**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

**Suggested Citation:**"APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT-COST ANALYSIS METHOD." National Academies of Sciences, Engineering, and Medicine. 2021.

*Guidelines for Cost-Effective Safety Treatments of Roadside Ditches*. Washington, DC: The National Academies Press. doi: 10.17226/26127.

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.

A-1 APPENDIX A. MATHEMATICAL DESCRIPTION OF THE BENEFIT- COST ANALYSIS METHOD

A-2 INTRODUCTION As indicated in the main sections, the tentatively developed benefit-cost analysis (BCA) method is a workable method intended to strike a balance between three key goals given the available resources, data, and modeling capability in hand. These three goals are as follows: To provide a reasonably wide coverage of the current design practice on roadside ditch geometry. To incorporate an encroachment characteristics model that has a reasonable degree of fidelity to reproduce the real-world encroachment conditions, especially those resulting in high-severity events, so that the crash cost can be estimated with good accuracy. To keep the computational time required to perform the analysis manageable. The execution of the CarSim component model in the BCA method is computationally intensive. Moreover, the number of simulation runs can quickly become unmanageable if the total number of combinations of roadside geometry and encroachment characteristics is not properly restrained. Thus, one of the challenges to the development is to decide on how many and which combinations of ditch geometry and encroachment characteristics to simulate with the CarSim model. Another venue to lower the computational time, which is discussed later, is to select a subset of roadside design variables as baseline variables and treat the rest of the variables as adjustment variables. To reduce computational time, stronger assumptions are imposed on adjustment variables regarding their effects on crash outcomes. This appendix provides a mathematical description of the BCA method and procedures involved. Both cost estimation and guideline development are covered. It has been constructed to serve two purposes. The first purpose is to add precision and clarity to the description provided in the main sections, especially on the procedural and computational aspects of the method. The second purpose is to provide the mathematical details for developing a software tool to perform the required computations (beyond the CarSim simulation runs). The presentation reflects the researchersâ current thinking on (a) which variables are important to consider and, to keep the computational time manageable, what range of variables could be considered; and (b) on which variables should be treated as baseline variables and which as adjustment variables. The mathematical exposition in this appendix is, however, quite general and can be revised as needed (e.g., to allow for more variables to be considered or to expand or limit the studied range of each variable). An important feature of the tentatively developed BCA method is that it provides a coherent way of developing guidelines for both the design and treatment of roadside ditches. Another feature is its capability to take into consideration a large number of roadway, roadside, traffic, and encroachment variables, as required by the project, in a systematic and integrated manner. The mathematical description of the BCA method allows these features to be more clearly explained. The remaining appendix is organized as follows: Variable Notations and Abbreviations. Baseline and Adjustment Design Variables. Baseline Design Condition. Crash Costs for Baseline Design Condition.

A-3 Preferred Design Envelopes for Baseline Design Condition. Preferred Design Envelopes for Non-Baseline Conditions. Grand Envelope Adjustment Factor. VARIABLE NOTATIONS AND ABBREVIATIONS Table A.1 provides a list of variables and associated variable names used in this appendix, including roadway, roadside, ditch treatments, encroachment characteristics, injury severity, and vehicle performance variables. A range of values for each variable in consideration and a description of what each value represents are also given in the table. For example, the variable for highway type is named âhtâ and has two valuesâ1 and 2ârepresenting two-lane two-way undivided highways and four-lane divided highways, respectively. As another example, ditch bottom width is abbreviated as âbtwâ and has three valuesâ1, 2, and 3ârepresenting 0 ft (V-ditch), 4 ft, and 10 ft, respectively. The following expression is used often in this appendix: Variable Name(variable #1, variable #2, â¦.). It is used to indicate that a variable named âVariable Nameâ is dependent on or is a function of a set of variables listed in the parentheses, which are âvariable #1,â âvariable #2,â and so forth. For example, ER(ht,psl,aadt,hc,vg) is used to indicate the encroachment rate (ER) of a particular highway type (ht), posted speed limit (psl), traffic volume (aadt), horizontal curvature (hc), and vertical grade (vg), which are all defined in Table A.1.

