МИНИСТЕРСТВОТРАНСПОРТНОГО СТРОИТЕЛЬСТВАСССРГОСУДАРСТВЕННЫЙ ВСЕСОЮЗНЫЙ ДОРОЖНЫЙНАУЧНО-ИССЛЕДОВАТЕЛЬСКИЙ ИНСТИТУТСОЮЗДОРНИИПРЕДЛОЖЕНИЯПО РАСЧЕТУ УСТОЙЧИВОСТИОТКОСОВ ВЫСОКИХ НАСЫПЕЙИ ГЛУБОКИХ ВЫЕМОКМОСКВА 1966

With this method the stability coefficient is determined by the formula:

FS=∑

1

m

[ N i tan φ i+c i li+T iud ]cosφi

cos (α i−φi )

∑1

m

T isdv

cosφ i

cos (α i−φi )

(24)

where:

Ni normal component of soil weight

Tiud tangential component (shear strength) of soil weight in passive zone

Tisdv tangential component (shear strength) of soil weight in active zone

The calculation is performed in the following sequence.

Let’s examining options for scheduled across the sliding surface..

Each restricted presumed slip surface compartment is divided into separate units by vertical planes, conducted in the field by fracture of the sliding surface, so that each block is uniform according to its ground base shear characteristics.

Are determined the values of Ni normal components and Ti tangential components of the weight force on blocks, and also traction ci · li

Stability coefficient calculated total collapse compartment.

Comparison of simulation results, obtained by the options considered, will be set to the minimum stability coefficient corresponding to the position of the critical slip surface. Example of calculation is given in Appendix 10.

If the slip curve at the end portions (within the first and last blocks (fig. 12)) is not predetermined by the geological structure of the massifs, then is not necessary to estimate the most advantageous position.

The first and last blocks outline several possible positions sliding surfaces at different angles of inclination to the horizontal α (Fig. 12).

For each of the identified variants are calculated values E horizontal force interaction between the adjacent blocks.

The magnitude acting between the first and the second blocks is calculated by the formula:

(25)

Fig. 12. Design scheme to the method of prof. GM Shahunyantsaa - slip surface passes through the toe; b - sliding surface extends through the base of the slope

Value of E, acting between the last and penultimate units, should be determined taking into account different possible directions of sliding surfaces within the last block:

a) for upward with respect to the horizontal direction of a sliding surface:

(26)

b) at its downstream direction:

(27)

where n-factor stability of the slope, the value of which should be taken equally acceptable.

Comparing the results obtained, for settlement in the first block should take the position of the sliding surface which corresponds to the largest value of E1, and at last block - the one which produces the minimum value Em-1.

If the potential sliding surface is a plane, soil weight for this same plane and the direction of all forces acting on the block are vertically (fig. 13), stability coefficient is determined by the formula:

(28)

where Q - resultant of all forces acting on the sliding block;

α - angle of the sliding surface to the horizon

All remaining notation are the former ones.

Fig. 13. Design scheme for the plane sliding surface (after prof. univ. Shahunyantsa GM)

The minimum value of the stability coefficient corresponding critical slip surface (α), we find from (27);

where γ-soil bulk density, t / m;

H - scale height of the slope, m;

β - the angle of inclination of the slope to the horizontal surface

If Slope complex is a homogeneous draining soil, without cohesion (c=0), on a solid foundation, the stability coefficient is determined by the formula:

Where:

β - angle of the slope surface to the horizon;

φ - angle of internal friction of soil

Appendix 10

Slope stability calculation by the method of professor. GM Shahunyantsa

The figure shows a cross section of the embankment with a counter bench, backfilled on the hillside.

Name of soils of the embankment are at the base of the diagram. On contact of the embankment with the slope there is a layer of soft soil (loam with pebbles, which is in a fluid state), which predetermines the possibility of sliding surface (shown in the drawing with a dotted line). Highlighted landslide is divided into 11 blocks array of for computing procedure.

Procedure for calculating the stability of the sliding surface is presented in the table. Stability factor on embankment shift contact layer is defined by the formula (24)

Example diagram of slope stability calculation by method Shahunyantsa

Block number φ °

c

ton/ m 2

α°

(α –

φ) °

P = ωγ

Weight of block

sin α

cos

α

T sd

v = P

· sin

α,

t

Restraining forces

cos

(α –

φ)

cos

φ

[9+

11 +

12]

x [1

5]

[8] x

[15]

T ud

= P

· sin

α ,

t

N =

P· c

os α

, t

N · t

g φ,

t c · l

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 30 ° 0 68 ° 38 ° 5,07 0,927 0,375 4.7 - 1.9 1.1 - 0,788 0,866 1.10 1.21 5.22 7 ° 2,5 35 ° 28 ° 23,3 0,574 0,819 13.4 - 19.1 2.4 8.5 0,883 0,992 1.12 12.2 15.03 7 ° 2,5 30 °

