Embankment Widening and Grade Raising on Soft Foundation Soils, Phase 2

SCHOOL OFCIVIL ENGINEERING

INDIANA

DEPARTMENT OF HIGHWAYS

JOINT HIGHWAY RESEARCH PROJECT

FHWA/ IN/JHRP/ 9 2 / 19

Final Report

PHASE II - EMBANKMENT WIDENING ANDGRADE RAISING ON SOFT FOUNDATION SOILS

Scott J. Ludlow

Wai-Fah Chen

Philippe L. Bourdeau

C. William Lovell

UNIVERSITY


FHWA/ IN/JHRP/ 9 2 / 19

Final Report

PHASE II - EMBANKMENT WIDENING ANDGRADE RAISING ON SOFT FOUNDATION SOILS

Scott J. Ludlow

Wai-Fah Chen

Philippe L. Bourdeau

C. William Lovell

Digitized by the Internet Archive

in 2011 with funding from

LYRASIS members and Sloan Foundation; Indiana Department of Transportation

http://www.archive.org/details/embankmentwideniOOIudl

Final Report

Phase II - Embankment Widening and GradeRaising on Soft Foundation Soils

by

Scott J. LudlowGraduate Research Assistant

Wai-Fah ChenProfessor of Civil Engineering

Philippe L. BourdeauAssistant Professor of Civil Engineering

and

C. William LovellProfessor of Civil Engineering

Joint Highway Research ProjectProject No: C-36-36U

File No: 6-14-21

This study was conducted by theJoint Highway Research Project

in cooperation with theIndiana Department of Transportation

and theFederal Highway Administration

The contents of this report reflect the views of the authors whoare responsible for the facts and the accuracy of the datapresented herein. The contents do not necessarily reflect theofficial views or policies of the Federal Highway Administration orof the Indiana Department of Transportation. This report does notconstitute a standard, specification, or regulation.

Purdue UniversityWest Lafayette, Indiana 47907

April 13, 1993

TECHNICAL REPORT STANDARD TITLE PACE

1. Report No.

FHWA/IN/JHRP-92/1 9

7. Government Acce

4. Till* and Subtitle

Phase II - embankment widening and grade risingon soft foundation soils

3. Recipient's Catalog No.

5. Report Data

June 30, 1992

6. Performing Organization Code

7. Aufhor(s)

Scott J. Ludlow, Wai-Fah Chen, Philippe L. Bourdeauand C. William TovpI 1

8. Performing Organization Report No.

FHWA/IN/JHRP-92/1

9

9. Performing Organization Nome and Address

Joint Highway Research ProjectCivil Engineering BuildingPurdue UniversityWest Lafayette, IN 47907

10. Work Unit No.

11. Controcf or Grant No.

HPR-2031

12. Sponsoring Agency Nome and Addres"

Indiana Department of TransportationState Office Building100 North Senate AvenueIndianapolis, IN 46204

13. Type of Report and Period Covered

Final Report

14. Sponsoring Agency Code

15. Supplementary Notes

Prepared in cooperation with the U.S. Department of Transportation, FederalHighway Administration.

16. Abstract

The finite element technique using a cap elastic-plastic work-hardening soilbehavior model was applied to the analysis of embankments constructed on softfoundation soils. A procedure was provided to estimate the cap model parametersfrom conventional field and laboratory test results. A sensitivity analysisof the cap model parameters comparing the observed and calculated responses wasalso provided. Results indicate that the undrained shear strength and over-consolidation ratios were observed to have the most-significant influence on thepredicted model behavior.

The technique was then applied to the analysis of two examples. The examples werebased on actual highway projects in Indiana where information on these projectswas provided by INDOT personnel. Results of the analysis were used to determinethe influence of several factors on reinforced and unreinforced embankmentbehavior. The results indicated that the crust strength and foundationcompressibility had the most-significant influence on embankment fill and foundationsoil behavior and the potential benefit possible with reinforcement. Reinforcementtype/modulus also influenced the behavior of the embankment fill and foundationsoil but to a lesser extent when compared to crush strength. The use ofreinforcement for widening and grade raising of existing embankments appeared tobe beneficial in reducing lateral movement.

17. Key Words

Embankments, reinforcement, geotextilesfinite element method, soft foundations,settlements, cap plasticity model

18. Distribution Statement

No restrictions. This document is

available to the public through theNational Technical Information Service,Springfield, Virginia 22161

19. Security Clossii. (of this report)

Unclassified

2C Security Classif. (of this page)

Unclassified

21. No. of Poges

356

22. Price

Form DOT F 1700.7 is-ssi

Purdue University

WSchool of Civil Engineering

DRAFT FINAL REPORT

TO: Vincent P. Drnevich, Director

Joint Highway Research Project

FROM: W. F. Chen, Research Engineer


July 23, 1992

Project No: C-36-36U

File: 6-14-21

Attached is the draft final Report on the HPR Part II study entitled, "Embankment

Widening and Grade Raising on Soft Foundation Soils". The authors of the report are Scott J.

Ludlow, Wai-Fah Chen, Philippe L. Bourdeau and C. William Lovell.

Copies of this report will be submitted to the IDOH and FHWA for their review. My co-

authors and I look forward to receiving their comments on the report.

Respectfully submitted,

/)

W.F. Chen

Research Engineer

WFC/cak

Attachment

cc: A.G. Altschaeffl

D. Andrewski

P.L. Bourdeau

M.D. BowmanM.J. Cassidy

L.M. Chang

S. DiamondJ.J. Dillon

W.L. Dolch

V.P. Drnevich

A.R. Fendrick

J.D. Flicker

D.W. Halpin

K.R. Hoover

R.H. Lee

C.W. Lovell

R.H. LowryD.W. Lucas

B.G. McCullouch

B.K. Partridge

J.A. Ramirez

G.J. Rorbakken

C.F. Scholer

G.B. Shoener

K.C. Sinha

D.L. Tolbert

C.A. Venable

T.D. White

L.E. WoodJ.R. Wright

R.F. Wukasch

Civil Engineering Building . West Lafayette, IN 47907-1284 . FAX (317)49&-1 105

TABLE OF CONTENTS

Page No.

LIST OP TABLES i

LIST OP FIGURES ii

ABSTRACT iii

CHAPTER 1:INTRODUCTION 1

1 .

1

Motivation 1

1 .

2

General Background 3

1 .

3

Objective and Scope of Work 6

1 .

4

Report Organization 6

CHAPTER 2:A PROCEDURE FOR ESTIMATING CAP MODEL PARAMETERS FROMCONVENTIONAL FIELD AND LABORATORY TEST RESULTS 8

2 .

1

Introduction 8

2.2 Description of the Cap Model 102.3 A Procedure for Estimating Cap Model

Parameters for NFAP 142.3.1 Ultimate Failure Surface (3) 142.3.2 Elastic and Plastic Behavior (4) 222.3.3 Cap Parameters (2) 262.3.4 Initial Stress State (2) 322.3.5 Pore Pressure Response Factor (1) .... 3 2

2.4 Discussion of Sensitivity of CapModel Parameters 352.4.1 Poisson's Ratio (v) 372.4.2 Compression Index (Cc ) 382.4.3 Recompression Index (Cr ) 382.4.4 Pore Pressure Response Factor (P) .... 392.4.5 Angle of Internal Friction (0) 392.4.6 Undrained Shear Strength Ratio 4

2.4.7 Over-consolidation Ratio (OCR) 42 .

5

Limitations of the Cap Model 402 .

6

Summary 4 2

CHAPTER 3:APPLICATION OF THE FINITE ELEMENT METHODTO NFAP 44

3 .

1

Introduction 443 .

2

Modeling Foundation Soils 453 .

3

Modeling Embankment Fill 463 .

4

Modeling Reinforcement 483 .

5

Incremental Construction 513 .

6

Concluding Remarks 53

Page No.

CHAPTER 4:CASE STUDIES 56

4 .

1

Introduction 564 .

2

Example 1 564 . 3 Example 2 . 62

CHAPTER 5:

DESIGN GUIDELINES 75

5 .

1

Introduction 7 5

5.2 Design Analysis . . . 7 6

5.2.1 Deformation Behavior of Reinforced andUnreinforced Embankments 7 6

5.2.2 Effect of Crust Strength 775.2.3 Effect of Reinforcement on Stability . 795.2.4 Effect of Foundation Compressibility . 805 . 2 .

5

Effect of Embankment Width 815.2.6 Effect of Embankment Widening and

Grade Raising 815.2.7 Summary 83

5.3 Design Analysis Limitations of NFAP program . 855 .

4

Design Recommendations 90

CHAPTER 6:SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 91

6 .

1

Summary and Conclusions 916.1.1 General 916.1.2 Cap Plasticity Model 916.1.3 Case Studies 93

6 .

2

General Recommendations 95

APPENDICES:

Appendix A: List of References 97Appendix B: Interim Report: "A Sensitivity

Analysis of the Parameters fora Cap Plasticity Model," Report No.FHWA/JHRP/IN - 92-1 . 23 pp.

Appendix C: Interim Report: "Embankment Wideningand Grade Raising on Soft FoundationSoils: Example 1 - Indiana StateRoute 55 over Turkey Creek in LakeCounty, Indiana," Report No. FHWA/IN/JHRP - 91-18 . 56 pp.

Appendix D: Interim Report: "Embankment Wideningand Grade Raising on Soft FoundationSoils: Example 2 - Indiana StateRoute 1 over Ramsey Creek in FranklinCounty, Indiana," Report No. FHWA/IN/JHRP - 92-9 . 173 pp.

LIST OF TABLES

Table Page No,

2 • 1 Summary of Cap Model Parameters 15

2.2 Constants for Determiningthe Cap Model Parameters . 16

2.3 Summary of Sensitivity Analysis of the CapPlasticity Model Parameters . 3 6

4.1 General Soil Profile for State Route 55 58

4 .

2

Example 1 - Summary of Cases for Indiana StateRoute 55 over Turkey Creek/Lake County, Indiana . . 63

4.3 General Soil Profile for State Route 1 . 67

4 .

4

Example 2 - Summary of Cases for Indiana StateRoute 1 over Ramsey Creek/ FranklinCounty , Indiana 7

5.

1

Relative Influence of Reinforcement 77

5.2 Relative Influence of Crust Strength 78

5.3 Relative Influence of Foundation Compressibilityon Crust Strength 81

6 . 1 Summary of Procedures for Determining Cap Parametersfrom Conventional Laboratory and Field Tests ..... 92

11

LIST OF FIGURES

Figure Page No.

1 . 1 Flow Chart of NFAP ....... 5

2 .

1

Cap Plasticity Model ............................. 9

2.2 Response due to an increment of loading 13

2.3 Drucker-Prager and Mohr-Coulomb failure 18

2.4 Non-circular and Mohr-Coulomb failuresurfaces on deviatoric plane 21

2.5 Idealized response of soil to hydrostatic stress . 2 4

2.6 Cap model response for undrained shear 24

3 . 1 Typical finite element representation 47

4.1 Typical finite element representationfor State Route 55 59

4.2 Finite element representation for Case 5 60

4.3 Finite element representation for Cases 6 and 8 .. 61

4.4 Typical finite element representationfor State Route 1 68

5.1 Effective stress path predicted by cap modelfor samples that experience a reversal ofprincipal stresses 89

Ill

ABSTRACT

The finite element technique using a cap elastic-plastic work-hardening soil behavior model was applied to the analysis ofembankments constructed on soft foundation soils. A procedure wasprovided to estimate the cap model parameters from conventionalfield and laboratory test results. A sensitivity analysis of thecap model parameters comparing the observed and calculatedresponses was also provided. Results indicate that the undrainedshear strength and over-consolidation ratios were observed to havethe most-significant influence on the predicted model behavior.

The technique was then applied to the analysis of twoexamples. The examples were based on actual highway projects inIndiana where information on these projects was provided by INDOTpersonnel. Results of the analysis were used to determine theinfluence of several factors on reinforced and unreinforcedembankment behavior. The results indicated that the crust strengthand foundation compressibility had the most-significant influenceon embankment fill and foundation soil behavior and the potentialbenefit possible with reinforcement. Reinforcement type/modulusalso influenced the behavior of the embankment fill and foundationsoil but to a lesser extent when compared to crust strength. Theuse of reinforcement for widening and grade raising of existingembankments appeared to be beneficial in reducing lateral movement.

IMPLEMENTATION REPORT

PHASE II - EMBANKMENT WIDENING ANDGRADE RAISING OVER SOFT FOUNDATION SOILS

The Need

The draft final report for this project includes the study of

embankment widening and grade raising over soft foundation soils.

With the urgent need to upgrade our nation's highway system,

engineers face challenging problems of widening existing highway

embankments and raising new embankments, particularly over soft

foundation soils. For the analysis/design of these problems,

engineers often rely on a combination of limit equilibrium type

methods; simplified load/deformation responses; and experience.

However, as a result of the complex interaction of the embankment

and foundation soils, the above methods provide only approximate

information concerning the embankment and foundation deformations

and the serviceability of the embankment following construction.

To assess the performance of the embankment/ foundation system,

advanced modeling techniques are needed that account for nonlinear

and elastic-plastic material behavior. A technique that has gained

acceptance for the analysis of such problems is the finite element

method (FEM)

.

The FEM

Use of finite element techniques have the ability to allow us

to:

1. model complex geometry and deformation patterns;

2. simulate nonlinear material behavior, including yielding,using advanced constitutive models;

3. analyze the interaction of different materials; and

2

4. combine geometrical nonlinearity with materialnonlinearity.

Thus, the FEM can help engineers to:

1. improve their understanding of observed behavior in thefield;

2. model the response of an embankment/ foundation underincreasing load level;

3. analyze the effects of changes in the components of thesystem (i.e. the properties of the embankment andfoundation soil) ; and

4

.

assess the impact of construction procedures and theresponse of the system.

The finite element technique is well recognized as being a

powerful tool for design and analysis of embankment/ foundation

systems, and numerous examples can be found in the literature.

However, the use of finite element analysis for performing studies

which could answer the questions raised above is subject to some

important limitations.

If one is to model the response of the embankment in the range

of large deformations, then it is essential to employ a finite

element formulation and constitutive model which:

1. models the stress-dependent properties of the embankmentfill and foundation soil(s);

2. correctly models plastic yielding and flow in the filland foundation soil; and

3. allows for potential slip at the interface of soil andreinforcement in cases where geosythetics are used toimprove the stability of the embankment.

The Problem

It is unfortunate that such methodologies that have been

proposed in the geotechnical literature are still of very sparse

3

use in the practicing community. A major hurdle for the application

of the FEM to the analysis of such previously-mentioned problems is

the lack of practical guidelines and step-by-step procedures to

determine the material parameters required by the model and

interpret the results in a design context. The ultimate goal of the

present study is to bridge this gap. This study outlines a set of

procedures to enable engineers to make use of available technology,

and provides guidelines for the design and analysis of reinforced

and unreinforced embankments over soft foundation soils.

The Objective

The major objective of. this study (Phase II) has been to

outline a set of procedures to enable engineers to make use of the

available finite element methodology, and provide guidelines for

the design and analysis of reinforced and unreinforced embankments,

particularly over soft foundation soils. In order to achieve this

objective, it is necessary to provide:

1. a simple and reliable procedure to derive the capplasticity model parameters from data obtained byconventional field and laboratory soil tests;

2. background on the application/modeling techniques of theFEM to NFAP program;

3. complete case example studies in order to illustrate thecapabilities of the method for the analysis and design ofembankment widening and grade raising over soft founda-tion soils; and

4. practical guidelines for the analysis of embankmentwidening and grade raising over soft foundation soilsusing the FEM methodology.

4

Results and Guidelines for Practical Implementation

• Practical Procedure for Cap Parameter Determination

A straight forward procedure to determine the cap parameters

for normally consolidated soils from conventional laboratory and

field tests and a sensitivity study which examines the effect of

the input soil parameters was provided. The main input soil

properties are the compressibilities ( Cc , Cz ) , the effective

Mohr-Coulomb shear strength parameters ( <j>, c ) / the undrained

S„ ...shear strength ratio ( USR; «=-

) and the over-consolidation ratio

( OCR ) . Solutions are given in graphical form and equations

suitable for hand calculation.

The procedure was used to determine the cap parameters for an

impact-compacted lacustrine clay using results from isotropically-

consolidated undrained-compression (CIUC) tests. These parameters

were then used in a computer program called CAP to calculate

stress-strain curves, pore pressure response and effective stress

paths. Comparisons were made to observed test results. In

general, there was good agreement except for a discrepancy for pore

pressures and effective stress paths.

• Sensitivity Study of Cap Model Parameters

In addition, a sensitivity study was made of the effect of the

input soil properties on calculated CIUC triaxial sample behavior.

The results show which parameters have the greatest effect on

computed behavior and provide guidance in the selection and

adjustment of input soil properties to obtain a better fit between

the calculated and observed response. The USR and OCR were

5

observed to have the most-significant influence on the predicted

behavior.

• Practical Case Study

A comparative finite element study of two practical examples

involving a total of 30 cases was made using an incremental

procedure which simulated embankment construction. Embankment fill

and foundation soil behavior was represented with an isotropic,

strain-hardening cap-plasticity model.

The results indicated that the properties of a stiff or of a

tensile resistant reinforcement had the most significant influence

on embankment fill and foundation soil behavior. Reinforcement

type/modulus also influenced the behavior of the embankment fill

and foundation soil but to a lesser extent when compared to crust

strength

.

For compressible foundation soils and a relatively stiff

embankment, a high-modulus woven geotextile was very effective in

reducing displacements within the embankment fill and foundation

soils and increasing the stability near the embankment toe. For a

given embankment stiffness, the foundation compressibility directly

influences the effectiveness of the crust strength. The foundation

compressibility had only a limited influence on embankment behavior

and the benefit from using reinforcement was modest except for that

noted previously. The effect of increasing the embankment width by

the use of a stabilizing berm only slightly reduced the displace-

ments within the embankment fill and foundation soil and modestly

increased the stability at the toe.

6

The widening and grade raising of an embankment locally

altered the magnitude of the state of stress and stability but did

not alter the overall pattern. Displacements developed primarily

near the former shoulder and the widened toe (horizontally) and

along the sideslope of the former embankment (vertically)

.

In several cases, the local factors of safety within the

foundation soil near the centerline of the embankment and continu-

ing outwardly to the toe and slightly beyond the toe were yielding

(strain hardening) . The other portions of the embankment fill and

foundation soils were either experiencing stress states on the cap

(elastic-plastic), or within the cap (elastic state)

.

Limitations

• On Strain-Softening Soil

The cap model has some limitations which impact the calculated

response. One limitation which was observed during this study was

the inability of the model to predict the reduction in undrained

strength or increase in pore pressure that occurs after the peak

strength is reached in strain softening soil. Consequently, this

would overestimate the shear strength and underestimate the pore

pressures and deformations.

• On Over-consolidated Soils

A second limitation is the ability to model the behavior of

over-consolidated soils. The cap model correctly predicts the

undrained strength at large strains. However, pore pressures are

not predicted correctly and plastic strains which many over-

7

consolidated soils experience for stress changes in the elastic

region bounded by the cap and ultimate failure surfaces are not

accounted for. Also, strain softening after failure can not be

modeled. It is speculated that this would lead to underestimation

of deformations.

• On Trial-and-error Procedure

Another limitation which can be attributed to the model's

formulation is the need to perform a trial-and-error calculation to

estimate the best-fit of the observed response. Although time-

consuming, a reasonable fit of the observed response can be

obtained rather quickly by making a critical review of the OCR and

USR parameters and then modifying one of the other five (from the

previous sensitivity study) parameters, if necessary.

* On Long-term Behavior of Soils

Another limitation which requires considerable attention and

can also be attributed to the model's formulation is the model's

inability to account for long-term behavior (e.g. creep and

consolidation) . Presently, the model is best-suited for depicting

short-term behavior only. To account for creep and consolidation,

the model would require extensive modifications.

Further Research Needs

The work presented in the study aims to facilitate the

application of the cap plasticity model to practical problems and

in design of embankments constructed on soft foundation soils.

However, there are some aspects of unreinforced and reinforced

embankment analysis that require further development of the

8

software program for this study. The ability of the cap model to

accurately predict soil behavior could be extended to a wider range

of soils, stress paths and drainage conditions. The following are

additional research needs:

1. A parametric study of the effect of reinforcement shouldbe made using NFAP for a wider range of embankmentgeometries and soil properties. It should includenarrower embankments, different crust thicknesses anddifferent reinforcement moduli. The tensile forces inthe reinforcement should be analyzed in this study.

2. The cap plasticity model should be extended to improveits ability to model soil behavior during rotation ofprincipal stresses and the behavior of over-consolidatedsoils.

3. Compacted cohesive soils are more often used as fillmaterial for embankments. The behavior of embankmentswith cohesive fill (which include the effects of compac-tive prestress) should be evaluated with the cap plastic-ity model.

4. Presently, the cap plasticity model is better at accomo-dating either drained or undrained conditions. The modelshould be extended to partially drained conditions andthen evaluated.

5. Slippage often occurs at or near the interface of theembankment fill/geosynthetic/foundation soil interface.NFAP should be modified to include slippage consider-ations.

6. To provide beneficial use to practicing engineers, NFAPshould be modified to include pre- and post-processinggraphics capabilities.

7. Post construction effects need to be considered, particu-larly since the original embankment and widening/raisinghave different chronologies. Foundation creep andembankment settling upon wetting in service are examples.

CHAPTER 1

INTRODUCTION

1.1 Motivation

With the urgent need to upgrade our nation's highway system,

engineers face challenging problems of widening existing highway

embankments and raising new embankments, particularly over soft

foundation soils. For the analysis/design of these problems,

engineers often rely on a combination of limiting equilibrium type

methods; simplified load/deformation responses; and experience.

