COMPARISONS OF PREDICTED AND OBSERVED FAILURE MECHANISMS IN MODEL REINFORCED SOIL WALLS

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1 Technical Paper by E.M. Palmeira and R.C. Gomes COMPARISONS OF PREDICTED AND OBSERVED FAILURE MECHANISMS IN MODEL REINFORCED SOIL WALLS ABSTRACT: This paper presents comparisons of predicted stability analyses to measured and observed results of model reinforced soil walls using theoretical design methods. The model walls were failed under a footing surcharge at different locations on the fill surface. These surcharges were compared to results that were predicted using the rigid wall approach for the reinforced soil mass, plane and circular failure surfaces and the two-wedge failure mechanism analyses. The latter method provided the best predictions in terms of surcharge values at failure and failure surfaces. The results also suggest that theoretical design methods that are not capable of accurately predicting the failure surface can severely overestimate the failure loads when the surcharge is the dominant factor for wall instability. KEYWORDS: Geosynthetics, Reinforced soil walls, Model reinforced walls, Two-Wedge failure, Circular failure, Plane failure, Limit equilibrium. AUTHORS: E.M. Palmeira, Associate Professor of Civil Engineering, University of Brasília, Brasília, DF, Brazil, Telephone: 55/ , Telefax: 55/ , palmeira@guarany.cpd.unb.br; and R.C. Gomes, Associate Professor of Civil Engineering, Federal University of Ouro Preto, Ouro Preto, MG, Brazil, Telephone: 55/ , Telefax: 55/ , romero@em.ufop.br. PUBLICATION: Geosynthetics International is published by the Industrial Fabrics Association International, 345 Cedar St., Suite 800, St. Paul, Minnesota , USA, Telephone: 1/ , Telefax: 1/ Geosynthetics International is registered under ISSN DATES: Original manuscript received 23 January 1996, revised version received 28 April 1996 and accepted 5 May Discussion open until 1 March REFERENCE: Palmeira, E.M. and Gomes, R.C., 1996, Comparisons of and Failure Mechanisms in Model Reinforced Soil Walls, Geosynthetics International, Vol. 3, No. 3, pp

2 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls 1 INTRODUCTION Geosynthetics are used extensively as reinforcing elements in reinforced soil walls and steep embankment slopes. Limit equilibrium methods are typically used during the design stage for the stability analysis of these types of structures. For retaining walls in particular, the standard stability analysis procedure requires internal and external stability analyses to be verified. Internal stability is commonly investigated by assuming plane failure surfaces in the reinforced mass. For external stability analyses, the reinforced mass is assumed to be a rigid body which is loaded by external sources such as the unreinforced surrounding soil mass and external surcharges. Additional stability analyses must also be performed, such as the bearing capacity of the foundation soil, eccentricity of the load at the wall base, and other possible general failure mechanisms. Rankine earth pressure theory is commonly used in the design of geosynthetic reinforced soil walls. It is acknowledged that this design approach leads to conservative solutions for walls with, or without, uniform distributed surcharges along the entire length of the fill surface. When localized surcharges are present, elastic and plastic solutions are combined to estimate reinforcement loads despite the conceptual contradiction of these solutions. In this study, model reinforced soil walls were loaded to failure using localized surface loads. The results of these tests are compared to theoretical predictions made by the following commonly used analyses methods that can be found in most soil mechanics and geosynthetics text books: reinforced wall as a rigid retaining wall; plane failure surface; a circular failure surface; and two-wedge failure mechanism. Despite the limitations caused by low stress levels in small models, they can still provide useful information regarding failure mechanisms. Model reinforced soil walls can also be used to calibrate theoretical design methods in a time saving, low cost and versatile manner, particularly for this study in which wall failure is caused predominantly by the footing load rather than the soil weight. Other similar, interesting studies can be found in the literature (Juran and Christopher 1989; Huang et al. 1994); however, these studies address other aspects of model geosynthetic reinforced soil structure testing. 2 EQUIPMENT, MATERIALS AND METHODOLOGY 2.1 Equipment A brief description of the model tests is presented below; for a detailed presentation the reader is referred to the paper by Gomes et al. (1994). The model reinforced soil wall tests were carried out in a rigid steel box with dimensions of 230 mm 300 mm 800 mm (Figure 1). One side of the box consisted of a thick perspex wall to allow for the observation of failure mechanisms in the soil mass. The rigid footing, that applied the localised stress on the fill surface (top of the wall), was 50 mm wide and covered the entire width of the box. The load was applied monotonically at a displacement rate of 0.76 mm/min. Load and displacement transducers were used to measure forces and displacements. Thin layers of coloured sand in the soil mass were used to locate failure mechanisms and surfaces. 330

