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Design Guidelines for Horizontal Sightline Offsets (2019)

Chapter: Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD

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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
×
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
×
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Suggested Citation:"Chapter 5 - Reliability Analysis Model for Horizontal Curves with Limited SSD." National Academies of Sciences, Engineering, and Medicine. 2019. Design Guidelines for Horizontal Sightline Offsets. Washington, DC: The National Academies Press. doi: 10.17226/25537.
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28 This chapter presents a reliability analysis model for horizontal curves with limited SSD due to sight obstructions on the inside of the curve. Any horizontal curve is likely to experi- ence more crashes than a tangent roadway section simply because of the presence of the curve (AASHTO 2010; AASHTO 2014) even in the absence of any sight distance limitation. The reliability analysis model developed in this research can determine the sight distance profile on any horizontal curve with a specific sight obstruction on the inside of the curve and can estimate the opportunities for crashes to occur due to limited SSD. The measure of oppor- tunities for crashes to occur is the number of events during a 1-year time period in which there is increased likelihood that a crash will occur in a portion of the roadway with limited SSD where there would be no increased likelihood of a crash if the horizontal sight obstruction were not present. The reliability model is applicable to the direction of travel closest to the inside of the horizontal curve. While the likelihood of sight-distance-related collisions in a particular direction of travel is greatest in the travel lane closest to the inside of the curve in that direction of travel, the model addresses the opportunities for sight-distance-related crashes in all travel lanes for a specific direction of travel. If the minimum ASSD is never less than the applicable DSSD at any point approaching or on the horizontal curve in question, then the opportunity for sight-distance-related crashes is zero. If the minimum ASSD is less than the applicable DSSD at any point approaching or on the horizontal curve, then the opportunity for sight-distance-related crashes is greater than zero. Most situations in which sight-distance-related crashes could occur do not result in crashes. Many factors make crashes unlikely even when the opportunity for a crash exists. For example, drivers may avoid crashes by decelerating more rapidly than assumed in design criteria or by making avoidance maneuvers, such as changing lanes or using the shoulder (if available). Also, the opportunity for a sight-distance-related crash may not persist for as long as assumed in the model. The model is purposely conservative in overestimating the opportunities for sight- distance-related crashes. Thus, the model estimates do not predict the number of crashes that will occur where a horizontal sight obstruction is present, but may be useful in estimating the relative frequency of opportunities for crashes at specific sites. At some sites, the opportunity for sight-distance-related crashes, even though nonzero, may be very small and may justify only low-cost mitigation measures. At other sites, the opportunity for sight-distance-related crashes may be substantially higher and may justify higher cost mitigation measures. The reliability analysis model considers two alternatives for measuring ASSD. The basic method incorporates the sight distance measurement assumptions used in the AASHTO Green Book (2011), including an assumed height of 2.0 ft (equivalent to vehicle taillight height) for the object to be seen and the assumption that sight distance is measured along the center of a given travel lane. The alternative method allows the user of the reliability model to specify an increased height for the object to be seen. For example, if the object to be seen is another vehicle, C H A P T E R 5 Reliability Analysis Model for Horizontal Curves with Limited SSD

Reliability Analysis Model for Horizontal Curves with Limited SSD 29 a more realistic assumption for the height of the object to be seen would be 3.5 ft, which is used as the assumed object height in passing sight distance design. In addition, the alternative method assumes that the lateral position of the driver’s eye within a given travel lane on a horizontal curve is at 75 percent of the distance from the inside to the outside edge of the travel lane on a curve to the right and at 25 percent of the distance from the inside to the outside edge of the travel lane on a curve to the left. These assumptions for the lateral position of the driver’s eye appear more realistic than the assumption that the driver’s eye is in the center of the lane. The reliability analysis model assumes that vehicles ahead on the roadway are visible to an approaching driver only when there is a direct line of sight from the driver’s eye to the other vehicle. This assumption represents daylight conditions. At night, atmospheric diffusion of light from the headlights or taillights of a vehicle ahead may make the presence of a vehicle apparent even when there is no direct line of sight from the driver’s eye to the other vehicle. Thus, the reliability analysis addresses the most critical condition. The input data, output results, and the reliability model logic are explained in this section. A spreadsheet tool has been developed to apply the reliability model. A users guide to this spreadsheet tool is presented in Appendix B with more details concerning both the input data and the output results. 5.1 Input Data for the Reliability Analysis Model The reliability analysis model is applied to a specific curve that has a potential horizontal sight obstruction and whose annual average daily traffic (AADT) and traffic volume are known. Input data for the reliability analysis model apply to the primary direction of travel (i.e., the direction of travel in which the horizontal sight obstruction is on the inside of the curve) and include: • Roadway type (freeway, ramp, rural multilane divided highway, rural multilane undivided highway, rural two-lane highway); • Number of travel lanes in primary direction of travel; • Average lane width (ft); • Design speed or operating speed (mph); • Direction of curve (right or left); • Radius of curve (ft); • Length of curve (mi); • Type of vertical geometry (straight grade/vertical curve); • Percent grade for straight grades; • Approach percent grade and departure percent grade for vertical curves; • Length of vertical curve (mi); • Distance to the sight obstruction from the inside edge of the traveled way (ft); • Type of sight obstruction (point obstruction, continuous obstruction); • Distance from the PC to the beginning of the sight obstruction (mi); • Distance from the PC to the end of the sight obstruction (mi); • AADT (veh/day) for the primary direction of travel; • Proportion of traffic in each lane for the primary direction of travel; and • Proportion of traffic in the peak hour (k). The reliability analysis model includes built-in tables of default values for the typical percent- age of traffic in each of the 24 hours of a typical day, as a function of the proportion of traffic in the peak hour (k). These default values can be replaced by the program user with agency-specific or site-specific values for the percentage of traffic in each of the 24 hours of a typical day. Figure 12 shows the main data entry screen for the reliability tool, illustrating the input data.

