Research Article  Open Access
Joseph Nkongho Anyi, Robert Nzengwa, Jean Chills Amba, Claude Valery Abbe Ngayihi, "Approximation of Linear Elastic Shells by Curved Triangular Finite Elements Based on Elastic Thick Shells Theory", Mathematical Problems in Engineering, vol. 2016, Article ID 8936075, 12 pages, 2016. https://doi.org/10.1155/2016/8936075
Approximation of Linear Elastic Shells by Curved Triangular Finite Elements Based on Elastic Thick Shells Theory
Abstract
We have developed a curved finite element for a cylindrical thick shell based on the thick shell equations established in 1999 by Nzengwa and Tagne (NT). The displacement field of the shell is interpolated from nodal displacements only and strains assumption. Numerical results on a cylindrical thin shell are compared with those of other wellknown benchmarks with satisfaction. Convergence is rapidly obtained with very few elements. A scaling was processed on the cylindrical thin shell by increasing the ratio (half the thickness over the smallest radius in absolute value) and comparing results with those obtained with the classical KirchhoffLove thin shell theory; it appears that results diverge at because of the significant energy contribution of the change of the third fundamental form found in NT model. This limit value of the thickness ratio which characterizes the limit between thin and thick cylindrical shells differs from the ratio 0.4 proposed by Leissa and 0.5 proposed by Narita and Leissa.
1. Introduction
In the field of structural mechanics the word shell refers to a spatial, curved structural member. The very high structural and architectural potential of shell structures is used in various fields of civil, architectural, mechanical, aeronautical, and marine engineering. The strength of the doublecurved structure is efficiently and economically used, for example, to cover large areas without supporting columns [1]. In addition to the mechanical advantages, the use of shell structures leads to aesthetic architectural appearance [2, 3]. Examples of shells used in civil and architectural engineering are shell roofs, liquid storage tanks, silos, cooling towers, containment shells of nuclear power plants, and arch dams. Piping systems, curved panels, pressure vessels, bottles, buckets, and parts of cars are examples of shells used in mechanical engineering [4]. In aeronautical and marine engineering, shells are used in aircrafts, space crafts, missiles, ships, and submarines. Because of the spatial shape of the structure the behavior of shell structures is different from the behavior of beam and plate structures [4].
The considerable effort in the development of rigorous shell theories dates back to the early twentieth century [5–7]. These shell theories reduce a basically threedimensional problem to a twodimensional one. Nevertheless, the analysis of shells with the aid of such theories involves complicated differential equations, which either cannot be solved at all [8–10], or whose solution requires the use of highlevel mathematics unfamiliar to structural engineers. Therefore, many approximate shell theories have been developed, mainly on the assumption that the shell is thin, and to obtain generic analysis tools obviously some accuracy had to be traded for convenience and simplicity [4, 11]. Hence, it is not surprising that the development of the numerical formulations since the 1950s has led to a gradual cessation of attempts to find closedform solutions to rigorous formulations [4]. But, with today’s availability of greatly increased computing power (also since the midtwentieth century), completeness rather than simplicity is given more emphasis.
In computational mechanics, many numerical techniques are developed to solve governing equations of shells. We can identify for this purpose the generalized differential quadrature finite elements (DGQFEM) [11–13], discrete singular convolution method (DSC) [14], the finite difference (FD), the discrete volume finite element (DVFEM), and the finite elements method (FEM) under which solidshells finite elements (SSFEM) and mixt finite element (MFEM) are found [15–18]. Our approach is found under the MFEM category in which threedimensional shell parameters are reduced to the midsurface of the shell of thickness . Some finite elements in this category describe well the behavior of shells under several loads. All these models of finite elements are link with either KirchhoffLove (KL) or ReissnerMindlin (RM) hypothesis which neglected the effect of curvature in elastic stiffness [11, 19]. This is one of the main reasons why KL and RM theories of shells are incomplete and unsuitable to handle various types of shells [20]. Structure engineers emphasis more on simplifying assumptions than limit analysis. Because of the third fundamental form that has been neglected in their theoretical approach of shells governing equations [21] and severe assumption on shell thickness, some mathematical terms without evident mechanic signification are introduced in the stiffness matrix in order to improve their efficiency [6, 22].
