PSI - Issue 58

Mikkel L. Larsen et al. / Procedia Structural Integrity 58 (2024) 73–79

78

6

M.L. Larsen et al. / Structural Integrity Procedia 00 (2024) 000–000 Table 1. RMSE and averaged prediction ratios for the experimental data and FE predictions for the original FE model and model updated model.

Model

Parameter Global gauges Global rosettes Local gauges

1.95 0.44 1.12 0.14

2.21 0.23 1.19 0.14

- -

P µ

Original model

RMSE

1.07 0.63

P µ

Updated model

RMSE

4. Non-proportionality quantification

Using the updated digital twin, the levels of non-proportionality can be determined. Many types of non proportionality quantification methods have been proposed in literature to take into account the increased fatigue damages caused by non-proportionality (Bolchoun et al. (2015); Pejkowski (2017)). Larsen et al. (2022a) developed an approach based on principal component analysis to determine the level of non-proportionality in welded joints. The approach is based on geometrical interpretation of the normal stresses σ x and shear stresses τ xy plotted in a σ x - τ xy graph, where a full circle corresponds to full non-proportionality or 90 degrees phase-shift and a straight line corre sponds to a proportional stress state with 0 degrees phase-shift. The principal components for a stress time series with normal and shear stresses can be found based on the covariance matrix: Cov = Cov( σ x ,σ x ) Cov( σ x ,τ xy ) Cov( σ x ,τ xy ) Cov( τ xy ,τ xy ) where: Cov( σ x ,τ xy ) = 1 n − 1 ( σ x , i − σ x )( τ xy , i − τ xy ) (1) Where σ x and τ xy denote the normal stress perpendicular to the weld toe and the corresponding shear stresses, respectively. It should be noted that the stress components need to be standardised (Z-score normalisation) before using equation 1. The standardisation ensures that the principal components can be found accurately, while also ensuring that the mean stress e ff ect is disregarded in the non-proportionality quantification (Larsen et al. (2022a)). From the covariance matrix, the principal components can be found using the eigenvalue problem in equation 2, where λ corresponds to the so-called principal components (PCs) and v are the corresponding directions of the PCs. I is the identity matrix. By weighing the PCs, the level of non-proportionality can be determined. The PCA-based approach will result in a value of NP PCA = 0 for a proportional stress state with no phase-shift. For stress states that are non-proportional by 90 degrees phase-shift, the PCA-based approach will result in a value of NP PCA = 1. Thus, maximum non-proportionality is defined as 1.0. n i = 1

min( λ ) max( λ )

( Cov − λ I ) v = 0

and NP PCA =

(2)

Based on the updated FE model, the PCA-based non-proportionality quantifier can be used to determine the non proportionality levels for various load cases without having to test them physically. Only four locations are of interest in the following case studies, namely the points denoted ”NP” in Fig. 3. These points are relatively highly loaded as compared to other places in the lifting arm, wherefore fatigue analysis is of special interest here. Based on the FE model, several di ff erent load combinations can be verified. As only the vertical loading direction has been validated in this paper, it should be noted that the results from horizontal loading F H in Fig. 2 may not be completely accurate. In total four load cases have been examined, where the F H and F V in Fig. 2 are modelled as constant amplitude sinusoidal loading with 0 degree, 22.5 degrees, 45 degrees and 90 degrees phase-shift respectively. The magnitude of the loading is not of interest, as the stress components are normalised before using them in equation 2. The non-proportionality levels of the four investigated points for the four di ff erent load scenarios are shown in Tab. 2.