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Mechanics of Advanced Materials and Structures
ISSN: 1537-6494 (Print) 1537-6532 (Online) Journal homepage: http://www.tandfonline.com/loi/umcm20
Free Vibration analysis of Reinforced Thin-WalledPlates and Shells through Various Finite ElementModels
E. Zappino, T. Cavallo & E. Carrera
To cite this article: E. Zappino, T. Cavallo & E. Carrera (2015): Free Vibration analysis ofReinforced Thin-Walled Plates and Shells through Various Finite Element Models, Mechanics ofAdvanced Materials and Structures, DOI: 10.1080/15376494.2015.1121562
To link to this article: http://dx.doi.org/10.1080/15376494.2015.1121562
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Free Vibration analysis of Reinforced Thin-WalledPlates and Shells through Various Finite Element
Models
E. Zappinoa∗, T. Cavalloa†, E. Carreraa,b‡,aMechanical and Aerospace Engineering Department, Politecnico di Torino, Italy.
bSchool of Aerospace, Mechanical and Manufacturing Engineering,RMIT University Melbourne, Australia.
Submitted toAuthor for correspondence:
E. Zappino, Research assistant,
Mechanical and Aerospace Engineering Department,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
10129 Torino, Italy,
tel: +39 011 090 6869,
fax: +39 011 090 6899,
e-mail: [email protected]
∗Research assistant, e-mail: [email protected]†PhD student, e-mail: [email protected]‡Professor of Aerospace Structures and Aeroelasticity, e-mail: [email protected]
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Abstract
This paper deals with free vibration analysis of thin-walled structures reinforced by longitudinal
stiffeners using refined one-dimensional 1D models. The 1D theory, which is used in the present
paper, has hierarchical features and it is based on the Carrera Unified Formulation (CUF). The
displacement field over the cross-section is obtained by means of Taylor (TE) or Lagrange (LE)
Expansions. Finite Element (FE) method is applied along the beam axis to obtain weak form
solutions of the related governing equations. The obtained results are compared with those from
classical finite element formulations based on plate and shell (2D), beam (1D) and solid (3D)
elements that are available in commercial software. When solid formulation is used to build the FE
solutions, stringers and skin are modelled with only 3D elements while, in the 2D-1D FE models,
shell and beam elements are used for skin and stringers, respectively. Three benchmark problems
are analyzed: a flat plate, a curved panel and a thin-walled cylinder. When TE models are used,
different orders of expansion, N, are considered, where N is a free parameter of the formulation. As
far as Lagrange expansions are concerned, four- (LE 4) and nine-node (LE 9) elements are used to
build different meshes on the cross-section. The results show that the present 1D models are able
to analyze the dynamic behaviour of complex structures and can detect 3D-effects as well as very
complex shell-like modes typical of thin-walled structures. Moreover, the 1D-CUF elements yield
accurate results with a low number of degrees of freedom.
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1 Introduction
Reinforced structures are used in various engineering fields. Spacecraft and aircraft consist of
an assembly ofreinforced shell. Figure1 shows part of the fuselage of an aircraft where both
longitudinal (stringers) and transversal (ribs) stiffeners are used. The use of stiffeners improves the
strength/weight ratio and makes the structure cost-effective.
However, the design and the production of this kind of structures are extremely complex pro-
cesses to undertake. Indeed, accurate optimization analyses are usually carried out in order to
define important structural parameters such as the geometry and the number of stiffeners.
Buckling, vibration characteristic and failure mechanisms are the driven features for a success-
ful design of reinforced shell structures. A large number of contributions have been made in the
last century on these topics. For example the books of Bruhn [1] and Bushless [2]. Some of these
contributions that are of interest for the present paper are here reviewed.
Analytical solutions were developed by Argiris and Kelsey [3] where the value of the shear
stress is considered constant within the panels while the stringers can withstand the normal stress.
Civil engineering analysts have largely contributed to the study of stiffened structures. Vecchio
and Collins[4] studied the response of rectangular reinforced concrete elements subject to in-plane
shear and axial stresses. Moreover by Schade [5], conceived an orthotropic model in which the
stiffened plate is considered like a single plate with a constant thickness; an attached layer was used
to simulate stiffeners. In other works, the stiffened plate was studied considering the plate and the
stiffeners behaviour separately; they are then combined by imposing conditions on displacement
between the plate and the stiffeners. In those cases in which closed form solutions are not possible,
numerical methods are used.
For instance, Rayleigh-Ritz method has been often used for the free-vibration analysis. Ap-
plications of this method can be found in the works of Liewat al.[6], where an orthogonal plate
function was applied on rectangular plates with different boundary conditions. Liewat al.[7] devel-
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oped a Mindlin-Engesser model to analyze the vibration of moderately thick plates with arbitrarily
oriented stiffeners. Xiang [8] used the Mindlin theory to investigate the free vibration of circular
and annular plates with concentric ring stiffeners. Jafari and Bagheri [9] studied the free vibration
analysis of cylindrical shells with circumferential stiffeners using the Ritz method in which the
stiffeners were treated as discrete elements. Other methods based on the smearing technique were
used by Luanat al.[10] to study the vibrations of cross-stiffened thin rectangular plates. Hemmat-
nezhadat al.[11] obtained an equivalent stiffness of the whole panel adding the stringer stiffness
to plate stiffness. In the same way, Edalatat al.[12], studied dynamic response and free vibration
analysis of a stiffened curved panel using an equivalent orthotropic shell parameter.
Nowadays, the most used method for the analysis of stiffened structures is the Finite Element
Method (FEM). Different types of structural elements can be used to simulate the structural be-
haviour of some structural components. Beam, shell and solid elements can usually be used
in most of the available commercial structures. The choice of a proper mathematical modeling
represents a crucial issue. Deb and Booton [13] analysed a stiffened plate subjected to transver-
sal load using Mindlin’s shear distortion theory. Mustafa and Ali [14] developed an eight-node
orthogonally-stiffened super finite element to study the free vibration of a stiffened cylindrical
shell. Edward and Samer [15] used a refined higher-order displacement model based on FEM, to
study the behaviour of concentrically and eccentrically stiffened laminated plates. Instead of using
the shear correction factor, the model incorporates non-linear variations of longitudinal displace-
ments through the thickness. As an alternative to FEMs, the element-free method, or mesh-free
method can be adopted. The mesh-free Galerkin method discretizes the domain of a problem with
a set of scattered points, without the need for meshes. Tamijani and Kapania[16] used the element-
free Galerkin method for vibration analysis of unitized structures, using the first-order shear de-
formation theory for the plate while Timoshenko beam theory was applied for the stringers. With
the same method, Pengat al.[17] carry out a static analysis of concentrically and eccentrically
stiffened plates using the first-order shear deformable theory.
