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Rainflow Counting Based Block Cycle Development for Fatigue Analysis using Nonlinear Stress_图文

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2013-01-1206 Published 04/08/2013 Copyright ? 2013 SAE International doi:10.4271/2013-01-1206

Rainflow Counting Based Block Cycle Development for Fatigue Analysis using Nonlinear Stress Approach
Weidong Zhang, Mingchao Guo and Sridhar Srikantan
Chrysler Group LLC
ABSTRACT An accurate representation of proving ground loading is essential for nonlinear Finite Element analysis and component fatigue test. In this paper, a rainflow counting based multiple blocks loading development procedure is described. The procedure includes: (1) Rainflow counting analysis to obtain the relationship between load range and cumulative repeats and the statistical relationship between load range and mean load; (2) Formation of preliminary multiple loading blocks with specified load range, mean load, and the approximate cycle repeats, and construction of the preliminary multiple loading blocks; (3) Calibration and finalization of the repeats for preliminary multiple loading blocks according to the equivalent damage rule, meaning that the damage value due to the block loads is equivalent to that from a PG loading. The multiple loading blocks simplifies the original PG loading into several typical load cases, which could be conveniently implemented in nonlinear stress approach based fatigue life estimations. An example study is made on a tie-down hook joint in a pickup truck cargo box. Several typical tie-down load cases are obtained after block cycle extraction. These blocking loads are then exploited to conduct the nonlinear analysis on a local joint assembly model, which includes contact interference and nonlinear material property. Fatigue lives are estimated based on nonlinear stresses/strains using linear damage accumulation rule. CITATION: Zhang, W., Guo, M. and Srikantan, S., "Rainflow Counting Based Block Cycle Development for Fatigue Analysis using Nonlinear Stress Approach," SAE Int. J. Mater. Manf. 6(2):2013, doi:10.4271/2013-01-1206. ____________________________________

In the past, automotive industry durability was determined through prototype field testing. Results from the field test would then be used by the engineer for future improvement. With technology advancement, the laboratory test and CAE simulation play a more and more important role in automotive development and validation today. However, the results of the laboratory test and CAE simulation are highly dependent on the input from proving ground loading (PG loading), which is collected from all instrumented vehicle when it runs over a series of predefined road profiles. Generally, PG loading varies with time and is complicated, random, dynamic and varies in amplitude. There are two types of durability lab testing, which are system level fatigue test and component level fatigue test. The system level test, such as a road test simulator (RTS) or four post test, is conducted by applying PG inputs from tire or tire spindles. System level testing is time consuming. It requires very sophisticated hydraulic control systems. Unlike a system level test, a component test is normally conducted on a component or a local assembly using a simplified

loading format, which could be implemented easily and does not lose the characteristics from its original PG loading. As computer hardware and software technology advance, CAE simulation is becoming a more and more important validation tool in automotive development. Similar to lab testing, there are two types of simulation based on PG loading, which are system level simulation approach and component level simulation approach. Vehicle system durability simulations highly depend on the proving ground (PG) loading. In order to exploit the multiple-channeled sophisticated time series loads, a normal durability simulation practice initiates linear pseudo stress analysis based on multiple unit load cases using inertia relief algorithm. The fatigue solvers could construct structural stress history results by superimposing PG loads with the corresponding pseudo stress results. But, the fatigue results based on linear FEA pseudo stress approach over contact area do not have much fidelity [1]. In order to increase confidence, nonlinear FEA analysis must be carried out to fully obtain the stress strain behavior over this area. However, it is not practical to apply PG time series loading directly in nonlinear FEA analysis. The problem becomes how to simplify the complicated PG


