Explain inelastic column buckling and the tangent modulus theory.

For intermediate columns where buckling stress exceeds proportional limit, Euler's formula overpredicts critical load. Tangent modulus theory (Engesser) uses instantaneous modulus Et at stress level: Pcr = π²Et×I/Le². Reduced modulus theory (Considere) accounts for load reversal during buckling. Modern codes use empirical formulas like Johnson's parabola or AISC curves that smoothly transition from Euler to yield failure.