Consideration of soil strata heterogeneity influence on differential foundation settlements of overpasses for high-speed railways

The implementation of projects for the construction of highspeed railways actualizes the search of effective approaches to accounting the influence of soil strata heterogeneity along the course of the track on differential foundation settlements of overpasses. Russian special technical conditions prescribe sufficiently stringent regulation limits of absolute values of overpasses' foundation soil settlements (20 mm for ballastless track) and angles of break in profile (the differential foundation soil settlement), which should not exceed 1 ‰ for ballastless track. These requirements make it necessary to develop the calculation method, which is based on the criterion of deformation. To ensure compliance of design solutions to the specified regulations it is appropriate to use the method of the predefined equated soil settlements for design of shallow foundations of overpasses for high-speed railways. Several features of application of this method are presented in this article.


Introduction
The solutions of tasks, related to transportation process, cannot be achieved without railway infrastructure construction [1]. Analysis of the world experience of high-speed railway realizations showed a widespread use of artificial structures [2]. The leading explanation for this is connected with necessity of taking into account the heterogeneity and complexity of geotechnical conditions along the route, which in a number of cases is led to expediency of overpasses use (instead of embankments) as a result of feasibility study [3].
According to the Russian regulations requirements (the project specific standard [4]), which are related to artificial structures design for the high-speed railway "Moscow-Kazan-Ekaterinburg", the maximum allowable value of foundation soil settlement for overpasses must be less than 30 mm and 20 mm for ballast and ballastless tracks respectively. Also, the angles of break in profile (the differential foundation soil settlement), respectively, must be less than 1,5 ‰ and 1‰. Similar requirements are represented in the building codes of other countries [5,6]. These fairly strict limitations cause the development of the calculation method, which is based on the criterion of foundation soil deformationthe allowable foundation soil settlement, which is determined by design engineer. This possibility is realized in the predefined equalized soil settlements method [7,8], based on the foundation soil settlement determination for a number of calculated foundations at the initial stage of design.

Theory section
Analysis of the question of foundation soil settlement determination in respect to the nonlinear stage of soil deformation has shown the applicability of two approaches, which can be broadly classified into two groups: numerical and analytical (approximate) methods. The first group includes solutions, which are based on the use of differential dependencies for soil state description (for example, plastic flow theory). The second group contains the approximate analytical (engineering) methods, which describes under given conditions the foundation soil settlement with sufficient accuracy. It is obvious that the use of first group methods allows obtaining the most accurate results; however, their application requires more time-consuming calculation procedures and obtaining additional experimental data [7]. The considered design method using the predefined equalized soil settlements relates to the second group methods and applies for calculation of the shallow foundations (using the natural bases). The main assumptions of this method are presented in the publications [7][8][9].
The traditional regulatory approach [10] to design of shallow foundations is to determine their size by bringing the average contact pressure under the bottom of foundations to the 0,9-0,95·R (Rthe design resistance of a soil), i.e. it is based on the pressures equalization. That is the reason of difference in volumes of soils V, which are involved in the work (V1≠V2), what promotes the development of unequal foundation soil settlements S1 and S2 (Fig. 1). Accordingly, this approach is limitedly applicable in changeable geotechnical conditions along the railway line. This is related to the fact that under similar loads N (N1≈N2) on the overpasses' foundations (generally) the different soils have the unequal values R (R1≠R2) and cause the unequal decisions on foundations' widths b (b1≠b2) and their settlements S (S1≠S2). The special feature of overpasses' foundations calculation using the predefined equalized soil settlements is the ability to select at the initial stage of design the value of the foundation soil settlement that matches the normative quantity. For determination of the foundation width for the certain soil settlement, degree of loading and ground conditions, the following formula is used [9]: k, nempirically determined coefficients, correspondingly equated 1,01 and 1,27 for weak soils, 0,96 and 1,04for moderately firm soils, 0,91 and 1for stiff soils; Nγ, Nq, Ncbearing capacity coefficients; N0vertical load at the edge of a foundation; Pin.crinitial critical pressure on a soil; γIunit weight of a soil below the foundation base; γ'Iunit weight of a soil above the foundation base; γavaverage unit weight of a soil and a foundation, usually equated 22 kN/m 3 ; ηaspect ratio of a foundation; ξγ, ξq, ξcfoundation form coefficients; ωstiffness coefficient of a foundation.
It is obvious that the solution (1) in the form of polynomial equation is difficult due to the introduction of empirical coefficients, which are necessary for correct consideration of soil deformation processes in conditions of different soil densities. Therefore, the calculations according to (1) must be carried out by means of successive approximation, which is implemented in the software "BRNL-FT" [11], developed for the automation of calculations according to the proposed method.

