# Crush Energy and Stiffness in Side Impacts.

INTRODUCTIONWhy the Study was Needed

A widely accepted technique for reconstructing crashes is based on the determination of the energy absorbed by both crash partners, as reflected in the damage done to both. Beyond the depth and extent of damage, the knowledge of the stiffness characteristics of the crushed structures is also required. This research focuses on vehicle side structure stiffness.

It is well known that stiffness characteristics vary widely among vehicles, even when they are lumped together in the same category (however that category is defined). It follows that the application of generalized class category stiffness parameters introduces variability into the calculation of crush energy for a specific vehicle in a specific crash.

The determination of vehicle-specific side structure stiffness parameters is complicated by the facts that both crash partners absorb energy, and both crash partners dissipate additional kinetic energy through post-impact motion. Perhaps for these reasons, the techniques used to derive published side structure stiffness parameters have either never been adequately described in the open literature, or have exhibited flaws in their methodology [1]. The derivation of commercially available side structure stiffness parameters has also not been published.

To provide transparency to the analysis, and to avoid skepticism of the results, accident reconstructionists need to utilize well-established engineering principles, publicly-available crash test data and commonly-used calculation techniques, such that the results can be reproduced by others--a fundamental requirement of engineering investigation.

BACKGROUND

Crush Energy Overview

With regard to automotive reconstruction, crush energy analysis can trace its roots back to the seminal work of Kenneth Campbell in 1974 [2]. Campbell's approach stemmed from the observation that for a given vehicle, the residual crush depth (the crush measurable after a crash) is linearly proportional to the closing speed of the impact, which means, following from fundamental principles of physics, that the residual crush is proportional to the resistive force of the vehicle structure [3]; i.e., the force-deflection relationship is also linear. Generalization of this approach to non-linear structural models has also been proposed [4, 5]. To explore the existence and nature of non-linearities, data points up and down the range of impact severities are required, and the repeated-test technique [6, 7] may be utilized to achieve those data points.

Among the reconstruction algorithms utilizing Campbell's work were Crash3 [8] and WinSMASH [9]. The purpose of these and similar programs was to provide a means to estimate accident severity for use in mass databases such as the National Accident Sampling System (NASS) [10]. As such, they were intended to provide unbiased crash severity statistics, as opposed to producing the best possible severity assessment of a specific accident [9]. Accordingly, the vehicle population was subdivided into categories, to each of which were assigned default values for computer program inputs. Among these default values were stiffness parameters for front, side, and rear structures in the various categories. The shortcomings of using default parameters, and applying Crash3 (as it was originally formulated) for reconstructing specific accidents, has been discussed previously [11].

Default side stiffness values for Crash3 were updated in 1987, based on a series of side impacts conducted by the National Highway Traffic Safety Administration (NHTSA) [12]. Default parameters were again updated by Siddall and Day in 1996 [13] and by Osterholt et al. in 2010 [14], but with no information provided in either case as to how the side stiffness values were derived. New categories were proposed and average parameter values were given by Lee et al. in 2014 [15], but again there was no explanation of how those values were determined.

Deriving Stiffness Coefficients from Impacts with Fixed Rigid Barriers

For vehicles involved in crashes, neither the crush nor the structural stiffness can be assumed to be uniform or homogeneous, in general. Therefore these quantities are expressed as functions of position on the vehicle. Crush energy (and force) are then obtained by a process of integration (and/or summation) across the crush profile.

For the most part, our discussion focuses on linear (also known as constant-stiffness) structures. In such cases, the crush force F per unit width may be written:

F = A + Bc, (1)

where F is in pounds per inch, A is the force offset (per unit width) in pounds per inch, B is the stiffness (per unit width) in pounds per square inch, and c = c(x) is the depth of crush in inches at location x (also in inches). In terms of the average crush [bar.c], we can write:

c(x) = [bar.c]f(x), (2)

where f(x) is a dimensionless shape function (identically equal to unity for the special case of uniform crush). In a differential slice of the crush profile, integrating the force-deflection curve over the residual crush c yields the differential crush energy:

d(CE) = (Ac + 1/2B[c.sup.2]+ G)dx, (3)

where G is a constant of integration in the units of pounds per inch per inch, or simply pounds. Assuming that A, B and G are uniform with respect to x (i.e., assuming the structure is homogeneous across the whole of the crush width), integration over the crush width L gives us:

CE = [L/2B[beta]][(A + B[beta][bar.c]).sup.2], (4)

where [beta] is a dimensionless form factor (describing the non-uniformity of the crush profile), given by:

[mathematical expression not reproducible] (5)

and where G is not independent, but depends on A and B according to the formula:

G = [[A.sup.2]/2B[beta]]. (6)

(If A, B and G are not constant over L, then the integrations have to be subdivided into sections over which they can be taken as constant.) Details of this derivation are found in Reference [3].

Equation (4) may be re-written as:

[mathematical expression not reproducible] (7)

The above relationship can be treated as a linear function between the left-hand side (dependent variable) and the independent variable [beta][bar.c]. A graph of this linear function will have the following attributes:

[mathematical expression not reproducible] (8)

Abscissa = [beta][bar.c] (9)

Slope = [square root of B] (10)

Intercept = [A/[square root of B]], (11)

where ECF is the Energy of Crush Factor. Such a graph is known as a crash plot and relates impacts of individual severities.

For a given crash test into a fixed rigid barrier, the crush energy is simply the loss in kinetic energy (initial minus rebound kinetic energies). For uniaxial (or co-linear) impacts, we can write:

CE = [DELTA]XE = 1/2[m.sub.v] (1 - [[epsilon].sup.2]) [V.sub.CL.sup.2], (12)

where [m.sub.v] is the mass of the vehicle, [epsilon] s the coefficient of restitution and [V.sub.CL.]is the initial impact velocity. If, for a given vehicle design, a number of barrier crash tests have been performed, and the impact velocity, restitution coefficient, form factor, average crush, and crush width are known for each of them, then each test may be represented by a data point on a crash plot. Subsequently, a regression line may be fitted among the points, from which A and B can be determined from Equations (10) and (11).

Deriving Stiffness Coefficients from Uniaxial Impacts with Rigid Moving Barriers

In the case where the rigid barrier has finite mass and can move, then momentum is conserved, in addition to energy. Enforcing these conditions produces the analog of Equation (12):

CE = [[m.sub.B][m.sub.V] (1-[[epsilon].sup.2]) [V.sub.CL.sup.2]/2([m.sub.B] + [m.sub.V])], (13)

where the subscripts B and V refer to the barrier and the vehicle, respectively, and where [V.sub.CL] is the initial closing velocity between the crash partners. A complete derivation can be found in Reference [3]. If we define the effective vehicle mass as:

[m.sub.EFF] = [[m.sub.V]/1 + [[m.sub.V]/[m.sub.B]]], (14)

then Equation (13) becomes:

CE = 1/2[m.sub.EFF] (1-[[epsilon].sup.2]) [V.sub.CL.sup.2], (15)

which in fact reduces to Equation (12) for the special case of an infinitely massive barrier. The crash plot analysis proceeds in the same way as described above.

The approach embodied in Equation (15) can be applied to side impacts, particularly perpendicular impacts by rigid moving barriers, such as the tests discussed in References [12] and [16]. The test mode described in those references did not result in significant post-impact vehicle rotations, but other modes may involve rotational energies that should be accounted for. In side impacts, some accounting may also be required for tire scrub as the struck vehicle slides sideways, particularly in low speed tests where energy losses from tire scrub are more significant with respect to the kinetic and crush energies.

What Happens when Both Crash Partners Absorb Energy

In a uniaxial impact between a vehicle and another vehicle, or between a vehicle and an MDB, both structures crush--and absorb energy in so doing. The two crushable structures are in series with each other and together may be viewed as a single crushable element. Equation (15) still applies, but now the left-hand side of the equation must be interpreted as the total crush energy for both crash partners. Thus, we can write:

[CE.sub.TOT] = [CE.sub.V] + [CE.sub.B] = 1/2[m.sub.EFF] (1-[[epsilon].sup.2]) [V.sub.CL.sup.2]. (16)

The fact that Equation (16) derives from Equation (7) opens the door to crash plot analysis of the vehicle structure, as long as [CE.sub.TOT] and [CE.sub.B] are known quantities. For such an approach to yield information on the vehicle structure, the total crush energy in the crash must be computed, the structural behavior of the MDB must be known, and the residual crush of both crash partners must be documented.

