Multi-Body Simulation Modeling of Vehicle Skidding and Roll over for Horizontal Curves on...

Multi-Body Simulation Modeling of Vehicle Skidding and Roll over 1

for Horizontal Curves on Longitudinal Grades 2

3

Amirarsalan Mehrara Molana, *

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a Researcher, Young Researchers and Elites Club, Science and Research Branch, Islamic Azad 5

University, Tehran, Iran. 6

Email: [email protected], Tel: +98-9126358911, Fax: +98-2188256371 7

*(Corresponding Author) 8

9

Ali Abdi Kordanib 10

b Assistant Professor, Department of Civil Engineering, Imam Khomeini International 11

University, Qazvin, Iran. 12

Email: [email protected], Tel: +98-9126085308 13

14

Submission Date: 27 July, 2013 15

Word Count: 5176 + 8 Table + 2 Figures = 7676 Word Equivalent 16

17

TRB 2014 Annual Meeting Paper revised from original submittal.

Abstract 18

The impact of combined horizontal curve and longitudinal grade on skidding and roll over of various 19

vehicles is investigated. A series of simulation tests are conducted using CarSim and TruckSim, which are 20

among the most popular multi-body simulation software packages in the vehicle industry. Two types of 21

behavior for the driving system are considered in the simulations: (1) the driver negotiates the curve at 22

constant speed or (2) the driver needs to brake while passing downgrades. Based on the results, a truck is 23

subjected to greater lateral accelerations on downgrades and lower lateral accelerations on upgrades. 24

Greater side friction factors are seen on downgrades for all vehicles when cornering. Braking has a 25

considerable effect on side friction factors, especially for a sedan car. Braking can be an important threat 26

to the safety of passenger cars (in terms of skidding) and trucks (in terms of rollover). 27

Keywords: Roadway safety; Skidding; Roll over; Horizontal curve; Longitudinal grade; Multi-body 28

simulation; Braking. 29

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1. Introduction 38

Horizontal curves have long been recognized as a considerable safety problem for vehicles, because 39

of factors such as centrifugal force and driver expectation, and are considered to be even more important 40

in this context than the sight distance issue, particularly on rural two-lane highways (Charlton 2007). 41

Horizontal curves pose two threats to vehicles: (1) skidding and (2) rollover toward the outer direction of 42

the curve (Awadallah 2006). The potential threat from horizontal curves can be more significant when 43

they are located on steep longitudinal grades. The latter, especially downgrades, may shift the center of 44

gravity of the vehicle and change the distribution of forces on its tires. Although road designers try to 45

avoid the combination of horizontal curves with grades, this is a common occurrence in mountainous 46

terrain and two-lane rural highways owing to economic issues and the time-consuming nature of the 47

construction projects in these areas. Moreover, this combination is also common on interchange ramps. 48

The Green Book (AASHTO 2011) considers a point-mass model for the basic curve equation. The 49

point-mass model is one of the simplest vehicle models, but has the following limitations: 50

(1) The effect of longitudinal grade is neglected and the differences in types of horizontal curves are 51

ignored. 52

(2) The distribution of forces acting on different tires is not taken into account in the determination of 53

side friction factors. Friction can vary significantly between tires when the vehicle is cornering 54

(MacAdam 1985). 55

(3) The point-mass model is independent of vehicle configuration and characteristics such as 56

suspension system, and the design criterion is based on an unsprung (rigid) vehicle. 57

Many researchers have pointed out that a more sophisticated model should be used in order to analyze 58

the cornering behavior of vehicles (Psarianos et al. 1998; Kontaratos et al. 1994; Bonneson 1999a). The 59

point-mass model might be suitable for modeling non-articulated vehicles, but it is insufficiently precise 60


for trucks (Bonneson 1999a). Varunjikar’s (2011) research showed that the transient bicycle, steady-state 61

bicycle and multi-body simulation models are better than the point-mass model for horizontal curve 62

design. 63

The primary objective of this paper is to investigate the impact of combined horizontal curve with 64

longitudinal grade on safety factors such as the side friction factor and lateral acceleration to determine 65

the stability of various vehicles against skidding and rollover. In addition to the key role of geometric 66

design on horizontal curve safety, adopting roadway design to vehicle characteristics appears to be 67

necessary. Vehicle engineering and roadway design are two main factors involved in attempts to improve 68

highway safety and decrease fatalities, and therefore compatibility between highway design and modern 69

vehicle design is essential (Chang 2001; Stine et al. 2010). 70

Although vehicle braking has been identified as a relevant issue in roadway safety, almost no 71

estimates of its effects on the side friction factor or rollover have been published. From the viewpoint of 72

vehicle dynamics, Gim et al. (2007), among others, have studied the properties of tires under braking 73

conditions, but they did not provide any conclusions specifically with regard to highway engineering. One 74

of the contributions of the present research is to take account of braking in the evaluation of friction 75

