TECHBRIEF Safety Evaluation of 

Horizontal Curve 

Realignment on Rural 

Two-Lane Roads

FHWA Publication No.: FHWA-HRT-17-067

FHWA Contact: Roya Amjadi, HRDS-20, (202) 493-3383, 

roya.amjadi@fhwa.dot.gov

This document is a technical summary of the Federal 

Highway Administration report Safety Evaluation of 

Horizontal Curve Realignment on Rural Two-Lane Roads

(FHWA-HRT-17-066).

Objective

The Federal Highway Administration (FHWA) established 

the Development of Crash Modification Factors (DCMF) 

Program in 2012 to address highway safety research 

needs for evaluating new and innovative safety strategies (improvements) by developing reliable quantitative 

estimates of their effectiveness in reducing crashes. The 

ultimate goal of the DCMF Program is to save lives by 

identifying new safety strategies that effectively reduce 

crashes and promote those strategies for nationwide 

implementation by providing measures of their safety 

effectiveness and benefit. This study evaluated realignment of horizontal curves on rural two-lane roads based 

on a before–after evaluation. This realignment strategy 

is intended to reduce lane departure crashes, especially 

run-off-road crashes. This evaluation developed crash 

modification factors (CMFs) for this strategy using stateof-the-art before–after empirical Bayes (EB) methods and 

then compared the results with previously developed 

CMFs from cross-sectional studies. The data included 39 

realigned locations from California, North Carolina, and 

Ohio, and 56 untreated reference curves.

Typically, researchers have based studies on the effect 

of horizontal curve radius on a cross-sectional analysis, 

where they compare the safety performance of curves 

Research, Development, and 

Technology

Turner-Fairbank Highway 

Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

www.fhwa.dot.gov/research
2

of differing radii. This methodology typically allows for a large potential sample of 

curves because there is no need to know a 

conversion/installation date as in a before–

after study. However, a before–after methodology is generally preferable to crosssectional for evaluating the safety effect of 

an infrastructure modification.(1)

Introduction

Several studies have produced CMFs and 

crash modification functions (CMFunctions) 

for curve radii on rural two-lane roads.

The Highway Safety Manual (HSM)’s chapter for rural two-lane roads includes a CMF 

for horizontal curvature.(2) The HSM based 

this CMF on regression analysis done by 

Zegeer et al.(3) The base condition of the 

HSM predictive model for rural two-lane 

roads is a tangent, and the CMF for a curve 

is calculated as shown in figure 1.

Where:

L = Length of horizontal curve in mi, 

including spiral transitions.

R = Radius of curve in ft.

S = 1 if spiral transition present; 0 if not; 0.5 

if present only on one end of curve.

A study by Bauer and Harwood focused 

on the interaction of horizontal curvature and vertical curvature and grade for 

rural two-lane highways.(4) The authors 

conducted a cross-sectional analysis of 

roadway segments from Washington 

State. They used 3,457 mi of road and 

6 years of crash data to produce a series of 

CMFunctions to quantify the safety performance of various combinations of vertical 

and horizontal alignment. Their study classified horizontal alignment of roadway segments as follows:

• Tangent.

• Horizontal curve.

Their study classified vertical alignment of 

roadway segments as follows:

• Level.

• Straight grade (constant percent grade 

of 1 percent or more).

• Type 1 or 2 crest vertical curve.

• Type 1 or 2 sag vertical curve.

Figure 2 shows one of the CMFunctions 

produced by Bauer and Harwood. The 

function calculates a CMF for fatal and 

injury crashes for a horizontal curve 

that occurs on a straight grade (i.e., no vertical curvature). The study produced other 

CMFunctions for other conditions (e.g., horizontal curve on a type 1 vertical crest).

Where:

G = Absolute value of percent grade 

(0 percent for level tangents, 

≥1 percent otherwise).

R = Curve radius (ft) (missing for tangents).

Lc = Horizontal curve length (mi) (n/a for 

tangents).

Figure 1. Equation. CMF for horizontal curvature on rural 

two-lane roads from the HSM.

Figure 2. Equation. Bauer and Harwood CMFunction for 

safety effect of horizontal curve on straight grade.(4)
3

Saleem and Persaud built on the work for 

two-lane highways to explore the development of a CMFunction where they explored 

designs to flatten an existing horizontal 

curve.(5) They focused on horizontal curves 

only on level grade (less than 3 percent). 

