Establishing design guidelines for compoundhorizontal curves on three-dimensional alignments

Said Easa and Essam Dabbour

Abstract: In current design guides, the minimum radii of compound horizontal curves are based on the design require-ments of simple horizontal curves for each arc on flat terrain. Such a design ignores the effects of compound curvatureand vertical alignment. This paper uses computer simulation software to establish the minimum radius requirements forcompound curves, considering these effects. The actual lateral acceleration experienced by a vehicle negotiating a two-dimensional (2-D) simple curve is recorded as a base scenario to facilitate the analysis of a compound curve on a flatterrain or combined with vertical alignment (three-dimensional (3-D) compound curves). The vertical alignments exam-ined include upgrades, downgrades, crest curves, and sag curves. Mathematical models for minimum radius require-ments were developed for flat and 3-D compound curves. Three types of design vehicles were used. The results showthat an increase in the minimum radius ranging from 5% to 26% is required to compensate for the effects of bothcompound curvature and vertical alignment.

Key words: highway geometric design, compound horizontal alignments, side friction, vehicle simulation, 3-D align-ments.

Résumé : Les guides de conception actuels indiquent que les rayons minimaux des courbes horizontales complexessont basés sur les exigences de conception de courbes horizontales simples pour chaque arc sur un terrain plat. Unetelle conception ignore les effets de l’alignement vertical et d’une courbure composée. Le présent article utilise un logi-ciel de simulation pour établir les exigences minimales de rayon pour les courbes composées, en tenant compte de ceseffets. L’accélération latérale réelle ressentie par un véhicule prenant un virage simple en deux dimensions est enregistréecomme référence afin de faciliter l’analyse d’une courbe composée sur un terrain plat ou combinée à l’alignement vertical(courbes composées tridimensionnelles). Les alignements verticaux étudiés comportent les pentes ascendantes et descen-dantes, les courbes de crête et d’affaissement. Les modèles mathématiques pour les exigences minimales de rayon ontété développés pour des courbes plates et tridimensionnelles composées. Trois types de véhicules de conception ont étéutilisés. Les résultats montrent qu’il faut augmenter le rayon minimal de 5 à 26 % pour compenser les effets à la foisde la courbe composée et de l’alignement vertical.

Mots clés : plan géométrique des autoroutes, alignements horizontaux composés, frottement latéral, simulation de véhi-cules, alignements tridimensionnels.

[Traduit par la Rédaction] Easa and Dabbour 626

Introduction

Compound horizontal curves are two horizontal arcs withtwo different radii in the same direction. These curves areused in different classes of highways and freeway inter-change ramps. Current design guides provide brief guide-lines for the design of compound horizontal curves, where amaximum ratio between the flatter and sharper arc radii isrecommended to be 1.5 by both the American Association ofState Highway and Transportation Officials (AASHTO

2001) and the Transportation Association of Canada (TAC1999). In addition, current design guides deal with vehiclestability on compound curves as two individual simple hori-zontal curves, each with its own design requirements. Thisapproach ignores the effect of compound curvature on theenergy-storing systems of a vehicle such as springs andshocks. Furthermore, if a compound curve overlaps with avertical alignment, current design guides ignore the effect ofthat vertical alignment. Consequently, ignoring both effectsmay underestimate the minimum radius required for botharcs of a compound curve.

Recent geometric design research related to compoundand other types of curves can be found in NCHRP Synthesis299 (Fitzpatrick and Wooldridge 2001). There are two im-portant geometric design aspects related to compound hori-zontal curves: sight distance and vehicle stability. Sightdistance for flat and three-dimensional (3-D) compoundcurves has been addressed by Easa (1993) and Hassan et al.(1996), respectively. Vehicle stability has been addressed inthe design guides for only simple horizontal curves. When avehicle negotiates a horizontal curve, it experiences a cen-trifugal force outward from the centre of the curve. Vehicle

Can. J. Civ. Eng. 32: 615–626 (2005) doi: 10.1139/L05-016 © 2005 NRC Canada

615

Received 7 July 2004. Revision accepted 9 February 2005.Published on the NRC Research Press Web site athttp://cjce.nrc.ca on 9 July 2005.

S. Easa1 and E. Dabbour. Department of Civil Engineering,Ryerson University, 350 Victoria St., Toronto, ON M5B 2K3,Canada.

Written discussion of this article is welcomed and will bereceived by the Editor until 31 December 2005.

1Corresponding author (e-mail: saideasa@acs.ryerson.ca orseasa13@yahoo.com).

stability is achieved by the frictional force and a componentof the vehicle weight. This approach simplifies corneringdynamics by reducing the vehicle into a point mass, withoutany length or width, travelling on a 2-D horizontal align-ment and is referred to as the point-mass (PM) model.

