I'm reading about buckling of timber members from Eurocode 5: Design of timber structures part 1.
Section 6.3.3 gives instructions to check the stability of beams subject to compression, bending or combination of both.
Formula 6.30 gives the relative slenderness of a member:
$$\lambda_{rel.m} = \sqrt{\frac{f_{m,k}}{\sigma_{m,crit}}}$$
where $\sigma_{m,crit}$ is the critical bending stress calculated according to the classical theory of stability, using 5-percentile stiffness values.
$f_{m,k}$ is the characteristic bending strength of the timber.
Formula 6.31 gives the equation for critical bending moment for lateral torsional buckling:
$$\sigma_{m,crit}=\frac{\pi \sqrt{E_{0.05}IG_{0.05}I_t}}{WL_{eff}}$$
Formula 6.32 gives the critical bending stress for solid rectangular cross-section:
$$\sigma_{m,crit}=\frac{0,78b^2}{hL_{ef}}E_{0,05}$$
I'm curious about this difference between formulae 6.31 and 6.32. The general formula 6.31 is the usual formula for lateral torsional buckling that I'm familiar with. For some reason, a different formula for rectangular cross sections are given. One thing that strikes me is that this formula does not include the shear modulus $G$, or the torsional stiffness $I_t$.
Why isn't the lateral torsional critical moment of rectangular cross sections dependent on shear modulus $G$ or torsional stiffness? Are the two formulas even equivalent (does the specific formula for rectangular sections follow from the general one?)