factors. The actual tire–pavement friction value during emergency or locked-wheel braking is a major 76

factor in accident analysis, but it is not known for most cases and a range of average friction values is 77

assumed for analysis (Heinrichs et al. 2003). 78

In this study, a total of 112 simulation tests were conducted to estimate vehicle stability on combined 79

horizontal curves with longitudinal grades using CarSim and TruckSim. These are multi-body simulation 80

software packages developed by Mechanical Simulation Corporation (MSC) to predict the performances 81

of various vehicles and analyze the dynamic behavior of vehicles in response to a given road geometry 82

with high accuracy. Several companies, such as Ford, Toyota and Opel, use CarSim for their simulation 83


tests. Although vehicle dynamics simulations have long been used for both vehicle design and crash 84

reconstruction, they have rarely been used by highway engineers for roadway design (Stine et al. 2010). 85

2. Review of the Literature 86

Kontaratos et al. (1994) included the effects of many factors such as grade to determine the minimum 87

horizontal curve radius. Their analysis concluded that there is a strong relationship between the radius of 88

the horizontal curve and grade, and that the minimum radius should increase on longitudinal grades at 89

higher vehicle speeds. 90

One of the studies that used simulations to analyze the safety of combined horizontal curves was 91

performed by Easa and Dabbour (2003). They compared the lateral acceleration between minimum flat 92

horizontal curve radii with the minimum radii on vertical alignments using VDM road software, and 93

recommended that the minimum radius of combined horizontal curves with vertical alignments be 94

increased by approximately 3–16% relative to that in a flat horizontal curve scenario. Similar research for 95

compounded horizontal curves by Easa and Dabbour (2005) revealed that the minimum radius should be 96

increased by 20% to achieve the same comfort limit that the driver experiences on a flat horizontal curve 97

with the same radius. Easa and Dabbour (2005) also recommended a further study to revise current values 98

of side friction, since these were determined many years ago and originally only for passenger cars 99

without considering heavy vehicles. Moreover, VDM has some limitations in modeling human factors 100

and driving behavior; for example, as Easa and Dabbour (2005) pointed out, it is not designed to 101

incorporate complex human factors such as braking. 102

Chang (2001) developed new models to determine the minimum radius, considering a sprung vehicle 103

for passenger cars and trucks. This research was important because the center of gravity shifts laterally 104

during cornering, a fact which is ignored in an unsprung vehicle model. Eq. (1) and (2) illustrate the 105

Chang's (2001) recommendation models: 106

Passenger cars: 𝑅 = 𝑣2/ 121(0.5𝑒 + 𝑓) (1) 107


Truck: 𝑅 = 𝑣2/122.5(0.75𝑒 + 𝑓) (2) 108

A key study was carried out by Bonneson (2000), who presented a series of friction factors and 109

vehicle dynamic data. Based on his research, a horizontal curve located on a downgrade is of great 110

concern because (1) drivers tend to brake to maintain a safe speed and (2) they also tend to brake an extra 111

amount to reduce speed when they get closer to the beginning of the curve. By considering the effect of 112

roadway grade on friction, Bonneson (2000) mentioned that the Green Book generally underestimates 113

side friction demand for most horizontal curves, particularly those that have steep grades, little 114

superelevation and a large radius. According to research by MacAdam et al. (1985), side friction demand 115

is about 15% more than the predicted value according to the Green Book owing to steering fluctuations 116

along the horizontal curve. 117

You et al. (2012) studied vehicle skidding and rollover as two failure modes and found that rollover 118

does not count as a major concern for passenger cars unless they collide with each other or with another 119

object. However, rollover is the main concern for trucks on horizontal curves. The rollover threshold for 120

passenger cars is significantly higher than for trucks: it is about 1.2 g for cars, whereas trucks might roll 121

over at just 0.34 g (Harwood et al. 1994). When the rollover threshold is more than the available side 122

friction, skidding will occur before rollover (Varunjikar 2011). Since the available side friction is usually 123

less than 0.9, skidding is the main concern for passenger cars. 124

Research was conducted by Eck and French (2002) to analyze the safety of some mountainous 125

horizontal curves. They found that when a sharp horizontal curve is constructed on a steep downgrade, 126

additional superelevation is required. Eck and French (2002) developed two new equations to 127

determinate the superelevation rate for passenger cars (Eq. 1) and articulated vehicles (Eq. 2), although it 128

was mentioned that more study needs to validate these equations. 129

𝑒 + 𝑓 = 𝑣2/ 15𝑅 + 𝐺 𝑠𝑖𝑛𝜃 (1) 130


𝑒 + 𝑓 = 𝑣2

15𝑅 + 𝐺𝑊 𝑡𝑟𝑎𝑐𝑡𝑜𝑟 ∗𝑠𝑖𝑛 𝜃𝑓𝑟𝑜𝑛𝑡 𝑤ℎ𝑒𝑒𝑙−𝑡𝑟𝑎𝑐𝑡𝑜𝑟 + 𝑊 𝑡𝑟𝑎𝑖𝑙𝑒𝑟 ∗𝑠𝑖𝑛 (𝜃𝑓𝑟𝑜𝑛𝑡 𝑤ℎ𝑒𝑒𝑙−𝑡𝑟𝑎𝑖𝑙𝑒𝑟 )

