Lateral Torsional Buckling in Steel Beams: A Complete Guide

A steel I-beam loaded in bending has a compression flange and a tension flange. The compression flange tends to buckle sideways (laterally) just as a column would buckle under axial load. Because the flanges are connected by the web, the beam does not simply deflect laterally — it twists simultaneously. This combined lateral-torsional movement is called lateral torsional buckling.

LTB reduces the beam's flexural capacity below the plastic moment M_p = F_y·Z_x. The reduction depends on the unbraced length L_b — the distance between points that restrain lateral movement of the compression flange.

What provides lateral restraint? Concrete slab (via shear connectors), cross-frames, purlins connected to the top flange, bridging, or bracing frames. The restraint must be stiff enough; AISC Appendix 6 quantifies the required brace stiffness.

Representative L_p and L_r Values for Common Sections (F_y=345 MPa)

Section	ry (mm)	Lp (m)	Lr (m)
W18×35	37.1	1.43	4.27
W18×50	40.6	1.56	4.60
W18×55	41.1	1.58	4.69
W21×68	43.2	1.66	4.98
W24×76	41.7	1.60	5.23
W14×48	58.7	2.26	7.09

Note that W14 sections (with their wider flanges relative to depth) have significantly higher L_p and L_r than W18/W21 sections — making them more efficient when lateral bracing is limited.

C_b accounts for the variation of moment along the unbraced segment. Uniform moment (worst case) has C_b=1.0. Non-uniform moment diagrams are less severe — C_b>1.0 increases the allowable M_n.

Loading Condition	Cb
Uniform moment (double curvature, equal end moments)	1.00
Uniform distributed load (simply supported)	1.14
Mid-span point load (simply supported)	1.32
Concentrated end moment one end (other end zero)	1.67
Reverse curvature (equal opposite end moments)	2.27

C_b is capped so that M_n ≤ M_p — the beam cannot exceed its plastic moment regardless of the moment diagram shape.

Step 1 — Classify section as compact

Check flange: λ_f = b_f/2t_f ≤ λ_pf = 0.38√(E/F_y) = 9.15 (F_y=345 MPa). Check web: λ_w = h/t_w ≤ λ_pw = 3.76√(E/F_y) = 90.6. Standard W-shapes are almost always compact.

Step 2 — Compute M_p and M_r

M_p = F_y·Z_x  |  M_r = 0.7·F_y·S_x

Step 3 — Determine L_b and compare with L_p, L_r

Step 4 — Compute M_n

• Zone 1 (L_b ≤ L_p): M_n = M_p
• Zone 2 (L_p < L_b ≤ L_r): M_n = C_b[M_p − (M_p−M_r)·(L_b−L_p)/(L_r−L_p)] ≤ M_p
• Zone 3 (L_b > L_r): M_n = F_cr·S_x ≤ M_p, where F_cr = C_b·π²E/[L_b/r_ts]²·√(1+0.078·J·(L_b/r_ts)²/(S_x·h_o))

Step 5 — Check φ_bM_n ≥ M_u (φ_b=0.90)

Given: W18×55, F_y=345 MPa, M_u=300 kN·m, L_b=4.5 m, C_b=1.25 (uniform distributed load between braces).

Section properties: Z_x=1,490 cm³, S_x=1,320 cm³, r_y=41.1 mm, r_ts=47.2 mm, J=70.6 cm⁴, h_o=432 mm.

Step 1 — M_p and M_r:

M_p = 345 × 1,490,000 = 514 kN·m  |  M_r = 0.7 × 345 × 1,320,000 = 319 kN·m

Step 2 — L_p and L_r:

L_p = 1.76 × 41.1 × √(200,000/345) = 1.76 × 41.1 × 24.08 = 1.74 m

L_r ≈ 5.21 m (computed from full formula)

Step 3 — Zone check: L_p=1.74 m < L_b=4.5 m < L_r=5.21 m → Zone 2 (Inelastic LTB)

Step 4 — M_n:

M_n = 1.25 × [514 − (514−319) × (4.5−1.74)/(5.21−1.74)]

= 1.25 × [514 − 195 × 2.76/3.47] = 1.25 × [514 − 155] = 1.25 × 359 = 449 kN·m < M_p=514 kN·m ✓

Step 5 — Check: φ_bM_n = 0.90 × 449 = 404 kN·m > M_u=300 kN·m ✓ (DCR=0.74)

Without C_b (conservative C_b=1.0): M_n=359 kN·m, φ_bM_n=323 kN·m — still passes but DCR=0.93. The C_b=1.25 factor saves about one section size.

• Brace at load points: Adding a brace at a concentrated load point splits the unbraced length in two, often recovering Zone 1 capacity entirely.
• Use C_b>1.0: Never default to C_b=1.0 unless you have uniform moment. For typical gravity loading, C_b=1.14–1.32 is common and can allow using a lighter section.
• Wide-flange vs deep sections: W14 sections have larger r_y (wider flanges relative to depth) and hence larger L_p. Where lateral bracing is limited, consider shallower wide-flange shapes.
• Composite action: A concrete slab connected to the top flange via shear studs effectively provides continuous lateral bracing to the compression flange — LTB does not govern for positive moment regions of composite beams under construction-stage loading only.
• Check the LTB-controlled weight: The steel cost of going up one section size is typically 5–10%. The benefit of keeping L_b ≤ L_p often outweighs the cost of an additional brace connection.