Formulas: doubly symmetric compact I-shaped members and channels bent about their major axis

Formula 1

Specifications prior to the 2005 AISC Specification, the residual stress, $F_{r}$ , for rolled and welded shapes was different, namely 10 ksi (69 MPa) and 16.5 ksi (110 MPa), respectively, while since the 2005 AISC Specification the residual stress has been taken as $0.3 F_{y}$ so that the value of $F_{y}-F_{r}=0.7 F_{y}$ is adopted.

F_{y}-F_{r}=0.7 F_{y}

Equation C-F2-1

F_{c r}=\frac{C_{b} \pi^{2} E}{\left(\frac{L_{b}}{r_{t s}}\right)^{2}} \sqrt{1+0.078 \frac{J c}{S_{x} h_{o}}\left(\frac{L_{b}}{r_{t s}}\right)^{2}}

Equation C-F2-2

is identical to Equation F1-13 in the 1999 LRFD Specification:

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

Equation C-F2-3

F_{c r}=\frac{C_{b} \pi^{2} E}{L_{b}^{2}} \frac{\sqrt{I_{y} C_{w}}}{S_{x}} \sqrt{1+\frac{G J}{E C_{w}}\left(\frac{L_{b}}{\pi}\right)^{2}}

Formula 5

$r_{t s}^{2}=\frac{\sqrt{I_{y} C_{w}}}{S_{x}}, C_{w}=\frac{I_{y} h_{o}^{2}}{4}$ , and $c=1$

r_{t s}^{2}=\frac{\sqrt{I_{y} C_{w}}}{S_{x}}, C_{w}=\frac{I_{y} h_{o}^{2}}{4}

Found in

DOUBLY SYMMETRIC COMPACT I-SHAPED MEMBERS AND CHANNELS BENT ABOUT THEIR MAJOR AXIS