 Research article
 Open Access
Best theory diagrams for multilayered structures via shell finite elements
 Marco Petrolo^{1}Email authorView ORCID ID profile and
 Erasmo Carrera^{1, 2}
https://doi.org/10.1186/s4032301901298
© The Author(s) 2019
 Received: 1 November 2018
 Accepted: 9 March 2019
 Published: 25 March 2019
Abstract
Composite structures are convenient structural solutions for many engineering fields, but their design is challenging and may lead to oversizing due to the significant amount of uncertainties concerning the current modeling capabilities. From a structural analysis standpoint, the finite element method is the most used approach and shell elements are of primary importance in the case of thin structures. Current research efforts aim at improving the accuracy of such elements with limited computational overheads to improve the predictive capabilities and widen the applicability to complex structures and nonlinear cases. The present paper presents shell elements with the minimum number of nodal degrees of freedom and maximum accuracy. Such elements compose the best theory diagram stemming from the combined use of the Carrera Unified Formulation and the Axiomatic/Asymptotic Method. Moreover, this paper provides guidelines on the choice of the proper higherorder terms via the introduction of relevance factor diagrams. The numerical cases consider various sets of design parameters such as the thickness, curvature, stacking sequence, and boundary conditions. The results show that the most relevant set of higherorder terms are thirdorder and that the thickness plays the primary role in their choice. Moreover, certain terms have very high influence, and their neglect may affect the accuracy of the model significantly.
Keywords
 Shell
 Composites
 Finite element
 Higherorder theories
 Best theory diagram
Introduction
The finite element method (FEM) is one of the most common tools for the design of structures and makes use of threedimensional (3D), 2D and 1D elements to solve a broad variety of linear and nonlinear structural problems. 2D and 1D elements, although less accurate than 3D, can lead to reduced computational costs. 2D models are referred to as shell and plate finite elements (FE) and can model metallic and composite thinwalled structures. 2D models available in commercial codes rely on the classical theories of structures [1–3]. In a 2D model, the primary unknown variables depend on two coordinates, x, and y. On the other hand, assumed fields define the unknown distributions along the thickness direction, z. A structural theory has a given expansion of the unknowns along z. Such expansions characterize the accuracy of a theory and its computational costs. For instance, in FEM, the expansion terms, referred to as generalized unknown variables, define the nodal degrees of freedom (DOF) of the model [4].
 1.
Moderately thick or thick structures, i.e., \(\frac{a}{h}\) < 50, where a is the characteristic length of the structure and h is the thickness.
 2.
Materials with high transverse deformability, e.g., common orthotropic materials, in which \(\frac{E_L}{E_T}\), \(\frac{E_L}{E_z} > 5\), and \(\frac{G}{E_L}<\frac{1}{10}\), where E and G are the Young and shear moduli and L is the fiber direction of the fiber and T, z are perpendicular to L.
 3.
Transverse anisotropy due, for instance, to the presence of contiguous layers with different properties.
The development of structural theories, i.e., the selection of the expansion terms, can follow two main approaches, namely, the axiomatic and asymptotic ones. The axiomatic method introduces expansions related to hypotheses on the mechanical behavior to reduce the mathematical complexity of the 3D differential equations of elasticity as in the case of classical theories [1–3]. The asymptotic method introduces a mathematically rigorous expansion having known accuracy if compared to the 3D exact solution [7, 8]. Axiomatic models are easier than asymptotic ones to implement but may miss fundamental expansion terms. Asymptotic models are more rigorous but the simultaneous consideration of multiple problem parameters, e.g., thickness and orthotropic ratio, may be cumbersome.
Over the last decades, the research activity has focused on the development of shell and plate models incorporating the effects mentioned above [9, 10]. Most recent efforts describe well the open research topics and refinement techniques related to shells, such as, improvements of classical models [11] and higherorder models [12–14]; asymptotic approaches [15]; improvement of FE performances regarding membrane and shear locking [16–21], mesh accuracy [22], and distortion [23]; improved modeling of the interlaminar shear stresses [24]; LayerWise (LW) models [25, 26]; Zig–Zag models [27, 28]; mixed formulations [29–31]; variable kinematics finite elements with multifield effects [32]; extensions to nonlinear problems [33, 34] and peridynamics [35]; innovative solution schemes such as the numerical manifold method [36].
Via the axiomatic/asymptotic method (AAM), this paper presents best theory diagrams (BTD) [37] providing the shell finite elements with the minimum computational cost and maximum accuracy for a given problem. In [37], the results stemmed from strongform solutions restricting the analysis concerning boundary conditions and stacking sequences. This paper is the first contribution based on shell finite elements allowing the generation of BTD for various boundary conditions and stacking sequences. Moreover, this paper presents a novel metric referred to as Relevance Factor (RF) to evaluate the influence of terms and outline guidelines for the proper choice of the expansion terms. This paper is organized as follows: the governing equations and the methodology are in “Finite element formulation” and “Best theory diagram” sections, then, the “Results” and “Conclusions” sections follow.
Finite element formulation
Best theory diagram
 1.
Definition of parameters such as geometry, boundary conditions, materials, and layer layouts.
 2.
Axiomatic choice of a starting theory and definition of the starting nodal unknowns. Usually, the starting theory provides 3Dlike solutions.
 3.
The CUF generates the governing equations for the theories considered. In particular, the CUF generates reduced models having combinations of the starting terms as generalized unknowns.
 4.
For each reduced model, the accuracy evaluation makes use of one or more control parameters, in this paper, the maximum transverse displacement.
Results
The numerical results focus on cases retrieved from [40]. The shell has a = b and \(\hbox {R}_{\alpha } = \hbox {R}_{\beta }\) = R. The load is bisinusoidal and applied on the top surface, \(\hbox {p}_z = \hat{p}_{z}\sin (\pi \alpha /\hbox {a})\sin (\pi \beta /\hbox {b})\). The material properties are \(\hbox {E}_1\)/\(\hbox {E}_2 = 25\), \(\hbox {G}_{12}\)/\(\hbox {E}_2 = \hbox {G}_{13}\)/\(\hbox {E}_2 = 0.5\), \(\hbox {G}_{13}\)/\(\hbox {E}_2 = 0.2\), \(\nu = 0.25\). The finite element model of a quarter of shell has a \(4\times 4\) mesh as this discretization provides sufficiently accurate results [40]. In all cases, the BTD vertical axis ranges from 5 to 15 since, more often than not, models with 4 or less DOF provide very high errors and are not of practical interest.
Simplysupported, 0/90/0
0/90/0, \(\overline{u}_{z}\ (\hbox {z} = 0) = 100u_{z}\) \(\hbox {E}_{T}\) h\(^3\)/(\(\overline{p}_z\) a\(^4\))
BTD models for 0/90/0, \(\hbox {R/a} = 5\), \(\hbox {a/h} = 100\)
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 1.00\)  \(\hbox {RF}_1 = 0.94\)  \(\hbox {RF}_2 = 0.64\)  \(\hbox {RF}_3 = 0.48\)  \(\hbox {RF}_4 = 0.27\) 
BTD models for 0/90/0, \(\hbox {R/a} = 5\), \(\hbox {a/h} = 10\)
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 1.00\)  \(\hbox {RF}_1 = 0.82\)  \(\hbox {RF}_2 = 0.58\)  \(\hbox {RF}_3 = 0.67\)  \(\hbox {RF}_4 = 0.27\) 
BTD models for 0/90/0, \(\hbox {R/a} = 5\), \(\hbox {a/h} = 5\)
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
7  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \) 
5  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 1.00\)  \(\hbox {RF}_1 = 0.73\)  \(\hbox {RF}_2 = 0.45\)  \(\hbox {RF}_3 = 0.76\)  \(\hbox {RF}_4 = 0.39\) 
Errors for all 14 DOF models, 0/90/0, \(\hbox {R/a} =5, \hbox {a/h} = 5\)
\({\varvec{R/a}} = 100\)  \({\varvec{R/a}} = 10\)  \({\varvec{R/a}} = 5\)  

