# A Stress Approach Model of Moderately Thick, Homogeneous Shells

Authors: Axel Fernando Domínguez Alvarado; Alberto Díaz Díaz
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

In structural engineering, models of plates and shells are necessary for carrying out reliable predictions of displacements and stresses without the excessive computational cost of three-dimensional solid finite element calculations. Displacement and stress approaches are commonly used as starting point for the construction of these models. In a displacement approach, the displacement field is approximated by linear combinations of friendly functions of the thickness coordinate such as polynomials. In a stress approach, an analogue approximation is made with the components of the 3D stress field. Additionally, in this last approach it is ideal that the approximate stress field verifies the 3D equilibrium equation and stress boundary conditions [1].In the family of linear elastic plate models, the Kirchhoff-Love theory [2] is the simplest one. In this theory, the plate thickness does not change during deformation and the in-plane displacements are approximated by first degree polynomials of the thickness coordinate while the out-of-plane displacement is uniform across the thickness. The coefficients of these polynomials depend only on the three displacements of the middle surface and their derivatives (these displacements can also be called generalized displacements). This model neglects out-of-plane shear forces and is usually applied to thin plates. A more complete plate model is Mindlin’s one [3] which considers the same polynomial degrees in the displacement approximation but includes two rotations in the coefficients of the in-plane displacement polynomials. Five generalized displacements are then involved in Mindlin’s model (three displacements and two rotations) and the linear out-of-plane strain remains zero as in Kirchhoff-Love theory. Another model of plates is Reissner’s model [4] which is based on a stress approach: the in-plane stresses are approximated by first degree polynomials while the out-of-plane shear and normal stresses are approximated by second degree and third degree polynomials, respectively. The stress approximation verifies the 3D equilibrium equation and the stress boundary conditions at the upper and lower faces of the plate. The coefficients of the stress polynomials are the stress resultants also called generalized forces. Owing to the shear correction factor introduced by Mindlin, the models of Reissner and Mindlin have similar plate equations: this explains the common denomination “Reissner-Mindlin” plate theory. These models provide accurate stiffness predictions of moderately thick plates and good but less precise results for thick shells. These models have been the starting point for the analysis of composite laminated plates and the development of other models such as higher-order models [5], zig-zag models [6], extensions of Reissner-Mindlin-like models [7, 8], and layerwise models [9]. In this last family of models, Pagano’s model [1], which proposes a polynomial stress approximation that verifies the 3D equilibrium in each layer, is worth citing. Pagano used Reissner’s variational mixed formulation [10] to build the model. No displacement approximation is made despite the use of a mixed formulation. The Hellinger-Reissner functional and the stress approximation helps to identify 2D generalized displacements. More recent developments of Pagano’s approach can be found in [11–13], where some simplifications are adopted to obtain more operational layerwise models. In [13], Alvarez-Lima et al. enhanced a layerwise model called M4-5N (originally developed by Chabot in [11]) in order to include the Poisson’s effect of out-of-plane normal stresses on in-plane strains which may have an important impact on plasticity and failure mechanisms. Most models of plates do not take into account this effect.Shell models can be classified depending on how the ratioη between the thickness to the smallest principal radius is considered in the model equations: thin and moderately thick shell models neglect any term multiplied by ηk for k≥1 and k≥2, respectively; thick shell models consider higher-degree terms. The simplest thin shell models are Love’s first approximation theories [2, 14–17] which neglect transverse shear deformation and are based on a displacement approach similar to that in Kirchhoff-Love plate theory. Other thin and moderately thick shell models apply a displacement approximation (in the curvilinear coordinate system) analogue to Mindlin’s plate model [18, 19]; for example, in [20], Toorani and Lakis proposed a generalization of the classical Reissner-Mindlin plate theory in order to include the curvature and transverse shear strains of the shell. These authors improved their model in [21] by enhancing the kinematic fields. In these displacement approach models, the 3D constitutive equations yield linear in-plane stresses and constant out-of-plane shear stresses across the thickness of the shell. For these models, the stress field does not verify the 3D equilibrium equations and the stress boundary conditions at the faces of the shell [22]. Moreover, the normal out-of-plane stresses and the linear out-of-plane strains are neglected and this may lead to important errors in thicker shells. These drawbacks have motivated the development of higher-degree displacement approximations, the use of better techniques to recover the 3D stress, and the construction of models based on stress or mixed stress/displacement approximations. In [23, 24], Tornabene and Viola used an approximation of the 3D displacement field to obtain all the generalized equations and generalized fields for shells of revolution. After solving the static problem, they applied the generalized differential quadrature method in order to approximate the spatial derivatives of the stress field. This approximation and Hooke’s law allowed recovering and correcting the 3D stress field which took into account the boundary conditions at the faces of the shell. One of the first stress approach models for shells is the one developed by Trefftz [25]; in this model, an approximation of in-plane stresses in terms of generalized forces is made, but transverse stresses are neglected. This implies that both 3D equilibrium equations and boundary conditions at the faces of the shell are not verified. In [26, 27], Synge and Chien proposed a mixed stress/displacement approach model for thin shells where stresses and displacements are approximated by polynomials of the thickness coordinate. In their theory, the boundary conditions at the faces of the shell are not fulfilled. In [28], Fang et al. developed a thick shell model using a mixed stress/displacement approach by applying Hellinger-Reissner functional [10]. In this mixed model, the stress field verifies the boundary conditions at the faces of the shell but not the 3D equilibrium equations. Shell models based on a pure stress approach are less usual than those based on a displacement approach; two noteworthy stress approach based models for cylindrical and spherical thick shells can be found in [29, 30], respectively. A summary and comparative study of several shell and plate theories was made by Leissa in [31, 32]. In Leissa’s work, all the shell models start with an approximation of the displacement field. There are few models based on a stress approach although these types of models have advantages over those based on displacement approximation.Many commercial finite element software packages offer surface elements to analyze shell structures. These elements are based on classical shell models. A family of these surface elements is the one introduced in the MITC (Mixed Interpolation of Tensorial Components) shell concept; this family is referred to in the literature as MITCn, where n is related to the number of nodes in the element [33]. Lee and Bathe [34] state that these elements development is based on the use of displacement interpolations and the selection of appropriate strain field interpolation. Finally, the element rigidity matrix is built using a stress-strain constitutive equation of classical shell models. Obviously, with this family of elements, the stress field does not verify the 3D equilibrium equations and the stress boundary conditions at the faces of the shell.In this paper, an original model of linear elastic moderately thick shells (η≤0.3) called SAM-H (Stress Approach Model of Homogeneous shells) is developed by making use of a method similar to that applied by Pagano in [1] and based on a stress approach which, in the limit case of a homogeneous plate, involves less generalized forces than Pagano’s model. Reissner’s variational mixed formulation [10] is applied to obtain the generalized fields and equations. The main contribution of the paper resides on the quality of the stress approximation and the simplicity of the model: the 3D equilibrium equations and the stress boundary conditions at the shell faces are verified, with the same number of generalized forces, moments, strains, and displacements as the linear elastic version of the classical model of general moderately thick shells proposed by Reddy [19]. Another originality of the model resides on taking into account and distinguishing the effect of stress vectors applied at the inner and outer faces of the shell. Finally, the inclusion of the Poisson’s effect of out-of-plane normal stresses on in-plane strains is another feature of the model that has not been found on any thin or moderately thick shell model.This paper is organized in six sections. In the first section, the adopted notations and assumptions are detailed. The second section recalls the Hellinger-Reissner variational principle of elastostatics. Then, in section three, the proposed stress approximation is shown. The generalized displacements, strains, equilibrium equations, and force edge conditions are obtained in section four. The generalized constitutive equations and displacement edge conditions are determined in section five. In the last section, the main differences between the SAM-H model and a classical shell model are discussed; a comparison of analytical and numerical results of these two models is also shown. Finally, an appendix containing further details of equations is included.


