**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

**Suggested Citation:**"Chapter 5 - Encroachment Relationships." National Academies of Sciences, Engineering, and Medicine. 2024.

*Development of Clear Recovery Area Guidelines*. Washington, DC: The National Academies Press. doi: 10.17226/27593.

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

35Â Â Encroachment Relationships The simulation results generated from the large vehicle encroachment simulation matrix were weighted using the marginal probabilities developed in ChapterÂ 4. This methodology permitted the probability of a discrete simulation to be determined through the application of observed crash data. The research team used the weighted simulation outcomes to develop various encroachment distance and severity relationships for 2U and 4D facility types. The developed relationships included rollover probability, lateral extent of encroachment and longitudinal distance traveled, impact speed, and impact angle at a given lateral distance. Statistical Model Specifications and Data Generation Process Since the planned risk analysis procedure was intended to evaluate a given clear zone distance for a roadway and roadside configuration of interest, the likelihood of reaching a certain lateral distance was a key aspect of the model development process. Determining the probability of an impact with hazards at the clear zone edge and the severity of such impacts was equally impor- tant to the risk analysis methodology. This involves modeling both longitudinal distance and speed at the defined lateral clear zone distance. Such data analysis processes call for complex models, such as mixed-effect models that take care of the repeated measures for the same simulation. Thus, the model development process involved structuring the encroachment simulation data for the mixed-effect models. Thus, for each simu- lation, the maximum lateral distance reached was determined, along with the vehicle speed and longitudinal distance traveled at different lateral offset distances ranging from 10Â ft to 70Â ft in 10Â ft increments. There are a total of 2,073,600 simulations, with each vehicle type having a total of 518,400 obser- vations. The data regeneration resulted in a total of 3,628,800 observations for rollover probability and the probability of reaching a certain clear zone distance. This was based on some simulations reaching multiple clear zone distances without steering back, stopping, or rolling over. Similarly, the following number of observations was available for speed and longitudinal distance modeling. â¢ Passenger sedan: 1,697,198 observations. â¢ Pickup truck: 1,892,359 observations. â¢ Crossover or CUV: 1,748,198 observations. â¢ SUV: 1,568,715 observations. The statistical models developed depended on the nature and distribution of the dependent vari- ables. The team utilized logistic regression for the probability of reaching a certain clear zone as well as the likelihood of rollover at a given lateral distance. This is due to the binary nature of the outcome C H A P T E R Â 5

36 Development of Clear Recovery Area Guidelines variable (rolloverâyes/no, and reaching a clear zone distanceâyes/no). On the other hand, the research team utilized a multiple linear regression for the speed distribution model and the impact angle model; a gamma regression was used for the longitudinal distance distribution model. Lateral Extent of Encroachment Models The lateral extent of encroachment defines how far a vehicle can travel in the lateral direction relative to the edge of the traveled way for a given set of roadway and roadside design conditions. As mentioned earlier, the research team utilized logistic regression models to predict the lateral extent of encroachment. TableÂ 11 through TableÂ 18 present the models for the likelihood of an encroaching vehicle reaching or crossing a given lateral distance (clear zone) threshold. The negative coefficient of the clear zone distance variable suggests that the probability of reaching a certain threshold declines as the lateral distance increases. The magnitude of the decrease of the probabilities varies by vehicle type. Other variables of interest that were found to be statistically significant in terms of the prob- ability of reaching a certain lateral distance include horizontal curvature, shoulder width, fore- slope width, backslope ratio, backslope width, and ditch bottom width. The model indicates that an increase in shoulder width, foreslope width, and ditch bottom width correspond to an increase in the probability of reaching a certain lateral distance. The model indicates that a flatter Estimate StandardError z-Value P-Value Intercept 0.080 0.069 1.16 0.246 Degree of Horizontal Curvature 0.729 0.027 26.54 <0.001 Shoulder Width 0.018 0.003 6.17 0.000 Foreslope Width 0.037 0.003 12.41 <0.001 Backslope 0.131 0.008 16.46 <0.001 Backslope Width â0.015 0.003 â5.06 0.000 Bottom Ditch Width 0.027 0.003 9.59 <0.001 Clear Zone Distance â0.046 0.001 â72.16 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 NOTE: AIC = Akaike information criteria. TableÂ 11. Lateral distance crossing probability model for 2U roads for CUV. Estimate StandardError z-Value P-Value Intercept â0.224 0.067 â3.33 0.001 Degree of Horizontal Curvature 0.850 0.027 31.34 <0.001 Shoulder Width 0.016 0.003 5.80 0.000 Foreslope Width 0.039 0.003 13.37 <0.001 Backslope 0.137 0.008 17.55 <0.001 Backslope Width â0.016 0.003 â5.48 0.000 Bottom Ditch Width 0.027 0.003 9.54 <0.001 Clear Zone Distance â0.044 0.001 â71.18 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 NOTE: AIC = Akaike information criteria. TableÂ 12. Lateral distance crossing probability model for 4D roads for CUV.

Encroachment Relationships 37Â Â Estimate Standard Error z-Value P-Value Intercept 0.292 0.068 4.31 0.000 Degree of Horizontal Curvature 0.843 0.027 31.20 <0.001 Shoulder Width 0.008 0.003 2.97 0.003 Foreslope Width 0.028 0.003 9.77 <0.001 Backslope 0.131 0.008 16.73 <0.001 Backslope Width â0.014 0.003 â4.99 0.000 Bottom Ditch Width 0.022 0.003 7.89 0.000 Clear Zone Distance â0.043 0.001 â69.03 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 NOTE: AIC = Akaike information criteria. TableÂ 13. Lateral distance crossing probability model for 2U roads for pickup. Estimate StandardError z-Value P-Value Intercept â0.062 0.066 â0.94 0.347 Degree of Horizontal Curvature 0.975 0.027 36.47 <0.001 Shoulder Width 0.008 0.003 3.02 0.003 Foreslope Width 0.031 0.003 11.05 <0.001 Backslope 0.137 0.008 17.81 <0.001 Backslope Width â0.014 0.003 â5.05 0.000 Bottom Ditch Width 0.023 0.003 8.50 <0.001 Clear Zone Distance â0.042 0.001 â69.00 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 NOTE: AIC = Akaike information criteria. TableÂ 14. Lateral distance crossing probability model for 4D roads for pickup. NOTE: AIC = Akaike information criteria. Estimate StandardError z-Value P-Value Intercept â0.150 0.071 â2.12 0.034 Degree of Horizontal Curvature 0.829 0.029 29.03 <0.001 Shoulder Width 0.020 0.003 6.77 0.000 Foreslope Width 0.034 0.003 11.27 <0.001 Backslope 0.154 0.008 18.68 <0.001 Backslope Width â0.008 0.003 â2.68 0.007 Bottom Ditch Width 0.039 0.003 13.19 <0.001 Clear Zone Distance â0.052 0.001 â76.68 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 15. Lateral distance crossing probability model for 2U roads for SUV.

