Highway Alignment: Three-Dimensional Problem and Three

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Highway Alignment Three-Dimensional Problem and Three-Dimensional Solution YASSER HASSAN, SAID M. EASA, AND A. O. ABD EL HALIM Highway geometric design has usually been considered in separate twodimensional (2-D) projections of horizontal and vertical alignments. Such a practice was followed mainly because three-dimensional (3-D) analysis of combined highway alignments was expected to be difficult. As a result, the effect of ignoring the 3-D nature of the highway alignment could not be quantified. With the long-term objective of developing 3-D design practice, a framework for 3-D highway geometric design was developed and 3-D sight distance was extensively studied as the first design basis. The status of sight distance in current design policies and previous research is summarized, and mainly 2-D analysis was considered. The five main tasks performed to cover the 3-D highway sight distance are presented. (a) As a preliminary step, the 2-D sight distance on complex separate horizontal and vertical alignments was modeled, and the finite element method was used for the first time in the highway geometric design. (b) The 2-D models were then expanded to cover the daytime and nighttime sight distances on 3-D combined alignments. (c) The analytical models were coded into computer software that can determine the available sight distance on actual highway segments. (d) The models were applied in 3-D design of combined horizontal and vertical curves in cut-and-fill sections, and preliminary design aids were derived. (e) Finally, a new concept of red zones was suggested to mark the locations on alignments designed according to current practices where the available sight distance will drop below that required. A comprehensive work on 3-D sight distance analysis has been compiled that should be of great importance for highway researchers and professionals.

Current highway geometric design policies in North America have always considered the highway alignment as a two-dimensional (2-D) problem (1,2). Each element is designed as an isolated element in a 2-D projection. However, because the highway is in reality a three-dimensional (3-D) object, following current 2-D practices does not guarantee a safe and pleasing design. For example, the Canadian design guide states “a section of road might be designed to meet these standards, yet the result can be a facility exhibiting numerous unsatisfactory or displeasing characteristics” (2). Although highway professionals and researchers have been aware of the 3-D nature of the highway alignment, the 2-D approach has been widely accepted because of its simplicity. For example, Mannering and Kilareski stated that “the alignment of a highway is a three dimensional problem. . . . However, in highway design practice, three-dimensional design computations are cumbersome” (3). As a result, 3-D highway geometric design has been a weak link in the overall highway design process (4). However, several researchers in the past decade have pointed out the need for 3-D analysis of highway alignments and for establishing 3-D design practice (5). Y. Hassan, Faculty of Engineering, Public Works Department, Cairo University, Giza, Egypt. S. M. Easa, Department of Civil Engineering, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1. A. El Halim, Carleton University, Ottawa, Ontario, Canada K1S 5B6.

With the long-term objective of establishing 3-D highway geometric design practice, extensive research was conducted to analyze highway alignments in 3-D. A framework was developed to comprehend the highway geometric design process and the required 3-D research. Figure 1 shows the framework that relates the constraints, bases, and elements of geometric design. In a 3-D design approach, all elements should be jointly designed to account for each basis within the limitations of the constraints. Previous research was found to have limited work related to 3-D highway alignments. That work involved mainly separate individual efforts that did not follow a long-term plan to develop a complete set of 3-D design practice. Following this review, extensive research work was carried out to analyze the design bases in a 3-D combined alignment, starting with sight distance and the related aspects. The review of the status of sight distance in current design guides and previous research is summarized, and the work and findings are compiled. It should be noted, however, that only the modeling of available sight distance is addressed. The sight distance required for safe and efficient highway operation is beyond the scope of this paper.

