Torbic et al.
1
2
3
SUPERELEVATION DESIGN FOR SHARP HORIZONTAL CURVES
ON STEEP GRADES
Darren J. Torbic, Ph.D. (Corresponding Author)
Principal Traffic Engineer
MRIGlobal
2332 Raven Hollow Road
State College, PA 16801
Phone: 814-237-8831
E-mail: [email protected]
Eric T. Donnell, Ph.D., P.E.
Associate Professor
Department of Civil and Environmental
Engineering
The Pennsylvania State University
231 Sackett Building
University Park, PA 16802
Phone: (814) 863-7053
Fax: (814) 863-7304
E-mail: [email protected]
Alexander Brown
Ph.D. Candidate
Department of Mechanical and Nuclear
Engineering
The Pennsylvania State University
127 Hammond Building
University Park, PA 16802
Phone: (814) 865-1345
E-mail: [email protected]
Mitchell K. O’Laughlin, EIT
Associate Traffic Engineer
MRIGlobal
425 Volker Boulevard
Kansas City, MO 64110
Phone: (816) 753-7600
E-mail: [email protected]
Karin M. Bauer
Principal Statistician
MRIGlobal
425 Volker Boulevard
Kansas City, MO 64110
Phone: (816) 360-5287
E-mail: [email protected]
Sean N. Brennan, Ph.D.
Associate Professor
Department of Mechanical and Nuclear
Engineering
The Pennsylvania State University
157E Hammond Building
University Park, PA 16802
Phone: (814) 863-2430
Fax: (814) 865-9693
E-mail: [email protected]
4
5
6
7
8
9
Submitted for presentation and publication consideration
Transportation Research Board 93rd Annual Meeting
Word Count: 6,098 Words + 6 figures/tables @ 250 words = 7,598 total words
10
November 2013
11
12
1
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
ABSTRACT
The objective of this research was to develop superelevation criteria for sharp horizontal curves
on steep grades. Field studies and vehicle dynamics simulations (point-mass, bicycle, and multibody) were undertaken to investigate combinations of horizontal curve and vertical grade design
criteria. The vehicle dynamics simulations used AASHTO design criteria, in combination with
the field-measured data, to investigate margins of safety against skidding and rollover for several
vehicle types on sharp horizontal curves with steep grades. Results of the research found that, for
a simple horizontal curve, the maximum rate of superelevation should not exceed 12 percent on a
downgrade. If considering a maximum superelevation rate greater than 12 percent, a spiral curve
transition is recommended. On upgrades of 4 percent and greater, the maximum superelevation
rate should be limited to 9 percent for minimum-radius curves under certain conditions. The
superelevation attained at the point of curve entry should be checked and compared to a lateral
friction margin condition so that the lateral friction margin on curve entry is not less than the
margin within the curve. On multilane highways the “STAY IN LANE” sign should be installed
in advance of sharp horizontal curves on steep downgrades.
INTRODUCTION
Geometric design policy for horizontal curves is established in the American Association of
State Highway and Transportation Officials’ (AASHTO) A Policy on Geometric Design of
Highways and Streets (1), commonly known as the Green Book, and by the design manuals of
individual highway agencies. These criteria are based on the physics of the interaction between
vehicles and the roadway, as well as consideration of driver behavior. Horizontal curves
designed in accordance with Green Book criteria, even minimum-radius curves, have been
shown to provide a substantial margin of safety with respect to both vehicle skidding and
rollover under normal circumstances for both passenger vehicles and trucks (2-4).
Geometric design criteria for horizontal curves are based on a simple model that
represents the vehicle as a point mass. Research (5) has shown that the vertical loads on tires, in
particular on trucks, and the side friction supplied between the tires and pavement surface when
traversing a horizontal curve vary dynamically and can be represented by more sophisticated
models. Braking and tractive forces associated with vehicle maneuvering on grades also lead to
variations between tires in vertical load and side friction supply. These variations in tire loads
and vertical forces may lead to skidding or rollover at lateral accelerations less than those
suggested by the point-mass model. Additional limitations of the point-mass model are noted by
Easa and Abd El Halim (6).
The variation in the side friction factor values and tire loads suggested by the point-mass
model in the AASHTO Green Book are expected to increase for horizontal curves on steep
grades, but this phenomenon has not been thoroughly investigated. Bonneson (7) conducted a
preliminary investigation of this issue, and the results have been incorporated into the Green
Book for horizontal curves on steep grades by suggesting that higher design speeds be assumed
on highways with steep downgrades and adjusting the superelevation accordingly, but additional
knowledge is needed to make such guidance more quantitative for specific combinations of
curvature and grade.
2
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
The objective of this research was to develop superelevation criteria for sharp horizontal
curves on steep grades. Both passenger cars and trucks were considered as were downgrades and
upgrades.
BACKGROUND
Point-Mass Model
AASHTO policy on horizontal curve design is based upon a point-mass model. From the laws of
physics, consider a point mass traveling on a curve with a constant radius (R) and constant
velocity (V). The point mass undergoes centripetal acceleration (ar) which acts towards the center
of curvature:
ar
V2
R
(1)
15
16
17
18
19
20
21
22
23
24
25
26
27
Assume the point mass is a vehicle. The acceleration is balanced by the side friction developed
between the vehicle's tires and the pavement surface, the component of vehicle's weight acting
parallel to the road due to superelevation, or a combination of both. Let the banking angle of the
roadway be α (radians.). The superelevation (e) is typically defined by the rise (change in
elevation) in feet per 100 feet across the road (i.e., in the transverse direction). Hence, e/100 =
tan α. Three forces act on the point mass including the normal reaction from the road (N), the
tire-pavement friction cornering force acting at the road towards the center of rotation (Fc), and
the vehicle weight (W = mg, where m is the mass of the vehicle and g is the gravitational
acceleration). Through a series of force balances, substitutions, and simplifications [refer to
Torbic et al. (8) for full point-mass derivation), the resulting point-mass model used in horizontal
curve design is as follows:
28
§
·
V2
R= ¨
¸
© g ˜ ( f +0.01˜ e) ¹
(2)
29
30
31
32
The limiting factor for road design is the side friction factor f. Also, the superelevation rate for a
curve will not exceed a maximum value selected by the designer. For a given design speed (VDS)
of a roadway, practical lower limits on the radius of curvature, Rmin, are given by:
33
Rmin
2
34
35
36
37
VDS
g ˜ f max 0.01 ˜ emax (3)
Here, fmax is the maximum demand side friction factor, and emax is the maximum superelevation
rate for a given design speed. AASHTO uses Equation 3 for determining the minimum radius of
curvature. The basic side friction formula can be obtained by rearranging terms:
3
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
f
VDS
0.01 ˜ e
g˜R
(4)
In AASHTO policy, the side friction factor (f) represents the portion of lateral acceleration not
balanced by superelevation. The term f represents a friction “demand” which must be resisted by
the available friction “supply” generated at the tire-pavement interface. In addition, the
unbalanced lateral acceleration creates an overturning moment on the vehicle that must be
resisted by the vehicle’s roll stability. An inherent assumption in the AASHTO point-mass policy
is that vehicles follow the curved path exactly.
