Load Spectra and Matrices

Abstract

As shown in Sect. 3.1, two-dimensional frequency distributions, so-called load spectra, are the results of one-parameter counting. The frequency of the amplitude or range (double amplitude) is thus represented graphically. The results of two-parameter counting are three-dimensional frequency matrices. With the use of a transition matrix, every half load cycle is recorded in a field of the matrix as a counting event, from a starting class all the way to a target class. From these min-max and max-min values, the amplitude of the stress as well as the mean value of each half cycle can be determined.

As shown in Sect. 3.1, two-dimensional frequency distributions , so-called load spectra, are the results of one-parameter counting . The frequency of the amplitude or range (double amplitude ) is thus represented graphically. The results of two-parameter counting are three-dimensional frequency matrices . With the use of a transition matrix , every half load cycle is recorded in a field of the matrix as a counting event, from a starting class all the way to a target class. From these min-max and max-min values, the amplitude of the stress as well as the mean value of each half cycle can be determined.

In this book, the two-parameter rainflow counting method is generally recommended for the evaluation and classification of measurements . This implies that a rainflow matrix should also constitute the basis for the analytical fatigue-life prediction . As a matter of principle, it is assumed that a mean value, that is, the mean stress , affects the allowable number of load cycles to failure. Proposals for performing the necessary calculations are presented in Chap. 9, “Analytical fatigue-life estimate ”.

For the purpose of evaluation, however, plotting of spectra may be more informative than the use of matrices . This approach is especially useful for:

• a comparison of measurements (for instance, the effect of different drivers over the same route of travel), Fig. 4.1,

• a plausibility test on measurements ,

• recognition of causes of stress , Fig. 4.2,

• appraisal of extrapolations , and

• appraisal of the damage in direct comparison with the S-N curve .

Fig. 4.1

Stress spectra for lateral force at front axle of an automobile , 14 drivers on same regular roadway

Fig. 4.2

Mixed spectrum for starting and stationary grinding operation of a bowl-mill crusher

In Fig. 4.1, stress spectra are plotted for lateral forces acting on the front axle of motor vehicles for 14 drivers travelling on the same route. Only slight scatter occurs at low amplitudes , which result especially from travel on a straight roadway. However, pronounced scatter is observed at high amplitudes caused by the stress associated with driving manoeuvres during travel over curves.

In many cases, causes of stress can be determined from the form of the spectrum. Thus, the spectrum plotted for a bowl-mill crusher (shaft torque), Fig. 4.2, can be attributed to starting processes and stationary grinding operation [Gehl92].

4.1 Description of Load Spectra

For representing a stress spectrum , the stress amplitude S_a is plotted as a function of the cumulative frequency H. If a spectrum is determined from measurements by counting methods, the frequency distributions are stepped with the same step spacing as that employed for counting. However, continuous distributions, that is, distributions with infinitesimal step spacing, can also be given. As a rule, these distributions must then be stepped in the course of an analytical fatigue-life estimate , for instance.

Spectra are usually plotted semilogarithmically, Fig. 4.3. A normalised plot of the stress is also possible.

Fig. 4.3

Stress spectrum in normalised representation

In the field of fatigue strength under variable stress amplitude , many terms and concepts were coined decades ago by E. Gassner and co-workers at the LBF (Laboratorium für Betriebsfestigkeit) Darmstadt, Germany [Buxb92, Haib06]. These include the term ‘Betriebsfestigkeit’ itself. A further example is the use of the term ‘spectrum’ for the frequency distribution of amplitudes (load spectrum). A literal or word-for-word translation of these terms is often not possible. By way of departure from the international convention for plotting statistical functions, the coordinates for the frequency and feature are interchanged in German-language literature in this field. At international congresses, this situation can easily become a cause of confusion and of lively discussions. The use of the cumulative frequency rather than the step frequency and also the use of a logarithmic plot for the purpose require considerable experience for interpretation.

In the simplest case, a spectrum can be described by three quantities:

• the maximum nominal stress for the spectrum, ${\widehat{S}}_a$

• the block length, H₀

• and the shape of the spectrum , ν.

Further important quantities include the mean stress S_m or the stress ratio R. The irregularity factor I of the stress-time function , the effective value , and the crest factor are often indicated, too, see Sect. 2.3.

