Mechanics of engineeringComprising statics and dynamics of solids: and the mechanics of the materials of constructions, or strength and elasticity of beams, columns, arches, shafts, etc . Hence by substitutingfor p and p0 in eq. (4) of § 318 we obtain z=z0= constant [Fig. 339] (7) for a parabolic linear arch. Therefore the depth of homo-geneous loading must be the same at all points as at thecrown ; , the load is uniformly distributed with respectto the horizontal. This result might have been antici-pated from the fact that a cord assumes the parabolicform when its load (as approximately t
Mechanics of engineeringComprising statics and dynamics of solids: and the mechanics of the materials of constructions, or strength and elasticity of beams, columns, arches, shafts, etc . Hence by substitutingfor p and p0 in eq. (4) of § 318 we obtain z=z0= constant [Fig. 339] (7) for a parabolic linear arch. Therefore the depth of homo-geneous loading must be the same at all points as at thecrown ; , the load is uniformly distributed with respectto the horizontal. This result might have been antici-pated from the fact that a cord assumes the parabolicform when its load (as approximately true for suspensionbridges) is uniformly distributed horizontally. See § 46in Statics and Dynamics. 321. Linear Arch for a Given Upper Contour of Loading, thearch itself being the unknown lower contour. Given theupper curve or limit of load and the depth zQ at crown, re-quired the form of linear arch which will be in equili-brium under the homogenous load between itself and thatupper curve. In Fig. 340 let MONhe the given uppercontour of load, z0 is given or assumed,*; and z are therespective ordinates of the two curves BA G and the eqation of BAG LINEAR ARCHES. 393. Fig. 340. Fig 341. As before, the loading is homogenous, so that theweights of any portions of it are proportional to thecorresponding areas between the curves. (Unity thick-ness 1 to paper.) Now, Fig. 341, regard two consecutivedss of the linear arch as two links or consecutive blocksbearing at their junction m the load dP — y (z + z} dx inwhich y denotes the heaviness of weight of a cubic unit ofthe loading. If T and T are the thrusts exerted on thesetwo blocks by their neighbors (here supposed removed)we have the three forces dP, T and Tt forming a systemin equilibrium. Hence from IX =0. T cos <p = T coe cp and IY=0 gives T sin cp— T sin <p = dP (1) (2) From (1) it appears that T cos <p is constant at all pointsof the linear arch (just as we found in § 318) and hence= the thrust at th
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