Inactive  Inactive  Inactive  
DOF  Error (%)  DOF  Error (%)  DOF  Error (%) 
\(\hbox {u}_{z5}\)  \(4.1 \times 10^{6}\)  \(\hbox {u}_{x3}\)  \(9.5 \times 10^{4}\)  \(\hbox {u}_{x3}\)  \(4.6 \times 10^{5}\) 
\(\hbox {u}_{z4}\)  \(1.5 \times 10^{4}\)  \(\hbox {u}_{x5}\)  \(2.1 \times 10^{3}\)  \(\hbox {u}_{x5}\)  \(3.8 \times 10^{3}\) 
\(\hbox {u}_{x3}\)  \(3.0 \times 10^{4}\)  \(\hbox {u}_{z4}\)  \(7.3 \times 10^{3}\)  \(\hbox {u}_{x2}\)  \(5.6 \times 10^{3}\) 
\(\hbox {u}_{x5}\)  \(4.0 \times 10^{4}\)  \(\hbox {u}_{y5}\)  \(8.5 \times 10^{3}\)  \(\hbox {u}_{y5}\)  \(9.4 \times 10^{3}\) 
\(\hbox {u}_{y5}\)  \(1.1 \times 10^{3}\)  \(\hbox {u}_{z5}\)  \(9.0 \times 10^{3}\)  \(\hbox {u}_{z4}\)  \(9.8 \times 10^{3}\) 
\(\hbox {u}_{y4}\)  \(1.2 \times 10^{3}\)  \(\hbox {u}_{y3}\)  \(2.6 \times 10^{2}\)  \(\hbox {u}_{y3}\)  \(4.0 \times 10^{2}\) 
\(\hbox {u}_{x4}\)  \(1.4 \times 10^{3}\)  \(\hbox {u}_{z2}\)  \(3.6 \times 10^{2}\)  \(\hbox {u}_{z5}\)  \(6.8 \times 10^{2}\) 
\(\hbox {u}_{y3}\)  \(3.2 \times 10^{3}\)  \(\hbox {u}_{y4}\)  \(9.2 \times 10^{2}\)  \(\hbox {u}_{z3}\)  \(2.0 \times 10^{1}\) 
\(\hbox {u}_{z2}\)  \(4.1 \times 10^{3}\)  \(\hbox {u}_{z3}\)  \(9.6 \times 10^{2}\)  \(\hbox {u}_{z2}\)  \(2.4 \times 10^{1}\) 
\(\hbox {u}_{z3}\)  \(4.5 \times 10^{3}\)  \(\hbox {u}_{x4}\)  11  \(\hbox {u}_{y4}\)  4.\(8 \times 10^{1}\) 
\(\hbox {u}_{y1}\)  76  \(\hbox {u}_{y1}\)  18  \(\hbox {u}_{x1}\)  9.5 
\(\hbox {u}_{x1}\)  87  \(\hbox {u}_{x2}\)  21  \(\hbox {u}_{y1}\)  10 
\(\hbox {u}_{y2}\)  93  \(\hbox {u}_{x1}\)  25  \(\hbox {u}_{x4}\)  18 
\(\hbox {u}_{x2}\)  95  \(\hbox {u}_{y2}\)  51  \(\hbox {u}_{y2}\)  29 
\(\hbox {u}_{z1}\)  100  \(\hbox {u}_{z1}\)  100  \(\hbox {u}_{z1}\)  100 