Experimental section
For the experimental verification of applicability of the considered method to overpasses' foundations design and for controlling the correctness of the entered assumptions [7][8][9] were conducted the laboratory small-scale (tray) plate-load (stamp) tests. Tests were carried out using the dust sand with layer-by-layer compaction for different presetted soil densities. The load on the stamps was transmitted through the lever device. Their settlement was determined by the dial gauge with scale division equated 0,01 mm.
Consider the problem of determining the dimensions of 2 stamps with the same soil settlement, which is equal to, for example, S1=S2=5 mm, and the different values of vertical loads (N1=0,095 kN, N2=0,13 kN). Then, using the expression (1), the widths of stamps are equal to b1=0,09 m and b2=0,067 m. For the cylindrical tray, as a rule, round stamps are used. It is necessary to take into account the equivalent area when defining their diameters: The calculated data are compared with the results of stamp tests, which are represented in the form of graph S=f(P) (Fig. 2). Analysis of this graph for the given soil settlement S1=S2=5 mm showed that the pressure under the stamp bottom (⌀1=0,1 m) was 13,7 kPa, for another stamp (⌀2=0,075 m) -27,3 kPa. It is obvious that deviations of the experimental pressure values from the calculated P1, P2 are minimal (≈10%), which confirms the correctness of the introduced assumptions [7][8][9] and the calculations carried out by the formula (1). Thus, it is possible to preset the allowable value of the soil settlement for a number of being designed foundations and determine their dimensions. The final justification of such decisions for the considered type of structure must be made based on adequacy assessment of the reliability coefficient of the soil base strength.

Calculation example
Consider the example of using the predefined equalized soil settlements method for the calculation of shallow foundations for high-speed railway overpasses. The initial data are presented in Fig. 3.   Fig. 3. The initial data for calculation example.
For the purpose of calculation results comparison using the traditional (regulatory) approach and the proposed method the widths of shallow foundations are defined (considering the vertical force N). To simplify the calculations, the horizontal forces and moments are excluded and the over-foundation structures are determined conditionally.
According to the requirements (the project specific standard [4]), the foundation soil settlement of the overpass shall be less than to 20 mm (the ballastless variant of track is accepted), the maximum allowable angle of break in profile -1 ‰. The results of the calculations are presented in the Table 1. Thus, the calculation with traditional (regulatory) method, using the elementary layer-by-layer summation of soil settlements, showed the excess of the absolute foundation soil settlement values (S=3,79 cm; 2,44 cm) and their irregularity values (dS/l=1,13‰) relative to the maximum allowable quantities for overpasses of high-speed railways. The use of the predefined equalized soil settlements is allowed obtaining the satisfactory solution (S=2 cm; dS/l=0) regarding requirements [4], however, with large widths of the foundations bases b and with increasing the reliability factor of foundation soil (base) strength Kr. This circumstance is connected with the consideration of the range of design pressures: according to the requirements [10]it is the Pi≤R; according to the proposed method -1,1·Pin.cr≤ Pi<Pult, which makes it possible to use the nonlinear stage of soil deformation. It should be noted that all calculation decisions are satisfactory in respect to the soil bearing capacity (Pi<Pult) and have sufficient reliability, which significantly exceeds the requirements [10] (Kr=1,2) for buildings of III geotechnical category.

Conclusions
1. Use of the predefined equated soil settlements method for design of shallow foundations of overpasses for high-speed railways allows ensuring compliance with sufficiently stringent regulatory requirements related to the absolute values of foundation soil settlements and their irregularity values. 2. Proposed design method takes into account the nonlinear stage of soil deformation, which provides greater range of possible values of presetted foundation soil settlements in comparison with the traditional (regulatory) approach. 3. Consideration of the nonlinear stage of soil deformation suggests the qualitative change in the design method of shallow foundations (using the natural bases) due to the more economical foundation design solutions, which promotes reducing the material and time consumption. 4. Accepting the final decision after the trial design stage in respect to the construction of shallow foundation using proposed method must be made based on adequacy assessment of the reliability coefficient of the soil base strength for the considered type of building.