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Overall stability of geosynthetic-reinforced embankments on soft soils
Geotextiles and Geomembranes 20 (2002) 395–421
Overall stability of geosynthetic-reinforced embankments on soft soils ! Jose! Leit*ao Borges*, Antonio Silva Cardoso Department of Civil Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Received 12 September 2001; received in revised form 3 March 2002; accepted 11 June 2002
Abstract Overall stability of geosynthetic-reinforced embankments on soft soils is analysed using two different methodologies: application of a numerical model based on the finite element method; use of a limit equilibrium method. These two methodologies are described and also applied on three geosynthetic-reinforced embankments on soft soils. One of the cases is a case history constructed up to failure. Considering the analysis of the results, some conclusions are formulated on the limit equilibrium method accuracy, namely regarding the critical slip surface, overall safety factor and overturning and resisting moments. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Reinforced embankment; Geosynthetic; Overall stability; Limit equilibrium; Finite element analysis
1. Introduction To design embankments on soft soils it is essential to take into account the multiple constructive techniques that allow to solve the problems usually associated with this kind of construction: overall stability deficiency and large settlements that develop slowly. The constructive solutions—usually based on both foundation soil properties improvement and construction procedures or fill properties alteration—provide one or more of the following effects: increase of overall stability, consolidation acceleration and decrease of long term settlements. *Corresponding author. Tel.: +351-22508-1928; fax: +351-22508-1440. E-mail address: [email protected] (J.L. Borges). 0266-1144/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 6 - 1 1 4 4 ( 0 2 ) 0 0 0 1 4 - 6
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With textile industry development in the 20th century, especially following the 1960s, geosynthetic reinforcement has been added to the list of possible solutions when embankments must be constructed on very soft foundations. In many cases, the use of one geotextile or geogrid can significantly increase the safety factor, improve performance in terms of displacements and reduce costs in comparison with more conventional solutions. Especially due to their simplicity, overall stability of reinforced embankments on soft soils is usually computed by limit equilibrium methods along potential slip circles. The reinforcement effect is considered by a resisting force due to the geosynthetic. Theoretically, however, because rigid–plastic behaviour is tacitly assumed for the materials (soil and reinforcement), the use of these methods may raise some reticences, as strains are not taken into account before overall failure, as well as stress redistribution caused by the geosynthetic. This often determines the use of the finite element method in the study of geosynthetic-reinforced embankments (Rowe, 1984; Humphrey, 1986; Soderman, 1986; Rowe and Soderman, 1987; Kwok, 1987; Rowe and Mylleville, 1990; Mylleville and Rowe, 1991; Russell, 1992; Borges, 1995). In the paper, overall stability of geosynthetic-reinforced embankments on soft soils is analysed adopting two different methodologies: (i) using the results obtained from a numerical model based on the finite element method (Borges, 1995); along each analysed slip circle, acting and resisting tangential forces are obtained from numerical results and strength characteristics of the materials; (ii) applying a limit equilibrium method, based on the formulation proposed by Kaniraj and Abdullah (1993) but with some improvements (see Section 3) performed by Borges (1995). The two methodologies are also applied on three geosynthetic-reinforced embankments on soft soils. The third embankment is a case history that was constructed up to failure (Quaresma, 1992).
2. Overall stability analysis using results from finite element method application 2.1. General characteristics of the finite element model Numerical analysis is simulated by a model, developed by Borges (1995), based on the finite element method, with the following theoretical hypotheses: (a) plane strain conditions; (b) coupled formulation of the flow and equilibrium equations, considering soil constitutive relations formulated in effective stresses (Biot’s consolidation theory) (Lewis and Schrefler, 1987; Britto and Gunn, 1987); this formulation is applied to all phases of the problem, both during the embankment construction and in the post-construction period; (c) utilisation of the p2q2y critical state model (Lewis and Schrefler, 1987; Britto and Gunn, 1987; Borges, 1995; Borges and Cardoso, 1998) to simulate constitutive behaviour of the foundation and embankment soils (see below); (d) utilisation of a hardening elasto-plastic model to simulate constitutive behaviour of the reinforcement; (e) simulation of constitutive behaviour of the soil–geosynthetic interfaces using a hardening elasto-plastic model.
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Geosynthetic and soil–geosynthetic interfaces are modelled by bar and interface elements, respectively. For 2D elements, triangular elements are used, namely sixnoded elements (at the vertices and middle of the edges) with 12 displacement degrees of freedom (for fill elements) or with three more excess pore pressure degrees of freedom at the vertices of the triangle (for foundation elements where consolidation analysis is considered). In order to verify accuracy of the numerical model used in this paper, Borges (1995) compared numerical and field results of two reinforced embankments on soft soils, one constructed up to failure (Quaresma, 1992) and the other observed until the end of the consolidation (Yeo, 1986). The accuracy was considered adequate in both cases, as numerical and field results are similar, namely in terms of settlements, pore pressures and strains or tensile forces in the reinforcement. Only some quantitative differences were observed in the horizontal displacements, despite an overall qualitative similarity too. 2.2. Evaluation of safety factors Using effective stresses obtained from numerical model application, one can compute overall safety factor and also partial safety factors of the different materials (reinforcement and foundation and fill soils) as described below (Borges, 1995; Borges and Cardoso, 1997). Firstly, for each potential slip circle, the intersection points of the circle with the edges of the finite elements of the mesh are determined. Therefore, the slip circle is divided into small line segments, each of them located inside of only one of the finite elements of the mesh. Afterwards, the average values of effective stresses (s0x ; s0y and txy ; normal and shear stresses in the directions of x- and y-axes) at each of those segments are computed extrapolating from the stresses at the Gauss points of the corresponding finite element. Mathematical procedures for this extrapolation depend on the type of the 2D finite element, i.e. position and number of nodes and Gauss points. For the 2D element used in this study—six-noded triangular element with seven Gauss points (see Fig. 1), whose strains vary linearly within the element—the following scheme was assessed adequate: (i) firstly, stresses at each node of the element are computed by linear extrapolation from two points—the
Node Gauss point
Line segment of the slip surface Fig. 1. 2D finite element.
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Gauss point at middle of the element and the Gauss point that is nearest the node; (ii) secondly, stresses at each extremity of the segment line of the slip circle are computed by linear interpolation from stresses at the two nearest nodes of the element; (iii) finally, stresses at each segment are the average values obtained from stresses at the two extremities of the segment. Considering the slip circle divided into line segments, partial safety factor of the soil along the slip surface (foundation and fill) is defined by PN qfi li Fs ¼ Pi¼1 ; ð1Þ N i¼1 ti li where li is the i-segment length; N is the number of 2D elements of the mesh intersected by the circle; ti is the average value of shear stress at i-segment (determined from effective stresses s0x ; s0y and txy ; known the angle that defines i-segment inclination); qfi is the average value of soil shear strength at i-segment. Limiting the sum only to the segments in the foundation or in the fill, the partial safety factors for corresponding soils along the slip circle can also be obtained with Eq. (1). Once the finite element method model uses a constitutive model, p2q2y critical state model, that formulates the soil strength or failure (critical state) by the Mohr– Coulomb criteria (see Fig. 2), soil shear strength at i-segment, qfi ; can be estimated by the equation qfi ¼ c0i cos f0i þ p0i sin f0i ;
ð2Þ stress and c0i and f0i are effective terms (s01i ; s03i
where p0i ¼ ðs0xi þ s0yi Þ=2 ¼ ðs01i þ s03i Þ=2 is effective volumetric the cohesion and friction angle of soil at i-segment defined in are the principal effective stresses; s0xi ; s0yi are the normal effective stresses in x- and y-directions).
Fig. 2. Yield and critical state surfaces of p2q2y critical state model in principal effective stress space.
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As shown by Borges and Cardoso (2001) and Borges (1995), if consolidation is not significant, the typical effective stress path in the foundation corresponds to low variations of p0 ¼ ðs01 þ s03 Þ=2 with significant increases of q ¼ ðs01 s03 Þ=2: As an approximation, p0 can be considered with the same value at acting stress state (obtained from finite element analysis) and at failure stress state, i.e. the corresponding Mohr circles can be considered concentric. Soil shear strength, qf ; is the radius of the failure Mohr circle, given by Eq. (2), as deduced in Fig. 3. The partial safety factor of the geosynthetic is determined by the equation Fg ¼
Tr ; Ta
ð3Þ
where Ta is the geosynthetic acting tensile force at the cut point (point I in Fig. 4), intersection of the slip circle with the geosynthetic (or geosynthetics, if there are several reinforcement layers) and Tr is the corresponding resisting tensile force. Ta is obtained from the numerical results interpolating from tensile stresses at the two nearest Gauss points of the bar element that contains point I: Tr is given by Tr ¼ minðTrg ; Trp Þ
τ
ð4Þ
τ =c'+σ'n tanφ'
qf =
(σ'n , τ)
qf c'
φ'
c'/tanφ'
σ'3f
σ'3
p'= (σ'1+σ'3)/2
p'
σ'1 σ'1f
σ3f
σ1f
p
failure Mohr circle acting Mohr circle
σ'n σn
c′ + p′ sin φ′ = tan φ′
= c′ cos φ ′ + p′ sin φ ′
total stress Mohr circle
Fig. 3. Relation between soil shear strength and effective cohesion and friction angle.
T A
geosynthetic
B
I θ
Fig. 4. Overall stability analysis.
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being Trg and Trp the maximum forces that can be mobilised at point I taking into account the geosynthetic strength and pull-out force of the soil–geosynthetic interfaces, respectively. So, Trg is a property of the geosynthetic and, because equilibrium has to be verified, Trp is calculated by the equation Trp ¼ minðTrp2AI ; Trp2IB Þ;
ð5Þ
where Trp2AI and Trp2IB correspond, respectively, to the integrals of the maximum adhesive and frictional stresses that can be mobilised in the soil–geosynthetic interfaces on the left, segment AI, and on the right, segment IB, from the cut point I (see Fig. 4): Z Trp2AI ¼ ½al þ au þ s0n ðtan dl þ tan du Þ dl; ð6aÞ AI
Trp2IB ¼
Z IB
½al þ au þ s0n ðtan dl þ tan du Þ dl;
ð6bÞ
where al ; dl are the adhesion and frictional angle of the lower interface; au ; du are the adhesion and frictional angle of the upper interface; and s0n is the normal stress acting on the interface. It should be noted that the values of s0n are known at the Gauss points of the elements that model the interface, so Eqs. (6a) and (6b) can be calculated approximately considering linear the variation of s0n between two contiguous Gauss points. This is an adequate approximation because the finite element meshes usually used in this kind of problems must have sufficiently small elements. Finally, overall safety factor is defined as follows: PN g PN i¼1 qfi li þ j¼1 ½Trj cosðcr yj Þ ; ð7Þ F ¼ PN PN g i¼1 ti li þ j¼1 ½Taj cosðcr yj Þ where Ng is the number of reinforcements (in most cases Ng ¼ 1), yj the angle between the reinforcement direction and the tangent to the slip circle at the cut point, cr a reduction coefficient (it varies between 0 and 1) related to the direction considered for the forces Trj and Taj (resisting and acting tensile forces of j-geosynthetic, as defined above), and li ; N; ti and qfi have the same meaning that in Eq. (1). It should be noted that the reinforcement force has been considered to act in its original orientation (usually horizontally) by some investigators (Broms, 1977; Tensar, 1982; Jewell, 1982; Ingold, 1982, 1983; Brakel et al., 1982; Duncan and Wong, 1984; Milligan and La Rochelle, 1984), tangentially to the circle by some others (Haliburton, 1981; Quast, 1983) assuming that the local deformations associated with the formation of failure surface result in a local reorientation of the geosynthetic, and between the above two directions by Huisman (1987). However, as indicated by Soderman (1986), the assumption that provides better agreement with observed embankment performance is, likely, dependent on the particular case. Because the assumption that assumes the geosynthetic remains in its original
J.L. Borges, A.S. Cardoso / Geotextiles and Geomembranes 20 (2002) 395–421 C'
401
C
COMPLEMENTARY SURFACE
INITIAL SURFACE
Fig. 5. Symmetric problem regarding one slip surface that does not include whole continuum.
orientation (cr ¼ 1) is the most conservative assumption, in doubt, one should consider it in design. Sometimes, because the problem is symmetric, the finite element mesh does not include the whole continuum. In this case, symmetry allows solving this question considering a complementary surface, as indicated in Fig. 5. In this second surface one can have all data and results regarding the part of initial surface that is out of the mesh limits.