For the stresses which occur during oscillation processes, the basic principles known from the theory of continuous random processes can be applied under certain conditions [Buxb79]. The frequency distribution for the stationary Gaussian random process associated with the level-crossing count can be described by the following equation, as indicated by S. O. Rice, see [Buxb92]:

$$ H(x)={H}_0\cdot \exp \left[-\frac{x^2}{2{\sigma}^2}\right] $$

(4.1)

H(x):

number of level crossings for horizon x,

H₀ :

number of zero or mean-load crossings, block length,

x:

in this case, normalised stress amplitude , see Fig. 4.3

With a = 0.5 σ² Eq. (4.1) can be rewritten as

$$ H(x)={H}_0\cdot \exp \left(- a\cdot {x}^2\right) $$

(4.2)

The stress -time function for a random process and the result of the level-crossing count are plotted on a linear scale in Fig. 4.4. For this random process with a stress ratio R = −1 and an irregularity factor I = 1 the level-crossing frequency is equal to the cumulative frequency H. With the parameter “a” the highness of a LTF can be adapted.

Fig. 4.4

Stress -time function and spectrum for a level-crossing count as well as step frequency of a random process

In this context, the spectrum can be interpreted as an approximation to a Gaussian normal distribution . The amplitude spectrum is represented by the upper portion of the bell curve. The standard distribution which was empirically derived from results of measurements by E. Gassner in 1948 corresponds largely to this distribution shape [Haib06].

Equation (4.2) can be transformed to a power function. For

$$ {S}_a={\widehat{S}}_a $$

or x = 1, one obtains H = 1. Thus,

$$ 1={H}_0\cdot \exp \left(- a\right) $$

(4.3)

With a = ln H₀, Eq. (4.2) becomes

$$ H(x)={H}_0\cdot \exp \left(- \ln {H}_0\cdot {x}^2\right) $$

(4.4)

With ln H(x) = ln H₀ (1–x²), the following antilogarithmic form is obtained:

$$ H(x)={H}_0^{1-{x}^2} $$

(4.5)

In the field of fatigue strength under variable stress amplitude , this distribution is designated as a standard or normal distribution .

For describing the shape of the spectrum, M. Hanke [Hank70] has proposed the substitution of the variable ν for the exponent “2” of x in Eq. (4.5), see also [Buxb92, Hück88, Seeg96]:

$$ H(x)={H}_0^{1-{x}^v} $$

(4.6)

$$ \begin{array}{l} For\;\frac{S_a}{{\widehat{S}}_a}=1\; becomes\; H=1\\ {} For\;\frac{S_a}{{\widehat{S}}_a}=0\; becomes\; H={H}_0\end{array} $$

With the shape parameter ν, the following distribution shapes are obtained, see Fig. 4.5.

ν = 0.8:

Concave distribution , interpretation as log-normal distribution possible: gusts of wind over an extended period [Buxb92]

ν = 1.0:

Exponential distribution which forms a straight line in a semilogarithmic plot: This distribution can be interpreted as a mixture of different normal distributions , for instance, during the measurement of stresses due to a rough sea over an extended period

ν = 2.0:

Standard or normal distribution which forms a parabola in a semilogarithmic plot: Frequency distribution from a stationary Gaussian random process ; measurement of stresses due to a rough sea or measurement of the vertical forces on a motor vehicle on a specific roadway without driving manoeuvres

ν = 4.0:

Convex distribution : Typical for crane and bridge construction

ν → ∞:

Rectangular spectrum (constant amplitude )

If an impact occurs only once, ν = 0.

Fig. 4.5

Related spectra for different shape parameters

In Fig. 4.6, two distribution shapes,—the normal or standard distribution and the exponential or straight-line distribution , are plotted differently, that is, on a linear and on a semilogarithmic scale, and as a cumulative frequency H (probability) as well as a step frequency h (probability density). From these plots, it is obvious that designations such as straight-line distribution are associated with specific representations. If it is assumed that the load cycles enter linearly into a damage accumulation analysis for estimating the fatigue life , the linear plot of the step frequency yields the more realistic representation of the respective damage content of a spectrum.