30 Design Guidelines for Horizontal Sightline Offsets 5.2 Output Data from the Reliability Analysis Model The outputs provided by the reliability model include: • Minimum ASSD at any point in each travel lane for the primary direction of travel; • Length of the sight-restricted area; • A color-coded indication whether the minimum ASSD for each travel lane is less than or greater than or equal to the AASHTO value of DSSD for the applicable design or operating speed; • An estimate of the total number of vehicles per year potentially affected by any sight distance restriction that is present, for each travel lane and for all lanes combined; • Total number of vehicles passing through the horizontal curve site in a year, for comparison to the previous measure; and • An estimate of the percentage of the total number of vehicles per year potentially affected by any sight distance restriction that is present, for each travel lane and for all lanes combined. Figure 13 illustrates the display of these results in the output spreadsheet tool. Figure 12. Example of input data entry screen in the reliability analysis spreadsheet tool.

Reliability Analysis Model for Horizontal Curves with Limited SSD 31 In addition, the model output includes the values of ASSD along the road from Station PC – S to PC + S, so that the point of minimum ASSD can be located and so that the designer can assess the distance over which the ASSD remains at a low level. Figure 14 shows an example of a sight distance profile plotted from the ASSD data in the model output. The number and percentage of vehicles potentially affected by the sight distance restrictions are measures of the opportunity for drivers to encounter a crash- or congestion-generated queue of vehicles in the sight-restricted area. The model assesses the probability that each driver approaching the horizontal sight obstruction over the course of a year will encounter a queue of stopped vehicles on a portion of the roadway which the horizontal sight obstruction restricts the driver from seeing. If the number or percentage of potentially affected drivers is small, the likelihood of crashes resulting is small. If the number or percentage of potentially affected drivers is large, the risk of collision is high. Thus, the reliability model provides a quantitative estimate of the opportunity for crashes to occur. Only a very small percentage of the potentially affected drivers may actually encounter crashes. Figure 13. Example of output results display in the reliability analysis spreadsheet tool. 0 100 200 300 400 500 600 700 800 -500 -300 -100 100 300 500 700 900 A va ila bl e Si gh t D is ta nc e (ft ) Longitudinal Distance Along Roadway Available SD EB (ft) Design SSD at 55 mi/h Design SSD at 50 mi/h PC PT Figure 14. Example of sight distance profile from model output.

32 Design Guidelines for Horizontal Sightline Offsets The reliability model can provide output for two different sets of assumptions concerning sight distance measurement. The first option uses the sight distance measurement assumptions exactly as presented in the AASHTO Green Book (AASHTO 2011). The assumptions are: • Sight distance is measured along the centerline of a travel lane; • The driver’s eye is located 3.5 ft above the roadway surface; and • The object ahead which the driver is expected to see has a height of 2 ft, equivalent to the taillight height of a typical passenger car. The model is also capable of providing output results for an alternative user-specified set of sight distance assumptions. Suggested values for those alternative assumptions are discussed here, but the model will permit the user to enter values for whatever set of alternative assumptions the user wishes: • The user may specify that sight distance be measured along a line other than the centerline of the travel lane. The alternative value entered for this option is the distance from the left edge of the travel lane to the driver’s eye location. The capability to consider noncenterline locations makes sense, because the driver’s eye location is unlikely to be in the center of the lane. The user may enter whatever driver’s eye location they wish, but the suggested value for the distance from the left edge of the travel lane to the driver’s eye is 25 percent of the lane width. These alternative assumptions are more realistic concerning the actual positioning of the vehicle within a lane and provide a sight distance advantage for the driver on curves to the right and a sight distance disadvantage to the driver on curves to the left, as compared to the Green Book assumptions. • The user may also specify an alternative value of driver eye height. The AASHTO value of 3.5 ft as a typical driver eye height for passenger cars is well established, and there is little need to vary this value for passenger cars unless an explicit study of lower height vehicles (e.g., sports cars) is being conducted. However, capability for changing the assumed driver eye height can be used to analyze SSD for trucks. The AASHTO Green Book recommends a value of 8.0 ft as the driver eye height for a truck. • The assumed object height of 2.0 ft used in the Green Book was based on research by Fambro et al. (1997). Fambro et al. found that very few collisions on the roadway involved objects less than 2.0 ft in height. Many, but not all, of the objects that vehicles collided with were other vehicles. The value of 2.0 ft for object height has been accepted because this corresponds to the typical taillight height for passenger cars, and seeing the taillights of vehicles ahead is certainly a desirable SSD criterion. Nevertheless, there are many horizontal curves, both to the right and to the left, where a roadside barrier constitutes a horizontal sight obstruction if sight distance is measured with the Green Book criteria, but the upper portion of each vehicle ahead is still visible above the barrier. Where this occurs, a highway agency may decide that removal or mitigation of the sight obstruction is not needed. The reliability model can identify this situation by assessing ASSD for an alternative value of object height, such as 3.5 ft, which allows the upper portion of a passenger car, from the driver’s eye to the roofline, to be seen. The structure of the reliability model and the logic employed in each component module of the reliability analysis model is presented in Sections 5.3 through 5.5. 5.3 Structure of Reliability Analysis Model The computations within the reliability analysis model are organized into two distinct modules: an ASSD module and a reliability analysis module. The available stopping sight distance module computes, for each travel lane, ASSD at a series of points along the roadway, determines the minimum value of ASSD, compares that minimum value of ASSD to the DSSD for the applicable design or operating speed, and determines whether there is a horizontal sight

