4
Eur. 1. Mech.•AlSolids. 16, nO 2, 295-306, 1997
Steady-state displacements of a beam on an elastic
half-space due to a uniformly moving constant load
H. A. DIETERMAN
* and
A. V. METRIKINE
**
AilsTRACT-. The steady-state displacements of an Euler-Bernoulli beam on an elastic half-space due to a
uniformly moving constant load are determined using the concept of the equivalent stiffness of the half-space. The
displacements are calculated for four different cases of beam and half-space parameters. The displacements in each
case are derived for five relevant load velocities. The lowest is small with respect to the shear (S)-wave velocity in
the half-space, the subsequent velocities are near the lowest critical velocity, between the critical velocity and the
Rayleigh (R)-wave velocity, between the R-wave and S-wave velocity and larger than S-wave velocity. The beam
displacement under the load is also determined for each case for all load velocities. Near the critical velocity the
effect of an external viscous damping along the beam on the displacement under the load is studied.
1. Introduction
Railway companies in Swiss (SBB), France (SNCF), Germany (DB), Holland (NS)
and Great-Britain(BR) have measured substantial increases of the vertical movements in
the track when trains move with a velocity in the order of the Rayleigh wave velocity
of the subsoil. This problem already takes place in traditional track on soft soils, such
as peat, at velocities as low as 140 kmlh. Mostly speed limits of the trains have been
ordered on these track parts to avoid the coupling between the train. and the lowest
critical velocity.
At high-speed track evasion of this phenomenon requests a substantial improvement
of the quality of the supporting subsoil.
The problem of a moving load over a beam on an elastic half-space has first been
studied by Filippov (1961) as a model for the train-subsoil interaction. It was shown
that the critical velocity of the train is approximately equal to the R-wave velocity.
Labra (1975) extended the results of Filippov taking into account the axial stresses in the
beam showing that the critical velocity is decreasing with increasing axial compression.
He confirmed the results found by Kerr (1972) who studied the beam on a Winkler
>I< Delft University of Technology, Holland, Faculty of Civil Engineering, Stevinweg 1.2628 CN Delft. Holland.
Tel. : 015 784281. FAX: 015 611465. E-mail: dieterman@dutctll.tudelft.nl.
** Mechanical Enginnering Institute Russian Academy of Sciences, Nizhny Novgorod, Russia. On post-doctoral
leave at the faculty of Civil Engineering. Delft University of Technology. Holland. Tel. : 8312 320576. FAX: 8312
320301. E-mail: A.Metrikine@mav.nnov.ru.
EUROPEAN JOURNAL OF MECHANICS. AlSOLlDS. VOL. 16. N° 2, 1997
0997-7538/97/02/$ 7.00/© Gauthier-Villars
296
H. A. DIETERMAN AND.k
V. METRIKINE
foundation. Both Filippov and Labra studied the loadvelocity range up tothe R-wave
velocity and did not derive the beam displacements due to the load.
A recent paper (Dieterman and .Metrikine, 1966) investigated the critical velocities of
a moving load over an Euler-Bernoulli beam in smooth contact with a half-space for all
load velocities, using the concept of equivalent stiffness of the half-space. It showed a
second critical velocity below the R-wave velocity.
In the present paper we calculate the steady-state displacements of the beam for
subseismic, transeisrnic and superseismic speed ranges of the uniformly moving load,
showing the amplifications of the displacements of the beam near the critical velocities.
To study the effect of damping an external viscous friction has been placed along the
beam.
The calculations are performed for two sets of half-space parameters, modelling soft
and stiff subsoil and for two sets of beam parameters, modelling small and large
track-ballast-embankment systems. Therefore four different cases are investigated.
in each case the beam displacements will be determined for five different load velocities.
The lowest velocity is small with respect to the shear (Sj-wave velocity in the half-space,
the subsequent velocities are near the lowest critical velocity, between the critical velocity
and the Rayleigh (R)-wave velocity, between the R-wave velocity and the Sswave velocity
and larger than S-wave velocity but smaller than thecompression (Pj-wave velocity.
Also the. beam displacement directly under the load is determined for each case for all
load velocities. This displacement is equal to the maximum displacement for small load
velocities only. For higher velocities, but smaller than the critical velocity, the maximum
displacement shifts behind the load due to the external viscous damping of the beam. For
velocities larger than the critical velocity the shift is more substantial due to the wave
radiation. To show this phenomenon the maximum displacements before and behind the
load. are determined for one of the cases.
Near the critical velocity the effect of the external viscous damping along the beam on
the displacement under the load is studied and shows an almost inverse proportionality
with the amount of damping.
The results given in the paper are an extension to the paper of Kenney (1954), who
calculated the beam displacements due to a uniformly moving constant load over an
Euler-Bernoulli beam on a visco-elastic Winkler foundation.
