Improved Design of Vertical Curves Using Sight

Improved Design of Vertical Curves Using Sight Distance Profiles Yasser Hassan, Assistant Professor, Department of Civil and Environmental Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada, K1S 5B6, Tel: (613) 520-2600X8625, Fax: (613) 520-3951, E-mail: [email protected]. Abstract. Design of vertical alignment is one of the main tasks in highway geometric design. This task requires, among others, the designer to ensure that drivers will always have a clear view of the road that would allow them to stop before hitting an unexpected object on the road. Therefore, the ability to determine the required and available stopping sight distance (SSD) at any point of the vertical alignment is essential for the design process. Current design guides in the US and Canada provide simple analytical models to determine the minimum length of a vertical curve that would satisfy the sight distance requirement. However, these models ignore the effect of grade on the required SSD. Alternative approaches and models have also been suggested but cover only special cases of vertical curves. In this paper, two specific models were expanded to determine the required SSD on crest and sag vertical curves. By comparing the profiles of available and required SSD on vertical curve examples, it was shown that the current North American design practices might yield segments of the vertical curve where the driver’s view is constrained to distance shorter than the required SSD. An alternative design procedure based the models developed in this paper was developed and used to determine the minimum length of crest and sag vertical. Depending on the approach grade, these new values of the minimum curve length might be greater or smaller than the values obtained through conventional design. Design aids were therefore provided in tabular form for designers’ easy and quick use. Keywords: vertical alignment, vertical curve length, stopping sight distance, braking distance Word count: Manuscript 4,001 Tables (6*250) 1,500 Figures (8*250) 2,000 ------Total 7,501 INTRODUCTION Vertical alignment is one of the main design elements in the highway geometric design process, where selection of a proper length of vertical curves is one of its main tasks. The North American design guides by the American Association of State Highway and Transportation Officials (AASHTO)(1) and the Transportation Association of Canada (TAC)(2) have identified sight distance as the main design basis for vertical curves. Similar to all design elements, a vertical curve must allow drivers to see ahead a distance that is at least enough for emergency stopping. Mathematical formulas are available in both guides to determine the required stopping sight distance (SSD), which in turn is used to determine the minimum length of vertical curves. While the formulas can account for the effect of grades on SSD, the conventional design practice as recommended by both guides is to ignore the effect of grades in the calculation. Alternative approaches may also be followed including the use of an average grade for the approach and departure grades. The average grade approach, however, ignores the braking maneuvers that take place over a portion of the curve where the average grade over the braking distance is not equal to the overall average grade of the curve. Therefore, other researchers developed mathematical models that can account for the actual grades along the braking distance (3,4). However, these models considered only special cases of vertical curves and usable design aids have not been developed. Furthermore, considerable research has been carried out on analysis of available sight distance on vertical curves. In this paper, previous research on required SSD and effect of grades is reviewed and expanded to develop comprehensive models on crest and sag vertical curves. In addition, these models are used along with models for available SSD to develop a new and improved procedure to determine the minimum length of vertical curves using sight distance profiles. First, the following section presents a review of current practice and previous research on required SSD and the minimum length of vertical curves. Mathematical models based on expanding previous research are then developed to determine the profile of required SSD on vertical curves. These models are

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used along with models for the profile of available SSD to determine the minimum length of vertical curves. Finally, the results are summarized in design tables and charts that can be easily used by designers. SIGHT DISTANCE ANALYSIS Sight distance on vertical curves has been analyzed from two distinct approaches. The first approach is analysis of required sight distance; that is to answer the question: how far ahead does the driver need to see for safe driving? The second approach is analysis of available sight distance; that is to answer the question: how far ahead can the driver actually see without being obstructed by the alignment features or side obstructions? The two approaches have to be combined together to determine the minimum vertical curve length as this length must ensure that the available SSD is greater than the required SSD at all points of the curve. Required SSD For a vehicle traveling at an initial speed V (km/h), the required SSD, Sreq, consists of a distance dPR traveled during the perception-reaction time P (s) and a braking distance dB traveled during the vehicle’s deceleration to a zerospeed; that is Sreq = dPR + dB. Both the perception-reaction and braking distances were modeled based on the laws of kinematics, and the required SSD was originally formulated as: V2 254 f where f = coefficient of longitudinal friction; value depends on the speed V. S req = 0.278 * V * P +

