This work is motivated by the increasing interest towards the application of the "Projection-by-Projection" (PbP) spectral method in finite element (FE) analysis of components under multiaxial random loadings. To help users and engineers in developing their software routines, this paper presents a set of numerical case studies to be used as a guideline to implement the PbP method. The sequence of analysis steps in the method are first summarized and explained. A first numerical example is then illustrated, in which various types of biaxial random stress are applied to three materials with different tension/torsion fatigue properties. Results of each analysis step are displayed explicitly to allow a plain understanding of how the PbP method works. The examples are chosen with the purpose to show the capability of the method to take into account the effect of correlation degree among stress components, and the relationship between material and multiaxial stress in relation to the tension/torsion fatigue properties. A case study is finally discussed, in which the method is applied to a FE structural durability analysis of a simple structure subjected to random excitations. The example describes the flowchart and the program by which to implement the method through Ansys APDL software. This final example illustrates how the PbP method is an efficient tool to analyze multiaxial random stresses in complex structures.

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Abstract. This work describes a new testing system for applying a coupled/uncoupled bendingtorsion loading in vibratory tests by a tri-axis shaker. The system is composed of a cylindrical specimen with eccentric tip masses, excited by horizontal and/or vertical base accelerations. The specimen tip is constrained by a lateral thin and flexible plate which impedes any bending when the specimen is excited horizontally, but which permits the specimen torsional rotation. This layout then allows torsional and bending deformations to be produced and controlled independently, when vertical and horizontal base accelerations are applied simultaneously. A finite element model is first used to estimate the system dynamic response and the stresses in the notched specimen section. The model is then validated through experimental tests under harmonic base accelerations. The strains at clamping system are also monitored to indirectly estimate the bending and torsion moment in the specimen. Comparison of numerical and experimental results showed a close correlation and proved that bending-torsion loading are truly uncoupled. Preliminary fatigue tests with harmonic bending loading (vertical base excitation) are finally compared to the constant amplitude S-N curve, showing a quite satisfactory agreement.

Abstract. In 2008, Allegri and Zhang published a study [Int. J. Fatigue. 2008, 30(6):967-977] in which they provided an exact analytical solution to the inverse scaling law for accelerated vibration tests of linear systems submitted to stationary Gaussian excitations By combining finite element analysis with multiaxial spectral methods defined in the frequency-domain, their solution generalised the simple inverse power law model suggested in some standards. The solution adopted the "equivalent von Mises stress" multiaxial criterion combined with the narrow-band damage expression. This work aims to propose a bandwidth correction to the original Allegri-Zhang solution to account for the actual spectral bandwidth of the local multiaxial stress. The corrected Allegri-Zhang solution is also extended to another multiaxial spectral method, namely the "Projection-by-Projection" criterion. A numerical example is finally discussed, in which the corrected solution is applied to an L-shaped beam submitted to random accelerations.

A new testing system for applying coupled/uncoupled bendingtorsion loading in vibratory tests by a tri-axis shaker has been recently developed at the Department of Engineering of University of Ferrara. The system is composed of a cylindrical specimen with eccentric tip masses, excited by horizontal and/or vertical base accelerations. A special design of the gripping system for specimen constraint allows torsional and bending deformations to be produced and controlled independently, when vertical and horizontal base accelerations are applied simultaneously by the shaker. The values of accelerations and strains in the tested specimens can be monitored continuously. This paper presents a first set of experimental results under harmonic bending and torsion, and under narrow-band combined random loading. Estimations from two frequency-domain approaches are also discussed.