However, as a result of the complex interaction of the embankment

and foundation soils, the above methods provide only approximate

information concerning the embankment and foundation deformations

and the serviceability of the embankment following construction.

To assess the performance of the embankment/ foundation system,

advanced modeling techniques are needed that account for nonlinear

and elastic-plastic material behavior. A technique that has gained

acceptance for the analysis of such problems is the finite element

method (FEM)

.

Use of finite element techniques have the ability to allow us

to:

1. model complex geometry and deformation patterns;

2. simulate nonlinear material behavior, including yielding,using advanced constitutive models; and

3. analyze the interaction of different materials.

Thus, the FEM can help engineers to:

1. improve their understanding of observed behavior in thefield;

2. model the response of an embankment/ foundation underincreasing load level;

3. analyze the effects of changes in the components of thesystem (i.e. the properties of the embankment andfoundation soil) ; and

4

.

assess the impact of construction procedures and theresponse of the system.

The finite element technique is well recognized as being a

powerful tool for design and analysis of embankment/ foundation

systems, and numerous examples can be found in the literature (e.g.

Boutrup and Holtz, 1983; Hird and Jewell, 1990; Kwok, 1987;

Leroueil et al., 1978; Rowe, 1982). However, the use of finite

element analysis for performing studies which could answer the

questions raised above is subject to some important limitations.

If one is to model the response of the embankment in the range

of large deformations, then it is essential to employ a finite

element formulation and constitutive model which:

1. models the stress-dependent properties of the embankmentfill and foundation soil(s);

2. correctly models plastic yielding and flow in the filland foundation soil; and

3. allows for potential slip at the interface of soil andreinforcement in cases where geosythetics are used toimprove the stability of the embankment.

It is unfortunate that such methodologies that have been

proposed in the geotechnical literature are still of very sparse

use in the practicing community. A major hurdle for the applica-

tion of the FEM to the analysis of such previously-mentioned

problems is the lack of practical guidelines and step-by-step

procedures to determine the material parameters required by the

3

model and interpret the results in a design context. The ultimate

goal of the present study is to bridge this gap. This study

outlines a set of procedures to enable engineers to make use of

available technology, and provides guidelines for the design and

analysis of reinforced and unreinforced embankments over soft

foundation soils.

1.2 General Background

Prior to widening or raising an embankment, it is common

practice to partially or totally undercut and replace soft or

otherwise unstable foundation soils with compacted "special" fill,

which is expensive (not only the fill but also the construction

techniques) . Alternatively, the foundation soils can be treated in

such a way as to accommodate the proposed loads. Numerous

techniques which are within the category of ground modification are

available, e.g. dewatering, grouting, preloading, compacting, etc.

However, these techniques are site specific, often costly and

generally time consuming. In this study, the focus with respect to

soil improvement is on the use of geosynthetic reinforcement

(specifically geotextile or geogrid) placed directly at the

interface of the soft ground and the new or added embankment. The

result of the embankment loading is to induce excess pore water

pressure in the underlying foundation soils which is subsequently

dissipated when consolidation occurs. During the critical stage

for the stability (short term) , the function of the geosynthetic is

of reinforcing and providing support (horizontally) to the

embankment. However, the situation may change somewhat with time

4

as the foundation soil gains strength through consolidation and

begins to support part, or all, of the embankment loading.

The software used in this study is based on a general purpose

finite element program named NFAP (nonlinear finite analysis

program) developed by Chang (1980) . NFAP was originally developed

for the use of nonlinear large deformation analysis of structures.

However, the program was later expanded and refined by Mizuno

(1981) and McCarron and Chen (1985) to include a cap plasticity

model to simulate the behavior of soils. Then, Humphrey (198 6)

condensed McCarron' s version of the program for use on personal

computers

.

During Phase I of this research study, Huang (1991) modified

and extended NFAP to include: double precision/ floating point

variables; foundations with sloping boundaries and a variable

groundwater level; and a procedure for determining the ini-

tial/existing stress state within an embankment. A flow chart for

the NFAP computer program is shown in Figure 1.1.

As previously stated, NFAP employs a cap plasticity model to

simulate the behavior of the soil. From a theoretical point of

view, the cap model employed herein is particularly appropriate in

modeling soil behavior, because it is capable of treating the

conditions of stress history, stress path dependency, dilatancy,

and the effect of the intermediate principal stress. At present,

however, many practicing engineers are not familiar with the

process of determining the material parameters of the cap model.

Often, existing procedures from well-known texts or more recent

f~ StartJ

Control data

Yes

No

Node and elementinformation

Node information

Construct initial

stress field

Element information

Initialize elementparameters

Assemble linear

stiffness

Calculate load vectorin all time steps

Start incrementload step

Calculate nodal forces

of linear-elastic elements

Assemble nonlinear

stiffness

Yes

No

Calculate fill

load vector

Equilibrium

iteration process

No

Calculate incremental

displacement

Print total displacement

and stress at each load step

Save information of the

last load step for restart

Stop

Figure 1.1 Flow Chart of NFAP

6

literature, which are mostly developed by mechanicians from the

engineering-mechanics viewpoint, are employed rather than using

data obtained from commonly performed soil tests.

1.3 Objective and Scope of Work

The major objective of this study (Phase II) has been to

outline a set of procedures to enable engineers to make use of the

available finite element methodology of NFAP, and provide guide-

lines for the design and analysis of reinforced and unreinforced

embankments, particularly over soft foundation soils. In order to

achieve this objective, it is necessary to provide:

1. a simple and reliable procedure to derive the capplasticity model parameters from data obtained byconventional field and laboratory soil tests;

2. background on the application/modeling techniques of theFEM to the NFAP program;

3. complete case example studies in order to illustrate thecapabilities of the method for the analysis and design ofembankment widening and grade raising over soft founda-tion soils; and

4

.

practical guidelines for the analysis of embankmentwidening and grade raising over soft foundation soilsusing the FEM methodology.

1.4 Report Organization

No theoretical or technical details are included beyond what

is absolutely necessary for following the report and making use of

the procedures and recommendations contained herein. Those

interested in acquiring a more thorough knowledge of the applica-

tions of the cap plasticity model and finite element formulation

and related work on soil reinforcement of soft embankment founda-

tions are referred to the references listed in Appendix A.

7

Chapter 2 discusses a proposed procedure for estimating cap

plasticity model parameters from conventional field and laboratory

test results. Often during a design process, many geotechnical

engineers are unable to perform a sufficient site investigation to

accurately estimate the soil parameters because of the project's

budgetary restraints. Consequently, the soil parameters are

obtained from empirical correlations using conventional field and

laboratory test results and/or engineering judgement. This chapter

employs these correlations and judgement to estimating cap

plasticity model parameters. In addition, a sensitivity study of

the cap parameters is performed.

Chapter 3 concentrates on the application and modeling

techniques of the FEM to the NFAP program.

Chapter 4 contains case studies of application of the

procedures outlined in Chapters 2 and 3. The cases in Chapter 4

are based on actual highway projects in Indiana where information

on these projects was provided by Indiana Department of Transporta-

tion (INDOT) personnel.

Chapter 5 presents the developed guidelines for the analysis

of embankment widening and grade raising over soft foundation soils

by using the NFAP methodology, based on the results obtained from

Chapter 4

.

Chapter 6 contains conclusions and recommendations, including

further work involving soil reinforcement, and expanding the

capabilities of the analytical model to better accommodate the

deformations of embankment/ foundations in service.

CHAPTER 2

A PROCEDURE FOR ESTIMATING CAP PLASTICITY MODEL PARAMETERS FROMCONVENTIONAL FIELD AND LABORATORY TEST RESULTS

2.1 Introduction

The deformational behavior of soils is primarily influenced by

stress-strain history, soil type and direction, rate, and magnitude

of loading. To successfully apply a soil model, it is essential

that the model expresses the significant characteristics of the

soil response for a particular state. One model that has received

acceptance in the area of finite element analysis of geotechnical

engineering problems is the cap plasticity (cap) model (e.g. Nelson

and Baladi 1977; Baladi and Rohani 1979; Chen and McCarron 1983;

Mizuno and Chen 1984; Daddazio et.al. 1987; McCarron and Chen

1987) .

In general, cap models describe the failure and yielding

behavior of soil with an ultimate yield surface that is fitted with

a movable end cap (see Figure 2.1). Both the ultimate or failure

and cap yield surfaces are symmetrical about the hydrostatic axis.

A "hardening rule" specifies the movement of the cap as a result

of the hardening or softening behavior of the soil when loaded.

Other cap models also permit the movement of the failure surface in

addition to the cap. Movement in these types of models are also

expressed by a hardening rule. It should also be noted that

strains are elastic for stress changes that lie within the region

of the failure and cap surfaces but are both elastic and plastic

for stress changes on the surfaces. The cap model utilized in this

study is an isotropic strain-hardening plasticity model with an

J1/2

2

Ultimate Yield

Surface \ ^s^" 1

fa

HorizontalTangent

(X-L)IR.^. Cap Yie

V"^ Surfac

K

x-z.

(a) 7,-jf Plane

9=j2xJ,

(b) Deviatoric Plane

Figure 2.1 Cap Plasticity Model

10

elliptical cap. This model is briefly described in the following

sections. Readers interested in acquiring more knowledge about cap

models are referred to a recent book by Chen and Mizuno (1990)

.

In this chapter, the reader is first given a brief description

of the cap model including a discussion of the yield functions

(ultimate and cap yield surfaces) and the hardening rule. Then, a

procedure for estimating the cap model parameters from conventional

field and laboratory test results is given. Finally, the results

of a sensitivity study are given which examine the effect of the

input soil properties on the calculated response.

2.2 Description of the Cap Model

The primary features of the cap model include the cone-shaped

ultimate/ failure surface and an elliptical-shaped cap. Figure 2.1

illustrates the model in an 1r-j\ lz space where ~ix

is the first

invariant of the effective stress tensor and J2 is the second

invariant of the deviatoric stress tensor. The invariants are

defined in terms of the principal stresses as:

Jx

= ax

+ a2

+ o3

(2.1)

J2 = \ t(°i ~ o2 )

2 + (o2

- o3 )

2 + (a3

- o x )2

] (2.2)

where a1,a

2 ,a 3 are the major, intermediate and minor effective

principal stresses, respectively.

The equation for the ultimate failure surface, ff , is based

on a generalization of the Mohr-Coulomb hypothesis (Drucker and

Prager, 1952) which is a smooth surface of the simple form:

11

ff = alx- J\n * K = (2.3)

where a and k are material parameters that are related to the

angle of internal friction and cohesion of the soil, respectively.

It should also be noted that the sign convention used herein is

positive for compressive stresses and strains, and therefore

consistent with soil mechanics sign convention.

The equation used to describe the shape of the cap is based on

a quarter of an ellipse:

fc = (Ja - L(l)) 2 + R2J2- (X - L(l)) 2 = (2.4)

where R is defined as the (aspect) ratio of the major and minor

radii in Figure 2.1. In addition, x and L(l) are defined as the

hardening rule and failure function, respectively. It should also

be recognized that x and L(l) define the l1-value at the inter-

sections of the elliptical cap with the Ix-axis and the failure

function, respectively. The location, x, of the cap is related to

the plastic volumetric strain, e£ / and this relation is assumed as

(DiMaggio and Sandler, 1971)

:

x = -lln(l - i?) (2.5)D W

where D is a curve fitting parameter with units of (stress)" 1 and W

is the limiting value of e£ at high stress.

Elastic volumetric and shear (distortion) strains are governed

by the bulk, K , and shear, G , moduli, respectively. Elastic

volumetric strain, e£ , is produced by changes in Jx

.

12

Jl (2.6)€v =(3J0

The elastic shear strain is given by:

e- = iJ (2.7)^ (2G)

where ei7- is the deviatoric strain tensor and 8^ is the deviatoric

stress tensor. To reiterate, stress changes in the region bounded

by the ultimate and cap yield surfaces produce only elastic

strains. However, loading on the ultimate yield surface or beyond

the initial location of the cap results in both elastic and plastic

strains.

In order to compute plastic strains for loading on the

ultimate and cap yield surfaces, an associated flow rule is

assumed. The flow rule defines the relationship between the next

increment of the plastic strain increment, de?j , and the present

state of stress, o_y , for an element of soil that is either on the

ultimate or cap yield surfaces and when subjected to further

loading. As indicated by Figure 2.2a, an increment of loading on

the cap causes it to expand and results in positive plastic

volumetric (contraction) strain and plastic shear strain. However,

loading on the ultimate yield surface produces negative plastic

volumetric (dilation) strain and plastic shear strain. As a result

of the dilation, the cap contracts (as shown in Figure 2.2b) by an

amount that is proportional to the hardening rule (Eguation 2.5).

For a stress state at the intersection between the cap and the

ultimate yield surface, the increment of plastic strain is taken to

(a)13

J?y(K ^^- (b)

— *?,

Initial Cap Position

~~""">

,

New Cap PositionN X\ \\ \\ \

\ ' 7

Figure 2.2 Response due to an increment of loading

(a) loading on cap yield surface

(b) Loading on ultimate yield surface

(c) loading at corner

14

be normal to the cap. Therefore, plastic shear strain occurs at a

constant volumetric strain (Figure 2.2c). In addition, this is

consistent with the observed behavior of soils at large strain.

2.3 A Procedure for Estimating Cap Model Parameters for NFAP

There are 12 cap model parameters and they are generally-

grouped into five categories. These categories include: 1) an

ultimate yield surface (3 parameters) ; 2) elastic and plastic

behavior (4); 3) cap surface (2); 4) initial stress state (2); and

5) pore pressure response factor (1) . Table 2.1 provides a summary

of the parameters, while the required constants for NFAP for

determining the model parameters are indicated on Table 2.2. The

difference between the required constants for NFAP and the

conventional parameters is in the descriptions of the elastic

behavior and hardening rule.

Parameters that are required to perform the analysis can be

obtained from soil properties of commonly performed laboratory and

field tests [e.g. consolidation (oedometer) test; unconsolidated

and consolidated undrained triaxial tests; standard penetration

test; field vane shear; and empirical correlations from index

property tests] . In some cases, it may be necessary to estimate a

few of these properties. The following procedures are given for

the use of estimating cap model parameters for NFAP. Chapter 4

will apply the procedures contained herein to estimate cap model

parameters from conventional field and laboratory test results.

2.3.1 Ultimate Failure Surface ( a,K,Tc )

As stated previously, the equation for the ultimate yield

15

Table 2 . 1 Summary of Cap Model Parameters

Ultimate yield surface

a slope of ultimate yield surface in ~lx-j\ 12 space

(degrees)

k slope intercept along j\12 axis in Y

x-j\ /2 space

(force per unit area)

Tc tension cut-off/crack potential (force per unitarea - also related to the minimum principal ten-sile strength of the soil)

Elastic and Plastic Behavior

Kmin minimum value of the elastic bulk modulus (forceper unit area)

CAr= r r total bulk modulus parameter

c 2.303 (l + e„)

A= r r elastic bulk modulus parametere 2.303 (l+e )

where ea is the initial void ratio and Cc and Cr arethe compression and recompression indices, respec-tively

v Poisson's ratio

Cap Surface

R cap aspect ratio (ratio of the major and minorradii in Figure 2.1)

OCR overconsolidation ratio

Initial Stress State

Y saturated or moist unit weight of the soil (forceper unit volume)

KQ initial coefficient of lateral earth pressure

Pore Pressure Response Factor

P a ratio relating the apparent bulk modulus of thefluids to the total stiffness (which includes boththe soil and fluid)

16

Table 2.2 Constants for Determining the Cap Model Parameters

Kîn minimum value of the elastic bulk modulus(force per unit area)

v Poisson's ratio

ea initial void ratio

<(> angle of internal friction (degrees)

c cohesion (force per unit area)

Cc compression index

Cr recompression index

S„undramed shear strength to effective over-burden pressureo.

OCR overconsolidation ratio

Tc tension cut-off (force per unit area)

Y saturated or moist unit weight of the soil(force per unit volume)

KQ coefficient of initial lateral earth pressure

P pore water pressure response factor

17

surface , ff , is based on a generalization of the Mohr-Coulomb

hypothesis (Drucker and Prager, 1952) which is:

ft = alx

- J\n * k = (2.8)

where a and k are material parameters that are related to the

angle of internal friction and cohesion of the soil, respectively.

However, before estimating the parameters, it is important to

recognize that there are several ways to approximate the Mohr-

Coulomb hexagonal failure surface by the Drucker-Prager circular

section in three-dimensional stress space as indicated in Figure

2.3. For geotechnical engineering applications, soil will either

experience some compression or extension depending primarily upon

the stress-strain history, soil type and direction of loading.

To simulate triaxial compression behavior of the soil (where

°2 =a 3 ) ' tne two sets of material constants ( a, k and c, <t> ) are

related by (Chen and Saleeb, 1982)

:

„ _ 2sin4> ._ _.a - —— (2.9)y/3(3 - Sin<j>)

6ccosd>K = —

—

T (2.10)v/3(3 - sin<|>)

To simulate triaxial extension behavior of the soil, the relations

are given by:

a - - 25in<t>(2.11)

^(3 + sin(J>)

K = ^ccosi_v/3(3 + sin<l>)

For plane strain conditions (e.g. a relatively long embankment)

,

18

Mohr-Coulomb

Drucker-Prager (matching

at 8 = 60°; Tensile Meridian)

Drucker-Prager

(matching at 8 = 0°;

Compressive Meridian)

Figure 2.3 Drucker-Prager and Mohr-Coulomb failure

criterion matched on compression and extension

meridians (after Chen and Saleeb, 1982)

19

the material constants are related by (Drucker and Prager, 1952)

:

_ tancfra 7 (2.13)

(9 + 12tan2<|>)

2

ic- 2£i (2.14)

(9 + I2tan2<|>)

2

However, for stress states within the soil mass other than triaxial

compression or extension, the match between the Mohr-Coulomb and

the Drucker-Prager criteria depends upon the Lode angle, , or

the intermediate principal stress at failure. Therefore, the

material constants are given by:

— sin<J>

a = — (2.15)[sin(8 + — )

- ^cos(6 + -£)sin<b]3 y/3

3

K = CCOS({>

;sin(6 + -J)" — (cos(9 + -£)sind>] (2.16)

3 ^3 3

where

9 = Acos-M^, „.„,2J2

2

J2 and J

3 in Equation 2.17 are the second and third invariants of

the deviatoric stress tensor, respectively.

When the stress state within the soil mass reaches the

intersection of the ultimate and cap yield surfaces, only shear

(distortion) strains ( y ) occur without plastic volume change

20

(i.e. de%.=0 )• As a result, the direction of the incremental

plastic strain must be normal to the elliptical cap rather than the

ultimate surface in order to satisfy normality conditions. For the

plane strain case with shear distortion allowed only at failure,

the material constants are derived in a similar manner by Drucker

and Prager (1952) and expressed as:

a = -sin<f> (2.18)

K = CC0S<(> (2 . 19)

Often, soils will exhibit a greater angle of internal friction

under plane strain conditions than those obtained from triaxial

test conditions (Lee, 1970) . This indicates that the failure

surface, which connects the triaxial compression and extension

zones on the deviatoric plane for the Mohr-Coulomb failure

criterion (see Figure 2.4), underestimates the actual behavior of

the soil. In order to represent the actual behavior of the soil

mass, Dafalias and Herrmann (1986) modified the material constants

with a non-circular cross section of the failure surface on the

deviatoric plane such that the parameters a and k depend on the

Lode angle, , where:

a = a c g(B,m) (2.20)

k = K. g(Q,m) (2.21)

and

21

Non-circular

Mohr-Coulomb

Figure 2.4 Non-circular and Mohr-Coulomb failure

surfaces on deviatoric plane

22

a c k_ 3 + sm<p

in which

2mg(Q,m)

1 + m - (1 - /n)sin(^ + 36) (2.23)

where the subscripts t and c denote the values with respect to

triaxial extension ( 8=60° ) and compression ( 0=0° ) • For a

condition of pure shear ( 0=30°)/ Equation 2.23 reduces to:

g(6,m) = -20- (2.24)1 + 777

In addition to the compressive stresses within the soil mass,

tensile stresses may occur. In NFAP, the tension potential is

evaluated using the minimum principal stress in the "in-plane"

stress state for the two-dimensional plane strain case. Since most

soils are not capable of supporting significant tensile stresses,

a small value of tension cut-off is generally assumed.

2.3.2 Elastic and Plastic Behavior ( Kmln ,A t,Ae , v)

Prior to discussing the elastic and plastic behavior of soils

within the cap model framework, it is important to realize that two

assumptions are incorporated herein. These include: (1) the one-

dimensional compression and rebound curves are parallel and linear

to the respective curves for hydrostatic loading; and (2) the ratio

of Cc and Cr to the specific volume, V , ( l +e ) is constant for

all values of p , the (mean) effective hydrostatic stress.