The tire-pavement interface can supply friction (ftire-pavement) to resist the tendency of the
vehicle to skid due to lateral acceleration as the vehicle traverses a curve. From the point-mass
model, the vehicle will skid if f > ftire-pavement, where ftire-pavement represents the maximum amount
of friction that can be generated at the tire-pavement interface to counteract lateral acceleration
and prevent skidding. Similarly, the vehicle will overturn if f > frollover, where frollover represents
the maximum lateral acceleration that a vehicle can experience without overturning. frollover is
referred to as the “rollover threshold” of the vehicle. Rollover thresholds are a characteristic of
vehicle design and loading.
Green Book design criteria for horizontal curves are not based on formal assumptions about
the magnitudes of ftire-pavement and frollover. Rather, horizontal curve design is based on limiting the
value of f to be less than or equal to a specified value, fmax, based on driver comfort levels (i.e.,
driver tolerance for lateral acceleration). A further assumption is that the values of fmax used in
design have been selected such that fmax < ftire-pavement and fmax < frollover. The first criterion, fmax <
ftire-pavement, is addressed in Green Book Figure 3-5, which shows that the values of fmax used in
design are less than the values of ftire-pavement. The second criterion, fmax < frollover, is asserted but
not demonstrated in the Green Book. Research (3,5) has shown that the assumptions of fmax < ftirepavement and fmax < frollover do appear to be generally applicable to both passenger vehicles and
trucks for horizontal curves designed in accordance with AASHTO policy.
Several research efforts have evaluated the adequacy of the point-mass model approach
to horizontal curve design. Harwood and Mason (3) evaluated the geometric design policy for
horizontal curves in the 1990 AASHTO Green Book which used separate friction factors for
high-speed and low-speed design and concluded that existing high-speed design policy provides
adequate margins of safety against skidding and rollover for both passenger vehicles and trucks
as long as the design speed of the curve is selected realistically. It was recommended, however,
that care be taken for curves with design speeds of 30 mph or less to assure that the selected
design speed will not be exceeded, particularly by trucks. For minimum-radius horizontal curves
designed in accordance with AASHTO low-speed criteria, AASHTO policy generally was found
to provide adequate margins of safety against skidding and rollover for passenger vehicles
traveling at the design speed, but for design speeds of 10 to 20 mph, minimum-radius curves may
not provide adequate margins of safety for trucks with poor tires on a poor, wet pavement or for
trucks with low rollover thresholds.
Bonneson (9) estimated statistical models of curve speed and side friction demand and
found that side friction demand decreased as curve approach speeds increased, while the side
friction demand increased as the speed reduction between the curve approach speed and the
speed at the mid-point of a horizontal curve increased. The side friction demand related to no
speed reduction between the approach tangent and mid-point of a horizontal curve was proposed
4
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
as the desirable upper limit on maximum design side friction factors (fmax), while a speed
reduction of 3 mph was proposed in consideration of traffic flow and construction cost.
To assess the margin of safety for the proposed side friction demand factors, Bonneson
(7) compared side friction supply for both slide and roll failure, to the proposed side friction
demand factors. Results showed that grades, particularly steep upgrades, reduced the margin of
safety, particularly for trucks. Other trends showed that roll failure is only observed in trucks on
low-speed curves and that slide failure will occur prior to roll failure for passenger vehicles at
any speed, and at higher speeds for trucks. Much of this research has formed the basis for
horizontal curve transition design policy in the 2011 AASHTO Green Book.
Eck and French (10) investigated issues faced by trucks traversing sharp horizontal
curves on steep vertical grades in West Virginia. The authors indicated that, on downgrades, a
portion of the friction supplied at the road-tire pavement interface is consumed by braking to
maintain a constant speed. This reduces the margin of safety against lateral skidding, and this
effect is more severe when braking. It was recommended that high rates of superelevation (e.g.,
up to 16 percent) make horizontal curves more forgiving on steep downgrades.
Side Friction Supply
Side friction supply (ftire-pavement) is the friction available between the pavement surface and
vehicle tires to prevent skidding on a horizontal curve. The maximum side friction supply is
utilized when a vehicle is at the point of impending sideways skid. Early work by Moyer (11)
found that side friction supply factors decrease as vehicle speed increases. On wet Portland
cement concrete pavements, Moyer showed that skidding side friction values ranged from
approximately 0.73 at speeds as low as 2.5 mph, to 0.6 at 40 mph speeds. More recently Fancher
et al. (12) reported average side friction supply for a variety of manufactured tires on dry and wet
concrete pavements. The average side friction supply was 0.54 on dry concrete pavements and
0.467 on wet concrete pavements, both at 40 mph speeds.
A wheel exhibits different performance characteristics depending on whether forces are
in the lateral direction or longitudinal direction, the condition of the tires, whether an anti-lock
braking system (ABS) is employed, and loading of the tires. The use of braking forces reduces
the available lateral (side) friction, and the use of lateral (side) force reduces the available
braking forces. This interrelationship between lateral and longitudinal forces is called the friction
ellipse and represents the operating range of tire forces per Equation 5 (13):
§ Fy
¨
¨F
© y ,max
2
· § Fx
¸ ¨
¸ ¨F
¹ © x ,max
·
¸
¸
¹
2
n2 d 1
(5)
Fx is the tire’s longitudinal (braking) force, Fy is the tire’s lateral (side or cornering) force, and
Fx,max and Fy,max are the maximum possible forces available in braking and cornering,
respectively. The term n represents the total utilized friction, and has a value of 1 when the tires
are at the friction limit. Values below 1 represent situations within the friction ellipse, whereas
values above 1 are beyond the tire’s force capabilities. Thus, as long as the value of n is less than
1, the operating point (i.e., tire forces in x and y direction) lies inside the friction ellipse (i.e., the
tire-pavement can generate required friction force). Equation 5 can be related to pavement
friction values in the lateral and longitudinal directions.
5
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
METHDOLOGY
The present study used a combination of field data, and analytical and vehicle dynamics
simulation models to evaluate geometric design criteria for sharp horizontal curves on steep
grades. The goal of the research was to develop recommended modifications to existing
AASHTO design policy to improve conditions that may generate concerns at sharp horizontal
curves on steep grades.
