Abstract
A more efficient procedure is proposed to speed up the Carpinteri-Spagnoli (CS) algorithm in numerical computations. The goal is accomplished by deriving the exact solution for the spectral moments and expected maximum peak of normal/shear stress in any rotated plane orientation. The algorithm then avoid the use of “for/end” loops to identify the five rotations that locate the critical plane in CS method. The procedure is especially advantageous if applied to three-dimensional finite element analysis, in which the stress spectra in thousands of nodes need to be processed iteratively. The procedure is based on theoretical results that have, however, a more general validity, being applicable to any multiaxial criterion that makes use of angular rotations to identify the critical plane.
Highlights
A more efficient procedure is proposed to speed up the Carpinteri-Spagnoli (CS) algorithm in numerical computations
CS spectral method has initially been turned into a numerical code
It seems unavoidable for the code to use several “for/end” loop to scan, in the three-dimensional space, all the planes before the critical plane is correctly located through five rotation angles
Summary
Multiaxial spectral methods are a special class of fatigue criteria which characterize a multiaxial random stress in the frequency-domain through its Power Spectral Density (PSD) matrix [1]. The first step in the CS method is to scan the angles φ and θ (in intervals 0 ≤ φ ≤ 2π and 0 ≤ θ ≤ π) in order to find that particular direction Z ′ (defined by φ ∗ and θ ∗ ) which maximizes the expected maximum peak of normal stress σz′ (t) in time T [2,3,4]:. Scanning the interval 0 ≤ γ ≤ 2π yields the value γ ∗ that maximizes the variance λ0,4′′4′′ = Var[τy′′ z′′ (t)] of the shear stress τy′′ z′′ (t) [2,3,4]: max [λ0,4′′4′′ ] = max ∫ G4′′4′′ (f) df 0≤γ≤2π After this last step, the critical plane is fully localized by the angles (φ ∗ , θ ∗ , ψ ∗ , δ ∗ , γ ∗ ). A narrow step improves the resolution in locating the critical plane, and increases the number of iterations, making the algorithm computationally expensive
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