Reliability Analysis Model for Horizontal Curves with Limited SSD 33 distance restriction. The reliability analysis model estimates the number and percentage of vehicles per year potentially affected by any sight distance restriction present. Aspects of the reliability analysis computation include: • Crash-generated queues in the sight-restricted area; • Congestion-generated queues in the sight-restricted area; and • Computation of potentially affected vehicles. Each module of the reliability analysis model is discussed in the following sections. 5.4 Available SSD Module The first module of the reliability model is the calculation of the ASSD along the curve where a horizontal sightline obstruction is present. The sightline obstruction either can be a point obstruction or a longitudinal obstruction. The tool calculates the minimum ASSD for each lane as well as the length of downstream roadway in which the ASSD is less than the AASHTO DSSD specified in the Green Book. Figure 15 shows how the sight-restricted area is determined. The beginning of the sight-restricted area is shown in Figure 15(a), where the ASSD begins to be less than the DSSD. The end of the sight-restricted area is shown in Figure 15(b), where the ASSD begins to be greater than the DSSD. The resulting length of downstream roadway in which the ASSD is less than the DSSD is shown in Figure 15(c). 5.4.1 Horizontal Sight Obstruction A horizontal sight obstruction can be defined as either a point obstruction or a longitudinal obstruction that extends along the roadside for a specified distance. For a point obstruction, the calculations are simply done in the x-y plane, because point obstructions are assumed to have a very tall height (such as a single tree or a corner of a building). If a point obstruction that is not very high needs to be evaluated, it can be treated as a very short longitudinal obstruction. For a longitudinal obstruction, the height of the obstruction is used in the calculation. The location of the obstruction relative to the roadway is treated as a fixed offset distance from the inside edge of the traveled way to the sight obstruction. 5.4.2 Roadway Alignment and Cross Section It is assumed that all horizontal curves are circular curves with no spiral transitions. In the case of longitudinal sight obstructions, the roadway profile is considered in the calculation of ASSD. The roadway profile can either consist of a straight grade or a vertical curve with specified approach and departure grades. The model is capable of considering a single horizontal curve, with the option of including a single vertical curve as well. The model does not address multiple horizontal curves or multiple vertical curves. The model assesses the ASSD separately for each travel lane, working in sequence from the inside lane to the outside lane on the curve. In the computation of ASSD for each successive travel lane, the distance to the horizontal sight obstruction and the radius of the horizontal curve are each increased by one lane width. The lane width considered is the average lane width; the model does not have the capability to consider different widths for each lane. 5.4.3 Procedure for ASSD Calculation ASSD cannot be determined with a single equation. ASSD as a function of the driver’s position on the roadway is not continuous from a point on the tangent upstream of the PC of a horizontal

