Dualne Aspekty Analizy Układów Prętowych
In the paper presented are these fundamental aspects of analysis which may be considered independently of the actual physical properties of the system considered. The discussion is thus confined to exclusively static and kinematic aspects which are independent and lead to two qualitatively different states (though complementing each other) of the same system. Determination of the statically and kinematically admissible states enables to formulate and analyze the problems of existence and uniqueness of those states by means of the known theorems of linear algebra, The systems were considered to be kinematically linear and loaded by forces applied to the nodes.
Using the suitably introduced notions of statically and kinematically admissible states determined are the mutual relations between the structure of the system considered and the structure of the corresponding matrix which appears in the equations of equilibrium and displacement compatibility. Mutual correspondence of the variables determining the statical and kinematical states is established, according to the role they play in the problem. The crucial one is the correspondence occuring between the variables which play analogous roles in the static and kinematic approaches to the same problem.
In view of the dual character of corresponding notions and mechanical quantities, parallel lines of reasoning are presented which make it possible to stress their mutual, dual character.
P. FUNK, Variationsrechnung und ihre Anwendung in Physik und Technik, Springer-Verlag, Berlin 1962.
M. J. SEWELL, On dual approximation principles and optimization in continuum mechanics, Phil. Trans. of the Royal Society of London, Sedes (A), Math. Phys. Sci., 1162, 265, 319-351, 1969.
W. R. SPILLERS, Network analogy for the truss problem, J. Eng. Mech. Div., Proc. ASCE, EM6, 33-40, 1962.
W. R. SPILLERS, Network analogy for linear structures, J. Eng. Mech. Div., Proc. ASCE, EM4, 21-29, 1963.
W. R. SPILLERS, Applications of topology in structural analysis, J. Struct. Div., Proc. ASCE, ST4, 301-313, 1963.
W. R. SPILLERS, Graph theory, switchinig theory and structural design, AMR, 24, 5, 1971.
N. C. LIND, Analysis of structures by system theory, J. Struct. Div., Proc. ASCE, ST2, 1-22, 1962.
S. J. FENVES, F. H. BRANIN, Network-topological formulation of structural analysis, J. Struct. Div., Proc. ASCE, ST4, 483-514, 1963.
S. P. MAUCH, S. J. FENVES, Releases and constraints in structural networks, J. Struct. Div., Proc. ASCE, STS, 401-417, 1967.
B. OLSZOWSKI, Obliczanie konstrukcji niewyznaczalnych metodą programowania dynamicznego, Arch. Inż. Ląd., 17, 2, 285-313, 1971.
J. H. ARGYRIS, Recent advances in matrix methods of structural analysis, Pergamon Press, Oxford 1964.
S. RAUTU, Considerations on the analysis of statically indeterminate structures, Rev. Roum. Sci. Techn.-Mec. Appl., 12, 6, 1317-1331, 1967.
M. SIMONNARD, Programmation lineaire, Dunod, Paris 1962.
C. LANCZ0S, The variational principles of mechanics, University of Toronto Press, Toronto 1962.