Advances in the formulation of the rotation-free basic shell triangle

Published in ''Computer Methods in Applied Mechanics and Engineering, vol. 194, 2406-2443, 2005''



== SUMMARY ==

A family of rotation-free three node triangular shell elements is presented. The simplest element of the family is based on an assumed constant curvature field expressed in terms of the nodal deflections of a patch of four elements and a constant membrane field computed from the standard linear interpolation of the displacements within each triangle. An enhanced version of the element is obtained by using a quadratic interpolation of the geometry in terms of the six patch nodes. This allows to compute an assumed linear membrane strain field which improves the in-plane behaviour of the original element. A simple and economic version of the element using a single integration point is presented. The efficiency of the different rotation-free shell triangles is demonstrated in many examples of application including linear and non linear analysis of shells under static and dynamic loads, the inflation and de-inflation of membranes and a sheet stamping problem.

==1 INTRODUCTION==

Triangular shell elements are very useful for the solution of large scale shell problems such as those occurring in many practical engineering situations. Typical examples are the analysis of shell roofs under static and dynamic loads, sheet stamping processes, vehicle dynamics and crash-worthiness situations. Many of these problems involve high geometrical and material non linearities and time changing frictional contact conditions. These difficulties are usually increased by the need of discretizing complex geometrical shapes. Here the use of shell triangles and non-structured meshes becomes a critical necessity. Despite recent advances in the field <span id='citeF-1'></span>[[#cite-1|[1]]]&#8211;<span id='citeF-6'></span>[[#cite-6|[6]]] there are not so many simple shell triangles which are capable of accurately modelling the deformation of a shell structure under arbitrary loading conditions.

A promising line to derive simple shell triangles is to use the nodal displacements as the only unknown for describing the shell kinematics. This idea goes back to the original attempts to solve thin plate bending problems using finite difference schemes with the deflection as the only nodal variable <span id='citeF-7'></span>[[#cite-7|[7]]]&#8211;<span id='citeF-9'></span>[[#cite-9|[9]]].

In past years some authors have derived a number of thin plate and shell triangular elements free of rotational degrees of freedom (d.o.f.) based on Kirchhoff's theory [10]&#8211;<span id='citeF-27'></span>[[#cite-27|[27]]]. In essence all methods attempt to express the curvatures field over an element in terms of the displacements of a collection of nodes belonging to a patch of adjacent elements. Oñate and Cervera [14] proposed a general procedure of this kind combining finite element and finite volume concepts for deriving thin plate triangles and quadrilaterals with the deflection as the only nodal variable and presented a simple and competitive rotation-free three d.o.f. triangular element termed BPT (for Basic Plate Triangle). These ideas were extended and formalized in <span id='citeF-29'></span>[[#cite-29|29]] to derive a number of rotation-free thin plate and shell triangles. The basic ingredients of the method are a mixed Hu-Washizu formulation, a standard discretization into three node triangles, a linear finite element interpolation of the displacement field within each triangle and a finite volume type approach for computing constant curvature and bending moment fields within appropriate non-overlapping control domains. The so called cell-centered and cell-vertex triangular domains yield different families of rotation-free plate and shell triangles. Both the BPT plate element and its extension to shell analysis (termed BST for Basic Shell Triangle) can be derived from the cell-centered formulation. Here the control domain is an individual triangle. The constant curvatures field within a triangle is computed in terms of the displacements of the six nodes belonging to the four elements patch formed by the chosen triangle and the three adjacent triangles. The cell-vertex approach yields a different family of rotation-free plate and shell triangles. Details of the derivation of both rotation-free triangular shell element families can be found in <span id='citeF-29'></span>[[#cite-29|[21]]].

An extension of the BST element to the non linear analysis of shells was implemented in an explicit dynamic code by Oñate ''et al.'' [26] using an updated lagrangian formulation and a hypo-elastic constitutive model. Excellent numerical results were obtained for non linear dynamics of shells involving frictional contact situations and sheet stamping problems [18,19,20,26].

A large strain formulation for the BST element using a total lagrangian description was presented by Flores and Oñate [24]. A recent extension of this formulation is based on a quadratic interpolation of the geometry of the patch formed by the BST element and the three adjacent triangles [27]. This yields a linear displacement gradient field over the element from which linear membrane strains and  constant curvatures  can be computed within the BST element.

In this paper the formulation of the BST element is revisited using an assumed strain approach. The content of the paper is the following. First some basic concepts of the formulation of the original BST element using an assumed constant curvature field are given. Next, the basic equations of the non linear thin shell theory chosen based on a total lagrangian description are presented. Then the non linear formulation of the BST element is presented. This is based on an assumed constant membrane field derived from the linear displacement interpolation and an assumed constant curvature field expressed in terms of the displacements of the nodes of the four element patch using a finite volume type approach. An enhanced version of the BST element is derived using an assumed linear field for the membrane strains and an assumed constant curvature field. Both assumed fields are obtained from the quadratic interpolation of the patch geometry following the ideas presented in [27]. Details of the derivation of the tangent stiffness matrix needed  for a quasi-static implicit solution are given for both the BST and EBST elements. An efficient version of the  EBST element using one single quadrature point for integration of the tangent matrix is  presented. An explicit  scheme adequate for dynamic analysis is  briefly described.

The efficiency and accuracy of the standard and enhanced versions of the BST element is validated in a number of examples of application including linear and non linear analysis of shells under static and dynamic loads, the inflation and de-inflation of membranes and a sheet stamping problem.

==2 FORMULATION OF THE BASIC PLATE TRIANGULAR USING AN ASSUMED CONSTANT CURVATURE FIELD==

Let us consider a patch of four plate three node triangles (Figure [[#img-1|1]]). The nodes 1, 2, and 3 in the main central triangle (M) are marked with circles while the external nodes in the patch (nodes 4, 5 and 6) are marked with squares. Mid side points in the central triangle are also marked with smaller squares. Kirchhoff's thin plate theory will be assumed to hold. The deflection is linearly interpolated within each three node triangle in the standard finite element manner as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>w=\sum _{i=1}^{3}L_{i}^{e}w_{i}^{e} </math>
|}
| style="width: 5px;text-align: right;" | (1)
|}

where <math display="inline">L_{i}^{e}</math> are the linear shape functions of the three node triangle, <math display="inline">w_{i}^{e}</math> are nodal deflections and superindex <math display="inline">e</math> denotes element values.

<div id='img-1'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig1.png|400px|Patch of three node triangular elements including the central triangle (M) and three adjacent triangles (1, 2 and 3)]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Patch of three node triangular elements including the central triangle (M) and three adjacent triangles (1, 2 and 3)
|}

The curvature field within the central triangle can be expressed in terms of a constant assumed curvature field as

<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \kappa }=\left\{ \begin{array}{c} \kappa _{xx}\\ \kappa _{yy}\\ \kappa _{xy} \end{array} \right\} =\hat{\boldsymbol \kappa } </math>
|}
| style="width: 5px;text-align: right;" | (2)
|}

where <math display="inline">{\boldsymbol \kappa }</math> is the curvature vector and <math display="inline">\hat{\boldsymbol \kappa }</math> is the assumed constant curvature field defined as

<span id="eq-3"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hat{\boldsymbol \kappa }={\frac{1}{A_{M}}}\int \int _{A_{M}}\left[ -{\frac{\partial ^{2} w}{\partial x^{2}}}-,{\frac{\partial ^{2}w}{\partial y^{2}}},-2{\frac{\partial ^{2}w}{\partial x\partial y}}\right] ^{T}dA </math>
|}
| style="width: 5px;text-align: right;" | (3)
|}

where <math display="inline">A_{M}</math> is the area of the central triangle in Figure [[#img-1|1]].

Integrating by parts Eq.([[#eq-3|3]]) and substituting the resulting expression for <math display="inline">\hat{\boldsymbol \kappa }</math> into Eq.([[#eq-2|2]]) gives the constant curvature field within the element as

<span id="eq-4"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \kappa }={\frac{1}{A_{M}}}   {\displaystyle \oint _{\Gamma _{M}}}  \left[ \begin{array}{cc} -n_{x} & 0\\ 0 & -n_{y}\\ -n_{y} & -n_{x} \end{array} \right] \left\{ \begin{array}{c} \dfrac{\partial w}{\partial x}\\ \dfrac{\partial w}{\partial y} \end{array} \right\} d\Gamma </math>
|}
| style="width: 5px;text-align: right;" | (4)
|}

where <math display="inline">\Gamma _{M}</math> is the boundary of the central triangle.

Eq.([[#eq-4|4]]) defines the assumed constant curvature field within the central triangle in terms of the deflection gradient along the sides of the triangle. Equation (4) can be found to be equivalent to the standard conservation laws used in finite volume procedures as described in [28,29].

The computation of the line integral in Eq.([[#eq-4|4]]) poses a difficulty as the deflection gradient is discontinuous along the element sides. A simple method to overcome this problem is to compute the deflection gradient at the element sides as the average values of the gradient contributed by the two triangles sharing the side <span id='citeF-29'></span>[[#cite-29|[21,29]]]. Following this idea the constant curvature field with the element is computed as

<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \kappa }={\frac{1}{A_{M}}}\sum _{j=1}^{3}{\frac{l_{j}^{M}}{2}}\left[ \begin{array}{cc} -n_{x}^{j} & 0\\ 0 & -n_{y}^{j}\\ -n_{y}^{j} & -n_{x}^{j} \end{array} \right] ^{M}\left[ {\nabla }L_{i}^{M}w_{i}^{M}+{\nabla }L_{i}^{j}w_{i} ^{j}\right] =\mathbf{B}_{b}\mathbf{w}^{p} </math>
|}
| style="width: 5px;text-align: right;" | (5)
|}

where <math display="inline">\mathbf{w}^{p}=[w_{1},w_{2},w_{3},w_{4},w_{5},w_{6}]^{T}</math> is the deflection vector of the six  nodes in the patch. In Eq.([[#eq-5|5]]) the sum extends over the three sides of the central element <math display="inline">M</math>, <math display="inline">l_{j}^{M}</math> are the lengths of the element sides and superindexes <math display="inline">M</math> and <math display="inline">j</math> refer to the central triangle and to each of the adjacent elements, respectively. The standard sum convention for repeated indexes is used.

Note that the constant curvature field is expressed in terms of the six nodes of the four element patch linked to the element <math display="inline">M</math>. The expression of the <math display="inline">3\times{6}</math> <math display="inline">\mathbf{B}_{b}</math> matrix can be found in [14,21].

The virtual work expression is written as

<span id="eq-6"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int \int _{A}\delta{\boldsymbol \kappa }^{T}\mathbf{m}\,dA=\int \int _{A}\delta w\,q\,dA </math>
|}
| style="width: 5px;text-align: right;" | (6)
|}

where '''m''' is the bending moment field related to the curvature by the standard constitutive equations

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{m}=[M_{xx},M_{yy},M_{xy}]^T = \mathbf{D}_{b}{\boldsymbol \kappa }\quad ,\quad \mathbf{D}_{b}={\frac{Eh^{3} }{(1-\nu ^{2})}}\left[ \begin{array}{ccc} 1 & \nu & 0\\ \nu & 1 & 0\\ 0 & 0 & \frac{1-\nu }{2} \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (7)
|}

In Eqs.(6) and (7) <math display="inline">h</math> is the plate thickness, <math display="inline">E</math> is the Young's modulus, <math display="inline">\nu </math> is the Poisson's ratio, <math display="inline">\delta{\boldsymbol \kappa }</math> and <math display="inline">\delta w</math> are the virtual curvatures and the virtual deflection, respectively, and <math display="inline">q</math> is a distributed vertical load.

Substituting the approximation for the vertical deflection and the assumed constant curvature field into ([[#eq-6|6]]) leads to the standard linear system of equations

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{K}\mathbf{w}=\mathbf{f}</math>
|}
| style="width: 5px;text-align: right;" | (8)
|}

where the stiffness matrix <math display="inline">\mathbf{K}</math> and the equivalent nodal force <math display="inline">\mathbf{f}</math> can be found by assembly of the element contributions given by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{K}^{e}=\int \int _{A^{e}}\mathbf{B}_{b}^{T}\mathbf{D}_{b}\mathbf{B}_{b}dA </math>
|}
| style="width: 5px;text-align: right;" | (9)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{f}^{e}=\int \int _{A^{e}}q\left\{ \begin{array}{c} L_{1}^e\\ L_{2}^e\\ L_{3}^e \end{array} \right\} dA </math>
|}
| style="width: 5px;text-align: right;" | (10)
|}

Note that <math display="inline">\mathbf{K}^{e}</math> is  a <math display="inline">6\times{6}</math> matrix, whereas <math display="inline">\mathbf{f}^{e}</math> has the same structure than for the standard linear triangle.

The explicit form of <math display="inline">\mathbf{K}^{e}</math> and <math display="inline">\mathbf{f}^{e}</math> can be found in [14].

The resulting Basic Plate Triangle (BPT) has one degree of freedom per node and a wider bandwidth than the standard three node triangles as each triangular element is linked to its three neighbours through Eq.([[#eq-5|5]]).

Examples of the good performance of the BPT element for analysis of thin plates can be found in [14,21]. The extension of the BPT element to the analysis of shells yields the Basic Shell Triangle (BST) <span id='citeF-29'></span>[[#cite-29|29]]. Different applications of the BST element to linear and non linear analysis of shells are reported in 14,18&#8211;21,24,26,27].

The ideas used to derive the BPT element will now be extended to derive two families of Basic Shell Triangles using a total lagrangian description.

==3 BASIC THIN SHELL EQUATIONS USING A TOTAL LAGRANGIAN FORMULATION==

===3.1 Shell kinematics===

A summary of the most relevant hypothesis related to the kinematic behaviour of a thin shell are presented. Further details may be found in the wide literature dedicated to this field [8,9].

Consider a shell with undeformed middle surface occupying the domain <math display="inline">\Omega ^{0}</math> in <math display="inline">R^{3}</math> with a boundary <math display="inline">\Gamma ^{0}</math>. At each point of the middle surface a thickness <math display="inline">h^{0}</math> is defined. The positions <math display="inline">\mathbf{x}^{0}</math> and <math display="inline">\mathbf{x}</math> of a point in the undeformed and the deformed configurations can be respectively written as a function of the coordinates of the middle surface <math display="inline">{\boldsymbol \varphi }</math> and the normal <math display="inline">\mathbf{t}_{3}</math> at the point as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math>\mathbf{x}^{0}\left( \xi _{1},\xi _{2},\zeta \right)  </math>
| <math>  ={\boldsymbol \varphi }^{0}\left( \xi _{1},\xi _{2}\right) +\lambda \mathbf{t}_{3}^{0}</math>
| style="width: 5px;text-align: right;" | (11)
|-
| style="text-align: right;" | <math> \mathbf{x}\left( \xi _{1},\xi _{2},\zeta \right)  </math>
| <math>  ={\boldsymbol \varphi }\left( \xi  _{1},\xi _{2}\right) +\zeta \lambda \mathbf{t}_{3} </math>
| style="width: 5px;text-align: right;" | (12)
|}
|}

where <math display="inline">\xi _{1},\xi _{2}</math> are curvilinear local coordinates defined over the middle surface of the shell, and <math display="inline">\zeta </math> is the distance in the undeformed configuration of the point to the middle surface. The product <math display="inline">\zeta \lambda </math> is the distance of the point to the middle surface measured on the deformed configuration. This implies a constant strain in the normal direction associated to the parameter <math display="inline">\lambda </math> relating the thickness at the present and initial configurations, i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\lambda =\frac{h}{h^{0}} </math>
|}
| style="width: 5px;text-align: right;" | (13)
|}