A-4 Table A.1. Key variables, abbreviated names, and values. Variable (Abbreviation in Parenthesis) Variable Name, Values, and Descriptions Roadway Type, Traffic, and Design Highway Type (ht) ht = 1 for two-lane two-way undivided highways = 2 for four-lane divided highways Posted Speed Limit (psl) psl = 1 for 55 mph = 2 for 65 mph Traffic Volume (aadt) aadt = 500 to 15,000 veh/day for ht=1 (representing Level of Service [LOS], from A to E) = 2,500 to 35,000 veh/day for ht=2 (representing LOS from A to D) Horizontal Curvature (hc) hc=1*,2, and 3 for 0 deg*, 4.5 deg, and 6 deg, resp., curve left Ranges represented: [0,3], (3,6), and â¥ 6 deg, resp. Vertical Grade (vg) vg=1*,2, and 3 for 0%*, 4%, and 6%, resp., downgrade Ranges represented: [0,2], (2,6), and â¥ 6%, resp. Roadside Design Shoulder Width (sw) sw = 1* for 6% cross slope, paved, width = 6 ft* = 2 for 6% cross slope, paved, width = 2 ft = 3 for 6% cross slope, paved+turf, width = 12 ft (5 ft paved and 7 ft turf) Ditch Geometry ï· Foreslope ratio (fs)* ï· Foreslope width (fw)* ï· Ditch bottom width (btw) ï· Backslope ratio (bs)* ï· Backslope width (bw)* ï· fs=1, 2, 3, and 4 for 10H:1V, 6H:1V, 4H:1V, and 3H:1V, resp. ï· fw=1 and 2 for 8 ft and 16 ft, resp. ï· btw=1*, 2, and 3 for 0 ft (V-ditch)*, 4 ft, and 10 ft, resp. ï· bs=1,2,...,5 for 10H:1V, 6H:1V, 4H:1V, 3H:1V, and 2H:1V, resp. ï· bw=1 and 2 for 8 ft and 16 ft, resp. Ditch Treatment Rounding (rnd) rnd = 1 for no-rounding* = 2 for level 1 rounding (e.g., 4 ft constant rounding of shoulder hinge point and/or bottom rounding) = 3 for level 2 rounding (e.g., optimum rounding) Surface Treatment (surf) surf= 1 for no surface treatment (w/ bare soil and weeds)* = 2 for type 1 treatment (e.g., short grass < 2 inches) = 3 for type 2 treatment (e.g., asphalt) Encroachment Characteristics Vehicle Type (vt) vt = 1 for 2,425-lb passenger car = 2 for 3,300-lb passenger sedan = 3 for 5,000-lb pickup truck = 4 for small sport utility vehicle Encroachment Speed (es) es = 1, 2, 3, 4, and 5 for 35, 45, 55, 65, and 75 mph, resp. Encroachment Angle (ea) ea = 1, 2, and 3, for 10, 20, and 30 deg, resp. * All values of the variable or a particular value of the variable is selected as the baseline design condition.

A-5 Table A.1. Key variables, abbreviated names, and values (continued). Variable (Abbreviation in Parenthesis) Variable Name, Values, and Descriptions Driver Control Input, Vehicle Tracking Status, and Perception-Reaction Time (PRT) Combinations (dci) (PRT: to be determined from the literature) dci = 1 for no control input (freewheeling), tracking, and w/ PRT (representing, e.g., distracted and drowsy drivers) = 2 for panic return-to-road steering, tracking, and w/ PRT = 3 for panic return-to-road steering, non-tracking with yaw rate of 15 deg/sec, and no PRT = 4 for combined return-to-road steering and full ABS braking, tracking, and no PRT = 5 for combined return-to-road steering and full ABS braking, non-tracking with yaw rate of 15 deg/sec, and no PRT Injury Severity Police-Reported Injury Severity Level (Defined as the Maximum Known Injury in a Crash) (pis) pis = 1 for no damage and no injury (no crash) = 2 for property damage only (PDO crash) = 3 for possible injury (type C crash) = 4 for non-incapacitating injury (type B crash) = 5 for incapacitating injury (type A crash) = 6 for fatal (type K crash) Vehicle Performance Measure Rollover (Roll) Roll = 0 for non-rollover = 1 for rollover BASELINE AND ADJUSTMENT DESIGN VARIABLES Baseline design condition is the starting roadway and roadside geometric design combinations used in the BCA method to estimate the expected crash cost per encroachment and determine the initial design envelope of foreslope ratio and backslope combinations. In Table A.1, the asterisk next to a particular value of a variable (e.g., hc=1*) indicates that the value (1) of that variable (hc) is selected as the baseline design condition. If the asterisk is next to a variable, for example, foreslope ratio (fs)* and foreslope width (fw)*, all values of that variable are considered as the baseline design condition. Such variables are called baseline (design) variables. All other roadway and roadside design variables are called adjustment (design) variables. All design combinations under the baseline design condition are simulated with the CarSim model and their expected crash cost is estimated. The estimated costs from the baseline design condition constitutes several sets of crash cost points for various foreslope-backslope combinations, which are used to select preferred design envelopes. The selected envelopes are then used as initial guidelines to further the analysis, comparison, and envelope adjustment for other non-baseline design configurations. Unlike the baseline design variables, adjustment variables are simulated individually and only for the baseline design condition associated with the preferred design envelopes. Thus, for baseline design variables, their individual effects on the crash outcome, as well as the pair-wise (or even higher-order) interactive effects of any two baseline design variables, can be evaluated and quantified when deemed necessary. The effects of adjustment variables, on the other hand, can only be analyzed individually, not in combination with other adjustment variables. In other words, the interactive effects between adjustment variables on crash outcomes are assumed to be nonexistent. This distinction between the two types of variables is explained in more detail later

