Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 4, pp. 763-775, Warsaw 2008

RESPONSE OF BEAM ON VISCO-ELASTIC FOUNDATION

TO MOVING DISTRIBUTED LOAD

Roman Bogacz

Cracow University of Technology and IPPT, Polish Academy of Sciences, Warsaw, Poland

e-mail: rbogacz@ippt.gov.pl

Włodzimierz Czyczuła

Cracow University of Technology, Poland

The paper is devoted to the study of several cases of stationary dyna-
mical problems in which motion is driven by a distributed load acting
on a beam on an elastic foundation at a moving position. The velocity
of motion is assumed constant. In particular, cases of a load described
by the Heaviside function (or its linear superposition) and a harmonic
function are studied. Some problems examined by the authors in their
previous investigations are reviewed.

Key words: dynamics, travelling load, wave propagation

1. Introduction

Thedevelopment of various kinds ofmodern technology, like explosive bonding
of layered materials or tracked high-speed transportation systems, becomes
more and more important. This makes a strong need for simplified but relia-
blemodels of continuous orhybrid systems in order to studyvariousdynamical
effects which influence durability of structures, damage of the environment or
comfort of transportation.Thefirst studyof beamson theWinkler foundation
subjected to a simple concentrated force moving with a constant speed was
initiated by Timoshenko (1926). The first stationary solution to a simple sta-
tionary case of theBernoulli-Euler beamonan elastic foundationwas properly
obtained by Ludwig (1938). The case of a moving and oscillating force was
formulated and partly solved byMathews (1958). The first proper solution to
theMathews problemwas given byBogacz andKrzyżyński (1986). There are



764 R. Bogacz, W. Czyczuła

various extensions of this classic problem towards more complicated but also
more realistic models of structures and loads. A great deal of new effects were
recognized by Bogacz et al. (1998) who examined the problem of an oscilla-
ting loadmoving along aperiodic (variable in space) structure.Thedynamical
effects for two or three-dimensional problems with moving loads have impor-
tant practical engineering applications (Bogacz and Frischmuth, 2008). Some
problems connected with a system of plates subjected to a traveling load can
be found in Bogacz (2008), Bogacz and Frischmuth (2008). An application of
the beammodel to the railway track mechanics is connected with taking into
account the axial force into the model (Kerr, 1972).
The aim of this paper is devoted to systematization and explanation of

some new effects related to the moving distributed and oscillating load.

2. Beam on the Winkler foundation subjected to a uniformly

distributed load acting on a segment

The problem of vibration of a flexibly supported beamwith the stiffness EI,
linearmass density mA, damping coefficient h andWinkler coefficient c, sub-
jected to the distributed loadmovingwith the velocity V0, can be composed of
the solution obtained for the limiting case of a load described by the following
Heaviside function F0H(x−V0t)

EIw,xxxx+Tw,xx+mAw,tt+hw,t+ cw=F0H(x−V0t) (2.1)

A similar case of thebeam(without the compression force, T =0)was studied
by Bogacz and Rozenbajgier (1979). The beam on an elastic foundation was
generalized there to the case of a beam on a visco-elastic semi-space. The
boundary conditions, equivalent to the condition of radiation, in the visco-
elastic case take the following form

limw(x) =











0 for x→∞
c

F0
for x→−∞

(2.2)

and in themoving coordinate system

X =x−V0t (2.3)

where thedisplacement w(X) aswell as itsderivatives w,X,w,XX, and w,XXX
are continuous at X =0.



Response of beam on visco-elastic foundation... 765

The equation of beam motion in the moving system of coordinates (2.3)
takes the form

EIw,XXXX +Tw,XX +mA(w,tt−2V0w,Xt+V 20w,XX)+
(2.4)

+h(w,t−V0w,X)+ cw=F0H(X)

In the stationary case, a characteristic equation ofEq. (2.4) takes the following
form

R4+4(Vq)2R2−8Vbq3R+4q4 =0 (2.5)

where

V =
V0
Vcr

Vcr =

√

√

√

√

√

4cEI

mA
− T
mA

q=

√

c

4EI
b=

h

2
√
cmA

Roots of Eq. (2.5) are

R1 =S1+iD1 R2 =S1− iD1 R3 =S2+iD2
R4 =S2− iD2 S1 =−S2

(2.6)

