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REPORT
This report is an archived publication and may contain dated technical, contact, and link information
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Publication Number:  FHWA-HRT-13-077    Date:  January 2014
Publication Number: FHWA-HRT-13-077
Date: January 2014

 

Safety Effects of Horizontal Curve and Grade Combinations on Rural Two-Lane Highways

CHAPTER 5—CRASH MODIFICATION FACTORS

CMFs used in the AASHTO HSM can be derived from the predictive models developed in chapter 4 of this current report. A CMF is a factor that represents the effect on crash frequency for a given crash severity level of varying geometric design or traffic control feature of interest (or a particular combination of geometric design or traffic control feature). Each CMF has a nominal value of 1.0 for a specified base condition. A CMF with a value greater than 1.0 represents a condition for which more crashes would be expected for the base condition. A CMF with a value less than 1.0 represents a condition for which fewer crashes would be expected than for the base condition. The base condition for all CMFs developed in this research is a level tangent roadway.

For each combination of alignment type (and for FI crashes and PDO crashes), CMFs were calculated as the ratio of the predicted crash frequency for a given horizontal curve and grade combination to the predicted crash frequency for the level tangent base condition. The following subsections provide equations for each CMF and figures illustrating the relationships of each roadway parameter to crash frequency and to the relevant CMF. Sample CMF calculations are also presented.

HORIZONTAL CURVES AND TANGENTS ON STRAIGHT GRADES

CMFs for horizontal curves and tangents on straight grades can be derived from figure 12 and figure 13 as follows shown in figure 39 and figure 40*:

*Modified on November 16, 2014

CMF subscript SG,FI equals open bracket exponent open bracket 0.044 times G plus 0.19 times natural logarithm of open parenthesis 2 times 5,730 divided by R closed parenthesis plus 4.52 times open parenthesis 1 divided by R closed parenthesis times open parenthesis 1 divided by L subscript C closed parenthesis closed bracket for horizontal curves; equals exponent open bracket 0.044 times G closed bracket for tangents on nonlevel grades; and equals 1.0 for level tangents (base condition) closed bracket.

Figure 39. Equation. FI CMF for horizontal curves and tangents on straight grades.

CMF subscript SG,PDO equals open bracket exponent open bracket 0.040 times G plus 0.13 times natural logarithm of open parenthesis 2 times 5,730 divided by R closed parenthesis plus 3.80 times open parenthesis 1 divided by R closed parenthesis times open parenthesis 1 divided by L subscript C closed parenthesis closed bracket for horizontal curves; equals exponent open bracket 0.040 times G closed bracket for tangents on nonlevel grades; and equals 1.0 for level tangents (base condition) closed bracket.

Figure 40. Equation. PDO CMF for horizontal curves and tangents on straight grades.

The functional relationships shown in figure 12 (crashes/mi/year) and figure 39 (CMF) for FI crashes are illustrated in figure 41 for combinations of horizontal curve lengths and percent grades. Curve radius ranged from 100 to 11,460 ft; AADT was fixed at 2,000 vehicles/day; the median traffic volume was for rural two-lane roadways in the database; the horizontal curve length was set at 0.05, 0.10, 0.15, and 0.20 mi; and grade was set at level (i.e., 0 percent) and 1 to 6 percent in increments of 1 percent. Similarly, figure 42 illustrates the relationships shown in figure 13 and figure 40 for PDO crashes.

This graph shows four plots of predicted fatal and injury (FI) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents on straight grades. The sets of plots correspond to different curve lengths of 0.05, 0.10, 0.15, and 0.20 mi. FI crashes/mi/year are shown on the left y-axis from zero to 2.50 crashes/mi/year, and the corresponding CMFs are shown on the right y-axis from zero to 8. The x-axis shows the radius from zero to 12,000 ft. Within each set of plots, seven lines are shown corresponding 
to grades ranging from level to grades of 1 through 6 percent. There is a dotted horizontal 
blue line that corresponds to a base condition tangent, an average annual daily traffic of 
2,000 vehicles/day, and a CMF of 1.0. All curves are exponential decay curves.

