Soil Mechanics for Unsaturated Soils phần 7 ppt

11.3 SLOPE STABILITY 325 Rearranging Eq. (1 1.63) yields tampb tan#b [ c' l.3 R + IN - u, P - tan#' - u, B (1 - s)] R tan#'] F, = (11.64) ALaL + c Wx - c Nf In the case where the pore-air pressure is atmospheric (Le., u, = 0), Eq. (11.64) has the following form: F,,, = ALaL + Wx - Nf (1 1.65) When the pore-water pressure is positive, the 9' value can be set equal to the 4' value. Equation (1 1 -64) can also be simplified for a circular slip surface (Fig. 1 1 SO) as follows: [c' 6 + {N - u, 8 - u, 8) tan#' IR F, = ALaL + c Wx (11.66) For a circular slip surface, the radius, R, is constant for all slices, and the normal force, N, acts through the center of rotation (i.e., f = 0). Factor of Wety with Respect to Force Equilibdum The factor of safety with respect to force equilibrium is derived from the summation of forces in the horizontal direction for all slices: - A, + C S,,, cosa - N sina = 0. (1 1.67) The horizontal interslice normal forces, EL and ER, can- cel when summed over the entire sliding mass. Substituting Eq. (11.55) for the mobilized shear force, S,, into Eq. (1 1.67) and replacing the (a,$) term with N give 1 - C [cf p cosa + {N tan#' - u, tan#' p Ff + (u, - u,) tan+b a}] = AL + N sina (11.68) where Ff = factor of safety with respect to force equilibrium. Rearranging Eq. (1 1.68) yields In the case where the pore-air pressure is atmospheric (i.e., u, = 0), Eq. (11.69) reverts to the following form: C[c'/3cosa + tan+' cosa] A, + N sina 'f - (11.70) When the pore-water pressure is positive, the +b value is equal to the 4' value. Equation (1 1.70) remains the same for both circular and composite slip surfaces. Interslice Force Function The interslice normal forces, EL and ER, am computed from the summation of horizontal forces on each slice (Fig. 11 S3): ER - EL = NCOW tans - S,,, COW. (11.71) Substituting Eq. (11.57) for the (N COW) term in Eq. (1 1.71) gives the following equation: ER - EL = {W - (X, - Xd - S, sina} tana - s,,, cosa. Rearranging Eq. (1 1.72) gives (1 1.72) (11.73) The interslice normal forces m calculated from Eq. (1 1.73) by integrating from left to right across the slope (Fig. 1 1 .53). The procedure is further explained in the next section. The left interslice normal force on the first slice is equal to any external water force which may exist, At, or it is set to zero when there is no water present in the tension crack zone. The assumption is made that the interslice shear force, X, can be related to the interslice normal fone, E, by a mathematical function (Morgenstern and Price, 1965): x = hf(x)E (11.74) (11.69) AL + N sina 326 I 1 PLASTIC AND LIMIT EQUILIBRIUM Direction for comDutation Figure 11.53 Convention for the designation of the interslice forces. where f(x) = a functional relationship which describes the manner in which the magnitude of X/E varies across the slip surface X = a scaling constant which represents the percentage of the function, f(x), used for solving the factor of safety equations. Some functional relationships, f(x), that can be used for slope stability analyses are illustrated in Fig. 11.54. Basi- cally, any shape of function can be assumed in the analysis. However, an unrealistic assumption of the interslice force function can result in convergence problems associated with solving the nonlinear factor of safety equations (Ching and Fredlund, 1983). Morgenstern and Price (1967) suggested that the interslice force function should be related to the shear and normal stresses on vertical slices through the soil mass. In 1979, Maksimovic (1979) used the finite element method and a nonlinear characterization of the soil to compute stresses in a soil mass. These stresses were then used in the limit equilibrium slope stability analysis. A generalized interslice force function, f(x), has been proposed by Fan et al. 1986. The function is based on twodimensional finite element analyses of a linear elastic continuum using constant strain triangular elements. The normal stresses in the x-direction and the shear stresses in the y-direction were integrated along vertical planes within a sliding mass in order to obtain normal and shear forces, respectively. The ratio of the shear force to the normal force was plotted for each vertical section to provide a distribution for the direction of the resultant interslice forces. Fig- ure 11.55 illustrates a typical interslice force function for one slip surface through a relatively steep slope. The analysis of many slopes showed that the interslice force function could be approximated by an extended form of an emr function equation. Inflection points were close to the crest and toe of the slope. The slope of the resultant interslice forces was steepest at the midpoint of the slope, and tended towards zero at some distances behind the crest and beyond the toe. The mathematical form for the empir- ical interslice force function can be written as follows: f(x) = Ke-(C"w")/2 (1 1.75) where e = base of the natural logarithm K = magnitude of the interslice force function at midslope (i.e., maximum value) C = variable to define the inflection points n = variable to specify the flatness or sharpness of cur- o = dimensionless x-position relative to the midpoint of vature of the function the slope. , f(x) = Constant , f(x) = Half-sine , f(x) = Clipped-sine , f(x) = Trapezoid 1 f(x) Specified 1 L X R L Left dimensional x-coordinate of the slip surface R = Right dimensional x-coordinate of the slip surface Figure 11.54 Various possible interslice force functions. 