Revision of calculation of stopping sight distance/ Sustojimo matomumo atstumo skaiciavimo modelio tobulinimas/ Apstasanas redzamibas distances aprekina parskatisana/ Peatumisnahtavuse kauguse arvutuse umberhindamine.
Cheng, Jianchuan ; Yuan, Hao ; Shi, Guifang 等
1. Introduction
Sight distance is a primary element in roadway geometric design.
From a geometric design standpoint, the min sight distance available on
a roadway should be long enough to allow a vehicle traveling at or
around the design speed to stop before reaching a stationary object in
its path. As a control index, the Stopping Sight Distance (SSD) is
relatively important among various sight distances. The sight distances
such as decesion sight distance and passing sight distance are all
calculated based on SSD. To simplify the relations of SSD with vehicle
and roadway, this paper takes traditional vehicle dynamics and roadway
geometry as consideration for SSD calculation, although some researches
verified that the real on-site SSDs are also affected by some factors
such as the Anti-lock braking system (ABS) and the roughness of roadway
surface, etc. (Bogdevicius, Vladimirov 2006; Durth, Bernhard 2000;
Greibe 2008; Mavromatis et al. 2005).
The current procedures for determining SSD are intended to allow a
normally alert passenger-car driver, traveling at or around the design
speed on wet pavement, to react and bring the vehicle to a stop before
striking a stationary object in its path. The basic calculation model
for this situation was formalized by American Association of State
Highway Officials (AASHO) in 1940. Although the calculation model has
remained unchanged, adjustments in calculation model parameters have
been made in several AASHO and American Association of State Highways
and Transportation Officials (AASHTO) publications over the past few
decades. Table 1 below is a summary of these changes since 1940.
The significant change in determining the SSD requirement is the
use of a comfortable deceleration rate rather than a friction factor at
A Policy on Geometric Design of Highways and Streets (2001) based on the
findings and recommendations reported in the NCHRP Report 400
Determination of Stopping Sight Distances (Fambro et al. 1997; Prosser
2005). The authors also made similar research work (Yuan et al. 2009)
although the current SSD calculation in almost all the other countries
of the
world including China's Design Specification for Highway
Alignment (JTG D20-2006) is still based on friction factor. However,
different opinions exist concerning the appropriateness of the
calculation model and the values of parameters used to determine the min
required SSD (Hall, Turner 1988; Olson et al. 1984). This paper analyzes
and revises the existing SSD calculation model which is based on driving
at straight section, not considering the influence of curve and cross
slope variation such as superelevation. Therefore, through refining the
physical calculation model with consideration of the influence of
driving at curve section, a more reasonable formula of SSD calculation
is presented.
2. The existing SSD calculation model and formulas
One of the most important requirements in highway design is to
provide adequate SSD at every place along the roadway. It is calculated
using basic principles of physics and relationships among the various
design parameters. As all known, SSD is the sum of two components, brake
reaction distance [S.sub.1] (distance traveled from the instant of
object detection to the instant the brake is applied) and braking
distance [S.sub.2] (distance traveled from the instant the brake is
applied to when the vehicle is decelerated to a stop).
Perception-reaction time for SSD is defined as the interval of time
between the instant that the driver recognizes the existence of an
object or hazard on roadway ahead and the instant that the driver
actually applies the brakes or makes an evasive maneuver. This interval
includes the time required to make the decision that a stop or path
correction is necessary. So the reaction distance can be expressed by
the Eq (1):
[S.sub.1] = Vt/3.6, (1)
where V--vehicle speed, km/h; t--perception-reaction time. For
approx 90% of drivers in the various studies mentioned, a reaction time
of 2.5 s was found to be adequate (Fambro et al. 1997; Johansson, Rumar
1971; Normann 1953; Shi et al. 2010), although some research works based
on field measurement suggested that a 2.0 s is enough (Durth, Bernhard
2000; Mavromatis et al. 2005).
The approx brake distance of a vehicle on a level terrain roadway
may be determined by the Eq (2):
[S.sub.2] = [V.sup.2]/25.92a, (2)
where V--vehicle speed, km/h; a--driver deceleration, m/[s.sup.2].
The existing formula adopts a physics calculation model in which
vehicles travel at a straight roadway section. Thus, under this
condition for level terrain, only the friction between tires and roadway
contributes to a stop (Fig. 1).
[FIGURE 1 OMITTED]
Friction between tires and roadway can be expressed by the Eq (3):
F = Gf = ma, (3)
where F--friction between tires and roadway, N; G--vehicle weight,
N; f--coefficient of friction between tires and roadway; m--vehicle
mass, kg; a--driver deceleration, m/[s.sup.2].
Then
a = gf, (4)
where g--acceleration of gravity, m/[s.sup.2].
Thus the existing Eqs for SSD calculation are:
[S.sub.2] = [V.sup.2]/25.92gf [approximately equal to]
[V.sup.2]/254f, (5)
S = [S.sub.1] + [S.sub.2] = Vt/3.6 + [V.sup.2]/254f. (6)
For driving at straight longitudinal slope sections, the Eq has a
little change as shown at Eq (7).
S = Vt/3.6 + [V.sup.2]/254([+ or -] i), (7)
where i--grade: + for upgrade, - for down grade, %.
Some countries like Austria, Germany and Greece use a slightly
different SSD model, which incorporates the effect of a speed-dependent
longitudinal friction factor and the aerodynamic drag force on the
decelerating vehicles, as shown in Eq (8) (Harwood et al. 1995).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)
where [f.sub.T](V)--speed dependent longitudinal friction factor;
[F.sub.L]--aerodynamic drag force, N.
3. Revision of SSD calculation model and formulas
The existing SSD calculation model is simple as it only considers
the situation of driving at straight section, ignoring the situation of
driving at curve section, the likely accident-prone area. Therefore,
there appeared documents revising the SSD application in practice. Among
these documents, 3D-alignment SSD analysis, field measurement based
parameters revision for perception-reaction time; deceleration rate and
coefficient of friction, as well as reliability-based SSD calculation
are most common (Arndt et al. 2010; Easa 2009; Greibe 2008; Nehate, Rys
2006; Sarhan, Hassan 2009). However, these revisions did not pay
attention to the SSD model itself, still taking driving at straight
section as default.
Obviously, driving at curve section with superelevation is less
safe and faces more safety concern caused by sight distance especially
at with small horizontal radii (Dissceti 2010). Therefore, driving at
curve section should be used for SSD calculation model.
As shown in Figs 2, 3, when vehicle is traveling at curve, the
physical calculation model of braking in limit state of slip can be
defined.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
Compared with Figs 1, 2 in critical condition of slip, that differs
from what takes place on straight and level sections (note: the
direction of friction is reverse to that of vehicle slip).
G sin [alpha] + F cos [beta] = ma cos [alpha], (9)
F sin [beta] = ma, (10)
cos [alpha] = 1, sin [alpha] [approximately equal to] tan [alpha] =
e has been assumed for [alpha] is generally tiny. Substituting for F =
Nf = G cos [alpha]f and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)
gf sin [beta] = a. (12)
Now Eq (11) can be written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)
then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)
Substituting for sin [beta] from Eq (14) in Eq (12), then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)
Compared Eq (15) with Eq (4), it is found when vehicles move on
curve sections, largest deceleration of vehicle is less than that on
level sections. It is also affected by vehicle speed V, curve radius R
and superelevation rate e. Moreover, if R = +[infinity] and e = 0 (level
terrain condition), Eq (15) will be same as Eq (4). So Eq (15) is
applicable to much more cases than Eq (4) which is a special case of Eq
(15).
Substituting for a from Eq (15) in Eq (2), then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)
So, the braking distance gap [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] by two models is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)
4. Case study
Based on Eq (17) Table 2 gives the revised braking distances at 5
max superelevation rates corresponding the min horizontal radii for
design speeds ranging from min speed 30 km/h to max speed 120 km/h in
comparison with those calculated using the 1994 Green Book (metric).
As shown in Figs 4 and 5, braking distance gap varies from 0.82 m
to 15.16 m with the max superelevation rates and design speeds. The gap
looks increasing longer SSDs with speed increases from 30 km/h to 110
km/h, except a decline at speed 120 km/h.
The braking distance gap is also illustrated graphically at Fig. 5.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
5. Conclusions
Based on finding out the critical condition of vehicles driving at
curves, a revised SSD calculation model and formulas were presented,
that are universal and cover the existing SSD model and formulas where R
= +[infinity] and no superelevation.
From a case study, the revised SSD calculation formula would result
basically increasing longer SSDs as speed varies from 30 km/h to 110
km/h, and a little decline at speed 120 km/h. This is intuitively
explained that driving at curves requires longer SSDs than those at
straights.
The revised SSD calculation model should take longitudinal slope as
consideration in the following research.
Field measurement experiment at the curve section with
superelevation should be designed and implemented to test the validity
of the revised SSD calculation model.
doi: 10.3846/bjrbe.2011.13
Acknowledgements
The authors would like to thank Mr. Wei Wu, senior engineer of
DELCAN Company, USA for his careful revisions, as well as the anonymous
reviewers for their feedbacks that definitely improved this paper.
