This simulator loads a slender compression specimen in a test frame so you can change its end restraints, unsupported length and weak-axis section to compare the ideal elastic Euler critical load against imperfection amplification as the applied compression approaches that threshold.
• A real-time 3D cutaway workbench with a compression specimen, pin/fixed base and top restraints, a rigid reaction frame, a load cell, a lateral displacement probe and a section coupon, with home view, focus-selected-part, toggleable full enclosure, exploded view, auto-rotate, expand and show/hide labels controls. • Adjustable unsupported column length (1-6 m), section width (80-160 mm), weak-direction section depth (20-80 mm), elastic modulus (70-210 GPa), target compression force (0-1000 kN), ideal end restraints (pinned-pinned K=1, fixed-fixed K=0.5, fixed-free/cantilever K=2), initial mode-shaped crookedness (0-10 mm) and selected elastic stress limit (100-400 MPa). • Playback/loading actions: ramp from zero to target, apply full target load, and release compression. • A Curves & measurements tab with an imperfection-amplification-below-Euler-load chart, the underlying model equations (A, I, r, Pcr, slenderness, amplification, stress indicator, mode shapes) and snapshot readouts (applied compression, Euler load, P/Pcr ratio, effective-length factor K, slenderness, Euler stress, axial stress, amplification, displacement, peak stress indicator). • An Experiments tab with four guided fixtures (pinned reference specimen, fixed ends, longer unsupported member, ramp toward instability) and a model verification bench that runs independent deterministic checks against a fresh model without disturbing your live trial, plus a timestamped event log and a copyable trial report. • A Learn & assess tab with guided lessons on effective length, instability versus material limits, imperfection amplification and boundary-condition effects, a knowledge-check quiz with reset, and a written model-scope statement with a technical reference link.
The Euler critical load is Pcr = π²EI/(KL)², where K is the effective-length factor set by the end restraints: K = 1 for pinned-pinned, K = 0.5 for ideal fixed-fixed, and K = 2 for a fixed-free (cantilever) column. Because Pcr scales with 1/(KL)², doubling the unsupported length divides the critical load by four, and switching from pinned to fixed-fixed ends quadruples it.
Below the threshold, an initially mode-shaped imperfection δ₀ is amplified according to δ = δ₀/(1 − P/Pcr) — a first-mode linear approximation, not a full nonlinear post-buckling solution. As P approaches Pcr, amplification grows without bound, and at or above the threshold the simulator switches to an illustrative eigenmode shape rather than inventing a finite stable post-buckling displacement.
Euler theory assumes purely elastic response. If the computed Euler stress Pcr/A exceeds the selected elastic stress limit, the ideal formula alone cannot predict the column's real failure mode — inelastic buckling or yielding may govern instead, and the simulator flags this condition without solving it.
This is an ideal Euler column model: weak-axis bending, constant EI, ideal end restraints, and a first-mode-shaped initial crookedness. It excludes inelastic buckling, residual stress, connection flexibility, self-weight, an eccentric-load solution, design safety factors and true post-buckling equilibrium; the stress indicator is approximate for general restraints, and above the critical threshold the drawn amplitude is illustrative only.
It divides the Euler critical load by four, since Pcr = π²EI/(KL)² is inversely proportional to the square of the effective length KL.
End restraints change the effective-length factor K, which appears squared in the Euler formula. An ideal fixed-fixed column (K = 0.5) has four times the Euler load of a pinned-pinned column (K = 1) of the same length, while a fixed-free cantilever (K = 2) has only a quarter of the pinned value.
No. Once the applied load reaches or exceeds the Euler critical load, the linear elastic model can no longer determine a finite stable post-buckling amplitude, so the simulator displays an illustrative eigenmode shape rather than a real collapse deformation.
No. It is an ideal Euler buckling model with constant EI and ideal end restraints that excludes inelastic buckling, residual stress, connection flexibility, self-weight, load eccentricity and design safety factors. A qualified structural engineer must perform any real column design.