It depends on the behavior you are after and the material makeup of the structure. If the behavior can be represented in terms of extension, bending, and torsion, and if the material is regular, then E-B beam theory is applicable to a beam with aspect ratio larger than 10. Larger should be better. Now, the E-B beam theory I am referring to a a beam theory without correspoinding ad hoc assumptions. Please refer to my recent work on beams.
Liu, X. and Yu, W.: "A Novel Approach to Analyze Beam-like Composite Structures Using Mechanics of Structure Genome," Advances in Engineering Software, vol. 100, 2016, pp. 238-251.
Peng, B.; A; Goodsell, J.; Pipes, R. B. and Yu, W.: "Generalized Free-Edge Stress Analysis Using Mechanics of Structure Genome," Journal of Applied Mechanics, vol. 83 (10), 2016, 101013.
Bernoulli principle of plane sections is no longer valid for small values L/h. The larger, the better as a rule, nevertheless the beam must have some flexural rigidity. Otherwise it would have become a string, which cannot support bending moments.
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