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Reliability Evaluation of Trailer Axles

-Larry McLean

Final Project for DESE-6070HV7

Statistical Methods for Reliability Engineering

Dr. Ernesto Gutierrez-Miravete

Rensselaer at Hartford

05Dec08

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TABLE of CONTENTS

TITLE: PG

Executive Summary…………………………...………………..3

Recommendations……………...…....…………………………3

Conclusions………………………………………………………3

Introduction………………………………………………………4

Component Description………………………………………..5

Component Reliability Structure……………………………..5

Component Test Description………………………………….7

Component Test Results………………………………………9

Component Test Data Reduction……...……………………10

Component Context (System Description)………………..17

System Reliability Analysis…………………...……………..18

Appendix 1: Component FMEA……………..………………26

Reliability Evaluation of Trailer Axles

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Executive SummaryA process for evaluating the in-service reliability of a trailer axle is described here.

Since the trailer axle has undergone some testing, it has been used to characterize the reliability behavior of the axle, based on some simplifying assumptions. A more complete analysis is recommended involving stress analysis of the component.

Using simplifying assumptions, a sample calculation is done to calculate the in-service reliability of the axle. Due to lack of data, the calculation, has no basis in reality, but would be useful as a model for performing a real analysis.

Maple, Minitab and ‘R’ were indispensable in manipulating and curve-fitting data and creating equations that made fundamental analysis possible.

Recommendations• On the occasion of further testing, place strain gauges at key locations on the axle.

• Finite Element stress analysis of the axle.

• Detailed information about the material properties of the axle would allow comparison between analysis and testing to be meaningful.

• With this information and the rig test, it could be determined if the component was attaining its expected life.

• It also would provide a vehicle for studying the design features with the twin aims of increasing reliability and reducing cost.

• Field testing to determine the environment that the axle is subjected to in-service.

• The objective of this testing would be to derive equations that could describe the service mission of the axle and allow a complete analysis of the reliability.

• Strain gauges coupled with trailer load information, road conditions and trailer speed are all important.

• A model of the trailer and associated suspension hardware.

• This would allow a prediction of the impact of varying environmental factors on the forces the axle sees.

• This would extend the usefulness of field testing and minimize the testing required.

Conclusions• The axle system reliability functions can be represented by using a single Weibull equation.

• The failure mode of the axle was consistent and demonstrated two symmetrical weak points on the axle.

• By generalizing Miner’s Rule, it is possible to create ‘mission’ formulas that can be used to estimate axle reliability.

• Monte Carlo methods are useful in taking these general equations and predicting in-service axle reliability.

• Using fitting techniques, the reliability curve can become the basis for calculating all the traditional reliability functions.

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Introduction: Many devices produced today have very high reliability under normal use. This makes reliability testing and analysis lengthy and expensive, unless reasonable shortcuts can be developed. In addition, this can be more complicated if the device is used under widely different conditions and missions, which is often the case. Axles used on tractor trailers have both these difficulties. Tractor trailers or ‘Big Rigs’ are used for hauling all sorts of cargo across North America and the world. In some cases the loads are extreme and in others the roads are poor or not existent. If not driven carefully over irregular surfaces, the axles can be subject to shocks. In addition, salt is commonly used on winter roads and hence corrosion becomes a consideration. For instance, figure #1 below shows a trailer being used to haul huge logs through a forest lined secondary road. In summary, to evaluate the reliability of trailer axles in service is a large challenge for many reasons:

Variability in road surface. Variability in load. Variability in driving practices. Irregular and unknown missions. Part corrosion. Part Wear. Shock loading System complexity. Etc…

Figure #1: In-service Trailer axles

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In addition, this problem is made more difficult, because, there are many manufacturers of these rigs, and for every manufacturer there are also third party aftermarket providers for components. These third party manufacturers do not necessarily have the resources nor the expertise necessary to do complex testing and analysis. What is needed is a reliable process that can be applied to get a reasonable result when data and resources are limited. For this project I have examined a set of endurance data typically gathered within the North American axle manufacturing industry to ensure the reliability of the product. Traditionally, the pass/fail criterion has been based on comparisons with proven successful designs. This approach does not give the manufacturer adequate analytical tools to understand the implications of novel designs, or evaluate new applications. In addition, the manufacturer is left without analytical capability for understanding the root cause of field problems. What has been lacking is the knowledge and the expertise needed to extrapolate this data to a consistent and accurate estimate of reliability. This would allow the manufacturer to understand his design margins and recognize when some other problem, l ike workmanship, misuse or other issues separate from normal wear.

