Rainflow Cycle Counting : A Historical Perspective

Darrell Socie Mechanical Engineering Department

University of Illinois at Urbana-Champaign

Abstract: This paper reviews the historical development of the rainflow cycle counting method invented by Tatsuo Endo in 1967. Key contributions that have extended the original work of Endo are briefly reviewed.

Keywords: Rainflow, Cycle Counting, Fatigue

1. Introduction

The local strain approach for fatigue analysis is now routinely used throughout the world for assessing the durability of structures and components. This approach was made possible by four key developments:

* Coffin's and Manson's work that established plastic strain as the controlling parameter for fatigue damage,

* Morrow's and Topper's work on simulating the stresses and strains of notched members,

* Commercial development of the closed loop servohydraulic materials testing machine by Johnson,and

* Endo's invention of the rainflow cycle counting method.

All of this work has stood the test of twenty-five years of careful scrutiny by researchers and engineers. In this conference, Fatigue Damage Measurement and Evaluation Under Complex Loadings, we honor the memory of Tatsuo Endo for his lasting contribution to science and industry. This paper briefly reviews the original work followed by a series of major events that have led to the widespread acceptance of the method.

2. Endo's Original Work

The original presentation of the rainflow method was contained in a series of two papers presented to the Kyushu District Meeting of the Japanese Society of Mechanical Engineers in November, 1967:

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T. Endo, K. Mitsunaga and H. Nakagawa, Fatigue of Metals Subjected to Varying Stress - Prediction of Fatigue Lives T. Endo, K. Mitsunaga, H. Nakagawa and K. Ikeda, Fatigue of Metals Subjected to Varying Stress - Low Cycle. Middle Cycle Fatigue.

This was followed by the third paper in March of 1968:

M. Matsuishi and T. Endo, Fatigue of Metals Subjected to Varying Stress - Fatigue Lives under Random Loading.

Three additional papers were published including the first English language publication of the method by the original authors:

T. Endo, M. Matsuishi, K. Mitsunaga, K. Kobayashi and K. Takahashi, Rain Flow Method - the Proposal and the Applications in Memoir Kyushu Institute Technical Engineering, 1974 T. Endo, K. Mitsunaga, K. Takahashi, K. Kobayashi, and M. Matsuishi, Damage Evaluation of Metals for Random or Varying Load - Three Aspects of Rain Flow Method in Proceedings of 1974 Symposium on Mechanical Behavior of Materials T. Endo, K. Kobayashi, K. Mitunaga and N. Sugimura, Numerical Comparison of the Cycle Count Methods for Fatigue Damage Evaluation, and Plastic-Strain Damping Energy of Metals under Random Loading presented at the 1975 Joint JSME-ASME applied Mechanics Western Conference.

These six papers form the basis of the rainflow cycle counting method. Details of the method will not be repeated here since they will be described in detail in the next paper in the symposium. The early papers all have diagrams of the stress strain response for simple variable amplitude loading histories. An example from the first paper is given below.

7- (kg/W)

El 5. fcfiM*^U**^*tf*&*? **&&&J $-&*& ' (5. SOtt **JX Figure 4 and 5 from Endo's first paper.

The cycle counting method was called " the effective range count method. H Endo writes " . . . the method corresponds to the work hardening properties of metals under complex change of strain. It is compatible with the abrupt change of tangent modulus which is conditionally observed under the process of reloading . . . " A cycle is identified when the material remembers its prior deformation history and changes its tangent stiffness to follow the original loading path. Only closed hysteresis loops that are formed during a short repeating history are shown in the first two papers as shown in this example.

12 fe. **-:& KM * %ru&. (3 7fc) fctt.-3MUB.Mfc fS 7(b) *3kfij5*>*!l

Figure 6 and 7 from Endo's first paper.

Half cycles are introduced in the third paper. All of the stress strain curves and hysteresis loops were plotted in terms of shear stresses and strains because all of the original experiments were conducted in torsion on a solid bar.

The first English language description is the Masters Thesis of Masanori Matsuishi in March of 1969. In his thesis, he described the effective range count method in great detail with many examples in about six pages of text. At the end, he describes an alternate procedure for obtaining the same ranges and called it rainflow. This description took two paragraphs and contains these familiar rules:

The rain-flow started from a maximum ( minimum ) peak stops when a following maximum ( minimum ) peak larger ( smaller ) than the peak appears.

If a rain-flow meets the rain-drop failed from a upper roof the rain-flow stops at that place.

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He implemented these rules in a FORTRAN program that required 350 lines of computer code for an IBM 360 computer.

Endo in the concluding remarks of the 1974 paper notes " In order to have better prediction of the effects of mean stress, the cyclic hardening or softening of metals, nonlinear damage law, the extension of the method to the crack propagation period etc. must be investigated." All of the have indeed been studied and with the exception of nonlinear damage laws are widely used in engineering practice.

3. Dowling's Confirming Experiments

Dowling [1] published the first description of rainflow in 1972 in the Journal of Materials. He conducted an extensive series of axial strain controlled fatigue tests with variable amplitude loading where he concluded " . . . the counting of all closed hysteresis loops as cycles by means of the rain flow counting method allows accurate life predictions. The use of any method of cycle counting other than range pair or rain flow methods can result in inconsistencies and gross differences between predicted and actual fatigue lives." He conducted tests to investigate the mean stress effect with a number of loading histories designed to differentiate between the average mean stress of the loading history and the mean stress of each cycle. An appendix of this paper describes the rainflow method in detail and gives examples of the half cycles identified by rainflow counting and the corresponding stress-strain hysteresis loops.