Figure 41. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and tangents on straight grades.

This graph shows four plots of property damage only (PDO) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents on straight grades. The four plots show different curve lengths of 0.05, 0.10, 0.15, and 0.20 mi. PDO crashes/mi/year are shown on the left y-axis from zero to 2.50 crashes/mi/year, and the corresponding CMFs are shown on the right y-axis from zero to 8. The x-axis shows the radius from zero to 12,000 ft. Seven lines are shown on the plots corresponding grades including level and grades 1 through 6. A dotted horizontal blue line corresponds to a base condition tangent with an average annual daily traffic of 2,000 vehicles/day and a CMF of 1.0.

Figure 42. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and tangents on straight grades.

In figure 41 and figure 42, crashes/mi/year are shown on the left y-axis, and the corresponding CMFs are shown on the right y-axis. The dotted blue line corresponds to a base condition tangent with an AADT of 2,000 vehicles/day and therefore has a CMF of 1.0.

To calculate CMF for FI or PDO crashes for a given horizontal curve on a level or nonlevel grade or a tangent on a nonlevel grade, G (percent), R (ft), and LC (mi) are substituted in figure 39 or figure 40. Example CMFs were calculated for rural two-lane roadways with a R of 1,433 or 5,730 ft; LC of 0.05, 0.10, and 0.50 mi; and G ranging from level to 6 percent. The results are shown in table 17 for rural two-lane roadways with AADTs from 200 to 26,000 vehicles/day.

Table 17. Example CMFs for FI and PDO crashes on horizontal curves and tangents on straight grades.

Grade
(Percent)

Tangent
on Nonlevel
Grade

R = 1,433 ft

R = 5,730 ft

Horizontal Curve Length (mi)

Horizontal Curve Length (mi)

0.05

0.10

0.50

0.05

0.10

0.50

CMFs for FI Crashes

Level
(< 1 percent)

1.00

1.57

1.53

1.49

1.15

1.15

1.14

1

1.04

1.64

1.59

1.56

1.20

1.20

1.19

2

1.09

1.71

1.67

1.63

1.25

1.25

1.25

3

1.14

1.79

1.74

1.70

1.31

1.31

1.30

4

1.19

1.87

1.82

1.78

1.37

1.36

1.36

5

1.25

1.95

1.9

1.86

1.43

1.42

1.42

6

1.30

2.04

1.98

1.94

1.49

1.49

1.48

CMFs for PDO Crashes

Level
(< 1 percent)

1.00

1.38

1.35

1.32

1.10

1.10

1.10

1

1.04

1.44

1.40

1.38

1.15

1.14

1.14

2

1.08

1.49

1.46

1.43

1.20

1.19

1.19

3

1.13

1.56

1.52

1.49

1.25

1.24

1.24

4

1.17

1.62

1.58

1.55

1.30

1.29

1.29

5

1.22

1.69

1.65

1.62

1.35

1.35

1.34

6

1.27

1.76

1.72

1.69

1.41

1.40

1.40

In any given column in table 17, the CMFs show the increasing effect on crashes of an increasing grade for a horizontal curve or tangent on a straight grade. In any given row for a given curve radius, the CMFs show the increasing effect on crashes of horizontal curve length. The decreasing effect on crashes of longer curve radii is shown by comparing any two columns for the same curve length between the two radii. The effect of a short and sharp curve (due to the last term in figure 39 and figure 40) is reflected in the high CMFs in the third column of the table. As the curve radius increases and becomes a tangent for all practical purposes, the CMFs for the curve approach in value those of a tangent with the same percent grade. When percent grade nears zero (level roadway) and the radius becomes infinite (tangent roadway), the roadway becomes a level tangent, and CMF becomes 1.

