finite element study of a geosynthetic-reinforced soil ...
Technical Paper by H.I. Ling, C.P. Cardany, L-X. Sun and H. Hashimoto
FINITE ELEMENT STUDY OF A GEOSYNTHETIC-REINFORCED SOIL RETAINING WALL WITH CONCRETE-BLOCK FACING ABSTRACT: This paper focuses on the results of finite element modeling of a full-scale, concrete-block, geosynthetic-reinforced soil retaining wall constructed at the Public Works Research Institute in Japan. A nonlinear hyperbolic geosynthetic model was incorporated into a computer program that is capable of simulating soil-structure interaction behavior. The soil was simulated using a hyperbolic model while the block-block and soil-block interactions were simulated using interface elements. Comparison of numerical and measured experimental results indicated that the finite element model is capable of simulating the construction behavior of concrete-block geosynthetic-reinforced soil structures. KEYWORDS: Geosynthetic, Geogrid, Reinforced soil retaining wall, Concrete block, Facing, Finite element method, Soil-structure interaction. AUTHORS: H.I. Ling, Associate Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University, 500 West 120th Street, New York, New York 10027, USA, Telephone: 1/212-854-1203, Telefax: 1/212-854-6267, E-mail: [email protected]; C.P. Cardany, Graduate Student, Department of Civil Engineering and Engineering Mechanics, Columbia University, 500 West 120th Street, New York, New York 10027, USA, and Senior Staff Engineer, Langan Engineering and Environmental Service, Inc., Telephone: 1/203-562-5771, Telefax: 1/203-789-6142, E-mail: [email protected]; L-X. Sun, Graduate Research Assistant, Department of Civil Engineering and Engineering Mechanics, Columbia University, 500 West 120th Street, New York, New York 10027, USA, Telephone: 1/212-854-4153, Telefax: 1/212-854-6267, E-mail: [email protected]; and H. Hashimoto, Research Engineer, Construction Engineering Division, Public Works Research Institute, Ministry of Construction, Tsukuba, Japan, Telephone: 81/298-64-4703, Telefax: 81/298-64-0564, E-mail: [email protected]. PUBLICATION: Geosynthetics International is published by the Industrial Fabrics Association International, 1801 County Road B West, Roseville, Minnesota 55113-4061, USA, Telephone: 1/651-222-2508, Telefax: 1/651-631-9334. Geosynthetics International is registered under ISSN 1072-6349. DATES: Original manuscript received 29 December 1999, revised version received 29 March 2000 and accepted 2 April 2000. Discussion open until 1 January 2001. REFERENCE: Ling, H.I., Cardany, C.P., Sun, L-X. and Hashimoto, H., 2000, “Finite Element Study of a Geosynthetic-Reinforced Soil Retaining Wall With Concrete-Block Facing”, Geosynthetics International, Vol. 7, No. 3, pp. 163-188.
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1
INTRODUCTION
Modular (segmental) blocks are widely used in North America for construction of geosynthetic-reinforced soil retaining walls (GRS-RWs) (Bathurst and Simac 1994; Collin 1997). Modular-block GRS-RW structures offer a wide range of aesthetic hardfaced finishes, while resulting in more economical structures compared to traditional gravity wall structures. There are numerous proprietary modular-block wall systems available that are commonly used for private and public sector retaining wall projects. In a modular-block wall, the geosynthetic layers are placed between the concrete blocks at regular vertical spacings. The blocks are used to retain the backfill. The integrity of the wall facing is maintained through the interaction between the blocks, blocks and backfill, as well as between the blocks and geosynthetic layers. Modular-block (segmental) units are typically connected to geosynthetic reinforcement layers using mechanical and/or frictional connection devices such as polymeric pins, inserts, clips, or concrete shear keys. The dead weight of the modular blocks contributes to the global stability of the GRSRW. The contribution of the facing was neglected in early designs; then, Bathurst and Simac (1994) and Leshchinsky (1993) proposed limit equilibrium design methodologies to account for the modular blocks. However, limit equilibrium methods do not allow the wall deformation and strain in the geosynthetic reinforcement to be evaluated directly during analysis and design. The present paper describes a finite element study of a well-instrumented geogridreinforced soil retaining wall constructed with a concrete-block facing. This study focuses on the numerical modeling technique used and adaptations that may apply to the study of modular-block GRS-RWs. The measured and predicted soil stresses, wall deformations, and tensile strain in the reinforcement layers are presented and compared. 2
GEOSYNTHETIC-REINFORCED SOIL RETAINING WALL AT THE PUBLIC WORKS RESEARCH INSTITUTE (PWRI WALL)
Three geosynthetic-reinforced soil retaining walls were constructed at the Public Works Research Institute (PWRI), Ministry of Construction, in Japan. Each wall had a different facing structure: concrete blocks, discrete panels, or expanded polystyrene (EPS) blocks. The walls were fully instrumented and subsequently failed by cutting the geosynthetic layers in stages (Miyatake et al. 1995; Tajiri et al. 1996). The present study focuses on the wall with concrete-block facing (hereafter referred to as the PWRI Wall) and its behavior during construction. The PWRI Wall is one of the very few full-scale geosynthetic-reinforced soil structures that has been fully instrumented. The material properties were also well defined. A brief summary of the PWRI Wall and instrumentation program is described in this section. The geometry of PWRI Wall is shown in Figure 1. It is 6 m high and 5 m wide and was constructed in a concrete test pit with a concrete floor. The side walls of the test facility were lubricated using grease and polymer sheets. A silty sand (mean particle diameter, D50 = 0.42 mm; unit weight, γ = 16.0 kN/m3) was used as the backfill. The particle size distribution of the backfill soil is shown in Figure 2. The stress-strain properties of the sand are described in Section 3.
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Secondary geogrid layers 350 3,500 450
Primary geogrid layers
6 5 5
1,000
4 4 5,000
3 3 2 2 1
1,000
270 550
1
1,000
1,000 1,000 250 Linear variable displacement transducer (LVDT) Load cell Strain gage All dimensions in mm
Figure 1. PWRI Wall: geometry and instrumentation.
A uniaxial geogrid (Tensar SR55), manufactured from extruded high-density polyethylene (HDPE), was used as the soil reinforcement. The spacings between the longitudinal and transverse ribs were 22 and 166 mm, respectively. The strength of the geogrid was approximately 55 kN/m. The PWRI Wall consisted of six primary and five secondary geogrid layers, 3.5 and 1.0 m long, respectively. The geogrid layers were bolted to the concrete blocks with the bolt and metal frame arrangement shown in Figure 3. A total of 12 concrete block rows were used to construct the wall face. Each block was 500 mm high and 350 mm wide (toe-to-heel dimension), except the top and bottom blocks, which were 450 and 550 mm high, respectively.
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Fraction passing by mass (%)
100 80 60 40 20 0 0.001
0.01
0.1
1.0
10
Particle diameter (mm) Figure 2. Particle size distribution of the backfill soil.
(a)
200 Load cell 500
75 1,000 350
(b) Load cell
500
Geogrid
All dimensions in mm
Metal frame
Bolt
Figure 3. Connection between the modular blocks and geosynthetic layers: (a) back of concrete block; (b) cross-section.