(1)

This model was recently revised based on the results of a major study sponsored by the National Cooperative Highway Research Program (NCHRP)(5). The new model replaces the coefficient of friction (f) with a deceleration rate (a) as follows: S req = 0.278 * V * P + 0.039

V2 a

(2)

A typical value of a = 3.4 m/s2 was suggested in the same study as most drivers, when asked to stop on wet pavement, selected a deceleration rate equal to or higher than this value. While the most recent edition of the TAC guide still uses the original model in Equation 1, the 2001 edition of AASHTO guide has adopted the newer model in Equation 2. Therefore, Equations 1 and 2 will be referred to in this paper as the TAC and AASHTO models, respectively. Finally, if braking is to take place over a section with a constant grade, both models can be adjusted to account for the effect of grades as follows: TAC model: S req = 0.278 *V * P +

V2 254( f + 0.01G )

V2 254(a / g + 0.01G ) where G = highway grade in percent (positive for upgrade and negative for downgrade); and g = gravitational acceleration (9.81 m/s2).

AASHTO model: S req = 0.278 *V * P +

(3) (4)

However, when braking occurs over a vertical curve, the instantaneous grade changes constantly over the entire length of the vertical curve. Different approaches have been suggested to account for the effect of grade in such a situation including using an average value of the approach and departure grades of the curve. However, both the AASHTO and TAC guides simplify the calculation of the required SSD through using a zero grade. That is the two models are reduced back to Equations 1 and 2. Length of Vertical Curve A crest vertical curve would typically limit the available sight distance by having the sight line tangent to the road surface. On the other hand, a sag vertical curve would limit the available sight distance to the distance covered by the vehicle headlights. Therefore, it is the highway designer’s task to select a length of the vertical curve that would provide an available SSD (Sav) greater than the required SSD (Sreq). Both AASHTO and TAC design guides recommend the following models to calculate the minimum length of vertical curve (L) as follows:

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For crest curves: L= 200

(

L=2S−

A S2 h1 + h2 200

(h

1

)

L S≤L

2

+ h2

A

)

(5)

2

L S>L

where S = required SSD; h1 = height of driver eye; h2 = object height; A = algebraic difference of vertical grades (percent) = G2 – G1; and G1 and G2 = approach and departure grades (percent), respectively. For sag curves: L=

A S2

L S≤L 200 (hh + S tan β) 200 (hh + S tan β) L=2S− L S>L A

(6)

where S = required SSD; hh = height of headlights; and β = angle of upward light spread. The design values recommended for each of the SSD parameters in the AASHTO and TAC guides are shown in Table 1. It should be noted, however, that these models can only ensure that the minimum Sav on a vertical curve is greater than Sreq on a level grade. Alternative approaches that would ensure that Sav is greater than Sreq on all points of the vertical curve have been suggested by Thomas et al. (3) and Taignidis and Kanellaidis (4). In the first approach by Thomas et al. (3), Sreq is calculated through integration of the braking distance over the vertical curve. For the case of a vertical curve longer than the SSD, the relationship between the braking distance and speed was formulated as (3):  Ad B2  V 2 = 254 * ( f + 0.01Gi )d B + 0.01  2 L   where Gi = tangent grade at the point of braking initiation, percent.

(7)

A procedure was also developed to determine the worst grade for braking initiation, which in turn can be used to calculate the minimum length of vertical curve. However, this procedure is valid only for the case of braking distance fully on the vertical curve. Taignidis and Kanellaidis (4) noted that the formula in Equation 7 could be reduced to the traditional braking distance formula in Equation 3 through using an average grade value of the two grades at the initiation and end of braking. Therefore, the second approach was based on calculating an average grade (Gm) over the braking distance. Using integration over crest vertical curves, the value of Gm was formulated for six different cases: (1) dB starts and ends before the curve; (2) dB starts before the curve and ends on the curve; (3) dB starts before the curve and ends after the curve; (4) dB starts and ends on the curve; (5) dB starts on the curve and ends after the curve; and (6) dB starts and ends after the curve. A procedure and algorithm were developed to compare the profiles of required and available SSD on a crest vertical curve and to determine the minimum length of the curve. However, the procedure was limited to crest vertical curves and no design aids were developed that can be easily used by designers and practitioners. REQUIRED SSD – EXTENSION OF PREVIOUS MODELS As mentioned earlier, the model by Thomas et al. (3) is limited to the case of a braking distance on the vertical curve while the model by Taignidis and Kanellaidis (4) is limited to crest vertical curves. However, the two models can be combined and extended to cover crest and sag vertical curves and the six different cases of braking distance mentioned earlier. The following should be noted in the model development: • The approach and departure (first and second) grades of the curve are G1 and G2, respectively. • Upgrades are positive and downgrades are negative. • The algebraic difference of vertical grades is A = G2 – G1. • Grades and algebraic difference of vertical grades are used as percents.

