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Force–displacement compatibility for reinforced embankments over soft clay
Geotextiles and Geomembranes 17 (1999) 147—156
Force—displacement compatibility for reinforced embankments over soft clay M. Srbulov SAGE Engineering Ltd, 1 Widcombe Parade, Bath BA2 4JT, UK Received 8 June 1998; received in revised form 19 September 1998; accepted 6 November 1998
Abstract Due to a complex soil-reinforcement interaction, an extension of conventional limit equilibrium method is suggested for the analysis of slope stability of reinforced embankments over soft clay. The advantage of the proposed method is in decreasing the number of material properties necessary for the analysis in comparison with a finite element method while at the same time force—displacement compatibility can be considered for slope stability. The method inherited the disadvantage of conventional limit equilibrium that a sliding mechanism must be assumed to govern the slope stability. 1999 Elsevier Science Ltd. Keywords: Limit; Equilibrium; Soil; Reinforcement; Interaction
1. Introduction Rowe and Mylleville (1994) reviewed the methods used for analysis and design of reinforced embankments on soft or weak foundations and stated that ‘‘Limit equilibrium calculations performed assuming an arbitrary cutoff strain (be it 2%, 5%, 10% or whatever) to determine the geosynthetic force cannot be expected to provide consistent results and, as shown by Rowe and Soderman (1987) and Rowe and Mylleville (1990), the level of error associated with this type of approach will vary from case to case.’’ Also . . . ‘‘caution should be exercised when selecting an allowable strain for use in designing reinforced embankments on brittle cohesive soils which are susceptible to progressive failure’’. The majority of methods assume that the reinforcement force acts in its original orientation, which is horizontal, and some methods assume that the force acts * Fax:#44 1225 447443; e-mail: [email protected] 0266—1144/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 2 6 6 — 1 1 4 4 ( 9 8 ) 0 0 0 3 1 — 4
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M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
tangential to the slip surface, based on the argument that local deformations at the slip surface result in a local reorientation of the reinforcement, tangential to the slip surface. The assumption that the reinforcement remains in its original (horizontal) orientation is conservative for global moment equilibrium. Although the application of finite elements may seem ideal for the consideration of deformation and the interaction of the various components of the reinforced embankments, Rowe and Mylleville (1994) stated that ‘‘careful consideration must be given to the type of finite element model and constitutive relationships which will be used to model the discrete components of the reinforced embankments. The validity of results provided by a particular formulation and program should be checked using either: (i) limiting analytical bench mark solutions, (ii) data from full scale embankments and/or (iii) data from centrifuge tests.’’ The use of finite element analysis involves cost, expertise and time, that may not be justified for routine design purposes. Therefore, an extension of a conventional limit equilibrium method could be more appropriate.
2. Method description Conventional limit equilibrium methods use a constant factor of safety F along a potential slip surface under the assumption that soil strength is mobilized at all places at the same (or similar) shear displacements. Factor of safety F is defined as the ratio between available shear strength q and the shear stress q necessary to maintain limit equilibrium F"q /q , (1) where q is equal to the peak strength q when F'1 or to the post-peak strength if yielding occurs. Using Coulomb—Mohr failure criterion, which relates shear and compressive stresses p (Fig. 1a), F can be expressed in terms of these stresses F"(c#p tan )/q , (2) where c is cohesion and angle of soil internal friction. Eq. (2) can be written in terms of normal N and shear ¹ force acting on a particular surface F"(cb#N tan )/¹,
(3)
where b is width of the surface. Knowing one of the components of the resultant forces acting along slip surface and a constant F, it is possible to calculate the other components, so that the number of unknown forces to be determined from available force and moment equilibrium equations is decreased. In the case of heterogeneous soil, when brittle soil (soil crust) can reach its peak value while the other softer zone (deep down) achieves only a part of its peak strength or when the brittle part yields to its residual strength, which can be considerably smaller than the peak strength, while the softer part achieves its peak strength at failure, the assumption of activating the peak strengths at all places (at a constant F) is unrealistic and on unsafe side. This problem can be solved by using only residual
M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
149
Fig. 1. (a) Shear strength q , q , q versus compressive stress p, (b) Shear stress q versus shear displacement d, N A P (c) shear zone thickness changed dt versus shear displacement d.
