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 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 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.