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Limit States

Limit States. Flexure Elastic Plastic Stability (buckling) Shear Deflection Fatigue Supports. Flexure. LRFD. ASD. Elastic Plastic Stability (buckling). Flexure - Elastic. S=I/c : Section Modulus (Tabulated Value). Flexure - Plastic. Flexure - Plastic. C=T A c f y =A t f y

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Limit States

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  1. Limit States • Flexure • Elastic • Plastic • Stability (buckling) • Shear • Deflection • Fatigue • Supports

  2. Flexure LRFD ASD Elastic Plastic Stability (buckling)

  3. Flexure - Elastic S=I/c : Section Modulus (Tabulated Value)

  4. Flexure - Plastic

  5. Flexure - Plastic C=T Acfy=Atfy Ac=At Mp = Acfy = Atfy = fy (0.5A) a = Mp=Zfy Mp/ My =Z/S For shapes that are symmetrical about the axis of bending the plastic and elastic neutral axes are the same Z=(0.5A)a : Plastic Section Modulus (Tabulated Value)

  6. Flexure - Stability A beam has failed when: Mp is reached and section becomes fully plastic Or Flange Local Buckling (FLB) Elastically or Inelastically Web Local Buckling (WLB) Elastically or Inelastically Lateral Torsional Buckling (LTB) Elastically or Inelastically

  7. bf tf h tw Lb Flexure - Stability Slenderness Parameter FLB l=bf/2tf WLB l=h/tw LTB l= Lb /ry

  8. Mp Mr Non Compact Compact Slender lp lr Flexure - Stability FLB and WLB (Section B5 Table B4.1) Evaluate Moment Capacity for Different l FLB l=bf/2tf WLB l=h/tw

  9. Slenderness Parameter - Limiting Values AISC B5 Table B4.1 pp 16.1-16

  10. Slenderness Parameter - Limiting Values AISC B5 Table B4.1 pp 16.1-17

  11. Slenderness Parameter - Limiting Values AISC B5 Table B4.1 pp 16.1-18

  12. Mp Mr Non Compact Compact Slender lp lr Flexure - Stability FLB and WLB (Section B5 Table B4.1) FLB l=bf/2tf WLB l=h/tw

  13. Bending Strength of Compact Shapes Lateral Torsional Buckling

  14. Bending Strength of Compact Shapes

  15. Bending Strength of Compact Shapes Laterally Supported Compact Beams

  16. Bending Strength of Compact Shapes

  17. Bending Strength of Compact Shapes Elastic Buckling

  18. L/4 L/4 L/4 L/4 Elastic Buckling Cb = factor to account for non-uniform bending within the unbraced length See AISC table 3-1 p 3.10 Mmax A B C

  19. Elastic Buckling

  20. Elastic Buckling

  21. Elastic Buckling Cb = factor to account for non-uniform bending within the unbraced length Rm= 1 for doubly symmetric cross sections and singly symmetric subject to single curvature

  22. Elastic Buckling Cb = factor to account for non-uniform bending within the unbraced length

  23. Elastic Buckling Cb = factor to account for non-uniform bending within the unbraced length ho = distance between flange centroids = d-tf

  24. Bending Strength of Compact Shapes

  25. Bending Strength of Compact Shapes Inelastic Buckling Linear variation between Mp and Mr

  26. Nominal Flexural Strength – Compact Shapes

  27. Nominal Flexural Strength – NON-Compact Shapes Most W- M- S- and C- shapes are compact A few are NON-compact NONE is slender Webs of ALL hot rolled shapes in the manual are compact FLB and LTB Built-Up welded shapes can have non-compact or slender webs FLB, WLB, LTB (AISC F4 and F5)

  28. Nominal Flexural Strength – NON-Compact Shapes WLB

  29. Design of Beams - Limit States • Flexure • Elastic • Plastic • Stability (buckling) • Shear • Deflection

  30. Large concentrated loads placed near beam supports Rigid connection of beams and columns with webs on the same plane Notched or coped beams Heavily loaded short beams Thin webs in girders Design for Shear

  31. Design for Shear V: Vertical shear at the section under consideration Q: First moment about of neutral axis of area of the cross section between point of interest and top or bottom of section (depends on y) I: Moment of inertia of section b: width of section at point of interest

  32. Design for Shear Small width b d/b=2 Error ~3% d/b=1 Error ~12% d/b=1/4 Error 100% Web fails before flanges Average Shear Stress Nominal Strength if no buckling:

  33. Design for Shear h/tw Failure of Web due to Shear: • Yielding • Inelastic Buckling • Elastic Buckling h/tw>260 Stiffeners are required Appendix F2

  34. Design for ShearAISC Specs G pp 16.1-64 Shear Strength must be sufficient to satisfy LRFD resistance factor for shear=0.9 maximum shear based on the controlling combination for factored loads nominal shear strength depends on failure mode ASD maximum shear based on the controlling combination for service loads Safety factor

  35. AISC Spec requirements for Shear Cv depends on whether the limit state is web yielding, web inelastic buckling or web elastic buckling

  36. AISC Spec requirements for Shear Special Case for Hot Rolled I shapes with Most W shapes with

  37. AISC Spec requirements for Shear Chapter G All other doubly and singly symmetric shapes except round HSS

  38. DEFLECTIONSAISC Specs Chapter L Serviceability Limit State Limiting Value Deflections due to Service Loads < Governing Building Code, IBC etc Use deflection formulas in AISC Part 3 Or standard analytical or numerical methods Calculate due to UNFACTORED (service) loads

  39. Design Shear is rarely a problem in rolled steel beams usual practice Design for Flexure and Check for Shear and Deflections Or Design for Deflections and Check for Flexure and Shear

  40. Design • Compute Required Moment Strength Mu or Ma • Weight of Beam can be assumed and verified or ignored and checked after member is selected • Select shape that satisfies strength requirements • Assume shape, compute strength, compare with required, revise if necessary or • Use beam design aids in Part 3 of the Manual • Check Shear and deflections

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