3D Equilibrium Equations Research Articles

Meshfree method for bending and free vibration analysis of laminated plates using the Reissner’s mixed variational theorem

Free vibration and bending behavior of laminated plates are investigated using a combination of Reissner’s mixed variational theorem and the meshless radial point interpolation method. By introducing Reissner’s mixed variational theorem, a refined transverse shear stress field is derived based on three-dimensional (3D) elasticity equations. The efficiency and accuracy of this model are demonstrated through numerical examples. It is found that accurate prediction of shear stresses significantly improves the prediction of natural frequency for composite laminated plates. The difference between shear stresses derived from 3D equilibrium equations and that from constitutive equations is significant.

3D electro-elastic static analysis of advanced plates and shells

The proposed three-dimensional (3D) coupled electro-elastic shell model allows the static analysis of smart structures embedding classical piezoelectric and functionally graded piezoelectric layers. Plates, cylindrical and spherical panels are investigated in both sensor and actuator configurations. The primary variables of the coupled model are displacement components and the electric potential. Therefore, displacements, stresses, strains, electric potential and electric displacements are calculated for smart structures used as sensors (applied mechanical load) and smart structures used as actuators (applied electric potential). In the case of spherical shells, 3D equilibrium equations are coupled with the 3D divergence electric displacement equation. The obtained coupled system is also valid for cylindrical shells and plates thanks to the use of a mixed curvilinear orthogonal reference system. The partial differential governing equations are solved using the Navier harmonic form and the exponential matrix method for simply supported structures. Preliminary assessments are proposed to validate the model and further benchmarks are analyzed to discuss the effects connected with the thickness ratio, the lamination scheme, the load conditions, the geometry of the structures and the employed materials. The main innovation point of the proposed model is the possibility to analyze several geometries including different materials and applied loads by means of a unique and general formulation where partial differential equations are solved in closed form. The use of functionally graded piezoelectric materials allows to eliminate all stress component discontinuities at each layer interface.

Three-dimensional vibration analysis of multilayered composite and functionally graded piezoelectric plates and shells

In the present paper, a coupled 3D exact electro-elastic shell model for Functionally Graded (FG) and composite piezoelectric structures is proposed. Primary variables of the electro-elastic model are the electric potential and the three displacement components. The model allows to evaluate the piezoelectric effect in terms of frequencies and vibration modes. Both closed and open circuit configurations are analyzed and compared. The 3D equilibrium equations and the 3D divergence electric displacement equation for spherical shells give the set of partial differential equations for the electro-elastic problem. The proposed model for spherical shells automatically degenerates into simpler models for plates and cylindrical shells via properly considerations on radii of curvature along in-plane directions. The orthogonal mixed curvilinear coordinates α, β and z are employed. The partial differential governing equations have constant coefficients considering fictitious layers and they are solved using the Navier harmonic form and the exponential matrix method. These features lead to an exact solution for simply-supported boundary conditions. Free vibration analyses are conducted and circular frequencies for the first three thickness vibration modes are computed. After a global assessment phase to verify the correctness of the developed model, new benchmarks are proposed: different thickness ratios and material configurations are investigated. The present work can be intended as a reference general solution for those scientists interested in the study of piezoelectric structures via 2D analytical and numerical formulations.

Longitudinal plane wave amplitude in N-type semiconductors with inviscid liquid loading and impedance boundary

The present analytical investigation focuses on longitudinal quasi-plane (qP) waves scattering in N-type piezoelectric semiconductor (PSC) halfspace, generating four reflected waves: qP, qSV, electro-acoustic, and carrier plane. It analyzes wave amplitudes under two conditions: inviscid liquid loading and impedance boundary conditions, deriving a secular equation and determining reflection/transmission coefficients. These calculations rely on 3D constitutive relations and equilibrium equations. Furthermore, the study also extends to the calculation of bulk energy flux. The outcomes of the study may provide insights into the applications of piezoelectric semiconductors in underwater navigation or detection systems and electromagnetic transducers.

Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium

– This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method’s effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: (1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; (2) the stress resultants are used to replace stress components in the latter equations; (3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; (4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach.

