Crest Vertical Curve Research Articles

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.

Required Stopping Sight Distance on Crest Vertical Curves

Required stopping sight distance (SSD), used to calculate the minimum rate of vertical curvature or the minimum length of a crest vertical curve (CVC), is usually calculated on the assumption that the grade of the braking section is zero (G = 0). This assumption is not the worst case and may lead to the selection of a CVC on which there is a segment with SSD restriction. On each CVC there are braking sections on one or both directions of travel, on which average grade is negative (G < 0). In the present paper a method for calculating required SSD, partly or wholly on a CVC, is developed, using the average grade over the braking distance. In addition, a computer program is compiled for calculating the value of the required SSD for all driver positions before or on a CVC. An SSD profile is drawn and it is proven that, for a significant length of the CVC, the required SSD is greater than the value used for calculating the minimum value of the rate of vertical curvature.

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.

Operating Speed Prediction on Crest Vertical Curves of Rural Two-Lane Highways in Nebraska

Since the repeal of the National Maximum Speed Limit by the U.S. Congress, individual states have been modifying existing speed limits on state and federal highways. Increases in speed limits have caused the inferred design speeds of many crest vertical curves along two-lane rural highways to be less than the posted and operating speeds according to accepted national guidelines. Past research has found that rate of curvature and inferred design speed can be used to predict operating speeds on crest vertical curves. Crest vertical curve, roadway, traffic, and speed characteristic data are included to determine the features that influence the speeds on crest vertical curves. Speed data were collected on 70 crest vertical curves across Nebraska. Multiple linear regressions were included to determine the crest vertical curve, roadway, traffic, and speed characteristics that affect speeds on crest vertical curvature. The posted speed of the highway was found to have the most influence on the operating speed. The inferred design speed of the vertical curves was not a significant factor.

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.

Operating Speed on Crest Vertical Curves with Limited Stopping Sight Distance

Horizontal and vertical elements of a highway are designed based on an assumed design speed. This concept was developed in the 1930s as a mechanism for designing rural alignments to permit most drivers to operate uniformly at their desired speed. In 1938, AASHO recognized that drivers select a speed influenced by the roadway environment instead of an assumed design speed. Recent research suggests that design speed is no longer the speed adopted by the faster group of drivers but that it has become a value used to establish the sharpness of horizontal and vertical design elements. The objective of this study was to establish the relationship between design and operating speeds for crest vertical curves with limited sight distance. Geometric data and 3,500 paired speeds (speeds at control and crest sections) were collected at 36 sites in 3 states. The results indicated that both the 85th percentile and the mean operating speeds were well above the inferred design speeds of the crest vertical curves for the range of conditions studied and that the lower the design speed the larger the difference between the 85th percentile speed and the design speed. The mean reductions in speed between the control and crest sections tend to increase as available sight distance is decreased; however, the reduction in speed is less than that suggested by current AASHTO criteria.

Safety Effects of Limited Stopping Sight Distance on Crest Vertical Curves

Previous studies that examined the relationship between stopping sight distance and safety have been inconclusive and inconsistent; however, the fact that a relationship has not been established does not necessarily imply that stopping sight distance is not a contributing factor in some crashes. Instead, it means that existing databases have failed to quantify any relationship that might exist. A promising approach to identifying such a relationship is a detailed examination or case study of crashes from a relatively large sample of limited sight distance roadways. If limited sight distance is a factor that contributes to crashes, it should show up in such a study. The objective of this study was to determine whether stopping sight distance was a contributing factor in crashes on roadway segments with limited sight distance crest vertical curves. This objective was accomplished by reviewing 439 narratives from crashes that occurred on 33 multilane and two-lane roadways with limited sight distance crest vertical curves. The findings suggest that the crash rates on rural two-lane highways with limited stopping sight distance are similar to the crash rates on all two-lane rural highways. They also suggest that the percentage of accidents involving large trucks and older drivers is similar on limited sight distance highways and all two-lane rural highways. Thus, for the range of conditions studied, limited stopping sight distance does not appear to be a safety problem.

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.