Component Description: Specifically, the component under investigation is an axle used to support cargo in a tractor trailer. Springs are mounted to the axel as shown in Figure #2 below. A properly distributed load in the trailer creates a mean downward load symmetrically on the axel at the spring locations. The axel is supported symmetrically at the wheel bearing locations. As the trailer moves, variations in speed, road surface and truck maneuvers results on oscillatory loads of varying amplitude. These are not necessarily symmetric, since the raod surface variations may not be. Other loading would be associated with acceleration and braking.

Component Reliability Structure: As shown in Figure #2, forces are applied to the system at two points. On average, these forces would be expected to be symmetric relative to the axel, therefore peak stresses relative to these inputs would also be symmetric and failure would be equally likely at two different points on the axel. Experience has shown that failures typically occur at the two symmetric locations where the plates holding the springs to the axel are welded to the axel body. Of course, when the component fails at one location, the component has failed, and therefore these failure events operate as a series system. This is shown below in Figure #3. In order to determine the reliability of the system, the component must be placed in context and the environment in which it will be used muist be understood. To characterize this system, a series of axels were subjected to cyclic loading until failure. The first task is to describe and analyze this data.

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Springs

Axle

Brakes

Wheels

44”

77.5” Bearing Span

Figure #2: The Axle Component

LHS Load Point Failure

RHS Load Point Failure

Other Failures

Figure #3: Axle Failure Event Tree

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Component Test Description: Based on his experience the manufacturer was concerned about the low cycle endurance capability of his axels. He constructed a test plan that corresponded with testing typically performed on these parts within the industry. There wa, howevrr, no plan to extend these results to a reliability of the part. Each trailer axel sample was seated on two end supports consisting of vertical beams, block bearings and shafts. The location of the end supports matched the bearing locations and the or ientation of the axel matched the in-service orientation. Two servo-hydraulic actuators equipped with 50 Kip calibrated load cells were placed under each axel and securely attached to match the spring loading locations of an in-situ axel. This test configuration allowed the engineer to apply a sinusoidal cyclic downward load of +400/+22500 at a frequency of 3 Hz. The cycle rate was chosen to match the capabilities of the equipment. The magnitude of the load was chosen to match service limits. Visual inspections were performed every 10,000 cycles. To help the engineer detect crack initiation, the parts were painted white. When cracks appeared, the parts were tested to failure and the progression of the distress was recorded. The original test plan called for testing until failure, to be censored at 200,000 cycles. This was modified as the test progressed; most parts were tested to failure, some past 200,000 cycles. Unfortunately, there were no strain gauges or instrumentation to aid correlation of the results with analysis. A photo of the rig set-up is shown in Figures 4 and 5.

Figure #4: General View of Test Rig

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Figure #5: Details of Test Set-up

Table #1: Table of Test Results

Sample No Hours tested LHS failed RHS failed

1 200,000 N N

2 153,348 Y N

3 192,595 Y N

4 218,787 N Y

5 179,968 Y N

6 226,100 N Y

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Component Test Results: A typical crack progession is illustrated in Figure 4a and 4b. Typically failures, when they occurred, consisted of cracks which initiated in the vicinity of the weld connecting the spring bracket to the axel. This is reported to be similar to what is seen in service. First the crack appears next to the weld as shown in Figure 4a. As cycling continues, the crack progresses circumferentially until the axel fails catastrophically to the applied load. Not surprisingly, these failures happened about equally on the left and the right bracket, since there is no theoretical difference between them. A summary of the results is presented in Table #1. Failures occurred over a time centering around 200,000 cycles. In order to allow correlation of the results with analytical data the failure rates at the two locations must be treated separately. It is implicitly assumed that no other failure mode of the system is significant relative to that considered here.

