Accelerated Fatigue Test For Automotive Chassis Parts Design: An Overview

Pauline Beaumont, Peugot Citroen Fabrice Guérin, University of Angers Pascal Lantieri, ParisTech Matteo L. Facchinetti, Peugot Citroen Guy Martin Borret, PhD Peugot Citroen

Key Words: Automotive Chassis parts, Fatigue, Wöhler, Locati, Staircase, Statistical Estimation, Simulation

SUMMARY & CONCLUSIONS

Fatigue strength is of prior concern for automotive chassis parts design. In a reliability framework, its assessment still requires long and expensive test campaigns. In particular, high-cycle endurance tests usually involve several specimens undergoing million-cycle procedures.

In order to reduce the specimen number (e.g. expensive prototypes) and to shorten testing procedures (i.e. time-to-result delay), Accelerated Life Tests (AFT) are developed under the constraint of the needed high-reliability assessment. These methods do not involve high frequency load cycling, but are mainly based on stress magnification and fatigue damage equivalence.

In this paper, at first a review of the fatigue test procedures currently used in the automotive industry (e.g. StrairCase, Locati and StairCase-Locati) is addressed. Thus, numerical simulations are used in order to assess the efficiency of these test procedures and compare their performances.

1 INTRODUCTION

Automotive chassis parts are usually designed with respect to multiple requirements, such as stiffness, noise and vibrations, mechanical strength under static and dynamic loads, wear, etc. Among all these requirements, reliability under cyclic loads is of prior concern. In this framework, this paper address metallic parts undergoing high cycle fatigue, e.g. 106 load cycles: the aim of this study is to optimize fatigue test plans with high-reliability assessment.

Fatigue tests, in which cyclic stresses are applied during a given number of cycles to failure, are designed to guarantee a risk of failure lower than a threshold (e.g. failure probability of 10-6) which can be related to fatigue strength during a reference lifespan (e.g. 106 load cycles).

The testing method is supposed to enable to assess the required performance as determined by a probabilistic approach: the Stress-strength method [1, 2, 3, and 4] leads to set the predicted strength distribution taking into account the

stress distribution and an acceptance risk, as shown in figure 1.

Figure 1: Stress-strength method – μR and σR, target characteristic of the strength, as a result of the stress

distribution (μC,σC) and an acceptance risk

The testing results give an experimental mesure of the resistance distribution with an average called mR, and a standard deviation called sR, which have to be compared with the required values μR, and σR respectively.

Fatigue test procedures currently used at PSA Peugeot Citroën for at least a decade are Locati, StairCase and StairCase-Locati procedures. At present, they involve some shortcuts: • Series of test runs can last up to 3 months. As

development and qualification time schedules are hardly reduced, procedure robustness should be checked as some hypothesis are not met anymore.

• The number of test runs at PSA and its suppliers due to fatigue tests is very large (e.g. 3.108 cycles / year), so that their reduction would imply cost and delay to be lower.

• Finally, the procedures have not been updated for 15 years, while academic researches on accelerate testing methods have been pursued with success, in particular for the electronic industry [5]. Hence, in this paper we present the surrent state of PSA

Stress Strength μC ,σC

μR ,σR

μC μR

f (Risk)

target

978-1-4577-1851-9/12/$26.00 ©2012 IEEE

fatigue test procedures (§2) then analyze them by numerical simulations (§3-4-5).

2 TESTING METHODS USED AT PSA

2.1 StairCase Method

The StairCase method [6] is a fatigue test procedure used to calculate the fatigue resistance for a sampling of parts. The method is based on an iterative principle, several parts are successively tested. The test on the current part depends on the result of the previous one, as shown on the following figure:

Figure 2: StairCase method algorithm

where: • Fd0 is the first stress level for the first part of the sample • d is the stress step between two consecutive levels.