A-6 when the effects of foreslope width and backslope width (two of the baseline design variables) on the crash cost are discussed. BASELINE DESIGN CONDITION As a summary of Table A.1, the following combinations of main-lane alignment, shoulder width, ditch geometries, and ditch treatment options are selected as the baseline design condition: A straight and flat highway section (hc=1 and vg=1). 6-ft paved shoulder width (sw=1). All four levels of foreslope ratios (fs=1,2,3, and 4; baseline variable). Both foreslope widths (fw=1 and 2; baseline variable). V-ditch (btw=1, i.e., ditch bottom width = 0 ft). All five levels of backslope ratios (bs=1,2,â¦,5; baseline variable). Both levels of backslope widths (bw=1 and 2; baseline variable). No-rounding (rnd=1). No surface treatment (surf=1). Thus, fs, fw, bs, and bw are baseline design variables (since all values of each variable are considered). The total number of design combinations under the baseline condition is: (1Ã1)Ã1Ã4Ã2Ã1Ã5Ã2Ã1Ã1 = 80 As indicated in the main sections, some of these baseline ditch configurations do not meet the 2-ft minimum depth requirement from the shoulder hinge point for cut slope sections and are not permitted in many states. Depending on how one defines cut and filled slope ditch sections and how one limits the maximum backslope height to study, 14 to 30 of these 80 configurations need not be considered. As also discussed in the main sections, the selection of a cut or a filled ditch section is often dictated by terrain features and to some extent by soil stability consideration. As part of the revised work plan for Task 7, a study will take place to separate all ditch configurations in consideration into cut and filled slopes, determine the maximum height of the backslope to be simulated for cut slope sections, and decide whether there is a need to have separate analyses and guidelines for cut and filled slopes. In the following discussion, we assume there are 80 ditch configurations in the baseline design condition, even though the actual number of configurations to be evaluated will be less than 80 for the reason stated above. Also, in many of the mathematical expressions to be presented next, a variable followed by an asterisk is used to emphasize that a particular variable is set at its baseline design condition: hc*, vg*, sw*, btw*, rnd*, and surf*. CRASH COSTS FOR BASELINE DESIGN CONDITION This section presents the method to be used to estimate crash costs for the baseline design condition. These estimated crash costs are used to determine preferred design envelopes for the baseline configurations in the next section. Encroachment Rate Model Given main-lane characteristics, the encroachment rate component model determines the expected number of encroached vehicles on a relatively homogeneous stretch of highways over a long period. The rate is typically expressed in number of encroachments per million vehicle

