F2Doubly symmetric compact I-shaped members and channels bent about their major axis

This section applies to doubly symmetric I-shaped members and channels bent about their major axis, having compact webs and compact flanges as defined in Section B4.1 for flexure.

User Note: For $F_{y}=50 \mathrm{ksi}(345 \mathrm{MPa})$, all current ASTM A6/A6M W, S, M, C, and MC shapes except $\mathrm{W} 21 \times 48, \mathrm{W} 14 \times 99, \mathrm{W} 14 \times 90, \mathrm{W} 12 \times 65, \mathrm{W} 10 \times 12$, $\mathrm{W} 8 \times 31, \mathrm{W} 8 \times 10, \mathrm{W} 6 \times 15, \mathrm{W} 6 \times 9, \mathrm{W} 6 \times 8.5$, and $\mathrm{M} 4 \times 6$ have compact flanges. For $F_{y} \leq 70 \mathrm{ksi}(485 \mathrm{MPa})$, all current ASTM A6/A6M W, S, M, HP, C, and MC shapes have compact webs.

The nominal flexural strength, $M_{n}$, shall be the lower value obtained according to the limit states of yielding (plastic moment) and lateral-torsional buckling.

F2.1 Yielding

Mn = Mp = Fy Zx (F2-1)

where

$F_{y}=$ specified minimum yield stress of the type of steel being used, ksi (MPa)

$\mathrm{Z}_{x}=\mathrm{plastic}$ section modulus about the $x$-axis, in. ${ }^{3}\left(\mathrm{~mm}^{3}\right)$

F2.2 Lateral-Torsional Buckling

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

(c) When $L_{b}>L_{r}$

(F2-3)

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)

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

(F2-4)

$E \quad=$ modulus of elasticity of steel

= 29,000 ksi (200 000 MPa)

$J \quad=$ torsional constant, in. ${ }^{4}\left(\mathrm{~mm}^{4}\right)$

$S_{x}$ = elastic section modulus taken about the x-axis, in. ${ }^{3}$ (mm ${ }^{3}$ )

• $h_{o}=$ distance between the flange centroids, in. (mm)

User Note: The square root term in Equation F2-4 may be conservatively taken as equal to 1.0.

User Note: Equations F2-3 and F2-4 provide identical solutions to the following expression for lateral-torsional buckling of doubly symmetric members that has been presented in past editions of this Specification:

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

The advantage of Equations F2-3 and F2-4 is that the form is very similar to the expression for lateral-torsional buckling of singly symmetric I-shaped members given in Equations F4-3 and F4-5.

Lp, the limiting laterally unbraced length for the limit state of yielding, in. (mm), is

(F2-5)

$L_{r}$, the limiting unbraced length for the limit state of inelastic lateral-torsional buckling, in. (mm), is

where

$\begin{aligned} r_{y} & =\text { radius of gyration about } y \text {-axis, in. (mm) } \\ r_{t s}^{2} & =\frac{\sqrt{I_{y} C_{w}}}{S_{x}}\end{aligned}$ (F2-7)

and the coefficient $c$ is determined as follows:

(1) For doubly symmetric I-shapes

(F2-8a)

(2) For channels

(F2-8b)

where

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

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

Ish r 25

$r_{t s}$ may be approximated accurately to conservatively as the radius of gyration of the compression flange plus one-sixth of the web:

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