2011 Leon Sleepers

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A procedure to develop scalable models for the transient response of sleepersin conventional and high-speed railway lines and implementation to thevertical vibration mode

E. Leon, D.C. Rizos n, J.M. Caicedo

Department of Civil and Environmental Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208, USA

a r t i c l e i n f o

Article history:Received 23 June 2010

Received in revised form

12 November 2010

Accepted 17 November 2010

a b s t r a c t

This paper presents a procedure to develop scalable reduced models for the through-the soil interactionand traveling wave effects of distant sleepers in a long railway track. For development purposes, and,

without loss of generality, the geometry of the sleepers is consistent with the UIC-60 track system

commonly usedin European high speed railand the vertical vibration mode is considered. Theballastand

the effects of soil layering are not considered in the present paper; however, it is the subject of ongoing

research. The proposed reduced models are based on B-Spline impulse response functions (BIRF) of the

sleepersonly as computed through boundary elementmethod (BEM) solutionsof thefull model,preserve

the frequency content of the full models, and they are highly accurate within the assumptions of linear

isotropic and homogeneous soil media. They are expressed in a scalable form with respect to soil

propertiesand sleeperspacing.In particular, theBIRFs of distantsleepers can be accuratelyapproximated

by appropriate scaling operations of time and amplitude of a reference sleeper BIRF while retaining all

dynamic characteristics of the full model. Three main scaling parameters are proposed: (i) the apparent

propagation velocity,(ii) the geometric dampingcoefficient, and(iii) the soil propertiesof a reference soil

(i.e., the shear modulus and shear wave velocity). The models are validated through comparisons with

other BEMsolutions,and the accuracy andefficiency areestablished. Theproposed models are developed

as part of an NSF funded research on vibrations inducedby high-speed rail traffic and are consistent with

the associated train and rail models and a multi-system interface coupling (MSIC) technique that were

developed as a part of the project and presented in companion papers. The proposed procedure forms the

framework for developing scaled reduced models for other vibration modes and different sleeper

geometries and can be generalized to include any foundation type or layered soil profiles.

Published by Elsevier Ltd.

1. Introduction

High-speed trains (HST) is a popular transportation mode in the

world and plans are being made in the United States for several high-

speed rail systems to be constructed in different areas of the country

over the next several decades. The definition of HSTs typically

suggests a passenger train traveling in excess of 160 km/h(100 mph), though HSTs in service may travel faster than 320 km/h

(200 mph). As the HST speeds continue to increase, potential

problems arise when the train travels over soft soils with inherently

lower Rayleighand shear wavevelocities.In such cases, thewave field

generated by the HST propagates at speeds lower than the speed of

the trains that generated them, creating a phenomenon equivalent to

the sonic boom and supersonic travel, as described for example in

Ref. [1]. The characteristics of the wave propagation due to the

passage of HST, e.g., frequency content and peak particle velocity

differ from the vibrations caused by conventional trains. Although

these wavefields do notcause large strains andsoil nonlinearities, the

vibrations could be potentially damaging to both the trains and the

infrastructure facilities and cause annoyance to passengers and

residents impacting negatively the public perception on the benefitsof HSR. It is, therefore, evident that an accurate prediction of HST

inducedvibrations needs to account forthetraveling waveeffects and

the through the soil interaction between sleepers at relatively large

distances in addition to the rail, track and vehicle dynamics. Such

tasks are among the most computationally expensive and require

efficient approaches and procedures.

The finite element method (FEM), boundary element method

(BEM), and hybrid FEMBEM are among the most popular techni-

ques for transient analysis and wave propagation and suitable for

such studies. General literature reviews on time and frequency

domain BEM, FEM, and coupled BEFE methods for problems in

transientanalysishave beenpresented in Refs. [24] among others.

Contents lists available at ScienceDirect

journal homepage:w ww.elsevier.com/locate/soildyn

Soil Dynamics and Earthquake Engineering

0267-7261/$- see front matter Published by Elsevier Ltd.

doi:10.1016/j.soildyn.2010.11.006

n Corresponding author. Tel.: +1 803 777 6166; fax: + 1 803 777 0670.

E-mail addresses: [email protected] (E. Leon),[email protected] (D.C. Rizos),

[email protected] (J.M. Caicedo).

Soil Dynamics and Earthquake Engineering 31 (2011) 502511
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Standard FEM procedures for wave propagation in infinite media

truncate the infinite extents by introducing fictitious boundaries

and cannot accurately represent the radiation condition in the soil

region. Proposed alternatives include (i) modeling of large compu-

tational domain with material damping, e.g.[5], (ii) use of springs

and visco-elastic dampers along the fictitious boundaries, e.g. [6],

and (iii) development of special types of elements that model the

infinite extents at the fictitious boundaries, e.g. [710]. Such

alternatives are either less accurate, or computationally expensiveor suitable for frequency domain analysis only. BEM for wave

propagation and sleeper soil interaction analysis are developed in

the frequency or time domain. The frequency-domain techniques

are based on the calculation of transfer functions followed by

suitable convolution operations to obtain the response to the train-

induced moving load[1117]. Time domain BEM formulations for

general 3-D wave propagation problems in the direct time can be

classified in view of the associated fundamental solutions in three

major types: (i) Dirac-d fundamental solutions, e.g.[18], (ii) step

impulse fundamental solutions, e.g. [19], and (iii) high order B-

Spline fundamental solutions, e.g.[20,21].