11.3 SLOPE STABILITY 327 1 :2 Slope 1 1.501 .oo Dimensionless x-coordinate K 14.79 25.00 Center of X = 30.00 20.00 rotation Y = 23.00 Radius = 15.50 Dimensions in meters 30.00 x-coordinate 38.00 \\ 10.00 y-coordinate 10.00 Figure 11.55 The interslice force function for a deep-seated slip surface through a one horizontal to two vertical slope. Figure 11.56 shows the definition of the dimensionless distance, w. The factor, K, in the interslice force function equation [Le., Eq. (11.75)], is a variable related to the average inclination of the slope and the depth factor, Of, for the slope surface under consideration: K = exp {Di + Ds(Dr - 1.0)) (11.76) where Df = depth factor (defined in Fig. 11 37) Di = the natural logarithm of the intercept on the ordi- D, = slope of the depth factor versus K relationship for nate when Or = 1.0 a specific slope Inflection Inflection r’ point point 0 I -2.0 l:o 2.0 m x - Distance Figure 11.56 Definition of the dimensionless distance. w. 2.0 b \ a Df - ‘0 0‘. 1 .o 0.9 1.2 Slope Vertical slope 0. - 0.5 al:1 Slope 2:l Slope I I 1 I 04.0 112 1.4 1.6 1.8 2.0 Depth factor. Dt Figure 11.57 The interslice side force ratio, K, at midslope versus the depth factor, Or. Eq. (1 1.76) is shown graphically in Fig. 11.57. Slip surfaces passing through or below a vertical slope are considered as a special case. The relationship between the factor, K, and depth factor, Of, for vertical slopes is also shown in Fig. 11 S7. The “finite element” based functions have been computed for slip surfaces which are circular. However, the shape of the function should, in general, be satisfactory for composite slip surfaces. The magnitude of “C” varies with the slope angle, as does the variable “n” (Fig. 11 -58). Procedures for Solving the Fmtom of we@ Equation The factor of safety equations with respect to moment and force equilibriums (i.e., Eqs. (11.64) and (11.69), respectively) are nonlinear. The factors of safety, F,,, or F,, ap- pear on both sides of the equations, with the factor of safety being included through the normal force equation [Le., Eq. (1 1 .a)]. The nonlinear factor of safety equations can be solved using an iterative technique. The factors of safety with respect to moment and force equilibriums can be calculated when the normal force, N, on each slice is known. The computation of the normal force [i.e., Eq. (11.60)] requires a magnitude for the interslice shear foms, X, and X,, and an estimate of the factor of safety, F. For the first iteration, the factor of safety, F, in the normal force equation can be set to 1 .O or estimated from the Ordinary method (Fredlund, 198%). The interslice shear forces can be set to 0.0 for the first iteration when com- 328 11 PLASTIC AND LIMIT EQUILIBRIUM Slope angle (a) Tangent of slope angle (b) Figure 11.58 Values of the “C” and “n” coefficients versus the slope angle. (a) C coefficient versus the slope angle; (b) n coefficient versus the tangent of the slope angle. puting the normal force, The computed normal force is then used to calculate the factors of safety with respect to moment and force equilibriums (Le., F, and Ff in Eqs. (1 l .64) and (1 l.69), respectively). This results in initial values for the factors of safety. The next step is to compute the interslice normal forces, E, and ER, in accordance with Eq. (1 1.73). There are two sets of interslice force calculations: one associated with moment equilibrium, and the other associated with force equilibrium. The interslice force calculation with respect to moment equilibrium uses the moment equilibrium factor of safety, F,, in computing the mobilized shear force, S,, in Eq. (1 1.55). On the other hand, the interslice force calculation associated with force equilibrium uses the force equilibrium factor of safety, Ff, in computing the mobilized shear force, S,, in Eq. (1 1.55). The interslice shear forces in Eq. (11.60) are also set to zero for the first iteration. The computation commences from the first slice on the left- hand side of the slope (Le., at the crest), and proceeds across the slope to the last slice at the toe (Fig. 11 S3). The right interslice normal force, ER, on the last