3 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls Perspex Displacement transducer Reinforcement Figure 1. Schematic profile view of the model reinforced soil wall tests performed. 2.2 Materials The soil used to build the model walls was a uniform micaceous sand; soil properties are summarised in Table 1. The sand was pluviated using a hopper in order to achieve a uniform and dense state. For the stress range observed in the model tests, the internal friction angle of the sand varied markedly with stress level and this variation was taken into account in the analyses performed. Figure 2 presents the variation of the peak internal friction angle, Ô, and dilation angle, ψ, of the sand with vertical stress obtained from direct shear tests. The dilation angle of the sand was obtained using a modified direct shear box, as described by Jewell (1980). Table 1. Properties of the sand used in the model reinforced soil wall tests. Property Value Unit weight, γ (kn/m 3 ) 16 Particle specific gravity, G (dimensionless) 2.63 Void ratio, e ( dimensionless) 0.60 Mean particle diameter, d 50 (mm) 1.27 Coefficient of uniformity, C u (dimensionless)

4 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls Internal friction angle, Ô _ ( ) Dilation angle, ψ Normal stress (kn/m 2 ) Figure 2. Internal friction and dilation angles versus normal stress for the sand used in the model reinforced soil walls. To obtain the expected mobilised internal friction angle of the sand in the region of the failure surface being investigated, the mean stress level, between the footing base and the lowest point of the failure surface, was estimated assuming a trapezoidal stress distribution with depth (spreading angle of 26.6_ from the vertical). Data from Figure 2 was then used to obtain the internal friction angle. For the soil mass outside of the trapezoidal region the internal friction angle corresponding to the lowest stress level was used. It is acknowledged that some inaccuracies may arise from this approach; however, results from previous studies (Lanz 1992; Gomes et al. 1994) demonstrated that this procedure was acceptable for the main purpose of the present work, which is to compare different methods of analysis under the same conditions. Lanz (1992) and Gomes et al. (1994) used the same sand and performed back-analyses of surcharge values at failure, q, using different stability analysis methods, and obtained a good fit with the observed failure mechanisms. Similar methodologies to estimate the mobilised internal friction angle have been used by other researchers while studying model reinforced soil structures (Burd 1986; Juran and Christopher 1989). It could also be argued that plane strain internal friction angles based on coaxiality between stress and strain increment directions for the sand should be preferred to direct shear internal friction angles, as suggested by Jewell and Wroth (1987). However, coaxiality between principal stress and strain increments is unlikely to occur under the footing due to large rotations of the principal stress directions. 332

