Part 13

361. We next come to the important practical subject of the strength of timber when supporting a transverse strain; that is, when used as a beam. The nature of a transverse strain may be understood from Fig. 51, which represents a small beam, strained by a load at its centre. Fig. 52 shows two supports 40" apart, across which a rod of pine 48" × 1" × 1" is laid; at the middle of this rod a hook is placed, from which a tray for the reception of weights is suspended. A rod thus supported, and bearing weights, is said to be strained transversely. A rafter of a roof, the flooring of a room, a gangway from the wharf to a ship, many forms of bridge, and innumerable other examples, might be given of beams strained in this manner. To this important subject we shall devote the remainder of this lecture and the whole of the next.

362. The first point to be noticed is the deflection of the beam from which a weight is suspended. The beam is at first horizontal; but as the weight in the tray is augmented, the beam gradually curves downwards until, when the weight reaches a certain amount, the beam breaks across in the middle and the tray falls.

For convenience in recording the experiments the tray chain and hooks have been adjusted to weigh exactly 14 lbs. (Fig. 52). A B is a cord which is kept stretched by the little weights D: this cord gives a rough measure of the deflection of the beam from its horizontal position when strained by a load in the tray. In order to observe the deflection accurately an instrument is used called the cathetometer (G). It consists of a small telescope, always directed horizontally, though capable of being moved up and down a vertical triangular pillar; on one of the sides of the pillar a scale is engraved, so that the height of the telescope in any position can be accurately determined. The cathetometer is levelled by means of the screws H H, so that the triangular pillar on which the telescope slides is accurately vertical: the dotted line shows the direction of the visual ray when the centre C of the beam is seen by the observer through the telescope.

Inside the telescope and at its focus a line of spider’s web is fixed horizontally; on the bar to be observed, and near its middle point C, a cross of two fine lines is marked. The tray being removed, the beam becomes horizontal; the telescope of the cathetometer is then directed towards the beam, so that the lines marked upon it can be seen distinctly. By means of a screw the telescope may be raised or lowered until the spider’s web inside the telescope is observed to pass through the image of the intersection of the lines. The scale then indicates precisely how high the telescope is on the pillar.

363. While I look through the telescope my assistant suspends the tray from the beam. Instantly I see the cross descend in the field of view. I lower the telescope until the spider’s web again passes through the image of the intersection of the lines, and then by looking at the scale I see that the telescope has been moved down 0"·19, that is, about one-fifth of an inch: this is, therefore, the distance by which the cross lines on the beam, and therefore the centre of the beam itself, must have descended. Indeed, even a simpler apparatus would be competent to measure the amount of deflection with some degree of precision. By placing successively one stone after another upon the tray, the beam is seen to deflect more and more, until even without the telescope you see the beam has deviated from the horizontal.

364. By carefully observing with the telescope, and measuring in the way already described, the deflections shown in Table XXIII. were determined. The scale along the vertical pillar was read after the spider’s web had been adjusted for each increase in the weight. The movement from the original position is recorded as the deflection for each load.

TABLE XXIII.—DEFLECTION OF A BEAM.

A rod of pine 48" × 1" × 1"; resting freely on supports 40" apart; and laden in the middle. +-------------+-----------+-------------+ | Number of | Magnitude | Deflection. | | Experiment. | of load. | | +-------------+-----------+-------------+ | 1 | 14 | 0"·19 | | 2 | 28 | 0"·37 | | 3 | 42 | 0"·55 | | 4 | 56 | 0"·74 | | 5 | 70 | 0"·94 | | 6 | 84 | 1"·13 | | 7 | 98 | 1"·35 | | 8 | 112 | 1"·61 | | 9 | 126 | 1"·95 | | 10 | 140 | 2"·37 | +-------------+-----------+-------------+

365. The first column records the number of the experiment. The second represents the load, and the third contains the corresponding deflections. It will be seen that up to 98 lbs. the deflection is about 0"·2 for every stone weight, but afterwards the deflection increases more rapidly. When the weight reaches 140 lbs. the deflection at first indicated is 2"·37; but gradually the cross lines are seen to descend in the field of the telescope, showing that the beam is yielding and finally it breaks across. This experiment teaches us that a beam is at first deflected by an amount proportional to the weight it supports; but that when two-thirds of the breaking weight is reached, the beam is deflected more rapidly.