## 2. Notations, Geometry Definition, and Assumptions

A doubly curved shell is considered. Its thicknessh is uniform. (ξ1,ξ2,ξ3) denote the orthogonal curvilinear coordinates where ξ1 and ξ2 are the lines of curvature at the middle surface (ξ3=0). The principal radii of curvature are R1 and R2; the principal curvatures are then κ1=1/R1 and κ2=1/R2. The Lamé coefficients are defined by(1)A1=a11+ξ3R1,A2=a21+ξ3R2,A3=1where a1=g112, a2=g222; gαβ (α,β=1,2) are the surface metric tensor components. In what follows, the next notations are adopted:(i)
Greek subscripts take the values 1 and 2;(ii)
Latin subscripts take the values 1, 2, and 3;(iii)
the shell lies in the volumeΩ; its middle surface is ω;(iv)
bold face characters denote vectors, matrices, and tensors;(v)
unit vectors in the directions of the lines of curvature aree1 and e2; the unit vector normal to the middle surface is e3=e1×e2, where × is the cross product;(vi)
the dot product of vectors and simple contraction of tensors is noted “·” while the double contraction of tensors and the tensor product are noted “:” and “⊗”, respectively;(vii)
an underline and a double underline below a bold face character (for example:U_ and M__) denote, respectively, vectors and second-order tensors of the 2D space defined by the basis e1,e2;(viii)
κ__=κ1e1⊗e1+κ2e2⊗e2 and κ__′=κ2e1⊗e1+κ1e2⊗e2 are curvature tensors;(ix)
the stress vectors applied on the outer (ξ3=h/2) and inner (ξ3=-h/2) faces are s+ and s-, respectively; these vectors are defined by(2)s+ξ1,ξ2=τ_++σ+e3ands-ξ1,ξ2=τ_-+σ-e3,whereτ_+=τ1+e1+τ2+e2andτ_-=τ1-e1+τ2-e2;in the above expressionσ+ and σ- are the applied normal stresses while τα+ and τα- are the applied shear stresses.(x)
f is the 3D body load vector and its components in the curvilinear coordinate system are fi;(xi)
n_0=n10e1+n20e2 is the outward-pointing normal vector to the edge of the middle surface ω;(xii)
σ and u denote the 3D stress tensor and displacement field;(xiii)
divσ denotes the 3D divergence operator applied to the second-order tensor σ:(3)divσ=1A1A2A3v1+v2where vectorsv1 and v2 take the array form:(4)v1=∂A2A3σ11∂ξ1+∂A1A3σ12∂ξ2+∂A1A2σ13∂ξ3∂A2A3σ21∂ξ1+∂A1A3σ22∂ξ2+∂A1A2σ23∂ξ3∂A2A3σ31∂ξ1+∂A1A3σ32∂ξ2+∂A1A2σ33∂ξ3and(5)v2=A2a2∂a1∂ξ2σ21-A1a1∂a2∂ξ1σ22+A2a1R1σ31A1a1∂a2∂ξ1σ12-A2a2∂a1∂ξ2σ11+A1a2R2σ32-A2a1R1σ11-A1a2R2σ22;(xiv)
grad__g_ is the 2D gradient of the 2D vector g_=g1e1+g2e2; its components take the following array form:(6)grad__g_=1a1∂g1∂ξ1+g2a1a2∂a1∂ξ21a2∂g1∂ξ2-g2a1a2∂a2∂ξ11a1∂g2∂ξ1-g1a1a2∂a1∂ξ21a2∂g2∂ξ2+g1a1a2∂a2∂ξ1;(xv)
grad_p=1/a1∂p/∂ξ1e1+1/a2∂p/∂ξ2e2 is the 2D gradient of the 2D scalar function p;(xvi)
div_M__ is the divergence of the 2D second-order tensor M__=Mαβeα⊗eβ (sum on α and β); its components have the following array form:(7)div_M__=1a1a2∂a2M11∂ξ1+∂a1M12∂ξ2+M21∂a1∂ξ2-M22∂a2∂ξ1∂a2M21∂ξ1+∂a1M22∂ξ2+M12∂a2∂ξ1-M11∂a1∂ξ2;(xvii)
the divergence of the 2D vector fieldv_=v1e1+v2e2 is defined by(8)divv_=1a1a2∂a2v1∂ξ1+∂a1v2∂ξ2.The following assumptions are adopted:(1)
small displacements and strains are assumed;(2)
the material is homogeneous and orthotropic, and one of its orthotropy directions is direction 3;S is the 3D, 4th-order compliance tensor, and its components are Sijkl; while handling plates and shells, it is useful to define the following compliance tensors and constants:(9)S____=∑α,β,γ,δ=12Sαβγδeα⊗eβ⊗eγ⊗eδ,Sc__=2∑α,β=12Sαβ33eα⊗eβ,SQ__=4∑α,β=12Sα3β3eα⊗eβandSσ=S3333;(3)
in all calculations the termsξ3/Rα and h/Rα are considered, but higher-order terms ξ3/Rαn and h/Rαn (n≥2) are neglected;(4)
the curvatures vary smoothly along the lines of curvature and their derivatives with respect toξα are neglected;(5)
in order to ease the generalized compliance calculations, the divergence of the shear stress vectors applied on the faces of the shell are assumed to be zero:(10)divτ_+=divτ_-=0;(6)
the body load vectorf is uniform through the thickness direction;(7)
no 3D displacement constraints are applied on the inner and outer faces of the shell.


## 3. Hellinger-Reissner Functional

On the boundaries∂Ωu and ∂Ωs of Ω, the body is subjected to the displacement vector ug and stress vector sg, respectively. In Figure 1, an example of the boundaries ∂Ωu and ∂Ωs is shown. The Hellinger-Reissner functional for elastic problems [10] applied to the shell is(11)H.R.u∗,σ∗=∫Ωσ∗:εu∗-f·u∗-we∗dΩ-∫∂Ωuσ∗·n·u∗-ugdS-∫∂Ωssg·u∗dSwhere n is the 3D outward-pointing normal vector to the boundary of Ω, u∗ is a piecewise C1 first order tensor field, σ∗ is a piecewise C1 second-order symmetric tensor field, ε is the Cauchy’s infinitesimal strain tensor, and we∗=1/2σ∗:S:σ∗ is the elastic energy density. In what follows, a field with an uppercase ∗ denotes a field that is not necessarily the solution of the problem and varies in its corresponding space. If a field (stress, generalized displacement, force, or moment) appears without uppercase ∗, this field is solution of the problem.Figure 1
Example of a shell with different regions of boundary conditions.In this paper, as in [35], an integration by parts is performed for the term containing σ∗:ε(u∗).(12)H.R.u∗,σ∗=-∫Ωdivσ∗·u∗+f·u∗+wedΩ+∫∂Ωuσ∗·n·ugdS+∫∂Ωsσ∗·n-sg·u∗dSReissner’s theorem [10] states that:(i)
the solution to the mechanical problem is the 3D displacement field/stress tensor field coupleu,σ that makes the H.R. functional stationary (notice that no uppercase ∗ appears for the solution);(ii)
the stationarity of H.R. with respect tou∗ yields the equilibrium equations and the boundary conditions on the stress vector while its stationarity with respect to σ∗ yields the constitutive equations and the boundary conditions on the displacement field.In this paper, a stress approach is proposed. The introduction of the stress approximation in the H.R. functional in (12) allows identifying the generalized displacements in the thickness integral of the term divσ∗·u∗. The stress approximation will lead to 5 generalized displacements (3 displacements and 2 rotations).


## 4. Stress Approximation

### 4.1. Polynomial Expressions

Let us define the following polynomial basis of 5-th degreeξ3-polynomials:(13)P0ξ3=1,P1ξ3=ξ3h,P2ξ3=-6ξ3h2+12,P3ξ3=-2ξ3h3+310ξ3h,P4ξ3=-143ξ3h4+ξ3h2-140andP5ξ3=ξ3h5-518ξ3h3+5336ξ3h.This polynomial basis is orthogonal:(14)∫-h/2h/2Pmξ3Pnξ3dξ3=0ifm≠n.The following stress approximations are selected:(15)σαβ∗ξ1,ξ2,ξ3=∑n=01σαβn∗ξ1,ξ2Pnξ3(16)σα3∗ξ1,ξ2,ξ3=∑n=03σα3n∗ξ1,ξ2Pnξ3(17)σ33∗ξ1,ξ2,ξ3=∑n=04σ33n∗ξ1,ξ2Pnξ3where σijn∗ are stress coefficients which will be expressed as linear combinations of the generalized forces. The polynomial degree of each stress component was chosen so as to verify the 3D equilibrium equation as will be shown in Section 4.3.


### 4.2. Generalized Forces

Let us define the components of the membrane forcesN__∗, shear forces Q_∗, and moments M__∗ as follows:(18)Nαβ∗=∫-h/2h/2σαβ∗1+ξ3R3-βdξ3(19)Qα∗=∫-h/2h/2σα3∗1+ξ3R3-αdξ3(20)Mαβ∗=∫-h/2h/2ξ3σαβ∗1+ξ3R3-βdξ3where the subscript 3-β is 2 if β=1, and 1 if β=2. The same can be said for the subscript 3-α. Let us point out that N__∗ and M__∗ may be nonsymmetric, while the stress tensor is always symmetric. Owing to assumption (3) and to the symmetry of the 3D stress tensor, N21 and M21 are related to N12 and M12 as follows:(21)N21=N12+δκM12andM21=h212δκN12+M12.Here δκ=κ2-κ1. By making use of assumption (3) in Section 2 and introducing the stress approximation defined in (15) and (16), one obtains the following.(22)Nαβ∗=hσαβ0∗+h212R3-βσαβ1∗(23)Qα∗=hσα30∗+h212R3-ασα31∗(24)Mαβ∗=h312R3-βσαβ0∗+h212σαβ1∗In the limit case of a plate,N__∗ and M__∗ are symmetric and thus the model contains 8 generalized forces and moments, 2 less than Pagano’s model [1] applied to a homogeneous plate. The generalized forces and moments of SAM-H are the same as those in the Reissner-Mindlin plate theory.


