Closed form implementation of an N-point unequally spaced approximation of vehicle shortening...

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International Journal of Impact Engineering 31 (2005) 1235–1252

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Closed form implementation of an N-point unequally spacedapproximation of vehicle shortening applied to various

force-shortening models

J. Singh�

Biomechanical Engineering Analysis & Research, Inc., 2060-D Avenida De Los Arboles, No. 487, Thousand Oaks,

California 91362, USA

Received 18 September 2003; received in revised form 20 July 2004; accepted 22 July 2004

Available online 6 October 2004

Abstract

Force-shortening models have been ubiquitously utilized for the quantification of dissipated energy formotor vehicles involved in real world collisions. Inherent in the use of force-shortening models is thenecessity for modeling of the shortening geometry. The methodologies presented in the subject studyincrease the utility of force-shortening models beyond the traditional three, four or six-point equally spacedapproximations and the recently published N-point equally spaced approximation of a continuum motorvehicle shortening profile. Closed form solutions for the four most commonly utilized force-shorteningmodels incorporating an N-point unequally spaced linearly interpolated model of the shortening geometryare provided. The specific solutions for the three, four and six point models are also presented.r 2004 Elsevier Ltd. All rights reserved.

Keywords: Crashworthiness; Collision reconstruction; Interpolation; Force-shortening models

1. Introduction

Structural modeling methodologies combined with the appropriate numerical methods play acritical role in the quantitative assessment of the crashworthiness of motor vehicles under both

see front matter r 2004 Elsevier Ltd. All rights reserved.

ijimpeng.2004.07.014

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ress: [email protected] (J. Singh).

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J. Singh / International Journal of Impact Engineering 31 (2005) 1235–12521236

research and field settings. The choice of the appropriate vehicle model is incumbent upon theavailability of sufficient data for characterization of the model parameters. Three commonlyused methods of analyzing vehicle crashworthiness can thus be viewed in light of thisrequirement. The finite element method (FEM) is the most complex of such modelingmethodologies. Concomitant with the complexity of FEM models is the necessity of detailedquantification of vehicle part geometries, constituent part material characteristics andconstituent part interconnectivity [1–3]. The use of multibody models in which a vehicle inquestion is modeled by a number of positive valued mass elements, with defined connectivity,represents a second modeling methodology [4]. The structural impact simulation and modelextraction (SISAME) program developed by the National Highway Transportation SafetyAdministration (NHTSA) is an example of the implementation of this method [5,6]. The lastmethod, which is the simplest in terms of formulation and parameter characterization, is theplanar model in which the vehicle structure is homogenized, the geometry is reduced to themaximum transverse section area and the structural response is characterized by a force-shortening relationship instead of a force-displacement relationship. Both the FEM and themultibody models are well suited for use in the research setting in which the not insignificantrequirements for parameter characterization can be readily achieved. The information availablefor field applications (i.e. the engineering reconstruction of a motor vehicle collision) is oftenlimited to the post-collision damage present on the involved vehicles. This dearth of informationcoupled with the time, cost and computational restraints involved in the evaluation of fieldcollisions enforces the usage of the simplest crashworthiness models. An example of the use ofsuch modeling methodologies is manifested in the quantitative information provided from fieldinvestigations of collisions by the NHTSA Crash Injury Research and Engineering Network(CIREN).

2. Objectives

The objective of this paper is to derive a closed form solution for an N-point unequally spacedlinearly interpolated approximation to a continuum shortening profile and to apply this solutionto the most commonly utilized force-shortening models.

3. Force-shortening models

A force-shortening analysis requires a discretization of the continuum shortening profile bydetermining the depth of shortening present at certain locations (nodal values) across theshortening width. The regions of the continuum between the nodal values have historically beenassumed to be linear and thus for all current force-shortening models are approximated by linearinterpolation functions. The three-dimensional vehicle structure is approximated in two-dimensional space. The variations in the shortening depth at nodal values as a function of theheight of the vehicle are accounted for depending on the location of the damage [7,8]. Theresultant of this process is a planar representation of the vehicle shortening profile in which theshortening depth can be defined as a function of the location along the shortening width.

