RICERCA ITALIANA     

Structural durability of mechanical components under random loading

Università degli Studi di Ferrara

Abstract

The research project is based on the design and structural reliability assessment of mechanical components under random loadings.
The project intends to develop both fatigue damage assessment procedures under random variable amplitude loading and the different aspects related to the dynamic structural behaviour of components and mechanical systems subjected to that actions.
This problems can be found in many civil and mechanical engineering applications: as an example, (i) either in vehicles or structures subjected to actions related to atmospheric variability; (ii) in either machines, electrical, electronic or electromechanical devices inside or linked to aforementioned situations.
Traditionally, we adopt a time domain approach in which the external actions are identified through suitable (experimental or simulated) temporal records, on which cycle counting, damage estimation and life prediction are performed.
The project aims to deepen the methodologies developed in the frequency domain and to compare their results to other consolidated, but more expensive, time domain methodologies. In fact, if compared to the classical time domain approach, a frequency domain method is potentially more efficient and rigorous. In this case, external actions are completely identified by their frequency content; structures are analysed in terms of dynamic response and both internal stress and internal strains are defined in the frequency domain. The advantages are both in the possibility of describing random phenomena in a more rigorous and complete way and in the capability of improving the quality of structural analysis and the accuracy of results reducing, at the same time, the computational complexity.

The project intends to deepen and to extend the frequency domain approach by considering all the aspects which, at present, are not either well developed or excluded in the current scientific research activity, i.e.:

- estimation of the cycle distribution and fatigue damage assessment in applications having a wide-band frequency spectrum;
- validation of the accuracy of damage accumulation criteria in the case of random loadings;
- extension of assessment procedures to applications involving non-Gaussian and non-stationary processes, and considering phenomena depending on sequence effects, as crack propagation;
- investigation of the structural reliability problem under multiaxial external and/or internal stress actions;
- development of methods for numerical analysis of the structural dynamic behaviour of components and mechanical systems, both using consolidated approaches as the Finite Element Method and the Multi-body analysis and more innovative numerical approaches, as the Cell Method;
- validation of the applicability of developed methodologies to cases of relevant industrial interest.


The project aims to follow an articulated method, based on theoretical and methodological investigations, on simulations and numerical calculations and on experimental validations of results.


In its more practical aspects, the project involves four research units (University of Ferrara, Palermo, Perugia and Trieste). The project sounfly founded on a strong interaction among all the units, with the aim to develop an organic design methodology and a fruitful exchange of research results. The areas of interest of research units are summarised as follows:

Un. Ferrara: fatigue cycle distribution assessment under random loadings;
Un. Palermo: analysis of the fatigue damage under random loadings;
Un. Perugia: numerical methods for dynamic and reliability analysis of components and mechanical systems;
Un. Trieste: new numerical methods for dynamic analysis of mechanical components. <<<

Principal Investigator

Roberto TOVO Università degli Studi di FERRARA

Research Objectives

The principal objective of the proposed research project is to develop a method suitable for predicting, during the initial stage of the design process, durability and structural reliability of mechanical components subjected to in service aleatory loadings.
In particular, the project deals with the problem of studying and developing those methodologies which are capable of describing the load histories damaging real components in the frequency domain. Moreover, the problem of the dynamic behaviour of mechanical components will be addressed by using numerical methods. Both the theoretical proposals and the numerical methods will systematically be validated by ad-hoc experimental investigations.

As highlighted in the section named "National or International Scientific Background", it is common practise to use methods addressing the aforementioned problems in the time domain. At present, the use of in frequency domain based approaches is restricted to few peculiar applications, even though it is recognised to be more accurate and rigorous. The main limitation in applying such approaches is that exhaustive theories have not been completely developed yet, so that, there are no sound and rigorous methods capable of predicting the fatigue damage by studying the problem just in the frequency domain. The state of the art shows that validity and accuracy of such approaches have been checked just considering systems having one degree of freedom subjected to narrow band Gaussian spectra under the hypothesis of a linear accumulation of the fatigue damage.

The main aim of the research is the to extend the use of this approach to situations of practical interest. For this reason, the different salient aspects of the problem will be address by the different research units. In particular, the main points which will be addressed are the following ones:

First level objectives:
1) to investigate the properties of wide band stationary processes in terms of both cycle distribution and fatigue damage accumulation;
2) to collect experimental data generated both gathering load histories from real components and testing samples under aleatory loading;
3) to study in deep the classical numerical methods (FEM and MBS) to investigate the dynamic behaviour of mechanical structures;
4) to investigate the applicability of new numerical methods (Cell Method) to the static and dynamic behaviour of mechanical components;
5) to apply the developed theories to virtual and real prototypes to predict fatigue damage of mechanical structure subjected to an external aleatory load which generates uniaxial stress states at the critical points by comparing the obtained results to the ones obtained applying in time domain methods.

