to these experiments, be 1745, that of the cast steel column 2518, of the column of Dantzic oak 108.8, and of the column of red deal 78.5.

Effect of drying on the strength of columns of timber. — It results from these experiments, that the strength of short columns of wet timber to resist crushing is not one half that of columns of the same dimensions of dry timber.

E

TORSION.

432. The elasticity of torsion.

Let ABCD represent a solid cylinder, one of whose transverse sections AEB is immoveably fixed, and every other displaced in its own plane, about its centre, by the action. of a pressure P applied, at a given distance a from the axis, to the section CD of the cylinder in the plane of that section and round its centre; the cylinder is said, under these circumstances, to be subjected to torsion, and the forces opposed to the alteration of its form, and to its rupture, constitute its resistance to torsion.

Let aabß be any section of the cylinder whose distance from the section AEB is represented by x, and let aß represent that diameter of the section aabß which was parallel to the diameter AB before the torsion commenced; let ab be the projection of the diameter AB upon the section aabß, and let the angle aca be represented by 0.

Now the elastic forces called into action upon the section aabß are in equilibrium with the pressure P. But these elastic forces result from the displacement of the section aabß upon its immediately subjacent section. Moreover, the actual displacement of any small element AK of the section aabß, upon the subjacent section, evidently depends partly upon the angular displacement of the one section upon the other, and partly upon the distance p of the element in ques

tion from the axis of the cylinder. Now the angle aca or is evidently the sum of the angular displacements of all the sections between aabß and AEB upon their subjacent sections; and the angular displacement of each upon its subjacent section is the same, the circumstances affecting the displacement of each being obviously the same: also the number of these sections varies as x, and the sum of their angular displacements is represented by ; therefore the angular displacement of each section upon its subjacent section Ө and the actual displacement of the small element

varies as

Ө

AK of the section aabß varies as P. Now the material being elastic, the pressure which must be applied to this element in order to keep it in this state of displacement varies as the amount of the displacement (Art. 345.), or asp. Let its actual amount, when referred to a unit of surface, be repre

0

sented by G-p, where G is a certain constant dependant for its amount on the elastic qualities of the material, and called the modulus of torsion; then will the force of torsion required to keep the element AK in its state of displacement be represented by G-PAK, and its moment about the axis

0

Ꮖ

of the cylinder by G-PAK. So that the sum of the moments of all such forces of torsion in respect to the whole

section aabß will be represented by G-Zp2AK, or by G-I, if I represent the moment of inertia of the section about the axis of the cylinder. Now these forces are in equilibrium with P; therefore, by the principle of the equality of moments,

If r represent the radius of the cylinder, I=r1 (Art. 85.). Substituting this value, representing by L the whole

length of the cylinder, and by ✪ the angle through which its extreme section CD is displaced or through which OP is made to revolve, called the angle of torsion, and solving in respect to C,

Thus, then, it appears that when the dimensions of the cylinder are given, the angle of torsion varies directly as the pressure P by which the torsion is produced; whence, also, it follows (Art. 97.) that if the cylinder, after having been deflected through any distance, be set free, it will oscillate isochronously about its position of repose, the time T of each oscillation being represented in seconds (equation 76) by the formula

pression (a) represents the length of the path described by the point P from its position of repose, so that the moving force upon the point P, when the pressure producing torsion is removed, varies as the path described by it from its position of repose.

The above is manifestly the theory of Coulomb's Torsion Balance. W represents in the formula the weight of the mass supposed to be carried round by the point P, and the inertia of the cylinder itself is neglected as exceedingly small when compared with the inertia of this weight.

The torsion of rectangular prisms has been made the subject of the profound investigations of MM. Cauchy†, Lamé et Clapeyron, and Poisson.§ It results from these investi

* Illustrations of Mechanics, art. 37.
† Exercises de Mathematique, 4° année.
Crelle's Journal.

§ Mémoires de l'Academie, tome viii.

gations that if 6 and e be taken to represent the sides of the rectangular section of the prism, and the same notation be adopted in other respects as before, then

M. Cauchy has shown the values of the constant G to be related to those of the modulus of elasticity E by the formula

G=E..... (693).

In using the values of G deduced by this formula from the table of moduli of elasticity, all the dimensions must be taken in inches, and the weights in pounds.

433. ELASTICITY OF TORSION IN A SOLID HAVING A CIRCULAR SECTION OF VARIABLE DIMENSIONS.

Let ab represent an element of the solid contained

by planes, perpendicular to the axis, whose distance from one another is represented by the exceedingly small increment Ax of the distance of the section ab from the fixed section AB, and let its radius be represented by y; and suppose the whole of the solid except this single element to become rigid, a supposition by which the conditions of the equilibrium of this particular element will remain unchanged, the pressure P remaining the same, and being that which produces the torsion of this single element. Whence, representing by A the angle of torsion of this element, and considering it a cylinder whose length is Ar, we have by equation (689), substituting for I its value

Passing to the limit, and integrating between the limits 0 and

* Navier, Resumé des Leçons, &c. Art. 159.

L, observing that at the former limit =0, and at the latter

If the sides AC and BD of the solid be straight lines, its form being that of a truncated cone, and if r, and r, represent its diameters AB and CD respectively; then

434. THE RUPTURE OF A CYLINDER BY TORSION.

It is evident that rupture will first take place in respect to those elements of the cylinder which are nearest to its surface, the displacement of each section upon its subjacent section being greatest about those points which are nearest to its circumference. If, therefore, we represent by T the pressure per square inch which will cause rupture by the sliding of any section of the mass upon its contiguous section*, then will T represent the resistance of torsion per square inch of the section, at the distance r from the axis, at the instant when rupture is upon the point of taking place, the radius of the cylinder being represented by r. Whence it follows that the displacement, and therefore the resistance to torsion per square inch of the section, at any other distance p from the axis, will be represented at that distance by T,

Τρ the

*Or the pressure per square inch necessary to shear it across (Art. 406.).