3. Application of a limit equilibrium method on the overall stability design Most of the design methods used to assess the overall stability of reinforced embankments on soft soils are limit equilibrium methods which use circular failure surfaces and consider a resisting force due to the reinforcement (Rowe, 1984; Jewell, 1982; Ingold, 1982; Haliburton, 1981; Brakel et al., 1982; Fowler, 1982; Milligan and LaRochelle, 1984). Overall safety factor, F ; can be computed by MR F¼ ; ð8Þ MO where MR is the resisting moment and MO the overturning moment, which can be given by (Fig. 6): MO ¼ Wx; MR ¼
X
ð9Þ tri Dli R þ Tg R cosðy aÞ;
ð10Þ
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x
y
R
W
Tg
geosynthetic
α
∆l i τri
θ
Fig. 6. Potential failure surface: rotational stability analysis.
where W is the weight of the fill mass inside the slip circle, tri the resisting shear stress along Dli long arc of the slip circle, and Tg the maximum force that reinforcement can mobilise at the intersection point with the circle (as seen in Section 2.2, this force depends on geosynthetic strength and the pull-out force that can be developed along the soil–geosynthetic interfaces). Alternatively to Eq. (8), in agreement with more recent design methodologies, partial safety factors can be used in overall failure analysis. These methodologies consider: (i) factors that multiply actions, Zg for unit weight of the fill soil and Zn for eventual variable actions; (ii) factors that divide materials properties, Gc and Gf for cohesion and friction angle of the fill soil, Gsu for undrained shear strength of the foundation soil, Gr for reinforcement strength, and Ga and Gd for adhesion and frictional angle of the soil–reinforcement interfaces. As known, using partial factors, one has to verify the inequation: MOd pMRd ;
ð11Þ
where MOd and MRd are, respectively, design overturning and resisting moments, i.e. determined with the above partial factors. Of course, if all partial factors of actions and materials are imposed equal to 1, an overall safety factor can be obtained as defined by Eq. (8). As said in Section 1, a limit equilibrium method is used in this paper, based on the assumptions proposed by Kaniraj and Abdullah (1993) but with some improvements implemented by Borges (1995). These improvements are: (a) introduction of partial safety factors; (b) correct calculation of the moment due to resisting forces in the foundation soil along the slip surface (which is done approximately by Kaniraj and Abdullah (1993), depending on the number of layers that the foundation is divided); (c) calculation of the pull-out force of the geosynthetic taking into account that its
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403
Fig. 7. Slip circle in a reinforced embankment on soft soil (Kaniraj and Abdullah, 1993).
value, due to equilibrium reasons, is the minimum of the two values computed on the left and on the right of the cut point (intersection of the reinforcement with the slip circle). The equations used to calculate MOd and MRd are presented in appendix assuming the geometry as shown in Fig. 7, which shows an arbitrary slip circle in a reinforced embankment. The origin of (X ;Y )-axes is assumed to be at the toe.
4. Overall stability analysis of geosynthetic-reinforced embankments on soft soils 4.1. Description of the illustrative cases In this section overall stability of three geosynthetic-reinforced embankments on soft soils is analysed using the two methodologies described in Sections 2 and 3. Case 1 is a 28-day continuous construction of a 2 m height symmetric embankment, with a 10.6 m crest width and 23 (V/H) inclined slopes. The foundation is a 5 m thick saturated clay layer lying on a rigid and impermeable soil, which constitutes the lower boundary. The clay is lightly overconsolidated to 1.8 m depth and normally consolidated from 1.8 to 5 m. One geosynthetic-reinforcement level is considered between the foundation and embankment soils. Taking into account the high permeability of the geosynthetic and of the embankment soil, the upper foundation surface is a drainage boundary. Fig. 8 shows the finite element mesh used in the numerical analysis; only 2D elements are represented; y-axis is the symmetry line.
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Fig. 8. Finite element mesh for Case 1.
Table 1 Geotechnical properties of the foundation and of the embankment for Case 1 g (kN/m3) n0
Foundation 17 Embankment 20
c0 (kPa) f0 (1) kx (m/s) ky (m/s) p2q2y critical state model
0.25 0 0.30 0
30 35
10 —
9
10 —
9
l
k
G
N
0.22 0.03
0.02 0.005
3.26 1.80
3.40 1.817
Table 2 At rest earth pressure coefficient, k0 ; and over-consolidation ratio, OCR, in the foundation for Case 1 Depth (m)
k0
OCR
0–1 1–1.8 1.8–5
0.7 0.7–0.5 0.5
2.43 2.43–1 1
The constitutive relations of both the embankment and foundation soils were simulated using the p2q2y critical state model (Lewis and Schrefler, 1987; Britto and Gunn, 1987; Borges, 1995; Borges and Cardoso, 1998) with the parameters indicated in Table 1 (l; slope of normal consolidation line and critical state line; k; slope of swelling and recompression line; G; specific volume of soil on the critical state line at mean normal stress equal to 1 kPa; N; specific volume of normally consolidated soil at mean normal stress equal to 1 kPa). Table 1 also shows other geotechnical properties: g; unit weight; n0 ; Poisson’s ratio for drained loading; c0 and f0 ; cohesion and angle of friction defined in effective terms; kx and ky ; coefficients of permeability in x- and y-directions. Table 2 indicates the variation with depth of the at rest earth pressure coefficient, k0 ; and over-consolidation ratio (OCR), in the foundation. All these parameters were defined taking into account typical experimental values for this kind of soils (Borges, 1995; Lambe and Whitman, 1969).
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FORCE (kN/m)
200 150 100 50 0 0
5
10
15
20
ε
STRAIN (%)
(a) τ
σn = 15kPa
SHEAR STRESS (kPa)
10
5
0
(b)
0
1
2 4 6 3 5 7 RELATIVE DISPLACEMENT (mm)
8
∆s
Fig. 9. Constitutive curve for Case 1 of the: (a) geosynthetic; (b) soil–geosynthetic interface for normal stress of 15 kPa.
Fig. 9 shows the constitutive curves of the geosynthetic and soil–geosynthetic interfaces. The geosynthetic thickness is 2 mm and its elastic modulus is 1.5 106 kPa. Normal and tangential stiffnesses of soil–reinforcement interfaces are 2.0 107 and 1.6 104 kPa/m, respectively. Geosynthetic strength is 200 kN/m. Adhesion, a, and frictional angle, d; for both upper and lower soil–geosynthetic interfaces are 0 kPa and 33.71, respectively. Mechanical behaviour of the reinforcement and soil–reinforcement interfaces were simulated in the numerical analysis by elasto-plastic models that incorporate the following hardening law (Prevost and Hoeg, 1975; Owen and Hinton, 1980; Thomas, 1984): Y ðhÞ ¼ c1 þ
c 2 h þ c 3 h2 ; 1 þ c 4 h þ c 5 h2
ð12Þ
where h is the hardening parameter and c1 ; c2 ; c3 ; c4 and c5 are parameters that characterise the material. For the geosynthetic, the yielding function is expressed by f ¼ s Y ðhÞ;
ð13Þ
where s is the tensile stress, Y ðhÞ ¼ sY ðep Þ is the hardening law, h ¼ ep the hardening parameter, sY the yielding stress and ep the plastic tensile strain.
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Table 3 Values of the hardening law parameters for Case 1 c1 Geosynthetic Soil–geosynthetic interface
c2 4
7.5 10 (kPa) 0.333
6
8.824 10 (kPa) 417.094 (m1)
c3
c4
c5
0 (kPa) 0 (m2)
35.29 1251.408 (m1)
0 0 (m2)
Table 4 Shear strength in the foundation for Case 1 Depth from ground surface (m)
Shear strength, qf (kPa)
0 1.0 1.8 5.0
8.49 8.49 4.725 13.125
Shear strength varies linearly between two contiguous shear strength values.
For the soil–geosynthetic interfaces, the yielding function is: f ¼ jtj sn Y ðhÞ;
ð14Þ
where t and sn are, respectively, P the tangential and normal stresses, Y ðhÞ ¼ tan dðsp Þ is the hardening law, h ¼ jsp j the hardening parameter, d the interface friction angle and sp the plastic tangential relative displacement. Table 3 shows parameters c1 ; c2 ; c3 ; c4 and c5 adopted. For the foundation clay, the correspondence between friction angle f0 (used in the numerical analysis) and shear strength qf at the end of the embankment construction (used in the limit equilibrium method) was determined by the application of Eq. (2). As said in Section 2 (Borges and Cardoso, 2001; Borges, 1995), if consolidation is not significant, one can consider approximately p0 ¼ p00 (where p00 is the initial effective volumetric stress), because the typical effective stress path during construction corresponds to low variations of effective volumetric stress with significant increases of shear stress. Being c0 ¼ 0; Table 4 shows the values of shear strength qf in the foundation for Case 1, obtained by the application of the above considerations, except from 0 to 1.0 m depth, where significant increases of effective volumetric stress, by consolidation, were observed during construction (about 14 kPa in average terms, as shown by the element finite analysis). To also consider the consolidation effect in the limit equilibrium method application, shear strength in the foundation, from 0 to 1.0 m, was determined applying Eq. (2) at the middle of the layer (0.5 m depth) with p0 ¼ p00 þ 14 kPa. A transition layer is considered from 1 to 1.8 m. Case 2 is a 28-day continuous construction of a 5 m height symmetric 1 embankment, with a 30.26 m crest width and 1:23 (V/H) inclined slopes. The foundation is a 21 m thick saturated clay layer lying on a rigid and impermeable soil. The clay is lightly overconsolidated to 7 m depth and normally consolidated from 7 to 21 m. The variation with depth of the at rest earth pressure coefficient, k0 ; and OCR in the foundation are indicated in Table 5. One geosynthetic-reinforcement
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Table 5 At rest earth pressure coefficient, k0 ; and over-consolidation ratio, OCR, in the foundation for Case 2 Depth (m)
k0
OCR
0–4.5 4.5–7.0 7.0–21.0
0.7 0.7–0.5 0.5
2.43 2.43–1 1
Symmetry line
0
5m
Fig. 10. Finite element mesh for Case 2.
level is considered in the embankment soil at 0.9 m height. Geotechnical properties of the embankment and foundation soils are considered with the same values as in Case 1 (indicated in Table 1), as well as properties of geosynthetic and soil–geosynthetic interfaces. Taking into account the high permeability of the embankment soil, the ground surface is a drainage boundary. Fig. 10 shows the finite element mesh used in the numerical analysis; only 2D elements are represented. Table 6 shows the values of shear strength qf in the foundation for Case 2, obtained by the same assumptions as in Case 1, i.e. application of Eq. (2) with p0 ¼ p00 from 7 to 21 m depth (where the numerical analysis showed that consolidation is not significant). Considering the increases of effective volumetric stress, by consolidation, during construction in the foundation from 0 to 4.5 m depth (about 63 kPa from 0 to 1 m and 16 kPa from 1 to 4.5 m, in average terms, as shown by the element finite analysis), p0 ¼ p00 þ 63 and p00 þ 16 kPa were the values adopted at middle of the corresponding layers. A transition layer is considered from 4.5 to 7 m. Case 3 is a case history of one asymmetrical embankment on soft soils, constructed 1 up to failure (8.75 m height), with a 19 m crest width and 1:23 (V/H) inclined slopes, which is documented in detail by Quaresma (1992). From the in situ and laboratorial tests, Quaresma (1992) identified the following layers in the foundation: an overconsolidated clay to 1 m depth, which becomes less overconsolidated from 1 to 3m; an organic soft soil from 3 to 7 m; and a lightly organic clay from 7 to 24 m lying on a gravel layer. The element finite mesh used in the numerical analysis is shown in Fig. 11; only 2D elements are represented. The lower boundary is at 24 m depth and the ground surface is a drainage boundary. Fig. 12 illustrates the
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Table 6 Shear strength in the foundation for Case 2 Depth from ground surface (m)
Shear strength, qf (kPa)
0–1.0 1.0–4.5 4.5–7.0 7.0–21.0
33.0 16.0 16.0–18.375 18.375–55.125
EMBANKMENT HEIGHT (m)
Fig. 11. Finite element mesh for Case 3.