Data from the field studies were used as both inputs (calibration) to the vehicle dynamics
simulation models and to validate the simulation model output. The field studies included:
x
x
x
Speed and vehicle maneuver studies;
Instrumented vehicle studies; and
Friction testing.
Speed and vehicle maneuver studies were completed at 20 sharp horizontal curves on
grades of 4 percent or more, on freeways, other divided highways, and undivided highways. Data
collection sites were identified in California, Maryland, Pennsylvania, Washington, and West
Virginia. For site selection purposes a sharp horizontal curve was generally defined as a
horizontal curve that, under current AASHTO policy, would require superelevation of at least 6
percent when designed with criteria applicable to a maximum superelevation rate of 8 percent
(i.e., sites with above minimum-radius curves were included in the field studies). Sharp
horizontal curves were also identified for inclusion in the field studies based on the presence of
curve warning and/or advisory speed signs. Roadway features of the speed data collection sites
are summarized in Table 1.
Free-flow operating speed data for passenger cars and trucks were collected using laser
guns. In general, speeds were measured approximately 500 ft prior to the beginning of a
horizontal curve through the mid-point of the curve. Descriptive statistics of the speed data for
both passenger cars and trucks were computed for each site. At multi-lane site locations, video
data were recorded to document the direction, frequency (proportion of total traffic), and
duration of vehicle lane change maneuvers.
At five data collection sites a range of data were collected using an instrumented vehicle
with a defense-grade Global Positioning System (GPS) coupled to a ring-laser-gyro Inertial
Measurement Unit (IMU) that gives accurate absolute measures of position and orientation. In
addition, a roof-mounted LIDAR sensor was used to collect cross-section measurements of the
road surface. A camera was mounted to the dashboard of the vehicle and aligned so that the
vanishing point of straight-line driving corresponds roughly to the center of the image. And
finally, a steering angle sensor captured the driver’s steering inputs directly. The purpose of the
instrumented vehicle study was to:
1. Measure the road geometry (i.e., grade, curvature, and cross-slope) of each site to
confirm whether the instrumented vehicle-based road measurements were in agreement
with information collected from roadway inventory files, roadway plan and profiles
sheets, and field measurements (e.g., superelevation and vertical grades).
6
TRB 2014 Annual Meeting
Paper revised from original submittal.
TRB 2014 Annual Meeting
Paper revised from original submittal.
2
1
MD
MD1
PA
WA
WA
WA
WA
PA2
WA1
WA2
WA3
WA4
WV
WV
WV3
WV4
Kern
Kittitas
Kittitas
Kittitas
Grant
Jefferson
Washington
Washington
Washington
Garrett
I-64 (EB)
I-77 (NB)
I-79 (SB)
I-68 (WB)
I-77 (SB)
US 2 (EB)
I-90 (EB)
Kanawha
Kanawha
Kanawha
Monongalia
Mercer
King
Kittitas
US 97 (NB) Kittitas
I-82 (EB)
I-82 (WB)
I-82 (WB)
I-90 (WB)
I-80 (EB)
I-79 (NB)
I-68 (WB)
I-68 (WB)
I-68 (WB)
SR 17 (SB) Santa Cruz
SR 17 (NB) Santa Clara
I-5 (NB)
County
Hancock West
Hancock East
Friendsville
Scotts Valley
Los Gatos
Lebec
Nearest city
Vantage
Brookville
49.7-50.5
76.5-78.0
2.05-2.5
9.9-10.6
20.6-21.4
Institute
Cabin Creek
Mink Shoals
Cheat Lake
Camp Creek
Skykomish
Ellensburg
131.48131.69
60.0-60.7
Ellensburg
Ellensburg
Ellensburg
162.7-163
21.75-22.5
4.00-4.63
15.14-15.94 Ellensburg
137.5-138
79.5-80.5
Interchange
Washington
I-70/I-79
72.5-73.5
74.5-75.0
5.5-7.0
10.3-9.7
2.0-3.0
1.6-2.1
MP
Compound curve with four radii: 430 ft, 230 ft, 150 ft, and 310 ft.
Upgrade sites.
WV
WV
WV2
WV5
WV
WA
WA
WV1
WA7
2
WA6
WA5
WA
PA
PA1
2
MD
MD3
MD
CA
CA3
2
CA
CA2
MD2
CA
CA1
Site
Route
State (direction)
7
Freeway
Freeway
Freeway
Freeway
Freeway
Multilane
Freeway
Two-lane
Freeway
Freeway
Freeway
Freeway
Freeway
Ramp
Freeway
Freeway
Freeway
Multilane
Multilane
Freeway
Roadway
type
Table 1. Data Collection Sites and Site Characteristic Information.
Torbic et al.
−5.0
−5.2
−3.7
−5.7
−4.9
5.9
−2.9
6.0
−3.8
−5.0
−5.0
−4.9
−4.0
−5.0
−5.7
6.0
−4.1
−6.3
−6.2
−3.1
0.58
> 1.0
0.75
> 1.0
> 1.0
> 1.0
> 1.0
0.86
0.6
> 1.0
> 1.0
> 1.0
0.67
1.0
0.21
> 1.0
0.78
0.25
0.25
> 1.0
Length
Grade of grade
(%)
(mi)
1,637
1,041
1,146
1,909
1,206
577
2,800
1,637
1,600
2,400
1,600
955
1,637
Comp
1
1,900
1,909
1,909
575
537
2,000
Curve
radius
(ft)
0.33
0.26
0.05
0.49
0.50
0.25
0.33
0.19
0.33
0.19
0.24
0.23
0.27
0.19
0.32
0.42
0.31
0.13
0.21
0.47
Curve
length
(mi)
7.2
8
8
7.8
8
10
7
2
5.8
7
10
9.3
8.3
6.25
4.5
5.5
6
8.8
12
2
Max.
super
(%)
Present
Present
Present
Present
Present
Present
Absent
Absent
Absent
Absent
Absent
Present
Present
Absent
Absent
Absent
Absent
Absent
Absent
Absent
Spiral
Left
Right
Left
Left
Left
Left
Right
Left
Right
Right
Left
Right
Left
Right
Right
Right
Left
Left
Right
Left
Curve
direction
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
2. Obtain in-vehicle dynamics measurements (i.e., roll/pitch/yaw angles and rates) for
comparison with simulation outputs to check the fidelity of the vehicle simulation
software.
3. Measure continuous speed profiles of vehicles traversing the entire lengths of the data
collection site (i.e., along the entire grade and curve) since laser gun measurements were
collected primarily on the tangent approaching the curve and through the curve.
The instrumented vehicle was driven behind free-flow vehicles in the traffic stream chosen
randomly.
