Bar element functions¶

Bar elements are available for one, two, and three dimensional analysis.

bar1e¶

Purpose:

Compute element stiffness matrix for a one dimensional bar element.

Syntax:

Ke = bar1e(ex, ep)
[Ke, fe] = bar1e(ex, ep, eq)

Description:

bar1e provides the element stiffness matrix \(\bar{\mathbf{K}}^e\) for a one dimensional bar element. The input variables

ex\(= [x_1 \;\; x_2]\) \(\qquad\) ep\(= [E \; A]\)

supply the element nodal coordinates \(x_1\) and \(x_2\), the modulus of elasticity \(E\), and the cross section area \(A\).

The element load vector \(\bar{\mathbf{f}}_l^e\) can also be computed if a uniformly distributed load is applied to the element. The optional input variable

eq\(= [q_{\bar{x}}]\)

contains the distributed load per unit length, \(q_{\bar{x}}\).

Theory:

The element stiffness matrix \(\bar{\mathbf{K}}^e\), stored in Ke, is computed according to

\[\begin{split}\bar{\mathbf{K}}^e = \frac{D_{EA}}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}\end{split}\]

where the axial stiffness \(D_{EA}\) and the length \(L\) are given by

\[D_{EA} = EA; \quad L = x_2 - x_1\]

The element load vector \(\bar{\mathbf{f}}_l^e\), stored in fe, is computed according to

\[\begin{split}\bar{\mathbf{f}}_l^e = \frac{q_{\bar{x}} L}{2} \begin{bmatrix} 1 \\ 1 \end{bmatrix}\end{split}\]

bar1s¶

Purpose:

Compute normal force in a one dimensional bar element.

Syntax:

es = bar1s(ex, ep, ed)
es = bar1s(ex, ep, ed, eq)
[es, edi] = bar1s(ex, ep, ed, eq, n)
[es, edi, eci] = bar1s(ex, ep, ed, eq, n)

Description:

bar1s computes the normal force in the one dimensional bar element bar1e.

The input variables ex and ep are defined in bar1e and the element nodal displacements, stored in ed, are obtained by the function extract_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

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 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

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

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

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

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

in which \(D_{EA}\), \(L\), and \(q_{\bar{x}}\) are defined in bar1e and

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

bar1we¶

Purpose:

Compute element stiffness matrix for a one dimensional bar element with elastic support.

Syntax:

Ke = bar1we(ex, ep)
[Ke, fe] = bar1we(ex, ep, eq)

Description:

bar1we provides the element stiffness matrix \(\bar{\mathbf{K}}^e\) for a one dimensional bar element with elastic support.

The input variables

ex\(= [x_1\;\; x_2]\) \(\qquad\) ep\(= [E\; A\; k_{\bar{x}}]\)

supply the element nodal coordinates \(x_1\) and \(x_2\), the modulus of elasticity \(E\), the cross section area \(A\) and the stiffness of the axial springs \(k_{\bar{x}}\).

The element load vector \(\bar{\mathbf{f}}_l^e\) can also be computed if a uniformly distributed load is applied to the element.

The optional input variable

eq\(= [q_{\bar{x}}]\)

contains the distributed load per unit length, \(q_{\bar{x}}\).

Bar element with distributed load

Theory:

The element stiffness matrix \(\bar{\mathbf{K}}^e\), stored in Ke, is computed according to

\[\bar{\mathbf{K}}^e = \bar{\mathbf{K}}^e_0 + \bar{\mathbf{K}}^e_s\]

where

\[\begin{split}\bar{\mathbf{K}}^e_0 = \frac{D_{EA}}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}\end{split}\]

\[\begin{split}\bar{\mathbf{K}}^e_s = k_{\bar{x}} L \begin{bmatrix} \frac{1}{3} & \frac{1}{6} \\ \frac{1}{6} & \frac{1}{3} \end{bmatrix}\end{split}\]

where the axial stiffness \(D_{EA}\) and the length \(L\) are given by

\[D_{EA} = EA; \qquad L = x_2 - x_1\]

The element load vector \(\bar{\mathbf{f}}_l^e\), stored in fe, is computed according to

\[\begin{split}\bar{\mathbf{f}}_l^e = \frac{q_{\bar{x}} L}{2} \begin{bmatrix} 1 \\ 1 \end{bmatrix}\end{split}\]

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

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

bar2e¶

Purpose:

Compute element stiffness matrix for a two dimensional bar element.

Syntax:

Ke = bar2e(ex, ey, ep)
[Ke, fe] = bar2e(ex, ey, ep, eq)

Description:

bar2e provides the global element stiffness matrix \({\mathbf{K}}^e\) for a two dimensional bar element.

The input variables

ex\(= [x_1 \;\; x_2]\) \(\qquad\) ey\(= [y_1 \;\; y_2]\) \(\qquad\) ep\(= [E \; A]\)

supply the element nodal coordinates \(x_1\), \(y_1\), \(x_2\), and \(y_2\), the modulus of elasticity \(E\), and the cross section area \(A\).

The element load vector \({\mathbf{f}}_l^e\) can also be computed if a uniformly distributed load is applied to the element. The optional input variable

eq\(= [q_{\bar{x}}]\)

contains the distributed load per unit length, \(q_{\bar{x}}\).

