# How does the imperfection factor in buckling include the effect of residual stresses?

Let's say we have a beam with pinned connections at both ends. The beam is subjected to axial load $$P$$. The beam has a cross sectional area $$A$$, radius of gyration $$r$$, distance from the neutral axis to the extreme fiber $$z$$, length $$L$$ and theoretical Euler buckling load $$P_e$$.

It is impossible to manufacture beams that are perfectly straight, so we assume the beam as an initial maximum deflection at the midpoint, $$d$$.

The maximal stress occurs at the midpoint of the beam, on the extreme fiber looking from the neutral axis. This stress is the sum of direct compressive stress $$\sigma_{c}$$ and bending stress $$\sigma_{b}$$:

$$\sigma_{max}=\sigma_{c}+\sigma_{b}=\frac{P}{A}+\frac{Mz}{Ar^2}=\frac{P}{A}+\frac{Pz}{Ar^2}\frac{d}{\frac{P_e}{P}-1}=\frac{P}{A}+\frac{P\mu}{\frac{P_e}{P}-1}$$

where $$\mu$$ is an imperfection factor that captures the effect initial imperfection.

We cannot predict $$\mu$$ exactly as it is related to manufacturing but we can assume its maximum values. In Eurocode it is assumed that $$\mu=\alpha(\lambda-0.2)$$

where $$\lambda=\frac{L}{r}$$, also called slenderness and $$\alpha$$ is a constant chosen from cross sectional properties. The assumption therefore is, that the imperfection factor is proportional to slenderness of the beam with some constant of proportionality $$\alpha$$. It is reasonable assumption; longer and thinner the beam is, more "crookedness" can be expected to be present after fabrication of the beam.

$$\alpha$$ is said to, among other things, to capture the effect of residual stresses due to manufacturing. Whether the beam is welded together or hot rolled affects the choice of $$\alpha$$. I don't see why this is a logical way of including residual stresses. The imperfection factor $$\mu$$ appears only in $$\sigma_b$$, which implies that if the initial deflection is negligeable, $$\sigma_{max}$$ equals just $$\frac{P}{A}$$ and the residual stress doesn't matter. But surely the residual stress matters in simple compression too?

Shouldn't a more logical way be something like this:

$$\sigma_{max}=\sigma_{c}+\sigma_{b}+\sigma_{residual}$$

with some appropriate expression for $$\sigma_{residual}$$ involving similar factor to $$\alpha$$? I get that the existing method does work, but I don't understand why.

The $$\alpha$$ factors in the Eurocodes is empirical, and are based on test data gathered over many years.