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 :numref:`planestress2dsummary`. The generalization of this element, that can be positioned arbitrarily in space is :param:`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 :numref:`linquad3dplanestresssummary`. .. _Planestress2dfig: .. tikz:: PlaneStress2d element. Node numbering, edge numbering and definition of local edge c.s.(a). \begin{tikzpicture}[scale=6,>=stealth] \tikzstyle{elemnode} = [draw,circle,inner sep=1,fill=white] \newcommand{\lcoordsys}[2]{ \begin{scope}[transform canvas={shift={#2},scale=0.5,rotate=#1}] \draw[->] (0,0.05) ++(-0.02,0) -- ++(0.2,0) node[above] {$x$}; \draw[->] (0,0.05) ++(0,-0.02) -- ++(0,0.1) node[right] {$y$}; \end{scope} } \draw[->] (-0.05,0) -- (0.8,0) node[above] {$x_g$}; \draw[->] (0,-0.05) -- (0,0.5) node[right] {$y_g$}; \draw[thick,xshift=-2] (0.2,0.1) -- (0.7,0.15) coordinate[midway] (e1) node[below,midway,blue] {1} node[elemnode] {} node[below right] {2} -- (0.8,0.45) coordinate[midway] (e2) node[right,midway,blue] {2} node[elemnode] {} node[above] {3} -- (0.3,0.5) coordinate[midway] (e3) node[above,midway,blue] {3} node[elemnode] {} node[above] {4} -- (0.2,0.1) coordinate[midway] (e4) node[left,midway,blue] {4} node[elemnode] {} node[below] {1}; \lcoordsys{ 7}{(e1)}; \lcoordsys{ 70}{(e2)}; \lcoordsys{175}{(e3)}; \lcoordsys{255}{(e4)}; \end{tikzpicture} .. table:: planestress2d element summary :name: planestress2dsummary +--------------------------+----------------------------------------------------------------------------------------------+ | Keyword | planestress2d | +--------------------------+----------------------------------------------------------------------------------------------+ | Description | 2D quadrilateral element for plane stress analysis | +--------------------------+----------------------------------------------------------------------------------------------+ | Specific parameters | :optelemparam:`NIP{in}` | +--------------------------+----------------------------------------------------------------------------------------------+ | Parameters | :param:`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 | | | :param:`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. (:ref:`Planestress2dfig`)). | +--------------------------+----------------------------------------------------------------------------------------------+ | 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:: linquad3dplanestress element summary :name: linquad3dplanestresssummary +--------------------------+----------------------------------------------------------------------------------------------+ | Keyword | linquad3dplanestress | +--------------------------+----------------------------------------------------------------------------------------------+ | Description | 3D quadrilateral element for plane stress analysis | +--------------------------+----------------------------------------------------------------------------------------------+ | Specific parameters | :optelemparam:`NIP{in}` | +--------------------------+----------------------------------------------------------------------------------------------+ | Parameters | :param:`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 | | | :param:`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. (:ref:`Planestress2dfig`)). | +--------------------------+----------------------------------------------------------------------------------------------+ | 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. (:ref:`qplanstrssfig`). The element features are summarized in :numref:`qplanestress2dsummary`. .. _qplanstrssfig: .. tikz:: QPlaneStress2d element - node numbering. \begin{tikzpicture}[scale=6,>=stealth] \tikzstyle{elemnode} = [draw,circle,inner sep=1,fill=white] \newcommand{\lcoordsys}[2]{ \begin{scope}[transform canvas={shift={#2},scale=0.5,rotate=#1}] \draw[->] (0,0.05) ++(-0.02,0) -- ++(0.2,0) node[above] {$x$}; \draw[->] (0,0.05) ++(0,-0.02) -- ++(0,0.1) node[right] {$y$}; \end{scope} } \draw[->] (-0.05,0) -- (0.8,0) node[above] {$x_g$}; \draw[->] (0,-0.05) -- (0,0.5) node[right] {$y_g$}; \draw[thick,xshift=-2] (0.2,0.1) node[elemnode] {} node[below] {1} to[out=-10,in=210] coordinate[midway] (e1) node[elemnode,midway] {} node[below,midway] {5} (0.7,0.15) node[elemnode] {} node[below right] {2} to[out=110,in=-100] coordinate[midway] (e2) node[elemnode,midway] {} node[right,midway] {6} (0.7,0.4) node[elemnode] {} node[above] {3} to[out=190,in=-10] coordinate[midway] (e3) node[elemnode,midway] {} node[above,midway] {7} (0.2,0.4) node[elemnode] {} node[above] {4} to[out=-100,in=100] coordinate[midway] (e4) node[elemnode,midway] {} node[left,midway] {8} (0.2,0.1); \lcoordsys{ 7}{(e1)}; \lcoordsys{ 90}{(e2)}; \lcoordsys{180}{(e3)}; \lcoordsys{275}{(e4)}; \end{tikzpicture} .. table:: qplanestress2d element summary :name: qplanestress2dsummary +--------------------------+----------------------------------------------------------------------------------------------+ | Keyword | qplanestress2d | +--------------------------+----------------------------------------------------------------------------------------------+ | Description | 2D