Dynamic Interaction of Various Beams With the Underlying Soil-finite and Infinite, Half-spance and...

8/6/2019 Dynamic Interaction of Various Beams With the Underlying Soil-finite and Infinite, Half-spance and Winkler Models

1/26

European Journal of Mechanics A/Solids 27 (2008) 933958

Dynamic interaction of various beams with the underlying soil finite and infinite, half-space and Winkler models

L. Auersch

Federal Institute for Materials Research and Testing, D 12200 Berlin, Germany

Received 6 February 2007; accepted 12 February 2008

Available online 10 March 2008

Abstract

Various beams lying on the elastic half-space and subjected to a harmonic load are analyzed by a double numerical integration

in wavenumber domain. The compliances of the beamsoil systems are presented for a wide frequency range and for a number

of realistic parameter sets. Generally, the soil stiffness G has a strong influence on the low-frequency beam compliance whereas

the beam parameters EI and m are more important for the high-frequency compliance. An important parameter is the elasticlength l = (EI/G)1/4 of the beamsoil system. Around the corresponding frequency l = vS/ l, the wave velocity of the combinedbeamsoil system changes from the Rayleigh wave vR vS to the bending wave velocity vB and the combined beamsoil wave hastypically a strong damping. The interaction frequency l is found not far from the characteristic frequency 0 = (G/m)1/2 wherean amplification compared to the static compliance is observed for special parameter constellations. In contrast, real foundation

beams show no resonance effects as they are highly damped by the radiation into the soil. At medium and high frequencies,asymptotes for the compliance of the beamsoil system are found, u/P (vPai)3/4 in case of the dominating damping andu/P (m2)3/4 for high frequencies. The low-frequency compliance of the coupled beamsoil system can be approximatedby u/P 1/Gl , but it also depends weakly on the width a of the foundation. All numerical results of different beamsoil systemsare evaluated to yield a unique relation u/P0 = f(a/l). The integral transform method is also applied to ballasted and slab tracksof railway lines, showing the influence of train speed on the deformation of the track beam. The presented results of infinite beams

on half-space are compared with results of finite beams and with infinite beams on a Winkler support. Approximating Winkler

parameters are given for realistic foundation-soil systems which are useful when vehicle-track interaction is analyzed for the

prediction of railway induced vibration.

2008 Elsevier Masson SAS. All rights reserved.

Keywords: Beam dynamics; Beamsoil interaction; Bending waves; Rayleigh wave; Railway track vibration; Elastic length; Wavenumber integrals

1. Introduction

Soilstructure interaction is a wide field of mechanics. The oldest and maybe simplest model is a beam on a

Winkler support (Zimmermann, 1888). The Winkler support (Winkler, 1867) is a simplified model of the soil which

reacts locally and linearly to a load. A more realistic soil model, the elastic half-space, was introduced by Boussinesq

* Tel.: +49 30 8104 3290; fax: +49 30 8104 1727.

E-mail address: [email protected].

0997-7538/$ see front matter

2008 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechsol.2008.02.001


2/26

934 L. Auersch / European Journal of Mechanics A/Solids 27 (2008) 933958

Nomenclature

a width of the beam

b mass length of the beamsoil system (37)

c

viscous Winkler support constant

D material damping of the soil

ex , ey , ez unit vectors

E elasticity modulus of the beam

EI bending stiffness of the beam

f frequency

f0 fundamental frequency of the beamsoil

system (38)

G shear modulus of the soil

h height of the beam

H compliance of the soil in frequency

wavenumber domain (13)

HS compliance of the soil for strip wave excita-tion (20), (22)

HBS compliance of the beamsoil system in fre-

quencywavenumber domain (25)

i imaginary unit

I moment of inertia of the beam

I identity matrix

k wave number

kS, kP shear/compressional wave number (10)

k elastic Winkler support constantKB stiffness of the beam in frequencywave-

number domain (23)KBS stiffness of the beamsoil system in frequen-

cywavenumber domain (24)

KW Winkler support stiffness

l elastic length of the beamsoil system (36)

m

mass per length of the beam

P force

P contact force per lengthqS, qP vertical shear/compressional wave number

of the soil (10)

t time

tx , ty , tz stress components

u displacement (vertical displacement of the

beam)

uB vertical displacement of the beam

uS vertical displacement of the soil surface

u displacement of the soil

v velocity of the strip wavevS, vP velocity of the shear/compressional wave (2)

vT velocity of the moving load

x coordinate across the beam

y coordinate along the beam

z vertical coordinate downwards

Poissons ratio of the soil

mass density

circular frequency

l interaction circular frequency (38)

0 fundamental circular frequency (38)

1 characteristic circular frequency (38)

(1885) for static and Lamb (1904) for dynamic loading. The static beam-half-space interaction was first examined by

Biot (1937) and later by Rvachev (1958), Vesic (1961) and others, see Selvadurai (1979) for details and Wang et al.

(2005) for a recent review.

Dynamic soilstructure interaction has been intensely studied starting in the 70s with a special focus on rigid

foundations on continuum soils and computer methods (Gazetas, 1983; Wolf, 1985). At the same time, flexible track

models are dynamically studied on Winkler support (Fryba, 1972; Grassie et al., 1982; Knothe and Grassie, 1993).

Especially in the last decade, dynamically loaded beams on continuum soils are investigated as models for tracksunder moving loads (e.g. Dietermann and Metrikine, 1996, 1997; Lieb and Sudret, 1998; Grundmann et al., 1999;

Sheng et al., 1999; Kononov and Wolfert, 2000). While moving loads became an attractive research topic, the basics

of dynamic beam-half-space interaction are rarely found in literature.

