The detailed definitions of local stress and strain in a body will come later, but for now, we can still use failure theories. In short, we will evaluate geometric properties and internal forces, which we will use to determine maximum stresses, which we will compare to material properties in order to predict yielding.
For circular cross sections, the area, second moment of area, and the second polar moment of area are as follows.
$A=\frac\pi 4(d_o^2-d_i^2)=\pi(r_o^2-r_i^2)$
$I_z=I_y=\frac\pi{64}(d_o^4-d_i^4)=\frac\pi 4(r_o^4-r_i^4)$
$J=I_y+I_z$
For rectangular cross section with dimension $b$ along the $z$ direction, $h$ along the $y$ direction, and with a cavity of dimension $b_i\times h_i$
$I_z=\frac1{12}bh^3-\frac1{12}b_ih_i^3$
$I_y=\frac1{12}b^3h-\frac1{12}b_i^3h_i$
$c_y=\frac h2$
$c_z=\frac b2$
These describe the maximum stress induced in a beam of a circular cross section in terms of internal forces. We will compare these stresses to material properties to predict failure. ($\hat{x}$ denotes maximum $x$)
$\sigma_{axial}=\frac{N_x}{A}$
$\hat{\sigma}_{bending}=\frac{r_o}{I}\sqrt{M_y^2+M_z^2}$
$\hat{\tau}_{torsion}=\frac{T_xr_o}{J}$
For more complex loading, we may utilize the Von Mises stress, which considers the all possible internal forces. It’s derivation requires the use of stress tensors, which are introduced at the end of the course. It can be written in terms of the above stresses, or internal forces directly.
$$ \hat{\sigma}{vm}=\sqrt{(\sigma{a}+\hat{\sigma}{b})^2+3\hat{\tau}{}^2}=\sqrt{(\frac{|N_x|}{A}+\frac{r_o}{I}\sqrt{M_y^2+M_z^2})^2+3(\frac{T_xr_o}{J})^2} $$
The below equations are the analogs for rectangular cross sections with corners at $(\pm c_y,\pm c_z)$, Internal cavity of area $A_i$ and cross sectional area of $A=A_o-A_i$. $A_i>>A\Rarr\eta=1.0$
$\sigma_{axial}=\frac{N_x}{A}$