Use of the 3D Equilibrium Equations in the Free-Edge Analyses for Laminated Structures with the Variable Kinematics Approach

This paper compares out-of-plane stresses evaluated with Hooke’s Law and the stress recovery technique, focusing on the free edges of composite plates and shells. The Carrera Unified Formulation and the finite element method are adopted to derive the governing equations. Lagrange polynomials are implemented in the equivalent single-layer, layer-wise, and variable kinematics approaches. The latter is used to refine structural models locally and reduce computational overheads. Laminated plates and shells subjected to uniaxial tension are considered. The out-of-plane stresses are compared with references from the existing literature for most cases. The results demonstrate that the stress recovery technique effectively calculates stresses and improves the accuracy of equivalent single-layer models. Furthermore, layer-wise models are needed for accurate results near the free-edge zone. Finally, variable kinematics theories are helpful in accurately detecting local phenomena along the structure’s thickness.

A coupled hygro-elastic 3D model for steady-state analysis of functionally graded plates and shells

Abstract This 3D coupled hygro-elastic model proposes the three-dimensional (3D) equilibrium equations associated with the 3D Fick diffusion equation for spherical shells. The primary unknowns of the problem are the displacements and the moisture content. This coupled 3D exact shell model allows to understand the effects of the moisture field in relation with the elastic field on stresses and deformations in different plates and shells. This model is specifically developed for configurations including functionally graded material (FGM) layers. Four different geometries are analyzed using an orthogonal mixed curvilinear reference system. The main advantage of this reference system for spherical shells is the degeneration of the equations to those for simpler geometries. The solving method is the exponential matrix method in the thickness direction. The closed-form solution is possible because of simply supported sides and harmonic forms for displacements and moisture content. The moisture content amplitudes are directly applied at the top and bottom outer faces through steady-state hypotheses. The final system is based on a set of coupled homogeneous second-order differential equations. The moisture field effects are evaluated for the static analysis in terms of displacement, strain, and stress components. After preliminary validations, used to better understand how to properly define the calculation of the curvature-related terms and FGM properties, four new benchmarks are proposed for several thickness ratios, geometrical data, FGM configurations, and moisture values imposed at the external surfaces. From the results, it is clear the accordance between the uncoupled hygro-elastic model and this new coupled hygro-elastic model when the 3D Fick diffusion law is employed. Both effects connected with the thickness layer and the embedded material are included in the 3D hygro-elastic analyses proposed. The 3D coupled hygro-elastic model is simpler than the uncoupled one because the 3D Fick diffusion law does not have to be separately solved.

Hygro-Elastic Coupling in a 3D Exact Shell Model for Bending Analysis of Layered Composite Structures

In this work, a 3D fully coupled hygro-elastic model is proposed. The moisture content profile is a primary variable of the model’s displacements. This generic fully coupled 3D exact shell model allows the investigations into the consequences arising from moisture content and elastic fields in terms of stresses and deformations on different plate and shell configurations embedded in composite and laminated layers. Cylinders, plates, cylindrical and spherical shells are analyzed in the orthogonal mixed curvilinear reference system. The 3D equilibrium equations and the 3D Fick diffusion equation for spherical shells are fully coupled in a dedicated system. The main advantage of the orthogonal mixed curvilinear coordinates is related to the degeneration of the equations for spherical shells to simpler geometries thanks to basic considerations of the radii of curvature. The exponential matrix method is used to solve this fully coupled model based on partial differential equations in the thickness direction. The closed-form solution is related to simply supported sides and harmonic forms for displacements and the moisture content. The moisture content amplitudes are directly applied at the top and bottom outer faces through steady-state hypotheses. The final system is based on a set of coupled homogeneous second-order differential equations. A first-order differential equation system is obtained by redoubling the number of variables. The moisture field implications are evaluated for the static analysis of the plates and shells in terms of displacement and stress components. After preliminary validations, new benchmarks are proposed for several thickness ratios, geometrical and material data, lamination sequences and moisture values imposed at the external surfaces. In the proposed results, there is clearly accordance between the uncoupled hygro-elastic model (where the 3D Fick diffusion law is separately solved) and this new fully coupled hygro-elastic model: the differences between the investigated variables (displacements, moisture contents, stresses and strains) are always less than 0.3%. The main advantages of the 3D coupled hygro-elastic model are a more compact mathematical formulation and lower computational costs. Both effects connected with the thickness layer and the embedded materials are included in the conducted hygro-elastic analyses.