38 Development of Clear Recovery Area Guidelines NOTE: AIC = Akaike information criteria. Estimate StandardError z-Value P-Value Intercept â0.459 0.070 â6.56 0.000 Degree of Horizontal Curvature 0.983 0.029 34.46 <0.001 Shoulder Width 0.020 0.003 6.94 0.000 Foreslope Width 0.035 0.003 11.63 <0.001 Backslope 0.151 0.008 18.68 <0.001 Backslope Width â0.008 0.003 â2.57 0.010 Bottom Ditch Width 0.037 0.003 12.79 <0.001 Clear Zone Distance â0.051 0.001 â76.08 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 16. Lateral distance crossing probability model for 4D roads for SUV. NOTE: AIC = Akaike information criteria. Estimate StandardError z-Value P-Value Intercept 0.051 0.069 0.74 0.461 Degree of Horizontal Curvature 0.892 0.028 32.11 <0.001 Shoulder Width 0.014 0.003 4.72 0.000 Foreslope Width 0.027 0.003 9.10 <0.001 Backslope 0.148 0.008 18.55 <0.001 Backslope Width â0.019 0.003 â6.44 0.000 Bottom Ditch Width 0.022 0.003 7.50 0.000 Clear Zone Distance â0.045 0.001 â70.67 <0.001 Model Summary Number Of Observations 3,628,800 AIC 18 TableÂ 17. Lateral distance crossing probability model for 2U roads for sedan. NOTE: AIC = Akaike information criteria. Estimate StandardError z-Value P-Value Intercept â0.255 0.068 â3.76 0.000 Degree of Horizontal Curvature 1.042 0.028 37.63 <0.001 Shoulder Width 0.012 0.003 4.25 0.000 Foreslope Width 0.030 0.003 10.21 <0.001 Backslope 0.145 0.008 18.51 <0.001 Backslope Width â0.019 0.003 â6.50 0.000 Bottom Ditch Width 0.020 0.003 7.13 0.000 Clear Zone Distance â0.044 0.001 â70.05 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 18. Lateral distance crossing probability model for 4D roads for sedan.

Encroachment Relationships 39Â Â backslope ratio is also associated with a greater probability of reaching a given lateral distance, while a larger backslope width results in a lower likelihood of reaching that lateral distance. An increase in the degree of horizontal curvature (i.e., a sharper curve) results in an increased prob- ability of reaching a certain lateral distance. Longitudinal Distance Models To predict the occurrence of crashes with xed objects (e.g., trees) at the clear zone edge, it is necessary to understand the longitudinal distance (along the roadway segment) that an encroaching vehicle has traveled when it reaches the given lateral oset. is permits the inter- action to be predicted between the encroaching vehicles and the hazards dened along the clear zone edge. us, the encroachment simulation trajectories were used to develop models for longitudinal distance distributions as a function of the lateral extent of encroachment. Although the process described in this section was followed to develop longitudinal distance models for each of the vehicle types and facility types being considered under this study, the plots presented for discussion correspond to the pickup data weighted to represent 2U roadways. FigureÂ 23 presents the distributions of longitudinal distances observed for selected ranges of lateral distance for pickup trucks corresponding to a sample of 50,000 simulated trajectories. It FigureÂ 23. Longitudinal distance marginal distributions for pickups on 2U roads. The x-axis is the longitudinal distance from the departure point in feet. Lat_D = lateral distance.

40 Development of Clear Recovery Area Guidelines can be seen that, in general, these distributions start with an accentuated spike on their leftmost side and then quickly decay as the longitudinal distance increases. A key feature observed in the data is that the spread of the longitudinal distance distribution increases with increasing lateral distances. This is logical given that the influence of encroachment speed, angle, and driver inputs (e.g., steering) on the trajectory of an encroaching vehicle will increase with lateral distance traveled. Another key feature is a noticeable softening of the leftmost spike as the lateral distance increases, moving the mode of the distribution slightly to the right for greater lateral distances (most notable for lateral distances beyond 45Â ft). In other words, three features are apparent: pos- itive skewness of the distributions in general, increasing spread of the distribution with increas- ing lateral distances, and high kurtosis (i.e., peakness) of the distributions that decreases with increasing lateral distances. Next, the research team fitted a preliminary model of the gamma distribution, where the scale parameter varies with the design variable, and the dispersion parameter is estimated as a flat average. The purpose of this exercise was to compare the marginal distributions from the raw data for lateral distance classes slightly different than in FigureÂ 23, to the distributions produced by the model. The comparison result is shown in FigureÂ 24. The general trend of the marginal distributions is well-captured, with the mode of each distri- bution roughly in the correct location. However, regarding the kurtosis observed in the marginal distributions, a mismatch is observed at the smaller lateral distances that tends to dissipate at larger lateral distances. The comparison, however, is not necessarily expected to be direct, as the marginal distributions from the raw data represent the aggregate of all simulations that had data within the range of lateral distances, while the model conditional distribution is calculated at the average lateral distance of each interval shown. The exploratory analysis identified the shape, location, and spread of the distributions change with lateral distance. A model that parameterizes only the scale parameter responds well to the need to model varying spread and location but seems rigid in capturing changes in shape. The research team determined that parameterizing both scale and dispersion parameters in terms of the design variables would provide more flexibility to the modeling process to capture the changes observed in the distributions. Modeling the Scale and Shape of Lateral Distance Distributions As an initial step of the modeling effort, the research team took a random sample of 12,000 simulated runs from the pickup truck dataset (containing approximately 44,500 longitudinal distances at different lateral distance thresholds). The team programmed an iterative estimation algorithm to estimate both distributional parameters for the conditional gamma distribution. TableÂ 19 shows the parameter estimates for the scale submodel fitted to the sample of simulated pickup encroachments. It can be seen that lateral distance, vertical grade, horizontal curvature, shoulder width, foreslope ratio, foreslope width, backslope ratio, backslope width, and ditch bottom width all play a role in determining the reach and spread of the longitudinal distance distribution. In addition to the parameter estimates, TableÂ 19 also summarizes the estimated variation between simulation runs that remain unaccounted for among the parameters, compared to the residual variation in the data beyond the model estimates. Similarly, TableÂ 20 shows the estimates corresponding to the shape parameter of the longitu- dinal distance distribution. It can be seen that only a subset of the factors that influence scale has an impact on the shape of the distribution. Most notably, lateral distance, horizontal curvature,

Encroachment Relationships 41Â Â backslope ratio, and backslope width. Shoulder width, foreslope ratio, and foreslope width have only a mild impact on the shape of the distribution. To have an initial visualization of how this complex model performs in capturing the observed longitudinal distances, the research team prepared a set of graphs, shown in FigureÂ 25, comparing the model-predicted distribution shapes and the observed longitudinal distances for a few randomly sampled simulation runs in the data. As expected, the observations tend to fall both on the lower and upper half of the distribu- tions, except for one case for which the observation is further to the right of the range. However, this plot does not provide further insights into the correctness of the shape and location of the Fr eq ue nc y longitudinal distance (ft) longitudinal distance (ft) longitudinal distance (ft) longitudinal distance (ft) longitudinal distance (ft) longitudinal distance (ft) longitudinal distance (ft) longitudinal distance (ft) longitudinal distance (ft) longitudinal distance (ft) Fr eq ue nc y Fr eq ue nc y Fr eq ue nc y Fr eq ue nc y Pr ob ab ilit y D en si ty Pr ob ab ilit y D en si ty Pr ob ab ilit y D en si ty Pr ob ab ilit y D en si ty Pr ob ab ilit y D en si ty FigureÂ 24. Longitudinal distance marginal distributions compared to the conditional distributions from a preliminary model of longitudinal distances for pickups and 2U roads.