Figure #6a: Sample 3 (LHS) Crack Initiation

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Figure #6b: Sample 3 (LHS) Failure Components Test Data Reduction: The data was reduced in two ways, with the ultimate objective of simplifying the resulting equations to make the more readily useful for a system reliability study. First, the two failure modes of the axle are fitted as separate but equal. Using right censoring for the data and concatenating all the data into one set (treating the left and right stress points as equivalent) we get the data fits shown in Figure #8 (Separate Modes). Clearly the two parameter Weibull is simple and satisfactory, with a correlation coefficient of 0.992. Looking more closely at the fitting curve (see Figure #9), we can see that the hazard function rises with time as would be expected with this kind of data. The curve is slightly skewed, but otherwise is bell shaped. Similarly, the data was plotted assuming the axle had a single failure mode (Merged Modes). The results of this analysis are shown in figure #10. For the merged results, the Weibull fit is chosen for simplicity and accuracy. Figure #11 shows the details of a Weibull curve-fit and Figure #12 shows the details of a Normal curve fit. Normal curve-fitting allows for an understanding of data mean and standard deviation which might prove useful in some discussions. Maple was used to create a model of the Axle system behavior from the “Separate” modes data to allow direct comparison with the “merged” mode results. Table #2 shows a comparison of the resulting eqiuations. And Figure #7 plots the two results on the same graph. Clearly the results are almost indistinguishable, and the simpler form will be used.

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Figure #7: Comparison of “Merged” vs “Separate” Probability Density Distributions

Separate(R1=R2):

Shape: 6.89321

Scale: 233,763

Rss=R1*R2

MTTFss=197,593 cycles

Merged:

Shape: 6.83966

Scale: 212,004

MTTFsm=198,076 cycles

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Figure 8: Probability Plots of Failure Data (Separate Modes)

Figure 9: Weibull Plots of Failure Data (Separate Modes)

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Weibull0.991

Lognormal0.977

Exponential*

Normal0.983

Correlation Coefficient

Probability Plot for timesLSXY Estimates-Censoring Column in censores

Weibull Lognormal

Exponential Normal

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Correlation 0.991

Shape 6.89321Scale 233763Mean 218495StDev 37239.7Median 221659IQR 49996.3Failure 5Censor 7AD* 19.689

Table of StatisticsProbability Density Function

Survival Function Hazard Function

Distribution Overview Plot for timesLSXY Estimates-Censoring Column in censores

Weibull

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Figure 10: Probability Plots of Failure Data (Merged Modes)

Figure 11: Weibull Plots of Failure Data (Separate Modes)

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Weibull0.991

Lognormal0.974

Exponential*

Normal0.981

Correlation Coefficient

Probability Plot for timemLSXY Estimates-Censoring Column in censorm

Weibull Lognormal

Exponential Normal

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Correlation 0.991

Shape 6.83966Scale 212004Mean 198076StDev 34004.7Median 200942IQR 45675.2Failure 5Censor 1AD* 3.479

Table of StatisticsProbability Density Function

Survival Function Hazard Function

Distribution Overview Plot for timemLSXY Estimates-Censoring Column in censorm

Weibull

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Mean 199125StDev 33909.9Median 199125IQR 45743.8Failure 5Censor 1AD* 3.500Correlation 0.981

Table of StatisticsProbability Density Function

Survival Function Hazard Function

Distribution Overview Plot for C2LSXY Estimates-Censoring Column in censor

Normal

Figure 12: Normal Plots of Failure Data (Separate Modes)

Figure #13 shows a typical S-N characteristic for iron based materials. Plotted on a Log-Log scale, it is usually conservative to assume the S-N relationship is linear as shown. To use the experimental data, without doing a stress analysis on the system, it has been assumed that there exists a force “Fe” which correstonds to “Se” on the S-N diagram and a force “Fu” that corresponds to “Su”. It is further assumed that the relationship between Fu and Fe is the same as the relationship between Su and Se. In addition, since the S-N curve is not a unique characteristic of the material, it is assumed that the characteristic probability distributions trace a series of parallel lines on the S-N diagram as illustrated in the figure. Using these assumptions, it is possible to locate the experimental results on the diagram; and it is possible to construct the diagram in terms of forces applied in the experiment. This is also illustrated on Figure #13. Using a Monte-Carlo method, it was possible to create a bN distribution that would reproduce the “N” distribution that has already been constructed from the data. Figure #14 shows probability plots of N and bN data from an MC analysis. 60000 points were used. The match with the original Weibull distribution for N is very good, and the bN Weibull fit would be expected satisfactorily recreate the original N distribution.

