Algorithms of the Method of Statically Admissible Discontinuous Stress Fields (SADSF) — Part III - BODASZEWSKI, W. (Engineering Transactions, 2005)
SUMMARY OF THE WHOLE PAPER: By now, the SADSF method is practically the only tool of shape design of complex machine elements that provides an effective solution even to the problems of 3D distribution of the material, and at the same time it is still enough user-friendly to be useful for engineers. This unique property of the method is due to the existence of its simple application version. When using it, a design engineer does not need to solve by oneself any statically admissible field – which could be very difficult – but obtains such a solution by assembling various ready-made particular solutions. The latter are in general obtained by means of individual and complex analyses and provided to a designer in a form of libraries. The algorithms presented in this paper break up with the individual approach to a particular field. The algorithms are the first ones of general character, as they apply to the fundamental problems of the method. The algorithms enable solving practically any boundary problem that one encounters in constructing 2D statically admissible, discontinuous stress fields, first of all the limit fields. In the presented approach, one deals first with the fields arising around isolated nodes of stress discontinuity lines (Parts II and III), then integrates these fields into 2D complex fields (Part IV). The software, created on the basis of the algorithms, among other things, allows one to find all the existing solutions of the discontinuity line systems and present them in a graphical form. It gives the possibility of analysing, updating and correcting these systems. In this way, it overcomes the greatest difficulty of the SADSF method following from the fact that the systems of discontinuity lines are not known a priori, and appropriate relationships are not known either, so that they could be found only in an arduous way by postulating the line systems and verifying them. Application version of the SADSF method is not described in this paper; however, a reference is given to inform the reader where it can be found. SUMMARY OF PART III: Boundary problems, characteristic for the already-known fields around convex and concave corners, are used in this part of the paper as the examples to present juxtaposition of conditions, and to obtain a solution for general conditions of the system – important for the fields that appear around nodes. The presented variants of these systems and the sets of unknowns, after minor completion, give the basis for deriving series of elementary problems, which are necessary to create the algorithm for solving arbitrary boundary problems, such as those encountered in the fields around nodes. The algorithm created on such a basis does not require formulating any particular relationships, and its implementation makes it possible to find any solution to the field around the node. The solution, presented in an illustrative graphical form, can then be easily edited. In effect, it becomes possible to test, almost instantaneously, admissibility of the structures, and verify the existence of solutions on the physical plane. The paper also presents short description of properties of the fields around nodes that facilitates interpretation of the results. It is particularly useful in the cases when one obtains surprising results, for example when structural degenerations (collapses) appear. It is worth mentioning that, with boundary conditions formulated for fields around both the above-mentioned types of corners, one obtains not only fields identical with the prototype, but also a whole variety of other fields that until now have been treated as different ones. Actually, these are the fields being solutions to the same boundary problem.