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Analyses of stability of geogrid reinforced steep slopes and retaining walls
Computers and Geotechnics 28 (2001) 255±268 www.elsevier.com/locate/compgeo
Analyses of stability of geogrid reinforced steep slopes and retaining walls M. Srbulov * SAGE Engineering Ltd, Newark House, 26±45 Cheltenham Street, Bath BA2 3EX, UK Received 9 February 2000; received in revised form 15 November 2000; accepted 17 November 2000
Abstract The results of measurements of axial strains in geogrids of two reinforced steep slopes and two retaining walls were uniformly interpreted. The stabilities of slopes and walls are analyzed using a method based on limit equilibrium. The method of analysis takes into account strains along boundaries of rigid wedges in addition to the forces considered by classical methods of limit equilibrium. However, the results obtained by the method remain only approximate due to necessity to introduce a number of simplifying assumptions. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Stability analysis; Geogrid; Reinforced; Slopes; Retaining walls
1. Introduction The use of natural ®bres to improve small tensile strength of in situ ground and man made ®lls date to pre-historic times. However, the rational approach to the problem of soil strength improvement by reinforcement is rather recent. In the late ®fties, Henri Vidal, a French architect and engineer, launched a new civil engineering material known as Reinforced Earth. With rapid development of technology, the initial metal strips are replaced by geotextiles and geogrids, which are almost exclusively used these days. Jewel [1] presented a rather comprehensive review of the methods used for the design of reinforced soil. The potential failure surfaces, most widely used for the analysis of internal equilibrium of steep reinforced soil slopes and walls in connection with the limit equilibrium * Tel.: +44-1225-486-500; fax: +44-1225-486-597. E-mail address: milutin@sage- uk.com 0266-352X/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-352X(00)00032-X
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method, include the logarithmic spiral and two-part wedge mechanism. These two failure mechanisms were found to be the simplest and at the same time the most appropriate representatives of the true failure modes throughout the reinforced part. These ®ndings will be re-examined in this paper for a number of cases of instrumented geogrid reinforced steep slopes and retaining walls. 2. Description of the extended method Conventional limit equilibrium methods use a constant factor of safety FS along a potential slip surface (and sometimes interfaces) under the assumption that soil strength is mobilized at all places at the same (or similar) shear strains. Factor of safety FS is usually de®ned as the ratio between available shear strength a and the shear stress e necessary to maintain limit equilibrium FS a = e
1
a is equal to the peak strength p when FS >1 or to the post-peak strength if yielding occurs. Using Mohr±Coulomb failure criterion, which relates shear and compressive stresses (Fig. 1a), FS can be expressed in terms of these stresses FS c tan e
2
Fig. 1. (a) Shear strength versus compressive stress, (b) shear stress versus shear strain, (c) speci®c thickness change versus shear strain.
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where c is cohesion and angle of soil internal friction. Eq. (2) can be written in terms of normal N and shear T force acting on a particular surface FS cb Ntan =T
3
where b is width of the surface. Knowing one of the components of the resultant forces acting along a slip surface and a constant FS, 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. However, shear strains are seldom uniform in the ®eld even within a homogeneous soil and at relatively small stress levels. When soil shear strength at large shear strain is smaller than its peak value then localized and propagating (progressive) failure may occur if induced shear strains are large enough. Similarly, if soil is heterogeneous and some parts mobilize and loose their peak shear strength while the other parts are still on the way to mobilize the peak strength then again localized and progressive failures may occur. The problem of analysing progressive failure may be avoided if only soil residual strength is considered. Reinforced soil also experiences a progressive failure. However, when soil reinforcement is loaded beyond its ultimate strength it tends to break and completely loose all the strength as its residual strength is zero. In the later case, the use of residual strength only is pointless because it would lead to consideration of unreinforced soil. Dierent methods have been proposed to solve the problem of propagating (progressive) failure using complete stress±strain solutions by ®nite and discrete/distinct elements. This article describes a procedure for the consideration of the local and progressive failures within the framework of the limit equilibrium method when applied to analysis of stability of reinforced steep slopes and retaining walls. Srbulov [2] used the method to analyze the progressive failures of actual but unreinforced slopes in brittle soil. It has been shown that the extended method predicted well the instability of the slopes, while a classical method provided FS much greater than 1 for the peak shear strength and much smaller than 1 for the residual shear strength alone. The activation of shear stresses a ; e is accompanied by development of shear strains (Fig. 1b) and, therefore, shear stress/strength can be expressed as a function of both compressive stress and shear strain . The function can be determined from soil shear tests. In its simplest form, when only the shape but not the value of function is assumed independent of (Fig. 1b), the function becomes C k
4
where C, k41 are soil constants determined by curve ®tting from laboratory test results. This assumption can be considered reasonable for a rather large stress range only for cohesive soil under undrained conditions and reinforced soil when reinforcement strength has a greater in¯uence on the behaviour of composite material. The stress dependent shapes of the function may be introduced but on
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account of the use of an additional iterative procedure that must be applied until the dierences between initially assumed stress levels are close to the calculated s stress levels within desired tolerance. An alternative approach would be division of a soil zone into subzones each corresponding to appropriate stress level as it has been done in the case when a nonlinear shear strength envelope is linearized within chosen stress intervals. Using Eq. (4), factor of safety at the surface i can be written in the form k Fi i;ak = i;e
5
k is the shear strain corresponding to available shear stress a at a surface i where i;a k and i;e is the shear strain corresponding to mobilized shear stress e at the surface i. Similarly at another surface j, k1 k1 Fj j;a = j;e
6
where the exponent k1 is dierent from k in the case when dierent soil types exist at the places i and j. From Eqs. (5) and (6) it follows k1 k k k1 = i;a
i;e = j;e Fj Fi j;a
7
Unknown Fi can be determined from available equilibrium equations similarly to a constant FS in conventional methods. Mobilized shear strains can be determined indirectly from Eq. (6)
j;e j;a Fj
l=kl
8
kl k The values of j;a and j;a can be determined from soil shear strength tests. It should k1 k k k1 be noted that the ratios j;a = i;a and i;e = j;e but not particular values of shear strains are necessary to calculate the local factor of safety at any surface j. This has several useful implications. If soil is tested in a direct shear apparatus, for example, then the ratio of shear strains at available strength can be replaced by the ratio of measured k horizontal displacements k1 j;a =i;a instead. If soil shear tests are performed using a triaxial apparatus then the results are presented as the deviatoric stress versus axial strain". If the properties of all soil types involved in an analysis are measured and presented consistently in terms of axial strain than it is possible to use the ratio k k1 k "k1 j;a ="i;a in place of the ratio j;a = i;a The reinforcement strength is usually described as a function of axial strain. If a soil zone is divided into wedges and the wedges between their boundaries do not change their volumes (they are rigid) except that some local yielding is allowed k k1 at their tips then the ratio j;e = i;e between the magnitudes of actual shear strains at two boundaries (i, j) will be the same to the ratio ki;e =k1 j;e of magnitudes of potential shear strains along these boundaries, because and will be directly proportional ( B, where B is an unknown constant). For rigid wedges, tangential displacements along a particular boundary will be constant. In undrained conditions, there
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will be no volumetric changes along wedge boundaries and potential shear strain can be de®ned as the ratio between potential tangential displacement along the boundary and the boundary thickness. For a unit thickness, the shear strains will be equal to the tangential displacements. Potential tangential displacements, illustrated in Fig. 2 for the case of zero soil volume change in undrained conditions, can be de®ned starting with a unit tangential displacement along the base of the ®rst wedge. Proceeding along the external boundaries so that the known tangential displacement is the vectorial sum of the displacement in the direction of the interface and base of the following wedge it is possible to de®ne all other potential displacements. The assumption of no soil volume change along a boundary is not correct for drained conditions nor is it correct to assume that the wedges are rigid. However, the volumetric changes due to axial stresses should be small for sti soil and at relatively small stresses acting within steep slopes. When they are likely to be signi®cant such in soft clay and loose sand then the assumption of no volumetric strain due to axial stresses in drained condition is less acceptable. Volumetric strain "v (i.e. speci®c thickness dt) change along the boundaries with shear strain change can be taken into account in the construction of strain diagram in Fig. 2 as the inclinations of strain vectors with respect to the boundaries. From calculated Fj using Eq. (7) it is possible to back calculate j;e using Eq. (8). The function of volumetric strain (speci®c thickness dt) change versus shear strain (Fig. 1c) can be determined from soil shear tests. The angle of inclination j of shear strain vector with respect to the boundary is simply j arctan dtj;e = j;e 9 If local over stressing occurs at face j (which means that Fj tends to become less than 1 and the mobilized strength greater than the peak value, which is impossible)
Fig. 2. Possible tangential displacements 0 s along boundaries of wedges.
