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Lateral Torsional Buckling Formulae and Calculator

Lateral Torsional Buckling Equation and Calculator

During bending, one half of a beam is thrown into compression, and this can cause buckling in a similar manner to the buckling of a strut. This is known as lateral torsional buckling or LTB and it is illustrated in Figure 1. Unlike a strut, half the beam will be in tension, and the tensile force will help to restrain buckling. This restraint will cause the beam to twist as it buckles and the beam’s torsional stiffness will resist this twisting. Tubular members have a high torsional stiffness and therefore do not normally experience LTB. However, I-section beams have very low torsional stiffness and are therefore highly
susceptible.

LTB is responsible for a large proportion of collapses of steel-framed structures. It is a particular problem during construction, when the steel may not be fully restrained against sideways movement. Site workers have been killed when temporary restraint against LTB has not been installed. Since this problem is not always obvious, it is the responsibility of the designer to communicate to the construction team the need for temporary restraints.

The main factors that affect LTB are as follows:
1. Lateral restraint. If the compression flange is restrained against sideways movement, then LTB will not occur. It is important to appreciate that (a) restraining the tension flange will not prevent LTB and (b) the compression flange is not always the top flange.
2. Torsional stiffness. Open cross sections, like I- and H-sections, have low torsional stiffness and therefore have little ability to resist twisting. Conversely, closed sections (hollow sections) have high torsional stiffness and are much less prone to LTB.
3. Beams in which the major axis second moment of area are much greater than the minor axis second moment of area are particularly vulnerable to LTB, i.e., I-sections.

Elastic critical buckling moment. For a beam, the relationship between the torsional moment (T) and the angle of twist (θ) is

Eq. 1

${\displaystyle T=H{I}_t\frac{d\mathrm{\theta <!--\; \theta \; -->}}{dx}}$

Figure 1
Lateral torsional buckling caused by vertical loading to a cantilever (note the twisting).

The product of GI_t is known as the torsional stiffness. For open cross sections, like I- and H-sections, an extra term to account for warping is included (see Figure 2) and the equation becomes

Eq. 2

${\displaystyle T=G{I}_t\frac{d\mathrm{\theta <!--\; \theta \; -->}}{dx}-<!--\; -\; -->E{I}_w\frac{d^3\mathrm{\theta <!--\; \theta \; -->}}{d{x}^3}}$

During LTB, deformation occurs about the x-, y- and z-axes and these deformations are interrelated in the form of three simultaneous differential equations, the solution of which is known as the elastic critical buckling moment, given as

Eq. 3

${\displaystyle M_{cr}=C\frac{\mathrm{\pi <!--\; \pi \; -->}}{L_{cr}}\sqrt{\frac{E{I}_z}{1-<!--\; -\; -->I_z/I_y}(G{I}_t+\frac{E{I}_w{\mathrm{\pi <!--\; \pi \; -->}}^2}{L_{cr}^2})}}$

Alternatively

Eq. 4

${\displaystyle M_{cr}=\frac{\mathrm{\pi <!--\; \pi \; -->}}{L_{cr}}\sqrt{E{I}_{z}G{I}_t}\sqrt{1+\frac{{\mathrm{\pi <!--\; \pi \; -->}}^2}{L_{cr}^2}\frac{E{I}_w}{G{I}_t}}}$

where

I_y is the major axis second moment of area.
I_z is the minor axis second moment of area.
I_t is the torsional constant
I_w is the warping constant.
L_cr is the effective length.
C is the equivalent uniform moment factor
E is the Modulus of elasticity
G is the shear modulus
x is position along the beam.
G is the shear modulus.

Simplification of the M_cr formula. In most practical situations, C makes little difference to strength. Therefore, in the interests of simplicity, it can be set as equal to 1.0 for all end conditions and therefore eliminated from the design process. In addition, the resistance to warping at the ends of the beam can also be neglected with only a slight loss of efficiency. These changes lead to the following simpler expression:

${\displaystyle M_{cr}=\frac{\mathrm{\pi <!--\; \pi \; -->}}{L_{cr}}\sqrt{\frac{E{I}_{z}G{I}_t}{1-<!--\; -\; -->I_z/I_y}}}$

Effective Length, L_cr. It is vital to use the correct effective length when designing laterally unrestrained beams, since this critically affects the load capacity.

Figure 2 Warping at the ends of a beam due to twisting.

Figure 3 Equivalent uniform moment factor, C

Source:

• Structural Design First Principles, Michael Byfield

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