Static Analysis of Reinforced Thin-Walled Plates and Shells by

means of Finite Element Models

E. Carreraa∗, E. Zappinoa†, T. Cavalloa‡,aDepartment of Aeronautic and Space Engineering, Politecnico di Torino,

Corso Duca degli Abruzzi 24, 10129 Torino, Italy.

Submitted to International Journal for Computational

Methods in Engineering Science & Mechanics

Author for correspondence:E. Carrera, Professor of Aerospace Structures and Aeroelasticity,Department of plate3DAeronautic and Space Engineering,Politecnico di Torino,Corso Duca degli Abruzzi 24,10129 Torino, Italy,tel: +39 011 090 6836,fax: +39 011 090 6899,e-mail: erasmo.carrera@polito.it

∗Professor of Aerospace Structures and Aeroelasticity, e-mail: erasmo.carrera@polito.it†Research assistant, e-mail: enrico.zappino@polito.it‡PhD student, e-mail: tommaso.cavallo@polito.it

1

Abstract

In this paper, variable kinematic one-dimensional (1D) structural models have been used to analyze thin-walled

structures with longitudinal stiffeners and static loads. These theories have hierarchical features and are based

on the Carrera Unified Formulation (CUF). CUF describes the displacement field of a slender structure as the

product of two function expansions, one over the cross-sectional coordinates, Taylor (TE) or Lagrange (LE)

expansions were used here, and one along the beam axis. The results obtained using the refined 1D models

have been compared with those from classical finite element analyses that make use of plates/shells and solids

elements. The performances of classical and refined structural models have been compared in terms of accuracy

and computational costs. The results show that the use of the Lagrange expansion over the cross-section allows

the strain/stress fields to be evaluated accurately for all the structural components. The comparisons with the

results obtained using the classical models highlight how, the use of 1D refined models, allows the number of

degrees of freedom to be reduced, meanwhile, the accuracy of the results can be preserved.

2

1 Introduction

Thin-walled structures are used extensively in structural engineering. These structures are built by means

of a thin skin reinforced with longitudinal and transversal stiffeners, such as ribs, stringers, or spars (see

Fig. 1). Examples include aircraft structures, but also nautical hulls, bridges and metallic structures used in

civil engineering. Reinforced structures are complex systems and they can be considered important because

they meet the requirements of lightness and strength. Reinforced structures are composed of one-dimensional

(rod/beam) and two-dimensional (plate/shell) structural elements, which have to be properly assembled.

Many models and techniques have been proposed to analyse the static response of these structures. Beam

models offer some advantages, because they reduce the problems to a set of variables that only depend on

the beam axis coordinate and they can be used in the analysis of the stiffeners. 2D models are also suitable

for thin-walled structure analysis, where the thickness of the structure is negligible. Classical approaches use

both, one- and two-dimensional models, to investigate reinforced thin-walled structures even though their

performances are limited by their fundamental assumptions.

Classical beam theories, such as the Euler-Bernoulli [1] and de Saint-Venant [2] theories do not consider

transverse shear deformation, while the Timoshenko beam theory [3] predicts a uniform shear distribution over

the beam cross-section. Mucichescu [4] has shown how, neither Euler-Bernoulli nor Timoshenko models, are

able to deal with non-classical effects, such as out- and in- plane deformations, warping, localized boundary

conditions (BCs), and both geometrical and mechanical torsional-bending coupling. These effects are usually

due to small slenderness ratios, thin-walled structures and the anisotropy of the materials. Different solutions

have been proposed to overcome the limitations of the 1D classical models and to extend the use of these

models to any beam geometry with any boundary condition. An example can be found in the book by

Novozhilov [5]. Typical approaches introduce correction factors, as in the books by Timoshenko and Goodier

[6] and by Sokolnikoff [7]. El Fatmi [8, 9] instead introduced a warping function φ to improve the description

of the normal and shear stresses over the cross-section. Another approach, based on an asymptotic solution,

was used by Berdichevsky at al. [10], in which a characteristic parameter (e.g., the cross-section thickness

of a beam) is exploited to build an asymptotic series. It is also appropriate to recall the work by Schardt

[11, 12], which developed generalized beam theories (GBT), which were then widely used by Silvestre [13] for

the analysis of thin-walled structures.

The theories proposed by Kirchhoff-Love [14] and Reissner-Mindlin [15, 16] have to be mentioned when

2D classical models are considered. The former model neglects the contribution of the transverse shear while

the latter considers a constant shear through the thickness. Both the models do not consider any variation in

the thickness and can not, therefore, be considered reliable when advanced materials, as sandwich material,

are analysed. Many refined 2D theories have been proposed to increase the accuracy of these models. An

example is the one by Reddy [17] which introduced a refined kinematic through the thickness.

The aim of this work is to compare the performances of different modelling approaches for the static

analysis of thin-walled structures with longitudinal stiffeners. Classical and refined 1D models are taken into

account. The classical approaches considered are based on the models available in the commercial codes and

are based on the structural theories introduced above. In particular two classical finite elements models are

introduced: the former uses a fully three-dimensional model built using solid elements while the latter uses

3

shell and beam elements, where the shells are used to represent the skin of the structure and the beams are used

to represent the stiffeners. An alternative modelling approach, based on a refined 1D model, have been used

in the present work and compared with classical approaches. This model has been derived using the Carrera

Unified Formulation, a tool that allows the FEM matrices to be written using a compact and iterative form.

The CUF first appeared in the work of Carrera [18] where it was used to derive a class of two-dimensional

theories using a compact formulation. The CUF was first applied to multi-layered, anisotropic, composite

plates and shells [19, 20], and later to study one-dimensional models [21]. CUF allows one-dimensional models

to be derived using different kinematic assumptions. The expansions over the beam cross-section can be in fact

an arbitrary choice, in this work two examples of approximations over the cross-section are used: the Taylor

expansion (TE) and the Lagrange expansion (LE). In the first case, the expansion is done around a generic

point over the cross-section. This kinematic model was proposed in the work of Carrera and Giunta [22] and

Carrera et al. [21]. A number of works have been published using this model, considering both static [23] and

dynamic [24] analyses. The Lagrange expansion (LE) uses the Lagrange functions to approximate the solution

over the cross-section. In this case, a number of 2D elements are used to approximate the cross-section, and

the solution is expanded around the nodes of these elements. When an LE model is considered, the unknowns

are only the nodal displacements. The LE based model was introduced by Carrera and Petrolo [25], and was

used to perform analyses of complex structures, as shown in [26] and [27]. Both models were obtained with no

need for ad hoc formulations, and the expansion order N was a free parameter of the analysis, that is, N is one

of the inputs. The Euler-Bernoulli and Timoshenko beam models can be obtained as particular cases of the TE

model. In both cases, the finite element method was used to handle arbitrary geometries and geometrical and

mechanical boundary conditions. In this work, the 1D structural models, derived and assessed in previous

works, are used in the analysis of complex reinforced structures and the results are compared with those

obtained using classical approaches. The performances of the different numerical models, in the evaluation of

the displacement and stress fields, have been compared considering different structures. Firstly a reinforced

panel has been analysed, then, a half cylinder and an entire cylinder have been taken into account.

