User:Eml4500.f08.qwiki.bishop/Lecture 8

=Lecture 8 - Friday, September 12, 2008=
 * Global Force-Displacement relation:
 * $$\mathbf{K_{(nxn)}d_{(nx1)} = F_{(nx1)}}$$
 * For this example, n=6

$$\begin{bmatrix} k_{11} & k_{12} & k_{13} & k_{14} & k_{15} & k_{16}\\ k_{21} & k_{22} & k_{23} & k_{24} & k_{25} & k_{26}\\ k_{31} & k_{32} & k_{33} & k_{34} & k_{35} & k_{36}\\ k_{41} & k_{42} & k_{43} & k_{44} & k_{45} & k_{46}\\ k_{51} & k_{52} & k_{53} & k_{54} & k_{55} & k_{56}\\ k_{61} & k_{62} & k_{63} & k_{64} & k_{65} & k_{66}\\ \end{bmatrix}\begin{pmatrix} d_1\\ d_2\\ d_3\\ d_4\\ d_5\\ d_6\\ \end{pmatrix}=\begin{pmatrix} F_1\\ F_2\\ F_3\\ F_4\\ F_5\\ F_6\\ \end{pmatrix}$$


 * In compact notation, this matrix can be written in the form:

$$\begin{bmatrix} k_{ij}\\ \end{bmatrix}_{(6x6)}\begin{pmatrix} d_j\\ \end{pmatrix}_{(6x1)}=\begin{pmatrix} F_i\\ \end{pmatrix}_{(6x1)}$$


 * (More generally, nxn)

$$\sum_{j=1}^{6}{K_{ij}d_j}=F_i, i=1,...,6$$

The above global matrices can be define as follows:


 * $$K_{(nxn)}=[K_{ij}]_{(nxn)}$$ = Global Stiffness Matrix
 * $$d_{(nx1)}=(d_j)_{(nxn)}$$ = Global Displacement Matrix
 * $$F_{(nx1)}=(f_i)_{(nxn)}$$ = Global Force Matrix

When dealing with a single element, the matrices can be written as:


 * $$K_{(4x4)}^{(e)}=[K_{ij}^{(e)}]_{(4x4)}$$ = Element Stiffness Matrix
 * $$d_{(4x1)}^{(e)}=(d_j^{(e)})_{(4x4)}$$ = Element Displacement Matrix
 * $$F_{(4x1)}^{(e)}=(f_i^{(e)})_{(4x4)}$$ = Element Force Matrix

An assembly process is used to go form an element matrix to a global matrix. Identify the correspondence between element displacement degrees of freedom and global displacement degrees of freedom. The following examples show how to go from the global level to the element level for each node.


 * Examples for node 1:
 * $$d_1=d_1^{(1)}$$
 * $$d_2=d_2^{(1)}$$


 * Examples for node 2:
 * $$d_3=d_3^{(1)}=d_1^{(2)}$$
 * $$d_4=d_4^{(1)}=d_2^{(2)}$$


 * Examples for node 3:
 * $$d_5=d_3^{(2)}$$
 * $$d_6=d_4^{(2)}$$