longitudinal dynamics of train collisions - crash analysis

8/13/2019 Longitudinal Dynamics of Train Collisions - Crash Analysis

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7TH MINI CONF. ON VEHICLE SYSTEM DYNAMICS, IDENTIFICATION AND ANOMALIES, BUDAPEST, NOV. 6-8, 2000

89

LONGITUDINAL DYNAMICS OF TRAIN COLLISIONS

- CRASH ANALYSIS

Istvn ZOBORY*, Hans G. REIMERDES**, Elemr BKEFI*,

Jens MARSOLEK**and Istvn NMETH** Budapest University of Technology

and EconomicsDepartment of Railway Vehicles

H-1521 Budapest, Hungary

** RWTH AachenDepartment of Aerospace Structures

-Light Weight ConstructionWllnerstr. 7. D-52062 Aachen, Germany

Received: November 8, 2000

ABSTRACT

The paper takes an attempt to develop a computation method for predicting the dynamics of train collisions in case

trains consisting of vehicles equipped with crash elements to amortise the impact energy. The initial step of the method

is a time domain simulation of the train collision occurring when two non-linear train models run against each other. In

the collision the inter-vehicle connection forces will temporary exceed the yielding limit of the buffer and coupler

components. As a first approximation it is assumed that in the yielding process the transmitted compressive force

remains constant for the whole plastic contact of the impacting vehicles. This assumption makes it possible to

determine the approximate variation with time of the intervehicle buffer forces. With the knowledge of the received

time dependent force functions FEM based crash dynamical computations can be carried out to determine the time

dependent elastic/plastic deformation processes of the cylindrical-shall-form crash elements built into the vehicle

underframes just behind the front beams. As a result of the FEM computations a second approximation of the plastic-

deformation-dependent inter-vehicle force transfer conditions can be evaluated in form of a non-linear connection force

vs. plastic deformation law, which can be built into the non-linear longitudinal dynamics analysis to be carried out

again in the time domain by using the familiar simulation methods.

Keywords: train dynamics simulation, collision dynamics, crash elements, crash analysis

1. INTRODUCTION

The number of collision connected railway accidents shows world-wide an increasing

tendency year by year. The ever increasing operation velocities cause an increasing

degree of the grave consequences both in loss of human life and severe damage to the

vehicles and other railway equipment. In the technical literature very few number of

publications can be found that are dealing with investigations into the train collision

processes to predict the level of forces and deformations realising in the course of

accidental collisions/crashes. The shortage of the literature sources can be explained bythe extremely complicated character of the dynamics of train crashes. On the one hand

the existing system models in the longitudinal train dynamics remain in the elastic

region concerning the connection forces arising in the intervehicle connections, so the

treatment of the plastic deformation processes makes it unavoidable to develop ap-

propriate methods for describing the intervehicle force transfer in the plastic region, as

well. On the other hand, the crash analysis of aerospace structures has already elabor-

ated investigation models and computation processes of tolerable accuracy, that on the

basis of shell dynamics are proper to be connected with the existing non-linear models

used in the longitudinal dynamics of trains. The paper takes an attempt to develop an

iterative computation methodfor predicting the dynamics of train collisions/crashes.


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The initial step of the iterative processis a time domain analysis of the train collision

occurring when two trains run against each other. The intervehicle connection forces

will achieve the yielding limit after the exhaustion of the stroke of the buffer- and

draw gears, and as a first approximation it is assumed that in the yielding process the

force compressive transmitted remains constant for the whole contact of the impacting

vehicles. This assumption makes it possible to determine the approximate variation

with time of the intervehicle buffer forces. With the knowledge of the received time

dependent force functions it is possible to carry out FEM based crash dynamical com-

putations to determine the time dependent elastic/plastic deformation processes of the

cylindrical-shall-form crash elements built into the vehicle underframes just behind the

front beams. As a result of the FEM computations a second approximation of the plas-

tic-deformation-dependent intervehicle force transfer conditions can be evaluated forming

a non-linear connection force vs. plastic deformation law, which can be built into the

longitudinal dynamics analysis to be carried out again in the time domain. The con-nection force vs. time functions provided by the time domain analysis can be fed back

again into the FEM dynamics, and the iterative process descriptioncan be realised.