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A total of 52 strain gages were used to measure elongation of the geogrid, i.e. seven and two strain gages on each layer of primary and secondary reinforcement, respectively. The horizontal displacement of the wall face was measured at 11 locations using linear variable displacement transducers (LVDTs). The lateral pressure acting against the back of the facing blocks was measured using 11 load cells installed at the mid-height of each concrete block. The vertical and horizontal loads acting at the toe of the facing were also measured using load cells. The vertical pressure due to the backfill soil was measured at six locations along the base of the soil mass. 3
FINITE ELEMENT STUDY OF PWRI WALL
The finite element method has been used to analyze different types of geotechnical structures, such as earth dams, embankments, shallow and deep foundations, slopes, and retaining walls. The application of the finite element method to reinforced soil structures is relatively recent. Reinforced soil is a complex system that involves interactions between different structural components and soil. Since the procedure itself is very sophisticated, the application to design is rare. However, finite element analysis renders additional information compared to traditional limit-equilibrium analysis, such as deformation and tensile load in the reinforcement layers, that are necessary for understanding the performance of reinforced soil structures. The results obtained from finite element analyses may be used to guide the development of more accurate limit-equilibrium design procedures. Finite element analyses of reinforced soil structures can be conducted using computer programs that simulate soil-structure interaction, i.e. programs that have the relevant soil and structural elements and material models. The program should be able to simulate the construction sequences, such as backfilling and the installation of reinforcement layers and wall facings. In the early 1970s, when finite element analyses were limited to only mainframe computers at high computing costs, a composite approach was proposed to model soil-reinforcement systems (Romstad et al. 1976; Shen et al. 1976). Modern finite element analysis methods use a discrete approach because computing costs have significantly reduced. In a discrete analysis, the reinforcement, soil, facing, and their interactions are modeled separately. Previous finite element studies of reinforced soils focused on numerical parametric studies (Kaliakin and Xi 1992; Rowe and Ho 1997). There were also attempts to reproduce the results of laboratory models and full-scale tests (Al-Hussaini and Johnson 1978; Collin 1986; Seed et al. 1986; Wu and Monley 1989; Bathurst et al. 1992; Chew and Mitchell 1994; Zornberg and Mitchell 1994; Cai and Bathurst 1995; Karpurapu and Bathurst 1995; Ling et al. 1995). A collection of finite element analysis predictions for a reinforced soil structure can be found in the proceedings of symposium that was edited by Wu (1991). Nevertheless, there have been a very limited number of well-instrumented, full-scale tests leading to the verification of finite element procedures. With the exception of the study by Cai and Bathurst (1995), the finite element study of reinforced soil walls with a modular-block facing, which is a relatively new type of reinforced soil system, has not been previously reported. In the present study, the analysis of the PWRI Wall was conducted using the finite element program M-CANDE (Ling and Tatsuoka 1991; Ling et al. 1995; Ling and
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Leshchinsky 1995), which is a modified version of the program CANDE (Katona 1976) that was developed for the design and analysis of buried culverts. The program carries out plane strain analyses under static loading conditions. The program can simulate the construction sequence and the inclusion of reinforcement and interface elements in each solution increment. In each increment of analysis, the solution is iterated to a userselected convergence criterion at the element level. A geosynthetic model was added to CANDE following previous analyses of reinforced soil structures (Ling et al. 1995). Creep (viscoelastic) phenomenon cannot be directly modeled using the current code. The finite element mesh (1,618 nodes) and interface elements used to simulate the PWRI Wall are shown in Figures 4a and 4b, respectively. The foundation, backfill, and facing blocks were represented by quadrilateral elements: 315 elements for the foundation, 925 elements for the backfill, and 37 elements for the concrete facing blocks. A total of 127 elements, corresponding to the primary and secondary geosynthetic reinforcement layers, were included in the analysis. A total of 110 interface elements were used to simulate the interactions between different materials at the wall facing. The reinforcement layers were assumed to be fully bonded to the backfill soil. The analysis was conducted in 38 steps to simulate wall construction. 4
MATERIAL MODELS AND CALIBRATION
4.1
Backfill Soil
The backfill was modeled using a nonlinear elastic soil model (Duncan et al. 1980) with a variable Young’s modulus and a constant Poisson’s ratio. The tangent Young’s modulus is expressed as follows: E tan = Ei
E i = K pa (σ 1 − σ 3 )f =
σ1 − σ3 1 − Rf (σ 1 − σ 3 ) f
σ3 pa
2
(1)
n
(2)
2c cos Ô + 2σ3 sin Ô 1 − sin Ô
σ Ô = Ô o − ∆Ô p 3 a
(3)
(4)
where: Etan = tangent Young’s modulus; Ei = initial Young’s modulus; K = modulus number; n = modulus exponent; Rf = failure ratio; Ô = internal friction angle of the soil; Ôo = initial value of the internal friction angle of the soil; ∆Ô = change of the Ô value with a ten-fold increase in the confining stress; pa = atmospheric pressure; c = soil cohesion; (σ1 - σ3 )f = deviator stress at failure; and σ1 and σ3 = major and minor principal stresses, respectively. Triaxial compression tests were conducted at Columbia Univer-
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(a)
1.50
2.50
1.00
0.35 0.45
5.00
All dimensions in m
0.55 0.27
3.00
10.00 (b)
Soil Concrete blocks Geosynthetic Interface
Interface Figure 4. PWRI Wall: (a) finite element mesh; (b) interface elements at the wall facing.
sity (New York, New York, USA) using dry specimens replicating the test wall condition. Hyperbolic model parameters were selected to fit triaxial test results and are summarized in Table 1. A comparison of measured and predicted triaxial test results is shown in Figure 5.
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Table 1. Material properties used in the finite element analyses of the PWRI Wall. Backfill γ (kN/m3)
Ôo (_)
16.0
45
∆Ô (_)
K
n
Rf
ν
5
207.2
0.5
0.81
0.42
Concrete (foundation and facing) γ (kN/m3)
E (kPa)
ν
-
2.0 × 106
0.17
23
2.0 × 106
0.17
Geogrid reinforcement Jo (kN/m)
Tf (kN/m)
λ
826.45
54.6
0.47
Notes: γ = unit weight of the soil; Ôo =initial value for the internal friction angle of the soil; ∆Ô = change of φ with a 10-fold increase in the confining stress; K = modulus number; n = modulus exponent; Rf = failure ratio for soil strength; ν = Poisson’s ratio; E = Young’s modulus of concrete; Jo = initial stiffness of reinforcement; Tf = reinforcement tensile load per unit width at failure; λ = failure ratio for tensile strength of reinforcement.
Deviator stress, q = σ1 --- σ3 (kPa)
σ3 = 100 kPa
Hyperbolic model Measured σ3 = 50 kPa
σ3 = 25 kPa
K = 207.2, n = 0.5 Ôo = 45_, ∆Ô = 5_, Rf = 0.81
Axial strain, ε (%) Figure 5. Predicted (hyperbolic model) and measured triaxial test results.
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No provision was made to measure the volume change during triaxial testing. It is well known that the hyperbolic model is suitable for analysis under working stress conditions. The model implemented in the M-CANDE code expresses the deviator stressaxial strain relationships satisfactorily, but does not account for the dilatant behavior of soil. In addition, compaction stresses cannot be modelled directly. A constant Poisson’s ratio (ν = 0.42) was selected for the soil to account for possible compaction-induced lateral stresses behind the wall facing blocks. 4.2
Foundation and Concrete Facing
A linear elastic model was used for the concrete foundation and concrete blocks. The elastic properties of the concrete adopted from Ling et al. (1995) are summarized in Table 1. 4.3
Geosynthetic Reinforcement
The geosynthetic reinforcement was simulated using one-dimensional elements having nonlinear material properties. The tangent stiffness of the geosynthetic reinforcement, Jtan , is obtained from the following expression, which is based on the hyperbolic load-strain, T - ε, relationship (Ling et al. 1995): J tan = Jo
1−λ T Tf
2
(5)
where: Jo = initial stiffness; T = reinforcement tensile load; Tf = failure tensile load; and λ = failure ratio. The load-strain behavior of HDPE geogrid is affected by strain rate. The strength was observed to increase by 30% when the strain rate was increased from 1 to 20% per minute in wide-width tensile tests. The current analysis did not include time-dependent behavior of the geogrid. The results obtained from the test conducted at a strain rate of 1% per minute were used to calibrate the model. The comparison between the measured and predicted results of the hyperbolic expression (Equation 5) is shown in Figure 6. 4.4
Interactions at the Wall Facing
Three-node interface elements (Katona 1983) were used to model the interactions at the concrete block-concrete block and concrete block-backfill soil interfaces (Figure 4b). Note that this interface element was formulated using the constraint approach and is numerically more stable than the commonly used Goodman interface element, which is based on the stiffness method (Goodman et al. 1968). The slippage or debonding of an interface element is based on the Coulomb failure criterion. The model requires two property values: the interface friction angle, δ, and the tensile strength normal to the interface, Tc . Because of the node-to-node contact for each interface element, connection failure between the geosynthetic and modular blocks, either by pullout or rupture of the reinforcement, can be simulated using the interface tensile strength. The midnode is used to store the energy of the element during computation.
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Reinforcement load per unit width, T (kN/m)
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Measured Predicted
Axial strain, ε (%) Figure 6. Predicted (Equation 5) and measured load strain behavior of the geogrid.
The concrete block-concrete block and concrete block-backfill soil interactions were studied at PWRI using a large direct shear device. The direct shear device has a cross-section area of 300 mm by 300 mm, and the tests were conducted at a displacement rate of 1 mm per minute. The test results are shown in Figure 7. The angle of friction and tensile strength of these interfaces are summarized in Table 2. Note that values for soil-geogrid interaction were not used in the present study because the geogrids were assumed to be fully bonded to the soil. Since the geogrid layers were bolted to the blocks, the connection strength was assumed to be the failure strength of the geogrid, i.e. Tc = 55 kN/m. For long-term performance, a reduction factor may be necessary to account for the creep to rupture of the connection (Bathurst and Simac 1997). A small tensile strength, Tc = 0.5 kN/m, was used to improve the convergence of the solution for the newly placed interface elements under the initial very low normal stress. 5
COMPARISON OF RESULTS
Several measured quantities are compared with the numerical analysis results and discussed in the following section.
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σn = 100 kPa 70 kPa
Shear stress, τ (kPa)
40 kPa
0
50
100
150
200
Shear displacement (mm) Figure 7. Direct shear testing of different interfaces (σn = normal stress).
Table 2. Interface friction angle and tensile strength values from direct shear tests. Interface
Interface friction angle, δ (_)
Concrete-Concrete (no slippage)
45.0
Tensile strength, Tc (kN/m) 50
Concrete-Concrete (slippage)
19.6
0.5
Concrete-Soil (with geogrid)
16.5
55
Concrete-Soil (without geogrid)
16.5
0.5
Soil-Geogrid (not used in analysis)
22.0
-
5.1
Facing Horizontal Displacement
Figure 8 shows the comparison between the predicted and measured results for the horizontal displacement of the wall facing. Physical measurements are available for the fill height above 2 m. The agreement is less satisfactory before the fill reaches a height of 3 m. Thereafter, the agreement is considered satisfactory. The analysis underestimated the displacement at the bottom of the facing. In the analysis, the bottom row of
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PWRI Wall
Wall elevation, h (m)
Fill height, H
Predicted Measured 0
10
20
30
40
50
Horizontal displacement of facing, ∆h (mm) Figure 8. Predicted (finite element) and measured horizontal displacement of the wall facing.
blocks was assumed to be placed directly on the concrete foundation without considering the attachment for the load cells at the base of the wall facing column. The largest displacement at the end of construction is approximately 30 mm, which occurs at mid-height of the wall. This magnitude of lateral wall displacement, corresponding to 0.5% of the wall height, would be acceptable for most applications because it is unlikely to cause any wall or facing instability. 5.2
Lateral Stress
The lateral stress acting at the wall face is shown in Figures 9a to 9f for backfill heights of 1 to 6 m, respectively. At the wall face, the lateral stresses obtained in the concrete blocks and in the soil were different. The blocks gave lateral tensile stresses that could be due to the presence of tensile reinforcement forces because the load cells were not modeled separately in the analyses. Although the predicted results fluctuate along the height of the wall, the values obtained by averaging the stresses in the block and soil were in reasonable agreement with the measured stresses. The results show that the lateral stress prediction improves toward the end of construction. It should be noted that, while the finite element method is well suited for stress-deformation analysis, the primary unknown is displacement. The stresses are considered secondary unknowns, which are interpolated from the nodal displacements of each element; thus, stress results are less reliable than displacement results. The results could improve with the use of higher-order finite elements.