9 8 7 6 5 4 3
Real sequence Simulated sequence
2 1 0
5 15 25 35
135 145 155 165 175 185 195 205 215 225 235 245 255 265 275 285 295 305 315 325 335 345 355
TIME (hour)
Fig. 12. Embankment construction sequence for Case 3.
construction sequence: firstly, the construction of a 4 m height global platform was undertaken and, secondly, embankment itself, up to 8.75 m height. One geosynthetic-reinforcement level was considered in the embankment soil at 0.9 m height. Adopting the average values defined by Quaresma (1992), Tables 7 and 8 summarize the geotechnical properties of the foundation and embankment soils. Geosynthetic elastic modulus is 1818 kN/m and normal and tangential stiffnesses of the soil– reinforcement interfaces are 2.0 107 and 1.28 104 kPa/m, respectively. Geosynthetic strength is 200 kN/m. Adhesion and frictional angle for both upper and lower soil–geosynthetic interfaces are 0 kPa and 30.961, respectively. Table 9 shows parameters c1 ; c2 ; c3 ; c4 and c5 adopted in the numerical analysis for elasto-plastic models that simulate the geosynthetic and soil–geosynthetic interfaces. 4.2. Analysis of the results Fig. 13 shows the critical slip circles for Case 1—and corresponding overall safety factors (F )—determined by the methodologies described in Section 2 using finite
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Table 7 Geotechnical properties of the foundation for Case 3 Depth (m) l 0–1 1–3 3–7 7–18 18–24
0.217 0.261 0.651 0.304 0.304
k
N
G
0.011 0.013 0.065 0.019 0.019
3.046 3.374 7.052 3.440 3.440
3.189 3.546 7.459 3.638 3.638
f (1) n0 30 33 32 31 31
0.25 0.25 0.25 0.25 0.25
g (kN/m3) kx ¼ ky (m/s) qf ¼ su (kPa) k0 18 16.6 13.5 17 17.5
10
43 31 30 32 32
6.5 10 6.5 1010 7 1010 2 1010 2 1010
OCR
1.740 22.5 0.712 3.067 0.616 1.964 0.534 1.270 0.534 1.270
Table 8 Geotechnical properties of the embankment for Case 3 Layer
Height (m)
l
k
G
N
f (1)
n0
g (kN/m3)
k0
OCR
1 2
0–1 1–8.75
0.021 0.010
0.002 0.001
1.8000 1.7689
1.8132 1.7752
33 35
0.20 0.30
17.2 21.9
0.455 0.426
1 1
Table 9 Values of the hardening law parameters for Case 3
Geosynthetic Soil–geosynthetic interface
c1
c2
c3
c4
c5
60 150 (kPa) 0.267
1 804 432.003 (kPa) 413.1437 (m1)
0 (kPa) 0 (m2)
119.9915 1240.6716 (m1)
0 0 (m2)
element analysis (slip circle A) and in Section 3 by the limit equilibrium method (slip circle B). For slip circles A and B, Table 10 shows the values of the overturning moment (MO ), total resisting moment (MR ) and partial resisting moments due to the reinforcement (MRR ) and due to the forces in the embankment (MRE ) and in the foundation (MRF ). Fig. 14 and Table 11 illustrate the corresponding results for Case 2. In order to compare results obtained by the two methodologies, Tables 10 and 11 also show the results obtained by the limit equilibrium method along slip circle A: The results of Case 3 are shown for two embankment heights: (i) 7 m, which still corresponds to an equilibrium situation, before overall failure; and (ii) 8.75 m, the embankment failure height, as reported by Quaresma (1992). According to all field measurements, the embankment began to slip approximately at 7.5 m height when strength of the soil along the slip surface was reached. The slipping and the increase of the embankment height caused the increase of the geosynthetic strains and determined its failure—and overall failure—at 8.75 m height. For 7 m height, the critical slip circles obtained by the two methodologies and corresponding values of the moments are illustrated in Fig. 15 and Table 12. For 8.75 m height, as expected, because the embankment is no longer in an equilibrium situation, an adequate numerical solution could not be reached for this height. Fig. 16 shows the slip circle obtained by the limit equilibrium method (slip circle B)
410
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2m 8.49
geosynthetic
8.49 4.725
5m
Slip circle B (F=1.74)
Slip circle A (F=1.88)
13.125
Variation of shear strength with depth (kPa)
HARD STRATRUM
Fig. 13. Critical slip circles for Case 1 and corresponding overall safety factors (F ) obtained from finite element analysis (slip circle A) and determined by the limit equilibrium method (slip circle B).
Table 10 Slip circles for Case 1: overall safety factors (F ) and overturning and resisting moments X0 Y0 R (m) (m) (m)
Overturning Resisting moment F moment, MO Total Foundation Embankment Geosynthetic (kN m/m) MR MRF MRE MRR (kN m/m) (kN m/m) (kN m/m) (kN m/m)
Slip circle Aa 1.0 1.0 5.74 553.71 Slip circle Bb 1.0 1.0 5.24 521.66 Slip circle Ab 1.0 1.0 5.74 631.38 a b
1039.75 907.37 1114.70
754.11 647.12 849.25
85.64 60.25 65.45
200.00 200.00 200.00
1.88 1.74 1.77
Determined by the element finite method. Determined by the limit equilibrium method.
and the slip surface estimated from field measurements (Quaresma, 1992). Table 13 illustrates the corresponding values of the moments for slip circle B. For the three embankments, all partial safety factors, Zg ; Gc ; Gf ; Gsu ; Gr ; Ga and Gd ; were imposed equal to 1, and the geosynthetic force was considered acting horizontally (the most conservative assumption). The analysis of the results allows to point out that the slip circles obtained by the two methodologies are similar (except for Case 3 at 7m height, where a larger discrepancy can be observed), as well as the values of the overall safety factor. In spite of this overall similarity—which is important in practical terms and can be considered as an adequate accuracy of the limit equilibrium method to assess the overall safety factor in this kind of problems, the values shown in Tables 10–12 point out that the most important differences between the two methods are related to the values of the overturning and resisting moments. In fact, these moments are substantially smaller in the methodology based on the numerical analysis (due to the
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EMBANKMENT 33.00
411
5m
geosynthetic
16.00
Slip circle B
18.375
(F=1.15) Slip circle A
21 m
(F=1.22) SOFT SOILS
55.125
Variation of shear strength with depth (kPa)
HARD STRATUM
Fig. 14. Critical slip circles for Case 2 and corresponding overall safety factors (F ) obtained from finite element analysis (slip circle A) and determined by the limit equilibrium method (slip circle B).
Table 11 Slip circles for Case 2: overall safety factors (F ) and overturning and resisting moments X0 (m)
Slip circle Aa Slip circle Bb Slip circle Ab a b
2.0 3.0 2.0
Y0 (m)
5.0 8.0 5.0
R (m)
12.94 15.04 12.94
Overturning moment, MO (kN m/m)
5694.88 9540.50 7667.20
F
Resisting moment Total MR (kN m/m)
Found. MRF (kN m/m)
Embank. MRE (kN m/m)
Geosyn. MRR (kN m/m)
6966.52 10972.20 9133.30
5894.40 8368.50 7360.60
252.12 1183.70 952.73
820.00 1420.00 820.00
1.22 1.15 1.19
Determined by the element finite method. Determined by the limit equilibrium method.
forces in the embankment and in the foundation), as can be seen from the results of the two methodologies along slip circle A. This indicates that inside of the soil mass some stress redistribution occurs causing an effect that decreases overturning and resisting moments. In fact, as shown by Borges (1995) and Borges and Cardoso (2001), during the construction period, the external forces (weight of fill soil), which are equilibrated by the internal stresses in the materials (soil and geosynthetic), determine that critical state (strength of the soil) is reached in some areas of the foundation earlier than in others. For those areas, additional external forces due to the increase of the embankment height cannot be equilibrated by stress increments. This determines that stresses have to migrate to areas where critical state has not
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412
Embankment
7m
43 31
geotextile
30
24 m
32
Soft soils
Slip circle A (F=1.15) Variation of shear strength with depth (kPa)
Slip circle B (F=1.19) Hard stratum
Fig. 15. Case 3 Critical slip circles for 7 m height and corresponding overall safety factors (F ) obtained from finite element analysis (slip circle A) and determined by the limit equilibrium method (slip circle B).
Table 12 Case 3–Slip circles for 7 m height: overall safety factors (F ) and overturning and resisting moments X0 (m)
Slip circle Aa Slip circle Bb Slip circle Ab a b
4.80 3.80 4.80
Y0 (m)
8.0 15.0 8.0
R (m)
21.83 31.47 21.83
Overturning moment, MO (kN m/m)
20638.96 63870.00 34166.03
F
Resisting moment Total MR (kN m/m)
Found. MRF (kN m/m)
Embank. MRE (kN m/m)
Geosyn. MRR (kN m/m)
23793.62 75754.50 42695.51
22255.68 68087.00 36451.01
117.94 4847.50 4824.50
1420.00 2820.00 1420.00
1.15 1.19 1.25
Determined by the element finite method. Determined by the limit equilibrium method.
been reached yet. This effect is caught by the finite element analysis but it is not obviously considered in the simplified hypotheses that support the limit equilibrium method. The results also indicate that the differences between the values of the moments in the two methods are larger for embankments with smaller values of the overall safety factor, which is the case of Cases 2 and 3 compared with Case 1. This clearly corroborates that smaller values of the overall safety factor are obviously related to larger stress redistribution inside of the soil mass, which implies larger differences between the moments obtained by the two methodologies.
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413
Embankment 43 31
geotextile
30
32
Slip surface estimated from field measurements (Quaresma,1992)
24 m Soft soils Slip circle B determined by the limit equilibrium method (F=0.99)
Variation of shear strength with depth (kPa)
Hard stratum
Fig. 16. Case 3 Critical slip circle for 8.75 m height and corresponding overall safety factor (F ) determined by the limit equilibrium method (slip circle B) and slip surface estimated from field measurements.
Table 13 Case 3 Slip circle for 8.75 m height determined by the limit equilibrium method: overall safety factor (F ) and overturning and resisting moments X0 (m)
Slip circle B
4.80
Y0 (m)
14.0
R (m)
28.80
F
Overturning moment, MO (kN m/m)
Resisting moment Total MR (kN m/m)
Found. MRF (kN m/m)
Embank. MRE (kN m/m)
Geosyn. MRR (kN m/m)
65116.60
64784.50
56419.10
5745.30
2620.10
0.99
5. Conclusions Overall stability of geosynthetic-reinforced embankments on soft soils was analysed with two different methodologies described in the paper: (i) application of a numerical model based on the finite element method (Borges, 1995); and (ii) use of a limit equilibrium method, based on the assumptions proposed by Kaniraj and Abdullah (1993) and Borges (1995). These two methodologies were applied on three geosynthetic-reinforced embankments on soft soils; one of them is a case history that was constructed up to failure. Considering the analysis of the results, the following conclusions can be formulated: The critical slip circles obtained by the two methodologies are similar for the three cases, as well as the values of the safety factor, which determines an adequate accuracy of the limit equilibrium method to assess overall safety factor in reinforced embankments on soft soils.
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414
The most important differences between the two methodologies are related to the values of the overturning and resisting moments, which are smaller in the methodology based on the numerical analysis. These differences which are larger in embankments with smaller values of the safety factor are related to some stress redistribution that occurs inside of the soil mass, simulated by the finite element method but not considered in the simplified hypotheses of the limit equilibrium method.