A-7 miles traveled (enc/MVMT) or per mile per year (enc/mi/yr). Unless indicated otherwise, the latter expression (i.e., enc/mi/yr) is used in the discussion. The encroachment rate of a particular highway type (ht), posted speed limit (psl), traffic volume (aadt), horizontal curvature (hc), and vertical grade (vg) is expressed as ER(ht,psl,aadt,hc,vg). In the section titled Scenarios to Study and Supporting Data in Chapter 4, the best available rates are provided for relative straight (hc=1) and flat (vg=1) sections, and the rates for other hc and vg values are provided as encroachment rate adjustment (ERA) factors. That is: ER(ht,psl,aadt,hc,vg) = ER(ht,psl,aadt,hc=1,vg=1)ÃERA(hc)ÃERA(vg) The data used to estimate the encroachment rate models employed by this study have an average AADT of about 4,500 veh/day for two-lane undivided highways and 21,500 veh/day for four- lane divided highways. These average traffic volumes are expressed as aadt_avg in the following discussion. The expected encroachment rates at aadt_avg for highways with relatively straight and flat sections for the four highway type-posted speed limit combinations are: ER(ht=1,psl=1,aadt= aadt_avg=4,500,hc=1,vg=1) = 1.78 enc/mi/yr. ER(ht=1,psl=2,aadt= aadt_avg=4,500,hc=1,vg=1) = 1.25 enc/mi/yr. ER(ht=2,psl=1,aadt= aadt_avg=21,500,hc=1,vg=1) = 3.28 enc/mi/yr. ER(ht=2,psl=2,aadt= aadt_avg=21,500,hc=1,vg=1) = 2.78 enc/mi/yr. Encroachment Characteristics Model For a specific roadway and roadside condition, each encroached vehicle has some probability of resulting in a crash of a certain police-reported injury severity (pis) level, depending on its encroachment characteristics. The six pis levels are listed in Table A.1, including a level where no damage and no injury occur (pis=1). The encroachment characteristics component model represents real-world encroachment conditions at the point of departure and a driverâs subsequent vehicle maneuvering behaviors during the traversal. Specifically, this model specifies how often, in terms of relative frequency, an encroachment that possesses certain encroachment characteristics is expected to occur in the field. The set of encroachment characteristics currently considered by this study include four vehicle types (vt), five encroachment speeds (es), three encroachment angles (ea), and five driver control inputâvehicle tracking statusâperception-reaction time (dci) combinations (which are listed in Table A.1). The probability for an encroachment to have a certain combination of encroachment characteristics is expressed as an encroachment characteristics probability (ECP) distribution function symbolized as ECP(ht,psl,vt,es,ea,dci), which is dependent on highway type (ht) and posted speed (psl) since the probability distributions of es and ea vary by ht and psl, as discussed in Scenarios to Study and Support Data in Chapter 4. Given ht and psl, and by assuming the four encroachment characteristics are independent, we have: ECP(ht,psl,vt,es,ea,dci) = ECP(vt)ÃECP(ht,psl,es)ÃECP(ht,psl,ea)ÃECP(dci) where the best available data for ECP(vt), ECP(ht,psl,es), ECP(ht,psl,ea), and ECP(dci) are provided in Scenarios to Study and Supporting Data in Chapter 4. Given a specific highway type-posted speed limit combination (e.g., ht=1 and psl=2), there are a total of 300 (=4Ã5Ã3Ã5) possible combinations of encroachment characteristics for each encroachment, and each combination has a certain probability of occurrence. The sum of probabilities over the 300 possible combinations equals 1. That is, for each of the four ht-psl combinations:

A-8 âvt âes âea âdci ECP(ht,psl,vt,es,ea,dci) = 1. These 300 probability values stratified by vt, es, ea, and dci constitute the probability-weight matrix referenced in the main section. Ditch Traversal and Impact Model (CarSim Model) For each of the 80 design configurations considered in the baseline condition, all 300 possible combinations of the encroachment characteristics discussed above are simulated with the CarSim model to represent all possible encroachment characteristics of each encroachment. Thus, the total number of CarSim simulation runs is 24,000 (=80Ã300). As indicated earlier, some of the baseline ditch configurations do not meet the 2 ft minimum ditch depth requirement for cut slope sections, and these configurations need not be considered. In addition, some limit on the maximum height of the backslope is determined for cut slope sections, which will further reduce the number of baseline configurations to consider. Thus, the actual number of simulation runs is less than 24,000. For every combination of the encroachment characteristics, CarSim is programmed to output key vehicle performance measures during ditch traversal, which are used by the next component model to determine crash injury severity distribution. These performance measures include the extent of lateral and longitudinal encroachments, an indicator variable indicating whether a full recovery of the vehicle is attained (assuming no collision with another vehicle as the vehicle returns to travelway), vehicle stability (e.g., rollover or non-rollover), vehicleâs path, speed, and acceleration as a function of time. Collectively, the various vehicle performance measures (VPMs) obtained from the CarSim model for the baseline design condition is expressed as: VPM(hc*,vg*,sw*,fs,fw,btw*,bs,bw,rnd*,surf*,vt,es,ea,dci) Example VPM is vehicle rollover status: Roll(hc*,vg*,sw*,fs,fw,btw*,bs,bw, rnd*,surf*,vt,es,ea,dci): 1 if rollover and 0 if non- rollover. Each VMP has 24,000 outcome values from the CarSim, and each represents a measure of vehicle performance for a particular combination of ditch configuration and encroachment characteristics. Impact-Severity Model This component model uses the VPMs from the CarSim model, such as vehicle stability, severity index, to decide whether a reportable crash occurs, and, if it occurs, predict injury severity distribution. Two separate sets of police-reported injury severity level distributions are developed, one for rollover and one for non-rollover crashes. If a rollover encroachment is predicted by CarSim, then a simple model is to assume that the severity distribution is the same as the distribution of real-world ditch-initiated rollover crashes, which was provided in Chapter 5. On the other hand, if a non-rollover encroachment is predicted by CarSim, a probability distribution of injury severity is determined using the severity index and police-reported severity distribution, also described in Chapter 5. Essentially, two impact-severity relationships are adopted in the model, one for rollover and one for non-rollover crashes. Each relationship takes the VPMs and produces an impact-

