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Safety performance functions incorporating design consistency variables
Accident Analysis and Prevention 74 (2015) 133–144
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Safety performance functions incorporating design consistency variables Alfonso Montella ∗ , Lella Liana Imbriani 1 Department of Civil, Architectural and Environmental Engineering, University of Naples Federico II, Via Claudio 21, 80125 Naples, Italy
a r t i c l e
i n f o
Article history: Received 13 June 2014 Received in revised form 15 September 2014 Accepted 16 October 2014 Keywords: Safety performance functions Design consistency Continuous speed profiles Consistency in driving dynamics Operating speed consistency Inertial speed consistency
a b s t r a c t Highway design which ensures that successive elements are coordinated in such a way as to produce harmonious and homogeneous driver performances along the road is considered consistent and safe. On the other hand, an alignment which requires drivers to handle high speed gradients and does not meet drivers’ expectancy is considered inconsistent and produces higher crash frequency. To increase the usefulness and the reliability of existing safety performance functions and contribute to solve inconsistencies of existing highways as well as inconsistencies arising in the design phase, we developed safety performance functions for rural motorways that incorporate design consistency measures. Since the design consistency variables were used only for curves, two different sets of models were fitted for tangents and curves. Models for the following crash characteristics were fitted: total, singlevehicle run-off-the-road, other single vehicle, multi vehicle, daytime, nighttime, non-rainy weather, rainy weather, dry pavement, wet pavement, property damage only, slight injury, and severe injury (including fatal). The design consistency parameters in this study are based on operating speed models developed through an instrumented vehicle equipped with a GPS continuous speed tracking from a field experiment conducted on the same motorway where the safety performance functions were fitted (motorway A16 in Italy). Study results show that geometric design consistency has a significant effect on safety of rural motorways. Previous studies on the relationship between geometric design consistency and crash frequency focused on two-lane rural highways since these highways have the higher crash rates and are generally characterized by considerable inconsistencies. Our study clearly highlights that the achievement of proper geometric design consistency is a key design element also on motorways because of the safety consequences of design inconsistencies. The design consistency measures which are significant explanatory variables of the safety performance functions developed in this study are: (1) consistency in driving dynamics, i.e., difference between side friction assumed with respect to the design speed and side friction demanded at the 85th percentile speed; (2) operating speed consistency, i.e., absolute value of the 85th percentile speed reduction through successive elements of the road; (3) inertial speed consistency, i.e., difference between the operating speed in the curve and the average operating speed along the 5 km preceding the beginning of the curve; and (4) length of tangent preceding the curve (only for run-off-the-road crashes). © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Highway design which ensures that successive elements are coordinated in such a way as to produce harmonious and homogeneous driver performances along the road is considered consistent
∗ Corresponding author. Tel.: +39 0817683941; fax: +39 081768394. E-mail addresses: [email protected] (A. Montella), [email protected] (L.L. Imbriani). 1 Tel.: +39 0817683375; fax: +39 0817683946. http://dx.doi.org/10.1016/j.aap.2014.10.019 0001-4575/© 2014 Elsevier Ltd. All rights reserved.
and safe. On the other hand, an alignment which requires drivers to handle high speed gradients and surprising events and does not meet drivers’ expectancy is considered inconsistent and produces higher crash frequency (Anderson et al., 1999; Awatta et al., 2006; Cafiso et al., 2007; Camacho-Torregrosa et al., 2013; Lamm et al., 1999; Montella, 2005, 2009, 2010; Montella et al., 2008, 2012). As a consequence, the identification of proper design consistency measures and the development of quantitative relationships between design consistency and crash frequency are crucial for several steps of the highway safety management process, such as the design of new highways, the hotspots identification, the design
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of improvements of existing highways, and the evaluation of the safety effects of the design decisions. Several design consistency measures have been successfully used as explanatory variables in safety performance functions fitted on two-lane rural highways (Anderson et al., 1999; Awatta et al., 2006; Cafiso et al., 2010; Camacho-Torregrosa et al., 2013; ˜ et al., 2014; Ng and Sayed, 2004). Design consistency de Ona measures are generally related to the operating speed profile and the speed differential from an approach tangent to an horizontal curve (Montella et al., 2014a,b; TRB, 2011). The best known consistency criteria are those proposed by Lamm et al. (1995, 1999), that defined three design classes (good, fair, and poor) basing on three consistency measures: (1) Vd , design speed consistency (absolute value of the difference between design speed and operating speed); (2) V85 , operating speed consistency (absolute value of the 85th percentile speed reduction through successive elements of the road); and (3) fr , consistency in driving dynamics (difference between side friction assumed with respect to the design speed and side friction demanded at the 85th percentile speed). To increase the usefulness and the reliability of the existing safety performance functions incorporating design consistency variables, there is the need of further studies not limited to two-lane rural highways (Montella et al., 2008) as well as of studies taking into account the location of any geometric element in relation to the overall road alignment and the distance from the other geometric elements (Camacho-Torregrosa et al., 2013; Findley et al., 2012). Aim of the paper is to fill these research gaps developing safety performance functions for rural motorways that incorporate design consistency measures and take into account the location of any geometric element in relation to the overall road alignment. The design consistency parameters in this study are based on operating speed models developed in previous research through an instrumented vehicle equipped with a GPS continuous speed tracking (Montella et al., 2014a) from a field experiment conducted on the same motorway where the safety performance functions were fitted. These operating speed models are more accurate than models based on spot speed studies since they are not based only on spot speed data collected at the centre of the horizontal curve and at the midpoint of the preceding tangent. The remainder of paper is organized as follows: Section 2 describes the study data; Section 3 describes the methodology used to develop the safety performance functions and the variables introduced in the models; Section 4 explains and discusses the results, with a specific focus on the comparison with results of previous studies; finally, conclusions are drawn in the last Section.
2. Data description 2.1. Geometric data The study site is the section Naples–Candela of the motorway A16 Naples–Canosa (L = 255.0 km, i.e., 127.5 km per carriageway). It is part of the Trans European Road Network (Road E841), is located in the south of Italy and links up west coast (Motorway A1) and east coast (Motorway A14). It is a divided highway with two lanes for each direction (lane width = 3.75 m, right shoulder width = 0.50–3.50 m, median width = 2.00 m), access control, and interchanges. Median safety barriers include wbeam, double w-beam, thrie-beam, and concrete New Jersey shaped barriers. Because of the topographical constraints and the mountainous terrain, the freeway is characterised by a bending alignment, with several low radius curves and many design inconsistencies, and by sections with high longitudinal grades (Table 1). It is worthwhile to point out that, because of the presence of numerous mountains, in Italy several motorways have geometric characteristics similar to the study site. The corridor is connected to the road network by 11 interchanges, with 21 exit ramps (13 on curves, 8 on tangents) and 20 entrance ramps (11 on curves, 9 on tangents). Part of the route is in mountainous terrain with 11 tunnels (L = 4.03 km) and 38 bridges (L = 8.11 km). Two climbing lanes are located in east direction (L = 10.48 km) and one climbing lane is located in west direction (L = 3.83 km). Statutory speed limit is 130 km/h but posted speed limits equal to 80 km/h are installed in both travel directions (L = 50.25 km in east carriageway, L = 26.60 km in west carriageway). Segmentation was carried out to obtain homogeneous segments with respect to average annual daily traffic and curvature. First, all horizontal curves and tangents were separated. Then, segments with homogeneous curvature were further separated when changes in traffic volumes occurred inside the segment due the presence of an interchange. Based on the horizontal alignment characteristics and the traffic flow volumes, a segmentation into 652 homogeneous sections (326 for each carriageway) was carried out. Length of the segments varies between 62 and 3509 m. Radius of the horizontal curves varies between 245 and 4000 m. Spiral transitions are not present. Deflection angle varies between 5 and 109 gon. Superelevation mean is equal to 3.25%. Radius of the vertical curves varies between 3000 m (sag curve) and 30,000 m (crest curve). Maximum longitudinal grade is equal to 6.35%. Sight distance is often less than the stopping sight distance and ranges between 62 and 840 m.
Table 1 Summary statistics of geometric data. Parameter
Mean
Standard deviation
Minimum
Lengtha (m) Radius of horizontal curvesb (m) Deflection angle (gon)c Superelevation (%) Radius of sag vertical curvesd (m) Radius of crest vertical curvese (m) Longitudinal grade (%) Sight distance (m) Right shoulder width (m)
394.10 790.77 34.57 3.25 9,000.00 10,888.89 2.41 342.87 2.10
348.12 575.09 20.75 1.09 3,233.58
61.51 245.00 4.88 1.30 3,000.00
1.70 173.53 0.89
0.00 62.27 0.50
a b c d e
652 segments. 332 horizontal curves. The dimension gon corresponds to 400 angle units in a circle instead of 360 degrees. 92 sag curves. 76 crest curves.
Maximum 3.509,58 4.000,00 109.07 6.42 20,000.00 30,000.00 6.35 840.00 3.50
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Table 2 Summary statistics of traffic data (AADT). Year
East direction Mean
2007 2008 2009 2010 2011
12,567 12,492 12,628 12,531 11,997
West direction Standard deviation
Minimum
3,664 3684 3613 3595 3459
8,185 8,077 8,283 8,249 7,886
Maximum
Mean
16,712 16,715 16,952 16,923 16,185
2.2. Traffic data Average annual daily traffic (AADT) data were provided by the motorway management agency for the period 2007–2011. They are disaggregated for each carriageway and for each section between successive interchanges. AADT ranges between 7886 and 17,255 vehicles per day (Table 2). It is worthwhile to observe that since 2009 a significant reduction of traffic volume is observed, mainly because of the economic recession.
