Flange Climb Derailment Criteria and Wheel/Rail Profile Management and Maintenance Guidelines for Transit Operations (2005)

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C-1 APPENDIX C: Investigation of Wheel Flange Climb Derailment Criteria for Transit Vehicles (Phase II Report)

C-3 SUMMARY This research investigated wheel flange climb derailment to develop a general flange-climb-distance criterion for transit vehicles in Phase II of the project. The inves- tigations used computer simulations of single wheelsets and representative transit vehi- cles. The Phase I work investigated the relationships between flange angle, flange length, axle AOA, and distance to climb. Based on these simulations flange-climb- distance equations were developed for some specific wheel profiles. Based on single wheelset simulation results, Phase II proposed a general flange- climb-distance criterion for transit vehicle wheelsets. The general flange-climb- distance criterion was validated by the flange-climb-distance equations in the Phase I report for each of the wheel profiles with different flange parameters. Phase II also proposed a biparameter flange-climb-distance criterion for vehicles with an AAR-1B wheel/136-pound rail profile combination. The bilinear characteristics between the transformed climb distance and the two parameters, AOA and lateral-over-vertical (L/V) ratio, were obtained through a nonlinear transformation. The accuracy of the fitting formula was further improved by using a gradual linearization methodology. The bipa- rameter distance criterion based on the simulation results was validated by comparison with the research team’s TLV test data. The application to two AAR Chapter XI performance acceptance tests and limitations of the biparameter distance criterion are also presented. The following conclusions were drawn from the Phase II work: • A general flange-climb-distance criterion taking the AOA, the maximum flange angle, and flange length as parameters is proposed for transit vehicles: D A B Len B Len < + * * *AOA INVESTIGATION OF WHEEL FLANGE CLIMB DERAILMENT CRITERIA FOR TRANSIT VEHICLES (PHASE II REPORT)

C-4 where AOA is in mrad and A and B are coefficients that are functions of the max- imum flange angle Ang (degrees) and flange length Len (in.): • The general flange-climb-distance criterion is validated by the flange-climb- distance equations in the Phase I report (Appendix B) for each of the wheel pro- files with different flange parameters. • Application of the general flange-climb-distance criterion to a test of a passenger car with an H-frame truck undergoing Chapter XI tests shows that the criterion is less conservative than the Chapter XI and the 50-msec criteria. • A biparameter flange-climb-distance criterion, which takes the AOA and the L/V ratio as parameters, was proposed for vehicles with AAR-1B wheel/136-pound rail profile: where AOA is in mrad. • A study of the flange-climb-distance criterion, which takes the friction coefficient as another parameter besides the L/V ratio and the AOA, is recommended for future work. • The biparameter distance criterion is validated by comparison with TLV test data. Since the running speed of the TLV test was only 0.25 mph, its validation for the biparameter distance criterion is limited. A trial test for validation is recommended. • Application of the biparameter distance criterion to a test of a passenger car with an H-frame truck undergoing Chapter XI tests shows that the biparameter distance crite- rion is less conservative than the Chapter XI criteria, including the 50-msec criterion. • Application of the biparameter distance criterion to an empty tank car derailment test results show that the biparameter distance criterion can be used as a criterion for the safety evaluation of wheel flange climb derailment. Application limitations of the biparameter distance criterion include the following: • The L/V ratio in the biparameter distance criterion must be higher than the L/V limit ratio corresponding to the AOA. No flange climb can occur if the L/V ratio is lower than the limit ratio. • The biparameter distance criterion is obtained by fitting in the bilinear data range where AOA is larger than 5 mrad. It is conservative at AOA less than 5 mrad due to the nonlinear characteristic. • The biparameter distance criterion was derived based on the simulation results for the AAR-1B wheel on AREMA 136-pound rail. It is only valid for vehicles with this combination of wheel and rail profiles. • For each of the different wheel profiles listed in Table B-2 of the Phase I report, individual biparameter flange-climb-distance criteria need to be derived based on the simulation results for each wheel and rail profile combination. D 0.001411 * AOA (0.0118 * AOA 0.1155) * L/V - 0.0671< + + 1 B Len Ang Len = − + +  + − − 10 21 157 2 1052 0 05 10 0 2688 0 0266 5 . . . * . . Len Ang Ang − − + − + 1 0 0092 1 2152 39 031 1 232 . ( ) . . . A Ang = − + +     100 1 9128 146 56 3 1 . . . *

C-5 CHAPTER 1 A GENERAL FLANGE-CLIMB-DISTANCE CRITERION The research team investigated wheel flange climb derail- ment to develop a general flange-climb-distance criterion for transit vehicles in Phase II of the project. The investigations used computer simulations of single wheelsets and represen- tative transit vehicles. The Phase I work investigated the rela- tionships between flange angle, flange length, axle AOA, and distance to climb. Based on these simulations, flange-climb- distance equations were developed for some specific wheel profiles. Based on single wheelset simulation results, Phase II pro- posed a general flange-climb-distance criterion for transit vehicle wheelsets. The general flange-climb-distance crite- rion was validated by the flange-climb-distance equations in Appendix B, the Phase I report, for each of the wheel profiles with different flange parameters. Flange-climb-distance criteria were developed for each of the rail/wheel profiles, as published in Table B-2 of Appen- dix B, the Phase I report. Since the wheel and rail profiles vary widely within transit systems, it was desirable to develop a general flange-climb-distance criterion with the maximum flange angle and flange length as parameters for different wheel profiles. The effects of the maximum flange angle and flange length on climb distance were further analyzed through single wheelset simulations by using 16 wheel profiles with differ- ent maximum flange angle and flange length combinations, as listed in Table C-1. The wheel maximum flange angle and flange length were deliberately varied using AutoCAD. The flange root and flange tip were kept the same shape with no restrictions on flange height and thickness. As shown in Figure C-1, the flange length is defined as the sum of the maximum flange angle length and the flange tip arc length from the maximum flange angle to 26.6 degrees. Figure C-2 shows the simulation results of these 16 wheel profiles on 115-pound AREMA rail profiles experiencing lat- eral and vertical forces, which produce an applied L/V ratio of about 1.99. Results are similar to the test (1) and simula- tion results in the Phase I report and show that the flange- climb distance decreases with increasing AOA. Results also show that the relationship between climb distance D and AOA is nonlinear, with climb distances converging asymp- totically to similar values for large AOA. To develop a general flange-climb-distance criterion with multiple parameters, a methodology was adopted in which the nonlinear relationship between the climb distance and parameters was linearized. This was achieved by using the Length of Maximum Flange Angle FaceL0 (in.) Maximum Flange Angle Ang (degrees) 0.252 0.352 0.452 0.552 63 degrees degrees degrees degrees W1 W2 W3 W4 68 W5 W6 W7 W8 72 W 9 W10 W11 W12 75 W13 W14 W15 W16 TABLE C-1 Wheel Profiles Designed by AutoCAD 75° Flange Length Len Maximum Flange Angle Length L0 Figure C-1. Definition of the flange length and maximum flange angle length. Figure C-2. Effect of AOA on flange-climb distance for different wheel profiles. 0 5 10 15 20 25 30 35 0 2 4 6 8 10 12 AOA (mrad) Cl im b D is ta n ce (f ee t) W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12 W13 W14 W15 W16