2. SYSTEM MODELS USED IN LONGITUDINAL

DYNAMICS OF TRAINS

In the traditional problems of longitudinal dynamics of trains in the majority finite degree-

of-freedom lumped parameter models are applied. Several dynamical train models

have been developed with very different degree of system decomposition. The

simplest train models do not consider the vehicles to be interconnected from the

masses of the wheelsets, the bogie-frames and the vehicle bodies, but they take the

train consisting of single mass-points concentrated in the gravity point of the vehicles

considered. The interaction between the adjacent vehicles - in the model between the

mass points introduced - is treated by special connection force diagrams. The latter

diagrams are generated by connection force vs. relative displacement and relative

velocity functions defined for each vehicle connection.

The most frequently used springing element in buffer- and draw gears is the ring-spring

construction, having piece-wise linear elastic restoring characteristics as a function of

the relative displacement and dry friction damping property, the intensity of which

depends almost exclusively also on the relative displacement.In case of problems of traditional longitudinal dynamics it is enough to consider the

individual displacements and relative velocities of the adjacent vehicles, but if train

collision is to model, also the absolute position of each vehicle should be followed in

the course of the simulations. Of course, in the latter case the relative displacements

and relative velocities of the adjacent vehicles should also be sequentially computed to

generate the connection force functions.

The structure of the familiar dynamical model with absolute vehicle position co-ordinates

is shown in Fig. 1. In the Figure the vehicle connections are indicated by parallelly

connected spring and friction damper elements, but the force transfer properties of the

elements in question are strongly non-linear. The basic properties of the mentioned con-

nection force transfer is characterised in the following Chapter.


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m n m i m 2 m 1F n-1,n F i+1, i F i-1,i F 2,3 F 1,2

vL O C O

xnx i

x2x1

0

Fig.1. Traditional lumped parameter train model for longitudinal dynamics

3. TREATMENT OF CONNECTION FORCES

BETWEEN THE ADJACENT VEHICLES

For the numerical treatment of the intervehicle connection forces one should prescribe:

the piece-wise linear elastic component Fe= Fe(x) by giving the co-ordinate

pairs of the break-points

the piece-wise linear frictional limit force Ff= Ff( x x, & ) by giving the co- ordinate pairs of the break-points for &x>0.On the basis of the specified force-relative displacement co-ordinate pairs force com-

ponent Fe(x) can be computed by linear interpolation for arbitrary x. Similarly,

with the knowledge of the co-ordinate pairs of the values of friction limit force vs.

displacement function Ff( x x, & ) under the condition &x>0 can be numerically

determined for each relative motion state of the connection. The practical numericaltreatment of Ff( x x, & ) was carried out by using the program FRICON elaborated atthe Department of Railway Vehicles of the BUTE to compute the force transfer of dry

friction connections on the basis of the principles introduced in [1]. Program FRICON

takes sideration the characteristic short distance memoryof the frictional connection.

The resultant force F transmitted is obtained by sum F = Fe(x) + Ff( x x, & ). As anexample, the extended characteristics of a single buffer gear (ring-spring with con-

siderable dry friction) for the treatment of the ideal plastic deformation of the under-

frame is shown in Fig.2. The positive x value means compression, while the positive

F value means compressive force

DEFORMATION

FORCE

F

d

d

YIELDING LIMIT

PROPORTIONALITY

0

LIMIT

dd

0

t

t

IDEAL PLASTIC DEFORMATION

TRANSMITTED

IF


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In accordance with the conditions plotted in Fig. 3, the two buffer gears are connected

in-parallel per loco, while the buffer gears of the impacting two locos come into in-

series connection.

LOCO 2 LOCO 1

in parallel

in series

Fig. 3 Connection of four buffers in case of frontal collision of two locos

4. NUMERICAL SIMULATION OF TRAIN COLLISION

BY SIMPLE MODELS

The above introduced extended buffer-force diagram developed for the collision dy-

namics of railway vehicles was tested first in case of the frontal collision process of

two identical locos of 80 t mass. The initial position and velocity conditions are shown

in Fig. 4.

m2

m1

Loco1Loco2

v2 v1s2

s1

0

Fig. 4 Two locos running against each other on straight track

The set of non-linear motion equations formulated by using the state space method

was solved numerically by the Euler's method. The computations were carried out by

methodical selection the the initial velocities of the two locos on a discrete partition of

closed interval [-20, 20] km/h. The results of the numerical simulations have been

included into 3D diagrams.