2.3. Crash data Since the Italian national crash database maintained by the National Institute of Statistics presents major issues related to the crash report form, the crash classification, the crash location, and the crash severity, a new database was developed according to a framework based on the critical review of the crash databases in Australasia, the European Union and the United States (Montella et al., 2013). Crash data were collected through analysis of police reports and were integrated with detailed site inspections. Crash data covered 2007–2011 (Table 3). Crashes at the interchanges ramps, at rest areas, and at tollbooths were excluded because of the location
12,877 12,773 12,891 12,774 12,209
Standard deviation 3,522 3,611 3,528 3,513 3,357
Minimum 8,590 8,396 8,589 8,535 8,164
Maximum 17,132 17,005 17,255 17,055 16,342
descriptions of the police reports. In the analysis period, 2660 total crashes occurred. Crash data were classified according to the alignment (tangent and curve), collision type (single-vehicle run-off-the-road, other single-vehicle, and multi-vehicle), lighting conditions (daytime and nighttime), weather conditions (rainy and non-rainy), pavement conditions (dry and wet), and crash severity (property damage only (PDO), slight injuries, and severe or fatal injuries). The crash severity classification was based on the most severe injury to any person involved in the crash. According to the agreed international definition, a traffic crash fatality was every single person that dies in the crash or within the 30 days following it. As far as non-fatal injuries, we classified severe and slight injuries. An injury was classified severe if the person is detained in hospital as an “in-patient”, or suffers any of the following injuries whether or not is detained in hospital: fractures, concussion, internal injuries, crushing, significant burns (second and third degree burns over 10% or more of the body), severe cuts and lacerations, severe general shock requiring medical treatment, partial disability for at least 30 days, or the injuries causing death 30 or more days after the crash. Any injuries that are evident at the scene of the crash, other than fatal or serious injuries, such as sprains, bruises or cuts which are not judged to be severe or
Table 3 Summary statistics of crash data. Mean
Curves Total ROR SV OT MV Daytime Nighttime Non rainy Rainy Dry Wet PDO Slight injury Severe injury Tangents Total ROR SV OT MV Daytime Nighttime Non rainy Rainy Dry Wet PDO Slight injury Severe injury
Standard deviation
Minimum
Maximum
Sites with zero crashes N
(%)
3.80 1.75 1.44 0.61 2.65 1.14 2.95 0.84 2.43 1.37 2.88 0.70 0.21
4.92 3.53 1.85 0.99 3.81 1.98 3.39 2.42 2.89 3.18 3.65 1.41 0.54
0 0 0 0 0 0 0 0 0 0 0 0 0
43 37 14 7 38 19 23 30 22 38 31 11 4
58 141 136 206 84 173 67 220 85 166 74 214 275
17.47 42.47 40.96 62.05 25.30 52.11 20.18 66.27 25.60 50.00 22.29 64.46 82.83
4.38 1.26 2.15 0.97 3.08 1.29 3.79 0.58 3.24 1.13 3.57 0.63 0.18
5.12 2.35 2.54 1.70 3.79 1.95 4.41 1.50 3.83 2.25 4.01 1.26 0.52
0 0 0 0 0 0 0 0 0 0 0 0 0
32 27 15 13 27 16 28 18 23 26 25 9 3
56 154 104 182 82 145 66 220 74 166 66 211 280
17.50 48.13 32.50 56.88 25.63 45.31 20.63 68.75 23.13 51.88 20.63 65.94 87.50
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slight shocks requiring roadside attention were classified slight injuries.
2 (Miaou et al., Two goodness-of-fit measures were calculated: R˛ 1996) and the Akaike information criterion (AIC) (Akaike, 1987). 2 , a dispersion parameter-based R2 , is calculated as follows: R˛
3. Method 3.1. Model description
2 R˛ =1−
Following common practice (Lord and Mannering, 2010; Mannering and Bhat, 2014), generalized linear modelling techniques were used to fit the models and a negative binomial distribution error structure was assumed. The selected model form is as follows: n
ˆ ) = L × ea0 +a11 ×ln(AADT) × e i=1 E(Y
bi ×xi
(1)
where Eˆ (Y) is the predicted annual crash frequency, L is the segment length (m), AADT is the segment average annual daily traffic (veh/day), a0 , a1 and bi are the model parameters, and xi are the explanatory variables other than AADT. The distribution of the crash frequency around Eˆ (Y) = is a negative binomial with variance of: Var(Y ) = + × 2
3.2. Goodness-of-fit measures
(2)
where is the dispersion parameter of the negative binomial distribution. This model form logically estimates zero crashes if one of the two exposure variables (AADT or L) is equal to zero. Segment length is an offset variable. It captures the logical requirement that if N crashes are expected to occur on 1 km of road, 2 N crashes should be expected to occur on an identical road that is 2 km long. The model parameters and the dispersion parameter of the negative binomial distribution were estimated by the maximum likelihood method using the GENMOD procedure in SAS. The models were developed by the stepwise forward procedure, adding one explanatory variable at each step. The decision on whether or not to keep a variable in the model was based on two criteria. The first is whether the t-ratio of the variable’s estimated coefficient is significant at the 5% level. The second criterion is based on the improvement of the goodness-of-fit measures of the model that includes that variable. Since the design consistency variables were used only for curves, two different sets of models were fitted for tangents and curves. Models for the following crash characteristics were fitted: total, single-vehicle run-off-the-road (ROR), other single vehicle (SV OT), multi vehicle (MV), daytime, nighttime, non-rainy weather, rainy weather, dry pavement, wet pavement, property damage only (PDO), slight injury, and severe injury (including fatal). To evaluate the model form, cumulative residual analysis (Hauer, 2004; Hauer and Bamfo, 1997; Lord and Persaud, 2000) was performed. The residual is equal to the difference between the observed and predicted values of the annual crash frequency. Assuming that residuals are normally distributed with expected value equal to 0 and a variance equal to *, it is possible to calculate the variance of the expected value as the square of the cumulate residuals. The trend in the residuals can be evaluated relative to the variance to qualitatively assess goodness of fit. The cumulative residual plot is used to examine whether the chosen functional form fits each explanatory variable along the entire range of its values represented in the data. In general, a good cumulative residuals plot is one that oscillates around 0. A bad cumulative residuals plot is one that is entirely above or below 0 (except at the edges). Thus, it is desirable that the plot of cumulative residuals should oscillate between over and under prediction and not stray beyond the ±2* boundaries.
(3)
max
where max is the largest possible dispersion parameter that is obtained by having no covariates in the model (by assuming that all sites have an identical prediction estimate equal to the mean over all sites) and is the dispersion parameter for the fitted model. This measure is bound between 0 (when no covariate is included) and 1 (when covariates are perfectly specified). The AIC value is calculated as follows: AIC = −2 × ML + 2 × p
(4)
where ML is the maximum log-likelihood of the fitted model, and p is the number of parameters in the model. A smaller AIC value reflects a better model. The first term in the AIC equation measures the goodness of fit, or bias, when the maximum likelihood estimates of the parameters are used. The second term measures the complexity of the model, thus penalizing the model for using more parameters. The minimisation of the AIC value allows to reduce over fitting, since it permits to choose the best fit with the least complexity. 3.3. Variables selection Explanatory variables related to traffic volume, design consistency, horizontal alignment, vertical alignment, sight distance, and roadside context were considered. Below, a brief description of the significant explanatory variables is reported. Table 4 shows descriptions and analytical details of the variables. 3.3.1. Traffic volume The relationship between crashes and traffic volume is frequently non-linear. Thus, traffic volume was not considered as an offset variable. Natural logarithm of the average annual daily traffic (AADT), which is an aggregate measure of the traffic volume, was considered. 3.3.2. Design consistency The consistency measures included in the models for the curves were: (a) V85 , operating speed consistency, i.e., absolute value of the 85th percentile speed reduction through successive elements of the road. Operating speed (V85 ) was assessed using the predictive models developed in previous research through an instrumented vehicle equipped with a GPS continuous speed tracking (Montella et al., 2014a) from a field experiment conducted on the same motorway where the safety performance functions were fitted, i.e., the motorway A16 in Italy. The following models were fitted: V858c = 135.490 −
7.483 − 1.290 × Gu − 0.080 × CCR2 − 14.427 R
× Tunnel − 4.083 × Bridge
V85t = 139.543 + 1.751 × Lt − ×Gu − 0.068 × CCR2
(5)
4.983 − 2.270 − 2.507 Rbefore Rafter (6)
where V85c is the 85th percentile of the operating speed on curves (km/h), V85 t is the 85th percentile of the operating speed on tangents (km/h), R is the radius of the curve (Km), Rbefore is the radius
A. Montella, L.L. Imbriani / Accident Analysis and Prevention 74 (2015) 133–144 Table 4 SPFs: Significant explanatory variables. Variable Traffic volume ln(AADT) [veh/day] Design consistency V85 [km/h]
fr [−]
V85,i –V85,5 [km/h]
Lt [km] Horizontal alignment 1/R2 [1/km2 ] 1/R before [1/km] 1/R after [1/km] Vertical alignment Gd Gu Gd × Ld Gu × Lu Roadside context Tunnel Yearly effects Year10 Year11
Description Natural logarithm of the average annual daily traffic Absolute value of the operating speed difference in the tangent-to-curve transition (Eqs. (5) and (6)) Difference between side friction assumed with respect to the design speed and side friction demanded at the 85th-percentile speed (Eq. (7)) Difference between the 85th percentile of the operating speed and the mean in the 5 km preceding the beginning of the curve of the 85th percentile of the operating speed Length of the tangent preceding the curve Square of horizontal curvature Horizontal curvature of the curve preceding the tangent Horizontal curvature of the curve following the tangent Equivalent downgrade Equivalent upgrade Interaction between equivalent downgrade and length of downgrade segment Interaction between equivalent upgrade and length of upgrade segment Binary variable, equal to 1 if the segment is on a tunnel Binary variable, equal to 1 in 2010 Binary variable, equal to 1 in 2011
of the curve preceding the tangent (Km), Rafter is the radius of the curve following the tangent (Km), CCR2 is the curvature change ratio of the 2 km preceding the geometric element (gon/km), Lt is the length of the tangent (km), Gu is the equivalent upgrade (%), Bridge is a binary variable equal to 1 if the segment is on a bridge, and Tunnel is a binary variable equal to 1 if the segment is on a tunnel. (b) fr , consistency in driving dynamics, i.e., difference between side friction assumed with respect to the design speed and side friction demanded at the 85th percentile speed:
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change rate. Since spiral transitions are not present, the square of curvature (1/R)2 was selected. In the model for tangents we included two variables which take into account the spatial relationships between the tangent and the curves before and after: (1) 1/Rbefore where Rbefore is the radius of the curve preceding the tangent and (2) 1/Rafter where Rafter is the radius of the curve following the tangent. 3.3.4. Vertical alignment Longitudinal slope is not always constant in the segments. To take into account this circumstance, for each segment an equivalent downgrade (Gd ) and upgrade (Gu ) were assessed. Equivalent grade is obtained by weighing each gradient in relation to the segment length. In the vertical curves, two situations occur. If the different grades have the same direction, half of the vertical curve is added to each grade. If the grades have opposite direction, a quarter of the vertical curve is added to each grade. Furthermore, the interactions between the equivalent grade and length of the segments (Gd × Ld and Gu × Lu ) were introduced in the models. 3.3.5. Roadside context To take into account the safety effect of the roadside context, the dummy variable tunnel was introduced. 3.3.6. Yearly effects Dummy variables were introduced to capture the potential nonrandom yearly effects. 3.3.7. Correlations between variables The Pearson product moment correlation (Pearson’s correlation) was assessed to identify correlations between the explanatory variables. Pearson’s correlation reflects the degree of linear relationship between two variables. A correlation of +1 (−1) means that there is a perfect positive (negative) linear relationship between variables. Variables are considered strongly correlated if the absolute value of Pearson’s correlation is higher than 0.85 and the estimated correlation has p-value less than 0.05. Variables included in the models are not highly correlated. 4. Results and discussion 4.1. Curves
where fra is the side friction assumed with respect to the design speed, frd is the side friction demanded at the 85th percentile speed, Vd is the design speed (km/h) calculated according to the Italian Geometric Design Standards (Italian Ministry of Infrastructures and Transports, 2001), V85 is the 85th percentile of the speed distribution in the curve (km/h) calculated with the equation (5), R is the curve radius (m), and e is the curve superelevation. (c) V85,i –V85,5 , inertial speed consistency, i.e., difference between the operating speed in the curve and the average operating speed along the 5 km preceding the beginning of the curve (weighted to element length) (km/h). (d) Lt , length of the tangent preceding the curve (km), which takes into account the distance from the previous curve.
