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Technical Report Documentation Page

1. Report No.

FHWA-HIF-17-004

2. Government Accession No. 3. Recipient’s Catalog No.

4. Title and Subtitle

Limit Equilibrium Design Framework 

Extensible Reinforcement

for MSE Structures with 

5. Report Date

 October 2016

6. Performing Organization Code

7. Principal Investigator(s): See Acknowledgements for Authors and 

Contributors 

Dov Leshchinsky, Ph.D1 1 , Ora Leshchinsky, P.E. , 

Brian Zelenko, P.E., John Horne, Ph.D., P.E.

8. Performing Organization Report 

No.

9. Performing Organization Name and Address

Parsons Brinckerhoff

1015 Half Street, SE, Suite 650

Washington, DC 20003

1ADAMA Engineering, Inc., 12042 SE Sunnyside Rd., 

Clackamas, OR 97015 

Suite 711, 

10. Work Unit No. (TRAIS)

11. Contract or Grant No.

DTFH6114D00047-5010

12. Sponsoring Agency Name and Address

Federal Highway Administration

HIBT-20

Office of Bridge Technology

1200 New Jersey Avenue, SE

Washington, DC 20005

13. Type of Report and Period 

14. Sponsoring Agency Code

15. Supplementary Notes

FHWA COR – Silas Nichols, P.E.

FHWA Alt. COR – Khalid Mohamed, P.E. 

16. Abstract

Current design of reinforced soil structures in the U.S. distinguishes between slopes and walls using the 

batter angle as a criterion. Using a unified approach in limit state design of reinforced ‘walls’ and 

‘slopes’ should diminish confusion while enabling a wide and consistent usage in solving geotechnical 

problems such as complex geometries and soil profiles. Limit equilibrium (LE) analysis has been used 

successfully in the design of complex and critical (e.g., tall dams) for many decades. Limit state 

analysis, including LE, assumes that the design strength of the soil is mobilized. Presented is a LE 

framework, limited to extensible reinforcement, which enables the designer to find the tensile force 

distribution in each layer required at a limit state. This approach is restricted to Allowable Stress Design

(ASD). Three example problems are presented.

17. Key Words

Mechanically Stabilized Earth Wall Design, MSE Wall

Design, Limit Equilibrium, Geotechnical, Extensible 

reinforcement

18. Distribution Statement

No restrictions.

19. Security Classif. (of this report)

UNCLASSIFIED

20. Security Classif. (of this 

UNCLASSIFIED

21. No. of Pages

120

22. Price

Form DOT F 1700.7(8-72) Reproduction of completed page authorized
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APPROXIMATE CONVERSIONS TO SI UNITS 

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in 

ft 

yd 

mi 

in2 

ft2 

yd2 

ac 

mi2 

fl oz 

gal

ft3 

yd3 

oz 

lb 

T 

OF 

fc 

fl 

lbf 

lbf/in2 

LENGTH 

inches 25.4 millimeters 

feet 0 .305 meters 

yards 0.914 meters 

miles 1.61 kilometers 

AREA 

square inches 645 .2 square millimeters 

square feet 0.093 square meters 

square yard 0.836 square meters 

acres 0.405 hectares 

square miles 2.59 square kilometers 

VOLUME 

fluid ounces 29.57 milliliters 

gallons 3.785 liters 

cubic feet 0 .028 cubic meters 

cubic yards 0 .765 cubic meters 

NOTE: volumes greater than 1000 l shall be shown in m3 

MASS 

ounces 28.35 grams 

pounds 0.454 kilograms 

short tons (2000 lb) 0.907 megagrams (or "metric ton") 

TEMPERATURE (exact degrees) 

Fahrenheit 5 (F-32)/9 Celsius 

or (F-32)/1.8 

ILLUMINATION 

foot-candles 10.76 lux 

foot-lamberts 3.426 candela/m2 

FORCEandPRESSUREorSTRESS 

poundforce 4.45 newtons 

poundforce per square inch 6 .89 kilopascals 

mm 

m 

m 

km 

2 mm 

m2 

m2 

ha 

km2 

ml 

l 

m3 

m3 

g 

kg 

Mg (or"t") 

oc 

Ix 

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APPROXIMATE CONVERSIONS FROM SI UNITS 

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mm 

m 

m 

km 

2 mm 

m2 

m2 

ha 

km2 

ml 

l 

m3 

m3 

g 

kg 

Mg (or "t") 

oc 

Ix 

cd/m2 

N 

kPa 

LENGTH 

millimeters 0 .039 inches 

meters 3 .28 feet 

meters 1.09 yards 

kilometers 0.621 miles 

AREA 

square millimeters 0.0016 square inches 

square meters 10.764 square feet 

square meters 1.1 95 square yards 

hectares 2.47 acres 

square kilometers 0.386 square miles 

VOLUME 

milliliters 0 .034 fluid ounces 

liters 0.264 gallons 

cubic meters 35.314 cubic feet 

cubic meters 1.307 cubic yards 

MASS 

grams 0.035 ounces 

kilograms 2.202 pounds 

megagrams (or "metric ton") 1.103 short tons (2000 lb) 

TEMPERATURE (exact degrees) 

Celsius 1.8C+32 Fahrenheit 

ILLUMINATION 

lux 0.0929 foot-candles 

candela/m2 0.2919 foot-lamberts 

FORCEandPRESSUREorSTRESS 

newtons 0.225 poundforce 

kilopascals 0.145 poundforce per square inch 

in 

ft 

yd 

mi 

in2 

ft2 

yd2 

ac 

mi2 

fl oz 

gal

ft3 

yd3 

oz 

lb 

T 

OF 

fc 

fl 

lbf 2 lbf/in

*SI is the symbol for the International System of Units. Appropriate rounding should be made to comply with Section 4 ofASTM E380. 

(Revised March 2003) 
TABLE OF CONTENTS
 

2.0 COMMON METHODS OF ANALYSIS........................................................................ 8
 

3.0 LE APPROACH INCLUDING THE SAFETY MAP TOOL .................................... 10
 

4.0 COMPARISON OF LE PREDICTIONS WITH PHYSICAL MODEL TESTS...... 18
 

4.1 Extensible Reinforcement......................................................................................... 18
 

4.1.1 Concluding Remarks............................................................................................. 28
 

4.2 Inextensible Reinforcement ...................................................................................... 29
 

5.0 COMPARISON OF LE PREDICTIONS WITH FE/FD RESULTS......................... 30
 

5.1 Extensible Reinforcement......................................................................................... 30
 

5.1.1 Concluding Remarks............................................................................................. 38
 

5.2 Inextensible Reinforcement ...................................................................................... 39
 

6.0 LE AND LRFD................................................................................................................ 40
 

7.0 USE OF LE IN DESIGN OF MSE WALLS ................................................................ 42
 

7.1 General...................................................................................................................... 42
 

7.2 The European Standard: Eurocode 7—Geotechnical Design................................... 42
 

7.3 The German Approach: EBGEO .............................................................................. 43
 

7.4 The British Standard: BS 8006 ................................................................................. 44
 

7.5 The Japan Rail Approach.......................................................................................... 45
 

8.0 RETROSPECTIVE: LIMIT STATE ANALYSIS OF REINFORCED WALLS .... 47
 

9.0 FRAMEWORK FOR LE ANALYSIS.......................................................................... 49
 

9.1 Basics of Soil Reinforcing ........................................................................................ 49
 

9.2 Bishop LE Formulation............................................................................................. 51
 

9.3 Overview of the Modified LE Procedure.................................................................. 54
 

9.4 Concept of Rupture and Pullout in LE Analysis....................................................... 56
 

9.5 Top-down Approach Yielding the Baseline Solution ............................................... 58
 

9.6 Facing Effects on Calculated Reinforcement Load at Limit State ........................... 60
 

9.7 Commentary.............................................................................................................. 62
 

10.0 INSTRUCTIVE DESIGN EXAMPLES ....................................................................... 65
 

10.1 Overview of Top-down Procedure............................................................................ 65
 

10.2 Example 1: Simple Wall Problem Compared with AASHTO 2002......................... 67
 

10.2.1 AASHTO Approach.............................................................................................. 67
 

10.2.2 LE Framework ...................................................................................................... 69
 

10.2.3 Closely Spaced Reinforcement ............................................................................. 78
 

10.2.4 Seismicity and Closely Spaced Reinforcement .................................................... 82
 

10.2.5 Secondary Reinforcement..................................................................................... 87
 

10.3 Example 2: Simple Slope Problem ........................................................................... 92
 

i 


 

1.0 INTRODUCTION............................................................................................................. 1
 

1.1 MSE Structures: Limit Equilibrium Concept in a Nutshell ........................................ 4
 

1.2 Overall Organization of Report and Rationale ........................................................... 6
 
10.4 Example 3: Tiered Reinforced System ..................................................................... 99
 

10.5 Commentary............................................................................................................ 105
 

11.0 CONCLUDING REMARKS ....................................................................................... 109
 

12.0 REFERENCES.............................................................................................................. 112
 

13.0 SOFTWARE REFERENCES...................................................................................... 117
 

APPENDIX A: DERIVATION OF BISHOP EQUATION FOR REINFORCED SOIL .. 118
 

ii 
FIGURES 

Figure 1-1. Result of missing reinforcement: close-up (left) and general view (right) 

[January 2008, Salt Lake City, Utah] ...........................................................................2 

Figure 1-2. Metallic reinforced wall: limit state shallow failure [June 2010, Monterrey,
 

Mexico, Hurricane Alex]..............................................................................................3 

Figure 1-3. Geosynthetic reinforced wall: limit state shallow failure [I-85 near Exit 70, 

Cusseta, Alabama; photo taken February 2007] ..........................................................3 

Figure 1-4. Geosynthetic rupture in reinforced tall slope [Yeager Airport, Charleston, 

WV, March 2015].........................................................................................................4 

Figure 1-5. Basic statics: (a) test body ABC representing the summation of all 

reinforcements along AB as ΣT; (b) free body diagram and force polygon .................5 

Figure 3-1. Available tensile resistance along reinforcement in current design............................11 

Figure 3-2. The basic problem of multi-tiered slope/wall .............................................................12 

Figure 3-3. Safety map for the unreinforced problem using Bishop’s analysis.............................13 

Figure 3-4. Safety map for the reinforced problem using Bishop’s analysis.................................14 

Figure 3-5. Safety map using Spencer’s two-part wedge translational surfaces ...........................15 

Figure 3-6. Safety map using three-part wedge surfaces...............................................................16 

Figure 3-7. Effects of replacing the foundation soil with select fill (two-part wedge)..................17 

Figure 4-1. The Norwegian wall: geometry and geogrid layout (Fannin and Hermann, 

1990)...........................................................................................................................19 

Figure 4-2. The Norwegian wall: traces of critical slip circles considering the impact of
 

soil strength and offset ...............................................................................................20 

Figure 4-3. The Norwegian wall: (a) loads as function of φ and offsets and (b) loads as
 
 

function of φ and reinforcement length in upper tier .................................................22 

Figure 4-4. Failed geotextile-reinforced slope/wall in centrifuge modeling (courtesy Prof. 

Zornberg)....................................................................................................................22 

Figure 4-5. General layout and dimensions of tested centrifugal models (after Zornberg et
 

al., 1998).....................................................................................................................23 

Figure 4-6. Prototype at failure deduced from centrifugal model test (left) and predicted 

and measured locus of Tmax-i (right) ...........................................................................23 

Figure 4-7. Configuration of centrifuge tiered wall model (after Mohamed et al., 2013).............24 

Figure 4-8. LE predicted slip surface vs. observed from centrifugal test (after Yang et al., 

2009)...........................................................................................................................25 

Figure 4-9. LE predicted failure acceleration vs. measured value (after Yang et al., 2009) .........25 

Figure 4-10. Cross-section of wrapped-face RMC test wall (after Bathurst et al., 2006) .............26 

Figure 4-11. max(Tmax), Tmaxmax, at various surcharge levels (after Yang et al., 2013) .................27 

Figure 4-12. Section of FHWA wall (reproduced after Allen and Bathurst, 2001).......................28 

Figure 4-13. FHWA wall: (a) Flat and (b) backsloped crest (Leshchinsky et al., 2014)...............28 

Figure 5-1. (a) Safety map and Bishop critical slip circle (Fs=1.00). (b) Maximum shear
 

zone using FLAC (Fs=0.98) (Leshchinsky and Han, 2004) ......................................31 

Figure 5-2. (a) Bearing capacity using Spencer 3-part wedge (Fs=0.92). (b) Shear zone
 

for bearing failure using FLAC (Fs=0.86) (Leshchinsky and Han, 2004).................32 

Figure 5-3. Slip surfaces predicted by FE and LE versus the location of ruptured 

geotextiles (after Mohamed et al., 2014)....................................................................33 

iii 
Figure 5-4. Mobilized tensile load (FE) at various accelerations and location of failure 

surface predicted by LE (after Mohamed et al., 2014)...............................................34 

Figure 5-5. Sum of Tmax versus the uniform surcharge on RMC wall (after Yang et al., 

2013)...........................................................................................................................35 

Figure 5-6. Effect of cohesion wall facing on Fs (after Han and Leshchinsky, 2010)..................36 

Figure 5-7. Critical surfaces predicted by LE and FD (after Han and Leshchinsky, 2010) ..........37 

Figure 5-8. max(Tmax) versus W/H (after Han and Leshchinsky, 2010) ........................................37 

Figure 5-9. Defined constraint reinforced space (after Leshchinsky et al., 2004).........................38 

Figure 7-1. Potential slip surfaces through and around the wall (after EBGEO, 2011) ................43 

Figure 7-2. External stability modes of failure (after BS 8006, 2010) ..........................................44 

Figure 7-3. Internal wedge stability (after BS 8006, 2010) ...........................................................45 

Figure 7-4. Internal log spiral stability—coherent method (after BS 8006, 2010)........................45 

Figure 9-1. Schematics of reinforcement layer required to stabilize steep slope ..........................51 

Figure 9-2. Circular test body used in the Bishop Method ............................................................52 

Figure 9-3. Forces acting on Slice i ...............................................................................................53 

Figure 9-4. Required force in layer i defined by front and rear pullout resistance........................56 

Figure 9-5. Length of layer i: Ideal (lower left), excessive (upper right), short (lower 

right) ...........................................................................................................................57 

Figure 9-6. Example of top-down numerical procedure................................................................58 

Figure 9-7. Example for determining the required connection capacity [Leshchinsky et
 

al., 2014].....................................................................................................................60 

Figure 9-8. Model to consider impact of facing on limit state.......................................................61 

Figure 10-1. Sketch of analyzed wall ............................................................................................67 

Figure 10-2: Example 1: Graphical representation of Tables 10-1 & 10-2 ...................................70 

Figure 10-3. Example 1: Baseline Treq-i(x) rendered by LE top-down approach...........................71 

Figure 10-4. Example 1: Treq for Layer 10 restricted by rear end pullout .....................................72 

Figure 10-5. Example 1: Treq for Layer 9 restricted by rear end pullout .......................................72 

Figure 10-6. Example 1: Treq for Layer 10 restricted by rear and front end pullout......................73 

Figure 10-7. Example 1: Distribution of tensile resistance needed for LE....................................74 

Figure 10-8. Example 1: Critical circle using LTDS corresponding to Tmax-i in Table 10-2..........75 

Figure 10-9. Global stability: Actual LTDS distribution along some reinforcement layers ..........76 

Figure 10-10. Safety map generated for the design example using Bishop analysis.....................77 

Figure 10-11. Safety map generated using Spencer analysis considering 2-part wedge ...............78 

Figure 10-12. Graphical representation of Tables 10-2 & 10-3.....................................................80 

Figure 10-13. Close reinforcement: Color coded map of distribution of tensile resistance ..........81 

Figure 10-14. Close reinforcement: Computed Treq along each layer ...........................................81 

Figure 10-15. Safety map generated for the closely spaced reinforcement...................................82 

Figure 10-16. Graphical representation of Tables 10-3 & 10-4.....................................................84 

Figure 10-17. Computed Treq along each layer for PGA=0.30g ....................................................85 

Figure 10-18. Color coded map of distribution of tensile resistance for PGA=0.30g ...................85 

Figure 10-19. Safety map for PGA=0.3g running Bishop’s circular arc analysis .........................86 

Figure 10-20. Safety map for PGA=0.3g running Spencer 2-part wedge analysis........................87 

Figure 10-21. Graphical representation of Tables 10-3 & 10-5.....................................................89 

Figure 10-22. Computed Treq(x) considering 5 ft long secondary reinforcement..........................90 

Figure 10-23. Color coded map of Treq(x) considering 5 ft long secondary reinforcement...........91 

Figure 10-24. Safety map for the case of secondary reinforcement ..............................................92 

iv 
Figure 10-25. Graphical representation of Tables 10-2 & 10-6.....................................................94 

Figure 10-26. Baseline Treq-i(x) for Example 2 ..............................................................................95 

Figure 10-27. Color coded map of distribution of tensile resistance for Example 2 .....................96 

Figure 10-28. Safety map generated for Example 2 using Bishop analysis ..................................96 

Figure 10-29. Safety map for adjusted strength of reinforcement.................................................97 

Figure 10-30. Safety map for adjusted length of reinforcement....................................................98 

Figure 10-31. Design section for Example 3 .................................................................................99 

Figure 10-32. Graphical representation of results in Table 10-7 .................................................101 

Figure 10-33. Baseline Treq-i(x) for Example 3 ............................................................................102 

Figure 10-34. Color coded map of distribution of tensile resistance for Example 3 ...................103 

Figure 10-35. Conventional stability analysis: Critical circle and Fs for LTDS=703 lb/ft..........104 

Figure 10-36. Example 3: Safety map for the designed problem ................................................105 

Figure 10-37. Link between LE results and possible design specification ..................................107 

Figure A-1. Forces acting on slice i .............................................................................................118
 

v 
TABLE 

Table 5-1. Summary Fs calculated by LE and FD for a given geometry and ∑Tmax 

(Leshchinsky et al., 2004) ..........................................................................................38
 

Table 10-1. Calculated loads and factors of safety for strength and pullout .................................68
 

Table 10-2. Example 1: Values produced by the LE framework ..................................................69
 

Table 10-3. Impact of decreased spacing of reinforcement layers ................................................79
 

Table 10-4. Wall with closely spaced reinforcement subjected to PGA=0.3g ..............................83
 

Table 10-5. Impact of secondary reinforcement ............................................................................88
 

Table 10-6. Computed values for Example 2 produced by the LE framework .............................93
 

Table 10-7. Computed Tmax and To for Example 3.......................................................................100
 

vi 
 

 

 

 

 

 

 

 

LIST OF ABBREVIATIONS AND SYMBOLS
 

Ka Active lateral earth pressure 

Ko At-rest pressure 

AASHTO American Assoication of State Highway and Transportation 

Officials 

ASD Allowable Stress Design 

To_i Connection capacity 

To-I Connection loads 

EA Effective block area 

Rv Effective weight 

CEN European Committee for Standardization or Comite Europeen de 

Normalisation 

F Factor of safety 

FHWA Federal Highway Assocaition 

FD Finite difference 

FE Finite element 

δ Interface friction 

JR Japan Railways 

LA Limit analysis 

LE Limit equilibrium 

LL Live load 

LRFD Load and Resistance Factor Design 

LTDS Long term design strength 

Tmax Maximum load 

Tmax-I Maximum load in each reinforcement layer 

MSE Mechanically stabilized earth 

vii 
M-C Mohr-Coulomb failure criterion 

FE/FD Numerical models 

R Radius 

Treq(x) Required reinforcement resistance 

SF Safety factor 

viii 
 

 

 

 

 

 

 

 

 

 

1.0 INTRODUCTION 

Measured field data generally indicates that under typical conditions, the maximum mobilized 

reinforcement force, particularly for geosynthetics in mechanically stabilized earth (MSE) walls 

and slopes, is less than predicted in current design. While reasonable conservatism in design is 

prudent, it seems that the one major reason for what appears as an overly-conservative design is 

the lack of distinction between typical and atypical field conditions. Atypical conditions may 

temporarily exist during the lifespan of the structure, and these conditions are addressed in any 

sound design. These conditions should consider situations such as heavy rainfalls, earthquakes, 

and vehicular collision impact. Measured field data related to reinforcement force and 

corresponding to, for example, high degree of backfill saturation is rare although numerous 

failures have occurred after heavy rainfall events. That is, measured data is usually for typical 

field conditions thus may not represent atypical conditions. 

Leshchinsky and Tatsuoka (2013) pointed out that the apparent small mobilization of 

reinforcement force under typical or normal conditions is mainly due to three reasons. One is the 

frictional strength of soil used in design is often significantly lower than the actual value when 

one considers typical select, well-compacted backfill. For example, AASHTO (2002, 2007) 

limits φ to a maximum value of 40°, allowing a default value of 34° if no shear testing is 

performed. Consequently, while for the specified gradation and compaction level the φ value 

could be much more than 40°, in design, a typical value of 34° is used. This alone may result in 

approximately doubling the predicted load in geosynthetic reinforcement. A second contributor 

to underestimation of reinforcement load is neglecting toe restraint. Frictional resistance 

generated along a 1 ft wide leveling pad or bottom facing block/panel can reduce the 

reinforcement load by as much as 50% (e.g. Huang et al., 2010; Leshchinsky and Vahedifard, 

2012). A third contributor is potential soil suction which generates an apparent cohesion in 

seemingly acceptable granular backfill. Such apparent cohesion which can be diminished with 

change in moisture content has a large impact on stability (e.g., Ling et al., 2009) thus rendering 

the reinforcement, in many cases, nearly inactive or dormant. However, even if all these factors 

are considered, experimental data shows that AASHTO’s design could still be overlyconservative. 

As an example, consider the failure of a wall designed based on AASHTO (2002) and shown in 

Figure 1-1. It occurred simultaneously on both sides of the rounded corner. Forensic study 

indicated that, layer by layer, sectors of geogrid reinforcement were not placed near the curved 

corner in the upper tier thus forming a vertical prism of unreinforced soil. The obvious 

expectation is that failure should have occurred during construction since unreinforced 

cohesionless soil, meeting AASHTO’s specification, could not stay stable as a vertical slope. 

However, collapse occurred about a year after the end of construction. The soil moisture content 

increased to a point where soil suction, and subsequently apparent cohesion, diminished to a 

level where missing reinforcements led to a predictable failure. Viewed differently, adequate 

installation of geogrids in this case would have rendered the reinforcement mostly dormant, i.e., 

hardly mobilized during and in the months following construction. However, as the apparent 

cohesion was diminishing with increased moisture, the geogrid strength and facing resistance 

would have been activated to a level needed to maintain the stability of the structure designed for 

cohesionless backfill. This case demonstrates that measuring the reinforcement load under 

1 
 

 

 

 

 

 

 

 

 

normal conditions must be assessed carefully as it may lead to unsafe conclusions regarding 

realistic, perhaps atypical, limit state conditions. 

Figure 1-1. Result of missing reinforcement: close-up (left) and general view (right) 

[January 2008, Salt Lake City, Utah] 

Sound geotechnical design needs to look at conditions which are likely to occur during the 

lifespan of the structure. Under these conditions the structure should be in static equilibrium. 

Limit equilibrium, LE, is one of several methods of analysis that can be used to assess limit state. 

It is noted that limit state failure in MSE structures is not a hypothetical situation; it has 

happened (e.g., Koerner and Koerner, 2013, Valentine, 2013). In structures where failure occurs 

along a zone extending from the reinforced into the retained soil (i.e., compound failure), the 

rupture of the reinforcement is not directly visible as the reinforcement is buried and entangled in 

a large mass of collapsed soil. However, often the facing fails as the shear strength of the backfill 

is fully mobilized along a shallow slip surface. It is accompanied by rupture of the reinforcement 

at the connection with backfill falling and slipping or sloughing along the front end of the 

reinforcement. Examples of such limit state situations are shown in Figure 1-2 and Figure 1-3. 

While it is not known whether the actual designs of these walls followed AASHTO, it is 

apparent that water played an important role leading to collapse of apparently stable structures. 

Figure 1-4 shows a ‘landslide’ of a tall reinforced slope, seemingly designed following FHWA 

(2002) procedure. The 30 layers of geogrid within the 90 feet high head scarp ruptured, all far 

from the slope face. The actual reasons for the failures depicted in these three figures are not 

fully known. However, the fact is that a strength limit state failure occurred sometime after 

construction, resulting in rupture of geosynthetic layers. The tradition of assessing stability of 

geotechnical structures while considering feasible long term conditions is clearly needed when 

designing MSE structures. 

The main objective of this report is to introduce a rational design framework suitable for 

determining the limit state conditions, enabling the designer to establish adequate margins of 

safety against such state. This rational approach uses LE analysis to produce baseline solution 

for a given problem of geosynthetic reinforced walls and slopes. This unified approach does not 

use an artificial distinction between reinforced ‘walls’ and ‘slopes.’ The modified LE approach 

allows for consistent results in solving geotechnical problems having complex geometries and 

soil profiles. At this stage the LE framework is limited to extensible reinforcement and is 

restricted to Allowable Stress Design (ASD). 

2 

 


 

Figure 1-2. Metallic reinforced wall: limit state shallow failure [June 2010, Monterrey, 

Mexico, Hurricane Alex]
 

Figure 1-3. Geosynthetic reinforced wall: limit state shallow failure [I-85 near Exit 70, 

Cusseta, Alabama; photo taken February 2007]
 

3 
Figure 1-4. Geosynthetic rupture in reinforced tall slope [Yeager Airport, Charleston, WV, 

March 2015] 

1.1 MSE Structures: Limit Equilibrium Concept in a Nutshell 

Details of a LE methodology suitable for design of MSE structures are provided in Chapters 9 

and 10. Aspects such as connection and reinforcement loads, pullout resistance, mixed length 

and spacing of reinforcements (e.g., secondary layers), are considered to be mutually dependent. 

However, to facilitate the understanding of the review presented in the following chapters, the 

LE concept as related to soil reinforcing is briefly described below. 

Consider the problem in Figure 1-5a (Leshchinsky, 2009). For simplicity, a planar potential slip 

surface AB is assumed. Rather than showing the reinforcement force at each intersection with 

AB, the summation or the resultant of these forces, denoted as ΣT, is shown at some elevation. 

The assumed AB fully defines a test body ABC. Figure 1-5b shows a free body diagram for ABC. 

In this simple illustrative case, the problem is statically determinate and solving the force 

equilibrium equations (i.e., see force polygon), one can find ΣT without resorting to statical 

assumptions. The line of action of ΣT can be determined based on the distribution of T. Imposing 

moment equilibrium for the test body makes it possible to find the location of the frictional 

resultant R; this is not usually done as it is not practically needed but it indicates that moment 

equilibrium is implicitly satisfied too. Repeating the calculations for a different test body (i.e., 

different angle θ) will yield different ΣT. The maximization process is repeated until max(ΣT) is 

found. The value of max(ΣT) is then distributed amongst all reinforcing layers using an assumed 

distribution function such as triangular, uniform or trapezoidal (e.g., Leshchinsky et al., 2010b). 

4 
 

 

 

 

 

 

While globally such an approach is consistent, an assumed distribution may result in some layers 

being locally overstressed. Chapter 9 introduces a methodology or framework in which global 

equilibrium is satisfied while potential local overstressing is addressed rationally. 

Figure 1-5. Basic statics: (a) test body ABC representing the summation of all 

reinforcements along AB as ΣT; (b) free body diagram and force polygon 

It is noted that while the problem in Figure 1-5 considers for simplicity a planar surface, more 

critical surface geometries (i.e., surfaces requiring larger max(ΣT)) may exist. Commonly, the 

planar surface that render max(ΣT) defines an active wedge. The shear strength of the soil along 

AB (Figure 1-5) is assumed to be fully mobilized. Full mobilization of soil strength in a limit 

state is feasible even when dissimilar materials are involved provided that these materials exhibit 

plastic strength (e.g., soil) or having ductile behavior while appropriate shear strengths were 

selected for analysis. A classic example would be a layered soil profile such as a granular soil 

layer over cohesive soil layer. However, this may not be the case for reinforcement that is 

incompatible with the embedding soil. Hence, a discussion on this potential issue is presented. 

AASHTO classifies reinforcement as either ‘extensible’ or ‘inextensible’ without providing an 

explicit definition of what constitutes extensible or inextensible. It implies that such 

classification is manifested by two empirical criteria. The first criterion is by determining the 

equivalent lateral earth pressure coefficient, Kr, for a given reinforcement at a working load 

condition. The second criterion is by examining the locus of Tmax through the reinforcement 

layers. This locus, often called ‘failure surface,’ defines an active wedge, although it is measured 

at working load conditions. These criteria serve as the basis for AASHTO’s design. The 

5 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

empirical data for establishing the classification of reinforcement is scattered, and thus requires 

statistical interpretation. 

