Car size and injury risk: a model for injury risk in frontal collisions

Accident Analysis and Prevention 34 (2002) 93 – 99 www.elsevier.com/locate/aap

Car size and injury risk: a model for injury risk in frontal collisions Denis P. Wood *, Ciaran K. Simms Denis Wood Associates, Isoldes Tower, 1 Essex Quay, Dublin 8, Ireland Received 13 November 2000; received in revised form 13 November 2000; accepted 29 November 2000

Abstract Empirical studies have established that when pairs of similar cars collide, the relative injury risk between pairs of different size is inversely related to their mass ratio. Further empirical studies have shown that in frontal collisions between dissimilar cars, relative injury risk is inversely proportional to mass ratio raised to the power of n. The value of the exponent n increases with impact speed, with n:1 at low speeds and n ranging from 2.81 to 3.74 for fatalities. In this paper a theory is derived which explains relative injury risk in terms of three parameters: length (or size) ratio, mass ratio and the ratio of collision energy absorption between the colliding vehicles. It is proposed that the ratio of collision energy absorption between colliding vehicles is a function of the structural collapse forces imposed at maximum dynamic crush. The theory shows that the fundamental factor in collisions between pairs of similar cars is size, i.e. length. For collisions between two dissimilar cars, Monte-Carlo simulations using generalised characterisations for the car population yield theoretical predictions that match empirical findings ranging from minor injuries (AIS1 +) to fatal (AIS6) injuries. © 2001 Elsevier Science Ltd. All rights reserved. Keywords: Injury risk; Size; Mass; Dissimilar cars; Energy absorption

1. Introduction Crashworthiness reflects the degree of protection provided by a vehicle to its occupants. It is well established that the occupants of larger and heavier cars are at lower risk than are the occupants of smaller and lighter cars (Evans and Frick, 1993; Evans, 1994; Joksch et al., 1998). This introduces a conflict between environmental and traffic management policies which both favour smaller, lighter cars and safety considerations which indicate that such changes could significantly increase the overall injury risk of the car population (Evans et al., 2000). Various researchers (Ernst et al., 1991; Evans, 1994; Evans et al., 2000) have demonstrated that in collisions between pairs of similar vehicles (vehicles of similar mass) the relative injury risk between pairs of vehicles of different sizes is inversely related to mass ratio (Mr ). Wood (1997) has shown that in collisions between pairs * Corresponding author. Tel.: + 353-1-6704566; fax: +353-16704560.

of similar cars the fundamental parameter for relative injury risk is car size, i.e. length, and the apparent empirical inverse mass ratio relation is due to the close correlation between car length and mass for the car population where, M(kg)=3.89L(m)2.48

(1)

and RIR1/2 =

M2 = Mr. M1

n

(2)

But fundamentally RIR1/2 =

L2 L1

2.5

,

(3)

where RIR1/2 is the relative injury risk in the pair of smaller vehicles compared to a pair of larger vehicles and L1, L2 are the lengths of the smaller and larger cars, respectively. Empirical studies (Evans and Frick, 1993; Joksch et al., 1998) have shown that relative fatality risk in collisions between dissimilar cars is a power function of mass ratio (Mr ) as follows,

0001-4575/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 1 - 4 5 7 5 ( 0 1 ) 0 0 0 0 3 - 3

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RIR 8 M nr .

(4)

In frontal collisions between cars the value of the exponent n for fatalities is in the range 2.8– 3.7, depending on the particular study. Others have shown that the magnitude of n varies with the level of injury risk considered, with n :1 for AIS1 + injury levels (Mizuno et al., 1997; Mizuno and Kajzer, 1999), and n ranging from 1.51 to 2.67 for AIS3+ injury levels (Mizuno et al., 1997; Wood, 1997; Mizuno and Kajzer, 1999). Wood (1997) addressed the theoretical considerations underlying relative injury risk in collisions between dissimilar cars but only provided a partial explanation for the empirical findings. This paper reformulates the basic theory of injury risk in terms of the parameter a¯L (the product of mean acceleration and vehicle length) and proposes a relation between the energy sharing characteristics of dissimilar vehicles and the interface forces at the instant of maximum dynamic crush. This theory is combined with the characteristics of the car population to determine the relative injury risk (RIR) characteristics of the car population in terms of the vehicle mass ratio (Mr ) and the DV of the smaller vehicle.