strength of brittle soil but at the expense of greater cost of such a solution. Similarly with reinforced soil, when reinforcement reaches its peak tensile strength while the surrounding softer soil mobilizes only a part of its peak strength and when reinforcement yields or fails when the softer soil reaches its peak strength, the assumptions of simultaneously activating the peak strengths in reinforcement and soil (at the same F) is unrealistic and on unsafe side. This problem is greater for reinforced slopes because reinforcement can fail and loose all its strength before the surrounding soil reaches its peak shear strength. Different methods have been attempted to solve the problem of propagating (progressive) failure using complete stress—strain (finite element) solutions or local (partial) factors of safety. This article describes a procedure for definition of the local factors of safety within the framework of limit equilibrium method. The activation of shear stresses q , q is accompanied by development of shear displacements d , d (Fig. 1b) at the peak and mobilized shear stress respectively and therefore shear stress/ strength can be expressed as a function of both compressive stress p and shear displacement d. The function can be determined from controlled displacement direct/simple/ring shear tests depending on the engineering judgement and specific problem at hand. In its simplest form, when the shape of function q!d is assumed independent of p, the function becomes q"CpdI,
(4)
where C, k)1 are soil constants determined by curve fitting from the test results. This assumption can be considered reasonable for rather large p stress range only for cohesive soil under undrained conditions. The p stress dependent shapes of the function may be introduced but on account of the use of an additional iterative procedure which must be applied until the differences between initially assumed p stress levels are close to the calculated p stress levels within desired tolerance. An alternative approach would be division of a soil zone into subzones each corresponding to appropriate p stress level as it has been done in the case when a nonlinear shear
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M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
strength envelope is linearized within chosen p stress intervals. Using Eq. (4), factor of safety at the surface i can be written in the form F "d /d , (5) where d is shear displacement corresponding to the available shear stress q at a surface i and d is shear displacement corresponding to mobilized shear stress q at limit equilibrium and the surface i. Similarly at another surface j, F "d / d , (6)
where the superscript m is different from k in the case if different soil types exist at the places i and j. From Eqs. (5) and (6) it follows F "F d /d d /d . (7)
Unknown F can be determined from available equilibrium equations similarly to a constant F in conventional methods. The values of d and d are determined from
soil shear strength tests. It should be noted that the ratio d /d and not separate values of shear displacements at mobilized stresses is necessary to define for the calculation of local factor of safety at any surface j. If regions between distinct sliding surfaces can be considered to be without volume changes (perfectly rigid) then the ratio d /d between shear displacements ds at two places (i, j) will be the same to the ratio D /D of only kinematically admissible sliding D’s along these surfaces, because d and D will be directly proportional (d"BD, where B is a constant). Only kinematically admissible sliding D’s can be determined from the kinematics of a sliding body motion (Fig. 2) and can be defined as magnitudes of vectors starting with a unit vector of sliding along the base of the first part and proceeding along the slip surface resolving previous known vectors into directions of the interface and base of the following part. The assumption of no volume change within a domain is convenient for well compacted soil in embankments and saturated cohesive soil in undrained (short term)
Fig. 2. Kinematically only admissible displacement Ds of a sliding body divided into n parts.
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conditions, which exist in the foundation. For normally consolidated (soft) clay, the short term condition is more critical than the long term state because such clay consolidates and increases the strength with time. Volume » (thickness t) changes d» (dt) of shear zones during shearing can be taken into account in the construction of displacement diagram in Fig. 2 as the inclinations of displacement vectors with respect to the surfaces. From calculated F using Eq. (7) it
is possible to back calculate d using Eq. (6). The function of volume (thickness)
change d» (dt) versus shear displacement (Fig. 1c) can be determined from shear tests. The angle of inclination a of shear displacement vector with respect to the surface is
a "arctan (dt /d ). (8)
If local overstressing occurs at face j (which means that F tends to become less than
1 and the mobilized strength greater than the peak value, which is impossible) then the yielding, accompanied by an increase in shear displacements d , can be simulated by
the increase of d , in Eq. (7) until F becomes equal to 1. Such simulation is necessary
because fixed sliding D , dependent on kinematics only, are used instead of actual shear displacements d . With post peak increase (yielding) of shear displacement d , brittle soil will soften and therefore the corresponding decrease in the shear
strength parameters c, with the increase in d (Fig. 1a — dashed line) is taken into
account in Eq. (3). A change (increase) in a factor of safety F will cause corresponding change in ¹ forces and all other forces because of the need to satisfy the equilibrium equations. Table 1 contains the list of unknown values, available equations and their numbers for a sliding body consisting of n different parts. It can be noted that the use of the local factors of safety F increased the number of unknown values for 2n!2 as well as
that the number of available equations type (7) increased for 2n!2 in comparison Table 1 Values, available equations, and their numbers for n parts of a sliding body Unknown values
Number
Available equations
Normal forces N at bases Location of N at the last base (at other bases assumed in the middle of bases) Normal forces N at interfaces Locations of N at interfaces Shear forces ¹ at bases Shear forces ¹ at interfaces Factor of safety F Local factors of safety F :
at bases (except at i) at interfaces
n
Forces equilibrium in horizontal direction Forces equilibrium in vertical direction Moments equilibrium F "F d /d D /D :
at bases(except at i) at interfaces ¹ "(c b #N tan )/F :
at bases at interfaces
Total
7n!3
1
n!1 n!1 n n!1 1
Number
n n n n!1 n!1 n n!1
n!1 n!1 Total
7n!3
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M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
with the number of unknown values and available equations in conventional methods of limit equilibrium. It is also evident that the positions of normal forces Ns at the bases (except at the last base) are assumed to be in the middle of bases. Various assumptions have to be introduced in all procedures based on the limit equilibrium method due to excessive number of unknown values in comparison with available limit equilibrium equations. Such assumptions cause that the solutions obtained by the methods are only approximate ones and not necessarily correct with regard to other stress—strain constitutive laws. The system of 3n equilibrium equations is nonlinear due to unknown F in the denominators of the coefficients of equations. It is possible to apply an iterative procedure by choosing an initial F ("1), solving 3n!1 linear equations, checking the 3nth equation and gradually changing (increasing) F in steps until all 3n equilib rium equations are satisfied to a specified tolerance. Several iterations will be necessary for each step if local yielding occurs and therefore the coefficients of the equations must be readjusted. For an unstable domain, the equilibrium equations cannot be satisfied and the stepping procedure will continue until permitted by the user. For a stable domain, an average factor of safety of domain stability F . can be calculated from the formula F ."& (q b )/& (q b /F ),
(9)
where j"1, 2 , n, and used for a comparison with a constant F from conventional methods. The results of a conventional and extended method were compared for different cases of non-reinforced ground (Srbulov, 1987, 1991, 1995, 1997). The expression relating forces and displacements follows from Eqs. (3) and (7) (c b #N tan ) H H . ¹" H H
F d /d D /D
(10)
Reinforcement can be treated as a thin soil layer, having specific q!d function, and therefore the compatibility of deformations of reinforcement and soil is possible to consider using the extended method for a heterogeneous soil. The orientation of the force in reinforcement at a shear zone varies with varying stress and displacement. At collapse, the orientation of the reinforcement can be near tangential to the sliding surface. A linear variation from the initial as built state at zero stress/strain to the tangential orientation at the ultimate strength is assumed in the examples.