A-9 severity probability (ISP) for the encroached vehicle to result in each police-reported injury severity level, i.e., pis, where pis=1,2, â¦, 6. The two relationships are expressed as follows: ISP_Roll(hc*,vg*,sw*,fs,fw,btw*,bs,bw, rnd*,surf*,vt,es,ea,dci,pis) = probability of resulting in an injury severity level âpisâ if rollover occurs, where pis=1,2,â¦,6; and set to 0 if rollover does not occur. Note that, for each of the 24,000 combinations of fs, fw, bs, bw, vt, es, ea, and dci variables,âpis ISP_Roll(â¦,pis) = 1 if rollover occurs and that âpis ISP_Roll(â¦,pis) = 0 if rollover does not occur. ISP_NoRoll(hc*,vg*,sw*,fs,fw,btw*,bs,bw, rnd*,surf*,vt,es,ea,dci,pis) = probability of resulting in an injury severity level of âpisâ if rollover does not occur, where pis=1,2,â¦,6; and set to 0 if rollover occurs. Note that, for each of the 24,000 combinations of fs, fw, bs, bw, vt, es, ea, and dci variables, âpis ISP_NoRoll(â¦,pis) = 1 if rollover does not occur and âpis ISP_NoRoll(â¦,pis) = 0 if rollover occurs. Crash Cost Model Ditch-initiated crash costs per crash (CCS) by police-reported injury severity level are provided in Chapter 4. These costs are estimated from crash data for rollover and non-rollover crashes, expressed as CCS_Roll(pis) and CCS_NoRoll(pis), respectively. The expected crash cost per encroachment (ECC) is estimated as the probability-weighted sum of CCS over pis, vt, es, ea, and dci, as follows: ECC(ht,psl,hc*,vg*,sw*,fs,fw,btw*,bs,bw, rnd*,surf*) = âpis âvt âes âea âdci {Roll(â¦) Ã CCS_Roll(pis)ÃISP_Roll(â¦, pis) + (1-Roll(â¦))ÃCCS_NoRoll(pis) ÃISP_NoRoll(â¦, pis)} Ã ECP(ht,psl,vt,es,ea,dci) where Roll(â¦) is the 0-1 rollover indicator variable defined before, ECP(â¦) is the encroachment characteristics probability distribution discussed above, and ht=1 or 2, psl=1 or 2, fs=1 to 4, fw=1 or 2, bs=1 to 5, and bw=1 or 2. Thus, there are 320 (=2Ã2Ã4Ã2Ã5Ã2) expected crash costs (per encroachment) for the baseline design condition. Effects of Foreslope and Backslope Widths There are four foreslope widthâbackslope width combinations: {(fw,bw): (1,1), (1,2), (2,1), and (2,2)} or {(FW,BW): (8,8), (8,16), (16,8), and (16,16)}, where FW and BW are foreslope and backslope widths (in ft), respectively . To simplify the relationships and allow the effects of these widths on crash costs to be extended beyond the simulated widths, the following expected crash cost adjustment (ECCA) factors, relative to the combination (fw,bw)=(1,1), are first derived. ï· Expected crash cost adjustment factor for (fw, bw) = (2,1): ECCA(ht,psl, fw=2, bw=1) = âfs âbs ECC(ht,psl,hc*,vg*,sw*,fs, fw=2,btw*,bs, bw=1, rnd*,surf*)/âfs âbs ECC(ht,psl,hc*,vg*,sw*,fs, fw=1,btw*,bs, bw=1, rnd*,surf*). ï· Expected crash cost adjustment factor for (fw, bw) = (1,2): ECCA(ht,psl, fw=1,bw=2) = âfs âbs ECC(ht,psl,hc*,vg*,sw*,fs, fw=1,btw*,bs, bw=2, rnd*,surf*)/âfs âbs ECC(hc*,vg*,sw*,fs,fw=1,btw*,bs,bw=1, rnd*,surf*). ï· Expected crash cost adjustment factor for (fw, bw) = (2,2): ECCA(ht,psl, fw=2, bw=2) = âfs âbs ECC(ht,psl,hc*,vg*,sw*,fs, fw=2,btw*,bs, bw=2, rnd*,surf*)/âfs âbs ECC(ht,psl,hc*,vg*,sw*,fs, fw=1,btw*,bs, bw=1, rnd*,surf*).

