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Analysis of back-to-back mechanically stabilized earth walls
Geotextiles and Geomembranes 28 (2010) 262–267
Contents lists available at ScienceDirect
Geotextiles and Geomembranes journal homepage: www.elsevier.com/locate/geotexmem
Analysis of back-to-back mechanically stabilized earth walls Jie Han a, *, Dov Leshchinsky b a b
Associate Professor, Department of Civil, Environmental, and Architectural Engineering, the University of Kansas, Lawrence, KS 66045, USA Professor, Department of Civil and Environmental Engineering, the University of Delaware, Newark, DE 19716, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 May 2008 Received in revised form 11 December 2008 Accepted 18 May 2009 Available online 26 November 2009
Back-to-back Mechanically Stabilized Earth (MSE) walls are commonly used for embankments approaching bridges. However, available design guidelines for this wall system are limited. The distance between two opposing walls is a key parameter used for determining the analysis methods in FHWA Guidelines. Two extreme cases are identified: (1) reinforcements from both sides meet in the middle or overlap, and (2) the walls are far apart, independent of each other. However, existing design methodologies do not provide a clear and justified answer how the required tensile strength of reinforcement and the external stability change with respect to the distance of the back-to-back walls. The focus of this paper is to investigate the effect of the wall width to height ratio on internal and external stability of MSE walls under static conditions. Finite difference method incorporated in the FLAC software and limit equilibrium method (i.e., the Bishop simplified method) in the ReSSA software were used for this analysis. Parametric studies were carried out by varying two important parameters, i.e., the wall width to height ratio and the quality of backfill material, to investigate their effects on the critical failure surface, the required tensile strength of reinforcement, and the lateral earth pressure behind the reinforced zone. The effect of the connection of reinforcements in the middle, when back-to-back walls are close, was also investigated. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Critical failure surface Maximum tension Factor of safety Limit equilibrium MSE wall Reinforcement
1. Introduction
mobilized so that the active thrust should be reduced. Here Di is defined as the interaction distance and expressed by
Back-to-back mechanically stabilized earth (MSE) walls are commonly used for embankments approaching bridges to raise elevations. Design of such walls is considered as a special situation, which has a complex geometry in the FHWA Demonstration Project 82 (Elias and Christopher, 1997). In this FHWA design guideline, two cases are considered based on the distance of two back-to-back or opposing walls, D, as illustrated in Fig. 1. When D is greater than H tan(45 f/2), full active thrust to the reinforced zone can be mobilized and the walls can be designed independently, where H is the height of the walls and f is the friction angle of the backfill. For this case, the typical design method for MSE walls can be used. When D is equal to 0, two walls are still designed independently for internal stability but no active thrust to the reinforced zone is assumed from the backfill. In other words, no analysis for external stability is needed. However, the later publication by Elias et al. (2001) indicated that an overlap of 0.3H is required for no active thrust. Both guidelines indicate that when D is less than H tan(45 f/2), active thrust to the reinforced zone cannot be fully
Di ¼ Htan 45 f=2
* Corresponding author. Tel.: þ1 785 864 3714; fax: þ1 785 864 5631. E-mail address: [email protected] (J. Han). 0266-1144/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2009.09.012
(1)
However, the FHWA guideline (Elias and Christopher, 1997) did not provide any method on how to consider the reduction of the active thrust when D < Di. Elias et al. (2001) provided the following recommendation for this intermediate configuration ‘‘the active earth thrust may be linearly interpolated from the full active case to zero’’. No justification was provided for this recommendation. A full-scale back-to-back geosynthetic-reinforced wall was constructed to evaluate the effect of the geosynthetic type on the internal deformation of the wall (Won and Kim, 2007). Due to the large distance of D ( ¼ 0.88H), no interaction was observed from two sides of walls. However, no investigation was conducted to evaluate the effect of the distance D. Therefore, an analysis is needed to evaluate the internal and external stability of back-toback MSE walls under a static condition. Both limit equilibrium and numerical methods have been successfully used to evaluate the stability of MSE walls (for example, Leshchinsky and Han, 2004; Han and Leshchinsky, 2006b; Han and Leshchinsky, 2007) that yielded close results in terms of factors of safety and critical failure surfaces. Leshchinsky et al.