When dealing with LE, extensible and inextensible reinforcement can be defined more rationally. 

That is, reinforcement that is sufficiently ductile to allow the soil to mobilize its shear strength 

would be considered as extensible. If AASHTO’s granular compacted backfill, meeting 

gradation requirements, is considered for walls, reinforcement rupture strains greater than, say, 

approximately 3% will allow the soil to fully mobilize its strength. Based on this “rule of thumb” 

criterion, most geosynthetics will be considered as extensible allowing the soil to mobilize its 

strength while contributing tensile resistance, well below rupture, to sustain a test body in limit 

equilibrium state. In design, geosynthetic reinforcement then must have sufficient long-term 

strength to allow for LE state with a prescribed margin of safety. Ductility of reinforcement 

allows for load shedding (e.g., Leshchinsky et al., 2010a) meaning possible redistribution of 

loads amongst reinforcement layers. 

Using the 3% rupture strain as a criterion, metallic reinforcement would generally be considered 

as inextensible. It means that prior to a LE state in which the soil strength is fully mobilized, the 

metal may be overstressed and rupture, and thus cease resistance contribution needed for 

stability. In reality, inextensible reinforcement restrains the formation of an active wedge and 

since the soil does not contribute its full strength, the reinforcement then must carry loads higher 

than extensible reinforcement to satisfy equilibrium of the wedge. While it may not be a problem 

meeting the needed strength of metallic reinforcement, its potential rupture after exceeding its 

yield strength could be incompatible with soil strength, thus requiring careful evaluation when 

considering inextensible reinforcement in LE analysis. As shown in the following chapters, a fair 

amount of experimental and numerical work has been conducted to establish the limit state for 

extensible reinforcement while similar work with inextensible reinforcement is seriously lacking. 

1.2 Overall Organization of Report and Rationale 

Experience indicates that reinforced soil structures are safe and economical. The intention of 

showing strength limit state failures in Chapter 1 is not to intimidate designers from using good 

technology. It is just to demonstrate that if conditions that may occur during the life of the 

structure are not adequately considered in strength limit state design, failure may occur. Such 

conditions include, for example, water percolation or seepage, seismic events, and material 

degradation. Limit state failures may be due to inadequate design and/or poor construction 

violating design assumptions. What appears as conservative design under typical conditions 

may fail under atypical but predictable conditions. The objective of this report is to provide a 

framework for adequate design strength limit state failure. It can deal with aspects such as water, 

seismicity, complex strata, and complex structure geometry. 

To develop this document, a sequence of tasks had to be accomplished. The presentation follows 

this sequence. First, an overview of common design approaches is briefly presented in Chapter 2. 

Second, a general LE approach is discussed (Chapter 3) including a detailed example using the 

safety map tool. It is shown that this tool can be effective in determining the layout and strength 

of reinforcement. However, this approach (termed ‘conventional’) requires the designer to 

specify the long term strength of the reinforcement including its connection. While the strength 

can be ‘guessed’ through trial and error, the connection strength is assumed. To determine 

6 
 

 

 

 

 

 

 

 

 

whether the LE design approach is warranted, comparisons of its limit state predictions with 

experimental test results (Chapter 4) and with continuum mechanics-based numerical methods 

results (Chapter 5) were conducted. This literature review revealed that, generally, the 

comparisons with structures reinforced with extensible inclusions were reasonably good. 

However, strength limit state comparisons with inextensible reinforcement are scarce. 

Consequently, the scope of this report is limited to the development of a design framework of 

geosynthetic reinforced structures. 

When formulating LE in the context of LRFD, it became apparent that there are fundamental 

conflicts between LRFD and LE applied to soil (i.e., material possessing frictional strength). 

Resolving these conflicts (Chapter 6) is beyond the scope of this project. Hence, the scope was 

limited to ASD thus enabling the use of standard LE analysis in the framework. 

Design standards in a few other countries were reviewed (Chapter 7). Use of LE in design of 

MSE walls in codes such as the German EBGEO adds confidence in developing a design tool 

based on LE for use in the USA. Since LE is already in use for design of reinforced slopes (i.e., 

face batter greater than 20 degrees), a brief retrospective of applicability of limit state analysis in 

reinforced walls is provided in Chapter 8. Such analysis is for the strength limit state where the 

reinforcement is sufficiently ductile to allow the soil to mobilize its strength well before it 

ruptures. It is noted that ASD design requires a minimum factor of safety on the soil strength, 

generally between 1.3 and 1.5. Such design artificially reduces the soil strength resulting in 

longer and stronger reinforcement. Hence, the actual structure is not at the brink of failure; it has 

a safety margin, similar to engineered ordinary slopes. 

The original contribution in this report starts in Chapter 9. A methodology is presented 

providing the designer with a baseline solution for the reinforced problem. That is, for a given 

layout of reinforcement, soil strata, a simple or complex geometry, surcharge, and possible future 

events (earthquake, flooding), the designer gets the force in the reinforcement, including at the 

connection. This force varies along each reinforcement layer ensuring that a prescribed factor of 

safety on soil strength is essentially the same everywhere within the reinforced mass. That is, a 

safety ‘map’ is produced where the soil is equally mobilized everywhere through adjustment of 

the reinforcement force. It is computationally extensive and the methodology presented ‘scans’ 

the mass with numerous trial slip circles using a top-down approach. The method of analysis 

selected in the framework is Bishop’s analysis. 

At first, the framework methodology may appear abstract, difficult to follow. To ease the 

understanding of the methodology and realize its practical implications, instructive examples are 

presented in Chapter 10. It includes comparison with AASHTO ASD, assessment of spacing 

effects, secondary reinforcement, a seismic problem, and a hybrid complex problem comprised 

of a reinforced ‘wall’ topped with a reinforced ‘slope’, carrying live load, and reinforcement 

length limited by cemented sandstone ‘retained soil’. It is shown that the baseline solution 

enables one to select adequate reinforcement and specify minimum connection strength. The 

example problems demonstrate that ‘traditional’ LE stability assessment is needed after selecting 

the reinforcement to ensure that the designed structure is indeed stable for other possible modes 

of failure. The implications of limit state connection loads are discussed in the context of design 

of critical structures where movement is severely restricted. In the concluding remarks (Chapter 

11), issues discussed before are summarized. 

7 
 

 

 

 

 

 

 

 

 

	 

	 

	 

2.0 COMMON METHODS OF ANALYSIS 

The general objective in designing MSE structures is to determine the required strength and 

layout of the reinforcement so as to satisfy prescribed performance criteria. For a given facing, 

the layout and strength of the reinforcement are coupled, possibly leading to many potential 

solutions yielding satisfactory performance. The selected solution should be economical, 

considering the cost of facing, reinforcement, backfill, and construction. In Chapters 9 and 10 an 

analytical framework is presented helping the designer in making a rational consideration of all 

factors. 

The following is a brief review of several design methodologies including semi-empirical, 

analytical, and complex numerical approaches: 

1.	 Lateral Earth Pressure: Most design methods determine the reinforcement load based on 

calculated lateral earth pressures (e.g., the ‘Simplified Method’ as in AASHTO 2002, 2007). 

The approach is semi-empirical (e.g., lateral earth pressure coefficient and connection load 

are empirically selected or arbitrarily imposed). The main advantage of this approach is its 

simplicity. However, experience shows that it could lead to overly conservative selection of 

reinforcement strength when max(Tmax) is used as the selection criterion. Also important is its 

limitation for wall face batter to a maximum of 20° as well as its applicability to only very 

simple geometry and to homogeneous backfill. It has limited consideration of the interaction 

amongst reinforcement layers, and therefore, it offers little insight in terms of producing an 

optimized design which may include intermediate reinforcement layers at some elevations. In 

fact, the empirical coefficients used in this approach were determined at working load 

conditions, not at a limit state. Nevertheless, frequently the locus of Tmax is termed failure 

surface thus implying the logical conclusion that it was established for a limit state. 

Furthermore, it is often confused with LE analysis where actually LE does not deal explicitly 

with lateral earth pressures. This confusion is promoted by the fact that for geosynthetics the 

locus of Tmax is assumed to coincide with the failure plane rendered by Coulomb analysis. 

While Coulomb’s is one of several possible LE analyses, it does not necessarily imply for 

uniformly spaced reinforcement a linear increase in force with depth as done, for example, in 

AASHTO for extensible reinforcement. 

2.	 Continuum Mechanics: This approach is numerically based (finite element, FE, and finite 

difference, FD). It is comprehensive in a sense that basic rules of mechanics are considered 

while accounting for boundary conditions and detailed material constitutive behavior 

including nonlinearity. It is valid for slopes, walls, stratified soil, water, and more. To obtain 

reliable results at working load conditions (e.g., displacement), quality field data is needed, 

more than would generally be used in typical design. FE and FD results at a limit state could 

be reliable, not requiring an a priori assumption regarding the shape of the critical failure 

surface. However, FE/FD requires a designer with understanding of the numerical tool 

including its potential limitations and pitfalls (e.g., Cheng et al., 2007; Shukha and Baker, 

2003; Duncan 1996). 

3.	 Limit Analysis (LA): Numerical upper bound in LA, based on the theory of soil plasticity, 

yields kinematically admissible failure mechanisms; i.e., it is not necessary to assume 

arbitrarily a failure mechanism, which is an advantage when complex problems are 

considered. Numerical LA can deal with layered soil, complex geometries, water, seismicity, 

8 
 

 

 

 

 

 

 

 

 

	 

etc. It is valid for reinforced slopes and walls. However, limited familiarity of practicing 

engineers with LA hinders its current use in routine design. 

4.	 Limit Equilibrium (LE): LE can be applicable for complex problems including walls, slopes, 

compound geometries, layered strata, water, seismicity, etc. Its application to reinforced soil 

problems is merely an extension of an approach that has been used for decades in other, some 

critical, geotechnical problems. One concern with LE (and LA) is its lack of direct 

consideration of compatibility between dissimilar materials. However, in unreinforced soil 

problems, consideration is given to the selection of material properties and prevailing failure 

mechanisms when vastly different soil layers exist. A thirty-year experience shows that the 

use of LE in conjunction with soil and geosynthetics, both “ductile” materials, is not much of 

an issue. However, material properties for limit state design should be carefully selected as 

stated in Section 1.1. Chapters 4 and 5 indicate that there is limited work done in 

conjunction with limit state of inextensible reinforced structures. Hence, the presented LE 

framework is currently limited to extensible reinforcement. 

The mechanics of LE is tangible and simple to apply. If properly used, it can yield reasonably 

good agreement with experimental data (e.g., see Chapter 4) as well as with results obtained 

from numerical analysis methods such as FE and FD (discussed in Chapter 5). Current LE design 

of geosynthetic structures is concerned mainly with global instability. The framework presented 

in Chapters 9 and 10 extends its use to local conditions thus yielding the required reinforcement 

strength along each layer considering a given layout of the reinforcement, geometry of the 

structure, backfill types, seismicity, toe resistance, cohesion, and other relevant factors. 

9 
 

 

 

 

 

3.0 LE APPROACH INCLUDING THE SAFETY MAP TOOL 

Duncan and Wright (2005) provide an overview of LE analysis of reinforced slopes and walls. 

There are numerous papers on reinforced slope stability analysis published in journals and 

conference proceedings since the late 1970’s. In fact, one may consider the design of reinforced 

slopes using LE as common nowadays. However, LE analysis that is general enough to deal with 

the specifics of MSE walls is quite limited (e.g., Leshchinsky et al., 1995; Leshchinsky, 1997; 

Leshchinsky et al., 2010b; Leshchinsky and Vahedifard, 2012). The papers by Han and 

Leshchinsky (2006) and Leshchinsky et al. (2014) deal with specifics such as required tensile 

resistance along the reinforcement, providing a design framework. Practical modification of this 

framework is presented in Chapters 9 and 10. 

Since stability of reinforced soil structures is a subset of slope stability problems, some design 

codes allow for LE-based design of such structures (see Chapter 7). FHWA and AASHTO allow 

for LE design of reinforced slopes, arbitrarily defining it as having a maximum inclination of 

70°, while requiring LE assessment of global stability of reinforced walls (i.e., inclination ≥70°) 

as a final design step. LE analysis is recognized by FHWA and AASHTO as a legitimate 

strength limit state design tool; however, its implementation in walls is lacking. 

In this chapter, the safety map methodology using LE analysis to select satisfactory layout of 

reinforcement is discussed, mainly through instructive demonstrations. It is based on safety 

factors in Allowable Stress Design (ASD) being applicable to any reinforced soil structure. The 

detailed demonstration is presented here, not just referenced, since the design framework in 

Chapters 9 and 10 is a special case of the safety map. That is, for any potential (circular) slip 

surface in the framework, the strength of the reinforcement is determined so as to render a 

constant spatial factor of safety; i.e., the likelihood of failure at any location within the reinforced 

soil mass is the same. 

Some generic slope stability software packages (e.g., ReSSA, Slide, Slope/W) offer a diagnostic 

tool that facilitates optimal design of reinforced soil structures. This tool, formally introduced by 

Baker and Leshchinsky (2001), is termed “safety map”. It is a color-coded map that, for a 

specific failure mechanism, shows the spatial distribution of the safety factors within the soil 

mass, thus indicating zones where the margin of safety is too low or where it is excessively high. 

The safety map, in the context of reinforced soil, indicates whether the assumed strength and 

length of reinforcement produces adequate stability. The specified strength of reinforcement 

along its length is illustrated in Figure 3-1. Note that at any location along the reinforcement, its 

strength is limited by either its intrinsic rupture strength or its pullout resistance, whichever value 

is smaller. Pullout resistance depends on the overburden pressure, reinforcement anchorage 

length, and reinforcement-soil interface properties. At the front side, pullout is superimposed on 

the connection strength when moving from the front into the backfill soil. In conventional LE 

analysis this connection strength has a known value using, for example, the method in AASHTO. 

The pullout resistance shown in Figure 3-1 varies linearly thus reflecting a simple problem with 

zero batter and a horizontal crest. For more complex boundary conditions, the pullout resistance 

distribution will not be linear (further details are given in Chapters 9 and 10). 

10 
QI 

() 

... >, .E ;:: 

() 

~ 111 a. 

(/) 111 

C: () 

~ 

Connectio~ 

strength l_ 

Pullout failure 

(front side) 

Reinforcement 

i/nsic strength 

Puii:ut' fallurJ: - (back side): 

I 

I 

I 

T Reinforcement length 

Front end w/ 

facing unit 

Rear end 

Figure 3-1. Available tensile resistance along reinforcement in current design 

The factor of safety, Fs, used in current LE methods is related to the soil shear strength. It 

signifies the value by which the soil strength should be reduced to attain equilibrium at a limit 

state. It means that the reciprocal value of Fs (i.e., 1/Fs) signifies the average level of 

mobilization of the soil strength. The safety factor, SF, has similar meaning to Fs except that its 

value at any location within the mass is larger than Fs unless SF is examined on the trace of the 

critical slip surface where it degenerates to its minimum value of Fs. That is, for each analyzed 

potential slip surface there is an associated safety factor, SF, and the factor of safety, Fs, 

corresponds to the lowest SF; i.e., Fs=min(SF). The most likely (i.e., the critical) slip surface is 

associated with Fs. 

It is noted that Baker and Leshchinsky (2001) introduction of the safety map is done 

mathematically rather than intuitively. That is, they showed that contour lines of SF associated 

with numerous analyzed potential slip surfaces do not intersect with each other thus making the 

concept of safety map physically valid. The practical implications of the safety map in the 

context of current LE design of reinforced soil is best demonstrated through an example 

problem. The following example problem was generated using program ReSSA. It was originally 

published by Leshchinsky (2005) with additional elaboration by Leshchinsky (2011). 

Consider the problem of stability of a multi-tiered slope/wall adjoining a rock slope as detailed in 

Figure 3-2. The design objective is to efficiently determine the required layout and strength of 

reinforcement to ensure sufficient minimum margin of safety Fs. The slope of the lower tier is 

2(v):1(h) while the top tier is at 20(v):1(h). According to AASHTO’s definition, this is a case of 

tiered walls over reinforced slope and, as such, there is no clear design methodology. Note that 

the foundation soil is comprised of an 8.2-ft thick layer of residual soil possessing drained shear 

strength of φ=15°. Also note that the bedrock defines a slender slope/wall structure while 

effectively limiting the depth of potential slip surfaces. 

11 
3.3 ft (1:3) 

$ 

16.4 ft (20:1) 

·r 

19.7 ft (10:1) 

le 1 

23.0 ft (5:1) 

L 1 

29.6 ft (2:1) 

Select Fill: 

y=127 pcf 

4>=34° 

Residual Soil: 

y=127 pcf 

4>=15° 

I q=210 psf I 

Bedrock: 

y=150 pcf 

ct>=4QO 

c=10,000 psf 

Figure 3-2. The basic problem of multi-tiered slope/wall 

For cohesionless unreinforced slopes, the critical slip surface coincides with the steepest slope 

surface. That is, if circular arcs are considered, the critical circle will have its center far from the 

slope rendering an arc that degenerates to planar surface coinciding with the slope surface. The 

corresponding Fs for the unreinforced problem then is trivial. As it is in an infinite cohesionless 

slope, its value is equal to tan(φ)/tan(β) where β is the angle of the steepest slope which is the 

upper tier in this example; i.e., the steepest slope is 20(v):1(h) meaning that 

Fs=tan(34°)/20=0.034. Figure 3-3 illustrates rotational failure surfaces as rendered by Bishop’s 

analysis. It shows the location of the critical circle. By itself, this surface is of little value when 

designing for reinforcement. However, the red zone in the safety map shows that, practically, 

most of the granular backfill needs to be reinforced since the safety factor, SF, is less than 1.3 

nearly everywhere. Note that for reinforced slopes, FHWA requires Fs≥1.3. The safety map, 

Figure 3-3, indicates visually the zones within which the SF is unsatisfactory. 

As a first iteration in the design process, the reinforcement layout shown in Figure 3-2 is 

specified. The long-term design strength of the reinforcement, LTDS, for the bottom tier is 5482 

lb/ft; for the second tier, it is 3426 lb/ft; for the third tier, it is 2056 lb/ft; and for the top tier, it is 

548 lb/ft. The connection strength is assumed to equal the design strength of the reinforcement; 

interface strength coefficient of reinforcement-backfill was taken as 0.8tan(φ) [i.e., F*α = 

0.8tan(φ)]; coverage ratio is Rc = 100%. Re-running the reinforced problem using Bishop’s 

analysis yields the safety map shown in Figure 3-4; Fs now is 1.29 (i.e., practically acceptable) 

and its corresponding critical circle is limited by the bedrock. The safety map implies the 

following: 

12 
 

 

 

 

 

	 

	 

	 

	 

	 

1.	 SF everywhere meets the required minimum of 1.3. The map shows that for a rather large 

zone, the range of safety factors is between 1.3 and 1.5 (i.e., an economical range for safety 

values). Hence, the selected strength and length of the reinforcement is adequate to resist 

rotational failure, in a seemingly economical arrangement. 

2.	 The red zone extends into the residual soil and is restricted by the bedrock. Hence, although 

the red zone in Figure 3-3 indicates an economical selection of reinforcement, it also signals 

different potential failure mechanisms that can adapt to the given geology, producing a more 

critical situation. 

Figure 3-3. Safety map for the unreinforced problem using Bishop’s analysis 

Figure 3-5 shows the safety map employing a two-part wedge translational failure mechanism 

combined with Spencer’s stability analysis. Slip surfaces along the interface with the foundation, 

as well as along each reinforcement layer, are examined. The corresponding safety map implies 

the following: 

1.	 Fs for the initially assumed reinforcement is 0.9, much lower than the acceptable value of 

1.3. As can be seen, the critical slip surface propagates along the interface with the 

foundation (top of residual soil), extending beyond all reinforcement layers and limited by 

the bedrock. 

2.	 The red zone signifies the range in which the safety factors are less than 1.3, i.e., 

unacceptable values. As seen, there are zones within each tier in which SF values are 

unacceptable as they are still less than 1.3. 

3.	 Clearly, the reinforcement for the top tier must be stronger. It also must be stronger for the 

tiers below. 

13 
 

 

 

	 

	 

	 

	 

4.	 While stronger reinforcement will improve stability against failures within the reinforced soil 

zones in all four tiers, it will not resolve the problem of failure around the reinforcement. 

Lengthening the reinforcement layers in the second and, perhaps, the third tier can solve this 

problem as reinforcement layers intersect the critical two-part slip surface. 

Figure 3-4. Safety map for the reinforced problem using Bishop’s analysis 

The lesson from using the two-part wedge translational mechanism combined with the initially 

assumed layout shows that one needs to increase both the strength and length of reinforcement. 

The depth of the red zone in the safety map suggests the extent to which the reinforcement 

should be lengthened; the existence of the red zone within the reinforced zone implies the need 

for increase in reinforcement strength. One can now lengthen and strengthen the reinforcement 

until the factor of safety is 1.3. Figure 3-6 shows the safety map employing three-part wedge 

failure mechanism combined with Spencer’s stability analysis. Translational failure mechanisms 

within the problematic zone, the foundation soil, are examined. The safety map implies that: 

1.	 Fs for the initially assumed reinforcement is 0.7, much lower than the permissible value of 

1.3. As can be seen, the critical slip surface propagates within the residual soil foundation, 

extending beyond all layers and limited by the bedrock. 

2.	 The red zone signifies the range in which the safety factors are less than 1.3, i.e., 

unacceptable values. As seen, there is one such zone extending between the rear of the 

reinforcement and the bedrock as well as within the entire foundation soil zone. 

3.	 The safety map implies that while increasing the strength of the reinforcement may narrow 

the red zone, it is not likely to eliminate it altogether. Lengthening of the reinforcement in the 

three upper tiers may help but not likely to render a safe and economical design. 

14 
 

 

 

4.	 The safety map indicates that the residual soil creates a zone which decreases stability 

significantly. A logical effective solution in this case could involve ground improvement 

such as replacement of the residual soil before construction of the tiered system starts. This 

will also increase the resistance to direct sliding failure depicted by the critical two-part 

wedge in Figure 3-5. 

Figure 3-5. Safety map using Spencer’s two-part wedge translational surfaces 

The safety map corresponding to the three-part wedge mechanism implies that replacing the 

residual soil may produce a satisfactory solution. Recalculating the same problem but with 

foundation soil that is the same as the reinforced one (i.e., φ = 34°; not shown) yields Fs = 1.26 

(as compared to 1.3) for the three-part wedge (figure not provided), a nearly acceptable value. 

The safety map shown in Figure 3-7 is a result of rerunning the problem for the two-part wedge 

translational mechanism. As seen, the problem associated with the foundation soil is resolved 

also for the two-part wedge (in fact, the safety factor along the foundation now is 1.37, see color 

code). The red zones in which safety factors less than 1.3 are within the second, third, and fourth 

tiers. These zones indicate that only a slight increase in reinforcement strength is needed; the 

length is adequate. Such an increase in reinforcement strength in the three upper tiers should 

produce a rather economical utilization of the reinforcement as the range of the safety factors 

will be mainly between 1.3 and 1.5. 

While an attempt to demonstrate the usefulness of the safety map in reinforced slope design is 

presented, one realizes that using LE analysis, the aspects of ‘internal stability’ (i.e., 

reinforcement strength, pullout, and connection) as well as ‘external stability’ (i.e., sliding along 

the reinforcement or along the foundation) are implicitly examined albeit globally. In fact, 

compound stability as well as ‘bearing capacity’ (i.e., foundation or deep-seated failure) are 

considered too while accounting for the given soil profile, and the layout and strength of the 

reinforcement. Chapters 9 and 10 provide a framework for baseline solution so that the minimum 

15 
 

 

 

required strength of reinforcement can be determined locally (as opposed to globally) within the 

mass. It should supplement the global approach which has been demonstrated in the example 

here. In fact, the framework in Chapters 9 and 10 use the safety map in ‘reverse’. It considers a 

design Fs (=SF) that is constant for any circular surface within the reinforced mass by changing 

iteratively the reinforcement strength required to render such constant Fs. That is, the long-term 

strength of the reinforcement is not given but rather is computed so that Fs = constant anywhere 

within the mass. It produces the baseline solution ensuring prescribed stability locally and, in its 

final stage, globally. 

Figure 3-6. Safety map using three-part wedge surfaces 

16 
F• Range: 

>2.47 

2.34 

2.21 

2.08 

1.95 

1.82 

1.69 

1.56 

1.43 

- 1.30 

1.17 

Residual soil replaced 

with select fill soil 

Figure 3-7. Effects of replacing the foundation soil with select fill (two-part wedge) 

17 
 

 

 

 

 

 

 

 

4.0 COMPARISON OF LE PREDICTIONS WITH PHYSICAL MODEL TESTS 

There has been notable research attempting to verify the limit state of reinforced earth structures 

using extensible reinforcement. However, reported work on limit state that deals with global 

internal or compound failures of MSE walls reinforced with inextensible inclusions is scarce. 

Physical modelling of MSE walls includes full-scale testing as well as small-scale modeling that 

obey similitude laws (e.g., centrifuge modelling or use of material with heavier unit weight such 

as done by Leshchinsky and Lambert, 1991). While small-scale models obeying similitude laws 

could be simple to construct and relatively inexpensive, large scale testing is expensive and 

usually suffers from practical limitations such as inducing failure and 3D effects that may affect 

the results in relation to limit state and 2D LE analysis. For example, 3D effects increase stability 

(i.e., decrease load in the reinforcement) possibly leading to unconservative conclusions when 

dealing with 2D problems, as commonly considered in design (Zhang et al., 2014). Inducing a 

limit state in a large scale structure is usually done by increasing a uniformly distributed 

surcharge load acting on the crest. Such surcharge increases both pullout resistance and the 

maximum load in the reinforcement, Tmax. This leads to a different evolution of failure for most 

reinforced earth structures as typically the predominant factor in failure is the self-weight of the 

reinforced backfill (i.e., gravity). Furthermore, significant soil suction (e.g., Yoo and Jung, 2007; 

Yoo, 2013; Kim and Borden, 2013) could complicate interpretation of data as it creates apparent 

cohesion; see Ling et al. (2009) to realize values of apparent cohesion in sand and its impact on 

stability of steep slopes. It is objectively difficult to conduct flawless large scale tests at a limit 

state. 

4.1 Extensible Reinforcement 

A 2(v):1(h) geogrid reinforced slope/wall, termed here as the Norwegian ‘wall’, was 

instrumented and tested by Fannin and Hermann (1990), as illustrated in Figure 4-1. Note that 

the drawing also shows a 2(v):1(h) upper slope which was termed by Fannin and Hermann 

(1990) as permanent berm surcharge. A literature survey identified a follow-up publication, 

Fannin (2001), where it is stated that the berm surcharge was indeed 2(v):1(h) but it does not 

detail the offset of this berm; however, a photo in Fannin (2001) shows a small offset, probably 

in the order of two feet. Through personal communications with Fannin, an approximately two 

feet offset was confirmed as reasonable. It was also confirmed that the upper steep slope was 

reinforced by biaxial geogrid; however, the exact layout of the reinforcement in the berm 

surcharge could not be recalled. Technically, per FHWA or AASHTO classification, the 

structure in Figure 4-1 is a two-tier ‘slope’ as its inclination is 63.4° (<70°). As a facing, a wire 

grid was used. The reported moist unit weight (γ) was 108 pcf and the plane strain residual 

strength friction (φps-residual) was 38o. Loads in the geogrid were measured using specially devised 

metallic load cells. 

