Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/333

]  §11. Change of Length due to Change of Temperature.—The change in the dimensions of structures due to a change of temperature exercises a material influence on the durability and strength of structures, and must not be lost sight of in the design of any bridge of more than common dimensions. The following short table gives the coefficient by which the length of a bar of each material measured in any unit must be multiplied to obtain the increase in length (in the same unit) resulting from a rise of temperature of 1 centigrade. VII.—Coefficient of Linear Expansion per degree cent. Cust-iron. o-ooooii Wrought Iron. 000012 Stone (paving or granite.) 000008 to 000009 Sandstone (Craitfleitli). 000012 Slate (Penryn). Brick (best Stock). 00001 0000055 Brick (Fire). 000005 Dry Deal, in direction of grain. 0000043

II.

§12. External Forces.—The beams or girders of bridges are subject to vertical loads, and they are supported by vertical reactions at piers or abutments. The sum of the loads is, therefore, necessarily equal to the sum of the reactions at the points of support ; or calling P and Pj the weights borne by two abutments (when the girder has no other support), and u, w.^ w.^, etc., the loads on various parts of the girder, we have—

1 P + P! = Sw. Fig. 7. Let L be the distance between the points of support, fig. 7, and let # x. 2, x 3 be the distances of the several loads w-^, w 2 , w s from the abutment bearing the weight P ; then, taking moments round the point of support at the above abutment, we have the upward reaction at the pier x span ? the sum of the products of each weight into its distance from the point of support, or

2. . PjL = &quot;Siiox, and similarly PL = 2tt0 (L - x). When the distribution of the loads is known, equation 2 gives the weight borne by each abutment. Applied to the case of a single load W rolling from end to end of a beam, calling x the distance of the load from the abutment sup porting the weight P, equation 2 gives

Wo; and P = W(L-.r) Applied to the case of a uniform advancing load, such as a railway train gradually covering the whole beam (fig. 8), calling x the distance covered by the train measured from the pier bearing load P, and w the weight of the load per unit of length, equation 2 gives

4. . . . P 1 = ZLi 21j These equations express the fact that the beam, as used in a bridge, is as a whole in equilibrium under a system of parallel vertical forces which may be called the external forces, and which are all determinate so soon as the dis tribution of load and the span are given. Fig. 8.

§13. Internal Forces.—The external forces call into play certain internal forces. A beam of given design will be properly proportioned if each part has just those dimensions which are suitable to bear the maximum inter nal stress which any distribution of load can bring to bear upon it ; and the beam will be properly designed if its form is such as to enable it to bear the load with the least possible quantity of material. A complete analysis of the internal forces in a loaded beam would in any case be exceedingly difficult and with most designs impos sible, but it is found by experiment that a beam will bear a given load if we provide strength enough to resist the horizontal components of those internal forces which tend to extend or compress the beam in the direction of its length, and if we also provide against the vertical com ponents which tend to shear the beam across in planes perpendicular to the longitudinal axis of the beam. The nature of the horizontal and vertical forces due to the elastic reaction of the material will be understood by refer ence to figs. 9, 10, 11. Fig. 9. Let a model be made of four stiff light frames of vrood A, A p A 2, A 3, each say 18 inches deep, 18 inches long, and 6 inches wide (fig. 9). Let these be connected with one another Ly smallcylinders of india-rubber, ab cd, a l b 1 c-^ d v and a 2 6 2 c 2 cZ 2. These cylinders must be so attached to the frames as to be capable of resisting both extension and com pression. The whole structure will now have somewhat the appearance of a beam, but if it is placed between two sup ports Q, N, it will be found unsuited to carry even its own weight, because the middle frames will tend to slip down between the two others. Fig. 10. This tendency will be still better seen when a load is placed on the imperfect beam. Fig. 1 shows the tendency to shear off the loaded from the unloaded part of the beam. The frame A,, is forced down below the frame A 3 by a shearing stress resisted by the india-rubber in a very 