SECTION 8.-Summary of the Principles of the Strength of Machines.

58. Nature and Division of the Subject. (A. M., 244).—The present section contains a very brief summary of the application of the principles of the strength of materials to the most simple questions which arise in designing machines. The rules are given without demonstration, in as small compass as possible, in order to save the necessity of referring, in ordinary cases, to more bulky treatises; and are almost all abstracted and abridged from a treatise on Applied Mechanics, Part II., Chapter III.

The load, or combination of external forces, which is applied to any piece, moving or fixed, in a machine, produces stress amongst the particles of that piece, being the combination of forces which they exert in resisting the tendency of the load to disfigure and break the piece, which is accompanied by strain, or alteration of the volumes and figures of the whole piece, and of each of its particles. If the load is continually increased, it at length produces either fracture, or (if the material is very tough and ductile) such a disfigurement of the piece as is practically equivalent to fracture, by rendering the piece useless.

The Ultimate Strength of a body is the load required to produce fracture in some specified way. The Proof Strength is the load required to produce the greatest strain of a specific kind consistent with safety; that is, with the retention of the strength of the material unimpaired. A load exceeding the proof strength of the body, although it may not produce instant fracture, produces fracture eventually by long-continued application and frequent repetition.

The Working Load on each piece of a machine is made less than the proof strength in a certain ratio determined by practical experience, in order to provide for unforeseen contingencies.

Each solid has as many different kinds of strength as there are different ways in which it can be strained or broken, as shown in the following classification :

59. Factors of Safety.-A factor of safety is the ratio in which the load that is just sufficient to overcome instantly the strength of a piece of material is greater than the greatest safe ordinary working load.

The following table gives examples of the values of those factors which occur in machines:

The great factor of safety, 40, is for shafts in millwork which transmit very variable efforts.

Almost all the experiments hitherto made on the strength of materials give the ultimate strength only. In using those data for the designing of structures and machines, the most convenient process of calculation is to multiply the intended working load of a piece by the proper factor, so as to find the breaking load, and to make the ultimate strength of the piece equal to that breaking load.

60. The Proof or Testing by experiment of the strength of a piece of material is to be conducted in two different ways, according to the object in view.

I. If the piece is to be afterwards used, the testing load must be so limited that there shall be no possibility of its impairing the strength of the piece; that is, it must not exceed the proof strength, being from one-third to one-half of the ultimate strength. About double of the working load is in general sufficient. Care should be taken to avoid vibrations and shocks when the testing load approaches near to the proof strength.

II. If the piece is to be sacrificed for the sake of ascertaining the strength of the material, the load is to be increased by degrees until the piece breaks, care being taken, especially when the breaking point is approached, to increase the load by small quantities at a time, so as to get a sufficiently precise result.

The proof strength requires much more time and trouble for its determination than the ultimate strength. One mode of approxiinating to the proof strength of a piece is to apply a moderate load and remove it, apply the same load again and remove it, two or three times in succession, observing at each time of application of the load, the strain or alteration of figure of the piece when loaded, by stretching, compression, bending, distortion, or twisting, as the case may be. If that alteration does not sensibly increase by repeated applications of the same load, the load is within the limit of proof strength. The effects of a greater and a greater load being successively tested in the same way, a load will at length be reached whose successive applications produce increasing disfigurements of the piece; and this load will be greater than the proof

F

strength, which will lie between the last load and the last load but one in the series of experiments.

It was formerly supposed that the production of a set, that is, a disfigurement which continues after the removal of the load, was a test of the proof strength being exceeded; but Mr. Hodgkinson showed that supposition to be erroneous, by proving that in most materials a set is produced by almost any load, how small

soever.

The strength of bars and beams to resist breaking across, and of axles to resist twisting, can be tested by the application of known weights either directly or through a lever.

To test the tenacity of rods, chains, and ropes, and the resistance of pillars to crushing, more powerful and complex mechanism is required. The apparatus most commonly employed is the hydraulic press. In computing the stress which it produces, no reliance ought to be placed on the load on the safety valve, or on a weight hung to the pump handle, as indicating the intensity of the pressure, which should be ascertained by means of a pressure gauge. This remark applies also to the proving of boilers by water pressure. From experiments by Messrs. Hick and Lüthy it appears that, in calculating the stress produced on a bar by means of a hydraulic press, the friction of the collar may be allowed for by deducting a force equivalent to the pressure of the water upon an area of a length equal to the circumference of the collar, and one-eightieth of an inch broad.

The measurement of tension and compression by means of the hydraulic press is but a rough approximation at the best. It may be sufficient for an immediate practical purpose; but for the exact determination of general laws, although the load may be applied at one end of the piece to be tested by means of a hydraulic press, it ought to be resisted and measured at the other end by means of a combination of levers such as that used at Woolwich Dockyard, and described by Mr. Barlow.

61. Tenacity (A. M., 265, 268, 269).-The ultimate strength or breaking load of a bar exposed to direct and uniform tension is the product of the area of cross-section of the bar into the tenacity of the material. Therefore, let

P denote the breaking load, in pounds;
S the area of section, in square inches;

f the tenacity, in pounds on the square inch; then

The following are some of the most useful values of the tenacity of materials used in machinery, in lbs. on the square inch :

per pound weight to the fathom,..... 4,480

Leathern belts, working tension,...

.285*

62. Cylindrical Boilers and Pipes.-Let r denote the radius of a thin hollow cylinder, such as the shell of a high pressure boiler; t the thickness of the shell;

f the tenacity of the material, in pounds per square inch;

p the intensity of the pressure, in pounds per square inch, required to burst the shell. This ought to be taken at SIX TIMES the effective working pressure-effective pressure meaning the excess of the pressure from within above the pressure from without, which last is usually the atmospheric pressure, of 147 lbs. on the square inch or thereabouts.

Then

ft

p

.(1.)

and the proper proportion of thickness to radius is given by the formula

The tenacity of good wrought iron boiler plates has been stated as 51,000 lbs. per square inch. That of a double rivetted joint, per square inch of the iron left between the rivet holes, is the same; that of a single rivetted joint somewhat less, as the tension is not uniformly distributed. It is convenient in practice to state the tenacity of rivetted joints in lbs. per square inch of the entire plate; and it is so stated in the annexed table, in which the results for rivetted joints are from the experiments of Mr. Fairbairn, and that for a welded joint from an experiment by Mr. Dunn. The joints of plate iron boilers are single rivetted; but from the manner in which the plates break joint, analogous to the bond in masonry, the tenacity of such boilers is considered to approach more nearly to that of a double rivetted joint than that of a single rivetted joint.

Wrought iron plate joints, double rivetted, the diameter of each hole being of the distance from

Wrought iron retort, with a welded joint,........

30,750

Cast iron boilers, cylinders, and pipes (average

British iron),.........

16,500

63. Spherical Shells, such as the ends of "egg-ended” cylindrical boilers, the tops of steam domes, &c., are twice as strong as cylindrical shells of the same radius and thickness.

Suppose a shell of the figure of a segment of a sphere to have a circular flange round its base, through which it is bolted to a flange upon a cylindrical shell, or upon another spherical shell.

Let r denote the radius of the sphere, in inches;

r', the radius of the circular base of the segmental shell, in inches; p, the bursting pressure, in lbs. on the square inch;

then the number and dimensions of the bolts by which the flange is held should be such, that the load required to tear them asunder all at once shall be

and the flange itself should require, in order to crush it, the following thrust in the direction of a tangent to it :

If the segment is a complete hemisphere, r, and the last

=

expression becomes

0.