2 Reinforced structure analysis approaches

The analysis of reinforced structures requires to include the contributions of two different structural elements

in the model, the thin skin and the stiffeners. These two structural components have to withstand different

loads. The skin is subjected to in-plane loads, in particular shear loads, while the stringers are supposed to

be loaded by axial stress. Figure 2 shows an example of a flat plate reinforced with three stringers.

The analysis of this structures can be performed using different approaches. Figure 3 shows different

models that can be used to study a part of the structure depicted in Figure 2. The use of solid elements, see

Figure 3 on the left, allows the model to be derived without introducing any assumptions, except those of the

discretization. This approach provides accurate results, but it is computationally expensive. Moreover, the

aspect ratio of the solid elements should not exceed a limit value to preserve the accuracy of the results, and

a large number of elements is therefore required even though simple structures are considered.

The geometry of reinforced structures suggests the use of shell and beam elements, the former on the skin

and the latter on the stringers. Figure 3 shows an example of this approach. The beam elements and the

4

shell elements must be connected at some nodes. Beam elements usually require an offset in order to respect

the geometry of the structure.

This approach reduces the computational costs of the structural analysis, but increases the complexity

of the pre-processing phase. Furthermore, these models do not provide an accurate solution over the whole

physical domain. In fact, they only solve the problem over the mid-surface of the skin and along the axis of

the stringers.

A third approach, based on refined one-dimensional models, is used in this work. The use of refined one-

dimensional models allows the structure to be analyzed using only one structural model, as in the case of a

solid element, but also allows the computational cost to be reduced as for shell/beam models. This approach

splits the model into two contributions, a FE discretization on the axis (one-dimensional) and an expansion

on the cross-section, which includes both the stringers and the skin, see Figure 3 on the right. A number

of different refined one-dimensional models can be derived using different kinematic approximations on the

cross-section. In the present work two models are considered, one based on the Taylor expansion and one

based on the Lagrange functions, many others have been presented by Carrera et al. in [28] but in this work

have not been considered.

3 Refined one-dimensional models

The main aspects of the refined 1D elements used in the present work are presented in this section. For such

of brevity some details are here omitted but can be found in the book by Carrera et al. [29]. The formulation

of the classical models that has been used in the analysis can be find in the documentation of the NASTRAN

code.

3.1 Preliminaries

The coordinate frame adopted is shown in figure 2, where y-axis is the beam axis. The beam boundaries over

y are 0 ≤ y ≤ L, where L is the beam length. The displacement vector is

u(x, y, z) =

ux uy uzT

(1)

The superscript ′T ′ denotes transposition. Stress, σ, and strain, ǫ, components are grouped as follows:

σ =

σxx σyy σzz σxy σxz σyz

T(2)

ǫ =

ǫxx ǫyy ǫzz ǫxy ǫxz ǫyzT

(3)

Linear strain-displacement relations are used,

ǫ = Du

where D is a differential operator. The explicit formulation of D can be found in Appendix A, more details

are reported in the book by Carrera et al.[29]. The materials is considered elastic and isotropic, therefore the

stresses can be derived using the Hook’s law:

σ = Cǫ

The explicit form of C in the case of isotropic materials can be found in Appendix B, more details can be

found in the books by Tsai [30] or Reddy [31].

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3.2 One-dimensional approximation

In the framework of the Carrera Unified Formulation [19, 20, 32, 29], the displacement field is assumed as an

expansion in terms of generic functions, Fτ :

u = Fτ (x, z)uτ (y), τ = 1, 2, ....,M (4)

where the Fτ functions are defined above the cross-section. uτ is the displacement vector and M stands for

the number of terms of the expansion of order N . In this work two different expansions are considered, the

first based on Taylor expansion, TE, and the second based on the Lagrange functions, LE. TE models are

based on polynomial expansions, xi zj, of the displacement field over the cross-section of the structure, where

i and j are positive integers. A generic N -order displacement field is then expressed by:

ux =

N∑

Ni=0

(

Ni∑

M=0

xN−M zM uxN(N+1)+M+12

)

uy =N∑

Ni=0

(

Ni∑

M=0

xN−M zM uxN(N+1)+M+12

)

uz =

N∑

Ni=0

(

Ni∑

M=0

xN−M zM uxN(N+1)+M+12

)

(5)

The order N of the expansion is arbitrary and is set as an input of the analysis. The choice of N for a given

structural problem is usually made through a convergence study. Only full-order TE models are considered

in this paper since, as N is fixed, all the terms of the corresponding expansion are taken into account. LE

models exploit Lagrange polynomials to build 1D higher-order theories. In this paper two type of cross-

section polynomial sets are adopted: nine-point elements, L9, and four-point elements, L4. The isoparametric

formulation is exploited to deal with arbitrary shaped geometries. The L9 interpolation functions are given

by [33]:Fτ = 1

4 (r2 + r rτ )(s

2 + s sτ ) τ = 1, 3, 5, 7

Fτ = 12s

2τ (s

2 − s sτ )(1− r2) + 12r

2τ (r

2 − r rτ )(1 − s2) τ = 2, 4, 6, 8

Fτ = (1− r2)(1− s2) τ = 9

(6)

where r and s range from −1 to +1. More details on refined beam models can be found in the book by

Carerra et al.[34] The Finite Element Method is used to approximate the displacement over the beam axis.

The displacement field can be written introducing the classical FEM one-dimensional shape functions,Ni:

u = Fτ (x, z)Ni(y)qiτ (7)

where index i indicates the node of the element. In the present work cubic shape functions are used. Therefore

each element is considered to have four nodes. The original three-dimensional field has, now, a new form

composed by two contributions: the cross-section expansion Fτ and the axial expansion Ni. While the two

expansions are assumed a priori the constants qiτ are the only unknowns of the problem.