𝑊 𝑜𝑣𝑒𝑟𝑎𝑙𝑙

(2)

Where: 131

e = Rate of superelevation 132

f = Side friction factor 133

v = Velocity (mph) 134

R = Radius (ft) 135

ө = Deflection angle between path of vehicle 136

and tires 137

G = Grade (decimal form) 138

W = Weight (Pound) 139

Moreover, the Transportation Research Board (TRB) and AASHTO technical committees on 140

geometric design, at a joint meeting in June 2004, declared the current superelevation issue to be one of 141

the five highest priorities (TRB Website 2010). To fill the gap in current superelevation design policy, 142

they initiated a project entitled ‘Superelevation criteria for sharp horizontal curves on steep grades’ 143

(Project 15-39). This project is still in progress and the results are yet to be published. 144

3. Simulation Process 145

The simulation process was developed on the basis of CarSim and TruckSim software, which are 146

among the most popular dynamics simulation packages in the vehicle industry. Any parameter, from 147

roadway geometric design to inertial properties and characteristics of vehicles, can be defined by software 148

users (Stine et al. 2010). 149

3.1 Vehicles 150

CarSim supports two-axle passenger cars and trucks with one- or two-axle trailers. TruckSim can also 151

simulate the dynamic behavior of heavy vehicles with two, three or four axles on the lead unit, dual tires 152

and various type of loading on trailer trucks. For this study, two different kinds of passenger cars and a 153


truck were used in the simulation: an E-class sedan, an E-class SUV and a two-axle conventional loaded 154

truck. 155

3.2 Road Environment (Geometry) 156

The performances of the vehicles were evaluated for different conditions of road geometry at various 157

speeds to study vehicle stability on horizontal curves. Design speeds of 40, 70, 100 and 130 km/h for the 158

passenger cars and 40, 70 and 100 km/h for the truck were considered. The speed of 130 km/h was 159

ignored in the truck simulation: it is not possible for a typical two-axle truck to pass a steep upgrade at a 160

speed of 130 km/h, and since the software is based on the real behavior of vehicles, it was unable to 161

consider this speed in the simulation process. An 8% maximum rate of superelevation was selected and 162

the minimum radii of horizontal curves were calculated from the AASHTO basic formula as 41, 168, 394 163

and 832 m, respectively, for each speed design. Note that the rate of superelevation was taken as 8% for 164

all tests to provide the same condition during the simulation, although sometimes more typically a rate of 165

4% or 6% is used for lower speeds. Also, for each speed design, a tangent-to-curve transition was 166

considered, with respective lengths of superelevation runoff Lr of 41, 52, 65 and 82 m and lengths of 167

tangent runoff Lt of 10.25, 13, 16.25, 20.5 m. Note that the proportion of runoff length placed on the 168

tangent was 0.8, which means that 20% of the runoff length was applied on the curved section. All of the 169

simulated curves turned to the right. Friction between tire and pavement depends on many factors, such as 170

type of tire, weather conditions, and type and age of pavement surface. It is common in studies to 171

consider a coefficient of 0.9 or 0.8 for the maximum road friction. For example, Furtado et al. (2002) used 172

a value of 0.9, while Wang et al. (2012) used 0.8. In this study, the maximum road friction coefficient was 173

assumed to be 0.8. Finally, in order to take account of the vertical alignment, longitudinal grades of −9%, 174

−6%, −3%, 0%, +3%, +6%, +9% for the passenger cars and −6%, −3%, 0%, +3%, +6% for the truck were 175

included in the simulation process. The speeds of heavy vehicles decrease significantly on sharp 176

horizontal curves on upgrades (Mavromatis and Psarianos 2003), resulting in undesirable speed 177


fluctuations on steep upgrades and downgrades, which was why the ±9% grades were ignored in the truck 178

simulation. 179

3.3 Driver Behavior 180

Two types of behavior for the driving system have been defined: (1) the driver negotiates the curve at 181

constant speed, or (2) the driver needs to use the brake while passing downgrades. The first type of 182

behavior (no braking) can produce the desired results for upgrades and light downgrades, although this 183

kind of driving is almost impossible on steep downgrades and drivers usually use the brake to decelerate 184

and prevent the vehicle from lateral offsetting. Bonneson (1999b) stated that 87% of drivers reduced 185

speed on the curve. Braking inputs in CarSim are defined by the pressure applied to the brake system. 186