A convective coordinate system is defined at each point as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{g}_{i}\left( \mathbf{\xi }\right) =\frac{\partial \mathbf{x}}{} {\partial \xi _{i}}\qquad i=1,2,3 </math>
|}
| style="width: 5px;text-align: right;" | (14)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: left;" | <math>\mathbf{g}_{\alpha }\left( \mathbf{\xi }\right)   =\frac{\partial \left( \mathbf{\boldsymbol \varphi }\left( \xi _{1},\xi _{2}\right) +\zeta \lambda \mathbf{t}_{3}\right) }{\partial \xi _{\alpha }}={\boldsymbol \varphi }_{^{\prime }\alpha }+\zeta \left( \lambda \mathbf{t}_{3}\right) _{^{\prime }\alpha }\quad \alpha=1,2</math>
| style="width: 5px;text-align: right;" | (15)
|-
| style="text-align: left;" | <math> \mathbf{g}_{3}\left( \mathbf{\xi }\right)   =\frac{\partial \left( \mathbf{\boldsymbol \varphi }\left( \xi _{1},\xi _{2}\right) +\zeta \lambda \mathbf{t}_{3}\right) }{\partial \zeta }=\lambda \mathbf{t}_{3} </math>
| style="width: 5px;text-align: right;" | (16)
|}
|}

This can be particularized for the points on the middle surface as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: left;" | <math>\mathbf{a}_{\alpha } =\mathbf{g}_{\alpha }\left( \zeta=0\right) ={\boldsymbol \varphi  }_{^{\prime }\alpha }</math>
| style="width: 5px;text-align: right;" | (17)
|-
| style="text-align: left;" | <math> \mathbf{a}_{3}   =\mathbf{g}_{3}\left( \zeta=0\right) =\lambda  \mathbf{t}_{3} </math>
| style="width: 5px;text-align: right;" | (18)
|}
|}

The covariant (first fundamental form) and contravariant metric tensors of the middle surface are

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>a_{\alpha \beta }=\mathbf{a}_{\alpha }\cdot \mathbf{a}_{\beta } </math>
|}
| style="width: 5px;text-align: right;" | (19)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>a^{\alpha \beta }=\mathbf{a}^{\alpha }\cdot \mathbf{a}^{\beta }={\tilde{\boldsymbol \varphi }}_{^{\prime }\alpha }\cdot{\tilde{\boldsymbol \varphi }}_{^{\prime }\beta } </math>
|}
| style="width: 5px;text-align: right;" | (20)
|}

The Green-Lagrange strain vector of the middle surface points (membrane strains) is defined as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{\boldsymbol \varepsilon }_{m}=[\varepsilon _{m_{11}},\varepsilon _{m_{12}},\varepsilon _{m_{12}}]^{T} </math>
|}
| style="width: 5px;text-align: right;" | (21)
|}

with

<span id="eq-22"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\varepsilon _{m_{ij}}=\frac{1}{2}(a_{ij}-\delta _{ij}) </math>
|}
| style="width: 5px;text-align: right;" | (22)
|}

The curvatures (second fundamental form) of the middle surface are obtained by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\kappa _{\alpha \beta }=\frac{1}{2}\left( {\boldsymbol \varphi }_{^{\prime }\alpha } \cdot \mathbf{t}_{3^{\prime }\beta }+{\boldsymbol \varphi }_{^{\prime }\beta }\cdot  \mathbf{t}_{3^{\prime }\alpha }\right) =- \mathbf{t}_{3}\cdot{\boldsymbol \varphi }_{{\prime }\alpha \beta }\quad , \quad \alpha ,\beta=1,2 </math>
|}
| style="width: 5px;text-align: right;" | (23)
|}

The deformation gradient tensor is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\mathbf{F=} [{x}_{{\prime }1},{x}_{{\prime }2},{x}_{{\prime }3}]=\left[ \begin{array}{ccc} {\boldsymbol \varphi }_{^{\prime }1}+\zeta \left( \lambda \mathbf{t}_{3}\right) _{^{\prime  }1} & {\boldsymbol \varphi }_{^{\prime }2}+\zeta \left( \lambda \mathbf{t}_{3}\right) _{^{\prime }2} & \lambda \mathbf{t}_{3} \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (24)
|}

The product <math display="inline">\mathbf{F}^{T}\mathbf{F=U}^{2}=\mathbf{C}</math> (where <math display="inline">\mathbf{U}</math> is the right stretch tensor, and <math display="inline">\mathbf{C}</math> the right Cauchy-Green deformation tensor) can be written as

<span id="eq-25"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\mathbf{U}^{2}=\left[ \begin{array}{ccc} a_{11}+2\kappa _{11}\zeta \lambda & a_{12}+2\kappa _{12}\zeta \lambda & 0\\ a_{12}+2\kappa _{12}\zeta \lambda & a_{22}+2\kappa _{22}\zeta \lambda & 0\\ 0 & 0 & \lambda ^{2} \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (25)
|}

In the derivation of expression ([[#eq-25|25]]) the derivatives of the thickness ratio <math display="inline">\lambda _{^{\prime }a}</math> and the terms associated to <math display="inline">\zeta ^{2}</math> have been neglected.

Eq.([[#eq-25|25]]) shows that <math display="inline">\mathbf{U}^{2}</math> is not a unit tensor at the original configuration for curved surfaces (<math display="inline">\kappa _{ij}^{0}\neq{0}</math>). The changes of curvature of the middle surface are computed by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\chi _{ij}=\kappa _{ij}-\kappa _{ij}^{0} </math>
|}
| style="width: 5px;text-align: right;" | (26)
|}

Note that <math display="inline">\delta \chi _{ij}=\delta \kappa _{ij}</math>.

For computational convenience the following approximate expression (which is exact for initially flat surfaces) will be adopted

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\mathbf{U}^{2}=\left[ \begin{array}{ccc} a_{11}+2\chi _{11}\zeta \lambda & a_{12}+2\chi _{12}\zeta \lambda & 0\\ a_{12}+2\chi _{12}\zeta \lambda & a_{22}+2\chi _{22}\zeta \lambda & 0\\ 0 & 0 & \lambda ^{2} \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (27)
|}

This expression is useful to compute different lagrangian strain measures. An advantage of these measures is that they are associated to material fibres, what makes it easy to take into account material anisotropy. It is also useful to compute the eigen decomposition of <math display="inline">\mathbf{U}</math> as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{U=}\sum _{\alpha=1}^{3}\lambda _{\alpha } \mathbf{r}_{\alpha } \otimes \mathbf{r}_{\alpha } </math>
|}
| style="width: 5px;text-align: right;" | (28)
|}

where <math display="inline">\lambda _{\alpha }</math> and <math display="inline">\mathbf{r}_{\alpha }</math> are the eigenvalues and eigenvectors of <math display="inline">\mathbf{U}</math>.

The resultant stresses  (axial forces and moments) are obtained by integrating across the original thickness the second Piola-Kirchhoff stress vector <math display="inline">{ \boldsymbol \sigma }</math> using the actual distance to the middle surface for  evaluating the bending moments, i.e.

<span id="eq-29"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \sigma }_{m}\equiv \lbrack N_{11},N_{22},N_{12}]^{T}=\int _{h^{0}}{\boldsymbol \sigma } d\zeta </math>
|}
| style="width: 5px;text-align: right;" | (29)
|}

<span id="eq-30"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \sigma }_{b}\equiv \lbrack M_{11},M_{22},M_{12}]^{T}=\int _{h^{0}}{\boldsymbol \sigma  }\lambda \zeta  d\zeta </math>
|}
| style="width: 5px;text-align: right;" | (30)
|}

With these values the virtual work can be written as

<span id="eq-31"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int \int _{A^{0}}\left[ \delta{\boldsymbol \varepsilon }_{m}^{T}{\boldsymbol \sigma }_{m}+\delta{\boldsymbol \kappa  }^{T}{\boldsymbol \sigma }_{b}\right] dA=\int \int _{A^{0}}\delta \mathbf{u}^{T}\mathbf{t}dA </math>
|}
| style="width: 5px;text-align: right;" | (31)
|}

where <math display="inline">\delta \mathbf{u}</math> are virtual displacements, <math display="inline">\delta{\boldsymbol \varepsilon }_{m}</math> is the virtual Green-Lagrange membrane strain vector, <math display="inline">\delta{\boldsymbol \kappa }</math> are the virtual curvatures and <math display="inline">\mathbf{t}</math> are the surface loads. Other load types can be easily included into ([[#eq-31|31]]).

===3.2 Constitutive models===

In order to treat plasticity at finite strains an adequate stress-strain pair must be used. The Hencky measures will be adopted here. The (logarithmic) strains are defined as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{E}_{\ln }\mathbf{=}\left[ \begin{array}{ccc} \varepsilon _{11} & \varepsilon _{21} & 0\\ \varepsilon _{12} & \varepsilon _{22} & 0\\ 0 & 0 & \varepsilon _{33} \end{array} \right] =\sum _{\alpha=1}^{3}\ln \left( \lambda _{\alpha }\right) \mathbf{r}_{\alpha }\otimes \mathbf{r}_{\alpha } </math>
|}
| style="width: 5px;text-align: right;" | (32)
|}

Two types of material models are considered here: an elastic-plastic material associated to thin rolled metal sheets and a hyper-elastic material for rubbers.

In the case of metals, where the elastic strains are small, the use of a logarithmic strain measure reasonably allows to adopt an additive decomposition of elastic and plastic components as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{E}_{\ln }\mathbf{=E}_{\ln }^{e}+\mathbf{E}_{\ln }^{p} </math>
|}
| style="width: 5px;text-align: right;" | (33)
|}

A constant linear relationship between the (plane) Hencky stresses and the logarithmic elastic strains is  adopted giving

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{T}=\mathbf{CE}_{\ln }^{e} </math>
|}
| style="width: 5px;text-align: right;" | (34)
|}

These constitutive equations are integrated using a standard return algorithm. The following Mises-Hill [30] yield function with non-linear isotropic hardening is chosen here

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left( G+H\right) \;T_{11}^{2}+\left( F+H\right) \;T_{22}^{2} -2H\;T_{11}T_{22}+2N\;T_{12}^{2}=\sigma _0\left(e_{0}+e^{p}\right) ^{n} </math>
|}
| style="width: 5px;text-align: right;" | (35)
|}

where <math display="inline">F, G, H</math> and <math display="inline">N</math> define the non-isotropic shape of the yield surface and the parameters <math display="inline">\sigma _{0}</math>, <math display="inline">e_{0}</math> and <math display="inline">n</math> define its size as a function of the effective plastic strain <math display="inline">e^{p}</math>.

The simple Mises-Hill yield function  allows, as a first approximation, to treat rolled thin metal sheets with planar and transversal anisotropy.

For the case of rubbers, the Ogden [31] model extended to the compressible range is considered. The material behaviour is characterized by the strain energy density per unit undeformed volume defined as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\psi =\frac{K}{2}\left( \ln J\right) ^{2}+\sum _{p=1}^{N}\frac{\mu _{p}}{} {\alpha _{p}}\left[ J^{-\frac{\alpha _{p}}{3}}\left( \sum _{i=1}^{3}\lambda  _{i}^{\alpha _{p}-1}\right) -3\right] </math>
|}
| style="width: 5px;text-align: right;" | (36)
|}

where <math display="inline">K</math> is the bulk modulus of the material, <math display="inline">J</math> is the determinant of <math display="inline">\mathbf{U}</math>, <math display="inline">N</math>, <math display="inline">\mu _{i}</math> and <math display="inline">\alpha _{i}</math> are material parameters, <math display="inline">\mu _{i}\,,\,\alpha _{i}</math> are real numbers such that <math display="inline">\mu _{i}\alpha _{i}>0</math> <math display="inline"> (\forall i=1,N)</math> and <math display="inline">N</math> is a positive integer.

The stress measures associated to the principal logarithmic strains are denoted by <math display="inline">\beta _{i}</math>. They can be computed noting that

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\beta _{i}=\frac{\partial \psi \left( \Lambda \right) }{\partial \left( \ln \lambda _{i}\right) }=K\left( \ln J\right) +\lambda _{i}\sum _{p=1}^{N} \mu _{p}J^{-\frac{\alpha _{p}}{3}}\left( \lambda _{i}^{\alpha _{p}-1}-\frac{1}{} {3}\frac{1}{\lambda _{i}}\sum _{j=1}^{3}\lambda _{j}^{\alpha _{p}}\right) </math>
|}
| style="width: 5px;text-align: right;" | (37)
|}

we define now

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>a^{p}=\sum _{j=1}^{3}\lambda _{j}^{\alpha _{p}} </math>
|}
| style="width: 5px;text-align: right;" | (38)
|}

which gives

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\beta _{i}=K\left( \ln J\right) +\sum _{p=1}^{N}\mu _{p}J^{-\frac{\alpha _{p} }{3}}\left( \lambda _{i}^{\alpha _{p}}-\frac{1}{3}a_{p}\right) </math>
|}
| style="width: 5px;text-align: right;" | (39)
|}

The values of <math display="inline">\beta _{i}</math>, expressed in the principal strains directions, allow to evaluate the Hencky stresses in the convective coordinate system as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{T}=\sum _{i=1}^{3}\beta _{i}\;\mathbf{r}_{i}\otimes \mathbf{r}_{i} </math>
|}
| style="width: 5px;text-align: right;" | (40)
|}

The Hencky stress tensor <math display="inline">\mathbf{T}</math> can be easily particularized for the plane stress case.

We define the rotated Hencky and second Piola-Kirchhoff stress tensors as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math>\mathbf{T}_{L}  </math>
| <math>  =\mathbf{R}_{L}^{T}\;\mathbf{T\;R}_{L}</math>
| style="width: 5px;text-align: right;" | (41)
|-
| style="text-align: right;" | <math> \mathbf{S}_{L}  </math>
| <math>  =\mathbf{R}_{L}^{T}\;\mathbf{S\;R}_{L} </math>
| style="width: 5px;text-align: right;" | (42)
|}
|}

where <math display="inline">\mathbf{R}_{L}</math> is the rotation tensor obtained from the eigenvectors of <math display="inline">\mathbf{U}</math> given by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{R}_{L}=\left[ \begin{array}{ccc} \mathbf{r}_{1} & \mathbf{r}_{2} & \mathbf{r}_{3} \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (43)
|}

The relationship between the rotated Hencky and Piola-Kirchhoff stresses is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math>\left[ S_{L}\right] _{\alpha \alpha }  </math>
| <math>  =\frac{1}{\lambda _{\alpha }^{2} }\left[ T_{L}\right] _{\alpha \alpha }</math>
|-
| style="text-align: right;" | <math> \left[ S_{L}\right] _{\alpha \beta }  </math>
| <math>  =\frac{\ln \left( \lambda _{\alpha  }/\lambda _{\beta }\right) }{\frac{1}{2}\left( \lambda _{\alpha }^{2} -\lambda _{\beta }^{2}\right) }\left[ T_{L}\right] _{\alpha \beta } </math>
|}
| style="width: 5px;text-align: right;" | (44)
|}

The second Piola-Kirchhoff stress tensor can be computed by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{S}=\mathbf{R}_{L}\;\mathbf{S}_{L}\mathbf{\;R}_{L}^{T} </math>
|}
| style="width: 5px;text-align: right;" | (45)
|}

The second Piola-Kirchhoff stress vector <math display="inline">{\boldsymbol \sigma }</math> of Eqs.([[#eq-29|29]]&#8211;[[#eq-30|30]]) can be readily extracted from the <math display="inline">\mathbf{S}</math> tensor.

==4 TOTAL LAGRANGIAN FORMULATION OF THE BASIC SHELL TRIANGLE==

===4.1 Definition of the element geometry and discretization of the displacement field===

The rotation-free BST element has three nodes with three displacement degrees of freedom at each node. As before an element patch is defined by the central triangle  and the three adjacent elements (Figure [[#img-1|1]]). This patch helps to define the curvature field within the central triangle (the BST element) in terms of the displacement of the six patch nodes.

The node-ordering in the patch is the following (see Figure [[#img-1|1]])

* The nodes in the main element (M) are numbered locally as 1, 2 and 3. They are defined counter-clockwise around the positive normal.

* The sides in the main element are numbered locally as 1, 2, and 3. They are defined by the local node opposite to the side.

* The adjacent elements (which are part of the cell) are numbered with the number associated to the common side.

* The extra nodes of the cell are numbered locally as 4, 5 and 6, corresponding to nodes on adjacent elements opposite to sides 1, 2  and 3 respectively.

* The connectivities in the adjacent elements are defined beginning with the extra node as shown in Table 1.


