In this paper, the first in a series of three, a procedure based on semi-analytical finite elements is presented for constructing Saint-Venant solutions for extension, bending, torsion, and flexure of a prismatic cylinder with inhomogeneous, anisotropic cross-sectional properties. Extension-bending-torsion involve stress fields independent of the axial coordinate and their displacements may be decomposed into two distinct parts which are called the primal field and the cross-sectional warpages herein. The primal field embodies the essence of the kinematic hypotheses of elementary bar and beam theories and that for unrestrained torsion. The cross-sectional warpages are independent of the axial coordinate and they are determined by testing the variationally derived finite element displacement equations of equilibrium with the primal field. For flexure, a restricted three-dimensional stress field is in effect where the stress can vary at most linearly along the axis. Integrating the displacement field based for extension-bending-torsion gives that for the flexure problem. The cross-sectional warpages for flexure are determined by testing the displacement equations of equilibrium with this displacement field. In the next paper, the cross-sectional properties such as the weighted-average centroid, center of twist and shear center are defined based on the Saint-Venant solutions established in the present paper and numerical examples are given. In the third paper, end effects or the quantification of Saint-Venant’s principle for the inhomogeneous, anisotropic cylinder is considered.
On Saint-Venant’s Problem for an Inhomogeneous, Anisotropic Cylinder—Part I: Methodology for Saint-Venant Solutions
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Oct. 7, 1999; final revision, July 21, 2000. Associate Editor: J. W. Ju. Discussion on the paper should be addressed to the Editor, Professor Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
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Dong, S. B., Kosmatka, J. B., and Lin, H. C. (July 21, 2000). "On Saint-Venant’s Problem for an Inhomogeneous, Anisotropic Cylinder—Part I: Methodology for Saint-Venant Solutions ." ASME. J. Appl. Mech. May 2001; 68(3): 376–381. https://doi.org/10.1115/1.1363598
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