unife_cristofori

International CAE Conference 2014 27-28 October. 2014

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An innovative approach for CAE based analysis of complex fatigue loadings

Alessandro Cristoforia, Denis Benasciuttib

a Università di Ferrara, Dipartimento di Ingegneria, Ferrara, Italy b Università di Udine, Dip. Ingegneria Elettrica Gestionale e Meccanica (DIEGM), Udine, Italy

Email: [email protected]

Web: http://endif.unife.it/it

Summary This paper presents an approach, called "Projection-by-Projection" (PbP) criterion, which is suitable for the CAE-based fatigue analysis of complex structures subjected to random loadings. The method framework allows a time domain, as well as a frequency domain analysis of random multiaxial stress. A CAE-based design of a structural component is discussed as an example, to show the capabilities of PbP method.

Keywords multi-axial fatigue, FEM analysis, random stress, Power Spectral Density, spectral method

Introduction Mechanical components and structures are often subjected to random loadings (some examples are shown in Fig. 1), which often lead to multiaxial random stresses in the most critical point of the structure. Contrary to deterministic loading, which are exactly known from present or past values, random loadings (or stresses) are inherently uncertain and future values can only be estimated in terms of probability. For example, typical questions in structural durability could be: which is the probability that a load will exceed a given threshold "x" in future time T? How many cycles have amplitudes higher than a given value? How many cycles will eventually lead to final fatigue failure?

The probabilistic approach to fatigue analysis of random loadings can be developed by two different philosophies: time-domain or frequency-domain approach [1,2]. From the designer's point of view, the matter is to understand which approach (time- or frequency-domain) is best suited for fatigue analysis of multi-axial random stresses, especially at the design phase within a FE environment.

The time-domain approach is based on consolidated deterministic algorithms (e.g. rainflow counting, Palmgren-Miner damage rule), which are applied to measured or simulated time-histories to estimate the fatigue damage and the component service life. Instead, the frequency-domain approach characterizes the random signals in the frequency-domain and it applies analytical expressions to estimate the fatigue damage and the service life directly from Power Spectral Density (PSD) data.


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Figure 1: Examples of random loadings or stresses and possible approaches to fatigue analysis: time-domain and frequency-domain.

Although mathematically more simple and intuitive, the time-domain approach is generally time-consuming and computationally demanding when it is applied to long time-histories, especially those simulated in finely discretized or 3D finite element models having hundreds of thousands of nodes. This issue becomes even more critical in multiaxial fatigue analysis (e.g. search of the critical plane). The frequency-domain approach, instead, is computationally more convenient, as it is based on analytical expressions that relate the fatigue damage to the PSD, which in turn can easily be calculated by a normal FE suite. The frequency-domain approach allows a strong reduction of the overall computational time, especially when adopted in multiaxial fatigue analysis. Which approach is the most appropriate depends on several factors related to the field of application and on the designer confidence in the applied method.

This paper gives an overview on a new procedure, the "Projection by Projection" (PbP) method, which allows estimating the fatigue damage both in the time-domain as well in the frequency-domain. In fact, the PbP theoretical framework adopts the same steps in both analysis domains. However, the PbP method is particularly useful for CAE-based fatigue analysis with random loadings, as it can take advantage of spectral analysis with finite elements. A CAE-based design of an L-shaped beam under random excitation is presented as an illustrative example. A spectrum analysis is first used to compute the stress PSD matrix at every node in the structure. Then, the local multiaxial fatigue damage is estimated by the proposed PbP criterion. MatLab® routines have been used to post-process nodal PSD matrix obtained by ANSYS®. This simple illustrative example is very useful to point out the main advantages of using the PbP method in the frequency-domain.

The "Projection-by-Projection" (PbP) criterion The PbP method was originally formulated in the time-domain [3,4] and then extended to the frequency domain [5,6]. The theoretical basis of the method has been widely discussed in Refs. [3-6], while its accuracy has been proved against experimental data for complex multiaxial loadings [3,4] and also compared with other methods.

The main steps to perform the damage evaluation are summarized in Fig. 2, where a biaxial stress is considered. At any point in a mechanical component, a multiaxial stress can be represented by a time-varying stress vector s(t), which for a biaxial stress has only two components s(t)=(s1(t), s2(t)). During a periodic loading, the tip of vector s(t) traces a closed curve � (called loading path). A simple biaxial stress, given for example by combined bending and torsion, results into a two-dimensional loading path, whose shape depends on the loading type: in case of constant amplitude sinusoidal loadings with


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the same frequency and phase-shift, the loading path is an ellipse, while in case of a biaxial random stress, the path is a random curve, see Fig. 2.

