plani4s¶

Purpose:

Compute stresses and strains in a 4 node isoparametric element in plane strain or plane stress.

Syntax:

[es,et,eci]=plani4s(ex,ey,ep,D,ed)

Description:

plani4s computes stresses es and the strains et in a 4 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 plani4e. The vector \(\mathbf{ed}\) contains the nodal displacements \(\mathbf{a}^e\) of the element and is obtained by the function extract as

\[\mathbf{ed} = (\mathbf{a}^e)^T = [\,u_1\;\; u_2\;\; \dots \;\; u_8\,]\]

The output variables

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

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

\[\begin{split}\mathbf{eci} = \begin{bmatrix} x_1 & y_1 \\ x_2 & y_2 \\ \vdots & \vdots \\ x_{n^2} & y_{n^2} \end{bmatrix}\end{split}\]

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. plani4e. 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 plani4e, and where the integration points are chosen as evaluation points.