<div class="center" style="font-size: 75%;">
'''Table 1'''. Element numbering and nodal connectivities of the four elements patch of Figure 1.</div>

{| class="wikitable" style="text-align: left; margin: 1em auto;"
|- style="border-top: 2px solid;"
| '''Element''' 
| N1 
| N2 
| N3
|- style="border-top: 2px solid;"
|  '''M''' 
| 1 
| 2 
| 3
|- style="border-top: 2px solid;"
|  '''1''' 
| 4 
| 3 
| 2
|- style="border-top: 2px solid;"
|  '''2''' 
| 5 
| 1 
| 3
|- style="border-top: 2px solid;border-bottom: 2px solid;"
|  '''3''' 
| 6 
| 2 
| 1

|}

The following local cartesian coordinate system can be defined for the patch. In the main element the unit vector <math display="inline">\mathbf{t}_{1}</math>(associated to the local coordinate <math display="inline">\xi _{1}</math>) is directed along side 3 (from node 1 to node 2), <math display="inline">\mathbf{t}_{3}</math> (associated to the coordinate <math display="inline">\zeta </math>) is the unit normal to the plane, and finally <math display="inline">\mathbf{t}_{2}=\mathbf{t}_{3}\times \mathbf{t}_{1}</math> (associated to the coordinate <math display="inline">\xi _{2}</math>).

The coordinates and the displacements are linearly interpolated within each three node triangle in the mesh in the standard manner, i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \varphi } = \sum \limits _{i=1}^{3} L_{i}^e {\boldsymbol \varphi }_{i} = \sum \limits _{i=1}^{3} L_{i}^e ({\boldsymbol \varphi }^{0}_{i} + \mathbf{u}_{i}) </math>
|}
| style="width: 5px;text-align: right;" | (46)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{u}=\left\{ \begin{array}{c} u_{1}\\ u_{2}\\ u_{3} \end{array} \right\} =\sum \limits _{i=1}^{3}L_{i}^e\mathbf{u}_{i}\quad ,\quad \mathbf{u}_{i}=\left\{ \begin{array}{c} u_{1}\\ u_{2}\\ u_{3} \end{array} \right\} _{i} </math>
|}
| style="width: 5px;text-align: right;" | (47)
|}

In above <math display="inline">{\boldsymbol \varphi }_{i}</math> and <math display="inline">\mathbf{u}_{i}</math> contain respectively the three coordinates and the three displacements of node <math display="inline">i</math>.

===4.2 Computation of the membrane strains===

The Green-Lagrange membrane strains are expressed by substituting the linear displacement interpolation into Eq.([[#eq-22|22]]). This gives

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \varepsilon }_{m}=\frac{1}{2}\left[ \begin{array}{c}\boldsymbol \varphi _{^{\prime }1}\cdot \boldsymbol \varphi _{^{\prime  }1}-1 \\ \boldsymbol \varphi _{^{\prime }2}\cdot \boldsymbol \varphi _{^{\prime  }2}-1 \\ 2\boldsymbol \varphi _{^{\prime }1}\cdot \boldsymbol \varphi _{^{\prime }2} \end{array}\right] </math>
|}
| style="width: 5px;text-align: right;" | (48)
|}

The membrane strain field is constant within each triangle similarly as in the standard CST element. The variation of the membrane strains is simply obtained by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\delta{\boldsymbol \varepsilon }_{m}=\mathbf{B}_{m}\delta \mathbf{a}^{e} </math>
|}
| style="width: 5px;text-align: right;" | (49)
|}

with

<span id="eq-50"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{B}_{m}=[\mathbf{B}_{m_{1}},\mathbf{B}_{m_{2}},\mathbf{B}_{m_{3}} ]\quad ,\quad \mathbf{a}^{e}=\left\{ \begin{array}{c} \mathbf{u}_{1}\\ \mathbf{u}_{2}\\ \mathbf{u}_{3} \end{array} \right\} </math>
|}
| style="width: 5px;text-align: right;" | (50)
|}

and

<span id="eq-51"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{c} \\ \mathbf{B}_{m_{i}}\\ 3\times{3} \end{array} =\left[ \begin{array}{c} L_{i,1}^M\boldsymbol \varphi _{^{\prime }1}^{T}\\ L_{i,2}^M\boldsymbol \varphi _{^{\prime }2}^{T}\\ L_{i,1}^M\boldsymbol \varphi _{^{\prime }2}^{T}+L_{i,2}^M\boldsymbol \varphi _{^{\prime }1}^{T} \end{array} \right]  </math>
|}
| style="width: 5px;text-align: right;" | (51)
|}

===4.3 Computation of bending strains (curvatures)===

We will assume the following constant curvature field within each element

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\kappa _{\alpha \beta }=\hat{\kappa }_{\alpha \beta } </math>
|}
| style="width: 5px;text-align: right;" | (52)
|}

where <math display="inline">\hat{\kappa }_{\alpha \beta }</math> is the assumed constant curvature field defined by

<span id="eq-53"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hat{\kappa }_{\alpha \beta }=-\frac{1}{A_{M}^{0}}\int _{A_{M}^{0}}\mathbf{t}_{3}\cdot{\boldsymbol \varphi }_{^{\prime }\beta \alpha }\;dA^{0} </math>
|}
| style="width: 5px;text-align: right;" | (53)
|}

where <math display="inline">A_{M}^{0}</math> is the area (in the original configuration) of the central element in the patch.

Substituting Eq.(53) into (52) and integrating by parts the area integral gives the curvature vector within the element in terms of the following line integral

<span id="eq-54"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \kappa }=\left\{ \begin{array}{c} \kappa _{11}\\ \kappa _{22}\\ 2\kappa _{12} \end{array} \right\} =\frac{1}{A_{M}^{0}}   {\displaystyle \oint _{\Gamma _{M}^{0}}}  \left[ \begin{array}{cc} -n_{1} & 0\\ 0 & -n_{2}\\ -n_{2} & -n_{1} \end{array} \right] \left[ \begin{array}{c} \mathbf{t}_{3}\cdot{\boldsymbol \varphi }_{^{\prime }1}\\ \mathbf{t}_{3}\cdot{\boldsymbol \varphi }_{^{\prime }2} \end{array} \right] d\Gamma </math>
|}
| style="width: 5px;text-align: right;" | (54)
|}

where <math display="inline">n_{i}</math> are the components (in the local system) of the normals to the element sides in the initial configuration <math display="inline">\Gamma _{M}^{0}</math>.

For the definition of the normal vector <math display="inline">\mathbf{t}_{3}</math>, the linear interpolation over the central element is used. In this case the tangent plane components are

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \varphi }_{^{\prime }\alpha } = \sum _{i=1}^{3} L_{i,\alpha }^M {\boldsymbol \varphi }_{i} \quad ,\quad \alpha=1,2 </math>
|}
| style="width: 5px;text-align: right;" | (55)
|}

<span id="eq-56"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{t}_{3}=\frac{{\boldsymbol \varphi }_{\prime{1}}\times{\boldsymbol \varphi }_{\prime{2}}}{\left\vert {\boldsymbol \varphi }_{\prime{1}}\times{\boldsymbol \varphi }_{\prime{2}}\right\vert }=\lambda \;_{1}\times{\boldsymbol \varphi }_{2} </math>
|}
| style="width: 5px;text-align: right;" | (56)
|}

From these expressions it is also possible to compute in the original configuration the element area <math display="inline">A^{0}_{M}</math>, the outer normals <math display="inline">\left( n_{1} ,n_{2}\right) ^{i}</math> at each side and the side lengths <math display="inline">l_{i}^{M}</math>. Eq.([[#eq-56|56]]) also allows to evaluate the thickness ratio <math display="inline">\lambda </math> in the deformed configuration and the actual normal <math display="inline">\mathbf{t}_{3}</math>.

In order to compute the line integral of equation ([[#eq-54|54]]) the averaging procedure described in Section 2 is used. Hence along each side of the triangle the average value of <math display="inline">{\boldsymbol \varphi }_{^{\prime }\alpha }</math> between the main triangle and the adjacent one is taken leading to

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \kappa }=\frac{1}{A^{0}_{M}}\sum _{I=1}^{3}\left[ \begin{array}{cc} -n_{1}^{i} & 0\\ 0 & n_{2}^{i}\\ n_{2}^{i} & -n_{1}^{i} \end{array} \right] \left[ \begin{array}{c} \mathbf{t}_{3}\cdot \frac{1}{2}\left( \mathbf{\boldsymbol \varphi }_{^{\prime }1} ^{M}+\mathbf{\boldsymbol \varphi }_{^{\prime }1}^{i}\right)\\ \mathbf{t}_{3}\cdot \frac{1}{2}\left( \mathbf{\boldsymbol \varphi }_{^{\prime }2} ^{M}+\mathbf{\boldsymbol \varphi }_{^{\prime }2}^{i}\right) \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (57)
|}

where the sum extends over the three elements adjacent to the central triangle <math display="inline">M</math>.

Noting that <math display="inline">\mathbf{t}_{3}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }\alpha }^{M}=0</math> in the main triangle it can be found <span id='citeF-24'></span>[[#cite-24|[24]]]

<span id="eq-58"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \kappa }=\sum _{i=1}^{3}\left[ \begin{array}{cc} L_{i,1}^M & 0 \\         0 & L_{i,2}^M \\ L_{i,2}^M & L_{i,1}^M  \end{array} \right] \left[ \begin{array}{c} \mathbf{t}_{3}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }1}^{i}\\ \mathbf{t}_{3}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }2}^{i} \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (58)
|}

This can be seen as the projection of the local derivatives in the adjacent triangles <math display="inline">\mathbf{\boldsymbol \varphi }_{^{\prime }\alpha }^{i}</math> (where index <math display="inline">i</math> denotes values associated to the adjacent elements) over the normal to the main triangle <math display="inline">\mathbf{t}_{3}</math>. As the triangles have a common side, <math display="inline">\mathbf{t}_{3}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }s}^{i}=0</math>, where <math display="inline">\mathbf{\boldsymbol \varphi }_{^{\prime } s}^{i}</math> is the derivative along the side. Hence only the derivative along the side normal (<math display="inline">\mathbf{\boldsymbol \varphi }_{^{\prime }n}^{i}</math>) has non-zero component over <math display="inline">\mathbf{t}_{3}</math>. This gives

<span id="eq-59"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[ \begin{array}{c} \mathbf{t}_{3}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }1}^{i}\\ \mathbf{t}_{3}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }2}^{i} \end{array} \right] =\left( \mathbf{t}_{3}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime  }n}^{i}\right)\mathbf{n}^{i} </math>
|}
| style="width: 5px;text-align: right;" | (59)
|}

An alternative form to express the curvatures, which is useful when their variations are needed, is to define the vectors

<span id="eq-60"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{h}_{ij}=\sum _{k=1}^{3}\frac{1}{2}\left( L_{k,i}^{M}{\boldsymbol \varphi  }_{^{\prime }j}^{k}+L_{k,j}^{M}{\boldsymbol \varphi }_{\prime i}^{k}\right) </math>
|}
| style="width: 5px;text-align: right;" | (60)
|}

This gives

<span id="eq-61"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\kappa _{ij}=\mathbf{h}_{ij}\cdot \mathbf{t}_{3}</math>
|}
| style="width: 5px;text-align: right;" | (61)
|}

The last expression allows to interpret the curvatures as the projections of the vectors <math display="inline">\mathbf{h}_{ij}</math> over the normal of the central element. The variation of the curvatures can be obtained as

<span id="eq-62"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\delta{\boldsymbol \kappa }=\sum _{i=1}^{3}\left\{ \left[ \begin{array}{cc} L_{i,1}^{M} & 0\\ 0 & L_{i,2}^{M}\\ L_{i,2}^{M} & L_{i,1}^{M} \end{array} \right] \sum _{J=1}^{3}\left[ \begin{array}{c} L_{j,1}^{i}(\mathbf{t}_{3}\cdot \delta \mathbf{u}_{j}^{i})\\ N_{j,2}^{i}(\mathbf{t}_{3}\cdot \delta \mathbf{u}_{j}^{i}) \end{array} \right] -2\left[ \begin{array}{c} (L_{i,1}^{M}\rho _{11}^{1}+L_{i,2}^{M}\rho _{11}^{2})\\ (L_{i,1}^{M}\rho _{22}^{1}+L_{i,2}^{M}\rho _{22}^{2})\\ (L_{i,1}^{M}\rho _{12}^{1}+L_{i,2}^{M}\rho _{12}^{2}) \end{array} \right] (\mathbf{t}_{3}\cdot \delta \mathbf{u}_{i}^{M})\right\} </math>
|}
| style="width: 5px;text-align: right;" | (62)
|}

where the projections of the vectors <math display="inline">\mathbf{h}_{ij}</math> over the contravariant base vectors <math display="inline">\tilde{\boldsymbol \varphi }_{^{\prime }\alpha }</math> have been included

<span id="eq-63"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho _{ij}^{\alpha }=\mathbf{h}_{ij}\cdot \tilde{\boldsymbol \varphi }_{^{\prime }\alpha } \quad ,\quad \alpha ,i,j=1,2</math>
|}
| style="width: 5px;text-align: right;" | (63)
|}

with

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math>\mathbf{\tilde{\boldsymbol \varphi }}_{^{\prime }1}  </math>
| <math>  =\lambda \;\mathbf{\boldsymbol \varphi }_{^{\prime }2}\times \mathbf{t}_{3}</math>
|}
| style="width: 5px;text-align: right;" | (64)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math> \mathbf{\tilde{\boldsymbol \varphi }}_{^{\prime }2}  </math>
| <math>  =-\lambda \;\mathbf{\boldsymbol \varphi  }_{^{\prime }1}\times \mathbf{t}_{3} </math>
|}
| style="width: 5px;text-align: right;" | (65)
|}


In above expressions superindexes in <math display="inline">L_{j}^k</math> and <math display="inline">\delta \mathbf{u}_{j}^k</math> refer to element numbers whereas subscripts denote node numbers. As usual the superindex <math display="inline">M</math> denotes values in the central triangle (Figure [[#img-1|1]]). Note that as expected the curvatures (and their variations) in the central element are a function of the nodal displacements of the six nodes in the four elements patch. Note also that

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\lambda ={\frac{h}{h^{0}}}={\frac{A_{M}^{0}}{A_{M}}} </math>
|}
| style="width: 5px;text-align: right;" | (66)
|}

Details of the derivation of Eq.([[#eq-62|62]]) can be found in [27].

Eq.([[#eq-62|62]]) can be rewritten in the form

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\delta{\boldsymbol \kappa }=\mathbf{B}_{b}\delta \mathbf{a}^{p} </math>
|}
| style="width: 5px;text-align: right;" | (67)
|}

where

<span id="eq-68"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{c} \\ \delta \mathbf{a}^{p}\\ 18\times{1} \end{array} =[\delta \mathbf{u}_{1}^{T},\delta \mathbf{u}_{2}^{T},\delta \mathbf{u}_{3} ^{T},\delta \mathbf{u}_{4}^{T},\delta \mathbf{u}_{5}^{T},\delta \mathbf{u}_{6}^{T}]^{T}</math>
|}
| style="width: 5px;text-align: right;" | (68)
|}

is the virtual displacement vector of the patch

<span id="eq-69"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{B}_{b}=[\mathbf{B}_{b1},\mathbf{B}_{b2}\cdots ,\mathbf{B}_{b6}]</math>
|}
| style="width: 5px;text-align: right;" | (69)
|}

is the curvature matrix relating the virtual curvatures within the central element and the 18 virtual displacements of the six nodes in the patch.

The form of matrix <math display="inline">\mathbf{B}_{b}</math> is given in the Appendix.

==5 ENHANCED BASIC SHELL TRIANGLE==

An enhanced version of the BST element (termed EBST) has been recently proposed by Flores and Oñate [27]. The main features of the element formulation are the following:

<ol>

<li>The geometry of the patch formed by the central element and the three adjacent elements is ''quadratically interpolated'' from the position of the six nodes in the patch. </li>

<li>The membrane strains are assumed to vary ''linearly'' within the central triangle and are expressed in terms of the (continuous) values of the deformation gradient at the mid side points of the triangle. </li>

<li>The assumed ''constant curvature'' field within the central triangle is obtained by expression ([[#eq-54|54]]) using now twice the values of the (continuous) deformation gradient at the mid side points. </li>

</ol>

Details of the derivation of the EBST element are given below.

===5.1 Definition of the element geometry and computation of membrane strains===

As mentioned above a quadratic approximation of the geometry of the four elements patch is chosen using the position of the six nodes in the patch. It is useful to define the patch in the isoparametric space using the nodal positions given in the Table 2 (see also Figure 2).


