Users:General FEM Analysis/Elements Reference/Shell10

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== Theory ==
 
== Theory ==
  
−
The theory and finite element formulation is originally based on the work of Linhard
+
The theory and finite element formulation is originally based on the work of Linhard <ref name=Lin2007">Linhard, J.; Wüchner, R.; Bletzinger K.U.: "Upgrading" membranes to shells — The CEG rotation free shell element and its application in structural analysis, Finite Elements in Analysis and Design, Elsevier, volume 44, pages 63-74, 2007</ref>. A detailled derivation of the presented formulas and additional information to the implementation is shown by Both <ref name="Both10">Both, T.: Formulation and object oriented implementation of a nonlinear rotation free Kirchhoff shell element, Masterthesis, Chair of structural analysis, Technische Universität München, 2010</ref>. A short introduction into theory, modelling aspects and behaviour is given in the following.
−
<ref name=Lin2007">Linhard, J.; Wüchner, R.; Bletzinger K.U.: "Upgrading" membranes to shells — The CEG rotation free shell
+
−
element and its application in structural analysis, Finite Elements in Analysis and Design, Elsevier, volume 44, pages 63-74, 2007</ref>
+
−
 
+
−
For the correct use of the Shell10 and the interpretation of related results, the following aspects should be considered:
+
  
 
=== Decomposition of virtual work ===
 
=== Decomposition of virtual work ===
  
−
For the derivation the principle of virtual work [[File:Formula_pvd.png|175px]] is splitted into membrane and bending parts:  [[File:Formula_splitup.png|200px]]
+
For the derivation the principle of virtual work [[File:Formula_pvd.png|175px]] is splitted into membrane and bending parts:  [[File:Formula_splitup.png|225px]]
−
The bending part is calculated with [[File:Formula_bendingpart.png|175px]] consisting out of moments acting on changes of curvature. With an ideal linear elastic material law, e.g. St. Venant-Kirchhoff model, these moments can be related to the change of curvature with: [[File:Formula_stvenantkirchhoff.png|300px]]. Finally the change of curvature can be calculated with: [[File:Formula_change_of_curvature.png|300px]]. The interesting part is how curvature related quantites on C0-continious meshes.
+
The bending part is calculated with [[File:Formula_bendingpart.png|200px]] consisting out of moments acting on changes of curvature. With an ideal linear elastic material law, e.g. St. Venant-Kirchhoff model, these moments can be related to the change of curvature with: [[File:Formula_stvenantkirchhoff.png|300px]]. Finally the change of curvature can be calculated with: [[File:Formula_change_of_curvature.png|175px]]. The calculation of curvature related quantites on C0-continious meshes is shown next.
  
 
=== Curvature on C0 continous finite element meshes ===
 
=== Curvature on C0 continous finite element meshes ===
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The basic idea is to extend the normal vector at the nodes with information of the neighbouring elements. The so called "nodal director" is the averaged normalized normal vector at each node and calulated with: [[File:Formula.png|175px]]. The weighting factors are the approximated area of the corresponding element. It can be shown that for a two dimensional discretized half-circle these weighting factors exactly reproduce the surface normals.
 
The basic idea is to extend the normal vector at the nodes with information of the neighbouring elements. The so called "nodal director" is the averaged normalized normal vector at each node and calulated with: [[File:Formula.png|175px]]. The weighting factors are the approximated area of the corresponding element. It can be shown that for a two dimensional discretized half-circle these weighting factors exactly reproduce the surface normals.
  
−
[[File:Weighting_factor.png|600px|up|Different weighting factors]]  
+
[[File:Weighting_factor.png|400px|up|Different weighting factors]]  
  
 
==== CIN: curvature through interpolated normal vectors ====
 
==== CIN: curvature through interpolated normal vectors ====
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For the curvature calculation the straightforward approach is to use the curvature tensor introduced in classical differential geometry on the discretized geometry. Since the normal vector is needed throughout the whole element the nodal director is interpolated with the same shape functions used for the isoparametric element. This formulation results in a deficiency shown in the following figure.
 