Using boundary conditions (2.2), the continuity conditions at X =0, one can
obtain the following kind of solution before and behind the front of the load:

— for X < 0

W1(X)=
F0
c
+exp(nX)

{

A1 sin
[(

2V 2+n2−
2Vh

n

)

X
]

+

(2.7)

+A2cos
[(

2V 2+n2− 2Vh
n

)

X
]}

— for X > 0

W2(X)= exp(−nX)
{

A3 sin
[(

2V 2+n2+
2Vh

n

)

X
]

+

(2.8)

+A4cos
[(

2V 2+n2+
2Vh

n

)

X
]}

where n is the positive root of the equation

n6+2V 2n4+(V 4−1)n2−V 2b2 =0 (2.9)



766 R. Bogacz, W. Czyczuła

and

A1 =−
F0
2Kc

n

2V 2+n2− 2Vh
n

[

2V 2(V 2+n2)−3
(Vh

n

)2

− Vh
n
(V 2+3n2)

]

A2 =−
F0
2Kc

[

2n2(V 2+n2)+
(Vh

n

)2

+
Vh

n
(V 2+3n2)

]

(2.10)

A3 =−
F0
2Kc

n

2V 2+n2+ 2Vh
n

[

2V 2(V 2+n2)−3
(Vh

n

)2

+
Vh

n
(V 2+3n2)

]

A4 =−
F0
2Kc

[

−2n2(V 2+n2)−
(Vh

n

)2

+
Vh

n
(V 2+3n2)

]

K =2n2(V 2+n2)−
(Vh

n

)2

The solution for the purely elastic case can be obtained from Eqs. (2.7) and
(2.8) for h → 0. The solution has a different form for the sub-critical and
super-critical case. The solution for the sub-critical case (V < 1) behind the
front of the load W1(X) and before the front W2(X) is described by following
formulas

W1(X)=
F0
2c

[

2− exp(
√

1−V 2X)
][ V 2√
1−V 4

sin(
√

1+V 2X)+cos(
√

1+V 2X)
]

(2.11)

W2(X)=−
F0
2c
exp(
√

1−V 2X)
[ V 2√
1−V 4

sin(
√

1+V 2X)− cos(
√

1+V 2X)
]

For the super-critical case (V > 1) the displacements are as follows

W1(X)=
F0
2c

[

2−
(

1+
V 2√
V 4−1

)

cos[(
√

1+V 2−
√

V 2−1)X]
]

(2.12)

W2(X)=−
F0
2c

[(

−1+
V 2√
V 4−1

)

cos[(
√

1+V 2+
√

V 2−1)X]
]

It is visible that in the stationary elastic case, for a super-critical velocity of
loadmotion V >Vcr, thewaves before andbehind the front of the load donot
decay for |X| → ∞. Shorter waves with the phase velocity smaller than the
group velocity propagate before the front of the load, and longer waves with
the phase speed higher than the group velocity propagate behind the front of
the load. The displacements in the sub-critical case are shown in Fig.2.
In the linear case, superposition of the obtained solution for the Heviside

function allows one to obtain various kinds of piece-wise constant loads distri-
buted on a finite-length segment. For example, if we describe a load with a



Response of beam on visco-elastic foundation... 767

Fig. 1.Wave velocity V0 versus wave number k for the Bernoulli-Euler beam on the
Winkler foundation subjected to longitudinal force T

Fig. 2. Displacements of the Bernoulli-Euler beam on theWinkler foundation in the
sub-critical case (V =0.8) for various damping coefficients

given value F1 distributed between x = 0 and x = L at t = 0, it is then
possible to write the load as follows

F(x,t)=F1[H(x−V0t)−H(x−L−V0t)] (2.13)

In such a case, the solution must fulfill conditions (2.2) and, additionally, the
continuity of displacements and derivatives w,X,w,XX, and w,XXX at X =0
and X =L.