A Layer-Wise Coupled Thermo-Elastic Shell Model for Three-Dimensional Stress Analysis of Functionally Graded Material Structures

In this work, a coupled 3D thermo-elastic shell model is presented. The primary variables are the scalar sovra-temperature and the displacement vector. This model allows for the thermal stress analysis of one-layered and sandwich plates and shells embedding Functionally Graded Material (FGM) layers. The 3D equilibrium equations and the 3D Fourier heat conduction equation for spherical shells are put together into a set of four coupled equations. They automatically degenerate in those for simpler geometries thanks to proper considerations about the radii of curvature and the use of orthogonal mixed curvilinear coordinates α, β, and z. The obtained partial differential governing the equations along the thickness direction are solved using the exponential matrix method. The closed form solution is possible assuming simply supported boundary conditions and proper harmonic forms for all the unknowns. The sovra-temperature amplitudes are directly imposed at the outer surfaces for each geometry in steady-state conditions. The effects of the thermal environment are related to the sovra-temperature profiles through the thickness. The static responses are evaluated in terms of displacements and stresses. After a proper and global preliminary validation, new cases are presented for different thickness ratios, geometries, and temperature values at the external surfaces. The considered FGM is metallic at the bottom and ceramic at the top. This FGM layer can be embedded in a sandwich configuration or in a one-layered configuration. This new fully coupled thermo-elastic model provides results that are coincident with the results proposed by the uncoupled thermo-elastic model that separately solves the 3D Fourier heat conduction equation. The differences are always less than 0.5% for each investigated displacement, temperature, and stress component. The differences between the present 3D full coupled model and the the advantages of this new model are clearly shown. Both the thickness layer and material layer effects are directly included in all the conducted coupled thermal stress analyses.

A neural network-assisted 3D theoretical thermoelastic solution for laminated liquid crystal elastomer plate used in restoring cardiac mechanical function

Some atrial contractile assist devices applied on the heart surface can be regarded as a laminated Liquid crystal elastomer (LCE) plate under steady temperature loads and a contact mechanical force. An exact solution for the deformation of the laminated LCE plate under combined thermal and mechanical loads is derived by solving the three-dimensional (3D) equilibrium equations including heat conduction and thermoelastic theory. The validity of mathematical formula and computer programming is proved by convergence and comparison examples with finite element method (FEM). In order to simplify the complex calculation of exact solution, a back propagation neural network (BPNN) is further trained with a database containing 9504 sets of thermo-mechanical load conditions and their corresponding deformation which is solved by the exact solutions. Then the deformations of LCE plate subject to combined thermo-mechanical load can be predicted by this BP neural network instead of complex numerical calculation. Moreover, it is also applied to inverse the contact mechanical force at the bottom surface of LCE plate with a given deformation and temperature conditions. The results show that: (1) The results from the exact theoretical solution are in consistence with that from FEM but have a higher computational efficiency and stability; (2) The deformation of the laminated plate is more sensitive to the layered thickness of LCE than the variation of the temperature; (3) 3-D elasticity solutions of a laminated LCE plate under the combined thermos-mechanical load can be effectively predicted by a trained BP neural network.

A uniform framework for the dynamic behavior of linearized anisotropic elastic rods

A uniform framework for the dynamic behavior of a rod composed of a linearized anisotropic material is constructed based on an asymptotic reduction method. Taylor–Young series expansions of the displacement vector and stress tensor are adopted at the beginning. By substituting the series expansions into the three-dimensional (3D) equilibrium equations and the lateral traction condition followed by proper manipulations, four scalar rod equations are obtained for the leading-order displacement vector and the twist angle of the central line. The associated one-dimensional (1D) boundary conditions are obtained through the virtual work principle. One key feature of the present theory is the establishment of recursive relations so that most of the unknowns from the expansion can be eliminated. Also, all the remainders are retained, so the pointwise error estimate of the equations is established. One important advantage of the present theory is that no ad hoc assumption on the forms of displacement and external loading is adopted. Therefore, the rod theory provides a uniform framework to study dynamic behaviors. As an illustrative example, the natural vibration of a linearized isotropic rod with fixed-free boundary conditions is studied. The natural frequency is calculated by applying this uniform framework for the bending, torsional, and longitudinal modes. The obtained results are compared with those of the 3D simulations, Euler–Bernoulli, and Timoshenko beam theories. It turns out that the present theory is accurate enough for moderate and long rods in the bending vibration, and provides explicit results with high accuracy for either long or short rods in torsional and longitudinal vibrations.

