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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.

**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.

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