Coupled formulations based on thecombination of the FEMand

BEM methods have also been presented, e.g.[2225], providing an

attractive numerical toolfor problemspertaining to coupled media.

Recent research work on the analysis of ground vibrations pro-

duced by HST has been reported in the frequency domain[2629],

and in the time domain[3033]. Time domain simulations of the

full length of a railroad track using FEM or BEM or combined

FEMBEM models are often time consuming since they involve a

significantly high number of degrees of freedom. A number of

techniques for reducing the size of the models have been reported

in the literature. One of the earliest and most popular reduction

methods used in FEManalysis is the staticor Guyanreduction [34],

but it is not exactfor dynamic analysisdue tothe presence of inertia

forces in the system. Model reduction techniques reported in the

literature for dynamic problems include the dynamic reduction

method [3537], theimprovedreductionsystem (IRS) method [38],

the iterative IRS technique[39], the component mode synthesis

(CMS) method[40], andthe system equivalent reduction expansionprocess (SEREP) method[41], among others. The aforementioned

techniques operate directly on the system matrices and are either

less accurate or require modal analysis to identify potential modes

that can be eliminated (or ignored). The B-Spline impulse response

techniques[21]present an alternate approach in model reduction

and are based on establishing signature responses of systems

subjected to impulses of B-Spline modulation using the BEM

reported in Ref. [20]. Once these characteristic responses are

established the time history of the system response to arbitrary

excitations is computed by a mere superposition. In the case of a

long straight railroad track these characteristic impulse responses

of the sleepers are repetitive for any two sourcereceiver sleeper

systems andscalable fordifferent soil types.Theyhavebeenusedto

generate the B-Spline impulse response function (BIRF) matrix of alarge system based on the analysis of much smaller systems

consisting of only a few sleepers [1]. Even though this approach

is more efficient than implementing the BEM on the entire track,

the calculation of the characteristic responses can be time con-

suming as the distance between the sleepers increases and further

optimization is in order.

This paper presents a procedure for developing scalable reduced

models of distant sleepers ina long railway track accounting forthe

through-the soil interaction and traveling wave effects. For devel-

opment purposes and without loss of generality the geometry of

the sleepers is consistent with the UIC-60 track system commonly

used in European High Speed Rail, and the vertical vibration mode

is considered. The effects of the ballast and soil layering are not

considered in the present paper; however, it is the subject of

ongoing research. The proposed reduced models are based on

B-Spline impulse response functions (BIRF) of the sleepers only as

computed through boundary element method (BEM) analysis of

the full model, preserve the frequency content and all dynamic

characteristics of the associated full models, and they are highly

accurate within the assumptions of linear isotropic and homo-

geneous soil media. The output informationof the full model can be

retrieved routinely on an as needed basis. The B-Spline BEM

formulation within the framework of B-Spline impulse responsetechniques [20] presents an innovative and efficient way to

simulate in almost real time the wave propagation in soils

accounting for kinematic interaction effects with the sleepers.

Inertia interaction effects are addressed through coupling with

other solution techniques as reported in Refs. [1,24]. The reduced

models are expressed in a scalable form with respect to soil

properties and sleeper spacing. In particular, the BIRFs of distant

sleepers can be accurately approximated by appropriate scaling

operations of time and amplitude of a reference sleeper BIRF while

retaining all dynamic characteristics of the full model. Three main

scaling parameters are introduced: (i) the apparent propagation

velocity, (ii) the geometric damping coefficient, and (iii) soil

properties of a reference soil (i.e., the shear modulus and shear

wave velocity). The proposed models are developed as part of an

NSF funded research on vibrations induced by high-speed rail

traffic and are consistent with the associated train and rail models

and a multi-system interface coupling (MSIC) technique developed

by the research team and presented in companion papers. The

proposed approach forms the framework for developing scaled

reduced models for other vibration modes and different sleeper

geometries and can be generalized to include any foundation type

or layered soil profiles. In the following sections, the proposed

technique is discussed. First, the BIRF techniques are introduced

briefly, thena description of the proposed model reduction concept

follows and, the scaling operations that lead to the proposed

scalable forms are presented. The validation of the proposed

closed-form solutions is provided and, finally for a more compre-

hensive use of these expressions, the reader is referred to an

implementation example.

