Abstract
This paper is devoted to the modeling of planar slender beams undergoing large displacements and finite rotations. Transverse shear deformation of beams that is trivial for most slender beams is neglected in the present model, though within the framework of the geometrically exact beam theory proposed by Reissner. A weak form quadrature element formulation is proposed which is characterized by highly efficient numerical integration and differentiation, thus minimizing the number of elements as well as the total degrees-of-freedom. Several typical examples are presented to demonstrate the effectiveness of the beam model and the weak form quadrature element formulation.
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References
Reissner E.: On one-dimensional finite-strain beam theory: the plane problem. J. Appl. Math. Phys. (ZAMP) 23, 795–804 (1972)
Jelenic G., Crisfield M.A.: Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput. Methods. Appl. Mech. Eng. 171, 141–171 (1999)
Auricchio F., Carotenuto P., Reali A.: On the geometrically exact beam model: a consistent, effective and simple derivation from three-dimensional finite elasticity. Int. J. Solids Struct. 45, 4766–4781 (2008)
Zupan E., Saje M., Zupan D.: The quaternion-based three-dimensional beam theory. Comput. Methods Appl. Mech. Eng. 198, 3944–3956 (2009)
Simo J.C.: A finite strain beam formulation: the three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49, 55–70 (1985)
Simo J.C., Vu-Quoc L.: A three-dimensional finite-strain rod model. Part II: computational aspects. Comput. Methods Appl. Mech. Eng. 58, 79–116 (1986)
Santos H.A.F.A., Pimenta P.M., MoitinhoDe Almeida J.P.: Hybrid and multifield variational principles for geometrically exact three-dimensional beams. Int J Non-linear Mech. 45, 809–820 (2010)
Češarek P., Saje M., Zupan D.: Kinematically exact curved and twisted strain-based beam. Int. J. Solids Struct. 49, 1802–1817 (2012)
Saje M.: A variational principle for finite planar deformation of straight slender elastic beams. Int. J. Solids Struct. 26(8), 887–900 (1990)
Gerstmayr J., Shabana A.A.: Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 45, 109–130 (2006)
Irschik H., Gerstmayr J.: A continuum mechanics based derivation of Reissner’s large-displacemnt finite-strain beam thory: the case of plane deformation of originally straight Bernoulli–Euler beams. Acta Mech. 206, 1–21 (2009)
Zhao Z., Ren G.: A quaternion-based formulation of Euler–Bernoulli beam without singularity. Nonlinear Dyn. 67, 1825–1835 (2012)
Zhong H., Yu T.: Flexural vibration analysis of an eccentric annular Mindlin plate. Arch. Appl. Mech. 77, 185–195 (2007)
Zhong H., Yu T.: A weak-form quadrature element method for plane elasticity problems. Appl. Math. Model. 33(10), 3801–3814 (2009)
Zhong H., Wang Y.: Weak form quadrature element analysis of Bickford beams. Eur. J. Mech. /Solids. 29(5), 851–858 (2010)
Zhong H., Gao M.: Quadrature element analysis of planar frameworks. Arch. Appl. Mech. 80, 1391–1405 (2010)
Zhong H., Zhang R., Yu H.: Buckling analysis of planar frameworks using the quadrature element method. Int. J. Struct. Stab. Dyn. 11(2), 363–378 (2011)
**ao N., Zhong H.: Nonlinear quadrature element analysis of planar frames based on geometrically exact beam theory. Int. J. Non-linear Mech. 47(5), 481–488 (2012)
Striz, A.G., Chen, W.L., Bert, C.W.: High accuracy plane stress and plate elements in the quadrature element method. In: Proceedings of the 36th AIAA/ ASME/ ASCE/ AHS/ ASC957-965 (1995)
Striz A.G., Chen W.L., Bert C.W.: Free vibration of plates by the high accuracy quadrature element method. J. Sound Vib. 202, 689–702 (1997)
Bellman R.E., Casti J.: Differential quadrature and long term integration. J. Math. Anal. Appl. 34, 235–238 (1971)
Bert C.W., Malik M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49, 1–27 (1996)
Shu C.: Differential quadrature and its application in engineering. Springer, London (2000)
Crisfield M.A.: Non-linear finite element analysis of solids and structures. Wiley, New York (1991)
Love A.E.H.: A treatise on the mathematical theory of elasticity. Dover, New York (1944)
Mattiasson, K.: Numerical results from elliptic integral solutions of some elastic problems of beams and frames[R]. Publ. 79, 10 (1979). Department of Structural mechanics, Chalmers University of Technology, Goteborg
Argyris J.H., Symeonides Sp.: A sequel to: nonlinear finite element analysis of elastic systems under nonconservative loading—natural formulation. Part I. Quasistatic problems. Comput. Methods Appl. Mech. Eng. 26, 373–383 (1981)
Betsch P., Steinmann P.: Frame indifferent beam finite elements based upon the geometrically exact beam theory. Int. J. Numer. Methods Eng. 54, 1775–1788 (2002)
Zupan D., Saje M.: Finite element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures. Comput. Methods Appl. Mech. Eng. 192, 5209–5248 (2003)
Saje M., Turk G., Kalagasidu A., Vratanar B.: A kinematically exact finite element formulation of elastic-plastic curved beams. Comput. Struct. 67, 197–214 (1998)
Coulter B.T., Miller R.E.: Loading, unloading and reloading of a generalized plane plastica. Int. J. Numer Methods Eng. 28, 1645–1660 (1989)
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Zhang, R., Zhong, H. Weak form quadrature element analysis of planar slender beams based on geometrically exact beam theory. Arch Appl Mech 83, 1309–1325 (2013). https://doi.org/10.1007/s00419-013-0748-3
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DOI: https://doi.org/10.1007/s00419-013-0748-3