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(a)
(b)
Wall elevation, h (m)
Fill height = 1 m
Fill height = 2 m
Measured
Measured
Finite element (concrete block)
Finite element (concrete block)
Finite element (soil element)
Finite element (soil element)
(c)
(d) Fill height = 4 m
Wall elevation, h (m)
Fill height = 3 m
Measured
Finite element (soil element)
Finite element (concrete block) Finite element (soil element)
(e)
Measured Finite element (concrete block)
(f)
Wall elevation, h (m)
Fill height = 5 m
Measured Finite element (soil element)
Fill height = 6 m
Measured Finite element (soil element) Finite element (concrete block)
Finite element (concrete block)
Lateral stress, σh (kPa)
K = 0.2 K = 0.26
Lateral stress, σh (kPa)
Figure 9. Predicted (finite element) and measured lateral stress for different fill heights: (a) H = 1 m; (b) H = 2 m; (c) H = 3 m; (d) H = 4 m; (e) H = 5 m; (f) H = 6 m.
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At the end of construction (Figure 9f), the earth pressure coefficient, K, was measured to be approximately 0.2 (from the earth pressure measurements) and predicted to be 0.26. The values calculated using the Jaky equation and Rankine solution for Ô = 45_ are Ko = 0.29 (at-rest earth pressure coefficient) and Ka = 0.17(active earth pressure coefficient), respectively. Thus, the measured and predicted results lie between the at-rest and active earth pressure values. 5.3
Vertical Stress
Figure 10 shows the vertical stress distributions at the base of the backfill and at the bottom of the facing blocks. The measured and predicted values show similar trends where large vertical stresses occur at the front end of the wall due to the larger unit weight of the concrete blocks compared to the soil. In the measured results, the vertical stress exceeds the value of the overburden pressure for the presented fill heights. The measurements also show a nonuniform stress distribution along the bottom of the wall, whereas the analysis predicts a uniform distribution. A more reasonable agreement between the predicted and measured vertical stress distribution could be realized by considering the nonlinear behavior of the foundation (for example, using a nonlinear model instead of linear model for concrete/soil). In the study by Ling et al. (2000), the vertical stress distribution at the base of the reinforced soil mass was also found to be affected by the presence of reinforcement, i.e. reinforcement length and stiffness.
300
Fill height, H 1m 2m 3m 4m 5m 6m
Vertical stress, σv (kPa)
250
200
Measured
150
Finite element 100
50
0
5
4 3 2 1 0 Distance from back of facing (m)
End of facing
Figure 10. Predicted (finite element) and measured vertical stresses at the wall base.
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5.4
Strain in Geosynthetic Layers
The strain distributions in the six primary and five secondary geogrid layers, at several stages of construction, are shown in Figures 11 and 12, respectively. The distances indicated in Figures 11 and 12 are measured from the connection end. All strain gages, except the second one measured from the connection in Layer 2, functioned satisfactorily. The analysis predicts the trend in measured strain distribution and that the strain increases with increasing fill height. At the end of construction, the strain is over-predicted at the connection end of the primary reinforcement Layer 1, whereas it is under-predicted for Layers 4 and 5. Under-prediction is also noticed for the secondary reinforcement Layer 4. The exact reasons for over-prediction and under-prediction of strains are not known. The strain gage measured the “point” strain whereas the strain from numerical analyses was the average value for an element of geogrid reinforcement. The inconsistency between local and global strains, especially for a geogrid embedded in the soil, has been raised by several researchers, including Bathurst (1991) and Min et al. (1995). The under-prediction could also be related to the creep behavior concentrated at the connections, while over-prediction could be due to a possible loose connection between the reinforcement and block. 6
CONCLUSIONS
Finite element analyses were conducted to simulate the construction response of a geosynthetic-reinforced soil retaining wall with a concrete-block facing. Comparisons between measured and predicted behavior are presented for the wall deformation, vertical and lateral stresses, and strains in the geogrid layers. The finite element model was able to give satisfactory agreement between the measured and predicted results. The verification presented herein enabled the program to be used for a series of parametric studies that will be reported in a future publication. Simulations of additional well-instrumented walls should be attempted. REFERENCES Al-Hussaini, M.M. and Johnson, L.D., 1978, “Numerical Analysis of a Reinforced Earth Wall”, Proceedings of the ASCE Symposium on Earth Reinforcement, ASCE, Pittsburgh, Pennsylvania, USA, April 1978, pp. 98-126. Bathurst, R.J., 1991, “Case Study of a Monitored Propped Panel Wall”, GeosyntheticReinforced Soil Retaining Walls, Balkema, 1992, Proceedings of the International Symposium on Geosynthetic-Reinforced Soil Retaining Walls, Denver, Colorado, USA, August 1991, pp. 159-166. Bathurst, R.J., Jarrett, P.M. and Benjamin, D.J.R.S., 1993, “A Database of Results from an Incrementally Constructed Geogrid-Reinforced Soil Wall Tests”, Proceedings of Soil Reinforcement: Full Scale Experiments of the 80’s, ISSMFE/ENPC, Paris, France, November 1993, pp. 401-430.
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(c)
(a)
Geogrid strain (%)
1.0
1.0 Primary layer 1 H=1m
0.8 0.6
0.8 0.6
Measured Finite element
0.4
0.2 4
3
2
1
0
0.0
Distance from back of facing (m) 0.8 (b)
Geogrid strain (%)
Primary layer 2
0.6
1.0
Geogrid strain (%)
Measured Finite element
0.4
0.2 0.0
Primary layer 1 H=3m
Primary layer 1 H=2m
0.8 0.6
0.4 0.2
Measured Finite element
0.4
0.0
0.2
0.8
0.0
0.6 Primary layer 2
0.8
Primary layer 3
0.4
0.6
0.2
0.4
0.0
4 3 2 1 0 Distance from back of facing (m)
0.2 0.0 4
3
2
1
0
Distance from back of facing (m) Figure 11. Predicted (finite element) and measured strains in the primary geogrid layers: (a) H = 1 m; (b) H = 2 m; (c) H = 3 m
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(d) 1.0
Geogrid strain (%)
Geogrid strain (%)
1.0 Primary layer 1
0.8
H=4m Measured Finite element
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 2
0.8
Primary layer 4
0.8
4
3
2
1
0
Distance from back of facing (m)
0.6 0.4 0.2
Geogrid strain (%)
0.0 Primary layer 3
0.8 0.6 0.4 0.2 0.0
4
3
2
1
0
Distance from back of facing (m) Figure 11 continued. (d) H = 4 m
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(e)
Geogrid strain (%)
Geogrid strain (%)
Geogrid strain (%)
1.0
1.0 Primary layer 1 H=5m Measured Finite element
0.8 0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 2
0.8
Primary layer 5
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 3
0.8
Primary layer 4
0.8
4
3
2
0.4 0.2 4
3
2
1
0
Distance from back of facing (m) Figure 11 continued. (e) H = 5 m
180
0
Distance from back of facing (m)
0.6
0.0
1
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(f)
Geogrid strain (%)
Geogrid strain (%)
Geogrid strain (%)
1.0
1.0 Primary layer 1 H=6m Measured Finite element
0.8 0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 2
0.8
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 3
0.6
0.4
0.4
0.2
0.2 4
3
2
Primary layer 6
0.8
0.6
0.0
Primary layer 5
0.8
0.6
0.8
Primary layer 4
0.8
1
0
0.0
Distance from back of facing (m)
4
3
2
1
0
Distance from back of facing (m)
Figure 11 continued. (f) H = 6 m.
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(a)
(c)
Geogrid strain (%)
1.0
1.0 Secondary layer 1 H=2m
0.8
0.8
Measured Finite element
0.6
0.6
0.4
0.4
0.2
0.2
0.0
1.0
0.8 0.6 0.4 0.2
0.0
0.0
Distance from back of facing (m) 0.8 (b)
Geogrid strain (%) Geogrid strain (%)
0.6
Secondary layer 1 H=3m Measured Finite element
0.4
0.4 0.2 0.0
0.2
0.8
0.0
0.6
0.8
Secondary layer 2
Secondary layer 3
0.4 0.2
0.6
0.0
0.4
1.0 0.8 0.6 0.4 0.2 0.0 Distance from back of facing (m)
0.2 0.0
Secondary layer 2
0.6
1.0 0.8
Secondary layer 1 H=4m Measured Finite element
1.0 0.8 0.6 0.4 0.2 0.0 Distance from back of facing (m)
Figure 12. Predicted (finite element) and measured strains in the secondary geogrid layers: (a) H = 2 m; (b) H = 3 m; (c) H = 4 m
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(d)
Geogrid strain (%)
1.0 0.8
1.0 Secondary layer 1 H=5m
0.6
0.6
0.4
0.4
0.2
Measured Finite element
0.0 Geogrid strain (%)
0.8
0.8
Secondary layer 4
0.2 0.0
Secondary layer 2
1.0 0.8 0.6 0.4
0.2 0.0
Distance from back of facing (m)
0.6 0.4 0.2
Geogrid strain (%)
0.0 0.8
Secondary layer 3
0.6 0.4 0.2 0.0
1.0 0.8 0.6 0.4 0.2 0.0 Distance from back of facing (m)
Figure 12 continued. (d) H = 5 m
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(e)
Geogrid strain (%)
1.0 0.8
1.0 Secondary layer 1 H=6m
0.6
0.6
0.4
0.4
0.2
Measured Finite element
Geogrid strain (%)
0.0
Geogrid strain (%)
0.8
0.8
0.2 0.0
Secondary layer 2
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.8
Secondary layer 4
Secondary layer 3
Secondary layer 5
1.0 0.8 0.6 0.4
Distance from back of facing (m)
0.6 0.4 0.2 0.0
1.0 0.8 0.6 0.4 0.2 0.0 Distance from back of facing (m)
Figure 12 continued. (e) H = 6 m.