### 4.3. Determination of Coefficients in the Stress Approximation

Owing to (3), the 3D equilibrium equation yields the following.(25)v1+v2+A1A2f=0Introducing (4)-(5) and (15)–(17) in (25) yields(26)∑q=02η1qξ1,ξ2Pqξ3+∑q=04λ1qξ1,ξ2Pqξ3+rf1ξ1,ξ2∑q=02η2qξ1,ξ2Pqξ3+∑q=04λ2qξ1,ξ2Pqξ3+rf2ξ1,ξ2∑q=03η3qξ1,ξ2Pqξ3+∑q=05λ3qξ1,ξ2Pqξ3+rf3ξ1,ξ2=000where r=a1a21+ξ3(κ1+κ2) (the terms in καξ32 are neglected owing to assumption (3) in Section 2) and ηiq are nonnegligible linear combinations of the stress coefficients σijn∗ and their derivatives with respect to ξ1 and ξ2 (none of the terms in ηiq are multiples of καh2). The terms λiq are also a linear combination of the stress coefficients but they are neglected because they are multiples of καh2. By making use of the linear independence of polynomials Pq, the values of the stress coefficients are selected so as to obtain (i)
ηα2=0; this is ensured by selecting σα33∗ judiciously; in this manner, as mentioned at the end of Section 3, the terms ηα0 and ηα1 will provide two generalized equilibrium equations and two generalized displacements for each α value;(ii)
η31=-a1a2h(κ1+κ2)f3 (in order to obtain η31P1+rf3=0), η32=0, and η33=0; this is ensured by a suitable selection of the values of σ332∗, σ333∗, and σ334∗; in this manner, the term in η30 will yield one more generalized equilibrium equation and another generalized displacement.Owing to (22)–(24), assumptions (3) and (4) in Section 2, and the definition of the stress vectors applied at the inner and outer faces, the obtained stress coefficients σijn∗ are expressed in terms of the generalized forces and applied shear and normal stresses at the faces of the shell. In (A.1)–(A.3) in Appendix A, B, and C, the expression of these stress coefficients is shown. Let us point out that, with these stress coefficients, the 3D boundary conditions at the faces of the shell are verified:(27)∀ξ1,ξ2∈ω:τα+ξ1,ξ2=σα3ξ1,ξ2,h2,τα-ξ1,ξ2=-σα3ξ1,ξ2,-h2,σ+ξ1,ξ2=σ33ξ1,ξ2,h2andσ-ξ1,ξ2=-σ33ξ1,ξ2,-h2.The obtained stress field in a classical model of moderately thick shells (such as that based on a displacement approach and presented by Reddy in [19]) does not verify these conditions.


## 4.1. Polynomial Expressions

Let us define the following polynomial basis of 5-th degreeξ3-polynomials:(13)P0ξ3=1,P1ξ3=ξ3h,P2ξ3=-6ξ3h2+12,P3ξ3=-2ξ3h3+310ξ3h,P4ξ3=-143ξ3h4+ξ3h2-140andP5ξ3=ξ3h5-518ξ3h3+5336ξ3h.This polynomial basis is orthogonal:(14)∫-h/2h/2Pmξ3Pnξ3dξ3=0ifm≠n.The following stress approximations are selected:(15)σαβ∗ξ1,ξ2,ξ3=∑n=01σαβn∗ξ1,ξ2Pnξ3(16)σα3∗ξ1,ξ2,ξ3=∑n=03σα3n∗ξ1,ξ2Pnξ3(17)σ33∗ξ1,ξ2,ξ3=∑n=04σ33n∗ξ1,ξ2Pnξ3where σijn∗ are stress coefficients which will be expressed as linear combinations of the generalized forces. The polynomial degree of each stress component was chosen so as to verify the 3D equilibrium equation as will be shown in Section 4.3.


## 4.2. Generalized Forces

Let us define the components of the membrane forcesN__∗, shear forces Q_∗, and moments M__∗ as follows:(18)Nαβ∗=∫-h/2h/2σαβ∗1+ξ3R3-βdξ3(19)Qα∗=∫-h/2h/2σα3∗1+ξ3R3-αdξ3(20)Mαβ∗=∫-h/2h/2ξ3σαβ∗1+ξ3R3-βdξ3where the subscript 3-β is 2 if β=1, and 1 if β=2. The same can be said for the subscript 3-α. Let us point out that N__∗ and M__∗ may be nonsymmetric, while the stress tensor is always symmetric. Owing to assumption (3) and to the symmetry of the 3D stress tensor, N21 and M21 are related to N12 and M12 as follows:(21)N21=N12+δκM12andM21=h212δκN12+M12.Here δκ=κ2-κ1. By making use of assumption (3) in Section 2 and introducing the stress approximation defined in (15) and (16), one obtains the following.(22)Nαβ∗=hσαβ0∗+h212R3-βσαβ1∗(23)Qα∗=hσα30∗+h212R3-ασα31∗(24)Mαβ∗=h312R3-βσαβ0∗+h212σαβ1∗In the limit case of a plate,N__∗ and M__∗ are symmetric and thus the model contains 8 generalized forces and moments, 2 less than Pagano’s model [1] applied to a homogeneous plate. The generalized forces and moments of SAM-H are the same as those in the Reissner-Mindlin plate theory.


## 4.3. Determination of Coefficients in the Stress Approximation

Owing to (3), the 3D equilibrium equation yields the following.(25)v1+v2+A1A2f=0Introducing (4)-(5) and (15)–(17) in (25) yields(26)∑q=02η1qξ1,ξ2Pqξ3+∑q=04λ1qξ1,ξ2Pqξ3+rf1ξ1,ξ2∑q=02η2qξ1,ξ2Pqξ3+∑q=04λ2qξ1,ξ2Pqξ3+rf2ξ1,ξ2∑q=03η3qξ1,ξ2Pqξ3+∑q=05λ3qξ1,ξ2Pqξ3+rf3ξ1,ξ2=000where r=a1a21+ξ3(κ1+κ2) (the terms in καξ32 are neglected owing to assumption (3) in Section 2) and ηiq are nonnegligible linear combinations of the stress coefficients σijn∗ and their derivatives with respect to ξ1 and ξ2 (none of the terms in ηiq are multiples of καh2). The terms λiq are also a linear combination of the stress coefficients but they are neglected because they are multiples of καh2. By making use of the linear independence of polynomials Pq, the values of the stress coefficients are selected so as to obtain (i)
ηα2=0; this is ensured by selecting σα33∗ judiciously; in this manner, as mentioned at the end of Section 3, the terms ηα0 and ηα1 will provide two generalized equilibrium equations and two generalized displacements for each α value;(ii)
η31=-a1a2h(κ1+κ2)f3 (in order to obtain η31P1+rf3=0), η32=0, and η33=0; this is ensured by a suitable selection of the values of σ332∗, σ333∗, and σ334∗; in this manner, the term in η30 will yield one more generalized equilibrium equation and another generalized displacement.Owing to (22)–(24), assumptions (3) and (4) in Section 2, and the definition of the stress vectors applied at the inner and outer faces, the obtained stress coefficients σijn∗ are expressed in terms of the generalized forces and applied shear and normal stresses at the faces of the shell. In (A.1)–(A.3) in Appendix A, B, and C, the expression of these stress coefficients is shown. Let us point out that, with these stress coefficients, the 3D boundary conditions at the faces of the shell are verified:(27)∀ξ1,ξ2∈ω:τα+ξ1,ξ2=σα3ξ1,ξ2,h2,τα-ξ1,ξ2=-σα3ξ1,ξ2,-h2,σ+ξ1,ξ2=σ33ξ1,ξ2,h2andσ-ξ1,ξ2=-σ33ξ1,ξ2,-h2.The obtained stress field in a classical model of moderately thick shells (such as that based on a displacement approach and presented by Reddy in [19]) does not verify these conditions.