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J. Singh / International Journal of Impact Engineering 31 (2005) 1235–1252 1237

An important distinction to note is the difference between shortening and deflection. Theformer represents a static value that is present following both the loading phase during forceapplication and the unloading phase during structural rebound. The latter refers to the dynamicparameter that occurs during the loading phase of force application. With the accounting ofrestitution, the dynamic deflection will always be greater than the static shortening.Following the quantification of the shortening profile, the appropriate force-shortening model

can be applied to determine the energy dissipated in shortening causation. For all of the currentlyutilized force-shortening models, the vehicle structure (front, side and rear) is modeled as beinghomogeneous. This homogeneity is either present globally across the full shortening width orpresent locally across a given division of the shortening width [9]. There are currently five majorforce-shortening models that have been presented in the literature. The constant stiffness per unitshortening width model [10–12], which is the force-shortening model incorporated into theCRASH3 [13–15] damage algorithm, was the first such model developed. A review of this modelhas been provided by Day and Hargens [16]. The saturation force model [17,18] followed in whichthe force-shortening response was modeled by a region of constant stiffness per unit width ofshortening up to a critical value of shortening followed by a region of constant force per unitwidth of shortening. The same authors proposed a constant force model in which the forcenormalized per unit shortening width remained constant regardless of the shortening depth. Theconstant force model, being overtly simplistic in its formulation and somewhat unrealistic in theassumption of a constant force development for all depths of shortening beyond the criticalshortening value, will not be covered further. The bilinear constant stiffness model [18–20]modeled the force-shortening response by a region of constant stiffness per unit width ofshortening up to a critical value of shortening followed by a second region of differing constantstiffness per unit width of shortening. The differing stiffness regimes were again separated by acritical value of shortening. Woolley [21,22] has recently suggested the use of a nonlinear powerlaw model. Implemented examples for each model, utilizing an equally spaced linearlyinterpolated approximation to the continuum shortening profile, can be found in the citedreferences.

3.1. Constant stiffness model

Campbell [9] and Mason and Whitcomb [11] separately hypothesized that the shorteningpresent to a vehicle in a full-engagement frontal impact with a fixed, rigid massive barrier waslinearly related to the velocity of the vehicle at impact. The impact speed into the barrier is nowreferred to as the barrier equivalent velocity (BEV) or the equivalent barrier speed (EBS), [20,23].The slope-intercept form of this relationship is given in Eq. (1) where b0 is the damage offsetspeed, b1 is either the linear curve fit parameter for the relationship or the system frequency for thesingle degree of freedom lumped mass-linear spring vehicle model [24,25] and c[l] is the shorteningdepth defined explicitly as a function of the location, l, along the shortening width.

BEV ¼ b0 þ b1c½l�: (1)

Campbell [12] subsequently indicated that the force per unit width of shortening required forthe development of a certain valuation of shortening was linearly related to the shortening

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F/l

c[l]

A

B

G

Fig. 1. Force-shortening relationship for the constant stiffness model. The term A represents the force applied to the

vehicle structure per unit width of shortening. The term B represents the slope of the linear relationship between the

force applied to the vehicle structure per unit width of shortening and the shortening depth. The term G represents the

energy dissipated prior to onset of shortening.


present. This relationship, using the terminology of CRASH3, is given by Eq. (2).

F

l¼ A þ Bc½l�: (2)

In Eq. (2), the term F refers to the applied force, l refers to the shortening width, A is the modelparameter measuring the shortening resistance with units of force/length and B is the modelparameter representing the slope of the force per unit shortening width versus shorteningrelationship with units of force/length2. The force-shortening relationship for the constantstiffness model is depicted in Fig. 1.The coefficients from Eqs. (1) and (2) can be related to each other by conservation of energy.

The change in kinetic energy associated with a barrier impact at a given BEV can be equated tothe work done on the vehicle by the barrier by integrating Eq. (2) over the shortening depth andthe shortening width [26]. The conservation of energy can be expressed mathematically as shownin Eq. (3).

W ¼ DE ¼1

2mðBEVÞ2 ¼

Z L

0

Z C

0

ðA þ Bc½l�Þdcdl: (3)

Performing the integration operation denoted by the inner integral (over the shortening depth)results in Eq. (4).

E ¼

Z L

0

Ac½l� þBc½l�2

2þ G

� �dl: (4)

The ‘‘G’’ term in Eq. (4) is a constant that represents the energy dissipated prior to the onset ofshortening. The CRASH3 stiffness coefficients A and B can in turn be related to the Prasadcoefficients d0 and d1 [27–29] as described by McHenry and McHenry [30].

3.2. Saturation force model

The saturation force model consists of a constant stiffness region extending from zeroshortening to some critical value of shortening, ccrit[l], where the force-shortening response

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F/l

c[l]

A B

G

ccrit

[F/l]crit

Fig. 2. Force-shortening relationship for the saturation force model. The terms A, B and G have the same meaning as

for the constant stiffness model. This force shortening model provides for a constant value of applied force per unit

width of shortening, [F/l]crit, for shortening values equal to or exceeding a critical shortening depth, ccrit.