3.3 Governing equations

The governing equations can be derived using the principle of virtual displacements in the static formulation:

δLint = δLext (8)

6

where Lint stands for the strain energy, and Lext is the work of the external loadings. δ stands for the virtual

variation. The internal work can be written as:

δLint =

∫

V

ǫTσdV (9)

Introducing the formulation of the stress and strain presented in Equations 2 and 3, and using the displacement

field introduced in Eq 7, the variation of the internal work becomes:

δLint =δqTsj

∫

V

Fs(x, z)Nj(y) DT C D Ni(y)Fτ (x, z)dV qτi =

=δqTsjk

ijτsqτi

(10)

kijτs is the stiffness matrix in form of “fundamental nucleus”, a 3× 3 matrix with an fixed form. The whole

stiffness matrix of the element can be evaluated varying the indexes i, j, τ and s. The explicit formulation of

the fundamental nucleus can be found in appendix C, a more exhaustive discussion can be found in [29].

The variation of the external work, δLext, can be derived in the case of a concentrated load as:

δLext = δuTP = δqTsjFs|PNj|PP = δqT

sjpsj (11)

where Fs|P and Nj |P are the values of the shape functions evaluated in the point there the load P is applied.

The terms psj is the fundamental nucleus of the loading vector.

4 Numerical analysis and discussion

The static analysis of different refined beam models has been performed in this section. The analyses have

the aim to show the capabilities of the present models in the analysis of stiffened structures and to compare

their performances with those provided by classical approaches. Three main problems, related to thin-walled

stiffened structures, are considered. The first problem is the static analysis of a stiffened plate loaded by a

static load. A ”very low” aspect-ratio has been used because this is the worst condition for classical beam

elements. The second case concerns the static analysis of stiffened panels with two asymmetric static loads.

Finally, the third problem considers the static analysis of a cylindrical thin-walled structure with longitudinal

stiffeners under two opposite static loads. If not otherwise stated, the material used is aluminium with a

Young modulus, E, of 75 GPa and a Poisson ratio, ν, equal to 0.3. The geometrical boundary conditions are

considered as clamped-clamped for all the structures, and neither displacements nor rotations are therefore

allowed at either the ends of the beam. All the analyses were performed using different expansions over the

cross-section, that is, Taylor (TE) and Lagrange (LE) expansions were considered.

Different orders of expansion, N, were considered in the TE models, while, different meshes were considered

over the cross-section when the LE models are used, both 4- and 9-node elements were considered. Only

cubical elements (B4) were used in the axial discretization. The nodes were uniformly distributed along the

longitudinal axis (y − axis). Tab.1 shows the properties of the LE models used in the following sections.

This table shows the name of the model in the first column, where P stands for Plate, S for Shell and C for

Cylinder. The second column contains the number of elements on the cross-section, the third column reports

the number of nodes of the cross-sectional elements, the fourth column shows the number of B4 elements on

the y − axis and the figures in which the cross-sectional meshes are shown are reported in the last column.

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4.1 Static analysis of reinforced flat panel

The static response of a stiffened plate with three longitudinal stringers has been investigated in this section

to assess the present model. The geometry of the structure is shown in Figure 4. Sides a and b are both

equal to 2 m, and the plate thickness is 0.02 m. The geometry of the cross-section, on the x-z plane, is shown

in Figure 5a , the dimensions are reported in Table 2. The structure is subjected to a concentrated load,

P, which is applied at point (C ) with magnitude of 20000 N (see Fig 6). The results are compared with

those obtained from a solid model built by means of the MSC NASTRANr commercial code. The vertical

displacements were evaluated at four points on the top of the plate (C, B, A′

and B′

, see Figure 4), while

the stress analysis was computed at four points through the thickness of the central stringers (α, β, γ, δ, see

Figure 5b). The point positions are reported in Table 3. Two solid models were compared with the redined

1D models. The first, called FEM3D (see Figure 8), has a comparable number of DOFs to the present beam

model, while, a refined 3D FE model, called FEM3D−REF (see Figure7) has a more refined mesh. Two TE

models were used: an eighth, TE8, and a tenth, TE10, order model. Four LE models were considered, P − 1,

P − 2, P − 3 and P − 4. Their cross-section are shown in Figures 9, 10 and 11, while the node distributions

are presented in Figures 12, 13 and 14, respectively. Models P − 4 and P − 2 have the same cross-section, but

the first has 8-B4 elements on the y − axis, as can be seen in Table 1, while the P − 2 has 12-B4 elements.

FEM3D and the P − 2 models have the same number of DOFs.

4.1.1 Displacement analysis

The vertical displacement in the x−z and z−y planes are reported in Figures 15a and b, respectively. Table 4

shows the displacements evaluated using the different models. The first column contains the models that were

used, and the results of each model are given in terms of displacements at the selected points shown in Figure

4. The refined solid FE model (FEM3D−REF ) is shown in the first row, while the coarse solid FE model

(FEM3D) is given in the second. The results of the TE models are shown in rows three and four in table 4.

The LE model results are given in rows five to eight. Two different elements were used on the cross-section;

model P − 1 was built using four-node elements, while models P − 2, P − 3 and P − 4 were built using

nine-node elements. The percentage difference for FEM3D−REF model is shown in superscript. The results

from the TE models seem to converge slowly with respect to the results from FEM3D−REF as the order of

the expansion N increases, but a tenth order model is not able to reach a good solution; the error in fact is

still higher than 10% in almost all the considered points. Figures 15a, b show that as the order N increases,

the vertical displacement, evaluated using the TE models, comes closer to the FEM3D−REF solution. Even

though the number of DOFs, when the TE10 model is used, is only 0.7% of FEM3D−REF , the results are

still inaccurate in the analysis of reinforced structures. As a consequence, the TE results were omitted in the

following analyses. The results of the P − 1 model show the limitations of the four-node elements; its poor

kinematic requires a refined mesh and a correction of the Poisson locking and higher order elements were

therefore preferred. Models P − 2, P − 3 and P − 4 provide an accurate solution. They show a difference from

FEM3D−REF that is close to 2%, using a number of DOFs that is less than 3% of those used by the reference

model. The results show that the accuracy of a refined beam model obtained with the present approach, e.g.

P − 2, is higher than the accuracy of a 3D FE model, see FEM3D, with the same number of DOFs.