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4

A crest vertical curve has a negative value of A, and a sag vertical curve has a positive value of A. The horizontal coordinate of any point on the curve is x, where the zero coordinate is at the beginning of vertical curve (BVC) (x is negative before BVC and positive after BVC). Braking starts at point x1 and ends at the point x2 (dB = x2 – x1). Depending on the values of x1 and x2, six possible cases for the positioning of the braking distance relative to the vertical curve (Figure 1). Although the illustration in Figure 1 is based on a crest vertical curve, it is equally applicable to sag vertical curves

The models developed in the following sub-sections are based first on the coefficient of longitudinal friction, and can be considered as an extension to the TAC model. Then, models based on deceleration rate are also suggested and can be considered as an extension to the AASHTO model. Formulas Using Coefficient of Friction Based on the model by Thomas et al. (3) the braking distance can be obtained from the following integration: x2

0

∫

∫

v dv = − g ( f + 0.01G x ) dx

vi

(8)

x1

where vi = initial speed at start of braking (m/s); g = gravitational acceleration = 9.81 m/s2; and Gx = instantaneous grade at any point x. By integration, Equation 8 can be re-written as: 2   0 − vi  2 

x2   = − gf ( x 2 − x1 ) − 0.01g G x dx   x1

∫

(9)

The instantaneous grade at any point x can be formulated as (4): L x≤0 G1  Ax G x = G1 + L 0< x< L L  L L≤x G 2

(10)

Combining Equations 9 and 10 and noting that dB = x2 – x1, the results of the integration can be formulated for the six cases as follows: 1.

Braking starts and ends before the curve (x1 < 0 and x2 ≤ 0): x

2 v i2 = gf d B + 0.01g G1 dx 2 x

∫

(11)

1

2.

3.

4.

Braking starts before the curve and ends on the curve (x1 < 0 and 0 < x2 ≤ L): x2 0 v i2 Ax     = gf d B + 0.01g ∫ G1 dx + ∫  G1 +  dx 2 L   x 0 1  Braking starts before the curve and ends after the curve (x1 < 0 and x2 > L): x2 L 0  v i2 Ax   = gf d B + 001g  ∫ G1 dx + ∫  G1 +  dx + ∫ G 2 dx  2 L  x  0 L 1  Braking starts and ends on the curve (x1 ≥ 0 and x2 ≤ L):

(12)

(13)

x

2 v i2 Ax   = gf d B + 0.01g ∫  G1 +  dx 2 L  x  1

5.

Braking starts on the curve and ends after the curve (x1 ≥ 0 and x2 > L):

(14)

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6.

5

x2 L  v i2 Ax   = gf d B + 0.01g  ∫  G1 +  dx + ∫ G 2 dx 2 L  x   L 1  Braking starts and ends after the curve (x1 ≥ L and x2 > L):

(15)

x

2 v i2 = gf d B + 0.01g ∫ G 2 dx 2 x

(16)

1

By integrating Equations 11 through 16 and replacing the initial speed at start of braking (vi, m/s) with the design speed (V, km/h), the following model for braking distance can be obtained:  V2   254( f + 0.01G1 )  V2  2     f + 0.01G1 + Ax 2  254   200 Ld B    2  V   254 f + 0.01G + A(2 x1 − L)  2   200d B    dB =  2 V   A( x1 + x 2 )    254 f + 0.01G1 +  200 L     V2    A( L − x1 ) 2   + − f G 254 0 . 01  2  200Ld B    2  V   254( f + 0.01G 2 ) 