The PM model has been adopted by the North Americandesign guides because of its simplicity. However, the modelhas the following limitations (Hassan et al. 1998; Easa andDabbour 2003):(1) The model does not account for vehicle size or charac-

teristics.(2) The model does not consider distribution of frictional

forces between different vehicle tires.(3) The model does not account for vehicle stability on com-

plex alignments, such as compound and reverse curves.(4) The model ignores the effect of vertical alignment alto-

gether.(5) The model is based on comfort level because of lateral

acceleration and does not account for rollover, whichmay be a potential risk, especially for trucks with highercentres of mass.

If a vehicle travels along a horizontal curve combined (oreven not combined) with a vertical alignment, it may be sub-jected to actual forces that exceed those determined by thePM model, and the vehicle favourable resistive forces mayalso be less than those determined by the North Americandesign guides. Some research work has been conducted inCanada to evaluate vehicle stability on simple horizontalcurves under different conditions. A study in Calgary foundthat while the design values of side friction provided highermargin of safety for passenger cars and pick-up trucksagainst skidding on dry pavement, the margin of safety onwet and icy conditions was questionable (Morrall andTalarico 1995; Talarico and Morrall 1994). The study alsofound that basing horizontal curve guidelines on ballbankangles, the base concept of point-mass model, might not bea conservative approach to highway design because skidmay occur at low speeds before discomfort is reached.

Another experimental study has been conducted using ac-tual five-axle semitrailer as a design vehicle with more con-sideration given to the evaluation of rollover threshold(Garcia et al. 2003). The study found that the vehicle carry-ing less-than-truck-load displayed the highest propensity torollover with some recorded values of lateral accelerationclose to 80% to 90% of the corresponding rollover threshold.The roll performance of tractor–semitrailer combinations hasbeen examined using computer simulation (Blue andKulakowski 1991). The study was dedicated to simple hori-zontal curves to evaluate the effect of transition spiral curveson vehicle stability. It was found that spiral curves resultedin smoother changes in lateral acceleration and roll angleand less need for driver correction when the truck is enteringthe curve.

Revised minimum radius requirements for simple horizon-tal curves on 3-D alignments based on vehicle stability havebeen developed (Easa and Dabbour 2003). It was found thatcombining a vertical alignment with a simple horizontalcurve results in greater lateral acceleration. Therefore, an in-crease in the minimum radius is required to maintain thesame level of lateral acceleration specified by design guide-lines (AASHTO 2001; TAC 1999). The increase ranges from

7% to 19%. Vehicle stability on reverse horizontal curvescombined with vertical alignments was also examined. Itwas found that an increase in the minimum radius of thesharper arc is required to compensate for the effects of bothreverse curvature and vertical alignment (Dabbour et al.2004). The required increase ranges from 3% to 27%.

This paper presents design guidelines for minimum radiusrequirements of compound horizontal curves based on vehi-cle stability for trucks, considering the effects of compoundcurvature and vertical alignment. The following sectionspresent a brief description of the vehicle dynamics model,experimental design, and analysis methodology. The resultsfor minimum radius requirements are then presented, fol-lowed by an application example and conclusions.

Vehicle dynamics model

Vehicle dynamic models roadway analysis and design(VDM RoAD) or simply vehicle dynamic model (VDM) is asimulation software developed at the University of MichiganTransportation Research Institute (Sayers 1999). The VDMuses a sophisticated vehicle dynamics model that replacesthe conventional point-mass model currently used by NorthAmerican design guides. The VDM can accurately simulatea vehicle travelling through a user-defined 3-D alignment,taking into account vehicle characteristics such as suspen-sion system, body roll, pitch, yaw, and lateral weight distri-bution. The software also accounts for different externalforces such as braking, steering, hitching, and load distribu-tion on different tires. The VDM can analyze lateral acceler-ation, directional control, roll stability, and stopping sightdistance. It generates a set of 3-D equations to simulate ve-hicle stability at every time interval selected by the user.These include vehicle equations (in both yaw and rollplanes) and models for hitch, tire, steering system, suspen-sion, and braking. The flexibility of VDM allows for onebody element or unsprung mass and up to six axles orsprung masses.

The simulation programs within VDM solve the variousdifferential equations that define vehicle motion and associ-ated forces and reactions. There are three equations that gov-ern vehicle behaviour in the yaw plan. Summing forcesalong both X and Y directions result in two equations, whilethe third equation can be derived by summing the momentsabout the vehicle centre of mass. The yaw behaviour is alsoaffected by the rotary motion of the vehicle body in roll andpitch. Vehicle stability in the roll plane may be determinedby balancing the moments about the centre of mass of thevehicle. When tires lift off the ground, the forces are zero onone side of the axle. When lift-off has occurred for morethan a critical number of axles, roll stability cannot beachieved and the vehicle will potentially rollover. This con-dition is considered to be the onset of rollover. Thus, the ba-sic factors for predicting stability against rollover are thevertical tire forces and their points of application relative tothe centre of mass of the vehicle, which changes as the vehi-cle rolls.