Appendix A. Equations of the limit equilibrium method A.1. Design overturning moment The design overturning moment MOd is contributed by the soil mass ABCEI (Fig. 7). As deduced by Kaniraj and Abdullah (1993), and adding partial safety factors, it is given by He MOd ¼ Zg K1 for Xe pn1 H þ b; ðA:1Þ 2 gHe gðH He Þ K2 MOd ¼ Zg K1 þ ðA:2Þ for Xe > n1 H þ b; 2 2 where He2 þ ðn1 X0 þ Y0 ÞHe 3 2 þ XI 2XI X0 2YI Y0 þ YI2 ;
K1 ¼ ðn21 þ 1Þ
ðA:3Þ
K2 ¼ ½b þ ðn1 þ n2 ÞHðb þ n1 H n2 He 2X0 Þ þ ðn1 þ n2 ÞX0 ðH þ He Þ þ
n22 n21 2 ðH þ He2 þ He HÞ; 3
ðA:4Þ
where XI and YI are the coordinates of point I (in this case YI ¼ 0); instead of this point, the coordinates of any other point of the circle can be taken. Xe and He are the coordinates of point E; extremity of the slip circle as shown in Fig. 7 (0pHe pH). A.2. Design resisting moment The design resisting moment MRd depends on the materials properties and can be expressed as MRd ¼ MREd þ MRFd þ MRRd ;
ðA:5Þ
where MREd is the moment due to resisting forces in the embankment along slip arc EI; MRFd the moment due to resisting forces in the foundation along slip arc MJI ; MRRd the moment due to the reinforcement resisting force that can be mobilised at point G:
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A.2.1. Design resisting moment MREd MREd can be given by (Low, 1989): c tan f þ lgHe MREd ¼ R ye R; Gc Gf
ðA:6Þ
where R is the radius of the slip circle, l is an averaging coefficient for normal stress along arc EI; He is the Y -coordinate of point E (intersection of slip circle with slope or crest) and ye (angle IOE in Fig. 7) is computed by ye ¼ sin1
Y0 ðY0 He Þ : sin1 R R
ðA:7Þ
As proposed by Low (1989), limiting its value approximately to 0.5, l can be expressed by l ¼ 0:19 þ
0:02n1 ðR Y0 Þ=He
ðA:8Þ
where n1 defines slope inclination. A.2.2. Design resisting moment MRFd The design resisting moment MRFd is computed considering the foundation divided into a number of layers with the variation of undrained shear strength su being linear in each layer along vertical direction (see Fig. 17).
Y 0 ( X0 , Y0 )
R θAi θ Bi
X
(0,0) su -A
i
A 'i
Ai
layer i B i'
Bi
su - B
i
Fig. 17. Linear variation of undrained shear strength along vertical direction in each foundation layer.
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416
Therefore, the partial resisting moment MRFdi due to the forces in the ith layer (acting along arc Ai Bi and its symmetric A0i B0i ) is given by Z yAi su MRFdi ¼ 2R R dy: ðA:9Þ yBi Gsu Integrating this equation (note that su does not vary linearly along arc Ai Bi ) and considering contribution of all layers, MRFd is obtained by (Borges, 1995): MRFd ¼
N X
2R2 ½mi Rðsin yAi sin yBi Þ
i¼1
þ ðmi Y0 þ bi ÞðyAi yBi Þ
1 ; Gsu
ðA:10Þ
where yAi ¼ cos1
Y0 YAi ; R
Y0 YBi ; R suAi suBi ; mi ¼ YAi YBi yBi ¼ cos1
bi ¼ suAi þ mi YAi
ðA:11Þ ðA:12Þ ðA:13Þ ðA:14Þ
with YAi ; YBi being the Y -coordinates at top and bottom of ith layer, suAi ; suBi the undrained shear strength at top and bottom of ith layer, N the number of layers in the foundation intersected by the slip circle. A.2.3. Design resisting moment MRRd Assuming that the resisting force T due to the geosynthetic acts at an angle a to the horizontal (as said above, a ¼ 0 is the most conservative assumption), MRRd is given by (Fig. 7) MRRd ¼ TR cosðyr aÞ;
ðA:15Þ
where Y0 Yc R and Xc and Yc are the coordinates of the cut point G: T is defined by the equation Tg ; Tp ; T ¼ min Gr yr ¼ cos1
ðA:16Þ
ðA:17Þ
where Tg is the geosynthetic strength and Tp the maximum force that can be mobilised at point G taking into account the strength of soil–geosynthetic interfaces reduced by their partial factors.
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417
b W2 n1 M
n2
W1
1
1 G( X c , Y c )
l1
l2
N
H
Z
geosynthetic
Fig. 18. Lengths of soil–geosynthetic interfaces conditioning maximum force mobilised at point G:
Because equilibrium has to be verified, Tp is calculated by the equation Tp ¼ minðTp1 ; Tp2 Þ;
ðA:18Þ
where Tp1 and Tp2 are computed from shear stress of soil–geosynthetic interfaces along segments MG and GN (Fig. 18). Therefore, Tp1 ¼
al þ au tan dl þ tan du l 1 þ W1 ; Ga Gd
ðA:19Þ
Tp2 ¼
al þ au tan dl þ tan du l 2 þ W2 ; Ga Gd
ðA:20Þ
where al ; dl are the adhesion and frictional angle of the lower soil–geosynthetic interface, au ; du the adhesion and frictional angle of the upper soil–geosynthetic interface, l1 the length of segment MG (Fig. 18), l2 the length of segment GN (Fig. 18), W1 the weight of fill soil on the segment MG; W2 the weight of fill soil on the segment GN: If g is the unit weight of the embankment material, then W1 and W2 can be given by W1 ¼ 0:5
ðXc n1 Yc Þ2 g for Xc pn1 H; n1
W1 ¼ 0:5½2Xc n1 ðH þ Yc ÞðH Yc Þg W1 ¼ W 0:5
ðA:21Þ for n1 HoXc on1 H þ b;
ðl þ n1 Yc Xc Þ2 g for Xc Xn1 H þ b; n2
W 2 ¼ W W1 ;
ðA:22Þ ðA:23Þ ðA:24Þ
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where W is the total weight of fill soil on the geosynthetic and l is the geosynthetic length (length of segment MN in Fig. 18) and can be expressed by W ¼ 0:5ðl þ bÞðH Yc Þg;
ðA:25Þ
l ¼ ðn1 þ n2 ÞðH Yc Þ þ b:
ðA:26Þ
If there are N number of reinforcement levels, Eqs. (A.17)–(A.26) are applied to each reinforcement, and MRRd is the sum of all moments MRRdi due to those reinforcements MRRd ¼
N X
MRRdi ¼
i¼1
N X
Ti R cosðyri ai Þ:
ðA:27Þ
i¼1
A.3. Sequence of computation Taking into account the above equations, the rotational stability analysis of a geosynthetic-reinforced embankment on soft soils can be computed as the following sequence: 1. Reading of data (see Fig. 19): (a) Geometry of the problem: n1 ; n2 ; b; H and D; (b) Data on the potential slip circles: X0 min ; Y0 min ; X0 max ; Y0 max ; Dxy and Dr (for each point O of the mesh of circle centres, several concentric circles are analysed with radius varying from a minimum value Rmin to a maximum value Rmax ; Rmin can be considered as the longer value of the distances from point O to the two extremities of the segment AB, and Rmax as the smaller value of the distances
Dxy
Y O
(X0max , Y 0max )
Dxy b C
B
(X0min , Y 0min )
n1 1
geosynthetic
n2 1
H Z
A
X
D
Hard stratum Fig. 19. Geometry of reinforced embankment on soft soils.
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419
from point O to point Z and to the hard stratum under the soft soils; Dr is radius length increment for concentric circles); (c) Properties of the foundation and embankment materials: * fill soil: unit weight, cohesion and friction angle; * foundation soils: number of levels where s is indicated (with linear variation u between two contiguous levels along vertical direction); value of su and Y coordinate for each level; * geosynthetic: number of reinforcement levels and, for each level, Y -coordinate, geosynthetic strength, adhesion and frictional angle of the lower and upper soil–geosynthetic interfaces; (d) Partial safety factors: Zg , Gc ; Gf ; Gsu ; Gr ; Ga andGd (if all these factors are imposed equal to 1, an overall safety factor is obtained); (e) Coefficient ai =yri : definition of geosynthetic force inclination; 2. For each potential slip circle, calculate: Design overturning moment MOd (Eqs. (A.1)–(A.4)); Design resisting moment MREd (Eqs. (A.7), (A.8) and (A.6)); Design resisting moment MRFd (Eqs. (A.11)–(A.14) and (A.10)); Design resisting moment MRRd (Eqs. (A.21)–(A.26), (A.19), (A.20), (A.18), (A.17), (A.16) and (A.15) (or A.27)); (e) Total design resisting moment MRd (Eq. (A.5)); (f) Value of E given by
(a) (b) (c) (d)
E ¼ MRd =MOd X1
ðA:28Þ
which is the excess of safety in relation to that assumed by the use of partial safety factors (if these factors are all equal to 1, E is the overall safety factor); 3. Selection of the critical slip circle corresponding to the smallest value of E obtained.
References Borges, J.L., 1995. Geosynthetic-reinforced embankments on soft soils. Analysis and design. Ph.D. Thesis in Civil Engineering, Faculty of Engineering, University of Porto, Portugal, 1995 (in Portuguese). Borges, J.L., Cardoso, A.S., 1997. Stability of geosynthetic-reinforced embankments on soft soils. Revista Geotecnia, 80, 43–57 (in Portuguese). Borges, J.L., Cardoso, A.S., 1998. Numerical simulation of the p–q–y critical state model in embankments on soft soils. Revista Geotecnia, 84, 39–63 (in Portuguese). Borges, J.L., Cardoso, A.S., 2001. Structural behaviour and parametric study of reinforced embankments on soft clays. Computers and Geotechnics 28/3, 209–233. Brakel, J., Coppens, M., Magdenberg, A.C., Risseuw, P., 1982. Stability of slopes constructed with polyester reinforcing fabric, test section at Almere–Holland, 79. Proceedings of the Second International Conference on Geotextiles, Las Vegas, Vol. 3, 727–732. Britto, A.M., Gunn, M.J., 1987. Critical Soil Mechanics Via Finite Elements. Ellis Horwood Limited, England.