5 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls Progressive failure mechanisms may also occur in model testing with sand which may then affect the value of the mobilised friction angle to be used in the analysis. The internal friction angle obtained in direct shear tests is likely to be already affected by a progressive failure mechanism since this is known to occur in direct shear tests on dense sands (Palmeira 1987). The extreme case of using the constant volume internal friction angle in the analysis demonstrated that the predicted values of footing pressures at failure were much smaller than the values observed, suggesting that progressive failure may not have been severe in the model tests. Huang and Tatsuoka (1994) discuss the influence of progressive failure and other factors on the internal friction angle to be used in stability analyses of model sand slopes. Nine different reinforcement materials were used in this study, as reported in Gomes et al. (1994). To avoid repetition of similar test results for types of reinforcement with similar characteristics, five out of the nine types of reinforcement originally tested were chosen for the present analyses based on characteristics such as tensile strength, stiffness and adherence between the soil and reinforcement. Table 2 summarises the main reinforcement characteristics of the five chosen representative types of reinforcement. The reinforcement terminology used by Gomes et al. (1994) is used in the current study for consistency. From Table 2 it can be observed that a wide range of reinforcement tensile strengths (0.1 to 1.1 kn/m) and stiffnesses (0.16 to kn/m) were investigated. These results were obtained from wide-width strip tensile tests (ASTM D 4595) conducted at 20_C and at a strain rate of 3%/minute using 200 mm specimens. The mean adherence factors between the soil and reinforcement, f, (ratio between the soil-reinforcement interface friction angle, Ô sr, and the soil friction angle, Ô) arealso presented in Table 2. These values varied between 0.51 and 0.71 depending on the type of reinforcement. Assuming a scale factor of 20, the model walls simulate 4.8 m high prototype walls with reinforcement tensile strengths varying between 40 and 440 kn/m, which is well within the typical range for geosynthetic reinforcement in practice. Table 2. Model wall reinforcement characteristics. Type of reinforcement Thickness (mm) Mass per unit area (g/m 2 ) Tensile strength (1) (kn/m) J (2) (kn/m) f (3) Aluminium foil Paper sheet Plastic A Plastic C Notes: PVC = polyvinyl chloride. (1) From wide-width strip tensile tests at a strain rate of 3%/minute (ASTM D 4595). (2) J = secant stiffness corresponding to 50% of the strain at failure. (3) Adherence factor, f = tanô sr /tanô. 333

6 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls 2.3 Methodology The model walls used were 240 mm high and the reinforcement length and spacing were 150 mm and 40 mm, respectively (Figure 1). The construction technique used to build the walls was similar to the construction techniques used in real works, except for the difference in soil compaction. The layers of reinforcement were laid on the soil with one end resting on a temporary vertical supporting plate that covered the entire box width. The height of the plate was 1.6 times the reinforcement spacing and prevented the sand and reinforcement end from falling during pluviation. After the sand layer was completed, the reinforcement end was folded over and the vertical plate lifted to allow for preparation of the next soil layer until the final wall height was achieved. All walls were constructed on the rigid base of the box. Tests performed by Gomes (1993), using different lubrication techniques for the box side walls, demonstrated that the influence of side friction on the test results was negligible. The distance between the footing edge and the wall face, d, was varied to provide three different footing positions that are referred to as d/b ratios of 0.2, 0.67 and 1.2 (where B is equal to wall width). Figure 3 shows typical results of footing pressure, q, versus the ratio between footing displacement, w, and footing width, b, for tests with d/b = 0.2. Table 3 presents the peak values of q for all the tests performed. Additional information on the test results have been reported by Gomes (1993) and Gomes et al. (1994). d b q w q (kn/m ) 2 B Figure 3. Typical results of footing pressures, q, versus the ratio of footing displacement and width, w/b,ford/b =

7 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls Table 3. Summary of peak surcharge values for all of the tests performed. Reinforcement Aluminium sheet d/b q mes (kn/m 2 ) q pred for different analysis methods (kn/m 2 ) Rigid wall Plane Two-wedge (1) Circular Paper sheet Plastic A Plastic C Notes: q mes = measured surcharge on the top of the wall; q pred = predicted surcharge on the top of the wall. (1) The dash separates values of q pred for δ =0andδ = Ô, respectively, where δ = wedge interface friction angle. 3 THEORETICAL DESIGN METHODS As mentioned earlier the aim of this study is to compare model wall test results to predictions made by commonly available design procedures. Four different theoretical design approaches were investigated and are presented schematically in Figures 4a to 4d. The main characteristics of these theoretical design approaches are described in this section. For all of the theoretical design methods employed in this study, the reinforcement layers were assumed to remain horizontal during the loading stages. Computer programs were written to perform the necessary calculations. The results obtained are discussedinsection Reinforced Wall as a Rigid Retaining Wall (Figure 4a) In this case the reinforced wall was considered to be a rigid retaining wall (Figure 4a) and its internal and external stability were assessed as described in soil mechanics and geosynthetic text books (John 1987; Koerner 1990) and some geosynthetic manufacturers manuals. The external stability was investigated in terms of factors of safety against sliding along the wall base, and overturning. Soil active earth pressures were calculated using Rankine theory. Increments of horizontal stress, due to footing loads, were calculated using elastic solutions for a uniform strip load applied to the surface of a semi-infinite medium (Poulos and Davis 1974), as shown in Figure 4a. Internal stability was verified by estimating the reinforcement loads to be equal to the section of the horizontal stress diagram area corresponding to each reinforcement layer. This load was then compared to the reinforcement tensile stress and bond strength available along the rein- 335