8

4.1.2 Stress analysis

The stress fields computed with the present models have been compared with those from a solid model. σyy

and τyz are considered. σyy was evaluated through the thickness of the central stringer for x =a

2and y =

b

2,

while τyz was measured for x =a

2, y =

3

4b. Figure 16a shows the σyy distribution through the thickness of

the structure, from the top of the plate to the bottom of the stringer. Figure 16b shows the τyz distribution

through the thickness of the stringers. Table 5 shows the value to the axial stress evaluated using the different

models considered. The shear stresses are reported in Table 6. Despite the FEM3D and P − 2 models having

the same DOFs, the P − 2 model provides a more accurate solution. All the LE models provide an accurate

solution, even though the LE DOFs are about 3% of the FEM3D−REF DOFs. Figure 16b shows the τyz

value at the center of the central stringer. In this case, the error made by the P − 2 model is about 14% of

the FEM3D−REF solution, but the coarse FEM3D model is much less accurate. Both the P − 3 and P − 4

models give very accurate solutions, and very close to the value provided by the FEM3D−REF model. The

evaluation of the shear stress requires a more refined model with respect the evaluation of the axial stress. In

this case only P-3 and P-4 models are able to provide accurate results. The results show that the use of 1D

refined model in the analysis of plate reinforce panels allows accurate results to be achieved, the use of 2D

elements requires an higher number of DOFs to provide the same accuracy.

4.2 Analysis of a reinforced cylindrical panel

The shell structure shown in Figure 17 has been investigated in this section. Figure 18 shows the geometry

of the cross-section at y = b/2 and a loading configuration. Two asymmetric loads of 500 N were considered,

both applied at the mid-point of the beam length. Load F1 is applied at point A and was directed from the

top to the bottom of the panel, while load F2 was applied at point E and was directed from the bottom to

the top of the structure. The cross-section was characterized by a radius r of 1 m, and a shell thickness of

0.002 m. Five longitudinal stringers, whit a rectangular cross-section were introduced where the larger side m

measured 0.054 m, while side p measured 0.02 m (see Figure18). The beam length, b, measured 1.5 m. Two

different FE models, built using the NASTRANr commercial code, were considered as references. Figure 19

shows the FE model built using only solid elements. A second FE model was created using shell and beam

elements (FEM2D−1D). Shells were used for the skin and beams were used for the stringers, as shown in

figure 20. Figures 21, 22, 23 and 24 show the cross-sectional meshes of the four different LE models, S − 1,

S − 2, S − 3 and S − 4, respectively. The four models have respectively 1, 2, 3 and 6 elements between two

stringers. Only nine-node elements were used to build the LE models. It should be noted that the number

of elements used on the cross-section has two effects on the model, the first is a refinement of the kinematics

model, while the other is an improvement of the geometrical approximation of the curved panel. When a

lower number of elements is used, the geometry of the model does not in fact accurately represent the real

geometry.

4.2.1 Displacement analysis

Figure 25 shows the vertical displacement over the x − z plane at the mid-length point. The figure includes

the results obtained with both the solid and shell-beam models. An amplification factor, × 1000, was used

to plot the displacements. Table 7 contains the displacements evaluated using the considered structural

9

models. Figure 18 shows the points where both the ux and uz displacement components were evaluated. The

displacement in the x-direction at the point A is denoted with Ax , while the displacement in the z-direction

with Az, the same notation is used in the other points. The percentage difference from the FEM3D model

is reported in superscript. The refined S − 4 model shows an error lower than 9% on both the ux and uz

components at both of the evaluated points. These results are very impressive because the S − 4 model has

only 2% of the DOFs of the FEM3D DOF model. FEM2D−1D also provides accurate results with a lower

number of DOFs but, as shown in the following section, is not able to properly describe the stress field. The

LE models, S − 1 and S − 2, provide inaccurate results because of the coarse mesh used on the cross-section.

4.2.2 Stress analysis

The axial stress has been evaluated on the stringer where the F1 load was applied. Four points were considered

through the thickness at this stringer, that is, points A, B, C and D, as shown in Figure 18. The error made

using LE dependeds on the mesh refinement, as expected. However LE gives a 3D stress field while the same

stress components are not available when the combined beam/shell model is used. The σyy stress distribution

is shown in Figure 26. The results are compared with those of the 3D and 2D − 1D FE models. When the

S − 4 model is used the solution is very close to the 3D distribution. All the LE models gave better results

than the 2D − 1D model. Table 8 shows the numerical results in terms of stress. The S − 4 model provides

very accurate results; the error is always below 9%. The stress analysis shows that the present model is able

to drastically reduce the computational costs while a good accuracy is preserved. In this case, the 2D − 1D

is computationally cheaper than the present model, but is not able to properly predict the stress field.

4.3 Analysis of a reinforced cylinder with 1D CUF elements

The static analysis of a stiffened cylinder with eight longitudinal stringers has been performed in this section.

The geometry of the structure is shown in Figure 27; the radius is equal to 1 m, the skin thickness is 0.002

m, and the cylinder length is 15 m. The boundary condition are still clamped-clamped and the structure is

subjected to two concentrated forces applied in the half length (b/2), in x=a/2 and in z=±1.001 m. The

magnitude of forces F1 and F2 is 500 N. F1 acts in a negative z-direction, while F2 acts accordingly with

the positive z direction, as can be seen in Figure 28. The stringers are in the longitudinal direction, and are

characterized by a rectangular cross-section with a base of 0.02 m and a height of 0.054 m. Two different FE

models were considerate as references. The first model was built using only solid elements, as shown in figure

30. Figure 31 shows the second FE model, which was built using shell and beam elements. An appropriate

offset had to be considered to make a representative model of the stringers. Figure 28 shows nine points, from

A to M, along the y−axis where, the vertical displacements were evaluated. The results were evaluated at two

points over the cross-section, α and β (see Figure 29), and at two different axial positions, y=b/2 and y=b/4.

Four LE models, with different cross-sectional meshes were considered. Table 1 shows that models C − 1 and

C − 3 have the same cross-sectional mesh (see Figure 32), but a different number of axial elements, while the

C − 2 and C − 4 models have a refined cross-sectional mesh (see Figure 33). As far as the curved panel is

concerned, the increasing of the number of elements over the cross-section improves both the kinematic model

and the geometrical approximation.