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then the yielding, accompanied by an increase in shear strains j;e can be simulated by the increase of j;a in Eq. (7) until Fj becomes equal to 1. Such simulation is necessary because ®xed value j;e , dependent on geometry only, is used instead of actual shear strain j;e . With post peak increase (yielding) of shear strain j;e , brittle soil will soften and, therefore, the corresponding decrease in the shear strength parameters c, with the increase in j;e is taken into account in Eq. (3). A change (increase) in a factor of safety Fj will cause a corresponding change in Tj force and all other forces because of the need to satisfy the equilibrium equations. The eect of the rate of soil shear strength decrease after the peak value has been investigated for two simple wedges before [3]. It has been found that there will be an evident dierence in the calculated factors of safety of the slope stability if soil decreases its peak strength towards the residual value more gradually then abruptly. In the former case the use of an instant jump from the peak to the residual strength is not appropriate. The calculated factors of safety by the extended method and by a classical method were also in a good agreement if soil exhibit an abrupt shear strength drop after the peak. However, the classical method yields smaller factors of safety than the extended method for gradually decreasing post peak shear because the mobilized shear strength is somewhere between its peak and the residual value. Two wedges, non brittle soil with shear strength parameters c=0, 0 24 and soil unit weight of 22 kN/m3 are used for an investigation of the eect of change in volumetric strain (speci®c thickness) with shearing. A function describing the speci®c thickness change with shear strain (Fig. 1c) is adopted in the form which is appropriate for both the contracting and dilating phases of soil volume changes up to its constant value at the steady-state dt 1=2 a ln b
10
where a,b are constants or functions of axial stress which can be determined by ®tting experimental data. The results of computations are shown in Table 1. The eect of volumetric strain changes with shearing is not very important for the majority of slightly dilatant/contractant soil but could be very important for heavily overconsolidated clay and very dense sand. Table 2 contains the list of unknown values, available equations and their numbers for n wedges. It can be noted that the use of the local factors of safety Fj increased the number of unknown values for 2n 2 as well as that the number of available equation types [7] increased for 2n 2 in comparison with the number of unknown values and available equations in conventional methods of limit equilibrium. It is also evident Table 1 The eects on factor of safety of soil dilatation ( ) and contraction (+) with shearing of soil along boundaries of the two wedges Peak volume/peak shear strain ratio Factor of safety
0.6
0.4
0.2
3.17
1.80
1.73
0
+0.2
+0.4
+0.6
1.67
1.63
1.59
1.51
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Table 2 Values, available equations, and their numbers for n wedges Unknown values
Number
Available equations
Number
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 T at bases Shear forces T at interfaces Local factors of safety Fj at bases (except at i) at interfaces
n l
Forces equilibrium in horizontal direction Forces equilibrium in vertical direction Moments equilibrium
n n n
n 1 n 1 n n 1
k1 k k1 = i;a ki;e = j;e : Fj Fi j;a at bases (except at i) at interfaces Tj cj bj Nj tan j =Fj : at bases at interfaces
n 1
Total
7n 3
Total
7n 3
n 1 n 1
n 1 n 1 n
that the positions of normal forces N's 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 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 Fi in the denominators of the coecients of equations. It is possible to apply an iterative procedure by choosing an initial Fi(=1), solve 3n 1 linear equations, check the 3nth equation and gradually change (increasing) Fi in steps until all 3n equilibrium equations are satis®ed to a speci®ed tolerance. Several iterations will be necessary for each step if local yielding occurs and therefore the coecients of the equations must be readjusted. For an unstable wedge assembly, the equilibrium equations cannot be satis®ed and the stepping procedure will continue until permitted by the user. For a stable wedge group, an average factor of safety of the group stability Favr . can be calculated from the formula Favr: a;j bj = a;j bj =Fj 11 where j=1...n; Favr is used for comparison with a constant FS from conventional methods and for the assessment of global stability of a group of wedges. The expression relating forces and shear strains follows from Eqs. (3) and (7) cj bj Nj tan j Tj 12 k Fi k1 j;a = i;a ki;e =k1 j;e k1 k All local Fsj will be the same if soil is homogeneous j;a i;a and the ratios ki;e =k1 j;e 1 at the bases of wedges along a plane or a circular cylinder with the interfaces passing through the centre of the circle.