Optimum Vertical Curves for Highway Profiles

Information on the vertical alignment of existing old highways is often not available. To evaluate the highway alignment, it is necessary to fit vertical curves (parabolic curves and associated tangents) to the observed profile data. This paper presents a linear optimization model that determines the best parabolic curve/tangents that fit a given profile. The decision variables of the model are the grades of the first and second tangents and the location of the start and end of the parabolic curve. The objective function of the model minimizes the sum of the absolute deviations between the observed profile and the vertical curve. The model includes a constraint to make the point of vertical intersection lie in the middle of the curve, thus ensuring that the estimated grades are indeed tangents to the parabolic curve. Extensions of the model to accommodate another objective and multiple vertical curves are presented. The application of the model was illustrated using a numerical example involving a crest vertical curve. The proposed model can be easily solved using optimization software that are readily available in practice.

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.

Aspects of Stopping-Sight Distance on Crest Vertical Curves

In the present study a simple methodology is developed, by which the values of available sight distance, for any driver location on crest vertical curves (CVCs), can be precisely and directly calcu...

Design Considerations of Sight Distance Red Zones on Crest Curves

The American Association of State Highway and Transportation Officials (AASHTO) presents general controls for the combination of horizontal and vertical alignments. Excellence in alignment coordination enhances safety, operation, and appearance. This paper quantifies one of the AASHTO general controls that is related to introducing a horizontal curve at or near the top of a crest vertical curve. It establishes the locations where a horizontal curve should not start in relation to a vertical curve because the available sight distance is less than (1) the stopping-sight distance (SSD); or (2) the preview sight distance (PVSD) that allows the driver to detect the horizontal curve and react properly and comfortably. The range of these locations, called the red zone, is established for a selected alignment using three-dimensional (3D) analysis of sight distance. Both daytime and nighttime cases are discussed. The sensitivity of the red zones for PVSD to the superelevation rate, turning direction of horizontal curve, vertical curvature, and spiral curves is examined. The effect of spiral curves and location of cut slopes on the red zone for daytime SSD is also examined. This paper shows how 3D analysis can be used to identify red zones for existing combined alignments or to avoid them for new alignments. As such, it should be of interest to the design engineers concerned with highway safety.

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 Considerations for Combined Highway Alignments

According to current highway design standards, horizontal and vertical alignments are treated separately in two-dimensional (2D) projections. In this paper, a model developed by the writers is used to study the effects of considering three-dimensional (3D) alignment on design requirements for sight distance. First, the paper examines the required minimum radius of a horizontal curve combined with a crest or sag vertical curve in a cut section. Also, the required minimum length of a crest vertical curve combined with a horizontal curve in a fill section is studied. The results show that the 3D design requirements may differ significantly from those in separate 2D projections. Depending on the vertical alignment and on the cut side-slope, 2D design may underestimate or overestimate the required radius. In fill sections, the 2D alignment overestimates the required length of the crest vertical curve when combined with long horizontal curves. In summary, the 2D alignment can compromise both safety and economics of highways. As a result, it is concluded that 3D-based design standards should be developed and used in highway geometric design.

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.

Sight distance evaluation on complex highway vertical alignments

Sight distance (stopping, passing, and decision) is a key element in highway geometric design. Existing models for evaluating sight distance on vertical alignments are applicable only to simple, isolated elements such as a crest vertical curve, a sag vertical curve, and a reverse vertical curve (a sag curve following a crest curve, or vice versa). This paper presents an analytical methodology for evaluating sight distance on complex vertical alignments that involve any combination of vertical alignment elements. The methodology can be used for evaluating passing sight distance on two-lane highways, and stopping sight distance and decision sight distance on all highways. Sight distance controlled by the headlight beam can also be evaluated. The locations of sight-hidden dips, which may develop when a sag vertical curve follows a crest vertical curve with or without a common tangent, can be determined. Also, sight distances obstructed by overpasses are evaluated. A profile of the available sight distance can be established and used to evaluate sight distance deficiency and the effect of alignment improvements. A software was developed and can be used for determining the available sight distance accurately. The software may replace the current field and graphical practice for establishing the no-passing zones and evaluating stopping and decision sight distances on complex vertical alignments. Key words: sight distance, vertical alignment, highway, passing zones.