42 Development of Clear Recovery Area Guidelines Variable Estimate StandardError z-Value P-Value Intercept 4.63E+00 2.32E-02 199.791 < 2e-16 Lateral Distance 3.04E-02 2.02E-04 150.714 < 2e-16 Lateral Distance Squared â1.40E-04 1.21E-06 â115.709 < 2e-16 Vertical Grade 4.13E-05 1.81E-04 0.228 0.81927 Degree of Horizontal Curvature â1.89E-01 4.29E-03 â44.108 < 2e-16 Shoulder Width â8.32E-04 4.56E-04 â1.826 0.06785 Foreslope Squared 9.42E-04 3.43E-04 2.748 0.00599 Foreslope â1.50E-02 5.33E-03 â2.815 0.00488 Foreslope Width â3.57E-03 4.25E-04 â8.391 < 2e-16 Bottom Ditch Width â3.79E-03 4.09E-04 â9.265 < 2e-16 Backslope 5.72E-03 3.07E-03 1.864 0.06239 Backslope Width â1.74E-03 1.03E-03 â1.681 0.09273 Lateral Distance* Vertical Grade 1.64E-05 2.13E-06 7.680 1.59E-14 Lateral Distance* Shoulder Width â5.10E-05 5.38E-06 â9.477 < 2e-16 Backslope* Foreslope 1.21E-03 4.75E-04 2.539 0.01110 Backslope Width * Foreslope â1.40E-04 1.59E-04 â0.882 0.37795 Lateral Distance* Backslope â6.86E-04 3.64E-05 â18.872 < 2e-16 Lateral Distance* Backslope Width 5.05E-05 5.55E-06 9.107 < 2e-16 Lateral Distance* Foreslope â1.56E-04 2.37E-05 â6.610 3.85E-11 Lateral Distance* Backslope* Foreslope 1.21E-05 5.37E-06 2.258 0.02397 Random Effects and Residual Variation Variance StandardDeviation Groups Run_Number 0.018241 0.13506 Residual 0.006338 0.07961 NOTE: *indicates interaction between the variables. TableÂ 19. Parameter estimates for the scale of longitudinal distance distribution for pickups at 2U roadways. NOTE: *indicates interaction between the variables. Variable Estimate StandardError z-Value P-Value Intercept â4.002880 0.181441 â22.062 < 2e-16 Lateral Distance â0.051700 0.005941 â8.703 < 2e-16 Lateral Distance Squared 0.000717 6.71E-05 10.679 < 2e-16 Degree of Horizontal Curvature â0.669750 0.058879 â11.375 < 2e-16 Shoulder Width â0.009770 0.005658 â1.726 0.08430 Foreslope â0.008140 0.018731 â0.434 0.66400 Foreslope Width â0.012430 0.005825 â2.135 0.03280 Backslope â0.091920 0.023145 â3.972 7.15E-05 Backslope Width* Backslope 0.000721 0.001383 0.521 0.60210 Lateral Distance* Foreslope â0.000490 0.000431 â1.140 0.25450 TableÂ 20. Parameter estimates for the shape of longitudinal distance distribution for pickups at 2U roadways. predicted distributions. To help with this, a graph was made of 850 predictions from the distri- butional model (i.e., the overlapping predicted distributions for 850 randomly picked pickup simulations). In FigureÂ 25, each graph represents a randomly selected encroachment simulation. The vertical line represents the observed longitudinal travel distance for the encroachment. The colored area is the distribution generated by the model for the design condition associated with the selected encroachment. The closer the observed is to the mean of the predicted distribution, the better the correlation. The overlapped distributions should roughly correspond to the spread and concentration areas of the corresponding observed longitudinal distances. Next, for a more formal assessment against the fit of the predicted distributions, the research team calculated the percentile of each observation from the corresponding predicted (conditional)

Encroachment Relationships 43Â Â Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) Pr ob ab ilit y D en si ty longitudinal distance (ft) FigureÂ 25. Longitudinal distance predicted distributions compared to actually observed longitudinal distance for pickup simulations at 2U roads selected at random.

44 Development of Clear Recovery Area Guidelines distribution and made a histogram and quantile-to-quantile (q-q) plot. The expectation is that the histogram should look relatively flat from the minimum of zero on the left to the maximum of 1.0 on the right, and the q-q plot should fall along the 1:1 line if the observed and predicted quantiles correspond to each other. This result is shown in FigureÂ 26. There is an obvious area of discrepancy on the left extreme of the two plots between 0 and about 0.15. This gap corresponds to the gap noticed in the superposition of distributions noted in FigureÂ 26. Similarly, the q-q plot on the right shows a poor fit between the distributions at the lower percentiles, but the distributions catch up to each other eventually, as the plot tends toward the expected line of 1:1 (the dotted line) with increasing percentiles. Post Hoc Adjustment of Predicted Distribution Given the mismatch between the observed and predicted distributions of longitudinal distances, the research team implemented an algorithm that applies a post hoc adjustment to the predicted dis- tribution to attempt to close the gap between the predicted and observed distributions. The follow- ing potential adjustments were included in the search for an optimized estimate in this algorithm. 1. An initial right shift with exponential decay to the domain of the distribution (meaning, the predicted longitudinal distance quantiles are shifted to the right by a threshold that decays with larger longitudinal distance quantiles). The adjustment is such that the order of pre- dicted quantiles does not change. This adjustment is controlled by new parameters defined as the right shift (a), and the rate of decay (b). 2. A flat shift in the predicted quantile equivalent to relocating the origin of the reference system, denoted by a new parameter (c). FigureÂ 26. Histogram (left) and q-q plot (right) for observed versus predicted longitudinal distance quantiles for pickups at 2U roads. Note: In the histogram, the blue line represents the density plot for the distribution.

Encroachment Relationships 45Â Â 3. To try to address the kurtosis and left-tail issues, the algorithm also searched for a truncation point of the predicted distributions at a given flat percentile (defined as parameter trun). The new quantiles are calculated from the remaining truncated distribution, discarding the lower portion of the predicted distribution. A genetic algorithm was implemented to find the best combinations of the adjustments above, stated as an optimization problem that minimizes the deviation between the predicted and observed percentiles. The graphs in FigureÂ 27 show the results after optimizing the above adjustments to different lateral distances. It can be seen in the plots in FigureÂ 27 that roughly the same combination of adjustments to the predicted distributions yields a significant improvement in the correspondence between the predicted and the observed percentiles. Sum.Sq.Res = 30.44232 trun = 0.151 Sum.Sq.Res = 30.44232 a = trun = 0.151 Sum.Sq.Res = 30.44232 trun = 0.151 Sum.Sq.Res = 30.44232 trun = 0.151 Sum.Sq.Res = 30.44232 FigureÂ 27. Q-q plots for observed versus adjusted predicted longitudinal distance distributions (sample model) for pickups at 2U roads. Sum.Sq.Res. 5 sum of the squares of the residuals (estimate of the error between the observed and predicted).

46 Development of Clear Recovery Area Guidelines Next, the research team refitted the distributional model to the complete dataset of pickup simula- tions, with the expectation that a more comprehensive database would yield a better prediction. The results are shown in FigureÂ 28. When applying the adjustments described above to the predictions of this model, it is apparent that the performance of the adjusted models is marginal compared to the adjusted predictions on the model fitted to a large random sample of the data. Upon reviewing the comparative performances, the research team decided to move forward with using the model fitted to the sample of runs. Models and Adjustments for Facility Types and Additional Vehicles The research team followed a similar process to the one described in the previous sections to arrive at longitudinal distance models and adjustments for the rest of the vehicle types for both a = 87.687 b = 0.009 c = 68.62 trun = 0.289 Sum.Sq.Res = 75.69593 a = 87.687 b = 0.009 c = 68.62 trun = 0.289 Sum.Sq.Res = 84.71386 a = 87.687 b = 0.009 c = 68.62 trun = 0.289 Sum.Sq.Res = 78.24212 a = 87.687 b = 0.009 c = 68.62 trun = 0.3 Sum.Sq.Res = 71.73792 a = 87.687 b = 0.009 c = 78.64 trun = 0.3 Sum.Sq.Res = 112.50291 FigureÂ 28. Q-q plots for observed versus adjusted predicted longitudinal distance distributions (full data model) for pickups at 2U roads. Sum.Sq.Res. 5 sum of the squares of the residuals (estimate of the error between the observed and predicted).