‘‘saturates’’ and transitions to a constant force response. A single equation closed-form solutionfor this response can be written with the use of the indicator function, which has a unity valueover the specified region and is zero otherwise. The closed form solution of the force-shorteningrelationship for the saturation force model is shown in Eq. (5).

F

l¼ ðA þ Bc½l�ÞI 0;ccrit½ �ðc½l�Þ þ

F

l

� �crit

I ½ccrit;cmax½l��ðc½l�Þ: (5)

The force-shortening relationship for the saturation force model is depicted in Fig. 2.Because of the continuity condition between the constant stiffness regime and the saturation

force regime at the critical value of shortening, ccrit, the term (F/l)crit in Eq. (5) can be solved forusing the constant stiffness regime of the saturation force relationship.

F

l

� �crit

¼ A þ Bccrit: (6)

Substitution of Eq. (6) into Eq. (5) followed by integration over the appropriate limits as definedby the indicator function in Eq. (5) results in a relationship defining the energy dissipated as afunction of the force-shortening parameters and the shortening parameters.

E ¼

Z L

0

Z ccrit

0

ðA þ Bc½l�Þdcdl þ

Z L

0

Z cmax½l�

ccrit

ðA þ BccritÞdcdl; (7a)

E ¼

Z L

0

Acmax½l� þ Bccritcmax½l� �Bc2crit2

þ G

� �dl: (7b)

3.3. Bilinear constant stiffness model

The bilinear constant stiffness model consists of two regions of differing constant stiffnessseparated by a critical value of shortening. The force per unit width of shortening—shortening

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relationship for this model is defined by Eq. (8).

F

l¼ ðA þ Bc½l�ÞI ½0;ccrit�ðc½l�Þ þ A� þ B�c½l�ð ÞI ½ccrit;cmax½l��ðc½l�Þ; (8)

Where A and B are the coefficients corresponding to the first constant stiffness region of the force-shortening model for shortening between 0 and the critical value of shortening and A* and B* arethe coefficients corresponding to the second constant stiffness region of the force-shorteningmodel for shortening between the critical value of shortening and a maximal value of shorteningthat is greater than the critical shortening value. The force-shortening relationship is depictedin Fig. 3.For shortening values less than the critical shortening value, Eq. (8) simplifies to the

constant stiffness model. For shortening values exceeding the critical shortening value, inte-gration of Eq. (8) over the appropriate limits as defined by the indicator function is given byEq. (9).

E ¼

Z L

0

Accrit þBc2crit2

þ G

� �dl þ

Z L

0

A�cmax½l� þB�cmax½l�

2

2

� �dl

�

Z L

0

A�ccrit þB�c2crit

2

� �dl; ð9Þ

Eq. (9) can be expanded and subsequently simplified to arrive at Eq. (10).

E ¼

Z L

0

ðA � A�Þccrit þðB � B�Þ

2c2crit þ G

� �dl þ

Z L

0

A�cmax½l� þB�

2cmax½l�

2

� �dl: (10)

F/l

c[l]

A B

G

ccrit

[F/l]crit

A*

B*

Fig. 3. Force-shortening relationship for the bilinear constant stiffness model. The terms A, B and G have the same

meaning as for the constant stiffness model. This model provides for a change in the relationship between the applied

force per unit width of shortening and the shortening depth for shortening depths exceeding a critical value, ccrit. The

force-shortening regime for shortening values exceeding ccrit is characterized by the parameters A* and B*, which are

the consistent, from a definitional perspective, with A and B respectively. It should be noted that the relationship

B*oB, as shown, is not ubiquitously true.

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3.4. Power law model

The power law model defines the force normalized per unit shortening width in terms of areference force per unit width, f0, a reference stiffness coefficient, k0, the shortening depth as afunction of position along the shortening width, c[l], a vehicle shortening offset, X0 and the power,n, which usually is valued between 0 and unity. This relationship is shown mathematically inEq. (11).

F

l¼ f 0

k0ðc½l� þ X 0Þ

f 0

� �n

¼ f 1�n0 kn

0ðc½l� þ X 0Þn: (11)

The force-shortening relationship is depicted in Fig. 4.The energy dissipated in shortening causation is given by Eq. (11) which then simplifies to

Eq. (12b).