30 ¢23 ° 30 ¢

64,9 0,508 0,862 33,0 - 56,0 6.9 9 0,917 0,992 1.08 17.2 35,7

4 7 ° 2,5 29 ° 22 ° 133,1 0,485 0,875 64,5 - 116,5 14.3 24,5 0,927 0,992 1.07 41,5 69,05 7 ° 2,5 24 ° 17 ° 254.0 0,407 0,914 103,2 - 232.0 28,5 47,0 0,956 0,992 1.04 89.0 107.06 7 ° 2,5 0 ° - 7 ° 197.0 0 1.00 - 197.0 24,2 31,8 0,992 0,992 1.00 56,0 07 7 ° 2,5 8 °

30 ¢1 °

30 ¢151.0 0,148 0,989 22,3 - 149.0 18,3 28,2 1.00 0,992 0,992 46,1 22,1

8 7 ° 2,5 0 ° - 7 ° 62,0 0 1.00 - - 62,0 7.6 11.0 0,992 0,992 1.00 18,6 -9 0 ° 4,4 0 ° 0 ° 45,5 0 1.00 - - 45,5 - 18.5 1.00 1.00 1.00 18.5 -10 0 ° 4,4 - 10 ° - 10 ° 32,6 0,174 0,985 - 5.7 32,1 - 34,0 0,985 1.00 1.02 40,5 -11 0 ° 4,4 - 14 ° - 14 ° 4.4 0,242 0,970 - 1.1 4.3 - 29,5 0,970 1.00 1.03 31,6 -

Σ 372,4

Σ 254,0

Weight of block: P = ωγ, Where ω = area of block and γ = specific weight of soil (ton/m3)If we consider the red system of coordinates then the cut vertical planes

which define the border ob block i are defined by:- downward: height hi = yof slip plane i - y of terrain i- upward: height hi-1 = yof slip plane i-1 - y of terrain i-1and ω = 0.5*(hi + hi-1)*(xi-1 - xi) and so the β angle of slope is inserted

indirectly.l = length of block’s slip plane = (xi-1 – xi)/cos α

ОЦІНКА СТІЙКОСТІ СХИЛУ НА ДІЛЯНЦІ ОГЛЯДОВОГО МАЙДАНЧИКА (UCRAINA)For comparison, calculations assess the stability of the slope at the site observation deck performed an

additional calculation method blocks method (GM Shahunyantsa).

In the calculations considered surface of sliding, which is more dangerous, for each of them was determined safety factor of stability. The method of calculating the stability of slopes, GM Shahunyantsa is based on solving a given cross-sectional plane in the presence of an array of fixed-slopes surface of sliding.

Slopes stability calculation by this method is performed in the following order:

- Using surveying data and geotechnical surveys data about materials shear area, were builded section of the landslide body by this vertical section;

- Break cut landslide body vertical lines into separate compartments so that the line slip displacement in each compartment had a constant slope.

Each compartment is determined by the formula of the weight:

Pihi li , (1)

where P i - weight of i-th compartment; li - line length slip in the compartment;

hi - height of the compartment

Next the normal and tangential acting forces on each block are computed by the formulas:

T i Pі sin i (2)N iP i cos i , (3)

where αi- the angle of the i-th compartment to the horizontal, deg.

Slopes stability factor by the method of G. Shahunyantsa [3] is given by:

(4)

where R - retaining force (friction force Nitg φi and adhesion cili):

(5)

F - shifting forces (tangential component of gravity of block Ti, Tci seismic and hydrodynamic forces Fw):

(6)

where φ i - angle of internal friction of soil within the i-th compartment, deg;

ci - specific adhesion of soil within the i-th compartment kPa;

li - the length of the slip surface within the i-th compartment, m;

Tci-component seismic shifting of forces is determined by the formula:Тсі= Ті .К, (7)

where K - the coefficient of seismicity; Fw - hydrodynamic force which is given by:

Fw =ρgAicp, (8)

where A - area of the seepage area; Icp - average pressure gradient seepage above the curve slip.

Also, when calculating the defined movable pressure acting on each separate compartment and sliding neighborhood as a whole [4]:

Еal =F-R/Kst, (9)

where Kst - normalized coefficient of stability that the effects of class (responsibility) SS3 observation deck is [Kst] = 1,25.

Evaluation method of slope stability with a graphical polygon forces

(in РЕКОМЕНДАЦИИ ПО КОЛИЧЕСТВЕННОЙ ОЦЕНКЕ

УСТОЙЧИВОСТИ ОПОЛЗНЕВЫХ СКЛОНОВ

Москва Стройиздат 1984

Evaluation of the sustainability of forces polygon method proposed by GM Shahunyantsa M1 is performed as follows:

1. Profile is plotted and the slope is determined by the estimated surface displacement given the geological conditions.

Fig. 26. The scheme builds on the graphic method polygon forces

Array located above the sliding surface is broken into a number of vertical blocks by vertical lines. The size and number of sections is selected in accordance with the geological structure, the morphology of the slope and shape of the surface displacement. Within each section is considered a flat sliding surface (Fig. 26).

