New Approach to Computational Analysis of Mixed-Boundary-Value Stress Problems of Curved Bodies

Abstract

This research addresses a new method of analysis for the curved structural members under various types of loadings and supporting conditions at the boundary surfaces. In this work, theory of elasticity, a new potential function, constitutive relations for coordinate transformations and finite-difference computational algorithm are integrated to develop a new computational method for stress analysis of curved structural elements which include curved beams, deep arches, finite rings etc. Specific contributions are follows: A new elasticity formulation is developed for stress analysis of mixed boundary-value stress problems of curved bodies. A new scheme of reduction of unknowns is used to develop the single scalar function formulation. More specifically, the scheme reduces the problem to finding a single field variable governed by a single partial differential equation of equilibrium. In the present formulation, a potential function is treated as the field variable which is defined in terms of radial and circumferential components of displacement, and the resulting formulation is called Displacement Potential Field Formulation. The formulation is superior to the standard stress function formulation in the sense that it is capable of satisfying all modes of physical conditions at the bounding surfaces appropriately, whether they are specified in terms of loadings or restraints or any combination of them. A finite-difference computational algorithm is developed for obtaining numerical solutions of the elastic field of curved structural bodies in terms of the displacement potential. The displacement potential computation method finds a single unknown at each nodal point, whereas the existing methods find at least two variables at each node of the plane computational domain and hence a tremendous saving in computational effort is achieved through the proposed approach. The application of the method is demonstrated through the numerical solutions of a number of structural problems with curved boundaries. The soundness and accuracy of the single variable computational scheme is verified through the comparison of results with those obtained by conventional computational and analytical approaches where available.