The Probabilistic Study of Heat Treatment Process for Railroad Wheels Using ANSYS/PDS

Kexiu Wang Griffin Wheel Company

Abstract

This paper outlines a probabilistic analysis performed on a Griffin Wheel CJ36 freight car wheel design to demonstrate how ANSYS Probabilistic Design System (PDS) can be used to understand the production process uncertainties and the parameter variations in a manufacturing process. Compared to the traditional deterministic approach, the probabilistic method is a more reliable means to account for uncertainties, when optimizing manufacturing processes as well as the product designs. The ANSYS/PDS provides an efficient tool to assess the interactions, effects, and sensitivities between input parameters and output variability.

After casting or forging, railroad wheels are heat-treated to induce the desirable circumferential compressive residual stress in the upper rim. However, the heat treatment process also generates an axial tensile stress that could contribute to Vertical Split Rims, which is a catastrophic wheel failure mode. To investigate the effects of different parameters in the heat treatment process and to identify those that have the greatest impact on the residual stress field of the railroad wheels, a probabilistic study was conducted by using the ANSYS/PDS tool. The results revealed that residual stress is highly sensitive to the creep material property. Therefore, this factor significantly affects the residual stress field of railroad wheels during heat treatment.

Introduction

Probabilistic Method We live in an uncertain world. Uncertainty, variability, and incomplete information are inherent in all fields ranging from engineering to economy. The concept of uncertainty has long been associated with gambling and games. Bernstein (1998) provides an extensive history of uncertainty in the context of probability theory and risk management. In engineering, the operational loads, geometry, manufacturing processes, computer models, material properties, and operational environments, as well as testing are all inherently subjected to scatter, variability and uncertainty in the real world. These uncertainties lead to uncertainty in product development and manufacturing.

The traditional deterministic approach that accounts for uncertainties in product development is based on safety factors that are applied to the loads. The specifications of safety factors are generally based on the empirical design guidelines established from years of testing and experience. It does not directly account for the random nature of design parameters. The consequence of treating parameters such as material properties, geometry, environment, and loads as singly determined values is lack of predictable reliability. Without a way to measure reliability, it is unlikely that the level of performance will be consistent. Moreover, the most common practice is to use the assumed worst case scenario and to apply safety factors to scale loads. In reality, a worst case scenario is rarely identifiable. Furthermore, the question of how these uncertainties affect product quality cannot be answered accurately.

Compared to the traditional deterministic approach, the probabilistic method is a more reliable means to account for uncertainties when optimizing the manufacturing processes and the product designs. In general, manufacturing processes and product designs can be treated as a system with uncertain inputs and outputs. The input parameters to be analyzed are treated as variables. The parameter uncertainties are statistically characterized in terms of Probability Density Functions (PDFs). Therefore, the probabilistic method provides a means to quantify the inherent risks in a system and also provides the ability to evaluate the sensitivities of the resulting output performances to all input parameters. It can identify parameters that drive risk so that action can be taken to minimize the risk. The effects of realistic variability in input parameters can be determined. The probabilistic method requires a more detailed analysis and it leads to a

more comprehensive understanding of a system. Eventually, the information gained from understanding the interactions, effects, and sensitivities of parameters can lead to a more efficient and reliable system. The sensitivity analysis is related to the uncertainty analysis. Its primary relation to the uncertainty analysis is to identify the parameters that have the greatest influence on the system output by evaluating the impact of the parameter uncertainties on the system.

The probabilistic approach has been commonly used in the aerospace industry and many probabilistic studies have been carried out by using this approach. Its application in other industries, especially in the railroad industry has been limited. The deterministic approach is still the method of choice in the railroad industry. One of the barriers to applying probabilistic method is the additional complexity, resulting in extensive calculations compared to the deterministic method. However, the advances in the computer software and hardware industry over the past decade are bound to help.