Fig. 5 Maximum forces arisen in the course of frontal collision of two locos


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In Fig. 5 the maximum forces arisen in the course of the frontal collision of the two

locos considered is shown as a bivariate function of the initial velocities.

Looking at the diagram it is obvious that for initial velocities v1>v2the two locos will

not impact, so the force of impact is identically zero. On the other hand, in case of realcollision the elasicity of the system buffer-gear underframe is quickly exhausted if

the relative velocity of the impact exceeds the limit 10 km/h. In Fig. 6 the the bivariate

function of the plastic deformation of the two impacting half-underframes is shown.

Fig.6 Ideal plastic deformation of the two impacting underframes

In Fig. 7 the bivariate diagram of the dissipated energy can be seen, also for the impact

of two locomotives. The erepresented energy values contain the friction caused dis-

sipation as well as the dissipation belonging to the ideally plastic deformation of the

half-underframes.

Fig.7 The dissipated energy in case of collision of two locos

The second test of the extended buffer-force diagram developed for the collision dy-namics of railway vehicles was carried out in case of simulation of the frontal collision


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process of two identical short trains consisting of a loco of 80t and two carriages of 50t

each. The initial position and velocity conditions of the two short trains are shown in

Fig. 8.

v2

Loco2m2

s1

v1

m 1 Loco1m 22 Car21Car23m 23 m 12 Car12 m 13 Car13

0

s2

x

Fig. 8. Two short trains running against each other on straight track

For the exact treatment of the intervehicle forces arising in the train it was necessary to

build up a bivariate connection force diagram describing the force transmitted in the

vehicle connections as a function of the relative position and relative velocity of the

adjacent vehicles in the trains. In Fig. 9 the connection force diagram merging in itself

the draw-gear diagram, the buffer-gear diagram and the underframe characteristics in

the elastic and ideally plastic domain is shown. It is to be emphasised that in case of

such a large relative displacement, which goes with achieving the plasticity limit, the

connection force remains constant (ideal plasticity) whilst the relative velocity of the

adjacent vehicles is positive (&x>0). If the relative displacement after achieving amaximum turns to decrease, i.e. the relative velocity becomes negative ( &x


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),,( tpFxfx =& , 0000 )(,)( pp tt FFxx == ,

where vector x(t) is the state vector containing the velocities, the preceding velocities,the displacements and the preceding displacements of the mass gravity points

modelling the vehicles in the two colliding trains. The preceding velocities anddisplacements are the values of the quantities mentioned belonging to the preceding

computation step. The necessity of considering also the preceding values is connected

with theshort distance memoryof the system containing components with dry friction,

namely the buffer and draw gears. As for the initial value vector x0 , it contains thenumerical values of the velocities, the preceding velocities, the displacements and the

preceding displacements of the vehicle mass gravity points belonging to initial time

point t0. Vector Fp(t) stands for the preceding force transmitted in the dry-friction con-nections, i.e. in case of time step h vector Fp(t) contains the friction force values be-longing to time point t-h. Accordingly, the initial value vector Fp0 stands for the

friction force values belonging to time point t0-h.The initial value problem introduced above was solved numerically for several initial

velocities. The initial preceding velocities were taken to be identical to the initial

velocities used. The initial positions were always the same, whilst the initial preceding

displacements were selected in accordance to the actual initial velocity and the com-

putation step h. Initial vector Fp0was selected to be zero vector, since the role of thisinitial vector was confined for the starting of the computation process. The further

values of the components of vector Fp(t) were sequentially generated with the knowl-edge of the values of the preceding velocity and displacement coordinates belonging to

the preceding time pint t-h.

The numerical simulation carried out by using the program elaborated in MATLAB

resulted in the time functions of the force arising in a single buffer gear in the course

of frontal train collision of different initial velocities. In Figs. 10, 11, 12 and 13.the

diagrams of the buffer-gear forces are shown. As it can be read off the diagrams, the

underframe plasticity limit is exhausted already in case of initial train velocities v2=-

v1=7.5 km/h. After the exchange of impulse in the course of the collision, the trains

change their original motion direction into their opposite, and at a reduced velocity

leave the place of collision, if no derailment has occurred.It is interesting torecognise thatin contrast with the oscillatory character of the collision caused buffer forces received

for moderate impact velocities, in case of higher initial velocities, e.g. for v2=-v1=30km/h the large remaining plastic deformation swept away the oscillatory variations.

