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

References
1.
Love
,
A. E. H.
,
1944
,
A Treatise on the Mathematical Theory of Elasticity
,
Courier Dover Publications
,
New York
.
2.
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
.
3.
von Dombrowski
,
S.
,
2002
, “
Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates
,”
Multibody Syst. Dyn.
,
8
(
4
), pp.
409
432
.
4.
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
.
5.
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
.
6.
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
.
7.
Pai
,
P. F.
,
2014
, “
Problems in Geometrically Exact Modeling of Highly Flexible Beams
,”
Thin-Walled Struct.
,
76
, pp.
65
76
.
8.
Rubin
,
B.
,
2000
,
Cosserat Theories: Shells, Rods and Points
,
Springer
, Dordrecht,
The Netherlands
.
9.
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
.
10.
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
.
11.
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
.
12.
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.
13.
Stuelpnagel
,
J.
,
1964
, “
On the Parametrization of the Three-Dimensional Rotation Group
,”
SIAM Rev.
,
6
(
4
), pp.
422
430
.
14.
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
.
15.
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
.
16.
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
.
17.
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.
18.
Shabana
,
A. A.
,
2015
, “
Definition of ANCF Finite Elements
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
5
), p.
054506
.
19.
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
.
20.
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
.
21.
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
.
22.
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
.
23.
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
.
24.
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
.
25.
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
.
26.
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
.
27.
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
.
28.
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
.
29.
Manta
,
D.
, and
Gonçalves
,
R.
,
2016
, “
A Geometrically Exact Kirchhoff Beam Model Including Torsion Warping
,”
Comput. Struct.
,
177
, pp.
192
203
.
30.
Hodges
,
D. H.
,
2006
, “
Nonlinear Composite Beam Theory
,”
Progress in Astronautics and Aeronautics
,
American Institute of Aeronautics and Astronautics
,
Reston, VA
.
31.
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
.
32.
Bauchau
,
O. A.
, and
Trainelli
,
L.
,
2003
, “
The Vectorial Parameterization of Rotation
,”
Nonlinear Dyn.
,
32
(
1
), pp.
71
92
.
33.
Bauchau
,
O. A.
, and
Han
,
S. L.
,
2014
, “
Interpolation of Rotation and Motion
,”
Multibody Syst. Dyn.
,
31
(
3
), pp.
339
370
.
34.
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
.
35.
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
.
36.
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
.
37.
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
.
38.
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
.
39.
Reissner
,
E.
,
1972
, “
On One-Dimensional Finite-Strain Beam Theory: The Plane Problem
,”
J. Appl. Math. Phys.
,
23
(
5
), pp.
795
804
.