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Probabilistic assessment of wind related accidents of road vehicles: A reliability approach
Journal of Wind Engineering and Industrial Aerodynamics 74—76 (1998) 1079—1090
Probabilistic assessment of wind related accidents of road vehicles: A reliability approach Ragnar Sigbjo¨rnsson, Jo´nas Tho´r Snæbjo¨rnsson* Engineering Research Institute, University of Iceland, Hjardarhagi 2-6, IS-107 Reykjavik, Iceland
Abstract A general probabilistic model for assessment of road vehicle accidents in windy environments is presented. The limit states of safe performance are outlined and the accident point is defined in the space of basic variables. The probability of accident is evaluated using a so-called safety index approach. The theory outlined is used to analyse a particular bus accident by setting up scenarios based on available information. The methodology presented has several potential applications, such as in accident analysis, and in improving the design of roads and highways by pointing out potential accident spots as well as in devising preventive measures to improve traffic safety in windy environments. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Accident risk; Aerodynamics; Dynamics; Reliability; Road vehicles; Saftey index; Wind engineering
1. Introduction In recent years wind-induced accidents of road vehicles have been an increasing problem in Iceland. These accidents seem most common at exposed locations where topographical features of the landscape magnify the wind effects. Further, the most notable ones involve high-sided vehicles like buses and trucks with trailers. This paper deals with such an accident on October 1995 when a bus with 41 passengers went off the road in a high cross wind and rolled over. This resulted in two passengers being killed and leaving the others with minor to severe injuries. The objective of this paper is to analyse, in general probabilistic settings, this particular incident.
* Corresponding author. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 9 9 - 3
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2. Modelling accidents The modelling of accidents can be approached in different ways. Herein, a straightforward methodology based on probabilistic mechanics is suggested to model windrelated accidents of road vehicles. The methodology consists of the following main steps: (1) the basic variables are defined, a mechanical model of the vehicle and its environment is constructed and the equations of motion derived; (2) suitable stability criteria are selected and the limit states of safe performance are determined; and (3) the accident is quantified in terms of a so-called accident index and an accident point. 2.1. On the basic equations of motion The equations of motion of a road vehicle in windy environments have been put forward by Baker (see, for instance, Ref. [5]). The basic equations consider the mechanical and aerodynamic properties of the environment and the vehicle, but a model for the psycho-physiological characteristics of the driver can also be included. The equations are somewhat lengthy and are therefore omitted herein. The basic forces of mechanical origin accounted for are: f Inertia forces, due to changes in vehicle speed and direction. These include the so-called ‘centrifugal’ forces induced when driving in curves. f Gravity forces, which are important when defining the tire reaction forces. They also contribute directly to the stability of vehicles. f Frictional forces, acting between the tires and the road surface, provide the necessary stability for safe performance. f Aerodynamic forces, induced by the relative motion of the vehicle and the air. Referring to Fig. 1, these forces are obtained by the following expressions: F "1oC (u)A »2 and i 2 Fi i R
M "1oC (u)A d »2 , i 2 Mi i i R
(1a)
» "J(R#» cos(0))2 and u"tan~1(» sin(0)/(R#» cos(0))). (1b) R Here, » is the wind velocity; 0 is the angle of incidence of the wind measured relative to the direction of the vehicle; R is the speed of the vehicle; u is the relative angle of incidence of the air hitting the vehicle; F and M are the aerodynamic force i i and moment referred to axis i3Mx, t, zN; C and C are the corresponding dimenFi Mi sionless force and moment coefficient, A is the reference area and d is the i i corresponding moment arm. f Elastic and damping forces, related to the suspension system and the tires of the vehicle. The quantities governing the motion of the vehicle are to a large extent uncertain. Depending on the degree of uncertainty, these quantities are herein either modelled as stochastic variables or deterministic parameters. The stochastic variables constitute the basic variables: X"MX , X , X ,2,X N. 1 2 3 n
(2)
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Fig. 1. Definitions of (a) aerodynamic forces acting on a vehicle in motion and (b) velocity vectors (plane view).
