Axisymmetric Elements¶

Implementation relies on elements located exclusively in \(x,y\) plane. The coordinate \(x\) corresponds to radius, \(y\) is the axis of rotation. Approximation of displacement functions \(u,v\) is carried out on a particular finite element. Nonzero strains read

\[\begin{split}\varepsilon_{x}=\varepsilon_{r}&=&\frac{\partial u}{\partial x}\\ \varepsilon_{y}=\varepsilon_{z}&=&\frac{\partial v}{\partial y}\\ \varepsilon_{\theta}&=&\frac{u}{r}\\ \gamma_{xy}=\gamma_{rz}&=&\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\end{split}\]

Stress components can be computed from elasticity matrix. Note that this matrix corresponds to a submatrix of the full 3D elasticity matrix.

\[\begin{split}\left\{\begin{array}{c} \sigma_x \\ \sigma_y \\ \sigma_\theta \\ \sigma_{xy} \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)}\left[ \begin{array}{cccc} 1-\nu & \nu & \nu & 0 \\ \nu & 1-\nu & \nu & 0 \\ \nu & \nu & 1-\nu & 0 \\ 0 & 0 & 0 & (1-2\nu)/2 \end{array} \right] = \left\{\begin{array}{c} \varepsilon_x \\ \varepsilon_y \\ \varepsilon_\theta \\ \gamma_{xy} \end{array} \right\}\end{split}\]

In OOFEM, the strain vector is arranged as \(\{\varepsilon_{x}, \varepsilon_{y}, \varepsilon_{\theta}, 0, 0, \gamma_{xy}\}^T\) and the stress vector \(\{\sigma_{x}, \sigma_{y}, \sigma_{\theta}, 0, 0, \tau_{xy}\}^T\). Implementation assumes a segment of 1 rad.

Axisymm3d element¶

Implementation of triangular three-node finite element for axisymmetric continuum. Each node has 2 degrees of freedom. Node numbering and edge position is the same as in Fig. TrPlaneStress2d element - node and side numbering.. The element features are summarized in Table Table 29.

Table 29 Axisymm3d element summary¶
Keyword	Axisymm3d
Description	Triangular axisymmetric linear element
Specific parameters	[`NIP` (in)] [`NIPfish` (in)]
Parameters	`NIP`: allows to set the number of integration points (possible completions are 1 (default), 4 and 7 point integration rule).
Unknowns	Two dofs (u-displacement, v-displacement) are required in each node.
Approximation	Linear approximation of displacement and geometry.
Integration	The integration can be altered using `NIP` paramter (default is 1 point integration).
Features
CS properties
Loads	Boundary and body loads are supported.
Nlgeo
Status
Tests/Examples	sm/axisymm06.in, sm/axisymm04.in, sm/axisymm05.in

Q4axisymm element¶

Implementation of quadratic isoparametric eight-node quadrilateral - finite element for axisymmetric 3d continuum. Each node has 2 degrees of freedom. The element features are summarized in Table 30.

Table 30 Q4axisymm element summary¶
Keyword	Q4axisymm
Description	Quadratic isoparametric eight-node quadrilateral for axisymmetric analysis
Specific parameters	[`NIP` (in)] [`NIPfish` (in)]
Parameters	`NIP`: allows to set the number of integration points for integration of terms corresponding to \(\varepsilon_x\) and \(\varepsilon_y\) strains (possible completions are 1, 4 (default), 9, and 16). `NIPfish`: allows to set the number of integration points for integration of remain terms (corresponding to \(\varepsilon_\theta\) and \(\gamma_{rz}\)) (Supported values include 1 (default), 4, 9, and 16 integration point formula).
Unknowns	Two dofs (u-displacement, v-displacement) are required in each node.
Approximation	Quadratic approximation of displacement and geometry.
Integration	The integration of terms corresponding to \(\varepsilon_x\) and \(\varepsilon_y\) strains can be altered using `NIP` parameter (default is 4 point formula). The remaining terms (creesponding to \(\varepsilon_\theta\) and \(\gamma_{rz}\)) are integrated by default using 1 point formula (see `NIPfish` parameter).
Features
CS properties
Loads	No boundary and body loads are supported.
Nlgeo
Status

L4axisymm element¶

Implementation of isoparametric four-node quadrilateral axisymmetric finite element with linear interpolations of displacements \(u, v\). Node numbering and edge position is the same as in Fig. PlaneStress2d element. Node numbering, edge numbering and definition of local edge c.s.(a).. The element features are summarized in Table 31.

Table 31 L4axisymm element summary¶
Keyword	L4axisymm
Description	Isoparametric four-node quadrilateral element for axisymmetric analysis
Specific parameters	[`NIP` (in)]
Parameters	`NIP`: allows to set the number of integration points for integration of terms corresponding to \(\varepsilon_x\) and \(\varepsilon_y\) strains (possible completions are 1, 4 (default), 9, and 16).
Unknowns	Two dofs (u-displacement, v-displacement) are required in each node.
Approximation	Linear approximation of displacement and geometry.
Integration	The integration of \(\varepsilon_x\) and \(\varepsilon_y\) strains can be altered using `NIP` parameter (possible completions are 1, 4 (default), 9 or 16 point integration rule). The remaining strain components (\(\varepsilon_\theta\) and \(\gamma_{rz}\)) are integrated using one point integration formula.
Features
CS properties
Loads	Boundary and body loads supported.
Nlgeo
Status
Tests/Examples	sm/axisymm03.in, sm/axisymm01.in, sm/axisymm02.in

Axisymmetric Elements¶