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Se; Fe

Su; Fu

Se = Endurance Limit of Material

Su = Ultimate Tensile Strength of the Material

St = Peak tensile stress (somewhere on the axle) during testing

Fe = Load applied to the axle when the peak stress is Se (somewhere on the axle)

Fu = Load applied to the axle when the peak stress is Su (somewhere on the axle)

Ft = Load applied to the axle when the peak stress is St (somewhere on the axle)

Log (S;F)

Log NNe=10^6Nu=10^3 N t = MTTFsm

=198,076 cycles

St; Ft

Probability Density Distribution around S-N curve (derived from test data)3sigma lines

Calculation of MTTF of Log-Log Line for S-N Diagram

S-N Mean Curve:

N=(F/bN)^1/mN

bN=153008

mN=-0.10034

Figure 13: Calculation of axle S-N Characteristic

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Figure #14: Weibull Probability Plots of Monte Carlo Analysis to determine bN distribution to match Test Data

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Component Context (System Description): As stated above, this manufacturer is fabricating axels for tractor trailers. This represents a small subsystem of a larger system with layers of complexity. To place the task in context, Figure #15 diagrams the major components of the truck/trailer system where these axels would be used. It is clear that a complete reliability study of the system would be a complex task. Even understanding the reliability of the axel as a subsystem involves understanding its relationship to the whole system which this line diagram begins to accomplish. Figure #16 shows the main activities associated with the design of the axel itself (separate from any of the components attached to it). Central to this is an understanding of the, in-use requirements. This might be covered in a well constructed specifications document, but a more valuable approach is a detailed investigation of “Mission Requireemnts”, the field conditions under which the axel is used. I have covered this under the heading of “Mission Requirements”. Mission requirements would include, trailer loading specifications, suspension (isolation of the axel from shocks), environmental conditions (temperature, corrosive elements…), driver experience and training (will he drive in the manner expected relative to the driving conditions), road conditions, and finally maintenance activities associated with the axel and running gear. To avoid problems, the axel designer must develop some understanding of all these influences.

Truck/Trailer System

Truck

Storage ContainerFrame

Bearings Brakes Wheels Turning System

Support Structure

Trailer

Running Gear

axles

FunctionStructure

Figure #15: System Diagram

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System Reliability Analysis: Figure #17, is an overview of the main steps required to determine the reliability of an axle in service. The rig work, that has already been described, can be used to characterize the axle; this is represented in the blue box on the diagram. A complete reliability study requires that information be gathered, calculated or assumed on in-service conditions. As shown in the figure, the mission branch of the analysis involves at least some testing of the field conditions that the axle must operate in. Mission Inputs: Fundamentally, what is needed is a relationship between the in-service use of the axle and the associated stress levels. It is useful to think of this is terms of ‘missions’ that the axle will encounter in service. One way to identify realistic stress profiles, is to instrument a trailer and run it through simulated or real test missions. Ideally, using knowledge of the axle, strain gauges could be placed strategically to capture the actual stress levels at significant locations. In choosing a mission, the worst or most severe average conditions are relevant here, since reliability is normally associated with overall life. Overloading situations resulting in faster failures must also be considered. The variations in conditions should be captured to the extent possible. In choosing important mission parameters, considerations might be given to the factors identified on TABLE #1 of the FMEA (see Appendix 1). Of course, to extent possible, global mission parameters, such as static load and vehicle speed, should be documented along with strain gauge information to allow proper correlation. Alternately, an analytical model of the system must be made and stress levels determined analytically.

Static Loading

Dynamic Loading and

Structural Analysis of attachments (Brakes, bearings, cams…etc)

axle Structural Evaluation

Figure #16: Structural Requirements of axle

Material Selection and Fabrication Process

Mission Specification

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In the case of a new design or a new application, some of this information must be specified, assumed or calculated. In summary, the data, analytical and measured, must create a load frequency distribution that accounts for all the variations that are expected in service due to cargo load, road surface variations, vehicle speed and other relevant mission variations. In this report mission information will be assumed. Figure #17, shows conceptually what is involved in linking the mission information to the axle characteristics, which have already been derived. The damage associated with cycles at any load level must be linked, not to the number of cycles to failure at a single load level, but somehow to the cumulative damage associated with cycles at all different load levels. One common practice is to use Miner’s Rule. The form of it used here is also illustrated on figure #18. For the purposes of this report, some simplifying assumption are made which are described below Figure #18-20.