The knowledge of variation with time of the dynamic forces loading the front buffer

gears in case of train collision makes it theoretically possible to carry out further

computer based analysis to determine, e.g. cylindrical shell form crash elemenst(im-

pact energy amortiser elements) with appropriate characteristics, which elements can

be axially mounted just behind the buffer gears or into the underframe structure behind

the front beam coaxially with the buffer gears. The mentioned crash elements can take

the role of undergoing large plastic deformation in the course of their partial or total

collapse (loss of stability), the large plastic deformation means the total deterioration

of the crash element, but the collision damage to the underframe or to the other

structural components of the vehicle can be avoided or at least significantly reduced.


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Fig. 10 Front buffer-gear force vs. time

v2=-v1=5 km/h.Fig. 11 Front buffer-gear force vs. time

v2=-v1=7.5 km/h.

Fig. 14 Front buffer-gear force vs. time

v2=-v1=10 km/h.Fig. 15 Front buffer-gear force vs. time

v2=-v1=30 km/h.

As an important goal of the investigations into the longitudinal dynamics of train col-

lisions can be the elaboration of a reliable dimensioning method of the above men-tioned crash elements. It is clear that in case of built in crash elements the intervehicle

connection forces will reflect the force vs. deformation characteristics stepping into

action in the course of the loss of stability (plastic collapse) process. The

investigations into such a combined intervehicle connection force vs. deformation

relationship, the detailed analysis of the force vs. deformation and the energy

amortisation conditions of the cylindrical or another form shell structures is of

paramount importance.

In the follwing chapters the FEMbased analyses and their backgrounds will be intro-

duced, that were carried out to get preliminary information on the dynamical and en-

ergetic behaviour of the cylindrical steel crash elements esteemed proper by theauthors for railway applications in carriages or traction units.


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5. APPLICATION OF "CRASH-ELEMENTS" IN VEHICLE STRUCTURES

In the special field of vehicle engineering there are several applications of crash ele-

ments. Considering that the role of the latters is to ensure artificially increased deform-

ation energy amortisation capacity in the vehicle structure, it is clear that beyond thesphere of applications in railway vehicle technology, all types of vehicles that are

obliged to face and to reckon with the danger of collisions are obviously interested in

the regular usage of crash elements. Accordingly, besides the applications in railway

vehicle technology, there have been old traditions of building in crash elements in the

aircraft/helicopter structures, epecially in sub-passenger floor sections for decreasing

the risk of crash landings. In case of automobiles crash elements are regularly used in

bumpers and also in the chassis, e.g. folding beam sections are built into the front part

of the longitudinal beam to absorb the energy in case of frontal collisions.

In concordance with the above depicted scinario, there are three main purposes ofapplying crash elements in vehicle structures. At the first palce the controlled reduc-

tion of kinetic crash energyshould be mentioned. Secondly, the limitation to possible

minimum of critical passenger accelerations and the prevention of severe sinjuries

should be ensured. Finally, the localisation of large deformations in the crash ele-

ments should be realised, with regard to the fact that the avoidance/limitation of de-

formation/damage to the main vehicle structure can lead to considerably reduced

repair and insurance costs

6. ENERGY ABSORPTION PROCESS OF METALLIC

CYLINDRICAL SHELLS

The folding process of axially loaded, metallic cylindrical shells can be summarised in

the following three main points:

Buckling of shell after critical buckling load (stability load) is exceeded

Growing of one buckle, formation of first plastic fold

Progressive formation of further folds (Ring or diamond shapedfolding patterns are possible)

The main factors on which the buckling and folding process as well as the related e-

nergy absorption depend are as follows:

Geometry (dimensions of shell, imperfections) Material properties (stiffness, plasticity, strain-rate sensitive material

behaviour)

Boundary conditions at the ends of the shell

Load introduction (time dependency, load distribution

In order to visualise the folding process and the general force-deflection curve belong-

ing to the shell buckling phenomenon, in Fig. 17 the characteristic phases are plotted.