In the literature there are many ways to estimate the fatigue limit with the StairCase results [7,8]; at PSA, the Dixon & Mood method [6] is applied. Thus, the mean and the standard deviation of the fatigue limit are calculated as follows:

012e

Am F dN

⎡ ⎤⎛ ⎞= + ±⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦ 1)

2

21.62 0.029e

e

N B As dN

⎛ ⎞−= +⎜ ⎟

⎝ ⎠ 2)

where the latter is applied only if: 2

2 0.3e

e

N B AN

−> (3)

with: • F0 : the lowest stress level used in the test • Ne : total number of the least frequent event (specimen

failed or not) for the overall test • A = Σ ini : used in the average calculation • B = Σ i2 ni : used in the standard deviation calculation • i : integer counter identifying the stress level, the 0

level corresponds to F0 • ni : occurrence of the least frequent event at the i level • - 1/2 : if the failure is the least frequent event • + 1/2 : if the failure is the most frequent event

Please note that PSA usually needs 7 parts in order to identify the fatigue limit mean, whereas the standard deviation of the fatigue limit is actually not determined with this testing method.

2.2 Locati Method

The Locati method [9] is usually applied when there are few specimens, theoretically one is enough. The aim is the same as StairCase’s, but the Locati procedure is based on additional physical hypothesis. PSA uses it to determine the average and the standard deviation of the fatigue limit with few specimens (e.g. 3 parts).

The method is based on a damage law (e.g. Miner’s rule [10]) and depends on Wöhler curve model choice (e.g. Basquin regression [10]).

The principle of such a test campaign is the following: the first part is tested during L cycles, at an F stress level. At the end of this step, the level is increased by a stress increment λ, and the same part keeps being stressed during L cycles. This scheme is used until the failure appears. The fatigue strength of the part at the reference lifespan N is defined as follows:

1

31 21 2 3

bb b b

eNN NF F F F

N N N⎛ ⎞= + +⎜ ⎟⎝ ⎠

(4)

with Fe the equivalent fatigue strength for that part at N cycles, Ni the number of cycles at each Fi stress level and b the Basquin parameter.

The test on the following part depends on the strength of the previous ones. The first stress level for the following part is:

2F Fm λ= − (5) Fm corresponds to the mean of all the previous parts’

strength Fe and also corresponds to the mean of the fatigue strength at the end of the procedure:

em Fm F= = (6) The standard deviation of the fatigue strength is the

standard deviation of all the equivalent stress Fe. This method gives a good approximation of the mean and

the standard deviation of the fatigue limit. Number of cycles L is chosen as 3.105 cycles by PSA. Moreover, it is also used during a StairCase pattern in order to make all the parts failed instead of censured data at the end of the StairCase. It gives further useful information: that is the StairCase-Locati method.

2.3 StairCase-Locati Method

Each part of the StairCase pattern which has not failed before the target number of cycles (censured data) can be carried on with a Locati procedure, without altering the StairCase results. Basically, it’s a StairCase procedure and all the parameters of the Locati are based on the StairCase procedure: • The first stress level of the Locati (after the StairCase) is

Stress level of the first part Fd0

Failure before N

cycles

No failure before N

cycles

Stress level of the next part

= Stress level of the

current part + d

Stress level of the next part

= Stress level of the

current part - d

the level of the StairCase increased by an increment d • The stress’ step of the Locati is the same as the StairCase

(λ=d) When a part doesn’t fail after N cycles, it is tested at an

increased stress level during L cycles, and so on with increased levels until the failure appears. The fatigue limit distribution is then calculated as the Locati method.

Hence, this leads to a better estimation of the standard deviation than the StairCase’s.

3 METHODS’ ANALYSIS BY NUMERICAL SIMULATION

In order to assess the efficiency of the previous methods, numerical simulations are built up: this approach provides a means of comparing them with respect to the quality of the estimation (convergence and bias) taking into account the cost (e.g. number of parts) and the time (e.g. number of cycles) needed for every test plan. Moreover, we’ll check if some parameters or hypothesis affect the final result in an unreliable way.

