Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.022210

ARTICLE

Static Analysis of Doubly-Curved Shell Structures of Smart Materials and Arbitrary Shape Subjected to General Loads Employing Higher Order Theories and Generalized Differential Quadrature Method

Department of Innovation Engineering, School of Engineering, University of Salento, Lecce, 73100, Italy

*Corresponding Author: Francesco Tornabene. Email: francesco.tornabene@unisalento.it

Received: 28 February 2022; Accepted: 11 May 2022

Abstract: The article proposes an Equivalent Single Layer (ESL) formulation for the linear static analysis of arbitrarily-shaped shell structures subjected to general surface loads and boundary conditions. A parametrization of the physical domain is provided by employing a set of curvilinear principal coordinates. The generalized blending methodology accounts for a distortion of the structure so that disparate geometries can be considered. Each layer of the stacking sequence has an arbitrary orientation and is modelled as a generally anisotropic continuum. In addition, re-entrant auxetic three-dimensional honeycomb cells with soft-core behaviour are considered in the model. The unknown variables are described employing a generalized displacement field and pre-determined through-the-thickness functions assessed in a unified formulation. Then, a weak assessment of the structural problem accounts for shape functions defined with an isogeometric approach starting from the computational grid. A generalized methodology has been proposed to define two-dimensional distributions of static surface loads. In the same way, boundary conditions with three-dimensional features are implemented along the shell edges employing linear springs. The fundamental relations are obtained from the stationary configuration of the total potential energy, and they are numerically tackled by employing the Generalized Differential Quadrature (GDQ) method, accounting for non-uniform computational grids. In the post-processing stage, an equilibrium-based recovery procedure allows the determination of the three-dimensional dispersion of the kinematic and static quantities. Some case studies have been presented, and a successful benchmark of different structural responses has been performed with respect to various refined theories.

Keywords: 3D honeycomb; anisotropic materials; differential quadrature method; general loads and constraints; higher order theories; linear static analysis; weak formulation

New advances in many engineering applications are based on the employment of smart innovative materials and appliances characterized by non-conventional shapes [1,2]. Very complex mechanical properties are very frequently required, together with an increased need for lightweight materials [3]. As a matter of fact, when unconventional materials and geometries are adopted, the structural behaviour is not easy to be predicted a priori with simple but effective models due to the absence of symmetry planes [4,5].

In this perspective, laminated structures are widespread solutions for achieving unconventional and optimized structural performances. As it is well-known, classical composite materials usually induce an orthotropic behaviour of the overall structure [6]. Nevertheless, new solutions can be found in literature where very complex deformation mechanisms can be induced, together with coupling effects. Moreover, several engineering manufacturing processes employ architectured geometries that have been conceived for the fulfillment of different aesthetic, ergonomic, and aerodynamic requirements [7]. An important reason comes from the endeavor to obtain the best form under fixed load cases and external constraints layup [8]. In fact, an effective reduction of the material is sought in this way. All shell structures are spatially distributed volumes [9,10]. For this reason, advanced design skills are required for a correct mathematical modelling of their shape and material properties. Moreover, the accuracy of a correct modelling is crucial for its reliability since the presence of curvature significantly orients the structural behaviour, as well as the constitutive relationships [11].

