# Global Buckling Resistance of Cold-Formed Steel Beams with Omega-Shaped Sections
## 2 Normative Procedures for Assessing Buckling Safety
### ABNT NBR 14762:2010
#### 2.2.1 Global Buckling  
\[M_{Re}=Wf_{y}\ \ \ \ \ \left(for\ \lambda_{0}\leq 0.6\right) \tag{4}\]  
\[M_{Re}=1.1(1-0.278\lambda_{0}^{2})Wf_{y}\ \ \ \ \ \left(for\ 0.6<\lambda_{0}\leq 1.336\right) \tag{5}\]  
\[M_{Re}=\frac{Wf_{y}}{\lambda_{0}^{2}}\ \ \ \ \ \left(for\ \lambda_{0}\geq 1.336\right) \tag{6}\]  
with  
\[\lambda_{0}=\sqrt{\frac{W\cdot f_{y}}{M_{e}}} \tag{7}\]  
where \(M_{Re}\) is the resistant bending moment to global buckling, \(\lambda_{0}\) is the reduced slenderness ratio related to the global buckling mode, \(W\) is the elastic section modulus about the bending axis, \(f_{y}\) is the yield strength of the steel, and \(M_{e}\) is the critical elastic bending moment for global buckling. To determine the design resistance moment for global buckling, it is necessary to consider the distribution of bending moments along the laterally restrained segment. To this end, one should multiply the resistant bending moment to global buckling \(\left(M_{Re}\right)\) by the modification factor for non-uniform bending moment diagram \(\left(C_{b}\right)\), given by:  
\[C_{b}=\frac{12.5M_{max}}{2.5M_{max}+3M_{A}+4M_{B}+3M_{C}}R_{m}\leq 3.0 \tag{8}\]where \(M_{max}\) is the value of the maximum bending moment acting on the unhraced length; \(M_{A}\) is the value of the bending moment acting at a quarter of the unhraced length, measured from the left end; \(M_{B}\) is the absolute value of the bending moment acting at the central section of the unhraced length; \(M_{C}\) is the absolute value of the bending moment acting at three-quarters of the unhraced length, measured from the left end; and \(R_{m}\) is a parameter that takes into account the asymmetry of the cross-sectional shape, equal to \(0.5+2\left(I_{yc}/I_{y}\right)\) for sections with one symmetry axis, subject to bending about a non-symmetry axis and experiencing reverse curvature, and is equal to 1.0 in all other cases, where \(I_{yc}\) is the moment of inertia of the compressed flange about the symmetry axis. To obtain the exact value of \(M_{e}\), since the numerical analysis of elastic stability provides an approximate result, it is recommended to use the following equation for cold-formed profiles with monosymmetric sections, derived for loading applied at the position of the torsion center and subject to bending about an axis perpendicular to the symmetry axis (Annex E\(-\)ABNT NBR 14762:2010 [6]):  
\[M_{e}=\frac{C_{s}N_{ex}}{C_{m}}\left[j+C_{s}\sqrt{j^{2}+r_{0}^{2}\left(\frac {N_{ez}}{N_{ex}}\right)}\right] \tag{9}\]  
where \(C_{s}=+1\) if the bending moment causes compression in the section part with a negative x-coordinate, i.e., on the same side as the torsion center; \(C_{s}=-1\) if the moment causes tension in the section part with a negative x-coordinate, i.e., on the same side as the torsion center; \(C_{m}\) is the parameter depending on loading conditions; \(r_{0}\) is the polar radius of gyration of the gross section about the torsion center; and j pertains to the factor representing the mono-symmetry of the profile.