First introduced in ANSYS 5.7, the ANSYS/PDS is a probabilistic analysis tool that is an integral part of the ANSYS Finite Element Analysis (FEA) program. It is an efficient tool for assessing the effects of uncertainty. ANSYS/PDS allows engineers to develop a better understanding of the product’s behavior in real life conditions, by quantifying the uncertainties. The input variables are studied simultaneously in an automated manner. The probabilistic characteristics of random variables such as boundary conditions, geometry, and material properties can be specified. The process is typically completed by using a certain type of statistical sampling method, such as Monte Carlo or Response Surface. The parametric FEA model is invoked repeatedly for deterministic analyses performed for a set of input parameters that are generated according to their PDFs to produce a set of output samples. The statistical characteristics of the output performance variables are then deduced. Using the parallel processing feature of ANSYS/PDS, the running time can be significantly reduced by distributing the analysis tasks among multiple computers on the local area network.

Railroad Freight Car Wheels

Freight car wheels, as shown in Figure 1, have the toughest job among all the components of a freight car. The wheels must support the weight of the car and steer it on the rails. The wheels also act as brake drums. The brake shoes are applied directly to the wheel tread in order to stop or slow down the train. The static load on a 36” wheel under a loaded 110-ton freight car is 35750 lb. The wheels must withstand tremendous amounts of abuse from extreme thermal and mechanical stresses caused by factors such as brake shoe friction and high dynamic loading.

Figure 1. A railroad fright car wheel

Griffin Wheel Company uses castings to manufacture its wheels. After casting is complete, the wheels are reheated to 1700 oF in a homogenization furnace to remove any undesired residual stress that remains in the wheels. Once the wheels are homogenized, the rims are quenched with water spray on the tread surface for four minutes. Following the quenching process, the wheels are placed in a tempering furnace at 950 oF for two hours to permit partial relief of the residual stress. After the wheels are removed from the tempering furnace, they go through the hub cooling process to meet the wheel-mounting requirement, in which water is sprayed on the wheel bore surface for 3.4 minutes. Afterwards, the wheels are exposed to ambient conditions as they cool to room temperature. The rim-quenched wheels strengthen the steel, improve the wear resistance, and induce the beneficial circumferential residual compressive stresses in the wheel’s upper rim. The compressive stresses are known to help prevent the formation of rim fatigue cracks and therefore they are important to the train’s safety. However, the heat treatment process also generates an axial tensile stress in the lower part of the rim, which could contribute to Vertical Split Rims as shown in Figure 2, a rare but catastrophic wheel failure mode. Fractures tend to propagate from the tread to the front rim fillet. The axial tensile stress could cause rapid fracture that is exasperated by the impact load during the late stages of crack propagation.

Figure 2. Vertical Split Rims and the residual axial tensile stress

For a Class U wheel, the carbon content varies from 0.67% to 0.77% as specified by the Association of American Railroads (AAR). As a result of the 0.1% carbon content variation combined with other chemical composition variations such as Mn, S, P, Si, etc., the material properties of wheels could range widely. A large range of values for coefficient of thermal expansion and stress-strain parameters of the wheel steels has been documented in literature. Previous researches have revealed that the residual stress varies according to many different parameters. A series of experiments performed by Perfect (1986), revealed that the residual stresses were very sensitive to variations in material properties, particularly the coefficient of thermal expansion. Perfect’s model predicted that stress levels decrease as draw temperatures increase. The work of Kuhlman, Sehitoglu, and Gallagher (1988) extended Perfect’s analysis to include the effects of creep. It indicated that creep is an important consideration when estimating the residual stress of the quenched wheels. It also concluded that the changes in the magnitude of thermal conductivity moderately affect the residual stresses. The study conducted by Gordon and Perlman (2003) showed modest variation in the residual stress with changes in the draw furnace temperature, quenching time, and coefficient of thermal expansion. It also supported Kuhlman’s conclusion that accurate prediction of as-manufactured residual stress distribution in wheels requires accounting for creep. Some studies indicated that as the film coefficient was increased, the residual stresses increased in magnitude. The focus of the previous studies

was the residual stress in the circumferential direction. Those investigations were based on the traditional deterministic approach. Not enough work using the probabilistic method has been accomplished to be able to analyze the heat treatment process of railroad wheels.

