The free vibration analysis of thin- and thick-walled layered structures via a refined one-dimensional (1D) approach is addressed in this paper. Carrera unified formulation (CUF) is employed to introduce higher-order 1D models with a variable order of expansion for the displacement unknowns over the cross section. Classical Euler–Bernoulli (EBBM) and Timoshenko (TBM) beam theories are obtained as particular cases. Different kinds of vibrational modes with increasing half-wave numbers are investigated for short and relatively short cylindrical shells with different cross section geometries and laminations. Numerical results of natural frequencies and modal shapes are provided by using the finite element method (FEM), which permits various boundary conditions to be handled with ease. The analyses highlight that the refinement of the displacement field by means of higher-order terms is fundamental especially to capture vibrational modes that require warping and in-plane deformation to be detected. Classical beam models are not able to predict the realistic dynamic behavior of shells. Comparisons with three-dimensional elasticity solutions and solid finite element solutions prove that CUF provides accuracy in the free vibration analysis of even short, nonhomogeneous thin- and thick-walled shell structures, despite its 1D approach. The results clearly show that bending, radial, axial, and also shell lobe-type modes can be accurately evaluated by variable kinematic 1D CUF models with a remarkably lower computational effort compared to solid FE models.

References

References
1.
Anderson
,
J. D.
,
2010
,
Fundamentals of Aerodynamics
,
5th ed.
,
McGraw-Hill
, New York.
2.
Librescu
,
L.
, and
Na
,
S.
,
1998
, “
Dynamic Response of Cantilevered Thin-Walled Beams to Blast and Sonic-Boom Loadings
,”
Shock Vib.
,
5
(
1
), pp.
23
33
.10.1155/1998/526216
3.
Fung
,
Y. C.
,
1993
,
Biomechanics: Mechanical Properties of Living Tissues
,
2nd ed.
,
Springer
,
New York
.
4.
Au
,
F. T. K.
,
Cheng
,
Y. S.
, and
Cheung
,
Y. K.
,
2001
, “
Vibration Analysis of Bridges Under Moving Vehicles and Trains: An Overview
,”
Prog. Struct. Eng. Mater.
,
3
(
3
), pp.
299
304
.10.1002/pse.89
5.
Qiu
,
X.
,
Deshpande
,
V. S.
, and
Fleck
,
N. A.
,
2003
, “
Finite Element Analysis of the Dynamic Response of Clamped Sandwich Beams Subject to Shock Loading
,”
Eur. J. Mech. A/Solids
,
22
(
6
), pp.
801
814
.10.1016/j.euromechsol.2003.09.002
6.
Lindeburg
,
M. R.
, and
McMullin
,
K. M.
,
2008
,
Seismic Design of Building Structures
,
9th ed.
,
Professional Publications
, Belmont, CA.
7.
Marur
,
S. R.
, and
Kant
,
T.
,
1997
, “
On the Performance of Higher Order Theories for Transient Dynamic Analysis of Sandwich and Composite Beams
,”
Comput. Struct.
,
65
(
5
), pp.
741
759
.10.1016/S0045-7949(96)00427-0
8.
Flügge
,
W.
,
1934
,
Statik und Dynamik der Schalen
,
Springer-Verlag
,
Berlin
.
9.
Lur'e
,
A. I.
,
1940
, “
The General Theory of Thin Elastic Shells
,”
Prikl. Mat. Mekh.
,
4
(
2
), pp.
7
34
.
10.
Byrne
,
R.
,
1944
, “
Theory of Small Deformations of a Thin Elastic Shell
,”
Univ. Calif., Publ. Math.
,
2
(
1
), pp.
103
152
.
11.
Love
,
A. E. H.
,
2011
,
A Treatise on the Mathematical Theory of Elasticity
,
4th ed.
,
Dover
, New York.
12.
Sanders
,
J. L.
,
1963
, “
Nonlinear Theories for Thin Shells
,”
Q. Appl. Math.
,
21
(
1
), pp.
21
36
.
13.
Qatu
,
M. S.
,
2002
, “
Recent Research Advances in the Dynamic Behavior of Shells: 1989–2000, Part 1: Laminated Composite Shells
,”
ASME Appl. Mech. Rev.
,
55
(
4
), pp.
325
350
.10.1115/1.1483079
14.
Qatu
,
M. S.
,
2002
, “
Recent Research Advances in the Dynamic Behavior Of Shells: 1989–2000, Part 2: Homogeneous Shells
,”
ASME Appl. Mech. Rev.
,
55
(
5
), pp.
415
434
.10.1115/1.1483078
15.
Qatu
,
M. S.
,
Sullivan
,
R. W.
, and
Wenchao
,
W.