Let us now consider a more complicated model of the beam on an elastic
foundation which takes into account shear deformation and rotary inertia of
the cross-section – theTimoshenkobeam.Thecase of theTimoshenkobeamon
an elastic foundation subjected touniformlydistributedmoving loadshasbeen
studied by several authors (Fryba, 1972; Bogacz et al., 1989). The equation of
motion of the Timoshenko beam takes the following form



768 R. Bogacz, W. Czyczuła

EIϕ,xx+k
′AG(w,,x−ϕ)−mAIϕ,tt =0

(2.14)

k′AG(w,xx−ϕ,x)−mAw,tt−hw,t− cw=−F0H(x−V0t)

where ϕ is thebeamrotation due to pure shear, k′ – shear coefficient, G–mo-
dulus of shear elasticity, A – cross-sectional area, and h – damping coefficient.
Thefirst stationary solution obtained for the case of theTimoshenkobeam

on an elastic foundation was obtained by Achenbach and Sun (1965). The
shape of displacement in this solution is shown in Fig.3.

Fig. 3. Displacements of the Timoshenko beam on theWinkler foundation for
various values of the load speed. The results are found for a parameter range

similiar to that used by Achenbach and Sun (1965)

The solution obtained by Achenbach and Sun (1965) is qualitatively dif-
ferent from that shown in Fig.4. Looking for the limiting stationary case, the
set of equations (2.14) can be reduced to the following fourth-order equation
with respect to displacement W(X) = w(X)

√

A/I, where I is the moment
of inertia of the cross-section

Q(V 2)WIV +2Vh(V 2−V 2E)W ′′′+[V 2(V 2E +1)−V 2G]W ′′+
(2.15)

+2VV 2EhW
′+V 2GW =F[V

2
GH(X)+(V

2−V 2E)H′′(X)]

where

F =F0
( A

Ic2

)

Q(V 2)= (V 2−V 2G)(V 2−V 2E)

V 2E =
EA2

Ic
V 2G =

k′GA2

Ic



Response of beam on visco-elastic foundation... 769

Fig. 4. Displacements of the Bernoulli-Euler beam on theWinkler foundation in the
super-critical case (V =1.2) for various damping coefficients

Making use of the above equation, we can determine the discontinuity
values of the derivatives of W(X) and rotation at the point X = 0. The
solution to the problem consists in determination of displacements W(X) and
rotation that satisfy equations (2.15). We shall obtain them by applying the
Fourier transformation to the equations of motion. To investigate the effect
of the load speed on the qualitative character of the solutions for the elastic
system, i.e. h→ 0, let us consider two sets of parameters.

Case I

V 2E >V
2
G(V

2
G+1) (2.16)

In this case, there exist threemain ranges of the load speed inwhich there are
three corresponding different solutions like those obtained by Achenbach and
Sun (1965) and shown in Fig.3.

Within range No. 1, for |V | < V1, the solutions tend to the asymptotes
W = 0 and W = 1 in monotonous ways. Within range No. 2 for V1 < V <
V2, the solution vanishes monotonously before the load front and oscillates
periodically around the value W = 1 behind the load front. Within range
No. 3 for V > V2, the displacement and rotation before the load front are
equal to zero, and behind the load the solution consists of superposition of
two particular periodic solutions. This solution, unobservable in Fig.3, can be
seen in Fig.6.



770 R. Bogacz, W. Czyczuła

Fig. 5. Phase velocity Vf versus wave number k for two qualitatively different cases
of the solution

Case II

V 2E <V
2
G(V

2
G+1) (2.17)

In this case, which is illustrated in Fig.6, we have four ranges of velocities
with qualitatively different solutions. At the critical speed V 2 = V 20 , the di-
splacement and rotation increase infinitely. To the speed range V 2 <V 20 there
corresponds a solution with properties being characteristic for range No. 1 in
case I. This case is represented by the curve V 2 =3 in Fig.6. In the range of
speed V0 <V <V1, the solution substantially changes in its quantitative fe-
ature. Namely, the solution consists then of two visible waves; thewave with a
small amplitude andwavelength before the load front and amuch greater am-
plitude andwavelength behind the front of the load. The solution in this case
is represented by the curve V 2 =4.5 (Fig.6).Within the range V1 <V <V2,
the periodic character of the wave behind the load front remains periodic but
before it the displacement vanishesmonotonously with distance from the load
front. This case is represented by the solution for V 2 = 6. The solution for
V 2 > V1 has a similar feature as in case I. The solution shown in Fig.6 for
V 2 =15 illustrates qualitative behaviour of the beam in this region.

