bar1e¶

Purpose:

Compute element stiffness matrix for a one dimensional bar element.

Syntax:

Ke = bar1e(ex, ep)
[Ke, fe] = bar1e(ex, ep, eq)

Description:

bar1e provides the element stiffness matrix \(\bar{\mathbf{K}}^e\) for a one dimensional bar element. The input variables

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

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

The element load vector \(\bar{\mathbf{f}}_l^e\) can also be computed if a uniformly distributed 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 \(\bar{\mathbf{K}}^e\), stored in Ke, is computed according to

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

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

\[D_{EA} = EA; \quad L = x_2 - x_1\]

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

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