2. Model and method of solution
=
We consider a constant load P moving uniformly (x == V t, Y 0, Z = 0) along an
Euler-Bernoulli beam resting on an elastic half-space as depicted in Figure 1.
It is assumed that the contact between the beam and the half-space is smooth, hence
the shear stresses Ta:z and Tyz are zero at the interface. It is further assumed that the
normal stresses between the beam and the half-space are uniformly distributed over the
width of the beam. Then, as it was shown in Dieterman and Metrikine (1996), the
EUROPEAN JOURNAL OF MECHANICS, AlSOLlDS, VOL.
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BEAM ON HALF-SPACE UNDER MOVING LOAD
Fig. I. - Model and reference system.
steady-state displacement. of the beam due to the moving load is given as (adding an
external viscous friction to the beam)
p
W (x, t) = - 27r
(1)
1
exp (ik (x - V t))
EI k4 _ mk2 V2 _ ik 8V + 27rJ..LX(k V, k) dk,
00
-00
where EI is the bending stiffness of the beam, m is the mass per unit length of the beam,
.5 is the viscosity coefficient, J..Lis the Lame constant of the half-space and 2 7rJ..LX(w, k)
is the equivalent.stiffness of the half-space (wand k are the frequency and wave number
of the waves in the beam).
The expression for X according to Dieterman and Metrikine (1996) for k > 0 has
been rewritten in a more convenient form for the numerical calculations as presented in
the Appendix. The real part of X is symmetrical and the imaginary part is asymmetrical
with respect to k, i.e.
k)) = Re(x(-kV,
Im (X(kV, k)) = -lm(x(-kV,
Re (X (KV,
1/
-k)),
-k)).
Using the symmetry of the equivalent stiffness and introducing the relative velocity
= VICt, we rewrite (1) as
(2)
w = Wsym + Wasym,
where
svm
r'c (k4 _
(3)
W·
(~) = -:-2A
10
(4)
W·
a.~Ynl(t)
., = 2 A
10
roo
(k4 - 0:2 1/2 k2 + (32 Re (X)) COS (k ~)
0:2 1/2 k2 + {32 Re (X))2 + (k 1/"{ _ {32 1m (X))2 dk
(k1/"{-{32Im·(x))sin(k~)
(k4 _ 0:2 1/2 k2 + {32 Re (X))2 + (k 1/"{ _ {32 1m (X))2 dk
= P127r EI, 0:2 = mc;1 EI, {32 = 27rJ..LIEI, "{ = 8 cd EI, X
- V t is the distance from the moving load.
Here A
~=x
= X (kv,
Expressions (3) and (4) describe the symmetrical wsym and asymmetrical
of the beam displacement with respect to the location of the moving load.
EUROPEAN JOURNAL OF MECHANICS. NSOLIDS. VOL.
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k) and
wasym
parts
r
298
H, kDIETERMAN
AND A, V, METRIKINE
3. Results
The calculations have been performed for two sets of half-space parameters (stiff
and soft) and two sets of beam parameters (small and large). These parameters are
shown inTable Land Table II. Parameters of the half-space are denoted as follows: E is
Young's modulus, p is the density, f-L is the Lame constartt, ci, Ct, CR are the velocities
of compressional, shear and Rayleigh waves respectively). The values of the parameters
are chosen to describe stiff and soft subsoil of railroad tracks.
TABLE
E(N/m2)
I..stiff
2. Soft
p(kg/m3)
1500
2000
108
107
I.
IJ.(N/1II2)
C!
c« (m/s)
300
3.8*107
3,8*106 .
TABLE
(m/s)
CR
160
51
95
(m/s)
150
47
II.
m (kg/m)
I. Small
2. Large
7500
50000
i.3*IOS
9"'109
The beam is actually an equivalent beam the parameters of which include the properties
of the ballast, track-and embankment. The Poisson ratio of the half-space 'istaken 0.3.
Parameters a2, (32, "'I and a are given in Table m. The value of "'I is taken to be
about 10% of the critical viscosity. This is defined .as the viscosity at which there is
no amplification of the beam displacement for V
vcr with respect to the quasistatic
case (V <t:: Ct).
=
TABLE
Half-space-Beam
"'1. Stiff-Small
2. Stiff~Large
3. Soft-Small
4.. Soft-Large
0:2
1.48
0.14
0.15
om
III.
rP
1.84
0.027
0.184
0.003
a
001
0.004
0.01
0.0004
3
5
3
5
In the Figure 2 the displacement of the beam under the load is depicted as a function
of the relative load velocity V/ Ct for all cases considered.
Analysing the figures the following conclusions can be drawn.
1, The largest displacement under the load takes place when the half-space is soft
and the beam is small;
2. The difference between vcr and CR is very small for the soft half-space and is about
10% for the stiff half-space .. This difference is' increasing as the beam becomes larger.