As an initial phase, the cylindrical shell undergoing mild axial compressive load is in

an elastic state of stress. The combined effect of the geometrical (length, diameter, wall-thick-

ness and random imperfections in the cylender geometry) and the elasticity conditionsdetermine the critical load when buckling occurs. The initial phase of buckling is


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major elastic, but with increasing loads the plastic fold formation appears. The actual

form of the initial folding at the proximity of the contact boundary is considerably in-

fluenced by the actual conditions of contact (e.g. radial dry ftiction constraint or axial

guiding rim, etc.) between the shell element and the impactor or the static support. In

case of further load increase, a progressive formation of ring shaped or diamond shaped

foldings can be experienced. In [7] it was shown that with the help of a suitable load

introduction certain diamond-shaped folding modes can also be prescribed to the folding

process of shells. In the course of formation of folding the plastic processes take the

decisive role. The energy amortisation is very effective in case of ring shaped foldings.

Fig. 17 Characteristic force-deformation diagram in case of folding process

In Fig 17. it can be clearly seen that prior to the bickling a definit elastic region of re-

lative high linear stiffness rules the conditions, and after the buckling a very steeply

descenting section comes, after that begins the fluctuation if restoring force in

correlation with the successive appearenceing of the foldings. In Fig. 18 the two char--

ring shaped folding diamond shaped folding

Fig. 18 Typical folding patterns in case of buckling of cylindrical shells


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acteristic cases, namely the ring shaped and the diamond shaped folding formation pro-

cesses are visualised.

7. NUMERICAL SIMULATION OF FOLDING PROCESS

WITH FE-CODE LS-DYNA3D

The numerical simulations to obtain quantitative prediction about the dynamic buck-

ling ang collapsing process of cylindrical shell elements have been carried out by

using the finite element code LS-DYNA3D. The latter software is an explicit, non-

linear finite element code, which has originally been developed for the multi purpose

numerical simulation based analyses of continuum mechanical problems emerging in

structural dynamics. Accordingly, its application for simulation of large deformation

processes in the time domain, especially for structures under crash and impact loading

is very fruitful. The selection of the software in question was motivated by the fact that

in the foregoing there were several successful applications for the simulation of crashphenomena in the automotive industry, also under commercional conditions.

In the following, the most crucial features of the FE-model used for simulation of

railway application purpose crash elements will be dealt with. The selection of the FE

partition of the cylindrical shell element under consideration was based on the

Belytschko-Tsay shell elements. The resulted mesh consisted of 2500 partition el-

ements. The modelling of (in the reality always existing) geometric imperfections was

realised by selecting small random perturbations in the node locations of the mesh. In

order to have possibly good approximation with the simulations, the elastic-plastic

behaviour of the applied mild steel material St14 by a piece-wise linear stress-strain

curve. In the simulations the strain rate dependency of material behaviour was takeninto account by the Cowper-Symonds law. The boundary conditions (constraints), the

detailed conditions of the contact problem arising in the course of the folding process,

and the outputs of the simulation procedure were as follows:

Displacements and rotations were fixed at one end on the shell

Impact with a rigid wall with mass m=90 t and different initial velocitieswere reckoned with,

Regular contact algorithm between the rigid wall and shell and frictional selfcontact algorithm for the buckled cylinder jacket was used,

The outputs were the reaction forces and the decelerations of the rigid walland the deformation of shell.

In Figs.19 and 20 the proceeding of the dynamic buckling and the force-deformation char-

acteristic are shown for the considered cylindrical crash element. The dimensions of the

crash element are indicated in the figure explanation text, where L stands for the unde-

formed length of the cylindric shell element, D for the nominal diameter of the cylinder,

while t for the wall thickness. The dimensions of the considered crash-element were

selected with the intent to satisfy the UIC requirements concerning the energy absorbtion

up to full collapse achieved through sequentional plastic folding process. The UIC pre-

scription specifies 0.5 MJ for a single crash-element to be built into the structure

behind a side buffer of a passenger carriage. In case of locomotives the demand forenergy absorbtion of a crash-element built behind a single side buffer gear is 1 MJ.


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Fig. 19 Collapse of the cylindrical crash element: L = 800, D = 300, t = 6, impact mass

90 t, initial velocity v = 30 km/h. The kinetic energy of the mass was not absorbed.

0

500

1000

1500

2000

0 100 200 300 400 500 600 700

Deformation [mm]

Force[kN]

Fig. 20 Computed restoring force-deformation diagram belonging to Fig. 19

(Shell: L = 800, D = 300, t = 6, impact mass 90 t, initial velocity v = 30 km/h)

In Figs. 21 and 22 the proceeding of the dynamic buckling and the force-deformation

characteristic are shown for the considered cylindrical crash element. The impact velo-

city was 10 km/h.

