Horizontal curves should be properly designed to maintain vehicle stability. As a vehicle .... autos and recreational vehicles ($/1000 vehicle-km). The vehicle ...
5th Transportation Specialty Conference of the Canadian Society for Civil Engineering 5e Conférence spécialisée en génie des transports de la Société canadienne de génie civil
Saskatoon, Saskatchewan, Canada June 2-5, 2004 / 2-5 juin 2004
ECONOMIC BENEFITS OF IMPLEMENTING NEW MINIMUM RADIUS GUIDELINES FOR SIMPLE HORIZONTAL CURVES 1
2
A.O. Abd El Halim , and S.M. Easa 1. Stantec, Kitchener, Ontario Canada N2H 6M7 2. Department of Civil Engineering, Ryerson University, Toronto, Ontario Canada M5B 2K3
Abstract: A new and improved design approach for determining the minimum radius of simple horizontal curves has been developed using advanced vehicle simulation software. The approach, which considers vehicle stability and three-dimensional alignments, showed that the minimum radius requirements of the current design guides of the Transportation Association of Canada and the American Association of Highway and Transportation Officials are inadequate. The new approach requires larger radii of simple horizontal curves than those presented in the design guides. The larger radii (flatter curves) of the new approach satisfy the driver comfort criterion and provide a smoother transition between the tangents and horizontal curves. Also, the required larger radius results in a reduction in the overall length of the alignment. In turn, this reduction in alignment results in savings in construction, collision, maintenance, and vehicle operating costs. This paper evaluates the costs and benefits associated with implementing the new design guidelines for minimum radius of simple horizontal curves. Two different cases were evaluated: (1) a new alignment involving construction of the flatter horizontal curve based on the new guidelines relative to TAC or AASHTO curve design and (2) an existing alignment for which a flatter horizontal curve is used to improve an existing horizontal curve. It was found that increasing the required minimum radius of the current design guides is generally cost-effective. Therefore, while satisfying vehicle stability requirements, the flatter curves of the new design approach are more economical than the curves of the current design guides.
1. INTRODUCTION The current North American guidelines establish the minimum radius of horizontal curves without adequately considering vehicle stability requirements or the effect of 3D-alignment. Easa and Dabbour (2003) developed a new and improved design approach for simple horizontal curves that considers vehicle stability and the effect of 3D alignment on minimum radius requirements. To develop the new curves, the Vehicle Dynamics Models Roadway Analysis and Design (VDM RoAD) simulation software was used. The model considers driving dynamics, braking inputs, and the lateral acceleration of a vehicle as it navigates a highway alignment. The results of the study showed that an increase in the minimum radius is required to satisfy vehicle stability and highway safety. As the radius of a horizontal curve increases, the curve becomes flatter, and a smoother transition between the tangent and curved sections of highway is provided. Increasing the radius of a horizontal curve is one of the most effective means of decreasing collision rates on horizontal curves, and thus increasing the level of safety of a highway (Lamm et al. 1999). Several benefits and costs arise from
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increasing curve radius. As the radius increases, the overall alignment length is reduced, resulting generally in savings in construction, collision, maintenance, and vehicle operating costs. 2. LITERATURE REVIEW Several studied have revealed that a large proportion of the total traffic-related fatalities occur on curved sections, especially on horizontal curves of two-lane rural highways (Lamm et al. 1999). A study conducted by Talarico and Morrall (1993) reported that curved highway sections increase the demands placed on the drivers, and these highway sections have significantly higher collision rates than tangent sections. Wooldridge et al (2000) conducted a study that evaluated the effects of horizontal curvature on driver visual demand and found that visual demand was significantly higher on curves with sharper radii. Collision rates on curves have been found to be 1.5 to 4 times greater than those on similar tangent sections. Statistics have also shown that sharper curves are associated with higher collision rates than milder ones. Horizontal curves should be properly designed to maintain vehicle stability. As a vehicle traverses a horizontal curve, the vehicle and driver experience a centripetal acceleration that acts towards the centre of the horizontal curve. The centrifugal force is in the direction and is proportional to the curve radius. For the vehicle to maintain stability as it navigates the curve, the centrifugal force is resisted by a component of the vehicle weight that acts perpendicular to the roadway superelevation and by the side friction developed between vehicle tires and pavement surface. The frictional forces created by the interaction of the tire with the pavement depend on the condition of the tire, road surface, weather, and vehicle kinematics. This highly simplified approach to modeling a vehicle on horizontal curves is based on the laws of mechanics and kinematics, and is referred to as the point-mass model. This model was adopted by the North American design guides due to its simplicity (Hassan et al. 1998). Current research has focussed on the use of vehicle simulation software to examine the effects of 3D alignments on vehicle stability. VDM RoAD (Vehicle Dynamics Models Roadway Analysis and Design) has been used to develop new and improved guidelines for simple horizontal curves (Easa and Dabbour 2003). The VDM RoAD, one of the most advanced vehicle stability models, was developed at the University of Michigan with the intention of being included within the Interactive Highway Safety Design Model developed by the FHWA. VDM RoAD is a user-friendly integrated set of computer tools for simulating and analyzing braking and handling behaviour of vehicles on selected highway designs. It simulates a vehicle running through a specified highway alignment, and can analyse lateral acceleration, directional control, roll stability, and stopping sight distance. The following sections present the geometry of TAC and new (flatter) horizontal curves and the economic analysis methodology. The cost-effectiveness of the new guidelines for simple horizontal curves is then evaluated, followed by a sensitivity analysis and conclusions.
