I am studying the book "Structural Analysis with Applications to Aerospace Structures" by Craig. In Chapter 8 titled "Thin-Walled Beams", the authors derived the governing differential equation
$$ \frac{\partial n}{\partial x_1} + \frac{\partial f}{\partial s} = 0 $$
where, $n$ is the axial flow, $f$ is shear flow, $x_1$ is axial coordinate, and $s$ is curvilinear coordinate along cross-section.
Substituting $n = \sigma_{xx} \times t(s)$, and $\sigma_{xx}$ from Euler-Bernoulli Beam theory, we have
$$ f = c + \frac{Q_3(s)H_{23}^c - Q_2(s)H_{33}^c}{H_{22}^c H_{33}^c - H_{23}^c H_{23}^c} V_3 - \frac{Q_3(s)H_{22}^c - Q_2(s)H_{23}^c}{H_{22}^c H_{33}^c - H_{23}^c H_{23}^c} V_2$$
with
$Q_2 = \int E x_3 t ds$ and $Q_3 = \int E x_2 t ds$.
where, $c$ is an arbitrary constant which is zero at open end of cross-sections, $H_{ij}^c$ are centriodal bending stiffnesses, $V_i$ are shear forces.
Another way to determine shear stress in a beam cross-section is to use Timoshenko beam theory.
Question: What are the difference between the shear stresses obtained from the two methods? Do both approach capture the same effect?