Delivered scenarios: Euler buckling of a pin-ended column with load P(t) = R·t and load factor Pcr/P, and beam deflection δ(t) = F(t)·L³/(48EI) under a ramped central load.

Pcr = π²·E·I / (K·L)² [N]; load factor = Pcr / P(t), P(t) = R·t.

Column with an initial half-sine bow e₀: EI·y″ + P·(y + e₀ sin(πx/Le)) = 0 ⇒ y_mid = e₀·P/(Pcr − P), growing without limit as P → Pcr.

Beam, central load F: δ = F·L³/(c·E·I), c = 48 (simply supported), 192 (fixed–fixed); cantilever tip load: c = 3. Curvature y″ = M(x)/EI.

Honest simplifications. Linear-elastic, slender, prismatic member; small deflections; K values are the textbook effective-length factors (fixed–pinned ≈ 0.7). No yielding, local buckling, shear deformation, self-weight or dynamic effects, and the load factor is not a design approval. The end-fixity, initial bow and support choices are extensions of the delivered scenarios. Preset values and captions are those of the original simulations.

Column Buckling and Beam Deflection Under a Rising Load: Euler Stability and Stiffness Case-Study Lab

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