Analyzing Lateral Crash Tests with Moving Deformable Barriers

Beginning in 1997, the US Government began performing side impact crash tests as part of the New Car Assessment Program (NCAP). These tests were not like those in References [12] and [16], which specifically focused on stiffness parameters for the reconstruction of side impacts. Rather, Side Impact NCAP (SINCAP) tests are focused on occupant crash protection and simulate a perpendicular intersection crash with both vehicles moving. The embodiment of this concept is a crabbed MDB striking the side of a stationary vehicle placed at an angle to the tow track, but perpendicular to the MDB [17], as seen in Figure 1, copied from Reference [17].

This crash mode complicates the analysis of structural behavior due to the presence of an additional energy-absorbing structure and the angles involved. However, in US-market vehicles, it is nearly the only source for side impact data. Even so, the SINCAP tests (and the similarly configured FMVSS 214D compliance tests) have produced a wealth of data, which if analyzed properly, could yield valuable insights into vehicle structural stiffness. First, the necessary analytical tools must be developed, which is the subject of this research.

Previous Research

As mentioned above, a key step in developing a methodology for analyzing SINCAP tests is acquiring knowledge of the stiffness parameters A and B for the MDB, so that the MDB contribution to the crush energy can be determined, based on the measured crush post-test. In 1984 the NHTSA published a dynamic force-deflection curve for the MDB [18], and various researchers [19] claimed that its A and B values were thus known quantities. However, the MDB A and B values presented in Reference [18] were based on square-on frontal barrier tests that do not produce the combined compressive and transverse loads on the crushable honeycomb barrier face that are present in the SINCAP tests. The structural differences have a significant effect on the prediction of the energy absorbed by the barrier.

An excellent discussion of the state of affairs in 1998 was provided by Strother et al. [20]. In 2001 a study by Struble et al. [1] analyzed some SINCAP-style crash tests of advanced instrumentation MDBs [21] that properly loaded the barrier face, and from which its A and B values could be determined. As a result, the MDB stiffness values were found to be A = 54.3 pounds per inch, B = 224.4 pounds per square inch, and the saturation crush [C.sub.S] was 9.54 inches. (There will be more discussion about saturation later on.) The damage onset speed was determined to be 0.04 miles per hour. By contrast, the values used in the sample calculation of Reference [19] were A = 357 pounds per inch, and B = 172 pounds per square inch, corresponding to a damage onset speed of 4.5 miles per hour. This model also neglects any possible effects of force saturation in the barrier.

Once the MDB was characterized, its crush energy and the force acting on it during a crash test could be calculated. The next step in the analysis as presented in Reference [1] required a careful accounting of the kinetic energy of both crash partners as they emerged from the impact. Energy conservation was enforced by subtraction: the initial kinetic energy minus the kinetic energies at separation, tire friction, and the MDB crush energy. This process produced the crush energy absorbed by the struck vehicle.

Enforcing energy conservation and force balance provided the two conditions by which the vehicle side stiffness parameters A and B could be discovered (through an iterative process). The iterative process converged most of the time, but not always.

In this method, there was no accounting in the analysis for the no-damage threshold of the side structure. Consequently, while the derived A and B parameters fit the crash test data point (i.e., they produced proper valuations of crush energy and force in the test), they did not consider the implications for structural behavior at low impact severities.

Rigorous accounting for rotational effects is a process fraught with complications, requiring analysis of eight channels of acceleration data per test (X- and Y-accelerations from each of two accelerometer packs mounted on each crash partner). For the side impacts analyzed in Reference [1], rotational kinetic energy at separation averaged about 10 per cent of the translational kinetic energy. Based on the variability of A and B, it is not clear that teasing out the rotational effects and exactly matching force levels between the crash partners actually improved the results. An alternative approach to determining both crash partner's post-impact yaw rate would be to analyze the overhead video, although that process too can be fraught with complications.

Nature of the Vehicle Fleet

The 2010 Osterholt study [14] addressed the obsolescence of the vehicle population studied by Siddall and Day [13] in 1996, at least for frontal impacts. Again, generic results were published for the same 11 vehicle categories specified in the earlier study, rather than specific vehicles. Reference [14] confirmed the trend that many observers have seen over the years: namely, frontal stiffness of vehicles continues to increase with time [22]. However, there was a lot of scatter within each category. The B value averages, taken as their own population, had a standard deviation of just under 15 pounds per square inch. This was less than the standard deviation within any of the individual categories. In other words, there was more variation within a given category than between categories. Almost the opposite was true for A (possibly because of fairly uniform assumptions for no-damage threshold speeds). The variables defining the categories (wheel base and vehicle type) do not seem reliable predictors of either A or B for reconstruction purposes. It is better to use calculated parameters for a specific vehicle, compared to relying on averaged data from broad vehicle categories. Publicly-available specific crash test reports are available for most high-volume vehicles.

The 2014 update by Lee et al. [15] re-defined the categories in terms of Department of Energy fuel economy guidelines. Market trends were used to select the vehicles to represent the categories. Again, stiffness was shown to be increasing over time, and much scatter was indicated in frontal A and B values. Rear and side stiffness parameters were also shown (with notably less scatter for the side A values), but no information was provided as to how they were derived. Credibility of the results requires an understanding of where the underlying parameters came from.

Approach for this Study

1. Determine a means of extracting the vehicle crush energy from the MDB crush data and the vehicle dynamics at separation, when the vehicle is struck in the side by a moving deformable barrier (MDB).

2. Extend the theoretical basis of crash plots to impacts in which both collision partners develop crush energy.

3. Develop a crush energy computation protocol suitable for accident reconstruction, compatible with the crush measurements documented in crash test reports.

4. To the extent possible, discern the no-damage threshold of crash severity in side impacts, i.e. find the test speed below which no residual damage is seen, and above which permanent damage is incurred.

5. Determine the vehicle stiffness parameters that correctly predict the crush energy in a crash test, while producing an appropriate no-damage threshold. For a crash test, such a procedure effectively reverses the crush energy calculations used in reconstructions, and may require iterative methods.

6. Explore the nature and effects of force saturation on the determination of stiffness parameters.

7. Apply this procedure to a representative sample of side impact crash tests to discern what trends may be present, if any.

METHODOLOGY

The present approach uses Equation (16) to analyze SINCAP test results, since both crash partners absorb energy in structural crush. Rotational effects are assumed to be small, in keeping with the findings of Reference [1]. (After all, the impacted area of the struck vehicle straddles its center of mass.) Separation velocities are obtained by integrating the appropriate accelerometer channels, from which the restitution coefficient (not accounted for in Crash3) is easily calculated. In accordance with Equation (16), the residual energy that gets dissipated in crush (total crush energy) can be expressed in terms of impact velocity, restitution coefficient, and the masses of the vehicle and barrier. Energy dissipation due to the lateral scuffing of the struck vehicle tires is not included, since Reference [1] found the contribution to be negligible in SINCAP tests (typically less than one percent of the total kinetic energy of the event).

The linear structural model developed in Reference [1] for the MDB is used to calculate the crush energy absorbed by the MDB, based on the reported bumper-level crush "...because the MDB bumper is so much stiffer than the main honeycomb stack. Despite that stiffness, the mid-bumper crush is easily the highest, because the bumper initially protrudes 4 inches forward of the rest of the honeycomb..." [1]. The MDB crush energy is subtracted from the residual energy to obtain the crush energy assignable to the vehicle.

A crash plot is then constructed with a straight line drawn between the crash test data point and an intercept based on a no-damage threshold of 2.1 mile per hour change in velocity, or [DELTA]V. (The threshold is based on the results of repeated side impact tests, discussed later.) A and B are then calculated using Equations (10) and (11).

Obtaining Separation Velocities and Restitution

Kinetic energy loss is computed by comparing the MDB kinetic energy at impact versus the kinetic energies of the vehicle and the MDB when they separate. Rotational kinetic energies have generally been found to be small compared to crush energy and the translational kinetic energies at separation, so the fundamental kinetic energy formula KE=1/2m[V.sup.2] is applied to each crash partner at separation, where V is the resultant velocity. Each resultant is computed from the vector sum of the longitudinal and lateral velocity components.

