On fatigue cycle distribution in non-stationary switching loadings with Markov chain structure

The non-stationary loadings experienced by structural members can be often modelled as a sequence of stationary load states, with different mean and variance levels, where load switches are controlled by an underlying random process (regime process). The structural integrity of structures and mechanical components under such complex loadings requires the assessment of the statistical distribution of rainflow cycles counted within single load states, as well as of transition cycles caused by load state switches. Under the assumption of a constant mean for the switching loading, the authors showed in a previous paper that a linear combination of single loading spectra would provide a fairly good estimation of the overall loading spectrum of the switching loading. However, when mean value differences among load states are large, additional fatigue cycles caused by load state transitions would be present, besides cycles counted within load states. This work presents a comprehensive approach to assess the overall loading spectrum in switching loadings with variable mean value, when modelling the underlying regime process as a stationary Markov chain. For large relative mean value differences compared to load state variances, transition cycles are found by rainflow counting the regime process. The rainflow matrix for transition cycles is then estimated from the transition probability matrix of regime process by a method available in the literature. The distribution of the ranges of transition cycles is finally estimated from the statistical distribution of the largest peak and the lowest valley within load states. An illustrative example is finally discussed to show the accuracy of the proposed method.     

An improved multiaxial rainflow algorithm for non-proportional stress or strain histories – Part II: The Modified Wang–Brown method

The objective of this work is to develop a simple multiaxial rainflow algorithm that allows the proper calculation of multiaxial damage in NP histories. Enclosing surface methods are usually employed to obtain the equivalent ranges necessary for damage calculation, as discussed in Part I of this 2-part paper. Part I also presented a new approach to evaluate equivalent ranges in NP histories, called the Moment Of Inertia (MOI) method. This second and last part presents a multiaxial rainflow counting algorithm that allows the MOI and enclosing surface methods to be generalized to non-periodic NP histories and to periodic NP histories formed by complex blocks with multiple cycles each. It is shown that Wang–Brown’s (WB) multiaxial rainflow algorithm has a few idiosyncrasies that can lead to non-conservative predictions, incorrectly filtering out significant events within a multiaxial loading cycle. An improved multiaxial rainflow algorithm is proposed, called Modified Wang–Brown (MWB). It has two main improvements over the WB algorithm. First, the criterion to choose the point where the count is started is modified. Examples are shown to prove that the original criterion can overlook the most damaging event from the history, as opposed to the modified version. And second, the algorithm implementation is significantly simplified when formulated in a reduced five-dimensional Euclidean space. Under plane stress conditions, the algorithm is further simplified using a three-dimensional Euclidean space based on the deviatoric stresses or strains. A simple pseudo-code is presented in a flowchart to efficiently implement the multiaxial count, allowing a fast and efficient calculation of fatigue damage even for very long non-periodic NP histories.     

Fatigue life prediction based on natural frequency changes for spot welds under random loading

Fatigue tests were carried out on tensile-shear spot-welded specimens under random loading; at the same time, natural frequencies at different life stages were measured. Test results were compared with that under constant amplitude loading. The relationship between damage and natural frequency change ratio established under constant amplitude loading is here modified. The nonlinear damage evolution equation was established using the damage represented by frequency change for spot welds. The fatigue life of spot welds under random loading was predicted with this damage evolution relationship using the rainflow counting method results, as well as other life prediction methods with different counting methods. Comparison of life prediction results showed that the method using changes of natural frequency can be an on-line approach and the predicted lives have a good agreement with the experimental data.     

Rainflow counting algorithm for very long stress histories

A computer algorithm for rainflow cycle counting of very long stress histories is presented. The procedure enables the rainflow counting to be performed without prior knowledge of the whole stress history being analysed. The stress history can be read and analysed by the computer one block at a time without reading the whole history at once. This will enable small computers to analyse very long stress histories because the length of one block can be adjusted to the computer capabilities. The algorithm does not require any rearrangment of the stress history before the start of counting and the peaks and troughs can be analysed as they were recorded in service.     