The two-tier problem was analyzed using Bishop LE method. The peak strength was assumed to 

be 42o to 43o based on common differences between peak and residual strength reported in the 

literature (e.g., Lee and Seed, 1967, Bolton, 1986). Coincidentally, testing and analysis of 

exhumed materials from the Norwegian site was conducted at the University of British Columbia 

25 years after its construction. A report on the testing (Quinteros, 2014) concludes that the 

mobilized friction angle (i.e., suitable for LE analysis of the Norwegian wall with a horizontal 

18 
 

 

 

 

crest) is 41±1°. This is close to the assumed peak plane strain value of 42o to 43o for a 

surcharged reinforced slope. Figure 4-2 shows the results of a sensitivity analysis on the effects 

of φ and offset on the critical slip circles while ignoring possible reinforcement of the upper tier 

as well as excluding potential slip surfaces emerging at the face of the upper tier. The critical 

circles shown are for LE state using program ReSSA (3.0) seeking the geogrid tensile force for 

Fs=1.0. Note in Figure 4-2 that for the offsets and soil strengths used, the critical Bishop circle 

tends to intercept only primary reinforcement layers. Also, the trace of the critical slip surface 

relative to the reinforcement layers is not sensitive considering the parameters used. Moreover, 

for large offsets, the critical circle is relatively shallow in the upper tier thus is likely to be 

affected by reinforcement in the lightly-reinforced upper tier. In fact, Figure 4-2 implies a 

difficulty in representing the upper tier as ‘pure’ surcharge load as often done in practice. That is, 

the slip circle propagating through the upper tier mobilizes soil resistance, an aspect ignored if 

one replaces that tier by simple surcharge; i.e., surcharge is a conservative approximation as soil 

resistance is ignored. Conversely, the mass of the sliding portion in the upper tier could be larger 

than reflected by simple uniform surcharge implying that the driving load could be larger; i.e., 

surcharge is unconservative approximation. LE analysis avoids the dilemma often associated 

with approximation that could be conservative or unconservative. 

Figure 4-1. The Norwegian wall: geometry and geogrid layout (Fannin and Hermann, 

1990) 

19 
Offset= 1 ft. 

$=38° 

y 

X 

Offset= 1 ft. 

$=40° 

y 

X 

Offset= 1 ft. 

$=42° 

y 

X 

I 

I 

Offset=3 ft. 

$=38° 

y 

X 

Offset=3 ft. 

$=40° 

y 

X 

Offset=3 ft. 

$=42° 

y 

X 

/ 

Ii 

I 

I 

I 

L 

/ 

Offset=5 ft. 

$=38° 

y 

X 

Offset=5 ft. 

$=40° 

y 

X 

Offset=5 ft. 

$=42° 

y 

X 

Figure 4-2. The Norwegian wall: traces of critical slip circles considering the impact of soil 

strength and offset 

Figure 4-3 compares the measured and predicted max(Tmax). The left figure corresponds to 

Figure 4-2; it shows the large impact φ and offset have. However, it ignores the reinforcement 

(albeit weak) in the upper tier. The figure on the right uses an offset of two feet showing the 

effects of assumed length of reinforcement, L, of five to seven feet in the upper tier, placed at a 

vertical spacing of two feet (see inset). For all φ and L values used in this figure, the trace of the 

critical slip circle goes around the reinforcements in the upper tier (see inset) thus making it 

20 
 

 

 

 

 

 

needed to stabilize the upper berm only, not affecting max(Tmax) in the lower tier. That is, even 

weak but short reinforcement in the upper tier will ‘push’ the critical circle deeper, around it. 

For φ=43o, the calculated max(Tmax) is about 45% larger than measured if L=5 feet and about 

10% larger for L=7 feet. For φ=42o, the calculated max(Tmax) is about 70% larger than measured 

if L=5 feet and about 45% larger for L=7 feet. While the predictions by LE using Bishop’s 

analysis are conservative relative to reported measured values, the deviations are deemed 

reasonable when peak soil strength is used, especially when considering the complexity of the 

reported tiered problem. 

It can be argued that the Norwegian wall was not in a LE state. The work by Quinteros (2014) 

implies that the mobilized soil strength, 41±1°, for the horizontal crest case is close to the 

assumed plane strain peak strength. For the two-tier system, Fannin and Hermann (1990) report 

noticeable increase in reinforcement loads implying that the mobilization of soil strength was 

even ‘closer’ to peak strength. Experience indicates that when an ‘active’ state develops (i.e., soil 

strength is mobilized), external geometrical deformations are hardly visible (Leshchinsky et al. 

2009), the same as in conventional retaining walls. Hence, the Norwegian case is considered 

relevant for LE comparisons. It is assumed in back-calculations that apparent cohesion due to 

suction was negligible in the Norwegian wall. However, if there was some viable apparent 

cohesion, the LE verification could be considered unconservative. While a field large-scale test 

is insightful, there are objective uncertainties associated with it (e.g., seasonal changes in 

properties of the soil mass, within different zones of the soil mass, such as unit weight, moisture 

content/saturation level, and shear strength) even if the field tested structure is well-instrumented 

and well-executed. Due to uncertainties, interpretation of field tests requires an idealization as 

does ‘theoretical’ analysis such as LE, LA, FE, or FD. 

Zornberg et al. (1998) report the results of centrifugal tests on dry sand that was placed by 

pluviation (i.e., there was no apparent cohesion due to moisture). Geotextile meeting centrifugal 

similitude modeling requirements was used; acceleration was increased until collapse occurred 

(Figure 4-4). The tested 2(v):1(h) slope/wall had wrapped face; the re-embedded tails of the 

geotextile can be considered as secondary layers. 

21 
 

 

Figure 4-4. Failed geotextile-reinforced slope/wall in centrifuge modeling (courtesy Prof. 

Zornberg) 

Figure 4-5 shows the general layout of reinforcement in the model tested by Zornberg et al. 

(1998). Figure 4-6 (left) shows the dimensions of the prototype corresponding to the centrifugal 

22 
 

 

model at failure. Using the LE methodology in Chapter 9 (but with log spiral method), 

Figure 4-6 (right) shows the locus of predicted and measured location of Tmax-i. The exact locus 

of Tmax-i is rather insensitive in the LE calculations and hence, the agreement is considered good. 

The computed Tmax-i in the prototype was 654 lb/ft; the ultimate strength of the geotextile in the 

prototype as reported by Zornberg et al. (1998) was 662 lb/ft. Clearly, the calculated locus of 

Tmax-i and the actual values Tmax-i, both relevant to design, calculated using peak plane strain 

φ=39.5° (Figure 4-6), are in as good of an agreement as one can expect. 

Figure 4-5. General layout and dimensions of tested centrifugal models (after Zornberg et 

al., 1998) 

Figure 4-6. Prototype at failure deduced from centrifugal model test (left) and predicted 

and measured locus of Tmax-i (right) 

Mohamed et al. (2013) used centrifugal models to study the stability of two-tiered walls with 

different offsets (Figure 4-7). Geotextile reinforcement and poorly graded sand with a peak plane 

strain friction angle of 42.3° were used. For analysis, the Spencer method, combined with 

general shaped slip surface (i.e., not a predetermined slip surface geometry), was employed using 

23 
 

 

 

 

 

 

 

 

 

 

the program Slide. The analysis sought Tmax at failure as well as the trace of the critical slip 

surface. 

Figure 4-7. Configuration of centrifuge tiered wall model (after Mohamed et al., 2013) 

For their layout of reinforcement, Mohamed et al. (2013) suggested to use in LE analysis 

uniform Tmax with depth. Such distribution is commonly used in LE analysis including FHWA 

(Berg et al., 2009) when dealing with reinforced slopes. Note that the baseline solution produced 

in the LE design framework in Chapters 9 and 10 (as well as the distributions in Figure 4-6) 

Tmax-i are the spatial strength required for a limit state; its peak values might be uniform with 

depth depending on factors such as the reinforcement layout. Furthermore, the reported location 

of the critical noncircular failure surface predicted by their LE analysis agreed well with the 

actual locations of the failure surfaces observed experimentally. It is noted that the reported 

traces of critical noncircular traces of slip surfaces are not very different from circular ones. It is 

likely that circular surfaces would have rendered only a slightly less critical Tmax values. Their 

conclusions are in agreement with Leshchinsky and Han (2004) work that compared LE and FD 

(FLAC) for tiered walls. In fact, in a follow-up paper by Mohamed et al. (2014), a comparison 

with FE analysis (Plaxis) was conducted, showing good agreement with the LE analysis they 

used; the findings of this follow-up paper are discussed in the Chapter 5. 

Yang, Gupta, and Zornberg (2009) studied the stability of a geosynthetic reinforced wall within a 

narrow space. They conducted centrifugal tests and compared the results to LE analysis. Spencer 

method with noncircular slip surface (program UTEXAS4) was used. Several centrifugal tests 

were conducted with increasingly limited space. The agreement between predicted slip surfaces 

and measured one was good in all cases. Figure 4-8 is just one example. 

24 
 

 

 

Figure 4-8. LE predicted slip surface vs. observed from centrifugal test (after Yang et al., 

2009) 

An indication about the agreement Tmax or pullout, whichever is smaller for each layer, can be 

realized in Figure 4-9. As can be seen, the predicted acceleration (analogous to prototype’s wall 

height) by LE versus the measured value is close. 

Figure 4-9. LE predicted failure acceleration vs. measured value (after Yang et al., 2009) 

25 
 

 

Yang et al. (2013) compared LE and FE analyses with an RMC wall (Bathurst et al. 2006) as 

illustrated in Figure 4-10. Comparisons with measured Tmax values were made under different 

uniform surcharge loads. 

Figure 4-10. Cross-section of wrapped-face RMC test wall (after Bathurst et al., 2006) 

Circular arc combined with Bishop’s stability analysis (program STEDwin) and φ=42° were 

used. With constant Tmax for all layers, the location of the locus of Tmax was nearly linear, 

agreeing well with the LE predicted slip surface. While this reference includes many 

comparisons (most are not relevant for this work), it is interesting to show the comparison in 

Figure 4-11. When the surcharge is zero, the agreement between Tmax from LE and the measured 

max(Tmax) is good. In fact, within a range of typical surcharge loads, the agreement is still 

reasonably good. It is noted that the main significance of max(Tmax) is that it is typically used to 

select the required reinforcement long term strength. 

26 
 

12 ; 

; 

11 Rankine ; 

; 

10 ~ ;; Coulomb / 

; 

~ . • / Li~it Equilibrium ; 

9 ,. 

,. / '../ 

- ; 

E 8 ; / / 

; / ....... ,. / z 7 ,. / 

.:.:: ; / / 

- 6 ,. 

X / (0 

E 5 / X 

(0 

~E 4 

3 Range of Measured 

Values 

2 Finite Element 

1 

0 

0 20 40 60 80 100 

Surcharge Load (kPa} 

Figure 4-11. max(Tmax), Tmaxmax, at various surcharge levels (after Yang et al., 2013) 

FHWA wall at Algonquin was analyzed by Leshchinsky et al. (2014) using LE and data reported 

by Allen and Bathurst (2001), superseded by Allen and Bathurst (2003), as illustrated in 

Figure 4-12. This wall had relatively large facing blocks (2–ft thick). Studies by Huang et al., 

(2010), and Leshchinsky and Vahedifard (2012) indicate that toe resistance along its interface 

with the foundation could be substantial. No measured data regarding this resistance was 

reported for the Algonquin wall. Hence, three values of interface friction between the leveling 

pad and the foundation soil, δb-f, were assumed: 0o

, 30o, and 43o. Computed results using the LE 

framework (with log spiral analysis) are shown in Figure 4-13. Figure 4-13a shows that for the 

case of horizontal crest, the measured Tmax-i reasonably corresponded to δb-f between 30° and 43°

although with some scatter. Figure 4-13b shows Tmax-i for the backslope surcharge. Similar to the 

case of the horizontal crest, the measured Tmax-i reasonably corresponds to δb-f between 30° and 

43°. Accurate comparisons in this case history are not warranted as the available data about toe 

resistance can only be speculated. The main value of this example is in demonstrating the impact 

of toe resistance, an aspect that is ignored in design and in most LE analyses (i.e., δb-f is assumed 

as zero). While toe resistance adds to structural redundancy, it also poses a problem in 

interpreting much of the existing field data, regardless whether the wall is at working load 

conditions or near collapse. 

27 
Figure 4-12. Section of FHWA wall (reproduced after Allen and Bathurst, 2001) 

(a) (b) 

Figure 4-13. FHWA wall: (a) Flat and (b) backsloped crest (Leshchinsky et al., 2014) 

4.1.1 Concluding Remarks 

Not all reported cases in the literature that compare LE with the performance of physical models 

of MSE structures are summarized here. However, the summarized cases are based on modelling 

28 
 

 

 

 

 

that is deemed meaningful in terms of both experimental and analytical (i.e., LE) aspects. It is 

clear that for wrapped face MSE structures, whether a reinforced steep slope or wall, LE exhibits 

good agreement with tested physical models. This agreement is in terms of both predicted critical 

slip surface and strength of ductile/extensible reinforcement. It demonstrates that correct LE 

analysis for reinforced soil structures is simply a subset of slope stability analysis, commonly 

used in design of geotechnical structures. However, combined factors such as toe resistance, 

apparent cohesion, and underestimation of soil strength (using φ=34° rather than, say, 50°) may 

increase stability by 2 times or more. When any of these factors exists, the required 

reinforcement strength for limit state could be about half or less the value predicted by LE. 

Current practice does not take advantage of toe resistance, ignores apparent cohesion, and limits 

frictional strength to 40°. However, LE methodology can readily be applied in cases where these 

restrictions are relaxed. Such inclusion would provide a useful tool for forensic studies as well as 

be instructive for designers. 

4.2 Inextensible Reinforcement 

It seems that little work exploring limit state, as related to global slip surfaces, has been done on 

MSE wall reinforced with inextensible, axially stiff material. It appears that in practice 

(Anderson et al., 2012) the reinforced soil mass may be modelled as a block, using a high 

cohesion value and forcing critical failure surfaces to be outside the structure. For example, they 

suggest that the reinforced soil mass can be represented by φ=34° and c=1460 psf in 

conventional global LE analysis. 

A major difficulty in conducting limit state analysis through the reinforced mass is the possible 

incompatibility between soil and inextensible, stiff metal (‘brittle’ relative to soil) leading to 

difficulty in selecting adequate strength of the soil for use in LE analysis. This is not a trivial 

problem and good experimental work could shed light on proper strength selection. Rupture of 

reinforcement before the soil mobilizes its strength could happen leading to structural collapse. 

Since usually inextensible reinforcement is relatively strong and carries high loads, LE for low 

coverage ratio, Rc, will typically indicate that slip surfaces will mobilize mainly the pullout 

resistance of the reinforcement, an aspect that is in agreement with LE analysis. However, this 

observation must be first substantiated experimentally. It is interesting to note that there are some 

papers dealing with limit state using physical and analytical modelling of soil nail reinforcement; 

e.g., Jacobsz (2013). However, such walls are designed and built differently than MSE walls. 

Based on the review in this chapter, the design framework in this report is restricted to 

geosynthetic (extensible) reinforcement. Future research should indicate whether and its scope 

can be extended to include metallic (inextensible reinforcement). 

29 
 

 

 

 

 

 

 

 

 

 

5.0 COMPARISON OF LE PREDICTIONS WITH FE/FD RESULTS 

Numerous publications reported comparisons of predictions by LE and numerical analysis, 

mainly for unreinforced slopes. There are quite a few papers on the subject for extensible 

reinforcement. While a considerable amount of numerical comparative work was conducted on 

MSE walls with inextensible reinforcement, very little was done in a way relevant to limit state 

or global stability as implied by LE. 

Numerical analysis (FE/FD) is based on continuum mechanics. Such analysis of geotechnical 

structures produces much information, some of which is ignored in design. The proposed 

Strength Reduction Method, SRM, by Zeinkiewicz et al. (1975) made it possible to consider limit 

state in FE/FD analysis in a format with which geotechnical engineers are familiar; i.e., it can 

produce the conventional factor of safety on soil strength, Fs. In the strength reduction 

technique, a series of trial factors of safety are used to adjust the actual cohesion, c, and the 

internal angle of friction,φ, of the soils as follows: 

ctrial =c/Fstrial (5.1a) 

φtrial = arctan[tan(φ)/Fstrial] (5.1b) 

The adjusted strength parameters of soil layers are used in the analysis iteratively, satisfying 

boundary conditions and equilibrium anywhere within the continuum. The strength reduction 

values change by increments until the adjusted cohesion and friction angle render the soil 

structure unstable, being on the verge of failure. The amount of strength reduction needed to 

reach this ‘numerically’ unstable state is, by definition, the same factor of safety as in LE 

analysis; i.e., it signifies the margin of safety against collapse of a structure. Producing Fs 

reduces the FE/FD complete solution into a singular useful number in the context of limit state 

design. FE/FD can consider dissimilar materials utilizing constitutive laws of the various 

elements involved while following the principles of mechanics. It is another tool, a very 

instructive one, which could be useful, especially in complex problems and where accurate data 

is available. Today’s user friendly FE/FD software combined with affordable cost made it 

popular, especially amongst researchers. 

The geometry of the critical surface in FE/FD is not postulated as typically done in LE. It is 

difficult in FE/FD to determine the safety factor along surfaces other than the critical one which 

may be less critical than the SRM solution but still require consideration for good engineering 

practice (Cheng et al., 2007). Perhaps future implementation of the safety map may resolve this 

limitation when it comes to reinforced soil design. 

5.1 Extensible Reinforcement 

Leshchinsky and Han (2004) studied the stability of geosynthetic reinforced multitiered walls. 

They used FD (program FLAC) and compared the results with LE (program ReSSA) using 

Bishop’s analysis as well as Spencer’s for 2-part and 3-part wedges. The reinforcement strength 

was taken as uniform at each elevation unless pullout resistance at the intersection with the 

analyzed surface was smaller (i.e., conventional approach in reinforced slope stability analysis). 

The parametric study included elements such as the impact of offset, reinforcement length, 

30 
 

 

 

 

 

 

 

 

 

strength and stiffness, surcharge, and water. Wrapped facing was considered. For each case the 

reinforcement strength was selected so that Fs=1.0 in LE. Using this strength, FD analysis of the 

problem then was conducted comparing the resulting Fs and its associated failure surface. For 

most cases the resulting Fs in FD was within 2% of that obtained in LE. 

Figure 5-1 shows the critical Bishop’s circle for LE and the spatial distribution of the safety 

factor. It also shows the FD maximum shear zone (which implies the location of the slip surface). 

As seen, both Fs values and the failure zone are very close. 

(a) (b) 

Figure 5-1. (a) Safety map and Bishop critical slip circle (Fs=1.00). (b) Maximum shear 

zone using FLAC (Fs=0.98) (Leshchinsky and Han, 2004) 

Figure 5-2 shows a foundation failure. Here the largest deviation of Fs was observed, about 6%, 

and the traces of the failure zones are quite close. It is likely that a multi-polygonal surface 

combined with Spencer would have reduced Fs slightly; however, such a difference for low Fs is 

practically insignificant. Note that the figure showing Spencer’s 3-part wedge includes the trace 

of the critical slip surface and its corresponding line of thrust. This thrust line shows the location 

where interslice force resultants are acting indicating the reasonableness of the static solution. 

For the analyzed problem the line of thrust is considered reasonable. The use of LE to assess 

foundation stability instead of classical bearing capacity for rigid footing is discussed in Chapter 

10. It can be stated that adequate LE stability analysis would identify the most critical 

mechanism for a given problem without a priori assuming that it must initiate at the rear end of 

the bottom reinforced soil zone as implicitly assumed in traditional bearing capacity for MSE 

structures. In-depth discussion on traditional bearing capacity of MSE walls versus LE/LA is 

given by Leshchinsky et al., (2012). 

31 
Une of Thrust 

Fs = 0.92 

Spencer Method: 3 -P11rt Wedge Surfllce 

.__ ___________ = ==--' a 

(a) (b) 

Figure 5-2. (a) Bearing capacity using Spencer 3-part wedge (Fs=0.92). (b) Shear zone for 

bearing failure using FLAC (Fs=0.86) (Leshchinsky and Han, 2004) 

It is interesting to note that Leshchinsky and Han (2004) changed the reinforcement stiffness to 

represent that of metals. The LE predictions were still close, suggesting that the problem of 

incompatible materials in terms of stiffness may not be an issue when using LE. However, as 

discussed earlier, it could be an issue when the metallic reinforcement ruptures before the soil 

mobilizes its strength, which was not considered in their study. Potential rupture possibility may 

require a strain-based selection of strengths as noted in Section 1.1. 

Mohamed et al. (2014) compared FE (Plaxis), LE using Spencer noncircular slip surface 

(program Slide), and centrifugal models considering two-tiered walls. Figure 5-3 shows one 

typical case where the predicted failure surfaces by FE and LE were close to each other as well 

as the surface inferred from tested centrifugal model. Figure 5-4 shows the mobilized tensile 

force in the reinforcement at various elevations as well as the trace of the failure surface 

predicted by LE. The mobilized force is approximately uniform with height, agreeing with the 

common approach used in global LE. 

32 
• Reinforcement tea rs I 

(by centrifuge test) I 

I Failure surface I 

{by LE) I 

-Incremental shear strain I 

{by FE) I 

I 

Maximum tension line ~ 

(by FHWA) 

Figure 5-3. Slip surfaces predicted by FE and LE versus the location of ruptured 

geotextiles (after Mohamed et al., 2014) 

Mohamed et al., 2014, conclude that excellent agreement was obtained among the centrifuge 

models, FE, and LE in locating the failure surface. The calculated max(Tmax) values obtained 

from FE analyses agreed with the results obtained from the LE, assuming uniform distribution of 

reinforcement with height; i.e., the distribution used in conventional global LE of reinforced 

slopes. 

33 
o.u 

..., 

O.o< 

O.o< o.os o.u 0.1' 0., 

o.u 

o.os 

.... O.o< 0., 

o.u Upper tier ..., 

.... O.o< o.os I 0 .U 0 .16 0., 

Mobilized reinforcement o.u 

tensile load {by FEM) Laye r 11 

o.u 0.1' o, 

Layer9 

Failure surface 

{by LEM) 0 .16 0., 

o, 

... Layer 6 

o.u 0 .16 0., 

Lower tier Layer 4 

1 ..., o.u 0.1' o, 

o.u 

o.os 

O.o< Layer 2 

,' O.o< o.os o.u 0.1' 0., 

Tensile load o.u ,' 

{kN/m) ..., ,' .... Laye r 1 

O.o< ..., o.u 0.1' o, 

Distance from wall face (m) 

Figure 5-4. Mobilized tensile load (FE) at various accelerations and location of failure 

surface predicted by LE (after Mohamed et al., 2014) 

Yang et al. (2013) compared FE (program Plaxis) and LE (Bishop using STEDwin) considering 

the tested RMC wall (Figure 4-10; Bathurst et al. 2006). Figure 5-5 shows the summation of 

Tmax, ∑Tmax, versus the uniform surcharge applied on this wall (reported experimental results are 

for zero and 1671 psf surcharge load). Yang et al. (2013) suggested that some ‘facing’ effects 

could have played a role and, therefore, arbitrarily represented a facing as a cohesive soil layer 

having an assumed cohesion of 210 psf. See Figure 5-6 for the impact of facing (or cohesion) 

when compared to FE predictions. Yang et al. (2013) conclude that their study “... demonstrated 

that the modeling of facing stiffness in the LE analysis can improve the prediction of Tmax.” The 

RMC wall in this case was a wrapped face and therefore one may ask what facing stiffness such 

a system has. Since there was no physical facing, presentation of the experimental data as done 

by Yang et al. (2013) is not relevant in the context of this report; however, a comparison 

between FE (albeit with imaginary ‘facing’) and LE with and without ‘facing.’ Without 

speculation about the RMC wall, the FE analysis, not the experimental work, demonstrates the 

effect of 210 psf cohesion in lieu of ‘facing’. It produces similar results to LE with the same 

cohesion as facing. 

34 
90 

80 

70 

_60 

E z' 50 

.:.:: 

-

Rankine ,; ,; 

LimitEquilibnum '-...,_ ,; ,; 

without Facing ~ ,; 

LimitEquilibnum ~,; 

with Facing ,; ,; 

,; 

,; / 

,; 

,; 

,; 

,; 

,; 

,; 

,; -- ~::,...,-

20 ,; / . ~ • -· . / "' 

10 

/ . .. ~ Finite Element 

0 

0 20 40 60 

Surcharge Load (kPa) 

80 100 

Figure 5-5. Sum of Tmax versus the uniform surcharge on RMC wall (after Yang et al., 

2013) 

Han and Leshchinsky (2010) studied the stability of back-to-back walls using FD (program 

FLAC) and LE (program ReSSA with Bishop circular arc). To eliminate facing effects (as done 

in current design) as well avoid numerical difficulties associated with the FD analysis, they 

represented the facings as high cohesion soil so as to exclude from consideration slip surfaces 

emerging through the face; i.e., forcing toe failures. Figure 5-6 shows that for given allowable 

reinforcement and soil strengths, an increase in facing cohesion results in an asymptotic Fs. That 

is, the slip surface then is forced to emerge at a weak zone having zero cohesion and same 

friction as the soil; this zone was the lowest elevation of a facing unit. Note that for both 

reinforced backfills, φ=25o and φ=34o, and at zero cohesion for facing, the required strength of 

the reinforcement for Fs=1.0 is about 3 times larger for the lower φ. However, as the cohesion 

representing the facing increases while holding the reinforcement strength constant, Fs for the 

lower φ attains higher value. This is the effect of using much stronger reinforcement combined 

with forced toe failure. 

35 
1.25 - -------------------------~ 

1.2 1· ~ 1.15 q> = 25° 

Ta= 30.2 kN/m 

.... "' 

t 

~ .... 

Ill 

1.1 

Ill 

0 

.... 

.. 

{J 

6 

Ill 

1.05 

~ q> = 34° 

1 Ta= 11.2 kN/m 

0.95 

0.9 +-----~----~-----~----~------! 

0 200 400 600 800 1000 

Cohesion of wall facing (kPa) 

Figure 5-6. Effect of cohesion wall facing on Fs (after Han and Leshchinsky, 2010) 

Figure 5-7 shows the critical surfaces predicted by LE and FD. It is noted that the LE approach 

used does not consider the impact of the right wall on the left wall. Figure 5-8 shows the 

predicted max(Tmax) as a function of the distance between the two opposing facings, W (L/H used 

was 0.7). It is seen that when this distance is large, the two walls are independent of each other 

and the agreement between FD and LE is good. However, when it gets smaller, the walls affect 

each other; this was not considered in the LE used by Han and Leshchinsky (2010), thus leading 

to relatively conservative LE predictions. For realistic backfill, the differences are not 

significant. 

36 
 

 

Figure 5-7. Critical surfaces predicted by LE and FD (after Han and Leshchinsky, 2010) 

Figure 5-8. max(Tmax) versus W/H (after Han and Leshchinsky, 2010) 

Leshchinsky et al. (2004) used LE (program ReSSA with Bishop’s circular arc) and FD 

(program FLAC) to study the effect of narrow backfill space on the required strength of 

reinforcement. The constraint reinforced space was defined as shown in Figure 5-9. 

37 
Table 5-1 shows the factors of safety calculated by FD and by LE. For the same reinforcement 

strength and layout, Fs values are very close. 

Table 5-1. Summary Fs calculated by LE and FD for a given geometry and ∑Tmax 

(Leshchinsky et al., 2004) 

Parameters 

B/H m φ ∑Tmax-i Fs Fs 

[deg.] [kN/m] (FLAC) (ReSSA) 

0.2 2 20 122.5 0.99 1.00 

0.2 3 20 120.3 0.98 1.00 

0.2 5 20 116 0.98 1.00 

0.2 10 20 111.0 0.99 1.00 

0.1 10 20 92.9 0.96 1.00 

0.1 10 30 64.9 1.06 1.00 

0.1 10 40 45.3 1.09 1.00 

0 5 20 78.3 0.97 1.00 

0 5 30 55.5 0.99 1.00 

0 5 40 38.2 1.06 1.00 

0.1 10 35 54.5 1.02 1.00 

5.1.1 Concluding Remarks 

The agreement between LE and FE/FD predictions in terms of both required strength of 

extensible reinforcement and location of critical slip surface is generally good. It should be noted 

that there are numerous uncited papers dealing with FD/FE on MSE walls reinforced with 

38 
 

 

 

 

 

 

 

 

extensible reinforcement. Most of these papers include realistic FE/FD simulations of wall 

systems that also consider facing effects. In some of these papers it is argued that LE is overlyconservative; however, the ‘LE’ used is as in AASHTO. While facing is usually ignored in 

design, its effects are relevant when comparing predictions. Since many of these papers 

implicitly consider facing effects while comparing results to AASHTO where facing is ignored, 

they were excluded from the comparison with LE here as they are deemed irrelevant in the 

context of this work. Finally, the LE analysis producing good agreement in the comparisons 

assumes that, at a LE state, Tmax is the same in each of the reinforcement layers. However, FE/FD 

may produce Tmax that is not exactly constant amongst layers but with max(Tmax) close to that 

predicted in LE analysis. Note that the numerical process is terminated just before the system 

collapses, indicating numerical divergence or instability. It means that since the system is 

approaching failure, numerically exceeding max(Tmax) in FE/FD will render a domino effect 

where layers quickly become overloaded, theoretically leading to a collapse in a rapid 

progressive manner. Therefore, the value of max(Tmax) in both LE and FE/FD is not only 

practically important in selecting an adequate reinforcement, it also signifies in the analyses the 

theoretical collapse of a system thus making this value in the comparisons meaningful. It is noted 

that in the LE methodology presented in Chapters 9 and 10, the distribution of Tmax with height is 

not necessarily uniform; it depends on the reinforcement layout, pullout and backfill properties. 