Human factors can also be modelled to some extent inVDM. This is represented by an input variable for the lane-change manoeuvre (single-lane or multiple). However, VDMwas not designed to handle more complex human factors,

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616 Can. J. Civ. Eng. Vol. 32, 2005

such as braking as a reaction to increasing lateral accelera-tion when it exceeds driver comfort limit or steering as a re-action to increasing roll angle. In addition, aerodynamicforces are not considered in truck models of VDM. TheVDM has been validated by comparing the VDM predictedresults and the actual lateral acceleration (and other mea-sures of vehicle dynamics) for a vehicle negotiating a spe-cific alignment (Sayers 1999).

Experimental design

The input parameters for the model include vehicle char-acteristics, highway geometry, lateral offset, braking, andspeed control. Hence, many parameters should be selected,including test alignment configurations, design vehicle,cargo load distribution, and other considerations. This sec-tion provides a brief description of the criteria used to selecteach parameter. A full description for the criteria used forthe selection of different parameters has been presented in aprevious research paper (Easa and Dabbour 2003). A seriesof alignments were produced in a 2-D mode as a basic con-trol group based on Canadian design guide (TAC 1999).Superelevation runoff and tangent runout were applied forthis control group and all other test alignments. Deflectionangles in all cases were selected as 45°. A 45-degree deflec-tion angle resulted in a sufficiently long curve for maximumsuperelevation to be achieved. Maximum superelevationrates of 0.04 and 0.06 were selected for evaluation.

The maximum vertical grades of +6% and –6% were se-lected since this grade was the controlling limit used on Ca-nadian rural highways (TAC 1999). The minimum lengths ofvertical curves were selected as to satisfy the maximum rate-of-change of grade, r. The inverse of r is denoted by K,which is the curve length corresponding to a change in gradeof 1%. For crest vertical curves, K was selected to satisfy theminimum stopping sight distance, whereas headlight controlwas the limiting factor for sag curves. It was found that themost conservative results were obtained when the verticalcurve length extended along the entire test segment.

The VDM has a comprehensive vehicle library. Hence,three different design vehicles were selected from the libraryfor use in the experimental work. The first vehicle, WB-15,represents the moderate truck on North American roads. Thesecond vehicle, WB-20, is the largest design truck inAASHTO. The third design vehicle, 3a/2a, is real heavytruck combination on North American roads whose proper-ties are measured in the field to support research at the Na-tional Highway Traffic Safety Administration (NHTSA). Aspart of the experimental design, single and multiple lane-change manoeuvres were examined, along with the tradi-tional case of constant lateral offset. It was found that themost conservative results corresponded to multiple lane-change manoeuvres, where the vehicle inertia resulted inmore lateral shift of the vehicle.

Analysis methodology

Geometric characteristicsA compound curve consists of two circular arcs (with dif-

ferent radii) in the same direction. Let RATIO be defined as

the ratio between the radii of the flatter arc (R2) and thesharper arc (R1). That is

[1] RATIO = R2/R1

As mentioned earlier, the guidelines for compound curve de-sign in the TAC and AASHTO design guides are brief. Theguides advise the designer to use caution when using com-pound curves. The guides recommend that RATIO shouldnot be more than 1.5. The TAC design guide recommends aratio of 1.25 to be more desirable on high-speed roads wherethe speeds are at or near the maximum for the curvature. Noparticular reason or reference was given for the recommen-dation regarding the maximum ratio of 1.5 or 1.25, whichseems to be based on practice to ensure providing adequatesight distance.

To determine the maximum value of RATIO to be used inthe analysis, operating speeds, V85, were calculated for botharcs, and the difference between operating speeds on botharcs, ∆V, was calculated. To achieve design consistency, ∆Vshould not exceed 20 km/h (Lamm et al. 1999). The follow-ing speed models were used for flat horizontal curves, hori-zontal curves on downgrades, and horizontal curves on sagvertical curves, respectively, (Fitzpatrick et al. 1997):

[2] VR

85 104.823574.51= − (Flat horizontal curves)

[3] VR

85 102.103077.13= −

(Horizontal curves on downgrades)

[4] VR

85 105.323438.19= −

(Horizontal curves on sag vertical curves)

The operating speeds, V85, for different design speeds onboth sharper and flatter arcs as well as the difference be-tween them, ∆V, are shown in Table 1 for the case of flatalignment and the maximum superelevation, emax, is 0.06. Itwas found that for flat horizontal compound curves,RATIO = 1.5 was the maximum value that maintained ∆V ≤20 km/h for all design speeds, except for the very low designspeed (V = 40 km/h) where the use of (RATIO = 1.5) associ-ated with this speed resulted in ∆V = 23.16 km/h. For hori-zontal curves combined with downgrades or vertical sagcurves, a ratio of 1.5 was found to be the maximum valuethat maintained ∆V ≤ 20 km/h for all design speeds. Hence,the largest ratio selected for the purpose of this study was1.5, which encompasses the maximum ratios suggested bythe design guides, except for low-design speeds that are nor-mally used for sharp interchange ramps, where a smaller ra-tio should be used.