2. At any selected compartment described in above manner (for example, slot number 2, see Fig. 26), act T2 - the tangential components of normal force N2, Q2 - weight of the volume of rock prism cover, and E1 and E2 the forces of

neighboring blocks, and on the slip surface of the possible failure of block are reactions from the surface equal to the forces of adhesion c2l2 and friction R2tgφ2, and normal reaction R2. Resultant friction force and the normal reaction will be equal to the surface:

and the tilt angle to the normal of the surface displacement is the angle φ2. Forces-parallel with sliding surface will be the tangential components Ti of Qi forces, if they are directed towards the possible shift of the array. All other forces are the forces or not directly affect the stability (strength Ni), or resist the opportunity to shift.

By stability coefficient understands ratio of all the tangential forces Tud, who resist to displacement, to all tangential forces Tsdv seeking to cause displacement.

In order for the entire array had stability coefficient K, it is necessary that each compartment satisfy the condition:

Тud – КТsdv = 0.

3. In compiling the equilibrium conditions instead of the tangential component of force Ti, Qi - accept value KTi, if the force tends to move the array T (compartments number 1, 2, 4), and does not change the value of the force Ti, if it resists to displacement (compartment number 3). It is difficult to determine the direction of forces E. The simplest is the assumption that the force E directed horizontally. You can take the angle of inclination to the horizontal forces E, equal to the angle of inclination of the surface of a possible displacement in the compartment, in which is the centre of gravity of the entire array.

If you know the direction of the reaction supporting structures, the slope to the horizontal direction of the forces E should be taken as the slope of the reaction structure.

4. For each compartment magnitude and direction of forces Ni, Ti, ci, li and Ti-1 are known. The force Ei-1 is known by solving the problem of the previous compartment, bearing in mind that in the first compartment force Ei-1 is absent. Magnitude of the forces Ei and Si, given their directions are unknown and are a result of plotting for each section of the polygon of forces. Problem of determining the pressure at the base of the landslide for a given coefficient of the slope stability is solved for each compartment consistent building polygons forces similar to those shown in Fig. 26. Strength Ei-1 are common sides of polygons for this compartment and the last, and when moving from one compartment to another should only change the direction of force Ei-1, which is part of the reaction compartment to the previous one, on the contrary, as in this case, the force Ei-1 transmitted from the previous compartment is active. Force Ei defined for the last compartment is a complete landslide pressure that develops at the base of the slope for a given factor of stability.

Note. If the construction of the polygon of forces in some i-th compartment Ei force becomes equal to zero, or took a negative value, ie disappeared or become a force instead of supporting shear (which is impossible if the soil can not occur tensile stress), it shows that part of the array from the beginning to the border somewhere inside this compartment has not only given the stability coefficient, but even excess.

In the case where the force E is in the i-th compartment is toward the direction of displacement, in the construction of a polygon of forces for the next compartment i + 1, Ei in this polygon is not included, if the tensile stress is not allowed, the assumption is taken into account only the tensile stress of the force Ei at which the tensile stresses do not exceed the permissible.

5 . To determine the stability of the slope K , should be asking different values of K, determine the value of E for the last compartment by constructing polygons forces. According to information received build change curve E (for the

last compartment ) . Degree of stability of the slope will correspond to such stability coefficient K, at which E = 0, ie unknown quantity stability factor is the abscissa of the intersection curve E = f (K) with the horizontal axis K. To construct the curve E = f ( K) should take these values K, at which the value of E was both positive and negative.

See also STANCIU & LUNGU – 2000.

РЕКОМЕНДАЦИИ ПО ВЫБОРУ МЕТОДОВ РАСЧЕТА

КОЭФФИЦИЕНТА УСТОЙЧИВОСТИ СКЛОНА И ОПОЛЗНЕВОГО ДАВЛЕНИЯ

ЦЕНТРАЛЬНОЕ БЮРО НАУЧНО-ТЕХНИЧЕСКОЙ ИНФОРМАЦИИ

МОСКВА - 1986

2.6. Method of tangential forcesEmployed the method name and its variants [14, 15, 17, 19, 24, 30, 38,

39]: the usual method, GM Shahunyantsa, leaning against the method of sections, method of Peterson, algebraic summation method, the method of flat surfaces shift method of algebraic addition of forces, leaning against the slope method, etc.

Method is most often used when the sliding surface in any of the available methods is clearly defined throughout. For example, when the talus slips on the rock, and the roof is taken as the last sliding surface. In such a case it is convenient to consider actual shear forces directed tangentially to the sliding surface. When this sliding surface are composed of a series of flat portions, i.e. in the form of a broken line.

GM Shahunyants, for example, proposed to use for determining the stability coefficient an array of soil sliding down at a fixed sliding surface, the formula derived for the circular cylindrical surface:

(71)Such a method has been accessed and many other authors, although

mathematically it is not quite strict: in this case sometimes develops multidirectional strength.