These include, wind velocity and direction, vehicle speed and direction, mass of the vehicle (including driver, passengers, luggage and fuel) and location of centre of gravity, frictional coefficient, road camber and spatial curvature of the vehicle path. The geometrical quantities and material properties of the vehicle are treated as deterministic. 2.2. On the limit states of safe performance — stability criteria The limit states of safe performance can be defined in terms of loss of controllability and stability. Typically, loss of controllability will result in difficulties to follow a specific lane on the road, while loss of stability implies overturning or loss of road grip, resulting in side slip. The solution of the equations of motion along with the stability criteria defines the limit states of safe performance in terms of a hyper-surface in the space of basic variables. This can be written formally as f (X)"f (X , X , X ,2,X )"0. 1 2 3 n
(3)
In fact, this hyper-surface of safe performance divides the space of basic variables in a safe domain, D "Mx:f (x)'0N, and an unsafe or accident domain, 4 D "Mx:f (x)(0N. ! 2.3. Probabilistic definition of accidents The probability of an accident can be assessed as follows within the framework of the theory of reliability, adopting the geometrical safety index method [9,10]. The basic variables are transformed into normalised Gaussian space as follows assuming independent basic variables: u "(U~1(F (x )). i Xi i
(4)
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Here, U~1 denotes the inverse standardised Gaussian distribution function, F is the Xi cumulative distribution of the basic variable X and u is the state variable correspondi i ing to the stochastic basic variable no. i, that is º , that hence can be treated as i normally distributed with zero mean and unit standard deviation. The limit state hyper-surface can be expressed formally in the normalised space as g(U )"g(º ,º ,º ,2,º )"0. (5) 1 2 3 n The point on the limit state surface with the highest probability density is defined as the accident point, u , that is the ‘most probable’ point on the surface. In the ! normalised space this is the point on the limit state surface that is closest to the origin. The Euclidean norm of the basic variables at the accident point, computed in the normalised space, is a measure of the reliability or the odds of an accident. This leads to the definition of an accident index, b (equivalent to the so-called safety index, see ! for instance Ref. [9]), i.e. b "sign(n "u )Jua"ua for na"n(u )3Mn: n"!+g(u)N ! ! ! ! and u 3Mu: g(u)"0N, (6) ! where n denotes the normal to the limit state hyper-surface, " is the inner product operator and +g denotes the partial derivatives of g. It should be stressed that the accident index can be both positive and negative. Positive b -values correspond to ! 0)P (0.5, b "0 gives P "0.5 and negative b -values yield 0.5(P )1, where, ! ! ! ! ! P , is an approximation to the probability of accident, i.e. ! P +U(!b ), (7) ! ! where U denotes the standardised Gaussian probability density. In general, it is difficult to formulate the limit state of safe performance in terms of a single function. In fact, it is convenient to model the hyper-surface of safe performance as a set of functions where each function corresponds to a particular stability criterion. From a computational view this may lead to one accident index for each limit state function. The lowest of those accident indices corresponds to the point on the hyper-surface in the normalised space with the highest probability density. 3. The case The theory outlined has been applied to analyse an accident on October 1995, when a bus with 41 passengers went off the road and rolled over. The elastic and damping forces in the equations of motion are omitted and the bus is treated as a rigid body. Further, the response characteristics of the driver are also ignored. Hence, only the purely mechanical quantities of the bus and the environment are included. 3.1. The accident The accident occurred when the bus was driving towards north alongside a narrow fjord in the northwest part of Iceland, approaching a gentle curve towards right. In
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this area it is fair to assume that the wind direction was changing from a northerly to a northeasterly one, due to local topographical features of the surroundings. The consequence was increasing cross-wind hitting the bus from the right. Approaching the curve, it seems that the bus was heading to much to the right, possibly due to disturbances induced by the growing cross-wind. The driver corrected the course by turning to the left away from the right edge of the road. After doing so he seemed to lose control of the vehicle, which continued towards the left edge and went off the road on the left side, diverging roughly 1° from the right course. 3.2. The road near the site of the accident The road has two lanes carrying traffic in opposite directions. The total width of the road is about 7 m. The radius of curvature of the road at the site of the accident is approximately 4 km. The camber of the road is in the range of 3—4% towards both sides, also in the curve. At the location of the accident there are about 3 m high embankments on both sides of the road, as the road crosses a small gulch. 3.3. Weather conditions The weather conditions at the time of the accident can be described as very windy, with sleet, visibility about 2 km and temperature between 1°C and 2°C, i.e. slightly over the freezing point. The surface of the road was wet and alongside it was a thin layer of wet snow. Based on temperature recordings as well as eyewitnesses it can be assumed that the road surface was slippery, but without ice. The result is a substantial reduction of available frictional forces necessary to ensure safe performance. According to information from the Icelandic Meteorological Office, the mean wind velocity was probably in the range 15—25 m/s. Based on turbulence measurements carried out by the Engineering Research Institute in this area the turbulence intensity was judged to have been near 15%. The measurements were carried out under near-neutral conditions using an ultrasonic anemometer located on the roof top of a minibus. The height of the sensor was about twice the height of the car. Fig. 2 shows an example of the recorded relative horizontal air velocity and angle of incidence for a stationary car (R"0) and a moving car (R"7.5 m/s). The mean values of the measurements verify the relationship given in Eq. (1b), to within 5% for the wind velocity and 1° for the wind direction (see also Ref. [11]). Analysis of the available weather data as well as topographical data suggests that the mean wind direction was changing along the fjord from north towards northeast near the accident location. This leads to a mean angle of incidence, relative to the direction of the bus, around 45°. 3.4. The bus The basic data of the bus is given in Table 1. No malfunction of the bus is reported and its operational conditions are judged to be in accordance with specifications. The
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Fig. 2. Wind velocity and wind direction as recorded about one car height above the roof of a stationary car at point A, a car moving against the wind from point A to B and a stationary car at point B.
