beam3eΒΆ

Purpose:

Compute element stiffness matrix for a three dimensional beam element.

_images/beam3e.svg
Syntax:

Ke = beam3e(ex, ey, ez, eo, ep)
[Ke, fe] = beam3e(ex, ey, ez, eo, ep, eq)
Description:

beam3e provides the global element stiffness matrix \({\mathbf{K}}^e\) for a three dimensional beam element.

The input variables

ex\(= [x_1 \;\; x_2]\) \(\qquad\) ey\(= [y_1 \;\; y_2]\) \(\qquad\) ez\(= [z_1 \;\; z_2]\) \(\qquad\) eo\(= [x_{\bar{z}} \;\; y_{\bar{z}} \;\; z_{\bar{z}}]\)

supply the element nodal coordinates \(x_1\), \(y_1\), etc. as well as the direction of the local beam coordinate system \((\bar{x}, \bar{y}, \bar{z})\). By giving a global vector \((x_{\bar{z}}, y_{\bar{z}}, z_{\bar{z}})\) parallel with the positive local \(\bar{z}\) axis of the beam, the local beam coordinate system is defined.

The variable

ep\(= [E \;\; G \;\; A \;\; I_{\bar{y}} \;\; I_{\bar{z}} \;\; K_v]\)

supplies the modulus of elasticity \(E\), the shear modulus \(G\), the cross section area \(A\), the moment of inertia with respect to the \(\bar{y}\) axis \(I_{\bar{y}}\), the moment of inertia with respect to the \(\bar{z}\) axis \(I_{\bar{z}}\), and St. Venant torsion constant \(K_v\).

The element load vector fe can also be computed if uniformly distributed loads are applied to the element. The optional input variable

eq\(= [q_{\bar{x}} \;\; q_{\bar{y}} \;\; q_{\bar{z}} \;\; q_{\bar{\omega}}]\)

then contains the distributed loads. The positive directions of \(q_{\bar{x}}\), \(q_{\bar{y}}\), and \(q_{\bar{z}}\) follow the local beam coordinate system. The distributed torque \(q_{\bar{\omega}}\) is positive if directed in the local \(\bar{x}\)-direction, i.e. from local \(\bar{y}\) to local \(\bar{z}\). All the loads are per unit length.

Theory:

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

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

where

\[\begin{split}\bar{\mathbf{K}}^e = \left[ \begin{array}{cccccccccccc} \frac{D_{EA}}{L} & 0 & 0 & 0 & 0 & 0 & -\frac{D_{EA}}{L} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{12D_{EI_{\bar z}}}{L^3} & 0 & 0 & 0 & \frac{6D_{EI_{\bar z}}}{L^2} & 0 & -\frac{12D_{EI_{\bar z}}}{L^3} & 0 & 0 & 0 & \frac{6D_{EI_{\bar z}}}{L^2} \\ 0 & 0 & \frac{12D_{EI_{\bar y}}}{L^3} & 0 & -\frac{6D_{EI_{\bar y}}}{L^2} & 0 & 0 & 0 & -\frac{12D_{EI_{\bar y}}}{L^3} & 0 & -\frac{6D_{EI_{\bar y}}}{L^2} & 0 \\ 0 & 0 & 0 & \frac{D_{GK}}{L} & 0 & 0 & 0 & 0 & 0 & -\frac{D_{GK}}{L} & 0 & 0 \\ 0 & 0 & -\frac{6D_{EI_{\bar y}}}{L^2} & 0 & \frac{4D_{EI_{\bar y}}}{L} & 0 & 0 & 0 & \frac{6D_{EI_{\bar y}}}{L^2} & 0 & \frac{2D_{EI_{\bar y}}}{L} & 0 \\ 0 & \frac{6D_{EI_{\bar z}}}{L^2} & 0 & 0 & 0 & \frac{4D_{EI_{\bar z}}}{L} & 0 & -\frac{6D_{EI_{\bar z}}}{L^2} & 0 & 0 & 0 & \frac{2D_{EI_{\bar z}}}{L} \\ -\frac{D_{EA}}{L} & 0 & 0 & 0 & 0 & 0 & \frac{D_{EA}}{L} & 0 & 0 & 0 & 0 & 0 \\ 0 & -\frac{12D_{EI_{\bar z}}}{L^3} & 0 & 0 & 0 & -\frac{6D_{EI_{\bar z}}}{L^2} & 0 & \frac{12D_{EI_{\bar z}}}{L^3} & 0 & 0 & 0 & -\frac{6D_{EI_{\bar z}}}{L^2} \\ 0 & 0 & -\frac{12D_{EI_{\bar y}}}{L^3} & 0 & \frac{6D_{EI_{\bar y}}}{L^2} & 0 & 0 & 0 & \frac{12D_{EI_{\bar y}}}{L^3} & 0 & \frac{6D_{EI_{\bar y}}}{L^2} & 0 \\ 0 & 0 & 0 & -\frac{D_{GK}}{L} & 0 & 0 & 0 & 0 & 0 & \frac{D_{GK}}{L} & 0 & 0 \\ 0 & 0 & -\frac{6D_{EI_{\bar y}}}{L^2} & 0 & \frac{2D_{EI_{\bar y}}}{L} & 0 & 0 & 0 & \frac{6D_{EI_{\bar y}}}{L^2} & 0 & \frac{4D_{EI_{\bar y}}}{L} & 0 \\ 0 & \frac{6D_{EI_{\bar z}}}{L^2} & 0 & 0 & 0 & \frac{2D_{EI_{\bar z}}}{L} & 0 & -\frac{6D_{EI_{\bar z}}}{L^2} & 0 & 0 & 0 & \frac{4D_{EI_{\bar z}}}{L} \end{array} \right]\end{split}\]

