Verification of the method of equivalent amplitude determination based on two - parameter fatigue characteristic

In the most causes the loads which are affected on structural components are various over time and their character changes is stochastic. The stochastic character of operational loads of construction elements in various machine types is depended on many factors, included : work forces variability, environmental conditions, physical properties of components etc. Fatigue life calculation for this type of loads are conducted on the basis of determined sinusoidal cycles set through to use of the cycles counting method. The cycles which are contained to the sinusoidal cycles set are characterized by extensive range of amplitude Sai variation and mean values Smi. Application of Sa-N curve in fatigue life calculations caused disregard of the cycle mean value. This may affect the accuracy of calculations. Taking into account the cycle mean value Sm in the calculations may be realized by determining a substitute cycle with an average value Sm=0 and a substitute amplitude Saz≠Sa. Abbreviations N number of cycles (fatigue life), N0 base number of cycles requires fatigue limit (N0 = 106), R stress ratio (expressed by the formula R = Smin/Smax), Sa stress amplitude of the sinusoidal cycle in MPa, Saz equivalent stress amplitude of the sinusoidal cycle with Sm i Sa parameters in MPa, ) T ( ) R ( a S stress amplitude of the sinusoidal cycle with given value of a stress ratio R determining a line of constant fatigue life (N = const.) in MPa, Sf (-1) fatigue limit of the sinusoidal cycles with stress ratio R=-1 for number of cycles N0 w MPa, Si value of local stressat the i-th load level in MPa, Sm average value of sinusoidal cycle stress in MPa, ) T ( ) R ( m S the mean value of the sinusoidal cycle stress with given value of a stress ratio R determining a line of constant fatigue life (N = const.) in MPa, Smax maximum value of stress in a sinusoidal cycle in MPa, Su material tensile strength in MPa, Sy yield point of material w MPa, k exponent in the equation N = N-k, m(-1) exponent of in the formula describing fatigue (Wöhler) graph for the stress ratio R=-1, ψN the sensitivity coefficient of the material for the asymmetry of the cycle for N ≠N0, Corresponding author: karolina.karolewska@utp.edu.pl © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 182, 02022 (2018) https://doi.org/10.1051/matecconf/201818202022 17th International Conference Diagnostics of Machines and Vehicles


Introduction
Fatigue life assessment is one of the main criteria during the design process of devices elements. On the basis of the fatigue life calculations under random loads conditions the design features selection of machine elements are made. Fatigue life determination for operational loads requires the use of characteristic describing the material properties, cumulative damage model and load spectrum developed for indicated loads. Operational loads can be divided into two groups: narrow or broad spectrum. Due to fatigue life between the mentioned loads significant differences refer to cycles participation from a given range of the stress ratio R values and relate to occurrence of cycles with large amplitude values, where the maximum amplitude Sa ≈ Smax.
Calculations of fatigue life for load chracterized by narrow spectrum are carried out by use of block load spectra (the spectra 1D) developed on the basis of determined cycles set by using one of the cycles counting method. The realization of fatigue life calculations in cause of board spectrum loads is based on loads specta describing variability of cycles parameters Sm and Sai (the spectra 2D). The use of this type of spectrum is connected with cycles stress ratio values -∞ < R < +∞ occuarncy in cycles loads and their impact on fatigue life [1].
Replacement of load two-parameter spectrum (the spectra 2D) by one-parametre spectrum (the specta 1D) is based on determination substitite amplitude method Saz with the use of two-parameter fatigue charakteristic [2]. For the purposes of the paper two-parameter characteristics denoted as : model IM (Fig. 2a), and model II (Fig.2b) were estabilished.
The aim of the paper is to compare determining fatigue life methods with the use of load spectrum. The load spectra were obtained using one of the cycles counting method, which were full cycles counting method, and also by using two-parameter characteristic: the IM model and II model.
The scope of the paper contains the presentation of determination substitute amplitude methods, comparison of fatigue life calculation results with the experimental test results for indicated operational loads.