3. Examples The first example of a reinforced slope instability analysis is for St. Alban (Quebec, Canada) test embankment described by Schaefer and Duncan (1988). They found that ‘‘The results of the finite element analyses were in good agreement with the measured field behaviour during early stages of construction, when the embankment was stable.
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153
In the later stage of construction, as the embankment approached failure, the agreement between the finite element analysis and the field measurements was not as good’’. Soil profile at the St. Alban test site consists of about 0.3 m of topsoil, a 1.5 m thick weathered clay crust, an 8 m thick layer of soft, very sensitive, cemented silty clay, known locally as Champlain Clay. The results of numerous in situ vane shear and cone penetration tests at the site during various research projects have demonstrated that the clay is very uniform across the site. The undrained shear strength of this clay is rather uniform within depth range from 1 m to 4.5 m and varies in the range from 8 to 16 kPa, and reaches up to 48 kPa within the top 1 m. Although, the clay exhibits brittle stress—strain behaviour typical of sensitive soil, only the residual strength of 10 kPa at failure, available from the field tests, will be considered in the analysis. The shear strain at failure e "10% and the exponent in Eq. (6) m"0.5 are adopted
based on the stress—strain curves of unconsolidated undrained tests of samples of Champlain clays published by La Rochelle et al. (1974). The soil used for the fill was a uniform, medium to coarse grained sand containing about 10% fine sand and 10% gravel. The sand was placed in a loose state with minimal compaction effort. The friction angle of the sand as placed was estimated to be 34° by Busbridge et al. (1985). The shear strain at failure e "20% and the
exponent m"0.1 are adopted based on experience. The reinforcement used in the test embankment was Tensar SR2, a high strength, high density polyethylene geogrid, manufactured by punching holes in a polymer sheet and then stretching it in one direction to align the long chain molecules. An upper load limit of 79 kN has been determined from tests run at a very rapid strain rate of 23%/min. The load-strain-time behaviour of Tensar geogrids has been described by McGown et al. (1984). The exponent m"0.5 is adopted from the published load-strain curve. Shear strain e is used in place of shear displacement d because
the test results on the geogrid are reported in terms of strain and because only the ratio between displacements or strains is important and not their values. The behaviour of the test embankment was monitored by an extensive series of geotechnical instruments. The behaviour of the Tensar geogrid was monitored using a specially designed load cell (Busbridge et al., 1985) to measure the tensile load in the geogrid while strains in the geogrid were measured using Bison strain gages. The failure of the embankment was sudden, occurring during 30—60 seconds, with no prior sign of distress. The rate of embankment failure justifies the use of the reinforcement strength at a rapid strain rate.The embankment failed in a rotational slip mode with the fill and foundation soil moving as a solid block, at the height of the embankment of 6m. The reinforcement load, measured in a load cell located under the midslope of the embankment, was relatively small at failure, only 15.4 kN. Larger load can be expected at the shear zone and near the centreline. The geometry of the actual sliding surface is not reported in the paper but its depth below the original ground surface can be inferred to about 4—5 m based on the results of reported measurements of the horizontal ground movement at the toe of the embankment. Several sliding mechanisms were analyzed and the most critical mechanism found is shown in Fig. 3. It is calculated that the available strength of the geogrid is not sufficient to maintain the limit equilibrium. Therefore, the ultimate strength has
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M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
Fig. 3. The geometry of the sliding mechanism considered for St. Alban test embankment.
been increased until it has been found that the tensile force in geogrid is 170 kN at failure (F ."1), which is greater than its tensile strength of 79 kN. The discrepancy may be the result of neglecting the beneficial effect on stability of the crust. This 1.5m thick crust can support lateral force of the order of 1.5 times its average undrained strength of 30 kPa;2 which is equal to the force of 90 kN, i.e. the difference between the calculated force and the maximal force. This stabilizing force can not be taken into account with confidence, however, because it may not exist if the crust is deformed so much before failure that it has buckled. If the Bishop method (1955) is modified to include line loads and the method of steepest descent is used for the search of the slip circle with minimum F ("1) among the group of circles tangential to a predefined horizontal plane (Maksimovic, 1988) then the horizontal line load at the ground surface necessary to maintain limit equilibrium would be 330 kN, which is almost double the value calculated by the extended limit equilibrium method. This force could decrease to 240 kN if a possible crust effect is taken into consideration. If the inclination of geogrid at failure is assumed tangential to the failure surface than the required force would be 210 kN without taking into account the beneficial effect of the upper soil crust. The second example of the analysis of instability of a reinforced slope is for test embankment at Almere, Holland, in 1979 (Brakel et al., 1982). The subsoil conditions consists of a homogeneous soft clay—peat layer 3—4.5 m thick, with undrained strength—cohesion of about 10 kPa. The shear strain at failure e "10% and the exponent m"0.5 are adopted from experience. Hydraulically
filled material was used for a quick construction of embankment. The shear strain at failure e "20% and the exponent m"0.1 are adopted from experience.
