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RELIABILITY BASED DESIGN OF HORIZONTAL CURVES CONSIDERING THE EFFECTS OF GRADES
ABSTRACT
Current horizontal curve geometric design policy, contained in the American Association of State Highway and Transportation Officials’ A Policy on Geometric Design of Highways and Streets, is not performance-based, establishing only minimum criteria and limiting values centered on the design speed concept. The current model used for horizontal curve design considers demand side friction as a passenger comfort level, and does not account for the effects of longitudinal friction demands (such as vertical grade or vehicle deceleration). This research uses a modified version of the point-mass model, with data collected at 99 horizontal curves in 7 states, to develop a probabilistic approach for horizontal curve design. Pavement friction supply and demand are compared using first order reliability methods. A reliability index is established for each of the observed horizontal curves and its association to crash frequency is assessed. Results indicate that trade-offs in design values (such as curve radius or superelevation rate) are quantifiable for a performance based design approach. The results show that vertical grade and driver deceleration should be considered, especially in high-speed design. An alternative design methodology is proposed as part of this research.
TABLE OF CONTENTS
List of Figures……………………………………………………………………………………………………….. vi
List of Tables…………………………………………………………………………………………………………. ix
Acknowledgements……………………………………………………………………………………………….. xiii
Key Definitions and Terms……………………………………………………………………………………… xiv
Chapter 1 Introduction ………………………………………………………………………………………………. 1
1.1 Background …………………………………………………………………………………………………. 1
1.2 Research Objectives ……………………………………………………………………………………… 4
1.3 Significance of Research ……………………………………………………………………………….. 5
1.4 Organization of Thesis ………………………………………………………………………………….. 6
Chapter 2 Literature Review ………………………………………………………………………………………. 10
2.1 Horizontal Curve Design Policy …………………………………………………………………….. 10
2.1.1 Design Speed ……………………………………………………………………………………… 11
2.1.2 Design Side Friction ……………………………………………………………………………. 11
2.1.3 Design Superelevation Rate …………………………………………………………………. 13
2.1.4 Minimum Horizontal Curve Radius ………………………………………………………. 13
2.2 Design Side Friction ……………………………………………………………………………………… 14
2.3 Horizontal Curve Design……………………………………………………………………………….. 20
2.4 Accommodation of Heavy Trucks ………………………………………………………………….. 31
2.5 Pavement Friction Supply and Demand …………………………………………………………… 37
2.6 Reliability in Highway Engineering ………………………………………………………………… 44
2.7 Safety Modeling …………………………………………………………………………………………… 52
Chapter 3 Data Collection and Characterization ……………………………………………………………. 55
3.1 Data Collection ……………………………………………………………………………………………. 56
3.1.1 Operating Speed Data Collection ………………………………………………………….. 563.1.2 Roadway Friction Data Collection ………………………………………………………… 653.1.3 Vehicle Acceleration/Deceleration Data ………………………………………………… 73
3.1.4 Vehicle Characteristics for Rollover ……………………………………………………… 75
3.2 Data Distributions ………………………………………………………………………………………… 77
3.2.1 Operating Speed …………………………………………………………………………………. 773.2.2 Pavement Friction ………………………………………………………………………………. 80
3.2.3 Vehicle Acceleration/Deceleration ………………………………………………………… 84
3.3 Correlations among Random Variables …………………………………………………………… 87
3.4 Safety Data ………………………………………………………………………………………………….. 89Chapter 4 Methodology …………………………………………………………………………………………….. 94
4.1 Friction Demand ………………………………………………………………………………………….. 94
4.1.1 Longitudinal Forces …………………………………………………………………………….. 95
4.1.2 Lateral Forces …………………………………………………………………………………….. 96
4.1.3. Forces Associated with Rollover ………………………………………………………….. 98
4.2 Reliability Analyses ……………………………………………………………………………………… 99
4.3 Limit State Functions ……………………………………………………………………………………. 107
4.4 Safety Modeling Methodology ………………………………………………………………………. 110
4.4.1 Poisson Regression ……………………………………………………………………………… 110
4.4.2 Negative Binomial Regression ……………………………………………………………… 111
Chapter 5 Reliability Based Design …………………………………………………………………………….. 112
5.1 Point-Mass Model Results …………………………………………………………………………….. 112