Referring to the decomposition of forces shown in Fig. 9, and, taking into account the seismic action we obtain:

- In the absence of groundwater:

                                         (72-а)- by simply flooding the slope:

-                                      (72-б)- the slope under the influence of seepage flow:

-                                      (72-в)As before, the weight of the soil is taken into account with a weighting number (if the cut length of the slope portion, equal to one):

Pвi = γilihср.i - γωlihi = (γihср.i - γωhi)li.                                       (73)Weight of the soil without weighing:

Pi = γilihср.i.                                                                   (74)

Fig. 9. Method of tangential forces:a - the case of plane sliding surface, b - case sloping sliding surface

If within the compartment being considered (on the surface or on the edge of the slope) are any buildings, their weight should be added to the weight of the soil in this compartment. In the particular case where fixed sliding surface of the whole landslide flat as shown in Fig. 9, and in written formulas trigonometric function of the angle of inclination of the surface to the horizon (sin α and cos α) can be taken out outside the summation sign. Then in the case of a homogeneous soil (constant shear characteristics) for the whole landslide landslide pressure prism formula has a simple form (in the absence of groundwater):

Eоп = Kу(Psin α + Qс) - tg φPcos α - cL,                                  (75)where P - weight of the sliding prism; L - length of the slip plane.More frequently, in practice, it happens that the sliding surface is not flat, but may be provided from individual sections having different inclinations to the horizontal. In this case it is convenient to define landslide pressure for individual compartments, and then build it changes epure.Assume that the landslide mass moving over the surface abcde (Fig. 9b).To determine the stability coefficient and the magnitude of the whole landslide pressure, divide the whole mass of landslide into a number of sliding blocks so that each of them has a flat (linear, plane) sliding surface. Further define the weight of each compartment Pi and decompose it into normal and tangential components to the slip plane of each block. To determine the pressure of landslide consider the equilibrium conditions of individual sections of the landslide, taking the sum of the projections of external forces on the direction of motion of each compartment. For convenience, it will begin with consideration of the upper compartment 1, then move on to the next two, etc. To cover one should take the sum of the projections of all forces on the slip plane ab, including the unknown pressure from the neighbouring compartment 2, and equate it to zero. Thus we find the magnitude of the reaction E1, which should be applied to compartment 1 by compartment 2 in the direction ab, so that the 1 compartment is in equilibrium. Value E1 is landslide pressure. Adopting the notation indicated in Fig. 9b, we find (initially excluding groundwater and seismic forces):

E1 + tg φ1N1 + c1l1 - Q1 = 0;

E1 + tg φ1P1cos α1 + c1l1 - P1sin α1 = 0,whence we get:

E1 = P1sin α1 - tg φ1P1cos α1 - cl1.When considering the balance compartment 2 is necessary to consider the action of force E1, but with the opposite sign. Similarly, is considered the equilibrium condition for all other compartments. In general, to determine the pressure of any sliding compartment of the landslide, the expression will be:

Ei = Pisin αi - tg φiPicos αi - cili + Ei-1,                                    (76)Where:Ei-1 - landslide pressure projection of previous compartment to the sliding direction of the compartment being considered.To determine the magnitude of the landslide pressure for the individual compartments of the landslide from the data build epure landslide pressures (Fig.

9b), which is required to select the holding place for laying a structure along the length of a landslide, is rationally to choose that block with a minimum of Ei. To obtain the required safety factor when calculating landslide pressure, as before, the shear force is multiplied by the calculated stability factor Ku. In "normal" method of calculation often Ei-1 are equal to the tangent of the resultant force in the previous compartment rather than its projection on the direction of the sliding compartment being considered. The difference in the angles of inclination to the horizontal lines of the sliding surface in adjacent compartments less than 10% error in this method of calculation is obtained insignificant. At sharp fluctuations slope slip surface this difference can not be neglected. However, since we consider the "Method of tangential forces," write the formula for determining the pressure of landslide in the case of simple summation of tangential forces. If you do not forget that the summation must necessarily be sequential, starting with the upper section (so that Ei-1 included in the sum), the formula for determining landslide pressure will be as follows:In the absence of groundwater:

Eal = [Kuэ(Pisin αi + Qсi) - (Picos αitg φi + cili)].                  (77-а)

In a simple flooding slope:

Eal = [Kуэ(Pвisin αi + Qсi) - (Pвicos αitg φвi + cвili)].           (77-б)

When exposed to the slope of seepage:

Eоп = [Kуэ(Pisin αi + Qсi) - (Pвicos αitg φвi + cвili)].            (77-в)

2.7. Analytical method GM ShahunyantsThis method [23, 36, 39], as well as the previous one, it is most convenient to use when the configuration of the sliding surface all through is already installed. The GM Shahunyants’ method generally is similar to the method of tangential forces, but in this case are more strictly observed the laws of structural mechanics. As before, the landslide is thought for calculations as an array of blocks which divide the landslide into a number of compartments. Usually take compartments such that no practical loss of accuracy can be within them to take over the plane and

the surface order to the ground conditions, slope shape, the effect of external forces, etc. were practically uniform.