3D hygro-elastic shell model for the analysis of composite and sandwich structures

The present work proposes the study of the hygrometric loading effects in the static analysis of multilayered composite and sandwich plates and shells. The employed model is based on a general exact 3D shell theory valid for one-layered and multilayered sandwich and composite plates, cylinders and cylindrical/spherical shell panels. The employed 3D equilibrium equations are developed in an orthogonal mixed curvilinear coordinate system for spherical shells in order to obtain the other simpler geometries as particular cases. A layer-wise approach is employed and equilibrium equations are solved in the thickness direction via the exponential matrix method. The partial derivatives in the in-plane directions are exactly calculated by assuming simply-supported edges and harmonic forms for the variables. The presented 3D shell model is extended to hygro-elastic cases by considering moisture content amplitudes imposed at the external surfaces of the structures in steady-state conditions. The moisture content profile through the thickness of the structures can be included in the 3D shell model using three different methodologies: it can be calculated by solving the steady-state 3D or 1D version of the Fick diffusion law, or it can be “a priori” assumed as linear through the thickness. Therefore, the developed system includes three non-homogeneous second order differential equilibrium equations that are solved after the introduction of opportune mathematical layers to consider the curvature effects through the thickness. The reduction to a first order differential equation system is obtained by redoubling the number of variables. The exponential matrix method is employed to accurately define both general and particular solutions. The implemented 3D hygro-elastic shell model is firstly validated and then it is employed with confidence for new benchmarks where the effects of thickness ratio, geometry, lamination scheme, material, thickness layer and imposed moisture content values at the external surfaces are investigated for composite and sandwich plates and shells. Showed results demonstrate the importance of an accurate developing of the elastic part of the 3D shell model combined with an appropriate evaluation of the moisture content profile through the thickness. Assumed linear moisture content profiles are correct only for thin one-layered isotropic structures, the use of the 1D Fick diffusion law allows only the correct definition of the material layer effect, the use of its 3D version allows the correct definition of both material and thickness layer effects.

A curved beam model with the asymptotic expansion method

In our previous work (Ferradi, 2015, 2016) [7,10], a model reduction technique was proposed for straight beams with a constant cross-section, based on the asymptotic expansion method applied to the general 3D equilibrium equations. An iterative scheme was then deduced, where 2D cross section modes, representing the general form of the kinematic, were derived at each order. The following paper is an extension of the work in Ferradi (2015, 2016) [7,10] to the curved beam elements, where in addition to the asymptotic expansion method, the differential geometry tools are used. Thus, a new efficient and locking free beam model is derived from the asymptotic expansion method, capable of representing accurately the full 3D stress and the displacement field of the curved beam, even in the vicinity of external forces. In each example, the results are compared to those from a reference shell model of the same beam geometry.

Interlaminar stress analysis of functionally graded graphene reinforced composite laminated plates based on a refined plate theory

Nowadays, graphene is considered as one of the most ideal reinforcements for composite materials due to the outstanding mechanical properties. From the literature survey, it is found that the researches on interlaminar stress analysis of functionally graded graphene reinforced composite (FG-GRC) laminated structures are scarce in literature. If transverse shear deformations are unable to be described accurately, the reasonable design of FG-GRC laminated structures will meet severe challenges due to the differentiation of material properties at the adjacent layers. Thereby, such issue is less studied by using the efficient models, so an advanced mixed-form plate theory is to be proposed for transverse shear stress analysis of FG-GRC laminated plates. The proposed theory can meet beforehand compatible conditions of transverse shear stresses at the interfaces of adjacent laminates and only contains seven independent displacement variables. By applying the three- dimensional (3D) equilibrium equations and the Reissner mixed variational theorem (RMVT), the accurate transverse shear stresses are obtained. The 3D elasticity solutions and the results computed by using the chosen models are selected to appraise the capability of the proposed model. Numerical results show that the proposed plate model can yield accurately transverse shear stresses without any post-processing procedure. In addition, the effects of graphene volume fraction, graphene distribution pattern, lamination sequence, span-to-thickness ratio and aspect ratio on the displacements and stresses of the FG-GRC laminated plates are thoroughly investigated.

