bar3e¶

Purpose:

Compute element stiffness matrix for a three dimensional bar element.

Syntax:

Ke = bar3e(ex, ey, ez, ep)
[Ke, fe] = bar3e(ex, ey, ez, ep, eq)

Description:

bar3e provides the global element stiffness matrix \({\mathbf{K}}^e\) for a three dimensional bar element.

The input variables

ex\(= [x_1 \;\; x_2]\) \(\qquad\) ey\(= [y_1 \;\; y_2]\) \(\qquad\) ez\(= [z_1 \;\; z_2]\) \(\qquad\) ep\(= [E \; A]\)

supply the element nodal coordinates \(x_1\), \(y_1\), \(z_1\), \(x_2\), \(y_2\), and \(z_2\), the modulus of elasticity \(E\), and the cross section area \(A\).

The element load vector fe can also be computed if a uniformly distributed axial load is applied to the element. The optional input variable

eq\(= [q_{\bar{x}}]\)

contains the distributed load per unit length, \(q_{\bar{x}}\).

Theory:

The 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

\[\begin{split}\bar{\mathbf{K}}^e = \frac{D_{EA}}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \qquad \mathbf{G} = \begin{bmatrix} n_{x\bar{x}} & n_{y\bar{x}} & n_{z\bar{x}} & 0 & 0 & 0 \\ 0 & 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} & n_{z\bar{x}} \end{bmatrix}\end{split}\]

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

\[D_{EA} = EA \qquad L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

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

\[n_{x\bar{x}} = \frac{x_2 - x_1}{L} \qquad n_{y\bar{x}} = \frac{y_2 - y_1}{L} \qquad n_{z\bar{x}} = \frac{z_2 - z_1}{L}\]

The element load vector \(\mathbf{f}_l^e\), stored in fe, is computed according to

\[\mathbf{f}_l^e = \mathbf{G}^T \; \bar{\mathbf{f}}_l^e\]

where

\[\begin{split}\bar{\mathbf{f}}_l^e = \frac{q_{\bar{x}} L}{2} \begin{bmatrix} 1 \\ 1 \end{bmatrix}\end{split}\]