beam2te

Purpose:

Compute element stiffness matrix for a two dimensional Timoshenko beam element.

Two dimensional beam element
Syntax:

Ke = beam2te(ex, ey, ep)
[Ke, fe] = beam2te(ex, ey, ep, eq)
Description:

beam2te provides the global element stiffness matrix \({\mathbf{K}}^e\) for a two dimensional Timoshenko beam element.

The input variables

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

supply the element nodal coordinates \(x_1\), \(y_1\), \(x_2\), and \(y_2\), the modulus of elasticity \(E\), the shear modulus \(G\), the cross section area \(A\), the moment of inertia \(I\) and the shear correction factor \(k_s\).

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

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

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

Uniformly distributed load

Uniformly distributed load

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 \(\mathbf{G}\) is described in beam2e, and \(\bar{\mathbf{K}}^e\) is given by

\[\begin{split}\bar{\mathbf{K}}^e = \begin{bmatrix} \frac{D_{EA}}{L} & 0 & 0 & -\frac{D_{EA}}{L} & 0 & 0 \\ 0 & \frac{12D_{EI}}{L^3(1+\mu)} & \frac{6D_{EI}}{L^2(1+\mu)} & 0 & -\frac{12D_{EI}}{L^3(1+\mu)} & \frac{6D_{EI}}{L^2(1+\mu)} \\ 0 & \frac{6D_{EI}}{L^2(1+\mu)} & \frac{4D_{EI}(1+\frac{\mu}{4})}{L(1+\mu)} & 0 & -\frac{6D_{EI}}{L^2(1+\mu)} & \frac{2D_{EI}(1-\frac{\mu}{2})}{L(1+\mu)} \\ -\frac{D_{EA}}{L} & 0 & 0 & \frac{D_{EA}}{L} & 0 & 0 \\ 0 & -\frac{12D_{EI}}{L^3(1+\mu)} & -\frac{6D_{EI}}{L^2(1+\mu)} & 0 & \frac{12D_{EI}}{L^3(1+\mu)} & -\frac{6D_{EI}}{L^2(1+\mu)} \\ 0 & \frac{6D_{EI}}{L^2(1+\mu)} & \frac{2D_{EI}(1-\frac{\mu}{2})}{L(1+\mu)} & 0 & -\frac{6D_{EI}}{L^2(1+\mu)} & \frac{4D_{EI}(1+\frac{\mu}{4})}{L(1+\mu)} \end{bmatrix}\end{split}\]

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

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

and where

\[\mu = \frac{12 D_{EI}}{L^2 G A k_s}\]

The element loads \(\mathbf{f}^e_l\) stored in the variable fe are computed according to

\[\mathbf{f}^e_l = \mathbf{G}^T \bar{\mathbf{f}}^e_l\]

where

\[\begin{split}\bar{\mathbf{f}}^e_l = \begin{bmatrix} \frac{q_{\bar{x}} L}{2} \\ \frac{q_{\bar{y}} L}{2} \\ \frac{q_{\bar{y}} L^2}{12} \\ \frac{q_{\bar{x}} L}{2} \\ \frac{q_{\bar{y}} L}{2} \\ -\frac{q_{\bar{y}} L^2}{12} \end{bmatrix}\end{split}\]