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Broms, B.B., 1977. Polyester fabric as reinforcement in soil. Proceedings of the International Conference on the Use of Fabrics in Geotechnics, Paris, Vol. 1, pp. 129–135. Duncan, J.M., Wong, K.S., 1984. STABGM: a computer solution for slope stability analysis with circular slip surfaces and geogrid reinforcement. Microcomputer version. A Tensar Corporation User’s Manual. Fowler, J., 1982. Theoretical design considerations for fabric reinforced embankments. Proceedings of the Second International Conference on Geotextiles, Las Vegas, Vol. 2, pp. 665–670. Haliburton, T.A., 1981. Use of engineering fabric in road and embankment construction. Seminar on the Use of Synthetic Fabrics in Civil Engineering, November 19, Toronto, pp. 66–94. Huisman, M.J.H., 1987. Design guideline for reinforced embankments on soft soils using Stabilenka reinforcing mats. Enka Technical Report, Arnhem. Humphrey, D.N., 1986. Design of reinforced embankments. Report FHWA/IN/JHRP-86/17 Joint Highway Research Project, School of Civil Engineering, Purdue University, West Lafayette, Indiana, 423pp. Ingold, T.S., 1982. An analytical study of geotextile reinforced embankments. Proceedings of the Second International Conference on Geotextiles, Las Vegas, Vol. 2, pp. 683–688. Ingold, T.S., 1983. Some factors in the design of geotextile reinforced embankments. In Improvement of Ground. Proceedings of the Eighth European Conference on Soil Mechanics and Foundation Engineering, Helsinki, Vol. 2, pp. 503–508. Jewell, R.A., 1982. A limit equilibrium design method for reinforced embankments on soft foundations. Proceedings of the Second International Conference on Geotextiles, Las Vegas 3, 671–676. Kaniraj, S.R., Abdullah, H., 1993. Rotational stability of narrow-crested reinforced embankments on soft soils. Geotextiles and Geomembranes 12, 599–614. Kwok, C.M., 1987. Finite elements studies of reinforced embankments on soft ground. Ph.D. Thesis, University of Sheffield, Sheffield. Lambe, T.W., Whitman, R.V., 1969. Soil Mechanics. Wiley, New York. Lewis, R.W., Schrefler, B.A., 1987. The Finite Element Method in the Deformation and Consolidation of Porous Media. Wiley, New York. Low, B.K., 1989. Stability analysis of embankments on soft ground. Journal of Geotechnical Engineering, Division, ASCE 115 (2), 211–227. Milligan, V., La Rochelle, P., 1984. Design methods for embankments over weak soils. Symposium on Polymer Grid Reinforcement in Civil Engineering, Institute of Civil Engineers, London, UK, Paper No. 3.4. Mylleville, B.L.J., Rowe, R.K., 1991. On the design of reinforced embankments on soft brittle clays. Geosynthetics 91 Conference, Atlanta, USA, pp. 395–408. Owen, D.R.J., Hinton, E., 1980. Finite Elements in Plasticity: Theory and Practice. Pineridge Press Limited, Swansea. Prevost, J.H., Hoeg, K., 1975. Effective stress–strain-strength model for soils. Journal of Geotechnical Engineering, Division, ASCE 101 (GT3), 259–278. Quaresma, M.G., 1992. Behaviour and modelling of an embankment over soft soils reinforced by geotextile. Ph.D. Thesis, Universite Joseph Fourier, Grenoble I (in French). Quast, P., 1983. Polyester reinforcement fabric mats for the improvement of the embankment stability. In improvement of ground. Proceedings of the Eighth European Conference on Soil Mechanics and Foundation Engineering, Helsinki, Vol. 2, pp. 531–534. Rowe, R.K., 1984. Reinforced embankments: analysis and design. Journal of Geotechnical Engineering, ASCE 110 (2), 231–246. Rowe, R.K., Mylleville, B.L.J., 1990. Implications of adopting an allowable geosynthetic strain in estimating stability. Proceedings of the Fourth International Conference on Geotextiles, Geomembranes and Related Products, The Hauge, Netherlands, pp. 131–136. Rowe, R.K., Soderman, K.L., 1987. Stabilisation of very soft soils using high strength geosynthetics: the role of finite elements analysis. Geotextiles and Geomembranes 6 (1), 53–80. Russell, D., 1992. Finite element analysis of embankments on soft ground incorporating reinforcement and drains. Ph.D. Thesis, University of Sheffield, Sheffield.
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Soderman, K.L., 1986. The behaviour of geotextile reinforced embankments. Ph.D. University of Western Ontario, Ontario. Tensar Corporation, 1982. Designing with Tensar. Morrow, Georgia and Toronto, Ont., 17pp. Thomas, J.N., 1984. An improved acelerated initial stress procedure for elasto-plastic finite element analysis. International Journal for Numerical and Analytical Methods in Geomechanics 8, 359–379. Yeo, K.C., 1986. Simplified foundation data to predictors. Proceedings of the Prediction Symposium on a Reinforced Embankment on Soft Ground, King’s College, London.
Overall stability of geosynthetic-reinforced embankments on soft soils ! Jose! Leit*ao Borges*, Antonio Silva Cardoso Department of Civil Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Received 12 September 2001; received in revised form 3 March 2002; accepted 11 June 2002
Abstract Overall stability of geosynthetic-reinforced embankments on soft soils is analysed using two different methodologies: application of a numerical model based on the finite element method; use of a limit equilibrium method. These two methodologies are described and also applied on three geosynthetic-reinforced embankments on soft soils. One of the cases is a case history constructed up to failure. Considering the analysis of the results, some conclusions are formulated on the limit equilibrium method accuracy, namely regarding the critical slip surface, overall safety factor and overturning and resisting moments. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Reinforced embankment; Geosynthetic; Overall stability; Limit equilibrium; Finite element analysis
1. Introduction To design embankments on soft soils it is essential to take into account the multiple constructive techniques that allow to solve the problems usually associated with this kind of construction: overall stability deficiency and large settlements that develop slowly. The constructive solutions—usually based on both foundation soil properties improvement and construction procedures or fill properties alteration—provide one or more of the following effects: increase of overall stability, consolidation acceleration and decrease of long term settlements. *Corresponding author. Tel.: +351-22508-1928; fax: +351-22508-1440. E-mail address: [email protected] (J.L. Borges). 0266-1144/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 6 - 1 1 4 4 ( 0 2 ) 0 0 0 1 4 - 6
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With textile industry development in the 20th century, especially following the 1960s, geosynthetic reinforcement has been added to the list of possible solutions when embankments must be constructed on very soft foundations. In many cases, the use of one geotextile or geogrid can significantly increase the safety factor, improve performance in terms of displacements and reduce costs in comparison with more conventional solutions. Especially due to their simplicity, overall stability of reinforced embankments on soft soils is usually computed by limit equilibrium methods along potential slip circles. The reinforcement effect is considered by a resisting force due to the geosynthetic. Theoretically, however, because rigid–plastic behaviour is tacitly assumed for the materials (soil and reinforcement), the use of these methods may raise some reticences, as strains are not taken into account before overall failure, as well as stress redistribution caused by the geosynthetic. This often determines the use of the finite element method in the study of geosynthetic-reinforced embankments (Rowe, 1984; Humphrey, 1986; Soderman, 1986; Rowe and Soderman, 1987; Kwok, 1987; Rowe and Mylleville, 1990; Mylleville and Rowe, 1991; Russell, 1992; Borges, 1995). In the paper, overall stability of geosynthetic-reinforced embankments on soft soils is analysed adopting two different methodologies: (i) using the results obtained from a numerical model based on the finite element method (Borges, 1995); along each analysed slip circle, acting and resisting tangential forces are obtained from numerical results and strength characteristics of the materials; (ii) applying a limit equilibrium method, based on the formulation proposed by Kaniraj and Abdullah (1993) but with some improvements (see Section 3) performed by Borges (1995). The two methodologies are also applied on three geosynthetic-reinforced embankments on soft soils. The third embankment is a case history that was constructed up to failure (Quaresma, 1992).
2. Overall stability analysis using results from finite element method application 2.1. General characteristics of the finite element model Numerical analysis is simulated by a model, developed by Borges (1995), based on the finite element method, with the following theoretical hypotheses: (a) plane strain conditions; (b) coupled formulation of the flow and equilibrium equations, considering soil constitutive relations formulated in effective stresses (Biot’s consolidation theory) (Lewis and Schrefler, 1987; Britto and Gunn, 1987); this formulation is applied to all phases of the problem, both during the embankment construction and in the post-construction period; (c) utilisation of the p2q2y critical state model (Lewis and Schrefler, 1987; Britto and Gunn, 1987; Borges, 1995; Borges and Cardoso, 1998) to simulate constitutive behaviour of the foundation and embankment soils (see below); (d) utilisation of a hardening elasto-plastic model to simulate constitutive behaviour of the reinforcement; (e) simulation of constitutive behaviour of the soil–geosynthetic interfaces using a hardening elasto-plastic model.
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Geosynthetic and soil–geosynthetic interfaces are modelled by bar and interface elements, respectively. For 2D elements, triangular elements are used, namely sixnoded elements (at the vertices and middle of the edges) with 12 displacement degrees of freedom (for fill elements) or with three more excess pore pressure degrees of freedom at the vertices of the triangle (for foundation elements where consolidation analysis is considered). In order to verify accuracy of the numerical model used in this paper, Borges (1995) compared numerical and field results of two reinforced embankments on soft soils, one constructed up to failure (Quaresma, 1992) and the other observed until the end of the consolidation (Yeo, 1986). The accuracy was considered adequate in both cases, as numerical and field results are similar, namely in terms of settlements, pore pressures and strains or tensile forces in the reinforcement. Only some quantitative differences were observed in the horizontal displacements, despite an overall qualitative similarity too. 2.2. Evaluation of safety factors Using effective stresses obtained from numerical model application, one can compute overall safety factor and also partial safety factors of the different materials (reinforcement and foundation and fill soils) as described below (Borges, 1995; Borges and Cardoso, 1997). Firstly, for each potential slip circle, the intersection points of the circle with the edges of the finite elements of the mesh are determined. Therefore, the slip circle is divided into small line segments, each of them located inside of only one of the finite elements of the mesh. Afterwards, the average values of effective stresses (s0x ; s0y and txy ; normal and shear stresses in the directions of x- and y-axes) at each of those segments are computed extrapolating from the stresses at the Gauss points of the corresponding finite element. Mathematical procedures for this extrapolation depend on the type of the 2D finite element, i.e. position and number of nodes and Gauss points. For the 2D element used in this study—six-noded triangular element with seven Gauss points (see Fig. 1), whose strains vary linearly within the element—the following scheme was assessed adequate: (i) firstly, stresses at each node of the element are computed by linear extrapolation from two points—the
Node Gauss point
Line segment of the slip surface Fig. 1. 2D finite element.
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Gauss point at middle of the element and the Gauss point that is nearest the node; (ii) secondly, stresses at each extremity of the segment line of the slip circle are computed by linear interpolation from stresses at the two nearest nodes of the element; (iii) finally, stresses at each segment are the average values obtained from stresses at the two extremities of the segment. Considering the slip circle divided into line segments, partial safety factor of the soil along the slip surface (foundation and fill) is defined by PN qfi li Fs ¼ Pi¼1 ; ð1Þ N i¼1 ti li where li is the i-segment length; N is the number of 2D elements of the mesh intersected by the circle; ti is the average value of shear stress at i-segment (determined from effective stresses s0x ; s0y and txy ; known the angle that defines i-segment inclination); qfi is the average value of soil shear strength at i-segment. Limiting the sum only to the segments in the foundation or in the fill, the partial safety factors for corresponding soils along the slip circle can also be obtained with Eq. (1). Once the finite element method model uses a constitutive model, p2q2y critical state model, that formulates the soil strength or failure (critical state) by the Mohr– Coulomb criteria (see Fig. 2), soil shear strength at i-segment, qfi ; can be estimated by the equation qfi ¼ c0i cos f0i þ p0i sin f0i ;
ð2Þ stress and c0i and f0i are effective terms (s01i ; s03i
where p0i ¼ ðs0xi þ s0yi Þ=2 ¼ ðs01i þ s03i Þ=2 is effective volumetric the cohesion and friction angle of soil at i-segment defined in are the principal effective stresses; s0xi ; s0yi are the normal effective stresses in x- and y-directions).
Fig. 2. Yield and critical state surfaces of p2q2y critical state model in principal effective stress space.
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399
As shown by Borges and Cardoso (2001) and Borges (1995), if consolidation is not significant, the typical effective stress path in the foundation corresponds to low variations of p0 ¼ ðs01 þ s03 Þ=2 with significant increases of q ¼ ðs01 s03 Þ=2: As an approximation, p0 can be considered with the same value at acting stress state (obtained from finite element analysis) and at failure stress state, i.e. the corresponding Mohr circles can be considered concentric. Soil shear strength, qf ; is the radius of the failure Mohr circle, given by Eq. (2), as deduced in Fig. 3. The partial safety factor of the geosynthetic is determined by the equation Fg ¼
Tr ; Ta
ð3Þ
where Ta is the geosynthetic acting tensile force at the cut point (point I in Fig. 4), intersection of the slip circle with the geosynthetic (or geosynthetics, if there are several reinforcement layers) and Tr is the corresponding resisting tensile force. Ta is obtained from the numerical results interpolating from tensile stresses at the two nearest Gauss points of the bar element that contains point I: Tr is given by Tr ¼ minðTrg ; Trp Þ
τ
ð4Þ
τ =c'+σ'n tanφ'
qf =
(σ'n , τ)
qf c'
φ'
c'/tanφ'
σ'3f
σ'3
p'= (σ'1+σ'3)/2
p'
σ'1 σ'1f
σ3f
σ1f
p
failure Mohr circle acting Mohr circle
σ'n σn
c′ + p′ sin φ′ = tan φ′
= c′ cos φ ′ + p′ sin φ ′
total stress Mohr circle
Fig. 3. Relation between soil shear strength and effective cohesion and friction angle.