## 5. Generalized Fields and Equations

### 5.1. Generalized Displacements, Equilibrium Equations, Strains, and Stress Boundary Conditions

The application of the stress approximation in the Hellinger-Reissner functional in (12) makes the term ∫Ω‍divσ∗·u∗+f·u∗dΩ become(28)∫ωdiv_N__∗+κ__·Q_∗+h2κ1+κ2τ_+-τ_-+τ_++τ_-·U_∗dω+∫ωdivQ_∗+σ+-σ-2hκ1+κ2+σ++σ--N11∗κ1-N22∗κ2U3∗dω+∫ωdiv_M__∗-Q_∗+h24κ1+κ2τ_++τ_-+h2τ_+-τ_-·Φ_∗dω+∫ωhf_·U_∗+hf3U3∗+h3κ1+κ212f_·Φ_∗dωwhere f_=f1e1+f2e2,(29)U3∗=∫-h/2h/2P0hu3∗dξ3,U_∗ and Φ_∗ are 2D vectors whose components are defined on the basis of lines of curvature as follows:(30)U_∗=U1∗U2∗,U1∗=∫-h/2h/2P0hu1∗dξ3,U2∗=∫-h/2h/2P0hu2∗dξ3,Φ_∗=ϕ1∗ϕ2∗,ϕ1∗=∫-h/2h/212P1h2u1∗dξ3,ϕ2∗=∫-h/2h/212P1h2u2∗dξ3.Expression (28) allows identifying the following 5 generalized displacements: U1∗, U2∗, U3∗, ϕ1∗, and ϕ2∗ (the two latter can be regarded as rotations). Let us point out that in the stress approach, the approximation of the 3D displacement field u is not required. Nevertheless, one can state that its components take the following array form:(31)u=U1ξ1,ξ2P0ξ3+hϕ1ξ1,ξ2P1ξ3+δu1ξ1,ξ2,ξ3U2ξ1,ξ2P0ξ3+hϕ2ξ1,ξ2P1ξ3+δu2ξ1,ξ2,ξ3U3ξ1,ξ2P0ξ3+δu3ξ1,ξ2,ξ3where δu1, δu2, and δu3 verify(32)∫-h/2h/2Piδuαdξ3=0fori≥2and∫-h/2h/2Pjδu3dξ3=0forj≥1.The stationarity of the H.R. functional (in (12)) with respect to the generalized displacements leads to the following generalized equilibrium equations:(33)div_N__+κ__·Q_+h2κ1+κ2τ_+-τ_-+τ_++τ_-+hf_=0_,(34)divQ_+σ+-σ-2hκ1+κ2+σ++σ--N11κ1-N22κ2+hf3=0,(35)div_M__-Q_+h24κ1+κ2τ_++τ_-+h2τ_+-τ_-+h3κ1+κ212f_=0_.Let us point out that in other shell models such as that in [19], a supplementary algebraic equilibrium equation is considered:(36)M12κ2-M21κ1=N21-N12.In SAM-H model, this equation is not considered as an equilibrium equation. Equations (21) will be considered in the constitutive equations (C.4) of SAM-H model and, with this consideration, it can be easily verified that condition (36) is automatically satisfied by making use of assumption (3).The aforementioned stationarity for the boundary integral(37)∫∂Ωsσ∗·n-sg·u∗dSin the H.R. functional provides the boundary conditions on the generalized forces. Let us point out that the term σ∗·n-sg is zero on the inner and outer faces of the shell due to (27). Thus, the boundary integral reduces to an integration on the lateral boundaries (not the faces) of the structure. At these lateral boundaries, the following assumptions are adopted for the stress vector sg: the first and second components of this vector (s1g and s2g) are first degree ξ3-polynomials, while the third component (s3g) is uniform through the thickness of the shell. Owing to these assumptions and to assumption (3) in Section 2, the aforementioned stationarity yields the boundary conditions on the generalized forces:(38)N__·n_0=F_g,Q_·n_0=F3g,M__·n_0=C_g.Here F_g, F3g, and C_g are the given membrane force vector, shear force, and bending moment vector at the shell edges, respectively. Their components are(39)Fαg=∫-h/2h/2P0+hn102κ1+n202κ2P1sαgdξ3,F3g=∫-h/2h/2P0+hn102κ1+n202κ2P1s3gdξ3andCαg=∫-h/2h/2h212n102κ1+n202κ2P0ξ3sαgdξ3+∫-h/2h/2hP1ξ31-h6n102κ1+n202κ2sαgdξ3.Once the generalized displacements are identified, by means of an integration by parts the term-∫Ω‍divσ∗·u∗+f·u∗dΩ in the H.R. functional becomes(40)∫ωN__∗:ε__∗t+Q_∗·d_∗+M__∗:χ__∗tdω+Twhere the superscript t denotes the transposition of a tensor; T is a term involving the generalized forces and displacements evaluated at the edges of the middle surface ω; ε__∗, χ__∗, and d_∗ are the generalized strain tensors defined as follows:(41)ε__∗=grad__U_∗+U3∗κ__,χ__∗=grad__Φ_∗,d_∗=grad_U3∗+Φ_∗-κ__·U_∗.


### 5.2. Generalized Constitutive Equations and Displacement Boundary Conditions

Let us now introduce the stress approximation in the expression of the volume elastic energy densitywe∗ in (12). The integral of this energy can be written as follows:(42)∫Ωwe∗dΩ=∫ωwes∗+wen∗+weQ∗+wec∗dωwhere wes∗, wen∗, weQ∗, and wec are the surface densities of elastic energy due to in-plane stresses σαβ, elastic energy due to normal out-of-plane stresses σ33, elastic energy due to out-of-plane shear stresses σα3, and elastic energy due to coupling between in-plane stresses and normal out-of-plane stresses, respectively. Let us point out that in moderately thick plate models and shell models, wec is neglected; this energy accounts for the 3D Poisson’s effect of normal stresses σ33 on in-plane strains. Owing to the definitions of compliance tensors and constants in (9), these surface densities of energy are defined by(43)wes∗=12∫-h/2h/2σ__∗:S____:σ__∗1+ξ3κ11+ξ3κ2dξ3wen∗=12∫-h/2h/2σ33∗Sσσ33∗1+ξ3κ11+ξ3κ2dξ3,weQ∗=12∑α=12∑β=12∫-h/2h/2σα3∗SαβQσβ3∗1+ξ3κ11+ξ3κ2dξ3andwec∗=12∫-h/2h/2σ__∗:Sc__σ33∗1+ξ3κ11+ξ3κ2dξ3.Let us express the surface densities of energy as algebraic functions of the generalized forces, the third component of the body load (f3), and the components of the stress vectors applied at the shell faces (s+ and s-). Let us first apply in the equations above the stress approximation detailed in (15)–(17) and its corresponding stress coefficients determined in (A.1)–(A.3) in Appendix A, B, and C. Some terms in these stress coefficients yield directly the desired algebraic combinations, and the others involve the following divergences:(i)
divQ_∗, hdivκ__′·Q_∗, hdivκ__·Q_∗,(ii)
div_M__∗·κ__′, hdivdiv_M__∗·κ__′,(iii)
divτ_+, divτ_-, divκ__′·τ_+, divκ__·τ_+, divκ__′·τ_- and divκ__·τ_-.These divergences can be expressed as linear algebraic functions of the generalized forces andf3. In this manner, one obtains that we∗ is a quadratic function of the algebraic values of the components of f, s+, s- and the generalized forces.By making use of the stationarity of the H.R. functional with respect to the generalized forces, one obtains the generalized constitutive equations in a tensor form. It is easier to write the generalized constitutive equations in terms of the following engineering vectors of generalized forcesq and strains ϵ:(44)q=N^M^,ϵ=ε^χ^,where(45)N^=N11N22N12N21,M^=M11M22M12M21,ε^=ε11ε22ε12ε21,andχ^=χ11χ22χ12χ21.In this manner, the constitutive equations can be grouped in two vector equations:(46)ϵ=Cq+σ+c++σ-c-+f3c3andd_=DQ_+D+τ_++D-τ_-,where(i)
C is an 8×8 compliance matrix,(ii)
c+, c-, and c3 are 8-component compliance vector,(iii)
D, D+, and D- are 2×2 compliance matrices.The components of these matrices and vectors are provided in AppendixB.AppendixC proves that there is an alternative form of the constitutive equations which expresses the generalized forces as linear combinations of the generalized strains and the applied loads.The constitutive equations can be grouped in the following two vector equations:(47)q=Kϵ+σ+k++σ-k-+f3k3,Q_=Ld_+L+τ_++L-τ_-.The expressions of the above matrices (K, L, L+, and L-) and vectors (k+, k-, and k3) are shown in Appendix C. The resulting N21 and M21 generalized forces obtained by means of (47) verify (21) and the supplementary algebraic “equilibrium” (36) included in Sanders model [14].The stationarity of the H.R. functional (in (11)) with respect to the generalized forces also provides in the -∫∂Ωuσ∗·n·u∗-ugdS term the generalized displacement boundary conditions on the edge of the shell:(48)U_=∫-h/2h/2P0hu1ge1+u2ge2dξ3,U3=∫-h/2h/2P0hu3gdξ3,Φ_=∫-h/2h/212P1h2u1ge1+u2ge2dξ3.