5.1.1 Point-Mass Model Reliability Results for Passenger Cars ………………………… 1135.1.2 Point-Mass Model for Heavy Trucks …………………………………………………….. 130
5.1.3 Reconciliation of Passenger Car and Heavy Truck Design Radii ………………. 132
5.2 Modified Point-Mass Model Results ………………………………………………………………. 133
5.2.1 Modified Point-Mass Model Results for Passenger Cars ………………………….. 135
5.2.2 Modified Point-Mass Model Results for Heavy Trucks……………………………. 144
5.3 Rollover Model Anlaysis ………………………………………………………………………………. 151
5.4 Summary …………………………………………………………………………………………………….. 157
Chapter 6 Safety Analysis ………………………………………………………………………………………….. 160
6.1 Pavement Friction Supply Model ……………………………………………………………………. 160
6.2 Safety Performance Functions ……………………………………………………………………….. 163
6.3 Summary …………………………………………………………………………………………………….. 174
Chapter 7 Conclusions and Recommendations ……………………………………………………………… 175
7.1 Conclusions from This Research ……………………………………………………………………. 175
7.2 Applications of This Research ……………………………………………………………………….. 177
7.3 Recommendations for Future Research …………………………………………………………… 181
Chapter 8 References ………………………………………………………………………………………………… 184
Appendix A Inferred Design Speed Calculations ………………………………………………….. 191
Appendix B Point-Mass FORM and SORM Design Radii ……………………………………… 195
Appendix C Scatter Plots of Safety Data ……………………………………………………………… 196
Chapter 1
Introduction
1.1 Background
The American Association of State Highway and Transportation Officials’ (AASHTO) A
Policy on Geometric Design of Highways and Streets (2011) (herein referred to as the Green Book) contains geometric design criteria for new and major reconstruction. These criteria are based on the design speed concept, which was established in the 1930s. Minimum or limiting values of geometric design criteria are based on the design speed. In the case of horizontal curve design, maximum design side friction factors are limited by driver comfort threshold, which is considered to cover the wide array of pavement surface conditions that could normally be expected to occur on roadways “with acceptable surface conditions attained at a reasonable cost” (AASHTO 2011). However, there are several fundamental issues with this argument:
- Driver comfort is subjective and is speed and environmentally dependent. The design side friction value in the Green Book has been found to be approximately equivalent to the 85th percentile of the operating speed distribution (Fitzpatrick et al. 2003), which means that 15 percent of drivers are typically willing to accept friction thresholds greater than design, which does not correlate to skidding or rollover failure. However, driver comfort (in terms of operating speed) can also be affected by environmental conditions, such as weather, daylight, level of fatigue, and presence of roadside objects.
- Driver characteristics and vehicle technologies change over time. Drivers today and in the future may be willing to accept different levels of comfort thresholds when compared to drivers of the 1930s. Similarly, advances in vehicle technology may
change how drivers perceive the forces acting on a vehicle traveling in curvilinear motion, thus comfort can be improved independent of side friction.
- “Acceptable” roadway surface conditions may be perceived differently in different contexts, especially between southern and northern climates within the United States. What represents acceptable roadway surface conditions may ignore the kinematic relationships that occur when adverse weather conditions are present during traffic operations, especially in northern climates. The Green Book defines unacceptable conditions as those that lack “in reasonable skid-resistant properties” such as glazed or bleeding roadways (AASHTO 2011). In northern and mountainous regions, it is quite reasonable to expect that unacceptable skid-resistant conditions can occur on horizontal curves with steep vertical grades.
The Green Book currently uses the point mass model (assuming a vehicle is an unsprung, singular point mass) for the geometric design of horizontal curves. The point mass model
(Equation 1-1) is written as:
V 2
e f (1-1)
15R
where e = superelevation rate (ft/ft),
= design side friction value (unitless),
V = design speed (in mph), and
R = horizontal curve radius (ft).
