Plane Stress Elements

Plane Stress Elements

PlaneStress2d

Represents isoparametric four-node quadrilateral plane-stress finite element. Each node has 2 degrees of freedom. 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 6.

The generalization of this element, that can be positioned arbitrarily in space is linquad3dplanestress element. This element requires 3 displacement degrees of freedon in each node and assumes, that the element geometry is flat, i.e. all nodes are in the same plane. The element features are summarized in Table Table 7.

Fig. 5 PlaneStress2d element. Node numbering, edge numbering and definition of local edge c.s.(a).

Table 6 planestress2d element summary

Keyword	planestress2d
Description	2D quadrilateral element for plane stress analysis
Specific parameters	[NIP (in)]
Parameters	NIP: allows to set the number of integration points
Unknowns	Two dofs (u-displacement, v-displacement) are required in each node.
Approximation	Linear approximation of displacements and geometry.
Integration	Integration of membrane strain terms using Gauss integration formula in 1, 4 (default), 9 or 16 integration points. The default number of integration points used can be overloaded using NIP parameter. Reduced integration for shear terms is employed. Shear terms are always integrated using the 1-point integration rule.
Features	Nonlocal constitutive support, Geometric nonlinearity support.
CS properties	cross section thickness is required.
Loads	Body loads are supported. Boundary loads are supported and computed using numerical integration. The side numbering is following. Each i-th element side begins in i-th element node and ends on next element node (i+1-th node or 1-st node, in the case of side number 4). The local positive edge x-axis coincides with side direction, the positive local edge y-axis is rotated 90 degrees anti-clockwise (see fig. (PlaneStress2d element. Node numbering, edge numbering and definition of local edge c.s.(a).)).
Nlgeo	0, 1.
Status	Reliable
Tests/Examples	sm/idm08.in, sm/idm11.in, sm/materOrient01.in, sm/concrete_fcm_visco.in, sm/EC2creep_casting.in, sm/control_switch_1.in, sm/control_switch_2.in, sm/hangingnode01.in, sm/hangingnode02.in, sm/hangingnode03.in (and 23 more)

Table 7 linquad3dplanestress element summary

Keyword	linquad3dplanestress
Description	3D quadrilateral element for plane stress analysis
Specific parameters	[NIP (in)]
Parameters	NIP: allows to set the number of integration points
Unknowns	Three dofs (u-displacement, v-displacement, w-displacement) are required in each node.
Approximation	Linear approximation of displacements and geometry.
Integration	Integration of membrane strain terms using Gauss integration formula in 1, 4 (default), 9 or 16 integration points. The default number of integration points used can be overloaded using NIP parameter. Reduced integration for shear terms is employed. Shear terms are always integrated using the 1-point integration rule.
Features	Nonlocal constitutive support, Geometric nonlinearity support.
CS properties	cross section thickness is required.
Loads	Body loads are supported. Boundary loads are supported and computed using numerical integration. The side numbering is following. Each i-th element side begins in i-th element node and ends on next element node (i+1-th node or 1-st node, in the case of side number 4). The local positive edge x-axis coincides with side direction, the positive local edge y-axis is rotated 90 degrees anti-clockwise (see fig. (PlaneStress2d element. Node numbering, edge numbering and definition of local edge c.s.(a).)).
Nlgeo	0, 1.
Status	Basic functionality tested, element loads need further testing.
Tests/Examples	sm/patch303.in

QPlaneStress2d

Implementation of quadratic isoparametric eight-node quadrilateral plane-stress finite element. Each node has 2 degrees of freedom. The node numbering is anti-clockwise and is explained in fig. (QPlaneStress2d element - node numbering.). The element features are summarized in Table 8.

Fig. 6 QPlaneStress2d element - node numbering.

Table 8 qplanestress2d element summary

Keyword	qplanestress2d
Description	2D quadratic isoparametric plane stress element
Specific parameters	[NIP (in)]
Parameters	NIP: allows to set the number of integration points
Unknowns	Two dofs (u-displacement, v-displacement) are required in each node.
Approximation	Quadratic approximation of displacements and geometry.
Integration	Full integration using Gauss integration formula in 4 (the default), 9 or 16 integration points. The default number of integration points used can be overloaded using NIP parameter.
Features	Adaptivity support.
CS properties	Cross section thickness is required.
Loads	Body and boundary loads are supported.
Nlgeo	0, 1.
Status	Stable
Tests/Examples	sm/deadweight01.in

TrPlaneStress2d

Implements an triangular three-node constant strain plane-stress finite element. Each node has 2 degrees of freedom. The node numbering is anti-clockwise. The element features are summarized in Table 9.

