#### 2.6 Plate & Shell Elements

##### 2.6.1 DKT Element

Implementation of Discrete Kirchhoff Triangle (DKT) plate element. This element is suitable for thin plates, as the traswerse shear strain energy is neglected. The structure should be defined in x,y plane, nodes should be numbered anti-clockwise (positive rotation around z-axis). The element features are summarized in Table 18.

Keyword

dktplate

Description

2D Discrete Kirchhoff Triangular plate element

Specific parameters

-

Unknowns

Three dofs (w-displacement, u and v - rotations) are required in each node.

Approximation

Quadratic approximation of rotations, cubic approximation of displacement along the edges. Note: there is no need to define interpolation for displacement on the element.

Integration

Default integration of all terms using three point formula.

Features

Layered cross section support.

CS properties

Cross section thickness is required.

Output

On output, the generalized shell strain/force momentum vectors in global coordinate system are printed, with the following meaning:

 sε = {εx,εy,εxz,κx,κy,κxy,γxz,γyz}, sσ = {nx,ny,nxy,mx,my,mz,mxy,qxz,qyz}
where εxyxy are membrane in plane normal deformations, γzxxz are (out of plane and in plane) shear componets, κxyxy are curvatures, nx,ny,nxy,qxz,qyz are integral forces (normal and shear forces), and mx,my,mxy are bending moments. Please note, for example, that bending moment mx is defined as mx = σxz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo

0.

Status

Reliable

Reference

J.L.Batoz, K.J.Bathe, L.W.Ho: A study of three-node triangular plate bending elements, IJNME, 15(12):1771-1812, 1980

Table 18: DKTplate element summary

##### 2.6.2 QDKT Element

Implementation of Discrete Kirchhoff Theory plate quad element (QDKT). This element is suitable for thin plates, as the traswerse shear strain energy is neglected. The structure should be defined in x,y plane, nodes should be numbered anti-clockwise (positive rotation around z-axis). The element features are summarized in Table 19.

Keyword

qdktplate

Description

2D Discrete Kirchhoff Quad plate element

Specific parameters

-

Unknowns

Three dofs (w-displacement, u and v - rotations) are required in each node.

Approximation

Quadratic approximation of rotations, cubic approximation of displacement along the edges. Note: there is no need to define interpolation for displacement on the element.

Integration

Default integration of all bending terms using four point formula.

Features

Layered cross section support.

CS properties

Cross section thickness is required.

Output

On output, the generalized shell strain/force momentum vectors in global coordinate system are printed, with the following meaning:

 sε = {εx,εy,εxz,κx,κy,κxy,γxz,γyz}, sσ = {nx,ny,nxy,mx,my,mz,mxy,qxz,qyz}
where εxyxy are membrane in plane normal deformations, γzxxz are (out of plane and in plane) shear componets, κxyxy are curvatures, nx,ny,nxy,qxz,qyz are integral forces (normal and shear forces), and mx,my,mxy are bending moments. Please note, for example, that bending moment mx is defined as mx = σxz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo

0.

Status

Reliable

Reference

J.L.Batoz, K.J.Bathe, L.W.Ho: A study of three-node triangular plate bending elements, IJNME, 15(12):1771-1812, 1980

Table 19: QDKTplate element summary

##### 2.6.3 CCT Element

Implementation of constant curvature triangular element for plate analysis. Formulation based on Mindlin hypothesis. The structure should be defined in x,y plane. The nodes should be numbered anti-clockwise (positive rotation around z-axis). The element features are summarized in Table 20.

Keyword

cctplate

Description

2D constant curvature triangular plate element

Specific parameters

-

Unknowns

Three dofs (w-displacement, u and v - rotations) are required in each node.

Approximation

Linear approximation of rotations, quadratic approximation of displacement.

Integration

Integration of all terms using one point formula.

Features

Layered cross section support.

CS properties

Cross section thickness is required.

Output

On output, the generalized shell strain/force momentum vectors in global coordinate system are printed, with the following meaning:

 sε = {εx,εy,εxz,κx,κy,κxy,γxz,γyz}, sσ = {nx,ny,nxy,mx,my,mz,mxy,qxz,qyz}
where εxyxy are membrane in plane normal deformations, γzxxz are (out of plane and in plane) shear componets, κxyxy are curvatures, nx,ny,nxy,qxz,qyz are integral forces (normal and shear forces), and mx,my,mxy are bending moments. Please note, for example, that bending moment mx is defined as mx = σxz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo

0.

