0
Research Papers

# A New Locking-Free Formulation of a Three-Dimensional Shear-Deformable Beam

[+] Author and Article Information
W. Fan

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering, University of Maryland, Baltimore,
1000 Hilltop Circle,
Baltimore, MD 21250

W. D. Zhu

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland, Baltimore,
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 28, 2016; final manuscript received February 14, 2017; published online May 26, 2017. Assoc. Editor: Marco Amabili.

J. Vib. Acoust 139(5), 051001 (May 26, 2017) (13 pages) Paper No: VIB-16-1519; doi: 10.1115/1.4036210 History: Received October 28, 2016; Revised February 14, 2017

## Abstract

A new locking-free formulation of a three-dimensional shear-deformable beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope that is related to the rotation of a corresponding cross section and stretch and shear strains. The rotation is parameterized by a rotation vector, which has a clear and intuitive physical meaning. Taylor polynomials are used for certain terms that have zero denominators to avoid singularity in numerical implementation. Since the rotation vector can have singular points when its norm equals $2mπ$, where $m$ is a nonzero integer, a rescaling strategy is adopted to resolve the singularity problem when there is only one singular point at a time instant, which is the case for most engineering applications. Governing equations of the beam are obtained using Lagrange's equations for systems with constraints, and several benchmark problems are simulated to show the performance of the current formulation. Results show that the current formulation does not suffer from shear and Poisson locking problems that the absolute nodal coordinate formulation (ANCF) can have. Results from the current formulation for a planar static case are compared with its exact solutions, and they are in excellent agreement with each other, which verifies accuracy of the current formulation. Results from the current formulation are compared with those from commercial software abaqus and recurdyn, and they are in good agreement with each other; the current formulation uses much fewer numbers of elements to yield converged results.