SIGHT DISTANCE: BACKGROUND Topographical and economical constraints usually make it impractical to ensure that any object on the pavement surface is visible to the drivers within the normal eye sight distance. For example, the driver’s line of sight on horizontal curves may be blocked by lateral obstructions such as trees, buildings, and cut side slopes. On crest vertical curves, the line of sight may be obstructed by the vertical curve itself. Also, nighttime sight distance on sag vertical curves may be limited to the farthest point covered by the vehicle headlights. Furthermore, overpasses represent sight obstructions for the traffic below. Therefore, designers have to check the available sight distance so the designed roads will provide drivers with at least the minimum sight distance required for safe and efficient operation. Although a clear correlation is lacking, current design guides emphasize the relationship between sight distance and traffic safety (1,2). According to current design policies, minimum sight distance, sufficient to bring a vehicle to a complete stop before hitting an unexpected object on the road ahead, must be provided at every point of any highway. Also, a sight distance sufficient to pass slow vehicles is required frequently on two-lane highways. In addition, a sight distance, usually longer than that required for stopping, may be required on sections where the proper driving decision is not clear or easy to perceive. An example for this is to provide enough sight distance to detect and react to a change in the horizontal alignment from tangent to curve. To aid designers, current design guides offer analytical models to analyze the available sight distance. The following sections present these models available in current design guides and previous research work.

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is used, the exact portion of the highway with deficient sight distance cannot be determined. Moreover, the AASHTO formulas can be used only in the case of a simple vertical curve with long tangents. As for reverse and unsymmetrical vertical curves, the available sight distance and sight-hidden dips, which may develop if a crest curve is followed by a sag curve, have been modeled. The designer can check the existence of a sight-hidden dip and determine the portion of the highway experiencing such a sight-hidden dip. Sight Distance on 3-D Combined Alignments

FIGURE 1

Framework for 3-D highway geometric design.

A common shortcoming among all the aforementioned models for available sight distance is that the highway is considered as separate 2-D projections of horizontal and vertical alignments. Although computer programs for 3-D highway visualization have been available for a long time, none has been directed at the 3-D analysis of sight distance. With the recent advancements of computers, more software packages have been developed and have been available to highway professionals. Nevertheless, as stated by Jull and Murray, “as with all computer programs, ITEDS (a computer software for highway design) does not provide any new revelations in the theory of highway design” (10). In a recent study, the interaction between the sight distance and the 3-D combined alignment of interchange connectors was studied by a graphical technique using the software InRoads (11). Although the methodology was successful in achieving the objectives of the study, it is time-consuming because the available sight distance is determined graphically (not analytically). Therefore, such a methodology cannot be used to establish a 3-D geometric design policy.

Sight Distance on 2-D Horizontal Alignments AASHTO developed a formula to relate the available sight distance to the lateral clearance on the inside of horizontal circular curves. Although the AASHTO formula is easy and direct, it is “of limited practical value except on long curves” (1). It is applicable only on long circular curves (L > S, where S = sight distance and L = curve length) and with constant lane width and lateral clearance. In addition, a single value of the minimum available sight distance instead of the full profile can be determined. Later research showed that the lateral clearance calculated by the AASHTO formula is needed only from S/2 after the point of the curve to S/2 before the point of the tangent (6). Therefore, the AASHTO guide recommends that the designer must use graphical methods to check sight distance on horizontal curves. Analytical models were also developed for the case of a single lateral obstruction on a simple horizontal curve where S > L (7–9). The case of a single lateral obstruction on compound and reverse curves has also been modeled. Yet, the case of a continuous obstruction (e.g., a cut side slope) had not been addressed nor was the case of a complex horizontal alignment built of successive tangents, spiral curves, and circular curves.

3-D SOLUTION The literature review, summarized above, showed that very little work has been conducted following a 3-D approach. In addition, the analysis of sight distance in 2-D had considered mainly an isolated curve with long tangents. Thus, the need has been established for 3-D analysis of sight distance, which was the first design basis to be studied. The following tasks were planned and commenced in the summer of 1994: 1. Developing analytical models for sight distance on 2-D complex alignments; 2. Expanding the 2-D models to 3-D combined alignments; 3. Coding the models into computer programs that can determine the available sight distance on real highway segments; 4. Applying the models in design; and 5. Applying the models to determine the “red zones” where a combined horizontal and vertical curve, designed using the 2-D practice, may compromise highway safety. Analytical Models: 2-D Complex Alignments

