Usually $\kappa = \frac{\text{average shear strain on section}}{\text{shear strain at centroid}}$ but the problem with this definition is that, it depends on external loading so you can't actually make table for $\kappa$ values. However we could make an assumption and define $\kappa$ purely in terms the shape of section. $\kappa_i = \frac{1}{\alpha_i}$. Then we can calculate from this expression. I won't go into details of the terms such as $C_{ij}$ and $A_{ij}$. You could find details for it in this document Shear correction factors in Timoshenko’sbeam theory for arbitrary shaped cross–sections. They also made a program for calculation of $\alpha_i$. Read the document if you have any problem, hit me up.

Usually $\kappa = \dfrac{\text{average shear strain on section}}{\text{shear strain at centroid}}$ but the problem with this definition is that it depends on external loading so you can't actually make a table for $\kappa$ values. However, we could make an assumption and define $\kappa$ purely in terms of the shape of the section: $\kappa_i = \frac{1}{\alpha_i}$. Then we can calculate from this expression:

$$\begin{align} \alpha_y &= \dfrac{A}{Q_y}\left[\dfrac{A_{\varphi\bar{y}}A_{\bar{z}\bar{z}} - A_{\varphi\bar{z}}A_{\bar{y}\bar{z}}}{A_{\bar{y}\bar{y}}A_{\bar{z}\bar{z}} - A^2_{\bar{y}\bar{z}}} + \dfrac{\nu}{2(1+\nu)}\dfrac{C_{zz}A_{\bar{z}\bar{z}} - C_{yy}A_{\bar{y}\bar{z}}}{A_{\bar{y}\bar{y}}A_{\bar{z}\bar{z}} - A^2_{\bar{y}\bar{z}}}\right] \\ \alpha_z &= \dfrac{A}{Q_z}\left[\dfrac{A_{\varphi\bar{z}}A_{\bar{y}\bar{y}} - A_{\varphi\bar{y}}A_{\bar{y}\bar{z}}}{A_{\bar{y}\bar{y}}A_{\bar{z}\bar{z}} - A^2_{\bar{y}\bar{z}}} + \dfrac{\nu}{2(1+\nu)}\dfrac{C_{yy}A_{\bar{y}\bar{y}} - C_{zz}A_{\bar{y}\bar{z}}}{A_{\bar{y}\bar{y}}A_{\bar{z}\bar{z}} - A^2_{\bar{y}\bar{z}}}\right] \end{align}$$

I won't go into details of the terms such as $C_{ij}$ and $A_{ij}$. You could find details for it in this document Shear correction factors in Timoshenko’sbeam theory for arbitrarily shaped cross–sections. They also made a program for the calculation of $\alpha_i$. Read the document if you have any problem, hit me up.