The parameter estimates and the measures of goodness of fit for each model are reported in Table 5. Non-significant parameters are 2 values of the models range between 19% and not reported. The R˛ 69% and all the parameters have logical and expected signs. Cumulative residuals plots against AADT are reported in Figs. 1 and 2. It can be observed that the plots of cumulative residuals oscillate around 0 and do not stray beyond the ±2* boundaries, apart a few exceptions where residuals slightly encroach the boundaries in specific points. The same kind of pattern can be seen for all the variables. Most of the models have many significant variables. The model with the greater number of significant variables is the model for total crashes, which has seven significant explanatory variables other than AADT related to the design consistency, the vertical alignment, and the yearly effects. To avoid over fitting, we explicitly penalized complex models selecting the model with the lower AIC value, i.e., a goodness-of-fit measure that takes into account the complexity of the model penalizing the model for using more parameters.
3.3.3. Horizontal alignment Literature suggests that the geometric design parameter which most affects road safety is the horizontal curvature or the curvature
4.1.1. Design consistency The most important finding from this study is that design consistency significantly affects the safety of horizontal curves on rural
f r = fra − frd = 0.6 × 0.925 × (0.59 − 4.85 × 10−3 × Vd
+ 1.51 × 10−5 × Vd2 ) −
2 V85
127 × R
−e
(7)
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Table 5 SPFs: parameter estimates and goodness of fit measures for curves. Total
ROR
SV OT
MV
Daytime
Nighttime
Non rainy
Rainy
Dry
Wet
PDO
Slight injury
Severe injury
Dispersion
1.101 (0.101) −15.425 (1.372)
1.637 (0.175) −15.807 (1.699)
2.139 (0.227) −12.199 (1.644)
0.427 (0.196) −24.553 (1.809)
1.347 (0.131) −16.123 (1.476)
2.078 (0.247) −17.225 (1.680)
0.991 (0.108) −15.096 (1.371)
3.265 (0.605) −14.890 (3.024)
1.131 (0.125) −16.570 (1.421)
2.232 (0.236) −14.577 (1.744)
1.188 (0.119) −14.169 (1.435)
1.338 (0.230) −26.102 (1.797)
3.006 (0.660) −23.241 (2.036)
0.935 (0.147)
0.823 (0.184)
0.519 (0.176)
1.707 (0.192)
0.943 (0.158)
1.012 (0.180)
0.889 (0.147)
0.609 (0.343)
1.047 (0.152)
0.654 (0.186)
0.754 (0.154)
1.885 (0.191)
1.500 (0.216)
0.037 (0.010) −4.692 (1.046) − 0.018 (0.008)
0.042 (0.011) −6.075 (1.857)
0.059 (0.020)
0.029 (0.009) −3.384 (0.935)
0.041 (0.011)
0.052 (0.009) −4.400 (0.937)
0.045 (0.011) −11.913 (1.237)
−13.837 (1.245)
Constant Traffic volume ln(AADT) Design consistency V85 fr V85,i –V85,5
Horizontal alignment 1/R2
Gu
0.116 (0.020) 11.799 (2.856) 7.529 (2.755)
Year11 R˛2 AIC
−0.207 (0.098) −0.247 (0.100) 0.38 3740
0.053 (0.013)
0.050 (0.013)
5.740 (2.499)
16.263 (3.397) 11.629 (3.532)
0.762 (0.231)
0.690 (0.220)
7.128 (2.524)
– Roadside context Tunnel Yearly effects Year10
−5.112 (1.322) − 0.051 (0.009)
−2.478 (1.058) − 0.033 (0.007)
0.304 (0.105)
Lt
Vertical alignment Gd
0.064 (0.010) −4.656 (0.966)
12.838 (2.891) 11.556 (2.929)
9.397 (3.029)
12.764 (2.900) 6.918 (2.721)
0.178 (0.023)
0.174 (0.013)
14.487 (4.598)
12.960 (2.560)
11.984 (2.862) 12.394 (2.827)
5.833 (2.637)
0.56 1728
0.33 3258
0.62 1265
5.368 (2.215)
− 0.292 (0.105)
0.57 2225
0.19 2183
0.69 1553
Note: standard deviations of the parameter estimates are reported in parenthesis.
0.36 3066
0.40 1813
0.36 3304
0.62 1318
0.33 2958
0.41 991
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Dependentvariable
A. Montella, L.L. Imbriani / Accident Analysis and Prevention 74 (2015) 133–144
Fig. 1. Cumulative residuals vs. AADT (curves: models for total, ROR, other single-vehicle, multi-vehicle, daytime, nighttime, rainy and non-rainy crashes).
139
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Fig. 2. Cumulative residuals vs. AADT (curves: models for wet, dry, PDO, slight injury, and severe injury crashes).
motorways. Consistency in driving dynamics (fr ) is significant in nine models (total, ROR, daytime, nighttime, non-rainy, dry, PDO, slight injury, and severe injury crashes), operating speed consistency (V85 ) is significant in eight models (total, ROR, daytime, rainy, dry, wet, PDO, and slight injury crashes), inertial speed consistency (V85,i –V85,5 ) is significant in three models (total, nighttime, and non-rainy crashes), and length of tangent preceding the curve (Lt ) is significant only in the model for run-off-the-road crashes. In most models, more design consistency variables are significant. As an example, in the model for total crashes consistency in driving dynamics, operating speed consistency, and inertial speed consistency are all significant. This result indicates that all the consistency
variables introduced in this study should be carefully checked in the design stage and that the safety effects of the design depend on the combination of more consistency measures. As the difference between friction demand and friction supply (fr ) decreases, crash frequency is expected to increase (i.e., the estimated coefficient of the parameter consistency in driving dynamics is negative). Given that in existing motorways curve superelevation is frequently lower than the superelevation required by the geometric design standards and guidelines, and that superelevation adjustment is a feasible and quick measure, this result has a relevant practical effect on the selection of safety measures.
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The estimated coefficient of the parameter operating speed consistency (V85 ) has a positive value. This implies that an higher speed reduction between two consecutive elements is related to an increase of the crash frequency and is therefore a visible indicator of inconsistency in geometric design. The relationship between the crash frequency and the operating speed reduction indicates therefore that the greater the speed reduction on a horizontal curve, the greater the curve’s crash experience. This result is consistent with previous studies on two-lane rural highways (Anderson et al., 1999; ˜ et al., Cafiso et al., 2010; Camacho-Torregrosa et al., 2013; de Ona 2014; Ng and Sayed, 2004) and implies that the geometric design should minimise the speed reduction from tangents to horizontal curves. The statistically significant negative value of the estimated coefficient of the inertial speed consistency (V85,i –V85,5 ) shows that the expected crash frequency increases with the difference between the average operating speed in the 5 km preceding the curve and the operating speed in the curve. All else being equal, an increase in the operating speed of the segment before the curve (5 km) has a positive effect on crash frequency whereas a decrease in the operating speed of the segment before the curve (5 km) has a negative effect on crash frequency. Based on the hypothesis that drivers’ long-term expectation at a curve depends on the average operating speed at the previous road segment, the significance of the inertial speed consistency variable indicates that long-term driver’s expectancy has a significant safety effect and drivers are surprised by curves with operating speed lower than the operating speed of the previous five kilometres. This results is consistent with a recent study carried out on two-lane rural roads in Spain by García et al. (2013) and implies that the curve operating speed should be compared with the operating speed along the 5 km before the curve. Thus, a curve that follows a 5 km segment with high operating speed should be a curve that allows high operating speed. The positive value of the estimated coefficient of the length of the tangent preceding the curve (Lt ) for the run-off-the-road crashes model shows that neighbouring curves have few predicted crashes than curves more distant to each other, as reported in the literature (Brenac, 1996; Findley et al., 2012). The length of the tangent preceding the curve indicates that short-term driver’s expectancy has a significant safety effect. A curve that is a part of a series of curves turns out to be safer than a curve which is isolated from other curves. As far as the collision type, consistency variables are significant for single-vehicle run-off-the-road crashes but are not significant for the other single-vehicle crashes and for multi-vehicle crashes. This result shows that highway geometry which does not meet drivers’ expectancy has higher effect on single-vehicle run-off-theroad crashes. 4.1.2. Horizontal alignment The square curvature estimated coefficient (1/R)2 has a positive sign, that is, the smaller the radius, the more the expected crash frequency. Square curvature is significant for ROR, non-ROR single vehicle, multi-vehicle, rainy, and wet crashes. 4.1.3. Vertical alignment Vertical alignment explanatory variables are significant in all the models. In five models (total, multi-vehicle, daytime, non-rainy, and PDO crashes), both equivalent downgrade (Gd ) and equivalent upgrade (Gu ) are significant and have positive value, with the estimated coefficient of equivalent downgrade greater than the coefficient of equivalent upgrade. Thus, the greater the longitudinal grade, the greater the crash frequency. The effect of longitudinal grade on crash frequency is higher in downgrade sections: 12% increase in total crashes for 1% increase in longitudinal
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downgrade vs. 8% increase in total crashes for 1% increase in longitudinal upgrade. In the models for ROR, nighttime, rainy, wet, and severe injury crashes only the equivalent downgrade is significant (positive value). In the models for non-ROR single vehicle and dry crashes only the equivalent upgrade is significant (positive value). 4.1.4. Roadside context The variable tunnel is significant for non-ROR single vehicle and multi-vehicle crashes. It has a positive value, i.e., in road tunnels more non-ROR single vehicle and multi-vehicle crashes are expected. 4.1.5. Yearly effects A significant decrease in total crashes is expected in 2010 and 2011, and in dry crashes in 2010. 4.2. Tangents The parameter estimates and the measures of goodness-of-fit for each model are reported in Table 6. Non-significant parame2 values of the models range between ters are not reported. The R˛ 47% and 93% and all the parameters have logical and expected signs. Cumulative residuals plots against the significant explanatory variables show that the plots of cumulative residuals oscillate around 0 and do not stray beyond the ±2* boundaries, apart a few exceptions where residuals slightly encroach the boundaries in specific points. 4.2.1. Horizontal alignment The curvature of the horizontal curves preceding (1/R before ) and following (1/R after ) the tangent are statistically significant in most models (in twelve models 1/R after and in six models 1/R after ). The values of all the estimated coefficients have a positive sign. Thus, the smaller the radius of the curves before and after the tangent, the greater the expected crash frequency in the tangent. In six models (total, ROR, daytime, rainy, wet, and PDO crashes), both the curvature before and after the tangent have a significant effect on crash frequency but, apart for crashes on wet pavement, the impact of the curvature after the tangent is greater than the impact of the curvature before the tangent. 4.2.2. Vertical alignment Vertical alignment explanatory variables are significant in all the models except in the nighttime crashes model. The comparison between the estimated coefficients of the vertical alignment variables in the models for curves and tangents shows that the effect of longitudinal grade on crashes is much higher on horizontal curves than on tangents. On horizontal curves, a 1% increase in longitudinal grade procedures a 12% increase in estimated total crashes in the downgrade sections vs. a 8% increase in total crashes in the upgrade section. On tangents, 1% increase in longitudinal grade procedures a 6% increase in estimated total crashes in the downgrade sections vs. no significant effect in the upgrade section. The interaction between the equivalent downgrade and the length of the downgrade segment (Gd × Ld ) is significant for wet crashes while the interaction between the equivalent upgrade and the length of the upgrade segment (Gu × Lu ) is significant for non-run-off-the-road single vehicle, daytime and non-rainy crashes. 4.2.3. Yearly effects As for the curves, a significant decrease in crash frequency is expected in 2010 (total, nighttime, non-rainy, PDO, and slight injury
