PSI - Issue 75

Benjamin Causse et al. / Procedia Structural Integrity 75 (2025) 205–218 Author name / Structural Integrity Procedia (2025) Drawing at each time the points (p(t); ‖τ( ⃗, ) ‖ ) on the Dang Van plan of abscise p and ordinate || || gives the Load-Path. Load-Path is then compared to the linear equation || ||=  −  .p( t ). If the points are under this line, multiaxial fatigue analysis is conclusive, if they are above, fatigue criterion is not reached (see Fig. 1 in Causse et al. (2024) for pedagogic drawing). Causse et al. (2024) also showed that Eurocode S-N Curves can be used to draw linear limits: line || || =  −  .p( t ) becoming || || = =0 √ 3 −  .p( t ) with  R=0 =  0 =f(N,  C ) the stress amplitude at failure for N cycles for detail category  C . For example, for N=2×10 6 cycles,  0 =0.5×  C . The calculations of the local shear stress amplitude || || in 2D is quite simple to evaluate since we can then write that || ( t )|| = ||T max ( t ) – T av || (with T max ( t ) maximum total shear stress at time t, and T av average shear stress over the considered event). By the way note that in 2D, concerning shear stress we are mathematically in 1D, so we can easily calculate the absolute value of the algebraic subtraction T max ( t ) – T av . However, fast calculation of local recentered shear stress amplitude || || in 3D are way more complex and are still today a subject of research (Aygül et al. (2012), Fauvin et al. (2020), Agard et al. (2022)). 1.5. Limits of our work and plan As recall in §1.3 and 1.4 above, multiaxial fatigue analysis with FKM or Dang Van are highly technical methods, taking time and expertise to implement correctly. When the stress tensor is multiaxial, is there another quick way to evaluate multiaxial fatigue lifetime? We developed a simple tool using charts (abacus) to help the engineer to quickly determine depending on the parameters of bi-axiality of the signal (k=  II /  I ) , average stress over amplitude (  m /  R ), variation of loading angle (  ) the difference in fatigue lifetime between an estimation by Eurocode (N E ) (uniaxial) or by Dang Van (N DV ) (multiaxial). We then discuss acceptable N DV / N E lifetime variation values to be applied by the cableway sector profession for future standardisation work. In the present research work, in order to initiate this method, we limited our investigation to two dimensions, with a proportional multiaxial signal, in order to meet the practical need of the engineer in the field of cable installation design. 2. Material & Method 2.1. General In the field, in the cableway sector, the engineer works with uniaxial or biaxial strain gauges (rosettes). We therefore focus on biaxial signals, which are in practice the data recovered from extensometric tests with rosettes carried out on cableway installations, in accordance with state of the art (EN 13796-1 (CEN/TC 242, 2017) and EN 1993-1 9 (CEN/TC 250, 2005)). Stress tensor looks like A in the fixed rosette’s referential or D in the eigen referential (see Nomenclature). For the arches hoops of chairlifts (with fixed or detachable grip), preliminary analysis of the signals from loaded vehicles as they pass the towers (the most frequent fatigue event on a ski lift line) shows that the general shape of the biaxial stress signals is proportional with a mean tensile stress that can be quite high (around 60 MPa). Theoretical calculations for the creation of charts (abacuses) will therefore be carried out on proportional signals, in phase. For all calculations :  C = 160 : the chosen detail category is set at 160 (generally used in the manufacture of chairlift hoops).  = 200 MPa stress range, that is to say an amplitude  R = 0.5*  = 100 MPa  m = 100 MPa : the average stress is set to 100 MPa so that the calculations are representative of a Ratio R=0 (noted R 0 ) and covering the range observed in the field on the chairlift arches hoops. Evaluations of service lives in number of cycles N E are made according to EN 1993-1-9 (CEN/TC 250, 2005) and N DV with a Dang Van criterion calibrated in R 0 with S-N curves of the Eurocode as a reference, according to Causse et al. (2024).

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