Fig. 10. Analytical method GM Shahunyantsa

Will determine the sustainability of the unit at an arbitrary surface potential displacement (Fig. 10). Consider the equilibrium condition for any i-th compartment (eg, second). All external active forces (weight of the soil compartment, external load, etc.) acting on the i-th compartment leads to resultant Pi. This force decompose at the point its applications into components: normal Ni and tangential to the possible shift plane compartment Qi. GM Shahunyants generally accepts that the resultant of the external forces active inclined to the vertical at an angle θi. We will simplify the argument to consider the case where the force Pi is vertical, ie the angle θi = 0, then:

Ni = Picos αi Qi = Pisin αi.                                           (78)The slope of sliding surfaces within each compartment in the direction of possible displacement unit values αi are taken with the plus sign in the fall in the direction of sliding surfaces and with a minus sign in the opposite direction. Following the basic laws of structural mechanics by providing a separate consideration i-th compartment, we must replace the influence on him of the unit overlying force Ei-1, and the influence of the underlying - the same force Ei. In general, GM Shahunyants accepts that the strength Ei-1 is directed at an angle ηi-1 to the horizon, the strength Ei - at an angle ηi to the horizon, etc. We assume in this case that the forces Ei directed along a line parallel to the direction of the reaction of the retaining structure, which will take the horizontal (as it is usually taken at the vertical edges of contact with the ground containment structure). Therefore, considering all the forces Ei oriented horizontally, ie ηi = 0. Resisting shear of i-th compartment at its base plane inclined at an angle αi to the horizon, is the cohesion force cili and friction force Si

ntg φi (where Sin is the normal reaction

of the base). Since the magnitude of E i-1 is, as will be shown below, the known rate of the previous compartment, the unknown forces are only Si

n and Ei. To find them, simply we are using two equations of statics. Projecting all the forces on the normal to the base compartment and the direction of the base, we obtain:

Sin = Ni + (Eisin αi - Ei-1sin αi);                                     (79)

Qi = cili + Sintg φi + (Eicos αi - Ei-1cos αi).                             (80)

Substituting the value of Sin in the expression for Qi and increasing in recent

times Kuэ, obtain an equation for Ei:

KuэQi = Nitg φi + cili + [Ei(cos αi + tg φisin αi) - Ei-1(cos αi + tg φisin αi)].       (81)

We Increase the value Qi times Kuэ in order to ensure that every i-th compartment

is in stabilized against shift its base with the given stability factor Kuэ . In this case

all possible shift block will have the same general coefficient of stability. Subsequently the tangential component Qi of the external active forces Pi, if it tends to move the compartment at its base, is denoted by Qi-sd. If the same tangential component Qi is directed to the unit side opposite to the direction of possible displacement, it will have the force to hold the compartment from potential displacement, and it should not be increased in the value times Ku

э. We denote Qi , in this case, by Qi-ud .Previous equation can be simplified by remembering that:

cos αi + tg φisin αi = (cos αicos φi + sin αisin φi)/cos φi = cos(αi - φi)/cos φi.      (82)After these explanations we find the value Ei:

                         (83)

For compartments in which Qi = Qi-sd in formula (83) values Qi-ud will be taken zero, where Qi = Qi-ud, the value Qi-sd should be zero. Since in most practical cases, the direction of incidence of the sliding surface along the entire length coincides with the possible displacement of the landslide block (ie a monotone), in the subsequent formulas we will be featured only Qi, which we mean as Qi-sd, and Qi-ud will assumed to be zero. However, in each case when performing the calculations should be aware of the possibility of forces Qi-ud.Ei reaction of each block, situated beloww the i-th compartment, in the general case can be determined by successive calculations, going from the first compartment where Ei-1 is zero, to the last one. Sequentially determined forces Ei, is especially useful when it is impossible to calculate without advance on separate stable part of the slope across from the unstable (?!).The first of the sections for which Ei got zero or even negative, separates the overlying stable part of the unit (including himself) of the underlying. While avoiding tensile stresses in the soil the underlying part should be considered separately. Analyzing consistently values Ei, it is easy to install places the gaps of the soil (the place of transition from stable to unstable parts of the unit), place expedient arrangement of retaining structures (for example, places the lowest values Ei and moderate values of the biasing layer thickness), etc. When making Qi-ud = 0 last written our formula becomes:

                                     (84)For the first compartment Ei-1 = 0, so

For the second compartment

In general, the pressure of the landslide is equal to:

                                         (85)It is obvious that the slope of the entire thrust force Eop supporting the last compartment, must be zero (since nothing supports this last compartment). On this basis, it is possible to get a free slope stability coefficient equating Eop = 0 (in this case, when it becomes the degree of stability of the slope, and not on the value of the stock of its consolidation, it will be Ku not Ku

э).

