U.S. Department of Transportation
Federal Highway Administration
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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
SUMMARY REPORT 
This summary report is an archived publication and may contain dated technical, contact, and link information 

Publication Number: FHWAHRT13078 Date: January 2014 
Publication Number:
FHWAHRT13078
Date: January 2014 
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Report  FHWAHRT13077  Summary Report  FHWAHRT13078  Excel Calcuator Tool  HRTM 2130 
HTML Version of Errata for FHWAHRT13078
Location  Incorrect Values  Corrected Values 
Page 3, Column 1, Bullet 7  (A; A = abs(G_{1} − G_{2})  (A; A = abs(G_{1} − G_{2})) 
Page 3, Column 1, Crash Modifications, Paragraph 1, Sentence 2, parenthesis 
design or traffic control  Design and traffic control 
Page 3, Column 2, (full) Paragraph 1, Sentence 2 
twoway  twolane 
Page 3, Column 2, CMFs for Horizontal Curves and Tangents on Straight Grades, Paragraph 1, Sentence 1  estimated as follows  provided in figures 2 and 3 
Page 3, Column 2, CMFs for Horizontal Curves and Tangents at Type 1 Crest Vertical Curves, Paragraph 1, Sentence 1  estimated as follows  provided in figures 4 and 5 
Page 4, Column 1, CMFs for Horizontal Curves and Tangents at Type 1 Sag Vertical Curves, Paragraph 1, Sentence 1  were estimated as follows  are shown in figures 6 and 7 
Page 4, Column 1, CMFs for Horizontal Curves and Tangents at Type 2 Crest Vertical Curves, Paragraph 1, Sentence 1  were estimated as follows  are shown in figures 8 and 9 
Page 4, Column 2, CMFs for Horizontal Curves and Tangents at Type 2 Sag Vertical Curves, Paragraph 1, Sentence 1  were estimated as follows  are shown in figures 10 and 11 
Page 4, Applications of CMFs, Paragraph 1, Sentence 1, “To calculate CMF values…”  To calculate CMF values  To calculate the CMF values 
Page 4, Applications of CMFs, Bullet 1, Sentence 1, “Any horizontal R…”  Any horizontal R less than 100 ft  Any horizontal curve R less than 100 ft 
Page 6, Column 2, Item 5  CurvesHCurve VCurve  HCurve VCurve 
Page 6, Column 2, Procedure 1  1. Open the Microsoft Excel® calculation tool workbook.  1. Open the Microsoft Excel® tool at http://www.fhwa.dot.gov/publications/ research/safety/hrtm2130/index.cfm 
Page 8, For More Information, Paragraph 2, Sentence 1  For more information about HSIS, contact Carol Tan, Ph.D., FHWA HSIS Program Manager, HRDS, 202493 3315, carol.tan@dot.gov.  For more information about HSIS, contact Carol Tan, Ph.D., FHWA HSIS Program Manager, HRDS, 202493 3315, carol.tan@dot.gov, or Ana Maria Eigen, D.Sc., FHWA HSIS Program Manager, HRDS, 2024933315, ana.eigen@dot.gov. 
The safety effects of horizontal curves and grades on rural twolane highways have been quantified in the American Association of State Highway and Transportation Officials (AASHTO) Highway Safety Manual (HSM), but it was not previously known whether and how the safety performance of horizontal curves and grades interact.^{(1)} Furthermore, there are no established safety effects for crest and sag vertical curves, and it is unknown whether and how the safety performance of crest or sag vertical curves is affected by the presence of horizontal curves.
The objective of this study was to quantify the combined safety effects of horizontal curves and grade combinations and express the results as crash modification factors (CMFs) that can be considered for use in the AASHTO HSM.^{(1)}
Design criteria for horizontal and vertical alignment are presented in chapter 3 of the AASHTO Policy on Geometric Design of Highways and Streets, commonly known as the Green Book.^{(2)} Many State highway agencies have their own design manuals, but in terms of horizontal and vertical alignment, they closely resemble the AASHTO Green Book.