3. GEOMETRY OF TAC AND NEW HORIZONTAL CURVES To reduce collision rate and improve the level of safety of a highway alignment, several different options are available to highway engineers and designers. One of the most effective methods of improving the safety of a highway alignment is to increase the radius of an existing sharp horizontal curve. This technique is referred to as “Curve Flattening”, and involves reconstructing a horizontal curve to make it less sharp, longer in length, and with a lower degree of curvature. As mentioned previously, the proposed new minimum radius is flatter than the existing TAC radius for specific conditions. The new flatter curves would be associated with lower collision rates. Therefore, the new guidelines, which improve the safety of a highway alignment, may be viewed as “Curve Flattening” with respect to the existing TAC guidelines. The new horizontal curve is less sharp and longer. Although the length of the new curve (Ln) is longer than the TAC curve (Lo), the overall length of the new alignment is reduced (Figure 1). The figure shows the effect of increasing the radius of a simple horizontal curve. The reduced alignment length produces
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several benefits. From Figure 1, it is observed that by increasing the radius (R1) of the sharper curve to that of the new curve (R2), the resulting length (L2) becomes longer than the TAC curve. However, the degree of curvature DC2 of the new curve is less than that of the TAC curve, DC1. It is important to note that the two curves share common tangents, and that the central angle of the TAC curve is equal to that of the new curve. As the degree of curvature DC is reduced, the collision rate decreases, and the level of safety of the alignment is improved. When the radius of a sharp horizontal curve is increased, the curve length increases as indicated by the following equation [2]: [1] [2]
5729.6 R DC1 L1 L2 = DC2 DC =
where L1 = length of the old (sharper) curve (m) L2 = length of the new (flatter) curve (m) o DC1 = degree of curvature of the old curve, /100m o DC2 = degree of curvature of the new curve, /100m
β
For: β=γ R 2 > R1 DC2 > DC1 L2 > L1
R1
T
Case 1: New Curve Case 2: Old Curve
L1 DC1 R1
T
L2
γ 100m
DC2 R2
R2
γ
Figure 1 Geometry and Effect of Curve Flattening
Despite the increase in the new curve length, the total length of the alignment for the TAC curve is longer since a portion of the tangent section on either end of the TAC curve must be considered in constructing
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the TAC curve. Assuming that the new horizontal curve has the same tangents as the TAC curve, the length of the tangent section can be derived as:
[3]
T =
5729.6 1 1 γ tan − 1000 DC2 DC1 2
where T = tangent length eliminated due to curve flattening (km) and γ = central angle (o). Therefore, the length of the TAC alignment, Lo (km) is: [4]
Lo = L1 + 2T
and the length of the new alignment, Ln (km) is: [5]
Ln = L2
When the radius of a horizontal curve is increased, the total length of the alignment is reduced by an amount ∆L, given by
[6]
1 1 γ − ∆L = 11.4592 tan − 0.10γ − 2 DC DC 2 1
where ∆L= change in length (km). 4. ANALYSIS METHODOLOGY To examine the cost-effectiveness of the new minimum radius requirements for simple horizontal curves (Easa and Dabbour 2003), it is necessary to compare the costs and benefits of constructing the old and new curves. The methodology used in this analysis is similar to that used by Talarico and Morrall (1993) who conducted a comprehensive study to investigate the cost-effectiveness of curve flattening in Alberta. Their methodology and approach for evaluating the cost-effectiveness of curve flattening has been used in this analysis. Some of the parameters used in the methodology have been revised to reflect Ontario conditions.