To investigate the effects of different parameters on the heat treatment process and to identify those that have the greatest impact on the residual stress field of the railroad wheels, a probabilistic study was conducted using the ANSYS/PDS analysis tool. The study was aimed at identifying and quantifying the key sources of variability, which affect the residual stress variations, especially the residual axial stress. The analysis determined the variations in the residual stresses when given the uncertainty of the manufacturing process parameters, boundary conditions, and material properties. The assumption was that it would eventually lead to optimizing the wheel residual stress field, and therefore improving the wheel reliability. A probabilistic analysis using ANSYS/PDS was successfully employed. This paper outlines the probabilistic analysis performed on a Griffin Wheel CJ36 freight car wheel design. It illustrates how ANSYS/PDS can be used to understand the production process uncertainties and the parameter variations in a manufacturing process.

Procedure A deterministic decoupled thermo-mechanical analysis on a baseline model was conducted first. It estimated the as-manufactured residual stress in a CJ36 fright car wheel. The analysis consisted of two parts, a nonlinear transient thermal analysis and a nonlinear static structural analysis. The thermal analysis determined the temperature distribution that varies over time. It included the seven time spans of pre-quench, quench, pre-draw, draw, pre-hub cooling, hub cooling, and air-cooling. To simulate the seven time spans, a time history of the loading was generated by using the concept of load step within ANSYS. The temperature distribution of the wheel, which was calculated during thermal analysis, was used as an input load for the structural analysis. The creep effect was included in the structural analysis to represent a stress relaxation phenomenon that occurs in a stressed material held at an elevated temperature for an extended period.

For the probabilistic analysis, the decoupled thermo-mechanical analysis was specified in an analysis file created by using ANSYS Parametric Design Language (APDL). With the analysis file, a PDS loop file was generated to perform the PDS analysis loops. The FEA model was set up parametrically. The input parameters included times, temperatures, boundary conditions, and material properties, which were thought to affect the residual stress field. These input parameters were defined as random input variables and were characterized by their distribution type and distribution parameters. Latin Hypercube sampling with 200 loops was used. For the same accuracy, Latin Hypercube sampling provides samples that ensure coverage of the full range of input parameters with far fewer simulation loops than the direct Monte Carlo method. The output parameters are the maximum residual stress in the radial, axial, and both the maximum and minimum circumferential stress.

Analysis

Mesh Generation Because of axisymmetric nature of both the geometry and load, a two-dimensional axisymmetric model of the wheel cross section was constructed by using four-node 2-D isoparametric elements, as shown in Figure 3. The model contained 813 nodes and 716 quadrilateral elements. The model was oriented in space with the wheel radius aligned with global X and the axis of the wheel aligned with global Y. The origin of the global coordinate system was at the center of the back hub face. All curves with small radii were removed to simplify the model. The same finite element mesh was used for both the thermal and mechanical analyses. However, different ANSYS element types were used: PLANE 55 for the thermal analysis and PLANE 182 for the structural analysis.

Figure 3. The FEA model

Material Properties The heat-treated steel wheels are classified into three classes by AAR based on the chemical compositions. Class A materials are those with 0.47-0.57% carbon content, Class B materials contain 0.57-0.67% carbon, and Class C materials consist of 0.67-0.77% carbon. Currently, Class C material dominates manufactured steel wheels. The non-heat treated steel wheels are classified as Class U and they have the same chemical composition as Class C. All the material properties used in the analysis were based on Class U steel. The material property data were obtained from the previous work by Kuhlman, Sehitoglu, and Gallagher (1988).

Table 1. Thermal Properties of Class U Materials Temperature, oF Specific Heat,

Btu/(lb⋅ oF)Thermal Conductivity,

Btu/(hr⋅in⋅ oF)Density, lb/in3

70 0.10929 2.3993100 0.11112 2.3788200 0.11725 2.3094300 0.12337 2.2381400 0.12949 2.1648500 0.13561 2.0897600 0.14174 2.0126700 0.14786 1.9337800 0.15398 1.8528900 0.16011 1.7701

1000 0.16623 1.68531100 0.17235 1.59871200 0.17848 1.51021300 0.18460 1.41981400 0.44273 1.32741500 0.15174 1.21351600 0.15369 1.24771700 0.15564 1.2819

0.283

An isotropic material was used for the analysis. A creep material model combined with bilinear kinematic hardening was used for the mechanical analysis. The creep strain-rate is dependent on the local values of temperature and stress and it is determined from the following equation, which was originally developed by Sehitoglu and Morrow (1983) based on AISI 1070 steel.