HORIZONTAL CURVES AND TANGENTS AT TYPE 1 CREST VERTICAL CURVES

CMFs for horizontal curves and tangents at type 1 crest vertical curves can be derived from figure 18 and figure 19 as follows shown in figure 43 and figure 44 *:

*Modified on November 16, 2014

CMF subscript C1,FI equals open bracket exponent open bracket 0.0088 times open parenthesis 5,730 divided by R closed parenthesis times L subscript VC divided by K closed bracket for horizontal curves; equals 1.0 for tangents at type 1 crests; and equals 1.0 for level tangents (base condition) closed bracket.

Figure 43. Equation. FI CMF for horizontal curves and tangents at type 1 crest vertical curves.

CMF subscript C1,PDO equals open bracket exponent open bracket 0.0046 times open parenthesis 5,730 divided by R closed parenthesis times L subscript VC divided by K closed bracket for horizontal curves; equals 1.0 for tangents t type 1 crests; and equals 1.0 for level tangents (base condition) closed bracket.

Figure 44. Equation. PDO CMF for horizontal curves and tangents at type 1 crest vertical curves.

The functional relationships shown in figure 18 (crashes/mi/year) and figure 43 (CMF) for FI crashes are illustrated in figure 45 for selected values of K. Curve radius ranged from 100 to 11,460 ft, AADT was fixed at 2,000 vehicles/day, and LVC was fixed at 500 ft, the median LVC in the database. Values of K were set at 250, 125, 83, 63, and 50, which correspond to a grade difference of 2, 4, 6, 8, and 10 percent, respectively, for a curve length of 500 ft. Similarly, figure 46 illustrates the relationships shown in figure 19 and figure 44 for PDO crashes.

To calculate CMF for FI or PDO crashes for a given horizontal curve at a type 1 crest vertical curve, the actual values of R (ft), LVC (ft), and parameter K (ft/percent) are substituted in figure 43 or figure 44. Example CMFs were calculated for rural two-lane roadways with an LVC of 500 ft, R of 1,433 or 5,730 ft, and K values of 250, 125, 83, 63, and 50 ft/percent using figure 43 and figure 44. The results are shown in table 18 for rural two-lane roadways with AADTs from 200 to 26,000 vehicles/day and LVC of 500 ft.

Table 18. Example CMFs for FI and PDO crashes on horizontal curves and tangents at type 1 crest vertical curves.

K

Tangent
at Crest

FI CMFs

PDO CMFs

R = 1,433 ft

R = 5,730 ft

R = 1,433 ft

R = 5,730 ft

250

1.0

1.07

1.02

1.04

1.01

125

1.0

1.15

1.04

1.08

1.02

83

1.0

1.24

1.05

1.12

1.03

63

1.0

1.33

1.07

1.16

1.04

50

1.0

1.42

1.09

1.20

1.05

This graph shows predicted fatal and injury (FI) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents at 
type 1 crest vertical curves. The base condition is a level tangent, and the length of the vertical curve is 500 ft. FI crashes/mi/year is on the left y-axis from zero to 1.1, and CMF is on the right y-axis from zero to 4.00. Radius is on the x-axis from zero to 12,000 ft. There are five lines plotted on the graph corresponding to K parameters of 250, 125, 83, 63, and 50. There is a dotted horizontal blue line that corresponds to a base condition tangent with an average annual daily traffic of 2,000 vehicles/day and a CMF of 1.0. All curves are exponential decay curves.

Figure 45. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and tangents at type 1 crest vertical curves.

This graph shows predicted property damage only (PDO) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents at type 1 crest vertical curves. The base condition is a level tangent, and the length of the vertical curve is 500 ft. PDO crashes/mi/year is on the left y-axis from zero to 0.9, and CMF is on the right y-axis from zero to 2.00. Radius is on the x-axis from zero to 12,000 ft. There are five lines plotted on the graph corresponding to K parameters of 250, 125, 83, 63, and 50. There is a dotted horizontal blue line that corresponds to a base condition tangent with an average annual daily traffic of 2,000 vehicles/day and a CMF of 1.0. All curves are exponential decay curves.