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When only one layer of reinforcement exists, such as in reinforced embankments, then reinforcement can be treated as a thin soil layer, Srbulov [4]. However, when many reinforced layers are present it is necessary to apply a composite material concept. Ingold [5] referred to the work of Long et al. [6] who observed that above a certain threshold value of applied con®ning pressure in triaxial apparatus there was a constant increase in applied vertical stress at failure in samples with reinforcement at a given tensile strength and spacing, like in Fig. 1a. Failure of the reinforced samples was very brittle (like in Fig. 1b), with a drastic decrease in strength when the peak was passed. The brittleness was less severe at higher applied con®ning pressures or in less heavily reinforced samples. Post-failure inspection of dismantled samples consistently showed that the reinforcement had failed in tension. It was concluded that since, for tensile reinforcement failure, the failure envelopes of both the reinforced and unreinforced sand are parallel, and therefore exhibit the same angle of internal shearing resistance, the additional strength imparted by the reinforcement could be represented by an apparent cohesion. In addition, the orientation of reinforcement is likely to change with respect to a wedge boundary, from as-built position to an almost parallel orientation at the failure. Jewel [7], proposed that the improvement in shearing resistance Ps, resulting from a reinforcement force, Pr, can be expressed by the equilibrium equation of forces Ps Pr sin costan 0
13
where is the angle between the reinforcement direction and a normal to the boundary. Eq. (13) also de®nes the degree of anisotropy of shearing resistance of reinforced soil with respect to the angle . For 0 close to 30 it follows from the above equation that almost a constant tensile force greater or equal to Pr acts for the range of between 30 and 90 . In the examples which follow, it will be assumed that an apparent cohesion is given by the expression c Pr =b
14
where is applied to all reinforced layers crossing a particular considered wedge boundary if the distance to the end of reinforcement is sucient for activation of reinforcement tensile strength. It also means that reinforced soil shear strength is considered isotropic, independent of the angle . 3. Case histories 3.1. Case 1.1Ðlarge scale model of a reinforced slope Kutara et al. [8] described a large-scale model of six meter high instrumented embankment, with the slope inclination of 1 vertical to 1 horizontal, made of compacted silty sand, reinforced by geogrid. The embankment was submerged and the
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water level was rapidly lowered in order to estimate the embankment stability using a rotational slip surface and limit equilibrium method. The embankment sand contained 30% of silt. The shear strength parameters obtained by triaxial compression test and used in the stability analysis by the authors were c0 =10 kPa and 0 32 . However, the test results have not been presented in the paper and the cohesion intercept may be a result of the envelope curvature or the way of interpretation and will not be used for the back analysis. The dry unit weight was 13 kN/m3 and the relative density of 87%. The reinforcement used was Tensar SS2 biaxially stretched polymer grid, placed in four 6 m long layers at spacing of 1.5 m. The tensile strength speci®ed in the paper is 15 kN/m at the peak axial strain of 5%. The slope surface was encapsulated with the grid. The behaviour of the embankment and the geogrid was monitored by strain gages attached to the geogrids at interval of 0.5 m, by measurement of the settlement of the embankment crest and its interior, by the measurement of horizontal and vertical strains of the slope and by the measurement of water level inside the embankment. In addition a surcharge was placed on top of the embankment. The surcharge was placed at a distance of 4 m from the slope crest. A 3 m high ®ll made of river sand was used as a surcharge of 43.5 kN/m2. The submergence of the embankment was achieved by water up to the height of 5.25 m. The water level had been rapidly lowered with the rate of 0.75 m/h in front of the slope after the deformation and the strain of the geogrid ceased to increase through several days of submergence. The measured geogrid strain and embankment settlement at the end of rapid water level decrease in front of embankment are shown in Fig. 3. The authors also presented the strain diagrams for the construction phase, submergence and dierent water level fall cases. Their shapes generally coincide with the ®nal shapes of the strain diagrams. The face of slope also bulged greatly between the geogrid layers up to the maximum of 38 cm.
Fig. 3. Case 1.1 Ð geometry, ground water level, measured geogrid strain and embankment settlement.
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Using a rotational slip surface and the equilibrium of moments, the authors obtained the safety factor of 1.62 at the end of embankment construction, 1.32 at the end of surcharge addition, 1.38 at the end of submergence and 0.97 at the end of sudden water drawdown. However, they did not give details about the method of stability analysis and the analyzed critical slip surface. For the stability analysis by the method presented in this paper, the location of a potential slip surface and the interfaces are determined from the positions of the peaks in the geogrid strain diagrams and the shape of top settlement diagram shown in Fig. 3. An almost rotational slip surface shape with inclined interfaces was inferred from Fig. 3. The value of axial strain at failure of 5% and the exponent k=0.5 are adopted for silty sand based on experience. The eect of the assumptions made appeared to be of no great in¯uence on the results. The exponent k=0.5 for the composite material (reinforced soil) was adopted based on the tensile strength-strain diagram shown in the paper. The apparent cohesion used for the sides No. 1, 5, 6 was 0 kPa, No. 2, 4 was 10 kPa, No. 3, 11 was 5 kPa, No. 7, 8, 10 was 4 kPa and No 9 was 8.5 kPa. The unit weight of soil used was 19 kN/m3. Calculated Favr =1.04 by the extended method is close to one and the real slope state. The local factors of safety along the boundaries 1±6 are all equal to 1 and the average factor slightly greater than one is the result of the local factors of safety along the interfaces (Fs,7=1.5, Fs,9=1.04, F1,11=1.09). Mobilized axial strains are calculated according to the equation "j;e "j;a Fi
1=k1
15
At the boundaries 1±4 they are all 17.3% and at the boundaries 5±6 they are 19.4%. However, the corresponding axial displacements can not be calculated due to the lack of a referent length. 3.2. Case 1.2Ðlarge scale model of a steep reinforced slope This case was described by the same authors as Case 1.1. The only dierence was that the geogrid spacing was 0.75 m and that seven layers were used instead of four. As a result of denser reinforcement, the maximum bulging of the embankment slope between the reinforcement after rapid water level draw-down was 25 cm instead of 38 cm and the top settlement was more uniform. Also, the authors of the paper reported the safety factor of 1.7 at the end of embankment construction, 1.38 at the end of surcharge addition, 1.44 at the end of submergence and 1.01 at the end of sudden water draw-down. They gave no details about the rotational slip surface and the method used for the calculation of factor of safety. The position of potential slip surface and the interfaces was inferred from the peaks of measured geogrid strain diagrams and the shape of top settlements shown in Fig. 4. From Fig. 4, it can be seen that the geogrid strain is sometimes uniform over a considerable length. This is the result of superposition of strain when several slip surfaces cross the geogrid at rather close distances. The apparent cohesion used for the sides No. 1, 6, 7, 8 was 0 kPa, No. 2, 5, 9, 15 was 15 kPa, No. 3, 4, 10, 14 was
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Fig. 4. Case 1.2 Ð geometry, ground water level, measured geogrid strain and embankment settlement.