quadratic isoparametric plane stress element | +--------------------------+----------------------------------------------------------------------------------------------+ | Specific parameters | :optelemparam:`NIP{in}` | +--------------------------+----------------------------------------------------------------------------------------------+ | Parameters | :param:`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 :param:`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 :numref:`trplanestress2dsummary`. .. _TrPlanestressfig: .. tikz:: TrPlaneStress2d element - node and side numbering. \begin{tikzpicture}[scale=6,>=stealth] \tikzstyle{elemnode} = [draw,circle,inner sep=1,fill=white] \newcommand{\lcoordsys}[2]{ \begin{scope}[transform canvas={shift={#2},scale=0.5,rotate=#1}] \draw[->] (0,0.05) ++(-0.02,0) -- ++(0.2,0) node[above] {$x$}; \draw[->] (0,0.05) ++(0,-0.02) -- ++(0,0.1) node[right] {$y$}; \end{scope} } \draw[->] (-0.05,0) -- (0.8,0) node[above] {$x_g$}; \draw[->] (0,-0.05) -- (0,0.5) node[right] {$y_g$}; \draw[thick] (0.2,0.1) node[elemnode] {} node[below] {1} -- (0.7,0.2) node[elemnode] {} node[below] {2} node[blue,midway,below] {1} coordinate[midway] (e1) -- (0.4,0.5) node[elemnode] {} node[above] {3} node[blue,midway,above right] {2} coordinate[midway] (e2) -- (0.2,0.1) node[blue,midway,above left] {3} coordinate[midway] (e3); \lcoordsys{ 12}{(e1)}; \lcoordsys{135}{(e2)}; \lcoordsys{243}{(e3)}; \end{tikzpicture} .. table:: trplanestress2d element summary :name: trplanestress2dsummary +--------------------------+----------------------------------------------------------------------------------------------+ | 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. (:ref:`TrPlanestressfig`)). | +--------------------------+----------------------------------------------------------------------------------------------+ | 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. (:ref:`qtrplanstressfig`). The element features are summarized in Table :numref:`qtrplstrsummary`. .. _qtrplanstressfig: .. tikz:: QTrPlStr element - node and side numbering. \tikzstyle{elemnode} = [draw,circle,inner sep=1,fill=white] \begin{tikzpicture}[scale=6,>=stealth] \draw[->] (-0.05,0) -- (0.8,0) node[above] {$x_g$}; \draw[->] (0,-0.05) -- (0,0.5) node[right] {$y_g$}; \draw[thick,xshift=-2] (0.2,0.1) node[elemnode] {} node[below] {1} to[out=-10,in=210] node[elemnode,midway] {} node[below,midway] {4} node[blue,below,near end] {1} (0.7,0.2) node[elemnode] {} node[below right] {2} to[out=110,in=-20] node[elemnode,midway] {} node[above right,midway] {5} node[blue,above right,near end] {2} (0.4,0.5) node[elemnode] {} node[above] {3} to[out=-90,in=40] node[elemnode,midway] {} node[above left,midway] {6} node[blue,above left,near end] {3} (0.2,0.1); \end{tikzpicture} .. table:: qtrplstr element summary :name: qtrplstrsummary +--------------------------+----------------------------------------------------------------------------------------------+ | Keyword | qtrplstr | +--------------------------+----------------------------------------------------------------------------------------------+ | Description | 2D quadratic triangular plane stress element | +--------------------------+----------------------------------------------------------------------------------------------+ | Specific parameters | :optelemparam:`NIP{in}` | +--------------------------+----------------------------------------------------------------------------------------------+ | Parameters | :param:`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 :param:`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 :numref:`trplanestrrotsummary`. The generalization of this element, that can be positioned arbitrarily in space is :param:`trplanestrrot3d` element. This element requires 6 degrees of freedon in each node. The element features are summarized in :numref:`trplanestrrot3dsummary`. 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 :math:`\omega = \del{1}{2}(\der{v}{x}-\der{u}{y}) = \nabla_u\mbf{u}`. The following form of potential energy functial is assumed: .. math:: \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 :math:`\mbf{\tau}` is pseudo-stress (component of anti-symmetric stress tensor) working on dislocation :math:`(\nabla_u\mbf{u}-\omega)`; the following constitutive relation foris assumed: :math:`\mbf{\tau} = G(\nabla_u\mbf{u}-\omega)`, where :math:`G` is elasticity modulus in shear. .. table:: trplanestrrot element summary :name: trplanestrrotsummary +--------------------------+----------------------------------------------------------------------------------------------+ | Keyword | trplanestrrot | +--------------------------+----------------------------------------------------------------------------------------------+ | Description | 2D linear triangular plane stress element with rotational DOFs | +--------------------------+----------------------------------------------------------------------------------------------+ | Specific parameters | :optelemparam:`NIP{in}` :optelemparam:`NIPRot{in}` | +--------------------------+----------------------------------------------------------------------------------------------+ | Parameters | :param:`NIP`: allows to set the number of integration points for integration of membrane | | | terms. :param:`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 :param:`NIP` parameter). Integration of strains associated | | | with rotational field integration using 1 point is default (4 and 7 points rules can be | | | specified using :param:`NIPRot` parameter). | +--------------------------+----------------------------------------------------------------------------------------------+ | Features | - | +--------------------------+----------------------------------------------------------------------------------------------+ | CS properties | Cross section thickness is required. | +--------------------------+----------------------------------------------------------------------------------------------+ | Loads | - | +--------------------------+----------------------------------------------------------------------------------------------+ | Nlgeo | 0. | +--------------------------+----------------------------------------------------------------------------------------------+ | Status | - | +--------------------------+----------------------------------------------------------------------------------------------+ | Tests/Examples | `sm/patch150.in | | | `_ | +--------------------------+----------------------------------------------------------------------------------------------+ .. table:: trplanestrrot3d element summary :name: trplanestrrot3dsummary +--------------------------+----------------------------------------------------------------------------------------------+ | Keyword | trplanestrrot3d | +--------------------------+----------------------------------------------------------------------------------------------+ | Description | 3D linear triangular plane stress element with rotational DOFs | +--------------------------+----------------------------------------------------------------------------------------------+ | Specific parameters | :optelemparam:`NIP{in}` :optelemparam:`NIPRot{in}` | +--------------------------+----------------------------------------------------------------------------------------------+ | Parameters | :param:`NIP`: allows to set the number of integration points for integration of membrane | | | terms. :param:`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 :param:`NIP` parameter). Integration of strains associated | | | with rotational field integration using 1 point is default (4 and 7 points rules can be | | | specified using :param:`NIPRot` parameter). | +--------------------------+----------------------------------------------------------------------------------------------+ | Features | - | +--------------------------+----------------------------------------------------------------------------------------------+ | CS properties | Cross section thickness is required. | +--------------------------+----------------------------------------------------------------------------------------------+ | Loads | - | +--------------------------+----------------------------------------------------------------------------------------------+ | Nlgeo | 0. | +--------------------------+----------------------------------------------------------------------------------------------+ | 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 :numref:`TrPlaneStressRotAllmansummary`. The generalization of this element, that can be positioned arbitrarily in space is :param:`trplanestressrotallman3d` element. This element requires 6 degrees of freedon in each node. The element features are summarized in Table :numref:`trplanestressrotallman3dsummary`. 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 .. math:: 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} where :math:`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: .. math:: 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}\\ .. table:: trplanestressrotallman element summary :name: TrPlaneStressRotAllmansummary +--------------------------+----------------------------------------------------------------------------------------------+ | 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 | 0. | +--------------------------+----------------------------------------------------------------------------------------------+ | Status | - | +--------------------------+----------------------------------------------------------------------------------------------+ | Tests/Examples | `sm/patch108.in | | | `_, `sm/beam44.in | | | `_ | +--------------------------+----------------------------------------------------------------------------------------------+ .. table:: trplanestressrotallman3d element summary :name: trplanestressrotallman3dsummary +--------------------------+----------------------------------------------------------------------------------------------+ | 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 | 0. | +--------------------------+----------------------------------------------------------------------------------------------+ | Status | - | +--------------------------+----------------------------------------------------------------------------------------------+