The present research was motivated by earlier work on different flexible foundations (Auersch, 1991, Fig. 1),

namely for plate-soil interaction (Auersch, 1994b, 1996). A number of similarities between plate and beam on half-

space are observed. In addition, the coupled finite-element boundary-element method (FEBEM, Auersch and Schmid,

1990) was used to calculate more detailed 3-dimensional railway track models (Auersch, 2005a).

The infinite beam on the continuum soil is calculated by integral transform methods (ITM) in the present contri-

bution. The solution is presented as a Fourier integral which can be solved analytically (Dietermann and Metrikine,

1996), numerically either by discrete Fourier transform algorithms (Sheng et al., 1999) or with an additional wavelet

transform (Lieb and Sudret, 1998). The present contribution uses a simple integration scheme. Compared to a plate

on half-space (Auersch, 1994b), the beam on half-space needs more numerical effort because two infinite integrals


3/26

L. Auersch / European Journal of Mechanics A/Solids 27 (2008) 933958 935

Fig. 1. The interaction of flexible foundations with the soil: beams, plates and railway tracks.

must be solved per frequency whereas it is only one integral in case of an infinite plate. There is also one additional

parameter, the width of the beam, so that the beam problem is more complex in the numerical and physical aspect.

The present contribution investigates the normal behaviour of a beam on half-space under a fixed dynamic load,

which is rarely found in literature. Moreover, a number of asymptotic expressions and approximations are found

which help to get a better understanding of the beam on half-space system. Namely, the possibility of a Winkler

approximation is discussed in detail.

The contribution consists of three main parts. The first part consists of the fundamentals and the methods (Sections 2

and 3). The main part consists of the presentation and discussion of the results for a beam on half-space, wavenumber

domain results in Sections 4 and 5 and most important the frequency dependent compliances of various beam

soil systems (Section 6). The third part is concerned with approximations (Section 7) and applications to railway

tracks (Sections 810). In these last sections, the displacement and load distributions along the beam are discussed

(Section 9) as well as the moving load effect for a track beam (Section 10).

2. Fundamentals

The soil is considered as an elastic continuum, which is described by the field equation for the displacement u

G

div grad u+ 11 2 graddiv u

= u (1)


4/26


and by the material constants G shear modulus, Poissons ratio, mass density, which yield the wave velocities

vS=

G

and vP =

2 21 2 vS (2)

of the shear and compressional wave.

The vertical displacements u of the beam fulfill the differential equation

EIu +mu= P (3)with EI the bending stiffness, m the mass per length, P the force per length. The contact force P between thebeam and the soil is distributed uniformly across the width a of the beam. There are no horizontal contact stresses

(relaxed boundary conditions)

tz|x| a/2= P/a,

tx = ty = 0.tz|x|> a/2= 0, (4)

The compatibility between the displacements of the beam and the soil can be stated in two ways:

The displacements of the soil and the beam are the same at the centre line of the beam (weak coupling)

uS(x = 0, y) = uB (y), (5)or the average displacements of the soil across the foundation width are equal to the displacements of the beam

(average coupling)

1

a

a/2a/2

uS(x,y)dx = uB (y). (6)

The beam is excited by a vertical harmonic point load and the vertical displacements u at the point of excitation

are calculated as a function u/P(f) of the frequency.

3. Methods of calculation

The solution for the beam on half-space is found in the wavenumber domain in the same way as in Dietermann

and Metrikine (1996), Grundmann et al. (1999), and Sheng et al. (1999). The vertical displacements due to a vertical

harmonic point load P= P ei t can be represented as an integral over the wavenumber ky

u(y,)= 12

HBS(ky, )P(ky, )eiky y dky (7)

with the compliance HBS of the coupled beamsoil system in wave number domain and u(y,) the complex amplitude

of the harmonic response.

3.1. Compliance of the soil for plane waves

The compliance of the soil is found by assuming a harmonic vertical stress wave

tz = tz(kx , ky )ei(kx x+ky y+t ) (8)which travels with the wave velocity

v =/ k and k =

k2x + k2yalong the surface. The corresponding displacements

u

=aPe

i(kx x+ky yqPz+t )

+aSe

i(kx x+ky yqSz+t ) (9)

vary with z and we have from Eq. (1)


5/26


qS=i

k2 k2S with kS=/vS,

qP =i

k2 k2P with kP =/vP (10)for the compressional wave and the shear wave, of which the amplitudes are restricted to

aP kx ex + ky ey qPeZ and aS kex + ky ey qSez. (11)The stress of these waves is evaluated according to

T=G

grad u+ grad uT + 21 2 div uI

. (12)

The wave amplitudes aP and aS are fitted to the boundary conditions, and the vertical compliance of the soil

H (kx , ky , ) =qPk

2S

iG det K(13)

is established with

det K

= k2S

2k22 + 4k

2qSqP. (14)

Asymptotic expressions of this compliance can be given for two cases. For small wave numbers k kP < kS, theroots of Eq. (10) are simply qP = kP and qS= kS so that

H= kPiGk2S

= 1ivP

(15)

which is the imaginary frequency dependent compliance of a damper. For large wave numbers k kS > kP, the rootsof Eq. (10) can be given as

qP ,S=ik(1 P ,S/2) with P ,S= k2P ,S/ k2 (16)yielding an elastic (real) compliance

H= k2S2Gk(k2S k2P)

= 1 Gk

. (17)

