Steel Beam Lateral-Torsional Buckling (AISC 360 F2)
The moment an unbraced compression flange allows before it buckles sideways, below the plastic moment.
Example
You enter
- Yield stress Fy (ksi) 50
- Plastic section modulus Zx (in³) 101
- Elastic section modulus Sx (in³) 88.9
- Weak-axis radius of gyration ry (in) 1.65
- Effective radius for LTB rts (in) 1.98
- Torsional constant J (in⁴) 1.24
- Distance between flange centroids ho (in) 17.4
- Unbraced length Lb (ft) 10
- LTB modification factor Cb 1
You get
- Lp (ft) 5.828
- Lr (ft) 16.946
- Mn (kip-ft) 360.2
- ASD allowable Mn/Omega 215.7 kip-ft
- LRFD design phi_b*Mn 324.2 kip-ft
Checked against published worked examples from: AISC.
Details, formula, and sources
ScopeAISC F2 grades the unbraced length against Lp = 1.76 ry sqrt(E/Fy) and the F2-6 Lr; between them Mn falls linearly, beyond Lr the elastic Fcr governs. A W18x50 braced every 10 ft drops from 421 to 360 kip-ft, and at 20 ft to 200 - less than half the braced number, the cliff that forces a brace or a heavier shape. Doubly-symmetric compact I-shapes, entered Cb. A design aid, not the engineer of record's stamped design.
Formula or tableMp = Fy Zx; Lp = 1.76 ry sqrt(E/Fy); Lr per AISC Eq. F2-6 (c = 1); Lb <= Lp: Mn = Mp; Lp < Lb <= Lr: Mn = Cb[Mp - (Mp - 0.7 Fy Sx)(Lb - Lp)/(Lr - Lp)] <= Mp; Lb > Lr: Fcr = Cb pi^2 E/(Lb/rts)^2 sqrt(1 + 0.078 Jc/(Sx ho)(Lb/rts)^2), Mn = Fcr Sx <= Mp. ASD = Mn/1.67; LRFD = 0.90 Mn. (E = 29,000 ksi)
Source and editionThe AISC 360-22 Section F2 lateral-torsional buckling provisions for a doubly-symmetric compact I-shape bending about its strong axis (Lp, the Eq. F2-6 Lr, the linear inelastic branch, and the Eq. F2-4 elastic critical stress with c = 1), by name; E = 29,000 ksi and the Omega_b = 1.67 / phi_b = 0.90 factors are named constants.
Free accessAISC 360 is free to read at aisc.org (Specification for Structural Steel Buildings); the shape properties are in the AISC Steel Construction Manual tables.
What governsEstimate. AHJ and licensed professional govern.
API field namesfy, zx, sx, ry, rts, j, ho, lb_ft, cb, lp_ft, lr_ft, mn_kipft, ma_kipft, phi_mn
- Bracing limits Lp = 1.76 ry sqrt(E/Fy); Lr from the Eq. F2-6 closed form with c = 1 for a doubly-symmetric shapeAISC 360-22 F2.2
- Three zones full Mp below Lp, linear interpolation to 0.7 Fy Sx at Lr, elastic Fcr Sx beyond, all capped at Mp and scaled by CbAISC 360-22 Eq. F2-1..F2-4
- Scope compact doubly-symmetric I-shapes about the strong axis; section properties from the AISC ManualAISC 360-22 F2 user note