beam2de¶

Purpose:

Compute element stiffness, mass and damping matrices for a two dimensional beam element.

Syntax:

[Ke, Me] = beam2de(ex, ey, ep)
[Ke, Me, Ce] = beam2de(ex, ey, ep)

Description:

beam2de provides the global element stiffness matrix \({\mathbf{K}}^e\), the global element mass matrix \({\mathbf{M}}^e\), and the global element damping matrix \({\mathbf{C}}^e\), for a two dimensional beam element.

The input variables ex and ey are described in beam2e, and

ep\(= [ E,\; A,\; I,\; m,\; [a_0,\; a_1] ]\)

contains the modulus of elasticity \(E\), the cross section area \(A\), the moment of inertia \(I\), the mass per unit length \(m\), and the Rayleigh damping coefficients \(a_0\) and \(a_1\). If \(a_0\) and \(a_1\) are omitted, the element damping matrix Ce is not computed.

Theory:

The element stiffness matrix \(\mathbf{K}^e\), the element mass matrix \(\mathbf{M}^e\) and the element damping matrix \(\mathbf{C}^e\), stored in the variables Ke, Me and Ce, respectively, are computed according to

\[\mathbf{K}^e = \mathbf{G}^T \bar{\mathbf{K}}^e \mathbf{G} \qquad \mathbf{M}^e = \mathbf{G}^T \bar{\mathbf{M}}^e \mathbf{G} \qquad \mathbf{C}^e = \mathbf{G}^T \bar{\mathbf{C}}^e \mathbf{G}\]

where \(\mathbf{G}\) and \(\bar{\mathbf{K}}^e\) are described in beam2e.

The matrix \(\bar{\mathbf{M}}^e\) is given by

\[\begin{split}\bar{\mathbf{M}}^e = \frac{mL}{420} \begin{bmatrix} 140 & 0 & 0 & 70 & 0 & 0 \\ 0 & 156 & 22L & 0 & 54 & -13L \\ 0 & 22L & 4L^2 & 0 & 13L & -3L^2 \\ 70 & 0 & 0 & 140 & 0 & 0 \\ 0 & 54 & 13L & 0 & 156 & -22L \\ 0 & -13L & -3L^2 & 0 & -22L & 4L^2 \end{bmatrix}\end{split}\]

and the matrix \(\bar{\mathbf{C}}^e\) is computed by combining \(\bar{\mathbf{K}}^e\) and \(\bar{\mathbf{M}}^e\):

\[\bar{\mathbf{C}}^e = a_0 \bar{\mathbf{M}}^e + a_1 \bar{\mathbf{K}}^e\]