3.2. Compliance of the soil for strip waves

Next, a time harmonic strip load

tz(x,y,t)= Pei(ky y+t )p1(x) (18)with

p1(x) =

1/a for |x|< a/2,0 else

(19)

is considered as the excitation of the soil. The corresponding displacements of the strip are calculated for both typesof coupling conditions (5) and (6). The response of the centre line of the strip, which is used for the weak coupling, is

given by the Fourier integral

u

P(ky , ) =

1

2

+

H (kx , ky ,)p1(kx ) dkx =HS(ky , ) =1

KS(ky , )(20)

with the Fourier transform of the strip load

p1(kx )=sin kx a/2

kx a/2. (21)

For the average coupling (6), the compliance HS

of the soil is given by the average of the displacements across the

foundation width


6/26


HS(ky , ) =1

2a

a/2a/2

+

H (kx , ky ,)p1(kx )eikx x dkx dx

=1

2a

+

H (kx , ky ,)p1(kx )

a/2

a/2

e

ikx x

d x d kx

= 12

+

H (kx , ky ,)p1(kx )sin kx a/2

kx a/2dkx

= 12

+

H (kx , ky ,)p21(kx ) dkx . (22)

The latter (averaging) method is used throughout this contribution as it yields smoother high-frequency results (see

Steenbergen and Metrikine, 2007) with the same numerical effort.

3.3. Stiffness of the beam for wave excitation and beamsoil interaction

The stiffness of the beam KB follows directly from the differential Eq. (3)

KB (ky , ) =EIk4y m2. (23)To yield the stiffness of the coupled beamsoil system, the stiffness of the beam and the soil must be added

KBS(ky , ) =KB +KS (24)and

HBS(ky , ) =1

KB+

KS. (25)

The displacements of the track are found by the inverse Fourier transformation of the displacements u in frequency-

wavenumber domain

u(ky , ) =HBS(ky ,)P,

u(y,)= P2

+

HBS(ky ,)eiky y dky (26)

for a point load on the beam which has the constant wavenumber transform P. The integral is further simplified for

the point of excitation (y = 0)

u(0, ) =P

2

+

HBS(ky , ) d ky . (27)

The integrals (22) and (27) are solved numerically. The integration steps in wavenumber domain are fitted to the

compliance of the soil so that the poles of the compliance are well represented. A realistic damping of the soil

D = 2.5% with G=G0(1+ i2D)is introduced so that the integrand is well defined for all frequencies and wavenumbers. Special care is necessary for

the case = 0, where all poles are at k = 0. A sufficiently good approximation is achieved by evaluating the integrandt a very low frequency f= 0 or by using an integration scheme without the lower bound of the interval.

The same formulas can be used for a beam under a moving load. The transfer function HBS must be evaluated for

shifted frequencies

ky vT instead of


7/26


for a load moving with speed vT in positive y-direction. The results are in the moving coordinate system, (see Di-

etermann and Metrikine, 1996; Grundmann et al., 1999; Auersch, 2008, for the derivation). Moreover, the method

can be applied to horizontally layered soil by introducing the corresponding compliance functions H of a layered soil

(Auersch, 1994a).

3.4. Finite-element boundary-element method

Beams of finite length or more detailed track systems are calculated by the combined finite-element boundary-

element method which is described in Auersch (2005a). The method is based on point load solutions (Greens

functions) of homogeneous or layered soils which are used to construct the soil stiffness matrix of the foundation

points. The dynamic stiffness matrix of the soil is added to the global dynamic stiffness matrix of the finite element

model of the structure (the beam, the track). The combined finite-element boundary-element matrix is solved for

given external loads or displacements (Auersch, 1988). An example result of beams with finite length is discussed in

Section 7.5.

4. Results about the soil vibrations under strip wave loading

At first, the response of the soil to a strip wave of velocity v is discussed. The compliance of the soil to such a strip

wave is presented in Figs. 2 and 3 as a function of the relative wave velocity v/vS for different width a of the strip

Fig. 2. Compliance of the soil with shear wave velocity vS

to a strip wave excitation of velocity v as a function of the relative wave velocity v/vS(variation off), ak =1 0.5, ! 1, P 2, +4.


8/26


Fig. 3. Compliance of the soil with shear wave velocity vS to a strip wave excitation of velocity v as a function of the relative wave velocity v/vS(variation ofk), akS= a/vS=1 0.5, ! 1, P 2, +4.

wave. Fig. 2 shows the compliance when the frequency is varied and the wavenumber is fixed. It can be seen that the

compliance approaches the value

HS(v/vS)=1

ivPa. (28)

If the wavenumber is increased in Fig. 3 and that means that the wave velocity is decreased, the compliance of the soil

tends to the value

HS(v/vS 0)=1 Gka

. (29)

Both asymptotes are reached earlier for wider strip waves. The asymptotes (28) and (29) of the soil for the strip wave

are the same as for the plane wave (15) and (17), but divided by the strip width a.

Between the two asymptotes for low and high wave velocities, there is a maximum of displacements close to the

value v/vS= 1 that is at the velocity of the Rayleigh wave, the free wave of the soil. The amplitude at this maximum islimited due to the material damping of the soil and to the averaging of different waves in case of the strip load. Besides

the dominating effect of the Rayleigh wave there is also a minor effect (small oscillations) at v/vS=

2=

vP

/vS

due

to the compressional wave.


9/26


Fig. 4. Compliance of the beamsoil system to strip wave excitation as a function of the wave velocity v/vS, f =1 5, ! 10, P 20, +40, 80,E 160 Hz.