slice will become zero when overall force equilibrium in the horizontal direction is fully satisfied. The computed interslice normal forces, E, and ER, can then be used in the calculation of the interslice shear forces, X, and XR, for all slices, in accordance with Eq. (1 1.74). An interslice force function,f(x), can be assumed from one of the functions shown in Fig. 11.54 or calculated using Eq. (11.75). The selected interslice force function, along with a specified h value, is used for the entire iterative procedure until convergence is achieved. For the next iteration, the computed moment equilibrium factor of safety, F,, and the corresponding interslice forces are used to recalculate new values for the normal force, N, and the moment of equilibrium factor of safety, F,,,. The updated values for the normal force, N, and the moment equilibrium factor of safety, F,, are then used to revise the interslice normal forces and interslice shear forces (i.e., E,, ER, X,, and XR) associated with moment equilibrium. The computed force equilibrium factor of safety, Ff, and the corresponding interslice forces from the first iteration are used to revise the magnitudes of the following variables: N, Ff, EL, ER, x,, and XR, associated with force equilibrium. The revised factor of safety values, F, and Ff, are then compared with the corresponding values from the previous iteration. Calculations are stopped when the difference in the factor of safety between two successive iterations is less than the desired tolerance. If the difference in either factor of safety, Ff or F,, is greater than the tolerance, the above pmedure is repeated until convergence is attained for both factors of safety. When the solution has converged, moment and force equilibrium factors of safety corresponding to the selected interslice force function, f(x), and the selected A value are obtained. The analysis can proceed using the same interslice force function, f(x), but varying the h value. Several factor of safety values, F, and Ff, associated with different h values can be obtained and plotted as shown in Fig. 1 1.59. The moment equilibrium factor of safety, F,, does not vary significantly with respect to the h values as compared to the force equilibrium factor of safety, Ff. Curves joining the F, and Ff data intersect at a point where total equilibrium (Le., moment and force equilibrium) is satisfied. Pore- Water Pressure Designation Pore-water pressures are often designated in terms of a pore pressure coefficient, r,,, for analysis purposes (Bishop and Morgenstern, 1960): (11.77) where r,, = water pore pressure coefficient hi = thickness of each soil layer pi = density of each soil layer. The pore pressure coefficient is generally considered as a positive value. However, it can also be used to represent 11.3 SLDPE STABILITY 329 o.900 u-l-l-u 0.2 0.4 0.6 0.8 1.0 h Figure 11.59 Variation of moment and force equilibrium factors of safety with respect to lambda, A. negative pore-water pressures, as well as pore-air pressures: (11.78) where rua = air pore pressure coefficient. Figure 11.60 illustrates how the pore pressure coefficient can be used when the pore-water pressure is negative. In this case, the pore pressure coefficient is also negative. The water pore pressure coefficient at the phreatic line is equal to zero. A water pore pressure coefficient of +0.5 indicates the verge of artesian pressure conditions since the density of water is approximately one half the density of soil. At points above the phreatic line, the pore-water pressure becomes increasingly negative. At the same time, the overburden pressure is decreasing. As a result, it is possible for the pore pressure coefficient to become highly negative. Let us assume that the pore-water pressure is -200 kPa at a depth of 1 m (Le., pgh = 20 Wa for p = 2000 kg/m3). This gives rise to a water pore pressure coefficient of - 10. The water pore pressure coefficient can tend to a negative, infinite number as ground surface is approached. In other words, the water pore pressure coefficient becomes a highly variable tern as ground surface is approached. Figure 11.60 shows the cross section of a dam under steady-state seepage conditions. One equipotential line is selected, and the water pore pressure coefficients are computed at various depths and plotted in Fig. 1 1.61. There is essentially a linear change in the pore pressure coefficient until the ground surface is approached. At this point, the coefficient becomes highly negative. The nonlinearity of the water pore pressure coefficient near ground surface somewhat limits its use as a means of designating negative pore-water pressures. The air pore pressure coefficient in natural soil deposits is always close to zero due to its contact with the atmo- sphere. In compacted earth fills, the pore-air pressures may become positive due