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loading history into a simple loading format, which is not only suitable for nonlinear FEA analysis, but also capable of correctly representing the characteristics of the original PG loading. An accurate representation of PG loading is critical for component level nonlinear structure FEA analysis and component fatigue tests. There are two ways to simplify a complicated PG loading history. The first one is the equivalent constant amplitude fatigue load, and another one is the equivalent multi-block fatigue load. As its name implies, the equivalent constant amplitude fatigue load approach is to represent the original dynamic and variable amplitude PG loading by using one constant amplitude loading block. But, equivalent constant amplitude loading block approach is too simple to precisely cover all the characteristics of a variable amplitude PG loading history. From this perspective, the equivalent multi-block fatigue load becomes a more favorable load representation for sophisticated PG loading. In order to obtain a precise description about the PG loading, a statistic at analysis or counting process is necessary to extract the cycle distribution of an original PG loading. Several cycle counting techniques have been developed to perform this task [2]. These approaches include level crossing cycle counting, peak-valley cycle counting, range-pair counting, range-mean, and rainflow counting. The above techniques convert a complicated variable amplitude loading history into a number of discrete simple constant amplitude loading events. Level crossing cycle counting, peak-valley cycle counting, and range-pair counting are categorized into one parameter counting techniques. As the name implies, one parameter counting only conduct the cycle counts based on one parameter. However, the one parameter counting method is unsatisfactory for the purpose of accurately depicting PG loading properties. Unlike one parameter counting, twoparameter counting technique counts the cycles according to two parameters, which have range-mean format, max-min format or from-to format. The rainflow counting is a well known two parameters' counting technique, initially proposed by M. Matsuiski and T. Endo in 1968 to count the cycles or the half cycles of strain-time signals. There are two types of rainflow cycle extraction approaches. The first one is a three point counting method, which determines an inner cycle using three consecutive points in a load-time history. Another one is four point counting method, which uses four consecutive points to define an inner cycle. The stress amplitude and mean stress are two key parameters to determine the fatigue life. Accordingly, the load range and mean load together with their distribution, which directly affect the stress amplitude and mean stress, become indispensable parameters for forming the equivalent multiblock fatigue loads. From this perspective, the rainflow counting technique is the most ideal counting technique for such purposes. This paper develops a procedure to construct an equivalent multi-block fatigue load based on the rainflow cycle counting technique. The first step in this procedure is to

construct a preliminary multiple-block loading cycle based on the rainflow statistical analysis. The relationship between the loading range and exceedance cumulative cycles is first obtained from rainflow analysis on a PG loading history. Based on such relationship, preliminary multiple load range blocks and corresponding repeats could be approximated. Preliminary multi-blocks loading is then generated after pairing the load range with a corresponding mean load, which is determined from a statistical analysis on mean load distribution at the corresponding load range. The corresponding preliminary block cycle is then constructed. The second step is to calibrate the preliminary block loading using the equivalent damage rule, meaning that the damage value due to block loading should be equivalent to that from PG loading. The calibration process is carried out on the same linear global stress results to finalize the repeats for equivalent multi-block fatigue loading. The equivalent multi-block loading simplifies a complex variable loading time history into a series of constant amplitude load blocks, which could be implemented easily in nonlinear FEA analysis. Unlike linear methodology, nonlinear FEA analysis could consider contact interference, material plasticity with hardening, loading sequence, and nonlinear geometry thoroughly. Therefore, more accurate stress and strain behavior could be achieved on the interference zone. These results could then be directly used to estimate the damage and life without any plastic strain manipulation. So, the fatigue analysis based on the true strain and stress could eliminate the potential errors caused by the plasticity correction used in the linear FEA approach. This paper demonstrates a nonlinear FEA analysis example on a tie-down joint in a pickup truck cargo box. Two loading sequences, ascending loading sequence and descending loading sequence, are studied in this paper to investigate the fatigue influence from the loading sequence as well.

In order to predict the structure life behavior using PG loading, a linear FEA approach is widely adopted to obtain pseudo stress for every single unit load case which locates in the corresponding position of PG loading channel by using inertia relief algorithm. As mentioned earlier, a linear FEA model and analysis could not include contact interference, nonlinear geometry, and nonlinear material property. These limitations result in low confidence around contact area. Fig. 1 is a cargo box with gross vehicle weight (GVW) FEA model. In this figure, (A), (B), (C), and (D) are four upper attachments to the payload from tie-down straps. A tie-down strap is used to hold and stabilize the payload on the floor bed of a cargo box. On the lower side of each strap, the strap is tied to the tie-down hook wire as shown in Fig. 2. When vehicle moves, the strap is subjected to a dynamic pulling force caused by the payload inertia, whose loading time history data is recorded in a load cell attached to tie down

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strap from proving ground. In the global FEA model, the tiedown strap is removed, and replaced with a pair of equal but opposite directional forces as shown in Fig. 2. Generally, the tie-down hook wire is subjected to the dynamic pulling force P(t) from the strap during the vehicle moving as shown in Fig. 3.

