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A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
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| I. I. Ivanchenko, "Dynamics of bridge and track structures under the action of the railway moving load," Mech. Solids. 40 (4), 127-140 (2005) |
| Year |
2005 |
Volume |
40 |
Number |
4 |
Pages |
127-140 |
| Title |
Dynamics of bridge and track structures under the action of the railway moving load |
| Author(s) |
I. I. Ivanchenko (Moscow) |
| Abstract |
Vertical vibrations of a bridge structure during the motion of a series of carriages
along a beam based (through discrete viscoelastic inertial supports)
on the bridge span and embankment sections are studied. The embankment
sections near the bridge are assumed to be rigid. To study the beam response
to a simple loading by a moving load, two basic methods are commonly used.
(These methods are utilized also for other structures and loads.)
The first method is based on the expansion of the deflection in terms of
natural shapes of the beam. The coefficients of this expansion are treated as
the generalized coordinates, and the model of the system is reduced
to a system of ordinary differential equations with variable coefficients [1-3].
In the second method, the system beam-load is decomposed, as was the case
for the problem of impact of a load against a beam [4], and the problem is reduced
to an integral equation for the dynamic response of the load [5, 6].
In the approach of [1-3] an increase in the number of shapes preserved in the expansion
leads to an increase in the order of the system of governing differential equations.
The approach of [5, 6] leads to difficulties in solving the integral equations.
These difficulties are associated with the conditional stability of step methods.
The approach of [7, 8] proposed for beams and frameworks combines the approaches
of [1-3] and [5, 6] and removes the drawbacks mentioned above.
This approach enables one to take into account any necessary number of shapes in the expansion of
the displacement and leads to the system of governing equations with a minimal number of variables.
This system of equations can be integrated by means of an unconditionally stable scheme, as is the
case for the integral equation method [5, 6]. In the present paper the last approach is developed
for combined systems that form a discrete model for the track, the train, and bridges. The
development of methods for calculations on the basis of continuous and discrete railway track
models, analysis of joint vibrations of the track and artificial structures under the action of a
moving load remain rather topical issues, especially in view of the increase in the velocity of
trains [9-14, 17].
This problem is proposed to be solved on the basis of special beam boundary elements.
The element consists of two parallel long beams connected by means of vertical nonlinear
coupling members that model the work of ties, tie plates, and the ballast layers on the embankment
and bridge span. The upper beam models the rail and the lower one the bridge span.
The removal of the lower beam and the imposition of constraints on the displacement
of the coupling members leads to the element that models the track outside of the bridge.
The boundary elements matching in length model the work of the track on the bridge
and outside of the bridge.
These elements are involved in the set of boundary elements that have been
proposed previously [8, 15], which enables unsteady vibrations in the system
train-track-bridge (TTB) to be investigated with various irregularities of the rail track
being taken into account. High degree of spatial discretization when studying
the vibration of the rail track and the framework that models the bridge is provided
by involving distributed rods in the model. The displacements in this model are
approximated by linear functions and trigonometric series (Fourier series), which
(as has been mentioned) enables one to increase the number of degrees of freedom
of the system under consideration, keeping unchanged the order of the system
of governing equations.
To solve the problem we utilize the time step procedure of [7] and the method
of [15] to take into account the moving load. This approach enables one to
investigate the stress-strain state in a TTB system with the track irregularity
and subsidence under ties being taken into account. The proposed numerical
procedure provides a unified approach for solving linear and nonlinear differential
equations that model the system of ties lying on a ballast layer with plates
between the ties and the rails. This approach minimizes the number of unknowns
in the system of governing equations, only the nodal accelerations (at the places
of matching of the boundary elements) and the vertical accelerations of moving nodes
(at the points of contact of wheels with the rails) being preserved.
In the previous studies of the problem under consideration, the authors
confined themselves to either infinitely long beams on lumped supports under
fairly simple loads [12] or finite-length beams to model the track in a nonlinear
approximation in the case of a simplified moving load [11] or the investigation
of vibrations of beam bridges in the linear approximation on the basis of
the traditional finite element method implying a power-law approximation
of displacements of load bearing structures of TTB systems [13]. The method
proposed in the present paper is based on the trigonometric approximation
of displacement in the beam modeling a rail. Such an approach extends
the potentials of the finite element method by enabling this method to
be applied to the investigation of dynamic response in a TTB system with
various imperfections including short irregularities and subsidence of the track. |
| References |
| 1. | C. E. Inglis, Mathematical Treatise on Vibrations in Railway Bridges,
Univ. Press, Cambridge, 1934. |
| 2. | V. V. Bolotin, "The problem of vibration of bridges under the action of a moving load,"
Izv. AN SSSR. OTN. Mekhanika i Mashinostroenie, No. 4, pp. 109-115, 1961. |
| 3. | N. G. Bondar', Yu. G. Koz'min, V. P. Tarasenko, et al.,
Interaction of Railway Bridges wits Trains [in Russian],
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| 4. | S. P. Timoshenko, Vibrations in Engineering [in Russian],
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| 5. | A. P. Filippov and S. S. Kokhmanyuk, Dynamic Influence of Moving Loads on Rods
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| 15. | I. I. Ivanchenko, "Analysis of the behavior of distributed rods subject to moving
loads," Prikladnaya Mekhanika, Vol. 24, No. 9, pp. 109-118, 1988. |
| 16. | L. A. Rozin, Variational Statements of Problems for Elastic Systems
[in Russian], Izd-vo LGU, Leningrad, 1978. |
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Structural Dynamics EURODYN 2002, Vol. 2, pp. 1173-1178, The Netherlands, Balkema, 2002. |
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Structural Mechanics: Dynamics and Stability of Structures
[in Russian], Stroiizdat, Moscow, 1984. |
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Russian], Gostekhizdat, Moscow, 1955. |
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Dynamics of a Carriage
[in Russian], Transport, Moscow, 1991. |
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Russian], Transport, Moscow, 1998. |
| 23. | N. N. Kudryavtsev, "Analysis of dynamics of unsprung masses in carriages," in
Transactions of VNIIZhT [in Russian], Issue 287, 1965. |
|
| Received |
19 June 2003 |
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