J. Han, D. Leshchinsky / Geotextiles and Geomembranes 28 (2010) 262–267
Notations Basic SI units are given in parentheses. f friction angle of fill (degree) n Poisson’s ratio D distance between the back of two opposing walls (m) interaction distance defined by the FHWA design Di guideline (m) E Young’s modulus (MPa) h height of reinforcement (m) H height of wall (m) elevation of wall (m) hz L reinforcement length (m) lateral earth pressure behind reinforced zone (kPa) ph theoretical active Rankine lateral thrust (kN) Ph maximum tensile stress in reinforcement (kN/m) Tmax allowable tensile strength of reinforcement (kN/m) Ta W distance between two opposing wall facings (m) Abbreviations FHWA Federal Highway Administration LE Limit Equilibrium MSE Mechanically Stabilized Earth
(2009) demonstrated through shaking table tests that geocellreinforced earth walls under seismic loading failed in a rotational or translational mode, which could be modeled using a limit equilibrium method. In this study, the limit equilibrium and numerical methods were also adopted to investigate the effect of the width to height ratio of the wall (W/H) and the quality of backfill material on the critical slip surface, the required tensile strength of reinforcement, and the active thrust to the reinforced zone. In addition, the effect of the connection of reinforcements in the middle, when back-to-back walls are close, was also investigated. 2. Method of analysis 2.1. Limit equilibrium method Bishop’s simplified method, utilizing a circular arc slip surface, is probably the most popular limit equilibrium method. Although Bishop’s method is not rigorous in a sense that it does not satisfy horizontal force limit equilibrium, it is simple to apply and, in many practical problems, it yields results close to rigorous limit equilibrium methods. In this study, Bishop’s simplified method was
263
modified to include reinforcement as a horizontal force intersecting the slip circle, which is incorporated in ReSSA Version 3.0 software, developed by ADAMA Engineering (2008). This modified formulation is consistent with the original formulation by Bishop (1955). The mobilized reinforcement strength at its intersection with the slip circle depends on its long-term strength, its rear-end pullout capacity (or connection strength), and the soil strength. The analysis assumes that when the soil strength is reduced by a factor, a limit equilibrium state is achieved (i.e., the system is at the verge of failure). The slip circle for which the lowest factor (i.e., the largest mobilized soil strength) exists is the critical slip surface for which the factor of safety is rendered. Under this state, when the factor of safety at unity, the soil and reinforcement mobilize their respective strengths simultaneously. ReSSA Version 3.0 also provides a factor of safety map, which identifies the critical zone based on the criterion set up by the user. 2.2. Numerical method The finite difference program, FLAC 2D Version 5.0 (Itasca Consulting Group, Inc., 2006), was adopted in this study. A shear strength reduction technique proposed by Zienkiewicz et al. (1975) was adopted in this program to solve for a factor of safety of stability. In this technique, a series of trial factors of safety are used to adjust the cohesion, c and the friction angle, 4, of soil. Adjusted cohesion and friction angle of soil layers are re-inputted in the model for limit equilibrium analysis. The factor of safety is sought when the specific adjusted cohesion and friction angle make the slope unstable from the verge of a stable condition (i.e., limit equilibrium). Details about the slope stability analysis using the FLAC software can be found in Dawson et al. (1999). The critical slip surface often can be identified based on the contours of the maximum shear strain rate. 3. Modeling 3.1. Baseline case The geometry and material properties of the baseline model used in this study are shown in Fig. 2. Since the factor of safety is determined based on a state of yield, or verge of failure, it is insensitive to the selected elastic parameters: Young’s modulus (E) and Poisson’s ratio (n) when using FLAC. If the system contains soils with largely different elastic parameters, it will take longer time to solve the factor of safety; however, the effects on this factor would be small since it depends mainly on Mohr-Coulomb strength parameters. Hence, constant values of E ¼ 100 MPa and n ¼ 0.3 were used for the reinforced fill in FLAC. High-strength parameters were assumed for the foundation to prevent any possible failure. The effect of wall facing cohesion on the required tensile strength of reinforcement in the numerical analysis will be discussed in the next section. Mohr-Coulomb failure criteria were used for strength W = 12 m 4.2 m
4.2 m
0.6m
H = 6m 1.5m 1m Fig. 1. Back-to-back MSE wall and definitions.
Blocks (γ = 18kN/m3 , c =1000kPa, φ = 340 ) Reinforced & retained fill (γ = 18kN/m3 , c =0kPa, φ = 340) Reinforcement Weak zone
Foundation (γ =18kN/m3 , c =1000kPa, φ =00 ) Fig. 2. Dimensions and parameters of the baseline case.
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between stacked blocks, the reinforced and retained fill, and the foundation soil. Reinforcement was modeled as a cable with grouted interface properties between cable and soil. The bond strength between reinforcement and reinforced fill was assumed equal to 80% the fill strength, same as in the limit equilibrium analysis when considering pullout resistance. A weak zone at the toe of the MSE wall with a dimension of 0.3 m wide and 0.4 m high, which has zero cohesion but the same friction angle as the fill, was assumed to ensure that the critical failure surface passes through the toe of the MSE wall. Details about the effect of wall facing cohesion will be discussed later. In this baseline case, the back-to-back wall width (W) to height (H) ratio is equal to 2.0 and the distance at back of two walls, D is equal to 3.6 m, which is slightly greater than H tan(45 f/ 2) ¼ 3.2 m. Based on the FHWA design guideline, a typical design method for a single wall can be adopted. The reinforcement length, L ¼ 4.2 m, was selected based on the typical reinforcement length/ wall height ratio of 0.7 recommended by the FHWA design guideline. The height of MSE walls is fixed at 6 m. Two important parameters, the back-to-back wall width to height ratio and the quality (i.e., friction angle) of backfill material, were selected in this study to investigate their influence on the critical failure surface, the required tensile strength of reinforcement, and the active thrust to the reinforced zone. In addition to W/H ¼ 2.0 for the baseline case, two other W/H ratios (1.4 and 3.0) were used in the parametric study. One parameter in the baseline was changed at a time while all others were unchanged. The same models were used in numerical and limit equilibrium analyses. The required tensile strength of reinforcement was determined to ensure the factor of safety of the MSE wall equal to 1.0.
