Analysis of objectives:
Towards a conceptual unification of exising approaches to structural analysis,
including aspects of application in engineering and mathematical theory aspects
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Sensitiveness to initial imperfections and
necessity of non linear approaches
A structure intended to bear loads denotes a limit up to which loads can not be increased. This limit is indicated as "failure" or "collapse" and is particulary interesting from an engineering point of view. Any failure phenomenon can be studied through stability analysis. In many cases, it is the consequence of phenomena which can be investigated by analyzing the capability of the structure to maintain its shape without inflections or bucklings. In these cases, the stability analysis concerns the shape of the structure.
The first example of this type of analysis dates back to Euler, who studied a beam loaded at its extremities, with the loads generating compression. Euler introduced, perpahs without full awareness, a postulate of platonic nature, taking into account beams of perfectly rectilinear shape, subject to loads perfectly alligned to the beam itself, i.e., abstract cases in comparison to those which can be realised even with the most sophisticated techniques. Considering a generic beam, and using a type of analysis which is not properly of stability but can be brought back to it, Euler has investigated the configurations which can be caused by potential deformations, both in the case where the simplification of linearities between displacements and deformations is introduced, to be viewed as a postulate, and in the case where such simplification is not introduced.
The linearised analysis demonstrates the existence, in the spaces defined by the linearity postulate, of descrete load levels associated to configurations of neutral equilibrium, where large deformations can happen with equal load, and whose loci consitute branches of the equilibrium curve. Such behaviours, which are not realistic because they are coherent only with linearity postulate, and in particular the minimun loads associated to them, solve the engineering problem of identifying an approximation of the collpase load. A very similar situation is encountered when trying to determine the collapse loads of flat panels subject to various load conditions: compression, cut, biaxial compresssion and other combinations of crossbeams, subject to bending, which can avoid the load through torsion rotations and in many other cases.
Perhaps due to chance, engeneering problems occorring until 1930 where such that it was considered as a straightforward solution to address the stability limits of structures, as far as the relationships between displacements and deformations isare concerned, using geometric perfection postulates and behaviour linearity. As a matter of fact, even the most advanced literature of that time did not even mention the need of taking into account initial shape defects, nor the possibility of adopting non linear approaches when addressing the move from displacements to internal deformations. In this cultural environment, Lundquist presented his experimental results of testing thin layer cylenders under compression, obtaining collapse loads equivalent to one fifth of the result foreseen by the theoretical approaches of that time. The scientific community working on the problem, despite ten years of efforts devoted to explaining these discrepancies between theory and experimental results, could not solve the problem, due to the widely accepted believe that linear theories and the postulate of geometric perfection were adequate for the case at hand.
It is von Karman's merit, togenther with one of his students, to have brought back the scientific conmmunity to considering using non linear analisys, which is suitable for examining also large displacememts, in order to investigate other behaviours, behind deformatins, which can determine equilibrium crises and therefore postcritical behaviours. The main result of von Karman's contribution is pointing out the potential of non linear approaches to explain, in the relationships between displacements and tensions, of sudden instabilities, a phenomenon which was already known, but could not be explained theretically at that time, in thin cylinder compression. The same non linear analysis was subsequently shown to be adequate also to correctly account for the effects of initial shape defects, which considerably decrease the critical load in comparison to the perfect shape. Thinking along these lines can be followed by considering the behaviour of a mechanism with two degrees of freedom. Such behaviours are typical of structures similar to compressed thin cylinders, and are particularly well suited to display the possibilities introduced in the mathematical models by various postulates. A result of the topology, by the Japanese scientist Yoshimura, which outlines how the lateral surface of a cylinder and a diamond surface have the same development, can help understand the full implications of von Karman's work.