Part 12

The wheel A is turned by the pinion B which contains 25 teeth. In the actual use of the crane, the axle which carries this pinion would be turned round by means of a handle; but for the purpose of experiments upon the relation of the power to the load, the handle would be inconvenient, and therefore we have placed upon the axle of the pinion a wheel C containing a groove in its circumference. Around this groove a string is wrapped, so that when a weight G is suspended from the string it will cause the wheel to revolve. This weight G will constitute the power by which the load may be raised.

333. Let us compute the velocity ratio of this machine before commencing experiments upon its mechanical efficiency. The effective circumference of the barrel D is found by trial to be 14"·9. Since there are 200 teeth on A and 25 on B, it follows that the pinion B must revolve eight times to produce one revolution of the barrel. Hence the wheel C at the circumference of which the power is applied must also revolve eight times for one revolution of the barrel. The effective circumference of C is 43"; the power must therefore have been applied through 8 × 43" = 344", in order to raise the load 15"·9. The velocity ratio is 344 ÷ 14·9 = 23 very nearly. We can easily verify this value of the velocity ratio by actually raising the load 1', when it appears that the number of revolutions of the wheel B is such that the power must have moved 23'.

334. The mechanical efficiency is to be found as usual by trial. 56 lbs. placed at F is raised by 3·1 lbs. at G; hence the mechanical efficiency deduced from this experiment is 56 ÷ 3·1 = 18. The percentage of useful effect is easily shown to be 78 by the method of Art. 323. Here, then, we have a machine possessing very considerable efficiency, and being at the same time economical of energy.

TABLE XXI.—THE CRANE.

Circumference of wheel to which the power is applied, 43"; train of wheels, 25 ÷ 200; circumference of drum on which rope is wound, 14"·9; velocity ratio, 23; mechanical efficiency, 18; useful effect, 78 per cent.; formula, _P_ = 0·0556 _R_. +-----------+--------+--------+----------+--------------+ | | | | P. |Difference of | | Number of | R. |Observed|Calculated| the observed | |Experiment.| Load | power | power |and calculated| | | in lbs.| in lbs.| in lbs. | values. | +-----------+--------+--------+----------+--------------+ | 1 | 14 | 0·9 | 0·8 | -0·1 | | 2 | 28 | 1·6 | 1·6 | 0·0 | | 3 | 42 | 2·4 | 2·3 | -0·1 | | 4 | 56 | 3·1 | 3·1 | 0·0 | | 5 | 70 | 3·8 | 3·9 | +0·1 | | 6 | 84 | 4·5 | 4·7 | +0·2 | | 7 | 98 | 5·3 | 5·5 | +0·2 | | 8 | 112 | 6·2 | 6·2 | +0·0 | +-----------+--------+--------+----------+--------------+

335. A series of experiments made with this crane is recorded in Table XXI., and a comparison of the calculated and observed values will show that the formula _P_ = 0·0556 _R_ represents the experiments with considerable accuracy.

336. It may be noticed that in this formula the term independent of _R_, which we frequently meet with in the expression of the relation between the power and the load, is absent. The probable explanation is to be found in the fact that some minute irregularity in the form of the barrel or of the wheel has been constantly acting like a small weight in favour of the power. In each experiment the motion is always started from the same position of the wheels, and hence any irregularity will be constantly acting in favour of the power or against it; here the former appears to have happened. In other cases doubtless the latter has occurred; the difference is, however, of extremely small amount. The friction of the machine itself when without a load is another source for the production of the constant term; it has happened in the present case that this friction has been almost exactly balanced by the accidental influence referred to.

337. In cranes it is usual to provide means of adding a second train of wheels, when the load is very heavy. In another model we applied the power to an axle with a pinion of 25 teeth, gearing into a wheel of 200 teeth; on the axle of the wheel with 200 teeth is a pinion of 30 teeth, which gears into a wheel of 180 teeth; the barrel is on the axle of the last wheel. A series of experiments with this crane is shown in Table XXII.

TABLE XXII.—THE CRANE FOR HEAVY LOADS.

Circumference of wheel to which power is applied, 43"; train of wheels, 25 ÷ 200 × 30 ÷ 180; circumference of drum on which rope is wound, 14"·9; velocity ratio, 137; mechanical efficiency, 87; useful effect, 63 per cent.; formula, _P_ = 0·185 + 0·00782 _R_. +-----------+--------+--------+----------+--------------+ | | | | P. |Difference of | | Number of | R. |Observed|Calculated| the observed | |Experiment.| Load | power | power |and calculated| | | in lbs.| in lbs.| in lbs. | values. | +-----------+--------+--------+----------+--------------+ | 1 | 14 | 0·30 | 0·29 | -0·01 | | 2 | 28 | 0·40 | 0·40 | 0·00 | | 3 | 42 | 0·50 | 0·51 | +0·01 | | 4 | 56 | 0·60 | 0·62 | +0·02 | | 5 | 70 | 0·75 | 0·73 | -0·02 | | 6 | 84 | 0·85 | 0·84 | -0·01 | | 7 | 98 | 0·95 | 0·95 | 0·00 | | 8 | 112 | 1·05 | 1·06 | +0·01 | +-----------+--------+--------+----------+--------------+

The velocity ratio is now 137, and the mechanical efficiency is 87; one man could therefore raise a ton with ease by applying a power of 26 lbs. to a crane of this kind.