Thus, another driving situation was defined by applying a 5 MPa braking on the curve and releasing the 187

brake 0.5 s after activating it (i.e., a reaction time of 0.5 s). The braking reaction time does not affect the 188

maximum side friction factor, and the most critical friction occurs rapidly after braking. According to 189

Varunjikar’s (2011) research, when a vehicle brakes during cornering the side friction factor increases to 190

a maximum value and then decreases, since the centripetal acceleration decreases as the speed of the 191

vehicle falls. Consequently, it was not essential to consider any greater a braking reaction time for the 192

simulation. 193

Note that the brake systems of the sedan and SUV are anti-braking systems (ABS), whereas the 194

conventional truck uses a 10 kN-m capacity hydraulic brake system. 195

4. Simulation Outcomes 196

When a vehicle negotiates a horizontal curve, two kinds of failure might occur, namely skidding (due 197

to side friction) and rollover (due to lateral acceleration). Thus, in this investigation, friction factor and 198

lateral acceleration were considered the two main criteria for safety. 199

200


4.1 Side Friction Factor on Upgrades 201

To calculate the cornering friction factor (demand) of vehicles on different positive grades, the forces 202

acting on the tires were obtained by simulation. The forces for the truck are shown in Table 1. Based on 203

these results and the work by Kontaratos et al. (1994), as a vehicle moves up the grade, it is subjected to 204

greater longitudinal forces, and consequently a greater longitudinal friction factor and a lower side 205

friction factor are expected to arise. It should be noted that all of the curves in the simulation turn to the 206

right, and so, for all of tests, the greatest vertical force acts on the left tire of the front axle and the least 207

vertical force acts on the right tire of the rear axle. Generally, the left side of the vehicle experiences 208

greater lateral and vertical forces than the right side during cornering, while on a straight road (no 209

cornering) the left tire is at a higher elevation than the right tire and is therefore expected to be subjected 210

to a slightly lower reaction. Lateral load transfer can occur during cornering, owing to the lateral 211

acceleration, torque generated by the engine, geometry or a combination of these. Therefore, the vertical 212

forces acting on the inner and outer tires are not the same (Varunjikar 2011). For example, on a flat curve, 213

62% of the lateral forces and 59% of the vertical forces act on the left side of the truck at a design speed 214

of 100 km/h. The lower weight on the inner tires is also a cause of rollover. 215

By dividing the lateral forces by the vertical forces, side friction factors were obtained and are shown 216

in Table 2. According to this table, the side friction factors of the sedan decrease as the grade increases. 217

For the SUV, in all the tests, the side friction factors vary only slightly with grade. However, for the 218

truck, the side friction factors are higher on upgrades than on a flat grade. Based on AASHTO (2011) and 219

Bonneson (2000), the side friction demand is greater on steep upgrades owing to the tractive forces. From 220

Table 2, a greater side friction demand arises only for the truck. Consequently, it seems that the tractive 221

forces appear only for the truck. Also, according to the results of Kontaratos et al. (1994), vehicles are 222

subjected to lower side friction on upgrades. 223

224


4.2 Side Friction Factor on Downgrades 225

4.2.1 Without Braking 226

The forces on the truck tires according to TruckSim are shown in Table 3. These results are for 227

vehicles traveling at constant speed without braking. The side friction factors of vehicles cornering on 228

downgrades were obtained by dividing the lateral forces by the vertical forces, and are shown in Table 4, 229

from which it is clear that, in all the tests, the side friction factors increase as the longitudinal grade 230

decreases. From the point of skidding, the sedan experiences the most critical situation compared with the 231

other vehicles (except at a speed of 40 km/h, where the truck is in the most critical situation). As with the 232

results for upgrades, the side friction factors of the SUV vary only slightly with grade. It seems that, at 233

constant speed, the side friction factor of the SUV is independent of longitudinal grade. Also, since 234

AASHTO recommends side friction factors of 0.23, 0.15, 0.12 and 0.08 at design speeds of 40, 70, 100 235

and 130 km/h, respectively, it can be seen that there is compatibility between the simulation results for the 236

SUV and the AASHTO side friction values. 237

4.2.2 With Braking 238

Many key points were seen in the simulation tests with braking. The side friction factor of the 239

vehicles fluctuated during braking, as shown in Fig. 1 for the sedan at a speed of 130 km/h on a −9% 240

grade during the braking reaction time. Therefore, owing to the presence of these fluctuations, the 241

maximum value of the side friction was taken to determine the worst situation with regard to the effect of 242

braking. As Fig. 1 shows, for the sedan at 130 km/h, the maximum value of the side friction factor 243

occurred 0.1 s after commencement of braking, while the minimum value occurred 0.1 s after release of 244

the brake in most of the tests. 245

The results for side friction factors in the braking case are presented in Table 5. As already 246

mentioned, because of the fluctuations of the side friction factor during the braking time (as shown in Fig. 247