PROCESSINGPROCESSINGPROCESSINGINPUTsINPUTsINPUTs

random stresses

s 1(t)

time

s 2(t)

uncorrelated stress projectionsuncorrelated stress projections

OUTPUTsOUTPUTsOUTPUTs

Rotated FrameDamage of projection di

DamageD=D ( di)

Life estimate

Figure 2: Analysis steps and quantities involved in the PbP spectral method

The path � can be projected along specified directions, which results into a set of projected paths �p,i that completely develop along each direction. From a geometrical point of view, the path � is fully represented by its projections �p,i. Note that each projection can formally be expressed as a function of time �p,i=�p,i(t), which means that each projected path is a uniaxial stress time-history. Therefore, the study of any complex loading path can simply be taken back to the study of its projections, leading to damage expressions in the form:

lineSNtfd ipip ;,, (1)

The fatigue damage due to each projection is calculated by defining a suitable reference SN fatigue line in a modified Wöhler diagram. This SN line depends on the material properties as well on the actual loading conditions and it is completely located by the reference fatigue strength SA,ref at NA cycles (typically NA =2×106) and the inverse slope kref.

In PbP criterion, the overall damage caused by the loading path � is calculated as a suitable non-linear sum of the different damage contributions given by each projected path �p,i(t):

2

3

1i

2

p,i

ref

ref)(

k

kdd

(2)

This accumulation rule employs a non-linear combination rule to handle damaging events occurring at the same time instant on different projected loadings and it was recognized to be appropriate by many authors, since it accounts for the out-of-phase among stress components.

Case study: CAE-based fatigue analysis of L-shaped beam This section presents a case study of a thin structure under plane stress condition and subjected to a random excitation, which has been thoroughly discussed in [6]. The structure is a L-shaped steel beam with a hole and two lateral notches, clamped at both ends (see Fig. 3). The beam is discretized by a total of 599 shell finite elements. Each clamp is subjected to band-limited white noise acceleration along the vertical direction, perpendicular to the beam plane. Two different sets of fatigue properties were considered for the material of the beam (Material 1 and Material 2).


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The analysis was based on a two-step approach, where MatLab® routines were used to post-process nodal FE results (PSD matrices) obtained by ANSYS®. As shown in Fig. 3, the location of the critical areas radically changes, depending on the different material considered. This example then shows the ability of PbP method to capture the material sensitivity to the local stress state in the structure. This result is of great relevance in engineering applications, where very often different materials have to be compared for the same component geometry, or when, instead, it is necessary to identify which material properties are the most suitable for a given application.

100

101

102

108

1010

1012

1014

1016

frequency (Hz)

stre

ss P

SD

, (P

a)2 *s

)

SxxSyySxy

Figure 3: Geometry, FE mesh and stress PSDs at a selected node. Damage maps obtained by applying the PbP approach considering 2 different materials (Reprinted from Ref. [6])

Conclusions The PbP method gives sound advantages in CAE-based design of structural component subjected to fatigue loading. The example discussed in this paper emphasizes that PbP spectral method provides an accurate framework to design and analyze complex structures under multiaxial random loadings. The results show the capability of the method to take into account the degree of correlation between stress components, as well as its capability to account for different material behaviors under different types of multiaxial stress states. The application of PbP method with commercial FEM software ANSYS® gives a very efficient tool for the analysis of a structural component subjected to fatigue loading.

References [1] N.W.M. Bishop, F. Sherratt, "Finite element based fatigue calculations, NAFEMS, 2000"

[2] Benasciutti D.: "Fatigue analysis of random loadings. A frequency-domain approach", LAP Lambert Academic Publishing, 2012

[3] Cristofori A.: "A new perspective in multiaxial fatigue damage estimation, Ph.D. Thesis, Dept. of Engineering, University of Ferrara, Italy, 2007

[4] Cristofori A., Susmel L., Tovo R.: "A stress invariant based criterion to estimate fatigue damage under multiaxial loading", Int. J. Fatigue, Vol. 30, No. 9, 2008, pp. 1646–1658

[5] Cristofori A., Benasciutti D., Tovo R.: "A stress invariant based spectral method to estimate fatigue life under multiaxial random loading", Int. J. Fatigue, Vol. 33, No. 7, 2011, pp. 887–899

[6] Cristofori A., Benasciutti D.: ""Projection-by-Projection" approach: a spectral method for multiaxial random fatigue", SAE Technical Paper 2014–01–0924.

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