THE MECHANICS OF A BRIDGE

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LECTURE XIV. THE MECHANICS OF A BRIDGE.

Introduction.—The Girder.—The Tubular Bridge.—The Suspension Bridge.

INTRODUCTION.

447. Perhaps it may be thought that the structures we have been lately considering are not those which are most universally used, and that the bridges which are generally referred to as monuments of engineering skill are of quite a different construction. Every one is familiar with the arch, and most of us have seen suspension bridges and the celebrated Menai tube. We must therefore allude further to some of these structures, and this we propose to do in the present lecture. It will only be possible to take a very slight survey of an extensive subject to which elaborate treatises have been devoted.

We shall first give a brief account of the use of iron in the arts of construction. We shall then explain simply the principle of the tubular bridge, and also of the suspension bridge. The more complex forms are beyond our scope.

THE GIRDER.

448. A horizontal beam supported at each end, and perhaps at intermediate points, and designed to support a heavy load is called a girder. Those rods upon which we have performed experiments, the results of which have been given in Table XXIV., are small girders; but the term is generally understood to relate to structures of iron: the greatest girders for railway bridges are made of bars or plates of iron riveted together.

449. We shall first consider the application of cast iron to girders, and show what form they should assume.

450. A beam of cast iron, supposing its section to be rectangular, has its strength determined by the same laws as the beams of pine. Thus, supposing the section of two beams to be the same, their strengths are inversely proportional to their lengths, and the strength of a beam placed edgewise is to its strength placed flatwise in the proportion of the greater dimension of its section to the less dimension. These laws determine the strength of every rectangular beam of cast iron when that of one beam is known, and we must perform an experiment in order to find the breaking load in a particular case.

451. I take here a piece of cast iron, which is 2' long, and 0"·5 × 0"·5 in section. I support this beam at each end upon a frame; the distance between the supports is 20". I attach the tray to the centre of the beam and load it with weights. The ends of the beam rest freely upon the supports, but I have taken the precaution of tying each end by a piece of wire, so that they may not fly about when the fracture occurs. Loading the tray, I find that with 280 lbs. the crash comes.[Pg 220]

452. Let us compare this result with No. 8 of Table XXIV. (p. 190). There we find that a piece of pine, the same size as the cast iron, was broken with 36 lbs.: the ratio of 280 to 36 is nearly 8, so that the beam of cast iron is about 8 times as strong as the piece of pine of the same size. This result is a little larger than we would have inferred from an examination of tables of the strength of large bars of cast iron; the reason may be that a very small casting, such as this bar, is stronger in proportion than a larger one, owing to the iron not being so uniform throughout the larger mass.

453. I hold here a bar of cast iron 12" long and 1" × 1" in section. I have not sufficient weights at hand to break it, but we can compute how much would be necessary by our former experiment.

454. In the first place a bar 12" long, and 0"·5 × 0"·5 of section, would require 20 × 280 ÷ 12 = 467 lbs. by the law that the strength is inversely as the length. We also know that one beam 12" × 1" × 1" is just as strong as two beams 12" × 1" × 0"·5, each placed edgewise; each of these latter beams is twice as strong as 12" × 1" × 0"·5 placed flatwise, because the strength when placed edgewise is to the strength when placed flatwise, as the depth to the breadth, that is as 2 to 1: hence the original beam is four times as strong as one beam 12" × 1" × 0"·5 placed flatwise: but this last beam is twice as strong as a beam 12" × 0"·5 × 0"·5, and hence we see that a beam 12" × 1" × 1" must be 8 times as strong as a beam of 12" × 0"·5 × 0"·5, but this last beam would require a load of 467 lbs. to break it, and hence the beam of 12" × 1" × 1" would require 467 × 8 = 3736 lbs. to produce fracture. This amounts to more than a ton and a half.[Pg 221]

455. It is a rule sometimes useful to practical men that a cast iron bar one foot long by one inch square would break with about a ton weight. If the iron be of the same quality as that which we have used, this result is too small, but the error is on the safe side; the real strength will then be generally a little greater than the strength calculated from this rule. What we have said (Art. 403) with reference to the precaution for safety in bars of wood applies also to cast iron. The load which the beam has to bear in ordinary practice should only be a small fraction of that which would break it.

456. In making any description of girder it is desirable on very special grounds that as little material as possible be uselessly employed. It will of course be remembered that a girder has to support its own weight, besides whatever may be placed upon it: and if the girder be massive, its own weight is a serious item. Of two girders, each capable of bearing the same total load, the lighter, besides employing less material, will be able to bear a greater weight placed upon it. It is therefore for a double reason desirable to diminish the weight. This remark applies especially to such a material as cast iron, which can be at once given the form in which it shall be capable of offering the greatest resistance.