As a matter of fact, three-dimensional (3D) solid simulations based on the widespread Finite Element Method (FEM) are nowadays the most adopted strategy for the structural assessment of curved and layered shells [12,13]. Accordingly, such models lead to a very demanding computational cost for the simulation [14]. For this reason, two-dimensional (2D) alternative approaches have been derived. The most famous strategy is Equivalent Single Layer (ESL) [15–17], which assesses the entire structural problem on a reference surface whose generic point is located in the middle thickness of the solid. In particular, in reference [16] an extensive study with higher order theories for doubly-curved shell structures is provided, and in reference [17] an ESL higher order model is developed for laminated composite curved structures. In the same way, the Layer-Wise (LW) approach [18–22] introduces a 2-manifold for each layer of the stacking sequence. In reference [19] the dynamic behaviour of anisotropic doubly-curved shells is investigated with a higher order LW approach, whereas in reference [20] an equivalent LW model is developed, accounting for an efficient description of the kinematic field, referring to the intrados and extrados of the structure. In the so-called Sub-Laminate Theory (SLT) a series of reference surfaces are defined for a specific group of the entire lamination scheme [23–25]. For the LW and the SLT, the compatibility conditions between the various reference surfaces are taken into account during the global matrices assembly. Generally speaking, LW provides more accurate results than the ESL approach, but it can be very high computationally demanding since the overall number of Degrees of Freedom (DOFs) comes to be proportional to the layers within the laminate [26]. Within the ESL framework, the number of unknown variables is not dependent on the actual stacking sequence. Moreover, suppose a higher order set of axiomatic assumptions is included in the model. In that case, the accuracy of ESL 2D formulations can be compared to that of the 3D FEM and LW solutions even for generally anisotropic lamination schemes [27].

As it is well known, classical finite element simulations are characterized by a local pre-determined polynomial approximation of the unknown field variable between two adjacent points. On the other hand, when shell elements with curvatures are considered, some issues like the shear locking can emerge, thus invalidating the simulation [28,29]. For this reason, very refined meshes are implemented so that very accurate results are requested. In this way, the geometric and simulation discretization error is reduced, but the computational cost increases a lot.

In contrast, the IsoGeometric Approach (IGA) accounts for the employment of the actual geometry of the structure directly taken from the Computer Aided Design (CAD) process in the theoretical modelling [30–34]. As a consequence, arbitrary interpolating functions are developed. Throughout literature, IGA has been employed for both the strong and weak implementations of disparate structural problems with curved edges and shapes. In particular, in reference [34] the IGA approach has been adopted for the buckling analysis of 3D plates, taking into account 2D computation of Non-Uniform Rational B-Spline (NURBS) curves, as well as the enforcement of boundary conditions. Furthermore, the hybrid meshfree collocation method is employed in reference [35], and a computationally efficient formulation is obtained, which is characterized by the advantages of IGA and meshfree methods.

When laminated doubly-curved shells are investigated by means of 2D models, a key aspect is the parametrization of the reference surfaces. It is feasible that a set of principal curvilinear coordinates are provided starting from the geometric properties of the structure. In the case of shells of arbitrary shape, a distortion algorithm is proposed so that the physical domain is described with a rectangular computational interval. To this end a set of generalized blending functions are developed [36,37]. They are based on the geometric description of the shell edges and the location of the corners of the structure within the physical domain. When IGA methodology is followed, NURBS curves are employed for this task. In references [38,39], an interesting insight into the computation of such curves is provided.

