J c}{S_{x} h_{o}}}+\sqrt{\left(\frac{J c}{S_{x} h_{o}}\right)^{2}+6.76\left(\frac{0.7 F_{y}}{E}\right)^{2}}

c=\frac{h_{o}}{2} \sqrt{\frac{I_{y}}{C_{w}}}

where

$I_{y}=$ moment of inertia about the y-axis, in. ${ }^{4}\left(\mathrm{~mm}^{4}\right)$

For doubly symmetric I-shapes with rectangular flanges, $C_{w}=\frac{I_{y} h_{o}^{2}}{4}$ , and, thus, Equation F2-7 becomes

C_{w}=\frac{I_{y} h_{o}^{2}}{4}

r_{i s}=\frac{b_{f}}{\sqrt{12\left(1+\frac{1}{6} \frac{h t_{w}}{b_{f} t_{f}}\right)}}