Fig. 7 TrPlaneStress2d element - node and side numbering.

Table 9 trplanestress2d element summary

Keyword	trplanestress2d
Description	2D linear triangular isoparametric plane stress element
Unknowns	Two dofs (u-displacement, v-displacement) are required in each node.
Approximation	Linear approximation of displacements and geometry.
Integration	Integration of membrane strain terms using one point gauss integration formula.
Features	Nonlocal constitutive support, Edge load support, Geometric nonlinearity support, Adaptivity support.
CS properties	Cross section thickness is required.
Loads	Body loads are supported. Boundary loads are supported and are computed using numerical integration. The side numbering is following. Each i-th element side begins in i-th element node and ends on next element node (i+1-th node or 1-st node, in the case of side number 3). The local positive edge x-axis coincides with side direction, the positive local edge y-axis is rotated 90 degrees anti-clockwise (see fig. (TrPlaneStress2d element - node and side numbering.)).
Nlgeo	0, 1.
Status	Reliable
Tests/Examples	sm/deadweight03.in, sm/adapt01.in, sm/patch104.in, sm/setprops01.in, sm/patch130.in, sm/patch102.in, sm/distancebasedaveraging.in, sm/stressbasedaveraging.in, sm/layeredcs01.in

QTrPlStr

Implementation of quadratic six-node plane-stress finite element. Each node has 2 degrees of freedom. Node numbering is anti-clockwise and is shown in fig. (QTrPlStr element - node and side numbering.). The element features are summarized in Table Table 10.

Fig. 8 QTrPlStr element - node and side numbering.

Table 10 qtrplstr element summary

Keyword	qtrplstr
Description	2D quadratic triangular plane stress element
Specific parameters	[NIP (in)]
Parameters	NIP: allows to set the number of integration points
Unknowns	Two dofs (u-displacement, v-displacement) are required in each node.
Approximation	Quadratic approximation of displacements and geometry.
Integration	Full integration using gauss integration formula in 4 points (the default) or in 7 points (using NIP parameter).
Features	Adaptivity support (error indicator).
CS properties	Cross section thickness is required.
Loads	Boundary loads are supported.
Nlgeo	0, 1.
Status
Tests/Examples	sm/patch140.in (and 1 more)

TrPlaneStrRot

Implementation of triangular three-node plane-stress finite element with independent rotation field. Each node has 3 degrees of freedom. The element features are summarized in Table 11.

The generalization of this element, that can be positioned arbitrarily in space is trplanestrrot3d element. This element requires 6 degrees of freedon in each node. The element features are summarized in Table 12.

The implementation is based on the following paper: Ibrahimbegovic, A., Taylor, R.L., Wilson, E. L.: A robust membrane qudritelar element with rotational degrees of freedom, Int. J. Num. Meth. Engng., 30, 445-457, 1990. The rotation field is defined as \(\omega = \del{1}{2}(\der{v}{x}-\der{u}{y}) = \nabla_u\mbf{u}\). The following form of potential energy functial is assumed:

\[\Pi = \del{1}{2}\int_{\Omega}\mbf{\sigma}^T\mbf{\varepsilon}\ d\Omega + \int_{\Omega}\mbf{\tau}^T(\nabla_u\mbf{u}-\omega)\ d\Omega - \int_\Omega \mbf{X}^T\mbf{u}\ d\Omega\]

where \(\mbf{\tau}\) is pseudo-stress (component of anti-symmetric stress tensor) working on dislocation \((\nabla_u\mbf{u}-\omega)\); the following constitutive relation foris assumed: \(\mbf{\tau} = G(\nabla_u\mbf{u}-\omega)\), where \(G\) is elasticity modulus in shear.