a¯ =

F , M

(7)

Eq. (7) can be rewritten in terms of basic dimensions by substituting for F and M using (8)

F=|A, and M= zAL,

(9)

where | is stress, z is density and A is the cross-sectional area of the vehicle structure. Combining Eqs. (7)–(9) yields, a¯L=



| . z

(10)

For a given car population, vehicles are of broadly similar design and are constructed from similar materials. Studies of the strength and dimensional characteristics of the car population indicate that the stress–density ratio (|/z) is uniform (independent of size) and is therefore a function of impact severity only (Hofferberth and Tommassoni, 1974; Schmidt, 1976). Rearranging Eq. (10) differently shows that average acceleration (a¯ ) for the car population during crushing is inversely proportional to overall length (L). Substitution into Eq. (5) shows,



 n 1 L

2.5

2. Theory

IR8 a¯ 2.5 8 f(I.S)

2.1. Injury risk

with f(I.S) indicating that the stress–density ratio (|/z) is a function of impact severity. The manner in which the stress–density ratio varies with impact severity can be investigated empirically using full width barrier test data. Wood (1992) has shown that the equivalent energy speed (EES) of the car population can be related to the normalised crush depth (Cd/L), where EES is defined as the square root of the kinetic energy of the vehicle divided by the empty (curb) mass. The energy equivalent speed is a surrogate for the specific energy (kinetic energy/mass) absorbed by the vehicle structure up to maximum dynamic crush. Analysis of the same data set used by Wood (1992), see Fig. 1a, shows that a¯L varies with EES in a linear manner (r 2 = 0.79). Kullgren (1996) and Kullgren et al. (1999) published mean acceleration (a¯ ) data in Swedish car-to-car real life frontal collisions derived from onboard crash pulse recorders. The mean mass of the cars examined by Kullgren was 1250 kg (Kullgren, 2000). Kullgren’s data was transformed to a¯L terms using Eq. (1), and is shown as a function of DV in Fig. 1b. Kullgren’s data set also shows a linear dependence of a¯L. A comparison of these two data sets shows that the magnitudes of a¯L are similar. EES and DV have different physical meanings. EES is calculated from the crush energy and vehicle mass, whereas DV is derived from momentum considerations.

The theoretical basis of this paper is the fundamental injury risk relation of Gadd (1966). Gadd found that injury risk (IR) is related to impact duration times mean body acceleration raised to the power of 2.5. It is proposed that for a given vehicle population which includes a mix of occupant protection level, seat belt usage, etc., the mean occupant acceleration in a collision is proportional to the mean acceleration imposed on the vehicle occupant compartment (a¯ ). Further, as the impact durations are identical when considering a colliding pair, then injury risk can be considered as being solely a function of mean acceleration of the occupant compartment, IR 8a¯ 2.5

(5)

and relative injury risk (RIR) is defined as RIR =

n a¯1 a¯2

2.5

(6)

2.2. Characteristics of car population Newtonian mechanics dictates that at a given impact severity, the mean acceleration (a¯ ) of a vehicle of mass M and crushing force F are related by

(11)

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The values of EES and DV are similar but not identical. The average acceleration imposed on the vehicles during crushing is the energy absorbed by the structure divided by the product of mass and the displacement of the centre of gravity, and EES is therefore the relevant parameter in this analysis. Linear regression of a¯ L with EES using the Kullgren et al. (1999) data set (as this is more recent) yields, a¯ L (m2/s2)= 68.99+32.12EES (m/s)

(12)

with N= 144 and r 2 =0.84.

2.3. Injury risk equations Substitution of Eq. (12) into Eq. (6) shows, RIR =



n n 2.5

68.99+ 32.12EES1 68.99+ 32.12EES2

·

L2 L1

2.5

(13)

and when substituting Eq. (1), RIR =



n

68.99+ 32.12EES1 68.99+ 32.12EES2

2.5

· [Mr ]

(14)

Despite the Mr term in Eq. (14), the fundamental parameter is the length ratio (L2/L1). In collisions between pairs of similar cars Eq. (13) reduces to Eq. (3). Eq. (13) shows that the relative injury risk between pairs of similar cars of different size is independent of impact severity level, as has been found empirically (Evans, 1994; Evans et al., 2000).