following nonlinear transformation function to transform the AOA and distance (D) in Figure C-3 to (x, y) as shown in Figure C-4 for wheelset W1: (C-1) The transformed simulation results in Figure C-4 were then fit with high accuracy (R2 of 0.998) in linear form, shown as “fit” in Figure C-4. The linear fit was then trans- formed back and plotted in Figure C-3. It is clearly shown in Figure C-4 that the relationship between 1/D and AOA is linear after the nonlinear transfor- mation of Equation C-1. The linear fitting result can be writ- ten in the following form: (C-2) The coefficients a and b for the W1 profile are shown in Figure C-4; i.e., a = 0.0427 and b = 0.3859. The correspond- y x= +a b X AOA Y 1/D = ={ C-6 ing nonlinear fitting result shown in Figure C-3 can be writ- ten in the following form: (C-3) where the two coefficients m and n can be calculated as: The highly accurate fitting Equation C-3 is obtained, as shown in Figure C-3, due to the benefit of the linear rela- tionship through the transformation. By using this methodology, 16 formulas were obtained through high accuracy fitting (R2 > 0.97) based on the simu- lation data at an L/V ratio of 1.99 for each of these wheel pro- files listed in Table C-1. Correlation analysis between the two coefficients m and n and the maximum flange angle and flange length were conducted to generate a general function expression. The coefficient n is decomposed as: where Len is defined as the flange length (in.) from the maximum flange angle Ang to 26.6 degrees as shown in Figure C-1, and B is a coefficient. Correlation analysis shows that the relation between the coefficient B and the maximum flange angle parameter Ang is roughly linear, as shown in Figure C-5. Based on the relationship shown in Figure C-5, coefficient B can be expressed in a linear form: B KB Ang CB= +* n B Len= * m a n b a = = 1 , D m AOA n = + Figure C-4. Linear relation between 1/D and AOA, W1 profile, 1.99 L/V ratio. y = 0.0427x + 0.3859 R2 = 0.998 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2 4 6 8 10 12 AOA (mrad) 1/ D (1/ fee t) Simulation fit Figure C-3. Effect of AOA on climb distance, W1 profile, 1.99 L/V ratio. 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 AOA (mrad) Cl im b Di st an ce (f ee t) Simulation fit Figure C-5. Effect of maximum flange angle on coefficient B for different wheel profiles, maximum flange angle length L0 in Table 1. 0 5 10 15 20 25 30 62 64 66 68 70 72 74 76 Maximum Flange Angle (Degree) Co ef fic ie n t B L0=0.252 L0=0.352 L0=0.452 L0=0.552

Coefficient KB and CB are obtained through linear fitting of the lines in Figure C-5. As shown in Figure C-6, the rela- tionship between KB and the flange length Len is nonlinear. To get a highly accurate fitting result, the linearization methodology is applied again at this step. First, the nonlinear relationship must be transformed to a linear one. However, no general method was found to construct a transformation func- tion; therefore, a trial and error method was used in this report. This resulted in the following transformation function: (C-4) A linear relationship was generated by using this nonlin- ear transformation function to transform the (Len, KB) in Figure C-6 to X, Y. See Figure C-7. The same methodology is applied to the coefficient CB to obtain a linear expression between CB and the flange length parameter Len. X Y 10/(KB 0.05) = = −{ Len C-7 Coefficient m was decomposed as: where B is another coefficient. Correlation analysis shows that the relation between the coefficient A and the flange angle parameter Len is roughly linear, as shown in Figure C-8. The linearization methodology is used to obtain an expres- sion between the coefficient A and flange parameters Ang, Len. Based on the above analysis of the coefficients, a general flange-climb-distance formula with the following AOA and flange parameters is proposed: (C-5) where A and B are coefficients that are functions of maxi- mum flange angle Ang (degrees) and flange length Len (in.): The corresponding general flange-climb-distance criterion is proposed as: D A B Len AOA B Len < + * * * Ang Len + − − 10 0 2688 0 0266 5 . . B Len = − + + 1021 157 2 1052 0 05. . . * Len Ang Ang − − + − + 1 0 0092 1 2152 39 031 1 232 . ( ) . . . A Ang = − + +     100 1 9128 146 56 3 1 . . . * D A B Len AOA B Len = + * * * m A B Len= * * Figure C-6. Nonlinear relationship between KB and the flange length. -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Flange Length (inch) Co e ffi c ie n t K B Simulation y = -21.157x + 2.1052 R2 = 1 -14 -12 -10 -8 -6 -4 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Flange Length (inch) Co ef fic ie n t 1 0/ (K B -0 .0 5) Simulation Linear (Simulation) Figure C-7. Linear relationship between the transformed KB and flange length. Figure C-8. Effect of flange length on coefficient A for different wheel profiles. 0 2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Flange Length (inch) Co ef fic ie n t A Ang=63Deg Ang=68Deg Ang=72Deg Ang=75Deg

