Research Papers

Three-Dimensional Beam Theory for Flexible Multibody Dynamics

[+] Author and Article Information
Olivier A. Bauchau, Shilei Han

University of Michigan-Shanghai Jiao Tong
University Joint Institute,
Shanghai 200240, China

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received April 16, 2013; final manuscript received October 23, 2013; published online July 11, 2014. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 9(4), 041011 (Jul 11, 2014) (12 pages) Paper No: CND-13-1082; doi: 10.1115/1.4025820 History: Received April 16, 2013; Revised October 23, 2013

In multibody systems, it is common practice to approximate flexible components as beams or shells. More often than not, classical beam theories, such as the Euler–Bernoulli beam theory, form the basis of the analytical development for beam dynamics. The advantage of this approach is that it leads to simple kinematic representations of the problem: the beam's section is assumed to remain plane and its displacement field is fully defined by three displacement and three rotation components. While such an approach is capable of accurately capturing the kinetic energy of the system, it cannot adequately represent the strain energy. For instance, it is well known from Saint-Venant's theory for torsion that the cross-section will warp under torque, leading to a three-dimensional deformation state that generates a complex stress state. To overcome this problem, sectional stiffnesses are computed based on sophisticated mechanics of material theories that evaluate the complete state of deformation. These sectional stiffnesses are then used within the framework of a Euler–Bernoulli beam theory based on far simpler kinematic assumptions. While this approach works well for simple cross-sections made of homogeneous material, inaccurate predictions may result for realistic configurations, such as thin-walled sections, or sections comprising anisotropic materials. This paper presents a different approach to the problem. Based on a finite element discretization of the cross-section, an exact solution of the theory of three-dimensional elasticity is developed. The only approximation is that inherent to the finite element discretization. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions, as expected from Saint-Venant's principle.

