Last Updated on March 11, 2026 by Maged kamel
Introduction to Local Buckling-part-1.
We will start a new subject: local buckling of columns.
We have covered local buckling of beams, discussed the local buckling coefficient for both flanges and webs, and discussed the general condition. There are three types: compact, noncompact, and slender sections.
Classification of a compression member into stiffened and unstiffened parts.
In Local Buckling-Part 1, we will examine a cross-section of a column showing a bulged flange due to localized buckling. Also, there is another picture showing the flange, and the other part is for web buckling in the local buckling theory.
Any column consists of a series of intersection plates; one plate resembles the upper flange, the second plate is for the web, and the third plate is for the lower flange part. The following images show the flange and web of a slender column that exhibits local buckling.
The upper hatched part on the right side is part of the flange and is called an unstiffened element. The reason for this is that it is free on the edge but supported by the web on the other side.
In Local Buckling-part-1, the flange differs from the web, which is stiffened on both sides. That is why we have the definitions of stiffened and unstiffened for columns.
The site at the link, Chapter 6 – Buckling Concepts, Local Buckling, explains that general buckling applies to the whole column and to all beams.
The second picture shows the behavior of the stiffened plate under compression; because it is fixed on both sides, it deflects in the longitudinal direction. At the same time, for the unstiffened, it is warped in the longitudinal direction according to the Condition of the edges, whether hinged-free, free, or fixed. This is the picture link.
The third shape is for the unstiffened elements, for the wide flange I beam the unstiffened is shown with the white color at the left side, half of the bf flange, and with the thickness of t flange.
The second shape is for the C channel, for which the unstiffened section width is half of the flange width, and its thickness is the flange thickness. The third shape is for the T section; the unstiffened part is half of the flange width. The fourth shape is the same.
The vertical plate, web, can also be unstiffened, since it is attached on one side to the upper flange and is free on the bottom side. Also, the angle is an example of an unstiffened element.
Local buckling-part-1 stiffened parts of the steel column.
This is the link to the picture of the unstiffened elements for a clearer view.
For the shapes of the stiffened parts included in the web of W section, the section is fixed from the top flange and bottom flange, the length is the height of the web minus the two kd values, where kd is the distance from the rounded part to the other rounded part, and the thickness is the web thickness.
For the box section, two parts are considered stiffened, with different widths and lengths, marked in white. The W flange has a plate welded to the top flange; the plate can be regarded as a stiffened plate, the last shape for a box with two cells, for which the white mark indicates a stiffened part with width b from the right and left.
This is the picture view from the source.
Introduction to Plate theory.
The following image shows the behavior of two plates: one has a lower b/t, and its carrying capacity is higher than that of the one with a higher b/t.The plate with higher b/t has a stress distribution that is maximum at the edges and decreases toward the center, and exhibits higher post-buckling strength.
The following image shows the forces acting on a plate with their notations. Since we are discussing the buckling of plates, we are interested in the force acting perpendicular to the shorter edge, which we denote by Nxx.
The deflection of the plate is based on the assumption that there are two sinusoidal deflections in the two directions that can be expressed as a function of both a and b in sine waves, where a is the longer direction and b is the shorter dimension.
Develop the expression for Nxx and Fcr.
The load acting on the edge of the plate [er unit length can be solved by introducing the differential equation of EIw””+-Nw”=0. That is the same equation we used to derive the expression for a column’s critical load.
After solving the differential equation and applying the boundary condition for the rectangular plate, we get the value of Nxx=k*π^2*D/b^2, where D is the flexural rigidity for the plate, which is similar to E*I, but I -t^3/12 since the plate width is 1 unit. To get Fcr value, divide Nxx/area, which is equal to Fcr=K*π^2*D*t^2/b^2*(1/(12*(1-ν^2)).
The difference in flexural rigidity between a beam and a plate.
The following image shows the difference in flexural rigidity between a beam and a plate of width 1 unit. For a plate, there is Poisson’s ratio for the lateral strain that has to be considered, and the width of the plate is equal to 1.
