D8.A.2.6 Bending

The design bending moment capacity of a section is primarily dependent on whether the member is laterally supported or unsupported.

You can control the lateral support condition of the member by the use of LAT parameter. The type of member (i.e., cantilever, simply supported, or general) is specified using the CAN parameter.

If the member is laterally supported, then the design strength is calculated as per the provisions of the section 8.2.1 of IS 800:2007, based on the following factors:

Whether section with webs susceptible to shear buckling before yielding
Ratio of shear force to design shear strength
Section classification

If the member is laterally unsupported, then the design strength is calculated as per the provisions of the section 8.2.2 of IS 800:2007, based on the following factors:

Lateral Torsional Buckling
Section Classification

Laterally unsupported sections of a solid rod are considered as laterally supported as mentioned in Cl. 8.2.2(b). The plastic moment of inertia, Z_p, is calculated as D³/6.

Working Stress Design

Actual bending stress values are given by, about major (Z) and minor (Y) axes, respectively:

f_bcz = M_z/Z_ecz

f_btz = M_z/Z_etz

f_bcy = M_y/Z_ecy

f_bty = M_y/Z_ety

The permissible bending stress is given as follows:

For laterally supported beams:
- F_abc = F_abt = 0.66·F_y for Plastic or Compact sections
- F_abc = F_abt = 0.60·F_y for Semi-compact sections
where
F_y
=
Yield strength of steel, indicated by the FYLD parameter.

For laterally unsupported beams:

About the major axis:

f_abcz = 0.60·M_d/Z_ecz

f_abtz = 0.60·M_d/Z_etz

where

M_d	=	Design Bending Strength as per Clause 8.2.2 = β_b · Z_pz · f_bd
f_bd	=	χ_LT · F_y / γ_mo
Z_ez	=	Elastic Section Modulus of the Section
Z_pz	=	Plastic Section Modulus of the Section
α_LT	=	0.21 for Rolled Steel Section and 0.49 for Welded Steel Section
β_b	=	1.0 for Plastic and Compact Section or Z_ez/Z_pz for Semi-Compact Section
λ_LT	=	Non-dimensional slenderness ratio
λ_LT	=	(β_b · Z_pz · F_y / M_cr)^1/2 ≤ (1.2 · Z_ez · F_y / M_cr )^1/2
ϕ_LT	=	0.5 · ( 1 + α_LT · ( λ_LT – 0.2 ) + λ_LT ²)
χ_LT	=	The Bending Stress Reduction Factor to account for Lateral Torsional Buckling
χ_LTZ	=	$\frac{1}{ϕ_{L T Z} + \sqrt{ϕ_{L T Z}^{2} - λ_{L T Z}^{2}}}$
Z_ecz	=	Elastic Section Modulus of the section about Major Axis for the compression side
Z_etz	=	Elastic Section Modulus of the section about Major Axis for the tension side
M_cr	=	$M_{c r} = \sqrt{\frac{π^{2} E I_{y}}{L_{L T}^{2}} (G I_{t} + \frac{π^{2} E I_{w}}{L_{L T}^{2}})}$
I_y	=	Moment of inertia about the minor axis
L_LT	=	Effective length for lateral torsional buckling as determined using either the `KX` or `LX` parameters
I_t	=	Torsional constant of the section
I_t	=	Warping constant of the section
G	=	Shear modulus of the material

About the minor axis, the permissible bending stress is calculated as for a laterally supported section.

Slender Sections

For member with slender section subjected to bending, moment is taken by flanges alone. Design bending strength should be calculated with effective elastic modulus disregarding the contribution of web of the section.

Z_ez = 2·[B_f · t_f ³/12 + (B_f · t_f) · (D/2 - t_f/2)² )] ⁄ (0.5 · D)

Z_ey = 2·(B_f · t_f ³/12) ⁄ (0.5 · B_f)

Where:

where

Z_ez	=	Elastic Section modulus about major principal axis
Z_ey	=	Elastic Section modulus about minor principal axis
B_f	=	Width of flange
T_f	=	thickness of flange
D	=	Overall depth of section

The Moment Capacity will be M_d = Z_e· f_y/γ_m0 for "Laterally Supported" condition.

The Moment Capacity will be M_d = Z_e· f_bd/γ_m0 for "Laterally Un-Supported" condition.

Where, f_bd is defined in clause 8.2.2 of IS:800-2007 (described in previous Working Stress Design section).

Note: Slender section can only attain elastic moment capacity and cannot reach to plastic moment capacity.