bar1ws¶

Purpose:

Compute normal force in a one dimensional bar element with elastic support.

Syntax:

es = bar1ws(ex, ep, ed)
es = bar1ws(ex, ep, ed, eq)
[es, edi] = bar1ws(ex, ep, ed, eq, n)
[es, edi, eci] = bar1ws(ex, ep, ed, eq, n)

Description:

bar1ws computes the normal force in the one dimensional bar element bar1ws.

The input variables ex and ep are defined in bar1we and the element nodal displacements, stored in ed, are obtained by the function _ed. If distributed load is applied to the element, the variable eq must be included.

The number of evaluation points for normal force and displacement are determined by n. If n is omitted, only the ends of the bar are evaluated.

The output variables are:

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\(= \begin{bmatrix} 0 \\ \bar{x}_2 \\ \vdots \\ \bar{x}_{n-1} \\ L \end{bmatrix}\)

These 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 local coordinates are given by

\[\begin{split}\mathbf{\bar{a}}^e = \begin{bmatrix} \bar{u}_1 \\ \bar{u}_2 \end{bmatrix}\end{split}\]

The transpose of \(\mathbf{\bar{a}}^e\) is stored in ed.

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

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

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

\[u_p(\bar{x}) = \frac{k_{\bar{x}}}{D_{EA}} \left[ \frac{\bar{x}^2 - L\bar{x}}{2} \quad \frac{\bar{x}^3 - L^2\bar{x}}{6} \right] \mathbf{C}^{-1} \mathbf{\bar{a}}^e - \frac{q_{\bar{x}}}{D_{EA}} \left( \frac{\bar{x}^2}{2} - \frac{L\bar{x}}{2} \right)\]

\[N_p(\bar{x}) = k_{\bar{x}} \left[ \frac{2\bar{x} - L}{2} \quad \frac{3\bar{x}^2 - L^2}{6} \right] \mathbf{C}^{-1} \mathbf{\bar{a}}^e - q_{\bar{x}} \left( \bar{x} - \frac{L}{2} \right)\]

in which \(D_{EA}\), \(L\), \(k_{\bar{x}}\) and \(q_{\bar{x}}\) are defined in bar1we and

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