Engineering Transactions, 53, 2, pp. 119–-131, 2005

Algorithms of the Method of Statically Admissible Discontinuous Stress Fields (SADSF) — Part IV: Integration of Fields Around Nodes into Planar Complex Fields

W. BODASZEWSKI
Kielce University of Technology
Poland

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 applicatory 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 particu¬lar 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 quickly 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.
Applicatory 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 IV: In this part of the paper, the author presents a general concept of algorithms of two program modules that integrate component fields around nodes into planar complex fields. The first module, the auxiliary one called module C, is used to construct the objects of incidence of the regions and the lines that are generated automatically, based on a freehand sketch of the field structure – which can be drawn, for example, on the monitor screen by means of a mouse. The proper integration, however, is performed by the second module, called module IL, which utilises both the incidences brought in by module C, and the solutions of component systems of stress discontinuity lines around nodes – the latter obtained using the module called A. The individual partial problems are still demonstrated here, and the example of the already known solution of the field type 190 is used for this purpose.
Attention is also focussed on the effects of partial autonomy that are revealed in the in-tegration problems. These effects consist in decoupling of two systems of conditions: the one defined on geometrical parameters, and that based on stress parameters. The conditions are utilised, for example, in the algorithms of the application version software for finding par-ticularly complicated fields that could contain as much as several dozens of homogeneous regions. An example of such a solution is shown along with its application to shaping com¬plex elements of a structure. The example confirms once again great potential of the SADSF method.
Keywords: shape design; limit analysis; numerical methods
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References

W. SZCZEPECISKI, Plastic design of machine parts [in Polish], PWN, Warszawa 1968.

W. BODASZEWSKI, Stress discontinuity line structures in yield zones satisfying the Huber Mises condition [in Polish], Engineering Transactions, part 1 36, 1, 29–54, 1988, part 2 and 3 37, 3, 415–442, 1989.

W. BODASZEWSKI, The software package KNPN for approximated shaping of complex plastic structures, VIII-th Symposium on Stability of Structures, Łódź University of Technology, Poland, 25–30, 1997.

I. MARKIEWICZ, Numerical determination of statically admissible link systems of stress discontinuity lines in limit Huber-Mises's fields [in Polish], Doctoral Thesis, Warsaw University of Technology, 1996.

Internet website: www.sadsf.net.




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