                                          (86)If the surface potential displacement - a plane with the same characteristics φ and c breakdown into compartments is not required, the landslide pressure is determined by the formula:

                                           (87)Thus it is possible to determine not only the active earth pressure on retaining structure (opposite the direction of structural response) Ea = Eop, but passive pressure (resistance) Eo = Eop. Since the tangential force in rebuffing force becomes Q, resisting shear, this fact should be taken when Ku = 1. The maximum value of Ea and the minimum value of Eo can be found from the condition

- a negative value of the second derivative in the first case and positive in the second.For the particular case of the ordinary Coulomb pressure (prism collapse at retaining wall with horizontal backfill surface) at Ku = 1 we obtain in this manner well-known formula:

Eоп = (1/2)γH2tg2(45 ± φ/2) + [ptg(45 ± φ/2) - c]Htg(45 ± φ/2),               (88)where p - vertical load on the backfill behind the wall. Here the upper signs refer to the case Eop = Ea, and the lower - to the case Eop = Eo.Thus, the method of G. Shahunyantsa is one for calculating the coefficients of free slope stability (slope), and for determining the force transferred by the retaining structure.The magnitude of the filter strength j, in the presence of hydrodynamic pressure, and the magnitude of seismic forces Qc, at the location of the slope in the seismic area, defined as before, by the formulas:

ji = γωωisin βфi Qci = μPi.                                             (89)This method will also assume approximately (in stock) that filtration and seismic forces are shearing forces (ie, directed parallel to the base compartment). Remembering the expression (78) components of Ni and Qi through the main force Pi, we obtain the final formula for determining the coefficient of slope stability and landslide pressure values. In the absence of groundwater:

                               (90-а)

               (91-а)In normal water saturation slope:

                          (90-б)

          (91-б)When exposed to the slope of seepage:

                                 (90-в)

(91-в)Often, in practice, there are cases of exposure to jet slope groundwater flow (eg, on the southern coast of the Crimea or the slopes of the Caucasus mountains). In this case, the hydrodynamic pressure is necessary to consider and weigh the soil - do not consider as a continuous saturation of soil slope is missing. That is, the formula should be used only on the basis of the filtration pressure:

                                (90-г)

          (91-г)For the derivation of formulas for determining the pressure and landslide stability coefficient were used two equations of statics. The third static condition (momentum equation) makes it possible to determine the point of application of the reaction Eop over the retaining structures (or force Ei for any i-th compartment). This solution is exact for circular cylindrical sliding surfaces and rough in others. The shown method is described by considering the G.M Shahunyants method of polygon forces.Written formulas can also be used for the scheme, taking into account that the force E inclined from the horizontal by an angle η, is constant for all compartments. Since the angle of limited quantities η 0 <η <ψ) (where ψ - the angle of shear), it can be assumed to be approximately 0,5 ψ. For this scheme, the calculation formulas in the expression

(corresponding to η = 0) is replaced by

where η = const.

2.8. Graphoanalytical method of polygons of forces GM Shahunyantsa

If the surface of the possible displacement is known, for example, predetermined geological structure of the slope, set in the performance of engineering surveys or adopted using different theoretical methods, stability analysis is often appropriate to maintain the array method of polygons of forces [23, 38, 39].The basis of the calculation is preserved hypothesis of solidified body. This hypothesis is broken, if the surface is not displacement plane and circular cylinder surface (which actually overlying the array can be shifted as one unit) as well as with any other surface during displacement of the outline of the array having the local tension. But these local stresses can create an array of motion purely local effect in the form of individual cracks or break local soil compaction. Since the calculation is to determine the conditions for stability of the array, it is possible to save as a working hypothesis about solidified body. This assumption lies in the conventional framework of the assumptions that have been adopted in almost conventional calculations of structural mechanics. In many cases of building settlements item is considered as a whole and is calculated on the common tension. If required, further include the effect of local stresses. This method will show the example of determining the pressure on the retaining structure, based on the assumption that no interaction with the structure the array will be stable or unstable enough.Fig. 11 and presented an array of drum on which there are any structures (not shown). An array landslide protectionsupported by structure is in equilibrium. Is necessary to determine earth pressure on retaining structure, equal in magnitude and opposite in direction of the reaction of the building.