<>

## References

Love, A. E. H. , 1944, A Treatise on the Mathematical Theory of Elasticity, Courier Dover Publications, New York.
Coleman, B. D. , Dill, E. H. , Lembo, M. , Lu, Z. , and Tobias, I. , 1993, “ On the Dynamics of Rods in the Theory of Kirchhoff and Clebsch,” Arch. Ration. Mech. Anal., 121(4), pp. 339–359.
von Dombrowski, S. , 2002, “ Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates,” Multibody Syst. Dyn., 8(4), pp. 409–432.
Zhao, Z. H. , and Ren, G. X. , 2012, “ A Quaternion-Based Formulation of Euler–Bernoulli Beam Without Singularity,” Nonlinear Dyn., 67(3), pp. 1825–1835.
Fan, W. , Zhu, W. D. , and Ren, H. , 2016, “ A New Singularity-Free Formulation of a Three-Dimensional Euler–Bernoulli Beam Using Euler Parameters,” ASME J. Comput. Nonlinear Dyn., 11(4), p. 041013.
Fan, W. , and Zhu, W. D. , 2016, “ An Accurate Singularity-Free Formulation of a Three-Dimensional Curved Euler–Bernoulli Beam for Flexible Multibody Dynamic Analysis,” ASME J. Vib. Acoust., 138(5), p. 051001.
Pai, P. F. , 2014, “ Problems in Geometrically Exact Modeling of Highly Flexible Beams,” Thin-Walled Struct., 76, pp. 65–76.
Rubin, B. , 2000, Cosserat Theories: Shells, Rods and Points, Springer, Dordrecht, The Netherlands.
Cao, D. Q. , Song, M. T. , Tucker, R. W. , Zhu, W. D. , Liu, D. S. , and Huang, W. H. , 2013, “ Dynamic Equations of Thermoelastic Cosserat Rods,” Commun. Nonlinear Sci. Numer. Simul., 18(7), pp. 1880–1887.
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.
Simo, J. C. , 1985, “ A Finite Strain Beam Formulation. The Three-Dimensional Dynamic Problem—Part I,” Comput. Methods Appl. Mech. Eng., 49(1), pp. 55–70.
Ren, H. , 2014, “ A Computationally Efficient and Robust Geometrically-Exact Curved Beam Formulation for Multibody Systems,” 3rd Joint International Conference on Multibody System Dynamics (IMSD) and the 7th Asian Conference on Multibody Dynamics (ACMD), Busan, Korea, June 30–July 3, Paper No. 0228.
Stuelpnagel, J. , 1964, “ On the Parametrization of the Three-Dimensional Rotation Group,” SIAM Rev., 6(4), pp. 422–430.
Zhu, W. D. , Fan, W. , Mao, Y. G. , and Ren, G. X. , 2017, “ Three-Dimensional Dynamic Modeling and Analysis of Moving Elevator Traveling Cables,” Proc. Inst. Mech. Eng., Part K, 231(1), pp. 167–180.
Zhang, Z. G. , Qi, Z. H. , Wu, Z. G. , and Fang, H. Q. , 2015, “ A Spatial Euler–Bernoulli Beam Element for Rigid-Flexible Coupling Dynamic Analysis of Flexible Structures,” Shock Vib., 2015, p. 208127.
Ibrahimbegović, A. , Frey, F. , and Kožar, I. , 1995, “ Computational Aspects of Vector-Like Parametrization of Three-Dimensional Finite Rotations,” Int. J. Numer. Methods Eng., 38(21), pp. 3653–3673.
Shabana, A. A. , 1996, “ An Absolute Nodal Coordinate Formulation for the Large Rotation and Deformation Analysis of Flexible Bodies,” University of Illinois at Chicago, Chicago, IL, Report No. MBS96-1-UIC.
Shabana, A. A. , 2015, “ Definition of ANCF Finite Elements,” ASME J. Comput. Nonlinear Dyn., 10(5), p. 054506.
Gerstmayr, J. , and Shabana, A. A. , 2006, “ Analysis of Thin Beams and Cables Using the Absolute Nodal Co-Ordinate Formulation,” Nonlinear Dyn., 45(1–2), pp. 109–130.
Dufva, K. , Sopanen, J. , and Mikkola, A. , 2005, “ A Two-Dimensional Shear Deformable Beam Element Based on the Absolute Nodal Coordinate Formulation,” J. Sound Vib., 280(3), pp. 719–738.
García-Vallejo, D. , Mikkola, A. M. , and Escalona, J. L. , 2007, “ A New Locking-Free Shear Deformable Finite Element Based on Absolute Nodal Coordinates,” Nonlinear Dyn., 50(1–2), pp. 249–264.
Schwab, A. , and Meijaard, J. , 2010, “ Comparison of Three-Dimensional Flexible Beam Elements for Dynamic Analysis: Classical Finite Element Formulation and Absolute Nodal Coordinate Formulation,” ASME J. Comput. Nonlinear Dyn., 5(1), p. 011010.
Sopanen, J. T. , and Mikkola, A. M. , 2003, “ Description of Elastic Forces in Absolute Nodal Coordinate Formulation,” Nonlinear Dyn., 34(1–2), pp. 53–74.
Gerstmayr, J. , Sugiyama, H. , and Mikkola, A. , 2013, “ Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 8(3), p. 031016.
Zhu, W. D. , Ren, H. , and Xiao, C. , 2011, “ A Nonlinear Model of a Slack Cable With Bending Stiffness and Moving Ends With Application to Elevator Traveling and Compensation Cables,” ASME J. Appl. Mech., 78(4), p. 041017.
Huang, J. L. , and Zhu, W. D. , 2014, “ Nonlinear Dynamics of a High-Dimensional Model of a Rotating Euler–Bernoulli Beam Under the Gravity Load,” ASME J. Appl. Mech., 81(10), p. 101007.
Li, L. , Zhu, W. D. , Zhang, D. G. , and Du, C. F. , 2015, “ A New Dynamic Model of a Planar Rotating Hub-Beam System Based on a Description Using the Slope Angle and Stretch Strain of the Beam,” J. Sound Vib., 345, pp. 214–232.
Ren, H. , Zhu, W. D. , and Fan, W. , 2016, “ A Nonlinear Planar Beam Formulation With Stretch and Shear Deformations Under End Forces and Moments,” Int. J. Non-Linear Mech., 85, pp. 126–142.
Manta, D. , and Gonçalves, R. , 2016, “ A Geometrically Exact Kirchhoff Beam Model Including Torsion Warping,” Comput. Struct., 177, pp. 192–203.
Hodges, D. H. , 2006, “ Nonlinear Composite Beam Theory,” Progress in Astronautics and Aeronautics, American Institute of Aeronautics and Astronautics, Reston, VA.
Orzechowski, G. , and Shabana, A. A. , 2016, “ Analysis of Warping Deformation Modes Using Higher Order ANCF Beam Element,” J. Sound Vib., 363, pp. 428–445.
Bauchau, O. A. , and Trainelli, L. , 2003, “ The Vectorial Parameterization of Rotation,” Nonlinear Dyn., 32(1), pp. 71–92.
Bauchau, O. A. , and Han, S. L. , 2014, “ Interpolation of Rotation and Motion,” Multibody Syst. Dyn., 31(3), pp. 339–370.
Campanelli, M. , Berzeri, M. , and Shabana, A. A. , 2000, “ Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems,” ASME J. Mech. Des., 122(4), pp. 498–507.
Ren, H. , 2015, “ A Simple Absolute Nodal Coordinate Formulation for Thin Beams With Large Deformations and Large Rotations,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 061005.
Omar, M. , and Shabana, A. , 2001, “ A Two-Dimensional Shear Deformable Beam for Large Rotation and Deformation Problems,” J. Sound Vib., 243(3), pp. 565–576.
Humer, A. , 2011, “ Elliptic Integral Solution of the Extensible Elastica With a Variable Length Under a Concentrated Force,” Acta Mech., 222(3–4), pp. 209–223.
Kane, T. , Ryan, R. , and Banerjee, A. , 1987, “ Dynamics of a Cantilever Beam Attached to a Moving Base,” J. Guid. Control Dyn., 10(2), pp. 139–151.
Reissner, E. , 1972, “ On One-Dimensional Finite-Strain Beam Theory: The Plane Problem,” J. Appl. Math. Phys., 23(5), pp. 795–804.

## Figures

Fig. 1

Geometrical description of a three-dimensional shear-deformable beam

Fig. 2

Configurations of the cantilever beam under concentrated bending moments, where dots indicate nodes of elements of the beam

Fig. 3

Schematic of a cantilever beam with (a) one and (b) two concentrated forces at its free end

Fig. 4

Vertical displacements from different formulations with different numbers of elements when α=0.1 and SL=10−4

Fig. 5

(a) X and (b) Y displacements at the tip of the beam

Fig. 6

(a) Longitudinal and (b) transverse deflections at the tip of the beam

Fig. 7

(a) Stretch and (b) shear strains at the midpoint of the centroid line of the beam

Fig. 8

Schematic of a rotating beam, where O-XY is the global coordinate system and O-xy is a rotating frame

Fig. 9

(a) Longitudinal and (b) transverse deflections at the tip of the rotating beam

Fig. 10

Longitudinal deflection at the tip of the rotating beam with material damping

Fig. 11

(a) X, (b) Y, and (c) Z displacements at the tip of the rotating beam under the gravity effect

Fig. 12

Norm of the rotation vector at the pinned end of the rotating beam under the gravity effect

Fig. 13

Angular velocities of the cross section at the tip of the beam

Fig. 14

Kinematic description of a planar extensible and shear-deformable beam

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections