bar2gs¶

Purpose:

Compute normal force in a two dimensional bar element with geometric nonlinearity.

Syntax:

es, Qx = cfc.bar2gs(ex, ey, ep, ed)
es, Qx, edi = cfc.bar2gs(ex, ey, ep, ed, n)
es, Qx, edi, eci = cfc.bar2gs(ex, ey, ep, ed, n)

Description:

bar2gs computes the normal force, axial force and displacements in the two dimensional bar element bar2ge.

The input variables ex, ey, and ep are defined in bar2ge and the element nodal displacements, stored in ed, are obtained by the function extract_ed. The number of evaluation points for section forces and displacements are determined by n. If n is omitted, only the ends of the bar are evaluated.

The output variable Qx contains the axial force \(Q_{\bar{x}}\) and the output variables

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

contain the normal force, the displacement, and the evaluation points on the local \(\bar{x}\)-axis. \(L\) is the length of the bar element.

Theory:

The nodal displacements in global coordinates are given by

\[\mathbf{a}^e = \left[\; u_1\;\; u_2\;\; u_3\;\; u_4 \;\right]^T\]

The transpose of \(\mathbf{a}^e\) is stored in ed. The nodal displacements in local coordinates are given by

\[\bar{\mathbf{a}}^e = \mathbf{G} \mathbf{a}^e\]

where the transformation matrix \(\mathbf{G}\) is defined in bar2ge. The displacements associated with bar action are determined as

\[\begin{split}\bar{\mathbf{a}}^e_{\text{bar}} = \left[ \begin{array}{r} \bar{u}_1 \\ \bar{u}_3 \end{array}\right]\end{split}\]

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

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

\[N(\bar{x}) = D_{EA} \mathbf{B} \bar{\mathbf{a}}^e_{\text{bar}}\]

where

\[\mathbf{N} = \left[\begin{array}{rr} 1 & \bar{x} \end{array}\right] \mathbf{C}^{-1} = \left[\begin{array}{rr} 1-\frac{\bar{x}}{L} & \frac{\bar{x}}{L} \end{array}\right]\]

\[\mathbf{B} = \left[\begin{array}{rr} 0 & 1 \end{array}\right] \mathbf{C}^{-1} = \frac{1}{L}\left[\begin{array}{rr} -1 & 1 \end{array}\right]\]

where \(D_{EA}\) and \(L\) are defined in bar2ge and

\[\begin{split}\mathbf{C}^{-1} = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{1}{L} & \frac{1}{L} \end{array}\right]\end{split}\]

An updated value of the axial force is computed as

\[Q_{\bar{x}} = N(0)\]