The general criterion was derived from simulation results of the 16-wheel profiles listed in Table C-1. The general equations presented above are considered to be conservative and adequate for use for wheel profiles with flange angles in the normal range of 60 to 75 degrees. Table C-2 lists a range of limiting flange-climb-distance values computed using Equation C-5 for a specified range of flange angles, flange lengths, and AOAs. Table C-2 indicates C-8 that at lower AOA of 5 mrad, the limiting flange-climb dis- tance increases as the wheel flange angle and flange length do. At higher AOA of 10 mrad, flange length has more effect on the distance limit than flange angle. In summary, considering that flange climb generally occurs at a higher AOA, increasing wheel flange angle can increase the wheel L/V ratio limit required for flange climb, and increasing flange length can increase the limiting flange climb distance. TABLE C-2 Limiting flange-climb distance computed using Equation C-5 AOA = 5 mrad AOA = 10 mrad Flane Angle deg 63 deg 2.0 2.4 3.2 2.2 2.6 3.5 2.4 2.9 3.7 2.3 2.8 4.3 68 deg 72 deg 75 deg 63 deg 68 deg 72 deg 75 deg Flange Length (inch) 0.4 inch 1.5 1.5 1.5 1.9 0.52 inch 1.8 1.8 1.8 2.1 0.75 inch 2.3 2.3 2.2 2.4

C-9 CHAPTER 2 VALIDATION OF THE GENERAL FLANGE-CLIMB-DISTANCE CRITERION Six different wheel profiles from several transit systems were analyzed in the Phase I report. Using the maximum flange angle and flange length from these wheels, the rela- tionship between climb distance and AOA was derived from the general Equation C-5 and plotted in Figure C-9. The cor- responding climb distance formulas for each wheel profile are shown in the figure. In the Phase I report, flange climb formulas were devel- oped for these same wheels based on an AOAe for various degrees of curvature. Results were shown in Figure B-32 of the Phase I report and are repeated here in Figure C-10. The shapes of the curves are very similar in nature, with climb distances converging asymptotically to similar values at high AOAs and increasingly sharp curves. The AOAe for transit and passenger vehicles in curves was derived from the curve radius, based on an assumption that these vehicles do not have significant wheelset misalignments within their trucks and do not have significant wheelset steering angles. Equation C-5 is derived based on the simulation results when the wheelset was experiencing a 1.99 L/V ratio. As shown in Chapter 7 of this appendix, the average 1.99 L/V ratio (not the peak value) lasting for more than 1 foot is rare according to practical test results. Compared with the mea- sured L/V ratio in practice, the L/V ratio of 1.99 is consid- ered to be conservative enough for transit cars. The general flange-climb-distance criterion is recom- mended for use with transit and commuter cars. It is conser- vative at a lower L/V ratio (< 1.99) and less conservative when the L/V ratio is close to 1.99. Figure C-9. Climb distance generated from Equation C-5 for different wheel profiles. 0 5 10 15 20 25 0 2 4 6 8 10 AOA (mrad) Cl im b D is ta n ce (fe e t) Wheel 1 Wheel 2 Wheel 3 Wheel 4/5 Wheel 6 Wheel 1: D=39.17944/(AOA+6.941152) Wheel 2: D=29.8701/(AOA+8.365168) Wheel 3: D=38.0625/(AOA+8.713917) Wheel 4/5: D=22.18793/(AOA+1.71472) Wheel 6: D=26.32589/(AOA+1.20198) Figure C-10. Climb distance for different wheel profiles. 0 5 10 15 20 25 0 2 4 6 8 10 Curve Curvature (degrees) Cl im b Di st an ce (fe et ) Wheel 1 Wheel 2 Wheel 3 Wheel 4/5 Wheel 6 Freight Car[3] Wheel 1: D=5/(0.13*AOAe+1) Wheel 2: D=4.1/(0.16*AOAe+1) Wheel 3: D=4.2/(0.136*AOAe+1) Wheel 4/5: D=28/(2*AOAe+1.5) Wheel 6: D=49/(2*AOAe+2 2)

CHAPTER 3 A BIPARAMETER DISTANCE CRITERION FOR FLANGE CLIMB DERAILMENT The flange-climb-distance criterion proposed in the research team’s previous research work (2, 3) for freight cars was based on single-wheelset simulations at a 2.7 L/V ratio for a range of different AOAs. An L/V ratio of 2.7 was considered conserva- tive for freight cars. The general flange-climb-distance criterion in Chapter 2 of this appendix was derived from simulation results at a fixed L/V ratio of 1.99 for different AOAs, which was considered conservative for transit cars. Both criteria were conservative at low L/V ratios, but not conservative enough at L/V ratios higher than the fixed L/V ratio used in the simula- tions, although the chance of encountering sustained L/V ratios this high is rare in practice. To avoid this dilemma, it is desir- able to include the L/V ratio as a variable parameter in the flange-climb-distance criterion. Results from testing (1) and simulations in the Phase I report show the flange-climb distance decreases with increas- ing L/V ratio. No flange climb happens (the climb distance is infinite) if the L/V ratio is lower than Nadal’s value. Since the L/V ratio is another important factor affecting flange climb besides the AOA, a criterion including the L/V ratio and AOA is expected to reveal more about the physical nature of flange climb and produce more accurate results, although the multi- variables fit is more complicated than that of a single variable. 3.1 THE BILINEAR CHARACTERISTIC BETWEEN 1/D AND THE PARAMETER’S AOA AND L/V RATIO In the following section, a combination of AAR-1B wheels and AREMA 136-pound rail profiles were used in simulations to develop a multivariable fit formula. Figure C-11 shows the simulation results of a single wheelset climbing at different L/V ratios and AOAs. Figure C-11 shows that the relationship between the climb distance D and the L/V ratio is nonlinear. Through a nonlin- ear transformation similar to that described in Chapter 1, a linear relationship between 1/D and the L/V ratio was devel- oped (Figure C-12). Due to the effect of AOA on the creep force, the wheel L/V ratios shown in Figures C-11 and 12 were not the same value for different AOAs even though the same group of lateral and vertical forces was applied to the wheelset. For example, when a 21,700-pound lateral force and 6,000-pound vertical force were applied to the wheelset at different AOAs, the C-10 wheel L/V ratios varied with the AOA as shown in the fol- lowing tabulations: AOA(mrad) L/V Ratio (Average value during climb) 0 2.87 2.5 2.82 5 2.78 10 2.73 20 2.61 Figure C-11. Effect of L/V ratio at different AOA, 75-degree, AAR-1B wheel, 136-pound rail. Figure C-12. The linear relationship between 1/D and L/V ratio, AAR-1B wheel, 136-pound rail. 0 2 4 6 8 10 12 14 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 Wheel L/V Ratio Cl im b D is ta n c e D (f e e t) AOA=0mrad AOA=2.5mrad AOA=5mrad AOA=10mrad AOA=20mrad 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 Wheel L/V Ratio 1/ D (1/ fe et ) AOA=0mrad AOA=2.5mrad AOA=5mrad AOA=10mrad AOA=20mrad