T A
geosynthetic
B
I θ
Fig. 4. Overall stability analysis.
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being Trg and Trp the maximum forces that can be mobilised at point I taking into account the geosynthetic strength and pull-out force of the soil–geosynthetic interfaces, respectively. So, Trg is a property of the geosynthetic and, because equilibrium has to be verified, Trp is calculated by the equation Trp ¼ minðTrp2AI ; Trp2IB Þ;
ð5Þ
where Trp2AI and Trp2IB correspond, respectively, to the integrals of the maximum adhesive and frictional stresses that can be mobilised in the soil–geosynthetic interfaces on the left, segment AI, and on the right, segment IB, from the cut point I (see Fig. 4): Z Trp2AI ¼ ½al þ au þ s0n ðtan dl þ tan du Þ dl; ð6aÞ AI
Trp2IB ¼
Z IB
½al þ au þ s0n ðtan dl þ tan du Þ dl;
ð6bÞ
where al ; dl are the adhesion and frictional angle of the lower interface; au ; du are the adhesion and frictional angle of the upper interface; and s0n is the normal stress acting on the interface. It should be noted that the values of s0n are known at the Gauss points of the elements that model the interface, so Eqs. (6a) and (6b) can be calculated approximately considering linear the variation of s0n between two contiguous Gauss points. This is an adequate approximation because the finite element meshes usually used in this kind of problems must have sufficiently small elements. Finally, overall safety factor is defined as follows: PN g PN i¼1 qfi li þ j¼1 ½Trj cosðcr yj Þ ; ð7Þ F ¼ PN PN g i¼1 ti li þ j¼1 ½Taj cosðcr yj Þ where Ng is the number of reinforcements (in most cases Ng ¼ 1), yj the angle between the reinforcement direction and the tangent to the slip circle at the cut point, cr a reduction coefficient (it varies between 0 and 1) related to the direction considered for the forces Trj and Taj (resisting and acting tensile forces of j-geosynthetic, as defined above), and li ; N; ti and qfi have the same meaning that in Eq. (1). It should be noted that the reinforcement force has been considered to act in its original orientation (usually horizontally) by some investigators (Broms, 1977; Tensar, 1982; Jewell, 1982; Ingold, 1982, 1983; Brakel et al., 1982; Duncan and Wong, 1984; Milligan and La Rochelle, 1984), tangentially to the circle by some others (Haliburton, 1981; Quast, 1983) assuming that the local deformations associated with the formation of failure surface result in a local reorientation of the geosynthetic, and between the above two directions by Huisman (1987). However, as indicated by Soderman (1986), the assumption that provides better agreement with observed embankment performance is, likely, dependent on the particular case. Because the assumption that assumes the geosynthetic remains in its original
J.L. Borges, A.S. Cardoso / Geotextiles and Geomembranes 20 (2002) 395–421 C'
401
C
COMPLEMENTARY SURFACE
INITIAL SURFACE
Fig. 5. Symmetric problem regarding one slip surface that does not include whole continuum.
orientation (cr ¼ 1) is the most conservative assumption, in doubt, one should consider it in design. Sometimes, because the problem is symmetric, the finite element mesh does not include the whole continuum. In this case, symmetry allows solving this question considering a complementary surface, as indicated in Fig. 5. In this second surface one can have all data and results regarding the part of initial surface that is out of the mesh limits.
3. Application of a limit equilibrium method on the overall stability design Most of the design methods used to assess the overall stability of reinforced embankments on soft soils are limit equilibrium methods which use circular failure surfaces and consider a resisting force due to the reinforcement (Rowe, 1984; Jewell, 1982; Ingold, 1982; Haliburton, 1981; Brakel et al., 1982; Fowler, 1982; Milligan and LaRochelle, 1984). Overall safety factor, F ; can be computed by MR F¼ ; ð8Þ MO where MR is the resisting moment and MO the overturning moment, which can be given by (Fig. 6): MO ¼ Wx; MR ¼
X
ð9Þ tri Dli R þ Tg R cosðy aÞ;
ð10Þ
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x
y
R
W
Tg
geosynthetic
α
∆l i τri
θ
Fig. 6. Potential failure surface: rotational stability analysis.
where W is the weight of the fill mass inside the slip circle, tri the resisting shear stress along Dli long arc of the slip circle, and Tg the maximum force that reinforcement can mobilise at the intersection point with the circle (as seen in Section 2.2, this force depends on geosynthetic strength and the pull-out force that can be developed along the soil–geosynthetic interfaces). Alternatively to Eq. (8), in agreement with more recent design methodologies, partial safety factors can be used in overall failure analysis. These methodologies consider: (i) factors that multiply actions, Zg for unit weight of the fill soil and Zn for eventual variable actions; (ii) factors that divide materials properties, Gc and Gf for cohesion and friction angle of the fill soil, Gsu for undrained shear strength of the foundation soil, Gr for reinforcement strength, and Ga and Gd for adhesion and frictional angle of the soil–reinforcement interfaces. As known, using partial factors, one has to verify the inequation: MOd pMRd ;
ð11Þ
where MOd and MRd are, respectively, design overturning and resisting moments, i.e. determined with the above partial factors. Of course, if all partial factors of actions and materials are imposed equal to 1, an overall safety factor can be obtained as defined by Eq. (8). As said in Section 1, a limit equilibrium method is used in this paper, based on the assumptions proposed by Kaniraj and Abdullah (1993) but with some improvements implemented by Borges (1995). These improvements are: (a) introduction of partial safety factors; (b) correct calculation of the moment due to resisting forces in the foundation soil along the slip surface (which is done approximately by Kaniraj and Abdullah (1993), depending on the number of layers that the foundation is divided); (c) calculation of the pull-out force of the geosynthetic taking into account that its
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403
Fig. 7. Slip circle in a reinforced embankment on soft soil (Kaniraj and Abdullah, 1993).
value, due to equilibrium reasons, is the minimum of the two values computed on the left and on the right of the cut point (intersection of the reinforcement with the slip circle). The equations used to calculate MOd and MRd are presented in appendix assuming the geometry as shown in Fig. 7, which shows an arbitrary slip circle in a reinforced embankment. The origin of (X ;Y )-axes is assumed to be at the toe.
4. Overall stability analysis of geosynthetic-reinforced embankments on soft soils 4.1. Description of the illustrative cases In this section overall stability of three geosynthetic-reinforced embankments on soft soils is analysed using the two methodologies described in Sections 2 and 3. Case 1 is a 28-day continuous construction of a 2 m height symmetric embankment, with a 10.6 m crest width and 23 (V/H) inclined slopes. The foundation is a 5 m thick saturated clay layer lying on a rigid and impermeable soil, which constitutes the lower boundary. The clay is lightly overconsolidated to 1.8 m depth and normally consolidated from 1.8 to 5 m. One geosynthetic-reinforcement level is considered between the foundation and embankment soils. Taking into account the high permeability of the geosynthetic and of the embankment soil, the upper foundation surface is a drainage boundary. Fig. 8 shows the finite element mesh used in the numerical analysis; only 2D elements are represented; y-axis is the symmetry line.
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Fig. 8. Finite element mesh for Case 1.
Table 1 Geotechnical properties of the foundation and of the embankment for Case 1 g (kN/m3) n0
Foundation 17 Embankment 20
c0 (kPa) f0 (1) kx (m/s) ky (m/s) p2q2y critical state model
0.25 0 0.30 0
30 35
10 —
9
10 —
9
l
k
G
N
0.22 0.03
0.02 0.005
3.26 1.80
3.40 1.817
Table 2 At rest earth pressure coefficient, k0 ; and over-consolidation ratio, OCR, in the foundation for Case 1 Depth (m)
k0
OCR
0–1 1–1.8 1.8–5
0.7 0.7–0.5 0.5
2.43 2.43–1 1
The constitutive relations of both the embankment and foundation soils were simulated using the p2q2y critical state model (Lewis and Schrefler, 1987; Britto and Gunn, 1987; Borges, 1995; Borges and Cardoso, 1998) with the parameters indicated in Table 1 (l; slope of normal consolidation line and critical state line; k; slope of swelling and recompression line; G; specific volume of soil on the critical state line at mean normal stress equal to 1 kPa; N; specific volume of normally consolidated soil at mean normal stress equal to 1 kPa). Table 1 also shows other geotechnical properties: g; unit weight; n0 ; Poisson’s ratio for drained loading; c0 and f0 ; cohesion and angle of friction defined in effective terms; kx and ky ; coefficients of permeability in x- and y-directions. Table 2 indicates the variation with depth of the at rest earth pressure coefficient, k0 ; and over-consolidation ratio (OCR), in the foundation. All these parameters were defined taking into account typical experimental values for this kind of soils (Borges, 1995; Lambe and Whitman, 1969).
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FORCE (kN/m)
200 150 100 50 0 0
5
10
15
20
ε
STRAIN (%)
(a) τ
σn = 15kPa
SHEAR STRESS (kPa)
10
5
0
(b)
0
1
2 4 6 3 5 7 RELATIVE DISPLACEMENT (mm)
8
∆s
Fig. 9. Constitutive curve for Case 1 of the: (a) geosynthetic; (b) soil–geosynthetic interface for normal stress of 15 kPa.
Fig. 9 shows the constitutive curves of the geosynthetic and soil–geosynthetic interfaces. The geosynthetic thickness is 2 mm and its elastic modulus is 1.5 106 kPa. Normal and tangential stiffnesses of soil–reinforcement interfaces are 2.0 107 and 1.6 104 kPa/m, respectively. Geosynthetic strength is 200 kN/m. Adhesion, a, and frictional angle, d; for both upper and lower soil–geosynthetic interfaces are 0 kPa and 33.71, respectively. Mechanical behaviour of the reinforcement and soil–reinforcement interfaces were simulated in the numerical analysis by elasto-plastic models that incorporate the following hardening law (Prevost and Hoeg, 1975; Owen and Hinton, 1980; Thomas, 1984): Y ðhÞ ¼ c1 þ
c 2 h þ c 3 h2 ; 1 þ c 4 h þ c 5 h2
ð12Þ
where h is the hardening parameter and c1 ; c2 ; c3 ; c4 and c5 are parameters that characterise the material. For the geosynthetic, the yielding function is expressed by f ¼ s Y ðhÞ;
ð13Þ
where s is the tensile stress, Y ðhÞ ¼ sY ðep Þ is the hardening law, h ¼ ep the hardening parameter, sY the yielding stress and ep the plastic tensile strain.
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Table 3 Values of the hardening law parameters for Case 1 c1 Geosynthetic Soil–geosynthetic interface
c2 4
7.5 10 (kPa) 0.333
6
8.824 10 (kPa) 417.094 (m1)
c3
c4
c5
0 (kPa) 0 (m2)
35.29 1251.408 (m1)
0 0 (m2)
Table 4 Shear strength in the foundation for Case 1 Depth from ground surface (m)
Shear strength, qf (kPa)
0 1.0 1.8 5.0
8.49 8.49 4.725 13.125
Shear strength varies linearly between two contiguous shear strength values.