The reinforcing fabric used was a woven polyester — Stabilenka 200, with ultimate strength of 220 kN at 9% strain. Maximum measured tensile stress in the fabric was 95 kN at approximately 8—10 m distance from the toe of the retaining embankment. The exponent m"1 is adopted from published test result (Brakel et al, 1982). The fill reached 2.75 m height above ground level, with the retaining bank shifting 1m towards the toe ditch (2 m deep) in 26 hours of filling, before a deep seated failure occurred.
M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
155
Fig. 4. The geometry of the sliding mechanism used for Almere test embankment.
The geometry of the actual sliding surface is not reported in the paper. Several sliding mechanisms were analyzed and the most critical mechanism found is shown in Fig. 4. The stability analysis showed that reinforcement is not needed in the case of a sliding type failure, i.e. a compressive rather than tensile force is obtained at the reinforcement location. The measured tensile force and its distribution at the fabric is probably the result of squeezing out of soft clay-peat layer into the ditch. Modified Bishop method requires the force in a horizontal reinforcement of only 13 kN at failure, F"1. Brakel et al. (1982) mentioned that ‘‘... the embankment itself was still stable on October 4, 1979 at 08:00 a.m. despite an ongoing horizontal displacement of the retaining bank and simultaneous squeezing out of the soft subsoil. In the reinforcing fabric there was then a maximum tensile stress of 95 kN/m’’. ‘‘The reinforced section finally failed on 4, October 1979 at about 11.00 a.m., due to circular sliding’’. ‘‘When the fabric was dug out, it was found that the fabric had torn’’. This description is in accordance with the finding of the analysis because the circular failure of the embankment could happen without the influence of the reinforcement which failed (due to dragging by squeezing soil) before the soil reached its peak strength at much greater shear strain incompatible with the reinforcement failure strain. Delayed failure of the embankment could be the result of a gradual propagation of rupture in the geotextile from the initial location at failed embankment section.
4. Conclusion The preceding examples indicated that the introduction of force—displacement relationships into the limit equilibrium method could improve the results. Such results remain only approximate due to the need of introduction of various assumptions such as that the shape of force—displacement function is independent of the axial stress level. This assumption can be avoided on account of introduction of an additional iterative procedure or the effect of the assumption can be decreased if axial stress level dependent zonation is introduced.
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M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
The method is simple to use and requires less input data than complete methods. It inherited the same disadvantages of conventional limit equilibrium methods that a sliding mechanism must be assumed to govern slope stability. The proposed method could be used for the stability analysis of reinforced slopes providing that it is thoroughly validated against the existing case histories and bench mark cases.
Acknowledgement Professor Maksimovic kindly provided computer software for the comparative analysis based on extended Bishop method.
References Bishop, A.W., 1955. The use of slip circle for stability analysis. Geotechnique 5 (1), 7—17. Brakel, J., Coppens, M., Maagdenberg, A.C., Risseeuw, P., 1982. Stability of slopes constructed with polyester reinforcing fabric, test section at Almere — Holland, ’79. In: Proc. Internat. Conf. on Geotextiles, Las Vegas, USA, pp. 727—732. Busbridge, J.R., Chan, P., Milligan, V., La Rochelle, P., Lefebrve, L.D., 1985. The effect of geogrid reinforcement on the stability of embankments on a soft sensitive Champlain clay deposit. Report to Transport Development Centre, Montreal, Quebec. La Rochell, P., Trak, B., Tavenas, F., Roy, M., 1974. Failure of a test embankment on a sensitive Champlain clay deposit. Canadian Geotechnical Journal 11, 142—164. Maksimovic, M., 1988. General slope stability software package for micro computers. Proc. 6th Intern. Conf. on Numerical Methods in Geomechanics, Insbruck 3, 2145—2150. McGown, A., Paiine, N., DuBois, D.D., 1984. Use of geogrid properties in limiit equilibrium analysis. Proc. Sympos. on Polymer Grid Reinforc. in Civil Engineering, London, Paper No. 1.4. Rowe, R.K., Soderman, K.L., 1987. Reinforcement of embankments on soils whose strength increases with depth. In: Proc. Conf. on Geosynthetics 1987, New Orleans, USA, p. 266. Rowe, R.K., Mylleville, B.L.J., 1990. Implications of adopting an allowable geosynthetic strain in estimating stability. Proc. 4th Intern. Conf. on Geotextiles, Geomembranes and Related Products, Hague, Netherlands, p. 131. Rowe, R.K., Mylleville, B.L.J., 1994. Analysis and design of reinforced embankments on soft or weak foundations. Chapter 7 in Soil Structure Interaction: Numerical Analysis and Modelling. John W. Bull (Ed.), E & FN Spon Chapman Hall, London, pp. 230—260. Schaefer, V.R., Duncan, J.M. 1988. Finite element analysis of the St. Alban test embankment. Geosynthetics for Soil Improvement. R.D.Holt (Ed.), Gt. Spec. Publ. No. 18, ASCE, pp. 158—177. Srbulov, M., 1987. Limit equilibrium method with local factors of safety for slope stability. Canadian Geotechnical Journal 24 (4), 652—656. Srbulov, M., 1991. Bearing capacity of a strip footing on brittle rock. Rock Mechanics and Rock Engineering 24 (4), 53—59. Srbulov, M., (1995). A simple method for the analysis of stability of slopes in brittle soil. Soils and Foundation 35 (4), 123—127. Srbulov, M., 1997. On the influence of soil strength brittleness and nonlinearity on slope stability. Computers and Geotechnics 20 (1), 95—104.