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Table 6 SPFs: parameter estimates and goodness of fit measures for tangents. Total
ROR
SV OT
MV
Daytime
Nighttime
Rainy
Non Rainy
Wet
Dry
PDO
Slight injury
Severe injury
Dispersion
0.393 (0.057) −18.145 (1.138)
0.983 (0.156) −25.765 (1.675)
0.787 (0.112) −10.526 (1.420)
0.171 (0.093) −30.213 (1.639)
0.565 (0.080) −18.564 (1.337)
0.594 (0.131) −16.915 (1.511)
1.813 (0.304) −26.417 (1.963)
0.356 (0.060) −16.188 (1.186)
1.185 (0.176) −25.031 (1.769)
0.435 (0.071) −15.679 (1.218)
0.454 (0.069) −16.697 (1.208)
0.551 (0.163) −29.024 (1.863)
1.479 (0.451) −28.653 (2.217)
1.834 (0.176)
0.363 (0.151)
2.364 (0.172)
1.214 (0.141)
0.999 (0.159)
1.765 (0.207)
1.023 (0.125)
1.743 (0.187)
0.959 (0.129)
1.054 (0.128)
2.148 (0.195)
1.977 (0.232)
0.161 (0.050)
0.336 (0.059) 0.568 (0.061)
0.131 (0.041)
0.394 (0.054) 0.296 (0.053)
0.123 (0.042)
0.092 (0.043) 0.195 (0.043)
0.343 (0.053)
0.357 (0.059)
11.411 (2.816)
6.269 (2.049)
5.250 (2.066)
5.355 (2.014)
8.825 (2.713)
−0.335 (0.095) −0.221 (0.093) 0.72 3079
−0.261 (0.091) −0.370 (0.095) 0.68 3288
−0.528 (0.146) 0.81 1074
Constant Traffic volume ln(AADT)
1.224 (0.120) Horizontal alignment 1/R before 0.083 (0.040) 0.221 1/R after (0.040) Vertical alignment 5.827 Gd (1.876) Gd × Ld
0.146 (0.055) 0.462 (0.054)
Year11 R˛2 AIC
0.123 (0.049)
9.149 (2.538)
8.280 (2.380)
0.002 (0.001) −0.232 (0.084) −0.370 (0.089) 0.72 3587
9.414 (2.240)
0.003 (0.001)
Gu × Lu Yearly effects Year10
0.090 (0.045) 0.220 (0.046)
−0.420 (0.130) 0.64 1773
0.53 2547
0.002 (0.001)
0.93 1353
−0.229 (0.096) 0.66 3025
Note: Standard deviations of the parameter estimates are reported in parentheses.
0.001 (0.001) −0.325 (0.115) −0.662 (0.132) 0.73 1798
−0.821 (0.169) 0.67 1044
−0.217 (0.087) −0.302 (0.090) 0.75 3315
−1.011 (0.153) 0.68 1611
0.47 926
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crashes) and 2011 (total, ROR, daytime, nighttime, non-rainy, rainy, dry, wet, PDO, and slight injury crashes). 5. Conclusions Study results show that geometric design consistency has a significant effect on safety of rural motorways. Previous studies on the relationship between geometric design consistency and crash frequency focused on two-lane rural highways since these highways have the higher crash rates and are generally characterized by considerable inconsistencies. Our study clearly highlights that the achievement of proper geometric design consistency is a key design element also on motorways because of the safety consequences of design inconsistencies. The safety performance functions incorporating design consistency measures allow quantitative evaluations of the safety effects of the design decisions that can be used for the design of new highways, the hotspot identification, and the design of improvements of existing highways. These safety performance functions allow to complement geometric design based on threshold values of the geometric parameters introduced in the geometric design standards and carry out safety-based design. According to the standards safety is nominal, i.e., any design is safe if parameters comply with defined thresholds and is not safe if parameters do not satisfy these threshold. However, highway safety changes with continuity in relation to the design parameters and the fulfilment of drivers’ expectations has direct consequences on crashes that can be quantitatively assessed by safety performance functions incorporating design consistency variables. The consistency measures that resulted significant explanatory variables of the safety performance functions developed in this study are: (1) consistency in driving dynamics, i.e., difference between side friction assumed with respect to the design speed and side friction demanded at the 85th percentile speed; (2) operating speed consistency, i.e., absolute value of the 85th percentile speed reduction through successive elements of the road; (3) inertial speed consistency, i.e., difference between the operating speed in the curve and the average operating speed along the 5 km preceding the beginning of the curve; and (4) length of tangent preceding the curve (only in the model for run-off-the-road crashes). As a consequence, to reduce crashes the geometric design should: (1) minimise the difference between friction demand and friction supply on horizontal curves, using superelevation based on predicted operating speed instead of superelevation based on design speed; (2) minimise the operating speed reduction from tangents to horizontal curves, meeting short-term driver’s expectancy; (3) minimise the difference between the curve operating speed and the operating speed along the 5 km before the curve, meeting longterm driver’s expectancy; and (4) avoid low-operating speed curves following long tangents. Furthermore, greater longitudinal downgrades and upgrades significantly increase crash frequency and the safety effect is higher on horizontal curves than on tangents. Thus, it is essential to carefully take into account the vertical alignment and his safety consequences. These findings highlight that the geometric design of both new motorways and improvements of existing motorways should explicitly consider the safety effects of consistency in driving dynamics, operating speed, and inertial speed. In existing motorways, the improvement of geometric design to reduce inconsistencies is not always feasible because of budget restrictions or other constraints. In these cases, we recommend to implement speed management policies aimed at obtaining operating speeds consistent with the highway geometry. Recently, very effective results in speed reduction have been obtained by the point-to-point speed enforcement (Lynch et al., 2011;
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Montella et al., 2012; Soole et al., 2013), named also average speed enforcement or section speed enforcement, which is a relatively new technological approach to traffic law enforcement that has increased in use in a number of highly motorized countries in the last decade. Unlike traditional spot-speed enforcement, which measures the speed of a vehicle at one point, pointto-point enforcement involves the calculation of the average speed over a section and encourages compliance over a greater distance. References Akaike, H., 1987. Factor analysis and AIC. Psycometrika 52, 317–332. Anderson, I.B., Bauer, K.M., Harwood, D.W., Fritzpatrick, L., 1999. Relationship to safety of geometric design consistency measures for rural two-lane highways. J. Transp. Res. Board 1658, 43–51, http://dx.doi.org/10.3141/1658-06. Awatta, M., Hassan, Y., Sayed, T., 2006. Quantitative evaluation of highway safety performance based on design consistency. Adv. Transp. Stud. 9, 29–44. Brenac, R., 1996. Safety at curves and road geometry standards in some European countries. Transp. Res. Record 1523, 99–106, http://dx.doi.org/10.3141/ 1523-12. Cafiso, S., La Cava, G., Montella, A., 2007. Safety index for evaluation of two-lane rural highways. Transp. Res. Rec. 2019, 136–145, http://dx.doi.org/10.3141/2019-17. Cafiso, S., Di Graziano, A., Di Silvestro, G., La Cava, G., Persaud, B., 2010. Development of comprehensive accident models for two-lane rural highways using exposure, geometry, consistency and context variables. Accid. Anal. Prev. 42, 1072–1079, http://dx.doi.org/10.1016/j.aap.2009.12.015. Camacho-Torregrosa, F.J., Pérez-Zuriaga, A.M., Campoy-Ungría, J.M., García, A., 2013. New geometric design consistency model based on operating speed profiles for road safety evaluation. Accid. Anal. Prev. 61, 33–42, http://dx.doi.org/10.1016/ j.aap.2012.10.001. ˜ J., Garach, L., Calvo, F., García-Munoz, ˜ de Ona, T., 2014. Relationship between predicted speed reduction on horizontal curves and safety on two-lane rural roads in Spain. J. Transp. Eng. 140 (3), http://dx.doi.org/10.1061/(ASCE)TE. 1943-5436.0000624. Findley, D.J., Hummer, J.E., Rasdorf, W., Zegeer, C.V., Fowler, T.J., 2012. Modeling the impact of spatial relationships on horizontal curve safety. Accid. Anal. Prev. 45, 296–304, http://dx.doi.org/10.1016/j.aap.2011.07.018. García, A., Llopis-Castelló, D., Camacho-Torregrosa, F.J., Pérez-Zuriaga, A.M., 2013. New consistency index based on inertial operating speed. Transp. Res. Rec. 2319, 105–112, http://dx.doi.org/10.3141/2391-10. Hauer, E., 2004. Statistical road safety modeling. Transp. Res. Rec. 1897, 81–87, http://dx.doi.org/10.3141/1897-11. Hauer, E., Bamfo, J., 1997. Two tools for finding what function links the dependent variable to the explanatory variables. In: Proceedings of ICTCT 97 Conference, Lund, Sweden. Italian Ministry of Infrastructures and Transports, 2001. Guidelines for the Design of Road Infrastructures: D.M. n. 6792, 5/11/2001. Rome. Lamm, R., Guenther, A.K., Choueiri, E., 1995. Safety module for highway geometric design. Transp. Res. Rec. 1512, 7–15. Lamm, R., Psarianos, B., Mailaender, T., Choueiri, E.M., Heger, R., Steyer, R., 1999. Highway Design and Traffic Safety Engineering Handbook. McGraw-Hill, New York. Lynch, M., White, M., Napier, R., 2011. Investigation into the use of point-to-point speed cameras. NZ Transport Agency Research Report 465, Wellington. Lord, D., Mannering, F., 2010. The statistical analysis of crash-frequency data: a review and assessment of methodological alternatives. Transp. Res. Part A 44 (5), 291–305, http://dx.doi.org/10.1016/j.tra.2010.02.001. Lord, D., Persaud, B., 2000. Accident prediction models with and without trend: application of the generalized estimating equations procedure. Transp. Res. Rec. 1717, 102–108, http://dx.doi.org/10.3141/1717-13. Mannering, F., Bhat, C.R., 2014. Analytic methods in accident research: methodological frontier and future directions. Anal. Methods Accid. Res. 1, 1–22, http://dx.doi.org/10.1016/j.amar.2013.09.001. Miaou, S.P., Lu, A., Lum, H.S., 1996. Pitfalls of using R2 to evaluate goodness of fit of accident prediction models. Transp. Res. Rec. 1542, 6–13, http://dx.doi.org/10.3141/1542-02. Montella, A., 2005. Safety reviews of existing roads: quantitative safety assessment methodology. Transp. Res. Rec. 1922, 62–72, http://dx.doi.org/10.3141/1922-09. Montella, A., 2009. Safety evaluation of curve delineation improvements: empirical bayes observational before and after study. Transp. Res. Rec. 2103, 69–79, http://dx.doi.org/10.3141/2103-09. Montella, A., 2010. A comparative analysis of hotspot identification methods. Accid. Anal. Prev. 42 (2), 571–581, http://dx.doi.org/10.1016/j.aap.2009.09.025. Montella, A., Colantuoni, L., Lamberti, R., 2008. Crash prediction models for rural motorways. Transp. Res. Rec. 2083, 189, http://dx.doi.org/10.3141/2083-21. Montella, A., Persaud, B., D’Apuzzo, M., Imbriani, L.L., 2012. Safety evaluation of an automated section speed enforcement system. Transp. Res. Rec. 2281, 16–25, http://dx.doi.org/10.3141/2281-03. Montella, A., Andreassen, D., Tarko, A., Turner, S., Mauriello, F., Imbriani, L.L., Romero, M., 2013. Crash databases in Australasia, the European Union, and the United States: review and prospects for improvement. Transp. Res. Rec. 2386, 128–136, http://dx.doi.org/10.3141/2386-15.
