## 2 Mechanical Behavior of Metal Anchors
### Characteristic Resistance of Anchors under Shear Loading  
Analogously to the anchoring systems subjected to tensile loading, the analytical formulation for anchors under shear stresses was determined according to the possible failure modes. In this case, the predictive analytical formulas aim at estimating the characteristic strength of a single anchor under shear loading, \(V_{Rk}\) (N).  
The types of failure modes associated with shear are shown in Figure 3: (a) steel failure of the anchor without lever arm; (b) steel failure of the anchor with lever arm; (c) substrate pyro-out failure; and (d) edge failure. It is noted that steel failure is usually complemented by crushing and spalling of the substrate material close to the anchor. For real-life images of the failure modes obtained in this work, the reader is referred to Section 4.  
Regarding the steel failure under shear load without lever arm, herein called T8, the characteristic resistance of the anchor can be estimated as [18]:  
\[V_{Rk}=0.50\cdot A_{s}\cdot f_{ik}, \tag{11}\]  
On the other hand, for the steel failure under shear loading with lever arm, herein referred to as T9, the characteristic resistance of the anchor is defined by [18]:  
\[V_{Rk}=\alpha_{M}\cdot M_{Rk}/l, \tag{12}\]  
where \(\alpha_{M}\) (-) is a factor that takes into account the degree of restraint of the fastener at the side of the load application (\(\alpha_{M}=1\) if the anchor can rotate or \(\alpha_{M}=2\) for full restraint), \(M_{Rk}\) (N\(\cdot\)mm) is the characteristic bending resistance, and \(l\) (mm) is the lever arm or distance between the substrate surface and the load. The characteristic bending resistance can be calculated as [18]:  
\[M_{Rk}=M_{Rk}^{0}\cdot(1-N_{Sd}/N_{Rd}). \tag{13}\]  
where \(M_{Rk}^{0}\) (N\(\cdot\)mm) is the characteristic bending resistance of a single anchor, \(N_{Sd}\) (N) is the design value for tensile load, and \(N_{Rd}\) (N) is the design resistance for tensile loading. The characteristic bending resistance of a single anchor is usually provided by the manufacturer and is defined as [18]:  
\[M_{Rk}^{0}=1.2\cdot W_{el}\cdot f_{uk}, \tag{14}\]  
where \(W_{el}\) (mm\({}^{3}\)) is the elastic section modulus of the fastener calculated from the effective cross-section of steel (\(\pi d^{3}/32\) for a round section of diameter \(d\)).  
In Equation (13), the design resistance for tension loading can be obtained by:  
\[N_{Rd}=N_{Rk}/\gamma_{M\times}, \tag{15}\]  
where \(N_{Rk}\) (N) is the characteristic strength of an anchor or group of anchors, obtained with the substrate cone failure formula, and \(\gamma_{M\times}\) (-) is the partial safety factor of the material, which is set to 1.25 for failure with lever arm [18].  
The substrate pyro-out failure, herein labelled T10, is associated with the rotation of the anchor under shear loading with lever arm. For this failure mode, the characteristic resistance can be calculated as [18]:  
\[V_{Rk}=k_{3}\cdot N_{Rk}, \tag{16}\]  
where \(k_{3}\) (-) is a semi-empirical factor, and \(N_{Rk}\) (N) is the characteristic strength of an anchor or group of anchors, obtained from the substrate cone failure formulation. According to  
Figure 3: Failure modes under shear loading: (**a**) steel failure without lever arm; (**b**) steel failure with lever arm; (**c**) substrate pyro-out failure; and (**d**) edge failure. Adapted from [17].  
EOTA TR 029 [18], the parameter \(k_{3}\) is equal to 1 for an effective embedment length \(h_{ef}<60\) mm, or 2 in the case of \(h_{ef}\geq 60\) mm.