Table 1 Characteristic quantities of the vehicle. Spatial dimensions are based on measurements while the mass is in accordance with official registration certificate Quantities
Size
Overall width 2.50 Overall length 11.50 Length at the top 10.90 Slope of the front window 8 Overall height measured from road level 3.25 Distance between the road and the sides of the bus 0.50 Distance to front axle measured from front 1.50 Distance to rear axle measured from aft 3.65 Distance between front and rear axles 6.30 Width between front tires (centre to centre) 2.05 Width between rear tires (double set of tires) 1.84 Mass of the unloaded vehicle 10 500
Unit m m m ° m m m m m m m kg
speed of the bus just before the accident was about 85 km/h, according to an automatic recorder in the bus. The aerodynamic properties of the bus selected for this analysis are based on wind tunnel studies of vehicles of similar size, approximating the pressure distribution and the derived force and moment coefficients as deterministic [1—8].
5000 5000 5000
50 70 90
0.1 0.1 0.1
MV (kg)
MV (km/h) CV
0.05 0.05 0.05
CV
Weight on front axle
Vehicle speed
8100 8100 8100
MV (kg) 0.05 0.05 0.05
CV
Weight on rear axle
0.2 0.3 0.4
MV 0.1 0.1 0.1
CV
Coeff. of friction
Camber of road
0 0 0
0.5 0.5 0.5
0 0 0
1.5 1.5 1.5
14 18 22
0.15 0.15 0.15
CV
Wind velocity
MV (1/km) SD (1/km) MV (%) SD (%) MV (m/s)
Turning curvature
Table 2 Distribution parameters of basic variables. MV stands for mean value, CV for coefficient of variation, and SD for standard deviation
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4. Computational assessment of accident indices Considering the case outlined, the limit state functions of safe performance have been constructed using side slip of the front wheels or the rear wheels as stability criteria. The following seven quantities are treated as independent stochastic basic variables: Wind speed, vehicle speed, weight on the front and rear axle of the vehicle, turning curvature, camber of the road and frictional coefficient. All these variables are assumed to be Gaussian except the wind velocity which is treated as Rayleighian. Other parameters involved are approximated as deterministic, including the mean wind direction relative to the direction of the bus which is taken as 45°, in accordance with the discussion above. The parameters quantifying the basic variables are given in Table 2. It should be noted that three different mean values are given for the wind velocity, frictional coefficient and vehicle speed. The purpose of this, is to assess the sensitivity of the accident index to these parameters. The parameters in the table are assumed to represent scenarios where the vehicle is driving on a straight road, that is before entering the curve. The theory outlined has been applied to compute the conditional accident indices and the corresponding probabilities of accident for these scenarios. The results of the computations are summarised in Table 3. It is seen that the probability of an accident is rather high in most of the cases considered. In fact, the results indicate that an accident is highly likely to occur under the given conditions. Fig. 3 indicates how the different basic variables contribute relatively to the accident index, for four cases in Table 3. Fig. 3a shows the relative contribution of the basic variables for a chosen reference set of parameters. In Fig. 3b the mean value of Table 3 Conditional accident indices and corresponding probability of side slip, for a given mean surface friction, mean wind velocity and mean vehicle speed. The mean wind direction is 45° and intensity of turbulence 15%. See Table 2 for other parameters Mean surface friction coefficient
Mean vechicle speed (km/h)
Accident index and probability of side slip at mean wind velocity 14 m/s
0.2
0.3
0.4
50 70 90 50 70 90 50 70 90
18 m/s
22 m/s
b !
U (!b ) ! (%)
b !
U (!b ) ! (%)
b !
U (!b ) ! (%)
3.43 2.21 1.11 5.18 3.85 2.47 6.36 5.05 3.54
0.03 1.35 13.45 0.00 0.01 0.68 &0! &0 0.02
1.69 0.70 !0.15 3.33 2.23 1.12 4.50 3.35 2.12
4.59 24.27 56.10 0.04 1.28 13.07 &0 0.04 1.71
0.08 !0.79 !1.45 1.64 0.69 !0.23 2.72 1.73 0.70
46.88 78.49 92.65 5.01 24.45 59.20 0.33 4.16 24.14
!Probabilities less than 1/1000 are indicated with &0.
