beam2gxsΒΆ

Purpose:

Compute section forces in a two dimensional geometric nonlinear beam element with exact solution.

Two dimensional geometric nonlinear exact beam element
Syntax:

[es,Qx] = beam2gxs(ex, ey, ep, ed, Qx)
[es,Qx] = beam2gxs(ex, ey, ep, ed, Qx, eq)
[es,Qx,edi] = beam2gxs(ex, ey, ep, ed, Qx, eq, n)
[es,Qx,edi,eci] = beam2gxs(ex, ey, ep, ed, Qx, eq, n)
Description:

beam2gxs computes the section forces and displacements in local directions along the geometric nonlinear beam element beam2gxe.

The input variables ex, ey, ep, Qx and eq, are described in beam2gxe. 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

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

The displacement \(u(\bar{x})\) is computed from

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

where

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

where \(L\) is defined in beam2gxe 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 rotation \(\theta(\bar{x})\), the bending moment \(M(\bar{x})\) and the shear force \(V(\bar{x})\) are computed from

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

For an axial compressive force (\(Q_{\bar{x}} < 0\)) we have

\[\mathbf{N}_{\text{beam}} = \begin{bmatrix} 1 & \bar{x} & \cos k \bar{x} & \sin k \bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[\frac{d\mathbf{N}_{\text{beam}}}{dx} = \begin{bmatrix} 0 & 1 & -k \sin k \bar{x} & k \cos k \bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[\mathbf{B}_{\text{beam}} = \begin{bmatrix} 0 & 0 & -k^2 \cos k \bar{x} & -k^2 \sin k \bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[\frac{d\mathbf{B}_{\text{beam}}}{dx} = \begin{bmatrix} 0 & 0 & k^3 \sin k \bar{x} & -k^3 \cos k \bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]
\[v_p(\bar{x}) = \frac{q_{\bar{y}}L^4}{2D_{EI}} \left[ \frac{1 + \cos kL}{(kL)^3 \sin kL}(-1 + \cos k \bar{x}) -\frac{1}{(kL)^3} \sin k \bar{x} + \frac{1}{(kL)^2} \left(\frac{\bar{x}^2}{L^2}-\frac{\bar{x}}{L}\right) \right]\]
\[\theta_p(\bar{x}) = \frac{q_{\bar{y}}L^3}{2D_{EI}} \left[ -\frac{1 + \cos kL}{(kL)^2 \sin kL} \sin k \bar{x} -\frac{1}{(kL)^2} \cos k \bar{x} + \frac{1}{(kL)^2} \left(\frac{2\bar{x}}{L}-1\right) \right]\]
\[M_p(\bar{x}) = \frac{q_{\bar{y}}L^2}{2} \left[ -\frac{1 + \cos kL}{(kL) \sin kL} \cos k \bar{x} +\frac{1}{(kL)} \sin k \bar{x} + \frac{2}{(kL)^2} \right]\]
\[\begin{split}V_p(\bar{x}) = Q_{\bar{x}} \begin{bmatrix} 0 \\ 0 \\ 2\bar{x} - L \\ 3\bar{x}^2 - \frac{9L^2}{10} \end{bmatrix}^T \mathbf{C}^{-1}_{\text{beam}} \mathbf{\bar{a}}^e_{\text{beam}} - q_{\bar{y}}\left(\bar{x} - \frac{L}{2}\right)\end{split}\]

in which \(D_{EI}\), \(L\), and \(q_{\bar{y}}\) are defined in beam2gxe and

\[\begin{split}\mathbf{C}^{-1}_{\text{beam}} = \begin{bmatrix} k (kL \sin kL+\cos kL-1) & -kL \cos kL+\sin kL & -k (1-\cos kL) & -\sin kL+kL \\ - k^2 \sin kL & -k (1-\cos kL) & k^2 \sin kL & -k (1-\cos kL) \\ -k(1-\cos kL) & kL \cos kL-\sin kL & k (1-\cos kL) & \sin kL-kL \\ k\sin kL & kL \sin kL+\cos kL-1 & -k \sin kL & 1-\cos kL \end{bmatrix}\end{split}\]

An updated value of the axial force is computed as

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

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

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