For the soil–geosynthetic interfaces, the yielding function is: f ¼ jtj sn Y ðhÞ;
ð14Þ
where t and sn are, respectively, P the tangential and normal stresses, Y ðhÞ ¼ tan dðsp Þ is the hardening law, h ¼ jsp j the hardening parameter, d the interface friction angle and sp the plastic tangential relative displacement. Table 3 shows parameters c1 ; c2 ; c3 ; c4 and c5 adopted. For the foundation clay, the correspondence between friction angle f0 (used in the numerical analysis) and shear strength qf at the end of the embankment construction (used in the limit equilibrium method) was determined by the application of Eq. (2). As said in Section 2 (Borges and Cardoso, 2001; Borges, 1995), if consolidation is not significant, one can consider approximately p0 ¼ p00 (where p00 is the initial effective volumetric stress), because the typical effective stress path during construction corresponds to low variations of effective volumetric stress with significant increases of shear stress. Being c0 ¼ 0; Table 4 shows the values of shear strength qf in the foundation for Case 1, obtained by the application of the above considerations, except from 0 to 1.0 m depth, where significant increases of effective volumetric stress, by consolidation, were observed during construction (about 14 kPa in average terms, as shown by the element finite analysis). To also consider the consolidation effect in the limit equilibrium method application, shear strength in the foundation, from 0 to 1.0 m, was determined applying Eq. (2) at the middle of the layer (0.5 m depth) with p0 ¼ p00 þ 14 kPa. A transition layer is considered from 1 to 1.8 m. Case 2 is a 28-day continuous construction of a 5 m height symmetric 1 embankment, with a 30.26 m crest width and 1:23 (V/H) inclined slopes. The foundation is a 21 m thick saturated clay layer lying on a rigid and impermeable soil. The clay is lightly overconsolidated to 7 m depth and normally consolidated from 7 to 21 m. The variation with depth of the at rest earth pressure coefficient, k0 ; and OCR in the foundation are indicated in Table 5. One geosynthetic-reinforcement
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Table 5 At rest earth pressure coefficient, k0 ; and over-consolidation ratio, OCR, in the foundation for Case 2 Depth (m)
k0
OCR
0–4.5 4.5–7.0 7.0–21.0
0.7 0.7–0.5 0.5
2.43 2.43–1 1
Symmetry line
0
5m
Fig. 10. Finite element mesh for Case 2.
level is considered in the embankment soil at 0.9 m height. Geotechnical properties of the embankment and foundation soils are considered with the same values as in Case 1 (indicated in Table 1), as well as properties of geosynthetic and soil–geosynthetic interfaces. Taking into account the high permeability of the embankment soil, the ground surface is a drainage boundary. Fig. 10 shows the finite element mesh used in the numerical analysis; only 2D elements are represented. Table 6 shows the values of shear strength qf in the foundation for Case 2, obtained by the same assumptions as in Case 1, i.e. application of Eq. (2) with p0 ¼ p00 from 7 to 21 m depth (where the numerical analysis showed that consolidation is not significant). Considering the increases of effective volumetric stress, by consolidation, during construction in the foundation from 0 to 4.5 m depth (about 63 kPa from 0 to 1 m and 16 kPa from 1 to 4.5 m, in average terms, as shown by the element finite analysis), p0 ¼ p00 þ 63 and p00 þ 16 kPa were the values adopted at middle of the corresponding layers. A transition layer is considered from 4.5 to 7 m. Case 3 is a case history of one asymmetrical embankment on soft soils, constructed 1 up to failure (8.75 m height), with a 19 m crest width and 1:23 (V/H) inclined slopes, which is documented in detail by Quaresma (1992). From the in situ and laboratorial tests, Quaresma (1992) identified the following layers in the foundation: an overconsolidated clay to 1 m depth, which becomes less overconsolidated from 1 to 3m; an organic soft soil from 3 to 7 m; and a lightly organic clay from 7 to 24 m lying on a gravel layer. The element finite mesh used in the numerical analysis is shown in Fig. 11; only 2D elements are represented. The lower boundary is at 24 m depth and the ground surface is a drainage boundary. Fig. 12 illustrates the
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Table 6 Shear strength in the foundation for Case 2 Depth from ground surface (m)
Shear strength, qf (kPa)
0–1.0 1.0–4.5 4.5–7.0 7.0–21.0
33.0 16.0 16.0–18.375 18.375–55.125
EMBANKMENT HEIGHT (m)
Fig. 11. Finite element mesh for Case 3.
9 8 7 6 5 4 3
Real sequence Simulated sequence
2 1 0
5 15 25 35
135 145 155 165 175 185 195 205 215 225 235 245 255 265 275 285 295 305 315 325 335 345 355
TIME (hour)
Fig. 12. Embankment construction sequence for Case 3.
construction sequence: firstly, the construction of a 4 m height global platform was undertaken and, secondly, embankment itself, up to 8.75 m height. One geosynthetic-reinforcement level was considered in the embankment soil at 0.9 m height. Adopting the average values defined by Quaresma (1992), Tables 7 and 8 summarize the geotechnical properties of the foundation and embankment soils. Geosynthetic elastic modulus is 1818 kN/m and normal and tangential stiffnesses of the soil– reinforcement interfaces are 2.0 107 and 1.28 104 kPa/m, respectively. Geosynthetic strength is 200 kN/m. Adhesion and frictional angle for both upper and lower soil–geosynthetic interfaces are 0 kPa and 30.961, respectively. Table 9 shows parameters c1 ; c2 ; c3 ; c4 and c5 adopted in the numerical analysis for elasto-plastic models that simulate the geosynthetic and soil–geosynthetic interfaces. 4.2. Analysis of the results Fig. 13 shows the critical slip circles for Case 1—and corresponding overall safety factors (F )—determined by the methodologies described in Section 2 using finite
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Table 7 Geotechnical properties of the foundation for Case 3 Depth (m) l 0–1 1–3 3–7 7–18 18–24
0.217 0.261 0.651 0.304 0.304
k
N
G
0.011 0.013 0.065 0.019 0.019
3.046 3.374 7.052 3.440 3.440
3.189 3.546 7.459 3.638 3.638
f (1) n0 30 33 32 31 31
0.25 0.25 0.25 0.25 0.25
g (kN/m3) kx ¼ ky (m/s) qf ¼ su (kPa) k0 18 16.6 13.5 17 17.5
10
43 31 30 32 32
6.5 10 6.5 1010 7 1010 2 1010 2 1010
OCR
1.740 22.5 0.712 3.067 0.616 1.964 0.534 1.270 0.534 1.270
Table 8 Geotechnical properties of the embankment for Case 3 Layer
Height (m)
l
k
G
N
f (1)
n0
g (kN/m3)
k0
OCR
1 2
0–1 1–8.75
0.021 0.010
0.002 0.001
1.8000 1.7689
1.8132 1.7752
33 35
0.20 0.30
17.2 21.9
0.455 0.426
1 1
Table 9 Values of the hardening law parameters for Case 3
Geosynthetic Soil–geosynthetic interface
c1
c2
c3
c4
c5
60 150 (kPa) 0.267
1 804 432.003 (kPa) 413.1437 (m1)
0 (kPa) 0 (m2)
119.9915 1240.6716 (m1)
0 0 (m2)
element analysis (slip circle A) and in Section 3 by the limit equilibrium method (slip circle B). For slip circles A and B, Table 10 shows the values of the overturning moment (MO ), total resisting moment (MR ) and partial resisting moments due to the reinforcement (MRR ) and due to the forces in the embankment (MRE ) and in the foundation (MRF ). Fig. 14 and Table 11 illustrate the corresponding results for Case 2. In order to compare results obtained by the two methodologies, Tables 10 and 11 also show the results obtained by the limit equilibrium method along slip circle A: The results of Case 3 are shown for two embankment heights: (i) 7 m, which still corresponds to an equilibrium situation, before overall failure; and (ii) 8.75 m, the embankment failure height, as reported by Quaresma (1992). According to all field measurements, the embankment began to slip approximately at 7.5 m height when strength of the soil along the slip surface was reached. The slipping and the increase of the embankment height caused the increase of the geosynthetic strains and determined its failure—and overall failure—at 8.75 m height. For 7 m height, the critical slip circles obtained by the two methodologies and corresponding values of the moments are illustrated in Fig. 15 and Table 12. For 8.75 m height, as expected, because the embankment is no longer in an equilibrium situation, an adequate numerical solution could not be reached for this height. Fig. 16 shows the slip circle obtained by the limit equilibrium method (slip circle B)
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2m 8.49
geosynthetic
8.49 4.725
5m
Slip circle B (F=1.74)
Slip circle A (F=1.88)
13.125
Variation of shear strength with depth (kPa)
HARD STRATRUM
Fig. 13. Critical slip circles for Case 1 and corresponding overall safety factors (F ) obtained from finite element analysis (slip circle A) and determined by the limit equilibrium method (slip circle B).
Table 10 Slip circles for Case 1: overall safety factors (F ) and overturning and resisting moments X0 Y0 R (m) (m) (m)
Overturning Resisting moment F moment, MO Total Foundation Embankment Geosynthetic (kN m/m) MR MRF MRE MRR (kN m/m) (kN m/m) (kN m/m) (kN m/m)
Slip circle Aa 1.0 1.0 5.74 553.71 Slip circle Bb 1.0 1.0 5.24 521.66 Slip circle Ab 1.0 1.0 5.74 631.38 a b
1039.75 907.37 1114.70
754.11 647.12 849.25
85.64 60.25 65.45
200.00 200.00 200.00
1.88 1.74 1.77
Determined by the element finite method. Determined by the limit equilibrium method.
and the slip surface estimated from field measurements (Quaresma, 1992). Table 13 illustrates the corresponding values of the moments for slip circle B. For the three embankments, all partial safety factors, Zg ; Gc ; Gf ; Gsu ; Gr ; Ga and Gd ; were imposed equal to 1, and the geosynthetic force was considered acting horizontally (the most conservative assumption). The analysis of the results allows to point out that the slip circles obtained by the two methodologies are similar (except for Case 3 at 7m height, where a larger discrepancy can be observed), as well as the values of the overall safety factor. In spite of this overall similarity—which is important in practical terms and can be considered as an adequate accuracy of the limit equilibrium method to assess the overall safety factor in this kind of problems, the values shown in Tables 10–12 point out that the most important differences between the two methods are related to the values of the overturning and resisting moments. In fact, these moments are substantially smaller in the methodology based on the numerical analysis (due to the
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EMBANKMENT 33.00
411
5m
geosynthetic
16.00
Slip circle B
18.375
(F=1.15) Slip circle A
21 m
(F=1.22) SOFT SOILS
55.125
Variation of shear strength with depth (kPa)
HARD STRATUM
Fig. 14. Critical slip circles for Case 2 and corresponding overall safety factors (F ) obtained from finite element analysis (slip circle A) and determined by the limit equilibrium method (slip circle B).
Table 11 Slip circles for Case 2: overall safety factors (F ) and overturning and resisting moments X0 (m)
Slip circle Aa Slip circle Bb Slip circle Ab a b
2.0 3.0 2.0
Y0 (m)
5.0 8.0 5.0
R (m)
12.94 15.04 12.94
Overturning moment, MO (kN m/m)
5694.88 9540.50 7667.20
F
Resisting moment Total MR (kN m/m)
Found. MRF (kN m/m)
Embank. MRE (kN m/m)
Geosyn. MRR (kN m/m)
6966.52 10972.20 9133.30
5894.40 8368.50 7360.60
252.12 1183.70 952.73
820.00 1420.00 820.00
1.22 1.15 1.19
Determined by the element finite method. Determined by the limit equilibrium method.
forces in the embankment and in the foundation), as can be seen from the results of the two methodologies along slip circle A. This indicates that inside of the soil mass some stress redistribution occurs causing an effect that decreases overturning and resisting moments. In fact, as shown by Borges (1995) and Borges and Cardoso (2001), during the construction period, the external forces (weight of fill soil), which are equilibrated by the internal stresses in the materials (soil and geosynthetic), determine that critical state (strength of the soil) is reached in some areas of the foundation earlier than in others. For those areas, additional external forces due to the increase of the embankment height cannot be equilibrated by stress increments. This determines that stresses have to migrate to areas where critical state has not
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412
Embankment
7m
43 31
geotextile
30
24 m
32
Soft soils
Slip circle A (F=1.15) Variation of shear strength with depth (kPa)
Slip circle B (F=1.19) Hard stratum
Fig. 15. Case 3 Critical slip circles for 7 m height and corresponding overall safety factors (F ) obtained from finite element analysis (slip circle A) and determined by the limit equilibrium method (slip circle B).