The average L/V ratio for a wheelset being subjected to the same group of lateral and vertical forces at different AOAs, L/Va, for this example is calculated as follows: L/Va = (2.87+2.82+2.78+2.73+2.61)/5 = 2.76 The L/Va ratio was used to further describe the relationship between climb distance and AOA for different L/V ratios. The relationship between the climb distance D and the AOA is nonlinear, as shown in Figure C-13. Again, a similar nonlinear transformation was performed, as described in Chapter 1, with results shown in Figure C-14. The figure shows that there is an approximately linear relationship between 1/D and the AOA higher than 5 mrad. However, it can be seen that the relationship between 1/D and the AOA lower than 5 mrad is nonlinear. 3.2 THE BIPARAMETER CLIMB DISTANCE FORMULA AND CRITERION Due to the bilinear characteristics between the function of 1/D and the two variables shown in Figures C-13 and C-14, a gradual linearization methodology including two steps described below was developed to obtain an accurate fitting formula. First, the least squares fitting method for two vari- ables was used to fit the simulation result. Since the relation- ship between the function of 1/D and the L/V ratio is linear for all L/V ratios in the simulations (shown in Figure C-12), the fitting data range for the L/V ratio is the whole data range. But the fitting data range for AOAs is from 5 mrad to 20 mrad to cut off the nonlinear relationship at lower AOAs (< 5 mrad), as shown in Figure C-14. The fitting formula is thus conserv- ative for those AOAs less than 5 mrad, which have a steeper slope than that of the fitting range (5 mrad<AOA<20 mrad). The resulting two-parameter fitting equation in the first step is as follows: 1/D = a1*L/V + a2*AOA + a3 (C-6) C-11 The fitting accuracy of Equation C-6 may not be satisfac- tory depending on the simulation model, wheel/rail profile, and the data fitting range. To improve the fitting accuracy, a refinement through further linearization corresponding to the gradual linearization methodology is used in the second step. The wheelset AOA was kept constant by constraining the axle yaw motions in the simulation. However, the L/V ratio varied during flange climb. The average L/V ratio during flange climb was used in the fitting process. Therefore, in Equation C-6, the coefficient a1 is less accurate than a2 due to the variation of the L/V ratio. Further transformation is performed as follows: Y = 1/D − a2*AOA (C-7) The simulation results were collected as different groups, according to the AOA. For the same AOA simulation group, an accurate fitting equation (R2>0.99) was obtained in the following linear form: Y = b1*L/V + b2 (C-8) The correlation analysis between the coefficient b1, b2, and the AOA for different groups shows that the coefficients b1 and b2 are linear functions of the AOA (R2>0.999): b1 = Kb1*AOA + Cb1 (C-9) b2 = Kb2*AOA + Cb2 (C-10) Substituting Equations C-8 to C-10 to Equation C-7, the resulting fitting formula is as follows: (C-11) Correspondingly, the biparameter flange-climb-distance cri- terion, which takes the AOA and the L/V ratio as parameters, D [0.001411 * AOA (0.0118 * AOA 0.1155) * L/V 0.0671] = + + − 1 Figure C-13. Effect of AOA at different L/V ratios, AAR-1B wheel, 136-pound rail. Figure C-14. Linear relation between 1/D and AOA, AAR-1B, 136-pound rail. 0 2 4 6 8 10 12 14 0 5 10 15 20 25 AOA (mrad) Cl im b Di st an ce D (f ee t) L/Va=1.54 L/Va=1.67 L/Va=1.86 L/Va=1.97 L/Va=2.76 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 5 10 15 20 25 AOA (mrad) 1/ D (1 /fe et ) L/Va=1.54 L/Va=1.67 L/Va=1.86 L/Va=1.97 L/Va=2.76

was proposed for vehicles with AAR-1B wheel/136-pound rail profile: where AOA is in mrad. Table C-3 shows the comparison of fitting errors between Equation C-6 and Equation C-11. The fitting accuracy was greatly improved through the “gradual linearization” methodology. The fitting error in Table C-3 is defined as: Based on the above derivation process, some application limitations of the biparameter distance criterion are as follows: • The L/V ratio in the criterion must be higher than the L/V limit ratio corresponding to the AOA, because no flange climb can occur if the L/V ratio is lower than the limit ratio. • The biparameter distance criterion is obtained by fitting in the bilinear data range where AOA is larger than Fitting Error Formula Value Simulation Value Simulation Value = − D [0.001411 * AOA (0.0118 * AOA 0.1155) * L/V 0.0671] < + + − 1 C-12 5 mrad. It is conservative at AOA less than 5 mrad due to the nonlinear characteristic. • The biparameter distance criterion was derived based on the simulation results for the AAR-1B wheel on 136-pound rail. It is only valid for vehicles with this combination of wheel and rail profiles. • For each of the different wheel profiles listed in Table B-2 of the Phase I report, individual biparame- ter flange-climb-distance criteria need to be derived based on the simulation results for each wheel and rail profile combination. Cases L/VRatio AOA (mrad) Fitting Error of Equation C-6 (%) Gradual Linearization Fitting Error (Equation C-11) (%) 1.58 1 1.69 1.87 1.98 1.67 1.83 1.94 1.63 1.79 5 5 5 10 10 10 20 20 20.70 1.68 −8.12 16.91 1.82 −6.64 19.31 4.76 2 3 4 5 6 7 8 1.23 1.31 −1.24 −0.89 −1.01 0.92 −0.20 9 1.89 20 −1.38 0.97 TABLE C-3 Fitting errors of Equation C-6 and Equation C-11

