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 , and the general appearance of the models thus produced may be seen from  and .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 , while a load of 2 cwt. is suspended from it. Thus the clamped rods bear a direct .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 .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 .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 ). By this means we shall ascertain whether the load has permanently injured the elasticity of the structure (). We begin by testing the deflection when a load is distributed uniformly, as the weights are disposed in the case of . 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 . 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  that a double load would be necessary. ); 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  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  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 ).A BRIDGE WITH FOUR STRUTS.429. The same principles that we have employed in the construction of the bridge of  may be extended further, as shown in the diagram of .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  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  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 ) 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 ., 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 . 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  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., 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 , 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 , where the weight is applied at a single point.446. With eight stone ranged along it, the bridge of  did not indicate an appreciable deflection.

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