Construction Costs and Savings When a horizontal curve is flattened, the total length of an alignment is decreased by an amount ∆L given by [6]. This reduction in alignment length can be considered as a construction savings, as it produces a shorter roadway alignment. Considering that the average cost of constructing a 9.0-m and 11.0-m wide highway in Ontario is $720 000/km and $880 000/km (MTO 2003 dollars), respectively, any reduction in alignment length can be considered a benefit. The cost of constructing a simple horizontal curve can be estimated from the following equation: [7]
Cs =
Rγ Π *U 180
where Cs is the construction cost ($), R is the curve radius (m), γ is the central angle ( ), and U is the unit cost ($/km). The cost savings attributed to constructing a larger radius horizontal curve can be calculated from: o
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[8]
C = ∆L * U
where C is the cost savings ($), ∆L is calculated from [6]. Collision Savings Research has shown that a relationship exists between the degree of curvature and the collision rate of a horizontal curve. The general trend outlined by Lamm et al. (1999) states that for an increasing degree of curvature (decreasing curve radius), the collision rate increases. When increasing the curve radius of a sharper horizontal curve, or flattening an existing designed horizontal curve, the degree of curvature decreases. With this decrease in DC, the collision rate on the alignment decreases and the level of safety is improved. To quantify the benefits attributed to a reduction in collisions, consider the following collision model developed by Glennon et al. (1985) and used by Talarico and Morrall (1993):
[9]
∆At = ARS (∆L) AADT + 0.0102( DC1 − DC 2 ) AADT
where ∆At is the annual reduction in collisions, ARs is the collision rate on a tangent section (collisions per million vehicle kilometres), ∆L is the change in length (km) calculated from [6], AADT is the traffic volume in year t (in millions), and DC1 and DC2 are the degrees of curvature of the old and new curve, respectively. The annual savings due to collision reduction, Kt, can be calculated as,
[10]
Kt = ∆At U
where ∆At is the annual reduction in collisions calculated from [9] and U is the average cost per collision ($). The cost of a collision varies with the severity of the collision. Also, of all collisions, generally 2% are fatal, 12.5% cause moderate injury, and 73% cause property damage only. In Alberta, the costs (adjusted in 2003 dollars) are as follows (Howery, 1991): fatality, $ 875,513; serious injury, $586,036; moderate injury, $5,730; and property damage, $2,285. Using these collision costs and data, a weighted average cost per collision of $93,149 was calculated.
Vehicle Running Costs Savings Vehicle operating cost is a function of vehicle type and speed, and also depends on the vertical profile of a highway alignment. The net savings per thousand vehicles in each year due to reduced vehicle running cost can be calculated from [11]
Wt = [ Pht ( H ht ) + Psub ( H sub ) + Parv ( H arv )](∆L)
where W t is the savings due to decreased running cost (2003 dollars per 1000 vehicles), Psub is the percentage of single unit trucks and busses, Parv is the percentage of autos and recreational vehicles, Hht is the running cost of single-unit trucks and buses ($/1000 vehicle-km), and Harv is the running cost of autos and recreational vehicles ($/1000 vehicle-km). The vehicle running costs (2003 dollars) in Alberta are estimated as (Howery, 1991) are; autos = $147 per 1000veh-km; single unit trucks and busses = $440 per 1000 veh-km; and heavy trucks = $375 per 1000 veh-km. Costs Associated with Speed Changes The costs associated with speed changes vary with vehicle type. Talarico and Morrall (1993) used the models developed by Howery (1991) to estimate the speed change cost using the following models:
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For Heavy Trucks, [12]
Z ht = (0.327Vi + 2.8)1.74 − (0.327V f + 2.8)1.74
For Single-Unit Trucks and Busses, [13]
Z sub = (0.1936Vi + 1.8)1.9 − (00.1936V f + 1.8)1.9
For Automobiles and Recreational vehicles, [14]
Z arv = (0.0588Vi + 1.4)1.9 − (0.0588V f + 1.4)1.9
where Zht, Zsub, Zarv are the speed change costs (adjusted to 2003 dollars per 1000 vehicles) for heavy trucks, single unit trucks and busses, and automobiles and recreational vehicles respectively, Vi and Vf (km/h) are the speeds of the new and TAC curves respectively. The speeds of the new or TAC curve is given by: [15]
V85 = 104.82 −
3574.51 R
th
where V85 is the 85 percentile operating speed and R is curve radius. Net Present Value of the Annual Savings The savings outlined in the previous section are annual savings during the life (duration) of the project and must be discounted to the start of the project duration. The net present value of the annual operating savings can be calculated from: N
[16]
1 t t =1 (1 + i )
NPVS & B = ∑
where NPVS & B is the net present value after t-years, i is the interest rate, and t is the duration of the project. The present value of a series of uniform end-of-period savings can be calculated by using the series present-worth factor which is given by: [17]
(1 + i ) N − 1 P=A i (1 + i ) N
where P is the present value of a series of savings or benefits, A is the annual savings. For this analysis, a 20-year project life and an interest rate of 3% were assumed.
