# Parametric Study of Three Types of Timber Connections with Metal Fasteners Using Eurocode 5
## 3 Simplified Equations in Designed Timber Connections
For a steel-to-timber connection (S-W-S) the characteristic load-carrying capacity depends on the thickness of the steel plates, Eurocode 5 part 1-1 [18], which are given by the Equations (5) and (6). The thickness of steel plates less than or equal to 0.5 multiplied by the dowel diameter are classified as thin plates, and greater than or equal to the diameter are classified as thick plates [18]. The characteristic load-carrying capacity of the connections with steel plate thickness between a thin and a thick plate must be calculated by linear interpolation between the previous values [18]. For a double shear connection, the characteristic load-carrying capacity using thick steel plates as the outer members is given by the Equation (5):  
\[F_{v,Rk}=min\left\{ \begin{aligned} 0.5f_{h,2,k}&\;t_{2}\;d \\ 2.3\sqrt{2M_{y,Rk}\;f_{h,2,k}\;d}&+\frac{F_{ax,Rk}}{4} \end{aligned} \right. \tag{5a}\]  
In the studied connections, the external steel plate has a thickness equal to 3 mm, being classified as a thin plate. For this condition, when steel plates are the outer members of a double shear connection, the characteristic load-carrying capacity is obtained using the Equation (6). \[F_{v,RK}=min\left\{ \begin{aligned} 0.5f_{h,2,k}\ t_{2}\ d \\ 1.15\sqrt{2M_{y,RK}\ f_{h,2,k}\ d}+\frac{F_{ax,RK}}{4}\end{aligned} \right.\] (6a) where: \[f_{h,i,k}\ (f_{h,1,k}\ \text{or}\ f_{h,2,k})\] is the characteristic embedment strength in timber member \[i\] ; \[M_{y,RK}\] is the characteristic yield moment of the fastener, \[\beta\] is the ratio between the embedment strength of the members considered equal to 1, and \[F_{ax,RK}\] represents the characteristic axial withdrawal capacity of the fastener, according to Eurocode 5 part 1-1 [18]. The three different failures modes are shown in Figure 3. The failure mode in (a) represents the wood embedment obtained from Equations (5a) and (6a), the failure mode (b) is the fastener bending from Equation (5b), and failure mode (c) is the combination of the previous failure modes obtained from Equation (6b), [18]. The value of the characteristic embedment strength in the timber member \(i\) parallel to the grain with pre-drilled holes is obtained due to the value of the dowel diameter between 6 mm and 30 mm, and the characteristic wood density \(\rho_{k}\), as defined in Equation (7). \[f_{h,1,k}=f_{h,2,k}=0.082(1-0.01\ d)\ \rho_{k} \tag{7}\]  
The value of \(M_{y,RK}\) is calculated according to the dowel diameter d, and its material characteristic tensile strength \(f_{u,k}\), [25]. \[M_{y,RK}=0.3\ f_{u,k}\ d^{2,6} \tag{8}\]  
After calculating the value of \(F_{v,RK}\), it is necessary to determine the design value of the characteristic load-carrying capacity, which is obtained from the Equation (9), in which two safety factors are considered, defined in accordance with Eurocode 5 part 1-1 [18]. \[F_{v,Rd}=\frac{F_{v,Rk}\ k_{mod}}{\gamma M} \tag{9}\]  
Finally, with the calculation from \(F_{v,Rd}\), it is possible to obtain the number of dowels, which is given by Equation (10), [25]. \[N\geq\frac{F_{d}}{F_{v,Rd}} \tag{10}\]  
To reduce the risk of failure modes, the minimum edge, and spacing criteria for connections with dowel were calculated [17]. The arrangement between dowels is according to the calculated spacing, following the Equations (11)-(14): parallel to the fastener grain and within a row \(a_{1}\), perpendicular to the grain and between rows \(a_{2}\), the distance between the fasteners and the loaded end \(a_{3,t}\) and unloaded edge \(a_{4},c\), which varies according to the dowel diameter [18; 25]. The parameter \(\alpha\) represents the angle, between the applied tensile load and the wood grain direction. \[i\]0  
\[i\]1  
\[i\]2  
\[i\]3  
According to Eurocode 5 part 1-1 [18], the arrangements of the dowels in the studied connections are represented in \[i\]4