Fig. 4.6

Cumulative and step frequency for two spectra, NV (ν = 2) and GV (ν = 1), in linear and semilogarithmic plots

From all four plots, it is obvious that the normal distribution represents a considerably harder spectrum than the straight-line distribution . Consequently, this distribution is preferentially employed for fatigue-strength testing because of the associated saving in time required for testing . The two plots of the step frequency at the bottom can be interpreted as damage distributions of the load horizons, if the slope of the S-N curve is k = 0, and the Miner elementary modification is applied.

If lengthy measurements are performed for different stationary processes (rough seas, roadways), a straight-line distribution often results. This distribution is obtained as an overall spectrum by superposition of normally distributed partial spectra which differ with respect to the spectral maximum and the cumulative frequency H₀, Fig. 4.7.

Fig. 4.7

Superposition of three normally distributed partial spectra to yield a straight-line distribution

Spectra which result from measurements on motor vehicles , aeroplanes, ships , machine plants , etc. under operational conditions differ drastically. From the wide variety of the spectra thus obtained, typical shapes are determined for these spectra and then idealised (standardized spectra). This approach has in fact been applied for a long time. Spectral shapes similar to those plotted in Fig. 4.5 have already been reported in [Gass64]. In this work, the so-called p-value spectra , which are important for crane construction , were also proposed. The weight of the empty trolley in the middle of the crane causes the occurrence of stress amplitudes during operation, and these frequently exceed a certain threshold value. In a relative plot, Fig. 4.8, the so called p-value denotes the ratio of this threshold value for H = 1000 to the maximum for the spectrum for a variety of possible spectrum shapes [Rada07].

Fig. 4.8

p-value spectra

Besides the shape parameter v, a different parameter v is also available for describing the shape of the spectrum [Häne99], and is defined as follows:

$$ \mathrm{v}=\sqrt[k]{\frac{\sum_{i=1}^n{h}_i\cdot {\left(\frac{S_{ai}}{{\widehat{\mathrm{S}}}_a}\right)}^k}{H_0}} $$

(4.7)

The range of values for v lies between zero and one. On this scale, constant-amplitude stress corresponds to the maximal degree of fullness , v = 1. The value also depends on the slope k of the S-N curve , in addition to the shape of the spectrum . In [Wirt87] and [Euli08], a value of k = 6 is proposed; as specified in the FKM Guideline [Häne99, Häne03], k = 5 applies in the case of bending, and k = 8 applies to torsion.

Three parameters are thus available for describing the shape of the spectrum : the shape parameter ν, the p-value , and the v parameter. Terms such as spectrum hardness or spectrum fullness are often employed for providing a qualitative description. The hardness or fullness of a spectrum increases progressively as the shape of the spectrum approaches a rectangular shape, which corresponds to a one-step stress .

4.2 Extrapolation

For measurements on machine plants , the duration of measurement necessary for obtaining a representative spectrum is always a question of vital importance. Of course, the stress spectrum which results from measurements over a limited period will never be sufficient for accurately predicting the stresses over a long design life . However, a representative spectrum can be expected to become established after a certain period of measurement , which depends on the operating conditions . Consequently, this question reduces to a problem of extrapolation . The spectra measured over a day, over a week, and over 3 months on a high-pressure grinding -roll mill and on a shredder are plotted in the following, Figs. 4.9 and 4.10 [Gehl92]. In the case of the high-pressure grinding -roll mill, hardly any changes in the operational state are observed, and only slight alterations in the ground stock (cement clinker) occur in the course of time. As indicated in Fig. 4.9, a useful spectrum is available after a measuring period of 1 month. After a shift to the right by a factor of 3, the result would nearly coincide with the spectrum obtained after 3 months [Gehl92].

Fig. 4.9

Torque spectra for a high-pressure grinding -roll mill after different measuring periods, see also basic sketch, Fig. 2.3 [Gehl92]

Fig. 4.10

Torque spectra for a shredder , measuring point s, above, after different measuring periods [Gehl92]

In the case of a shredder , the rotor torque is highly dependent on the type of scrap to be processed and also on the type of feed system. The shape of the spectrum obtained after 111 days cannot be inferred from the shapes of the spectra obtained after shorter periods of operation, Fig. 4.10 [Gehl92]. In this case, an attempt must be made to estimate a representative spectrum by means of a statistical analysis of the expected operational states (type of scrap), as shown in Sect. 10.2.