E ¼

Z L

0

Z c½l�

0

f 1�n0 kn

0ðc½l� þ X 0Þn dcdl; (12a)

E ¼

Z L

0

f 1�n0 kn

0ðc½l� þ X 0Þnþ1

n þ 1dl: (12b)

4. Interpolation methodology

The solution for the two, four and six point equally spaced linearly interpolated approximationto the continuum shortening profile applied to the constant stiffness model are ubiquitously foundin the accident reconstruction engineering literature [31]. Singh et al. [32] have developed a threepoint equally spaced linearly interpolated approximation to the continuum shortening profile,which has particular application in the derivation of older NHTSA vehicle front to barrier fullengagement tests in which shortening data is provided at the left end, centerline and right end ofthe test vehicle. Subsequently, Singh et al. [33] developed a closed form solution for modeling thecontinuum shortening profile with N equally spaced nodal values with the internodal regions

F/l

c[l]

n = 0

n = 1

0 < n < 1

X0

f0

Fig. 4. Force-shortening relationship for the power law model. The shortening offset is denoted by X0. The reference

force per unit width of shortening, f0, can be determined by setting the power equal to zero. The power law model

reduces to the constant stiffness model when the power is equal to unity.

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being fitted by linear interpolation. This general solution is given as Eq. (13).

c½l� ¼XN�1

i¼1

ci þ ðciþ1 � ciÞðN � 1Þl

L� ði � 1Þ

� �� I ði�1ÞL

N�1; iLN�1

� �½l�: (13)

In Eq. (13), the terms ci and ci+1 represent successive nodal values of shortening (i.e. shorteningdepths at successive shortening indices), N is the total number of approximation points, l is thevariable denoting the position along the shortening width and L is the total width of shortening.A similar closed-form relationship can be derived for any N number of unequally spaced nodal

shortening values with the internodal regions being fitted piecewise with linear interpolationfunctions. The closed form solution for the N-point unequally spaced linearly interpolatedshortening profile is given by Eq. (14) subject to the condition that l1=0.

cðlÞ ¼XN�1

i¼1

ci þ ðciþ1 � ciÞl � li

liþ1 � li

� �� I ½li ;liþ1�ðlÞ: (14)

5. Application to force-shortening models

5.1. Constant stiffness model

The energy dissipated in shortening causation, as indicated by the constant stiffness modelutilized by CRASH3, can be determined by the substitution of Eq. (14) into Eq. (4).

E ¼XN�1

i¼1

Z liþ1

li

A ci þ ðciþ1 � ciÞl � li

liþ1 � li

� �� þ

B ci þ ðciþ1 � ciÞl�li

liþ1�li

22

þ G

0B@

1CAdl

264

375:(15)

The simplification of Eq. (15) is facilitated by noting that the CRASH3 A stiffness coefficientterm is a constant multiplied by the integration of the dependant variable, l and the CRASH3 Bstiffness coefficient term is a constant multiplied by the integration of the square of the dependantvariable, l. The final form for the energy dissipated equation thus becomes:

E ¼XN�1

i¼1

A

2ðciþ1 � ciÞðliþ1 � liÞ þ

B

6c2i þ ci ciþ1 þ c2iþ1

� �ðliþ1 � liÞ þ Gðliþ1 � liÞ

� �: (16)

Eq. (16) can be written for any number of interpolation points. Eq. (17a)–(17c) list theseequations for the three, four and six interpolation points.

E3-Point ¼A

2ðc1l2 þ c2l3 þ c3ðl3 � l2ÞÞ

þB

6ðc21l2 þ c22l3 þ c23ðl3 � l2Þ þ c1c2l2 þ c2c3ðl3 � l2ÞÞ þ G � l3 ð17aÞ

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E4-Point ¼A

2ðc1l2 þ c2l3 þ c3ðl4 � l2Þ þ c4ðl4 � l3ÞÞ

þB

6

c21l2 þ c22l3 þ c23ðl4 � l2Þ þ c24ðl4 � l3Þ

þc1c2l2 þ c2c3ðl3 � l2Þ þ c3c4ðl4 � l3Þ

!

þ Gl4 ð17bÞ

E6-Point ¼A

2ðc1l2 þ c2l3 þ c3ðl4 � l2Þ þ c4ðl5 � l3Þ þ c5ðl6 � l4Þ þ c6ðl6 � l5ÞÞ

þB

6

c21l2 þ c22l3 þ c23ðl4 � l2Þ þ c24ðl5 � l3Þ þ c25ðl6 � l4Þ þ c26ðl6 � l5Þ

c1c2l2 þ c2c3ðl3 � l2Þ þ c3c4ðl4 � l3Þ þ c4c5ðl5 � l4Þ þ c5c6ðl6 � l5Þ

!