The factor Kc is a factor that is related to m, the number of half-sin waves, and the a and b values.
The minimum value of kc occurs when m=1, i.e., for a square plate, which is the case of a plate pinned in all directions. Please have a look at the relation between a/b and K values for different m from the following slide.
For an individual plate critical stress, it has to be equal to or higher than the yield stress for a column to control the local buckling width/thickness ratio. Introducing a new factor λ. The square of λ= Fy/Fcr. Readjust the equation of fcr as Fcr=Fy/λ^2, which should be bigger than or equal to =K*π^2*D*t^2/b^2*(1/(12*(1-ν^2)). Adjust the terms to define the limiting b/t.
For a Poisson ratio of 0.30 for steel, we get b/t = 0.9506*λc*sqrt (Kc*E/Fy).
λc is taken as equal to 0.70, so as to match the actual behavior of the plate represented by the transition curve. Please refer to the next slide.
The final expression for b/t in terms of kc, Fy, and E.
After introducing the lambda value =0.70, the limiting value of b/t=0.666*sqrt(Kc*E/Fy). The next step is to find the value of kc based on the end conditions for plates.
The different values of k based on the end conditions of plates.
There are five cases for plates loaded in compression. Nxx, the two sides where the load is applied are both pinned. Consider the ratio of the longer side to the shorter side, a/b > 5.
Case A, where the two unloaded sides are both fixed, the Kc value equals 6.97; Case B, where one unloaded side is fixed while the other is pinned, Kc=5.42.
In Case C, where both unloaded sides are pinned, the Kc value equals 4.00.
In Case D, where one unloaded side is fixed while the other is Free, Kc=1.277.
In Case E, where one unloaded side is pinned while the other is Free, Kc=0.425.
The previous values of kc for the different end conditions are shown in the table below. Two cases are for unsiffened elements, while the remaining three cases are for stiffened elements.
What are the values for λr for unsiffened elements?
Based on Aisc-360-16, the required k value for the unsiffened elements, for rolled flange K=0.70. For angle leg K=0.425, while for stenm of Tee K=1.277.
For rolled flange- I sections, select k=one third the distance between pinned and fixed, which gives λr=0.666*sqrt(K*E/Fy)=0.66*sqrt(0.7*Ey/Fy)=0.56*sqrt(E/Fy). The remaining two items are shown in the image on the next slide.
What are the values for λr for siffened elements?
Based on Aisc-360-16, the required k-value for the stiffened elements is K = 5.0 for the rolled web. For the HSS wall, K = 4.40; for the other, K = 5.00.
For Hss wb, select k= 4.40, which gives λr=0.666*sqrt(K*E/Fy)=0.66*sqrt(4.4*Ey/Fy)=1.40*sqrt(E/Fy). The rolled web k value=5.00 , which gives λr=0.666*sqrt(K*E/Fy)=0.66*sqrt(5.0*Ey/Fy)=1.49*sqrt(E/Fy) as shown in the image on the next slide.
The PDF file for this post and the following post can be viewed and downloaded from the following link.
The next post is Number 10a-Easy study of Local Buckling-part-2.
For a good A Beginner’s Guide to the Steel Construction Manual, 14th ed. Chapter 7 – Concentrically Loaded Compression Members.
For a good A Beginner’s Guide to the Steel Construction Manual, 15th ed. Chapter 7 – Concentrically Loaded Compression Members.
For a good A Beginner’s Guide to the Steel Construction Manual, 16th ed. Chapter 7 – Concentrically Loaded Compression Members.
For this post, the resources are from Steel Structures: Design and Behavior by Charles S. Salmon, fifth edition.
Solid Mechanics Part I: An Introduction to Solid Mechanics
Solid Mechanics Part II: An Introduction to Solid Mechanics
localbuckling-limits-report_aisc_adhoctg.
Gerard and Becker- stability book.