After a brief presentation of Nzengwa and Tagne (NT) kinematic equations of elastic thick shells, we develop a curved cylindrical 3node finite element based on both KL and NT’s shell models using strain tensor interpolation assumption [23]. Next, through numerical implementation on classical benchmark, convergence of NT’s shell model is investigated. The deviation and the limit thickness ratio between the two theoretical approaches of thin shells and thick shells are established and discussed by applying successive scaling of a sample of thin shell.
2. Materials and Methods
2.1. NT’s TwoDimensional Elastic Thick Shells Model
The NT’s twodimensional model for linear elastic thick shells has been deduced from the threedimensional problem without any ad hoc assumption whether of geometrical or mechanical nature. The twodimensional equations are deduced by applying asymptotic analysis on a family of variational equations obtained from an abstract scaled shell through multiple scaling of the initial threedimensional equations. The theoretical elastic thick shells governing equations and limits displacements obtained from this approach (at the limit analysis) are more general because they contain additional terms to those found in KirchhoffLove model. The model also completes the thick shells theory of ReissnerMindlin [24].
For a threedimensional shell, the following bases are dual bases and defined on the midsurface also constitute dual bases of shell if . Let be the threedimensional displacement field vector; it is defined as follows in the local coordinate system: The strain tensor for a threedimensional shell readsFrom the limit analysis NT demonstrated that the displacements field satisfies the equation and the unique solution iswhere are local membrane displacement components and the transvers displacement:The general shell deformation tensor derived from the kinematics defined above iswhere is the membrane strain tensor, is the bending strain tensor, and is the tensor of Gauss curvature.
The strain tensor contains the change of the third fundamental form:
In KirchhoffLove thin shells theory, the shell is assumed to be thin. Following that assumption, the threedimensional displacements field vector is taken equal to the midsurface displacements field:It should be noted that the terms in the current model of thick shells disappear in the classical theory of thin shells because it is proportional to and considered small. The thin shell theory, with the above assumption, cannot perform good shells behavior because the thickness has a considerable influence on the shells behavior. The thick shell model used in this survey appears to be suitable to handle both thin and thick shells because of its completeness.
2.2. Curved Shell Elements
Curved shell elements are suitable to model the midsurface geometry more accurately. In the case of certain surfaces such as cylindrical shells, it means the exact description of the original surface. For more complicated cases, similar to the displacement field, the curvatures of the surface are approximated by interpolation functions. In this respect such elements belong to the parametric element types [25].
2.3. Triangular Curve Shell Element
The triangular shell element is described in Figure 1; it can be doublecurved (e.g., spherical structures) or singlecurved element (cylindrical structures). In accordance with the NT’s equations of the theory of elastic thick shells, the geometrical properties of the cylindrical shell are the following:where and are curvilinear coordinates, and , cylinder radius, and are global coordinates.
2.4. Displacement Field
Let be the global displacement field vector defined as follows: With respect to the cylinder’s geometry, the angles of rotations and the Gaussian curvatures are developed in function of membrane displacements and and the derivatives of the transverse displacement including curvature components :which leads towhere , and are geometrical properties of the midsurface of a shell within the respective coordinatelines and .
In the case of a cylindrical shell they are defined as follows [25]:The third component of rotation is added for a smooth transfer of element stiffness matrix from one system of coordinates to another.
2.5. Compatibility
We recall the strain components in thick shell theory found in (3) as follows:
The membrane deformation tensor and bending tensor are widely used in thin shell theory. But the tensor is neglected [24]. In classical thin shell and thick shell computation analysis, the contribution of this tensor (Gauss curvature tensor) to the deformation energy disappears in the stiffness matrix. As the shell becomes thicker, the contribution of Gauss curvature tensor in terms of energy is no more negligible as compared to that of the first two tensors used in classical thin shells. So this model is able to handle thick shells as well as the classical RM models.