)1()1(5.2823E 46053712

12.542-

hre T +−•

××= σε

where

σ: von Mises stress, psi

T: temperature, oF

The thermal and mechanical properties of Class U materials are listed in Table 1 and Table 2, respectively. The values of specific heat, thermal conductivity, modulus of elasticity, tangent modulus, and yield strength are shown as a function of temperature. Except for density and Poisson’s ratio, the values given in the tables were also used as the mean values in the probabilistic analysis.

Table 2. Mechanical Properties of Class U Materials Temperature, oF 70 400 600 800 1300 1600 Modulus of Elasticity, x106 psi 29.181 28.899 28.139 26.568 17.115 6.150 Yield Strength, ksi 57.990 56.505 53.841 47.122 15 9.813 Tangent Modulus, x106 psi 2.187 3.273 3.089 2.526 0.722 0.2092 Poisson’s Ratio 0.3 Coefficient of Thermal Expansion, in/(in⋅oF)

9.44 x 10-6

Boundary Conditions The nonlinear transient thermal analysis was broken down into seven steps, as shown in Table 3. At time zero the model was initialized to a temperature of 1650 oF. Three constant film coefficients were used, for the tread, hub, and other surfaces. The first coefficient, from wheel to water, was applied along the quenched surface for the quenching step only. The second one, from wheel to air, was applied on all unquenched surfaces. The last coefficient was applied on the wheel bore surface for the hub cooling step, which is assumed to equal one fourth of the one applied on the tread quenching surface. The boundary conditions of each load step are listed in Table 3. The boundary conditions of the quenching and hub cooling steps are graphically shown in Figure 4 and Figure 5 respectively. Again, the film coefficients and the furnace temperature listed in Table 3 as well as the initial temperature were also used as the mean values in the probabilistic analysis.

For the structural analysis, the transient temperature field, which was determined during thermal analysis, was used as the time-dependent input loading. The transient temperatures at the element nodes were applied as body loads. At the front left corner of the wheel hub, a single constraint in Y direction was applied. This boundary condition restrains the model from rigid body motion.

Figure 4. The boundary conditions of the quenching step

Figure 5. The boundary conditions of the hub cooling steps

Table 3. Boundary Conditions of Thermal Analysis Film Coefficient, Btu/(hr⋅in2⋅oF) Bulk Temperature, oF Step Process Duration Tread Hub Other Surfaces

1 pre-quench 2 min 3.47 x 10-2 3.47 x 10-2 3.47 x 10-2 70 2 quench 4 min 2.16 3.47 x 10-2 3.47 x 10-2 70 3 pre-draw 2 min 3.47 x 10-2 3.47 x 10-2 3.47 x 10-2 70 4 draw 2 hr 3.47 x 10-2 3.47 x 10-2 3.47 x 10-2 950 5 pre-hub-cooling 40 sec 3.47 x 10-2 3.47 x 10-2 3.47 x 10-2 70 6 hub-cooling 3.4 min 3.47 x 10-2 0.54 3.47 x 10-2 70 7 cooling 10 hr 3.47 x 10-2 3.47 x 10-2 3.47 x 10-2 70

The Input Parameters The input parameters consisted of 14 random input variables that could cause variation in the residual stress. A list of these random input variables, the distribution they were subjected to, and their distribution parameters are provided in Table 4. As mentioned in the Material Properties section, specific heat, thermal conductivity, modulus of elasticity, yield strength, and tangent modulus are temperature-dependent material properties. The following additive equation was used to describe the randomness by separating the temperature dependency from scatter effect.

)2()()1()( TMTM rand δ+=

where

M (T): the mean value of material properties as a function of temperature

δ: a random variable describing the scatter of material properties

Table 4. Input Parameters Input Parameters Description Distribution Type and Parameters

TIME_Q, hr quenching time Uniform, Xmin=6.667 x10-2, Xmax = 6.667 x10-3

TEMP_FUR, oF draw furnace temperature Lognormal, µ = 950, σ= 20 TEMP_INI, oF initial temperature Truncated Normal, Xmin =1450,

Xmlv=1650, Xmax =1700 CFC_AIR, Btu/(hr⋅in2) film coefficient, steel to air Lognormal , µ = 3.47 x10-2,