In Fig. 21 and 22 the dynamic collapse process and the restoring force vs. deformation

performance curve shown that in this case the initial plastic stroke of the considered


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crash element is exhausted. As it is visible, after the folding process caused exhaustion

of the initial plastic stroke, the longitudinal stiffness of the severely deformed element

takes a considerable almost constant value.

Fig. 21 Collapse of the cylindrical crash element: Shell: L = 800, D = 300, t = 6,

impact mass 90 t, initial velocity v = 10 km/h. The kinetic energy of the impacting

mass was completely absorbed.

0

500

1000

1500

2000

0 100 200 300 400 500 600 700

Deformation [mm]

Force[kN]


(Shell: L = 800, D = 300, t = 6, impact mass 90 t, initial velocity v = 10 km/h)

In Figs. 23 and 24 the dynamic collapse process and the restoring force vs. defor-

mation diagram is shown for impact velocity 7,5 km/h.


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Fig. 23 Collapse of the cylindrical crash element: Shell: L = 800, D = 300, t = 6, impact mass

90 t, initial velocity v = 7,5 km/h kinetic energy of impacting mass completely absorbed

0

500

1000

1500

2000

0 100 200 300 400 500 600 700

Deformation [mm]

Force[kN]


(Shell: L = 800, D = 300, t = 6, impact mass 90 t, initial velocity v = 7,5 km/h)

In Figs. 25 and 26 the dynamic collapse process and the restoring force vs. defor-

mation diagram is shown for impact velocity 5 km/h.


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Fig. 25 Collapse of the cylindrical crash element: Shell: L = 800, D = 300, t = 6, impact mass

90 t, initial velocity v = 5 km/h kinetic energy of impacting mass completely absorbed

For the sake of comparison of the restoring force vs. deformation diagrams belonging

to the collapse processes under different initial (impact) velocities a common diagram

was constructed, see Fig. 27. Since the variation of restoring force with deformation is

very similar for the chosen material within the treated range of all velocities up to a

limit belonging to the exhaustion of the plastic stroke of the crash-element, as an ap-

0

500

1000

1500

2000

0 100 200 300 400 500 600 700

Deformation [mm]

Force[kN]

Fig. 26 Computed restoring force-deformation diagram belonging to Fig. 25(Shell: L = 800, D = 300, t = 6, impact mass 90 t, initial velocity v = 5 km/h)


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proximation it can be accepted that a unified diagram can be derived by means of av-

eraging of the diagrams belonging to the concidered initial velocities. Obviously, after the

exhaustion if the plastic stroke the collapsed crash-element can be considered as a

tubular thick bar having considerable stiffness. The mentioned stiffness can be clearly

seen in Fig. 27, namely the steeply increasing final section of the restoring force vs.deformation characteristic belonging to 30 km/h can be referred.

0

100

200

300

400

500

0 100 200 300 400 500 600 700

Deformation [mm]

AbsorbedE

nergy[kJ

v = 5 km/h

v = 7,5 km/h

v = 10 km/h

v = 30 km/h

Fig. 28 Common diagram of all energy absorbtion vs. deformation characteristics

0

500

1000

1500

2000

0 100 200 300 400 500 600 700

Force[kN]

v = 5 km/h

v = 7,5 km/h

v = 10 km/h

v = 30 km/h

Fig. 27 Common diagram of all restoring force vs. deformation characteristics of the

cylindrical crash element considered


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In Fig. 28 the energy absorbtion vs. deformation diagrams are visualised in a common

figure. As it can be seen, the variation of the absorbed energy is almost proportional to

the (plastic) deformation up to the limit characterised by the axhaustion of the plastic

stroke

8. EXTENDED INTERVEHICLE CONNECTION FORCE CHARACTERISTICS

FOR COLLISION DYNAMICS

On the basis of the results received in Chapter 7, the similar quasi-congruent varia-

tion of the force vs. deformation characteristics, and the quasi-covering increase of

the energy absorption vs. deformation characteristics of the cylindrical crash elements

in case of different impact velocities, it is possible to elaborate such a formulation of

the intervehicle connection force characteristics, that the classical lumped parameter

models of longitudinal train dynamics can be preserved. In Fig. 29 the side elevation

of a railway carriage is plotted in order to give a sketch on the allocation of the crashelements at the front and rear end of the longitudinal main beams of the underframe

just behind the side buffers mounted on the front and rear lateral beams.