95

2.4. Car-to-car interaction In order to apply the relative injury risk equations to collisions between dissimilar cars it is necessary to determine the EES values and the manner in which the collision energy is shared between both cars. When cars of different size and mass collide, the extent of crushing of each car depends on the size, mass, geometric and structural characteristics of both vehicles. As cars collide, the forces at the interface between the two structures are equal and opposite. These interface forces have two elements: structural collapse forces and inertia forces due to the deceleration of the mass components of the collapsed portions of each vehicle. During crushing the inertia forces are indeterminate. However, at the instant of maximum mutual dynamic crush, there is no relative velocity at the crush interface and the interface force consists solely of structural collapse forces. Wood et al. (1997) and Park et al. (1999) have shown that the structural collapse force of vehicles increases progressively with increasing crush depth up to a peak force or plateau. At larger crush depths the structural collapse force decreases. This decrease is generally associated with crushing of the vehicle occupant compartment. Analysis of the full width barrier data from Wood (1992) shows that the car population has the following mean structural collapse force (SCF) characteristic, SCF =

  

 n

M C C 9184 d − 17187 d L L L

2

(15)

where M (kg) is the curb mass of the vehicle, L (m) the overall length and Cd (m) the dynamic crush. It is proposed that at maximum dynamic crush the structural collapse forces are equal and opposite. The forces at maximum dynamic crush can be matched using Eq. (15) up to a maximum dynamic crush ratio for the smaller car of (Cd/L)1 = 0.267 at which point the peak value of the SCF is reached. At higher values of (Cd/L)1, it is assumed that the ratio of the collision energy absorbed by the respective vehicles remains constant.

2.5. Summary

Fig. 1. Empirical relationship between a¯ L and EES/DV; (a) Wood (1992) and (b) Kullgren et al. (1999).

This model is based on a number of premises: the fundamental basis is that injury risk is related to the ratio of the car body acceleration raised to the power of 2.5. A further premise is that the normalised average acceleration during impact (a¯ L) is independent of car size and is a linear function of energy equivalent speed (EES). For collisions between dissimilar cars the structural collapse forces of the colliding cars at the instant of maximum dynamic crush are matched, and this is used to determine the distribution of collision energy between the cars. At high crush levels (above the peak

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structural collapse force for the smaller vehicle) the ratio of collision energy distribution is assumed to remain the same as at the peak structural collapse force point. Finally, it is assumed that the car population can be characterised in terms of a number of generalised relationships: between mass and length, energy equivalent speed (EES) and normalised crush (Cd/L), and structural collapse force (SCF) and normalised crush (Cd/L).

3. Results

3.1. Calculation procedure The equations characterising the car population are detailed in Appendix A. At each specified value of EES for the lighter car, the combination of Eq. (14), the characterisation of the car population (see Appendix A) and Eq. (15) were used to calculate the variation of relative injury risk (RIR) with mass ratio (Mr ). Monte Carlo simulation methods were used to analyse collisions between pairs of cars, allowing for the variability within the car population with 40 000 iterations (collisions) per EES value.

3.2. Car-to-car collisions The fundamental mass characteristic in vehicle collisions is the mass ratio. This can be estimated if a statistical distribution of car mass is known. Wood (1992) has estimated the mass distribution characteristics of the car population for sedan cars as having a mean mass of 1141.9 kg, a standard deviation (s) of 317.1 kg, and maximum and minimum values of 3335 and 505 kg, respectively. This distribution is used to characterise the representative mass ratio (Mr ) pattern in car-to-car collisions. Monte Carlo methods were used to randomly select each colliding pair, and Mr for each pair was defined as the mass of heavier car/mass of lighter car. Fig. 2a,b shows the resulting probability density and cumulative frequency distributions for Mr. The 50th percentile value of Mr is 1.31 while the 5th and 95th percentile values are 1.02 and 2.35, respectively.

Fig. 2. Probability density and cumulative frequency of Mr derived from Monte-Carlo simulations.

increases with DV, becoming asymptotic to n :3 at high DV values. Fig. 3b shows a strong correlation between ln(Mr ) and ln(RIR) above DV =5 m/s. However this correlation does not reflect the high degree of variability of RIR about the trend line. Fig. 4 shows a typical

3.3. Results Fig. 3a,b shows the variation in n (the exponent relating Mr to RIR, see Eq. (4)) and its coefficient of determination (r 2) with DV predicted by the theory. The values of n were obtained by linear regression of the natural logarithm (ln) of RIR against the natural logarithm of Mr data for each simulation. At low DV values, RIR is proportional to Mr, as is the case in collisions between pairs of similar cars. The exponent n

Fig. 3. Variation of the exponent n and the coefficient of determination of n with DV.