From a measurement under operational conditions during a limited period of time, a measurement of the maximal operational load value cannot be expected. Instead, it is highly probable that the observed maximum for the spectrum during the entire design life will be decidedly higher. The ratio between the maximum expected for the spectrum during the design life and the maximum which actually occurs during the measurement depends on the duration of the measurement, as referred to the design life and the amount of load scatter , among other factors.

One possibility for estimating the maximum for the spectrum is the multiplication of the measured amplitudes by a safety factor . This factor is indicated as a function of the risk and consequences of damage in series of technical rules ; see also Chap. 10. However, the aforementioned decisive parameters, that is, the duration of measurement and the load scatter , are usually not considered.

4.2.1 Analytical Extrapolation with the Shape Parameter

The problem of deciding how to perform an extrapolation is illustrated with the use of the following typical example: Results are available from a representative measurement under operational conditions during a limited period of time. From these results, the spectrum must be determined for the entire design life , see Fig. 4.11.

Fig. 4.11

Normalised spectra for the duration of measurement and for the design life

The stress spectrum for the operational measurement with the cumulative frequency H_M can be shifted toward the right, and the cumulative frequency H_N for the design life can thus be attained. In the special case of a load limitation , an increase in the maximum for the spectrum of measured values ${\widehat{S}}_{aM}$ would not be possible. That is, the spectrum for the entire design life would be limited from above by a horizontal line at the level of ${\widehat{S}}_{aM}$ (truncation ). In the more general case, however, amplitudes S_aN which exceed ${\widehat{S}}_{aM}$ will occur during the design life . In many cases, therefore, it can be expected that the shape parameter ν will not change. In the past, a graphical extrapolation was performed with the use of a curve template. As indicated in [Hink11], an analytical extrapolation can also be performed with the use of a constant shape parameter. It should be pointed out that the measured spectrum can also be a mixture, that is, a combination of partial spectra for different operational states. Hence, a prerequisite for the application of the method described in the following is the assumption that the fractional composition of the mixed spectrum does not change during the design life .

The starting point for the following consideration is Eq. (4.7) or a normalised spectrum such as that plotted in Fig. 4.5. In Fig. 4.11, a normalised spectrum has been plotted for the duration of measurement and for the useful life with the use of a shape parameter ν.

For this purpose, an extrapolation factor

$$ e=\frac{H_N}{H_M} $$

(4.8)

and an incremental factor

$$ E=\frac{{\widehat{S}}_{aN}}{{\widehat{S}}_{aM}} $$

(4.9)

can be defined. For Eq. (4.6), this yields

$$ H={H}_0^{1-{\left(\frac{S_a}{{\widehat{\mathrm{S}}}_{\mathit{\mathrm{aN}}}}\right)}^v} $$

(4.10)

In correspondence with log H = log H_N − log H_M, or

$$ H=\frac{H_N}{H_M} $$

(4.11)

and H₀ = H_N, the following is obtained

$$ \frac{H_N}{H_M}={H}_N^{1-{\left(\frac{{\widehat{\mathrm{S}}}_{aM}}{{\widehat{\mathrm{S}}}_{\mathit{\mathrm{aN}}}}\right)}^v} $$

(4.12)

In a logarithmic representation, this yields

$$ \log {H}_N- \log {H}_M= \log {H}_N- \log {H}_N{\left(\frac{{\widehat{S}}_{aM}}{{\widehat{S}}_{aN}}\right)}^v $$

or

$$ \frac{ \log {H}_M}{ \log {H}_N}={\left(\frac{{\widehat{S}}_{aM}}{{\widehat{S}}_{aN}}\right)}^v $$

(4.13)

After a transformation, the following is obtained:

$$ \frac{{\widehat{S}}_{aN}}{{\widehat{S}}_{aM}}={\left(\frac{ \log {H}_N}{ \log {H}_M}\right)}^{\frac{1}{v}} $$

(4.14)

After insertion of e and E, this yields

$$ E={\left(\frac{ \log \left( e\cdot {H}_M\right)}{ \log {H}_M}\right)}^{\frac{1}{v}} $$

(4.15)

That is, the incremental factor for the maximum of the spectrum depends on the extrapolation factor , on the shape of the spectrum , and on the cumulative frequency of the measured spectrum. The latter can be interpreted as the duration of the measurement .