5.2 Inextensible Reinforcement 

There is voluminous literature on FE/FD studies of metallic MSE walls. Studies such as Damians 

et al. (2013) are for MSE walls at working load conditions or for a particular overstressed 

element (Damians et al., 2015). It seems that studies in terms of both LE and FE exploring these 

structures at a limit state for global collapse are scarce. This is surprising as global instability of 

metallic MSE walls under atypical conditions may happen, albeit rarely. Such studies could be 

particularly instructive as FE/FD may help in selecting appropriate strengths, perhaps strainbased, of soil and reinforcement to be used in LE analysis. 

The review in this chapter leads to a similar conclusion to that in Chapter 4. That is, studies 

related to limit state of metallic MSE walls are scarce. Consequently, the design framework in 

this report is restricted to geosynthetic (extensible) reinforcement. Future research is needed to 

further evaluate metallic (inextensible) reinforcement. 

39 
 

 

 

 

 

 

 

 

 

	 

	 

	 

	 

6.0 LE AND LRFD 

The basic premise in load and resistance factors deign, LRFD, is that the loads acting on a 

structure and its resistance to collapse are independent of each other. The implication of this 

approach is that the statistical variability of these components in a structure can be identified in a 

straightforward manner. Equilibrium then can consider factored loads and resistances, assumed 

to be mutually exclusive, at a limit state leading to rational and economical design. 

The LRFD approach is well-suited for structural engineering. In the context of LRFD, 

implementation in slope stability engineering is more difficult: 

•	 There are difficulties in establishing the statistical variability of the load and resistance 

factors, especially as related to a particular site. An acceptable method by AASHTO is to 

‘calibrate’ the factors so that the LRFD calculations outcome would be similar to that of the 

equivalent allowable stress design, ASD. With evolving research and experience these 

factors could be readjusted or refined to better reflect actual variability. 

•	 Frictional material, such as soil, does not lend itself in a straightforward manner to the basic 

premise of LRFD. Consider Mohr-Coulomb (M-C) failure criterion: τ=c+σ tan(φ) where τ

and σ are the shear and normal stress at a point on a failure surface, respectively, and c and φ

are the cohesion and internal angle of friction, respectively, of the soil at the same point. In 

LE, M-C is employed over the slip surface when assessing limit state. Increase in load may 

increase σ which then will increase the shear resistance,τ, thus directly linking load and 

resistance. Conversely, decreasing the resistance means a decrease in c and/or φ. This may 

result in an increased load. 

•	 Soil weight is considered as a load in LRFD. When deep-seated failures (i.e., when the slip 

surface also provides passive resistance) are likely, applying a load factor on soil weight will 

simultaneously increase and decrease stability. That is, the ‘active’ zone will decrease 

stability while in the ‘passive’ zone the load factor will increase stability. 

•	 Pullout resistance enables the reinforcement to mobilize its strength and, therefore, is an 

important aspect of LE analysis. Pullout is directly related to the normal stress acting along 

the common interface between the embedded reinforcement and the confining soil. Hence, 

increasing the weight of the soil by a load factor increases pullout resistance. Once more, 

contrary to the basic premise of LRFD, load (soil weight) and resistance are coupled. 

Seemingly, this issue has been artfully resolved by AASHTO where the load factor for 

pullout calculations is taken as 1.0 while a resistance factor is applied on the entire pullout 

equation and not its actual resistive components. However, the weight of the same reinforced 

soil as related to calculation of the force in the reinforcement is increased by a load factor of 

1.35. 

Perhaps for these reasons, implementing LRFD in LE (or limit state) analysis is a challenge. It is 

noted that AASHTO requires LE assessment of the global stability of MSE walls. In this 

assessment, the load factor is taken as 1.0 and the resistance factor is the calculated as (1/Fs). 

AASHTO suggests considering (1/Fs), the resistance factor, as a formal design criterion. For 

example, if Fs ≥1.3, the target design then is for (1/Fs) ≤ 0.77. Essentially, this is ASD which, in 

40 
 

 

 

this case, limits the average mobilization of the soil strength along the critical slip surface to a 

maximum of 77%. The German code (EBGEO 2011) terms this value as ‘Utilization Factor’. 

The objective of this work is to develop a generic and complete LE approach for designing MSE 

walls in a limit state. Considering the difficulties of associating LRFD with LE concepts, the 

scope of this approach in the context of design is limited to ASD. That is, limit state will be 

assessed for the design M-C shear strength of the soil while the long-term strength of the 

reinforcement will be factored for safety. In the current AASHTO design specification, the 

unfactored design strength of soil in MSE walls is used; i.e., the resistance factor on soil strength 

is unity. This is analogous to using Fs=1.0 in ASD, with long-term limit state stability 

determined by factors applied to the reinforcement. It is recommended that, in lieu of LRFD, 

future development of limit state design will include stochastic stability analysis. That is, 

probabilistic LE analysis is quite established and it can be used within the framework presented 

in this report to consider the stochastic nature of data used in design. 

41 
 

 

 

 

 

 

 

 

 

7.0 USE OF LE IN DESIGN OF MSE WALLS 

7.1 General 

AASHTO and FHWA define a slope when the face inclination is less than 70°; otherwise it is a 

wall. This boundary of 70° seems arbitrary. However, lateral earth pressures in walls are 

associated with planar slip surfaces resulting in simple formulae. It can be verified that as the 

slope angle turns shallower (i.e., as the batter increases), the planar surface is progressively less 

critical than a curved one, thus yielding potentially unsafe pressures. Consequently, the 70° is 

considered as a practical acceptable limit when calculating lateral earth pressures. 

When dealing with the internal stability of reinforced structure, the current FHWA approach may 

render a paradoxical situation. That is, the reinforcement strength for a 70° degree ‘wall’ could 

be about twice that required for a 69.9° slope. This jump or discontinuity stems from using LE 

analysis with uniform Tmax amongst reinforcement layers for slopes while for walls, lateral earth 

pressure, linearly increasing with depth, is used. That is, setting the 70° face angle has resulted 

with inconsistent analyses leading to a contradictory outcome. This is unnecessarily confusing. 

To ensure smooth transition one could use the ReSlope method developed for the US Army 

Corps of Engineers (Leshchinsky, 1997). However, while this approach extends rationally wall 

results to slopes, it is recognized that the outcome could be conservative, as is the case for 

current AASHTO and FHWA design procedures for walls. Conversely, as shown in Chapters 5 

and 6, adequately modified LE analysis will reasonably agree with physical models as well as 

numerical analysis predictions dealing with strength limit state. 

Proper LE analysis can deal consistently with all modes of failure, thus rendering the current 

artificial separation into internal stability (reinforcement strength, connection strength, and 

pullout), external stability (sliding, eccentricity, and bearing capacity), and global/compound 

failure, unnecessary. The framework of such analysis is presented in Chapter 9. However, some 

of these elements and details are currently used by FHWA (Berg et al., 2009) in its design 

guidelines for reinforced slopes; thus, this report represents an evolutionary process. 

7.2 The European Standard: Eurocode 7—Geotechnical Design 

Eurocode 7 (2004), EC 7, provides a generic framework which members of the European 

Committee for Standardization or Comite Europeen de Normalisation (CEN) follow after 

establishing specific local national rules. As such, EC 7 is unspecific, especially as related to 

MSE walls; i.e., Chapter 9 in EC 7 deals with retaining walls but without explicit reference to 

MSE walls. It states that “Calculations for limit states shall establish that equilibrium can be 

achieved using the design actions or effects of actions and the design strengths or resistances, as 

specified in clause 2.4. Compatibility of deformations shall be considered in assessing design 

strengths or resistances.” Clause 2.4 (i.e., Section 2.4) provides very general guidelines for 

geotechnical calculations. When assessing limit state, EC 7 allows any method of calculation, 

such as LE, LA, FE, or FD, provided that consideration of relevant aspects of equilibrium are 

included using the correct strength values while realistic modes of failure are examined. Since 

EC 7 is short on details, it seems to be not very useful for this study except that it requires the 

assessment of strength or ultimate limit state in design where LE can be used. 

42 
 

 

 

 

 

7.3 The German Approach: EBGEO 

The German code, EBGEO (2011), was specifically developed for geosynthetic reinforcement. 

The EBGEO (2011) approach complies with the design framework established by EC 7 while 

having specific details developed in Germany. It states that in limit state, one should consider 

“...all possible failure mechanisms and slip planes intersecting reinforcement layers (previously: 

analysis of internal stability), not intersecting reinforcement layers (previously: analysis of 

external stability) and where the sliding body moves directly on reinforcement are investigated.” 

The geosynthetic resistance along a slip surface intersecting a reinforcement layer is the lesser 

value of pullout resistance or the reinforcement strength. However, connection strength is 

assumed, designed to be equal to the reinforcement strength; it is not designed based on 

structural demand as done in Chapters 9 and 10. It is interesting to note that EBGEO does not 

distinguish between walls and slope; i.e., the 70° boundary has been removed. This is possible 

since a unified and consistent LE analysis is adopted in this design code. 

EBGEO (2011) states that all possible slip planes should be considered until the critical failure 

mechanism is identified. A variety of surfaces and their locations are shown in Figure 7-1. 

Generally, the following geometries of slip surfaces are suggested: circular or log spiral surfaces 

as well as composite surfaces “with at least two failure” wedges. EBGEO (2011) clearly 

indicates that LE analyses should be conducted for limit state. 

Figure 7-1. Potential slip surfaces through and around the wall (after EBGEO, 2011) 

EBGEO (2011) uses 2-part wedge (LE) for sliding while considering the reinforcement mass to 

remain intact or coherent. It calculates the ratio of driving force to resisting force using the term 

‘Utilization Factor’ for this ratio. While still a LE analysis, it could have been done using the 

general LE approach used for other surfaces where, for consistency, the soil mobilization, 1/Fs, 

is used as a measure instead of a ratio of forces. Furthermore, it uses conventional bearing 

capacity approach to assess the ability of the foundation soil to carry the reinforced coherent 

mass. Once more, for consistency, bearing capacity could be assessed using general LE 

approach, as discussed in Leshchinsky et al. (2012) as well as Chapter 10. In general, the 

consistent LE analysis used in EBGEO, specifies a partial factor (‘resistance factor’) of 1.25 on 

the soil strength. This means that the soil strength allowed to be mobilized (utilized) is 1/1.25, or 

80%, of its full strength. Generally, the load and resistance factors are not consistent with 

AASHTO and FHWA. 

Numerous geosynthetic reinforced walls have been designed in Germany and Austria following 

the EBGEO guidelines. It is believed that these walls have performed well. 

43 
 

 

 

 

7.4 The British Standard: BS 8006 

The British Standard, BS 8006 (2010) “Code of practice for strengthened/reinforced soils and 

other fills,” is a comprehensive document. Like FHWA, it covers reinforced walls and slopes 

separately. It uses partial load and resistance factors; it is interesting to note that the factor on 

φpeak at the limit state is 1.0, similar to AASHTO but different from EBGEO. Similar to 

AASHTO, internal, external, and global stability are assessed separately. For internal stability, 

two methods are presented: the tieback approach and the coherent gravity method (which is for 

metallic walls). Similar to AASHTO, the two approaches are based on lateral earth pressure. 

Limit state has specific load and resistance factors. For external stability, Figure 7-2 shows the 

failure mechanisms to be considered. Similar to AASHTO (but with different load and resistance 

factors), the Meyerhof method is used for bearing capacity. In global stability, potential slip 

surfaces, passing partially through the structure, should be considered; like AASHTO, LE global 

stability analysis is conducted on the reinforced mass. 

In addition to a lateral earth pressure approach, and unlike AASHTO, BS 8006 suggests that 

wedge stability analysis be made (especially if the structure is not very simple) as shown in 

Figure 7-3. It is for extensible reinforcement where slip surfaces are examined to verify stability 

with respect to Tmax at a limit state. This approach is LE-based, similar to ReSlope (Leshchinsky, 

1997). The line of Tmax for inextensible reinforcement (associated with the coherent gravity 

method) and its rupture failure can be determined using the log spiral mechanism as shown in 

Figure 7-4. BS 8006 also suggests an alternative, simpler mechanism using bilinear surfaces. 

Although BS 8006 is explicit on details, it is not clear how incompatibility of inextensible 

reinforcement is accounted for in stability calculations. 

Figure 7-2. External stability modes of failure (after BS 8006, 2010) 

It is interesting to quote the following from BS 8006 (2010) (Section 7, p. 135) “Slopes with face 

angles within 20° of the vertical may be designed in accordance with the procedures in Section 6 

if desired.” Section 7 deals with the design of reinforced slopes while Section 6 with the design 

44 
 

 

of MSE walls. The words “if desired” imply that slopes having an inclination greater than 70°

may be designed using LE, as done in BS 8006 for reinforced slopes. 

Figure 7-3. Internal wedge stability (after BS 8006, 2010) 

Figure 7-4. Internal log spiral stability—coherent method (after BS 8006, 2010) 

7.5 The Japan Rail Approach 

Japan took a leadership role in developing reinforced soil walls during the implementation of its 

high speed rail network (i.e. bullet train) in the early 1970’s. Numerous laboratory and field 

testing efforts supported this development. In the 1990’s Japan Railways (JR) incorporated 

geosynthetic reinforced walls with rigid facing into their infrastructure. JR uses these walls as 

true bridge abutments and as retaining walls supporting the tracks for the critically important 

45 
 

 

 

bullet train. To date, over about 100 miles of walls, sometimes called “Tatsuoka Walls,” have 

been constructed, surviving some serious earthquakes, and are performing very well. 

The limit state design of JR geosynthetic MSE walls is based on LE, considering static and 

seismic loadings (Design Standard for Railway Earth Structures, 2013; Tatsuoka, et al., 2010). 

Usually one and two-part wedge analysis is conducted. The seismic aspect follows a modified 

M-O formulation. It considers strain softening of the compacted backfill. That is, the peak soil 

strength is used to determine the location of the critical slip surface under static conditions. 

However, under seismic conditions the soil loosens and its strength degrades to its residual value, 

acting along the surface determined statically. 

Japan Highway Department is also using the LE-based design approach, but is different than 

JR’s method. It appears that design of metallic walls follows the methodology developed by the 

wall supplier Terre Armee. It includes LE global stability analysis considering also failure within 

the reinforced soil and assuming circular arc surfaces. Details related to Japan Highway 

Department design were not identified. 

46 
 

 

 

 

 

 

 

 

 

	 

8.0 RETROSPECTIVE: LIMIT STATE ANALYSIS OF REINFORCED WALLS 

Limit state analysis of reinforced soil slopes and walls is based on the premise that: 

•	 Reinforcement tensile resistance produces stability with a prescribed margin of safety; i.e., 

stability at a limit state hinges on the reinforcement being sufficiently strong. Therefore, 

tensile force in the reinforcement is mobilized to ensure that stability. 

•	 At a limit state, the soil contributes its full shear strength along a potential slip surface. 

Depending on the reinforcement stiffness and/or its spacing, its length, its pullout resistance, and 

its tensile load, the reinforced soil might be substantially restrained from movement, thus not 

allowing for the soil shear strength along a potential slip surface to be fully mobilized. 

Considering the smaller resistance contributed by the soil, the reinforcement then must 

contribute larger resistance to render a system in static equilibrium. However, since LE assumes 

mobilization of an allowable soil strength, the calculated load in the reinforcement might be 

smaller than actually needed when the soil is restrained from movement; i.e., LE would 

underestimate the reinforcement load. This is completely equivalent to the case of active lateral 

earth pressure (signified by Ka) and at-rest pressure (signified by Ko) in the design of 

conventional retaining walls. That is, when, say, a cantilever wall moves outwards a little, an 

active wedge is formed while the retained soil mobilizes its strength, and the lateral resultant 

force acting on the wall then corresponds to Ka. However, if this wall does not move, the retained 

soil does not mobilize its full strength, and the lateral resultant force on the wall will correspond 

to Ko, producing a load that typically could be 50% to 100% larger than for the active state. 

With ‘brittle’ reinforcing material (i.e., reinforcement that reaches its yield strength at strains 

lower than those of the backfill), overstressing of the reinforcement due to lack of soil movement 

might result in rupture, its strength dropping to zero, essentially rendering an unreinforced, 

globally collapsed structure. However, with ductile reinforcement, overstressing could result in 

its local straining which allows soil movement, subsequently leading to load relaxation in the 

reinforcement as, simultaneously, the soil mobilizes and contributes more of its strength. That is, 

load redistribution may occur as an ‘active’ state of stresses develops. 

Practically, ductility of geosynthetics that rupture at strains larger than, say, about 3 to 5% would 

be sufficient to allow for a limit state in which the strength of compacted cohesionless select-soil 

is fully mobilized. As to metallic reinforcement, this possibility has yet to be established for 

different soils. While it might be the case with high-quality and very well-compacted backfill, it 

is possibly not the case with ‘less than perfect’ backfill. As shown in previous chapters, there is 

not enough data to support the use of inextensible reinforcement with LE analysis. Consequently, 

the scope of this work is limited to geosynthetic reinforced slopes and walls. 

There is an overwhelming number of publications dealing with MSE walls and slopes. The 

following retrospective conclusions can be drawn from the previous chapters: 

•	 The LE safety map is a useful design tool in finding the optimal layout of the reinforcement 

meeting a certain limit state performance criterion (i.e., ensure that a minimum Fs is met). 

However, it is based on global stability analysis, and thus not being able to reveal local 

stability aspects such as mobilization of connection strength or overstressing of some 

reinforcement layers, while understressing others. It could be a complete diagnostic tool of 

47 
 

 

 

 

 

 

 

 

 

 

 

	 

	 

	 

	 

	 

the limit state of the reinforced mass once local conditions have been assured by LE analysis 

(see Chapter 10); i.e., it would ideally be used as a final check. 

•	 Limit state comparisons between predictions by LE analysis and physical model testing with 

geosynthetic reinforcement show good agreement. Each layer in the LE analysis carried the 

same Tmax. The agreement was demonstrated for wrapped face walls. Toe resistance and 

facing units may reduce the load in the reinforcement significantly. 

•	 Limit state comparisons between predictions by LE analysis and continuum mechanics-based 

numerical models (FE/FD) show good agreement. However, each layer in the global LE 

analysis carried the same Tmax while in FE/FD most layers were at about Tmax level of load 

and some were still understressed. The limit state predictions are not overly sensitive to the 

exact distribution of Tmax amongst layers as (numerical) failure follows a domino collapse 

process; i.e., exceedance of certain Tmax at some layers may trigger a ‘numerical’ collapse. 

Numerical models imply the effect of facing/toe restraint and this can be implemented in LE 

analysis as well (see Chapter 9). 

•	 FHWA defines walls as having a face inclination ≥70°. It allows design of reinforced slopes 

solely based on LE analysis; however, it uses lateral earth pressures for walls. The two 

approaches yield incompatible results; for a wall at 70°, max(Tmax) could be about twice the 

value needed for a slope at 69.9°. Moreover, the lateral earth pressure approach is perhaps 

applicable for cases similar to the ones it was calibrated for. Calibration-based approach 

limits its application to more complex geometries and structures. 

•	 The German and Japan Rail codes use LE analysis in design of geosynthetic reinforced walls. 

The German code does not distinguish between walls and slopes. 

•	 LE approach is not consistent with the principles of LRFD. Therefore, in this work the 

traditional ASD approach is maintained. The methodology of LE stability analysis introduced 

in this work can be expanded into a stochastic analysis (i.e., probabilistic approach). That is, 

in lieu of an LRFD approach, a stochastic rather than deterministic slope stability analysis 

can be used while being consistent with the principles of soil mechanics. However, while 

stochastic approach to limit state is increasingly being used in practice, it is beyond the scope 

of the presented deterministic work. 

Consequently, to be consistent with existing knowledge and experience, Chapters 9 and 10 are 

limited to the ASD approach and to extensible reinforcement only. Chapter 9 provides a 

methodological approach to determine the mobilization of reinforcement strength considering 

local and global conditions. That is, it produces the baseline distribution of minimum required 

reinforcement strength and connection capacity so that everywhere the soil is uniformly 

mobilized. This modified LE approach is capable of considering such aspects as toe restraint, 

facings, and short intermediate reinforcement layers. Chapter 10 provides instructive examples 

of design using the modified LE approach. 

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9.0 FRAMEWORK FOR LE ANALYSIS 

9.1 Basics of Soil Reinforcing 

Slopes in this work are inclusive of MSE walls. When referring to slopes, it is implicitly assumed 

that such a structure is steep, having insufficient stability to function as intended unless 

reinforced. The objective of the design then is to determine the required layout and strength of 

the reinforcement so as to ensure everywhere within the mass a prescribed minimum margin of 

safety. 

Figure 9-1 presents a schematic of a typical reinforced slope/wall. For clarity only one layer of 

reinforcement is shown; however, the concept is applicable to multiple reinforcement layers. A 

test body, bounded by the slope surface and an assumed slip surface, is considered at a state of 

limit equilibrium (i.e., limit state). The mass would be unstable unless the reinforcement 

develops some force at the location of the slip surface. The amount of force developed renders a 

system in limit equilibrium/limit state. Commonly it is called an “active wedge”; however, here 

it could be along any slip surface. For each such test body the reinforcement restores stability to 

a limit state. That is, at each location a different force is mobilized in the reinforcement to 

produce a state of static limit equilibrium of the selected test body. Many test bodies and their 

associated slip surfaces can be analyzed to produce the distribution of required force along the 

reinforcement, as illustrated in Figure 9-1. 

It is noted that if the shear strength of the soil along a potential slip surface is not fully mobilized, 

the force in the reinforcement needed to maintain stability will be larger when compared with the 

postulated case described before. That is, to maintain static equilibrium of the free body, less 

resistance contributed by the soil requires larger force to be contributed by the reinforcement. 

This phenomenon means that, at a state commonly known as “working load conditions,” the 

reinforcement’s restraining force would be larger than at a limit state. That is, soil movement is 

restrained by the reinforcement, and therefore, it cannot fully mobilize its strength. This 

phenomenon is typical to inextensible (relatively stiff) reinforcement and, in the context of limit 

equilibrium, could be a concern if the reinforcement is not sufficiently ductile. For this reason, 

the present scope of this work is limited to geosynthetics only, where rupture strain is in excess 

of about 3%. 

The capacity of the reinforcement to mobilize its intrinsic tensile resistance, producing 

equilibrium for the test body, is enabled by its interaction with the confining soil along common 

interfaces. Such interaction is due to interface friction, adhesion bonding, and passive resistance. 

Adhesion bonding is usually not considered in long term design, and generally, is not relevant to 

MSE wall backfill. Passive resistance requires stiff protruding elements transverse to the 

direction of force mobilized in the reinforcement. Generally, passive resistance is not a 

significant element when dealing with planar geosynthetic reinforcement. Hence, the main 

contributor to interface interaction is friction. In geosynthetics this friction is usually defined as: 

δ = tan−1[CCii tan(φ)] (9.1a) 

49 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

where φ is the internal angle of friction of the embedding soil and Ci is the interaction coefficient, 

determined experimentally from standard pullout tests. It is common to specify for Ci 

conservative default values without conducting pullout tests. A typical range of values for 

reinforcing geosynthetics is 0.5 to 1.0. It is noted that pullout in AASHTO and FHWA is 

characterized by two parameters F* and α: 

CCii = FF∗ αα⁄tan φ (9.1b) 

where F* is the pullout resistance factor and α is the scale effect correction factor. Combining 

Eqns. 9.1a and 9.1b: 

δ = tan−1(FF∗ αα) (9.1c) 

AASHTO provides default design values for F* and α which can be converted to δ using Eqn. 

9.1c. However, note that these values are suitable to well-defined backfill material as specified in 

AASHTO. 

Actual pullout resistance can be calculated by dividing the length of the reinforcement into small 

segments dddd and integrating (i.e., summing) the incremental pullout resistance dPr calculated as: 

ddddrr = RRcc σσ tan δδ dddd (9.2) 

at every point dd along the length of the reinforcement where σσ is the overburden pressure acting 

over the respective dddd, and RRcc is the coverage ratio. The values of σσ and δ are a function of 

location dd and may therefore be variables. Commonly, the overburden pressure at dd is taken as 

σσ = γγ ZZ where ZZ is the vertical distance to the soil surface above point dd. External surcharge, if 

acting above dd, is added to σσ. 

The physical meaning of integration or summation of Eqn. 9.2 is illustrated in Figure 9-1. Two 

pullout envelops are sketched, each starting at an opposing end of the reinforcement. On the right 

side, the available pullout resistance linearly increases from zero along (LL − dd) where L is the 

total length of the embedded reinforcement layer. The linear increase in the depicted case is due 

to constant overburden pressure generated by a horizontal ground surface. Physically, this 

pullout resistance can be meaningful or valid up to the location where the considered slip surface 

intersects the reinforcement. On the left side, the integrated or summed pullout resistance 

initially assumes a parabolic shape followed by a linear increase. The reason for the parabolic 

rate of increase of pullout resistance on the left is due to changing overburden pressure when dd is 

under the slope. Once dd is under the horizontal ground surface, the overburden pressure is 

constant, resulting in a linear increase rate in pullout capacity. As with the pullout available from 

the right side, on the left side it is physically meaningful up to the intersection with the analyzed 

slip surface. Note that at this stage, the available front-end pullout does not consider anchorage 

or connection to the facing. Imposing the need for front end pullout so that the reinforcement can 

50 
 

 

 

 

 

 

mobilize its required resistance should produce the required connection capacity; this is 

discussed in section 9.4. 

If the required intrinsic tensile resistance needed for limit state (LE) is above the pullout 

envelopes, the reinforcement cannot mobilize sufficient resistance as pullout mode of failure will 

prevail. Physically, the intrinsic tensile resistance due to strength of the reinforcement at any 

point dd can develop only if it is below or at the pullout resistance envelops. 

Current AASHTO design considers pullout resistance capacity only at the rear end or rear side of 

the reinforcement. However, as presented herein, sufficient pullout resistance on both sides of 

point dd is needed to enable mobilization of the required tensile force at any point dd along the 

reinforcement. This approach enables the development of the framework leading to a rational 

determination of the reinforcement strength required at any point and elevation considering the 

layout of reinforcement. Furthermore, for each layer it produces the required capacity of the 

connection to the facing. 

Figure 9-1. Schematics of reinforcement layer required to stabilize steep slope 

9.2 Bishop LE Formulation 

Generally, LE problems are statically indeterminate since, for a test body, there are more 

unknowns than available equations. Hence, one cannot solve the problem without making some 

assumptions. Consequently, there are many LE methods, with each based on a unique set of 

assumptions. Methods such as Spencer, Janbu, and Morgenstern-Price, are rigorous in a sense 

that, for 2D problems, all three limit equilibrium equations are explicitly solved for a general 

shape slip surface. Other methods use specific surface geometry such as log spiral or planar 

which, for a homogeneous soil profile, satisfy all LE equations implicitly. Bishop (1955) method 

51 
 

 

 

 

 

 

 

uses circular arc surface satisfying only moment equilibrium and vertical force equilibrium for a 

given test body. The assumptions in statics made by Bishop render a numerically stable solution 

with ease, usually close to results obtained from complex rigorous methods. Unlike the log spiral 

approach, the Bishop Method can deal with realistic geotechnical problems such as an 

inhomogeneous soil profile. When the soil profile is homogeneous, however, the trace of the 

critical circle as well as the resulting factor of safety are nearly identical to that of the critical log 

spiral. Also, the circular arc can nearly degenerate to a planar surface if this defines the critical 

test body. Being simple and often used in practice, the Bishop Method is selected in this work. 

However, the computational scheme presented as a framework in this report can be replicated 

with any LE slope stability method. 

Consider a test body defined by a circular arc and the soil surface as shown in Figure 9-2. The 

mass is divided into n slices. Note that a surcharge load acting on Slice i is represented as a 

resultant force having the components Qvi (vertical) and Qhi (horizontal). Reinforcement layer i 

intersects the base of the circle within Slice i. Other reinforcement layers may or may not 

intersect Slice i and/or other slices. Figure 9-3 shows the forces acting on Slice i as a free body 

extracted from the test body in Figure 9-2. For clarity, not all possible forces on this slice are 

shown (e.g., force due to pore water pressure, pseudostatic forces). Also not shown are the 

locations of forces although these locations affect the LE equations. Bishop (1955) considered 

such forces on a slice (without reinforcement or seismic forces) in assembling the global moment 

equilibrium for the test body while also satisfying vertical force equilibrium. Based on an 

assumption related to interslice horizontal forces, his formulation yields the associated safety 

factor for each analyzed test body defined by a circle. His solution requires an iterative 

computation process. 

