Variational and approximate solutions

In chapter 1, the fundamental equations of linear elasticity are developed, and fifteen equations fully describe the mechanics of deformable bodies. Unfortunately, these governing equations are partial differential equations in three dimensions, and although of first order only, their solution cannot be completed in closed form for most practical problems. Open form or series solutions have been developed for a limited number of applications, but no general approach exists for solving these equations in closed form. A barrier to the development of closed-form solutions to partial differential equations is the fact the arbitrary integration constants involved in the solution of ordinary differential equations are now replaced by arbitrary integration functions. Consequently, boundary conditions often play a greater role in the solution process for partial differential equations.

A very successful approach for dealing with complex problems is to reduce their geometric dimensionality from three to one, thereby replacing the governing partial differential equations by ordinary differential equations for which general solution procedures are available. A important example of this dimensional reduction procedure is beam theory, which leads to the ordinary differential equations presented in chapters 4 through 8. The reduction is based on the assumption that long, slender beams have one dimension, their span, which is much larger than the cross-sectional dimensions. Another important example of dimensional reduction is plate theory, presented in chapter 16, which transforms the three-dimensional elasticity equations into two-dimensional partial differential equations. In this case, the basic assumption is that one of the plate’s geometric dimensions, its thickness, is much smaller than the other two.