Incoprea Research · Supplied Round 3 · 30 October 2026

A supplied Euler calculation: an ideal column’s critical stress crosses a chosen reference

Evidence status: Passed Euler stress arithmetic with a narrowed mechanical interpretation; actual limb failure order and maximum length are untested.

The supplied calculation gives an Euler critical stress equal to 120 MPa at slenderness approximately 38.4765 when Young’s modulus is assumed to be 18 GPa. That arithmetic is correct for an ideal pinned column. It does not establish that an actual limb buckles before bending failure, because the calculation contains no applied bending moment, limb geometry or comparison of the relevant loads.

What the posted hypothesis asks

The limb-design hypothesis proposes that a ratio of bending strength to buckling load sets maximum limb length, so slender limbs fail by buckling before bending. The supplied script compares a calculated axial buckling stress with a fixed material stress value. That is a narrower comparison than the hypothesis’s loading and biological claims.

To preserve the verified part, this note treats 120 MPa as a chosen reference stress. The source calls it cortical-bone bending yield strength, but those material values were not measured or independently validated for a specific specimen. They are inputs, not results of this test.

The Euler relation and its assumptions

For a perfectly straight, uniform, linearly elastic column with pinned ends and centered axial compression, the classical critical load is P_cr=π²EI/L². Here E is elastic modulus, I is the second moment of area and L is the pinned span. With area A and radius of gyration r_g=√(I/A), define slenderness λ=L/r_g.

Dividing load by area gives σ_cr=π²E/λ². This is the source script’s formula. Its implicit effective-length factor is one, corresponding to the stated pinned-end interpretation; other end restraints change the critical load. The formula is an elastic stability model, not a universal material-failure equation.

The verified crossover

Setting σ_cr equal to a reference σ_ref gives λ*=π√(E/σ_ref). With E=18×10⁹ Pa and σ_ref=120×10⁶ Pa, their ratio is 150 and λ*=π√150≈38.4765. Slenderness is dimensionless because both L and r_g are lengths, and the modulus-to-stress ratio is also dimensionless.

QuantitySupplied value or relation
Elastic modulus18 GPa, assumed
Reference stress120 MPa, assumed
Critical axial stressπ²E/λ²
Stress crossoverλ≈38.4765
Effect of doubling slendernessOne quarter of the Euler stress

The script evaluates 400 slenderness settings from 5 to 120. Euler stress ranges from about 7,106 MPa to 12.337 MPa on that grid. The large stresses at short slenderness are formal formula outputs; they are not evidence that a real material remains elastic or survives to those stresses.

Why bending failure is not decided

Axial critical stress and bending stress are different loading quantities. Comparing Euler stress numerically with a material value does not compute the bending moment that would cause yield. To decide which physical event occurs first, one must specify the applied force and moment, section geometry, restraints, imperfections and the material’s behavior under those conditions.

No such load history or bending moment appears in the supplied code. Nor does slenderness alone determine a maximum length without a cross-sectional scale and loading specification. A real limb adds taper, curvature, anisotropy, joints, muscle forces and varying loads that this ideal column does not represent.

Established mechanics and provenance

David Parks’s MIT mechanics notes give established Euler-column context, including the importance of boundary conditions and the distinction between buckling and yield. The calculation here illustrates that classical relation; it is not a novel buckling law or a validated biological failure prediction.

The supplied DeepSeek-assisted Round 3 script and its numeric outputs reproduced byte-for-byte in an isolated copy. Codex checked units and the mechanical scope and prepared this corrected publication. No new specimen, experiment, material estimate or loading model was introduced. Grid settings are not measured limbs, and repeated equal limits in the source JSON are not statistical confidence intervals. The unchanged hypothesis remains at Incoprea Research. No external peer review is claimed.