13 kPa, No. 11, 13 was 11kPa and No. 12 was 15kPa. The extended method failed to satisfy equilibrium conditions. 3.3. Case 2Ðlarge scale model of a reinforced retaining wall The same authors (Kutara et al., [8]) investigated and presented the results of measurements and stability analyses for a retaining wall 6 m high, with slope inclined 1 vert. to 0.2 hor. with precast concrete panels over the surface. The panels were joined to the reinforcement by reinforcing bars on the rear sides. The ®ll soil was silty sand as for the slopes. The reinforcement used was Tensar SR55 uniaxially stretched polymer grid, placed in six 4 m long layers at spacing of 1 m. The tensile strength speci®ed in the paper is 55 kN/m at the peak axial strain of 14%. To strengthen the wall face, the reinforcement was interposed by reinforcing material of the same type with length of 1 m. The results of measured geogrid strain and the surface strains are shown in Fig. 5. The authors also presented the strain diagrams for the construction phase, submergence and dierent water level fall cases. Their shapes generally coincide with the ®nal shapes of the strain diagrams. For the back analysis, the location of a potential slip surface and the interfaces are determined from the positions of the peaks in the geogrid strain diagrams and the shapes of surface displacements diagram shown in Fig. 5. The authors used a two-wedge failure mechanism with a vertical smooth interface between the wedges for the analysis of wall stability by limit equilibrium method. They calculated the safety factor at the end of sudden water decrease FS=1.9 if the ®ll cohesion was taken into account and less than 1 if it was ignored using the equilibrium of horizontal forces only. When the failure mechanism shown in Fig. 5 by thick dashed line is considered and silty sand cohesion is ignored then the exten-
266
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Fig. 5. Case 2 Ð geometry, ground water level, measured geogrid strain and wall displacements.
ded method gave Fave =1.36. More detailed results of the latter analyses are shown in Table 3. From this table it can be seen that the local factors of safety at the failure surface are all equal to 1 and that the Favr >1 is the result of larger factors of safety along the interfaces. 3.4. Case 3Ðlarge scale model of a reinforced retaining wall Thamm et al. [9] described the results of measurements and calculations performed for a 3.6 m high retaining wall loaded by top surcharge to failure. The soil used for the ®ll was a gravely sand with density of 1.95 g/cm3 and the angle of internal friction 0 =39 determined from direct shear tests within the range of normal stresses from 40 to 120 kPa. The strain at failure of 2% and the exponent k=0.5 for gravelly sand were adopted based on the experience. The reinforcement was combined of two dierent geogrids. The main reinforcement was made of Tensar SR2 geogrid with the peak strength of 67 kN/m at the axial strain of 11%. The geogrid was 2.7 m long, placed in three bottom layers at the Table 3 Detailed results of the analyses No.
c0 (kPa)
N (kN)
T (kN)
M/Na (m)
T/N
Fs,1
"e (%)
1 2 3 4 5 6 7
0 40 35 0 40 28 40
17.5 153 69 251 80 134 169
7.5 51 30.6 155.5 50 61 45.5
0 0 0 0.08 1.73 2.36 0.78
0.43 0.33 0.44 0.62 0.62 0.46 0.27
1 1 1 1 1.96 1 3.06
64.4 19.6 19.6 47.6 3.6 14 1.5
a
M/N is the eccentricity of the resultant force acting on a surface.
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spacing of 0.8 m and the two top layers at the spacing of 1 m. The additional reinforcement was 0.5 m long Tensar SS2 grid (with the peak strength of 31,5 kN/m at the strain of 10.5%) near the wall surface, which was cascaded with the steps 0.2 m at each geogrid level. These two geogrids were connected by a HDPE rope. TERRAM 2000 nonwoven geotextile was placed along the wall face to prevent the loss of soil from the part of the wall. The loading was applied over a reinforced concrete slab (2.40.9 m) placed 0.5 m apart from the wall top edge. The authors used two-wedge sliding mechanisms with a vertical interface and a slip circle surface to calculate failure loads of 950, 620, and 1296 kN respectively. The load was applied in steps and the ®nal failure load that caused excessive settlements of the concrete slab and after some time the ®nal failure was 1065 kN. For the load just prior the failure of 1026 kN, the geogrid strain reached the maximum value of 2.9% in the top reinforcement. For the lower two geogrid layers the maximum measured strains were about 1.7% (layer three) and 0.9% (layer two). The failure strain reached was 5%. The shapes of the geogrid strain at dierent load steps were very similar. The graphs of geogrid strain at the load level of 1026 kN together with the wall horizontal strains are shown in Fig. 6. The position of potential failure surface was inferred from the peaks in the strain diagrams and coincides well with the measured pattern of wall strain. The position of the interface is adopted under assumptions that the interface would not cross the reinforcement, which would provide greater resistance to shearing and prevent the establishment of the minimum potential energy at failure. The apparent cohesion for the sides No. 1 and 2 was 67 kPa while for all the other sides 0 kPa. For the load of 1026 kN, the extended method indicated failure of the wall.
Fig. 6. Case 3 Ð geometry, load, measured geogrid strain and wall displacements.
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4. Conclusion From preceding examples it is evident that the positions of the peak values of strain in the geogrids are aligned along distinct and almost planar interfaces. From Figs. 3±6 it can be seen that the smaller spacing between the reinforcement provides smoother slip surfaces and that typical shapes for the steep slopes were circular while for the retaining walls were ``y'' shaped wedges with inclined interfaces. The extended method with introduced force-strain relationships could provide more realistic solutions than classical limit equilibrium methods, which cannot account for potential progressive failure in brittle soil-geogrid composites. These solutions remain only approximate due to the need of introduction of various assumptions such as that the resultant forces along bases of wedges except the last base act in the middle of the bases, that volumetric strains caused by axial stresses are zero and that the shape of shear stress±strain function is independent of the axial stress level. This later assumption can be avoided on account of introduction of an additional iterative procedure or the eect of the assumption can be decreased if axial stress level dependent zonation is introduced. The method is simple to use and requires less input data than complete stress± strain methods. It inherited the same disadvantages of conventional limit equilibrium methods that a sliding mechanism must be assumed to govern slope stability. References [1] Jewel RA. Soil reinforcement with geotextiles. CIRIA Special Publication 1996;123:45±6. [2] Srbulov M. A simple method for the analysis of stability of slopes in brittle soil. Soils and Foundations 1995;4:123±7. [3] Srbulov M. On the in¯uence of soil strength brittleness and nonlinearity on slope stability. Computers and Geotechnics 1997;20(1):95±104. [4] Srbulov M. Force-strain compatibility for reinforced embankments over soft clay. Journal Geotextiles and Geomembranes 1999;17(3):147±56. [5] Ingold TS. Reinforced Earth. London: Thomas Telford Ltd, 1982 9±10. [6] Long NT, Guegan Y, Legeay G. Etude de la terre armee a l'appareil triaxial. Rapport de Recherche 1972;17:LCPC 6. [7] Jewel, R.A., Strength and deformation in reinforced soil design. In: Hoedt D., editor. Proc. 4th International Conference on Geotextiles, Geomembranes and Related Products. The Hague, Netherlands, Den Hoedt (Ed.), Balkema, Rotterdam, 1990, Vol. 3, 913±946. [8] Kutara K, Miki H, Kudoh K, Nakamura K, Minami T, Iwasaki K, Nishimura J, Fukuda N, Taki M. Experimental study on prototype polymer grid reinforced retaining wall. In: Hoedt, D. (Ed.), Proc. 4th International Conference on Geotextiles, Geomembranes and Related Products. The Hague, Netherlands, Balkema, Rotterdam, 1990, 1, 73±78. [9] Thamm BR, Krieger B,. Krieger J. Full-scale test on a geotextile reinforced retaining structure. In: Hoedt, D. (Ed.), Proc. 4th International Conference on Geotextiles, Geomembranes and Related Products. The Hague, Netherlands, Balkema, Rotterdam, 1990, Vol. 1, 3±8.
