Comparative assessment of the calculation of the sediment base of foundations by the SP method and the method of NRU "Geotechnika" MGSU and their analysis

. The article provides a comparative assessment of methods for calculating the sediment of foundations of finite width with and without taking into account the horizontal movements of layers, as well as taking into account the self-weight of layers. As a calculation, to estimate the stress state of the base, the Flemann formulas are used (the plane problem), which allow us to determine the stress components Ãx, Ãz, Äxz, as well as the average stress Ãm = (Ãxp + Ãzp)(1 + ν)/3, depending on the load intensity p = const acting on the surface of the half-space along the strip b = 2a. In addition, this article provides formulas for determining the surface precipitation of a linearly deformable half-space S(x, 0) = f(E, ν, b = 2a).


Introduction
It is known that for the calculation of the precipitation of the bases of the final width at the present time, the normative document SP (set of rules) 22.13330.2016(the main regulatory document of the Russian Federation in the field of geotechnics, further SP) recommends using the method of layer-by-layer compaction of soil layers in the compressible base layer in the absence of the possibility of lateral expansion and under the influence of the compacting load at an intensity equal to the maximum value Ãzp = p0(z) along the entire length of the layer.
It is assumed that this assumption under the condition of •x = •y = 0 compensates for the omission of the influence of horizontal displacements and stresses Ãxp and Ãyp on the sediment of the layer and therefore the sediment of the layer i according to the SP (set of rules) is determined by the formula where Ãzpi = Ãmaxzp(x) on condition •x = •y = 0; E0i -modulus of total deformation; ³(Ài) = 0.8 -coefficient, which is a constant independent of the real value of the Poisson's ratio of the soil Ài.
Based on (1), it is recommended to determine the base sediment by the formula: = ∑ ÿ ÿ=Ā ÿ=1 (2) where n -the number of layers in the compressible thickness of the base.Thus, the calculation of the foundation base precipitation is simplified as much as possible and is reduced to determining the minimum number of parameters Ãzp,i and E0i, while this method of calculating the layer precipitation with the restriction of horizontal movements in all types of dependence • -Ã, including Hencky, inevitably leads to a decrease in the sediment of layers.Note that the use of a compression curve to determine the modulus of deformation of the layers at each load stage is an approximate method.According to the results of three-axis tests, the determination of the strain modulus for compression compression conditions is problematic, since the strain modulus depends on the average stress Ãm.
The method of calculating the sedimentation of layers using the system of Hencky equations [1], which takes into account the nonlinear dependence, is devoid of these disadvantages • -Ã, including the dependence of the shear G and volume K strain modules on the stress state, moreover, with a linear dependence • -Ã the system of Hencky equations passes to the system of Hooke equations.In addition, taking into account the possibility of horizontal displacements of layers makes it possible to use various nonlinear models of soils.
This article is devoted to a comparative assessment of the calculation of precipitation by SP and by REC as well as the justification of the method for calculating the sedimentation of foundations of finite width using the Hencky equation system, which takes into account all three stress components including the average stress Ãm.This method is based on the idea of representing the linear deformation of the soil layer •z = ∆S / ∆z as the sum of the volume •z,À and the shear •z,´ components of the linear deformations •z in the form: Indeed, the transformation of the system of linear Hooke equations leads to the dependence of • -Ã to the form where G and K are the modules of the shear and volume deformation of the layer, and Note that the formula (4) was obtained by Z. G. Ter-Martirosyan [2] independently of the generalized Hencky equations [1].It can be shown, that the usual Hooke equation is obtained from (4) if G, K, and Ãm from ( 5) are substituted into it.Formula (4) allows the use of G and K independently of each other, which are known to be determined based on standard triaxial tests (Figure 1).In addition, these tests allow you to determine the strength parameters of the soil.To determine the precipitation of a layer with a thickness of ∆zi on the basis (4), as in SP (set of rules), it is necessary to use the method of layer-by-layer summation of the sediment of the layers.Then we get: where ̄ÿ,ÿ -weighted average value of the plot ̄ÿ , i.e.
where p and bthe intensity and width of the load within b = 2a.

Comparative evaluation of the methods for calculating the precipitation of the elementary layer by sp and by nru mgsu, considered in this article
In contrast to the SP in the REC, the calculation scheme includes the average values ̄ÿ and ̄ÿ( ̄ÿ) (Figure 2), and also ̄ý , G(Ãm) and K(Ãm), moreover, b is the width of the loading band, p is the intensity of the load on the band.Below are the calculated schemes for determining the precipitation of the elementary layer with thickness ∆z by SP and by NRU (Figures 2a, 2b).A schematic representation of the change in the shape of the loadbearing column (conditionally) under the foundation is also given when using the SP (set of rules), method and the NRU method (Figures 2b, 2g).Comparative evaluation of JV and NRU methods: -in the SP method, you must define ÿĀ , ÿ , ý = þ = 0 ⋅ ÿ , 0 , 0 ; -in the NRU method, you must define ̄ÿĀ , ̄ÿ , ̄ý , ̄ÿ, ( ÿ ), ( ÿ ); -in the SP method, one parameter is calculated -average layer sediment according to the formula △ ̄ÿ = ÿ ⋅△ ÿ ⋅ Ā( ÿ )/ 0ÿ , on condition ³(Ài) = 0.8; -several parameters are calculated in the NRU method △ ̄ÿ, △ ý (if necessary), moreover -in the SP (set of rules) method, the calculation of the layer precipitation always ends with stabilization, both in the linear and nonlinear formulation due to the condition •x = •y = 0; -in the NRU method, the calculation of the sediment of layers can be stabilized in the linear setting, and in the elastic-plastic setting it can not be stabilized and continue to develop up to the undamped values of the sediment.The obvious difficulties in the NRU method are justified by the fact that it allows you to calculate the base subsidence at p > R (where R is the calculated resistance of the base soil) within the specified limits p.Such a forecast is necessary when designing the foundations of buildings and structures, because it allows you to use the reserves of load-bearing capacity and take the optimized dimensions of the foundation at a given precipitation.Figure 3 shows a schematic representation of the dependencies ∆S -∆Ãz by the SP method and by the NRU method.The next section of this paper provides a computational and theoretical justification of the curves ∆S -∆ ̄ÿ including ∆S´ and ∆SÀ.  3 Theoretical foundations of the forecast of the sedimentation of the foundations of the foundations by the rec method (flat problem)

