Columns & buckling on the PE Mechanical exam

Buckling is a failure mode of slender compression members, and the exam tests whether you can find the critical load and recognize when a column is slender enough for Euler to govern. It's a small, formula- driven topic that rewards the slenderness check.

What you actually need to own

• Euler's critical load: P_cr = π^2 · E · I / (K·L)^2, where I is the least moment of inertia (columns buckle about the weak axis), L the unsupported length, and K the end-condition factor.
• Effective-length factor K (end conditions):
  • Pinned–pinned: K = 1.0
  • Fixed–fixed: K = 0.5
  • Fixed–pinned: K ≈ 0.7
  • Fixed–free (cantilever): K = 2.0
• Slenderness ratio: K·L / r, with r = √(I/A) the radius of gyration. High slenderness → Euler (elastic) buckling governs.
• When Euler doesn't apply: below a transition slenderness, the column is "intermediate" and fails by inelastic buckling/yielding — use the Johnson (parabolic) relation or the yield check instead. Know to compare against the slenderness limit before blindly using Euler.

Where people lose points

• Using the wrong moment of inertia — buckling is about the weak axis (least I), not the strong one.
• Wrong K for the end conditions (the fixed–free K = 2.0 is the most-missed).
• Applying Euler to a non-slender column that actually fails by yielding/inelastic buckling.
• Mixing L and K·L, or radius-of-gyration slips.

How to study it

Lock in the four K values and the Euler equation, then always run the slenderness check first to decide Euler vs Johnson/yield. Drill a handful of problems across end conditions and both axes so picking the least I and the right K is automatic.

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