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In classical commercial codes, two approaches can be used to simulate a reinforced structure.
The first approach uses 2D and 1D elements together, in order to represent the physical model
properly the use of offsets is often required, and this makes the pre-processing phase more com-
plex. The second approach requires the use of solid (3D) elements but the computational cost
increases dramatically. When reinforced structures are built up via the Computer Numerical Con-
trol (CNC) process (Figure2a), the continuity between skin and reinforcements is verified. 2D-1D
FE models could be not accurate in the analysis of such components, in fact, these models in-
troduce an “artificial” separation between stringer and skin and do not ensure continuity in the
displacement field. Otherwise, in welded or riveted component (Figure2b), stringers and skin are
different structural elements which are welded or riveted to obtain the whole panel. In this paper
a refined one-dimensional model is used to overcome the limitations of the classical FE models in
the analysis of reinforced structures.
Refined one-dimensional (1D) beam models are very attractive when a slender body is analyzed.
They can reduce the number of degrees of freedoms (DOFs) providing the same accuracy of the
classical models. Classical 1D models, such as Euler-Bernoulli and Timoshenko, have been proved
to be a reliable framework, but these are difficult and not very reliable to be used in the analysis
of thin-walled structures with longitudinal stiffeners. The classical models are, in fact, not able
to predict cross-section distortion and the effects of the local stiffeners. Refined 1D hierarchical
variable kinematic theories are used in this paper to analyse walled structures reinforced with
longitudinal stiffeners. These are based on the Carrera Unified Formulation (CUF), developed
initially for plate/shell models [18], [19], [20], and later used for 1D models [21] with different
kinematic assumptions. Two different expansions over the beam cross-section are used in this
work: Taylor Expansion (TE) or Lagrange Expansion (LE). Carrera and Giunta [21] and Carrera
et al. [22] introduced TE, and applied it to various problems on both static and dynamic analysis,
as [23] and [24]. When a TE approximation is used over the cross-section, a global expansion
about the centroid is considered, while, using the LE approach a local expansion around the nodes
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introduced over the cross-section is considered. Carreraet al. [25, 26] used LE to perform analyses
of complex structures.
The present TE and LE 1D models were obtained with no need for ad hoc formulations. When
a TE model is used, the expansion over the cross-section is characterized by the order of the Taylor
expansion N. It is a free-parameter of the analysis, therefore, the solution can be improved easily
by increasing the order N. Euler-Bernoulli and Timoshenko models can be obtained as a particular
case of TE models. When a LE approach is used, the cross-section is divided into a number of
2D elements in which the Lagrange polynomials are defined. The accuracy of the results depends
on both the type of elements (4 or 9 nodes) and the number of elements used to represent the
cross-section. In both models, the FEM is used to obtain the solution over the beam axis.
This work uses TE and LE models for the analysis of flat and curved structures, and it compares
the results to FE models usually used in commercial codes. Plate and cylindrical thin-walled
reinforced structures are considered. The stringers are placed along the beam axis. The same
model are used to describe both panels and stiffeners without the need to use two different models.
Results are proposed in terms of natural frequencies and modal shapes. Particular attention is
given to the comparison of different FE models obtained from the commercial FE code MSC
NASTRANr using solid FE (3D) models and combined shell-beam FE (2D-1D) models. Section
2 introduces the TE and LE theoretical models. Section 3 shows the results obtained using different
approaches for all considered structures. Finally, in section 4 the main conclusions are reported
and outlook of the present work.
2 Refined one-dimensional models
The structural models used in the present work are derived using the CUF. The one-dimensional
formulation is considered. More details about FE models derived via CUF can be found in the
book by Carreraet al. [27].
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2.1 Adopted TE and LE displacement fields
Figure3 shows the adopted coordinate frame,y is the beam-axis, and the beam length isL. The
displacement vector can be written as:
u(x, y, z) ={
ux uy uz
}T
(1)
The superscriptT denotes transposition. Similarly for stress,σ, and strain,ε, components can be
introduced as follows:
σ =
{σxx σyy σzz σxy σxz σyz
}T
(2)
ε ={εxx εyy εzz εxy εxz εyz
}T
(3)
Linear strain-displacement relations are used,
ε = Du =([Dy] + [DΩ]
)u (4)
D is a differential operator and its explicit form can be found in [27]. Dy andDΩ represent the
differential operator on beam axisy and on the beam cross-section, respectively. The material is
isotropic, the Hooke’s law can be written as:
σ = Cε (5)
In the framework of the CUF [28, 27], for a 1D problem, the displacement fieldu is written as the
product of two functions, a cross-sectional expanding functionFτ and the generalized displacement
vector unknownuτ on the y-axis:
u(x, y, z) = Fτ(x, z)uτ(y) (6)
where indexτ ranges from 1 to the number of terms of the expansion of orderN.
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2.1.1 Taylor expansion models (TE)
Using TE 1D models, Eq.6 consists of an expansion that uses 2D polynomialsxm zn, asFτ, where
m andn are positive integers. For instance, second-order displacement field reads:
ux = ux1 + x ux2 + z ux3 + x2 ux4 + xz ux5 + z2 ux6
uy = uy1 + x uy2 + z uy3 + x2 uy4 + xz uy5 + z2 uy6
uz = uz1 + x uz2 + z uz3 + x2 uz4 + xz uz5 + z2 uz6
(7)
Timoshenko theory (TBT) can be derived as a particular case by considering only the terms:
ux = ux1
uy = uy1 + x uy2 + z uy3 (8)
uz = uz1
It can be shown that Euler-Bernoulli beam theory (EBBT) can be derived by introducing a penal-
ization on the shear terms of the TBT model (see [29]).