Force—displacement compatibility for reinforced embankments over soft clay M. Srbulov SAGE Engineering Ltd, 1 Widcombe Parade, Bath BA2 4JT, UK Received 8 June 1998; received in revised form 19 September 1998; accepted 6 November 1998
Abstract Due to a complex soil-reinforcement interaction, an extension of conventional limit equilibrium method is suggested for the analysis of slope stability of reinforced embankments over soft clay. The advantage of the proposed method is in decreasing the number of material properties necessary for the analysis in comparison with a finite element method while at the same time force—displacement compatibility can be considered for slope stability. The method inherited the disadvantage of conventional limit equilibrium that a sliding mechanism must be assumed to govern the slope stability. 1999 Elsevier Science Ltd. Keywords: Limit; Equilibrium; Soil; Reinforcement; Interaction
1. Introduction Rowe and Mylleville (1994) reviewed the methods used for analysis and design of reinforced embankments on soft or weak foundations and stated that ‘‘Limit equilibrium calculations performed assuming an arbitrary cutoff strain (be it 2%, 5%, 10% or whatever) to determine the geosynthetic force cannot be expected to provide consistent results and, as shown by Rowe and Soderman (1987) and Rowe and Mylleville (1990), the level of error associated with this type of approach will vary from case to case.’’ Also . . . ‘‘caution should be exercised when selecting an allowable strain for use in designing reinforced embankments on brittle cohesive soils which are susceptible to progressive failure’’. The majority of methods assume that the reinforcement force acts in its original orientation, which is horizontal, and some methods assume that the force acts * Fax:#44 1225 447443; e-mail: [email protected] 0266—1144/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 2 6 6 — 1 1 4 4 ( 9 8 ) 0 0 0 3 1 — 4
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M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
tangential to the slip surface, based on the argument that local deformations at the slip surface result in a local reorientation of the reinforcement, tangential to the slip surface. The assumption that the reinforcement remains in its original (horizontal) orientation is conservative for global moment equilibrium. Although the application of finite elements may seem ideal for the consideration of deformation and the interaction of the various components of the reinforced embankments, Rowe and Mylleville (1994) stated that ‘‘careful consideration must be given to the type of finite element model and constitutive relationships which will be used to model the discrete components of the reinforced embankments. The validity of results provided by a particular formulation and program should be checked using either: (i) limiting analytical bench mark solutions, (ii) data from full scale embankments and/or (iii) data from centrifuge tests.’’ The use of finite element analysis involves cost, expertise and time, that may not be justified for routine design purposes. Therefore, an extension of a conventional limit equilibrium method could be more appropriate.
2. Method description Conventional limit equilibrium methods use a constant factor of safety F along a potential slip surface under the assumption that soil strength is mobilized at all places at the same (or similar) shear displacements. Factor of safety F is defined as the ratio between available shear strength q and the shear stress q necessary to maintain limit equilibrium F"q /q , (1) where q is equal to the peak strength q when F'1 or to the post-peak strength if yielding occurs. Using Coulomb—Mohr failure criterion, which relates shear and compressive stresses p (Fig. 1a), F can be expressed in terms of these stresses F"(c#p tan )/q , (2) where c is cohesion and angle of soil internal friction. Eq. (2) can be written in terms of normal N and shear ¹ force acting on a particular surface F"(cb#N tan )/¹,
(3)
where b is width of the surface. Knowing one of the components of the resultant forces acting along slip surface and a constant F, it is possible to calculate the other components, so that the number of unknown forces to be determined from available force and moment equilibrium equations is decreased. In the case of heterogeneous soil, when brittle soil (soil crust) can reach its peak value while the other softer zone (deep down) achieves only a part of its peak strength or when the brittle part yields to its residual strength, which can be considerably smaller than the peak strength, while the softer part achieves its peak strength at failure, the assumption of activating the peak strengths at all places (at a constant F) is unrealistic and on unsafe side. This problem can be solved by using only residual
M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
149
Fig. 1. (a) Shear strength q , q , q versus compressive stress p, (b) Shear stress q versus shear displacement d, N A P (c) shear zone thickness changed dt versus shear displacement d.