52 

 

Ts(x) 

Figure 9-3. Forces acting on Slice i 

Following step-by-step the derivation by Bishop (1955), it can be shown that inclusion of the 

reinforcement as a known force yields the safety factor, SF, associated with a ‘test body’ defined 

by a circular arc having radius R, as: 

′ ቂccii′ ∆xxii+൫WWii+QQvvii−uuii ∆xxii൯ቃ tttttt φφii ∑ ቊ ቋ mmααii SSFF = (9.3a) QQhiiddii ∑ቂ(WWii+QQvvii) sinααii + − TTii RRcc cos ααiiቃ RR

and 

mαi= cos αi + (sin αi tan φi’)/SF (9.3b) 

Many of the variables in Eqn. 9.3a are marked in Figure 9-7 and are self-explanatory. Refer to 

Appendix A which provides a derivation of the Bishop Equation for reinforced soil. Note that 

∆ddii is the width of slice i, uuii is the average porewater pressure acting on the base of slice i,

ccii

′ aaaadd φφii′ are the effective cohesion and friction angle available at the base of slice i, and ddii is 

the vertical distance of the horizontal force component of surface surcharge from the center of 

the analyzed circle. Coverage ratio of the reinforcement is denoted as Rc. While cohesion is 

shown in Eqn. 9.3a, it is not recommended to count on this shear strength parameter within the 

53 
 

 

 

 

 

 

 

 

 

 

 

reinforced soil in long term design. Note in Eqn. 9.3a that the reinforcement force Ti, at the base 

of slice i is considered to act horizontally. Such an assumption is common in slope stability of 

geosynthetic reinforced slopes. 

It is noted that Eqn. 9.3a is written for a circular slip surface. In fact, it represents the global 

resisting moments divided by the global driving moments for a circle having a radius R. 

However, since the numerator and denominator are multiplied by R to obtain moments, this 

constant value cancels out and the Bishop’s equation misleadingly appears as a ratio of summed 

forces treated as scalars. Consideration of force vectors and the statics in Bishop’s formulation 

does not allow for an arbitrary inclusion of Ti (or its moment) in the numerator or having an 

opposite sign to the one in the denominator in Eqn. 9.3a. Furthermore, the safety factor relates to 

the soil strength alone; it is not applied to the force in the reinforcement. The factor of safety on 

the reinforcement load is applied later after its required load for limit state has been determined. 

There is no physical reason to impose the same safety factor on soil and the reinforcement. 

The objective of the conventional LE analysis is to determine the critical test body; i.e., the 

circular arc that yields the lowest safety factor, the factor of safety Fs. This process is conducted 

through examination of many test bodies using their respective safety factors in the minimization 

process. Looking at the process differently, the factor of safety determined using LE procedures 

represents a surface along which the soil shear strength is mobilized the most: 

FFFF = min(SSFF) (9.3c) 

Physically, the value of (1/Fs) implies the average degree of mobilization of soil strength along 

the critical slip surface. That is, if Fs=1, the full shear strength of the soil is mobilized and 

failure is imminent unless the reinforcement is sufficiently strong. A value of Fs=1.5 means that, 

in an average sense, 67% of the soil strength is mobilized. However, a value of Fs=0.5 (i.e., 

Fs<1.0) means that 200% of the soil strength is mobilized, an impossibility. Consequently, in the 

context of reinforced soil, the resistance of reinforcements that is needed to maintain stability at 

an acceptable state of LE is sought. That is, the required mobilized force in the reinforcement is 

sought so to ensure that the design strength of the soil is not exceeded; i.e., Fs≥1.0. 

9.3 Overview of the Modified LE Procedure 

The following sections provide details of the modified analysis that forms the framework for LE 

design, producing the baseline requirement for reinforcement. For a given problem, including 

layout of reinforcement, the framework produces the required reinforcement resistance, TTrrrrrr (dd), 

at any location dd along each layer. This enables the designer to select reinforcement with 

adequate long term strength as well as assess the required connection strength. Stated differently, 

the framework helps the designer assess the demand, not the capacity, aspects for reinforcement. 

In this section an overview of the analytical process is provided, facilitating the understanding of 

the details shown later. 

Recall the concept of safety map introduced in Chapter 3. The safety map is a diagnostic tool 

showing the distribution of the safety factor in zones within the soil mass, providing a visual tool 

identifying under or over designed slopes or walls. If Bishop Method is used, Eqn. 9.3a is solved 

for many test bodies and potential circular slip surfaces, to determine the smallest safety factor 

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within the mass. Rather than examining only the critical surface along which the factor of safety, 

Fs, from Eqn. 9.3c, as done, the ‘state of stability’ is examined using the safety map at relevant 

zones. Such representation is an objective and efficient tool to optimize the layout of the 

reinforcement. 

The modified LE procedure that renders TTrrrrrr (dd) is, in a sense, an antithesis to the safety map. 

The objective now is to produce the same safety factor everywhere within the mass by adjusting 

systematically Ti for each test body (circle) in Eqn. 9.3a. That is, if the full strength of the soil is 

allowed, then Fs=1.0 everywhere and the required reinforcement resistance at each location, 

TTrrrrrr (dd), can be determined by solving Eqn. 9.3a using a trial and error process. Such an 

approach produces a ‘virtual’ structure where failure is likely to occur at any location within the 

mass as SF=Fs everywhere and not only along a singular surface. The term ‘virtual’ structure is 

used to indicate in reality that the reinforcement strength is dictated by a specified product, not 

varying along dd as produced by the modified procedure. Hence, the procedure determines the 

required minimum capacity of the reinforcement at any location. The following can be stated 

about the procedure: 

•	 When assessing TTrrrrrr (dd) along each layer, its value at the intersection between the slip circle 

and the respective layer cannot exceed the pullout resistance of the rear end of that layer (see 

Section 9.1). 

•	 Initially, front end pullout can be exceeded when calculating TTrrrrrr (dd). However, the front end 

pullout resistance then must be adjusted to ensure that the TTrrrrrr (dd) can indeed be mobilized. 

This adjustment produces the required minimum capacity of the connection. That is, the 

pullout resistance resulting from the integration of Eqn. 9.2 along dd must equal or 

exceed TTrrrrrr (dd). This can be achieved by introducing additional resistance at dd = 0. The 

additional resistance is the connection capacity TToo = TTrrrrrr (dd = 0) where TToo is the 

connection load or the required connection minimum strength. Once TTrrrrrr (dd) has been 

determined, the front end pullout curve has to be shifted or translated vertically until it is 

tangent to the demand curve TTrrrrrr (dd) produced by the LE analysis; the amount of vertical 

shift is TToo and, mathematically, it ensures that TTrrrrrr (dd) can indeed develop. Further 

discussion in following sections should help explain this concept. 

•	 While for long and equally spaced reinforcement each layer will carry the same maximum 

load, Tmax-i, practical layout of reinforcement may affect the distribution of Tmax-i with depth. 

The maximum load mobilized in each layer and its location is a result of the analysis, not a 

priori assumed. 

•	 The basic premise of the analysis is that without reinforcement, the mass is unstable. In such 

a case the analysis will render TTrrrrrr (dd) ≥ 0 everywhere. However, within any mass for 

practical reasons reinforcement is installed also within stable zones. In these zones the 

analysis will yield TTrrrrrr (dd) < 0. Negative required tensile resistance implies that the 

reinforcement needs to destabilize the mass to reach the prescribed Fs; it implies that 

reinforcement in such zones is not needed for stabilization. 

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9.4 Concept of Rupture and Pullout in LE Analysis 

To determine the required tensile resistance along reinforcement i, Treq_i(x), each layer is 

discretized into small segments. Each test body, defined by a circular arc passing through a 

particular segment, will require Treq_i(x) which is determined by solving Eqn. 9.3a for a target SF 

equal to, say Fs=1.0, rendering a limit state or LE where the design strength of the soil is fully 

utilized. The circle requiring the maximum value of Treq_i(x) at each segment, explained in the 

next section, renders the prevailing value of required tensile resistance at that location; i.e., 

smaller values rendered by many other circles passing through this segment are irrelevant. While 

LE equilibrium requires a certain value of Treq_i(x) to meet the target Fs, the capacity of some 

layers could be limited by its pullout resistance (see Section 9.1 for the relevance of pullout 

resistance in soil reinforcing). In such a case, reinforcement layers below and possibly above 

(depending on reinforcement layout) should compensate for the lesser load carried by that layer. 

While this compensation may work for insufficient pullout resistance at the rear end of the 

reinforcement, at the front end pullout resistance needs to be increased by adjusting the 

connection capacity To_i. Figure 9-4 illustrates the synergy between the available pullout 

resistance and Treq_i(x) along a layer (see Sections 9.1 and 9.3 for additional insight). Note that 

for simplicity both ends’ pullouts are shown as linear, a situation corresponding to surcharge-free 

vertical slope with horizontal crest. The actual pullout distribution is calculated by integrating 

Eqn. 9.2 considering the overburden pressure as well as the properties of the interface soilreinforcement. Figure 9-4 shows that the value of To_i is determined by translating the front end 

pullout resistance curve, parallel to itself, up to a point where the envelop is tangent to Treq_i(x). 

The amount of translation at x=0 is the required connection capacity (or minimum required 

strength), To_i. Such shifting of pullout resistance at the front end ensures that the intrinsic 

resistance of the reinforcement in the ‘active’ mass indeed can develop as calculated at a limit 

state. However, Treq_i(x) calculated by the many circles passing through all segments for LE state 

can be mobilized only up to the magnitude of the rear pullout capacity. 

Figure 9-4. Required force in layer i defined by front and rear pullout resistance 

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The synergy of pullout and required tensile resistances shown in Figure 9-4 implies that each 

reinforcement layer could be either too short or too long or have an ideal length. Figure 9-5 

illustrates these three possibilities. If the reinforcement is too short, there will be insufficient 

pullout resistance to enable the development of Treq_i(x) at its rear (e.g., this is likely to happen 

along secondary short reinforcement layers that facilitate construction or reduce connection 

tension load). Consequently, other layers intersecting relevant circles should make up for this 

deficit in available tensile resistance which is needed for LE state. If the reinforcement is too 

long, there is excessive pullout resistance, rendering portion of the reinforcement dormant (e.g., 

this is likely to happen with lower layers when the foundation soil is competent). If the length of 

a layer is ideal, the pullout resistance at the rear exactly enables the development of Treq_i(x). In 

such a case, the rear end pullout resistance curve is ideally tangent to Treq_i(x). Note that 

imposing pullout capacity which enables Treq_i(x) to develop creates a zone in which the 

reinforcement load capacity is slightly more than needed for a LE state. In the ideal length case, 

this excess or redundancy is at both front and rear zone areas, as indicated at the lower left of 

Figure 9-5. 

Figure 9-5. Length of layer i: Ideal (lower left), excessive (upper right), short (lower right) 

57 
 

 

 

9.5 Top-down Approach Yielding the Baseline Solution 

The numerical procedure to determine Treq_i(x) considering the rear end pullout is illustrated by 

using a set of figures. Note that the connection load or the minimum required connection 

capacity, To_i, is determined only at the end of the iterative computational procedure. 

Mathematically, To_i is an auxiliary parameter that must enable the development of Treq_i(x) in the 

front portion of the reinforcement. It is noted that the process leads to the minimum required 

reinforcement and connection strengths at a limit state and is termed baseline solution. Equipped 

with this information, the designer can assess an actual problem ensuring that it produces 

sufficient margins of safety. 

For a given geometry and reinforcement layout, select point 1 for a circle emerging below the 

top layer, denoted reinforcement 1 in Figure 9-6. Initially, the emerging point is selected above 

or at reinforcement 2. Each test body defined by a circle is considered at an LE state by solving 

Eqn. 9.3a for a target SF=Fs value. The only unknown in this equation is the required tensile 

resistance of reinforcement 1 at its intersection with the analyzed circle. Many circles passing at 

the same point (or same segment that is approximated as a point) on reinforcement 1 and 

emerging at point 1 are considered. The prevailing value at each point (segment) is the maximum 

required tensile resistance, i.e., it is the result of a numerical maximization process. Repeating 

the process for all segments on reinforcement 1, a curve of Treq_i(x) is obtained (see Figure 9-6). 

Its value ensures that any test body passing through any point along reinforcement 1 and 

emerging at point 1 has the same Fs; i.e., equal likelihood of failure. Also note that the pullout 

capacity of the top layer at this stage is excessive thus enabling the full development of Treq_i(x). 

That is, for circles emerging at point 1, the reinforcement is dormant at its rear end. 

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Next, select emerging point 2, below reinforcement 2 and above or at reinforcement 3. The 

process to establish Treq_i(x) is repeated; however, this time, it is for reinforcement 1 and 2. 

Circles intersecting two layers of reinforcement are initially assumed to mobilize equal force of 

each layer unless that required strength is limited by insufficient pullout resistance capacity. 

However, if the evenly distributed required strength (dotted line in Figure 9-6) at a point 

(segment) is less than the value required for previous emerging point 1 (value A in Figure 9-6), 

the higher value prevails as it is required for LE state for a circle emerging at point 1. 

Subsequently, selecting for the two layers an equal mobilized force value of A means excessive 

strength for reinforcement 2. Hence, while the (previous) value A is kept for reinforcement 1, a 

reduced value of B is assigned to reinforcement 2 by solving Eqn. 9.3a for a prescribed Fs while 

using T1=A. For the specific circle used to determine B (“failure surface 1”), a state of LE exists 

as reinforcement 1 carries A>B. As seen in Figure 9-6, for a certain circle (“failure surface 2”) 

the required resistance from the two layers is equal to each other, C=D. To the right of C, the 

required strength of reinforcement 1 is increased from its calculated value for the previous 

emerging point 1, dictated to be equal to that of reinforcement 2 rendering LE state for all test 

bodies. Figure 9-6 indicates that rear end pullout of the top layer is nearly exceeded (“failure 

surface 3”); in fact, at this stage, reinforcement 1 has an “ideal” length. 

The process is repeated for emerging point 3. In Figure 9-6, for failure surface 1 the required 

tensile resistance in reinforcement 3 is the same as in 2 but smaller than in 1 (i.e., A>B=C). For 

failure surface 2, all layers require equal resistance; i.e., D=E=F. While failure surface 3 also 

renders equal load G=H=I, deeper surfaces than surface 3 require load contribution by 

reinforcement 1 that is larger than its pullout resistance enables. Consequently, in zones to the 

right of surface 3, the load in reinforcement 2 and 3 is increased to ensure an LE state by 

compensating for insufficient pullout resistance of the top layer. 

The top-down process continues down to the toe elevation, generating Treq_i(x) for all layers 

considering its rear pullout capacity. However, the required connection capacity or To_i, enabling 

Treq_i(x) at the front portion of the reinforcement has not yet been determined. Refer to the 

illustrative example in Figure 9-7, originally presented by Leshchinsky et al. (2014) for a log 

spiral failure mechanism. However, for such a simple problem, Treq_i(x) from Bishop’s analysis 

will be close to the log spiral with nearly indistinguishable traces of respective surfaces. Hence, 

the slip surfaces and numbers are attributed here, as an approximation, to Bishop. In Chapter 10 

specific detailed examples for Bishop’s analysis are presented. Figure 9-7shows the results for a 

wall having H=13.1 ft. with a batter of 20o

, γ = 127 psf, and φ=30o and a given layout of 

reinforcement L=0.7H=9.2 ft., Sv= 3.3 ft., and Rc=100%. For this example, the required Treq_i(x) 

was calculated using the top-down process. Pullout resistance is considered as a function of 

simplified overburden pressure (i.e., σσ = γγ ZZ where Z is measured vertically between the 

reinforcement elevation and the soil surface – see Eqn. 9.2) and the interaction coefficient Ci 

(here Ci =0.8). For batter greater than zero, the slope angle will affect the overburden pressure 

near the connection. As a result, the front end pullout resistance is not linear, as illustrated in 

Figure 9-7 (see discussion in Section 9.1). In this problem, the overburden at the rear end of the 

reinforcement is uniform and, as implied by summation of Eqn. 9.2 over segments starting at the 

end, should result in pullout resistance that varies linearly with its distance from the end of the 

reinforcement. As can be seen, the front end pullout offers nonlinear resistance curves which 

were shifted (i.e., copied parallel to themselves) so as to be tangent to the Bishop’s calculated 

59 
 

 

 

 

 

 

 

Treq_i(x). Such shift renders To_i for each of the layers enabling the calculated mobilized force in 

the reinforcement. Practically, the point of tangency could be sensitive to numerical inaccuracy. 

If the reinforcement is equally spaced and sufficiently long, the top-down limit state approach 

will result in the same required maximum tensile resistance Tmax-i for each layer. Such 

requirement is approximately implied through interpretation of some experimental work 

(Zornberg et al., 1998). This distribution is also assumed in geotechnical practice of reinforced 

slope stability analysis. 

Figure 9-7. Example for determining the required connection capacity [Leshchinsky et al., 

2014] 

9.6 Facing Effects on Calculated Reinforcement Load at Limit State 

To examine the impact of facings, the simple model shown in Figure 9-8 was used. Given the 

dimensions and bulk unit weight of the facing units, typically termed as blocks, γu, the effective 

weight, Rv, of the effective block area, EA (see Figure 9-8), above a desired elevation can be 

calculated as Rv=γu EA. Only the weight of facing above the interface at a desired elevation (i.e., 

the shaded area in Figure 9-8) is calculated. This might be conservative, especially for small 

batters (say, <10°), however, the simplicity of it is attractive for the purpose of this work. Once 

Rv is calculated at a certain elevation, the shear resistance value at that elevation, Rh, can be 

assessed using a given value of interface friction. For block to block this friction is denoted as δbb and for block (or leveling pad) and foundation it is denoted by δb-f. That is, the horizontal limit 

state frictional resistance could be Rh=Rv tan(δb-b) or Rh=Rv tan(δb-f). Bishop’s moment 

equilibrium, Eqn. 9.3a, written about the center of the analyzed circle can now be modified to 

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include the moment due to the shear force Rh at the point of emergence of the respective circles. 

This moment would be (D Rh) where D is the vertical distance between the respective Rh and the 

center of the analyzed circle. This additional moment resistance modifies the denominator in 

Eqn. 9.3a. Note that the vertical force Rv is not included as a modifier to Eqn. 9.3a, as it is 

considered to be part of the test body; i.e., an internal force that generates Rh in a limit state. 

Figure 9-8. Model to consider impact of facing on limit state 

Parametric studies conducted by Leshchinsky et al. (2014) show (for simple wall geometry) that 

changing δb-b from 0 to 50° only slightly decrease the reinforcement tension in its front end while 

having small effect on Tmax-i, i.e., δb-b has local effect on the reinforcement, all in the vicinity of 

its connection to the facing. However, δb-f, the facing to foundation shear resistance (i.e., toe 

resistance), when combined with large δb-b, has large impact on the required tensile resistance. 

Changing its δb-f value from 0 to 50°, while keeping δb-b relatively high, decreases max(Tmax-i) by 

over 50%. This impactful toe resistance is produced by a typical 1 ft. thick block. The large 

impact of toe resistance on Tmax was discussed by Ehrlich and Becker (2010), Huang et al. 

(2010), and Leshchinsky and Vahedifard (2012). The implication stemming from LE is that 

while high shear resistance facing may prevent limit state occurring through the facing (as is the 

case with conventional gravity walls), failure through the toe is feasible. This toe failure then is 

resisted by toe resistance and reinforcement tension. In the event that toe resistance is ignored, as 

commonly done in design, not only Tmax-i is affected very little by the facing shear resistance, but 

also the required connection capacity To-i is not affected much. This is a consequence of Treq(x) 

distribution being predominantly affected by slip surface emerging through the non-resisting toe. 

This connection capacity is required to enable the development of Treq(x) within the sliding or 

active mass. 

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It is recommended to ignore the effects of narrow facing in design. This recommendation is due 

to possible uncertainty in the long term availability of toe resistance. However, the ability to 

account for facing is useful when massive facing is used (e.g., 28 inch wet cast concrete or 36 

inch gabions) or in forensic studies and in interpretation of field data. 

9.7 Commentary 

Presented is a LE-based framework that produces the baseline distribution of Treq. This 

distribution provides a rational basis for limit state design. To assess the reasonableness of the 

proposed framework, exhaustive parametric studies using log spiral surfaces were conducted and 

presented by Leshchinsky et al. (2014). In this report the basic formulation and procedure for 

Bishop’s circular arc analysis is provided, generalizing the framework to deal with realistic 

design problems as shown in the next chapter. The following commentary, concluding this 

chapter, briefly summarizes the presented approach: 

1.	 The distribution of Treq is based on ‘free bodies’ at LE state, each uniformly mobilizing the 

design strength of the reinforced soil. Subsequently, a state of LE at any zone within the 

reinforced mass is attained by assigning local reinforcement force so as to ensure a 

prescribed margin of safety. However, if that zone is stable without reinforcement, an LE 

state cannot be attained there; i.e., it is not relevant to soil reinforcing. 

2.	 The maximum load in each reinforcement layer, Tmax-i, is implied by the baseline solution of 

Treq-i. In design, a geosynthetic is selected so as to have long term strength that enables 

max(Tmax-i) considering durability, installation damage, and creep. These aspects are 

accounted for by using established relevant reduction factors (see Chapter 10). 

3.	 AASHTO requires that the long term connection capacity would be based on an arbitrary 

(and safe) equality To-i=Tmax-i. Conversely, the framework finds the connection load based on 

front end pullout resistance that enables the reinforcement to mobilize its strength thus 

rendering a LE state. Parametric studies (Leshchinsky et al., 2014) show that for a typical 

layout of reinforcement, the connection load would be less than about 0.6Tmax-i. For vertical 

walls, the connection load at lower elevations is very small since the high overburden 

pressure results in high front end pullout resistance. The same studies indicate that To-i/Tmax-i 

for large batter may increase with depth since the pullout resistance at the front end could be 

low due to lower overburden pressures under the slope. When closer spacing is used in a 

given problem, the top-down methodology yields Tmax-i which gets progressively smaller 

while front end pullout of individual reinforcement is unaffected by spacing. Such a 

situation leads, at a certain spacing, to a rapidly diminishing connection load, To-i. That is, the 

soil-reinforcement interaction next to the face is such that there is hardly any need for facing 

support at a limit state. Such interaction could be referred to as soil arching occurring in the 

horizontal direction (Leshchinsky 1997). It is interesting to note that the phenomenon of 

negligible connection load has been demonstrated in GRS-IBS where the geosynthetic 

spacing is close, typically about 8 inches. 

4.	 There are two possible consistent concepts that can be utilized in applying the presented 

framework in design. One is consistent with conventional LE analysis. The second is 

consistent with AASHTO perspective. Here is a brief of each concept: 

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–	 In conventional LE analysis, a design factor of safety, Fs, is applied on the soil strength. 

This means that soils used in the actual analysis have reduced strength: 

φm=tan-1[tan(φ)/Fs] and cm=c/Fs. That is, limit state or LE is conducted on ‘artificial’ 

soils having reduced strength of φm and cm. Consequently, for any Fs>1.0, the resulting 

Tmax values, as well as length of reinforcement, will be larger than needed for a true limit 

state where Fs=1.0. For reinforced slope design, FHWA recommends Fs≥1.3 (Berg et al., 

2009). The minimum value of Fs=1.3 compared with Fs=1.0 results in an increased value 

Tmax by a non-constant number, typically more than 1.5, depending on the soil strength. 

Hence, when assessing Tult in the process of selecting geosynthetics, it is not common to 

further factor max(Tmax) by an additional factor of safety. Such an approach is common in 

the design practice of reinforced slopes where the long term design strength, LTDS, of the 

geosynthetics is used in the stability analysis. Furthermore, increased value max(Tmax) 

also means increase in the required resistance along the reinforcement, Treq(x), leading to 

an increased connection load To. Hence, similar to max(Tmax), To resulting from LE 

analysis when Fs>1.0 increase the connection load compared with the value for Fs=1.0. 

Also note that connection load is derived from an upwards shifted front end pullout 

curves (see Figure 9-4, Figure 9-5, and Figure 9-7). Pullout resistance is affected by the 

relevant parameters F* and αα or Ci, as well as the soil strength (Eqns. 9.1). Hence, a 

decrease in design φ due to Fs>1.0 renders flatter pullout curves resulting in larger 

required upwards shifting and subsequently, larger To. 

–	 In AASHTO max(Tmax) is multiplied by a factor of safety of Fs-strength=1.5 applied on 

LTDS so as to select reinforcement with adequate Tult. Also in AASHTO, the peak soil 

strength is used to determine max(Tmax); i.e., no Fs is applied on the strength of the 

reinforced soil. Furthermore, the minimum length of reinforcement is constrained to 

L/H=0.7. Finally, in AASHTO, a factor of safety on pullout, Fs-po=1.5, is directly 

imposed on the resistance equation. In the context of the LE framework, the design shear 

strength of soil can be used as in AASHTO. It yields max(Tmax) which should be 

multiplied by 1.5 and by reduction factors to determine the required minimum Tult -- see 

Chapter 10. Also, a factor of safety of 1.5 on pullout resistance (Eqn. 9.2) should be 

applied in the LE analysis of limit state. This factor on pullout will result in an increase of 

To beyond its limit state value, as required in design. 

5.	 In this work, it is recommended to use above option as in AASHTO since it is more tangible 

than considering an artificial soil where the outcome depends on a single factor of safety 

applied to the soil strength. Although such a design should be safe, it is a significant leap 

from current design methodology for reinforced slopes. Therefore, a system initially 

designed by the top-down approach using the second option in 4 must also be checked for 

suitable global LE stability. Hence, the first option in 4 above should be used to check the 

top-down designed reinforced structure to ascertain sufficient global stability; i.e., to verify 

that the specified structure has global Fs greater than, say, 1.3.. This requirement is 

completely analogous to FHWA and AASHTO requirements for assuring global stability 

checking compound and deep-seated/foundation stability for a structure that has been 

synergistically designed based on internal and external stability. 

6.	 FHWA (Berg et al., 2009) and AASHTO require that global stability of MSE structures be 

checked using LE analysis to ensure that the factor of safety on soil strength is Fs≥1.3. In this 

check one should assess stability against compound and deep-seated/foundation failures as 

63 
	 

well as potential failures corresponding to other mechanisms. The same design requirement 

exists for global stability analysis of the structure produced by the LE top-down framework 

presented herein. This is particularly important since the LE framework should produce, 

generally, less conservative reinforced mass than AASHTO. Stability aspects such as 

‘bearing capacity’ and ‘sliding’ required in AASHTO in ‘external stability’ can be evaluated 

by using existing, commercially available tools for LE slope stability. Global stability 

assessment will be demonstrated and discussed in Chapter 10. 

7.	 Although the LE approach can consider the shear strength of facing units as well as toe 

resistance, it is recommended to ignore it in design (see Section 9.6). Perhaps there is 

justification for considering in design the impact of large facing units; however, it is beyond 

the scope of this work. 

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10.0 INSTRUCTIVE DESIGN EXAMPLES 

Presented are several design examples which are based on the Framework for LE Analysis 

discussed in the previous chapter. Since no commercial software for this LE framework is 

available yet, program ReSSA has been modified ad hoc for demonstration purposes. The 

framework presented in this report is transparent and detailed. Therefore, it can be implemented 

by software developers in existing or new codes, using Bishop or other LE methods. 

Development of commercial computer codes is beyond the scope of this report. Note that for 

completeness, global LE stability features in ReSSA were used as well. However, such global 

stability can be assessed by any suitable commercial slope stability software and the use of 

ReSSA for this analysis is solely due to convenience. 

10.1 Overview of Top-down Procedure 

Before dealing with specific example problems, here is a step-by-step recap of the top-down 

computation process used to generate the baseline results (i.e., Tmax-i and To-i): 

1. The designer initially develops a trial layout or arrangement of 

reinforcement layers; i.e., vertical spacing and length behind the face of 

the slope. 

