Lateral Torsional Buckling Explained — Lb, Lp, Lr, Cb Deep Dive

Lateral-torsional buckling (LTB) is the failure mode that separates steel beam design from a simple flexural stress check. A beam that would easily carry its bending moment if fully braced can suddenly buckle sideways and twist when the compression flange is not restrained. LTB has caused real structural failures — typically during construction when temporary bracing is inadequate. This guide explains LTB from first principles, walks through every parameter in the AISC 360-22 Chapter F equations, and provides worked examples for all three regions of the LTB curve: plastic, inelastic, and elastic.

What is lateral-torsional buckling?

When a beam bends about its strong axis, the top flange is in compression and the bottom flange is in tension. The compression flange, like any compression member, wants to buckle. But unlike a column that can buckle in any direction, the compression flange of a beam is restrained by the web and the tension flange. Instead of simply buckling sideways, the beam undergoes a combined lateral displacement and twist — lateral-torsional buckling.

The beam cross-section translates laterally (in the weak-axis direction) and rotates about the shear centre simultaneously. The phenomenon is fundamentally a stability problem: below a critical moment Mcr, the beam is stable; at Mcr, it buckles. The AISC Specification quantifies this through the nominal flexural strength Mn, which varies with the unbraced length Lb.

The key parameters: Lb, Lp, Lr

AISC 360-22 Chapter F divides the LTB behaviour into three zones based on the unbraced length of the compression flange:

Computing Lp and Lr per AISC F2

These are section properties, computed from the cross-section geometry:

Lp = 1.76 * ry * sqrt(E / Fy)              (AISC Eq F2-5)

Lr = 1.95 * rts * (E / (0.7*Fy)) *
     sqrt( J*c / (Sx*ho) + sqrt( (J*c/(Sx*ho))^2 + 6.76*(0.7*Fy/E)^2 ) )
                                                     (AISC Eq F2-6)

where:
  ry  = weak-axis radius of gyration
  rts = effective radius of gyration for LTB
      = sqrt( sqrt(Iy*Cw) / Sx )  for doubly-symmetric I-shapes
  J   = torsional constant
  c   = 1.0 for doubly-symmetric I-shapes
  ho  = distance between flange centroids = d - tf

Example: computing Lp and Lr for a W21x50

W21x50 properties (A992, Fy=50 ksi):
  ry = 1.30 in, Sx = 94.5 in^3, J = 1.14 in^4
  Iy = 24.9 in^4, Cw = 1070 in^6, ho = 20.3 in

rts = sqrt( sqrt(24.9 * 1070) / 94.5 ) = sqrt( sqrt(26643) / 94.5 )
    = sqrt( 163.2 / 94.5 ) = sqrt(1.727) = 1.314 in

Lp = 1.76 * 1.30 * sqrt(29000/50) = 1.76 * 1.30 * 24.08
   = 55.1 in = 4.59 ft

c = 1.0 (doubly-symmetric)

Lr = 1.95 * 1.314 * (29000 / (0.7*50)) *
     sqrt( 1.14/(94.5*20.3) + sqrt( (1.14/(94.5*20.3))^2 + 6.76*(35/29000)^2 ) )

  = 1.95 * 1.314 * 828.6 *
    sqrt( 5.95e-4 + sqrt( 3.54e-7 + 6.76 * (1.207e-3)^2 ) )

  = 2,124 * sqrt( 5.95e-4 + sqrt( 3.54e-7 + 9.83e-6 ) )
  = 2,124 * sqrt( 5.95e-4 + 0.00319 )
  = 2,124 * sqrt( 0.00379 )
  = 2,124 * 0.0615
  = 131 in = 10.9 ft

For a W21x50 in A992 steel: Lp = 4.59 ft, Lr = 10.9 ft. These two numbers define the entire LTB behaviour of this section. Let us now compute Mn in each region.

Region 1: Plastic (Lb <= Lp) — no LTB reduction

When the unbraced length is less than or equal to Lp, the compression flange is adequately restrained. The section can reach its full plastic moment Mp:

Mn = Mp = Fy * Zx

For W21x50 (Zx = 104 in^3):
  Mp = 50 * 104 = 5,200 kip-in = 433 kip-ft

LRFD: phi_b * Mn = 0.90 * 433 = 390 kip-ft

Applicable when Lb <= 4.59 ft.