To determine the time of separation (i.e., the end of the contact phase), the velocity-time (V-t) curves of the vehicle and the MDB are compared to see when the local extrema occur; usually they closely coincide in time. If a local minimum exists for the MDB x-axis V-t curve, that time value is chosen; otherwise, the local maximum of the vehicle y-axis V-t curve may be used. In either case, the selected time value is applied to both crash partners.

The V-t curves are obtained by using Signal Browser, one of the applications included in the NHTSA Signal Analysis Software package, available on the NHTSA website (1). To use it, the appropriate accelerometer channel(s) for the test of interest are selected from a structured list of channel identifiers. The data are immediately graphed (at CFC 1000). The accelerometer traces are then filtered to the appropriate Channel Frequency Class (CFC) according to the test protocol (vehicle and barrier accelerometer data are typically filtered to Class 180 before integration [23]) by clicking on the appropriate numeric icon.

To obtain the V-t curve(s), the integration function is used. Velocity and time coordinates can be displayed, which allows the user to pinpoint a velocity at a desired time value (separation, for example).

The resultant velocities and kinetic energies are then readily calculated. Relative velocities between the vehicle and MDB at impact and separation, lateral to the vehicle, are then computed. By treating the crash test as an essentially uniaxial collision, the restitution coefficient can be calculated from these relative velocity components.

Findings from Repeated Impact Tests In 1991 Prasad [16] reported on a test program in which each of five vehicles was subjected to a series of perpendicular side impacts by a rigid moving barrier shaped similarly to an MDB, but full crash test reports were not available for those tests, and the data presented in [16] was insufficient to perform a full analysis. Similar repeated test series of two additional vehicles were found in the NHTSA cash test database. These tests were of a 1984 Audi 5000: DOT 1644 - 1647 (four repeated impacts at 15.2, 20.1, 20.0, and 40.1 miles per hour) and a 1985 Ford Escort: DOT 1823 - 1826 (four repeated impacts at 20.0, 20.0, 20.0, and 34.7 miles per hour). Analysis of these tests provides insight in the force-deflection behavior of these vehicles across a broad spectrum of severities, particularly on the low end.

Repeated-impact tests are analyzed in References [3] and [6]. The speed of a single test S that is equivalent in crush energy to R repeated impacts in a test series and is given by:

[mathematical expression not reproducible] (17)

The various equivalent impact speeds are determined by setting R = 2... 3 or 4 in Equation (17) for these series. The various restitution coefficients were calculated from Signal Browser information, which gives the speeds in kilometers per hour. For the four Audi 5000 tests, the restitution coefficients were 0.173, 0.151, 0.114, and 0.015--a decreasing trend with increasing closing speed that many observers have reported. It appears that a reasonable restitution coefficient in a no-damage impact would be given by:

[[epsilon].sub.0] = 0.2. (18)

If a regression line were fitted to the resulting data points for the Audi 5000, it would have a typically high correlation coefficient. For this vehicle, however, we would see a definite bend in the data point pattern at the second test, as illustrated in Figure 2. This is a typical indicator of force saturation, wherein the force stops increasing at some level of crush. In other words, the test data may be explained better by a structural model that is constant stiffness only up to a certain point--the saturation crush--beyond which the force remains constant--the saturation force [5, 20].

Instead, consider another crash plot, one containing an imaginary "no-damage" data point, as shown in Figure 3. Its coordinates would be derived from the y-intercept of a line running through the data points corresponding to the first two tests: (5.00, 57.40) and (8.59, 89.11). The resulting point would have an ordinate of 13.3178 (and a corresponding abscissa of zero, as there is zero average crush). The test speed required to produce this "no-damage" condition (the closing speed between the stationary target vehicle and the moving rigid contoured barrier) would be 3.53 miles per hour (5.7 kilometers per hour). The [DELTA]V of the struck vehicle works out to be 2.10 miles per hour. While the no-damage threshold varies from vehicle to vehicle, as seen in the few repeated-impact test series available, it is representative of the values typically assumed by analysts. Thus a no-damage threshold [DELTA]V of 2.1 miles per hour was utilized for all side impacts (including those by an MDB). The closing speed corresponding to this [DELTA]V would vary according to the masses of the collision partners in the various crash tests.

The saturation crush [C.sub.S] for this vehicle/test series was 7.1 inches, which was the average crush corresponding to the bend in the crash plot at the second impact. Again, this value was chosen to be representative of all side impacts (including those by an MDB). Sensitivity to this choice is discussed in Appendix B. The A and B values derived from the first two actual tests represented on the crash plot were 117.5 pounds per inch and 77.8 pounds per square inch, respectively.

Figure 3 shows that a force saturation model can provide adherence with low-speed crash behavior, while still retaining fidelity in high-severity impacts. The maximum error magnitude in predicting the crash test ECF was 5% (for Test 1646); the error magnitudes in the remaining tests were less than 21/2%.

The crash plot resulting from the side impact series for the 1985 Ford Escort appears in Figure 4. In this case, any force saturation is not sufficiently pronounced to determine a saturation crush value, in view of the data point for the second actual crash test being below the regression line. The regression fit through all five data points yields A and B values of 121.2 pounds per inch and 47.4 pounds per square inch, respectively. This is representative of other tests in which no force saturation is apparent.

In the analysis in Reference [1], the saturation crush for side structures was set sufficiently high (20 inches) to cause all the crush profiles to be entirely unsaturated (thus making crash plots a valid analysis technique). However, there were some tests in which the crushable barrier face saturated over at least part of its crush profile. It seemed entirely likely, therefore, that some vehicle structures would be saturating as well, particularly if the door attachments were compromised. It was therefore decided to analyze each side structure two ways: one unsaturated, and one with a saturation crush of 7.1 inches--the average crush seen in the second Audi 5000 test. The analytical method is discussed in Appendix A. The [DELTA]V for no damage was set at 2.10 miles per hour for both saturated and non-saturated analytical models.

Crash Plots for Individual Crash Tests

The actual crash test and a "fictitious" test, reflecting no damage at the no-damage threshold speed, are plotted (at least conceptually) on a graph of ECF vs. abscissa, as previously illustrated in connection with Equations (8) and (9), and shown in Figure 5. To calculate the intercept of the plot, we first compute the closing velocity that would be required to produce a [DELTA]V of 2.1 miles per hour, given the masses of the MDB and the vehicle in the test being analyzed. Momentum conservation gives us:

[V.sub.nodam] = [2.10/1 + [[epsilon].sub.0]] [[m.sub.V] + [m.sub.B]/[m.sub.B]], (19)

where [epsilon].sub.0] is the restitution coefficient at the no-damage threshold.

Next, we calculate the Energy of Crush Factor [ECF.sub.0] for an impact at this closing speed. We begin by noting that if [m.sub.EFF] is in slugs (pound-seconds squared per foot), G is in feet per seconds squared, [L.sub.V] is the crush width from the actual test in inches (although by definition, there is no crush and therefore no crush width; this reflects the damage region in the actual test), and [V.sub.nodam] is in miles per hour, Equations (8) and (15) yield:

[mathematical expression not reproducible] (20)

See Reference [3] for the derivation of the conversion constant 0.8954. From the functional form of a straight line (and Equations (10) and (11)), we can calculate the stiffnesses A and B:

[mathematical expression not reproducible] (21)

and

[mathematical expression not reproducible] (22)

where ECF and Abscissa refer to the values from the actual crash test.

Non-Linear Analyses and Force Saturation

The approach leading to Equations (21) and (22) is a very convenient one because it leads to explicit expressions for A and B. When force saturation, or any other nonlinear condition, exists in the force-deflection relation, however, a crash plot will deviate from the straight line seen in Figure 5, and as discussed in Reference [5]. Since Campbell's crash-plot analysis is based on linearity, a new methodology must be developed to incorporate any observed non-linearity of side deformation.

We start with the realization that if the force-deflection relation is linear up to the saturation crush [C.sub.S], and is constant-force beyond that point, the crush energy can still be calculated if A, B, and [C.sub.S] are known. It is only necessary to subdivide the crush profile into saturated and non-saturated portions, analyze them separately, and then sum the results. See Reference [3], which illustrates how to do the calculations with a spread sheet. In this case, we desire to invert the process, determining A and B (and [C.sub.S]) from the known deformation and crush energy rather than calculating crush energy from stiffness and deformation, but this inversion cannot be done because of the nonlinear nature of the problem.