2.1.2 Lagrange expansion models (LE)
In the case of LE models, Lagrange polynomials are used to build 1D higher-order theories. Two
types of cross-section polynomial set are adopted in this paper: nine-point LE 9, and four-point LE
4. The isoparametric formulation is exploited to deal with arbitrary cross-section shaped geome-
tries. For instance, LE 4 interpolation functions are:
F1 =14 (1− ξ) (1− η) F2 =
14 (1+ ξ) (1− z)
F3 =14 (1+ ξ) (1+ η) F4 =
14 (1− ξ) (1+ η)
(9)
Whereξ andη are the coordinates in the natural reference system. Equation9 coincides with the
linear Lagrange polynomial in two dimensions. Using LE the unknowns are only the displacements
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of the cross-sectional nodes. Figure4 shows how a stringer can be introduced in the skin using
the appropriate mesh over the cross-section. The nodes over the stringer section (EL 4) must be
connected with the nodes on the skin; therefore the element 2 (EL 2) is used to connect the two
structural components.
2.1.3 Finite Elements (FE) solution
The FEM is used to approximate the displacement along the beam axis. In particular, the general-
ized displacement vector,uτ(y), is linearly interpolated using the classical shape functions. When
a beam element withNNE nodes along the axis is considered, the generalized displacement vector
becomes:
uτ(y) = Ni(y)qiτ; i = 1 . . .NNE (10)
In Eq. 10 the indexi ranges from 1 to the number of nodes per elementNNE. Therefore, the
displacement field can be written as follows:
u(x, y, z) = Fτ(x, z)Ni(y)qiτ (11)
Where indexi indicates the node of the element along they-axis. Three- and four-node refined
beam elements were used along they-axis in the present work.
2.2 Governing equations
The governing equations can be derived using the Principle of Virtual Displacements (PVD) that
in the dynamic case assumes the following form:
δLint = δLext− δLine (12)
WhereLint stands for the strain energy,Lext is the work of the external loadings, andLine is the
work of the inertial loadings.δ denotes the virtual variation. In the case of free vibration analysis,
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the work of the external loadings is equal to zero, Eq.12becomes:
δLint + δLine = 0 (13)
The internal work can be written as:
δLint =
∫
VδεTσdV (14)
By introducing the constitutive equations and the geometrical relations given respectively in Eq.3
and Eq.2 and introducing the displacement field given in Eq.11, the variation of the internal work
becomes:
δLint = δqTs j
∫
V
[Ni(y)Fτ(x, z) DT C D Fs(x, z)Nj(y)
]dVqτi (15)
The differential matrixD operates onFτ, Fs, Ni andNj. By splitting the integral on the volumeV
in two sub-integrals, one on the cross-sectionΩ and one along the beam lengthl, Eq. 15becomes:
(16)
WhereKi jτs is the stiffness matrix expressed in form of “fundamental nucleus” that is a 3×3 array.
qτi is the vector of the nodal unknown andδqs j is its virtual variation.I is the identity matrix. The
explicit form of the fundamental nucleus can be found in [27].
The virtual variation of the work of the inertial loadings is:
δLine =
∫
VδuT ρ u dV (17)
where u stands the acceleration vector andρ is the density of the material. The Eq.17 can be
rewritten using Eq.4:
(18)
Where the mass matrixM i jτs is a 3× 3 fundamental nucleus, where the indices have the same
meaning of those of stiffness matrix.
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In conclusion, the PVD can be written as following:
δqTs j
(Ki jτs qτi + M i jτs qτi
)= 0 (19)
The global stiffness and mass matrices are obtained by means of the classical FE assembling
procedure, as shown in [27]. The equation for a free undamped problem can be written as:
Mq+ Kq = 0 (20)
Whereq is the global unknowns vector. Due to linearity of the problem, harmonic solutions are
possible and the natural frequencies,ωk, are obtained by solving the following eigenvalues prob-
lem:
(−ωk
2M + K)qk = 0 (21)
Whereqk is thek-theigenvector, andk ranges form 1 to total numbers of DOF of the structures.
3 Numerical Analysis and Discussion
This section aims to show the capabilities of different models in the dynamic analysis of stiffened
thin-walled structures. Three different structures have been considered.
• A flat stiffened plate with three longitudinal stringers.
• A curved-panel with five longitudinal stringers.
• A stiffened cylinder with eight longitudinal stringers.
When TE functions are used, different orders of the expansion (N) can be considered. On the
other hand, different meshes over the cross-section are adopted for LE expansions. The results
are compared with those from the commercial FE code MSC NASTRANr. For comparisons
purposes, a FE solid (3D) model was used for the stiffened plate, whereas a FE model made of
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both shell (2D) and beam (1D) elements was considered for the stiffened cylinder. Both 3D and
2D-1D FE models were adopted to compare the results of the third analysis case.
The considered material is aluminium with Young modulusE = 75 GPa, Poisson ratioν = 0.3,
and densityρ = 2700kg/m3.
3.1 Flat Stiffened Plate
The geometry of the structure is shown in Figure5. It is a square plate with sidel = 2 m and
thicknesst=20 mm. The stringers have a rectangular cross-section with dimensions 20× 50 mm.
The edges that are perpendicular to the stringers are clamped whereas the others are free. Eight
four-node beam elements have been used along the y-axis. Two different meshes were used for the
LE model, a simple mesh and a refined mesh (denoted with+).
Table1 shows the first 10 natural frequencies evaluated using different models.EBBT is not
able to detect shell-like (S) modes. When the order of expansion of the displacement field in-
creases, the frequencies converge to the reference values. The use of 15 terms (TE 4) leads to a
good agreement with the solid model. Figure6 shows the first 10 modes related toN = 7. Refined
CUF 1D theories detect complex modes; such as mode 8 (see Figure6h) that is a shell-like mode
with two half-waves along both directions. Bending (mode 1) and torsion (mode 2) modes (see
Figure6a and b, respectively) are also detected.
