Sag Vertical Curves Research Articles

Consideration of Sight Distance in Placement of Concrete Barriers on Horizontal Curves of Roads

Sight distance is an important criterion in highway geometric design. Many design elements, such as horizontal and vertical curvatures and offsets to lateral obstructions, must be checked so that drivers have sufficient sight distance to operate safely. One of the obstacles that can restrict sight distance is the roadside or median concrete barriers that are frequently placed along the road edges for safety. There are common types and dimensions for such barriers, such as New Jersey, F-shape, and constant-slope concrete barriers. Such barriers may cause a significant reduction in the available sight distance on horizontal curves, especially in the lane closest to the barriers. However, there is a clear lack of a reliable, accurate tool to determine the available sight distance when the sight distance is restricted by road barriers in three-dimensional combined alignments. This paper uses a software package previously developed by the authors to examine closely the trends of available stopping sight distance and the effects of the overlapping vertical alignment. The results confirm previous findings that the available sight distance depends on the type of the vertical alignment and the curvature of crest or sag vertical curves overlapping on the horizontal curve. In addition, it is shown that the recommendations in current design guides could lead to barrier placement that compromises the available sight distance. Charts were prepared as an easy-to-use tool to estimate the available stopping sight distance on horizontal curves when overlapping with a specific vertical alignment.

Length requirements for single-arc asymmetrical sag vertical curve for highways

This paper presents the length requirements for a single-arc asymmetrical vertical curve (sag type) for highways that complements two other papers on the curve development and the length requirements (crest type) that have been published. The single-arc, asymmetrical curve takes the form of a cubic instead of parabolic function and its rate of change in grade gradually varies between the start and the end of the curve. The paper presents the length requirements of the single-arc asymmetrical curve based on the headlight, comfort, and overpass criteria. For the headlight criterion, sight distance (SD) relationships are first developed for the new curve considering different locations of the driver and object. The SD profile and minimum sight distance are then examined. Design aids are then established. The results show that the length requirements based on the comfort criteria are about 60% of those of the headlight criteria. The new single-arc asymmetrical curve improves curve characteristics, and as such would be of interest to the highway design community.

Improved Sight Distance Model for Sag Vertical Curves with Overpasses

On sag vertical curves underpassing a structure, the fascia of the structure may cut the line of sight and limit the available sight distance. The current AASHTO model for symmetric sag curves with overpasses is intended for those that lie close to the point of vertical intersection on long sag curves. This paper presents an improved model that provides the exact minimum sight distance ( Sm) for any conditions including non-centered overpasses and short sag curves, which are typical in urban areas. The model explicitly includes as variables the overpass location, the overpass width, and the configuration of the first and second grades of the sag vertical curve. The model is formulated by using mathematical optimization that involves an objective function and constraints. The objective function minimizes the available sight distance subject to constraints involving the geometry of the curve and the overpass. The model is spreadsheet based and is solved with powerful optimization software. Application of the model is illustrated with some examples and the results show that the AASHTO model may underestimate Sm by 10% to 30%. A closed-form model for Sm on single-arc asymmetric sag curves is also presented for Sm > L. The proposed model provides a friendly, flexible, and accurate tool for sight distance analysis and should be of interest to highway designers and practitioners.

Collision Prediction Models for Three-Dimensional Two-Lane Highways

Collision prediction models for three-dimensional (3-D) alignments of two-lane rural highways are lacking in the literature. This paper presents such models with vehicle collision data on 5,760 km (3,600 mi) of two-lane rural highways in Washington State collected from 2002 through 2005. Five statistical models were developed for different combinations of 3-D alignments to establish the relationship between collision frequency and the relevant variables. The alignment combinations are (a) horizontal curves combined with crest vertical curves, (b) horizontal curves combined with sag vertical curves, (c) horizontal curves combined with multiple vertical curves, (d) horizontal curves combined with grades of less than 5%, and (e) horizontal curves combined with grades of more than 5%. For each combination, four different statistical techniques were explored: Poisson, negative binomial, zero-inflated Poisson, and zero-inflated negative binomial. For validation, two models were selected and estimated by using the first 3 years of collision data and validated with the last-year data. The results show that the most significant predictors for collisions on horizontal curves on 3-D alignments are the degree of curvature, roadway width (lanes plus shoulders), access density, product of grade value and grade length, road section length, and average annual daily traffic. The results of this study should be useful in evaluating road safety on 3-D alignments and optimizing their design based on the substantive safety approaches.