6...10m1m0.1m

RAILWAY VEHICLE EQUIPPED WITH CRASH ELEMENTS

Buffer GearCrash Element

Vehicle Body

Underframe

6...10m 1m 0.1m

Crash ElementBuffer Gear

AT BOTH ENDS OF THE UNDERFRAME

Fig. 29. Allocation of the crash elements behind the buffer-gears

It is interesting to recognise that there is an order of magnitude difference between the

possible deformation of the buffer-gears, the crash elements. Similar relation is in

force between the crash element and the half-underframe deformations.

In Fig. 30 the extended force vs. deformation characteristics of the combined buffer-

gear crash-element - half-underframe system is shown. As it can be read off the fig-ure, for small deformations the restoring force obeys the law determined by the buffer

gears characteristics. If the sprung stroke of the buffer-gear is exhausted, the restoring

force will steeply increase in accordance with the in series connected stiffnesses of the

unbuckled crash element and the half-underframe. The mentioned increase in restoring

force is limited by the occurrence of buckling and collapsing process of the crash ele-

ment. In the course of the mentioned loss of stability process large deformation occurs

in the plastic domain and a characteristic fluctuating variation with descent mean in

the restoring force can be indicated for positive deformation velocities, in accordance

with the results of Chapter 7. After the exhaustion of the full plastic stroke of the crash


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element the large remaining deformation of the half underframe comes after, accom-

panied by irregular restoring force fluctuations of larger wavelength.

F

x

BUFFER

GEAR

CRAS

H

ELEMEN

T DEMAGE TO UNDERFRAMEAND VEHICLE BODY

~1m ~4...8 m~0.1m

Fig. 30 The extended force vs. deformation characteristics of the combined buffer-

gear crash-element half-underframe system

For the sake of explaining the case of the backwards motion after having achieved the

maximum full deformation, consider the right boundary of the shaded area. As it can

be seen, the restoring force steeply falls down due to the remaining elasticity of thelargely deformed structure. If the restoring force diminishes down to the limit deter-

mined by the lower branch of the dry friction damped buffer-gear diagram, and in

case of damage free survival of the buffer gear the further decrease in restoring force

follows the lower buffer-gear diagram branch mentioned, down to vanishing. In case

of further motion of negative velocity, the restoring force remains identically zero.

The mathematical modelling of the restoring force arising in case of the above de-

scribed process is elaborated under the assumption that the plastic collapsing process

of the shell element and the large deformation process of the in series connected beam

can be described by an appropriate non-negative continuous functiong(x). Letf*(x) be

the not neccesserily linear function describing the elastic deformation of the axially

loaded shell-beam system. Function f*(x) is assumed to obbey condition f*(0)=0. If

the deformation of the system starting from the unloaded zero position is x(t) then con-

sider first the maximum deformation occurred in time span [0,t], which is determined

by the following non-linear operation:

{ })(max)(0

xtyt

= .

In order to get a restoring force function defined also for negative deformations (i.e.

for the case when the system is unloaded) let us take function

{ })(,0max)( xfxfRx

= .


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Under the above premise the bivariate restoring force function elaborated for the com-

bined shell-beam system takes the following form (see [4], the so called dream

formula):

{ }))))(),((min((),(min),( 1 yfygfyxfxgyxF = ,

where g(x) stands for the shifted version of the restoring force function s(x) occurring

in case of the plastic deformation (loss of stability) of the cylindrical shell element in

the system. The measure of shifting is determined by the deformation belonging to

the maximum still elastic displacement of the system prior to shell-buckling. The for-

mula:

{ })(,min)( 0 = xsgxg

where g0is the critical buckling load. The elaborated extended restoring force function

F(x,y) has been tested for different variations with time of the deformation process

x(t). In Fig. 31 an expediently constructed double spiral form phase trajectory is shown.

The initial monotonic increase in deformation and deformation velocity (drawn in full

line) turns into gradual decrease (drawn in dashed line) in the mentioned quantities.

This selection of the kinematic conditions makes it possible to model the process of

after another realised sequential impacts causing sequentially increasing plastic de-

formation in the crash element. In Fig. 32 the variation with time of the restoring

force in question is shown over the time function of the deformation process

simulated.