Figure 46. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and tangents at type 1 crest vertical curves.

In any given column in table 18, the CMFs show the increasing effect on crashes of a decreasing K (steeper crest) of a horizontal curve at a type 1 crest vertical curve. This effect is less pronounced for PDO crashes than for FI crashes. The combined effect of a sharp horizontal curve on a steep vertical crest is reflected in the last rows of the third and fifth columns. When K becomes infinite (i.e., level roadway) and R becomes infinite (i.e., tangent roadway), the roadway becomes a level tangent, and CMF becomes 1.0.

HORIZONTAL CURVES AND TANGENTS AT TYPE 1 SAG VERTICAL CURVES

CMFs for horizontal curves and tangents at type 1 sag vertical curves can be derived from figure 25 and figure 26 as follows shown in figure 47 and figure 48*:

*Modified on November 16, 2014

CMF subscript S1,FI equals open bracket exponent open bracket 10.51 times 1 divided by K plus 0.011 times open parenthesis 5,730 divided by R closed parenthesis times L subscript VC divided by K closed bracket for horizontal curves; equals exponent open bracket 10.51 times 1 divided by K closed bracket for tangents at type 1 sags; and equals 1.0 for level tangents (base condition) closed bracket.

Figure 47. Equation. FI CMF for horizontal curves and tangents at type 1 sag vertical curves.

CMF subscript S1,PDO equals open bracket exponent open bracket 8.62 times 1 divided by K plus 0.010 times open parenthesis 5,730 divided by R closed parenthesis times L subscript VC divided by K closed bracket for horizontal curves; equals exponent open bracket 8.62 times 1 divided by K closed bracket for tangents at type 1 sags; and equals 1.0 for level tangents (base condition) closed bracket.

Figure 48. Equation. PDO CMF for horizontal curves and tangents at type 1 sag vertical curves.

The functional relationships shown in figure 25 (crashes/mi/year) and figure 47 (CMF) for FI crashes are illustrated in figure 49 for K values of 250, 125, 83, 63, and 50. R ranged from 100 to 11,460 ft, AADT was fixed at 2,000 vehicles/day, and LVC was fixed at 500 ft. Similarly, figure 50 illustrates the relationships shown in figure 26 and figure 48 for PDO crashes.

To calculate CMF for FI or PDO crashes for a given horizontal curve at a type 1 sag vertical curve, the actual values of R (ft), LVC (ft), and parameter K (ft/percent) are substituted in figure 47 or figure 48. Example CMFs were calculated for rural two-lane roadways with a vertical curve length of 500 ft, an R of 1,433 or 5,730 ft, and K values of 250, 125, 83, 63, and 50 using figure 47 and figure 48. The results are shown in table 19 for rural two-lane roadways with AADTs from 200 to 26,000 vehicles/day.

This graph shows predicted fatal and injury (FI) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents at 
type 1 sag vertical curves. The base condition is a level tangent, and the length of the vertical curve is 500 ft. FI crashes/mi/year is on the left y-axis from zero to 1.1, and CMF is on the right y-axis from zero to 4.00. Radius is on the x-axis from zero to 12,000 ft. There are five lines plotted on the graph corresponding to K parameters of 250, 125, 83, 63, and 50. There is a dotted horizontal blue line that corresponds to a base condition tangent with an average annual daily traffic of 2,000 vehicles/day and a CMF of 1.0. All curves are exponential decay curves.

Figure 49. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and tangents at type 1 sag vertical curves.