Other Considerations It is important to note that many other benefits such as time savings, fuel economy benefits, and winter maintenance savings (de-icing agents) exist. However, these were not included in the analysis because either the savings are difficult to quantify or are very small. Also, costs such as land acquisition have not been considered in this analysis as they are highly variable and depend on several different factors that are difficult to quantify (Talarico and Morrall 1993).
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5. COST-EFFECTIVESS OF NEW GUIDELINES In this section, the cost-effectiveness of the improved simple horizontal curve design approach developed by Easa and Dabbour (2003) that proposed flatter minimum radii than those of TAC will be evaluated. The benefits and costs of the new design approach will be compared with the traditional TAC horizontal curve design approach (TAC 1999). For this scenario, the benefits of a newly constructed simple horizontal curve will be examined. As previously mentioned, it is assumed that the two curves will share common tangents, thus the central angle of the new and TAC curves are equal.
Case S1: New Alignment To compare the benefits and costs of the new simple horizontal curve design approach with the TAC design approach, consider a simple horizontal curve with design speed VD = 110 km/h, superelevation rate o emax = 0.04, and a central angle of 45 . The TAC guidelines recommend a minimum radius of 680 m, while the new design approach suggests a minimum radius of 810 m (Easa and Dabbour 2003). When the radius is increased from Rmin = 680 m to Rmin = 810 m, the horizontal curve radius is increased by 130 m. There are several benefits associated with the larger radius of the horizontal curve. For horizontal o o curves of radii 680 m and 810 m, the degree of curvature according to [1] is 8.43 and 7.07 , respectively. When the radius is increased, the total length of an alignment is decreased by ∆L. By substituting DC1 = o o o 8.43 , DC2 = 7.07 , and ? = 45 into [6], ∆L becomes 0.0429 km, or 42.9 m. Therefore, by using the flatter radius, the total length of the alignment decreases by 42.9 m. This reduction in alignment length produces several benefits. The cost of constructing the two horizontal curves can be estimated from the alignment length and the highway unit construction cost ($/km). Considering that the average cost of a 9.0-m wide highway in Ontario is $720,000/km (MTO 2003), constructing the larger radius horizontal curve results in a construction savings of $30,917. By using the collision model developed by Glennon et al. (1985) where ∆At is the annual reduction in collisions, and substituting ∆L = 0.0429 km, DC1 = 8.43, DC2 = 7.07, ARs = 0.560, and AADT = 7,000 into [9], the reduction in the total number of annual collisions can be calculated as 0.09681. The annual savings due to collision reduction, Kt, can be calculated from [10] where ∆At is the annual reduction in collisions calculated from [9], and U is the average cost per collision ($). Assuming that the average cost of a collision is $93,149 (in 2003 Dollars), the annual savings due to collision reductions in the first year is $9006. When the radius of a horizontal curve is increased, the resulting alignment length decreases resulting in savings due to reduced running costs. Assuming that autos make up 75% of the total vehicle traffic, single unit trucks and busses make up 7%, and heavy trucks make up 18% of the total share, the net savings per thousand vehicles in each year due to reduced vehicle running costs calculated from [11] is $8.95 per 1000 vehicles. Therefore, the savings due to operating cost reductions for the first year is $22,859 (AADT = 7000). The estimated speeds for the new and TAC design curves calculated from [11] are 100.4 km/h and 99.6 km/h respectively. The speed change cost savings achieved in each year for the three vehicle classes can be calculated. The estimated speed change cost savings for heavy trucks, single-unit trucks and buses, and autos and recreational vehicles is $12.33 per 1000 vehicles, $8.85 per 1000 vehicles, and $1.03 per 1000 vehicles, respectively. The net present value of the annual operating cost savings can be calculated by multiplying the annual savings and benefits by the present-worth factor of [17]. Assuming a project life of 20 years, and an interest rate of 3%, the present-worth factor is equal to 14.88. Therefore, over the 20-year life of the project, $134,005 would be saved in collision cost savings, $137,274 would be saved due to speed change cost savings, $340,141 would be saved due to reduced running costs, and $30,917 would be saved in construction costs due to the reduced alignment length. In total, assuming a 20-year project life, and an interest rate of 3%, $642,337 in savings would be produced by constructing the new flatter simple horizontal curve (VD = 110 km/h, Rmin = 810 m, emax = 0.04) compared with the TAC design curve (VD = 110 km/h, Rmin = 680 m, emax = 0.04).