Status

Reliable

Table 20: cctplate element summary

##### 2.6.4 CCT3D Element

Implementation of constant curvature triangular element for plate analysis. Formulation based on Mindlin hypothesis. The element could be arbitrarily oriented in space. The nodes should be numbered anti-clockwise (positive rotation around element normal). The element features are summarized in Table 21.

Keyword

cctplate3d

Description

Constant curvature triangular plate element in arbitray position

Specific parameters

-

Unknowns

Six dofs (u,v,w-displacements and u,v,w rotations) are in general required in each node.

Approximation

Linear approximation of ratations, quadratic approximation of displacement.

Integration

Integration of all terms using one point formula.

Features

Layered cross section support.

CS properties

Cross section thickness is required.

Output

On output, the shell force (sf), shell strain (ss), shell momentum (sm), and shell curvature (sc) tensors in global coordinate system are printed as vector form with 6 components, with the following meaning:

 sf = {nx,ny,nz,vyz,vxz,vxy}, ss = {εx,εy,εz,γyz,γxz,γxy}, sm = {mx,my,mz,myz,mxz,mxy}, sc = {κx,κy,κz,κyz,κxz,κxy}
where εxyz are membrane normal deformations, γzyzxxy are (out of plane and in plane) shear componets, κxyzyzxzxy are curvatures, nx,ny,nz,vyz,vxz,vxy are integral forces (normal and shear forces), and mx,my,mz,myz,mxz,mxy are bending moments. Please note, for example, that bending moment mx is defined as mx = σxz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo

0.

Status

Reliable

Table 21: cctplate3d element summary

##### 2.6.5 RerShell Element

Combination of CCT plate element (Mindlin hypothesis) with triangular plane stress element for membrane behavior. The element curvature can be specified. Although element requires generally six DOFs per node, no stiffness to local rotation along z-axis (rotation around element normal) is supplied. The element features are summarized in Table 22.

Keyword

rershell

Description

Simple shell based on combination of CCT plate element (Mindlin hypothesis) with triangular plane stress element. element can be arbitrary positioned in space.

Specific parameters

-

Unknowns

Six dofs (u,v,w-displacements and u,v,w rotations) are in general required in each node. Note, that although element it requires generally six DOFs per node, no stiffness to local rotation along z-axis (rotation around element normal) is supplied.

Approximation

Linear approximation of ratations, quadratic approximation of displacement.

Integration

Integration of all terms using one point formula.

Features

Layered cross section support.

CS properties

Cross section thickness is required.

Output

On output, the shell force (sf), shell strain (ss), shell momentum (sm), and shell curvature (sc) tensors in global coordinate system are printed as vector form with 6 components, with the following meaning:

 sf = {nx,ny,nz,vyz,vxz,vxy}, ss = {εx,εy,εz,γyz,γxz,γxy}, sm = {mx,my,mz,myz,mxz,mxy}, sc = {κx,κy,κz,κyz,κxz,κxy}
where εxyz are membrane normal deformations, γzyzxxy are (out of plane and in plane) shear componets, κxyzyzxzxy are curvatures, nx,ny,nz,vyz,vxz,vxy are integral forces (normal and shear forces), and mx,my,mz,myz,mxz,mxy are bending moments. Please note, for example, that bending moment mx is defined as mx = σxz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo

0.

Status

Reliable

Table 22: rershell element summary

##### 2.6.6 tr_shell01 element

Combination of CCT3D plate element (Mindlin hypothesis) with triangular plane stress element for membrane behavior. It comes with complete set of 6 DOFs per node. The element features are summarized in Table 23.

Keyword

tr_shell01

Description

Triangular shell element combining CCT3D plate element (Mindlin hypothesis) with triangular plane stress element with rotational DOFs

Specific parameters

-

Unknowns

Six dofs (u,v,w-displacements and u,v,w rotations) are in general required in each node.

Approximation

See description of cct and trplanstrrot elements

Integration

Integration of all terms using one point formula.

Features

Layered cross section support.

CS properties

Cross section thickness is required.

