bar2ge¶

Purpose:

Compute element stiffness matrix for a two dimensional bar element with geometric nonlinearity.

Syntax:

Ke = cfc.bar2ge(ex, ey, ep, Qx)

Description:

bar2ge provides the global element stiffness matrix \({\mathbf{K}}^e\) for a two dimensional bar element with geometric nonlinearity.

The input variables

ex\(= [x_1 \;\; x_2]\) \(\qquad\) ey\(= [y_1 \;\; y_2]\) \(\qquad\) ep\(= [E \; A]\)

supply the element nodal coordinates \(x_1\), \(y_1\), \(x_2\), and \(y_2\), the modulus of elasticity \(E\), and the cross section area \(A\).

The input variable

Qx\(= [Q_{\bar{x}}]\)

contains the value of the axial force, which is positive in tension.

Theory:

The global element stiffness matrix \(\mathbf{K}^e\), stored in Ke, is computed according to

\[\mathbf{K}^e = \mathbf{G}^T\,\bar{\mathbf{K}}^e\,\mathbf{G}\]

where \(\bar{\mathbf{K}}^e\) is given by

\[\bar{\mathbf{K}}^e = \bar{\mathbf{K}}^e_0 + \bar{\mathbf{K}}^e_{\sigma}\]

with

\[\begin{split}\bar{\mathbf{K}}^e_0 = \frac{D_{EA}}{L} \begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\end{split}\]

\[\begin{split}\bar{\mathbf{K}}^e_{\sigma} = \frac{Q_{\bar{x}}}{L} \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 \end{bmatrix}\end{split}\]

\[\begin{split}\mathbf{G} = \begin{bmatrix} n_{x\bar{x}} & n_{y\bar{x}} & 0 & 0 \\ n_{x\bar{y}} & n_{y\bar{y}} & 0 & 0 \\ 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} \\ 0 & 0 & n_{x\bar{y}} & n_{y\bar{y}} \end{bmatrix}\end{split}\]

where the axial stiffness \(D_{EA}\) and the length \(L\) are given by

\[D_{EA} = EA \qquad L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

and the transformation matrix \(\mathbf{G}\) contains the direction cosines

\[n_{x\bar{x}} = n_{y\bar{y}} = \frac{x_2 - x_1}{L} \qquad n_{y\bar{x}} = -n_{x\bar{y}} = \frac{y_2 - y_1}{L}\]