In Fig. 4.12, the incremental factor is plotted for three spectra (straight-line distribution , standard distribution, and ν = 4). A linear relationship is obtained for the straight-line distribution .

Fig. 4.12

Enhancement of the spectral maximum with analytical extrapolation

4.2.2 Statistical Extrapolation for Stationary Processes

Methods from the field of extreme-value statistics are also applied for extrapolation in the case of stationary processes . For instance, methods of this kind are also applied for performing estimates in order to deal with a “wave of the century” or extremely severe earthquakes. For extreme-value statistics as described by Gumbel [Gumb58], see also [Buxb92], it is assumed that the maxima for each series of measurements correspond to a log-normal distribution for n series of measurements (random samples) during a stationary random process . Hence, the measurement on an operational state (for instance, travel of a motor vehicle over a distance of 1000 km on a roadway) can be subdivided into n = 20 equal intervals each 50 km in length, and the maximum can then be determined for each interval. The maxima for the individual intervals i are sorted in decreasing order of their values, and a probability of exceeding a given value is assigned to each event (in accordance with Rossow ) [Ross64], as shown in Table 4.1.

Table 4.1 Subdivision and order of the values

The individual pairs of values (maximum, level-crossing probability) can be plotted on a Gaussian probability grid and evaluated to determine the mean value and standard deviation .

From the total distance travelled during the design life and the length of a measured interval, the theoretical number of load intervals n_tot is then determined for the entire design life . The level-crossing probability for the greatest load event to be expected during the design life (i = 1) is thus:

$$ P=2/\left(3{n}_{tot}+1\right) $$

(4.16)

The maximum for the spectrum can thus be determined from the probability grid , see Fig. 4.13. The essential characteristic values from this evaluation are the mean value and the standard deviation s of the maxima for the individual intervals.

$$ \log {\overline{x}}_{\max }=\frac{1}{n}\sum_{i=1}^n \log {x}_{\max, i} $$

(4.17)

$$ s={\left[\frac{1}{n-1}\sum_{i=1}^n{\left( \log {x}_{\max, i}- \log {\overline{x}}_{\max}\right)}^2\right]}^{\frac{1}{2}} $$

(4.18)

Level-crossing probabilities have been determined for the extreme values , but the published results differ somewhat [Buxb92, Gude95]. Among other factors, these results are based on extended safety and reliability considerations. For ensuring the consistency of the equations employed here, the method of estimation due to Rossow , which is preferred for normally distributed values, is applied for the probability of exceeding a given value.

Fig. 4.13

Distribution of the extreme values on probability grid

Stationary processes are a prerequisite for this method. Consequently, extrapolation is permissible only for partial spectra; that is, the method is not applicable to mixed spectra. A graphical plot of the maxima on a probability grid is well suited for determining whether a stationary process or a mixed distribution is involved. If the values are not well ordered along a straight compensation line in a logarithmic plot of the maximum, the method is not applicable. Two possible reasons can be given for this restriction: The maxima may be associated with loads due to different physical causes, or load limitations may be present.

An example of loads due to different physical causes is the longitudinal force on the wheel of a motor vehicle . The longitudinal force can be caused by drive/braking processes, by rolling over bumpy roadways, or by inertial forces during longitudinal wheel oscillations .

Load limitations include, for instance, slipping clutches or predetermined breaking points , safety valves or bursting discs on pressure vessels , spring-excursion limiters, or the traction limit (on the wheels) of motor vehicles . Load limitations are often not attained during the measurement of operational loads . Consequently, the results obtained by extrapolation methods must always be checked for plausibility.

4.3 Derivation of Spectra from Matrices

The purpose of the following chapter is to show how the results of one-parameter counting methods can be derived from the results of two-parameter counting methods . As a rule, the objective is first to determine the matrices for the two-parameter counting methods and then to extract the results of one-parameter counting for the sake of easier visualisation. This approach ensures that the information content is maximised and that the advantages of the one-parameter counting methods can be utilised, see also [Haib06] and [Krüg88].

4.3.1 Transition Matrix

The following explanation is based on the transition matrix known from Sect. 3.4.2, Fig. 4.14. The positive slopes of the stress -time function have been entered above the diagonal if the associated target class is larger than the start class. The negative slopes of the stress-time function have been entered below the diagonal if the associated target class is smaller than the start class, Fig. 4.15. The peak values per class are shown in Fig. 4.16.

