Cross Sectional Properties Calculated by HyperBeam
The beam cross section is always defined in a y,z plane.
The xaxis is defined along the beam axis. The coordinate system you define is called the local coordinate system; the system parallel to the local coordinate system with the origin in the centroid is called the centroidal coordinate system; the system referring to the principal bending axes is called the principal coordinate system.
 Area
 $A={\displaystyle \int dA}$
 Area Moments of Inertia
 ${I}_{yy}={\displaystyle \int {z}^{2}dA}$
 Area Products of Inertia
 ${I}_{zz}={\displaystyle \int {y}^{2}dA}$
 Radius of Gyration
 ${R}_{g}=\sqrt{\frac{{I}_{\mathrm{min}}}{A}}$
 Elastic Section Modulus
 ${E}_{y}=\frac{{I}_{yy}}{{z}_{\mathrm{max}}}$
 Max Coordinate Extension
 ${y}_{\mathrm{max}}=\mathrm{max}\lefty\right$
 Plastic Section Modulus
 ${P}_{y}{\displaystyle \int \leftz\rightdA}$
 Torsional Constant

 Solid
 ${I}_{t}={I}_{yy}+{I}_{zz}+{\displaystyle \int \left(z\frac{\partial \omega}{\partial y}y\frac{\partial \omega}{\partial z}\right)}dA$
 Shell open
 ${I}_{t}=\frac{1}{3}{\displaystyle \int {t}^{3}ds}$
 Shell closed
 ${I}_{t}=2{\displaystyle \sum {A}_{mi}{F}_{si}}$
 Elastic Torsion Modulus

 Solid
 ${E}_{t}=\frac{{I}_{t}}{\mathrm{max}}\left({y}^{2}+{z}^{2}+z\frac{\partial \omega}{\partial y}y\frac{\partial \omega}{\partial z}\right)$
 Shell open
 ${E}_{t}=\frac{{I}_{t}}{\mathrm{max}\text{\hspace{0.17em}}t}$
 Shell closed
 ${E}_{t}=\frac{{I}_{t}}{\mathrm{max}\left(\frac{{F}_{si}}{t}\right)}$
 Shear Center
 ${y}_{s}=\frac{{I}_{yz}{I}_{y\omega}{I}_{zz}{I}_{z\omega}}{{I}_{yy}{I}_{zz}{I}_{yz}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{y\omega =}{\displaystyle \int y\omega dA,{I}_{z\omega}={\displaystyle \int z\omega dA}}$
 Warping Constant (normalized to the shear center)
 ${I}_{\omega \omega}={\displaystyle \int {\omega}^{2}dA}$
 Shear deformation coefficients
 ${\alpha}_{zz}=\frac{1}{{Q}_{y}^{2}}{\displaystyle \int \left({\tau}_{xy}^{2}{}_{{}_{{Q}_{z}=0}}+{\tau}_{xz}^{2}{}_{{}_{{Q}_{z}=0}}\right)}dA$
 Shear stiffness factors
 ${k}_{yy}=\frac{1}{{\alpha}_{zz}}$
 Shear stiffness
 ${S}_{ii}={k}_{ii}GA$
 Warping Function
 ${\nabla}^{2}\omega =0$
Nastran Type Notation
$/1={I}_{zz}$
$/2={I}_{yy}$
$/12={I}_{yz}$
$K1={K}_{yy}$
$K2={K}_{zz}$