[en] Based on a formulation on the special Euclidean group SE(3), a geometrically exact thin-walled beam with an arbitrary open cross-section is proposed to deal with the finite deformation and rotation issues. The beam strains are based on a kinematic assumption where warping deformation and Wagner effects are included such that the nonlinear behavior of a thin-walled beam is predicted accurately, particular under large torsion. To reduce the nonlinearity of rigid motion, static and dynamic equations are derived in the SE(3) framework based on the local frame approach. As the value of the iteration matrix, including the Jacobian matrix of inertial and internal forces, is invariable under arbitrary rigid motion, the number of updates required during the computation process decreases sharply, which drastically improves the computational efficiency. Furthermore, the isogeometric analysis (IGA) based on the non-uniform rational B-splines (NURBS) basis functions, which promotes the integration of computer-aided design (CAD) and computer-aided engineering (CAE), is adopted to interpolate the displacement, rotation, and warping fields separately. The interpolated strain measures satisfy the objectivity by removing the rigid motion of the reference point. To obtain the symmetric Jacobian matrix of internal forces, the linearization operation is conducted based on the previously converged configuration. A Lie group SE(3) extension of the generalized-α time integration method is utilized to solve the equations of motion for thin-walled beams. Finally, the proposed formulation is successfully tested and validated in several static and dynamic numerical examples.
Disciplines :
Mechanical engineering
Author, co-author :
Rong, Jili
Wu, Zhipei
Liu, Cheng
Bruls, Olivier ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Language :
English
Title :
Geometrically exact thin-walled beam including warping formulated on the special Euclidean group SE(3)
Publication date :
2020
Journal title :
Computer Methods in Applied Mechanics and Engineering
Reissner, E., On one-dimensional finite-strain beam theory: the plane problem. J. Appl. Math. Phys. 23 (1972), 795–804, 10.1007/BF01602645.
Reissner, E., On one-dimensional large-displacement finite-strain beam theory. Stud. Appl. Math., 1973, 10.1002/sapm197352287.
Simo, J.C., A finite strain beam formulation. The three-dimensional dynamic problem, Part I. Comput. Methods Appl. Mech. Engrg. 49 (1985), 55–70, 10.1016/0045-7825(85)90050-7.
Simo, J.C., Vu-Quoc, L., A three-dimensional finite-strain rod model. part II: Computational aspects. Comput. Methods Appl. Mech. Engrg. 58:1 (1986), 79–116, 10.1016/0045-7825(86)90079-4.
Simo, J.C., Vu-Quoc, L., On the dynamics in space of rods undergoing large motions - A geometrically exact approach. Comput. Methods Appl. Mech. Engrg. 66 (1988), 125–161, 10.1016/0045-7825(88)90073-4.
Cardona, A., Géradin, M., A beam finite element non-linear theory with finite rotations. Int. J. Numer. Methods Engrg. 26 (1988), 2403–2438, 10.1002/nme.1620261105.
Ibrahimbegović, A., Frey, F., Kozar, I., Computational aspects of vector-like parametrization of three-dimensional finite rotations. Int. J. Numer. Methods Engrg. 38 (1995), 3653–3673, 10.1002/nme.1620382107.
Crisfield, M.A., Jelenić, G., Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 455:1983 (1999), 1125–1147, 10.1098/rspa.1999.0352.
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. Engrg. 171 (1999), 141–171, 10.1016/S0045-7825(98)00249-7.
Angeles, J., Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms. 2003, Springer-Verlag, Berlin, 10.1007/b97597.
Selig, J.M., Geometric Fundamentals of Robotics. 2005, Springer, New York, 10.1007/b138859.
Murray, R.M., Li, Z., Sastry, S., A Mathematical Introduction To Robotic Manipulation. 1994, CRC Press, Boca Raton, 10.1201/9781315136370.
Sonneville, V., Cardona, A., Brüls, O., Geometrically exact beam finite element formulated on the special euclidean group SE(3). Comput. Methods Appl. Mech. Engrg. 268 (2014), 451–474, 10.1016/j.cma.2013.10.008s.
Borri, M., Bottasso, C.L., Trainelli, L., Integration of elastic multibody systems by invariant conserving/dissipating algorithms I formulation. Comput. Methods Appl. Mech. Engrg. 190 (2001), 3669–3699, 10.1016/S0045-7825(00)00286-3.