Sight Distance on 2-D Vertical Alignments AASHTO has developed formulas to relate the available sight distance on a simple crest or sag vertical curve to the curve parameters (1). These formulas are intended for designing vertical curves to satisfy the stopping-sight distance (SSD) and can also be used to evaluate the passing sight distance requirements on crest curves on two-lane highways. However, using these formulas to evaluate the available sight distance would produce only the minimum sight distance instead of the sight distance profile. Therefore, only the conclusion of whether a sight distance deficiency exists can be reached. Unless a graphical method

First, analytical models were developed to analyze daytime and nighttime sight distances on 2-D separate horizontal and vertical alignments using exact mathematical formulation for each element in the alignment (12–14). In horizontal alignments, the design elements are straight segments, circular curves, and spiral curves; in vertical alignments, they are straight segments and parabolic curves. Another element that is encountered in the vertical alignment of existing Canadian roads is the spline grade, which is a curve defined by a number of points with known stations and elevations but with no explicit mathematical formulation. The developed models cover

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virtually all types of sight obstructions including single and lateral horizontal obstructions in the horizontal alignment as well as crest curves and overpasses in the vertical alignment. The models were mainly iterative, where a small value of the available sight distance is assumed, and then increased until a farther small increment (which is the modeling accuracy) will produce a constrained sight distance. Closed-form relationships were also provided for the case of continuous obstruction on simple horizontal curves. However, the models could not consider the case of variable lateral clearance or lane width in the horizontal alignment or spline grades in the vertical alignment. More importantly, it was difficult to expand these models to 3-D combined alignments. Therefore, the finite element method was used to model 2-D separate alignments (15). According to this model, complex highway alignments and sight obstructions are simulated by a series of elements that can be modeled mathematically. Every element is defined by a number of points with known coordinates, which are referred to as nodes. The use of elements and nodes produces an exact modeling of straight segments and parabolic curves. On the other hand, the modeling of spline grades and circular curves is approximate, and small elements should be used to increase the modeling accuracy. Then, through an iterative procedure, the available sight distance can be determined to the required accuracy. Analytical Models: 3-D Combined Alignments The 2-D finite element model was expanded to 3-D combined alignments for both daytime and nighttime sight distances (16,17). Similar to the 2-D model, the alignment and the sight obstructions are simulated by a net of elements where each element is defined by nodes. For daytime, the model determines the available sight distance on the combined alignment from a specific driver’s eye height to a specific object height. By changing the object height, the model can be used to determine the available passing or SSD. For nighttime, the light beams produced by the vehicle’s headlights are modeled by the outer surfaces of two pyramids. The model determines the sight distance that is covered by the vehicle’s headlights. In addition, the model can account for lateral obstructions and crest curves.

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Computer Software Each of the developed models was coded into a computer program to evaluate the available sight distance. The programs for 2-D sight distance were verified by using artificial alignments on which the sight distance was measured graphically. For the 3-D sight distance, a verification was carried out with field measurements on an overpass that has a horizontal curve combined with a crest vertical curve. The verification results suggested that the developed models and software can accurately determine the available sight distance. By comparing the results of the exact and approximate 2-D modeling, it was also found that the approximate modeling in the finite element technique does not compromise the accuracy of the available sight distance. The software for 2-D and 3-D daytime sight distance (MARKS and MARKC, respectively) were used to determine the passing and nopassing zones in a case study involving a 7-km highway section of Highway 61 (Ontario, Canada). Figure 2 shows the profiles of 2-D and 3-D available passing sight distance, and Table 1 shows the marking of passing and no-passing zones by using the profiles of required and 3-D available passing sight distance. The application showed that the developed software can determine the profile of available sight distance on actual highway segments, where most of the required input data are currently available in the highway agencies. Using the software to determine the passing and no-passing zones on two-lane highways would save time, effort, and expenses and would provide highway designers with the flexibility to change the proposed alignment and check the gains or losses in the passing zones. It was also found that the 2-D analysis of sight distance can underestimate or overestimate the available sight distance, where the 2-D sight distance ranges from 50 to 110 percent of the 3-D sight distance. Application: 3-D Design The developed 3-D models were used in 3-D design of combined horizontal and vertical alignments. By adding iterative loops to the computer software, the minimum length of a vertical curve or the

FIGURE 2 Profile of available passing sight (PSD) distance on segment of Highway 61.