8 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls (a) K (b) (c) A (d) Figure 4. The different theoretical design methods used in the stability analyses: (a) reinforced mass as a rigid retaining wall; (b) plane failure surface; (c) two-wedge failure mechanism; (d) circular failure surface. 336

9 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls forcement anchorage length (the segment of the reinforcement beyond the Rankine critical plane). The smallest value of q that caused sliding of the wall, overturning, or reinforcement tensile or anchorage failure was considered to be the critical surcharge for this method. 3.2 Plane Failure Surface (Figure 4b) This approach considers that failure will occur along a single plane. The plane orientation is varied to find the plane which gives the minimum factor of safety, while obeying equilibrium requirements for the wedge defined by the sliding plane and the wall boundaries (Figure 4b). The factor of safety is defined as the ratio between resisting and driving forces acting on the surface of the plane. The resisting force of each layer of reinforcement was chosen to be the smaller value of tensile strength, or bond strength along the anchorage length. 3.3 Two-Wedge Failure Mechanism (Figure 4c) This mechanism is shown in Figure 4c and assumes that the failure surface comprises two linear segments. The factor of safety is defined as the ratio between the available shear strength and the mobilised shear stress along the failure surface that closes the force polygon for both wedges. Several failure surfaces were investigated by varying the position of point A along an established mesh, and by varying the inclination of the base of the wedges to the horizontal, θ 1 and θ 2 (Figure 4c). The factor of safety also depends on the inclination, δ, of the thrust, E, between the wedges and the horizontal. Cases where δ = 0 (conservative) and δ = Ô were investigated, where Ô is the mobilised soil internal friction angle. 3.4 Circular Failure Surface (Figure 4d) For this case the modified Bishop s method was employed to calculate the factor of safety for circular failure surfaces (Figure 4d). The normal and tangential forces in the reinforcement (either the tensile, or the bond strength, whichever is smallest) were incorporated into the equilibrium equations for each corresponding slice. A large number of circles, with centres on an established mesh, and varying radii were investigated to determine the critical circular failure surface. Increments of vertical stress along the reinforcement anchorage length, due to the surcharge on the surface, were estimated using an elastic solution for bond strength calculations, as mentioned in Section COMPARISON OF PREDICTED AND MEASURED RESULTS 4.1 Reinforced Mass as a Rigid Retaining Wall Figure 5 and Table 3 summarise the comparison of q mes and q pred values on the top of the wall at failure, assuming a reinforced mass as a rigid retaining wall. The results show that the predicted values are closer to the measured values for d/b = 0.2 when the rein- 337

10 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls 2 q pred (kn/m ) Aluminium sheet Paper sheet Plastic A Plastic C q mes (kn/m 2 ) Figure 5. Comparison of predicted and measured surcharge failure values using the rigid retaining wall theoretical design method. forcement length is large in comparison to the size of the region affected by the failure mechanism (i.e. a large anchorage length). If λ is defined as the ratio of q pred to q mes values, it can be observed that the test values typically fall within the range 0.7 λ 1.3, and for most cases within 0.7 λ 1. However, for values of d/b > 0.2, the q pred values severely overestimated the q mes values at failure due to the difference between the actual failure surface and the predicted Rankine failure plane. For all cases where d/b = 0.2 or 0.67, the instability mechanism predicted by this method was tension failure of the top reinforcement layers. For d/b = 1.2, the instability mechanism predicted was wall overturning. With regard to the values in Table 3, it is interesting to note that the q pred values are closer to the q mes values for stiffer types of reinforcement, which is consistent with limit equilibrium theory. 4.2 Plane Failure Surface The comparison of q pred and q mes surcharge values at failure, assuming plane failure surfaces, are presented in Table 3 and Figure 6. Again, the q pred values are closer to the q mes values for the ratio d/b = 0.2, and the larger the d/b ratio the more this design procedure overestimates the value of q at failure. The cause of this overestimation can be seen 338

