What is the expression, according to Euler’s theory, for the elastic critical compressive stress in a long column in terms of modulus E and slenderness ratio (L_eff/r)?

Introduction / Context:
Euler buckling links the maximum elastic compressive stress a column can sustain without lateral buckling to its slenderness ratio. This stress form is convenient for comparing different sections and lengths on the same basis.

Given Data / Assumptions:

• E = modulus of elasticity of steel.
• L_eff/r = slenderness ratio λ (effective length over least radius).
• Prismatic, straight member; concentric loading; elastic behavior.

Concept / Approach:
Load form: P_cr = (pi^2 * E * I) / (L_eff^2). Divide by area A and use r^2 = I/A to get stress: sigma_cr = P_cr / A = (pi^2 * E) / (L_eff/r)^2.

Step-by-Step Solution:
Start with P_cr = (pi^2 * E * I) / (L_eff^2).Use r^2 = I/A ⇒ I = A * r^2.sigma_cr = P_cr / A = (pi^2 * E * A * r^2) / (L_eff^2 * A) = (pi^2 * E) / (L_eff/r)^2.

Verification / Alternative check:
The expression is dimensionally correct and appears in all standard steel design texts and handbooks.

Why Other Options Are Wrong:

• Linear denominator or missing pi^2 understates/overstates capacity.
• Using E alone ignores geometry (slenderness).

Common Pitfalls:
Mistaking P_cr (load) for sigma_cr (stress); using actual length instead of effective length.

Final Answer:
sigma_cr = (pi^2 * E) / (L_eff/r)^2