CONCLUSION.

338. It will be useful to contrast the wheel and axle on which we have experimented (Art. 304) with the differential pulley (Art. 209). The velocity ratio of the former machine is nearly double that of the latter, and its mechanical efficiency is nearly four times as great. Less than half the applied power is wasted in the wheel and axle, while more than half is wasted in the differential pulley. This makes the wheel and axle both a more powerful machine, and a more economical machine than the differential pulley. On the other hand, the greater compactness of the latter, its facility of application, and the practical conveniences arising from the property of not allowing the load to run down, do often more than compensate for the superior mechanical advantage of the wheel and axle.

339. We may also contrast the wheel and axle with the screw (Art. 277). The screw is remarkable among the mechanical powers for its very high velocity ratio, and its excessive friction. Thus we have seen in Art. 291 how the velocity ratio of a screw-jack with an arm attached amounted to 414, while its mechanical efficiency was little more than one-fourth as great. No _single_ wheel and axle could conveniently be made to give a mechanical efficiency of 116; but from Art. 337 we could easily design a _combination_ of wheels and axles to yield an efficiency of quite this amount. The friction in the wheel and axle is very much less than in the screw, and consequently energy is saved by the use of the former machine.

340. In practice, however, it generally happens that economy of energy does not weigh much in the selection of a mechanical power for any purpose, as there are always other considerations of greater consequence.

341. For example, let us take the case of a lifting crane employed in loading or unloading a vessel, and inquire why it is that a train of wheels is generally used for the purpose of producing the requisite power. The answer is simple, the train of wheels is convenient, for by their aid any length of chain can be wound upon the barrel; whereas if a screw were used, we should require a screw as long as the greatest height of lift. This screw would be inconvenient, and indeed impracticable, and the additional circumstance that a train of wheels is more economical of energy than a screw has no influence in the matter.

342. On the other hand, suppose that a very heavy load has to be overcome for a short distance, as for example in starting a ship launch, a screw-jack is evidently the proper machine to employ; it is easily applied, and has a high mechanical efficiency. The want of economy of energy is of no consequence in such an operation.

LECTURE XI. _THE MECHANICAL PROPERTIES OF TIMBER._

Introduction.—The General Properties of Timber.—Resistance to Extension.—Resistance to Compression.—Condition of a Beam strained by a Transverse Force.

INTRODUCTION

343. In the lectures on the mechanical powers which have been just completed, we have seen how great weights may be raised or other large resistances overcome. We are now to consider the important subject of the application of mechanical principles to _structures_. These are fixtures, while machines are adapted for motion; a roof or a bridge is a structure, but a crane or a screw-jack is a machine. Structures are employed for supporting weights, and the mechanical powers give the means of raising them.

344. A structure has to support both its own weight and also any load that is to be placed upon it. Thus a railway bridge must at all times sustain what is called the permanent load, and frequently, of course, the weight of one or more trains. The problem which the engineer solves is to design a bridge which shall be sufficiently strong, and, at the same time, economical; his skill is shown by the manner in which he can attain these two ends in the same structure.

345. In the four lectures of the course which will be devoted to this subject it will only be possible to give a slight sketch, and therefore but few details can be introduced. An extended account of the properties of different materials used in structures would be beyond our scope, but there are some general principles relating to the strength of materials which may be discussed. Timber, as a building material, has, in modern times, been replaced to a great extent by iron in large structures, but timber is more capable than iron of being experimented upon in the lecture room. The elementary laws which we shall demonstrate with reference to the strength of timber, are also, substantially the same as the corresponding laws for the strength of iron or any other material. Hence we shall commence the study of structures by two lectures on timber. The laws which we shall prove experimentally will afterwards be applied to a few simple cases of bridges and other actual structures.

THE GENERAL PROPERTIES OF TIMBER.

346. The uses of timber in the arts are as various as its qualities. Some woods are useful for their beauty, and others for their strength or durability under different circumstances. We shall only employ “pine” in our experimental inquiries. This wood is selected because it is so well known and so much used. A knowledge of the properties of pine would probably be more useful than a knowledge of the properties of any other wood, and at the same time it must be remembered that the laws which we shall establish by means of slips of pine may be generally applied.

