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

Note: For S16-19, the exception for single angle profiles is included per Cl. 13.5 D) I.
For Class 4 sections (Cl. 13.5(c) ):

Mr = ϕ·S_e· F_y

Note: For S16-19, the exceptions for circular hollow profiles per Cl. 13.5 C) IV and for single angle profiles per Cl. 13.5 d) ii) is included.

where

S_e

the effective section modulus determined using an effective flange width,b_e , of

670 t / \sqrt{F_{y}}

for flanges along two edges parallel to the direction of stress and an effective flange width,b_e of

200 t / \sqrt{F_{y}}

for flanges supported along one edge parallel to the direction of stress. For flange supported along one edge, b_eI_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{matrix} ϕ M_{u} when M_{u} \leq 0.67 M_{p} \\ 1.15 ϕ M_{p} (1 - \frac{0.28 M_{p}}{M_{u}}) \leq ϕ M_{p} when M_{u} > 0.67 M_{p} \end{matrix}

where

M_u	=	$\frac{ω_{2} π}{L} \sqrt{E I_{y} G J + {(\frac{π E}{L})}^{2} I_{y} C_{w}}$
ω₂	=	$1.75 + 1.05 κ + 0.3 κ^{2} \leq 2.5$
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).

Note: The value for ω₂ can be specified using the CB parameter. Otherwise, it is calculated as indicated here.

Note: For S16-19, Cl. 13.6 H) I pertaining to cantilevers with different bracing conditions is included.

For doubly symmetric Class 3 and Class 4 sections –except closed square and circular sections– and for channels:
$M_{r} = {\begin{matrix} ϕ M_{u} when M_{u} \leq 0.67 M_{y} \\ 1.15 ϕ M_{y} (1 - \frac{0.28 M_{y}}{M_{u}}) \leq ϕ M_{p} when M_{u} > 0.67 M_{y} \end{matrix}$
but not greater than ϕM_y for Class 3 sections and the value specified in Cl.13.5(c)(iii) for Class 4 sections.
Note: For S16-19, Cl. 13.6 H) II pertaining to cantilevers with different bracing conditions is included in accordance with Cl. 13.6(b), except for M_u and ω₂, which depend on Cl. 13.6 H) I.

For singly symmetric (monosymmetric) Class 1, Class 2, or Class 3 sections and T-shape 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} = ϕ [M_{p} - (M_{p} - M_{y r}) (\frac{L - L_{u}}{L_{y r} - L_{u}})] \leq ϕ M_{p}

where

M_yr	=	0.7S_xF_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} = ϕ M_{u}

where

M_u	=	the critical elastic moment of the unbraced section = $\frac{ω_{3} π^{2} E I_{y}}{2 L^{2}} [β_{x} + \sqrt{β_{x}^{2} + 4 (\frac{G J L^{2}}{π^{2} E I_{y}} + \frac{C_{w}}{I_{y}})}]$
β_x	=	asymmetry parameter for singly symmetric beam = $0.9 (d - t) (\frac{2 I_{y c}}{I_{y}} - 1) [1 - {(\frac{I_{y}}{I_{x}})}^{2}]$ For CSA S16-19, the approximate formula given is only valid for cases where I_x > 2I_y and where 0.1 < I_y/(I_yc + I_yt) < 0.9 and it is not valid for T sections.
I_yc	=	moment of inertia of the compression flange about the y-axis
I_yt	=	moment of inertia of the tension flange about the y-axis

when singly symmetric beams are in single curvature,

ω₃ = ω₂ for beams with two flanges, = 1.0 for T-sections

in all other cases,

ω₃ = ω₂[0.5 + 2(I_yc/Iy)²] f, but ≤ 2.0 for T-Sections

Note: For single angles (asymmetric sections) designed per S16-19, Cl. 13.6 G (I) and (II) are used.