,
2010
, “
Recent Research Advances in the Dynamic Behavior of Composite Shells: 2000–2009
,”
Compos. Struct.
,
93
(
1
), pp.
14
31
.10.1016/j.compstruct.2010.05.014
16.
Herrmann
,
G.
, and
Mirsky
,
I.
,
1956
, “
Three-Dimensional and Shell-Theory Analysis of Axially Symmetric Motions of Cylinders
,”
ASME J. Appl. Mech.
,
23
(
4
), pp.
563
568
.
17.
Mirsky
,
I.
, and
Herrmann
,
G.
,
1957
, “
Nonaxially Symmetric Motions of Cylindrical Shells
,”
J. Acoust. Soc. Am.
,
29
(
10
), pp.
1116
1123
.10.1121/1.1908716
18.
Gazis
,
D. C.
,
1959
, “
Three-Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders. I. Analytical Foundation
,”
J. Acoust. Soc. Am.
,
31
(
5
), pp.
568
573
.10.1121/1.1907753
19.
Armenàkas
,
A. E.
,
Gazis
,
D. C.
, and
Herrmann
,
G.
,
1969
,
Free Vibrations of Circular Cylindrical Shells
,
Pergamon, Oxford
,
UK.
20.
Mirsky
,
I.
,
1963
, “
Radial Vibrations of Thick-Walled Orthotropic Cylinders
,”
AIAA J.
,
1
(
2
), pp.
487
488
.10.2514/3.1578
21.
Soldatos
,
K. P.
,
1994
, “
Review of Three Dimensional Dynamic Analyses of Circular Cylinders and Cylindrical Shells
,”
ASME Appl. Mech. Rev.
,
47
(
10
), pp.
501
516
.10.1115/1.3111064
22.
Soldatos
,
K. P.
, and
Hadjigeorgiou
,
V. P.
,
1990
, “
Three-Dimensional Solution of the Free Vibration Problem of Homogeneous Isotropic Cylindrical Shells and Panels
,”
J. Sound Vib.
,
137
(
3
), pp.
369
384
.10.1016/0022-460X(90)90805-A
23.
Bhimaraddi
,
A.
,
1984
, “
A Higher Order Theory for Free Vibration Analysis of Circular Cylindrical Shells
,”
Int. J. Solids Struct.
,
20
(
7
), pp.
623
630
.10.1016/0020-7683(84)90019-2
24.
Timarci
,
T.
, and
Soldatos
,
K. P.
,
2000
, “
Vibrations of Angle-Ply Laminated Circular Cylindrical Shells Subjected to Different Sets of Edge Boundary Conditions
,”
J. Eng. Math.
,
37
(
1–3
), pp.
211
230
.10.1023/A:1004794513444
25.
Soldatos
,
K. P.
, and
Timarci
,
T.
,
1993
, “
A Unified Formulation of Laminated Composite, Shear Deformable, Five-Degrees-of-Freedom Cylindrical Shell Theories
,”
Compos. Struct.
,
25
(
1–4
), pp.
165
171
.10.1016/0263-8223(93)90162-J
26.
Mofakhami
,
M. R.
,
Toudeshky
,
H. H.
, and
Hashemi
,
S. H.
,
2006
, “
Finite Cylinder Vibrations With Different End Boundary Conditions
,”
J. Sound Vib.
,
297
(
1–2
), pp.
293
314
.10.1016/j.jsv.2006.03.041
27.
Alijani
,
F.
, and
Amabili
,
M.
,
2014
, “
Nonlinear Vibrations and Multiple Resonances of Fluid Filled Arbitrary Laminated Circular Cylindrical Shells
,”
Compos. Struct.
,
108
(1), pp.
951
962
.10.1016/j.compstruct.2013.10.029
28.
Toorani
,
M. H.
, and
Lakis
,
A. A.
,
2006
, “
Free Vibrations of Non-Uniform Composite Cylindrical Shells
,”
Nucl. Eng. Des.
,
236
(
17
), pp.
1748
1758
.10.1016/j.nucengdes.2006.01.004
29.
Khalili
,
S. M. R.
,
Davar
,
A.
, and
Malekzadeh Fard
,
K.
,
2012
, “
Free Vibration Analysis of Homogeneous Isotropic Circular Cylindrical Shells Based on a New Three-Dimensional Refined Higher-Order Theory
,”
Int. J. Mech. Sci.
,
56
(
1
), pp.
1
25
.10.1016/j.ijmecsci.2011.11.002
30.
Bathe
,
K.
,
1996
,
Finite Element Procedure
,
Prentice Hall
,
Upper Saddle River, NJ.
31.
Euler
,
L.
,
1744
,
De curvis elasticis
,
Bousquet
,
Lausanne/Geneva, Switzerland
.