34 Design Guidelines for Horizontal Sightline Offsets (a) (b) (c) Figure 15. Determination of the sight-restricted area.

Reliability Analysis Model for Horizontal Curves with Limited SSD 35 curve to a point where ASSD is no longer limited by the sight obstruction. Rather, ASSD as a function of the driver’s position on the roadway must be analyzed with a set of equations for a specific scenario applicable to that position on the roadway. The scenarios that must be addressed separately in the computations include each unique combination of the following factors: • The driver’s eye location either before or after the PC of the horizontal curve; • The driver’s eye location either before the vertical point of curvature (VPC), between the VPC and vertical point of tangency (VPT), or after the VPT of the vertical curve (applicable only if a vertical curve is present); • The location of the point along the sight obstruction to which the driver’s sightline is tangent (applicable only to longitudinal obstructions); and • The location of the point where the sightline intersects the downstream roadway, either before or after the PT of the horizontal curve. The equations used to compute ASSD for each scenario and an explanation of how the equations are used are presented in Potts et al. (2018). 5.5 Reliability Module The computations in the reliability module are explained in the following sections. 5.5.1 Estimating the Number of Vehicles That May Encounter a Queue in a Sight-Restricted Area If a Stopped Vehicle Is Present The reliability module estimates the total number of vehicles in a year that may potentially encounter a queue present in the sight-limited area. The queue of concern occurs on the roadway where the obstruction limits the view of the stopped vehicles such that the ASSD is less than the DSSD. The reliability module considers two events that may generate queues behind a stopped vehicle: crashes and congestion. Later sections deal with the probability of a stopped vehicle due to a crash or congestion under the flow conditions present during a particular hour of the day. This discussion addresses the maximum number of drivers that may approach a stopped vehicle from the rear and not be able to see the stopped vehicle or the queue that has formed behind it. The model logic assumes that the crash- or congestion-generated queue will not be cleared until after the queue has spilled back out of the site-restricted area. The number of vehicles that may potentially be affected when a queue occurs is dependent on the length of roadway that falls within the sight-restricted area as well as the hourly flow rate. Where multiple lanes are present, these factors vary depending on which lane is being analyzed. The sight-restricted area is divided for computational purposes into 25-ft segments, where 25 ft represents the length of a passenger car, plus a typical spacing between stopped vehicles. Let jy represent the number of 25-ft segments in the sight-restricted area in lane y. Figure 16 shows a lane in which the sight-restricted area is divided into 25-ft segments. If a stopped car is present in Segment 1, then a vehicle traveling toward the stopped vehicle will not be able to see the stopped vehicle with an ASSD greater than or equal to DSSD. However, the second vehicle traveling toward the stopped vehicle will be able to see the first vehicle that is stopped behind the stopped vehicle with an ASSD greater than or equal to DSSD. In Figure 16, a vehicle is stopped in Segment 4. The first three approaching vehicles will occupy Segments 1 through 3 after they encounter the first vehicle and are forced to stop. The fourth vehicle will be the last vehicle whose driver will experience ASSD less than DSSD. The driver of any fifth or subsequent vehicle approaching the queue would be able to see the queue without any sight distance limitation. Thus, a maximum of four vehicles would encounter the stopped

36 Design Guidelines for Horizontal Sightline Offsets vehicle with the ASSD less than DSSD so that their drivers could not see the queue with the full DSSD available. If a stopped vehicle should be present in Segment 2, then the first two vehicles traveling toward the stopped vehicle will be in a situation with ASSD less than DSSD. This trend continues all the way until a stopped vehicle is present in the last segment of the sight-restricted area, Segment 9. The sum of all these scenarios can be expressed as: ∑= = Sum of potentially affected vehicles due to stopped vehicle in sight-restricted area (14) 1 n n jq where jy = number of 25-ft segments in sight-restricted area in lane y; qiy = flow rate (veh/hr/lane) during i in lane y; jq = smaller of jy and qiy; and n = index variable defined as increasing from 1 to jy. An additional situation to consider is that a stopped vehicle downstream of the sight-restricted area may generate a queue that spills back into the sight-restricted area. In this situation, the number of affected vehicles would be the total number of vehicles within the sight-restricted area, which is equal to jy, as illustrated in Figure 17. The vehicle that stops in Segment 9 is the last vehicle that cannot be seen by an approaching driver with ASSD greater than DSSD. The drivers of the next nine approaching vehicles experience ASSD less than the DSSD. Figure 16. Example of stopped vehicle in the sight-restricted area.

Reliability Analysis Model for Horizontal Curves with Limited SSD 37 The furthest downstream a stopped vehicle can queue back into the sight-restricted area during hour i is dependent on the flow rate, qiy, in lane y during hour i. Let this distance be expressed by xiy, the number of 25-ft segments downstream of the sight-limited area in lane y in which a crash could occur that would produce a queue long enough to enter the sight-limited area in hour i, determined as: = − (15)x q jiy iy y where xiy is the number of 25-ft segments downstream of the sight-limited area in lane y in which a crash could occur that would produce a queue long enough to enter the sight-limited area in hour i. A stopped vehicle in any 25-ft segment beyond the sight-restricted area will affect jy vehicles. The sum of all vehicles affected by a stopped vehicle in any of these downstream segments is computed as: = Sum of potentially affected vehicles due to stopped vehicle downstream of sight-restricted area (16)j xy pos where xpos is the larger of xiy and zero. Figure 17. Vehicles experiencing ASSD less than DSSD due to a queue backing into the sight-restricted area.

