## 2 Mechanical Behavior of Metal Anchors
### Characteristic Resistance of Anchors under Tensile Loading
#### 2.1.1 Adhesive Anchors  
For adhesive anchors (chemical and cementitious grout), the main types of failure modes associated with tensile loading are shown in Figure 1: (a) steel failure of the anchor; (b) pull-out or interface failure (either anchor/adhesive or adhesive/substrate); (c) substrate cone failure; (d) combined pull-out and substrate cone failure; (e) pull-out of one brick; (f) lateral cone or blow-out failure. For real-life images of the specific failure modes obtained in this work, the reader is referred to Section 4. In the case of steel failure, herein labelled T1, the characteristic resistance of an anchor can be defined as [18; 19]:  
\[N_{Rk}=0.75\cdot A_{s}f_{uk}, \tag{1}\]  
where \(A_{s}\) (mm\({}^{2}\)) is the effective cross-sectional area of the anchor, and \(f_{uk}\) (MPa) is the characteristic ultimate strength of steel. Alternatively, the characteristic resistance can be calculated as [20]:  
\[N_{Rk}=0.90\cdot A_{s}\cdot f_{yk}, \tag{2}\]  
where \(f_{yk}\) (MPa) is the characteristic yield strength of steel. It must be noted that steel failure is rarely observed in masonry. Nonetheless, it might occur in applications with considerable embedment length and high masonry strength [21]. Figure 1: Failure modes under tensile loading: (**a**) steel failure; (**b**) pull-out or interface failure; (**c**) substrate cone; (**d**) combined pull-out and cone; (**e**) pull-out of one brick; and (**f**) lateral cone or blow-out failure. Adapted from [17]. The second type of failure mode, i.e., pull-out or interface failure, herein called T2, may occur between the anchor and the adhesive or between the adhesive and the substrate [9]. For both cases, the tension model of uniform adhesion to describe the interface behavior is usually adopted [17; 18; 22]. Thus, the characteristic resistance for interface failure between the anchor and the adhesive can be estimated by [10]:  
\[N_{RK}=\tau_{k}\cdot\pi\cdot d\cdot h_{ef}, \tag{3}\]  
where \(\tau_{k}\) (MPa) is the adhesive bond strength of the anchor/adhesive interface, \(d\) (mm) is the anchor diameter, and \(h_{ef}\) (mm) is the effective embedment length. Conversely, the formula for the interface failure between the adhesive and the substrate is [11]:  
\[N_{RK}=\tau_{k0}\cdot\pi\cdot d_{0}\cdot h_{ef}, \tag{4}\]  
where \(\tau_{k0}\) (MPa) is the adhesive bond strength of the adhesive/substrate interface, and \(d_{0}\) (mm) is the drill hole diameter. Existing studies suggest that this type of failure can be influenced by several parameters, such as drill hole conditions during the application of the anchor, water absorption capacity of the substrate, adhesion effective length, injection time, or the presence of masonry joints [7]. For the substrate cone failure mode, herein labelled T3, the formulation developed for concrete is here extended to masonry. Therefore, the characteristic resistance of the anchor can be calculated as [18]:  
\[N_{RK}=k_{1}\cdot(f_{ck\;cube})^{0.5}\cdot h_{ef}^{1.5}\cdot\left(A_{c,N}/A_{c,N}^{0}\right), \tag{5}\]  
where \(k_{1}\) (N\({}^{0.5}\)/mm\({}^{0.5}\)) is a semi-empirical parameter, \(f_{ck\;cube}\) (MPa) is the characteristic compressive strength of concrete measured for 200 mm cube specimens, and \(A_{c,N}/A_{c,N}^{0}\) is a geometrical reduction factor defined, as shown in Figure 2. In particular, \(A_{c,N}\) (mm\({}^{2}\)) is the projected area of an anchor or group of anchors on the concrete surface, whereas \(A_{c,N}^{0}\) (mm\({}^{2}\)) is the projected area of a single anchor not limited by adjacent edges. Hence, for a single anchor not located on an edge, \(A_{c,N}/A_{c,N}^{0}=1\). For \(k_{1}\), EOTA TR 029 [18] proposes 7.2 N\({}^{0.5}\)/mm\({}^{0.5}\) and 10.1 N\({}^{0.5}\)/mm\({}^{0.5}\) for cracked and non-cracked concrete, respectively, whereas EN 1992-4 [17] and fib Bulletin 58 [22] propose 7.7 N\({}^{0.5}\)/mm\({}^{0.5}\) and 11.0 N\({}^{0.5}\)/mm\({}^{0.5}\) for cracked and uncracked concrete applications, respectively.