2. Overview of B-Spline impulse response techniques

A detailed formulation of the employed 3-D BEM is too

extensive and beyond the scope of this paper and can be found

in Ref. [1,42]. The BEM uses the time domain fourth-order B-Spline

fundamentalsolutions of the3-D full space along with higher order

spatial discretization of the boundary. These fundamental solu-

tions are developed assuming an excitation of the form of a fourth-

order B-Spline polynomials. The B-Spline polynomials are piece-

wise smooth polynomials and are the basis functions in function

interpolation techniques. The fourth-order polynomials, when

used as the excitation functions in the development of thefundamental solutions, satisfy the continuity requirements. The

B-Spline impulse response technique is based on: (1) the relation

between the B-Spline impulse excitation and the associated

B-Spline impulse response function (BIRF) of the elastodynamic

system as computed through a BEM method using the proposed

B-Spline fundamental solutions and (2) the representation of any

function as a linear combination of B-Spline polynomials. Conse-

quently,if the response of theelastodynamicsystem to an arbitrary

excitation function is sought after, the response can be computed

through a mere superposition of the BIRFs without requiring a

rigorous BEM solution to be performed anew for each arbitrary

excitation. The technique implies linearity of the problem.

The boundary integral equation associated to the Navier

Cauchy governing equations of motion is expressed in a discrete

E. Leon et al. / Soil Dynamics and Earthquake Engineering 31 (2011) 502511 503


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form yielding a system of algebraic equations at step Nrelating

displacementsu to forcesfat discrete boundary nodes in the BEM

model and at discrete time instants tjand tj, as

cutN XN 1

n 1

UtNftNn 2TtNutNn 2 N 1,2,. . .,Nend 1

where c is a discontinuity term, and U and T are the BEM

coefficient matrices associated with the B-Spline fundamental

solutions. Eq. (1) can be solved in a time marching scheme forNend time steps yielding theB-Spline impulse response (BIRF)of the

system. In order to derive equivalent, time-dependent flexibility

matrices associated with the loaded part of the BE model

(e.g. points of application of external excitation), a unit B-Spline

impulse force djf, perturbs each active degree of freedom,

j 1,y,NN, at a time and successive solutions of Eq. (1) yield the

B-Spline impulse response vectors, bNj utN. These BIRFs can be

collected in matrix form as

BN bN1 b

N2 . . . b

Nj . . . b

NNN

h i 2

Matrix B represents the BIRF associated to the displacements of

the BEM nodes due to concentrated unit forces applied in the

directions of thedegreesof freedom that vary in time according to a

B-Spline function, and captures the traveling wave characteristics.Subsequently, the response, uN, of the elastodynamic system

subjected to arbitrary external forces fP applied at the nodes

are computed at time step Nas

uN XN 1

n 1

BNPNn 2 3

By separating known from unknown quantities at time step N,

and assuming that PN 0, Eq. (3) can be expressed as

uN ZFN HN 4

where

Z 2B1 B2; HN XN 1

n 3

BnFNn 2B1FN1 5

Matrix Z represents the flexibility matrix of the loaded BEM

region and vector HN represents the influence of the response

history on the current step. The flexibility matrix is independent of

time; however, vector HNneeds to be evaluated at every time step.

The proposed method is implemented in two major phases. The

first phase calculates the BIRF matrices of the boundary of the

domain. The evaluation of the BIRF matrices, BN, is computationally

intensive; however, theyare independent of the externalexcitation

and need to be evaluated only once for a given model geometry.

The second phase calculates the response of the boundary of the

domain given a known excitation force time history. In the

presence of rigid bodies in contact with the free surface, such as

the case of the railroad sleepers considered in this work, the same

method canbe used along with the rigid surface boundary elementreported in Ref. [43]to compute the BIRF functions of the soil

sleeper interface.

3. Proposed model reduction

3.1. The concept

For the analysis of a real system consisting of a large number of

sleepers where all sleepers are loaded in an asynchronous pattern

due to themoving load, the calculationof theBIRF matrix, Eq.(2), is

a computationally expensive task if all sleepers and an extended

free field are to be considered simultaneously in the full model. In

such a complete approach, every sleeper in the system needs to be

loadedby a B-Spline impulse andthe response of all sleepers needs

to be computed. Although this approach is an accurate representa-

tion of the railsleeper system, it may be simplified by observing

that: (1) the response of each rigid sleeper is described by six

degrees of freedom,i.e., three translations andthree rotations of the

center of the sleeper, (2) onlythe vertical andhorizontal translation

degrees of freedom of the sleepers are coupled with the rail, and

(3) repetitive responses of the sleepers are expected when a single

representative sleeper is loadeddue to the assumption of a straighttrack. Therefore, only one sleeper should be loaded by a B-Spline

excitation and the BIRFs of all sleepers should be computed. Since

thefreefield effects arealready accounted forin thecomputed BIRF

functions of the sleepers, the size of the full model is therefore

reduced to the number of degrees of freedom of all sleepers in the

system. This represents a first level of model reduction. A second

level of reduction is attained next. In view of the wave attenuation

and the HST speed, only a finite number of sleepers are expected to

have a significant contribution to the response at any given time.