Analyses of stability of geogrid reinforced steep slopes and retaining walls M. Srbulov * SAGE Engineering Ltd, Newark House, 26±45 Cheltenham Street, Bath BA2 3EX, UK Received 9 February 2000; received in revised form 15 November 2000; accepted 17 November 2000
Abstract The results of measurements of axial strains in geogrids of two reinforced steep slopes and two retaining walls were uniformly interpreted. The stabilities of slopes and walls are analyzed using a method based on limit equilibrium. The method of analysis takes into account strains along boundaries of rigid wedges in addition to the forces considered by classical methods of limit equilibrium. However, the results obtained by the method remain only approximate due to necessity to introduce a number of simplifying assumptions. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Stability analysis; Geogrid; Reinforced; Slopes; Retaining walls
1. Introduction The use of natural ®bres to improve small tensile strength of in situ ground and man made ®lls date to pre-historic times. However, the rational approach to the problem of soil strength improvement by reinforcement is rather recent. In the late ®fties, Henri Vidal, a French architect and engineer, launched a new civil engineering material known as Reinforced Earth. With rapid development of technology, the initial metal strips are replaced by geotextiles and geogrids, which are almost exclusively used these days. Jewel [1] presented a rather comprehensive review of the methods used for the design of reinforced soil. The potential failure surfaces, most widely used for the analysis of internal equilibrium of steep reinforced soil slopes and walls in connection with the limit equilibrium * Tel.: +44-1225-486-500; fax: +44-1225-486-597. E-mail address: milutin@sage- uk.com 0266-352X/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-352X(00)00032-X
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method, include the logarithmic spiral and two-part wedge mechanism. These two failure mechanisms were found to be the simplest and at the same time the most appropriate representatives of the true failure modes throughout the reinforced part. These ®ndings will be re-examined in this paper for a number of cases of instrumented geogrid reinforced steep slopes and retaining walls. 2. Description of the extended method Conventional limit equilibrium methods use a constant factor of safety FS along a potential slip surface (and sometimes interfaces) under the assumption that soil strength is mobilized at all places at the same (or similar) shear strains. Factor of safety FS is usually de®ned as the ratio between available shear strength a and the shear stress e necessary to maintain limit equilibrium FS a = e
1
a is equal to the peak strength p when FS >1 or to the post-peak strength if yielding occurs. Using Mohr±Coulomb failure criterion, which relates shear and compressive stresses (Fig. 1a), FS can be expressed in terms of these stresses FS c tan e
2
Fig. 1. (a) Shear strength versus compressive stress, (b) shear stress versus shear strain, (c) speci®c thickness change versus shear strain.
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where c is cohesion and angle of soil internal friction. Eq. (2) can be written in terms of normal N and shear T force acting on a particular surface FS cb Ntan =T
3
where b is width of the surface. Knowing one of the components of the resultant forces acting along a slip surface and a constant FS, 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. However, shear strains are seldom uniform in the ®eld even within a homogeneous soil and at relatively small stress levels. When soil shear strength at large shear strain is smaller than its peak value then localized and propagating (progressive) failure may occur if induced shear strains are large enough. Similarly, if soil is heterogeneous and some parts mobilize and loose their peak shear strength while the other parts are still on the way to mobilize the peak strength then again localized and progressive failures may occur. The problem of analysing progressive failure may be avoided if only soil residual strength is considered. Reinforced soil also experiences a progressive failure. However, when soil reinforcement is loaded beyond its ultimate strength it tends to break and completely loose all the strength as its residual strength is zero. In the later case, the use of residual strength only is pointless because it would lead to consideration of unreinforced soil. Dierent methods have been proposed to solve the problem of propagating (progressive) failure using complete stress±strain solutions by ®nite and discrete/distinct elements. This article describes a procedure for the consideration of the local and progressive failures within the framework of the limit equilibrium method when applied to analysis of stability of reinforced steep slopes and retaining walls. Srbulov [2] used the method to analyze the progressive failures of actual but unreinforced slopes in brittle soil. It has been shown that the extended method predicted well the instability of the slopes, while a classical method provided FS much greater than 1 for the peak shear strength and much smaller than 1 for the residual shear strength alone. The activation of shear stresses a ; e is accompanied by development of shear strains (Fig. 1b) and, therefore, shear stress/strength can be expressed as a function of both compressive stress and shear strain . The function can be determined from soil shear tests. In its simplest form, when only the shape but not the value of function is assumed independent of (Fig. 1b), the function becomes C k
4
where C, k41 are soil constants determined by curve ®tting from laboratory test results. This assumption can be considered reasonable for a rather large stress range only for cohesive soil under undrained conditions and reinforced soil when reinforcement strength has a greater in¯uence on the behaviour of composite material. The stress dependent shapes of the function may be introduced but on
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account of the use of an additional iterative procedure that must be applied until the dierences between initially assumed stress levels are close to the calculated s stress levels within desired tolerance. An alternative approach would be division of a soil zone into subzones each corresponding to appropriate stress level as it has been done in the case when a nonlinear shear strength envelope is linearized within chosen stress intervals. Using Eq. (4), factor of safety at the surface i can be written in the form k Fi i;ak = i;e
5
k is the shear strain corresponding to available shear stress a at a surface i where i;a k and i;e is the shear strain corresponding to mobilized shear stress e at the surface i. Similarly at another surface j, k1 k1 Fj j;a = j;e
6
where the exponent k1 is dierent from k in the case when dierent soil types exist at the places i and j. From Eqs. (5) and (6) it follows k1 k k k1 = i;a
i;e = j;e Fj Fi j;a
7
Unknown Fi can be determined from available equilibrium equations similarly to a constant FS in conventional methods. Mobilized shear strains can be determined indirectly from Eq. (6)
j;e j;a Fj
l=kl
8
kl k The values of j;a and j;a can be determined from soil shear strength tests. It should k1 k k k1 be noted that the ratios j;a = i;a and i;e = j;e but not particular values of shear strains are necessary to calculate the local factor of safety at any surface j. This has several useful implications. If soil is tested in a direct shear apparatus, for example, then the ratio of shear strains at available strength can be replaced by the ratio of measured k horizontal displacements k1 j;a =i;a instead. If soil shear tests are performed using a triaxial apparatus then the results are presented as the deviatoric stress versus axial strain". If the properties of all soil types involved in an analysis are measured and presented consistently in terms of axial strain than it is possible to use the ratio k k1 k "k1 j;a ="i;a in place of the ratio j;a = i;a The reinforcement strength is usually described as a function of axial strain. If a soil zone is divided into wedges and the wedges between their boundaries do not change their volumes (they are rigid) except that some local yielding is allowed k k1 at their tips then the ratio j;e = i;e between the magnitudes of actual shear strains at two boundaries (i, j) will be the same to the ratio ki;e =k1 j;e of magnitudes of potential shear strains along these boundaries, because and will be directly proportional ( B, where B is an unknown constant). For rigid wedges, tangential displacements along a particular boundary will be constant. In undrained conditions, there
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will be no volumetric changes along wedge boundaries and potential shear strain can be de®ned as the ratio between potential tangential displacement along the boundary and the boundary thickness. For a unit thickness, the shear strains will be equal to the tangential displacements. Potential tangential displacements, illustrated in Fig. 2 for the case of zero soil volume change in undrained conditions, can be de®ned starting with a unit tangential displacement along the base of the ®rst wedge. Proceeding along the external boundaries so that the known tangential displacement is the vectorial sum of the displacement in the direction of the interface and base of the following wedge it is possible to de®ne all other potential displacements. The assumption of no soil volume change along a boundary is not correct for drained conditions nor is it correct to assume that the wedges are rigid. However, the volumetric changes due to axial stresses should be small for sti soil and at relatively small stresses acting within steep slopes. When they are likely to be signi®cant such in soft clay and loose sand then the assumption of no volumetric strain due to axial stresses in drained condition is less acceptable. Volumetric strain "v (i.e. speci®c thickness dt) change along the boundaries with shear strain change can be taken into account in the construction of strain diagram in Fig. 2 as the inclinations of strain vectors with respect to the boundaries. From calculated Fj using Eq. (7) it is possible to back calculate j;e using Eq. (8). The function of volumetric strain (speci®c thickness dt) change versus shear strain (Fig. 1c) can be determined from soil shear tests. The angle of inclination j of shear strain vector with respect to the boundary is simply j arctan dtj;e = j;e 9 If local over stressing occurs at face j (which means that Fj tends to become less than 1 and the mobilized strength greater than the peak value, which is impossible)
Fig. 2. Possible tangential displacements 0 s along boundaries of wedges.