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Bauchau, O. A., and Craig, J. I., 2009, Structural Analysis With Application to Aerospace Structures, Springer, Dordrecht, Netherlands.
Timoshenko, S. P., 1921, “On the Correction Factor for Shear of the Differential Equation for Transverse Vibrations of Bars of Uniform Cross-Section,” Philos. Mag., 41, pp. 744–746. [CrossRef]
Simo, J. C., 1985, “A Finite Strain Beam Formulation. The Three-Dimensional Dynamic Problem. Part I,” Comput. Methods App. Mech. Eng., 49(1), pp. 55–70. [CrossRef]
Cardona, A., and Géradin, M., 1988, “A Beam Finite Element Non-Linear Theory With Finite Rotation,’ Int. J. Numer. Methods Eng., 26, pp. 2403–2438. [CrossRef]
Wasfy, T. M., and Noor, A. K., 2003, “Computational Strategies for Flexible Multibody Systems,’ ASME Appl. Mech. Rev., 56(2), pp. 553–613. [CrossRef]
Giavotto, V., Borri, M., Mantegazza, P., Ghiringhelli, G., Carmaschi, V., Maffioli, G. C., and Mussi, F., 1983, “Anisotropic Beam Theory and Applications,” Comput. Struct., 16(1–4), pp. 403–413. [CrossRef]
de Saint-Venant, J. C.-B., 1855, “Mémoire sur la torsion des prismes,” Receuil des Savants Étrangers, 14, pp. 233–560.
Mielke, A., 1988, “Saint-Venant's Problem and Semi-Inverse Solutions in Nonlinear Elasticity,” Arch. Ration. Mech. Anal., 102, pp. 205–229. [CrossRef]
Mielke, A., 1990, “Normal Hyperbolicity of Center Manifolds and Saint-Venant's Principle,” Arch. Ration. Mech. Anal., 110, pp. 353–372. [CrossRef]
Mielke, A., 1991, Hamiltonian and Lagrangian Flows on Center Manifolds with Applications to Elliptic Variational Problems (Lecture Notes in Mathematics), Vol. 1489, Springer, Berlin, Germany.
Zhong, W. X., 1995, A New Systematic Methodology for Theory of Elasticity, Dalian University of Technology Press, Dalian, China.
Zhong, W. X., Xu, X. S., and Zhang, H. W., 1996, “Hamiltonian Systems and the Saint-Venant Problem in Elasticity,” Appl. Math. Mech., 17(9), pp. 827–836. [CrossRef]
Yao, W. A., Zhong, W. X., and Lim, C. W., 2009, Symplectic Elasticity, World Scientific, Singapore.
Morandini, M., Chierichetti, M., and Mantegazza, P., 2010, “Characteristic Behavior of Prismatic Anisotropic Beam Via Generalized Eigenvectors,” Int.l J. Solids Struct., 47, pp. 1327–1337. [CrossRef]
Berdichevsky, V. L., 1982, “On the Energy of an Elastic Rod,’ Prikl. Mat. Mekh., 45(4), pp. 518–529.
Hodges, D. H., 1990, “A Review of Composite Rotor Blade Modeling,’ AIAA J., 28(3), pp. 561–565. [CrossRef]
Atilgan, A. R., Hodges, D. H., and Fulton, M. V., 1991, “Nonlinear Deformation of Composite Beams: Unification of Cross-Sectional and Elastical Analyses,” Appli. Mech. Rev., 44(11), pp. S9–S15 [CrossRef]
Atilgan, A. R., and Hodges, D. H., 1991, “Unified Nonlinear Analysis for Nonhomogeneous Anisotropic Beams With Closed Cross Sections,” AIAA J., 29(11), pp. 1990–1999. [CrossRef]
Hodges, D. H., 2006, Nonlinear Composite Beam Theory, AIAA, Reston, VA.
Hughes, T. J. R., 1987, The Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ.
Bathe, K. J., 1996, Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ.
Lanczos, C., 1970, The Variational Principles of Mechanics, Dover, New York,
Borri, M., and Merlini, T., 1986, “A Large Displacement Formulation for Anisotropic Beam Analysis,” Meccanica, 21, pp. 30–37. [CrossRef]
Zhong, W. X., 2004, Duality System in Applied Mechanics and Optimal Control, Kluwer, Boston, MA.
Hochstadt, H., 1964, Differential Equations, Dover, New York.
Bauchau, O. A., 2011, Flexible Multibody Dynamics, Springer, Dordrecht, Netherlands.
Popescu, B., and Hodges, D. H., 2000, “On Asymptotically Correct Timoshenko-Like Anisotropic Beam Theory,” Int. Jo. Solids Struct., 37(3), pp. 535–558. [CrossRef]
Simo, J. C., and Vu-Quoc, L., 1986, “A Three-Dimensional Finite Strain Rod Model. Part II: Computational Aspects,” Comput. Methods Appl. Mech. Eng., 58(1), pp. 79–116. [CrossRef]
Betsch, P., and Steinmann, P., 2002, “A DAE Approach to Flexible Multibody Dynamics,’ Multibody Syst. Dyn., 8, pp. 367–391. [CrossRef]
Yu, W. B., Hodges, D. H., Volovoi, V. V., and Fuchs, E. D., 2005, “A Generalized Vlasov Theory for Composite Beams,” Thin-Walled Struct., 43(9), pp. 1493–1511. [CrossRef]


Grahic Jump Location
Fig. 1

Configuration of a straight beam

Grahic Jump Location
Fig. 2

Illustration of Saint-Venant's principle

Grahic Jump Location
Fig. 4

Shear stresses at points A and B. Analytical solution: solid line (point B), dashed line (point A); present solution: symbols ▽ (point B), △ (point A).

Grahic Jump Location
Fig. 5

Configuration of the I-section beam

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Fig. 6

Warping of the I-section beam

Grahic Jump Location
Fig. 3

Nondimensional torsional stiffness versus aspect ratio. Analytical solution: solid line; present solution: symbols ○.



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