Last Updated on September 14, 2026 by Maged kamel
Regions for Lateral-torsional buckling for beams-2/2
Lateral-torsional buckling in beams has three regions, with shapes similar to the previous zones for local Flange buckling and web buckling in steel beams.
However, instead of using the λ factor on the horizontal axis as λp (plastic) and λr (elastic), we use Lp (plastic length) and Lr (unbraced length), which mark the boundary between elastic and inelastic lateral-torsional buckling, in Feet.
The first of the three Regions for lateral-torsional buckling starts from Lb=0, where Lb stands for bracing length, and goes to Lb = Lp.
Lp is the plastic length between bracings. The second point is Lr for the unbraced elastic Moment, which has a stress equal to 0.7Fy*Sx, where Sx is the elastic section Modulus, and the inclined line has a Zone of inelastic buckling. While the last curved Zone is for elastic buckling, every line has a reference equation.
Each formula starts from F2-1 to F2-3. Our Chapter is Chapter 3, Part 2, covering the three regions of Lateral-torsional buckling for beams, as shown in the next slide image with the relevant equation.
This is a reference from the AISC Specification for Yielding Structural Steel Buildings.
Cb-coefficient of bending.
A new factor will be considered: the Cb= bending coefficient, defined in equation 5. 0. This factor appears in the three Regions for Lateral-torsional buckling of beams in equations F2-1, F2-2, and F2-3.
The recent equation for Cb equals 12.50 *M-max/(2.50 Mmax+3Mb+4Mb+3Mc)*Rm<=3.0.
The factor 12.5 in the numerator equals the sum of 2.50+3+4+3 in the denominator; the term Mmax is the maximum Moment in the Beam, divided into three points: A, B, and C.
We will discuss Cb later with solved examples, while the old equation was Cb=(1.75+1.05*M1/M2+0.30*(M1/M2)^2)<=2.30.
The first part of the curve is for Cb=1, but Cb, due to bracing, can be 1. The three Regions for Lateral-torsional buckling for beams can be seen. The second inclined line joins Mp with Mr = 0.70*Fy*Sx, which applies when Cb> 1.
The horizontal line extends horizontally, and the inclined line will be displaced in case of Cb>1.
How do you evaluate Mn for lateral-torsional buckling?
The most important step is determining the relationship between the bracing length and Lp and Lr. How can we estimate Lp and r? How do we get the Value of Mn in the case of lateral-torsional buckling?
For the case of Cb=1, the horizontal line has a Value of Mp, while the inclined line joining two points has a first point of Mp at the unbraced length of Lp and a second point of 0.70*Fy*Sx.
Lp Value according to E, Fy, and ry.
There is a relation between Lp and ry, or the radius of Gyration in the y direction, and Fy, which is the yield stress of steel based on the designated ASTM.
Mn for any point, if we have a bracing length =Lb between Lp and Lr, can be evaluated as Mn= MP—slope by the difference, then Mn=(Mp—0.70*Fy*Sx) /Lr- Lp all *(Lb -LP)*Cb, Lp Value =ry *300/ sqrt (F ). The Fy values are listed along with their corresponding Lp values.
The following slide lists different Fy values, starting with yield stresses of 36 ksi, 42 ksi, 45 ksi, and 50 ksi and ending with Fy = 65 ksi.
For instance, for Fy = 36 ksi, Lp=50*ry, where ry is the radius of Gyration about the y-axis. Or Fy = 42 ksi, then Lp = 46.247*ry.
For Fy = 45 ksi, then Lp=44.68*ry. For Fy = 65 ksi, then Lp=37.175*ry.
Find Lp for W12x30 A992 steel.
For A992 steel, we need the Lp Value for W12x30. We use Table 2-4 to find Fy for A992 steel; that table indicates that Fy = 50 ksi. The Value of Lp equals 42.386*ry.
We use Table 1-1, Part 2, for W12x30 to find the radius of Gyration about the y-axis, which is 1.52 inches. The final Value of Lr is 64.427 inches, or 5.3 or 5.37. To determine Lr, you need the values of rts, c, and h0.
How do we get the Lr Value for W12x30?
To find the Value of Lr, we use a lengthy equation for Lr. Please proceed to the slide for the Lr estimate. As shown, Lr equals 187.24 inches and can be approximated as 15.60 ft.
Confirm the Value of Lp and Lr from Table 3-2.
We can confirm Lp and Lr for W12x30 using Table 3-2, where Zx sorts the sections.
You can review or download the PDF for the data in this post and the previous one.
Here is the Link to Chapter 8, ” Bending Members—A Beginner’s Guide to the Steel Construction Manual, 14th ed.
Here is the Link to Chapter 8, “Bending Members.” A Beginner’s Guide to the Steel Construction Manual, 15th ed.
Here is the Link to Chapter 8, “Bending Members.” A Beginner’s Guide to the Steel Construction Manual, 16th ed.
The next post, post 11, is a Beam Solved problem 4-5: How to design a steel Beam?