This graph shows predicted property damage only (PDO) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents at type 1 sag vertical curves. The base condition is a level tangent, and the length of the vertical curve is 500 ft. PDO crashes/mi/year is on the left y-axis from zero to 1.7, and CMF is on the right y-axis from zero to 4.00. Radius is on the x-axis from zero to 12,000 ft. There are five lines plotted on the graph corresponding to K parameters of 250, 125, 83, 63, and 50. There is a dotted horizontal blue line that corresponds to a base condition tangent with an average annual daily traffic of 2,000 vehicles/day and a CMF of 1.0. All curves are exponential decay curves.

Figure 50. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and tangents at type 1 sag vertical curves.

Table 19. Example CMFs for FI and PDO crashes on horizontal curves and tangents at type 1 sag vertical curves.

K

FI CMFs

PDO CMFs

Tangent at Sag

R = 1,433 ft

R = 5,730 ft

Tangent at Sag

R = 1,433 ft

R = 5,730 ft

250

1.04

1.14

1.07

1.04

1.12

1.05

125

1.09

1.30

1.14

1.07

1.25

1.11

83

1.13

1.49

1.21

1.11

1.39

1.17

63

1.18

1.68

1.29

1.15

1.55

1.24

50

1.23

1.93

1.38

1.19

1.74

1.31

In any given column in table 19, CMFs show the increasing effect on crashes of decreasing K (sharper sag) for a horizontal curve or tangent at a type 1 sag vertical curve. This effect is slightly less pronounced for PDO than for FI crashes. When K becomes very large (i.e., level roadway), and R becomes infinite (i.e., tangent roadway), the roadway becomes a level tangent, and CMF becomes 1.0.

HORIZONTAL CURVES AND TANGENTS AT TYPE 2 CREST VERTICAL CURVES

CMFs for horizontal curves and tangents at type 2 crest vertical curves can be derived from figure 32 and figure 33 as follows shown in figure 51 and figure 52*:

*Modified on November 16, 2014

CMF subscript C2,FI equals open bracket exponent open bracket 0.20 times natural logarithm of open parenthesis 2 times 5,730 divided by R closed parenthesis closed bracket for horizontal curves; equals 1.0 for tangents at type 2 crests; and equals 1.0 for level tangents (base condition) closed bracket.

Figure 51. Equation. FI CMF for horizontal curves and tangents at type 2 crest vertical curves.

CMF subscript C2,PDO equals open bracket exponent open bracket 0.10 times natural logarithm of open parenthesis 2 times 5,730 divided by R closed parenthesis closed bracket for horizontal curves; equals 1.0 for tangents at type 2 crests; and equals 1.0 for level tangents (base condition) closed bracket.

Figure 52. Equation. PDO CMF for horizontal curves and tangents at type 2 crest vertical curves.

The functional relationships shown in figure 32 (crashes/mi/year) and figure 51 (CMF) for FI crashes are illustrated in figure 53 for an R range from 100 to 11,460 ft and AADTs of 1,000, 2,000, 4,000, and 7,000 vehicles/day (corresponding approximately to the 25th, 50th, 75th, and 90th percentile AADTs for rural two-lane highways in the database). Similarly, figure 54 illustrates the relationships shown in figure 33 and figure 52 for PDO crashes.

To calculate CMF for FI or PDO crashes for a given horizontal curve at a type 2 crest vertical curve, the actual value of R (ft) is substituted in figure 51 or figure 52. Example CMFs were calculated for rural two-lane roadways with an R of 1,433, 5,730, or 11,460 ft using figure 51 and figure 52. The results are shown in table 20 for rural two-lane roadways with AADTs from 200 to 26,000 vehicles/day.

This graph shows four plots of predicted fatal and injury (FI) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents at type 2 crest vertical curves. The four plots show different average annual daily traffics of 1,000, 2,000, 4,000, and 7,000 vehicles/day. FI crashes/mi/year are shown on the left y-axis from zero to 2.2 crashes/mi/year for all four plots, and the corresponding CMFs are shown on the right y-axis from zero to 16, zero to 7.5, zero to 3.5, and zero to 2.0 for the four plots, respectively. The x-axis shows the radius from zero to 12,000 ft for all four plots. There is a dotted horizontal blue line that corresponds to a base condition tangent with a CMF of 1.0. All curves are exponential decay curves.

