IS 800:2007 Section 8 Design of members subjected to bending

SECTION 8 DESIGN OF MEMBERS SUBJECTED TO BENDING 8.1 General 8.2 Design Strength in Bending (Flexure) 8.2.1 Laterally Supported Beam 8.2.2 Laterally Unsupported Beams 8.3 Effective Length of Compression Flanges 8.4 Shear ------------------------------------------------------------------------------------------------- 8.5 Stiffened Web Panels 8.5.1 End Panels design 8.5.2 End Panels designed using Tension field action 8.5.3 Anchor forces 8.6 Design of Beams and Plate Girders with Solid Webs 8.6.1 Minimum Web Thickness 8.6.2 Sectional Properties 8.6.3 Flanges Cont...

SECTION 8 DESIGN OF MEMBERS SUBJECTED TO BENDING 8.7 Stiffener Design 8.7.1 General 8.7.2 Design of Intermediate Transverse Web Stiffeners 8.7.3 Load carrying stiffeners 8.7.4 Bearing Stiffeners 8.7.5 Design of Load Carrying Stiffeners 8.7.6 Design of Bearing Stiffeners 8.7.7 Design of Diagonal Stiffeners 8.7.8 Design of Tension Stiffeners 8.7.9 Torsional Stiffeners 8.7.10 Connection to Web of Load Carrying and Bearing Stiffeners 8.7.11 Connection to Flanges 8.7.12 Hollow Sections 8.8 Box Girders 8.9 Purlins and sheeting rails (girts) 8.10 Bending in a Non-Principal Plane

RESPONSE OF BEAMS TO VERTICAL LOADING Plastic hinge formation Lateral deflection and twist Local buckling of i) Flange in compression ii) Web due to shear iii) Web in compression due to concentrated loads Local failure by i) Yield of web by shear ii) Crushing of web iii) Buckling of thin flanges

LOCAL BUCKLING AND SECTION CLASSIFICATION OPEN AND CLOSED SECTIONS Strength of compression members depends on slenderness ratio

LOCAL BUCKLING (a) (b) Local buckling of Compression Members Beams – compression flange buckles locally Fabricated and cold-formed sections prone to local buckling Local buckling gives distortion of c/s but need not lead to collapse

BASIC CONCEPTS OF PLASTIC THEORY w Collapse mechanism L Plastic hinges Plastic hinges Mp Mp Bending Moment Diagram Bending Moment Diagram Formation of a Collapse Mechanism in a Fixed Beam First yield moment My Plastic moment Mp Shape factor S = Mp/My Rotation Capacity (a) at My (b) My < M<Mp (c) at Mp Plastification of Cross-section under Bending

SECTION CLASSIFICATION Mp Rotation  My y u Slender Semi-compact Compact Plastic Section Classification based on Moment-Rotation Characteristics

SECTION CLASSIFICATION BASED ON WIDTH -THICKNESS RATIO My Mp 1 2 3 =b/t Semi- Compact Slender Plastic Moment Capacities of Sections For Compression members use compact or plastic sections

Table 2 Limits on Width to Thickness Ratio of Plate Elements Type of Element Type of Section Class of Section Plastic (1) Compact (2) Semi-compact (3) Outstand element of compression flange Rolled b/t  9.4 b/t  10.5 b/t  15.7 Welded b/t  8.4 b/t  9.4 b/t  13.6 Internal element of compression flange bending b/t  29.3 b/t  33.5 b/t  42 Axial comp. not applicable b/t  42 Web NA at mid depth d/t  84.0 d/t  105 d/t  126 Angles bending   b/t  9.4   b/t  10.5 b/t  15.7 Axial comp.   not   applicable b/t  15.7 (b+d)/t  25 Circular tube with outer diameter D   D/t  442 D/t  632 D/t  882

Condition for Beam Lateral Stability 1 Laterally Supported Beam The design bending strength of beams, adequately supported against lateral torsional buckling (laterally supported beam) is governed by the yield stress 2 Laterally Unsupported Beams 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

Design Strength in Bending (Flexure) The factored design moment, M at any section, in a beam due to external actions shall satisfy 8.2.1 Laterally Supported Beam Type 1 Sections with stocky webs d / tw  67 The design bending strength as governed by plastic strength, Md, shall be found without Shear Interaction for low shear case represented by V <0.6 Vd

8.2.1.3 Design Bending Strength under High Shear V exceeds 0.6Vd Md = Mdv Mdv= design bending strength under high shear as defined in section 9.2

Definition of Yield and Plastic Moment Capacities

8.2 Design Strength in Bending (Flexure) The factored design moment, M at any section, in a beam due to external actions shall satisfy 8.2.1 Laterally Supported Beam The design bending strength as governed by plastic strength, Md, shall be taken as Md = b Z p fy / m0  1.2 Ze fy / m0 8.2.1.4 Holes in the tension zone (Anf / Agf)  (fy/fu) (m1 / m0 ) / 0.9