σ= 1.5 x10-3 CFC_WAT, Btu/(hr⋅in2) film coefficient, steel to water Lognormal, µ = 2.16, σ= 0.1 RAND_C the random variable for specific heat Normal, µ = 0, σ= 0.05 RAND_K the random variable for thermal

conductivity Normal, µ = 0, σ= 0.05

RAND_E the random variable for modulus of elasticity

Normal, µ = 0, σ= 0.05

RAND_Y the random variable for yield strength Normal, µ = 0, σ= 0.05 RAND_TE the random variable for tangent Normal, µ = 0, σ= 0.05 ALPHA, in/(in⋅oF) coefficient of thermal expansion Lognormal, µ = 9.44 x10-6,

σ= 4 x10-7 C1 the random variable for the constant of

creep strain rate equation Lognormal, µ = 1, σ= 0.05

C2 the stress exponent of creep strain rate equation

Lognormal, µ = 12.5, σ= 0.3125

C3 the activation energy related constant of creep strain rate equation

Lognormal, µ = 53712, σ= 1500

To evaluate the effect of creep material property variation on the residual stress, equation (1) is written in a parametric format as follows:

)3(4603

21

+•

××= TC

C eAC σε

where

C1: a random variable describing the variation of the constant A, and A=5.2823x10-42

C2: the stress exponent

C3: the activation energy related constant

Analysis Results & Discussion The distribution of the residual axial stress for the baseline model is illustrated in Figure 6. It shows that the maximum axial tensile stress reaches 14.28 ksi, which occurs at the upper middle rim. Figure 7 shows the residual circumferential stress for the baseline model. The minimum compressive stress on the tread surface is in the order of –26.17 ksi.

Figure 6. The residual stress distribution in the axial direction

A summary of the statistical characteristics of output parameters is provided in Table 5, with a confidence level of 0.95. It demonstrates that all stress components vary in range widely. The scatter of the input parameters causes larger variability of the outputs. Compared with all input parameters with less than 5% of the coefficients of variation, the coefficients of variation of the output stresses are at least 21%. The axial stress shows the largest variation among all stresses. The maximum residual stress in the axial direction has a mean of 13.65 ksi and a standard deviation of 4.27 ksi, which results in a coefficient of variation of 31%. The scatter of the maximum axial stress is illustrated in a histogram shown in Figure 8. The large variations of the output stresses can help explain why few wheels fail but most wheels do not.

Figure 7. The residual stress distribution in the circumferential direction

Figure 8. The histogram of the maximum residual axial stress

Table 5. The Statistical Characteristics of Output Parameters Parameters Mean Value, ksi Standard Deviation, ksi Coefficient of Variation

Max Radial Stress 23.63 4.92 21%

Max Axial Stress 13.65 4.27 31%

Max Circumferential Stress 22.52 4.81 21%

Min Circumferential Stress -27.14 7.28 27%

As mentioned in the Introduction section, the maximum axial stress may contribute to the Vertical Split Rim, a wheel failure mode. The cumulative distribution curve for the max axial stress is shown in Figure 9, with a confidence level of 0.95. The max axial stress for a given probability can be obtained from Figure 9, and it is summarized in Table 6. There is a 0.5% probability that the maximum axial stress is greater than 21.23 ksi, which is significantly larger than that of the baseline model. The maximum axial stress for the baseline model is 14.28 ksi.

Figure 9. The cumulative distribution curve of the maximum residual axial stress

The minimum circumferential stress, which is crucial in preventing fatigue crack initiation, is also provided in Table 6. When brake shoes are applied to the wheel tread, the tread surface temperature increases as a result of friction. Severe thermal input into a wheel occurs when a loaded train travels on a downhill slop for an extended period. In addition, the failure of the brake mechanism may keep the brake locked on the tread of the wheels. On these occasions, the wheel rim is heated to a high temperature for an extended period. As a result, the beneficial as-manufactured compressive residual stress could be altered to the tensile stress.

As shown in Table 6, there is a 0.5% probability that the absolute magnitude of the circumferential compressive stress may drop bellow 14.71 ksi, which is much less than the result from the baseline model. The minimum circumferential stress for the baseline model is –26.17 ksi. Consequently, the upper rim stress state can be reversed from compression to tension at even lower temperatures because of low compressive circumferential stress.