10

4.3.1 Displacements analysis

Figures 34 and 35 show the deformation of the cross-section at y = b/2 and y = b/4, respectively, evaluated

using the different structural models. An amplification factor, × 100, was used to plot the displacements.

Table 9 shows the displacements evaluated at different points (see Figure 29). The displacements evaluated

using model C − 2 has an error lower than 10% with respect to the reference model. These results are very

impressive, considering that the C−2 model only has 1.5% of the DOFs of the FEM3D model. The refinement

of the axial mesh, see the results for the Model C-4, does not have a strong impact on the displacement values.

Figure 36 shows the vertical displacement in the y − z plane. Tables 9 and 10 show that as the number of

elements on the cross-section is increased, the displacement accuracy increases. Even though, FEM2D−1D

makes a lower error of 3% using very few DOFs, the present model is able to predict the complete stress field.

4.3.2 Stress analysis

The stress results are given in terms of σyy. The results for y = b/2 are shown in Table 11. When the C-1

model is considered the results are not accurate. The refinement of the cross-sectional approximation, see the

Model C-2, or the axial discretization, see model C-3, improves the accuracy but still the error with respect to

the 3D model is higher than 19%. When bot, the axial discretization and the cross-sectional approximation

are improved, as in the case of the Model C-4, the error decreases and becomes closer to 10%. Model C-4

provides more accurate results than the mixed 2D-1D model with a computational cost 30 times lower than

the 3D model.

5 Concluding Remarks

In the present work, the performances of various finite element models have been compared in the static

analysis of thin-walled reinforced structures. Four different FEM models have been used in the analyses: two

1D refined structural models derived using the CUF formulation, and two classical models built using the MSC

NASTRANr commercial code. When refined 1D models were considered two different formulations, based

on Taylor and Lagrange expansions, were used. Otherwise, when classical models were used two different

approaches have been considered, the first used only 3D elements while the second included 1D and 2D

elements (the beams for the stiffeners and the shells for the skin).

Three different thin-walled structures have been considered in the analyses: a flat plate, a cylindrical plate

and a hollow cylinder. Each structure was reinforced using longitudinal stiffeners. Only isotropic materials

were considered. The results show that:

• The 1D models are able to provide a quasi-3D solution of the displacements and stress fields. This

makes these models very attractive with respect to the combined beam/shell model, which is not able

to provide accurate results in terms of stresses;

• The results, in terms of displacements, show a good agreement with those of the solid model. The TE

models are not accurate in the analysis of thin-walled structures, while the LE models provide accurate

results in terms of stress and displacement if an appropriate cross-sectional mesh is adopted.

11

• The refined 1D models, if compared with 3D models having the same number of DOFs, provide more

accurate results in terms of displacements and stresses.

In short, the models here presented appear very attractive for the analysis of reinforced structures. These

models provide and accuracy comparable with those obtained with the classical approaches, but with a lower

computational cost. Application to dynamics as well as to more complex structures, such as aircraft and

launcher structures, could be the subject of future research.

References

[1] L. Euler. De curvis elasticis. methodus inveniendi lineas curvas maximi minimive proprietate gaudentes,

sive solutio problematis iso-perimetrici lattissimo sensu accepti. 1744. Bousquet, Geneva.

[2] A. de Saint-Venant. Memoire sur la flexion des prismes, sur les glissements transversaux et longitudinaux

qui l accompagnent lors- quelle ne sopere pas uniformement ou en arc de cercle, et sur la forme courbe

affectee alors par leurs sections transversales primitivement planes. J. Math. Pures Appl., 1:89–189, 1856.

In French.

[3] S. P. Timoshenko. On the correction for shear of the differential equation for transverse vibrations of

prismatic bars. Philosophical Magazine., 41:744–746, 1921. New York.

[4] D. T. Mucichescu. Bounds for stiffness of prismatic beams. Int. J. Solids Struct., 36(10):1523–1540, 1984.

[5] V. V. Novozhilov. Theory of elasticity. Pergamon., Oxford, UK. 1961.

[6] S. P. Timoshenko and J. N. Goodier. Theory of elasticity. McGraw-Hill., 1970.

[7] I.S. Sokolnikoff. Mathematical theory of elasticity. McGraw–Hill., 1956. New York.

[8] R. El Fatmi. Nonuniform warping including the effects of torsion and shear forces. part i: A general

beam theory. Int. J. Solids Struct., 44:18–19, 2007a. DOI: 5912–5929.

[9] R. El Fatmi. Nonuniform warping including the effects of torsion and shear forces. part ii: Analytical

and numerical applications. Int. J. Solids Struct., 44:18–19. DOI: 5930–5952.

[10] V. L. Berdichevsky, E. Armanios, and A. Badir. Theory of anisotropic thin-walled closed-cross-section

beams. Composites Eng., 2(5–7):411–432, 1992.

[11] R. Schardt. Eine erweiterung der technischen biegetheorie zur berech- nung prismatischer faltwerke. Der

Stahlbau, 35:161–171, 1966.

[12] R. Schardt. Verallgemeinerte technische biegetheorie. Springer-Verlag, 1989. Berlin.

[13] N. Silvestre and D. Camotim. First-order generalised beam theory for arbitrary orthotropic materials.

Thin-Walled Struct., 40(9):791–820, 2002.

[14] A E H Love. The Small Free Vibrations and Deformation of a Thin Elastic Shell. Philosophical Transac-

tions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 179:491–546,

1888.

12

[15] E Reissner. The effect of transverse shear deformation on the bending of elastic plates. ASME Journal

of Applied Mechanics, 12:68–77, 1945.

[16] R D Mindlin. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. ASME

Journal of Applied Mechanics, 18:31–38, 1951.

[17] J. N. Reddy and C. F. Liu. A higher-order shear deformation theory of laminated elastic shells. Inter-

national Journal of Engineering Science, 23(3):319–330, 1985.

[18] E. Carrera. A class of two dimensional theories for multilayered plates analysis. Atti Accademia delle

Scienze di Torino, Memorie Scienze Fisiche, 19-20:49–87, 1995.

[19] E. Carrera. Theories and finite elements for multilayered, anisotropic, composite plates and shells.

Archives of Computational Methods in Engineering, 9(2):87–140, 2002.

[20] E. Carrera. Theories and finite elements for multilayered plates and shells: a unified compact formula-

tion with numerical assessment and benchmarking. Archives of Computational Methods in Engineering,

10(3):216–296, 2003.

