Euler's critical load

*Fig. 1: Critical stress vs slenderness ratio for steel, for E = 200 GPa, yield strength = 240 MPa*.

Euler's critical load or Euler's buckling load is the compressive load at which a slender column will suddenly bend or buckle. It is given by the formula:^[1]

$P_{cr}={\frac {\pi ^{2}EI}{(KL)^{2}}}$

where

$P_{cr}$ , Euler's critical load (longitudinal compression load on column),
$E$ , Young's modulus of the column material,
$I$ , second moment of area of the cross section of the column (area moment of inertia),
$L$ , unsupported length of column,
$K$ , column effective length factor

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 critical load puts the column in a state of unstable equilibrium. A load beyond the critical load causes the column to fail by buckling. As the load is increased beyond the critical load the lateral deflections increase, until it may fail in other modes such as yielding of the material. Loading of columns beyond the critical load are not addressed in this article.

Around 1900, J. B. Johnson showed that at low slenderness ratios an alternative formula should be used.

^ "Column Buckling | MechaniCalc". mechanicalc.com. Retrieved 2020-12-27.
^ Euler, Leonard (1744). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti [A method of finding curved lines enjoying the maximum-minimum property, or the solution of the isoperimetric problem in the broadest sense] (in Latin). Geneva, Switzerland: Marc Michel Bousquet et Cie. pp. 267–268. From pp. 267-268: "37. Quae ante de specie prima sunt annotata inservire possunt viribus columnarum dijudicandis. […] contra vero si pondus P fuerit majus, columna incurvationi resistere non poterit." (37. Those [things] which have been noted before about the first type can serve in judging the strength of columns. So let the column AB be placed vertically on the base A, bearing the load P. For if the column is already arranged in such a way that it cannot slide [away] from the load P, [then] if [the load] will have been too great, there will be nothing else to fear except the bending of the column; in this case, therefore, the column may be regarded as endowed with elasticity. Therefore let the absolute elasticity of the column = Ekk, and its height AB = 2f = a; and [in] §25 above [p. 261] we have seen that the required bending force on this column or the minimum [load that is required to bend this column] = ππ Ekk/4ff = ππ Ekk/aa. Thus, unless the load P being carried is greater than Ekk/aa, absolutely no bending will have to be feared; but on the other hand, if the load P will have been greater, the column will not be able to resist bending.)

[1] "Column Buckling | MechaniCalc". mechanicalc.com. Retrieved 2020-12-27.

[2] Euler, Leonard (1744). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti [A method of finding curved lines enjoying the maximum-minimum property, or the solution of the isoperimetric problem in the broadest sense] (in Latin). Geneva, Switzerland: Marc Michel Bousquet et Cie. pp. 267–268. From pp. 267-268: "37. Quae ante de specie prima sunt annotata inservire possunt viribus columnarum dijudicandis. […] contra vero si pondus P fuerit majus, columna incurvationi resistere non poterit." (37. Those [things] which have been noted before about the first type can serve in judging the strength of columns. So let the column AB be placed vertically on the base A, bearing the load P. For if the column is already arranged in such a way that it cannot slide [away] from the load P, [then] if [the load] will have been too great, there will be nothing else to fear except the bending of the column; in this case, therefore, the column may be regarded as endowed with elasticity. Therefore let the absolute elasticity of the column = Ekk, and its height AB = 2f = a; and [in] §25 above [p. 261] we have seen that the required bending force on this column or the minimum [load that is required to bend this column] = ππ Ekk/4ff = ππ Ekk/aa. Thus, unless the load P being carried is greater than Ekk/aa, absolutely no bending will have to be feared; but on the other hand, if the load P will have been greater, the column will not be able to resist bending.)

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