The transformation equations for the CTRVEC and CTRCCV commands.

\(\quad \alpha = \text{Angle between global X and local X}\)
\(\quad t = \text{Shell thickness}\)
\(\quad E = \text{Shell material Young's modulus}\)
\(\quad \nu = \text{Shell material Poisson ratio}\)

\( \quad S_{x_G} = S_{x_L}*cos^2(\alpha)+2*S_{xy_L}*sin(\alpha)*cos(\alpha)+S_{y_L}sin^2(\alpha) \) \( \quad S_{y_G} = S_{x_L}*sin^2(\alpha)-2*S_{xy_L}*sin(\alpha)*cos(\alpha)+S_{y_L}cos^2(\alpha) \) \( \quad S_{xy_G} = S_{xy_L}*(cos^2(\alpha)-sin^2(\alpha))+(S_{y_L}-S_{x_L})*sin(\alpha)*cos(\alpha) \) \( \quad M_{x_G} = M_{x_L}*cos^2(\alpha)+2*M_{xy_L}*sin(\alpha)*cos(\alpha)+M_{y_L}sin^2(\alpha) \) \( \quad M_{y_G} = M_{x_L}*sin^2(\alpha)-2*M_{xy_L}*sin(\alpha)*cos(\alpha)+M_{y_L}cos^2(\alpha) \) \( \quad M_{xy_G} = M_{xy_L}*(cos^2(\alpha)-sin^2(\alpha))+(M_{y_L}-M_{x_L})*sin(\alpha)*cos(\alpha) \)

\( \quad N_{x_G} = N_{x_L}*cos^2(\alpha)+2*N_{xy_L}*sin(\alpha)*cos(\alpha)+N_{y_L}sin^2(\alpha) \) \( \quad N_{y_G} = N_{x_L}*sin^2(\alpha)-2*N_{xy_L}*sin(\alpha)*cos(\alpha)+N_{y_L}cos^2(\alpha) \) \( \quad N_{xy_G} = N_{xy_L}*(cos^2(\alpha)-sin^2(\alpha))+(N_{y_L}-N_{x_L})*sin(\alpha)*cos(\alpha) \) \( \quad M_{x_G} = M_{x_L}*cos^2(\alpha)+2*M_{xy_L}*sin(\alpha)*cos(\alpha)+M_{y_L}sin^2(\alpha) \) \( \quad M_{y_G} = M_{x_L}*sin^2(\alpha)-2*M_{xy_L}*sin(\alpha)*cos(\alpha)+M_{y_L}cos^2(\alpha) \) \( \quad M_{xy_G} = M_{xy_L}*(cos^2(\alpha)-sin^2(\alpha))+(M_{y_L}-M_{x_L})*sin(\alpha)*cos(\alpha) \) \( \quad Q_{x_G} = Q_{x_L}*cos(\alpha)+Q_{y_L}*sin(\alpha) \)
\( \quad Q_{y_G} = -Q_{x_L}*sin(\alpha)+Q_{y_L}*cos(\alpha) \)

$$ \varepsilon_x = \frac{\sigma_x}{E}-\nu \frac{\sigma_y}{E} \qquad \varepsilon_y = \frac{\sigma_y}{E}-\nu \frac{\sigma_x}{E} \qquad \varepsilon_{xy} = \frac{\tau_{xy}}{E}*2(1+\nu) $$ $$\varepsilon_{1,2} = \frac{\varepsilon_x+\varepsilon_y}{2} \pm \frac{\sqrt{(\varepsilon_x-\varepsilon_y)^2+\varepsilon_{xy}^2}}{2}$$

$$ \sigma_x=S_{x_G} \pm \frac{6*M_{x_G}}{t^2} \qquad \sigma_y=S_{y_G} \pm \frac{6*M_{y_G}}{t^2} \qquad \tau_{xy}=S_{xy_G} \pm \frac{6*M_{xy_G}}{t^2}$$

$$ \sigma_x=\frac{N_{x_G}}{t} \pm \frac{6*M_{x_G}}{t^2} \qquad \sigma_y=\frac{N_{y_G}}{t} \pm \frac{6*M_{y_G}}{t^2} \qquad \tau_{xy}=\frac{N_{xy_G}}{t} \pm \frac{6*M_{xy_G}}{t^2}$$