50 ■ Transportation Research Record 1737Paper No. 00-1627

Prediction and estimation of speeds on two-lane rural highways are ofenormous significance to planners and designers. Estimation of speedson curves may be easier than prediction of speeds on tangent sectionsbecause of the strong correlation of speeds with a few defined and limit-ing variables, such as curvature, superelevation, and the side-frictioncoefficients between road surface and tires. On tangent sections, however,the speed of vehicles is dependent on a wide array of roadway character-istics, such as the length of the tangent section, the radius of the curvebefore and after the section, cross-section elements, vertical alignment,general terrain, and available sight distance. Few studies have dealt withthis issue because a considerable database is necessary to identify anysignificant trends and substantial modeling effort is required. Researchanalyzed the variability of the operating speeds on 162 tangent sections oftwo-lane rural highways, and models were developed for prediction ofoperating speed based on the geometric characteristics available. A one-model approach was used initially. Because of the low R2 values, a familyof models was developed that better predicted operating speeds.

Speed on two-lane rural highways is strongly associated with the qual-ity of flow and level of service on the roadway (1). In addition, thevariability of speeds reduces driver comfort and causes crash rates toincrease (2, 3). For safety and operational reasons, it is almost alwaysnecessary to know the compatibility among the three functionalspeeds of geometric design and traffic engineering: design speed,operating speed, and posted speed. The prediction and estimationof speeds on two-lane rural highways, therefore, are of enormoussignificance to planners and designers.

Prediction of speeds on horizontal curves may be easier than pre-diction of speeds on tangent sections because of the correlation ofspeeds to a few defined and limiting variables, such as curvature,superelevation, and the side-friction coefficients between road sur-face and tires. Several studies found that on curves, the radius (ordegree of curve) is the most significant variable for predicting speedsat the midpoint of the curve (4–6). Generally, a change in these vari-ables is controlled by designers during the planning and designprocess; therefore, operating speeds may be predicted and evaluated.It is possible, then, to determine the speed limit as a selected per-centage of the distribution of the operating speed, for example, the85th percentile.

On tangent sections of two-lane rural highways, however, thespeed of vehicles is dependent on a wide array of roadway charac-teristics, such as the length of the tangent section, the radius of the

Predicting Operating Speeds on TangentSections of Two-Lane Rural Highways

Abishai Polus, Kay Fitzpatrick, and Daniel B. Fambro*

horizontal curve before and after the section, cross-sectional ele-ments, vertical alignment, general terrain, and available sight dis-tance. Additionally, the operating speed on tangents largely dependson driver attitude, as well as on the acceleration and decelerationcapabilities of the vehicle composition on a specific road. Therefore,it is a more complex task to predict operating speeds on tangent sec-tions; a large database is necessary to identify any significant trends,and substantial modeling effort is required. Few studies have dealtwith this issue.

BACKGROUND

Although the issue of speeds on tangent sections has not been studied extensively, several U.S. and international researchershave studied operating speeds and average running speeds, designspeeds, speed limits, and the impact of geometric and traffic mea-sures on speeds. Van Winkle (7 ) in the United States discusseddesign speed and operating speed and concluded that the formershould remain consistent with posted speed, established by mea-suring driver behavior. Cielecki and Wieckowski (8) in Polandconducted an extensive study of factors that affect vehicle speedson two-lane rural roads. They concluded that the factors of highestimportance were traffic volume, road curvature, and intersectiondensity. In their study, hilliness was found to be strongly correlatedwith curvature, and, therefore, it was excluded from the analyses.

Polus et al. (9) in Israel analyzed the effect of traffic and geomet-ric measures on average highway running speed. They found that theaverage curvature, average hilliness, and traffic volume each had amoderate negative correlation with the average running speed. Theeffect of road width on speeds was found to be insignificant; how-ever, a strong correlation was found between vertical and horizontalalignment, except on level terrain with many constraints that requireshigher curvature. According to this study, driver speeds behaveddifferently in the low-flow range (below 200 vph) and in the higher-flow range, where volume had some impact on the measured averagerunning speeds.