A-10 Now, assuming that the effects are relatively smooth beyond the simulated data points, a general model can be formulated as follows: ECCA(ht,psl, FW,BW)= exp(b1(ht,psl)Ã(FW-8)+b2Ã(BW-8)+b12(ht,psl)Ã(FW-8)Ã(BW-8)) where, b1(ht,psl) = (1/8) Ã ln(ECCA(ht,psl, fw=2, bw=1)), b2(ht,psl) = (1/8) Ã ln(ECCA(ht,psl, fw=1, bw=2)) , b12(ht,psl)= (1/64) Ã ln(ECCA(ht,psl, fw=2, bw=2)/{ECCA(ht,psl, fw=2,bw=1)ÃECCA(ht, psl, fw=1,bw=2)}, and ln(.) is the natural logarithm. As before, FW and BW are foreslope width and backslope width, respectively, in ft. As can be checked, the adjustment factor for the reference combination (fw=1,bw=1) is oneâthat is, ECCA(ht,psl, FW=8,BW=8)= ECCA(ht,psl, fw=1,bw=1) = 1. As mentioned earlier, if both FW and BW are treated as adjustment variables, then the interactive effect between the two variables cannot be estimated, which is equivalent to assuming that the coefficient b12(ht,psl) is zero. The applicable range of the model, especially the extension beyond 16 ft, needs to be determined after the mode is constructed. Also, the model can be further reduced if the differences between the four ht-psl combinations are relatively small. Technically, a more sophisticated model involving the interaction between slopes and widths can be developed if the set of coefficientsâb1(ht,psl), b2(ht,psl), and b12(ht,psl)â obtained through the averaging of the crash cost over foreslope and backslope ratios is not adequate. This interaction between slopes and widths will be tested after the simulated data become available. Normalized Expected Crash Costs (NECC) With the ECCA model above, the ECC (expected crash cost/enc) with respect to the various foreslope width and backslope width combinations can now be simplified as follows: ECC(ht,psl,hc*,vg*,sw*,fs, FW,btw*,bs ,BW, rnd*,surf*) = ECC(ht,psl,hc*,vg*,sw*,fs, fw=1,btw*,bs, bw=1, rnd*,surf*)* ECCA(ht,psl, FW,BW) That is, only the ECC for the configurations involving fw=1 and bw=1 need to be maintained, and the costs of other combinations can be regarded as âadjustedâ costs of this basic set of costs. The set of baseline design configurations involving fw=1 and bw=1 is referred to as the âdefault design configurations or conditionsâ in the following discussion. For a highway section with one mile in length and with traffic volume aadt, the expected crash cost per mile per year is calculated as follows: ECC(ht,psl, aadt,hc*,vg*,sw*,fs, FW,btw*,bs ,BW, rnd*,surf*) = ER(ht,psl, aadt,hc*,vg*)Ã ECC(ht,psl,aadt,hc*,vg*,sw*,fs, fw=1,btw*,bs, bw=1, rnd*,surf*)Ã ECCA(ht,psl, FW,BW) where ER(ht,psl, aadt,hc*,vg*) is the encroachment rate (in enc/mi/yr) when the AADT is at level aadt, as discussed earlier. The normalized expected crash cost per mile per year (NECC) is computed as follows: NECC(ht,psl,aadt,hc*,vg*,sw*,fs, FW,btw*,bs ,BW, rnd*,surf*) = ECC(ht,psl,aadt,hc*,vg*,sw*,fs, FW,btw*,bs ,BW, rnd*,surf*)/{CC_RealWorld Ã ER(ht,psl,aadt_avg,hc*,vg*)} = {ER(ht,psl,aadt,hc*,vg*) Ã