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Bathurst, R.J., Karpurapu, R. and Jarrett, P.M., 1992, “Finite Element Analysis of Geogrid Reinforced Soil Walls”, Grouting, Soil Improvement and Geosynthetics, Geotechnical Special Publication No. 30, Borden, R.H., Holtz, R.D and Juran, I., Editors, ASCE, 1992, Vol. 2., New Orleans, Louisiana, USA, February 1992, pp. 1213-1224. Bathurst, R.J. and Simac, M.R., 1994, “Geosynthetic Reinforced Segmental Retaining Wall Structures in North America”, Proceedings of the Fifth International Conference on Geotextiles, Geomembranes and Related Products, Keynote lecture, Singapore, September 1994, pp. 31-54. Bathurst, R.J. and Simac, M.R., 1997, “Design and Performance of the Facing Column for Geosynthetic Reinforced Segmental Walls”, Mechanically Stabilized Backfill, Wu, J.T.H., Editor, Balkema, Proceedings of the International Symposium on Mechanically Stabilized Backfill, Denver, Colorado, USA, February, 1997 pp. 193-208. Cai, Z. and Bathurst, R.J., 1995, “Seismic Response Analysis of Geosynthetic Reinforced Soil Segmental Retaining Walls by Finite Element Method”, Computers and Geotechnics, Vol. 17, No. 4, pp. 523-546. Chew, S.H. and Mitchell, J.K., 1994, “Deformation Evaluation Procedure for Reinforced Soil Walls”, Proceedings of the Fifth International Conference on Geotextiles, Geomembranes and Related Products, Vol. 1, Singapore, September 1994, pp. 171-192. Collin, J.G., 1986, “Earth Wall Design”, Ph.D. Thesis, University of California, Berkeley, Berkeley, California, USA, 440 p. Collin, J.G., Editor, 1997, “Design Manual for Segmental Retaining Walls”, Second Edition, National Concrete Masonry Association, Herndon, Virginia, USA, 289 p. Duncan, J.M., Byrne, P., Wong, K.S. and Mabry, P., 1980, “Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analyses of Stresses and Movements in Soil Masses”, Report No. UCB/GT/80-01, Department of Civil Engineering, University of California, Berkeley, California, USA, 70 p. Goodman, R.E., Taylor, R.L. and Brekke, T.L., 1968, “A Model for the Mechanics of Jointed Rock”, Journal of Soil Mechanics and Foundations Division, Vol. 94, No. 3, pp. 637-659. Kaliakin, V.N. and Xi, F., 1992, “Modeling of Interfaces in Finite Element Analyses of Geosynthetically Reinforced Walls”, Earth Reinforcement Practice, Ochiai, H., Hayashi, S. and Otani, J., Editors, Balkema, 1992, Proceedings of the International Symposium on Earth Reinforcement Practice, Vol. 1, Kyushu University, Fukuoka, Japan, November 1992, pp. 351-356. Karpurapu, R. and Bathurst, R., 1995, “Behavior of Geosynthetic Reinforced Soil Retaining Walls Using the Finite Element Method”, Computers and Geotechnics, Vol. 17, No. 3, pp. 279-299. Katona, M.G., 1983, “A Simple Contact-Friction Interface Element with Applications to Buried Culverts”, International Journal of Numerical and Analytical Methods in Geomechanics, Vol. 7, pp. 371-384.
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Katona, M.G., Smith, J.M., Odello, R.S. and Allgood, J.R., 1976, “A Modern Approach for the Structural Design and Analysis of Buried Culverts”, FHWA-RD-77-5, Federal Highway Administration, US Department of Transportation, Washington, DC, USA, 475 p. Leshchinsky, D., 1993, “Geosynthetic-Reinforced Steep Slopes and Walls: Effects of Facing Blocks”, Proceedings of the International Seminar on Slope Stability Engineering, Tokushima, Japan, pp. 95-133. Ling, H.I., Cardany, C.P. and Sun, L-X., 2000, “Finite Element Analysis and Parametric Study of Modular-Block Facing GRS-RW”, manuscript submitted for review. Ling, H.I. and Leshchinsky, D., 1995, Unpublished Report to The Tensar Corporation. Ling, H.I. and Tatsuoka, F., 1991, “Nonlinear Analysis of Reinforced Soil Structures by Modified CANDE (M-CANDE)”, Geosynthetic-Reinforced Soil Retaining Walls, Wu, J.T.H., Editor, Balkema, 1992, Proceedings of the International Symposium on Geosynthetic-Reinforced Soil Retaining Walls, Denver, Colorado, USA, August 1991, pp. 279-296. Ling, H.I., Tatsuoka, F. and Tateyama, M., 1995, “Simulating Performance of GRS-RW by Finite-Element Procedure”, Journal of Geotechnical Engineering, Vol. 121, No. 4, pp. 330-340. Min, Y., Leshchinsky, D., Ling, H.I. and Kaliakin, V.N., 1995, “Effects of Sustained and Repeated Loads on Geogrid Embedded in Sand”, Geotechnical Testing Journal, Vol. 18, No. 2, pp. 204-225. Miyatake, H., Ochiai, Y., Maruo, S., Nakane, A., Yamamoto, M., Terayama, T., Maejima, T. and Tsukada, Y., 1995, “Full-Scale Failure Experiments on Reinforced Earth Wall with Geotextiles (Part 2) - Facing with Concrete Blocks”, Proceedings of 30 th Annual Conference of Geotechnical Engineering, Kanazawa, Japan, pp. 2427-2430. Romstad, K.M., Herrmann, L.R. and Shen, C-K., 1976, “Integrated Study of Reinforced Earth - I: Theoretical Formulation”, Journal of Geotechnical Engineering, Vol. 102, No. GT5, pp. 457-471. Rowe, R.K. and Ho, S.K., 1997, “Continuous Panel Reinforced Soil Walls on Rigid Foundations”, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 123, No. 10, pp. 912-920. Seed, R.B., Collin, J.G. and Mitchell, J.K., 1986, “FEM Analyses of Compacted Reinforced Soil Walls”, Numerical Models in Geomechanics, Pande, G.N. and Van Impe, W.F., Editors, M. Jackson & Son Publishers, Proceedings of the International Symposium on Numerical Models in Geomechanics, Ghent, Belgium, March 1986, pp. 553-562. Shen, C-K., Romstad, K.M. and Herrmann, L.R., 1976, “Integrated Study of Reinforced Earth - II: Behavior and Design”, Journal of Geotechnical Engineering, Vol. 102, No. GT6, pp. 577-590. Tajiri, N., Sasaki, H., Nishimura, J., Ochiai, Y. and Dobashi, K., 1996, “Full-Scale Failure Experiments of Geotextile-Reinforced Soil Walls with Different Facings”, Earth Reinforcement, Ochiai, H., Yaufuku, N. and Omine, K., Editors, Balkema, 1997, Proceedings of the International Symposium on Earth Reinforcement, Vol. 1, Fukuoka, Kyushu, Japan, November 1996, pp. 525-530.