Friction data were collected at eight data collection sites. Twenty-one friction
measurements recorded at each site – nine on the approach tangent and twelve within the limits
of the horizontal curve. The purpose of the friction testing was to establish friction supply (ftirepavement) values for tires on both passenger vehicles and trucks suitable for modeling a vehicle’s
expected behavior on steep grades and through sharp horizontal curves. The friction data were
used in conjunction with the simulation analyses to determine the difference between the
AASHTO design friction curves (fmax) and the friction supply and demand (f based on operating
speed) on representative grades and horizontal curves. Testing was performed in accordance with
ASTM E1911-09a, Standard Test Method for Measuring Pavement Surface Frictional
Properties Using the Dynamic Friction Tester (14). The friction data were used to generate
friction supply curves for a range of speeds, vehicle types, and pavement types and conditions.
The vehicle dynamics simulations were carried out in 11 sequential steps (Table 2). Three
classes of passenger cars and three classes of trucks were considered in the analysis:
Passenger Vehicles:
Trucks:
x
x
x
x
x
x
E-class sedan (i.e.,mid-class sedan)
E-class SUV (i.e., mid-size SUV)
Full-size SUV
Single-unit truck
Tractor semi-trailer truck
Tractor semi-trailer/full-trailer truck (double)
In referring to Table 2, several key terms must be defined, including:
x
x
x
Lateral Friction Margin: the difference between the available tire-pavement friction
and the friction demand of the vehicle as it tracks the curve [i.e., side friction supply
(ftire-pavement) − side friction factor (f)]. A positive margin indicates a vehicle can
undergo additional lateral acceleration without skidding, while a negative margin
indicates the vehicle tires will skid.
Rollover Margin: Rollover margin is defined in the two ways:
o The difference between the current lateral acceleration and the maximum
lateral acceleration that a vehicle can experience without overturning.
o The proximity of the load-transfer ratio to an absolute value of unity (e.g., how
close an axle is to experiencing wheel lift). In both cases, a value of 0 indicates
the onset of wheel lift.
Transient Vehicle Behavior: when a driver changes the steering input on a vehicle
(e.g., during transition from an approach tangent to a horizontal curve), the vehicle will
8
TRB 2014 Annual Meeting
Paper revised from original submittal.
TRB 2014 Annual Meeting
Paper revised from original submittal.
Define vehicles and maneuvers
to use in non-point-mass models
Predict wheel lift using quasistatic models
Predict skidding of individual
axles during steady-state
behavior on a curve
Predict skidding of individual
axles during braking and lanechange maneuvers on a curve
Predict skidding of individual
axles during transient steering
maneuvers and severe braking
Predict skidding of individual
wheels
Predict wheel lift of individual
wheels during transient
maneuvers
Analysis of upgrades
5
6
7
8
9
10
11
Description
Define basic tire-pavement
interaction model and estimate
lateral friction margins against
skidding in AASHTO’s current
horizontal curve policy
Define road geometries and
variable ranges for use in
subsequent steps
Develop side friction demand
curves and calculate lateral
friction margins against skidding
considering grade using the
modified point-mass model
4
3
2
1
Step
9
Based on areas of concern (small positive or negative margins of safety) from Steps
3, 6, 7, and 8, use vehicle dynamics simulation software to predict skidding of
individual vehicle wheels.
Based on areas of concern (small positive or negative margins of safety) from Steps
3, 6, 7, and 8, use vehicle dynamics simulation software to predict wheel lift of
individual vehicle wheels.
Transient bicycle and multi-body models used to assess margins of safety against
skidding and rollover. Passenger cars were assumed to maintain constant speed on
the upgrade, while trucks were assumed to reach a crawl speed on the upgrade.
Model from Step 7 was modified to determine if severe braking while traversing a
sharp horizontal curve affects the ability of a vehicle to traverse a curve without
skidding.
Vehicle inertia properties, vehicle dimensions, and suspension data for six vehicle
classes [mid-size sedan, mid-size SUV, full-size SUV, single-unit truck, tractor semitrailer truck, and tractor semi-trailer/full-trailer truck (double)]. Additionally, four
deceleration scenarios (0, 3, 11.2 and 15 ft/sec2) were considered, as were various
lane change maneuvers from field data.
Analytical derivation of rollover margins for six vehicle classes based on horizontal
curve, vertical grade, and superelevation scenarios defined in Step 2.
Analytical derivation of friction demand for individual vehicle axles, accounting for
vehicle type, vertical grade, superelevation and rate of deceleration. Both constant
speed and constant braking (i.e., decelerating when approaching and within a curve)
were considered.
Same as step 6 input, except that braking, lane changes, and non-steady steering
maneuvers were considered. Data from vehicle speed, maneuver, and instrumented
vehicle studies were used in the simulations.
Comparison of friction demand from models to Step 1
friction supply curves.
Comparison of friction demand from models to Step 1
friction supply curves.
Comparison of friction demand from models to Step 1
friction supply curves.
Comparison of friction demand resulting from models to
friction supply computed in Step 1. Additionally, the
lateral skidding distances were computed, if negative
margins of safety were determined.
Comparison of friction demand resulting from models to
friction supply computed in Step 1.
Comparison of friction demand resulting from models to
friction supply computed in Step 1.
Rollover thresholds for vehicles considered in this
research.
Determination that, in subsequent simulation steps, that
braking maneuvers would be initiated after the onset of
the curve.
Demand side friction curves from hypothetical
geometries were compared to friction supply curves
generated in Step 1 to compute margins of safety
against vehicle skidding.
Hypothetical geometries/scenarios to be considered in
vehicle dynamics simulation models.
Published literature; range of design speed, minimum radius, superelevation, vertical
grade, and levels of vehicle deceleration from AASHTO Green Book; and field data.
Hypothetical geometries in Step 2.
Friction supply curves for different tire types, vehicle
speeds, and pavement surface type and condition.
Key Output / Results
Field friction measurements from DF tester and CT meter.
Input
Table 2. Vehicle Dynamics Simulation Modeling Process.
Torbic et al.
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
x
enter the curve with motions that are initially unsteady (i.e., the spin of the vehicle, the
yaw rate, will not at first match that of the curve), which is referred to as a “transient
response.”
Steady-State Vehicle Behavior: at the conclusion of the period of transient response
resulting from a steering input change, the yaw rate of the vehicle will become
constant, which is referred to as “steady-state response.”
DATA
Data needed to discuss the results of the analytical and simulation models are presented in this
section of the paper. The reader is referred to Torbic et al. (8) for a complete summary of the
speed, lane-change maneuver, and instrumented vehicle studies. This section of the paper
describes the friction supply curves developed from the friction study, as well as the range of
inputs considered in the hypothetical analytical and simulation modeling scenarios.
