# Why isn't lateral torsional buckling of rectangular timber members dependent on torsional stiffness?

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?)

• Since I would normally be defined as (b*h^3)/12, What would It be defined as? My guess is the the general terms G,W and It manage to somehow cancel each other out and simplify down to the formula in 6.32 Commented Jun 13, 2022 at 18:58
• @ForwardEd Thank you. I find it unintuitive that such cancellation is possible, as G and E are material properties and the rest of the parameters are geometric ones. Cancelling out material parameters with ones that depend only on geometry (such as I) does not seem possible.. Commented Jun 14, 2022 at 18:01
• You need to go back and look how formula 6.31 was derived. There may be something in that formula build up that cancels out G under certain geometric conditions. Not saying this is the case, just a possibility. I do not have the derivation equation to test with. What I do see though is the I, G, It and W terms disappear but the b and the h terms appear which would point me towards some sort of simplification. Commented Jun 14, 2022 at 18:11
• Any possibility that the torsion consideration is negligible for a solid rectangle so that part of the equation is eliminated and simplified and errs on the conservative side? Commented Jun 14, 2022 at 18:12
• @ForwardEd It could be so. I have some texts on the theory which I could inspect further. Commented Jun 14, 2022 at 20:32