扭力轴多工况二维随机疲劳载荷谱的编制

扭力轴用来减轻自行火炮在行驶过程中地面对车体的冲击。针对扭力轴扭矩载荷谱多工况和非对称的特点，进行单一工况雨流法计数载荷循环、单一工况载荷的分布规律与检验、各工况频次向寿命周期扩展、载荷分级与级循环次数的计算、载荷谱列表。编谱过程基本上包括了多工况非对称型载荷编谱的所有步骤，对编制其它受多工况非对称型载荷的构件的疲劳载荷谱也有参考价值。     

Improved modelling of the loading spectra using a mixture model approach

In order to perform a fatigue-life analysis of structures the parameters of the structure loading spectra must be assessed. If the load time series are counted using a two-parametric rainflow counting method, the structure loading spectrum provides a probability for the occurrence of a load-cycle with certain amplitude and mean values. It is beneficial for the prediction of the fatigue life to describe the loading spectrum by a continuous function. We have previously discovered that mixtures of Gaussian probability density functions can be used to model the loading spectra. The main problems of this approach that have not been satisfactorily resolved before are related to the estimation of the number of components in the applied mixture models, and to the modelling of the load-cycle distributions with relatively fat tails. In this article, we describe a method for estimating the parameters of mixture models, which allows automatic determination of the number of components in a mixture model. The presented method is applied for modelling simulated and measured loading spectra using mixtures of the multivariate Gaussian or t probability density functions. In the article we also show that the mixture of t probability density functions sometimes better describes the loading spectra than the mixture of Gaussian probability density functions.     

Fatigue reliability in welded joints of offshore structures

Metal fatigue in welded joints in offshore structures is considered. Due to the considerable variability of conditions, a probabilistic approach is used. Theoretical studies of various aspects of the fatigue reliability problem in welded joints are presented. These include a study of the Palmgren-Miner rule, a modified linear model on S/N data, the use of the rainflow method of counting, and a closed form expression for the probability of fatigue failure. A probability model is derived as a suggested basis for an approach to fatigue design.     

Frequency based fatigue analysis and temperature effect

This paper proposes a temperature modified Dirlik method to estimate the high cycle fatigue damage for uniaxial loadings caused by random vibrations directly from a power spectral analysis. Besides, the methodology for combining the frequency based fatigue analysis with the temperature effect is represented. This approach is based on a new definition of loading as a random Gaussian process. The fatigue damage estimation of the high pressure die-cast aluminium alloy AlSi₉Cu₃ is investigated at elevated temperatures. Finally, numerical simulations on the known power spectral densities with different shapes at different temperatures are performed in order to establish proper dependence between the temperature modified Dirlik method, the rainflow cycle counting, the linear cumulative fatigue damage and the spectral bandwidth parameters. The proposed method enables computationally fast fatigue damage estimation for the random loadings and the temperature histories.     

On rainflow cycles and the distribution of the number of interval crossings by a Markov chain

This paper treats the number of crossings of an interval by a Markov chain observed for finite time. The exact marginal distribution is derived, in the form of its probability generating function. A numerical example is presented. Interval crossings have an important application in fatigue of materials. For analysing complex fatigue loads, the so-called rainflow cycle counting method is widely used, which is equivalent to counting crossings of intervals.     

雨流循环均值-幅值二维联合概率密度的实用算法

在实际工程结构疲劳损伤计算中，经常忽略雨流循环均值，而只考虑幅值的影响。这对计算结果会带来一定的误差，尤其是那些对均值影响比较敏感的材料来说更是如此。若考虑雨流循环均值的影响，在时域计算中可以很方便地实现，但在频域中则必须得到雨流循环均值-幅值二维联合概率密度。本文针对工程中常常遇到的平稳Gauss随机过程，提出由结构响应的PSD得到雨流循环均值-幅值条件概率的简便计算公式，进而结合Dirlik经验公式直接计算出雨流循环均值-幅值二维联合概率密度函数的实用方法。     

FATIGUE LIFE ESTIMATION OF WELDED JOINTS OF AN ALUMINIUM ALLOY UNDER SUPERIMPOSED RANDOM LOAD WAVES: APPLICATION OF A 2-DIMENSIONAL RAINFLOW METHOD

Abstract— Fatigue tests were carried out on welded T-joints of JIS A5083P-O Al-Mg alloy under constant amplitude and superimposed random load waves. Each random wave was generated by superimposing a secondary zero-mean random process on a constant amplitude pulsating trapezoidal wave, which simply simulated the GAG (ground-air-ground) stress cycles, or on a constant d.c. (direct current) component. It is proved that the 1-dimensional rainflow method is insufficient as the mean stress effects are not considered. From a procedure using a 2-dimensional rainflow method and the original and a modified Goodman diagram, it was concluded that the mean stress effects on secondary waves are larger in the case of GAG than d.c. Considering this, a life estimation procedure is proposed that gives consistent and acceptable estimation results.     