Output

On output, the shell force (sf), shell strain (ss), shell momentum (sm), and shell curvature (sc) tensors in global coordinate system are printed as vector form with 6 components, with the following meaning:

 sf = {nx,ny,nz,vyz,vxz,vxy}, ss = {εx,εy,εz,γyz,γxz,γxy}, sm = {mx,my,mz,myz,mxz,mxy}, sc = {κx,κy,κz,κyz,κxz,κxy}
where εxyz are membrane normal deformations, γzyzxxy are (out of plane and in plane) shear componets, κxyzyzxzxy are curvatures, nx,ny,nz,vyz,vxz,vxy are integral forces (normal and shear forces), and mx,my,mz,myz,mxz,mxy are bending moments. Please note, for example, that bending moment mx is defined as mx = σxz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo

0.

Status

Reliable

Table 23: tr_shell01 element summary

##### 2.6.7 tr_shell02 element

Combination of thin-plate DKT plate element with plane stress element (TrPlanestressRotAllman). This element comes with complete set of 6 DOFs per node. The element features are summarized in Table 24.

Keyword

tr_shell02

Description

Triangular shell element combining DKT plate element with triangular plane stress element with rotational DOFs

Specific parameters

-

Unknowns

Six dofs (u,v,w-displacements and u,v,w rotations) are in general required in each node.

Approximation

See description of cct and trplanstrrot elements

Integration

4 integration points necessary, use ”NIP 4” in element record.

CS properties

Cross section thickness is required.

Output

On output, the shell force (sf), shell strain (ss), shell momentum (sm), and shell curvature (sc) tensors in global coordinate system are printed as vector form with 6 components, with the following meaning:

 sf = {nx,ny,nz,vyz,vxz,vxy}, ss = {εx,εy,εz,γyz,γxz,γxy}, sm = {mx,my,mz,myz,mxz,mxy}, sc = {κx,κy,κz,κyz,κxz,κxy}
where εxyz are membrane normal deformations, γzyzxxy are (out of plane and in plane) shear componets, κxyzyzxzxy are curvatures, nx,ny,nz,vyz,vxz,vxy are integral forces (normal and shear forces), and mx,my,mz,myz,mxz,mxy are bending moments. Please note, for example, that bending moment mx is defined as mx = σxz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo

0.

Status

-

Table 24: tr_shell02 element summary

This class implements an quadrilateral, bilinear, four-node Mindlin plate. This type of element exhibit strong shear locking (thin plates exhibit almost no bending). Implements the lumped mass matrix. The element features are summarized in Table 25.

Keyword

Description

Specific parameters

[NIP #(in)]

Unknowns

Three dofs (w-displacement, u and v - rotation) are required in each node.

Approximation

Linear for all unknowns.

Integration

Default uses 4 integration points. No reduced integration is used, as it causes numerical problems.

Features

Layered cross section support.

CS properties

Cross section thickness is required.

Output

On output, the generalized shell strain/force momentum vectors in global coordinate system are printed, with the following meaning:

 sε = {εx,εy,εxz,κx,κy,κxy,γxz,γyz}, sσ = {nx,ny,nxy,mx,my,mz,mxy,qxz,qyz}
where εxyxy are membrane in plane normal deformations, γzxxz are (out of plane and in plane) shear componets, κxyxy are curvatures, nx,ny,nxy,qxz,qyz are integral forces (normal and shear forces), and mx,my,mxy are bending moments. Please note, for example, that bending moment mx is defined as mx = σxz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo

0.

Reference

[1]

Status

Experimental

##### 2.6.9 Tr2Shell7 Element

This class implements a triangular, quadratic, six-node shell element. The element is a so-called seven parameter shell with seven dofs per node – a displacement field (3 dofs), an extensible director field (3 dofs) and a seventh dof representing inhomogenous thickness strain. This last parameter is included in the model in order to deal with volumetric/Poisson lock effects.

The element features are summarized in Table 25.

