Problem $1.$ The reduced stiffness matrix can be expressed either in the materials' $($local$)$ coordinate system $("1-2")$ or in the global coordinate system $("x-y")$ in the following forms:

$\left[\begin{array}{c}{}_1\\ {}_2\\ {}_{12}\end{array}\right]=\left[\begin{array}{ccc}{Q}_{11}& Q_{12}& 0\\ Q_{12}& Q_{22}& 0\\ 0& 0& Q_{66}\end{array}\right]\left[\begin{array}{c}{}_1\\ {}_2\\ {}_{12}\end{array}\right],\left[\begin{array}{c}{}_x\\ {}_y\\ {}_{v1}\end{array}\right]=\left[\begin{array}{ccc}\frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{11}& \frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{12}& \frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{16}\\ \frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{12}& \frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{22}& \frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{26}\\ \frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{16}& \frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{26}& \frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{66}\end{array}\right]\left[\begin{array}{c}{}_x\\ {}_y\\ {}_x\end{array}\right]$

Derive the following transformation equations for Q matrix between local and global coordinate systems, where s$=$sin$($ and cocos$(),$ is the angle between global and local coordinate systems.

$\frac{?}{\text{b}\text{a}\text{r}}(Q{)}_{11}=Q_{11}{c}^4+Q_{22}{s}^4+2(Q_{12}+2Q_0)s^2{c}^2.\frac{?}{b}$

$ar(Q{)}_n=(Q_n+Q_z-4Q_w)s^2{c}^2+Q_n(c^4+s^2).\frac{?}{b}$

$ar({)}_{1s}=(Q_{11}-Q_{12}-2Q_{})e^{\text{'}}s-(Q_2-Q_{12}-2Q_1)s^{3}c\frac{?}{b}$

$ar(Q{)}_{22}=Q_{12}{s}^4+Q_{22}{c}^4+2(Q_{12}+2Q_m)s^2{c}^2.\frac{?}{b}$

$ar(Q{)}_{s0}=(Q_n-Q_{n1}-2Q_{en})c{s}^{\text{'}}-(Q_n-Q_{12}-2Q_{ns})c^{\text{'}}{s}_s\frac{?}{b}$

$ar(Q{)}_{es}=(Q_{11}+Q_{22}-2Q_{12}-2Q_e)s^2{c}^2+Q_{es}(s^4+c^t)$