5. Beamsoil interaction: compliance for strip waves, dispersion and damping

The beam displacements under wave excitation can be given explicitly as

u

P= 1

EIk4 m2 . (30)

For high frequencies, the inertia of the beam dominates giving the asymptote

1/m2, whereas for high wave

numbers, the bending stiffness dominates with the asymptote 1/EIk4. Between these asymptotes, there is a maximumat the dispersive wave velocity of the beam

vB = 4

EI

m

. (31)

The combined compliance of beam and soil is shown in Fig. 4 for an example beam (see parameters (33)). The

compliance is given for certain frequencies as a function of the relative wave velocity v/vS as in Fig. 2. Different

parts of the beamsoil system determine the behaviour for different frequencies and different wave velocities. The

strongest influence is found at low wave velocities where the bending stiffness of the beam yields a strong increase

with v/vS. The soil has a dominating influence at low frequencies and a medium range of v/vS, where a weaker

increase with v/vS is found. Close to v/vS= 1, the low-frequency compliances have a clear maximum which belongsto the Rayleigh wave of the soil. For higher wave velocities, the compliances are complex and the radiation damping

of the soil is predominant. At high frequencies and high wave velocities, the mass of the beam becomes the most


10/26


important parameter which reduces the compliance of the beamsoil system. A clear maximum arises which belongs

to the bending wave of the free beam.

The dispersion of the beamsoil system can be concluded from these results as follows: At low frequencies, the

combined system has the same free wave as the soil the Rayleigh wave with v = vR vS. At high frequencies, thecombined system behaves like the free beam v

=vB . In the transition zone between these two regimes, the maxima of

the compliance functions are very small. This is due to the high damping of the beamsoil interaction which is foundbetween 20 and 80 Hz. This highly damped frequency range is above the coincidence frequency

fC =1

2

m

EIv2S= 13 Hz (32)

of the two (beam and soil) wave velocities. Similar observations are made for the combined plate-soil system (Auersch,

1994b).

6. Beamsoil interaction: Compliance of various beamsoil systems for harmonic point-load

The dynamic behaviour of the beam on the soil is discussed for a standard beam with the following properties:

a = 0.5 m, h = 0.5 m, E = 3 1010 N/m2, = 2.5 103 kg/m3, I= ah3/12 = 5.2 103 m4, m = a h= 625 kg/m. (33)

The standard soil is defined by

G= 8 107 N/m2, = 2 103 kg/m3, = 0.33, D = 2.5%,and that means

vS= 200 m/s and vP = 400 m/s.The relevant parameters of the beamsoil system are varied and the influence of these variations on the beam behaviour

is discussed in the following sections.

6.1. Variation of the stiffness of the soil

At first, the stiffness of the soil is varied and the corresponding compliances of the beamsoil system are shown

in Fig. 5. A strong influence of the soil is found at zero frequency. The static stiffness K0 is almost proportional

to the soil stiffness, the relation is near to K0 G0.75. The influence of the soil is considerably reduced at highfrequencies. The four amplitude curves of different soil stiffnesses are close together so that it is concluded that beam

properties determine the behaviour at high frequencies. No resonance maximum occurs for these beamsoil systems.

The compliances decrease monotonously with increasing frequency and the phase delays increase to values of more

than 100

. These amplitude and phase changes with frequency are strongest for the softest soil. In contrast, the stiffest

soil has almost constant amplitudes and the weakest and a constantly increasing phase delay.

6.2. Variation of the height of the beam

Fig. 6 shows the effects if the height of the beam is varied. This variation means mainly an increase of the bending

stiffness, but also a minor increase of the beam mass. All corresponding compliance curves in Fig. 6 show the same

frequency dependent behaviour. The amplitudes are reduced for the higher beams but the reduction is small compared

to the strong variation of the stiffness of the beam. Whereas the beam stiffness EI is increased by a factor of 17, the

static stiffness K0 of the beamsoil system is increased only by a factor near to 2, yielding a relation K0 EI0.25.The relative changes are stronger at higher frequencies where the displacements vary between 0.25 to 1 mm/MN. The

differences between the different phase curves are small and an asymptotic value of about

130

can be observed at

high frequencies.


11/26


Fig. 5. Compliance of the standard beam on different soils, vS=1 100, ! 150, P 200, +300 m/s.

6.3. Variation of the width of the beam and the foundation

If the width of the beam is varied, there are changes in the bending stiffness of the beam as well as in the beam mass,

thus keeping the velocity of the bending wave constant. Besides the variation of the beam stiffness and mass, there is

also a change in the soil stiffness. The effect of this more complex variation is shown in Fig. 7. There is a considerable

influence of the beam width on the static stiffness. The widest beam has the highest stiffness (lowest compliance), and

it has also the strongest phase delay. The strong decrease of the phase turns into a weak decrease at about 40 Hz for

the widest beam with a = 4 m, and at about 70 Hz for a = 2 m. Therefore, all phase curves come close together forhigh frequencies above 100 Hz. If the width of the beam is increased even stronger (Fig. 8), the compliances have a

strong decrease in the whole frequency range. The phases rapidly reach values of80 to 100 degrees and are nearlyconstant in the following frequency range. These are typical aspects of 2-dimensional systems on the homogeneous

half-space: infinite static displacements and the dominant behaviour of a viscous damper (Auersch, 1991).

The next variation tries to extract the influence of the width of the foundation. The foundation width is studied for

extreme values from very small a = 0.1 m and rather wide a = 3 m and no other parameter than the foundation widthis changed. With the width a, the influence of the soil is modified yielding quite different characteristics (Fig. 9).

