bar1we¶

Purpose:

Compute element stiffness matrix for a one dimensional bar element with elastic support.

Syntax:

Ke = bar1we(ex, ep)
[Ke, fe] = bar1we(ex, ep, eq)

Description:

bar1we provides the element stiffness matrix \(\bar{\mathbf{K}}^e\) for a one dimensional bar element with elastic support.

The input variables

ex\(= [x_1\;\; x_2]\) \(\qquad\) ep\(= [E\; A\; k_{\bar{x}}]\)

supply the element nodal coordinates \(x_1\) and \(x_2\), the modulus of elasticity \(E\), the cross section area \(A\) and the stiffness of the axial springs \(k_{\bar{x}}\).

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, \(q_{\bar{x}}\).

Bar element with distributed load

Theory:

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

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

where

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

\[\begin{split}\bar{\mathbf{K}}^e_s = k_{\bar{x}} L \begin{bmatrix} \frac{1}{3} & \frac{1}{6} \\ \frac{1}{6} & \frac{1}{3} \end{bmatrix}\end{split}\]

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

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

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

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