beam2weΒΆ

Purpose:

Compute element stiffness matrix for a two dimensional beam element on elastic support.

Two dimensional beam element with elastic support
Syntax:

Ke = beam2we(ex, ey, ep)
[Ke, fe] = beam2we(ex, ey, ep, eq)
Description:

beam2we provides the global element stiffness matrix \({\mathbf{K}}^e\) for a two dimensional beam element with elastic support.

The input variables

ex\(= [x_1 \;\; x_2]\) \(\qquad\) ey\(= [y_1 \;\; y_2]\) \(\qquad\) ep\(= [E\;\; A\;\; I\;\; k_{\bar{x}}\;\; k_{\bar{y}}]\)

supply the element nodal coordinates \(x_1\), \(x_2\), \(y_1\), and \(y_2\), the modulus of elasticity \(E\), the cross section area \(A\), the moment of inertia \(I\), the spring stiffness in the axial direction \(k_{\bar{x}}\), and the spring stiffness in the transverse direction \(k_{\bar{y}}\).

The element load vector \({\mathbf{f}}_l^e\) can also be computed if uniformly distributed loads are applied to the element. The optional input variable

\[\text{eq} = [q_{\bar{x}}\;\; q_{\bar{y}}]\]

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

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

\[\bar{\mathbf{K}}^e = \bar{\mathbf{K}}^e_0 + \bar{\mathbf{K}}^e_s\]
\[\begin{split}\bar{\mathbf{K}}^e_0 = \begin{bmatrix} \frac{D_{EA}}{L} & 0 & 0 & -\frac{D_{EA}}{L} & 0 & 0 \\ 0 & \frac{12 D_{EI}}{L^3} & \frac{6 D_{EI}}{L^2} & 0 & -\frac{12 D_{EI}}{L^3} & \frac{6 D_{EI}}{L^2} \\ 0 & \frac{6 D_{EI}}{L^2} & \frac{4 D_{EI}}{L} & 0 & -\frac{6 D_{EI}}{L^2} & \frac{2 D_{EI}}{L} \\ -\frac{D_{EA}}{L} & 0 & 0 & \frac{D_{EA}}{L} & 0 & 0 \\ 0 & -\frac{12 D_{EI}}{L^3} & -\frac{6 D_{EI}}{L^2} & 0 & \frac{12 D_{EI}}{L^3} & -\frac{6 D_{EI}}{L^2} \\ 0 & \frac{6 D_{EI}}{L^2} & \frac{2 D_{EI}}{L} & 0 & -\frac{6 D_{EI}}{L^2} & \frac{4 D_{EI}}{L} \end{bmatrix}\end{split}\]
\[\begin{split}\bar{\mathbf{K}}^e_s = \frac{L}{420} \begin{bmatrix} 140k_{\bar{x}} & 0 & 0 & 70k_{\bar{x}} & 0 & 0 \\ 0 & 156k_{\bar{y}} & 22k_{\bar{y}}L & 0 & 54k_{\bar{y}} & -13k_{\bar{y}}L \\ 0 & 22k_{\bar{y}}L & 4k_{\bar{y}}L^2 & 0 & 13k_{\bar{y}}L & -3k_{\bar{y}}L^2 \\ 70k_{\bar{x}} & 0 & 0 & 140k_{\bar{x}} & 0 & 0 \\ 0 & 54k_{\bar{y}} & 13k_{\bar{y}}L & 0 & 156k_{\bar{y}} & -22k_{\bar{y}}L \\ 0 & -13k_{\bar{y}}L & -3k_{\bar{y}}L^2 & 0 & -22k_{\bar{y}}L & 4k_{\bar{y}}L^2 \end{bmatrix}\end{split}\]
\[\begin{split}\mathbf{G} = \begin{bmatrix} n_{x\bar{x}} & n_{y\bar{x}} & 0 & 0 & 0 & 0 \\ n_{x\bar{y}} & n_{y\bar{y}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} & 0 \\ 0 & 0 & 0 & n_{x\bar{y}} & n_{y\bar{y}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

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

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

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

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

The element loads \(\mathbf{f}^e_l\) stored in the variable fe are computed according to

\[\mathbf{f}^e_l = \mathbf{G}^T \bar{\mathbf{f}}^e_l\]

where

\[\begin{split}\bar{\mathbf{f}}^e_l = \begin{bmatrix} \dfrac{q_{\bar{x}}L}{2} \\ \dfrac{q_{\bar{y}}L}{2} \\ \dfrac{q_{\bar{y}}L^2}{12} \\ \dfrac{q_{\bar{x}}L}{2} \\ \dfrac{q_{\bar{y}}L}{2} \\ -\dfrac{q_{\bar{y}}L^2}{12} \end{bmatrix}\end{split}\]