A weak resonance occurs at 60 Hz in case of the very narrow foundation with a = 0.1 m. This reflects the fact thata considerable mass is supported by a small soil area. For higher frequency, the amplitudes decrease rather strongly

and the curve comes close to the other curves. For larger foundation widths, the curves decrease more smoothly

from the static value and they show no resonance effect, although the phase delays reach values higher than 90

. The

phase decrease is higher for larger foundation areas and that means that the damping by the soil is increased with the


12/26


Fig. 6. Compliance of beams with different height h=1 0.5, ! 0.7, P 1.0, +1.3 m on a soil with vS= 200 m/s.

foundation area. The phases at high frequencies are in reverse order; the widest foundation has the lowest phase delay.

This is once again an effect of the different damping of the different foundation areas which results in lower phase

delays above the resonance frequency for wider foundations.

Besides the strong damping effect of the foundation width, there is also a minor effect on the static stiffness. While

the foundation width is increased by a factor of 30, the resulting displacements are doubled. This effect could be

presented as a potential factor

u

a0.25

or more complete with the other parameters as

uG0.81EI0.19 a0.25. (34)This holds for the realistic range of parameters that are examined in this contribution. The potential law u G0.8was also found for finite 3-dimensional models of railway tracks (Auersch, 2005a).

6.4. Variation of the mass of the beam

In Fig. 10, the mass of the beam is varied between 0 and = 104 kg/m3, that is four times as heavy as a concretebeam. Whereas massless and light beams show a slow decrease of displacement amplitudes, the beams with high mass

yield clear resonance effects with a strong decrease at higher frequencies. The resonance maximum coincides with a

phase of60 and occurs at 30, 40 and 55 Hz for the three highest mass factors. The lighter (normal) beams pass


13/26


Fig. 7. Compliance of beams with different width a =1 0.5, ! 1, P 2, +4 m on a soil with vS= 200 m/s.

through this phase value without any resonance maximum, whereas the massless beam does not pass this resonance

phase. The maximum phase delay, which is reached by the heaviest beam, is 135.

6.5. Dimensionless rules for the beamsoil compliances

The results have been presented for various specific parameters. These results can be transformed into dimen-

sionless laws and thereby transferred to other situations and combinations of parameters. The problem is described

completely by the following set of parametersu= f (P,G,,,EI, m,a,). (35)

It is useful to define an elastic length

l = 4

EI

G(= 1.2 m), (36)

a mass length

b = 2

m

(= 0.56 m) (37)

and corresponding frequencies

l = vS/ l (= 2 27 Hz), 0 = vS/b (= 2 55 Hz), 1 = vS/a (= 2 64 Hz). (38)


14/26


Fig. 8. Compliance of beams with different width a =1 5, ! 10, P 15, +20 m on a soil with vS= 200 m/s.

The values for the standard beam on standard soil (33) are given in brackets. The following dimensionless parameters

u = uGlP

or u = uGaP

= uGP

,

l = lvS

or a = avS

or b = bvS

, (39)

m=

m

l 2or m

=m

al =m

lor m

=m

a 2

can be used to present the solution as a function of 8 4 = 4 dimensionless parameters, for example asu = f (l, a / l , m, ) or u = f (a, a / l , m, ) or u = f (b,a/b,a/l,). (40)

Simpler laws are found for special cases:

For a static load, the normalized displacements are

u = f(a/l,). (41)This function of the normalized width a/ l is shown in Fig. 11 for the standard value = 0.33 and all calculationsperformed. All results fit well in one curve which shows a slight decrease of the displacements with increasing relative

width a/ l and which can be described roughly by

u = f (a/l, = 0.33)= 0.3 0.09ln(a/l). (42)


15/26


Fig. 9. Compliance of a standard beam on a soil with vS= 200 m/s and for varied foundation width a =1 0.1, ! 0.3, P 1, +3 m.

For the results with a/l < 1, the static relation can further be simplified to

u = f (a/l, = 0.33)Gl(a/l)0.25

which perfectly agrees with the results of Vesic (1961).

For a massless beam, the mass parameter m is zero and the solution

u = f (l, a / l , ) (43)

is a function of three dimensionless parameters.A beam with an infinite width (a plate excited by a line load P) has not the parameter a and therefore no a or a

can appear in the dimensionless formulas. This leads to a function

u = f (l, m,). (44)Due to the fact, that the parameter EI/a (similar to the bending stiffness B of a plate) must be used instead ofEI, the

definition ofl and the dimensionless parameters are modified to

l2 = 3

EI

aG 3

B

Gand l = l2

vS. (45)

The modified parameters u and m are given in Eq. (39).The present dimensionless functions are defined in the same way as it is usual in soil dynamics. They are well

suited for the low-frequency behaviour of the beamsoil system. For the high-frequency behaviour, other normalized


16/26


Fig. 10. Compliance of beams with different mass density =1 0, ! 1,P 2.5, +5, 10 103 kg/m3 on a soil with vS= 200 m/s.

Fig. 11. Static compliance of all calculated beamsoil systems as a functions of the width ratio a/ l.


17/26


displacements or compliances are more suitable, which are related to the radiation damping of the soil or the mass of

the beam, see Section 7.2.