[21] Giunta G. Nali P. Carrera, E. and M. Petrolo. Refined beam ele-ments with arbitrary cross-section

geometries. Comput. Struct., 88(5–6):283–293, 2011.

[22] E. Carrera and G. Giunta. Refined beam theories based on a unified formulation. Int. J. Appl. Mech.,

2(1):117–143, 2010.

[23] Petrolo M. Carrera, E. and E. Zappino. Performance of cuf approach to analyze the structural behavior

of slender bodies. J. Struct. Eng., 138:285–297, 2012.

[24] Petrolo M. Carrera, E. and P. Nali. Unified formulation applied to free vibrations finite element analysis

of beams with arbitrary section. Shock Vib., 18(3):485–502, 2011.

[25] E. Carrera and M. Petrolo. Refined beam elements with only displace- ment variables and plate/shell

capabilities. Meccanica, 47(3):537–556, 2012.

[26] M. Petrolo E. Carrera, A. Pagani. Classical, refined and component-wise analysis of reinforced-shell

structures. AIAA Journal, 51(5):1255–1268, 2013.

[27] M. Petrolo E. Carrera, A. Pagani. Component-wise method applied to vibration of wing structures.

Journal of Applied Mechanics, 88(4):041012–1–041012–15, 2013.

[28] E Carrera, M Filippi, and E Zappino. Free Vibration Analysis of Laminated Beam by Polynomial,

Trigonometric, Exponential and Zig-Zag Theories. Journal of Composite Materials.

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Unified Formulation. John Wiley & Sons, 2014. In press.

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13

[31] J. N. Reddy. Mechanics of laminated composite plates and shells. theory and analysis. CRC Press, 2nd

ed., 2004. Boca Raton, FL.

[32] E. Carrera and G. Giunta. Refined beam theories based on a unified formulation. International Journal

of Applied Mechanics, 2(1):117–143, 2010.

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Barcelona, Spain, 2009.

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& Sons Ltd., 2011. ISBN 9780470972007.

A Geometrical relation

The matrix [D] is a differential operator which describes the relation between the displacement, u, and the

strain, ǫ.

D =

∂∂x 0 00 ∂

∂y 0

0 0 ∂∂z

∂∂y

∂∂x 0

∂∂z 0 ∂

∂x

0 ∂∂z

∂∂y

(12)

B Constitutive equations

The coefficients for isotropic materials are:

C =

C11 C12 C12 0 0 0C21 C11 C12 0 0 0C21 C21 C11 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C44

(13)

where

C11 = 2G+ λ,C12 = C21 = λ,C44 = G (14)

and

G =E

2 (1 + ν), λ =

νE

(1 + ν) (1− 2ν)(15)

E is the Young modulus, G is the shear modulus and ν is the Poisson ratio. λ and G are also known as Lame

coefficients.

C Fundamental nucleus

The 9 terms of the fundamental nucleus of the stiffness matrix are:

14

kijτsxx = C22

∫

Ω

Fτ,xFs,xdΩ

∫

l

NiNjdy + C66

∫

Ω

Fτ,zFs,zdΩ

∫

l

NiNjdy+

C44

∫

Ω

FτFsdΩ

∫

l

Ni,yNj,ydy

kijτsxy = C23

∫

Ω

Fτ,xFsdΩ

∫

l

NiNj,ydy + C44

∫

Ω

FτFs,xdΩ

∫

l

Ni,yNjdy

kijτsxz = C12

∫

Ω

Fτ,xFs,zdΩ

∫

l

NiNjdy + C66

∫

Ω

Fτ,zFs,xdΩ

∫

l

NiNjdy

kijτsyx = C44

∫

Ω

Fτ,xFsdΩ

∫

l

NiNj,ydy + C23

∫

Ω

FτFs,xdΩ

∫

l

Ni,yNjdy

kijτsyy = C55

∫

Ω

Fτ,zFs,zdΩ

∫

l

NiNjdy + C44

∫

Ω

Fτ,xFs,xdΩ

∫

l

NiNjdy+

C33

∫

Ω

FτFsdΩ

∫

l

Ni,yNj,ydy(16)

kijτsyz = C55

∫

Ω

Fτ,zFsdΩ

∫

l

NiNj,ydy + C13

∫

Ω

FτFs,zdΩ

∫

l

Ni,yNjdy

kijτszx = C12

∫

Ω

Fτ,zFs,xdΩ

∫

l

NiNjdy + C66

∫

Ω

Fτ,xFs,zdΩ

∫

l

NiNjdy

kijτszy = C13

∫

Ω

Fτ,zFsdΩ

∫

l

NiNj,ydy + C55

∫

Ω

FτFs,zdΩ

∫

l

Ni,yNjdy

kijτszz = C11

∫

Ω

Fτ,zFs,zdΩ

∫

l

NiNjdy + C66

∫

Ω

Fτ,xFs,xdΩ

∫

l

NiNjdy+

C55

∫

Ω

FτFsdΩ

∫

l

Ni,yNj,ydy

The components of the stiffness matrix are products of two contributes: the first is an integral along the axis,

the second an integral above the cross-section Ω.

15

Tables

Model NEl NNodes−El NB4 FigReinforced Plate

P − 1 14 4 8 Fig.9P − 2 36 9 8 Fig.10P − 3 36 9 8 Fig.11P − 4 36 9 12 Fig.10

Reinforced ShellS − 1 14 9 4 Fig.21S − 2 18 9 4 Fig.22S − 3 22 9 4 Fig.23S − 4 34 9 4 Fig.24

Reinforced CylinderC − 1 40 9 4 Fig.32C − 2 64 9 4 Fig.33C − 3 40 9 8 Fig.32C − 24 64 9 8 Fig.33

Table 1: Properties of the LE models used in the present work.

a 2.00c 0.30d 0.70e 0.70f 0.30g 0.05h 0.02i 0.02

Table 2: Stiffened Plate: cross-section geometry [m].

a

16

Point x[m] y[m] z[m]A a/2 0 h/2B a/2 b/4 h/2C a/2 b/2 h/2

D a/23

4b h/2

E a/2 b h/2A’ 0 b/2 h/2B’ a/4 b/2 h/2

D’3

4a b/2 h/2

E’ a b/2 h/2

Table 3: Stiffened Plate: Position of reference points.