1), the maximum values of the factors were considered in this study. From a comparison of the results in 248


Tables 4 and 5, the effects of braking on the side friction factor can be seen to be most significant for the 249

sedan. 250

Also, according to Varunjikar (2011), the side friction factors obtained using the transient bicycle 251

model show good agreement with the CarSim results during braking. Thus, the braking results from 252

CarSim can be taken as reliable when designing downgrades. 253

4.3 Lateral Acceleration on Upgrades 254

Another important factor affecting vehicle stability and safety is the lateral acceleration to which it is 255

subjected. Two expressions commonly occur when dealing with the lateral accelerations and lateral forces 256

that arise on cornering: centrifugal force and centripetal acceleration. The centrifugal force is an 257

imaginary force that presents only when a vehicle moves on a circular path and is opposite in direction to 258

the centripetal acceleration (Awadallah 2006). The latter is provided by the acceleration due to 259

superelevation and tire–pavement friction (Bonneson 1999a). 260

In this part of the study, the lateral acceleration of the vehicle’s center of gravity was determined using 261

the simulation software and the centripetal acceleration was calculated from the values already found for 262

the side friction factor (see the previous section). A comparison between the lateral acceleration and the 263

centripetal acceleration is presented in this section. Figure 2 shows a diagram of the lateral acceleration 264

acting on the truck while it is traveling on a flat grade at a speed of 100 km/h. Note that in Fig. 2, the 265

starting point of the curve (PC) is at approximately 6 s of motion and the end point (PT) is at 26

s. As the 266

figure shows, the maximum lateral acceleration acts at the start of the horizontal curve and, after a while, 267

the vehicle undergoes the same lateral acceleration for the entire distance of the curve. The lateral 268

acceleration for most of the curve is 5–10% lower than the maximum value. According to Bonneson 269

(1999a), vehicles tend to shift laterally on entering a sharp horizontal curve owing to the unbalanced 270

lateral accelerations that act on them at this point. Reports of accidents involving vehicles veering off the 271

road at horizontal curves indicate that this generally happens near to the start of the curve, which is 272


known to be the most hazardous area of the curve. The value of the lateral acceleration is constant (in a 273

steady state) for the entire distance of the curve after the starting point, which is probably due to the 274

constant steering angle during cornering after the curve has been entered. Finally, the lateral acceleration 275

decreases at the end of the horizontal curve and becomes zero at the straight section of road. Note that the 276

direction of the measured lateral acceleration is on the outward the curve and opposite to the centripetal 277

acceleration. 278

Table 6 gives the lateral and centripetal accelerations of vehicles on upgrades during cornering. It can 279

be seen from this table that the longitudinal grade does not significantly affect the lateral acceleration for 280

either of the passenger cars or for the truck. However, a difference between lateral acceleration and 281

centripetal acceleration is found in some of the tests. The lateral accelerations of the SUV at a speed of 282

130 km/h and the truck at all speeds are slightly higher than the respective centripetal accelerations. This 283

difference between centrifugal and centripetal accelerations provides an acceleration causing a lateral 284

shift that could result in the vehicle encroaching on an adjacent lane or shoulder (Bonneson 1999a). The 285

lateral accelerations acting on the sedan and the SUV are almost the same, but the truck is subjected to a 286

slightly greater lateral acceleration. Because of the higher values of the centripetal acceleration for the 287

sedan, it is more stable against lateral accelerations than the SUV and the truck. Another observation from 288

Table 6 is that the lateral and centripetal accelerations have higher values at low speeds. 289

4.4 Lateral Acceleration on Downgrades 290

As with the friction factors, the lateral accelerations of vehicles have been studied for two different 291

driving behaviors, namely with and without braking. 292

4.4.1 Without Braking 293

Table 7 gives the lateral and centripetal accelerations of vehicles on downgrades. As was the case 294

with the upgrades, the downgrades have only a slight effect on the lateral accelerations of the passenger 295


cars. However, slight differences between lateral and centripetal accelerations are seen in some of the 296

tests for the truck. 297

4.4.2 With Braking 298

Table 8 gives the lateral and centripetal accelerations of vehicles on downgrades during cornering 299

with a braking force applied. As in the case of the friction factors (Table 5), because of fluctuations 300

during the braking period, maximum accelerations were considered in this study. 301

The lateral acceleration of the truck is seen to be close to the static stability factor (SSF) in a few of 302

the tests. The SSF of a vehicle is calculated by dividing half of the track width by the height of the center 303

of gravity, and once the vehicle’s lateral acceleration exceeds this ratio, it is likely to roll over (Farmer 304

and Lund 2002). The SSFs of the sedan and the SUV are 1.593 and 1.303, respectively, while that of the 305

truck without any payload is 1.039 and that of the loaded truck is 0.677. This also shows the effect of 306

loading on the stability of heavy vehicles. Many rollover crashes occur because of overloaded cargo that 307

increases the height of the center of gravity (Huang and Fan 2010). The simulation gave the maximum 308

lateral acceleration of the truck, with a speed of 40 km/h on a −6% downgrade during braking, as 309