EUROPEAN JOURNAL OF MECHANICS, AlSOLlDS, VOL. 16. N° 2. 1997
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BEAM ON HALF-SPACE UNDER MOVING LOAD
10.00
(a)
(b)
4.00
0.00
0.00
Stiff -Small
Stiff -Large
·4.00
-10.00
0.00
0.00
0.40
0.80
1.20
1.60
0.40
0.80
1.20
1.60
(c)
50.00
2,00
2.00
(d)
-50.00
Soft-Large
Soft -Small
-100.00
0.00
0.40
0.80
1.20
1.60
2.00
0.00
0.40
0.80
1.20
1;60
2.00
Fig. 2. - Displacement of the beam under the load.
3_The displacement of the beam remains finite for. vcr (the external viscosity is not
equal to zero) and becomes infinite when the velocity of the load tends to the Rayleigh
wave velocity in the half-space (the vertical dotted lines in the figures);
!
When the velocity of the load is larger than the Rayleigh wave velocity there is an
interval' of the load velocities for which the displacement under the load is directed
upwardly (qualitatively the same result was obtained in Lansing
(1966) for the case of a point load moving along a half-space). This intervalis wider
for a stiff half-space.
It is important.to note that the displacement under the load is not always equal to
the maximum displacement of the beam. This is seen in Figure 3a. where the beam
displacement under the load (curve 1), maximum displacement before (curve 2) and
behind (curve 3) the load are depicted for the "stiff-small" case for V < CR. The physical'
EUROPEAN JOURNAL OF MECHANICS. AlSOLlDS. VOL.. 16, N° 2, 1997
300
H. A. DIETERMAN AND A. V. METRIKINE
reasons for this phenomenon are the viscosity of the beam for subcritical velocities and
the wave radiation and the viscosity for supercritical load velocities.
To illustrate the influence of the beam viscosity, the beam displacement under the load .
is shown in Figure 3 b for. different values of the. viscosity in the interval of the load
velocities [OJCRJ. The "stiff-small" case is elaborated. It shows the well-known influence
of the viscosity on the amplitude of the resonance vibrations.
0.00
-20.00
20.00
-40.00
3
30.00 +--.---.---.--,--r--,-.--,--.,...-,
0.00
(a)
Vic .
60.00+-r--,-.--,----r--r--r-r--r-, t
0.00 0.20 0.40 0.60 0.80 1.00
.
(b)
Fig. 3. - (a) Displacement under the load (I), maximum displacement before (2)
and behind (3) the load; (b) displacement underthe load for different viscosities.
In Figures 4-7 the relative beam displacement (W* 2 7f' 108/ P) shapes versus relative
velocity v/ Ct for the load at the origin of the reference, moving with the load velocity, are
shown for the "stiff-small", "stiff-large", "soft-small" and "soft-large" cases respectively.
The Figure 4a-7a are related to the quasistaticcase V =:: 0.2Ct. The Figures4k-7b are
related to a load velocity slightly smaller than vcr. The Figures 4 c-7 c are related to a
load velocity located between vcr and the Rayleigh wave velocity. The figures 4 d-l d
are related to a load velocity slightly larger than the Rayleigh wave velocity. In the
figures 4 e-7 e the load velocity is located between the shear and the compressional wave
velocities in the half-space (V 1.6 Ct).
=
4. Conclusions
In this paper the steady-state wave forms in an Euler-Bernoulli beam in smooth contact
with an elastic half-space due to a uniformly moving load have been derived. The beam
has additionally an external viscous type of damping (about 10% of the critical).
For load velocities smaller than the lowest critical velocities the wave forms travelling
with the load are almost symmetrical. The small asymmetry seen at velocities slightly
before the lowest critical velocity is due to the viscous damping. As the load velocity
approaches the critical velocity the maximum displacement grows to values which are
EUROPEAN JOURNAL OF MECHANICS, AlSOLlDS, VOL.
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',"'.
305
BEAM ON HALF·SPACE UNDER MOVING LOAD
substantially larger than at speeds relative small with respect to the S-wave velocity.
The maximum displacement at the critical velocity is increasing with a decrease of the
viscous damping in the system and a decrease of the elasticity modulus. Hereby we have
to keep in mind that no radiation occurs at subcritical velocities, so the viscous damping
models the frictional losses in the subsoi1.
The second critical load velocity is located at Rwave velocity showing a infinite
(steady-state) amplification. This is explained by the zero equivalent stiffness of the
half-space at this velocity.
The shift ofthe lowest critical velocity to a relative lower value for the large beam on
the stiffer half-space is not In line, with expectations. However this is explained, by the
smooth contact model and the increase of the beam mass.