7. The beam on a Winkler soil, asymptotes and approximations

7.1. The solution for a beam on a Winkler soil

The Winkler foundation is a much simpler foundation model than the continuum soil. The soil is replaced by a

visco-elastic support which is easily described by

KW(,k)=KW() = k + ciwith a spring k and a damper c constant per length. A beam on a Winkler support has the simple static solution

u

P= 1

2

2EI1/4 k3/4(46)

which can be extended to the dynamic solution asu

P= 1

2

2EI1/4(k + ci m2)3/4. (47)

This explicit solution shows that the influence of the beam stiffness is constant u/P EI1/4 for all frequencies.The second part of the Winkler solution has some similarity with a single-degree-of-freedom system. The factor

k+ ci m2 is the same, but it is set to the power of 3/4. Therefore, the frequency dependent effects are reduced,for example the resonance amplification and the phase delay which is 67.5 at resonance and 135 at the high-frequency limit instead of 90 and 180 for the SDOF-system. Examples of Winkler solutions are given in Fig. 12.

7.2. High-frequency asymptotes for half-space and Winkler model

The Winkler model at medium and high frequencies can be expressed as

u

P= 1

2

2 EI1/4(ci m2)3/4. (48)

A similar expression holds for the high-frequency asymptote of the beam on half-space system

u

P= 1

2

2 EI1/4(vPai m2)3/4(49)

if the asymptote (15) or (28) of the soil is used in the integral (27). The Winkler as well as the half-space asymptote

include two special cases: The special case of a free beam without any soil

u

P= 1

2

2 EI1/4(m2)3/4(50)

and the special case of a massless beam on the soil

u

P= 1

2

2 EI1/4(vPai)3/4. (51)

The formulas describe the amplitude reduction with increasing frequency

u

P 1

(

m2)3/4

3/2 or uP 1

(vPai)3/43/4 (52)

and the corresponding phase delays of 135 and 67.5.


18/26


Fig. 12. Compliance of the standard beam on Winkler support with stiffness k =1 3, ! 7,P 12, +27 107 N/m2 and damping c =1 15, ! 23,P 30, +45 104 N s/m2.

7.3. The approximation by a Winkler foundation

Besides the similar high-frequency asymptotes of the Winkler and continuum soil, it was found that the static

half-space solution is very close to the static Winkler solution

u

P 1

EI1/4 G3/4. (53)

If the minor influence of a is disregarded, the compliance can completely approximated by the dynamic Winkler

solution. The parameters of the Winkler approximation are expected at

c = vPa,k =G G/(1 )

according to the asymptotic values (15) and (17) for the soil in wavenumber domain. In fact, the Winkler parameters

are a bit lower for the damping

vSa c vPa,

and the static stiffness k is in the range

G k 1.5G for 0.1 a/ l 1.


19/26


Fig. 13. Compliance of the standard beam of finite length on different soils, vS=1 100, ! 150, P 200, +300 m/s.

The approximate solution is shown in Fig. 12 which is in good agreement with the exact half-space solution in Fig. 5.

A similar approximation was suggested in Auersch (1991) for foundation beams and in Auersch (2005b) for railway

tracks.

7.4. The limiting case of an infinitely wide beam

If the width of the beam and foundation is increased, the mass, bending stiffness of the beam and the damping of

the soil are proportional to the width. Only the static stiffness of the soil does not increase linearly with the foundation

width. The results of the wide beams can also be presented as uGa/P to show the approach to the 2-dimensional

solution uG/P of an infinitely wide beam under a line-load excitation P. The four beams of 5 to 20 m width of Fig. 8would give one identical curve. Only close to the zero frequency, there are considerable deviations which are due to

the fact that for the infinitely wide beam as for all 2-dimensional loadings of the homogeneous half-space the static

displacements tend to infinity whereas the displacements are finite for the finitely wide beams. The high-frequency

asymptotes are the same for 2 and 3 dimensions, but they are reached at lower frequency for the 2-dimensional case.

Similar observations were made for infinitely wide beams of different finite lengths in Auersch (1991).


20/26


Fig. 14. Compliance of a ballast-track beam on different soils, vS=1 100, ! 150, P 200, +300 m/s, beam width as track width.

7.5. Beams of finite or infinite length

As an example, a standard beam of ten meter length is investigated in Fig. 13 (Auersch, 1991). The compliances for

different soil stiffnesses are very similar to those of infinite beams in Fig. 5. The only exception is a weak resonance

around 100 Hz. To discuss this resonance, the eigenfrequencies of a free beam without any soil contact are given

fB1 = 18 Hz, fB2 = 50 Hz (antimetric), fB3 = 98 Hz.

The resonance frequencies f0 of the rigid beams on the different soils are

f0 = 28, 41, 55, 83 Hz for vS= 100, 150, 200, 300 m/s.

No bending modes can exist at frequencies below the fundamental frequency f0. Instead of this, the elastic modes of

the free beam are shifted to frequencies higher than f0. The frequency range around f0 is highly damped. Therefore,

the first bending mode of fB1 is highly damped and can not be seen in the compliances. The third mode, however,

is less damped and can be seen for the softest soil where the difference 100 Hz = fB3 f0 = 28 Hz between thefree and the fundamental eigenfrequency is sufficiently large. So it may be concluded that in many cases, the dynamic

compliances of finite and infinite beams on the soil are very similar. This is also true for finite and infinite plates on

the soil (Auersch, 1996).


21/26


Fig. 15. Compliance of a ballast-track beam on different soils, vS=1 100, ! 150, P 200, +300 m/s, beam width adjusted to track foundationarea.

