bar2s¶

Purpose:

Compute normal force in a two dimensional bar element.

Syntax:

es = bar2s(ex, ey, ep, ed)
es = bar2s(ex, ey, ep, ed, eq)
[es, edi] = bar2s(ex, ey, ep, ed, eq, n)
[es, edi, eci] = bar2s(ex, ey, ep, ed, eq, n)

Description:

bar2s computes the normal force in the two dimensional bar element bar2e.

The input variables ex, ey, and ep are defined in bar2e and the element nodal 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 bar are evaluated.

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

\[\mathbf{a}^e = \begin{bmatrix} u_1 & u_2 & u_3 & u_4 \end{bmatrix}^T\]

are also shown in bar2e. The transpose of \(\mathbf{a}^e\) is stored in ed.

The nodal displacements in local coordinates are given by

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

where the transformation matrix \(\mathbf{G}\) is defined in bar2e.

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

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

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

where

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

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

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

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

where \(D_{EA}\), \(L\), \(q_{\bar{x}}\) are defined in bar2e and

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