This is why beams are typically braced by the floor or roof deck at close intervals: keeping Lb below Lp allows the full plastic capacity to be used. For a 30 ft beam, providing intermediate bracing at 4 ft centres keeps Lb = 4 ft < Lp = 4.59 ft, eliminating LTB as a design concern.

Region 2: Inelastic LTB (Lp < Lb <= Lr)

In the inelastic region, the beam can still reach stresses above the yield stress in some fibres but not the full plastic moment. The nominal moment is linearly interpolated between Mp (at Lb = Lp) and 0.7 _ Fy _ Sx (at Lb = Lr):

Mn = Cb * [ Mp - (Mp - 0.7*Fy*Sx) * (Lb - Lp)/(Lr - Lp) ]  <=  Mp
                                                      (AISC Eq F2-2)

For W21x50 with Lb = 8.0 ft (Cb = 1.0 for now):
  Mp             = 5,200 kip-in
  0.7*Fy*Sx       = 0.7 * 50 * 94.5 = 3,308 kip-in
  Lb - Lp        = 96 - 55.1 = 40.9 in
  Lr - Lp        = 131 - 55.1 = 75.9 in

  Mn = 1.0 * [ 5200 - (5200 - 3308) * 40.9/75.9 ]
     = 5200 - 1892 * 0.539
     = 5200 - 1020
     = 4,180 kip-in = 348 kip-ft

LRFD: phi_b * Mn = 0.90 * 348 = 314 kip-ft

Compared to the plastic capacity of 390 kip-ft:
  LTB reduction at Lb = 8.0 ft: 314/390 = 0.81 (19% reduction)

Region 3: Elastic LTB (Lb > Lr)

When the unbraced length exceeds Lr, the beam buckles elastically — the critical moment is below the yield moment. The capacity drops rapidly with increasing Lb:

Mn = Fcr * Sx  <=  Mp             (AISC Eq F2-3)

Fcr = (Cb * pi^2 * E) / (Lb/rts)^2 *
      sqrt( 1 + 0.078 * J*c / (Sx*ho) * (Lb/rts)^2 )
                                                 (AISC Eq F2-4)

For W21x50 with Lb = 20.0 ft = 240 in (Cb = 1.0):
  Lb/rts = 240 / 1.314 = 182.6

  Cb * pi^2 * E / (Lb/rts)^2 = 1.0 * pi^2 * 29000 / 182.6^2
                              = 286216 / 33343 = 8.58 ksi

  sqrt term = sqrt( 1 + 0.078 * 1.14 * 1.0 / (94.5*20.3) * 182.6^2 )
            = sqrt( 1 + 0.078 * 1.14 / 1918 * 33343 )
            = sqrt( 1 + 1.544 )
            = sqrt(2.544) = 1.595

  Fcr = 8.58 * 1.595 = 13.69 ksi

  Mn = 13.69 * 94.5 = 1,294 kip-in = 108 kip-ft

LRFD: phi_b * Mn = 0.90 * 108 = 97 kip-ft

Compared to plastic capacity of 390 kip-ft:
  LTB reduction at Lb = 20 ft: 97/390 = 0.25 (75% reduction)

At 20 ft unbraced length, the W21x50 retains only 25% of its fully braced flexural capacity. This is why long unbraced beams require either larger sections (with higher Iy and Cw), intermediate bracing, or a reduced moment demand. The elastic LTB region is where section selection matters most: a W21x50 at Lb=20 ft has Mn=108 kip-ft, while a W21x68 (heavier, wider flanges) has significantly higher LTB capacity.

The Cb factor: moment gradient effect

Cb is the lateral-torsional buckling modification factor for non-uniform moment diagrams. It accounts for the beneficial effect of a moment gradient: a beam with a linearly varying moment (e.g., simply supported with end moments) has higher LTB resistance than a beam with a uniform moment, because the compression flange is only highly stressed over a portion of the span.