## 5.1. Generalized Displacements, Equilibrium Equations, Strains, and Stress Boundary Conditions

The application of the stress approximation in the Hellinger-Reissner functional in (12) makes the term ∫Ω‍divσ∗·u∗+f·u∗dΩ become(28)∫ωdiv_N__∗+κ__·Q_∗+h2κ1+κ2τ_+-τ_-+τ_++τ_-·U_∗dω+∫ωdivQ_∗+σ+-σ-2hκ1+κ2+σ++σ--N11∗κ1-N22∗κ2U3∗dω+∫ωdiv_M__∗-Q_∗+h24κ1+κ2τ_++τ_-+h2τ_+-τ_-·Φ_∗dω+∫ωhf_·U_∗+hf3U3∗+h3κ1+κ212f_·Φ_∗dωwhere f_=f1e1+f2e2,(29)U3∗=∫-h/2h/2P0hu3∗dξ3,U_∗ and Φ_∗ are 2D vectors whose components are defined on the basis of lines of curvature as follows:(30)U_∗=U1∗U2∗,U1∗=∫-h/2h/2P0hu1∗dξ3,U2∗=∫-h/2h/2P0hu2∗dξ3,Φ_∗=ϕ1∗ϕ2∗,ϕ1∗=∫-h/2h/212P1h2u1∗dξ3,ϕ2∗=∫-h/2h/212P1h2u2∗dξ3.Expression (28) allows identifying the following 5 generalized displacements: U1∗, U2∗, U3∗, ϕ1∗, and ϕ2∗ (the two latter can be regarded as rotations). Let us point out that in the stress approach, the approximation of the 3D displacement field u is not required. Nevertheless, one can state that its components take the following array form:(31)u=U1ξ1,ξ2P0ξ3+hϕ1ξ1,ξ2P1ξ3+δu1ξ1,ξ2,ξ3U2ξ1,ξ2P0ξ3+hϕ2ξ1,ξ2P1ξ3+δu2ξ1,ξ2,ξ3U3ξ1,ξ2P0ξ3+δu3ξ1,ξ2,ξ3where δu1, δu2, and δu3 verify(32)∫-h/2h/2Piδuαdξ3=0fori≥2and∫-h/2h/2Pjδu3dξ3=0forj≥1.The stationarity of the H.R. functional (in (12)) with respect to the generalized displacements leads to the following generalized equilibrium equations:(33)div_N__+κ__·Q_+h2κ1+κ2τ_+-τ_-+τ_++τ_-+hf_=0_,(34)divQ_+σ+-σ-2hκ1+κ2+σ++σ--N11κ1-N22κ2+hf3=0,(35)div_M__-Q_+h24κ1+κ2τ_++τ_-+h2τ_+-τ_-+h3κ1+κ212f_=0_.Let us point out that in other shell models such as that in [19], a supplementary algebraic equilibrium equation is considered:(36)M12κ2-M21κ1=N21-N12.In SAM-H model, this equation is not considered as an equilibrium equation. Equations (21) will be considered in the constitutive equations (C.4) of SAM-H model and, with this consideration, it can be easily verified that condition (36) is automatically satisfied by making use of assumption (3).The aforementioned stationarity for the boundary integral(37)∫∂Ωsσ∗·n-sg·u∗dSin the H.R. functional provides the boundary conditions on the generalized forces. Let us point out that the term σ∗·n-sg is zero on the inner and outer faces of the shell due to (27). Thus, the boundary integral reduces to an integration on the lateral boundaries (not the faces) of the structure. At these lateral boundaries, the following assumptions are adopted for the stress vector sg: the first and second components of this vector (s1g and s2g) are first degree ξ3-polynomials, while the third component (s3g) is uniform through the thickness of the shell. Owing to these assumptions and to assumption (3) in Section 2, the aforementioned stationarity yields the boundary conditions on the generalized forces:(38)N__·n_0=F_g,Q_·n_0=F3g,M__·n_0=C_g.Here F_g, F3g, and C_g are the given membrane force vector, shear force, and bending moment vector at the shell edges, respectively. Their components are(39)Fαg=∫-h/2h/2P0+hn102κ1+n202κ2P1sαgdξ3,F3g=∫-h/2h/2P0+hn102κ1+n202κ2P1s3gdξ3andCαg=∫-h/2h/2h212n102κ1+n202κ2P0ξ3sαgdξ3+∫-h/2h/2hP1ξ31-h6n102κ1+n202κ2sαgdξ3.Once the generalized displacements are identified, by means of an integration by parts the term-∫Ω‍divσ∗·u∗+f·u∗dΩ in the H.R. functional becomes(40)∫ωN__∗:ε__∗t+Q_∗·d_∗+M__∗:χ__∗tdω+Twhere the superscript t denotes the transposition of a tensor; T is a term involving the generalized forces and displacements evaluated at the edges of the middle surface ω; ε__∗, χ__∗, and d_∗ are the generalized strain tensors defined as follows:(41)ε__∗=grad__U_∗+U3∗κ__,χ__∗=grad__Φ_∗,d_∗=grad_U3∗+Φ_∗-κ__·U_∗.


## 5.2. Generalized Constitutive Equations and Displacement Boundary Conditions

Let us now introduce the stress approximation in the expression of the volume elastic energy densitywe∗ in (12). The integral of this energy can be written as follows:(42)∫Ωwe∗dΩ=∫ωwes∗+wen∗+weQ∗+wec∗dωwhere wes∗, wen∗, weQ∗, and wec are the surface densities of elastic energy due to in-plane stresses σαβ, elastic energy due to normal out-of-plane stresses σ33, elastic energy due to out-of-plane shear stresses σα3, and elastic energy due to coupling between in-plane stresses and normal out-of-plane stresses, respectively. Let us point out that in moderately thick plate models and shell models, wec is neglected; this energy accounts for the 3D Poisson’s effect of normal stresses σ33 on in-plane strains. Owing to the definitions of compliance tensors and constants in (9), these surface densities of energy are defined by(43)wes∗=12∫-h/2h/2σ__∗:S____:σ__∗1+ξ3κ11+ξ3κ2dξ3wen∗=12∫-h/2h/2σ33∗Sσσ33∗1+ξ3κ11+ξ3κ2dξ3,weQ∗=12∑α=12∑β=12∫-h/2h/2σα3∗SαβQσβ3∗1+ξ3κ11+ξ3κ2dξ3andwec∗=12∫-h/2h/2σ__∗:Sc__σ33∗1+ξ3κ11+ξ3κ2dξ3.Let us express the surface densities of energy as algebraic functions of the generalized forces, the third component of the body load (f3), and the components of the stress vectors applied at the shell faces (s+ and s-). Let us first apply in the equations above the stress approximation detailed in (15)–(17) and its corresponding stress coefficients determined in (A.1)–(A.3) in Appendix A, B, and C. Some terms in these stress coefficients yield directly the desired algebraic combinations, and the others involve the following divergences:(i)
divQ_∗, hdivκ__′·Q_∗, hdivκ__·Q_∗,(ii)
div_M__∗·κ__′, hdivdiv_M__∗·κ__′,(iii)
divτ_+, divτ_-, divκ__′·τ_+, divκ__·τ_+, divκ__′·τ_- and divκ__·τ_-.These divergences can be expressed as linear algebraic functions of the generalized forces andf3. In this manner, one obtains that we∗ is a quadratic function of the algebraic values of the components of f, s+, s- and the generalized forces.By making use of the stationarity of the H.R. functional with respect to the generalized forces, one obtains the generalized constitutive equations in a tensor form. It is easier to write the generalized constitutive equations in terms of the following engineering vectors of generalized forcesq and strains ϵ:(44)q=N^M^,ϵ=ε^χ^,where(45)N^=N11N22N12N21,M^=M11M22M12M21,ε^=ε11ε22ε12ε21,andχ^=χ11χ22χ12χ21.In this manner, the constitutive equations can be grouped in two vector equations:(46)ϵ=Cq+σ+c++σ-c-+f3c3andd_=DQ_+D+τ_++D-τ_-,where(i)
C is an 8×8 compliance matrix,(ii)
c+, c-, and c3 are 8-component compliance vector,(iii)
D, D+, and D- are 2×2 compliance matrices.The components of these matrices and vectors are provided in AppendixB.AppendixC proves that there is an alternative form of the constitutive equations which expresses the generalized forces as linear combinations of the generalized strains and the applied loads.The constitutive equations can be grouped in the following two vector equations:(47)q=Kϵ+σ+k++σ-k-+f3k3,Q_=Ld_+L+τ_++L-τ_-.The expressions of the above matrices (K, L, L+, and L-) and vectors (k+, k-, and k3) are shown in Appendix C. The resulting N21 and M21 generalized forces obtained by means of (47) verify (21) and the supplementary algebraic “equilibrium” (36) included in Sanders model [14].The stationarity of the H.R. functional (in (11)) with respect to the generalized forces also provides in the -∫∂Ωuσ∗·n·u∗-ugdS term the generalized displacement boundary conditions on the edge of the shell:(48)U_=∫-h/2h/2P0hu1ge1+u2ge2dξ3,U3=∫-h/2h/2P0hu3gdξ3,Φ_=∫-h/2h/212P1h2u1ge1+u2ge2dξ3.


