ENCE 710 Design of Steel Structures V LateralTorsional

ENCE 710 Design of Steel Structures V. Lateral-Torsional Buckling of Beams C. C. Fu, Ph. D. , P. E. Civil and Environmental Engineering Department University of Maryland

Introduction Following subjects are covered: n Lateral Torsional Buckling (LTB) n Flange Local Buckling (FLB) n Web Local Buckling (WLB) n Shear strength n Lateral Bracing Design Reading: n Chapters 9 of Salmon & Johnson n AISC LRFD Specification Chapters B (Design Requirements) and F (Design of Members for Flexure) 2

Introduction A beam can fail by reaching the plastic moment and becoming fully plastic (see last section) or fail prematurely by: 1. LTB, either elastically or inelastically 2. FLB, either elastically or inelastically 3. WLB, either elastically or inelastically If the maximum bending stress is less than the proportional limit when buckling occurs, the failure is elastic. Else it is inelastic. For bending b. Mn( b = 0. 9) 3

Design of Members for Flexure (about Major Axis) 4

Lateral Torsional Buckling (LTB) n n n Compact Members (AISC F 2) Failure Mode Plastic LTB (Yielding) Inelastic LTB Elastic LTB Moment Gradient Factor Cb 5

Lateral Torsional Buckling (cont. ) n Failure Mode A beam can buckle in a lateral-torsional mode when the bending moment exceeds the critical moment. 6

Lateral Torsional Buckling (cont. ) n Nominal Flexural Strength Mn n plastic when inelastic when and and Mn C b = 1. 0 Mp plastic inelastic Mr Lb Lp Lr 7

Lateral Torsional Buckling (cont. ) I-Beam in a Buckled Position 8

Lateral Torsional Buckling (cont. ) n Elastic LTB n coupled differential equations for rotation and lateral translation (8. 5. 10) where Mz z G J E Cw = = = = moment at location z along member axis along member length angle of twist shear modulus torsional constant (AISC Table 1 -1 for torsional prop. ) modulus of elasticity warping constant (AISC Table 1 -1 for warping) 9

Lateral Torsional Buckling (cont. ) n Plastic LTB (Yielding) n Flexural Strength (AISC F 2 -1) where Z= plastic section modulus & Fy= section yield stress n Limits n Lateral bracing limit (AISC F 2 -5) n Flange and Web width/thickness limit (AISC Table B 4. 1) (Note: Lpd in Salmon & Johnson Eq. (9. 6. 2) is removed from AISC 13 th Ed. ) 10

Lateral Torsional Buckling (cont. ) n Inelastic LTB n Flexure Strength (straight line interpolation) (9. 6. 4) or (AISC F 2 -2) 11

Lateral Torsional Buckling (cont. ) n Elastic LTB n Flexure Strength (AISC F 2 -3) (AISC F 2 -4) (The square root term may be conservatively taken equal to 1. 0) (c in AISC F 2 -8 a, b for doubly symmetric I-shape, and channel, respectively) n Limit (AISC F 2 -6) (AISC F 2 -7) 12

Lateral Torsional Buckling (cont. ) n Moment Gradient Factor Cb n n The moment gradient factor Cb accounts for the variation of moment along the beam length between bracing points. Its value is highest, Cb=1, when the moment diagram is uniform between adjacent bracing points. When the moment diagram is not uniform (9. 6. 3) (AISC F 1 -1) where Mmax= absolute value of maximum moment in unbraced length MA, MB, MC= absolute moment values at one-quarter, one-half, and three-quarter points of unbraced length 13

Cb for a Simple Span Bridge 14

Nominal Moment Strength Mu as affected by Cb 15

Flange Local Buckling (FLB) n n n Compact Web and Noncompact/Slender Flanges (AISC F 3) Failure Mode Noncompact Flange Slender Flange Nominal Flexural strength, Mn = Min (F 2, F 3) 16

Flange Local Buckling (cont. ) n Failure Mode The compression flange of a beam can buckle locally when the bending stress in the flange exceeds the critical stress. 17