Instead, to analyze a crash test with force saturation in the struck vehicle, we run the analysis procedure in a forward direction, and systematically vary A and B (holding [C.sub.S] constant) until the no-damage threshold and (most importantly) the side structure crush energy are predicted. To simultaneously vary A and B to satisfy these two conditions, a multivariate search function, such as the Solver tool available for Excel[R], can be used.

Excel's Solver utility accommodates multiple unknowns, multiple equations, and multiple constraints. In this case, the target crush energy in the side-struck vehicle is obtained through energy conservation in the side impact test, as discussed previously. The actual vehicle crush energy is calculated in the usual way based on A, B, [C.sub.S], and the crush profile. When Solver is invoked, it changes A and B until the target is hit, subject to the constraint that A and B produce a no-damage threshold [DELTA]V of 2.10 miles per hour.

The development of the constraint condition, and its implementation in Solver, are discussed in Appendix B.

This iterative procedure could be utilized even in the absence of force saturation. In fact, when there is no saturation anywhere on the crush profile, a closed form solution exists and yields identical results. (MDB force saturation can happen independently of saturation in the vehicle, but only the MDB crush energy as a whole--not its components--is used in the vehicle calculations.)

Selecting the Appropriate Crush Profiles

We have been speaking of crush as a function of x--the location of a given crush measurement along the crush profile. In reality, crush is a function not only of x, but of z--the water line, or height above the ground. Consequently, the SINCAP test procedure [17] calls for measuring vehicle crush at five water lines located relative to the vehicle body, and MDB crush measurements at four water lines relative to the ground. In general, the vehicle and MDB water lines do not coincide.

At the mid-bumper (Level 1), the MDB protrudes the farthest and has the highest stiffness. It usually experiences the greatest crush, and for these reasons figures to absorb the most energy. It also figures to inflict the most damage on the test vehicle. Thus the mid-bumper crush was used to characterize the MDB in Reference [1], in which it was shown that the mid-bumper MDB crushed shape matched up well with the post-test vehicle side structure at the level of deepest penetration.

In field accidents, it is also customary to measure vehicle crush at deepest penetration. For consistency with field measurement techniques, and in consideration of all of the above, we thus decided to characterize the vehicle structure using the profile with the greatest crush. For passenger cars and vans, this level is usually at the occupant H-point. Depending on specific vehicle geometry, the H-point can be Level 2 or Level 3, but the specific levels are indicated in each crash test report.

For SUVs and pickups, the MDB bumper often matches up best with the top of the sill (Level 1), where the greatest crush is virtually always found. Therefore Level 1 was the default choice for characterizing the side structures of these vehicles. The general rule is to select the level of deepest penetration on the vehicle to find A and B for its side structure.

Vehicles Examined in this Study

Test reports for about 350 SINCAP tests were downloaded for this study. Analysis was performed on over 100 of these. To maximize the paper's interest and applicability in the near future, the range of results reported herein was narrowed to reflect recent best-selling vehicles. A list was found for the top 20 vehicle sales volumes for the first 11 months of the 2015 calendar year [24]. For each vehicle on that list, the production run was identified, along with any essentially identical make/models [25]. All tests meeting the selection criteria were targeted for analysis. Multiple tests were available in many cases. Of the 81 tests in this group, 68 involved Model Years 2010 or later.

Of the 81 tests, 16 had fatal errors (that could not be rectified or worked around, precluding analysis): 12 with electronic signal issues, and four with post-test measurement or reporting errors. There were another 14 with structural interaction problems during the test, such that any results could not be accepted as valid. The underride issues will be discussed below. Fifty-one tests remained in which analysis could be expected to produce reasonable results outright, plus another two in which some obvious data errors could be corrected via minor adjustments.

RESULTS

Table 1 shows the top 20 best-selling vehicles for 2015, their sales volumes, plus summary results from analyzing an applicable SINCAP test for a proximate Model Year. As has been often noted, the top three "cars" sold in the U.S. were actually pickup trucks. The stiffness values shown were calculated including the effects of force saturation of the vehicle structure. Most of the time, ignoring these effects produced only slightly different results, as will be discussed below.

It should also be noted that results were not obtained for four of the vehicles (five including the GMC Sierra). With the Hyundai Elantra, only one test was available (DOT 7508), but for that one the NHTSA Signal Browser reported an error with the vehicle y-axis accelerations. The Chevrolet Cruze also had only one test (DOT 7160), and in that test the vehicle CG y-axis accelerometer signal was lost at about 22 milliseconds. For the Chevrolet Silverado and Ram pickups, a total of eight applicable tests were found--four for each vehicle. In every such test, however, override of the barrier was present, as discussed below. While A and B could be calculated mathematically, the results could not be considered valid.

Other tests were encountered with obvious post-test measurement or reporting errors (MDB crush measurements in particular), but generally there were alternative applicable tests to choose from.

More detailed results for Model Year 2015 are presented in Appendix C.

DISCUSSION

Underride

For the Silverado and Ram pickups in Table 1, eight SINCAP tests were conducted. In every test, the MDB mid-bumper water level aligned best with the top of the vehicle sill, and the deepest penetration on the vehicle was recorded at that level. However, the MDB clearly underrode the test vehicle. The first indication in the test report is Data Sheet 9, "Maximum Static Crush of Honeycomb Impact Face," which presents the maximum crush for each of the four MDB levels. The "center of bumper" should show the highest crush value, not the lowest. Similarly, the "MDB Exterior Static Crush Measurements" in Data Sheet 12 should show the same pattern. (Surprisingly, these two data sources occasionally disagree.)

The most obvious evidence of underride is in the post-test photographs, though. Looked at from the side, if a rectangular block of honeycomb has been turned into triangle, as can be seen in Figure 6, one has visual proof that the MDB crushable face has not been exercised similarly to the impact tests from which its A and B parameters were derived. Calculations performed for the vehicle using these parameters for the MDB cannot be considered valid in such cases. Aside from structural mismatch, tuck-under of the side of the struck vehicle may also be playing a role.

Similar behavior was found in several of the SINCAP tests of Ford F-150 pickups. In all cases, the best match- up with the MDB bumper was the struck vehicle top of sill. It is safe to say that when the vehicle crush level nearest to 432 mm (the MDB mid-bumper height) is Level 1 (top of sill), the test should be scrutinized very carefully for underride. This caution applies to some SUVs as well as pickups.

This is not to say that such tests are invalid for their original purpose--assessment of occupant protection. For reconstruction purposes, however, a different test protocol may function better for characterizing side structures of high-profile vehicles.

Variability

There is a lot of variability seen in the A and B parameters for side structures. However, for any given test, two separate and independent methods produce similar results, even when force saturation is present. When there is no force saturation, identical results are obtained. So the variability does not arise from the way the calculations are done, nor could so much variability arise from the usual scatter seen in experimental endeavors, in our opinion. Systematic post-test measurement errors produce data anomalies that are obvious to investigators, and many of them should have been addressed through quality control. Rather, the variability in results seems to be real.

The implication is that A and B values for a particular vehicle cannot be predicted on the basis of class, wheelbase, weight, interior volume, or any other general characteristic that has been proposed. One has a much better chance of reconstruction accuracy if one simply analyzes the crash tests for the specific vehicle(s) involved (or sisters/clones), or in the absence of test data for a specific vehicle, test data for a substantially similar vehicle, according to engineering judgment.

Similarity in names can be misleading, though. The Jeep Cherokee and the Jeep Grand Cherokee are both listed in Table 1 because the sales information did not differentiate between them. However, they are completely different vehicles from one another, as manifested in their dissimilar stiffness values.

Some stiffness variability could be attributable to differences in the engagement of hard points such as pillars and wheels, although the SINCAP test protocol tends to minimize such effects. For example, the 2014 Corolla has a much higher B value than does the 1015 Camry. Is that because the smaller car has a more complete engagement of the C-Pillar by the MDB bumper? Pictures in the test reports are not definitive, as there are no door-open pictures of the left side pillars and sill. However, the crush data are clear. The Corolla had an average crush of only 5.53 inches; none of the crush profile was in saturation. By contrast, the Camry average crush was 8.34 inches, with 84% of the crush profile in saturation.

When the MDB bumper matches up best with the struck vehicle sill and the sill profile is used instead that for the H-point, the resulting stiffness values can be expected to be higher. The sill is a primary structure supported by structures such as the floor pan and cross-members and figures to be stiffer.

