Abstract
This paper deals with finite rotations, and finite strains of three-dimensional space-curved elastic beams, under the action of conservative as well as nonconservative type external distributed forces and moments. The plausible deformation hypothesis of “plane sections remaining plane” is invoked. Exact expressions for the curvature, twist, and transverse shear strains are given; as is a consistent set of boundary conditions. General mixed variational principles, corresponding to the stationarity of a functional with respect to the displacement vector, rotation tensor, stress-resultants, stress-couples, and their conjugate strain-measures, are stated for the case when conservative-type external moments act on the beam. The momentum-balance conditions arising out of these functionals, either coincide exactly with, or are equivalent to, those from the “static method”. The incremental variational functionals, governing both the Total and Updated Lagrangian incremental finite element formulations, are given. An example of the case of the buckling of a beam subject to axial compression and non-conservative type axial twisting couple, is presented and discussed.
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Argyris, J. H.; Balmer, H.; Doltsinis, J. St.; Dunne, P. C.; Haase, M.; Kleiber, M.; Malejannakis, G. A.; Mlejnek, H. P.; Muler, M.; Scharpf, D. W. (1979): Finite element method-The natural approach. Comp. Meth. Appl. Mech. Eng. 17/18, 1–106
Atluri, S. N.; Murakawa, H. (1977): On hybrid-finite element models in nonlinear solid mechanics. Finite elements in nonlinear mechanics. (ed. by P. G. Bergan et al.), Vol 1, 3–41. Norway: Tapir Press
Atluri, S. N. (1979): On rate principles for finite strain analysis of elastic and inelastic nonlinear solids. Recent research on mechanical behavior of solids, pp. 79–107 (Prof. H. Miyamoto's Anniversary Volume), University of Tokyo Press
Atluri, S. N. (1980): On some new general and complementary energy theorems for the rate problems in finite strain, classical elastoplasticity. J. Struct. Mech., 8 (1), 61–92
Atluri, S. N. (1984): Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitely deformed solids, with application o plates and shells — 1, theory. Computers and Struct. 18, No. 1, 93–116
Cowper, G. R. (1966): The shear coefficient in Timoshenko's beam theory. J. Appl. Mech. 33, 335–340
Ericksen, J. L.; Truesdell, C. (1958): Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal. 1, 295–323
Fraeijs de Veubeke, B. (1972): A new variational principle for finite elastic displacements. Int. J. Eng. Sci. 10, 745–763
Hibbit, H. D. (1986): Practical aspects of finite elements computations in solid mechanis. Appl. Mech. Rev. 39, 11, 1678–1681
Kane, T. R.; Likins, P. W.; Levinson, D. A. (1983): Spacecraft dynamics. New York: McGraw-Hill
Kondoh, K.; Atluri, S. N. (1987): Large-deformation, elasto-plastic analysis of frames under nonconservative loading, using explicitly derived tangent stiffness based on assumed stresses. Comput. Mech. 2, 1–25
Murakawa, H.; Atluri, S. N. (1978): Finite elasticity solutions using hybrid finite elements based on a complementary energy principle. J. Appl. Mech. 45, 539–547
Parker, D. F. (1979): The role of Saint Venant's solutions in rods and beam theories. J. Appl. Mech. 46, 861–866
Pietraszkiewicz, W.; Badur, J. (1983): Finite rotations in the description of continuum deformation. Int. J. Eng. Sci. 21, No. 9, 1097–1115
Pleus, P.; Sayir, M. (1983): A second order theory for large deflections of slender beams. J. Appl. Math. Phy. (ZAMM) 34, 192–217
Punch, E.; Atluri, S. N. (1986): Large displacement analysis of plates by a stress-based finite element approach. Computers and Struct. 24, No. 1, 107–117
Reissner, E. (1973): On one-dimensional large-displacement finite-strain beam theory. Studies Appl. Math. 52, 87–95
Reissner, E. (1981): On finite deformations of space-curved beams. J. Appl. Math. Phy. (ZAMP) 32, 734–744
Reissner, E. (1982): Some remarks on the problem of column buckling. Ing.-Arch. 52, 115–119
Simmonds, J. G.; Danielson, D. A. (1972): Non-linear shell theory with finite rotation and stress function vectors. J. Appl. Mech. 39, 1085–1090
Stuelpnagel, J. (1964): On the parametrization of the three-dimensional rotation group. SIAM Rev. 6, No. 4, 422–430
Timsohenko, S. P.; Gere, J. M. (1961): Theory of elastic stability. 2nd ed. New York: McGraw-Hill
Washizu, K. (1982): Variational methods in elasticity and plasticity. 3rd ed. New York: Pergamon Press
Ziegler, H. (1982): Argument for and against Engesser's buckling formula. Ing.-Archiv 52, 105–113
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Communicated by G. Yagawa, March 2, 1987
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Iura, M., Atluri, S.N. On a consistent theory, and variational formulation of finitely stretched and rotated 3-D space-curved beams. Computational Mechanics 4, 73–88 (1988). https://doi.org/10.1007/BF00282411
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DOI: https://doi.org/10.1007/BF00282411