For the curvature calculation the straightforward approach is to use the curvature tensor introduced in classical differential geometry on the discretized geometry. Since the normal vector is needed throughout the whole element the nodal director is interpolated with the same shape functions used for the isoparametric element. This formulation results in a deficiency shown in the following figure.
  
−
[[File:Deficiency.png|600px|up|Deficiency of CIN operator]]  
+
[[File:Deficiency.png|400px|up|Deficiency of CIN operator]]  
  
 
Due to the linear interpolation of the nodal directors, no change in the normal field will occur, hence no change of the curvature
 
Due to the linear interpolation of the nodal directors, no change in the normal field will occur, hence no change of the curvature
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against the tangential plane of the specific isoparametric element, although it should be normal to it. In order to repair this discrepancy, additional rotations are introduced, which can be determined by the inclination of the nodal director against the tangential plane. These rotations are then interpolated using suitable shape functions. The enhanced curvature of a CEG element consists of the actual curvature of the isoparametric surface and the additional curvature obtained by the derivation of the introduced additional rotation field w.r.t the surface coordinates.
 
against the tangential plane of the specific isoparametric element, although it should be normal to it. In order to repair this discrepancy, additional rotations are introduced, which can be determined by the inclination of the nodal director against the tangential plane. These rotations are then interpolated using suitable shape functions. The enhanced curvature of a CEG element consists of the actual curvature of the isoparametric surface and the additional curvature obtained by the derivation of the introduced additional rotation field w.r.t the surface coordinates.
  
−
[[File:Discrete_rotations.png|600px|up|Discrete rotations on node 3 of a quadrilateral element]]  
+
[[File:Discrete_rotations.png|400px|up|Discrete rotations on node 3 of a quadrilateral element]]  
  
  
 +
=== Modelling aspects ===
  
 +
==== Clamped edges ====
  
−
=== Degrees of Freedom, Rotational DOFs, Moment Loads ===
+
[[File:Modelling_clampededge.png|400px|up|Clamped edges with the Shell10 element]]
  
−
The degrees of freedom consist of the three displacements of the midsurface (according to standard shell theory). This has to be taken into account at the formulation of BCs and Loads, especially at clamped supports.
+
For the additional stiffness at the free degrees of freedom (red nodes) one additional fixed element row is sufficient. Only for curved geometries a second row is necessary in order to interpolate the curvature through the first fixed element row correctly. In the lower part of the figure the situation for the deformed geometry is depicted. The geometry approximated through the nodal directors is displayed with a green dotted line and one sees the deformation in the first fixed element row. One trick to influence the direction of the nodal director is to model the fixed element rows really small, since the area of the element enters the  formula of the nodal director as a weighting factor. Therewith the unwanted deformation in the fixed element row becomes
 +
smaller.
  
 +
==== Symmetry ====
  
−
=== Deformation in Thickness Direction ===
+
[[File:Symmetry.png|400px|up|Symmetry with the Shell10 element]]
  
−
The three components of the deformation of the director also comprise a possible deformation in thickness direction (change of thickness). The three DOFs ''Dir_Diff_Shell8_X'', ''Dir_Diff_Shell8_Y'' and ''Dir_Diff_Shell8_Z'' could be converted in two rotations and one deformation in thickness direction. However, the thickness deformation does not necessarily coincide with the 6th degree of freedom (''Dir_Diff_Shell8_Z'')! This would only be true, if the shell midplaine would be parallel to the xy-plane. In general, the thickness deformation is that component of the difference vector, which is parallel to the director (i.e. a linear combination of all three cartesian components).
+
The appliciation of symmetry is not implemented for this element, because contributing elements behind the symmetry planes are missing. For a better understanding see in the above figure the rectangular sheet of metal subjected to selfweight, while both short ends are supported. Modelling only half it, the dark grey elements would result and the additional boundary condition is the rollersupport preventing the elements to deform above the symmetry plane. The first thing which is not possible is to keep the rotations at the symmetry plane fixed, therefore disconuities can evolve at the symmetry plane. The second thing are missing stiffness contributions, marked at the red dots in the first sketch. Elements located behind the symmetry plane adding their stiffness contributions are missing. In the second and third sketch these missing elements are drawn. Exemplarily Masterelements (red) with their neighboring patch/influence zone (green) are shown and one can clearly see how stiffness contributions are simply neglected.
  
