The first term is the central deflection piedicted by the simple bending theory. The second term is the correction to include deflection due to shear. As indicated by the form of eqn. (8.101) the latter is small when the span/depth ratio is large, but is more significant for deep beams. By combining equations (8.100) and (8.101) the displacements u and v can be obtained at any point (x, y) in the beam.

8.27.3. The use of polar coordinates in two dimensions

Many engineering components have a degree of axial symmetry, that is they are either rotationally symmetric about a central axis, as in a circular ring, disc and thick cylinder, or contain circular holes which dominate the stress field, or yet again are made up from parts of hollow discs, like a curved bar. In such cases it is advantageous to use cylindrical polar coordinates (r, 6, z), where r and 6 are measured from a fixed origin and axis, respectively and t is in the axial direction. The equilibrium equations for this case are given in eqns. (8.40) and (8.41).

The form of applied loading for these components need not be restricted to the simple rotationally symmetric cases dealt with in earlier chapters. In fact the great value of the stress function concept is that complex loading patterns can be adequately represented by the use of either cos nQ and/or sin nO, where n is the harmonic order.

A two-dimensional stress field (arr, am. r^) is again used for these cases. That is plane stress or plane strain is assumed to provide an adequate approximation of the three-dimensional problem. The next step is to transform the biharmonic eqn. (8.87) to the relevant polar form, namely:

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