<div class="center" style="font-size: 75%;">
'''Table 2'''. Isoparametric coordinates of the six nodes in the patch of Figure 2.</div>

{| class="wikitable" style="text-align: left; margin: 1em auto;"
|- style="border-top: 2px solid;"
| 1 
| 2 
| 3 
| 4 
| 5 
| 6
|- style="border-top: 2px solid;"
| <math display="inline">\xi </math> 
| 0 
| 1 
| 0 
| 1 
| -1 
| 1
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| <math display="inline">\eta </math> 
| 0 
| 0 
| 1 
| 1 
| 1 
| -1

|}

The quadratic interpolation is defined by

<span id="eq-70"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \varphi }=\sum _{i=1}^{6}N_{i}{\boldsymbol \varphi }_{i}</math>
|}
| style="width: 5px;text-align: right;" | (70)
|}

with (<math display="inline">\zeta=1-\xi-\eta</math>)

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{ccc} N_{1}=\zeta{+\xi}\eta &  & N_{4}=\frac{\zeta }{2}\left( \zeta{-1}\right) \\ N_{2}=\xi{+\eta}\zeta &  & N_{5}=\frac{\xi }{2}\left( \xi{-1}\right) \\ N_{3}=\eta{+\zeta}\xi &  & N_{6}=\frac{\eta }{2}\left( \eta{-1}\right) \end{array} </math>
|}
| style="width: 5px;text-align: right;" | (71)
|}

This interpolation allows to compute the displacement gradients at selected points in order to use an assumed strain approach. The computation of the gradients is performed at the mid side points of the central element of the patch denoted by <math display="inline">G_{1}</math>, <math display="inline">G_{2}</math> and <math display="inline">G_{3}</math> in Figure [[#img-2|2]]. This choice has the following advantages.

<div id='img-2'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig2.png|600px|Patch of elements in the isoparametric space.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Patch of elements in the isoparametric space.
|}

* Gradients at the three mid side points depend only on the nodes belonging to the two elements adjacent to each side. This can be easily verified by sampling the derivatives of the shape functions at each mid-side point.

* When gradients are computed at the common mid-side point of two adjacent elements, the same values are obtained, as the coordinates of the same four points are used. This in practice means that the gradients at the mid-side points are independent of the element where they are computed. A side-oriented implementation of the finite element will therefore lead to a unique evaluation of the gradients per side.

The cartesian derivatives of the shape functions are computed at the original configuration by the standard expression

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[ \begin{array}{c} N_{i,1}\\ N_{i,2} \end{array} \right] =\mathbf{J}^{-1}\left[ \begin{array}{c} N_{i,\xi } \\ N_{i,\eta } \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (72)
|}

where the Jacobian matrix at the original configuration is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{J=}\left[ \begin{array}{cc} \mathbf{\boldsymbol \varphi }_{^{\prime }\xi }^{0}\cdot \mathbf{t}_{1} & \mathbf{\boldsymbol \varphi  }_{^{\prime }\eta }^{0}\cdot \mathbf{t}_{1}\\ \mathbf{\boldsymbol \varphi }_{^{\prime }\xi }^{0}\cdot \mathbf{t}_{2} & \mathbf{\boldsymbol \varphi  }_{^{\prime }\eta }^{0}\cdot \mathbf{t}_{2} \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (73)
|}

The deformation gradients on the middle surface, associated to an arbitrary spatial cartesian system and to the material cartesian system defined on the middle surface are related by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[ {\boldsymbol \varphi }_{^{\prime }1},\mathbf{\boldsymbol \varphi }_{^{\prime }2}\right] =\left[ \mathbf{\boldsymbol \varphi }_{^{\prime }\xi },\mathbf{\boldsymbol \varphi }_{^{\prime }\eta }\right]  \mathbf{J}^{-1} </math>
|}
| style="width: 5px;text-align: right;" | (74)
|}

The Green-Lagrange membrane strains within the central triangle are now obtained using a linear assumed membrane strain field <math display="inline">\hat{\boldsymbol \varepsilon }_{m}</math>, i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \varepsilon }_{m}=\hat{\boldsymbol \varepsilon }_{m} </math>
|}
| style="width: 5px;text-align: right;" | (75)
|}

with

<span id="eq-76"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hat{\boldsymbol \varepsilon }_{m}=(1-2\zeta ){\boldsymbol \varepsilon }_{m}^{1}+(1-2\xi ){\boldsymbol \varepsilon  }_{m}^{2}+(1-2\eta ){\boldsymbol \varepsilon }_{m}^{3}=\sum _{i=1}^{3}\bar{N}_{i} {\boldsymbol \varepsilon }_{m}^{i}</math>
|}
| style="width: 5px;text-align: right;" | (76)
|}

where <math display="inline">{\boldsymbol \varepsilon }_{m}^{i}</math> are the membrane strains computed at the three mid side points <math display="inline">G_{i}</math> (<math display="inline">i=1,2,3</math>  see Figure [[#img-2|2]]). In Eq.([[#eq-76|76]]) <math display="inline">\bar{N}_{1}=(1-2\zeta )</math>, etc.

The gradient at each mid side point is computed from the quadratic interpolation ([[#eq-70|70]]):

<span id="eq-77"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left( {\boldsymbol \varphi }_{^{\prime }\alpha }\right) _{G_{i}}={\boldsymbol \varphi }_{^{\prime  }\alpha }^{i}=\left[ \sum _{j=1}^{3}N_{j,\alpha }^{i}{\boldsymbol \varphi }_{j}\right] +N_{i+3,\alpha }^{i}{\boldsymbol \varphi }_{i+3}\quad ,\quad \alpha=1,2\quad ,\quad  i=1,2,3</math>
|}
| style="width: 5px;text-align: right;" | (77)
|}

Substituting Eq.([[#eq-22|22]]) into ([[#eq-76|76]]) and using Eq.([[#eq-22|22]]) gives the membrane strain vector as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \varepsilon }_{m}=\sum _{i=1}^{3}\frac{1}{2}\bar{N}_{i}\left\{ \begin{array}{c} {\boldsymbol \varphi }_{^{\prime }1}^{i}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }1}^{i}-1\\ {\boldsymbol \varphi }_{^{\prime }2}^{i}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }2}^{i}-1\\ 2{\boldsymbol \varphi }_{^{\prime }1}^{i}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }2}^{i} \end{array} \right\} </math>
|}
| style="width: 5px;text-align: right;" | (78)
|}

and the virtual membrane strains as

<span id="eq-79"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\delta{\boldsymbol \varepsilon }_{m}=\sum _{i=1}^{3}\bar{N}_{i}\left\{ \begin{array}{c} {\boldsymbol \varphi }_{^{\prime }1}^{i}\cdot \delta \mathbf{\boldsymbol \varphi }_{^{\prime }1}^{i}\\ {\boldsymbol \varphi }_{2}^{i}\cdot \delta \mathbf{\boldsymbol \varphi }_{^{\prime }2}^{i}\\ \delta{\boldsymbol \varphi }_{^{\prime }1}^{i}\cdot \mathbf{\boldsymbol \varphi }_{^{\prime }2} ^{i}+{\boldsymbol \varphi }_{^{\prime }1}^{i}\cdot \delta \mathbf{\boldsymbol \varphi }_{2}^{i} \end{array} \right\} </math>
|}
| style="width: 5px;text-align: right;" | (79)
|}

We note that the gradient at each mid side point <math display="inline">G_{i}</math> depends only on the coordinates of the three nodes of the central triangle and on those of an additional node in the patch, associated to the side <math display="inline">i</math> where the gradient is computed.

Combining Eqs.([[#eq-79|79]]) and ([[#eq-77|77]]) gives

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\delta{\boldsymbol \varepsilon }_{m}=\mathbf{B}_{m}\delta \mathbf{a}^{p} </math>
|}
| style="width: 5px;text-align: right;" | (80)
|}

where <math display="inline">\delta \mathbf{a}^{p}</math> is the patch displacement vector (see Eq.([[#eq-68|68]])) and <math display="inline">\mathbf{B}_{m}</math> is the membrane strain matrix. An explicit form of this matrix is given in the Appendix.

Differently from the original BST element the membrane strains within the EBST element are now a function of the displacements of the six patch nodes.

===5.2 Computation of curvatures===

The constant curvature field assumed for the BST element is chosen again here. The numerical evaluation of the line  integral in Eq.([[#eq-54|54]]) results in a sum over the integration points at the element boundary which are, in fact, the same points used for evaluating the gradients when computing the membrane strains. As one integration point is used over each side, it is not necessary to distinguish between sides (<math display="inline">i</math>) and integration points (<math display="inline">G_{i}</math>). In this way the curvatures can be computed by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \kappa }=2\sum _{i=1}^{3}\left[ \begin{array}{cc} L_{i,1}^M & 0\\ 0         & L_{i,2}^M \\ L_{i,2}^M & L_{i,1}^M  \end{array} \right] \left[ \begin{array}{c} \mathbf{t}_{3}\cdot{\boldsymbol \varphi }_{^{\prime }1}^{i}\\ \mathbf{t}_{3}\cdot{\boldsymbol \varphi }_{^{\prime }2}^{i} \end{array} \right] </math>
|}
| style="width: 5px;text-align: right;" | (81)
|}

In the standard BST element <span id='citeF-21'></span><span id='citeF-24'></span>[[#cite-21|[21,24]]] the gradient <math display="inline">\mathbf{\boldsymbol \varphi  }_{\prime \alpha }^{i}</math> is computed as the average of the linear approximations over the two adjacent elements (see Section 4.3). In the enhanced version, the gradient is evaluated at each side <math display="inline">G_{i}</math> from the quadratic interpolation

<span id="eq-82"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[ \begin{array}{c} {\boldsymbol \varphi }_{\prime{1}}^{i}\\ {\boldsymbol \varphi }_{\prime{2}}^{i} \end{array} \right] =\left[ \begin{array}{cccc} N_{1,1}^{i} & N_{2,1}^{i} & N_{3,1}^{i} & N_{i+3,1}^{i}\\ N_{1,2}^{i} & N_{2,2}^{i} & N_{3,2}^{i} & N_{i+3,2}^{i} \end{array} \right] \left[ \begin{array}{c} {\boldsymbol \varphi }_{1}\\ {\boldsymbol \varphi }_{2}\\ {\boldsymbol \varphi }_{3}\\ {\boldsymbol \varphi }_{i+3} \end{array} \right]  </math>
|}
| style="width: 5px;text-align: right;" | (82)
|}

Note again than at each side the gradients depend only on the positions of the three nodes of the central triangle and of an extra node (<math display="inline">i+3</math>), associated precisely to the side (<math display="inline">G_{i}</math>) where the gradient is computed.

Direction '''t'''<math display="inline">_{3}</math> in Eq.([[#eq-82|82]]) can be seen as a reference direction. If a different direction than that given by Eq.([[#eq-56|56]]) is chosen, at an angle <math display="inline">\theta </math> with the former, this has an influence of order <math display="inline">\theta ^{2}</math> in the projection. This justifies Eq.([[#eq-56|56]]) for the definition of '''t'''<math display="inline">_{3}</math> as a function exclusively of the three nodes of the central triangle, instead of using the 6-node isoparametric interpolation.

The variation of the curvatures can be obtained as

<span id="eq-89"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math>\delta{\boldsymbol \kappa } </math>
| <math>  =2\sum _{i=1}^{3}\left[ \begin{array}{cc} L_{i,1}^M & 0\\ 0         & L_{i,2}^M\\ L_{i,2}^M & L_{i,1}^M \end{array} \right] \left\{ \sum _{i=1}^{3}\left[ \begin{array}{c} N_{j,1}^{i}(\mathbf{t}_{3}\cdot \delta \mathbf{u}_{j})\\ N_{j,2}^{i}(\mathbf{t}_{3}\cdot \delta \mathbf{u}_{j}) \end{array} \right] +\left[ \begin{array}{c} N_{i+3,1}^{i}(\mathbf{t}_{3}\cdot \delta \mathbf{u}^{i+3})\\ N_{i+3,2}^{i}(\mathbf{t}_{3}\cdot \delta \mathbf{u}^{i+3}) \end{array} \right] \right\} -</math>
|-
| style="text-align: right;" | 
| <math>  -\sum _{i=1}^{3}\left[ \begin{array}{c} (L_{i,1}^M\rho _{11}^{1}+L_{i,2}^M\rho _{11}^{2})\\ (L_{i,1}^M\rho _{22}^{1}+L_{i,2}^M\rho _{22}^{2})\\ (L_{i,1}^M\rho _{12}^{1}+L_{i,2}^M\rho _{12}^{2}) \end{array} \right] (\mathbf{t}_{3}\cdot \delta \mathbf{u}_{i})=\mathbf{B}_{b} \delta \mathbf{a}^{p}</math>
|}
| style="width: 5px;text-align: right;" | (89)
|}

where the definitions ([[#eq-61|61]]) and ([[#eq-63|63]]) still hold but with the new definition of <math display="inline">\mathbf{h}_{ij}</math> given by <span id='citeF-27'></span>[[#cite-27|[27]]]

<span id="eq-90"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{h}_{ij}=\sum _{k=1}^{3}\left( L_{k,i}^M{\boldsymbol \varphi }_{\prime j}^{k} +L_{k,j}^M{\boldsymbol \varphi }_{^{\prime }i}^{k}\right) </math>
|}
| style="width: 5px;text-align: right;" | (90)
|}

In Eq.([[#eq-89|89]])

<span id="eq-91"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{B}_{b}=[\mathbf{B}_{b_{1}},\mathbf{B}_{b_{2}},\cdots ,\mathbf{B}_{b_{6}}]</math>
|}
| style="width: 5px;text-align: right;" | (91)
|}

The expression of the curvature matrix <math display="inline">\mathbf{B}_b</math> is given in the Appendix. Details of the derivation of Eq.([[#eq-89|89]]) can be found in [27].

===5.3 The EBST1 element===

A simplified and yet very effective version of the EBST element can be obtained by using ''one point quadrature'' for the computation of all the element integrals. This element is termed EBST1. Note that this only affects the membrane stiffness matrices and it is equivalent to using a assumed constant membrane strain field defined by an average of the metric tensors computed at each side.

Numerical experiments have shown that both the EBST and the EBST1 elements are free of spurious energy modes.

==6 BOUNDARY CONDITIONS==

Elements at the domain boundary, where an adjacent element does not exist, deserve a special attention. The treatment of essential boundary conditions associated to translational constraints is straightforward, as they are the natural degrees of freedom of the element. The conditions associated to the normal vector are crucial in this formulation for bending. For clamped sides or symmetry planes, the normal vector <math display="inline">\mathbf{t}_{3}</math> must be kept fixed (clamped case), or constrained to move in the plane of symmetry (symmetry case). The former case can be seen as a special case of the latter, so we will consider symmetry planes only. This restriction can be imposed through the definition of the tangent plane at the boundary, including the normal to the plane of symmetry <math display="inline">\boldsymbol \varphi _{^{\prime }n}^{0}</math> that does not change during the process.