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Fig. 3. The relative contribution of basic variables to the accident index for different sets of parameters given on the histograms and in Table 2. Variable 1 is the front axle weight, 2 is the aft axle weight, 3 is the coefficient of friction, 4 is the vehicle speed, 5 is the turning curvature, 6 is the road camber and 7 is the wind velocity.
surface friction is increased, in Fig. 3c the mean vehicle speed is increased and in Fig. 3d the mean wind velocity is increased. As can be seen in Table 3, b , and thereby safety, increases with increased surface ! friction, whereas b decreases with increased vehicle speed and wind velocity. This is ! also demonstrated in Fig. 3, where the relative contribution of the variables is seen to increase somewhat with an increase in parameter value. The wind velocity is seen to
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play an important role. On the other hand, the direct contribution of vehicle speed seems to be relatively small, and the effect of increasing vehicle speed is primarily observed in an increased contribution from turning curvature. It should be mentioned that almost without exception the instability is due to side slip of the front wheels, which is quite serious as loss in controllability is inevitable under such conditions. This is partly seen in Fig. 3, where the contribution of the weight of the front axle is seen to be considerably greater than the contribution from the rear axle. Fig. 4, exemplifies four different two-dimensional views of the limit state surface in the neighbourhood of the accident point for the chosen reference set of parameters shown in Fig. 3a. Each view indicates the relation between two of the seven basic variables. It is seen that the surface is fairly linear in the area around the accident point, at least for the variables shown. This implies that the probabilities derived from
Fig. 4. Four different two-dimensional views of the limit state surface in the neighbourhood of the accident point (?), for the chosen reference set of parameters shown in Fig. 3a. Each view indicates the relation between two of the seven basic variables.
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the geometrical accident index are reasonable approximations to the exact computational probabilities. It is also worth mentioning that for high probability values the choice of distribution for the different variables was found to have small effect on the results.
5. Concluding remarks A general probabilistic model using reliability approach for assessment of road vehicle accidents in windy environments has been presented and applied in analysing a particular incident. Probabilistic analysis of reconstructed scenarios representing the environmental and driving condition at the time of the accident indicate that there was a very high probability of accident. The analysis also demonstrate that this particular accident cannot be assigned to a single variable or parameter, but is the consequence of a combination of basic variables as represented by the accident point. Other parameters, such as the aerodynamic properties of the bus, which are not optimal, coupled with short-term changes in wind direction are also of great importance. The presented methodology should therefore be extended to include these effects more accurately. The study suggests that available methods of probabilistic mechanics and theory of reliability can be of value for analysis of wind-related traffic accidents. Using such methods to set up computer simulations to analyse scenarios can be helpful in post-evaluation of accidents and to improve the design of roads and highways. Furthermore, they can be of use in devising and evaluating preventive measures to improve traffic safety.
Acknowledgements The authors would like to acknowledge the financial support provided by the University of Iceland Research Fund. The encouragement and advice of Professor C.J. Baker at Nottingham University are also greatly appreciated.
References [1] C.J. Baker, A simple analysis of various types of wind-induced road vehicle accidents, J. Wind Eng. Aerodyn. 22 (1) (1986). [2] C.J. Baker, Measures to control vehicles movements at exposed sites during windy periods, J. Wind Eng. Ind. Aerodyn. 25 (2) (1987). [3] C.J. Baker, High sided articulated vehicles in strong cross winds, J. Wind. Eng. Aerodyn. 31 (1) (1988). [4] C.J. Baker, Ground vehicles in high cross winds — Part I: Steady aerodynamic forces — Part II: Unsteady aerodynamic forces — Part III: The interaction of aerodynamic forces and the vehicles system. J. Fluid Struct. 5 (1991). [5] C.J. Baker, The quantification of accident risk for road vehicles in cross wind, J. Wind Eng. Ind. Aerodyn. 52 (1994) 93—107. [6] C.J. Baker, Private communication, 1996.
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[7] S.A. Coleman, C.J. Baker, High sided road vehicles in cross wind, J. Wind Eng. Ind. Aerodyn. 36 Part 2 (1990) 1383—1392. [8] S.A. Coleman, C.J. Baker, An experimental study of the aerodynamic behaviour of high sided lorries in cross winds, J. Wind Eng. Ind. Aerodyn. 53 (3) (1994) 401—429. [9] O. Ditlevsen, Uncertainty Modelling with Application to Multidimensional Civil Engineering Systems, McGraw-Hill, New York, 1981. [10] O. Ditlevsen, H.O. Madsen, Structural Reliability Methods, Wiley, New York, 1996. [11] S. Watkins, J.W. Saunders, P.H. Hoffmann, Turbulence experienced by moving vehicles — Part I : Introduction and turbulence intensity, J. Wind Eng. Ind. Aerodyn. 57 (1) (1995) 1—17.