347. A transverse section of a tree shows a number of rings, each of which represents the growth of wood in one year. The age of the tree may sometimes be approximately found by counting the number of distinguishable rings. The outer rings are the newer portions of the wood.

348. When a tree is felled it contains a large quantity of sap, which must be allowed to evaporate before the wood is fit for use. With this object the timber is stored in suitable yards for two or more years according to the purposes for which it is intended; sometimes the process of seasoning, as it is called, is hastened by other means. Wood, when seasoning, contracts; hence blocks of timber are often found split from the circumference to the centre, for the outer rings, being newer and containing more sap, contract more than the inner rings. For the same reason a plank is found to warp when the wood is not thoroughly seasoned. The side of the plank which was farthest from the centre of the tree contracts more than the other side, and becomes concave. This can be easily verified by looking at the edge of the plank, for we there see the rings of which it is composed.

349. Timber may be softened by steaming. I have here a rod of pine, 24" × 0"·5 × 0"·5, and here a second rod cut from the same piece and of the same size, which has been exposed to steam of boiling water for more than an hour: securing these at one end to a firm stand, I bend them down together, and you see that after the dry rod has broken the steamed rod can be bent much farther before it gives way. This property of wood is utilized in shaping the timbers of wooden ships. We shall be able to understand the action of steam if we reflect that wood is composed of a number of fibres ranged side by side and united together. A rope is composed of a number of fibres laid together and twisted, but the fibres are not coherent as they are in wood. Hence we find that a rod of wood is stiff, while a rope is flexible. The steam finds its way into the interstices between the fibres of the wood; it softens their connections, and increases the pliability of the fibres themselves, and thus, the operation of steaming tends to soften a piece of timber and render it tractable.

350. The structure of wood is exhibited by the following simple experiment:—Here are two pieces of pine, each 9" × 1" × 1". One of them I easily snap across with a blow, while my blows are unable to break the other. The difference is merely that one of these pieces is cut against the grain, while the other is with it. In the first case I have only to separate the connection between the fibres, which is quite easy. In the other case I would have to tear asunder the fibres themselves, which is vastly more difficult. To a certain extent the grained structure is also found in wrought iron, but the contrast between the strength of iron with the grain and against the grain is not so marked as it is in wood.

RESISTANCE TO EXTENSION.

351. It will be necessary to explain a little more definitely what is meant by the strength of timber. We may conceive a rod to be broken in three different ways. In the first place the rod may be taken by a force at each end and torn asunder by pulling, as a thread may be broken. To do this requires very great power, and the strength of the material with reference to such a mode of destroying it is called its resistance to extension. In the second place, it may be broken by longitudinal pressure at each end, as a pillar may be crushed by the superincumbent weight being too large; the strength that relates to this form of force is called resistance to compression: finally, the rod may be broken by a force applied transversely. The strength of pine with reference to these different applications of force will be considered successively. The rods that are to be used have been cut from the same piece of timber, which has been selected on account of its straightness of grain and freedom from knots. They are of different rectangular sections, 1" × 0"·5 and 0"·5 × 0"·5 being generally used, but sometimes 1" × 1" is employed.

352. With reference to the strength of timber in its capacity to resist extension, we can do but little in the lecture room. I have here a pine rod A B, of dimensions 48" × 0".5 × 0"·5, Fig. 49. Each end of this rod is firmly secured between two cheeks of iron, which are bolted together: the rod is suspended by its upper extremity from the hook of the epicycloidal pulley-block (Art. 213), which is itself supported by a tripod; hooks are attached to the lower end of the rod for carrying the weights. By placing 3 cwt. on these hooks and pulling the hand chain of the pulley-block, I find that I can raise the weight safely, and therefore the rod will resist at all events a tension of 3 cwt. From experiments which have been made on the subject, it is ascertained that about a ton would be necessary to tear such a rod asunder; hence we see that pine is enormously strong in resisting a force of extension. The tensile strength of the rod does not depend upon its length, but upon the area of the cross section. That of the rod we have used is one-fourth of a square inch, and the breaking weight of a rod one square inch in section is about four tons.

353. A rod of any material generally elongates to some extent under the action of a suspended weight; and we shall ascertain whether this occurs perceptibly in wood. Before the rod was strained I had marked two points upon it exactly 2 feet apart. When the rod supports 3 cwt. I find that the distance between the two points has not appreciably altered, though by more delicate measurement I have no doubt we should find that the distance had elongated to an insignificant extent.