þ Gl6: ð17cÞ

5.2. Saturation force model

The energy dissipated according to the saturation force model can be determined bysubstitution of Eq. (14) into Eq. (7b).

E ¼XN�1

i¼1

Z liþ1

li

A ci þ ðciþ1 � ciÞl�li

liþ1�li

þBccrit ci þ ðciþ1 � ciÞ

l�li

liþ1�li

�

Bc2crit

2þ G

0B@

1CAdl: (18)

Simplification of Eq. (18) results in Eq. (19).

E ¼XN�1

i¼1

A2ðci þ ciþ1Þðliþ1 � liÞ

þ B2

ccritðci þ ciþ1Þðliþ1 � liÞ �B2

c2critðliþ1 � liÞ þ Gðliþ1 � liÞ

" #: (19)

The solution for the three, four and six point approximations are given as Eqs. (20a)–(20c).

E3-Point ¼A

2ðc1l2 þ c2l3 þ c3ðl3 � l2ÞÞ

þB

2ccritðc1l2 þ c2l3 þ c3ðl3 � l2ÞÞ �

B

2c2critl3 þ Gl3 ð20aÞ

E4-Point ¼A

2ðc1l2 þ c2l3 þ c3ðl4 � l2Þ þ c4ðl4 � l3ÞÞ

þB

2ccritððc1l2 þ c2l3 þ c3ðl4 � l2Þ þ c4ðl4 � l3ÞÞ �

B

2c2critl4 þ Gl4 ð20bÞ

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E6-Point ¼A

2ððc1l2 þ c2l3 þ c3ðl4 � l2Þ þ c4ðl5 � l3Þ þ c5ðl6 � l4Þ þ c6ðl6 � l5ÞÞ

þB

2ccritððc1l2 þ c2l3 þ c3ðl4 � l2Þ þ c4ðl5 � l3Þ

þ c5ðl6 � l4Þ þ c6ðl6 � l5ÞÞ �B

2c2critl6 þ Gl6: ð20cÞ

5.3. Bilinear constant stiffness model

The energy dissipated according to the bilinear constant stiffness model can be determined bysubstitution of Eq. (14) into Eq. (10)

E ¼XN�1

i¼1

R L

0 ðA � A�Þccrit þðB�B�Þ

2c2crit þ G

dl

þR L

0 A� ci þ ðciþ1 � ciÞl�li

liþ1�li

h iþ B�

2ci þ ðciþ1 � ciÞ

l�li

liþ1�li

h i2� �dl

2664

3775: (21)


E ¼XN�1

i¼1

Accrit þB

2c2crit þ G

� �ðliþ1 � liÞ

þA�

2ðci þ ciþ1 � 2ccritÞ þ

B�

6ðc2i þ ciciþ1 þ c2iþ1 � 3c2critÞ

� �ðliþ1 � liÞ

26664

37775: (22)


E3-Point ¼ Accrit þB

2c2crit þ G

� �l3 þ

A�

2ððc1 � c3Þl2 þ ðc2 þ c3 � 2ccritÞl3Þ

þB�

6ððc21 þ c1c2 � c2c3 � c23Þl2 þ ðc22 þ c2c3 þ c23 � 3c2critÞl3Þ: ð23aÞ


2c2crit þ G

� �l4 þ

A�

2ððc1 � c3Þl2 þ ðc2 � c4Þl3

þ ðc3 þ c4 � 2ccritÞl4Þ þB�

6ððc21 þ c1c2 � c2c3 � c23Þl2

þ ðc22 þ c2c3 � c3c4 � c24Þl3 þ ðc23 þ c3c4 � c4c5 � 3c2critÞl4Þ ð23bÞ


2c2crit þ G

� �l6

þA�

2ðc1 � c3Þl2 þ ðc2 � c4Þl3 þ ðc3 � c5Þl4 þ ðc4 � c6Þl5 þ ðc5 þ c6 � 2ccritÞl6Þ

þB�

6

ðc21 þ c1c2 � c2c3 � c23Þl2 þ ðc22 þ c2c3 � c3c4 � c24Þl3 þ ðc23 þ c3c4 � c4c5 � c25Þl4

þðc24 þ c4c5 � c5c6 � c26Þl5 þ ðc25 þ c5c6 þ c26 � 3c2critÞl6

!

ð23cÞ

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5.4. Power law model

The energy dissipated according to the power law model can be determined by substitution ofEq. (14) into Eq. (12b).