8. Compliances of different railway tracks

Railway tracks as shown in Fig. 1 are not as simple beams as considered so far. They have been calculated with

detailed finite-element boundary-element models (Auersch, 2005a). The geometric parameters are given in Fig. 1 and

the material properties are

UIC 60 rails:

EI= 2 6.4106 N m2, mR = 2 60 kg/m,and concrete sleepers:

mS= 340 kg/0.6 m (0.6 m the sleeper distance), m =mR +mS= 680 kg/m.The main difference between a 3-dimensional FEBEM calculation and a track beam model is the way of coupling the

track to the soil. The track is coupled to the soil via the sleepers. That means that less than the half of the foundation

area is coupled to the soil whereas the areas between two sleepers have no contact with the soil. Two track beam

models are calculated to approximate the dynamic track behaviour. One assumes the total foundation area in contact

with the soil, the beam width is a = 2.6 m. The other model has an equivalent widtha

=(0.34/0.6)

2.6 m

=1.5 m

which gives the same reduced contact area as for the track on sleepers.


22/26


Fig. 16. Compliance of a slab track (d= 0.2 m) on different soils, vS=1 100, ! 150, P 200, +300 m/s.

The mass m of the track is almost the same as for the standard beam whereas the bending stiffness is about tentimes lower. Therefore, the elastic lengths of the tracks are shorter, for example l = 0.73 m for the ballasted trackon standard soil. The greater width a indicates a stronger damping of the tracks compared to the foundation beams

whereas the characteristic frequencies

f0 = 26, 40, 53, 79 Hz for vS= 100, 150, 200, 300 m/s

are almost the same.The dynamic compliances of the ballasted tracks on different soils are presented in Fig. 14 for the total width

and in Fig. 15 for the equivalent width. In Fig. 16, also a slab track with an additional plate of 0.2 m thickness is

considered. It is clear that the slab track is coupled to the soil by the total foundation area or track width. All tracks,

ballasted tracks with total or reduced width and slab tracks, as well as the 3-dimensional FEBEM-models of Auersch

(2005a), show similar results. There are great differences at low frequencies due to the different soils, but only small

differences at high frequencies. At high frequencies, the properties of the track dominate, especially for the stiffer slab

track where the four different systems end in a single amplitude curve. The phase curves start with a strong decrease,

but at medium frequencies, there is a clear change. A weaker phase decrease starts at

f= 30, 45, 60, 90 Hz for vS= 100, 150, 200, 300 m/s (Fig. 14).The different widths of the beam models for the ballasted track yield different static displacements where the model

with the total width has 40% lower amplitudes than the model with the equivalent width. The slab track gives an


23/26


24/26


Fig. 18. Load distribution under the standard beam for an excitation with f=1 0, ! 25, P 50, +75, 100 Hz.

Fig. 19. Deformation of track beams on different soils with vS=1 100, ! 150, P 200, +300 m/s.


25/26


Fig. 20. Moving load effect for a standard track on soft soil vS= 100 m/s, train speed vT =1 0, ! vR , P vS, +1.5vS m/s.

10. Tracks under moving loads

The wave velocity of the softest soil vS= 100 m/s can be reached by high-speed trains at vT = 360 km/h. Theeffect of such a high-speed load on the track deformation is studied in Fig. 20. The deformation is given for v = 0standing train, v = vR Rayleigh wave velocity, v = vS shear wave velocity and v = 1.5vS supercritical speed.

With increasing train speed, the maximum of downward displacements is shifted behind the load and a new upward

maximum appears in front of the load. This secondary maximum is shifted towards the load with increasing train

speed. The highest amplitude of the track is found at vT = vR . This maximum is twice as high as for the standingtrain. The change of the deformation pattern results in an increasing slope for the running wheel and that means an

increase of the elastic drag and the necessary power supply of the trains. All these findings are in agreement with the

results of Dietermann and Metrikine (1996, 1997), Grundmann et al. (1999), Sheng et al. (1999).The moving load effect the increase of the track amplitudes at critical speed is stronger if the ballast is included

simply as an additional mass of the track beam model. A corresponding higher beam mass ofm = 2000 kg/m yieldsa moving load factor of 2.5.

In contrast, a layered soil with a stiffer underlying half-space can destroy the moving load effect considerably

(Auersch, 2008).

11. Conclusions

Various beams lying on the elastic half-space and subjected to a harmonic load are analyzed by the integral trans-

formation method (ITM). Two infinite integrals in wavenumber domain are solved numerically: one integral for the

compliance of the soil to strip wave excitation, one integral to get the displacements of the beamsoil system un-der a harmonic point-load excitation. The influence of the soil and beam parameters on the dynamic compliance is

investigated in detail. An important parameter is the elastic length l = (EI/G)1/4 of the beamsoil system whichcompares the stiffnesses of the beam and the soil. Generally, the soil has a strong influence on the low-frequency

beam compliance whereas the beam parameters, namely the beam mass, are more important for the high-frequency

compliance. The beam stiffness has a minor influence which is constant in the whole frequency range. The foundation

width mainly determines the foundation damping and very limited the foundation stiffness. The weak influence on

the static compliance is identified as the function u/P0 = f(a/l) which is evaluated from all numerical examples. Athigher frequencies, asymptotes for the compliance of the beamsoil system are found, u/P (vPai)3/4 in caseof the dominating damping and u/P (m2)3/4 for very high frequencies.

The beam on the elastic half-space model is compared with other models, namely with the Winkler model of the

visco-elastically supported beam and its explicit solution. The Winkler model yields similar results and has almost the

same asymptotes, if the Winkler parameter k 1.5G and c 1.5vSa are used.


26/26


The present method is also applied to ballasted and slab tracks for railway lines. The detailed integral method

and the approximate Winkler method can be used for vehicle-track interaction analysis (Auersch, 2005a) and the

prediction of railway induced vibration (Rcker et al., 2005). Moving loads can easily be treated by the same integral

method, and their effects on the beam deformation are given for realistic soil and track parameters.