<div id='img-3'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_5749_Fig3.jpeg|600px|Local cartesian system for the treatment of symmetry boundary conditions]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Local cartesian system for the treatment of symmetry boundary conditions
|}

The tangent plane at the boundary (mid-side point) is expressed in terms of two orthogonal unit vectors referred to a local-to-the-boundary Cartesian system (see Figure [[#img-3|3]]) defined as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[\boldsymbol \varphi _{^{\prime }n}^{0},\;\bar{\boldsymbol \varphi }_{^{\prime }s}\right] </math>
|}
| style="width: 5px;text-align: right;" | (92)
|}

where vector <math display="inline">\boldsymbol \varphi _{^{\prime }n}^{0}</math> is fixed during the process while direction <math display="inline">\bar{\boldsymbol \varphi }_{^{\prime }s}</math> emerges from the intersection of the symmetry plane with the plane defined by the central element (<math display="inline">M</math>). The plane (gradient) defined by the central element in the selected original convective Cartesian system (<math display="inline">\mathbf{t}_{1},\mathbf{t}_{2} </math>) is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[\boldsymbol \varphi _{^{\prime }1}^{M},\;\boldsymbol \varphi _{^{\prime  }2}^{M}\right] </math>
|}
| style="width: 5px;text-align: right;" | (93)
|}

the intersection line (side <math display="inline">i</math>) of this plane with the plane of symmetry can be written in terms of the position of the nodes that define the side (<math display="inline">j </math> and <math display="inline">k</math>) and the original length of the side <math display="inline">l_{i}^{M}</math>, i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\boldsymbol \varphi _{^{\prime }s}^{i}=\frac{1}{l_{i}^{M}}\left(\boldsymbol  \varphi _{k}-\boldsymbol \varphi _{j}\right) </math>
|}
| style="width: 5px;text-align: right;" | (94)
|}

That together with the outer normal to the side <math display="inline">\mathbf{n}^{i} = \left[n_{1},n_{2}\right]^{T}=\left[\mathbf{n\cdot t}_{1},\mathbf{n\cdot t}_{2}\right]^{T}</math> (resolved in the selected original convective Cartesian system) leads to

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[ \begin{array}{c}\boldsymbol \varphi _{^{\prime }1}^{iT} \\ \boldsymbol \varphi _{^{\prime }2}^{iT} \end{array}\right]=\left[ \begin{array}{cc}n_{1} & -n_{2} \\ n_{2} & n_{1} \end{array}\right]\left[ \begin{array}{c}\boldsymbol \varphi _{^{\prime }n}^{iT} \\ \boldsymbol \varphi _{^{\prime }s}^{iT} \end{array}\right] </math>
|}
| style="width: 5px;text-align: right;" | (95)
|}

where, noting  that <math display="inline">\lambda </math> is the determinant of the gradient, the normal component of the gradient <math display="inline">\boldsymbol \varphi _{^{\prime }n}^{i}</math> can be approximated by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\boldsymbol \varphi _{^{\prime }n}^{i}=\frac{\boldsymbol \varphi _{^{\prime }n}^{0} }{\lambda |\boldsymbol \varphi _{^{\prime }s}^{i}|} </math>
|}
| style="width: 5px;text-align: right;" | (96)
|}

In this way the contribution of the gradient at side <math display="inline">i</math> to vectors <math display="inline">\mathbf{ h}_{\alpha \beta }</math> (equations [[#eq-60|60]] and [[#eq-90|90]]) results in

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[ \begin{array}{c}\mathbf{h}_{11}^{T} \\ \mathbf{h}_{22}^{T} \\ 2\mathbf{h}_{12}^{T} \end{array}\right]^{i}=2\left[ \begin{array}{cc}L_{i,1}^{M} & 0 \\ 0 & L_{i,2}^{M} \\ L_{i,2}^{M} & L_{i,1}^{M} \end{array}\right]\left[ \begin{array}{c}\boldsymbol \varphi _{^{\prime }1}^{iT} \\ \boldsymbol \varphi _{^{\prime }2}^{iT} \end{array}\right]=2\left[ \begin{array}{cc}L_{i,1}^{M} & 0 \\ 0 & L_{i,2}^{M} \\ L_{i,2}^{M} & L_{i,1}^{M} \end{array}\right]\left[ \begin{array}{cc}n_{1} & -n_{2} \\ n_{2} & n_{1} \end{array}\right]\left[ \begin{array}{c}\boldsymbol \varphi _{^{\prime }n}^{iT} \\ \boldsymbol \varphi _{^{\prime }s}^{iT} \end{array}\right] </math>
|}
| style="width: 5px;text-align: right;" | (97)
|}

For the computation of the curvature variations, the contribution from the gradient at side <math display="inline">i</math> is now (see Ref. <span id='citeF-27'></span>[[#cite-27|[27]]])

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math> \delta \left[ \begin{array}{c} \mathbf{h}_{11}^{T} \\ \mathbf{h}_{22}^{T} \\ 2\mathbf{h}_{12}^{T} \end{array} \right]^{i} =2\left[ \begin{array}{cc} L_{i,1}^{M} & 0 \\ 0 & L_{i,2}^{M} \\ L_{i,2}^{M} & L_{i,1}^{M} \end{array} \right]\left[ \begin{array}{cc} n_{1} & -n_{2} \\ n_{2} & n_{1} \end{array} \right]\left[ \begin{array}{c} \mathbf{0} \\ \frac{1}{L_{o}}\left[\delta \mathbf{u}_{k}-\delta \mathbf{u}_{j}\right]^{T} \end{array} \right]</math>
|}
| style="width: 5px;text-align: right;" |  (92a)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math> =\frac{2}{l_{i}^{M}}\left[ \begin{array}{c} -L_{i,1}^{M}n_{2} \\ L_{i,2}^{M}n_{1} \\ L_{i,1}^{M}n_{1}-L_{i,2}^{M}n_{2} \end{array} \right]\left[\delta \mathbf{u}_{k}-\delta \mathbf{u}_{j}\right]^{T}</math>
|}
| style="width: 5px;text-align: right;" |  %
(92b)
|}

where the influence of variations in the length of vector <math display="inline">\boldsymbol \varphi  _{^{\prime }n}</math> has been neglected.

For a simple supported (hinged) side, the problem is not completely defined. The simplest choice is to neglect the contribution to the side rotations from the adjacent element missing in the patch in the evaluation of the curvatures via eq.([[#eq-54|54]]) <span id='citeF-29'></span>[[#cite-29|29]]. This is equivalent to assume that the gradient at the side is equal to the gradient in the central element, i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[\boldsymbol \varphi _{^{\prime }1}^{i},\;\boldsymbol \varphi _{^{\prime }2}^{i} \right]=\left[\boldsymbol \varphi _{^{\prime }1}^{M},\;\boldsymbol \varphi  _{^{\prime }2}^{M}\right] </math>
|}
|}

More precise changes can be however introduced to account for the different natural boundary conditions. One may assume that the curvature normal to the side is zero, and consider a contribution of the missing side to introduce this constraint. As the change of curvature parallel to the side is zero along the hinged side, both things lead to zero curvatures in both directions. Denoting the contribution to curvatures of the existing sides (<math display="inline">j </math> and <math display="inline">k</math>) by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[ \begin{array}{c}\kappa _{11} \\ \kappa _{22} \\ \kappa _{12} \end{array} \right]^{j-k} </math>
|}
|}

It can be easily shown that to set the normal curvature to zero the contribution of the simple supported side (<math display="inline">i</math>) should be

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[ \begin{array}{c}\kappa _{11} \\ \kappa _{22} \\ \kappa _{12} \end{array} \right]^{i}=-\left[ \begin{array}{ccc}\left(n_{1}\right)^{4} & \left(n_{1}\right)^{2}\left(n_{2}\right)^{2} & \left(n_{1}\right)^{3}n_{2} \\ \left(n_{1}\right)^{2}\left(n_{2}\right)^{2} & \left(n_{2}\right)^{4} & n_{1}\left(n_{2}\right)^{3} \\ 2\left(n_{1}\right)^{3}n_{2} & 2n_{1}\left(n_{2}\right)^{3} & 2\left( n_{1}\right)^{2}\left(n_{2}\right)^{2} \end{array} \right]\left[ \begin{array}{c}\kappa _{11} \\ \kappa _{22} \\ \kappa _{12} \end{array} \right]^{j-k} </math>
|}
| style="width: 5px;text-align: right;" | (93)
|}

For the case of a triangle with two sides associated to hinged sides, the normal curvatures to both sides must be zero. Denoting by <math display="inline">\mathbf{n}^{i}</math> and <math display="inline">\mathbf{n}^{j}</math> the normal to the sides, and by <math display="inline">\mathbf{m}^{i}</math> and <math display="inline"> \mathbf{m}^{j}</math> the dual base (associated to base <math display="inline">\mathbf{n}^{i}-</math> <math display="inline">\mathbf{ n}^{j}</math>), the contribution from the hinged sides (<math display="inline">i</math> and <math display="inline">j</math>) can be written as a function of the contribution of the only existing side (<math display="inline">k</math>):

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\left[ \begin{array}{c}\kappa _{11} \\ \kappa _{22} \\ \kappa _{12} \end{array} \right]^{i-j}=-\left[ \begin{array}{c}m_{1}^{i}m_{1}^{j} \\ m_{2}^{i}m_{2}^{j} \\ m_{1}^{i}m_{2}^{j}+m_{2}^{i}m_{1}^{j} \end{array} \right]\left[ \begin{array}{ccc}2n_{1}^{i}n_{1}^{j} & 2n_{2}^{i}n_{2}^{j} & n_{1}^{i}n_{2}^{j}+n_{2}^{i}n_{1}^{j} \end{array} \right]\left[ \begin{array}{c}\kappa _{11} \\ \kappa _{22} \\ \kappa _{12} \end{array} \right]^{k} </math>
|}
| style="width: 5px;text-align: right;" | (94)
|}

For a free edge the same approximation can be used but due to Poisson's effect this will lead to some error. The curvature variations of these contributions can be easily computed.

For the membrane formulation of element EBST, the gradient at the mid-side point of the boundary is assumed equal to the gradient of the main triangle.

==7 IMPLICIT SOLUTION SCHEME==

For a step <math display="inline">n</math> the configuration <math display="inline">\mathbf{\boldsymbol \varphi }^{n}</math> and the plastic strains <math display="inline">{\boldsymbol \varepsilon }_{p}^{n}</math> are known. The configuration <math display="inline">\mathbf{\boldsymbol \varphi }^{n}</math> is obtained by adding the total displacements to the original configuration <math display="inline"> \mathbf{\boldsymbol \varphi }^{n}=\mathbf{\boldsymbol \varphi }^{0} +\mathbf{u}^{n}</math>. The stresses are computed at each triangle using a single sampling (integration) point at the center and <math display="inline">N_{L}</math> integration points (layers) through the thickness. The plane stress state condition of the classical thin shell theory is assumed, so that for every layer three stress components are computed, (<math display="inline">\sigma _{11}</math>,<math display="inline">\sigma _{22}</math>, and <math display="inline">\sigma _{12}</math>) referred to the local cartesian system.

The computation of the incremental stresses is as follows:

<ol>

<li>Evaluate the incremental displacements: <math display="inline">\Delta \mathbf{u}^{n}=\mathbf{K}_{T}^{n}\mathbf{r}^{n}</math> where <math display="inline">\mathbf{K}_{T}</math> is the tangent stiffness matrix and '''r''' is the residual force vector  defined by for each element

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{r}^e_i =\int \int _A L_i {t}\, dA - \int \int _{A^\circ } ({B}_{m_i}^T {\boldsymbol \sigma }_m + {B}_{b_i}^T {\boldsymbol \sigma }_b)dA </math>
|}
| style="width: 5px;text-align: right;" | (95)
|}</li>

The expression of the tangent stiffness matrix for the element is given below. Details of the derivation can be found in <span id='citeF-24'></span>[[#cite-24|[24]]],<span id='citeF-27'></span>[[#cite-27|[27]]].

<li>Generate the actual configuration <math display="inline">\mathbf{\boldsymbol \varphi }^{n+1} =\mathbf{\boldsymbol \varphi }^{n}+\Delta \mathbf{u}^{n}</math> </li>

<li>Compute the metric tensor <math display="inline">a_{\alpha \beta }^{n+1}\mathbf{ }</math>and the curvatures <math display="inline">\kappa _{\alpha \beta }^{n+1}</math> </li>

<li>Compute the total and elastic deformations at each layer <math display="inline">k</math> </li>

<span id="eq-96"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math>{\boldsymbol \varepsilon }_{k}^{n+1} </math>
| <math>  ={\boldsymbol \varepsilon }_{m}^{n+1}+z_{k}{\boldsymbol \chi } ^{n+1}</math>
| style="width: 5px;text-align: right;" | (96)
|-
| style="text-align: right;" | <math> \left[ {\boldsymbol \varepsilon }_{e}\right] _{k}^{n+1} </math>
| <math>  ={\boldsymbol \varepsilon  }_{k}^{n+1}-\left[ {\boldsymbol \varepsilon }_{p}\right] _{k}^{n} </math>
|}
|}

<li>Compute the trial elastic stresses at each layer <math display="inline">k</math>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\boldsymbol \sigma } _{k}^{n+1}=\mathbf{C}\left[ {\boldsymbol \varepsilon }_{e}\right] _{k}^{n+1} </math>
|}
| style="width: 5px;text-align: right;" | (97)
|}</li>

<li>Check the plasticity condition and return to the plasticity surface. If necessary correct the plastic strains <math display="inline">\left[{\boldsymbol \varepsilon } _{p}\right] _{k}^{n+1}</math> at each layer (small strain plasticity) </li>

<li>Compute the generalized stresses

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math>{\boldsymbol \sigma }^{n+1}_{m}  </math>
| <math>  =\frac{h^{0}}{N_{L}}\sum _{k=1}^{N_{L}}\boldsymbol \sigma _{k} ^{n+1} w_{k}</math>
|-
| style="text-align: right;" | <math> {\boldsymbol \sigma }^{n+1}_{b}  </math>
| <math>  =\frac{h^{0}}{N_{L}}\sum _{k=1}^{N_{L}}\boldsymbol \sigma _{k} ^{n+1}z_{k} w_{k} </math>
|}
| style="width: 5px;text-align: right;" | (98)
|}</li>

Where <math display="inline"> w_{k}</math> is the weight of the through-the-thickness integration point. Recall that <math display="inline">z_{k}</math> is the current distance of the layer to the mid-surface and not the original distance. However, for small strain plasticity this distinction is not important.

This computation of stresses is adequate for an implicit scheme independent of the step size and it is exact for an elastic problem.

<li>Compute the residual force vector. The contribution for the <math display="inline">M</math>th element is given by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>(\mathbf{r}^{M})^{n+1}=-A_{M}^{0}\left[ \begin{array}{cc} \mathbf{B}_{m}^{T} & \mathbf{B}_{b}^{T} \end{array} \right] ^{n+1}\left[ \begin{array}{c} \boldsymbol \sigma _{m}\\ \boldsymbol \sigma _{b} \end{array} \right] ^{n+1} </math>
|}
| style="width: 5px;text-align: right;" | (99)
|}</li>

</ol>

===7.1 Tangent stiffness matrix===

As usual the tangent stiffness matrix is split into material and geometric components. The material tangent stiffness matrix is simply computed from the integral

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{K}^{M}=\int \int _{A^{M}}\mathbf{B}^{T}\mathbf{C}\mathbf{B}dA </math>
|}
| style="width: 5px;text-align: right;" | (100)
|}

where <math display="inline">\mathbf{B}=\mathbf{B}_{m}+\mathbf{B}_{b}</math> includes:<br/>

* '''-'''  a membrane contribution <math display="inline">\mathbf{B}_{m}</math> given by Eq.([[#eq-51|51]]) or Eq.(80).

* '''-'''  a bending contribution <math display="inline">\mathbf{B}_{b}</math> given by Eq.([[#eq-69|69]]) or Eq.([[#eq-91|91]])  which is constant over the element.

<br/>

A three point quadrature is used for integrating the stiffness terms <math display="inline">\mathbf{B}_{m}^{T}\mathbf{C}\mathbf{B}_{m}</math> (recall that for the EBST element the membrane strains vary linearly within the element) whereas one point quadrature is chosen for the rest of the terms in <math display="inline">\mathbf{K}^{M}</math>.