Figure 53. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and tangents at type 2 crest vertical curves.

This graph shows four plots of predicted property damage only (PDO) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents at type 2 crest vertical curves. The four plots show different average annual daily traffics of 1,000, 2,000, 4,000, and 7,000 vehicles/day. PDO crashes/mi/year are shown on the left y-axis from zero to 2.2 crashes/mi/year for all four plots, and the corresponding CMFs are shown on the right y-axis from zero to 9, zero to 5.0, zero to 2.5, and zero to 1.4 for the 
four plots, respectively. The x-axis shows the radius from zero to 12,000 ft for all four plots. There is a dotted horizontal blue line that corresponds to a base condition tangent with a CMF 
of 1.0. All curves are exponential decay curves.

Figure 54. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and tangents at type 2 crest vertical curves.

Table 20. Example CMFs for FI and PDO crashes on horizontal curves and tangents at type 2 crest vertical curves.

FI CMFs

PDO CMFs

R = 1,433 ft

R = 5,730 ft

R = 11,460 ft

R = 1,433 ft

R = 5,730 ft

R = 11,460 ft

1.52

1.15

1.00

1.23

1.07

1.00

Only R of the horizontal curve at a type 2 vertical crest has an effect on crash rates, with increasing radii producing smaller CMFs. The effect of R is more pronounced for FI crashes than for PDO crashes. When R becomes infinite, the roadway becomes a tangent, and CMF becomes 1.0 regardless of the vertical curve characteristics.

Because this CMF lacks any measure of vertical alignment, consideration may be given to replacing this CMF with a CMF based on figure 10 and figure 11.

HORIZONTAL CURVES AND TANGENTS AT TYPE 2 SAG VERTICAL CURVES

CMFs for horizontal curves and tangents at type 2 sag vertical curves can be derived from figure 37 and figure 38 as follows shown in figure 55 and figure 56*:

*Modified on November 16, 2014

CMF subscript S2,FI equals open bracket exponent open bracket 0.188 times natural logarithm of times open parenthesis 2 times 5,730 divided by R closed parenthesis closed bracket for horizontal curves; equals 1.0 for tangents at type 2 sags; and equals 1.0 for level tangents (base condition) closed bracket.

Figure 55. Equation. FI CMF for horizontal curves and tangents at type 2 sag vertical curves.

CMF subscript S2,PDO equals open bracket exponent open bracket 0.022 times open parenthesis 5,730 divided by R closed parenthesis times A closed bracket for horizontal curves; equals 1.0 for tangents at type 2 sags; and equals 1.0 for level tangents (base condition) 
closed bracket.

Figure 56. Equation. PDO CMF for horizontal curves and tangents at type 2 sag vertical curves.

The functional relationships shown in figure 37 (crashes/mi/year) and figure 55 (CMF) for FI crashes are illustrated in figure 57 for an R range from 100 to 11,460 ft and AADTs of 1,000, 2,000, 4,000, and 7,000 vehicles/day. Similarly, figure 58 illustrates the relationships shown in figure 38 and figure 56 for PDO crashes for K values of 250, 125, 83, 63, and 50; curve radii ranging from 100 to 11,460 ft; and AADT fixed at 2,000 vehicles/day.

To calculate CMF for FI crashes for a given horizontal curve at a type 2 sag vertical curve, the actual value of R (ft) is substituted in figure 55. To calculate CMF for PDO crashes, the actual value of R (ft), LVC, and K are substituted in figure 56. Example FI CMFs were calculated for rural two-lane roadways with an R of 1,433, 5,730, or 11,460 ft using figure 55. Example PDO CMFs were calculated using figure 56 for rural two-lane roadways with an R of 1,433 or 5,730 ft, an LVC of 500 ft, and A values ranging from 2 to 10 percent. The results are shown in table 21 for FI crashes and table 22 for PDO crashes for rural two-lane roadways with AADTs from 200 to 26,000 vehicles/day.

