Ebook A Finite Element Method For The Plastic Bending Analysis Of Structures
The description of plastic behavior presents some basic difficulties to the structural analyst. These difficulties are associated with a proper description of the material phenomenon and the nonlinear nature of the resulting governing equation. Thus, the mathematical formulation of the plasticity problem makes an analysis of all but the simplest structures a very formidable, if not an impossible, task. Consequently, considerable attention has been given recently to the extension of finite element techniques to include the effects of plastic behavior (References 1-5). These techniques have the advantage of being capable of treating the effects of plasticity in complex structures by utilizing various algorithms for linearizing the basic nonlinear nature of the problem.
Most of the current effort concerned with the application of finite element techniques to the plasticity problem has been limited to the treatment of structures in states of membrane stress. In addition, the methods generally neglect the effects of geometric nonlinearity. These limitations are too restrictive for many important aerospace structures. Consequently, it is the purpose of the present paper to extend the methods already developed to provide for a plastic bending analysis that accounts for membrane stress states and geometric non-linearity.
The present method makes use of a governing linear matrix equation that relates the applied loading to the nodal displacement and initial strains. For the purpose of a plasticity analysis, the plastic strains are interpreted as initial strains. Use of the initial strain concept, to treat the effects of plasticity, requires the development of appropriate matrix relations based on assumptions for the distribution of both displacements and initial (plastic) strain within a finite element. The specification of a distribution for plastic strain within a finite element forms the basis on which the present plasticity analysis depends.
Inclusion of the effects of geometric nonlinearity is primarily of concern in problems involving thin beams, plates, and shells in the plastic, as well as elastic, range. A finite element method that utilizes an incremental procedure requiring a successive modification of the element stiffness properties has been discussed in Reference 6. This method requires the introduction of an additional stiffness matrix to account for the effects on the bending stiffness of the membrane stresses generated as a consequence of geometric nonlinearity. In addition, the effect that changes in geometry have on subsequent deformations is taken into account. This incremental procedure is incorporated into the plastic bending analysis to treat the combined effects of material and geometric nonlinearity.
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