Elastic Lateral-Torsional Buckling Moment
Elastic Lateral-Torsional Buckling Moment
A slender steel beam bent about its strong axis can fail not by yielding but by lateral-torsional buckling: the compression flange, unrestrained over some unbraced length, kicks sideways and twists the whole section out of plane before the material itself is overstressed. This is why beam design always pairs a bending strength check with a check (or a bracing requirement) against this buckling mode.Long unbraced lengths — a beam with no intermediate lateral bracing along its top flange, for instance during erection before the deck is placed — are exactly when LTB governs over yielding, and it can reduce the usable moment capacity dramatically compared to what the cross-section alone could carry if fully braced.
The elastic critical buckling moment for a simply supported beam with pure lateral-torsional buckling is M_cr = (pi/L_b)·sqrt(E·I_y·G·J), a simplified form neglecting warping stiffness. where E is the modulus of elasticity, I_y is the weak-axis moment of inertia, G is the shear modulus, J is the torsional constant, L_b is the unbraced length, and M_cr is the resulting elastic critical buckling moment.
Inputs
E := 200 GPa=200000 MPa
I_y := 1.8e-5 m^4=1.8e-05 m^4
G := 77 GPa=77000 MPa
J := 6e-7 m^4=6e-07 m^4
L_b := 4 m=4 m
This closed-form expression combines the beam's resistance to sideways bending (E·I_y) with its resistance to twisting (G·J) to get the moment at which the two effects couple into buckling.
M_cr := (pi/L_b)·sqrt(E·I_y·G·J)=320303.82 Nm
Results
The critical moment found here (about 320 kN·m) sets an upper bound on usable bending strength for this unbraced length; the actual code-allowable moment will be lower once a safety factor or resistance factor is applied. Because M_cr scales inversely with L_b, shortening the unbraced length with an intermediate brace point is usually the most effective way to raise capacity — far more so than upsizing the section, which mainly helps through I_y and J. This simplified form omits the warping (Cw) term that real wide-flange sections have, so it understates M_cr somewhat compared to full code formulas (like AISC F2), making it a conservative first estimate.
Assumptions & limitations
- Simply supported beam ends with no warping restraint, and a uniform moment along the unbraced length.
- Warping torsional stiffness (Cw term) is neglected — a simplified, conservative form.
- Section is assumed doubly symmetric (no monosymmetry effects).
- Only elastic buckling is checked; inelastic buckling near full plastic moment is a separate code case.