E ¼XN�1

i¼1

Z liþ1

li

f 1�n0 kn

0 ci þ ðciþ1 � ciÞl�li

liþ1�li

þ X 0

nþ1

n þ 1dl

264

375: (24)


E ¼XN�1

i¼1

ðliþ1 � liÞf1�n0 kn

0ððciþ1 þ X 0Þ2þn

� ðci þ X 0Þ2þn

Þ

ðciþ1 � ciÞð1þ nÞð2þ nÞ

� �: (25)


E3-Point ¼f 1�n0 k0

ð1þ nÞð2þ nÞ

ðl2Þððc2þX 0Þ2þn

�ðc1þX 0Þ2þn

Þ

c2�c1ð Þ

þðl3�l2Þððc3þX 0Þ

2þn�ðc2þX 0Þ

2þnÞ

ðc3�c2Þ

24

35; (26a)


ð1þ nÞð2þ nÞ


�ðc1þX 0Þ2þn

Þ

ðc2�c1Þ


2þn�ðc2þX 0Þ

2þnÞ

ðc3�c2Þ


2þn�ðc3þX 0Þ

2þnÞ

ðc4�c3Þ

26664

37775; (26b)


ð1þ nÞð2þ nÞ


�ðc1þX 0Þ2þn

Þ

ðc2�c1Þ


2þn�ðc2þX 0Þ

2þnÞ

ðc3�c2Þ


2þn�ðc3þX 0Þ

2þnÞ

ðc4�c3Þ


2þn�ðc4þX 0Þ

2þnÞ

ðc5�c4Þ


2þn�ðc5þX 0Þ

2þnÞ

ðc6�c5Þ

266666666664

377777777775: (26c)

6. Implemented example

United States (US) NHTSA Federal Motor Vehicle Safety Standard (FMVSS) 214Dcompliance testing [34] provides for a source of data for deriving vehicle specific constantstiffness model parameters for use in field studies [35]. In short, this methodology utilizes theknown quantities of test vehicle and moving deformable barrier (MDB) pre-collision dimensions,post-collision shortening profiles, inertial properties, a priori MDB constant stiffness modelparameters and specified MDB crab angle in conjunction with Newton’s Third Law of Motion todetermine the test vehicle constant stiffness model parameters. The total force applied to theMDB per unit width of shortening is determined by summation of applied force per zone (region

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between adjacent nodes of measured or interpolated shortening) of direct damage, which in turn isdetermined by the implementation of the trapezoidal approximation of Eq. (2). By equating themagnitude of the force applied to the MDB to the magnitude of the force applied to the testvehicle, the constant stiffness parameters for the side of the test vehicle can be determined. Thisformulation works under the assumption of constrained shortening along the lateral axis of thetest vehicle, does not account for the induced damage region on the test vehicle, does not includetransverse or interzonal (i.e. interelement) shear and does not account for tire traction during thecollision. Furthermore, the complexities associated with curved vehicle geometries are mitigatedby assuming linear approximations thereof. The width of the shortening profile of the MDB is1600mm, which is symmetrically positioned about the centerline with the shortening measuredevery 100mm. This, in turn, leaves 38mm on the lateralmost aspect of both the right and leftbarrier face over which the shortening is not specified. Shortening on the test vehicle is measuredevery 150mm. Full inclusion of the shortening width of the MDB requires the use of 11 measuredshortening depths across the direct damage region of the test vehicle and one additionalmeasurement of shortening at the closest location aft of this region (i.e. the shortening depth onthe test vehicle corresponding to 1600mm in the global coordinate system is interpolated from themeasured shortening depths at 1500 and 1650mm). A preliminary investigation on the effects offineness of the interpolation mesh on calculated constant stiffness coefficients has been conductedby the author [36].The subject example involves the evaluation of NHTSA test number v3341—the FMVSS214D

compliance test evaluation for a 2000 Ford Focus 3-door hatchback [37]. In this test, the leftside of a 1396.5kg test vehicle was struck by a 1362.5 kg MDB traveling at 14.8m/s with a leftcrab angle of 271. The total shortening profiles of both the test vehicle and the MDB are shownin Fig. 5.In that the location of measured shortening of the test vehicle and the MDB in the global