Second level objectives:
6) to extend the study of the statistical properties of both fatigue cycles and damage to more complex load histories, like non-Gaussian, non-stationary phenomena analysing the material cracking behaviour under these complex conditions;
7) to address the problem of fatigue damage under externally applied multiaxial fatigue loading. In particular, it will be studied those situations for which fatigue failures occur at sites where the stress state is either uniaxial or, at least, describable using an equivalent uniaxial stress;
8) to implement the methods suitable for predicting fatigue damage and durability of mechanical components in numerical algorithms suitable for virtually simulating the evolution of the fatigue damage;
9) to perform the fatigue assessment of both components having simple shape and standard specimens to check the accuracy of the proposed methods considering situations of practical interest.

Third level objectives:
10) to address the problem of assessing real components under multiaxial fatigue loading, some aleatory multiaxial fatigue processes will be simulated both to evaluate their statistical properties and to check the applicability of the multiaxial fatigue criteria suitable for estimating the fatigue damage under variable amplitude multiaxial loading.
11) to virtually simulate both the dynamic behaviour and the fatigue damage of components subjected to multiple random load histories having an high degree of multiaxiality.

The principal result which is expected to be obtained from this research activity is the formulation of a new methodology suitable for predicting the structural reliability of mechanical components damaged by stress states which are, at the critical sites, mainly uniaxial. This methodology should be based on new numerical tools capable of numerically predicting the dynamic behaviour of components as well as suitable for performing the statistical analysis of the fatigue damage in the frequency domain.
In parallel, this research should lay the basis for addressing in a more rigorous way the problem of assessing components subjected to aleatory multiaxial fatigue loading. This could represent an important preliminary result to be developed to check the accuracy of the multiaxial fatigue criteria suitable for assessing components under variable amplitude multiaxial fatigue loading.
To conclude, it is important to highlight the fact that, all the proposed methodologies will be validated in order to check their applicability to situation of practical interest in a way which is compatible with the industrial needs. <<<

First Results

The first phase is based on the following intermediate results:
- the definition of the types of the PSD's to be used in the research;
- the identification of the mechanical components to be analysed by numerical techniques;
- the identification of the materials and the components to be analysed by experimental tests;
- the effective draft of calculus and experimental capabilities of units involved in.Optimal results expected from phase 2 are the achievement of five tasks of first level already mentioned as the main task of this research project.Concerning phase three, the main expected results are the achievements of the four tasks of the second level already given in the research tasks overview.
Moreover we expect to address also the tasks of level three; but, in this moment, it is not possible to asses if the higher level tasks can be completely full-filled within this project.Hopefully the main scientific results should be already been achieved during phase two and three.
In this last phase the predicted results are the systematic revision of obtained experimental data. Hence we expect to compare all the theoretical and numerical approaches developed with the experimental data obtained during the research. <<<

