This paper presents the analytical solutions for bilaterally infinite strings and infinite beams on which a point force is initially applied, which then moves on the structure at a constant velocity. The solutions are sought by first applying the Fourier transform to the spatial coordinate dependence, and then the Laplace transform to the time variable of dependence, of the governing equations of motion. For the strings, it is necessary to distinguish between the case of a sonic load (a force moving at the phase velocity of transverse waves) and the cases of subsonic and supersonic loads. This is achieved by a suitable expansion in polynomial ratios of the Laplace transform, before going back to the original Fourier transform, whose inverse is obtained by exact calculations of the integrals over the complex infinite domain. For the Euler-Bernoulli beam, the same process leads to the closed-form (exact) formula for the displacement, from which the stress can be deduced. The displacement consists of the sum of two integrals: one representing the transient part, and the other, the stationary part of the solution. The stationary part is observed in the vicinity of the force for a very long travel time. The transient part is observed at a finite position coordinate, in relative proximity to the starting point of the moving force. For the Timoshenko beam, the final step in the calculation of the displacement and rotation, which requires a numerical evaluation of the integrals, leads to Fourier cosine and sine transforms. The response of the beam depends on the load velocity, relative to the two characteristic velocities: those of shear waves and longitudinal waves. This demonstrates that the transient parts of the solutions, in the Euler-Bernoulli beam or in the Timoshenko beam, are quasi identical. However, classical theory fails to forecast high frequency responses, occurring with velocities of the load exceeding twenty per cent of the bar velocity. For a velocity greater than the velocity of the shear waves, classical theory wrongly forecasts the response. In addition, according to the Euler-Bernoulli beam theory, the flexural waves are able to exceed the bar velocity, which is not realistic. If the load moves for a long period, the solution in the vicinity of the load tends towards a stationary solution. It is important to note that the solution to the stationary problem must be completed by the solution to the associated homogeneous system to represent the physical stationary solution.

References

1.
Yin
,
S.-H.
, and
Tang
,
C.-Y.
, 2011, “
Identifying Cable Tension Loss and Deck Damage in a Cable-Stayed Bridge Using a Moving Vehicle
,”
J. Vibr. Acoust.
,
133
, pp.
021007
-1–021007-
11
.
2.
Cartmell
,
M. P.
, and
McKenzie
,
D. J.
, 2008, “
A Review of Space Tether Research
,”
Prog. Aerosp. Sci.
44
, pp.
1
21
.
3.
Felszeghy
,
S. F.
, 1996, “
The Timoshenko Beam on an Elastic Foundation and Subject to a Moving Step Load. Part 1: Steady-State Response
,”
ASME. J. Vibr. Acoust.
,
118
, pp.
277
284
.
4.
Fryba
,
L.
,
Vibration of Solids and Structures Under Moving Loads
(
Noordhoff International, Groningen
,
Netherlands
, 1977).
5.
Lambert
,
Q.
,
Langlet
,
A.
,
Renard
,
J.
, and
Eches
,
N.
, 2008, “
Dynamique en Flexion de Tubes Parcourus à Grandes Vitesses
,”
Mécanique & Industries
,
9
(
6
), pp.
559
569
.
6.
Oni
,
S. T.
, and
Omolofe
,
B.
, “
Dynamic Response of Prestressed Rayleigh Beam Resting on Elastic Foundation and Subjected to Masses Traveling at Varying Velocity
,”
J. Vibr. Acoust.
,
133
, pp.
041005
-1–041005-
15
.
7.
Felszeghy
,
S. F.
, 1996, “
The Timoshenko Beam on an Elastic Foundation and Subject to a Moving Step Load. Part 2: Transient Response
,”
ASME J. Vibr. Acoust.
118
, pp.
285
291
.
8.
Steele
,
C. R.
, 1968, “
The Timoshenko Beam with a Moving Load
,”
ASME J. Appl. Mech.
35
(
3
), pp.
481
488
.
9.
Stadler
,
W.
, and
Shreeves
,
R. W.
, 1970, “
The Transient and Steady-State Response of the Infinite Bernoulli-Euler Beam with Damping and an Elastic Foundation
,”
Q. J. Mech. Appl. Math.
,
XXIII
(
2
), pp.
197
208
.
10.
Flaherty
,
F. T.
, Jr.
, 1968, “
Transient Resonance of an Ideal String Under a Load Moving with Variable Speed
,”
Int. J.Solids Struct.
,
4
, pp.
1221
1231
.
11.
Kanninen
,
M. F.
, and
Florence
,
A. L.
, 1967, “
Traveling Forces on Strings and Membranes
,”
Int. J.Solids Struct.
,
4
, pp.
143
154
.
12.
Shultz
,
A. B.
, 1968, “
Large Dynamic Deformations Caused by a Force Travelling on an Extensible String
,”
Int. J.Solids Struct.
,
4
, pp.
799
809
.
13.
Tianyun
,
L.
, and
Qingbin
,
L.
, 2003, “
Transient Elastic Wave Propagation in an Infinite Timoshenko Beam on Viscoelastic Foundation
,”
Int. J.Solids Struct.
40
, pp.
3211
3228
.
14.
Cowper
,
G. R.
,1966, “
The Shear Coefficient in Timoshenko’s Beam Theory
,”
J. Appl. Mech.
,
33
, pp.
335
340
.
15.
Florence
,
A. L.
, “
Traveling Force on a Timoshenko Beam
,”
ASME. J. Appl. Mech.
,
32
(
2
), pp.
351
358
.
16.
Renard
,
J.
, and
Taazount
,
M.
, 2002, “
Transient Responses of Beams and Plates Subject to a Travelling Load
,”
European Journal of Mechanics A/Solids
21
, pp.
301
322
.
17.
Langlet
,
A.
,
Girault
,
G.
, and
Renard
J.
, 2006, “
Interaction d’un Liquide avec une Plaque Soumise à une Explosion: Validation Expérimentale d’un Modèle Linéaire
,”
Mécanique & Industries
,
7
, pp.
97
106
.
18.
Girault
,
G.
, 2006, “
Réponse d’une Plaque Couplée à un Liquide et Soumise à une Pression Mobile. Aspects Théoriques et Expérimentaux en Détonique
,” Ph.D. thesis, l’Université d’Orléans, France.
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