ProceduresTwo different schemes were used to evaluate vehicle sta-

bility on compound curves (Fig. 1). The first scheme was toinvestigate the effect of compound curvature by itself, whilethe second scheme was to investigate the effect of being su-perimposed by a vertical alignment. A description for eachscheme is described herein to examine its individual andcombined effects.

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Easa and Dabbour 617

To quantify the effect of compound curvature, a flat com-pound curve is used. The following procedure is followedusing different combinations of RATIO, design vehicles, de-sign speed V, and emax (Fig. 2):• Input a simple horizontal curve with radius R1 as a basis

for comparison. The radius R1 is based on V and emax.• Run simulation for that simple horizontal curve (sce-

nario 1). Read the lateral acceleration and rollover (if any).• Introduce another horizontal curve in the same direction

of curvature (with R2) to form a compound curve consist-ing of two horizontal arcs with radii (R1 and R2). The ra-dius R2 is calculated based on eq. [1].

• Run simulation for that flat compound horizontal curve(scenario 2) and read the lateral acceleration and rollover(if any).

• Increase R1 and in turn R2 to maintain the selected ratio.Re-read the lateral acceleration and rollover (if any). Con-tinue to increase the radii until the results obtained fromscenario 2 match those obtained from scenario 1.

• Record R1 from scenario 2. This is the minimum radiusthat achieves the same lateral acceleration (comfort level)as the flat simple horizontal curve.To quantify the effect of vertical alignment, the compound

curves used in scenario 2 were superimposed by different

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618 Can. J. Civ. Eng. Vol. 32, 2005

Design speed(km/h)

Design sharperradiusa (m)

Required sharperradius, R1

b (m)Required flatterradius R2

c (m)

Operating speedon sharper arcd,

V85 (sharper) (km/h)

Operating speed onflatter arcd, V85 (flatter)

(km/h)

Difference inoperating speede,

∆V (km/h)

40 55 58 77.2 35.3 58.5 23.2f

50 90 97 120.4 60.3 75.1 14.960 130 141 192.4 77.0 86.2 9.370 190 207 293.2 86.5 92.6 6.180 250 275 422.8 92.1 96.4 4.290 340 377 581.2 95.6 98.7 3.1

100 440 490 768.4 97.8 100.2 2.3110 600 672 984.4 99.4 101.2 1.8

aBased on AASHTO formula for minimum radius of a horizontal curve.bFrom Table 2.cR2 = R1 × RATIOdUsing mathematical model 2.e∆V = V85 (flatter) – V85 (sharper).fExceeds the maximum acceptable value (20 km/h).

Table 1. Operating speeds on sharper and flatter arcs for compound curves on flat alignments (emax = 0.06 and RATIO = 1.5).

Fig. 1. Different scenarios involving horizontal and vertical alignments: (a) simple horizontal curve on flat grade (raduis R1), (b) com-pound horizontal curve on flat grade with two arcs radii (R1 and R2, and (c) compound horizontal curve with two arcs radii (R1 andR2) superimposed by with vertical alignments (3-D compound curve).

types of vertical alignments (upgrade, downgrade, crestcurve, and sag curve) as shown in scenario 3 of Fig. 1. Theprocedure is as follows:• Superimpose the compound alignment (obtained from sce-

nario 2) by a specific vertical alignment.• Run simulation for that 3-D alignment and re-read the lat-

eral acceleration and rollover (if any).• Increase R1 and in turn R2 to maintain the selected ratio.

Re-read the results (scenario 3). Continue to increase theradii until the results obtained from scenario 3 matchthose obtained from scenario 2.

• Record R1 from scenario 3. This is the minimum radiusthat achieves the same lateral acceleration (comfort level)as the flat simple horizontal curve.Note that in all cases, the vehicle runs from the flatter

curve (R2) to the sharper curve (R1). This case was found tobe more critical because it resulted in a maximum lateral ac-celeration greater than that for a vehicle moving in the oppo-site direction. An example of the change in lateralacceleration along a compound curve for a WB-20 designvehicle is shown in Fig. 3. The geometric data for this par-ticular alignment are R1 = 380 m, R2 = 475 m (RATIO =1.25), V = 90 km/h, emax = 0.04, and 6% downgrade. Asnoted, the actual lateral acceleration along the straight seg-ments of the road is identical to that assumed by the PMmodel. The actual lateral acceleration starts to fluctuate fromstation 80 m (20 m before the first flatter arc starts at station100 m) to station 160 m and from station 310 m (the secondsharper arc starts at station 330 m) to station 400 m. At the

end of the compound curve (station 550 m), the actual lat-eral acceleration fluctuates until station 640 m, after which itbecomes identical to the assumed value. These results showthat the fluctuation starts almost 20 m before the start of thefirst arc until about 90 m after the end of the second arc.

