Euler's burden

The Euler load is in the appendix on Elastic curves indicated the 1744 published work (E065). Leonhard Euler calculates nine different classes of bending lines of straight bars that are only loaded at the ends and made of ideal linear-elastic material with small or large deformations or curvatures. ${\displaystyle P_{Euler}}$

The first class includes bars that are bent or warped in an infinitesimal sinusoidal shape by the action of a load or that remain straight. For them, the simplification can be made that the rod length is equal to the length of the straight line connecting the rod ends even in the deformed state. ${\displaystyle P}$ ${\displaystyle s}$

To the bending line of the rod

${\displaystyle w(x)=w\cdot \sin(\pi \cdot x/s)}$

belongs to the curvature of the rod

${\displaystyle 1/r(x)=w(x)''=-\pi ^{2}/s^{2}\cdot w\cdot \sin(\pi \cdot x/s).}$

From the curvature of the rod, the bending stiffness results in the resisting bending moment ${\displaystyle EI}$

${\displaystyle M_{R}(x)=-EI/r(x)=-EI\cdot w(x)''.}$

The load causes the bending moment in the member ${\displaystyle P}$

${\displaystyle M_{E}(x)=P\cdot w(x).}$

In the equilibrium state of the rod,, the relationship holds ${\displaystyle M_{E}(x)=M_{R}(x)}$

${\displaystyle P\cdot w\cdot \sin(\pi \cdot x/s)=EI\cdot \pi ^{2}/s^{2}\cdot w\cdot \sin(\pi \cdot x/s)}$.

It is fulfilled with for loads of any size and with for ${\displaystyle w=0}$ ${\displaystyle P}$ ${\displaystyle w>0}$

${\displaystyle P=EI\cdot \pi ^{2}/s^{2}=P_{Euler}.}$

After translation of Truesdell and the used there names and Euler has found this fact in § 25 and described as follows: ... the force required to produce this infinitely small curvature of the band is a finite quantity, P = ¼ pi² • B / f² . That is, if the ends A and B are tied together by a thread AB, then this thread is pulled by the force ¼ pi² • B / f². ${\displaystyle EI=B}$ ${\displaystyle s=2\,f}$

Euler uses this knowledge later in § 37 to specify the load-bearing capacity of a loaded column of the length . ... If the weight P is not too great, then the most to be feared is a bending of the column. And further: … we have seen that the force needed to bend the column ever so little is pi² • B / s². Therefore unless the weight P to be supported satisfies P> pi² • B / s² there is no fear of bending… . ${\displaystyle P}$ ${\displaystyle l=s}$

Today, Euler's load is understood as the buckling load of the column articulated on both sides (Euler case 2), at which the stability failure of the column occurs. ${\displaystyle P_{Euler}}$

The nine classes of bending lines must not be confused with Euler's buckling cases.

See also

• Beam theory

literature

• Leonhard Euler: About the elastic curves . 1744
  • Translation into German in: H. Linsenbarth: Treatises on the equilibrium and the oscillations of the plane elastic curves . Leipzig, 1910, Ostwald's classic.
• Clifford Truesdell : The Rational Mechanics of Flexible or Elastic Bodies, 1638-1788 . Introduction to Leonhardi Euleri Opera Omnia Vol. X et XI Seriei secundae. Füssli, Birkhäuser, Zurich, Basel, 1960, ISBN 978-3-7643-1441-5 .