Sommario
Si considerano strutture elasto-plastiche discrete e discretizzate, soggette a carichi esterni di tipo conservativo e a distorsioni assegnate. Si suppone che il legame costitutivo del materiale sia associato e la condizione di plasticità e le leggi di incrudimento vengano linearizzate a tratti.
Per un'assegnata configurazione di equilibrio di un sistema di questo tipo, si dimostra una condizione necessaria e sufficiente per la positività del lavoro plastico del second'ordine; si mostra che esso è positivo se e solo se una certa matrice risulta positiva definita. Le implicazioni di questo evento con la stabilità della configurazione sono brevemente discusse.
L'asserto dimostrato ha un duplice interesse. In primo luogo approfondisce il significato di alcune ipotesi alla base di certe tecniche numeriche per l'analisi elasto-plastica; in secondo luogo porta a procedimenti operativi per l'analisi della stabilità che possono rivelarsi utili in alcuni casi, come è illustrato con riferimento alla determinazione del carico di collasso di telai elasto-plastici in presenza di effetti del II ordine.
Summary
The paper considers discrete or discretized elastic-plastic structures subjected to conservative external forces and given imposed strains. Material flow laws are assumed to be associated; the plasticity condition and the hardening rule are piecewise linearized.
For a given equilibrium configuration of such systems, a necessary and sufficient condition for the positivity of second order plastic work is proved. It is shown that this work is positive if and only if a certain matrix is positive definite. Connections with the kinematic stability of the configuration are discussed.
The interest of the statement is two-fold. Firstly it provides a deeper understanding on the assumptions on which some techniques for elastic-plastic analysis are based. Secondly, it leads to operative procedures for stability analysis which may prove efficient in some cases, as it is illustrated with reference to the evaluation of the collapse load for elastic-perfectly plastic frames in the presence of second order geometrical effects.
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References
MAIER G.,A Matrix Structural Theory of Piecewise Linear Elastoplasticity with Interacting Yield Planes, Meccanica, 5, p. 54, 1970.
KOITER W. T.,General Theorems for Elastic Plastic Solids, in “Progress in Solid Mechanics”, Vol. 1, Amsterdam, p. 167, 1960.
MAIER G.,A Quadratic Programming Approach for Certain Classes of Nonlinear Structural Problems, Meccanica, 3, p. 121, 1968.
MAIER G.,Incremental Elastoplastic Analysis in the Presence of Large Displacements and Physical Instabilizing Effects, International Journal of Solids and Structures, 7, p. 345, 1971.
CORRADI L., DE DONATO O., MAIER G.Inelastic Analysis of Reinforced Concrete Frames, Journal of the Structural Division, Proceedings A.S.C.E., 100, p. 1925, 1974.
CORRADI L., MAIER G.,Extremum Theorems for Large Displacement Analysis of Discrete Elastoplastic Structures with Piecewise Linear Yield Surfaces, Journal of Optimization Theory and Applications, 15, p. 51, 1975.
CORRADI L.,Mathematical Programming Methods for Displacement Bounds in Elasto-Plastic Dynamics, Nuclear Engineering and Design, 37, p. 161, 1976.
DRUCKER D. C.,A More Fundamental Approach to Stress Strain Relations, Proceedings of the First National Congress for Applied Mechanics, A.S.M.E., p. 487, 1951.
DRUCKER D. C.,On the Postulate of Stability of Material in the Mechanics of Continua, Journal de Mécanique, 3, p. 235, 1964.
MANDEL J.,Conditions de Stabilité et Postulate de Drucker, Proceedings of the Symposium of the IUTAM, Springer Verlag, p. 58, 1966.
ARGYRIS J. H.,Continua and Discontinua, First Conference on Matrix Methods in Structural Mechanics, Wright-Patterson AFB, Dayton, Ohio, 1967.
DRUCKER D. C., MAIER G.,Effects of Geometry Change on Essential sential Features of Inelastic Behavior, Journal of the Engineering Mechanics Division, Proceedings A.S.C.E., 99, p. 819, 1973.
MAIER G., DE DONATO O., CORRADI L.,Inelastic Analysis of Reinforced Concrete Frames by Quadratic Programming, in “Inelasticity and Non-Linearity in Structural Concrete”, University of Waterloo Press, p. 265, 1972.
ANGL A. H. S., LOPEZ L. A.,Discrete Model Analysis of Elastic Plastic Plates, Journal of the Engineering Mechanics Division, Proceedings A.S.C.E., 94, p. 271, 1968.
SCHNOBRICH W. C., PECKNOLD D. A.,The Lumped Parameter or Bar-Node Model Approach to Thin Shell Analysis, in “Numerical and Computer Methods in Structural Mechanics”, Academic Press, p. 377, 1973.
LIN T. H.,Theory of Inelastic Structures, John Wiley and Sons, 1968.
HOHN F. E.,Elementary Matrix Algebra, McMillan, 1964.
GANTMACHER F.R.,The theory of Matrices, Vol. I, Chelsea, New York, 1960.
COTTLE R. W.,Manifestations of the Schur Complement, Linear Algebra and its Applications, 8, p. 189, 1974.
HILL R.,A General Theory of Uniqueness and Stability in Elastic-Plastic Solids, Journal of the Mechanics and Physics of Solids, 6, p. 236, 1958.
DRUCKER D. C., ONAT T.,On the Concept of Stability of Inelastic Systems, Journal of Aeronautical Sciences, p. 543, 1954.
CORRADI L.,Sulla Condizione di Stabilità der Strutture Elasto-Plastiche con Parti Instabili, Rendiconti dell'Istituto Lombardo di Scienze e Lettere, 102 A, p. 942, 1968.
DE DONATO O., CORRADI L.,QPFRAM — Finite Element Inelastic Analysis of Frames by Quadratic Programming, Computer Report N. 4, Istituto di Scienza e Tecnica delle Costruzioni del Politecnico di Milano, 1973.
HORNE M. R., MERCHANT W.,The Stability of Frames, Pergamon Press, 1965.
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Corradi, L. On a stability condition for elastic plastic structures. Meccanica 12, 24–37 (1977). https://doi.org/10.1007/BF02172204
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DOI: https://doi.org/10.1007/BF02172204