Determine the displacement field for the flexure problem of a beam of circular section given in Example 9.8. Starting with the stress solution (9.9.9), integrate the strain-displacement relations and use boundary conditions that require the displacements and rotations to vanish at z = 0. Compare the elasticity results with strength of materials theory. Also investigate whether the elasticity displacements indicate that plane sections remain plane.

Data from example 9.8

 

Equation 9.9.9

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Consider the flexure of an elastic beam of circular section, as shown in Fig. 9.20. The end loading (Px = 0, Py = P) passes through the center of the section, which coincides with the centroid and center of twist. Thus, for this problem there will be no torsion (a = 0), and so = 0 and F = V. It is convenient to use polar coordinates for this problem, and the governing equation (9.8.11) can then be written as V P 1+vIx 8y = while the boundary condition (9.8.11)2 becomes 1 0 1 P = The solution to (9.9.1) can then be taken as -r cos 0 Pa sin0 on r=a 21x P V * = [ - 1 1 - - cos0] 61+v x (9.9.1) (9.9.2) (9.9.3)