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.

References

References
1.
Bauchau
,
O. A.
, and
Craig
,
J. I.
,
2009
,
Structural Analysis With Application to Aerospace Structures
,
Springer
,
Dordrecht
, Netherlands.
2.
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
.
3.
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
.10.1016/0045-7825(85)90050-7
4.
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
.10.1002/nme.1620261105
5.
Wasfy
,
T. M.
, and
Noor
,
A. K.
,
2003
, “
Computational Strategies for Flexible Multibody Systems
,’
ASME Appl. Mech. Rev.
,
56
(
2
), pp.
553
613
.10.1115/1.1590354
6.
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
.10.1016/0045-7949(83)90179-7
7.
de Saint-Venant
,
J. C.-B.
,
1855
, “
Mémoire sur la torsion des prismes
,”
Receuil des Savants Étrangers
,
14
, pp.
233
560
.
8.
Mielke
,
A.
,
1988
, “
Saint-Venant's Problem and Semi-Inverse Solutions in Nonlinear Elasticity
,”
Arch. Ration. Mech. Anal.
,
102
, pp.
205
229
.10.1007/BF00281347
9.
Mielke
,
A.
,
1990
, “
Normal Hyperbolicity of Center Manifolds and Saint-Venant's Principle
,”
Arch. Ration. Mech. Anal.
,
110
, pp.
353
372
.10.1007/BF00393272
10.
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.
11.
Zhong
,
W. X.
,
1995
,
A New Systematic Methodology for Theory of Elasticity
,
Dalian University of Technology Press
,
Dalian
, China.
12.
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
.10.1007/BF00127182
13.
Yao
,
W. A.
,
Zhong
,
W. X.
, and
Lim
,
C. W.
,
2009
,
Symplectic Elasticity
,
World Scientific
,
Singapore
.
14.
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
.10.1016/j.ijsolstr.2010.01.017
15.
Berdichevsky
,
V. L.
,
1982
, “
On the Energy of an Elastic Rod
,
’ Prikl. Mat. Mekh.
,
45
(
4
), pp.
518
529
.
16.
Hodges
,
D. H.
,
1990
, “
A Review of Composite Rotor Blade Modeling
,
’ AIAA J.
,
28
(
3
), pp.
561
565
.10.2514/3.10430
17.
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
10.1115/1.3121379
18.
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
.10.2514/3.10829
19.
Hodges
,
D. H.
,
2006
,
Nonlinear Composite Beam Theory
,
AIAA
,
Reston, VA
.
20.
Hughes
,
T. J. R.
,
1987
,
The Finite Element Method
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
21.
Bathe
,
K. J.
,
1996
,
Finite Element Procedures
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
22.
Lanczos
,
C.
,
1970
,
The Variational Principles of Mechanics
,
Dover
,
New York
,
23.
Borri
,
M.
, and
Merlini
,
T.
,
1986
, “
A Large Displacement Formulation for Anisotropic Beam Analysis
,”
Meccanica
,
21
, pp.
30
37
.10.1007/BF01556314
24.
Zhong
,
W. X.
,
2004
,
Duality System in Applied Mechanics and Optimal Control
,
Kluwer
,
Boston
, MA.
25.
Hochstadt
,
H.
,
1964
,
Differential Equations
,
Dover
,
New York
.
26.
Bauchau
,
O. A.
,
2011
,
Flexible Multibody Dynamics
,
Springer
,
Dordrecht
, Netherlands.
27.
Popescu
,
B.
, and
Hodges
,
D. H.
,
2000
, “
On Asymptotically Correct Timoshenko-Like Anisotropic Beam Theory
,”
Int. Jo. Solids Struct.
,
37
(
3
), pp.
535
558
.10.1016/S0020-7683(99)00020-7
28.
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
.10.1016/0045-7825(86)90079-4
29.
Betsch
,
P.
, and
Steinmann
,
P.
,
2002
, “
A DAE Approach to Flexible Multibody Dynamics
,
’ Multibody Syst. Dyn.
,
8
, pp.
367
391
.10.1023/A:1020934000786
30.
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
.10.1016/j.tws.2005.02.003
You do not currently have access to this content.