Gattis and Duncan (10) in Arkansas emphasized the importance ofsight distance in their study. They suggested that ample preview sightdistance is needed for safe and comfortable operation. Preliminarymeasurements showed that the required preview time should be in theorder of 1.3 to 1.7 s. Krammes and Glascock in their work on designconsistency and crash experience on two-lane rural highways (11)also identified the importance of sight distance in allowing drivers tohave sufficient time to increase their attention levels and performnecessary control functions.

Tarris et al. (5) predicted the operating speeds on low-volume urbancollector streets in Pennsylvania. A model for predicting the 85th per-

*Deceased. A. Polus, Department of Civil Engineering and TransportationResearch Institute, Technion–Israel Institute of Technology, Haifa 3200, Israel.K. Fitzpatrick, Texas Transportation Institute, Texas A&M University, College Sta-tion, TX 77843-3135. D. B. Fambro, Department of Civil Engineering, TexasA&M University, College Station, TX 77843-3135.

Polus et al. Paper No. 00 -1627 51

centile speed based on the desired speed and curvature (1/R) was pre-sented. Earlier, Roszell and Braaksma (12) in Canada studied thecauses of variation in car speeds along a parkway and concluded thatamong other factors (such as age and sex of driver), route geome-try, particularly horizontal geometry, had a significant effect on thelocation of speed changes.

A Texas study on design speed, operating speed, and posted speedconducted by Fitzpatrick et al. (13) found that the posted speed limit,based on the 85th percentile speed, exceeded the roadway designspeed in some locations because the calculated design speed incor-porates significant safety margin because of the use of lower frictioncoefficients than actual values. Posted speed and cross section werefound to influence factors in driver selection of operating speeds.

Fink studied the effect of tangent length and sight distance on crashrates at horizontal curves on two-lane rural highways (14). The mostsignificant variable in predicting crash rates was the degree of curve.Tangent length and sight distance had no practical significance.

Parker examined the effect of raising and lowering speed limits(15). He found the change to be insignificant; driver speeds changedslightly. Driver violations of the speed limit increased, however, whenthe speed limit was lowered. Another study investigated whetherroadway elements, roadside elements, or selected traffic controldevices influence operating speed on suburban arterials (16 ). Theposted speed limit was the best factor for explaining the variability inoperating speed. It explained more than 50 percent of the variability.

OBJECTIVE

The objective of this research was to analyze the variability of oper-ating speeds on tangent sections of two-lane rural highways and todevelop models for its prediction based on the geometric character-istics of a planned (or existing) road. Numerous speed observationsare needed for various tangent sections of two-lane rural roads in dif-ferent geographic regions to develop the models. The models shoulduse variables readily available to the designer. Data were availablefrom a recent FHWA study to allow exploration into predicting speedon tangent sections (6).

DATA

Collection

Data were collected in six states: Minnesota, New York, Pennsyl-vania, Oregon, Washington, and Texas. The general criteria used toidentify sites resulted in a database that represents the most commonconditions found in the United States. For example, the databaseincluded roads with few access points (driveways and intersections),posted speeds between 75 and 115 km/h, and lane widths between 2.7and 3.7 m. Lower-volume facilities, those with fewer than 2,000 vehi-cles per day, were preferred to reduce the potential for restricted vehi-cle flow. The sites selected could not be close to towns or developedareas that could significantly affect the speed patterns on the curves.

The speed data were collected between July 1996 and January 1997during daylight, off-peak periods, and under dry weather conditions.At least 100 observations were recorded at each site. The speed datawere collected by using radar meters and on-pavement piezoelectricsensors connected to traffic counter-classifiers. Speeds were mea-sured on the curve and on the preceding tangent. Those vehicles thatwere not at free-flow speed (determined as having less than a 5-s head-

way between vehicles) were eliminated. After removal of non-free-flow vehicles, speeds for passenger cars, trucks, and recreationalvehicles were separated into three categories on the basis of thevehicle axle spacings. All analyses documented in this paper wereon passenger cars because of the limited data available for trucksand recreational vehicles.