A layerwise, stress approach model of laminated shells

In this work, layerwise SAM-L model for thin to moderately thick laminated shells is introduced. Generalized equations are obtained from a stress approximation and from Hellinger–Reissner functional (Reissner, 1950). Stress approximation verifies the 3D equilibrium equations and boundary conditions on shell faces. Hellinger–Reissner functional allows identifying 5 generalized displacements per layer. The equations of the model are obtained using Reissner’s variational principle. The model takes into account in its constitutive equations the Poisson’s effect of out-of-plane normal stresses on in-plane strains. To prove the accuracy of the model, SAM-L is applied to the analysis of laminated cylinders and one catenoid. SAM-L results are compared to exact solutions, solid finite elements, and other models available in the literature. SAM-L shows high quality in predicting the stress and displacement fields, even for thick shells. Edge effects on interlaminar stresses are assessed with high precision. SAM-L model proves to be an alternative to solid finite elements and other higher computing cost theories for structural analysis of doubly curved laminated shells.

A Stress Approach Model of Functionally Graded Shells

This paper presents a model called SAM-FG (stress approach model of functionally graded shells) for linear elastic, thin, and moderately thick shells made of functionally graded materials. The model is an extension of the SAM-H model, originally created for homogeneous shells. Assuming that the material is orthotropic and that one of its orthotropic directions is the thickness direction, the extension consists in considering that the 3D compliance tensor may depend on the thickness coordinate. The model starts with a tunable polynomial approximation of the 3D stress field that contains the same generalized forces as SAM-H. This stress approximation verifies the 3D equilibrium equations and the stress boundary conditions at the faces of the shell. As in SAM-H, 5 generalized displacements appear in SAM-FG. By applying the Hellinger–Reissner functional and Reissner’s variational method, the generalized forces, strains, and equations in SAM-FG turn out to be the same as in SAM-H, except for the generalized constitutive equations. To prove the accuracy of the model, SAM-FG is first applied to a simply supported, functionally graded plate and its results are compared to other models. To validate the model for shell-like structures, SAM-FG results are compared to those obtained with solid finite element calculations for three case studies of structures subjected to an internal pressure. The first one deals with a hollow sphere made of an isotropic functionally graded material. The second case considers a hollow cylinder made of an orthotropic functionally graded material. In the last case, a catenoid with an isotropic functionally graded material is studied. In all cases, the mean displacements are correctly predicted, even if the main purpose of the SAM-FG model is not to calculate these fields accurately. The stress field approximations are very accurate, and since the implementation of the shell model in a finite element code would imply 5 degrees of freedom per node, SAM-FG is a good alternative to solid finite element calculations for the structural analysis of functionally graded shells with a reasonable computational cost.

Free vibration analysis of laminated composite and sandwich plates based on a mixed zigzag theory

In this article, an accurate mixed zigzag theory (HZZTM) is proposed for free vibration analysis of laminated composite and sandwich plates. In-plane displacements in the proposed zigzag theory nonlinearly distribute through the thickness and exhibit an abrupt discontinuity of slope at interfaces, which coincide with three-dimensional analysis. In order to obtain the accurate transverse shear stresses, a preprocessing approach based on the three-dimensional (3D) equilibrium equations and the Reissner mixed variational theorem (RMVT) is used. It is significant that the second-order derivatives of in-plane displacement variables have been removed from the transverse shear stress fields, such that the finite element implementation is greatly simplified. Thus, based on the proposed zigzag model, a computationally efficient C0-type three-node triangular plate element with linear interpolation function is proposed for free vibration analysis of thick laminated composite and sandwich plates. Moreover, the accurate transverse shear stresses are introduced in the dynamic equilibrium equation by employing the Hamilton’s principle, which can actively improve the accuracy of natural frequencies of the laminated composite plates with different thickness and material properties at each ply. Performance of the proposed model is assessed by comparing with several benchmark solutions. Agreement between the present results and the reference solutions is very good, and the proposed model only includes the seven displacement variables which can demonstrate the accuracy and effectiveness of the proposed model.

