Contact problems of multilayer slabs interaction on an elastic foundation

A mathematical model and a method to assess the internal force factors in multilayer strip – slab on elastic foundation under various static loads were developed in the paper. A detailed review of well-known publications on assessing the stress-strain state and dynamic behavior of various structures interacting with the base is given. A closed system of integro-differential equations describing the strain process in multilayer slabs on an elastic foundation was derived. The problem under consideration was reduced (using the Chebyshev polynomial) to the solution of infinite systems of algebraic equations. The regularities of infinite systems of algebraic equations were proved and the corresponding estimates were obtained. A number of terms of the Chebyshev polynomial for obtaining a solution to the problem with the required accuracy was determined. The efficiency of the method was shown on the example of test problems.


Introduction
At present, through the efforts of researchers, many different methods for calculating structures on a deformable foundation are developed; the properties of these structures are described by various physical models. These include building foundations, airfield and road surfaces, slabs, hydro-technical structures, rails and railway sleepers, etc. In most of the methods developed, an attention is paid to the analysis of the relationship between the contacting structural elements with the soil base [1][2][3][4]. In [5][6][7][8], various laws of strain and structural destruction of soil were considered, as well as seismic wave propagation, and the rigid body-soil interaction. There are a number of works in which the joint work of the structure with the base is considered: -in [9], the contact problems of the indentation of a rectangular stamp with a flat base into an elastic rough half-space in the presence of Coulomb friction with unknown zones of cohesion and slippage were considered; -in [10]the statement and mathematical methods for solving problems of hydro-elasticity of three-layer structural elements were presented; -in [11][12][13][14][15], various studies were presented to assess the strength and dynamics of earth dams, both taking/and not taking into account the base under static and dynamic effects; -in [16] the interaction of a nonlinear system, i.e. the earth base, with the gravity support was studied. An ideal elastoplastic model was used in the superstructure model, and the Winkler's model was adopted for the foundation; ICECAE 2020 IOP Conf. Series: Earth and Environmental Science 614 (2020) 012089 IOP Publishing doi: 10.1088/1755-1315/614/1/012089 2 -the behavior of a structure resting on a foundation was investigated in [17] during an earthquake. It was assumed that the surface of soil and foundation is a set of discrete non-linear elements, consisting of springs, attachment units and clearance elements; -in [18], the problems of evaluating the critical stress and strain in a hinge-supported rectangular slab beyond the elastic limit was considered and the stability of a bent slab was evaluated. Besides, there is a number of publications devoted to the dynamics, where the work of the structure is considered together with the foundation using artificial boundary conditions at the boundary of the foundation final area: in [19], the solution of the plane problem of wave propagation from a stamp located on the surface of a half-space was considered; -in [20], the problem of axisymmetric vibrations of a flexible ring lying on a viscoelastic layered foundation was studied. The damping properties of the system were analyzed under various excitation frequencies; -in [21] a linear problem of interaction of a Rayleigh surface wave propagating in a sandy medium with a rigid structure partially buried in soil was solved; -in [22], vibrations, stress state and stability of road foundations under machines were considered, taking into account the sub-soil base; -in [23], the dynamic behavior of concrete earth dams was estimated together with the foundation. Therefore, the development of mathematical models and methods for assessing the stress-strain state of multilayer strip-slabs lying on an elastic foundation, taking into account their geometrical and physical features, is a relevant task. In the present study, a closed system of integro-differential equations was derived to describe the strain process of multilayer slabs on an elastic foundation. The considered problems were solved by the expansion in a series of reactive pressure on the foundation by orthogonal polynomials and were reduced to the study of infinite systems of algebraic equations. Moreover, the regularity of infinite systems of algebraic equations was proved and the corresponding estimates were obtained. To show the effectiveness of the methods, a number of test problems were solved.  Consider n -layered strip-slabs lying on an elastic foundation (Fig. 1) , ,..., n q q q , respectively, constant loads along the slab, arbitrary loads across it. Assume that there is an elastic filler between the strips, and the response of the filler is proportional to the difference in deflections of the connecting strips. The lower strip, tightly fit to the base, in addition to external loads and the response of the upper strip filler, is also influenced by the reactive response of the base.