When LE models are considered, the results show that the element with 4 nodes shows a slow
convergence due to the bilinear displacement field, see [30]. Fourteen elements are used on the
cross-section for the coarseLE4 mesh, whereas only 1 element is used along the thickness of the
plate. The first two frequencies show differences of about 7% with respect to the solid FE model,
while the differences increase, as Table2 shows, for higher number frequencies. A refined mesh
(LE4+) is introduced considering 128 elements on the cross-section. In this case, the use of two
elements through the thickness of the plate, improve the accuracy of the model. TheLE9+ model
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has 36 elements with 9 nodes on the cross-section; Figure7 shows the first 10 modes evaluated
with this model. When a coarse mesh (LE9) is considered, only 12 elements are used on the cross-
section. The modes in Figure7 highlight that very complex modes are detected. Mode 9 (Figure
7i) involves a large cross-sectional deformation, it is characterized by 1 half-wave along they-axis
and 3 half-waves on the cross-section, also mode 8 (Figure7h) shows a complex shape that is
usually detected only using 2D or 3D models.
Table2 shows the percentage difference between the TE and LE with respect to 3D FE in terms
of DOFs. When TE is used, the increase of the order,N, has two consequences: 1) the theory is
able to identify new modes on the cross-section, i.e. the third frequency can only be found when
N = 7 is used, and 2) the agreement with the reference solution is improved, even though for
complex shell-like modes the error remain until 20%. In some cases, the result is very close to
solid results, as in the fourth frequency withN = 5 andN = 6, where the error is only 1 %. On the
other hand, the number of DOFs withN = 7 is only the 4.2% of the DOFs required for the solid
FE solution. When theLE4 model is used, the first two frequencies show an error of only 7% if
compared to the solid solution, but for higher frequencies, the results are not accurately detected.
Better results are obtained using theLE4+ model, indeed the differences of the first frequencies
are lower than 1.5%. In theLE9 case, all frequency differences are lower than 5% compared to the
solid model, and they are only 0.8 % in the first 2 frequencies. It should be noted that the number
of DOFs is only 9% compared to the solid DOFs. WhenLE9+ is used, the percentage error of
the first 5 frequencies is lower than 2%, but the DOFs are only about 23% of the reference model.
Neither LE model is able to detect the tenth frequency, which requires a more refined discretization
over the cross-section. The analysis suggests the following remarks:
1. theT E4 model leads to a good agreement, compared with thesolid FE models, in the com-
putation of bending frequencies.
2. when the number of terms in TE increases, the model is able to detect shell-like modes.
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3. LE9 provides excellent results and with a lower number of DOFs with respect to the solid FE
models.
3.2 Reinforced Curved Panel
The geometry of the curved panel is shown in Figure8. Two different length values (b) were used
along they-axes: 1.5 m and 3.0 m. The cross-section is shown in Figure9, where the radius is
R = 1 m, the thickness ist = 0.002 m, and the stringers are located with an angular distance,α
equal to 45◦. Five longitudinal stringers were considered withP = 0.02 m andHs = 0.054 m.
The structure is clamped at both ends perpendicular to the stiffeners. Three-node refined beam
elements have been used along they-axes for both the TE and LE cases. Two different FE models
were provided withNAS TRANr commercial code. The first model was made with solid elements
and the second one consisted of a coupling ofshell/beamelements, where the shells were used
to simulate the skin, while the beam elements, with a correct off-set, were used for the stringers.
Three different meshes were considered on the cross-section using theLE9 model with 18, 22 and
34 elements.
Tables3 and4 show the first 15 frequencies evaluated by means of different structural models
for both the geometries considered. Figure10 shows the first 10 modal shapes withT E6 and
b = 3.0 m. When a TE model is used, andN is increases, a convergent solution is obtained. The
convergence is faster in the case ofb = 3, and a good agreement with the shell/beam solution
is obtained. Table3 shows that the bending and torsional modes do not appear in the first 30
frequencies for eitherN = 6 or N = 7, and only shell-like modes appear, in according with
the MSCNAS TRANr results. However, only whenN = 7 is used, the modes are all shell-
likes as showed in Table4. Moreover, whenb is increased, the bending modes appear at lower
frequencies, as expected. When LE is used, only shell-like modes are found. Better convergence
can be expected for higher values ofb. The number of LE elements on the cross-section plays an
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important role in the accuracy of the solution. In fact, as shown in Table5, whenLE34 is used, the
first 5 modes are found, whereas only the first 2 frequencies were detected withLE22.
Figure11shows how the first modal shape changes when different meshes are used on the cross-
section. In fact, when 18 elements are used (Figure11a) the geometry is far from that of a curved
panel and the first modal shape is not accurate. The first 10 modal shapes withLE34 andb = 3.0
m are depicted in Figure12. Table5 shows that, whenLE34 is used, the results are accurate for
both values ofb but are still better forb = 3, as previously mentioned. Forb = 1.5, see Table5,
the number of LE elements on the cross-section has an important effect on the solution, because
shell-like modes appear in the first frequencies. Whenb = 3 is considered, both the refined and
coarse meshes give accurate results for the first four frequencies because they are related to the
bending phenomena. The reduction in the number of DOFs of the model in the case ofLE22 and
LE34, with respect to the classical approaches, is shown in Table5. The following considerations
can be made:
1. The orderN = 7 assures a good agreement with the commercial code for the computation of
the shell-like frequencies.
2. LE models are more accurate than TE models for the study of thin-walled structures.
3. The geometry of the structure and the cross-section shape are important parameters that
should be considered in the choice of the model.
3.3 Reinforced Cylinder
The geometry of the considered cylindrical structure is shown in Figure13, where the length along
they-axis isb = 15 m. Figure14 shows the cross-section of the structure. The radius,R, of the
circular cross-section, is 1 m, and the thickness value ist = 2 mm. The stringers are placed with
α = 45◦, and they are characterized by a rectangular cross-section, with baseP = 0.02 m, and
height Hs = 0.054 m. The diameter/thickness ratio is equal to 1000. The structure is clamped
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at both ends. Seven four-node refined beam elements are used along they-axes. The results are
compared with those of ashell/beamFE model made using the commercial codeNAS TRANr.
When TE is used, differentN orders of expansion are considered, whereas both four and nine node
elements and two different meshes - i.e. 24 and 32 elements respectively - are considered on the
cross-section for the LE models. The first 30 modes have been investigated, and, in the case of
shell-like modes, the frequencies have been characterized using two parameters,L andM, that are,
the numbers of waves that can be identified in the longitudinal direction and on the circumferential
direction, respectively.