Straight road sections with no horizontal curvatures are generally referred to as "tangents" because such straight road sections are generally tangent to any horizontal curves that they adjoin.
The key design parameters for horizontal curves include the following:
The safety effects of both radius and length of horizontal curves are addressed in CMFs developed in this current study. The safety effects of superelevation and transition design are outside the scope of the study because no data concerning these features were available at the time this study was conducted.
The fundamental design parameter for vertical alignment is the percent grade. A road section with constant percent grade, regardless of its horizontal alignment, is generally referred to as a straight grade. Where the grade of the roadway changes, the straight grade sections are normally joined by a parabolic vertical curve. Figure 1 illustrates the four types of vertical curves (two types of crest vertical curves and two types of sag vertical curves) that are used in highway design. Key design parameters for vertical curves include the following:
The safety effects of each of these design parameters for vertical alignment are addressed in CMFs developed in the current study.
Source: AASHTO. Used by permission.
Figure 1. Illustration. Types of vertical curves.^{(2)}
Research was undertaken to quantify the safety effects of horizontal and vertical alignment combinations and to present them as CMFs. The complete results of this research are documented in the full report, Safety Effects of Horizontal Curve and Grade Combinations on Rural TwoLane Highways.^{(3)}
The research was performed with the Highway Safety Information System (HSIS) data for State highways in Washington. This is the only data source that includes systemwide data on curve and grade geometry that can be linked to systemwide roadway characteristics, traffic volume data, and crash data. Several roadway types were considered, but only rural twolane highways in Washington had sufficient data for which modeling efforts appeared promising.
Crash data for a 6year period (2003 to 2008) were obtained and used in the analysis. Each crash was assigned to a particular roadway segment with particular horizontal and vertical alignment based on its assigned milepost location. Since the results of this research are intended for use in the roadway segment procedures of the AASHTO HSM, only nonintersection crashes were considered.^{(1)}
Of the 6,944 mi of roadway in the entire Washington HSIS database, 4,785 mi (69 percent) are on rural twolane highways. Of these, 3,457 mi were used for analysis. Rural twolane highways with passing or climbing lanes and segments with missing or obviously incorrect alignment data (e.g., overlapping curves) were excluded from the study. Roadway length (miles), exposure (million vehicle miles traveled (MVMT) in the 6year period), crash frequencies, and crash rates per MVMT are shown in table 1 for specific horizontal and vertical alignment for rural twolane highways.
Table 1. Roadway length, exposure, crash frequency, and crash rates
for rural twolane highways in the Washington HSIS database.
Alignment Type 
Roadway Element 
Roadway Length (mi) 
Exposure (MVMT)^{a} 
Crash Frequency^{a}  Crash Rate per MVMT  

Fatal and Injury (FI) 
Property Damage Only (PDO) 
FI  PDO  
Horizontal  Tangent  2,472.1  16,675.2  7,360  10,519  0.441  0.631 
Curve  985.0  6,194.2  3,659  4,758  0.591  0.768  
Total  3,457.1  22,869.5  11,019  15,277  N/A  N/A  
Vertical  Straight grade  2,260.7  14,847.0  7,347  10,222  0.495  0.688 
Type 1 crest  364.5  2,616.4  1,168  1,498  0.446  0.573  
Type 2 crest  300.8  1,870.5  826  1,264  0.442  0.676  
Type 1 sag  252.1  1,772.6  896  1,154  0.505  0.651  
Type 2 sag  279.1  1,762.9  782  1,139  0.444  0.646  
Total  3,457.1  22,869.5  11,019  15,277  N/A  N/A 
^{a}For years 2003 to 2008.
N/A = Not applicable.