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Wu, J.T.H., Editor, 1991, “Geosynthetic-Reinforced Soil Retaining Walls”, Balkema, 1992, Proceedings of the International Symposium on Geosynthetic-Reinforced Soil Retaining Walls, Denver, Colorado, USA, August 1991, 375 p. Wu, T.H. and Monley, G.J., 1989, “Effectiveness of Tensile Reinforcement in Alleviating Bridge Approach”, Proceedings of Geosynthetics ’89, IFAI, Vol. 1, San Diego, California, USA, February 1989, pp. 104-111. Zornberg, J.G. and Mitchell, J.K., 1994, “Effect of Sloping Backfills on Geosynthetically Reinforced Walls”, Proceedings of the Fifth International Conference on Geotextiles, Geomembranes and Related Products, Vol. 1, Singapore, September 1994, pp. 201-206. NOTATIONS Basic SI units are given in parentheses. c D50 E Etan , Ei H h Jtan , Jo K Ko , Ka n pa q Rf T Tc Tf ∆h ∆Ô δ ε γ λ ν Ô Ôo
= = = = = = = = = = = = = = = = = = = = = = = = =
cohesion (Pa) mean particle diameter (m) Young’s modulus of concrete (Pa) tangent and initial Young’s modulus of soil (Pa) total wall/backfill height (m) height along the wall (m) tangent and initial stiffness of reinforcement (N/m) modulus number (dimensionless) at-rest and active earth pressure coefficients (dimensionless) modulus exponent (dimensionless) atmospheric pressure (Pa) deviator stress = σ1 - σ3 (Pa) failure ratio for soil strength (dimensionless) reinforcement tensile load per unit width (N/m) tensile strength normal to interface (N/m) reinforcement tensile load per unit width at failure (N/m) horizontal displacement of wall facing (m) change of φ with 10-fold increase in confining stress (_) interface friction angle (_) axial strain (Figure 5) (%) unit weight of soil (N/m3) failure ratio for tensile strength of reinforcement (dimensionless) Poisson’s ratio (dimensionless) internal friction angle of soil (_) initial internal friction angle of soil (_)
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σn σh , σv σ1 , σ3 (σ1 - σ3 ) f τ
188
= = = = =
normal stress (Pa) lateral and vertical stress, respectively (Pa) major and minor principal stresses, respectively (Pa) deviator stress at failure (Pa) shear stress (Pa)
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FINITE ELEMENT STUDY OF A GEOSYNTHETIC-REINFORCED SOIL RETAINING WALL WITH CONCRETE-BLOCK FACING ABSTRACT: This paper focuses on the results of finite element modeling of a full-scale, concrete-block, geosynthetic-reinforced soil retaining wall constructed at the Public Works Research Institute in Japan. A nonlinear hyperbolic geosynthetic model was incorporated into a computer program that is capable of simulating soil-structure interaction behavior. The soil was simulated using a hyperbolic model while the block-block and soil-block interactions were simulated using interface elements. Comparison of numerical and measured experimental results indicated that the finite element model is capable of simulating the construction behavior of concrete-block geosynthetic-reinforced soil structures. KEYWORDS: Geosynthetic, Geogrid, Reinforced soil retaining wall, Concrete block, Facing, Finite element method, Soil-structure interaction. AUTHORS: H.I. Ling, Associate Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University, 500 West 120th Street, New York, New York 10027, USA, Telephone: 1/212-854-1203, Telefax: 1/212-854-6267, E-mail: [email protected]; C.P. Cardany, Graduate Student, Department of Civil Engineering and Engineering Mechanics, Columbia University, 500 West 120th Street, New York, New York 10027, USA, and Senior Staff Engineer, Langan Engineering and Environmental Service, Inc., Telephone: 1/203-562-5771, Telefax: 1/203-789-6142, E-mail: [email protected]; L-X. Sun, Graduate Research Assistant, Department of Civil Engineering and Engineering Mechanics, Columbia University, 500 West 120th Street, New York, New York 10027, USA, Telephone: 1/212-854-4153, Telefax: 1/212-854-6267, E-mail: [email protected]; and H. Hashimoto, Research Engineer, Construction Engineering Division, Public Works Research Institute, Ministry of Construction, Tsukuba, Japan, Telephone: 81/298-64-4703, Telefax: 81/298-64-0564, E-mail: [email protected]. PUBLICATION: Geosynthetics International is published by the Industrial Fabrics Association International, 1801 County Road B West, Roseville, Minnesota 55113-4061, USA, Telephone: 1/651-222-2508, Telefax: 1/651-631-9334. Geosynthetics International is registered under ISSN 1072-6349. DATES: Original manuscript received 29 December 1999, revised version received 29 March 2000 and accepted 2 April 2000. Discussion open until 1 January 2001. REFERENCE: Ling, H.I., Cardany, C.P., Sun, L-X. and Hashimoto, H., 2000, “Finite Element Study of a Geosynthetic-Reinforced Soil Retaining Wall With Concrete-Block Facing”, Geosynthetics International, Vol. 7, No. 3, pp. 163-188.
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1
INTRODUCTION
Modular (segmental) blocks are widely used in North America for construction of geosynthetic-reinforced soil retaining walls (GRS-RWs) (Bathurst and Simac 1994; Collin 1997). Modular-block GRS-RW structures offer a wide range of aesthetic hardfaced finishes, while resulting in more economical structures compared to traditional gravity wall structures. There are numerous proprietary modular-block wall systems available that are commonly used for private and public sector retaining wall projects. In a modular-block wall, the geosynthetic layers are placed between the concrete blocks at regular vertical spacings. The blocks are used to retain the backfill. The integrity of the wall facing is maintained through the interaction between the blocks, blocks and backfill, as well as between the blocks and geosynthetic layers. Modular-block (segmental) units are typically connected to geosynthetic reinforcement layers using mechanical and/or frictional connection devices such as polymeric pins, inserts, clips, or concrete shear keys. The dead weight of the modular blocks contributes to the global stability of the GRSRW. The contribution of the facing was neglected in early designs; then, Bathurst and Simac (1994) and Leshchinsky (1993) proposed limit equilibrium design methodologies to account for the modular blocks. However, limit equilibrium methods do not allow the wall deformation and strain in the geosynthetic reinforcement to be evaluated directly during analysis and design. The present paper describes a finite element study of a well-instrumented geogridreinforced soil retaining wall constructed with a concrete-block facing. This study focuses on the numerical modeling technique used and adaptations that may apply to the study of modular-block GRS-RWs. The measured and predicted soil stresses, wall deformations, and tensile strain in the reinforcement layers are presented and compared. 2
GEOSYNTHETIC-REINFORCED SOIL RETAINING WALL AT THE PUBLIC WORKS RESEARCH INSTITUTE (PWRI WALL)
Three geosynthetic-reinforced soil retaining walls were constructed at the Public Works Research Institute (PWRI), Ministry of Construction, in Japan. Each wall had a different facing structure: concrete blocks, discrete panels, or expanded polystyrene (EPS) blocks. The walls were fully instrumented and subsequently failed by cutting the geosynthetic layers in stages (Miyatake et al. 1995; Tajiri et al. 1996). The present study focuses on the wall with concrete-block facing (hereafter referred to as the PWRI Wall) and its behavior during construction. The PWRI Wall is one of the very few full-scale geosynthetic-reinforced soil structures that has been fully instrumented. The material properties were also well defined. A brief summary of the PWRI Wall and instrumentation program is described in this section. The geometry of PWRI Wall is shown in Figure 1. It is 6 m high and 5 m wide and was constructed in a concrete test pit with a concrete floor. The side walls of the test facility were lubricated using grease and polymer sheets. A silty sand (mean particle diameter, D50 = 0.42 mm; unit weight, γ = 16.0 kN/m3) was used as the backfill. The particle size distribution of the backfill soil is shown in Figure 2. The stress-strain properties of the sand are described in Section 3.
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Secondary geogrid layers 350 3,500 450
Primary geogrid layers
6 5 5
1,000
4 4 5,000
3 3 2 2 1
1,000
270 550
1
1,000
1,000 1,000 250 Linear variable displacement transducer (LVDT) Load cell Strain gage All dimensions in mm
Figure 1. PWRI Wall: geometry and instrumentation.
A uniaxial geogrid (Tensar SR55), manufactured from extruded high-density polyethylene (HDPE), was used as the soil reinforcement. The spacings between the longitudinal and transverse ribs were 22 and 166 mm, respectively. The strength of the geogrid was approximately 55 kN/m. The PWRI Wall consisted of six primary and five secondary geogrid layers, 3.5 and 1.0 m long, respectively. The geogrid layers were bolted to the concrete blocks with the bolt and metal frame arrangement shown in Figure 3. A total of 12 concrete block rows were used to construct the wall face. Each block was 500 mm high and 350 mm wide (toe-to-heel dimension), except the top and bottom blocks, which were 450 and 550 mm high, respectively.
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Fraction passing by mass (%)
100 80 60 40 20 0 0.001
0.01
0.1
1.0
10
Particle diameter (mm) Figure 2. Particle size distribution of the backfill soil.
(a)
200 Load cell 500
75 1,000 350
(b) Load cell
500
Geogrid
All dimensions in mm
Metal frame
Bolt
Figure 3. Connection between the modular blocks and geosynthetic layers: (a) back of concrete block; (b) cross-section.
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A total of 52 strain gages were used to measure elongation of the geogrid, i.e. seven and two strain gages on each layer of primary and secondary reinforcement, respectively. The horizontal displacement of the wall face was measured at 11 locations using linear variable displacement transducers (LVDTs). The lateral pressure acting against the back of the facing blocks was measured using 11 load cells installed at the mid-height of each concrete block. The vertical and horizontal loads acting at the toe of the facing were also measured using load cells. The vertical pressure due to the backfill soil was measured at six locations along the base of the soil mass. 3
FINITE ELEMENT STUDY OF PWRI WALL
The finite element method has been used to analyze different types of geotechnical structures, such as earth dams, embankments, shallow and deep foundations, slopes, and retaining walls. The application of the finite element method to reinforced soil structures is relatively recent. Reinforced soil is a complex system that involves interactions between different structural components and soil. Since the procedure itself is very sophisticated, the application to design is rare. However, finite element analysis renders additional information compared to traditional limit-equilibrium analysis, such as deformation and tensile load in the reinforcement layers, that are necessary for understanding the performance of reinforced soil structures. The results obtained from finite element analyses may be used to guide the development of more accurate limit-equilibrium design procedures. Finite element analyses of reinforced soil structures can be conducted using computer programs that simulate soil-structure interaction, i.e. programs that have the relevant soil and structural elements and material models. The program should be able to simulate the construction sequences, such as backfilling and the installation of reinforcement layers and wall facings. In the early 1970s, when finite element analyses were limited to only mainframe computers at high computing costs, a composite approach was proposed to model soil-reinforcement systems (Romstad et al. 1976; Shen et al. 1976). Modern finite element analysis methods use a discrete approach because computing costs have significantly reduced. In a discrete analysis, the reinforcement, soil, facing, and their interactions are modeled separately. Previous finite element studies of reinforced soils focused on numerical parametric studies (Kaliakin and Xi 1992; Rowe and Ho 1997). There were also attempts to reproduce the results of laboratory models and full-scale tests (Al-Hussaini and Johnson 1978; Collin 1986; Seed et al. 1986; Wu and Monley 1989; Bathurst et al. 1992; Chew and Mitchell 1994; Zornberg and Mitchell 1994; Cai and Bathurst 1995; Karpurapu and Bathurst 1995; Ling et al. 1995). A collection of finite element analysis predictions for a reinforced soil structure can be found in the proceedings of symposium that was edited by Wu (1991). Nevertheless, there have been a very limited number of well-instrumented, full-scale tests leading to the verification of finite element procedures. With the exception of the study by Cai and Bathurst (1995), the finite element study of reinforced soil walls with a modular-block facing, which is a relatively new type of reinforced soil system, has not been previously reported. In the present study, the analysis of the PWRI Wall was conducted using the finite element program M-CANDE (Ling and Tatsuoka 1991; Ling et al. 1995; Ling and
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Leshchinsky 1995), which is a modified version of the program CANDE (Katona 1976) that was developed for the design and analysis of buried culverts. The program carries out plane strain analyses under static loading conditions. The program can simulate the construction sequence and the inclusion of reinforcement and interface elements in each solution increment. In each increment of analysis, the solution is iterated to a userselected convergence criterion at the element level. A geosynthetic model was added to CANDE following previous analyses of reinforced soil structures (Ling et al. 1995). Creep (viscoelastic) phenomenon cannot be directly modeled using the current code. The finite element mesh (1,618 nodes) and interface elements used to simulate the PWRI Wall are shown in Figures 4a and 4b, respectively. The foundation, backfill, and facing blocks were represented by quadrilateral elements: 315 elements for the foundation, 925 elements for the backfill, and 37 elements for the concrete facing blocks. A total of 127 elements, corresponding to the primary and secondary geosynthetic reinforcement layers, were included in the analysis. A total of 110 interface elements were used to simulate the interactions between different materials at the wall facing. The reinforcement layers were assumed to be fully bonded to the backfill soil. The analysis was conducted in 38 steps to simulate wall construction. 4
MATERIAL MODELS AND CALIBRATION
4.1
Backfill Soil
The backfill was modeled using a nonlinear elastic soil model (Duncan et al. 1980) with a variable Young’s modulus and a constant Poisson’s ratio. The tangent Young’s modulus is expressed as follows: E tan = Ei
E i = K pa (σ 1 − σ 3 )f =
σ1 − σ3 1 − Rf (σ 1 − σ 3 ) f
σ3 pa
2
(1)
n
(2)
2c cos Ô + 2σ3 sin Ô 1 − sin Ô
σ Ô = Ô o − ∆Ô p 3 a
(3)
(4)
where: Etan = tangent Young’s modulus; Ei = initial Young’s modulus; K = modulus number; n = modulus exponent; Rf = failure ratio; Ô = internal friction angle of the soil; Ôo = initial value of the internal friction angle of the soil; ∆Ô = change of the Ô value with a ten-fold increase in the confining stress; pa = atmospheric pressure; c = soil cohesion; (σ1 - σ3 )f = deviator stress at failure; and σ1 and σ3 = major and minor principal stresses, respectively. Triaxial compression tests were conducted at Columbia Univer-
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(a)
1.50
2.50
1.00
0.35 0.45
5.00
All dimensions in m
0.55 0.27
3.00
10.00 (b)
Soil Concrete blocks Geosynthetic Interface
Interface Figure 4. PWRI Wall: (a) finite element mesh; (b) interface elements at the wall facing.