===7.2 Geometric tangent stiffness matrix===

The geometric stiffness is written as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{K}^{G}=\mathbf{K}_{m}^{G}+\mathbf{K}_{b}^{G} </math>
|}
| style="width: 5px;text-align: right;" | (101)
|}

where subscripts <math display="inline">m</math> and <math display="inline">b</math> denote as usual membrane and bending contributions. For the BST element the membrane part is the same than for the standard constant strain triangle, leading to

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math>\delta \mathbf{u}^{T}\mathbf{K}_{m}^{G}\mathbf{\;}\Delta \mathbf{u} </math>
| <math> =A^{M}\sum _{i=1}^{3}\sum _{j=1}^{3}\left\{ \delta \mathbf{u}_{i}\;\left[ \begin{array}{cc} L_{i,1}^{M} & L_{i,2}^{M} \end{array} \right] \left[ \begin{array}{cc} N_{11} & N_{12}\\ N_{21} & N_{22} \end{array} \right] \left[ \begin{array}{c} L_{j,1}^{M}\\ L_{j,2}^{M} \end{array} \right] \Delta \mathbf{u}_{j}\right\} </math>
|-
| style="text-align: right;" | 
| 
|}
| style="width: 5px;text-align: right;" | (104)
|}

While for the EBST element the membrane part is computed as the sum of the contributions of the three sides, i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | <math>\delta \mathbf{u}^{T}\mathbf{K}_{m}^{G}\mathbf{\;}\Delta \mathbf{u} </math>
| <math> =\frac{A^{M}}{3}\sum _{k=1}^{3}\sum _{i=1}^{6}\sum _{j=1}^{6}\left\{ \delta \mathbf{u}_{i}\;\left[ \begin{array}{cc} N_{i,1}^{k} & N_{i,2}^{k} \end{array} \right] \left[ \begin{array}{cc} N_{11}^{k} & N_{12}^{k}\\ N_{21}^{k} & N_{22}^{k} \end{array} \right] \left[ \begin{array}{c} N_{j,1}^{k}\\ N_{j,2}^{k} \end{array} \right] \Delta \mathbf{u}_{j}\right\} </math>
|-
| style="text-align: right;" | 
| 
|}
| style="width: 5px;text-align: right;" | (107)
|}

where <math display="inline">N_{ij}={\sigma _{m}}_{ij}</math> are the axial forces defined in Eq.(29).

The geometric stiffness associated to bending moments is much more involved and can be found in  [27]. Numerical experiments have shown that the bending part of the geometric stiffness is not so important and can be disregarded in the iterative process.

Again three and one point quadratures are used for computing the membrane and bending contributions to the geometric stiffness matrix. We note that for elastic-plastic problems a uniform one point quadrature has been chosen for integration of both the membrane and bending stiffness matrices.

==8 EXPLICIT SOLUTION SCHEME==

For simulations including large non-linearities, such as frictional contact conditions on complex geometries or large instabilities in membranes, convergence is difficult to achieve with implicit schemes. In those cases an explicit solution algorithm is typically most advantageous. This scheme provides the solution for dynamic problems and also for static problems if an adequate damping is chosen.

The dynamic equations of motion to solve are of the form

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{r}(\mathbf{u}) + \mathbf{C} \dot{\mathbf{u}} + \mathbf{M} \ddot{\mathbf{u}} = 0 </math>
|}
| style="width: 5px;text-align: right;" | (108)
|}

where <math display="inline">\mathbf{M}</math> is the mass matrix, <math display="inline">\mathbf{C}</math> is the damping matrix and the dot means the time derivative. The solution is performed using the ''central difference method''. To make the method competitive a diagonal (lumped) <math display="inline">\mathbf{M}</math> matrix is typically used and <math display="inline">\mathbf{C}</math> is taken proportional to <math display="inline">\mathbf{M}</math>. As usual, mass lumping is performed by assigning, one third of the triangular element mass to each node in the central element.

The explicit solution scheme can be summarized as follows. At each time step <math display="inline">n</math> where displacements have been computed:

<ol>

<li>Compute the internal forces <math display="inline">\mathbf{r}^{n}</math>. This simply follows the same steps (2-8) described for the implicit scheme in the previous section. </li>

<li>Compute the accelerations at time <math display="inline">t_{n}</math>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\ddot{\mathbf{u}}^{n} = {M}_d^{-1} [ \mathbf{r}^{n} - \mathbf{C} \dot{\mathbf{u}}^{n-1/2} ]  </math>
|}
|}</li>

where <math display="inline">{M}_d</math> is the diagonal (lumped) mass matrix.

<li>Compute the velocities at time <math display="inline">t_{n+1/2}</math>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\dot{\mathbf{u}}^{n+1/2} = \dot{\mathbf{u}}^{n-1/2} \ddot{\mathbf{u}}^{n} \delta t  </math>
|}
|}</li>

<li>Compute the displacements at  time <math display="inline">t_{n+1}</math>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\mathbf{u}^{n+1} = \mathbf{u}^{n} +\dot{\mathbf{u}}^{n+1/2} \delta t  </math>
|}
|}</li>
<li>Update the shell geometry </li>
<li>Check frictional contact conditions </li>

</ol>

Further details of the implementation of the standard BST element within an explicit solution scheme can be found in [26].

==9 EXAMPLES==

In this section several examples are presented to show the good performance of the rotation-free shell elements (BST, EBST and EBST1). The first five static examples are solved using an implicit code. The rest of the examples are computed using the explicit dynamic scheme. For the explicit scheme the  EBST element is always integrated using one integration point per element (EBST1 version) although not indicated.

===9.1 Patch tests===

The three elements considered (BST, EBST and EBST1) satisfy the membrane patch test defined in Figure [[#img-4|4]]. A uniform axial tensile stress is obtained in all cases.

<div id='img-4'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_1361_Fig4.jpeg|300px|Patch test for uniform tensile stress]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' Patch test for uniform tensile stress
|}

<div id='img-5'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_8502_Fig5.jpeg|400px|Patch test for uniform torsion]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 5:''' Patch test for uniform torsion
|}

The element bending formulation does not allow to apply external bending moments (there are not rotational DOFs). Hence it is not possible to analyse a patch of elements under loads leading to a uniform bending moment. A uniform torsion can be considered if a point load is applied at the corner of a rectangular plate with two consecutive free sides and two simple supported sides. Figure [[#img-5|5]] shows three patches leading to correct results both in displacements and stresses. All three patches are structured meshes. When the central node in the third patch is shifted from the center, the results obtained with the EBST and EBST1 elements are not correct. This however does not seems to preclude the excellent performance of these elements, as proved in the rest of the examples analyzed. On the other hand, the BST element  gives correct results in all torsion patch tests if natural boundary conditions are imposed in the formulation. If this is not the case, incorrect results are obtained even with structured meshes.

===9.2 Cook's membrane problem===

This example is used to assess the membrane performance of the EBST and EBST1 elements and to compare it with the standard linear triangle (constant strain) and the quadratic triangle (linear strain). This example involves important shear energy and was proposed to assess the distortion capability of elements. Figure [[#img-6|6]].a shows the geometry and the applied load. Figure [[#img-6|6]].b plots the vertical displacement of the upper vertex as a function of the number of nodes in the mesh. Results obtained with other isoparametric elements have also been  plotted for comparison. They include the constant strain triangle (CST), the bilinear quadrilateral (QUAD4) and the linear strain triangle (LST). Note that as this is a pure  membrane problem  the BST and the CST elements give identical results.

<div id='img-6'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig6a.png|300px|]]
|
[[File:Draft_Samper_226033773_1597_Fig6b.jpg|300px|Cook membrane problem (a) Geometry (b) Results]]
|-
|style="text-align: center; font-size: 75%;padding:10px;"|(a)
|style="text-align: center; font-size: 75%;padding:10px;"|(b)
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 6:''' Cook membrane problem (a) Geometry (b) Results
|}

From the plot shown it can be seen that the enhanced element with three integration points (EBST) gives values slightly better that the constant strain triangle for the most coarse mesh (only two elements). However, when the mesh is refined, a performance similar to the linear strain triangle is obtained that is dramatically superior that the former. On the other hand, if a one point quadrature is used (EBST1) the convergence in the reported displacement is notably better that for the rest of the elements.

===9.3 Cylindrical roof===

In this example an effective membrane interpolation is of primary importance. The geometry is a cylindrical roof supported by a rigid diaphragm at both ends and it is loaded by a uniform dead weight (see Figure [[#img-7|7]].a.). Only one quarter of the structure is modelled due to symmetry conditions. Unstructured and structured meshes are considered. In the latter case two orientations are possible (Figure [[#img-7|7]] shows orientation B).

Tables [[#table-3|3]], [[#table-4|4]] and [[#table-5|5]] present the normalized vertical displacements at the crown (point A) and at the midpoint of the free side (point B) for the two orientations of the structured meshes and for the non-structured mesh. Values used for normalization are <math display="inline">u_{A}=0.5407</math> y <math display="inline">u_{B}=-3.610</math> that are quoted in reference [32].

<div id='img-7'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_7308_Fig7a.jpeg|400px|]]
|[[Image:draft_Samper_226033773-fig7b.png|400px|]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 7:''' Cylindrical roof under dead weight. <math>E=3 \times 10^{6}</math>, <math>\nu=0.0</math>, Thickness =3.0, shell weight =0.625 per unit area.
|}


{| class="wikitable" style="text-align: left; margin: 1em auto;"
|+ <span id='table-3'></span>Table. 3 Cylindrical roof under dead weight. Normalized vertical displacements for mesh orientation A
|- style="border-top: 2px solid;"
| [c]|r|r|r|r|r|r|r| 
| colspan='3' style="text-align: (null);border-left: 2px solid;border-right: 2px solid;" | Point-A
| colspan='3' style="text-align: (null);border-left: 2px solid;border-right: 2px solid;" | Point-B
|- style="border-top: 2px solid;"
|  NDOFs 
| EBST 
| EBST1 
| BST 
| CBST 
| EBST1 
| BST
|-
| 16 
| 0.65724 
| 0.91855 
| 0.74161 
| 0.40950 
| 0.70100 
| 1.35230
|-
| 56 
| 0.53790 
| 1.03331 
| 0.74006 
| 0.54859 
| 1.00759 
| 0.75590
|-
| 208 
| 0.89588 
| 1.04374 
| 0.88491 
| 0.91612 
| 1.02155 
| 0.88269
|-
| 800 
| 0.99658 
| 1.01391 
| 0.96521 
| 0.99263 
| 1.00607 
| 0.96393
|- style="border-bottom: 2px solid;"
| 3136 
| 1.00142 
| 1.00385 
| 0.99105 
| 0.99881 
| 1.00102 
| 0.98992

|}


{| class="wikitable" style="text-align: left; margin: 1em auto;"
|+ <span id='table-4'></span>Table. 4 Cylindrical roof under dead weight. Normalized vertical displacements for mesh orientation B
|- style="border-top: 2px solid;"
| [c]|r|r|r|r|r|r|r| 
| colspan='3' style="text-align: (null);border-left: 2px solid;border-right: 2px solid;" | Point-A
| colspan='3' style="text-align: (null);border-left: 2px solid;border-right: 2px solid;" | Point-B
|- style="border-top: 2px solid;"
|  NDOFs 
| EBST 
| EBST1 
| BST 
| CBST 
| EBST1 
| BST
|-
| 16 
| 0.26029 
| 0.83917 
| 0.40416 
| 0.52601 
| 0.86133 
| 0.89778
|-
| 56 
| 0.81274 
| 1.10368 
| 0.61642 
| 0.67898 
| 0.93931 
| 0.68238
|-
| 208 
| 0.97651 
| 1.04256 
| 0.85010 
| 0.93704 
| 1.00255 
| 0.86366
|-
| 800 
| 1.00085 
| 1.01195 
| 0.95626 
| 0.99194 
| 1.00211 
| 0.95864
|- style="border-bottom: 2px solid;"
| 3136 
| 1.00129 
| 1.00337 
| 0.98879 
| 0.99828 
| 1.00017 
| 0.98848

|}


{| class="wikitable" style="text-align: left; margin: 1em auto;"
|+ <span id='table-5'></span>Table. 5 Cylindrical roof under dead weight. Normalized vertical displacements for non-structured mesh
|- style="border-top: 2px solid;"
| [c]|r|r|r|r|r|r|r| 
| colspan='3' style="text-align: (null);border-left: 2px solid;border-right: 2px solid;" | Point-A
| colspan='3' style="text-align: (null);border-left: 2px solid;border-right: 2px solid;" | Point-B
|- style="border-top: 2px solid;"
|  NDOFs 
| EBST 
| EBST1 
| BST 
| EBST 
| EBST1 
| BST
|-
| 851 
| 0.97546 
| 0.8581 
| 0.97598 
| 0.97662 
| 1.0027 
| 0.97194
|-
| 3311 
| 0.98729 
| 0.9682 
| 0.98968 
| 0.98476 
| 1.0083 
| 0.98598
|- style="border-bottom: 2px solid;"
| 13536 
| 0.99582 
| 0.9992 
| 1.00057 
| 0.99316 
| 0.9973 
| 0.99596

|}

Plots in Figure [[#img-7|7]].b show the normalized displacement of point-B for structured meshes as a function of the number of degrees of freedom for each case studied. An excellent convergence for the EBST element can be seen. The version with only one integration point (EBST1) presents a behavior a little more flexible and converges from above for structured meshes. Table [[#table-5|5]] shows that both the EBST and the EBST1 elements have an excellent behavior for non structured meshes.

===9.4 Open semi-spherical dome with point loads===

The main problem of finite elements with initially curved geometry is the so called membrane locking. The EBST element  has a quadratic interpolation of the geometry, then it may suffer from this problem. To assess this we resort to an example of inextensional bending. This is an hemispherical shell of radius <math display="inline">r=10</math> and thickness <math display="inline">h=0.04</math> with an 18<math display="inline">^{o}</math> hole in the pole and free at all boundaries, subjected to two inward and two outward forces 90<math display="inline">^{o}</math> apart. Material properties are <math display="inline">E=6.825\times{10}^{7}</math> and <math display="inline">\nu=0.3</math>. Figure [[#img-8|8]].a shows the discretized geometry (only one quarter of the geometry is considered due to symmetry).

<div id='img-8'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_7798_Fig8a.jpeg|400px|]]
|[[Image:draft_Samper_226033773-fig8b.png|400px|Pinched hemispherical shell with a hole, (a)geometry, (b)normalized displacement]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 8:''' Pinched hemispherical shell with a hole, (a)geometry, (b)normalized displacement
|}

In Figure [[#img-8|8]].b the displacements of the points under the loads have been plotted versus the number of nodes used in the discretization. Due to the orientation of the meshes chosen, the displacement of the point under the inward load is not the same as the displacement under the outward load, so in the figure an average (the absolute values) has been used. Results obtained with other elements have been included for comparison: two membrane locking free elements, namely the original linear BST element and a transverse shear-deformable quadrilateral (QUAD) [33]; a transverse shear deformable quadratic triangle (TRIA) [2] that is vulnerable to locking and an assumed strain quadratic triangle (TRIC) [3] that does not exhibit membrane locking.

From the plotted results it can be seen that the EBST element presents slight membrane locking in bending dominated problems with initially curved geometries. This locking is much less severe than in a standard quadratic triangle. Membrane locking disappears when only one integration point is used (EBST1 element).

===9.5 Inflation of a sphere===

The example is the inflation of a spherical shell under internal pressure. An incompressible Mooney-Rivlin constitutive material has been considered. The Ogden parameters are <math display="inline">N=2</math>, <math display="inline">\alpha _{1}=2</math>, <math display="inline">\mu _{1}=40</math>, <math display="inline">\alpha _{2}=-2</math>, <math display="inline">\mu _{2}=-20</math>. Due to the simple geometry an analytical solution exists [34] (with <math display="inline">\gamma =R/R^{0}</math>):

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math> p=\frac{h^{0}}{R^{0}\gamma ^{2}}\frac{dW}{d\gamma }=\frac{8h^{0} }{R^{0}\gamma ^{2}} \left( \gamma ^{6}-1\right) \left( \mu _{1}-\mu _{2}\gamma ^{2}\right) </math>
|}
|}

In this numerical simulation the same geometric and material parameters used in Ref. <span id='citeF-23'></span>[[#cite-23|[23]]] have been adopted: <math display="inline">R^{0}=1</math> and <math display="inline">h^{0}=0.02</math>. The three meshes of EBST1 element considered to evaluate convergence are shown in Figure [[#img-9|9]].a. The value of the actual radius as a function of the internal pressure is plotted in Figure [[#img-9|9]].b for the different meshes and is also compared with the analytical solution. It can be seen that with a few degrees of freedom it is possible to obtain an excellent agreement for the range of strains considered. The final value corresponds to a  ratio of <math display="inline">h/R=0.00024</math>.

