Modelling the dynamic force of bridge crossings

. Viscoelastic elements of Kelvin-Feugt type are supposed to be used for modelling dynamic influences from the transport side and dynamic behaviour of separate elements and the whole structure as a whole. Periodic monitoring of bridge crossings using the considered approaches will make it possible to detect the presence of defects in the span itself, associated with the formation of cracks, spalling, because in the presence of such defects the estimated frequencies change.


Introduction
Not only high-speed (highways) and high-speed roads are essential for the development of domestic transport systems, but also ordinary public roads of regional, intermunicipal and local importance, which are owned by the constituent entities of the Russian Federation or municipalities and financed from the respective budgets.
There are peculiarities of operation and monitoring of low-water bridges which are found on regional, inter-municipal, local roads as well as on private roads. Low-water bridges are built to overcome an obstacle of low height above its surface and the low-water level can be taken as the main water level position, i.e. the spans of such artificial structures can be flooded when the water level of the passing water is high.
The design scheme is most often based on a girder system or a truss system with a bottom drive. The approach under consideration can be applied when developing algorithms for monitoring, diagnostics and certification of such bridge crossings, which is topical because of difficulties of visual inspection of bridge structures from water.

Research methods
The present study implements an approach that allows taking into account the anisotropic properties of the span, associated with different reinforcement along and across the bridge carriageway, and presents the design scheme of the span not as a beam supported at the edges by hinges or viscoelastic dampers, but as a plate which can have different fixation conditions along the entire contour [1-3].
Consider a transversally isotropic plate of constant thickness, which in an undeformed form occupies the region { } The approximate equation of transverse oscillations of such a plane element in partial derivatives of the fourth order has the hyperbolic form [4]: (1) here W -vertical displacement of the points of the plate median plane,  -Laplace operator, , wherematerial density of the superstructure, b -transverse (shear) wave velocity, -anisotropy coefficients. The plate, which simulates the span of the bridge crossing on two opposite sides, has an unloaded edge and is elastically supported on both sides by the transom beams, so the boundary conditions can be represented by the following relations [5][6][7][8]: where D, M define the reduced cylindrical stiffness of the plate and the bending moment at its edge, respectively; index "op", in the lower case of the value indicates that the value refers to the elastic damper on which the flat deck is supported, but the physical meaning remains the same [9,10]. The solution to equation (1) is proposed to be found in the following form: where   natural frequency of the plate.
If we substitute (4) in (1) and use the new designations for the functional coefficients, we obtain the defining equation in the following form: the following designations are used here For the convenience of mathematical transformations, we introduce dimensionless coordinates and deflection functions: Introducing a new designation , determining the aspect ratio of the span, equation (5) with reference to relations (7) can be written as To solve this equation, we propose an approach (decomposition method) which consists in splitting the complex problem into three simpler auxiliary problems: 2) 3 When formulating the auxiliary tasks, it is assumed that the following relations are fulfilled at given points of a plate element [11][12][13]: (12) In expressions (9) -(11) -arbitrary functions, which in general are represented as: here -are arbitrary constants and the index i takes the values 1 or 2.
A common solution for all three auxiliary problems is proposed to be found in the form of: ∑ where () and () are the arbitrary functions depending on dimensionless coordinates.
Arbitrary functions in expression (14) for can be determined by using the ratios (9): (15) by Slope , get .
Arbitrary functions from the expression for in system (14) can be determined using the relations (9): by the by the (17) Substituting expressions (16) and (17) into the decomposition method relations (11) and (12), we obtain the defining system of equations The system (18) can be rewritten in a more compact form using the following designations ( ) Equations of the system (19) have zero right part and hence nontrivial solution of these relations is possible only under the condition of the solution of the characteristic equation concerning the natural frequencies [14][15][16][17], which is equality to zero of the main system determinants (19). The solution of the characteristic equation determines the frequencies of natural vibrations of the plate structure, which can be shown as graphical curves for various mechanical and geometric parameters of the bridge [18][19][20][21]. As an example, a reinforced concrete bridge crossing with a span length of 18 m, a span width of 6 m and a reduced thickness h = 0.6 m has been considered, the parameters of the support unit have been taken as follows:  op = 0.35,  op = 2500kg/m 3 , M = 2 kNм. Fig. 1 shows the eigenfrequency of a slab span as a function of the reduced stiffness of the span bearing on the transom beam. The solid curve in Fig. 1 corresponds to the third form of natural frequencies, the dashed curve to the second form and the dashed curve to the first form. The obtained graphical dependences have a pronounced non-linear character with a rigid non-linearity characteristic, i.e. the convexity of the curves is directed downward. The format of the obtained results corresponds to the format necessary to check the conditions of normative documents about the fall of the first form of natural frequencies of the vibration of the bridge spans in the interval between the upper and lower value of the relevant limit [22][23][24][25]. This will allow for the correct selection of span lengths and vehicle speeds during construction, reconstruction or overhaul of artificial structures.

Conclusions
The results obtained using the proposed models and approaches are especially relevant for low-water bridges, for which there is often no possibility of visual inspection or instrumental examination from the underside of the supporting part of the span. Natural vibration frequency values can also be used to estimate the water level above the interstice and to predict flooding situations during which the carriageway of a low-water bridge can be flooded.