Encroachment Relationships 47Â Â facility types of interest after confirming very similar trends from the raw data. The following sections document the performance of the adjusted distributional predictions. FigureÂ 29 shows the performance of the final adjusted models for pickup trucks on 4D roads. Similar to the performance at 2U sites, the expectation is that roughly the same set of adjustments should produce acceptable performance for the model, regardless of the lateral distance. FigureÂ 30 shows the performance of the final adjusted models for sedans on 2U roads, and FigureÂ 31 presents the corresponding performance for 4D roads. FigureÂ 32 shows the per- formance of the final adjusted models for CUVs on 2U roads, while FigureÂ 33 presents the corresponding performance for 4D roads. FigureÂ 34 shows the performance of the final adjusted models for SUVs on 2U roads, while FigureÂ 35 presents the corresponding performance for 4D roads. Similar to the performances for the pickup truck, the expectation is that roughly the same set of adjustments should produce acceptable performance for the model, regardless of the lateral distance. a = 39.494 b = 0.004 c = -24.03 trun = 0.044 Sum.Sq.Res = 15.77297 a = 39.494 b = 0.004 c = -24.03 trun = 0.044 Sum.Sq.Res = 13.785 a = 39.841 b = 0.004 c = -24.03 trun = 0.044 Sum.Sq.Res = 12.33115 a = 39.494 b = 0.004 c = -24.03 trun = 0.044 Sum.Sq.Res = 12.36913 a = 39.494 b = 0.004 c = -24.03 trun = 0.044 Sum.Sq.Res = 14.33208 Predicted percentiles O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es Predicted percentiles Predicted percentiles Predicted percentiles Predicted percentiles O bs er ve d pe rc en til es O bs er ve d pe rc en til es FigureÂ 29. Q-q plots for observed versus adjusted predicted longitudinal distance distributions (sample model) for pickups at 4D roads. Sum.Sq.Res. 5 sum of the squares of the residuals (estimate of the error between the observed and predicted).

48 Development of Clear Recovery Area Guidelines a = 44.844 b = 0.011 c = -18.71 trun = 0.155 Sum.Sq.Res = 3.72597 a = 44.844 b = 0.011 c = -18.71 trun = 0.155 Sum.Sq.Res = 10.60382 a = 47.935 b = 0.011 c = 2.703 trun = 0.204 Sum.Sq.Res = 5.58145 a = 47.935 b = 0.011 c = 2.703 trun = 0.204 Sum.Sq.Res = 4.00706 a = 47.935 b = 0.011 c = 2.703 trun = 0.204 Sum.Sq.Res = 1.25847 Predicted percentiles Predicted percentiles Predicted percentiles Predicted percentiles Predicted percentiles O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es FigureÂ 30. Q-q plots for observed versus adjusted predicted longitudinal distance distributions (sample model) for sedans at 2U roads. Sum.Sq.Res. 5 sum of the squares of the residuals (estimate of the error between the observed and predicted).

Encroachment Relationships 49Â Â a = 42.564 b = 0.043 c = 23.295 trun = 0.229 Sum.Sq.Res = 10.07684 a = 42.564 b = 0.043 c = 23.295 trun = 0.229 Sum.Sq.Res = 10.95607 a = 42.564 b = 0.043 c = 23.295 trun = 0.229 Sum.Sq.Res = 10.18595 a = 42.564 b = 0.043 c = 23.295 trun = 0.229 Sum.Sq.Res = 11.31545 a = 42.564 b = 0.043 c = 23.295 trun = 0.229 Sum.Sq.Res = 12.15359 Predicted percentilesPredicted percentiles Predicted percentiles Predicted percentiles Predicted percentiles O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es FigureÂ 31. Q-q plots for observed versus adjusted predicted longitudinal distance distributions (sample model) for sedans at 4D roads. Sum.Sq.Res. 5 sum of the squares of the residuals (estimate of the error between the observed and predicted).

50 Development of Clear Recovery Area Guidelines a = 88.925 b = 0.038 c = 39.381 trun = 0.101 Sum.Sq.Res = 16.39958 a = 88.925 b = 0.038 c = 39.381 trun = 0.101 Sum.Sq.Res = 7.6189 a = 88.925 b = 0.038 c = 39.381 trun = 0.101 Sum.Sq.Res = 8.27455 a = 88.925 b = 0.038 c = 39.381 trun = 0.101 Sum.Sq.Res = 9.42885 a = 88.925 b = 0.038 c = 39.381 trun = 0.101 Sum.Sq.Res = 8.98358 Predicted percentiles Predicted percentiles Predicted percentilesPredicted percentilesPredicted percentiles O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es FigureÂ 32. Q-q plots for observed versus adjusted predicted longitudinal distance distributions (sample model) for CUVs at 2U roads. Sum.Sq.Res. 5 sum of the squares of the residuals (estimate of the error between the observed and predicted).

Encroachment Relationships 51Â Â a = 7.295 b = 0.005 c = 39.879 trun = 0.23 Sum.Sq.Res = 5.35658 a = 7.295 b = 0.005 c = 39.879 trun = 0.23 Sum.Sq.Res = 5.99412 a = 7.295 b = 0.005 c = 39.879 trun = 0.23 Sum.Sq.Res = 6.07357 a = 7.295 b = 0.005 c = 39.879 trun = 0.23 Sum.Sq.Res = 5.24419 a = 7.295 b = 0.005 c = 39.879 trun = 0.23 Sum.Sq.Res = 5.56634 Predicted percentiles Predicted percentiles Predicted percentiles Predicted percentiles Predicted percentiles O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es FigureÂ 33. Q-q plots for observed versus adjusted predicted longitudinal distance distributions (sample model) for CUVs at 4D roads. Sum.Sq.Res. 5 sum of the squares of the residuals (estimate of the error between the observed and predicted).

52 Development of Clear Recovery Area Guidelines a = 85.977 b = 0.02 c = 24.485 trun = 0.187 Sum.Sq.Res = 6.97588 a = 85.977 b = 0.02 c = 24.485 trun = 0.187 Sum.Sq.Res = 6.18441 a = 85.977 b = 0.02 c = 24.485 trun = 0.187 Sum.Sq.Res = 7.91525 a = 85.977 b = 0.02 c = 24.485 trun = 0.187 Sum.Sq.Res = 7.59335 a = 85.977 b = 0.02 c = 24.485 trun = 0.187 Sum.Sq.Res = 7.16276 Predicted percentilesPredicted percentiles Predicted percentiles Predicted percentiles Predicted percentiles O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es FigureÂ 34. Q-q plots for observed versus adjusted predicted longitudinal distance distributions (sample model) for SUVs at 2U roads. Sum.Sq.Res. 5 sum of the squares of the residuals (estimate of the error between the observed and predicted).

Encroachment Relationships 53Â Â a = 71.859 b = 0.05 c = 39.889 trun = 0.156 Sum.Sq.Res = 7.87364 a = 71.859 b = 0.05 c = 39.889 trun = 0.156 Sum.Sq.Res = 6.83467 a = 71.859 b = 0.05 c = 39.889 trun = 0.156 Sum.Sq.Res = 6.8945 a = 71.859 b = 0.05 c = 39.889 trun = 0.156 Sum.Sq.Res = 6.4468 a = 71.859 b = 0.05 c = 39.889 trun = 0.156 Sum.Sq.Res = 6.7219 Predicted percentiles O bs er ve d pe rc en til es Predicted percentiles Predicted percentiles Predicted percentilesPredicted percentiles O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es O bs er ve d pe rc en til es FigureÂ 35. Q-q plots for observed versus adjusted predicted longitudinal distance distributions (sample model) for SUVs at 4D roads. Sum.Sq.Res. 5 sum of the squares of the residuals (estimate of the error between the observed and predicted).

54 Development of Clear Recovery Area Guidelines Impact Speed Models The Impact Speed Models determine the speed at which an encroaching vehicle would hit an object at different clear zone distances. This information is subsequently used within the severity module to determine the severity of the impact. The simulated encroachment data was further analyzed to develop speed distributions for the encroaching vehicles that reach a given lateral offset. As noted, the speed distributions will be used to estimate crash severity (i.e., injury probability) for an impact with a fixed object encountered at the lateral distance of interest. FigureÂ 36 and FigureÂ 37 present the density and boxplot distribution of resultant speed for the encroaching vehicles that reach the prescribed lateral distance. It can be observed in FigureÂ 36 that five peaks exist, which are defined by the values of encroachment speed used in the simulation matrix. The simulation matrix involved five encroachment speeds: 25 mph (40 kph), 35 mph (56 kph), 45 mph (72 kph), 55 mph (89 kph), and 65 mph (105 kph). FigureÂ 37 shows a boxplot of the absolute speeds of the vehicles at different lateral offsets. It can be observed that overall, the median value ranged between 39 mph (62 kph) and 41 mph (66 kph). Incorporating Posted Speed Limit in the Impact Speed Models The research team also worked on incorporating posted speed limits into the Impact Speed Models. The intent is to be able to distinguish clear zone guidance by posted speed limit along with other roadway and roadside design variables. First, the team developed a relationship between the departure speed and the posted speed limit using the crash database from NCHRP Web-Only Document 341 (3). The purpose was FigureÂ 36. Absolute speed distribution.

