Stability of dam slopes taking into account anisotropy

. This paper presents the results of experimental studies on the deformation and strength anisotropy of gravel-pebble soils used in soil dams. During stabilometric tests, varying shear angles were obtained for stone at different orientations of soil particles in the working chamber, whether horizontal or vertical. As the hydrostatic pressure in the working chamber increased to 0.8 MPa, a reduction of shear angles of a stone by 8-12 degrees depending on the direction of layering of soil was observed. The authors compiled a calculation program called OTKOS_Ani (Delphi), which considers the strength anisotropy of pebble soils for calculating the stability of slopes of soil dams and slopes. The slope calculations were carried out for different design and height of stone and stone-soil dams, and correction factors for taking into account the strength anisotropy in slope stability calculations, 𝑘 ан , were obtained depending on the type, height, and density of gravel-pebble soil stacking in the dam body. The results showed that considering the anisotropy of strength properties in the calculations of slope stability of earth dams reduces the safety factors by 3-11%. This should be taken into account when selecting the design of earth dams with prisms of gravel-pebble soils.


Introduction
Construction of earth-rock dams has been carried out all over the world for many years, especially in the republics of Central Asia, South-East Asia, and China, in mountainous conditions and in high seismicity zones.The most economical structures with a compressed profile can be achieved through a high degree of compaction of the embankment.For example, the gravel-pebble soil density at Nurek dam dumping reached almost 22.5 kN/m3 due to special compaction technology.Despite this, the compacted embankment exhibits unusual behavior in terms of differences in mechanical properties along and across the layering.The term "layering" of material usually refers to non-uniform interlayers formed at the contact surface with the construction equipment.During compaction of gravel and pebble soil, another kind of layering appears, which is associated with a change in orientation of stone particles in the backfill when they are compacted layer by layer.Research has shown that heterogeneity of physical properties arises when "flakiness" of a stone, expressed in the ratio of the maximum and minimum size of a particle, is more than 3.Such material -channel pebbles -is commonly used in construction and has been used in the dumping of the Nurek and Rogun dams.Compacting gravel and subsequent forces cause the particles to arrange with larger contact planes along the compaction direction.It is natural to assume that the work of each separate particle will proceed differently depending on the direction of the acting load, which imparts anisotropic properties to homogeneous material as a whole.

Methods
In geomechanics, the phenomena of material anisotropy are well known and accounted for in calculations.Sedimentary soils in natural conditions have properties of filtration, strain, and strength anisotropy [1,2] associated with the natural layering of rocks.It is known that the filtration coefficient in rolled cores of soil dams in different directions may differ by 10 or more times [3].In soil dams, the difference in deformation and strength characteristics of the ground along the directions of layering leads to the redistribution of stresses.Considering this factor in calculations of the stress-strain state of dams and the stability of slopes allows for increased reliability in justifying their constructions.
The problem of stress distribution in foundations, taking into account the anisotropic properties of soils, has a long history of research by many scientists, including Lekhnitsky S.G. [4], Zenkevich O. [5], K. Wolf, L. Barden, Abelev M.Yu.[6], Boldyrev G.G. and Idrisov I.Kh.[7,8], Goldstein M.N. and Lapkin V.B. [9], Nabokov I.M. [10], School [11], Ter-Martirosyan Z.G. [12,13], Korobova O.A. [14], and others.Studies of slope stability, taking into account the anisotropic structure of soil and rock massifs, are presented in the works of Zerkal and Fomenko [15], Kremnev, Vishnyakov, and Sedun [16], and others.Most of these works consider rock massifs (slopes) with different angles of stratification of weaker in strength layers, lenses, along which different strength parameters of the material are set apart from the whole massif.The aim of the present investigation of anisotropic properties of gravel-pebble soils was to obtain differences in strength properties of bulk soil depending on its stress-strain state, in particular, depending on the position of the main stress area in the soil with respect to the direction of the layering axis of the gravel laid into the dam body.
Considering anisotropy in calculations of strength and stability of soil dam slopes is based on experimental studies of anisotropy of deformation and strength properties of soils.For this purpose, three series of experiments were conducted:  in a two-stamp odometer using an Instron 3382 universal electromechanical machine at pressures up to 7 MPa at the Research Center of IS MGSU † ;  in a vacuum stabilometer at pressures up to 0.9 atm. at the Department of GIGS MGSU;  automated test complex "ASIS" for testing coarse-clastic soils in three-axial compression at lateral compression pressures σ3 in the range of 0.2 ÷ 0.8 MPa in the Geotechnical Research Center at MGSU ‡ .
To reveal the anisotropic properties of gravel-pebble soil, the experiments were carried out with horizontal and vertical orientation of particles by a larger diameter when putting them into the working chamber, thus simulating horizontal and vertical layering.The minimum size of oriented particles with an aspect ratio greater than 3 was 10 mm on the smaller dimension.In both the compression and triaxial experiments, the main stress  у was directed vertically.Thus, the experiments determined the mechanical properties of the soil along and across the layering axis.Each experiment was performed five times.The obtained results of strain anisotropy tests for gravel-pebble soil in all experiments gave higher strain moduli for vertically placed soil particles (along the stratification axis) than strain moduli for soil particles placed perpendicularly to the acting load.The anisotropy coefficient determined from the strain modulus ratio as  =     (for the initial strain moduli) in the odometric tests varies in the range from 1.2 to 1.7 [17].The anisotropy coefficient of gravel-pebble soil was tested using a vacuum stabilometer, which yielded results ranging from 1.7 to 1.9 [18].Another test conducted using a large triaxial stabilometer at a compression range of 0.2 to 0.8 MPa showed that the anisotropy coefficient ranged from 1.2 to 1.4 [19,20].
It is a well-known fact that the shear strength of soils depends on the applied stresses, with the shear angle ϕ reaching its maximum value at low stresses σ and decreasing as the stresses increase [21,22].Based on the triaxial test results, Mohr's circles were constructed for different soil stacking schemes, and the resulting data is presented in the form of a soil strength passport in Figures 1 and 2.

