plani8s¶

Purpose:

Compute stresses and strains in an 8 node isoparametric element in plane strain or plane stress.

Syntax:

[es,et,eci]=plani8s(ex,ey,ep,D,ed)

Description:

plani8s computes stresses es and the strains et in an 8 node isoparametric element in plane strain or plane stress.

The input variables \(\mathbf{ex}\), \(\mathbf{ey}\), \(\mathbf{ep}\) and the matrix \(\mathbf{D}\) are defined in plani8e. The vector \(\mathbf{ed}\) contains the nodal displacements \(\mathbf{a}^e\) of the element and is obtained by the function extract as

\[\mathbf{ed} = (\mathbf{a}^e)^T = [\,u_1\;\; u_2\;\; \dots \;\; u_{16}\,]\]

The output variables

\[\begin{split}\mathrm{es} = \boldsymbol{\sigma}^T = \begin{bmatrix} \sigma^1_{xx} & \sigma^1_{yy} & [\sigma^1_{zz}] & \sigma^1_{xy} & [\sigma^1_{xz}] & [\sigma^1_{yz}] \\ \sigma^2_{xx} & \sigma^2_{yy} & [\sigma^2_{zz}] & \sigma^2_{xy} & [\sigma^2_{xz}] & [\sigma^2_{yz}] \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \sigma^{n^2}_{xx} & \sigma^{n^2}_{yy} & [\sigma^{n^2}_{zz}] & \sigma^{n^2}_{xy} & [\sigma^{n^2}_{xz}] & [\sigma^{n^2}_{yz}] \end{bmatrix}\end{split}\]

\[\begin{split}\mathrm{et} = \boldsymbol{\varepsilon}^T = \begin{bmatrix} \varepsilon^1_{xx} & \varepsilon^1_{yy} & [\varepsilon^1_{zz}] & \gamma^1_{xy} & [\gamma^1_{xz}] & [\gamma^1_{yz}] \\ \varepsilon^2_{xx} & \varepsilon^2_{yy} & [\varepsilon^2_{zz}] & \gamma^2_{xy} & [\gamma^2_{xz}] & [\gamma^2_{yz}] \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \varepsilon^{n^2}_{xx} & \varepsilon^{n^2}_{yy} & [\varepsilon^{n^2}_{zz}] & \gamma^{n^2}_{xy} & [\gamma^{n^2}_{xz}] & [\gamma^{n^2}_{yz}] \end{bmatrix} \qquad \mathbf{eci} = \begin{bmatrix} x_1 & y_1 \\ x_2 & y_2 \\ \vdots & \vdots \\ x_{n^2} & y_{n^2} \end{bmatrix}\end{split}\]

contain the stress and strain components, and the coordinates of the integration points. The index \(n\) denotes the number of integration points used within the element, cf. plani8e. The number of columns in es and et follows the size of D. Note that for plane stress \(\varepsilon_{zz} \neq 0\), and for plane strain \(\sigma_{zz} \neq 0\).

Theory:

The strains and stresses are computed according to

\[\boldsymbol{\varepsilon} = \mathbf{B}^e\,\mathbf{a}^e\]

\[\boldsymbol{\sigma} = \mathbf{D}\;\boldsymbol{\varepsilon}\]

where the matrices \(\mathbf{D}\), \(\mathbf{B}^e\), and \(\mathbf{a}^e\) are described in plani8e, and where the integration points are chosen as evaluation points.