Efficient Method for Estimating Globally Optimal Simple Vertical Curves

Different methods for estimating simple vertical curves that optimally fit observed profile data have been developed. In 1999, the author developed a linear programming (LP) method for estimating simple vertical curves using LINGO optimization software. To obtain the global optimal solution, the LP formulation was manually solved for different combinations of the two unknown nonlinear variables (using 5m increments). In 2004, an improved method that automates the iterations using Visual-Basic in Excel Solver was published. The global optimal solution required 10h for an increment of 0.1m. This technical note presents an extension of the previously developed LP formulation that converges to the global optimal solution in a minute. The formulation involves no iterations of the nonlinear variables. Instead, the start and end points of the parabolic curve were modeled using three binary variables, and the resulting mixed-integer nonlinear model was solved using LINGO global option that has been recently developed. The proposed method, which is applicable to both crest and sag vertical curves, should be of interest to surveying professionals.

Assessment of Sag Curve Design Criteria considering Modern Headlamp Performance

For more than 50 years, headlight sight distance has been one method for calculating the length of sag vertical curves; the others are passenger comfort, drainage control, and general appearance. The headlight sight distance is based on a 1° upward divergence of the headlamp beam. Over the past 20 years, headlamps have transitioned from being 100% sealed beam to modern replaceable bulb headlamps that project less light above the horizontal plane than the sealed-beam headlamps that are the basis for sag curve design. This paper describes theoretical and field measurements of the illuminance levels provided by sealed-beam and modern headlamps. The results of the analysis indicate that modern headlamps provide significantly less light above the horizontal than sealed-beam headlamps, indicating a potential need to modify the design equations for sag vertical curves. According to the theoretical analysis, the upward divergent headlamp angle used in the sag curve design equation should be reduced from 1° to between 0.75° and 0.90°. The field analysis indicated a significant difference in illuminance levels from the theoretical analysis, but the results also indicate a need to reduce the headlamp angle used in sag curve design.

How Modern Headlamp Performance Impacts Sag Vertical Curve Design

For over 50 years, the basis for calculating the length of sag vertical curves has been the sight distance based on a 1° upward divergence of the headlamp beam. Over the last 20 years, headlamps have transitioned from being 100% sealed beam to modern replaceable bulb headlamps that project less light above the horizontal plane compared to the sealed beam headlamps that are the basis for the sag curve design. This paper analyzes the performance of modern headlamps with respect to sag vertical curve design by calculating the amount of illumination reaching the pavement in a sag curve. The analysis indicates that the headlamp beam angle used to calculate sag curve length should change from 1 to 0.75° to provide an illuminance level with modern headlamps that is equal to the illuminance provided by sealed beam headlamps. Such a change would increase the K values for sag curve design from 13, 30, and 55 to 15, 37, and 70 for design speeds of 110, 80, and 50 km/h, respectively.

Improved speed-profile model for two-lane rural highways

The speed-profile model has been suggested as a valuable tool for evaluating geometric design consistency for two-lane rural highways with isolated and combined horizontal and vertical alignments. The model determines the operating speeds on the speed-change (SC) segment, which is the distance between speed-limiting curves. The speed-limiting curves are the horizontal curves and the limited sight-distance crest vertical curves on horizontal tangents, where the sight distance required by the design guides is not satisfied. The model assumes that deceleration begins where required, which implies that the next curve is visible when deceleration starts. This paper presents an extension to the speed-profile model to incorporate the effect of sight obstruction on operating speeds and deceleration rates. The SC segment may include nonlimited sight–distance crest vertical and sag vertical curves. These curves may present sight obstruction. A procedure to determine whether the sight line is obstructed is developed. If it does, simple formulas are applied for revising the operating-speed profile. The extended model is suitable for inclusion in the design consistency module of the interactive highway safety design model.Key words: speed profile, model, two-lane highways, alignments, design consistency.