Encroachment Relationships 55Â Â two-fold: improve the prediction of impact speeds for the analysis and provide sensitivity to speed limit ranges in the risk calculations. A total of 373 right-departure, fixed-object crashes was used to estimate the corresponding model presented in TableÂ 21. The relationship associated with the model is shown graphically in FigureÂ 38. The developed relationship was then used to predict the posted speed limit given the depar- ture speed and other parameters used in the simulation analysis. The predicted speed limit is treated as âpointâ data, meaning it is the average of all predicted speeds given the existing condi- tions. TableÂ 22 presents the distribution of the predicted mean posted speed limit by facility type. It can be observed that the predicted mean posted speed limit is slightly higher for 4D facilities than for 2U facilities. Further, given the developed speed model, the research team simulated the speed limit to pro- duce a weighted speed limit. This enables the model to cover a wider range of data than simply relying on the mean predicted speed limit. The simulated posted speed limit data were divided FigureÂ 37. Boxplot of the absolute speed by lateral distances. Parameter Estimate StandardError z-Value P-Value Significance Intercept 45.160170 3.188638 14.163 < 2e-16 *** Departure Speed 0.346342 0.088586 3.910 0.00011 *** Shoulder Width 2.094883 1.020895 2.052 0.04088 * Departure Speed Squared â0.000800 0.000548 â1.452 0.14743 ââ Unknown Shoulder Condition 10.464750 4.457715 2.348 0.01943 * Departure Speed*No Shoulder Condition â0.137420 0.029335 â4.684 3.96E-06 *** Departure Speed Squared*4D 0.000611 0.000258 2.364 0.01859 * NOTE: * in the left column indicates interaction between the variables; *** = statistically significant at the 99% confidence interval; * in the table = statistically significant at the 95% confidence interval; â = not statistically significant at the 90% confidence interval. TableÂ 21. Posted speed model derived from NCHRP Web-Only Document 341 data (3).

56 Development of Clear Recovery Area Guidelines into three categories of speed limits: low speed limit (less than 45 mph), medium speed limit (45â55 mph), and high speed limit (greater than or equal to 60 mph). TableÂ 23 through TableÂ 30 present the resulting Impact Speed Models. Two ranges of posted speed limits were incorporated into the model: the medium speed limit of 45â55 mph and the high speed limit greater than or equal to 60 mph. It can be observed that an increase in speed limit is associated with increased impact speed. The crash database in NCHRP Web-Only Docu- ment 341 did not support additional categorization of posted speed (3). It can be noted that the magnitude of impact speed increase varies by facility type (i.e., 2U and 4D) and the posted-speed-limit category. It can be observed that an increase in impact speed is indicated by the positive coefficients associated with the 45â55 mph and 60 mph speed limit categories. Vertical grade, a decrease in foreslope rate, an increase in foreslope, and ditch bottom width increase the impact speed. Conversely, horizontal curvature, an increase in shoulder width, and increasing lateral and longitudinal distance from the point of departure are associated with a decrease in impact speed. FigureÂ 38. Actual versus predicted posted speed. Facility Minimum 1stQuartile Median Mean 3rd Quartile Maximum 2U 38.75 46.11 51.35 50.84 57.01 61.14 4D 39.37 47.91 52.46 53.36 59.01 65.32 TableÂ 22. Distribution of predicted mean posted speed limit (mph) by facility type.

Fixed Effects Estimate StandardError z-Value Intercept 6.38E+01 1.48E-01 431.552 Vertical Grade 5.35E-02 2.36E-03 22.673 Horizontal Curve Radius â1.33E+00 5.82E-02 â22.841 Shoulder Width â1.69E+00 1.48E-02 â114.647 Foreslope 4.65E-02 1.73E-02 2.688 Foreslope Width 3.02E-02 6.13E-03 4.925 Bottom Ditch Width 2.63E-02 5.96E-03 4.415 Lateral Distance â1.68E-01 6.29E-04 â267.679 Longitudinal Distance â5.65E-02 2.59E-04 â217.810 Simulated Speed Limit 45â55 mph 2.59E+01 5.66E-02 456.937 Simulated Speed Limit â¥ 60 mph 3.16E+01 6.80E-02 464.491 Shoulder Width*Foreslope â4.75E-03 2.24E-03 â2.118 Lateral Distance*Longitudinal Distance 6.24E-04 3.56E-06 175.157 Random Effects Variance StandardDeviation Intercept 247.87 15.744 Residual 15.94 3.993 Number of Observations 1,697,198 NOTE: *indicates an interaction between the variables. TableÂ 23. Sedan speed model for 2U facility. NOTE: *indicates interaction between the variables. Fixed Effects Estimate StandardError z-Value Intercept 7.05E+01 1.64E-01 429.604 Vertical Grade 5.26E-02 2.64E-03 19.899 Horizontal Curve Radius â1.56E+00 6.51E-02 â23.975 Shoulder Width â9.89E-01 1.64E-02 â60.293 Foreslope 9.58E-02 1.94E-02 4.948 Foreslope Width 2.13E-02 6.87E-03 3.102 Bottom Ditch Width 2.07E-02 6.68E-03 3.092 Lateral Distance â1.64E-01 6.46E-04 â254.536 Longitudinal Distance â5.18E-02 2.43E-04 â213.482 Simulated Speed Limit 45â55 mph 6.56E+00 3.42E-02 192.130 Simulated Speed Limit â¥ 60 mph 1.73E+01 5.77E-02 300.377 Shoulder Width*Foreslope â9.49E-03 2.51E-03 â3.774 Lateral Distance*Longitudinal Distance 5.57E-04 3.38E-06 164.893 Random Effects Variance StandardDeviation Intercept 261.28 16.164 Residual 16.44 4.054 Number of Observations 1,697,198 TableÂ 24. Sedan speed model for 4D facility. NOTE: *indicates interaction between the variables. Fixed Effects Estimate StandardError z-Value Intercept 6.60E+01 1.47E-01 448.269 Vertical Grade 6.53E-02 2.33E-03 28.049 Horizontal Curve Radius â1.65E+00 5.73E-02 â28.787 Shoulder Width â1.84E+00 1.46E-02 â126.184 Foreslope 4.61E-02 1.71E-02 2.695 Foreslope Width â1.32E-03 6.05E-03 â0.218 Bottom Ditch Width 4.62E-03 5.89E-03 0.785 Lateral Distance â1.75E-01 6.61E-04 â264.428 Longitudinal Distance â7.11E-02 2.79E-04 â254.737 Simulated Speed Limit 45â55 mph 2.68E+01 5.69E-02 470.423 Simulated Speed Limit â¥ 60 mph 3.36E+01 6.93E-02 484.217 Shoulder Width*Foreslope â8.98E-03 2.21E-03 â4.067 Lateral Distance*Longitudinal Distance 9.20E-04 3.72E-06 247.676 Random Effects Variance StandardDeviation Intercept 250.19 15.818 Residual 21.48 4.635 Number of Observations 1,892,359 TableÂ 25. Pickup speed model for 2U facility.

