D4.E.6.2 Members Subject to Bending
The laterally unsupported length of the compression
flange for the purpose of computing the factored moment resistance is specified
in
STAAD.Pro through the
UNT and
UNB parameters (Refer to
D4.E.7
Design Parameters). The
LAT parameter is used to specify if the member is
laterally supported against lateral torsional buckling.
 The factored moment
resistance, Mr, developed by a member subjected to uniaxial bending moments
about a principal axis where effectively continuous lateral support is provided
to the compression flange or where the member has no tendency to buckle
laterally, is calculated as:

For Class 1 and Class 2 sections (Cl. 13.5(a) ):
Mr = ϕ·Z· F_{y} = ϕ·M_{p}

For Class 3 sections (Cl. 13.5(b) ):
Mr = ϕ·S· F_{y} = ϕ·M_{y}

For Class 4 sections (Cl. 13.5(c) ):
Mr = ϕ·S_{e}· F_{y}
 S_{e}
=  the effective section modulus determined using an effective flange width,b_{e} , of $670t/\sqrt{{F}_{y}}$ for flanges along two edges parallel to the direction of stress and an effective flange width,b_{e} of $200t/\sqrt{{F}_{y}}$ for flanges supported along one edge parallel to the direction of stress. For flange supported along one edge, b_{e}I_{e}/t shall not exceed 60.

 For laterally unsupported
members, flexural resistance is calculated as follows:
 For doubly symmetric
Class 1 and Class 2 sections (Cl 13.6(a) ):
${M}_{r}=\{\begin{array}{c}\varphi {M}_{u}\text{when}{M}_{u}\le 0.67{M}_{p}\\ 1.15\varphi {M}_{p}(1\frac{0.28{M}_{p}}{{M}_{u}})\le \varphi {M}_{p}\text{when}{M}_{u}0.67{M}_{p}\end{array}$where
 M_{u}
=  $\frac{{\omega}_{2}\pi}{L}\sqrt{E{I}_{y}GJ+{\left(\frac{\pi E}{L}\right)}^{2}{I}_{y}{C}_{w}}$
 ω_{2}
=  $1.75+1.05\kappa +0.3{\kappa}^{2}\le 2.5$
 κ
=  ratio of smaller factored moment to the larger factored moment at opposite ends of the unbraced length (positive for double curvature and negative for single curvature).
 For doubly symmetric
Class 3 and Class 4 sections –except closed square and circular sections– and
for channels:
${M}_{r}=\{\begin{array}{c}\varphi {M}_{u}\text{when}{M}_{u}\le 0.67{M}_{y}\\ 1.15\varphi {M}_{y}(1\frac{0.28{M}_{y}}{{M}_{u}})\le \varphi {M}_{p}\text{when}{M}_{u}0.67{M}_{y}\end{array}$but not greater than ϕM_{y} for Class 3 sections and the value specified in Cl.13.5(c)(iii) for Class 4 sections.
 For singly symmetric
(monosymmetric) Class 1, Class 2, or Class 3 sections and Tshape sections,
lateral torsional buckling strength shall be checked separately for each flange
under compression under factored loads at any point along its unbraced length:
 when
M_{u} >
M_{yr}
:
${M}_{r}=\varphi [{M}_{p}({M}_{p}{M}_{yr}\left)\right(\frac{L{L}_{u}}{{L}_{yr}{L}_{u}}\left)\right]\le \varphi {M}_{p}$where
 M_{yr}
=  0.7S_{x}F_{y} , with S_{x} taken as the smaller of the two potential values
 L_{yr}
=  length L obtained by setting M_{u} = M_{yr}
 L_{u}
=  $1.1{r}_{t}\sqrt{E/{F}_{y}}=\frac{490{r}_{t}}{\sqrt{{F}_{y}}}$
 r_{t}
=  $\frac{{b}_{c}}{\sqrt{12(1+\frac{{h}_{c}w}{3{b}_{c}{t}_{c}})}}$
 h_{c}
=  depth of the web in compression
 b_{c}
=  width of the compression flange
 t_{c}
=  thickness of the compression flange

when M ≤ M_{yr} :${M}_{r}=\varphi {M}_{u}$where
 M_{u}
=  the critical elastic moment of the unbraced section =
$\frac{{\omega}_{3}{\pi}^{2}E{I}_{y}}{2{L}^{2}}[{\beta}_{x}+\sqrt{{\beta}_{x}^{2}+4(\frac{GJ{L}^{2}}{{\pi}^{2}E{I}_{y}}+\frac{{C}_{w}}{{I}_{y}})}]$ β_{x}
=  asymmetry parameter for singly symmetric beam =
$0.9(dt)(\frac{2{I}_{yc}}{{I}_{y}}1)[1{\left(\frac{{I}_{y}}{{I}_{x}}\right)}^{2}]$ I_{yc}
=  moment of inertia of the compression flange about the yaxis
 I_{yt}
=  moment of inertia of the tension flange about the yaxis
when singly symmetric beams are in single curvature,
 ω_{3} = ω_{2} for beams with two flanges, = 1.0 for Tsections
in all other cases,
 ω_{3} = ω_{2}[0.5 + 2(I_{yc}/Iy)^{2}] f, but ≤ 2.0 for TSections
 when
M_{u} >
M_{yr}
:
 For doubly symmetric
Class 1 and Class 2 sections (Cl 13.6(a) ):