beam2ts¶

Purpose:

Compute section forces in a two dimensional Timoshenko beam element.

Syntax:

es = beam2ts(ex, ey, ep, ed)
es = beam2ts(ex, ey, ep, ed, eq)
[es, edi, eci] = beam2ts(ex, ey, ep, ed, eq, n)

Description:

beam2ts computes the section forces and displacements in local directions along the beam element beam2te.

The input variables ex, ey, ep and eq are defined in beam2te. 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

\[\begin{split}\begin{aligned} \mathrm{es} &= \left[\; \mathbf{N} \; \mathbf{V} \; \mathbf{M}\; \right] \\ \mathrm{edi} &= \left[\; \mathbf{u} \; \mathbf{v} \; \boldsymbol{\theta} \; \right] \\ \mathrm{eci} &= \left[ \mathbf{\bar{x}} \right] \end{aligned}\end{split}\]

consist of column matrices that contain the section forces, the displacements and rotation of the cross section (note that the rotation \(\theta\) is not equal to \(\frac{d\bar v}{d\bar x}\)), and the evaluation points on the local \(\bar{x}\)-axis. The explicit matrix expressions are

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\(= \begin{bmatrix} u_{1} & v_{1} & \theta_1 \\ u_{2} & v_{2} & \theta_2 \\ \vdots & \vdots & \vdots \\ u_{n-1} & v_{n-1} & \theta_{n-1}\\ u_{n} & v_{n} & \theta_n \end{bmatrix}\) \(\quad\) eci\(= \begin{bmatrix} 0 \\ \bar{x}_2 \\ \vdots \\ \bar{x}_{n-1} \\ L \end{bmatrix}\)

where \(L\) 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

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

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

\[u(\bar{x}) = \mathbf{N}_{\mathrm{bar}} \mathbf{\bar{a}}^e_{\mathrm{bar}} + u_p(\bar{x})\]

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

where

\[\mathbf{N}_{\mathrm{bar}} = \begin{bmatrix} 1 & \bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\mathrm{bar}} = \begin{bmatrix} 1-\frac{\bar{x}}{L} & \frac{\bar{x}}{L} \end{bmatrix}\]

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

\[u_p(\bar{x}) = -\frac{q_{\bar{x}}}{D_{EA}}\left(\frac{\bar{x}^2}{2}-\frac{L\bar{x}}{2}\right)\]

\[N_p(\bar{x}) = -q_{\bar{x}}\left(\bar{x}-\frac{L}{2}\right)\]

in which \(D_{EA}\), \(L\), and \(q_{\bar{x}}\) are defined in beam2te and

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

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

\[v(\bar{x}) = \mathbf{N}_{\mathrm{beam},v} \mathbf{\bar{a}}^e_{\mathrm{beam}} + v_p(\bar{x})\]

\[\theta(\bar{x}) = \mathbf{N}_{\mathrm{beam},\theta} \mathbf{\bar{a}}^e_{\mathrm{beam}} + \theta_p(\bar{x})\]

\[M(\bar{x}) = D_{EI} \frac{d\theta}{dx} = D_{EI} \frac{d\mathbf{N}_{\mathrm{beam},\theta}}{d\bar{x}} \mathbf{\bar{a}}^e_{\mathrm{beam}} + M_p(\bar{x})\]

\[V(\bar{x}) = D_{GA} k_s \left(\frac{d v}{dx} - \theta \right) = D_{GA} k_s \left(\frac{d\mathbf{N}_{\mathrm{beam},v}}{d\bar{x}} - \mathbf{N}_{\mathrm{beam},\theta} \right) \mathbf{\bar{a}}^e_{\mathrm{beam}} + V_p(\bar{x})\]

where

\[\mathbf{N}_{\mathrm{beam},v} = \begin{bmatrix} 1 & \bar{x} & \bar{x}^2 & \bar{x}^3 \end{bmatrix} \mathbf{C}^{-1}_{\mathrm{beam}}\]

\[\frac{d\mathbf{N}_{\mathrm{beam},v}}{d\bar{x}} = \begin{bmatrix} 0 & 1 & 2\bar{x} & 3\bar{x}^2 \end{bmatrix} \mathbf{C}^{-1}_{\mathrm{beam}}\]

\[\mathbf{N}_{\mathrm{beam},\theta} = \begin{bmatrix} 0 & 1 & 2\bar{x} & 3\bar{x}^2 + 6\alpha \end{bmatrix} \mathbf{C}^{-1}_{\mathrm{beam}}\]

\[\frac{d\mathbf{N}_{\mathrm{beam},\theta}}{d\bar{x}} = \begin{bmatrix} 0 & 0 & 2 & 6\bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\mathrm{beam}}\]

\[v_p(\bar{x}) = \frac{q_{\bar{y}}}{D_{EI}}\left(\frac{\bar{x}^4}{24} - \frac{L\bar{x}^3}{12} + \frac{L^2\bar{x}^2}{2}\right) + \frac{q_{\bar{y}}}{D_{GA}k_s}\left(-\frac{\bar{x}^2}{2} + \frac{L\bar{x}}{2}\right)\]

\[\theta_p(\bar{x}) = \frac{q_{\bar{y}}}{D_{EI}}\left(\frac{\bar{x}^3}{6} - \frac{L\bar{x}^2}{4} + \frac{L^2\bar{x}}{12}\right)\]

\[M_p(\bar{x}) = q_{\bar{y}}\left(\frac{\bar{x}^2}{2} - \frac{L\bar{x}}{2} + \frac{L^2}{12}\right)\]

\[V_p(\bar{x}) = -q_{\bar{y}}\left(\bar{x} - \frac{L}{2}\right)\]

in which \(D_{EI}\), \(D_{GA}\), \(k_s\), \(L\), and \(q_{\bar{y}}\) are defined in beam2te and

\[\begin{split}\mathbf{C}^{-1}_{\mathrm{beam}} = \frac{1}{L^2 + 12\alpha} \begin{bmatrix} L^2 + 12\alpha & 0 & 0 & 0 \\ -\frac{12\alpha}{L} & L^2 + 6\alpha & \frac{12\alpha}{L} & -6\alpha \\ -3 & -2L - \frac{6\alpha}{L} & 3 & -L + \frac{6\alpha}{L} \\ \frac{2}{L} & 1 & -\frac{2}{L} & 1 \end{bmatrix}\end{split}\]

with

\[\alpha = \frac{D_{EI}}{D_{GA} k_s}\]