NOTE: *indicates interaction between the variables. Fixed Effects Estimate StandardError z-Value Intercept 7.24E+01 1.61E-01 448.770 Vertical grade 6.48E-02 2.58E-03 25.118 Horizontal Curve Radius â2.01E+00 6.33E-02 â31.812 Shoulder Width â1.15E+00 1.60E-02 â71.803 Foreslope 1.06E-01 1.90E-02 5.595 Foreslope Width â1.36E-02 6.70E-03 â2.025 Bottom Ditch Width â2.82E-03 6.52E-03 -0.433 Lateral Distance â1.67E-01 6.67E-04 â250.449 Longitudinal Distance â6.64E-02 2.60E-04 â255.769 Simulated Speed Limit 45â55 mph 7.44E+00 3.53E-02 210.541 Simulated Speed Limit â¥ 60 mph 1.99E+01 5.97E-02 333.097 Shoulder Width*Foreslope â1.53E-02 2.45E-03 â6.270 Lateral Distance*Longitudinal Distance 8.48E-04 3.48E-06 243.764 Random Effects Variance StandardDeviation Intercept 312.18 17.669 Residual 22.85 4.781 Number of Observations 1,892,359 TableÂ 26. Pickup speed model for 4D facility. NOTE: *indicates interaction between the variables. Fixed Effects Estimate StandardError z-Value Intercept 6.32E+01 1.47E-01 428.670 Vertical Grade 5.61E-02 2.34E-03 23.997 Horizontal Curve Radius â1.24E+00 5.74E-02 â21.583 Shoulder Width â1.88E+00 1.47E-02 â127.952 Foreslope 1.00E-01 1.72E-02 5.820 Foreslope Width 3.96E-02 6.07E-03 6.521 Bottom Ditch Width 4.23E-02 5.91E-03 7.164 Lateral Distance â2.28E-01 6.95E-04 â328.279 Longitudinal Distance â5.92E-02 2.37E-04 â250.007 Simulated Speed Limit 45â55 mph 2.78E+01 5.80E-02 479.124 Simulated Speed Limit â¥ 60 mph 3.51E+01 7.11E-02 493.317 Shoulder Width*Foreslope â8.07E-03 2.22E-03 â3.637 Lateral Distance*Longitudinal Distance 6.75E-04 3.65E-06 185.091 Random Effects Variance StandardDeviation Intercept 247.2 15.722 Residual 17.9 4.231 Number of Observations 1,568,715 TableÂ 27. SUV speed model for 2U facility. NOTE: *indicates interaction between the variables. Fixed Effects Estimate StandardError z-Value Intercept 7.00E+01 1.61E-01 433.890 Vertical Grade 5.51E-02 2.58E-03 21.356 Horizontal Curve Radius â1.65E+00 6.32E-02 â26.070 Shoulder Width â1.21E+00 1.61E-02 â75.090 Foreslope 1.51E-01 1.90E-02 7.974 Foreslope Width 1.88E-02 6.70E-03 2.806 Bottom Ditch Width 2.89E-02 6.52E-03 4.437 Lateral Distance â2.22E-01 6.97E-04 â318.562 Longitudinal Distance â5.34E-02 2.14E-04 â249.141 Simulated Speed Limit 45â55 mph 8.07E+00 3.68E-02 219.384 Simulated Speed Limit â¥ 60 mph 2.12E+01 6.18E-02 343.382 Shoulder Width*Foreslope â1.29E-02 2.45E-03 â5.265 Lateral Distance*Longitudinal Distance 5.90E-04 3.34E-06 176.451 Random Effects Variance StandardDeviation Intercept 306.47 17.51 Residual 19.09 4.37 Number of Observations 1,568,715 TableÂ 28. SUV speed model for 4D facility.

Encroachment Relationships 59Â Â NOTE: *indicates interaction between the variables. Fixed Effects Estimate StandardError z-Value Intercept 6.48E+01 1.51E-01 428.044 Vertical Grade 5.61E-02 2.41E-03 23.332 Horizontal Curve Radius â7.87E-01 5.90E-02 -13.335 Shoulder Width â1.71E+00 1.50E-02 â114.116 Foreslope 2.24E-02 1.77E-02 1.265 Foreslope Width 2.75E-02 6.25E-03 4.402 Bottom Ditch Width 2.64E-02 6.08E-03 4.339 Lateral Distance â1.73E-01 6.59E-04 â263.036 Longitudinal Distance â6.67E-02 2.69E-04 â247.665 Simulated Speed Limit 45â55 mph 2.53E+01 5.72E-02 441.401 Simulated Speed Limit â¥ 60 mph 3.09E+01 6.80E-02 453.800 Shoulder Width*Foreslope â3.77E-03 2.28E-03 â1.653 Lateral Distance*Longitudinal Distance 6.73E-04 3.48E-06 193.387 Random Effects Variance StandardDeviation Intercept 265.32 16.289 Residual 16.19 4.023 Number of Observations 1,748,198 TableÂ 29. CUV speed model for 2U facility. NOTE: *indicates interaction between the variables. Fixed Effects Estimate StandardError z-Value Intercept 7.18E+01 1.68E-01 427.135 Vertical Grade 5.55E-02 2.70E-03 20.539 Horizontal Curve Radius â1.18E+00 6.61E-02 â17.801 Shoulder Width â9.82E-01 1.67E-02 â58.902 Foreslope 6.99E-02 1.99E-02 3.518 Foreslope Width 1.67E-02 7.02E-03 2.377 Bottom Ditch Width 2.10E-02 6.83E-03 3.074 Lateral Distance â1.59E-01 6.80E-04 â232.993 Longitudinal Distance â6.43E-02 2.55E-04 â252.066 Simulated Speed Limit 45â55 mph 5.84E+00 3.29E-02 177.520 Simulated Speed Limit â¥ 60 mph 1.62E+01 5.66E-02 286.801 Shoulder Width*Foreslope â8.60E-03 2.56E-03 â3.360 Lateral Distance*Longitudinal Distance 5.87E-04 3.33E-06 176.146 Random Effects Variance StandardDeviation Intercept 339.0 18.413 Residual 17.4 4.171 Number of Observations 1,748,198 TableÂ 30. CUV speed model for 4D facility. Impact Angle Models The impact angle model determines the angle at which a vehicle would hit an object at dif- ferent clear zone distances. This information is subsequently used as part of the determination of impact probability with fixed objects at the clear zone edge by defining a vehicle projection envelope based on the impact angle, vehicle CG position, and vehicle width. To determine the impact angle at a given distance, the team selected the vehicle CG coordi- nates at three points along the vehicle trajectory at or near the prescribed lateral offset distance. Statistical models were then developed to predict the impact angles for 2U and 4D facilities. TableÂ 31 through TableÂ 38 present the model results for 2U and 4D facilities for the four different vehicle types (crossover or CUV, SUV, pickup, and sedan). Design variables associated with greater impact angles are wider foreslope and ditch bottom, wider shoulder, flatter backslope, and horizontal

