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This formula was derived in 1744 by the Swiss mathematician Leonhard Euler. [2] The column will remain straight for loads less than the critical load. The critical load is the greatest load that will not cause lateral deflection (buckling). For loads greater than the critical load, the column will deflect laterally.
The starting point is the relation from Euler-Bernoulli beam theory = Where is the deflection and is the bending moment. This equation [7] is simpler than the fourth-order beam equation and can be integrated twice to find if the value of as a function of is known.
Johnson's formula interpolates between the yield stress of the column material and the critical stress given by Euler's formula. It creates a new failure border by fitting a parabola to the graph of failure for Euler buckling using = () There is a transition point on the graph of the Euler curve, located at the critical slenderness ratio.
The theory of the behavior of columns was investigated in 1757 by mathematician Leonhard Euler. He derived the formula, termed Euler's critical load, that gives the maximum axial load that a long, slender, ideal column can carry without buckling. An ideal column is one that is:
These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification = ˙ ˙. We conclude that the function ψ {\displaystyle \psi } is the value of the minimizing integral A {\displaystyle A} as a function of the upper end point.
Initially created for stability problems in column buckling, the Southwell method has also been used to determine critical loads in frame and plate buckling experiments. The method is particularly useful for field tests of structures that are likely to be damaged by applying loads near the critical load and beyond, such as reinforced concrete ...
In Itô calculus, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations named after Leonhard Euler and Gisiro Maruyama. The ...
The Euler equations were among the first partial differential equations to be written down, after the wave equation. In Euler's original work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible flow.