Rate-independent counting methods

Data reduction methods in time domain are significantly more effective than methods in the frequency domain, at least as long as resonance behavior can be neglected. Damage based data reduction and the principles of mechanics of solid materials even lead to rate-independent methods, which is the motivation of the data pre-processing in the next section.

Most of the well known methods such as rainflow or Markov counting (two-parameter methods), or level crossing and range pair counting (one-parameter methods) are rate-independent methods. These methods and others are presented in detail in [DIN 45667] [ASTM E 1049-85] or [SAE AE-10].

The following figure shows the relationship among some of the methods. The arrows indicate that some results can be derived from others without using the time series again.

Derivation of Counting Methods

In the following, the rainflow counting methods are described.

Data pre-processing

All methods need a type of pre-processing consisting of a peak/valley filter followed by a hysteresis filter and a discretization. After pre-processing, the data merely contains a series of peaks and valleys in a finite number of bins. The details are as follows:

Peak/valley filtering

For rate-independent materials, all that is needed for rainflow counting is the load reversal points, or peaks and valleys in the time history. The time history is scanned, and any points that are not load reversal points are discarded from the time history. The peak-picked history of figure Stress-Time History is shown in the figure Peak-Picked Stress-Time History below.

Stress-Time History

Peak-Picked Stress-Time History

Hysteresis filtering

All points of the peak/valley sequence corresponding to a change in amplitude smaller than a certain threshold are discarded. As explained in the section on rainflow counting, this eliminates small hysteresis cycles from the stress-strain path. Typically the threshold is chosen as the width of one bin.

Hysteresis Filter

Discretization

The domain in the range space L of the load histories, which is given by a minimum and a maximum value, is divided into a certain number N of equally spaced bins. Then each point of the peak/valley sequence is replaced by the index of its corresponding bin. Points lying on the separation line of two bins are counted in the upper bin, points lying below the lower bound are counted in the first bin and points lying at or above the upper bound are counted in the highest bin.

Successive points lying in the same bin are reduced to one point. This can only happen if the threshold of the hysteresis filtering is smaller than one bin or if points are lying outside of the domain.

In Specialist Durability, you can enter the number of bins to be used by the fatigue solver in the Rainflow Bins durability simulation object. In the standard analysis templates, it is set as an implicit simulation object. If you need to change it, you must create a new analysis template.

Rainflow cycle counting

Rainflow cycle counting is a technique for determining closed hysteresis loops in the stress-strain response of a material. As mentioned previously, a closed hysteresis loop represents a cycle of loading, which is the basic unit for which fatigue damage is calculated. In this section, uniaxial or one-dimensional rainflow cycle counting will be discussed.

An observation of the behavior of metals is that when loaded past the yield point of the material, unloaded, then re-loaded to an even higher stress level, the material "remembers" the original loading curve and follows it until the next unloading event, as illustrated in the figure Illustration of a Hysteresis Loop below. The unloading and reloading path is called a hysteresis loop. It is possible to even have many small loops "nested" inside of larger loops, with each unloading point remembered until the load is reversed and the loop is closed.

Illustration of a Hysteresis Loop

Although these hysteresis loops are easily visualized by examining the stress-strain response of the material, it is only the change in the load, stress, or other independent quantity that determines if the load reverses and loops remain open or closed. Therefore, an analysis of only the load history can identify hysteresis loops.

The most important advantage of rainflow counting is the consistency in which it may be related to fatigue damage. This is a consequence of the physical interpretation of the terms of the closed hysteresis cycles.

Rainflow and related methods have been known since 1967 when Endo [Endo 1992] published an algorithm for counting closed hysteresis cycles in a stress-strain path. The name rainflow results from a comparison of this method to the flow of rain falling on a pagoda and running down the edges of the roof. In 1969, De Jonge independently developed the range pair range method, and in 1986 Clormann-Seeger published a variant based on the principles of mechanics of solids. There are also standard methods of rainflow counting given by ASTM [ASTM E 1049-85] and SAE [SAE AE-10].

Here, another variant called four-point counting will be described, which was developed by Specialist Durability. This version is intended to allow an easy implementation as well as a complete reconstruction of time series from the rainflow data. It is identical to the French AFNOR standard.

All of these variants essentially count closed hysteresis cycles in the stress-strain path that is obtained if the time series is taken as a strain history, a stress-strain curve is given, and the Masing and memory rules are applied to construct a stress-strain-path.

The four-point algorithm

The four-point algorithm works with a stack of four points used to find closed loops and is updated as it passes through the time series. It is assumed that the time series has been preprocessed and is represented as a sequence of peaks and valleys. The stack is initially filled with the first 4 points of the history. The residue RES, which will remain after the counting procedure, is also initialized with these four points.

If the second and third point are not contained in the interval spanned by the first and the fourth point, then the first point is stored in the residue and removed from the stack. The stack is filled with the next point of the history.

Otherwise a closed loop starting at point 2, reaching point 3 and arriving back at point 2 is found and stored in the N x N matrix RFM at the component RFM (i,j), where i = point 2 and j = point 3. Here N is the number of bins of the time series. The corresponding points 2 and 3 are removed from both the stack and the residue RES.

Now the stack has to be filled again. If possible, it is filled with the last two points already stored in the residue keeping the original order of the points in the time series. If the residue contains no points at all or respectively only one point, then the stack is filled with the next points of the time series, again keeping the order of points.

This is repeated as long as there are points remaining in the time series. The result of this method is the residue RES, which is a vector with a maximum length of 2*N-1 and the N x N matrix RFM, where the number of loops starting in bin i, reaching bin j and arriving back at bin i is stored at position (i,j). This form is called a from-to-matrix. The following figure illustrates the algorithm. The pairs of numbers in parenthesis indicate the closed loops counted.

The Four-Point Algorithm

This variant allows a complete reconstruction and is very easy to implement. Another important feature is its generality: the Clormann-Seeger variant, the definition of ASTM and the simplified definition of ASTM in [ASTM E 1049] can all be deduced from RFM and RES.

Clormann-Seeger correction

Neither the four-point algorithm nor the ASTM definition counts all closed hysteresis cycles. This is corrected by the Clormann-Seeger algorithm. There the time series starts at the zero level. This correction is added in Specialist Durability by counting if

where y1, y2, y3, and y4 define the four-point stack. If these loops are added to RFM, we obtain the Clormann-Seeger data.

These cycles are included in the damage calculation for the first run through a load history, only.

To get the ASTM definition, you simply have to count each range (RES(i),RES(i+1)) in the residue RES with the weight ½ and add it to RFM at position (RES(i), RES(i+1)).

The from-to rainflow matrix construction is outlined schematically in the following figure.

Schematic of From-to Matrix Construction

For typical variable amplitude loading histories, there are often hysteresis loops that do not close. These open loops are called the residue of the rainflow matrix. If the loading history is a block that repeats, these open loops will close on the next pass of the time history, and will then contribute to the damage.

Therefore, damage results are then given based on the following:

Damage = (Damage of the Rainflow Matrix) if Damage > limit damage sum

Damage = (Damage of the residue + damage of the Rainflow matrix)/(limit damage sum + damage of the residue) if Damage of the rainflow matrix <= limit damage sum.

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