457. The principles which will guide us in ascertaining the proper form to give a cast iron girder, are easily deduced from what we have laid down in Lectures XI. and XII. We have seen that depth is very desirable for a strong beam. If therefore we strive to attain great depth in a light beam, the beam must be very thin. Now an extremely thin beam will not be safe. In the first place it would be flexible and liable to displacement sideways; and, in the second place, there is a still more fatal difficulty. We [Pg 222]have shown that when a beam of wood is supporting a weight, the fibres at the bottom of the beam are extended, the tendency being to tear them (Art. 376). The fibres on the top of the beam are compressed, while the centre of the beam is in its natural state. The condition of strain in a cast iron beam is precisely similar; the bottom portions are in a state of extension, while the top is compressed. If therefore a beam be very thin, the material at the lower part may not be sufficient to withstand the forces of extension, and fracture is produced. To obviate this, we strengthen the bottom of the beam by placing extra material there. Thus we are led to the idea of a thin beam with an excess of iron at the bottom.

Fig. 64.

458. e f (Fig. 64) is the thin iron beam along the bottom of which is the stout flange shown at c d; rupture cannot commence at the bottom unless this flange be torn asunder; for until this happens it is clear that fracture cannot begin to attack the upper and slender part of the beam e f.

459. But the beam is in a state of compression along its upper side, just as in the wooden beams which we have already considered. If therefore the upper parts were not powerful enough to resist this compression, they would be crushed, and the beam would give way. The remedy for this source of weakness is obvious; a second flange runs along the top of the beam, as shown at a b. If this be strong enough to resist the compression, the stability of the beam is ensured.[Pg 223]

460. The upper flange is made very much smaller than the lower one, in consequence of a property of cast iron. This metal is more capable of resisting forces of compression than forces of extension, and it is only necessary to use one-sixth of the iron on the upper flange that is required for the lower. When the section has been thus proportioned, the beam is equally strong at both top and bottom; adding material to either flange without strengthening the other, will not benefit the girder, but will rather prove a source of weakness, by increasing the weight which has to be supported.

461. I have here a small girder made of what we are familiar with under the name of “tin,” but which is of course sheet iron thinly covered over with tin. It has the shape shown in Fig. 64, and it is 12" long. I support it at each end, and you see it bears two hundred weight without apparent deflection.

THE TUBULAR BRIDGE.

462. I shall commence the description of the principle of this bridge by performing some experiments upon a tube, which I hold in my hand. The tube is square, 1" × 1" in section, and 38" long. It is made of “tin,” and weighs rather less than a pound.

463. Here is a solid rod of iron of the same length as the tube, but containing considerably more metal. This is easily verified by weighing the tube and the rod one against the other. I shall regard them as two girders, and experiment upon their strength, and we shall find that, though the tube contains less substance than the rod, it is much the stronger.

464. I place the rod on a pair of supports about 3' apart; I then attach the tray to the middle of the rod: 14 lbs. produce a deflection of 0"·51, and 42 lbs. bends down the rod through 3"·18. This is a large [Pg 224]deflection; and when I remove the load, the rod only returns through 1"·78, thus showing that a permanent deflection of 1"·40 is produced. This proves that the rod is greatly injured, and demonstrates its unsuitability for a girder.

465. Next we place the tube upon the same supports, and treat it in the same manner. A load of 56 lbs. only produces a deflection of 0"·09, and, when this load is removed, the tube returns to its original position: this is shown by the cathetometer, for a cross is marked on the tube, and I bring the image of it on the horizontal wire of the telescope before the load of 56 lbs. is placed in the tray. When the load is removed, I see that the cross returns exactly to where it was before, thus proving that the elasticity of the tube is unimpaired. We double the load, thus placing 1 cwt. in the tray, the deflection only reaches 0"·26, and, when the load is removed, the tube is found to be permanently deflected by a quantity, at all events not greater than 0"·004; hence we learn that the tube bears easily and without injury a load more than twice as great as that which practically destroyed a rod of wrought iron, containing more iron than the tube. We load the tube still further by placing additional weights in the tray, and with 140 lbs. the tube breaks; the fracture has occurred at a joint which was soldered, and the real breaking strength of the tube, had it been continuous, is doubtless far greater. Enough, however, has been borne to show the increase of strength obtained by the tubular form.

466. We can explain the reason of this remarkable result by means of Fig. 64. Were the thin portion of the girder e f made of two parts placed side by side, the strength would not be altered. If we then imagine the flange a b widened to the width of c d, and the two parts which form e f [Pg 225]opened out so as to form a tube, the strength of the girder is still retained in its modified form.