Within the ESL 2D simulations, since the 3D shell structure is reduced to its equivalent reference surface as described in references [40,41] the three-dimensional unknown field variable should be expressed in terms of a set of predetermined through-the-thickness shape functions [42–45], each of them arranged employing a generalized matrix formulation [46,47]. Accordingly, reference [42] employs power polynomials for the assessment of all thickness functions set and derives a closed-form analytical solution, whereas in references [43,44] a higher order power assumption is adopted only for in-plane displacement components. As far as ESL theories is concerned, both polynomial and trigonometric functions can be adopted along each shell principal direction [48–50]. As the order of the kinematic expansion increases, the so-called Higher Order Shear Deformation Theories (HSDTs) come out [16]. The above-mentioned generalized approach also embeds classical theories like the First Order Shear Deformation Theory (FSDT) [51,52] and the Third Order Shear Deformation Theory (TSDT) [53–55] which considers a first and a third-order polynomial expansion for the in-plane kinematic field, respectively. The out-of-plane direction is characterized by a constant distribution of the displacement field instead, thus neglecting any kind of stretching and shear effect. The employment of non-uniform through-the-thickness assumptions is crucial for modelling the actual deflection of the structure. Actually, in reference [56] it is shown that the out-of-plane stretching plays an important role in the deformation response, whereas in reference [57] the shear contribution in the total deflection of composite laminated structure is outlined. Even though an axiomatic selection of thickness functions can be assessed, some refined theories have been derived for orthotropic laminated structures in which the through-the-thickness hypotheses are based on the mechanical characteristics of the constituent materials [58–61]. In particular, in reference [58] the classical FSDT theory is refined with the introduction of a shear thickness function derived from the actual lamination scheme of the selected plate. Furthermore, in reference [59] the same approach has been applied to mono-dimensional (1D) beams. In the case of laminated shells the unknown field variable dispersion should fulfil the interlaminar continuity since a piecewise transverse displacement field and shear stress distribution should be modelled. Moreover, it is feasible that the shear effect is embedded in the formulation, since experimental tests show that its contribution turns out to be prominent in laminated structures [62,63]. For this reason, the LW approach is the most suitable strategy. On the other hand, the ESL approach can provide good results too if the so-called zigzag functions are adopted for the field variable description. In reference [64] an interesting review for multilayered theories is reported, where the selection of thickness functions accounts for continuous transverse stresses and a discontinuous first order derivative of the same quantities for both in-plane and out-of-plane components. In references [65,66] instead, a discontinuous first order derivative of the in-plane displacement components is provided in an effective way from the introduction of a piecewise function at the first order of the kinematic expansion.

Some considerations should be made on the treatment of external surface loads distributions within a 2D formulation. Classical solutions for plate problems [67,68] account in general for a mere computation of normal external loads within the equilibrium equations. In the case of higher order assumptions on the displacement field variable, the external loads should be inserted within the 2D approach. An effective procedure for the assessment of external loads is based on the Static Equivalence Principle [16,69]. The same methodology can be applied to classical refined models when a higher order displacement field assumption is taken along each in-plane field variable. Throughout literature, a huge variety of works can be found where the static performance of plates and shells is evaluated under a uniform distribution of normal and tangential loads [70–75]. In reference [75] a consistent method is proposed for concentrated loads within an ESL model directly in a strong form. In particular, a smooth continuous function is taken into account, and the calibration of the shape and position parameters can lead to a proper modelling of the singularity induced by point and line loads. On the other hand, there are only few works from literature dealing with a general distribution of surface pressure. A different group of studies investigates the problem of concentrated and line loads, starting from the numerical approximation methods of the Delta-Dirac function. Among others, the interested reader can refer to the articles by Eftekhari et al. [76–78], where the Dirac-Delta function is discretized with success taking into account the employed numerical method. In particular, in reference [76] moving concentrated loads have been applied to 1D structures starting from the definition of the Dirac-Delta function. Then, in reference [77] the formulation has been extended to 2D structural problems. Moving load problems have been treated in reference [78] with the well-known Heaviside function. Generally speaking, the approximating function of a singularity problem should fulfill some characteristic properties derived from the theoretical definition of concentrated loads; otherwise, it should be properly normalized. These kinds of pressures are singularities in a differential continuum model, such that various numerical techniques have been proposed for their approximation with smooth functions [79–81]. According to the author’s knowledge, there are some works accounting for a Gaussian distribution (see reference [82] among others). An interesting problem related to concentrated loads on curved structures is the so-called pinched cylinder. In references [83–85] a series of theoretical developments have been provided for a circular cylinder subjected to a point load applied at the diameter of the surface.