Fig. 11. Method of polygons of forces:a - the design scheme of the slope, b - a diagram of one of the compartments of a polygon of forces, v - detail

sole compartment, g - general polygon of forces

We divide the entire array into a number of compartments. In weight derived compartments Pi we include weight of soil and structures located on each compartment. In each compartment acts tangential Qi and normal Ni components of the weight compartment Pi. Cut out mentally any compartment of the array (for example, 3 - see Fig. 11b) and apply to it, other than those of forces, the forces of E2 and E3, replacing effect on him neighboring compartments. On the surface of AB possible shift compartment are reactions from the surface equal to the limit equilibrium conditions, adhesion and friction forces on the surface (with the full manifestation of these forces), and a normal reaction. Adhesive force is denoted by C3 = c3l3. Friction force T3 and the normal reaction S3

н on surface AB replace their resultant S3 (see Fig. 11v). Friction force T3 = S3

нtg φ3, and the resultant

S3 =                         (92)

The angle of inclination of the resultant S3 to the normal to the surface AB is thus equal to the angle φ3 internal friction of soil on the soil surface AB (see Fig. 11v), as the tangent of this angle is the ratio of the friction force T3 = tg φ3S3

н to normal reaction S3

н. The entire block will have a predetermined coefficient Kу of stability if for each compartment will be sustained the same requirement.Forces, shearing the entire massif on the sliding surface will be tangential components Qi of forces Pi, if they are directed towards the possible shift of the block. All other forces, including reaction and retaining structures, forces are either not directly affect the stability (as strength Ni), or resist the opportunity to displacement. Consider the stability factor for the possible displacement of each compartment on the sliding surface (received for each compartment in the plane) as the ratio of the tangential forces holding array (Rуд), to all tangential forces trying to cause its displacement (Qсдв). Then we can write for a given coefficient of stability it is necessary to:

Rуд - KуQсдв = 0 (Kу = Rуд/Qсдв; Rуд = Kу/Qсдв).                        (93)In other words, given the task of ensuring the stability coefficient Ku can be reduced to the usual equilibrium conditions, when considering the forces instead of shearing tangential forces take power in Ku times greater in magnitude and acting in the same direction. So we take in compiling the equilibrium conditions, instead of the tangential component of the value Qi of forces Pi , the Kу Qi value, if Qi force tends to move the block (as is the case in compartments 1, 2 and 4). Force value Qi is accepted without changes if it resists to displacement (as is the case in the compartment 3). In the calculations assume that Ei force is directed along a line parallel to the direction of the reaction of the retaining structures Eоп. In Fig. 11, the reaction of structure to landslide thrust Eоп is directed horizontally, so all forces Ei are horizontally oriented. Solving the problem of relatively independently of each compartment (see Fig. 11b) we should consider for each compartment forces Qi, Ci and Ei-1 as known. The force Ei-1 is known by solving the problem of the previous compartment, bearing in mind that in the first compartment force is absent. Unknown forces in each compartment are therefore forces Ei and Si.

Leaving for the moment the issue of determining the force application points, solve the problem of determining the force magnitudes Ei and Si, which are determined by their directions. There are enough two statics conditions. This problem can be solved either analytically (see the previous method of calculation), or constructing the polygon of forces. The latest decision will usually provide sufficient accuracy. It was shown for the third compartment in Fig. 11, b. For the entire slope problem is solved graphically from compartment to compartment, building for each block the polygon of forces. In Fig. 11 g these polygons are shown continuously following one after the other with common side of polygons Ei - for current and previous compartment. In the transition from one compartment to another should only change the direction of force Ei-1 (which is part of the reaction compartments to the previous) on the reverse, as in this case, the force Ei-1 is transmitted from the previous compartment active. Strength E4, which is defined for the last compartment, will be the complete reaction, and must ensure retaining structure in order to the landslide has the given stability factor Kу.In this regard, the calculation of the the retaining structures should be required only to customary conditions of equilibrium for the stability coefficient already taken into account in determining the strength of Eоп. Reaction point of the retaining structures, Eоп being already known in magnitude and direction, can be found using the remaining third equilibrium condition. To do this, find the center of the curve, as close as possible to the actual matching sliding curve (in Fig. 11, and replacing the arc circle of radius r with center O shown in dashed lines). The arm from the center O to the direction of the force Eоп - reaction the retaining structures - is denoted by z.Taking moment of all forces about the center O, we obtain:

Eопz = r(KуΣQiсдв - ΣCi - ΣSisin φi - ΣNitg φi - ΣQiуд);

z = (KуΣQiсдв - ΣCi - ΣSisin φi - ΣNitg φi - ΣQiуд)r/Eоп.                         (94)Polygon of forces also allows you to define the best location and the retaining structures, and places most likely breaks the array and places of possible occurrences mounds and terraces bulging when sliding the array (ie Ku ≤ 1), and a number of other problems which in practice often great interest.From consideration of Figure 11, z is seen, for example, that would be a more rational arrangement of the retaining structure at the border of the fourth and the third compartment, as this would lower the reaction Eop, since it is equal to E3, but not E4 (a polygon of forces can be seen that the E3 <E4).In constructing the polygon of forces must ensure that it become not in a compartment i-th strength equal to zero or of opposite sign, that is, not if it disappears or becomes instead whether the shear forces supporting force. Last physically can not be (if the soil can not occur tensile stress) and show only that part of the array from the beginning to the border somewhere inside this compartment has not only given the stability coefficient, but even excess. Thus, the construction of the polygon of forces directly determines the stability boundary portions of the array. When there is a negative force value (when it is in the direction of i-th compartment in the displacement direction) and a force for drawing the polygon the next compartment i + 1 strength Ei in this polygon is excluded (provided to prevent tensile stress, the tensile stress on the assumption may be considered a that part of the strength Ei, where the tensile stresses do not exceed the permitted).