Based on the above assumptions, an idealized response of the

23

one-dimensional consolidation behavior (for simulating isotropic

consolidation conditions) is shown in Figure 2.5. The equation for

the virgin isotropic consolidation line is:

Cc = - . /1

de^ (2.25)c d(logp)

where e is the void ratio and Cc is the compression index. Recall

that p is the effective hydrostatic stress where:

p = ii (2.26)3

Based on the assumption that one-dimensional consolidation behavior

represents isotropic consolidation conditions, the total volumetric

strain, de£ , is then a function of the change in the void ratio

where

:

Cfe$ = " -d+

S

e(2.27)

Substituting Equation 2.27 into Equation 2.25 and rearranging

results:

C dec - ^ (2.28)

or

1 + e d(logp)

v _ dp _ p

where Kt is the total bulk modulus and is a function of the

effective hydrostatic stress, p , and Ac is a constant which has

the form:

24

Virgin isotropic

consolidation

Unloading

-reloading

Lift)

Figure 2.5 Idealized response of soil to hydrostatic stress

Stress Path A ('v^,)

Figure 2.6 Cap model response for undrained shear

25

A. = - (2.30)c 2.303(1 + eo )

The elastic bulk modulus, Ke , can be derived in the same

manner and expressed as:

with

A = - (2.32)2.303(1 + ej

where Cr is the recompression index and e* is the elastic or

recoverable volumetric strain. The total volumetric strain

consists of elastic/recoverable strains and plastic/ irrecoverable

strains.

e£ = < + e£ (2.33)

Based on Equations 2.29 and 2.31, it is apparent that both

bulk moduli, Kt and Kg , are proportional to the effective hydro-

static stress. Therefore, Km±n , which is the minimum value of the

elastic bulk modulus for a specific effective hydrostatic stress or

possibly a free stress state is required and can be estimated from

elastic moduli relations.

As mentioned earlier, the other elastic constant that is

required is the shear modulus, G , which can be evaluated from

Poisson's ratio, v , which must also be determined or approximated:

26

C - 3*» (1 " 2V)(2.34)

2(1 + v)

2.3.3 Cap Parameters ( i? , OCR )

The aspect ratio, R , is generally evaluated from shear

tests on normally consolidated soil where the loading path causes

the cap to expand. It is assumed that R remains constant as the

state of stress moves from the initial to the failure condition and

that R is independent of the initial hydrostatic stress state.

The initial state of stress ( j j\'o) for normally consoli-

dated soil is on the cap and the failure state of stress

( Tlf , j\f2) is at the intersection of the cap and ultimate failure

surface as shown in Figure 2.6 (page 24). The initial state of

stress is calculated from the effective vertical consolidation

stress ~o vo and the at-rest normally-consolidated coefficient of

lateral earth pressure KQ which is given by:

K = £fi£ (2.35)

where "aho is the horizontal effective stress. Equations 2.1 and

2.2 are used to calculate J1o and j\'J- for "ox= a vo and

Or, = (J, = Oho

I±a = m (l + 2Ka) (2.36)

j\o = -ô vo (l - KQ ) (2.37)

However, at failure, Equation 2.3 is used to relate the final

states of stress ( f' j\f2)

27

- _ (J2f k) (2.38)if ~

The undrained shear strength ratio (USR) is introduced as a

normalized measure of the shear stress at failure. At final state,

the USR can be expressed as:

DBB, 3L = <°i- q3> f( 2 .39)

where (c^ - o3 ) f is the principal stress difference at failure. At

failure, the second invariant of the deviatoric stress tensor J2f

is:

**t = 3^1 " °3>i (2.40)

or from Equation 2.39: .

J?f -^(2 t7Si?o vo )(2.41)

Substitution of Equation 2.41 into Equation 2.8 gives:

Ilf = - (-K + 2^2 USR) (2.42)

Using these relations and Equation 2.29, the elastic volumet-

ric strain change can be evaluated as:

deev = Ae Jlf ~_Jla

(2.43)(Ilf * Jla )/2

For undrained behavior, de£ = -del • From Equations 2.36, 2.42 and

2.43, cfe£ can also be written as:

28

(-=£- + 2-^5) - (1 + 2iC )a

d< = -2ê

^2 j£_ (2.44)(-JL + 2-^) + (1 + 2K )a

The hardening rule as described earlier by Equation 2.5 will

not be used herein. However, a modification as proposed by Huang

and Chen (1991) describing the evolution of the cap will be used.

Since the hardening rule describes the evolution of subsequent

loading or yielding surfaces, it will be necessary to determine the

initial yielding surface, and then any subsequent yielding surface

due to additional loading. From Equations 2.29 and 2.31, the

plastic volumetric strain change, cfe£ , can then be related to the

effective hydrostatic stress change, dp , as:

de£ = del ' <K = U, - Ae )M (2.45)c e p

and assuming K -consolidation in which:

X = (1 + 2K )p (2.46)

and

rearranging yields:

dX - (1 + 2K )dp (2.47)

d< = -(At- Ae)4r (2.48)

Then, the evolution of the cap, dX , can be approximated by

Equation 2.48 as:

29

dX = .

Xa- cfe£ (2.49)

where Xa is the initial cap position (see Figure 2.6) for a stress

state at point A which can also be evaluated for subsequent

loadings as follows.

The equations for the ultimate failure (Equation 2.3) and cap

(Equation 2.4) surfaces are used to determine La .

(1 - a 2R 2 )L 2

a- 2(Jla

+ a<R 2 )La+ (Jâ + R 2J2a - R 2k2

) = (2.50)

Then, the cap position Xa at A is:

Xa= La

+ (k + aLa )R (2.51)

and the position of the cap at state B , Xb is:

Xb » Xa+ dX (2.52)

where dX and Xa can be obtained respectively from Equations 2.49

and 2.51. The value of the cap aspect ratio is:

R =X„ - Iif

(2.53)

Jif

From the above equations, the cap aspect ratio can then be

evaluated through a trial-and-error procedure. However, if K

equals one, the stress state experiences a hydrostatic condition

and the cap aspect ratio can be evaluated after substitution:

30

2A{ m+ 2

USR _3)

= y/J/n _ J. + 3/3{

_« /3a ,

(2#54)2aC75i? a 2J75i? ,A - & )

(M + 2 USR + 3)

where -m = =— . When k = , Equation 2.54 simplifies to:

2Ae(2-^-3)

i? = -1 + 3_j/I_[i - j£E«] (2.55)

2USR<A

t-A.)(2-^ + 3>

/3a

It should be noted that the foregoing discussion is limited to

a triaxial condition only. For the plane strain case, the initial

stress state at A can be expressed as:

Ola " "°Vo (2.56)

(2.57)

o 2a= v(o

1+ o3 ) = v(l + # )a vo (2.58)

Assuming undrained conditions (v = 0.5) , o 2a becomes:

o 2a= 0.5(1 + K )a vo (2.59)

. — l

From Equations 2.56, 2.57 and 2.58, the values of J. and Tl at1

<~>2

point A are obtained as:

Jla = 1.5(1 * 2Ko)0 vo (2.60)

J~\ = 0.5(1 - K )o vo(2 ' 61)

The same procedure to evaluate the cap aspect ratio under

plane strain conditions can be obtained in a similar manner as the

triaxial condition.

31

As for the overconsolidation ratio, OCR , it is used to

specify the initial location of the cap along the hydrostatic axis.

If only elastic deformation occurs during unloading, the initial

cap location for the over-consolidated soil will be n times less

than the currently over-consolidated state. The n- value is

determined as:

( 1 + O XT \

= 1± t±2RL CR (2.62)(1 + 2Kov )

where Kon and Kov are respectively the initial coefficients of

lateral earth pressure under normally consolidated and over-

consolidated conditions.

For determining the initial location and shape of the cap, KQ , Su

and OCR are required. A simple first-order approximation of KQ

for normally consolidated (cohesive) soils was proposed by Jaky

(1944) :

K = 1 - sin<(> (2.63)

The undrained shear strength, Su , can be evaluated from

undrained triaxial compression tests. The over-consolidated

behavior can be evaluated through consolidation test results and

correlated to KQ . Additional references for the evaluation of

soil properties for the determination of the initial location and

shape of the cap can be found in the books mentioned previously or

in papers and reports by: Ladd et al. (1977); Mayne (1980); Mayne

and Kulhawy (1982); and Kulhawy and Mayne (1990).

32

2.3.4 Initial Stress State ( y,K )

From the boundary geometry, location of the water level and

unit weight and coefficient of at-rest lateral earth pressure of

the embankment and foundation soils, the initial stress field

within the embankment and foundation soils can be constructed. The

stress field within the embankment and foundation soils is

represented in NFAP as:

ax = Ax

+ Bxy + Cxz (2.64)

ay

= A2* B

2y + C2 z (2.65)

a z= A, + B3y + C

3z (2.66)

ayz = A, + BiY + C4 2 (2.67)

where oy and a z are respectively the normal stresses in the

horizontal and vertical directions, oyz is the shear stress, and ox

is the normal stress in the longitudinal direction. Constants,

Aif Bit Ct are dependent on the boundary geometry, location of the

ground water level and K

2.3.5 Pore Pressure Response Factor ( p )

For soils experiencing either drained or undrained conditions,

it is convenient to consider a common formulation to include the

effect of pore pressure (e.g. Naylor, et al., 1981; Herrmann, et

al., 1982). This can be achieved by superimposing a large bulk

modulus over the soil stiffness if slight compressibility of soil

is recognized for an undrained condition. For a drained condition

(i.e. no pore pressure development), the bulk modulus can be

33

assumed to be zero while a partially-saturated soil can be managed

as a value which is a function of the Henkel pore pressure

parameter (Naylor, et al., 1981).

The total stiffness, Dt , is given by:

DC= DS

+ Df(2.68)

where Dc

is the total stiffness, D3is the soil stiffness and Df

is the stiffness contributed by the pore fluid/ solid system. Df

has the form [K^] where Ktj are the components of the stiffness

matrix. Since fluids cannot resist shear distortions, Df has the

form:

Kf Kf KfKf Kf Kf

5f= Kf KfKf (2>69)

where Kf is the apparent bulk modulus of pore fluid/ solid system.

For a given displacement field, the resistance provided by the soil

is

[a] = Ds [€] (2.70)

where the transpose of a is

[a-

]T =[a'a'o'irxTl (2.71)1 J L x "y "z xy k yz w zx J * '

and the transpose of e is

[e]T = lex ey ez y^ yyz y zx] (2.72)

The pore pressure developed is

34

\i = Kt ev (2.73)

where ev is the total volumetric strain given by

ev = ex + ey+ ez (2.74)

The total stress [o] is then

[a] = [o] + [ill] (2.75)

where

[I] T= [1110 0] (2.76)

The apparent bulk modulus K of the pore fluid/ solid

particle system for saturated undrained conditions is (Nay lor, et

al., 1981)

1 = JL + 1 ~ n(2.77)K Kw Ks

where Kw , K3 are respectively the bulk modulus of the water and

solid particles and n is the porosity. Since Ks- 3 0JCW (Lambe

and Whitman, 1969), the second term in Equation 2.77 can be

neglected with only a few percent error, therefore,

K = -^ (2.78)n

The bulk modulus of water is 2.1xl0 6 kPa (43xl0 5psf; 300,000

psi) . It is convenient to express Kf in terms of the bulk modulus

of the soil skeleton K using a pore pressure response factor p

which is defined as:

p = £k (2.79)

35

For partially saturated conditions Kf is related to the

Henkel pore pressure parameter (Naylor, et al., 1981)

K„=-JX- (2.80)

where

An -**(-££) +Ab (3J2 )$ (2.81)

Ah and Bh are the Henkel pore pressure parameters (Henkel,

1960; Henkel and Wade, 1966). Bh is zero for dry soil and

approaches 1 for saturated soils. Comparison of Equations 2.79 and

2.80 shows that

B = —^— (2.82)

It may therefore be possible to use p to simulate partially

drained conditions.

2.4 Discussion of Sensitivity of Cap Model Parameters

A sensitivity study was performed to show the effect of

varying the input soil properties on calculated stress-strain

curves, pore pressure response and effective stress paths. The

results show which parameters have the greatest effect on computed

behavior and provide guidance in the selection and adjustment of

input soil properties to obtain a better fit between the calculated

and observed response. Sample response calculated using the

parameters shown in Table 2.3 and for test SW23-276 (Nwabuokei,

1984) was used as the basis for comparison. Test SW23-276

8<3\ a\ C* ff> en en

^ <* CN*? «* «tf «* *r i"

co

o i-H rH rH H rH rH1-1

rH CN

wEh

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£9

CN CN CN CN CN CN rH CO CN

KCN CN CN CN CN CM CO «* CN

0.

t> O d o O d odd o

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Eh o O O O O o O o dM>MEHMCO2uco CN P> CN O CN CN CN CM CN

fa

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w O O O o O d d o oJ00<H

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§ o<&

B3 . -0 <D 73s <0 .C t/1 -H

Appendix

ity

Model

rs

remain

-P 4->

rH >3 (0

-P Ul

rH

Q)

14-1

CD

rH JJ

-P _Ul o

_ U 0) rH QJ-H

C -H(-J -p id•H JJ JJ f0

udy

Plas

he

o •H4->

(0rH

O Ul

-P -

n (0 cU

itivity

for

a

Ca

ed

while

c

wUl

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Ul

J3 ul

cd -hrH

Ohd)

5^t/i a)

•r-1 <D _ -pC U rH £1 e

the

se

Paramete]

r

was

va

t

for

t

ned

frc

ed

for

A •H Ul

4h QJ0) fD 3

d) +J

rC d)

-p

014J g QJ u(0

e

result

lysis

of

one

pararH

0)

-P(1)

etd

rH

(0

a

*"«ui <y>

aoUl

a) cC r0 u_irH rO

*S° rCrj

rGT3 4->

igures

of

nsitivity

the

value

a)

<rH

s

(observe

5

rather

bility.

Ul

a)

3rHn Q) in 0) * rC

w «r > 3 . +jo> S rH O 01

-c < 2 •p (0 CJJ s <w^ c > <W-H

id

+* 0) u c

.p Q) <-i

tn m (0

c §crH (0 (0 n

U -P 0) 4-> o -H « ^0) fo 0)

<4-i > e01 -H 01

MH -P rH C a)

<D C O JC 0) 3CC <D En tH rH < C

36

37

consisted of an impact-compacted lacustrine clay soil. Compaction

was performed in accordance with Standard AASHTO procedures and

wet-of-optimum. Following compaction, the clay soil was subjected

to a second level of consolidation pressure (isotropically-

consolidated undrained-compression conditions; CIUC) of 276 kPa (40

psi) after saturation (via back-pressure) and, then sheared in an

undrained condition. The soil properties v, Cc , Cr , <J>, USR, and OCR

were varied individually and the effect on the calculated response

was observed. The reference values of the required soil properties

for the lacustrine clay are summarized as:

v = 0.49Cc = 0.162Cz = 0.044

<J>= 24

USR = 0.22OCR = 1.49

The influence of p was also investigated. The figures of the

results of the sensitivity study are presented in Appendix B as

part of an earlier report (Ludlow, et al., 1992).

2.4.1 Poisson's Ratio ( v )

Three values of Poisson's Ratio, v : 0.15, 0.3 and 0.4 5 were

investigated. From Equation 2.34, Poisson's ratio will affect the

elastic shear modulus. A lower v-value results in an increase of

elastic shear modulus, which in turn yields a steeper initial

stress-strain and excess pore pressure response, as shown in Figure

A. 1.1 of Appendix B. However, the values have little to no effect

at higher axial strain. The effective stress path is independent

of the variation of v (Figure A. 1.3). This is because the

38

determination of the cap shape is not related to v .

2.4.2 Compression Index ( Cc )

The effect of the compression index with a ± 50% variation

was examined. Figure A. 2.1 illustrates that a lower compression

index results in a slightly-higher calculated stress-strain

response and a moderately-lower excess pore pressure. Within 2%

axial strain, the effects are insignificant. A comparison of the

calculated and observed response indicates relatively good

agreement for the stress-strain relation with the "best-fit"

occurring within the range of 0.107 and 0.162. However, the

calculated excess pore pressure in all cases is considerably less

than the observed response. The influence on the effective stress

path was relatively small.

2.4.3 Recompression Index ( Cr )

The influence of the recompression index with a ± 50%

variation was investigated. From Eguations 2.31 and 2.32, Cr

will affect the elastic bulk modulus, which results in a consider-

able change in the stress-strain relation and pore pressure

response (Figure A. 3.1). A lower Cr results in a steeper initial

stress-strain response and a higher pore pressure. However, at

larger strains, lower Cr -values result in a slightly lower

stress-strain response. A comparison of the calculated and

observed response indicates relatively good agreement for the

stress-strain relation with the "best-fit" occurring within the

range of 0.044 and 0.066. However, the calculated excess pore

pressure in all cases is considerably less than the observed

39

response. The influence on the effective stress path or also cap

shape was relatively small (Figure A. 3. 3).

2.4.4 Pore Pressure Response Factor ( p )

The influence of the pore pressure response factor ( p : 1,

5 and 10) on the calculated response was examined. It should be

noted that a p -value of 10 is recommended by Nay lor, et al.

(1983) for simulating an undrained condition in practice.

Lower values of p result in a higher stress-strain response

and a lower pore pressure (Figure A. 4.1). A comparison of the



occurring near 5. However, the calculated excess pore pressure in

all cases is considerably less than the observed response. For

smaller p -values in which partial drainage is permitted, the

results indicate a lower pore pressure response which also shifts

the effective stress path to the right (Figure A. 4. 3).

2.4.5 Angle of Internal Friction (<J> )

The influence of varying the effective stress angle of

internal friction was studied. The parameters corresponding to a

± 25% variation of<J>

are shown in Figures A. 5.1 and A. 5. 3.

Observations indicate that increasing 4> results in a

slightly flatter stress-strain curve, causes higher pore pressures

at failure and shifts the effective stress path to the left which

results in a greater aspect ratio of the cap. A comparison of the



40

occurring within the range of 24 and 3 degrees. However, the

calculated excess pore pressure in all cases is considerably less

than the observed response.

2.4.6 Undrained Shear Strength Ratio ( USR )

The effect of the USR with values of 0.22, 0.31 and 0.43 was

investigated; other parameters are given in Table 2.3. Figures

A. 6.1 and A. 6. 3 indicate a significant change in the calculated

response of the USR . A higher USR results in a lower aspect

ratio of the cap which shifts the effective stress path to the

right (Figure A. 6. 3). For a higher value of USR , a shorter

path is required and a smaller pore pressure is induced at failure.

2.4.7 Over-consolidation Ratio ( OCR )

The variation of the OCR with values of 1.0, 1.49 and 2.2 5

was examined. The calculated response for describing the over-

consolidated (describes the initial location of the cap along the

hydrostatic axis) condition is shown on Figures A. 7.1 and A. 7. 3.

Observations of the figures indicate that an increasing OCR

causes a significantly steeper and higher stress-strain response.

A comparison of the calculated and observed response indicates

relatively good agreement for the stress-strain relation with the

"best-fit" occurring near an OCR value of 1.49. However, the

calculated excess pore pressure is considerably less than the

observed response except in the case where the OCR has a value of

2.25 and limited in strain.

2 . 5 Limitations of the Cap Model

The cap model has some limitations which impact the calculated

41

response. One limitation which was observed during this study was

the inability of the model to predict the reduction in undrained

strength or increase in pore pressure that occurs after the peak

strength is reached in strain softening soil. Consequently, this

would overestimate the shear strength and underestimate the pore

pressures and deformations.

A second limitation is the ability to model the behavior of

over-consolidated soils. The cap model correctly predicts the

undrained strength at large strains. However, pore pressures are

not predicted correctly and plastic strains which many over-

consolidated soils experience for stress changes in the elastic

region bounded by the cap and ultimate failure surfaces are not

accounted for. Also, strain softening after failure can not be

modeled. It is speculated that this would lead to underestimation

of deformations.

Another limitation which can be attributed to the model's

formulation is the need to perform a trial-and-error calculation to

estimate the best-fit of the observed response. Although time-

consuming, a reasonable fit of the observed response can be

obtained rather quickly by making a critical review of the OCR and

USR parameters and then modifying one of the other five (from the

previous sensitivity study) parameters, if necessary.

Another limitation which requires considerable attention and

can also be attributed to the model's formulation is the model's

inability to account for long-term behavior (e.g. creep and

consolidation) . Presently, the model is best-suited for depicting

42

short-term behavior only. To account for creep and consolidation,

the model would require extensive modifications.

2.6 summary



field tests and a sensitivity study which examines the effect of

the input soil parameters was provided. The main input soil

properties are the compressibilities ( Cc , Cz ) , the effective

Mohr-Coulomb shear strength parameters ( <j), c ) , the undrained

Su ...shear strength ratio ( USR; =-=

) and the over-consolidation ratio

( OCR ) . Solutions are given in graphical form and equations

suitable for hand calculation.








pressures and effective stress paths. The discrepancy is at least

partially because this formulation of the cap model does not allow

plastic volumetric strain for stress changes within the region

bounded by the cap and ultimate failure surfaces.




43





behavior.

44

CHAPTER 3

APPLICATION OF THE FINITE ELEMENT METHOD TO NFAP

3.1 Introduction

The software used in this study is based on a general purpose

finite element program named NFAP (nonlinear finite analysis

program) developed by Chang (1980) . NFAP was originally developed

for the use of nonlinear large deformation analysis of structures.

However, the program was later expanded and refined by Mizuno

(1981) and McCarron and Chen (1985) to include a cap plasticity

model to simulate the behavior of soils. Then, Humphrey (1986)

condensed McCarron' s version of the program for use on personal

computers

.

During Phase I of this research study, Huang (1991) modified

and extended NFAP to include: double precision/ floating point

variables; foundations with sloping boundaries and a variable

groundwater level; and a procedure for determining the ini-

tial/existing stress state within an embankment.

NFAP employs a cap plasticity model, as discussed in the

previous chapter, to simulate the behavior of the soil. From a

theoretical point of view, the cap model employed herein is

particularly appropriate in modeling soil behavior, because it is

capable of treating the conditions of stress history, stress path

dependency, dilatancy (positive and negative) , and the effect of

the intermediate principal stress.