Table 12 Case 3–Slip circles for 7 m height: overall safety factors (F ) and overturning and resisting moments X0 (m)
Slip circle Aa Slip circle Bb Slip circle Ab a b
4.80 3.80 4.80
Y0 (m)
8.0 15.0 8.0
R (m)
21.83 31.47 21.83
Overturning moment, MO (kN m/m)
20638.96 63870.00 34166.03
F
Resisting moment Total MR (kN m/m)
Found. MRF (kN m/m)
Embank. MRE (kN m/m)
Geosyn. MRR (kN m/m)
23793.62 75754.50 42695.51
22255.68 68087.00 36451.01
117.94 4847.50 4824.50
1420.00 2820.00 1420.00
1.15 1.19 1.25
Determined by the element finite method. Determined by the limit equilibrium method.
been reached yet. This effect is caught by the finite element analysis but it is not obviously considered in the simplified hypotheses that support the limit equilibrium method. The results also indicate that the differences between the values of the moments in the two methods are larger for embankments with smaller values of the overall safety factor, which is the case of Cases 2 and 3 compared with Case 1. This clearly corroborates that smaller values of the overall safety factor are obviously related to larger stress redistribution inside of the soil mass, which implies larger differences between the moments obtained by the two methodologies.
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413
Embankment 43 31
geotextile
30
32
Slip surface estimated from field measurements (Quaresma,1992)
24 m Soft soils Slip circle B determined by the limit equilibrium method (F=0.99)
Variation of shear strength with depth (kPa)
Hard stratum
Fig. 16. Case 3 Critical slip circle for 8.75 m height and corresponding overall safety factor (F ) determined by the limit equilibrium method (slip circle B) and slip surface estimated from field measurements.
Table 13 Case 3 Slip circle for 8.75 m height determined by the limit equilibrium method: overall safety factor (F ) and overturning and resisting moments X0 (m)
Slip circle B
4.80
Y0 (m)
14.0
R (m)
28.80
F
Overturning moment, MO (kN m/m)
Resisting moment Total MR (kN m/m)
Found. MRF (kN m/m)
Embank. MRE (kN m/m)
Geosyn. MRR (kN m/m)
65116.60
64784.50
56419.10
5745.30
2620.10
0.99
5. Conclusions Overall stability of geosynthetic-reinforced embankments on soft soils was analysed with two different methodologies described in the paper: (i) application of a numerical model based on the finite element method (Borges, 1995); and (ii) use of a limit equilibrium method, based on the assumptions proposed by Kaniraj and Abdullah (1993) and Borges (1995). These two methodologies were applied on three geosynthetic-reinforced embankments on soft soils; one of them is a case history that was constructed up to failure. Considering the analysis of the results, the following conclusions can be formulated: The critical slip circles obtained by the two methodologies are similar for the three cases, as well as the values of the safety factor, which determines an adequate accuracy of the limit equilibrium method to assess overall safety factor in reinforced embankments on soft soils.
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414
The most important differences between the two methodologies are related to the values of the overturning and resisting moments, which are smaller in the methodology based on the numerical analysis. These differences which are larger in embankments with smaller values of the safety factor are related to some stress redistribution that occurs inside of the soil mass, simulated by the finite element method but not considered in the simplified hypotheses of the limit equilibrium method.
Appendix A. Equations of the limit equilibrium method A.1. Design overturning moment The design overturning moment MOd is contributed by the soil mass ABCEI (Fig. 7). As deduced by Kaniraj and Abdullah (1993), and adding partial safety factors, it is given by He MOd ¼ Zg K1 for Xe pn1 H þ b; ðA:1Þ 2 gHe gðH He Þ K2 MOd ¼ Zg K1 þ ðA:2Þ for Xe > n1 H þ b; 2 2 where He2 þ ðn1 X0 þ Y0 ÞHe 3 2 þ XI 2XI X0 2YI Y0 þ YI2 ;
K1 ¼ ðn21 þ 1Þ
ðA:3Þ
K2 ¼ ½b þ ðn1 þ n2 ÞHðb þ n1 H n2 He 2X0 Þ þ ðn1 þ n2 ÞX0 ðH þ He Þ þ
n22 n21 2 ðH þ He2 þ He HÞ; 3
ðA:4Þ
where XI and YI are the coordinates of point I (in this case YI ¼ 0); instead of this point, the coordinates of any other point of the circle can be taken. Xe and He are the coordinates of point E; extremity of the slip circle as shown in Fig. 7 (0pHe pH). A.2. Design resisting moment The design resisting moment MRd depends on the materials properties and can be expressed as MRd ¼ MREd þ MRFd þ MRRd ;
ðA:5Þ
where MREd is the moment due to resisting forces in the embankment along slip arc EI; MRFd the moment due to resisting forces in the foundation along slip arc MJI ; MRRd the moment due to the reinforcement resisting force that can be mobilised at point G:
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A.2.1. Design resisting moment MREd MREd can be given by (Low, 1989): c tan f þ lgHe MREd ¼ R ye R; Gc Gf
ðA:6Þ
where R is the radius of the slip circle, l is an averaging coefficient for normal stress along arc EI; He is the Y -coordinate of point E (intersection of slip circle with slope or crest) and ye (angle IOE in Fig. 7) is computed by ye ¼ sin1
Y0 ðY0 He Þ : sin1 R R
ðA:7Þ
As proposed by Low (1989), limiting its value approximately to 0.5, l can be expressed by l ¼ 0:19 þ
0:02n1 ðR Y0 Þ=He
ðA:8Þ
where n1 defines slope inclination. A.2.2. Design resisting moment MRFd The design resisting moment MRFd is computed considering the foundation divided into a number of layers with the variation of undrained shear strength su being linear in each layer along vertical direction (see Fig. 17).
Y 0 ( X0 , Y0 )
R θAi θ Bi
X
(0,0) su -A
i
A 'i
Ai
layer i B i'
Bi
su - B
i
Fig. 17. Linear variation of undrained shear strength along vertical direction in each foundation layer.
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Therefore, the partial resisting moment MRFdi due to the forces in the ith layer (acting along arc Ai Bi and its symmetric A0i B0i ) is given by Z yAi su MRFdi ¼ 2R R dy: ðA:9Þ yBi Gsu Integrating this equation (note that su does not vary linearly along arc Ai Bi ) and considering contribution of all layers, MRFd is obtained by (Borges, 1995): MRFd ¼
N X
2R2 ½mi Rðsin yAi sin yBi Þ
i¼1
þ ðmi Y0 þ bi ÞðyAi yBi Þ
1 ; Gsu
ðA:10Þ
where yAi ¼ cos1
Y0 YAi ; R
Y0 YBi ; R suAi suBi ; mi ¼ YAi YBi yBi ¼ cos1
bi ¼ suAi þ mi YAi
ðA:11Þ ðA:12Þ ðA:13Þ ðA:14Þ
with YAi ; YBi being the Y -coordinates at top and bottom of ith layer, suAi ; suBi the undrained shear strength at top and bottom of ith layer, N the number of layers in the foundation intersected by the slip circle. A.2.3. Design resisting moment MRRd Assuming that the resisting force T due to the geosynthetic acts at an angle a to the horizontal (as said above, a ¼ 0 is the most conservative assumption), MRRd is given by (Fig. 7) MRRd ¼ TR cosðyr aÞ;
ðA:15Þ
where Y0 Yc R and Xc and Yc are the coordinates of the cut point G: T is defined by the equation Tg ; Tp ; T ¼ min Gr yr ¼ cos1
ðA:16Þ
ðA:17Þ
where Tg is the geosynthetic strength and Tp the maximum force that can be mobilised at point G taking into account the strength of soil–geosynthetic interfaces reduced by their partial factors.
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b W2 n1 M
n2
W1
1
1 G( X c , Y c )
l1
l2
N
H
Z
geosynthetic
Fig. 18. Lengths of soil–geosynthetic interfaces conditioning maximum force mobilised at point G:
Because equilibrium has to be verified, Tp is calculated by the equation Tp ¼ minðTp1 ; Tp2 Þ;
ðA:18Þ
where Tp1 and Tp2 are computed from shear stress of soil–geosynthetic interfaces along segments MG and GN (Fig. 18). Therefore, Tp1 ¼
al þ au tan dl þ tan du l 1 þ W1 ; Ga Gd
ðA:19Þ
Tp2 ¼
al þ au tan dl þ tan du l 2 þ W2 ; Ga Gd
ðA:20Þ
where al ; dl are the adhesion and frictional angle of the lower soil–geosynthetic interface, au ; du the adhesion and frictional angle of the upper soil–geosynthetic interface, l1 the length of segment MG (Fig. 18), l2 the length of segment GN (Fig. 18), W1 the weight of fill soil on the segment MG; W2 the weight of fill soil on the segment GN: If g is the unit weight of the embankment material, then W1 and W2 can be given by W1 ¼ 0:5
ðXc n1 Yc Þ2 g for Xc pn1 H; n1
W1 ¼ 0:5½2Xc n1 ðH þ Yc ÞðH Yc Þg W1 ¼ W 0:5
ðA:21Þ for n1 HoXc on1 H þ b;
ðl þ n1 Yc Xc Þ2 g for Xc Xn1 H þ b; n2
W 2 ¼ W W1 ;
ðA:22Þ ðA:23Þ ðA:24Þ
418
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where W is the total weight of fill soil on the geosynthetic and l is the geosynthetic length (length of segment MN in Fig. 18) and can be expressed by W ¼ 0:5ðl þ bÞðH Yc Þg;
ðA:25Þ
l ¼ ðn1 þ n2 ÞðH Yc Þ þ b:
ðA:26Þ
If there are N number of reinforcement levels, Eqs. (A.17)–(A.26) are applied to each reinforcement, and MRRd is the sum of all moments MRRdi due to those reinforcements MRRd ¼
N X
MRRdi ¼
i¼1
N X
Ti R cosðyri ai Þ:
ðA:27Þ
i¼1
A.3. Sequence of computation Taking into account the above equations, the rotational stability analysis of a geosynthetic-reinforced embankment on soft soils can be computed as the following sequence: 1. Reading of data (see Fig. 19): (a) Geometry of the problem: n1 ; n2 ; b; H and D; (b) Data on the potential slip circles: X0 min ; Y0 min ; X0 max ; Y0 max ; Dxy and Dr (for each point O of the mesh of circle centres, several concentric circles are analysed with radius varying from a minimum value Rmin to a maximum value Rmax ; Rmin can be considered as the longer value of the distances from point O to the two extremities of the segment AB, and Rmax as the smaller value of the distances
Dxy
Y O
(X0max , Y 0max )
Dxy b C
B
(X0min , Y 0min )
n1 1
geosynthetic
n2 1
H Z
A
X
D
Hard stratum Fig. 19. Geometry of reinforced embankment on soft soils.
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419
from point O to point Z and to the hard stratum under the soft soils; Dr is radius length increment for concentric circles); (c) Properties of the foundation and embankment materials: * fill soil: unit weight, cohesion and friction angle; * foundation soils: number of levels where s is indicated (with linear variation u between two contiguous levels along vertical direction); value of su and Y coordinate for each level; * geosynthetic: number of reinforcement levels and, for each level, Y -coordinate, geosynthetic strength, adhesion and frictional angle of the lower and upper soil–geosynthetic interfaces; (d) Partial safety factors: Zg , Gc ; Gf ; Gsu ; Gr ; Ga andGd (if all these factors are imposed equal to 1, an overall safety factor is obtained); (e) Coefficient ai =yri : definition of geosynthetic force inclination; 2. For each potential slip circle, calculate: Design overturning moment MOd (Eqs. (A.1)–(A.4)); Design resisting moment MREd (Eqs. (A.7), (A.8) and (A.6)); Design resisting moment MRFd (Eqs. (A.11)–(A.14) and (A.10)); Design resisting moment MRRd (Eqs. (A.21)–(A.26), (A.19), (A.20), (A.18), (A.17), (A.16) and (A.15) (or A.27)); (e) Total design resisting moment MRd (Eq. (A.5)); (f) Value of E given by
(a) (b) (c) (d)
E ¼ MRd =MOd X1
ðA:28Þ
which is the excess of safety in relation to that assumed by the use of partial safety factors (if these factors are all equal to 1, E is the overall safety factor); 3. Selection of the critical slip circle corresponding to the smallest value of E obtained.
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