2. Using a "top‐down" order, conduct a search and analyze failure surfaces 

emerging at the face of the slope, from the top of the slope down to, but 

not crossing, the elevation of the second layer of reinforcement, using 

search limits extending sufficiently behind the face of the slope while 

considering realistic failure modes. For each failure surface evaluated in 

the search, adjust the reinforcement force so as to obtain the target factor 

of safety. Record the resulted force at its location along the length of the 

reinforcement, where the failure surface crosses the reinforcement. This 

force indicates the tensile force that must be mobilized in a LE state. Its 

location, relative to the rear of the reinforcement, indicates the embedded 

length over which pullout resistance behind the failure surface can 

develop. The mobilized force cannot exceed rear end pullout resistance. 

The uppermost layer must have sufficient embedded length to develop 

the required pullout resistance since there are no overlying layers to 

share the responsibility of ensuring stability as prescribed by the target 

factor of safety. For each location along the reinforcement, the largest 

reinforcing force needed to obtain the target factor of safety for all failure 

surfaces crossing that point determines the required tensile force at that 

location. The completion of the stability analyses for this step produces 

the largest required force as a function of location along the length of the 

top reinforcement. 

3. Repeat Step 2 for the upper two layers of reinforcement by searching from the top 

of the slope down to, but not crossing, the elevation of the third layer of 

reinforcement. If the failure surface under consideration only crosses a single 

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layer of reinforcement, the layer must have sufficient embedded length behind the 

surface to develop the required pullout resistance. Surfaces crossing multiple 

layers of reinforcement are stabilized by the tension developed in, or shared by 

each layer to obtain the target factor of safety on soil strength. The combined 

stabilizing resistance (e.g., moment) provided by the reinforcement forces is 

distributed to each layer by initially assuming equal mobilization of force (which 

produces unequal moments in each layer due to the differences in moment arm 

about the center of the circular failure surface). If the initial computational 

assumption of equal mobilization of force results in a layer’s force that exceeds 

the available pullout resistance behind the surface, assign that layer the 

maximum available pullout resistance and increase the forces in the remaining 

layer(s) assuming equal force mobilization such that the stabilizing moment to 

obtain the target factor of safety is achieved. The force determined for each layer 

of reinforcement at the location where the failure surface under consideration 

crosses is compared to the forces determined at the same location for all surfaces 

evaluated in previous steps. Each layer present in previous steps is assigned the 

highest required force in previous steps if that force exceeds the force determined 

in the current step, assuming equal force mobilization. Forces are recalculated in 

the remaining layers crossed by the failure surface, assuming equal force 

mobilization among layers not assigned a force determined from previous steps 

such that the combined stabilizing moment provided by all of the reinforcement 

layers crossed by the failure surface under consideration produces the target 

factor of safety. For a given failure surface, this process m a y result in lower 

required forces in layers that were not required to mobilize higher forces in 

previous steps, compared to the initial assumption of equal force mobilization in 

all layers. As in Step 2, for each location along each layer of reinforcement, the 

largest reinforcing force needed to obtain the target factor of safety for all failure 

surfaces crossing that point determines the required tensile force at that location. 

4. Repeat the processes described in Steps 2 and 3 down to the toe elevation. Now 

you have the required force in each reinforcement layer, considering rear pullout, 

needed to render the same factor of safety nearly anywhere within the reinforced 

soil mass. 

5. Determine connection loads (To-i) so that there is sufficient "front end" pullout 

capacity to enable the reinforcement to mobilize the required resistance, calculated 

in Steps 2-4, to produce the prescribed factor of safety. This is done by adjusting 

the front pullout envelope so that the calculated required force in each layer is 

below this envelope. The amount of adjustment or shift from zero resistance at 

the slope is the minimum connection load at each elevation. 

6. Based on Tmax-i and To-i determine the minimum required long-term rupture 

strength of the reinforcement and the required connection capacity. 

7. Adjust the lengths of reinforcement layers and repeat Steps 2 to 6 to achieve an 

economical design. 

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In general, the top-down process seeks the reinforcement force at each location needed to 

produce the same factor of safety at any location with the reinforced mass. That is, produce a 

safety map with constant factor of safety. The baseline solution for a given problem then enables 

the designer to rationally establish the baseline ‘demand’ for reinforcement. Such a process 

leads then to selection of reinforcement and connection that has the necessary long term 

capacity. 

10.2 Example 1: Simple Wall Problem Compared with AASHTO 2002 

10.2.1 AASHTO Approach 

To compare the framework design approach with AASHTO, it would be best to select a simple 

problem. Since the LE design is based on ASD, AASHTO’s ASD is selected (AASHTO 2002). 

It is noted that the static LRFD in the current AASHTO for a simple problem will render similar 

design outcome to ASD. Hence, the comparison with AASHTO ASD, which is selected for 

clarity (i.e., avoiding irrelevant load and resistance factors and their integration in design), is also 

valid for the outcome of LRFD-based design. AASHTO design here was conducted using 

program MSEW (3.0). 

Consider a wall having 8° batter and 20 ft high; i.e., ωω=8° and H=20 ft, as illustrated in 

Figure 10-1. 

Figure 10-1. Sketch of analyzed wall 

Ten layers of equally spaced geosynthetics are placed at Sv=2 ft with the bottom layer (#1) at 1 ft 

above the toe elevation. Following AASHTO, L/H=0.7 is used, corresponding to reinforcement 

length L=14 ft long for all layers. The coverage ratio Rc=100%. For simplicity and without 

affecting the generality of the conclusions, the wall embedment depth is taken as zero. 

No seismicity is considered. AASHTO suggests using an arbitrary connection capacity of 

To-i=Tmax-i. Generally, when multiple layers exist, the strength of the connection has little effect 

on global stability, especially if it exceeds the minimum required by the LE baseline solution. 

The relevant soil properties are (1) Reinforced soil: γ=125 pcf, φ=34°, c=0; (2) Retained soil: 

γ=120 pcf, φ=30°, c=0; and (3) Foundation soil: γ=110 pcf, φ=28°, c=0. 

67 
 

 

 

For global stability, the long term design strength, LTDS, of the reinforcement is needed. The 

reduction factor for durability, RFd, and for installation damage, RFid, are each taken as 1.1. The 

reduction factor for creep, RFc is 1.66. Hence, the reduction factor on the ultimate strength, 

yielding the long term strength, RF=1.1×1.1×1.66≅2.0. The ultimate strength of the 

reinforcement is Tult=4050 lb/ft and, therefore, LTDS=4050/(1.1×1.1×1.66)=2,016 lb/ft. 

The pullout resistance factor and scale correction factor are taken as F*=0.8tan(φ) and α=0.8, 

respectively. Using Eqn. 9.1b, the interaction coefficient for reinforced soil having φ=34° is 

Ci=0.64. Such interaction coefficient corresponds to pullout mechanism having an interface 

friction angle of δ=23.3°, Eqn. 9.1c. For sliding mechanism, the friction between reinforcement 

and reinforced soil is taken as ρ=28°. 

Using AASHTO (2002) for the given structure, in external stability one gets: 

• Actual maximum eccentricity e/L=0.073 < 0.167 maximum allowed value. 

• Actual minimum factor of safety against sliding 2.09 >1.5 min. required value. 

• Bearing capacity factor of safety 4.18 > 2.0 min. required value. 

• Clearly, using L=14 ft in this case satisfies external stability. 

The values in Table 10-1 show the calculated Tmax-i. It also shows along each layer the calculated 

factor of safety on strength considering LTDS (i.e., Fs-strength=LTDS/ Tmax) and the calculated 

factor of safety against pullout, Fs-po. 

Table 10-1. Calculated loads and factors of safety for strength and pullout 

Layer i Height above Toe 

[ft] 

Tmax-i 

[lb/ft] Fs-strength Fs-po 

1 1 1343 1.50 16.1 

2 3 1202 1.68 15.9 

3 5 1060 1.90 15.3 

4 7 919 2.19 14.3 

5 9 777 2.59 13.3 

6 11 636 3.17 12.3 

7 13 495 4.08 11.3 

8 15 353 5.71 10.3 

9 17 212 9.51 9.4 

10 19 71 28.5 8.4 

Note that layer 1 (bottom layer) has Fs-strength=1.50, the minimum value required in design. At 

the top the available strength of the reinforcement is hardly utilized. However, as expected, the 

pullout resistance at the top layer is the lowest thus rendering the lowest Fs-po although the load 

is the smallest. 

68 
 

 

 

 

 

 

 

 

 

From internal stability calculations it may be concluded that geosynthetic having Tult=4,050 lb/ft 

combined with RF=2.0 and a factor of safety on Tmax of Fs-strength=1.50 is adequate based on 

AASHTO (2002) criteria. 

10.2.2 LE Framework 

To use the LE framework with Bishop’s analysis, pullout resistance at each location was reduced 

by Fs-po=1.5. This value is consistent with AASHTO ASD. Running Bishop modified to be 

consistent with the LE framework (top-down approach) resulted in the values shown in Table 

10-2. 

Table 10-2. Example 1: Values produced by the LE framework 

Layer i 

Height above 

Toe [ft] 

Tmax-i 

[lb/ft] 

xi where Tmax-i 

[ft] 

To-i 

[lb/ft] To-i/Tmax-i 

1 1 665 1.0 439 0.66 

2 3 708 2.0 315 0.44 

3 5 708 3.5 281 0.40 

4 7 708 4.8 226 0.32 

5 9 708 5.9 192 0.27 

6 11 708 6.7 137 0.19 

7 13 708 7.4 123 0.17 

8 15 708 8.0 89 0.13 

9 17 708 8.3 103 0.15 

10 19 676 7.2 363 0.54 

AASHTO and the LE framework yield max(Tmax) of 1,343 lb/ft and 708 lb/ft, respectively. This 

means that with Fs-strength=1.5, Tult=4,050 lb/ft and Tult=(1.5×2.0×708)=2,124 lb/ft for 

AASHTO and LE, respectively. That is, for the same specified factor of safety on strength, same 

factor of safety on pullout, and same soil strength, the LE approach yield required reinforcement 

strength which is about 53% of AASHTO’s value. The calculated ratio To-i/Tmax-i is significantly 

smaller than the ratio of 1.0 required in AASHTO. Comparing Table 10-1 and Table 10-2, the 

actual value of To is significantly larger in LE than in AASHTO at the top layer while it is 

significantly lower for all layers below. See discussion regarding connection in Section 10.5. It 

should be noted that for tall walls, different strengths of reinforcement could be selected along 

the height of wall, using Tmax at proper elevations from AASHTO’s calculations. This may 

result in a more economical structure relative to the singular strength value in Section 10.2.1. 

However, in such a case LE analysis must be used to ascertain that weaker reinforcement in 

upper layers will still provide a structure with sufficient margin of safety. That is, LE must be 

used to ensure that a prescribed margin of safety is available locally considering strength limit 

state. 

Figure 10-2 compares the results presented in Table 10-1 and Table 10-2. It can be seen that 

Tmax-i for LE is nearly uniform with depth while AASHTO’s values increase linearly with depth. 

The connection load, To-i, from LE is substantially lower than AASHTO’s values for all layers 

except the top one. Use of secondary reinforcement (Section 10.2.5) may further reduce the 

69 
connection load at the top layer. Traditional global LE approach or AASHTO cannot rationally 

render the connection load as is produced by the LE framework. 

Figure 10-2: Example 1: Graphical representation of Tables 10-1 & 10-2 

Figure 10-3 through Figure 10-11 help in visualizing the framework methodology as applied to 

this design example. Figure 10-3 shows the baseline Treq-i(x) rendered from the LE top-down 

approach using Bishop’s circular arc approach. It is drawn to scale which can be determined by 

using the Tmax-i values in Table 10-2. The displayed curves are based on involved numerical 

process and approximations and therefore, are not smooth as one would expect. 

70 
Toe point y 

X 

Figure 10-3. Example 1: Baseline Treq-i(x) rendered by LE top-down approach 

As can be seen, the upper 4 layers are affected by rear end pullout thus shedding loads to layers 

below. Figure 10-4 and Figure 10-5 show Treq distribution for the upper two layers restricted by 

the rear end pullout capacity of the embedded reinforcement for Fs-po=1.5. Note the increase in 

the gradient of the pullout envelop as the embedment depth increases from the top layer to the 

layer below. 

71 
 

 

Figure 10-4. Example 1: Treq for Layer 10 restricted by rear end pullout 

Figure 10-5. Example 1: Treq for Layer 9 restricted by rear end pullout 

72 
 

 

 

 

 

 

 

 

Figure 10-6 shows Treq distribution for the top layer combined with front end pullout. The darker 

bars represent the unadjusted available front end pullout resistance with Fs-po=1.5 while the 

green bars show the vertically upwards shifted or translated pullout envelope enabling the 

development of Treq obtained from Bishop. This translation is to a level where the shifted pullout 

curve is tangent to the Treq curve, enabling the reinforcement to mobilize the resistance required 

for limit state. The amount of vertical translation is the connection load To-10 (i.e., 439 lb/ft in 

Table 10-2). Examining Figure 10-4 and Figure 10-6 one realizes that the top layer is controlled 

by pullout on either end; i.e., its mobilized tensile resistance is limited by pullout on either end. 

Figure 10-6. Example 1: Treq for Layer 10 restricted by rear and front end pullout 

Figure 10-7 shows the distribution of the required or mobilized force in each layer needed to 

render a state of LE anywhere within the mass. In the segments where the color bars imply zero, 

the reinforcement is not needed considering Bishop’s analysis examining slip circles emerging 

along the face; i.e., it is dormant in these zones. Figure 10-4 and Figure 10-7 imply that the locus 

or line of Tmax-i passing through layers is not necessarily a singular slip surface as commonly 

assumed in, say, AASHTO. To better illustrate this phenomenon, refer to Figure 10-8, which 

shows the critical slip surface and its associated Fs as identified for the given problem. To 

produce Figure 10-8, the LTDS of each layer was set to equal to the respective Tmax-i value given 

in Table 10-2, which is based on the LE top-down approach with specified Fs=1.00. That is, the 

conventional global LE methodology for reinforced slopes, as described in Chapter 3 where the 

LTDS is as shown in Figure 3-1, was used. The factor of safety for this circle is 1.02, somewhat 

73 
 

 

 

 

 

 

 

 

larger than 1.00. While the critical circle approximately passes through all Tmax for layers 1 

through 9 (these maximum values are within the red bars in Figure 10-7), its trace at the top is 

slightly to the right of Tmax for layer 10. Similarly, it is to the left of Tmax for layer 1. That is, for 

getting Fs=1.00 with a singular circle, the specified LTDS for the top and bottom layers needs to 

be slightly less than the value used based on Tmax-1 and Tmax-10 from Table 10-2. This ‘excess’ in 

LTDS, as well as the fact that the calculated Tmax for layers 1 through 9 is not exactly on a 

singular circle, renders Fs=1.02>1.00. However, the values of Tmax are needed for ‘local 

stability’ as it renders Fs=1.0 for a different circle emerging at the face. In summary, in ‘virtual 

reality’ where the reinforcement LTDS is a variable in space, such as indicated in Figure 10-3 

and Figure 10-7 [i.e., Treq(x)], Fs for any circle passing through the colored zone would be near 

1.0. Hence, Treq(x) was termed baseline solution enabling the designer to rationally and 

economically select reinforcement considering the layout of reinforcement and soil strata. 

Figure 10-7. Example 1: Distribution of tensile resistance needed for LE 

74 
SCALE [ft.] 

30 

• 

20 

10 

0 

I I .____ _ __J 

Toe point Y 

X 

Fs=1.02 

Figure 10-8. Example 1: Critical circle using LTDS corresponding to Tmax-i in Table 10-2 

As stated in Section 9.7, the results of LE top-down design must also be checked for global 

stability to ensure that Fs≥1.3. In the computerized global stability analysis, the problem 

specified used LTDS based on RFd=1.1, RFid=1.1, RFc=1.66, or RF=2.0 and Tult=2,124 lb/ft. 

Subsequently, LTDS=2,124 /(2.0)≅1,065 lb/ft. In global stability, no factor of safety is applied 

on the reinforcement strength as done on the results of the top-down approach where 

Fs-strength=1.5 was used. Rather, safety is conveyed by a singular value Fs which is related to 

the soil strength – see Eqns. 9.3a-c. That is, factor of safety, Fs, that is greater than unity (say, 

equal 1.3) implies that the actual LTDS is larger than needed for stability in a limit state for the 

actual backfill and hence, there is an implicit margin of safety on the allowable strength of the 

reinforcement (see Chapter 9). The required long term connection capacity, meeting the values in 

Table 10-2, needs to be adjusted to design values (see Section 10.5). However, for stability 

analysis, each connection capacity in Table 10-2, To-i, was input to represent the lowest long term 

capacity. All data (e.g., F* and α) were identical to those used in Example 1. 

As can be seen in Figure 10-9, the actual LTDS in global stability is also dependent on pullout 

resistance in the front and rear ends of the reinforcement; i.e., dark green color signifies the 

segment along which pullout controls the available strength for stability analysis. Available 

connection resistance (dark green at the face) is considered. For the bottom layer (Layer 1), the 

LTDS of 1,065 lb/ft is predominant along the length of the reinforcement. Conversely, for Layer 

10 pullout resistance controls the available strength anywhere along the reinforcement (i.e., the 

full strength, LTDS, cannot mobilize). In global stability analysis, the location of intersection of 

the analyzed slip surface and the reinforcement determines the available resistance to contribute 

to stability. 

75 
Layer10 / 

LayerB / 

Layer 1 / 

Figure 10-9. Global stability: Actual LTDS distribution along some reinforcement layers 

Figure 10-10 displays the safety map produced by Bishop Analysis. As can be seen, the critical 

slip circle intersects the bottom reinforcement layer having Fs=1.31, greater than the minimum 

requirement of 1.30. The red zone here is ‘good’ as it shows the zone within which the factor of 

safety is in a desired range between 1.31 and 1.50. Note that the factor of safety for any internal 

and compound failures is in excess of 1.31 indicating that for rotational failure the reinforcement 

is sufficiently strong and long. In fact, the more the reinforced zone is in ‘red’ the more efficient 

is the design; the reinforcement layout and strength then is near practically optimal as Fs is 

between 1.30 and 1.50. It is interesting to note that the foundation/deep-seated slip surface 

resembles a ‘bearing capacity’ mode of failure. A factor of safety of 1.3 applied to soil strength 

(i.e., LE) will typically render factor of safety in the traditional realm of conventional bearing 

capacity greater than 2.0. Note that the factor of safety in traditional bearing capacity reflects the 

value of ultimate bearing load divided by the actual load (i.e., it does not consider factor of safety 

on the soil strength). In fact, for the given problem, AASHTO produces a Meyerhof bearing 

76 
 

 

 

 

 

 

capacity factor of safety (Section 10.2.1) of 4.18 versus the LE value of 1.31. For different 

problems both factors of safety may change. LE global stability addresses indirectly the bearing 

capacity aspect typical to AASHTO design. However, while AASHTO consider the reinforced 

mass as a coherent block, LE considers ‘bearing capacity’ for potential failures going through the 

reinforcement, the retained soil, and stratified foundation, possibly identifying more critical 

conditions than the traditional bearing capacity approach used for MSE structures. In other 

words, LE considers ‘bearing capacity’ as the stability of an earthen embankment, albeit 

reinforced, resting over foundation soil whereas AASHTO idealize the reinforced soil mass as a 

rigid shallow eccentric footing supported by a foundation soil. The versatility of LE analysis of 

foundation failure by far exceeds that of traditional bearing capacity when dealing with design of 

MSE structures. Leshchinsky et al. (2012) provides an in-depth review of bearing capacity 

predictions produced by AASHTO versus values obtained from LE or LA stability analysis. 

Figure 10-10. Safety map generated for the design example using Bishop analysis 

Finally, refer to Figure 10-11. Here sliding along the foundation as well as along reinforcement 

layers was assessed using the Spencer Method considering a 2-part wedge mechanism. The 

presented safety map corresponds to surfaces that produce statically valid results. That is, 

wedges yielding unacceptable results from statical standpoint (e.g., interslice thrust line outside 

the sliding body) were excluded. Sliding along the full length of the reinforced mass has very 

large factor of safety (i.e., in excess of 1.8). Two-part wedges within the reinforced soil zone 

with the front wedge sliding along a base-interface having a weak friction angle of ρρ of 28◦ were 

77 
 

considered. Such surfaces yielded minimum Fs=1.36. The safety map indicates that the 

reinforcement layout and strength exceed the minimum requirements considering translational 

global stability. It is noted, however, that for stratified foundation soil, more complex failure 

mechanisms should be considered. 

Figure 10-11. Safety map generated using Spencer analysis considering 2-part wedge 

10.2.3 Closely Spaced Reinforcement 

To realize the impact of spacing on reinforcement and connection loads, the original spacing of 

reinforcement is reduced from 2 ft to 1 ft while preserving the original length of 14 ft, which is 

needed to resist global deep-seated failure. This increases the number of layers from 10 to 19. 

Table 10-3 shows the results. 

78 
 

 

 

Table 10-3. Impact of decreased spacing of reinforcement layers 

Layer i 

Height above 

Toe [ft] 

Tmax-i 

[lb/ft] 

xi where 

Tmax-i 

[ft] 

To-i 

[lb/ft] To-i/Tmax-i 

1 1 355 0.8 206 0.58 

2 2 355 1.5 199 0.56 

3 3 355 2.2 164 0.46 

4 4 355 2.9 164 0.46 

5 5 355 3.6 164 0.46 

6 6 355 4.1 130 0.37 

7 7 355 4.6 130 0.37 

8 8 355 5.0 116 0.33 

9 9 355 5.5 89 0.25 

10 10 355 5.9 89 0.25 

11 11 355 6.3 41 0.12 

12 12 355 6.6 48 0.14 

13 13 355 6.9 48 0.14 

14 14 355 7.0 Negligible Negligible 

15 15 355 7.3 Negligible Negligible 

16 16 355 7.4 Negligible Negligible 

17 17 355 7.6 Negligible Negligible 

18 18 355 7.6 Negligible Negligible 

19 19 355 7.7 41 0.12 

When comparing Table 10-2 and Table 10-3, increasing the number of layers from 10 to 19 

results in approximately halving of max(Tmax); i.e., it dropped from 708 lb/ft to 355 lb/ft. This 

corresponds to Tult = (1.5×2.0×355) = 1,065 lb/ft. whereas before it was 2,124 lb/ft. While this 

reduction is not surprising, the connection load dropped dramatically. For many connections the 

required capacity is small or even negligible. This is a consequence of the connection load being 

affected mainly by slip circles that are close to the face; i.e., shallow surfaces (see the color 

coded map Figure 10-13 as well the actual distribution of Treq(x) in Figure 10-14). Such surfaces 

do not require much tensile resistance by the reinforcement for stabilization. The fact that the 

upper layer is slightly mobilized is enough to render most surfaces considered in the top-down 

approach, emerging at lower elevations, stable having Fs>1.0 (i.e., no reinforcement is needed at 

some locations as the design criterion of Fs is exceeded). Stated differently, as the reinforcement 

gets closer, the amplitude of the calculated Treq(x) is smaller while the front end pullout envelope 

is unaffected. Consequently, the amount of shifting of the front end pullout envelope needed to 

enable Treq(x) gets smaller resulting in diminishing value of connection load To-i. Comparing the 

two tables shows that for 2 ft spacing the required connection capacity of the top layer is about 

360 lb/ft whereas with 1 ft spacing it is about 40 lb/ft; i.e., about an order of magnitude lower 

connection load in a zone where, for some systems, it would be difficult to get high connection. 

Note that lower layers have modest connection load; this is a result of compound slip surface that 

requires resistance by layers intersected near the connection. This non-intuitive drop in 

connection load will diminish if shorter reinforcement is used. Conversely, lengthening the 

79 
 

 

reinforcement will decrease the connection load at the bottom as more layers will resist 

compound failure. 

Figure 10-12 compares the results presented in Table 10-2 and Table 10-3. Tmax-i appears to be 

proportional to the spacing. When considering upper layer, To-i decreases dramatically as the 

spacing gets closer. 

Figure 10-12. Graphical representation of Tables 10-2 & 10-3 

Finally, for reinforcement having LTDS = Tult/RF = 1,065/2 = 533 lb/ft (for the 2-ft spaced 

reinforcement LTDS = 1,065 lb/ft was used), global stability indicates Fs=1.31 as illustrated in 

Figure 10-15, corresponding to foundation failure, the same as in the previous problem (Figure 

10-10). The critical slip circle passes through several layers of reinforcement. Internal and 

compound circular arc failure surfaces have Fs well in excess of 1.4 thus producing satisfactory 

design. In fact, the red zone is over a large area of the reinforced soil indicating an efficient 

layout and strength of reinforcement. Note that the problem has two minima for the safety 

factors. The absolute minimum is the circle for which Fs=1.31; however, within the reinforced 

zone there is a local minimum (of about 1.4) as implied by the safety map and the gap in the 

internal red zone. 

80 
Figure 10-13. Close reinforcement: Color coded map of distribution of tensile resistance 

Figure 10-14. Close reinforcement: Computed Treq along each layer 

81 
Results of Bishop Analysis 

RESULTS DISPLAY PLOT / PRINT CAPTURE SAFETY MAP 

Color Code: Safety Factors 

SCALE [ft.] 

< E3 30 

4 

E 6.00 

E 20 

7 5.00 

E 4.00 

~ 3.00 10 

10 2.50 11 

12 2.00 

13 

-

1.50 0 

H 1.30 

15 

16 

l7 Re~un ... 

18 

19 

Hide Scale 

Minimum factor of safety = 1. 31 X = N/A 

Radius = 26.49 ft. Y = N/A 

. 

Right-dick to change Type 

or Length of reinf. or soil 

parameters. OK 

Figure 10-15. Safety map generated for the closely spaced reinforcement 

10.2.4 Seismicity and Closely Spaced Reinforcement 

This section considers the effect of seismic loading of MSE walls with the closely spaced 

reinforcement. The example problem considers a peak ground acceleration of PGA=0.3g. 

Following common practice in slope stability analyses, the design value of the horizontal seismic 

coefficient in a pseudostatic analysis is taken as Kh = (PGA/2)/g = 0.15. In fact, FHWA (Berg et 

al., 2009) recommends the same fraction of PGA for a pseudostatic analysis of reinforced slopes. 

Using large shaking table tests, Leshchinsky et al. (2009) demonstrated that for geosynthetic 

reinforced slopes 0.33PGA to 0.4PGA can be used since seismic loads are not true static (i.e., not 

sustained load over time). Bishop’s formulation, Eqn. 9.3a, can be modified to also include a 

‘static’ force component, Kh×W, for each slice. Applying Kh of 0.15 results in Table 10-4. 

Comparing Table 10-4 and Table 10-3, one sees a substantial increase in the required max(Tmax) 

from 355 lb/ft to 590 lb/ft, an increase by a factor of 1.66. AASHTO does not consider creep as a 

strength time-degradation phenomenon. Consequently, during a seismic event, creep strain 

cannot practically evolve and the creep reduction factor is not applied on Tult (RFcr in Example 1 

is 1.66). Therefore, Tult=1,065 lb/ft (established under static conditions) should be reduced only 

by RFd and RFid (i.e., 1.1×1.1=1.21) to yield the design strength under seismic condition; i.e., 

1,065/1.21=880 lb/ft will be adequate for stability analysis under seismic excitation. The 

required connection capacity (ignoring possible inertia of facings) is marginally affected as slip 

surfaces under seismic conditions tend to be deeper and away from the face. 

82 
 

Table 10-4. Wall with closely spaced reinforcement subjected to PGA=0.3g 

Layer i 

Height above 

Toe [ft] 

Tmax-i 

[lb/ft] 

xi where 

Tmax-i 

[ft] 

To-i 

[lb/ft] To-i/Tmax-i 

1 1 590 1.3 226 0.38 

2 2 590 2.8 199 0.34 

3 3 590 4.3 219 0.37 

4 4 590 5.6 247 0.42 

5 5 590 7.1 247 0.42 

6 6 590 8.4 212 0.36 

7 7 590 9.9 178 0.30 

8 8 590 11.2 192 0.33 

9 9 590 12.7 185 0.31 

10 10 590 14.0 164 0.28 

11 11 580 14.2 Negligible Negligible 

12 12 564 14.6 Negligible Negligible 

13 13 537 15.0 Negligible Negligible 

14 14 494 14.3 Negligible Negligible 

15 15 473 15.0 Negligible Negligible 

16 16 440 14.9 Negligible Negligible 

17 17 408 14.4 Negligible Negligible 

18 18 398 8.7 Negligible Negligible 

19 19 398 8.6 55 0.14 

Figure 10-16 compares the results shown in Table 10-3 and Table 10-4. Tmax-i is not affected very 

much. However, considering the given length of reinforcement, the rear end pullout prevails 

with increased seismicity resulting in reinforcement layers that cannot mobilize their intrinsic 

strength. Subsequently, lower layers compensate for the upper layers by carrying higher load. 

