CHAPTER 6
Development and Validation of Impact Loading Methodology

The objective of the work presented in this chapter was to develop methodologies that engineers can use to calculate realistic derailment impact loading for use in bridge design. The methodologies account for both vertical and horizontal loading. The research team selected the best methodologies from the ones they developed with the goal of meeting the following criteria (subsequent to each criterion are corresponding explanations):

1. The methodology should be geared towards practitioners.
   • Easy to implement
   • Uses tools that bridge designers and/or analysts already use
   • Accounts for the fact that bridge designers/analysts have less information about rail cars
2. The methodology should be sufficiently conservative to use in bridge design.
   • Accounts for variability in rail cars, bridges, structural components of interest, and derailment scenarios
3. The methodology should be accurate across a range of derailment scenarios.
   • Validated by high-fidelity simulations presented in the previous chapters
   • Based on engineering principles

To strike a balance between these three criteria, two types of methodologies were developed for vertical impact loading. The first methodology calculates an equivalent static load that can be used to represent the dynamic effects of derailment impact in static analysis models. This methodology is focused on meeting the first two criteria—it is the easiest to implement and provides a conservative approach consistent with current bridge design tools. The second methodology calculates a force–time history that can be applied to the bridge deck in dynamic analysis models. This methodology is focused on the third criterion—although it is not as easy to implement as the first methodology since it requires dynamic analysis, it has better accuracy and potentially less conservatism. Bridge designers may use the first methodology (i.e., static) when a simpler analysis method is needed and/or when more conservatism is desired. Bridge designers may use the second methodology (i.e., dynamic) when providing more conservatism is not possible (e.g., when evaluating an existing structure). A comprehensive discussion and detailed technical background on the derivations of these methods are provided in the Appendix.

6.1 Static Vertical Impact Loading Methodology

In this section, a simplified approach to obtaining the dynamic response of the bridge due to train derailment impact is proposed. The static vertical impact load is intended to be applied to one truck from an actual train—it is not intended to be applied to the LRT-16 notional live load given in the AASHTO LRFD Guide. Therefore, to calculate the static vertical impact load, the bridge designer needs to know the truck load magnitude, which would include the weight of the truck itself and the tributary weight of the carbody (a truck load is typically equal to two rail car

Long Description.

The table includes two columns: Bridge Natural Frequency in hertz and lambda subscript t subscript d. The first row shows frequency less than or equal to 1.9 hertz with a lambda subscript t subscript d value of 1.00. The second row shows 2.0 to 2.9 hertz with a value of 0.90. The third row shows 3.0 to 4.4 hertz with a value of 0.80. The fourth row shows 4.5 to 5.9 hertz with a value of 0.65. The fifth row shows greater than or equal to 6.0 hertz with a value of 0.50. The table shows that as natural frequency increases, the lambda subscript t subscript d value decreases.

Long Description.

The table includes two columns: Impact Location and lambda subscript e. For midspan, the energy restitution factor is 1.1. For quarter span, it is 1.2. For impact at pier, it is 1.3. The values increase with the proximity of impact to the pier location.

axle loads). The derailment impact equivalent static vertical load is illustrated in Figure 36. This load is defined by the following equation:

(6.1)

where P_t is the static vertical load of a single truck and DAF is the dynamic amplification factor. The following equation provides the static vertical load DAF that estimates the bridge structural response due to derailment impacts:

(6.2)

where f_n is the bridge natural frequency (Hz), H is the drop height, g is gravity, λ_td is the pulse duration reduction factor (Table 7), λ_e is the energy restitution factor (Table 8), and λ_m is the system mass reduction factor (Table 9). The equivalent static vertical load defined in Equation 6.1 is applied to the bridge at the location of impact in a static analysis model, independent of all other loads or initial conditions.

6.2 Dynamic Vertical Loading Methodology

In this section, an alternative approach to obtaining the dynamic response of the bridge due to train derailment impact is proposed. This dynamic loading approach involves applying a force–time history to a bridge model at the location of impact. This approach can be used in any structural or FEA software that supports dynamic analysis. A bridge designer may use

P_t is the static vertical truck load (the same value used in Equation 6.1).