and

\[\begin{split}\mathbf{G} = \left[ \begin{array}{cccccccccccc} n_{x\bar{x}} & n_{y\bar{x}} & n_{z\bar{x}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ n_{x\bar{y}} & n_{y\bar{y}} & n_{z\bar{y}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ n_{x\bar{z}} & n_{y\bar{z}} & n_{z\bar{z}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} & n_{z\bar{x}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & n_{x\bar{y}} & n_{y\bar{y}} & n_{z\bar{y}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & n_{x\bar{z}} & n_{y\bar{z}} & n_{z\bar{z}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} & n_{z\bar{x}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & n_{x\bar{y}} & n_{y\bar{y}} & n_{z\bar{y}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & n_{x\bar{z}} & n_{y\bar{z}} & n_{z\bar{z}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & n_{x\bar{x}} & n_{y\bar{x}} & n_{z\bar{x}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & n_{x\bar{y}} & n_{y\bar{y}} & n_{z\bar{y}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & n_{x\bar{z}} & n_{y\bar{z}} & n_{z\bar{z}} \end{array} \right]\end{split}\]

where the axial stiffness \(D_{EA}\), the bending stiffness \(D_{EI_{\bar z}}\), the bending stiffness \(D_{EI_{\bar y}}\), and the St. Venant torsion stiffness \(D_{GK}\) are given by

\[D_{EA} = EA; \quad D_{EI_{\bar z}} = EI_{\bar z}; \quad D_{EI_{\bar y}} = EI_{\bar y}; \quad D_{GK} = GK_v\]

The length \(L\) is given by

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

The transformation matrix \(\mathbf{G}\) contains direction cosines computed as

\[\begin{split}\begin{aligned} n_{x\bar{x}} &= \frac{x_2 - x_1}{L} \qquad n_{y\bar{x}} = \frac{y_2 - y_1}{L} \qquad n_{z\bar{x}} = \frac{z_2 - z_1}{L} \\ n_{x\bar{z}} &= \frac{x_{\bar{z}}}{L_{\bar{z}}} \qquad n_{y\bar{z}} = \frac{y_{\bar{z}}}{L_{\bar{z}}} \qquad n_{z\bar{z}} = \frac{z_{\bar{z}}}{L_{\bar{z}}} \\ n_{x\bar{y}} &= n_{y\bar{z}} n_{z\bar{x}} - n_{z\bar{z}} n_{y\bar{x}} \\ n_{y\bar{y}} &= n_{z\bar{z}} n_{x\bar{x}} - n_{x\bar{z}} n_{z\bar{x}} \\ n_{z\bar{y}} &= n_{x\bar{z}} n_{y\bar{x}} - n_{y\bar{z}} n_{x\bar{x}} \end{aligned}\end{split}\]

where

\[L_{\bar{z}} = \sqrt{x_{\bar{z}}^2 + y_{\bar{z}}^2 + z_{\bar{z}}^2}\]

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

\[\begin{split}\bar{\mathbf{f}}_l^e = \begin{bmatrix} \dfrac{q_{\bar{x}}L}{2} \\ \dfrac{q_{\bar{y}}L}{2} \\ \dfrac{q_{\bar{z}}L}{2} \\ \dfrac{q_{\bar{\omega}}L}{2} \\ -\dfrac{q_{\bar{z}}L^2}{12} \\ \dfrac{q_{\bar{y}}L^2}{12} \\ \dfrac{q_{\bar{x}}L}{2} \\ \dfrac{q_{\bar{y}}L}{2} \\ \dfrac{q_{\bar{z}}L}{2} \\ \dfrac{q_{\bar{\omega}}L}{2} \\ \dfrac{q_{\bar{z}}L^2}{12} \\ -\dfrac{q_{\bar{y}}L^2}{12} \end{bmatrix}\end{split}\]