38 Design Guidelines for Horizontal Sightline Offsets The estimated number of vehicles potentially affected when a stopped vehicle is present during hour i in lane y, Viy, is the sum of Equations (11) and (13) divided by the total num- ber of 25-ft segments where a vehicle could be stopped. This estimated number of vehicles potentially affected is computed as: ∑ ( ) ( )= + + − = + + = 0.5 1 (17)1V n j x j q j j j j x q iy y pos n j y iy y q y y pos iy q where Viy is average number of vehicles affected when a stopped vehicle is present during hour i in lane y. 5.5.2 Frequency of Crash-Generated Queues Crashes are one of the two types of events considered in the model that may generate queues of stopped vehicles in a sight-restricted area. Using safety performance functions (SPFs) from the Highway Safety Manual (AASHTO 2010; AASHTO 2014), the predicted average annual crash frequency, N25, can be estimated for a 25-ft section of roadway. Since the SPFs from the Highway Safety Manual estimate crash frequencies for two-way roadways, it is assumed that the crash frequency for a 25-ft section of roadway in the direction of travel being evaluated is half of the predicted value from the SPF. The predicted average annual crash frequency for the 25-ft interval, N25, can be multiplied by the k-factor for hour i, ki, and the percentage of traffic in lane y, Uy, to estimate the number of average annual crashes in hour i in lane y, Niy,25, as follows: N N k Uiy i y (18),25 25= where Niy,25 = predicted number of annual crashes during hour i in lane y; N25 = predicted annual crash frequency for a 25-ft segment in the analysis direction of travel; ki = percentage of AADT during hour i; and Uy = percentage of vehicles using lane y in analysis direction. The predicted average annual crash frequency during hour i in lane y along the entire length of roadway in which a stopped vehicle will have an impact in the sight-limited area can be estimated as: = (19),25N N qiy iy iy where Niy is predicted average annual crash frequency during hour i in lane y along the entire length of roadway in which a stopped vehicle will have an impact in the sight-limited area. 5.5.3 Frequency of Congestion-Generated Queues Congestion is the second type of event considered in the model that may generate queues of stopped vehicles in a sight-restricted area. Queues can form on roadways where volumes are near capacity. The cumulative distribution function of the shifted lognormal distribution with a mean value of 1.1609 and a standard deviation of 0.4906 was used to determine the probability of a breakdown in traffic flow based on average headway. These values for the log normal distri- bution are those recommended by Jia et al. (2010) for a similar application. It was assumed that the queueing dynamics were the same for all facilities. The cumulative distribution function is

Reliability Analysis Model for Horizontal Curves with Limited SSD 39 shifted, however, based on the capacity of the facility. The probability of a breakdown in flow resulting in a congestion-related queue during any particular hour with a specified flow rate is computed as: ∫= s π ( )− −µ s −∞ 1 2 (20), ln 2 2 2P x econgestion iy xx = −3600 (21)x q shift iy = 3600 (22)shift c where: Pcongestion,iy = probability of a queue forming due to congestion; x = shifted average headway (sec/veh/lane); s = standard deviation of lognormal distribution, 0.4906; µ = mean of lognormal distribution, 1.1609; and c = capacity (veh/hr/lane). 5.5.4 Total Potentially Affected Vehicles As stated in the previous discussions, two events that can lead to a stopped vehicle and queue- ing in a particular travel lane are crashes and congestion (flow rates nearing capacity). During an hour of a given day, it is assumed that if a crash occurs, then queueing due to congestion is not going to occur. This is because the crash in the traveled way has already produced a queue. A second, independent queueing event is unlikely during that same hour. The total number of vehicles potentially affected per year on a curve with a horizontal sight obstruction is determined by multiplying the average number of vehicles affected when a queueing event occurs by the probable annual number of queueing events summed over all 24 hours of the day and all lanes of the roadway, as follows: ∑∑ [ ][ ]( )= + − == Total number of affected vehicles per year 365 (23), 1 24 1 V N N Piy iy iy congestion iy iy L where L = total number of lane in analysis direction; y = lane number; and i = hour of the day. 5.6 Application of Reliability Analysis to Curves with Horizontal Sight Obstructions Since the crash frequency estimates to support benefit-cost analysis are so uncertain, an alternative approach to assessment of horizontal sight obstructions that is not dependent on crash history data is needed. The reliability analysis model described earlier can be used for this purpose. The model considers the stream of vehicles approaching a horizontal curve with a sight obstruction on the inside of the curve and assesses the average number and percentage of