## 6. Results and Discussion

### 6.1. Main Differences with a Classical Model of Shells

In [19], Reddy builds a model of homogeneous general shells (from thin to moderately thick) which generalizes Sanders shell theory [14]. The generalization consists of including von Kármán nonlinear strains. In order to compare the linear SAM-H model to Reddy’s model, the nonlinear terms are not considered herein. Hereafter, the linear version of Reddy’s model will be referred to as the classical shell model (CS model). The CS model is based on an approximation of the 3D displacement field components in the curvilinear coordinate system:(49)uαξ1,ξ2,ξ3=Uαcξ1,ξ2+hϕαcP1ξ3u3ξ1,ξ2,ξ3=U3cξ1,ξ2which involves 5 generalized displacements (U1c, U2c, U3c, ϕ1c, ϕ2c). The first 3 generalized displacements are the 3D displacements at the middle surface. The 5 generalized displacements can be expressed as thickness integrals of the 3D displacements in a similar manner to that in (29) and (30). In this sense, the generalized displacements of the SAM-H model are equivalent to those of the classical shell model (CS model). SAM-H model does not propose an approximation of the 3D displacements but it has the same number of generalized displacements as the CS model. From SAM-H generalized displacements, one may obtain some characteristics of the 3D displacement field. By applying the 3D gradient to the 3D displacement field in (49), one obtains the 3D strains of the CS model. Let us point out that the 3D linear normal strain component ε33 is zero. These 3D strains can be expressed as linear algebraic combinations of the generalized strains of the CS model. The generalized strains are related to the 2D gradients of the generalized displacements of the CS model in a similar manner as in the SAM-H model (see (41)):(50)ε__c=grad__Uc_+U3cκ__,χ__c=grad__Φc_,d_c=grad_U3c+Φ_c-κ__·Uc_.Let us point out that the SAM-H model does not propose any approximation of the 3D strain tensor. By applying Hamilton’s variational principle to the shell, Reddy obtains the generalized equilibrium equations of the CS model [19].(51)div_Nc__+κ__·Qc_+feq_=0divQc_-N11cκ1-N22cκ2+f3eq=0div_Mc__-Qc_=0where Nc__, Qc_, and Mc__ are the generalized forces of the CS model and have a definition similar to those of the SAM-H model in (18) to (20); feq_ and f3eq are the equivalent middle surface in-plane and out-of-plane loads, respectively. The additional equilibrium equation (36) in Sanders model [14] is also considered.The equivalent middle surface loads can model the effect of the body loads and the loads applied at the faces of the shell. Let us point out that in SAM-H model, the body loads and the applied stresses on the faces appear explicitly in its equations. Besides, in SAM-H model, in some terms of the equilibrium equations, these loads and stresses are multiplied by the principal curvatures, and the loads applied on the inner and outer faces of the shell are distinguished.In the CS model, after obtaining the approximate 3D strains, the 3D constitutive equations for a homogeneous shell yield the stress field approximation. The CS model neglects out-of-plane normal stressesσ33. An integration through the thickness yields the generalized constitutive equations:(52)Nc^Mc^=AcBcBcDc·εc^χc^andQc_=Kc__·dc_,where Nc^, Mc^, εc^, and χc^ have definitions analogue to those in (45); Ac, Bc, Dc, and Kc__ are stiffness matrices. Let us point out that in the CS model an “ABBD” stiffness matrix appears while an “ABCD” stiffness matrix is expected in the SAM-H model (see the expression of the generalized compliance matrices in (B.1)). Another difference is the fact that in the CS model, the constitutive equations do not involve the body loads and the loads applied at the faces of the shell; in SAM-H model, these loads appear explicitly in the constitutive equations. The Poisson’s effects of the out-of-plane normal stresses on in-plane strains are considered in the SAM-H model but not in the CS model.


### 6.2. Analytical Results

Let us now compare the analytical solutions of the CS and SAM-H models in the case of a hollow sphere subjected to an internal pressurep. The thickness of the sphere is h; the radius of the middle surface is R. The material is isotropic, and its Young’s modulus and Poisson’s ratio are, respectively, E and ν. Vector e3 points from inside to outside the sphere. The analytical resolution of the 3D problem with spherical symmetry leads to the following radial displacement (u3) and normal stress σ33:(53)u33Dξ¯3=pRE1-η/2321-2ν1+ηξ¯33+1+ν1+η/2321+η/23-1-η/231+ηξ¯32,andσ333Dξ¯3=p1-η/231+η/23-1-η/231-1+η/21+ηξ¯33,where η=h/R is the thickness to radius ratio and ξ¯3=ξ3/h is the normalized position through the thickness. The analytical resolution of the equations of the CS model yields the generalized displacement and the normal stress:(54)U3CS=pRE1-ν2ηandσ33CS=0.The generalized displacement and normal stress of the SAM-H model are(55)U3SAM=U3CS-pRE1-2ν2+3140η+13ν30η+ν10η2,andσ33SAMξ¯3=p-1-η2+ξ¯3-2ξ¯32η+η28-η2ξ¯322.Let us first compare the displacement solutions of the CS and SAM-H models with the mean 3D displacement(56)U33D=∫-h/2h/2P0hu33Dξ3hdξ3=pRE21-η/231-2ν+1-η/221+η/221+ν21+η/23-1-η/23.Let us define the relative errors(57)δU3SAM=U3SAM-U33DU33DandδU3CS=U3CS-U33DU33Din the evaluation of the mean displacement for the CS and SAM-H models. These errors only depend on η and ν. In Figures 2(a) and 2(b), the relative errors δU3SAM and δU3CS are plotted against the thickness to radius ratio η for five values of the Poisson’s ratio (-0.4, -0.2, 0, 0.2, and 0.4), respectively. The maximum value of the thickness to radius ratio considered in this example is 0.5; for moderately thick shells, this ratio is less than 0.3 (this limit will be justified below). The absolute value of the relative error δU3SAM of the SAM-H model increases as η and ν increase but is less than 4% for moderately thick shells and less than 14% for thicker shells. The relative error δU3CS of the CS model increase as η increases and ν decreases; it may be more than 50% for moderately thick shells and more than 100% for thicker shells. The SAM-H model provides a far better approximation than the CS model for the mean normal displacement. Let us point out that SAM-H accuracy in the prediction of U3 is mainly due to the consideration of the elastic energy wec (see (43)) corresponding to the coupling between in-plane stresses and normal out-of-plane stresses. In Figure 3, the relative error δU3SAM in the calculation of U3 without taking into account wec is plotted against η for five values of the Poisson’s ratio (-0.4, -0.2, 0, 0.2, and 0.4). Neglecting wec can make this error greater than 20% for the thickest moderately thick shell (η=0.3); this value is 5 times greater than that considering wec. This confirms the utility of taking into account wec (the Poisson’s effect of the out-of-plane normal stresses on in-plane strains). In what follows, this energy is taken into account in the SAM-H model results.Figure 2
Relative displacement errors vs thickness to radius ratio for the SAM-H (a) and CS (b) models.
(a) (b)Figure 3
Relative displacement error vs thickness to radius ratio for the SAM-H without taking into accountwec.Let us now analyze the normal stressesσ33. In the CS model, these stresses are zero. The SAM-H model predicts nonzero σ33 stresses and in order to compare the SAM-H model results to those of the 3D case, the normalized stresses σ33/p are analyzed. In Figure 4, the normalized stresses calculated by the SAM-H model and the 3D model are plotted against the dimensionless position ξ¯3 for three values of the thickness to radius ratio: η=0.1, 0.3, and 0.5. One can observe that the accuracy of the SAM-H results decreases as η increases. In spite of this, the SAM-H results are very accurate even for the thickest case (η=0.5) and the boundary conditions at the inner (ξ¯3=-0.5) and outer (ξ¯3=0.5) faces of the shell are verified.Figure 4
Normalized 3D and SAM-H normal stressesσ33/p vs dimensionless position ξ¯3.


