=103. Stresses in Circular Ring=—In Fig. 81_a_ the loads shown indicate the distribution ordinarily assumed in sewer design, the forces being uniformly distributed across the diameter. To find the bending moment in the pipe caused by this loading, let _ab_ in Fig. 81_b_ represent a section of a pipe loaded with equally distributed horizontal and vertical forces. Then the vertical component on a strip of differential length _ds_ is _wds_ cos Θ and the horizontal component is _wds_ sin Θ and resolving, the resultant normal to the surface is _wds_, in which _w_ is the intensity per unit length of the horizontal and vertical forces and Θ is the angle which the tangent to _ds_ makes with the horizontal. Thus the loading of the nature shown in Fig. 81_b_ is equivalent to a loading of equally distributed normal forces which give no moment in the ring. [Illustration: FIG. 81.—Distribution of Stresses on Buried Pipe. ] Considering a ring subjected to vertical forces only, the moments will be as shown in Fig. 81_c_ and if loaded with horizontal forces only, the moments will be as shown in Fig. 81_d_. Because of the symmetry of the figure, moment (1) equals moment (4) but is opposite in direction and moment (2) equals moment (3) but is opposite in direction. When the horizontal and vertical forces are combined on the same ring as in Fig. 81_b_ these moments cancel each other as has been proven. Therefore moment (1) equals moment (2) and moment (3) equals moment (4). Then in Fig. 81_e_, _M_{a}_ = _M_{b}_. Now ∑_M_ = _O_ for conditions of equilibrium, therefore _M_{a}_ + _M_{b}_ + (_W_⁄2)(_d_⁄4) = _O_ and solving _M_{a}_ = (_Wd_)⁄16. This moment occurs at the ends of the horizontal and vertical diameters and causes tension on the inside of the pipe at the top and on the outside at the ends of the horizontal diameter. There will also be compression at each end of the horizontal diameter equal to one-half of the total load on the pipe. If the material of the pipe is homogeneous, the maximum fiber stress _f_ can be found through the expression _f_ = (_My_)⁄_I_ ± _P_⁄_A_ in which _M_ is the bending moment, _y_ is the distance from the neutral axis to the extreme fiber of a cross-section of the shell of the pipe of unit length, _I_ is the moment of inertia of this cross-section about its neutral axis, _P_ is one-half the total load on the pipe, and _A_ is the area of the cross-section. For reinforced concrete, the standard formulas should be used with this expression for _M_. The stresses in a circular ring subjected to other distributions of loads are shown in Table 48. An exhaustive study of the stresses in circular rings was published by Prof. A. N. Talbot in Bulletin No. 22 of the Engineering Experiment Station at the University of Illinois, 1908. TABLE 48 MAXIMUM STRESS IN FLEXIBLE RINGS DUE TO DIFFERENT LOADINGS