General Structural Principles I

The previous chapters developed the analysis of particular structural systems by the consistent use of the twin concepts of compatibility and equilibrium. In each case, we derived a set of governing differential equations, integrated them, and then determined the constants of integration by satisfying the appropriate boundary and compatibility conditions. This approach will work in every case, but as a practical tool it suffers from a number of drawbacks. Primary among these is that the solution can become very cumbersome when more than one integration region is involved. It may also happen that the governing differential equations cannot be integrated in a closed form. When approximations are used the differential equations are not in a suitable form for direct manipulation. For example, we obtained the approximate geometric matrix in Chapter 5 by first obtaining its exact form, and then using a Taylor series expansion to obtain the linearized version. But if the cross-sectional area varies, for example, this approach is not feasible at all; it would be much more convenient if we could go directly to the approximate solution.