Improved Design of Vertical Curves with Sight Distance Profiles

Design of vertical alignment is one of the main tasks in highway geometric design. This task requires, among other things, that the designer ensure drivers always have a clear view of the road so they can stop before hitting an unexpected object in 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 United States and Canada provide simple analytical models for determining 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. Two specific models were expanded to determine the required SSD on crest and sag vertical curves. By comparing profiles of available SSD and required SSD on examples of vertical curves, it was shown that current North American design practices might yield segments of the vertical curve where the driver’s view is constrained to a distance shorter than the required SSD. An alternative design procedure based on the models was developed and used to determine the minimum lengths of crest and sag vertical curves. Depending on the approach grade, these new values of minimum curve length might be greater than or less than values obtained through conventional design procedures. Design aids were therefore provided in tabular form for designers’ easy and quick use.

Influence of Vertical Alignment on Horizontal Curve Perception: Phase II: Modeling Perceived Radius

Phase I of a study established that perception of horizontal curvature depends on the overlapping vertical alignment. A horizontal curve radius will be perceived incorrectly if the curve overlaps with a crest or sag vertical curve. Phase II quantified the extent of such erroneous perception. An experiment involving a three-dimensional static presentation was carried out to determine the mean value of perceived horizontal curve radius for different sets of alignment parameters. The results of the experiment were analyzed statistically to examine the effects of the alignment parameters on the perceived radius. The parameters that were found to significantly affect the perceived radius were the type of overlapping vertical curve, actual radius, and turning direction (on crest and sag curves) and sight distance (on sag curves). In addition, simple statistical models were developed using regression analysis to estimate the perceived radius of any horizontal curve in a combined alignment.

Prediction of Operating Speed on Three-Dimensional Highway Alignments

Achieving consistent geometric design is an important goal in highway design to ensure obtaining safe, economical, and smooth traffic operation. Existing operating speed models for design consistency in North America and Europe are mainly based on two-dimensional (2D) analysis of highway horizontal alignments. This paper develops operating speed models for two-lane rural highways that account for the three-dimensional (3D) nature of highways. The models will help highway designers to predict operating speed and evaluate design consistency more accurately, and thus aid highway safety. Two types of 3D combinations were considered: a horizontal curve combined with a sag vertical curve and a horizontal curve combined with a crest vertical curve. Regression analysis was used to develop the operating speed models based on data collected on Highway 61 and Highway 102 in Ontario. The results show that there is a significant difference between the predicted operating speed using the 2D and 3D models. Therefore, it is recommended that the developed 3D models be used in highway consistency analysis and evaluation.

Development of transitioned vertical curve I Properties

Introducing spiral curves before and after horizontal circular curves has been widely accepted to enhance traffic safety, highway esthetics, sight distance, and driver comfort. Though, vertical curves are still designed as parabolic curves that are connected directly to the tangent (without transitions). In this paper, a cubic polynomial is used to develop a vertical transition curve before and after the parabolic vertical curve. The resulting curve, called transitioned vertical curve, consists of transition-parabolic-transition segments. Detailed mathematical formulation and derivation of the instantaneous elevation, grade, rate of curvature, and offset from the first tangent at any point are presented. The highest (or lowest) point on a transitioned crest (or sag) vertical curve, where the instantaneous grade equals 0, is determined as it is of particular importance in highway drainage design. The minimum length of a transition curve is derived based on the criterion of driver comfort. In addition, guidelines are provided to identify the conditions where the drainage of surface water on transitioned curves can be a concern. Finally, the layout of the transitioned vertical curve is described and illustrated using two numerical examples. The new transitioned vertical curve, which exhibits striking similarities to the spiraled horizontal curve, should enhance the design of highway vertical alignments.