W_br is the effective weight of the bridge and may be taken as half the weight of the bridge span on which the impact occurs. If the impact occurs at a pier or abutment, then the weight of the pier or abutment should be added to W_br.

Long Description.

The table contains two columns: Train to Bridge Mass Ratio defined as P subscript t over W subscript b r and lambda subscript m. When the ratio is less than 0.05, lambda subscript m is 1.00. When the ratio is between 0.05 and 0.10, lambda subscript m is 0.95. When the ratio is greater than 0.10, lambda subscript m is 0.90. The values show that as the train to bridge mass ratio increases, the lambda subscript m factor decreases.

this approach to obtain a more accurate bridge response after identifying the governing impact location (e.g., by using the static vertical loading methodology described in Section 6.1). The derailment impact dynamic vertical load is shown in Figure 37. The force–time history is a half-sine pulse force function shown in Figure 38. The function is defined by the following equation:

(6.3)

where P₀ is the pulse amplitude, t is time, and t_d is the pulse duration. Setting t_d as 0.15 s and using the following equation to estimate P₀ are recommended:

(6.4)

where P_t is the static load carried by a single impacting truck, H is the height of the rail, and λ_e and λ_m are the energy restitution and system mass reduction factors. Recommended values for λ_e and λ_m are provided in Tables 8 and 9, respectively.

Long Description.

The graph depicts a time history of dynamic load labeled P subscript V comma d of t plotted against time t. The vertical axis represents load amplitude, and the horizontal axis represents time. The curve starts at zero, rises smoothly to a maximum amplitude marked as P 0 at the midpoint, and then symmetrically decreases back to zero. The total width of the curve along the horizontal axis is labeled pulse duration t subscript d. The shape of the curve is symmetric and represents a single pulse event applied over time.

6.3 Validation of Vertical Impact Loading Methodologies

This section describes the case studies, inputs, and metrics used to validate both the static and dynamic vertical impact loading methods. The static vertical loading method and recommendations presented in Section 6.1 were tested using the case studies presented in Chapters 4 and 5. The full height of the rail minus the wheel flange thickness was considered as the drop height in all analyses to facilitate the comparisons with the results from the impact analyses in Chapters 4 and 5. Additionally, a half-sine pulse with a duration of 0.15 s was considered. A COR of 0.1 was used, given the simplified bridge model represents the effective stiffness and mass at midspan (Figure 10). All other inputs to Equation 6.3 are shown in Tables 10 and 11.

To test the dynamic vertical loading method from Section 6.2, analyses were conducted for all six derailment cases considered in Chapter 5, which included three different impact locations, two different train vehicles, and two distinct bridges. Table 12 summarizes the performed analyses. Details on the bridge models and the derailment scenarios considered were provided in Sections 5.1.1 and 5.1.2, respectively. In all dynamic vertical loading analyses, the half-sine pulses were applied to the plinth in the area of impact based on the results of the impact analyses described in Chapter 5 to facilitate the comparison with those results. The load application area

Long Description.

The table has seven columns labelled Vehicle Type, Passenger Load Case, Static Truck Load in kilonewtons, Bridge Natural Frequency in hertz, Pulse Duration Factor, System Mass Factor, and Energy Restitution Factor. For Low-Floor LRV with AW0 load, the static truck load is 154, bridge natural frequencies are 2.0, 2.5, and 3.0, the corresponding pulse duration factors are 0.9, 0.9, and 0.8, the system mass factor is 0.95, and the energy restitution factor is 1.1. For Low-Floor LRV with AW4 load, the static truck load is 222, the system mass factor is 0.93, and the energy restitution factor is 1.1. For High-Floor LRV with AW0 load, the static truck load is 159, the system mass factor is 0.95, and the energy restitution factor is 1.1; with AW4 load, the static truck load is 218, the system mass factor is 0.93, and the energy restitution factor is 1.1. For Single-Level Commuter with AW0 load, the static truck load is 171, the bridge natural frequency is 2.0, the pulse duration factor is 0.9, the system mass factor is 0.95, and the energy restitution factor is 1.1. For Bi-Level Commuter with AW0 load, the static truck load is 214, the system mass factor is 0.93, and the energy restitution factor is 1.1.

Long Description.