−
In case of a fixed support, also the thickness deformation is fixed. This is correct in the context of real physical clamped supports, but it is wrong in the context of clamped supports that model symmetry conditions.
+
==== Free edges ====
  
 +
[[File:Modelling_freeedge.png|400px|up|Free edges with the Shell10 element]]
  
−
=== EAS Enhancement, 7-parameter formulation ===
+
Free edges do not need special attention in the modelling, but an interesting aspect should be highlighted. For curved geometries the representation of curvature at free edges is not correct. Imagine half of a Cylinder as depicted in the above figure. The curvature on the continous surface (green) is everywhere the same. For the discretized geometry (grey) the nodal directors match the normal vectors of the surface, but not at the free edge. Due to missing information how the surface would continue a correct representation of the curvature is not possible. The nodal directors at the free edge (red) are simply normal to the element. This effect vanishes in the limit where the elementsize tends towards zero.
  
−
In case of a nonzero Poisson ratio, the 6 nodal DOFs are not sufficient in order to describe a shell theory which is assymptotically correct for bending situations. That's why a seventh parameter is introduced via the EAS method. The interpolation of this parameter within the element is described by value 3 of the parameter EAS of the property block. It has to be at least constant, i.e. the value must be at least =1. In general, for quadrilateral element a bilinear interpolation is recommended ( i.e. EAS = x,x,4,x,x), for triangular elements a constant or linear interpolation is suggested (see table above). 
+
== Tested examples ==
  
 +
The following benchmarks show the good behaviour of the Shell10 element.
  
−
=== Element Technology ===
+
=== Shell Obstacle Course ===
  
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The are different kind of locking effects for shells. An overview is given at <ref name="Bi99" />.
+
The intention of the shell obstacle course by Belytschko et al. <ref name="bely85"> BELYTSCHKO T., Stolarski H., Liu W., Carpenter N. and Ong J.: Stress projection for membrane and shear locking in shell finite elements, Computer Methods in Applied Mechanics and Engineering, 51:221–258, 1985</ref> is to test the ability of elements to handle inextensional bending modes of deformation, rigid-body motion without straining, and complex membrane states of stress. It consists of three problems, namely the Scordelis-
 +
Lo Roof, the pinched cylinder with diaphragm and the hemispherical shell.
  
−
* (Bi-)linear elements (3 noded, 4-noded) are almost completely free of <u>membrane locking</u>, thus EAS is not really necessary here. For this reason, these elements are in the context of shells superior to elements of higher order. EAS does not completely eliminate membrane locking, but pushes it back to values of a higher critical parameter (higher slenderness)
+
==== Scordelis-Lo Roof ====
  
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* <u>transverse shear locking</u> must always be tackled, either by EAS or by ANS. Only in the context of extremely thick structures, this can be turned down
+
[[File:Obstacle_scoloroof.png|400px|up|Scordelis-Lo roof]]
  
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* Also <u>thickness locking</u> must always be tackled, either by EAS or by ANS.
+
The Scordelis-Lo roof is a section of a cylindrical shell subjected to a uniform gravity load. Exemplarily deformation figures are shown below. The normalized convergence curve is shown here: [[File:Conv_ScoLoRoof.pdf|600px|up|Convergence Curve Scordelis-Lo roof]]
  