40 Design Guidelines for Horizontal Sightline Offsets vehicles per year that will potentially encounter a stopped vehicle or a queue of stopped vehicles, due to a crash or due to flow in excess of capacity, in the sight-restricted area. Approaching vehicles are treated as potentially affected by the sight distance limitation only if the crash- involved vehicle or the stopped vehicle at the rear of the queue is not visible to the approaching driver over a distance greater than or equal to the applicable value of DSSD. The measures provided by the reliability analysis model that can assist designers in assessing the priority of removing or mitigating a particular sight obstruction include: • Minimum ASSD in any point in each travel lane for the primary direction of travel; • A color-coded indication whether the minimum ASSD for each travel lane is less than or greater than or equal to the AASHTO value of DSSD for the applicable design or operating speed; • ASSD values at user-specified intervals from Station PC – S to Station PT + S; • The total length of the sight-restricted area for which ASSD is less than DSSD; • An estimate of the total number of vehicles per year potentially affected by any sight distance restriction that is present, for each travel lane and for all lanes combined; • Total number of vehicles passing through the horizontal curve site in a year, for comparison to the previous measure; and • An estimate of the percentage of the total number of vehicles per year potentially affected by any sight distance restriction that is present, for each travel lane and for all lanes combined. The reliability analysis model has been incorporated in a spreadsheet tool for application by designers to assess horizontal sight obstructions. A users guide to the spreadsheet tool is presented in Appendix B. The reliability analysis model was used to perform a sensitivity analysis of three key measures—minimum ASSD, number of potentially affected vehicles per year, and percentage of potentially affected vehicles per year. This sensitivity analysis considered: • Five design scenarios (rural two-lane highway curve to the right/rural two-lane highway curve to the left/urban six-lane freeway curve to the right/rural four-lane freeway curve to the right/urban one-lane exit ramp curve to the right); • Six representative curve radii, curve length, and design speed combinations for each design scenario; • Nine representative AADT levels for each design scenario; and • Six representative offset distances from the obstruction to the inside edge of the traveled way. The results address three additional design scenarios because, for urban freeways, rural freeways, and ramps, the sensitivity analysis results for curves to the left would be identical to those for curves to the right. Rather than presenting the results of all the sensitivity analyses that were performed, Table 3 shows the results for just the smallest curve radius considered for each design scenario. The smallest curve radius was selected for presentation in the table, because it results in the largest values for the number and percentage of potentially affected vehicles. The smallest radius curves are generally below the Green Book minimum radii for the applicable design speed. The sensitivity analysis results for rural two-lane highways show very little likelihood of approaching vehicles encountering crash-involved vehicles or queues of stopped vehicles in the sight-restricted area. For a curve to the right on a two-lane highway with the largest AADT considered (10,000 veh/day/lane or 20,000 veh/day in both directions combined), the maximum number of potentially affected vehicles is 412 vehicles per year or 0.011 percent of the total yearly flow. For the curve to the left in the opposing direction of travel on such a road, the maximum number of potentially affected vehicles is 387 vehicles per year, also equivalent to

Rural two-lane highway curve to the right, 250-ft radius, 0.20-mi curve length, 60-mph design speed AADT per lane (veh/day) Offset of obstruction from inside edge of traveled way (ft) Minimum ASSD (ft) Number of potentially affected vehicles Percentage of total vehicles that are potentially affected 0 2 5 10 15 20 0 2 5 10 15 20 0 2 5 10 15 20 500 110 127 149 180 206 230 1 1 1 1 1 0 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 1,000 110 127 149 180 206 230 2 2 2 2 2 2 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 1,500 110 127 149 180 206 230 7 6 6 6 6 6 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 2,000 110 127 149 180 206 230 13 13 13 13 12 12 0.002% 0.002% 0.002% 0.002% 0.002% 0.002% 2,500 110 127 149 180 206 230 22 21 21 21 20 20 0.002% 0.002% 0.002% 0.002% 0.002% 0.002% 4,000 110 127 149 180 206 230 61 60 59 58 56 55 0.004% 0.004% 0.004% 0.004% 0.004% 0.004% 5,000 110 127 149 180 206 230 97 96 95 93 90 89 0.005% 0.005% 0.005% 0.005% 0.005% 0.005% 7,500 110 127 149 180 206 230 228 224 221 217 210 207 0.008% 0.008% 0.008% 0.008% 0.008% 0.008% 10,000 110 127 149 180 206 230 412 406 400 393 380 374 0.011% 0.011% 0.011% 0.011% 0.010% 0.010% Rural two-lane highway curve to the left, 250-ft radius, 0.20-mi curve length, 60-mph design speed AADT per lane (veh/day) Offset of obstruction from inside edge of traveled way (ft) Minimum ASSD (ft) Number of potentially affected vehicles Percentage of total vehicles that are potentially affected 0 2 5 10 15 20 0 2 5 10 15 20 0 2 5 10 15 20 500 195 206 221 244 266 286 1 1 1 0 0 0 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 1,000 195 206 221 244 266 286 2 2 2 2 2 2 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 1,500 195 206 221 244 266 286 6 6 6 6 6 6 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 2,000 195 206 221 244 266 286 12 12 12 12 12 12 0.002% 0.002% 0.002% 0.002% 0.002% 0.002% 2,500 195 206 221 244 266 286 21 20 20 20 19 19 0.002% 0.002% 0.002% 0.002% 0.002% 0.002% 4,000 195 206 221 244 266 286 57 56 56 55 54 53 0.004% 0.004% 0.004% 0.004% 0.004% 0.004% 5,000 195 206 221 244 266 286 92 90 90 87 86 84 0.005% 0.005% 0.005% 0.005% 0.005% 0.005% 7,500 195 206 221 244 266 286 214 210 210 203 200 196 0.008% 0.008% 0.008% 0.007% 0.007% 0.007% 10,000 195 206 221 244 266 286 387 380 380 367 361 355 0.011% 0.010% 0.010% 0.010% 0.010% 0.010% Urban six-lane freeway curve to the right, 750-ft radius, 0.20-mi curve length, 60-mph design speed AADT right lane (veh/day) Offset of obstruction from inside edge of traveled way (ft) Minimum ASSD, Right Lane (ft) Number of potentially affected vehicles, right lane Percentage of total vehicles that are potentially affected, right lane 0 2 5 10 15 20 0 2 5 10 15 20 0 2 5 10 15 20 6,800 190 219 257 310 356 396 99 97 94 89 84 79 0.004% 0.004% 0.004% 0.004% 0.003% 0.003% 8,500 190 219 257 310 356 396 163 160 155 147 138 130 0.005% 0.005% 0.005% 0.005% 0.004% 0.004% 10,200 190 219 257 310 356 396 247 243 235 222 210 197 0.007% 0.007% 0.006% 0.006% 0.006% 0.005% 11,900 190 219 257 310 356 396 354 348 336 318 300 282 0.008% 0.008% 0.008% 0.007% 0.007% 0.006% 13,600 190 219 257 310 356 396 489 481 465 440 415 390 0.010% 0.010% 0.009% 0.009% 0.008% 0.008% 17,000 190 219 257 310 356 396 960 943 911 861 812 762 0.015% 0.015% 0.015% 0.014% 0.013% 0.012% 20,400 190 219 257 310 356 396 2,938 2,888 2,787 2,635 2,483 2,331 0.039% 0.039% 0.037% 0.035% 0.033% 0.031% 25,500 190 219 257 310 356 396 27,579 27,103 26,149 24,717 23,282 21,845 0.296% 0.291% 0.281% 0.266% 0.250% 0.235% 30,600 190 219 257 310 356 396 94,588 92,951 89,675 84,754 79,826 74,891 0.847% 0.832% 0.803% 0.759% 0.715% 0.671% Table 3. Results of sensitivity analysis with the reliability analysis model for key design scenarios (Potts et al. 2018). (continued on next page)