L (1) x1 < 0, x 2 ≤ 0 L (2) x1 < 0, 0 < x 2 ≤ L

L (3) x1 < 0, x 2 > L

(17) L (4) x1 ≥ 0, x 2 ≤ L

L (5) x1 ≥ 0, x 2 > L

L (6) x1 ≥ L, x 2 > L

It should be noted that the calculation of the braking distance in some cases of this model may depend on the location of the point at end of braking, and in these cases a simple iterative procedure can be followed to solve for dB. Once the braking distance is calculated, the required SSD can be calculated by adding the perception-reaction distance, dPR. Because the developed model is based on the coefficient of longitudinal friction, it can be considered an extension of the TAC model. Also, it should be noted that although the model in Equation 17 was developed based on the model by Thomas et al. (3), the formulas are similar to those based on the by Taignidis and Kanellaidis (4). However, the model presented here is applicable to both crest and sag vertical curves. Formulas Using Deceleration Rate By comparing the TAC and AASHTO models in Equations 3 and 4, it can be shown that the coefficient of friction in the TAC model is in effect replaced by the ratio of a/g in the AASHTO model. Furthermore, the denominator in each of the six cases in Equation 17 can be regarded as a summation of coefficient of friction and an average grade. As a result, the extension to the TAC model can also be adopted into an extension of the AASHTO model by replacing f in Equation 17 with a/g. That is, the braking distance for the six cases can be formulated as:

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 V2   254 a + 0.01G  1 g     V2   2    254 a + 0.01G + Ax 2  1   200 Ld B  g  V2    A(2 x1 − L)   254 a + 0.01G 2 +  g 200d B    dB =  V2   A( x1 + x 2 )  a   254 + 0.01G1 + 200 L   g  V2  a  A( L − x1 ) 2   254 + 0.01G 2 − 200 Ld B   g  V2  a    254 g + 0.01G 2    

6

L (1) x1 < 0, x 2 ≤ 0

L (2) x1 < 0, 0 < x 2 ≤ L

L (3) x1 < 0, x 2 > L

L (4) x1 ≥ 0, x 2 ≤ L

L (5) x1 ≥ 0, x 2 > L

L (6) x1 ≥ L, x 2 > L

(18)

DESIGN CONSIDERATIONS Evaluation of Current Design Practice – Conventional Design To examine the implication of the new models presented in the previous section, Table 2 shows the parameters of four curve examples that were designed based on the conventional design procedure in AASHTO and TAC guides and assuming the SSD design parameters shown in Table 1. All curve examples are assumed on two-lane highways where the vertical profile is the same for the two opposing directions of travel. Therefore, the analysis of sight distance is carried out for the two directions referred to as the right lane (RL) and the left lane (LL), which correspond to the direction of increasing stations and the direction of decreasing stations, respectively. The profiles of required SSD on these curve examples were determined based on the models developed in this paper (Equations 17 and 18), and the profiles of available SSD were determined based on previous research (6,7). As shown in Figure 2, the required SSD exceeded the available SSD on the left lane of all curves. Expectedly, the left lane corresponded to the lane with the largest downgrade (refer back to Table 2). Therefore, the curves designed according to AASHTO and TAC recommendations did not provide available SSD greater than the required SSD at all points of the curve. Consequently, there is a need to develop new and improved values for the minimum length of a vertical curve that would ensure that the available SSD at any point of the curve would always exceed the required SSD. Such values can be developed by utilizing the profiles of available and required SSD. Minimum Length of Vertical Curve – New Approach An iterative procedure that utilizes the profiles of available and required SSD was developed in this paper and is illustrated in a flowchart format in Figure 3. As shown in the figure, the procedure also ensures that the length of the vertical curve in meters would not be less than the absolute minimum value recommended by AASHTO (0.6*V) or TAC (1.0*V), where V is the design speed in km/h. By using this procedure, the minimum curve length was determined for crest and sag vertical curves with algebraic difference of vertical grades A ranging from 2 to 12% and first tangent grade G1 ranging from –6% to +6%. For each curve, the length was determined based on both AASHTO and TAC guidelines, where the design parameters were taken as shown in Table 1. The available and required SSD were compared at a 5-m incremental step along the curve, and the curve length was determined to the nearest higher integer (λ = 1.0 m).