A-11 ECC(ht,psl,aadt,hc*,vg*,sw*,fs, fw=1,btw*,bs, bw=1, rnd*,surf*) Ã ECCA(ht,psl, FW,BW)}/{CC_RealWorld Ã ER(ht,psl,aadt_avg,hc*,vg*)} where ECC(â¦) is the expected crash costs per mile per year obtained above, CC_RealWorld is the real-world per-crash cost associated with ditch-initiated crashes (about $127,000 in 2010 dollars; see Severity and Cost of Ditch-Initiated Crashes in Chapter 3), and ER(ht,psl,aadt_avg,hc*,vg*) is the encroachment rate evaluated at the average traffic volume of the data discussed earlier in the Encroachment Rate Model subsection. PREFERRED DESIGN ENVELOPES FOR NON-BASELINE CONDITIONS The same envelope adjustment factor (EAF) concept and method developed in the last section for the baseline design condition can be extended to select preferred design envelopes for non-baseline design conditions. Crash cost adjustment factors are developed for the foreslope width and backslope width for the baseline design condition in the last section. Additional adjustment factors need to be developed for this extension to take place. Specifically, additional ECCA factors need to be developed for the following design variables for non-baseline conditions: horizontal curvature (hc), vertical grade (vg), shoulder width (sw), ditch bottom width (btw), rounding treatment (rnd), and surface treatment (surf). These variables are treated as adjustment variables, as defined in Section A.3. As in the baseline design condition, the CarSim simulations are performed to support the development of such adjustment factors. However, the number of simulation runs is a lot smaller for two reasons: (1) adjustment variables are considered individually (as discussed in Section A.3), and (2) only the foreslope-backslope ratios on the preferred envelopes in the default design condition are considered. About five design combinations of foreslope ratio and backslope ratio from each of the four preferred envelopes are selected for the additional CarSim simulation runs. The five selected foreslope ratio and backslope ratio combinations from the preferred design envelopes for the default design condition are denoted by (fs^, bs^) in the following discussion. The same method and notations as those used for the baseline condition in the last section are used to develop ECCA factors for each of the non-baseline design variables. What follows is a list of calculations and number of simulation runs needed to obtain the ECCA for the following six non-baseline design variables: horizontal curvature (hc), vertical grade (vg), shoulder width (sw), ditch bottom width (btw), rounding treatment (rnd), and surface treatment (surf). As can be seen, a total of 36,000 (=6Ã6,000) additional CarSim simulation runs are needed to obtain VPMs for these non-baseline variables. ECCA for Horizontal Curvature (hc) The horizontal curvature for the baseline condition is 0 deg (hc = 1) (i.e., a straight section). To obtain the adjustment factor for hc = 2 and 3 (i.e., 4.5 deg and 6 deg, respectively), the following steps need to be taken: ï· Obtain VPM(ht,psl,hc,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*,vt,es,ea,dci). The number of CarSim simulation runs required are: 2 (ht)Ã2 (psl) Ã2 (hc) Ã 5 (fs^,bs^) Ã300 (vt,es,ea,dci) = 6,000. ï· Obtain ISP_Roll(ht,psl,hc,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*,vt,es,ea,dci,pis) if rollover occurs, and ISP_NoRoll(ht,psl,hc,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*,vt,es,ea,dci,pis) if rollover does not occur, where pis=1,2,â¦,6. ï· Compute ECC(ht,psl,hc,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*).

A-12 ï· Compute ECCA(ht,psl, hc) = â(fs^,bs^) ECC(ht,psl,hc,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*)/â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*). ECCA for Vertical Grade (vg) The vertical grade for the baseline condition is 0% (vg = 1) (i.e., a flat section). To obtain the adjustment factor for vg = 2 and 3 (i.e., 4% and 6%, respectively), the following steps need to be taken: ï· Obtain VPM(ht,psl,hc*,vg,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*,vt,es,ea,dci). The number of CarSim simulation runs required are: 2 (ht)Ã2 (psl) Ã2 (vg) Ã 5 (fs^,bs^) Ã300 (vt,es,ea,dci) = 6,000. ï· Obtain ISP_Roll(ht,psl,hc*,vg,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*,vt,es,ea,dci,pis) if rollover occurs, and ISP_NoRoll(ht,psl,hc*,vg,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*,vt,es,ea,dci,pis) if rollover does not occur, where pis=1,2,â¦,6. ï· Compute ECC(ht,psl,hc*,vg,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*). ï· Compute ECCA(ht,psl,vg) = â(fs^,bs^) ECC(ht,psl,hc*,vg,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*)/â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*). ECCA for Shoulder Width (sw) The baseline shoulder width is 6 ft (sw = 1). To obtain the adjustment factor for sw = 2 and 3 (i.e., 2 ft and 12 ft, respectively), the following steps need to be taken: ï· Obtain VPM(ht,psl,hc*,vg*,sw,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*,vt,es,ea,dci). The number of CarSim simulation runs required are: 2 (ht)Ã2 (psl) Ã2 (sw) Ã 5 (fs^,bs^) Ã300 (vt,es,ea,dci) = 6,000. ï· Obtain ISP_Roll(ht,psl,hc*,vg*,sw,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*,vt,es,ea,dci,pis) if rollover occurs, and ISP_NoRoll(ht,psl,hc*,vg*,sw,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*,vt,es,ea,dci,pis) if rollover does not occur, where pis=1,2,â¦,6. ï· Compute ECC(ht,psl,hc*,vg*,sw,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*). ï· Compute ECCA(ht,psl,sw=2) = â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw=2,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*)/â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*). ï· Develop the formula to extend the adjustment factor beyond the simulated points: ECCA(ht,psl, SW)= exp(b(ht,psl)Ã(SW-6)), where SW is the shoulder with in ft, SW is between 0 and 6 ft, and b(ht,psl) = (1/4) Ã ln(ECCA(ht,psl, sw=2)). See the discussion in the last section on the effects of foreslope and backslope widths regarding the method. ï· Compute ECCA(ht,psl,sw=3) = â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw=3,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*)/â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*). ï· Develop the formula to extend the adjustment factor beyond the simulated points: ECCA(ht,psl, SW)= exp(b(ht,psl)Ã(SW-6)), where SW is the shoulder with in ft, SW is between 6 and a wider shoulder greater than 12 ft (to be determined), and b(ht,psl) = (1/6) Ã ln(ECCA(ht,psl, sw=3)).

