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Algorithms of the method of statically admissible discontinuous stress fields (SADSF). P. 3 Software implementations and example problems formulated for fields around nodes

Bodaszewski W.

Engineering Transactions

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2005

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Vol. 53, iss. 1

15-30

EN

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 bove-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.

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Algorithms of the method of statically admissible discontinuous stress fields (SADSF) . P. 4 Inegration of fields around nodes into planar complex fields

Bodaszewski W.

Engineering Transactions

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2005

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Vol. 53, iss. 2

119-131

EN

In this part of the paper, the author presents a general concepts 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 B, 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 f90 is used for this purpose. Attention is also focussed on the effects of partial autonomy that are revealed in the integration 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 particularly 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 forming complex elements of a structure. The example confirms once again great potential of the SADSF method.

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Algorithms of the method of statically admissible discontinuous stress fields (SADSP) : [Part I]

Bodaszewski W.

Engineering Transactions

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2004

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Vol. 52, iss.3

175-193

EN

As a form of introduction, the author presents in a broad outline the general concept of the SADSF method, its fundamentals, basic conditions, typical formulation of boundary problems and the essence of difficulties in problem solving. These are not only nonlinearities, singularities, and conditions expressed by complex functions (i.e. given in a form of algorithms), but first of all the structure of the condition system - that is not a priori known - which cause difficulties. As it turns out, the structure depends on boundary conditions, but the relations that express this dependence are not known. In the physical space, it refers to the problems with unknown discretization that is dependent on boundary conditions. Also, as a form of introduction, it will be shown that the investigation of mentioned dependence is possible, anyway. However, it is possible not for two-dimensional fields as a whole, but at least for their component fields around nodes. Thereby fundamental ideas of algorithms are presented in broad outline. Such algorithms are discussed in detail in Parts II and III. The application version of this method, attractive for practical purposes, is briefly described in this part. Numerical examples, also included here, illustrate the position and the possibilities of the method in solving practical problems of shape design of complex thin-wall structures.

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Algorithms of the method of statically admissible discontinuous stress fields (SADSF) - Part II

Bodaszewski W.

Engineering Transactions

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2004

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Vol. 52, iss.4

231-252

EN

In the paper, the author introduces the sets of conditions that create the algorithms of the functions on which one defines the boundary problems met in the search for discontinuous limit fields existing around isolated nodes. Among those, there are functions describing states of stress in the component homogeneous regions, the parameters of lines that separate these regions, and, first of all, the formulae for determining the domains based on the general conditions of existence. These formulae play a key role in numerical implementations of the method. The fields satisfying the Huber-Mises yield condition are of primary choice however, the derived relationships have a general meaning. To emphasise this fact one presents not only the areas of existence valid for the Huber-Mises condition, but also the areas obtained for several other yield conditions applicable to plastically homogeneous materials. The knowledge of the areas opens the possibility of developing the method of search for the fields that obey these conditions, and for algorithmizing this method. This could be applied even for the fields that are characterised by arbitrary, admissible states of stress. One also presents, basing on a mathematically complete set of conditions, typical formulations of problems concerning the fields around the nodes. One discusses the balance between the set of conditions and the unknowns, as well as the transformations into global systems connected with complex fields. One consequently applies parametrisation of the yield conditions, which not only reduces the number of unknowns and leads to simple, recursive forms of the formulae, but, first of all, makes it possible to find the formulae for generation of domains, without which numerical solution of the fields and algorithmization of the method would not be possible at all.