Last Updated on September 20, 2026 by Maged kamel
Stiffness reduction factor for inelastic columns.
Introduction to the Tangent Modulus of elasticity.
This is a new subject, and we will discuss the difference between the Modulus of elasticity E and the other Modulus of elasticity, Et (the tangent Modulus of elasticity), and when we use Et.
The second point, τb, or the stress-reduction factor, applies to short or medium columns.
In front of you is the slide, quoted from Prof. Segui’s book, page no.114, in brief. Long columns reach only the Proportional limit stress, which is a portion of Fy.
He pointed out in solved example 4-1 that the stress at Buckling is Pcr/A, where Pcr is the critical Load, and A is the Area.
The Value is =19.10 ksi, which is well below the proportional limit for any grade of structural steel. Thus, for long columns, the stress in that column will be nearly equal to, or less than, the proportional limit.
For inelastic columns, such as short or intermediate columns, temperature changes during fabrication induce residual stress, resulting in stress variations within the structural steel column.
Accordingly, the stress-strain graph for a column changes: it is linear up to the proportional limit, then curves until failure.
This introduces a new Value of E, called Et.
The question then is how to estimate the Modulus of elasticity for short or intermediate columns. Prof. Fredric Engesser has proposed several trials using the slope values between the proportional limit and the failure points. He has proposed a new tangent Modulus of elasticity for Short and Intermediate columns.
The Euler equation Pcr=π^2EI/L^2,. For elastic failure, the Expression can be modified by replacing E with Et: Pcr=π^2Et*I/L^2, since Et is smaller than the slope in the Euler equation, or the initial Modulus of elasticity.
For any material, the critical Buckling stress can be plotted as a function of slenderness as shown in Figure 4.6. Later studies showed that we can use this method as follows for the first part of the L/r versus Fcr graph.
If we extend the slope of Fcr = π^2*E/(l/r)^2 to the point where Fcr = Fy/2.25, it marks the point where we differentiate between long and short columns.
As we remember, at Kl/r, or le/r=4.71*sqrt(E/fy), there is another curve, for which we estimate the proposed critical stress using the AISC equation.
Suppose we fit the two curves, one for elastic Buckling and the other for inelastic Buckling, so that they become tangent to each other. When the curves are tangent, the left-hand curve represents the inelastic columns, and we can write Fcr as π^2* Et/(l/r)^2.
Now, for any Value of Kl/r, compare the values estimated by Fcr from the Euler equation and the Value calculated using the Engesser equation.
We will find that Fcr inelastic is < Fcr from Euler’s equation; then Fcr inelastic/Fcr elastic is called τ and is < 1.
The modified Value of the stiffness ratio at a joint in terms of E and the stiffness reduction factor.
How to use this information for the alignment chart for columns?
For instance, if we have a column line A’AB, column AB is elastic, while the intersecting beams are elastic. The column intersects with two beams, EB and BF. The G Value at joint B is represented as Gb =∑EI/L column/=∑EI/L Beam. The E will be eliminated.
Let Et/E = τb, the stiffness reduction factor. The column behaves inelastically, as indicated by the yellow part of the graph, with E = Et, and the G Value will be modified to Be G inelastic.
Consider elastic columns and elastic beams. E will cancel each other; then G elastic = (Ic/Ib)*(Lb/Lc), while G inelastic = τb*(Ic/Ib)*(Lb/Lc) = G elastic*τb. τb = Et/E < 1.
At P-117, we state that the column Buckling problem can also be formulated as a fourth-order differential equation. This proves convenient when dealing with boundary conditions other than pinned ends. Hence, we can evaluate the k Value in the equation.
Pcr at elastic failure=Pcr=π^2E*A/(k*L)^2, or for inelastic columns, Pcr=π^2Et*A/(k*L)^2. Columns given by 4.6b, when divided by the cross-sectional Area, provide the Buckling stress.
The FCR Value is obtained by dividing PCR by the Area. Fcr inelastic =π^2Et*/(k*L)^2, and G- inelastic=G-elastic*Et/E which is τb.
The next slide shows the Johnson graph for which Fcr=Fy-Fy^2*(Kl/r)^2*(1/4*pi^2*E).
The term (λ^2) is introduced as Fy/Fe, which modifies the Johnson equation to Fcr = Fy (1-λ^2/4); for the tau Value at the point where λ^2=2.0, Fe=Fcr inelastic, and the tau Value τb = 1.00, which is applied for the LRFD Design.
From Galambos’ two expressions, Fcr inelastic = = (1-λc^2/4) Fy, and the second Expression is Fcr in = Fy/λc^2, where λc^2 = Fy/FE, and FE is the Euler stress according to the elastic curve. The Value of λc is shown in the next slide.
In the next slide, we see the different methods used to develop expressions for column Design equations. We will focus on the first two equations derived from the CRc curves, with λc^2 = 2.00.
We will continue discussing the equations for inelastic columns in the next Post.
The PDF file used to illustrate this Post and the following Post is available for viewing or download below.
The following Post, 25a, will be titled “Stiffness reduction factor for inelastic columns – 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.