The program performs an incremental load-displacement

analysis. After each increment of load is applied, the

45

displacement field is modified using an iterative procedure (The

Newton-Raphson or the modified Newton-Raphson can be selected.)

until an equilibrium configuration is reached. Convergence is

based on the difference between two successive displacement norms,

and 1% is used in the case studies of Chapter 4

.

NFAP also incorporates a large strain deformation analysis

which is based on the assumption of small strain and large

rotation, for which the structural stiffness of the global

equilibrium equations is symmetric. For solution of the

equilibrium equations, gaussian quadrature using second or third

order integration is recommended. In addition, a node-renumbering

system in NFAP is introduced to minimize the band-width of the

global stiffness matrix and improve solution efficiency. It should

also be noted that, presently, NFAP is only capable of accepting a

finite element mesh with 500 or fewer nodes. However, there are no

restrictions to the number of elements. If necessary, NFAP can be

easily modified to accommodate a mesh that may have more than 500

nodes. This can be achieved by changing the FORTRAN code "common"

statement within the subroutine "GIBB" to the number of nodes that

are required. Presently, the size of the one-dimensional arrays

are established at 500.

3.2 Modeling Foundation Soils

In NFAP, the foundation soils are modeled with two-dimensional

4-, 6- or 8-node isoparametric (implying that the shape functions

for the displacements are the same as those for coordinate

transformation) plane strain continuum elements. Figure 3.1

46

illustrates a typical finite element mesh. Soil behavior is

represented by the strain-hardening cap model (which is time and

temperature independent) described in the previous chapter or a

linear elastic response by specifying Young's modulus and Poisson's

ratio. In addition, cap parameters for the foundation soils were

chosen using the procedure given in Chapter 2. The linear elastic

response was not discussed herein since the behavior of soils is

often non-linear and inelastic and because one of the major thrusts

of this study was to analyze practical engineering problems using

the cap model. However, it should be noted that the solutions of

linear elastic models are often used as a benchmark to evaluate

other results from more sophisticated models (e.g. cap plasticity

model)

.

3.3 Modeling Embankment Fill

The embankment fill is also modeled with two-dimensional 4-,

6- or 8-node isoparametric plane strain continuum elements. Again,

soil behavior is represented by the strain-hardening cap model

described in the previous chapter or a linear elastic response by

specifying Young's modulus and Poisson's ratio. Just as in the

case of the foundation soil, use of the cap model would also

require knowledge of the compaction-induced preconsolidation

pressure or compactive prestress (stiffness related) and at-rest

lateral earth pressure coefficient in the compacted fill. For

compacted clay fills, these are generally unknown. Therefore,

plastic strains within the soil mass may not be accurately

predicted by the model. In addition, finite element studies (e.g.

....•

I

CD>»TO

.-=

Oco i- <N co **

node

truss

element

Typical

four-node

subparametric

Foundation

\

plane

strain

element

/;•\

i

i

1

1

1

1

\' j

k.CD

oICO /o /Q- />> /hr /

CCD

EScCOXI

ELU

/ y N{, ^f

i /1

'(

1

1.

» i

1

ECD V y

\' VmCOCO

ECD

1CDCD

3 / N

Q.

» 1

fK

>

y^^ S 1

!

i

> <

1

1

V !/

cCD

Q.

CO

g'o.

CO

0)w3

il

47

cd

FCO cm CD

Q. Ho ci)

<f>CD

CD c8 sc

to

X CD

m cmu.

COoQ.><

E2cCD

s.b

XI3 "cd

to cCD CO

E toc

CDX cCO CO

n8Q.>•

48

Rowe, 1982; Kwok, 1987; Hird and Jewell, 1989) have shown that an

increase in the embankment stiffness can modestly reduce the

deformations in the soft foundation soils.

3.4 Modeling Reinforcement

The reinforcement is modeled with 2 or 3 node one-dimensional

truss elements which can only sustain axial tensile load. One- or

multiple- layer reinforcement can be modeled during embankment

construction. The reinforcement behavior is modeled as a linear

elastic or non-linear elastic response where the stress-strain

relation of the reinforcement is specified. For this study, a non-

linear elastic response of the reinforcement was used in the case

studies of Chapter 4

.

Slippage at the embankment fill-reinforcement or

reinforcement-foundation soil interfaces cannot be explicitly

modeled by NFAP (i.e. assumes perfect adherence between the

soil/fill and reinforcement) . This assumption is adequate

provided: (1) the shear stresses developed at the interface are

less than the interface strength; or (2) the soil-reinforcement

interface shear strength equals or exceeds the soil-soil shear

strength. The latter case was assumed for this study. It implies

that slip will occur in the adjacent soil when the shear strength

is exceeded rather than at the interface. This may underestimate

the interface movement. Experimental studies reported in the

literature indicate soil-reinforcement interface resistance is

often greater than 80% of the soil shear strength.

However, if the interaction between the soil mass (embankment

49

fill and foundation soil) and the reinforcement became a concern

for a particular application, three approaches could be considered

of which two would require extensive modification to the existing

program. The approaches are: (1) the use of joint elements (Al-

Hussaini and Johnson, 1978) ; (2) nodal-compatibility slip elements;

or (3) substructuring which involves refinement of the mesh and

soil element properties at and near the interface of the

reinforcement (applicable to NFAP)

.

Common approaches to modeling the soil reinforcement interface

by the joint element involve three nodes at each point along the

reinforcement; one attached to the soil above the reinforcement,

one on the reinforcement, and one to the soil below the

reinforcement. The nodal-compatibility slip element which may be

formulated initially in terms of normal and tangential (shear)

springs with very high stiffnesses:

1. ensures compatible displacement between a pair of dualnodes until some failure criterion is reached; and

2. replaces the compatibility conditions by a failurecondition and dilatancy equation once the interfacestrength is exceeded.

Joint elements allow relative deformation of the soil and

reinforcement, prior to failure of the interface. This is based on

some assumed constitutive relationship of what is in effect on the

interface layer between the reinforcement and the soil continuum.

The joint element may be modelled as a pair of normal and

tangential springs. As the stiffness of a joint element increases,

it tends to a nodal-compatibility slip element. The distinction

between the two is related to the question of whether a discrete

50

interface layer exists or whether the deformations at the interface

(prior to failure) are simply due to the interaction between the

reinforcement and the soil on the either side of the reinforcement.

According to Rowe (1988), any modeling of the interface

behavior must consider three possible mechanisms of failure as

noted below.

1. If there is insufficient anchorage capacity, failure willoccur at the soil reinforcement interface above and belowthe reinforcement as the reinforcement is pulled out ofthe soil. This "pullout" mode involves displacement ofthe reinforcement relative to the soil on both sides ofthe reinforcement.

2. If the shear strength of the soil reinforcement is lessthan the shear strength of the soil alone, then failuremay occur by sliding of the soil along the upper surfaceof the reinforcement and the upper soil mass movesrelative to both the reinforcement and the underlyingsoil.

3. The soil below the reinforcement (for cases of softfoundation soils) may be "squeezed out" laterally frombeneath the lowest reinforcement layer. In this case,the lower soil may move relative to the reinforcement andthe overlying soil.

In the case of geotextiles, since the reinforcement is in the

form of a sheet, which completely separates the soil above and

below the reinforcement, the interface resistance can be readily

determined by direct shear tests (Rowe et.al., 1985). In this

case, provision for slip at the interface is the same irrespective

of the mechanism of failure (that is, direct shear or pullout)

.

This difference, in addition to boundary effects and stress

concentrations, makes the pull-out test results difficult to

analyze. It is noted that during direct shear tests the geotextile

reinforcement experiences simple shear deformation instead of

51

tensile deformation developed in the pull-out mode. However, if

the reinforcement consists of a geogrid, with openings that are

large compared to the grain size of the soil, or if the

reinforcement consists of separate reinforcing strips, then special

care is required to correctly model the failure mechanism.

For planar reinforcement, independent movement of the soil may

occur above and below the reinforcement following either a direct

shear or pullout failure. On the hand, for strip reinforcement,

independent movement of the soil above and below the plane of

reinforcement can only occur during a direct shear mode of failure

(Rowe et.al., 1985). Pullout of strips is really a three

dimensional phenomenon in which the strips move relative to the

soil around them but the soil between strips remains continuous.

As noted by Naylor and Richards (1978) , the common approach of

using a conventional joint element (or nodal compatibility element)

implicitly treats the strips as an equivalent two dimensional sheet

and will cause error since it interrupts the transfer of shear

stress through the soil.

3.5 Incremental Construction

Construction of the embankment is simulated using an

incremental loading technique (McCarron, 1985) which does not

directly account for compactive prestress during embankment

construction. The embankment is usually divided into horizontal

layers each represented by a row of elements with zero initial

stress. During each increment, the gravity stresses in the next

layer are increased from zero to their full value in one or more

52

subincrements (which are specified by the user) . Additional layers

are then applied until construction of the embankment is complete

and/or failure of the side slope or foundation occurs.

There is a trade-off between layer thickness and the number of

subincrements. Thicker layers require more subincrements to

achieve a stable solution (Naylor, et al., 1981), so there is

little savings in solution time by using excessively thick layers.

Thicker layers also result in a stiffer embankment response. It is

not necessary to represent each construction lift (typically 6 to

8 inches thick) by a layer of elements. Provided the thickness of

the layers is chosen so that 4 to 5 layers are applied prior to

embankment failure, computed behavior is approximately the same as

obtained with thinner layers. This recommendation should be re-

examined for layers that are many times thicker than the

construction lift.

As stated previously, the incremental loading technique used

in NFAP does not directly account for stresses produced by the

compaction equipment during construction of the embankment.

Compaction prestress which is analgous to preconsolidation stress

represents the fraction of the compaction energy which is

effectively transmitted to the soil matrix due to plastic

straining/deformations. To account for compactive prestresses,

knowledge of the compressibility and shear strength responses would

be required within defined layers of the embankment. Often, this

is not known. One approach, again which would require knowledge of

the compressibility and shear strength responses of the fill (which

53

is limited to cohesive type soils only) , is SHANSEP (Soil History

and Normalized Soil Engineering Properties)

.

3

.

6 Concluding Remarks

The following comments are also related to the application of

the FEM and should be regarded as general guidelines for the setup

and analysis of practical engineering problems. These points

include:

1. Create a general subsurface (soil/groundwater) profilefrom the test boring information and appropriatelaboratory test results;

2. Determine the cap parameters from the procedure outlinedin the previous chapter;

3

.

Input the parameters from step 2 into the CAP program andperform the analysis;

4. Compare the computed (from the CAP program) and observed(from the laboratory test results) results and iterate toobtain the "best-fit";

5. Draw a finite element mesh showing the geometricboundaries and the different soil layers (regions)

;

• Draw the elements as square as possible keeping anaspect ratio (length to height) of the element lessthan 2L:1H;

• Avoid inverted elements (interior angle of anyelement must be less than 180 degrees)

;

• Avoid triangle elements in a 4-node element scheme(if needed, use them in a region away from the zoneof interest)

;

• Keep the mesh fine near the region of interest orwhere changes in shear stress and strain aresignificant;

• Avoid sudden jumps in element size;

• Ideally model the subsurface profile to arelatively incompressible layer (e.g. rock or densesand and gravel) or model the bottom foundationlayer to a depth of two to three times the

54

6.

7.

embankment height and/ or width;

• Ideally model the foundation layer to a distanceaway from the embankment toe of two to four timesthe embankment width;

• Take advantage of embankment symmetry, if possible;

• Minimize bandwidth of global element stiffnessmatrix by decreasing the maximum difference of thenode numbering in each element;

• When performing an incremental analysis, simulatethe construction seguencing as much as possible;and

• Keep it simple (i.e. do not try to model everydetail of the soil-embankment system)

;

Setup the data file and check the data input; and

Perform the finite element analysis and interpret andcheck the results. Verification of the solution shouldinclude:

A. A check against a well-documented case history orphysical model;

B. Simple hand calculation; and

C. Reasonable checks of output including:

effective vertical stress;

effective horizontal stress (for rough caseonly)

;

stress level;

boundary conditions and deformed mesh;

shear strain; and

orientation and magnitude of principalstresses (best check)

.

A few additional notes on the application of the FEM include:

1. The FEM is relatively good at predicting stresses butsomewhat poor at estimating strains and displacements;

The FEM formulation over-estimates the global element

55

stiffness matrix because of the assumption thatdisplacements across an element are defined by someinterpolation function of the displacements at the nodesof the element; and

Four primary errors associated with the FEM include:

• error due to numerical integration (slightly under-estimates the stiffness)

;

• error due to above-mentioned item 2 (over-estimatesthe stiffness)

;

• FEM used to analyze boundary value problem (i.e.error from not accurately modeling system geometryand boundary conditions) ; and

• error due to constitutive law from not modeling thesoil behavior (dominant)

.

56

CHAPTER 4

CASE STUDIES

4.1 Introduction

This chapter summarizes the case studies contained in this

report and describes the implementation procedures, which are

outlined in Chapters 2 and 3, that were used to setup each example.

The examples in this chapter are based on actual highway projects

in Indiana where information on these projects was provided by

INDOT personnel. These examples include: (1) State Route 55 over

Turkey Creek in Lake County; and (2) State Route 1 over Ramsey

Creek in Franklin County.

The influence of several factors on the short term behavior of

reinforced and unreinforced embankments was studied. The influence

of reinforcement was assessed by comparing behavior of reinforced

and unreinforced embankments. A total of 3 cases (12 cases for

State Route 55 and 18 cases for State Route 1) including various

embankment geometries and soil properties was considered. The

effect of the crust strength, foundation thickness and compress-

ibility and embankment width and widening and grade raising were

examined.

4.2 Example l - State Route 55 over Turkey Creek in Lake County

This example involved a 13-ft high and 60-ft wide (at the top)

approach embankment for State Route 55 over Turkey Creek in Lake

County, Indiana. A representative section was assumed near Station

52+00 along the centerline of the proposed roadway. Again,

information regarding this example was provided by INDOT personnel

57

[Project No. ST-4145(B), Structure No. 55-45-7366] and summarized

in a geotechnical report prepared by Engineering Testing Service

(1989) . For this case, engineering properties of the foundation

soils and embankment fill were limited and as a result, many of the

cap model parameters were estimated from empirical correlations.

The rationale for estimating these parameters is contained in

Appendix C (entitled Interim Report: "Embankment Widening and

Grade Raising on Soft Foundation Soils: Example 1 - Indiana State

Route 55 over Turkey Creek in Lake County, Indiana") and a general

soil profile is summarized in Table 4.1.

As stated previously, 12 cases were analyzed for this example.

In all cases, the groundwater level was assumed to be at the ground

surface. Typically, the drainage conditions during simulation of

the embankment construction were assumed to be undrained except for

case 2 where drainage of the embankment fill was permitted.

In all of the cases except for three, the finite element

representation of the embankment fill and foundation soil consisted

of 349 nodes and 102 eight-node isoparametric plane strain

continuum elements (Figure 4.1). The exceptions consisted either

of: (1) 355 nodes and 104 eight-node isoparametric plane strain

continuum elements [case 5 - which included the construction of a

stabilizing berm (Figure 4.2)]; or (2) 397 nodes and 118 eight-node

isoparametric plane strain continuum elements [cases 6 and 8 -

which included embankment widening and grade raising (Figure 4.3) ] .

It should also be noted that the embankment construction was

simulated with eight 1.625-ft (0.5 m) thick lifts (or sixteen 0.81-

58

Table 4.1. General Soil Profile for State Route 5 5

Cap ModelParameter

Foundation Layer/Value 1

1 2 3 4

-Kmin

300 ksf 300 ksf 300 ksf 300 ksf

V0.45 0.45 0.45 0.45

eo

0.700 0.700 0.700 0.700

*33° 33° 33° 33°

Cpsf psf psf psf

<?c

0.246 0.246 0.246 0.246

Cr0.0246 0.0246 0.0246 0.0246

0.36 0.36 0.36 0.36

OCR3.0 2.5 1.5 1.0

Y125 pcf 125 pcf 125 pcf 125 pcf

K°0.71 0.66 0.54 0.46

P10 10 10 10

Notes; l. Refer to Figure 4.1 for location of foundationlayers.

2

.

Refer to Geotechnical Investigation of State Route55 (1989) for additional soils information.

3. Refer to Chapter 2 for additional informationregarding the cap plasticity model.

4. Refer to Appendix C for rationale for estimatingthe cap model parameters.

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ft thick subincrements) . Figures 4.1 through 4.3 illustrate the

finite element representation of the embankment/ foundation soil

geometries

.

Cases 3, 4, 8, 10, 11 and 12 involved the use of either a low-

modulus nonwoven geotextile (LMNG) or a high-modulus woven

geotextile (HMWG) placed either: at the mid-height of the

embankment and embankment/ foundation interface; or at the embank-

ment/foundation interface only. The geotextile was modeled with a

3-node one-dimensional truss element (non-linear elastic stress-

strain relation) as discussed in Chapter 3. A "non-linear elastic

stress-strain response" implies that the strains are recoverable

and proportional to the non-linear response of the material.

Stress-strain relations of the geotextile were obtained from

Koerner (1990). Refer to Table 4.2 for a summary of the cases.

4.3 Example 2 - State Route l over Ramsey Creek in Franklin County

This example involved a 90-ft high and 50-ft wide (at the top)

embankment for State Route 1 in Franklin County, Indiana. A

representative section was assumed near Station 122 6+90 along the

centerline of the proposed roadway. Information regarding this

example was provided by INDOT personnel [Project No. RS-5124(2)]

and summarized in a geotechnical report prepared by The H.C.

Nutting Company (1986) . For this case, engineering properties of

the foundation soils were available from laboratory tests (e.g.

isotropically-consolidated undrained compression and consolidation

test results) and as a result, many of the cap model parameters

were estimated from the test results. After obtaining the

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66

appropriate information from the laboratory test results, a

comparison of the calculated and observed (from laboratory test

results) responses was performed using the CAP program. Several

iterations were performed to obtain the "best-fit" with results

shown in Appendix D (entitled Interim Report: "Embankment Widening

and Grade Raising on Soft Foundation Soils: Example 2 - Indiana

State Route 1 over Ramsey Creek in Franklin County, Indiana") . A

general soil profile is also provided for review in Table 4.3.

As stated previously, 18 cases were analyzed for this example.

In all cases, the groundwater level was assumed to be at 11.5 ft

below the existing ground surface except for case 3 where the

groundwater level was assumed to be at 7.5 ft. Typically, the

drainage conditions during simulation of the embankment construc-

tion were assumed to be undrained and drained for the foundation

soils and embankment fill, respectively. Based on the geotechnical

report, rock fill from roadway excavations would be used to

construct the embankment fill.

In all of the cases, the finite element representation of the

embankment fill and foundation soil consisted of 460 nodes and 409

four-node subparametric plane strain continuum elements. It should

also be noted that the embankment construction was simulated with

eighteen 5.0-ft (1.52 m) thick lifts. Figure 4.4 illustrates the

finite element representation of the embankment/ foundation soil

geometries. This example focuses more on the effect of the

reinforcement and the foundation soil properties rather than the

issue of embankment widening and grade raising. This is because

67

Table 4.3. General Soil Profile for State Route 1

Cap Model Foundation Layer/Value 1

Parameter1 2 3 4 5 6 7

-^min

3002 3002 3002 3002 3002 3002 3002

V0.40 0.40 0.40 0.40 0.40 0.40 0.40

So0.700 0.700 0.700 0.700 0.700 0.700 0.700

*21 21 21 21 21 21 21

c0.7202 0.7202 0.7202 0.7202 0.720 2 0.720 2 0.720 2

Cc0.230 0.250 0.250 0.250 0.250 0.250 0.124

Cr0.039 0.050 0.050 0.050 0.050 0.020 0.020

in°

1.58 1.46 1.07 0.96 0.87 0.80 0.63

OCR8.0 7.2 4.9 4.3 3.8 3o4 2.5

Y0.125 2 0.1202 0.1202 0.1202 0.1202 0.1202 0.1202

Ko1.50 1.44 1.23 1.17 1.11 1.06 0.94

P10 10 10 10 10 10 10

Notes; 1. Refer to Figure 4.4 for location of foundationlayers.

2. Units of ksf.3

.

Refer to Geotechnical Investigation of State Route1 (1986) for additional soils information.

1 B

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69

the number of nodes used in this example was near the maximum

number (500) permitted by NFAP. Further modification of the mesh

for embankment widening and grade raising would require additional

nodes which would have exceeded the capacity of the program.

Cases 4, 5, 6, 7, 8, and 12 through 17 involved the use of

either a LMNG or a HMWG placed either: at the 3 0-ft and 60-ft

height of the embankment and at the embankment/ foundation inter-

face; or at the embankment/ foundation interface only. The

geotextile was modeled with a 2 node one-dimensional truss element

(non-linear elastic stress-strain relation) . Again, stress-strain

relations of the geotextile were obtained from Koerner (1990) .

Refer to Table 4.4 for a summary of the cases.

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75

CHAPTER 5

DESIGN GUIDELINES

5.1 Introduction

This chapter investigates the effect of reinforcement, crust

strength, stability, foundation compressibility, embankment width

and embankment widening and grade raising on short term deforma-

tions and stresses to identify which aspects of embankment geometry

and foundation properties have the greatest effect on the overall

behavior. To evaluate these effects, 3 cases as described in the

previous chapter were considered. Although a significant number of

cases were analyzed with NFAP, the effect of the above-mentioned

aspects, which are discussed herein, provides merely a gualitative

assessment of the behavior. During the design process, the results

of NFAP should be only used to assess/check the design and not be

used as a "stand alone" design tool.

