Материал: Russian Journal of Building Construction and Architecture

BASES AND FOUNDATIONS,UNDERGROUND STRUCTURES

DOI 10.25987/VSTU.2018.41.1.001

UDC 539.3 : 624.073.2

A. A. Sedaev1

PRESENTATION OF A KERNEL AND TRANSFORMANTS

OF A NONCLASSICAL ELASTIC FOUNDATION

THROUGH ITS INHOMOGENEITY FUNCTION

Voronezh State Technical University Russia, Voronezh, e-mail: sed@vmail.ru

1D. Sc. in of Physics and Mathematics, Prof. of the Dept. of Applied Mathematics and Mechanics

Statement of the problem. The paper is devoted to the question of practical applicability of a mathematical model by Aleinikov-Snitko which describes the contact interaction of a subbase and a shallow foundation. The main results of this method were announced with no proof in 2009 in a joint article [3].

Results. In this paper, based on the methods of integration and the Hankel transform, all the formulas and tools of the model by Aleinikov-Snitko are fully proven and optimized. Here we calculate and compare the results obtained by means of this method and the classical Mindlin method based on the theory of elasticity for a foundation whose elastic modulus is given by a power function. Interestingly, the results obtained are similar in form, but differ in magnitude. Using these results, we obtain simple approximate formulas for finding the precipitation of the base surface from the action of a point vertical load for both methods.

Conclusions. The formulas proved in the article can be useful in modeling the interaction of the soil base and the foundation, which is crucial in construction and design.

Keywords: inhomogeneous linearly deformable half-space, influence matrix, base kernel, transformant, base non-uniformity function, quasitransformant, Bessel function, Hankel transform.

Introduction. The problem of mathematical modeling of the interaction of a foundation with a subbase is a quite challenging one and has no universal solutions for all types of soils. The simplest for engineering calculations is the known Winkler model with one foundation coefficient. The main disadvantage of this model is that is fundamentally impossible to reflect a distribution capacity of soil in transmitting a vertical load in a horizontal direction while involving those layers of soil that are beyond the loading area [11, 13].

© Sedaev А. А., 2019

* The work is supported by the Russian Fund for Basic Research, grant № 17-01-00138.

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Russian Journal of Building Construction and Architecture

Unlike the Winkler model, homogeneous elastic half-space modeling a subbase foundation using the elasticity theory causes an increase in the distribution capacity of actual soil.

In practice in order to mitigate the above disadvantages, the Winkler model is supplemented by new parameters (extra foundation coefficients) and a model of an elastic half-space is assumed to have elastic characteristics (elasticity modulus or the Poisson coefficient) that change as does the depth of a foundation point.

In [1, 3] S. М. Аleynikob following N. K. Snitko’s suggestion [15] developed a calculation scheme for the interaction of a foundation and heterogeneous foundation base. In this paper an analytical aspect of the method is developed and compared with the classical approach based on the methods of the elasticity theory and the Fourier transformation. The problem of identifying the main part of the deposit of the surface of a soil foundation under the effect of a concentrated force is discussed as well as the options for practical implementation of the method.

1. Statement of the problem and preliminary data. A model of a linearly deformed foundation (half space) will be used where settling w and distributed along the area on a daytime surface of a half space and the load q are connected with a ratio

where G(x y u v) is a so-called nuclear of an elastic foundation [12]. The latter is an influence function that equals the displacement of the pointP(x y) of a daytime surface of an elastic halfspace caused by a single vertical concentrated force applied to the pointQ(u v) of the surface.

Note that for the half-space which is isotropic in a horizontal area when its deformation characteristics depend only on the vertical coordinate z , nuclear G(x, y,u,v) depend only on the distance r between P and Q . It is the case that will be dealt with as we proceed.

1.1. Representation of the influence function based on the methods of the elasticity theory.

Let a single load be concentrated at the beginning of the coordinate system XOY on a daytime

surface and thus r x2 y2 . The use of the methods of the elasticity theory and Fourier transformation lead to the following formula [10, 12, 14] while searching for an influence function:

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Note that the integrals (2) and (3) included in the formulas are nothing but a direct and reverse

1.2. Calculation of the nuclear using the function of heterogeneity of a foundation. Following the example of N.K. Snitko [15], S. М. Аleynikov developed an alternative method for identifying the function that determines the nuclear for an isotropic, heterogeneous linearly deformed half-space. Let E(z) be the elasticity modulus of a foundation depending on the depth z, and is the Poisson coefficient of the half-space that will be considered constant. Then according to [1, 3], settling of a daytime surface caused by a single load can be specified with the equation

is generated by the law of changeE(z) and called the function of heterogeneity of a foundation. For a homogeneous space when E(z) E0 const , the formulas (2) and (5) yield the same results while in other cases (e.g., whenE(z) En zn ) these formulas yield similar-looking but different expressions for (r).

The comparison of the formulas (2) and (5) shows that the functions c(s) and e(R) play a similar role, i.e. they generate a nuclear of an elastic foundation. Certainly there is an issue with identifying a function similar to c(s) that corresponds with a new method. I.e. using the specified function e(R)we will be identifying such a function C(s) so that with the formula (2) it yielded the same result that the formula (5). Such a function C(s) will be called a quasi-transform one.

2. Main results

2.1. Presentation of a quasi-transform using a function of heterogeneity of a foundation.

The first formula is obtained by inserting the expression (5) into the formula (3):

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