## 1 Introduction  
Welding procedures have allowed the simple realization of many complex structural connections in several different applications, such as bridges, cranes, offshore structures, and buildings [1]. Besides the optimization of the process parameters [2; 3; 4], researchers have focused on the fatigue behavior of welded components. All the aforementioned applications are in fact subjected to repetitive loading and hence the fatigue of welded structures has become one of the greatest matters of concern [5]. Welds represent local regions of higher stresses (i.e., stress concentrators) and fatigue failure is widely known to be sensitive to local peak stresses. Thus, over the years, several methods have been developed to assess the fatigue behavior of welds, and among these the Notch Stress Intensity Factor (NSIF) approach is worth mentioning [6; 7; 8]. In fact, since experimental observations have revealed that the most used welding procedures, such as arc and laser welding, are characterized by a very small weld toe radius, it is acceptable to model it as a sharp notch and the fatigue behavior can thus be correlated to the NSIFs [9; 10]. Fatigue life can be divided into two different phases; i.e., nucleation and growth of cracks--and NSIFs were found to be able to predict either the initiation phase or the whole fatigue life [11; 12; 13; 14]. This holds when the majority of the propagation phase has been spent within the region governed by the stress singularity. Several studies proved this behavior. [PERSON] demonstrated that in transverse, non-load-carrying joints manufactured via different welding procedures the nucleation of a 0.1 mm long crack required up to 40% of the whole fatigue life [15]. In addition, [PERSON] et al. reported that in load-carrying AISI 304L fillet joints up to 70% of the fatigue life was spent propagating the original lack of a penetration crack of 0.5 mm [16; 17]. However, the results are highly dependent of the on the stress distribution ahead of the point of singularity and, since time consuming FE analyses are required, this has limited the diffusion of the NSIFs approach in the assessment of real structures. Furthermore, the main limitation of this method consists in its geometry dependency. Its unit depends in fact on the notch opening angle according to [PERSON] formulations [18]. Different notch geometries are not thus comparable but ad-hoc material properties are required for each geometry. A breakthrough in the fatigue assessment of welded joints is represented by the strain energy density (SED) method, capable of overcoming all the limitations of the NSIFs approach [19]. The SED is evaluated averaging the range value of the strain energy in a cylindrical volume sector of radius \(R_{c}\) centered at the notch tip. According to this approach, fatigue failure occurs then this scalar quantity reaches its critical value \(\Delta W_{c}\). Both the radius \(R_{c}\) and the critical energy range \(\Delta W_{c}\). depend only on material properties [11] and shall be assumed to be independent of the load mode. Using this approach, [PERSON] and [PERSON] [20] summarized more than 600 experimental data obtained from different welded joints within a single narrow SED scatter-band. Failures occurred either at the weld root or at the weld toe. This work has then been extended by [PERSON] and [PERSON] [21], including hollow section joints, and now more than 900 fatigue data are summarized in a single scatter band as depicted in Figure 1.  
Moreover, the SED approach has been also reported to be a reliable tool in multi-axial fatigue assessment.