Expansion of experimental mode shape from operational modal analysis and virtual sensing for fatigue analysis using the modal expansion method

Marius Tarpø*, Bruna Nabuco, Christos T. Georgakis, Rune Brincker

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


Offshore structures are exposed to fatigue damage due to fluctuating environmental and operational forces. The actual stress history of structures in operation is an important parameter for the remaining fatigue lifetime. Unfortunately, the fatigue critical locations are often difficult to reach for inspection of cracks or installation of sensors and these direct locations lead to corruption of sensors due to the environmental effects. Virtual sensing enables full-field stress/strain estimation that facilitates fatigue analyses of offshore structures in operation. In the literature, various virtual sensing methods exist but this paper focuses on modal expansion. This technique uses the mode shapes (the linear subspace) of the structure to transform the measured system response into the full-field stress/strain response. The applied mode shapes either hail from a finite element model or an expansion of experimental mode shapes from an operational modal analysis. This paper compares the use of mode shapes from a finite element model to the use of expanded experimental mode shapes. Modal expansion is applied to a scaled offshore platform in the laboratory to estimate the strain response using the finite element mode shapes and expanded experimental mode shapes using different expansion techniques: System Equivalent Reduction Expansion Process (SEREP), local correspondence principle, and the new leave-p-out cross validation implementation of the local correspondence principle introduced in this paper. The results are analysed with different metrics and this paper introduces a new quality measurement – the normalised error of fatigue damage – for strain estimation intended for fatigue analysis. Based on this study, expansion of experimental mode shapes has the potential to reduce errors in stress/strain estimation. The expansion is, however, a fitting process and, thus, it contains fitting error dependent on the case. In this study, the finite element mode shapes outperform some of the mode shape expansion techniques due to this fitting errors. Therefore, expansion of experimental mode shapes can improve stress/strain estimation but it should be used with care to avoid an overfit.
Original languageEnglish
Article number105280
JournalInternational Journal of Fatigue
Number of pages12
Publication statusPublished - 2020


  • Stress estimation
  • Mode shape expansion
  • Virtual sensing
  • Fatigue analysis
  • Structural health monitoring

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