to the weight of the overlying soil layers. The air pore pressure coefficient will'be positive, but generally quite small. The water and air pore pressure coefficients are similar in form to the B pore pressure parameten developed in Chapter 8. However, there are some differences. First, the pore pressure coefficients are generally used in conjunction with relating field experience. Second, they are used to compute changes in the pore pressures referenced to the total overburden pressure, as opposed to being referenced to changes in the principal stresses. Third, the pore pressure coefficient has been used in two ways. The above description has defined the pore pressure coefficient with respect to a point in an earth mass. However, the pore pressure coefficient is also used as an average value for an entire soil region, Bishop and Morgenstern (1960) suggested procedures for obtaining an average pore pressure coefficient over a region. This value was then used to compute the pore-water pressure in a slope stability analysis. There are other pmedures which can be used to designate pore-water pressures for a slope stability analysis. Pore pressure changes can also be written in terms of A and B pore pressure parameters. The B pore pressure parameter represents the change in pore pressure due to isotropic or all-around loading. The A pore pressure parameter repre- Phreatic line (zero pressure) / Reservoir level I l(ln Figure 11.60 Steady-state seepage conditions in an earth-fill dam. 330 11 PLASTIC AND LIMIT EQUILIBRIUM p’ Figure 11.61 Water-pore pressure coefficient along the 7 m equipotential line. sents the change in pore pressure due to deviatoric loading. The pore pressure parameters for the water and air phases are presented in Chapter 8 (Le., B,, A,, B,, and AJ. Hilf s analysis (1948) and Bishop’s analysis (1956) also provide relationships between pore-water pressure and overburden pressure. These analyses make use of the compressibility of air and its solubility in water, along with the compressibility of the soil structure, to obtain what is essentially a nonlinear pore pressure coefficient (see Chapter 8). A grid of pore-water pressure values can be superimposed over a cross section under consideration (Fig. 11.62). The pore-water pressures can be either positive or negative, and an interpolation technique can be used in order to obtain the porn-water pressure at any designated point. Negative pow-water pressures can also be contoured as a series of lines (Fig. 11.63). An interpolation procedure can be used to obtain the pore-water ptessure for points between the contours. Some of the above procedures are later described in further detail. Piezometric lines can also be used to designate the pore- water pressures in a slope (Fig. 11.64). The vertical distance from the piezometric line down to a point below the line is equal to the positive pore-water pressure head (i.e., u, = h, p,g). On the other hand, the vertical distance from the piezometric line up to a point above the line can be considered as the negative pore-water pressure head (i.e., u, (-) = h, p,g). When slopes are steep and the gradient along the water table is high, this procedure can lead to pore-water pressures which are in considerable er- ror. 11.3.3 Other Limit Equilibrium Methods The General Limit Equilibrium (GLE) method can be spe- cialized to correspond to various limit equilibrium methods. The various methods of slices can be categorized in terms of the conditions of statical equilibrium satisfied and the assumption used with respect to the interslice forces. Table 11.2 summarizes the conditions of statical equilibrium satisfied by the various methods of slices. The statics used in each of the methods of slices for computing the factor of safety are summarized in Table 11.3. Most methods use either moment equilibrium or force equilibrium in the calculation for the factor of safety. The Ordinary and Simplified Bishop methods use moment equilibrium, while the Janbu Simplified, Janbu Generalized, Lowe and Kara- fiath, and the Corps of Engineers methods use force equilibrium in computing the factor of safety. On the other hand, the Spencer and Morgenstern-Price methods satisfy both moment and force equilibriums in computing the factor of safety. In this respect, these two methods are similar in principle to the GLE method which satisfies force and moment equilibriums in calculating the factor of safety. The GLE method can be used to simulate the various methods of slices by using the appropriate interslice force assumption, The interslice force assumptions used for sim- ulating the various methods are given in Table 1 1.3. P 200 - Bedrock 8 I I I I 1 1400 1500 1600 1700 1800 1900 2000 2100 Distance (m) Mgure 11.62 Grid of pore pressures heads superimposed over the geometry. 11.3 SLOPE STABILITY 331 .