The described method of constructing polygons forces can be applied to solve the problem: Does this array stability factor of not less than specified? If the array does not have a predetermined stability coefficient Ku, Eop strength is equal, for example, in the case considered, E4, which must be attached to the end of the last compartment to provide the coefficient array, will focus on maintaining the array. If the actual rate stability equal to or more than the array specified, the force Eop = E4 will be either zero or directed toward the possible shift of the array. This shows that in order to actually reduce the high stability coefficient to a given, you need to make shearing force Eop = E4. In the example above the array has given stability factor Ku, since the force Eop = E4 in the polygon of forces was aimed at maintaining the slope. If there is necessary accounting seismic impact force Qsi and a slope or groundwater ji, they are added directly to the forces Qi and thus also involved in the construction of the graphics. Weigh ground considered analogous manner as described in the previous calculation methods. As can be seen from the above, the method considered polygons forces allows to directly determine the magnitude of the landslide pressure to which shall be calculated landslide retaining structure.

2.9. Rapid method of calculation by GM ShahunyantsaAs practice shows calculations performed in the design antilandslide retaining structures deep foundations, analytical method GM Shahunyantsa is very effective. Buildings designed for landslide pressure calculated by this method are normally operated for several years. Especially convenient application written using the above formulas digital computer. In this case, calculations on the developed model programs require minimal labor costs. Often, however, still have to perform the calculations by hand (in the absence of a computer, when the rough calculations, comparing options, when the settlement prikidok directly in a nature during the examination of landslides, when deciding on the possibility of installing mechanisms on a slope, etc.) when required operational expense and there is no sense or ability to go on the car. In this case the formula G. Shahunyantsa advantageous to use a modified form of several proposed by the author of this work [3]. Let us write the slope shown in Fig. 12, the previously derived equation (91 g).

Substituting into this equation the values: Pi = γiaihср.i (for γi should take into account not only the volumetric weight of the soil, but also the weight of the external load on the compartment when it is available); Qсi = γωaihср.iμ; ji = γωaihisin βфi, after simple transformations we obtain:

                           (95)There are designated:

                                         (96)For these three ratios using digital computer graphics can be constructed by the type presented in Figures 13, 14, 15, in order to facilitate the calculations.

Fig. 12. Rapid method of calculation by GM Shahunyantsa:a - overall design scheme of the slope, b - pressure diagram landslide

From the graph in Fig. 13 interesting thing is clear: with increasing surface slip angle α effect of the change of the angle of internal friction φ decreases. This is natural, because with increasing angle α main role is played by the force of gravity, do not depend on the strength characteristics of soils. If you are doing real particular slope calculations will not keep within the above assumption of monotony and the sliding surface in a compartment she would not falling and rising, it may be considered the adoption eoi from Fig. 13 at a negative angle αi. At continuous saturated soil, where it is required, it should be physical and mechanical properties of rocks with regard to their soaking water: γvi, cvi, φvi. In this case the seismic force, not only the weight of the ground, but the similar manner, by weight water.In addition to the graphs, the author of this work composed Table. 1 - 3 (see Appendix), contributing to the implementation of sufficiently accurate calculations. Furthermore, in the tables are data necessary for broader than in the graphs, the range of input parameters. When the location of the slope in the non-seismic adopted μ = 0, in the absence of groundwater flow - hi = 0. In this case, formula (95) takes the form:

                                       (97-а)or what is the same

                                                (97-б)

Fig. 13. Graphics depending eoi = Kуэsin αi - cos αitg φi

When using the given expressions should be remembered that to determine the total pressure of the landslide at the end of i-th compartment of the landslide of the block located above the slope, you need to consistently summarize all landslide pressure from overlying each of the compartments, starting with the first (uppermost)

Fig. 14. Graphics depending coi = ci/(hср.icos αi)

Fig. 15. Graphics depending λi = cos φi/cos (αi - φi)

                                                        (98)Using these formulas constructed diagram landslide pressure along the length of the entire slope (see Fig. 12b). Calculations it is recommended to be performed using a blank tab. 2 and 3. Several simplified by this method may be, formula (90-i) for determining the stability of a slope factor. After simple transformations, it takes the following form. In the general case:

                        (99-а)the absence of groundwater in non-seismic areas:

                                             (99-б)Calculating the coefficient of stability is recommended that using pre Table. 4 and 5.

Т а б л и ц а 2Calculation landslide pressure formula (95)

Т а б л и ц а 3Calculation landslide pressure formula (97-а)

Т а б л и ц а 4

Compute the slope stability by the formula (99-а)

Т а б л и ц а 5Compute the slope stability by the formula (99-б)