Table 6. The Maximum Axial Stress for a Given Probability Probability, % 0.5 5 10

Max Axial Stress, ksi >21.23 >19.53 >18.85

Min Circumferential Stress, ksi >-14.71 >-16.32 >-18.73

The sensitivity analysis indicates that all stress components are highly sensitive to the creep material property. It is derived by the Spearman rank-order correlation coefficients that are provided in Table 7, with a significance level of 5%. The critical value of the correlation coefficient is 0.138 for the sample size of 200 with a significance level of 5%, which can be found in the critical correlation coefficient table in a statistics book such as Wackerly, Mendenhall, and Scheaffer (1995). Accordingly, all max stress components show a strong negative correlation with the stress exponent and moderate positive correlation with the activation energy related constant. In addition, all stresses show minor sensitivity to tangent modulus. Except for the maximum circumferential stress, all other stress components have a weak correlation with thermal conductivity. The draw furnace temperature has a weak influence on all stress components except for the radial stress. The radial stress also has a weak correlation with modulus of elasticity and the film coefficient from wheel to water. The absolute magnitudes of all those correlation coefficients are below 0.212. No significant correlation with other input parameters was found. Figure 10 shows the sensitivity plot of the maximum axial stress. Clearly, the stress exponent (C2) is the dominant input variable. The strong correlation between the axial stress and the stress exponent of creep strain-rate equation can also be illustrated by scatter plots, which are shown in Figure 11.

Figure 10. The sensitivities of the maximum residual axial stress

Figure 11. The scatter plot of the maximum axial stress vs. the stress exponent

Table 7. The Correlation Coefficients between the Stress Components and Input Parameters C2, the Stress Exponent of

Creep Strain Rate Equation C3, the Activation Energy Related Constant of the Creep Strain Rate Equation

Max Radial Stress -0.823 0.374

Max Axial Stress -0.923 0.330

Max Circumferential Stress -0.925 0.310

Min Circumferential Stress 0.912 -0.310

Conclusions 1. The small variations in input parameters of the heat treatment process, including the process

parameters, boundary conditions, and material properties can cause a larger variation of the residual stress field in the railroad wheels. With the coefficient of variation being less than 5% of the 14 input parameters, it reaches at least 21% of the coefficient of variation for output stresses.

2. The maximum residual stress in the axial direction has a mean of 13.65 ksi and a standard deviation of 4.27 ksi, with a confidence level of 0.95. Its coefficient of variation is 31%, which is the largest variation among all stress components.

3. There is a 0.5% probability that the maximum axial stress may exceed 21.23 ksi, which is significantly greater than 14.28 ksi for the baseline model.

4. There is a 0.5% probability that the absolute magnitudes of circumferential compressive stress may drop below 14.71 ksi, which is significantly less than 26.17 ksi for the baseline model. Therefore, even moderate but not severe thermal input due to brake heating could reverse the residual stress state of the wheel’s upper rim from compression to tension.

5. The creep material property is the dominant variable in the residual stress field of the railroad wheel. All stress components have a strong correlation to the stress exponent of the creep strain-rate equation. The absolute magnitudes of all correlation coefficients are at least 0.823, with a sample size of 200 and a significance level of 5%.

6. The correlation coefficient between the maximum axial stress and the stress exponent of the creep strain-rate equation is -0.923.

Recommendation The creep strain rate in equation (1) for the wheel heat treatment simulation, which was developed by Sehitoglu and Morrow (1983), is widely used in the railroad industry. The samples used for testing purposes were made from AISI 1070 steel but not from actual wheel steels. The constants in the steady state creep equation were determined based on only one set of testing data. The testing temperature was in the range of 572-842 oF. The stress amplitude was differentially changed in the range of 29-51ksi. The creep material property tested on the wheel samples in a wider range of temperature and stress is not currently available. The residual stress field of the railroad wheel is most sensitive to the creep material, which is the primary finding of this paper. Therefore, a statistically based creep material property test based on samples from both non heat treated wheel (Class U steel) and as-manufactured wheel (Class C steel) in a wider temperature and stress range is recommended.

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