Furthermore, depending on the geometry and spacing of sleepers,

cross interaction effects are expected to be important for only the

first twoto four sleepers adjacentto the loadedone, as discussed in

Ref. [1] andverified in this work. Therefore, it is feasible to generate

BIRF matrices of a large system based on the analysis of much

smaller systems consisting of only a few sleepers. Consider, for

example, the system shown in Fig. 1 that consists of a source

sleeper,s , and m receiver sleepers on each side of the source. As

proposed in Ref. [1] and assuming for demonstration purposes that

the cross-interaction effects are significant within two sleepers

away of the source, two groups of sleepers are identified. The first

group (Group A) pertains to the source and the two adjacent

sleepers oneachside.For this groupthe BIRF is computedthrougha

rigorous BEMsolution. The second group (Group B) consists of pairs

of source/receiver sleepers with the receiver sleeper not belonging

to the first group. In this case all other sleepers between the source

and receiver (light shaded sleepers in Fig. 1) are omitted and it is

assumed that cross-interaction effects and wave scattering due to

neighboring sleepers are negligible. This simplifying assumption is

expected to produce conservative results since the sleepersbetween source and receiver act also as wave barriers. In Ref.[1]

the BIRFs for this group are also computed numerically through a

rigorous BEM solution with all sleepers between source and

receiver being omitted. Subsequently, the BIRF of theentire system

is formed by combining the BIRFs of the two groups. However, as

the model considers larger number of sleepers, the BEM solution

even for group B will become computationally expensive, since

large regions of the free field should be modeled to account for the

surface waves correctly.

The proposed reduced model can be efficiently obtained for the

systems in group B if the BIRF of any receiver sleeper in the group

can be accurately approximated by an appropriate scaling of the

BIRF ofa reference sleeper that is computedbased onBEM solutions

of the full model. To demonstrate this concept, a representativesourcereceiver sleeper system in Group B (e.g. S-j in Fig. 1) is

considered as the reference system. The distance between the

source and receiver system isDrefand the BIRF is computed by the

BEMmethod forthisreference system. Inthe present work theBIRF

of another sleeper at distance DDi, may be computed through

appropriate scaling of the reference system with respect to time

and amplitude.

3.2. The features of BIRFs of 2-sleeper systems

In order to develop the proposed reduced models, the char-

acteristics of the B-Spline impulse response functions (BIRF) need

to be explored first. To this end, the BEM methodology outlined in

E. Leon et al. / Soil Dynamics and Earthquake Engineering 31 (2011) 502511504


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section 2 is used to calculate the BIRFs of a series of source-receiver

sleeper systems spaced at variable distances.Fig. 2illustrates such

BIRF responses for six different sourcereceiver spacings for the

vertical vibration mode.

The intercept of each BIRF indicates the distance between thesource and receiver while the actual amplitude of the BIRF is

omitted.The repetitive nature of the BIRFs is evident. Thevariations

in amplitude, and number of high amplitude peaks with distance

from the source areapparent. The observed horizontal shift in time

is due to the quiescent past of the response while the amplitude

decrease is due to attenuation.Despite these variations, a similarity

in the overall shape of the response curves is observed and can be

interpreted by considering the characteristics of the wave propa-

gationfromthe source towardthe receiversleepers. The BIRF of any

sleeper is affected by both body waves (Pressure P- and Shear S-)

and surface waves (Rayleigh R-) propagating through the

semi-infinite soil. The general characteristics of the stress wave

propagation theory are observed in the sleeper BIRFs illustrated in

Fig. 2. Additional features inherent to the wave generation and

propagation in soil due to the particular sleepers arrangement are

observed:

1. The motion between the first P- and S-wave arrivals becomes

more pronounced and the number of peaks between these

arrival times increases as the distance from the excitation

source increases. This part of the motion includes contributions

from: (a) the incident P-wave field emanated at the source

sleeper and propagating with velocity cp; and (b) interaction of

the scattered P-waves with the S-wave incident field.

2. The part of the motion between the first S- and R-wave arrival

includes contributions from, (a) the incident S-wave field

emanated at the source sleeper and propagating at cS; and

(b) interaction of the scattered S-waves propagating at cSwith

the R-wave incident field propagating atcR. The contribution of

scatteredP-waves is notanticipated to be significant in thistime

interval.

3. The response of the sleeper following the first R-wave arrival

seems to be composed mainly of the incident R-waves ema-

nated at the source sleeper propagating with velocity cR. The

most likely contribution might be that of the scattered S-waves

since their velocity is only about 10% higher than the velocity of

R-waves.

Comparing the BIRFs of the receivers in any 2-sleeper systems,time and amplitude of a reference receiver BIRF can be scaled with

respect to distance and the soil properties to obtainaccurate match

of the response of the other.

3.3. Scaling of time

During any time intervalDtthe group of waves of the same type

emanated from the sourcesleeper travels a distance cDt, where cis

the wave velocity of the medium. If the group of waves reaches

receiver sleeperjat timetjand receiver sleeperiat timetithen the

horizontal distance between the two sleepers is covered within

Dt titj DiDj

c 6

Fig.2. Comparisonof thevertical componentof displacement of receiverslocated at

increasing distances from the source as indicated by the intercept. The source

function is a B-Spline impulse.

Fig. 1. An m-sleeper system and the computation of the BIRF function.