Friction Data
The goal of this analysis was to define reasonable estimates of the friction supply, ftire-pavement, as
a function of speed, and to represent the values in a manner easily interpreted in terms of lateral
friction margins against skidding. Vehicle dynamics literature contains an array of tire-pavement
models, and the choice of the LuGre model (15) is a tradeoff between its comparatively high
accuracy versus modest computational demands. The LuGre tire model requires input from the
DF tester and CT meter (macro- and microtexture of the pavement surface), and converts these
data to a longitudinal and lateral friction supply measure. In Figure 1, the mean and two standard
deviations below the mean for the wet pavement maximum (i.e. point of impending skid)
braking-only and cornering-only passenger car friction curves are shown compared to the
AASHTO maximum side friction values (fmax) used in horizontal curve design, along with the
same friction distribution curves for skidding friction during braking and corning. Figure 1 also
shows the maximum and skidding friction supply curves for trucks. In all cases for the passenger
car, the mean minus two standard deviation cornering friction supply curves are at least 0.3
above the AASHTO maximum design friction curves, while the mean cornering friction supply
curves are at least 0.4 above the AASHTO maximum design friction curves. For trucks, the mean
minus two standard deviation cornering friction supply curves are at least 0.28 above the
AASHTO maximum design friction curves, while the mean cornering friction supply curves are
at least 0.4 above the AASHTO maximum design friction curves.
Road Geometries Used in Analytical and Simulation Models
In Step 2 of the analytical and simulation modeling process, the range of superelevations,
horizontal curve radii, side friction levels, and grades to be considered in the analysis were
defined. Table 3-7 (Minimum Radius Using Limiting Values of e and f) in the 2011 Green Book
provides a range of design values for consideration in this research, for example, design speeds
range from 10 to 80 mph; maximum superelevations range from 4 to 12 percent; and maximum
side friction factors range from 0.08 to 0.38. At minimum, it was important to investigate the full
range of design values to sufficiently address the scope of this research. While the Green Book
provides a range of hypothetical conditions to consider in the analytical and simulation models,
10
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
1
Skidding Friction
Maximum Friction
0.9
mean braking
0.9
0.7
0.8
2stddev braking
mean cornering
Friction value (unitless)
Friction value (unitless)
0.8
0.6
2stddev cornering
0.5
0.4
0.3
0.2
AASHTO
0.1
30
40
1
0.5
2stddev braking
2stddev cornering
0.4
0.3
AASHTO
0.1
70
0
80
30
40
AASHTO
50
60
Speed (mph)
70
80
Passenger Vehicles
2
3
1
1
Maximum Friction
Skidding Friction
0.9
0.9
mean braking
0.8
0.7
0.6
Friction value (unitless)
Friction value (unitless)
0.8
mean cornering
2stddev braking
2stddev cornering
0.5
0.4
0.3
0.2
AASHTO
0.1
0
30
40
4
0.7
0.6
70
mean cornering
2stddev braking
2stddev cornering
0.4
0.3
AASHTO
AASHTO
AASHTO
50
60
Speed (mph)
mean braking
0.5
0.2
AASHTO
5
6
7
8
9
10
11
12
13
14
15
mean cornering
AASHTO
AASHTO
50
60
Speed (mph)
mean braking
0.6
0.2
AASHTO
0
0.7
0.1
0
80
30
40
AASHTO
50
60
Speed (mph)
70
80
Trucks
Figure 1. Wet-Pavement Friction Levels.
(Left panels: Mean and two standard deviations below the mean for maximum wet-tire friction for braking and
cornering directions based on field measurements; Right panels: Mean and two standard deviations below the mean
for skidding wet-tire friction for braking and cornering directions based on field measurements)
11
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
2
3
4
5
geometric features of the field study sites (see Table 1) provided additional insights concerning
the range of design practice.
The range of design values considered in the analytical and simulation modeling
included:
x
x
x
x
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Curve radius (R):
Curve design speed (VDS)
(also, entry speed):
Superelvation (e):
Grade (G):
Rmin, 0.8Rmin
25 to 85 mph (5-mph interval)
0, 4% to 16% (1% interval)
0, 4% to 9% (1% interval) (downgrades and upgrades)
For analysis of the hypothetical geometries, speeds/decelerations of the vehicles also had to be
defined. Four speeds/deceleration levels were selected for analyses:
x
x
x
x
No deceleration (0 ft/s2) (i.e., constant speed)
Curve entry deceleration (−3 ft/s2) based on research by Bonneson (7) and confirmed in
the speed field studies for the present study
Deceleration rates used in calculating stopping sight distance (i.e., −11.2 ft/s2)
Deceleration rates assumed for emergency braking maneuvers (i.e., −15 ft/s2) (analyzed
for select cases)
For the variations in the minimum design radius, reducing the design radius from Rmin to 80
percent of Rmin can either be analyzed as a geometric change, a speed change, or a friction
change. This is best understood by considering the point-mass model on which the AASHTO
policy is based, and considering a horizontal curve with no superelevation. From Equation 3, if
the speed is kept the same while the radius is reduced by 80 percent, then the acceleration would
be balanced if the demanded friction and superelevation are both increased by a factor of 1/0.8,
or 1.25. Similarly, the effect of decreasing the radius by 0.8 and keeping the speed fixed is
equivalent to increasing the speed by 11.8 percent (the square-root of 1/0.8) and keeping the
radius fixed. For horizontal curves without superelevation, the right-side of the equation is
simply the demanded friction. In this case, a decrease in radius by 0.8 requires an increase in
demand friction by 1.25. This corresponds to forcing a systematic downward shift of the friction
margins for the Rmin case to the 0.8 Rmin. Thus interpretation of the reduced-radius case can be
used to understand outcomes for situations of overspeed, low superelevation, or reduced friction
margins.
FINDINGS
Several general results from the study are presented here, as well as the specific findings that
were used to develop geometric design guidance for sharp horizontal curves on steep grades.
General Findings
With regards to general findings, the following results were reached for all modeling scenarios
considered:
12
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
x
x
x
x
x
x
The AASHTO Green Book maximum side friction factors (fmax) used in horizontal curve
design are below friction supply curves for lateral (cornering) and longitudinal (braking)
directions, for both passenger vehicles and trucks, for design speeds greater than 20 mph.
Thus, current horizontal curve design policy appears to provide reasonable lateral friction
margins against skidding in most situations. However, the more complex vehicle
dynamics models (i.e., the transient bicycle and multi-body models) indicate the pointmass model generally overestimates the margins of safety against skidding and rollover
across all vehicle types.
There is no concern of a passenger vehicle rolling over while traveling at the design
speed on a sharp horizontal curve with a steep downgrade, when designed according to
current AASHTO Green Book policy.
