beam2wsΒΆ

Purpose:

Compute section forces in a two dimensional beam element with elastic support.

_images/beam2s.svg
Syntax:

es = beam2ws(ex, ey, ep, ed)
es = beam2ws(ex, ey, ep, ed, eq)
[es, edi, eci] = beam2ws(ex, ey, ep, ed, eq, n)
Description:

beam2ws computes the section forces and displacements in local directions along the beam element beam2we.

The input variables ex, ey, ep and eq are defined in beam2we, and the element displacements, stored in ed, are obtained by the function extract_ed. If distributed loads are applied to the element, the variable eq must be included. The number of evaluation points for section forces and displacements are determined by n. If n is omitted, only the ends of the beam are evaluated.

The output variables

es\(= \begin{bmatrix} N(0) & V(0) & M(0) \\ N(\bar{x}_2) & V(\bar{x}_2) & M(\bar{x}_2) \\ \vdots & \vdots & \vdots \\ N(\bar{x}_{n-1}) & V(\bar{x}_{n-1}) & M(\bar{x}_{n-1}) \\ N(L) & V(L) & M(L) \end{bmatrix}\) \(\quad\) edi\(= \begin{bmatrix} u(0) & v(0) \\ u(\bar{x}_2) & v(\bar{x}_2) \\ \vdots & \vdots \\ u(\bar{x}_{n-1}) & v(\bar{x}_{n-1}) \\ u(L) & v(L) \end{bmatrix}\) \(\quad\) eci\(= \begin{bmatrix} 0 \\ \bar{x}_2 \\ \vdots \\ \bar{x}_{n-1} \\ L \end{bmatrix}\)

contain the section forces, the displacements, and the evaluation points on the local \(\bar{x}\)-axis. \(L\) is the length of the beam element.

Theory:

The nodal displacements in local coordinates are given by

\[\begin{split}\bar{\mathbf{a}}^e = \begin{bmatrix} \bar{u}_1 \\ \bar{u}_2 \\ \bar{u}_3 \\ \bar{u}_4 \\ \bar{u}_5 \\ \bar{u}_6 \end{bmatrix} = \mathbf{G} \mathbf{a}^e\end{split}\]

where \(\mathbf{G}\) is described in beam2we and the transpose of \(\mathbf{a}^e\) is stored in ed. The displacements associated with bar action and beam action are determined as

\[\begin{split}\bar{\mathbf{a}}^e_{\text{bar}} = \begin{bmatrix} \bar{u}_1 \\ \bar{u}_4 \end{bmatrix} \qquad \bar{\mathbf{a}}^e_{\text{beam}} = \begin{bmatrix} \bar{u}_2 \\ \bar{u}_3 \\ \bar{u}_5 \\ \bar{u}_6 \end{bmatrix}\end{split}\]

The displacement \(u(\bar{x})\) and the normal force \(N(\bar{x})\) are computed from

\[u(\bar{x}) = \mathbf{N}_{\text{bar}} \bar{\mathbf{a}}^e_{\text{bar}} + u_p(\bar{x})\]
\[N(\bar{x}) = D_{EA} \mathbf{B}_{\text{bar}} \bar{\mathbf{a}}^e + N_p(\bar{x})\]

where

\[\mathbf{N}_{\text{bar}} = \begin{bmatrix} 1 & \bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\text{bar}} = \begin{bmatrix} 1-\frac{\bar{x}}{L} & \frac{\bar{x}}{L} \end{bmatrix}\]
\[\mathbf{B}_{\text{bar}} = \begin{bmatrix} 0 & 1 \end{bmatrix} \mathbf{C}^{-1}_{\text{bar}} = \begin{bmatrix} -\frac{1}{L} & \frac{1}{L} \end{bmatrix}\]
\[u_p(\bar{x}) = \frac{k_{\bar{x}}}{D_{EA}} \begin{bmatrix} \frac{\bar{x}^2-L\bar{x}}{2} & \frac{\bar{x}^3-L^2\bar{x}}{6} \end{bmatrix} \mathbf{C}^{-1}_{\text{bar}} \bar{\mathbf{a}}^e_{\text{bar}} - \frac{q_{\bar{x}}}{D_{EA}}\left(\frac{\bar{x}^2}{2}-\frac{L\bar{x}}{2}\right)\]
\[N_p(\bar{x}) = k_{\bar{x}} \begin{bmatrix} \frac{2\bar{x}-L}{2} & \frac{3\bar{x}^2-L^2}{6} \end{bmatrix} \mathbf{C}^{-1}_{\text{bar}} \bar{\mathbf{a}}^e_{\text{bar}} - q_{\bar{x}}\left(\bar{x}-\frac{L}{2}\right)\]

in which \(D_{EA}\), \(k_{\bar{x}}\), \(L\), and \(q_{\bar{x}}\) are defined in beam2we and

\[\begin{split}\mathbf{C}^{-1}_{\text{bar}} = \begin{bmatrix} 1 & 0 \\ -\frac{1}{L} & \frac{1}{L} \end{bmatrix}\end{split}\]