Components of the stress state in the base
It is known that the components of the stress state in the ground half-space under the action of the load on the band b = 2a are determined by the Flemann formulas in the form: On the basis of these formulas in the work of V. A. Florin (1959), a table is compiled for Ãx/p and for Ãz/p.However, determining the weighted average value ̄ÿ , acting on the layer results in false expressions: where b and p are the width of the foundation and the intensity of the load under the foundation; At the same time, the view of the curve Ãzp (x) (Figure 4) given in the work of V. A. Florin (Figure 4) is similar to the Gaussian probability curve.This allows the curve Ãzp (x) to be represented as: where p0(z) -the maximum intensity of the load on the layer at x = 0 and at depth z, and at z = 0, p0(z) = p.
α -a curve parameter that can be determined from the equilibrium condition (11) in the form This integral is known and the formula (13) takes the form:

Calculated models of soil bases
In the case of linear deformation from (17) we obtain: moreover, the system of Hencky's equations passes into the system of Hooke's equations [2].
We also assume that in the process of loading, the elastic-plastic properties of clay soil during shear are described by the formula of S. P. Timoshenko [5], which, as applied to soils, has the form: where Äi, Ä*i -the effective and limit values of the intensity of tangential stresses (Figure 5b), and: where φi, ci are the limit values of strength parameters determined by the limit line in the planeÄi -Ãm (Figure 1); Ge -is the shear modulus at Äi → 0, and ā ÿ þ = .In the presence of over-compacted soils, it is necessary to use the residual stress in the overcompacted soils, determined by the results of compression tests by the Casagrande method [6].

Calculation of the sediment of a linearly deformable base using the NRU method
In the simplest case of a linear relationship between stresses and deformations with the parameters G and K, the draft can be determined by an analytical solution for the z axis (x = 0).Then we can write This is a transcendental equation with respect to the unknown lateral pressure Ãx = Ãz as a function of Ãz.Defining them using the PC MathCad and substituting in (25), we get the compression curve •(Ã).Figure 6 shows the results of calculating the dependencies •z(Ãz), •z,À(Ãz), •z,´(Ãz), as well as Ãx(Ãz) for loading and unloading on the basis of (25) using a PC MathCad.From the analysis of the curves, it can be seen that •z(Ãz) = •z,À(Ãz) + •z,´(Ãz).In addition, their residual values are recorded during elastic unloading.The residual Ãx is also fixed on the Ãx -Ãz curve.

Conclusions
In this paper, a comparative assessment and analysis of the methods for calculating the sediment of the SP and NRU "Geotechnika" MGSU is given, during which the following points are noted: 1.The SP method is based on the condition of compressive compression of the layers (•x = •y = 0 which, for any law of deformation of the layers in the compressible thickness, leads to the stabilization of the sediment, i.e., with an increase in Ãz the parameter •z → 0. 2. The method of NRU "Geotechnika" MGSU is based on the condition of free horizontal displacements of layers from the compressible thickness of the foundation soil and, depending on the parameters of the volumetric and shear deformations of the Hencky, it can have both a damped and non-damped character of double curvature, including the total sediment of the base of foundations of the same width.The method of NRU "Geotechnika" MGSU allows you to predict the base sediment beyond R > p calc, but the SP method does not. 3. Along with the SP (set of rules) method, the method of NRU "Geotechnika" MGSU should be added to the regulatory document as an additional method necessary for predicting the sediment of the foundations of buildings and structures of increased responsibility

Fig. 1 .
Fig. 1.Schematic representation of the results of standard three-axis tests of soils under the kinematic loading mode ( 1 = const and ̇1 = const).

Fig. 2 .
Fig.2.Calculation schemes for determining the precipitation of a layer with a thickness of ∆z according to the SP method (a) and the NRU MGSU, method (b) and the shape of the supporting pillar under the foundation according to the SP method (b) according to the NRU method (g).

Fig. 5 .
Fig. 5. General view of dependencies •i -Ãm (a) and Äi -´i (b)In this case, the secant modulus of the volume strain K can be determined by dividing in the expression (16) •*, by Ãm i.e.

Fig. 8 .
Fig. 8.The dependence of the sediment is the load (S -p) of a non-linearly deformable foundation base of finite width, taking into account (1) (the NRU model) and not taking into account (2) (the SP model) horizontal displacements of layers in the compressible thickness of the base, calculated by formulas (35) and (36), as well as by the method of layer-by-layer summation.