We express the strain components using the parameters of the cylindrical shell’s midsurface. In detail, for the case of a cylindrical shell, we have
2.6. Interpolation
The approach we are using here is based on strain interpolation. This method consists in solving a system of differential equations from strain assumption. We have to find the displacements functions which perfectly capture the rigidlike motion; then particular displacements are calculated from approximation of the deformations components. These components are approximated such that membrane, shearing, and bending behavior are decoupled. So a pure bending state or a pure shearing state can be well represented. In this section we consider and
The rigidbodylike motion is defined for . It means that , , and .Let be an arbitrary function that can satisfy the following equations:Since our displacement field is of six components, three rotations and three translations, we need six independent parameters to define our rigid bodylike motion [18, 25].When (23) and (24) are taken into (18) and (19), it comes that the rigid bodylike motion is well handled by the following displacement components:The displacements found in (25) are expressed in matrix form:where . is the vector of unknown coefficient which captures the rigid bodylike motion: is the matrix of interpolation functions.
The general solution displacements field is the sum of the displacements field of rigid bodylike motion and that of a particular solution of the deformation :where
The strain components are interpolated in such a way that interference between shear strains and bending is not possible. The following displacement components satisfy strains defined above:with is the interpolation functions matrix related to the particular solution of displacements field.
From the total displacement vector field expression above, we have 18 unknowns. It then requires 18 displacement parameters to find all the above unknowns (see Figure 2).
So the total displacement vector over the triangular element is In the local (curvilinear) system of coordinates, each node has six (06) components of displacement.
2.7. Stiffness Matrix
In order to calculate the stiffness matrix, we have to formulate the variational problem over a domain. Let be the border of the domain; then let be partitioned in two and the border of the shell with , and . We define and . We suppose here that the shell is clamped on and loaded by volume and surface forces as stated above; the threedimensional variational equation related to the equilibrium is is volume forces in the domain and is surface forces in the domain .
The constitutive law of the linear elastic homogenous material is where and are Lamé coefficients which depend on intrinsic properties of materials.
Replacing (34) in (33), the problem is stated as follows:The strain tensor is defined at (7), if substitution is done in the left side of the above equation, it can now be written:Naturally at the limit analysis. Then at the midsurface, . By using Taylor’s expansion on , and truncating at , we obtain the best firstorder twodimensional
Remark 1. For Taylor’s expansion used above, if we truncate (37) for , the bilinear continuous form will take the value:which is found in the literature to be KL shell deformation energy. It is included in which is the NT shell deformation energy. In what follows, we compute both and which are, respectively, stiffness matrix related to and .
For the computing aim, is transformed For this purpose, components of strain tensor are expressed as a product of the matrix of interpolation functions’ gradient and that of the total displacement vector over the domain. From (39), the stress tensor components are deduced in function of the product above.Also, the bending stress and strain are expressed:Finally as previous, the Gauss curvature tensor is expressed asTotal displacement is stated as the sum of particular solution and homogeneous solution of our system of strain differential equations.where is the matrix of interpolation functions andThe lefthand side of the variational problem readsThis is written as follows:One can deduct from (47) what follows: where is the global stiffness matrix. The midsurface is symmetric in shell thickness: that is, the midsurface is at from the top surface and at from the bottom one.The righthand side of the best firstorder twodimensional variational equation readswhere and are defined as in [24] and is the force vector from distributed load .By equating to , we obtain the structural equilibrium equation over the whole area as follows:Over a singlecurved triangular element of area the elementary stiffness matrix and elementary force vectors can be locally implemented.By regular assembling process of elementary stiffness matrix and elementary force vectors , we then expressed the global stiffness matrix and force vector over the whole area as follows: is total number of elements.
3. Numerical Study
A numerical study which aims to quantitatively investigate the accuracy of different thick shell theories for the linear elastic analysis of cylindrical structures and to provide a basis for the longterm analysis is presented.
One of the problems frequently used to evaluate the performances of a shell element is that of the cylindrical roof (of Scordelislo) [4, 26] subjected to its selfweight described in Figure 3. The flat rims are free and the curved edges rest on rigid diaphragms in their plans. The geometrical and mechanical characteristics are indicated for = 0.01, = 200. The length of the cylinder = 600 cm; its radius is = 300 cm; the angle subtended by the roof is = 80 deg; the thickness is cm; Young’s modulus is Pa and Poisson’s ratio is .
3.1. Convergence
The quarter of the roof is discretized by considering triangular singlecurved elements with elements on edges and . The transverse displacements at and are plotted (Figure 4) and compared with other models of finite elements: see Figure 5.
