# Stability of solid rectangular wood member

I'm reading about buckling of timber members from Eurocode 5: Design of timber structures part 1.

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 6.32 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 last formula, how does it result as the critical bending stress from classical theory of stability? I'm used to seeing the flexural buckling calculated with the Euler formula:

$$\sigma_{crit}=\frac{\pi^2EI}{hbL_{eff}^2}$$

Simplifying the $$I$$, using the normal formula $$I=\frac{bh^3}{12}$$ for rectangular member I get:

$$\sigma_{crit}=\frac{\pi^2Eh^2}{12L_{eff}^2}$$

The result does not look similar to what is provided in Eurocode. Some difference in constant for example would not be so surprising as some corrective factors might be used, but the form is completely different.

So, I wonder if someone has better information on the critical bending stress for a rectangular section? Where is my derivation going wrong?

Thank you!

• What is the diameter of a rectangular cross section? the largest dimension? or the smaller? Commented May 29, 2022 at 15:12
• Commented May 29, 2022 at 15:39
• @SolarMike As always, h is the dimension in the direction of bending and b is the width of the beam, perpendicular to the bending direction. The article you provided doesn't address my question, it calculates critical load in the same way I have. Commented May 29, 2022 at 17:02
• But the article does give a better explanation which is what you ask for or need. Commented May 29, 2022 at 17:15