To discern these effects, computer-generated contour plots of

the horizontal and vertical displacements, principal stress ratios

and local factors of safety were created. These plots for examples

1 (State Route 55) and 2 (State Route 1) are provided in Appendices

C and D, respectively.

After executing the program NFAP, an output file known as

"TTT" is generated in ASCII format. The file contains information

regarding coordinates of the nodal points (original mesh) , nodal

connectivity, coordinates of the deformed mesh and local factors of

safety (defined as the ratio between the available shear strength

and mobilized shear stress at the same average compressive stress,

76

Z ) and principal stresses (and angle of rotation) of each

element in sequential order. A post-processing program written in

BASIC language interfaces with the TTT file and a commercial

software package (Surfer, 1989) which was used to generate the

contour plots.

5.2 Design Analysis

5.2.1 Deformation Behavior of Reinforced and UnreinforcedEmbankments

In this section a detailed examination is presented of the

behavior of reinforced and unreinforced embankments in relationship

with the examples discussed previously in Chapter 4. A comparison

of similar cases in Appendices C and D involving reinforced and

unreinforced embankments indicates that the reinforcement locally

alters the magnitude of the short term displacements but does not

alter the overall pattern of displacements. Horizontal displace-

ments at the embankment toe and maximum settlement (not in the

sense of consolidation but rather displacements/deformations during

undrained shear) at the base of the embankment were slightly

reduced with single and multiple layers of reinforcement and with

the type/modulus of the reinforcement. However, in comparison of

cases 7 and 12 of State Route 55, a considerable reduction of the

horizontal and vertical displacements at the toe of the embankment

(in the foundation soil) and beneath the shoulder (embankment fill)

were observed. The reduction was about 3 3% at each location. It

should be noted that the over-consolidation ratio of the foundation

soil was reduced to yield a softer response for cases 7 and 12 and

a high-modulus woven geotextile was placed at the interface of the

77

embankment fill and foundation soil for case 12 to permit the

comparison.

In general, comparisons indicate that the largest horizontal

displacements develop beneath the shoulder/ slope and at the toe

(more so at the toe than the shoulder) . These findings are in

general agreement with the results of previous studies (e.g., Ohta,

et al., 1980; Andrawes, et al., 1980, 1982; McGown, et al., 1981;

Rowe, 1982; Boutrup and Holtz, 1982, 1983; McCarron, 1985; and

Humphrey, 1986) . In cases where reinforcement was used, the

deformations were decreased and, as will be discussed in Section

5.2.3, stability was increased. However, the reinforcement did not

have much of an effect at other sections within the embankment and

at greater depths in the foundation. The location of the maximum

displacements and the embankment stability is believed to depend

primarily on the reinforcement type/modulus, embankment width,

foundation depth and compressibility and the relative stiffness of

the embankment fill and foundation soils. The beneficial effects

of reinforcement on the short term deformation patterns are

summarized in Table 5.1.

Table 5.1. Relative Influence of Reinforcement

Stiff Embankment Soft Embankment

Stiff Foundation Little Effect Little Effect

Soft Foundation- Significant Effect Little to ModerateEffect

5.2.2 Effect of Crust Strength

Displacements for embankments on foundations with strong and

78

weak crust are compared on the basis of the examples as discussed

previously in Chapter 4. It should be noted that for these cases

that the crust was modeled as a moderately to highly over-consoli-

dated soil as discussed in Section 3.2 of Chapter 3. A comparison

of similar cases with crust reported in Appendices C and D

indicates that the crust strength has a significant effect on the

overall performance (short term only) of the embankment fill and

foundation soils. In comparison of cases 7 and 9 of State Route

55, the displacements are reduced by almost 80% for a given

crust/ foundation thickness and an increase in crust strength,

especially when the cohesion was increased. A strong crust

strength alters the magnitude of the displacements and causes the

maximum horizontal displacement to develop at a slightly greater

depth, directly beneath the embankment slope and in the foundation

soil rather than near the toe for the weaker case. Again, similar

conclusions as discussed in the previous section regarding

reinforcement can be made here. The beneficial effects of crust

strength are summarized in Table 5.2.

Table 5.2. Relative Influence of Crust Strength

Stiff Embankment Soft Embankment

Strong Crust Moderate to SignificantEffect

Moderate Effect

Weak Crust - Little Effect Little Effect

Based on the information in this section and the previous, a

strong crust strength appears to significantly enhance the overall

performance of the embankment fill and foundation soil and is

79

somewhat analogous to reinforcement. However, the beneficial

effects of crust strength appear to outweigh those of reinforcement

for the range of crust/ foundation thicknesses ratio (i.e. 20% to

25%) considered in the examples.

5.2.3 Effect of Reinforcement on Stability

The effect of reinforcement on stability was investigated. A

comparison of similar cases in Appendices C and D involving

reinforced embankments indicates that the reinforcement locally

alters the magnitude of the safety factor but does not influence

the overall pattern of stability. Safety factors in the zone of

the embankment toe and sideslope were slightly increased by the

presence of reinforcement and more significantly by the presence of

multiple layers of reinforcement. The potential benefit was more

noticeable when using a high-modulus woven geotextile. In a

comparison of cases 7 and 12 of State Route 55, a pronounced

increase of the stability by about 2 0% at the toe (foundation soil)

and by nearly 8% beneath the shoulder (embankment fill) was

observed. It should be noted that the over-consolidation ratio of

the foundation soil was reduced to yield a softer response for

cases 7 and 12 and a high-modulus woven geotextile was placed at

the interface of the embankment fill and foundation soil for case

12 to allow comparison. The effects of reinforcement on stability

are extremely beneficial in situations where a relatively stiff

embankment is established on very weak and compressible foundation

soils. An interesting point in the comparison of cases 10 and 14

of State Route 1 indicates an increase by about 2 0% in the

80

embankment stability near the toe upon incorporation of a high-

modulus woven geotextile placed at the interface of the embankment

fill and foundation soil.

5.2.4 Effect of Foundation Compressibility

The effect of foundation compressibility was investigated on

the basis of the examples discussed in Tables 4.2 and 4.4 of

Chapter 4 . A comparison of similar cases in Appendices C and D

indicates that the foundation compressibility locally alters the

magnitude of the short term displacements but does not alter the

overall pattern of displacements. In general, comparisons indicate

that the largest horizontal displacements develop at the toe.

However, the largest vertical displacements develop near the

shoulder and to a lesser extent at the center of the embankment and

migrate to the sideslope with increasing foundation compressibili-

ty. Recall that a strong crust strength alters the magnitude of

the short term displacements and causes the maximum horizontal

displacement to develop at a slightly greater depth, directly

beneath the embankment slope and in the foundation soil rather than

near the toe for the weaker case. For a given embankment stiff-

ness, the foundation compressibility directly influences the

effectiveness of the crust strength. The effects of the foundation

compressibility on the short term performance of the crust strength

are summarized in Table 5.3.

31

Table 5.3. Relative Influence of Foundation Compressibilityon Crust Strength

Normal Compressibility High Compressibility

Strong Crust Little Effect Significant Effect

Weak Crust Moderate to SignificantEffect

Significant Effect

5.2.5 Effect of Embankment Width

The effect of embankment width was investigated by comparing

cases 1 and 5 of State Route 55. The embankment width of case 5

[originally 45 ft (13.7 m) in case 1] was increased by the

placement of a 10-ft (3 m) wide and 3.25-ft (1 m) high stabilizing

berm of similar soil properties as the embankment fill. The effect

of increasing the embankment width slightly reduces the short term

displacements within the embankment fill and foundation soil.

However, at a distance of approximately 20 ft (6.1 m) to 3 ft (9.1

m) from the toe of the berm, foundation heave slightly increases.

In addition, the stability increases modestly near the toe of the

embankment. Although the height of the berm is relatively low, the

beneficial effect is somewhat modest in comparison to the use of

reinforcement and/ or an increase of crust strength.

5.2.6 Effect of Embankment Widening and Grade Raising

The effect of embankment widening and grade raising was

investigated by comparing cases 1, 6 and 8 of State Route 55. Case

6 involves the widening (10 ft; 3 m) and grade raising (3.25 ft; 3

m) of the existing embankment in case 1. Case 8 includes the

placement of a low-modulus nonwoven geotextile at the interface of

82

the embankment fill and foundation soil. Case 8 is similar to case

6 except for the incorporation of a geotextile.

Comparisons of cases 1 and 6 indicate that the horizontal

displacements increase by as much as 0.4 in. (1 cm) to 0.6 in. (1.5

cm) following widening and grade raising. Settlements increase by

as much as 1 in. (2.5 cm). Displacements develop primarily near

the former shoulder and the widened toe (horizontally) and along

the sideslope of the former embankment (vertically) . Because of

the loading and undrained conditions, some displacements develop

causing movement of the existing embankment fill and foundation

soil toward and upward along the centerline of the embankment.

However, these displacements are relatively small.

The widening and raising locally alters the magnitude of the

state of stress and stability but does not alter the overall

pattern. Recall that the local factor of safety is defined as the

ratio between the available shear strength (which is defined on the

ultimate yield surface) and mobilized shear stress [which is

defined by the stress state either on the ultimate yield surface

(failure) , within the cap (for overly-consolidated soils/elastic

response) or on the cap (for normally-consolidated soils/elastic-

plastic response)] at the same hydrostatic stress state. In review

of the local factors of safety of cases 1 and 6, a greater portion

of the foundation soil near the centerline of the embankment is

yielding in case 1. By definition, a factor of safety of 1 implies

that the strength is fully mobilized [i.e. the stress state is

located at the corner of the ultimate and cap yielding surfaces

83

(see Figure 2.2c)]. As the embankment is widened, foundation soils

near the centerline of the embankment and outwardly from the toe

experience slight increases and decreases of the stability,

respectively. In other words for the foundation soil near the

centerline, the mobilized shear stress is slightly reduced upon

widening and hence the stability is somewhat increased. While

outwards from the toe, the widening of the embankment causes a

slight increase in the mobilized shear stress in the foundation

soil and hence the stability is somewhat decreased.

As stated previously, case 8 involves the incorporation of a

low-modulus nonwoven geotextile at the interface of the embankment

fill and foundation soil prior to widening and grade raising.

Comparisons of cases 6 and 8 indicate only localized beneficial

effects by the reinforcement. Again, the location of the maximum

displacements and the embankment stability is believed to depend

primarily on the reinforcement type/modulus, embankment width,

foundation depth and compressibility and the relative stiff-

ness/strength of the embankment fill and foundation soils.

5.2.7 Summary

Finite element analyses of two practical examples involving a

total of 3 cases were performed. An incremental procedure was

used in these computations to simulate embankment construction.

The responses of the modeled embankment fill and foundation soils

were represented using an isotropic, strain-hardening cap-plastici-

ty model.

The results indicated that the presence of a stiff superficial

84

crust or of high tensile modulus geosynthetic reinforcement had the

most significant influence and potential benefit on the system's

short term performance. Reinforcement type/modulus also influenced

the behavior of the embankment fill and foundation soil but to a

lesser extent when compared to crust strength.







compressibility had only a limited influence, on embankment behavior











.




(strain-hardening) . The other portions of the embankment fill and

85


(compressive hardening) , or within the cap (elastic state) as

discussed in Chapter 2

.

5.3 Design Analysis Limitations of NFAP program

NFAP is an extremely useful tool in gaining insight on the

short-term behavior of embankments constructed on soil foundation

soils. However, to fully appreciate the program's unique capabili-

ties as a practical analysis tool requires a basic knowledge of

such items as: the cap plasticity model (Chapter 2) ; the applica-

tion of the FEM to NFAP (Chapter 3) ; and interpretation of the

results as discussed in this chapter. NFAP does not have the

capability to answer such questions as: "Is the design accept-

able?"; and "How will the embankment behave during long-term

conditions (e.g. consolidation, creep and certain environmental

conditions/changes)?". Nevertheless, these are questions that the

engineer must answer during the design process. These types of

questions usually require engineering judgement or experience on

the part of the engineer and there may not be one "right" solution

but rather several.

As discussed in Chapter 2, NFAP employs a cap plasticity model

to simulate the behavior of the soil. From a theoretical point of

view, the cap model employed herein is particularly appropriate in

modeling soil behavior, because it is capable of handling the

conditions of stress history, stress path dependency, dilatancy,

and the effect of the intermediate principal stress. However, the

model is incapable of treating such time-dependant phenomena as

86

consolidation and creep. Rather than using the cap model to

simulate quasi-time-dependant effects, it is recommended that the

engineer: (1) modify the output subroutine of NFAP to yield

information regarding excess pore pressures or utilize the CAP

program to estimate excess pore pressures based on loading

conditions similar to that at critical locations within the

embankment. In either circumstance estimate the consolidation

settlements using an approximate coefficient permeability of the

soils. Another approach to simulate long term time-dependant

effects (e.g. consolidation) could be the use of isotropically-

consolidated drained compression (CIDC) tests. Although embank-

ments are typically constructed in a relatively short time frame,

this solution will provide a reasonable upper bound to the overall

performance (deformation and stability) of the embankment/ founda-

tion system. In the case of creep, use a variable moduli model

(e.g. Duncan and Chang, 1970) however, the Duncan and Chang model

is not based on plasticity theory and nor will it adapt to NFAP.

At present, the FEM is the only technique available to the

geotechnical engineering community which allows a detailed

assessment of deformations within the embankment fill and founda-

tion soils during and following construction. The FEM technique is

based on continuum mechanics. In the continuum theory of soil

mechanics that includes the mathematical theory of elasticity,

plasticity and viscosity, the following three basic sets of

equations must be applied:

1. Equations of equilibrium of motion for a static ordynamic analysis, respectively;

87

2. Conditions of geometry or the compatibility of strainsand displacements; and

3. Material constitutive laws or stress-strain relations.

As a result of the formulation, NFAP and for that matter most

other FEM-based approaches are incapable of producing a physical

crack and the associated propagation of the crack (violates item 2

above) . However, NFAP is capable of evaluating the crack poten-

tial. By reviewing the output, the user is able to locate zones of

potential tensile stresses (e.g. at the embankment toe or shoul-

der) . Presently, the crack potential is evaluated in NFAP using

the minimum principal stress in the in-plane stress state for two-

dimensional plane strain analysis. When tension is recognized, the

program performs a stress re-distribution (Chen, 1982) . Upon

completion of the analysis, the output of the program provides the

computed stress at the Gauss points as well as the stress situation

for each element, including the potential for tension. Hence, the

user can identify where such a potential crack exists. From a

practical standpoint, this could lead the analyst to modify the

geometry/soil characteristics and re-analyze the problem, or

possibly to perform a limit eguilibrium analysis that includes a

tension crack.

Another point of particular interest and related to the

previous discussions is the aspect of reversal of principal

stresses. Reversal of principal stresses develops when the minor

principal stress is increased to failure or the major principal

stress is decreased to failure. This occurs for KQ consolidated

samples sheared on axial extension and lateral compression stress

88

paths (e.g. near the toe of an embankment) . The state of stress

begins on the cap, however, the shear stress in initially reduced

and the stress state moves into the elastic region. The cap model

predicts a stress path having the shape shown in Figure 5.1. For

convenience, states of stress where ~o v < ~oh are plotted below the

~T. axis. The path moves vertically downward until it reaches the

cap and only elastic strains occur. Then, the path follows the cap

and the cap expands causing plastic strains. Failure occurs when

the state of stress reaches the ultimate yield surface. This

stress path differs significantly from the behavior of many soils

i i

(Figure 5.1) and often T 2 is less than T 2 . It is not possibleu2f u2o

_i

to calibrate the model for this situation. However, when , 2 isd2o

1

less than T 2 the model can be calibrated to yield the correctd2£

jL

T 2 , although predicted pore pressures will likely be in error.Lj2f

The inability of the model to represent this behavior is

because stress changes within the region bounded by the cap and

ultimate yield surfaces cause only elastic strains and the cap and

ultimate yield surfaces are symmetric about the Jx

axis. This is

not the case for real soils.

It should also be recognized during the design process that

NFAP does not account for three-dimensional effects. NFAP is based

on a two-dimensional formulation where the tensorial and engineer-

ing shear strains are set egual to zero. In the practical sense,

this may not be so where the edge of a slip/ failure surface

intersects the longitudinal axis of an embankment.

89

,1/2

Initial Position of Cap

I A' Failure Position of Cap

Figure 5.1 Effective stress path predicted by cap model for

samples that experience a reversal of principal stresses

90

5.4 Design Recommendations

Examples 1 and 2 involved the use of empirical correlations

and laboratory test data, respectively, to determine the cap

plasticity model parameters. Once an initial estimate of the

parameters were determined for Example 2 from laboratory test

results, an analysis using the CAP program was performed. Results

from the CAP program were then compared to the observed behavior to

assess the "fit". Once a reasonable fit of the observed data was

attained for Example 2, the data file was constructed. However in

Example 1, no comparison of the calculated and observed behavior

was performed since no laboratory tests were performed. Develop-

ment of the data file also reguires the construction of a finite

element mesh which represents the geometry and soil conditions of

the specific problem. Once the mesh and data file are prepared, an

analysis of the design can be performed using NFAP. Following the

analysis, it is recommended that the results be carefully re-

viewed/evaluated with respect to those points discussed in Chapter

3.

91

CHAPTER 6

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

6.1 Summary and Conclusions

6.1.1 General

The finite element technique using the cap soil behavior model

was applied to the analysis of embankments constructed on soft

foundation soils. A procedure was provided to estimate the cap

model parameters from conventional field and laboratory test

results. A sensitivity analysis of the cap model parameters

comparing the observed and calculated responses was also provided.

The technique was then applied to the analysis of two examples.

The examples were based on actual highway projects in Indiana where

information on these projects was provided by INDOT personnel.

Results of the analysis were used to determine the influence of

several factors on reinforced and unreinforced embankment behavior.

6.1.2 Cap Plasticity Model



field tests is proposed. Table 6.1 on the following page summariz-

es the procedure for determining the cap parameters from conven-

tional laboratory and field tests. A sensitivity study which

examines the effect of the input soil parameters was performed.

The main input soil properties are the compressibilities

( Cc , Cr ) , the effective Mohr-Coulomb shear strength parameters

g( <|>, c ) , the undrained shear strength ratio ( USR; -=J±- ) and the

over-consolidation ratio ( OCR ) . Solutions are given in

92

Table 6.1. Summary of Procedures for Determining Cap Parameters fromConventional Laboratory and Field Tests

Cap ModelParameter

Procedure

Cohesive Soil Granular Soil

^ninCIUC/CIDC/Correlations Correlations/

Standard Penetration Test(SPT) /PressuremeterTesting ( PMT ) /Vacuum

Triaxial

VCIUC/CIDC/Correlations Correlations/Vacuum

Triaxial

eCIUC/CIDC/Correlations/Consolidation Testing

Correlations/SPT/ Index Property

Testing

«F

CIUC/CIDC/Correlations Correlations/Vacuum Tri-axial/SPT/Cone Penetrometer

Testing (CPT)/Dilatometer Testing

(DMT) /PMT

cCIUC/CIDC/Correlations -

CcCorrelations /Consolidation

TestingCorrelations/Consolidation

Testing

CrCorrelations/Consolidation

TestingCorrelat ions/Consolidation

Testing

CIUC/CIDC/Correlations/VaneShear/CPT

Assume a fictitious valueand iterate using the CAPprogram comparing results

OCRCorrelations/Consolidation

TestingCorrelat ions/ConsolidationTesting/Plate Load Tests

YCorrelat ions / Index

Property TestsCorrelations/ IndexProperty Testing

K Correlations/CIUC/CIDCPMT/DMT/CPT

Correlations/PMT/DMT/CPT/SPT

93

graphical form and equations suitable for hand calculation.








pressures and effective stress paths. The discrepancy is at least

partially because this formulation of the cap model does not allow

plastic volumetric strain for stress changes within the region

bounded by the cap and ultimate failure surfaces.








behavior.

6.1.3 Case Studies

A comparative finite element study of two practical examples

involving a total of 30 cases was made using an incremental

procedure which simulated embankment construction. Embankment fill

and foundation soil behavior was represented with an isotropic,

strain-hardening cap-plasticity model.

94

The results indicated that the properties of a stiff or of a

tensile resistant reinforcement had the most significant influence

on embankment fill and foundation soil behavior. Reinforcement

type/modulus also influenced the behavior of the embankment fill

and foundation soil but to a lesser extent when compared to crust

strength

.







compressibility had only a limited influence on embankment behavior











.




95

(strain hardening) . The other portions of the embankment fill and


(elastic-plastic) , or within the cap (elastic state)

.

6.2 General Recommendations

The work presented herein aims to facilitate the application

of the cap plasticity model to practical problems and in design of

embankments constructed on soft foundation soils. However, there

are some aspects of unreinforced and reinforced embankment analysis

that reguire further development of the NFAP program. The ability

of the cap model to accurately predict soil behavior could be

extended to a wider range of soils, stress paths and drainage

conditions. The following are recommended as additional research

topics:

1. A parametric study of the effect of reinforcement shouldbe made using NFAP for a wider range of embankmentgeometries and soil properties. It should includenarrower embankments, different crust thicknesses anddifferent reinforcement moduli. The tensile forces inthe reinforcement should be analyzed in this study.

2

.

The cap plasticity model should be extended to improveits ability to model soil behavior during rotation ofprincipal stresses and the behavior of over-consolidatedsoils.

3. Compacted cohesive soils are more often used as fillmaterial for embankments. The behavior of embankmentswith cohesive fill (which include the effects of compac-tive prestress) should be evaluated with the cap plastic-ity model.