^ Eauiwtential I I 1 1 I I I I I I I 0 5 10 15 20 25 30 35 40 45 52 x-coordinate (m) Fiure 11.63 Contours used to designate pore-water pressure heads. Note: Pore-water masure. u is \ 'd+) Y\b Figure 11.64 Piezometric line for designating pore-water pressures. Table 11.2 Elements of Statical Equilibrium Satislied by Various Limit Equilibrium Methods Force Equilibrium 1st Directiona 2nd Dimtion' Moment Method (e.g., Vertical) (e.g., Horizontal) Equilibrium Ordinary or Fellenius Bishop's Simplified Janbu's Simplified Janbu's Generalized Spencer Morgenstern-Price Corps of Engineers Lowe-Karafiath Yes Yes Yes Yes Yes Yes Yes Yes No No Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes No No b 'Any of two orthogonal directions can be selected for the summation of forces. bMoment equilibrium is used to calculate interslice shear forces. 332 11 PLASTIC AND LIMIT EQUILIBRIUM Table 11.3 Comparison of Commonly Used Methods of Slices ~ ~ ~-~ Factors of Safety Moment Force Interslice Force Method Equilibrium Equilibrium Assumption Ordinary X X = 0, E = 0 Bishop's Simplified X X=O,ErO Janbu's Simplified X X=O,EzO Janbu's Generalized Spencer X X X/E = tan Ob Morgenstern-Price X X X/E = Af (x) Lowe-Karafiath Corps of Engineers X X/E = Average ground surface slope X XR = ER tan a, - (ER - EL)tR/ba X X/E = Average slope of ground and slip surface 'a, = angle between the line of thrust across a slice and the horizontal. tR = vertical distance from the base of the slice to the line of thrust on the right side of the slice. = angle of the resultant interslice force from the horizontal. 11.3.4 Numerical Difficulties Associated with the Limit Equilibrium Method of Slices Most problems associated with nonconverging solutions can be traced to one of three possible conditions (Ching and Fredlund, 1983). First, an unrealistic assumption regarding the shape of the slip surface can produce mathematical instability. Second, high cohesion values may result in a negative normal force and produce mathematical instability. Third, the assumption used to render the analysis determinate may impose unrealistic conditions and prevent convergence. The normal force at the base of a slice [Eq. (1 1.60)] may become unreasonable due to the unrealistic value of m, (Whitman and Bailey, 1967). Unrealistic ma values commonly occur as a result of an assumed slip surface, which is inconsistent with the earth pressure theory. When the m, term approaches zero, the normal force at the base of a slice will tend to infinity [Eq. (11.60)]. An unreasonably large normal force will affect the calculation of the factor of safety. The m, problem can be resolved by limiting the inclination of the slip surface at the crest of the slope (Le., the active zone) to the maximum obliquity for the active state (Fig. 11.65): (11.79) Similarly, the inclination of the slip surface at the toe of the slope (Le., the passive zone) should be limited to a maximum angle in accordance with the passive state (Fig. 11.65): 9' amax = 45" - - 2' 9' = 45" + 1. (11.80) A vertical tension crack zone at the crest of the slope can also be used to limit the inclination of the slip surface in order to alleviate the rn, problem. The slip surface will terminate at the base of the tension crack zone. The problem of negative normal forces at the base of a slice is caused by a high cohesion value, and is relevant to slopes with highly negative pore-water pressures. This problem is particularly significant for relatively shallow slip surfaces where the cohesive component dominates the shear strength of the soil. Figure 11.66 illustrates the effect on the normal stress along the slip surface of increasing cohesion on a steep slope with a shallow slip surface. The cohesion can be considered to increase with increasing matric suction (i.e., c = c' + (u, - u,) tan&. The increase in cohesion has been shown in Fig. 11.66 to result in negative normal stresses. The negative normal r of rotation I Cente Tension crack 450 + - 9' 450 - - Circular slip 1 surface CY=O I I I 8 I Active earth I Passive earth pressure zone , pressure zone Figure 11.65 Limiting inclination angles for the slip surface at the crest and toe of the slope. 