On downgrades, the lowest margins of safety against skidding and rollover generally
occur at design speeds of 40 mph and lower for all vehicle types. This appears to be the
result of higher side friction factors used in design for horizontal curves with lower
design speeds.
When maintaining a vehicle operating speed at or near the design speed on a horizontal
curve, grade and maximum superelevation rate (emax) appear to have little effect on the
margins of safety against skidding and rollover for all vehicle types.
Based on current AASHTO Green Book horizontal curve-superelevation design policy, a
vehicle that performs an emergency braking maneuver (−15 ft/s2 deceleration) on a steep
downgrade-horizontal curve combination will likely skid off the roadway in many cases
if the vehicle is not equipped with anti-lock braking systems (ABS).
The method used in the current AASHTO Green Book policy to distribute superelevation
and side friction on tangent-curve transitions is adequate and produces positive margins
of safety against skidding and rollover for all vehicle types on horizontal curves designed
using maximum superelevation and minimum curve radii. However, the superelevation
attained at the point of curve entry should be checked and compared to the lateral friction
margin condition below so that the lateral friction margin on the curve entry is not less
than the margin within the curve:
e
1
V2
u
100 1 ptangent gR
(6)
31
32
33
34
35
36
37
where: e
= superelevation at PC of horizontal curve;
ptangent = proportion of the maximum superelevation attained at the PC of
horizontal curve;
V
= design speed (ft/s);
g
= gravitational constant (32.2 ft/s2);
R
= radius of horizontal curve (ft).
38
39
40
41
42
43
If the condition presented above is met, the superelevation transition may be placed as
indicated in Green Book Table 3-18. If the condition presented above is not met,
designers should reduce the proportion of the maximum superelevation attained at the PC
of the horizontal curve, or introduce a spiral transition curve between the approach
tangent and simple horizontal curve. Based on a sensitivity analysis, the condition above
is satisfied for maximum superelevation/minimum radius curves for all design speeds.
13
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
However, the condition above may be violated when using greater than minimum
horizontal curve radii. If the condition is not met, it is recommended that a lower
proportion of the superelevation runoff (e.g., 70 percent) be introduced prior to horizontal
curve entry.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Specific Findings
Several findings from the analytical and simulation modeling suggest that certain vehicle types
or driver behavior (i.e., rate of deceleration entering horizontal curves or lane-change
maneuvers) produce negative margins of safety against skidding or rollover on sharp horizontal
curves with steep grades. A discussion of these findings is provided below:
x
x
Steep vertical downgrade-sharp horizontal curve combinations that necessitate braking to
maintain a constant speed (and maintain lane position) from the approach tangent through
a horizontal curve for a passenger car sedan have large margins of safety against skidding
(> 0.33) for design speeds ranging from 25 to 85 mph (Figure 2). Similarly, positive
margins of safety against skidding (> 0.23) for passenger cars that decelerate at a rate of
−3 ft/s2 (curve entry deceleration) or at a rate of −11.2 ft/s2 (stopping sight distance
deceleration) exist for all design speed-downgrade combinations considered in the
present study. Deceleration rates of −15 ft/s2 (emergency braking) produce negative
margins of safety for many design speeds for vertical downgrade-sharp horizontal curve
combinations when the passenger car sedan enters the horizontal curve. Similar findings
were produced for the mid-size and full-size SUV vehicle types. The emergency braking
maneuver, however, does not seem likely to occur with sufficient frequency to constitute
a reasonable basis for design.
Steep vertical downgrade-sharp horizontal curve combinations that necessitate braking
for a single-unit truck to maintain a constant speed (and maintain lane position) from the
approach tangent through a horizontal curve have large margins of safety against
skidding (> 0.25) for design speeds ranging from 25 to 85 mph (Figure 3). Similarly,
margins of safety against skidding for the single-unit truck that decelerates at a rate of −3
ft/s2 exceed 0.10 for all design speeds for vertical downgrade-sharp horizontal curve
combinations considered in the present study. Based upon the steady-state and transient
bicycle models, when single-unit trucks must decelerate at a rate of −11.2 ft/s2 (stopping
sight distance braking) or a rate equivalent to emergency braking (−15 ft/s2), significant
negative margins of safety against skidding result across all design speed-downgrade
combinations considered in the present study. However, based on multi-body model
analyses for deceleration rates of −11.2 ft/s2 and −15 ft/s2 by a single-unit truck on a
curve, the single-unit truck is able to maintain control on the curve when equipped with
antilock braking systems (ABS).
40
14
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
Constant Speed, ax = 0 ft/s 2
Constant Speed, ax = 0 ft/s 2
0.4
Curve Entry Decel, ax = -3 ft/s 2
Minimum Fy Margin
Minimum Fy Margin
0.4
0.3
SSD Decel, ax = -11 ft/s 2
0.2
Emergency Decel, ax = -15 ft/s 2
0.1
0
-0.1
40
0.3
SSD Decel, ax = -11 ft/s 2
Emergency Decel, ax = -15 ft/s 2
0.2
0.1
0
Grade = 0%
Intermediate grades
Grade = -9%
30
Curve Entry Deceleration, ax = -3 ft/s 2
Grade = 0%
Intermediate grades
Grade = -9%
E-Class Sedan, e = 0%
50
60
70
-0.1
80
30
70
80
0.4
Curve Entry Decel, ax = -3 ft/s 2
Minimum Fy Margin
Minimum Fy Margin
60
Constant Speed, ax = 0 ft/s 2
0.4
0.3
SSD Decel, ax = -11 ft/s 2
0.2
0.1
Emergency Decel, ax = -15 ft/s 2
0
-0.1
50
Speed (mph)
Constant Speed, ax = 0 ft/s 2
30
40
Curve Entry Decel, ax = -3 ft/s 2
0.3
SSD Decel, ax = -11 ft/s 2
0.2
Emergency Decel, ax = -15 ft/s 2
0.1
0
Grade = 0%
Intermediate grades
Grade = -9%
Grade = 0%
Intermediate grades
Grade = -9%
E-Class Sedan, e = 16%
50
60
70
-0.1
80
Speed (mph)
2
3
4
5
6
7
8
40
Speed (mph)
1
E-Class Sedan, e = 0%
30
40
E-Class Sedan, e = 16%
50
60
70
80
Speed (mph)
Figure 2. Lateral Friction Margins from Steady-State Bicycle (left) and Transient Bicycle (right)
Models for E-Class Sedan (G = 0 to −9%, e = 0 and 16%) (ax = 0, −3, −11,
and −15 ft/s2)
15
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