4. Presently, the cap plasticity model is better at accomo-dating either drained or undrained conditions. The modelshould be extended to partially drained conditions andthen evaluated.

5. Slippage often occurs at or near the interface of theembankment fill/geosynthetic/ foundation soil interface.NFAP should be modified to include slippage consider-

96

ations.

To provide beneficial use to practicing engineers, NFAPshould be modified to include pre- and post-processinggraphics capabilities.

Post construction effects need to be considered, particu-larly since the original embankment and widening/raisinghave different chronologies. Foundation creep andembankment settling upon wetting in service are examples.

Appendix A

List of References

97

LIST OF REFERENCES

Al-Hussaini, M.M. , and Johnson, L.D. (1978), "NumericalAnalysis of Reinforced Earth Wall," Proceedings from a

Symposium on Earth Reinforcement , ASCE, Pittsburgh, pp. 98-126.

Andrawes, K.Z., McGown, A., Mashhour, M.M. , and Wilson-Fahmy,R.F. (1980), "Tension Resistant Inclusions in Soils," Journalof Geotechnical Engineering Division . ASCE, Vol. 106, No.GT12, December, pp. 1313-1326.

Andrawes, K.Z., McGown, A., Wilson-Fahmy, R.F., and Mashhour,M.M. (1982) , "The Finite Element Method of Analysis Applied toSoil-Geotextile Systems," Proceedings of the Second Interna-tional Conference on Geotextiles , Las Vegas, Nevada, August,Vol. Ill, pp. 695-700.

Baladi, G.Y., and Rohani, B. (1979), "An Elastic-PlasticConstitutive Model for Saturated Sand Subjected to Monotonicand/or Cyclic Loadings," Third International Conference onNumerical Methods in Geomechanics , Aachen, Germany, pp. 3 89-404.

Bjerrum, L. (1972), "Embankments on Soft Ground," Proceedingsof the Specialty Conference on Performance of Earth and Earth-Supported Structures , ASCE, Vol. II, pp. 1-54.

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104

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Appendix B:

Interim Report: "A Sensitivity Analysis of the Parameters fora Cap Plasticity Model," Report No. FHWA/JHRP/IN - 92-1 , 2 3

pp.


Draft Interim ReportFHWA/JHRP/IN - 92/1

A SENSITIVITY ANALYSIS OF THEPARAMETERS FOR A CAPPLASTICITY MODEL

Scott J. Ludlow,Wai-Fah Chen,Philippe L. Bourdeau, andC. William Lovell

Interim Report: "A Sensitivity Analysisof the Parameters for a Cap Plasticity Model"

by




and


Joint Highway Research ProjectHPR 2031

Project No.: C-36-36UFile No. : 6-14-21

This study is proposed to be conducted by

the Joint Highway Research Project

in Cooperation with

the Indiana Department of Highways

and the

U.S. Department of TransportationFederal Highway Administration

The contents of this report reflect the views of the authors whoare responsible for the facts and the accuracy of the datapresented herein.


Date: January 23, 1991

TABLE OP CONTENTS

LIST OF TABLES .

LIST OF FIGURES

APPENDIX A -

APPENDIX BAPPENDIX C

A.lA.

2

A.

3

A.

4

A.

5

A.

6

A.

7

Page

i

i

APPENDICES

CaseCaseCaseCaseCaseCaseCase

Computer disk with input data filesList of References

LIST OF TABLES

Table Page

1. Summary of Sensitivity Analysis of the CapPlasticity Model Parameters 1

LIST OF FIGURES

Figure

A. 1.1 Case 1 - Effect of Poisson's Ratio onthe Principal Stress Difference andExcess Pore Pressure vs. Axial Strain . . 2

A. 1.2 Case 1 - Effect of Poisson's Ratio onthe Principal Stress Ratio vs. AxialStrain 3

A. 1.3 Case 1 - Effect of Poisson's Ratio on thelocation of the Cap in the J2 1 ^ 2 ~liSpace and on the g-p ' Diagram 4

A. 2.1 Case 2 - Effect of the Compression Index onthe Principal Stress Difference andExcess Pore Pressure vs. Axial Strain . . 5

A. 2. 2 Case 2 - Effect of the Compression Index onthe Principal Stress Ratio vs. AxialStrain 6

A. 2. 3 Case 2 - Effect of the Compression Index onlocation of the Cap in the J2 1/,2-I

iSpace and on the g-p ' Diagram 7

A. 3.1 Case 3 - Effect of the Recompression Index onthe Principal Stress Difference andExcess Pore Pressure vs. Axial Strain . . 8

A. 3. 2 Case 3 - Effect of the Recompression Index onthe Principal Stress Ratio vs. AxialStrain 9

A. 3. 3 Case 3 - Effect of the Recompression Index onlocation of the Cap in the J2 1/,2-I iSpace and on the g-p' Diagram 10

A. 4.1 Case 4 - Effect of the Pore Pressure ResponseFactor on the Principal StressDifference and Excess Pore Pressure vs.Axial Strain 11

A. 4. 2 Case 4 - Effect of the Pore Pressure ResponseFactor on the Principal Stress Ratio vs.Axial Strain 12

A. 4. 3 Case 4 - Effect of the Pore Pressure ResponseFactor on location of the Cap in theJ2 1 / 2 -!! Space and on the g-p' Diagram .. 13

11

LIST OF FIGURES cont'd

Figure Page

A. 5.1 Case 5 - Effect of the Angle of InternalFriction on the Principal StressDifference and Excess Pore Pressure vs.Axial Strain 14

A. 5. 2 Case 5 - Effect of the Angle of InternalFriction on the Principal Stress Ratiovs. Axial Strain 15

A. 5. 3 Case 5 - Effect of the Angle of InternalFriction on location of the Cap in the^2

1 ^ 2~^-i sPace an^ on tne ^"P '

Diagram .. 16

A. 6.1 Case 6 - Effect of the Undrained Shear StrengthRatio on the Principal StressDifference and Excess Pore Pressure vs.Axial Strain 17

A. 6.2 Case 6 - Effect of the Undrained Shear StrengthRatio on the Principal Stress Ratiovs . Axial Strain 18

A. 6. 3 Case 6 - Effect of the Undrained Shear StrengthRatio on location of the Cap in theJ2

1 ^ 2 -I1Space and on the q-p 1 Diagram .. 19

A. 7.1 Case 7 - Effect of the Over-consolidationRatio on the Principal StressDifference and Excess Pore Pressure vs.Axial Strain 2

A. 7. 2 Case 7 - Effect of the Over-consolidationRatio on the Principal Stress Ratiovs . Axial Strain 21

A. 7. 3 Case 7 - Effect of the Over-consolidationRatio on location of the Cap in theJ21 ^2_Ii Space and on the q-p 1 Diagram .. 2 2

APPENDIX A

A.l - Case 1

e<n CM ffl en en ffi Q o> m° «tf CN*T ^ «* >* •*!• **

! ° rH rH iH rH H rH ^ rH CN

wOSEdEhEd CN CN CN CN CN CNHM CN2ia.

CN

OCN

OCN

0'

CN

0'

CN

Ocn m tO 0' 0'

CN

O

jEd

ao2!hEh

e-if «* «* rf en •* t T

O CN CN CN CN rH CN HI CN CN

MEh«<JD4

Pi<OEd ca O O O rH^° O O OX rH iH rH rH rH !-H

HfaO

MwXij>* <* T cm «tf 10 <* <* rf Ws |y

"* "* CN <*• U3 •31 ^ *t «tf

rt O O O OOO O O O O>H O O 000' O O O OEhM>MEhMw2CdW CN r- cn CN CN CN CN CN

in O lO^ vO lO in lO 10

En O rt HHM rH rH rH rH rH

O OOO O' O O O»«!S

2DW

•ir> O in in in in in in in

> <-h n >* «* >* <* •* «* «*

Ed OOO O O O O OmrtEh

«EdEhEd

EdW < ca < ca u < ca rt CD O < ca < ca < mri! rlHH CN CN CN n n n in in in lOifliO r- r- r»

OSO

<a,

40Principal Stress Difference vs. Axial Strain

Observed Response

v = 0.45

15 20 25 30

Axial Strain (%)

35 40

Excess Pore Pressure vs. Axial Strain40

35 -

30 -

25 .^^^" '—*— ^_

j/* "* Observed Response

20 // i/ = 0.15/ v = 0.30

15 - / . ,, - n 4g

10

5

n i 1 1 F 1 I 1 1 1

10 15 20 25

Axial Strain (%)

30 35 40

Figure A. 1.1 - Effect of Poisson's Ratio on the Principal StressDifference and Excess Pore Pressure vs. AxialStrain

4Principal Stress Ratio vs. Axial Strain

v = 0.15V = 0.30

,/ = n is

3

2/ nhsprverf Rpspnnso

1

n I I i I i i i

10 15 20 25

Axial Strain (%)

30 35 40

Figure A. 1.2 - Effect of Poisson's Ratio on the Principal StressRatio vs. Axial Strain

1/2J2

(psi)

30

25

20

10

20 40

1/2J2

vs. I

1/ = 0.15

i/ = 0.30

i> = 0.45

60 80

Ij (psi)

100 120 140

q

(psi)

40

35

30

25

20

15

10

5

10

q vs. p

15 20

P' (psi)

25 30 35 40

Figure A. 1.3 - Effect of Poisson's Ratio on the location of theCap in the J2 1/

^

2 ~ Ii

sPace and on the q-p * Diagram

A. 2 - Case 2

40

35

Principal Stress Difference vs. Axial Strain

C = 0.107

- Observed Response

C„ = 0.162

15 20 25 30Axial Strain (%)

35 40

40

35

30

£ 10

Excess Pore Pressure vs. Axial Strain

Observed Response

C„ = 0.163

15 20 25Axial Strain (%)

30 35 40

Figure A. 2.1 - Effect of the Compression Index on the PrincipalStress Difference and Excess Pore Pressure vs.Axial Strain

Principal Stress Ratio vs. Axial Strain

C„ = 0.162

Observed Response


30 40

Figure A. 2. 2 - Effect of the Compression Index on the PrincipalStress Ratio vs. Axial Strain

30

25

20

1/2J2 15

(psi)

10

5 -

1/2Jq vs. I-

20 40 60 80 100 120 140

h (Psi)

q vs. p40

35

30

25

q 20

(psi)15

10

5

10 15 20 25

P' (psi)

30 35 40

Figure A. 2. 3 - Effect of the Compression Index on location of theCap in the J2 1//2-I i sPace and on the q-p 1 Diagram

A. 3 - Case 3


5 10

Cr= 0.066

Observed Response

15 20 25 30 35 40Axial Strain (%)

40

35

30

01

£ 20

a io -


Observed Response


30 35 40

Figure A. 3.1 - Effect of the Recompression Index on the PrincipalStress Difference and Excess Pore Pressure vs.Axial Strain


Cr= 0.029 c

r= 0.044= C„ = 0.066

10

Observed Response

_i i i i_


30 35 40

Figure A. 3. 2 - Effect of the Recompression Index on the PrincipalStress Ratio vs. Axial Strain

10

30

25

20

1/2J2 15

(psi)

10

20

1/2J2 vs. Ij

C^ = 0.066

40 60 80Ij (psi)

100 120 140

40

35

30

25

q 20

(psi)15

10

5

q vs. p

10 15 20P' (psi)

25 30 35 40

Figure A. 3. 3 - Effect of the Recompression Index on location ofthe Cap in the J2 1 -I

iSpace and on the q-p

'

Diagram

A. 4 - Case 4

11


10 15 20 25 30Axial Strain (%)

35 40



Figure A. 4.1 - Effect of the Pore Pressure Response Factor on thePrincipal Stress Difference and Excess PorePressure vs. Axial Strain

12


jr

/J = 10

Observed Response

1 !

-"""^/S = 1

fi= 5

I1

i


JO 35 40

Figure A. 4. 2 - Effect of the Pore Pressure Response Factor on thePrincipal Stress Ratio vs. Axial Strain

13

(psi)

60 80II (psi)

120 140

40

35

30

25

q 20

(PS1)15

10

5

q vs. p

-

-

= 5

= 1

= 10 -*S=B-^1

11 1 1

/10 15 20 25

P' (psi)

30 35 40

Figure A. 4. 3 - Effect of the Pore Pressure Response Factor onlocation of the Cap in the J^l *i Space and onthe q-p" Diagram

A. 5 - Case 5

14


<p = 30°

Observed Response

p = 24°


30 35 40



Figure A. 5.1 - Effect of the Angle of Internal Friction on thePrincipal Stress Difference and Excess PorePressure vs. Axial Strain

15



Figure A. 5. 2 - Effect of the Angle of Internal Friction on thePrincipal Stress Ratio vs. Axial Strain

16

(psi)

40 60 80

h (psi)

140

q vs. p

15 20 25

P' (psi)

Figure A. 5. 3 - Effect of the Angle of Internal Friction onlocation of the Cap in the ^"^"li Space andthe q-p' Diagram

on

A. 6 - Case 6

17

60

a 50


Observed Response

10 15 20 25 30 35 40Axial Strain (%)

60

cr 50

u 403

&h 30


Observed Response

USR = 0.22

USR = 0.31


30 35 40

Figure A. 6.1 - Effect of the Undrained Shear Strength Ratio onthe Principal Stress Difference and Excess PorePressure vs. Axial Strain

18


usr = 0.43

5

Observed Response

10 15 20 25 30 35 40Axial Strain (%)

Figure A. 6. 2 - Effect of the Undrained Shear Strength Ratio onthe Principal Stress Ratio vs. Axial Strain

19

30

25

20 -

1/2J2 15

(psi)

10

1/2J2 vs. I

x

-

USR = 0.43

-

USR = 0.31

USR = 0.22

i i

20 40 60 80 100 120 140

h (psi)

40

35

30

25

q 20

(psi)15

10

5

q vs. p

10 20 30

P' (psi)40 50 60

Figure A. 6. 3 - Effect of the Undrained Shear Strength Ratio onlocation of the Cap in the J2 1/

' 2 ~I 1 sPace and onthe q-p 1 Diagram

A. 7 - Case 7

20


Observed Response

OCR = 1.49

10 15 20 25 30 35 40Axial Strain (%)

Excess Pore Pressure vs. Axial Strain40

35 -

30 -

25

0- 20

0. 15

01w

u 10Xw

-

OCR = 2.25

jT Observed Response

- /OCR = 1.0

-/^~~~~

OCR 1 10

'iii i i i


30 35 40

Figure A. 7.1 - Effect of the Over-consolidation Ratio on thePrincipal Stress Difference and Excess PorePressure vs. Axial Strain

21


OCR = 1.49

Observed Response

10 15 20 25 30 35 40Axial Strain (%)

Figure A. 7. 2 - Effect of the Over-consolidation Ratio on thePrincipal Stress Ratio vs. Axial Strain

22

60

50

u 2vs. h

-

40 -

1/2J2 30 OCR = 2.25

(psi)"1

20OCR = 1.49

OCR = 1.0 —•i

10

n i i i

1

\

20 40 60 80 100 120 140 160lj (psi)

40

35

30

25

q 20

(psi)

15

10

5

q vs. p

-

-OCR = 2.25

-

OCR= 1.49 /OCR = 1.0 N.

/

-

i

120 40

P' (psi)

60 80

Figure A. 7. 3 - Effect of the Over-consolidation Ratio on locationof the Cap in the J2 1//2

~li Space and on the q-p 1

Diagram

APPENDIX B

Computer disk with input data files

APPENDIX C

List of References

LIST OP REFERENCES 2 3

Holtz, R.D., and Kovacs, W.D. (1981), An Introduction toGeotechnical Engineering , Prentice-Hall, Inc., New Jersey, 733

pp.

Huang, T.K., and Chen, W.F. (1990), "Simple Procedure forDetermining Cap-Plasticity-Model Parameters," Journal ofGeotechnical Engineering , ASCE, Vol. 116, No. 3, pp. 492-513.

Huang, T.K., and Chen, W.F. (1991), "Embankment Widening andGrade Raising on Soft Foundation Soils: Computer ProgramImplementation," Report No. CE-STR-91-3 , School of CivilEngineering, Purdue University, West Lafayette, Indiana,January, 244 pp.

Huang, T.K., Chen, W.F., and Chameau, J.L. (1990), "TheApplication of Cap-Plasticity-Model to Embankment Problems,"Report No. CE-STR-90-21 , School of Civil Engineering, PurdueUniversity, West Lafayette, Indiana, 37 pp.

Humphrey, D.N. (1986), "Design of Reinforced Embankments,"Report No. JHRP-86-16 , Joint Highway Research Project, Schoolof Civil Engineering, Purdue University, West Lafayette,Indiana, October, 423 pp.

Kulhawy, F.H., and Mayne, P.W. (1990), "Manual on EstimatingSoil Properties for Foundation Design," Research Project 1493-6, Prepared for Electric Power Research Institute, Prepared byCornell University, Geotechnical Engineering Group, Ithaca,New York, August.

Ladd, C.C., Foote, R. , Ishihara, K., Schlosser, F., andPoulos, H.G., (1977), "Stress-Deformation and StrengthCharacteristics," State-of-the-Art Report, Proceedings of theNinth International Conference on Soil Mechanics andFoundation Engineering , Tokyo, Vol. 2, pp. 421-494.

Ludlow, S.J., Personal data base - "Engineering Properties ofSoils,".

Ludlow, S.J., Chen, W.F., Bourdeau, P.L., and Lovell, C.W.,(1991) , "Interim Report: Embankment Widening and Grade Raisingon Soft Foundation Soils: Example 1 - Indiana State Route 55over Turkey Creek in Lake County, Indiana," Report No. JHRP-91-18 , Joint Highway Research Project, School of CivilEngineering, Purdue University, West Lafayette, Indiana,December, 56 pp.

Nwabuokei, S.O., (1984), "Compressibility and Shear StrengthCharacteristics of Impact Compacted Lacustrine Clay," Ph.D.Thesis, Purdue University, West Lafayette, Indiana.

Appendix C:

Interim Report: "Embankment Widening and Grade Raising on SoftFoundation Soils: Example 1 - Indiana State Route 55 overTurkey Creek in Lake County, Indiana," Report No. FHWA/IN/JHRP- 91-18 , 56 pp.


FHWA/IN/JHRP - 91-18

Interim Report

Embankment Widening and GradeRaising on Soft Foundation Soils

Scott J. LudlowWai-Fah Chen

Philippe L. BourdeauC. William Lovell

Interim Report: "Embankment Widening and GradeRaising on Soft Foundation Soils:

Example 1 - Indiana State Route 55 overTurkey Creek in Lake County/ Indiana"

by




and






in Cooperation with


and the




Date: December 23, 1991

TABLE OP CONTENTS

Page

LIST OF TABLES

LIST OF FIGURES

APPENDICES

APPENDIX A -


Example 1 - Indiana State Route 55 over TurkeyCreek in Lake County, IndianaA.l - Rational for estimating cap/soil model

parametersA. 2 - Case 1

A. 3 - Case 2

A. 4 - Case 3

A. 5 - Case 4

A. 6 - Case 5

A. 7 - Case 6

A. 8 - Case 7

A. 9 - Case 8

A. 10 - Case 9

A. 11 - Case 10A. 12 - Case 11A. 13 - Case 12Computer disk with input data filesList of References

LIST OP TABLES

Table

1. Example 1 - Summary of Cases for Indiana StateRoute 55 over Turkey Creek/Lake County, Indiana

Page

LIST OF FIGURES

Figure

A. 2.1 Case 1 - Contours of Horizontal DisplacementA. 2. 2 Case 1 - Contours of Vertical Displacement ..

A. 2. 3 Case 1 - Contours of Principal Stress Ratio .

A. 2. 4 Case 1 - Contours of Local Factors of Safety


A. 3. 3 Case 2 - Contours of Principal Stress Ratio ,


A. 4.1 Case 3 - Contours of Horizontal DisplacementA. 4. 2 Case 3 - Contours of Vertical Displacement .,

A. 4. 3 Case 3 - Contours of Principal Stress Ratio ,



A. 5. 3 Case 4 - Contours of Principal Stress Ratio .


Case 5 - Contours of Horizontal DisplacementCase 5 - Contours of Vertical Displacement .

,

Case 5 - Contours of Principal Stress Ratio .

Case 5 - Contours of Local Factors of Safety


,

Case 6 - Contours of Principal Stress Ratio ,



,




,



A. 6.

A. 6.

A. 6.

A. 6,

A. 7.

A. 7,

A. 7,

A. 7,

A. 8.

A. 8.

A. 8,

A. 8,

A. 9.

A. 9.

A. 9,

A. 9,

9

1011

12131415

16171819

20212223

24252627

28293031

32333435

36373839

LIST OP FIGURES cont'd

Figure

A. 10 .1 Case 9

A. 10 2 Case 9

A, 10 3 Case 9

A, 10 4 Case 9

A. 11 1 Case 10A. 11 2 Case 10A. 11 3 Case 10A. 11 4 Case 10

A. 12 1 Case 11A. 12 2 Case 11A. 12 3 Case 11A. 12 4 Case 11

A. 13 1 Case 12A. 13 2 Case 12A. 13 3 Case 12

A. 13 4 Case 12

Contours of Horizontal DisplacementContours of Vertical Displacement .

.

Contours of Principal Stress Ratio ,

Contours of Local Factors of Safety-


.

Contours of Principal Stress Ratio ,

Contours of Local Factors of Safety


,

Contours of Principal Stress Ratio .



.

Contours of Principal Stress Ratio .