Reduction

The geometric data from the data collection sheets were summarizedinto spreadsheet format. Descriptive statistics of the speed data werecalculated by using SAS software. These statistics included meanspeed, 85th percentile speed, and standard deviation. The descriptivestatistics were merged with the geometric data to create a master file.Variables available included inferred design speed; horizontal curveradius, deflection angle, and length; grade; tangent length; preced-ing curve radius and length; vertical curve characteristics; lane andpavement width; superelevation; terrain type; and posted speed.

It was necessary to assume that all sections at all sites were de-signed according to the design-speed concept as proposed and dis-cussed in the AASHTO guidelines (17). Consequently, all radii in thedata set were assumed to be equal to or larger than the minimumradius for the (unknown) section design speeds. This assumption isnecessary for the later development of general prediction models andelimination of the possibility of predicting for extreme conditions,such as inappropriate small radii.

Correlation Between Variables

Correlation may exist between certain geometric variables associatedwith a two-lane highway. For example, design speed and radius arestrongly correlated, and therefore they should not appear together inone model. Clearly, it is necessary to identify the most importantindependent variables for use in a model. Sometimes, however, vari-ables that intuitively appear to be correlated are actually not corre-lated at all. One such example is the preceding horizontal radius (R1)and the following horizontal radius (R2). It may be expected that adesigner would select a similar radius before and after a tangent sec-tion. However, when these two variables were plotted (see Figure 1),no trend was identified, and consequently the variables were notfound to be correlated (R2 = 0.0016). Therefore, it was necessary todevelop a geometric measure that uses both these radii as an inputbecause of their strong contribution to operating speeds on a tangentsection bounded by two curves.

MODEL FORMULATION

Desirably, a single model would be developed to predict speed on tan-gent sections of rural two-lane highways. This model would includeprimary variables and secondary variables that notably improvedthe model predicting capabilities. To identify the appropriate formof model, the primary and secondary variables expected to be rele-vant were examined, and preliminary forms of the model were inves-tigated. For this study, a one-model approach was initially used.Because of the low R2 values, another approach was used to determineif a family of models could be developed that would better predictoperating speeds. These approaches were termed geometry measuremodel and group models.

52 Paper No. 00-1627 Transportation Research Record 1737

Primary Variables

It is easier to estimate speed on curves than on tangents because theformer depend largely on a limited number of parameters, such asthe curvature, the superelevation, and the side-friction coefficient.The number of variables that influence speed on tangents is consid-erably larger. Perhaps the most important variable is the tangentlength, because of its direct impact on speed: on short tangents,speed cannot be fully developed; as tangent length increases, speedalso may increase. Other relevant variables include the speed limitand enforcement level, curvature before and after the tangent, vehi-cle acceleration and deceleration characteristics, grade of the roador the general terrain, the width of the roadway, the side slopes, andthe existence of spirals on the adjacent curves.

Still another significant variable that may affect speed on tangentsections reflects the characteristics of the entire section: both pre-ceding (R1) and succeeding (R2) radii and the length of the tangentsection between. In situations in which the length exceeds a giventhreshold (t), above which it is likely that the length of the tangentsection affects the speed of vehicles, it is advantageous to includetangent length in the geometric measure.

Several geometric measures were explored. The form in Equa-tion 1 was selected for sites where the tangent length exceeds agiven threshold:

where

GML = geometric measure of tangent section and attached curvesfor long tangent lengths (m2);

R1, R2 = previous and following curve radii (m);TL = tangent length (m); and

t = selected threshold for tangent length (m).

GM TL R R

TL t

L = × ×( )[ ]≥

1 21 2 100 1( )

for

The form in Equation 2 was selected for short tangent lengths:

where GMS is the geometric measure for short tangent lengths (m).Trends that were identified in the data set showed a sharp initial

increase in 85th percentile speed with an increase in the length ofthe tangent section and with an increase in both geometric measures(GML and GMS). The sharp initial increase transitioned to a lessermarginal increase at higher TL and GM values.