### 6.3. Numerical Results

In this subsection, SAM-H model equations are solved by means of the finite element method implemented in COMSOL Multiphysics software (5.3 version). To test the accuracy of SAM-H model, its results are compared to those obtained with(i)
solid finite elements (SFE) available in COMSOL Multiphysics;(ii)
the classical shell model (CS model); its equations are implemented in COMSOL Multiphysics in a similar manner to how SAM-H equations were treated;(iii)
MITC6 shell finite elements [33] (triangular elements) available in COMSOL Multiphysics.Let us consider a 2m tall hollow cylinder. The radius of the midsurface and the wall thickness are 1m and 0.3m, respectively. A 1MPa pressure is applied on a 0.25m tall circumferential section at the outer face of the cylinder. The resolution of the problem using solid finite elements (SFE) can be simplified due to symmetries as shown in Figure5(a). The shell-type computations (SAM-H, CS, and MITC6) use the mesh shown in Figure 5(b). The material is isotropic and its Young’s modulus and Poisson’s ratio are 200GPa and 0.3.Figure 5
Constriction of a thick hollow cylinder: modelling using solid finite elements (a) and mesh for shell models (b).
(a) (b)The resolution of the generalized equations of SAM-H and CS models allows building the approximate 3D stress field and plotting the stress components on the cross section of the cylinder drawn in Figure5. COMSOL Multiphysics offers the possibility of obtaining from the MITC6 results the 3D stress field in a given position through the thickness of the solid. In Figure 6, the normal stresses σrr calculated by solid finite elements (SFE), the shell models (SAM-H and CS), and MITC6 shell elements are plotted on the cross section. The σrr stresses calculated by the CS model and MITC6 method are neglected whereas that calculated by SAM-H varies across the thickness in a similar manner to the SFE stress. The pressure application on a delimited portion of the outer face provokes the existence of a σrz shear stress. In Figure 7, the color maps of the calculated σrz stresses by SFE, SAM-H, CS, and MITC6 methods are plotted on the cross section of the cylinder. The CS model predicts a constant through thickness shear stress whereas SAM-H model yields a variable shear stress that verifies the boundary conditions at the inner and outer faces, in a similar manner to the MITC6 method. The values of the σrz stress calculated with SAM-H model seem more to the SFE results than those calculated with the CS model. In Figure 8, the von Mises stresses obtained by SFE, SAM-H, CS, and MITC6 are plotted on the cross section of the cylinder. Once again, one can note that SAM-H results are more accurate than those of the CS model and MITC6 method. The maximum von Mises stress at the inner face of the cylinder calculated by SFE, SAM-H, CS, and MITC6 methods is 2.15MPa, 2.10MPa, 1.91 MPa, and 1.63MPa, respectively. In Figure 8, it can be seen that the location of the maximum value of the von Mises stress is at the inner face of the shell for SFE and SAM-H models while that for CS model is at the outer face of the shell. SAM-H provides a better stress approximation than the CS and MITC6 methods for this mechanical problem.Figure 6
Normal stressσrr (in MPa) in the cross section of a constricted cylinder (SFE, SAM-H, CS, and MITC6 results).Figure 7
Shear stressσrz (in MPa) in the cross section of a constricted cylinder (SFE, SAM-H, CS, and MITC6 results).Figure 8
Von Mises stress (in MPa) in the cross section of a constricted cylinder (SFE, SAM-H, CS, and MITC6 results).


## 6.1. Main Differences with a Classical Model of Shells

In [19], Reddy builds a model of homogeneous general shells (from thin to moderately thick) which generalizes Sanders shell theory [14]. The generalization consists of including von Kármán nonlinear strains. In order to compare the linear SAM-H model to Reddy’s model, the nonlinear terms are not considered herein. Hereafter, the linear version of Reddy’s model will be referred to as the classical shell model (CS model). The CS model is based on an approximation of the 3D displacement field components in the curvilinear coordinate system:(49)uαξ1,ξ2,ξ3=Uαcξ1,ξ2+hϕαcP1ξ3u3ξ1,ξ2,ξ3=U3cξ1,ξ2which involves 5 generalized displacements (U1c, U2c, U3c, ϕ1c, ϕ2c). The first 3 generalized displacements are the 3D displacements at the middle surface. The 5 generalized displacements can be expressed as thickness integrals of the 3D displacements in a similar manner to that in (29) and (30). In this sense, the generalized displacements of the SAM-H model are equivalent to those of the classical shell model (CS model). SAM-H model does not propose an approximation of the 3D displacements but it has the same number of generalized displacements as the CS model. From SAM-H generalized displacements, one may obtain some characteristics of the 3D displacement field. By applying the 3D gradient to the 3D displacement field in (49), one obtains the 3D strains of the CS model. Let us point out that the 3D linear normal strain component ε33 is zero. These 3D strains can be expressed as linear algebraic combinations of the generalized strains of the CS model. The generalized strains are related to the 2D gradients of the generalized displacements of the CS model in a similar manner as in the SAM-H model (see (41)):(50)ε__c=grad__Uc_+U3cκ__,χ__c=grad__Φc_,d_c=grad_U3c+Φ_c-κ__·Uc_.Let us point out that the SAM-H model does not propose any approximation of the 3D strain tensor. By applying Hamilton’s variational principle to the shell, Reddy obtains the generalized equilibrium equations of the CS model [19].(51)div_Nc__+κ__·Qc_+feq_=0divQc_-N11cκ1-N22cκ2+f3eq=0div_Mc__-Qc_=0where Nc__, Qc_, and Mc__ are the generalized forces of the CS model and have a definition similar to those of the SAM-H model in (18) to (20); feq_ and f3eq are the equivalent middle surface in-plane and out-of-plane loads, respectively. The additional equilibrium equation (36) in Sanders model [14] is also considered.The equivalent middle surface loads can model the effect of the body loads and the loads applied at the faces of the shell. Let us point out that in SAM-H model, the body loads and the applied stresses on the faces appear explicitly in its equations. Besides, in SAM-H model, in some terms of the equilibrium equations, these loads and stresses are multiplied by the principal curvatures, and the loads applied on the inner and outer faces of the shell are distinguished.In the CS model, after obtaining the approximate 3D strains, the 3D constitutive equations for a homogeneous shell yield the stress field approximation. The CS model neglects out-of-plane normal stressesσ33. An integration through the thickness yields the generalized constitutive equations:(52)Nc^Mc^=AcBcBcDc·εc^χc^andQc_=Kc__·dc_,where Nc^, Mc^, εc^, and χc^ have definitions analogue to those in (45); Ac, Bc, Dc, and Kc__ are stiffness matrices. Let us point out that in the CS model an “ABBD” stiffness matrix appears while an “ABCD” stiffness matrix is expected in the SAM-H model (see the expression of the generalized compliance matrices in (B.1)). Another difference is the fact that in the CS model, the constitutive equations do not involve the body loads and the loads applied at the faces of the shell; in SAM-H model, these loads appear explicitly in the constitutive equations. The Poisson’s effects of the out-of-plane normal stresses on in-plane strains are considered in the SAM-H model but not in the CS model.