The connection load in the upper layers is not affected much by seismicity as deeper slip surfaces 

develop affecting very little the facing. Note that inertia of the facing is ignored. 

83 
20 

18 ~ Tmax 

PGA=O .... To 

16 ...,._Tmax 

14 ._To 

.....,. 

~ 12 

<I) 

0 

10 

I-

<I) 

> 0 

8 

.J:J 

"' 6 

.E 

0) PGA=0.3g ·.; 4 

I 

2 

0 

0 100 200 3)() 400 500 600 700 

Tmax-i, To-i [lb/ft] 

Figure 10-16. Graphical representation of Tables 10-3 & 10-4 

Figure 10-17 shows the computed distribution Treq along layers. Compared with the static 

distribution in Figure 10-14 one realizes the shift of Tmax towards the rear of the reinforced zone 

with magnitude restricted by rear end pullout. The color coded map in Figure 10-18, especially 

when compared with the static case in Figure 10-13, provides another perspective of the impact 

of seismicity. The peak zone signifies compound stability effects. 

84 
 

 

Figure 10-17. Computed Treq along each layer for PGA=0.30g 

Figure 10-18. Color coded map of distribution of tensile resistance for PGA=0.30g 

Under seismic conditions, the allowable design strength of the reinforcement (ignoring creep) is 

880 lb/ft. Running global stability, one gets the safety maps shown in Figure 10-19 and Figure 

10-20. Under pseudostatic limit state conditions, it is common to allow Fs of about 1.1 (Berg et 

85 
 

 

al., 2009). The safety map shows a potential problem with respect to foundation/deep-seated 

failure where Fs=1.04 (Figure 10-19). This issue can be resolved by lengthening a few 

reinforcement layers at the bottom. In the lower zone of the reinforced mass, low (but 

acceptable) safety factors are between 1.1 and 1.3, signifying a compound mode of potential 

failure. This compound type of potential failure is also implied by Figure 10-15 and Figure 

10-17. In fact, seismic excitation larger than 0.3g will require lengthening the reinforcement 

beyond the L/H value of 0.7 used in this problem. Indeed, high seismicity often results in 

reinforcement that requires L/H of 1.0 or even more. 

86 
10.2.5 Secondary Reinforcement 

In Section 10.2.3 the effect of decreasing the vertical distance between reinforcement layers was 

examined; i.e., Sv was changed from 2 ft in 10.2.2 to 1 ft in 10.2.3 while keeping all other data 

the same including uniform length of reinforcement of 14 ft. The decrease in spacing resulted in 

a proportional decrease in the required max(Tmax) simultaneous with significant decrease in 

connection load. Practically, while closer reinforcement increase the amount of reinforcement 

used (in the current example almost doubled), it tends to result in better construction outcome as 

compaction is facilitated, especially near the facing. It also facilitates the use of systems counting 

on frictional connection. Furthermore, closely spaced reinforcement improves the seismic 

performance of MSE structures (Ling et al. 2005). These results suggest that there may be 

benefit to using secondary (or intermediate) reinforcement. Such short layers may shed some 

load off the primary reinforcement layer making the front end of the structure more robust. 

To assess the impact of secondary reinforcement, consider 5 ft long layers, located midway 

between the primary layers in Section 10.2.2. The layout then is similar to that shown in Section 

10.2.3 with alternating layers starting from 2 ft above the toe being 5 ft rather than 14 ft long. 

87 
Table 10-5. Impact of secondary reinforcement 

xi where 

Height above Tmax-i Tmax-i To-i 

Layer i Toe [ft] [lb/ft] [ft] [lb/ft] To-i/Tmax-i 

1 1 483 1.3 281 0.59 

2 2 483 2.4 260 0.54 

3 3 483 3.4 212 0.44 

4 4 483 4.3 212 0.44 

5 5 483 5.1 178 0.37 

6 6 483 5.0 164 0.34 

7 7 483 6.5 130 0.27 

8 8 419 5.6 137 0.33 

9 9 483 7.6 110 0.23 

10 10 366 5.8 103 0.28 

11 11 483 8.6 48 0.10 

12 12 323 6.1 89 0.28 

13 13 483 9.3 34 0.07 

14 14 291 6.3 14 0.05 

15 15 483 9.7 Negligible Negligible 

16 16 269 6.3 Negligible Negligible 

17 17 483 10.1 Negligible Negligible 

18 18 248 5.7 Negligible Negligible 

19 19 462 10.3 41 0.09 

 

Figure 10-21 compares the results presented in Table 10-3 and Table 10-5. For the same 

spacing, secondary (short) reinforcement may have the same impact on connection load as long 

reinforcement. The oscillating Tmax-i for upper secondary layers is a result of these layers being 

too short, out of the locus of Tmax-i for the primary layers. Figure 10-22 illustrates this point. 

88 
.,..... 

Table 10-5 shows the results for Tmax and To. Compared with Table 10-3 (i.e., results for closely 

spaced reinforcement), the max(Tmax) for the secondary reinforcement case is 483/355 = 1.36 

higher. That is, for secondary reinforcement the required strength would be 36% larger than for 

the close full length reinforcement. However, compared with Table 10-2, which is for 2 ft 

spacing, the required strength would be 483/708 = 0.68; i.e., the required strength drops by 32% 

because of using the short secondary reinforcement. As to the connection load of the top layer, 

for both closely spaced and secondary reinforcement the resulting value is the same: To-19 = 41 

lb/ft. The bottom layer, however, would have connection load, To-1, of 206 and 281 lb/ft for the 

closely spaced and secondary reinforcement layout, respectively. Hence, at the bottom, the 

effectiveness of secondary reinforcement at the connection is not significant as at the top. 

However, the bottom layer can carry substantial load as can be seen in Figure 10-22. It is noted 

that for the original 2 ft spacing, Table 10-2 indicates that To is 439 and 368 lb/ft at the bottom 

and top, respectively. Hence, the top-down LE analysis indicates that the impact of close 

reinforcement/secondary reinforcement on the connection load in upper layers is significant. 

89 
Toe point y 

X 

Figure 10-22. Computed Treq(x) considering 5 ft long secondary reinforcement 

Notice in Figure 10-22 that by virtue of being short, the mobilized resistance of the secondary 

reinforcement is limited by its rear end pullout capacity. Clearly, bottom secondary layers loads 

are as high as those for the primary reinforcement. 

Figure 10-23 shows the color coded map of the distribution of required tensile resistance along 

each layer. Comparing with Figure 10-13, one can see the impact of the secondary reinforcement 

near its rear end while for full length close spacing the force mobilization is quite uniform with 

height. 

90 
I 

I 

I 

Tensile Resistance 

Ra Color Code: nge of T -required 

Values in [ lb / ft I 

0.00 

40.00 >• 

90.00 >• 

140.00 > 

190.00 > 

240.00 >• 

290.00 > 

340.00 > 

390.00 > 

440.00 > 

490.00 >• 

Figure 10-23. Color coded map of Treq(x) considering 5 ft long secondary reinforcement 

Based on the LE computed max(Tmax) = 483 lb/ft, and using an overall reduction factor of 

RF=2.0 and Fs-strength=1.5, the required ultimate strength would be Tult = 1,500 lb/ft. Hence, 

for global stability of a system having LTDS =Tult/RF = 1,500/2 = 725 lb/ft the safety map in 

Figure 10-24 is displayed. Note that the critical slip surface extends from the reinforced soil into 

the retained soil (i.e., compound) rendering Fs=1.21. In fact, the entire red zone has Fs<1.30. 

Compared with Figure 10-15, one sees that the full length, closely spaced reinforcement, albeit 

weaker, rendered Fs>1.30 everywhere. Clearly, while secondary reinforcement may have an 

important impact on connection loads, its contribution to global stability could be limited. The 

safety map indicates that lengthening the lower secondary reinforcements will intercept the 

critical slip surface, increasing the minimum safety factor for this problem. One may increase 

LTDS of the lowest five primary reinforcement layers and thus increase the minimum factor of 

safety in global stability to be greater than 1.30. Alternatively, a more laborious way is to run 

the top-down approach using specified Fs that is slightly larger than 1.0 to ensure that in global 

stability, as depicted in Figure 10-24, the factor of safety is larger than 1.30. 

91 
Results of Bishop Analysis 

RESULTS DISPLAY PLOT / PRINT CAPTURE SAFETY MAP 

Color Code: Safety Factors 

SCALE [ft.] 

~ 10.00 

< 30 7.00 

4 6.00 

E 5.00 

E 20 

7 4.00 

E 3.00 

~ 2.50 

10 

11 

2.00 10 

12 1.50 

13 

-

1.30 

H 1.00 0 

15 

16 

l7 Re~un ... 

18 

19 

Minimum factor of safety = 1.21 X = N/A 

Radius = 108.87 ft. Y = N/A 

Right-dick to change Type 

or Length of reinf. or soil 

parameters. 

? 

OK 

Figure 10-24. Safety map for the case of secondary reinforcement 

10.3 Example 2: Simple Slope Problem 

AASHTO and FHWA define MSE walls as reinforced structures having batter ≤20 o

 (slope 

inclination ≥70 o

) and reinforced slopes having batter >20 o (slope inclination <70 o). Different 

design methods are used for ‘slopes’ and ‘walls,’ leading to significantly disjointed outcomes for 

two similar geotechnical structures. Conversely, the LE framework suggests a unified approach 

for any slope reinforced with geosynthetics. Consequently, it would be instructive to demonstrate 

the LE approach for a 2:1 slope (inclination of 63.4 o) using the same basic data as in Example 1. 

92 
 

 

 

 

Table 10-6. Computed values for Example 2 produced by the LE framework 

Layer i Height above Toe [ft] 

Tmax-i 

[lb/ft] 

xi where Tmax-i 

[ft] 

To-i 

[lb/ft] To-i/Tmax-i 

1 1 350 2.0 165 0.47 

2 3 387 3.7 116 0.30 

3 5 387 6.0 103 0.27 

4 7 387 7.7 89 0.23 

5 9 387 9.1 75 0.19 

6 11 387 10.3 62 0.16 

7 13 387 11.0 48 0.12 

8 15 387 11.6 7 0.02 

9 17 387 12.0 34 0.09 

10 19 387 12.2 226 0.58 

The numbers in Table 10-6 were generated by running the LE framework. Recall that in 

Example 1 the wall had a batter of 8o. The framework yielded max(Tmax) = 708 lb/ft for that wall 

whereas it yields max(Tmax) = 387 lb/ft for the slope. Considering a factor of safety on the 

reinforcement strength of Fs-strength =1.5, and using the combined reduction factor RF=2.0, Tult 

= (1.5×2.0×387) = 1,161 lb/ft compared with Tult =2,124 lb/ft for the wall. That is, the ultimate 

strength of the reinforcement decreases by about 45% due to a change in slope inclination from 

82o

 to 63.4o, a decrease in slope by about 19 o. Comparing Table 10-6 and Table 10-2, the actual 

value of To has also dropped significantly for the slope. Figure 10-25 helps visualizing the 

results presented in these two tables. 

93 
20 

-+-Tmax 

18 

-II-To 

16 -.-rmax 

._To 

14 

.....,. 

~ 12 

<I) 10 0 

I-

<I) 

> 0 

8 

.J:J 

"' .E 

6 

0) 

·.; 

I 4 

2 

0 

0 100 200 300 400 500 600 700 800 

Tmax-i, To-i [lb/ft] 

Figure 10-25. Graphical representation of Tables 10-2 & 10-6 

Figure 10-26 shows the required distribution of tensile resistance, Treq(x), for Fs=1.0, as 

generated by the framework. Figure 10-26 is drawn to scale which can be determined from the 

Tmax-i values in Table 10-6. Note that none of the layers is affected by rear end pullout, indicating 

full utilization of the reinforcement intrinsic resistance. 

Figure 10-27, similar to Figure 10-7, signifies the level of mobilized force along each layer 

rendering a state of LE nearly anywhere within the mass. In segments where the color bars are 

not shown, the reinforcement is not needed based on Bishop’s analysis when examining slip 

circles emerging along the face; i.e., it is dormant along these segments. The commentary 

regarding Figure 10-7 is valid here as well. 

94 
Figure 10-26. Baseline Treq-i(x) for Example 2 

95 
 

Figure 10-27. Color coded map of distribution of tensile resistance for Example 2 

Figure 10-28. Safety map generated for Example 2 using Bishop analysis 

96 
 

Based on the LE baseline solution, geosynthetics having Tult = 1,161 lb/ft combined with RFd, 

RFid, and RFc, were specified for assessment of global stability. Running Bishop, the safety map 

displayed in Figure 10-28 was obtained. The ‘red zone’ in this map indicates zones within which 

safety factors are lower than 1.3. Unlike the case for Example 1, here there are two apparent 

problems that need attention. First, the minimum safety factor of Fs=1.20 for foundation/deep 

seated stability is unacceptable. Second, there is an internal narrow red zone within the 

reinforced zone. While the factor there is less than 1.3, it is larger than 1.20; in fact, it is 1.28. 

This internal red zone indicates that the reinforcement needs a minor strength adjustment to meet 

an absolute minimum factor of 1.30. Consequently, Tult was increased from 1,161 lb/ft to 1,200 

lb/ft, eliminating the internal red zone by increasing the ‘internal’ factor of safety to 1.30. Figure 

10-29 shows the safety map for the adjusted strength. 

Figure 10-29. Safety map for adjusted strength of reinforcement 

While the adjusted strength eliminates the unsatisfactory internal global stability, the foundation 

problem remains as the critical circle propagates mainly around the reinforcement. In such a case 

some ground improvement can be helpful. However, one simple solution would be to lengthen 

all four bottom layers intersecting the critical circle beyond the red zone. To check the 

effectiveness of such a solution, these four layers were lengthened from 14 to 18 ft. For 

simplicity, the reinforced soil mass was also increased to 18 ft. Figure 10-30 now shows a green 

zone for which the safety factors are between 1.30 and 1.50. As can be seen, much of the mass is 

‘green,’ indicating a nearly optimal design in terms of both length and strength of reinforcement. 

97 
 

 

 

 

Using LE analyses for local and global stability provided a rational method for developing an 

effective design. 

Figure 10-30. Safety map for adjusted length of reinforcement 

It is interesting to compare Figure 10-10 and Figure 10-29. While the deep seated factor of safety 

for the wall in Example 1 is 1.31, in Example 2 it is 1.20. In both cases the reinforcement layout 

and the reinforced mass are identical. This might be counterintuitive as the reinforced mass in 

Example 1 has the same footprint but is ‘heavier’ by virtue of having a batter of 8° whereas 

Example 2 has a batter of nearly 27°. However, this phenomenon is related to the fact that 

reinforced soil is not a ‘coherent mass’ or a footing, as commonly assumed to simplify the design 

of MSE walls. It is an earthen embankment, albeit reinforced, placed on top of foundation soil. 

Consequently, failures may occur through the reinforced mass. Furthermore, the stresses inside 

the foundation soil which are induced by the reinforced mass are smaller under the slope for 

Example 2 than for Example 1. Consequently, although the shear stress along a potential slip 

surface increases in Example 1 due to the weight of the reinforced mass, the soil’s frictional 

shear resistance increases as well. Hence, it is difficult to a priori state which increasing element, 

shear resistance or destabilizing load, will have greater impact. Stability analysis for a specific 

case should provide a clear answer, be it intuitive or not. 

98 
 

Additional stability analyses for other modes of failure, such as sliding, might be warranted in 

the design of slopes as well. 

10.4 Example 3: Tiered Reinforced System 

Figure 10-31 presents the geometry of a tiered reinforced ‘system.’ The lower tier is a reinforced 

‘wall’ (with batter of about 2.8°) while the upper tier is a reinforced ‘slope’ (with batter of 24° or 

inclination of 66°). Hence the term reinforced ‘system’. 

The interaction parameters used in the framework are F* = 0.8tan(φ) and α = 0.8. The factor of 

safety on pullout was specified as Fs-po =1.50. Note that the ‘retained’ soil is a sandstone that 

has large cohesion value and therefore, does not need a retaining system to remain stable. The 

system is subjected to live load (LL) of 250 psf acting over a 24-ft base, representing a two lane 

highway. The LL does not contribute to pullout resistance; however, it affects Treq(x) and hence, 

Tmax-i. Neither AASHTO nor FHWA provide explicit tools for designing similar ‘hybrid’ 

reinforced structures. 

99 
xi where TmaxHeight above Tmax-i i To-i To-i/Tmax-i 

Layer i Toe [ft] [lb/ft] [ft] [lb/ft] 

1 1.00 703 0.9 240 0.34 

2 2.33 703 2.1 75 0.11 

3 3.67 703 3.3 55 0.08 

4 5.00 703 4.3 27 0.04 

5 6.33 703 5.4 14 0.02 

6 7.67 703 6.4 14 0.02 

7 9.00 703 7.4 14 0.02 

8 10.33 703 8.3 21 0.03 

9 11.67 703 9.3 27 0.04 

10 13,00 703 10.2 62 0.09 

11 14.33 703 11.1 89 0.13 

12 15.67 703 11.8 41 0.06 

13 17.00 703 12.7 55 0.08 

14 18.34 703 13.5 14 0.02 

15 19.67 703 14.1 130 0.18 

16 21.00 703 14.9 55 0.08 

17 23.00 703 15.9 110 0.16 

18 25.00 703 16.8 89 0.13 

19 27.00 703 17.7 315 0.45 

20 29.00 697 17.4 486 0.70 

Table 10-7. Computed Tmax and To for Example 3 

Table 10-7 shows the computed Tmax-i and To-i values. The layout of the reinforcement seems 

efficient as all have similar values of Tmax. The connection load for the top layer is quite high as 

it has larger ‘tributary’ area that is subjected also to a broken backslope. For the same reason, 

layer 15, located at the top of the lower tier, is also subjected to a relatively higher connection 

load. 

Figure 10-32 displays the results presented in Table 10-7. The connection load for the top is 

substantially higher than for lower layers. Note that it has large ‘tributary’ area, it is under a 

backslope, and is subjected to live load, LL (i.e., possible increase in pullout resistance due to LL 

is ignored; only the extra reinforcement load is considered). 

100 
3S 

~ Tmax 

3) 

... -II-To 

- ' 

2S 

~ 20 > <I) 

0 

I- 1S ~ <I) 

> 0 

.J:> 

"' 10 • 

.c 

-

0) 

·.; 

I s 

- 0 

0 100 200 soo 600 700 800 

Tmax-i, To-i [lb/ft] 

Figure 10-32. Graphical representation of results in Table 10-7 

Figure 10-33 shows the baseline Treq-i(x) for Example 3. Clearly, the sandstone resists the 

development of compound failures (i.e., restricts potential failure to the reinforced soil) and, 

therefore, has a major impact on the mobilized force in the reinforcement. Pullout affects the top 

layer as well as layer 15. Note the double peak in the mobilized force in layers 12 through 15. 

This is a result of the setback between the two tiers combined with the impact of the upper tier. 

While this phenomenon might be intuitive, the LE framework provides a tool to quantify it. 

101 
Toe point y 

X 

Figure 10-33. Baseline Treq-i(x) for Example 3 

102 
Tensile Resistance ? 

Ra Color Code: nge of T ~equired 

Values in [ lb / ft) 

0.00 

150.00 >• .HU,UU >• 360.00 > 410.00 > 460.00 > 510.00 > 560.00 > 6 10.00 > 660.00 > 710.00 >• 

Close 

Figure 10-34. Color coded map of distribution of tensile resistance for Example 3 

The color coded map in Figure 10-34 complements Figure 10-34. It shows that the reinforcement 

is highly mobilized between the two tiers next to the bench in the sandstone. The greenish zone 

in the lower tier shows the area where the second (lower) peak of reinforcement force is 

mobilized. It is located under the slope of the upper tier as well as under the inner edge setback. 

Recall that in this computation, the soil strength is fully mobilized; i.e., the target Fs is 1.0. For a 

complex case like Example 3, it is recommended to use the LE framework in ‘reverse’. That is, 

check whether for reinforcement with LTDS which is equal to Tmax-i the resulted Fs is 

approximately 1.0 (i.e., the same value based on which Treq was computed). Using Bishop 

analysis, Figure 10-35 shows that for LTDS = 703 lb/ft, the actual Fs=1.00. Note that the 

computed value is for assumed circular slip surfaces. In complex strata, multi-polygonal surfaces 

could be more critical. It is recommended to check other possible surfaces to ascertain that for 

the specified LTDS value indeed Fs=1.00 and not substantially lower. Commercially available 

software which considers pullout resistance while enabling the use of general shaped slip surface 

for reinforced soil can be used for this purpose. 

103 
Results of Bishop Analysis ? 

RESULTS DISPLAY PLOT / PRINT CAPTURE SAFETY M A P 

·-

Strip load 

-

_l SCALE [ft.] 

_l_ 30 

--"-

_i 

• --"-

_i_ ,-- I I 

-1.. 20 I I I 

_i_ I I I 

C I I I --"- I _jQ I I 

-11. , 

_.ll 10 , 

, 

-11 , 

, _ll I 

__.1§_ / 

/ 

_.1§_ 0 / 

Re~un y / ___E X3··>X4 Toe point / 

....ll ~ I 

/ 

....ll A / 

~ 4 @11>1 X 

Hide Scale I ~ II Gridlines I ~ ~ Minimum factor of safety = 1.00 X = N/A Right-dick to change Type 

l c lick to display distribution of Radius = 85. 77 ft. Y = N/A or Length of reinf. or soil I OK available tensile resistance parameters. I 

Figure 10-35. Conventional stability analysis: Critical circle and Fs for LTDS=703 lb/ft 

Finally, max(Tmax) = 703 lb/ft (Table 10-7); selecting reduction factor of RF=2.0 (see Example 1) 

which accounts for installation damage, durability and creep, and a factor of safety on strength 

Fs-strength =1.5, the required minimum ultimate strength of the geosynthetic would be Tult = 

703×2.0×1.5 = 2,109 lb/ft. This implies long term design strength in global stability calculations 

of LTDS = 2,109/2 = 1,055 lb/ft. Using this long term design value in Bishop global stability 

analysis while considering connection capacity and front and rear pullouts, one gets the safety 

map shown in Figure 10-36. It can be seen that Fs = 1.36 (i.e., >1.30). The zone within which the 

safety factor is between 1.36 and 1.50 is rather narrow, tending to be in the rear of the reinforced 

zone in the upper half of the structure. In the lower portion of the wall, the reinforcement is 

hardly mobilized in its back side (see also Figure 10-34). 

104 
	 

Results of Bishop Analysis ? 

RESULTS DISPLAY PLOT / PRINT CAPTURE SAFETY MAP 

·-

Strip load 

- Color Code: Safety Factors _l 

E3 SCALE [ft.] 

_l_ 30 

---t-••••~•J .... 

-"-

_i 3.00 

• -"- 2.00 

_i_ r•IJI,-;.. IJII 

-1.. 1.50 20 I A • 

_i_ 1.30 / J ! 

C .'-' ' -"- #j _jQ 

I }: -11. 10 . :;,. _.ll . 

. .. -11 D .. 

.. _ll ·~ 

__.1§_ .. 

~- _.1§_ 0 ,' ~- ___E Re~un Toe point y ~- X3-·>X4 .• ~- _j§_ . .• ~- r _ll A ,. r 

~ 4 @11>1 . X 

Hide Scale I ~ II Gridlines I ~ ~ Minimum factor of safety = 1. 36 X = N/A Right-dick to change Type 

l click to display distribution of Radius = 40. 53 ft. Y = N/A or Length of reinf. or soil I OK available tensile resistance parameters. I 

Figure 10-36. Example 3: Safety map for the designed problem 

While Figure 10-36 indicates that the reinforcement layout and strength is satisfactory, it is based 

on rotational circular failures. One needs to ascertain that other failure mechanisms are not more 

critical. 

10.5 Commentary 

As stated in Section 9.7 and demonstrated in this chapter, the LE-based design involves two 

stages: 

1.	 First, use the top-down framework to compute the baseline Treq(x) for each layer i and then 

determine Tmax-i. This is done for a specified layout of reinforcement, utilizing the peak 

design strength (e.g., φ) of the soil. Use of full (peak) strength is consistent with AASHTO 

where φ is limited to maximum of 40°. The 40° limitation is an artifact of existing practice 

where the calibrated semi-empirical method has been used. Such limitation in the context of 

LE design is not needed thus enabling a geotechnical engineer to take full advantage of well 

compacted good quality selected backfill. The process leads to the selection of (ductile or 

extensible) geosynthetics for a ‘true’ limit state. That is, stability at this state is hinging on 

the ultimate strength of the reinforcement, 

105 
Tult ≥ (Fs-strength ×Tmax-i) RFd×RFid×RFc (10.1) 

Written differently, 

LTDS = Tult /(RFd×RFid×RFc) ≥ (Fs-strength ×Tmax-i) (10.2) 

Tult, and the reduction factors related to installation damage (RFid), durability (RFd), and 

creep (RFc), are leading to the long term design strength, LTDS, of the reinforcement: 

2.	 Second, check the preliminary design rendered by the baseline solution for global stability 

using target Fs on the peak strength of the soil that is ≥1.3. This Fs is standard in 

conventional LE analysis of reinforced or unreinforced slopes. In this analysis, LTDS is 

considered. At the intersection of any slip surface with a reinforcement layer, either LTDS or 

pullout resistance, whichever is smaller, is used (Figure 3-1) in the LE equations. Note that 

the factor of safety on the reinforcement strength, Fs-strength, is not used in this stage; it is 

used in the first stage. In the second stage, one global factor of safety, Fs≥1.3, applied to the 

soil strength is used. If the actual Fs is less than the minimum required value (e.g., 1.3), the 

designer needs to increase the strength and/or length of reinforcement until global stability is 

adequate. As demonstrated in this chapter, the safety map may facilitate this process. 

There is seemingly a conflict; i.e., no explicit factor of safety on the strength of reinforcement 

used in the second stage. However, LTDS is utilized for soils having reduced shear strength by 

Fs {i.e., φdesign = tan-1[tan(φ)/Fs] and cdesign = c/Fs}. Reduced soil strength results in required 

longer and stronger reinforcement than for the baseline where the soil actual strength is fully 

mobilized (i.e., Fs=1.0). Note that Tult contains Fs-strength and, when conducting the second 

stage, the ultimate strength should not be smaller than the value selected in the first stage. This 

means that if, due to unforeseen circumstances, φ is fully mobilized reaching the idealized LE 

state in the first stage (i.e., an ‘active’ wedge is formed), Fs-strength would be ≥1.5. The first 

stage design should ensure this Fs-strength. If the initially selected LTDS renders Fs≥1.3 in the 

second stage, Fs-strength≥1.5 is assured in actual limit state condition. However, if the initial 

LTDS must be increased and/or the reinforcement lengthened to meet the Fs≥1.3 requirement in 

the second stage than a new ultimate or long term design strength must be selected, 

Tult=(LTDS×RFd×RFid×RFc), and it should be larger than the original value produced in the first 

stage. This requires a simple trial and error process that needs to satisfy global stability (e.g., see 

discussion in Section 10.3). Alternatively, specifying Fs>1.0 can be done in the first stage. It is 

noted that an increase in LTDS and/or length of reinforcement in the second stage could also be 

due to foundation failure (see Section 10.3) and it is not directly related to the reinforced mass 

itself. The top-down (first stage) approach does not consider directly the foundation effects (i.e., 

it assume a competent foundation) counting on the second stage (global stability assessment) to 

rectify possible deep-seated instability. However, if warranted, the top-down approach can be 

extended to include the impact of the foundation in the first stage. 

Generally, the connection loads, To-i, produced by the LE framework at a limit state are small 

relative to AASHTO’s design method. However, the connection load at the top layer could be 

large, especially if the spacing or backslope is large. The following comments are made: 

106 
 

 

 

	 

	 

	 

	 

1.	 The LE methodology is for the complete structure. However, during construction, as layers 

are placed sequentially, each layer will serve temporarily as a ‘top’ layer. Hence, the short 

term connection strength for each layer below the final top layer could exceed the connection 

load at the top, even if for the final structure the load at the lower connections decrease. Of 

course, if the connection load of the lower layers is higher than that at the top, these loads 

prevail. This construction perspective combined with flexible facing should not affect the 

outcome of design. 