Results

Minimum radius for flat compound curvesFollowing the preceding procedures (scenario 2), the re-

quired increase in the sharper arc radius (R1) correspondingto different values of RATIO are shown in Tables 2 and 3 foremax = 0.04 and 0.06, respectively. Note that the values ofthe required minimum radius and the increase shown in thetables correspond to the sharper arc with radius R1. The re-quired radius of the flatter arc, R2, would simply be calcu-lated as the product of RATIO and R1.

The tables show that the compound curvature alone hassome impact on truck stability. An increase in the TAC orAASHTO minimum radius is needed to compensate for theeffect of compound curvature on flat alignments. The re-quired increase in R1 ranges from 0% to 12.8%, dependingon the value of RATIO. More increase is generally requiredas RATIO increases. It should be noted that the increase re-quired to compensate for the effect of compound curvatureis generally smaller than what was found to be required forreverse curvature (Dabbour et al. 2004).

As shown in the tables, the required increase in R1 alsodepends on both design speed and superelevation rate. The

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Fig. 2. Procedure for analysis of flat compound curves.

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620 Can. J. Civ. Eng. Vol. 32, 2005

RATIO = 1.5 RATIO = 1.25 RATIO = 1.05

Designspeed (km/h)

AASHTO designminimum radius (m)

Minimumrequiredradius (m)

Requiredincrease(%)

Minimumrequiredradius (m)

Requiredincrease(%)

Minimumrequiredradius (m)

Requiredincrease(%)

WB-1540 60 62 3.3 61 1.7 60 0.050 100 105 5.0 104 4.0 101 1.060 150 158 5.3 157 4.7 152 1.370 200 213 6.5 211 5.5 204 2.080 280 300 7.1 296 5.7 287 2.590 380 409 7.6 403 6.1 394 3.7

100 490 530 8.2 523 6.7 509 3.9110 680 738 8.5 728 7.1 708 4.1

WB-2040 60 64 6.7 63 5.0 62 3.350 100 108 8.0 106 6.0 105 5.060 150 163 8.7 161 7.3 159 6.070 200 219 9.5 216 8.0 214 7.080 280 307 9.6 304 8.6 301 7.590 380 419 10.3 414 8.9 412 8.4

100 490 543 10.8 535 9.2 533 8.8110 680 757 11.3 746 9.7 741 9.0

NHTSA 3a/2a40 60 62 3.3 62 3.3 60 0.050 100 106 6.0 105 5.0 103 3.060 150 160 6.7 158 5.3 155 3.370 200 215 7.5 211 5.5 209 4.580 280 303 8.2 297 6.1 296 5.790 380 408 7.4 405 6.6 403 6.1

100 490 528 7.8 524 6.9 522 6.5110 680 735 8.1 729 7.2 728 7.1

Table 2. Required minimum radius of the sharper arc, R1, for flat compound curves (emax = 0.04).

Fig. 3. Example of the change in lateral acceleration along a compound horizontal curve (WB-20, R1 = 380 m, RATIO = 1.25, V =90 km/h, emax = 0.04, and 6% downgrade).

increase for emax = 0.06 is greater than that for emax = 0.04.For example, for emax = 0.04 and V = 60 km/h, the assumedmaximum side friction, fmax, is 0.15 and minimum curvatureradius, Rmin, is 150 m (TAC 1999). Whereas, for emax = 0.06and the same design speed, Rmin = 130 m (86.7% of Rmin re-quired for emax = 0.04). If the actual side friction (maintain-ing the same margin of safety) was more than fmax (say0.18), then Rmin will be 130 and 120 m for emax values of0.04 and 0.06, respectively, for the same V = 60 km/h. Inthis case, Rmin required for emax = 0.06 is 92.3% of that re-quired for emax = 0.04. Note also that the variation in loaddistribution on different tires increases with the increase insuperelevation, where the load increases on the inner tiresand decreases on the outer tires. The variation in load distri-bution results in more variation in the side friction demandat various tires of trucks.

Minimum radius for three-dimensional compoundcurves

Following the preceding procedures (scenario 3), the re-sults for the required minimum radius and the required in-crease in curve radius R1 are shown in Tables 4 and 5 foremax = 0.04 and 0.06, respectively. It should be noted that

this increase is the overall increase to compensate for the ef-fects of both compound curvature and vertical alignment.

For a given design speed and RATIO, the required mini-mum radius shown in the tables represents the maximum ofthe values of the required minimum radius obtained from allstudy configurations of vertical alignments (i.e., flat grade,upgrade, downgrade, crest curve, and sag curve). It wasfound that the most critical cases occur when the compoundcurve is superimposed by either a sag curve or a downgradewhere both the vehicle weight and the traction forces are inthe same direction. The results show that the overall increasein R1, because of the combination of compound curvatureand vertical alignment, ranges from 3.3% to 25.8%.