As shown in equation 1-1, grade effects are not considered in the design of horizontal curves. The Green Book has no specific additional criteria for the design of horizontal curves on steep grades; however, it does state that:
“On long or fairly steep grades, drivers tend to travel faster in the downgrade than in the upgrade direction. Additionally, research has shown that side friction demand is greater on both downgrades (due to braking forces) and steep upgrades (due to the tractive forces). Some adjustment in superelevation rates should be considered for grades steeper than 5 percent. This adjustment is particularly important on facilities with high truck volumes and low-speed facilities with intermediate curves using high levels of friction demand.” (AASHTO 2011)
The Green Book goes on to note that adjustments can be readily made to the design speed for divided, one-way roadways, but is more complex for undivided roadways. However, there are no quantitative guidelines for the adjustment in geometry of a curve to include the effect of a downgrade (Varunjikar 2011).
The purpose of this research is to develop a probabilistic approach for horizontal curve design considering the effects of the vertical alignment while incorporating a wide array of factors that are typically assumed in geometric design. These include distributions of random input variables (as opposed to deterministic design values), adverse pavement conditions, and the interaction of forces (tire-pavement friction) in the longitudinal and lateral directions in relation to the vehicle. Within the assumptions currently made for highway design, worst-case scenarios are typically considered for every deterministic design variable. Utilizing probabilistic methods allows the design to consider a more accurate representation of scenarios, rather than considering design scenarios that may be grossly over-conservative.
According to the 2009 Fatality Analysis Reporting System (FARS) data (NHSTA 2010), there were 30,797 fatal crashes in the United States. Approximately 70.6 percent of these crashes occurred on flat profiles (grade less than 3 percent). However, 41.3 percent of horizontal curve crashes occurred on grades steeper than 3 percent. This seems to suggest that grades may have a stronger association with crashes on horizontal curves than on tangents. This is inherently intuitive, since the friction demand would be much greater for horizontal curves than on horizontal tangents, for equivalent vertical grades. Table 1-1 shows a pattern for the percentage of fatal crashes on horizontal curves with grades versus level (no) grades for different pavement conditions. In each case, the percentage of fatal crashes is always higher on grades than on a level surface for the adverse pavement conditions shown (i.e., wet, snow, ice). The percentage of fatal crashes is always higher on level horizontal curves under dry pavement conditions (compared to horizontal curves on grades).
Table 1-1. Percentage of fatal crashes on horizontal curves by pavement condition.
| Year | 2010 | 2009 | 2008 | 2007 | ||||
| Condition | Level | Grade | Level | Grade | Level | Grade | Level | Grade |
| Dry | 84.4 | 81.5 | 83.0 | 80.2 | 83.6 | 80.3 | 84.7 | 82.0 |
| Wet | 13.5 | 14.7 | 13.0 | 14.7 | 12.8 | 14.0 | 11.5 | 13.3 |
| Snow | 1.0 | 3.0 | 1.1 | 1.6 | 1.1 | 2.2 | 1.6 | 1.9 |
| Ice | 1.1 | 0.8 | 1.8 | 2.2 | 1.5 | 2.3 | 1.1 | 1.6 |
This four-year consistent trend in fatality statistics appears to suggest that adverse weather conditions have a greater impact on horizontal curve safety on vertical grades than on horizontal curves that have a level profile. This is intuitive since tire-pavement friction is more critical as the vertical grade increases, since a vehicle will need to use available friction in the direction of travel (for braking or maintaining speed) as well as to traverse a horizontal curve. This simultaneous demand in two dimensions has been shown to follow an elliptical shape since the friction supply is not equivalent in the longitudinal and lateral directions (see section 2.3). For this reason, the effects of adverse weather conditions must be considered in the analysis of pavement friction supply and demand for horizontal curves on vertical grades.
1.2 Research Objectives
The objective of this research is to investigate the design of horizontal curves over the range of possible vertical grades through reliability analyses of both skidding and rollover failures for passenger cars and heavy trucks. Reliability analysis is frequently used to estimate the probability of failure for demand that exceeds available supply. This approach is commonly used in structural engineering and has been used recently in stopping sight distance research (Ismail and Sayed 2010). Design variables are considered random instead of deterministic to incorporate variations that occur in the field (e.g., driver heterogeneity or differences in tire performance characteristics) into the analysis. The friction ellipse equation is used to account for the interaction between longitudinal (i.e. tractive or braking) forces in combination with lateral forces under a variety of superelevation and vertical grade combinations. Some geometric design variables are considered as random variables (e.g. available pavement friction), along with their appropriate distributions, for reliability analysis. A correlation between the estimated reliability and reported crash data is examined for several case study horizontal curves in Pennsylvania, Maryland, West Virginia, California, Virginia, North Carolina, and Washington.