Probabilistic assessment of wind related accidents of road vehicles: A reliability approach Ragnar Sigbjo¨rnsson, Jo´nas Tho´r Snæbjo¨rnsson* Engineering Research Institute, University of Iceland, Hjardarhagi 2-6, IS-107 Reykjavik, Iceland
Abstract A general probabilistic model for assessment of road vehicle accidents in windy environments is presented. The limit states of safe performance are outlined and the accident point is defined in the space of basic variables. The probability of accident is evaluated using a so-called safety index approach. The theory outlined is used to analyse a particular bus accident by setting up scenarios based on available information. The methodology presented has several potential applications, such as in accident analysis, and in improving the design of roads and highways by pointing out potential accident spots as well as in devising preventive measures to improve traffic safety in windy environments. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Accident risk; Aerodynamics; Dynamics; Reliability; Road vehicles; Saftey index; Wind engineering
1. Introduction In recent years wind-induced accidents of road vehicles have been an increasing problem in Iceland. These accidents seem most common at exposed locations where topographical features of the landscape magnify the wind effects. Further, the most notable ones involve high-sided vehicles like buses and trucks with trailers. This paper deals with such an accident on October 1995 when a bus with 41 passengers went off the road in a high cross wind and rolled over. This resulted in two passengers being killed and leaving the others with minor to severe injuries. The objective of this paper is to analyse, in general probabilistic settings, this particular incident.
* Corresponding author. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 9 9 - 3
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R. Sigbjo~ rnsson, J.T. Snæbjo~ rnsson/J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 1079–1090
2. Modelling accidents The modelling of accidents can be approached in different ways. Herein, a straightforward methodology based on probabilistic mechanics is suggested to model windrelated accidents of road vehicles. The methodology consists of the following main steps: (1) the basic variables are defined, a mechanical model of the vehicle and its environment is constructed and the equations of motion derived; (2) suitable stability criteria are selected and the limit states of safe performance are determined; and (3) the accident is quantified in terms of a so-called accident index and an accident point. 2.1. On the basic equations of motion The equations of motion of a road vehicle in windy environments have been put forward by Baker (see, for instance, Ref. [5]). The basic equations consider the mechanical and aerodynamic properties of the environment and the vehicle, but a model for the psycho-physiological characteristics of the driver can also be included. The equations are somewhat lengthy and are therefore omitted herein. The basic forces of mechanical origin accounted for are: f Inertia forces, due to changes in vehicle speed and direction. These include the so-called ‘centrifugal’ forces induced when driving in curves. f Gravity forces, which are important when defining the tire reaction forces. They also contribute directly to the stability of vehicles. f Frictional forces, acting between the tires and the road surface, provide the necessary stability for safe performance. f Aerodynamic forces, induced by the relative motion of the vehicle and the air. Referring to Fig. 1, these forces are obtained by the following expressions: F "1oC (u)A »2 and i 2 Fi i R
M "1oC (u)A d »2 , i 2 Mi i i R
(1a)
» "J(R#» cos(0))2 and u"tan~1(» sin(0)/(R#» cos(0))). (1b) R Here, » is the wind velocity; 0 is the angle of incidence of the wind measured relative to the direction of the vehicle; R is the speed of the vehicle; u is the relative angle of incidence of the air hitting the vehicle; F and M are the aerodynamic force i i and moment referred to axis i3Mx, t, zN; C and C are the corresponding dimenFi Mi sionless force and moment coefficient, A is the reference area and d is the i i corresponding moment arm. f Elastic and damping forces, related to the suspension system and the tires of the vehicle. The quantities governing the motion of the vehicle are to a large extent uncertain. Depending on the degree of uncertainty, these quantities are herein either modelled as stochastic variables or deterministic parameters. The stochastic variables constitute the basic variables: X"MX , X , X ,2,X N. 1 2 3 n
(2)
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Fig. 1. Definitions of (a) aerodynamic forces acting on a vehicle in motion and (b) velocity vectors (plane view).
