. The Encyclopaedia Britannica; ... A dictionary of arts, sciences and general literature. 1^ .a;j. Fig. 7. and let a;,, x^, x^ be the distances of the several loads !Cj, Wo,«>3 from the abutment bearing the weight P; then, takingmoments round the point of support at the above abutment,we have the upward reaction at the pier x span = thesum of the products of each weight into its distance fromthe point of support, or 2. PjL = ^wx, and similarly PL = 2w (L - a;). When the distribution of the loads is known, equation 2gives the weight borne by each abutment. Applied to thecase of a single loa
. The Encyclopaedia Britannica; ... A dictionary of arts, sciences and general literature. 1^ .a;j. Fig. 7. and let a;,, x^, x^ be the distances of the several loads !Cj, Wo,«>3 from the abutment bearing the weight P; then, takingmoments round the point of support at the above abutment,we have the upward reaction at the pier x span = thesum of the products of each weight into its distance fromthe point of support, or 2. PjL = ^wx, and similarly PL = 2w (L - a;). When the distribution of the loads is known, equation 2gives the weight borne by each abutment. Applied to thecase 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 W(L-a:)L Pr = ^^andP. Applied to the case of a uniform advancing load, such as arailway train gradually covering the whole beam (fig. 8),. Fig. 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. Pi = 2L and P = wa: 2L-Z2L • These equations express the fact that the beam, as used ina bridge, is as a whole in equilibrium under a system ofparallel vertical forces which may be called the externalforces, and which are all determinate so soon as the dis-tribution of load and the span are given. § 13. Internal Forces.—The external forces call into playcertain internal forces. A beam of given design willbe properly proportioned if each part has just thosedimensions which are suitable to bear the maximum inter-nal stress which any distribution of load can bring tobear upon it; and the beam will be properly c/eai^detf ifits form is such as to enable it to bear the load with theleast possible quantity of material. A complete analysisof the internal forces in a loaded beam would in anycase be exceedingly difficult and with mos
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