Remarks for Two Dimensional Solutions for Straight and Circular Beams

Concluding Remarks

Other texts such as, for example, Timoshenko’s Theory of Elasticity, give or refer to many other two-dimensional solutions for other cases of straight and curved beams. These are primarily for distributed loads which can, no matter how complicated, be represented by Fourier series to any accuracy.

However, one purpose in developing a complete field theory such as elas-ticity is to determine when it is not necessary. Throughout the chapter it is emphasized, perhaps excessively, that even for short deep beams, approximate analysis based on the assumption of plane sections turns out to be perfectly adequate for engineering purposes. Although elasticity includes transverse stresses and gives shear exactly, these refinements are not significant. Thus, for simple or complicated loading, one can proceed with the methods of determinate or indeterminate structural analysis based on shear and bending moment diagrams computing stresses, rotations, and displacements from elementary straight beam or the Winkler formulae.

Moreover, throughout the chapter we have referred to elasticity as more advanced, more sophisticated while strength-of-materials-type analysis is called elementary. A designer would argue the reverse. Elasticity is a com-plete field theory and mathematically sophisticated, but for beams it is lim-ited. The two-dimensional solutions apply only to beams of constant curvature and rectangular cross-section.

Thus elastic solutions are physically elementary when it comes to beams. Straight-beam or Winkler analysis in comparison is much more sophisti-cated. Any symmetric cross-section is fine, e.g., I-Beams, T-Beams, circular, or elliptic sections, solid or hollow. They can have any variation in curvature or, for that matter, changing depth and thickness. To designers, so-called ele-mentary analysis is the much more powerful alternative.

Consequently, books on structural mechanics here diverge from elasticity and go on to develop this highly versatile approach in all its complexity. For straight and curved beams, the idea of shear flow and the concept of shear center are developed for nonsymmetric and for thin-walled sections. Buckling and stability and the interaction effects of combined bending and axial forces become all important in design and the “elementary” approach becomes any-thing but simple. However, rather than follow that route, we will go on to analysis and design problems where elasticity is not only preferable, but often essential.