PSI - Issue 39

Davide Leonetti et al. / Procedia Structural Integrity 39 (2022) 9–19 Author name / Structural Integrity Procedia 00 (2019) 000–000

11 3

complex loading condition is applied, as in the case of residual stress distributions in welded joints, as done in Leonetti (2020). The weight functions for an inclined edge crack in a semi plane were derived by Beghini (1999), using a two term power expansion as a mathematical form for the weight functions. Moreover, Beghini (2013) later employed these weight functions for the analysis of RCF cracks under a line load following a Hertzian stress distribution approximated by a step function. In addition to this, also the effect of liquid entrapped in the crack was quantified using the weight functions. In Fett (1997) the geometric functions for an inclined edge crack in a finite width plate under both remote membrane and bending stresses are reported. The mechanical scheme is showed in Figure 1. On the basis of these two reference cases, approximate weight functions were derived in Fett (1997).

a

x

σ b

σ m

n t

ω

W

Fig. 1 Inclined edge crack in a finite width plate under the reference load cases, i.e. tension and bending.

The present paper presents a procedure for estimating the mode I and mode II SIFs of RCF cracks under a moving contact load, modelled as a distributed normal and tangential line load, in a rail, modelled as a plane problem. The geometry consists of an inclined edge crack in a finite width plate which is supported by an elastic foundation, the latter being a typical scheme used to model the behaviour of ballasted railway track, see Esveld (2001) and Lichtberger (2011). The weight functions for an inclined edge crack in a finite width plate are proposed and validated by Fett (1997), and were derived using the direct adjustment method proposed by Petrosky and Achenbach (1978), by making use of the Green’s functions derived in unpublished literature. Section 2 presents the weight functions preproposal in Fett (1997). Secondly, the approach to determine the state of stress in the uncracked structure is presented, involving the superposition of stresses resulting from the local contact condition and the internal forces and moments, which are derived integrating the elastic line of the beam on the elastic foundation for the considered loading. In Section 3 the stress state in the uncracked body determined using the proposed approach is compared with the stress state resulting from a finite element model developed in ANSYS framework, with the aim of validating the approach. In addition, a comparison of the SIFs obtained using the weight functions is presented and compared with results from literature. 2. Models and methods This section provides a description of the proposed procedure to calculate the SIFs for an edge crack in a beam supported by an elastic foundation under the action of normal and tangential contact forces. A representation of the mechanical scheme is given in Figure 2, including the relevant parameters. At first, the weight function method is introduced and the weight functions for an edge crack in a finite width plate are presented. Then, the solutions of the stress fields resulting from the patch loading and the internal forces and moments in the beam are reported. The SIFs are calculated, following the superposition principle, as the sum of the SIFs due to the stress fields resulting from the patch loading and the internal forces and moments in the beam.