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Clearly, the simple horizontal curve developed by Easa and Dabbour (2003) is more cost-effective to construct and maintain than the TAC design curve, as it is shorter in overall length, and has a smaller degree of curvature. As well, several benefits such as time cost savings, vehicle operating costs, and collision cost savings are produced when using the larger radius horizontal curve. The costs and benefits for other design speeds (60 km/h to 110 km/h) are presented in Table 1. Case S2: Existing Alignment In this section, the effects of increasing the radius of an existing TAC simple horizontal curve to that of the newly designed curve developed by Easa and Dabbour (2003) will be evaluated. To compare the benefits and costs of the new and TAC curves, consider a similar simple horizontal curve with design speed, VD = o 110 km/h, superelevation rate, emax = 0.04, and a central angle of 45 . The existing simple curve has a radius of 680 m that will be increased to 810 m. This represents an increase of 130m. The cost of flattening the TAC curve can be estimated by determining the length of the new curve. The alignment length of the larger 810 m horizontal curve is 0.636 km, and thus the cost of constructing the new larger radius curve, calculated from [7], is $458,044. In order to determine whether or not it is feasible to increase the radius of the existing horizontal curve, the net present worth of the project must be calculated. The net present worth can be calculated from: [18]
NPVtotal = -C + NPVs + NPVA
Assuming a project life of 20 years, and an interest rate of 3%, the present-worth factor is equal to 14.88. Therefore, over the 20-year life of the project, the savings in collision cost savings, speed change cost savings, and reduced running costs would be the same as in case S1 ($134,006, $137,274, and $340,141, respectively). The cost of increasing the radius of the existing curve calculated from [7] is $458,044. Therefore, the net present worth of the project can be calculated as: NPVtotal = - 458,044 + 134,006 + 137,274 +340,141 = $153,377 Since the net present value is positive, increasing the radius of the existing horizontal curve is a costeffective and is economically feasible. A designer must conduct a comprehensive and detailed study focussing on the net benefits and costs of constructing the new alignment. If the net benefits are greater than the cost of constructing the alignment, increasing the radius of the curve may be a feasible option. On the other hand, if the engineer or designer finds that the costs of constructing the new alignment far outweighs the net benefits then increasing the radius of the existing curve may not be a feasible alternative. Table 1 Benefits of Implementing Larger Radius of Simple Horizontal Curve in New and Existing Alignments over 20-year Project Life (AADT = 7,000)
VD (km/h)
Improved TAC Rmin (1) Design Rreq(2)
TAC DC1
Improved Design DC2
Total Vehicle Operating Total Speed Construction Existing Savings Wt Savings Zt Construction New Design Design NPW Cost (20) (20) Savings Existing Curve NPW of of Benefits ($) ($) (New Curve) ($) ($) Benefits ($) ($)
60
150
170
38.20
33.70
52329.39
397773.30
4756.46
96132.65
630284.39
529395.28
70
200
230
28.65
24.91
78494.09
346731.00
7134.68
130061.83
586987.51
449791.00
80
280
320
20.46
17.90
104658.78
246308.59
9512.91
180955.58
479076.87
288608.38
90
380
440
15.08
13.02
156988.17
202822.78
14269.37
248813.93
487652.86
224569.57
100
490
580
11.69
9.88
235482.26
181685.01
21404.05
327982.00
563067.27
213681.22
110
680
810
8.43
7.07
340141.04
137274.04
30916.97
458043.82
642337.03
153376.24
6. SENSITIVITY ANALYSIS It is important to note that the net present worth of any project highly depends on the AADT of the alignment. It was found that the net benefits increased as AADT increases. For lower AADT values (e.g.,