<div id='img-9'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_226033773_5233_Fig9.jpeg|500px|]]
|-
|[[File:Draft_Samper_226033773_9697_Fig9b.jpeg|300px|Inflation of sphere of Mooney-Rivlin material. (a) Meshes of EBST1 elements used in the analysis (b) Change of radius as a function of the internal pressure.]]
|- style="text-align: center; font-size: 75%;"
| '''Figure 9:''' Inflation of sphere of Mooney-Rivlin material. (a) Meshes of EBST1 elements used in the analysis (b) Change of radius as a function of the internal pressure.
|}

===9.6 Clamped spherical dome under impulse pressure loading===

The geometry of the dome and the material properties chosen are shown in Figure [[#img-10|10]]. A uniform pressure load of 600 psi is applied to the upper surface of the dome. The different meshes used in the analysis are shown in Figure [[#img-11|11]]. One fourth of the dome is considered only due to symmetry. Two different analyses under elastic and elastic-plastic conditions were carried out. The number of thickness layers in eq.([[#eq-96|96]]) is four. Numerical experiments show that this suffice to provide an accurate solution for large elastic-plastic problems [26]. Results are obtained using the explicit scheme.

<div id='img-10'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig10.png|400px|Spherical dome under impulse pressure. Geometry and material]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 10:''' Spherical dome under impulse pressure. Geometry and material
|}

<div id='img-11'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_1877_Fig11.jpeg|600px|Spherical dome under impulse pressure. Meshes used in the analysis. Mesh-1 with 338 elements, Mesh-2 with 1250 elements, and Mesh-3 with 2888 elements]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 11:''' Spherical dome under impulse pressure. Meshes used in the analysis. Mesh-1 with 338 elements, Mesh-2 with 1250 elements, and Mesh-3 with 2888 elements
|}

<div id='img-12'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig12.png|600px|Spherical dome under impulse pressure. History of central deflection for elastic material]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 12:''' Spherical dome under impulse pressure. History of central deflection for elastic material
|}

<div id='img-13'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig13.png|600px|Spherical dome under impulse pressure. History of central deflection for elastic-plastic material]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 13:''' Spherical dome under impulse pressure. History of central deflection for elastic-plastic material
|}

Figure [[#img-12|12]] shows results for the time history of the central deflection using different meshes and ''elastic material properties'' for both  BST and EBST1 elements. Results are almost identical for mesh-2 and mesh-3, showing the excellent convergence properties. The coarsest mesh shows some differences between both elements, but for the finer meshes the results are almost identical. Figure [[#img-13|13]] shows similar results but now for an ''elastic-plastic material''. The excellent convergence  of the BST and EBST elements is again noticeable.

Results obtained with the present elements compare very well with published results using fine meshes. See for example ABAQUS Explicit example problems manual <span id='citeF-35'></span>[[#cite-35|[35]]] and WHAMS-3D manual [36], showing plots comparing results using different shell elements.

A summary of results for the central deflection at significant times is given in Tables [[#table-6|6]] and [[#table-7|7]]. Further details on the solution of this problem with the standard  BST element can be found in [26].


{| class="wikitable" style="text-align: right; margin: 1em auto;"
|+ <span id='table-6'></span>Table. 6 Spherical dome. Elastic material. Comparison of the central deflection values at the mid point obtained with the BST and EBST1  elements for different meshes
|- style="border-top: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  Element/mesh 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>t = 0.2 ms</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>t = 0.4 ms</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>t = 0.6 ms</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>t = 0.8 ms</math>
|- style="border-top: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |   BST Coarse  
| style="border-left: 2px solid;border-right: 2px solid;" | -0.05155 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.09130 
| style="border-left: 2px solid;border-right: 2px solid;" | 0.04414 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.08945 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | BST Medium  
| style="border-left: 2px solid;border-right: 2px solid;" | -0.04542 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.09177 
| style="border-left: 2px solid;border-right: 2px solid;" | 0.03863 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.08052 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | BST Fine    
| style="border-left: 2px solid;border-right: 2px solid;" | -0.04460 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.09022 
| style="border-left: 2px solid;border-right: 2px solid;" | 0.03514 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.08132 
|- style="border-top: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" |  EBST1 Coarse  
| style="border-left: 2px solid;border-right: 2px solid;" | -0.05088 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.08929 
| style="border-left: 2px solid;border-right: 2px solid;" | 0.04348 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.08708 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | EBST1 Medium  
| style="border-left: 2px solid;border-right: 2px solid;" | -0.04527 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.09134 
| style="border-left: 2px solid;border-right: 2px solid;" | 0.03865 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.07979 
|- style="border-bottom: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | EBST1 Fine    
| style="border-left: 2px solid;border-right: 2px solid;" | -0.04453 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.09004 
| style="border-left: 2px solid;border-right: 2px solid;" | 0.03510 
| style="border-left: 2px solid;border-right: 2px solid;" | -0.08099 

|}


{| class="wikitable" style="text-align: left; margin: 1em auto;"
|+ <span id='table-7'></span>Table. 7 Spherical dome. Elastic-plastic material. Comparison of the central deflection values at the mid point obtained with the BST and EBST1  elements for different meshes
|- style="border-top: 2px solid;"
| [c]|l|r|r|r|r| Element/mesh 
| <math>t = 0.2 ms</math>
| <math>t = 0.4 ms</math>
| <math>t = 0.6 ms</math>
| <math>t = 0.8 ms</math>
|- style="border-top: 2px solid;"
|   BST Coarse  
| -0.05888 
| -0.05869 
| -0.02938 
| -0.06521 
|-
| BST Medium  
| -0.05376 
| -0.06000 
| -0.02564 
| -0.06098 
|-
| BST Fine    
| -0.05312 
| -0.05993 
| -0.02464 
| -0.06105 
|- style="border-top: 2px solid;"
|  EBST1 Coarse  
| -0.05827 
| -0.05478 
| -0.02792 
| -0.06187 
|-
| EBST1 Medium  
| -0.05374 
| -0.05884 
| -0.02543 
| -0.06080 
|- style="border-bottom: 2px solid;"
| EBST1 Fine    
| -0.05317 
| -0.05935 
| -0.02458 
| -0.06085 

|}

===9.7 Cylindrical panel under impulse loading===

The geometry of the cylinder and the material properties are shown in Figure [[#img-14|14]]. A prescribed initial normal velocity of <math display="inline">v_{o}=-5650</math> in/sec is applied to the points in the region shown modelling the effect of the detonation of an explosive layer. The panel is assumed clamped along all the boundary. One half of the cylinder is discretized only due to symmetry conditions. Three different meshes of <math display="inline">6\times{12}</math>, <math display="inline">12\times{32}</math> and <math display="inline">18\times{48}</math>  BST elements were used for the analysis. The deformed configurations for <math display="inline">time =1 msec</math> are shown for the three meshes in Figure [[#img-15|15]].

<div id='img-14'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig14.png|600px|Cylindrical panel under impulse loading. Geometry and material properties]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 14:''' Cylindrical panel under impulse loading. Geometry and material properties
|}

<div id='img-15'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig15.png|600px|Impulsively loaded cylindrical panel. Deformed meshes for time =1 msec]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 15:''' Impulsively loaded cylindrical panel. Deformed meshes for <math>time =1 msec</math>
|}

<div id='img-16'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig16.png|600px|Cylindrical panel under impulse loading. Time evolution of the displacement of two points along the crown line. Comparison of results obtained with BST and EBST1 elements (mesh 1: 6×12 elements and mesh 3: 18×48 elements) and experimental values ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 16:''' Cylindrical panel under impulse loading. Time evolution of the displacement of two points along the crown line. Comparison of results obtained with BST and EBST1 elements (mesh 1: <math>6\times{12}</math> elements and mesh 3: <math>18\times{48}</math> elements) and experimental values 
|}

The analysis was performed assuming an elastic-perfect plastic material behaviour (<math display="inline">\sigma _y = k</math> <math display="inline">k'=0</math>). A study of the convergence of the solution with the number of thickness layers showed again that four layers suffice to capture accurately the non linear material response [26].

A comparison of the results obtained with both elements using the coarse mesh and the finer mesh is shown in Figure [[#img-16|16]] where experimental results reported in <span id='citeF-37'></span>[[#cite-37|[37]]] have also been plotted for comparison purposes. Good agreement between the numerical and experimental results is obtained. Figures [[#img-16|16]] show the time evolution of the vertical displacement of two reference points along the center line located at <math display="inline">y=6.28</math>in and <math display="inline">y=9.42</math>in, respectively. For the finer mesh results between both elements are almost identical. For the coarse mesh it can been seen again that the element BST is more flexible than element EBST1.

The numerical values of the vertical displacement at the two reference points obtained with the BST and EBST1  elements after a time of 0.4 ms using the <math display="inline">16\times{32}</math> mesh are compared in Table [[#table-8|8]]  with a numerical solution obtained by Stolarski ''et al.'' [38] using a curved triangular shell element and the <math display="inline">16\times{32}</math> mesh. Experimental results reported in [37] are also given for comparison. It is interesting to note the reasonable agreement of the results for <math display="inline">y=6.28</math>in. and the discrepancy of present and other published numerical solutions with the experimental value for <math display="inline">y=9.42</math>in.


{| class="wikitable" style="text-align: left; margin: 1em auto;"
|+ <span id='table-8'></span>Table. 8 Cylindrical panel under impulse load. Comparison of vertical displacement values of two central points for <math>t=0.4</math> ms
|- style="border-top: 2px solid;"
| [c]|l|r|r|r|r| 
| colspan='2' style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | Vertical displacement (in.)
|- style="border-top: 2px solid;"
|  element/mesh                
| <math>y=6.28</math>in 
| <math>y=9.42</math>in 
|- style="border-top: 2px solid;"
|  BST  (<math display="inline"> 6\times 12</math> el.)    
| -1.310     
| -0.679      
|-
| BST  (<math display="inline">18\times 48</math> el.)    
| -1.181     
| -0.587      
|-
| EBST1 (<math display="inline"> 6\times 12</math> el.)    
| -1.147     
| -0.575      
|-
| EBST1 (<math display="inline">18\times 48</math> el.)    
| -1.171     
| -0.584      
|-
| Stolarski ''et al.'' [38] 
| -1.183     
| -0.530      
|- style="border-bottom: 2px solid;"
| Experimental [37] 
| -1.280     
| -0.700      

|}

<div id='img-17'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig17.png|600px|Cylindrical panel under impulse loading. Final deformation (t=1 msec) of the panel at the cross section y=6.28 in Comparison with experimental values. ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 17:''' Cylindrical panel under impulse loading. Final deformation (<math>t=1 msec</math>) of the panel at the cross section <math>y=6.28 in</math> Comparison with experimental values. 
|}

<div id='img-18'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig18.png|600px|Cylindrical panel under impulse loading. Final deformation (t=1 msec) of the panel at the crown line (x=0.00 in). Comparison with experimental values. ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 18:''' Cylindrical panel under impulse loading. Final deformation (<math>t=1 msec</math>) of the panel at the crown line (<math>x=0.00 in</math>). Comparison with experimental values. 
|}

The deformed shapes of the transverse section for <math display="inline">y=6.28</math>in. and the longitudinal section for <math display="inline">x=0</math> obtained with the both elements for the coarse and the fine meshes after 1ms. are compared with the experimental results in Figures [[#img-17|17]] and [[#img-18|18]].  Excellent agreement is observed for the fine mesh for both elements.

===9.8 Airbag Membranes===

===Inflation/deflation of a circular airbag===

This example has been taken from Ref.[23] where it is shown that the final configuration is mesh dependent due to the strong instabilities leading to a non-uniqueness of the solution. In [23]  it is also discussed the important regularizing properties of the bending energy, that when disregarded leads to massive wrinkling in the compressed zones.

The airbag geometry is initially circular with an undeformed radius of <math display="inline">0.35</math>.  The constitutive material is a linear isotropic elastic one with modulus of elasticity <math display="inline">E=6\times 10^{7}</math>Pa, Poisson's ratio <math display="inline">\nu =0.3</math> and density <math display="inline">\rho = 2000</math>kg/m<math display="inline">^3</math>.  Arbitrarily only one quarter of the geometry has been modelled.  Only the normal displacement to the original plane is constrained along the boundaries.  The thickness considered is <math display="inline">h=0.0004</math>m and the inflation pressure is <math display="inline">5000</math>Pa. Pressure is linearly increased from 0 to the final value in <math display="inline">t=0.15</math> sec.

Figure 19 shows the final deformed configurations for a mesh with 10201 nodes and 20000 EBST1 elements.  The figure on the left (a) corresponds to an analysis including full bending effects and the right figure (b) is a pure membrane analysis.

We note that when the bending energy is included a more regular final pattern is obtained.  Also the final pattern is rather independent of the discretization (note that the solution is non unique due to the strong instabilities), and a massive wrinkling appears in the center of the modelled region.  On the other hand, the pure membrane solution shows a wrinkling pattern where the width of the wrinkle is the length of the element.

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_2615_Fig19.jpeg|600px|Inflation of a circular airbag. Deformed configurations for final pressure. (a) bending formulation; (b) membrane formulation.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 19:''' Inflation of a circular airbag. Deformed configurations for final pressure. (a) bending formulation; (b) membrane formulation.
|}

<div id='img-20'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_9407_Fig20.jpeg|600px|Inflation and deflation of a circular air-bag.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 20:''' Inflation and deflation of a circular air-bag.
|}

<div id='img-21'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_9850_Fig21.jpeg|600px|Inflation and deflation of a closed  tube. L=5, D=1, h=5×10⁻³.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 21:''' Inflation and deflation of a closed  tube. <math>L=5</math>, <math>D=1</math>, <math>h=5\times 10^{-3}</math>.
|}

<div id='img-22'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_8861_Fig22.jpeg|600px|Inflation and deflation of a closed  tube. L=6, D=1, h=3×10⁻³.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 22:''' Inflation and deflation of a closed  tube. <math>L=6</math>, <math>D=1</math>, <math>h=3\times 10^{-3}</math>.
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_226033773_6495_fig23.jpg|600px|Inflation of a square airbag against an spherical object. Deformed configurations for different times. Left figure: results obtained with the full bending formulation. Right figure: results obtained with a pure membrane solution.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 23:''' Inflation of a square airbag against an spherical object. Deformed configurations for different times. Left figure: results obtained with the full bending formulation. Right figure: results obtained with a pure membrane solution.
|}

Figure 20 shows the results obtained for the de-inflation process.  Note that the spherical membrane falls down due to the self weight.  The final configuration is of course non-unique.

The next problem is the study of the inflating and de-inflating of a tube with a semi-spherical end cap.  The tube diameter is <math display="inline">D=1</math>, its total length is <math display="inline">L=5</math>m and the thickness <math display="inline">h=5\times 10^{-3}</math>m.  The material has the following properties <math display="inline">E=4\times 10^{8}</math>Pa, <math display="inline">\nu =0.35 </math>, <math display="inline">\rho =5\times 10^{4}</math>kg/m<math display="inline">^3</math>.  The tube is inflated fast until a pressure of <math display="inline">10^4</math> and then is de-inflated under self weight.  The analysis is performed with a mesh of 4176 EBST1 elements and 2163 nodes modelling a quarter of the geometry.  The evolution of the tube walls during the de-inflating process can be seen in Figure 21.  Note that the central part collapses as expected, while the semi-spherical cap remains unaltered.

The same analysis is repeated for a longer and thinner tube (<math display="inline">L=6</math>m and <math display="inline">h=3\times 10^{-3}</math>m).  The same material than in the previous case was chosen with a higher density (<math display="inline">\rho =7.5\times 10^{4}</math>kg/m<math display="inline">^3</math>).  The evolution of the tube walls is shown in Figure 22.  Note that the central part collapses again but in a less smoother manner due to the smaller thickness.

The last example of this kind is the inflation of a square airbag supporting a spherical object.  This example resembles a problem studied (numerically and experimentally) in Ref.[39], where fluid-structure interaction is the main subject.  Here the fluid is not modelled, and a uniform pressure is applied over all the internal surfaces.  The lower surface part of the airbag is limited by a rigid plane and on the upper part a spherical dummy object is set to study the interaction between the airbag and the object.

The airbag geometry is initially square with an undeformed side length of 0.643m.  The constitutive material chosen is a linear isotropic elastic one with <math display="inline">E=5.88\times 10^8</math>Pa, <math display="inline">\nu =0.4</math> and a density of <math display="inline">\rho = 1000</math> kg/m<math display="inline">^3</math>.  Only one quarter of the geometry has been modelled due to symmetry.  The thickness <math display="inline">h=0.00075</math>m and the inflation pressure is 250000Pa.  Pressure is linearly incremented from 0 to the final value in <math display="inline">t=0.15</math>sec.  The spherical object has a radius <math display="inline">r=0.08</math>m and a mass of 4.8kg (one quarter), and is subjected to gravity load during all the process.

