Structural & Civil Engineering
Vlasov Open I-Beam Warping Constant C_w and Torsion Length Formula
Calculate open I-beam Saint-Venant torsion J_sv, Vlasov warping constant C_w = t_f*b_f^3*h_0^2 / 24, and characteristic warping length a_warp = SQRT(E*C_w / (G*J_sv)).
Classify pure Saint-Venant vs Vlasov warping torsion via L / a_warp
Open wide-flange I-beams have tiny Saint-Venant torsional stiffness J_sv = (2 * b_f * t_f^3 + h_0 * t_w^3) / 3, yet welded end-plates or continuous spans restrain longitudinal flange warping and resist torque via differential lateral flange bending! Vlasov (1940) governs this coupling via warping constant C_w = (t_f * b_f^3 * h_0^2) / 24 and characteristic decay length a_warp = SQRT((E * C_w) / (G * J_sv)).
Check by hand for a steel wide-flange girder: flange width b_f = 0.20 m (B2), flange thickness t_f = 0.012 m (B3), centroid-to-centroid flange spacing h_0 = 0.30 m (B4), web thickness t_w = 0.008 m (B5), E = 200 GPa (B6), G = 80 GPa (B7), and span L = 3.0 m (B8). Here J_sv = 28.1600 cm^4 (2.816e-7 m^4), C_w = 360,000 cm^6 (3.600e-7 m^6), and a_warp = SQRT((200 * 3.6) / (80 * 2.816)) = 1.7877 m (L / a_warp = 1.6781 -> Mixed Warping-Dominated Torsion!).
| I-Beam Torsion Metric | Vlasov Formula | Exact Output |
|---|---|---|
| Saint-Venant J_sv (cm^4) | 10^8 * (2*b_f*t_f^3 + h_0*t_w^3) / 3 | 28.1600 cm^4 |
| Vlasov Warping Constant C_w (cm^6) | 10^12 * (t_f * b_f^3 * h_0^2) / 24 | 360,000 cm^6 |
| Characteristic Length a_warp (m) | SQRT((E * C_w_m6) / (G * J_sv_m4)) | 1.7877 m |
| Slenderness Ratio L / a_warp | L / a_warp (<0.5 Pure Warping; >10 Pure SV) | 1.6781 (Mixed Regime) |
Vlasov Characteristic Warping Length a_warp (m) Formula
Evaluate J_sv in m^4 and C_w in m^6 inside LET so GPa units cancel cleanly in E / G.
=LET(b_fl, B2, t_fl, B3, h_ct, B4, t_wb, B5, e_gp, B6, g_gp, B7, IF(OR(b_fl <= 0, t_fl <= 0, h_ct <= 0, t_wb <= 0, e_gp <= 0, g_gp <= 0), "Invalid Section Input", LET(j_sv, (2 * b_fl * (t_fl^3) + h_ct * (t_wb^3)) / 3, c_wp, (t_fl * (b_fl^3) * (h_ct^2)) / 24, ROUND(SQRT((e_gp * c_wp) / (g_gp * j_sv)), 4))))AISC Torsion Design-Regime Classifier Formula
Classify L / a_warp following AISC Design Guide 9: L / a_warp <= 0.50 is "Pure Warping", >= 10 is "Pure Saint-Venant", and intermediate values are "Mixed Vlasov-SV Torsion".
=LET(rat_l, B8 / E4, IF(rat_l <= 0.5, "Pure Warping", IF(rat_l >= 10, "Pure Saint-Venant", "Mixed Vlasov-SV Torsion")))Try the example
Use the original values or change them to test your understanding.