These include, wind velocity and direction, vehicle speed and direction, mass of the vehicle (including driver, passengers, luggage and fuel) and location of centre of gravity, frictional coefficient, road camber and spatial curvature of the vehicle path. The geometrical quantities and material properties of the vehicle are treated as deterministic. 2.2. On the limit states of safe performance — stability criteria The limit states of safe performance can be defined in terms of loss of controllability and stability. Typically, loss of controllability will result in difficulties to follow a specific lane on the road, while loss of stability implies overturning or loss of road grip, resulting in side slip. The solution of the equations of motion along with the stability criteria defines the limit states of safe performance in terms of a hyper-surface in the space of basic variables. This can be written formally as f (X)"f (X , X , X ,2,X )"0. 1 2 3 n
(3)
In fact, this hyper-surface of safe performance divides the space of basic variables in a safe domain, D "Mx:f (x)'0N, and an unsafe or accident domain, 4 D "Mx:f (x)(0N. ! 2.3. Probabilistic definition of accidents The probability of an accident can be assessed as follows within the framework of the theory of reliability, adopting the geometrical safety index method [9,10]. The basic variables are transformed into normalised Gaussian space as follows assuming independent basic variables: u "(U~1(F (x )). i Xi i
(4)
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Here, U~1 denotes the inverse standardised Gaussian distribution function, F is the Xi cumulative distribution of the basic variable X and u is the state variable correspondi i ing to the stochastic basic variable no. i, that is º , that hence can be treated as i normally distributed with zero mean and unit standard deviation. The limit state hyper-surface can be expressed formally in the normalised space as g(U )"g(º ,º ,º ,2,º )"0. (5) 1 2 3 n The point on the limit state surface with the highest probability density is defined as the accident point, u , that is the ‘most probable’ point on the surface. In the ! normalised space this is the point on the limit state surface that is closest to the origin. The Euclidean norm of the basic variables at the accident point, computed in the normalised space, is a measure of the reliability or the odds of an accident. This leads to the definition of an accident index, b (equivalent to the so-called safety index, see ! for instance Ref. [9]), i.e. b "sign(n "u )Jua"ua for na"n(u )3Mn: n"!+g(u)N ! ! ! ! and u 3Mu: g(u)"0N, (6) ! where n denotes the normal to the limit state hyper-surface, " is the inner product operator and +g denotes the partial derivatives of g. It should be stressed that the accident index can be both positive and negative. Positive b -values correspond to ! 0)P (0.5, b "0 gives P "0.5 and negative b -values yield 0.5(P )1, where, ! ! ! ! ! P , is an approximation to the probability of accident, i.e. ! P +U(!b ), (7) ! ! where U denotes the standardised Gaussian probability density. In general, it is difficult to formulate the limit state of safe performance in terms of a single function. In fact, it is convenient to model the hyper-surface of safe performance as a set of functions where each function corresponds to a particular stability criterion. From a computational view this may lead to one accident index for each limit state function. The lowest of those accident indices corresponds to the point on the hyper-surface in the normalised space with the highest probability density. 3. The case The theory outlined has been applied to analyse an accident on October 1995, when a bus with 41 passengers went off the road and rolled over. The elastic and damping forces in the equations of motion are omitted and the bus is treated as a rigid body. Further, the response characteristics of the driver are also ignored. Hence, only the purely mechanical quantities of the bus and the environment are included. 3.1. The accident The accident occurred when the bus was driving towards north alongside a narrow fjord in the northwest part of Iceland, approaching a gentle curve towards right. In
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this area it is fair to assume that the wind direction was changing from a northerly to a northeasterly one, due to local topographical features of the surroundings. The consequence was increasing cross-wind hitting the bus from the right. Approaching the curve, it seems that the bus was heading to much to the right, possibly due to disturbances induced by the growing cross-wind. The driver corrected the course by turning to the left away from the right edge of the road. After doing so he seemed to lose control of the vehicle, which continued towards the left edge and went off the road on the left side, diverging roughly 1° from the right course. 3.2. The road near the site of the accident The road has two lanes carrying traffic in opposite directions. The total width of the road is about 7 m. The radius of curvature of the road at the site of the accident is approximately 4 km. The camber of the road is in the range of 3—4% towards both sides, also in the curve. At the location of the accident there are about 3 m high embankments on both sides of the road, as the road crosses a small gulch. 3.3. Weather conditions The weather conditions at the time of the accident can be described as very windy, with sleet, visibility about 2 km and temperature between 1°C and 2°C, i.e. slightly over the freezing point. The surface of the road was wet and alongside it was a thin layer of wet snow. Based on temperature recordings as well as eyewitnesses it can be assumed that the road surface was slippery, but without ice. The result is a substantial reduction of available frictional forces necessary to ensure safe performance. According to information from the Icelandic Meteorological Office, the mean wind velocity was probably in the range 15—25 m/s. Based on turbulence measurements carried out by the Engineering Research Institute in this area the turbulence intensity was judged to have been near 15%. The measurements were carried out under near-neutral conditions using an ultrasonic anemometer located on the roof top of a minibus. The height of the sensor was about twice the height of the car. Fig. 2 shows an example of the recorded relative horizontal air velocity and angle of incidence for a stationary car (R"0) and a moving car (R"7.5 m/s). The mean values of the measurements verify the relationship given in Eq. (1b), to within 5% for the wind velocity and 1° for the wind direction (see also Ref. [11]). Analysis of the available weather data as well as topographical data suggests that the mean wind direction was changing along the fjord from north towards northeast near the accident location. This leads to a mean angle of incidence, relative to the direction of the bus, around 45°. 3.4. The bus The basic data of the bus is given in Table 1. No malfunction of the bus is reported and its operational conditions are judged to be in accordance with specifications. The
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Fig. 2. Wind velocity and wind direction as recorded about one car height above the roof of a stationary car at point A, a car moving against the wind from point A to B and a stationary car at point B.