0.44861, while its SSF, as already mentioned, is 0.677. However, for passenger cars negotiating a 310

downgrade horizontal curve, braking is not a threat from the point of view of acceleration. Passenger cars 311

are resistant to rollover because of their high rollover thresholds, and a passenger car will skid before 312

rollover can occur, whereas the rollover thresholds of trucks are low because of their higher centers of 313

gravity and different suspension systems, and they usually roll over before they are likely to skid 314

(Harwood et al. 2003). 315

5. Conclusions 316

For a vehicle, the most dangerous part of a horizontal curve in a road is at the start of the curve, 317

because of the fluctuations in the steering angle and the more negative lateral acceleration (centrifugal 318

acceleration). At this point, the lateral acceleration to which the vehicle is subjected is 5–10% greater than 319


elsewhere along the curve. Since the steering angle is related to the lateral acceleration, it is essential to 320

control this angle during cornering. 321

In the simulations for upgrades, a greater side friction factor (demand) is produced for the truck, 322

although the side friction factor of the sedan is decreased slightly and the SUV had a constant side friction 323

factor. It seems that the tractive forces affect the side friction factors only of heavy vehicles such as 324

trucks, with no effect on those of passenger cars. 325

The side friction factor increases as the downgrade increases at constant speed (no braking), which 326

means that a greater side friction factor is produced on steep downgrades in comparison with a flat grade 327

and mild downgrades. Thus, the margin of safety decreases on steep downgrades. It should also be noted 328

that Bonneson (2000) has pointed out that the AASHTO generally underestimates the side friction factor 329

for most horizontal curves, particularly those that have steep grades. 330

Upgrades and downgrades do not have a significant effect on negative lateral accelerations 331

(centrifugal accelerations) for passenger cars. However, according to the simulations, the lateral 332

accelerations of the truck increase slightly on downgrades and decrease on upgrades. Thus, lateral 333

acceleration cannot be taken to be a threat for passenger cars for any longitudinal grade, although it seems 334

that rollover is a serious problem for trucks because of their high static stability factor. According to the 335

simulations, the maximum lateral acceleration of the truck at a speed of 40 km/h on a −6% downgrade is 336

0.44861 during the braking, while its static stability factor is 0.677. Therefore, this aspect should be 337

considered in the design of rural highways carrying a high percentage of trucks. 338

The effect of braking on the side friction factors of vehicles was found to be significant. Another 339

important finding regarding the braking is the presence of large fluctuations in the side friction factors 340

throughout the braking period. According to Table 8, braking can be a serious problem for the safety of 341

passenger cars (in terms of skidding) and trucks (in terms of rollover). 342

343


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Table 1. Forces acting on the truck tires during cornering on upgrades according to TruckSim

Design

speed

(km/h)

AASHTO

design

radius (m)

Grade

(%)

Lateral Forces on Tires (N) Longitudinal Forces on Tires (N) Vertical Forces on Tires (N)

L1 L2 R1 R2 L1 L2 R1 R2 L1

L2-i

L2-o

R1

R2-i

R2-o

Conventional Truck Loaded

40 41 0 9298.4 11668 3603.0 6060.6 303.34 9030.0 98.921 1164.5 33221

26219

28672

12620

13933

11481 40 41 3 9309.2 11710 3713.4 6181.5 295.16 2995.6 101.15 3257.9 34222 26678

29189

12622 14067

11544

40 41 6 9755.1 13052 3844.9 6193.4 228.66 538.73 24.40 254.12 34552 26869 29598

12812 14115 11552

70 168 0 5463.2 6413.6 2361.8 4398.5 278.41 946.30 152.26 1107.5 29611 23332 24764

16700 16162 14710

70 168 3 5615.4 6555.6 2381.2 4461.9 266.13 1869.7 136.77 2036.7 29664 23479

24853

16396 16192

14736 70 168 6 5627.4 6561.6 2426.4 4501.3 242.9 1938.9 107.88 2102.5 29823 23557

25098

16004 16232

14787

100 394 0 4223.2 4884.5 1891.2 3640.0 289.53 1570.9 183.63 1710.4 27933 22684

23808

18008 17051

15927

100 394 3 4405.4 5100.9 1951.6 3770.0 273.14 1866.1 154.57 2010.9 28031 22766 23911

17357 17005 15848

100 394 6 4455.0 5151.2 1958.8 3817.0 218.79 2001.1 105.45 2145.6 28379 22793

23955

17494 16989

15836

Note: L2-i = Rear inner tire of the left side, L2-o = Rear outer tire of the left side, R2-i = Front inner tire of the right side, R2-o = Rear outer

tire of the right side

Table 2. Side friction factors of vehicles cornering on upgrades according to CarSim and TruckSim

