ACS SASSI Documentation

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====== NAS Command Transformations ======

The transformation equations for the [[com:NASVEC]] and [[com:NASCCV]] commands.    

===== Stress Transformation Equations =====

#;;
\(\quad \alpha = \text{Angle between global X and local X}\)\\
\(\quad t = \text{Shell thickness}\)\\
\(\quad E = \text{Shell material Young's modulus}\)\\
\(\quad \nu = \text{Shell material Poisson ratio}\)\\
#;;
==== Thin Shell ====
#;;
\( \quad S_{x_G} = S_{x_L}*cos^2(\alpha)+2*S_{xy_L}*sin(\alpha)*cos(\alpha)+S_{y_L}sin^2(\alpha) \)
\( \quad S_{y_G} = S_{x_L}*sin^2(\alpha)-2*S_{xy_L}*sin(\alpha)*cos(\alpha)+S_{y_L}cos^2(\alpha) \)
\( \quad S_{xy_G} = S_{xy_L}*(cos^2(\alpha)-sin^2(\alpha))-(S_{y_L}+S_{x_L})*sin(\alpha)*cos(\alpha) \)
\( \quad M_{x_G} = M_{x_L}*cos^2(\alpha)+2*M_{xy_L}*sin(\alpha)*cos(\alpha)+M_{y_L}sin^2(\alpha) \)
\( \quad M_{y_G} = M_{x_L}*sin^2(\alpha)-2*M_{xy_L}*sin(\alpha)*cos(\alpha)+M_{y_L}cos^2(\alpha) \)
\( \quad M_{xy_G} = M_{xy_L}*(cos^2(\alpha)-sin^2(\alpha))-(M_{y_L}+M_{x_L})*sin(\alpha)*cos(\alpha) \)
#;;
==== Thick  Shell ====
\( \quad N_{x_G} = N_{x_L}*cos^2(\alpha)+2*N_{xy_L}*sin(\alpha)*cos(\alpha)+N_{y_L}sin^2(\alpha) \)
\( \quad N_{y_G} = N_{x_L}*sin^2(\alpha)-2*N_{xy_L}*sin(\alpha)*cos(\alpha)+N_{y_L}cos^2(\alpha) \)
\( \quad N_{xy_G} = N_{xy_L}*(cos^2(\alpha)-sin^2(\alpha))-(N_{y_L}+N_{x_L})*sin(\alpha)*cos(\alpha) \)
\( \quad M_{x_G} = M_{x_L}*cos^2(\alpha)+2*M_{xy_L}*sin(\alpha)*cos(\alpha)+M_{y_L}sin^2(\alpha) \)
\( \quad M_{y_G} = M_{x_L}*sin^2(\alpha)-2*M_{xy_L}*sin(\alpha)*cos(\alpha)+M_{y_L}cos^2(\alpha) \)
\( \quad M_{xy_G} = M_{xy_L}*(cos^2(\alpha)-sin^2(\alpha))-(M_{y_L}+M_{x_L})*sin(\alpha)*cos(\alpha) \)
\( \quad Q_{x_G} = Q_{x_L}*cos(\alpha)+Q_{y_L}*sin(\alpha) \) \\
\( \quad Q_{y_G} = -Q_{x_L}*sin(\alpha)+Q_{y_L}*cos(\alpha) \)
===== Strain  Equations =====
$$ \varepsilon_x = \frac{\sigma_x}{E}-\nu \frac{\sigma_y}{E} \qquad \varepsilon_y = \frac{\sigma_y}{E}-\nu \frac{\sigma_x}{E} \qquad \varepsilon_{xy} = \frac{\tau_{xy}}{E}*2(1+\nu)  $$
$$\varepsilon_{1,2} = \frac{\varepsilon_x+\varepsilon_y}{2} \pm \frac{\sqrt{(\varepsilon_x-\varepsilon_y)^2+\varepsilon_{xy}^2}}{2}$$
==== Thin Shell ====
$$ \sigma_x=S_{x_G} \pm \frac{6*M_{x_G}}{t^2} \qquad \sigma_y=S_{y_G} \pm \frac{6*M_{y_G}}{t^2} \qquad \tau_{xy}=S_{xy_G} \pm \frac{6*M_{xy_G}}{t^2}$$

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

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