Page

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APPENDIX A

Example l - Indiana State Route 55 over TurkeyCreek in Lake County, Indiana

Rational for estimating cap/ soil model parameters

Objective: Estimate cap/ soil model parameters from indexproperty tests (test results summarized in a reportprepared by ETS of Indianapolis, Indiana on May 20,

1989) . For an explanation of the cap modelparameters, refer to Huang and Chen (1991)

.

I. Foundation Soil.

A. Determine the effective stress parameters for theultimate failure surface (a,K, Tc) .

Using a relationship developed by Mitchell fromisotropically consolidated, undrained triaxialcompression (CIUC) tests with pore water pressuremeasurements, (appropriate for normally-consolidated andremolded clays only)

sin<t>cv=0 . 8-0 . 094lnPJ

(first-order approximation)

where, ^cv , is the critical void ratio friction angle forinsensitive, uncemented normally-consolidated clay soilsand, PI , is the plasticity index. From the report, PIis 13. Therefore, 4^ is about 33°.

Based on the geology of northwestern Indiana, the type ofclay encountered at the site is likely lacustrine inorigin. A personal data base by the author for claysoils in this area indicates that the PI is likely nearthe lower bound for lacustrine clay soils in northwesternIndiana. The plasticity index typically ranges from 12to 18 averaging about 15. In addition, the liguid limittypically ranges from 30% to 38% averaging about 34%.

The ultimate failure surface used herein is described bythe Drucker-Prager criterion (which is a circular surfacein section unlike the Mohr-Coulomb surface which ishexagonal) . For triaxial compression, the materialconstants, a and k are related to <j> and c by:

a _ 2sin<t> 6ccos<{)

^(3-sin<t>) ''

v/3 (3-sin<t>)

For a plane strain condition, soils often display aslightly greater angle of internal friction than thoseobtained during a triaxial compression test. To simulatethis behavior, the material constants depend on the Lodeangle, 6 , (Dafalias and Herrmann, 1986) , according to:

a=a cg-(e,i7?) , K=Kcg(d,m)

and - „4„xm 3-sinpm= :—

r

3+sin<p

g(B,m)=- 2w1+/7?

Assume k=0 and a=0.209 from the above-mentionedrelationships

.

B. Determine the elastic and plastic parameters (Kmln ,A t ,Ae^v)

Km±n ; Because of the problems with the laboratorydetermination of undrained elastic modulus,^ , andbecause large-scale field loading tests are expensive, itis common to assume that£"u is somehow related to theundrained shear strength. For example, Bjerrum (1972)proposed that the ratio Eu/Su ranges from 500 to 1500,with Su determined by the vane shear test. However,there appears to be much scatter about the data. Anotherway to estimate îs from !?u test results. Based onthe laboratory test results, the undrained elasticmodulus varies from 9.4 to 25 ksf^ In addition, theundrained shear strength ratio, Su/a ol varies from 0.2 3

(in Boring TB-4) to . 8 (in Boring TB-2) . Eu appears tobe considerably low and is likely a result of sampledisturbance. In addition, the Cutests were performed onsplitspoon samples. Based on the author's personal database, Qu tests performed on lacustrine clays from similardepths and moisture contents, in which a piston sampler(Osterberg type) was used, indicate Rvalues as high as150 ksf. Use an Eu of 90 ksf (Eu for soft claystypically ranges from 100 to 500 ksf) . Therefore,

p.

3 (l-2v)

^ = 300 ksf

assuming v is 0.45.

A t ,A e ; No laboratory consolidation tests were performed,Recall that:

C*'" 2.303 (l + e )

and

A -Cr

e 2.303(l+eJ

Estimate e : eo=Z2_£; Gs typically ranges from

2.71 to 2.75. Say wc is about 26%.Therefore, eQ is approximately 0.7.

Estimate Cc ; Cc :

For Chicago clays: Cc=0.01(wc ) therefore, 0.26.

For undisturbed clays with low to mediumplasticity: Cc=0 . 009 (LL-10) ; LL implies the liguidlimit, approximately 34%. Therefore, Cc is 0.216.

Author's personal data base: Cc ranges from 0.2 to0.3, averaging about 0.246.

Use 0.246. Therefore, At is 0.0625.

Say Cr is 10% of Ce Therefore, 0.00625.

Values of Cr outside the range of 0.005 to 0.05should be considered guestionable.

C. Estimate the cap surface (R, OCR)

.

R; implies the cap aspect ratio (ratio of the majorand minor radii of an ellipse) . The cap aspectratio, R, can be evaluated provided that theDrucker-Prager constants, a, k, Kq , Su/a Q are known.Presently, NFAP (nonlinear finite element analysisprogram) makes provisions to evaluate R and thiswill not be discussed herein.

OCR; overconsolidation ratio. To accuratelyestimate the OCR of the lacustrine clay soils, aconsolidation test(s) should have been performed.Based on a comparison of the moisture contentvalues and Atterberg limits, the clay soils do notappear to be overly-consolidated. Although notsite specific, results from consolidation testsperformed on lacustrine clay soils fromnorthwestern Indiana indicate that the clay soilsmay be slightly over-consolidated. Refer to thedata input for the OCR values for analysis.

D. Initial stress state; (y 3imiK )

y SiW= 125 pcf

K" =(l-sin<t>) therefore, 0.44

Ko=0. 44+0. 42 (PI/100) therefore, 0.49.

Say 0.5.

Pore pressure response factor; (P)

Say 10 for undrained conditions otherwise fordrained. Note NFAP does not account fordissipation of pore water pressure. The porepressure response factor is actually a ratio of theapparent bulk modulus of the fluids to the total(soil and water) bulk modulus.

gUndrained shear strength ratio; (--£)

For normally- to slightly over-consolidated clays:

^=0.11 +0.0037 (PI)

S°•=£ typically ranges from 0.16 to 0.6.

Based on Boring TB-1 (sample depth/ interval; 3 3.5to 35 ft), say 0.36 (first-order approximation).Another technigue in estimating the undrained shearstrength ratio is to use SHANSEP (Ladd etal)

.

II. Embankment Fill.

For a first approximation of the embankment behavior,assume the soil conditions of the fill are similar tothose of the foundation soil near the surface.

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APPENDIX C

List of References

56

LIST OF REFERENCES

Geotechnical Investigation, (1989), Project No. St-4145(B),Structure No. 55-45-7366, "Indiana State Route 55 over TurkeyCreek in Lake County, Indiana," prepared by ETS, Inc. ofIndianapolis, Indiana, May, 10 pp.

Holtz, R.D., and Kovacs, W.D. (1981), An Introduction toGeotechnical Engineering , Prentice-Hall, Inc., New Jersey, 733pp.

Huang, T.K., and Chen, W.F. (1990), "Simple Procedure forDetermining Cap-Plasticity-Model Parameters," Journal ofGeotechnical Engineering . ASCE, Vol. 116, No. 3, pp. 492-513.

Huang, T.K. , and Chen, W.F. (1991), "Embankment Widening andGrade Raising on Soft Foundation Soils: Computer ProgramImplementation," Report No. CE-STR-91-3 . School of CivilEngineering, Purdue University, West Lafayette, Indiana,January, 244 pp.

Huang, T.K. , Chen, W.F., and Chameau, J.L. (1990), "TheApplication of Cap-Plasticity-Model to Embankment Problems,"Report No. CE-STR-90-21 . School of Civil Engineering, PurdueUniversity, West Lafayette, Indiana, 37 pp.

Humphrey, D.N. (1986), "Design of Reinforced Embankments,"Report No. JHRP-86-16 , Joint Highway Research Project, Schoolof Civil Engineering, Purdue University, West Lafayette,Indiana, October, 42 3 pp.

Kulhawy, F.H. , and Mayne, P.W. (1990), "Manual on EstimatingSoil Properties for Foundation Design," Research Project 1493-6, Prepared for Electric Power Research Institute, Prepared byCornell University, Geotechnical Engineering Group, Ithaca,New York, August.

Ladd, C.C., Foote, R. , Ishihara, K. , Schlosser, F. , andPoulos, H.G., (1977), "Stress-Deformation and StrengthCharacteristics," State-of-the-Art Report, Proceedings of theNinth International Conference on Soil Mechanics andFoundation Engineering . Tokyo, Vol. 2, pp. 421-494.



Appendix D:

Interim Report: "Embankment Widening and Grade Raising on SoftFoundation Soils: Example 2 - Indiana State Route 1 overRamsey Creek in Franklin County, Indiana," Report No

.

FHWA/IN/JHRP - 92-9 . 173 pp.


FHWA/IN/JHRP 92/9

Interim Report: Embankment Wideningand Grade Raising on Soft Foundation

SoilsExample 2 - Indiana State Route 1 Over

Ramsey Creek in Franklin County

Philippe Bourdeau,Wai-Fah Chen,C. William Lovell, andScott J. Ludlow

Interim Report: "Embankment Widening and GradeRaising on Soft Foundation Soils:

Example 2 - Indiana State Route 1 overRamsey Creek in Franklin County, Indiana"

by




and






in Cooperation with


and the




Date: March 31, 1991

TABLE OF CONTENTS

LIST OF TABLES ,

LIST OF FIGURES

Page

i

i

APPENDIX A

APPENDICES

Example 2 - Indiana State Route 1 over Ramsey Creekin Franklin County, IndianaA.l


Estimated Soil ResponsePlasticity ModelCase 1

2

3

4

5

6

7

8

9

101112131415161718

Computer disk with input data filesList of References

using Cap

A.

2

A.

3

A.

4

A.

5

A.

6

A.

7

A.

8

A.

9

A. 10A. 11A. 12A. 13A. 14A. 15A. 16A. 17A. 18A. 19

CaseCaseCaseCaseCaseCaseCaseCaseCaseCaseCaseCaseCaseCaseCaseCaseCase

LIST OF TABLES

Table

1.

Page

Example 2 - Summary of Cases for Indiana StateRoute 1 over Ramsey Creek/Franklin County, Indiana

LIST OF FIGURES

Figure

A .1 .1

A .2 .1A .2 .2A .2 .3A .2 .4

A..3 .1A,.3 .2

A,.3 .3

A,.3 .4

A..4 .1A,,4,.2A.,4..3A..4,.4

A.,5,.1

A.,5..2A.,5.,3A. 5.,4

A. 6.,1

A. 6.,2

A. 6..3

A. 6. 4

A. 7. 1A. 7 = 2A. 7. 3

A. 7. 4

A. 8. 1A. 8. 2

A. 8. 3

A. 8. 4

A. 9.1

Predicted Response - Principal Stress Difference andExcess Pore Pressure vs. Axial Strain 7

Case 1 - Contours of Horizontal Displacement ....... 13Case 1 - Contours of Vertical Displacement 14Case 1 - Contours of Principal Stress Ratio 15Case 1 - Contours of Local Factors of Safety 16

Case 2 - Contours of Horizontal Displacement ....... 22Case 2 - Contours of Vertical Displacement ......... 23Case 2 - Contours of Principal Stress Ratio ........ 24Case 2 - Contours of Local Factors of Safety 25

Case 3 - Contours of Horizontal Displacement ....... 31Case 3 - Contours of Vertical Displacement 32Case 3 - Contours of Principal Stress Ratio 3 3

Case 3 - Contours of Local Factors of Safety 34

Case 4 - Contours of Horizontal Displacement 40Case 4 - Contours of Vertical Displacement 41Case 4 - Contours of Principal Stress Ratio 42Case 4 - Contours of Local Factors of Safety ....... 43

Case 5 - Contours of Horizontal Displacement 50Case 5 - Contours of Vertical Displacement 51Case 5 - Contours of Principal Stress Ratio 52Case 5 - Contours of Local Factors of Safety 53



Case 8 - Contours of Horizontal Displacement 78

11


Figure

A. 9.

2

Case 8

A. 9.

3

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4

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A. 10.1 Case 9A. 10.2 Case 9

A. 10.

3

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A. 10.

4

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A. 11.1 Case 10A. 11.2 Case 10A. 11.

3

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A. 12.1 Case 11A. 12.2 Case 11A. 12.3 Case 11A. 12.

4

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A. 13.1 Case 12A. 13.

2

Case 12A. 13.

3

Case 12A. 13.

4

Case 12

A. 14.1 Case 13A. 14.

2

Case 13A. 14.

3

Case 13A. 14.

4

Case 13

A. 15.1 Case 14A. 15.2 Case 14A. 15.

3


A. 16.1 Case 15A. 16.2 Case 15A. 16.

3

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4

Case 15

A. 17.1 Case 16A. 17.2 Case 16A. 17.3 Case 16A. 17.4 Case 16

A. 18.1 Case 17A. 18.2 Case 17A. 18.

3


Pags

Contours of Vertical Displacement 79Contours of Principal Stress Ratio 80Contours of Local Factors of Safety 81

Contours of Horizontal Displacement 87Contours of Vertical Displacement 88Contours of Principal Stress Ratio 89Contours of Local Factors of Safety 9

Contours of Horizontal Displacement ....... 96Contours of Vertical Displacement 97Contours of Principal Stress Ratio 98Contours of Local Factors of Safety 99






Contours of Horizontal Displacement ...... 150Contours of Vertical Displacement 151Contours of Principal Stress Ratio 152Contours of Local Factors of Safety 153


Ill


Figure Page

A. 19.1 Case 18 - Contours of Horizontal Displacement 168A. 19.2 Case 18 - Contours of Vertical Displacement 169A. 19. 3 Case 18 - Contours of Principal Stress Ratio ....... 170A. 19. 4 Case 18 - Contours of Local Factors of Safety 171

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APPENDIX A

Indiana State Route 1 over Ramsey Creekin Franklin County, Indiana

A.l Estimated Soil Response using a CapPlasticity Model

Indiana State Route 1 over Ramsey Creek Franklin County, Indiana (10 psi)

1. 0.40 0.75 21. 5.0 0.25 0.05 .63 2.5 0.01 10.0 .730 10

900

0. 0.020 0. 0.

0. 0. 0. 0.


Observed Response

Calculated Response

Observed Calculated

Cc 0.25 0.25

Cr

0.05 0.05

fi 10 10

<t' 21° 21°

USB 63 63

OCR 25 2.5

Confining Pressure 10 psi

10 15 20 25

Axial Strain (%)

30 35 40

20

15


Calculated Response

Observed Response

15 20 25

Axial Strain (%)

30 35 40

Computer Program InputFigure A. 2.1 Case 1 - Contours of Horizontal DisplacementFigure A. 2. 2 Case 1 - Contours of Vertical DisplacementFigure A. 2. 3 Case 1 —Contours of Principal Stress RatioFigure A. 2. 4 Case 1 - Contours of Local Factors of Safety

State Route 1 over Ramsey Creek - Franklin County, Indiana Case 1

460 1 1 18 1

1 15 . 01

,0 o.: 9.0

1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

13 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

34 I625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

459 40.000 85.000

460 25.000 90.000

8

1

0. 640. -7.5 0. 0. 0.

0.

1

0. 0. -11.5 0. 125 0.0626 21

1

0. 640. -11.5 -7.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -27.5 -23.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -11.5 0. 130 0.0676 21

3

0. 295. 0. 90. 0. 0.

0.1 0.50

2 409 4 10 8 13 4 2 16

1

300. 0.4 0.7500 21. 0,.720 0.230 .039

8.0 0.01 0.000 10. 2.

2

300. 0.4 0.7500 21. 0,,720 0.250 .050

7.20 0.01 0.000 10. 2.

3

300. 0.4 0.7500 21. 0. 720 0.250 .050

4.9 0.01 0.000 10. 2.

4

300. 0.4 0.7500 21. 0. 720 0.250 0,.050

4.30 0.01 0.000 10. 2.

5

300. 0.4 0.7500 21. 0. 720 0.250 0,,050

3.8 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0. 720 0.250 0..050

3.40 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0..020

2.5 0.01 0.000 10. 2.

8

300. 0.4 0.7500 60. 0. 000 0.124 0. 020

20.0

1

0.01

4 1

0.125

1 8

0.

0.

2.

9 1 2 10

43 4 1 1 0.

345 337 338 346

44 4 1 2 8 0.

10 2 3 11

86 4 1 2 0.

8.0

7.20

4,9

4.30

3.8

3.40

2,5

1.58

1.46

1.07

0.96

0.87

0.80

0.63

1.58

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 384 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

418 417 401 402

354 4 1 8 2.

405 418 402 403

355 4 1 8 2.

405 405 403 404

356 4 1 8 3.

365 356 355 364

357 4 1 8 3.

420 365 364 406

358 4 1 8 1 3.

421 420 406 407

367 4 1 8 3.

430 429 415 416

368 4 1 8 3.

419 430 416 417

369 4 1 8 3.

419 419 417 418

370 4 1 8 4.

366 357 356 365

371 4 1 8 4.

432 366 365 420

372 4 1 8 1 4.

433 432 420 421

379 4 1 8 4.

440 439 427 428

380 4 1 8 4.

431 440 428 429

381 4 1 8 4.

431 431 429 430

382 4 1 8 5.

367 358 357 366

383 4 1 8 5.

442 367 366 432

384 4 1 8 1 5.

443 442 432 433

389 4 1 8 5.

448 447 437 438

390 4 1 8 5.

441 448 438 439

391 4 1 8 5.

10

346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

11

441 441 439

392 4 1

368 359 358

393 4 1

450 368 367

394 4 1

451 450 442

397 4 1

454 453 445

398 4 1

449 454 446

399 4 1

449 449 447

400 4 1

369 360 359

401 4 1

456 369 368

402 4 1

45^ 456 450

403 4 1

458 457 451

404 4 1

455 458 452

405 4 1

455 455 453

406 4 1

370 361 360

407 4 1

460 570 369

408 4 1

459 460 456

409 4 1

459 459 457

440

8 6.

367

8 6.

442

8 1 6.

443

8 6.

446

8 6.

447

8 6.

448

8 7.

368

8 7.

450

8 7.

451

8 7.

452

8 7.

453

8 7.

454

8 8.

369

8 8.

456

8 8.

457

8 8.

458

12

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Computer Program InputFigure A. 3.1 Case 2 - Contours of Horizontal DisplacementFigure A. 3. 2 Case 2 - Contours of Vertical DisplacementFigure A. 3. 3 Case 2 - Contours of Principal Stress RatioFigure A. 3. 4 Case 2 - Contours of Local Factors of Safety


17

460 1 1 13 1

1 15 . 01

,0 0.! 9.0

1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

459 40.000 85.000

460 25.000 90.000

8

1

0. 640. -7.5 0. 0. 0.

0. 0. 0. -11.5 0. 125 0.0626 21

1

0. 640. -11.5 -7.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -27.5 -23.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -11.5 0. 130 0.0676 21

3

0. 295. 0. 90. 0, 0.

0.1 0.50

2 409 4 10 8 13 4 2 16

1

300. 0.4 0.7500 21. 0,.720 0.230 0.039

8.0 0.01 0.000 10. 2.

2

300. 0.4 0.7500 21. .720 0.250 0.050

7.20 0.01 0.000 10. 2

3

300. 0.4 0.7500 21. 0. 720 0.250 0.050

4.9 0.01 0.000 10. 2.

4

300. 0.4 0.7500 21. 0. 720 0.250 0.050

4.30 0.01 0.000 10. 2.

5

300. 0.4 0.7500 21. 0, 720 0.250 0.050

3.8 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.40 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0, 720 0.124 0.020

2.5 0.01 0.000 10. 2.

8

300. 0.4 0.7500 60. 0. 000 0.124 0.020

20.0

1

0.01

4 1

0.125

1 8

0.

0.

2.

9 1 2 10

43 4 1 1 0.

345 337 338 346

44 4 1 2 8 0.

10 2 3 11

86 4 1 2 0.

18

8.0

7.20

4.9

4.30

3,8

3.40

2.5

1.58

1.46

1.07

0.96

0.87

0.80

0.63

1.58

346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

19

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 384 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

418 417 401 402

354 4 1 8 2.

405 418 402 403

355 4 1 8 2.

405 405 403 404

356 4 1 8 3.

365 356 355 364

357 4 1 8 3.

420 365 364 406

358 4 1 8 1 3.

421 420 406 407

367 4 1 8 3.

430 429 415 416

368 4 1 8 3.

419 430 416 417

369 4 1 8 3.

419 419 417 418

370 4 1 8 4.

366 357 356 365

371 4 1 8 4.

432 366 365 420

372 4 1 8 1 4.

433 432 420 421

379 4 1 8 4.

440 439 427 428

380 4 1 8 4.

431 440 428 429

381 4 1 8 4.

431 431 429 430

382 4 1 8 5.

367 358 357 366

383 4 1 8 5.

442 367 366 432

384 4 1 8 1 5.

443 442 432 433

389 4 1 8 5.

448

390

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441 448 438 439

391 4 1 8 5.

20

441 441 439 440

392 4 1 8 6.

368 359 358 367

393 4 1 8 6.

450 368 367 442

394 4 1 8 1 6.

451 450 442 443

397 4 1 8 6.

454 453 445 446

398 4 1 8 6.

449 454 446 447

399 4 1 8 6.

449 449 447 448

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369 360 359 368

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1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

2? 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

544 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

459 40.000 85.000

460 25.000 90.000

8

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0. 640. -7.5 0. 0. 0.

0. 0. 0. -7.5 0. 125 0.0626 21

0. 640. -11.5 -7.51 0. 0.

0. 0. 0. -7.5 0. 120 0.0576 21

2

0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -7.5 0. 120 0.0576 21

2

0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -7.5 0. 120 0.0576 21

2

0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -7.5 0. 120 0.0576 21

2

0. 640. -27.5 -23.51 0. 0.

0. 0. 0. -7.5 0, 120 0.0576 21.

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -7.5 0. 130 0.0676 21,

3

0. 295. 0. 90. 0. 0.