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First, Figure 4 and Figure 5 show the minimum length of crest vertical curves designed according to AASHTO and TAC guidelines, respectively. In either figure, each part of the figure corresponds to a specific value of algebraic difference of vertical grades A and each series corresponds to a specific value of first tangent grade G1. A series corresponding to the conventional design (Equation 5) is also shown in each part of either figure. As shown in the figures, for low design speeds, there is only a small or no difference between the conventional design and that based on the profiles of SSD. Moreover, for the case of a low design speed and small algebraic difference of vertical grades, the minimum length of the vertical curve is usually controlled by the absolute minimum recommended in the design guide. However, as the design speed increases, the difference between the conventional and new designs also increases, where the conventional design may underestimate or overestimate the minimum curve length depending on the value of approach grade. Similarly, Figure 6 and Figure 7 show the minimum length of sag vertical curves designed according to AASHTO and TAC guidelines, respectively. In this case, the series corresponding to conventional design is based on Equation 6. As shown in the figures, for the low values of A, the absolute minimum length controls the curve length, and therefore there is no difference between the two designs. However, as the value of A increases the trend observed in the case of crest curves becomes more evident. That is as the design speed increases, the difference between the conventional design and the design based on the profiles of SSD increases, where the conventional design may underestimate or overestimate the minimum curve length depending on the value of approach grade. While Figure 4 through Figure 7 can provide a good overall view of the difference between the conventional and new designs as well as the effect of the approach grade, re-presenting the results in a tabular format will provide better design aids. Table 3 through Table 6 provide the design values of the minimum length of vertical crest and sag curves based on AASHTO and TAC design parameters. It should be noted that these lengths are based on comparing the available and required SSD in one direction of travel only. Thus, these lengths could be used in designing multi-lane divided highways, where the vertical profile of either direction of travel is designed separately. Furthermore, they can be used in designing two-lane or multi-lane undivided highways, where both travel directions have the same vertical profile. In this latter case, the minimum vertical curve length should correspond to the greater length for both directions. For example, from Table 3 and for a 90-km/h design speed, the AASHTO crest curve with G1 = +4% and G2 = 0% will have a 140-m minimum curve length if the curve is on a multi-lane divided highway with a large median. However, if the same curve is designed on a two-lane highway, the minimum curve length should be the larger of the two values corresponding to a curve with G1 = +4% and G2 = 0% and a curve with G1 = 0% and G2 = –4%. That is the minimum curve length is 175 m. Evaluation of New Design This section evaluates the new design methodology and the minimum curve lengths developed in the previous section through a close re-examination of the four curve examples after being designed using the new procedure. For a 90km-h design speed and using the design tables (Table 3 though Table 6), the minimum length for Curves I though IV were determined as 161, 244, 154, and 175 m, respectively. Then, the profiles of available and required SSD on each curve were determined as shown in Figure 8. As illustrated in the figure, using the new design procedure has ensured that the available SSD is greater than the required SSD at every point of the two directions of the curve. Using the minimum length yields an optimum design where the profile of required SSD is tangent to the profile of available SSD for at least one direction of the curve. CONCLUDING REMARKS In this paper, available models for required stopping sight distance and effect of grade were reviewed. Two specific models were then expanded to determine the required SSD on crest and sag vertical curves. The developed models can be regarded as extensions to the analytical models available in current North American design guides. Using the developed models along with other models for available SSD on vertical curves, it was shown that the current design practice (conventional design) might yield segments of the vertical curve that violate the sight distance design basis; that is the required SSD exceeds the available SSD. An alternative design procedure based on using the profiles of available and required SSD was then developed and used to determine the minimum length of crest and sag vertical curves according to the AASHTO and TAC design guides. These new values of the minimum curve length might be considerably longer than the values obtained through conventional design. The largest differences of curve length were found to correspond to the higher range of design speed, which are typically used at the higher road classifications. That is the greatest deficit caused by the conventional design would be encountered on the most

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important links of a road network. On the other hand, on multi-lane divided highways where the median is large enough to design each direction separately, the conventional design may overestimate the required vertical curve length in one direction and underestimate the length in the other direction depending on the approach grades. Finally, design aids were provided in tabular form for easy and quick use by designers and practitioners. ACKNOWLEDGEMENTS Financial support by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. REFERENCES 1. 2. 3. 4. 5. 6. 7.

