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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.

Loads

Body loads are supported. Boundary load support is beta.
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 εx,εy,εxy are membrane in plane normal deformations, γzx,γxz are (out of plane and in plane) shear componets, κx,κy,κxy 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.

Loads

Body loads are supported.

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 εx,εy,εxy are membrane in plane normal deformations, γzx,γxz are (out of plane and in plane) shear componets, κx,κy,κxy 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.

Loads

Body loads are supported. Boundary loads are not supported now.

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 εx,εy,εxy are membrane in plane normal deformations, γzx,γxz are (out of plane and in plane) shear componets, κx,κy,κxy 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.

Loads

Body loads are supported. Boundary loads are not supported now.

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 εx,εy,εz are membrane normal deformations, γzy,γzx,γxy are (out of plane and in plane) shear componets, κx,κy,κz,κyz,κxz,κxy 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.

Loads

Body loads are supported. Boundary loads are not supported now.

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 εx,εy,εz are membrane normal deformations, γzy,γzx,γxy are (out of plane and in plane) shear componets, κx,κy,κz,κyz,κxz,κxy 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.

PIC

Figure 11: Geometry of tr_shell01 element.

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.

Loads

Body loads are supported. Boundary loads are supported (only surface loads).

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 εx,εy,εz are membrane normal deformations, γzy,γzx,γxy are (out of plane and in plane) shear componets, κx,κy,κz,κyz,κxz,κxy 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.

Loads

Body loads are supported. Boundary loads are supported (only surface loads).

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 εx,εy,εz are membrane normal deformations, γzy,γzx,γxy are (out of plane and in plane) shear componets, κx,κy,κz,κyz,κxz,κxy 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

2.6.8 Quad1Mindlin Element

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

quad1mindlin

Description

Quadrilateral, bilinear, four-node Mindlin plate

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.

Loads

Dead weight loads, and edge loads are supported.

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 εx,εy,εxy are membrane in plane normal deformations, γzx,γxz are (out of plane and in plane) shear componets, κx,κy,κxy 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


Table 25: quad1mindlin element summary

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.

Loads

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, εx,εy,εz are membrane normal deformations, γzy,γzx,γxy are (out of plane and in plane) shear componets, κx,κy,κz,κyz,κxz,κxy 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

2.6.12 quad1plateSubsoil Element

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]


Table 28: quad1platesubsoil element summary

2.6.13 Tria1PlateSubSoil Element

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

tria1platesubsoil

Description

Tringular, three-node sub-soil plate element with linear interpolation

Specific parameters
Unknowns

One dof (w-displacement) is required in each node.

Approximation

Linear for transwersal displacement.

Integration

1 integration points.

Loads

Surface load support.

Note

Requires material model with 2dPlateSubSoil mode support.

Reference

[2]


Table 29: tria1platesubsoil element summary

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