Mesh-adapted stress analysis of multilayered plates using a layerwise model

This paper proposes a new finite-element modelling of a recent layerwise model for multilayered plates. This layerwise model is built from a specific 3D stress-field expansion along the thickness direction and involves, in particular, interlaminar transverse shear and out-of-plane stresses as generalized stresses. Its main feature is that 3D equilibrium equations and free-edge boundary conditions are directly taken into account into the stress-based construction of the model. A dual displacement-based finite-element discretization is implemented using the FEniCS software package and a remeshing strategy is proposed based on a novel error indicator. The error indicator is built based on the 3D stress field directly deduced from the layerwise generalized stresses and compared to a reconstructed stress field based on the model generalized displacements. The proposed error indicator is shown to identify the most critical parts of a laminate structure associated with complex 3D stress fields such as boundaries or stress concentration/singularity regions (near free-edges or delamination fronts). Through the combination of thickness discretization and in-plane mesh refinement in regions of interest, the proposed framework therefore offers an attractive alternative to 3D solid finite elements for an accurate prediction of stress states in composite laminates.

Active tuning and maximization of natural frequency in three-dimensional functionally graded shape memory alloy composite structures using meshless local Petrov–Galerkin method

A mesh-less method based on the local Petrov–Galerkin approach is developed and numerically implemented for active tuning and three-dimensional (3D) analysis of free vibration response of functionally graded (FG) shape memory alloy (SMA) sandwich composites. The local symmetric weak formulation is derived based on the 3D equilibrium equations of elasticity and the field variables are interpolated using the moving-least squares approach. Two main goals are followed. Firstly, free vibration analysis of a sandwich plate made of a SMA fiber reinforced composite core and two FG face sheets are presented. A parametric study is performed to investigate the influences of design parameters on the fundamental natural frequency (FNF) of composite structure and also maximize that. Secondly, SMA wires are embedded to fiber reinforced (FR) composite structures to enable precisely active control of their dynamic characteristics and change that in a required manner. To increase the controllability level and tune the complex exhibits, functionally graded shape memory alloys are considered which have the advantage of combining the functionalities of the shape memory effect and those of FG structures. This paper uses multi-layer activation patterns and the polynomial graded orientation angle of SMA wire along the structure thickness to furnish the FR composite designer with a powerful free vibration control tool. Finally, two activation techniques of SMA wires, the active property tuning (APT) and the active strain energy tuning (ASET) techniques, are compared. It is concluded from all investigated cases that the obtained FNF of ASET technique is always larger than the APT technique.

A Stress Approach Model of Moderately Thick, Homogeneous Shells

This paper presents the theoretical development of a new model of shells called SAM-H (Stress Approach Model of Homogeneous shells) and adapted for linear elastic shells, from thin to moderately thick ones. The model starts from an original stress polynomial approximation which involves the generalized forces and verifies the 3D equilibrium equations and the stress boundary conditions at the faces of the shell. Hellinger-Reissner functional and Reissner’s variational method are applied to determine the generalized fields and equations. The generalized forces and displacements are the same as those obtained in a classical, moderately thick shell model (CS model). The equilibrium and constitutive equations have some differences from those of a CS model, mainly in consideration of applied stress vectors at the upper and lower faces of the shell and the stiffness matrices. Another feature of the SAM-H model is the inclusion of the Poisson’s effect of out-of-plane normal stresses on in-plane strains. As a first application example to test the accuracy of the model, the case of a pressurized hollow sphere is considered. The analytical results of stresses and displacements of the SAM-H and CS models are compared to those of an exact 3D resolution. In this example, SAM-H model proves to be much more accurate than the CS model and its approximation of the normal out-of-plane stress is very precise. Finally, an implementation of the SAM-H model equations in a finite element software is performed and a case study is analyzed to show the advantages of using the SAM-H model.