American Association of State Highway and Transportation Officials. A Policy on Geometric Design of Highways and Streets. Washington, D.C., 2001. Transportation Association of Canada. Geometric Design Guide for Canadian Roads. Ottawa, Ontario, 1999. Thomas, N.E., B. Hafeez, and A. Evans. Revised Design Parameters for Vertical Curves. Journal of Transportation Engineering, American Society of Civil Engineers, ASCE, Vol. 124(4), pp. 326-334, 1998. Taignidis, I. and G. Kanellaidis. Required Stopping Sight Distance on Crest Vertical Curves. Journal of Transportation Engineering, American Society of Civil Engineers, ASCE, Vol. 127(4), pp. 275-282, 2001. Fambro, D.B., K. Fitzpatrick, and R.J. Koppa. Determination of Stopping Sight Distances. NCHRP Report 400, Transportation Research Board, Washington, DC, 1997. Easa, S.M. and Y. Hassan. Analysis of Headlight Sight Distance on Separate Highway Alignments: A New Approach. Canadian Journal of Civil Engineering, Vol. 24(6), December 1997, pp. 1007-1018. Easa, S.M., A.O. Abd El Halim, and Y. Hassan. Sight Distance Evaluation on Complex Highway Vertical Alignments. Canadian Journal of Civil Engineering, Vol. 23(3), June 1996, pp. 577-586.

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LIST OF TABLES Table 1. SSD design parameters in North American guides. Table 2. Design parameters of vertical curve examples. Table 3. Design length of crest vertical curves based on AASHTO guidelines. Table 4. Design length of crest vertical curves based on TAC guidelines. Table 5. Design length of sag vertical curves based on AASHTO guidelines. Table 6. Design length of sag vertical curves based on TAC guidelines.

LIST OF FIGURES Figure 1. Positioning of braking distance relative to vertical curve. Figure 2. Profiles of required and available SSD on curve examples (station of BVC = 1000 m; RL = right lane; and LL = left lane). Figure 3. Procedure flowchart to determine minimum length of vertical curve. Figure 4. Minimum length of crest vertical curves based on AASHTO guidelines. Figure 5. Minimum length of crest vertical curves based on TAC guidelines. Figure 6. Minimum length of sag vertical curves based on AASHTO guidelines. Figure 7. Minimum length of sag vertical curves based on TAC guidelines. Figure 8. Profiles of required and available SSD on curve examples with lengths based on new design.

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Table 1. SSD design parameters in North American guides. Parameter h1 (m) h2 (m) hh (m) β (°)

Crest curves AASHTO (1) TAC (2) 1.08 1.05 0.60 0.38 ---------

Sag curves AASHTO (1) TAC (2) --------0.60 0.60 1 1

Table 2. Design parameters of vertical curve examples. Parameter Curve I Curve II Curve III Curve type Crest Crest Sag Design guidelines AASHTO (1) TAC (2) AASHTO (1) G1 (%)§ 4 4 2 G2 (%)§ 0 0 6 A (%) 4 −4 −4 Sreq (m)† 155.5 168.8 155.5 Lmin (m) †† 146.4 211.7 145.2 L (m)‡ 147.0 212.0 146.0 § Grades correspond to direction of increasing stations. † Based on V = 90 km/h; f = 0.3; a = 3.4 m/s2. †† Calculated curve length. ‡ Rounded curve length.

Curve IV Sag TAC (2) 2 6 4 168.8 160.3 161.0

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Table 3. Design length of crest vertical curves based on AASHTO guidelines. G1 (%) G2 (%)

40

50

60

-4 -2 0 2 4 6

-6 -4 -2 0 2 4

24 24 24 24 24 24

30 30 30 30 30 30

36 36 36 36 36 36

-2 0 2 4 6

-6 -4 -2 0 2

24 24 24 24 24

30 30 30 30 30

36 36 36 36 36

0 2 4 6

-6 -4 -2 0

24 24 24 24

30 30 30 30

69 63 58 54

2 4 6

-6 -4 -2

24 24 24

53 49 46

95 89 84

4 6

-6 -4

32 30

68 64

119 111

6

-6

42

82

142

V (km/h) 70 80 90 A = 2%. 42 48 54 42 48 54 42 48 54 42 48 54 42 48 54 42 48 54 A = 4%. 64 117 175 56 106 161 49 97 150 43 89 140 42 81 130 A = 6%. 116 178 262 108 165 242 101 154 225 95 144 210 A = 8%. 155 238 350 144 220 323 135 205 299 A = 10%. 193 297 437 180 275 403 A = 12%. 231 356 525

