*2.2. Methods*

The present study proposes a methodology for the analysis of tubular cantilever beams containing (circumferential) through thickness U-notches by applying the TCD. This requires completing experimental tests and FE simulations.

Regarding the experimental program, three of the tubes mentioned above were conducted to failure through bending tests, but previously both fracture and tensile tests were performed in order to characterize the material. Fracture and tensile specimens were machined from the remnant fourth tube (with an outer diameter of 260 mm and 5 mm thickness).

Three tensile tests were conducted according to the ASTM E8M standard [26]. Figure 1 shows the dimensions of the samples that were machined in the longitudinal direction. The tests were carried out with a loading rate of 5 mm/min.

**Figure 1.** Tensile test specimens. Dimensions in mm.

Subsequently, the fracture behavior of AL6060-T66 was characterized. A total of nine SENB specimens were tested following ASTM E1820 [27]. Three specimens for each notch radii were obtained in LC orientation: the opening stresses act in the longitudinal direction of the pipe, and the defect propagates circumferentially. The notch radii considered in this work are 0 mm (crack-like defect), 1 mm, and 2 mm. Notches of finite radius (1 mm and 2 mm) were obtained by machining, whereas crack-like defects were generated by fatigue pre-cracking according to ASTM E1820. Figure 2 shows a schematic of the specimen used in the fracture characterization. It can be noticed that the width is slightly smaller than the tube thickness due to the need to have prismatic samples. The loading rate was 10 mm/min.

**Figure 2.** Schematic of fracture single edge notched bend (SENB) specimens. Dimensions in mm.

To conclude with the experimental works, the remnant three tubular cantilever beams were prepared to be tested. Through-thickness circumferential U-notches were machined at a distance of approximately 350 mm from one of the tube ends. In order to obtain a fixed support, the same tube end was introduced 330 mm in reinforced concrete. Figure 3 shows an image of the experimental setup, Figure 4 represents a schematic of the notched tubular cantilever beams, and Table 2 gathers the geometry of both the tubes and the notches.

**Figure 3.** Experimental setup.

**Figure 4.** Schematic of the tubular cantilever beams containing a U-notch close to the fixed support.

**Table 2.** Geometrical parameters of the tubes and their corresponding U-notch: *Ø*, outer diameter; *B*, tube thickness; *D*, distance from concrete support to notched section; *L*, distance from applied load to notched section; *2a*, defect length; ρ, defect radius. Dimensions in mm.


In order to obtain the experimental critical loads (load-bearing capacity), the tubes were set up in the testing bench, ensuring that the solid concrete block was totally fixed with screws avoiding any kind of movement. A single vertical load was applied at the free edge with a testing rate of 10 mm/min, while a calibrated laser comparator measured the resulting deflection.

As explained above, the application of the PM requires the stress field around the defect tip to be determined. With this aim, FE analyses were carried out. The simulations were performed in linear elastic conditions using the finite element software ANSYS 19.2 (Ansys Inc, Canonsburg, PA, USA) both in the SENB specimens and the cantilever beams.

The simulation of the SENB specimens (Figure 5a) was performed using a structured mesh composed by 20-node hexahedron elements, as shown in Figure 5b. The area surrounding the notch tip was discretized using a much finer mesh, because of the higher stress gradient generated in that zone. For each notch radius, the stress–distance curves were finally obtained in the middle line of the fracture section and for the corresponding average value of the critical loads. The stresses used in the analyses are the corresponding maximum principal stresses, which, in these particular structural conditions, act in the longitudinal direction of the tubular beams.

**Figure 5.** Geometry of the model used in finite element (FE) simulations, showing the middle line on the fracture section (**a**) and the generated mesh (**b**).

Once the stress–distance curves for each notch radius were determined, the PM was applied to calibrate the material parameters. When PM is used, it is su fficient to obtain the cuto ff point between the di fferent curves, which theoretically corresponds to the coordinates (*L*/2, <sup>σ</sup>*o*), as shown in Figure 6.

**Figure 6.** Obtaining theory of critical distances (TCD) parameters using the stress–distance curves.

Finally, the three cantilever tubular beams were modeled (see an example in Figure 7). Again, a structured mesh composed by 20-node hexahedron elements was used. The notch region was partitioned in order to generate a refined mesh, also ensuring 20 elements along the tube thickness. The simulation was performed with just a half of the tube because of the symmetry conditions, applying the load at the free end of each beam. The part of the tube fixed in the concrete block had all the movements restricted, and the points of the tube located in the symmetry plane could only have displacements in such a plane. Here, it is important to notice that the critical load of the complete tubular beam is twice the critical load of the model. A path was created on each tube, starting at the notch root, at half of the tube thickness and along the circumferential direction. Thus, the stress–distance curve (along the corresponding path) was obtained for each tube under the load being applied. Finally, the estimated critical load (or load-bearing capacity) was that for which Equation (2) was satisfied.

**Figure 7.** (**a**) Mesh employed in the FEA (finite element analysis) of the tubular beams; (**b**) detail of the notch tip.