 Keyword tr2shell7 Description Triangular, quadratic, six-node shell with 7 dofs/node Specific parameters [NIP #(in)] Unknowns Seven dofs (displacement in u, v and w-direction; change in director field in u, v and w-direction; and inhomgenous thickness stretch) are required in each node. Approximation Quadratic for all unknowns. Integration Default uses 6 integration points in the midsurface plane. Number of integration points in the thickness direction is determined by the Layered cross section. Features Layered cross section support. CS properties This element must be used with a Layered cross section. Loads Edge loads, constant pressure loads and surface loads are supported. Nlgeo Not applicable. The implementation is for large defomrations and hence geometrical nonlinearities will always be present, regardless the value of Nlgeo. Reference [3] Status Experimental

Table 26: tr2shell7 element summary

##### 2.6.10 MITC4Shell Element

A four-node quadrilateral shell element formulated using three-dimensional continuum mechanics theory degenerated to shell behaviour. The element is applicable to thick and thin shells as the “mixed interpolation of tensorial components” (MITC) approach is used to remove shear locking. The implementation is based on the following paper: Dvorkin, E.N., Bathe, K.J.: A continuum mechanics based four-node shell element for general non-linear analysis, Eng.Comput., Vol.1, 77-88, 1984.

Although element requires generally six DOFs per node, no stiffness to local rotation along z-axis (rotation around director vector) is supplied. The element features are summarized in Table 27.

Keyword

mitc4shell

Description

Quadrilateral, bilinear, four-node shell element using the MITC technique.

Specific parameters

[NIP #(in)] [NIPZ #(in)] [directorType #(in)]

Parameters

NIP: allows to set the number of integration points in local x-y plane (default 4).

NIPZ: allows to set the number of integration points in local z-direction (default 2).

directorType: allows to set director vectors. Director vectors can be set as normal to the plane (directorType = 0, default), or calculated for each node as an average of neighbouring elements of same crosssection (directorType = 1), or can be specified at crosssection (directorType =2).

Unknowns

Six dofs (u,v,w-displacements and u,v,w rotations) are in general required in each node. Note, that although element requires generally six DOFs per node, no stiffness to local rotation along z-axis (rotation around director vector) is supplied.

Approximation

Linear approximation of displacements and rotations.

Integration

Integration of all terms using Gauss integration formula in 8 points (default) or specified using NIP and NIPZ parameters.

Features

Variable cross section support.

CS properties

Cross section thickness is required (measured along director vector). Director vectors components may be specified [directorx #(in)][directory #(in)][directorz #(in)] in case of directorType 2.

Body and boundary loads are supported.

Output

On output, the shell force (sf), shell momentum (sm), shell strain (ss), shell curvature (sc), strain (ε), and stress (σ) tensors in global coordinate system are printed as vector form with 6 components, with the following meaning:

 sf = {nx,ny,nz,vyz,vxz,vxy}, sm = {mx,my,mz,myz,mxz,mxy}, ss = {εx,εy,εz,γyz,γxz,γxy}, sc = {κx,κy,κz,κyz,κxz,κxy} ε = {εx,εy,εz,γyz,γzx,γxy}, σ = {σx,σy,σz,σyz,σxz,σxy}.
where nx,ny,nz,vyz,vxz,vxy are integral forces (normal and shear forces), and mx,my,mz,myz,mxz,mxy are bending moments, εxyz are membrane normal deformations, γzyzxxy are (out of plane and in plane) shear componets, κxyzyzxzxy are curvatures. Please note, for example, the bending moment mx is defined as mx = σxz dz, so it acts along the y-axis and positive value causes tension in bottom layer (positive z-coordinate). The shell force (sf), shell momentum (sm), shell strain (ss), and shell curvature (sc) tensors are evaluated at the midplane of the element (thus are constant along the thickness) while the strain (ε), and stress (σ) tensors are evaluated at each Gausspoint.
Nlgeo

0.

Status

-

Table 27: mitc4shell element summary

##### 2.6.11 Sub-soil Elements

This class implements an quadrilateral, bilinear, four-node plate subsoil element. Typically this element is combined with suitable plate element with quadrilateral geometry to model plate element on (elastic) subsoill foundation, but it can be used alone. The element geometry should be define in xy plane. The element features are summarized in Table 28.

 Keyword quad1plateSubsoil Description Quadrilateral, bilinear, four-node sub-soil plate element Specific parameters Unknowns One dof (w-displacement) is required in each node. Approximation Linear for transwersal displacement. Integration 4 integration points. Loads Surface load support. Note Requires material model with 2dPlateSubSoil mode support. Reference [2]