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 . ght = H)we have. £=>+2> •• • ^ as a relation holding good for any point of the linear archwhich is to be in equilibrium under the load includedbetween itself and the given curve whose ordinates are z,Fig. 340. 322. Example of Preceding. Upper Contour a Straight Line.—Fig. 342. Let the upper contour be a right line and hor-izontal ; then the z of eq. 5 becomes zero at all points ofON. Hence drop the acce
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 . ght = H)we have. £=>+2> •• • ^ as a relation holding good for any point of the linear archwhich is to be in equilibrium under the load includedbetween itself and the given curve whose ordinates are z,Fig. 340. 322. Example of Preceding. Upper Contour a Straight Line.—Fig. 342. Let the upper contour be a right line and hor-izontal ; then the z of eq. 5 becomes zero at all points ofON. Hence drop the accent of z in eq. (5) and we have d2z zdx2 dl Multiplying which by dz we obtain dzf^=±zdz (6) dx* a2 w This being true of the z, dz, d2z and dx of each element ofthe curve OB whose equation is desired, conceive it writ-ten out for each element between 0 and any point m, andput the sum of the left-hand members of these equations= to that of the right-hand members, remembering thata2 and dx2 are the same for each element. This gives L f 4A-I fzdz; , * **J:\**\x> I a dx* 2 at 2 2 J «y tte=0 %J z=z0 (i) aaz _n \ *o/ .... (7. dx= ——= a VJ-zl <© LINEAR AKCHES. 395. Fig. 342. Fig 343 Integrating (7.) between 0 and any point m [? /, , x=a -•O-lsV®-1) P_l. /r ^© j or z- i[f+f\ (8)(8.)(9.) This curve is called the transformed catenary since we mayobtain it from a common catenary by altering all the ordi-nates of the latter in a constant ratio, just as an ellipsemay be obtained from a circle. If in eq. (9) a were = zQthe curve would be a common catenary. Supposing z0 and the co-ordinates xY and zx of the pointB (abutment) given, we may compute a from eq. 8 by put-ting x —xx and z = z{, and solving for a. Then the crown-thrust H = ya2 becomes known, and a can be used in eqs.(8) or (9) to plot points in the curve or linear arch. Fromeq. (9) we have Xr- X —X -1 1- X —X-l (10) area00mn j- ==J0zdx=2j0 [_e dx+e dx \~ 2 |_e _e J Fig. 343. Call this area
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectenginee, bookyear1888