Fig. 4.14

Transition matrix for the stress -time function Fig. 3.15 taken as example

Fig. 4.15

Transition matrix with ranges for positive and negative slopes

Fig. 4.16

Transition matrix with a range for the number of peaks in class 7

The cross-hatched area in Fig. 4.17 contains the step frequency h of the maxima which occur in class 7. As a result, the same values are indicated as those which resulted from one-parameter peak counting , Fig. 4.17. A corresponding result applies to minima , Fig. 4.18. The cross-hatched area contains the total number n₁ of evaluated maxima for the stress -time function , Fig. 4.19.

Fig. 4.17

Transition matrix with ranges for positive slopes

Fig. 4.18

Transition matrix with ranges for negative slopes

Fig. 4.19

Transition matrix with a range for maxima (above and below the zero line)

4.3.1.1 Determination of the Irregularity Factor I

For the application of counting methods, the irregularity factor I of a stress -time function is often regarded as an important criterion. It is defined as the ratio of the number of mean-value crossings in one and the same direction n₀ to the number of maxima n₁ [ISO13]. From the number of mean-value crossings in the positive direction n₀ (cross-hatched area), the irregularity factor I = n₀/n₁ can be determined, Fig. 4.20.

Fig. 4.20

Transition matrix with mean-value crossings in the positive direction

In this context, it is important to note that an irregularity factor I = 1 designates a regular load sequence, that is, without mean-load fluctuations. In a strict sense, therefore, the irregularity factor I really designates the regularity of a load sequence.

For the stress -time function taken as example, the resulting irregularity factor I is equal to 4/6 = 0.67, Fig. 4.21. The procedure for the minima is analogous.

Fig. 4.21

Transition matrix with irregularity factor I for the stress -time function taken as example

Equal stress ranges are located on diagonals which are parallel to the main diagonal . The result of range counting can be obtained with this procedure, Fig. 4.22.

Fig. 4.22

Transition matrix with ranges of equal size

The results obtained for the stress -time function taken as example corresponds to those of one-parameter range counting , Fig. 4.23, see Sect. 3.3.3.

Fig. 4.23

Transition matrix with ranges of equal size

The same procedure is also applicable in the case of range-mean counting , Figs. 4.24 and 4.25. The lines of equal mean value are perpendicular to the main diagonal . The result is the matrix already known from Sect. 3.4.1, Fig. 4.26.

Fig. 4.24

Transition matrix with main diagonal and lines of equal mean values

Fig. 4.25

Transition matrix with lines of equal mean values

Fig. 4.26

Result of range-mean-counting

In the negative range, a similar result can be indicated underneath the main diagonal . For the present matrix , the area indicated in Fig. 4.27 includes the crossings of level a. For the stress -time function taken as example, the resulting cumulative frequencies for the level crossings are indicated in Fig. 4.28, see also Fig. 3.8.

Fig. 4.27

Transition matrix with a level

Fig. 4.28

Transition matrix with the result of level-crossing counting

This result can also be compared to that of level-crossing counting , see Sect. 3.3.2, Fig. 3.8.

Comment

Besides the irregularity factor , range counting , range-mean counting , and level-crossing counting can be derived from the transition matrix , but not range-pair counting .

4.3.2 Rainflow Matrix

The following considerations are referred to rainflow half-matrices for the sake of clarity; similar considerations apply to full matrices . The result matrix from Sect. 3.4.4 is employed as an example, Fig. 4.29. In each case, the number of complete cycles is entered between the minima and maxima .

Fig. 4.29

Rainflow matrix for the stress -time function as example

Lines for cycles of equal range are oriented in parallel with the diagonal. The stress range increases in the direction indicated by the arrow, Fig. 4.30.

Fig. 4.30

Rainflow matrix with a diagrammatic representation of the increasing range of the cycles

Lines of equal mean value are perpendicular to the main diagonal ; the mean values themselves increase in the direction of the arrow, Fig. 4.31.

Fig. 4.31

Rainflow matrix with a diagrammatic representation of the increasing mean values of the cycles

The following one-parameter counting methods can be derived from the rainflow matrix : range-pair counting , range-pair-mean counting , level-crossing counting , range counting , and peak counting .