Model DOF Point− C Point−B Point−A′ Point−B′

FEM3D−REF 724299 -4.216 -2.047 +0.239 -2.055

FEM3D 14325 -3.819(−10.4%) -2.258(+10.3%) +0.242(+1.7%) -1.768(−14.0%)

TE 8 3375 -3.273(−23.2%) -1.557(−23.9%) -0.590(−346.9%) -1.961(−4.3%)

TE 10 4950 -3.665(−14.0%) -1.752(−14.4%) +0.138(−42.3%) -2.026(−1.4%)

P − 1 2325 -2.192(−48.6%) -0.991(−51.6%) -1.499(−727.2%) -1.528(−25.6%)

P − 2 14325 -4.186(−1.8%) -2.004(−2.1%) +0.247(+3.3%) -2.023(−1.6%)

P − 3 16275 -4.216(−1.1%) -2.014(−1.6%) +0.238(−0.4%) -2.017(−1.8%)

P − 4 21201 -4.203(−1.4%) -2.018(−1.4%) +0.244(+2.1%) -2.034(−1.0%)

( )(∗%) : ∗ percentage different with respect to FEM3D−REF Model.

Table 4: Stiffened Plate: Vertical displacement × 10−3 [m].

Model DOF Point− α Point− β Point− γ Point− δx = a/2 , y = b/2

FEM3D−REF 724299 -53.14 +13.05 +74.23 +118.13

FEM3D 14325 -24.07 (−54.7%) +10.81 (−17.2%) +39.87 (−46.3%) +70.53 (−40.3%)

P − 2 14325 -50.77 (−4.5%) +12.24 (−6.2%) +72.89 (−1.8%) +121.11 (+2.5%)

P − 3 16275 -56.49 (+6.3%) +14.17 (+8.6%) +71.15 (−4.1%) +121.06 (+2.5%)

P − 4 21201 -52.86 (−0.5%) +12.69 (−2.8%) +72.70 (−2.1%) +120.19 (+1.7%)

( )(∗%) : ∗ percentage different with respect to FEM3D−REF Model.

Table 5: Stiffened Plate: value of σyy [MPa] at the selected points.

Model DOF Point− γx = a/2 , y = (3/4)b

FEM3D−REF 724299 -2.95

FEM3D 14325 -5.12 (+73.6%)

P − 2 14325 -2.54 (−13.9%)

P − 3 16275 -3.03 (+2.7%)

P − 4 21201 -2.98 (+1.0%)

( )(∗%) : ∗ percentage different with respect to FEM3D−REF Model.

Table 6: Stiffened Plate: values of τyz [MPa] at the central stringer.

17

Model DOF Ax Ex Az Ez

ux/z × 10−3[m]FEM3D 346437 +0.1710 +0.1710 -0.1710 +0.1710

FEM2D−1D 636 +0.2050(−19.9%) +0.2050(+19.9%) -0.2050(+19.9%) +0.2050(+19.9%)

S − 1 2349 +0.0167(−90.2%) +0.0167(−90.2%) -0.0170(−90.1%) +0.0170(−90.1%)

S − 2 2997 +0.0707(−58.7%) +0.0707(−58.7%) -0.0703(−58.9%) +0.0703(−58.9%)

S − 3 3645 +0.1242(−27.4%) +0.1242(−27.4%) -0.1245(−27.2%) +0.1245(−27.2%)

S − 4 5589 +0.1586(−7.3%) +0.1586(−7.3%) -0.1586(−7.3%) +0.1586(−7.3%)

( )(∗%) : ∗ percentage different to FEM3D

Table 7: Stiffened Shell: displacement at the selected points at y = b/2.

Model DOF A B C DFEM3D 346437 -4975 -4222 +1187 +5758

FEM2D−1D 636 -1593(−68.0%) -927 (−78.0%) ** **

S − 1 2349 -1181(−76.3%) -946 (−77.6%) +80 (−93.3%) +889 (−84.6%)

S − 2 2997 -2998(−39.7%) -2412(−42.9%) +259 (−78.4%) +2719(−52.8%)

S − 3 3645 -4116(−17.3%) -3475(−17.7%) +629 (−47.0%) +4598(−20.1%)

S − 4 5589 -4542(−8.7%) -3900(−7.6%) +1135(−4.4%) +6189(7.5%)

( )(∗%) : ∗ percentage different to FEM3D

∗∗ : Not expected from the model

Table 8: Stiffened Shell: value of σyy in [KPa] along the stringer at y = b/2.

Model DOF βx βz αz

y = b/2FEM3D 986976 -1.659 +1.5450 -5.489

FEM2D−1D 4992 -1.621(−2.3%) +1.505(−2.6%) -5.6020(+2.1%)

C − 1 9360 -1.547(−6.7%) +1.390(−10.0%) -4.225(−23.0%)

C − 2 14976 -1.617(−2.5%) +1.499(−2.94%) -5.124(−6.63%)

C − 3 18000 -1.552(−6.4%) +1.392(−9.9%) -4.244(−22.7%)

C − 4 28800 -1.623(−2.1%) +1.503(−2.7%) -5.151(−6.2%)

y = b/4FEM3D 986976 -0.907 +0.763 -1.432

FEM2D−1D 4992 -0.929 (+2.4%) +0.781 (+2.4%) -1.475 (−5.7%)

C − 1 9360 -0.801(−11.7%) +0.658 (−13.74%) -1.350 (0.6%)

C − 2 14976 0.874(−3.7%) +0.732(−4.0%) -1.440 (−7.5%)

C − 3 18000 -0.820(−9.6%) +0.672(−11.9%) -1.324 (−2.5%)

C − 4 28800 -0.881(−2.9%) +0.734(−3.3%) -1.396 (−6.2%)

( )(∗%) : ∗ percentage different to FEM3D

ux × 10−3 [m]

Table 9: Stiffened Cylinder: Displacements at the selected point at y=b/2.

Model FEM3D FEM2D−1D C − 1 C − 2 C − 3 C − 4DOF 986976 4992 9360 14976 18000 28800

A 0.000 +0.0000(+0.0%) 0.000 (0.0%) 0.000 (0.0%) 0.000 (0.0%) 0.000 (0.0%)

B -0.277 -0.2840(+2.5%) -0.257 (−7.4%) -0.251 (−9.2%) -0.265 (−4.5%) -0.269 (−2.8%)

C -0.961 -0.9870(+2.7%) -0.907 (−5.7%) -0.948 (−1.4%) -0.905 (−5.8%) -0.940 (−2.1%)

D -2.012 -2.0760(+3.2%) -1.726 (−14.2%) -1.830 (−9.0%) -1.822 (−9.4%) -1.947 (−3.2%)

E -3.700 -3.7980(+2.6%) -3.089 (−16.5%) -3.576 (−3.4%) -2.909 (−21.4%) -3.478 (−6.0%)

F -5.489 -5.6020(+2.1%) -4.225 (−23.0%) -5.125 (−6.6%) -4.245 (−22.7%) -5.151 (−6.1%)

( )(∗%) : ∗ Percentage different to FEM3D

Note : Simmetric structure

Table 10: Stiffened Cylinder: vertical displacement uz × 10−3[m] at point α along the beam axis.