References

Auersch, L., 1988. Wechselwirkung starrer und flexibler Strukturen mit dem Baugrund insbesondere bei Anregung durch Bodenerschtterungen.

Forschungsbericht 151 (Thesis), Bundesanstalt fr Materialforschung und -prfung, Berlin.

Auersch, L., 1991. Dynamic soilstructure interaction of rigid and flexible foundations. In: Proc. 5th Int. Conf. on Soil Dyn. and Earthq. Eng.,

Karlsruhe, pp. 467478.

Auersch, L., 1994a. Wave propagation in layered soil, theoretical solution in wavenumber domain and experimental results of hammer and railway

traffic excitation. J. Sound Vib. 173, 233264.

Auersch, L., 1994b. Zur Dynamik einer unendlichen Platte auf dem Halbraum Fundamentnachgiebigkeit und Wellenfeld bei harmonischer

Punktlast. Arch. Appl. Mech. 64, 346356.

Auersch, L., 1996. Dynamic plate soil interaction finite and infinite, flexible and rigid plates on homogeneous, layered or Winkler soil. Soil

Dyn. Earthq. Eng. 15, 5159.

Auersch, L., 2005a. Dynamics of the railway track and the underlying soil, the boundary-element solution, theoretical results and their experimental

verification. Vehicle System Dynamics 43, 671695.Auersch, L., 2005b. Simplified methods for wave propagation and soilstructure interaction: The dispersion of layered soil and the approximation

of FEBEM results. In: Proc. 6th Int. Conf. on Structural Dynamics. EURODYN 2005. Millpress, Rotterdam, pp. 13031309.

Auersch, L., 2008. Moving-load and reduction effects of railway tracks on soft soil and isolation elements. J. Sound Vib. 310, 587607.

Auersch, L., Schmid, G., 1990. A simple boundary element formulation and its application to wavefield excited soilstructure interaction. Earthq.

Eng. Struct. Dyn. 19, 931947.

Biot, M., 1937. Bending of an infinite beam on an elastic foundation. J. Appl. Mech. (ASME) 4, A1A7.

Boussinesq, J., 1885. Application des potentiels ltude de lquilibre et mouvement des solides lastiques. Gauthier-Vilars, Paris.

Dietermann, H., Metrikine, A., 1996. The equivalent stiffness of a half-space interacting with a beam. Critical velocities of a load moving along a

beam. Eur. J. Mech. A Solids 15, 6790.

Dietermann, H., Metrikine, A., 1997. Steady state displacements of a beam on an elastic half-space due to a uniformly moving constant load. Eur.

J. Mech. A Solids 16, 295306.

Fryba, L., 1972. Vibrations of Structures under Moving Loads. Noordhof Int. Publishing, Groningen.

Gazetas, G., 1983. Analysis of machine foundation vibrations: state of the art. Soil Dyn. Earthq. Eng. 2, 241.

Grassie, S., Gregory, R., Harrison, D., Johnson, K., 1982. The dynamic response of railway track to high frequency vertical excitation. J. Mech.Eng. Sci. 24, 7790.

Grundmann, H., Lieb, M., Trommer, E., 1999. The response of a layered half-space to traffic loads moving along its surface. Arch. Appl. Mech. 69,

5567.

Knothe, K., Grassie, S., 1993. Modelling of railway track and vehicle/track interaction at high frequencies. Vehicle System Dynamics 22, 209262.

Kononov, A., Wolfert, R., 2000. Load motion along abeam on a viscoelastic half-space. Eur. J. Mech. A Solids 19, 361371.

Lamb, H., 1904. On the propagation of tremors over the surface of an elastic solid. Philos. Trans. Roy. Soc. London Ser. A 203, 142.

Lieb, M., Sudret, B., 1998. A fast algorithm for soil dynamics calculation by wavelet decomposition. Arch. Appl. Mech. 68, 147157.

Rcker, W., Auersch, L., Gerstberger, U., Meinhardt, C., 2005. A practical method for the prediction of railway vibration. In: Proc. 6th Int. Conf.

on Structural Dynamics. EURODYN 2005. Millpress, Rotterdam, pp. 601606.

Rvachev, V., 1958. On the bending of an infinite beam on an elastic half-space. J. Appl. Math. Mech. 22, 984988.

Selvadurai, A., 1979. Elastic Analysis of SoilFoundation Interaction. Elsevier, Amsterdam.

Sheng, X., Jones, C., Petyt, M., 1999. Ground vibration generated by a harmonic load acting on a railway track. J. Sound Vib. 225, 328.

Steenbergen, M., Metrikine, A., 2007. The effect of the interface conditions on the dynamic response of a beam on a half-space to a moving load.

Eur. J. Mech. A Solids 26, 3354.

Vesic, A., 1961. Bending of beams resting on isotropic elastic solid. J. Eng. Mech. Div., ASCE.

Wang, Y., Tham, L., Cheung, Y., 2005. Beams and plates on elastic foundations: a review. Progr. Struct. Eng. Mater. 7, 174182.

Winkler, E., 1867. Die Lehre von der Elastizitt und Festigkeit. Dominicus, Prag.

Wolf, J., 1985. Dynamic SoilStructure Interaction. Prentice-Hall, Englewood Cliffs, NJ.

Zimmermann, H., 1888. Die Berechnung des Eisenbahnoberbaus. Ernst & Korn, Berlin.

Date post:	08-Apr-2018
Category:	Documents
Upload:	bao-chunxiao
View:	216 times
Download:	0 times

Dynamic Interaction of Various Beams With the Underlying Soil-finite and Infinite, Half-spance and...

Documents