# Parametric Study of Three Types of Timber Connections with Metal Fasteners Using Eurocode 5
## 3 Simplified Equations in Designed Timber Connections
His theory characterizes three distinct failure modes but does not consider potential brittle modes [29], only considers ductile failure modes [14]. The equations derived by [PERSON] [10] for connection strength are dependent on geometry, fastener strength, wood embedment strength, and bending strength [24]. In the derivation of these equations, however, the friction forces between the connected members were ignored, as well as the withdrawal resistance of the fasteners [24]. To include these effects, the characteristic load-carrying capacity of a fastener is given by the Equations (3)-(6) [10; 22; 24], according to Eurocode 5 part 1-1 [18]. For symmetrical connections in double shear, the characteristic load-carrying capacity per shear plane and fastener will be the minimum value of these three equations, and the failure mode will be the mode associated with each equation, Figures 1-3. Using all the simplified equations proposed from Eurocode 5, part 1-1 [18], the characteristic load-carrying capacity per shear plane and fastener \(F_{v,Rk}\), in case the timber connection has an internal steel plate of any thickness (W-S-W) are given by the Equation (3). Figure 1: Steel-to-timber connections in double shear (W-S-W): (**a**) wood embedding; (**b**) fastener bending; (**c**) combination of failure modes. Figure 3: Steel-to-timber connections in double shear (S-W-S): (**a**) wood embedding; (**b**) fastener bending; (**c**) combination of failure modes. Figure 2: Timber-to-timber connections in double shear (W-W-W): (**a**) bearing failure of the edge members; (**b**) central member bearing failure; (**c**) plastic hinge on the fastener inside the central member and bearing failure of all the members; (**d**) plastic hinges on the fastener and bearing failure on all members. \[F_{v,Rk}=min\left\{ \begin{aligned} f_{h,1,k}&\;t_{1}\;d \\ f_{h,1,k}&\;t_{1}\;d\end{aligned} \right. \tag{3a}\] \[F_{v,Rk}=min\left\{ \begin{aligned} f_{h,1,k}&\;t_{1}\;d \\ f_{h,1,k}&\;t_{1}\;d\end{aligned} \right.\left[\sqrt{2+\frac{4 M_{y,Rk}}{f_{h,Rk}\;d\;t_{1}^{2}}}-1\right]+\frac{F_{ax,Rk}}{4}\] (3b) \[\;2.3\;\sqrt{M_{y,Rk}\;f_{h,1,k}\;d}+\frac{F_{ax,Rk}}{4} \tag{3c}\]  
For the double shear connection with a steel plate of any thickness as the central member, the different failure modes are represented in Figure 1, relative to the characteristic load-carrying capacity per shear plane and fastener, calculated by the Equation (3a-c). The ratio between the side width and the dowel diameter is often used to explain the failure mode of a connection [27]. In Figure 1 the failure mode in (a) represents embedment of the wood, in (b) it represents the bending of the fastener and in (c) it is the combination of the previously mentioned modes [18]. For a timber-to-timber connection (W-W-W) with dowel in double shears, the characteristic load-carrying capacity is given by the Equation (4). \[F_{v,Rk}=min\left\{ \begin{aligned} f_{h,1,k}&\;t_{1}\;d \\ & 0.5\;f_{h,2,k}&\;t_{2}\;d\\ & 1.05\;\frac{f_{h,1,k}\;t_{1}\;d}{2+\beta}\left[\sqrt{2\beta(1+ \beta)+\frac{4\beta(2+\beta)M_{y,Rk}}{f_{h,Rk}\;d\;t_{1}^{2}}-\beta}\right]+ \frac{F_{ax,Rk}}{4}\\ & 1.15\sqrt{\frac{2\beta}{1+\beta}}\;\sqrt{2M_{y,Rk}\;f_{h,1,k}\;d }+\frac{F_{ax,Rk}}{4}\end{aligned} \right. \tag{4d}\]  
The four modes of failure of the timber-to-timber connections are illustrated in Figure 2, which correspond, respectively, to the four characteristic load-carrying capacities given by the Equation (4a-d). The failure mode (a) represents the bearing failure of the edge members, in (b) it represents the bearing failure of the central member, in (c) it represents the simultaneous formation of the plastic hinge on the fastener inside the central member and bearing failure of all the members and (d) it represents the simultaneous plastic hinge formation in the fastener within all members and bearing failure of all members [18; 24].