Explicitly determine the bending stress x for the problem in Example 8.4. For the case l/c

Explicitly determine the bending stress σ_x for the problem in Example 8.4. For the case l/c = 3, plot this stress distribution through the beam thickness at x = l/2, and compare with strength of materials theory. For long beams (l >> c), show that the elasticity results approach the strength of materials predictions.

Data from example 8.4

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Consider the simply supported beam carrying a sinusoidal loading along its top edge as shown in Fig. 8.6. The boundary conditions for this problem can be written as a(0, y) = a(l,y)=0 Txy(x, c) = 0 ay(x - c) = 0 ay(x, c) = -qo sin(x/I) [ty(0, y)dy = -9,1/ [Exy(1, y)dy = qol/T (8.2.8) Note that these conditions do not specify the pointwise distribution of shear stress on the ends of the beam, but rather stipulate the resultant condition based on overall problem equilibrium. Thus, we again are generating a solution valid away from the ends that would be most useful for the case where / >> c. Because the vertical normal stress has a sinusoidal variation in x along y = c, an appropriate trial solution from the general case is p = sin 6x[(A + Cay) sinh By +(B+ Day) cosh by] The stresses from this trial form are 0x = 3 sin 6x A sinh By + C(By sinh By+ 2 cosh y) +B cosh By+D(By cosh by +2 sinh By)] oy=- sin 3x[(A + CBy) sinh By+ (B+ Day) cosh By] Txy = 8 cos 8x[A+ cosh by + C(By cosh By+sinh By) + B sinh By+D(By sinh by + cosh py)] == (8.2.10) (8.2.9)