**1. Introduction**

Tubular sections are widely used in engineering applications such as transport systems (e.g., tubes, pipelines), naval and aeronautical engineering, o ffshore equipment, or lifting systems (e.g., cranes), among others. These structural solutions have been demonstrated to have grea<sup>t</sup> strength against di fferent types of loading conditions such as torsion, compression, or multiaxial bending. In addition, their specific shape has proved to be an adequate solution for structures exposed to wind, water, or wave loads, and they are less prone to corrosion processes, as they do not tend to generate local accumulations of water. With all this, it is of grea<sup>t</sup> importance from an engineering perspective to be able to estimate the critical loads of this kind of structures, especially when they contain defects.

In the field of structural integrity, the analysis of defects plays an important role in ensuring the safety of structural components. Structural integrity procedures (e.g., [1–3]) are able to evaluate components containing cracks, combining fracture and plastic collapse analyses. However, in many cases, the structures present defects with finite radii on the tip. These defects are generally named notches, and if they are assessed as crack-like defects using standard methodologies (traditionally based on fracture mechanics [4,5]), the results tend to be over-conservative. This is caused by the fact that notches generate more relaxed stress fields at their tip (when compared to those generated by cracks). Apparently, the material develops a higher fracture resistance (usually referred to as the apparent fracture toughness) than that developed in cracked conditions (fracture toughness). Therefore, it is necessary to provide structural assessment methodologies that are capable of taking the notch e ffect into account, providing accurate predictions of the resulting critical loads. Di fferent works (e.g., [6–11]) dealing with the structural integrity of tubular sections may be found in literature, although failure processes (e.g., plastic collapse, buckling) and defect types (e.g., cracks, cutouts) are di fferent to those considered in this work.

In this sense, when dealing with notch assessments, there are two main types of criteria: the global criterion (based on the use of a notch stress intensity factor, analogously to ordinary fracture mechanics), and local criteria (based on the study of stress or strain fields around the notch tip). Among the latter, the theory of critical distances (TCD) stands out, and its applicability in fracture assessments has been widely reported in the literature for a variety of materials (such as polymers [12,13], metals [14,15], composites [16], or ceramics [17,18]). Moreover, the TCD has also been validated to analyze phenomena such as fatigue [19] or environmentally assisted cracking [20] and has been applied to different length scales [19,21,22].

The TCD is actually a group of methodologies initially proposed in the mid-twentieth century by Neuber [23] and Peterson [24] to predict the fatigue behavior of structural components containing notches. All these methodologies have in common the use of two additional parameters: a material length parameter called the critical distance (*L*), which is defined by Equation (1), and a material strength parameter named the inherent strength (<sup>σ</sup>*0*). In fracture analysis, both parameters are directly related with the material fracture resistance (*Kmat*) through Equation (1).

$$L = \frac{1}{\pi} \left(\frac{K\_{\text{mat}}}{\sigma\_0}\right)^2 \tag{1}$$

For brittle materials (e.g., ceramics) or quasi-brittle materials (e.g., many fiber-reinforced composites), the inherent material strength is equal or very close to the corresponding ultimate tensile strength (<sup>σ</sup>*u*). Otherwise, σ*0* tends to be higher than σ*u*, with this tendency being more pronounced as long as plasticity is developed in the vicinity of the notch. In such cases, σ*o* has to be determined (calibrated) through experimental tests of specimens containing notches with different radii, or through a combination of experimental tests and finite element (FE) modeling.

Within the different approaches proposed by the TCD, the point method (PM) stands out for its simplicity, and provides similar results to other TCD methodologies, such as the line method, the area method, or the volume method, among others [19]. According to the PM criterion, fracture occurs when the stress equates the inherent strength, σ*0*, at a distance equal to *L*/2 from the defect tip. The mathematical expression is given by Equation (2):

$$
\sigma \left( \frac{L}{2} \right) = \,\,\sigma\_0 \tag{2}
$$

Thus, the PM allows the fracture behavior of notched components to be analyzed by simply knowing *L* together with the (linear elastic) stress field at the notch tip. The evolution of FE tools allows the stress distribution at a stress concentrator to be more easily determined, something that has allowed extensive validation of the TCD methodologies [19]. However, this validation has been strongly focused on fracture mechanics notched specimens (e.g., CT and single edge notched bend (SENB) samples). In this context, this paper attempts to validate the application of the TCD (coupled with FE analyses) on a larger scale in real structural components (in this case, tubular cantilever beams containing U-notches).

With all of this, Section 2 presents the material and methods, Section 3 gathers the results obtained experimentally and through the TCD-FE analysis, together with the corresponding discussion, and Section 4 summarizes the main conclusions.