366. It is a question of the utmost importance to ascertain the greatest load a beam can sustain without injury to its strength. This subject is to be studied by examining the effect of different deflections upon the fibres of a beam. A beam is always deflected whatever be the load it supports; thus by looking through the telescope of the cathetometer I can detect an increase of deflection when a single pound is placed in the tray: hence whenever a beam is loaded we must have some deflection. An experiment will show what amount of deflection may be experienced without producing any permanently injurious effect.

367. A pine rod 40" × 1" × 1" is freely supported at each end, the distances between the supports being 38", and the tray is suspended from its middle point. A fine pair of cross lines is marked upon the beam, and the telescope of the cathetometer is adjusted so that the spider’s line exactly passes through the image of the intersection. 14 lbs. being placed in the tray, the cross is seen to descend; the weight being removed, the cross returns precisely to its original position with reference to the spider’s line: hence, after this amount of deflection, the beam has clearly returned to its initial condition, and is evidently just as good as it was before. The tray next received 56 lbs.; the beam was, of course, considerably deflected, but when the weight was removed the cross again returned,—at all events, to within 0"·01 of where the spider’s line was left to indicate its former position. We may consider that the beam is in this case also restored to its original condition, even though it has borne a strain which, including the tray, amounted to 70 lbs. But when the beam has been made to carry 84 lbs. for a few seconds, the cross does not completely return on the removal of the load from the tray, but it shows that the beam has now received a permanent deflection of 0"·03. This is still more apparent after the beam has carried 98 lbs., for when this load is removed the centre of the beam is permanently deflected by 0"·13. Here, then, we may infer that the fibres of the beam are beginning to be strained beyond their powers of resistance, and this is verified when we find that with 28 additional pounds in the tray a collapse ensues.

368. Reasoning from this experiment, we might infer that the elasticity of a beam is not affected by a weight which is less than half that which would break it, and that, therefore, it may bear without injury a weight not exceeding this amount. As, however, in our experiments the weight was only applied once, and then but for a short time, we cannot be sure that a longer-continued or more frequent application of the same load might not prove injurious; hence, to be on the safe side, we assume that one-third of the breaking weight of a beam is the greatest load it should be made to bear in any structure. In many cases it is found desirable to make the beam much stronger than this ratio would indicate.

369. We next consider the condition of the fibres of a beam when strained by a transverse force. It is evident that since the fracture commences at the lower surface of the beam, the fibres there must be in a state of tension, while those at the concave upper surface of the beam are compressed together. This condition of the fibres may be proved by the following experiment.

370. I take two pine rods, each 48" × 1" × 1", perfectly similar in all respects, cut from the same piece of timber, and therefore probably of very nearly identical strength. With a fine tenon saw I cut each of the rods half through at its middle point. I now place one of these beams on the supports 40" apart, with the cut side of the beam upwards. I suspend from it the tray, which I gradually load with weights until the beam breaks, which it does when the total weight is 81 lbs.

If I were to place the second beam on the same supports with the cut upwards, then there can be no doubt that it would require as nearly as possible the same weight to break it. I place it, however, with the cut downwards, I suspend the tray, and find that the beam breaks with a load of 31 lbs. This is less than half the weight that would have been required if the cut had been upwards.