The method of determining substitute cycles of load
Calculations of fatigue life under operational loads require replacing a random cycles load (Fig1.a) with sinusoidal cycles. To determine load equivalent cycles, one of the cycles counting method is used: peak counting method, simple-range counting method, full cycles counting method, range-pair counting method and rainflow counting method [4].
By means of one of the cycles counting method, sinusoidal cycles are obtained. The recived cycles vary in the stress amplitude values Sai and mean values Smi (Fig.1b). Sets of cycles with different amplitude values Sai and mean value Smi are replaced by equivalent cycles with a mean value Smi=0 and equivalent amplitude Sazi on the basis of two-parameter characteristics. Model IM (Fig. 2a) was created on the basis of the model I presented in paper [4]. This model generalizes the Serensen's stress limit plot on the range of high cycle and low cycle fatigue.
Model IM (Fig. 2a) is divided into four zones, which are determined by required values of stress ratio R. A line of constant fatigue life (N = const.) is plotted taking into account the stress amplitude Sai and the average value Smi for sinusoidal cycle, and on this basis the determination of the equivalent amplitude Sazi is performed. Selection of stress amplitude value Sai on the ordinate axis, and mean value Smi on the abscissa axis determines the G point. This point is interpolated to point C, in which the line of constant fatigue life intersects with the ordinate axis, determining the load equivalent cycle with the average value Sm = 0. Several load cycles, which are located on the same line of constant fatigue life, have the same equivalent stress amplitude Sazi and are characterized by the stress ratio R = -1 regardless of in which zone of two-parameter characteristic they are located.
In the model IM the line of constant fatigue life is described by the following formulas for four zones denoted from 1 to 4 (Fig.2b), which characterized by stress ratio variability.  (Fig. 2b) for determining the equivalent load amplitude is a generalization of the Goodman's formula over the limiting fatigue life ranges low cycle and high cycle fatigue. This model is based on the Wöhler fatigue diagram for the stress ratio R = -1. Under load conditions characterized by stress ratio R = -1, point A corresponds to the tensile strength of the material Su, while point C corresponds to the fatigue limit Sf(-1). The ACE line is the constant durability line (N = const.) For the sinusoidal load for the range of the cycle asymmetry coefficient -∞ <R <1 with variable stress amplitudes Sai and mean values Smi is described by the following equation: On the basis of the described methods above of determination equivalent stress amplitudes, the block spectra of load for the applied load run were created. By means of the obtained sinusoidal cycles sets using the presented cycles counting methods, load spectra were developed.

Operational load
In Figure 3 is presented run of stress changes in a structural element, which were recorded in operational load conditions. Due to use in calculations of fatigue life, run of stress changes is presented in the relative form Si/Smax, where Smax is the maximum value occurring in the load run. The load run presented in Figure 3 provide to determination of sinusoidal cycles set by use of the full cycle counting method. This method uses the assumption, that all ranges occurring in pairs within a definitive scope form cycles, that form closed hysteresis loops. This method schematization is based on progressive filtering of cycles with increasing amplitude value [5].
The obtained data sets contained sinusoidal cycles with Sai/ Smax and Smi /Smax parameters from the variability of the stress ratio -∞ < R <1, which is characteristic for broad spectrum loads. By means of determined data were appointed cycle set characterized by mean value Sm = 0 and equivalent amplitude Saz received by the use of two-parameter fatigue characteristic. Method of substitute amplitude Saz determination, described in point 2, were applied. For assumed operational run determined block load spectra. The basis for the spectra development was the set of sinusoidal cycles determined by the adopted cycles counting methods. The S355J0 steel was used to perform calculations. The block load spectra (Fig. 5), used in fatigue life calculations, were determined for Smax values = 500 MPa for stress changes, taking into account model IM and model II of two-parameter fatigue characteristics. Figure 5a presents a block spectrum allowing the determined substitute amplitudes Saz based on the IM model of a two-parameter fatigue characteristic. Whereas, Figure 5b illustrates a block spectrum determined by means on the II model of two-parameter fatigue characteristics.

Material properties
The presented two-parameter fatigue characteristics were evolved for S355J0 steel. This material is a low-alloy steel with increased strength. Structure of the steel is composed of ferrite and pearlite grains. The S355J0 steel has the following static properties: tensile strength Su = 678 MPa and yield point Sy = 499 MPa, and Young's modulus E = 208159 MPa. Cyclic properties of S355J0 steel on the basis of the calculations were made: m(-1) = 12,33, C(-1) = 1.

Calculations terms
Fatigue life calculations was carried by means on the Palmgren-Miner's hypothesis on linear fatigue damage summation expressed as follows: The fatigue life results were presented in the form of a number of accomplished load blocks, computed from the formula bellow:  (Fig. 6). The chart presents calculations carrying out on the basis of substitute load spectra for the two-parameter fatigue characteristics with the results of experimental tests.

Analysis of research results and summary
The results of fatigue life calculations based on the adopted cycle counting method and selected two-parameter fatigue characteristics (model IM and model II) are characterized by discrepancy.
Due to calculations using two-parameter fatigue characteristics, the stress average value Sm is taken into account. As opposed to fatigue life calculations by means on Wohler diagram, which ignored the impact of mean value Sm. This has a significant impact to assess the fatigue life of structural element. In particular, element that is exposed to board spectrum of operational loads.
On the fatigue life diagrams is presented that the highest durability is found in the results obtained on the basis of the substitute load spectra for the IM model of the twoparameter fatigue characteristic. The lowest fatigue life was obtained for calculations based on the model II of the two-parameter fatigue characteristic. Fatigue life for model II is similar to the durability determined during experimental tests. Meaning differences in the results of fatigue life calculations between model IM and model II of two-parameter characteristics may be affected the range of using chosen characteristics [3].
Model II is used for the stress ratio -∞ < R < -1, ignored zone 4 ( Fig.2) with range of stress ratio 1 < R < + ∞. It can be said that the selection of the appropriate model of the two-parameter fatigue characteristic has a significant impact on the fatigue life calculations results. The choice of the proper model is related to the cyclic properties of researched material. .
In addition to the fatigue characteristics models, the cycle counting method also affects the fatigue life results. In cause that different cycle counting method is used, the calculated durability values would be different from each other.