The table has columns labeled Bridge Model, Bridge Natural Frequency in hertz, Vehicle Type, Impact Location, Static Truck Load in kilonewtons, Pulse Duration Factor, System Mass Factor, and Energy Restitution Factor. For the Bulb-Tee Girder with a frequency of 3.4 hertz, Low-Floor LRV is considered with impact locations at midspan, quarter span, and pier, each with a static truck load of 154 kilonewtons. The pulse duration factor is 0.80, the system mass factor is 0.95, and the energy restitution factors are 1.1, 1.2, and 1.3, respectively. For the Bi-Level Commuter vehicle at midspan on the same bridge, the truck load is 214 kilonewtons, the pulse duration factor remains 0.80, the system mass factor is 0.95, and the energy restitution factor is 1.1. For the U-Girder with a frequency of 6.0 hertz and Low-Floor LRV, the midspan and quarter span locations have a truck load of 154 kilonewtons, a pulse duration factor of 0.50, and energy restitution factors of 1.1 and 1.2, respectively, with the system mass factor of 0.95.

Long Description.

The table contains columns for Bridge Model, Vehicle Type, Impact Location, Static Truck Load in kilonewtons, Energy Restitution Factor, Pulse Amplitude in kilonewtons, and Pulse Duration in seconds. For the Bulb-Tee Girder with a Low-Floor LRV, impact locations are midspan, quarter span, and pier, with a static load of 154 kilonewtons. The energy restitution factors are 1.1, 1.2, and 1.3, respectively, resulting in pulse amplitudes of 300, 327, and 354 kilonewtons. For the Bi-Level Commuter at midspan on the same girder, the static load is 214 kilonewtons, the restitution factor is 1.1, and the pulse amplitude is 416 kilonewtons. For the U-Girder with Low-Floor LRV, midspan and quarter span impacts have the same 154 kilonewtons truck load with restitution factors of 1.1 and 1.2, producing pulse amplitudes of 300 and 327 kilonewtons. The pulse duration for all cases is 0.15 seconds.

was specified such that it encapsulates the impacting wheel location for the prescribed duration of the pulse, starting with the wheelʼs first contact with the plinth. Furthermore, the gravity loads of bridges and trains were excluded; only the pulse load was included in the analyses described in this chapter.

For all case studies, the research team estimated the DAF safety margin predicted by its proposed impact loading methods over the obtained DAF using the detailed derailment impact analyses described in Chapters 4 and 5 (considered as ground truth). The safety margin is defined as:

(6.5)

where DAF_p and DAF_a represent the DAFs provided by the impact loading method and detailed derailment FEA, respectively. The DAFs derived from the input in Tables 10 and 12 are presented in Sections 6.4.1 and 6.4.2 and compared to the results obtained through detailed analyses.

6.4 Results of Vertical Impact Methodology Validation Analyses

The results of the static and dynamic vertical impact methodology validation analyses are presented in this section. Section 6.4.1 is dedicated to describing the results of the simplified bridge model and detailed bridge derailment scenarios using the static vertical impact loading method. In Section 6.4.2, the results of the detailed bridge model are shown using the dynamic approach.

6.4.1 Static Impact Loading Analyses

Tables 13 and 14 summarize the DAFs estimated using the static vertical impact loading method outlined in Section 6.1 for the LRV vehicles and the commuter cars using the parameters in Tables 10 and 11. In addition, the safety margins (Equation 6.5) were calculated using the DAFs obtained via detailed derailment analyses as ground truth (Sections 4.2 and 5.2) and were provided in Tables 13 and 14.

For simplified bridge derailment scenarios, the proposed static impact loading has provided positive margins for all considered cases except for the single-level commuter train, which underestimates the observed DAF by 142%. For the detailed bridge scenarios, all except the low-floor LRV impacting the bulb-tee girder bridge at the pier (–2%) have positive safety margins.

Long Description.