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then the yielding, accompanied by an increase in shear strains j;e can be simulated by the increase of j;a in Eq. (7) until Fj becomes equal to 1. Such simulation is necessary because ®xed value j;e , dependent on geometry only, is used instead of actual shear strain j;e . With post peak increase (yielding) of shear strain j;e , brittle soil will soften and, therefore, the corresponding decrease in the shear strength parameters c, with the increase in j;e is taken into account in Eq. (3). A change (increase) in a factor of safety Fj will cause a corresponding change in Tj force and all other forces because of the need to satisfy the equilibrium equations. The eect of the rate of soil shear strength decrease after the peak value has been investigated for two simple wedges before [3]. It has been found that there will be an evident dierence in the calculated factors of safety of the slope stability if soil decreases its peak strength towards the residual value more gradually then abruptly. In the former case the use of an instant jump from the peak to the residual strength is not appropriate. The calculated factors of safety by the extended method and by a classical method were also in a good agreement if soil exhibit an abrupt shear strength drop after the peak. However, the classical method yields smaller factors of safety than the extended method for gradually decreasing post peak shear because the mobilized shear strength is somewhere between its peak and the residual value. Two wedges, non brittle soil with shear strength parameters c=0, 0 24 and soil unit weight of 22 kN/m3 are used for an investigation of the eect of change in volumetric strain (speci®c thickness) with shearing. A function describing the speci®c thickness change with shear strain (Fig. 1c) is adopted in the form which is appropriate for both the contracting and dilating phases of soil volume changes up to its constant value at the steady-state dt 1=2 a ln b
10
where a,b are constants or functions of axial stress which can be determined by ®tting experimental data. The results of computations are shown in Table 1. The eect of volumetric strain changes with shearing is not very important for the majority of slightly dilatant/contractant soil but could be very important for heavily overconsolidated clay and very dense sand. Table 2 contains the list of unknown values, available equations and their numbers for n wedges. It can be noted that the use of the local factors of safety Fj increased the number of unknown values for 2n 2 as well as that the number of available equation types [7] increased for 2n 2 in comparison with the number of unknown values and available equations in conventional methods of limit equilibrium. It is also evident Table 1 The eects on factor of safety of soil dilatation ( ) and contraction (+) with shearing of soil along boundaries of the two wedges Peak volume/peak shear strain ratio Factor of safety
0.6
0.4
0.2
3.17
1.80
1.73
0
+0.2
+0.4
+0.6
1.67
1.63
1.59
1.51
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Table 2 Values, available equations, and their numbers for n wedges Unknown values
Number
Available equations
Number
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 T at bases Shear forces T at interfaces Local factors of safety Fj at bases (except at i) at interfaces
n l
Forces equilibrium in horizontal direction Forces equilibrium in vertical direction Moments equilibrium
n n n
n 1 n 1 n n 1
k1 k k1 = i;a ki;e = j;e : Fj Fi j;a at bases (except at i) at interfaces Tj cj bj Nj tan j =Fj : at bases at interfaces
n 1
Total
7n 3
Total
7n 3
n 1 n 1
n 1 n 1 n
that the positions of normal forces N's 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 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 Fi in the denominators of the coecients of equations. It is possible to apply an iterative procedure by choosing an initial Fi(=1), solve 3n 1 linear equations, check the 3nth equation and gradually change (increasing) Fi in steps until all 3n equilibrium equations are satis®ed to a speci®ed tolerance. Several iterations will be necessary for each step if local yielding occurs and therefore the coecients of the equations must be readjusted. For an unstable wedge assembly, the equilibrium equations cannot be satis®ed and the stepping procedure will continue until permitted by the user. For a stable wedge group, an average factor of safety of the group stability Favr . can be calculated from the formula Favr: a;j bj = a;j bj =Fj 11 where j=1...n; Favr is used for comparison with a constant FS from conventional methods and for the assessment of global stability of a group of wedges. The expression relating forces and shear strains follows from Eqs. (3) and (7) cj bj Nj tan j Tj 12 k Fi k1 j;a = i;a ki;e =k1 j;e k1 k All local Fsj will be the same if soil is homogeneous j;a i;a and the ratios ki;e =k1 j;e 1 at the bases of wedges along a plane or a circular cylinder with the interfaces passing through the centre of the circle.