Even so, the Ford F-series pickup results derived from the sill crush are not noticeably different from other vehicles in which the H-point profile was used. Average crush ranged from about 8.3 to 8.6 inches, with saturation occurring over 68% to 75% of the profile. The Nissan Rogue was the only other vehicle in Table 1 in which the stiffness analysis was based on the sill profile. In that case, the average crush was only 4.4 inches, with none of the profile in saturation. Naturally, its B value turned out to be very high.

The underlying design issues would make a very interesting study, but were beyond the scope of this research project.

Repeatability

The test of repeatability would come from conducting nominally identical tests on nominally identical vehicles, and then comparing the results. The desire to perform SINCAP tests on the widest possible range of vehicles within a finite budget constrains such efforts. Nevertheless, there are some groups of tests that suggest a considerable degree of repeatability, even though crash tests are sudden and violent events that are full of opportunities for data acquisition errors. Consider, for example, the two tests (DOT 9474 and 9545) of 2016 Ford F-250 pickups. The cabs are different (SuperCab vs. SuperCrew), but the stiffness remains unaffected. The two A values are 330 and 334, and the two B values are 194 and 192. Similarly, consider the two tests (DOT 8078 and 8219) of 2013 Ford Fusions. The two power plants are different, but as one might expect, there is no effect on side stiffness. The two A values are 231 and 230, and the two B values are 186 and 188. While anecdotal, such evidence suggests high repeatability may be seen whenever tests of identical vehicles are conducted.

Occasionally, two tests of similar or identical vehicles are found in a given model year. However, one of the tests is usually plagued by data or measurement issues, which may account for why it was tested more than once.

CONCLUSIONS AND RECOMMENDATIONS

The variability of A and B for vehicles of similar weight and wheelbase means that the vehicle categories proposed to date cannot be used to predict what the stiffness values will be for a specific vehicle. Default or categorical values were not designed for use in specific crashes. Stiffness parameter values can be purchased for a specific vehicle, but the means by which the values were obtained would not be known.

On the other hand, an analytical methodology, using widely available computational tools, has been presented by which the reconstructionist can develop information based on publicly-available test data. The source data and the analysis are subject to examination, and confirmation, by all interested parties.

Sample calculations have been presented, along with summary results for popular recent-model vehicles.

Various simplifications (such as ignoring force saturation) are available that in most cases will allow a good approximation to more rigorously-derived results. The most rigorous solution accounts for force saturation, but does not permit a closed-form solution. Rather, it requires an iterative method, which can be easily implemented using a spreadsheet-based multivariate solver routine, such as the Solver function within Microsoft Excel[R].

Recommendations for using the present methodology for characterizing vehicle side structures from SINCAP test data are as follows:

1. Use the vehicle interchange list to identify the production run of the vehicle in question, and all similar vehicles. Search the NHTSA crash test data base for applicable tests, with an eye to analyzing them all. If results cannot be obtained for similar vehicles throughout the production run, analyze tests of the vehicles closest in similarity, as this is still preferable to using default stiffness values.

2. Choose the vehicle crush profile on the basis of deepest penetration, paying attention to how the selected profile height compares to the MDB mid-bumper height.

3. Check for MDB underride of the vehicle using post-test photos and by comparing the MDB crush at various levels, especially when the top of the vehicle sill (the lowest level) provides the closest match-up with the MDB bumper.

4. Obtain the vehicle and MDB separation velocity components by using NHTSA Signal Browser software. Filter the CG acceleration components to CFC 180, and then integrate to obtain the V-t curves.

5. Choose the time of separation (crush duration), usually corresponding to the time of minimum MDB x-velocity. Read the four velocity components that exist at that time value.

6. Look for anomalies in the electronic data. Issues that have been found include: no signal, signal spikes in the first 200 msec, wrong sign (polarity), wrong initial velocities, and possible data channel mis-labeling.

7. Look for crush measurement anomalies. Surprisingly, all of these found so far have been in the MDB, not the vehicle. They appear to result from procedural errors, resulting in wrong sign, unrealistically high or low values, and inconsistencies among the tables in the crash test report.

8. As discussed below, be on the lookout for calculated target vehicle crush energies below about 15,000 foot-pounds.

The details in Table 2 of Appendix C are included to assist in identifying anomalies.

It is also recommended that:

1. Low-severity side impacts be conducted to learn more about the onset of residual crush and whether the choice of linear models is justified.

2. Side structural characterization of high-profile vehicles be improved. This could be done by either:

a. Conducting SINCAP-type tests utilizing a rigid barrier face (that avoids deforming into the shape of a ramp) and that extends high enough (the height of a typical pickup hood, for example) to fully engage the side structure.

b. Characterizing the existing barrier face for underride crashes, using tests similar to those in Reference [21], except that the flat rigid barrier in those tests would be replaced by one that is shaped so as to exercise the upper layers of the honeycomb stack.

LIMITATIONS

Cautionary Notes

1. To calculate the residual kinetic energy in the crash event (that is converted to crush energy in the crash partners), the kinetic energy calculations have been simplified and made more reliable by treating the crash uniaxially and treating any rotational effects as negligible. Nevertheless, there are a few tests in which the calculated residual kinetic energy (total crush energy) is impossibly small. Such situations can arise from problems in calculating the exit velocities by integrating the acceleration data. Instrumentation issues may be at work, and data traces should be examined for anomalies.

2. Sometimes the MDB crush measurements are unrealistically high. In such occasions, the MDB winds up hogging all the crush energy calculated to be available. This situation could arise from MDB crush measurement error, or structural mismatch. Either way, the target vehicle crush energy can be unrealistically low. Such situations can lead to A and B values that can be either unrealistically high or low, depending on the amount of vehicle crush. Target crush energies for the vehicle that are below about 15,000 foot-pounds are cause to question the results. MDB crush profiles should be compared with reported crush maxima, and with photographs.

3. The assignment of crush width affects the results. The authors' standard practice is to extrapolate the MDB crush profile to the width of the MDB (since obviously the entire width is crushed), even though it is not practical to measure the crush at the very edge of the MDB deformable face. Similarly, the vehicle profile is extrapolated to the point(s) of zero crush at the end(s) of its profile (if doing so produces a zero crossing within the next 150-millimeter crush measurement interval). Both practices produce profile widths that are somewhat wider than those obtained from the numbers in the crush tables as they are but are more consistent with what would be measured in the field. After all, the crash test labs mark the crush measurement points before the test, so they cannot know just where the vehicle crush profile is going to end. Improved consistency between the test facility and the field is to be encouraged when it comes to measuring crush profiles.

4. The crush width used to calculate the no-damage condition for a particular vehicle is the same as it was in the actual crash test.

5. The computational problems of structural mismatch are alleviated (partially, but not always totally) by using the vehicle crush profile that represents the line of deepest crush, or at least close to it. For automobiles and vans, this is usually the occupant H-point level, which is usually either Level 2 or 3. For trucks and SUVs, the deepest crush is often at the top of the sill, Level 1.

6. If the major MDB crush is above the bumper (usually due to a high struck-vehicle ride height), the calculated stiffness results cannot be considered valid, since the MDB structural characterization is based on bumper-level crush.

REFERENCES

(1.) Struble, D., Welsh, K. and Struble, J., "Side Impact Structural Characterization from FMVSS 214D Test Data," SAE Technical Paper 2001-01-0122, 2001, doi:10.4271/2001-01-0122.

(2.) Campbell, K., "Energy Basis for Collision Severity," SAE Technical Paper 740565, 1974, doi:10.4271/740565.

(3.) Struble, D., "Automotive Accident Reconstruction: Practices and Principles," (Boca Raton, CRC Press, 2014), ISBN 978-1-4665-8837-0.

(4.) Strother, C., Woolley, R., James, M. and Warner, C., "Crush Energy in Accident Reconstruction," SAE Technical Paper 860371, 1986, doi:10.4271/860371.

(5.) Wolley, R., "Non-Linear Damage Analysis in Accident Reconstruction," SAE Technical Paper 2001-01-0504, 2001, doi:10.4271/2001-01-0504.

(6.) Warner, C., Allsop, D. and Germane, G., "A Repeated Crash-Test Technique for Quantifying Vehicle Crush Coefficients," SAE Technical Paper 860208, 1986, doi:10.4271/860208.

