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Functional Data Analysis for Residual Stress Modelling

Introduction

Many codes and standards use probabilistic approaches to assess performance and safety, with the aim of eliminating the need for conservatism through the use of accurate probabilities. Statistical methods can include leeway for uncertainties such as material or welding process variations into the calculations without using direct measurements, potentially saving time, cost or operational disruption. Probability thresholds can be added to these measurements to provide additional conservatism.

One area where probabilistic modelling is used is for residual stress. This is trapped or locked secondary stress created as a result of welding processes. Residual stress is important for measuring the structural integrity of an engineering component or asset. It can be either a tensile stress or a compressive stress and can form in any direction with respect to the weld. However, for the purpose of engineering critical assessments (ECA), we are most concerned with tensile residual stress transverse to the direction of the weld and how these stresses change through the weld thickness. Although residual stress can never be eliminated, it can be mitigated with appropriate surface treatment (e.g. shot peening), welding processes, and the application of post-weld heat treatment (PWHT).

Understanding residual stress from a probabilistic perspective is of interest because, (1) it can be used to produce upper bound curves either for independent use or to validate other such curves already in use and (2) it can be used in probabilistic engineering critical assessments (ECA) to generate randomised residual stress values at a given depth. Both of these uses can be tailored to specific welding procedures and conditions, given available data.

Objectives

Current methods of modelling residual stress have an undefined level of conservatism and cannot be adjusted for welding procedure, geometry, or level of conservatism. To address these challenges, TWI undertook a study of two methods for transforming discrete data points into functional data, and created a statistical framework for analysing this data. The first method is an interpolation method based on cubic smoothing splines while the second method is an extrapolation method based on the self-equilibrium of residual stress.

By applying these two methods to low heat input residual stress data, residual stress data after post-weld heat treatment, and electron beam residual stress data, we aimed to provide a clearer picture of residual stress modelling for industry, creating a more accurate probabilistic model with less inherent conservatism.

Figure 1. The mean curve for low heat input residual stress along with the 95% one-sided confidence interval plotted with the BS7910 level 2 upper bound fit
Figure 1. The mean curve for low heat input residual stress along with the 95% one-sided confidence interval plotted with the BS7910 level 2 upper bound fit

Methods

We applied the proposed probabilistic modelling techniques to three residual stress databases: arc welds for low heat inputs, arc welds post-weld heat treatment, and electron beam welds. The probabilistic models were calculated and applied to all three sets of data and any model assumptions were tested for suitability.

 

Conclusions

It was determined that residual stress data can be interpolated using cubic smoothing splines or, alternatively, can be extrapolated using Fourier curves. The use of Fourier curves was shown to be less conservative for some datasets – especially for data at the inner and outer surfaces.

Irrespective of the profile-fitting method used, the residual stress profiles appear to follow a normal distribution and residual stress data can be used to effectively estimate mean and standard deviation curves, which can be used to calculate confidence intervals or as input in probabilistic assessment.

The research showed that the BS 7910 level 2 curve for low heat inputs is conservative with respect to 95% one-sided confidence. Similarly, when PWHT has been applied, the BS 7910 guidance to use 20% of the lesser of the yield strengths of the weld or parent metal is conservative with respect to 95% one-sided confidence.

Overall, this research showed an improved and systematic approach to probabilistic modelling of residual stress compared to previous, conventional methods.

Figure 2. The mean curve for post-weld heat treated residual stress along with the 95% one-sided confidence interval plotted with the BS7910 advice to use 20% of yield strength
Figure 2. The mean curve for post-weld heat treated residual stress along with the 95% one-sided confidence interval plotted with the BS7910 advice to use 20% of yield strength
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