11.3 SLOPE STABILITY 333 Center of rotation Center of rotation X= 160m Y = 185m R=211m 4’ = 30° - cc 0 25 50 Distance (m) (a) 120 r Distance (m) -40 - (b) Figure 11.66 Effect of increasing cohesion values on the normal stress distribution. (a) Steep slope with a shallow slip surface; (b) normal stress distribution along the slip surface. forces are the result of having mobilized a large shearing force, S,,,, due to the high cohesion values. The shearing force has a positive sign, indicating an opposite direction to the sliding direction. In order for the force polygon to close (or force equilibrium to be satisfied), the normal force has to become negative. Spencer (1968, 1973) suggested that a tension crack zone should be located at the crest of the slope in order to reduce the large mobilized shearing force. The depth of the tension crack zone may extend through the region with negative normal forces. Nonconvergence can be encounted with any limit equilibrium method which uses an unrealistic assumption regarding the interslice conditions. This problem appears to be attributable to unreasonable assumption regarding the line of thrust (Ching, 1981). The moment equilibrium equation can be used to generate the equivalent of an interslice force function based on an assumed “line of thrust.” The shape of the resulting function can be unrealistic when compared to an elastic analysis (Fan, 1983) as illustrated in Fig. 11.67. The steepness of the function at the ends produces high interslice shear forces which may exceed the weight of the slice. It is suggested that the assumption used in a slope stability analysis should be somewhat consistent with the stresses resulting from gravity. 11.3.5 Effects of Negative Pore-Water Pressure on Slope Stability All of the components associated with performing slope stability analyses in situations where the pore-water pressures are negative, have been discussed. One procedure C 10- slip surface 0. 0 10 20 30 40 50 Distance (m) (a) Functbn generated from the assumed line of thrust Gradient of 0 10 20 30 40 50 Distance (m) (b) Figure 11.67 Illustration of an interslice force function generated from an unrealistic assumption for the line of thrust. (a) Ge- ometry showing the line of thrust; (b) possible henlice force functions. which can be used for performing slope stability analysis involves the incorporation of the matric suction into the cohesion of the soil (Ching et al. 1984). This will be re- ferred to as the “total cohesion” method. The second pm- cedure involves deriving the factor of safety equations in order to accommodate both positive and negative pore- water pressures (Fredlund, 1987a, 1989; Fredlund and Barbour, 1990; Rahardjo and Fdund, 1991). A nonlinear shear strength versus matric suction relationship can also be incorporated in the slope stability analysis (Rahardjo, Predlund and Vanapalli, 1992). The “Total Cohesion” Method In the “total cohesion” method, the soil cohesion, c, is considered to increase as the matric suction of the soil in- creases. The increase in the cohesion due to matric suction (i.e., (u, - u,,,) tan &’) is illustrated in Fig. 11.68 for various gb angles. Another form for the relationship between negative pore- water pressures and cohesion is illustrated in Fig. 11.69. Here, matric suctions are presented as a percentage of the hydrostatic negative pote-air pressures above the water table. Matric suction is multiplied by (tan $9 to give an equivalent increase in the cohesion for a I#I~ angle of 15’ (Fig. 11.69). The increase in the factor of safety due to negative pore- water pressures (or matric suction) is illustrated in Figs. 1 1.70 and 1 1.71. The shear strength contribution from matric suction is incorporated into the designation of the cohe- 334 11 PLASTIC AND LIMIT EQUILIBRIUM Matric suction, (u. - u,) (kPa) Figure 11.68 The component of cohesion due to matric suction for various I$’ angles. sion of the soil (Le., c = cr + (u,, - u,) tan &’). It can readily be appreciated that the factor of safety of a slope can decrease significantly when the cohesion due to matric suction is decreased during a prolonged wet period. Two Examples Using the “Total Cohesion” Method The following two example problems illustrate the appli- cation of the “total cohesion” method in analyzing slopes with negative pore-water pressures. The example problems involve studies of steep slopes in Hong Kong. The soil stratigraphy was determined from numerous borings. The shear strength parameters (Le., c’, t#/, and r$’) were obtained through the testing of undisturbed soil samples in the laboratory. Negative pore-water pressures were mea- sured in situ using tensiometers. Slope stability analyses were performed to assess the effect of matric suction changes on the factor of safety. Also, parametric-type analyses were conducted using various percentages of the hydrostatic negative pore-water pressures. Example no. 1. The site plan of example no. 1 is shown in Fig. 11.72. The site consists of a row of residen- tial buildings with a steep cut slope at the back. The slope has an average inclination angle of 60” to the horizontal and a maximum height of 35 m. The slope has been pro- tected from infiltration of surface water by a layer of soil / ?’ 15.7 kN/m3 , c‘=OkPa / 0 4b = 200 .’ 1.8 Cohesion corresponding to 1 atm db= 15O I suction I I I I I 5 10 15 20 . 25 Cohesion intercept (kPa) 0.8k I Matric suction, (u. - u,)(kPa) Cohesion due to matric suction, [(u. - u,) tan VI (kPa) Figure 11.69 Equivalent incmse in cohesion for various matric suction profiles. I 1 I 1 1 I I I 0 10 20 30 40 50 60 70 Matric suction, (u. - u,) (kPa) Figure 11.70 Factor of safety versus matric suction for a simple slope. [...]