A-13 ECCA for Ditch Bottom Width (btw) The baseline ditch bottom width is 0 ft (btw = 1) (i.e., a V-ditch). To obtain the adjustment factor for btw = 2 and 3 (i.e., 4 ft and 10 ft, respectively), the following steps need to be taken: ï· Obtain VPM(ht,psl,hc*,vg*,sw*,fs^,fw=1, btw,bs^,bw=1, rnd*,surf*,vt,es,ea,dci). The number of CarSim simulation runs required are: 2 (ht)Ã2 (psl) Ã2 (btw) Ã 5 (fs^,bs^) Ã300 (vt,es,ea,dci) = 6,000. ï· Obtain ISP_Roll(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw,bs^,bw=1, rnd*,surf*,vt,es,ea,dci,pis) if rollover occurs, and ISP_NoRoll(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw,bs^,bw=1, rnd*,surf*,vt,es,ea,dci,pis) if rollover does not occur, where pis=1,2,â¦,6. ï· Compute ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw,bs^,bw=1, rnd*,surf*). ï· Compute ECCA(ht,psl,btw=2) = â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw=2,bs^,bw=1, rnd*,surf*)/â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*). ï· Develop the formula to extend the adjustment factor beyond the simulated points: ECCA(ht,psl, BTW)= exp(b(ht,psl)ÃBTW), where BTW is the ditch bottom with in ft, BTW is between 0 and 4 ft, ad b(ht,psl) = (1/4) Ã ln(ECCA(ht,psl, btw=2). See the discussion in the previous section on the effects of foreslope and backslope widths for the methodology. ï· Compute ECCA(ht,psl,btw=3) = â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw=3,bs^,bw=1, rnd*,surf*)/â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*). ï· Develop the formula to extend the adjustment factor beyond the simulated points: ECCA(ht,psl, BTW)= exp(b(ht,psl)Ã(BTW-4)), where BTW is the shoulder with in ft, BTW is from 4 ft to a wider bottom width greater than 10 ft (to be determined), and b(ht,psl) = (1/6) Ã ln(ECCA(ht,psl, btw=3). ECCA for Rounding Treatment (rnd) The baseline condition assumes no-rounding treatment (rnd = 1). To obtain the adjustment factor for the level of rounding considered in rnd = 2 and 3, the following steps need to be taken: ï· Obtain VPM(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd,surf*,vt,es,ea,dci). The number of CarSim simulation runs required are: 2 (ht)Ã2 (psl) Ã2 (rnd) Ã 5 (fs^,bs^) Ã300 (vt,es,ea,dci) = 6,000. ï· Obtain ISP_Roll(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd,surf*,vt,es,ea,dci,pis) if rollover occurs, and ISP_NoRoll(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd,surf*,vt,es,ea,dci,pis) if rollover does not occur, where pis=1,2,â¦,6. ï· Compute ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd,surf*). ï· Compute ECCA(ht,psl,rnd) = â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd,surf*)/â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*).

A-14 ECCA for Surface Treatment (surf) The baseline condition assumes no surface treatment (surf = 1). To obtain the adjustment factor for the level of surface treatments considered in surf = 2 and surf = 3, the following steps need to be taken: ï· Obtain VPM(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf,vt,es,ea,dci). The number of CarSim simulation runs required are: 2 (ht)Ã2 (psl) Ã2 (surf) Ã 5 (fs^,bs^) Ã300 (vt,es,ea,dci) = 6,000. ï· Obtain ISP_Roll(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf,vt,es,ea,dci,pis) if rollover occurs, and ISP_NoRoll(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf,vt,es,ea,dci,pis) if rollover does not occur, where pis=1,2,â¦,6. ï· Compute ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf). ï· Compute ECCA(ht,psl,surf) = â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf)/â(fs^,bs^) ECC(ht,psl,hc*,vg*,sw*,fs^,fw=1,btw*,bs^,bw=1, rnd*,surf*).