Methods of Extending the Range of Frequency Counters

The various methods of extending the range of frequency counters are discussed and some of the shortcomings of each are pointed out. A new method that minimizes these short-comings is in the range of frequencies that can be counted entirely by the frequency limitations of AND gates; also it requires no tuning and its counting precision is the same as that of the basic counter.     

Simple rainflow counting algorithms

Two simple algorithms for performing rainflow counting are presented in this paper. The second algorithm is suitable for microcomputer devices that are placed in vehicles to record field data.     

Fatigue life estimation under random loading using the energy criterion

A procedure for estimating the useful life of a component for a given (admissable) probability of fatigue fracture origination under random loading is presented. The method uses material constants obtained from the S/N and cyclic stress/strain curves, standard deviation and probability density distribution of the loading process and a macroblock of harmonic cycles obtained by applying the rainflow cycle counting method to the random loading process. Theoretical and experimental lives are found to exhibit good agreement.     

一般尺度法的振动疲劳强化系数分析

在Dirlik模型的基础之上,结合一般尺度法（General scaling law）提出适用于平稳宽带随机过程的疲劳强化系数模型。一般尺度法认为结构在原始载荷和强化后载荷下的应力响应功率谱各阶谱矩成简单的线性关系;与Dirlik宽带疲劳损伤模型结合,便得到适用于平稳宽带随机过程的疲劳强化系数模型。为了验证模型的有效性,分别在原始加速度功率谱和经一般尺度法强化后的加速度功率谱载荷下;对悬臂梁采用雨流计数(Rainflow counting)和Miner损伤准则进行疲劳寿命预测,得到它们的疲劳强化系数,并与疲劳强化系数模型得到的疲劳强化系数进行对比。结果表明：提出的疲劳强化系数模型精度为99.7 %,由此可验证疲劳强化系数模型的有效性。     

Cycle counting for fatigue crack growth analysis

Fatigue crack propagation tests were carried out on an Al-Cu alloy under specially designed complex load sequences. Electron fractography of the fatigue fracture surfaces suggests that rainflow cycle counting is applicable to the analysis of fatigue crack growth under complex load sequences.     

Uncertainty of the fatigue damage arising from a stochastic process with multiple frequency modes

Predicting the variance of the fatigue damage due to a stochastic load process is a difficult classical problem that dates back to the 1960s. For many years, the available analytical methods for tackling this problem have been limited to the linear oscillator response under Gaussian white noise excitation. In a recent prior work, the author developed an improved method for calculating the damage variance for a general narrowband Gaussian process. From a fatigue uncertainty perspective, a narrowband process is particularly crucial as the amplitude correlation magnifies the variance. This paper extends the analysis to a multimodal process comprising two or more narrowband components. The proposed method is tractable, involving a single summation for an arbitrary spectral density. Moreover, closed form expressions are available for two special cases, i.e. the components are all linear oscillator responses or the spectral density of each is rectangular. The equations also yield insight on the multilayered correlation mechanisms produced by different narrowband components. The accuracy of the method is verified by rainflow counting of simulated time history stresses.     

Fatigue properties of jointed wood composites Part II Life prediction analysis for variable amplitude loading

A method of predicting lifetime to failure for any wood composite system subjected to a complex load–time history has been developed. The prediction first requires the generation of a simple model to characterize the fatigue response of the particular composite system and a rainflow analysis breakdown of the load–time history under investigation. Once the models are derived they can be used to predict lifetimes to failure for any load–time history using a modified Palmgren–Miner damage summation rule. Variable amplitude fatigue testing of sample material using the same load–time histories allowed a comparison to be made between predicted and actual lifetimes to failure and was useful in verifying and refining the life prediction models. © 1998 Kluwer Academic Publishers     

Fatigue life prediction for GRP subjected to variable amplitude loading

A method of predicting lifetime to failure for any glass fibre reinforced composite system subjected to a complex load–time history has been developed. The prediction first requires the generation of a model to characterize the general fatigue response of the particular composite system. Once the models are derived they can be used to predict lifetimes to failure for any load–time history using a modified Miner’s damage summation rule and rainflow analysis. Variable amplitude fatigue testing of different GRP materials, using the same load–time history, allowed a comparison to be made between predicted and actual lifetimes to failure and was useful in verifying the accuracy and robustness of the life prediction methodology employed. The results from the life prediction models for three very different GRP systems showed themselves to be accurate predictors of fatigue behaviour for the variable amplitude loading condition investigated.