A finite element-transfer matrix procedure developed for determination of Saint-Venant decay rates of self-equilibrated loading at one end of a semi-infinite prismatic elastic rod of general cross section, which are the eigenvalues of a single repeating cell transfer matrix, is applied to the case of a rectangular cross section. First, a characteristic length of the rod is modelled within a finite element code; a superelement stiffness matrix relating force and displacement components at the master nodes at the ends of the length is then constructed, and its manipulation provides the transfer matrix, from which the eigenvalues and eigenvectors are determined. Over the range from plane stress to plane strain, which are the extremes of aspect ratio, there are always eigenmodes which decay slower than the generalized Papkovitch-Fadle modes, the latter being largely insensitive to aspect ratio. For compact cross sections, close to square, the slowest decay is for a mode having a distribution of axial displacement reminiscent of that associated with warping during torsion; for less compact cross sections, slowest decay is for a mode characterized by cross-sectional bending, caused by self-equilibrated twisting moment.
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Brief Notes
Saint-Venant Decay Rates for the Rectangular Cross Section Rod
N. G. Stephen,
N. G. Stephen
School of Engineering Sciences, Mechanical Engineering, The University of Southampton, Highfield, Southampton SO17 1BJ, UK
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P. J. Wang
P. J. Wang
School of Mechanical, Materials, Manufacturing Engineering and Management, The University of Nottingham, University Park, Nottingham NG7 2RD, UK
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N. G. Stephen
School of Engineering Sciences, Mechanical Engineering, The University of Southampton, Highfield, Southampton SO17 1BJ, UK
P. J. Wang
School of Mechanical, Materials, Manufacturing Engineering and Management, The University of Nottingham, University Park, Nottingham NG7 2RD, UK
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, Jan. 22, 2003; final revision, Oct. 28, 2003. Associate Editor: K. R. Rajagopal.
J. Appl. Mech. May 2004, 71(3): 429-433 (5 pages)
Published Online: June 22, 2004
Article history
Received:
January 22, 2003
Revised:
October 28, 2003
Online:
June 22, 2004
Citation
Stephen , N. G., and Wang , P. J. (June 22, 2004). "Saint-Venant Decay Rates for the Rectangular Cross Section Rod ." ASME. J. Appl. Mech. May 2004; 71(3): 429–433. https://doi.org/10.1115/1.1687794
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