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Hence, the problem of calculating a scaled time depends on the

availability of the wave propagation velocities, c. At each time

instant,t, the group of waves reaching the receiver sleeper can be

thought of as combination of P-, S-, and R-waves. In lieu of

considering the separate wave velocities, a parameter henceforth

calledapparent propagation velocity Cis introduced to describe the

propagation velocity of the wave packet of different wave types.

The apparent velocity is a function of time, in contrast to the wave

velocity of a particular wave type. This parameter captures theeffects of the geometry of the excitation source and the reflections

from the receiver sleeper which, as discussed in the previous

section, are perceived as factors that contribute to the complexity

of the wave packet that travels from the source to the receiver.

Assuming that C is known, the desired scaled time tscaled is a

function of the apparent propagation velocity itself and is readily

calculated as

tscaled tref DDrefC 7

The procedure to estimate the apparent propagation velocity is

discussed in section 4.2

3.4. Scaling of amplitude

The attenuation of amplitude of vibration with distance attrib-

uted only to geometric (radiation) damping may be described by

Bornitzs equation[44]

wi wj DjDi

n8

where wi and wj are vibration amplitudes at distance Di and Dj from

a source of vibration and n is a geometric damping coefficient.

Values ofn have been provided in the literature[45]for various

types of excitation sources and types of propagating waves. For

example, the amplitude of body waves at distance R from the

source, decreases in proportion to 1/R (n1) except along the

surface of the elastic half-space where the amplitude decreases in

proportion to 1/R2

(n2). The amplitude of the Rayleigh wavesdecreases in proportion to 1/R0.5 (n 0.5). Nevertheless, the values

ofnfor the system inFig. 1are anticipated to differ from the latter

ones due to theeffects of thegeometryof the excitation sourceand

the reflections from the receiver sleeper. Therefore, for the accurate

prediction of the wave attenuation due to applied loading in the

current problem,nneeds to be quantified as a function of time for

the specificgeometryof theproblem.The procedure to estimate the

geometric attenuation coefficient is discussed in section 4.3. Using

the estimated n values, the desired scaled amplitude wscaled is

readily calculated as

wscaled wref Dref

D

n9

3.5. Soil Property scaling

Scaled time and amplitude, as defined in Eqs. (7) and (9),

respectively, are associated with the soil medium of a reference

system. This implies that thesame soil medium properties exist for

the reference system and the system whose scaled response is

desired. However, different soil medium properties can be con-

sidered for the two systems. Considering that the arbitrary system,

i, consists of sleepers of the same dimensions as the reference

system andGi,cs,i,riand Gref,cs,ref,rrefare the shear modulus, shearwave velocity and mass density of the arbitrary and the reference

system, respectively, the scaled response at distance D can be

computed as

witi wscaled GrefGi

wscaled cs,refcs,i

2rrefri

, ti tscaled cs,refcs,i

10

The proposed scaling method is only valid for sleepers having

the same widths and is applicable only in the absence of

connecting rails.

4. Parameter estimation and development of scalable BIRF

In order to obtain the closed-form solutions for the BIRF of a

receiver sleeper as a function of distance from the source, the

scaling functions in Eqs. (7) and (9) involve the estimation of

the scaling parameters, i.e., the apparent propagation velocity andthe geometric damping coefficient. This section discusses the

procedure to estimate these parameters and presents the scaling

functions to obtainthe BIRF solutions of thereducedmodel. It must

be noted that the scaled model should capture all the significant

information of the numerically obtainedBIRFs. To this end,the BIRF

of a receiver sleeper is obtained numerically for a number of

sourcereceiver distances. Subsequently, discrete values of the

apparent velocity and damping coefficient are obtained at char-

acteristic time instants from the BIRFs. Finally, the apparent

velocity and damping functions are established through a regres-

sion analysis of the discrete values.

4.1. Sleeper BIRF function

The sleeper BIRF functions are calculated for the track geometry

presented in Ref.[1]. The dimensionsof each sleeperare 0.285 m 2.5

m witha centerto centerspacingof 0.955 m, yielding an edge-to-edge

spacing of 0.67 m. The sleepers are assumed rigid and massless, and

therefore, only kinematic interaction effects are accounted for.

Complete interaction effects are readily accommodated through

coupling of the proposed formulation with the FEM solution for

the track-rail model where the inertia interaction due to sleepers

mass is accounted for. The sleepersare assumed to rest directlyon the

horizontal free surface of a linear elastic half-space with which they

remain always in contact. A reference soil having a relatively low

shear wavevelocity is usedin theanalysis andthe assumedproperties

are shown inTable 1. As shown inFig. 3, the boundary of the half-

space is discretizedinto 8-nodeboundary elements, andthe motionofthe sleeper is expressed by the 3 translations and 3 rotations of a

reference node, R, at the sleeper center. Since only the vertical

vibration mode is of interest in this work, only the vertical degree

of freedom (DOF) of the source sleeper is excited with a fourth-order

Table 1

Input data for BEM rigorous solution.

