Safety Effects of Horizontal Curve and 

Grade Combinations on Rural Two-Lane 

Highways

The safety effects of horizontal curves and grades on rural two-lane 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)

Background

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:

• Radius of curvature.

• Length of curve.

• Superelevation.

• Transition design.

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:

• Algebraic difference (A) between the initial (G1) and final (G2) grades.

• Length of curve (LVC).

• K, the ratio of LVC and A, which represents the measure of sharpness of the 

vertical curve.

The safety effects of each of these design parameters for vertical alignment are addressed in 

CMFs developed in the current study.

An Analysis 

Tool for Making 

Informed Safety 

Decisions

SUMMARY REPORT
2

Methodology

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 Two-Lane Highways.

(3)

Database Used

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 

system-wide data on curve and grade geometry that 

can be linked to system-wide roadway characteristics, 

traffic volume data, and crash data. Several roadway 

types were considered, but only rural two-lane highways in Washington had sufficient data for which 

modeling efforts appeared promising.

Crash data for a 6-year 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 

two-lane highways. Of these, 3,457 mi were used for 

analysis. Rural two-lane 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 6-year period), crash frequencies, 

and crash rates per MVMT are shown in table 1 for 

specific horizontal and vertical alignment for rural 

two-lane highways. 

Analysis Approach

The safety effects of horizontal curve and grade combinations were estimated based on a cross-sectional 

analysis using a generalized linear model approach 

Table 1. Roadway length, exposure, crash frequency, and crash rates for rural two-lane highways in the Washington HSIS database.

ALIGNMENT 

TYPE

ROADWAY 

ELEMENT

ROADWAY 

LENGTH (MI)

EXPOSURE

(MVMT)a

CRASH FREQUENCYa 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

aFor years 2003 to 2008.

N/A = Not applicable.

Figure 1. Illustration. Types of vertical curves.(2)

Source: AASHTO. Used by permission.
3

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:

• Average annual daily traffic (averaged across 

all 6 years).

• Segment length.

• Horizontal curve radius (R).

• Absolute value of percent grade (G).

• Horizontal curve length (LC).

• Vertical curve length (LVC).

• Algebraic difference between the initial (G1) 

and final (G2) grades (A; A = abs(G1 − G2).

• Measure of the sharpness of vertical curvature 

(K; K = LVC/A).

• Relevant interactions of selected parameters.

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)

Crash Modification Factors

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 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. 

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)

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 

two-way highways.

CMFs for Horizontal Curves and Tangents 

on Straight Grades

The CMFs for horizontal curves and tangents on 

straight grades were estimated as follows:

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 LC (mi) must be substituted in 

figure 2 or figure 3.

CMFs for Horizontal Curves and Tangents 

at Type 1 Crest Vertical Curves

CMFs for horizontal curves and tangents at type 1 

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.

Figure 4. Equation. FI CMF for horizontal curves and tangents at 

type 1 crest vertical curves.
4

crest vertical curves (C1) were estimated as follows:

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) must be substituted in figure 4 or figure 5.

CMFs for Horizontal Curves and Tangents 

at Type 1 Sag Vertical Curves

The CMFs for horizontal curves and tangents at type 1 

sag vertical curves (S1) were estimated as follows:

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) must be substituted in figure 6 or figure 7.

CMFs for Horizontal Curves and Tangents 

at Type 2 Crest Vertical Curves

The CMFs for horizontal curves and tangents at type 2 

crest vertical curves (C2) were estimated as follows:

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.

CMFs for Horizontal Curves and Tangents 

at Type 2 Sag Vertical Curves

The CMFs for horizontal curves and tangents at type 2 

sag vertical curves (S2) were estimated as follows:

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.

Application of CMFs

To calculate CMF values using the equations in 

figure 2 through figure 11, the following guidelines 

should be applied:(3)

• Any horizontal R less than 100 ft should be 

treated as equal to 100 ft. This implements 

guidance currently presented in the 

AASHTO HSM.(1)

• If R for a horizontal curve is greater than 

Figure 5. Equation. PDO CMF for horizontal curves and tangents at 

type 1 crest vertical curves.

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. 

Figure 8. Equation. FI CMF for horizontal curves and tangents at 

type 2 crest vertical curves. 

Figure 9. PDO CMF for horizontal curves and tangents at type 2 

crest vertical curves.

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. 
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)

5

or equal to 11,460 ft, CMF applicable to 

tangents (on either level or nonlevel grades, 

as appropriate) should be used rather than 

CMF for a horizontal curve.

• For either a tangent or a horizontal curve, if 

the percent G for a straight grade is between 

-1.0 and +1.0 percent, the CMF applicable to a 

level grade (G = 0) should be used.