In all cases, no more than six DOF are necessary to provide errors lower than 1%.

The analysis of all combinations shows that for thin shells there is a significant gap between models providing acceptable accuracies and those with errors larger than 70%. On the other hand, as the thickness increases, the distribution has fewer accuracy gaps. As shown in Table 5, the zeroth and firstorder terms affect the gap width to a great extent. In thin shells, their role is predominant, whereas, in thick shells, higherorder terms gain relevance. A more regular accuracy distribution is an indication of more relevance of higherorder terms.

According to the distributions of accuracy from all combinations, the introduction of new terms in an expansion is ineffective if a very relevant term is not present. For instance, u\(_{x4}\) gains significance as the thickness increases.

For thin shells, the FSDT provides higher accuracy with less DOF than the BTD due to the correction of the Poisson locking. For moderately thick shells, a/h = 10, the FSDT matches the BTD but with moderate accuracy. The use of 6 DOF improves the accuracy to a great extent. As a/h decreases further, the FSDT is no longer on the BTD.

The \(\hbox {N} = 3 \) is always on the BTD, whereas the \(\hbox {N} = 2\) is a BTD only for thin shells.

The thickness ratio influences the BTD more than curvature.

The zerothorder terms are active in each BTD independently of the thickness, i.e., \(\hbox {RF}_0 = 1\).

The relevance of first and secondorder terms decreases as the thickness increases.