11 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls 2 q pred (kn/m ) Aluminium sheet Paper sheet Plastic A Plastic C q mes (kn/m 2 ) Figure 6. Comparison of predicted and measured surcharge failure values using the plane failure surface theoretical design method. in Figure 7 where the predicted failure planes are compared to the observed failure mechanisms. It should be pointed out that the observed shear zones in the models were typically 15 to 20 mm thick; however, for the sake of clarity, only the center of the shear zones are labelled using a straight line (Figure 7). For d/b > 0.2, the observed failure mechanism is very poorly simulated using a plane failure surface, which leads to errors in the q pred values. 4.3 Two-Wedge Failure Mechanism Figures 8 and 9, and Table 3 present comparisons of q pred and q mes footing surcharge values and failure mechanisms assuming a two-wedge failure design procedure. The bi-linear failure surface in this approach, although still rather crude, provides more freedom to capture more critical failure mechanisms than the methods described in Sections 4.1 and 4.2. It can also define failures along the reinforcement plane, which occurred in some of the models tested. From Figure 8 it can be observed that 0.7 λ 1 in most cases. The increase in angle δ from zero to the soil friction angle, Ô, appears to increase the accuracy of the q pred values (λ is closer to unity) which is typically assumed in earth pressure or slope stability analyses using two-wedge solutions. However, depending on 339

12 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls (a) Paper (predicted) Aluminium (predicted) Plastics A and C (predicted) PVC (predicted Aluminium and paper sheets and plastics A and C (b) Aluminium and paper sheets and plastic C (c) Aluminium sheet Figure 7. Comparison of predicted and observed failure mechanisms using the plane failure surface theoretical design method: (a) d/b =0.2;(b)d/B = 0.67; (c) d/b =1.2. the inclination of the failure surface to the horizontal, the value of Ô and the reinforcement tensile strength, negative values of E can be obtained which may lead to numerical instability or erroneous values of q at failure. When δ = Ô, the value of E reduces even more, making the situation worse. This problem was observed in a few of the cases analysed and was mainly caused by high sand internal friction angles and low stress levels in combination with a strong reinforcement (e.g. aluminium sheet). Even though fullscale reinforced walls are likely to be less affected by this problem, the value of E (i.e. whether it is positive or negative) should always be assessed before concluding that the calculated factor of safety of the structure is reasonable when this design methodology is used. Figure 9 shows that, for all of the different types of reinforcement and d/b values analysed, a reasonably good agreement is observed between the predicted two-wedge failure mechanisms and the observed failure mechanisms. Good agreement between the 340

13 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls Aluminium sheet Paper sheet Plastic A Plastic C 2 q pred (kn/m ) q mes (kn/m 2 ) Figure 8. Comparison of predicted and measured surcharge failure values using the two-wedge theoretical design method. q pred and q mes values using this design method has also been reported by Tei (1993) (centrifuge tests) and by Lanz (1992) and Gomes et al. (1994) for walls with uniformly distributed surcharges on the top surface. 4.4 Circular Failure Surface Figures 10 and 11, and Table 3 summarise the results obtained for the analyses using a circular failure surface design method (modified Bishop s method). Good agreement between q pred and q mes values at failure (Figure 10) were obtained only for the cases where the predicted failure surfaces compared well to the observed failure surfaces. This occurred for the case of d/b = 0.2, as shown in Figure 11. For the case of d/b =1.2, the predicted failure surface models the shape of the observed failure surface, but the q pred values still overestimate the q mes values. The more the predicted failure surface differs from the observed failure mechanism the greater the overestimation of q values at failure. Figure 12 shows comparisons between predicted and observed failure mechanisms for plane, circular and bi-linear surfaces. As mentioned in Section 4.3, the two-wedge 341

