Makendran.2

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1 Design of steel flexural members Prepared by C.MAKENDRAN M.E.,(Ph.D) Assistant Professor,

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2 Real world steel beams

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3 Prismatic members loaded along their axes Load is carried by tension and compression

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4 Load is carried by moment + shear BM diagram SF diagram Concentrated load

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5 Most bridge girders are simply supported

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6 Beam under uniformly distributed load( moment + shear) BM diagram SF diagram

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7 FIXED BEAM BM diagram SF diagram

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8 Rectangular cross section Bending stresses Shear stresses Stresses in cross sections under bending Tension Compression

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9 Rectangular cross section Bending stresses I sections under bending Tension Compression Shear stresses

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10 Deep Beam or Timoshenko Beam Thin beam or Euler Bernoulli beam Flexure dominates Shear dominates

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LIMIT BEHAVIOUR OF LATERALLY RESTRAINED BEAMS AND ITS DESIGN

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Lateral-torsional buckling Flexural yielding TYPES OF BEAM BEHAVIOUR

Various modes of beam failure:

Various modes of beam failure

Laterally supported beams:

Laterally supported beams

Limit states for LR beams:

Limit states for LR beams Limit state of flexure Limit state of shear Limit state of bearing Limit state of serviceability

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M Radius of curvature Curvature of bending (a) (b) A z M h c Deflected shape N A

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Stress 1 strain 2 3 4 f y Plastic range Elastic range Idealised elasto- plastic stress stain curve for the purpose of design Idealised stress strain curve f

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 1 <  y  2 =  y f 2 =f y f 1 <f y Strain and stress distributions in the elastic range 2 1

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 3 >  y f 3 =f y f 3 <f y Strain and stress distributions in plastic range  4 >>  y f 4 =f y 4 A C A T Z c Z T B 3 f y T d t d 0 d 0 z c z T Stress Strain Strain Stress

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1 2 4 3 f y f y f y (b) Plastification of cross section <f y M y M P M (a) BM diagram 2 1 4 2 1 (c) Curvature Diagram Curvature max at collapse 3 4 3

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1 2 3 4 Plastic Hinge Simply supported beam and its deflection at various stages

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2 1 3 4 Design strength of different class of beams

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2.0 1.7 1.27 1.14 1.5 Some typical shape factor

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Flexural Capacities of LR beams

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25 FIXED BEAM BM diagram SF diagram

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26 Rectangular cross section Bending stresses I sections under bending Tension Compression Shear stresses

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27 FIXED BEAM – External BM SF diagram BM diagram 27 Internal moment and shear resistance

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Stress 1 strain 2 3 4 f y Plastic range Elastic range Idealised elasto- plastic stress stain curve for the purpose of design Idealised stress strain curve f

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 2 =  y f 2 =f y Strain and stress distributions in the elastic range 2

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30 Rectangular cross section Bending stresses I sections under bending Stage II Tension Compression Shear stresses My

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 3 >  y f 3 =f y f 3 <f y Strain and stress distributions in plastic range B 3 f y T d t d 0 d 0 z c z T Stress Strain

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32 Rectangular cross section Bending stresses I sections under bending stage III Tension Compression Shear stresses >My <Mp

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Strain and stress distributions in plastic range  4 >>  y f 4 =f y 4 A C A T Z c Z T Strain Stress

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34 Rectangular cross section Bending stresses I sections under bending Stage IV Tension Compression Shear stresses Mp

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35 Rectangular cross section Bending stresses Stress distribution for low shear problems Tension Compression Shear stresses Mp

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36 Rectangular cross section Bending stresses Stress distribution for high shear problems Tension Compression Shear stresses <Mp

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IS:800 Draft-22.04.04 37 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, M d , shall be taken as M d =  b Z p f y /  m0  1.2 Z e f y /  m0 PROVIDED THE SHEAR IS LOW OTHERWIDE REDUCE Md

Combined Shear and Bending:

Combined Shear and Bending a) Plastic or Compact Section b) Semi-compact Section

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Other Limit States of beams

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Combined bending and shear in beams Elastic Bending stress Elastic Shear stress Plastic range a b c LIMIT STATE OF SHEAR

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41 EQUATIONS FOR PLASTIC SHEAR CAPACITY

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42 BUCKLING OF WEBS

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43 Shear Buckling of web

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Shear buckling of a plate BUCKLING OF WEB PLATES IN SHEAR  cr

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45 8.4 Shear The factored design shear force, V , in a beam due to external actions shall satisfy V  V d V d = design strength calculated as , V d = V n / γ m0 8.4.1 The nominal plastic shear resistance under pure shear is given by: V n = V p A v = shear area Cont…

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46 45 0 d / 2 d / 2 b 1 n 1 Effective width for web buckling LIMIT STATE OF WEB BUCKLING

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47 b 1 n 2 1:2.5 slope Root radius Effective width of web bearing Web Crippling in beams LIMIT STATE OF WEB CRIPPLING

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