As far as the numerical implementation is concerned, several articles can be found in which spectral collocation methods are adopted within IGA framework. Among them, the Generalized Differential Quadrature (GDQ) method [86–90] has demonstrated to be a reliable tool since it overcomes many computational drawbacks. Based on polynomial functional approximation methods, the GDQ has been successfully applied to a series of structural problems [91,92]. In particular, some works can be found where GDQ has been adopted for the dynamic analysis of smart structures, as well as multifield problems [93,94]. It has been shown that it is an extension of pseudospectral collocation methods since the similar levels of accuracy are obtained when the same computational grid is adopted [95–97]. On the other hand, the generalized procedure for the calculation of weighting coefficients based on Lagrange polynomials for the global interpolation is not based on the actual selection of discrete points and the derivation order. In particular, the GDQ method provides the best performance among all the numerical techniques belonging to the weighted residual class. In reference [98] the outcomes of the free vibration analysis of thin-walled beams calculated by means of the GDQ method are compared to those coming from a 1D Ritz formulation. Other works [99–102] provide the theoretical assessment together with some parametric static and dynamic analyses of shells of different curvatures characterized by innovative material like Carbon Nanotubes (CNTs) and Functionally Graded Materials (FGMs). In reference [103] FGM structures have been investigated by employing a semi-analytical model. In reference [104] an effective homogenization algorithm is developed for the mechanical assessment of sandwich shells with both honeycomb and re-entrant patterns. Referring to pantographic lattice structures, the widespread method for the computation of equivalent elastic properties is the well-known Neumann’s principle [105], accounting for the computation of generalized stiffnesses starting from simple independent load cases [106–108]. The accuracy of the solution lies in the theoretical hypothesis considered for the numerical modelling. In particular, some considerations should be made on the nodal area within the honeycomb cell, as well as the bending and shear effects of vertical and inclined struts [109–111].

Once the 2D solution is provided on the reference surface of a doubly-curved shell, the post-processing algorithm is crucial for the reconstruction of both static and kinematic quantities along the three-dimensional solid. Actually, both the ESL and the LW formulation account for the assessment of the generalized unknown field variables lying on the reference surface. Accordingly, in the case of simulations performed with the FEM, the solution is calculated at the discrete set of nodes of the computational grid. Then, a reconstruction along the continuum domain should be assessed. Referring to the latter, there are three main classes of methodologies for the solution extrapolation, namely the Superconvergent Patch Recovery (SPR) [112,113], the Averaging Method (AVG) [114,115] and the Projection Method (PROJ) [116]. As far as the through-the thickness equilibrium-based recovery procedure is concerned, the generalized constitutive relationship can be adopted as usual, but a correction of shear and membrane stresses is essential so the boundary conditions related to applied loads can be fulfilled [117–120].

In the present work, an ESL theoretical formulation is proposed for generally anisotropic doubly-curved shells of arbitrary shapes by means of the ESL method. The reference surface of the structure is described employing the main outcomes of differential geometry, setting an orthogonal curvilinear set of principal coordinates. A NURBS based generalized set of blending functions is employed for the distortion of the physical domain in the case of arbitrarily-shaped structures. The displacement field component vector is intended to be the unknown variable of the problem, and it is expressed via the employment of a generalized set of thickness functions. Actually, a higher order kinematic expansion is performed together with a consistent zigzag function to properly catch all the possible stretching and warping effects within each layer, as well as coupling issues at the interlaminar stage. A generalized weak expression of the field variable is introduced, accounting for higher order shape functions for the interpolation alongside the reference surface [121]. A general distribution of surface loads has been considered in the model, setting a Gaussian and a Super-Elliptic distribution for the application of external loads along each principal direction of the shell. In particular, the calibration of distribution parameters leads to model different load cases. The fundamental governing equations, derived from the Minimum Potential Energy Principle, are numerically tackled by the GDQ method. The Generalized Integral Quadrature (GIQ) method has been adopted for the computation of integrals [16]. The GIQ moves from the above-discussed GDQ procedure, and it allows performing numerical integrations following an effective quadrature approach.

