## 2 Materials and Methods
### Design Models  
Several design proposals can be found in the literature to predict the splitting capacity of timber connections loaded perpendicular to the grain. However, there is much discussion on the parameters influencing the splitting behaviour, and no solid proposal has yet been developed that is widely accepted by the research community. A comprehensive review of most of the existing models, their theoretical basis, and their scope in the design of timber connections can be found in [34; 60]. A summary of simple analytical models and the relationships between them is compiled in [61]. The three design models used in this work to predict the splitting failure load of the European beadopted. Since the expression of the code is only valid for softwoods, in the present work it was decided to apply the original relationship and determine the \(C_{1}\) value for European beech instead of \(C_{1}=14\) N/mm\({}^{1.5}\), by means of the shear modulus \(G\) and critical fracture energy \(G_{\rm c}\) of the material obtained experimentally, resulting in \(C_{1}=23.27\) N/mm\({}^{1.5}\). The second design model that was applied to estimate European beech's carrying capacity in the present work is the model formulated by [PERSON] [46]. It relies on [PERSON]'s beam theory for elastic foundations (BEFs) and the quasi-non-linear fracture mechanics. For its development, a beam is considered with a cracking layer modelled by springs to which the material fracture properties are assigned. Upon cracking, the beam below this layer is considered to be resting on elastic Wrinkler springs connected to the upper part, which is assumed to be infinitely rigid. For a single load acting far from the beam end and small crack lengths, Equations (18)-(22) result:  
\[P_{u}=\gamma\,P_{u,LEFM} \tag{18}\]  
\[P_{u,LEFM}=2 bC_{1}\sqrt{\frac{h_{e}}{1-\frac{h_{e}}{h}}} \tag{19}\]  
\[\gamma=\frac{\sqrt{2\zeta_{r}+1}}{\zeta+1} \tag{20}\]  
\[\zeta=\frac{C_{1}}{f_{t}}\sqrt{10\frac{G}{E}\frac{1}{h_{e}}} \tag{21}\]  
\[C_{1}=\sqrt{\frac{5}{3}GG_{f}} \tag{22}\]  
with \(P_{\rm u}\) being the splitting load of the connection, \(E\) the longitudinal modulus of elasticity, \(G\) the shear modulus of elasticity, and \(G_{f}\) the fracture energy. One of the differentiating features of this model compared to previous proposals is that it takes into account the perpendicular tensile strength of the timber, \(f_{t}\). The third of the design approaches selected to estimate the theoretical splitting failure load of European beech is the one postulated by [PERSON] and [PERSON] [47] given by Equations (23)-(26). The design approach was developed through experimental investigations and finite element numerical simulations considering numerous connection arrangements, in the framework of non-linear fracture mechanics. It is one of the most comprehensive models in the literature, which for the first time considers mixed-mode I + II (tension and shear) failure criteria that represent a more realistic situation and not only mode I, as most of the previous models conservatively do:  
\[F_{90}=\frac{b\cdot 10^{3}}{\left(\frac{C_{1,\,\rm norm}}{C_{\rm lc}}+\frac{C_{ \rm II,\,\rm norm}}{C_{\rm llc}}\right)}\cdot k_{r} \tag{23}\]  
\[G_{\rm I,\,\rm norm}=e^{(h^{-1}(200-10h_{e}h^{-0.25}-a_{r}))} \tag{24}\]  
\[G_{\rm II,\,\rm norm}=\left(0.05+0.12\frac{h_{e}}{h}+1\cdot 10^{-3}a_{r}\right) \tag{25}\]  
\[k_{r}=\left\{ \begin{array}{ll}1&\mbox{for $n=1$\ down}\\ 0.1+\left(\arctan(n)\right)^{0.6}&\mbox{for $n>1$\ down}\end{array} \right. \tag{26}\]  
with \(F_{90}\) being the splitting load of the connection, \(G_{\rm lc}\) and \(G_{\rm llc}\) the critical fracture energy of the material in mode I and II, respectively, and \(G_{\rm I,\,\rm norm}\) and \(G_{\rm ll,\,\rm norm}\) the normalised fracture energy in mode I and II, respectively. The model also includes other parameters concerning the connection layout, such as the width of the connection (\(a_{r}\)) and the number of rows of fasteners parallel to the grain through the \(k_{r}\) factor.