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When only one layer of reinforcement exists, such as in reinforced embankments, then reinforcement can be treated as a thin soil layer, Srbulov [4]. However, when many reinforced layers are present it is necessary to apply a composite material concept. Ingold [5] referred to the work of Long et al. [6] who observed that above a certain threshold value of applied con®ning pressure in triaxial apparatus there was a constant increase in applied vertical stress at failure in samples with reinforcement at a given tensile strength and spacing, like in Fig. 1a. Failure of the reinforced samples was very brittle (like in Fig. 1b), with a drastic decrease in strength when the peak was passed. The brittleness was less severe at higher applied con®ning pressures or in less heavily reinforced samples. Post-failure inspection of dismantled samples consistently showed that the reinforcement had failed in tension. It was concluded that since, for tensile reinforcement failure, the failure envelopes of both the reinforced and unreinforced sand are parallel, and therefore exhibit the same angle of internal shearing resistance, the additional strength imparted by the reinforcement could be represented by an apparent cohesion. In addition, the orientation of reinforcement is likely to change with respect to a wedge boundary, from as-built position to an almost parallel orientation at the failure. Jewel [7], proposed that the improvement in shearing resistance Ps, resulting from a reinforcement force, Pr, can be expressed by the equilibrium equation of forces Ps Pr sin costan 0
13
where is the angle between the reinforcement direction and a normal to the boundary. Eq. (13) also de®nes the degree of anisotropy of shearing resistance of reinforced soil with respect to the angle . For 0 close to 30 it follows from the above equation that almost a constant tensile force greater or equal to Pr acts for the range of between 30 and 90 . In the examples which follow, it will be assumed that an apparent cohesion is given by the expression c Pr =b
14
where is applied to all reinforced layers crossing a particular considered wedge boundary if the distance to the end of reinforcement is sucient for activation of reinforcement tensile strength. It also means that reinforced soil shear strength is considered isotropic, independent of the angle . 3. Case histories 3.1. Case 1.1Ðlarge scale model of a reinforced slope Kutara et al. [8] described a large-scale model of six meter high instrumented embankment, with the slope inclination of 1 vertical to 1 horizontal, made of compacted silty sand, reinforced by geogrid. The embankment was submerged and the
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water level was rapidly lowered in order to estimate the embankment stability using a rotational slip surface and limit equilibrium method. The embankment sand contained 30% of silt. The shear strength parameters obtained by triaxial compression test and used in the stability analysis by the authors were c0 =10 kPa and 0 32 . However, the test results have not been presented in the paper and the cohesion intercept may be a result of the envelope curvature or the way of interpretation and will not be used for the back analysis. The dry unit weight was 13 kN/m3 and the relative density of 87%. The reinforcement used was Tensar SS2 biaxially stretched polymer grid, placed in four 6 m long layers at spacing of 1.5 m. The tensile strength speci®ed in the paper is 15 kN/m at the peak axial strain of 5%. The slope surface was encapsulated with the grid. The behaviour of the embankment and the geogrid was monitored by strain gages attached to the geogrids at interval of 0.5 m, by measurement of the settlement of the embankment crest and its interior, by the measurement of horizontal and vertical strains of the slope and by the measurement of water level inside the embankment. In addition a surcharge was placed on top of the embankment. The surcharge was placed at a distance of 4 m from the slope crest. A 3 m high ®ll made of river sand was used as a surcharge of 43.5 kN/m2. The submergence of the embankment was achieved by water up to the height of 5.25 m. The water level had been rapidly lowered with the rate of 0.75 m/h in front of the slope after the deformation and the strain of the geogrid ceased to increase through several days of submergence. The measured geogrid strain and embankment settlement at the end of rapid water level decrease in front of embankment are shown in Fig. 3. The authors also presented the strain diagrams for the construction phase, submergence and dierent water level fall cases. Their shapes generally coincide with the ®nal shapes of the strain diagrams. The face of slope also bulged greatly between the geogrid layers up to the maximum of 38 cm.
Fig. 3. Case 1.1 Ð geometry, ground water level, measured geogrid strain and embankment settlement.
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Using a rotational slip surface and the equilibrium of moments, the authors obtained the safety factor of 1.62 at the end of embankment construction, 1.32 at the end of surcharge addition, 1.38 at the end of submergence and 0.97 at the end of sudden water drawdown. However, they did not give details about the method of stability analysis and the analyzed critical slip surface. For the stability analysis by the method presented in this paper, the location of a potential slip surface and the interfaces are determined from the positions of the peaks in the geogrid strain diagrams and the shape of top settlement diagram shown in Fig. 3. An almost rotational slip surface shape with inclined interfaces was inferred from Fig. 3. The value of axial strain at failure of 5% and the exponent k=0.5 are adopted for silty sand based on experience. The eect of the assumptions made appeared to be of no great in¯uence on the results. The exponent k=0.5 for the composite material (reinforced soil) was adopted based on the tensile strength-strain diagram shown in the paper. The apparent cohesion used for the sides No. 1, 5, 6 was 0 kPa, No. 2, 4 was 10 kPa, No. 3, 11 was 5 kPa, No. 7, 8, 10 was 4 kPa and No 9 was 8.5 kPa. The unit weight of soil used was 19 kN/m3. Calculated Favr =1.04 by the extended method is close to one and the real slope state. The local factors of safety along the boundaries 1±6 are all equal to 1 and the average factor slightly greater than one is the result of the local factors of safety along the interfaces (Fs,7=1.5, Fs,9=1.04, F1,11=1.09). Mobilized axial strains are calculated according to the equation "j;e "j;a Fi
1=k1
15
At the boundaries 1±4 they are all 17.3% and at the boundaries 5±6 they are 19.4%. However, the corresponding axial displacements can not be calculated due to the lack of a referent length. 3.2. Case 1.2Ðlarge scale model of a steep reinforced slope This case was described by the same authors as Case 1.1. The only dierence was that the geogrid spacing was 0.75 m and that seven layers were used instead of four. As a result of denser reinforcement, the maximum bulging of the embankment slope between the reinforcement after rapid water level draw-down was 25 cm instead of 38 cm and the top settlement was more uniform. Also, the authors of the paper reported the safety factor of 1.7 at the end of embankment construction, 1.38 at the end of surcharge addition, 1.44 at the end of submergence and 1.01 at the end of sudden water draw-down. They gave no details about the rotational slip surface and the method used for the calculation of factor of safety. The position of potential slip surface and the interfaces was inferred from the peaks of measured geogrid strain diagrams and the shape of top settlements shown in Fig. 4. From Fig. 4, it can be seen that the geogrid strain is sometimes uniform over a considerable length. This is the result of superposition of strain when several slip surfaces cross the geogrid at rather close distances. The apparent cohesion used for the sides No. 1, 6, 7, 8 was 0 kPa, No. 2, 5, 9, 15 was 15 kPa, No. 3, 4, 10, 14 was
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Fig. 4. Case 1.2 Ð geometry, ground water level, measured geogrid strain and embankment settlement.