They used 440 curves from Washington 

State with radii between 100 and 11,000 

ft to conduct a cross-sectional analysis 

and develop CMFunctions. Their functions showed that CMFs decreased (fewer 

crashes occurred) for scenarios in which a 

large radius was entertained. Furthermore, 

they found that the level of annual average 

daily traffic (AADT) on the subject segment 

did not influence the effect.

Previous studies have generally shown that 

a high curve radius is associated with few 

crashes. However, all published studies 

that have examined the safety effect of 

horizontal curvature were based on crosssectional studies, which may not always 

have provided reliable CMFs. This illustrates 

the need for before–after studies, even if 

researchers must base those studies on a 

small sample to verify the results the crosssectional studies have obtained.

Data and Methodology

The evaluation used data from 

California, North Carolina, and Ohio. The 

evaluation included the following crash 

types:

• Total crashes.

• Injury and fatal crashes.

• Run-off-road and fixed object crashes.

• Crashes during dark conditions.

• Wet crashes.

The evaluation included 39 realigned 

curves and 56 reference curves that were 

not realigned. Table 1 shows the summary 

statistics for the realigned (treatment) and 

reference sites.

The evaluation applied the EB methodology for observational before–after 

studies.(6) The project team estimated safety 

performance functions (SPFs) used in the 

EB methodology through generalized linear 

modeling assuming a negative binomial 

error distribution, which is consistent with 

the state of research in developing these 

models. In specifying a negative binomial 

error structure, the project team estimated 

an over-dispersion parameter that was used 

in the EB methodology.

The full report includes a detailed explanation of the methodology and development 

of SPFs, including a description of how estimates of safety effects for each crash type 

was calculated. 

Results

Table 2 provides the estimates of expected 

crashes in the after period without treatment, the observed crashes in the after 

period, and the estimated CMF and its 

standard error for all crash types considered. Although all crash categories had 

low sample sizes of crashes, the project 

team did not analyze head-on and sideswipe crashes because of the low number 

of crashes of these types in the study sites. 

The project team based the CMFs reported 

in table 2 on all 39 realigned curve locations. The project team also estimated the 

CMFs after removing the three locations the 

States realigned to straight segments, and 

the resulting CMFs were almost identical 

to the ones reported in table 2. The results 

indicate large reductions that were statistically significant at the 95-percent confidence level for all crash types analyzed.
4

Results pertained to a range of site characteristics, the most important of which 

was the range of before–after curve radii. 

Average degrees of curve in the before–

after periods were 18.1 and 6.9, respectively 

(the degree of curve is mentioned here 

instead of radius because three curves were 

realigned to straight segments, which had 

undefined radius values but had a degree of 

0.0). The average central angle of the curves 

was approximately 42 degrees, the average 

AADT at these sites was about 3,500, and 

*Statistically significant at the 95-percent confidence level.

Metric Total Injury and 

Fatal

Run-Off-Road 

plus Fixed Object Dark Wet-Road

EB estimate of crashes expected 

in after period without strategy 91.43 38.11 50.29 21.90 33.88

Count of crashes observed 

in after period 29 10 11 13 7

Estimate of CMF 0.315* 0.259* 0.216* 0.584* 0.204*

Standard error of estimate of CMF 0.064 0.086 0.068 0.176 0.079

Table 2. Crash modification factors.

Variables Treatment Sites 

Before

Treatment Sites 

After Reference Sites

Number of sites 39 39 56

Years of data 220 130 610

Mi-years 33.2 18.8 55.8

Total crashes per year 198 29 214

Injury and fatal crashes per year 109 10 86

Run-off-road plus fixed object crashes per year 132 11 104

Head on crashes per year 10 0 5

Sideswipe crashes per year 11 2 17

Dark crashes per year 61 13 53

Wet crashes per year 88 7 78

Table 1. Summary statistics.
5

the average length of the segments that 

States realigned was 0.1455 mi.

One intent of this evaluation was to compare results of before–after evaluations 

with published results from cross-sectional 

studies. The project team accomplished this 

using two different approaches. The first 

approach involved entering the average 

radii in the before–after periods along with 

the average segment length of the study 

sites to estimate CMFs based on the equations given in Zegeer et al. (for total crashes) 

and Bauer and Harwood (for injury and fatal 

crashes).(3,4) Based on the equation given 

in Zegeer et al.—and assuming no spiral 

transition curves—the ratio of the CMF for 

a 6.9-degree curve to an 18.1-degree curve 

was 0.672 for total crashes.(3) For the same 

conditions, the CMFunction in Bauer and 

Harwood estimated a CMF of 0.784 for 

injury and fatal crashes.(4)

The first approach was simplistic in that 

it did not specifically account for the fact 

that the relationship between CMF and 

curve radius and segment length was nonlinear in both Zegeer et al. and Bauer and 

Harwood.(3,4) To address this limitation, the 

project team identified a second approach. 