The theoretical formulation proposed in this manuscript has been implemented in the DiQuMASPAB software [122], a research code based on the GDQ method accounting for static and dynamic simulations on doubly-curved shell structures. All the numerical examples of mechanical properties have been included in the built-in material library. The graphic user interface provided by the software allows to implement the lamination scheme, the geometry of the structure, as well as external load and boundary conditions features. The structures that are considered for the numerical validation account for different mechanical issues, namely the presence of a single and double curvature, the number of layers and the presence of the soft core. Different loading conditions have been adopted. The equilibrium-based 3D reconstruction of mechanical quantities has been demonstrated to be very efficient in accordance with the predictions of refined 2D and 3D solutions. The same level of accuracy has been outlined in the case of lamination schemes with a lattice layer characterized by a macro mechanical orthotropic behaviour. Any performance reduction has been observed when a mapping of the physical domain has been required.

2 Geometrical Description of a Doubly-Curved Shell

According to the ESL formulation, a doubly-curved shell should be referred to its reference surface whose points are located at the middle thickness of the structure. It should be remarked that a generic three-dimensional solid can be expressed with respect to a Cartesian global reference system

being

where the

More specifically, Eq. (2) describes a doubly-curved shell when all the variables vary in a closed interval. In particular, it should be stated that

where

In addition, the well-known Lamè parameters

Starting from Eqs. (1) and (2), the parameters

Accordingly, in the case of straight shells along the

In the present manuscript, doubly-curved shells of variable thickness [16] have been considered; therefore, a generalized analytical expression of

being

In the present formulation, the expressions of

When an arbitrary curve is considered lying on the reference surface

Since the curve at issue lies on

3 NURBS Blending of the Physical Domain

In the previous paragraph, the geometry of a generic doubly-curved shell has been described starting from the reference surface

In Eq. (12), the symbols

being

where

Starting from Eq. (13), it is possible to derive an expression for partial derivatives with respect to in-plane directions

where the definitions

If

Accordingly, the inverse relation of Eq. (17) accounts as follows:

From a direct comparison between Eqs. (16) and (19), the definition of coefficients

For the sake of completeness, first order partial derivatives

Following a similar procedure of that adopted in Eqs. (16)–(20), the chain rule can also be employed for the computation of second order derivatives with respect to

being

In addition, one gets the following expression for pure second order derivatives with respect to

where the coefficients

In the present work, the

4 ESL Kinematic Equations Employing Generalized Shape Functions

We now deal with the definition of the ESL assessment of the displacement field variable employing a generalized set of shape function for the weak formulation of the structural problem. The 3D displacement field component vector

Employing the unified notation of Eq. (26), a generalized displacement field component vector

Referring to

As a matter of fact, the employment of the unified notation of Eq. (26) lets us to perform a systematic analysis with different ESL theories, since the theoretical formulation is independent from the actual selection of the thickness function analytical expression. Nevertheless, since various polynomials orders can be selected in Eq. (27), a nomenclature is adopted for a smarter identification of the axiomatic through-the-thickness assumptions of the simulation. In particular, the acronym

Once the ESL assessment of the displacement field component vector

Since all quantities in Eq. (26) are evaluated in each point of the discrete computational grid, quantities

Once column vectors of Eq. (30) have been correctly defined the following definition can be stated, accounting for all the values assumed by the generalized displacement field component vector in each point of the computational grid:

Thus, it is possible to introduce the discrete form of Eq. (26) employing a global interpolation algorithm. Accordingly, the following relation can be assessed for each

In a more expanded form, one gets [16]:

where

Starting from Eq. (33) it is possible to define the generalized shape function matrix

In Eq. (35), the Lagrange interpolating polynomials introduced in Eq. (34) are collected in a

In the same way, vector

Since the Kronecker tensorial product, denoted with

To sum up, the ESL assessment of the kinematic field introduced in Eq. (26) is rearranged accounting for the interpolation procedure of Eq. (32), leading to the following expression:

