beam2sΒΆ

Purpose:

Compute section forces in a two-dimensional beam element.

_images/beam2s.svg
Syntax:

[es] = beam2s(ex, ey, ep, ed)
[es] = beam2s(ex, ey, ep, ed, eq)
[es, edi] = beam2s(ex, ey, ep, ed, eq, n)
[es, edi, eci] = beam2s(ex, ey, ep, ed, eq, n)
Description:

beam2s computes the section forces and displacements in local directions along the beam element beam2e.

The input variables ex, ey, ep and eq are defined in beam2e.

The element displacements, stored in ed, are obtained by the function extract_ed. If a distributed load is applied to the element, the variable eq must be included. The number of evaluation points for section forces and displacements is 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:

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

where \(\mathbf{G}\) is described in beam2e and the transpose of \(\mathbf{a}^e\) is stored in ed.

The displacements associated with bar action and beam action are determined as:

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

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

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

where:

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

where \(D_{EA}\), \(L\), and \(q_{\bar{x}}\) are defined in beam2e, 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:

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

where:

\[\mathbf{N}_{\text{beam}} = \begin{bmatrix} 1 & \bar{x} & \bar{x}^2 & \bar{x}^3 \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[\mathbf{B}_{\text{beam}} = \begin{bmatrix} 0 & 0 & 2 & 6\bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[\frac{d\mathbf{B}_{\text{beam}}}{d\bar{x}} = \begin{bmatrix} 0 & 0 & 0 & 6 \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[v_p(\bar{x}) = \frac{q_{\bar{y}}}{D_{EI}} \left(\frac{\bar{x}^4}{24} - \frac{L \bar{x}^3}{12} + \frac{L^2 \bar{x}^2}{24}\right)\]
\[M_p(\bar{x}) = q_{\bar{y}} \left(\frac{\bar{x}^2}{2} - \frac{L \bar{x}}{2} + \frac{L^2}{12}\right)\]
\[V_p(\bar{x}) = -q_{\bar{y}} \left(\bar{x} - \frac{L}{2}\right)\]

where \(D_{EI}\), \(L\), and \(q_{\bar{y}}\) are defined in beam2e, and:

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