32.
Timoshenko
,
S.
,
1921
, “
On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars
,”
Philos. Mag.
,
41
(245)
, pp.
744
746
.10.1080/14786442108636264
33.
Carrera
,
E.
, and
Giunta
,
G.
,
2010
, “
Refined Beam Theories Based on a Unified Formulation
,”
ASME Int. J. Appl. Mech.
,
2
(
1
), pp.
117
143
.10.1142/S1758825110000500
34.
Carrera
,
E.
,
Giunta
,
G.
, and
Petrolo
,
M.
,
2011
,
Beam Structures: Classical and Advanced Theories
,
Wiley
, Chichester, UK.
35.
Yu
,
W.
,
Volovoi
,
V.
,
Hodges
,
D.
, and
Hong
,
X.
,
2002
, “
Validation of the Variational Asymptotic Beam Sectional Analysis (VABS)
,”
AIAA J.
,
40
(
10
), pp.
2105
2113
.10.2514/2.1545
36.
Silvestre
,
N.
, and
Camotim
,
D.
,
2002
, “
Second-Order Generalised Beam Theory for Arbitrary Orthotropic Materials
,”
Thin-Walled Struct.
,
40
(
9
), pp.
791
820
.10.1016/S0263-8231(02)00026-5
37.
Kapania
,
K.
, and
Raciti
,
S.
,
1989
, “
Recent Advances in Analysis of Laminated Beams and Plates, Part II: Vibrations and Wave Propagation
,”
AIAA J.
,
27
(
7
), pp.
935
946
.10.2514/3.59909
38.
Carrera
,
E.
, and
Varello
,
A.
,
2012
, “
Dynamic Response of Thin-Walled Structures by Variable Kinematic One-Dimensional Models
,”
J. Sound Vib.
,
331
(
24
), pp.
5268
5282
.10.1016/j.jsv.2012.07.006
39.
Carrera
,
E.
,
Brischetto
,
S.
, and
Nali
,
P.
,
2011
,
Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis
,
Wiley
, Chichester, UK.
40.
Carrera
,
E.
,
Giunta
,
G.
,
Nali
,
P.
, and
Petrolo
,
M.
,
2010
, “
Refined Beam Elements With Arbitrary Cross-Section Geometries
,”
Comput. Struct.
,
88
(
5–6
), pp.
283
293
.10.1016/j.compstruc.2009.11.002
41.
Carrera
,
E.
, and
Petrolo
,
M.
,
2012
, “
Refined One-Dimensional Formulations for Laminated Structure Analysis
,”
AIAA J.
,
50
(
1
), pp.
176
189
.10.2514/1.J051219
42.
Carrera
,
E.
,
Varello
,
A.
, and
Demasi
,
L.
,
2013
, “
A Refined Structural Model for Static Aeroelastic Response and Divergence of Metallic and Composite Wings
,”
CEAS Aeronaut. J.
,
4
(
2
), pp.
175
189
.10.1007/s13272-013-0063-2
43.
Carrera
,
E.
,
Petrolo
,
M.
, and
Varello
,
A.
,
2012
, “
Advanced Beam Formulations for Free Vibration Analysis of Conventional and Joined Wings
,”
J. Aerosp. Eng.
,
25
(
2
), pp.
282
293
.10.1061/(ASCE)AS.1943-5525.0000130
44.
Jones
,
R.
,
1999
,
Mechanics of Composite Materials
,
2nd ed.
,
Taylor & Francis
,
Philadelphia, PA
.
45.
Almroth
,
B. O.
,
1966
, “
Influence of Edge Conditions on the Stability of Axially Compressed Cylindrical Shells
,”
AIAA J.
,
4
(
1
), pp.
134
140
.10.2514/3.3396
46.
Carrera
,
E.
,
Zappino
,
E.
, and
Petrolo
,
M.
,
2012
, “
Analysis of Thin-Walled Structures With Longitudinal and Transversal Stiffeners
,”
ASME J. Appl. Mech.
,
80
(
1
), p.
011006
.10.1115/1.4006939
47.
Carrera
,
E.
, and
Brischetto
,
S.
,
2008
, “
Analysis of Thickness Locking in Classical, Refined and Mixed Multilayered Plate Theories
,”
Compos. Struct.
,
82
(
4
), pp.
549
562
.10.1016/j.compstruct.2007.02.002
48.
Carrera
,
E.
,
Petrolo
,
M.
, and
Nali
,
P.
,
2011
, “
Unified Formulation Applied to Free Vibrations Finite Element Analysis of Beams With Arbitrary Section
,”
Shock Vib.
,
18
(3)
, pp.
485
502
.10.1155/2011/706541
You do not currently have access to this content.