Comparative resultsA comparison of the effects of compound curvature and

vertical alignment is shown in Fig. 4 for WB-20, emax =0.06, and RATIO = 1.5. The results show that the effect ofvertical alignment on the required increase in minimum ra-dius is comparable to that of compound curvature. For ex-ample, for V = 60 km/h the increases corresponding tocompound curvature and vertical alignment are 9.2% and11.6% (combined effect is 20.8%). It is also noted that the

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Easa and Dabbour 621

RATIO = 1.5 RATIO = 1.25 RATIO = 1.05

Design speed(km/h)

AASHTO designminimum radius (m)

Minimumrequiredradius (m)

Requiredincrease(%)

Minimumrequiredradius (m)

Requiredincrease(%)

Minimumrequiredradius (m)

Requiredincrease(%)

WB-1540 55 58 5.5 57 3.6 56 1.850 90 96 6.7 94 4.4 92 2.260 130 140 7.7 137 5.4 134 3.170 190 206 8.4 201 5.8 196 3.280 250 272 8.8 265 6.0 260 4.090 340 372 9.4 362 6.5 355 4.4

100 440 484 10.0 470 6.8 462 5.0110 600 664 10.7 645 7.5 633 5.5

WB-2040 55 59 7.3 58 5.5 57 3.650 90 98 8.9 96 6.7 95 5.660 130 142 9.2 140 7.7 139 6.970 190 209 10.0 206 8.4 204 7.480 250 279 11.6 272 8.8 270 8.090 340 381 12.1 371 9.1 369 8.5

100 440 495 12.5 481 9.3 479 8.9110 600 677 12.8 658 9.7 655 9.2

NHTSA 3a/2a400 55 58 5.5 57 3.6 56 1.8

50 90 97 7.8 95 5.6 93 3.360 130 141 8.5 138 6.2 136 4.670 190 207 8.9 203 6.8 200 5.380 250 275 10.0 268 7.2 266 6.490 340 377 10.9 367 7.9 363 6.8

100 440 490 11.4 476 8.2 472 7.3110 600 672 12.0 651 8.5 646 7.7

Table 3. Required minimum radius of the sharper arc, R1, for flat compound curves (emax = 0.06).

effect of vertical alignment significantly increases betweenV = 50 and 60 km/h, where the overall increase jumps from11.1% to 20.8%.

The effect of superelevation on the required increase inminimum radii is shown in Fig. 5 for WB-20 and RATIO =1.5. As noted, the effect for emax = 0.06 is greater than thatfor emax = 0.04. This finding is true for the cases of flat and3-D compound curves. The effect of different superelevationrates is not as significant as other effects, ranging only from1% to 3%.

The effect of different design vehicles on the required in-crease in minimum radius is shown in Fig. 6 for emax = 0.06and RATIO = 1.5. The increase for WB-20 is generallygreater than that for WB-15. Since WB-20 is the largest de-sign vehicle in AASHTO (2001), it is expected to have thelargest variation in tire forces and reactions between frontand rear tires, as well as between inside and outside tires.The required increase for NHTSA 3a/2a truck lies betweenthose of WB-20 and WB-15. The effect of RATIO on the re-quired increase in the minimum radius of the sharper arc R1were previously shown in Tables 2–5. The ratio between theflatter and sharper radii clearly affects vehicle stability oncompound curves. However, the effect of this ratio for com-

pound curves is not as pronounced as that found for reversehorizontal curves (Dabbour et al. 2004).

Mathematical models

The values in Tables 2–5 represent conservative valuesbased on all test alignments and configurations, where sagcurves and downgrades are generally controlled in mostcases. Thus, these values will be more conservative for up-grades and crest curves. To determine the minimum radiusrequirements based on specific vertical alignment configura-tions, which are not explicitly included in the tables, a math-ematical model was developed and may be used for specificalignment configurations. The model was calibrated usingthe simulation results obtained from VDM. The requiredminimum radius for the sharper arc (dependent variable) wasrelated to design speed and alignment data (independentvariables). Many combinations of independent variableswere examined, and the final model was selected based onthe same criteria used before for both simple and reversecurves. A total of 540 observations were used to calibratethe mathematical models. The observations corresponded tothe 3a/2a design vehicle, which as previously mentioned rep-

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622 Can. J. Civ. Eng. Vol. 32, 2005

RATIO = 1.5 RATIO = 1.25 RATIO = 1.05

Design speed(km/h)

AASHTO designminimum radius (m)

3-D minimumrequiredradius (m)

Requiredincrease (%)

3-D minimumrequiredradius (m)

Requiredincrease (%)

3-D minimumrequiredradius (m)

Requiredincrease (%)

