Chapter 14Design of Members Subjected to Bending

14.1 General

Members subjected to predominant bending shall have adequate design strength to resist bending moment, shear force, and concentrated forces imposed upon and their combinations. Further, the members shall satisfy

the deflection limitation presented in Section 6(e), as serviceability criteria. Member subjected to other forces in addition to bending or biaxial bending shall be designed in accordance with Section 6(j).

The design bending strength of beam, adequately supported against lateral torsional buckling (laterally supported beam) is governed by the yield stress (see 14.2.1). When a beam is not adequately supported against lateral buckling (laterally un-supported beams) the design bending strength may be governed by lateral torsional buckling strength (see 14.2.2).

The factored design moment, $M$ at any section, in a beam due to external actions, shall satisfy

where

$M_{\mathrm{d}} =$ design bending strength of the section, calculated as given in 14.2.1.2.

14.2.1 Laterally Supported Beam

A beam may be assumed to be adequately supported at the supports, provided the compression flange has full lateral restraint and nominal torsional restraint at supports supplied by web cleats, partial depth end plates, fin plates or continuity with the adjacent span. Full lateral restraint to compression flange may be assumed to exist if the frictional or other positive restraint of a floor connection to the compression flange of the member is capable of resisting a lateral force not less than 2.5 percent of the maximum force in the compression flange of the member. This may be considered to be uniformly distributed along the flange, provided gravity loads constitute the dominant loading on the member and the floor construction is capable of resisting this lateral force.

The design bending strength of a section which are not susceptible to web buckling under shear before yielding (where $d / t_{\mathrm{w}} \leq 67e$) shall be determined according to 14.2.1.2.

14.2.1.1 Section with webs susceptible to shear buckling before yielding

When the flanges are plastic, compact or semi-compact but the web is susceptible to shear buckling before

yielding (see 14.4.2.1), the design bending strength shall be calculated using one of the following methods:

a) The bending moment and axial force acting on the section may be assumed to be resisted by flanges only and the web is designed only to resist shear (see 14.4).

b) The bending moment and axial force acting on the section may be assumed to be resisted by the whole section. In such a case, the web shall be designed for combined shear and normal stresses using simple elastic theory in case of semi-compact webs and simple plastic theory in the case of compact and plastic webs.

14.2.1.2 When the factored design shear force does not exceed $0.6 V_{d}$ , where $V_{d}$ is the design shear strength of the cross-section (see 14.4), the design bending strength, $M_{d}$ shall be taken as:

To avoid irreversible deformation under serviceability loads, $M_{d}$ shall be less than $1.2 Z_{e} f_{y} / \gamma_{m0}$ in case of simply supported and $1.5 Z_{e} f_{y} / \gamma_{m0}$ in case of cantilever beams;

where

$Z_{p}, Z_{e} = \text{plastic and elastic section moduli of the cross-section, respectively};$

$f_{\mathrm{y}} =$ yield stress of the material; and

14.2.1.3 When the design shear force (factored), $V$ exceeds $0.6V_{\mathrm{d}}$, where $V_{\mathrm{d}}$ is the design shear strength of the cross-section (see 14.4) the design bending strength, $M_{\mathrm{d}}$ shall be taken as:

where

$M_{dv} =$ design bending strength under high shear as defined in 15.2.

Resistance to lateral torsional buckling need not be checked separately (member may be treated as laterally supported, see 14.2.1) in the following cases:

a) Bending is about the minor axis of the section,

b) Section is hollow (rectangular/ tubular) or solid bars, and

c) In case of major axis bending, $\lambda_{\mathrm{LT}}$ (as defined herein) is less than 0.4.

The design bending strength of laterally unsupported beam as governed by lateral torsional buckling is given by:

where

$\beta_{\mathrm{b}} = 1.0$ for plastic and compact sections;

$= Z_{\mathrm{e}} / Z_{\mathrm{p}}$ for semi-compact sections;

$Z_{p}, Z_{e}=$ plastic section modulus and elastic section modulus with respect to extreme compression fibre;

$f_{bd}$ = design bending compressive stress, obtained as given below [see Tables 13(a) and 13(b)];

$\chi_{LT} =$ bending stress reduction factor to account for lateral torsional buckling, given by:

$\alpha_{\mathrm{LT}}$ , the imperfection parameter is given by:

$\alpha_{\mathrm{LT}} = 0.21$ for rolled steel section

$\alpha_{\mathrm{LT}} = 0.49$ for welded steel section

The non-dimensional slenderness ratio, $l_{\mathrm{LT}}$, is given by:

where

$M_{cr}$ = elastic critical moment calculated in accordance with 14.2.2.1, and

$f_{cr,b}$ = extreme fibre bending compressive stress corresponding to elastic lateral buckling moment (see 14.2.2.1 and Table 14).