strength of brittle soil but at the expense of greater cost of such a solution. Similarly with reinforced soil, when reinforcement reaches its peak tensile strength while the surrounding softer soil mobilizes only a part of its peak strength and when reinforcement yields or fails when the softer soil reaches its peak strength, the assumptions of simultaneously activating the peak strengths in reinforcement and soil (at the same F) is unrealistic and on unsafe side. This problem is greater for reinforced slopes because reinforcement can fail and loose all its strength before the surrounding soil reaches its peak shear strength. Different methods have been attempted to solve the problem of propagating (progressive) failure using complete stress—strain (finite element) solutions or local (partial) factors of safety. This article describes a procedure for definition of the local factors of safety within the framework of limit equilibrium method. The activation of shear stresses q , q is accompanied by development of shear displacements d , d (Fig. 1b) at the peak and mobilized shear stress respectively and therefore shear stress/ strength can be expressed as a function of both compressive stress p and shear displacement d. The function can be determined from controlled displacement direct/simple/ring shear tests depending on the engineering judgement and specific problem at hand. In its simplest form, when the shape of function q!d is assumed independent of p, the function becomes q"CpdI,
(4)
where C, k)1 are soil constants determined by curve fitting from the test results. This assumption can be considered reasonable for rather large p stress range only for cohesive soil under undrained conditions. The p stress dependent shapes of the function may be introduced but on account of the use of an additional iterative procedure which must be applied until the differences between initially assumed p stress levels are close to the calculated p stress levels within desired tolerance. An alternative approach would be division of a soil zone into subzones each corresponding to appropriate p stress level as it has been done in the case when a nonlinear shear
150
M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
strength envelope is linearized within chosen p stress intervals. Using Eq. (4), factor of safety at the surface i can be written in the form F "d /d , (5) where d is shear displacement corresponding to the available shear stress q at a surface i and d is shear displacement corresponding to mobilized shear stress q at limit equilibrium and the surface i. Similarly at another surface j, F "d / d , (6)
where the superscript m is different from k in the case if different soil types exist at the places i and j. From Eqs. (5) and (6) it follows F "F d /d d /d . (7)
Unknown F can be determined from available equilibrium equations similarly to a constant F in conventional methods. The values of d and d are determined from
soil shear strength tests. It should be noted that the ratio d /d and not separate values of shear displacements at mobilized stresses is necessary to define for the calculation of local factor of safety at any surface j. If regions between distinct sliding surfaces can be considered to be without volume changes (perfectly rigid) then the ratio d /d between shear displacements ds at two places (i, j) will be the same to the ratio D /D of only kinematically admissible sliding D’s along these surfaces, because d and D will be directly proportional (d"BD, where B is a constant). Only kinematically admissible sliding D’s can be determined from the kinematics of a sliding body motion (Fig. 2) and can be defined as magnitudes of vectors starting with a unit vector of sliding along the base of the first part and proceeding along the slip surface resolving previous known vectors into directions of the interface and base of the following part. The assumption of no volume change within a domain is convenient for well compacted soil in embankments and saturated cohesive soil in undrained (short term)
Fig. 2. Kinematically only admissible displacement Ds of a sliding body divided into n parts.
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M. Srbulov/Geotextiles and Geomembranes 17 (1999) 147—156
conditions, which exist in the foundation. For normally consolidated (soft) clay, the short term condition is more critical than the long term state because such clay consolidates and increases the strength with time. Volume » (thickness t) changes d» (dt) of shear zones during shearing can be taken into account in the construction of displacement diagram in Fig. 2 as the inclinations of displacement vectors with respect to the surfaces. From calculated F using Eq. (7) it
is possible to back calculate d using Eq. (6). The function of volume (thickness)
change d» (dt) versus shear displacement (Fig. 1c) can be determined from shear tests. The angle of inclination a of shear displacement vector with respect to the surface is
a "arctan (dt /d ). (8)
If local overstressing occurs at face j (which means that F tends to become less than
1 and the mobilized strength greater than the peak value, which is impossible) then the yielding, accompanied by an increase in shear displacements d , can be simulated by
the increase of d , in Eq. (7) until F becomes equal to 1. Such simulation is necessary
because fixed sliding D , dependent on kinematics only, are used instead of actual shear displacements d . With post peak increase (yielding) of shear displacement d , brittle soil will soften and therefore the corresponding decrease in the shear
strength parameters c, with the increase in d (Fig. 1a — dashed line) is taken into
account in Eq. (3). A change (increase) in a factor of safety F will cause corresponding change in ¹ forces and all other forces because of the need to satisfy the equilibrium equations. Table 1 contains the list of unknown values, available equations and their numbers for a sliding body consisting of n different parts. It can be noted that the use of the local factors of safety F increased the number of unknown values for 2n!2 as well as
that the number of available equations type (7) increased for 2n!2 in comparison Table 1 Values, available equations, and their numbers for n parts of a sliding body Unknown values
Number
Available equations
Normal forces N at bases Location of N at the last base (at other bases assumed in the middle of bases) Normal forces N at interfaces Locations of N at interfaces Shear forces ¹ at bases Shear forces ¹ at interfaces Factor of safety F Local factors of safety F :
at bases (except at i) at interfaces
n
Forces equilibrium in horizontal direction Forces equilibrium in vertical direction Moments equilibrium F "F d /d D /D :
at bases(except at i) at interfaces ¹ "(c b #N tan )/F :
at bases at interfaces
Total
7n!3
1
n!1 n!1 n n!1 1
Number
n n n n!1 n!1 n n!1
n!1 n!1 Total
7n!3
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with the number of unknown values and available equations in conventional methods of limit equilibrium. It is also evident that the positions of normal forces Ns at the bases (except at the last base) are assumed to be in the middle of bases. Various assumptions have to be introduced in all procedures based on the limit equilibrium method due to excessive number of unknown values in comparison with available limit equilibrium equations. Such assumptions cause that the solutions obtained by the methods are only approximate ones and not necessarily correct with regard to other stress—strain constitutive laws. The system of 3n equilibrium equations is nonlinear due to unknown F in the denominators of the coefficients of equations. It is possible to apply an iterative procedure by choosing an initial F ("1), solving 3n!1 linear equations, checking the 3nth equation and gradually changing (increasing) F in steps until all 3n equilib rium equations are satisfied to a specified tolerance. Several iterations will be necessary for each step if local yielding occurs and therefore the coefficients of the equations must be readjusted. For an unstable domain, the equilibrium equations cannot be satisfied and the stepping procedure will continue until permitted by the user. For a stable domain, an average factor of safety of domain stability F . can be calculated from the formula F ."& (q b )/& (q b /F ),
(9)
where j"1, 2 , n, and used for a comparison with a constant F from conventional methods. The results of a conventional and extended method were compared for different cases of non-reinforced ground (Srbulov, 1987, 1991, 1995, 1997). The expression relating forces and displacements follows from Eqs. (3) and (7) (c b #N tan ) H H . ¹" H H
F d /d D /D
(10)
Reinforcement can be treated as a thin soil layer, having specific q!d function, and therefore the compatibility of deformations of reinforcement and soil is possible to consider using the extended method for a heterogeneous soil. The orientation of the force in reinforcement at a shear zone varies with varying stress and displacement. At collapse, the orientation of the reinforcement can be near tangential to the sliding surface. A linear variation from the initial as built state at zero stress/strain to the tangential orientation at the ultimate strength is assumed in the examples.