3.2. Effect of wall facing cohesion The effect of wall facing cohesion was examined for the baseline case in this study. As shown in Fig. 3, the factor of safety of the backto-back MSE wall increases with an increase of the wall facing cohesion. However, it becomes constant after the cohesion is greater than 100 kPa. In this case, the potential failure surface was restricted to pass only through the toe of the MSE wall. Fig. 3 also shows that the effect of the wall facing cohesion for the case with low-quality backfill (4 ¼ 25 ) is more significant than that with high-quality backfill (4 ¼ 34 ). In all analyses discussed below, the 1.25 1.2
φ = 25o Ta = 30.2kN/m
Factor of safety
1.15
4.2m φ=34o
φ=25o
6m
φ=25
φ=34o o
Numerical LE
Fig. 4. Critical failure surfaces within walls at W/H ¼ 3.
cohesion of the wall facing was assumed to be 1000 kPa except a weak zone close to the toe. 4. Results 4.1. Critical failure surfaces The locations and shapes of critical failure surfaces of the backto-back walls at different wall width to height ratios (W/H) were determined based on the contours of shear strain rates in the numerical analysis and presented in Figs. 4–6. Fig. 4 shows that the critical failure surfaces in two opposing walls do not intercept each other, therefore, they behave independently. The critical failure surfaces by the LE method assuming only one side wall are also shown in Fig. 4, and they have slightly steeper angles than those by the numerical method. This conclusion is consistent with that from the previous study conducted by the authors (Han and Leshchinsky, 2006b). Since the LE method cannot analyze two side walls, which become important when the distance between the walls gets closer, the results presented below are based on the numerical analysis unless noted. Fig. 5 shows the critical failure surfaces within back-to-back walls at W/H ¼ 2 that intercept each other from two sides. More interactions occur for the case with a low-quality backfill. (i.e., 4 ¼ 25 ). For both cases, the critical failure surfaces do not enter the reinforced zone on the opposing side. In other words, the potential failure surface is constrained by the reinforced zone on the opposing side. The interaction distance, Di, based on the FHWA design guideline can be determined using Equation (1). Considering the backfill with the friction angles of 25 and 34 , the interaction distances are 0.64H and 0.53H, respectively. In other words, when the wall width to height ratio, W/H > 2.04 for the friction angle of the fill equal to 25 or W/H > 1.93 for the friction angle of the fill equal to 34 , the two back-to-back walls should perform independently. However, Fig. 5 shows that the back-toback walls still interact each other when W/H ¼ 2.0 > 1.93 for the
4.2m
1.1 1.05
4.2m
9.6m
3.6m
4.2m
φ = 34o Ta = 11.2kN/m
1
6m 0.95
φ = 34o φ = 25o
0.9 0
200
400
600
800
1000
Cohesion of wall facing (kPa) Fig. 3. Effect of wall facing cohesion.
Fig. 5. Critical failure surfaces within walls at W/H ¼ 2.0.
J. Han, D. Leshchinsky / Geotextiles and Geomembranes 28 (2010) 262–267
4.2m
connected when they meet in the middle of the walls. The numerical results show the interactions of critical failure surfaces in two opposing walls. In both cases, the failure surfaces enter the reinforced zone from another side. The comparisons of locations and shapes of critical failure surfaces at different W/H ratios but the same quality of fill are presented in Fig. 7. Fig. 7 shows that the locations and shapes of the critical failures are almost same for W/H ¼ 3 and 2. This result can be explained as the failure surfaces not entering the reinforced zone on the opposing side. For W/H ¼ 1.4, however, the locations and shapes of the critical failure surfaces deviate from others as the failure surfaces enter the reinforced zone on the opposing side.
4.2m
6m
φ = 34o
φ = 25o
4.2. Distribution of maximum tension with height
Fig. 6. Critical failure surfaces within walls at W/H ¼ 1.4.
friction angle of the fill equal to 34 . Apparently, this assumption in the FHWA design guideline is not supported by the numerical result. However, the FHWA assumption leads to more conservative results. Fig. 6 shows the critical failure surfaces developed within the back-to-back walls when there is no retained fill between these two walls (i.e., D ¼ 0 m). In both cases, reinforcement layers are not
a
4.2m
φ = 34o W/H = 3
6m
W/H = 2 W/H = 1.4
The distribution of the maximum tension, Tmax, on each reinforcement is plotted against the height of the reinforcement, h, for both backfill materials in Fig. 8. These distributions were determined from the numerical results based on the reinforcements having equal strength and the MSE wall having a factor of safety of 1.0. These distributions are trapezoidal and similar to those obtained by Han and Leshchinsky (2006a) using the limit equilibrium method. Fig. 8 shows that the maximum tension in the reinforcement in the MSE wall with low friction angle fill is much higher than that with high friction angle fill. The width to height ratio of the back-to-back wall has a slight effect on the distribution of the maximum tension in the reinforcement at height, i.e., the overall maximum tension increases with the increase of the width to height ratio. This result implies that it is slightly conservative to ignore the influence of the width to height ratio on the overall maximum tension in the internal stability analysis of the backto-back MSE wall. Fig. 8 shows that the overall maximum tension occurs from the bottom to 1/3 or ½ height of the wall for the low or high friction angle fill, respectively. It is noted that the maximum tension in the upper portion of the wall is slightly higher for W/H ¼ 1.4 than W/H ¼ 2.0 and 3.0. This result can be explained by the fact that the maximum tension in the reinforcement for W/H ¼ 1.4 is contributed by both sides of walls due to the interception of the failure surfaces in the upper portion.
φ = 34o 4.2m
b
φ = 25o 6m
265
W/H = 3 W/H = 2 W/H = 1.4
φ = 25o Fig. 7. Critical failure surfaces at different W/H ratios.
Fig. 8. Distribution of maximum tension in each reinforcement.
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a
6
Elevation of wall, hz (m)
5 W/H = 1.4 W/H = 2.0 W/H = 3.0 Rankine
4
3
2
1
0 0
10
20
30
40
Lateral earth pressure behind reinforced zone, ph (kPa)
b
6
Fig. 9. Required maximum tensile strength of reinforcement.