coordinate system are not coincident, one can either use the shortening regime of the MDB anddetermine the shortening at the corresponding coincident locations on the test vehicle byinterpolating between the two closest locations of measured shortening that bracket the desiredlocations on the same or start with the measured shortening on the test vehicle and determine theshortening at the coincident locations on the MDB. The former method is employed here. Theresultant MDB and corresponding test vehicle shortening profiles using the reported shortening, afour point unequally spaced and a six point equally spaced approximation are shown in Figs. 6and 7 respectively. The six point approximation for the equally spaced scheme is shown in that itrepresents the most commonly used method in current practice.It should be noted that the six point equally spaced approximation scheme, as with most of the

equally spaced approximation schemes (2pNp17) requires the determination of shortening at thedesired interior locations by means of position on the linear interpolation function between thetwo closest locations where the shortening was measured that bracket the desired location. This isnot a requirement when using an unequally spaced scheme in that the nodes can be assigned tolocations on the MDB where the shortening is actually measured.The constant stiffness model coefficients determined for the subject four node unequally spaced

approximation scheme can be compared to those based on the standard N point equally spacedapproximation scheme (2pNp17). For all iterations, the MDB constant stiffness coefficientswere taken as AMDB=62.52N/mm and BMDB=1.18 N/mm2 and the damage onset speed for the

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-150

-100

-50

0

50

100

150

200

250

300

350

-1500 -1000 -500 0 500 1000 1500 2000 2500 3000 3500

Longitudinal distance [mm]

Sho

rten

ing

[mm

]

Vehicle

Barrier

Fig. 5. Shortening profile for the test vehicle and MDB. The longitudinal distance represents the global coordinate

system longitudinal axis, which is coincident with the test vehicle fixed longitudinal axis. The MDB shortening depths

are plotted as having negative valuation in order to distinguish them from the test vehicle shortening values. It should

be noted that the shortening profiles for both the test vehicle and the MDB are discrete, but are shown as being

continuous for ease of visualization.

-140

-120

-100

-80

-60

-40

-20

0

-350 150 650 1150 1650

Longitudinal distance [mm]

Shor

teni

ng [m

m]

Barrier

Barrier (N = 4, unequal)

Barrier (N = 6, equal)

Fig. 6. Shortening profile for the MDB plotted on the global coordinate system axes. The longitudinal distance denotes

the location of shortening over 1600mm of the MDB face accounting for the 38mm offset from the impact point on the

test vehicle. It should again be noted that the shortening profile for the MDB is discrete, but is shown as being

continuous for ease of visualization. The internodal regions for both the four point unequally spaced and six point

equally spaced approximations are fitted linearly.


test vehicle was 0.894 m/s [35]. The comparison between the N point equally spacedapproximation scheme and the subject four node unequally spaced approximation scheme areshown in Figs. 8 and 9.

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0

50

100

150

200

250

300

350

-350 150 650 1150 1650Longitudinal distance [mm]

Shor

teni

ng [m

m]

Vehicle

Vehicle (N=4, unequal)

Vehicle (N = 6, equal)

Fig. 7. Shortening profile for the test vehicle plotted on the global coordinate system axes. The closest location of

shortening, within the direct damage region on the test vehicle, falls at a longitudinal distance from the impact point of

1500mm. In order to include a coincident value of shortening corresponding to the shortening measured at 1638mm on

the MDB, the test vehicle shortening at 1650mm is also included. The shortening for the test vehicle is discrete but is

shown as being continuous for ease of visualization. The internodal regions for both the four point unequally spaced

and six point equally spaced approximations are fitted linearly.

14.23

17.35

13.39

12.96

12.88

12.77

12.67

12.62

12.58

12.56

12.54

12.51

12.50

12.49

12.48

12.49

(4, 13.53)

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8 10 12 14 16 18Number of equally spaced nodes (N)

Con

stan

t stif

fnes

s m

odel

A c

oeff

icie

nt

(N/m

m)

Fig. 8. Constant stiffness model ‘‘A’’ coefficient shown as a function of equally spaced node numbers (2pNp17) and

for the subject four point unequally spaced approximation scheme. The former is shown as the calculated discrete

values connected in a continuous manner for ease of visualization. The latter is shown as the singular plotted value

labeled using both components of the graphical coordinate system and delineated by the labeling arrow with leader line.


The accuracy of the constant stiffness model can be assessed by reconstructing the collision. Inthat the MDB is not rotating about its center of mass during the approach, the impact velocityvectors located at the center of mass and at the impulse center are equal. The reconstructed MDBpre-impact velocity vector magnitude and direction of the impact force on the MDB are shown inFigs. 10 and 11 respectively.

