beam2e¶

Purpose:

Compute element stiffness matrix for a two-dimensional beam element.

Syntax:

Ke = beam2e(ex, ey, ep)
[Ke, fe] = beam2e(ex, ey, ep, eq)

Description:

beam2e provides the global element stiffness matrix \({\mathbf{K}}^e\) for a two-dimensional beam element.

The input variables:

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

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

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

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

contains the distributed loads 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:

\[\begin{split}\bar{\mathbf{K}}^e = \begin{bmatrix} \frac{D_{EA}}{L} & 0 & 0 & -\frac{D_{EA}}{L} & 0 & 0 \\ 0 & \frac{12D_{EI}}{L^3} & \frac{6D_{EI}}{L^2} & 0 & -\frac{12D_{EI}}{L^3} & \frac{6D_{EI}}{L^2} \\ 0 & \frac{6D_{EI}}{L^2} & \frac{4D_{EI}}{L} & 0 & -\frac{6D_{EI}}{L^2} & \frac{2D_{EI}}{L} \\ -\frac{D_{EA}}{L} & 0 & 0 & \frac{D_{EA}}{L} & 0 & 0 \\ 0 & -\frac{12D_{EI}}{L^3} & -\frac{6D_{EI}}{L^2} & 0 & \frac{12D_{EI}}{L^3} & -\frac{6D_{EI}}{L^2} \\ 0 & \frac{6D_{EI}}{L^2} & \frac{2D_{EI}}{L} & 0 & -\frac{6D_{EI}}{L^2} & \frac{4D_{EI}}{L} \end{bmatrix}\end{split}\]

\[\begin{split}\mathbf{G} = \begin{bmatrix} n_{x\bar{x}} & n_{y\bar{x}} & 0 & 0 & 0 & 0 \\ n_{x\bar{y}} & n_{y\bar{y}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} & 0 \\ 0 & 0 & 0 & n_{x\bar{y}} & n_{y\bar{y}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

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:

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

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:

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