The required overall maximum tensile tension or strengths of reinforcements for all the cases discussed above are presented in Fig. 9. The results from the LE method were based on the analyses of one side wall, therefore, no interaction of two opposing walls was considered. In other words, the required tensile strengths do not change with the W/H ratios. Fig. 9 clearly shows that a decrease of W/H ratio from 3 to 1.4 reduces the required maximum tensile strength of reinforcement. This result implies that the back-to-back walls for both backfill materials still interact at W/H ranging from 2.0 to 3.0. The LE method without considering the interaction of the opposing walls would provide conservative design of back-to-back MSE walls. The difference in the maximum tensile strength of reinforcement with and without considering the interaction is within 12% based on the cases investigated in this study. The required maximum tensile strengths can be used for the selection of reinforcements (such as geosynthetics) in the back-to-back MSE walls.
Elevation of wall, hz (m)
5
4.3. Required tensile strength
W/H = 1.4 W/H = 2.0 W/H = 3.0 Rankine
4
3
2
1
0
0
10
20
30
40
Fig. 10. Distribution of lateral earth pressure behind the reinforced zone.
4.4. Lateral earth pressure behind the reinforced zone Based on the FHWA guideline (Elias and Christopher, 1997; Elias et al., 2001), the lateral earth pressure or active thrust behind the reinforced zone for the external stability analysis should depend on the width to height ratio. Fig. 10 presents the numerical results of the lateral earth pressure, ph, behind the reinforced zone at a different elevation of the wall, hz. It is clearly shown that the lateral earth pressure exists behind the reinforced zone, even for the width to height ratio of W/H ¼ 1.4 (i.e., no retained fill). However, the FHWA guideline (Elias and Christopher, 1997) suggested that the lateral earth pressure for external analysis should be ignored if D ¼ 0 (i.e., W/H ¼ 1.4). Obviously, this suggestion would yield an unsafe design. Fig. 10 also shows that the average lateral earth pressure behind the reinforced zone is close to the active Rankine lateral earth pressure when the width to height ratio is large (for example, W/H ¼ 3.0). However, the lateral earth pressure decreases when the width to height ratio decreases. The percent of the active lateral thrust behind the reinforced zone to the theoretical active Rankine lateral thrust, Ph, is presented in Fig. 11, which shows the influence of the backfill friction angle and the width to
50
Lateral earth pressure behind reinforced zone, ph (kPa)
Fig. 11. Percent of lateral thrust behind the reinforced zone.
J. Han, D. Leshchinsky / Geotextiles and Geomembranes 28 (2010) 262–267
267
This interaction will change the location and shape of critical failure surface. The FHWA design guideline underestimates the interaction distance. When the distance of the two back-to-back walls gets closer, the required maximum tensile strength of reinforcement slightly decreases. Ignoring the effect of the width to height ratio on the overall maximum tensile strength of reinforcement results in a slightly conservative design for internal stability. However, ignoring the external lateral earth pressure on the reinforced zone, when the distance of the back-to-back walls, D, equals to 0, results in an unsafe design for external stability. The connection of reinforcements in the middle ensures the mobilization of the full strength of the reinforcement and reduces the maximum required tensile strength.
Acknowledgements
Fig. 12. Effect of reinforcement connection on the maximum tension.
height ratio on the mobilization of the lateral thrust. When a low friction angle backfill material is used, the percent of the lateral thrust is low. However, an increase of the width to height ratio from 1.4 to 3.0 increases the lateral thrust behind the reinforced zone. 4.5. Effect of reinforcement connection in the middle When the distance of two opposing walls, D, equals to 0, the reinforcements from both sides would meet in the middle. For an ease of construction, these reinforcement layers are often not connected. However, Fig. 12 shows that the connection of these reinforcements reduces the overall maximum tension in the wall because each reinforcement can mobilize its full strength when they are connected in the middle. The pullout from the middle of the walls becomes impossible. The maximum tension in the reinforcement decreasing with the height of the reinforcement for the unconnected case is because the mobilized tension in the reinforcement is limited by its pullout capacity at a higher height close to the top of the wall. 5. Conclusions The study using the numerical and limit equilibrium methods shows that two back-to-back walls perform independently when they are far apart and interact with each other when they are close.
This paper is based upon the work supported by the National Science Foundation under Grant No. 0442159. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.