The above case shows that there exists a set of parameters for the Ti-
moshenko beam which can be taken qualitatively as the limiting case, i.e.
transition to the Bernoulli- Euler beam. The change between case I and ca-
se II is connectedwith the change from the hyperbolic to parabolic type of the
equation. This is the reasonwhy the solution obtained byAchenbach and Sun
(1965) is valid in thewhole range of velocity, but only for the set of parameters
fulfilling inequality (2.16).



Response of beam on visco-elastic foundation... 771

Fig. 6. Displacements of the Timoshenko beam on theWinkler foundation for
various values of the load speed, in the case of parameters V 2E <V

2

G(V
2

G+1) (which
was not considered by Achenbach and Sun, 1965)

3. Beam on the Winkler foundation subjected to a harmonically

distributed moving load

In the case when the load is described by a continuous and oscillating (har-
monic) function, and is moving with a given velocity V0, the beam equation
is described as follows

EIw,xxxx+Tw,xx+m0w,tt+hw,t+ cw=F0 sin[k(x−V0t)] (3.1)

or in the moving system of coordinates

EIw,XXXX +Tw,XX +m0(w,tt−2V0w,Xt+V 20w,XX)+
(3.2)

+h(w,t−V0w,X)+cw=F0 sin(kX)

In the case of Eq. (3.2) the solution has the following form

w(x,t) =Ws sin[k(x−V0t)]+Wccos[k(x−V0t)] (3.3)

while in the case of Eq. (3.2), we can write it as follows

w(X,t) =Ws sin(kX) (3.4)



772 R. Bogacz, W. Czyczuła

In the elastic case (h=0), Wc =0 and the dependence between Ws and F0
takes the following form

f(V0)=
Ws
F0
=

1

EIk4−Tk2−m0V 20 k2+ c
=

1

m0k2(R
2
0−V 20 )

(3.5)

It is visible that Ws →∞ for given F0, k and V0 =
√

(EIk2−T + c/k2)/m0
greater than Vcr, which is described by the formula

Vcr =

√

4cEI

m0
− T
m0

R20 =
1

m0

(

EIk2−T + c
k2

)

(3.6)

An example of the dependence W0/F0 = f(V0) is presented in Fig.7a.We can
see that in the case of Bernoulli-Euler beam for |V0|=R0, similarly as in the
case of vibration resonance, the solution changes frombeing ”in phase” to ”out
of phase”. This solution is important for applications in railway engineering,
when the track dynamics can be studied as the Bernoulli-Euler beam on an
elastic or visco-elastic foundation subjected to a longitudinal force T . This
force can have a destabilizing character in the case of increasing temperature
that produces a compressing force in the rails.

Fig. 7. (a) Function W0/F0 = f(V0) for the Bernoulli-Euler beam, and (b) for the
Timoshenko beam

Let us consider a more complicated case – the Timoshenko beam on an
elastic foundation subjected to a continuous harmonic load F0 sin(kX)moving
with a given speed V0,X =x−V0t. The equation of motion is described now
by a set of equations similar to (2.15)

Q(V 2)WIV +2Vh(V 2−V 2E)W ′′′+[V 2(V 2G+1)−V 2E]W ′′+
(3.7)

+2VV 2GhW
′+V 2GW =F[V

2
G− (V 2−V 2E)k2]sin(kX)



Response of beam on visco-elastic foundation... 773

where

F =F0
( A

Ic2

)

Q(V 2)= (V 2−V 2G)(V 2−V 2E)

Now the ratio W/F = f1(V ) is given by the following equation

f1(V )=
V 2G− (V 2−V 2E)k2

k4Q(V 2)−k2[V 2(V 2
G
+1)−V 2

E
]
+V 2G (3.8)

where the velocities illustrated in Fig.7b are as follows

VL =OL=

√

1

2

(

β−
√

β2−4γ
)