The table includes six columns: Vehicle Type, Passenger Load Case, Bridge Natural Frequency in hertz, DAF Prediction, DAF Analysis from Chapter 4, and Safety Margin. For the Low-Floor LRV under load case AW0 at 2.0 hertz, the predicted DAF is 200 percent, the analyzed DAF is 126 percent, and the safety margin is 74 percent. At 2.5 and 3.0 hertz, the DAF predictions rise to 250 and 267 percent, the analyzed values are 146 and 163 percent, and the safety margins are 104 percent. Under load case AW4 at 2.0 hertz, DAF prediction is 196 percent, analysis is 168 percent, with a safety margin of 28 percent. For the High-Floor LRV at 2.0 hertz, AW0 case has a prediction of 200 percent, analysis of 129 percent, and margin of 71 percent; AW4 case shows slightly lower prediction at 196 percent and analysis at 135 percent, yielding a 61 percent margin. The Single-Level Commuter at 2.0 hertz has a DAF prediction of 199 percent but a very high analyzed value of 341 percent, resulting in a safety margin of negative 142 percent. Lastly, the Bi-Level Commuter at 2.0 hertz has a prediction of 178 percent, analysis of 166 percent, and a safety margin of 12 percent.

Long Description.

The table presents Dynamic Amplification Factor, DAF, predictions and Chapter 5 DAF analysis values for different bridge types, train vehicles, and impact locations, along with their corresponding safety margins. For the Bulb-Tee Girder with a Low-Floor LRV, the DAF at midspan predicted value is 172 percent, analyzed is 160 percent, with a 12 percent safety margin. At the quarter span, the predicted DAF is 302 percent, analyzed is 254 percent, giving a 48 percent margin. At the pier, the prediction is 159 percent, the analysis is 161 percent, resulting in a negative 2 percent safety margin. For the Bi-Level Commuter on the Bulb-Tee Girder at midspan, the predicted DAF is 286 percent, analyzed as 251 percent, with a 42 percent margin. On the U-Girder with a Low-Floor LRV, the DAF at midspan is predicted as 163 percent, analyzed as 91 percent, with a 72 percent margin, while at the quarter span, the prediction is 156 percent, the analysis is 116 percent, and the margin is 40 percent.

By fitting a Studentʼs t-distribution [25] to the combined margins of Tables 13 and 14, the probability of the margin being at or below that of the single-level commuter safety margin is 0.005 (0.027 if using the data from Table 13 only), implying that the single-level commuter safety margin is a statistical outlier. Excluding the safety margin of the single-level commuter, the safety margin for the simplified bridge cases has a mean of 67% and a probability of zero non-exceedance (i.e., negative margins) of 0.035. For the detailed bridge cases, the safety margin has a mean of 34% and a probability of zero non-exceedance of 0.13. Further statistical analyses of the results are presented in Section 6.5.

6.4.2 Dynamic Vertical Loading Analyses

Table 15 summarizes the DAFs estimated using the simplified dynamic vertical impact loading method outlined in Sections 6.1 and 6.2 using the pulse shapes described in Table 12. The safety margins were calculated (Equation 6.5) using the DAFs obtained via the detailed derailment analyses as ground truth (Section 5.2) and provided in Table 15. Figures 39 through 44 show the bridge deck deflection time histories at the point of impact due to the applied half-sine pulse, which were overlaid with the results of the detailed train derailment analyses described in Chapter 5. The time of the half-sine pulse loading response was shifted such that the first peak coincides with the peak of the results described in Chapter 5.

There are two cases of safety margins falling below zero: the low-floor LRV impacting the bulb-tee girder bridge at quarter span (–3%) and at the pier (–55%). However, the “at pier” impact response time history shows relatively high-frequency content compared to all other cases. The train derailment short-duration pulses [0.005–0.01 s (see the Appendix)] amplify higher-order modes, while the amplification of the higher-order modes due to the prolonged idealized pulse is negligible. Smoothing out the response using a 0.05-s (5-point) moving average results in a derailment DAF of 105% and an impulse DAF safety margin of 0.4% (Figure 38). Assuming a Studentʼs t-distribution, the safety margin has a mean of 11% and a probability of zero non-exceedance of 0.39. Considering the moving average “at pier” deflection response instead of the raw response, the safety margin mean rises to 22%, while the probability of zero non-exceedance drops to 0.2.