18

Model DOF β αFEM3D 986976 +1019481 -7690953

FEM2D−1D 4992 +1408061(+38.1%) -8951554(+16.4%)

C − 1 9360 +914739(−6.2%) -6219046(−19.1%)

C − 2 14976 +1166628(+19.7%) -7215811(−6.2%)

C − 3 18000 +731990 (−25.0%) -7316460(−4.9%)

C − 4 28800 +1105451 (+13.4%) -8459448(10.0%)

( )(∗%) : ∗ percentage different to FEM3D

Table 11: Stiffened Cylinder: values of σyy [Pa] at the selected points at y = b/2.

Figures

Skin

Rib

Stringers

Figure 1: Example of reinforced structures

Figure 2: Example of reinforced structure, on the right can be seen a single stiffener and a portion of skin.

19

Figure 3: Comparisons of different models used in the analysis of a portion of reinforced structure.

Figure 4: Three-dimensional plate model.

(a) Cross-section geometry.

0.0

7 [

m]

0.0

5 [

m]

0.0

25 [

m] δ

β

γ

αz

x

(b) Stringer geometry and reference points.

Figure 5: Cross-section geometrical properties.

20

P

z

yA B C D E

Figure 6: Beam configuration: Geometrical and mechanical boundary conditions.

(a) Global structure. (b) Local view.

Figure 7: Solid FE Refined Model, 724299 DOFs.

(a) Global structure. (b) Local view.

Figure 8: Solid FE Model, 14325 DOFs.

Figure 9: Model P-1 Cross-section.

Figure 10: Model P-2 and P-4 Cross-section.

Figure 11: Model P-3 Cross-section.

21

Figure 12: Model P-1: Nodes distribution.

Figure 13: Model P-2: Nodes distribution.

Figure 14: Model P-3: Nodes distribution.

-5

-4

-3

-2

-1

0

0 0.5 1 1.5 2

Ver

tical

Dis

plac

emen

t [m

m]

y-axis [mm]

FEM-3D_REF

FEM-3D

TE-10

Model P-1

Model P-4

(a) Axial Displacement at x = a/2.

-5

-4

-3

-2

-1

0

0 0.5 1 1.5 2

Ver

tical

Dis

plac

emen

t [m

m]

x-axis [mm]

FEM-3D_REF

FEM-3D

TE-10

Model P-1

Model P-4

(b) Cross-section displacement at y = b/2.

Figure 15: Stiffened plate: Displacements field.

22

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

-60 -40 -20 0 20 40 60 80 100 120 140

z-ax

is [m

m]

Sigmayy [MPa]

FEM-3DREF

FEM-3D

TE-10

Model P-1

Model P-4

(a) σyy in x = a/2 and y = b/2.

0

0.01

0.02

0.03

0.04

0.05

-6 -5 -4 -3 -2 -1 0

z-ax

is [m

m]

Tauyz [MPa]

FEM-3DREF

FEM-3D

TE-10

Model P-1

Model P-4

(b) τyz along the central stringer, in x = a/2 and y = (3/4)b.

Figure 16: Stiffened plate: Stress field through the thickness.

-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 17: 3D shell model

AB

δ

CD

F1

F2

r

E

δ

z

x

p

p2

m

Figure 18: Stiffened shell: geometry and load configuration in y=b/2.

23

(a) Global structure. (b) Local view.

Figure 19: Solid FE Model, 346437 DOFs.

(a) Global structure. (b) Local view.

Figure 20: Shell/Beam FE Model.

Figure 21: Model S-1: Cross-section and nodes distribution.

24

Figure 23: Model S-3: Cross-section and nodes distribution.

Figure 24: Model S-4: Cross-section and nodes distribution.

25

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-1.5 -1 -0.5 0 0.5 1 1.5

z-co

ordi

nate

x-coordinate

undeformed

FEM 3D

FEM 2D-1D

Model S-4

Model S-2

Figure 25: Stiffened shell: Displacment in y=b/2 (Amplificator factor × 1000).

0.665

0.67

0.675

0.68

0.685

0.69

0.695

0.7

0.705

0.71

-6 -4 -2 0 2 4 6 8

z [m

m]

Sigma_yy [MPa]

FEM 3D

FEM 2D-1D

Model S-1

Model S-2

Model S-3

Model S-4

Figure 26: Stiffened shell: Stress analysis along the stringer there is applied F1, y=b/2.

26

-2.5-2

-1.5-1

-0.5 0

0.5x 0

2 4

6 8

10 12

14 16

y

-1

-0.5

0

0.5

1

z

Figure 27: 3D cylinder model.

z

y

F

A B C D E

b

G H I L M

F1

F2

ε ε=1.5 [m]

Figure 28: Beam configuration: geometrical and mechanical boundary conditions.

27

β

α

δ=45°

Figure 29: Reference points on a quarter of cylinder.

(a) Global structure. (b) Local view.

Figure 30: Solid FE Model, 346437 DOFs.

(a) Global structure. (b) Local view.

Figure 31: Shell-Beam FE Model.

28

Figure 32: Model C-1: Cross-section and nodes distribution.

Figure 33: Model C-2: Cross-section and nodes distribution.

29

z-c

oord

inate

x-coordinate

FEM 3D

FEM 2D-1D

Model C-3

Model C-4

Figure 34: Stiffened Cylinder: Cross-sectional deformation in y=b/2, scale factor × 102.

z-c

oord

inate

x-coordinate

FEM 3D

FEM 2D-1D

Model C-3

Model C-4

Figure 35: Stiffened Cylinder: Cross-sectional deformation in y=b/4, scale factor × 302.

30

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0 2 4 6 8 10 12 14

z [m

]

y [m]

FEM 3D

FEM 2D-1D

Model C-1

Model C-2

Model C-3

Figure 36: Stiffened Cylinder: Vertical displacement at point α along the y-axis.

31