Dr S R Satish Kumar, IIT Madras1 IS 800:2007 Section 8 Design of members subjected to bending

Dr S R Satish Kumar, IIT Madras2 SECTION 8 DESIGN OF MEMBERS SUBJECTED TO BENDING 8.1General 8.2Design Strength in Bending (Flexure) 8.2.1Laterally Supported Beam 8.2.2Laterally Unsupported Beams 8.3Effective Length of Compression Flanges 8.4Shear Stiffened Web Panels End Panels design End Panels designed using Tension field action 8.5.3Anchor forces 8.6Design of Beams and Plate Girders with Solid Webs Minimum Web Thickness Sectional Properties FlangesCont...

Dr S R Satish Kumar, IIT Madras3 SECTION 8 DESIGN OF MEMBERS SUBJECTED TO BENDING 8.7Stiffener Design General Design of Intermediate Transverse Web Stiffeners Load carrying stiffeners Bearing Stiffeners 8.7.5Design of Load Carrying Stiffeners 8.7.6Design of Bearing Stiffeners Design of Diagonal Stiffeners Design of Tension Stiffeners Torsional Stiffeners Connection to Web of Load Carrying and Bearing Stiffeners Connection to Flanges Hollow Sections 8.8Box Girders 8.9 Purlins and sheeting rails (girts) 8.10 Bending in a Non-Principal Plane

Dr S R Satish Kumar, IIT Madras4 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 RESPONSE OF BEAMS TO VERTICAL LOADING

Dr S R Satish Kumar, IIT Madras5 LOCAL BUCKLING AND SECTION CLASSIFICATION OPEN AND CLOSED SECTIONS Strength of compression members depends on slenderness ratio

Dr S R Satish Kumar, IIT Madras6 (b)(a) Local buckling of Compression Members LOCAL BUCKLING 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

Dr S R Satish Kumar, IIT Madras7 L Bending Moment Diagram Plastic hinges MpMp Collapse mechanism Plastic hinges MpMp Formation of a Collapse Mechanism in a Fixed Beam w Bending Moment Diagram BASIC CONCEPTS OF PLASTIC THEORY First yield moment My Plastic moment Mp Shape factor S = Mp/My Rotation Capacity (a) at M y (b) M y < M<M p (c) at M p Plastification of Cross-section under Bending

Dr S R Satish Kumar, IIT Madras8 SECTION CLASSIFICATION MpMp Rotation  MyMy yy uu Slender Semi-compact Compact Plastic Section Classification based on Moment-Rotation Characteristics

Dr S R Satish Kumar, IIT Madras9 Moment Capacities of Sections MyMy MpMp 11 22 33  =b/t Semi- Compact SlenderPlasticCompact SECTION CLASSIFICATION BASED ON WIDTH -THICKNESS RATIO For Compression members use compact or plastic sections

Dr S R Satish Kumar, IIT Madras10 Type of ElementType 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. notapplicable b/t  42  WebNA at mid depth d/t  84.0  d/t  105  d/t  126  Anglesbending Axial comp. Circular tube with outer diameter D D/t  44  2 D/t  63  2 D/t  88  2 Table 2 Limits on Width to Thickness Ratio of Plate Elements b/t  9.4  b/t  10.5  b/t  15.7  notapplicable b/t  15.7  (b+d)/t  25 

Dr S R Satish Kumar, IIT Madras 11 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

Dr S R Satish Kumar, IIT Madras 12 Design Strength in Bending (Flexure) The factored design moment, M at any section, in a beam due to external actions shall satisfy Laterally Supported Beam Type 1 Sections with stocky webs d / t w  67  The design bending strength as governed by plastic strength, M d, shall be found without Shear Interaction for low shear case represented by V <0.6 V d

Dr S R Satish Kumar, IIT Madras 13 V exceeds 0.6V d M d = M dv M dv = design bending strength under high shear as defined in section Design Bending Strength under High Shear

Dr S R Satish Kumar, IIT Madras 14 Definition of Yield and Plastic Moment Capacities

Dr S R Satish Kumar, IIT Madras Design Strength in Bending (Flexure) The factored design moment, M at any section, in a beam due to external actions shall satisfy Laterally Supported Beam The design bending strength as governed by plastic strength, M d, shall be taken as M d =  b Z p f y /  m0  1.2 Z e f y /  m Holes in the tension zone (A nf / A gf )  (f y /f u ) (  m1 /  m0 ) / 0.9

Dr S R Satish Kumar, IIT Madras 16 Laterally Stability of Beams

Dr S R Satish Kumar, IIT Madras 17 BEHAVIOUR OF MEMBERS SUBJECTED TO BENDING

Dr S R Satish Kumar, IIT Madras 18 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.

Dr S R Satish Kumar, IIT Madras 19 SIMILARITY BETWEEN COLUMN BUCKLING AND LATERAL BUCKLING OF BEAMS ColumnBeam 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 buckling Pure flexural mode Function of slenderness Coupled lateral deflection and twist function of slenderness Both have tendency to fail by buckling in their weaker plane

Dr S R Satish Kumar, IIT Madras20 Beam buckling EI x >EI y EI x >GJ SIMILARITY OF COLUMN BUCKLING AND BEAM BUCKLING -1 M  u M Section B-B u P P B B B B Y X Z Column buckling

Dr S R Satish Kumar, IIT Madras21 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 (M cr )

Dr S R Satish Kumar, IIT Madras22 (a) ORIGINAL BEAM (b) LATERALLY BUCKLED BEAM M Plan Elevation l M Section (a) θ Lateral Deflection y z (b) Twisting x A A Section A-A

Dr S R Satish Kumar, IIT Madras23 M cr = [ (Torsional resistance ) 2 + (Warping resistance ) 2 ] 1/2 or EI y = flexural rigidity GJ = torsional rigidity E  = warping rigidity

Dr S R Satish Kumar, IIT Madras24 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 ?

Dr S R Satish Kumar, IIT Madras25 EQUIVALENT UNIFORM MOMENT FACTOR (m) Elastic instability at M’ = m M max (m  1) m = ß + 0.1ß 2 > 0.43 ß = M min / M max (-1.0  ß  1.0) M min M max M min Positive  M max M min Negative  M max also check M max  M p

Dr S R Satish Kumar, IIT Madras Laterally Unsupported Beams The design bending strength of laterally unsupported beam is given by: M d =  b Z p f bd f bd = design stress in bending, obtained as,f bd =  LT f y /γ m0  LT = reduction factor to account for lateral torsional buckling given by:  LT = 0.21 for rolled section,  LT = 0.49 for welded section Cont…

Dr S R Satish Kumar, IIT Madras 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

Dr S R Satish Kumar, IIT Madras28 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

Dr S R Satish Kumar, IIT Madras 29 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 BRACING REQUIREMENTS

Dr S R Satish Kumar, IIT Madras30 Other Failure Modes Shear yielding near support Web buckling Web crippling

Dr S R Satish Kumar, IIT Madras31 Web Buckling 45 0 d / 2 b1b1 n1n1 Effective width for web buckling

Dr S R Satish Kumar, IIT Madras32 Web Crippling b1b1 n2n2 1:2.5 slope Root radius Stiff bearing length

Dr S R Satish Kumar, IIT Madras33 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.