The model .used 'in this paper should' be improved to a non-smooth contact to get a
better, understanding of the influence of the contact between beam and half-space on
the critical velocities.
APPENDIX
2 ak
X (k V, k) ::::-ct V2 In j
where
~:
< CRi
CR < V < Cti
when V
n==
{
when
3,
when Ct
<V <
4,
when CI
< V,
Cli
II == 81 - 2Gl~ 2G2,
h == 82 - 2Gl - 2G2+ i83,'
13 == 82 - 2 Gl - 2 G3+ i 83 - 2 G4,
14 = 82 - 2 Gs + i 83 - 2 G6 - 2 G7,
81
= 8 (exp (-ak VI-Ph)
-1),
83 ::: 8 sin (ak
,',
8
= 1r 2 (13'k-
G2 =
+.
1) (2 (2 13'k-
-Joo
Gl-
.~
J
~
"
'~
82 =S(cos(ak
Vf3'k - 1) - 1),
V13'k- 1),
RI
pi) - (Rt 131/Rl + Rl 13'kIRt + 2 Rl Rt))'
, '
Qi(exp(-ak1])-l),
d1]
(2 (1 - rp) - 13;)2+ 4 (1 - r,z) Qi Qt 1]'
4(1-772)(Ql)'lQi(exp(-ak1])-1)
d1]
(2 (1 -772) - 13;)4 + 16(1-1]2) (QtP (Q/)2 77
EUROPEANJOURNALOF MECHANICS.NSOLIDS. VOL. 16. N° 2. 1997
306
H. A. DIETERMAN
G3
=-
r~
G4
= +Jo
+ ~2}(P,+)2 Pt- (exp (iak~) - 1)
(2 (1 + e2) "'"' 191)4+ 16{1 + e2)2 (Pt )2 (P,+)2
G5 = -)0
T'
QT (exp( -ak'l]) - 1)
d1]
(2 (1 _1]2) - 191)2+ 4 (1 -1]2) Qt QT -:;f'
I..jiJf=l
G6 == Jo
f·..;'iJf::],
G, = .~
d~
4 (1
'. {X,
.
A. V. METRIKINE
'4(1-1]2)(Qi"")2QT(exp(-ak1])-I)
d1]
(2 (1 - 1]2) - 191)4+16 (1- 1]2)(Qt)2 (Q1)2 1]
Jo
.r~
AND
P,- (exp (iak~) :- 1)
de
(2 (1+ ~2) - t9f)2 + 4 (1+~2)l't- P,-)
'I'
4(1
+ e)(p/)2 Pt- (exp (iak~)~
1)
(2(1 +~2)7 19;)4 +16(1+~2)2(Pt-)2 (Pz+)2
d~
T'
in which
-
RI,t
/ 2
2
t9R- .t9"t'
=V2/ cf, t, R"
=V
t9~t, R
cR, Ct and' c; are the velocities of the Rayleigh, shear and compressional waves in the
half-space respectively, 2a is the beam width.
The expressions above
the results (27)-(30) derived in Dieterman and Metrikine
(1996) which have been rewritten into a more convenient form for the numerical
calculations. The contribution of 80 (see Dieterman and Metrikine, 1996) has been
included into the integrals. This is possible' since So is the contribution of the pole
~ = 0 in the integral (25) in Dieterman and Metrikine (1996) and we can cancel it by
adding and subtracting "1" into the numerator of the .inregral (25) and then using the
equality
,= o.
'
are
J~oo~((ti 1
-, REFERENCES
DIETERMAIoI H. A'i METRIKINB A., 1996, The equivalent stiffness of a half-space interacting with a beam. Critical
velocities of a moving load along thebeam, European Journa/ of Mechanics, AlSo/ids,IS,
nO I; 67-90.
FILIPPOV A. P.; 1961, Steady-state vibrations of an infinite beam on elastic half-space subjected to a moving load,
Izvestija ANSSSR OTN Mehanikal Mashinostroenie, 6,97-105 (translated from Russian).
KENNEY J. T., 1954, Steady-state' vibrations of beam on elastic foundation for moving ..Ioad, Journal of Appl.
Mech., 76, 359-364.
KE~ A. D., 1972, The continuously supported rail subjected to an axial force and a moving load, Int. J. Mech.
Sci., 14, 71-78.
~
LABRA J. J., 1975, An axially stressed railroad track on an elastic continuum subjected to a moving load, Acta
Mechanica, 22, 113-129.
LANSING D; L., 1966, The .displacements in an elastic half-space due to a moving concentrated normal load,
NASA technical report, TR. R-238. '.
(Manuscript received June 6, 1995;
revised and accepted May IS, 1996.)
EUROPEAN JOURNAL OF MECHANICS, AlSOUDS. VOL. 16, N° 2, 1997