Fig. 1: Moment Capacity during the 3 stages of Lateral Torsional Buckling (L.T.B.)

### What is Lateral Torsional Buckling (L.T.B.)?

Lateral Torsional Buckling (LTB) is a failure criteria for beams in flexure. The AISC defines Lateral Torsional Buckling as: the buckling mode of a flexural member involving deflection normal to the plane of bending occurring simultaneously with twist about the shear center of the cross-section. LTB occurs when the compression portion of a beam is no longer sufficient in strength and instead the beam is restrained by the tension portion of the beam (which causes deflection or twisting to occur).

The best analogy I can provide is a girl on a tightrope. As the rope gets longer and longer it will become more and more difficult for the girl to remain balanced (since the swing/twist of the wire will become greater and greater). This also makes sense empirically; without LTB then a beam would have the same flexural strength whether it stretched 10 feet or 100 feet.

The best ways to prevent Lateral Torsional Buckling are to:

• Adequately brace the beam laterally at small intervals
• Use a larger section size (this will increase your ry which will increase the Lp shown in Figure 1 above)

## How to find your LTB limits:

Three limits exist when solving for lateral torsional buckling (shown in Fig. 1 above):

• Lb < Lp which defines when a member is not subject to LTB
• Lp < Lb < Lr which defines when a member is subject to inelastic LTB
• Lr < Lb which defines when a member is subject to elastic LTB

where:

Lb = Unbraced Length (bracing must resist displacement of the compression flange or twisting of the cross section)

### For Compact Cross Sections (most common):

 Solving for Lp: L_p = {1.76{r_y}\sqrt{\frac{E}{F_y}}}

where:

Lp = The limit between no LTB and Inelastic LTB
ry = The radius of gyration? about the weak axis of the cross section
E = The modulus of elasticity of the steel beam (e.g. 29000 ksi)
Fy = The yield strength of the steal beam (e.g. A36 has a yield strength of 36 ksi)
 Solving for Lr: L_r = 1.95r_{ts}\frac{E}{0.7F_y}\sqrt{\frac{Jc}{S_xh_O}}\sqrt{1+\sqrt{1+6.76\left(\frac{0.7F_y}{E}\frac{S_xh_O}{Jc}\right)^2}}

where:

Lr = The limit between Inelastic LTB and Elastic LTB (ft)
rts2 = \frac{\sqrt{I_yC_w}}{S_x} = \frac{I_yh_O}{2S_x}
Iy = The Moment of Inertia about the weak axis of the cross section (in4)
Cw = \frac{I_yh_O^2}{4} (for rectangular flanged doubly symmetric shapes)
Sx = Section Modulus? of the beam about the strong axis of the cross section (in3)
hO = distance between flange centroids = d - tf (in)
E = The modulus of elasticity of the steel beam (e.g. 29000 ksi)
Fy = The yield strength of the steal beam (e.g. A36 has a yield strength of 36 ksi)
J = torsional constant (in4)
c = 1 (for doubly symmetric I-shape) or \frac{h_O}{2}\sqrt{\frac{I_y}{C_w}} (for a channel)
Cw = warping constant (in6)

Note: For standard doubly symmetric I-shapes: rts, ry, J, Sx, hO, Iy, & Cw can all be found in Table 1 of the AISC 360 Steel Construction Manual[1]. This makes these calculations much easier.

### For Non-Compact Cross Sections (uncommon):

Fig. 2: Moment Capacity vs. Unbraced Length
 Solving for Mp' (non-compact section): M_p' = M_p - (M_p - 0.7F_yS_x) \frac{\lambda - \lambda_p'}{\lambda_r - \lambda_p}

where:

Mp' = The moment capacity with no LTB for a non-compact beam
Mp = The moment capacity with no LTB for a compact beam
Fy = The yield strength of the steal beam (e.g. A36 has a yield strength of 36 ksi)
Sx = Section Modulus? of the beam about the strong axis of the cross section (in3)
λ = \frac{b_f}{2t_f}
λp = the limiting slenderness for a compact flange (see AISC 360 Table B4.1) = 0.38\sqrt{\frac{E}{F_y}} (for flexure in the flanges of a rolled I-shape)
λr = the limiting slenderness for a non-compact flange (see AISC 360 Table B4.1) = 1.0\sqrt{\frac{E}{F_y}} (for flexure in the flanges of a rolled I-shape)
bf = the base of the flange (in)
tf = the thickness of the flange (in)

 Solving for Lp' (non-compact section): L_p' = L_p + (L_r - L_p) \frac{M_p - M_p'}{M_p - M_r}

where:

Lp' = The limit between no LTB and Inelastic LTB for a non-compact section
Lp = The limit between no LTB and Inelastic LTB for a compact section
Lr = The limit between Inelastic LTB and Elastic LTB (ft)
Mp' = The moment capacity with no LTB for a non-compact beam
Mp = The moment capacity with no LTB for a compact beam
Mr = The moment capacity at the transition between inelastic and elastic LTB = 0.7FySx

## Solving for different Lateral Torsional Buckling cross sections

Solving for the capacity of a steel beam (and thus checking to see if the beam is in the inelastic or elastic lateral torsional buckling range) is shape dependent. Above you found how to find those limits. Below you will find how to design a member and find if LTB is present.

## References:

1. American Institute of Steel Construction), "Steel Construction Manual 13th edition", 2005
• This Section is thoroughly covered in Part 3 of the AISC Steel Manual and in Chapter F of the Specifications (AISC 360)
• Table B4.1 in the Specifications (AISC 360) has also been referenced to find λp & λr.

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