sity (New York, New York, USA) using dry specimens replicating the test wall condition. Hyperbolic model parameters were selected to fit triaxial test results and are summarized in Table 1. A comparison of measured and predicted triaxial test results is shown in Figure 5.
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Table 1. Material properties used in the finite element analyses of the PWRI Wall. Backfill γ (kN/m3)
Ôo (_)
16.0
45
∆Ô (_)
K
n
Rf
ν
5
207.2
0.5
0.81
0.42
Concrete (foundation and facing) γ (kN/m3)
E (kPa)
ν
-
2.0 × 106
0.17
23
2.0 × 106
0.17
Geogrid reinforcement Jo (kN/m)
Tf (kN/m)
λ
826.45
54.6
0.47
Notes: γ = unit weight of the soil; Ôo =initial value for the internal friction angle of the soil; ∆Ô = change of φ with a 10-fold increase in the confining stress; K = modulus number; n = modulus exponent; Rf = failure ratio for soil strength; ν = Poisson’s ratio; E = Young’s modulus of concrete; Jo = initial stiffness of reinforcement; Tf = reinforcement tensile load per unit width at failure; λ = failure ratio for tensile strength of reinforcement.
Deviator stress, q = σ1 --- σ3 (kPa)
σ3 = 100 kPa
Hyperbolic model Measured σ3 = 50 kPa
σ3 = 25 kPa
K = 207.2, n = 0.5 Ôo = 45_, ∆Ô = 5_, Rf = 0.81
Axial strain, ε (%) Figure 5. Predicted (hyperbolic model) and measured triaxial test results.
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No provision was made to measure the volume change during triaxial testing. It is well known that the hyperbolic model is suitable for analysis under working stress conditions. The model implemented in the M-CANDE code expresses the deviator stressaxial strain relationships satisfactorily, but does not account for the dilatant behavior of soil. In addition, compaction stresses cannot be modelled directly. A constant Poisson’s ratio (ν = 0.42) was selected for the soil to account for possible compaction-induced lateral stresses behind the wall facing blocks. 4.2
Foundation and Concrete Facing
A linear elastic model was used for the concrete foundation and concrete blocks. The elastic properties of the concrete adopted from Ling et al. (1995) are summarized in Table 1. 4.3
Geosynthetic Reinforcement
The geosynthetic reinforcement was simulated using one-dimensional elements having nonlinear material properties. The tangent stiffness of the geosynthetic reinforcement, Jtan , is obtained from the following expression, which is based on the hyperbolic load-strain, T - ε, relationship (Ling et al. 1995): J tan = Jo
1−λ T Tf
2
(5)
where: Jo = initial stiffness; T = reinforcement tensile load; Tf = failure tensile load; and λ = failure ratio. The load-strain behavior of HDPE geogrid is affected by strain rate. The strength was observed to increase by 30% when the strain rate was increased from 1 to 20% per minute in wide-width tensile tests. The current analysis did not include time-dependent behavior of the geogrid. The results obtained from the test conducted at a strain rate of 1% per minute were used to calibrate the model. The comparison between the measured and predicted results of the hyperbolic expression (Equation 5) is shown in Figure 6. 4.4
Interactions at the Wall Facing
Three-node interface elements (Katona 1983) were used to model the interactions at the concrete block-concrete block and concrete block-backfill soil interfaces (Figure 4b). Note that this interface element was formulated using the constraint approach and is numerically more stable than the commonly used Goodman interface element, which is based on the stiffness method (Goodman et al. 1968). The slippage or debonding of an interface element is based on the Coulomb failure criterion. The model requires two property values: the interface friction angle, δ, and the tensile strength normal to the interface, Tc . Because of the node-to-node contact for each interface element, connection failure between the geosynthetic and modular blocks, either by pullout or rupture of the reinforcement, can be simulated using the interface tensile strength. The midnode is used to store the energy of the element during computation.
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Reinforcement load per unit width, T (kN/m)
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Measured Predicted
Axial strain, ε (%) Figure 6. Predicted (Equation 5) and measured load strain behavior of the geogrid.
The concrete block-concrete block and concrete block-backfill soil interactions were studied at PWRI using a large direct shear device. The direct shear device has a cross-section area of 300 mm by 300 mm, and the tests were conducted at a displacement rate of 1 mm per minute. The test results are shown in Figure 7. The angle of friction and tensile strength of these interfaces are summarized in Table 2. Note that values for soil-geogrid interaction were not used in the present study because the geogrids were assumed to be fully bonded to the soil. Since the geogrid layers were bolted to the blocks, the connection strength was assumed to be the failure strength of the geogrid, i.e. Tc = 55 kN/m. For long-term performance, a reduction factor may be necessary to account for the creep to rupture of the connection (Bathurst and Simac 1997). A small tensile strength, Tc = 0.5 kN/m, was used to improve the convergence of the solution for the newly placed interface elements under the initial very low normal stress. 5
COMPARISON OF RESULTS
Several measured quantities are compared with the numerical analysis results and discussed in the following section.
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σn = 100 kPa 70 kPa
Shear stress, τ (kPa)
40 kPa
0
50
100
150
200
Shear displacement (mm) Figure 7. Direct shear testing of different interfaces (σn = normal stress).
Table 2. Interface friction angle and tensile strength values from direct shear tests. Interface
Interface friction angle, δ (_)
Concrete-Concrete (no slippage)
45.0
Tensile strength, Tc (kN/m) 50
Concrete-Concrete (slippage)
19.6
0.5
Concrete-Soil (with geogrid)
16.5
55
Concrete-Soil (without geogrid)
16.5
0.5
Soil-Geogrid (not used in analysis)
22.0
-
5.1
Facing Horizontal Displacement
Figure 8 shows the comparison between the predicted and measured results for the horizontal displacement of the wall facing. Physical measurements are available for the fill height above 2 m. The agreement is less satisfactory before the fill reaches a height of 3 m. Thereafter, the agreement is considered satisfactory. The analysis underestimated the displacement at the bottom of the facing. In the analysis, the bottom row of
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PWRI Wall
Wall elevation, h (m)
Fill height, H
Predicted Measured 0
10
20
30
40
50
Horizontal displacement of facing, ∆h (mm) Figure 8. Predicted (finite element) and measured horizontal displacement of the wall facing.
blocks was assumed to be placed directly on the concrete foundation without considering the attachment for the load cells at the base of the wall facing column. The largest displacement at the end of construction is approximately 30 mm, which occurs at mid-height of the wall. This magnitude of lateral wall displacement, corresponding to 0.5% of the wall height, would be acceptable for most applications because it is unlikely to cause any wall or facing instability. 5.2
Lateral Stress
The lateral stress acting at the wall face is shown in Figures 9a to 9f for backfill heights of 1 to 6 m, respectively. At the wall face, the lateral stresses obtained in the concrete blocks and in the soil were different. The blocks gave lateral tensile stresses that could be due to the presence of tensile reinforcement forces because the load cells were not modeled separately in the analyses. Although the predicted results fluctuate along the height of the wall, the values obtained by averaging the stresses in the block and soil were in reasonable agreement with the measured stresses. The results show that the lateral stress prediction improves toward the end of construction. It should be noted that, while the finite element method is well suited for stress-deformation analysis, the primary unknown is displacement. The stresses are considered secondary unknowns, which are interpolated from the nodal displacements of each element; thus, stress results are less reliable than displacement results. The results could improve with the use of higher-order finite elements.
