The Perry Robertson formula is a mathematical formula which is able to produce a good approximation of buckling

Buckling

In science, buckling is a mathematical instability, leading to a failure mode.Theoretically, buckling is caused by a bifurcation in the solution to the equations of static equilibrium...

 loads in long slender beam

Beam

Beam may refer to:*Beam , a construction element*Beam , the most extreme width of a nautical vessel, or a point alongside the ship at the mid-point of its length*A narrow, propagating stream of particles or energy:...

s, and is the basis for the buckling formulation adopted in EN 1993

EN 1993

EN 1993 - Eurocode 3: Design of steel structures, is the part of the European building code that deals with the design of steel structures, using the limit state design philosophy...

.
The formula in question can be expressed in the following form:



with


where:

• is the average longitudinal stress in the beam's cross section
• is the material's elastic limit
• is the average tension measured in the cross section which correspond to the beam's Euler load
• the amplitude of the initial geometrical imperfection
• distance from the cross section's centroid
  Centroid
  In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
  
   to the section's most stressed fiber
• the section's radius of gyration
  Radius of gyration
  Radius of gyration or gyradius is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the objects' parts from either its center of gravity or an axis....
  
  

Robertson then proposed that , where represents the beam's slenderness.