THE PRINCIPLES OF FRAMEWORK

Experimental Mechanics by Robert S. Ball is part of the HackerNoon Books Series. You can jump to any chapter in this book here. THE PRINCIPLES OF FRAMEWORK

LECTURE XIII. THE PRINCIPLES OF FRAMEWORK.

Introduction.—Weight sustained by Tie and Strut.—Bridge with Two Struts.—Bridge with Four Struts.—Bridge with Two Ties.—Simple Form of Trussed Bridge.

INTRODUCTION.

415. In this lecture and the next we shall experiment upon some of the arts of construction. We shall employ slips of pine 0"·5 × 0"·5 in section for the purpose of making models of simple framework: these slips can be attached to each other by means of the small clamps about 3" long, shown in Fig. 56, and the general appearance of the models thus produced may be seen from Figs. 58 and 62.

Fig. 56.

416. The following experiment shows the tenacity with which these clamps hold. Two slips of pine, each 12" × 0"·5 × 0"·5, are clamped together, so that they overlap about 2", thus forming a length of 22": this composite rod is raised by a pulley-block as in Fig. 49, while a load of 2 cwt. is suspended from it. Thus the clamped rods bear a direct [Pg 204]tension of 2 cwt. The efficiency of the clamps depends principally upon friction, aided doubtless by a slight crushing of the wood, which brings the surfaces into perfect contact.

417. These slips of pine united by the clamps are possessed of strength quite sufficient for the experiments now to be described. Models thus constructed have the great advantage of being erected, varied or pulled down, with the utmost facility.

We have learned that the compressive strength, and, still more, the tensile strength of timber, is much greater than its transverse strength. This principle is largely used in the arts of construction. We endeavour by means of suitable combinations to turn transverse forces into forces of tension or compression, and thus strengthen our constructions. We shall illustrate the mode of doing so by simple forms of framework.

WEIGHT SUSTAINED BY TIE AND STRUT.

418. We begin with the study of a very simple contrivance, represented in Fig. 57.

a b is a rod of pine 20" long. In the diagram it is represented, for simplicity, imbedded at the end a in the support. In reality, however, it is clamped to the support, and the same remark may be made about some other diagrams used in this lecture. Were a b unsupported except at its end a, it would of course break when a weight of 10 lbs. was suspended at b, as we have already found in Art. 414.

419. We must ascertain whether the transverse force on a b cannot be changed into forces of tension and compression. The tie b c is attached by means of clamps; a b is sustained by this tie; it cannot bend downwards under the action of the weight [Pg 205]w, because we should then require to have on the same base and on the same side of it two triangles having their conterminous sides equal, but this we know from Euclid (I. 7) is impossible. Hence b is supported, and we find that 112 lbs. may be safely suspended, so that the strength is enormously increased. In fact the transverse force is changed into a compressive force or thrust down a b, and a tensile force on b c.

Fig. 57.

  420. The actual magnitudes of these can be computed. Draw the parallelogram c d e b; if b d represent the weight w, it may be resolved into two forces,—one, b c, a force of extension on the tie; the other, b e, a compressive force on a b, which is therefore a strut. Hence the forces are proportional to the sides of the triangle, a b c. In the present case

a b = 20",  a c = 18",   b c = 27";

therefore, when w is 112 lbs., we calculate that the force on a b is 124 lbs., and on c b 168 lbs. a b would require about 300 lbs. to crush it, and c b about 2,000 lbs. to tear it asunder, consequently the tie and strut can support [Pg 206]1 cwt. with ease. If, however, w were increased to about 270 lbs., the force on a b would become too great, and fracture would arise from the collapse of this strut.

421. When a structure is loaded up to the breaking point of one part, it is proper for economy that all the other parts should be so designed that they shall be as near as possible to their breaking points. In fact, since nothing is stronger than its weakest part, any additional strength which the remaining parts may possess adds no strength to the whole, and is only so much material wasted. Hence our structure would be just as strong, and would be more properly designed if the section of b c were reduced to one-fifth, for the tie would then break when the tension upon it amounted to 400 lbs. When w is 270 lbs. the compression on a b is 300 lbs., and the tension on b c is 405 lbs., so that both tie and strut attain their breaking loads together. The principle of duly apportioning the strength of each piece to the load it has to carry, involves the essence of sound engineering. In that greatest of mechanical feats, the construction of a mighty railway bridge across a wide span, attention to this principle is of vital importance. Such a bridge has to bear the occasional load of a passing train, but it has always to support the far greater load of the bridge materials. There is thus every inducement to make the weight of each part of the bridge as light as may be consistent with safety.