Cb = 12.5 * Mmax / (2.5*Mmax + 3*MA + 4*MB + 3*MC)    (AISC Eq F1-1)

where:
  Mmax = absolute maximum moment in the unbraced segment
  MA   = absolute moment at 1/4 point of unbraced segment
  MB   = absolute moment at 1/2 point
  MC   = absolute moment at 3/4 point

Typical Cb values:
  Uniform moment (constant):             Cb = 1.00  (worst case)
  Simply supported, UDL:                Cb = 1.14
  Simply supported, midspan point load:  Cb = 1.32
  End moments, M1/M2 = -1 (reverse):     Cb = 2.27
  End moments, M1/M2 = -0.5:             Cb = 1.67
  End moments, M1/M2 = 0:                Cb = 1.67
  End moments, M1/M2 = 0.5:              Cb = 1.30
  Cantilever (free end loaded):          Cb = 1.00 (use 1.0 per AISC F1)

Cb can dramatically increase the LTB capacity. For the W21x50 at Lb = 20 ft with Cb = 1.32 (simply supported, midspan point load): Mn = 1.32 * 108 = 142 kip-ft instead of 108 kip-ft. That is a 32% increase at no cost — simply the result of a more favourable moment distribution.

Complete LTB curve for W21x50

Plotting Mn as a function of Lb (with Cb = 1.0) shows the characteristic three-zone shape:

The drop-off in the elastic region is steep: doubling Lb from 15 ft to 30 ft reduces capacity by 73% (from 166 to 45 kip-ft). This exponential relationship — Mn is roughly proportional to 1/Lb^2 in the elastic region — is why long-span beams without intermediate bracing are uneconomical. The steel weight required to resist LTB grows faster than the span increase.

Why wider flanges help: W21x50 vs W21x68

Compare the W21x50 with the next heavier section in the same depth group, the W21x68:

W21x68 (A992): A=20.0 in^2, ry=1.80 in, Sx=140 in^3, Zx=160 in^3
  Iy=55.6 in^4 (vs 24.9), Cw=3550 in^6 (vs 1070), J=2.40 in^4 (vs 1.14)
  bf=8.27 in (vs 6.53 in) -- flange is 1.74 in wider

Lp = 1.76 * 1.80 * 24.08 = 76.3 in = 6.36 ft (vs 4.59 ft for W21x50)
Lr ≈ 16.2 ft (vs 10.9 ft)

At Lb = 20 ft with Cb=1.0:
  Mn ≈ 250 kip-ft (vs 108 kip-ft for W21x50)
  --> W21x68 has 2.3x the LTB capacity at the same unbraced length!

The 32% increase in flange width gives a 130% increase in LTB capacity at Lb=20 ft. This is why beam selection for unbraced conditions should focus on the Iy and Cw columns in Table 1-1 — the strong-axis properties (Zx, Ix) can be misleading when LTB controls. For sections with the same depth, always compare Iy, not Ix, when LTB is the governing limit state.

Practical strategies to improve LTB capacity

• Reduce Lb with intermediate bracing: This is the most effective strategy. Adding bracing at 5 ft centres instead of 10 ft can move the beam from the elastic region into the inelastic or plastic region, doubling or tripling capacity.
• Select a section with wider flanges: A W21x68 (bf = 8.27 in, Iy = 55.6 in^4) has significantly higher Iy and Cw than a W21x50 (bf = 6.53 in, Iy = 24.9 in^4). The LTB capacity scales approximately with Iy.
• Use Cb to your advantage: Do not default to Cb = 1.0. Calculating the actual moment gradient factor can increase capacity by 15–50% with no change to the beam section or bracing layout.
• Consider a box or HSS section: Closed sections have much higher torsional stiffness (J) and are far less susceptible to LTB. An HSS beam at the same weight may have 3–5x the LTB capacity of a W-shape.

Try the calculator

The Steel Calculator Beam Capacity Tool automatically computes Lp, Lr, and the Mn curve for any W, HSS, or channel section per AISC 360-22 Chapter F. Enter the unbraced length and moment gradient, and the calculator determines which LTB region applies and computes the capacity with full formula substitution and clause references.

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