Principles of highway engineering and traffic analysis

Chapter 1: Introduction to Highway Engineering and Traffic Analysis. 1.1 Introduction. 1.2 Highways and the Economy. 1.2.1 The Highway Economy. 1.2.2 Supply Chains. 1.2.3 Economic Development. 1.3 Highways, Energy and the Environment. 1.4 Highways and the Transportation System. 1.5 Highway Transportation and the Human Element. 1.5.1 Passenger Transportation Modes and Traffic Congestion.1.5.2 Highway Safety.1.5.3 Demographic Trends.1.6 Highways and Evolving Technologies.1.6.1 Infrastructure Technologies.1.6.2 Vehicle Technologies. 1.6.3 Traffic Control Technologies.1.7 Scope of Study. Chapter 2: Road Vehicle Performance. 2.1 Introduction. 2.2 Tractive Effort and Resistance. 2.3 Aerodynamic Resistance. 2.4 Rolling Resistance. 2.5 Grade Resistance. 2.6 Available Tractive Effort. 2.6.1 Maximum Tractive Effort. 2.6.2 Engine-Generated Tractive Effort. 2.7 Vehicle Acceleration. 2.8 Fuel Efficiency. 2.9 Principles of Braking. 2.9.1 Braking Forces. 2.9.2 Braking Force Ratio and Efficiency. 2.9.3 Antilock Braking Systems. 2.9.4 Theoretical Stopping Distance. 2.9.5 Practical Stopping Distance. 2.9.6 Distance Traveled During Driver Perception/Reaction. Chapter 3: Geometric Design of Highways. 3.1 Introduction. 3.2 Principles of Highway Alignment. 3.3 Vertical Alignment. 3.3.1 Vertical Curve Fundamentals. 3.3.2 Stopping Sight Distance. 3.3.3 Stopping Sight Distance and Crest Vertical Curve Design. 3.3.4 Stopping Sight Distance and Sag Vertical Curve Design. 3.3.5 Passing Sight Distance and Crest Vertical Curve Design. 3.3.6 Underpass Sight Distance and Sag Vertical Curve Design. 3.4 Horizontal Alignment. 3.4.1 Vehicle Cornering. 3.4.2 Horizontal Curve Fundamentals. 3.4.3 Stopping Sight Distance and Horizontal Curve Design. 3.5 Combined Vertical and Horizontal Alignment. Chapter 4: Pavement Design. 4.1 Introduction. 4.2 Pavement Types. 4.2.1 Flexible Pavements. 4.2.2 Rigid Pavements 95. 4.3 Pavement System Design: Principles for Flexible Pavements. 4.3.1 Calculation of Flexible Pavement Stresses and Deflections. 4.4 The AASHTO Flexible-Pavement Design Procedure. 4.4.1 Serviceability Concept. 4.4.2 Flexible-Pavement Design Equation. 4.4.3 Structural Number. 4.5 Pavement System Design: Principles for Rigid Pavements. 4.5.1 Calculation of Rigid-Pavement Stresses and Deflections. 4.6 The AASHTO Rigid-Pavement Design Procedure. 4.7 Measuring Pavement Quality and Performance. 4.7.1 International Roughness Index. 4.7.2 Friction Measurements. 4.7.3 Rut Depth. Chapter 5: Fundamentals of Traffic Flow and Queuing Theory. 5.1 Introduction. 5.2 Traffic Stream Parameters. 5.2.1 Traffic Flow, Speed, and Density. 5.3 Basic Traffic Stream Models. 5.3.1 Speed-Density Model. 5.3.2 Flow-Density Model. 5.3.3 Speed-Flow Model. 5.4 Models Of Traffic Flow. 5.4.1 Poisson Model. 5.4.2 Limitations of the Poisson Model. 5.5 Queuing Theory and Traffic Flow Analysis. 5.5.1 Dimensions of Queuing Models. 5.5.2 D/D/1 Queuing. 5.5.3 M/D/1 Queuing. 5.5.4 M/M/1 Queuing. 5.5.5 M/M/N Queuing. 