D.P. Wood, C.K. Simms / Accident Analysis and Pre6ention 34 (2002) 93–99

Fig. 4. Typical scatterplot from Monte-Carlo simulations (DV=9.5 m/s).

scatterplot, highlighting both the strong linear relation between ln(RIR) and ln(Mr ) and also the high variability about the trend line.

97

Fig. 6. Theoretical scatterplot predictions compared to Japanese empirical AIS3 + injury data from frontal (belted and unbelted) collisions (Mizuno et al., 1997).

4. Discussion

Mizuno et al. (1997) and Mizuno and Kajzer (1999) have published relative injury risk data for AIS3+ injury severity in frontal car-to-car collisions in Japan. Ernst et al. (1991) published similar information for AIS3 + injuries in both urban and rural frontal impact car-to-car collisions in Germany. The estimated DV values for 50% probability of AIS3+ injury are 17.3 and 18.1 m/s for the German and Japanese car populations respectively. Figs. 5 and 6 compare the theoretical scatterplots of ln(RIR) versus ln(Mr ) with the mean German data and Japanese data respectively. Evans and Frick (1993), Joksch et al. (1998) and Evans et al. (2000) have examined the variation in RIR in the US using the FARS data. Evans and Frick (1993) analysed all frontal collisions over the period 1975–1989. Joksch et al. (1998) examined all frontal collisions over the period 1991– 1994, while Evans et al. (2000) detailed the results for unbelted drivers over the period 1975–1998. Evans (1994) showed that the DV for a 50% likelihood of fatality in frontal car-to-car collisions is 25 m/s. Figs. 7– 9 show the theoretical scatterplots compared to these real life data sets.

There is good agreement between the real life data and the scatterplot predictions from the Monte Carlo simulations. All of the US fatality data and all of the Japanese AIS3+ injury data lie within the respective upper and lower 2.5% limits of the simulation data. The error band of the German AIS3+ data is also within the 2.5% confidence limits from the simulation. The simulations were carried out using the a¯ L versus EES regression derived from the data of Kullgren (1996) and Kullgren et al. (1999) of frontal collisions in Sweden. Further simulations using the a¯ L versus EES regression derived from full width barrier data (Wood, 1992, see Appendix A) gave similar results. The values of the exponent n (see Eq. (4)) computed from the Monte Carlo simulations differs from those obtained by Mizuno et al. (1997), Mizuno and Kajzer (1999), Evans and Frick (1993), Evans et al. (2000) and Joksch et al. (1998). Mizuno et al. (1997) obtained n= 2.67 for AIS3+ injuries. The corresponding value from the regression of the simulation data is 2.5. However, as shown in Fig. 5b, all of the real life data points lie within the 2.5% limits obtained from the simulation. Mizuno et al. (1997) and Mizuno and Kajzer (1999) also examined AIS1+ injuries and reported n values of 0.96 and 1.08 for unbelted and belted drivers respec-

Fig. 5. Theoretical scatterplot predictions compared to mean German empirical AIS3 + injury data from frontal collisions-urban, ; rural, (Ernst et al., 1991).

Fig. 7. Theoretical scatterplot predictions compared to empirical data (solid white squares): US Fatalities in (belted and unbelted) frontal collisions, based on FARS data, 1991 – 1994 (Joksch et al., 1998).

3.4. Comparison with real life data

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D.P. Wood, C.K. Simms / Accident Analysis and Pre6ention 34 (2002) 93–99

More importantly, for collisions between dissimilar cars the model predicts results for both serious injury and for fatalities similar to those found in real life. The relative injury risk relation (Eq. (13)) simplified by ignoring the constant terms can be restated in terms of the energy absorbed by the car structures, their masses and lengths, RIR = Fig. 8. Theoretical scatterplot predictions compared to empirical data (solid white squares): US Fatalities in (belted and unbelted) frontal collisions, based on FARS data 1975 –1989 (Evans and Frick, 1993).

