Last Updated on September 17, 2026 by Maged kamel
Euler’s formula-part 2.
In this Post, Steel Columns and Euler’s formula-part 2, we will discuss the different shapes of steel columns & how to derive Euler’s formula, and later, the modes of failure for columns.
Structural steel shapes and their use for steel columns.
The first shape is the single angle. The second shape in B is a double angle, back-to-back, used in T-sections and W-columns, as well as round shapes. He began discussing each element and its appropriate applications, for instance, a single-angle member section used as bracing and compression in a light truss. Equal angles are more economical than unequal angles because their least r values are greater for the same Area of steel.
The double angles are likely to be used in trusses as a top chord, which might consist of a pair of angles back to back, as in section B.
The angles are spaced to insert the gusset plate; they are back-to-back. An examination of this section will show that using unequal leg angles with long legs is probably desirable back-to-back.
For extra lateral support, use the C channel. W sections are suitable for use as columns in buildings and as compression members in highway bridges. The radius of Gyration values are different.
The next slide image shows more details about box sections.
Types of modes of failure.
Compression members can fail in three general modes of failure:
1-Flexural buckling, or Euler buckling, occurs when members buckle about the section’s least Gyrationf gyration.
2- Local buckling: Some parts of the cross-section will buckle locally in compression due to slender compression members.
3- Flexural-torsional buckling for members with a particular cross-section configuration.
Derivation of Euler’s formula.
We are all familiar with the double integration method. Euler’s formula depends on estimating the Moment caused by a compression force: P*y = (E*Iy”), where y” is the curvature.
1-Multiply by 2*d and integrate once to get the Expression for the Deflection, since this equation is in general form.
2- Then,, from the boundary condition, establish a value for C1 by considering- dy/dx=0 at L=L/2.
3—After several mathematical operations, we get the Value for the critical Load the column can carry. The critical Load depends on E, the Modulus of elasticity for steel (29000 ksi), I, the section inertia, and the column’s effective length.
The critical buckling Load
P =π^2 EI/L^2, where Pcr is the critical buckling Load, which is the maximum Load that the column can carry, and the stress Fe can be estimated by dividing the critical lLoad/Area, and recall that the Inertia I can be expressed as the multiplication of A*r^2.
The stress can be evaluated as Euler’s stress, Fe = π^2 EI/(L/r)^2, which can be written as Fe. This assumes that the curve starts at zero at the end support, increases to the delta at the middle, then follows that Expression.
In reality, several different end conditions exist, which is why the k Expression is introduced. As a result, the L can be rewritten as (kL).
Euler’s critical stress, Fe, is given by π^2 E/(KL/r)^2, where KL/r is the slenderness ratio, and K is the effective length factor that modifies the actual column length and support conditions.
The following slide image gives a better understanding of the failure modes of compression columns. Two columns are shown; the first has general or elastic buckling. The second column exhibits local buckling because the width-to-thickness ratio is small.
The modes of failure for short columns.
There are two modes of failure: ductile failure has a short, bulging shape, while brittle failure has a Shear failure with a slanted angle.
The next slide shows the different modes of failure for columns, based on flexural and Torsional buckling.
You can view or download the PDF for this Post and the previous Post from the following Link. This Post covers pages 4 to the end.
For the previous Post, the easy approach to Steel columns and Euler’s formula 1/2.
For the next Post 2: Buckling for columns – effective length factors.
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