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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.
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:
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
A column can buckle due to its own weight with no other direct forces acting on it, in a failure mode called self-buckling. In conventional column buckling problems, the self-weight is often neglected since it is assumed to be small when compared to the applied axial loads .
The elastica theory is a theory of mechanics of solid materials developed by Leonhard Euler that allows for very large scale elastic deflections of structures. Euler (1744) and Jakob Bernoulli developed the theory for elastic lines (yielding the solution known as the elastica curve ) and studied buckling.
Euler's identity is a direct result of Euler's formula, published in his monumental 1748 work of mathematical analysis, Introductio in analysin infinitorum, [16] but it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it.
Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices. These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments ) acting on the rigid body.