100

110

120

130

80 63 60 60 60 60

150 129 111 95 81 68

225 201 179 160 143 127

305 277 252 229 209 191

249 229 212 197 184

344 315 290 269 251

463 423 389 360 335

611 556 510 471 437

373 343 317 295

515 472 435 403

695 634 583 539

916 834 765 706

498 457 422

687 629 580

926 845 777

1221 1112 1019

622 571

858 786

1157 1055

1526 1389

746

1030

1388

1832

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Table 4. Design length of crest vertical curves based on TAC guidelines. G1 (%) G2 (%)

40

50

60

-4 -2 0 2 4 6

-6 -4 -2 0 2 4

40 40 40 40 40 40

50 50 50 50 50 50

60 60 60 60 60 60

-2 0 2 4 6

-6 -4 -2 0 2

40 40 40 40 40

50 50 50 50 50

60 60 60 60 60

0 2 4 6

-6 -4 -2 0

40 40 40 40

50 50 50 50

94 87 81 76

2 4 6

-6 -4 -2

40 40 40

67 63 59

125 116 109

4 6

-6 -4

40 40

83 79

156 145

6

-6

48

100

187

V (km/h) 70 80 90 A = 2%. 70 80 115 70 80 96 70 80 90 70 80 90 70 80 90 70 80 90 A = 4%. 111 182 270 101 165 244 91 151 222 84 139 204 77 129 189 A = 6%. 167 272 405 153 247 365 141 226 333 131 209 306 A = 8%. 222 363 540 203 329 487 188 301 444 A = 10%. 278 453 675 254 411 608 A = 12%. 333 544 810

100

110

120

130

203 177 155 136 119 105

306 270 241 217 195 176

416 366 326 294 268 245

553 486 432 388 352 322

411 367 331 302 277

611 540 482 436 398

831 732 652 588 535

1106 971 863 776 704

616 550 497 453

916 809 723 654

1246 1097 978 882

1658 1456 1294 1164

821 733 662

1221 1078 965

1661 1463 1304

2211 1941 1726

1027 916

1526 1348

2076 1828

2763 2426

1232

1831

2493

3317

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13

Table 5. Design length of sag vertical curves based on AASHTO guidelines. G1 (%) G2 (%)

40

50

60

-6 -4 -2 0 2 4

-4 -2 0 2 4 6

24 24 24 24 24 24

30 30 30 30 30 30

36 36 36 36 36 36

-6 -4 -2 0 2

-2 0 2 4 6

24 24 24 24 24

44 42 40 38 37

67 64 61 59 57

-6 -4 -2 0

0 2 4 6

47 45 44 42

73 71 68 66

105 101 98 94

-6 -4 -2

2 4 6

62 60 58

97 94 91

140 135 130

-6 -4

4 6

77 75

122 117

175 168

-6

6

93

146

210

V (km/h) 70 80 90 A = 2%. 42 48 54 42 48 54 42 48 54 42 48 54 42 48 54 42 48 54 A = 4%. 93 122 154 89 117 148 86 112 142 82 108 136 79 104 131 A = 6%. 143 185 232 137 177 222 132 170 213 127 164 205 A = 8%. 190 246 309 182 236 296 176 227 284 A = 10%. 237 308 387 228 295 370 A = 12%. 285 369 464

100

110

120

130

60 60 60 60 60 60

66 66 66 66 66 66

72 72 72 72 72 72

78 78 78 78 78 78

190 181 174 167 161

227 217 208 200 192

268 256 245 236 227

312 298 285 274 263

284 272 261 251

341 326 312 300

402 384 368 353

468 447 428 410

379 362 347

454 434 416

536 512 490

624 596 570

473 453

568 543

670 640

780 744

568

681

804

936

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Table 6. Design length of sag vertical curves based on TAC guidelines. G1 (%) G2 (%)