The mesh includes 8192 EBST1 elements and 4225 nodes on each surface of the airbag.  Figure 23 shows the deformed configurations for three different times.  The sequence on the left of the figure corresponds to an analysis including full bending effects and the sequence on the right is the result of a pure membrane analysis.  A standard penalty formulation is used for frictionless contact.

===9.9 S-rail sheet stamping===

The final problem corresponds to one of the sheet stamping benchmark tests proposed in NUMISHEET'96 <span id='citeF-40'></span>[[#cite-40|[40]]].  The analysis comprises two parts, namely, stamping of a S-rail sheet component and springback computations once the stamping tools are removed.  Figure [[#img-24|24]] shows the deformed sheet after springback.

<div id='img-24'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig_n1.png|600px|Stamping of a S-rail. Final deformation of the sheet after springback obtained in the simulation. The triangular mesh of the deformed sheet is also shown]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 24:''' Stamping of a S-rail. Final deformation of the sheet after springback obtained in the simulation. The triangular mesh of the deformed sheet is also shown
|}

The detailed geometry and material data can be found in the proceedings of the conference <span id='citeF-40'></span>[[#cite-40|[40]]] or in the web <span id='citeF-41'></span>[[#cite-41|[41]]]. The mesh used for the sheet has 6000 three  node triangular elements and 3111 points (Figure 24). The tools are treated as rigid bodies. The meshes used for the sheet and the tools are those provided by the  benchmark organizers. The material considered here is a mild steel (IF) with Young Modulus <math display="inline">E=2.06 GPa</math> and Poisson ratio <math display="inline">\nu=0.3</math>. Mises yield criterion was used for plasticity behaviour with non-linear isotropic hardening defined by <math display="inline">k(e^p) = 545(0.13+e^p)^{0.267} [MPa]</math>. A uniform friction of 0.15 was used for all the tools. A low (10kN) blank holder force was considered in this simulation.

Figure [[#img-25|25]] compares the punch force during the stamping stage obtained with both BST and EBST1 elements for the simulation and experimental values. Also for reference the average values of the simulations presented in the conference are included. Explicit and implicit simulations are considered as different curves. There is a remarkable coincidence between the experimental values and the results obtained with BST and EBST1 elements.

<div id='img-25'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig_n2.png|600px|Stamping of a S-rail. Punch force versus punch travel. Average of explicit and implicit results reported at the benchmark are also shown. ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 25:''' Stamping of a S-rail. Punch force versus punch travel. Average of explicit and implicit results reported at the benchmark are also shown. 
|}

Figure [[#img-26|26]] plots the <math display="inline">Z</math> coordinate along line B"&#8211;G" after springback stage. The top surface of the sheet does not remain plane due to some instabilities for the low blank holder force used. Results obtained with the simulations compare very well with the experimental results.

<div id='img-26'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_226033773-fig_n3.png|600px|Stamping of a S-rail. Z-coordinate along line B''&#8211;-G'' after springback. Average of explicit and implicit results reported at the benchmark are also shown. ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 26:''' Stamping of a S-rail. Z-coordinate along line B''&#8211;-G'' after springback. Average of explicit and implicit results reported at the benchmark are also shown. 
|}

==10 CONCLUDING REMARKS==

We have presented in the paper two alternative formulations for the rotation-free basic shell triangle (BST) using an assumed strain approach.  The simplest element of the family is based on an assumed constant curvature field expressed in terms of the nodal deflections of a patch of four elements and a constant membrane field computed from the standard linear interpolation of the displacements within each triangle. An enhanced version of the element is obtained by using a quadratic interpolation of the geometry in terms of the six patch nodes.  This allows to compute an assumed linear membrane strain field which improves the in-plane behaviour of the original element.  A simple and economic version of the element using a single integration point has been presented.  The efficiency of the different rotation-free shell triangles has been demonstrated in many examples of application including linear and non linear analysis of shells under static and dynamic loads, the inflation and de-inflation of membranes and a sheet stamping problem.

The enhanced rotation-free basic shell triangle element with a single integration point (the EBST1 element) has proven to be an excellent candidate for solving practical engineering shell and membrane problems involving complex geometry, dynamics, material non linearity and frictional contact conditions.

==ACKNOWLEDGEMENTS==

The problems analyzed with the explicit formulation were solved with the computer code STAMPACK <span id='citeF-42'></span>[[#cite-42|[42]]] where the rotation-free elements here presented have been implemented.  The support of the company QUANTECH (www.quantech.es) providing the code STAMPACK is gratefully acknowledged.

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==APPENDIX==

==11 Curvature matrix for the BST element==

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\delta{\boldsymbol \kappa }=\mathbf{B}_{b} \times \mathbf{t}_3 \delta \mathbf{a}^{p}  </math>
|}
|}

with

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{c} \\ \delta \mathbf{a}^{p}\\ 18\times{1} \end{array} =[\delta \mathbf{u}_{1}^{T},\delta \mathbf{u}_{2}^{T},\delta \mathbf{u}_{3} ^{T},\delta \mathbf{u}_{4}^{T},\delta \mathbf{u}_{5}^{T},\delta \mathbf{u}_{6}^{T}]^{T}  </math>
|}
|}

and 

<math>\mathbf{B}_{b}^{T}=</math>


{| class="wikitable" style="text-align: center; margin: 1em auto;"
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">+L^{M}_{2,1} L^{2}_{2,1}    +L^{M}_{3,1} L^{3}_{3,1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>+L^{M}_{2,2} L^{2}_{2,2}    +L^{M}_{3,2} L^{3}_{3,2} </math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>+L^{M}_{2,2} L^{2}_{2,1} +L^{M}_{2,1} L^{2}_{2,2}    +L^{M}_{3,2} L^{3}_{3,1} +L^{M}_{3,1} L^{3}_{3,2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> L^{M}_{1,1} L^{1}_{3,1}    +L^{M}_{3,1} L^{3}_{2,1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{1,2} L^{1}_{3,2}    +L^{M}_{3,2} L^{3}_{2,2} </math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{1,2} L^{1}_{3,1} +L^{M}_{1,1} L^{1}_{3,2}    +L^{M}_{3,2} L^{3}_{2,1} +L^{M}_{3,1} L^{3}_{2,2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> L^{M}_{1,1} L^{1}_{2,1}    +L^{M}_{2,1} L^{2}_{3,1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{1,2} L^{1}_{2,2}    +L^{M}_{2,2} L^{2}_{3,2} </math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{1,2} L^{1}_{2,1} +L^{M}_{1,1} L^{1}_{j,3}    +L^{M}_{2,2} L^{2}_{3,1} +L^{M}_{2,1} L^{2}_{3,2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L^{M}_{1,1} L^{1}_{1,1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L^{M}_{1,2} L^{1}_{1,2} </math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L^{M}_{1,2} L^{1}_{1,1} +L^{M}_{1,1} L^{1}_{1,3} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L^{M}_{2,1} L^{2}_{1,1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L^{M}_{2,2} L^{2}_{1,2} </math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L^{M}_{2,2} L^{2}_{1,1} +L^{M}_{2,1} L^{2}_{1,3} </math>
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L^{M}_{3,1} L^{3}_{1,1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L^{M}_{3,2} L^{3}_{1,2} </math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L^{M}_{3,2} L^{3}_{1,1} +L^{M}_{3,1} L^{3}_{1,3} </math>

|}

<math display="inline">-2</math> 
{| class="wikitable" style="text-align: center; margin: 1em auto;"
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> L^{M}_{1,1}\rho _{11}^{1}+L^{M}_{1,2}\rho _{11}^{2} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{1,1}\rho _{22}^{1}+L^{M}_{i,2}\rho _{22}^{2} </math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{1,1}\rho _{12}^{1}+L^{M}_{1,2}\rho _{12}^{2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> L^{M}_{2,1}\rho _{11}^{1}+L^{M}_{2,2}\rho _{11}^{2} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{2,1}\rho _{22}^{1}+L^{M}_{2,2}\rho _{22}^{2} </math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{2,1}\rho _{12}^{1}+L^{M}_{2,2}\rho _{12}^{2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> L^{M}_{3,1}\rho _{11}^{1}+L^{M}_{3,2}\rho _{11}^{2} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{3,1}\rho _{22}^{1}+L^{M}_{3,2}\rho _{22}^{2} </math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math> L^{M}_{3,1}\rho _{12}^{1}+L^{M}_{3,2}\rho _{12}^{2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">0</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">0</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">0</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>

|}

<br/><br/>

==12 Membrane strain matrix and curvature matrix for the EBST element==

===12.1 Membrane strain matrix===

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\delta {\boldsymbol \varepsilon }_m ={B}_m \delta {a}^p  </math>
|}
|}

<math>\mathbf{B}_{m}^{T}=\frac{1}{3}</math>


{| class="wikitable" style="text-align: center; margin: 1em auto;"
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{1}_{1,1}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }1}   + N^{2}_{1,1}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }1}   + N^{3}_{1,1}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> N^{1}_{1,2}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }2}   + N^{2}_{1,2}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }2}   + N^{3}_{1,2}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{1}_{2,1}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }1}   + N^{2}_{2,1}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }1}   + N^{3}_{2,1}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> N^{1}_{2,2}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }2}   + N^{2}_{2,2}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }2}   + N^{3}_{2,2}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{1}_{3,1}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }1}   + N^{2}_{3,1}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }1}   + N^{3}_{3,1}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> N^{1}_{3,2}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }2}   + N^{2}_{3,2}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }2}   + N^{3}_{3,2}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{1}_{4,1}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> N^{1}_{4,2}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }2} </math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{2}_{5,1}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> N^{2}_{5,2}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }2} </math>
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{3}_{6,1}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }1} </math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math> N^{3}_{6,2}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }2} </math>

|}


{| class="wikitable" style="text-align: center; margin: 1em auto;"
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{1}_{1,2}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }1}    +N^{1}_{1,1}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }2}    +N^{2}_{1,2}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }1}    +N^{2}_{1,1}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }2}    +N^{3}_{1,2}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }1}    +N^{3}_{1,1}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }2} </math> 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{1}_{2,2}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }1}    +N^{1}_{2,1}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }2}   + N^{2}_{2,2}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }1}    +N^{2}_{2,1}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }2}   + N^{3}_{2,2}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }1}    +N^{3}_{2,1}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }2} </math> 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{1}_{3,2}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }1}    +N^{1}_{3,1}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }2}    +N^{2}_{3,2}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }1}    +N^{2}_{3,1}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }2}    +N^{3}_{3,2}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }1}    +N^{3}_{3,1}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }2} </math> 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{1}_{4,2}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }1}    +N^{1}_{4,1}\mathbf{\boldsymbol \varphi }^{1}_{^{\prime }2} </math> 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{2}_{5,2}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }1}    +N^{2}_{5,1}\mathbf{\boldsymbol \varphi }^{2}_{^{\prime }2} </math> 
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline"> N^{3}_{6,2}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }1}    +N^{3}_{6,1}\mathbf{\boldsymbol \varphi }^{3}_{^{\prime }2} </math> 

|}

===12.2 Curvature matrix===

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\delta {\boldsymbol \kappa } ={B}_b \times \mathbf{t}_3 \delta {a}^p  </math>
|}
|}

<math>\mathbf{B}_{b}^{T}=2</math>


{| class="wikitable" style="text-align: center; margin: 1em auto;"
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{1,1}\left(N_{1,1}\right)_{G_{1}}   +L_{2,1}\left(N_{1,1}\right)_{G_{2}}   +L_{3,1}\left(N_{1,1}\right)_{G_{3}}</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L_{1,2}\left(N_{1,2}\right)_{G_{1}}   +L_{2,2}\left(N_{1,2}\right)_{G_{2}}   +L_{3,2}\left(N_{1,2}\right)_{G_{3}}</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{1,1}\left(N_{2,1}\right)_{G_{1}}   +L_{2,1}\left(N_{2,1}\right)_{G_{2}}   +L_{3,1}\left(N_{2,1}\right)_{G_{3}}</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L_{1,2}\left(N_{2,2}\right)_{G_{1}}   +L_{2,2}\left(N_{2,2}\right)_{G_{2}}   +L_{3,2}\left(N_{2,2}\right)_{G_{3}}</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{1,1}\left(N_{3,1}\right)_{G_{1}}   +L_{2,1}\left(N_{3,1}\right)_{G_{2}}   +L_{3,1}\left(N_{3,1}\right)_{G_{3}}</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L_{1,2}\left(N_{3,2}\right)_{G_{1}}   +L_{2,2}\left(N_{3,2}\right)_{G_{2}}   +L_{3,2}\left(N_{3,2}\right)_{G_{3}}</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{1,1}\left(N_{4,1}\right)_{G_{1}}</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L_{1,2}\left(N_{4,2}\right)_{G_{1}}</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{2,1}\left(N_{5,1}\right)_{G_{2}}</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L_{2,2}\left(N_{5,2}\right)_{G_{2}}</math>
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{3,1}\left(N_{6,1}\right)_{G_{3}}</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>L_{3,2}\left(N_{6,2}\right)_{G_{3}}</math>

|}


{| class="wikitable" style="text-align: center; margin: 1em auto;"
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{1,2}\left(N_{1,1}\right)_{G_{1}}+L_{1,1}\left(N_{1,2}\right)_{G_{1}}   +L_{2,2}\left(N_{1,1}\right)_{G_{2}}+L_{2,1}\left(N_{1,2}\right)_{G_{2}}   +L_{3,2}\left(N_{1,1}\right)_{G_{3}}+L_{3,1}\left(N_{1,2}\right)_{G_{3}}</math> 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{1,2}\left(N_{2,1}\right)_{G_{1}}+L_{1,1}\left(N_{2,2}\right)_{G_{1}}   +L_{2,2}\left(N_{2,1}\right)_{G_{2}}+L_{2,1}\left(N_{2,2}\right)_{G_{2}}   +L_{3,2}\left(N_{2,1}\right)_{G_{3}}+L_{3,1}\left(N_{2,2}\right)_{G_{3}}</math> 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{1,2}\left(N_{3,1}\right)_{G_{1}}+L_{1,1}\left(N_{j,3}\right)_{G_{1}}   +L_{2,2}\left(N_{3,1}\right)_{G_{2}}+L_{2,1}\left(N_{j,3}\right)_{G_{2}}   +L_{3,2}\left(N_{3,1}\right)_{G_{3}}+L_{3,1}\left(N_{j,3}\right)_{G_{3}}</math> 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{1,2}\left(N_{4,1}\right)_{G_{1}}+L_{1,1}\left(N_{4,3}\right)_{G_{1}}</math> 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{2,2}\left(N_{5,1}\right)_{G_{2}}+L_{2,1}\left(N_{5,3}\right)_{G_{2}}</math> 
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">L_{3,2}\left(N_{6,1}\right)_{G_{3}}+L_{3,1}\left(N_{6,3}\right)_{G_{6}}</math> 

|}

<math>-2</math>


{| class="wikitable" style="text-align: center; margin: 1em auto;"
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\left(L_{1,1}\rho _{11}^{1}+L_{1,2}\rho _{11}^{2}\right)</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\left(L_{1,1}\rho _{22}^{1}+L_{i,2}\rho _{22}^{2}\right)</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\left(L_{1,1}\rho _{12}^{1}+L_{1,2}\rho _{12}^{2}\right)</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\left(L_{2,1}\rho _{11}^{1}+L_{2,2}\rho _{11}^{2}\right)</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\left(L_{2,1}\rho _{22}^{1}+L_{2,2}\rho _{22}^{2}\right)</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\left(L_{2,1}\rho _{12}^{1}+L_{2,2}\rho _{12}^{2}\right)</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\left(L_{3,1}\rho _{11}^{1}+L_{3,2}\rho _{11}^{2}\right)</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\left(L_{3,1}\rho _{22}^{1}+L_{3,2}\rho _{22}^{2}\right)</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\left(L_{3,1}\rho _{12}^{1}+L_{3,2}\rho _{12}^{2}\right)</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">0</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">0</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">0</math> 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>0</math>

|}

In this last expression <math display="inline">L_{i,j} =L_{i,j}^{M}</math>