Parallel Axis

Consider the inertia tensor $\bm{I}_x$ defined by $$ \bm{I}_x \triangleq \int \bm{x} \cdot \bm{x} \, \mathbb{1} - \bm{x} \otimes \bm{x} \, \mathrm{d} \alpha $$ The transformed tensor $\bm{I}_{x+\rho}$ corresponding to a translation by $\bm{\rho}$ is obtained by replacing $\bm{x}$ with $\bm{x} + \bm{\rho}$, furnishing $$ \bm{I}_{x+\rho} = \bm{I}_x + 2 A \,\bm{\rho}\cdot \bar{\bm{\rho}} \, \mathbb{1} - A (\bar{\bm{\rho}}\otimes\bm{\rho} + \bm{\rho}\otimes\bar{\bm{\rho}}) + A (\bm{\rho}\cdot\bm{\rho}\,\mathbb{1} - \bm{\rho}\otimes\bm{\rho}) $$ where the centroid $\bar{\bm{\rho}}$ and area $A$ are defined as

Monosymmetry constants

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