Results
Research conducted using uniaxial compression apparatus, full compression apparatus, and large triaxial stabilometer indicated that the deformation behavior of gravel-pebble soil in a dam is dependent on the orientation of layering.Mohr circles were constructed from experimental data for two different layering orientations to determine the variation in soil shear angles with axial and normal pressures.The relationship between the shear angle φ and the principal stresses  1   3 , is determined by the Coulomb-More strength condition given by Equation ( 1): Using the obtained values of shear angles, approximate relationships were constructed for the two different layering orientations, which are presented in the form of Equations ( 2) and (3) below: For horizontal layering ( ┴ ): For vertical foliation ( ǁ ): here, σ is the normal stress applied on the shear pad, measured in MPa.
As can be seen, in contrast to the deformation properties, the value of shear angles for the gravel-pebble soil in the stratification direction  ǁ is lower than the values of shear angles in the direction across the stratification  ┴ (  <   ).The functions φ=f(σ) obtained in the experiments as a function of the normal pressure at the shear pad and the orientation of the foliation axis with respect to the direction of the main stress at the shear pad (Fig. 3) correlate with the experimental data presented in the works of Rasskazov L.N.
Based   The difference in strength characteristics of gravel-pebble soil depending on the position of the stratification axis in relation to the position of the principal stress area (PSA) at some point of the soil at the shear pad can be represented as a hodograph of shear angles changing according to the law of the ellipse.The values of shear angles of the material at the location of the main stress area along and perpendicular to the axis of soil layering are taken as the largest and smallest radius of the ellipse semi-axis.The direction of the layered material axis on the hodograph is assumed to be horizontal, along the x-axis.Thus, in experiments with a horizontal arrangement of particles, the principal stress vector (axial compression) is perpendicular to the stratification axis, and the MSA is parallel to the stratification axis of the material.In this experiment, the shear angle takes the highest value:   = , which is deposited on the large (vertical) semiaxis of the ellipse (Fig. 5).
In the experiment with vertical orientation of the particles in the working chamber, the shear angle is smaller:   = , and it is plotted on the horizontal axis (Fig. 5), with the MSA perpendicular to the layering axis of the material.
For an arbitrary direction of the main stress in relation to the axis of layering of the material, which is taken at an angle α, we can obtain an expression for the value of the shear angle based on the canonical equation of the ellipse, which in the Cartesian coordinate system is written in the form: where a and b are values of the major and minor semi-axes of the ellipse.Any material point on the surface of the ellipse (p.M) defines the direction of the principal stress in the form of a vector connecting the origin of coordinates (intersection of the major axes of the ellipse) with the point itself, and the angle between the direction of the principal stress and the axis of layering is defined as the angle α, where 0 ≤  ≤  2 .For p. M on the surface of the ellipse, the value of the shear angle is determined by the value of the radius-vector φ.To determine φ from the known angle α, we can write the equation of the ellipse in parametric form: After substituting ( 5) into (4), we obtain the expression for φ: Thus, the value of the shear angle φ, taking into account anisotropy depending on the direction of the vector of main stresses in relation to the direction of the layering of the material, i.e. depending on the angle α, can be determined by formula (6).The values of shear angles a and b included in expression ( 6) are determined by functional dependences ( 2) and ( 3), depending on normal stress at the shear area (Fig. 3).In order to determine the trajectories of main stresses in design elements of dams and solve problems about stressstrain state of the dam, including those taking into account deformation anisotropy of soil, the angle α can be obtained at each point of the collapse surface.
With the aim of developing a procedure for calculating the stability of soil slopes of dams with allowance for strength anisotropy, based on the experimental data obtained, the authors have developed mathematical models of changes in the strength properties of the material depending on the magnitude and direction of the main stress with respect to the layered material axis, developed a calculation procedure, and written a calculation program "OTKOS-Ani" using the method of circular cylindrical collapse surfaces in the Delphi 10 programming environment.
Calculations of dams of different designs and heights were made with and without taking into account anisotropic difference in strength characteristics of the material.On the basis of these calculations, correction factors (k_an) were obtained to account for anisotropic properties of gravel-pebble soil in calculations of slope stability of different types of dams according to the formula: where:   ан -coefficient of stability factor of dam slopes in calculation considering strength anisotropy properties of gravel-pebble soil; -coefficient of stability of dam slopes in calculations without taking into account anisotropic properties of soil.
The program was tested by comparing the obtained results with the results of calculations by "Otkos" program, developed and used in practice of calculations at GIS department of Moscow State Building University for more than 30 years [25].
The series of calculation problems included structures of homogeneous dams, dams with a central core, concrete faced soil dams and dams with a diaphragm.Typical dam constructions were considered in different heights: 20m, 50m, 150m, 200m, 250m, 300m and different slope ratios, which was taken equal for upstream and downstream prisms.1:1.6 and 1:2.4 slope ratios were considered.
Interface of the program complex with calculations is shown on pic.6 for rock fill dam with 200m high diaphragm and 1:2,4 slope ratio.Nomograms for determination of correction factor  ан for taking strength anisotropy into account in slope stability analysis of the groundwater dams were prepared by the results of the alculations in Fig. 7 (for concrete faced soil dams) and in Fig. 8 (for dam with central impervious element):