The observed shape of the relationship between the dependentvariable (85th percentile speed for each site in km/h) and the inde-pendent variables (GML and GMS, TL) led to the formulation of sev-eral nonlinear models. This nonlinearity may be reflected by twobasic forms, as shown in Equations 3 and 4. Equation 3 is for whenone of the GM variables is used:

Equation 4 is for use with tangent length:

Another simplified option for the model in Equation 3 is given inEquation 5; here it is not needed to estimate the maximum speedfrom the data:

In all models, SP is the 85th percentile speed (km/h), TL is the tan-gent length (m), GMi is as defined in Equation 1 or 2, and a, b, c, d,m, and n are constants. The parameter MS is the maximum “rea-sonable” 85th percentile speed that can be developed by reasonable

SP m n GMi= − ( )5

SP c TL d= × ( ) +ln ( )4

SP MS a e b GMi= − ×( ) ( )3

GM R R

TL t

S = +( )

<

1 2 2 2( )

for

FIGURE 1 Potential correlation between preceding and following horizontal curve radius.

Polus et al. Paper No. 00 -1627 53

drivers without severely compromising a driver’s safety (km/h).This maximum speed can be determined from the acceleration anddeceleration capabilities of a typical vehicle or from observations oftypical maximum speeds on a given tangent length.

Secondary Variables

Some additional but secondary geometric variables may contributeto driver “work load” and speed-choice decisions, although gener-ally these effects are limited in size compared with the effect of theprimary variables. For example, a crest vertical curve within the tan-gent section could induce speed changes. Previous research find-ings, however, showed that this type curve influences speeds onlywhen the available sight distance over the crest is less than approxi-mately 100 m, and even then the impact is minimal (18, 19).

The effect of longitudinal grade on the speed of passenger cars isnot clear. Moderate downgrades could increase speeds, but steepdowngrades or upgrades could impede performance and reducespeeds. The influence of grade was investigated and clear trendscould not be identified because of the variability of the data.

Other secondary variables that influence speeds include the cross-section characteristics; the presence of spirals, which affects theperceived horizontal curvature as well as driver comfort and, there-fore, the speed; the topography, which influences driver desire totravel at a certain assumed “safe” speed; and the overall alignmentcharacteristics of the road, such as average horizontal curvature andaverage slope.

RESULTS

The analysis is based on an examination of the entire data set, exceptfor a few sites for which there were not complete data records. Dur-ing the calibration of the preceding models, it was necessary to deter-

mine the value of the maximum reasonable speed for the two-laneroads under investigation. This was determined not by observing thehighest speeds of individual drivers, but by identifying the highestobserved 85th percentile speeds (for all sites with similar character-istics) and subsequently selecting a reasonable maximum value. Itshould be noted that this is not the maximum speed of an individualdriver; rather, it is a value that results from the distribution of speedsamong sites with similar characteristics.

The highest reasonable speed on short tangents (or small GML) isinfluenced primarily by the controlling geometry of the precedingand succeeding curves. On long tangents (or large GML), however,the speed also is influenced by supplemental factors, such as speedlimit, level of enforcement, and secondary geometric variables, suchas the roadway environment (cross section) and longitudinal slope.

The combination of all these variables makes the prediction of an85th percentile speed on tangents a relatively complex task. For thedevelopment of models in Equation 3, however, it may be possibleto adopt a maximum speed that is higher than the observed 85th per-centile at all sites studied. Alternatively, it is possible to eliminateone or two values (sites) by assuming that they do not represent the“family” of sites because of some unusual conditions. Conse-quently, observation of the data set led to the selection of values of102.5 km/h (for GML ≤ 200) and of 105 km/h (for GML ≤ 1000) forthe maximum reasonable 85th percentile speed.

Geometric Measure Models

The calibration of the model in Equation 3 resulted in two modelsfor the prediction of the 85th percentile speed (Equations 6 and 7).Figure 2 shows the data and the plot of the following predicted best-fit model for when the GML is less than 200:

SP e

GM

GM

L

L= −

≤

×( )102 5 37 34 6

200

0 00668. . ( ).

FIGURE 2 85th percentile speeds for tangents with GML < 200.

54 Paper No. 00-1627 Transportation Research Record 1737

The correlation coefficient (R2) was 0.332. If the threshold for theGML is increased to 1000, then obviously more variability is addedto the model, which is less sensitive to the initial rapid increase atsites for which GML is low. The maximum 85th percentile speed,then, was 105 km/h, and the resulting model assumed the form ofEquation 7:

Figure 3 shows the data and the plot of the model. The correlationcoefficient (R2) was 0.228. In Figure 3, the model was extrapolatedto higher GML values, to show the reasonable fit to that data althoughthe calibration was conducted only to a value of 1000.