371. What is the cause of this difference? The fibres being compressed together on the upper surface, a cut has no tendency to open there; and if the cut could be made with an extremely fine saw, so as to remove but little material, the beam would be substantially the same as if it had not been tampered with. On the other hand, the fibres at the lower surface are in a state of tension; therefore when the cut is below it yawns open, and the beam is greatly weakened. It is, in fact, no stronger than a beam of 48" × 0"·5 × 1", placed with its shortest dimension vertical. If we remember that an entire beam of the same size required about 140 lbs. to break it (Art. 366), we see that the strength of a beam is reduced to one-fourth by being cut half-way through and having the cut underneath.

372. We may learn from this the practical consequence that the sounder side of a beam should always be placed downwards. Any flaw on the lower surface will seriously weaken the beam: so that the most knotty face of the wood should certainly be placed uppermost. If a portion of the actual substance of a beam be removed—for example, if a notch be cut out of it—this will be almost equally injurious on either side of the beam.

373. We may illustrate the condition of the upper surface of the beam by a further experiment. I make two cuts 0"·5 deep in the middle of a pine rod 48" × 1" × 1". These cuts are 0"·5 apart, and slightly inclined; the piece between them being removed, a wedge is shaped to fit tightly into the space; the wedge is long enough to project a little on one side. If the wedge be uppermost when the beam is placed on the supports, the beam will be in the same condition as if it had two fine cuts on the upper surface. I now load the beam with the tray in the usual manner, and I find it to bear 70 lbs. securely. On examining the beam, which has curved down considerably, I find that the wedge is held in very tightly by the pressure of the fibres upon it, but, by a sharp tap at the end, I knock out the wedge, and instantly the load of 70 lbs. breaks the beam; the reason is simple—the piece being removed, there is no longer any resistance to the compressive strain of the upper fibres, and consequently the beam gives way.

374. The collapse of a beam by a transverse strain commences by fracture of the fibres on the lower surface, followed by a rupture of all fibres up to a considerable depth. Here we see that by a transverse force the fibres in a beam of 48" × 1" × 1" have been broken by a strain of 140 lbs. (Art. 366); but we have already stated (Art. 353) that to tear such a rod across by a direct pull at each end a force of about four tons is necessary. The breaking strain of the fibres must be a certain definite quantity, yet we find that to overcome it in one way four tons is necessary, while by another mode of applying the strain 140 lbs. is sufficient.

375. To explain this discrepancy we may refer to the experiment of Art. 28, wherein a piece of string was broken by the transverse pull of a piece of thread in illustration of the fact that one force may be resolved into two others, each of them very much greater than itself. A similar resolution of force occurs in the transverse deflection of the beam, and the force of 140 lbs. is changed into two other forces, each of them enormously greater and sufficiently strong to rupture the fibres. We need not suppose that the force thus developed is so great as four tons, because that is the amount required to tear across a square inch of fibres simultaneously, whereas in the transverse fracture the fibres appear to be broken row after row; the fracture is thus only gradual, nor does it extend through the entire depth of the beam.

376. We shall conclude this lecture with one more remark, on the condition of a beam when strained by a transverse force. We have seen that the fibres on the upper surface are compressed, while those on the lower surface are extended; but what is the condition of the fibres in the interior? There can be no doubt that the following is the state of the case:—The fibres immediately beneath the upper surface are in compression; at a greater depth the amount of compression diminishes until at the middle of the beam the fibres are in their natural condition; on approaching the lower surface the fibres commence to be strained in extension, and the amount of the extension gradually increases until it reaches a maximum at the lower surface.

LECTURE XII. _THE STRENGTH OF A BEAM._

A Beam free at the Ends and loaded in the Middle.—A Beam uniformly loaded.—A Beam loaded in the Middle, whose Ends are secured.—A Beam supported at one end and loaded at the other.

A BEAM FREE AT THE ENDS AND LOADED IN THE MIDDLE.