Borri, M., Trainelli, L., Bottasso, C.L., On representations and parameterizations of motion. Multibody Syst. Dyn. 4 (2000), 129–193, 10.1023/A:100-9830626597.
Sander, O., Geodesic finite elements for cosserat rods. Internat. J. Numer. Methods Engrg. 82 (2010), 1645–1670, 10.1002/nme.2814.
Kollbrunner, C.F., Basler, K., Torsion in Structures. 1969, Springer, Berlin, 10.1007/978-3-662-22557-8.
Gruttmann, F., Sauer, R., Wagner, W., A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections. Comput. Methods Appl. Mech. Engrg. 160 (1998), 383–400, 10.1016/S0045-7825(97)00305-8.
Simo, J.C., Vu-Quoc, L., A geometrically-exact rod model incorporating shear and torsion-warping deformation. Int. J. Solid Struct. 27:3 (1991), 371–393, 10.1016/0020-7683(91)90089-X.
Gruttmann, F., Sauer, R., Wagner, W., Theory and numerics of three-dimensi-onal beams with elastoplastic material behavior. Int. J. Numer. Methods Engrg. 48 (2000), 1675–1702, 10.1002/1097-0207(20000830)48:12¡1675::A-ID-NME957¿3.3.CO;2-Y.
Manta, D., Gonçalves, R., A geometrically exact Kirchhoff beam model including torsion warping. Comput. Struct. 177 (2016), 192–203, 10.1016/j-.compstruc.2016.08.013.
Saravia, C.M., Machado, S.P., Cortínez, V.H., A geometrically exact nonlinear finite element for composite closed section thin-walled beams. Comput. Struct. 89 (2011), 2337–2351, 10.1016/j.compstruc.2011.07.009.
Saravia, C.M., Machado, S.P., Cortínez, V.H., A composite beam finite element for multibody dynamics: Application to large wind turbine modeling. Eng. Struct. 56 (2013), 1164–1176, 10.1016/j.engstruct.2013.06.037.
Han, S., Bauchau, O.A., On the solution of Almansi-Michell's problem. Int. J. Solids Struct. 75-76 (2015), 156–171, 10.1016/j.ijsolstr.2015.08.010.
Han, S., Bauchau, O.A., Nonlinear, three-dimensional beam theory for dynamic analysis. Multibody Syst. Dyn. 41 (2016), 173–200, 10.1007/s11044-016-9554-3.
Kim, N.-I., Kim, M.-Y., Exact dynamic/static stiffness matrices of non-symm-etric thin-walled beams considering coupled shear deformation effects. Thin-Walled Struct. 43 (2005), 701–734, 10.1016/j.tws.2005.01.004.
Petrov, E., Géradin, M., Finite element theory for curved and twisted beams based on exact solutions for three-dimensional solids, Part 2: Anisotropic and advanced beam models. Comput. Methods Appl. Mech. Engrg. 165 (1998), 93–127, 10.1016/S0045-7825(98)00060-7.
Battini, J.M., Pacoste, C., Co-rotational beam elements with warping effects in instability problems. Comput. Methods Appl. Mech. Engrg. 191 (2002), 1755–1789, 10.1016/S0045-7825(01)00352-8.
Hirashima, M., Iura, M., Yoda, T., Finite displacement theory of naturtally curved and twisted thin-walled members. Proc. JSCE 1979 (1979), 13–27, 10.2208/jscej1969.1979.292-13.
Pi, Y.-L., Bradford, M.A., Uy, B., A spatially curved-beam element with warping and Wagner effects. Internat. J. Numer. Methods Engrg. 63 (2005), 1342–1369, 10.1002/nme.1337.
Pi, Y.-L., Bradford, M.A., Uy, B., Nonlinear analysis of members curved in space with warping and wagner effects. Int. J. Solids Struct. 42 (2005), 3147–3169, 10.1016/j.ijsolstr.2004.10.012.
De Lorenzis, L., Wriggers, P., Hughes, T.J.R., Isogeometric contact: a review. GAMM-Mitt. 37 (2014), 85–123, 10.1002/gamm.201410005.
Beirão da Veiga, L., Lovadina, C., Reali, A., Avoiding shear locking for the timoshenko beam problem via isogeometric collocation methods. Comput. Methods Appl. Mech. Engrg. 241-244 (2012), 38–51, 10.1016/j.cma.2012-.05.020.