Table 1 Characteristic quantities of the vehicle. Spatial dimensions are based on measurements while the mass is in accordance with official registration certificate Quantities
Size
Overall width 2.50 Overall length 11.50 Length at the top 10.90 Slope of the front window 8 Overall height measured from road level 3.25 Distance between the road and the sides of the bus 0.50 Distance to front axle measured from front 1.50 Distance to rear axle measured from aft 3.65 Distance between front and rear axles 6.30 Width between front tires (centre to centre) 2.05 Width between rear tires (double set of tires) 1.84 Mass of the unloaded vehicle 10 500
Unit m m m ° m m m m m m m kg
speed of the bus just before the accident was about 85 km/h, according to an automatic recorder in the bus. The aerodynamic properties of the bus selected for this analysis are based on wind tunnel studies of vehicles of similar size, approximating the pressure distribution and the derived force and moment coefficients as deterministic [1—8].
5000 5000 5000
50 70 90
0.1 0.1 0.1
MV (kg)
MV (km/h) CV
0.05 0.05 0.05
CV
Weight on front axle
Vehicle speed
8100 8100 8100
MV (kg) 0.05 0.05 0.05
CV
Weight on rear axle
0.2 0.3 0.4
MV 0.1 0.1 0.1
CV
Coeff. of friction
Camber of road
0 0 0
0.5 0.5 0.5
0 0 0
1.5 1.5 1.5
14 18 22
0.15 0.15 0.15
CV
Wind velocity
MV (1/km) SD (1/km) MV (%) SD (%) MV (m/s)
Turning curvature
Table 2 Distribution parameters of basic variables. MV stands for mean value, CV for coefficient of variation, and SD for standard deviation
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4. Computational assessment of accident indices Considering the case outlined, the limit state functions of safe performance have been constructed using side slip of the front wheels or the rear wheels as stability criteria. The following seven quantities are treated as independent stochastic basic variables: Wind speed, vehicle speed, weight on the front and rear axle of the vehicle, turning curvature, camber of the road and frictional coefficient. All these variables are assumed to be Gaussian except the wind velocity which is treated as Rayleighian. Other parameters involved are approximated as deterministic, including the mean wind direction relative to the direction of the bus which is taken as 45°, in accordance with the discussion above. The parameters quantifying the basic variables are given in Table 2. It should be noted that three different mean values are given for the wind velocity, frictional coefficient and vehicle speed. The purpose of this, is to assess the sensitivity of the accident index to these parameters. The parameters in the table are assumed to represent scenarios where the vehicle is driving on a straight road, that is before entering the curve. The theory outlined has been applied to compute the conditional accident indices and the corresponding probabilities of accident for these scenarios. The results of the computations are summarised in Table 3. It is seen that the probability of an accident is rather high in most of the cases considered. In fact, the results indicate that an accident is highly likely to occur under the given conditions. Fig. 3 indicates how the different basic variables contribute relatively to the accident index, for four cases in Table 3. Fig. 3a shows the relative contribution of the basic variables for a chosen reference set of parameters. In Fig. 3b the mean value of Table 3 Conditional accident indices and corresponding probability of side slip, for a given mean surface friction, mean wind velocity and mean vehicle speed. The mean wind direction is 45° and intensity of turbulence 15%. See Table 2 for other parameters Mean surface friction coefficient
Mean vechicle speed (km/h)
Accident index and probability of side slip at mean wind velocity 14 m/s
0.2
0.3
0.4
50 70 90 50 70 90 50 70 90
18 m/s
22 m/s
b !
U (!b ) ! (%)
b !
U (!b ) ! (%)
b !
U (!b ) ! (%)
3.43 2.21 1.11 5.18 3.85 2.47 6.36 5.05 3.54
0.03 1.35 13.45 0.00 0.01 0.68 &0! &0 0.02
1.69 0.70 !0.15 3.33 2.23 1.12 4.50 3.35 2.12
4.59 24.27 56.10 0.04 1.28 13.07 &0 0.04 1.71
0.08 !0.79 !1.45 1.64 0.69 !0.23 2.72 1.73 0.70
46.88 78.49 92.65 5.01 24.45 59.20 0.33 4.16 24.14
!Probabilities less than 1/1000 are indicated with &0.
