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In many sources, such as here, it is said that buckling failure is independent of the material's strength.

But in many building codes, yield stress of the material is a factor that changes the allowed buckling load. The Perry-Robertson formula for example, which is the basis for buckling curves in Eurocode has yield stress as a parameter. The formula is derived by assuming initial curvature on a beam and solving for the allowed load so that yield stress is not reached in the cross section of the beam. So clearly yield stress affects the magnitude of allowed load against buckling. So why is it correct to say that buckling has nothing to do with yielding?

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I think that your first link covers it. Stress or strain virus the -strength- of the materiel is not part of the buckling equation. This equation provides the upper limit, it is going to buckle at this point.

The strength can also be based on the yield strength if the material is stressed to that point before reaching the Euler buckling point. This is the limit based on material strength. If it is lower than the Euler, then it controls design

Then design strength will be lower because of factors of safety, etc

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A column can be considered "failed" if it buckles. The equation for the critical buckling load for a slender column contains the material's elastic modulus but not its yield strength. In this sense, the load-to-buckle is independent of the yield strength.

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