References ADAMA Engineering, Inc, 2008. ReSSA Software Version 3.0 Newark, Delaware, USA. Bishop, A.W., 1955. The use of the slip circle in the stability analysis of slopes. Geotechnique 5, 7–17. Dawson, E.M., Roth, W.H., Drescher, A., 1999. Slope stability analysis by strength reduction. Geotechnique 49 (6), 835–840. Elias, V., Christopher, B.R., 1997. Mechanically Stabilized Earth Walls and Reinforced Soil Slopes Design and Construction Guidelines Publication No. FHWA-SA-96–071. Elias, V., Christopher, B.R., Berg, R.R., 2001. Mechanically Stabilized Earth Walls and Reinforced Soil Slopes Design and Construction Guidelines Publication No. FHWA-NHI-00–043. Han, J., Leshchinsky, D., 2006a. General analytical framework for design of flexible reinforced earth structures. ASCE Journal of Geotechnical and Geoenvironmental Engineering 132 (11), 1427–1435. Han, J., Leshchinsky, D., 2006b. Stability analyses of geosynthetic-reinforced earth structures using limit equilibrium and numerical methods. In: Proc. of the 8th Int. Geosynthetics Conf., 18–22 Sept., Yokohama, Japan, pp. 1347–1350. Han, J., Leshchinsky, D., 2007. Stability analysis of back-to-back MSE walls. In: Proc.. of the 5th International Symposium on Earth Reinforcement (IS Kyushu’07), November 14–16, Fukuoka, Japan, pp. 487–490. Itasca Consulting Group, Inc, 2006. FLAC5.0 User’s Guide Minneapolis. Leshchinsky, D., Han, J., 2004. Geosynthetic reinforced multitiered walls. ASCE Journal of Geotechnical and Geoenvironmental Engineering 130 (12), 1225–1235. Leshchinsky, D., Ling, H.I., Wang, J.-P., Rosen, A., Mohri, Y., 2009. Equivalent seismic coefficient in geocell retention systems. Geotextiles and Geomembranes 27, 9–18. Won, M.-S., Kim, Y.-S., 2007. Internal deformation behavior of geosynthetic-reinforced soil walls. Geotextiles and Geomembranes 25, 10–22. Zienkiewicz, O.C., Humpheson, C., Lewis, R.W., 1975. Associated and non-associated visco-plasticity and plasticity in soil mechanics. Geotechnique 25 (4), 671–689.
Contents lists available at ScienceDirect
Geotextiles and Geomembranes journal homepage: www.elsevier.com/locate/geotexmem
Analysis of back-to-back mechanically stabilized earth walls Jie Han a, *, Dov Leshchinsky b a b
Associate Professor, Department of Civil, Environmental, and Architectural Engineering, the University of Kansas, Lawrence, KS 66045, USA Professor, Department of Civil and Environmental Engineering, the University of Delaware, Newark, DE 19716, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 May 2008 Received in revised form 11 December 2008 Accepted 18 May 2009 Available online 26 November 2009
Back-to-back Mechanically Stabilized Earth (MSE) walls are commonly used for embankments approaching bridges. However, available design guidelines for this wall system are limited. The distance between two opposing walls is a key parameter used for determining the analysis methods in FHWA Guidelines. Two extreme cases are identified: (1) reinforcements from both sides meet in the middle or overlap, and (2) the walls are far apart, independent of each other. However, existing design methodologies do not provide a clear and justified answer how the required tensile strength of reinforcement and the external stability change with respect to the distance of the back-to-back walls. The focus of this paper is to investigate the effect of the wall width to height ratio on internal and external stability of MSE walls under static conditions. Finite difference method incorporated in the FLAC software and limit equilibrium method (i.e., the Bishop simplified method) in the ReSSA software were used for this analysis. Parametric studies were carried out by varying two important parameters, i.e., the wall width to height ratio and the quality of backfill material, to investigate their effects on the critical failure surface, the required tensile strength of reinforcement, and the lateral earth pressure behind the reinforced zone. The effect of the connection of reinforcements in the middle, when back-to-back walls are close, was also investigated. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Critical failure surface Maximum tension Factor of safety Limit equilibrium MSE wall Reinforcement
1. Introduction
mobilized so that the active thrust should be reduced. Here Di is defined as the interaction distance and expressed by
Back-to-back mechanically stabilized earth (MSE) walls are commonly used for embankments approaching bridges to raise elevations. Design of such walls is considered as a special situation, which has a complex geometry in the FHWA Demonstration Project 82 (Elias and Christopher, 1997). In this FHWA design guideline, two cases are considered based on the distance of two back-to-back or opposing walls, D, as illustrated in Fig. 1. When D is greater than H tan(45 f/2), full active thrust to the reinforced zone can be mobilized and the walls can be designed independently, where H is the height of the walls and f is the friction angle of the backfill. For this case, the typical design method for MSE walls can be used. When D is equal to 0, two walls are still designed independently for internal stability but no active thrust to the reinforced zone is assumed from the backfill. In other words, no analysis for external stability is needed. However, the later publication by Elias et al. (2001) indicated that an overlap of 0.3H is required for no active thrust. Both guidelines indicate that when D is less than H tan(45 f/2), active thrust to the reinforced zone cannot be fully
Di ¼ Htan 45 f=2
* Corresponding author. Tel.: þ1 785 864 3714; fax: þ1 785 864 5631. E-mail address: [email protected] (J. Han). 0266-1144/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2009.09.012
(1)
However, the FHWA guideline (Elias and Christopher, 1997) did not provide any method on how to consider the reduction of the active thrust when D < Di. Elias et al. (2001) provided the following recommendation for this intermediate configuration ‘‘the active earth thrust may be linearly interpolated from the full active case to zero’’. No justification was provided for this recommendation. A full-scale back-to-back geosynthetic-reinforced wall was constructed to evaluate the effect of the geosynthetic type on the internal deformation of the wall (Won and Kim, 2007). Due to the large distance of D ( ¼ 0.88H), no interaction was observed from two sides of walls. However, no investigation was conducted to evaluate the effect of the distance D. Therefore, an analysis is needed to evaluate the internal and external stability of back-toback MSE walls under a static condition. Both limit equilibrium and numerical methods have been successfully used to evaluate the stability of MSE walls (for example, Leshchinsky and Han, 2004; Han and Leshchinsky, 2006b; Han and Leshchinsky, 2007) that yielded close results in terms of factors of safety and critical failure surfaces. Leshchinsky et al.
