This work deals with refined theories for beams with an increasing number of displacement variables. Reference has been made to the asymptotic and axiomatic methods. A Taylor-type expansion up to the fourth-order has been assumed over the section coordinates. The finite element governing equations have been derived in the framework of the Carrera unified formulation (CUF). The effectiveness of each expansion term, that is, of each displacement variable, has been established numerically considering various problems (traction, bending, and torsion), several beam sections (square, annular, and airfoil-type), and different beam slenderness ratios. The accuracy of these theories have been evaluated for displacement and stress components at different points over the section and along the beam axis. Error-type parameters have been introduced to establish the role played by each generalized displacement variable. It has been found that the number of terms that have to be retained for each of the considered beam theories is closely related to the addressed problem; different variables are requested to obtain accurate results for different problems. It has, therefore, been concluded that the full implementation of CUF, retaining all the available terms, would avoid the need of changing the theory when a problem is changed (geometries and/or loading conditions), as what happens in most engineering problems. On the other hand, CUF could be used to construct suitable beam theories in view of the fulfillment of prescribed accuracies.

1.
Yu
,
W.
,
Volovoi
,
V. V.
,
Hodges
,
D. H.
, and
Hong
,
X.
, 2002, “
Validation of the Variational Asymptotic Beam Sectional Analysis (VABS)
,”
AIAA J.
0001-1452,
40
, pp.
2105
2112
.
2.
Giunta
,
G.
,
Carrera
,
E.
, and
Belouettar
,
S.
, 2009, “
A Refined Beam Theory With Only Displacement Variables and Deformable Cross-Section
,”
Fiftieth AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
, Palm Springs, CA, May 4–7.
3.
Koiter
,
W. T.
, 1961, “
A Consistent First Approximation in the General Theory of Thin Elastic Shells
,”
Proc. Kon. Ned. Ak. Wet
,
B64
, pp.
612
619
. 0002-7820
4.
Koiter
,
W. T.
, 1960, “
A Systematic Simplification of the General Equations in the Linear Theory of Thin Shells
,”
Proceedings of the IUTAM Symposium ib Theory of Thin Elastic Shells
, Delft, pp.
12
33
.
5.
Koiter
,
W. T.
, 1967, “
Foundations and Basic Equations of Shell Theory: A Survey of Recent Progress
,”
Proceedings of the IUTAM Symposium on the Theory of Thin Elastic Shells
, Copenhagen, pp.
93
105
.
6.
Kraus
,
H.
, 1967,
Thin Elastic Shells
,
Wiley
,
New York
.
7.
Cicala
,
P.
, 1965,
Systematic Approximation Approach to Linear Shell Theory
,
Levrotto e Bella
,
Torino, Italy
.
8.
Gol’denweizer
,
A. L.
, 1962, “
Derivation of an Approximate Theory of Bending of a Plate by the Method of Asymptotic Integration of the Equations of the Theory of Elasticity
,”
Prikl. Mat. Mekh.
0032-8235,
26
, pp.
1000
1025
.
9.
Gol’denweizer
,
A. L.
, 1963, “
Derivation of an Approximate Theory of Shells by Means of Asymptotic Integration of the Equations of the Theory of Elasticity
,”
Prikl. Mat. Mekh.
0032-8235,
27
, pp.
903
924
.
10.
Gol’denweizer
,
A. L.
, 1966, “
The Principles of Reducing Three Dimensional Problems of Elasticity to Two Dimensional Problems of the Theory of Plates and Shells
,”
Proceedings of the 11th International Congress on Applied Mechanics
, Munich, pp.
306
311
.
11.
Carrera
,
E.
, 2002, “
Theories and Finite Elements for Multilayered, Anisotropic, Composite Plates and Shells
,”
Arch. Comput. Methods Eng.
1134-3060,
9
(
2
), pp.
87
140
.
12.
Kapania
,
R.
, and
Raciti
,
S.
, 1989, “
Recent Advances in Analysis of Laminated Beams and Plates, Part I: Shear Effects and Buckling
,”
AIAA J.
0001-1452,
27
(
7
), pp.
923
935
.
13.
Kapania
,
R.
, and
Raciti
,
S.
, 1989, “
Recent Advances in Analysis of Laminated Beams and Plates, Part II: Vibrations and Wave Propagation
,”
AIAA J.
0001-1452,
27
(
7
), pp.
935
946
.
14.
Savoia
,
M.
, and
Tullini
,
N.
, 1996, “
Beam Theory for Strongly Orthotropic Materials
,”
Int. J. Solids Struct.
0020-7683,
33
(
17
), pp.
2459
2484
.
15.