C-13 CHAPTER 4 COMPARISON BETWEEN THE SIMULATION DATA AND THE BIPARAMETER FORMULA The comparison between the simulation data and Equa- tion C-11 for all L/V ratios at different AOA is shown in Figure C-15. Overall, the results are consistent, especially at AOA greater than 5 mrad. Figures C-16 through C-20 compare the simulation results with results of Equation C-11 for a range of AOA. Figures C-16 and C-17 show that Equation C-11 is conser- vative for AOA less than 5 mrad, with calculated climb distance shorter than the corresponding values from the simulations. Above 5 mrad AOA, the simulations and Equation C-11 match very closely. Figure C-16. Comparison between the simulation and equation C-11, AOA = 0 mrad. Figure C-17. Comparison between the simulation and equation C-11, AOA = 2.5 mrad. Figure C-18. Comparison between the simulation and equation C-11, AOA = 5 mrad. Figure C-15. Comparison between the simulation and equation C-11 for all L/V ratios. 0 2 4 6 8 10 12 14 0 5 10 15 20 25 AOA (mrad) Cl im b Di st an ce (f ee t) Simulation Formula 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4 Wheel L/V Ratio Cl im b Di st an ce (f ee t) Simulation Formula 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 2.5 3 3.5 4 Wheel L/V Ratio Cl im b Di st an ce (f ee t) Simulation Formula 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 Wheel L/V Ratio Cl im b Di st an ce (f ee t) Simulation Formula

C-14 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 Wheel L/V Ratio Cl im b Di st an ce (f ee t) Simulation Formula Figure C-19. Comparison between the simulation and equation C-11, AOA = 10 mrad. Figure C-20. Comparison between the simulation and equation C-11, AOA = 20 mrad. 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Wheel L/V Ratio Cl im b Di st an ce (f ee t) Simulation Formula

C-15 CHAPTER 5 VALIDATION THROUGH TLV TEST The biparameter flange-climb-distance criterion was vali- dated with flange-climb test data from the TLV test on August 25, 1997 (1). The test was conducted on new rails. Since the climb distance is sensitive to AOA, the AOA values were cal- culated from the test data by the longitudinal displacements (channel ARR and ARL) of sensors installed on the right and left side of the wheelset by using the following equation: (C-12) where AOA is in mrad and ARL and ARR are in inches. The distance between the right and left sensor was 93.5 in. Figure C-21 shows the overall comparison between the test data and Equation C-11 for all L/V ratios at different AOA. Figures C-21 through C-25 compare the TLV test data with results from Equation C-11 for several of the controlled AOAs. Results of Equation C-11 are more consistent with the test data at higher AOA than at lower AOA. The difference between the TLV test and Equation C-11, as shown in Figures C-22 through C-25, is due to two main factors: the wheel/rail friction coefficients and the running speed. Equation C-11 was derived based on simulations of a single wheelset with 0.5 friction coefficient at 5 mph running speed. The TLV test was conducted at an average 0.25 mph running speed, and the test data (1) show that the friction AOA ARR ARL= + 93 5. coefficients during test varied from 0.29 to 0.54 for the dry flange face of the new rail. To demonstrate these differences, three TLV test cases at 32 mrad AOA were simulated by using the single-wheelset flange climb model. The friction coefficients in these simula- tions were derived from the instrumented wheelset L/V ratios. Simulation results show the L/V ratio converges to Nadal’s value when AOA is larger than 10 mrad. For these runs (runs 30, 31, and 32), the L/V ratio just before the wheel climb is Figure C-21. Comparison between the TLV test and Equation C-11 for all L/V ratios. Figure C-22. Comparison between the TLV test and Equation C-11, AOA = -2.8 mrad. 0 2 4 6 8 10 12 14 16 -5 0 5 10 15 20 25 30 35 AOA (mrad) Cl im b Di st an ce (f ee t) Formula TLV test 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4 Wheel L/V Ratio Cl im b Di st an ce (f ee t) TLV Test Formula Figure C-23. Comparison between the TLV test and Equation C-11, AOA = 4.4 mrad. 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 Wheel L/V Ratio Cl im b Di st an ce (f ee t) TLV Test Formula

1.57. The instrumented wheel profile is the 75-degree AAR- 1B wheel profile. The friction coefficient between wheel and rail is thus calculated as 0.32, according to Nadal’s formula. As can be seen in Figure C-25, the simulations with 0.32 fric- tion coefficient and 0.25-mph running speed show a good agreement between the simulation results and test data. Considering the running speed in practice, it is reasonable to use a 5-mph simulation speed rather than the actual 0.25-mph test speed for developing the flange climb criteria. A trend evident in Figures C-22 through C-25 was that the climb distance in the TLV test is shorter than that of Equa- tion C-11 with the increase of AOA. Besides the effect of the lower test speed and the lower friction coefficients in the runs of TLV, the effect of friction coefficients at different AOAs must be considered. In the Phase I report, simulation results show the following: • For AOAs greater than 5 mrad, the wheel climbed quickly over the maximum flange angle face and took most of the time to climb on the flange tip. • For AOAs less than 5 mrad, the wheelset took most of the time to climb on the maximum flange angle. C-16 Corresponding to these two situations, the effects of flang- ing friction coefficients differ: • For AOA greater than 5 mrad, the climb distance decreases with a decreasing flanging friction coeffi- cient µ because the lateral creep force changes direc- tion on the flange tip to resist the derailment. If µ is smaller, then the resisting force is smaller; thus, the wheelset derails faster than that with a higher friction coefficient. • For AOA less than or equal to 5 mrad, the climb dis- tance increased with a decreasing flanging friction coef- ficient µ. The lateral creep force helped the wheel to climb on the flange face and took less time to climb on the tip. In total, it took more time to derail than that with a higher friction coefficient. The effect of friction coefficients is much more compli- cated than that of the L/V ratio and the AOA. A study of the flange-climb-distance criterion, which takes the friction coef- ficient as another parameter besides the L/V ratio and the AOA, is recommended for future work. Figure C-24. Comparison between the TLV test and Equation C-11, AOA = 11 mrad. Figure C-25. Comparison between the TLV test and Equation C-11, AOA = 32 mrad. 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 Wheel L/V Ratio Cl im b Di st an ce (f ee t) TLV test Formula 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Wheel L/V Ratio Cl im b Di st an ce (f ee t) TLV Test Formula Simulation of TLV Test