The second approach involved the following steps:

1. For each treated site, calculate CMF 

based on Zegeer et al. and Bauer and 

Harwood.(3,4)

2. Multiply EB expected crashes in the 

after period for each site (expected 

crashes had the treatment not been 

implemented) with CMFs from Zegeer 

et al. and Bauer and Harwood.(3,4) 

This product gives the expected crashes 

with the treatment for each site based on 

CMFs from these two studies.

3. Calculate the sum of expected crashes 

with the treatment for all sites based on 

the two studies.

4. The ratio of EB expected crashes with 

the treatment to EB expected crashes 

without the treatment provides an 

approximate CMF associated with the 

equations from the two studies.

Based on the second approach, the CMF 

from Zegeer et al. was 0.699 (for total 

crashes), and the CMF from the Bauer and 

Harwood was 0.736 (for injury and fatal 

crashes).(3,4)

CMFs from the previous studies were 

higher than CMFs of 0.315 and 0.259 that 

were estimated using the before–after 

EB evaluation for total and injury and 

fatal crashes, respectively. This difference 

was consistent with previous studies of 

other treatments that seemed to indicate 

cross-sectional studies can sometimes 

underestimate the safety benefit of a 

treatment (i.e., overestimate the CMF). 

However, the project team based this 

before–after evaluation on a limited sample, 

and readers are encouraged to apply the 

results with prudence.

CMFunctions were estimated using results 

of the evaluation, and the full report details 

the approach. The CMFunctions indicated 

the CMF increased as AADT increased but 

decreased as the difference between the 

radii of the two curves increased and as the 

central angle increased.

CMFunctions are not presented here 

because the project team based them on 

a small site sample. However, the general 

indications of the CMFunction may be useful in a contemplated curve realignment—in 

particular, that safety benefits may increase 

for curves with a large central angle and 
6

where the difference in radius between the 

before–after period conditions is larger. This 

is consistent with previous cross-sectional 

studies (e.g., Zegeer et al. and Bauer and 

Harwood), which indicated safety benefits 

due to realignment would increase with a 

large increase in radius from the before–

after periods.(3,4)

Economic Analysis

The project team estimated the expected 

benefit due to the realignment as 1.495 

injury and fatal crashes per mi-year and 

1.825 property damage only (PDO) crashes 

per mi-year. For the benefit calculations, the 

project team used the most recent FHWA 

mean comprehensive crash costs disaggregated by crash severity and location 

type as a base. They developed these costs 

based on 2001 crash costs, and unit costs 

(in 2001 U.S. dollars) for injury and fatal 

crashes and PDO crashes in rural areas 

were $206,015 and $7,800, respectively.(7) 

The project team then updated this 

to 2015 dollars by applying the ratio of 

the U.S. Department of Transportation 

(USDOT) 2015 value of a statistical 

life of $9.4 million to the 2001 value of 

$3.8 million.(8) Applying this factor of 

2.47 to the unit costs resulted in aggregate 2015 unit costs of $510,446 for 

injury and fatal crashes and $19,326 for 

PDO crashes. The expected annual 

benefit due to a reduced crash rate after 

realignment was $798,507. Based on 

the suggestions from USDOT, the project team conducted sensitivity analyses 

to obtain a low and high value for the 

benefits and, consequently, a low and high 

value for the benefit–cost (B/C) ratios.

The project team estimated the annualized 

cost of the treatment using the equation 

shown in Figure 3.

Where:

C = Treatment cost; based on information 

from North Carolina, average realignment cost per mi was $3,121,599.

R = Discount rate (as a decimal) and 

assumed to be 0.07.

N = Expected service life (years) of 

30 years.

The annualized cost per year for realignment was $251,558. The project team calculated a B/C ratio of 3.17:1 as the ratio 

of the annual crash savings to the annualized treatment cost. The sensitivity analysis 

yielded a B/C range from 1.75:1 to 4.38:1.

Summary and Conclusions

Table 3 presents recommended CMFs for 

curve realignment on rural two-lane roads. 

Results pertained to a range of site characteristics, the most important of which was 

the range of before–after degrees of curve. 

Average degrees of curve in the before–

after periods were 18.1 (minimum of 3.2 

and maximum of 52.1) and 6.9 (minimum 

of 0.0 and maximum of 16.3), respectively. 

