Make a comparison of theory of elasticity and strength of materials shear stresses for the flexure of a beam of rectangular section from Example 9.9. For each theory, calculate and plot the dimensionless shear stress τ_yz(0, y) b²/P versus y/b for an aspect ratio b/a = 1.

Data from example 9.9

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Our second flexure example involves a beam of rectangular section with end loading (P = 0, Py = P) passing through the shear center, as shown in Fig. 9.21. The section dimensions are the same as those given in Fig. 9.12. As in the previous example, there is no torsion (a = 0), and so for this case p = 0 and F = . Formulation equations (9.8.11) then give 8y = For the rectangular section dy ds dy ds with the harmonic function f satisfying f(x, y) = = dy dy dy F V P 1+ v lx dx 1 P 2 1 1 P ===b, y = b 21x Based on these boundary relations we are motivated to select a solution of the form = [1 - 6 1 - , (x - 2 x) - 52 3] 61+ 2dx ds = = 0, x=ta -x in R 0, x= a V 6(1 + v) on S -(x-ax), y = b (9.9.11) (9.9.12) (9.9.13) (9.9.14)