C-17 CHAPTER 6 ESTIMATION OF AOA Fixed AOA was used in the single-wheelset flange climb simulations and the TLV test in order to investigate the effect of AOA on flange climb. Both the single-wheelset flange climb simulations and the TLV test have shown that the flange-climb distance is sensitive to AOA. However, the wheelset was not kept at a constant AOA but varied during the climb, as shown in the vehicle simulations. In most practical applications, measurement of instantaneous AOA is not possible. Therefore, to evaluate flange climb potential, an equivalent AOA (AOAe) has to be estimated on the basis of available information (e.g., vehicle type, track geometry, perturbation, suspension parameters) in order to use the biparameter flange-climb-distance crite- rion in practice. In the Phase I report, three kinds of representative vehicles corresponding to the Light Rail Vehicle Model 1 (LRV1), Light Rail Model 2 (LRV2), and Heavy Rail Vehicle (HRV) were evaluated. Further simulations, including a freight car with three-piece bogies, were made for these vehicles run- ning on a 10-degree curve, with 4 in. superelevation, and with the AAR Chapter XI Dynamic Curve perturbation. Sim- ulation results were used to estimate the AOAe during wheelset flange climb. Five running speeds of 12, 19, 24, 28, and 32 mph—cor- responding to a 3- and 1.5-in. underbalance and balance (respectively) and a 1.5- and 3-in. overbalance speed—were simulated to find the worst flange climb cases with the longest climb distances. Longitudinal primary suspension stiffness of the passenger trucks can have a significant effect on axle steering and axle AOA. Therefore, for each of these vehicles, two stiffness vari- ations, which were 50 percent lower and 150 percent higher than that of the designed longitudinal primary stiffness, were used to investigate the effect of suspension parameters on flange climb. Figure C-26 shows the effect of longitudinal primary sus- pension stiffness on AOAe, which was calculated as the average AOA during the flange climb. The warp stiffness of three-piece bogies has an important influence on the AOAe. As shown in Figure C-27, for the worn AAR-1B wheel/136-pound rail profiles, the average AOA during climb decreased with increasing warp stiffness corresponding to the worn truck, new truck, and stiff H-frame truck. For the new wheel/rail profile, the wheel did not climb on the rail due to improved steering resulting from the new profile having a larger RRD on the tread than that of the worn profile. In the Phase I report, an equivalent AOAe formula for the leading axle of a two-axle truck, based on the geometric analysis of the truck geometry in a curve, was derived as Figure C-26. Effect of longitudinal suspension stiffness on AOAe. 6 8 10 12 14 16 18 0.4 0.6 0.8 1 1.2 1.4 1.6 Longitudinal Primary Suspension Stiffness Ratio A O Ae (m ra d) LRV1 LRV2 HRV 6 8 10 12 14 16 18 20 22 4.00E-01 1.00E+07 2.00E+07 3.00E+07 4.00E+07 5.00E+07 Warp Stiffness (in-lb/rad) AO Ae (m rad ) New Profile Wron Profile Figure C-27. Effect of warp stiffness on AOAe.

Equation C-6. Table C-4 lists the constant c in Equation B-6 of Phase I report (Appendix B) for these four kinds of repre- sentative vehicles (LRV1, LRV2, HRV, Three-Piece Bogie) C-18 based on the simulation result of the maximum AOAe and axle spacing distance for each of them. Due to the track perturbations and the degrading of wheelset steering capability, the practical wheelset AOA could be higher than the value calculated by Equation 2.5. The following AOAe, which were considered conservative enough according to the simulation results and test data, were recommended in Table C-5 and shown in Figure C-28. When the vehicle runs on a curve with the curvature lower than 10 degrees and not listed in Table C-5, it is recom- mended that a linear interpolation value between the segment points in Table C-5 be used in the criterion, as shown in Figure C-28. Also, it is recommended that AOA statistical data from the wayside monitoring system be used in the cri- terion to take into account the many factors affecting AOAe if such systems are available. 0 5 10 15 20 25 30 0 42 86 10 12 Curvature (degree) A O Ae (m ra d) IRW Solid Wheelset Figure C-28. Recommended conservative AOAe for practical use. Vehicle and Truck Type Straight Lines 5-Degree Curves 10 Degree Curves Above 10 Degree Curves Vehicle with Independent Rolling Wheel or Worn Three-Piece Bogies Others 10 5 10 15 20 Equation C-6 (Appendix B) +10 15 Equation C-6 (Appendix B) +5 TABLE C-5 Conservative AOAe for practical use Vehicle Type Maximum AOAe (mrad) Axle Spacing Distance (in.) Constant c LRV1 16.8 15.6 12.1 74.8 3.08 LRV2 75 2.86 HRV 82 70 70 2.04 Freight Car with Three- Piece Bogies (New Bogie) 12.7 2.5 Freight Car with Three- Piece Bogies (Worn Bogie) 20.7 4.0 TABLE C-4 Estimation of AOAe

