PSI - Issue 57

Andrew Halfpenny et al. / Procedia Structural Integrity 57 (2024) 718–730

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Andrew Halfpenny / Structural Integrity Procedia 00 (2023) 000–000

Where σ ( t ) is the stress tensor time history at a particular critical node, P k ( t ) is the load time history for load case k , and s k is the stress tensor result for load case k obtained from FE analysis under a static unit input load.

3.1.2. Harmonic analysis Where the structure responds dynamically and steady-state, then the ROM is represented by a harmonic frequency response function H ( ω ) as illustrated in eq. 2. σ ( ω ) = k H k ( ω ) · F k ( ω ) (2) Where σ ( ω ) is the Fourier transform of the complex stress tensor, F k ( ω ) is the Fourier transform of the load time history for load case k , and H k ( ω ) is the harmonic transfer function obtained from FE analysis. ω is the frequency expressed in rad s . 3.1.3. Random PSD loading Where the structure is analysed using random dynamic loads expressed as PSD (Power Spectral Density) functions, the ROM is also represented by the harmonic frequency response function H ( ω ) as illustrated in eq. 3 and described in Halfpenny (1999).

k b = 1

k a = 1

H a ( ω ) · W ab ( ω ) · H b ( ω )

(3)

G ( ω ) =

Where G ( ω ) is the single-sided PSD stress tensor, W ab ( ω ) is the complex cross-power spectral density function of loading between load cases a and b , and H a ( ω ) and H a ( ω ) are the harmonic transfer function and its complex conjugate respectively.

3.1.4. Non-linear FEA In the case of non-linear FE models, the ROM may become more di ffi cult to define. In the worst case the entire FE model, along with the fatigue model, must be solved repeatedly within the Monte Carlo loop. This might necessitate having to reduce the number of simulations because of runtime / cost implications. However, it should be remembered that simulations are still much more cost-e ff ective than physical prototyping. In other cases, the non-linear ROM may be modelled using a non-linear regression technique. These are beyond the scope of this paper. For more information on nonlinear FE analysis, refer to Hinton (1992). For information on non linear structural dynamics and response, refer to Worden and Tomlinson (2001), Masri et al (2005) and Mohammad et al (1992). Statistical sampling techniques, known as DOE, are used to optimize the size and quality of the design space matrix. The benefit of DOE is to significantly reduce the number of simulations required. Halfpenny et al (2019) describes three di ff erent DOE methods depending on their application: 1. Design for reliability – this is concerned with exploring the statistical variability of design space. It is particularly useful for identifying the 50 th to 99 th percentile cases, such as quantifying warranty exposure. This application favours a DOE based on Latin Hypercube sampling. 2. Design for robustness – this is concerned with exploring the extremities of design space. It is particularly use ful for exploring abusive loads and safety events. This application favours a DOE based on either ‘Factorial sampling,’ or an iterative ‘Response Surface Model’. See ReliaSoft (2015a). 3.2. Design of Experiments (DOE)