14 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls (a) Aluminium sheet Paper sheet Plastics A and C (b) Plastic C (predicted) PVC (predicted) Aluminium and paper sheets and plastic C (c) Aluminium sheet Figure 9. Comparison of predicted and observed failure mechanisms using the twowedge theoretical design method, and δ =0:(a)d/B =0.2;(b)d/B = 0.67; (c) d/b =1.2. method remains the most versatile of the design methods used in this study with regard to predicting failure surfaces. 342

15 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls 2 q pred (kn/m ) Aluminium sheet Paper sheet Plastic A Plastic C q mes (kn/m 2 ) Figure 10. Comparison of predicted and measured surcharge failure values using the circular failure theoretical design method. 5 CONCLUSIONS This paper presents a comparison of observed failure mechanisms in reinforced model walls to predicted failure mechanisms based on simple stability analysis methods. The conclusions of this study are summarised below: S The type of failure mechanism observed is dependent on the position of the wall surface footing (d/b); some of the design methods employed do not properly consider this factor. When the reinforced mass was assumed to be a rigid retaining wall and the reinforcement length was long enough to ensure tension failure in the reinforcement, the best agreement between q pred and q mes values occurred for cases in which d/b = 0.2. For cases in which d/b = 0.67, this approach severely overestimated the value of q at failure, while for d/b =1.2,theq pred values were conservative. S plane failure surfaces differed considerably from the observed failure mechanisms for d/b = 0.67 and 1.2, and in both cases the q pred values also overestimated the q mes value. 343

16 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls (a) Paper (predicted) Aluminium (predicted) Plastics A and C (predicted) PVC (predicted) Aluminium and paper sheets and plastics A and C (b) Plastic C (predicted) PVC (predicted) Aluminium and paper sheets and plastic C (c) Aluminium sheets Figure 11. Comparison of predicted and observed failure mechanisms for a circular failure surface: (a) d/b =0.2;(b)d/B = 0.67; (c) d/b =1.2. S Similar to the results for plane surfaces, the same disparity between predicted and observed circular failure surfaces led to an overestimation of q values at failure. When the predicted circular failure surface simulated the observed failure surface the q pred values were close to the q mes values. S Predictions using the two-wedge design method were in general closer to the q mes values and the observed failure surfaces. This was due to the greater versatility of this approach to accommodate different types of failure surfaces in an approximate manner. However, the possibility of obtaining negative inter-wedge forces and numerical instabilities, that may occur in the analysis of problems with high soil friction angles or cohesive fill materials and strong reinforcements, must be avoided. Most of the q values obtained using this approach fell within the range 0.7 to 1 times the q mes values leading to conservative predictions when the inter-wedge force was horizontal. 344

17 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls (a) Aluminium sheets (b) Aluminium sheets (c) Aluminium sheets Plane Bi-linear Circular Figure 12. Comparison of predicted and observed failure mechanisms for aluminium and reinforcement sheets for plane, circular and bi-linear failure surfaces: (a) d/b =0.2;(b)d/B = 0.67; (c) d/b =1.2. S Despite the small number of test results obtained, the results suggest that any of the design methods used in this study may provide reasonably good preliminary predictions of factors of safety if the failure surface is accurately determined. Caution must be exercised when extrapolating model results to real structures, but the results of this study also suggest that the rigid wall, plane and circular failure surfaces may overestimate the value of q at failure for walls where the major cause of instability is the surcharge on the fill surface rather than the fill weight. For the preliminary stability analysis of these types of structures, the two-wedge method appears to be the most appropriate theoretical design method among those investigated in this study. 345