The influence of thirdorder terms increases as the thickness increases.

The fourthorder terms are the least influential, although, at a/h \(= 5\), the RF increases considerably to the level of secondorder terms.

Most of the zeroth, first and thirdorder terms present a regular pattern along a BTD table, i.e., as one of these terms becomes inactive, it does not appear in the BTD anymore. On the other hand, second and fourthorder terms have a more irregular pattern indicating that their influence depends on the activation or deactivation of other terms.
Simplysupported, 0/90/0/90
The second numerical case deals with a different stacking sequence to investigate the effect of an asymmetric lamination on the BTD. All other parameters remain as those of the previous case. Moreover, this section considers two additional R/a values, 100 and 50, for a more comprehensive analysis on the effect of the curvature. Table 6 presents the transverse displacement values with comparisons with other models from literature, when available.

The present case has more uniform accuracy distributions than the previous one indicating higher relevances of the higherorder terms. For the thick case, there are no relevant gaps up to 60%, and the proper choice of terms can provide any accuracy level. For a/h \(= 10\), there is an accuracy gap between 20 and 35% meaning that there are not structural models that can provide such level of accuracy.

As in the previous case, the FSDT validity is confirmed for the thin case, whereas its accuracy is not sufficient from a/h \(= 10\) and below.

Unlike the previous case, from a/h = 10 and below, some ten DOF are necessary to have errors lower than 1%.

As the thickness increases, the RF distributions are similar to the previous case with a slightly higher influence of the higherorder terms and lower for zeroth and firstorder ones.

For a/h \(= 5\) the influence of higherorder terms is of particular relevance. For instance, the 5 DOF BTD differs significantly from the FSDT and requires thirdorder terms.

The variation of the curvature leads to less significant modifications of the BTD than the thickness.
BTD models for 0/90/0/90, \(\hbox {R/a} = 5, \hbox {a/h }= 100\)
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 1.00\)  \(\hbox {RF}_1 = 0.94\)  \(\hbox {RF}_2 = 0.61\)  \(\hbox {RF}_3 = 0.48\)  \(\hbox {RF}_4 = 0.30\) 
BTD models for 0/90/0/90, \(\hbox {R/a} = 5, \hbox {a/h} = 10\)
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 1.00\)  \(\hbox {RF}_1 = 0.76\)  \(\hbox {RF}_2 = 0.52\)  \(\hbox {RF}_3 = 0.64\)  \(\hbox {RF}_4 = 0.42\) 
BTD models for 0/90/0/90, \(\hbox {R/a} = 5, \hbox {a/h} = 5\)
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 0.91\)  \(\hbox {RF}_1 = 0.79\)  \(\hbox {RF}_2 = 0.55\)  \(\hbox {RF}_3 = 0.67\)  \(\hbox {RF}_4 = 0.42\) 
BTD models for 0/90/0/90, \(\hbox {R/a} = 50, \hbox {a/h} = 10\)
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 1.00\)  \(\hbox {RF}_1 = 0.76\)  \(\hbox {RF}_2 = 0.55\)  \(\hbox {RF}_3 = 0.61\)  \(\hbox {RF}_4 = 0.42\) 
BTD models for 0/90/0/90, \(\hbox {R/a} = 100, \hbox {a/h} = 10\)
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 1.00\)  \(\hbox {RF}_1 = 0.73\)  \(\hbox {RF}_2 = 0.55\)  \(\hbox {RF}_3 = 0.61\)  \(\hbox {RF}_4 = 0.45\) 
0/90/0/90, \(\overline{u}_{z}\)(z = 0) = 100\(\hbox {u}_{z}\) \(\hbox {E}_{T}\) h\(^3\)/(\(\overline{p}_z\,a^4)\), clampedfree
Model  \({\varvec{R/a}} = 5\)  