• For either a tangent or a horizontal curve, if G1 

and G2 are between -1.0 and +1.0 percent, the 

CMF applicable to a straight grade that is level 

(G = 0) should be used rather than the CMF for 

a vertical curve.

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:

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.

The values used for PFI and PPDO must always sum 

to 1.0. The 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)

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 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.

CMF presented in figure 12 can be considered to 

Figure 12. Equation. CMF for combined crash severity level.
replace the combined effect of CMF3r for horizontal 

curves presented in AASHTO HSM equation 10-13 

and CMF5r for grades presented in AASHTO HSM 

table 10-11.(1) In other words, CMFTOT is a potential 

substitute for the product of CMF3r × CMF5r in 

AASHTO HSM equation 10-2, 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 10-2. 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 CMF4r, 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 

CMF4r should still be used even if CMF3r and CMF5r

are replaced by the new CMFs developed in this 

current study. 

CMF Calculation Tool

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:

1. Instructions: Description of each tab and its 

function.

2. Tgt StraightGrade: Calculation of CMFs for 

tangents on straight grades.

3. HCurve StraightGrade: Calculation of CMFs for 

horizontal curves on straight grades.

4. Tgt VCurve: Calculation of CMFs for tangents at 

vertical curves.

5. CurvesHCurve VCurve: Calculation of CMFs 

for horizontal curves at vertical curves.

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:

1. Open the Microsoft Excel® calculation tool 

workbook.

2. Click on the “Instructions” tab. Instructions on 

how to use the calculation tool and how to input 

data are provided and are detailed in steps 3–5.

6

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)
7

3. Select the appropriate CMF calculation worksheet 

from the four available worksheets (2–5) by clicking on the tabs at the bottom of the screen display. 

The worksheet selected should be appropriate for 

the specific combination of horizontal and vertical alignment for which a CMF is to be calculated. 

The alignment combinations include the following:

• Tangents on straight grades.

• Horizontal curves on straight grades.

• Tangents at vertical curves.

• Horizontal curves at vertical curves.

4. Enter the applicable input data describing the 

horizontal and vertical alignment in the data input 

table. The text immediately above the data input 

table on each worksheet gives guidance on typical 

ranges of input values. Default values based on 

AASHTO HSM Chapter 10 are provided for 

PFI and PPDO, which must sum to 1.0.(1) Users 

may substitute local values for the PFI and 

PPDO defaults.

5. Click “Run.” The computed CMF values, along 

with a summary of the input data, will appear on a 

new row added at the bottom of the results table. 

Users may choose to display multiple rows in the 

results table. Click “Reset” to refresh the results 

table by deleting all displayed rows.

Figure 15 shows a typical sheet from the Microsoft 

Figure 15. Screenshot. Sample screen for horizontal curve at type 1 crest CMF calculations.
The Highway Safety Information System (HSIS) is a safety database that contains crash, roadway inventory, and traffic 

volume data for a select group of States and cities. The participating States of California, Illinois, Maine, Minnesota, 

North Carolina, Ohio, and Washington and the city of Charlotte were selected based on the quality of their data, the range of 

data available, and their ability to merge the data from various files. The HSIS database also contains historic data from 

Michigan and Utah. The HSIS is issued by FHWA staff, contractors, university researchers, and others to study current 

highway safety issues, direct research efforts, and evaluate the effectiveness of crash countermeasures.

JANUARY 2014

FHWA-HRT-13-078

HRDS-20/12-13(750)E McLean, VA 22101-2298 6300 Georgetown Pike Turner-Fairbank Highway Research Center Research, Development, and Technology www.HSISinfo.org 

Excel® calculation tool with both input data and 

computed results displayed.(5)

References

1. American Association of State Highway and Transportation Officials. (2010). 

Highway Safety Manual, 1st Edition, AASHTO, Washington, DC.

2. American Association of State Highway and Transportation Officials. (2011). A 

Policy on Geometric Design of Highways and Streets, AASHTO, Washington, DC.

3. Bauer, K.M. and Harwood, D.W. (2013). Safety Effects of Horizontal Curve and 

Grade Combinations on Rural Two-Lane Highways, Report No. FHWA-HRT-13-077, 

Federal Highway Administration, Washington, DC.

4. SAS® Institute Inc. (2011). SAS® 9.3 User’s Guide, SAS Institute Inc., Cary, NC. 

5. Federal Highway Administration. Highway Safety Information System, U.S. 

Department of Transportation, Washington, DC. Obtained from: www.hsisinfo.org.

For More Information

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 Two-Lane 

Highways, is published as Report No. FHWA-HRT-13-077.(3)

For more information about HSIS, contact Carol Tan, Ph.D., Federal Highway 

Administration HSIS Program Manager, HRDS, 202-493-3315, carol.tan@dot.gov.