(a)
(b)
The computed deformed limit surface is shown in Figure 4. Here, displacementconvergence results are shown in Figure 5 and Table 1. In the diaphragm case, we monitor the displacements under the selfweight at points and . The convergence properties of the method are evident from Figure 5 and Table 1. As may be seen, the CSFE3 converges as well as both the semifinite elements (SFE) and the DKT elements for the displacements under the loads.

3.2. Scaling and Variation of Displacements
A successive scaling is carried out on the thickness ratio over the Scordelislo roof. We have used the following range of ratios: 0.1; 0.2; 0.3; 0.325; 0.4; 0.5, because it certainly belongs to both thin shells and thick shells ranges of thickness. The radius is constant while the thickness varies with the ratio. The scaled shell obtained here has been implemented using both KL theory and that proposed by NT. The results are plotted for each ratio as shown in Figure 6.
(a)
(b)
(c)
(d)
(e)
(f)
We observe from the diagrams shown in Figure 6 that for the thickness ratios = 0.1, 0.2, and 0.3, the transverse displacements at point are the same for singlecurved triangular computation of both KL and NT models (see Figures 6(a), 6(b), and 6(c)). One can verify that, at any other points of the roof structure, these same observations are satisfied. For thickness ratios greater or equal to 0.325, we observe that transverse displacements computed from the shells equations of KL and NT are not the same (see Figures 6(d), 6(e), and 6(f)). We can still investigate the exact value in from which displacements results issued from these two approaches of shells theories diverge in a cylindrical wall and probably other shells.
3.3. Deviations
NT’s equations for elastic thick shell are more general [24] and with regard to that, we assume the displacements issued from this model of thick shell to be more accurate. We can then investigate the deviations encountered when using KL model to evaluate a shell. The investigation consists in calculating the deviations (difference between KL displacement and NT displacement) for each meshstep (number of elements per half side) with respect to the series of thickness ratios = 0.1; 0.2; 0.3; 0.325; 0.4; and 0.5 at the same point. This investigation was carried out on two different points and (see Figure 3) and the results are plotted in Figures 7(a) and 7(b).
(a)
(b)
From diagrams shown in Figures 7(a) and 7(b), we observe that (i) at points and of our cylindrical roof, deviations are encountered from thickness ratios above 0.3; (ii) deviations increase with meshsteps at point and decrease at point .
4. Discussion
We have investigated in this framework the influence of the third fundamental form proportional to (with ) included in NT’s model of elastic thick shells. This model was implemented using cylindrical shell triangular finite elements (CSFE3). Based on the results obtained, the following comments can be made.
The convergence studies of CSFE3 show that this curved finite element of three nodes only converges as well as SFE, DKT12, and DKT18 which are robust and greedy (memory) [27]. The last three finite elements models are based on theoretical approach of KirchhoffLove (thin shells) or ReissnerMindlin (Thick shells) which neglect the third fundamental form in their shell kinematic equations. In addition to that, these classical shell theories are based on two powerful assumptions of KL or RM [28, 29].
The theoretical equations of thick and thin shells issued from these assumptions can be questioned because the transverse fiber (normal to the reference surface) has been forced to follow a specific behavior without any scientific or technical background that justifies their kinematic equations [20, 24, 26]. Consequently some quantities without evident mechanical significations are added in the formulation of finite elements models (like DKT, SFE …) in order to improve computational accuracy [7, 26].
In the contrary, our finite element (CSFE3) is based on NT’s theoretical model of thick shells which is deduced from = 0 obtained at limits analysis of 3dimensional shell equations [24]. Resultant kinematic equations of shell here are more general since they contain additional terms to those found in classical equations of shells. They are mathematically and technically justified without ad hoc assumption on transverse fiber’s behavior. This can explain why a simple 3node curved triangular finite element obtained efficient convergent result. The investigation with a 6node curved triangle will surely improve the convergence rate [7, 26].
The successive scaling over the benchmark structure has presented divergence of displacements results as increases for KL kinematic model of shells and that of NT. This is due to the influence of the additional terms found in NT’s shell theory in the deformation energy of the structure. For homogeneous and isotropic elastic shells with a midsurface between and , the total deformation energy contains membrane stiffness , bending stiffness , coupled stiffnesses and (membrane and Gaussian bending), and finally Gaussian stiffness .