Laterally Stability of Beams

BEHAVIOUR OF MEMBERS SUBJECTED TO BENDING Plastic Range Inelastic Elastic Mp My Mcr Unbraced Length, L Mo L Beam Buckling Behaviour

LATERAL BUCKLING OF BEAMS FACTORS TO BE CONSIDERED Distance between lateral supports to the compression flange. Restraints at the ends and at intermediate support locations (boundary conditions). Type and position of the loads. Moment gradient along the unsupported length. Type of cross-section. Non-prismatic nature of the member. Material properties. Magnitude and distribution of residual stresses. Initial imperfections of geometry and eccentricity of loading.

SIMILARITY BETWEEN COLUMN BUCKLING AND LATERAL BUCKLING OF BEAMS Both have tendency to fail by buckling in their weaker plane Column Beam Short span Axial compression & attainment of squash load Bending in the plane of loads and attaining plastic capacity Long span Initial shortening and lateral buckling Initial vertical deflection and lateral torsional Pure flexural mode Function of slenderness Coupled lateral deflection and twist function of slenderness

SIMILARITY OF COLUMN BUCKLING AND BEAM BUCKLING -1 P X Z B B B B M  P u u Section B-B Section B-B Column buckling Beam buckling EIx >EIy EIx >GJ

LATERAL TORSIONAL BUCKLING OF SYMMETRIC SECTIONS Assumptions for the ideal (basic) case Beam undistorted Elastic behaviour Loading by equal and opposite moments in the plane of the web No residual stresses Ends are simply supported vertically and laterally The bending moment at which a beam fails by lateral buckling when subjected to uniform end moment is called its elastic critical moment (Mcr)

(a) ORIGINAL BEAM (b) LATERALLY BUCKLED BEAM Plan Elevation l Section (a) θ Lateral Deflection y z (b) Twisting x A Section A-A

Mcr = [ (Torsional resistance )2 + (Warping resistance )2 ]1/2 EIy = flexural rigidity GJ = torsional rigidity E = warping rigidity

FACTORS AFFECTING LATERAL STABILITY Support Conditions effective (unsupported) length Level of load application stabilizing or destabilizing ? Type of loading Uniform or moment gradient ? Shape of cross-section open or closed section ?

EQUIVALENT UNIFORM MOMENT FACTOR (m) Elastic instability at M’ = m Mmax (m  1) m = 0.57+ 0.33ß + 0.1ß2 > 0.43 ß = Mmin / Mmax (-1.0  ß  1.0) Mmin Mmax Positive Negative also check Mmax  Mp

8.2.2 Laterally Unsupported Beams The design bending strength of laterally unsupported beam is given by: Md = b Zp fbd fbd = design stress in bending, obtained as ,fbd = LT fy /γm0 LT = reduction factor to account for lateral torsional buckling given by: LT = 0.21 for rolled section, LT = 0.49 for welded section Cont…

APPENDIX F ELASTIC LATERAL TORSIONAL BUCKLING 8.2.2.1 Elastic Lateral Torsional Buckling Moment APPENDIX F ELASTIC LATERAL TORSIONAL BUCKLING F.1 Elastic Critical Moment F.1.1 Basic F.1.2 Elastic Critical Moment of a Section Symmetrical about Minor Axis

EFFECTIVE LATERAL RESTRAINT Provision of proper lateral bracing improves lateral stability Discrete and continuous bracing Cross sectional distortion in the hogging moment region Discrete bracing Level of attachment to the beam Level of application of the transverse load Type of connection Properties of the beams Bracing should be of sufficient stiffness to produce buckling between braces Sufficient strength to withstand force transformed by beam before connecting

BRACING REQUIREMENTS Effective bracing if they can resist not less than 1) 1% of the maximum force in the compression flange 2) Couple with lever arm distance between the flange centroid and force not less than 1% of compression flange force. Temporary bracing

Shear yielding near support Other Failure Modes Shear yielding near support Web buckling Web crippling

Effective width for web buckling 450 d / 2 b1 n1 Effective width for web buckling

Web Crippling b1 n2 1:2.5 slope Root radius Stiff bearing length

SUMMARY Unrestrained beams , loaded in their stiffer planes may undergo lateral torsional buckling The prime factors that influence the buckling strength of beams are unbraced span, Cross sectional shape, Type of end restraint and Distribution of moment A simplified design approach has been presented Behaviour of real beams, cantilever and continuous beams was described. Cases of mono symmetric beams , non uniform beams and beams with unsymmetric sections were also discussed.