Now, the generalized form of the displacement field of Eq. (39) is adopted for the definition of the kinematic relations for a doubly-curved shell structure, according to the ESL approach. More specifically, the kinematic relations for a 3D structure in principal coordinates

being

In the same way,

being

Introducing in Eq. (40) the ESL assessment of the displacement field settled in Eq. (26), the kinematic relations turn into the following equation, accounting for an expansion up to the

In the previous equation, the strain vector

In Eq. (44) it has also been shown that the generalized strain component vector

where the kinematic operators

We recall that in Eq. (47) the Lagrange polynomials employed for the interpolation of the field variable along the

5 Generally Anisotropic Lamination Scheme Assessment

The present manuscript investigates the problem of the static structural response of a doubly-curved shell with generally anisotropic laminated structures characterized by general orientations and material syngonies. It should be remarked for the sake of completeness that a sequence of l laminae has been contemplated, whose

Since a linear elastic theoretical assessment of the structural problem is performed, the matrix

where

Nevertheless, Eq. (48) should be referred to the curvilinear principal reference system of the physical domain, accounting for the 3D stress and strain vectors

On the other hand Eq. (51), defined for the

where

whose generic component

being

Employing the kinematic relations performed in Eq. (46) utilizing the generalized interpolation, Eq. (52) gets into:

Thus, the generalized ESL assessment of the constitutive relationship performed in Eq. (56) can be expressed so that each component of

In Appendix A, the interested reader can find the complete expressions for each component of matrix

6 Equilibrium Relations in the Variational Form

In the present section, the equilibrium equations for an arbitrary doubly-curved shell are derived by employing an energy approach. The stationary configuration is specifically enforced to the total potential energy virtual variation

being

We focus on the total elastic strain energy

Employing the ESL assessment of the displacement field of Eq. (26), the through-the-thickness integration performed in Eq. (59) is avoided. Thus,

Taking into account the weak formulation expression of generalized strain resultant vector of Eq. (46) in the previous equation, the virtual variation of the total elastic strain energy reads as:

In Eq. (61) the global stiffness matrix

In Appendix B, an extended expression of each stiffness matrix coefficient

6.2 External Loads General Distributions

Let us consider two generic vectors

where symbols

For a constant loading distribution, it is:

In the present manuscript, two different distributions for

being

where

If a virtual variation

being

The static equivalence principle employs both Eqs. (68) and (69), so that [16]:

Starting from Eq. (70), the extended expressions of the

Since the generalized displacement field component vectors

In Eq. (72), the generalized external actions referred to each

The last contribution to the total potential energy occurring in Eq. (58) accounts for the influence of the elastic constraints distributed along the edges of the structure. In particular, a set of linear elastic springs of stiffness

where

being

As a matter of fact, the Super Elliptic distribution (S) of the

In the following, the expression of a Double–Weibull distribution (D) has been reported in terms of

where

Starting from Eqs. (74) and (75) which have been expressed for a 3D structure, the stresses coming from the linear elastic springs distributions should be computed in terms of generalized stress resultants, accounting for the ESL assessment of the displacement field. Substituting Eq. (26) in the previously cited relations, one gets for

Referring to the edges of the structure located at

Fundamental coefficients

Having in mind the ESL assessment of general distributions of boundary springs, the virtual work contribution

where

being

Thus, the generalized displacement vector

Note that the fundamental stiffness matrix

6.4 Fundamental Governing Equations

The final form of the fundamental governing equations of the static problem can be outlined from Eq. (58), taking into account the contributions of the elastic strain energy of Eq. (61), the virtual work of surface loads of Eq. (72) and that of external boundary springs of Eq. (85). Referring to the

In extended form, Eq. (86) can be assembled with respect to the displacement field ESL higher order model of Eq. (26), leading to the following expression:

For shells of arbitrary shapes, the employment of the generalized blending functions assessed in Eq. (13) with the Jacobian matrix

