bar3s¶

Purpose:

Compute normal force in a three dimensional bar element.

Syntax:

es = bar3s(ex, ey, ez, ep, ed)
es = bar3s(ex, ey, ez, ep, ed, eq)
[es, edi] = bar3s(ex, ey, ez, ep, ed, eq, n)
[es, edi, eci] = bar3s(ex, ey, ez, ep, ed, eq, n)

Description:

bar3s computes the normal force in a three dimensional bar element (see bar3e).

The input variables ex, ey, and ep are defined in bar3e 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 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\(= \begin{bmatrix} 0 \\ \bar{x}_2 \\ \vdots \\ \bar{x}_{n-1} \\ L \end{bmatrix}\) \(\qquad\)

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

\[\mathbf{a}^e = \begin{bmatrix} u_1 & u_2 & u_3 & u_4 & u_5 & u_6 \end{bmatrix}^T\]

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 bar3e.

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 bar3e and

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