Property Value SI (US) Property Value SI (US)

Sleeper length,L 2.5 m (98.425 in.) B-spline support, Dt 2 103 s

Sleeper width,b 0.285 m (11.220 in.) Time step, dt 5 104 s

Lames,l 5.19 107 KPa (7.52 106 lbf/in2) P-wave velocity,cp 246.07 m/s (9687.76 in/s)

Shear modulus,G 3.46 107 KPa (5,02 106 lbf/in2) S-wave velocity, cs 131.53 m/s (5178.31 in/s)

Density,r 2,000 Kg/m3 (0.002246 slugs/in3) R-wave velocity,cR 121.93 m/s (4800.28 in/s)



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B-Spline impulse of duration Dt2 103 s, shown in the inset

ofFig. 4. The associated time histories of the response of all DOFs

are computed using the BEMmethod [21] at discretetimes tnnDt/4.

The BIRF of the receiver sleepers pertaining to the vertical trans-

lation is monitored and shown inFig. 4. The six curves representsix different sourcereceiver sleeper systems with the following

source-receiver distances in increasing order: D22.85, D33.99,D44.845, D77.695, D10 10.545 and D1212.54 m, where sub-

scripts indicate the number of sleepers between the source and

receiver. It is noted that these distances are not always multiples of

the spacing between the adjacent sleepers in order to facilitate the

efficient discretization of the free field. However, this discrepancy

does not affect in any way the proposed scaling operations. The latter

is applicable to any spacing and not restricted to the UIC-60 track

system.

4.2. Determination of apparent propagation velocity

The first step in determining the apparent velocity as a function

of time is to compute discrete valuesbasedon the BIRFs obtained in

the previous section. To this end, one response at a time is

considered as the reference response where the sourcereceiver

spacing is Di. The remaining responses, with sourcereceiver

spacing Dj, are the target responses and are compared to the

reference in view of their characteristic peaks. Thus,the time lapse,

Dt, between the time,ti, of the reference response where a peak is

observed and the time, tj, of the target response where the same

peak takes place is measured. Subsequently, Eq. (6) can be solved

for the velocity, C. This velocity is a measure of the apparent

velocity at time ti for this pair of reference and target responses.

Therefore, discretevalues of the apparentvelocity canbe calculated

for all combinations of reference-target BIRFs. The calculated

apparent velocity is presented in a graph form in Fig. 5 for four

reference systems.

It is observedthat the calculated Cfunctions are piecewisesmoothfunctionsdefined in four timeintervals, i.e.[tp, tc1], [tc1, tc2], [tc2, ts]and

[ts,N) for each sourcereceiver distance. It is also observed that the C

values corresponding to tp, tc1, tc2 and ts do not depend on the distance

and are determined as Ctp cp 246m=s, Ctc1 2cpcs

228m=s, Ctc2 cs 132m=s and Cts cR 122m=s. Times tpand ts correspond to the first arrival of the P- and S-waves,

respectively, and tc1otc2 correspond to time instances where the

computed apparent velocity is significantly affected by the convo-

luted pressure and shear waves. A regression analysis,shown in Fig. 6,

reveals a linear relationship between tc1,tc2and the distance of the

reference sleeper from the source with an R2 value greater than 0.99.

Generalized expressions for any reference soil with cs,refare derived

tc1 1

cs,ref 0:579 Dref 0:552, Dref44m; tc1in s 11

tc2 1

cs,ref 0:934 Dref0:776, Dref44m; tc2in s 12

Fig. 6 also shows the characteristic times tc3 and tc4 that areused

in the estimate of the geometric damping coefficient discussed in

the next section. Because of definition tc1otc2, these linear

relationships apply only for Dref44 m (more than 3 sleepers

between source and reference sleeper). This is consistent with

the findings in Ref. [1] where it is reported that the number of

significant sleepers for cross-interaction effects is 24 on each side

of the source sleeper.

It is observed that in all but the secondinterval, [tc1, tc2],a linear

relationship accurately describes the variation of the apparent

Fig. 3. Discretized free field sleepersoil interface with 8 node boundary elements.

Fig. 4. Discrete BIRF functions for the vertical translation as obtained from a

rigorous BEM solution for six different sourcereceiver distances.

Fig. 5. Calculated apparent propagation velocity with respect to time for four

different distances of the reference sleeper from the source and the soilsleeper

system ofTable 1. Curve corresponding to reference sleeper at D10 is selected to

illustrate time limitsof thefourdiscrete parts that areobservedin allof thederived

curves.



7/10

velocity with time. The apparent velocity in the interval [tc1,tc2], is

better approximated by a power function. In summary, the

following closed-form expressions are derived for C

Ctref

cp,ref 2cp,refcs ,refcp,ref

tc1,reftp,ref treftp,ref tp,refotrefotc1,ref

a tbref

tc1,refotrefotc2,ref

cs,ref cR,refcs,refts,reftc2,ref

treftc2,ref tc2,refotrefots,ref

cR,ref ts,refotref

8>>>>>>>>>>>:

13

where a 2cp,refcs,ref=tbc1,ref

cs,ref=tbc2,ref

and b ln2cp,refcs,ref=cs,ref=lntc1,ref=tc2,ref are estimated by enforcing the con-

tinuity conditions at the ends of interval [tc1,tc2].