The displacement \(v(\bar{x})\), the bending moment \(M(\bar{x})\) and the shear force \(V(\bar{x})\) are computed from

\[v(\bar{x}) = \mathbf{N}_{\text{beam}} \bar{\mathbf{a}}^e_{\text{beam}} + v_p(\bar{x})\]
\[M(\bar{x}) = D_{EI} \mathbf{B}_{\text{beam}} \bar{\mathbf{a}}^e_{\text{beam}} + M_p(\bar{x})\]
\[V(\bar{x}) = -D_{EI} \frac{d\mathbf{B}_{\text{beam}}}{dx} \bar{\mathbf{a}}^e_{\text{beam}} + V_p(\bar{x})\]

where

\[\mathbf{N}_{\text{beam}} = \begin{bmatrix} 1 & \bar{x} & \bar{x}^2 & \bar{x}^3 \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[\mathbf{B}_{\text{beam}} = \begin{bmatrix} 0 & 0 & 2 & 6\bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[\frac{d\mathbf{B}_{\text{beam}}}{dx} = \begin{bmatrix} 0 & 0 & 0 & 6 \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[\begin{split}v_p(\bar{x}) = -\frac{k_{\bar{y}}}{D_{EI}} \begin{bmatrix} \frac{\bar{x}^4-2L\bar{x}^3+L^2\bar{x}^2}{24} \\ \frac{\bar{x}^5-3L^2\bar{x}^3+2L^3\bar{x}^2}{120} \\ \frac{\bar{x}^6-4L^3\bar{x}^3+3L^4\bar{x}^2}{360} \\ \frac{\bar{x}^7-5L^4\bar{x}^3+4L^5\bar{x}^2}{840} \end{bmatrix}^T \mathbf{C}^{-1}_{\text{beam}} \bar{\mathbf{a}}^e_{\text{beam}} + \frac{q_{\bar{y}}}{D_{EI}}\left(\frac{\bar{x}^4}{24}-\frac{L\bar{x}^3}{12}+\frac{L^2\bar{x}^2}{24}\right)\end{split}\]
\[\begin{split}M_p(\bar{x}) = -k_{\bar{y}} \begin{bmatrix} \frac{6\bar{x}^2-6L\bar{x}+L^2}{12} \\ \frac{10\bar{x}^3-9L^2\bar{x}+2L^3}{60} \\ \frac{5\bar{x}^4-4L^3\bar{x}+L^4}{60} \\ \frac{21\bar{x}^5-15L^4\bar{x}+4L^5}{420} \end{bmatrix}^T \mathbf{C}^{-1}_{\text{beam}} \bar{\mathbf{a}}^e_{\text{beam}} + q_{\bar{y}}\left(\frac{\bar{x}^2}{2}-\frac{L\bar{x}}{2}+\frac{L^2}{12}\right)\end{split}\]
\[\begin{split}V_p(\bar{x}) = k_{\bar{y}} \begin{bmatrix} \frac{2\bar{x}-L}{2} \\ \frac{10\bar{x}^2-3L^2}{20} \\ \frac{5\bar{x}^3-L^3}{15} \\ \frac{7\bar{x}^4-L^4}{28} \end{bmatrix}^T \mathbf{C}^{-1}_{\text{beam}} \bar{\mathbf{a}}^e_{\text{beam}} - q_{\bar{y}}\left(\bar{x}-\frac{L}{2}\right)\end{split}\]

in which \(D_{EI}\), \(k_{\bar{y}}\), \(L\), and \(q_{\bar{y}}\) are defined in beam2we and

\[\begin{split}\mathbf{C}^{-1}_{\text{beam}} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -\frac{3}{L^2} & -\frac{2}{L} & \frac{3}{L^2} & -\frac{1}{L} \\ \frac{2}{L^3} & \frac{1}{L^2} & -\frac{2}{L^3} & \frac{1}{L^2} \end{bmatrix}\end{split}\]