I have try to follow the description from Pro:Holz CLT Structural Design, see link.
The annex A2 gives formulas and some standard values but they don't really match.
http://www.binderholz.com/fileadmin/PDF/Basisprodukte/Brettsperrholz_BBS/201411_proHolz_Cross-Laminated_Timber_Structural_Design.pdf
Hello Pierre Landel,
the formulas given in A2 are derived based on the complementary energy approach as far as I can see. That means, the complementary energy with respect to the transverse shear stress for the Timoshenko beam is compared with the corresponding value by assuming certain approximations for the transverse shear stress with respect to the thickness coordinate. I assume that formally, these formulas are correct. But you may check these formulas by computing the bounds of the transverse shear stiffness by Voigt and Reuss estimates.
The problem you addressed may also be related to the accuracy of the Timoshenko beam theory rather than the shear correction factor. For laminates with shear compliant layers the Timoshenko theory may overestimate the deflection of the beam significantly, even if the shear correction factor is accurately computed. An example is a laminated glass beam with a compliant PVB core layer, discussed in our paper:
https://www.researchgate.net/publication/256733970_Analysis_of_laminated_glass_beams_for_photovoltaic_applications
Article Analysis of laminated glass beams for photovoltaic applications
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