Monte Carlo Axle Reliability Calculation

Nominal Mission

Definition

•Variation in Cargo Load

•Variation in Road Surface

•Variation in vehicle speed

•Driving habitsMission Inputs: Axle Load (or stress)

frequency distribution

Road Test Results

Axle Fatigue Performance Characteristics

Axle Strength Characteristics:

Axle F/N Diagrams

Output: Axle Reliability Figure #17: Reliability Analysis

Data Reduction and Analysis

Field Data

Rig Data

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Cargo Load (Mean Stress)

Dynamic Load (Variable Stress)

Frequency

Speed 1

Other Variables:

•Road: 1) Rural Road; 2) Secondary; Road 3) Highway

Mission Definition: Test Results

Speed 2

Table: Mission Points

Condition Time Speed Road Cargo Load

#1

#2

#3

etc

Figure #18: Mission Data

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“n”

Failure Limit (N)

Log F

Log N

Fe, Se

Fu, Su

•The integral of “n/N” over time gives an accumulated axle damage (D).

nR=n/N=damage done per hour of use

Mission Inputs

(n)

Se = endurance limit; Su=ultimate tensile strength

Figure #19: Linking Mission data to Axle Characteristics

Miner’s Rule:

If n1/N1+n2/N2+…+nr/Nr<1 then a failure will not occur.

“n” and “N” are each stochastic in nature. Representing n/N approximately as a continuous function, the above formula can be generalized to:

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But we know that N follows a Weibull distribution. Therefore we can calculate bN by stochastically generation. Nt=212004*(-Ln(1-F))^(1/6.83966)………………………(1) bN=45000*Nt^0.10034…………………………………….(2) n/N=Truck Life(in hours)*(bN/48246)^^(-1/0.10034)…...(3) If n/N<1; X=1 otherwise X=0. Reliability is the average value of ”X” over several “trucks”. Formulating the equations 1,2 and 3 into a Monte Carlo Analysis with 50,000 point by generating “0<F<1” as a random function and repeated enough times to create a reliability plot; the following reliability plot results(Figure #20): Note: The above formulation is possible with an additional simplifying assumption (mN=mn).

Figure #20: Reliability Plot for Axle

Sample Solution:

By assuming that n can be represented by a formula of the form:

The ratio “n/N” can be written:

And:

A plot of the “N” and the “n” curve according with the above assumptions is plot on a Log-Log chart in Figure #19:

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Using ‘R’ (this is a free downloadable statistics package), I was able to ‘fit’ a Weibull to the above data:

The MC generated data with the fitted curve is shown below as Figure #21:

Results of MC Analysis

By placing this equation into Maple, it is possible to generate all the usual Reliability functions. The results of this are shown in Figure #22.

Figure #21: Fitted Plot of Reliability Results

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Figure 22: Plots of Axle Reliability Functions Generated by Maple

MTTF=99064

Probability Density Function

Hazard Function

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List of References-Textbooks:

Faires, V.M., Design of Machine Elements (4th Ed), the MacMillan Company, 1965.

Shigley, J.E., Mechanical Engineering Design, McGraw-Hill, 1977.

Rausand, M and Hoyland, A., System Reliability Theory (2nd Ed), John Wiley and Sons, 2004.

Verzani, John, Using R for Introductory Statistics, Chapman and Hall/CRC, 2005

Abernathy, R.E.; Breneman, J.E.; Medlin, CH; Reinman, GL; Weibull Analysis Handbook; November 1983.

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Appendix 1: Component FMEA: Tables #1, #2 and #3 are a summary of the FMEA done on the axel. This study was done on the highest level. No attempt was made to drill down to a more detail in the modes, causes and effects, nor were the failure modes broadened to include associated hardware failures and structural context. Nevertheless, it appears that the axels are traditionally quite reliable but critical to the functioning of the system. Table #1 shows the high level failure modes of the system resulting from axel failures. The main purpose of this table is to identify occurence rates, which are summarized under Avg. Occ. Generally occurrences of axel failures are unusual.This information is transferred to Table #2, to develop a more complete information by calculating RPN numbers. It can be seen that the severity of these occurrences is high; this is particularly true on secondary roads, where general noise levels are high and the drivers ability to hear problems (like axels failures) is less likely. The scoring system used to create this FMEA is shown in Table #3.

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