−
* The in-plane bending behaviour of (bi-)linear elements is very bad. If such kind of deformations are expected to show up, then the use of EAS is recommended for the 4-noded element. However, this locking behaviour cannot be eliminated for the 3-noded triangular element. However, the related locking phenomena (<u>in-plane shear locking</u>, <u>volumetric locking</u>) are usually less critical for shells compared to the aforementioned ones.
+
{|
 +
|[[File:ScoLoRoof_full_quad_192.png|400px|thumb|up|192 quadrilateral elements]]
 +
|[[File:ScoLoRoof_full_tria_300.png|400px|thumb|up|300 triangular elements]]
 +
|}
  
 +
==== Pinched Cylinder ====
  
−
=== Mesh Generation ===
+
[[File:Obstacle_pinched.png|400px|up|Scordelis-Lo roof]]
  
−
As the DOF formulation is based on the difference vector, the physical meaning of the fourth up to sixth degree of freedom is dependent on the position and orientation of the director. Thus it has to be made sure that a director is uniquely defined at each node. Due to that it is important that the director is always pointing to one side of the shell surface.
+
The pinched cylinder with diaphragms is a cylindrical shell subjected to two opposite point loads located in the middle of the cylinder. Exemplarily deformation figures are shown below. The normalized convergence curve is shown here: [[File:Conv_PinchedCyl.pdf|600px|up|Convergence Curve Scordelis-Lo roof]]
  
 +
{|
 +
|[[File:PinchedCyl_quad_1824.png|400px|thumb|up|1824 quadrilateral elements]]
 +
|[[File:PinchedCyl_tria_231936.png|400px|thumb|up|231936 triangular elements]]
 +
|}
  
−
== A Full Example ==
+
The Pinched Cylinder with its regular mesh is prone to the previously mentioned zero-energy modes. Results of the unphysical deformation behaviour is shown below.
  
−
The following example describes a simple cantilever problem discretized by [[Users:General FEM Analysis/Elements Reference/Shell8|SHELL8]] elements. The respective input file can be found here:  
+
{|
 +
|[[File:PiCyl_quad_CIN_1824.png|400px|thumb|up|1824 quadrilateral elements, CIN operator, front view]]  
 +
|[[File:PiCyl_quad_CIN2_1825.png|400px|thumb|up|1824 quadrilateral elements, CIN operator, iso view]]
 +
|}
 +
[[File:PiCyl_tria_CIN2_3648.png|400px|thumb|up|3648 triangular elements, CIN operator, iso view]]
  
−
{{PathToBenchmarkExamples}}elements/shell8_quad_lin_canti_I/shell8_canti_2x20_elem_load.dat
 
  
 +
==== Hemispherical Shell ====
  
−
The problem computes three computation cases (dead load, snow load and pressure load). The [[Users:General FEM Analysis/BCs Reference|boundary conditions]] are visualized by the figure below.
+
[[File:Obstacle_hemishell.png |400px|up|Scordelis-Lo roof]]  
−
{|
+
 
−
|[[File:Shell8_canti_org_geom_01.png‎|frame|up|Simple cantilever problem using SHELL8 elements]]  
+
The hemisperical shell is subjected to four point loads acting every 90 degrees at the free bottom of the structure. Exemplarily deformation figures are shown below. The normalized convergence curve is shown here: [[File:Conv_Hemishell.pdf |600px|up|Convergence Curve Scordelis-Lo roof]]
−
|}
+
  
−
The basic goal of each linear static analysis is the computation of the displacement field. For load case 1 this result is depicted in the figure below, whereas the deformation in z-direction is additionally visualized by the color plot.
 
−
{|
 
−
|[[File:Shell8_canti_def_geo_disp.png‎|frame|up|Displacement of cantilever problem]]
 
−
|}
 
−
It can be seen that the support region does not show any deformation whereas the tip region deforms by a value of 8.76.
 