Table6 shows the first 30 frequencies evaluated with different models. When TE models are
considered, if the orderN increases, the results show a slow convergence if compared to the values
found with NAS TRANr. More terms are necessary in the expansions to accurately describe the
displacement field of such structures. Figure15 shows the first 10 modes obtained with the TE
models. Classical modes have been found, see number 1 (first-bending) and number 9 (second-
bending), but also shell-like modes such as mode number 5, withL = 2 andM = 2, and mode
number 8, withL = 3 andM = 2. On the other hand, when theLE9 model is used, the results
are very accurate. The first 10 modal shapes, evaluated usingLE9 and 24 elements on the cross-
section, are shown in Figure16. The values of theshell-like frequencies are reported in Table
7.
4 Concluding Remarks
Various finite elements have been compared in this paper for the free vibration analysis of thin-
walled reinforced structures. Two 1D refined models have been derived by means of CUF, the
former is based on Taylor expansions, the latter on Lagrange expansions. The results from three
representative thin-walled reinforced structures have been presented: a flat plate, a cylindrical
panel and a closed cylinder. The results have been compared with those obtained with solid as well
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as shell/beam finite element models using the commercial code MSCNAS TRANr.
The following features are worth noting:
• 1D CUF models are able to provide all modal shapes including those not attainable with
classical beam theories.
• 1D CUF models allow both thin-walled and stiffened structures to be analyzed with the same
FE formulation.
• 1D CUF approach successfully faces the need for matching different models (2D and 1D).
This drawback can be overcome in FE commercial codes by only using solids.
• In most of the considered cases, 1D CUF models provide the same accuracy as solid models
with fewer DOFs.
• When LE is used, the convergence of the solution depends on the type of element and the
mesh used on the cross-section. WhenLE9 is used with a refined mesh, the solution is very
close to theNAS TRANr solution.
In conclusion, the models proposed in this article appear attractive for the analysis of reinforced
structures. In fact, the results from free vibration analyses demonstrate that the present refined
1D model could represents a valuable tool and might substitute, in principle, classic FE analyses
because its computational efficiency.
Extensions to more complex cases, as well as the dynamic analysis of cylindrical structures
reinforced with transversal and longitudinal stringers such as a complete launcher structure, could
be the subject of future investigations.
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Tables
Table 1: First 10 natural Frequencies of the stiffened plate, aspect-ratio equal to 1.
f s [Hz] EBBT T E4 T E 5 T E 6 T E 7 LE4 LE4+ LE9 LE9+ FEM3D
1 51.48(B) 54.90(B) 54.48(B) 54.17(B) 53.90(B) 55.04(B) 52.28(B) 51.90(B) 51.64(B) 51.48(B)
2 144.01(B) 64.90(T) 64.15(T) 62.53(T) 61.55(T) 57.60(T) 54.98(T) 54.57(T) 54.30(T) 54.12(T)
3 282.10(B) 110.55(S) 104.77(S) 102.99(S) 94.23(S) 147.85(B) 104.85(S) 67.81(S) 67.48(S) 67.17(S)
4 465.92(B) 149.37(B) 146.5.6(B) 145.45(B) 131.01(S) 150.37(T) 138.20(B) 107.60(S) 105.20(S) 103.65(S)
5 695.54(B) 162.46(T) 159.78(T) 154.49(T) 144.15(B) 213.26(S) 140.59(T) 136.27(B) 133.70(B) 132.73(B)
6 971.41(B) 207.96(S) 205.48(S) 164.69(S) 151.98(T) 260.67(S) 169.04(S) 137.86(T) 135.63(T) 134.93(T)
7 1294.00(B) 258.68(S) 227.17(S) 202.09(S) 186.36(S) 279.88(B) 207.07(S) 147.58(S) 145.99(S) 144.39(S)
8 1317.62(A) 288.16(B) 277.10(B) 246.16(S) 216.96(S) 279.89(T) 250.72(S) 168.80(S) 165.38(S) 163.44(S)
9 1675.40(B) 297.23(T) 290.18(T) 273.46(B) 266.57(B) 359.46(S) 251.64(T) 193.48(S) 183.53(S) 234.90(B)
10 1893.61(A) 338.71(S) 324.45(S) 274.83(T) 268.71(T) 437.90(B) 251.97(B) 242.65(T) 237.09(T) 235.38(T)
(B) Bending, (T) Torsion, (S) Shell-Like, (A) Axial.
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Table 2: Comparison of the natural frequencies of the stiffened plate evaluated using differentstructural models.
f s [Hz] DOF Mode1 Mode2 Mode3 Mode4 Mode5 Mode6 Mode7 Mode8FEM3D 63531 51.48(B) 54.1(T) 67.1(S) 103.6(S) 132.7(B) 134.9(T) 144.3(S) 163.4(S)
T E 4 1125 54.9(+6.6%) 64.9(+19.9%) - 110.5(+6.6%) 149.3(+12.5%) 162.4(+20.4%) 207.9(+44.0%) 258.6(+58.2%)
T E 5 1575 54.4(+5.8%) 64.1(+18.5%) - 104.7(+1.0%) 146.5(+10.4%) 159.7(+18.4%) 205.4(+42.3%) 227.1(+38.9%)
T E 6 2100 54.1(+5.2%) 62.5(+15.5%) - 102.9(−0.6%) 145.4(+9.5%) 154.4(+14.5%) 164.6(+14.0%) 202.0(+23.6%)
T E 7 2700 53.9(+4.7%) 61.5(+13.7%) 94.2(+40.29%) 133.6(+28.9%) 144.1(+8.6%) 151.9(+12.6%) 186.3(+29.0%) 216.9(+32.7%)
LE 4 2325 55.0(+6.9%) 57.6(+6.4%) - - 147.8(+11.3%) 150.3(+11.4%) 213.2(+47.7%) 260.6(+59.4%)
LE 9 5625 51.9(+0.8%) 54.5(+0.8%) 67.8(+0.95%) 107.6(+3.8%) 136.2(+2.6%) 137.8(+2.1%) 147.5(+2.2%) 168.8(+3.2%)
LE 4+ 13050 52.2(+1.5%) 54.9(+1.5%) - 104.8(+1.1%) 138.2(+4.1%) 140.5(+4.1%) 169.0(+17.0%) 207.0(+26.6%)
LE 9+ 14400 51.6(+0.3%) 54.3(+0.3%) 67.4(+0.46%) 105.2(+1.5%) 133.7(+0.7%) 135.6(+0.5%) 145.9(+1.1%) 165.3(+1.1%)
(B) Bending, (T) Torsion,(S) Shell-Like.