Flange Local Buckling (cont. ) n Nominal Flexural Strength Mn n plastic when inelastic when and and Mn Mp compact noncompact slender Mr λp λr λ= bf tf 18

Flange Local Buckling (cont. ) n Noncompact Flange (straight line interpolation) n Flexure Strength (AISC F 3 -1) 19

Flange Local Buckling (cont. ) n Slender Flange n Flexure Strength (AISC F 3 -2) (kc shall not be less than 0. 35 and not greater than 0. 76) n Limit (AISC Table B 4. 1) 20

Web Local Buckling (WLB) n n n Compact or Noncompact Webs (AISC F 4) Failure Mode Compact Web (Yielding) Noncompact Web Slender Web Nominal Flexural Strength, Mn=min (compression flange yielding, LTB, compression FLB, tension flange yielding) 21

Web Local Buckling (cont. ) n Failure Mode The web of a beam can also buckle locally when the bending stress in the web exceeds the critical stress. 22

Web Local Buckling (cont. ) n Nominal Flexural Strength Mn n plastic when inelastic when and and 23

Web Local Buckling (cont. ) n Compression Flange Yielding n Flexural Strength (AISC F 4 -1) where Rpc= web plasticification factor (AISC F 4 -9 a, b) & Fy= section yield stress n Limits (AISC Tables B 4. 1) 24

Web Local Buckling (cont. ) n LTB (Inelastic) n Flexure Strength (AISC F 4 -12) where FL= a stress determined by AISC F 4 -6 a, b 25

Web Local Buckling (cont. ) n LTB (Elastic) n Flexure Strength (AISC F 4 -3) (AISC F 4 -5) n Limit (AISC Table B 4. 1) (AISC F 4 -8) 26

Web Local Buckling (cont. ) n Compression FLB (Noncompact Flange) n Flexure Strength (AISC F 4 -12) n Compression FLB (Slender Flange) n Flexure Strength (AISC F 4 -13) (kc shall not be less than 0. 35 and not greater than 0. 76) 27

Web Local Buckling (cont. ) n Tension Flange Yielding n Flexure Strength (AISC F 4 -14) Rpt = web plastification factor to the tension flange yielding limit (a) hc/tw ≤ pw (b) hc/tw > pw Rpt=Mp/Myt (AISC F 4 -15 a) (AISC F 4 -15 b) 28

Shear Strength Failure Mode n Shear-Buckling Coefficient n Elastic Shear Strength n Inelastic Shear Strength n Plastic Shear Strength For shear v. Vn( v = 0. 9 except certain rolled Ibeam h/tw≤ 2. 24√E/Fy, v = 1. 0) Vn=0. 6 Fy. Aw. Cv (AISC G 2 -1) n 29

Shear Strength (cont. ) n Failure Mode The web of a beam or plate girder buckles when the web shear stress exceeds the critical stress. 30

Shear Strength (cont. ) n Nominal Shear Strength Vn ( v = 0. 9) n n n plastic when inelastic when and and 31

Shear Strength (cont. ) n AISC G 2 Nominal Shear Strength Vn (a) For (AISC G 2 -4) (a) For (AISC G 2 -5) (AISC G 2 -3) 32

Lateral Bracing Design 33

Lateral Bracing Design AISC Provisions – Stability Bracing Design for Beams n n 1. For stiffness βreqd, βreqd = 2 βideal 2. For nominal strength Fbr, (a) Fbr = βideal (2Δ 0); (b) Fbr = βideal (0. 004 Lb) Where βideal = Pcr/Lb 34

Lateral Bracing Design (cont. ) AISC Provisions – LRFD Stability Bracing Design for Beams 1. Relative bracing Φ=0. 75 2. Nodal bracing 35

Lateral Bracing Design (cont. ) AISC Provisions – LRFD Stability Bracing Design for Columns 1. Relative bracing Φ=0. 75 2. Nodal bracing 36