Figure 4. A typical tie-down hook joint Since correct stress results cannot be guaranteed over contact region from the linear pseudo stress analysis, the fatigue results over this region are also hardly dependable and trustworthy. In Fig. 5, very low fatigue lives for spot welds as shown are obtained from linear stress approach. However, such results have no correlation with the experimental results, which do not have issues in this area. Questions are frequently raised about including the contact interference in FEA analysis.

Figure 1. Linear Cargo Box GVW FEA Model

Figure 5. The tie-down joint fatigue results using linear stress approach Figure 2. Tie-down strap model

Before deriving a multi-block fatigue loading, the rainflow counting technique is used to identify the statistical properties from the PG loading history. Fig. 6 is an illustration about a rainflow counting for a simple fluctuating loading history and its equivalent stress strain history. The four points' counting method is normally used to count an inner cycle and a residual cycle. For example, in Fig. 6, 1, 2, 3 and 4 are four consecutive loading turning points that include a close cycle (2, 3) and a residual cycle (1, 4). In this way, one inner cycle (2, 3), with the corresponding max load point and min load point, is identified. The (2, 3) loading cycle can also be described as the format of rangemean pair or from-to format. In the range-mean format, load range and mean load are two parameters. In from-to format, the loading sequences from “from” to “to” are two parameters. After an inner cycle is identified, this inner cycle

Figure 3. An example of tie down PG loading history P(t) The tie-down pulling force P(t) is then distributed to a typical tie-down hook joint as shown in the Fig. 4. In Fig. 4, (1) is a tie-down bracket, (2) is a hook wire, (3) are spot weld connections, and (4) is a box inner panel. The tie-down hook wire passes the tie-down force P(t) to the tie-down bracket and then tie down bracket distributes the tie down load to the surrounding parts through spot weld connections and contact bearing. However, the pseudo stress approach could only consider loading path through weld connections, and does not have a good way to include the contact bearing interference in the system analysis approach.

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will be counted and removed from the rainflow counting process. The residual cycle will remain in the rainflow counting process. In the same manner, 2-3, 5-6 and 4-7 are identified as three closed loops, and 1-8-9 is identified as a residual loop. In Fig. 6, these three cycles could be clearly identified from the stress-strain hysteresis curve. From this curve, cycle 2-3 and cycle 5-6 look to have an almost similar load range, but mean stresses from these two cycles are different. Therefore, the fatigue from these two cycles will be different due to the mean stresses effect. Expanding to the cyclic load description, both load range and mean load are also two indispensable parameters to define the variable amplitude cyclic loading property. As for the time series tiedown load history as shown in Fig. 3, it is of variable loading range and also of variable mean load along the time domain. Rainflow counting generates a matrix format with cycle counting data according to their range-mean or from-to pairs for the original PG loading. The histogram pad can visually bring out these data from the matrix of counted cycling that have been put into even or nonevenly spaced bins. Fig. 7 displays 3D histogram cycle distribution according to range mean pair, with one axis having bins split by load ranges (max-min) and the other axis with the bins split by the mean load ((max-min)/2). Fig. 8 is a 3D histogram cycle distribution according to from-to pair. This is similar to a max-min format with considering sequence. One axis contains the binned values for the “From” values-the first in the sequence, with the second axis containing the binned values for the “To” values, the second in the sequence.

A cumulative exceedance curve could then be generated from from-to pair matrics as shown in Fig. 9. The ratio 1 is used for manipulation. The x axis in Fig. 9 is the cumulative cycle exceedance in logarithm format, while y axis is the loading range. This diagram clearly describes the nature of the loading history by integrating loading range and its cumulative number of cycles. The plot visually relates the cumulative cycle exceedance to the loading range.