3. Examples The first example of a reinforced slope instability analysis is for St. Alban (Quebec, Canada) test embankment described by Schaefer and Duncan (1988). They found that ‘‘The results of the finite element analyses were in good agreement with the measured field behaviour during early stages of construction, when the embankment was stable.
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In the later stage of construction, as the embankment approached failure, the agreement between the finite element analysis and the field measurements was not as good’’. Soil profile at the St. Alban test site consists of about 0.3 m of topsoil, a 1.5 m thick weathered clay crust, an 8 m thick layer of soft, very sensitive, cemented silty clay, known locally as Champlain Clay. The results of numerous in situ vane shear and cone penetration tests at the site during various research projects have demonstrated that the clay is very uniform across the site. The undrained shear strength of this clay is rather uniform within depth range from 1 m to 4.5 m and varies in the range from 8 to 16 kPa, and reaches up to 48 kPa within the top 1 m. Although, the clay exhibits brittle stress—strain behaviour typical of sensitive soil, only the residual strength of 10 kPa at failure, available from the field tests, will be considered in the analysis. The shear strain at failure e "10% and the exponent in Eq. (6) m"0.5 are adopted
based on the stress—strain curves of unconsolidated undrained tests of samples of Champlain clays published by La Rochelle et al. (1974). The soil used for the fill was a uniform, medium to coarse grained sand containing about 10% fine sand and 10% gravel. The sand was placed in a loose state with minimal compaction effort. The friction angle of the sand as placed was estimated to be 34° by Busbridge et al. (1985). The shear strain at failure e "20% and the
exponent m"0.1 are adopted based on experience. The reinforcement used in the test embankment was Tensar SR2, a high strength, high density polyethylene geogrid, manufactured by punching holes in a polymer sheet and then stretching it in one direction to align the long chain molecules. An upper load limit of 79 kN has been determined from tests run at a very rapid strain rate of 23%/min. The load-strain-time behaviour of Tensar geogrids has been described by McGown et al. (1984). The exponent m"0.5 is adopted from the published load-strain curve. Shear strain e is used in place of shear displacement d because
the test results on the geogrid are reported in terms of strain and because only the ratio between displacements or strains is important and not their values. The behaviour of the test embankment was monitored by an extensive series of geotechnical instruments. The behaviour of the Tensar geogrid was monitored using a specially designed load cell (Busbridge et al., 1985) to measure the tensile load in the geogrid while strains in the geogrid were measured using Bison strain gages. The failure of the embankment was sudden, occurring during 30—60 seconds, with no prior sign of distress. The rate of embankment failure justifies the use of the reinforcement strength at a rapid strain rate.The embankment failed in a rotational slip mode with the fill and foundation soil moving as a solid block, at the height of the embankment of 6m. The reinforcement load, measured in a load cell located under the midslope of the embankment, was relatively small at failure, only 15.4 kN. Larger load can be expected at the shear zone and near the centreline. The geometry of the actual sliding surface is not reported in the paper but its depth below the original ground surface can be inferred to about 4—5 m based on the results of reported measurements of the horizontal ground movement at the toe of the embankment. Several sliding mechanisms were analyzed and the most critical mechanism found is shown in Fig. 3. It is calculated that the available strength of the geogrid is not sufficient to maintain the limit equilibrium. Therefore, the ultimate strength has
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Fig. 3. The geometry of the sliding mechanism considered for St. Alban test embankment.
been increased until it has been found that the tensile force in geogrid is 170 kN at failure (F ."1), which is greater than its tensile strength of 79 kN. The discrepancy may be the result of neglecting the beneficial effect on stability of the crust. This 1.5m thick crust can support lateral force of the order of 1.5 times its average undrained strength of 30 kPa;2 which is equal to the force of 90 kN, i.e. the difference between the calculated force and the maximal force. This stabilizing force can not be taken into account with confidence, however, because it may not exist if the crust is deformed so much before failure that it has buckled. If the Bishop method (1955) is modified to include line loads and the method of steepest descent is used for the search of the slip circle with minimum F ("1) among the group of circles tangential to a predefined horizontal plane (Maksimovic, 1988) then the horizontal line load at the ground surface necessary to maintain limit equilibrium would be 330 kN, which is almost double the value calculated by the extended limit equilibrium method. This force could decrease to 240 kN if a possible crust effect is taken into consideration. If the inclination of geogrid at failure is assumed tangential to the failure surface than the required force would be 210 kN without taking into account the beneficial effect of the upper soil crust. The second example of the analysis of instability of a reinforced slope is for test embankment at Almere, Holland, in 1979 (Brakel et al., 1982). The subsoil conditions consists of a homogeneous soft clay—peat layer 3—4.5 m thick, with undrained strength—cohesion of about 10 kPa. The shear strain at failure e "10% and the exponent m"0.5 are adopted from experience. Hydraulically
filled material was used for a quick construction of embankment. The shear strain at failure e "20% and the exponent m"0.1 are adopted from experience.