We generate samples, i.e. tests results based on 1. a theoretical fatigue limit distribution (μ,σ) 2. some hypothesis such as the number of parts, the a priori

distribution law, the Basquin’s parameter... Samples are simulated for the three methods, considering

7 up to 1000 specimens. The computer simulation is designed to analyze 1000 runs of each procedure, giving 1000 results of each sample size in order to provide a distribution of calculated averages and standard deviations.

First, these simulations are made according to “ideal” test conditions to check the convergence, which means a starting stress level equal to the fatigue strength average, and a step stress equal to the fatigue strength standard deviation. On top of these, for methods using Locati procedure (Locati and StairCase-Locati), “ideal” conditions mean the Basquin parameter equal to the right hypothetical value.

Secondly, in order to test any bias for each procedure, results are checked with offset test conditions, that is : 1. the starting stress level is different from the true average 2. the step stress value is different from the true standard

deviation 3. a wrong Basquin’s parameter value is used. In the sequel, each chapter relates the application of the numerical simulation analysis to a testing procedure.

4 THE NUMERICAL SIMULATION OF THE STAIRCASE

4.1 Convergence

The StairCase is launched with “ideal” test conditions, i.e. setting a priori, the true fatigue strength distribution (i.e. Normal distribution [μ=100 ; σ=8]). The simulation drives a StairCase protocol, which leads to the calculated fatigue strength distribution parameters: figure 3 shows the distribution of the mean estimation obtained for 1000 StairCase simulations.

We can assess the convergence of the method by plotting the mean of each parameter estimates (mean and standard deviation) as a function of the sample size (figures 4 and 5).

Figure 3: Mean distribution of 1000 runs of a 1000 sample size StairCase simulation

Figure 4: Error percentage on mean average

estimates as a function of the sample size

Figure 5:

Figure 5. Error percentage on Standard deviation average estimates function of the sample size

The figures show that the StairCase method is converging indeed; the error trends toward zero as the sample size increases, both for the mean and the standard deviation of the fatigue limit. For very small samples, the method gives acceptable results for mean estimates only. For a sample size

%

%

mean

of 7, we note an error on the average of the mean estimate lower than 0.1%, whereas the maximum error for the 1000 simulations is about 10% (figure 6 right side). Considering the standard deviation estimation we can see a more important error up to 50% on the 1000 standard deviation average, as shown on figure 6, left side.

Figure 6: Standard deviation and mean distribution of 1000 runs of a 7 sample size StairCase simulation

Thus, for the standard deviation estimation a more reliable estimator is supposed to be used.

4.2 Bias

The bias is checked as follows: 1. the effect on the average and standard deviation estimates

of the starting level (different from the true average) 2. the effect of the step stress value (different from the true

standard deviation) 3. the combined effect of the starting level and the step size. The effect of the starting level Several stress values were used:

up to by 0,5σinit step 1000 simulation runs are made for each initial stress level.

Figure 7: Box-and-whisker diagrams of average estimates versus starting stress value for a 7 sample

size

Figure 7 shows the result of a 7 sample size: the average estimates are represented by box-and-whisker diagrams, versus Fd0 values with μinit = 100, and σinit= 8. Here the lower

quartile is 25% and the upper is 75%. The same analysis is applied to the effect on the standard

deviation estimates, and results are shown in the same way. We can see that with very large samples (about 500 and

more) the first level of the testing procedure has no effect at all on the results, both for the mean and the standard deviation. But for smaller samples there is an impact indeed: the mean is slightly underestimated when the first level is lower than the theoretical value, and overestimated when it is greater. For the standard deviation estimates of smaller samples we can see that there is the same underestimate no matter what the first level is. The effect of step stress value

The starting stress level is set equal to the true fatigue strength average and several step stresses are used:

d = 3 up to 20 by a unit step 1000 simulation runs are made for each step stress value.

