. Journal. e of the shaft.* K is the semidiameter * If a force W acts perpendicu-larly to a plane section of any 4r- not through the centre of gravity of its area, it will produce differentstresses at different points of thesurface. The hypothesis universally adopt-ed, usually known as the law ofthe trapesium, or Naviers law,amounts to assuming that the pianosurface after deformation accordingto the law of elasticity will still bea plane. Let the point G, Fig. 4, be of gravity of the surface, andlet the force \V act through thepoint 0, ut a distance x from G. The weight W
. Journal. e of the shaft.* K is the semidiameter * If a force W acts perpendicu-larly to a plane section of any 4r- not through the centre of gravity of its area, it will produce differentstresses at different points of thesurface. The hypothesis universally adopt-ed, usually known as the law ofthe trapesium, or Naviers law,amounts to assuming that the pianosurface after deformation accordingto the law of elasticity will still bea plane. Let the point G, Fig. 4, be of gravity of the surface, andlet the force \V act through thepoint 0, ut a distance x from G. The weight W will be balancedby elementary stresses s acting onelements of area d n. Since thedeformed surface is still a plane,the stress s in the stress diagramfor any cross section parallel to ZZwill be of the form s + /3 z, wheres = stress in the same cross sectionin the line X the same reason s will be of the form a + ax, where a = the stress at thecentre of gravity of the area. •. s = <r+ a x + /8 2-. 158 STABILITY OF CHIMNEY SHAFTS. of the kernel of the section. The kernel of the octagon isshown on Figs. 2 and 3. So long as the resultant cuts the sectionwithin the kernel there is no tension. But, from the conditions of equilibrium, JVrZn = Wfxsdn = x Wfz sd fl = 2 Now if 6 = the angle that any axis Y Y makes with Z Z, Fig 5, we haveI = f Y2 da = f(z sin e — x cos 9
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