Figure 8. Histogram of from-to pair

Figure 9. Exceedance plot and preliminary block loading Figure 6. Loading sequence and stress- strain hysteresis The exceedance plot provides a good way to extract a series of meaningful and characteristic loading range block by following the exceedance curve. As shown in Fig. 9, a series of steps are created to fit with the exceedance curve. These steps are visualizations for determining preliminary multiple loading range blocks diagram. Each step represents one load range block. In each step, the horizontal line is associated with a specific load range and the vertical line indicates the cumulative cycles exceedance for this specific loading range. Thus, a series of long range pairs (Pi, ni) could be obtained, where Pi represents one load range together with corresponding cycle repeats (ni). However, a series of load range pairs (Pi, ni) are not sufficient for constructing block cycles because they have to pair with mean load. Fig. 10 shows one example of the mean load distribution for a specific load range. To be practical, it is always reasonable to pair the load range with most

Figure 7. Histogram of range-mean Pair

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dominate mean load. To make it simple, the statistic mean of mean load distribution on a specific load range should be a suitable option to couple with this specific load range. For example, mean loads at a specific load range Pi distributes in the following spaces (F1,N1), (F2,N2), ……(F1m,Nm). Fi denotes mean load, Ni indicates cycle repeats for each mean load Fi. The statistical mean of mean loads at a specific load range could then be calculated from Eq. (1).


Figure 12. Preliminary block loading

The preliminary block is still just an approximation from PG loading. In order to accurately represent its original PG loading, calibration is needed to ensure the preliminary block loading to have the same influence from its PG loading to the structure. The equivalent damage rule is a well known criteria to bridge the preliminary blocking loads and its original PG loading for such calibration. Among three parameters for Figure 10. An example of mean load distribution Fig. 11 is the relationship between load range and the mean of mean load, which is derived from the rainflow matrix using Eq. (1). This relationship could be easily used to determine the corresponding mean load for a typical load range. Eventually, a series of blocks (Pi, obtained, where Pi is ith loading range, mean load, and ni is the repeats for ith block. , ni) could be is ith mean of each block, (Pi, ) will remain fixed, while cycle repeats ni will be adjusted for calibration. In this paper, a global cargo box model is used to calibrate the repeats for the preliminary loading blocks by using the equivalent damage rule. First, the global model was used to obtain the damage results for elements around loading zone from original PG loading. Meanwhile, the same model is used to calculate the damage on the same group of elements under preliminary blocking loads. The damage difference between the two sets of loads is monitored until the suitable repeats ni are reached to make such damage difference to be close to 0. Fig. 13 is the illustration about such calibration process.

Figure 11. The relationship between range and mean of mean load Based on the above data, the preliminary multi-blocks fatigue load could be constructed as shown in Fig. 12. To be convenient, Pi is used to describe a loading block instead of using the pair (Pi, ).

Figure 13. The illustration for equivalent damage rule based on linear FEA models

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1. FEA Modeling
A partial cargo box model is cut from the original full cargo box model for tie down hook joint nonlinear stress analysis as shown in Fig. 15. In this model, the nodes along the cutting section as highlighted are constrained for DOFs 1 to 3. Because the tie-down location is far away from the cutting section, elemental mechanical behavior around tie down joints will not be significantly influenced by the constraints along the cutting section nodes. In other words, the partial modeling is sufficiently accurate for nonlinear stress/strain behavior around tie down joint under tie down loading. In this model, all the surfaces over the tie down area are defined to form contact pairs to carry over or distribute the tie down loads from the tie-down hook.

According to Palmgren-Miner rule, the mathematical cumulative damage from all the blocks could be described as: (3) Where D1, D2 … . . Dk = the damage from each typical loading block. By adjusting ni for each loading block, the damage will be changed accordingly. The practical way is to adjust repeats for one block each time. It takes a few iterations to calibrate the repeats for each block to meet the equivalent damage rule. The solution of ni of each loading block is not unique, the criteria is that the finalized loading block should also fit well with the cumulative cycle exceedance curve as shown in the Fig. 14. Fig. 14 exhibits the difference between the preliminary block loading and finalized block loading. The green dash lines represent the preliminary blocks step, and the red solid steps are the finalized block loading.

2. Material Models
The familiar Ramberg-Osgood equation is used to describe material nonlinear behavior as (4).

(4) Where ε = total strain, εe = elastic strain, εP = plastic strain, σ = total stress, K′ = cyclic strength coefficient, is a known as the material is selected, n′ = cyclic strain hardening exponent, is a known as the material is selected, and E = modulus of elasticity.

