814 



APPLIED MECHANICa 



[TOUSIVE BTREN 



of tin- section referred to, u greater than the number ill 

 a similar portion of the smaller bar, in proportion as 

 the iquare of the one diameter, or total area of the 

 larger, exceed* that of the unaller. Again, in the larger 

 bar the leverage of the aet of fibre* to resist the screw 

 elongation, or the amount of the elongation which tlu-y 

 hare to undergo, exceeds that of the smaller, in propor- 

 tion as the one diameter exceeds the other. Therefore, 

 upon the whole, we must conclude, as before, that the 

 resistance of the larger to torsion, is to that of the 

 smaller, as the cube of the one diameter is to that of the 



otlliT. 



Tno same principles of reasoning apply equally to otl er 

 tlniii circular forms of section. The toraive strengths of 

 square bars follow the sainu law ; as do those of any 

 other forms, provided they bo similar, or have all their 

 corresponding parts proportional. In comparing bars 



Fig. M. 



of circular with those of 

 square section of equal dia- 

 meter (Kig. 94), we readily 

 see that the torsive strength 

 of the square must exceed 

 the other, not only on ac- 

 count of the square having 

 the larger area, and, there- 

 fore, the greater number 

 of fibres, or points of cohe- 

 sion, but, also, because the 

 additional area is situated 

 at places farther from the 

 centre, and, therefore, re- 

 sisting with more leverage than any parts of the circular 

 section. On the other hand, we observe, that while all 

 the fibres of the circular section are supported against 

 being forced aside by those around them, the fibres in 

 the angles of the square section are not so supported, 

 and may be expected to yield more readily to the twist- 

 ing force. Upon the whole, it has been found that the 

 torsive strength of the square section is about one-fourth 

 greater than that of the circular. As the distance of any 

 fibre from the centre contributes much to its resisting 

 power, it may bo advantageous to remove some of the 

 material of a shaft from the inside, where it is of little 

 value to resist torsion, and place it on the outside, 

 where it acts with greatly increased leverage. Un this 

 principle, hollow or tubular shafts present the advantage 

 of economising material, and securing strength with 

 lightness ; for each of the fibres in the ring section of a 

 tube has so much greater leverage than those towards 

 the centre of a shaft, that their number, and consequently 

 the sect onal area of the ring, may bo considerably di- 

 minished without lessening the torsive strength. 



From a number of experiments, made with a view of 

 ascertaining the torsive strength of different materials, 

 we are enabled to compile a table, from which the 

 strength of a shaft of given materials and dimensions 

 may be calculated ; or, conversely, the dimensions may 

 be computed of a shaft destined to sustain a given 

 torsive strain. 



Table of torsive strengths of cylindrical shafts 1 inch in 

 diameter ; weight* acting at 1 fuot leverage. 



N : ! Permanent 



material. load. 



Brm .................... 150 1U. 



Copper ....... ........... 1SJ 



Oun-meul .............. 170 



Iroo (wrought, Kngluh) . . SJJ 

 Iron ( rough!, Swediah).. 230 



( 



man-rial. 

 Lead 



Steel (bli.ter) 

 Steel (cut) 

 Steel (tbcar) 



Tin.......: 



Permanent 

 load. 

 34 Ibi. 

 680 

 80 

 670 



47 ;; 



The table, col. 2, gives the weight in pound* placed 

 at the end of a lever 1 foot long, which a cylindrical 

 shaft, 1 inch in diameter, will permanently sustain 

 without injury; and the following rules embody the 

 method* of computing the torsive strengths of other 

 shaft*, according to their diameter. 



n tlu> iliami't'T of a cylindrical shaft, and the 

 li of lever at which the twisting weiyht acU, to find 

 what weight it will permanently sustain. 



-Multiply the mnnlier in the table three times 

 by the diameter (in inches), and divide by the length of 

 lever (iii feet). 



Example 1. Required the strain that can be applied 

 at the end of a lever 2 feet long to turn a cast-irou shaft 

 4 inches in diameter. 



330 X 4 X 4 X 



10,560 Ibs., nearly 4} tons. 



Ef ample 2. A rope passes round a pulley 3 foot in 

 diameter, fixed to a gun-metal spindle 2i inches in 

 diameter ; required the weight that may be attached to 

 the rope. 



Since the diameter of the pulley is 3 feet, the leverage 

 of the weight is 1 J ft. ; and 



II. When the shaft has a square section. 



Jiule. Add }th to the weight. 



Example. Required the weight that may bo hung to 

 a lever 6 feet long, fixed to a wrought-iron (English) 

 square shaft 2 inches broad. 



335 X 2 x 2 X 2 



= 536 Ibs. 



Add 1 = 134 



670 Ibs., nearly G cwt. 



III. Given the weight and lever, to find the diameter 

 of a cylindrical shaft. 



li\de. Multiply the weight (in Ibs.) by the length of 

 lever (in feet), divide by the number in the table, and 

 extract the cube root for the diameter (in inches). 



Example 4. A cast-steel spindle has to withstand the 

 torsion of 5 tons at the end of a lever 8 feet long : re- 

 quired its diameter. 



11 200 X 8 



5 tons = 11,200 Ibs., and ' 590 - 152 nearly. 



The cube root of 152 is about 5J inches. 



I V. When the shaft is square. 



Rule. Multiply the weight by the lever, divide by 

 the tabular number, deduct Jth of the result, and ex- 

 tract the cube root. 



Example 5. A square shaft of English wrought-iron 

 bos to sustain 6 cwt. at the end of a lever 5 feet long : 

 required the breadth. 



6 cwt. - 672 Ibs. X 5 



Deduct fcth- 2 



8, cube root 2 inches. 



The table and rules apply only to cases where a sii. 

 regular strain is applied. When shafts or spindles are 

 intended to convey motion to machinery, they are gene- 

 rally subjected to great irregularities of torsive strain ; 

 and though they may only convey upon the whole a 

 certain power, yet at different periods of their revolution 

 they may be subjected to strains ranging from nothing 

 up to many times the average strain due to the power 

 they convey. If we consider for a moment the condi- 

 tions under which the crank-shaft of a steam-engine 

 rotates, we shall see the great variation of strain to which 

 it is exposed. At two points of each revolution, what 

 are technically called the dead-centres, the steam power 

 has no effect to turn the shaft round ; but, on the con- 

 trary, the momentum stored up in the fly-wheel turns 

 the shaft, and through it gives movement to the parts of 

 the engine. At two other points, when the connecting- 

 rod of the engine is acting in the most favourable position, 

 the shaft receives a torsive strain through the crank, 

 greater than the total pressure of the steam on the piston ; 

 and the amount of this strain depends on the total 

 steam-pressure on the piston, the obliquity of the con- 

 necting-rod, and the len/th of the crank. But all 

 machinery is besides subject to accidental irre^'nlm. 

 of strain. For instance, in the engines of a steam-vessel 

 propelled by paddle-wheels, sometime* in a heavy sea, 