WB-1540 60 64 6.7 63 5.0 62 3.350 100 109 9.0 106 6.0 104 4.060 150 170 13.3 166 10.7 164 9.370 200 225 12.5 225 12.5 223 11.580 280 328 17.1 322 15.0 320 14.390 380 451 18.7 444 16.8 438 15.3

100 490 588 20.0 581 18.6 574 17.1110 680 826 21.5 815 19.9 810 19.1

WB-2040 60 65 8.3 64 6.7 63 5.050 100 110 10.0 107 7.0 106 6.060 150 176 17.3 172 14.7 168 12.070 200 238 19.0 231 15.5 227 13.580 280 336 20.0 330 17.9 327 16.890 380 462 21.6 457 20.3 452 18.9

100 490 599 22.2 595 21.4 591 20.6110 680 842 23.8 839 23.4 828 21.8

NHTSA 3a/2a40 60 65 8.3 63 5.0 62 3.350 100 109 9.0 106 6.0 105 5.060 150 172 14.7 168 12.0 165 10.070 200 231 15.5 227 13.5 225 12.580 280 332 18.6 326 16.4 324 15.790 380 459 20.8 450 18.4 442 16.3

100 490 594 21.2 588 20.0 582 18.8110 680 833 22.5 822 20.9 816 20.0

Table 4. Required minimum radius of the sharper arc, R1, for three-dimensional (3-D) compound curves (emax = 0.04).

resents the majority of trucks operating on North Americanroads. The final models are given by

[5] Rmin = 15.309 + 0.096(V-30)2 + 28.333I

+ 17.684RATIO

(Flat compound curves)

[6] Rmin = 13.165 + 0.102(V-30)2 + 28.200I

+ 18.785RATIO – 247.917g1

(3-D compound curves)

where Rmin is the required minimum radius of the sharperarc of the compound curve (m); V is the designspeed (km/h); I is the dummy variable (1 for emax = 0.04 and0 for emax = 0.06); and g1 is the first grade of vertical align-ment (in decimal), negative for downgrade and positive forupgrade. The coefficient of determination, R2, for the modelsof eqs. [5] and [6] were 0.992 and 0.988, respectively. Thestatistics of both models are shown in Table 6. Note that forRATIO = 1 (the case of a simple curve), this model becomesalmost identical to that developed previously for simplecurves (Easa and Dabbour 2003).

The models of eqs. [5] and [6] have logical explanationsfor the effect of each independent variable on Rmin. The pos-itive sign for the coefficient of V means that Rmin increaseswith the increase in V. The positive sign for I means thatRmin for emax = 0.04 is greater than that for emax = 0.06, asexpected. The negative sign for the coefficient of g1 meansthat Rmin increases when g1 is a downgrade. As in this caseboth vehicle weight and traction forces are in the same di-rection, it is expected that maximum variation in the loaddistribution on different vehicle tires would occur. Note thatg2 was rejected from the model as it was found to be statisti-cally insignificant. This is plausible because the largest in-crease in the actual lateral acceleration occurs in the firstsegment of the horizontal curve while the vehicle is still ne-gotiating the first grade of the vertical alignment.

The following example of the application highlights thebenefit of using the mathematical models of eqs. [5] and [6].For a truck negotiating an existing compound curve with de-sign speed (V = 60 km/h), maximum rate of superelevation(emax = 0.04), and arc radii of 150 and 195 m for the sharperand flatter arc radii, respectively (the radius 150 m is basedon AASHTO). Thus, RATIO = 195/150 = 1.3. Applying themathematical model for flat alignments, eq. [5], the requiredminimum radius for the sharper arc is 153.0 m. For a 6%

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Easa and Dabbour 623

RATIO = 1.5 RATIO = 1.25 RATIO = 1.05

Design speed(km/h)

AASHTO designminimum radius (m)

3-D minimumrequiredradius (m)

Requiredincrease (%)

3-D minimumrequiredradius (m)

Requiredincrease (%)

3-D minimumrequiredradius (m)

Requiredincrease (%)

WB-1540 55 58 5.5 58 5.5 57 3.650 90 98 8.9 96 6.7 94 4.460 130 152 16.9 145 11.5 143 10.070 190 225 18.4 216 13.7 213 12.180 250 301 20.4 292 16.8 286 14.490 340 410 20.6 403 18.5 393 15.6

100 440 537 22.0 529 20.2 522 18.6110 600 736 22.7 730 21.7 719 19.8

WB-2040 55 60 9.1 59 7.3 58 5.550 90 100 11.1 97 7.8 96 6.760 130 157 20.8 151 16.2 149 14.670 190 233 22.6 226 18.9 223 17.480 250 309 23.6 305 22.0 299 19.690 340 425 25.0 419 23.2 413 21.5

100 440 552 25.5 550 25.0 536 21.8110 600 755 25.8 751 25.2 742 23.7

NHTSA 3a/2a40 55 59 7.3 58 5.5 57 3.650 90 99 10.0 96 6.7 95 5.660 130 154 18.5 146 12.3 145 11.570 190 231 21.6 226 18.9 221 16.380 250 305 22.0 300 20.0 296 18.490 340 420 23.5 414 21.8 409 20.3

100 440 548 24.5 539 22.5 533 21.1110 600 750 25.0 744 24.0 732 22.0

Table 5. Required minimum radius of the sharper arc, R1, for three-dimensional (3-D) compound curves (emax = 0.06).