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Montella, A., Pariota, L., Galante, G., Imbriani, L.L., Mauriello, F., 2014a. Prediction of drivers’ speed behaviour on rural motorways based on an instrumented vehicle study. Transp. Res. Rec., in press. Montella, A., Galante, F., Imbriani, L.L., Mauriello, F., Pernetti, M., 2014b. Simulator evaluation of drivers’ behaviour on horizontal curves of two-lane rural highways. Adv. Transp. Studies: An Int. J. 34, 91–104, http://dx.doi.org/10.4399/ 97888548767439.
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Safety performance functions incorporating design consistency variables Alfonso Montella ∗ , Lella Liana Imbriani 1 Department of Civil, Architectural and Environmental Engineering, University of Naples Federico II, Via Claudio 21, 80125 Naples, Italy
a r t i c l e
i n f o
Article history: Received 13 June 2014 Received in revised form 15 September 2014 Accepted 16 October 2014 Keywords: Safety performance functions Design consistency Continuous speed profiles Consistency in driving dynamics Operating speed consistency Inertial speed consistency
a b s t r a c t Highway design which ensures that successive elements are coordinated in such a way as to produce harmonious and homogeneous driver performances along the road is considered consistent and safe. On the other hand, an alignment which requires drivers to handle high speed gradients and does not meet drivers’ expectancy is considered inconsistent and produces higher crash frequency. To increase the usefulness and the reliability of existing safety performance functions and contribute to solve inconsistencies of existing highways as well as inconsistencies arising in the design phase, we developed safety performance functions for rural motorways that incorporate design consistency measures. Since the design consistency variables were used only for curves, two different sets of models were fitted for tangents and curves. Models for the following crash characteristics were fitted: total, singlevehicle run-off-the-road, other single vehicle, multi vehicle, daytime, nighttime, non-rainy weather, rainy weather, dry pavement, wet pavement, property damage only, slight injury, and severe injury (including fatal). The design consistency parameters in this study are based on operating speed models developed through an instrumented vehicle equipped with a GPS continuous speed tracking from a field experiment conducted on the same motorway where the safety performance functions were fitted (motorway A16 in Italy). Study results show that geometric design consistency has a significant effect on safety of rural motorways. Previous studies on the relationship between geometric design consistency and crash frequency focused on two-lane rural highways since these highways have the higher crash rates and are generally characterized by considerable inconsistencies. Our study clearly highlights that the achievement of proper geometric design consistency is a key design element also on motorways because of the safety consequences of design inconsistencies. The design consistency measures which are significant explanatory variables of the safety performance functions developed in this study are: (1) consistency in driving dynamics, i.e., difference between side friction assumed with respect to the design speed and side friction demanded at the 85th percentile speed; (2) operating speed consistency, i.e., absolute value of the 85th percentile speed reduction through successive elements of the road; (3) inertial speed consistency, i.e., difference between the operating speed in the curve and the average operating speed along the 5 km preceding the beginning of the curve; and (4) length of tangent preceding the curve (only for run-off-the-road crashes). © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Highway design which ensures that successive elements are coordinated in such a way as to produce harmonious and homogeneous driver performances along the road is considered consistent
∗ Corresponding author. Tel.: +39 0817683941; fax: +39 081768394. E-mail addresses: [email protected] (A. Montella), [email protected] (L.L. Imbriani). 1 Tel.: +39 0817683375; fax: +39 0817683946. http://dx.doi.org/10.1016/j.aap.2014.10.019 0001-4575/© 2014 Elsevier Ltd. All rights reserved.
and safe. On the other hand, an alignment which requires drivers to handle high speed gradients and surprising events and does not meet drivers’ expectancy is considered inconsistent and produces higher crash frequency (Anderson et al., 1999; Awatta et al., 2006; Cafiso et al., 2007; Camacho-Torregrosa et al., 2013; Lamm et al., 1999; Montella, 2005, 2009, 2010; Montella et al., 2008, 2012). As a consequence, the identification of proper design consistency measures and the development of quantitative relationships between design consistency and crash frequency are crucial for several steps of the highway safety management process, such as the design of new highways, the hotspots identification, the design
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of improvements of existing highways, and the evaluation of the safety effects of the design decisions. Several design consistency measures have been successfully used as explanatory variables in safety performance functions fitted on two-lane rural highways (Anderson et al., 1999; Awatta et al., 2006; Cafiso et al., 2010; Camacho-Torregrosa et al., 2013; ˜ et al., 2014; Ng and Sayed, 2004). Design consistency de Ona measures are generally related to the operating speed profile and the speed differential from an approach tangent to an horizontal curve (Montella et al., 2014a,b; TRB, 2011). The best known consistency criteria are those proposed by Lamm et al. (1995, 1999), that defined three design classes (good, fair, and poor) basing on three consistency measures: (1) Vd , design speed consistency (absolute value of the difference between design speed and operating speed); (2) V85 , operating speed consistency (absolute value of the 85th percentile speed reduction through successive elements of the road); and (3) fr , consistency in driving dynamics (difference between side friction assumed with respect to the design speed and side friction demanded at the 85th percentile speed). To increase the usefulness and the reliability of the existing safety performance functions incorporating design consistency variables, there is the need of further studies not limited to two-lane rural highways (Montella et al., 2008) as well as of studies taking into account the location of any geometric element in relation to the overall road alignment and the distance from the other geometric elements (Camacho-Torregrosa et al., 2013; Findley et al., 2012). Aim of the paper is to fill these research gaps developing safety performance functions for rural motorways that incorporate design consistency measures and take into account the location of any geometric element in relation to the overall road alignment. The design consistency parameters in this study are based on operating speed models developed in previous research through an instrumented vehicle equipped with a GPS continuous speed tracking (Montella et al., 2014a) from a field experiment conducted on the same motorway where the safety performance functions were fitted. These operating speed models are more accurate than models based on spot speed studies since they are not based only on spot speed data collected at the centre of the horizontal curve and at the midpoint of the preceding tangent. The remainder of paper is organized as follows: Section 2 describes the study data; Section 3 describes the methodology used to develop the safety performance functions and the variables introduced in the models; Section 4 explains and discusses the results, with a specific focus on the comparison with results of previous studies; finally, conclusions are drawn in the last Section.
2. Data description 2.1. Geometric data The study site is the section Naples–Candela of the motorway A16 Naples–Canosa (L = 255.0 km, i.e., 127.5 km per carriageway). It is part of the Trans European Road Network (Road E841), is located in the south of Italy and links up west coast (Motorway A1) and east coast (Motorway A14). It is a divided highway with two lanes for each direction (lane width = 3.75 m, right shoulder width = 0.50–3.50 m, median width = 2.00 m), access control, and interchanges. Median safety barriers include wbeam, double w-beam, thrie-beam, and concrete New Jersey shaped barriers. Because of the topographical constraints and the mountainous terrain, the freeway is characterised by a bending alignment, with several low radius curves and many design inconsistencies, and by sections with high longitudinal grades (Table 1). It is worthwhile to point out that, because of the presence of numerous mountains, in Italy several motorways have geometric characteristics similar to the study site. The corridor is connected to the road network by 11 interchanges, with 21 exit ramps (13 on curves, 8 on tangents) and 20 entrance ramps (11 on curves, 9 on tangents). Part of the route is in mountainous terrain with 11 tunnels (L = 4.03 km) and 38 bridges (L = 8.11 km). Two climbing lanes are located in east direction (L = 10.48 km) and one climbing lane is located in west direction (L = 3.83 km). Statutory speed limit is 130 km/h but posted speed limits equal to 80 km/h are installed in both travel directions (L = 50.25 km in east carriageway, L = 26.60 km in west carriageway). Segmentation was carried out to obtain homogeneous segments with respect to average annual daily traffic and curvature. First, all horizontal curves and tangents were separated. Then, segments with homogeneous curvature were further separated when changes in traffic volumes occurred inside the segment due the presence of an interchange. Based on the horizontal alignment characteristics and the traffic flow volumes, a segmentation into 652 homogeneous sections (326 for each carriageway) was carried out. Length of the segments varies between 62 and 3509 m. Radius of the horizontal curves varies between 245 and 4000 m. Spiral transitions are not present. Deflection angle varies between 5 and 109 gon. Superelevation mean is equal to 3.25%. Radius of the vertical curves varies between 3000 m (sag curve) and 30,000 m (crest curve). Maximum longitudinal grade is equal to 6.35%. Sight distance is often less than the stopping sight distance and ranges between 62 and 840 m.
