Structural Analysis Prior to Computers

Basic principles of structural analysis—static equilibrium, trusses, and beam theory—were known long before computers, or airplanes, existed. Bridges, towers and other buildings, and ships were designed by a combination of experience and some amount of analysis—more so as designs became larger and more ambitious during and after the Industrial Revolution.

With airplanes came much greater emphasis on weight minimiza­tion. Massive overdesign was no longer an acceptable means to achieve structural integrity. More rigorous analysis and structural sizing was required. Simplifications allowed the analysis of primary members under simple loading conditions:

• Slender beams: axial load, shear, bending, torsion.

• Trusses: members carry axial load only, joined to other such members at ends.

• Simple shells: pressure loading.

• Semi-monocoque (skin and stringer) structures: shear flow, etc.

• Superposition of loading conditions.

With these simplifications, primary structural members could be sized appropriately to the expected loads. In the days of wood, wire, and fabric, many aircraft structures could be analyzed as trusses: exter­nally braced biplane wings; fuselage structures consisting of longerons, uprights, and cross braces, with diagonal braces or wires carrying tor­sion; landing gears; and engine mounts. As early as the First World War and in the 1920s, researchers were working to cover every required aspect of the problem: general analysis methods, analysis of wings, horizontal and vertical tails, gust loads, test methods, etc. The National Advisory Committee for Aeronautics (NACA) contributed significantly to the build­ing of this early body of methodology.[787]

Structures with redundancy—multiple structural members capable of sharing one or more loading components—may be desirable for safety, but they posed new problems for analysis. Redundant structures cannot be analyzed by force equilibrium alone. A conservative simplification, often practiced in the early days of aviation, was to analyze the struc­ture with redundant members missing. A more precise solution would require the consideration of displacements and "compatibility” condi­tions: members that are connected to one another must deform in such a manner that they move together at the point of connection. Analysis was feasible but time-consuming. Large-scale solutions to redundant ("statically indeterminate”) structure problems would become practical with the aid of computers. Until then, more simplifications were made, and specific types of solutions—very useful ones—were developed.

While these analysis methods were being developed, there was a lot of airplane building going on without very much analysis at all. In the "golden age of aviation,” many airplanes were built in garages or at small companies that lacked the resources for extensive analysis. "In many cases people who flew the airplanes were the same people who car­ried out the analysis and design. They also owned the company. There was very little of what we now call structural analysis. Engineers were brought in and paid—not to design the aircraft—but to certify that the aircraft met certain safety requirements.”[788]

Through the 1930s, as aircraft structures began to be formed out of aluminum, the semi-monocoque or skin-and-stringer structure became prevalent, and analysis methods were developed to suit. "In the 1930s, ’40s, and ’50s, techniques were being developed to analyze specific struc­tural components, such as wing boxes and shear panels, with combined bending, torsion, and shear loads and with stiffeners on the skins.”[789] A number of exact solutions to the differential equations for stress and strain in a structural member were known, but these generally exist only for very simple geometric shapes and very limited sets of loading conditions and boundary conditions. Exact solutions were of little prac­tical value to the aircraft designer or stress analyst. Instead, "free body diagrams” were used to analyze structures at selected locations, or "sta­tions.” The structure was considered to be cut by a theoretical plane at the station of interest. All loads, applied and inertial, on the portion of the aircraft outboard of the cut had to be borne (reacted) by the struc­ture at the cut.

In principle, this allowed the stress at any point in the structure to be analyzed—given the time to make an arbitrarily large number of these theoretical cuts through the aircraft. In practice, free body dia­grams were used to analyze the structure at key locations—selected fuselage stations, the root, and selected stations of wings and tail sur­faces. Structural members were left constant, or tapered appropriately, according to experience and judgment, between the analyzed sections. For major projects such as airliners or bombers, the analysis would be more thorough, and consequently, major design organizations had rooms full of people whose jobs were to perform the required calculations.

The NACA also utilized this brute-force approach to large calcu­lations, and the people who performed the calculations—overwhelm­ingly women—were called "computers.” Annie J. Easley, who worked at the NASA Lewis (now Glenn) Research Center starting in 1955, recalls:

. . . we were called computers until we started to get the machines, and then we were changed over to either math tech­nicians or mathematicians. . . . The engineers and the scien­tists are working away in their labs and their test cells, and they come up with problems that need mathematical compu­tation. At that time, they would bring that portion to the com­puters, and our equipment then were the huge calculators, where you’d put in some numbers and it would clonk, clonk, clonk out some answers, and you would record them by hand. Could add, subtract, multiply, and divide. That was pretty much what those big machines, those big desktop machines, could do. If we needed to find a logarithm or an exponential, we then pulled out the tables.[790]

After World War II, with jet engines pushing aircraft into ever more demanding flight regimes, the analytical community sought to keep up. The NACA continued to improve the methodologies for calculating loads on various parts of an aircraft, and some of the reports generated during that time are still used by industry practitioners today. NACA Technical Report (TR) 1007, for horizontal tail loads in pitch maneuvers, is a good example, although it does not cover all of the conditions required by recent airworthiness regulations.[791]

For structural analysis, energy methods and matrix methods began to receive more attention. Energy methods work as follows: one first expresses the deflection of a member as a set of assumed shape func­tions, each multiplied by an (initially unknown) coefficient; expresses the total strain energy in terms of these unknown coefficients; and finally, finds the values of the coefficients that minimize the strain energy. If the shape functions, from which the solution is built, satisfy the boundary conditions of the problem, then so does the final solution.

Energy methods were not new. The concept of energy minimization was introduced by Lord Rayleigh in the late 19 th century and extended by Walter Ritz in two papers of 1908 and 1909.[792] Rayleigh and Ritz were par­ticularly concerned with vibrations. Carlo Alberto Castigliano, an Italian engineer, published a dissertation in 1873 that included two important theorems for applying energy principles to forces and static displace­ments in structures.[793] However, in the early works, the shape functions were continuous over the domain of interest. The idea of breaking up (discretizing) a complex structure into many simple elements for numer­ical solution would lead to the concept of finite elements, but for this to be useful, computing technology needed to mature.