## 5. Assembly of system equation (cont'd)

 $\left(\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right)\left(\begin{array}{c}{w}_{1}\\ {w}_{2}\\ {w}_{3}\end{array}\right)=\left(\begin{array}{c}{f}_{1}\\ {f}_{2}\\ {f}_{3}\end{array}\right)$ and $\left(\begin{array}{ccc}{b}_{11}& {b}_{12}& {b}_{13}\\ {b}_{21}& {b}_{22}& {b}_{23}\\ {b}_{31}& {b}_{32}& {b}_{33}\end{array}\right)\left(\begin{array}{c}{w}_{2}\\ {w}_{3}\\ {w}_{4}\end{array}\right)=\left(\begin{array}{c}{g}_{2}\\ {g}_{3}\\ {g}_{4}\end{array}\right)\text{.}$ (Eqn. 11)

Now, since the $$w_2$$ and $$w_3$$ in Equation 11 are the same for both elements, we can combine the two equations as follows:
 $\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& {a}_{13}& 0\\ {a}_{21}& {a}_{22}+{b}_{11}& {a}_{23}+{b}_{12}& {b}_{13}\\ {a}_{31}& {a}_{32}+{b}_{21}& {a}_{33}+{b}_{22}& {b}_{23}\\ 0& {b}_{31}& {b}_{32}& {b}_{33}\end{array}\right)\left(\begin{array}{c}{w}_{1}\\ {w}_{2}\\ {w}_{3}\\ {w}_{4}\end{array}\right)=\left(\begin{array}{c}{f}_{1}\\ {f}_{2}+{g}_{2}\\ {f}_{3}+{g}_{3}\\ {g}_{4}\end{array}\right)\text{.}$ (Eqn. 12)

R. Funnell