plante¶

Purpose:

Compute element matrices for a triangular element in plane strain or plane stress.

Syntax:

Ke = plante(ex, ey, ep, D)
[Ke, fe] = plante(ex, ey, ep, D, eq)

Description:

plante provides an element stiffness matrix Ke and an element load vector fe for a triangular element in plane strain or plane stress.

The element nodal coordinates \(x_1, y_1, x_2, \ldots\) are supplied to the function by \(\mathbf{ex}\) and \(\mathbf{ey}\). The type of analysis ptype and the element thickness t are supplied by \(\mathbf{ep}\),

\[\begin{split}\begin{array}{l} ptype=1 \quad \text{plane stress} \\ ptype=2 \quad \text{plane strain} \end{array}\end{split}\]

and the material properties are supplied by the constitutive matrix D. Any arbitrary \(\mathbf{D}\)-matrix with dimensions from \(3 \times 3\) to \(6 \times 6\) may be given. For an isotropic elastic material the constitutive matrix can be formed by the function hooke, see Section Material functions.

\[\begin{split}\mathbf{ex} = [\,x_1 \;\; x_2 \;\; x_3\,] \\ \mathbf{ey} = [\,y_1 \;\; y_2 \;\; y_3\,] \\ \mathbf{ep} = [\,ptype \;\; t\,]\end{split}\]

\[\begin{split}\mathbf{D} = \begin{bmatrix} D_{11} & D_{12} & D_{13} \\ D_{21} & D_{22} & D_{23} \\ D_{31} & D_{32} & D_{33} \end{bmatrix} \quad \text{or} \quad \mathbf{D} = \begin{bmatrix} D_{11} & D_{12} & D_{13} & D_{14} & [D_{15}] & [D_{16}] \\ D_{21} & D_{22} & D_{23} & D_{24} & [D_{25}] & [D_{26}] \\ D_{31} & D_{32} & D_{33} & D_{34} & [D_{35}] & [D_{36}] \\ D_{41} & D_{42} & D_{43} & D_{44} & [D_{45}] & [D_{46}] \\ [D_{51}] & [D_{52}] & [D_{53}] & [D_{54}] & [D_{55}] & [D_{56}] \\ [D_{61}] & [D_{62}] & [D_{63}] & [D_{64}] & [D_{65}] & [D_{66}] \end{bmatrix}\end{split}\]

If uniformly distributed loads are applied to the element, the element load vector fe is computed. The input variable

\[\begin{split}\text{eq} = \begin{bmatrix} b_x \\ b_y \end{bmatrix}\end{split}\]

containing loads per unit volume, \(b_x\) and \(b_y\), is then given.

Theory:

The element stiffness matrix \(\mathbf{K}^e\) and the element load vector \(\mathbf{f}_l^e\), stored in Ke and fe, respectively, are computed according to

\[\mathbf{K}^e = (\mathbf{C}^{-1})^T \int_A \bar{\mathbf{B}}^T \mathbf{D} \bar{\mathbf{B}}\, t\, dA\, \mathbf{C}^{-1}\]

\[\mathbf{f}_l^e = (\mathbf{C}^{-1})^T \int_A \bar{\mathbf{N}}^T \mathbf{b}\, t\, dA\]

with the constitutive matrix \(\mathbf{D}\) defined by D, and the body force vector \(\mathbf{b}\) defined by eq.

The evaluation of the integrals for the triangular element is based on a linear displacement approximation \(\mathbf{u}(x, y)\) and is expressed in terms of the nodal variables \(u_1, u_2, \ldots, u_6\) as

\[\mathbf{u}(x, y) = \mathbf{N}^e \mathbf{a}^e = \bar{\mathbf{N}}\, \mathbf{C}^{-1} \mathbf{a}^e\]

where

\[\begin{split}\mathbf{u} = \begin{bmatrix} u_x \\ u_y \end{bmatrix} \quad \bar{\mathbf{N}} = \begin{bmatrix} 1 & x & y & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x & y \end{bmatrix}\end{split}\]

\[\begin{split}\mathbf{C} = \begin{bmatrix} 1 & x_1 & y_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x_1 & y_1 \\ 1 & x_2 & y_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x_2 & y_2 \\ 1 & x_3 & y_3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x_3 & y_3 \\ \end{bmatrix} \quad \mathbf{a}^e = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \\ u_6 \end{bmatrix}\end{split}\]

The matrix \(\bar{\mathbf{B}}\) is obtained as

\[\begin{split}\bar{\mathbf{B}} = \tilde{\nabla} \bar{\mathbf{N}} \quad \text{where} \quad \tilde{\nabla} = \begin{bmatrix} \dfrac{\partial}{\partial x} & 0 \\ 0 & \dfrac{\partial}{\partial y} \\ \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial x} \end{bmatrix}\end{split}\]

If a larger \(\mathbf{D}\)-matrix than \(3 \times 3\) is used for plane stress (\(ptype=1\)), the \(\mathbf{D}\)-matrix is reduced to a \(3 \times 3\) matrix by static condensation using \(\sigma_{zz} = \sigma_{xz} = \sigma_{yz} = 0\). These stress components are connected with the rows 3, 5 and 6 in the D-matrix respectively.

If a larger \(\mathbf{D}\)-matrix than \(3 \times 3\) is used for plane strain (\(ptype=2\)), the \(\mathbf{D}\)-matrix is reduced to a \(3 \times 3\) matrix using \(\varepsilon_{zz} = \gamma_{xz} = \gamma_{yz} = 0\). This implies that a \(3 \times 3\) \(\mathbf{D}\)-matrix is created by the rows and the columns 1, 2 and 4 from the original D-matrix.

Evaluation of the integrals for the triangular element yields

\[\mathbf{K}^e = (\mathbf{C}^{-1})^T \bar{\mathbf{B}}^T \mathbf{D} \bar{\mathbf{B}}\, \mathbf{C}^{-1}\, t\, A\]

\[\mathbf{f}_l^e = \frac{A t}{3} \begin{bmatrix} b_x & b_y & b_x & b_y & b_x & b_y \end{bmatrix}^T\]

where the element area \(A\) is determined as

\[\begin{split}A = \frac{1}{2} \det \begin{bmatrix} 1 & x_1 & y_1 \\ 1 & x_2 & y_2 \\ 1 & x_3 & y_3 \end{bmatrix}\end{split}\]