Theory:

The element stiffness matrix \(\mathbf{K}^e\), stored in Ke, is computed according to

\[\mathbf{K}^e = \mathbf{G}^T \; \bar{\mathbf{K}}^e \; \mathbf{G}\]

where

\[\begin{split}\bar{\mathbf{K}}^e = \frac{D_{EA}}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \qquad \mathbf{G} = \begin{bmatrix} n_{x\bar{x}} & n_{y\bar{x}} & 0 & 0 \\ 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} \end{bmatrix}\end{split}\]

where the axial stiffness \(D_{EA}\) and the length \(L\) are given by

\[D_{EA} = EA; \qquad L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

and the transformation matrix \(\mathbf{G}\) contains the direction cosines

\[n_{x\bar{x}} = \frac{x_2 - x_1}{L} \qquad n_{y\bar{x}} = \frac{y_2 - y_1}{L}\]

The element load vector \(\mathbf{f}_l^e\), stored in fe, is computed according to

\[\mathbf{f}_l^e = \mathbf{G}^T \; \bar{\mathbf{f}}_l^e\]

where

\[\begin{split}\bar{\mathbf{f}}_l^e = \frac{q_{\bar{x}} L}{2} \begin{bmatrix} 1 \\ 1 \end{bmatrix}\end{split}\]

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

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

bar2ge¶

Purpose:

Compute element stiffness matrix for a two dimensional geometric nonlinear bar.

Syntax:

Ke = bar2ge(ex, ey, ep, Qx)

Description:

bar2ge provides the element stiffness matrix \({\mathbf{K}}^e\) for a two dimensional geometric nonlinear bar element.

The input variables

ex\(= [x_1 \;\; x_2]\) \(\qquad\) ey\(= [y_1 \;\; y_2]\) \(\qquad\) ep\(= [E \; A]\)

supply the element nodal coordinates \(x_1\), \(y_1\), \(x_2\), and \(y_2\), the modulus of elasticity \(E\), and the cross section area \(A\).

The input variable

Qx\(= [Q_{\bar{x}}]\)

contains the value of the axial force, which is positive in tension.

Theory:

The global element stiffness matrix \(\mathbf{K}^e\), stored in Ke, is computed according to

\[\mathbf{K}^e = \mathbf{G}^T\,\mathbf{\bar{K}}^e\,\mathbf{G}\]

where

\[\begin{split}\mathbf{\bar{K}}^e = \frac{D_{EA}}{L} \begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} + \frac{Q_{\bar{x}}}{L} \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 \end{bmatrix}\end{split}\]

\[\begin{split}\mathbf{G} = \begin{bmatrix} n_{x\bar{x}} & n_{y\bar{x}} & 0 & 0 \\ n_{x\bar{y}} & n_{y\bar{y}} & 0 & 0 \\ 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} \\ 0 & 0 & n_{x\bar{y}} & n_{y\bar{y}} \end{bmatrix}\end{split}\]

where the axial stiffness \(D_{EA}\) and the length \(L\) are given by

\[D_{EA} = EA \qquad L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

and the transformation matrix \(\mathbf{G}\) contains the direction cosines

\[n_{x\bar{x}} = n_{y\bar{y}} = \frac{x_2 - x_1}{L} \qquad n_{y\bar{x}} = -n_{x\bar{y}} = \frac{y_2 - y_1}{L}\]

bar2gs¶

Purpose:

Compute axial force and normal force in a two dimensional bar element.

Syntax:

[es, Qx] = bar2gs(ex, ey, ep, ed)
[es, Qx, edi] = bar2gs(ex, ey, ep, ed, n)
[es, Qx, edi, eci] = bar2gs(ex, ey, ep, ed, n)

Description:

bar2gs computes the normal force in the two dimensional bar elements 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

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

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

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

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

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

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

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

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

bar3e¶

Purpose:

Compute element stiffness matrix for a three dimensional bar element.

Syntax:

Ke = bar3e(ex, ey, ez, ep)
[Ke, fe] = bar3e(ex, ey, ez, ep, eq)

Description:

bar3e provides the global element stiffness matrix \({\mathbf{K}}^e\) for a three dimensional bar element.

The input variables

ex\(= [x_1 \;\; x_2]\) \(\qquad\) ey\(= [y_1 \;\; y_2]\) \(\qquad\) ez\(= [z_1 \;\; z_2]\) \(\qquad\) ep\(= [E \; A]\)

supply the element nodal coordinates \(x_1\), \(y_1\), \(z_1\), \(x_2\), \(y_2\), and \(z_2\), the modulus of elasticity \(E\), and the cross section area \(A\).

The element load vector fe can also be computed if a uniformly distributed axial load is applied to the element. The optional input variable

eq\(= [q_{\bar{x}}]\)

contains the distributed load per unit length, \(q_{\bar{x}}\).

Theory:

The element stiffness matrix \(\mathbf{K}^e\), stored in Ke, is computed according to

\[\mathbf{K}^e = \mathbf{G}^T \; \bar{\mathbf{K}}^e \; \mathbf{G}\]

where

\[\begin{split}\bar{\mathbf{K}}^e = \frac{D_{EA}}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \qquad \mathbf{G} = \begin{bmatrix} n_{x\bar{x}} & n_{y\bar{x}} & n_{z\bar{x}} & 0 & 0 & 0 \\ 0 & 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} & n_{z\bar{x}} \end{bmatrix}\end{split}\]

where the axial stiffness \(D_{EA}\) and the length \(L\) are given by

\[D_{EA} = EA \qquad L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

and the transformation matrix \(\mathbf{G}\) contains the direction cosines

\[n_{x\bar{x}} = \frac{x_2 - x_1}{L} \qquad n_{y\bar{x}} = \frac{y_2 - y_1}{L} \qquad n_{z\bar{x}} = \frac{z_2 - z_1}{L}\]

The element load vector \(\mathbf{f}_l^e\), stored in fe, is computed according to

\[\mathbf{f}_l^e = \mathbf{G}^T \; \bar{\mathbf{f}}_l^e\]

where

\[\begin{split}\bar{\mathbf{f}}_l^e = \frac{q_{\bar{x}} L}{2} \begin{bmatrix} 1 \\ 1 \end{bmatrix}\end{split}\]

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

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