Table 1 Summary statistics of geometric data. Parameter
Mean
Standard deviation
Minimum
Lengtha (m) Radius of horizontal curvesb (m) Deflection angle (gon)c Superelevation (%) Radius of sag vertical curvesd (m) Radius of crest vertical curvese (m) Longitudinal grade (%) Sight distance (m) Right shoulder width (m)
394.10 790.77 34.57 3.25 9,000.00 10,888.89 2.41 342.87 2.10
348.12 575.09 20.75 1.09 3,233.58
61.51 245.00 4.88 1.30 3,000.00
1.70 173.53 0.89
0.00 62.27 0.50
a b c d e
652 segments. 332 horizontal curves. The dimension gon corresponds to 400 angle units in a circle instead of 360 degrees. 92 sag curves. 76 crest curves.
Maximum 3.509,58 4.000,00 109.07 6.42 20,000.00 30,000.00 6.35 840.00 3.50
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Table 2 Summary statistics of traffic data (AADT). Year
East direction Mean
2007 2008 2009 2010 2011
12,567 12,492 12,628 12,531 11,997
West direction Standard deviation
Minimum
3,664 3684 3613 3595 3459
8,185 8,077 8,283 8,249 7,886
Maximum
Mean
16,712 16,715 16,952 16,923 16,185
2.2. Traffic data Average annual daily traffic (AADT) data were provided by the motorway management agency for the period 2007–2011. They are disaggregated for each carriageway and for each section between successive interchanges. AADT ranges between 7886 and 17,255 vehicles per day (Table 2). It is worthwhile to observe that since 2009 a significant reduction of traffic volume is observed, mainly because of the economic recession.
2.3. Crash data Since the Italian national crash database maintained by the National Institute of Statistics presents major issues related to the crash report form, the crash classification, the crash location, and the crash severity, a new database was developed according to a framework based on the critical review of the crash databases in Australasia, the European Union and the United States (Montella et al., 2013). Crash data were collected through analysis of police reports and were integrated with detailed site inspections. Crash data covered 2007–2011 (Table 3). Crashes at the interchanges ramps, at rest areas, and at tollbooths were excluded because of the location
12,877 12,773 12,891 12,774 12,209
Standard deviation 3,522 3,611 3,528 3,513 3,357
Minimum 8,590 8,396 8,589 8,535 8,164
Maximum 17,132 17,005 17,255 17,055 16,342
descriptions of the police reports. In the analysis period, 2660 total crashes occurred. Crash data were classified according to the alignment (tangent and curve), collision type (single-vehicle run-off-the-road, other single-vehicle, and multi-vehicle), lighting conditions (daytime and nighttime), weather conditions (rainy and non-rainy), pavement conditions (dry and wet), and crash severity (property damage only (PDO), slight injuries, and severe or fatal injuries). The crash severity classification was based on the most severe injury to any person involved in the crash. According to the agreed international definition, a traffic crash fatality was every single person that dies in the crash or within the 30 days following it. As far as non-fatal injuries, we classified severe and slight injuries. An injury was classified severe if the person is detained in hospital as an “in-patient”, or suffers any of the following injuries whether or not is detained in hospital: fractures, concussion, internal injuries, crushing, significant burns (second and third degree burns over 10% or more of the body), severe cuts and lacerations, severe general shock requiring medical treatment, partial disability for at least 30 days, or the injuries causing death 30 or more days after the crash. Any injuries that are evident at the scene of the crash, other than fatal or serious injuries, such as sprains, bruises or cuts which are not judged to be severe or
Table 3 Summary statistics of crash data. Mean
Curves Total ROR SV OT MV Daytime Nighttime Non rainy Rainy Dry Wet PDO Slight injury Severe injury Tangents Total ROR SV OT MV Daytime Nighttime Non rainy Rainy Dry Wet PDO Slight injury Severe injury
Standard deviation
Minimum
Maximum
Sites with zero crashes N
(%)
3.80 1.75 1.44 0.61 2.65 1.14 2.95 0.84 2.43 1.37 2.88 0.70 0.21
4.92 3.53 1.85 0.99 3.81 1.98 3.39 2.42 2.89 3.18 3.65 1.41 0.54
0 0 0 0 0 0 0 0 0 0 0 0 0
43 37 14 7 38 19 23 30 22 38 31 11 4
58 141 136 206 84 173 67 220 85 166 74 214 275
17.47 42.47 40.96 62.05 25.30 52.11 20.18 66.27 25.60 50.00 22.29 64.46 82.83
4.38 1.26 2.15 0.97 3.08 1.29 3.79 0.58 3.24 1.13 3.57 0.63 0.18
5.12 2.35 2.54 1.70 3.79 1.95 4.41 1.50 3.83 2.25 4.01 1.26 0.52
0 0 0 0 0 0 0 0 0 0 0 0 0
32 27 15 13 27 16 28 18 23 26 25 9 3
56 154 104 182 82 145 66 220 74 166 66 211 280
17.50 48.13 32.50 56.88 25.63 45.31 20.63 68.75 23.13 51.88 20.63 65.94 87.50
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slight shocks requiring roadside attention were classified slight injuries.
2 (Miaou et al., Two goodness-of-fit measures were calculated: R˛ 1996) and the Akaike information criterion (AIC) (Akaike, 1987). 2 , a dispersion parameter-based R2 , is calculated as follows: R˛
3. Method 3.1. Model description
2 R˛ =1−
Following common practice (Lord and Mannering, 2010; Mannering and Bhat, 2014), generalized linear modelling techniques were used to fit the models and a negative binomial distribution error structure was assumed. The selected model form is as follows: n
ˆ ) = L × ea0 +a11 ×ln(AADT) × e i=1 E(Y
bi ×xi
(1)
where Eˆ (Y) is the predicted annual crash frequency, L is the segment length (m), AADT is the segment average annual daily traffic (veh/day), a0 , a1 and bi are the model parameters, and xi are the explanatory variables other than AADT. The distribution of the crash frequency around Eˆ (Y) = is a negative binomial with variance of: Var(Y ) = + × 2
3.2. Goodness-of-fit measures
(2)
where is the dispersion parameter of the negative binomial distribution. This model form logically estimates zero crashes if one of the two exposure variables (AADT or L) is equal to zero. Segment length is an offset variable. It captures the logical requirement that if N crashes are expected to occur on 1 km of road, 2 N crashes should be expected to occur on an identical road that is 2 km long. The model parameters and the dispersion parameter of the negative binomial distribution were estimated by the maximum likelihood method using the GENMOD procedure in SAS. The models were developed by the stepwise forward procedure, adding one explanatory variable at each step. The decision on whether or not to keep a variable in the model was based on two criteria. The first is whether the t-ratio of the variable’s estimated coefficient is significant at the 5% level. The second criterion is based on the improvement of the goodness-of-fit measures of the model that includes that variable. Since the design consistency variables were used only for curves, two different sets of models were fitted for tangents and curves. Models for the following crash characteristics were fitted: total, single-vehicle run-off-the-road (ROR), other single vehicle (SV OT), multi vehicle (MV), daytime, nighttime, non-rainy weather, rainy weather, dry pavement, wet pavement, property damage only (PDO), slight injury, and severe injury (including fatal). To evaluate the model form, cumulative residual analysis (Hauer, 2004; Hauer and Bamfo, 1997; Lord and Persaud, 2000) was performed. The residual is equal to the difference between the observed and predicted values of the annual crash frequency. Assuming that residuals are normally distributed with expected value equal to 0 and a variance equal to *, it is possible to calculate the variance of the expected value as the square of the cumulate residuals. The trend in the residuals can be evaluated relative to the variance to qualitatively assess goodness of fit. The cumulative residual plot is used to examine whether the chosen functional form fits each explanatory variable along the entire range of its values represented in the data. In general, a good cumulative residuals plot is one that oscillates around 0. A bad cumulative residuals plot is one that is entirely above or below 0 (except at the edges). Thus, it is desirable that the plot of cumulative residuals should oscillate between over and under prediction and not stray beyond the ±2* boundaries.