. This graph shows four plots of predicted fatal and injury (FI) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents at type 2 sag vertical curves. The four plots show different average annual daily traffics of 1,000, 2,000, 4,000, and 7,000 vehicles/day. FI crashes/mi/year are shown on the left y-axis from zero to 2.2 crashes/mi/year for all four plots, and the corresponding CMFs are shown on the right 
y-axis from zero to 16, zero to 7.5, zero to 3.5, and zero to 2.0 for the four plots, respectively. The x-axis shows the radius from zero to 12,000 ft for all four plots. There is a dotted horizontal blue line that corresponds to a base condition tangent with a CMF of 1.0. All curves are exponential decay curves.

Figure 57. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and tangents at type 2 sag vertical curves.

This graph shows the predicted property damage only (PDO) crashes/mi/year and crash modification factors (CMFs) for horizontal curves and tangents at type 2 sag vertical curves. PDO crashes/mi/year are shown on the left y-axis from zero to 
1.1 crashes/mi/year, and the corresponding CMFs are shown on the right y-axis from zero to 2.50. The x-axis shows the radius from zero to 12,000 ft for all four plots. Five lines are 
plotted for the following A terms: 2, 4, 6, 8, and 10. There is a dotted horizontal blue line 
that corresponds to a base condition tangent with an annual average daily traffic of 
2,000 vehicles/day and a CMF of 1.0. All curves are exponential decay curves.

Figure 58. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and tangents at type 2 sag vertical curves.

Table 21. Example CMFs for FI crashes on horizontal curves and tangents at type 2 sag vertical curves.

FI CMFs

R = 1,433 ft

R = 5,730 ft

R = 11,460 ft

1.48

1.14

1.00

Table 22. Example CMFs PDO crashes on horizontal curves and tangents at type 2 sag vertical curves.

A

PDO CMFs

R = 1,433 ft

R = 5,730 ft

R = 11,460 ft

All tangents

1.00

1.00

1.00

2

1.19

1.04

1.02

4

1.42

1.09

1.04

6

1.69

1.14

1.07

8

2.02

1.19

1.09

10

2.40

1.24

1.12

Only R of the horizontal curve at a type 2 vertical sag has an effect on FI crash rates, with increasing radii producing smaller CMFs (see table 21). When R becomes infinite, the roadway becomes a tangent and CMF becomes 1.0 regardless of the characteristics of the vertical curve. Only the interaction term of A and inverse R has an effect on PDO crash rates (see table 22), highlighting the joint effect of sharp horizontal curves at steep vertical curves on PDO crashes. When the initial and final grade difference, A, nears zero (i.e., level roadway) and the R becomes infinite (i.e., tangent roadway), the roadway becomes a level tangent, and CMF becomes 1.0.

Because this CMF (at least for FI crashes) lacks any measure of vertical alignment, consideration may be given to replacing this CMF with a CMF based on figure 10 and figure 11.

CMFs FOR COMBINED CRASH SEVERITY LEVELS

This study provides separate CMFs for FI and PDO crashes. At some future time, the AASHTO HSM may be structured to provide separate CMFs by severity level for all CMFs. In the meantime, if users want a CMF for total crashes (i.e., all crash severity levels combined), it can be computed from the results in this report as follows shown in figure 59*:

*Modified on November 16, 2014

CMF subscript TOT equals open bracket open parenthesis CMF subscript FI minus 1.0 closed parenthesis times P subscript FI plus open parenthesis CMF subscript PDO minus 1.0 closed parenthesis times P subscript PDO closed bracket plus 1.0.

Figure 59. Equation. CMF for combined crash severity level.