354. Let us contrast the resistance of a rod of timber to extension with the effect upon a rope under the same circumstances. I have here a rope about 0".25 diameter; it is suspended from a point, and bears a 14 lb. weight in order to be completely stretched. I mark points upon the rope 2' apart. I now change the stone weight for a weight of 1 cwt., and on measurement I find that the two points which before were 2' apart, are now 2' 2"; thus the rope has stretched at the rate of an inch per foot for a strain of 1 cwt., while the timber did not stretch perceptibly for a strain of 3 cwt.

355. We have already explained in Art. 37 the meaning of the word “tie.” The material suitable for a tie should be capable of offering great resistance, not only to actual rupture by tension, but even to appreciable elongation. These qualities we have found to be possessed by wood. They are, however, possessed in a much higher degree by wrought iron, which possesses other advantages in durability and facility of attachment.

RESISTANCE TO COMPRESSION.

356. We proceed to examine into the capability of timber to resist forces of longitudinal compression, either as a pillar or in any other form of “strut,” such for instance, as the jib of the crane represented in Fig. 17. The use of timber as a strut depends in a great degree upon the coherence of the fibres to each other, as well as upon their actual rigidity. The action of timber in resisting forces of compression is thus very different from its action when resisting forces of extension; we can examine, by actual experiment, the strength of timber under the former conditions, as the weights which it will be necessary to employ are within the capabilities of our lecture room apparatus.

357. The apparatus is shown in Fig. 50. It consists of a lever of the second order, 10' long, the mechanical advantage of which is threefold; the resistance of the pillar D E to crushing is the load to be overcome, and the power consists of weights, to receive which the tray B is used; every pound placed in the tray produces a compressive force of 3 lbs. on the pillar at D. The fulcrum is at A and guides at G. The lever and the tray would somewhat complicate our calculations unless their weights were counterpoised. A cord attached to the extremity of the lever passes over a pulley F; at the other end of this cord, sufficient weights C are attached to neutralize the weight of the apparatus. In fact, the lever and tray now swing as if they had no weight, and we may therefore leave them out of consideration. The pillar to be experimented upon is fitted at its lower end E into a hole in a cast iron bracket: this bracket can be adjusted so as to take in pieces of different lengths; the upper end of the pillar passes through a hole in a second piece of cast iron, which is bolted to the lever: thus our little experimental column is secured at each end, and the risk of slipping is avoided. The stands are heavily weighted to secure the stability of the arrangement

358. The first experiment we shall make with this apparatus is upon a pine rod 40" long and 0"·5 square; the lower bracket is so placed that the lever is horizontal when just resting upon the top of the rod. Weights placed in the tray produce a pressure three times as great down the rod, the effect of which will first be to bend the rod, and, when the deflection has reached a certain amount, to break it across. I place 28 lbs. in the tray: this produces a pressure of 84 lbs. upon the rod, but the rod still remains perfectly straight, so that it bears this pressure easily. When the pressure is increased to 96 lbs. a very slight amount of deflection may be seen. When the strain reaches 114 lbs. the rod begins to bend into a curved form, though the deflection of the middle of the rod from its original position is still less than 0"·25. Gradually augmenting the pressure, I find that when it reaches 132 lbs. the deviation has reached 0"·5; and finally, when 48 lbs. is placed in the tray, that is, when the rod is subjected to 144 lbs., it breaks across the middle. Hence we see that this rod sustained a load of 96 lbs. without sensibly bending, but that fracture ensued when the load was increased about half as much again. Another experiment with a similar rod gave a slightly less value (132 lbs.) for the breaking load. If I add these results together, and divide the sum by 2, I find 138 lbs. as the mean value of the breaking load, and this is a sufficiently exact determination.

359. Let us next try the resistance of a shorter rod of the same section. I place a piece of pine 20" long and 0"·5 square in the apparatus, firmly securing each end as in the former case. The lower bracket is adjusted so as to make the lever horizontal; the counterpoise, of course, remains the same, and weights are placed in the tray as before. No deflection is noticed when the rod supports 126 lbs.; a very slight amount of bending is noticeable with 186 lbs.; with 228 lbs., the amount by which the centre of the rod has deviated laterally from its original position is about 0"·2; and finally, when the load reaches 294 lbs., the rod breaks. Fracture first occurs in the middle, but is immediately followed by other fractures near where the ends of the rod are secured.

360. Hence the breaking load of a rod 20" long is more than double the breaking load of a rod of 40" long the same section; from this we learn that the sections being equal, short pillars are stronger than long pillars. It has been ascertained by experiment that the strength of a square pillar to resist compression is proportional to the square of its sectional area. Hence a rod of pine, 40" long and 1" square, having four times the section of the rod of the same length we have experimented on, would be sixteen times as strong, and consequently its breaking weight would amount to nearly a ton. The strength of a rod used as a _tie_ depends only on its section, while the strength of a rod used as a _strut_ depends on its length as well as on its section.

CONDITION OF A BEAM STRAINED BY A TRANSVERSE FORCE.