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AADT < 1000), the net present worth of the project would be negative, indicating that increasing the radius of the existing horizontal curve is not cost-effective. The sensitivity of the total net present worth of the new guidelines to AADT and the unit construction cost was conducted. Since most of the models of cost savings and benefits are functions of AADT, it was observed that AADT has a significant impact on the net present worth of a project. Figure 2 presents the net present worth (for various levels of AADT) of increasing the radius of an existing TAC simple horizontal curve to the radius of the new guidelines (emax = 0.04, VD = 60 km/h to 110 km/h) developed by Easa and Dabbour (2003). It is observed that as AADT increases, the net present worth of a project increases as well. When the design speed is less than 90 km/h, the NPW of the project is positive for AADT values between 3,000 and 13,000. For higher values of design speed (> 80 km/h), an increase in radius would involve some cost if low levels of AADT are present on the alignment. The unit cost of construction of a highway alignment is an important factor to consider when conducting a cost-effectiveness analysis. When increasing the radius of an existing horizontal curve (Case S2), the construction cost is a function of the alignment length. Therefore, any increases or decreases in the unit cost ($/km) will affect the costs of constructing the project. In this analysis, an estimated construction cost of $720,000/km was considered (MTO 2003 dollars). When a new curve is constructed (Case S1), any changes in the unit cost will affect the savings due to a reduced alignment length calculated from [8]. An important finding is that regardless of any changes in the unit construction cost, the improved horizontal curves are more cost-effective to implement due to the reduced alignment length. 7. CONCLUSIONS Several studies have illustrated that the collision rate on sharp horizontal curves are up to 4 times higher than that on straight tangent sections. Also, the collision rate was found to increase with increasing the degree of curvature. Increasing the radius of a sharp horizontal curve produces
1200000 1000
3000
5000
7000
9000
11000
13000
1000000
NPW Benefits($)
800000 600000 400000 200000 0 60.00
70.00
80.00
90.00
100.00
110.00
-200000 -400000 -600000 Design Speed (km/h)
Figure 2 NPW-Benefits of Increasing Radius of an Existing TAC Horizontal Curve for Various Design Speeds and Levels of AADT. several cost savings such as reduced alignment length, collision reduction, and vehicle operating savings. The new and improved simple horizontal curve design approach that considers the effects of vehicle stability and vertical alignment developed by Easa and Dabbour (2003) produces larger radii of horizontal curves than that of the TAC design guide. Regardless of the unit construction cost, the new horizontal curves are more cost-effective to construct, provided that the central angle of the new and TAC curves are
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the same. In addition to cost-effectiveness, the new simple horizontal curves address vehicle stability requirements and improve safety compared with the TAC guidelines. REFERENCES American Association of State Highway and Transportation Officials, AASHTO. A policy on geometric design of highways and streets. Washington, D.C., 2001. Easa, S.M. and Dabbour, E. Design radius requirements for simple horizontal curve on 3d alignments. Canadian Journal of Civil Engineering, 30(6), 2003, 1022-1033. Hassan, Y., Easa, S.M., and Abd El Halim, A.O., Highway alignment: Three dimensional problem and three dimensional solution. Transportation Research Record 1616, TRB, National Research Council, 1998, 17-25. Howery, K.E. and Applications Management Consulting Ltd. 1991. Benefit cost analysis. Report prepared for Alberta Transportation and Utilities. Lamm, R., Psarianos, B., Mailaender, T. Highway design and traffic safety engineering handbook. McGraw-Hill Companies, Inc., New York, NY. 1999. Talarico, R,J. and Morrall, J. F. The cost-effectiveness of curve flattening in Alberta. Canadian Journal of Civil Engineering, Vol. 21, 1993, 285-296 Woolridge, M, D., Fitzpatrick, K, Koppa, R., and Bauer, K. Effects of horizontal curvature on driver visual demand. Transportation Research Record 1737, TRB, Washington D.C., 2000, 71-77. Transportation Association of Canada. Geometric Design guide for Canadian roads, Ottawa, Ontario, 1999.
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