Figure 8 shows the effect of the step stress on the average estimates for a 20 sample size:

Figure 8: Box-and-whisker diagrams of average estimates versus step stress for a 20 sample size

To conclude on the step-stress effect, there is not a noticeable effect on the mean estimation whatever the sample is large or small. Conversely, on the standard deviation estimates we observe that for every sample size the standard deviation estimates increase if the error on the step stress is about twice the theoretical value. Moreover, for a sample smaller than 50 the best estimation of the standard deviation is provided when the step-stress is about twice the theoretical value of the standard deviation. The combined effect of the first level and the step stress

Let us now observe the effect of the starting stress level and the step stress errors. We use the same parameters’ variation values as shown before. The results are given in the form of 3D graphs representing the response surface of the calculated parameters’ averages of each 1000 simulations (Figure 9 & figure 10).

Whatever the first stress level is, the mean of the fatigue limit is erratic for d values smaller than the theoretical standard deviation. For small samples (e.g. 7, 20 and 50) we confirm that there is no effect on the mean estimation as long

initinitFd σμ 5,40 −= initinit σμ 5,4+

Average estimates

Step stress

meanmedianσinit

histogram theoritical values (μ and σ) calculated values (m and s)

Standard deviation mean

Average estimates

Starting stress level

meanmedianμinit

as the starting level is not too far from the theoretical mean value. But when the first stress level is moving away from the true value, we can ensure the error is the highest if we choose too small of a step.

Figure 9: Example on a 7 sample size, the average of the fatigue limit mean estimates versus d and Fd0

values

Figure 10: Example on a 500 sample size, the average of the fatigue limit standard deviation estimates

versus d and Fd0 values.

For the estimation of the standard deviation, the 3D analysis shows that for large samples (over 50) there is no impact on the estimation as long as the step is lower than twice the theoretical standard deviation. Over this limit the standard deviation is overestimated, whatever the first stress level. For smaller samples, we don’t observe the same phenomenon: the estimation is good only for a step value approximately equal to twice the theoretical standard deviation. When the step moves away from this value, the error on the standard deviation estimate increases or decreases respectively, no matter what the first stress level is.

5 THE NUMERICAL SIMULATION OF THE LOCATI

Please note that applying the usual numerical procedure analysis, the Locati method needs for the Basquin parameter to be fixed.

5.1 Convergence

The Locati was shown to be a convergent method for the mean and, in opposition with the StairCase method, it is also convergent for the standard deviation estimation. The mean estimates are as acceptable as the StairCase ones (e.g. an average error less than 0.2%). But the standard deviation estimation shows that the average error on the 1000 estimations is about 6% for the most critical case with 7 samples, while it is about 50% with a StairCase method. Moreover, we can see that the standard deviation is always underestimated, in a bigger way as the sample size decreases.

5.2 Bias

Starting level effect With a Locati method there is no impact at all of the first

stress level, both on the mean and the standard deviation estimation, while in a StairCase procedure, for small samples, we have seen a slight impact.

Moreover, on the standard deviation estimates we still notice the same underestimation whatever the first stress level is. Step stress effect

The step stress size has no impact at all, either on the mean or the standard deviation estimations, whereas with a StairCase procedure we have shown that if the step is twice the theoretical standard deviation we had a better estimation of it. Here, the underestimation on the standard deviation is the same no matter the size of the step.

6 THE NUMERICAL SIMULATION OF THE STAIRCASE-LOCATI

As for the Locati simulation, the StairCase-Locati method needs for the Basquin parameter to be fixed.

6.1 Convergence

The StairCase-Locati was proven to be a convergent method for the mean and the standard deviation estimates; they are as acceptable as the Locati ones. We observe an average error on the mean less than 0.2% and an average error on the standard deviation about 7% for the most critical case with 7 samples.

These results correspond to the Locati ones; in particular we note the underestimation of the standard deviation as long as the sample size decreases.

6.2 Bias

Starting level effect As the Locati results we can observe that there is no

impact at all of the first stress level, both on the mean and the standard deviation estimation, while we have seen a slight impact with a StairCase procedure for small samples.