(3)
max
where max is the largest possible dispersion parameter that is obtained by having no covariates in the model (by assuming that all sites have an identical prediction estimate equal to the mean over all sites) and is the dispersion parameter for the fitted model. This measure is bound between 0 (when no covariate is included) and 1 (when covariates are perfectly specified). The AIC value is calculated as follows: AIC = −2 × ML + 2 × p
(4)
where ML is the maximum log-likelihood of the fitted model, and p is the number of parameters in the model. A smaller AIC value reflects a better model. The first term in the AIC equation measures the goodness of fit, or bias, when the maximum likelihood estimates of the parameters are used. The second term measures the complexity of the model, thus penalizing the model for using more parameters. The minimisation of the AIC value allows to reduce over fitting, since it permits to choose the best fit with the least complexity. 3.3. Variables selection Explanatory variables related to traffic volume, design consistency, horizontal alignment, vertical alignment, sight distance, and roadside context were considered. Below, a brief description of the significant explanatory variables is reported. Table 4 shows descriptions and analytical details of the variables. 3.3.1. Traffic volume The relationship between crashes and traffic volume is frequently non-linear. Thus, traffic volume was not considered as an offset variable. Natural logarithm of the average annual daily traffic (AADT), which is an aggregate measure of the traffic volume, was considered. 3.3.2. Design consistency The consistency measures included in the models for the curves were: (a) V85 , operating speed consistency, i.e., absolute value of the 85th percentile speed reduction through successive elements of the road. Operating speed (V85 ) was assessed using the predictive models developed in previous research through an instrumented vehicle equipped with a GPS continuous speed tracking (Montella et al., 2014a) from a field experiment conducted on the same motorway where the safety performance functions were fitted, i.e., the motorway A16 in Italy. The following models were fitted: V858c = 135.490 −
7.483 − 1.290 × Gu − 0.080 × CCR2 − 14.427 R
× Tunnel − 4.083 × Bridge
V85t = 139.543 + 1.751 × Lt − ×Gu − 0.068 × CCR2
(5)
4.983 − 2.270 − 2.507 Rbefore Rafter (6)
where V85c is the 85th percentile of the operating speed on curves (km/h), V85 t is the 85th percentile of the operating speed on tangents (km/h), R is the radius of the curve (Km), Rbefore is the radius
A. Montella, L.L. Imbriani / Accident Analysis and Prevention 74 (2015) 133–144 Table 4 SPFs: Significant explanatory variables. Variable Traffic volume ln(AADT) [veh/day] Design consistency V85 [km/h]
fr [−]
V85,i –V85,5 [km/h]
Lt [km] Horizontal alignment 1/R2 [1/km2 ] 1/R before [1/km] 1/R after [1/km] Vertical alignment Gd Gu Gd × Ld Gu × Lu Roadside context Tunnel Yearly effects Year10 Year11
Description Natural logarithm of the average annual daily traffic Absolute value of the operating speed difference in the tangent-to-curve transition (Eqs. (5) and (6)) Difference between side friction assumed with respect to the design speed and side friction demanded at the 85th-percentile speed (Eq. (7)) Difference between the 85th percentile of the operating speed and the mean in the 5 km preceding the beginning of the curve of the 85th percentile of the operating speed Length of the tangent preceding the curve Square of horizontal curvature Horizontal curvature of the curve preceding the tangent Horizontal curvature of the curve following the tangent Equivalent downgrade Equivalent upgrade Interaction between equivalent downgrade and length of downgrade segment Interaction between equivalent upgrade and length of upgrade segment Binary variable, equal to 1 if the segment is on a tunnel Binary variable, equal to 1 in 2010 Binary variable, equal to 1 in 2011
of the curve preceding the tangent (Km), Rafter is the radius of the curve following the tangent (Km), CCR2 is the curvature change ratio of the 2 km preceding the geometric element (gon/km), Lt is the length of the tangent (km), Gu is the equivalent upgrade (%), Bridge is a binary variable equal to 1 if the segment is on a bridge, and Tunnel is a binary variable equal to 1 if the segment is on a tunnel. (b) fr , consistency in driving dynamics, i.e., difference between side friction assumed with respect to the design speed and side friction demanded at the 85th percentile speed:
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change rate. Since spiral transitions are not present, the square of curvature (1/R)2 was selected. In the model for tangents we included two variables which take into account the spatial relationships between the tangent and the curves before and after: (1) 1/Rbefore where Rbefore is the radius of the curve preceding the tangent and (2) 1/Rafter where Rafter is the radius of the curve following the tangent. 3.3.4. Vertical alignment Longitudinal slope is not always constant in the segments. To take into account this circumstance, for each segment an equivalent downgrade (Gd ) and upgrade (Gu ) were assessed. Equivalent grade is obtained by weighing each gradient in relation to the segment length. In the vertical curves, two situations occur. If the different grades have the same direction, half of the vertical curve is added to each grade. If the grades have opposite direction, a quarter of the vertical curve is added to each grade. Furthermore, the interactions between the equivalent grade and length of the segments (Gd × Ld and Gu × Lu ) were introduced in the models. 3.3.5. Roadside context To take into account the safety effect of the roadside context, the dummy variable tunnel was introduced. 3.3.6. Yearly effects Dummy variables were introduced to capture the potential nonrandom yearly effects. 3.3.7. Correlations between variables The Pearson product moment correlation (Pearson’s correlation) was assessed to identify correlations between the explanatory variables. Pearson’s correlation reflects the degree of linear relationship between two variables. A correlation of +1 (−1) means that there is a perfect positive (negative) linear relationship between variables. Variables are considered strongly correlated if the absolute value of Pearson’s correlation is higher than 0.85 and the estimated correlation has p-value less than 0.05. Variables included in the models are not highly correlated. 4. Results and discussion 4.1. Curves
where fra is the side friction assumed with respect to the design speed, frd is the side friction demanded at the 85th percentile speed, Vd is the design speed (km/h) calculated according to the Italian Geometric Design Standards (Italian Ministry of Infrastructures and Transports, 2001), V85 is the 85th percentile of the speed distribution in the curve (km/h) calculated with the equation (5), R is the curve radius (m), and e is the curve superelevation. (c) V85,i –V85,5 , inertial speed consistency, i.e., difference between the operating speed in the curve and the average operating speed along the 5 km preceding the beginning of the curve (weighted to element length) (km/h). (d) Lt , length of the tangent preceding the curve (km), which takes into account the distance from the previous curve.
The parameter estimates and the measures of goodness of fit for each model are reported in Table 5. Non-significant parameters are 2 values of the models range between 19% and not reported. The R˛ 69% and all the parameters have logical and expected signs. Cumulative residuals plots against AADT are reported in Figs. 1 and 2. It can be observed that the plots of cumulative residuals oscillate around 0 and do not stray beyond the ±2* boundaries, apart a few exceptions where residuals slightly encroach the boundaries in specific points. The same kind of pattern can be seen for all the variables. Most of the models have many significant variables. The model with the greater number of significant variables is the model for total crashes, which has seven significant explanatory variables other than AADT related to the design consistency, the vertical alignment, and the yearly effects. To avoid over fitting, we explicitly penalized complex models selecting the model with the lower AIC value, i.e., a goodness-of-fit measure that takes into account the complexity of the model penalizing the model for using more parameters.
3.3.3. Horizontal alignment Literature suggests that the geometric design parameter which most affects road safety is the horizontal curvature or the curvature
4.1.1. Design consistency The most important finding from this study is that design consistency significantly affects the safety of horizontal curves on rural
f r = fra − frd = 0.6 × 0.925 × (0.59 − 4.85 × 10−3 × Vd
+ 1.51 × 10−5 × Vd2 ) −
2 V85
127 × R
−e
(7)
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Table 5 SPFs: parameter estimates and goodness of fit measures for curves. Total
ROR
SV OT
MV
Daytime
Nighttime
Non rainy
Rainy
Dry
Wet
PDO
Slight injury
Severe injury
Dispersion
1.101 (0.101) −15.425 (1.372)
1.637 (0.175) −15.807 (1.699)
2.139 (0.227) −12.199 (1.644)
0.427 (0.196) −24.553 (1.809)
1.347 (0.131) −16.123 (1.476)
2.078 (0.247) −17.225 (1.680)
0.991 (0.108) −15.096 (1.371)
3.265 (0.605) −14.890 (3.024)
1.131 (0.125) −16.570 (1.421)
2.232 (0.236) −14.577 (1.744)
1.188 (0.119) −14.169 (1.435)
1.338 (0.230) −26.102 (1.797)
3.006 (0.660) −23.241 (2.036)
0.935 (0.147)
0.823 (0.184)
0.519 (0.176)
1.707 (0.192)
0.943 (0.158)
1.012 (0.180)
0.889 (0.147)
0.609 (0.343)
1.047 (0.152)
0.654 (0.186)
0.754 (0.154)
1.885 (0.191)
1.500 (0.216)
0.037 (0.010) −4.692 (1.046) − 0.018 (0.008)
0.042 (0.011) −6.075 (1.857)
0.059 (0.020)
0.029 (0.009) −3.384 (0.935)
0.041 (0.011)
0.052 (0.009) −4.400 (0.937)
0.045 (0.011) −11.913 (1.237)
−13.837 (1.245)
Constant Traffic volume ln(AADT) Design consistency V85 fr V85,i –V85,5
Horizontal alignment 1/R2
Gu
0.116 (0.020) 11.799 (2.856) 7.529 (2.755)
Year11 R˛2 AIC
−0.207 (0.098) −0.247 (0.100) 0.38 3740
0.053 (0.013)
0.050 (0.013)
5.740 (2.499)
16.263 (3.397) 11.629 (3.532)
0.762 (0.231)
0.690 (0.220)
7.128 (2.524)
– Roadside context Tunnel Yearly effects Year10
−5.112 (1.322) − 0.051 (0.009)
−2.478 (1.058) − 0.033 (0.007)
0.304 (0.105)
Lt
Vertical alignment Gd
0.064 (0.010) −4.656 (0.966)
12.838 (2.891) 11.556 (2.929)
9.397 (3.029)
12.764 (2.900) 6.918 (2.721)
0.178 (0.023)
0.174 (0.013)
14.487 (4.598)
12.960 (2.560)
11.984 (2.862) 12.394 (2.827)
5.833 (2.637)
0.56 1728
0.33 3258
0.62 1265
5.368 (2.215)
− 0.292 (0.105)
0.57 2225
0.19 2183
0.69 1553
Note: standard deviations of the parameter estimates are reported in parenthesis.
0.36 3066
0.40 1813
0.36 3304
0.62 1318
0.33 2958
0.41 991
A. Montella, L.L. Imbriani / Accident Analysis and Prevention 74 (2015) 133–144
Dependentvariable
A. Montella, L.L. Imbriani / Accident Analysis and Prevention 74 (2015) 133–144
Fig. 1. Cumulative residuals vs. AADT (curves: models for total, ROR, other single-vehicle, multi-vehicle, daytime, nighttime, rainy and non-rainy crashes).
139
140
A. Montella, L.L. Imbriani / Accident Analysis and Prevention 74 (2015) 133–144
Fig. 2. Cumulative residuals vs. AADT (curves: models for wet, dry, PDO, slight injury, and severe injury crashes).
motorways. Consistency in driving dynamics (fr ) is significant in nine models (total, ROR, daytime, nighttime, non-rainy, dry, PDO, slight injury, and severe injury crashes), operating speed consistency (V85 ) is significant in eight models (total, ROR, daytime, rainy, dry, wet, PDO, and slight injury crashes), inertial speed consistency (V85,i –V85,5 ) is significant in three models (total, nighttime, and non-rainy crashes), and length of tangent preceding the curve (Lt ) is significant only in the model for run-off-the-road crashes. In most models, more design consistency variables are significant. As an example, in the model for total crashes consistency in driving dynamics, operating speed consistency, and inertial speed consistency are all significant. This result indicates that all the consistency
variables introduced in this study should be carefully checked in the design stage and that the safety effects of the design depend on the combination of more consistency measures. As the difference between friction demand and friction supply (fr ) decreases, crash frequency is expected to increase (i.e., the estimated coefficient of the parameter consistency in driving dynamics is negative). Given that in existing motorways curve superelevation is frequently lower than the superelevation required by the geometric design standards and guidelines, and that superelevation adjustment is a feasible and quick measure, this result has a relevant practical effect on the selection of safety measures.