The safety effects of horizontal curve and grade combinations were estimated based on a crosssectional analysis using a generalized linear model approach assuming a negative binomial distribution of crash counts and an exponential model using the combined crash data from all 6 years and selected roadway geometrics. FI and PDO crashes were modeled separately for each of the five types of horizontal curves and grade combinations. All analyses were performed using a procedure for fitting generalized linear models of SAS^{®} Version 9.3.^{(4)}
The following parameters were considered for inclusion in each model:
For each type of horizontal curve and grade combination, the dataset used for modeling included the roadway segments for the relevant curve and grade combination but also all level tangents (i.e., no horizontal curvature and grade < 1 percent) to serve as the base condition. Details of the statistical analysis and resulting crash prediction models are presented in the full report.^{(3)}
CMFs used in the AASHTO HSM were derived from predictive
models.^{(1)}
CMF is a factor that represents the effect on crash frequency for a given crash
severity level of varying geometric design and 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.
CMF with a value greater than 1.0 represents a condition for which more crashes
would be expected than for the base condition. 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. This is consistent with the base conditions in the
current AASHTO HSM.^{(1)}
AMENDED February 8, 2016
For each alignment type combination (as well as each FI
and PDO crash), 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 the equations for CMF
in each of the five alignment categories for rural
twolane highways.
AMENDED February 8, 2016
The CMFs for horizontal curves and tangents on straight
grades were provided in figures 2 and 3:
AMENDED February 8, 2016
Figure 2. Equation. FI CMF for horizontal curves and tangents on straight grades.
Figure 3. Equation. PDO CMF for horizontal curves and tangents on straight grades.
Where SG = straight grade.
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, the absolute value of G (percent), R (ft), and L_{C} (mi) must be substituted in figure 2 or figure 3.
CMFs for horizontal curves and tangents at type 1 crest
vertical curves (C1) are provided in figures 4 and 5:
AMENDED February 8, 2016
Figure 4. Equation. FI CMF for horizontal curves and tangents at type 1 crest vertical curves.
Figure 5. Equation. PDO CMF for horizontal curves and tangents at type 1 crest vertical curves.
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), L_{VC} (ft), and parameter K (ft/percent) must be substituted in figure 4 or figure 5.
The CMFs for horizontal curves and tangents at type 1 sag
vertical curves (S1) were are shown in figures 6 and 7
AMENDED February 8, 2016
Figure 6. Equation. FI CMF for horizontal curves and tangents at type 1 sag vertical curves.
Figure 7. Equation. PDO CMF for horizontal curves and tangents at type 1 sag vertical curves.
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), L_{VC} (ft), and parameter K (ft/percent) must be substituted in figure 6 or figure 7.
The CMFs for horizontal curves and tangents at type 2
crest vertical curves (C2) are shown in figures 8 and 9:
AMENDED February 8, 2016
Figure 8. Equation. FI CMF for horizontal curves and tangents at type 2 crest vertical curves.
Figure 9. Equation. PDO CMF for horizontal curves and tangents at type 2 crest vertical curves.
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) must be substituted in figure 8 or figure 9.
The CMFs for horizontal curves and tangents at type 2 sag
vertical curves (S2) are shown in figures 10 and 11:
AMENDED February 8, 2016
Figure 10. Equation. FI CMF for horizontal curves and tangents at type 2 sag vertical curves.
Figure 11. Equation. PDO CMF for horizontal curves and tangents at type 2 sag vertical curves.
To calculate CMF for FI crashes for a given horizontal curve at a type 2 sag vertical curve, the actual value of R (ft) must be substituted in figure 10 or figure 11, and A must be substituted in figure 11.
To calculate the CMF values using the equations in figure 2 through figure 11, the following guidelines should be applied:^{(3)}
AMENDED February 8, 2016
The results presented in figure 2 through figure 11 provide separate CMFs for FI and PDO crashes. CMF for total crashes (i.e., all crash severity levels combined) can be computed as follows:
Figure 12. Equation. CMF for combined crash severity level.
Where:
CMF_{TOT} = CMF for total crashes (i.e., all severity levels combined).