... d(U,H u,) W (12.53) 12.3.2 Compressibdity Form In the preceding sections, the constitutive relations for an unsaturated soil were formulated using a linear elasticity form These constitutive equations can be rewritten in a compressibility form more common to soil mechanics The compressibility form of the constitutive equation for the soil structure of a saturated soil is written as de,, = m , , d ( ~ u,)... constitutive equations for an unsaturated soil A change in void ratio is commonly used as the deformation state variable for a saturated soil, giving rise to the following constitutive equation: de = a,d(u - u,) (12. 57) where a, = coefficient of compressibility For an unsaturated soil, void ratio and gravimetric water content can be used as the deformation state variables for the soil structure and water... the deformation state of the unsaturated soil One constitutive relationship was formulated for the soil structure, and the other 346 constitutive relationship was for the water phase Two independent stress variables were used in the formulations In total, four volumetric deformation coefficients were required to link the stress and deformation states Attempts to link the deformation behavior of an unsaturated. .. (12.12)] For a saturated soil, the overall volume change of the soil is equal to the water volume change since soil particles are essentially incompressible The constitutive relations for an unsuturared soil can be formulated as an extension of the equations used for a saturated soil, using the appropriate stress state variables (Fredlund and Morgenstern, 1 976 ; Fredlund, 1 979 ) Let us assume that the soil. .. compressibility form can be written for specific loading conditions Table 12.2 presents the m;, m;, my, and my coefficients for the loading conditions described in the previous section These coefficients are regarded as another form of the soil volumetric deformation coefficients 12.3.3 Volume-Mass Form (Soil Mechanics Terminology) The volume-mass properties of a soil can also be used in the formulation... Stress state variable Figure 12 .7 Tests for the uniqueness of the constitutive surface for an unsaturated soil ’ Stress state variable (a) (b) Figure 12.6 Constitutive surfaces for an unsaturated soil expressed using soil mechanics terminology (a) Threedimensional void ratio and water content constitutive surfaces, (b) twodimensional comparison showing volumetric deformation moduli in terms of the... state variables representing the soil structure deformation On the other hand, changes in water content can be considered as the deformation state variable for the water phase Refemng to continuum mechanics kinematics, there are several ways to describe the relative movement or deformation in a phase Only two of these descriptions are relevant to unsaturated soils For a referential description, the... and matric suction as stress variables for describing volume change behavior (Aitchison and Woodbum, 1969; Brackley, 1 971 ; and Aitchison and Martin, 1 973 ) The role of (a - u,) and (u, - u,) as stress state variables for an unsaturated soil was later demonstrated by Fredlund (1 974 ) and Fredlund and Morgenstem (1 977 ) A stress analysis based on multiphase continuum mechanics showed that any two of three... the soil structure in the x-, y-, and z-directions can be expressed as + + au e, = - ax Ey (12.6) E, Overall Volume Change The overall or total volume change of a soil refers to the volume change of the soil structure Consider a two-di- =aw =- (12 .7) 12.2.2 I' Undeformed element av (12.5) aY az Deformed element Figure 12.2 Translation and deformation of a two-dimensional element of unsaturated soil. .. between the volumes of the voids in the element before and after deformation, AV,, referenced to the initial volume of the element, Vo: (12.13) The volumetric strain, E,, can be used as a deformation state variable for the soil structure It defines the soil structure volume change resulting from deformation 351 12.2.3 Water and Air Volume Changes The unsaturated soil element shown in Fig 12.2 can be used . rainfalls of Distance (m) Figure 11 .79 Section A-A for example no. 2. Table 11 .7 Strength Properties for Soils of Example Problem 2 Soil Type Unit Weight C' 4'. shear forces, respectively. The ratio of the shear force to the normal force was plotted for each vertical section to provide a distribution for the direction of the resultant interslice forces corresponding interslice forces are used to recalculate new values for the normal force, N, and the moment of equilibrium factor of safety, F,,,. The updated values for the normal force, N, and

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