(5) The Eq. (5) is employed to define material plasticity properties at room temperature in Abaqus by using *Plastic keyword. Meanwhile, the combined hardening rule is employed. Figure 14. The comparison between the finalized block loading and preliminary block loading

3. Operation Load Cases
The equivalent multi-block fatigue loading simplifies a sophisticated PG loading into a series of typical load blocks, which could be easily converted into the corresponding load cases in Abaqus step definitions. Generally, one typical loading block will be decomposed into two loading cases, one is loading up to the maximum or load peak for this block, and then followed by another case which is unloading to the minimum load or load valley. According to Ref. [1], each load case should be applied in two cycles to obtain stable stress/strain results. For each block, there are four steps, loading, unloading, loading, and unloading. In order to study the influence caused by the block loading sequence, two sequences are constructed as shown in Fig. 16. The descending load sequence in Fig. 16a follows the sequence from big loading block to small loading block, and the ascending load sequence in Fig. 16b is arranged from small

Figure 15. The local fixed cargo box model

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loading block to big loading block. Abaqus 6.10.2 is employed for multiple load case nonlinear FEA analyses. Fig. 17 shows the stress strain relationship comparison under the two load sequences.

a. Descending load sequence

b. Ascending load sequence Figure 16. Load sequence cases

a. Comparison between all steps from two sequences

b. Comparison between 2nd loading and unloading in P1 from two sequences Figure 17. Stress-strain comparison between two loading sequences

Fig. 17 is stress-strain comparison between two loading sequences. Fig. 17a is a comparison between all steps from two sequences; and Fig. 17b is only the comparison between 2nd loading and unloading steps in P1 from two sequences. The tie-down block loading is not fully reversed, and tiedown force is always in tension. Therefore, Fig. 17a does not look like a characteristic stress strain hysteresis curve. However, Fig. 17a exhibits a very meaningful difference between two loading sequence. In the descending loading sequence, the biggest loading block is applied first; smaller and smaller block loading is applied in sequence. At the very first loading step for the biggest loading block, the maximum load causes an element to reach its extreme plasticity point. However, stress will linearly drop in the following unloading step, and at the end of unloading in the biggest loading block, the remaining plasticity causes the elemental stress in Fig. 17a to enter compression state. On the base of this initial stress and state, this element behaves linearly in the all the following steps. In Fig. 17a, the green solid line is a stress strain envelope for all the loading steps described in the ascending loading sequence. This green line clearly shows the nonlinear stress strain curve portion during the 1st big loading, and it also shows the linear stress-strain descending sloped line during its unloading. The mechanical behavior for the rest of the loading steps is hidden in the green slop line. As for the ascending loading sequence, the smallest loading block is applied first, and then the bigger loading follows until the biggest one is applied eventually. For every single block, its nonlinear stress and strain behavior could be clearly identified in its first loading steps. Also for the rest of the loading steps in every single block, the linear stress-strain behavior could be identified. In this linear stress strain curves, there are two curves from 2nd loading and 2nd unloading from each block hidden in the corresponding black dash slope line from 1st unloading. These two hidden curves form a closed cycle. As the loading block is switched to the first loading of a higher loading block, the higher loading will break the earlier linear status, and follows a new nonlinear stress strain behavior. When the corresponding plastic state for higher load is reached, a new linear material behavior is formed for its cyclic loading. The envelope of stress and strain relationship from ascending load sequence agrees well with the one from the descending load sequence. The minor difference between two indicates the cyclic material hardening behavior. Fig. 17b shows a clear difference between the 2nd loading and 2nd unloading at P1 loading block in two load sequences. According to this figure, the loading sequence causes significant difference for same loading block. Green line in this figure indicates a cycle caused by the P1 in descending loading sequence. And the black dash line indicates a cycle caused by the P1 block in ascending loading sequence. The stress-strain ranges from the two cycles look identical, but their locations are different. The black dashed cycle resides in

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the tension phase, but the green cycle resides in the compression phase.