For ReissnerMindlin and KL classical model of shells, the global stiffness matrix contains only and while in NT’s model, the global stiffness matrix contains additional terms () to those found in . The proportion of energy contributed by each stiffness matrix to the global deformation energy is as follows: ; ; ; and . We monitored that for thickness ratio less or equal to 0.3, that is, , the couple stiffness and Gaussian stiffness disappear in the global deformation energy because they are inversely proportional, respectively, to 1000 and 10.000. We then obtain . This explains why deviations are zero for thickness ratio less than 0.3 (see Figure 7). For thickness ratio greater or equal to displacements results encountered for the two models are different because the coupled stiffness and the Gaussian stiffness which are now inversely proportional, respectively, to 100 and 1000 cannot disappear anymore in the global deformation energy of the structure. This additive proportion of energy into the global deformation energy found in this model improves total stiffness and increases the rigidity of the structure.
5. Conclusion
A finite element model for predicting the elastic thick shell behavior of cylindrical structure has been developed in this work. It has been quantitatively shown that the thickness ratio (thickness on radius) plays important role in their elastic behavior and structural safety. Hence, a reliable control of this parameter is required. NT’s theoretical approach for the modelling of the displacement and stains in cylindrical structures have been examined and compared with that of KL for the case of selfweight loading cylindrical roof.
The comparison has revealed that NT’s shell theory is suitable for the analysis of cylindrical thin and thick structures from the perspectives of both accuracy and simplicity. A 3node cylindrical shell finite element has been developed for the simulation of static behaviour of shells in general, which requires the ability of well handling thickness ratios from the vicinity of zero to one.
The ability of the CSFE3 model to deal with a wide range of ratios has been demonstrated through numerical examples.
It has been found that the use of KL theoretical approach based on those assumptions leads to inaccurate results as the thickness becomes greater than the square root of 0.1.
It has also been found that above the value square root 0.1, the deviations linearly increase with the increase of the thickness ratio.
The issues addressed in this work highlight the need for NT’s approach to be implemented for the elastic shells analysis of cylindrical structures.
The analytical model and the numerical results presented here contribute to the establishment of a foundation of theoretical knowledge required for reliable analysis, effective design, and safe use of cylindrical walled structures.
Finally, the paper provides a platform for doublecurved finite elements model to be developed for studying the influence of thickness ratio on all shell geometries.
Competing Interests
The authors declare that they have no competing interests.
References
 H. Stolarski and T. Belytschko, “Membrane locking and reduced integration for curved elements,” Journal of Applied Mechanics, vol. 49, no. 1, pp. 172–176, 1982. View at: Publisher Site  Google Scholar
 J. F. Doyle, Nonlinear Analysis of ThinWalled Structures: Statics, Dynamics, and Stability, Springer Science & Business Media, 2013.
 J. R. Vinson, The Behavior of Thin Walled Structures: Beams, Plates, and Shells, Springer, Berlin, Germany, 2012.
 J. Hoefakker, Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks, Eburon Uitgeverij BV, 2010.
 W. T. Koiter and J. G. Simmonds, “Foundations of shell theory,” in Theoretical and Applied Mechanics, E. Becker and G. Mikhailov, Eds., pp. 150–176, Springer, Berlin, Germany, 1973. View at: Google Scholar
 D. Chapelle and K. J. Bathe, “Fundamental considerations for the finite element analysis of shell structures,” Computers & Structures, vol. 66, no. 1, pp. 19–36, 1998. View at: Publisher Site  Google Scholar
 D.N. Kim and K.J. Bathe, “A triangular sixnode shell element,” Computers & Structures, vol. 87, no. 2324, pp. 1451–1460, 2009. View at: Publisher Site  Google Scholar