C-19 CHAPTER 7 APPLICATION TO VEHICLE DYNAMIC PERFORMANCE ACCEPTANCE TESTS 7.1 APPLICATION TO A PASSENGER CAR TEST The general flange climb criterion (Equation C-5) and the biparameter distance criterion (Equation C-11) were applied to a passenger car with an H-frame truck undergoing dynamic performance tests at the FRA’s Transportation Technology Center, Pueblo, Colorado, on July 28, 1997. The car was running at 20 mph through a 5-degree curve with 2-in. vertical dips on the outside rail of the curve. The L/V ratios were calculated from vertical and lateral forces mea- sured from the instrumented wheelsets on the car. Table C-6 lists the five runs with L/V ratios higher than 1.0, exceeding the AAR Chapter XI flange-climb safety criterion. The rails during the tests were dry, with an estimated friction coeffi- cient of 0.6. The wheel flange angle was 75 degrees, result- ing in a corresponding Nadal value of 1.0. The climb distance and average L/V ratio (L/V ave) in Table C-6 were calculated for each run from the point where the L/V ratio exceeded 1.0. Figure C-29 compares the climb distances to the corresponding distances that are equivalent to a 50-msec time duration. As can be seen, all the climb dis- tances exceeded the 50-msec duration. However, there does not appear to be a direct correlation between test speed and climb duration. 7.1.1 Application of General Flange Climb Criterion The instrumented wheelset has the AAR-1B wheel profile with a 75.13-degree maximum flange angle and 0.62 in. flange length. By substituting these two parameters into the general flange climb criterion, the flange criterion for the AAR-1B wheel profile is as follows: The axle spacing distance for this passenger car is 102 in., 2.04 was adopted for the constant c since the vehicle and truck design is similar to the heavy rail vehicle in Table C-4. According to Equation B-6 published in the Phase I report (Appendix B), the AOAe is about 7.6 mrad for this passenger H-frame truck on a 5-degree curve. By substitut- ing the AOAe into the above criteria, the safe climb distance without derailment is 3 ft. According to Table C-5, the con- servative AOAe for a 5 degree curve should be 10 mrad. The conservative safe climb distance without derailment is 2.4 ft; however, the climb distance according to the 50-ms criterion is 1.4 ft. The wheel, which climbed a 2 ft distance in the run (rn046) with a 1.01 average L/V ratio (maximum L/V ratio 1.06), was running safely without threat of derailment according to the criterion. The other four runs were unsafe because their climb distances exceeded the criterion. 7.1.2 Application of Biparameters Distance Criterion Equation C-11 was used to calculate a climb distance criterion for each run, based on the measured L/V ratios, flange angle, and flange length from the test wheels. Because AOA was not measured during the test, the Equation C-11 D AOAe < + 26 33 1 2 . . Runs Speed L/V Maximum Average L/V Climb Distance rn023 20.39 mph 1.79 1.37 6.2 ft rn025 19.83 mph 2.00 1.43 7 ft rn045 19.27 mph 1.32 1.10 4 ft rn046 20.07 mph 1.06 1.01 2 ft rn047 21.45 mph 1.85 1.47 5.7 ft TABLE C-6 Passenger car test results: climb distance and average L/V (L/V ave) measured from the point where the L/V ratio exceeded 1.0 for friction coefficient of 0.6

calculation was made for several values of AOA. Results are compared to the 50-msec duration in Figure C-30. According to the biparameter distance criterion, the run with a 1.01 average L/V ratio (maximum L/V ratio 1.06) was acceptable even for the 20-mrad average AOA, which is an unlikely occurrence for an H-frame truck in a 5-degree curve. The run with a 1.1 average L/V ratio (maximum L/V ratio 1.32) was acceptable according to the new criterion, as shown in Figure C-30. It would be unacceptable if the AOAe was greater than 13 mrad. This result also means the bipara- meter distance criterion is less conservative than the general flange-climb-distance criterion. The other three test runs were unacceptable since they exceeded the new criterion for AOA greater than 7.6 mrad. The same conclusion can also be drawn by applying the criterion with a conservative 10-mrad AOA, according to Table C-5. As noted before, all the test runs exceed the 50-msec criterion. If a friction coefficient of 0.5 is assumed instead of 0.6, the corresponding climb distances, measured at an L/V ratio higher than Nadal’s value of 1.13, are listed in Table C-7. The run with the maximum L/V ratio 1.06 would then be acceptable because no climb was calculated when the L/V ratio was lower than Nadal’s value. The run with the maxi- C-20 mum 1.32 L/V ratio would be acceptable since the climb dis- tance was well below the 20-mrad AOAe criterion, as shown in Figure C-31. The other three runs would be considered unacceptable because their climb distances exceeded the 7.6-mrad AOAe criterion line. The same conclusion can also be drawn if the conservative AOAe (10 mrad) in Table C-5 is used. This passenger car test shows that Nadal’s value, the AAR Chapter XI criterion, and the 50-msec time-based criterion are more conservative than the new distance-based criterion for speeds of around 20 mph. This means that critical L/V values would be permitted for longer distances under the distance-based criterion at low speeds. 7.2 APPLICATION TO AN EMPTY TANK CAR FLANGE CLIMB DERAILMENT The biparameter distance criterion was applied to an empty tank car flange climb derailment that occurred during dynamic performance testing at TTCI on September 29, 1998. The car was running at 15 mph through the exit spiral of a 12-degree curve. The L/V ratios and wheel/rail contact positions on the tread, measured from the instrumented wheelsets on the car, are Figure C-29. Application of 50-msec climb-distance criterion. Figure C-30. Comparison of new criterion (Equation C-11) to the 50-msec criterion, 0.6 friction coefficient. 0 1 2 3 4 5 6 7 8 19 19.5 20 20.5 21 21.5 22 Running Speed (mph) Cl im b Di st an ce (f ee t) Measured 50 msec Criterion 0 1 2 3 4 5 6 7 8 9 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Average L/V Ratio during Climb Cl im b Di st an ce (f ee t) Measured Formula, 7.6 mrad AOA Formula, 10 mrad AOA Formula, 13 mrad AOA Formula, 20 mrad AOA 50 msec Criteria at 20mph TABLE C-7 Passenger car test results: distance measured from the L/V ratio higher than 1.13 for friction coefficient of 0.5 Runs Speed L/V Maximum Average L/V Ratio Climb Distance rn023 20.39 mph 19.83 mph 19.27 mph 21.45 mph 1.79 1.39 5.8 ft rn025 2.00 1.45 6.3 ft rn045 1.32 1.23 0.7 ft rn047 1.85 1.52 5 ft