6.5 Vertical Load Factors

Inherent uncertainty exists in derailment impact loading because of variation in rail vehicle and bridge characteristics. Additionally, as mentioned in Sections 5.1.4 and 5.3, DAF varies with structural components and the output considered. Therefore, a probabilistic evaluation of derailment impact loading was performed. Load factors that may be applied to the predicted DAFs were calculated using both the simplified static and dynamic vertical loading methods to ensure at most a 0.05 probability of DAF exceedance, given the available validation results described in Section 6.4. Given the limited number of observations, the safety margins (Equation 6.5) listed in Tables 13 through 15 were assumed to follow a Studentʼs t-distribution [25]. By multiplying the prediction DAF (DAF_p) by a load factor (γ), the updated safety margin mean and standard deviations are given by the following equations:

(6.6)

(6.7)

where µ_p and µ_a are the means and σ_p and σ_a are the standard deviations of the predicted (i.e., simplified) and actual (i.e., detailed FEA) DAFs, respectively. r is the correlation coefficient. Sample statistics were used for all parameters in Equations 6.6 and 6.7. The t-value at the zero safety margin is estimated using the following equation:

(6.8)

A threshold is specified using the critical one-tailed t-value at a 5% significance level using the vth degrees-of-freedom Studentʼs t-distribution where v is the sample size –1. The following condition must be satisfied to ensure a zero-safety margin exceedance probability of at most 0.05 based on the fitted distribution:

(6.9)

where t_0.05,v is the critical t-value. Finally, the adjustment factor γ is optimized such that t_M=0 ≈ t_0.05,v.

The results of the simplified and detailed bridge models presented in Tables 13 and 14 were combined to optimize the load factor of the static impact loading method. The single-level commuter train observation was excluded from the dataset because of its anomalous derailment pattern (see the Appendix), in addition to it being a statistical outlier with a safety margin p-value of 0.005. The results of the detailed bridge model presented in Table 15, including the moving average “at pier” result instead of the raw “at pier” data, were used to optimize the

Long Description.

The table presents a comparative summary of dynamic amplification factors and statistical safety margins for load factors of 1.00 and 1.04. For a load factor of 1.00, the mean of DAF subscript p is 2.11 and of DAF subscript a is 1.59, with standard deviations of 0.49 and 0.47, respectively. The correlation coefficient rho is 0.77, yielding a mean safety margin of 0.52, with a standard deviation of 0.33 and a zero-margin t-value of negative 1.60. The critical t-value at t subscript 0.05 comma 12 is negative 1.78, and the margin at this value is negative 0.06. For the load factor 1.04, the mean of DAF subscript p increases to 2.19 while DAF subscript a remains 1.59, and the standard deviations are 0.51 and 0.47. The safety margin improves to a mean of 0.60 with a standard deviation of 0.33, the zero-margin t-value becomes negative 1.81, and the margin at the critical t-value increases slightly to 0.01.

Long Description.

The table presents comparative statistical data for two load factors, 1.00 and 1.19. For each, the mean and standard deviation of predicted dynamic amplification factor, DAF subscript p, and actual dynamic amplification factor, DAF subscript a, are given. At a load factor of 1.00, the mean DAF subscript p is 1.83 and DAF subscript a is 1.63, with standard deviations of 0.72 and 0.73, respectively. At load factor 1.19, the means are 2.18 and 1.63, with standard deviations of 0.86 and 0.73. The correlation coefficient rho is 0.95. Safety margins increase from 0.20 to 0.55, with standard deviations of 0.22 and 0.27, between the two load factors. The zero-margin t-value for M equals 0 decreases from negative 0.93 to negative 2.03. The critical t-value at t subscript 0.05 comma 5 is negative 2.02. The margin at t subscript 0.05 comma 12 improves from negative 0.24 at load factor 1.00 to 0.00 at load factor 1.19.

adjustment factor for the dynamic vertical impact method. The DAF and safety margin statistics with and without the adjustment factor are summarized in Tables 16 and 17. Additionally, the fitted probability density and cumulative probability functions of DAF_p and DAF_a, respectively, and the safety margins for both the static and dynamic vertical impact analyses are shown in Figures 45 and 46. The load factors of 1.04 and 1.19 satisfy the condition in Equation 6.9 for the static and dynamic impact loading methods, respectively.