Moreover, on the standard deviation estimates, as the Locati results observation, we still notice the same underestimation no matter the first stress level. Step stress effect

As the Locati results we can see that the step stress size has no impact at all, either on the mean or the standard

μcalc

d value

Fd0 value

Fd0 value

σcalc

d value

deviation estimations, contrary to the StairCase estimations. Here, the underestimation on the standard deviation is the same regardless of the step size.

7 CONCLUSION AND PERSPECTIVES

The StairCase-Locati and the Locati methods are found to be insensitive to the first stress level and the step stress: thus they seem to be more robust than the StairCase.

The PSA testing methods choice seems to be justified by this analysis:

Figure 11: Mean (left) and standard deviation (right) distributions result with the 3 PSA methods for small

sample sizes

In fact the mean and standard deviation calculations by a StairCase and a StairCase-Locati respectively, give good estimation results with small samples (e.g. 7 sample size).

To conclude on the quality of estimation with a StairCase-Locati and Locati testing procedures, a complementary analysis is expected to further evaluate the Basquin parameter.

Moreover, further work should be done in order to assess Accelerated Life Tests quality of estimation. A cost function based on the number of specimens and the number of cycles required will help to evaluate the most efficient testing method.

REFERENCES

1. A. Bignonnet, « Dimensionnement en fatigue et conception dans l’automobile », PSA PEUGEOT CITROEN, internal document, 1999.

2. J.J. Thomas, A. Bignonnet « Conception et fiabilité dans l’industrie automobile », Congrès Français de Mécanique, 2001.

3. M. Lemaire, « Approche probabiliste du dimensionnement - Modélisation de l’incertitude et méthodes d’approximation », Techniques de l’Ingénieur, 2008.

4. M. Lemaire, « Structural Reliability », ISTE/Wiley, 2009 5. B. Dodson H. Schwab, « Accelerated Testing, a

Practitioners Guide to Accelerated and Reliability Testing », SAE International, 2006.

6. W.J. Dixon, A.M. Mood, « A method for obtaining and analyzing sensitivity data », Journal of the American Statistical Association, 43 (1948), 109.

7. Y.Z. Zhao, B.Yang, « Probabilistic measurements of the fatigue limit data from a small sampling up-and-down test method », International Journal of Fatigue 30 (2008), 2094-2103.

8. S.K. Lin, Y.L. Lee, M.W. Lu, « Evaluation of the staircase and the accelerated test methods for fatigue limit distributions », International Journal of Fatigue 23 (2001), 75 - 83.

9. B. V. Boitsov and E. P. Obolenskii, « Accelerated tests of determining the endurance limit as an efficient method of evaluating the accepted design and technological solutions » in « Strength of Materials » volume 15, 1983.

10. Lalanne C. « Vibrations et chocs mécaniques, Tome 4 : Dommage par fatigue ». Hermes Science Publications, Paris, 1999.

BIOGRAPHIES

Pauline BEAUMONT, PhD Student in PSA PEUGEOT CITROEN & LASQUO Laboratory VVA CC138 Route de Gisy 78140 Vélizy-Villacoublay, FRANCE

e-mail: pauline.beaumont@mpsa.com

Fabrice GUERIN, Ph. D, HdR I.S.T.I.A - Quality and Reliability department 62 av Notre-Dame du Lac 49000 Angers, FRANCE

e-mail: fabrice.guerin@univ-angers.fr

Pascal LANTIERI, Associate professor, ENSAM ParisTech 2 bd du Ronceray 49100 Angers, FRANCE

e-mail: Pascal.Lantieri@ensam.eu

Matteo L. FACCHINETTI, Guy MARTIN BORRET, PhD Engineers in Mechanics, PSA PEUGEOT CITROEN VVA CC138 Route de Gisy 78140 Vélizy-Villacoublay, FRANCE

e-mail: matteoluca.facchinetti@mpsa.com

Standard deviationmean

SC

Locati

SC-Locati