RESISTANCE TO TORSION.] 



APPLIED MECHANICS. 



813 



Kg. 90. 



of two separate plates put artificially together, we may 

 conceive, that at any 

 place where a solid 

 body might be shorn 

 across, the particles of 

 the body, or the grains 

 of which it is com- 

 posed, fit into the in- 

 terstices of each other; 

 and the two parts into 

 which the body may 

 be divided or shorn 

 are held firmly to- 

 gether by the cohesion of the particles over the whole 

 surface where the division takes place (Fig. 90). In 

 order to effect a separation by pressing the body in oppo- 

 site directions, as with shears, we should have to over- 

 come the cohesion, and the resistance which the asperities 

 present to the slip across the rough surface of separation. 

 The actual amount of cohesive attraction and the resist- 

 ance from asperity, must mainly depend upon the peculiar 

 nature of the material acted on ; but the relative amount 

 of resistance which two pieces of the same material of 

 different dimensions present, seems to depend simply 

 upon the number of particles separated ; that Ls to say, 

 on the area of the section or quantity of surface sepa- 

 rated. Thus, to separate crosswise, a piece of iron 2 

 inches square, we should expect to apply four times as 

 niii'-.'i force as would be required to separate a piece of 

 the =ame iron 1 inch square, because the piece 2 inches 

 square has an area of section, or a surface of separation, 

 four times as great as the piece 1 inch square. But, 

 farther, if we suppose the separation to be effected by 

 means of shears (Fig. 91), so arranged that the 2-inch 



Fig. 91. 



piece is placed twice as far from the pivot or fulcrum of 

 the lever, of which the shears form the arms, as the 

 1-inch piece, we should have to apply double the force to 

 divide the 2-inch piece, on account of the disadvantage 

 of leverage ; and, on the whole, in such a case we should 

 require eight times the force to divide the 2-inch piece 

 that we require for the 1-inch piece. If now, instead of 

 Fig. 02. shearing across a 



square bar, we apply 

 this principle to the 

 fracture of a cylin- 

 drical bar or shaft, 

 by torsion, the centre 

 of the shaft be- 

 comes the fulcrum of 

 i. the lever, which we 

 may assume to be 

 one foot long ; and 

 each portion of the sectional area of the shaft resists 

 separation with a force proportional to its area, and to 

 its leverage or distance from the centre C (Fig. 92). If 

 we take any small square of the section, such as A, and 

 suppose a ring traced inclosing it, it is clear that the re- 

 sistance of every part of that ring of particles, of the 

 same magnitude as A, offers the same resistance, because 

 it has the same leverage, or is equally distant from the 

 centre ; and that, therefore, the total resistance of the 

 ring is as its area multiplied by its leverage A C. But 



if the ring be very narrow, its area is nearly its circum- 

 ference multiplied by its breadth, and its circumference 

 is as its radius A C ; therefore its resistance is as its 

 breadth multiplied by its radius twice, or the square of 

 its radius. 



Were we to suppose, then, the whole circle divided 

 into numerous narrow rings of equal breadth, the resist- 

 ance of the whole to separation by torsion would be 

 the sum of the resistances of all the rings, the resistance 

 of each being as its breadth multiplied by the square of 

 its radius. Now, if we compare a circle of one radius, 

 such as 12 ins., with another, such as 6 ins. (Fig. 93), 

 dividing each radius into an equal number of parts, such 



Fig. 93. 



as 4, and tracing the rings marking them, for the sake of 

 distinctness, by large and small letters respectively, we 

 find that the breadth of ring A is double that of A, and 

 that the radius of A is also double that of A ; therefore 

 the resistance of A to torsion is 2 X 2 X 2, or 8 times 

 that of A. So we find the resistance of B to be 8 times 

 that of B, and so on ; therefore the resistance of the 

 whole of the larger circle is 8 times that of the smaller. 

 Were the radius of the one 3 times that of the other, we 

 should find, by a like process, that its resistance would 

 be 3 X 3 X 3, or 27 times ; and, generally, that the 

 resistance of any cylindrical shaft to torsion is as the 

 cube of its radius, or of its diameter (which is the 

 double of its radius). 



This general proposition is almost self-evident ; for if 

 we admit that the resistance of any part of the circular 

 section is as its area multiplied by the leverage at which 

 it acts, the resistance of the whole circular section must 

 be as its area multiplied by the mean or average leverage, 

 of all its parts. The areas of circles are as the squares of 

 their diameters, and in every circle, the mean leverage is 

 proportional to the diameter ; therefore the resistance to 

 torsion is as the area or square of the diameter multiplied 

 by the diameter that is to say, as the cube of the 

 diameter. 



The reasoning we have used is founded on the assump- 

 tion that the effect of excessive torsion is to divide or 

 shear a shaft across at some place where the material is 

 accidentally weaker than elsewhere. But in shafts of 

 uniform strength throughout their length, and of librous 

 texture, this is by no means the effect of destructive 

 torsion, ilr a wrought-iron shaft, for instance, the 

 fibres are made to twist, so that, in some cases, the 

 outer surface presents the screw-form of a cord or rope ; 

 and ultimately they separate and present a fracture 

 where parts seem to have been dragged out of their 

 place by this twisting action. In producing such a twist 

 of fibres, there must be a considerable elongation, par- 

 ticularly of the outer fibres ; for it is longer round the 

 thread of a screw than in a direct line from one end of a 

 bar to the other ; and the greater the diameter of the 

 thread, or the farther it is from the centre, the longer is 

 its course. If, then, we confine our attention to any set 

 of fibres similarly situated, in two bars of different dia- 

 meters subjected to torsion, we see, in the first place, 

 that in the larger bar the number of fibres in the portiou 