5.6 Traffic Analysis at Highway Bottlenecks. Chapter 6: Highway Capacity and Level-of-Service Analysis. 6.1 Introduction. 6.2 Level-of-Service Concept. 6.3 Level-of-Service Determination. 6.3.1 Base Conditions and Capacity. 6.3.2 Determine Free-Flow Speed. 6.3.3 Determine Analysis Flow Rate. 6.3.4 Calculate Service Measure(s) and Determine LOS. 6.4 Basic Freeway Segments. 6.4.1 Base Conditions and Capacity. 6.4.2 Service Measure. 6.4.3 Determining Free-Flow Speed. 6.4.4 Determining Analysis Flow Rate. 6.4.5 Calculating Density and Determining LOS. 6.5 Multilane Highways. 6.5.1 Base Conditions and Capacity. 6.5.2 Service Measure. 6.5.3 Determining Free-Flow Speed. 6.5.4 Determining Analysis Flow Rate. 6.5.5 Calculating Density and Determining LOS. 6.6 Two-Lane Highways. 6.6.1 Base Conditions and Capacity. 6.6.2 Service Measures. 6.6.3 Determining Free-Flow Speed. 6.6.4 Determining Analysis Flow Rate. 6.6.5 Calculate Service Measures. 6.6.6 Determine LOS. 6.7 Design Traffic Volumes. Chapter 7: Traffic Control and Analysis at Signalized Intersections. 7.1 Introduction. 7.2 Intersection and Signal Control Characteristics. 7.2.1 Actuated Control. 7.2.2 Vehicle Detection. 7.2.3 Typical Phase Operation. 7.2.4 Signal Controller Operation. 7.3 Analysis of Traffic at Signalized Intersections. 7.3.1 Concepts and Definitions. 7.3.2 Signalized Intersection Analysis with D/D/1 Queuing. 7.3.3 Signalized Intersection Analysis for Level of Service. 7.4 Optimal Traffic Signal Timing. 7.5 Development of a Traffic Signal Phasing and Timing Plan. 7.5.1 Select Signal Phasing. 7.5.2 Establish Analysis Lane Groups. 7.5.3 Calculate Analysis Flow Rates and Adjusted Saturation Flow Rates. 7.5.4 Determine Critical Lane Groups and Total Cycle Lost Time. 7.5.5 Calculate Cycle Length. 7.5.6 Allocate Green Time. 7.5.7 Calculate Change and Clearance Intervals. 7.5.8 Check Pedestrian Crossing Time. 7.6 Level-of-Service Determination. 7.7 Signal Coordination. 7.7.1 Fundamental Relationships. 7.7.2 Effective Green to Cycle Length Ratio (g/C). 7.7.3 Platoon Dispersion. 7.7.4 State of the Practice. 7.7.5 Progression Quality and Level of Service Analysis. 7.8 The Progression Adjustment Factor and Arrival Type. Chapter 8: Travel Demand and Traffic Forecasting. 8.1 Introduction. 8.2 Traveler Decisions. 8.3 Scope of the Travel Demand and Traffic Forecasting Problem. 8.4 Trip Generation. 8.4.1 Typical Trip Generation Models. 8.4.2 Trip Generation with Count Data Models. 8.5 Mode and Destination Choice. 8.5.1 Methodological Approach. 8.5.2 Logit Model Applications. 8.6 Highway Route Choice. 8.6.1 Highway Performance Functions. 8.6.2 User Equilibrium. 8.6.3 Mathematical Programming Approach to User Equilibrium. 8.6.4 System Optimization. 8.7 Traffic Forecasting in Practice. 8.8 The Traditional Four-Step Process. 8.9 The Current State of Travel Demand and Traffic Forecasting. Appendix 8A: Least Squares Estimation. Appendix 8B: Maximum-Likelihood Estimation. Appendix A: Metric Example Problems. Appendix B: Metric End-of-Chapter Problems. Index.