6.6 Horizontal Impact Methodology

In this section, a simplified approach to obtaining the dynamic response of the bridge due to horizontal loads from train derailment impact is proposed. Similar to the vertical impact loads, the horizontal impact load is intended to be applied to one truck from an actual train—it is not intended to be applied to the LRT-16 notional live load given in the AASHTO LRFD Guide. Equivalent static loads shall be applied in the transverse and longitudinal directions to account for the effects of derailment in accordance with Equation 6.10:

(6.10)

where DAF_H is the horizontal DAF, µ_f is the coefficient of friction, and P_t is the static load carried by the derailing truck (the same as in Equation 6.1). DAF_H is recommended to be taken as 2.0. This factor is consistent with the duration of force impulse being much longer than the natural period of the structure [24]. Bridges are typically much stiffer in the transverse and longitudinal directions than in the vertical direction, and the duration of frictional forces is much longer in the longitudinal and transverse directions than in the vertical direction. Lower values of DAF_H may be used if justified. The research team recommends taking µ_f as 0.6, which is consistent with previous research on ballasted track [26] [27]. The analysis results indicate that this factor

is conservative for a direct fixation track, and more recent research indicates that a factor of 0.6 is conservative for off-track ballasted trackbed conditions and appropriate for on-track conditions with brakes fully applied [23]. Lower values of µ_f may be used if justified, and F_H is intended to be applied on a single truck. Since these horizontal forces occur after the initial impact, F_H does not need to be applied coincidently with vertical impact loads. Although the analyses showed that friction forces reduced the transverse velocity of the front truck to zero shortly after derailment, off-track behavior is difficult to predict [26]. Therefore, for tracks without guardrails, structures on the bridge (e.g., walls) should be designed to withstand impact from the train. The research conducted for this project does not justify modifying the barrier wall loads prescribed by the AASHTO LRFD Guide.

6.7 Discussion

Two methodologies were developed to account for the dynamic effects of train derailment on bridge structures. The results of these methodologies were compared to the range of simulation results presented in Chapters 4 and 5. Based on these comparisons, load factors were developed for both methodologies to ensure that the 5% probability of exceedance is calculated on the condition that derailment occurs (i.e., assuming 100% probability of derailment). The static vertical impact loading methodology is essentially a modification to the current AASHTO LRFD Guide. Rather than using a constant DAF of 100%, the static methodology uses a formula to calculate a more accurate DAF based on the dynamic properties of the bridge. The static vertical impact loading methodology has sufficient conservatism to ensure a 5% probability of exceedance with a load factor of 1.0. The dynamic vertical loading methodology is different than the current AASHTO LRFD Guide. The primary difference is that with the dynamic vertical loading methodology, analysts apply a force–time history to the bridge deck to account for the derailment impact. This new methodology would require bridge designers to perform a dynamic analysis. The research for this project showed that it is impossible to completely decouple the train derailment impact from the bridge response—some interaction does exist between the bridge and rail car. Therefore, some factors were included in the force–time history to account for the dynamic properties of the bridge. The research teamʼs evaluation of the proposed methodologies found that the dynamic methodology can produce a better match to the full derailment simulation results. Consequently, less conservatism is inherent in the dynamic methodology. The static method has a 95% probability of non-exceedance with a load factor of 1.0, while the dynamic method would require a load factor of 1.2 to match a 95% probability of non-exceedance. Overall, the team found that the static methodology is easier for bridge designers to use since it is a direct replacement of the current AASHTO LRFD Guide and its results have sufficient conservatism for use in bridge design. Some bridge designers may select the dynamic methodology in circumstances in which greater accuracy and less conservatism are required.

In addition, the research team found that horizontal impact forces are relatively small. The dynamic bridge response is also much stiffer in the horizontal directions. Longitudinal derailment impact forces are small on concrete surfaces because of the wheel rotation. Longitudinal forces only increase when brakes are applied or off-track ballasted trackbed conditions are present. The teamʼs analysis found that transverse forces from derailment impact are consistent with the friction associated with static wheel loads—these friction forces are not increased by the vertical dynamic amplification associated with derailment impact. Governing horizontal impact forces do not occur simultaneously with governing vertical impact forces. Therefore, horizontal and vertical impact forces do not need to be applied to a design model simultaneously. Other horizontal loads currently required by the AASHTO LRFD Guide—braking loads in the longitudinal direction and barrier wall loads in the transverse direction—govern horizontal loads associated with derailment impact.