Constant Speed, ax = 0 ft/s 2
Minimum Fy Margin
Minimum Fy Margin
0.3
0.2
Curve Entry Decel, ax = -3 ft/s 2
0.1
(not shown) SSD Decel, ax = -11 ft/s 2 (from -0.11 to -0.36)
(not shown) Emergency Decel, ax = -15 ft/s 2 (from -0.14 to -0.55)
0
-0.1
Constant Speed, ax = 0 ft/s 2
0.4
0.4
40
0.2
Curve Entry Decel, ax = -3 ft/s 2
0.1
(not shown) SSD Decel, ax = -11 ft/s 2 (from -0.15 to -0.36)
(not shown) Emergency Decel, ax = -15 ft/s 2 (from -0.24 to -0.54)
0
Grade = 0%
Intermediate grades
Grade = -9%
30
0.3
Grade = 0%
Intermediate grades
Grade = -9%
Single Unit Truck, e = 0%
50
60
70
-0.1
80
30
40
50
0.4
0.2
Curve Entry Decel, ax = -3 ft/s 2
0.1
(not shown) SSD Decel, ax = -11 ft/s 2 (from -0.11 to -0.36)
(not shown) Emergency Decel, ax = -15 ft/s 2 (from -0.14 to -0.55)
-0.1
30
40
0.3
0.2
Curve Entry Decel, ax = -3 ft/s 2
0.1
(not shown) SSD Decel, ax = -11 ft/s 2 (from -0.24 to -0.34)
(not shown) Emergency Decel, ax = -15 ft/s 2 (from -0.30 to -0.50)
Grade = 0%
Intermediate grades
Grade = -9%
Single Unit Truck, e = 16%
50
60
80
Constant Speed, ax = 0 ft/s 2
0
70
-0.1
80
30
40
50
60
70
80
Speed (mph)
Speed (mph)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Minimum Fy Margin
Minimum Fy Margin
0.3
Grade = 0%
Intermediate grades
Grade = -9%
70
Single Unit Truck, e = 16%
Constant Speed, ax = 0 ft/s 2
0.4
0
60
Speed (mph)
Speed (mph)
1
Single Unit Truck, e = 0%
Figure 3. Lateral Friction Margins from Steady-State Bicycle (left) and Transient Bicycle (right)
Models for Single-Unit Truck (G = 0 to −9%, e = 0 and 16%) (ax = 0, −3, −11,
and −15 ft/s2).
16
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
x
x
Steep vertical downgrade-sharp horizontal curve combinations that necessitate braking
for a tractor semi-trailer to maintain a constant speed (and maintain lane position) from
the approach tangent through a horizontal curve have large margins of safety against
skidding (> 0.28) for design speeds ranging from 25 to 85 mph (Figure 4). Similarly,
margins of safety against skidding for a tractor semi-trailer that decelerates at a rate of −3
ft/s2 exceed 0.26 for all design speeds for vertical downgrade-sharp horizontal curve
combinations considered in the present study, and when a tractor semi-trailer must
decelerate at a rate of −11.2 ft/s2, the margins of safety exceed 0.11. For emergency
braking (−15 ft/s2), a tractor semi-trailer will experience negative lateral friction margins
at low design speeds (e.g., 35 mph or less). However, the emergency braking scenario
does not seem likely to occur frequently enough to constitute a reasonable basis for
design. The margins of safety against skidding were slightly higher for the tractor semitrailer/full-trailer truck when compared to the tractor semi-trailer
When vehicles change lanes towards the inside of a horizontal curve, the margins of
safety against skidding decrease considerably for all vehicle types. When lane changing
occurs during a stopping sight distance or emergency braking maneuver, all vehicles
exhibited negative margins of safety against skidding. For those situations (i.e.,
combinations of horizontal curvature, grade, and vehicle maneuvers) in which the
transient bicycle model predicted skidding (i.e., negative lateral friction margins), the
multi-body model showed that if a vehicle has ABS, and the driver properly responds to
minor lateral skidding, then the vehicle can maintain its intended path. The case of a
tractor semi-trailer without ABS need not be considered because all tractor semi-trailers
are mandated to have ABS.
CONCLUSIONS
The following conclusions were drawn from the combined field studies and analytical and
simulation modeling efforts in the present study:
x
x
For a simple horizontal curve, the maximum rate of superelevation (emax) should not
exceed 12 percent on a downgrade. If considering a maximum superelevation rate
greater than 12 percent, a spiral curve transition is recommended to increase the margins
of safety against skidding between the approach tangent and horizontal curve. On
upgrades of 4 percent and greater, the maximum superelevation rate should be limited to
9 percent for minimum-radius curves with design speeds of 55 mph and higher, to avoid
the possibility of wheel lift events. Alternatively, if it can be verified that the available
sight distance is such that deceleration at −11.2 ft/s2 is unlikely to be required on
upgrades of 4 percent or more (i.e., the available sight distance is greater than minimum
stopping sight distance design values), emax values up to 12 percent may be used for
minimum-radius curves.
For horizontal curve design, the superelevation attained at the point of curve entry
should be checked and compared to the lateral friction margin condition (i.e., Equation
6) so that the lateral friction margin on curve entry is not less than the margin within the
curve.
17
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
MarginsManyGradesOneEAllAccelsSS, 2-sig,no lanechange, ax=0 to 15,G=-9 to 0,e=0. TractorTrailer.
Constant Speed, ax = 0 ft/s 2
Constant Speed, ax = 0 ft/s 2
Curve Entry Decel, ax = -3 ft/s 2
0.4
SSD Decel, ax = -11 ft/s 2
0.3
Emergency Decel, ax = -15 ft/s 2
0.2
0.1
0
-0.1
30
40
0.3
SSD Decel, ax = -11 ft/s 2
0.2
Emergency Decel, ax = -15 ft/s 2
0.1
0
Grade = 0%
Intermediate grades
Grade = -9%
Grade = 0%
Intermediate grades
Grade = -9%
Tractor Trailer, e = 0%
50
60
70
-0.1
80
30
40
Minimum Fy Margin
Minimum Fy Margin
0.4
Curve Entry Decel, ax = -3 ft/s 2
0.3
SSD Decel, ax = -11 ft/s 2
0.2
0.1
Emergency Decel, ax = -15 ft/s 2
0
-0.1
40
50
60
70
80
Tractor Trailer, e = 16%
Curve Entry Decel, ax = -3 ft/s 2
0.3
SSD Decel, ax = -11 ft/s 2
0.2
Emergency Decel, ax = -15 ft/s 2
0.1
Grade = 0%
Intermediate grades
Grade = -9%
Tractor Trailer, e = 16%
70
-0.1
80
Speed (mph)
2
60
0
Grade = 0%
Intermediate grades
Grade = -9%
30
50
Constant Speed, ax = 0 ft/s 2
Constant Speed, ax = 0 ft/s 2
0.4
Tractor Trailer, e = 0%
Speed (mph)
Speed (mph)
1
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Minimum Fy Margin
Minimum Fy Margin
0.4
Curve Entry Decel, ax = -3 ft/s 2
30
40
50
60
70
80
Speed (mph)
Figure 4. Lateral Friction Margins from Steady-State Bicycle (left) and Transient Bicycle (right)
Models for Tractor Semi-Trailer (G = 0 to −9%, e = 0 and 16%) (ax = 0, −3, −11,
and −15 ft/s2).
x
x
For sharp horizontal curves (or near minimum-radius curves) on downgrades of 4
percent or more, the “STAY IN LANE” sign (R4-9) [MUTCD, 2009] should be
installed in advance of the curve on multilane highways. Consideration may also be
given to using solid white lane line markings to supplement the R4-9 sign.