New Stopping Sight Distance Model for Use in Highway Geometric Design

Stopping sight distance is an important design parameter in that it defines the minimum sight distance that must be provided at all points along the highway. Thus, it influences geometric design values, construction costs, and highway safety. Stopping sight distance is defined as the sum of two components—brake reaction distance and braking distance. The basic model for calculating stopping sight distances was formalized in 1940, and the model’s parameters have been altered to compensate for changes in eye height, object height, and driver behavior over the past 50 years. Recent studies, however, question whether the model’s parameters and assumptions represent real-world conditions. A new model for determining stopping sight distance requirements for geometric design of highways is presented. This model is based on parameters describing driver and vehicle capabilities that can be validated with field data and defended as safe driving behavior. More than 50 drivers, 3,000 braking maneuvers, 1,000 driver eye heights, and 1,000 accident narratives were used to develop the recommended parameter values for the new model. The recommended values are attainable by most drivers, vehicles, and roadways. This model results in stopping sight distances, sag vertical curve lengths, and lateral clearances that are between the current minimum and desirable requirements and crest vertical curve lengths that are shorter than current minimum requirements.

Impact of highway consistency on capacity utilization of two-lane rural highways

Consistent highway design is expected to provide safe, economical, and smooth traffic operation. Several studies have been performed to investigate the effect of highway consistency on traffic safety. However, the relationship between design consistency and highway capacity and level of service has not been addressed in current research work and design practices. In addition, the effect of the three-dimensional (3D) nature of highway alignments was not considered, and design consistency was studied based solely on two-dimensional (2D) analysis of highway horizontal alignments. This paper presents a methodology to determine the effect of highway design consistency on highway capacity utilization based on 3D analysis. This methodology will help road designers to estimate highway capacity more accurately. The study was performed on two-lane rural highways in Ontario, where two types of 3D combinations were considered: a horizontal curve combined with a sag vertical curve (sag combination) and a horizontal curve combined with a crest vertical curve (crest combination). An additional adjustment factor that reflects the effect of highway design consistency on capacity utilization was developed. Different statistical models are introduced to estimate this factor based on geometric or traffic data. In addition, typical values of the consistency factor were developed based on an overall consistency evaluation criterion and can be easily used in capacity analysis.Key words: three-dimensional, alignments, capacity, geometric design, operating speed, design consistency.

Evaluation of Zero-Length Vertical Curves

To evaluate the use of zero-length vertical curves with respect to Texas Department of Transportation design practice, construction results, and vehicle dynamics, and to compare zero-length vertical curves with minimum-length design vertical curves, 20 zero-length vertical curves and 15 minimum-length vertical curves were evaluated. Evaluations performed included examinations of sight distance for zero-length vertical curves, surveying roadway profiles, and measuring vertical accelerations. Sight distance criteria were found to be generally inapplicable below 2 percent for sag vertical curves. For crest vertical curves, sight distance criteria were inapplicable below 1 percent for design speeds of 90 km/h or less and inapplicable below 0.5 percent for design speeds of 100 to 120 km/h. The results of vertical acceleration testing showed that, below 0.5 percent grade change, no practical difference was found between zero-length and minimum-design-length vertical curves. Between 1.0 and 0.5 percent grade change, significantly higher accelerations were measured for high-speed tests on zero-length vertical curves. Finally, it was found that zero-length vertical curves were more likely to meet drainage grade requirements than minimum-design-length vertical curves were.

Sight distance red zones on combined horizontal and sag vertical curves

Coordination of highway horizontal and vertical alignments is based on subjective guidelines in current standards. This paper presents a quantitative analysis of coordinating horizontal and sag vertical curves that are designed using two-dimensional standards. The locations where a horizontal curve should not be positioned relative to a sag vertical curve (called red zones) are identified. In the red zone, the available sight distance (computed using three-dimensional models) is less than the required sight distance. Two types of red zones, based on stopping sight distance (SSD) and preview sight distance (PVSD), are examined. The SSD red zone corresponds to the locations where an overlap between a horizontal curve and a sag vertical curve should be avoided because the three-dimensional sight distance will be less than the required SSD. The PVSD red zone corresponds to the locations where a horizontal curve should not start because drivers will not be able to perceive it and safely react to it. The SSD red zones exist for practical highway alignment parameters, and therefore designers should check the alignments for potential SSD red zones. The range of SSD red zones was found to depend on the different alignment parameters, especially the superelevation rate. On the other hand, the results showed that the PVSD red zones exist only for large values of the required PVSD, and therefore this type of red zones is not critical. This paper should be of particular interest to the highway designers and professionals concerned with highway safety.Key words: sight distance, red zone, combined alignment.