Group Models

The relatively low correlation coefficients indicate that one speedprediction model could not be developed for considerably differentsection lengths or different preceding or succeeding radii. How toestablish the threshold value for the tangent length, or the geometricmeasure (GML or GMS) between any two geometry regimes (i.e., fam-ily of sites), while ensuring that the predicted speeds at this boundaryare similar was investigated extensively. The lack of a strong theoret-ical justification for the selection of the threshold led to the adoptionof an empirical approach. Models were fitted for different boundaryvalues between 150 m and 250 m that appeared from visual observa-tions (and previous studies) to provide the distinction between groups.Not much difference existed among several models tested, althoughthe value of TL = 215 m initially provided a slightly better separationbetween the two regimes.

To achieve the highest degree of reliability in predicting the 85thpercentile speed on tangent sections during the planning process, it isnecessary to establish the common characteristics of similar familiesof sites. For example, a long tangent following a large radius requires

SP e

GM

GM

L

L= −

≤

×( )105 00 21 30 7

1000

0 00092. . ( ).

a considerably different design and operational condition than a shorttangent followed (or preceded) by a small radius. Although eachalignment has its unique characteristics, it is necessary to group thedata by some common, easily identifiable features. After analysis ofthe database, the following four groups were suggested:

• Group 1: Small radii and small TL. Both preceding and suc-ceeding horizontal curves have a radius of 250 m or less, and theconnecting tangent is shorter than 150 m.

• Group 2: Small radii and intermediate TL. The radius for eachhorizontal curve is less than or equal to 250 m with a tangent lengthof between 150 m and 1000 m. The initial and final speeds (on thetangent) are then controlled by the small radii; however, vehiclescan accelerate and develop a higher speed at the midpoint of the tan-gent. To be included in this group, the maximum value of the GML

variable must be less than 1500.• Group 3: Intermediate radii and intermediate TL. The radius

for each horizontal curve is greater than 250 m with a tangent lengthof between 150 m and 1000 m. The speed is determined by the con-ditions prevailing on the tangent. To be included in this group, theminimum value of the GM variable is 1500 and the maximum valueis 7500.

• Group 4: Large TL and any reasonable radius. The tangent sec-tion is longer than 1000 m (1 km) and the radius is reasonable (i.e.,does not violate the minimum-radius criterion for the known orassumed design speed of the road). The impact of the radius on themidtangent speed is not significant, because vehicles can accelerateto almost any desirable speed on the tangent. No maximum valuefor GM was established, although values up to 15 000 were obtainedfrom the available data.

The 85th percentile tangent speed on Group 1 sections (smallradii and TL) is determined primarily by the combination of radii,i.e., by R1 and R2. Therefore, after exploring several measures thatincluded the radii but not the tangent section length, and their impacton the speeds, it was decided to use the simplest measure, which is

FIGURE 3 85th percentile speeds for tangents with GML < 1000.

Polus et al. Paper No. 00 -1627 55

the average of the two radii [i.e., GMS as presented in Equation 2].Several models were calibrated with reasonable success, but themodel selected assumed the form of Equation 5 and was calibratedas follows:

This nonlinear model reflects a pronounced increase in speed whenthe average radius is small, compared with a moderate increasewhen the radius is large. Its applicability is for the Group 1 sections.Figure 4 presents the data for the 23 pertinent sites.

It is reasonable to assume that the speed on Group 2 (small radiiand intermediate tangent length) sections is actually determined by acombination of the radius before and after the tangent section and thelength of the section itself. Therefore, the use of the GML measure,defined in Equation 1, appears both warranted and necessary.

