beam2gs¶

Purpose:

Compute section forces in a two dimensional nonlinear beam element with geometrical nonlinearity.

Syntax:

[es, Qx] = beam2gs(ex, ey, ep, ed, Qx)
[es, Qx] = beam2gs(ex, ey, ep, ed, Qx, eq)
[es, Qx, edi] = beam2gs(ex, ey, ep, ed, Qx, eq, n)
[es, Qx, edi, eci] = beam2gs(ex, ey, ep, ed, Qx, eq, n)

Description:

beam2gs computes the section forces and displacements in local directions along the geometric nonlinear beam element beam2ge.

The input variables ex, ey, ep, Qx and eq, are described in beam2ge. The element displacements, stored in ed, are obtained by the function extract_ed. If a distributed transversal load is 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 variable Qx contains \(Q_{\bar{x}}\) and 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 beam2ge 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 \(u(\bar{x})\) is computed from

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

where

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

where \(L\) is defined in beam2ge 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

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

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

\[M(\bar{x}) = D_{EI} \mathbf{B}_{\mathrm{beam}} \mathbf{\bar{a}}^e_{\mathrm{beam}} + M_p(\bar{x})\]

\[V(\bar{x}) = -D_{EI} \frac{d\mathbf{B}_{\mathrm{beam}}}{dx} \mathbf{\bar{a}}^e_{\mathrm{beam}} + V_p(\bar{x})\]

where

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

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

\[\mathbf{B}_{\mathrm{beam}} = \begin{bmatrix} 0 & 0 & 2 & 6\bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\mathrm{beam}}\]

\[\frac{d\mathbf{B}_{\mathrm{beam}}}{dx} = \begin{bmatrix} 0 & 0 & 0 & 6 \end{bmatrix} \mathbf{C}^{-1}_{\mathrm{beam}}\]

\[\begin{split}v_p(\bar{x}) = -\frac{Q_{\bar{x}}}{D_{EI}} \begin{bmatrix} 0 \\ 0 \\ \frac{\bar{x}^4}{12}-\frac{L \bar{x}^3}{6}+\frac{L^2 \bar{x}^2}{12} \\ \frac{\bar{x}^5}{20}-\frac{3L^2 \bar{x}^3}{20}+\frac{L^3 \bar{x}^2}{10} \end{bmatrix}^T \mathbf{C}^{-1}_{\mathrm{beam}} \mathbf{\bar{a}}^e_{\mathrm{beam}} + \frac{q_{\bar{y}}}{D_{EI}}\left(\frac{\bar{x}^4}{24}-\frac{L \bar{x}^3}{12}+\frac{L^2 \bar{x}^2}{24}\right)\end{split}\]

\[\begin{split}\theta_p(\bar{x}) = -\frac{Q_{\bar{x}}}{D_{EI}} \begin{bmatrix} 0 \\ 0 \\ \frac{\bar{x}^3}{3}-\frac{L \bar{x}^2}{2}+\frac{L^2 \bar{x}}{6} \\ \frac{\bar{x}^4}{4}-\frac{9L^2 \bar{x}^2}{20}+\frac{L^3 \bar{x}}{5} \end{bmatrix}^T \mathbf{C}^{-1}_{\mathrm{beam}} \mathbf{\bar{a}}^e_{\mathrm{beam}} + \frac{q_{\bar{y}}}{D_{EI}}\left(\frac{\bar{x}^3}{6}-\frac{L \bar{x}^2}{4}+\frac{L^2 \bar{x}}{12}\right)\end{split}\]

\[\begin{split}M_p(\bar{x}) = -Q_{\bar{x}} \begin{bmatrix} 0 \\ 0 \\ \bar{x}^2 - L\bar{x} + \frac{L^2}{6} \\ \bar{x}^3 - \frac{9L^2 \bar{x}}{10} + \frac{L^3}{5} \end{bmatrix}^T \mathbf{C}^{-1}_{\mathrm{beam}} \mathbf{\bar{a}}^e_{\mathrm{beam}} + q_{\bar{y}}\left(\frac{\bar{x}^2}{2}-\frac{L \bar{x}}{2}+\frac{L^2}{12}\right)\end{split}\]

\[\begin{split}V_p(\bar{x}) = Q_{\bar{x}} \begin{bmatrix} 0 \\ 0 \\ 2\bar{x} - L \\ 3\bar{x}^2 - \frac{9L^2}{10} \end{bmatrix}^T \mathbf{C}^{-1}_{\mathrm{beam}} \mathbf{\bar{a}}^e_{\mathrm{beam}} - q_{\bar{y}}\left(\bar{x} - \frac{L}{2}\right)\end{split}\]

in which \(D_{EI}\), \(L\), and \(q_{\bar{y}}\) are defined in beam2ge and

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

An updated value of the axial force is computed as

\[Q_{\bar{x}} = D_{EA} \begin{bmatrix} 0 & 1 \end{bmatrix} \mathbf{C}^{-1}_{\mathrm{bar}} \mathbf{\bar{a}}^e_{\mathrm{bar}}\]

The normal force \(N(\bar{x})\) is then computed as

\[N(\bar{x}) = Q_{\bar{x}} + \theta(\bar{x}) V(\bar{x})\]