(7.) Prasad, A., "Energy Dissipated in Vehicle Crush--A Study Using the Repeated Test Technique," SAE Technical Paper 900412, 1990, doi:10.4271/900412.

(8.) Noga, T. and Oppenheim, T., "CRASH3 User's Guide and Technical Manual," US Department of Transportation Report HS 805 732, (Washington, US Government Printing Office, 1981).

(9.) Sharma, D., Stern, S., Brophy, J. and Choi, E.-H., "An Overview of NHTSA's Crash Reconstruction Software WinSMASH," Proceedings of the 20th International Technical Conference on the Enhanced Safety of Vehicles (ESV), 2007.

(10.) Tumbas, N. and Smith, R., "Measurement Protocol for Quantifying Vehicle Damage from an Energy Point of View," SAE Technical Paper 880072, 1988, doi:10.4271/880072.

(11.) Struble, D., "Generalizing CRASH3 for Reconstructing Specific Accidents," SAE Technical Paper 870041, 1987, doi:10.4271/870041.

(12.) Willke, D. and Monk, M., "Crash III Model Improvements: Derivation of New Side Stiffness Parameters from Crash Tests, Vol. 2," US Department of Transportation Report HS 807 353, (Washington, US Government Printing Office, 1987).

(13.) Siddall, D. and Day, T., "Updating the Vehicle Class Categories," SAE Technical Paper 960697, 1996, doi:10.4271/960697.

(14.) Osterholt, G., Cummings, J., Biller, B. and Calhoun, V., "Updating Generic Crush Stiffness Coefficients for Accident Reconstruction," SAE Technical Paper 2010-01-1581, 2010, doi:10.4271/2010-01-1581.

(15.) Lee, E., Lee, P., Erickson, M. and Hayes, W., "Increase in Vehicle Front, Rear and Side Stiffness Coefficients in the Past Twenty Years Necessitates New Representative Database," SAE Technical Paper 2014-01-0351, 2014, doi:10.4271/2014-01-0351.

(16.) Prasad, A., "Energy Absorbed by Vehicle Structures in Side-Impacts," SAE Technical Paper 910599, 1991, doi:10.4271/910599.

(17.) National Highway Traffic Safety Administration, "Laboratory Test Procedure for FMVSS 214 'Dynamic' Side Impact Protection," US Department of Transportation Report TP-214D-05, (Washington, US Government Printing Office, 1999).

(18.) Hackney, J., Monk, M., Holliwell, W. and Willke, D., "Results of the National Highway Traffic Safety Administration's Thoracic Side impact Protection Research Program," SAE Technical Paper 840886, 1984, doi:10.4271/840886.

(19.) Neptune, J. and Flynn, J., "A Method for Determining Crush Stiffness Coefficients from Offset Frontal and Side Crash Tests," SAE Technical Paper 980024, 1998, doi:10.4271/980024.

(20.) Strother, C., Kent, R. and Warner, C., "Estimating Vehicle Deformation Energy for Vehicles Struck in the Side," SAE Technical Paper 980215, 1998, doi:10.4271/980215.

(21.) Trella, T., Samaha, R., Fleck, J. and Strassburg, G., "Moving Deformable Barrier with Dynamic Force and Deflection Measurement Capabilities for Full Scale Tests," SAE Technical Paper 2000-01-0637, 2000, doi:10.4271/2000-01-0637.

(22.) Swanson, J., Rockwell, T., Beuse, N., Summers, L.et al., "Evaluation of Stiffness Measures from the U.S. New Car Assessment Program," Proceedings of the 18th Technical Conference on the Enhanced Safety of Vehicles (ESV), 2003.

(23.) Marine, M. and Werner, S., "Delta-V Analysis from Crash Test Data for Vehicles with Post-Impact Yaw Motion," SAE Technical Paper 980219, 1998, doi:10.4271/980219.

(24.) "Top 20 U.S. Vehicle Sales: 2015, from www.motorintelligence.com," Wall Street Journal Online, wsj.com, 5 January 2016.

(25.) Anderson, G., "Vehicle Year & Model Interchange List (Sisters & Clones List)," (Madison, Scalia Safety Engineering, 2016).

CONTACT INFORMATION

John D. Struble

Exponent, Inc.

3440 Market Street, Suite 600

Philadelphia, PA 19104

(215) 594-8841

jstruble@exponent.com

APPENDIX

Appendix A. Implementing Calculations in a Spreadsheet

Documentation of crush is generally accomplished by a series of discrete measurements. The crush profile is then approximated by a piecewise-linear function constructed between the points. It is only natural to use a spreadsheet to perform the analysis, segment by segment. Accuracy and generality are enhanced by allowing the crush profile to more accurately follow the damage shape, something not usually accomplished with the traditional six arbitrarily placed crush measurements.

For crush energy calculations, the basic enabler of segment-by-segment analysis is the need to integrate over the length of the crush profile, as in Equations (4) and (5). The integrals are broken up into segments defined by the crush measurement locations. The i-th segment lies between x. and [x.sub.i+1] has a length [DELTA][x.sub.i], and has crush that varies linearly between [C.sub.i] and [C.sub.i+1] for which an average crush [[bar.C].sub.i] can be calculated. For the crush profile as a whole, the overall average crush [bar.C] is given by:

[mathematical expression not reproducible], (23)

where N is the number of measurements. Since the inner measurements border two segments compared to only one segment for the outer two measurements, this area average gives more weight to the inner crush measurement(s) compared to the outer ones, as can be seen by expanding Equation (23). (If the segments are uniform in length, the inner measurements get double the weight.) L is the width of the crush profile, given by:

[mathematical expression not reproducible] (24)

The form factor [beta] is obtained from Equation (5) by substituting a linear function for the N-1 segments and performing the indicated integration. The result is:

[mathematical expression not reproducible] (25)

where [(Coeff).sub.i], the contribution to the calculation of [beta] from the i-th segment, is given by:

[mathematical expression not reproducible] (26)

The contribution to crush energy from Segment i is found by applying the piece-wise linear crush function to Equation (3) and integrating over the segment length [DELTA][x.sub.i]. The result is:

[mathematical expression not reproducible] (27)

When the crush in the segment exceeds the saturation crush [C.sub.S], the force remains constant at the saturation force [F.sub.S], regardless of c or x. That force is given by:

[F.sub.S] = A + B[C.sub.S]. (28)

As a result, the segment crush energy is a linear function of [[bar.C].sub.i], and neither the form factor [beta] nor its components [(Coeff).sub.i]. appears in the analysis. Integrating Equation (3), we obtain:

[(CE).sub.i] =[A[C.sub.S] + 1/2B[C.sub.S.sup.2] + G + [F.sub.S] ([[bar.C].sub.i] - [C.sub.S])][DELTA][x.sub.i] [[bar.C].sub.i] [greater than or equal to] [C.sub.S] (29)

Note that for complete rigor, additional data points must be inserted by interpolation wherever the crush transitions between saturated and unsaturated behavior, so that either Equation (27) or (29) remains accurate throughout the segment. This is made possible by the fact that nowhere is there an assumption of six crush measurements, or that they be equally spaced. For further details, see Reference [3].

This flexibility in describing the crush profiles means that the crush data in SINCAP tests (typically 17 points for the MDB and a similar number for the vehicle) may be used as reported, and extrapolated if desired. Interpolated and extrapolated points may be inserted to reflect the full width of the MDB, the termination of contact between the vehicle and the MDB, or the locations of any transitions between saturated and unsaturated crush.

Typical crush energy calculations are shown in Figure 7 for a 2016 Chrysler 300 subjected to a SINCAP test, DOT 9497. The average crush [bar.C], the quotient in Equation (23), is calculated in cell AD10 (not shown in order to improve readability). Equations (24) and (25) are implemented in cells X26 and AD11, respectively. The results of Equation (26) appear in Row 29. Segment crush energies are calculated in Row 31: Columns D - J and U - V for unsaturated crush--Equation (27), and Columns K - T for saturated crush--Equation (29). Additional points were inserted by interpolation to accommodate crush transitions at [C.sub.S] = 7.10 inches (180.3 millimeters). Another point was inserted--also by interpolation--to account for the end of MDB contact at 66 inches (1676 millimeters). Finally, the crush measurement locations were extrapolated to zero crush, since those locations were not far beyond the ends of the reported crush.

Kinetic energy analysis was performed on a second spread sheet in the workbook, shown here in Figure 8.