1.3 Significance of Research
The present study will develop a performance-based horizontal curve design methodology, considering the design speed of the roadway, focused on friction supply and demand instead of a design friction value. Multiple objectives can be used for horizontal curve performance analysis, including the probability of skid or rollover failure, or the expected number of vehicles whose friction demand exceeds friction supply.
Additionally, this research incorporates a modified version of the point-mass model, allowing for longitudinal friction demand to be considered for horizontal curve design. In this case, the effects of the vertical grade are included along with demands associated with vehicle acceleration or deceleration upon entering the horizontal curve. The friction ellipse is used to incorporate the longitudinal friction demand.
The results of this research could be directly applicable to developing a design exception justification procedure when Green Book minimum values cannot be met. Design exceptions are not requests for permission to violate design policy; they document deliberate variances from controlling geometric design criteria (MODOT 2011). They arise when it is impractical or impossible to meet minimum (or limiting) design criteria. Documentation of a design exception is necessary for DOTs to defend themselves from possible litigation. The Federal Highway Administration (FHWA) must approve all design exceptions that involve the 13 controlling criteria (e.g., horizontal alignment) on National Highway System roadways. However, there are few tools for DOTs to use to estimate the safety or operational impacts of variations from the design policy. A survey of state transportation agencies by Mason and Mahoney (2003) found that the horizontal alignment is the most commonly cited criteria for requiring design exceptions.
1.4 Organization of Thesis
To accomplish the research objectives, several steps were completed. The analysis framework for this research is presented in Figure 1-1. The first set of analyses was used to determine the appropriate method for estimating the reliability for the limit state functions that are developed in Chapter 4. Since the limit state functions are not linear, first order reliability methods (FORM) are compared to second order reliability methods (SORM) to determine if curvature correction is necessary. If there is no effect due to curvature of the limit state function, then the results of the FORM and SORM analyses are identical. Curvature may also be introduced through inclusion of correlations between random variables in the models. Therefore correlations were considered when determining the best estimation method. Once the best method for analysis was determined, the point-mass, modified point-mass, and rollover models were used to determine the reliability based design parameters for horizontal curve design (radii which give a reliability index of three for each grade and superelevation combination).
Figure 1-1. Analysis Framework.
Once reliability-based values were determined for the vast array of generic conditions, the reliability index was estimated for each of the data collection curves for this study. Crash frequency data were analyzed for their statistical association with the reliability index under various pavement conditions and vehicle types. The crash frequency data were also analyzed considering the rollover reliability indices. Care was taken to include the effects of confounding variables such as average daily traffic (a measure of exposure) as well as other geometric roadway characteristics. The organization of this thesis is as follows:
- Literature Review (Chapter 2) – The extant literature is summarized in this section, including the current horizontal curve design methodology, factors related to horizontal curve design, design vehicles, and risk and reliability methods used in other research. Descriptions and magnitudes of variables considered in previous research are documented.
- Data Collection (Chapter 3) – A discussion of the data collected and the procedures used for this research are described in this section. A description of the data and data structures is provided. Detailed analyses were conducted to determine the appropriate distributions and parameters of each distribution for the input random variables.
- Methodology (Chapter 4) – A comprehensive description of the proposed reliability methods are documented and described in this section. Derivations of relevant equations are also presented.
- Reliability Based Design (Chapter 5) – This chapter presents the findings of the reliability based design analyses. Each of the limit state functions are examined and interpreted for their implications to geometric design criteria.
- Safety Analysis (Chapter 6) – Statistical models are presented comparing the reliability indices for the data collection sites to observed crash frequency. Negative binomial
models were used to determine if a statistical relationship exists between the reliability index and crash frequency for different crash types, vehicle types, and roadway pavement conditions.
- Conclusions and Recommendations for Future Work (Chapter 7) – This chapter summarizes the significant findings of this research and presents recommendations for future considerations in reliability based design of horizontal curves.