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(a)
(b)
Wall elevation, h (m)
Fill height = 1 m
Fill height = 2 m
Measured
Measured
Finite element (concrete block)
Finite element (concrete block)
Finite element (soil element)
Finite element (soil element)
(c)
(d) Fill height = 4 m
Wall elevation, h (m)
Fill height = 3 m
Measured
Finite element (soil element)
Finite element (concrete block) Finite element (soil element)
(e)
Measured Finite element (concrete block)
(f)
Wall elevation, h (m)
Fill height = 5 m
Measured Finite element (soil element)
Fill height = 6 m
Measured Finite element (soil element) Finite element (concrete block)
Finite element (concrete block)
Lateral stress, σh (kPa)
K = 0.2 K = 0.26
Lateral stress, σh (kPa)
Figure 9. Predicted (finite element) and measured lateral stress for different fill heights: (a) H = 1 m; (b) H = 2 m; (c) H = 3 m; (d) H = 4 m; (e) H = 5 m; (f) H = 6 m.
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At the end of construction (Figure 9f), the earth pressure coefficient, K, was measured to be approximately 0.2 (from the earth pressure measurements) and predicted to be 0.26. The values calculated using the Jaky equation and Rankine solution for Ô = 45_ are Ko = 0.29 (at-rest earth pressure coefficient) and Ka = 0.17(active earth pressure coefficient), respectively. Thus, the measured and predicted results lie between the at-rest and active earth pressure values. 5.3
Vertical Stress
Figure 10 shows the vertical stress distributions at the base of the backfill and at the bottom of the facing blocks. The measured and predicted values show similar trends where large vertical stresses occur at the front end of the wall due to the larger unit weight of the concrete blocks compared to the soil. In the measured results, the vertical stress exceeds the value of the overburden pressure for the presented fill heights. The measurements also show a nonuniform stress distribution along the bottom of the wall, whereas the analysis predicts a uniform distribution. A more reasonable agreement between the predicted and measured vertical stress distribution could be realized by considering the nonlinear behavior of the foundation (for example, using a nonlinear model instead of linear model for concrete/soil). In the study by Ling et al. (2000), the vertical stress distribution at the base of the reinforced soil mass was also found to be affected by the presence of reinforcement, i.e. reinforcement length and stiffness.
300
Fill height, H 1m 2m 3m 4m 5m 6m
Vertical stress, σv (kPa)
250
200
Measured
150
Finite element 100
50
0
5
4 3 2 1 0 Distance from back of facing (m)
End of facing
Figure 10. Predicted (finite element) and measured vertical stresses at the wall base.
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5.4
Strain in Geosynthetic Layers
The strain distributions in the six primary and five secondary geogrid layers, at several stages of construction, are shown in Figures 11 and 12, respectively. The distances indicated in Figures 11 and 12 are measured from the connection end. All strain gages, except the second one measured from the connection in Layer 2, functioned satisfactorily. The analysis predicts the trend in measured strain distribution and that the strain increases with increasing fill height. At the end of construction, the strain is over-predicted at the connection end of the primary reinforcement Layer 1, whereas it is under-predicted for Layers 4 and 5. Under-prediction is also noticed for the secondary reinforcement Layer 4. The exact reasons for over-prediction and under-prediction of strains are not known. The strain gage measured the “point” strain whereas the strain from numerical analyses was the average value for an element of geogrid reinforcement. The inconsistency between local and global strains, especially for a geogrid embedded in the soil, has been raised by several researchers, including Bathurst (1991) and Min et al. (1995). The under-prediction could also be related to the creep behavior concentrated at the connections, while over-prediction could be due to a possible loose connection between the reinforcement and block. 6
CONCLUSIONS
Finite element analyses were conducted to simulate the construction response of a geosynthetic-reinforced soil retaining wall with a concrete-block facing. Comparisons between measured and predicted behavior are presented for the wall deformation, vertical and lateral stresses, and strains in the geogrid layers. The finite element model was able to give satisfactory agreement between the measured and predicted results. The verification presented herein enabled the program to be used for a series of parametric studies that will be reported in a future publication. Simulations of additional well-instrumented walls should be attempted. REFERENCES Al-Hussaini, M.M. and Johnson, L.D., 1978, “Numerical Analysis of a Reinforced Earth Wall”, Proceedings of the ASCE Symposium on Earth Reinforcement, ASCE, Pittsburgh, Pennsylvania, USA, April 1978, pp. 98-126. Bathurst, R.J., 1991, “Case Study of a Monitored Propped Panel Wall”, GeosyntheticReinforced Soil Retaining Walls, Balkema, 1992, Proceedings of the International Symposium on Geosynthetic-Reinforced Soil Retaining Walls, Denver, Colorado, USA, August 1991, pp. 159-166. Bathurst, R.J., Jarrett, P.M. and Benjamin, D.J.R.S., 1993, “A Database of Results from an Incrementally Constructed Geogrid-Reinforced Soil Wall Tests”, Proceedings of Soil Reinforcement: Full Scale Experiments of the 80’s, ISSMFE/ENPC, Paris, France, November 1993, pp. 401-430.
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(c)
(a)
Geogrid strain (%)
1.0
1.0 Primary layer 1 H=1m
0.8 0.6
0.8 0.6
Measured Finite element
0.4
0.2 4
3
2
1
0
0.0
Distance from back of facing (m) 0.8 (b)
Geogrid strain (%)
Primary layer 2
0.6
1.0
Geogrid strain (%)
Measured Finite element
0.4
0.2 0.0
Primary layer 1 H=3m
Primary layer 1 H=2m
0.8 0.6
0.4 0.2
Measured Finite element
0.4
0.0
0.2
0.8
0.0
0.6 Primary layer 2
0.8
Primary layer 3
0.4
0.6
0.2
0.4
0.0
4 3 2 1 0 Distance from back of facing (m)
0.2 0.0 4
3
2
1
0
Distance from back of facing (m) Figure 11. Predicted (finite element) and measured strains in the primary geogrid layers: (a) H = 1 m; (b) H = 2 m; (c) H = 3 m
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(d) 1.0
Geogrid strain (%)
Geogrid strain (%)
1.0 Primary layer 1
0.8
H=4m Measured Finite element
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 2
0.8
Primary layer 4
0.8
4
3
2
1
0
Distance from back of facing (m)
0.6 0.4 0.2
Geogrid strain (%)
0.0 Primary layer 3
0.8 0.6 0.4 0.2 0.0
4
3
2
1
0
Distance from back of facing (m) Figure 11 continued. (d) H = 4 m
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(e)
Geogrid strain (%)
Geogrid strain (%)
Geogrid strain (%)
1.0
1.0 Primary layer 1 H=5m Measured Finite element
0.8 0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 2
0.8
Primary layer 5
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 3
0.8
Primary layer 4
0.8
4
3
2
0.4 0.2 4
3
2
1
0
Distance from back of facing (m) Figure 11 continued. (e) H = 5 m
180
0
Distance from back of facing (m)
0.6
0.0
1
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(f)
Geogrid strain (%)
Geogrid strain (%)
Geogrid strain (%)
1.0
1.0 Primary layer 1 H=6m Measured Finite element
0.8 0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 2
0.8
0.6
0.4
0.4
0.2
0.2
0.0
0.0 Primary layer 3
0.6
0.4
0.4
0.2
0.2 4
3
2
Primary layer 6
0.8
0.6
0.0
Primary layer 5
0.8
0.6
0.8
Primary layer 4
0.8
1
0
0.0
Distance from back of facing (m)
4
3
2
1
0
Distance from back of facing (m)
Figure 11 continued. (f) H = 6 m.
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(a)
(c)
Geogrid strain (%)
1.0
1.0 Secondary layer 1 H=2m
0.8
0.8
Measured Finite element
0.6
0.6
0.4
0.4
0.2
0.2
0.0
1.0
0.8 0.6 0.4 0.2
0.0
0.0
Distance from back of facing (m) 0.8 (b)
Geogrid strain (%) Geogrid strain (%)
0.6
Secondary layer 1 H=3m Measured Finite element
0.4
0.4 0.2 0.0
0.2
0.8
0.0
0.6
0.8
Secondary layer 2
Secondary layer 3
0.4 0.2
0.6
0.0
0.4
1.0 0.8 0.6 0.4 0.2 0.0 Distance from back of facing (m)
0.2 0.0
Secondary layer 2
0.6
1.0 0.8
Secondary layer 1 H=4m Measured Finite element
1.0 0.8 0.6 0.4 0.2 0.0 Distance from back of facing (m)
Figure 12. Predicted (finite element) and measured strains in the secondary geogrid layers: (a) H = 2 m; (b) H = 3 m; (c) H = 4 m
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(d)
Geogrid strain (%)
1.0 0.8
1.0 Secondary layer 1 H=5m
0.6
0.6
0.4
0.4
0.2
Measured Finite element
0.0 Geogrid strain (%)
0.8
0.8
Secondary layer 4
0.2 0.0
Secondary layer 2
1.0 0.8 0.6 0.4
0.2 0.0
Distance from back of facing (m)
0.6 0.4 0.2
Geogrid strain (%)
0.0 0.8
Secondary layer 3
0.6 0.4 0.2 0.0
1.0 0.8 0.6 0.4 0.2 0.0 Distance from back of facing (m)
Figure 12 continued. (d) H = 5 m
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(e)
Geogrid strain (%)
1.0 0.8
1.0 Secondary layer 1 H=6m
0.6
0.6
0.4
0.4
0.2
Measured Finite element
Geogrid strain (%)
0.0
Geogrid strain (%)
0.8
0.8
0.2 0.0
Secondary layer 2
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.8
Secondary layer 4
Secondary layer 3
Secondary layer 5
1.0 0.8 0.6 0.4
Distance from back of facing (m)
0.6 0.4 0.2 0.0
1.0 0.8 0.6 0.4 0.2 0.0 Distance from back of facing (m)
Figure 12 continued. (e) H = 6 m.
