PSI - Issue 52

A.D. Cummings et al. / Procedia Structural Integrity 52 (2024) 762–784 A. Cummings / Structural Integrity Procedia 00 (2023) 000–000

781

20

Appendix A. T-stress calculations in bending for low a / Bratios

The solutions for T-stress provided in Annex N of BS7910 (2019) for surface cracked plates in tension and bending are quoted to be valid for 0 ≤ a / B ≤ 0.8and0.2 ≤ a / c ≤ 0.8. The geometry range of interest in this report is a / B ≈ 0.01. Therefore, initial comparisons between cracked body models and the solutions for the geometry range of interest did not compare well. The original source of the polynomial expressions provided in BS7910 (2019) is due to Wang (2003). The polynomial expressions were fitted to numerically calculated T-stresses for bending and tension for a wide range of geometry as shown in Fig. A.1. Note, Wang (2003) denotes thickness as “t” but for consistency with the rest of this article (and BS7910 (2019)) “B” is referred to thickness. It is evident that in the case of bending, the polynomial curves at lower a / B ratios (a / t in the image) are nonlinear whilst the numerical data appears to be linear. This is the source of the discrepancy between initial comparisons.

(b)

(a)

Fig. A.1. Geometry description and bending T-stress results after Wang (2003) (a) Geometry (b) Results and empirical equation fit

A.1. Base T-stress verification

As the loading at the centre of the base is predominantly bending, only bending T-stresses have been considered. Verification was carried out in two stages:

1. Benchmarking against published data due to Wang (2003) 2. Extrapolating to the base geometry

Benchmarking was performed for a / B = 0.2 and a / B = 0.4 whilst a / c = 0.4 for both cases. Fig. A.2 shows the first model and results for a / B = 0.2. The model does not agree well at the free surface which is expected due to the loss of the r − 1 / 2 singularity Wang (2003). At the deepest point of the crack the model is in good agreement producing a T-stress / σ = -0.171 vs -0.174 due to Wang (2003). The second model, a / B = 0.4, produced excellent agreement T-stress / σ = 0.1380 vs 0.1377 due to Wang (2003). The modelling approach was therefore considered verified against the solutions in BS7910 (2019).

A.2. Base T-stress extrapolation

In the original publication Wang (2003) W / c = 16 throughout the study. To achieve the base geometry of a / B = 0.01 andW / c = 108 both W and B were gradually changed to show the e ff ects of extrapolation. For all models a / c = 0.4with a = 1mmand c = 2.5mm. W / c was varied progressively from 16 to 108 - the ratio between the half-width