 M. Jawad, Theory and Design of Plate and Shell Structures, Springer, Berlin, Germany, 2012.
 M. Z. Nejad, M. Jabbari, and M. Ghannad, “Elastic analysis of axially functionally graded rotating thick cylinder with variable thickness under nonuniform arbitrarily pressure loading,” International Journal of Engineering Science, vol. 89, pp. 86–99, 2015. View at: Publisher Site  Google Scholar
 H. Zeighampour and Y. Tadi Beni, “Cylindrical thinshell model based on modified strain gradient theory,” International Journal of Engineering Science, vol. 78, pp. 27–47, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 F. Tornabene, A. Liverani, and G. Caligiana, “General anisotropic doublycurved shell theory: a differential quadrature solution for free vibrations of shells and panels of revolution with a freeform meridian,” Journal of Sound and Vibration, vol. 331, no. 22, pp. 4848–4869, 2012. View at: Publisher Site  Google Scholar
 F. Tornabene, E. Viola, and D. J. Inman, “2D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annul plate structures,” Journal of Sound and Vibration, vol. 328, no. 3, pp. 259–290, 2009. View at: Publisher Site  Google Scholar
 E. Viola, F. Tornabene, and N. Fantuzzi, “Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape,” Composite Structures, vol. 106, pp. 815–834, 2013. View at: Publisher Site  Google Scholar
 E. Pindza and E. Maré, “Discrete singular convolution method for numerical solutions of fifth order kortewegde vries equations,” Journal of Applied Mathematics and Physics, vol. 1, no. 7, pp. 5–15, 2013. View at: Publisher Site  Google Scholar
 F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, Berlin, Germany, 2012.
 T. NguyenThoi, P. PhungVan, C. ThaiHoang, and H. NguyenXuan, “A cellbased smoothed discrete shear gap method (CSDSG3) using triangular elements for static and free vibration analyses of shell structures,” International Journal of Mechanical Sciences, vol. 74, pp. 32–45, 2013. View at: Publisher Site  Google Scholar
 R. Echter, B. Oesterle, and M. Bischoff, “A hierarchic family of isogeometric shell finite elements,” Computer Methods in Applied Mechanics and Engineering, vol. 254, pp. 170–180, 2013. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. NguyenXuan, L. V. Tran, C. H. Thai, and T. NguyenThoi, “Analysis of functionally graded plates by an efficient finite element method with nodebased strain smoothing,” ThinWalled Structures, vol. 54, pp. 1–18, 2012. View at: Publisher Site  Google Scholar
 F. Tornabene, A. Liverani, and G. Caligiana, “Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method,” International Journal of Mechanical Sciences, vol. 61, no. 1, pp. 71–87, 2012. View at: Publisher Site  Google Scholar
 Y.G. Fang, J.H. Pan, and W.X. Chen, “Theory of thickwalled shells and its application in cylindrical shell,” Applied Mathematics and Mechanics, vol. 13, no. 11, pp. 1055–1065, 1992. View at: Publisher Site  Google Scholar
 D. Chapelle and K.J. Bathe, The Finite Element Analysis of ShellsFundamentals, Springer, Berlin, Germany, 2010.
 M. Bernadou, P. G. Ciarlet, and B. Miara, “Existence theorems for twodimensional linear shell theories,” Journal of Elasticity, vol. 34, no. 2, pp. 111–138, 1994. View at: Publisher Site  Google Scholar  MathSciNet
 S. Sævik, “Theoretical and experimental studies of stresses in flexible pipes,” Computers & Structures, vol. 89, no. 2324, pp. 2273–2291, 2011. View at: Publisher Site  Google Scholar
 R. Nzengwa and B. H. T. Simo, “A twodimensional model for linear elastic thick shells,” International Journal of Solids and Structures, vol. 36, no. 34, pp. 5141–5176, 1999. View at: Publisher Site  Google Scholar
 I. Moharos, I. Oldal, and A. Szekrényes, Finite Element Methode, Typotex Publishing House, Budapest, Hungary, 2012.
 M. Bernadou, P. M. Eiroa, and P. Trouve, “On the approximation of general linear thin shell problems BYD. KT methods,” Computation and Applied Mathematics, vol. 10, article 103, 1991. View at: Google Scholar
 L. Della Croce and P. Venini, “Finite elements for functionally graded ReissnerMindlin plates,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 9–11, pp. 705–725, 2004. View at: Publisher Site  Google Scholar
 E. Carrera, G. Giunta, P. Nali, and M. Petrolo, “Refined beam elements with arbitrary crosssection geometries,” Computers & Structures, vol. 88, no. 56, pp. 283–293, 2010. View at: Publisher Site  Google Scholar
 E. Carrera, S. Brischetto, M. Cinefra, and M. Soave, “Effects of thickness stretching in functionally graded plates and shells,” Composites Part B: Engineering, vol. 42, no. 2, pp. 123–133, 2011. View at: Publisher Site  Google Scholar
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Copyright © 2016 Joseph Nkongho Anyi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.