J. Han, D. Leshchinsky / Geotextiles and Geomembranes 28 (2010) 262–267
Notations Basic SI units are given in parentheses. f friction angle of fill (degree) n Poisson’s ratio D distance between the back of two opposing walls (m) interaction distance defined by the FHWA design Di guideline (m) E Young’s modulus (MPa) h height of reinforcement (m) H height of wall (m) elevation of wall (m) hz L reinforcement length (m) lateral earth pressure behind reinforced zone (kPa) ph theoretical active Rankine lateral thrust (kN) Ph maximum tensile stress in reinforcement (kN/m) Tmax allowable tensile strength of reinforcement (kN/m) Ta W distance between two opposing wall facings (m) Abbreviations FHWA Federal Highway Administration LE Limit Equilibrium MSE Mechanically Stabilized Earth
(2009) demonstrated through shaking table tests that geocellreinforced earth walls under seismic loading failed in a rotational or translational mode, which could be modeled using a limit equilibrium method. In this study, the limit equilibrium and numerical methods were also adopted to investigate the effect of the width to height ratio of the wall (W/H) and the quality of backfill material on the critical slip surface, the required tensile strength of reinforcement, and the active thrust to the reinforced zone. In addition, the effect of the connection of reinforcements in the middle, when back-to-back walls are close, was also investigated. 2. Method of analysis 2.1. Limit equilibrium method Bishop’s simplified method, utilizing a circular arc slip surface, is probably the most popular limit equilibrium method. Although Bishop’s method is not rigorous in a sense that it does not satisfy horizontal force limit equilibrium, it is simple to apply and, in many practical problems, it yields results close to rigorous limit equilibrium methods. In this study, Bishop’s simplified method was
263
modified to include reinforcement as a horizontal force intersecting the slip circle, which is incorporated in ReSSA Version 3.0 software, developed by ADAMA Engineering (2008). This modified formulation is consistent with the original formulation by Bishop (1955). The mobilized reinforcement strength at its intersection with the slip circle depends on its long-term strength, its rear-end pullout capacity (or connection strength), and the soil strength. The analysis assumes that when the soil strength is reduced by a factor, a limit equilibrium state is achieved (i.e., the system is at the verge of failure). The slip circle for which the lowest factor (i.e., the largest mobilized soil strength) exists is the critical slip surface for which the factor of safety is rendered. Under this state, when the factor of safety at unity, the soil and reinforcement mobilize their respective strengths simultaneously. ReSSA Version 3.0 also provides a factor of safety map, which identifies the critical zone based on the criterion set up by the user. 2.2. Numerical method The finite difference program, FLAC 2D Version 5.0 (Itasca Consulting Group, Inc., 2006), was adopted in this study. A shear strength reduction technique proposed by Zienkiewicz et al. (1975) was adopted in this program to solve for a factor of safety of stability. In this technique, a series of trial factors of safety are used to adjust the cohesion, c and the friction angle, 4, of soil. Adjusted cohesion and friction angle of soil layers are re-inputted in the model for limit equilibrium analysis. The factor of safety is sought when the specific adjusted cohesion and friction angle make the slope unstable from the verge of a stable condition (i.e., limit equilibrium). Details about the slope stability analysis using the FLAC software can be found in Dawson et al. (1999). The critical slip surface often can be identified based on the contours of the maximum shear strain rate. 3. Modeling 3.1. Baseline case The geometry and material properties of the baseline model used in this study are shown in Fig. 2. Since the factor of safety is determined based on a state of yield, or verge of failure, it is insensitive to the selected elastic parameters: Young’s modulus (E) and Poisson’s ratio (n) when using FLAC. If the system contains soils with largely different elastic parameters, it will take longer time to solve the factor of safety; however, the effects on this factor would be small since it depends mainly on Mohr-Coulomb strength parameters. Hence, constant values of E ¼ 100 MPa and n ¼ 0.3 were used for the reinforced fill in FLAC. High-strength parameters were assumed for the foundation to prevent any possible failure. The effect of wall facing cohesion on the required tensile strength of reinforcement in the numerical analysis will be discussed in the next section. Mohr-Coulomb failure criteria were used for strength W = 12 m 4.2 m
4.2 m
0.6m
H = 6m 1.5m 1m Fig. 1. Back-to-back MSE wall and definitions.
Blocks (γ = 18kN/m3 , c =1000kPa, φ = 340 ) Reinforced & retained fill (γ = 18kN/m3 , c =0kPa, φ = 340) Reinforcement Weak zone
Foundation (γ =18kN/m3 , c =1000kPa, φ =00 ) Fig. 2. Dimensions and parameters of the baseline case.
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between stacked blocks, the reinforced and retained fill, and the foundation soil. Reinforcement was modeled as a cable with grouted interface properties between cable and soil. The bond strength between reinforcement and reinforced fill was assumed equal to 80% the fill strength, same as in the limit equilibrium analysis when considering pullout resistance. A weak zone at the toe of the MSE wall with a dimension of 0.3 m wide and 0.4 m high, which has zero cohesion but the same friction angle as the fill, was assumed to ensure that the critical failure surface passes through the toe of the MSE wall. Details about the effect of wall facing cohesion will be discussed later. In this baseline case, the back-to-back wall width (W) to height (H) ratio is equal to 2.0 and the distance at back of two walls, D is equal to 3.6 m, which is slightly greater than H tan(45 f/ 2) ¼ 3.2 m. Based on the FHWA design guideline, a typical design method for a single wall can be adopted. The reinforcement length, L ¼ 4.2 m, was selected based on the typical reinforcement length/ wall height ratio of 0.7 recommended by the FHWA design guideline. The height of MSE walls is fixed at 6 m. Two important parameters, the back-to-back wall width to height ratio and the quality (i.e., friction angle) of backfill material, were selected in this study to investigate their influence on the critical failure surface, the required tensile strength of reinforcement, and the active thrust to the reinforced zone. In addition to W/H ¼ 2.0 for the baseline case, two other W/H ratios (1.4 and 3.0) were used in the parametric study. One parameter in the baseline was changed at a time while all others were unchanged. The same models were used in numerical and limit equilibrium analyses. The required tensile strength of reinforcement was determined to ensure the factor of safety of the MSE wall equal to 1.0.