Grade (%)

Radius (m) and Design speed (km/h)

R = 41 and V = 40

fy

R = 168 and V = 70

fy

R = 394 and V = 100

fy

R = 832 and V = 130

fy

E-Class Sedan

0

3

6

9

E-Class SUV

0

3

6

9

Conventional

Truck Loaded

0

3

6

0.22661

0.22652

0.22617

0.22548

0.22722

0.22709

0.22669

0.22602

0.24281

0.24409

0.25362

0.17383

0.16174

0.15059

0.14534

0.14644

0.14613

0.14569

0.14514

0.14876

0.15172

0.15232

0.15389

0.14150

0.12916

0.11762

0.11849

0.11823

0.11763

0.11731

0.11673

0.12190

0.12262

0.12922

0.11847

0.10741

0.09736

0.07997

0.07984

0.07956

0.07920

-

-

-

Note: fy= Side friction factor


Table 3. Forces acting on the truck during cornering on downgrades according to TruckSim

Design

speed

(km/h)

AASHTO

design

radius

(m)

Grade

(%)

Lateral Forces on Tires (N) Longitudinal Forces on Tires (N) Vertical Forces on Tires (N)

L1 L2 R1 R2 L1 L2 R1 R2 L1

L2-i

L2-o

R1

R2-i

R2-o

Conventional Truck Loaded

40 41 0 9298.4 11768 3603.0 6180.6 303.34 9030.0 98.921 1164.5 34221

26219 28672

12622

13933 11481

40 41 -3 10693 13397 3907.0 6714.5 344.84 389.10 110.62 222.26 36171 25733

28256

11968 13144

10642 40 41 -6 11521 13870 3793.6 6680.8 388.33 573.13 112.79 260.65 37384 26795

29527

10816 13212

10523

70 168 0 5463.2 6413.6 2361.8 4398.5 278.41 946.30 152.26 1107.5 29661 23332

24764

16700 16162

14730

70 168 -3 6034.6 7202.4 2550.0 4735.7 481.68 720.32 337.07 542.69 31078 2334724895

16784 15603 14059

70 168 -6 4237.4 4906.0 1863.0 3649.0 296.75 276.06 187.05 139.27 32164 22331

23729

17465 15350

13961

100 394 0 4223.2 4884.5 1891.2 3640.0 289.53 1570.9 183.63 1710.4 27933 22684

23808

18008 17051

15927 100 394 -3 6525.3 6808.2 2481.5 4657.0 2945.5 3076.3 2802.4 2922.2 28619 223782

3483

18381 16849

15746 100 394 -6 4516.6 5359.6 2030.7 3920.9 2112.4 2335.0 1989.2 2189.8 30255 21984

23145

18902 16174

15016

Note: L2-i = Rear inner tire of the left side, L2-o = Rear outer tire of the left side, R2-i = Front inner tire of the right side, R2-o = Rear outer

tire of the right side

Table 4. Side friction factors of vehicles cornering on downgrades according to CarSim and TruckSim

Grade (%)


R = 41 and V = 40

fy

R = 168 and V = 70

fy

R = 394 and V = 100

fy

R = 832 and V = 130

fy

E-Class Sedan

0

-3

-6

-9

E-Class SUV

0

-3

-6

-9

Conventional

Truck Loaded

0

-3

-6

0.22651

0.22748

0.23418

0.23572

0.22722

0.22714

0.22379

0.21568

0.24263

0.27568

0.27964

0.17383

0.18418

0.19078

0.19564

0.14644

0.14658

0.14720

0.14761

0.14868

0.16318

0.16378

0.15389

0.16461

0.17265

0.17824

0.11849

0.11862

0.11884

0.11911

0.11673

0.11682

0.12614

0.12922

0.13834

0.14717

0.15320

0.07997

0.08009

0.08017

0.08035

-

-

-



Table 5. Maximum side friction factors of vehicles cornering on downgrades during braking according to

CarSim and TruckSim

Grade (%)


R = 41 and V = 40

fy

R = 168 and V = 70

fy

R = 394 and V = 100

fy

R = 832 and V = 130

fy

E-Class Sedan

0

-3

-6

-9

E-Class SUV

0

-3

-6

-9

Conventional

Truck Loaded

0

-3

-6

0.27099

0.26831

0.26521

0.25795

0.22809

0.22779

0.22462

0.21928

0.26728

0.31569

0.39230

0.24033

0.23644

0.23281

0.22844

0.16497

0.16645

0.17032

0.17715

0.15224

0.16186

0.17840

0.23140

0.22795

0.22294

0.21887

0.15235

0.15159

0.14867

0.14678

0.11972

0.11975

0.13283

0.21603

0.21146

0.20579

0.20102

0.11337

0.11335

0.11306

0.10935

-

-

-


Table 6. Accelerations of vehicles cornering on upgrades according to CarSim and TruckSim