Sharp horizontal curves (or near minimum-radius curves) on downgrades of 4 percent or
more should not be designed for low design speeds (i.e., 30 mph or less). In the event
that such situations cannot be avoided, warning signs to reduce speeds well in advance
of the start of the horizontal curve should be used.
The present study investigated a variety of horizontal curve-vertical grade combinations
that may be encountered on multi-lane divided and undivided highways, two-lane highways, and
ramps. Both high- and low-speed design scenarios were considered for a variety of vehicle and
pavement surface types and conditions. Future research should consider horizontal curve design
for low-speed design. Although not the primary focus of the present study, the vehicle dynamics
simulations performed in the study found that, for design speeds of 10 and 15 mph, the
18
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
maximum side friction factors (fmax) used for horizontal curve design (0.38 and 0.32 for 10 mph
and 15 mph, respectively) exceed truck rollover thresholds.
Future research should also include simulation modeling with tanker trailers. Existing
commercial vehicle simulation models do not have the capability to simulate the dynamic effects
of liquid sloshing in a tank trailer, so this research question is particularly challenging at present.
Future research should be directed at collecting information concerning the relative
propensity of emergency braking maneuvers under normal travel conditions to determine to what
extent these should be considered in horizontal curve-vertical grade geometric design policy.
Naturalistic driving studies may provide the opportunity to collect these data.
The margins of safety against skidding and rollover are generally lowest for horizontal
curves with low design speeds (40 mph or lower). This suggests that the difference between
friction demanded by vehicles in relation to design side friction factors (fmax) is lower at low
design speeds relative to higher design speeds. Because design side friction factors are based on
driver comfort levels, future research should be directed at reconciling comfort thresholds
acceptable to drivers on horizontal curves to the friction limits predicted by vehicle dynamic
simulations.
ACKNOWLEDGEMENT
The authors wish to acknowledge the National Cooperative Highway Research Program, which
sponsored this research under Project 15-39. Further, the authors wish to thank the State
Departments of Transportation of California, Maryland, Pennsylvania, Washington, and West
Virginia for their assistance in this research.
REFERENCES
1. American Association of State Highway and Transportation Officials. A Policy on Geometric
Design of Highways and Streets, Washington, D.C., 2011.
2. Harwood, D. W., J. M. Mason, W. D. Glauz, B. T. Kulakowski, and K. Fitzpatrick. Truck
Characteristics for Use in Highway Design and Operation. Vol. I. Research Report, Report
No. FHWA-RD-89-226, 1989.
3. Harwood, D. W. and J. M. Mason. Horizontal Curve Design for Passenger Cars and Trucks.
Transportation Research Record 1445, Transportation Research Board, Washington, D.C.,
1994, pp. 22-33.
4. Harwood, D. W., D. J. Torbic, K. R. Richard, W. D. Glauz, and L. Elefteriadou. NCHRP
Report 505: Review of Truck Characteristics as Factors in Roadway Design. Transportation
Research Board, Washington, D.C., 2003.
5. MacAdam, C.C., P.S. Fancher, and L. Segel. Side Friction for Superelevation on Horizontal
Curves. Final Technical Report Volume II, Report No. UMTRI-85-18-2, University of
Michigan Transportation Research Institute, 1985.
6. Easa, S.M. and A. Abd El Halim. Radius Requirements for Trucks on Three-Dimensional
Reverse Horizontal Curves with Intermediate Tangents. Transportation Research Record:
Journal of the Transportation Research Board, No. 1961, Transportation Research Board of
the National Academies, Washington, D.C., 2006, pp. 83-93.
7. Bonneson, J.A. NCHRP Report 439: Superelevation Distribution Methods and Transition
Designs. Transportation Research Board, Washington, D.C., 2000.
19
TRB 2014 Annual Meeting
Paper revised from original submittal.
Torbic et al.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
8. Torbic, D.J., M.K. O’Laughlin, D.W. Harwood, K.M. Bauer, C.D. Bokenkroger, L.M. Lucas,
J.R. Ronchetto, S. Brennan, E. Donnell, A. Brown, and T. Varunjikar. Superelevation
Criteria for Sharp Horizontal Curves on Steep Grades, Final Report NCHRP Project 15-39,
MRIGlobal, 2013.
9. Bonneson, J.A. Side Friction and Speed as Controls for Horizontal Curve Design. Journal of
Transportation Engineering, Vol. 125, No. 6, 1999, pp. 473-480.
10. Eck, R.W. and L.J. French. Effective Superelevation for Large Trucks on Sharp Curves and
Steep Grades. West Virginia Department of Transportation Research Project #153, West
Virginia University, Morgantown, WV, 2002.
11. Moyer, R.A. Skidding Characteristics of Automobile Tires on Roadway Surfaces and Their
Relation to Highway Safety. Bulletin No. 120, Ames, Iowa: Iowa Engineering Experiment
Station, 1934.
12. Fancher, P.S., R.D. Ervin, C.B. Winkler, and T.D. Gillespie. A Fact Book of the Mechanical
Properties of the Components for Single-Unit and Articulated Heavy Trucks. Report No.
DOT HS 807 125, National Highway Traffic Safety Administration, U.S. Department of
Transportation, 1986.
13. Wong, J.Y., Theory of Ground Vehicles, John Wiley and Sons, Inc. 2008.
14. American Society for Testing and Materials. Standard Test Method for Measuring Pavement
Surface Frictional Properties Using the Dynamic Friction Tester, E1911-09. Annual Book of
ASTM Standards, West Conshohocken, PA, 2005.
15. Velenis, E., P. Tsiotras, and C. Canudas-de Wit. "Extension of the lugre dynamic tire friction
model to 2d motion." Proceedings of the 10th IEEE Mediterranean Conference on Control
and Automation-MED. 2002.
20
TRB 2014 Annual Meeting
Paper revised from original submittal.