β=V 2G+V
2
E +
1

k2
(V 2G+1)

γ=V 2GV
2
E +
1

k2
V 2EV

2
G+
1

k4
V 2G VM =OM =

1

k

√

V 2
G
+k2V 2

E

(3.9)

As can be seen in Fig.7b, there exist two values of the speed VL and VM
for which some kind of resonance occurs. The critical speed VL =Vc and the
related wave number k0 as function of VE are shown in Fig.8.

Fig. 8. Critical speed Vc and related wave number k0 versus the value of
longitudinal wave VE for selected values of VG

For these relationships, the denominator of (3.8) is equal to zero, which
corresponds to the case when beamdisplacements tend to infinity. It is visible
that for a given value of VG, there exists a limiting value of VE determined
by inequality (2.17), where k0 is bounded. For a value greater than VE, the
phase velocity tends to VG for the wave number approaching infinity.
From the point of view of applications in railway engineering, it is par-

ticularly important to study the problem of response of a periodic structure
to a distributed load. Generalization of the problem investigated by Jezequel
(1981) andMead (1986) will be studied in a separate paper.



774 R. Bogacz, W. Czyczuła

4. Summary

Several cases of stationary dynamics of a continuous system are considered.
The investigated problem seems to be important for applications in railway
engineering. The considered one-dimensional continuous system is subjected
to a distributedmoving load. The load is described by the Heaviside function
(or its linear superposition) and by a moving load harmonically distributed
in space. The velocity of load motion is assumed to be constant. The results
obtained in this paper will be a basis for generalization of the problem of the
response of periodic structures to periodically distributed loads.

Acknowledgement

This paper was supported by the EUREKAProject E!2264 TOSIN.

References

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shenko beam, Int. J. Solid and Structures, 1, 353-370

2. Bogacz R., 1983, On dynamics and stability of continuous systems subjected
to distributed moving load, Ing. Archiv., 57-69

3. Bogacz R., 2008, Recent investigations in dynamics of continuous systems
subjected tomoving load, SymposiumVibrations in Physical Systems, Poznań-
Będlewo, 35-47

4. Bogacz R., Frischmuth K., 2008, Vibration of array of plates induced by
moving load, In: Simulation in R&D, L. Bobrowski, Z. Łukasik and Z. Strzy-
zakowski (Edit.), 41-46

5. Bogacz R., Krzyżyński T., 1986, On the Bernoulli-Euler beam on a visco-
elastic foundation undermoving oscillating load, Inst. Fund. Technol. Res. Re-
port, 38

6. Bogacz R., Krzyzynski T., Popp K., 1989, Influence of beam modelling
on solutions of generalized Mathews problem, Z. Angew. Math. Mech., 69, 5,
T320-T321

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8. BogaczR., Krzyzynski T., PoppK., 1998,Wave propagation in two dyna-
mically coupled periodic systems,Proc. International Symposium onDynamics
of Continua, Bad Honnef, Shaker Verlag, 55-64



Response of beam on visco-elastic foundation... 775

9. Bogacz R., Rozenbajgier Z., 1979, Steady state vibration of beam on a vi-
scoelastic semi-space undermoving load,Proc. ofWarsawUniv. of Technology,
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Drgania belki na lepko-sprężystym podłożu pod wpływem ruchomego,

rozłożonego obciążenia

Streszczenie

Niniejszy artykuł jest poświęcony badaniu wybranych przypadków stacjonarnych
zagadnień dynamicznych, w których belka na sprężystym podłożu poddana jest ru-
chomemu obciążeniu. Rozważono zagadnienie stałej prędkości ruchu obciążenia opi-
sanego w przestrzeni funkcją Heaviside’a (lub liniowej kombinacji tych funkcji) oraz
obciążenia harmonicznie zmiennego.Niektóre z zagadnieńbadanychwcześniej zostały
krytycznie omówione i uzupełnione.

Manuscript received July 4, 2008; accepted for print August 20, 2008