  
−
Often the stress distribution is visualized by color plots. Shell structures require the specification of the layer on which the stresses are computed. The picture below shows the first principle stress on the top of the cantilever.
 
 
{|
 
{|
−
|[[File:Shell8_canti_def_geo_stress1.png|frame|up|First principle stress of cantilever problem (top layer)]]  
+
|[[File:Hemi_quad_103.png|400px|thumb|up|103 quadrilateral elements]]
 +
|[[File:Hemi_tria_68.png|400px|thumb|up|231936 triangular elements]]  
 
|}
 
|}
−
It can be seen that the stresses show maximum values at the support region of the cantilever. At the tip they are nearly zero. In contrast to analytical results the stresses are not exactly zero in numerical models. The reason is the stress computation at the Gauss points. These points are situated inside the elements and not exactly at the tip of the cantilever.
+
 
 +
== Benchmarks ==
 +
 
 +
The main benchmark files in the ''Carat++''-repository are  
 +
 
 +
'../examples/benchmark_examples/elements/shell10_quad_CEG/cbm_ScoLoRoof_full_quad_struc_192.dat'
 +
 
 +
'../examples/benchmark_examples/elements/shell10_quad_CIN/cbm_HemiShell_full_quad_103.dat'
 +
 
 +
'../examples/benchmark_examples/elements/shell10_tria_CEG/cbm_PinchedCyl_full_tria_240.dat'
 +
 
 +
'../examples/benchmark_examples/elements/shell10_tria_CIN/cbm_ScoLoRoof_full_tria_struc_384.dat'
 +
 
 +
'../examples/benchmark_examples/elements/shell10_tria_nonlin/cbm_NONLN_Cyl_tria_8x8.dat'
  
 
== References ==
 
== References ==
  
 
<references/>
 
<references/>

Latest revision as of 15:16, 12 January 2017


Contents

General Description

Triangular Shell10 element
Quadrilateral Shell10 element


Element Type

  • This shell element is a rotationfree shell element with 3 (external) degrees of freedom per node.
  • It has Kirchhoff-Love kinematics.
  • Underlying element is the Membrane1 element and additional bending stiffness is added for each dof.
  • Curvature information is calculated out of neighboring elements.

File:Patch example-eps-converted-to.pdf

Degrees of Freedom

The degrees of freedom are:

  • the 3 translatoric components of the shell midplaine in the global cartesian frame (the 3 "classical" translatoric DOFs): They are named Disp_X, Disp_Y, Disp_Z.


Input Parameters

Parameter Description

Parameter Description

Compulsory Parameters
Parameter Values, Default(*) Description
MAT EL-MAT int Number for the used Material

e.g. MAT=EL-MAT 1

THICKNESS int Thickness of the shell
A_X, A_Y, A_Z

B_X, B_Y, B_Z

Definition of the principle directions of the surface. To define the prestress dirctions. (see Membrane1 element)
SIG11

SIG22 SIG12

Prestress state of the membrane. (see Membrane1 element)
CURV_TYPE_SHELL10 CEG, CIN Definition of curvature calculation type. CIN for curvature through interpolated normal field. CEG for curvature through enhanced geometry. (see description below)
Optional Parameters


Example of a Complete Input Block 

EL-PROP 1 : SHELL10
MAT=EL-MAT 1        
THICKNESS=0.1 
INT_TYPE_SHELL10 = FULL
CURV_TYPE_SHELL10 = CEG
PRESTRESS      SIG11=0.0     SIG22=0.0      SIG12=0.0
A_X=1.0     A_Y=0.0    A_Z=0.0
B_X=0.0    B_Y=1.0    B_Z=0.0


Element Loading

Pressure

Dead Load

Snow Load

Theory

The theory and finite element formulation is originally based on the work of Linhard [1]. A detailled derivation of the presented formulas and additional information to the implementation is shown by Both [2]. A short introduction into theory, modelling aspects and behaviour is given in the following.