CMF_{FI} = CMF for FI crashes.
CMF_{PDO} = CMF for PDO crashes.
P_{FI} = FI crashes expressed as a proportion of total crashes.
P_{PDO} = PDO crashes expressed as a proportion of total crashes.
The values used for P_{FI} and P_{PDO} must always sum to 1.0. The values of P_{FI} and P_{PDO} indicated for rural twolane highways in AASHTO HSM table 103 (P_{FI} = 0.321 and P_{PDO} = 0.679) may be used, or users may develop values for P_{FI} and P_{PDO} from their agencies’ data.^{(1)}
Figure 13 illustrates a typical comparison of CMFs for horizontal curves on straight grades developed in this study, as shown in figure 2 and figure 3 for FI and PDO crashes, respectively, to the combined HSM CMF. The length of horizontal curve and radius were kept constant, while the percent grade was varied.
Figure 13. 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)}
Figure 14 is an analogous plot, where the length of horizontal curve and percent grade were kept constant, while the radius of the horizontal curve varied. The plots show that the CMF for FI crashes developed in the current study is consistently larger than the CMF for PDO crashes developed in the current study. This represents an advance in knowledge compared to the AASHTO HSM, which treated the CMFs as equal for all severity levels.^{(1)} The plots also show that the new CMFs are generally larger than the combined HSM CMFs, except that the new CMF for PDO crashes is smaller than the existing CMFs for horizontal curves with short radii.
Figure 14. 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)}
CMF presented in figure 12 can be considered to replace the combined effect of CMF_{3r} for horizontal curves presented in AASHTO HSM equation 1013 and CMF_{5r} for grades presented in AASHTO HSM table 1011.^{(1)} In other words, CMF_{TOT} is a potential substitute for the product of CMF_{3r} × CMF_{5r} in AASHTO HSM equation 102, which currently represents the combined total of FI and PDO crashes.^{(1)} It is expected that future AASHTO HSM editions will model FI and PDO crashes separately so that CMFs for individual crash severity levels may be used directly in AASHTO HSM equation 102. A decision as to whether new CMFs presented in figure 2 through figure 12 should be incorporated in the AASHTO HSM will be made by AASHTO at some future time.
The AASHTO HSM also includes CMF_{4r}, which represents the safety effect of superelevation variance defined for any horizontal curve as the design superelevation rate for that curve recommended in the AASHTO Green Book minus the actual superelevation of the curve.^{(1,2)} The models developed in the current study do not account for superelevation variance, so CMF_{4r} should still be used even if CMF_{3r} and CMF_{5r} are replaced by the new CMFs developed in this current study.
A Microsoft Excel^{®} calculation tool has been developed to assist users in determining the values of CMFs using figure 2 through figure 12 for any horizontal curve and grade combination. The tool includes five worksheets. The names and function of each worksheet are as follows:
Worksheets 2–5 provide a table for data input by the user and a table that displays the calculated CMF values. CMF calculations are performed using figure 2 through figure 12.
Procedures for using the CMF calculation tool are as follows:
Figure 15 shows a typical sheet from the Microsoft Excel^{®} calculation tool with both input data and computed results displayed.^{(5)}
Figure 15. Screenshot. Sample screen for horizontal curve at type 1 crest CMF calculations.
This research was conducted by Karin M. Bauer and Douglas W. Harwood of MRIGlobal. Chris Fees of MRIGlobal programmed the CMF calculation tool. The final report, Safety Effects of Horizontal Curve and Grade Combinations on Rural TwoLane Highways, is published as FHWA Report No. FHWAHRT13077.^{(3)}
For more information about HSIS, contact Carol Tan, Ph.D., FHWA HSIS Program Manager, HRDS, 2024933315, carol.tan@dot.gov or Ana Maria Eigen, D.Sc., FHWA HSIS Program Manager, HRDS, 2024933315, ana.eigen@dot.gov.
AMENDED February 8, 2016