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TABLE 1

Marking Passing and No-Passing Zones on Segment of Highway 61

radius of a horizontal curve that is required to maintain a specific sight distance could be determined. Three main design applications were presented: (a) the required radius of a horizontal curve in a cut section, (b) the required length of a crest vertical curve in a fill section, and (c) the required length of a sag vertical curve in a fill or cut section. In the following sections, the 3-D value of a parameter refers to the value obtained through the 3-D analysis and the 2-D value refers to the value obtained by using the 2-D formulas presented in current design guides.

Minimum Radius of Horizontal Curve One of the design criteria for the radius of a horizontal curve, R, is to provide the required SSD if a lateral obstruction exists. An example of such a situation is a horizontal curve in a cut section where the cut side slope acts as a continuous lateral sight obstruction. By using the only relationship provided by AASHTO for this criterion, an average height between the driver’s eye and the object is used to calculate the lateral clearance, m (1). However, if the horizontal

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curve overlaps with a crest curve, the vertical curvature will cause the driver’s eye and the object to sink relative to the side slope, reducing the value of m, and vice versa on sag vertical curves. Thus, the required 3-D R can be longer or shorter than the 2-D value derived from the AASHTO formula if the horizontal curve overlaps with a crest or sag vertical curve, respectively. Table 2 shows the required 3-D R combined with a crest or sag vertical curve in a cut section where the side slope is 2;1 (horizontal to vertical) (18). The design speed is 110 km/h, and the other design parameters are taken according to the Canadian design guide (2). Minimum Length of Crest Vertical Curve On fill sections with no lateral obstructions, the road surface on a crest vertical curve is the main sight obstruction in the daytime. The formulas presented by AASHTO can be used to determine the minimum length of the vertical curve, L, or the minimum length per 1 percent change in grade, K (where K = L/A and A = algebraic difference of tangent grades), required for a specific daytime SSD. However, if the

TABLE 2

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crest curve overlaps with a horizontal curve, the horizontal curvature and the superelevation tend to increase the available 3-D sight distance relative to the 2-D value. Consequently, the required 3-D K may be less than the 2-D value. Table 3 shows the required 3-D K of a crest vertical curve combined with a horizontal curve in a fill section (18). The design speed is 110 km/h, and the other design parameters are taken according to the Canadian design guide (2). Minimum Length of Sag Vertical Curve At nighttime, sight distance on sag vertical curves is limited to the farthest point covered by the vehicle’s headlights regardless of the crosssection type (fill or cut). Assuming that the vehicle’s headlights can provide unlimited visibility, the AASHTO formulas can be used to determine the minimum L or K required for a specific nighttime SSD. However, if the sag curve overlaps with a horizontal curve, the horizontal curvature and the superelevation tend to reduce the available 3-D sight distance relative to the 2-D value. Consequently, the required 3-D K may be greater than the 2-D value. Figure 3 shows the

Required 3-D R of Horizontal Curve Combined with Vertical Curve

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TABLE 3

Required 3-D K of Crest Vertical Curve Combined with Horizontal Curve

required 3-D K of a sag vertical curve combined with a long horizontal curve (19). The design speed is 110 km/h, and the other design parameters are taken according to the Canadian design guide (2). In summary, by comparing the results of the traditional 2-D and the new 3-D designs, it was found that the required 3-D R can range from 30 to 130 percent of the 2-D value. Similarly, the required 3-D L (and 3-D K) of a crest curve may be as low as 50 percent of the 2-D value. On the other hand, the required 3-D L (and 3-D K) of a sag curve may be as large as 260 percent of the 2-D value. The highway superelevation and the degree of overlap between the horizontal and vertical alignments were found to affect the design significantly. Although the work has produced preliminary 3-D design aids, a complete set of such design aids is still needed. Red Zones The analysis of red zones is intended to identify the locations on a vertical (or horizontal) curve where a combined horizontal (or vertical) curve should not start. It can be an alternative to the 3-D design by providing a quantitative representation of the AASHTO general guidelines for combined horizontal and vertical curves, which currently rely on the designer’s judgment. In this case, the designer is assumed to have control over the relative positioning of the vertical and horizontal curves. On the basis of sight distance needs, two types of red zones can develop on an alignment designed by current 2-D practice: (a) preview sight distance (PVSD) red zones and (b) SSD red zones.