3.2. Effect of wall facing cohesion The effect of wall facing cohesion was examined for the baseline case in this study. As shown in Fig. 3, the factor of safety of the backto-back MSE wall increases with an increase of the wall facing cohesion. However, it becomes constant after the cohesion is greater than 100 kPa. In this case, the potential failure surface was restricted to pass only through the toe of the MSE wall. Fig. 3 also shows that the effect of the wall facing cohesion for the case with low-quality backfill (4 ¼ 25 ) is more significant than that with high-quality backfill (4 ¼ 34 ). In all analyses discussed below, the 1.25 1.2
φ = 25o Ta = 30.2kN/m
Factor of safety
1.15
4.2m φ=34o
φ=25o
6m
φ=25
φ=34o o
Numerical LE
Fig. 4. Critical failure surfaces within walls at W/H ¼ 3.
cohesion of the wall facing was assumed to be 1000 kPa except a weak zone close to the toe. 4. Results 4.1. Critical failure surfaces The locations and shapes of critical failure surfaces of the backto-back walls at different wall width to height ratios (W/H) were determined based on the contours of shear strain rates in the numerical analysis and presented in Figs. 4–6. Fig. 4 shows that the critical failure surfaces in two opposing walls do not intercept each other, therefore, they behave independently. The critical failure surfaces by the LE method assuming only one side wall are also shown in Fig. 4, and they have slightly steeper angles than those by the numerical method. This conclusion is consistent with that from the previous study conducted by the authors (Han and Leshchinsky, 2006b). Since the LE method cannot analyze two side walls, which become important when the distance between the walls gets closer, the results presented below are based on the numerical analysis unless noted. Fig. 5 shows the critical failure surfaces within back-to-back walls at W/H ¼ 2 that intercept each other from two sides. More interactions occur for the case with a low-quality backfill. (i.e., 4 ¼ 25 ). For both cases, the critical failure surfaces do not enter the reinforced zone on the opposing side. In other words, the potential failure surface is constrained by the reinforced zone on the opposing side. The interaction distance, Di, based on the FHWA design guideline can be determined using Equation (1). Considering the backfill with the friction angles of 25 and 34 , the interaction distances are 0.64H and 0.53H, respectively. In other words, when the wall width to height ratio, W/H > 2.04 for the friction angle of the fill equal to 25 or W/H > 1.93 for the friction angle of the fill equal to 34 , the two back-to-back walls should perform independently. However, Fig. 5 shows that the back-toback walls still interact each other when W/H ¼ 2.0 > 1.93 for the
4.2m
1.1 1.05
4.2m
9.6m
3.6m
4.2m
φ = 34o Ta = 11.2kN/m
1
6m 0.95
φ = 34o φ = 25o
0.9 0
200
400
600
800
1000
Cohesion of wall facing (kPa) Fig. 3. Effect of wall facing cohesion.
Fig. 5. Critical failure surfaces within walls at W/H ¼ 2.0.
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4.2m
connected when they meet in the middle of the walls. The numerical results show the interactions of critical failure surfaces in two opposing walls. In both cases, the failure surfaces enter the reinforced zone from another side. The comparisons of locations and shapes of critical failure surfaces at different W/H ratios but the same quality of fill are presented in Fig. 7. Fig. 7 shows that the locations and shapes of the critical failures are almost same for W/H ¼ 3 and 2. This result can be explained as the failure surfaces not entering the reinforced zone on the opposing side. For W/H ¼ 1.4, however, the locations and shapes of the critical failure surfaces deviate from others as the failure surfaces enter the reinforced zone on the opposing side.
4.2m
6m
φ = 34o
φ = 25o
4.2. Distribution of maximum tension with height
Fig. 6. Critical failure surfaces within walls at W/H ¼ 1.4.
friction angle of the fill equal to 34 . Apparently, this assumption in the FHWA design guideline is not supported by the numerical result. However, the FHWA assumption leads to more conservative results. Fig. 6 shows the critical failure surfaces developed within the back-to-back walls when there is no retained fill between these two walls (i.e., D ¼ 0 m). In both cases, reinforcement layers are not
a
4.2m
φ = 34o W/H = 3
6m
W/H = 2 W/H = 1.4
The distribution of the maximum tension, Tmax, on each reinforcement is plotted against the height of the reinforcement, h, for both backfill materials in Fig. 8. These distributions were determined from the numerical results based on the reinforcements having equal strength and the MSE wall having a factor of safety of 1.0. These distributions are trapezoidal and similar to those obtained by Han and Leshchinsky (2006a) using the limit equilibrium method. Fig. 8 shows that the maximum tension in the reinforcement in the MSE wall with low friction angle fill is much higher than that with high friction angle fill. The width to height ratio of the back-to-back wall has a slight effect on the distribution of the maximum tension in the reinforcement at height, i.e., the overall maximum tension increases with the increase of the width to height ratio. This result implies that it is slightly conservative to ignore the influence of the width to height ratio on the overall maximum tension in the internal stability analysis of the backto-back MSE wall. Fig. 8 shows that the overall maximum tension occurs from the bottom to 1/3 or ½ height of the wall for the low or high friction angle fill, respectively. It is noted that the maximum tension in the upper portion of the wall is slightly higher for W/H ¼ 1.4 than W/H ¼ 2.0 and 3.0. This result can be explained by the fact that the maximum tension in the reinforcement for W/H ¼ 1.4 is contributed by both sides of walls due to the interception of the failure surfaces in the upper portion.
