Rational Mechanics

Strategies for Elastic Analysis and Design

Rational Mechanics

Design and analysis are intertwined concepts for the engineer. The Theory of Elasticity is now complete with 15 equations and 15 unknowns and a pure mathematician, after some discussion of uniqueness and existence, would at this point go on to other things. To the engineer, the fun has just begun.

How can we solve these equations for specific structures-the classic 'boundary-value problem'? Better yet, can we control the internal stresses flowing through a structure and its displacements by adjusting its shape or composition or support conditions? This is the 'inverse problem' or 'design problem' and is fundamentally much more obscure, difficult, and rewarding.

For all intents and purposes, textbooks and the literature today deal with the first question-finding solutions to the boundary value problem.* It is more man-ageable, particularly with the increasing power of the computer with refined numerical models. Compared to 'the design problem' it may not be as much fun, but it is easy to do and easy to teach. Moreover, for most engineers, design is thought of exclusively as a process of trial and error using successive boundary-value solutions for different shapes, materials, or support conditions.

To imaginative engineers, the great designers, solving boundary value problems is a prelude necessary but not sufficient for creating new and better structures. For them the object is to understand solid mechanics at the more profound level of discovering 'how structures work.' Solving boundary-value problems can produce this fundamental understanding from which creativity springs but only if the mathematics is thought of as a means to explore the phys-ical truths hidden in the equations and only if the results are completed through plots of the most revealing field variables with sensitivity studies of the material properties and boundary parameters. All this is then coupled with comparison to simple cases, approximate solutions from strength of materials, and evidence from structural models tested in the laboratory or at full scale in the field.

This sort of visual approach is sometimes called 'Rational Mechanics.' In this and succeeding chapters, the approach of rational mechanics* will be empha-sized. This theme differentiates this text from others that cover the same material. We will not present an in-depth development of all the closed-form solution techniques for the field equations or a compilation of known results. There are many wonderful books on elasticity with hundreds of solutions and references to hundreds more.** The field of plasticity and limit analysis is nearly as rich. Nor will we do more than comment on the various numerical methods such as Finite Difference solutions for differential equations or finite modeling techniques such as the Boundary Element and Finite Element methods, which dominate the liter-ature today. These are already extremely powerful tools for analysis and becom-ing more so, but they require a clear understanding of the fundamentals of elasticity and plasticity to be appreciated and applied properly.

Similarly the entire subject of experimental analysis must be omitted even though observation of real behavior is of paramount importance in developing a physical feel for the flow of stress and strain through structures. Photographic results from models using such experimental methods as Moire, holographic interferometry, and photoelasticity will, on occasion, be given for comparison to analytic solutions. All these techniques produce optical, full-field interference patterns, which are actually contour maps of the various displacement or stress components (more precisely, strain components). Unfortunately, however, no real explanation of these wonderful methods is possible.***

Instead we will concentrate on a few classical structures such as beams, rings, and wedges primarily in their two-dimensional idealizations. We will explore their behavior in both the elastic and plastic range from both an anal-ysis and a design perspective. Because the regular geometry of such struc-tures can be described by simple coordinate systems, closed-form solutions can often be found that best illustrate strategies at a level where the student can appreciate the intricacy of the concepts involved and their interrelation-ships. Subsequent courses on numerical or physical modeling to consider structures with irregular geometry, time-dependent boundary or thermal loading, built perhaps of anisotropic, inhomogeneous, or nonlinear materials will then be based on a firm foundation.