Grade (%) Radius (m) and Design speed (km/h)

R = 41 and V = 40 R = 168 and V = 70 R = 394 and V = 100 R = 832 and V = 130

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

E-Class

Sedan

0 -0.30576 0.30651 -0.22808 0.25383 -0.19935 0.23389 -0.16001 0.20922

3 -0.30549 0.30662 -0.22738 0.24174 -0.19912 0.22150 -0.15993 0.19847

6 -0.30477 0.30617 -0.22684 0.23059 -0.19852 0.20916 -0.15948 0.18741

9

-0.30348 0.30548 -0.22583 0.22534 -0.19761 0.19762 -0.15873 0.17736

E-Class SUV

0 -0.30517 0.30722 -0.22749 0.22644 -0.19938 0.19849 -0.16006 0.15997

3 -0.30496 0.30709 -0.22719 0.22613 -0.19913 0.19823 -0.16002 0.15984

6 -0.30420 0.30669 -0.22652 0.22569 -0.19852 0.19783 -0.15964 0.15956

9

Conventional

Truck

Loaded

0

3

6

-0.30298

-0.33396

-0.32787

-0.32497

0.30602

0.32263

0.32408

0.33802

-0.22547

-0.23238

-0.23216

-0.23171

0.22514

0.22868

0.23177

0.23237

-0.19764

-0.20071

-0.20020

-0.19941

0.19731

0.19673

0.20181

0.20272

-0.15895

-

-

-

0.15920

-

-

-


Table 7. Accelerations of vehicles cornering on downgrades according to CarSim and TruckSim



Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

E-Class Sedan

0 -0.30576 0.30651 -0.22808 0.25383 -0.19935 0.23389 -0.16001 0.20922

-3 -0.30549 0.30748 -0.22750 0.26418 -0.19922 0.24461 -0.15973 0.21834

-6 -0.30415 0.31418 -0.22527 0.27078 -0.19905 0.25265 -0.15945 0.22717

-9

-0.29533 0.31572 -0.22132 0.27564 -0.19876 0.25824 -0.15911 0.23320

E-Class SUV

0 -0.30517 0.30722 -0.22749 0.22644 -0.19938 0.19849 -0.16006 0.15997

-3 -0.30492 0.30714 -0.22748 0.22658 -0.19927 0.19862 -0.15986 0.16009

-6 -0.30239 0.30379 -0.22469 0.22720 -0.19903 0.19884 -0.15938 0.16017

-9

Conventional

Truck Loaded

0

-3

-6

-0.29112

-0.32974

-0.32709

-0.32564

0.29568

0.32262

0.35568

0.35964

-0.22027

-0.23238

-0.23252

-0.23266

0.22761

0.22868

0.24344

0.24378

-0.19859

-0.20071

-0.20074

-0.20082

0.19911

0.19673

0.19682

0.20614

-0.15901

-

-

-

0.16035

-

-

-

Table 8. Maximum accelerations of vehicles cornering on downgrades during braking according to CarSim and

TruckSim



Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

Lateral

acceleration

of CG

Centripetal

acceleration

(e+f)

E-Class Sedan

0 -0.26748 0.35099 -0.21012 0.32033 -0.18616 0.31140 -0.15008 0.29603

-3 -0.26327 0.34831 -0.20759 0.31644 -0.18465 0.30795 -0.14981 0.29146

-6 -0.25934 0.34521 -0.20633 0.31281 -0.18442 0.30294 -0.14917 0.28579

-9

-0.25036 0.33795 -0.20422 0.30844 -0.18549

0.29887 -0.14978 0.28102

E-Class SUV

0 -0.31230 0.30809 -0.23873 0.24497 -0.23098 0.23235 -0.19347 0.19337

-3 -0.31185 0.30779 -0.23955 0.24645 -0.22963 0.23159 -0.19273 0.19305

-6 -0.30796 0.30462 -0.24238 0.25032 -0.22418 0.22867 -0.19187 0.19336

-9

Conventional

Truck Loaded

0

-3

-6

-0.30137

-0.30040

-0.34889

-0.44861

0.29928

0.34728

0.39569

0.47230

-0.27365

-0.22560

-0.23372

-0.26085

0.28056

0.23224

0.24186

0.25840

-0.25285

-0.20696

-0.20767

-0.21959

0.25139

0.19972

0.19975

0.21283

-0.18701

-

-

-

0.18935

-

-

-


Fig. 1 Side friction factors of the sedan (V = 130 km/h) on a −9% grade during the braking

reaction time (0 = commencement of braking, 0.5 = release of brake)

Fig. 2 Lateral acceleration of the truck cornering at a speed of 100 km/h on a flat horizontal

curve with a radius of 168 m

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6S

ide

Fri

ctio

n F

acto

rTime (s)

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 10 20 30

Lat

eral

Acc

eler

atio

n

Time (s)


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