()(∗%) : * percentage error with respect toFEM3D.
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Table 3: Reinforced cylindrical panel: first 15 natural frequencies consideringl = 1.5 m.
f s [Hz] FEM3D FEM2D−1D T E 5 T E 6 T E 7 LE 18 LE 22 LE 34DOF 58344 8268 567 756 972 2997 3645 5589
1 103.57(S) 97.17(S) 273.91(S) 201.75(S) 144.80(S) 140.79(S) 134.02(S) 112.86(S)
2 103.71(S) 97.20(S) 285.15(S) 216.98(S) 176.98(S) 143.97(S) 140.87(S) 115.68(S)
3 107.41(S) 103.78(S) 370.98(S) 312.46(S) 178.93(S) 157.72(S) 149.31(S) 122.74(S)
4 109.67(S) 104.55(S) 422.72(S) 351.50(S) 225.73(S) 176.69(S) 162.85(S) 123.90(S)
5 110.98(S) 107.00(S) 581.28(B) 397.69(S) 265.18(S) 203.56(S) 177.70(S) 130.24(S)
6 118.48(S) 114.75(S) 629.20(S) 450.63(S) 306.59(S) 203.78(S) 180.94(S) 135.08(S)
7 124.85(S) 120.84(S) 646.50(S) 521.70(S) 313.95(S) 241.91(S) 192.79(S) 138.69(S)
8 128.19(S) 124.22(S) 664.51(S) 532.63(S) 334.96(S) 276.43(S) 196.14(S) 144.87(S)
9 149.35(S) 141.16(S) 711.18(S) 549.11(S) 391.30(S) 285.23(S) 241.06(S) 155.72(S)
10 151.10(S) 143.30(S) 767.06(S) 560.56(S) 454.75(S) 317.62(S) 245.12(S) 165.53(S)
11 154.73(S) 147.84(S) 792.85(S) 590.02(S) 506.35(S) 353.68(S) 252.87(S) 169.94(S)
12 159.07(S) 150.12(S) 862.26(S) 661.63(S) 512.47(S) 365.49(S) 290.80(S) 172.86(S)
13 159.37(S) 155.51(S) 975.20(T) 664.27(S) 529.32(S) 400.64(S) 347.53(S) 190.42(S)
14 161.01(S) 156.10(S) 1098.45(S) 735.07(S) 544.40(S) 445.85(S) 365.00(S) 190.49(S)
15 167.87(S) 158.99(S) 1112.12(S) 767.81(S) 602.78(S) 495.08(S) 371.42(S) 199.81(S)
(B) Bending, (T) Torsion, (S) Shell-Like, (A) Axial.
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Table 4: Reinforced cylindrical panel: first 15 natural frequencies consideringl = 3.0m.
f s [Hz] FEM3D FEM2D−1D T E 5 T E 6 T E 7 LE 18 LE 22 LE 34DOF 106485 16038 567 756 972 2997 3645 5589
1 41.02(S) 39.76(S) 95.98(S) 80.74(S) 72.00(S) 46.31(S) 44.16(S) 44.01(S)
2 41.44(S) 40.17(S) 100.78(S) 88.33(S) 84.67(S) 46.46(S) 44.17(S) 44.65(S)
3 43.75(S) 42.12(S) 196.25(S) 144.96(S) 127.25(S) 65.68(S) 45.84(S) 46.07(S)
4 46.65(S) 44.85(S) 197.38(S) 174.84(S) 159.17(S) 67.78(S) 51.01(S) 50.65(S)
5 50.23(S) 48.57(S) 208.31(S) 184.60(S) 179.63(S) 90.96(S) 60.34(S) 55.57(S)
6 51.38(S) 50.41(S) 246.41(S) 236.92(S) 244.91(S) 93.23(S) 90.85(S) 61.16(S)
7 53.04(S) 51.69(S) 302.46(B) 259.75(S) 261.01(S) 97.01(S) 94.94(S) 63.93(S)
8 59.03(S) 57.58(S) 306.62(S) 284.46(B) 266.26(S) 118.59(S) 99.63(S) 69.56(S)
9 68.67(S) 67.03(S) 394.13(S) 350.37(S) 309.54(S) 133.62(S) 111.17(S) 78.90(S)
10 83.52(S) 82.48(S) 398.83(S) 355.18(S) 342.78(S) 135.47(S) 111.97(S) 95.93(S)
11 83.92(S) 82.79(S) 488.44(T) 369.10(S) 352.75(S) 150.56(S) 115.78(S) 105.77(S)
12 85.07(S) 82.97(S) 554.37(S) 423.16(S) 378.00(S) 159.17(S) 124.42(S) 107.59(S)
13 88.43(S) 86.90(S) 590.73(S) 430.14(S) 398.51(S) 171.01(S) 130.38(S) 109.34(S)
14 90.50(S) 88.33(S) 644.11(S) 447.64(S) 432.32(S) 198.17(S) 132.90(S) 112.64(S)
15 90.70(S) 89.27(S) 677.37(S) 481.74(T) 438.45(S) 199.61(S) 140.81(S) 115.72(S)
(B) Bending, (T) Torsion, (S) Shell-Like, (A) Axial.
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Table 5: Reinforced cylindrical panel: comparison of the first 5 natural frequencies with respect tothe solid FE model.
f s [Hz] FEM3D FEM2D−1D LE 22 LE 34DOF 106485 16038 3645 5589
b = 1.5 [m]1 103.57 97.17(−6.4%) 140.87(+36.0%) 116.19(+12.2%)
2 103.71 97.20(−6.6%) 134.02(+29.2%) 115.08(+11.0%)
3 107.41 103.78(−3.1%) ∗∗ 123.53(+15.0%)
4 109.67 104.55(−4.7%) ∗∗ 130.37(+18.9%)
5 110.98 107.00(−4.2%) ∗∗ 134.13(+20.9%)
b = 3.0 [m]1 41.02 39.76(−1.9%) 44.16(+7.9%) 44.01(+7.3%)
2 41.44 40.17(−3.5%) 44.17(+6.1%) 44.45(+7.3%)
3 43.75 42.12(−4.6%) 45.84(+4.9%) 46.07(+5.3%)
4 46.65 44.85(−3.5%) 51.01(+8.4%) 50.65(+8.6%)
5 50.23 48.57(−2.8%) 90.85(+81.4%) 55.57(+10.6%)
()(∗%) : * percentage error with respect toFEM3D.