Appendix B. Constructing a Constraint Condition for Force Saturation

At or below the severity of the no-damage test, there is obviously no force saturation, and thus no reason to assume any non-linearity in the force-deflection relation. Therefore in that region, the Campbell analysis applies. By definition, all crush measurements are zero, and because of the uniformity, the form factor [beta] is 1.000. The average crush [bar.C] is 0.00. Thus we can write:

[mathematical expression not reproducible] (30)

If L is in inches, A in pounds per inch, and B in pounds per square inch, then the above expression yields crush energy in inch-pounds.

On the other hand, momentum and energy conservation give us Equation (16). If [m.sub.EFF] is in slugs (pound-seconds squared per foot), and [V.sub.CL] is in feet per second, then CE is in foot-pounds (since [epsilon] is dimensionless). Equating the two expressions for CE in inch-pounds at the no-damage threshold, we have:

CE (in - lb) = [L[A.sub.2]/2B] = 12/2 [m.sub.EFF] (1 - [[epsilon].sub.0.sup.2]) [V.sub.cl.sup.2], (31)

or

B = [L[A.sub.2]/12[m.sub.EFF] (1 - [[epsilon].sub.0.sup.2]) [V.sub.cl.sup.2]]. (32)

where the subscript 0 denotes the no-damage condition. If A and B are being varied iteratively in order to satisfy energy conservation at the actual test speed, then Equation (32) can be considered a constraint imposed during the iterations by specifying the no-damage threshold [DELTA]V. For the particular test under consideration, the remaining factors are invariant during the iteration process. Therefore we can define a constant [gamma] as:

[gamma] = [L/12[m.sub.EFF] (1 - [[epsilon].sub.0.sup.2]) [V.sub.cl.sup.2]] (33)

and write the constraint equation as:

B = [gamma][A.sup.2] (34)

This equation is applied simultaneously while forcing the crush energy calculation for the crash test to match the requirement of energy conservation. Since spread sheet analysis is in use, it is natural to look for multivariate search tools in a commercial program (Microsoft Excel[R], in this case). Among Excel's add-in programs is a routine called Solver. Its functionality is like an extension of Goal Seek, except that the utility may not be present on the as-installed Excel menu. To rectify this situation, Excel Help may be invoked for instructions on how to install Solver. Once installed, the Solver command can be found in the Analysis group on the Data tab.

A typical implementation of the constraint condition with Solver is shown in Figure 9. The vehicle crush energy "CEveh" is a named variable in the spread sheet, as are "A", "B", and "gamma"; otherwise, cell row and column references would have to be used in the dialog box. "CEveh" is the crush energy resulting from applying A, B, and [C.sub.S] to the vehicle crush profile. The target value of the vehicle crush energy (83030) is obtained from the results of the energy conservation calculations, and entered manually. Upon clicking on "Solve", Solver seeks a solution and then informs the user whether one has been found.

Appendix C. More Detailed Sample Results

Table 2. Further detail on 2015 vehicles. Stiffnesses of 2015 Vehicles Make/Model Model DOT MY Vehicle run No. Test Ford F-series p/u Ford F-150 2015- 7352 2011 F-150 SuperCrew RWD Ford F-250 2008- 9474 2016 F-250SuperCab RWD 9545 2016 F-250 SuperCrew RWD Chevrolet Silverado p/u 2014- Ram p/u Toyota Camry 2007- 9001 2015 Camry Toyota Corolla/Matrix 2014- 8409 2014 Corolla Honda Accord 2013- 8033 2013 Accord 4dr Honda CR-V 2007- 9052 2015 CR-V Nissan Altima 2013- 7967 2013 Altima 4dr Honda Civic 2013- 8157 2013 Civic 2dr Toyota RAV4 2013- 9074 2015 RAV4 Ford Escape 2013- 7936 2013 Escape FWD Ford Fusion 2013- 8078 2013 Fusion Hybrid 8219 2013 Fusion Energi Nissan Rogue 2014- 8546 2014 Rogue FWD Chevrolet Equinox 2010- 6789 2010 Equinox Ford Explorer 2011- 9486 2016 Explorer FWD Hyundai Elantra 2011- Chevrolet Cruze 2011- GMC Sierra p/u 2007- Jeep Cherokee Jeep Grand Cherokee 2011- 7036 2011 Grand Cherokee Laredo 4WD Jeep Cherokee 2014- 8513 2014 Cherokee FWD Ford Focus 2013- 9084 2015 Focus Stiffnesses of 2015 Vehicles Total Test Vehicle Crush Crush Profile Make/Model Energy Energy Level ft-lb ft-lb Ford F-series p/u Ford F-150 102,545 80,487 1 Ford F-250 115,517 82,810 1 109,803 79,982 1 Chevrolet Silverado p/u Ram p/u Toyota Camry 56,035 32,812 2 Toyota Corolla/Matrix 68,169 35,638 2 Honda Accord 77,978 36,028 2 Honda CR-V 90,004 37,629 2 Nissan Altima 64,509 40,936 2 Honda Civic 69,440 34,538 2 Toyota RAV4 83,881 34,890 2 Ford Escape 71,579 25,961 2 Ford Fusion 85,519 55,256 2 92,073 62,366 2 Nissan Rogue 98,209 57,086 1 Chevrolet Equinox 95,628 51,439 2 Ford Explorer 88,644 32,664 2 Hyundai Elantra Chevrolet Cruze GMC Sierra p/u Jeep Cherokee Jeep Grand Cherokee 88,067 26,641 2 Jeep Cherokee 93,882 2 Ford Focus 69,313 44,463 2 Stiffnesses of 2015 Vehicles Test Vehicle Without saturation With saturation Make/Model A B A B lb/in lb/[in.sup.2] lb/in lb/[in.sup.2] Ford F-series p/u Ford F-150 271 138 283 196 Ford F-250 312 173 330 194 320 176 334 192 Chevrolet Silverado p/u ND ND Ram p/u ND ND Toyota Camry 158 100 161 104 Toyota Corolla/Matrix 216 232 216 232 Honda Accord 212 190 212 190 Honda CR-V 220 244 220 244 Nissan Altima 167 123 172 130 Honda Civic 162 140 163 142 Toyota RAV4 227 195 227 195 Ford Escape 194 155 194 155 Ford Fusion 227 179 231 186 222 174 230 188 Nissan Rogue 371 581 371 581 Chevrolet Equinox 237 245 237 246 Ford Explorer 222 128 224 130 Hyundai Elantra ND ND Chevrolet Cruze ND ND GMC Sierra p/u See Chevrolet Silverado Jeep Cherokee Jeep Grand Cherokee 190 111 191 112 Jeep Cherokee 411 592 411 591 Ford Focus 276 326 276 326 Stiffnesses of 2015 Vehicles MDB Crush Avg Make/Model Energy Crush ft-lb in Ford F-series p/u Ford F-150 22,058 5.66 Ford F-250 32,707 6.92 29,821 6.52 Chevrolet Silverado p/u Ram p/u Toyota Camry 23,223 5.81 Toyota Corolla/Matrix 32,531 6.92 Honda Accord 41,950 7.97 Honda CR-V 52,375 8.97 Nissan Altima 23,574 5.91 Honda Civic 34,902 7.12 Toyota RAV4 48,991 8.67 Ford Escape 45,617 8.34 Ford Fusion 30,263 6.73 29,706 6.68 Nissan Rogue 41,122 7.90 Chevrolet Equinox 44,188 8.29 Ford Explorer 55,980 9.28 Hyundai Elantra Chevrolet Cruze GMC Sierra p/u Jeep Cherokee Jeep Grand Cherokee 61,426 9.73 Jeep Cherokee 42,513 8.04 Ford Focus 24,850 6.10

Appendix D. Sensitivity Analysis

Can force saturation be ignored

If there is no force saturation anywhere in the vehicle crush profile, linearity assumptions apply, and the crash plot approach produces a closed-form solution and an accurate result. In other words, force saturation can be ignored in the calculations, with no loss of accuracy. But what about ignoring saturation (i.e., using the crash plot method) when saturation is in fact present? We would expect the greatest discrepancy to occur with the highest vehicle crush. Consider, for example, the 2008 Nissan Altima (DOT 6189), for which the maximum vehicle crush was 327 millimeters (12.87 inches). A width of 1792 millimeters of the 3052 millimeter crush profile was in force saturation (i.e., the crush in that region exceeded the assumed saturation crush [C.sub.S] of 7.1 inches). The results from including force saturation: A = 161 pounds per inch; B = 160 pounds per square inch. The results from ignoring force saturation: A = 150 pounds per inch; B = 137 pounds per square inch. On the other hand, consider the 2013 Honda Civic two-door coupe (DOT 8157). Here the maximum crush was 216 millimeters (8.50 inches) - a more typical value for current automobiles. The results from including force saturation: A = 163 pounds per inch; B = 142 pounds per square inch. The results from ignoring force saturation: A = 162 pounds per inch; B = 140 pounds per square inch. In the more-or-less worst case scenario of the 2008 Nissan Altima, ignoring saturation reduced the calculated values of A and B by 7% and 14%, respectively. The reader can make his/her own trade-off choice regarding computation method.