Rural four-lane freeway curve to the right, 1000-ft radius, 0.20-mi curve length, 75-mph design speed AADT right lane (veh/day) Offset of obstruction from inside edge of traveled way (ft) Minimum ASSD, Right Lane (ft) Number of potentially affected vehicles, right lane Percentage of total vehicles that are potentially affected, right lane 0 2 5 10 15 20 0 2 5 10 15 20 0 2 5 10 15 20 2,500 219 253 297 358 411 457 11 11 11 11 10 10 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 5,000 219 253 297 358 411 457 51 49 48 46 44 42 0.003% 0.003% 0.003% 0.003% 0.002% 0.002% 7,500 219 253 297 358 411 457 120 117 113 108 103 100 0.004% 0.004% 0.004% 0.004% 0.004% 0.004% 10,000 219 253 297 358 411 457 224 218 211 202 192 186 0.006% 0.006% 0.006% 0.006% 0.005% 0.005% 12,500 219 253 297 358 411 457 368 358 347 331 315 304 0.008% 0.008% 0.008% 0.007% 0.007% 0.007% 15,000 219 253 297 358 411 457 574 557 541 516 490 473 0.010% 0.010% 0.010% 0.009% 0.009% 0.009% 20,000 219 253 297 358 411 457 2,228 2,163 2,097 1,998 1,899 1,832 0.031% 0.030% 0.029% 0.027% 0.026% 0.025% 25,000 219 253 297 358 411 457 20,116 19,518 18,918 18,018 17,117 16,515 0.220% 0.214% 0.207% 0.197% 0.188% 0.181% 30,000 219 253 297 358 411 457 74,337 72,121 69,902 66,569 63,232 61,004 0.679% 0.659% 0.638% 0.608% 0.577% 0.557% Urban one-lane exit ramp curve to the right, 250-ft radius, 0.20-mi curve length, 60-mph design speed AADT (veh/day) Offset of obstruction from inside edge of traveled way (ft) Minimum ASSD (ft) Number of potentially affected vehicles Percentage of total vehicles that are potentially affected 0 2 5 10 15 20 0 2 5 10 15 20 0 2 5 10 15 20 2,000 110 127 149 180 206 230 16 16 16 16 15 15 0.002% 0.002% 0.002% 0.002% 0.002% 0.002% 4,000 110 127 149 180 206 230 63 62 61 60 58 57 0.004% 0.004% 0.004% 0.004% 0.004% 0.004% 6,000 110 127 149 180 206 230 133 131 129 127 123 121 0.006% 0.006% 0.006% 0.006% 0.006% 0.006% 8,000 110 127 149 180 206 230 225 221 218 214 208 204 0.008% 0.008% 0.007% 0.007% 0.007% 0.007% 10,000 110 127 149 180 206 230 337 331 326 321 311 305 0.009% 0.009% 0.009% 0.009% 0.009% 0.008% 12,500 110 127 149 180 206 230 507 499 491 483 467 459 0.011% 0.011% 0.011% 0.011% 0.010% 0.010% 15,000 110 127 149 180 206 230 746 734 722 710 687 675 0.014% 0.013% 0.013% 0.013% 0.013% 0.012% 20,000 110 127 149 180 206 230 4,161 4,094 4,027 3,960 3,827 3,760 0.057% 0.056% 0.055% 0.054% 0.052% 0.052% 25,000 110 127 149 180 206 230 44,892 44,168 43,444 42,720 41,270 40,544 0.492% 0.484% 0.476% 0.468% 0.452% 0.444% NOTE: Table addresses the smallest radius horizontal curve considered for each design scenario; assumed lane width = 12 ft; ASSD measured with Green Book assumptions Table 3. (Continued).

