Introduction

“Non-Linear Theory of Elasticity and Optimal Design” deals with developing and proving a new fundamental theory. Although the useful concepts and methods of the current Linear Theory of Infinitesimal Deformations remain, the basic physical concept of strength and elastic stability of a structure changes. The logical structure of the theory of elasticity, the concepts, the criterion of strength and elastic stability, the equation of deformation have been changed, and an equation for elastic stability was added. The method for optimizing the dimensions of a structure is new. The approach to mathematics in physical theory is changed. A new point of view on the role of logic in the construction of physical theory is presented. Logic becomes definitive.

The theory of elasticity is the foundation of structural design. An important characteristic of elastic relations is the limit of elasticity. The limit for an individual structure currently can be found only by testing the structure destructively. The reason is that linear theory by its nature cannot describe a limit, because a limit is not a property of a linear function. The non-linear theory presented in this book, on the other hand, describes limits for individual structures and allows optimization of structures.

A new concept of strength is associated with the non-linear theory of elasticity. The actual limit of elasticity of a structure, which reveals itself in the destruction of the structure, can be of different physical origins. It can be the limit of the material, but more often it is generated by the geometry of the structure. Both limits should be known for structural analysis and design to be successful. This book describes a simple non-destructive method of establishing minimal reliable dimensions of a structure.

This engineering problem is at the foundation of structural analysis and design. The safety and cost of a structure in the mechanical, civil and aerospace engineering fields depend on establishing minimal reliable dimensions for the structure. The problem, formulated in 1638 by Galileo, “is to find the form of the generating curves so that the resistance of a section may be exactly equal to the tendency to rupture at that place.” Galileo was unaware of the elastic properties of materials and did not describe relations mathematically.

The English physicist Robert Hooke discovered the existence of elastic properties of materials and structures in 1678. Since that time a continuous effort has been made to find a scientific method for predicting the limit of elastic relations and establishing safe dimensions of structures within that limit.

There are reasons for the fact that this optimization problem has not been solved although a mathematical method of optimization exists. If the problem could be solved using empirical, statistical and probabilistic methods, it would have been solved already, for there is no lack of empirical data. If it were possible to find a solution for the optimization problem within the framework of the established linear theory of elasticity, then it would have been done in the 19th century when the mathematical apparatus of linear theory was developed.

Solving the problem of structure optimization is possible only after a revision of the linear theory of elasticity, its logical structure, mathematical apparatus and physical foundation, which are presented in this book. Solving the problem is connected not only with the criticism of current theory, but also with the development of a new reliable method of construction and verification: the Non-Linear Theory of Elasticity (NLTE).

The new theory has a new criterion that designates fundamental changes. “If the actually formulated laws of our physics can be shown to undergo change themselves, it can only be in reference to something else which is constant in relation to them” (“A Preface to Logic”, Morris R. Cohen). Currently, the criterion for design calculations is the limit of elasticity of the material. NLTE factors in the rate of change of deformation. The main reason for choosing this criterion is that the limit of elasticity corresponds with a rapid increase of change in deformation. A mathematical description for the rate of change is missing in the linear theory. The new equation of elastic stability is obtained as a derivative of the basic equation of deformation. Not every description of deformation reflects elastic relations correctly. Here a new equation of deformation is presented and justified.

In the 20th century new technology to build high-rise buildings, airplanes, bridges, and the like, developed rapidly. But at the same time the science of structural design stagnated. Engineering disciplines such as “Strength of Materials” and the “Theory of Elasticity” have practically been closed to free independent scientific thought by standard-setting organizations that control scientific ideas and research, engineering publications and engineering practices. For example, one of the prominent standard-setting organizations, the American Institute of Steel Constructions (AISC), representing the interests of steel fabricators, has served as the link between the steel monopolies and the countless manufacturers and builders who use standard steel products. Scientific laws, which have a tendency to change, became the objects of governmental laws. Another reason for the slow development of science is the inertia of established theoretical principles.

Until now the theory of elasticity has been designed as the Linear Theory of Infinitesimal Deformations. The crisis of linear theory came to light at the end of the 19th century after the main principles and mathematical descriptions of the theory had been developed. “Except in very simple cases, the demonstrations are less rigorous than those which form the Mathematical Theory of Elasticity, an exact science which is unable to furnish solutions for the majority of the practical problems which present themselves to the engineer in the design of machines and structures.” (“Strength of Materials”, Arthur Morley, Eds. 1908 till 1954). Since the 1950s no major changes in this science have occurred despite the fact that the design process has become more complicated, uncertain and expensive.

Under pressure of the demands of steel construction technology and new ideas that have infiltrated the field, the AISC recently changed its manual and specifications. An allowable stress design (ASD) specification was substituted for the load resistance factor design (LRFD) specification. However, according to the AISC, in the new specification the “philosophy of design remains the same.” In the “Steel Design Handbook”, edited by Akbar R. Tamboll © 1997, the reason given for a new method is that “until recently engineers were basing the analysis and design of structures on a linear theory of elasticity. On the whole, the results have been satisfactory. The buildings and bridges have withstood the test of time. Why then should one be concerned with the LRFD method? Finally, elastic analysis of all but the simplest of structures is complicated. Obviously the net result is a waste of material. For structures such as aircraft, where weight is of prime importance, the results may be even more serious. Further, since such an analysis would have little rational basis, a true estimation of the safety factor would become virtually impossible.” This explanation still contains no rational basis for the LRFD method.

No theoretical foundation has been offered for the LRFD method. It has been maintained that the LRFD specification accounts for the factors that influence strength and loads by using a probabilistic basis and statistical methods. However, the probabilistic basis and a rational logical–mathematical deterministic basis are two different approaches. A statistical method has a rationale for its use when a reliable mathematical description of the elastic relations exists and statistics give the deviation of the empirical data from the mathematical description. Without sound theory a statistical method is just speculation. The problem with the linear theory of elasticity is that it distorts the relations it describes. No amount of statistical data and probabilistic method corrects that.

The need to revise the linear theory arises from the fact that this theory is fundamentally inconsistent with the experimental observations. Thus, linear theory identifies the elastic limit for a structure with the limit of elasticity of the material, while observations and experiments show that different structures made of the same material have significantly different limits depending on the geometry of the structure. In physics such disagreement raises doubt in the theory.

The Linear Theory of Infinitesimal Deformations is based on the assumption that because deformations are very small in comparison with the dimensions of a structure, the relationship between them can be described with a linear function. Here we will consider the Non-Linear Theory of Elasticity. One of the physical points of view in the new theory...