plants¶

Purpose:

Compute stresses and strains in a triangular element in plane strain or plane stress.

Syntax:

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

Description:

plants computes the stresses es and the strains et in a triangular element in plane strain or plane stress.

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

\[\mathbf{ed} = (\mathbf{a}^e)^T = [\, u_1\;\; u_2\;\; \dots \;\; u_6\,]\]

The output variables

\[\mathrm{es} = \boldsymbol{\sigma}^T = \left[\, \sigma_{xx}\; \sigma_{yy}\; [\sigma_{zz}]\; \sigma_{xy}\; [\sigma_{xz}]\; [\sigma_{yz}]\, \right]\]

\[\mathrm{et} = \boldsymbol{\varepsilon}^T = [\, \varepsilon_{xx}\; \varepsilon_{yy}\; [\varepsilon_{zz}]\; \gamma_{xy}\; [\gamma_{xz}]\; [\gamma_{yz}]\,]\]

contain the stress and strain components. The size of 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

\[\boldsymbol{\varepsilon} = \bar{\mathbf{B}}\, \mathbf{C}^{-1}\, \mathbf{a}^e\]

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

where the matrices \(\mathbf{D}\), \(\bar{\mathbf{B}}\), \(\mathbf{C}\) and \(\mathbf{a}^e\) are described in plante. Note that both the strains and the stresses are constant in the element.