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0.30

0.44

0.26

0.25

0.24

0.24

0.24

0.23

0.23

0.23

0.23

0.23

0.23

0.23

0.23

0.23

(4, 0.26)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Con

stan

t stif

fnes

s m

odel

B c

oeff

icie

nt

(N/m

m^2

)

Fig. 9. Constant stiffness model ‘‘B’’ coefficient shown as a function of equally spaced node numbers (2pNp17) and

for the subject four point unequally spaced approximation scheme. The former is shown as the calculated discrete

values connected in a continuous manner for ease of visualization. The latter is shown as the singular plotted value

labeled using both components of the graphical coordinate system and delineated by the labeling arrow with leader line.

(4, 14.8)

13.8

14

14.2

14.4

14.6

14.8

15

0 2 4 6 8 10 12 14 16 18

Number of equally spaced nodes (N)

Mag

nitu

de o

f M

DB

Pre

-im

pact

Vel

ocity

(m

/sec

)

Fig. 10. Reconstructed magnitude of the MDB pre-impact velocity. The solid line at 14.8m/s denotes the a priori (as

determined by speed trap) magnitude of the MDB velocity vector at impact. The plotted point labeled with both

components of the graphical coordinate system is the reconstructed velocity based upon the four point unequally

spaced approximation scheme. The continuous appearing plot that is based upon the discrete point locations shown in

this figure represents the solutions for the magnitude of the MDB pre-impact velocity vector as a function of the

number of equally spaced nodes used.


7. Discussion

The unequally spaced linearly interpolated forms of the various force-shortening models aresignificantly similar to their equally spaced linearly interpolated counterparts. The majordifference in the forms of the respective equations arises from the necessity of encoding eachinternodal region differentially based on the distances between the nodal shortening values for the

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(4, 22.8)

0

5

10

15

20

25

30


Prin

cipl

e D

irec

tion

of F

orce

on

the

MD

B(d

eg)

Fig. 11. Reconstructed principle direction of force on the MDB. The solid line at 271 denotes the a priori valuation.

The plotted point labeled with both components of the graphical coordinate system is the reconstructed principle

direction of force based upon the four point unequally spaced approximation scheme. The continuous appearing plot

that is based upon the discrete point locations shown in this figure represents the solutions for the principle direction of

force on the MDB as a function of the number of equally spaced nodes used.


unequally spaced formulation whereas for the equally spaced formulation, the internodal regionsare equal and can be defined as the total shortening width divided by N�1, where N is the numberof nodal approximations utilized. The utilization of an unequally spaced shortening profile for usein the constant stiffness model has been examined previously [38]. The piecewise approximationfor the unequally spaced linearly interpolated shortening profile (Eq. (3) of Struble [38]) isconsistent with the closed form solution derived by authors as shown in Eq. (14). The piecewisesolution derived by Struble, however, for the energy dissipated in shortening causation (Eq. (4))does not appear to be correct in its incorporation of the ‘‘G’’ term. Summation over the N–1intervals, as noted by Struble, would result in a final solution of 6G instead of the 1G that isexpected.The determination of the critical shortening depth for the saturation force and the bilinear

constant stiffness models is an important issue that arises in the utilization of these models.Neptune [39] has proposed that the critical shortening depth be that which separates enginecompartment shortening from occupant compartment shortening. Therefore, for a given degree ofuniform crush, values exceeding the length of the engine compartment with the initiation ofcrushing of the occupant compartment will result in localization on the second region of stiffnessfor the force-saturation or bilinear constant stiffness models. Non-uniform crush profiles, such asthose encountered following offset impacts, can be evaluated by first determining the equivalentuniform crush [40] and comparing this value to the critical crush depth.Broad residual damage profiles with areas of variable localized deformation are relatively

common in broadside type collisions for the vehicle which is struck in the side. For increasingcrush profile widths for such scenarios, a fixed value nodal crush depth measurement scheme willresult in an increasing loss of fineness of the geometry mesh. Conversely, increasing the number ofnodes for an N-point equally spaced approximation in order to retain the fineness of the geometrymesh may result in unnecessary node placement and crush depth determination over regions that

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could be fully modeled with reduced nodes (i.e. a region of the shortening profile that is broadlylinear). The subject interpolation methodology and the elucidation of closed form force-shortening models allows for the evaluation of large width shortening profiles while limiting thediscretization over regions of relatively constant geometry. The coupling of the interpolationmethodology with the closed form force-shortening models allows for the determination ofdissipated energy, as a first order solution for determining subsequent crashworthinessparameters, for the field data subset of motor vehicle collisions in which quantifiable vehicleshortening is present.

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