Home Of Metal Buildings Engineers

## Beam Elastic Line

The strainging actions of a Simple Beam at both Ends 1 & 2 can be evaluated using the following Matrix Form as a function in the beam elastic line above deformed shape: \begin{align*} \begin{Bmatrix} V_{1}\\ M_{1}\\ V_{2}\\ M_{2} \end{Bmatrix} & = \frac{EI}{L^{3}} \begin{pmatrix} 12 & 6L & -12 & 6L\\ 6L & 4L^{2} & -6L & 2L^{2}\\ -12 & -6L & 12 & -6L\\ 6L & 2L^{2} & -6L & 4L^{2} \end{pmatrix} \begin{Bmatrix} v_{1}\\ \theta _{1}\\ v_{2}\\ \theta _{2} \end{Bmatrix} \end{align*} The above now is in the form $$\{p\}=[K]\{d\}$$ The Beam stiﬀness matrix [K] is : $\left[ K \right] = \frac{EI}{L^{3}} \begin{pmatrix} 12 & 6L & -12 & 6L\\ 6L & 4L^{2} & -6L & 2L^{2}\\ -12 & -6L & 12 & -6L\\ 6L & 2L^{2} & -6L & 4L^{2} \end{pmatrix}$