**: Mode not found.
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Table 6: Reinforced cylinder: first 30 natural frequencies.
f s [Hz] FEM2D−1D T E 1 T E 2 T E 3 T E 4 T E 5 LE 424El LE 432El LE 924El LE 932El
1 6.42 51.11(F) 51.66(F) 46.97(F) 46.91(F) 46.88(F) 41.23(F) 40.75(S) 10.16(S) 7.04(S)
2 7.69 51.11(F) 51.66(F) 47.01(F) 46.93(F) 46.91(F) 41.61(F) 40.80(S) 10.17(S) 8.28(S)
3 7.69 108.95(T) 108.95(T) 81.46(S) 70.42(S) 54.37(S) 42.96(S) 41.62(F) 11.23(S) 8.40(S)
4 9.64 123.53(F) 124.51(F) 81.47(S) 70.42(S) 54.37(S) 42.96(S) 42.36(F) 17.02(S) 10.93(S)
5 11.73 123.53(F) 124.51(F) 107.95(F) 82.90(S) 69.97(S) 60.14(S) 54.88(S) 17.03(S) 13.18(S)
6 11.87 144.52(A) 176.43(A) 108.02(F) 82.90(S) 69.97(S) 60.20(S) 55.05(S) 17.14(S) 13.66(S)
7 11.87 212.90(F) 214.12(F) 108.95(T) 107.72(F) 99.91(S) 80.87(T) 80.41(S) 23.64(S) 14.13(S)
8 12.77 212.90(F) 214.12(F) 119.95(S) 107.76(F) 99.91(S) 90.29(F) 81.20(S) 23.71(S) 15.62(S)
9 12.77 217.90(T) 217.90(T) 119.95(S) 108.95(T) 107.63(F) 90.55(S) 84.54(T) 23.99(S) 15.72(S)
10 13.05 288.38(A) 312.61(F) 176.04(A) 109.32(S) 107.69(F) 90.79(S) 90.63(F) 31.28(S) 16.18(S)
11 13.05 311.34(F) 312.61(F) 177.85(S) 109.33(S) 108.95(T) 91.79(F) 91.50(S) 38.97(S) 16.33(S)
12 15.98 311.34(F) 326.85(T) 177.86(S) 146.91(S) 139.77(S) 93.09(S) 91.84(S) 40.87(F) 19.33(S)
13 15.98 326.85(T) 350.82(A) 180.46(F) 146.93(S) 139.77(S) 93.09(S) 92.78(S) 41.27(F) 20.49(S)
14 16.70 415.45(F) 416.70(F) 180.56(F) 176.03(A) 176.02(A) 96.88(S) 93.55(S) 41.90(S) 21.52(S)
15 18.09 415.45(F) 416.70(F) 217.90(T) 179.72(F) 179.55(F) 96.91(S) 93.82(S) 42.12(S) 23.34(S)
16 18.09 430.91(A) 435.81(T) 250.02(S) 179.79(F) 179.63(F) 107.33(S) 97.93(S) 42.20(S) 23.91(S)
17 18.37 435.81(T) 520.74(A) 250.04(S) 191.61(S) 185.39(S) 107.50(S) 98.43(S) 42.25(S) 24.06(S)
18 18.37 522.99(F) 524.22(F) 258.10(F) 191.63(S) 185.39(S) 120.29(S) 107.35(S) 47.24(S) 24.17(S)
19 18.91 522.99(F) 524.22(F) 258.21(F) 194.58(S) 187.72(S) 122.42(S) 107.87(S) 56.33(S) 24.33(S)
20 18.91 544.76(T) 544.76(T) 326.85(T) 194.59(S) 187.72(S) 126.07(S) 113.25(S) 62.59(S) 24.42(S)
21 18.94 571.43(A) 633.84(F) 332.63(S) 216.96(S) 190.34(S) 126.62(S) 114.86(S) 63.04(S) 24.58(S)
22 18.94 632.57(F) 633.84(F) 332.65(S) 216.96(S) 190.34(S) 128.87(S) 120.50(S) 65.39(S) 28.54(S)
23 20.04 632.57(F) 653.72(T) 337.58(F) 217.90(T) 196.12(S) 128.98(S) 121.04(S) 73.85(S) 29.62(S)
24 20.04 653.72(T) 683.05(A) 337.70(F) 240.43(S) 196.13(S) 129.43(S) 122.14(S) 73.96(S) 31.31(S)
25 20.17 709.32(A) 741.87(F) 349.75(A) 240.45(S) 206.28(S) 141.72(S) 122.69(S) 80.71(T) 31.42(S)
26 21.64 740.55(F) 741.88(F) 415.81(F) 253.26(S) 206.30(S) 147.30(F) 122.79(S) 83.21(S) 31.63(S)
27 21.64 740.56(F) 762.67(T) 415.94(F) 253.26(S) 217.90(T) 150.39(F) 126.39(S) 83.88(S) 34.38(S)
28 22.05 762.67(T) 774.74(F) 422.91(S) 256.15(F) 221.50(S) 152.90(S) 132.91(S) 88.66(F) 34.65(S)
29 22.30 775.04(F) 774.74(F) 422.93(S) 256.22(F) 221.53(S) 154.17(S) 142.55(S) 90.10(F) 35.06(S)
30 22.30 775.04(F) 833.30(F) 435.81(T) 291.34(S) 233.94(S) 161.76(T) 143.53(S) 101.34(S) 35.22(S)
(B) Bending, (T) Torsion, (S) Shell-Like, (A) Axial.
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Table 7: Reinforced Cylinder: shell-like natural frequencies in [Hz].