Auricchio, F., Beirão da Veiga, L., Kiendl, J., Lovadina, C., Reali, A., Locking-free isogeometric collocation methods for spatial Timoshenko rods. Comput. Methods Appl. Mech. Engrg. 263 (2013), 113–126, 10.1016/j.cma.2013.0-3.009.
Greco, L., Cuomo, M., An isogeometric implicit G1 mixed finite element for Kirchhoff space rods. Comput. Methods Appl. Mech. Engrg. 298 (2016), 325–349, 10.1016/j.cma.2015.06.014.
Bauer, A., Breitenberger, M., Philipp, B., Wüchner, R., Bletzinger, K.-U., Nonlinear isogeometric spatial Bernoulli beam. Comput. Methods Appl. Mech. Engrg. 303 (2016), 101–127, 10.1016/j.cma.2015.12.027.
Marino, E., Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams. Comput. Methods Appl. Mech. Engrg. 307 (2016), 383–410, 10.1016/j.cma.2016.04.016.
Marino, E., Locking-free isogeometric collocation formulation for three-dime-nsional geometrically exact shear-deformable beams with arbitrary initial curvature. Comput. Methods Appl. Mech. Engrg. 324 (2017), 546–572, 10.1016/j.cma.2017.06.031.
Choi, M.-J., Yoon, M., Cho, S., Isogeometric configuration design sensitivity analysis of finite deformation curved beam structures using Jaumann strain formulation. Comput. Methods Appl. Mech. Engrg. 309 (2016), 41–73, 10.1016/j.cma.2016.05.040.
Brüls, O., Cardona, A., On the use of Lie group time integrators in multibody dynamics. J Comput. Nonlin. Dyn., 5, 2010, 10.1115/1.4001370.
Brüls, O., Cardona, A., Arnold, M., Lie Group generalized-α time integration of constrained flexible multibody systems. Mech. Mach. Theory 48 (2012), 121–137, 10.1016/j.mechmachtheory.2011.07.017.
Giavotto, V., Borri, M., Mantegazza, P., Ghiringhelli, G., Carmaschi, V., Maoli, G.C., Mussi, F., Anisotropic beam theory and applications. Comput. Struct. 16:1–4 (1983), 403–413, 10.1016/0045-7949(83)90179-7.
Borri, M., Ghiringhelli, G.L., Merlini, T., Linear analysis of naturally curved and twisted anisotropic beams. Compos. Eng. 2:57 (1992), 433–456, 10.1016/0961-9526(92)90036-6.
Bauchau, O.A., Han, S., Three-dimensional beam theory for flexible multibody dynamics. J. Comput. Nonlin. Dyn., 9(4), 2014 041011-1, http://dx.doi.org/10.1115/1.4025820.
Han, S., Bauchau, O.A., Nonlinear three-dimensional beam theory for flexible multibody dynamics. Multibody Syst. Dyn. 34:3 (2015), 211–242, 10.1007/s11044-014-9433-8.
Ibrahimbegović, A., On the choice of finite rotation parameters. Comput. Methods Appl. Mech. Engrg. 149 (1997), 49–71, 10.1016/s0045-7825(97)00059-5.
Ibrahimbegović, A., Taylor, R.L., On the role of frame-invariance of structural mechanics models at finite rotations. Comput. Methods Appl. Mech. Engrg. 191 (2002), 5159–5176, 10.1016/s0045-7825(02)00442-5.
Gonçalves, R., A shell-like stress resultant approach for elastoplastic geometrically exact thin-walled beam finite elements. Thin-Walled Struct. 103 (2016), 263–272, 10.1016/j.tws.2016.01.011.
Ibrahimbegović, A., AlMikdad, M., Finite rotations in dynamics of beams and implicit time-stepping schemes. Internat. J. Numer. Methods Engrg. 41 (1998), 781–814, 10.1002/(SICI)1097-0207(19980315)41:5¡781::AID-NME308¿3.0.C-O;2-9.
Lens, E.V., Cardona, A., A nonlinear beam element formulation in the framework of an energy preserving time integration scheme for constrained multibody systems dynamics. Comput. Struct. 86 (2008), 47–63, 10.1016/j.com-pstruc.2007.05.036.
George, A., Liu, A., J. W, T., Computer solution of large sparse positive definite systems. SIAM Rev. 26 (1984), 289–291, 10.1137/1026055.
Ibrahimbegović, A., Mamouri, S., On rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3D geometrically exact beam model. Comput. Methods Appl. Mech. Engrg. 188 (2000), 805–831, 10.1016/S0045-7825(99)00363-1.