Timescale

24 months

National and international background

The structural reliability of mechanical as well as structural components depends on their capability of being resistant to in field applied loadings. Among the different physical mechanisms damaging components, fatigue must be mentioned, because it occurs in the presence of cyclic loadings, which are very common in practical applications.
Fatigue failures can also be generated by internal loads which do not vary in a deterministic way, since external actions depend on random phenomena. The most interesting applications are:
- automotive vehicles: for instance, vehicles (having two wheels or more) running on uneven paths are damaged by fatigue loadings which depend on the interaction between the wheels and the roughness of the path;
- naval structure and aircrafts: even in this case, the load history depends on the interaction between vehicle and environment and the applied loadings are random;
- structures subjected to environmental phenomena: for instance, either mechanical structures under wind loading or off-shore structures subjected to sea's action [46];
In these situations, the problem of the reliability involves both the principal structural elements and the different devices within the structures themselves. This makes it evident that this problem must be addressed also for either mechanical or electronic devices situated within the vehicles.
The fatigue assessment in the early stage of the design process involves:
- knowledge of the properties of the external actions;
- prediction of the structure dynamic behaviour under externally applied loadings;
- methods suitable for predicting the fatigue cumulative damage;
Frequently, this study is carried out in the time domain. First of all, experimentally measured or numerically simulated time histories are collected, in order to have a number of statistically meaningful data to properly describe the entire random process. Then, the dynamic behaviour of the structure is studied to determine either the internal loads or the stress and strain histories. Using prototypes, these two different parts of the assessment can coincide with the direct gathering of the internal forces damaging the studied components during its in field use. Using the collected load histories, the load cycles are determined by using appropriate counting methods (the most employed one is the so called "rainflow counting"). Finally, the total fatigue damage is estimated using Palmagren-Miner's rule [18].
According to the aforementioned methodology, the statistical properties of the applied loadings are evaluated by studying the cycle loading spectra to determine both their fundamental statistical properties (for instance, distribution, rate of occurrence) and the extreme events (as the most damaging events) [20].
The main advantage of the time domain approach is that it can be applied independently of the statistical properties of the external actions, as stationarity and Gaussianity. The disadvantages are the following ones: the lack of sound and exhaustive theories; the need of post-processing a large amount of experimental data (in terms of length of the time histories) in order to make up for the lack of a theoretical method suitable to determine the statistical properties; the need of working either on real prototypes or on numerical models to be analysed in the time domain; this process is definitively too cumbersome to apply in a industrial reality.
Following a completely different approach, these problems can be assessed in the frequency domain. The external actions are evaluated in terms of frequency contain (generally speaking, using the Power Spectral Density, PSD).
The dynamic behaviour of the structural component can be defined, in many cases, in terms of an harmonic transfer function, in order to obtain the frequency content of either stresses or internal actions [16]. Finally, the fatigue damage can directly be predicted, either in terms of the spectral density or by estimating the statistical distribution of the load cycles.
Such an approach is definitively much more efficient, much more sophisticated from a theoretical point of view and potentially more effective than a study performed in the time domain. It is important to highlight that, at present, this approach can be employed only in the presence of stationary loadings, which are Gaussian, narrow-band and applied to structures having a linear dynamic behaviour. In this situation, it is well-known, for instance, that the amplitudes of the counted cycles are characterised by a Rayleigh distribution. Unfortunately, due to such restrictions, this approach can be applied just to a reduced number o practical situations; in particular, some aspects are relevant in practical applications and are not sufficiently developed. Among these aspects, in the following, six areas are identified.

1) Several approximate methods have been proposed by various authors suitable for estimating the probability density functions of the fatigue cycles (or the fatigue damage) either directly from the PSD data (frequency domain methods). These methods include the spectral moment methods [28, 38, 47], Markov's methods (see, for instance, Refs. [4, 14, 27, 36]), methods valid for particular stress processes, methods having the PSD concentrated around two modal frequencies [15], or, finally, for particular stress states (uniaxial, biaxial [25], multiaxial [34, 29]). In practice, the statistical distribution of the fatigue cycles depends, in general, on the so called irregularity factor. Unfortunately, recent technical papers highlighted that, in order to correctly describe the statistical distribution of the fatigue cycles, the introduced methods must account for several additional parameters, which depend on the spectral moments of higher order [31, 32, 33, 42].

2) The aforementioned methods can be applied to Gaussian stationary processes; nevertheless, it can be noticed that, in practical applications, the measured loadings do not assured the Gaussian condition, due to non-linearity (for instance, loadings generated by either sea or wind) [45]. To evaluate the distribution of the fatigue cycles under non-Gaussian conditions, the influence of non-normality must be accounted for. Some techniques approximately extend the use of the narrow-bnad hypothesis to non-Gaussian situations (therefore, their use is restricted to non-Gaussian narrow-band processes) [37, 45]; moreover, they can account for non-normality only in terms of kurtosis. In any case, a more realistic characterisation of non-Gaussian phenomena should be based on methods which can be applied to broad-band processes and depending also on the skewness. The method proposed in Ref. [49] accounts for the effect of both the bandwidth and non-Gaussian phenomena by using an appropriate coefficient; the technique proposed in Ref. [3] is suitable for taking into account the statistical distribution of the load cycles under non-Gaussian conditions (both skewness and kurtosis) and it can be directly applied to broad-band random processes. The accuracy of this method was widely checked using load histories gathered from real components.

3) Another important aspect is that real practical situations are characterised by non-stationary conditions generated by the variability of the in field conditions (for instance, either variability of the environmental conditions when structures are damaged by wind or the variability of the speed in vehicles running along different kinds of paths). In these situations, it is fundamental to perform the fatigue assessment accounting for also non-stationary external actions. Interesting results have been obtained by using the so-called "switching process" model, i.e. a piecewise stationary process having only a finite number of stationary states, under the Markov's assumption.

