Direct limit analysis via rigid-plastic finite elements

Abstract This work concerns the limit analysis of rigid-perfectly plastic plane structures. The continuum is discretized into finite elements, thereby permitting the theoretical nonlinear problem to be transformed to an equivalent linear elastic problem for which the corresponding collapse load may be found in a reasonable number of analysis iterations; as such, the analysis may be termed a quasi-direct approach. The duality concepts for linear elastic-analysis by the finite element method are extended to the plasticity problem using classical variational principles; in this way, both primal and dual quasi-direct approaches to the limit analysis problem are identified. Generalization of the variational principles permits the analysis to be formulated for not only compatible elements or equilibrium finite elements, but also for hybrid elements having two independent fields (i.e. stress and strain). Some numerical examples concerning classical academic problems for plane structures illustrate the efficiency of the method. Finally, a new hybrid element with an arbitrary stress field in the interior and a quadratic displacement field on the boundary is described in detail in an appendix.

[1]  Ian P. King,et al.  Elastic-plastic analysis of two-dimensional stress systems by the finite element method , 1967 .

[2]  John Patrick Wolf,et al.  Generalized stress models for finite element analysis , 1974 .

[3]  P. B. Mellor,et al.  Plasticity for mechanical engineers , 1962 .

[4]  Theodore H. H. Pian,et al.  Basis of finite element methods for solid continua , 1969 .

[5]  E. Faccioli.  A variational theorem for Mises' solids based on stress potentials , 1969 .

[6]  D. C. Drucker,et al.  Soil mechanics and plastic analysis or limit design , 1952 .

[7]  Donald W. Taylor,et al.  Fundamentals of soil mechanics , 1948 .

[8]  Ted Belytschko,et al.  Numerical Methods for the Limit Analysis of Plates , 1968 .

[9]  Moshe F. Rubinstein,et al.  Elastic-Plastic Analysis by Quadratic Programming , 1972 .

[10]  T. Pian.  Derivation of element stiffness matrices , 1964 .

[11]  J. M. Alexander.  On complete solutions for frictionless extrusion in plane strain , 1961 .

[12]  E. Anderheggen,et al.  Finite element limit analysis using linear programming , 1972 .

[13]  Giulio Maier,et al.  Quadratic programming and theory of elastic-perfectly plastic structures , 1968 .

[14]  J. Heyman.  The stability of a vertical cut , 1973 .

[15]  J. M. Alexander,et al.  Manufacturing Properties of Materials , 1967 .

[16]  E. Faccioli,et al.  A finite element, linear programming methods for the limit analysis of thin plates , 1973 .

[17]  Aris Phillips,et al.  Plastic Analysis of Structures , 1959 .

[18]  S. Nakagiri,et al.  Elastic‐plastic analysis of Saint‐Venant torsion problem by a hybrid stress model , 1972 .

[19]  William Prager.  Variational principles for elastic plates with relaxed continuity requirements , 1968 .

[20]  T. Mura,et al.  Application of variational principles to limit analysis , 1963 .

[21]  Janusz S. Kowalik,et al.  Iterative methods for nonlinear optimization problems , 1972 .

[22]  Leonard R. Herrmann,et al.  Finite-Element Bending Analysis for Plates , 1967 .

[23]  J. Argyris,et al.  Elasto-plastic Matrix Displacement Analysis of Three-dimensional Continua , 1965, The Journal of the Royal Aeronautical Society.

[24]  B. D. Veubeke.  Displacement and equilibrium models in the finite element method , 1965 .

[25]  O. C. Zienkiewicz,et al.  Curved, isoparametric, “quadrilateral” elements for finite element analysis , 1968 .

[26]  O. C. Zienkiewicz,et al.  Elasto‐plastic stress analysis. A generalization for various contitutive relations including strain softening , 1972 .

[27]  Experiments on yielding of tension specimens with notches and holes , 1970 .

[28]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[29]  E. Reissner.  On a Variational Theorem in Elasticity , 1950 .

[30]  A general variational theorem for the rate problem in elasto-plasticity , 1972 .

[31]  H. E. Shull,et al.  Load-Carrying Capacities of Simply Supported Rectangular Plates , 1963 .

[32]  W. Prager,et al.  Theory of perfectly plastic solids , 1968 .

[33]  Walter Schumann,et al.  On limit analysis of plates , 1958 .

[34]  Y. Yamada,et al.  Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method , 1968 .

[35]  R. Hill.  The mathematical theory of plasticity , 1950 .

[36]  H. Hencky,et al.  Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen , 1924 .

[37]  Angelo Miele,et al.  The Calculus of Variations in Applied Aerodynamics and Flight Mechanics , 1962 .

[38]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[39]  O. C. Zienkiewicz,et al.  Elasto‐plastic solutions of engineering problems ‘initial stress’, finite element approach , 1969 .

[40]  J. H. Argyris,et al.  The LUMINA Element for the Matrix Displacement Method , 1968, The Aeronautical Journal (1968).

[41]  Alberto Franchi,et al.  A modified gradient method for finite element elastoplastic analysis by quadratic programming , 1973 .

[42]  L. Kachanov,et al.  Foundations Of The Theory Of Plasticity , 1971 .

[43]  Robert E. Jones.  A generalization of the direct-stiffness method of structural analysis , 1964 .

[44]  E. P. Unksov.  An engineering theory of plasticity , 1961 .

[45]  G. Maier.  A quadratic programming approach for certain classes of non linear structural problems , 1968 .

[46]  S. G. Mikhlin,et al.  Approximate methods for solution of differential and integral equations , 1970 .