The reinforcing fabric used was a woven polyester — Stabilenka 200, with ultimate strength of 220 kN at 9% strain. Maximum measured tensile stress in the fabric was 95 kN at approximately 8—10 m distance from the toe of the retaining embankment. The exponent m"1 is adopted from published test result (Brakel et al, 1982). The fill reached 2.75 m height above ground level, with the retaining bank shifting 1m towards the toe ditch (2 m deep) in 26 hours of filling, before a deep seated failure occurred.
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Fig. 4. The geometry of the sliding mechanism used for Almere test embankment.
The geometry of the actual sliding surface is not reported in the paper. Several sliding mechanisms were analyzed and the most critical mechanism found is shown in Fig. 4. The stability analysis showed that reinforcement is not needed in the case of a sliding type failure, i.e. a compressive rather than tensile force is obtained at the reinforcement location. The measured tensile force and its distribution at the fabric is probably the result of squeezing out of soft clay-peat layer into the ditch. Modified Bishop method requires the force in a horizontal reinforcement of only 13 kN at failure, F"1. Brakel et al. (1982) mentioned that ‘‘... the embankment itself was still stable on October 4, 1979 at 08:00 a.m. despite an ongoing horizontal displacement of the retaining bank and simultaneous squeezing out of the soft subsoil. In the reinforcing fabric there was then a maximum tensile stress of 95 kN/m’’. ‘‘The reinforced section finally failed on 4, October 1979 at about 11.00 a.m., due to circular sliding’’. ‘‘When the fabric was dug out, it was found that the fabric had torn’’. This description is in accordance with the finding of the analysis because the circular failure of the embankment could happen without the influence of the reinforcement which failed (due to dragging by squeezing soil) before the soil reached its peak strength at much greater shear strain incompatible with the reinforcement failure strain. Delayed failure of the embankment could be the result of a gradual propagation of rupture in the geotextile from the initial location at failed embankment section.
4. Conclusion The preceding examples indicated that the introduction of force—displacement relationships into the limit equilibrium method could improve the results. Such results remain only approximate due to the need of introduction of various assumptions such as that the shape of force—displacement function is independent of the axial stress level. This assumption can be avoided on account of introduction of an additional iterative procedure or the effect of the assumption can be decreased if axial stress level dependent zonation is introduced.
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The method is simple to use and requires less input data than complete methods. It inherited the same disadvantages of conventional limit equilibrium methods that a sliding mechanism must be assumed to govern slope stability. The proposed method could be used for the stability analysis of reinforced slopes providing that it is thoroughly validated against the existing case histories and bench mark cases.
Acknowledgement Professor Maksimovic kindly provided computer software for the comparative analysis based on extended Bishop method.
References Bishop, A.W., 1955. The use of slip circle for stability analysis. Geotechnique 5 (1), 7—17. Brakel, J., Coppens, M., Maagdenberg, A.C., Risseeuw, P., 1982. Stability of slopes constructed with polyester reinforcing fabric, test section at Almere — Holland, ’79. In: Proc. Internat. Conf. on Geotextiles, Las Vegas, USA, pp. 727—732. Busbridge, J.R., Chan, P., Milligan, V., La Rochelle, P., Lefebrve, L.D., 1985. The effect of geogrid reinforcement on the stability of embankments on a soft sensitive Champlain clay deposit. Report to Transport Development Centre, Montreal, Quebec. La Rochell, P., Trak, B., Tavenas, F., Roy, M., 1974. Failure of a test embankment on a sensitive Champlain clay deposit. Canadian Geotechnical Journal 11, 142—164. Maksimovic, M., 1988. General slope stability software package for micro computers. Proc. 6th Intern. Conf. on Numerical Methods in Geomechanics, Insbruck 3, 2145—2150. McGown, A., Paiine, N., DuBois, D.D., 1984. Use of geogrid properties in limiit equilibrium analysis. Proc. Sympos. on Polymer Grid Reinforc. in Civil Engineering, London, Paper No. 1.4. Rowe, R.K., Soderman, K.L., 1987. Reinforcement of embankments on soils whose strength increases with depth. In: Proc. Conf. on Geosynthetics 1987, New Orleans, USA, p. 266. Rowe, R.K., Mylleville, B.L.J., 1990. Implications of adopting an allowable geosynthetic strain in estimating stability. Proc. 4th Intern. Conf. on Geotextiles, Geomembranes and Related Products, Hague, Netherlands, p. 131. Rowe, R.K., Mylleville, B.L.J., 1994. Analysis and design of reinforced embankments on soft or weak foundations. Chapter 7 in Soil Structure Interaction: Numerical Analysis and Modelling. John W. Bull (Ed.), E & FN Spon Chapman Hall, London, pp. 230—260. Schaefer, V.R., Duncan, J.M. 1988. Finite element analysis of the St. Alban test embankment. Geosynthetics for Soil Improvement. R.D.Holt (Ed.), Gt. Spec. Publ. No. 18, ASCE, pp. 158—177. Srbulov, M., 1987. Limit equilibrium method with local factors of safety for slope stability. Canadian Geotechnical Journal 24 (4), 652—656. Srbulov, M., 1991. Bearing capacity of a strip footing on brittle rock. Rock Mechanics and Rock Engineering 24 (4), 53—59. Srbulov, M., (1995). A simple method for the analysis of stability of slopes in brittle soil. Soils and Foundation 35 (4), 123—127. Srbulov, M., 1997. On the influence of soil strength brittleness and nonlinearity on slope stability. Computers and Geotechnics 20 (1), 95—104.