## A quantity that is often used when designing a steel beam according to CSA Standard S16 Clause 13.6 is L_{u }, the longest unbraced length with which the beam will reach either M_{r} = φ_{ }M_{p} or M_{r} = φ_{ }M_{y} , depending on the class of the cross-section.

In other words, whenever the unsupported length of a beam is less than L_{u }, the member is considered laterally supported and will reach its full moment capacity. L_{u} also appears in several design tables in Part 5 of the Handbook of Steel Construction, such as the Beam Selection and Beam Load Tables.

As the S16 Standard does not provide an explicit expression for L_{u }, engineers sometimes wonder how it is derived. When calculating its value, it is assumed that the beam is simply supported and subject to a uniform bending moment, as shown in Figure 1.

Deriving the formula for L_{u} for doubly-symmetric wide-flange beams involves a bit of algebra. A Class 1 or 2 beam (maximum factored moment resistance, M_{r} = φ_{ }M_{p} = φ_{ }Z_{x }F_{y}) will be assumed in the following calculations.

The factored moment resistance of a laterally unsupported beam is given by S16:19 Clause 13.6(a):

Eq. 1

when M_{u} > 0.67 M_{p }. M_{u} is the elastic lateral-torsional buckling moment of a laterally unbraced beam of length, L, given by:

Eq. 2

and ω_{2} = 1.0 for a beam under uniform moment.

Setting M_{r} = φ M_{p} in Equation 1, and substituting the expression for M_{u} from Equation 2:

Eq. 3

Re-arranging Equation 3 so that the square root appears on the left-hand side:

Eq. 4

After squaring both sides and multiplying throughout by L^{2}, we get the following:

Eq. 5

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If we define X = L^{2}, Equation 5 is a quadratic expression of the form:

Eq. 6

whose solution is:

Eq. 7

If we identify A, B and C in Equation 6 with the corresponding terms in Equation 5, and retain the plus sign in front of the square root, Equation 7 yields the following solution for L_{u}:

Eq. 8