Last Updated on July 18, 2025 by Maged kamel
Stiffness reduction factor for inelastic columns-part 2.
In the previous post, we talked about the difference between Fcr from the Euler graph and the Fcr for inelastic columns. Please refer to this link for review.
How do we get the expression of the Stiffness reduction factor for inelastic columns?
Prof. Segui has introduced various equations to get an expression for τb, using Fcr inelastic term.
Using the λ^2=Fy/Fcrelastic based on Galambos’ second equation, then *(Fcr inel/Fcr inelastic), use the τb= Fcr in/Fcr elastic, then we get λ^2=τb*(Fy/Fcr inelastic).
We get the final Expression τb, the stiffness reduction factor τb = 4*(Fcr/Fy)*(1-Fcr/Fy).
If we use the stress expression, in terms of loads, the stiffness reduction factor will be τb =4*(Pn/Py)*(1-Pn/Py), Pn=Fcr*Ag, Py=Fy*Ag.
The previous expression was further modified by the load, which can be expressed in both the LRFD or the ASD format; another α factor was introduced.
There is a new factor, termed α; In the case of the LRFD design, α = 1, and for the ASD design,
α = 1.6. This is the code provision. First, check that α*Pr/Pns <=0.50. If we use Pns, this is considered Py in all cases, with an α factor of 1 for LRFD, and Pr = Pult = 1.2 D.L + 1.60L.
In the case of ASD then Pt=Pd+PL, refer to τb equation, for the pr expression (α*Ppr)=(1*Pult)=(1*Pult), the Pns =Py, in case of ASD,(α*Ppr)=(1.60*PT).
The new slide shows various types of curves, as quoted from the UMass link, which includes a handy illustration for compression steel members. The blue curve is the Euler curve, and the dotted curve in yellow is the inelastic column curve, which has Et<E.
While the dark blue color curve is the AISC code provision for inelastic columns, the Fcr equation can be divided into two parts. In the first part, the elastic columns, where Kl/r>4.71(sqrt(E/Fy) differentiate between the elastic and inelastic columns.
The value of Fcr for elastic columns is equal to 0.877 Fe, while Fcr for inelastic columns equals 0.658 raised to λ^2, all multiplied by Fy.
This is Table 4-13, which is used to determine the value of the stiffness reduction factor for inelastic columns, τb. The table is divided into the upper row for the different values of the yield stress Fy
On the other left side of the table, Pt/Ag, as in the case of ASD, if Pt=Pd+Pl, and in the case of LRFD, we use Pult/Ag. In the following post, we will introduce a solved p[problem 4-13 to estimate the stiffness reduction factor.
A handy external link is Chapter 7 – Concentrically Loaded Compression Members.
The next post will be a solved problem 4-13 for the stiffness reduction factor.