## 6.2. Analytical Results

Let us now compare the analytical solutions of the CS and SAM-H models in the case of a hollow sphere subjected to an internal pressurep. The thickness of the sphere is h; the radius of the middle surface is R. The material is isotropic, and its Young’s modulus and Poisson’s ratio are, respectively, E and ν. Vector e3 points from inside to outside the sphere. The analytical resolution of the 3D problem with spherical symmetry leads to the following radial displacement (u3) and normal stress σ33:(53)u33Dξ¯3=pRE1-η/2321-2ν1+ηξ¯33+1+ν1+η/2321+η/23-1-η/231+ηξ¯32,andσ333Dξ¯3=p1-η/231+η/23-1-η/231-1+η/21+ηξ¯33,where η=h/R is the thickness to radius ratio and ξ¯3=ξ3/h is the normalized position through the thickness. The analytical resolution of the equations of the CS model yields the generalized displacement and the normal stress:(54)U3CS=pRE1-ν2ηandσ33CS=0.The generalized displacement and normal stress of the SAM-H model are(55)U3SAM=U3CS-pRE1-2ν2+3140η+13ν30η+ν10η2,andσ33SAMξ¯3=p-1-η2+ξ¯3-2ξ¯32η+η28-η2ξ¯322.Let us first compare the displacement solutions of the CS and SAM-H models with the mean 3D displacement(56)U33D=∫-h/2h/2P0hu33Dξ3hdξ3=pRE21-η/231-2ν+1-η/221+η/221+ν21+η/23-1-η/23.Let us define the relative errors(57)δU3SAM=U3SAM-U33DU33DandδU3CS=U3CS-U33DU33Din the evaluation of the mean displacement for the CS and SAM-H models. These errors only depend on η and ν. In Figures 2(a) and 2(b), the relative errors δU3SAM and δU3CS are plotted against the thickness to radius ratio η for five values of the Poisson’s ratio (-0.4, -0.2, 0, 0.2, and 0.4), respectively. The maximum value of the thickness to radius ratio considered in this example is 0.5; for moderately thick shells, this ratio is less than 0.3 (this limit will be justified below). The absolute value of the relative error δU3SAM of the SAM-H model increases as η and ν increase but is less than 4% for moderately thick shells and less than 14% for thicker shells. The relative error δU3CS of the CS model increase as η increases and ν decreases; it may be more than 50% for moderately thick shells and more than 100% for thicker shells. The SAM-H model provides a far better approximation than the CS model for the mean normal displacement. Let us point out that SAM-H accuracy in the prediction of U3 is mainly due to the consideration of the elastic energy wec (see (43)) corresponding to the coupling between in-plane stresses and normal out-of-plane stresses. In Figure 3, the relative error δU3SAM in the calculation of U3 without taking into account wec is plotted against η for five values of the Poisson’s ratio (-0.4, -0.2, 0, 0.2, and 0.4). Neglecting wec can make this error greater than 20% for the thickest moderately thick shell (η=0.3); this value is 5 times greater than that considering wec. This confirms the utility of taking into account wec (the Poisson’s effect of the out-of-plane normal stresses on in-plane strains). In what follows, this energy is taken into account in the SAM-H model results.Figure 2
Relative displacement errors vs thickness to radius ratio for the SAM-H (a) and CS (b) models.
(a) (b)Figure 3
Relative displacement error vs thickness to radius ratio for the SAM-H without taking into accountwec.Let us now analyze the normal stressesσ33. In the CS model, these stresses are zero. The SAM-H model predicts nonzero σ33 stresses and in order to compare the SAM-H model results to those of the 3D case, the normalized stresses σ33/p are analyzed. In Figure 4, the normalized stresses calculated by the SAM-H model and the 3D model are plotted against the dimensionless position ξ¯3 for three values of the thickness to radius ratio: η=0.1, 0.3, and 0.5. One can observe that the accuracy of the SAM-H results decreases as η increases. In spite of this, the SAM-H results are very accurate even for the thickest case (η=0.5) and the boundary conditions at the inner (ξ¯3=-0.5) and outer (ξ¯3=0.5) faces of the shell are verified.Figure 4
Normalized 3D and SAM-H normal stressesσ33/p vs dimensionless position ξ¯3.


## 6.3. Numerical Results

In this subsection, SAM-H model equations are solved by means of the finite element method implemented in COMSOL Multiphysics software (5.3 version). To test the accuracy of SAM-H model, its results are compared to those obtained with(i)
solid finite elements (SFE) available in COMSOL Multiphysics;(ii)
the classical shell model (CS model); its equations are implemented in COMSOL Multiphysics in a similar manner to how SAM-H equations were treated;(iii)
MITC6 shell finite elements [33] (triangular elements) available in COMSOL Multiphysics.Let us consider a 2m tall hollow cylinder. The radius of the midsurface and the wall thickness are 1m and 0.3m, respectively. A 1MPa pressure is applied on a 0.25m tall circumferential section at the outer face of the cylinder. The resolution of the problem using solid finite elements (SFE) can be simplified due to symmetries as shown in Figure5(a). The shell-type computations (SAM-H, CS, and MITC6) use the mesh shown in Figure 5(b). The material is isotropic and its Young’s modulus and Poisson’s ratio are 200GPa and 0.3.Figure 5
Constriction of a thick hollow cylinder: modelling using solid finite elements (a) and mesh for shell models (b).
(a) (b)The resolution of the generalized equations of SAM-H and CS models allows building the approximate 3D stress field and plotting the stress components on the cross section of the cylinder drawn in Figure5. COMSOL Multiphysics offers the possibility of obtaining from the MITC6 results the 3D stress field in a given position through the thickness of the solid. In Figure 6, the normal stresses σrr calculated by solid finite elements (SFE), the shell models (SAM-H and CS), and MITC6 shell elements are plotted on the cross section. The σrr stresses calculated by the CS model and MITC6 method are neglected whereas that calculated by SAM-H varies across the thickness in a similar manner to the SFE stress. The pressure application on a delimited portion of the outer face provokes the existence of a σrz shear stress. In Figure 7, the color maps of the calculated σrz stresses by SFE, SAM-H, CS, and MITC6 methods are plotted on the cross section of the cylinder. The CS model predicts a constant through thickness shear stress whereas SAM-H model yields a variable shear stress that verifies the boundary conditions at the inner and outer faces, in a similar manner to the MITC6 method. The values of the σrz stress calculated with SAM-H model seem more to the SFE results than those calculated with the CS model. In Figure 8, the von Mises stresses obtained by SFE, SAM-H, CS, and MITC6 are plotted on the cross section of the cylinder. Once again, one can note that SAM-H results are more accurate than those of the CS model and MITC6 method. The maximum von Mises stress at the inner face of the cylinder calculated by SFE, SAM-H, CS, and MITC6 methods is 2.15MPa, 2.10MPa, 1.91 MPa, and 1.63MPa, respectively. In Figure 8, it can be seen that the location of the maximum value of the von Mises stress is at the inner face of the shell for SFE and SAM-H models while that for CS model is at the outer face of the shell. SAM-H provides a better stress approximation than the CS and MITC6 methods for this mechanical problem.Figure 6
Normal stressσrr (in MPa) in the cross section of a constricted cylinder (SFE, SAM-H, CS, and MITC6 results).Figure 7
Shear stressσrz (in MPa) in the cross section of a constricted cylinder (SFE, SAM-H, CS, and MITC6 results).Figure 8
Von Mises stress (in MPa) in the cross section of a constricted cylinder (SFE, SAM-H, CS, and MITC6 results).


## 7. Conclusions

To conclude, a new model of linear homogeneous shells (from thin to moderately thick) called SAM-H was developed in this paper. Hellinger-Reissner’s functional was applied to build the model equations and fields. The model is based on a polynomial approximation of the stress field across the shell thickness. The stress field verifies the 3D equilibrium equation, and the coefficients of the stress polynomials are the generalized forces (membrane and shear forces) and bending moments of the model. The 3D stress boundary conditions at the faces of the shell are also verified. Hellinger-Reissner’s functional allowed identifying the generalized displacements (3 displacements and 2 rotations) and strains. The stationarity of this functional with respect to the generalized displacements yielded 5 generalized equilibrium equations and the generalized boundary conditions on the forces at the edges of the shell. The stationarity with respect to the generalized forces provided the generalized constitutive equations and displacement boundary conditions at the shell’s edges. As compared to the classical linear model (CS model) of general moderately thick shells adapted from Reddy’s model in [19], SAM-H model involves the same number of generalized displacements, strains, and forces. The stress approximation of the CS model does not verify the 3D equilibrium equation and the 3D stress boundary conditions at the faces of the shell. SAM-H model involves, explicitly in its equilibrium and constitutive equations, the body loads and the applied stress vector at the shell faces, while the CS model involves only an equivalent middle surface load in the equilibrium equations. The constitutive equations of the SAM-H model are more accurate since they involve Poisson’s effects of out-of-plane loads on in-plane strains. To prove the accuracy of the SAM-H model, the case of a pressurized hollow sphere was considered and the analytical results of the SAM-H and CS models for displacements and σ33 stress were compared to a 3D solid model solution. The SAM-H proved to be very accurate for moderately thick shells having a thickness to radius ratio less than 0.3. Its results are by far more accurate than the CS model. SAM-H model equations were also implemented in a finite element commercial software, and the case of the constriction of a cylinder was analyzed. The numerical results were compared to those of solid finite elements (SFE), the classical shell model (CS model), and MITC6 shell finite elements. SAM-H proved to be very accurate, it gives a better approximation of SFE results than the other shell models.From a resolution point of view, the SAM-H model proposed in this paper is not more complex than a CS model (both have the same number of generalized displacements) but is more complete since it is able to calculate the normal out-of-plane stresses which should not be neglected in the failure prediction of shells. The higher polynomial degree of out-of plane stresses does not complicate the resolution of the SAM-H model equations; the calculation of these stresses is performed in a postprocessing stage. Another important advantage of the SAM-H model is that, while being applied to plates, it is equivalent to the extended Reissner’s plate model in [13] (adapted to a homogeneous plate) that includes the above-mentioned Poisson’s effects. For these reasons, the SAM-H model can be applied by engineers and implemented in commercial finite element software to obtain more accurate predictions of the structural behavior and strength of linear homogeneous shells having a moderate thickness (η≤0.3).In subsequent papers, the static model developed herein will be extended so as to include dynamic effects and calculate vibration modes and eigenfrequencies of shells. Also, the same stress approach will be applied to develop new models of laminated composite shells. The consideration of geometric nonlinearities is an important topic that will also be treated in a future work.



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*Source: 6141683-2018-12-31.xml*