284 BENDING OF RODS. [ART. 32

the summit is gradually increased, the column will remain erect, without bending, until the weight becomes nearly equal to a certain quantity depending on the flexibility and dimensions of the column.

Since the constant K is equal to Eco& (Art. 13) it follows that the bending weight, for columns of the same kind, varies as the fourth power of the diameter directly, and as the square of the length inversely. This result is usually called Euler's* law.

Columns yield under pressure in two ways, first the materials may he crushed, and secondly the column may bend and then break across. In some cases both effects may occur at once. If the column is short it follows from Euler's law that the bending weight is large, so that short columns yield by crushing. Long columns on the other hand break by bending and are not crushed.

Many experiments have been made to test the truth of Euler's law. The results have not been altogether confirmatory, possibly because Euler's law applies only to uniform thin columns, in which the central line in the unstrained state is a vertical straight line. For the details of these experiments we must refer the reader to works on engineering. See also Mr Hodgkinson's Experimental researches on the strength of pillars, Phil. Trans. 1840.

In this investigation we have supposed that the weight has been placed centri-eally over the axis of the column. The weight of the column itself has also been neglected and no allowance has been made for the shortening of the column due to the weight it has to support.

32. Heavy columns. Ex. 1. A vertical column in the form of a paraboloid of latus rectum 4m with its vertex upwards is fixed in the ground. Show that it will bend under its own weight when slightly displaced if the length be greater than TT (2Em/i0)4, where w is the weight of a unit of volume, E the weight which would stretch a bar of the same material and unit area to twice its natural length.

Ex. 2. A vertical cylindrical column of radius r is fixed in the ground. Show

that it will bend under its own weight if its length be greater than c3 ( r-pr— V >

\jibw/ where c is the least root of J_j (c) = 0.

Let A be the area, r the radius of a section of the column (supposed to be thin and straight) at a distance x from the base (7, then (Art. 13), K=EAh*. When the

* E tiler, Berlin Memoirs, 1757. Petersburg Commentaries, 1778. Lagrange, Acad. de Berlin, 1769. Poisson, Trait€ de Mecanique, 1833. See also Thomson and Tait, vol. i. Art. 611, where some figures are given. Also the Proceedings of the Roy. Irish Acad. 1873, where Sir E. Ball notes an error in Poisson's analysis. In the Proc. London Math. Soc. 1893, vol. xxiv., Prof. Love discusses the stability of columns. A discussion of Euler's theory is contributed to the Canadian Society of Civil Engineers, 1890, by C. F. Eindlay, C.B.