Modello : FEM2D−1D T E 5 LE 924El LE 932El
DOF : 9360 3465 9072 12096L = 1M = 2 18.37 54.37(+196.0%) 17.02(−7.3%) 15.62(−15.0%)
M = 3 7.69 187.72(+2341.1%) 10.16(+32.1%) 8.28(+7.7%)
M = 4 6.42 306.17(+4669.0%) 11.23(+74.9%) 7.04(+9.7%)
L = 2M = 2 38.52 69.97(+81.6%) 42.2(+9.6%) 43.16(+12.0%)
M = 3 22.3 190.34(+753.5%) 23.64(+6.0%) 24.06(+7.9%)
M = 4 11.73 320.11(+2629.0%) 17.14(+46.1%) 13.18(+12.4%)
L = 3M = 2 81.22 99.91(+23.0%) 73.85(−9.1%) 74.09(−8.8%)
M = 3 38.47 196.12(+409.8%) 41.9(+8.9%) 41.98(+9.1%)
M = 4 20.17 344.27(+1606.8%) 23.99(+18.9%) 23.34(+15.7%)
L = 4M = 2 119.74 139.77(+16.7%) 107.9(−9.9%) 110.59(−7.6%)
M = 3 53.99 206.28(+282.1%) 62.59(+15.9%) 64.82(+20.1%)
M = 4 30.01 379.23(+1163.7%) 31.28(+4.2%) 36.72(+22.4%)
L = 5M = 2 173.64 185.39(+6.8%) 141.0(−18.8%) 154.78(−10.9%)
M = 3 95.66 221.5(+131.5%) 83.21(−13.0%) 91.69(−4.2%)
M = 4 39.68 424.95(+970.9%) 38.97(−1.8%) 47.05(+18.6%)
L: half-waves along y-axis.
M: Lobes on cross-section.
()(∗%) : * percentage error with respect toFEM2D−1D.
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Figures
Figure 1: Example of reinforced structures.
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(a) Milled. (b) Welded.
Figure 2: Different approaches to build a stiffened panel.
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Figure 3: Coordinate frame.
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Figure 4: Construction of a stiffened plate using fourLE4.
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Figure 5: Flat Stiffened Plate: 3D model.
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(a) f s1 = 53.90Hz. (b) f s2 = 61.55Hz. (c) f s3 = 94.23Hz.
(d) f s4 = 131.01Hz. (e) f s5 = 144.15Hz. (f) f s6 = 151.98Hz.
(g) f s7 = 186.36Hz. (h) f s8 = 216.96Hz. (i) f s9 = 266.57Hz.
(j) f s10 = 268.71Hz.
Figure 6: Flat Stiffened Plate: First 10 Modes by the TE 7 model.
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(a) f s1 = 51.64Hz. (b) f s2 = 54.30Hz. (c) f s3 = 67.48Hz.
(d) f s4 = 105.20Hz. (e) f s5 = 133.70Hz. (f) f s6 = 135.63Hz.
(g) f s7 = 145.99Hz. (h) f s8 = 165.38Hz. (i) f s9 = 183.53Hz.
(j) f s10 = 237.09Hz.
Figure 7: Flat Stiffened Plate: First 10 Modes by the LE 9+ model.
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-1.5-1
-0.5 0
0.5 1
1.5x 0
0.5 1
1.5 2
2.5 3
y
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
z
Figure 8: Geometry of the reinforced cylindrical panel.
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H s
Figure 9: Cross-section geometry of the reinforced cylindrical panel.
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(a) f s1 = 80.74Hz. (b) f s2 = 88.33Hz. (c) f s3 = 144.96Hz.
(d) f s4 = 174.84Hz. (e) f s5 = 184.60Hz. (f) f s6 = 236.92Hz.
(g) f s7 = 259.75Hz. (h) f s8 = 284.46Hz. (i) f s9 = 350.37Hz.
(j) f s10 = 355.18Hz.
Figure 10: Reinforced Cylindrical Panel: first 10 modes by theT E6 model and considering l=3.0m.
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(a) f 1LE18 = 46.31Hz. (b) f 1LE22 = 44.16Hz. (c) f 1LE34 = 44.01Hz.
Figure 11: Reinforced cylindrical panel: first mode using different mesh on the cross-section andconsidering l=3.0 m.
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(a) f s1 = 32.34Hz. (b) f s2 = 44.17Hz. (c) f s3 = 45.54Hz.
(d) f s4 = 50.27Hz. (e) f s5 = 55.56Hz. (f) f s6 = 59.48Hz.
(g) f s7 = 62.86Hz. (h) f s8 = 69.33Hz. (i) f s9 = 79.95Hz.
(j) f s10 = 98.90Hz.
Figure 12: Reinforced cylindrical panel: first 10 modes byLE34 model and considering l=3.0 m.
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Figure 13: Global configuration of the reinforced cylinder.
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H S
Figure 14: Cross-section geometry of the reinforced cylinder.
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(a) f s1 = 46.88 Hz (b) f s2 = 46.91 Hz (c) f s3 = 54.37 Hz
(d) f s4 = 54.37 Hz (e) f s5 = 69.97 Hz (f) f s6 = 69.97 Hz
(g) f s7 = 99.91 Hz (h) f s8 = 99.91 Hz (i) f s9 = 107.63 Hz
(j) f s10 = 107.69 Hz
Figure 15: Reinforced Cylinder: First 10 modes by theT E5 model.
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(a) f s1 = 10.16 Hz (b) f s2 = 10.17 Hz (c) f s3 = 11.23 Hz
(d) f s4 = 17.02 Hz (e) f s5 = 17.03 Hz (f) f s6 = 17.14 Hz
(g) f s7 = 23.64 Hz (h) f s8 = 23.71 Hz (i) f s9 = 23.99 Hz
(j) f s10 = 31.28 Hz
Figure 16: Reinforced Cylinder: First 10 modes by theLE9 model with 24 elements on the cross-section
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![Free vibration analysis of piezoelectric cylindrical nanoshell ......vibration analysis of shallow shells [25]. Loy et al. presented the free vibration analysis of cylindrical shells](https://img.pdfslide.us/doc/110x75/5f736ba706f884064751d11e/free-vibration-analysis-of-piezoelectric-cylindrical-nanoshell-vibration.jpg)