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Fig. 3. The relative contribution of basic variables to the accident index for different sets of parameters given on the histograms and in Table 2. Variable 1 is the front axle weight, 2 is the aft axle weight, 3 is the coefficient of friction, 4 is the vehicle speed, 5 is the turning curvature, 6 is the road camber and 7 is the wind velocity.
surface friction is increased, in Fig. 3c the mean vehicle speed is increased and in Fig. 3d the mean wind velocity is increased. As can be seen in Table 3, b , and thereby safety, increases with increased surface ! friction, whereas b decreases with increased vehicle speed and wind velocity. This is ! also demonstrated in Fig. 3, where the relative contribution of the variables is seen to increase somewhat with an increase in parameter value. The wind velocity is seen to
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play an important role. On the other hand, the direct contribution of vehicle speed seems to be relatively small, and the effect of increasing vehicle speed is primarily observed in an increased contribution from turning curvature. It should be mentioned that almost without exception the instability is due to side slip of the front wheels, which is quite serious as loss in controllability is inevitable under such conditions. This is partly seen in Fig. 3, where the contribution of the weight of the front axle is seen to be considerably greater than the contribution from the rear axle. Fig. 4, exemplifies four different two-dimensional views of the limit state surface in the neighbourhood of the accident point for the chosen reference set of parameters shown in Fig. 3a. Each view indicates the relation between two of the seven basic variables. It is seen that the surface is fairly linear in the area around the accident point, at least for the variables shown. This implies that the probabilities derived from
Fig. 4. Four different two-dimensional views of the limit state surface in the neighbourhood of the accident point (?), for the chosen reference set of parameters shown in Fig. 3a. Each view indicates the relation between two of the seven basic variables.
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the geometrical accident index are reasonable approximations to the exact computational probabilities. It is also worth mentioning that for high probability values the choice of distribution for the different variables was found to have small effect on the results.
5. Concluding remarks A general probabilistic model using reliability approach for assessment of road vehicle accidents in windy environments has been presented and applied in analysing a particular incident. Probabilistic analysis of reconstructed scenarios representing the environmental and driving condition at the time of the accident indicate that there was a very high probability of accident. The analysis also demonstrate that this particular accident cannot be assigned to a single variable or parameter, but is the consequence of a combination of basic variables as represented by the accident point. Other parameters, such as the aerodynamic properties of the bus, which are not optimal, coupled with short-term changes in wind direction are also of great importance. The presented methodology should therefore be extended to include these effects more accurately. The study suggests that available methods of probabilistic mechanics and theory of reliability can be of value for analysis of wind-related traffic accidents. Using such methods to set up computer simulations to analyse scenarios can be helpful in post-evaluation of accidents and to improve the design of roads and highways. Furthermore, they can be of use in devising and evaluating preventive measures to improve traffic safety.
Acknowledgements The authors would like to acknowledge the financial support provided by the University of Iceland Research Fund. The encouragement and advice of Professor C.J. Baker at Nottingham University are also greatly appreciated.
References [1] C.J. Baker, A simple analysis of various types of wind-induced road vehicle accidents, J. Wind Eng. Aerodyn. 22 (1) (1986). [2] C.J. Baker, Measures to control vehicles movements at exposed sites during windy periods, J. Wind Eng. Ind. Aerodyn. 25 (2) (1987). [3] C.J. Baker, High sided articulated vehicles in strong cross winds, J. Wind. Eng. Aerodyn. 31 (1) (1988). [4] C.J. Baker, Ground vehicles in high cross winds — Part I: Steady aerodynamic forces — Part II: Unsteady aerodynamic forces — Part III: The interaction of aerodynamic forces and the vehicles system. J. Fluid Struct. 5 (1991). [5] C.J. Baker, The quantification of accident risk for road vehicles in cross wind, J. Wind Eng. Ind. Aerodyn. 52 (1994) 93—107. [6] C.J. Baker, Private communication, 1996.
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[7] S.A. Coleman, C.J. Baker, High sided road vehicles in cross wind, J. Wind Eng. Ind. Aerodyn. 36 Part 2 (1990) 1383—1392. [8] S.A. Coleman, C.J. Baker, An experimental study of the aerodynamic behaviour of high sided lorries in cross winds, J. Wind Eng. Ind. Aerodyn. 53 (3) (1994) 401—429. [9] O. Ditlevsen, Uncertainty Modelling with Application to Multidimensional Civil Engineering Systems, McGraw-Hill, New York, 1981. [10] O. Ditlevsen, H.O. Madsen, Structural Reliability Methods, Wiley, New York, 1996. [11] S. Watkins, J.W. Saunders, P.H. Hoffmann, Turbulence experienced by moving vehicles — Part I : Introduction and turbulence intensity, J. Wind Eng. Ind. Aerodyn. 57 (1) (1995) 1—17.