A BRIDGE WITH TWO STRUTS.

422. We shall next examine the structure of a type of bridge, shown in Fig. 58.

Fig. 58.

It consists of two beams, a b, 4' long, placed parallel to each other at a distance of 3"·5, and supported at each end; they are firmly clamped to the supports, and a roadway of short pieces is laid upon them. At the points of trisection of the beams c, d, struts c f and d e are clamped, their lower ends being supported by the framework: these struts are 2' long, and there are two of them supporting each of the beams. The tray [Pg 208]g is attached by a chain to a stout piece of wood, which rests upon the roadway at the centre of the bridge.

423. We shall first determine the strength of this bridge by actual experiment, and then we shall endeavour to explain the results in accordance with mechanical principles. We can observe the deflection of the bridge by the cathetometer in the manner already described (Art. 362). By this means we shall ascertain whether the load has permanently injured the elasticity of the structure (Art. 367). We begin by testing the deflection when a load is distributed uniformly, as the weights are disposed in the case of Fig. 62. A cross is marked upon one of the beams, and is viewed in the cathetometer. We arrange 11 stone weights along the bridge, and the cathetometer shows that the deflection is only 0"·09: the elasticity of the bridge remains unaltered, for when the weights are removed the cross on the beam returns to its original position; hence the bridge is well able to bear this load.

424. We remove the row of weights from the bridge and suspend the tray from the roadway. I take my place at the cathetometer to note the deflection, while my assistant places weights h h on the tray. 1 cwt. being the load, I see that the deflection amounts to 0"·2; with 2 cwt. the deflection reaches 0·43"; and the bridge breaks with 238 lbs.

425. Let us endeavour to calculate the additional strength which the struts have imparted to the bridge. By Table XXIV. we see that a rod 40" × 0"·5 × 0"·5 is broken by a load of 19 lbs.: hence the beams of the bridge would have been broken by a load of 38 lbs. if their ends had been free. As, however, the ends of the beams had been clamped down, we learn from Art. 411 that a double load would be necessary. [Pg 209]We may, however, be confident that about 80 lbs. would have broken the unsupported bridge. The strength is, therefore, increased threefold by the struts, for a load of 238 lbs. was required to produce fracture.

426. We might have anticipated this result, because the points c and d being supported by the struts may be considered as almost fixed points; in fact, we see that c cannot descend, because the triangle a c f is unalterable, and for a similar reason d remains fixed: the beam breaks between c and d, and the force required must therefore be sufficient to break a beam supported at the points c and d, whose ends are secured. But c d is one-third of a b, and we have already seen that the strength of a beam is inversely as its length (Art. 388); hence the force required to break the beam when supported by the struts is three times as large as would have been necessary to break the unsupported beam. Thus the strength of the bridge is explained.

427. As a load of 238 lbs. applied near the centre is necessary to break this bridge, it follows from the principle of Art. 408 that a load of about double this amount must be placed uniformly on the roadway before it succumbs; we can, therefore, understand how a load of 11 stone was easily borne (Art 423) without permanent injury to the elasticity of the structure. If we take the factor of safety as 3, we see that a bridge of the form we have been considering may carry, as its ordinary working load, a far greater weight than would have crushed it if unsupported by the struts and with free ends.

428. The strength of the bridge in Fig. 58 is greater in some parts than in others. At the points c and d a maximum load could be borne; the weakest places on the bridge are in the middle [Pg 210]points of the segments a c, d c, and d b. The load applied by the tray was principally borne at the middle of d c, but owing to the piece of wood which sustained the chain being about 18" long, the load was to some extent distributed.

The thrust upon the struts is not so easy to calculate accurately. That down c f for example must be less than if the part c d were removed, and half the load were suspended from c. The force in this case can be determined by principles already explained (Art. 420).

A BRIDGE WITH FOUR STRUTS.

