beam2ds¶

Purpose:

Compute section forces for a two dimensional beam element in dynamic analysis.

Syntax:

es = beam2ds(ex, ey, ep, ed, ev, ea)

Description:

beam2ds computes the section forces at the ends of the dynamic beam element beam2de.

The input variables ex, ey and ep are defined in beam2de. The element displacements, velocities, and accelerations, stored in ed, ev, and ea respectively, are obtained by the function extract_ed.

The output variable es contains the section forces at the ends of the beam:

\[\begin{split}es = \begin{bmatrix} N_1 & V_1 & M_1 \\ N_2 & V_2 & M_2 \end{bmatrix}\end{split}\]

Theory:

The section forces at the ends of the beam are obtained from the element force vector:

\[\bar{\mathbf{P}} = \begin{bmatrix} -N_1 & -V_1 & -M_1 & N_2 & V_2 & M_2 \end{bmatrix}^T\]

computed according to:

\[\bar{\mathbf{P}} = \bar{\mathbf{K}}^e \mathbf{G} \mathbf{a}^e + \bar{\mathbf{C}}^e \mathbf{G} \dot{\mathbf{a}}^e + \bar{\mathbf{M}}^e \mathbf{G} \ddot{\mathbf{a}}^e\]

The matrices \(\bar{\mathbf{K}}^e\) and \(\mathbf{G}\) are described in beam2e, and the matrices \(\bar{\mathbf{M}}^e\) and \(\bar{\mathbf{C}}^e\) are described in beam2d.

The nodal displacements:

\[\mathbf{a}^e = \begin{bmatrix} u_1 & u_2 & u_3 & u_4 & u_5 & u_6 \end{bmatrix}^T\]

shown in beam2de also define the directions of the nodal velocities:

\[\dot{\mathbf{a}}^e = \begin{bmatrix} \dot{u}_1 & \dot{u}_2 & \dot{u}_3 & \dot{u}_4 & \dot{u}_5 & \dot{u}_6 \end{bmatrix}^T\]

and the nodal accelerations:

\[\ddot{\mathbf{a}}^e = \begin{bmatrix} \ddot{u}_1 & \ddot{u}_2 & \ddot{u}_3 & \ddot{u}_4 & \ddot{u}_5 & \ddot{u}_6 \end{bmatrix}^T\]

Note that the transposes of \(\mathbf{a}^e\), \(\dot{\mathbf{a}}^e\), and \(\ddot{\mathbf{a}}^e\) are stored in ed, ev, and ea respectively.