2.	 To-i was obtained for limit state where some facing movement during construction is 

acceptable. However, this may cause a serviceability issue in critical applications such as 

bridge abutments or in cases where footings of buildings are located over the reinforced soil 

zone. That is, when high surcharge is applied after construction is complete. In such cases 

one may require that the connection capacity at each elevation to be equal to the limit state 

maximum load in the respective reinforcement layer; i.e., (To-i)imposed=Tmax-i. Figure 10-37 

schematically shows this requirement. 

3.	 Regardless of serviceability considerations mentioned in (2), firmly restraining the facing 

through increased connection capacity (Figure 10-37) should result in the following: 

–	 Enable better compaction next to the facing, an important element responsible for good 

performance of earth structures. 

107 
 

 

 

 

	 

	 

	 

	 

–	 Provide better confinement at the facing. Such confinement stiffens the reinforced soil 

mass as well as increase the soil strength, resulting in additional margin of safety for 

stability. 

–	 In lieu of firmly restraining the facing (i.e., use To for limit state rather than To=Tmax as in 

Figure 10-37), closely spaced reinforcement combined with high quality compacted 

backfill can be used. The restraining and confinement produced by such technique has a 

proven record in numerous constructed GRS walls. 

4.	 Most MSE structures are not sensitive to minor movement of the facing that develop during 

construction. Furthermore, experience indicates that in many systems with moderate 

connection capacity, existing construction techniques ensure acceptable overall performance. 

Consequently, connection capacities, considered as long term values, generated by the LE 

framework, are adequate provided that (1) above is satisfied. 

5.	 As demonstrated in this chapter, if needed, the connection load can be reduced if closely 

spaced reinforcement is used. Such reinforcement need not be through the entire structure nor 

need it be full length; i.e., it could be installed as secondary reinforcement in locations where 

needed. 

108 
 

 

 

 

 

 

 

 

 

11.0 CONCLUDING REMARKS 

Current design of reinforced earth structures in the US distinguishes between slopes and walls 

using the batter angle as a criterion. Batter ≤20° is defined as a reinforced wall and batter >20° is 

defined as a reinforced slope. Different design-oriented analyses are used for walls and slopes, 

generally leading to significantly different outcomes. Such distinction simplifies the design of 

walls by virtue of using a synergy of simple analyses that are tangible and metaphorically, except 

for global stability, can be done ‘on-the-back-of-the-envelope-calculations’. While the semiempirical approach to walls results in safe structures, it is not consistent with traditional and 

well-established geotechnical design of similar structures; i.e., slopes. Reinforced slopes and 

walls can be considered as a subset of slope stability problems where manmade reinforced soil 

slopes are constructed over foundation soil. Slope inclination (or batter) is just a typical design 

variable which is not an artifact of convenience of calculations. Using a unified approach in limit 

state design of reinforced ‘walls’ and ‘slopes’ should diminish confusion. It should enable a 

consistent approach to solving various geotechnical problems such as complex geometries and 

non-uniform soil profiles. 

Limit equilibrium (LE) analysis has been used successfully in the design of complex and critical 

structures (e.g., tall dams) for many decades. The LE formulation requires some assumptions in 

statics and/or geometry of failure. Hence, there are quite a few LE methods where each is based 

on somewhat different assumptions. Generally, and in a practical perspective, the difference 

introduced by properly using different assumptions is lesser than that associated with field 

characterization of properties. Reliability and simplicity of LE analyses keep its use common in 

the USA. 

Limit state analysis, including LE, assumes that the design strength of the soil is mobilized. The 

degree of mobilization (or utilization) signifies the margin or factor of safety. Reinforcement is 

installed in slopes that otherwise are inherently unstable. Therefore, any procedure assumes that 

at a limit state the design strength of the soil is fully mobilized (i.e. Fs=1.0) and stability now is 

hinging upon the mobilized tensile resistance of the reinforcement. At that state, design should 

ensure that the long term strength of the reinforcement would be available along the 

reinforcement. A silent assumption in this concept is that the reinforcement will not rupture as 

the soil deforms while mobilizing its strength. Often this phenomenon is referred to as 

compatibility. Geosynthetic reinforcement is ductile (‘extensible’), capable of developing 

substantial strain (typically >8%) before rupture. Such planar strains are much larger than those 

needed for granular compacted backfill to mobilize its strength (i.e., to form an ‘active’ mass). 

From this perspective, combined with experimental and numerical investigations reviewed in this 

report, the limit state concept in general and LE in particular are suitable for design of 

geosynthetic reinforced slopes/walls. This assertion is supported by the performance of 

numerous reinforced slopes designed and constructed over the past three decades. However, 

experimental and numerical confirmation of limit state of relatively brittle (i.e., ‘inextensible’) 

reinforcement is scarce. Hence, the question of compatibility of inextensible reinforcement in the 

context of limit state still remains. Consequently, at this stage the scope of this work is limited to 

extensible reinforcement. 

Presented herein is a LE framework which enables the designer to find the tensile force 

distribution in each layer required at a limit state. This distribution is a function of the geometry 

109 
 

 

 

 

 

 

 

 

 

 

 

of the slope, the soils comprising the slope, the layout of the reinforcement, the parameters 

characterizing the interaction of the reinforcement with the confining soil, seismicity, and other 

relevant aspects. The tensile distribution is determined by assessing test bodies at many locations 

while varying the mobilized force in the reinforcement, aiming to produce the same limit state at 

each location (i.e., any point within the reinforced mass has Fs=1.0 and hence, the same margin 

of safety as expressed by Fs). The end result of such analysis produces the maximum tensile 

force mobilized in each layer as well as the connection load between the reinforcement and the 

facing. The designer can then select geosynthetic (and facing) considering reduction factors for 

installation damage, durability, creep, as well as an applied factor of safety on the long term 

strength (i.e., standard procedure in selecting geosynthetic reinforcement with adequate ultimate 

strength). However, while this stage of design ensures ‘local’ stability considering the 

reinforcement force as a variable, it may not guarantee global stability where the soil strength 

mobilized is limited by a prescribed minimum factor of safety (e.g., Fs≥1.3). Hence, after 

selecting the reinforcement based on the first stage, in the second stage the designer needs to 

conduct a conventional LE stability assessment using commercially available software. 

Compound and foundation/deep seated failures need to be assessed while the reinforcement 

strength is limited by its actual long term strength. This strength is not subjected to an additional 

factor of safety (Fs-strength) since the target global factor of safety, Fs, de facto considers 

‘weaker’ soil and therefore, results in stronger and longer reinforcement than needed for actual 

limit state. This long term strength along the reinforcement is limited by pullout resistance at the 

rear and front ends of each reinforcement layer. Connection capacity plays a role in assessing the 

front end pullout capacity. 

Three example problems are presented. For a case comparable with AASHTO, it is shown that 

the reinforcement strength required using the LE framework is about half that computed using 

the AASHTO method. It is also shown that the so-called bearing capacity mode of failure is 

addressed through LE stability analysis considering foundation failure. While the discussion on 

traditional bearing capacity of MSE structures, such as AASHTO, versus LE stability analysis of 

the foundation is limited, in-depth review of the subject is presented in Leshchinsky et al. (2012). 

Sliding is assessed through two-part wedge LE analysis. The design aspects of eccentricity 

considered in AASHTO, cannot be conducted using LE in a straightforward manner. However, 

while eccentricity is a valid consideration in footing design, it has little relevance to earthen fill 

over foundation soil. That is, eccentricity and ‘overturning’ are typically assessed in rigid gravity 

wall analyses. Indeed overturning is a common mode of failure of gravity walls; however, there 

is little evidence of such mode of failure when dealing with reinforced walls/slopes. Besides, this 

issue can and should be dealt with in serviceability assessment where differential settlement of 

the foundation soil is assessed. This example also shows the impact of closely spaced 

reinforcement, secondary reinforcement, and seismicity. Closely spaced reinforcement reduces 

the required strength of reinforcement as well as the connection load. Secondary reinforcement 

has intermediate effects between large spacing and close spacing of reinforcement; i.e., it reduces 

the required strength of geosynthetics while significantly reducing the connection loads at some 

zones. Seismicity increases the required short term strength of the reinforcement with little effect 

on connection load if inertia of facing is ignored. The locus of maximum load in the 

reinforcement is pushed backwards as seismicity increases. This may result in increased length 

of reinforcement compared with the static case. 

110 
 

 

 

 

 

 

 

 

The second example is similar to the first one but has a batter that classifies it as a ‘slope’. The 

mobilized force in the reinforcement and connection drops relative to the first example. 

However, foundation instability could be an issue. It is shown that lengthening a few lower 

layers could resolve this problem. More importantly, a ‘slope’ and a ‘wall’ can be compared 

using the same design tool. This may lead to more objective decisions which relate to the 

economics of the alternatives. 

The third example is for a complex wall/slope geometry consisting of a two-tiered system where 

the upper tier is defined by AASHTO as a ‘slope’, and the lower one is defined as a ‘wall’. Each 

tier has its own backfill. The ‘retained’ material is a sandstone with large cohesion. The cut in 

the sandstone does not require a retaining wall to support it. The LE framework provides the 

required strength of reinforcement and connection without resorting to assumptions beyond those 

used for any other problem (such as in the first two examples). 

To use AASHTO terminology, the LE approach is restricted to Allowable Stress Design (ASD). 

Modifying it to Load and Resistance Factor Design (LRFD) is a challenge. Simply, the concept 

of LRFD requires complete separation between load and resistance. In limit analysis of soil 

(including LE), such separation is not straightforward; e.g., increase in weight of a sliding mass 

increases the destabilizing load while simultaneously increases the resistance of the frictional 

material. In lieu of LRFD, it is suggested that a future development would be a modification of 

the proposed approach to include stochastic (probabilistic) LE analysis. This is consistent with 

soil behavior and geotechnical practice. 

Finally, the top-down approach is applicable to slopes that do not possess sufficient stability as 

prescribed by a required minimum factor of safety and where extensible reinforcement may be 

used as a remedy. It is also applicable to embankment over soft soil where failures through the 

foundation can be controlled by base reinforcement. However, in such problems the top-down 

approach needs to consider slip surfaces emerging away from the toe; i.e., foundation/deepseated failures. Such problems are beyond the scope of this work. 

111 
12.0 REFERENCES 

AASHTO, Standard Specifications for Highway Bridges, 17th ed., AASHTO, Washington, DC, 

2002. 

AASHTO, LRFD bridge design specifications, 4th ed., AASHTO, Washington, DC, 2007. 

Allen, T.M., and Bathurst, R.J., “Prediction of soil reinforcement loads in mechanically 

stabilized earth (MSE) walls.” Rep. No. WARD-522.1, Washington State Department of 

Transportation, Seattle, 2001. 

Allen, T.M., and Bathurst, R.J., "Prediction of reinforcement loads in reinforced soil walls," Rep. 

No. WARD-522.2, Washington State Transportation Center, 2003. 

Anderson, P., Gladstone, R., and Sankey, J., “State of the Practice of MSE Wall Design for 

Highway Structures,” Proc. of GeoCongress 2012, ASCE, GSP 2012, 443-463. 

Baker, R., and Leshchinsky, D., “Spatial distributions of safety factors,” Journal of Geotechnical 

and Geoenvironmental Eng., ASCE, 127(2), 2001, 135–145. 

Bathurst, R. J., Vlachopoulos N., Walters, D. L., Burgess, P. G., and Allen, T. M., “The 

influence of facing stiffness on the performance of two geosynthetic reinforced soil retaining 

walls,” Canadian Geotechnical Journal, 43(12), 2006, 1225-1237. 

Berg, R., Christopher, B.R., and Samtani, N., Design of mechanically stabilized earth walls and 

reinforced soil slopes, Vols. I and II, Reports No. FHWA-NHI-10-024/025, Federal Highway 

Administration, 2009. 684 pp. 

Bishop, A. W., “The use of the slip circle in the stability analysis of slopes,” Géotechnique, 5(1), 

1955, 7-17. 

Bolton, M.D., “The strength and dilatency of sands,” Geotechnique, 1986, 36(1), 65-78. 

BS 8006, 2010. Code of Practice for Strengthened/Reinforced Soils and Other Fills. 

British Standard Institution, ISBN 978 0 580 53842 1, ICS 93.020, 260 p. 

Cheng, Y.M., Lansivaara, T., and Wei, W.B., “Two-dimensional slope stability analysis by limit 

equilibrium and strength reduction methods,” Computers and Geotechnics, 34, 2007, 137-150. 

Damians, I.P., Bathurst, R.J., Josa, A., and Lloret, A., “Numerical Analysis of an Instrumented 

Steel-Reinforced Soil Wall,” Int. Journal of Geomechanics, 15, 2015, 1-15. 

Damians, I.P., Bathurst, R.J., Josa, A., Lloret, A., and Albuquerque, P.J.R., “Vertical-Facing 

Loads in Steel-Reinforced Soil Walls,” Journal of Geotechnical and Geoenvironmental Eng., 

139(9), 2013, 1419-1432. 

112 
Design Standard for Railway Earth Structures, 2013, Railway Technical Research Institute. 

Maruzen, pp. 56-57 (in Japanese). 

Duncan, J.M., “State of the art: Limit equilibrium and finite-element analysis of slopes,” Journal 

of Geotechnical Engineering, 122(7), 1996, 577-596. 

Duncan, J.M., and Wright, S.G., Soil Strength and Slope Stability, 2005, John Wiley & Sons, 297 

pp 

EBGEO, 2011. Recommendations for Design and Analysis of Earth Structures Using 

Geosynthetic Reinforcement. German Geotechnical Society, Ernst & Sohn, GmbH & Co. KG 

(Chapter 7). 

Ehrlich, M., and Becker, L. D. B., “Reinforced soil wall measurements and predictions.” Proc., 

9th Int. Conf. on Geosynthetics: Geosynthetics for a challenging world, E. M. Palmeira, D. M. 

Vidal, A. S. J. F. Sayao, and M. Ehrlich, eds., Vol. 1, IGS Brasil and ABMS, Brazil, 2010, 547– 

559. 

Eurocode 7, 2004. Geotechnical design—Part 1: General rules, CEN—European committee for 

standardization, Brussels. 

Fannin, R. J., and Hermann, S., “Performance data for a sloped reinforced soil wall,” Canadian 

Geotechnical Journal, 27(5), 1990, 676–686. 

Fannin, R.J., 2001, “Long-Term Variations of Force and Strain in a Steep 

Geogrid-Reinforced Soil Slope”, Geosynthetics International, Vol. 8, No. 1, pp. 81-96. 

Han, J. and Leshchinsky, D., “General Analytical Framework for Design of Flexible Reinforced 

Earth Structures,” Journal of Geotechnical and Geoenvironmental Eng., 132(11), 2006, 1427– 

1435. 

Han, J., and Leshchinsky, D., “Stability Analysis of Back–to–Back MSE Walls,” Geotextiles and 

Geomembranes, 28(3), 2010, 262-267 

Huang, B., Bathurst, R. J., Hatami, K., and Allen, T. M., “Influence of toe restraint on reinforced 

soil segmental walls,” Canadian Geotechnical J., 47(8), 2010, 885–904. 

Jacobsz, S.W., “Centrifuge modelling of a soil nail retaining wall,” J. of the South Africa 

Institute of Civil Engineering, 55(1), 2013, 85-93. 

Kim, W.S., and Borden, R.H., “Numerical simulation of MSE wall behavior induced by surface 

water infiltration,” Journal of Geotechnical and Geoenvironmental Eng., ASCE, 139(12), 2013, 

2110-2124. 

Koerner, R. M. and Koerner G. R., “A data base, statistics and recommendations regarding 171 

failed geosynthetic reinforced mechanically stabilized earth (MSE) walls,” Geotextile and 

Geomembranes, 2013, Vol. 40, 20-27. 

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Lee, K.L. and Seed, H.B., “Drained Strength Characteristics of Sands,” Journal of the Soil 

Mechanics and Foundations Division, 1967, 93(6), 117-141. 

Leshchinsky, D., "Design Procedure for Geosynthetic Reinforced Steep Slopes," Geotechnical 

Laboratory, US Army Corps of Eng., Waterways Experiment Station, Report REMR-GT-23, 

1997, Vicksburg, Mississippi. 

Leshchinsky, D., “The Power of Software in Reinforcement Applications: Part I, Part II, Part III, 

Part IV,” Geotechnical Fabrics Report, 23, 2005, Numbers 5, 6, 7, 8. 

Leshchinsky, D., “On global equilibrium in design of geosynthetic reinforced walls,” Journal of 

Geotechnical and Geoenvironmental Engineering, ASCE, 135(3), 2009, 309-315. 

Leshchinsky, D., “Topics in Geosynthetic Reinforced Earth Structures,” Keynote Paper, Proc., 

Conferenze Di Geotechnica Di Torino, XXIII Cicloe: Earth Retaining Structures and Slope 

Stabilization: Theory, Design, and Application, 2011 

Leshchinsky, D., and Han, J., “Geosynthetic reinforced multitiered walls.” Journal of 

Geotechnical and Geoenvironmental Eng., ASCE, 130(12), 2004, 1225–1235. 

Leshchinsky, D., Hu, Y, and Han, J., “Limited Reinforced Space in Segmental Retaining Walls,” 

Geotextiles and Geomembranes, 22(6), 2004, 543-553. 

Leshchinsky, D., Imamoglu, B. and Meehan, C.L., “Exhumed Geogrid-Reinforced Retaining 

Wall,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 2010a, 136(10), 

1311-1323 

Leshchinsky, D., Kang, B.J., Han, J., and Ling, H.I., “Framework for Limit State Design of 

Geosynthetic-Reinforced Walls and Slopes,” Transportation Infrastructure Geotechnology, 

Springer, 1(2), 2014, 129-164 

Leshchinsky, D. and Lambert, G., "Failure of Cohesionless Model Slopes Reinforced with 

Flexible and Extensible Inclusions," Transportation Research Record, No. 1330, 1991, 54-63. 

Leshchinsky, D., Ling, H. I., and Hanks, G., “Unified design approach to geosynthetic reinforced 

slopes and segmental walls.” Geosynthetic International, 2(4), 1995, 845–881. 

Leshchinsky, D., Ling, H.I., Wang, J-P, Rosen, A., Mohri, Y., “Equivalent seismic coefficient in 

geocell retention systems,” Geotextiles and Geomembranes, 27(1) 2009, 9-18. 

Leshchinsky, D. and Tatsuoka, F., “Geosynthetic reinforced walls in the public sector: 

Performance, design, and redundancy” Geosynthetics Magazine, 31(3), 2013, 12-21. 

Leshchinsky, D. and Vahedifard, F., "Impact of Toe Resistance in Reinforced Masonry Block 

Walls: Design Dilemma," Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 

138(2), 2012, 236-240. 

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Leshchinsky, D., Vahedifard, F., and Leshchinsky, B.A., “Revisiting bearing capacity analysis of 

MSE walls,” Geotextiles and Geomembranes, 34, 2012, 100-107. 

Leshchinsky, D., Zhu F., and Meehan, C.L., “Required Unfactored Strength of Geosynthetic in 

Reinforced Earth Structures,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 

2010b, 136(2), 281-289. 

Ling, H.I., Mohri, Y., Leshchinsky, D., Burke, C., Matsushima, K. and Liu, H., “Large-Scale 

Shaking Table Tests on Modular-Block Reinforced Soil Retaining Wall,” Journal of Geotechnical 

and Geoenvironmental Engineering, ASCE, 2005,131(4), 465-476. 

Ling, H.I., Wu, M.-H., Leshchinsky, D., Leshchinsky, B. A., “Centrifuge Modeling of Slope 

Instability,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 2009, 135(6), 

758-767. 

Mohamed, S.B.A., Yang, K-H, Hung, W-Y, “Limit equilibrium analyses of geosyntheticreinforced two-tiered walls: Calibration from centrifuge tests,” Geotextiles and Geomembranes, 

41, 2013, 1-16. 

Mohamed, S.B.A., Yang, K-H, Hung, W-Y, “Finite element analyses of two-tier geosyntheticreinforced soil walls: Comparison involving centrifuge tests and limit equilibrium results,” 

Computers and Geotechnics, 61, 2014, 67-84. 

Quinteros, V. S., 2014, “Observations on the mobilization of strength in reinforced soil 

structures,” Master of Applied Science Thesis in the Faculty of Graduate and Postdoctoral 

Studies (Civil Engineering), The University of British Columbia, Vancouver, Canada. 

Shukha R, Baker, R., “Mesh geometry effects on slope stability calculation by FLAC strength 

reduction method—linear and nonlinear failure criteria,” 3rd International Conference on FLAC 

and numerical modeling in geomechanics, Ontario, Canada, 2003, 109–116. 

Tatsuoka, F., Koseki, J. and Tateyama, M., “Introduction to Japanese codes for reinforced soil 

design,” Panel Discussion on Reinforced Soil Design Standards, Proc. 9th International 

Conference on Geosynthetics, Brazil, 2010, pp. 245-255. 

Valentine, R.J., “An Assessment of the Factors that Contribute to the Poor Performance of 

Geosynthetic-Reinforced Earth Retaining Walls,” Proceedings of the International Symposium 

on Design and Practice of Geosynthetic-Reinforced Soil Structures, 14-16 October, 2013, 

Bologna, Italy, DEStech Publications, 751 pp. pp. 318-330 

Yang, K-H, Gupta, R., and Zornberg, J.G., “Location of Failure Plane within Narrow GRS Wall 

Systems,” Proc. Geosynthetics 2009, Salt Lake City, Utah, 329-339. 

Yang, K-H, Utomo, P., and Liu, T-L, “Evaluation of force-equilibrium and deformation-based 

design approaches for predicting reinforcement loads within geosynthetic reinforced soil 

structures,” Journal of GeoEngineering, 8(2), 2013, 41-54. 

115 
Yoo, C., “Effect of rainfall on performance of geosynthetic reinforced soil wall using stress-pore 

pressure coupled analysis,” Proc. of Geo-Congress 2013, ASCE, 566-573. 

Yoo, C. and Jung, H.Y., "Case History of Geosynthetics Reinforced Segmental Retaining Wall 

Failure", Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 132(12), 2007, 

1538-1548. 

Zhang, F., Leshchinsky, D., Gao, Y., and Leshchinsky, B. A., “Required Unfactored Strength of 

Geosynthetics in Reinforced 3D Slopes,” Geotextiles and Geomembranes, Elsevier, 42(6), 

November 2014, 576-585. 

Zienkiewicz, O. C., Humpheson, C., and Lewis, R. W., “Associated and nonassociated 

viscoplasticity and plasticity in soil mechanics,” Géotechnique, 25(4), 1975, 671–689. 

Zornberg, J. G., Sitar, N., and Mitchell, J. K., “Limit equilibrium as basis for design of 

geosynthetic reinforced slopes,” Journal of Geotechnical and Geoenvironmental Eng., ASCE, 

124(8), 1998, 684–694. 

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13.0 SOFTWARE REFERENCES 

A few computer programs are referenced in the report. Most of these programs were used by the 

individuals cited in the report. The details of software used can be found in the cited papers. 

Below is a list of the referenced programs and their availability: 

Flac, Itasca Consulting Group, http://www.itascacg.com/software/flac 

MSEW, ADAMA Engineering, www.MSEW.com 

Plaxis, Plaxis BV, www.plaxis.nl 

ReSlope, ADAMA Engineering, www.GeoPrograms.com 

ReSSA, ADAMA Engineering, www.GeoPrograms.com 

Slide, Rocscience, https://www.rocscience.com/rocscience/products/slide 

Slope/W, GeoSlope International, http://www.geo-slope.com/ 

STEDwin, Annapolis Engineering Software, www.stedwin.com 

UTEXAS4, Ensoft Inc., http://ensoftinc.com/ 

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APPENDIX A: DERIVATION OF BISHOP EQUATION FOR REINFORCED SOIL 

Eqn. 9.3 shows Bishop Equation for a safety factor corresponding to a soil mass defined by a 

circular arc as shown in Figure 9-2. This mass intersects reinforcement layers, all assumed to 

mobilize force in the horizontal direction. Many test bodies are examined in the optimization 

process until the one rending the lowest safety factor, Fs, is identified. 

In deriving Bishop equation, the original methodology presented by Bishop (1955) is preserved. 

Consider slice i as shown in Figure A-1. Notice typical forces acting on it; for simplicity forces 

due to external loading, water, and seismicity are not shown. A single reinforcement layer 

intersects this slice producing a certain force, Ti. The reinforcement force (perhaps at different 

magnitude) needs to be shown on the left side of the slice. However, this is an internal force 

that eventually cancels out. At the base of the slice, Ti is inclined at angle θ to the horizontal. 

Eqn. 9.3 takes θ=0; however, the derivation here allows for this force to be inclined. 

Figure A-1. Forces acting on slice i 

Taking moments about the center of a test circle having radius R and requiring limit state 

equilibrium ΣMo=0, result in elimination of the normal force Ni: 

Wi R sinαi – Si R + Ex+Δx (R cosαi - bx+Dx) – Ex (R cosαi – bx) + 

(T x+Δx – Tx) R sinαi – Ti R cos(αi -θ) = 0 (A.1) 

118 
 

 

 

 

 

 

 

 

 

 

 

 

 

 


 


 


 

Mohr-Coulomb failure criterion with the safety factor applied on soil strength:
 

Si = (ci Δxi /cosαi + Ni tan φi) / SF (A.2)
 

Substituting Eqn. A.2 into A.1 and dividing through by R gives us the forces on a slice in terms
 

of the safety factor SF: 

Wi sinαi – (ci Δxi /cosαi + Ni tanφi) / SF + Ex+Δx ( cosαi - bx+Δx/SF) – 

Ex (cosαi – bx/R) + (Tx+Δx – Tx) sinαi – Ti cos(αi - θ) = 0 (A.3) 

The definition of SF (taken as a constant along the circular surface) will not change if Eqn. A.3 is 

summed over all n slices. However, the moments due to interslice forces, E and T as well as 

internal reinforcement force (not shown in Figure A-1), will be cancelled by the moments of the 

same internal forces acting in opposite direct on the adjacent slice. Consequently, SF can be 

extracted from Eqn. A.3: 

SF = {Σ(ciΔxi /cosαi + Ni tanφi) } / { Σ[Wi sinαi - Ti cos(αi - θ)] } (A.4) 

To determine Ni, forces in the vertical direction are summed for slice i requiring equilibrium 

ΣNi=0: 

Wi – Ni cosαi – Si sinαi + Tx+Δx – Tx – Ti sin θ = 0 (A.5) 

Substituting Eqn. A.2 into A.5 and rearranging the terms, one gets the normal force as: 

Ni = {Wi – (ciΔxi tanαi)/SF + Tx – Tx+Δx – Ti sinθ} / {cosαi + (sinαi tanφi)/SF} (A.6) 

Use the following notation: 

mαi= cos αi + (sin αi tan φi)/SF 

Substitute Eqn. A.6 into A.4 and rearrange the terms using the notation of mαi: 

SF = Σ{ciΔxi / mαi + [(Wi – Ti sinθ) + (Tx – Tx+Δx)] tanφi / mαi } / 

{ Σ [ Wi sinαi -Ti cos(αi - θ)]} (A.7) 

This expression can be solved for SF provided that Tx and Tx+ Δx are known. The result then is 

rigorous (for a circular sliding surface). However, the values of Tx and Tx+ Δx are not 

known. From statics, recognizing that force equilibrium over all internal forces requires: 

Σ(Tx – Tx+Δx) = 0 

119 
 

 

 

 

 

 

 

 

To make the problem solvable, Bishop assumed that the following exists: 

Σ{(Tx – Tx+Δx) tanφi / mαi} = 0 (A.8) 

Using the assumption in Eqn. A.8, Eqn. A.7 degenerates to a single unknown F: 

∑ ൜

ൣccii ∆xxii+൫WWii+ TTii sin θθ ൯൧ tan φφii

mmααii SSFF = 

ൠ

(A.9) ∑[WWii sinααii − TTii cos(ααii−θθ)] 

where mαi= cos αi + (sin αi tan φi)/SF 

This equation is Bishop Equation modified to include reinforcement force acting at the base of 

slice i and inclined at angle θ to the horizontal. Eqn. 9.3 is a representation of Eqn. A.9 taking 

θ=0. Note that Eqn. A.9 is essentially a ratio of moments (commonly termed as resisting over 

driving moments) where the leverage arm R cancels by virtue of being a multiplier in both 

denominator and numerator. This fact is noted since one looking at Eqn. A.9 might conclude that 

it is a ratio of scalar summation of force vectors (it is not). The impact of other forces acting on 

slices can be integrated into this moment equation following the presented derivation. Consistent 

derivation eliminates the frequently stated dilemma whether the moment contribution by Ti 

should be in the denominator or numerator. It is not an arbitrary decision but rather a result of 

proper formulation in statics. 

120