Reddy
,
J. N.
, 1997, “
On Locking-Free Shear Deformable Beam Finite Elements
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
149
, pp.
113
132
.
16.
Kim
,
C.
, and
White
,
S. R.
, 1997, “
Thick-Walled Composite Beam Theory Including 3-D Elastic Effects and Torsional Warping
,”
Int. J. Solids Struct.
0020-7683,
34
(
31–32
), pp.
4237
4259
.
17.
Taufik
,
A.
,
Barrau
,
J. J.
, and
Lorin
,
F.
, 1999, “
Composite Beam Analysis With Arbitrary Cross Section
,”
Compos. Struct.
0263-8223,
44
, pp.
189
194
.
18.
Eisenberger
,
M.
, 2003, “
An Exact High Order Beam Element
,”
Comput. Struct.
0045-7949,
81
, pp.
147
152
.
19.
Goyal
,
V. K.
, and
Kapania
,
R. K.
, 2007, “
A Shear-Deformable Beam Element for the Analysis of Laminated Composites
,”
Finite Elem. Anal. Design
0168-874X,
43
, pp.
463
477
.
20.
Volovoi
,
V. V.
,
Hodges
,
D. H.
,
Berdichevsky
,
V. L.
, and
Sutyrin
,
V. G.
, 1999, “
Asymptotic Theory for Static Behavior of Elastic Anisotropic I—Beams
,”
Int. J. Solids Struct.
0020-7683,
36
, pp.
1017
1043
.
21.
Volovoi
,
V. V.
, and
Hodges
,
D. H.
, 2000, “
Theory of Anisotropic Thin-Walled Beams
,”
ASME J. Appl. Mech.
0021-8936,
67
(
3
), pp.
453
459
.
22.
Popescu
,
B.
, and
Hodges
,
D. H.
, 2000, “
On Asymptotically Correct Timoshenko-Like Anisotropic Beam Theory
,”
Int. J. Solids Struct.
0020-7683,
37
, pp.
535
558
.
23.
Yu
,
W.
,
Hodges
,
D. H.
,
Volovoi
,
V. V.
, and
Cesnik
,
C. E. S.
, 2002, “
On Timoshenko-Like Modeling of Initially Curved and Twisted Composite Beams
,”
Int. J. Solids Struct.
0020-7683,
39
, pp.
5101
5121
.
24.
Yu
,
W.
, and
Hodges
,
D. H.
, 2004, “
Elasticity Solutions Versus Asymptotic Sectional Analysis of Homogeneous, Isotropic, Prismatic Beams
,”
ASME J. Appl. Mech.
0021-8936,
71
, pp.
15
23
.
25.
Yu
,
W.
, and
Hodges
,
D. H.
, 2005, “
Generalized Timoshenko Theory of the Variational Asymptotic Beam Sectional Analysis
,”
Journal of the American Helicopter Society
,
50
(
1
), pp.
46
55
.
26.
Carrera
,
E.
, 2003, “
Theories and Finite Elements for Multilayered Plates and Shells: A Unified Compact Formulation With Numerical Assessment and Benchmarking
,”
Arch. Comput. Methods Eng.
1134-3060,
10
(
3
), pp.
215
296
.
27.
Carrera
,
E.
,
Giunta
,
G.
,
Nali
,
P.
, and
Petrolo
,
M.
, 2010, “
Refined Beam Elements With Arbitrary Cross-Section Geometries
,”
Comput. Struct.
0045-7949,
88
, pp.
283
293
.
28.
Tsai
,
S. W.
, 1988,
Composites Design
,
4th ed.
,
Think Composites
,
Dayton, OH
.
29.
Reddy
,
J. N.
, 2004,
Mechanics of Laminated Composite Plates and Shells. Theory and Analysis
,
2nd ed.
,
CRC
,
Boca, Raton, FL
.
30.
Carrera
,
E.
, and
Brischetto
,
S.
, 2008, “
Analysis of Thickness Locking in Classical, Refined and Mixed Multilayered Plate Theories
,”
Compos. Struct.
0263-8223,
82
(
4
), pp.
549
562
.
31.
Carrera
,
E.
, and
Brischetto
,
S.
, 2008, “
Analysis of Thickness Locking in Classical, Refined and Mixed Theories for Layered Shells
,”
Compos. Struct.
0263-8223,
85
(
1
), pp.
83
90
.
32.
Carrera
,
E.
, and
Giunta
,
G.
, 2010, “
Refined Beam Theories Based on Carrera’s Unified Formulation
,”
International Journal of Applied Mechanics
,
2
(
1
),
117
143
.
33.
Bathe
,
K. J.
, 1996,
Finite Element Procedure
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
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