Categorization of Buckling by Means of Spherical Geometry

Lack of stiffness may lead to buckling of structures. Slender metallic structures may buckle already at relatively low loading of the material. If buckling occurs, lack of statically redundant supports and scarcity of stabilizing tensile ties may cause great damage. This can lead to collapse of structures connected with danger for human lives. The overriding goal of the project is to improve the buckling behavior of structures by means of small changes of the original design, considering functional requirements and aesthetic demands. For this purpose, sensitivity analyses of the buckling behavior including the initial phase of the buckling process, frequently termed as (initial) postbuckling, will be carried out. The aim of these analyses is to convert originally imperfection-sensitive structures into imperfection-insensitive ones in order to avoid potentially disastrous consequences of buckling. A particularly desirable form of such a conversion is restricted to buckling from a membrane stress state. This suggests choosing the percentage bending energy of the total strain energy as the criterion for categorization of buckling according to different static initial states of the buckling process. Proof of computability of this term for arbitrary structures and arbitrary loading, in the frame of the Finite Element Method (FEM), without previous computation of technologically rather insignificant absolute energy quantities is of great epistemological as well as practical relevance. An important aspect of the intended categorization, representing sort of a central thread running through the project, are "hidden conditions" for identification of the mechanical situation in the prebuckling regime and at the onset of buckling, in the context with the FEM. Here, "hidden" means that these conditions do not routinely appear in FE analysis. This creates the impression that they are unknown. The mathematical tool for realization of the proposed research project is the so-called consistently linearized eigenproblem. The eigenvalues and eigenvectors depend on a dimensionless load factor which is increased proportionally from zero up to the stability limit. The geometrical interpretation of the vector function representing the normalized fundamental eigenvector is one of a surface curve on the unit sphere. The choice of mechanically useful quantities for the azimuth and the zenith angle is one of the challenges of the project. The course of the surface curve on the unit sphere allows a quantitative assessment of the extent of a possible percentage shift from membrane to bending energy in the prebuckling regime. The intellectual foundation of the project is the link of structural mechanics with spherical geometry in order to solve a demanding technological problem.

When designing metallic structures, which are frequently encountered e.g. in architecture and civil engineering, attention must be paid to sufficient safety against loss of stability, triggering off the buckling process. Buckling may have catastrophic consequences up to loss of human lives. The load-carrying capacity of structures after reaching the stability limit depends on the load in the prebuckling region, which is influenced by several mechanical and geometrical factors. In general, the load results in stretching and bending, which can be quantified in form of stretching energy in case of plates and shells also referred to as membrane energy and bending energy, stored in the structure. The categorization, named in the title of the research project, refers to the percentage split of the strain energy into the membrane part and the residual part. The latter is usually dominated by the bending part. The title is note of a novel symbiosis of structural mechanics and spherical geometry that is not only epistemologically significant. Not least, it is a technologically important goal, which is the basis for determination of the influence of bending action in the prebuckling region on the initial buckling behavior. The aforementioned split of the strain energy is reached with the help of differential geometry, representing a branch of mathematics that does not seem to be causally connected with this goal. The mathematical means to reach this goal is a specific linear eigenvalue problem, which has originally served a completely different purpose. The hypothesis for the split of the strain energy is based on this eigenproblem. It was verified exactly for the limiting case of a membrane stress state and, in form of an excellent approximation, for the limiting case of pure bending. However, for structures subjected to general stress states, it had to be modified. Verification of the new hypothesis by means of comparing the obtained numerical results to the ones from a conventional split of the strain energy, which are already available, is expected in the immediate future. What is particularly remarkable, is the attractive form of visualization of the results as a curve on the surface of an octant on the unit sphere. It permits a quick, energetically well-founded, quantitative assessment of the global load-carrying behavior of the investigated structure up to the stability limit and, thus, relativizes the significance of traditional qualitative statements about the mechanical behavior of structures. The numerical implementation of the demanding theoretical concept was a great challenge. This referred primarily to the numerical solution of the linear eigenproblem, where one of its two coefficient matrices had to be approximated as a difference quotient.