Reliability Analysis Model for Horizontal Curves with Limited SSD 43 0.011 percent of the total yearly flow. These results indicate that sight-distance-related crashes are highly unlikely on rural two-lane highways, as only a small fraction of the potentially affected vehicles would likely become involved in a crash. The sensitivity analysis results for six-lane urban freeways show that the number of poten- tially affected vehicles is 79 per year (0.003 percent of the total yearly flow) for the smallest AADT and the largest offset to obstruction considered. For the largest AADT and smallest obstruction to offset considered, the number of potentially affected vehicles increases to 94,588 per year (0.847 percent of the total yearly flow). Thus, the number of vehicles that may potentially encounter a crash-involved or stopped vehicle in the sight-restricted area increases by a factor of approximately 1,200 with changes in AADT and offset to obstruction. Similarly, the sensitivity analysis results for four-lane rural freeways show that the number of potentially affected vehicles is 10 per year (0.001 percent of the total yearly flow) for the smallest AADT and the largest offset to obstruction considered. For the largest AADT and smallest obstruction to offset considered, the number of potentially affected vehicles increases to 74,337 per year (0.679 percent of the total yearly flow). Thus, the number of vehicles that may potentially encounter a crash-involved or stopped vehicle in the sight-restricted area increases by a factor of approximately 7,400 with changes in AADT and offset to obstruction. The sensitivity analysis results for one-lane urban exit ramps show that the number of potentially affected vehicles is 15 per year (0.002 percent of the total yearly flow) for the smallest AADT and the largest offset to obstruction considered. For the largest AADT and smallest obstruction to offset considered, the number of potentially affected vehicles increases to 44,892 per year (0.492 percent of the total yearly flow). Thus, the number of vehicles that may potentially encounter a crash-involved or stopped vehicle in the sight-restricted area increases by a factor of approximately 3,000 with changes in AADT and offset to obstruction. These results show that the reliability analysis model can be useful in quantifying the potential opportunities for crashes at horizontal curves with sight restrictions on the inside of the curve. The percentage of vehicles that may encounter a crash-involved vehicle or a queue of stopped vehicles over the course of a year can range from essentially zero to a value approaching 1 percent of the total yearly flow. The number of crashes that will actually occur will be substan- tially smaller than the number of potentially affected vehicles, but this number and percentage of potentially affected vehicles provide a relative measure that can be used to prioritize sites for improvement. The reliability analysis model is a flexible tool that can be used by planners and designers to compare, in a relative sense, the need for sight distance improvements on specific horizontal curves. A key strength of the model is its ability to consider roadway alignment in three dimen- sions (i.e., horizontal and vertical alignment in combination). Limitations of the model include: • The model can access a site with at most one horizontal and one vertical curve; compound curves are not considered. • The model provides a conservative estimate of the number and percentage of vehicles poten- tially affected by a particular sight distance limitation and, thus, estimates a maximum value rather than an average value of these measures. • The number of crashes related to a particular sight distance limitation will likely be much smaller than the number of potentially affected vehicles. • The model does not quantify the increased number of vehicles potentially affected by a sight distance limitation if an intersection, driveway, ramp terminal, pedestrian crossing, or subsequent horizontal curve is located within the sight-restricted area.

Next: Chapter 6 - Assessing Removal or Mitigation of Horizontal Sight Obstructions »
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The distance between the driver’s line of sight along the roadway ahead on a horizontal curve and a sight obstruction on the inside of the curve is known as the horizontal sightline offset (HSO). Highway agencies can use NCHRP Research Report 910: Design Guidelines for Horizontal Sightline Offsets as guidance to address the types of sight distance restrictions that are most likely to be encountered on specific roadway types.

The relationship between stopping sight distance (SSD) and the frequency and severity of crashes has been difficult to quantify because the role of SSD in reducing crashes is highly situational. The design criteria for the horizontal component of SSD in what is known as AASHTO's Green Book are based on the maximum sightline offset that may be needed at any point along a curve with a given radius, which doesn't cover all possible situations.

Designers compensate for the limitations on driver sight distance in various ways, including: accepting shorter sightlines, lowering design speed, increasing shoulder width, or providing additional signage. There are advantages and disadvantages to the trade-offs; as a result, many highway agencies have used the design exception process to address the trade-offs for sight distance in such situations.

This project conducted research to evaluate these situations and determine what criteria or mitigation will provide acceptable solutions when impaired horizontal sightline offsets are encountered. The project includes a tool (an Excel spreadsheet) that may be used to calculate sight distance.

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