E4Torsional and flexural-torsional buckling of single angles and members without slender elements

This section applies to singly symmetric and unsymmetric members, certain doubly symmetric members, such as cruciform or built-up members, and doubly symmetric members when the torsional unbraced length exceeds the lateral unbraced length, all without slender elements. These provisions also apply to single angles with $b / t>0.71 \sqrt{E / F_{y}}$, where $b$ is the width of the longest leg and $t$ is the thickness.

The nominal compressive strength, $P_{n}$, shall be determined based on the limit states of torsional and flexural-torsional buckling:

(E4-1)

The nominal stress, $F_{n}$, shall be determined according to Equation E3-2 or E3-3, using the torsional or flexural-torsional elastic buckling stress, $F_{e}$, determined as follows:

• (a) For doubly symmetric members twisting about the shear center

(E4-2)

• (b) For singly symmetric members twisting about the shear center where y is the axis of symmetry

(E4-3)

User Note: For singly symmetric members with the $x$-axis as the axis of symmetry, such as channels, Equation E4-3 is applicable with $F_{e y}$ replaced by $F_{e x}$.

(c) For unsymmetric members twisting about the shear center, $F_{e}$ is the lowest root of the cubic equation

(E4-4)

where

(E4-5)

Figure description:

• Equation (E4-6: $F_{ey} = \frac{\pi^2 E}{\left(\frac{L_{cy}}{r_y}\right)^2}$
• Key Entities:
  • $F_{ey}$: Elastic buckling stress about the y-axis.
  • $E$: Modulus of elasticity of steel.
  • $L_{cy}$: Effective length of the member for buckling about the y-axis.
  • $r_y$: Radius of gyration about the y-axis.

(E4-7) (E4-8)

$\begin{aligned} I_{x}, I_{y} & =\text { moment of inertia about the principal axes, in. }^{4}\left(\mathrm{~mm}^{4}\right) \\ J & =\text { torsional constant, in. }^{4}\left(\mathrm{~mm}^{4}\right) \\ K_{x} & =\text { effective length factor for flexural buckling about } x \text {-axis } \\ K_{y} & =\text { effective length factor for flexural buckling about } y \text {-axis } \\ K_{z} & =\text { effective length factor for torsional buckling about the longitudinal } \\ & \text { axis }\end{aligned}$

$\begin{aligned} L_{c x} & =\text { effective length of member for buckling about } x \text {-axis, in. (mm) } \\ & =K_{x} L_{x}\end{aligned}$

$\begin{aligned} L_{c y} & =\text { effective length of member for buckling about } y \text {-axis, in. (mm) } \\ & =K_{y} L_{y}\end{aligned}$

$\begin{aligned} L_{c z} & =\text { effective length of member for buckling about longitudinal axis, } \\ & \text { in. (mm) } \\ & =K_{z} L_{z}\end{aligned}$

$\begin{aligned} L_{x}, L_{y}, L_{z} & =\text { laterally unbraced length of the member for each axis, in. (mm) } \\ \bar{r}_{o} & =\text { polar radius of gyration about the shear center, in. (mm) } \\ \bar{r}_{o}^{2} & =x_{o}^{2}+y_{o}^{2}+\frac{I_{x}+I_{y}}{A_{g}} \\ r_{x} & =\text { radius of gyration about } x \text {-axis, in. (mm) } \\ r_{y} & =\text { radius of gyration about } y \text {-axis, in. (mm) } \\ x_{o}, y_{o} & =\text { coordinates of the shear center with respect to the centroid, in. (mm) }\end{aligned}$

User Note: For doubly symmetric I-shaped sections, Cw may be taken as I h y o2 4, where ho is the distance between flange centroids, in lieu of a more precise analysis. For tees and double angles, the term with Cw may be omitted when computing Fez.

• (d) For doubly symmetric I-shaped members with minor-axis lateral bracing offset from the shear center

(E4-10)

where

• (e) For doubly symmetric I-shaped members with major-axis lateral bracing offset from the shear center

(E4-12)

where

• (f) For all other members with lateral bracing offset from the shear center, the elastic buckling stress, $F_{e}$, shall be determined by analysis.

User Note: Bracing offset from the shear center is often referred to as constrained-axis torsional buckling and is discussed further in the Commentary. Members that buckle in this mode will exhibit twisting because the braces restrain only lateral movement.