I'm reading this paper on structural members subject to both bending and compression.

In the first paragraph, in context of the equations 6.61 and 6.62, it is said:

The second two terms are modified by factors that allow for the interaction between the different modes of buckling.

What is meant by the interaction between different modes of buckling? How do buckling modes "interact"? I tried searching online but couldn't really find a good source. Checking a member against buckling I've understood that only the first mode is usually checked, with $N=1$ in Euler buckling equation.

$$\begin{align} \frac{N_{Ed}}{N_{b, y, Rd}} + k_{yy}\frac{M_{y, Ed}}{M_{b, Rd}} + k_{yz}\frac{M_{z, Ed}}{M_{c, z, Rd}} &\leq 1 \\ \frac{N_{Ed}}{N_{b, z, Rd}} + k_{zy}\frac{M_{y, Ed}}{M_{b, Rd}} + k_{zz}\frac{M_{z, Ed}}{M_{c, z, Rd}} &\leq 1 \\ \end{align}$$

  • $\begingroup$ answered here: engineering.stackexchange.com/a/39979/10902 $\endgroup$
    – Solar Mike
    Commented Jan 27, 2021 at 14:22
  • $\begingroup$ @SolarMike It's not the same question. Now I'm asking about the interaction of buckling modes, as in the quote "..second two terms are modified by factors that allow for the interaction between the different modes of buckling." $\endgroup$
    – S. Rotos
    Commented Jan 27, 2021 at 14:33
  • $\begingroup$ But the answer does talk about bending and then combining tension... which is the interaction you seem to be needing. $\endgroup$
    – Solar Mike
    Commented Jan 27, 2021 at 14:37
  • $\begingroup$ @SolarMike I don't see how. The equation in the previous question does not have those factors in front of the moment terms. That equation seems to be about simply combining compression the stresses caused by moments, not about buckling. In this one we seem to include buckling somehow, and its interactions. $\endgroup$
    – S. Rotos
    Commented Jan 27, 2021 at 14:50

1 Answer 1


The equation shown in the question looks like a unity check for strength with bi-axial bending and axial compression. I cannot say from which specification it was taken.

In this country the AISC Specification covers beam-columns in chapter H.

Members subjected to bending will deflect or displace normal to the plane of bending by an amount (d). If axial compression (P) is added, a second-order effect will occur. This will increase the bending moments all along the member to a maximum amount equal to P(d) at the point of maximum deflection. This additional bending will add more deflection and more P(d) moment continuing iteratively until equilibrium occurs or instability is indicated. This is a second order analysis. The AISC specification requires second-order structural analysis for both P-D and P-d moments. P-D moments are caused by eccentricity about structural nodes such as with frame sway. P-d moments are smaller and occur between nodes in individual members. Hence, adjustment factors for beam columns are no longer required in the unity equations and they are much simpler.


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