467. A tube of rectangular section has the advantage of greater depth than a solid rod of the same weight; and if the bottom of the tube be strong enough to resist the extension, and the top strong enough to resist the compression, the girder will be stiff and strong.

468. In the Menai Tubular Bridge, where a gigantic tube supported at each end bridges over a span of four hundred and sixty feet, special arrangements have been made for strengthening the top. It is formed of cells, as wrought iron disposed in this way is especially adapted for resisting compression.

469. We have only spoken of rectangular tubes, but it is equally true for tubes of circular or other sections that when suitably constructed they are stronger than the same quantity of material, if made into a solid rod.

470. We find this principle in nature; bones and quills are often found to be hollow in order to combine lightness with strength, and the stalks of wheat and other plants are tubular for the same reason.

THE SUSPENSION BRIDGE.

471. Where a great span is required, the suspension bridge possesses many advantages. It is lighter than a girder bridge of the same span, and consequently cheaper, while its singular elegance contrasts very favourably with the appearance of more solid structures. On the other hand, a suspension bridge is not so well suited for railway traffic as the lattice girder.

472. The mechanical character of the suspension bridge is simple. If a rope or a chain be suspended from two points to which its ends are [Pg 226]attached, the chain hangs in a certain curve known to mathematicians as the catenary. The form of the catenary varies with the length of the rope, but it would not be possible to make the chain lie in a straight line between the two points of support, for reasons pointed out in Art. 20. No matter how great be the force applied, it will still be concave. When the chain is stretched until the depression in the middle is small compared with the distance between the points of support, the curve though always a catenary, has a very close resemblance to the parabola.

473. In Fig. 65 a model of a suspension bridge is shown. The two chains are fixed one on each side at the points e and f; they then pass over the piers a, d, and bridge a span of nine feet. The vertical line at the centre b c shows the greatest amount by which the chain has deflected from the horizontal a d. When the deflection of the middle of the chain is about one-tenth part of a d, the curve a c d becomes for all practical purposes a parabola. The roadway is suspended by slender iron rods from the chains, the lengths of the suspension rods being so regulated as to make it nearly horizontal.

474. The roadway in the model is laden with 8 stone weights. We have distributed them in this manner in order to represent the permanent load which a great suspension bridge has to carry. The series of weights thus arranged produces substantially the same effect as if it were actually distributed uniformly along the length. In a real suspension bridge the weight of the chain itself adds greatly to the tension.

475. We assume that the chain hangs in the form of a parabola, and that the load is uniformly ranged along the bridge. The tension upon the chains is greatest at their highest points, and least at their lowest points, though the difference is small. The amount of the tension can be calculated when the load, span, and deflection are known. We cannot give the steps of the calculation, but we shall enunciate the result.

Fig. 65.

[Pg 228]476. The magnitude of the tension at the lowest point c of each chain is found by multiplying the total weight (including chains, suspension rods, and roadway) by the span, and dividing the product by sixteen times the deflection.

The tension of the chain at the highest point a exceeds that at the lowest point c, by a weight found by multiplying the total load by the deflection, and dividing the product by twice the span.

477. The total weight of roadway, chains, and load in the model is 120 lbs.; the deflection is 10", the span 108"; the product of the weight and span is 12,960; sixteen times the deflection is 160; and, therefore, the tension at the point c is found, by dividing 12,960 by 160, to be 81 lbs.

To find the tension at the point a, we multiply 120 by 10, and divide the product by 216; the quotient is nearly 6. This added to 81 lbs. gives 87 lbs. for the tension on the chain at a.

478. One chain of the model is attached to a spring balance at a; by reference to the scale we see the tension indicated to be 90 lbs.: a sufficiently close approximation to the calculated tension of 87 lbs.

479. A large suspension bridge has its chains strained by an enormous force. It is therefore necessary that the ends of these chains be very firmly secured. A good attachment is obtained by anchoring the chain to a large iron anchor imbedded in solid rock.

480. In Art. 45 we have pointed out how the dimensions of the tie rod [Pg 229]could be determined when the tension was known. Similar considerations will enable us to calculate the size of the chain necessary for a suspension bridge when we have ascertained the tension to which it will be subjected.

481. We can easily determine by trial what effect is produced on the tension of the chain, by placing a weight upon the bridge in addition to the permanent load. Thus an additional stone weight in the centre raises the tension of the spring balance to 100 lbs.; of course the tension in the other chain is the same: and thus we find a weight of 14 lbs. has produced additional tensions of 10 lbs. each in the two chains. With a weight of 28 lbs. at the centre we find a strain of 110 lbs. on the chain.

482. These additional weights may be regarded as analogous to the weights of the vehicles which the suspension bridge is required to carry. In a large suspension bridge the tension produced by the passing loads is only a small fraction of the permanent load.

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