40

50

60

-6 -4 -2 0 2 4

-4 -2 0 2 4 6

40 40 40 40 40 40

50 50 50 50 50 50

60 60 60 60 60 60

-6 -4 -2 0 2

-2 0 2 4 6

40 40 40 40 40

50 50 50 50 50

70 66 64 61 60

-6 -4 -2 0

0 2 4 6

44 43 42 41

73 70 68 66

109 105 101 97

-6 -4 -2

2 4 6

59 57 55

97 93 90

145 139 134

-6 -4

4 6

73 71

121 116

182 174

-6

6

88

145

218

V (km/h) 70 80 90 A = 2%. 70 80 90 70 80 90 70 80 90 70 80 90 70 80 90 70 80 90 A = 4%. 102 138 175 97 131 165 93 125 157 89 119 150 85 114 144 A = 6%. 156 208 262 148 198 249 142 188 236 136 180 226 A = 8%. 207 278 350 198 263 331 189 251 315 A = 10%. 259 347 437 247 329 414 A = 12%. 311 416 524

100

110

120

130

100 100 100 100 100 100

110 110 110 110 110 110

120 120 120 120 120 120

130 130 130 130 130 130

221 209 198 188 180

275 259 244 232 221

325 306 289 274 260

380 356 336 318 303

332 313 297 283

412 388 366 347

488 458 433 410

569 534 504 477

442 417 396

550 517 488

650 611 577

759 712 672

553 522

687 646

813 763

949 890

663

824

975

1138

Hassan

15

x=0 x1 dB x2

x=L

x=0 x1

dB

L

L

(1) x1 < 0, x2 ≤ 0.

x=0 x1

(2) x1 < 0, 0 < x2 ≤ L.

x=L x2

dB

x=0 x1

L

dB

x=L x2

L (4) x1 ≥ 0, x2 ≤ L.

(3) x1 < 0, x2 > L.

x=0 x1

x=L x2

x=L dB

x=0

L (5) x1 ≥ 0, x2 > L.

x=L x1 dB x2

x2

L (6) x1 ≥ L, x2 > L.

Figure 1. Positioning of braking distance relative to vertical curve.

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16

Sav (RL)

Sreq (RL)

Sav (RL)

Sreq (RL)

Sav (LL)

Sreq (LL)

Sav (LL)

Sreq (LL)

(a) Curve I (AASHTO, L = 147 m).

(b) Curve II (TAC, L = 212 m).

Sav (RL)

Sreq (RL)

Sav (RL)

Sreq (RL)

Sav (LL)

Sreq (LL)

Sav (LL)

Sreq (LL)

(c) Curve III (AASHTO, L = 146 m).

(d) Curve IV (TAC, L = 161 m).

Figure 2. Profiles of required and available SSD on curve examples (station of BVC = 1000 m; RL = right lane; and LL = left lane).

Hassan

17

Start

Define the curve parameters (G1, G2 & A)

Define the design speed and coefficient of friction or deceleration rate (V & f or a)

Define the accuracy required (λ)

Determine the absolute minimum length of vertical curve: AASHTO (1): L (m) = 0.6 V (km/h) TAC (2): L (m) = V (km/h)

Establish the profiles of available and required SSD

Sav ≥ Sreq

No

Increase curve length: L = L + λ

Yes Output: L

End

Figure 3. Procedure flowchart to determine minimum length of vertical curve.

Hassan

Figure 4. Minimum length of crest vertical curves based on AASHTO guidelines.

18

Hassan

Figure 5. Minimum length of crest vertical curves based on TAC guidelines.

19

Hassan

Figure 6. Minimum length of sag vertical curves based on AASHTO guidelines.

20

Hassan

Figure 7. Minimum length of sag vertical curves based on TAC guidelines.

21

Hassan

22

Sav (RL)

Sreq (RL)

Sav (RL)

Sreq (RL)

Sav (LL)

Sreq (LL)

Sav (LL)

Sreq (LL)

(a) Curve I (AASHTO, L = 161 m).

(b) Curve II (TAC, L = 244 m).

Sav (RL)

Sreq (RL)

Sav (RL)

Sreq (RL)

Sav (LL)

Sreq (LL)

Sav (LL)

Sreq (LL)

(c) Curve III (AASHTO, L = 154 m).

(d) Curve IV (TAC, L = 175 m).

Figure 8. Profiles of required and available SSD on curve examples with lengths based on new design.