377. In the preceding lecture we have examined some general circumstances in connection with the condition of a beam acted on by a transverse force; we proceed in the present to inquire more particularly into the strength under these conditions. We shall, as before, use for our experiments rods of pine only, as we wish rather to illustrate the general laws than to determine the strength of different materials. The strength of a beam depends upon its length, breadth, and thickness; we must endeavour to distinguish the effects of each of these elements on the capacity of the beam to sustain its load.

We shall only employ beams of rectangular section; this being generally the form in which beams of wood are used. Beams of iron, when large, are usually not rectangular, as the material can be more effectively disposed in sections of a different form. It is important to distinguish between the _stiffness_ of a beam in its capacity to resist flexure, and the _strength_ of a beam in its capacity to resist fracture. Thus the stiffest beam which can be made from the cylindrical trunk of a tree 1' in diameter is 6" broad and 10"·5 deep, while the strongest beam is 7" broad and 9"·75 deep. We are now discussing the strength (not the stiffness) of beams.

378. We shall commence the inquiry by making a number of experiments: these we shall record in a table, and then we shall endeavour to see what we can learn from an examination of this table. I have here ten pieces of pine, of lengths varying from 1' to 4', and of three different sections, viz. 1" × 1", 1" × 0"·5, and 0"·5 × 0"·5. I have arranged four different stands, on which we can break these pieces: on the first stand the distance between the points of support is 40", and on the other stands the distances are 30," 20", and 10" respectively; the pieces being 4', 3', 2', and 1' long, will just be conveniently held on the supports.

379. The mode of breaking is as follows:—The beam being laid upon the supports, an S hook is placed at its middle point, and from this S hook the tray is suspended. Weights are then carefully added to the tray until the beam breaks; the load in the tray, together with the weight of the tray, is recorded in the table as the breaking load.

380. In order to guard as much as possible against error, I have here another set of ten pieces of pine, duplicates of the former. I shall also break these; and whenever I find any difference between the breaking loads of two similar beams, I shall record in the table the mean between the two loads. The results are shown in Table XXIV.

TABLE XXIV.—STRENGTH OF A BEAM.

Slips of pine (cut from the same piece) supported freely at each end; the length recorded is the distance between the points of support; the load is suspended from the centre of the beam, and gradually increased until the beam breaks;

area of section × depth Formula, _P_ = 6080 ———————————————————————— span

+-------+---------------------+------------+----------+------------+ | | |Mean of the | P. |Difference | | | Dimensions. |observations|Calculated| of the | |No. of +-----+--------+------+ of the | breaking |observed and| |Experi-| | | | breaking | load | calculated | |ment. |Span.|Breadth.|Depth.|load in lbs.| in lbs. | values. | +-------+-----+--------+------+------------+----------+------------+ | 1 |40"·0| 1"·0 | 1"·0 | 152 | 152 | 0·0 | | 2 |40"·0| 0"·5 | 1"·0 | 77 | 76 | -1·0 | | 3 |40"·0| 1"·0 | 0"·5 | 38 | 38 | 0·0 | | 4 |40"·0| 0"·5 | 0"·5 | 19 | 19 | 0·0 | | 5 |30"·0| 1"·0 | 0"·5 | 59 | 51 | -8·0 | | 6 |30"·0| 0"·5 | 0"·5 | 25 | 25 | 0·0 | | 7 |20"·0| 1"·0 | 0"·5 | 74 | 76 | +2·0 | | 8 |20"·0| 0"·5 | 0"·5 | 36 | 38 | +2·0 | | 9 |10"·0| 1"·0 | 0"·5 | 154 | 152 | -2·0 | | 10 |10"·0| 0"·5 | 0"·5 | 68 | 76 | +8·0 | +-------+-----+--------+------+------------+----------+------------+

381. In the first column is a series of figures for convenience of reference. The next three columns are occupied with the dimensions of the beams. By span is meant the distance between the points of support; the real length is of course greater; the depth is that dimension of the beam which is vertical. The fifth column gives the mean of two observations of the breaking load. Thus for example, in experiment No. 5 the two beams used were each 36" × 1" × 0"·5, they were placed on points of support 30" distant, so the span recorded is 30": one of the beams was broken by a load of 58 lbs., and the second by a load of 60 lbs.; the mean between the two, 59 lbs., is recorded as the mean breaking load. In this manner the column of breaking loads has been found. The meaning of the two last columns of the table will be explained presently.