Decomposition of virtual work

For the derivation the principle of virtual work Formula pvd.png is splitted into membrane and bending parts: Formula splitup.png The bending part is calculated with Formula bendingpart.png consisting out of moments acting on changes of curvature. With an ideal linear elastic material law, e.g. St. Venant-Kirchhoff model, these moments can be related to the change of curvature with: Formula stvenantkirchhoff.png. Finally the change of curvature can be calculated with: Formula change of curvature.png. The calculation of curvature related quantites on C0-continious meshes is shown next.

Curvature on C0 continous finite element meshes

While isoparametric triangular elements are flat, quadrilateral elements allready bring a curvature along, but which is not sufficient for the calculation of a surface curvature.

The basic idea is to extend the normal vector at the nodes with information of the neighbouring elements. The so called "nodal director" is the averaged normalized normal vector at each node and calulated with: Formula.png. The weighting factors are the approximated area of the corresponding element. It can be shown that for a two dimensional discretized half-circle these weighting factors exactly reproduce the surface normals.

Different weighting factors

CIN: curvature through interpolated normal vectors

For the curvature calculation the straightforward approach is to use the curvature tensor introduced in classical differential geometry on the discretized geometry. Since the normal vector is needed throughout the whole element the nodal director is interpolated with the same shape functions used for the isoparametric element. This formulation results in a deficiency shown in the following figure.

Deficiency of CIN operator

Due to the linear interpolation of the nodal directors, no change in the normal field will occur, hence no change of the curvature can be detected. This represents a deformation, which can not be captured by the formulas and therefore will not enter the bending part of the internal virtual work. One therefore speaks of a zero-energy deformation mode. Performing analysis with this operator will in some cases produce this "zig-zag"-like results and the use of this curvature operator is not recommended. Examples of these unphysical deformation figures are shown below.


CEG: curvature calculation through enhanced geometry

In order to overcome these spurious deformation modes a second approach is implemented. The basic idea of this method of curvature calculation is to enhance the geometry with an additional rotation field. The nodal director of the surface is in general inclined against the tangential plane of the specific isoparametric element, although it should be normal to it. In order to repair this discrepancy, additional rotations are introduced, which can be determined by the inclination of the nodal director against the tangential plane. These rotations are then interpolated using suitable shape functions. The enhanced curvature of a CEG element consists of the actual curvature of the isoparametric surface and the additional curvature obtained by the derivation of the introduced additional rotation field w.r.t the surface coordinates.

Discrete rotations on node 3 of a quadrilateral element


Modelling aspects

Clamped edges

Clamped edges with the Shell10 element

For the additional stiffness at the free degrees of freedom (red nodes) one additional fixed element row is sufficient. Only for curved geometries a second row is necessary in order to interpolate the curvature through the first fixed element row correctly. In the lower part of the figure the situation for the deformed geometry is depicted. The geometry approximated through the nodal directors is displayed with a green dotted line and one sees the deformation in the first fixed element row. One trick to influence the direction of the nodal director is to model the fixed element rows really small, since the area of the element enters the formula of the nodal director as a weighting factor. Therewith the unwanted deformation in the fixed element row becomes smaller.

Symmetry

Symmetry with the Shell10 element

The appliciation of symmetry is not implemented for this element, because contributing elements behind the symmetry planes are missing. For a better understanding see in the above figure the rectangular sheet of metal subjected to selfweight, while both short ends are supported. Modelling only half it, the dark grey elements would result and the additional boundary condition is the rollersupport preventing the elements to deform above the symmetry plane. The first thing which is not possible is to keep the rotations at the symmetry plane fixed, therefore disconuities can evolve at the symmetry plane. The second thing are missing stiffness contributions, marked at the red dots in the first sketch. Elements located behind the symmetry plane adding their stiffness contributions are missing. In the second and third sketch these missing elements are drawn. Exemplarily Masterelements (red) with their neighboring patch/influence zone (green) are shown and one can clearly see how stiffness contributions are simply neglected.