How sensitive are the results to the value of the saturation crush CS

According to Reference [3], as the average crush in a crash plot exceeds saturation, the deviation from linearity occurs gradually (even if the entire profile becomes saturated all at once). So the actual value of [C.sub.S] cannot be determined with all that much precision from a crash plot, even when crash tests are available over a range of severities. Given the uncertainty, how much difference does the [C.sub.S] specification make?

If [C.sub.S] is greater than the maximum crush in the profile, then no saturation is present. In that case, the two methods yield the same result; namely, the value produced by the crash plot method (which is independent of [C.sub.S]). For the 2008 Nissan Altima (DOT 6189), the maximum crush was 327 millimeters (12.87 inches). Therefore, if [C.sub.S] [greater than or equal to] 12.87 inches, then A = 150 pounds per inch; B = 137 pounds per square inch either way they are calculated.

On the other hand, suppose [C.sub.S] for this vehicle is not 7.1 inches, but increased 40%, to 9.9 inches. Again, the crash plot method does not account for it, so there are no changes in the calculations. Accounting for force saturation, however, changes the calculated A and B values to 152 pounds per inch and 141 pounds per square inch, respectively, or decreases them by 6% and 12%, respectively. On the other hand, if [C.sub.S] is decreased 40%, from 7.1 to 4.3 inches, the amount of the crush profile in saturation changes from 1792 millimeters to 1986 millimeters (+11%). Accounting for force saturation changes the A and B calculations to 186 pounds per inch and 213 pounds per square inch--increases of 16% and 33%, respectively.

In summary, large changes in assumed value of [C.sub.S] produce relatively smaller changes in the A and B calculations, but the magnitude of the effect depends on the circumstances.

What is the effect of not adding transition data points to the crush profile

When the crush profile transitions between saturation and non-saturation, an additional data point is inserted by interpolation at each transition, so that the shape of the profile is left unchanged but all the crush in any given segment is either saturated or not. This is done because different formulas are used for the two situations. If these transition points are not inserted, some error can be expected in the crush energy calculation. How much?

One can expect the answer to depend on the average length of the segments being used. With about 17 segments, the effect should be smaller than if only 5 were being used. In the case of the 2008 Nissan Altima, leaving out the transition points changes the calculated vehicle crush energy from 70,520 ft-lb (which is the target amount) to 68,592 ft-lb. To rebalance the energy while maintaining the no-damage threshold, A and B must be changed from 161 to 164 pounds per inch, and from 160 to 165 pounds per square inch, respectively. These are small effects.

What is the effect of the no-damage restitution coefficient [[epsilon].sub.0]

Since the no-damage restitution coefficient [[epsilon].sub.0] shows up in the energy calculations in the form (1-[[epsilon].sub.0.sup.2]), and [[epsilon].sub.0] is small compared to unity, the effect of changing [[epsilon].sub.0] can be expected to be very small. Again in the case of the 2008 Nissan Altima, consider changing [[epsilon].sub.0] by 75%, from 0.20 to 0.35. Accounting for force saturation, the 75% change in [[epsilon].sub.0] results in A decreasing by 4%, from 161 to 155 pounds per inch. At the same time, B increases by 0.6%, from 160 to 161 pounds per square inch. The crash plot method (ignoring force saturation) produces similar results. The A and B values change from 150 to 144 pounds per inch (-4%), and from 137 to 138 pounds per square inch (+0.7%), respectively. Indeed, the proposed change in [[epsilon].sub.0] makes very little difference in the results.

What is the effect of increasing the crush width (by extrapolating the profile to zero crush)

Often in side impacts, one can see the crush measurements tending toward zero as the end of the profile is approached. If the crush never reaches zero, it may be that the actual ends of the crush were not documented. This may well be due to the fact that testing agencies mark the measurement points before the tests, when the eventual ends of the profile cannot be known, whereas in the field, the crush has already occurred by the time the investigator sees it. To encourage measurement consistency in the test facility and in the field, it is our practice to make the extrapolation in the test data, if so doing results in a crush profile zero crossing before the next measurement would have been located.

Because the length of the crush profile and the area of the crush are increased by this procedure, the overall average crush can be expected to decrease. We can expect A and B to be effected, but only slightly because the crush, by definition, is small because it is trending to zero.

For example, consider the 2016 Chrysler 300 (DOT 9497). One can see in Figure 7 that this profile has been extended on both ends. Doing so changes the profile length from 82.66 to 87.64 inches, and the calculated average crush is reduced from 6.79 to 6.44 inches. Stiffness A changes from 341 to 332 pounds per inch, and B changes from 380 to 382 pounds per square inch when force saturation is accounted for. Using the crash plot approach (ignoring force saturation) produces similar effects: A goes from 334 to 326 pounds per inch, and B goes from 366 to 368 pounds per square inch. Expectations are confirmed.

(1.) http://www.nhtsa.gov/Research/Databases+and+Software/Signal+Analysis+Software+for+Windows

John D. Struble

Exponent, Inc.

Donald E. Struble

Top 20 U.S. Vehicle Sales: 2015 Rank Make/Model YTD Sales thru DOT My Nov (*) No. Tested 1 Ford F-series p/u 6,95,143 7352 2011 9474 2016 9545 2016 2 Chevrolet Silverado p/u 5,37,552 3 Ram p/u 4,07,981 4 Toyota Camry 3,92,056 9001 2015 5 Toycta Corolla/Matrix 3,30,837 8409 2014 6 Honda Accord 3,20,501 8033 2013 7 Honda CR-V 3,14,462 9052 2015 8 Nissan- Altima 3,03,936 7967 2013 9 Honda Civic 3,02,588 8157 2013 10 Htc wta f!AV4 2,33,546 9074 2015 11 Ford Escape 2,78,538 7936 2013 12 Ford Fusion 2,74,594 8078 2013 8219 2013 13 Nissan Rogue 2,60,711 8546 2014 14 Chevrolet Equinox 2,55,762 6789 2010 15 Ford Explorer 2,27,584 9486 2015 16 Hyundai Elantra 2,27,464 17 Chevrolet Cruze 2,09,753 18 GMC Sierra p/u 1,96,701 19 Jeep Cherokee 1,96,211 Jeep Grand Cherokee 7036 2011 Jeep Cherokee 8513 2014 20 Ford Focus 1,91,473 9084 2015 Top 20 U.S. Vehicle Sales: 2015 Rank Vehicle A B lb/in lb/[in.sup.2] 1 F-150 SuperCrew RWD 233 196 F-250 SuperCab RWD 330 194 F-2S0 SuperCrew RWD 334 192 2 ND 3 ND 4 Camry 161 104 5 Carolla 216 232 6 Accord Adr 212 190 7 CR-V 220 244 8 Altima Adr 172 130 9 Ehfe 2dr 163 142 10 RAV4 227 195 11 Escape FWD 194 155 12 Fusion Hybrid 231 186 Fusion Energi 230 188 13 Rogue FWD 371 581 14 Equinox 237 246 15 Explorer FWD 224 130 16 ND 17 ND 18 See Chevrolet Silverado 19 Grand Cherokee Laredo 4WD 191 112 Cherokee FWD 411 591 20 Focus 276 326 (*) Source: www.motorintelligence.com, reported in Wall Street Journal Online, wsj.com, 1/5/2016

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Author: | Struble, John D.; Struble, Donald E. |
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Publication: | SAE International Journal of Transportation Safety |

Article Type: | Report |

Date: | Apr 1, 2017 |

Words: | 12476 |

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