D14.C.6.6 Bending

The laterally unsupported length of the compression flange for the purpose of computing the factored moment resistance is specified using the parameters UNB and UNT.

Laterally Supported Members

Per section 13.5, the factored moment resistance, M_r , developed by a member subjected to uniaxial bending moments about a principal axis and where continuous lateral support is provided to the compressive flange shall be taken as:

For class 1 and class 2 sections:

M_r = ϕZ_plf_y = ϕM_p
For class 3 sections:

M_r = ϕZ_ef_y = ϕM_y

Laterally Unsupported Members

Per section 13.6, where continuous lateral support is not provided to the compression flange of a member subjected to uniaxial strong axis bending, the factored moment resistance, M_r , may be taken as follows:

For doubly symmetric class 1 and class 2 sections, except closed square and circular sections:

When M_cr > 0.67M_p ,

$M_{r} = 1.15 ϕ M_{p} (1 - \frac{0.28 M_{p}}{M_{c r}})$

but not greater than ϕM_p .

When M_cr ≤ 0.67M_p ,

M_r = ϕM_cr

where

M_cr	=	= the critical elastic moment of the unbraced member, $= \frac{ω_{2} π}{K L} \sqrt{E I_{y} G J + {(\frac{π E}{K L})}^{2} I_{y} C_{w}}$
KL	=	= the effective length of the unbraced portion of the beam, in mm.
ω₂	=	= 1.75 + 1.05κ + 0.3κ² ≤ 2.5 for unbraced lengths subjected to end moments, or = 1.0 when the bending moment at any point within the unbraced length is larger than the end moment or when the re is no effective lateral support for the compression flange at one of the ends of the unsupported length. Note: The value for ω₂ can be specified using the `CB` parameter. Otherwise, it is calculated as indicated here.
κ	=	= the ratio of the smaller factored moment to the larger factored moment at opposite ends of the unbraced length, positive for double curvature and negative for single curvature.
C_w	=	= the ratio of the 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: Alternatively, E may be specified directly.

For doubly symmetric class 3 sections, except closed square and circular sections, and for channels:
1. When M_cr > 0.67M_p ,
  
  $M_{r} = 1.15 ϕ M_{y} (1 - \frac{0.28 M_{y}}{M_{c r}})$
  
  but not greater than ϕM_y for class 3 sections and the value given in 13.5(c)(iii) for class 4 sections.
2. When M_cr ≤ 0.67M_y ,
  
  M_r = ϕM_cr
  
  where M_cr and ω₂ are as defined in 13.6(a).
For closed sections and circular sections, M_cr shall be determined in accordance with section 13.5.
For biaxial bending, the member shall meet the following criterion:

$\frac{M_{u x}}{M_{r x}} + \frac{M_{u y}}{M_{r y}} \leq 1.0$

For monosymmetric sections, a rational method of analysis should be used.

Note: STAAD.Pro uses AISC LRFD guidelines for the design of channels, double angles, tees, and single angle sections.

For tees and double angles:

$= \frac{π \sqrt{E I_{y} G J}}{L_{b}} (B + \sqrt{1 + B^{2}})$ (AISC LRFD equation F1-15)

where

- 2.3 (d / UNL) \sqrt{I_{y} / J}

minus sign considered for conservative side

For channel sections:

When M_cr ≤ 0.67M_y :

$M_{r} = 1.15 ϕ M_{y} [1 - 0.28 \frac{M_{y}}{M_{c r}}] \leq 0.9 M_{y}$

When M_cr > 0.67M_y

M_r = 0.9M_cr

For angle sections:

When M_ob ≤ M_y :

M_{n} = M_{o b} [0.92 - 0.17 \frac{M_{o b}}{M_{y}}]

When M_ob > M_y

M_{n} = M_{y} [1.92 - 1.17 \sqrt{\frac{M_{y}}{M_{o b}}}] \leq 1.5 M_{y}

where

M_ob	=	$= 4.9 E \frac{I_{z}}{l^{2}} CB [\sqrt{β_{w}^{2} + 0.052 {(l t / r_{z})}^{2}} + β_{w}]$ for unequal leg angles. Note: β_w is conservatively assumed as zero. $= CB \frac{0.46 E b^{2} t^{2}}{l}$ for equal leg angles.
I_z	=	minor principal axis moment of inertia
r_z	=	radius of gyration for minor principal axis
β_w	=	0 (conservative assumption)
`CB`	=	The design parameter corresponding to ω₂ .
b	=	width of the angle leg
t	=	thickness of the angle