18 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls ACKNOWLEDGEMENTS This work was made possible by the financial and material support provided by the University of Brasília, Federal University of Ouro Preto, Brazilian Research Council (CNPq), CAPES/Brazilian Ministry of Education and Rhodia S.A. REFERENCES ASTM D 4595, Standard Test Method for Tensile Properties of Geotextiles by the Wide-Width Strip Method, American Society for Testing and Materials, West Conshohocken, Pennsylvania, USA. Burd, H.J., 1986, A Large Displacement Finite Element Analysis of a Reinforced Unpaved Road, D. Phil. Thesis, University of Oxford, UK, 221 p. Gomes, R.C., 1993, Interaction Between Soil and Reinforcement and Failure Mechanisms in Geosynthetic Reinforced Soil, Ph.D. Thesis, University of Sao Paulo, USP- SC, Brazil, 270 p. (in Portuguese) Gomes, R.C., Palmeira, E.M. and Lanz, D., 1994, Failure and Deformation Mechanisms in Model Reinforced Walls Subjected to Different Loading Conditions, Geosynthetics International, Vol. 1, No. 1, pp Huang, C. and Tatsuoka, F., 1994, Stability Analysis for Footings on Reinforced Sand Slopes, Soils and Foundations, Vol. 34, No. 3, pp Huang, C., Tatsuoka, F. and Sato, Y., 1994, Failure Mechanisms of Reinforced Sand Slopes Loaded with a Footing, Soils and Foundations, Vol. 34, No. 2, pp Jewell, R.A., 1980, Some Effects of Reinforcement on the Mechanical Behaviour of Soils, Ph.D. Thesis, University of Cambridge, Cambridge, UK, 320 p. Jewell, R.A. and Wroth, C.P., 1987, Direct Shear Tests on Reinforced Sand, Geotechnique, Vol. 37, No. 1, pp John, N.W.M., 1987, Geotextiles, Blackie and Son Ltd, Glasgow and London, UK, 347 p. Juran, I. and Christopher, B., 1989, Laboratory Model Study on Geosynthetic Reinforced Soil Retaining Walls, Journal of Geotechnical Engineering, Vol. 115, No. 7, pp Koerner, R.M., 1990, Designing with Geosynthetics, Second Edition, Prentice-Hall Inc., Englewood Cliffs, New Jersey, USA, 652 p. Lanz, D., 1992, A Study on the Deformation and Failure of Geotextile Reinforced Walls, M.Sc. Thesis, University of Brasília, Brazil, 150 p. (in Portuguese) Palmeira, E.M., 1987, The Study of Soil-Reinforcement Interaction by Means of Large Scale Laboratory Tests, Ph.D. Thesis, University of Oxford, UK, 238 p. Poulos, H.G. and Davis, E.H. 1974, Elastic Solutions for Soil and Rock Mechanics, John Wiley and Sons, New York, New York, USA, 411 p. Tei, K., 1993, A Study of Soil Nailing in Sand, Ph.D. Thesis, University of Oxford, UK, 256 p. 346

19 PALMEIRA AND GOMES D and Failures in Model Reinforced Soil Walls NOTATIONS Basic SI units are given in parentheses. B = width of the reinforced soil mass or reinforcement length (m) b = width of external surcharge (footing) (m) C u = soil coefficient of uniformity (dimensionless) d = distance between the front edge of the footing and the wall face (m) d 50 = mean particle diameter (m) E = inter-wedge or inter-slice force (N/m) e = void ratio (dimensionless) f = adherence factor between soil and reinforcement (= tanô sr /tanô) (dimensionless) G = soil particle specific gravity (dimensionless) H = wall height (m) J = secant stiffness of reinforcement corresponding to 50% of the strain at failure (N/m) K a = Rankine active earth pressure coefficient (dimensionless) q = surcharge on the top of the wall (N/m 2 ) q mes = measured surcharge on the top of the wall at failure (N/m 2 ) q pred = predicted surcharge on the top of the wall at failure (N/m 2 ) Q 1, Q 2 = resultant forces from surcharges on wedges 1 and 2, respectively (N/m) R = force at the base of a wedge or slice (N/m) S = reinforcement spacing (m) T = tensile force in the reinforcement (N/m) w = vertical footing displacement (m) W = total weight of soil wedge or slice per unit width (N/m) x, y = coordinates (m) α, β = angles for elastic stress increment calculations (_) γ = soil unit weight (N/m 3 ) δ = inclination of the force between wedges to the horizontal (interface friction angle) (_) λ = ratio between predicted and observed surcharge values (dimensionless) σ x = horizontal stress increment due to footing load (N/m 2 ) σ y = vertical stress increment due to footing load (N/m 2 ) θ 1, θ 2 = inclination of the base of the wedges to the horizontal (_) Ô = soil internal friction angle (_) Ô sr = interface friction angle between soil and reinforcement (_) ψ = soil dilation angle (_) 347

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