13 kPa, No. 11, 13 was 11kPa and No. 12 was 15kPa. The extended method failed to satisfy equilibrium conditions. 3.3. Case 2Ðlarge scale model of a reinforced retaining wall The same authors (Kutara et al., [8]) investigated and presented the results of measurements and stability analyses for a retaining wall 6 m high, with slope inclined 1 vert. to 0.2 hor. with precast concrete panels over the surface. The panels were joined to the reinforcement by reinforcing bars on the rear sides. The ®ll soil was silty sand as for the slopes. The reinforcement used was Tensar SR55 uniaxially stretched polymer grid, placed in six 4 m long layers at spacing of 1 m. The tensile strength speci®ed in the paper is 55 kN/m at the peak axial strain of 14%. To strengthen the wall face, the reinforcement was interposed by reinforcing material of the same type with length of 1 m. The results of measured geogrid strain and the surface strains are shown in Fig. 5. The authors also presented the strain diagrams for the construction phase, submergence and dierent water level fall cases. Their shapes generally coincide with the ®nal shapes of the strain diagrams. For the back analysis, the location of a potential slip surface and the interfaces are determined from the positions of the peaks in the geogrid strain diagrams and the shapes of surface displacements diagram shown in Fig. 5. The authors used a two-wedge failure mechanism with a vertical smooth interface between the wedges for the analysis of wall stability by limit equilibrium method. They calculated the safety factor at the end of sudden water decrease FS=1.9 if the ®ll cohesion was taken into account and less than 1 if it was ignored using the equilibrium of horizontal forces only. When the failure mechanism shown in Fig. 5 by thick dashed line is considered and silty sand cohesion is ignored then the exten-
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Fig. 5. Case 2 Ð geometry, ground water level, measured geogrid strain and wall displacements.
ded method gave Fave =1.36. More detailed results of the latter analyses are shown in Table 3. From this table it can be seen that the local factors of safety at the failure surface are all equal to 1 and that the Favr >1 is the result of larger factors of safety along the interfaces. 3.4. Case 3Ðlarge scale model of a reinforced retaining wall Thamm et al. [9] described the results of measurements and calculations performed for a 3.6 m high retaining wall loaded by top surcharge to failure. The soil used for the ®ll was a gravely sand with density of 1.95 g/cm3 and the angle of internal friction 0 =39 determined from direct shear tests within the range of normal stresses from 40 to 120 kPa. The strain at failure of 2% and the exponent k=0.5 for gravelly sand were adopted based on the experience. The reinforcement was combined of two dierent geogrids. The main reinforcement was made of Tensar SR2 geogrid with the peak strength of 67 kN/m at the axial strain of 11%. The geogrid was 2.7 m long, placed in three bottom layers at the Table 3 Detailed results of the analyses No.
c0 (kPa)
N (kN)
T (kN)
M/Na (m)
T/N
Fs,1
"e (%)
1 2 3 4 5 6 7
0 40 35 0 40 28 40
17.5 153 69 251 80 134 169
7.5 51 30.6 155.5 50 61 45.5
0 0 0 0.08 1.73 2.36 0.78
0.43 0.33 0.44 0.62 0.62 0.46 0.27
1 1 1 1 1.96 1 3.06
64.4 19.6 19.6 47.6 3.6 14 1.5
a
M/N is the eccentricity of the resultant force acting on a surface.
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spacing of 0.8 m and the two top layers at the spacing of 1 m. The additional reinforcement was 0.5 m long Tensar SS2 grid (with the peak strength of 31,5 kN/m at the strain of 10.5%) near the wall surface, which was cascaded with the steps 0.2 m at each geogrid level. These two geogrids were connected by a HDPE rope. TERRAM 2000 nonwoven geotextile was placed along the wall face to prevent the loss of soil from the part of the wall. The loading was applied over a reinforced concrete slab (2.40.9 m) placed 0.5 m apart from the wall top edge. The authors used two-wedge sliding mechanisms with a vertical interface and a slip circle surface to calculate failure loads of 950, 620, and 1296 kN respectively. The load was applied in steps and the ®nal failure load that caused excessive settlements of the concrete slab and after some time the ®nal failure was 1065 kN. For the load just prior the failure of 1026 kN, the geogrid strain reached the maximum value of 2.9% in the top reinforcement. For the lower two geogrid layers the maximum measured strains were about 1.7% (layer three) and 0.9% (layer two). The failure strain reached was 5%. The shapes of the geogrid strain at dierent load steps were very similar. The graphs of geogrid strain at the load level of 1026 kN together with the wall horizontal strains are shown in Fig. 6. The position of potential failure surface was inferred from the peaks in the strain diagrams and coincides well with the measured pattern of wall strain. The position of the interface is adopted under assumptions that the interface would not cross the reinforcement, which would provide greater resistance to shearing and prevent the establishment of the minimum potential energy at failure. The apparent cohesion for the sides No. 1 and 2 was 67 kPa while for all the other sides 0 kPa. For the load of 1026 kN, the extended method indicated failure of the wall.
Fig. 6. Case 3 Ð geometry, load, measured geogrid strain and wall displacements.
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4. Conclusion From preceding examples it is evident that the positions of the peak values of strain in the geogrids are aligned along distinct and almost planar interfaces. From Figs. 3±6 it can be seen that the smaller spacing between the reinforcement provides smoother slip surfaces and that typical shapes for the steep slopes were circular while for the retaining walls were ``y'' shaped wedges with inclined interfaces. The extended method with introduced force-strain relationships could provide more realistic solutions than classical limit equilibrium methods, which cannot account for potential progressive failure in brittle soil-geogrid composites. These solutions remain only approximate due to the need of introduction of various assumptions such as that the resultant forces along bases of wedges except the last base act in the middle of the bases, that volumetric strains caused by axial stresses are zero and that the shape of shear stress±strain function is independent of the axial stress level. This later assumption can be avoided on account of introduction of an additional iterative procedure or the eect of the assumption can be decreased if axial stress level dependent zonation is introduced. The method is simple to use and requires less input data than complete stress± strain methods. It inherited the same disadvantages of conventional limit equilibrium methods that a sliding mechanism must be assumed to govern slope stability. References [1] Jewel RA. Soil reinforcement with geotextiles. CIRIA Special Publication 1996;123:45±6. [2] Srbulov M. A simple method for the analysis of stability of slopes in brittle soil. Soils and Foundations 1995;4:123±7. [3] Srbulov M. On the in¯uence of soil strength brittleness and nonlinearity on slope stability. Computers and Geotechnics 1997;20(1):95±104. [4] Srbulov M. Force-strain compatibility for reinforced embankments over soft clay. Journal Geotextiles and Geomembranes 1999;17(3):147±56. [5] Ingold TS. Reinforced Earth. London: Thomas Telford Ltd, 1982 9±10. [6] Long NT, Guegan Y, Legeay G. Etude de la terre armee a l'appareil triaxial. Rapport de Recherche 1972;17:LCPC 6. [7] Jewel, R.A., Strength and deformation in reinforced soil design. In: Hoedt D., editor. Proc. 4th International Conference on Geotextiles, Geomembranes and Related Products. The Hague, Netherlands, Den Hoedt (Ed.), Balkema, Rotterdam, 1990, Vol. 3, 913±946. [8] Kutara K, Miki H, Kudoh K, Nakamura K, Minami T, Iwasaki K, Nishimura J, Fukuda N, Taki M. Experimental study on prototype polymer grid reinforced retaining wall. In: Hoedt, D. (Ed.), Proc. 4th International Conference on Geotextiles, Geomembranes and Related Products. The Hague, Netherlands, Balkema, Rotterdam, 1990, 1, 73±78. [9] Thamm BR, Krieger B,. Krieger J. Full-scale test on a geotextile reinforced retaining structure. In: Hoedt, D. (Ed.), Proc. 4th International Conference on Geotextiles, Geomembranes and Related Products. The Hague, Netherlands, Balkema, Rotterdam, 1990, Vol. 1, 3±8.