Issue 43

F. Berto et alii, Frattura ed Integrità Strutturale, 43 (2018) 1-32; DOI: 10.3221/IGF-ESIS.43.01 6 equal to 211 MPa mm 0.326 was found re-analyzing experimental results taking from the literature [10, 15-16]. For butt ground welds made of ferritic steels a mean value Δ σ A = 155 MPa (at N A = 5  10 6 cycles, with R =0) was employed for setting the method. Then, by introducing the above mentioned value into Eq. (5), one obtains for steel welded joints with failures from the weld toe R 0 =0.28 mm. By modelling the weld toe regions as sharp V-notches and using the local strain energy, more than 900 fatigue strength data from welded joints with weld toe and weld root failures were analyzed and the theoretical scatter band in terms of SED was obtained and recently updated with all the possible data available in the literature for which the local geometries were properly defined [16]. The geometry exhibited a strong variability of the main plate thickness (from 6 to 100 mm), the transverse plate (from 3 to 200 mm) and the bead flank (from 0 to 150 degrees). The synthesis of all those data is shown in Fig. 4, where the number of cycles to failure is given as a function of  1 W (the Mode II stress distribution being non-singular for all those geometries). The figure includes data obtained both under tension and bending loads, as well as from “as-welded” and “stress-relieved” joints. The scatter index T W , related to the two curves with probabilities of survival P S = 2.3% and 97.7%, is 3.3, to be compared with the variation of the strain energy density range, from about 4.0 to about 0.1 MJ/m 3 . T W =3.3 becomes equal to 1.50 when reconverted to an equivalent local stress range with probabilities of survival P S =10% and 90% ( T W =  3.3 /1.21 1.5 ). The final synthesis based on more than 900 experimental data is shown in Fig. 4 where some recent results from butt welded joints, three-dimensional models and hollow section joints have been included. A good agreement is found, giving a sound, robust basis to the approach when the welded plate thickness is equal to or greater than 6 mm. A DVANTAGES OF THE METHOD s opposed to the direct evaluation of the stress intensity factors (SIFs) or generalized notch stress intensity factors (NSIFs), which need very refined meshes, the mean value of the elastic SED on the control volume can be determined with high accuracy by using coarse meshes [78-81]. Very refined meshes are necessary to directly determine the NSIFs from the local stress distributions. Refined meshes are not necessary when the aim of the finite element analysis is to determine the mean value of the local strain energy density on a control volume surrounding the points of stress singularity. The SED in fact can be derived directly from nodal displacements, so that also coarse meshes are able to give sufficiently accurate values for it. Some recent contributions document the weak variability of the SED as determined from very refined meshes and coarse meshes, considering some typical welded joint geometries and provide a theoretical justification to the weak dependence exhibited by the mean value of the local SED when evaluated over a control volume centered at the weld root or the weld toe. On the contrary singular stress distributions are strongly mesh dependent. The NSIFs can be estimated from the local SED value of pointed V-notches in plates subjected to mode I, Mode II or a mixed mode loading. Taking advantage of some closed-form relationships linking the local stress distributions ahead of the notch to the maximum elastic stresses at the notch tip the coarse mesh SED-based procedure is used to estimate the relevant theoretical stress concentration factor K t for blunt notches considering, in particular, a circular hole and a U-shaped notch, the former in mode I loading, the latter also in mixed, I + II, mode [79, 80]. Other important advantages can be achieved by using the SED approach. The most important are as follows: • It permits consideration of the scale effect which is fully included in the Notch Stress Intensity Factor Approach [15, 16] • It permits consideration of the cycle nominal load ratio [15, 16]. • It overcomes the complex problem tied to the different NSIF units of measure in the case of different notch opening angles (i.e crack initiation at the toe (2 α =135°) or root (2 α =0°) in a welded joint) [10, 15-16] • It overcomes the complex problem of multiple crack initiation and their interaction on different planes. • It directly takes into account the T-stress and this aspect becomes fundamental when thin structures are analysed [82, 83]. • It permits consideration of the contribution of different Modes [70-76, 84-85]. • It directly includes three-dimensional effects and out-of-plane singularities not assessed by Williams’ theory as it will be described in the next section. Three-dimensional effects Dealing with 3d effects the SED is able to take into account coupled induced modes pioneering investigated by Sih, Pook and Kotousov [86-93]. A