ED4  0.0255  0.4206  1.1890 
a/h  100  10  5 
BTD models for 0/90/0/90, \(\hbox {R/a} = 5, \hbox {a/h} = 100\), clampedfree
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 1.00\)  \(\hbox {RF}_1 = 0.97\)  \(\hbox {RF}_2 = 0.61\)  \(\hbox {RF}_3 = 0.42\)  \(\hbox {RF}_4 = 0.33\) 
BTD models for 0/90/0/90, \(\hbox {R/a} = 5, \hbox {a/h} = 10\), clampedfree
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 0.91\)  \(\hbox {RF}_1 = 0.76\)  \(\hbox {RF}_2 = 0.55\)  \(\hbox {RF}_3 = 0.64\)  \(\hbox {RF}_4 = 0.48\) 
BTD models for 0/90/0/90, \(\hbox {R/a} = 5,\hbox {a/h} = 5\), clampedfree
DOF  \({\varvec{u}}_{x1}\)  \({\varvec{u}}_{y1}\)  \({\varvec{u}}_{z1}\)  \({\varvec{u}}_{x2}\)  \({\varvec{u}}_{y2}\)  \({\varvec{u}}_{z2}\)  \({\varvec{u}}_{x3}\)  \({\varvec{u}}_{y3}\)  \({\varvec{u}}_{z3}\)  \({\varvec{u}}_{x4}\)  \({\varvec{u}}_{y4}\)  \({\varvec{u}}_{z4}\)  \({\varvec{u}}_{x5}\)  \({\varvec{u}}_{y5}\)  \({\varvec{u}}_{z5}\) 

15  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \) 
14  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
13  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
12  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
11  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
10  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \) 
9  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
8  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
7  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \) 
6  \(\blacktriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
5  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\blacktriangle \)  \(\blacktriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \)  \(\vartriangle \) 
\(\hbox {RF}_0 = 0.97\)  \(\hbox {RF}_1 = 0.61\)  \(\hbox {RF}_2 = 0.58\)  \(\hbox {RF}_3 = 0.76\)  \(\hbox {RF}_4 = 0.42\) 
Clampedfree, 0/90/0/90
Analysis of the relevance of generalized displacement variables

Zerothorder terms As expected, these terms have very high influence and are almost always present in BTD. Just u\(_{x1}\) presents RF lower than unity in three cases in which the 5 DOF BTD requires higherorder terms as discussed in previous sections.

Firstorder terms Inplane components have unitary RF in most cases. On the other hand, the outofplane component has lower relevance and is consistent with the appearance of the FSDT model as 5 DOF BTD for thin and moderately thick shells.

Secondorder terms The influence of these terms varies consistently. u\(_{x3}\) has little relevance in the 0/90/0 case but higher in the asymmetric case, and such relevance tends to increase for higher thickness, and the curvature does not influence it. The u\(_{y3}\) relevance has smaller variations due to the thickness change. The thickness strongly influences the outofplane component and its influence decreases for thicker shells.

Thirdorder terms The inplane components have significant influence which increases for thicker shells. The outofplane influence is relevant in the symmetric case and increases for thicker shells.

Fourthorder terms These terms are the less influential except for u\(_{x5}\) in the clampedfree case. The relevance of these terms should increase as soon as the BTD considers stress distributions.
Conclusions

For the cases considered in this paper, the thickness and stacking sequence are the most important factors for the choice of the primary variables. For thin shells, six DOF are sufficient to obtain errors lower than 1%. For thick shells, ten DOF are necessary.

In most cases, the accuracy level obtainable from combinations of a given set of variables is not continuous as the DOF decrease. In other words, there are no structural models that can satisfy certain accuracy of the solution.

Accuracy gaps indicate the presence of very effective terms that must be present in the expansion to ensure satisfactory accuracies. For instance, for thin shells, these terms coincide with the FSDT expansions. However, as the presence of nonclassical effects due to asymmetries or high thickness increases, the relevance of higherorder terms increases and the accuracy gaps tend to disappear.

The FSDT and secondorder model are BTD only for thin shells. The thirdorder model is close to the BTD in most cases.

As the thickness increases, the relevance of thirdorder variables increases significantly, and these terms can be the most relevant together with the zerothorder ones.

The outofplane displacement variables tend to have less relevance than inplane ones. Such a relevance should increase significantly as soon as the analysis considers stress distributions as control parameters.

The set of variables composing a BTD model depends on the boundary conditions; however, such a dependency is weaker than the thickness one.
Declarations
Authors' contributions
EC provided the CUF FEM framework. MP developed the AAM and obtained the BTD. Both authors read and approved the final manuscript.
Acknowledgements
Erasmo Carrera acknowledges the Russian Science Foundation under Grant No. 181900092.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Data are available upon request.
Funding
This work was partially supported by the Russian Science Foundation under Grant No. 181900092.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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