φ = 34o 4.2m
b
φ = 25o 6m
265
W/H = 3 W/H = 2 W/H = 1.4
φ = 25o Fig. 7. Critical failure surfaces at different W/H ratios.
Fig. 8. Distribution of maximum tension in each reinforcement.
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a
6
Elevation of wall, hz (m)
5 W/H = 1.4 W/H = 2.0 W/H = 3.0 Rankine
4
3
2
1
0 0
10
20
30
40
Lateral earth pressure behind reinforced zone, ph (kPa)
b
6
Fig. 9. Required maximum tensile strength of reinforcement.
The required overall maximum tensile tension or strengths of reinforcements for all the cases discussed above are presented in Fig. 9. The results from the LE method were based on the analyses of one side wall, therefore, no interaction of two opposing walls was considered. In other words, the required tensile strengths do not change with the W/H ratios. Fig. 9 clearly shows that a decrease of W/H ratio from 3 to 1.4 reduces the required maximum tensile strength of reinforcement. This result implies that the back-to-back walls for both backfill materials still interact at W/H ranging from 2.0 to 3.0. The LE method without considering the interaction of the opposing walls would provide conservative design of back-to-back MSE walls. The difference in the maximum tensile strength of reinforcement with and without considering the interaction is within 12% based on the cases investigated in this study. The required maximum tensile strengths can be used for the selection of reinforcements (such as geosynthetics) in the back-to-back MSE walls.
Elevation of wall, hz (m)
5
4.3. Required tensile strength
W/H = 1.4 W/H = 2.0 W/H = 3.0 Rankine
4
3
2
1
0
0
10
20
30
40
Fig. 10. Distribution of lateral earth pressure behind the reinforced zone.
4.4. Lateral earth pressure behind the reinforced zone Based on the FHWA guideline (Elias and Christopher, 1997; Elias et al., 2001), the lateral earth pressure or active thrust behind the reinforced zone for the external stability analysis should depend on the width to height ratio. Fig. 10 presents the numerical results of the lateral earth pressure, ph, behind the reinforced zone at a different elevation of the wall, hz. It is clearly shown that the lateral earth pressure exists behind the reinforced zone, even for the width to height ratio of W/H ¼ 1.4 (i.e., no retained fill). However, the FHWA guideline (Elias and Christopher, 1997) suggested that the lateral earth pressure for external analysis should be ignored if D ¼ 0 (i.e., W/H ¼ 1.4). Obviously, this suggestion would yield an unsafe design. Fig. 10 also shows that the average lateral earth pressure behind the reinforced zone is close to the active Rankine lateral earth pressure when the width to height ratio is large (for example, W/H ¼ 3.0). However, the lateral earth pressure decreases when the width to height ratio decreases. The percent of the active lateral thrust behind the reinforced zone to the theoretical active Rankine lateral thrust, Ph, is presented in Fig. 11, which shows the influence of the backfill friction angle and the width to
50
Lateral earth pressure behind reinforced zone, ph (kPa)
Fig. 11. Percent of lateral thrust behind the reinforced zone.
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This interaction will change the location and shape of critical failure surface. The FHWA design guideline underestimates the interaction distance. When the distance of the two back-to-back walls gets closer, the required maximum tensile strength of reinforcement slightly decreases. Ignoring the effect of the width to height ratio on the overall maximum tensile strength of reinforcement results in a slightly conservative design for internal stability. However, ignoring the external lateral earth pressure on the reinforced zone, when the distance of the back-to-back walls, D, equals to 0, results in an unsafe design for external stability. The connection of reinforcements in the middle ensures the mobilization of the full strength of the reinforcement and reduces the maximum required tensile strength.
Acknowledgements
Fig. 12. Effect of reinforcement connection on the maximum tension.
height ratio on the mobilization of the lateral thrust. When a low friction angle backfill material is used, the percent of the lateral thrust is low. However, an increase of the width to height ratio from 1.4 to 3.0 increases the lateral thrust behind the reinforced zone. 4.5. Effect of reinforcement connection in the middle When the distance of two opposing walls, D, equals to 0, the reinforcements from both sides would meet in the middle. For an ease of construction, these reinforcement layers are often not connected. However, Fig. 12 shows that the connection of these reinforcements reduces the overall maximum tension in the wall because each reinforcement can mobilize its full strength when they are connected in the middle. The pullout from the middle of the walls becomes impossible. The maximum tension in the reinforcement decreasing with the height of the reinforcement for the unconnected case is because the mobilized tension in the reinforcement is limited by its pullout capacity at a higher height close to the top of the wall. 5. Conclusions The study using the numerical and limit equilibrium methods shows that two back-to-back walls perform independently when they are far apart and interact with each other when they are close.
This paper is based upon the work supported by the National Science Foundation under Grant No. 0442159. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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