NOTE: *indicates interaction between the variables. Fixed Effects Estimate Standard Error z-Value Intercept 4.16E+00 3.77E-03 1105.86 Foreslope Width 1.03E-02 1.23E-04 83.48 Bottom Ditch Width 4.80E-03 1.19E-04 40.23 Lateral Distance â9.16E-03 5.16E-05 â177.70 Longitudinal Distance â1.44E-02 1.68E-05 â858.75 Foreslope â3.95E-03 1.84E-04 â21.50 Shoulder Width 8.88E-03 1.20E-04 74.19 Vertical Grade â9.00E-05 8.51E-05 â1.06 Horizontal Curve Radius 1.74E-03 1.97E-03 0.88 Backslope 2.10E-02 3.32E-04 63.12 Backslope Width â3.20E-03 1.23E-04 â26.05 Lateral Distance*Longitudinal Distance 1.77E-04 2.70E-07 656.63 Horizontal Curve Radius*Vertical Grade â4.89E-04 1.12E-04 â4.36 Random Effects Variance Standard Deviation Intercept 0.05837 0.2416 Residual 0.11534 0.3396 Number of Observations 1,759,850 TableÂ 31. CUV impact angle model for 2U facility. NOTE: *indicates interaction between the variables. Fixed Effects Estimate Standard Deviation z-Value Intercept 4.02E+00 3.76E-03 1068.55 Foreslope Width 1.06E-02 1.24E-04 85.33 Bottom Ditch Width 4.81E-03 1.21E-04 39.86 Lateral Distance â7.02E-03 5.18E-05 â135.31 Longitudinal Distance â1.34E-02 1.56E-05 â856.15 Foreslope â4.60E-03 1.86E-04 â24.74 Shoulder Width 8.76E-03 1.21E-04 72.32 Vertical Grade â1.55E-04 8.64E-05 â1.79 Horizontal Curve Radius â1.67E-03 2.00E-03 â0.84 Backslope 2.03E-02 3.36E-04 60.54 Backslope Width â3.10E-03 1.24E-04 â25.00 Lateral Distance*Longitudinal Distance 1.63E-04 2.53E-07 643.17 Horizontal Curve Radius*Vertical Grade â3.86E-04 1.14E-04 â3.40 Random Effects Variance Standard Deviation Intercept 0.06585 0.2566 Residual 0.1182 0.3438 Number of Observations 1,759,850 TableÂ 32. CUV impact angle model for 4D facility. NOTE: *indicates interaction between the variables. NOTE: *indicates interaction between the variables. Fixed Effects Estimate Standard Deviation z-Value Intercept 3.87E+00 4.36E-03 887.67 Foreslope Width 1.10E-02 1.49E-04 74.15 Bottom Ditch Width 5.78E-03 1.45E-04 39.99 Lateral Distance â1.44E-02 5.82E-05 â248.37 Longitudinal Distance â1.31E-02 1.65E-05 â790.83 Foreslope 9.80E-04 2.22E-04 4.42 Shoulder Width 7.18E-03 1.45E-04 49.59 Vertical Grade 7.84E-04 1.04E-04 7.56 Horizontal Curve Radius 9.48E-02 2.38E-03 39.89 Backslope 1.82E-02 4.02E-04 45.36 Backslope Width â1.55E-03 1.48E-04 â10.45 Lateral Distance*Longitudinal Distance 1.87E-04 3.04E-07 615.46 Horizontal Curve Radius*Vertical Grade â9.72E-04 1.36E-04 â7.14 Random Effects Variance Standard Deviation Intercept 0.08844 0.2974 Residual 0.13978 0.3739 Number of Observations 1,578,153 Intercept 0.04965 0.2228 Residual 0.08892 0.2982 Number of Observations 1,900,968 TableÂ 33. SUV impact angle model for 2U facility.

NOTE: *indicates interaction between the variables. Fixed Effects Estimate Standard Deviation z-Value Intercept 3.64E+00 4.46E-03 815.67 Foreslope Width 1.20E-02 1.55E-04 77.33 Bottom Ditch Width 6.14E-03 1.50E-04 40.84 Lateral Distance â1.32E-02 5.78E-05 â228.03 Longitudinal Distance â1.14E-02 1.50E-05 â756.26 Foreslope 2.43E-04 2.31E-04 1.05 Shoulder Width 7.89E-03 1.51E-04 52.40 Vertical Grade 8.25E-04 1.08E-04 7.64 Horizontal Curve Radius 1.12E-01 2.48E-03 45.03 Backslope 1.78E-02 4.18E-04 42.67 Backslope Width â1.49E-03 1.54E-04 â9.67 Lateral Distance*Longitudinal Distance 1.67E-04 2.78E-07 601.59 Horizontal Curve Radius*Vertical Grade â1.07E-03 1.42E-04 â7.52 Random Effects Variance Standard Deviation Intercept 0.1078 0.3283 Residual 0.1455 0.3814 Number of Observations 1,578,153 TableÂ 34. SUV impact angle model for 4D facility. NOTE: *indicates interaction between the variables. Fixed Effects Estimate Standard Deviation z-Value Intercept 4.00E+00 3.29E-03 1216.49 Foreslope Width 9.00E-03 1.09E-04 82.31 Bottom Ditch Width 5.49E-03 1.06E-04 51.64 Lateral Distance â5.82E-03 4.02E-05 â144.85 Longitudinal Distance â1.35E-02 1.35E-05 â996.62 Foreslope â3.74E-04 1.64E-04 -2.29 Shoulder Width 8.40E-03 1.07E-04 78.79 Vertical Grade â1.78E-04 7.63E-05 â2.33 Horizontal Curve Radius 2.17E-02 1.76E-03 12.36 Backslope 2.28E-02 2.95E-04 77.07 Backslope Width â2.62E-03 1.09E-04 â23.97 Lateral Distance*Longitudinal Distance 1.55E-04 2.19E-07 706.86 Horizontal Curve Radius*Vertical Grade â1.12E-04 1.00E-04 â1.12 Random Effects Variance Standard Deviation Intercept 0.04965 0.2228 Residual 0.08892 0.2982 Number of Observations 1,900,968 TableÂ 35. Pickup impact angle model for 2U facility. NOTE: *indicates interaction between the variables. Fixed Effects Estimate Standard Deviation z-Value Intercept 3.85E+00 3.30E-03 1166.103 Foreslope Width 9.43E-03 1.11E-04 85.023 Bottom Ditch Width 5.61E-03 1.08E-04 51.997 Lateral Distance â4.15E-03 4.03E-05 â102.884 Longitudinal Distance â1.25E-02 1.26E-05 â988.250 Foreslope â9.85E-04 1.66E-04 â5.933 Shoulder Width 8.47E-03 1.08E-04 78.271 Vertical Grade â2.35E-04 7.76E-05 â3.023 Horizontal Curve Radius 2.65E-02 1.79E-03 14.844 Backslope 2.20E-02 3.00E-04 73.397 Backslope Width â2.52E-03 1.11E-04 â22.723 Lateral Distance*Longitudinal Distance 1.43E-04 2.04E-07 701.485 Horizontal Curve Radius*Vertical Grade â6.79E-05 1.02E-04 â0.667 Random Effects Variance Standard Deviation Intercept 0.0555 0.2356 Residual 0.0940 0.3066 Number of Observations 1,900,968 TableÂ 36. Pickup impact angle model for 4D facility.

62 Development of Clear Recovery Area Guidelines NOTE: *indicates interaction between the variables. Fixed Effects Estimate Standard Deviation z-Value Intercept 4.15E+00 4.46E-03 928.854 Foreslope Width 9.40E-03 1.51E-04 62.170 Bottom Ditch Width 3.15E-03 1.47E-04 21.467 Lateral Distance â9.13E-03 5.22E-05 â175.074 Longitudinal Distance â1.40E-02 1.79E-05 â783.619 Foreslope â7.57E-03 2.26E-04 â33.450 Shoulder Width 6.89E-03 1.48E-04 46.669 Vertical Grade â1.76E-04 1.06E-04 â1.664 Horizontal Curve Radius 2.76E-02 2.43E-03 11.381 Backslope 1.92E-02 4.09E-04 46.990 Backslope Width â3.13E-03 1.51E-04 â20.699 Lateral Distance*Longitudinal Distance 1.69E-04 2.86E-07 589.263 Horizontal Curve Radius*Vertical Grade â1.62E-05 1.39E-04 â0.117 Random Effects Variance Standard Deviation Intercept 0.1025 0.3202 Residual 0.1197 0.3460 Number of Observations 1,709,512 TableÂ 37. Sedan impact angle model for 2U facility. NOTE: *indicates interaction between the variables. Fixed Effects Estimate StandardError z-Value Intercept 3.96E+00 4.45E-03 889.466 Foreslope Width 9.93E-03 1.53E-04 65.065 Bottom Ditch Width 3.19E-03 1.48E-04 21.556 Lateral Distance â7.99E-03 5.18E-05 â154.195 Lateral Distance Squared â1.25E-02 1.65E-05 â761.250 Longitudinal Distance â8.12E-03 2.28E-04 â35.553 Foreslope 7.19E-03 1.49E-04 48.220 Shoulder Width â2.06E-04 1.07E-04 â1.929 Vertical Grade 3.60E-02 2.46E-03 14.672 Horizontal Curve Radius 1.84E-02 4.12E-04 44.595 Backslope â3.01E-03 1.52E-04 â19.767 Backslope Width 1.53E-04 2.65E-07 577.641 Horizontal Curve Radius*Vertical Grade â2.55E-05 1.40E-04 â0.182 Random Effects Variance Standard Deviation Intercept 0.1127 0.3357 Residual 0.1224 0.3499 Number of Observations 1,709,512 TableÂ 38. Sedan impact angle model for 4D facility. curvature. Increased lateral and longitudinal distance, flatter foreslope, wider backslope, and vertical grade are associated with smaller impact angles. The lateral and longitudinal distance traveled by the encroaching vehicle has a significant interactive relationship with the angle of impact. Rollover Probability Models The team developed logistic regression models to predict the likelihood of rollover before reaching a certain lateral threshold (clear zone distance). All roadway and roadside design vari- ables and various combinations thereof were considered in the model. Only variables that were statistically significant at a 95% confidence interval were retained in the final models. TableÂ 39 through TableÂ 46 present the models for rollover probability before reaching a given lateral distance for all vehicle types expressed in terms of statistically significant design variables

