beam1weΒΆ

Purpose:

Compute element stiffness matrix for a one dimensional beam element on elastic support.

_images/beam1w.svg
Syntax:

Ke = beam1we(ex, ep)
[Ke, fe] = beam1we(ex, ep, eq)
Description:

beam1we provides the global element stiffness matrix \({\mathbf{K}}^e\) for a one dimensional beam element with elastic support.

The input variables

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

supply the element nodal coordinates \(x_1\) and \(x_2\), the modulus of elasticity \(E\), the moment of inertia \(I\), and the spring stiffness in the transverse direction \(k_{\bar{y}}\).

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\(= [q_{\bar{y}}]\)

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

_images/beam1e_2.svg
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

\[\begin{split}\bar{\mathbf{K}}^e_0 = \frac{D_{EI}}{L^3} \begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix}\end{split}\]

and

\[\begin{split}\bar{\mathbf{K}}^e_s = \frac{k_{\bar{y}} L}{420} \begin{bmatrix} 156 & 22L & 54 & -13L \\ 22L & 4L^2 & 13L & -3L^2 \\ 54 & 13L & 156 & -22L \\ -13L & -3L^2 & -22L & 4L^2 \end{bmatrix}\end{split}\]

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

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

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

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