Table 11 trplanestrrot element summary

Keyword	trplanestrrot
Description	2D linear triangular plane stress element with rotational DOFs
Specific parameters	[NIP (in)] [NIPRot (in)]
Parameters	NIP: allows to set the number of integration points for integration of membrane terms. NIPRot: allows to set the number of integration points for integration of terms associated to rotational field.
Unknowns	Three dofs (u-displacement, v-displacement, z-rotation) are required in each node.
Approximation	Linear approximation of displacements and geometry.
Integration	Integration of membrane strain terms using gauss integration formula in 4 points (default) or using 1 or 7 points (using NIP parameter). Integration of strains associated with rotational field integration using 1 point is default (4 and 7 points rules can be specified using NIPRot parameter).
Features
CS properties	Cross section thickness is required.
Loads
Nlgeo
Status
Tests/Examples	sm/patch150.in

Table 12 trplanestrrot3d element summary

Keyword	trplanestrrot3d
Description	3D linear triangular plane stress element with rotational DOFs
Specific parameters	[NIP (in)] [NIPRot (in)]
Parameters	NIP: allows to set the number of integration points for integration of membrane terms. NIPRot: allows to set the number of integration points for integration of terms associated to rotational field.
Unknowns	Six dofs (u-displacement, v-displacement, w-displacement, x-rotation, y-rotation, z-rotation) are required in each node.
Approximation	Linear approximation of displacements and geometry.
Integration	Integration of membrane strain terms using gauss integration formula in 4 points (default) or using 1 or 7 points (using NIP parameter). Integration of strains associated with rotational field integration using 1 point is default (4 and 7 points rules can be specified using NIPRot parameter).
Features
CS properties	Cross section thickness is required.
Loads
Nlgeo
Status

TrPlaneStressRotAllman

Implementation of triangular three-node plane-stress with nodal rotations. Each node has 3 degrees of freedom. The element features are summarized in Table Table 13.

The generalization of this element, that can be positioned arbitrarily in space is trplanestressrotallman3d element. This element requires 6 degrees of freedon in each node. The element features are summarized in Table Table 14.

The implementation is based on the following paper: Allman, D.J.: A compatible triangular element including vertex rotations for plane elasticity analysis, Computers & Structures, vol. 19, no. 1-2, pp. 1-8, 1984. The element is based on plane stress element with quadratic interpolation. The displacements in midside nodes are expressed using vertex displacements and vertex rotations (for edge normal displacement component); the tangential component is interpolated from vertex values. For particular element side starting at i-th vertex and ending in j-th vertex the normal and tangential displacements at edge midpoint can be expressed as

\[\begin{split}u_n\vert_{l/2} &=& \del{u_{ni}+u_{nj}}{2}+\del{l}{8}(\omega_i-\omega_j)\\ u_t\vert_{l/2} &=& \del{u_{ti}+u_{tj}}{2}\end{split}\]

where \(l\) is edge length. This allows to express global displacements in element midside nodes using vertex displacements and rotations. For a single edge, one obtains:

\[\begin{split}u\vert_{l/2} &=&-\del{u_{ni}+u_{nj}}{2}+\del{l}{8}(\omega_i-\omega_j)\del{\Delta y_{ji}}{l}+(\del{u_{t1}+u_{t2}}{2})\del{\Delta x_{ji}}{l}\\ v\vert_{l/2} &=& \del{u_{ni}+u_{nj}}{2}+\del{l}{8}(\omega_i-\omega_j)\del{\Delta x_{ji}}{l}+(\del{u_{t1}+u_{t2}}{2})\del{\Delta y_{ji}}{l}\\\end{split}\]

Table 13 trplanestressrotallman element summary

Keyword	trplanestressrotallman
Description	2D linear triangular plane stress element with rotational DOFs
Unknowns	Three dofs (u-displacement, v-displacement, z-rotation) are required in each node.
Approximation	Linear approximation of geometry, quadratic interpolation of displacements.
Integration	Integration of membrane strain terms using gauss integration formula in 4 points.
Zero energy mode	The zero energy mode (equal rotations) is handled by adding additional energy term preventing spurious modes.
Features
CS properties	Cross section thickness is required.
Loads
Nlgeo
Status
Tests/Examples	sm/patch108.in, sm/beam44.in

Table 14 trplanestressrotallman3d element summary

Keyword	trplanestressrotallman3d
Description	2D linear triangular plane stress element with rotational DOFs
Unknowns	Six dofs (D_u, D_v, D_w, R_x, R_y, R_z) are required in each node.
Approximation	Linear approximation of geometry, quadratic interpolation of displacements.
Integration	Integration of membrane strain terms using gauss integration formula in 4 points.
Zero energy mode	The zero energy mode (equal rotations) is handled by adding additional energy term preventing spurious modes.
Features
CS properties	Cross section thickness is required.
Loads
Nlgeo
Status