4.3. Determination of geometric damping coefficient

The geometric damping coefficient that appears in Eq.(9) can be

determined in a manner similar to the apparent propagation

velocity described in the previous section. From the amplitudes

of the vertical displacement at characteristic peaks along two

discrete BIRF functions, the geometricdamping coefficient n can be

calculated from Eq. (8). Scaled in time responses are now selected

consecutively to represent the reference response and coefficient n

is calculated as the one required to match the amplitude values of

the reference response with those of the remaining target

responses simultaneously. The calculatedcoefficientn withrespect

to time is illustrated inFig. 7(a)(d).

Similarly to the closed-form expression derivation process for C,

distinct limits are selected for the derivation of the closed-form

expressions for n. Beginningfromthe left these are: [tp, tc1], [tc1, tc2],

[tc2, tc3], [tc3, ts], [ts, tR], [tR, tc4] an d [tc4, N). The n valuescorresponding totp,tc1,tc2,tc3,ts,tRand tc4do not depend on the

distanceand are determined as n(tp)1.5, n(tc1)1.5, n(tc2) 1.0,

n(tc3) 2.0n(ts)0.0,n(tR)0.7 andn(tc4) 1.0. Timestc1, tc2,

tc3, tc4, have been presented inFig. 6 and similarly to tc1 and tc2,

linear relationships between tc3, tc4 and the distance of the

reference sleeper from the source are derived

tc3 1

cs,ref 0:973 Dref0:789, Dref44m; tc3in s 14

tc4 1

cs,ref 1:026 Dref 1:960, Dref44m; tc4in s 15

In all intervals a linear relationship accurately describes the

variation of the geometric damping coefficient with time.In

Fig. 6. Arrival timetc1andtc2as-obtained from the discrete BIRF functions ofFig. 4

for six different sourcereceiver distances and the soilsleeper system ofTable 1.

Arrival time tc3 and tc4 discussed in Section 4.3 are also plotted. Best fit curves

indicate a linear relationship with respect to distance of the reference sleeper from

the source.

Fig. 7. Calculated geometric damping coefficient with respect to time at (a) D4;

(b)D7; (c)D10 and (d) D12. A smoother curve, discretized in seven time intervals,

representing the derived closed-form expressions for n is also illustrated.



8/10

summary, the following expressions are derived for n:

ntref

1:5 tp,refotrefotc1,ref

1:5 2:5tc2,reftc1,ref treftc1,ref tc1,refotrefotc2,ref

1:0 1:0tc3,reftc2,ref treftc2,ref tc2,refotrefotc3,ref

2:0 2:0ts,reftc3,ref treftc3,ref tc3,refotrefots,ref

0:7tR,refts,ref

trefts,ref ts,refotrefotR,ref

0:7 1:7tc4,reftR,ref treftR,ref tR,refotrefrtc4,ref

1:0 tc4,refotref

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

16

4.4. Closed-form expressions

Closed-form expressions for scaling of time and amplitude are

summarized in Tables 2 and 3, respectively. These expressions

provide a simple algorithm that needs only a reference response at

a distance Drefto predict the response at a greater distance D due to

thesame excitationforce.The reference response is associatedwith

the referencesystem whose geometrywas presented in Table 1 and

soil medium properties as represented by cp,ref, cs,ref, cR,ref. The

predicted response is associated with an arbitrary system of the

same geometry as the reference but different soil medium asrepresented by cs,i. With such expressions the computational effort

is significantly minimized since only the reference response at a

short distance Dref needs to be calculated through rigorous

boundary element analysis. For longer distances and the discrete

time intervals over theduration of the reference response shown in

Tables2 and 3, thecorresponding closed-form expressions are used

in lieu of boundaryelement analysis. The limits of the discretetime

intervals of the reference response are: (a) the arrival times of the

primary waves which are computed as: tp,refDref/cp,ref,ts,refDref/

cs,ref and tR,ref Dref/cR,ref; and (b) arrival times tc1,ref, tc2,ref,

tc3,ref, tc4,ref, which are calculated from Eqs. (11), (12), (14) and

(15), respectively. The proposed closed-form expressions are onlyvalid forDref44 m and for sleepers having the same geometry as

the ones of the reference system presented in Table 1.

5. Model validation and implementation

The performance of the closed-form solutions is evaluated

quantitatively on the actual response of a sleeper located at a

distance D D1616.245 m from the source. The source sleeper is

excited with a fourth-order B-Spline impulse force and the BIRF of

the receiver is computed numerically using the BEM method. It

should be emphasized that this BIRF was not considered in the

developmentof the scaling parameters and serves as a benchmark.

To validate the proposed model theBIRF of the receiver of thesame

system is computed by the scaling of the two reference responses,

DrefD77.695 m, and DrefD1212.54 m. A comparison of the

three BIRFs is shown in Fig. 8. The agreement of the BIRFs as scaled

from the two reference responses with the BIRF obtained numeri-

cally is evident.

