Euler buckling and the length penalty
A column is a compression member that resists axial loads. Thick, short columns can support enormous weights. A long, thin column buckles at surprisingly small loads. Euler's formula quantifies this: critical buckling load is proportional to stiffness (EI) and inversely proportional to length squared (L squared).
Double the length of a column and the buckling load drops to a quarter. This square relationship is harsh. Long thin columns are geometrically weak. This is why skyscrapers use massive columns and bracing to resist wind, and why a tree much taller than a building can break in a storm.
Real buckling and imperfections
Euler's formula assumes perfect alignment and material. Real columns are never perfect. Tiny manufacturing tolerances, microscopic material flaws, and minor misalignments all reduce the buckling load. A real column often fails at 50-80% of the ideal Euler load.
Engineers also account for intermediate buckling (inelastic buckling) for shorter columns. Very short columns do not buckle; they fail in compression. As length increases, Euler buckling takes over. In the middle range, a transition occurs. Building codes provide curves that match real materials and accounts for these real-world imperfections.