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Bathurst, R.J., Karpurapu, R. and Jarrett, P.M., 1992, “Finite Element Analysis of Geogrid Reinforced Soil Walls”, Grouting, Soil Improvement and Geosynthetics, Geotechnical Special Publication No. 30, Borden, R.H., Holtz, R.D and Juran, I., Editors, ASCE, 1992, Vol. 2., New Orleans, Louisiana, USA, February 1992, pp. 1213-1224. Bathurst, R.J. and Simac, M.R., 1994, “Geosynthetic Reinforced Segmental Retaining Wall Structures in North America”, Proceedings of the Fifth International Conference on Geotextiles, Geomembranes and Related Products, Keynote lecture, Singapore, September 1994, pp. 31-54. Bathurst, R.J. and Simac, M.R., 1997, “Design and Performance of the Facing Column for Geosynthetic Reinforced Segmental Walls”, Mechanically Stabilized Backfill, Wu, J.T.H., Editor, Balkema, Proceedings of the International Symposium on Mechanically Stabilized Backfill, Denver, Colorado, USA, February, 1997 pp. 193-208. Cai, Z. and Bathurst, R.J., 1995, “Seismic Response Analysis of Geosynthetic Reinforced Soil Segmental Retaining Walls by Finite Element Method”, Computers and Geotechnics, Vol. 17, No. 4, pp. 523-546. Chew, S.H. and Mitchell, J.K., 1994, “Deformation Evaluation Procedure for Reinforced Soil Walls”, Proceedings of the Fifth International Conference on Geotextiles, Geomembranes and Related Products, Vol. 1, Singapore, September 1994, pp. 171-192. Collin, J.G., 1986, “Earth Wall Design”, Ph.D. Thesis, University of California, Berkeley, Berkeley, California, USA, 440 p. Collin, J.G., Editor, 1997, “Design Manual for Segmental Retaining Walls”, Second Edition, National Concrete Masonry Association, Herndon, Virginia, USA, 289 p. Duncan, J.M., Byrne, P., Wong, K.S. and Mabry, P., 1980, “Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analyses of Stresses and Movements in Soil Masses”, Report No. UCB/GT/80-01, Department of Civil Engineering, University of California, Berkeley, California, USA, 70 p. Goodman, R.E., Taylor, R.L. and Brekke, T.L., 1968, “A Model for the Mechanics of Jointed Rock”, Journal of Soil Mechanics and Foundations Division, Vol. 94, No. 3, pp. 637-659. Kaliakin, V.N. and Xi, F., 1992, “Modeling of Interfaces in Finite Element Analyses of Geosynthetically Reinforced Walls”, Earth Reinforcement Practice, Ochiai, H., Hayashi, S. and Otani, J., Editors, Balkema, 1992, Proceedings of the International Symposium on Earth Reinforcement Practice, Vol. 1, Kyushu University, Fukuoka, Japan, November 1992, pp. 351-356. Karpurapu, R. and Bathurst, R., 1995, “Behavior of Geosynthetic Reinforced Soil Retaining Walls Using the Finite Element Method”, Computers and Geotechnics, Vol. 17, No. 3, pp. 279-299. Katona, M.G., 1983, “A Simple Contact-Friction Interface Element with Applications to Buried Culverts”, International Journal of Numerical and Analytical Methods in Geomechanics, Vol. 7, pp. 371-384.
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Katona, M.G., Smith, J.M., Odello, R.S. and Allgood, J.R., 1976, “A Modern Approach for the Structural Design and Analysis of Buried Culverts”, FHWA-RD-77-5, Federal Highway Administration, US Department of Transportation, Washington, DC, USA, 475 p. Leshchinsky, D., 1993, “Geosynthetic-Reinforced Steep Slopes and Walls: Effects of Facing Blocks”, Proceedings of the International Seminar on Slope Stability Engineering, Tokushima, Japan, pp. 95-133. Ling, H.I., Cardany, C.P. and Sun, L-X., 2000, “Finite Element Analysis and Parametric Study of Modular-Block Facing GRS-RW”, manuscript submitted for review. Ling, H.I. and Leshchinsky, D., 1995, Unpublished Report to The Tensar Corporation. Ling, H.I. and Tatsuoka, F., 1991, “Nonlinear Analysis of Reinforced Soil Structures by Modified CANDE (M-CANDE)”, Geosynthetic-Reinforced Soil Retaining Walls, Wu, J.T.H., Editor, Balkema, 1992, Proceedings of the International Symposium on Geosynthetic-Reinforced Soil Retaining Walls, Denver, Colorado, USA, August 1991, pp. 279-296. Ling, H.I., Tatsuoka, F. and Tateyama, M., 1995, “Simulating Performance of GRS-RW by Finite-Element Procedure”, Journal of Geotechnical Engineering, Vol. 121, No. 4, pp. 330-340. Min, Y., Leshchinsky, D., Ling, H.I. and Kaliakin, V.N., 1995, “Effects of Sustained and Repeated Loads on Geogrid Embedded in Sand”, Geotechnical Testing Journal, Vol. 18, No. 2, pp. 204-225. Miyatake, H., Ochiai, Y., Maruo, S., Nakane, A., Yamamoto, M., Terayama, T., Maejima, T. and Tsukada, Y., 1995, “Full-Scale Failure Experiments on Reinforced Earth Wall with Geotextiles (Part 2) - Facing with Concrete Blocks”, Proceedings of 30 th Annual Conference of Geotechnical Engineering, Kanazawa, Japan, pp. 2427-2430. Romstad, K.M., Herrmann, L.R. and Shen, C-K., 1976, “Integrated Study of Reinforced Earth - I: Theoretical Formulation”, Journal of Geotechnical Engineering, Vol. 102, No. GT5, pp. 457-471. Rowe, R.K. and Ho, S.K., 1997, “Continuous Panel Reinforced Soil Walls on Rigid Foundations”, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 123, No. 10, pp. 912-920. Seed, R.B., Collin, J.G. and Mitchell, J.K., 1986, “FEM Analyses of Compacted Reinforced Soil Walls”, Numerical Models in Geomechanics, Pande, G.N. and Van Impe, W.F., Editors, M. Jackson & Son Publishers, Proceedings of the International Symposium on Numerical Models in Geomechanics, Ghent, Belgium, March 1986, pp. 553-562. Shen, C-K., Romstad, K.M. and Herrmann, L.R., 1976, “Integrated Study of Reinforced Earth - II: Behavior and Design”, Journal of Geotechnical Engineering, Vol. 102, No. GT6, pp. 577-590. Tajiri, N., Sasaki, H., Nishimura, J., Ochiai, Y. and Dobashi, K., 1996, “Full-Scale Failure Experiments of Geotextile-Reinforced Soil Walls with Different Facings”, Earth Reinforcement, Ochiai, H., Yaufuku, N. and Omine, K., Editors, Balkema, 1997, Proceedings of the International Symposium on Earth Reinforcement, Vol. 1, Fukuoka, Kyushu, Japan, November 1996, pp. 525-530.
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Wu, J.T.H., Editor, 1991, “Geosynthetic-Reinforced Soil Retaining Walls”, Balkema, 1992, Proceedings of the International Symposium on Geosynthetic-Reinforced Soil Retaining Walls, Denver, Colorado, USA, August 1991, 375 p. Wu, T.H. and Monley, G.J., 1989, “Effectiveness of Tensile Reinforcement in Alleviating Bridge Approach”, Proceedings of Geosynthetics ’89, IFAI, Vol. 1, San Diego, California, USA, February 1989, pp. 104-111. Zornberg, J.G. and Mitchell, J.K., 1994, “Effect of Sloping Backfills on Geosynthetically Reinforced Walls”, Proceedings of the Fifth International Conference on Geotextiles, Geomembranes and Related Products, Vol. 1, Singapore, September 1994, pp. 201-206. NOTATIONS Basic SI units are given in parentheses. c D50 E Etan , Ei H h Jtan , Jo K Ko , Ka n pa q Rf T Tc Tf ∆h ∆Ô δ ε γ λ ν Ô Ôo
= = = = = = = = = = = = = = = = = = = = = = = = =
cohesion (Pa) mean particle diameter (m) Young’s modulus of concrete (Pa) tangent and initial Young’s modulus of soil (Pa) total wall/backfill height (m) height along the wall (m) tangent and initial stiffness of reinforcement (N/m) modulus number (dimensionless) at-rest and active earth pressure coefficients (dimensionless) modulus exponent (dimensionless) atmospheric pressure (Pa) deviator stress = σ1 - σ3 (Pa) failure ratio for soil strength (dimensionless) reinforcement tensile load per unit width (N/m) tensile strength normal to interface (N/m) reinforcement tensile load per unit width at failure (N/m) horizontal displacement of wall facing (m) change of φ with 10-fold increase in confining stress (_) interface friction angle (_) axial strain (Figure 5) (%) unit weight of soil (N/m3) failure ratio for tensile strength of reinforcement (dimensionless) Poisson’s ratio (dimensionless) internal friction angle of soil (_) initial internal friction angle of soil (_)
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LING, CARDANY, SUN AND HASHIMOTO D Geosynthetic-Reinforced Soil Retaining Wall
σn σh , σv σ1 , σ3 (σ1 - σ3 ) f τ
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= = = = =
normal stress (Pa) lateral and vertical stress, respectively (Pa) major and minor principal stresses, respectively (Pa) deviator stress at failure (Pa) shear stress (Pa)
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