4) Also the analysis, design and development of industrial components can be addressed following two different ways, both in time and frequency domain. The analysis is usually performed (e.g. in automotive and aerospace industries) in the time domain, through laboratory and on site tests performed on physical prototypes and through numerical simulation performed on virtual prototypes. In this situation, the possibility of predicting the component fatigue behaviour by means of either numerical simulations or virtual prototypes results in a reduction of both costs and time needed to develop the final product [7, 8]. The traditional approach in time domain is based on the transient dynamic analysis performed both using directly the finite element analysis (FEA) and using the dynamic multibody simulation (MBS), in this case with the necessary support of the FEA itself. The FEA combined with multibody dynamic simulation was capable of providing all the necessary information to evaluate the fatigue damage of mechanical components in time domain [5, 22], starting from stress and strain time histories recovered through both static approach and the so-called modal approach [1]. Two are the main problems that this approach shows: the significativity of the input loads time histories and the sampling and the integration modality of the numerical analysis. These problems can be overcome by performing a dynamic analysis of the system in the frequency domain using as inputs pieces of information provided in terms of PSD functions. The technical literature shows that the PSD based methods are usually used in conjunction with FE analyses [17]. This approach can be applied only if the statistical and dynamical interaction between the studied system and the environmental conditions is completely defined: this means that both the boundary conditions and the input PSD matrix must be correctly determined. The result is the Power Spectrum Density matrix S(w) (in general 6x6 matrix) with the stress PSD having auto-correlation and cross-correlation terms. The approach based on a study in the frequency domain hides two fundamental problems: the first one is the need of calculating the fatigue damage by using a tensor like the PSD matrix S(w); according to both the theoretical [26, 35, 48] and experimental [23] results published by different researchers, the cross-correlation terms of the PSD matrix play a fundamental role in properly describing the component fatigue behaviour. The second problem to be addressed is that, to estimate the fatigue damage, critical sites must be a priori determined by applying criteria based on in frequency domain analyses. The most accurate method suitable for define the position of critical sites has been extensively discussed in Ref. [8].

5) In alternative to time domain methods performed by the FEA, there are alternative approaches able to overcome computational problems. In fact, although the Finite Element Method is a very important and widely used tool, there is a motivation for the development of new numerical methods in order to improve the analysis results, with regard to both computation speed and solution accuracy. The Cell Method (CM), a recently developed numerical method [39, 40] that is currently applied in several fields, as in heterogeneous materials modelling [10, 11], biomechanics [12], diffusion, structural mechanics problems, etc., has proved to be particularly promising when considering the stress and strain analysis under dynamic loading [13]. The results achievable with the Cell Method are similar to those obtainable with the Finite Element Method, but the Cell Method doesn't use a differential formulation. Opposite to the FEM, that defines a single mesh, the CM makes use of two complexes of cells: while the primal complex is used to describe the configuration of the system, the dual complex is used to write the balance equations directly in a discrete form. In the fatigue ambit, the CM can be used to determine either the stress or strain state in components subjected to dynamic loading in order to perform its fatigue assessment.

6) Finally, among the problems connected to the fatigue damage estimation which are still far away to be solved, the following ones can be mentioned: influence of multiaxial fatigue loading, effect of non-zero mean stresses and assessment in the presence of non-linear fatigue damage accumulation. For instance, under variable amplitude multiaxial fatigue loading three sub problems can be mentioned: (i) the fatigue damage mechanisms active under this kind of loading are not clear; (ii) there is no a unique formulation suitable for defining cycles under complex fatigue stress states; (iii) an universal and reliable criterion must be still defined. The study of the fatigue damage mechanisms under variable amplitude multiaxial fatigue loading is complicated by the fact that the principal directions rotate during the load history. This results in a difficult definition of the material cracking behaviour during the early crack initiation stage [21]. Moreover, the state of the art shows that there are no universal method suitable for counting cycles under variable amplitude multiaxial fatigue loading. Some authors suggested to perform the cycle counting using as master channel the applied stress component which is assumed to be the most damaging one. Unfortunately, there are no criteria suitable to unambiguously single out the stress component to use as reference information to estimate the fatigue damage. Recently, to overcome this problem, Carpinteri et al. proposed a new method base on the use of weight functions suitable to define the average position of the principal stress directions [9]. Finally, it is important to highlight that, recently, Langlis et al. [26] formalised a method suitable for performing a cycle counting on the plane experiencing the maximum fatigue damage due to the shear stress components: this aspect is very interesting because this approach seems to lay the basis to apply the classical critical plane approaches to variable amplitude multiaxial fatigue situations. <<<