429. The same principles that we have employed in the construction of the bridge of Fig. 58 may be extended further, as shown in the diagram of Fig. 59.

Fig. 59.

We have here two horizontal rods, 48" × 0"·5 × 0"·5, each end being secured to the supports; one of these rods is shown in the figure. It is divided into five equal parts in the points b, c, c´, b´. We support the rod in these four points by struts, the other extremities of which are fastened to the framework. The points b, c, c´, b´ are fixed, as they are sustained by the struts: hence a weight suspended from p, which is to break the bridge, must be sufficiently strong to [Pg 211]break a piece c c´, which is secured at the ends; the rod a a´ would have been broken with 38 lbs., hence 190 lbs. would be necessary to break c c´. There is a similar beam on the other side of the bridge, and therefore to break the bridge 380 lbs. would be necessary, but this force must be applied exactly at the centre of c c´; and if the weights be spread over any considerable length, a heavier load will be necessary. In fact, if I were to distribute the weight uniformly over the distance c c´, it appears from Art. 408 that double the load would be necessary to produce fracture.

430. We shall now break this model. I place 18 stone upon it ranged uniformly, and the cathetometer tells me that the bridge only deflects 0"·1, and that its elasticity is not injured. Placing the tray in position, and loading the bridge by this means, I find with a weight of 2 cwt. that there is a deflection of 0"·15; with 4 cwt. the deflection amounts to 0'·72. We therefore infer that the bridge is beginning to yield, and the clamps give way when the load is increased to 500 lbs.

A BRIDGE WITH TWO TIES.

431. It might happen that circumstances would not make it convenient to obtain points of support below the bridge on which to erect the struts. In such a case, if suitable positions for ties can be obtained, a bridge of the form represented in Fig. 60 may be used.

a d is a horizontal rod of pine 40" × 0"·5 × 0"·5; it is trisected in the points b and c, from which points the ties b e and c f are secured to the upper parts of the framework. a d is then supported in the points b and c, which may therefore be regarded as fixed points. Hence, for the reasons we have already explained, the strength of the bridge should be increased nearly threefold. Remembering that the [Pg 212]bridge has two beams we know it would require about 70 lbs. or 80 lbs. to produce fracture without the ties, and therefore we might expect that over 200 lbs. would be necessary when the beams were supported by the ties. I perform the experiment, and you see the bridge yields when the load reaches 194 lbs.: this is somewhat less than the amount we had calculated; the reason being, I think, that one of the clamps slipped before fracture.

Fig. 60.

A SIMPLE FORM OF TRUSS.

432. It is often not convenient, or even possible, to sustain a bridge by the methods we have been considering. It is desirable therefore to inquire whether we cannot arrange some plan of strengthening a beam, by giving to it what shall be equivalent to an increase of depth.

433. We shall only be able to describe here some very simple methods for doing this. Superb examples are to be found in railway bridges all over the country, but the full investigation of these complex structures is a problem of no little difficulty, and one into which it [Pg 213]would be quite beyond our province to enter. We shall, however, show how by a judicious combination of several parts a structure can offer sufficient resistance. The most complex lattice girder is little more than a network of ties and struts.

Fig. 61.

434. Let a b (Fig. 61) be a rod of pine 40" × 0"·5" × 0"·5, secured at each end. We shall suppose that the load is applied at the two points g and h, in the manner shown in the figure. The load which a bridge must bear when a train passes over it is distributed over a distance equal to the length of the train, and the weight of the bridge itself is of course arranged along the entire span; hence the load which a bridge bears is at all times more or less distributed and never entirely concentrated at the centre in the manner we have been considering. In the present experiment we shall apply the breaking load at the two points g and h, as this will be a variation from the mode we have latterly used. e f is an iron bar supported in the loops e g and f h. Let us first try what weight will break the beam. Suspending the tray from e f, I find that a load of 48 lbs. is sufficient; much less would have done had not the ends been clamped. We have already applied a load in this manner in Art. 406.[Pg 214]

435. You observed that the beam, as usual, deflected before it broke; if we could prevent deflection we might reasonably expect to increase the strength. Thus if we support the centre of the beam c, deflection would be prevented. This can be done very simply. We clamp the pieces d a, d b, d c, on a similar beam, and it is evident that c cannot descend so long as the joints at a, b, d, c remain firmly secured. We now find that even with a weight of 112 lbs. in the tray, the bar is unbroken. An arrangement of this kind is frequently employed in engineering, for it seems to be able to bear more than double the load which is sufficient to break the unsupported beam.