Received 15 July 2009; accepted 10 March 2011
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Jianchuan Cheng (1), Hao Yuan (2), Guifang Shi (3), Xiaoming Huang
(4)
School of Transportation, Southeast University, 2 Sipailou,
Nanjing, 210096, China
E-mails: (1) jccheng@seu.edu.cn; (2) yhaohao@163.com; (3)
guifangshi@gmail.com; (4) huangxm@seu.edu.cn
Table 1. Changes in the 6 parameters used for SSD calculation
Reference Speed of Perception-
(year, title) calculation reaction time, s
1940, A Policy on Design speed 3.0
Sight Distance for at 50.7 km/h
Highways or 2.0
at 118.3 km/h
1954, A Policy on 85-95% of design 2.5
Geometric Design speed
of Rural Highways
1965, A Policy on 80-93% of design 2.5
Geometric Design speed
of Rural Highways
1971, A Policy on Min.--80 to 93% of 2.5
Geometric Design design speed
of Highways and Des.--design speed
Streets
1984 and 1990, A Min.--80 to 93% of 2.5
Policy on design speed
Geometric Design Des.--design speed
of Highways and
Streets
1994, A Policy on Min.--82 to 100% of 2.5
Geometric Design design speed
of Highways and Des.--design speed
Streets 2001 and
2004,
Policy on Design speed 2.5
Geometric Design
of Highways and
Streets
Reference Design Friction factors or
(year, title) pavement/stop deceleration rate
1940, A Policy on Dry/Locked- Ranges from 0.50 at
Sight Distance for wheel 50.7 km/h to 0.40 at
Highways 118.3 km/h
1954, A Policy on Dry/Locked- Ranges from 0.36
Geometric Design wheel at 50.7 km/h to
of Rural Highways 0.29 at 118.3 km/h
1965, A Policy on Dry/Locked- Ranges from 0.36 at
Geometric Design wheel 50.7 km/h to 0.27
of Rural Highways at 118.3 km/h
1971, A Policy on Dry/Locked- Ranges from 0.35 at
Geometric Design wheel 50.7 km/h to 0.27
of Highways and at 118.3 km/h
Streets
1984 and 1990, A Dry/Locked- Slightly higher at
Policy on wheel higher speeds than
Geometric Design 1970 values
of Highways and
Streets
1994, A Policy on Dry/Locked- Ranges from
Geometric Design wheel 0.40 at 30 km/h to
of Highways and 0.28 at 120 km/h
Streets 2001 and
2004,
Policy on Dry/Locked- Deceleration rate
Geometric Design wheel 3.4 m/[s.sup.2]
of Highways and
Streets
Reference Eye height, Object
(year, title) m height, m
1940, A Policy on 1.37 0.10
Sight Distance for
Highways
1954, A Policy on 1.37 0.10
Geometric Design
of Rural Highways
1965, A Policy on 1.14 0.15
Geometric Design
of Rural Highways
1971, A Policy on 1.14 0.15
Geometric Design
of Highways and
Streets
1984 and 1990, A 1.07 0.15
Policy on
Geometric Design
of Highways and
Streets
1994, A Policy on 1.07 0.15
Geometric Design
of Highways and
Streets 2001 and
2004,
Policy on 1.07 0.6
Geometric Design
of Highways and
Streets
Note: Min--minimum speed; Des--disarable speed.
Table 2. Distance gap between the existing braking distance
and the revised value
Braking Revised
Speed, distance, value, Distance
km/h e R, m m m gap, m
30 0.04 35 8.8 9.69 0.89
0.06 30 9.87 1.07
0.08 30 9.62 0.82
0.10 25 9.97 1.17
0.12 25 9.70 0.90
40 0.04 60 16.6 18.53 1.93
0.06 55 18.51 1.91
0.08 50 18.59 1.99
0.10 45 18.82 2.22
0.12 45 18.27 1.67
50 0.04 100 28.1 31.45 3.35
0.06 90 31.56 3.46
0.08 80 31.94 3.84
0.10 75 31.75 3.65
0.12 70 31.68 3.58
60 0.04 150 42.9 48.13 5.23
0.06 135 48.21 5.31
0.08 125 47.95 5.05
0.10 115 47.92 5.02
0.12 105 48.21 5.31
70 0.04 215 62.2 69.67 7.47
0.06 195 69.47 7.27
0.08 175 69.80 7.60
0.10 160 69.89 7.69
0.12 150 69.40 7.20
80 0.04 280 83.9 94.95 11.05
0.06 250 95.26 11.36
0.08 230 94.79 10.89
0.10 210 94.95 11.05
0.12 195 94.66 10.76
90 0.04 375 106.2 117.96 11.76
0.06 335 118.02 11.82
0.08 305 117.75 11.55
0.10 275 118.35 12.15
0.12 255 117.96 11.76
100 0.04 490 135.6 149.29 13.69
0.06 435 149.37 13.77
0.08 395 148.94 13.34
0.10 360 148.79 13.19
0.12 330 148.76 13.16
110 0.04 625 170.0 185.01 15.01
0.06 560 185.04 15.04
0.08 500 185.16 15.16
0.10 455 184.81 14.81
0.12 415 184.87 14.87
120 0.04 870 202.3 213.90 11.60
0.06 755 213.86 11.56
0.08 665 213.94 11.64
0.10 595 213.96 11.66
0.12 540 213.80 11.50