In addition to the threshold values for R1, R2, and TL, it was alsosuggested that the maximum value of the geometric measure (GML)for Group 2 sections should be less than some limiting value: GML

max ≤ 1500. Unfortunately, few data points (sites) were available inthis group; nevertheless, two preliminary models were suggested. Thefollowing models, in Equations 9 and 10, present the calibration of themodels in Equations 5 and 3, respectively:

If the maximum 85th percentile speed is established as 105 km/h, then

Figure 5 shows the models in Equation 9 and Equation 10. The lat-ter model seems to provide a better prediction than the former,because the initial speed increase (at small GML values between 250

SP e

R

GML= −

=

×( )105 00 28 107 10

0 742

0 00108

2

. . ( )

.

.

SP GM

R

L= −

=

98 405 3184 9

0 6842

. ( )

.

SP GM

R

S= −

=

101 11 3420 8

0 5532

. ( )

.

and 500) and later smoothing (near GML values of 1500) appear to fitthe data (albeit limited) quite well.

Although some preliminary models were suggested for Group 3(intermediate radii and tangent length), no successful models wereidentified, primarily because of the variability of the data set in thisgroup. Figure 6 shows this variability, as well as the best-fit linearmodel and its equation.

On Group 4 sections (large tangent length and any radius), thespeed may be determined by the combination of horizontal radii(R1 and R2) and tangent length; the use of the geometric measureGML is advised. It should be noted, however, that the impact ofother secondary variables (discussed previously) on unconstrainedsections may be more significant. Such variables may include thecross-section characteristics, the longitudinal slope, and the k-valueof the vertical curve (if one exists). Not enough data on these siteswere available for any meaningful analysis; however, a preliminarymodel assumed the form of Equation 11:

Figure 7 shows the data and the model.

CONCLUSIONS AND RECOMMENDATIONS

This paper presented models and a methodology for estimating theexpected 85th percentile speed on tangent sections of two-lane ruralhighways. The extensive database provided ample geometry infor-mation and knowledge on driver behavior and speed selection. Themodels developed are valid for two-lane rural highways where thevolume is rather low and does not affect speed.

The analyses showed that when determining 85th percentile speedsin the middle of a tangent section, it is necessary to observe a longersection, one that includes the preceding and succeeding curves,because these constitute the primary variables affecting speed. Theinfluence of other, secondary geometric variables was investigatedand found not to affect speed as much as do the primary variables.

SP e

R

GML= −

=

×( )105 00 22 953 11

0 838

0 00012

2

. . ( )

.

.

FIGURE 4 Model for R1 and R2 ≤ 250 m and tangent length ≤ 150 m.

56 Paper No. 00-1627 Transportation Research Record 1737

Several geometric measures characterizing the geometry of the entiresection (the tangent and attached curves) were developed, and the bestmeasure (GML) was adopted for the development of the predictionmodels.

After considerable examination of the 162 sites, the data wereassembled into four groups of similar characteristics: Group 1, smallradii and short tangent length; Group 2, short radii and intermediatetangent length; Group 3, intermediate radii and intermediate tangentlength; and Group 4, any radius and long tangent length. Separateprediction models for the 85th percentile speed were developed for

each of the four groups separately. The models for Group 1 andGroup 2 sections provided a good fit to the data and could be adaptedfor prediction purposes during the planning process for new two-lane highways. The models for Group 3 and Group 4 sections werepreliminary, and they clearly need additional data. Further researchis also suggested on the impact of some secondary variables, suchas the cross-section elements (lane width and roadside characteris-tics); the direction of the curves, especially for sites within Group 1(e.g., was a broken-back design used?); and the longitudinal slopeon the 85th percentile speed on two-lane rural highways.

FIGURE 5 Model for R1 and R2 < 250 m and tangent length between 150 and 1000 m.

FIGURE 6 Model for R1 and R2 > 250 m and tangent length between 150 and 1000 m.

Polus et al. Paper No. 00 -1627 57

ACKNOWLEDGMENTS

The data used in this paper are from a study funded by FHWA. Theauthors recognize other researchers whose efforts and discussionsassisted in this work, especially Mark Wooldridge, Paul Carlson, andJon Collins.

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The opinions, findings, conclusions, or recommendations expressed in this paperare those of the authors and do not necessarily reflect the views of FHWA.

Publication of this paper sponsored by Section on General Design.

FIGURE 7 Model for tangent length > 1000 m.