14.2.2.1 Elastic lateral torsional buckling moment

In case of simply supported, prismatic members with symmetric cross-section, the elastic lateral buckling moment, $M_{cr}$ , can be determined from:

$f_{cr,b}$ of non-slender rolled steel sections in the above equation may be approximately calculated from the values given in the Table 14, which has been prepared using the following equation:

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

The following simplified equation may be used in the case of prismatic members made of standard rolled I-sections and welded doubly symmetric I-sections, for calculating the elastic lateral buckling moment, $M_{cr}$ (see Table 14).

where

$I_{\mathrm{t}}$ = torsional constant $\sum b_{\mathrm{i}}t_{\mathrm{i}}^{3} / 3$ for open section;

$I_{\mathrm{w}}$ = warping constant;

$I_{y}, r_{y} = \text{moment of inertia and radius of gyration, respectively about the weaker axis;}$

$L_{LT}$ = effective length for lateral torsional buckling (see 14.3);

$h_{\mathrm{f}}$ = centre-to-centre distance between flanges; and

$t_{\mathrm{f}}$ = thickness of the flange.

$M_{cr}$ for different beam sections, considering loading, support condition, and asymmetric section, shall be more accurately calculated using the method given in Annex D. The above equations for calculating $M_{cr}$ may be conservatively used for channel sections.

14.3.1 For simply supported beams and girders of span length, L, where no lateral restraint to the compression flanges is provided, but where each end of the beam is restrained against torsion, the effective length $L_{LT}$ of the lateral buckling to be used in 14.2.2.1 shall be taken as in Table 15.

In simply supported beams with intermediate lateral restraints against lateral torsional buckling, the effective length for lateral torsional buckling, $L_{\mathrm{LT}}$ to be used in 14.2.2.1, shall be taken as the length of the relevant segment in between the lateral restraints. In the case of intermediate partial lateral restraints, the effective length, $L_{\mathrm{LT}}$ shall be taken as equal to 1.2 times the length of the relevant segment in between the partial lateral restraints.

Restraint against torsional rotation at supports in these beams may be provided by,

a) web or flange cleats, or

b) bearing stiffeners acting in conjunction with the bearing of the beam, or

c) lateral end frames or external supports providing lateral restraint to the compression flanges at the ends, or

d) their being built into walls.

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

14.3.2 For beams, which are provided with members giving effective lateral restraint to the compression flange at intervals along the span, in addition to the end torsional restraint required in 14.3.1, the effective length for lateral torsional buckling shall be taken as the distance, centre-to-centre of the restraint members in the relevant segment under normal loading condition and 1.2 times this distance, where the load is not acting on the beam at or above the shear centre and towards the shear centre, so as to have destabilizing effect during lateral torsional buckling deformation.

14.3.3 For cantilever beams of projecting length $L$, the effective length $L_{\mathrm{LT}}$ to be used in 14.2.2.1 shall be taken as in Table 16 for different support conditions.

14.3.4 Where a member is provided intermediate lateral supports to improve the lateral buckling strength, these restraints should have sufficient strength and stiffness to prevent lateral movement of the compression flange at that point, relative to the end

14.3.5 Purlins adequately restrained by sheeting need not be normally checked for the restraining forces required by rafters, roof trusses or portal frames that carry predominately roof loads provided there is bracing of adequate stiffness in the plane of rafters or roof sheeting which is capable of acting as a stressed skin diaphragm.

14.3.6 In case of beams with double curvature bending, adequate direct lateral support to the compression flange in the hogging moment region may be provided as given above for simply supported beam. The effect of support to the tension (top) flange in the hogging moment region on lateral restraint to the compression flange may be considered as per specialist literature.

The factored design shear force, $V$, in a beam due to external actions shall satisfy

where

$V_{\mathrm{d}} =$ design strength

where

$\gamma_{m0} = \text{partial safety factor against shear failure}$ (see 11.4.1).

The nominal shear strength of a cross-section, $V_{n}$ , may be governed by plastic shear resistance (see 14.4.1) or strength of the web as governed by shear buckling (see 14.4.2).

14.4.1 The nominal plastic shear resistance under pure shear is given by:

where

$A_{\mathrm{v}} =$ shear area; and

$f_{\mathrm{yw}} =$ yield strength of the web.

14.4.1.1 The shear area may be calculated as given below:

a) I-and channel sections:

Major Axis Bending:

Minor Axis Bending:

Hot-rolled or welded : 2b $t_{f}$

b) Rectangular hollow sections of uniform thickness:

Loaded parallel to depth (h) : A h/(b+h) Loaded parallel to width (b) : A b/(b+h)

c) Circular hollow tubes of uniform : 2A/π thickness

d) Plates and solid bars : A where

$A =$ cross-section area,

b = overall breadth of tubular section, breadth of I-section flanges,

$d =$ clear depth of the web between flanges,

$h =$ overall depth of the section,

$t_\mathrm{f} =$ thickness of the flange, and

$t_\mathrm{w} =$ thickness of the web.

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

NOTE — Fastener holes need not be accounted for in plastic design shear strength calculation provided that:

If $A_{\mathrm{vn}}$ does not satisfy the above condition, the effective shear area may be taken as that satisfying the above limit. Block shear failure criteria may be verified at the end connections. Section 6(j) may be referred to for design strength under combined high shear and bending.

14.4.2 Resistance to Shear Buckling

14.4.2.1 Resistance to shear buckling shall be verified as specified, when

$d / t_{\mathrm{w}} > 67\varepsilon_{\mathrm{w}}$ for a web without stiffeners, and

where

$K_{\mathrm{v}} =$ shear buckling coefficient (see 14.4.2.2), and