THE REACTIONS OF THE «BEAM – FOUNDATION» SYSTEM TO THE SUDDEN CHANGE OF THE BOUNDARY CONDITIONS

The authors constructed a mathematical model of a dynamic process in a loaded beam on the elastic Winkler foundation in a sudden formation of a defect in the form of a change in the boundary conditions. The solution of the static problem of bending of the beam pinched at the ends served as the initial condition for the process of forced vibrations hinged supported at the ends of a beam, which arose after a sudden break in the connections that prevented the rotation of the end sections. The authors determined the dynamic increments of stresses in a beam for various combinations of a beam and foundation parameters.


INTRODUCTION
An important problem of construction mechanics is the analysis of the sensitivity of load-bearing structures to structural rearrangements under load such as suddenly disconnected connections, cracks, fractures, etc. Obtaining such information for real constructions requires the development of special methods, since this problem cannot be solved by universal methods. From the standpoint of structural mechanics in these problems, it becomes necessary to calculate such systems as constructively nonlinear, changing the design scheme under load, i.e. with dynamic overloads, caused by sudden beyond projected effects.
In the present work, the task is to construct a mathematical model of transient dynamic processes in a beam on an elastic foundation when a defect is suddenly formed in the form of a change in the boundary conditions. Before the formation of a defect, the reaction of the structure is determined by a static action. The sudden formation of a defect leads to a reduction in the overall rigidity of the structure, which does not ensure the static equilibrium of the system. The inertial forces that have arisen cause a dynamic reaction, redistribution and growth of strains and stresses. As a result, there may be a violation of the regular functioning of the structure, or loss of load capacity and destruction.

STATEMENT OF THE PROBLEM
The elastic beam with flexural stiffness EI rests on the entire length l on the elastic Winkler foundation with stiffness coefficient k, rigidly clamped at the ends. The uniformly distributed intensity load q and the foundation reaction affect the outer layers of the beam. It is assumed that at some point in time 0 t  , the connections in a statically deformed beam, which prevent the rotation of the end sections in the supports, suddenly collapsed, forming hinges in the place of sealing. The static state of the loaded beam ceases to be in equilibrium and the beam will move into motion ( , ), x t  during which the deformations and stresses in the beam acquire dynamic increments.

SOLUTION OF THE PROBLEM
The problem is solved in the following sequence: 1) we determine the static deflection of a ("undamaged") beam with clamped ends on an elastic foundation, which is used subsequently as the initial condition of a dynamic process, which is initiated in the system by a sudden transformation of the boundary conditions; 2) we determine the frequencies and forms of bending vibrations of a ("damaged") beam with hinged ends on an elastic foundation; 3) we study the forced bending vibrations of a loaded beam. In this case, the load, the static deflection of the "undamaged" beam and the desired dynamic deflection are decomposed into series according to the modes of natural vibrations of the "damaged" beam. 3.
  are the initial parameters, respectively, the dimensionless bending moment and the shear force at the origin 0.

 
The dimensionless bending moment in a static state is determined by the function Figure 1 shows the diagrams of bending moments in a beam with clamped ends for various values of the generalized rigidity of the "beam-foundation" system 4 4 .

  
It is worth paying attention to the somewhat "unusual" form, which takes the moment epures with increasing rigidity of the systemthe moments in the central part of the beam are much lower than in the quarter of spans. This is the result of the combined effect of external unloading on the beam and the reaction of the elastic foundation.
is a parameter having a frequency dimension and, therefore, called a "conventional" frequency. Equation (4) describes the forced vibrations of a loaded beam. The Winkler model does not imply dynamic phenomena in the elastic foundation. The required eigenfunctions and frequencies of the problem will be obtained from equation (4) with the zeroed right-hand side, which after separation of the variables by representation is a dimensionless eigenfrequency of the bending vibrations of a beam on an elastic foundation.
Using the "conventional" frequency 0 ,  characterizing stiff and inertial properties of the "beam-foundation" system, and the known basic frequency of bending vibrations of a free beam supported in the same way (without foundation support) 2 we bring the equation to the form (6)   where It was shown in [3,4] that for a beam completely supported on the Winkler foundation, in the case of canonical boundary conditionspinch-pinch, pinch -hinge, hinge-hinge, consoleonly variant (10) is realized.
Using the initial parameters we write the relations characterizing the state of arbitrary cross-section  of the beam, using version (10), (11). In this case, the deflection function has the form where   ; The state of the arbitrary beam section is described by the matrix equation is a functional matrix of influence of initial parameters on the state of .
We will conduct an analysis of free frequencies and forms of flexural vibrations of a beam on an elastic foundation when the ends are hinged. In this case, the boundary conditions and the deflection function have the form Satisfying the second pair of boundary conditions (16), from function (17) and its second derivative, we obtain a system of two algebraic equations of relatively unknown initial parameters and 0 Equating the determinant of the system (18) to zero, we obtain the frequency equation

CONCLUSIONS
If we consider the transformation of the boundary conditions in the given "beamfoundation system" under the load as a defect, then the conducted study shows that the quasi-static formation of a defect, that is, a reduction in the rigidity of the end supports, leads to an insignificant increase in the maximum stresses in the beam (К st > 1) if there is no foundation ( = 0) and low values of the indicator of the "beam-foundation" system (0< ≤ 10 1,79 ). For beams resting on more rigid foundations (> 10 1,79 ), the formation of the same defect, on the contrary, leads to a decrease in the greatest stresses (К st < 1).
A sudden formation of a defect gives more than three times (К dyn = 3,614) an increase in the maximum stress in a free beam (= 0). For systems with higher rigidity, the effect of transforming the boundary conditions is reduced. There is a redistribution of stresses along the span, but the greatest stress at  >10 4 does not exceed the initial static value (К dyn = 1).
In addition, regardless of the rate of defect formation with increasing rigidity of the system, the greatest stresses move from the center of the beam to the periphery of the span.
The work was conducted within the framework of the basic part of the state task 1.5265.2017/BP (1.5265.2017/8.9)