4. Fatigue Predication based on Nonlinear Stress Results
1. E-N fatigue engine is used. Nonlinear strain and stress results are directly imported into fatigue engine for the further operation. 2. Duty Cycle Procedure. This is due to the fact of Abaqus stress and strain results. Results at the second cycle of each block loading are used with associated number of cycle repeats according to finalized block loading. 3. Damage accumulation. Calculate the damage for each loading block from E-N curve definitions based on mean stress correction. If the damage di is caused by the one repeats of loading block Pi based on the nonlinear stress strain results, where i=1, 2…..k, the overall damage Dblk,nonlinear can be obtained in terms of Miner's rule. (6) Where ni represent the corresponding cycle repeats for loading blockPi. The accumulated damage value was carried out based on the damage from each time step and repeats definition in the duty cycle.

makes the mean stress for P1 tension, which degrades fatigue life. That is the reason why the life results from the descending load sequence are better. Since PG loading is a random load with arbitrary loading sequence, the ascending load sequence is a conservative recommendation for such tiedown joint. The bearing contact helps smoothly distribute the tiedown load to the surrounding structure instead of the spot weld connections only in the linear stress approach, so the spot weld mechanical behavior in nonlinear stress approach is close to the real situation. Fig. 18 is spotweld fatigue based on the nonlinear stress approach. The fatigue results for these problematic spot welds meet fatigue life requirement and do not have the problems as shown from a linear stress approach in Fig. 5. The results from nonlinear stress approach agree very well with experiment in the tie down location.

In this paper, an equivalent multi-block fatigue loading procedure is demonstrated to extract multiple typical loading blocks from a sophisticated PG time history. The rainflow counting technique is first used to conduct the cycle distribution on a PG loading and generate the load range and cumulative cycles exceedance curve. Then, preliminary loading blocks are approximated by creating a series of steps to fit with cumulative cycles exceedance curve. The corresponding mean load for a specific load range is determined from a statistical mean calculation on a group of mean load distributions, which is also from rainflow counting matrix at a specific load range. The repeats for each typical block is finalized and calibrated using the equivalent damage rule on the same global model. The block cycles can then be created. The calibration process ensures the block loading to have the same influence to the structure as the original PG loading. A global cargo box model is used for this process. The damage from original PG loading is used as a target for calibrating the repeats for each typical block in block loading. Several iterations are needed to finalize the cycle repeats for block loading. The loading blocks are then converted to step definitions in the nonlinear stress strain analysis. Each block is converted to two steps, one is loading to the maximum load (the Peak load), and another one unloading to the minimum load (the Valley load). Each block is implemented twice in Abaqus to obtain stable results, which then is used for fatigue estimation. Two load sequences are studied for their influences to fatigue results. The results from ascending loading sequence are worse than the results from the descending loading sequence. This difference is caused by the totally different mean stress states due to two loading sequences. Tension mean stress state degrades fatigue file; and compression mean stress state helps fatigue life. The block cycles development could be extended for application in the laboratory component test.

Figure 18. Fatigue results from ascending load sequence According to fatigue results, the sheet metal fatigue results based on the nonlinear stress approach from two load sequences agree perfectly with the earlier analysis. Overall, the fatigue results from descending load sequence are much better than the results from the ascending load sequence. In Fig. 17b, the black dash line indicates the stress strain cycle for P1 loading in ascending load sequence, while the green solid line denotes the stress strain cycle for P1 in the descending load sequence. For the descending load sequence, the mean stress for P1 is compression, which is beneficial to fatigue. On another side, the ascending loading sequence

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In the future, the technique will be continuously modified and improved by means of more investigations on fatigue methodology usage, FEA technique usage and correlations to tests.

1. 2. 3. 4. 5. Guo, M., Bhosale, S., Srikantan, S., Munson, K. et al., “A Fatigue Life Estimation Technique for Body Mount Joints,” SAE Int. J. Mater. Manf. 5(1):226-234, 2012, doi:10.4271/2012-01-0733. Lee Y, Pan J., Hathaway R. & Barkey M., Fatigue Testing and Analysis (Theory and Practice), Elsevier Butterworth Heinemann, 2005. Abaqus User's Manual, Volume IV: Elements, Version 6.8, Dassault Systems, SIMULA, 2008 Abaqus User's Manual, Volume III: Material, Version 6.8, Dassault Systems, SIMULA, 2008. Design Life Online Manual, “Design Life User Guide 6.0”, HBM, 2010.

Weidong Zhang 248 944 1371 (office)

The authors would like to thank colleague Dr. Baizhong Lin for technical discussions. Also, the authors would like to appreciate HBM for their nCode fatigue solver, which provides a convenient platform for rainflow counting, blocks construction, and fatigue life estimation.

FEA - Finite Element Analysis E-N - Strain based fatigue calculation PG - Proving Ground RTS - Road Test Simulator GVW - Gross Vehicle Weight


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