# Structural Behaviour of Aluminium-Timber Composite Beams with Partial Shear Connections
## 2 Materials and Methods
### The Numerical Models  
Four 2D numerical models of aluminium-timber composite beams were developed in Abaqus software (Figure 3). The first model represented the composite beam with full shear connection. In this model, a slab was connected with a girder using a tie function. A continuous tie type of interaction is often adopted in the case of full composite action in composite beams [65]. The numerical models 2-4 represent composite beams with partial shear connections tested in the present study. Discrete connections were used to model shear connections and to take slip into account. Discrete connections are often used in numerical models because they provide reasonable accuracy in reflecting the behaviour of the real structure [66; 67]. At each point corresponding to the locations of the screws, the LVL slabs and the aluminium girders were connected using zero-length translator-type connectors. This type of connector makes it possible for the connected elements to move only in one direction (i.e., in the longitudinal direction). For each discrete connection, the forces and displacements from Figure 4 were used. The slip modulus \(k_{u}\) was calculated from Eurocode 5 [68]:  
\[k_{u}=n\frac{2}{69}\rho_{m}^{1.5}d \tag{1}\]  
where \(n\) is the number of connectors in one connection (2 in the analysed beams); \(\rho_{m}\) is the mean density of LVL; \(d\) is the diameter of the screw. The value of the slip modulus \(k_{0}\) was assumed to be 30 kN/mm for the slip ranging from 0 to 0.2 mm to take into account the low initial slip values observed in the beams in the laboratory tests. Additionally, steel elements representing the screws were embedded in the LVL slabs to consider the presence of the screws in the slabs. The loading and supporting steel plates, the aluminium girders, and the LVL slabs were modelled using 4-node bilinear plane stress quadrilateral elements with reduced integration and hourglass control (CPS4R). The screws were modelled using T2D2 linear line elements. The mesh size did not exceed 5 mm. The interaction between beam parts was captured by surface-to-surface \"hard\" contacts and friction contacts. In the case of the LVL-aluminium interface, the friction coefficient equal to 0.3 was adopted, just like for the LVL-steel frictional interaction [52; 69]. In the case of the steel-aluminium interface, a friction coefficient equal to 0.3 was used. Figure 5 presents the boundary conditions used in the numerical analyses. Figure 3: The numerical models: The composite beam with full shear connection (**a**), the B120 beam (**b**), the B160 beam (**c**), the B200 beam (**d**). The aluminium yielding was modelled by an elastic-isotropic hardening plastic constitutive law (\(E=70\) GPa, \(f_{y}=186.7\) MPa, \(f_{u}=210.2\) MPa). The steel was modelled as a bi-linear elastic-perfectly plastic material (\(E=210\) GPa, \(f_{y}=235\) MPa). The LVL was treated as an orthotropic material. The Hashin damage model was used to capture the failure of the LVL. The material properties of the LVL adopted in the numerical models are provided in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Properties of an Elastic Orthotropic Material (Type: Lamina)**} \\ \hline \(E_{1}\) & \(E_{2}\) & \(\
u_{12}\) & \(G_{12}\) & \(G_{13}\) & \(G_{23}\) \\ \([\)**MPa\(]\)** & \([\)MPa\(]\) & \([\)–\(]\(]\) & \([\)MPa\(]\) & \([\)MPa\(]\) & \([\)MPa\(]\) \\ \hline \(24\ 000\) & \(430\) & \(0.48\) & \(650\) & \(650\) &  
Figure 8: Slip distribution in the B120 composite beam. Figure 9: Slip distribution in the B160 composite beam. Figure 11 presents the comparison of the mean slip values in the interface of the analysed beams. The mean values of the slip were calculated considering the slip values from the left and right sides of the beams. The degree of shear connection had an impact on the slip value. Beam B200 had the highest slip value, which resulted from the lowest value of the degree of shear connection. Figure 11: The comparison of the mean values of the slip in the analysed beams.