PVSD Red Zones PVSD is used here to refer to the sight distance required for a driver to detect the existence of a horizontal curve ahead and react to it. It was suggested that PVSD should consist of a distance on the tangent when drivers can react after detecting the curve by using another distance on the curve. Also, because drivers depend on the road marking, especially at nighttime, to read the road ahead, it was suggested that the PVSD be measured from the driver’s eye or the vehicle’s headlights (at daytime or nighttime, respectively) to the road marking. A driver traveling on a crest curve (at daytime or nighttime) or on a sag curve (at nighttime) will have a limited sight distance, which may be less than the PVSD. In such a situation, a PVSD red zone is identified and the designer should not allow the horizontal curve to start within the red zone unless flatter curves are used. With the developed 3-D models and software, the range of PVSD red zones was determined where a horizontal curve should not start relative to a crest or a sag vertical curve. This range of red zone is defined by two distances d1 and d2 measured from the beginning of the vertical curve (BVC) and the end of the vertical curve (EVC), respectively (Figure 4). The horizontal curve should not start from BVC + d1 to EVC + d2. Table 4 shows a sample of actual values for the range of PVSD red zones on crest vertical curves; PVSD red zones were found to be insignificant if the horizontal curve overlaps with a sag curve.

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FIGURE 3 Required 3-D K of sag vertical curve combined with horizontal curve.

SSD Red Zones

CONCLUDING REMARKS

It has been shown that current 2-D practice may overestimate the available sight distance. Therefore, a combined alignment designed according to current practice may have locations where the available SSD is less than required. The marking of SSD red zones determines such locations where an overlap between a horizontal and a vertical curve should be avoided unless flatter curves are used. The range of red zone is also defined by two distances, d1 and d2, measured from BVC and EVC, respectively (Figure 5). However, on SSD red zones, the horizontal curve not only should not start, it also should not overlap from BVC + d1 to EVC + d2. It was found that SSD red zones can develop on either a sag or a crest vertical curve, but they are more significant in the former case. Table 5 shows a sample of actual values for the range of SSD red zones on sag vertical curves.

Although the results of this research have not been implemented yet, they can cast more light on the effect of ignoring the 3-D nature of highway alignments. The tasks performed have shown that current 2-D design practice may overestimate the design requirements, causing extra and unnecessary expenditure. More importantly, this practice can underestimate the design requirements, causing construction of questionable road sections. In addition, ignoring the 3-D nature of the highway alignment will result in highway sections with safe speeds higher and lower than the intended design speed depending on the type of combined alignment. As a result, constructed highways will lack the design consistency required to minimize the driver’s workload and will have frequent deviations from the driver’s expectations.

FIGURE 4

Red zones for PVSD.

TABLE 4

FIGURE 5

Nighttime PVSD Red Zones on Crest Curve

Red zones for SSD.

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TABLE 5

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Nighttime SSD Red Zones on Sag Curve