Average central angle of the curves was 

approximately 42 degrees (minimum of 

1 and maximum of 117). Average AADT 

at the treated sites was about 3,500 (minimum of 465 and maximum of 11,917). 

Average length of the realigned segments 

was 0.15 mi (minimum of 0.03 and maximum of 0.60).

Figure 3. Equation. Determination of annual cost.
7

Estimated CMFunctions showed that 

safety benefits may be greater for curves 

with a larger central angle and where the 

difference in radius between the before–

after period conditions is larger. In addition 

to developing CMFs using before–after 

evaluation, this study also compared total 

crash CMF with results from previous crosssectional studies. CMFs from this before–

after evaluation were lower than those 

estimated by two previous cross-sectional 

studies. As part of this effort, the North 

Carolina Department of Transportation 

conducted an analysis of crashes attributed 

to the curves in the study based on milepost crash reports. Their analysis revealed 

that relying solely on identifying crashes 

using the mileposts could potentially lead 

to significant undercounting. However, for 

this study, the effect of this undercounting 

on the CMF result was not significant. 

Further research is needed to understand 

this issue. The economic analysis yielded 

a B/C ratio of 3.17:1 with a range from 

1.75:1 to 4.38:1. There is a need for further 

research with a larger sample of sites to 

assess the reliability of CMFs obtained 

from this before–after evaluation.

References

1. Gross, F., Persaud, B., and Lyon, C. 

(2010). A Guide to Developing Quality 

Crash Modification Factors, Report 

FWHA-SA-10-032, Federal Highway 

Administration, Washington, DC.

2. AASHTO. (2010). Highway Safety 

Manual, First Edition, American Association of State Highway and Transportation Officials, Washington DC.

3. Zegeer, C., Stewart, J.R., Council, 

F.M., Reinfurt, D.W., and Hamilton, E. 

(1992). “Safety Effects of Geometric 

Improvements on Horizontal Curves,” 

Transportation Research Record, 

1356, Transportation Research Board, 

Washington, DC.

4. Bauer, K. M. and Harwood, D. W. (2014). 

Safety Effects of Horizontal Curve and 

Grade Combinations on Rural Two-Lane 

Highways, Report FHWA-HRT-13-077, 

Federal Highway Administration, 

Washington, DC.

5. Saleem, T. and Persaud, B. (2016). 

“Another Look at the Safety Effects 

of Horizontal Curvature on Rural 

Two-Lane Highways,” Transportation 

Research Board Annual Meeting Paper 

Compendium, Paper No. 16-6347, 

Transportation Research Board, 

Washington, DC. 

6. Hauer, E. (1997). Observational BeforeAfter Studies in Road Safety, Pergamon 

Press, Oxford, UK.

7. Council, F., Zaloshnja, E., Miller, T., 

and Persaud, B. (2005). Crash Cost 

*Statistically significant at the 95-percent confidence level.

Metric Total Injury and Fatal Run-off-Road plus 

Fixed Object Dark Wet-Road

Estimate of CMF 0.315* 0.259* 0.216* 0.584* 0.204*

Standard error of estimate of CMF 0.064 0.086 0.068 0.176 0.079

Table 3. Recommended CMFs.
8

NOVEMBER 2017 FHWA-HRT-17-067

HRDS-20/11-17(200)E

Researchers—Raghavan Srinivasan, Daniel Carter, Craig Lyon, and Matt Albee.

Distribution—This TechBrief is being distributed according to a standard 

distribution. Direct distribution is being made to the Divisions and Resource Center.

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Center by e-mail to report.center@dot.gov, fax to (814) 239-2156, phone to (814) 

239-1160, or online at http://www.fhwa.dot.gov/research.

Key Words—Horizontal curve, realignment, radius, degree of curve, safety 

improvements, safety evaluations, empirical Bayesian, run-off-road, crash 

modification functions.

Notice—This document is disseminated under the sponsorship of the U.S. 

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to ensure continuous quality improvement.

Estimates by Maximum Police-Reported 

Injury Severity Within Selected Crash 

Geometries, FHWA-HRT-05-051, Federal 

Highway Administration, Washington, 

DC.

8. U.S. Department of Transportation. 

(2015). Guidance on Treatment of the 

Value of a Statistical Life (VSL) in 

U.S. Department of Transportation 

Analyses—2015 Adjustment, Memorandum, Washington, DC. Available 

online: https://cms.dot.gov/sites/dot.

gov/files/docs/VSL2015_0.pdf, last 

accessed May 10, 2016.