tively. The detailed data was not reported. The DV for 50% likelihood of AIS1+ injuries in frontal collisions is low, typically in the region 3– 6 m/s (Hobbs and Mills, 1984). Fig. 3 shows that the theoretical values of n are close to 1.0 at low DV values. The model also shows that the correlation between mass ratio and relative injury risk is poor at low DV values. Evans and Frick (1993) reported n values of 2.81 for US fatalities in frontal collisions for car model years 1980+ , 3.74 for belted and unbelted fatalities over period 1975–1989. Evans et al. (2000) reported n =3.58 for fatalities of unbelted drivers in the period 1975– 1998, while Joksch et al. (1998) reported n = 3.2 for belted and unbelted drivers in frontal collisions over period 1991–1994. However, in the three reports where detailed data sets are available, these data lie within the 95th percentile range predicted by the Monte Carlo simulations. Similar to Wood (1997), this model also predicts that the relative injury risk between pairs of similar cars (i.e. a pair of smaller cars compared to a pair of larger cars) is inversely proportional to the ratio of car mass. The model also shows that this relationship is fundamentally due to the size of the cars.

Fig. 9. Theoretical scatterplot predictions compared to empirical data (solid white squares): US Fatalities in unbelted frontal collisions, based on FARS 1975 –1998 (Evans et al., 2000).

n  n n E1 E2

1.25

·

M2 M1

1.25

·

L2 L1

2.5

(16)

Eq. (16) shows that three vehicle factors determine relative injury risk: the collision energy absorption properties of the vehicle structures, the ratio of the vehicle masses and the ratio of the vehicle lengths or size. Comparison of Eq. (16) with the variation of the exponent n with DV obtained from the simulations for dissimilar cars indicates that at low speeds the collision energy absorbed by individual cars is proportional to their mass (n= 1.0). At high collision speeds, the collision energy absorbed approaches inverse proportionality with car mass (n= 3.5). This means, for example, in frontal collisions between cars of masses 750 and 1500 kg, the relative injury risk in the lighter car is 2.0 times that of the heavier car for low speed, minor collisions, but increases to 11.3 for high speed fatal crushes. Eq. (16) explains this increase in relative injury risk as being due to the manner in which the apportionment of collision energy between the colliding pair changes with impact speed. This apportionment is determined by the structural collapse force (SCF) characteristic of each car. Thus the theory indicates that the relative safety of small, lighter cars can be improved by the development of appropriate structural collapse force characteristics.

5. Conclusions A theory explaining relative injury risk in terms of length or size ratio, mass ratio and the ratio of collision energy absorption between colliding vehicles has been derived. The theory proposes that the ratio of collision energy absorption is related to the structural collapse forces imposed on the vehicles at maximum dynamic crush. The theory confirms that for collisions between pairs of similar cars the relative injury risk between pairs of different sizes is inversely related to the size of the cars (L2/L1)2.5. For collisions between dissimilar cars the Monte-Carlo simulations show a good comparison to empirical findings ranging from minor injuries (AIS2+) to fatal injuries (AIS6). This paper, therefore, explains the link between injury risk and relative car size and mass for impacts between dissimilar cars.

D.P. Wood, C.K. Simms / Accident Analysis and Pre6ention 34 (2002) 93–99

Appendix A Analysis of crash data from Wood (1992) yields the following regressions: EES = 0.8713+ 100.98

 

 

Cd Cd −75.6 L L

2

,

N= 224, r 2 =0.85

Constant (Cd/L) (Cd/L)2

Coefficient

S.E.

t-Statistic

0.8713 100.98 −75.63

0.63 7.36 21.15

1.38 13.72 −3.58

Also a¯ L (m2/s2)= 31.48+45.55EES (m/s) N= 224, r 2 =0.79.

Constant EES

Coefficient

S.E.

t-Statistic

31.48 45.5

24.07 1.59

1.31 28.58

Similar analysis of Kullgren et al. (1999) gives a¯ L (m2/s2)= 68.99+32.12EES (m/s) N= 144, r 2 =0.84.

Constant EES

Coefficient

S.E.

t-Statistic

68.99 32.12

8.20 1.16

8.41 27.76

The impact equations are CCS = and DV =





n

n

1+Mr · [EES 21 +MrEES 22] Mr

n

Mr · CCS, 1+ Mr

1/2

99

where CCS = collision closing speed; DV = velocity change of the smaller car; EES = energy equivalent speed; Mr = mass ratio (M2/M1), where M2 \ M1.

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