4. SAE Round Robin Test Program

The Society of Automotive Engineers (SAE) Fatigue Design and Evaluation Committee began a testing and analysis program to evaluate cumulative damage models. [2] A number of experiments were performed on a notched specimen with three variable amplitude loading histories that were obtained from ground vehicles on a test track. In addition to providing valuable data, the program served a common forum to discuss techniques for life predictions. Most of this work was being accomplished by industrial participants with real problems to solve rather than by researchers at the universities. Life prediction software developers were using the extensive data generated in this program to validate their codes. Soon a common feature of all successful models began to emerge. Rainflow counting or some equivalent counting method was used to determine the equivalent constant amplitude cycles from the variable amplitude loading histories.

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By 1975 minicomputers became commonplace in many industrial laboratories and life prediction software developed for them. Computers were taken out of the laboratory and into the field. These computers required more efficient algorithms to process the large amounts of data that could be collected in the field during a one or two hour test. Downing [3] was instrumental in this development. The first microprocessors became available and commercial products for rainflow counting became available in the late 1970's. Ten years after its first publication in the Japanese literature the rainflow counting method was in widespread use throughout the world for fatigue analysis.

5. ASTM Cycle Counting Standard

Many variations of the rainflow counting method have been proposed and used. Standard terminology and methodology was needed. The American Society for Testing and Materials (ASTM) Committee E9 on Fatigue adopted the first standard for cycle counting methods for fatigue analysis. The standard E1049 Standard Practice for Cvcle Counting in Fatiaue Analysis was developed and approved by ASTM in 1985 using the simple rules for rainflow cycle counting that were suggested by Downing.

6. Application for Multiaxial Loading

A strategy for fatigue life estimates for multiaxial nonproportional loading histories was proposed by Socie[4]. Three dimensional stress strain simulation is used to obtain the stresses from the measured strains after which a tensor rotation is used to determine the stress and strain history on any potential failure plane in the material. This results in a stress and strain history for each potential failure plane in the material as shown in this example from reference 4.

Multiaxial Stress and Strain Time History

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Fatigue damage is computed by rainflow counting the strain history to identify cycles. Subcycles are easily identified as the strain is increased from point A to point B. In multiaxial loading the corresponding stress-strain response have hysteresis loops which are on the outside of the hysteresis loop formed by the major cycle.

1000

CL 5

-1000

^

I 1 1 I

^

1 1 1 1 -0.005 0.005

Multiaxial stress-strain response

Compare this figure with Endo's figures. The normal strain history is counted on each plane and fatigue damage computed from a tensile damage model. The shear strain history is also counted and a shear damage model is used to compute shear damage. The plane experiencing the greatest damage is identified as the critical plane for crack nucleation.

7. Application to Crack Growth

Crack growth measurements were made during the SAE testing program. In 1976, Nelson and Fuchs[5] used the overall range method which is a counting method that is equivalent to rainflow to make good estimates of the crack propagation lives. Later, Socie and Kurath[6] conducted a series of tests with loading histories that were designed to differentiate between range and rainflow cycle counting methods. They concluded that the rainflow method gave the best life estimates and that range counting could lead to nonconservative results.

8. Application to Load History Reconstruction

Instrumentation for obtaining rainflow counted data is available to collect months of operating data because it is stored in the reduced histogram format. There is considerable interest in reconstructing a time history from this data for cycle-by-cycle damage analysis or to develop a loading sequence for fatigue testing. The new

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sequence is not the same as the original but does have the same fatigue damage. Perrett [7] concludes in his work, " In general the process of rainflow counting and reversed range-mean-pairs reconstruction produces sequences that create fatigue damage at the same rate that it accumulates in the original. In the majority of the tests described in this paper reconstituted lives were the same as the originating FALSTAFF lives in conditions dominated by crack initiation and crack growth and also in conditions where fretting forces are significant.

Bishop and Sherratt[8] provide a method for the estimations of rainflow range density functions using statistics computed directly from power spectral density data. The rainflow range mechanism is broken down into a set of events which can be analyzed using Markov process theory.

Summary

The Rainflow counting method has been shown to give the best fatigue life estimates. It has been applied for both fatigue crack initiation and fatigue crack propagation problems with great success. The method has been extended for multiaxial loading and to reconstructing fatigue test sequences.

References

1. Dowling, N.E. "Fatigue Failure Predictions for Complicated Stress-Strain Histories" J. of Materials, Vol 7, No 1 , 1972, 71-87

2. Wetzel, R.M. editor Fatigue Under Complex Loading; Analysis and Experiments SAE, AE-6, 1977

3. Downing, S.D. and Socie, D.F. "Simplified Rainflow Counting Algorithms" Int. Journal of Fatigue, Vol 4, No 1, 1982, 31-40

4. Socie, D.F. "Multiaxial Fatigue Damage Assessment" Low Cvcle Fatigue and Elasto-Plastic Behavior of Materials, 1987

5. Nelson, D.V. and Fuchs, H.O. "Prediction of Fatigue Crack Growth Under Irregular Loading" ASTM STP 595, 1976, 267-291

6. Socie, D.F. and Kurath, P. "Cycle Counting for Variable Amplitude Crack Growth" ASTM STP 791, 1983, II19-II32

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7. Perrett, B. "An Evaluation of a Method for Reconstructing Fatigue Test Loading Sequences From Load Data Acquired via Rainflow Counting" Proc. 14th ICAF Symposium, 1987

8. Bishop, N.W.M. and Sherratt, F. "A Theoretical Solution for the Estimation of Rainflow Ranges From Power Spectral Density Data" Fatigue and Fracture of Engineering Materials and Structures, 1990, 311-326

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