Structural Steel Stability, AISC 360 / Eurocode 3 Beams & LTB
Steel I-Beam Lateral-Torsional Buckling (Mcr) Formula
Calculate Pure St. Venant Torsional Buckling Term (kN·m), Combined Timoshenko Elastic Critical LTB Moment M_cr (kN·m) including Flange Warping C_w, and Warping Stiffness Boost (%).
Unbraced steel I-beams twist sideways long before yielding—and flange warping C_w adds +65% capacity on 6 m spans!
When a long slender steel wide-flange I-beam bends about its strong axis without continuous compression-flange bracing over unbraced length L_b (m = 1,000 * L_b mm), the top compression flange acts like an Euler column and kicks sideways while twisting the web in Lateral-Torsional Buckling (LTB)! Under Timoshenko’s exact elastic stability solution (AISC 360 Eq. F2-4, with steel E = 200,000 MPa, shear modulus G = 77,200 MPa, weak-axis inertia I_y, St. Venant torsion J, warping constant C_w, and moment-gradient modifier C_b >= 1.0), Elastic Critical Moment is M_cr = C_b * (PI() / L) * SQRT(E * I_y * G * J + ((PI() * E / L)^2) * I_y * C_w)! Notice St. Venant uniform torsion scales as 1 / L_b while differential flange warping scales as 1 / L_b^2!
Verify on a 6.0 m Unbraced Roof Girder—Unbraced_Length L_b = 6.00 m (B2), Moment_Gradient_Factor C_b = 1.14 (uniformly loaded simple span, B3), Weak_Axis_Inertia I_y = 18.0 (x 10^6 mm^4, B4), St_Venant_Torsion J = 520.0 (x 10^3 mm^4, B5), and Warping_Constant C_w = 1.25 (x 10^12 mm^6, B6): Pure St. Venant Torsion Term alone gives 226.92 kN·m (E2), Full Elastic Critical Moment M_cr hits 373.37 kN·m (E3), Flange Warping adds +64.54% extra buckling resistance (E4), and Design Buckling Resistance phi * M_cr (phi = 0.90) equals 336.03 kN·m (E5)!
| AISC F2 Elastic LTB KPI (Lb=6 m, Cb=1.14, Iy=18e6, J=520e3, Cw=1.25e12) | Formula | Verified Output |
|---|---|---|
| Pure St. Venant Term (kN·m) | Cb*(PI()/L)*SQRT(E*Iy*G*J) / 1e6 | 226.92 kN·m (151.28 @ Lb = 9 m) |
| Full Elastic Critical Mcr (kN·m) | Cb*(PI()/L)*SQRT(E*Iy*G*J + (PI()*E/L)^2*Iy*Cw)/1e6 | 373.37 kN·m (200.62 @ Lb = 9 m!) |
| Flange Warping Boost (%) | ((Mcr - Pure_StVenant) / Pure_StVenant) * 100 | 64.54% (32.62% @ Lb = 9 m) |
| Design Elastic Capacity 0.9*Mcr | 0.90 * Elastic_Critical_Mcr | 336.03 kN·m |
Timoshenko / AISC 360 Elastic Lateral-Torsional Critical Moment Formula
Validate Unbraced_Length L_b > 0.50 m, Moment_Gradient C_b in [1.0, 3.0], I_y > 0, J > 0, and C_w >= 0!
=LET(lb_m, B2, cb_f, B3, iy_6, B4, jt_3, B5, cw_12, B6, IF(OR(lb_m <= 0.5, cb_f < 1, cb_f > 3, iy_6 <= 0, jt_3 <= 0, cw_12 < 0), "Invalid LTB Input", LET(ln_m, lb_m * 1000, iy_m, iy_6 * 1000000, jt_m, jt_3 * 1000, cw_m, cw_12 * 1000000000000, sv_t, 200000 * iy_m * 77200 * jt_m, wp_t, (((PI() * 200000) / ln_m) ^ 2) * iy_m * cw_m, ROUND((cb_f * (PI() / ln_m) * SQRT(sv_t + wp_t)) / 1000000, 2))))Missing Midspan Bracing 9.0 m Span Steep Drop Check (Set Unbraced Length L_b = 9.0 m)
Omit a purlin brace so Unbraced_Length B2 increases 50% from 6.00 m to 9.00 m: because warping decays as 1 / L_b^2, Warping Boost E4 halves to 32.62% and Elastic Critical Moment E3 plunges 46.3% from 373.37 kN·m down to 200.62 kN·m!
=IF(ISTEXT(E3), E3, ROUND(0.9 * E3, 2))Try the example
Use the original values or change them to test your understanding.