A. Montella, L.L. Imbriani / Accident Analysis and Prevention 74 (2015) 133–144
The estimated coefficient of the parameter operating speed consistency (V85 ) has a positive value. This implies that an higher speed reduction between two consecutive elements is related to an increase of the crash frequency and is therefore a visible indicator of inconsistency in geometric design. The relationship between the crash frequency and the operating speed reduction indicates therefore that the greater the speed reduction on a horizontal curve, the greater the curve’s crash experience. This result is consistent with previous studies on two-lane rural highways (Anderson et al., 1999; ˜ et al., Cafiso et al., 2010; Camacho-Torregrosa et al., 2013; de Ona 2014; Ng and Sayed, 2004) and implies that the geometric design should minimise the speed reduction from tangents to horizontal curves. The statistically significant negative value of the estimated coefficient of the inertial speed consistency (V85,i –V85,5 ) shows that the expected crash frequency increases with the difference between the average operating speed in the 5 km preceding the curve and the operating speed in the curve. All else being equal, an increase in the operating speed of the segment before the curve (5 km) has a positive effect on crash frequency whereas a decrease in the operating speed of the segment before the curve (5 km) has a negative effect on crash frequency. Based on the hypothesis that drivers’ long-term expectation at a curve depends on the average operating speed at the previous road segment, the significance of the inertial speed consistency variable indicates that long-term driver’s expectancy has a significant safety effect and drivers are surprised by curves with operating speed lower than the operating speed of the previous five kilometres. This results is consistent with a recent study carried out on two-lane rural roads in Spain by García et al. (2013) and implies that the curve operating speed should be compared with the operating speed along the 5 km before the curve. Thus, a curve that follows a 5 km segment with high operating speed should be a curve that allows high operating speed. The positive value of the estimated coefficient of the length of the tangent preceding the curve (Lt ) for the run-off-the-road crashes model shows that neighbouring curves have few predicted crashes than curves more distant to each other, as reported in the literature (Brenac, 1996; Findley et al., 2012). The length of the tangent preceding the curve indicates that short-term driver’s expectancy has a significant safety effect. A curve that is a part of a series of curves turns out to be safer than a curve which is isolated from other curves. As far as the collision type, consistency variables are significant for single-vehicle run-off-the-road crashes but are not significant for the other single-vehicle crashes and for multi-vehicle crashes. This result shows that highway geometry which does not meet drivers’ expectancy has higher effect on single-vehicle run-off-theroad crashes. 4.1.2. Horizontal alignment The square curvature estimated coefficient (1/R)2 has a positive sign, that is, the smaller the radius, the more the expected crash frequency. Square curvature is significant for ROR, non-ROR single vehicle, multi-vehicle, rainy, and wet crashes. 4.1.3. Vertical alignment Vertical alignment explanatory variables are significant in all the models. In five models (total, multi-vehicle, daytime, non-rainy, and PDO crashes), both equivalent downgrade (Gd ) and equivalent upgrade (Gu ) are significant and have positive value, with the estimated coefficient of equivalent downgrade greater than the coefficient of equivalent upgrade. Thus, the greater the longitudinal grade, the greater the crash frequency. The effect of longitudinal grade on crash frequency is higher in downgrade sections: 12% increase in total crashes for 1% increase in longitudinal
141
downgrade vs. 8% increase in total crashes for 1% increase in longitudinal upgrade. In the models for ROR, nighttime, rainy, wet, and severe injury crashes only the equivalent downgrade is significant (positive value). In the models for non-ROR single vehicle and dry crashes only the equivalent upgrade is significant (positive value). 4.1.4. Roadside context The variable tunnel is significant for non-ROR single vehicle and multi-vehicle crashes. It has a positive value, i.e., in road tunnels more non-ROR single vehicle and multi-vehicle crashes are expected. 4.1.5. Yearly effects A significant decrease in total crashes is expected in 2010 and 2011, and in dry crashes in 2010. 4.2. Tangents The parameter estimates and the measures of goodness-of-fit for each model are reported in Table 6. Non-significant parame2 values of the models range between ters are not reported. The R˛ 47% and 93% and all the parameters have logical and expected signs. Cumulative residuals plots against the significant explanatory variables show that the plots of cumulative residuals oscillate around 0 and do not stray beyond the ±2* boundaries, apart a few exceptions where residuals slightly encroach the boundaries in specific points. 4.2.1. Horizontal alignment The curvature of the horizontal curves preceding (1/R before ) and following (1/R after ) the tangent are statistically significant in most models (in twelve models 1/R after and in six models 1/R after ). The values of all the estimated coefficients have a positive sign. Thus, the smaller the radius of the curves before and after the tangent, the greater the expected crash frequency in the tangent. In six models (total, ROR, daytime, rainy, wet, and PDO crashes), both the curvature before and after the tangent have a significant effect on crash frequency but, apart for crashes on wet pavement, the impact of the curvature after the tangent is greater than the impact of the curvature before the tangent. 4.2.2. Vertical alignment Vertical alignment explanatory variables are significant in all the models except in the nighttime crashes model. The comparison between the estimated coefficients of the vertical alignment variables in the models for curves and tangents shows that the effect of longitudinal grade on crashes is much higher on horizontal curves than on tangents. On horizontal curves, a 1% increase in longitudinal grade procedures a 12% increase in estimated total crashes in the downgrade sections vs. a 8% increase in total crashes in the upgrade section. On tangents, 1% increase in longitudinal grade procedures a 6% increase in estimated total crashes in the downgrade sections vs. no significant effect in the upgrade section. The interaction between the equivalent downgrade and the length of the downgrade segment (Gd × Ld ) is significant for wet crashes while the interaction between the equivalent upgrade and the length of the upgrade segment (Gu × Lu ) is significant for non-run-off-the-road single vehicle, daytime and non-rainy crashes. 4.2.3. Yearly effects As for the curves, a significant decrease in crash frequency is expected in 2010 (total, nighttime, non-rainy, PDO, and slight injury
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Table 6 SPFs: parameter estimates and goodness of fit measures for tangents. Total
ROR
SV OT
MV
Daytime
Nighttime
Rainy
Non Rainy
Wet
Dry
PDO
Slight injury
Severe injury
Dispersion
0.393 (0.057) −18.145 (1.138)
0.983 (0.156) −25.765 (1.675)
0.787 (0.112) −10.526 (1.420)
0.171 (0.093) −30.213 (1.639)
0.565 (0.080) −18.564 (1.337)
0.594 (0.131) −16.915 (1.511)
1.813 (0.304) −26.417 (1.963)
0.356 (0.060) −16.188 (1.186)
1.185 (0.176) −25.031 (1.769)
0.435 (0.071) −15.679 (1.218)
0.454 (0.069) −16.697 (1.208)
0.551 (0.163) −29.024 (1.863)
1.479 (0.451) −28.653 (2.217)
1.834 (0.176)
0.363 (0.151)
2.364 (0.172)
1.214 (0.141)
0.999 (0.159)
1.765 (0.207)
1.023 (0.125)
1.743 (0.187)
0.959 (0.129)
1.054 (0.128)
2.148 (0.195)
1.977 (0.232)
0.161 (0.050)
0.336 (0.059) 0.568 (0.061)
0.131 (0.041)
0.394 (0.054) 0.296 (0.053)
0.123 (0.042)
0.092 (0.043) 0.195 (0.043)
0.343 (0.053)
0.357 (0.059)
11.411 (2.816)
6.269 (2.049)
5.250 (2.066)
5.355 (2.014)
8.825 (2.713)
−0.335 (0.095) −0.221 (0.093) 0.72 3079
−0.261 (0.091) −0.370 (0.095) 0.68 3288
−0.528 (0.146) 0.81 1074
Constant Traffic volume ln(AADT)
1.224 (0.120) Horizontal alignment 1/R before 0.083 (0.040) 0.221 1/R after (0.040) Vertical alignment 5.827 Gd (1.876) Gd × Ld
0.146 (0.055) 0.462 (0.054)
Year11 R˛2 AIC
0.123 (0.049)
9.149 (2.538)
8.280 (2.380)
0.002 (0.001) −0.232 (0.084) −0.370 (0.089) 0.72 3587
9.414 (2.240)
0.003 (0.001)
Gu × Lu Yearly effects Year10
0.090 (0.045) 0.220 (0.046)
−0.420 (0.130) 0.64 1773
0.53 2547
0.002 (0.001)
0.93 1353
−0.229 (0.096) 0.66 3025
Note: Standard deviations of the parameter estimates are reported in parentheses.
0.001 (0.001) −0.325 (0.115) −0.662 (0.132) 0.73 1798
−0.821 (0.169) 0.67 1044
−0.217 (0.087) −0.302 (0.090) 0.75 3315
−1.011 (0.153) 0.68 1611
0.47 926
A. Montella, L.L. Imbriani / Accident Analysis and Prevention 74 (2015) 133–144
Variable
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crashes) and 2011 (total, ROR, daytime, nighttime, non-rainy, rainy, dry, wet, PDO, and slight injury crashes). 5. Conclusions Study results show that geometric design consistency has a significant effect on safety of rural motorways. Previous studies on the relationship between geometric design consistency and crash frequency focused on two-lane rural highways since these highways have the higher crash rates and are generally characterized by considerable inconsistencies. Our study clearly highlights that the achievement of proper geometric design consistency is a key design element also on motorways because of the safety consequences of design inconsistencies. The safety performance functions incorporating design consistency measures allow quantitative evaluations of the safety effects of the design decisions that can be used for the design of new highways, the hotspot identification, and the design of improvements of existing highways. These safety performance functions allow to complement geometric design based on threshold values of the geometric parameters introduced in the geometric design standards and carry out safety-based design. According to the standards safety is nominal, i.e., any design is safe if parameters comply with defined thresholds and is not safe if parameters do not satisfy these threshold. However, highway safety changes with continuity in relation to the design parameters and the fulfilment of drivers’ expectations has direct consequences on crashes that can be quantitatively assessed by safety performance functions incorporating design consistency variables. The consistency measures that resulted significant explanatory variables of the safety performance functions developed in this study are: (1) consistency in driving dynamics, i.e., difference between side friction assumed with respect to the design speed and side friction demanded at the 85th percentile speed; (2) operating speed consistency, i.e., absolute value of the 85th percentile speed reduction through successive elements of the road; (3) inertial speed consistency, i.e., difference between the operating speed in the curve and the average operating speed along the 5 km preceding the beginning of the curve; and (4) length of tangent preceding the curve (only in the model for run-off-the-road crashes). As a consequence, to reduce crashes the geometric design should: (1) minimise the difference between friction demand and friction supply on horizontal curves, using superelevation based on predicted operating speed instead of superelevation based on design speed; (2) minimise the operating speed reduction from tangents to horizontal curves, meeting short-term driver’s expectancy; (3) minimise the difference between the curve operating speed and the operating speed along the 5 km before the curve, meeting longterm driver’s expectancy; and (4) avoid low-operating speed curves following long tangents. Furthermore, greater longitudinal downgrades and upgrades significantly increase crash frequency and the safety effect is higher on horizontal curves than on tangents. Thus, it is essential to carefully take into account the vertical alignment and his safety consequences. These findings highlight that the geometric design of both new motorways and improvements of existing motorways should explicitly consider the safety effects of consistency in driving dynamics, operating speed, and inertial speed. In existing motorways, the improvement of geometric design to reduce inconsistencies is not always feasible because of budget restrictions or other constraints. In these cases, we recommend to implement speed management policies aimed at obtaining operating speeds consistent with the highway geometry. Recently, very effective results in speed reduction have been obtained by the point-to-point speed enforcement (Lynch et al., 2011;
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