The computer software that has been developed can help highway designers and professionals locate the actual substandard sections. At times when financial resources become more scarce, the funds available for improvements can be efficiently allocated to sections that need the improvement the most. The software can also be a valuable tool for determining passing and no-passing zones. It can provide designers with a great deal of flexibility to change the alignment and check the gains and losses in the passing zones. Thus, the level of service on the designed facility can be maximized. It should be noted, however, that a more comprehensive field verification is required before these findings are implemented. As the sight distance has been thoroughly researched, more research should be directed at 3-D analysis of other design bases—namely, vehicle stability, driver comfort, drainage, and aesthetics. ACKNOWLEDGMENT The financial assistance by the Natural Science and Engineering Research Council of Canada is gratefully acknowledged. REFERENCES 1. A Policy on Geometric Design of Highways and Streets. AASHTO, Washington, D.C., 1994. 2. Manual of Geometric Design Standards for Canadian Roads. Transportation Association of Canada, Ottawa, Ontario, 1986. 3. Mannering, F. L., and W. P. Kilareski. Principles of Highway Engineering and Traffic Analysis. John Wiley and Sons, Inc., New York, 1990. 4. Smith, B. L., and R. Lamm. Coordination of Horizontal and Vertical Alignment with Regard to Highway Aesthetics. In Transportation Research Record 1445, TRB, National Research Council, Washington, D.C., 1994, pp. 73–85. 5. Krammes, R. A., and M. A. Garnham. Review of Alignment Design Policies Worldwide. Presented at International Symposium on Highway Geometric Design Practices, Boston, Mass., 1995. 6. Neuman, T. R., and J. C. Glennon. Cost-Effectiveness of Improvements to Stopping Sight Distance. In Transportation Research Record 923, TRB, National Research Council, Washington, D.C., 1984, pp. 26–34. 7. Olson, P. L., D. E. Cleveland, P. S. Fancher, L. P. Koystyniuk, and L. W. Schneider. NCHRP Report 270: Parameters Affecting Stopping Sight Distance. NCHRP, National Research Council, Washington, D.C., 1984.

8. Waissi, G. R., and D. E. Cleveland. Sight Distance Relationships Involving Horizontal Curves. In Transportation Research Record 1122, TRB, National Research Council, Washington, D.C., 1987, pp. 96–107. 9. Easa, S. M. Lateral Clearance to Vision Obstacles on Horizontal Curves. In Transportation Research Record 1303, TRB, National Research Council, Washington, D.C., 1991, pp. 22–32. 10. Jull, D. E., and G. T. Murray. ITEDS: A Computer Aided Design System for Highway Engineering. Proceedings of the Annual Conference of the Canadian Society for Civil Engineering (CSCE), Halifax, Nova Scotia, May 23–25, 1984, pp. 431–445. 11. Sanchez, E. A 3-Dimensional Analysis of Sight Distance on Interchange Connectors. In Transportation Research Record 1445, TRB, National Research Council, Washington, D.C., 1994, pp. 101–108. 12. Hassan, Y., S. M. Easa, and A. O. Abd El Halim. Sight Distance on Horizontal Alignments with Continuous Lateral Obstructions. In Transportation Research Record 1500, TRB, National Research Council, Washington, D.C., 1995, pp. 31–42. 13. 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, No. 3, June 1996, pp. 577–586. 14. Easa, S. M., and Y. Hassan. Analysis of Headlight Sight Distance on Separate Highway Alignments: A New Approach. Canadian Journal of Civil Engineering, 1997, in press. 15. Hassan, Y., S. M. Easa, and A. O. Abd El Halim. Automation of Determining Passing and No-Passing Zones on Two-Lane Highways. Canadian Journal of Civil Engineering, Vol. 24, No. 2, April 1997, pp. 263–275. 16. Hassan, Y., S. M. Easa, and A. O. Abd El Halim. Analytical Model for Sight Distance Analysis on 3-D Highway Alignments. In Transportation Research Record 1523, TRB, National Research Council, Washington, D.C., 1996, pp. 1–10. 17. Hassan, Y., S. M. Easa, and A. O. Abd El Halim. Modeling Headlight Sight Distance on 3-D Highway Alignments. In Transportation Research Record 1579, TRB, National Research Council, Washington, D.C., 1997, pp. 79–88. 18. Hassan, Y., S. M. Easa, and A. O. Abd El Halim. Design Considerations for Combined Highway Alignments. Journal of Transportation Engineering American Society of Civil Engineers, Vol. 123, No. 1, Jan./Feb. 1997, pp. 60–68. 19. Hassan, Y., and S. M. Easa. Design of Sag Vertical Curves in ThreeDimensional Alignments. Journal of Transportation Engineering American Society of Civil Engineers, Vol. 124, No. 1, Jan./Feb. 1998, in press.

Publication of this paper sponsored by Committee on Geometric Design.