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624 Can. J. Civ. Eng. Vol. 32, 2005

Fig. 5. Effect of superelevation on required increase in minimum radius for 3-D compound curves (RATIO = 1.5 and WB-20).

Fig. 4. Effect of compound curvature and vertical alignment on required increase in minimum radius (RATIO = 1.5 and WB-20).

downgrade (or a sag curve with first grade of 6%), the re-quired minimum radius for the sharper arc, based oneq. [6], is 171.2 m. As noted, the increase required in thesharper arc radius for flat alignment is 3.0 m, which is 2%,whereas the increase required for the downgrade alignmentis 21.2 m, which is 14.1%. This example confirms the re-sults of the experimental work in which the effect of com-pound curvature is not as pronounced as that of verticalalignment.

Concluding remarks

Conventional design guidelines for minimum radius ofhorizontal curves are mainly based on flat simple horizontalcurves. The main objective of this paper was to establishminimum radius requirements for two types of compoundhorizontal curves: flat horizontal compound curves and 3-Dcompound curves (combined with vertical alignments).Based on this study, the following remarks are offered:

(1) For compound horizontal curves that overlap with verti-cal alignments, an increase in the minimum radius ofthe sharper arcs of the compound curves on is requiredto compensate for the effects of compound curvatureand vertical alignment. The overall increase ranges from5% to 26%, depending on design speed, design vehicle,ratio of flatter to sharper arcs, superelevation, and thetype of alignment (flat or three dimensional). The radiusof the flatter arc should also be increased based on theratio of the flatter to sharper arcs.

(2) Tables for determining the required minimum radius forflat and 3-D compound curves were established. Thevalues of the minimum radius are based on the mostconservative alignments tested, and therefore would beconservative for other alignments. To explicitly addressthe type of vertical alignment and other factors, mathe-matical models for minimum radius requirements weredeveloped for flat and 3-D alignments.

(3) The results of speed consistency analysis conducted inthis study support the maximum ratio of 1.5 between theflatter and sharper arcs of compound curves, as wassuggested by the TAC and AASHTO design guides.However, the ratio of 1.5 does not maintain speed con-sistency for low-design speeds (based on a speed differ-

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Easa and Dabbour 625

Independent Variable Coefficient t-Stat

Flat alignmentsIntercept 15.309 0.801(V-30)2 0.096 75.933I 28.333 5.283RATIO 17.684 1.214Coefficient of determination,

R2 = 0.992Significance of f statistic = 0.000

3-D alignmentsIntercept 13.165 1.189(V-30)2 0.102 138.816I 28.200 9.081RATIO 18.785 2.227g1 –247.917 –8.569Coefficient of determination,

R2 = 0.988Significance of f statistic = 0.000

Table 6. Statistics of the mathematical models for compoundcurves on flat and three-dimensional (3-D) alignments.

Fig. 6. Effect of vehicle type on required increase in minimum radius for 3-D compound curves (emax = 0.06 and RATIO = 1.5).

ence of 20 km/h). For such speeds, a slightly smallerratio might be recommended.

(4) No spiral (transition) curves were used in this study asthe effect of using spiral curves has been found to berelatively negligible. This finding was validated in thisstudy by evaluating vehicle stability using two identicalalignments, with and without spiral curves. The lateralacceleration values in both cases were almost identical.

(5) One of the major limitations in the design of horizontalcurves is that it is based on values of side friction thatwere developed over 60 years ago (Barnett 1936; Moyerand Berry 1940). Applying these values of side frictionfor passenger cars of today is questionable. In addition,these values were originally developed for passengercars and their application to truck fleet of today is defi-nitely questionable. Further studies to revise current val-ues of side friction are warranted.

(6) The results presented in this paper were based on sidefriction factors for open highways. These side frictionfactors are different from those for urban streets or in-terchange ramps. Minimum radius requirements for in-terchange ramps and urban streets based on vehiclestability should be developed. Future research shouldalso focus on establishing the rollover as a design con-trol for trucks in addition to the driver comfort currentlybeing used in the design guides for all types of vehicles.This paper has addressed vehicle stability for three typi-cal types of vehicles, and further research may examinevehicle stability of other types of vehicles such as sportsutility vehicles.

Acknowledgements

This study was financially supported by a discovery grantfrom the Natural Sciences and Engineering Research Coun-cil of Canada and an Ontario Graduate Studies Scholarship.

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