Exact Solution of Out-of-Plane Problems of an Arch with Varying Curvature and Cross Section
Publication: Journal of Engineering Mechanics
Volume 132, Issue 6
Abstract
This paper deals with the exact solution of the differential equations for the out-of-plane behavior of an arch with varying curvature and cross section. The differential equations include the shear deformation effect. The cross section of the arch is doubly symmetric. Due to the double symmetry, in-plane and out-of-plane behavior will be uncoupled. However, a coupling of the out-of-plane bending and the torsional response will exist and will be discussed in this study. The governing differential equations of planar arches loaded perpendicular to their plane are solved exactly by using the initial value method. The analytical expressions of the fundamental matrix can be obtained for some cases. It is also possible to use these analytical expressions in order to obtain the displacements and the stress resultants for an arch with any loading and boundary conditions. The examples given in the literature are solved and the results are compared. The analytical expressions of the results are given for some examples.
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References
Dasgupta, S., and Sengupta, D. (1988). “Horizontally curved isoparametric beam element with or without elastic foundation including effect of shear deformation.” Comput. Struct., 29(6), 967–973.
Ibrahimbegovic, A. (1995). “On finite element implementation of geometrically nonlinear Reissner’s beam theory: Three-dimensional curved beam elements.” Comput. Methods Appl. Mech. Eng., 122, 11–26.
Kang, K., Bert, C. W., and Striz, A. G. (1996). “Static analysis of a curved shaft subjected to end torques.” Int. J. Solids Struct., 33(11), 1587–1596.
Kapania, R. K., and Li, J. (2003). “A formulation and implementation of geometrically exact curved beam elements incorporating finite strains and finite rotations.” Comput. Mech., 30, 444–459.
Kim, Y. Y., and Kim, Y. (2002). “A one-dimensional theory of thin-walled curved rectangular box beams under torsion and out-of-plane bending.” Int. J. Numer. Methods Eng., 53, 1675–1693.
Love, A. E. H. (1944). A treatise on the mathematical theory of elasticity, Dover, New York.
MacNeal, R. H., and Harder, R. L. (1985). “A proposed standard set of problems to test finite element accuracy.” Finite Elem. Anal. Design, 1, 3–20.
Malekzadeh, P., and Karami, G. (2003). “Out-of-plane static analysis of circular arches by DQM.” Int. J. Solids Struct., 40, 6527–6545.
Naganarayana, B. P., and Prathap, G. (1990). “Consistency aspects of out-of-plane bending, torsion and shear in a quadratic curved beam element.” Int. J. Numer. Methods Eng., 30, 431–443.
Pantazopoulou, S. J. (1992). “Low-order interpolation functions for curved beams.” J. Eng. Mech., 118(2), 329–350.
Rattanawangcharoen, N., Bai, H., and Shah, A. H. (2004). “A 3D cylindrical finite element model for thick curved beam stress analysis.” Int. J. Numer. Methods Eng., 59, 511–531.
Rosen, A., and Abramovich, H. (1984). “Galerkin method as a tool to investigate the planar and nonplanar behavior of curved beams.” Comput. Struct., 18(1), 165–174.
Rubin, M. B. (2000). Cosserat theories: Shells, rods and points, Kluwer Academic, Dordrecht, The Netherlands.
Sengupta, D., and Dasgupta, S. (1997). “Static and dynamic applications of a five noded horizontally curved beam element with shear deformation.” Int. J. Numer. Methods Eng., 40, 1801–1819.
Sheikh, A. H. (2002). “New concept to include shear deformation in a curved beam element.” J. Struct. Eng., 128(3), 406–410.
Timoshenko, S., and Goodier, J. N. (1951). Theory of elasticity, 2nd Ed., McGraw–Hill, Tokyo.
Tufekci, E. (2004). “On finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures [1].” Comput. Methods Appl. Mech. Eng., 193, 4067–4068.
Zivkovic, M., Kojic, M., Slavkovic, R., and Grujovic, N. (2001). “A general beam finite element with deformable cross-section.” Comput. Methods Appl. Mech. Eng., 190, 2651–2680.
Zupan, D., and Saje, M. (2003). “Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures.” Comput. Methods Appl. Mech. Eng., 192, 5209–5248.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 132 • Issue 6 • June 2006
Pages: 600 - 609
Copyright
© 2006 ASCE.
History
Received: Nov 4, 2004
Accepted: Jul 22, 2005
Published online: Jun 1, 2006
Published in print: Jun 2006
Notes
Note. Associate Editor: Bojan B. Guzina
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