This note develops some of the theory supporting the ShearFiber finite element section model in OpenSees.
We look to motivate the definition of the monosymmetry constant $\beta_z$.
When higher order kinematic terms are incorporated in the formulation of a cross section, an additional term \(T_{\mathrm{wag}}\) contributes to the resultant torque \(T\):
$$ T_{\mathrm{wag}} \triangleq \vartheta' \int \sigma \tilde{r}^2\, \mathrm{d}\alpha $$where $\tilde{r}^2 \triangleq (y-\tilde{\rho}_y)^2 + (z-\tilde{\rho}_z)^2$ and
$$ \sigma = E\, \begin{bmatrix} 1 & z & -y & -\varphi \end{bmatrix} \begin{pmatrix} \varepsilon \\ \theta_y' \\ \theta_z' \\ \vartheta'' \end{pmatrix} $$so that
$$ T_{\mathrm{wag}} = \vartheta'\int E \tilde{r}^2 \, \begin{bmatrix} 1 & z & -y & -\varphi \end{bmatrix} \, \mathrm{d}\alpha \begin{pmatrix} \varepsilon \\ \theta_y' \\ \theta_z' \\ \vartheta'' \end{pmatrix} $$Thus, one might define
$$ C_z \triangleq \int y \tilde{r}^2 \, \mathrm{d}\alpha $$and simplify
$$ \begin{aligned} C_z &= \int\left(y-\tilde{\rho}_y\right)^2 y+\left(z-\tilde{\rho}_z\right)^2 y \, \mathrm{d}\alpha \\ &= \int y^3-2 y^2 \tilde{\rho}_y+\tilde{\rho}_y^2 y+z^2 y-2 \tilde{\rho}_z z y+\tilde{\rho}_z^2 y \,\mathrm{d}\alpha \\ & =\int y^3\,\mathrm{d}\alpha - 2 \tilde{\rho}_y I_{z z}+\left(\tilde{\rho}_y^2+\tilde{\rho}_z^2\right) A_z+\int z^2 y\,\mathrm{d}\alpha +2 \tilde{\rho}_z I_{z y} \\ & =\int y^3+z^2 yz \, \mathrm{d}\alpha - 2 \tilde{\rho}_y I_{z z}+2 \tilde{\rho}_z I_{z y}+\tilde{\rho}^2 A_z \end{aligned} $$where $I_{zy} \triangleq - \int zy\,\mathrm{d}\alpha$ and $\tilde{\rho}^2 \triangleq \tilde{\rho}_z^2 + \tilde{\rho}_y^2$. Thus, in terms of the monosymmetry constant $\beta_z$
$$ C_z = I_{zz} \beta_z + 2 \tilde{\rho}_z I_{z y}+\tilde{\rho}^2 A_z $$where
$$ \beta_z \triangleq \frac{1}{I_{zz}} \int y^3+z^2 yz \, \mathrm{d}\alpha - 2 \tilde{\rho}_y $$For examples of this section formulation, got to https://gallery.stairlab.io