The results of range-pair counting are obtained by summation of the cycles along the diagonal which is parallel to the main diagonal , Fig. 4.32.

Fig. 4.32

Rainflow matrix with range pairs of equal range

The rainflow matrix , Fig. 4.29, can be transformed to the plot for range-pair-mean counting , if the associated mean load is also indicated, in addition to the stress range, for the respective cycles, Fig. 4.33. In this process, the values 3.5, 4.5, and 7.5 have been rounded upward.

Fig. 4.33

Rainflow matrix and the included mean values of the range pairs

The results of peak counting can be calculated from the maxima and minima , Figs. 4.34 and 4.35.

Fig. 4.34

Rainflow matrix with maxima of equal size

Fig. 4.35

Rainflow matrix with minima of equal size

The number of maxima per class is obtained by summation of the horizontal rows; the number of minima is obtained by summation of the elements in the vertical columns.

Determination of the Irregularity Factor I

In Fig. 4.36, the rainflow matrix is shown with the maxima above the mean load , for instance, > class 4. The sum of these values corresponds to n₁ = 6; compare Fig. 4.19.

Fig. 4.36

Rainflow matrix with maxima above the zero line

The number of zero crossings, or of the crossings of a mean load which deviates from zero, n₀, is plotted in Fig. 4.37. As a rule, the mean value crossings are referred to the average operational load which occurs during the measurement .

Fig. 4.37

Rainflow matrix with zero crossings

The number of zero crossings is n₀ = 4; compare Figs. 4.37 and 4.20. For the stress -time function taken as example, the result is I = n₀/n₁ = 0.67.

Results of level-crossing counting can also be derived from the rainflow matrix . For instance, cycles which are entered into field 8/1 are crossings of levels 1–7 with the plotting procedure for level-crossing counting in which only positive slopes are evaluated. If the level crossings are summed in this manner for all fields of the matrix , the result shown in Fig. 4.38 is obtained with this counting method. This result is identical with the spectrum, Sect. 3.3.2, Fig. 3.8.

Fig. 4.38

Level-crossing counting from the rainflow matrix

Comment

Besides the irregularity factor , range-pair counting , range-pair-mean counting , and level-crossing counting can be derived from the rainflow matrix . Moreover, other counting methods, such as range counting and peak counting , can be derived from rainflow counting , although this is not described in more detail in the present chapter.

4.4 Standardised Load Sequences and Spectra

The purpose of standardised load sequences is the specification of typical load sequences and spectra for particular structural components and applications. The original idea was to define a representative load sequence for improving analytical fatigue life predictions or for performing comparative tests on structural components. However, the load sequence was not intended for providing a suitable load assumption for designing and dimensioning . The first standardized load sequence was the eight-step block program test defined by E. Gassner in 1939. This standard fatigue-strength test was applied for decades in Germany. With the widespread application of the servohydraulic testing technique, the restriction of applying variable amplitudes as a blocked load sequence was eliminated. As a result, load standards first became established in the aircraft and jet-engine industry in the 1970s, and later also in the machine-plant and automotive industries . The Gaussian standard was especially important for numerous applications because it allowed the performance of random tests with acceptable test duration. In the example shown in Fig. 4.39, a section of the Gaussian standard is illustrated with a diagrammatic representation of the range-pair spectrum. The crest factor is defined as C = 5.256 for this standard, see Sect. 2.3 [Fisc74, Have89, Heul98, Köbl76, Schü89, Schü94, Schü99]. During the first random tests , the set point setting was imposed with noise generators. Since the crest factor was not adjustable, the results of tests performed on different test stands were not mutually comparable.

Fig. 4.39

Standardised stress -time function taken as example

The first standards were intended for defining one-channel load sequences, which could be supplemented by temperature variations if necessary. However, more recent standards, especially the various car-loading standards (CARLOS ), represent multi-channel load sequences, which are correlated in time.

By way of departure from the original intention, standardised load sequences are employed as spectra for designing and dimensioning specific structural components nowadays. A prerequisite for this application is the development of appropriate methods for specifying the load level for the respective application in conjunction with the standardisation, in addition to the use of a representative load sequence as basis. These load standards can thus be applied instead of load assumptions and operational load measurements . A comprehensive set of standardised load sequences is indicated in chronological order in Table 4.2.

Table 4.2 Standardized load sequences in chronological order