Abstract
A consistent theory for linear elastic behavior in which the strains are small but the displacements and rotations can be large is applied to the bending and twisting of a rod or beam by end loads and by its own weight. The theory provides solutions for geometries and loadings between those for which the infinitesimal theory applies and those for which structural theories, such as Kirchhoff s theory for rods, can be used. The results confirm the validity of Kirchhoff's rod theory within the range of small-strain, linear elastic behavior.
Zusammenfassung
Eine konsistente Theorie, bei der linearelastisches Verhalten und Verzerrungen klein vorausgesetzt, jedoch große Verschiebungen und Drehungen zugelassen sind, wird auf die Verbiegung und Verdrehung eines Stabes unter Endlasten und Eigengewicht angewendet. Die Theorie ergibt Lösungen für Geometrien und Belastungen im mittleren Bereich zwischen dem Gültigkeitsbereich der infinitesimalen Theorie und jenem der strukturellen Theorien wie die Kirchhoff'sche Stabtheorie im Bereich der kleinen Verzerrungen und des linear-elastischen Verhaltens.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. E. H. Love,A Treatise on the Mathematical Theory of Elasticity. Cambridge Univ., 4th ed., 1927. Reprinted by Dover, New York 1944.
A. Rigolot,Déplacements finis et petites déformations des poutres droites. J. Méch. appl.1, 175–206 (1977).
D. F. Parker,An asymptotic analysis of large deflections and rotations of elastic rods. Int. J. Solids Struct.15, 361–377 (1979).
D. F. Parker,The role of Saint Venant's solutions in rod and beam theories. J. Appl. Mech.46, 861–866 (1979).
D. F. Parker,On the derivation of nonlinear rod theories from three-dimensional elasticity. J. Appl. Math. Phys. (ZAMP)35, 833–847 (1984).
J. L. Ericksen,Bending a prism to helical form, Arch. Rat. Mech. Anal.66, 1–18 (1977).
R. T. Shield,A consistent theory for elastic deformations with small strains. J. Appl. Mech.51, 717–723 (1984).
S. Im,Large deflections of structures with small elastic strains. Ph. D. Thesis, University of Illinois at Urbana-Champaign, October 1985. T. & A. M. Report No. 474, October 1985.
S. Im and R. T. Shield,Elastic deformations of strips and circular plates by uniform pressure. To appear in J. Appl. Mech.
C. O. Horgan and J. K. Knowles,Recent development concerning Saint-Venant's principle. Adv. Appl. Mech.23, 179–269 (1983).
R. Frisch-Fay,Flexible Bars. Butterworths, Washington 1962.
W. G. Bickley,The heavy elastica. Phil. Mag., Ser.7, 17, 603–622 (1934).
J. J. Stoker,Nonlinear Elasticity. Nelson, London 1968.
C. Truesdell and W. Noll,The Nonlinear Field Theory of Mechanics. Handb. Phys., Vol. III/3, Springer, Berlin 1965.
F. H. K. Chen and R. T. Shield,Conservation laws in elasticity of the J-integral type. J. Appl. Math. Phys. (ZAMP)28, 1–22 (1977).
D. H. Hodges and D. A. Peters,On the lateral buckling of uniform slender cantilever beams. Int. J. Solids Struct.11, 1269–1280 (1975).
R. T. Shield,Load-displacement relations for elastic bodies. J. Appl. Math. Phys. (ZAMP)18, 682–693 (1967).
S. P. Timoshenko and J. N. Goodier,Theory of Elasticity. McGraw-Hill, 3rd Ed., 1951.
R. T. Shield,An energy method for certain second-order effects with application to torsion of elastic bars under tension. J. Appl. Mech.47, 75–81 (1980).
P. Pleus and M. Sayir,A second order theory for large deflections of slender beams. J. Appl. Math. Phys. (ZAMP)34, 192–217 (1983).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shield, R.T., Im, S. Small strain deformations of elastic beams and rods including large deflections. Z. angew. Math. Phys. 37, 491–513 (1986). https://doi.org/10.1007/BF00945427
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00945427