User:Eml4500.f08.echo.lacroix/HW2

=Mon, Sept 15=

$$\mathbf{K}= \begin{bmatrix} k_{11}^{(1)} & k_{12}^{(1)} & k_{13}^{(1)} & k_{14}^{(1)} & 0 & 0\\k_{21}^{(1)} & k_{22}^{(1)} & k_{23}^{(1)} & k_{24}^{(1)} & 0 & 0\\k_{31}^{(1)} & k_{32}^{(1)} & k_{33}^{(1)}+k_{11}^{(2)} & k_{34}^{(1)}+k_{12}^{(2)} & k_{13}^{(2)} & k_{14}^{(2)}\\k_{41}^{(1)} & k_{42}^{(1)} & k_{43}^{(1)}+k_{21}^{(2)} & k_{44}^{(1)}+k_{22}^{(2)} & k_{23}^{(2)} & k_{24}^{(2)}\\0 & 0 & k_{31}^{(2)} & k_{32}^{(2)} & k_{33}^{(2)} & k_{34}^{(2)}\\0 & 0 & k_{41}^{(2)} & k_{42}^{(2)} & k_{43}^{(2)} & k_{44}^{(2)}\end{bmatrix}= \begin{bmatrix} K_{11} & K_{12} & K_{13} & K_{14} & K_{15} & K_{16}\\K_{21} & K_{22} & K_{23} & K_{24} & K_{25} & K_{26}\\K_{31} & K_{32} & K_{33} & K_{34} & K_{35} & K_{36}\\K_{41} & K_{42} & K_{43} & K_{44} & K_{45} & K_{46}\\K_{51} & K_{52} & K_{53} & K_{54} & K_{55} & K_{56}\\K_{61} & K_{62} & K_{63} & K_{64} & K_{65} & K_{66}\end{bmatrix}$$

$$K_{11}=k_{11}^{(1)}=\frac{9}{16}=0.5625$$

$$K_{12}=K_{21}=k_{12}^{(1)}=\frac{3\sqrt{3}}{16}=0.3248$$

$$K_{13}=K_{31}=k_{13}^{(1)}=-k_{11}^{(1)}=-0.5625$$

$$K_{14}=K_{23}=K_{32}=K_{41}=k_{14}^{(1)}=-k_{12}^{(1)}=-0.3248$$

$$K_{22}=k_{22}^{(1)}=\frac{3}{16}=0.1875$$

$$K_{24}=K_{42}=k_{24}^{(1)}=-k_{22}^{(1)}=-0.1875$$

$$K_{33}=k_{33}^{(1)}+k_{11}^{(2)}=	\begin{matrix} \frac{9}{16} \end{matrix}+\begin{matrix} \frac{5}{2} \end{matrix}=3.0625$$

$$K_{34}=k_{34}^{(1)}+k_{12}^{(2)}=	\begin{matrix} \frac{3\sqrt{3}}{16} \end{matrix}+\begin{matrix} \frac{-5}{2} \end{matrix}=-2.1752$$

$$K_{35}=K_{53}=k_{13}^{(2)}=-k_{11}^{(2)}=\frac{-5}{2}=-2.5$$

$$K_{36}=K_{45}=K_{54}=K_{63}=k_{14}^{(2)}=-k_{12}^{2}=2.5$$

$$K_{43}=K_{34}=k_{43}^{(1)}+k_{21}^{(2)}=k_{34}^{(1)}+k_{12}^{(2)}=-2.1752$$

$$K_{44}=k_{44}^{(1)}+k_{22}^{(2)}=\frac{3}{16}+\frac{5}{2}=2.6875$$

$$K_{46}=K_{64}=k_{24}^{(2)}=\frac{-5}{2}=-2.5$$

$$K_{55}=k_{33}^{(2)}=\frac{5}{2}=2.5$$

$$K_{56}=K_{65}=k_{34}^{(2)}=\frac{-5}{2}=-2.5$$

$$K_{66}=k_{44}^{(2)}=\frac{5}{2}=2.5$$

$$K_{15}=K_{16}=K_{25}=K_{26}=K_{51}=K_{52}=K_{61}=K_{62}=0$$

$$\mathbf{K}=\begin{bmatrix} 0.5625 & 0.3248 & -0.5625 & -0.3248 & 0 & 0\\0.3248 & 0.1875 & -0.3248 & -0.1875 & 0 & 0\\-0.5625 & -0.3248 & 3.0625 & -2.1752 & -2.5 & 2.5\\-0.3248 & -0.1875 & -2.1752 & 2.6875 & 2.5 & -2.5\\0 & 0 & -2.5 & 2.5 & 2.5 & -2.5\\0 & 0 & 2.5 & -2.5 & -2.5 & 2.5\end{bmatrix}$$