Where:
CMFTOT = CMF for total crashes (i.e., all severity levels combined).
CMFFI = CMF for FI crashes.
CMFPDO = CMF for PDO crashes.
PFI = FI crashes expressed as a proportion of total crashes.
PPDO = PDO crashes expressed as a proportion of total crashes.

Values used for PFI and PPDO must always sum to 1.0. Values of PFI and PPDO indicated for rural two-lane highways in AASHTO HSM table 10-3 (PFI = 0.321 and PPDO = 0.679) may be used, or users may develop values for PFI and PPDO from their agencies’ data.(1)

COMPARISON OF RESULTS TO EXISTING HSM CMFS

The current AASHTO HSM presents separate CMFs for horizontal curves and straight grades, as seen as CMFHC in figure 3 and CMFG in figure 5, respectively.(1) The combined effect of horizontal curves and grades is represented in the AASHTO HSM as the product of CMFHC and CMFG. Figure 60 illustrates a comparison of CMFs for horizontal curves on straight grades developed in this study, and shown in figure 39 and figure 40 for FI and PDO crashes, respectively, to the combined AASHTO HSM CMF, holding length of horizontal curve and radius constant while varying percent grade. Figure 61 is an analogous plot, keeping the length of horizontal curve and percent grade constant while varying the radius of horizontal curve.

The plots show that the new CMF for FI crashes is consistently larger than the new CMF for PDO crashes. This represents an advance in knowledge over the AASHTO HSM, which treats CMFs for all severity levels as equal.(1) The plots also show that the new CMFs are generally larger than the combined AASHTO HSM CMFs, except that the new CMF for PDO crashes is smaller than the existing CMFs for horizontal curves with short radii. No other comparisons between the HSM CMFs and the CMFs developed in this study are relevant because the AASHTO HSM does not address the safety effects of crest or sag vertical curves.

This graph shows a comparison of crash modification factors (CMFs) developed in this study to the combined American Association of State Highways and Transportation Officials (AAHSTO) Highway Safety Manual (HSM) CMFs for horizontal curves and grades for fixed radius and varying percent grades. CMF is on the y-axis from 1.0 to 3.0, and grade is on the x-axis from zero to 10 percent in increments of 1 percent. The curve length is 0.10 mi, and the curve radius is 
2,000 ft. There are three lines plotted: CMF for fatal and injury (FI) crashes, CMF for property damage only (PDO) crashes, and the combined HSM CMFs. All three curves are monotonically increasing with increasing percent grade. The curve for the combined HSM CMFs is lower than the curve for the PDO CMFs which in turn is lower than the curve for the FI CMFs.

Figure 60. Graph. Comparison of CMFs developed in this study to the combined AASHTO HSM CMFs for horizontal curves and grades for fixed radius and varying percent grades.(1)

This graph shows a comparison of crash modification factors (CMFs) developed in this study to the combined American Association of State Highways and Transportation Officials (AASHTO) Highway Safety Manual (HSM) CMFs for horizontal curves and grades for fixed percent grade and varying radii. CMF is on the y-axis from 1.0 to 3.0, and radius is on the x-axis from zero to 12,000 ft in increments of 2,000 ft. The length of the curve is 0.10 mi, and the grade is 2 percent. There are three lines plotted: CMF for fatal and injury (FI) crashes, CMF for property damage only (PDO) crashes, and the combined HSM CMFs. All three curves are exponential decay curves. The curve for FI CMFs is always higher than the other two curves for radii above approximately 800 ft where it is below the curve for the combined HSM CMFs but still above the curve for PDO CMFs. The two curves for PDO CMFs and combined HSM CMFs cross over when the radius is approximately 1,200 ft, with the curve for the PDO CMFs being lower than the curve for the combined HSM CMFs below the crossover point.

Figure 61. Graph. Comparison of CMFs developed in this study to the combined AASHTO HSM CMFs for horizontal curves and grades for fixed percent grade and varying radii.(1)

 

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