This note develops some of the theory supporting the ShearFiberβ z \beta_z β 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 w a g ≜ ϑ ′ ∫ σ r ~ 2 d α T_{\mathrm{wag}} \triangleq \vartheta' \int \sigma \tilde{r}^2\, \mathrm{d}\alpha T wag ≜ ϑ ′ ∫ σ r ~ 2 d α where r ~ 2 ≜ ( y − ρ ~ y ) 2 + ( z − ρ ~ z ) 2 \tilde{r}^2 \triangleq (y-\tilde{\rho}_y)^2 + (z-\tilde{\rho}_z)^2 r ~ 2 ≜ ( y − ρ ~ y ) 2 + ( z − ρ ~ z ) 2
σ = E [ 1 z − y − φ ] ( ε θ y ′ θ z ′ ϑ ′ ′ ) \sigma = E\, \begin{bmatrix}
1 & z & -y & -\varphi
\end{bmatrix}
\begin{pmatrix}
\varepsilon \\ \theta_y' \\ \theta_z' \\ \vartheta''
\end{pmatrix} σ = E [ 1 z − y − φ ] ε θ y ′ θ z ′ ϑ ′′ so that
T w a g = ϑ ′ ∫ E r ~ 2 [ 1 z − y − φ ] d α ( ε θ y ′ θ z ′ ϑ ′ ′ ) 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} T wag = ϑ ′ ∫ E r ~ 2 [ 1 z − y − φ ] d α ε θ y ′ θ z ′ ϑ ′′ Thus, one might define
C z ≜ ∫ y r ~ 2 d α C_z \triangleq \int y \tilde{r}^2 \, \mathrm{d}\alpha C z ≜ ∫ y r ~ 2 d α and simplify
C z = ∫ ( y − ρ ~ y ) 2 y + ( z − ρ ~ z ) 2 y d α = ∫ y 3 − 2 y 2 ρ ~ y + ρ ~ y 2 y + z 2 y − 2 ρ ~ z z y + ρ ~ z 2 y d α = ∫ y 3 d α − 2 ρ ~ y I z z + ( ρ ~ y 2 + ρ ~ z 2 ) A z + ∫ z 2 y d α + 2 ρ ~ z I z y = ∫ y 3 + z 2 y z d α − 2 ρ ~ y I z z + 2 ρ ~ z I z y + ρ ~ 2 A z \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} C z = ∫ ( y − ρ ~ y ) 2 y + ( z − ρ ~ z ) 2 y d α = ∫ y 3 − 2 y 2 ρ ~ y + ρ ~ y 2 y + z 2 y − 2 ρ ~ z z y + ρ ~ z 2 y d α = ∫ y 3 d α − 2 ρ ~ y I z z + ( ρ ~ y 2 + ρ ~ z 2 ) A z + ∫ z 2 y d α + 2 ρ ~ z I z y = ∫ y 3 + z 2 y z d α − 2 ρ ~ y I z z + 2 ρ ~ z I z y + ρ ~ 2 A z where I z y ≜ − ∫ z y d α I_{zy} \triangleq - \int zy\,\mathrm{d}\alpha I z y ≜ − ∫ z y d α ρ ~ 2 ≜ ρ ~ z 2 + ρ ~ y 2 \tilde{\rho}^2 \triangleq \tilde{\rho}_z^2 + \tilde{\rho}_y^2 ρ ~ 2 ≜ ρ ~ z 2 + ρ ~ y 2 monosymmetry constant β z \beta_z β z
C z = I z z β z + 2 ρ ~ z I z y + ρ ~ 2 A z C_z = I_{zz} \beta_z + 2 \tilde{\rho}_z I_{z y}+\tilde{\rho}^2 A_z C z = I z z β z + 2 ρ ~ z I z y + ρ ~ 2 A z where
β z ≜ 1 I z z ∫ y 3 + z 2 y z d α − 2 ρ ~ y \beta_z \triangleq \frac{1}{I_{zz}} \int y^3+z^2 yz \, \mathrm{d}\alpha - 2 \tilde{\rho}_y β z ≜ I z z 1 ∫ y 3 + z 2 y z d α − 2 ρ ~ y For examples of this section formulation, got to https://gallery.stairlab.io