Math 2.0

Mechanics of deformable solids Static problems and stability Proposition

Perry–Robertson formula

Builds on Euler's critical loadEuler–Bernoulli beam equation

Statement

Let a pin-ended linear elastic column have length LL, cross-section area AA, radius of gyration rr and extreme-fibre distance cc. Let it have an initial crookedness w0(x)=a0sin⁡(πx/L)w_0(x) = a_0 \sin(\pi x / L) with a0≥0a_0 \ge 0. Let σE=π2E/(L/r)2\sigma_E = \pi^2 E / (L/r)^2 be its Euler critical stress, let σy\sigma_y be the yield stress, and let η=a0c/r2\eta = a_0 c / r^2. Assume the column fails when the largest compressive stress first reaches σy\sigma_y. Then the mean axial stress σ=P/A\sigma = P/A at failure is σ=σy+(1+η)σE2−(σy+(1+η)σE2)2−σyσE.\sigma = \frac{\sigma_y + (1 + \eta)\sigma_E}{2} - \sqrt{\left( \frac{\sigma_y + (1 + \eta)\sigma_E}{2} \right)^2 - \sigma_y \sigma_E}.

Remarks

Under an axial load P<PEP < P_E the linearized beam equation amplifies the initial bow to the total amplitude a0/(1−P/PE)a_0 / (1 - P/P_E), where PE=σEAP_E = \sigma_E A. The bending moment at mid-length is Pa0/(1−σ/σE)P a_0 / (1 - \sigma/\sigma_E), so the largest stress is σ(1+η/(1−σ/σE))\sigma \left(1 + \eta / (1 - \sigma/\sigma_E)\right). Setting it equal to σy\sigma_y gives the quadratic (σy−σ)(σE−σ)=ησσE(\sigma_y - \sigma)(\sigma_E - \sigma) = \eta \sigma \sigma_E. The failure stress is its smaller root, which is the formula above.

For η=0\eta = 0 the formula gives σ=min⁡(σy,σE)\sigma = \min(\sigma_y, \sigma_E), the larger root of the quadratic being discarded, so it reduces to the yield and Euler limits. For η>0\eta > 0 the failure stress lies below both σy\sigma_y and σE\sigma_E. In design practice η\eta is chosen as a function of the slenderness ratio to represent unavoidable imperfections.

Going deeper

For a student

The formula is the classic way to turn Euler's critical load into a design rule. It replaces the bifurcation of a perfect column by a load–deflection curve of an imperfect one, the same idea that Koiter's theory makes systematic for general structures. Compared with Johnson's parabolic formula, it has a mechanical derivation and a single parameter η\eta that measures imperfection.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Buckling, concept, no evidence levelBucklingconceptEuler–Bernoulli beam equation, concept, no evidence levelEuler–Bernoulli beam …conceptSecond moment of area, concept, no evidence levelSecond moment of areaconceptYoung's modulus, concept, no evidence levelYoung's modulusconceptEuler's critical load, statement, no evidence levelEuler's critical loadstatementPerry–Robertson formula, statement, no evidence levelPerry–Robertson formu…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Buckling concept
  2. Euler–Bernoulli beam equation concept
  3. Second moment of area concept
  4. Young's modulus concept
  5. Euler's critical load statement
  6. Perry–Robertson formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Perry%E2%80%93Robertson_formula
About this page
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AboutBuckling
UsesEuler's critical load, Euler–Bernoulli beam equation
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Changed3 November 2026

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