382. We shall endeavour to elicit from these observations the laws which connect the breaking load with the span, breadth, and depth of the beam.

383. Let us first examine the effect of the span; for this purpose we bring together the observations upon beams of the same section, but of different spans. Sections of 0"·5 × 0"·5 will be convenient for this purpose; Nos. 4, 6, 8, and 10 are experiments upon beams of this section. Let us first compare 4 and 8. Here we have two beams of the same section, and the span of one (40") is double that of the other (20"). When we examine the breaking weights we find that they are 19 lbs. and 36 lbs.; the former of these numbers is rather more than half of the latter. In fact, had the breaking load of 40" been ¾ lb. less, 18·25 lbs., and had that of 20" been ½ lb. more, 36·5 lbs., one of the breaking loads would have been exactly half the other.

384. We must not look for perfect numerical accuracy in these experiments; we must only expect to meet with approximation, because the laws for which we are in search are in reality only approximate laws. Wood itself is variable in quality, even when cut from the same piece: parts near the circumference are different in strength from those nearer the centre; in a young tree they are generally weaker, and in an old tree generally stronger. Minute differences in the grain, greater or less perfectness in the seasoning, these are also among the circumstances which prevent one piece of timber from being identical with another. We shall, however, generally find that the effect of these differences is small, but occasionally this is not the case, and in trying many experiments upon the breaking of timber, discrepancies occasionally appear for which it is difficult to account.

385. But you will find, I think, that, making reasonable allowances for such difficulties as do occur, the laws on the whole represent the experiments very closely.

386. We shall, then, assume that the breaking weight of a bar of 40" is half that of a bar of 20" of the same section, and we ask, Is this generally true? is it true that the breaking weight is inversely proportional to the span? In order to test this hypothesis, we can calculate the breaking weight of a bar of 30" (No. 6), and then compare the result with the observed value; if the supposition be true, the breaking weight should be given by the proportion—

30" : 40" :: 19 : Answer.

The answer is 25·3 lbs.; on reference to the table we find 25 lbs. to be the observed value, hence our hypothesis is verified for this bar.

387. Let us test the law also for the 10" bar, No. 10—

10" : 40" :: 19 : Answer.

The answer in this case is 76, whereas the observed value is 68, or 8 lbs. less; this does not agree very well with the theory, but still the difference, though 8 lbs., is only about 11 or 12 per cent. of the whole, and we shall still retain the law, for certainly there is no other that can express the result as well.

388. But the table will supply another verification. In experiment No. 3 a 40" bar, 1" broad, and 0"·5 deep, broke with 38 lbs.; and in experiment No. 7 a 20" bar of the same section broke with 74 lbs.; but this is so nearly double the breaking weight of the 40" bar, as to be an additional illustration of the law, that _for a given section the breaking load varies inversely as the span_.

389. We next inquire as to the effect of the _breadth_ of the beam upon its strength? For this purpose we compare experiments Nos. 3 and 4: we there find that a bar 40" × 1" × 0"·5 is broken by a load of 38 lbs., while a bar just half the breadth is broken by 19 lbs. We might have anticipated this result, for it is evident that the bar of No. 3 must have the same strength as two bars similar to that of No. 4 placed side by side.

390. This view is confirmed by a comparison of Nos. 7 and 8, where we find that a 20" bar takes twice the load to break it that is required for a bar of half its breadth. The law is not quite so well verified by Nos. 5 and 6, for half the breaking weight of No. 5, namely 29·5 lbs., is more than 25, the observed breaking weight of No. 6: a similar remark may be made about Nos. 9 and 10.