shown in Figures C-32 through C-35. Positive contact positions indicate contact on the outside of the wheel taping line, while negative values indicate contact on the flange side of the taping line. Negative values approaching −2.0 indicate hard flange contact. This is shown for Wheel B, which derailed. C-21 The climb distance measured when the L/V ratio was greater than 1.13 (Nadal’s value for a 75-degree flange angle and a 0.5 friction coefficient) is 17.9 ft, as shown in Figure C-35. The average L/V ratio is 1.43 during the 17.9 ft climb distance. The data shown is for an instrumented wheelset that was in the leading position of the truck. The curvature of the spiral during the climb is about 9 degrees. The axle spacing distance for this tank car is 70 in. The constant c was adopted as 2.5, which represents a new bogie in Table C-4. According to Equation B-6 in the Phase I report, the AOAe is about 11 mrad for the three-piece bogie at this location in the spiral curve. According to Equation C-11, the climb distance is 3.3 ft for the 11-mrad AOAe. The corresponding 50-msec distance at 15 mph would be 1.1 ft. Since the measured climb distance exceeded the value of the biparameter distance criterion, the vehicle was running unsafely at that moment. Wheel B started climbing at 1,054.6 ft and derailed at 1,164 ft. Therefore, the actual flange-climb distance is longer than 17.9 ft. As shown in Figure C-35, the wheel climbed a longer distance on the flange tip, and the L/V ratio decreased due to the lower flange angle on the tip. The empty tank car derailment test results show that the biparameter distance criterion can be used as a criterion for the safety evaluation of wheel flange climb derailment. Figure C-31. Application of the new criterion (Equation C-11) for a friction coefficient of 0.5. Figure C-32. Contact position on tread of wheel A. 0 1 2 3 4 5 6 7 8 9 1.1 1.2 1.3 1.4 1.5 1.6 Average L/V Ratio during Climb Cl im b Di st an ce (f ee t) Measured Formula,7.6 mrad AOA Formula, 10 mrad AOA Formula, 20 mrad AOA -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 944 994 1044 1094 1144 Distance (feet) W he el A C on ta ct P os iti on (in ch es ) Figure C-35. L/V ratio of wheel B. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 944 994 1044 1094 1144 Distance (feet) W he el A L /V R at io Figure C-33. L/V ratio of wheel A. Figure C-34. Contact position on tread of wheel B. -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 944 994 1044 1094 1144 Distance (feet) W he el B C on ta ct P os iti on (in ch es ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 944 994 1044 1094 1144 Distance (feet) W he el B L /V R at io

CHAPTER 8 CONCLUSION The following findings were made: • A general flange-climb-distance criterion that uses the AOA, maximum flange angle, and flange length as pa- rameters is proposed for transit vehicles: where AOA is in mrad and A and B are coefficients that are functions of maximum flange angle Ang (degrees) and flange length Len (in.): • The general flange-climb-distance criterion is validated by the flange-climb-distance equations in the Phase I report for each of the wheel profiles with different flange parameters. • Application of the general flange-climb-distance criterion to a test of a passenger car with an H-frame truck under- going Chapter XI tests shows that the criterion is less con- servative than the Chapter XI and the 50-msec criteria. • A biparameter flange-climb-distance criterion, which uses the AOA and the L/V ratio as parameters, was pro- posed for vehicles with AAR-1B wheel and AREMA 136-pound rail profiles: where AOA is in mrad. D [0.001411 * AOA (0.0118 * AOA 0.1155) * L/V 0.0671] < + + − 1 Ang Len + − − 10 0 2688 0 0266 5 . . B Len = − + + 1021 157 2 1052 0 05. . . * Len Ang Ang − − + − + 1 0 0092 1 2152 39 031 1 232 . ( ) . . . A Ang = − + +     100 1 9128 146 56 3 1 . . . * D A B Len AOA B Len < + * * * C-22 • A study of the flange-climb-distance criterion that takes the friction coefficients as other parameters besides the L/V ratio and the AOA is recommended for future work. • The biparameter distance criterion has been validated by the TTCI TLV test data. Since the running speed of the TLV test was only 0.25 mph, one test’s validation for the biparameter distance criterion is limited. A trial test to validate the biparameter distance criterion is recommended. • Application of the biparameter distance criterion to a test of a passenger car with an H-frame truck under- going Chapter XI tests shows that the criterion is less conservative than the Chapter XI and 50-msec criteria. • Application of the biparameter distance criterion to an empty tank car derailment test results showed that the criterion can be used in the safety evaluation on the wheel flange climb derailment. Application limitations of the biparameter distance crite- rion include the following: • The L/V ratio in the biparameter distance criterion must be higher than the L/V limit ratio corresponding to the AOA, because no flange climb can occur if the L/V ratio is lower than the limit ratio. • The biparameter distance criterion is obtained by fitting in the bilinear data range where AOA is larger than 5 mrad. It is conservative at AOAs less than 5 mrad due to the nonlinear characteristic. • The biparameter distance criterion was derived based on the simulation results for the AAR-1B wheel on AREMA 136-pound rail. It is only valid for vehicles with this combination of wheel and rail profiles. • For each of the different wheel profiles listed in Table B-2 of the Phase I report, individual biparameter flange-climb-distance criteria must be derived based on the simulation results for each wheel and rail pro- file combination.

C-23 REFERENCES 1. Shust, W.C., Elkins, J., Kalay, S., and EI-Sibaie, M., “Wheel- Climb Derailment Tests Using AAR’s Track Loading Vehicle,” Report R-910, Association of American Railroads, Washington, D.C., December 1997. 2. Wu, H., and Elkins, J., “Investigation of Wheel Flange Climb Derailment Criteria,” Report R-931, Association of American Railroads, Washington, D.C., July 1999. 3. Elkins, J., and Wu, H., “New Criteria for Flange Climb Derail- ment,” Proceedings, IEEE/ASME Joint Railroad Conference, Newark, New Jersey, 2000.