Moving load

In structural dynamics this is the load that changes in time the place to which is applied. Examples: vehicles that pass bridges, trains on the track, guideways, etc. In computational models the load is usually applied as:

Numerous historical reviews concerning the moving load problem exist (for example,). Several publications deal with similar problems.

The fundamental monograph is devoted to massless loads. Inertial load in numerical models are described in Unexpected property of differential equations that govern the motion of the mass particle travelling on the string, Timoshenko beam, and Mindlin plate is described in. It is the discontinuity of the mass trajectory near the end of the span (well visible in string at the speed v=0.5c). The moving load significantly increases displacements. The critical velocity, at which the growth of displacements is the maximum, must be taken into account in engineering projects. Structures that carry moving loads can have finite dimensions or can be infinite and supported periodically or placed on the elastic foundation.

Let us consider simply supported string of the length l, cross-sectional area A, mass density ρ, tensile force N, subjected to a constant force P moving with constant velocity v. The motion equation of the string under the moving force has a form

Displacements of any point of the simply supported string is given by the sinus series

where

and the natural circular frequency of the string

In the case of inertial moving load the analytical solutions are unknown. The equation of motion is increased by the term related to the inertia of the moving load. A concentrated mass m accompanied by a point force P:

The last term, because of complexity of computations, is often neglected by engineers. The load influence is reduced to the massless load term. Sometimes the oscillator is placed in the contact point. Such approaches are acceptable only in low range of the travelling load velocity. In higher ranges both the amplitude and the frequency of vibrations differ significantly in the case of both types of a load.

The differential equation can be solved in a semi-analytical way only for simple problems. The series determining the solution converges well and 2-3 terms are sufficient in practice. More complex problems can be solved by the finite element method or space-time finite element method.

The discontinuity of the mass trajectory is also well visible in the Timoshenko beam. High shear stiffness emphasizes the phenomenon.

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