The time history of the response of the receiver atD16due to an

arbitrary load is shown inFig. 9. The load function is shown as an

inset in the figure. The response is computed numerically based on

the superposition scheme described by Eq. (3).

In the particular example Bis a scalar and is represented by the

two scaled BIRFs and the computed BIRF presented in Fig. 9,

whereasP is the load function. Comparison between the real and

the scaled responses indicate a high accuracy of the scaled BIRFs.

Therefore, it is demonstrated that the proposed method can

reproduce the BEM solutions accurately for all practical purposesin an extremely efficient manner.

6. Conclusions

Thephysical railsleepersystem that consists of a large number

of sleepers loaded in an asynchronous pattern due to the moving

Table 2

Closed-form expressions for scaling of time.

Reference time Time scaling

Apparent propagation velocity, C Scaled time,ti

trefotp,ref tref DDref

cs,refcs,i

tp,refotrefotc1,refcp,ref

2cp,refcs,refcp,reftc1,reftp,ref

treftp,ref tref DDref

V

cs,refcs,i

tc1,refotrefotc2,ref a tbref

a 2cp,refcs,ref

tbc1,ref

cs,ref

tbc2,ref

b ln 2cp,refcs,ref

cs,ref

=ln

tc1,reftc2,ref

tc2,refotrefots,ref cs,ref cR,refcs,refts,reftc2,ref

treftc2,ref

trefots,ref cR,ref

Table 3

Closed-form expressions for scaling of amplitude

Reference time Amplitude scaling

Geometric damping coefficient,n Scaled amplitude,wi

trefotp,ref 0.0wref

DrefD

n

cs,refcs,i

2rrefri

tp,refotrefotc1,ref 1.5

tc1,refotrefotc2,ref 1:5 2:5

tc2,reftc1,ref treftc1,ref

tc2,refotrefotc3,ref1:0

1:0

tc3,reftc2,ref treftc2,ref

tc3,refotrefots,ref2:0

2:0

ts,reftc3,ref treftc3,ref

ts,refotrefotR,ref 0:7

tR,refts,ref trefts,ref

tR,refotrefotc4,ref 0:7 1:7

tc4,reftR,ref treftR,ref

trefotc4,ref 1.0



9/10

train load is reduced to twosubsystems. The first one addresses thenear to the source group of sleepers and the through the soil cross-

interaction effects and has been presented in previous work of the

authors. The second subsystem addresses the far from the source

sleepers andthe travelingwave effects andis thefocus of this work.

This paper presented a procedure to develop scalable reduced

modelsin the time domain forthe vibrationresponse analysisof far

fromthe source railroad sleepers to a B-Splineimpulseexcitation. A

two sleeper system is considered consisting of a source sleeper

loaded by a vertical B-Spline impulse function and a receiver

sleeper located at a large distance from the source where the

B-Splineimpulse responsefunction will be computed. It is assumed

thatfor the far sleepers the cross-interactioneffectswith the source

are not important, and the scattering due to the presence of other

sleepers between the source and receiver will yield non-conserva-tive results. In order to compute the BIRF of any two sleepers a full

model that includes thefree surface in a discrete form is considered

and BEM solutions are computed. The number of degrees of

freedom of the full model is in the order of thousands. The BIRF

function of the receiver sleeper itself represents a reduced model

that can be used in lieu of the full modelto compute theresponseof

the receiver sleeper to any arbitrary excitation and can be readily

combined with the near the source sleeper subsystem. The BIRF

model preserves the frequency content and all dynamic character-

istics of the full model; however, it has a maximum of only six

degrees of freedom in general. Furthermore,it is demonstratedthat

this BIRF function model is scalable with respect to soil properties

and distance from the source and the proposed scaling procedure

introduced three scaling parameters, i.e., the soil shear wave

velocity,the apparentwave propagation velocity andthe geometric

damping coefficient. The soil shear wave velocityis used forscaling

the model with respect to the soil properties and is constant. The

apparent wave velocity and geometric damping coefficients are

used forscaling the model with respect to distance from thesource,

are piecewise smooth functions of time, and are presented in a

closed form through regression analysis. The scaled models

increase further the computational efficiency of the BIRF models

since only the BIRF of a reference sourcereceiver sleeper systemneeds to be computed. BIRF models for the remaining receiver

sleepers in the railroad track are accurately approximated through

the scaling procedures. The scaling operations however are not

valid for source receiver distances less than four sleepers apart.

Althoughthe development of the procedure focused on the vertical

vibration mode of the receiver sleeper, other important modes can

be approximated in a similar way. The reduced models are

validated through comparisons with other BEM solutions, and

theiraccuracy and efficiency are established. The present approach

can be extended to more complex problems like layered soil, effect

of topography or coupled vibrations of nearby structures, as

reported in forthcoming papers.

Acknowledgement

This work was supported by the National Science Foundation

under Grant# CMMI-0800414. Any opinions, findings and conclu-

sions or recommendations expressed in this material are those of

the authors and do not necessarily reflect those of the National

Science Foundation.

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