Mathematical models of the problem
To simulate the strain process of a multilayer strip-slab, we take n-layered beam slab cut with a width equal to one. This allows us to reduce the problem under consideration to the problem of n-layered beam slabs of length 2l , thickness 1 2 , ,..., n h h h and width equal to one (Fig. 2). Establishing the origin of the coordinate at the symmetrical center of the beam slabs with the abscissa on the segment > @ To simulate the strain process of n-layered beam slabs, a system of differential equations is written for the unknown deflections of beam slabs in the form: p p x reactive normal pressure in the base.
The equation relating the settlement of a homogeneous base ( ) V x to the reactive pressure ( ) p x under conditions of plane deformation according to [4] can be represented as: Here o E , o Q − are the modulus of elasticity and Poisson's ratio of the base material, respectively.
Further, it is assumed that there is a two-way link between the slab surface and the base, therefore, the contact conditions can be written in the following form: Now the problem under consideration can be formulated as follows: It is necessary to determine the deflections of multilayer beam slabs satisfying the system of equations (1) and the base settlement satisfying the equation (2)

Solution Methods
In what follows, a dimensionless coordinate x is introduced, equal to the ratio of the absolute coordinate to the half-length of the beam l .
The reactive pressure distribution law is sought in the form of a series using the Chebyshev polynomial [24]: where ( ) n T x is the Chebyshev polynomial of the first kind [16]; n A are the unknown coefficients.
The reactive pressures of the foundation ( ) p x must satisfy the equilibrium equation of the beam slabs, i.e.: where , P M are the sum of all vertical forces and the sum of their moments relative to the middle of the beam slabs, respectively. Substituting (4) into (5) and taking into account the orthogonality of Chebyshev polynomials with respect to the weight Substituting (4) into (2) and using the well-known relation [25] for the base settlement we get: If the remaining coefficients of series (4) are assumed to be zero, then this is the case of an absolutely rigid beam slab. The terms of series (4) for values of 2 n represent a correction that distinguishes the distribution of reactive pressure for the absolutely rigid beam slabs. For simplicity, consider a two-layer beam slab interacting with a homogeneous elastic foundation. Then the system of differential equations for beam slabs deflections (1) takes the form The general solution of the system of differential equations (8) taking into account (4) is represented in the following form 4 4   are Jacobi polynomials [24].
Using the explicit form ( ) cos( arccos ) n T x n x (19) of Chebyshev polynomials, it is possible to obtain an explicit form for the function ( )

q x q x q const
In this case, due to the symmetry of load, series (4) includes only even polynomials From the equilibrium equation (5), we have Expressions (11), (12) have the form 4 2 24 With formulas satisfying the boundary conditions and relative deflections of the slab Next, expressions (22) and (25)   To eliminate the singularity of integrals (30) and (31), integrating by parts, they are reduced to the following form: (2 )! 1 2 (2 1) The obtained formulas (32) and (33) are convenient and allow obtaining exact values of the integrals. Thus, the problem under consideration is reduced to the study of infinite systems of algebraic equations (29). It is known that regular infinite systems of algebraic equations have a unique bounded solution. A regular infinite system of algebraic equations can be solved by the reduction methods. Therefore, to apply the reduction method, it is necessary to prove the regularity of the system of infinite equations (29).

Substantiation of the solution method
To prove the regularity of infinite systems (29), coefficients n a 2 , Taking into account inequalities (39) and (40), the following estimates can be obtained from (52), (53)   Based on the reduction method, we restrict ourselves to the first four terms in series (20). Then the system of infinite equations (29) turns into a system of three equations with three unknown coefficients 2