Free edges

Free edges with the Shell10 element

Free edges do not need special attention in the modelling, but an interesting aspect should be highlighted. For curved geometries the representation of curvature at free edges is not correct. Imagine half of a Cylinder as depicted in the above figure. The curvature on the continous surface (green) is everywhere the same. For the discretized geometry (grey) the nodal directors match the normal vectors of the surface, but not at the free edge. Due to missing information how the surface would continue a correct representation of the curvature is not possible. The nodal directors at the free edge (red) are simply normal to the element. This effect vanishes in the limit where the elementsize tends towards zero.

Tested examples

The following benchmarks show the good behaviour of the Shell10 element.

Shell Obstacle Course

The intention of the shell obstacle course by Belytschko et al. [3] is to test the ability of elements to handle inextensional bending modes of deformation, rigid-body motion without straining, and complex membrane states of stress. It consists of three problems, namely the Scordelis- Lo Roof, the pinched cylinder with diaphragm and the hemispherical shell.

Scordelis-Lo Roof

Scordelis-Lo roof

The Scordelis-Lo roof is a section of a cylindrical shell subjected to a uniform gravity load. Exemplarily deformation figures are shown below. The normalized convergence curve is shown here: File:Conv ScoLoRoof.pdf

192 quadrilateral elements
300 triangular elements

Pinched Cylinder

Scordelis-Lo roof

The pinched cylinder with diaphragms is a cylindrical shell subjected to two opposite point loads located in the middle of the cylinder. Exemplarily deformation figures are shown below. The normalized convergence curve is shown here: File:Conv PinchedCyl.pdf

1824 quadrilateral elements
231936 triangular elements

The Pinched Cylinder with its regular mesh is prone to the previously mentioned zero-energy modes. Results of the unphysical deformation behaviour is shown below.

1824 quadrilateral elements, CIN operator, front view
1824 quadrilateral elements, CIN operator, iso view
3648 triangular elements, CIN operator, iso view


Hemispherical Shell

Scordelis-Lo roof

The hemisperical shell is subjected to four point loads acting every 90 degrees at the free bottom of the structure. Exemplarily deformation figures are shown below. The normalized convergence curve is shown here: File:Conv Hemishell.pdf


103 quadrilateral elements
231936 triangular elements

Benchmarks

The main benchmark files in the Carat++-repository are

'../examples/benchmark_examples/elements/shell10_quad_CEG/cbm_ScoLoRoof_full_quad_struc_192.dat'

'../examples/benchmark_examples/elements/shell10_quad_CIN/cbm_HemiShell_full_quad_103.dat'

'../examples/benchmark_examples/elements/shell10_tria_CEG/cbm_PinchedCyl_full_tria_240.dat'

'../examples/benchmark_examples/elements/shell10_tria_CIN/cbm_ScoLoRoof_full_tria_struc_384.dat'

'../examples/benchmark_examples/elements/shell10_tria_nonlin/cbm_NONLN_Cyl_tria_8x8.dat'

References

  1. ↑ Linhard, J.; Wüchner, R.; Bletzinger K.U.: "Upgrading" membranes to shells — The CEG rotation free shell element and its application in structural analysis, Finite Elements in Analysis and Design, Elsevier, volume 44, pages 63-74, 2007
  2. ↑ Both, T.: Formulation and object oriented implementation of a nonlinear rotation free Kirchhoff shell element, Masterthesis, Chair of structural analysis, Technische Universität München, 2010
  3. ↑ BELYTSCHKO T., Stolarski H., Liu W., Carpenter N. and Ong J.: Stress projection for membrane and shear locking in shell finite elements, Computer Methods in Applied Mechanics and Engineering, 51:221–258, 1985




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