Fig. 62.

436. Two frames of this kind, with a roadway laid between them, would form a bridge, or if the frames were turned upside down they would answer equally well, though of course in this case d a and d b would become ties, and d c a strut, but a better arrangement for a bridge will be next described.[Pg 215]

THE WYE BRIDGE.

437. An instructive bridge was erected by the late Sir I. Brunel over the Wye, for the purpose of carrying a railway. The essential parts of the bridge are represented in the model shown in Fig. 62, which as before is made of slips of pine clamped together.

438. Our model is composed of two similar frames, one of which we shall describe, a b is a rod of pine 48" × 0"·5 × 0"·5, supported at each extremity. This rod is sustained at its points of trisection d, c by the uprights d e and c f, while e and f are supported by the rods b e, f e, and a f; the rectangle d e f c is stiffened by the piece c e, and it would be proper in an actual structure to have a piece connecting d and f, but it has not been introduced into the model.

Fig. 63.

439. We shall understand the use of the diagonal c e by an inspection of Fig. 63. Suppose the quadrilateral a b c d be formed of four pieces of wood hinged at the corners. It is evident that this quadrilateral can be deformed by pressing a and c together, or by pulling them asunder. Even if there were actual joints at the corners, it would be almost impossible to make the quadrilateral stiff by the strength of the joints. You see this by the frame which I hold in my hand; the pieces are clamped together at the corners, but no matter how tightly I compress the clamps, I am able with the slightest exertion to deform the figure.

440. We must therefore look for some method of stiffening the frame. I [Pg 216]have here a triangle of three pieces, which have been simply clamped together at the corners; this triangle is unalterable in form; in fact, since it is impossible to make two different triangles with the same three sides, it is evident the triangle cannot be deformed. This points to a guiding principle in all bridgework. The quadrilateral is not stiff because innumerable different quadrilaterals can be made with the same four sides. But if we draw the diagonal a c of the quadrilateral it is divided into two triangles, and hence when we attach to the quadrilateral, which has been clamped at the four corners, an additional piece in the direction of one of the diagonals, it becomes unalterable in shape.

441. In Fig. 63 we have drawn the two diagonals a c and b d: one would be theoretically sufficient, but it is desirable to have both, and for the following reason. If I pull a and c apart, I stretch the diagonal a c and compress b d. If I compress a and c together, I compress the line a c and extend b d; hence in one of these cases a c is a tie, and in the other it is a strut. It therefore follows that in all cases one of the diagonals is a tie, and the other a strut. If then we have only one diagonal, it is called upon to perform alternately the functions of a tie and of a strut. This is not desirable, because it is evident that a piece which may act perfectly as a tie may be very unsuitable for a strut, and vice versâ. But if we insert both diagonals we may make both of them ties, or both of them struts, and the frame must be rigid. Thus for example, I might make a c and b d slender bars of wrought iron, which form admirable ties, though quite incapable of acting as struts.

442. What we have said with reference to the necessity for dividing a quadrilateral figure into triangles applies still more to a polygon with a large number of sides, and we may lay down the general principle that every such piece of framework should be composed of triangles.[Pg 217]

443. Returning to Fig. 62, we see the reason why the rectangle e d c f should have one or both of its diagonals introduced. A load placed, for example, at d would tend to depress the piece d e, and thus deform the rectangle, but when the diagonals are introduced this deformation is impossible.

444. Hence one of these frames is almost as strong as a beam supported at the points c and d, and therefore, from the principles of Art. 388, its strength is three times as great as that of an unsupported beam.

445. The two frames placed side by side and carrying a roadway form an admirable bridge, quite independent of any external support, except that given by the piers upon which the extremities of the frames rest. It would be proper to connect the frames together by means of braces, which are not, however, shown in the figure. The model is represented as carrying a uniform load in contradistinction to Fig. 58, where the weight is applied at a single point.

446. With eight stone ranged along it, the bridge of Fig. 62 did not indicate an appreciable deflection.

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