Discussions
The calculations have shown that the correction factor k_n^an is less than unity, so the slope stability coefficient with regard to anisotropic properties of soil decreases compared with the stability of isotropic soil: ан =  ан •   , (8) According to the nomograms, obtained, influence of strength anisotropy manifests itself in reduction of stability coefficient from 3% to 11% depending on type and height of dams As the height of the dam increases, correction factor decreases and the least value of  ан = 0,89 is obtained for upstream slope of dams with screen.

Conclusions
1.The investigations carried out in the uniaxial compression instrument, in the all-round compression instrument, in the large triaxial stabilometer have shown that the gravelpebble soil placed in the dam by the method of layer-by-layer compaction (horizontal layering) has a different character of deformation depending on the direction of the main stress, with   >   .The modulus ratio characterized by the strain anisotropy coefficient of soil  = , which decreases as the load increases.The maximum values of the anisotropy coefficient were obtained in the odometric tests of 1.9 and in the stabilometric tests of 1.4.2. The anisotropic properties of stratified compacted gravel-pebble soil also manifest themselves in strength anisotropy.It is obtained that the shear angles of the soil in the direction across the foliation are higher than along the foliation (  <   ).The difference in maximum shear angles is 2-5 degrees, and the difference in shear angles increases as stresses increase.3. A procedure for calculating the stability of slopes of stone and earth-rock dams taking into account the strength anisotropy of soils depending on the stress-strain state of the dam and the position of the main stress areas in the calculation region has been developed and implemented in the OTKOC-Ani software package.4. According to the results of calculations, it was found that the correction factor for taking into account anisotropic properties of gravel-pebble soil in slope stability calculations  ан depends on the type of dam (with screen, with diaphragm, with core), height and density of soil stacking in the dam buttress prisms.According to obtained diagrams on Fig. 9÷12, coefficient values change in the range of 0.89 -0.97, which gives reduction of slope stability coefficient by 3-11%.
This work was financially supported by the Ministry of Science and Higher Education of Russian Federation (grant # 075-15-2021-686).Tests were carried out using research equipment of The Head Regional Shared Research Facilities of the Moscow State University of Civil Engineering.
on the graphs of the   = () (2) and   = () (3), the function of the ratio of soil shear angles depending on the normal stress  ан = ), which graph is shown in Fig.4, can be obtained.

Fig. 3 .
Fig. 3. Shear angles of gravel-pebble soil with increasing effective stresses ϕ=f(σ) according to the results of triaxial tests for different soil stacking schemes.

Fig. 4 .
Fig. 4. Ratio of shear angles of gravel material     as a function of the normal stress at the shear site.

Fig. 6 .
Fig. 6.Calculation of stability of dam slopes with and without strength anisotropy for 200-m high dam.