Encroachment Relationships 63Â Â NOTE: AIC = Akaike information criteria. Fixed Effects Estimate StandardError z-Value P-Value Intercept â1.711 0.204 â8.37 <0.001 Shoulder Width â0.082 0.005 â15.35 <0.001 Foreslope â0.642 0.058 â11.03 <0.001 Foreslope Squared 0.035 0.004 7.71 <0.001 Foreslope Width â0.062 0.005 â11.75 <0.001 Backslope â0.276 0.015 â17.98 <0.001 Bottom Ditch Width â0.111 0.005 â20.54 <0.001 Lateral Distance 0.264 0.007 37.43 <0.001 Lateral Distance Squared â0.003 0.0001 â33.03 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 39. CUV rollover probability model for 2U roads. NOTE: AIC = Akaike information criteria. Fixed Effects Estimate StandardError z-Value P-Value Intercept â1.900 0.203 â9.35 <0.001 Shoulder Width â0.081 0.005 â15.27 <0.001 Foreslope â0.626 0.058 â10.78 <0.001 Foreslope Squared 0.034 0.004 7.54 <0.001 Foreslope Width â0.065 0.005 â12.41 <0.001 Backslope â0.279 0.015 â18.15 <0.001 Bottom Ditch Width â0.108 0.005 â20.10 <0.001 Lateral Distance 0.277 0.007 39.20 <0.001 Lateral Distance Squared â0.003 0.0001 â34.46 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 40. CUV rollover probability model for 4D roads. NOTE: AIC = Akaike information criteria. Fixed Effects Estimate StandardError z-Value P-Value Intercept â0.054 0.306 â0.18 0.860 Shoulder Width â0.076 0.008 â9.74 <0.001 Foreslope â1.005 0.092 â10.89 <0.001 Foreslope Squared 0.054 0.007 7.41 <0.001 Foreslope Width â0.070 0.008 â9.00 <0.001 Backslope â0.560 0.027 â20.72 <0.001 Bottom Ditch Width â0.163 0.009 â18.90 <0.001 Lateral Distance 0.238 0.010 23.70 <0.001 Lateral Distance Squared â0.003 0.0001 â20.95 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 41. Pickup rollover probability model for 2U roads.

64 Development of Clear Recovery Area Guidelines NOTE: AIC = Akaike information criteria. Fixed Effects Estimate StandardError z-Value P-Value Intercept â0.525 0.3050 â1.72 0.085 Shoulder Width â0.068 0.0080 â8.83 <0.001 Foreslope â1.005 0.0910 â11.01 <0.001 Foreslope Squared 0.055 0.0070 7.59 <0.001 Foreslope Width â0.069 0.0080 â9.01 <0.001 Backslope â0.579 0.0270 â21.39 <0.001 Bottom Ditch Width â0.162 0.0080 â19.12 <0.001 Lateral Distance 0.267 0.0100 25.66 <0.001 Lateral Distance Squared â0.003 0.0001 â22.59 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 42. Pickup rollover probability model for 4D roads. NOTE: AIC = Akaike information criteria. Fixed Effects Estimate StandardError z-Value P-Value Intercept â0.371 0.2330 â1.59 0.112 Shoulder Width â0.109 0.0060 â17.87 <2e-16 Foreslope â0.917 0.0670 â13.64 <2e-16 Foreslope Squared 0.050 0.0050 9.67 <2e-16 Foreslope Width â0.093 0.0060 â15.46 <2e-16 Backslope â0.527 0.0200 â26.98 <2e-16 Bottom Ditch Width â0.201 0.0070 â29.60 <2e-16 Lateral Distance 0.322 0.0090 35.99 <2e-16 Lateral Distance Squared â0.004 0.0001 â31.01 <2e-16 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 43. SUV rollover probability model for 2U roads. NOTE: AIC = Akaike information criteria. Fixed Effects Estimate StandardError z-Value P-Value Intercept â0.734 0.232 â3.16 0.002 Shoulder Width â0.107 0.006 â17.58 <0.001 Foreslope â0.889 0.0670 â13.24 <0.001 Foreslope Squared 0.049 0.0050 9.33 <0.001 Foreslope Width â0.095 0.0060 â15.81 <0.001 Backslope â0.504 0.0190 â26.12 <0.001 Bottom Ditch Width â0.195 0.0070 â29.06 <0.001 Lateral Distance 0.341 0.0090 37.32 <0.001 Lateral Distance Squared â0.004 0.0001 â32.11 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 44. SUV rollover probability model for 4D roads.

Encroachment Relationships 65Â Â for 2U and 4D roadways. Overall, results suggest that the probability of rollover increases as the lateral distance increases as indicated by the positive coefficients associated with the lateral dis- tance variable in TableÂ 39 through TableÂ 46. However, the coefficient of the square of the lateral distance variable is negative, indicating that at a certain distance, the rollover probability starts to decline. Upon further investigation, it was found that the rollover probabilities started to decline approximately 40Â ft from the edge of the roadway. This may be due to an associated reduction in speed or interaction with the ditch backslope. Other variables of interest that were found to be statistically significant in terms of rollover prediction include shoulder width, foreslope ratio, foreslope width, backslope ratio, and ditch bottom width. The presence of the square of the foreslope ratio indicates that the relationship is quadratic rather than linear. Irrespective of the vehicle type and facility type, a wider shoulder width is associated with a lower rollover probability. Likewise, the model indicates that a flatter foreslope or backslope ratio is also associated with a lower likelihood of rollover. NOTE: AIC = Akaike information criteria. Fixed Effects Estimate StandardError z-Value P-Value Intercept â0.694 0.2830 â2.45 0.014 Shoulder Width â0.084 0.0070 â11.87 <0.001 Foreslope â0.998 0.0820 â12.11 <0.001 Foreslope Squared 0.056 0.0060 8.61 <0.001 Foreslope Width â0.032 0.0070 â4.52 <0.001 Backslope â0.637 0.0250 â25.06 <0.001 Bottom Ditch Width â0.163 0.0080 â21.12 <0.001 Lateral Distance 0.270 0.0100 28.28 <0.001 Lateral Distance Squared â0.003 0.0001 â24.39 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 45. Sedan rollover probability model for 2U roads. NOTE: AIC = Akaike information criteria. Fixed Effects Estimate StandardError z-Value P-Value Intercept â1.113 0.2860 â3.89 <0.001 Shoulder Width â0.083 0.0070 â11.47 <0.001 Foreslope â0.969 0.0830 â11.62 <0.001 Foreslope Squared 0.054 0.0070 8.21 <0.001 Foreslope Width â0.036 0.0070 â5.04 <0.001 Backslope â0.595 0.0250 â23.72 <0.001 Bottom Ditch Width â0.152 0.0080 â19.82 <0.001 Lateral Distance 0.281 0.0100 29.03 <0.001 Lateral Distance Squared â0.003 0.0001 â24.97 <0.001 Model Summary Number of Observations 3,628,800 AIC 18 TableÂ 46. Sedan rollover probability model for 4D roads.