The ESL assessment of external surface load vector

As far as the definition of generalized boundary springs for arbitrarily-shaped structures is concerned, it is useful to refer to the local coordinate system introduced in Eq. (11) along each edge of the shell. As a consequence, boundary stresses can be expressed with respect to

Starting from the blending coordinates transformation of Eq. (13), the length

Since the natural coordinate set

Once the quantity

In order to compute the generalized stress resultants

In the same way, the generalized stress resultants

7 Numerical Implementation with the GDQ Method

In the present section we deal with the numerical assessment of the fundamental governing equations for the static problem derived in Eq. (87) in their assembled form. The employed numerical technique that has been adopted is the well-known GDQ method. According to this methodology, the discrete form of the derivatives is directly provided. Referring to a generic point

where

Eq. (96) is based on the interpolation of the unknown function employing the Lagrange interpolating polynomials

being

Based on Eq. (97), it is

where

A 2D Legendre-Gauss-Lobatto (LGL) grid of dimensions

In Eq. (100),

Since the fundamental governing relations (86) account for derivations along

being

The employment of the GDQ method allows to obtain the discrete form of the governing equations. When the assembly of the fundamental relations is performed alongside the entire

Performing a static condensation of Eq. (104) with respect to

The numerical integrations that are involved in the formulation of the present manuscript are performed by means of the GIQ procedure [16]. Referring to a generic univariate function

where

Then, they are collected in the matrix

All the terms

where

Referring to the bivariate function

For the sake of completeness, symbol

We recall that in Eq. (87) the global stiffness matrix of the shell has been provided, starting from a parametrization of the reference surface

being

Employing Eqs. (113) and (114), the stiffness matrix

In the following, the discrete form of the generalized external loads of Eq. (84) is provided by means of the GIQ method presented in Eq. (106):

In the same way, Eq. (116) becomes suitable for an arbitrary mapped domain. According to the nomenclature introduced in Eq. (12), it gives for

The edge characterized by

On the other hand, for

Finally, the edge identified with

Once the fundamental governing relations reported in Eq. (87) have been solved by means of the GDQ method, the solution in terms of the generalized displacement field component vector

being

Employing the unified ESL assessment of the displacement field introduced in Eq. (26) it is possible to derive the through-the-thickness distribution of the displacement field component vector

being

Referring to the generic

Nevertheless, the remaining stress components

where

It should be noted that the expression of

The first order differential relations (126) should fulfill a single boundary condition. Referring to the first lamina of the stacking sequence

being

For the surface loads

In the previous equation an index

Once the corrected values

Now, it is possible to solve the third differential relation in (126), while enforcing the stress compatibility condition for the surface external loads

In the same way, the following relation should be fulfilled at the interface level between the

Following a similar procedure to Eq. (132) for

Therefore, the load boundary conditions at the top surface of the shell read as [16]:

being

Starting from the main outcomes of Eqs. (131) and (137), the corrected values of the out-of-plane 3D deformations collected

where

From the inversion of the fundamental linear system of Eq. (138), the corrected values of

In the last Eq. (140),

The corrected strain values can be employed for the correction of membrane stresses

We now present a series of case studies to validate the proposed methodology. Different structures have been considered, characterized by a variety of lamination schemes and curvatures. More specifically, the linear static response of zero-, singly- and doubly-curved panels have been investigated. Geometries of arbitrary shape have been considered, accounting for the generalized mapping technique of Eq. (13). The results provided by the ESL methodology presented in the manuscript have been compared to predictions obtained from refined 3D FEM simulations with parabolic elements as provided by a commercial software. The numerical investigations check for the accuracy of the model for different displacement field axiomatic assumptions according to Eq. (26), governing parameters of the employed distributions of external loads, and boundary linear springs distributions.

9.1 Materials Employed in the Analyses

In the present section we describe the mechanical input properties of materials for numerical analyses. For each component, the stiffness matrix

In the following, the 3D stiffness matrix of the triclinic material

The trigonal material