Revised Design Parameters for Vertical Curves

A Policy on Geometric Design of Highways and Streets, by the American Association of State Highway and Transportation Officials (AASHTO), exhibited severe disparities between its recommended procedures for the computation of braking distances on grades and the design controls of vertical curves. The braking distances it advocated for the design of vertical curves are fully consistent with flat terrains. The grades of the tangents only influenced the curvatures needed to achieve braking distances and thus stopping sight distances on flat grades. A vehicle traveling on a crest vertical curve of Type II, or a sag vertical curve of Type IV, experienced at no time a flat grade. Types II and IV vertical curves join tangents with identical grade orientations. This study pinpoints the inconsistencies of the previous procedures and proposes conciliatory procedures. A new methodology computes simultaneously the worst grades of braking initiation, the associated braking and stopping sight distances, and the resultin...

Design of Sag Vertical Curves in Three-Dimensional Alignments

Headlight sight distance has been an important element in designing sag vertical curves, because nighttime drivers need sufficient sight distance for stopping and because they depend on the road marking to maintain uniform speed and proper lane positioning. Considering an isolated sag vertical curve in a two-dimensional projection based on the sight distance needs for stopping, the current design standards recommend a minimum length of the sag curve. However, little research work has been directed toward determining the available headlight sight distance in three-dimensional (3D) highway alignments. A new analytical model developed by the writers for 3D analysis of headlight sight distance has shown that the 2D analysis may overestimate the available headlight sight distance on sag vertical curves combined with horizontal curves. In this paper, a computer program based on the 3D analytical model is used to evaluate the current 2D standards for sag vertical curves on fill sections. The results show that the current 2D design standards overestimate the available headlight sight distance and therefore underestimate the required length of sag vertical curves combined with horizontal curves. In this paper, more accurate values for the required length of sag vertical curves combined with horizontal circular curves are developed. A more economical design is also suggested through the use of asymmetrical (compound) sag vertical curves.

Sight distance red zones on combined horizontal and sag vertical curves

Sight distance red zones on combined horizontal and sag vertical curves

Modeling Headlight Sight Distance on Three-Dimensional Highway Alignments

Sight distance is one of the major elements that must be considered in the geometric design to achieve safe and comfortable highways. Daytime sight distance has been extensively studied, and analytical models for two-dimensional (2-D) and three-dimensional (3-D) alignments have been developed. However, nighttime (headlight) sight distance has received less attention. Despite the higher accident rate during nighttime than during daytime, existing analytical models evaluating headlight sight distance are very primitive. Moreover, the interaction between the horizontal and vertical alignments has not been modeled. A four-phase analytical model for headlight sight distance on 3-D combined alignments is presented. The model is an application of the finite-element technique in highway geometric design. The model can determine the maximum distance that can be covered by the vehicle’s headlights and that is not obstructed by any sight obstructions including the road surface. On the basis of this analytical model, computer software was developed and used in a preliminary application for 3-D headlight sight distances on a sag or crest vertical curve combined with a horizontal curve. The application showed that the 3-D sight distance on sag vertical curves was generally lower than the corresponding 2-D value when the sag curve was overlapping with a horizontal curve. On the other hand, the overlapping of horizontal curves with crest vertical curves enhanced the 3-D sight distance. The difference between 2-D and 3-D sight distance values in both cases increased with a decrease in the horizontal curve radius and an increase in the pavement cross slope. The model was proved to be extremely valuable in establishing 3-D highway geometric design standards.