Fig. 31 Constructed phase trajectory for testing the restoring force function

In Fig. 33 the 3D diagram of the variation of the restoring force values are shown over

the constructed phase trajectory. The intersecting in-space curves should fit onto a

bivariate performance surface describing the extended restoring force arising in the

buffer crash-element underframe system.


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Fig. 32 Restoring force vs. time function over the constructed

displacement time function

Fig. 33 Restoring force rover the constructed phase trajectory

9. POSSIBILITY OF TIME DOMAIN ANALYSIS BY USING

THE EXTENDED CHARACTERISTICS

With the knowledge of the extended restoring force diagram for all vehicle

connections the numerical treatment of the longitudinal dynamical processes can be

quite similar to the traditional case.

The initial value problem has the form ( )t,xx=& , where function f contains the

extended connection force diagrams depending on the relative displacement and re-

lative velocity of the vehicles' gravity points. Starting from initial values00 xx =t

and Fp (t0) = Fp0, the familiar integration methods can be used, e.g. Heun, Runge-Kutta, MATLAB ODE, ODEs, etc.


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The variation with time of the connection forces can be determined by substituting the

solution received from the kinematic characteristics into the combined draw-gear ex-

tended restoring force functions defined for each intervehicle connection.

Numerical results concerning the variation with time of the above intervehicle forceswill be introduced in a subsequent publication

10. CONCLUSIONS

On the basis of the train of thoughts and the computer based numerical computation

results, as well as the evaluations introduced, the following conclusions can be drawn:

The simulation of train collision processes by using traditional

connection force diagrams with ideal plastic extension show

that over v2= -v1>5 km/h train velocities the forces arising

in the underframe achieve the yielding limit.

To avoid the damage to the underframes it is advantageous to

build in cylindrical shell form crash elements behind the buf-

fer gears.

The UIC demand of 0.5 MJ energy absorption per crash el-

ement for railway carriages can be satisfied by the developed

cylindrical shell form element.

The traditional connection force diagrams can be extended tak-

ing into account the force vs. deformation characteristics ofthe crash elements and the large deformations causing damages

to the underframe and the vehicle body

The non-linear dynamical models with their existing numerical

treatment can be used when the extended connection force

diagrams are included in the lumped parameter system.

Further research is necessary to analyse the behaviour of crash

elements in wider deformation velocity range and with other

geometry to achieve better dissipation behaviour and impact

direction insensitivity

11. REFERENCES

[1]Zobory, I.- Bkefi, E.: On Real-Time Simulation of the Longitudinal Dynamics ofTrains on a Specified Railway Line. Proceedings of the 4th Mini Conference on

Vehicle System Dynamics, Identification and Anomalies, held at the TU of

Budapest, 7-9 of November, 1994. p. 88-100.

[2] Zobory, I.- Bkefi, E.: Longitudinal Dynamics of Vehicle Systems and TrafficFlows. Proceedings of the 5th Mini Conference on Vehicle System Dynamics,

Identification and Anomalies, held at the TU of Budapest, 11-13 of November,1996. p. 81-93.


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[3] Horvth, K.- Gyrik, A.- Zobory, I.: Modellierungsmglichkeiten der durchtrockene Reibung gedmpften Schwingungen bei der lngsdynamischen Unter-

suchung von Eisenbahnzgen. Periodica Polytechnica, Transportation Engineer-

ing, Vol.7. No.2. 1979. p.149-161.

[4]Zobory, I.: On Mathematical Modelling of the Restoring Force Arising in Case ofImpact on in Series Connected Beam and Cylindrical Shell Element. Manuscript,

Budapest, 2001.

[5] Gmez Garcia, J.- Marsolek, J.- Reimerdes, H.-G.: Determination of the energyabsorption of cylindrical shells under axial loading by analysis of the dynamic

buckling and folding process, International Crashworthiness Conference IJCRASH

98, Dearborn, Michigen, USA, 1998

[6] Gmez Garcia, J.- Marsolek, J.- Reimerdes, H.-G.: Energieaufnehmeverhalten

metallischer Zylinderschalen unter axialer Stobeastung, DGLR-Jahrestagung,Dresden, 1996

[7] Marsolek, J.- Reimerdes, H.-G.: Numerische und experimentelle Untersuchungendes Faltvorgangs von zylinderschalenfrmigen, metallischen Energieabsorben, 18.

CAD-FEM Users Meeting, Friedrichshafen, 2000

longitudinal dynamics of train collisions - crash analysis

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