Determination of the own forms of vibration of the span of beam bridges on elastic supporting parts

. The authors consider the determination of the eigenforms of a beam on elastic support parts. Elastic support parts made of rubber and laminated metal plates find a wide use for seismic isolation of bridges in earthquake-resistant construction. Determination of eigenforms of structures is the main stage when designing the buildings and structures for resisting the seismic impacts. The span structure of girder bridges is considered as a system of uniformly distributed load, the free vibrations of which are described by a homogeneous fourth-order partial differential equation. By solving the differential equation, formulas for determining the eigenforms of a beam on elastic support parts were obtained. The reliability of the obtained formulas is confirmed with the substitution of other boundary conditions of the beam. The formulas obtained are considered as a general case, and from them the special cases for different ways of fixing of beams can be obtained.


Introduction
Buildings and structures in seismic areas are designed and constructed considering seismic impacts with a certain intensity depending on the scale of construction sites. The bridge spans are also designed for vertical moving loads.
Determination of the natural vibration modes of beams is the most important task when calculating spans for dynamic loads. Rubber bearings are widely used in the bridge construction [1]. When vertical and horizontal vibrations of the span are combined, the elastic support parts change shape and the natural vibration modes of beam, reducing the effects of dynamic loads on spans [2,3,4,5,6].
The rubber bearing parts are widely used in bridge construction and in this work the vertical transverse free vibrations along the y-axis of the span structure of a beam bridge on elastic supports are considered.
Horizontal transverse vibrations along the z-axis is considered in the same way by replacing y with z.
The method of structural mechanics, special part of construction dynamics, was applied to obtain the deflection, the angle of rotation, the shear force, and the moment in the sections of the beam with free vibration.
By placing the boundary conditions of the beam on the elastic support parts into these expressions, the natural modes of vibration are determined.
Simplification of the cumbersome formulas associated with the Krylov functions were carried out by methods of mathematical analysis.
In the analysis of the obtained formulas for the natural vibration modes on elastic support parts and in the transition from the general case to the special case, the function limit was used.

Results and Discussion
The bridge framework is considered as a beam fixed on both ends. It is assumed that each bridge spans are independent to each other. We will consider a beam with elastic supports (Fig. 1). Normal vibrations of a beam with a uniformly distributed mass without attenuation are described by a fourth-order partial differential equation [7 -12]. (1) where EJ is the beam's stiffness, ݉ ഥ is the mass of beam's length unit.
The solution (1) can be represented in Fourier form ‫,ݔ(‬ ‫)ݐ‬ = ‫)ݔ(ݕ‬ × ‫,)ݐ(ߔ‬ which leads to two ordinary differential equations The solution of the first equation (2) will be To determine the vibration mode we will assess A, B, C, D through initial boundary conditions у 0 , M 0 , М 0 , Q 0 , with x = 0.
For this purpose, we differentiate the expression (3) to the third order and obtain the following. (4) represent the Krylov's functions [9 -13].
By placing these dependencies into expression (4), we have The expression (5) defines deflection, rotation angle, torque, and transverse force, respectively, within beam's natural vibration. Using this expression, we determine the natural vibration modes of the beam installed on the elastic pillars.
We apply the following boundary condition = ⋅ and = , where is stiffness of the elastic support; , и are deflection, shear force and torque, respectively, at the origin of the coordinates.
We consider the first equation of expression (5) as followed [9 -13] By applying boundary conditions for beam with elastic supports from third and fourth equations (5) we obtain the following, provided that = and = − ⋅ .
The normal vibration modes will be Similarly, we can obtain the normal vibration modes for a beam with one rigid pillar and one elastic pillar.
From expression (8) we can get the normal vibration modes for a beam with rigid pillars where С 0 o∞ and for a beam with no pillars where С 0 = 0 (See Fig. 2).
For the first case, when С 0 o∞ ߙ = ߨ݅ and ‫݊݅ݏ‬ ߨ ݅ = 0, we get ‫)ݔ(ݕ‬ = ‫݊݅ݏ‬ The normal vibration modes for beams with different installation methods are illustrated in the fig. 3.
The obtained formulas for determining the natural vibration modes are used in calculating the span of a bridge with seismically insulating elastic supports for the effects of seismic and moving loads [14,15].
The table 1 specifies the natural vibration modes for beams with different pillars at the ends. The positions 1, 2, 3 in the table are taken from the previous works [2,9,10,11], and position 4 is proposed by the authors. Positions 1, 2 arise from position 4, when С 0 =0, С 0 =∞, respectively. The proposed formulas (8) and (9) can be used for horizontal transverse vibration of beams. Then in the formulas take ‫ݖ‬ ௫ instead of ‫ݕ‬ ௫ .
In [16,17], using the same boundary conditions, formulas were proposed to determine the frequencies of natural vibrations of beams on elastic support parts.

Conclusions
The proposed formulas (8), (9) can be used to calculate the bridge spans for seismic and moving loads, and formula (8) is used to calculate intermediate spans, and formula (9) is used for side spans.
Formula (8) can be used to determine the eigenmodes of the transverse vibration of the superstructure with seismic-insulating sliding bearing parts, replacing the sliding friction force of the bearing parts with the reduced stiffness of the elastic bearing parts from the condition that the work of the friction and elastic forces is equal.