0.1 0.50

2

1

409 4 10 8 13 4 2 16

1

300. 0.4 0.7500 21. .720 0.230 0.039

8.0 0.01 0.000 10. 2.

2

300. 0.4 0.7500 21. ,720 0.250 0.050

7.20 0.01 0.000 10. 2.

3

300. 0.4 0.7500 21. 0.,720 0.250 0.050

4.9 0.01 0.000 10. 2,

4

300. 0.4 0.7500 21. 0,,720 0.250 0.050

4.30 0.01 0.000 10. 2.

5

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.8 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0, 720 0.250 0.050

3.40 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0.020

2.5 0.01 0.000 10. 2.

8

300. 0.4 0.7500 60. 0. 000 0.124 0.020

20.0

1

0.01

4 1

0.125

1 8

0.

0.

2.

9 1 2 10

43 4 1 1 0.

345 337 338 346

44 4 1 2 8 0.

10 2 3 11

86 4 1 2 0.

27

8.0

7.20

4,9

4.30

3.8

3.40

2.5

1.58

1.46

1.07

0.96

0.87

0.80

0.63

1.58

346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

28

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 384 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

418 417 401 402

354 4 1 8 2.

405 418 402 403

355 4 1 8 2.

405 405 403 404

356 4 1 8 3.

365 356 355 364

357 4 1 8 3.

420 365 364 406

358 4 1 8 1 3.

421 420 406 407

367 4 1 8 3.

430 429 415 416

368 4 1 8 3.

419 430 416 417

369 4 1 8 3.

419 419 417 418

370 4 1 8 4.

366 357 356 365

371 4 1 8 4.

432 366 365 420

372 4 1 8 1 4.

433 432 420 421

379 4 1 8 4.

440 439 427 428

380 4 1 8 4.

431 440 428 429

381 4 1 8 4.

431 431 429 430

382 4 1 8 5.

367 358 357 366

383 4 1 8 5.

442 367 366 432

384 4 1 8 1 5.

443 442 432 433

389 4 1 8 5.

448 447 437 438

390 4 1 8 5.

441 448 438 439

391 4 1 8 5.

29

441 441 439 440

392 4 1 8 6.

36S 359 358 367

393 4 1 8 6.

450 368 367 442

394 4 1 8 1 6.

451 450 442 443

397 4 1 8 6.

454 453 445 446

398 4 1 8 6.

449 454 446 447

399 4 1 8 6.

449 449 447 448

400 4 1 8 7.

369 360 359 366

401 4 1 8 7.

456 369 368 450

402 4 1 8 7.

457 456 450 451

403 4 1 8 7.

458 457 451 452

404 4 1 8 7.

455 458 452 453

405 4 1 8 7.

455 455 453 454

406 4 1 8 8.

370 361 360 369

407 4 1 8 8.

460 370 369 456

408 4 1 8 8.

459 460 456 457

409 4 1 8 8.

459 459 457 458

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Computer Program InputFigure A. 5.1 Case 4 - Contours of Horizontal DisplacementFigure A. 5. 2 Case 4 - Contours of Vertical DisplacementFigure A. 5. 3 Case 4 - Contours of Principal Stress RatioFigure A* 5. 4 Case 4 - Contours of Local Factors of Safety


460 2 1 18 1

1 15 .01

0.0 0.5 9.0

1 1 0.000 0.000

2 1 0.000 -7.500 1

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000 8

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500 1

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500 8

338 625.000 -7.500

346 1 640.000 -7.500 1

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500 8

339 625.000 -11.500

20 25.000 -15.500 8

340 625.000 -15.500

21 25.000 -19.500 8

341 625.000 -19.500

22 25.000 -23.500 8

342 625.000 -23.500

23 25.000 -27.500 8

343 625.000 -27.500

24 1 25.000 -35.000 8

344 1 625.000 -35.000

353 1 0.000 10.000 1

361 1 0.000 90.000

362 12.500 10.000 1

370 12.500 90.000

371 25.000 10.000 1

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000 1

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000 1

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000 1

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000 1

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000 1

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000 1

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000 1

458 55.000 80.000

35

459 40.000 85.000

460 25.000 90.000

8

1

0. 640. -7.5 0. 0. 0.

0.

1

0. 0. -11.5 0. 125 0.0626 21

0. 640. -11.5 -7.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -27.5 -23.51 0, 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -11.5 0. 130 0.0676 21,

3

0. 295. 0. 90. 0. 0.

0.1 0.50

2 409 4 10 8 13 4 2 16

1

300. 0.4 0.7500 21. ,720 0.230 0.039

8.0 0.01 0.000 10. 2

2

300. 0.4 0.7500 21. 0.,720 0.250 0.050

7.20 0.01 0.000 10. 2.

3

300. 0.4 0.7500 21. 720 0.250 0.050

4.9 0.01 0.000 10. 2,

4

300. 0.4 0.7500 21. 0..720 0.250 0.050

4.30 0.01 0.000 10. 2.

5

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.8 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.40 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0.020

2.5 0.01 0.000 10. 2.

8

300. 0.4 0.7500 60. 0. 000 0.124 0.020

20.0

1

0.01

4 1

0.125

1 8

0.

0.

2.

9 1 2 10

43 4 1 1 0.

345 337 338 346

44 4 1 2 8 0.

10 2 3 11

86 4 1 2 0.

36

8.0

7.20

4.9

4.30

3.8

3.40

2.5

1.58

1.46

1.07

0.96

0.87

0.80

0.63

1.58

346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

37

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 384 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

418 417 401 402

354 4 1 8 2.

405 418 402 403

355 4 1 8 2.

405 405 403 404

356 4 1 8 3.

365 356 355 364

357 4 1 8 3.

420 365 364 406

358 4 1 8 1 3.

421 420 406 407

367 4 1 8 3.

430 429 415 416

368 4 1 8 3.

419 430 416 417

369 4 1 8 3.

419 419 417 418

370 4 1 8 4.

366 357 356 365

371 4 1 8 4.

432 366 365 420

372 4 1 8 1 4.

433 432 420 421

379 4 1 8 4.

440 439 427 428

380 4 1 8 4.

431 440 428 429

381 4 1 8 4.

431 431 429 430

382 4 1 8 5.

367 358 357 366

383 4 1 8 5.

442 367 366 432

384 4 1 8 1 5.

443 442 432 433

389 4 1 8 5.

448 447 437 438

390 4 1 8 5.

441 448 438 439

391 4 1 8 5.

38

441 441 439 440

392 4 1 8 6.

368 359 358 367

393 4 1 8 6.

450 368 367 442

394 4 1 8 1 6.

451 450 442 443

397 4 1 8 6.

454 453 445 446

398 4 1 8 6.

449 454 446 447

399 4 1 8 6.

449 449 447 448

400 4 1 8 7.

369 360 359 368

401 4 1 8 7.

456 369 368 450

402 4 1 8 7,

457 456 450 451

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346 1 640.000 -7.500

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19 25.000 11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

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24 1 25.000 -35.000

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371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 173.000 40.000

431 160.000 45.000

43? 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

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216 4 1 6 8 0.

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375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

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9 65 73

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11 81 89

12 89 97

13 97 105

14 105 113

15 113 121

16 121 129

17 129 137

18 137 145

19 145 153

20 153 161

48

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23 406 407 2

24 407 408 2

25 408 409 2

26 409 410 2

27 410 411 2

28 411 412 2

29 412 413 2

30 413 414 2

31 414 415 2

32 415 416 2

33 416 417 2

34 417 418 2

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36 367 442 3

37 442 443 3

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Computer Program InputFigure A. 7.1 Case 6 - Contours of Horizontal DisplacementFigure A. 7. 2 Case 6 - Contours of Vertical DisplacementFigure A .7, 3 Case 6 - Contours of Principal Stress RatioFigure A. 7. 4 Case 6 - Contours of Local Factors of Safety


460 2 1 18 1

1 15 .01

0.0 0,.5 9.0

1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

54

459 40.000 85.000

460 25.000 90.000

8

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0. 0. 0. -7.5 0. 120 0.0576 21

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0. 0. 0. -7.5 0. 120 0.0576 21

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0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -7.5 0. 130 0.0676 21.

3

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0.1 0.50

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409 4 10 8 13 4 2 16

1

300, 0.4 0.7500 21. .720 0.230 0.039

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7.20 0.01 0.000 10. 2

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4.9 0.01 0.000 10. 2

4

300. 0.4 0.7500 21. 720 0.250 0.050

4.30 0.01 0.000 10. 2

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300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.8 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.40 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0.020

2.5 0.01 0.000 10. 2.

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345 337 338 346

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1.07

0.96

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346 338 339 347

87 4 1 3 8 0.

11 3 4 12

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347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

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349 341 342 350

216 4 1 6 8 0.

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350 342 343 351

259 4 1 7 8 0.

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351 343 344 352

302 4 1 8 0.01

362 353 1 9

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371 362 9 17

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372 371 17 25

305 4 1 8 0.01

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374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

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0.0 0,.5 9.0

1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

459 40.000 85.000

460 25.000 90.000

8

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8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

459 40.000 85.000

460 25.000 90.000

8

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0. 640. -7.5 0. 0. 0.

0. 0. 0. -11.5 0. 125 0.0626 21

1

0. 640. -11.5 -7.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -27.5 -23.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -11.5 0. 130 0.0676 21

3

0. 295. 0. 90. 0. 0.

0.1 0.50

2 409 4 10 8 13 4 2 16

1

300. 0.4 0.7500 21. 0.,720 0.230 .039

8.0 0.01 0.000 10. 2.

2

300. 0.4 0.7500 21. 0..720 0.250 .050

7.20 0.01 0.000 10. 2.

3

300. 0.4 0.7500 21. 0. 720 0.250 0..050

4.9 0.01 0.000 10. 2.

4

300. 0.4 0.7500 21. 0. 720 0.250 0,.050

.4.30 0.01 0.000 10. 2.

5

300. 0.4 0.7500 21. 0. 720 0.250 0,.050

3.8 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0.720 0.250 0.,050

3.40 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0.,020

2.5 0.01 0.000 10. 2.

8

300. 0.4 0.7500 60. 0. 000 0.124 0.,020

8.0

1

0.01

4 1

0.125

1 8

0.

0.

2.

9 1 2 10

43 4 1 1 0.

345 337 338 346

44 4 1 2 8 0.

10 2 3 11

86 4 1 2 0.

110

8.0

7.20

4.9

4.30

3.8

3.40

2.5

1.58

1.46

1.07

0.96

0.87

0.80

0.63

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346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

111

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 384 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

418 417 401 402

354 4 1 8 2.

405 418 402 403

355 4 1 8 2.

405 405 403 404

356 4 1 8 3.

365 356 355 364

357 4 1 8 3.

420 365 364 406

358 4 1 8 1 3.

421 420 406 407

367 4 1 8 3.

430 429 415 416

368 4 1 8 3.

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369 4 1 8 3.

419 419 417 418

370 4 1 8 4.

366 357 356 365

371 4 1 8 4.

432 366 365 420

372 4 1 8 1 4.

433 432 420 421

379 4 1 8 4.

440 439 427 428

380 4 1 8 4.

431 440 428 429

381 4 1 8 4.

431 431 429 430

382 4 1 8 5.

367 358 357 366

383 4 1 8 5.

442 367 366 432

384 4 1 8 1 5.

443 442 432 433

389 4 1 8 5.

448 447 437 438

390 4 1 8 5.

441 448 438 439

391 4 1 8 5.

112

441 441 439 440

392 4 1 8 6.

368 359 358 367

393 4 1 8 6.

450 368 367 442

394 4 1 8 1 6.

451 450 442 443

397 4 1 8 6.

454 453 445 446

398 4 1 8 6.

449 454 446 447

399 4 1 8 6.

449 449 447 448

400 4 1 8 7.

369 360 359 368

401 4 1 8 7.

456 369 368 450

402 4 1 8 7.

457 456 450 451

403 4 1 8 7.

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370 361 360 369

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5 33 41

6 41 49

7 49 57

8 57 65

9 65 73

10 73 81

11 81 89

12 89 97

13 97 105

14 105 113

15 113 121

16 121 129

17 129 137

18 137 145

19 145 153

20 153 161

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460 2 1 18 1

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1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

118

459 40.000 85.000

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8

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0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

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0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

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0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21,

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0. 640. -27.5 -23.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21.

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -11.5 0. 130 0.0676 21.

3

0. 295. 0. 90. 0. 0.

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409 4 10 8 13 4 2 16

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300. 0.4 0.7500 21. 0..720 0.230 0.039

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300. 0.4 0.7500 21. 0.,720 0.250 0.050

4.9 0.01 0.000 10. 2.

4

300. 0.4 0.7500 21. 0. 720 0.250 0.050

4.30 0.01 0.000 10. 2.

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300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.8 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.40 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0.020

2.5 0.01 0.000 10. 2.

8

300. 0.4 0.7500 60. 0. 000 0.124 0.020

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1

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9 1 2 10

43 4 1 1 0.

345 337 338 346

44 4 1 2 8 0.

10 2 3 11

86 4 1 2 0.

119

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7.20

4.9

4.30

3.8

3.40

2.5

1.58

1.46

1.07

0.96

0.87

0.80

0.63

1.58

346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

120

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 384 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

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406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

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354 4 1 8 2.

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358 4 1 8 1 3.

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369 4 1 8 3.

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379 4 1 8 4.

440 439 427 428

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383 4 1 8 5.

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389 4 1 8 5.

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390 4 1 8 5.

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391 4 1 8 5.

121

441 441 439 440

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368 359 358 367

393 4 1 8 6.

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394 4 1 8 1 6.

451 450 442 443

397 4 1 8 6.

454 453 445 446

398 4 1 8 6.

449 454 446 447

399 4 1 8 6.

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400 4 1 8 7.

369 360 359 368

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457 456 450 451

403 4 1 8 7.

458 457 451 452

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455 455 453 454

406 4 1 8 8.

370 361 360 369

407 4 1 8 8.

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9 65 73

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11 81 89

12 89 97

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Computer Program InputFigure A. 15.1 Case 14 - Contours of Horizontal DisplacementFigure A. 15.2 Case 14 - Contours of Vertical DisplacementFigure A. 15.3 Case 14 - Contours of Principal Stress RatioFigure A. 15.4 Case 14 - Contours of Local Factors of Safety


127

460 2 1 18 1

1 15 .01

0.0 .5 9.0

1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

. 390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

459 40.000 85.000

460 25.000 90.000

8

11

0. 640. -7.5 0. 0. 0.

0.

1

0. 0. -11.5 0. 125 0.0626 21

0. 640. -11.5 -7.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21,

2

0. 640. -27.5 -23.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21.

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -11.5 0. 130 0.0676 21.

3

0. 295. 0. 90. 0. 0.

0.1 0.50

2

1

409 4 10 8 13 4 2 16

300. 0.4 0.7500 21. 0..720 0.230 0.039

10.0 0.01 0.000 10. 2

2

300. 0.4 0.7500 21. 0..720 0.250 0.050

7.20 0.01 0.000 10. 2.

3

300. 0.4 0.7500 21. 0. 720 0.250 0.050

4.9 0.01 0.000 10. 2.

4

300. 0.4 0.7500 21. 0. 720 0.250 0.050

•4.30 0.01 0.000 10. 2.

5

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.8 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.40 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0.020

2.5 0.01 0.000 10. 2.

8

300. 0.4 0.7500 60. 0. 000 0.124 0.020

20.0

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43 4 1 1 0.

345 337 338 346

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10 2 3 11

86 4 1 2 0.

128

10.0

7.20

4.9

4.30

3.8

3.40

2.5

1.90

1.46

1.07

0.96

0.87

0.80

0.63

1.58

346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

129

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 354 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

418 417 401 402

354 4 1 8 2.

405 418 402 403

355 4 1 8 2.

405 405 403 404

356 4 1 8 3.

365 356 355 364

357 4 1 8 3.

420 365 364 406

358 4 1 8 1 3.

421 420 406 407

367 4 1 8 3.

430 429 415 416

368 4 1 8 3.

419 430 416 417

369 4 1 8 3.

419 419 417 418

370 4 1 8 4.

366 357 356 365

371 4 1 8 4.

432 366 365 420

372 4 1 8 1 4.

433 432 420 421

379 4 1 8 4.

440 439 427 428

380 4 1 8 4.

431 440 428 429

381 4 1 8 4.

431 431 429 430

382 4 1 8 5.

367 358 357 366

383 4 1 8 5.

442 367 366 432

384 4 1 8 1 5.

443 442 432 433

389 4 1 8 5.

448 447 437 438

390 4 1 8 5.

441 448 438 439

391 4 1 8 5.

130

441 441 439 440

392 4 1 8 6.

368 359 358 367

393 4 1 8 6.

450 368 367 442

394 4 1 8 1 6.

451 450 442 443

397 4 1 8 6.

454 453 445 446

398 4 1 8 6.

449 454 446 447

399 4 1 8 6.

449 449 447 448

400 4 1 8 7.

369 360 359 368

401 4 1 8 7.

456 369 368 450

402 4 1 8 7.

457 456 450 451

403 4 1 8 7.

458 457 451 452

404 4 1 8 7.

455 458 452 453

405 4 1 8 7.

455 455 453 454

406 4 1 8 8.

370 361 360 369

407 4 1 8 8.

460 370 369 456

408 4 1 8 8.

459 460 456 457

409 4 1 8 8.

459 459 457 458

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3 17 25

4 25 33

5 33 41

6 41 49

7 49 57

8 57 65

9 65 73

10 73 81

11 81 89

12 89 97

13 97 105

14 105 113

15 113 121

16 121 129

17 129 137

18 137 145

19 145 153

20 153 161

131

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136

460 2 1 18 1

1 15 .01

0.0 .5 9.0

1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

459 40.000 85.000

460 25.000 90.000

8

1

0. 640. -7.5 0. 0. 0.

0. 0. 0. -11.5 0. 125 0.0626 21

0. 640. -11.5 -7.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21.

2

0. 640. -27.5 -23.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21,

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -11.5 0. 130 0.0676 21.

3

0. 295. 0. 90. 0. 0.

0.1 0.50

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1

409 4 10 8 13 4 2 16

1

300. 0.4 0.7500 21. .720 0.230 0.039

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7.20 0.01 0.000 10. 2

3

300. 0.4 0.7500 21. 0..720 0.250 0.050

4.9 0.01 0.000 10. 2.

4

300. 0.4 0.7500 21. 0. 720 0.250 0.050

-4.30 0.01 0.000 10. 2.

5

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.8 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0. 720 0.250 0.050

3.40 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0.020

2.5 0.01 0.000 10. 2.

8

300. 0.4 0.7500 60. 0. 000 0.124 0.020

20.0

1

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9 1 2 10

43 4 1 1 0.

345 337 338 346

44 4 1 2 8 0.

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86 4 1 2 0.

137

10.0

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4.9

4.30

3.8

3.40

2.5

1.90

1.46

1.07

0.96

0.87

0.80

0.63

1.58

346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

138

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 384 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

418 417 401 402

354 4 1 8 2.

405 418 402 403

355 4 1 8 2.

405 405 403 404

356 4 1 8 3.

365 356 355 364

357 4 1 8 3.

420 365 364 406

358 4 1 8 1 3.

421 420 406 407

367 4 1 8 3.

430 429 415 416

368 4 1 8 3.

419 430 416 417

369 4 1 8 3.

419 419 417 418

370 4 1 8 4.

366 357 356 365

371 4 1 8 4.

432 366 365 420

372 4 1 8 1 4.

433 432 420 421

379 4 1 8 4.

440 439 427 428

380 4 1 8 4.

431 440 428 429

381 4 1 8 4.

431 431 429 430

382 4 1 8 5.

367 358 357 366

383 4 1 8 5.

442 367 366 432

384 4 1 8 1 5.

443 442 432 433

389 4 1 8 5.

448 447 437 438

390 4 1 8 5.

441 448 438 439

391 4 1 8 5.

139

441 441 439 440

392 4 1 8 6.

368 359 358 367

393 4 1 8 6.

450 368 367 442

394 4 1 8 1 6.

451 450 442 443

397 4 1 8 6.

454 453 445 446

398 4 1 8 6.

449 454 446 447

399 4 1 8 6.

449 449 447 448

400 4 1 8 7.

369 360 359 368

401 4 1 8 7.

456 369 368 450

402 4 1 8 7.

457 456 450 451

403 4 1 8 7.

458 457 451 452

404 4 1 8 7.

455 458 452 453

405 4 1 8 7.

455 455 453 454

406 4 1 8 8.

3/0 361 360 369

407 4 1 8 8.

460 370 369 456

408 4 1 8 8.

459 460 456 457

409 4 1 8 8.

459 459 457 458

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2 9 17

3 17 25

4 25 33

5 33 41

6 41 49

7 49 57

8 57 65

9 65 73

10 73 81

11 81 89

12 89 97

13 97 105

14 105 113

15 113 121

16 121 129

17 129 137

18 137 145

19 145 153

20 153 161

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460 2 1 18 1

1 15 .01

0.0 ,5 9.0

1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

145

459 40.000 85.000

460 25.000 90.000

8

41

0. 640. -7.5 0. 0. 0.

0. 0. 0. -11.5 0. 125 0.0626 21

0. 640. -11.5 -7.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21.

2

0. 640. -27.5 -23.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21,

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -11.5 0. 130 0.0676 21.

3

0. 295. 0. 90. 0. 0.

0.1 0.50

2 409 4 10 8 13 4 2 16

1

300. 0.4 0.7500 21. 0.,720 0.230 0.039

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300. 0.4 0.7500 21. 0..720 0.250 0.050

3.00 0.01 0.000 10. 2.

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300. 0.4 0.7500 21. 0. 720 0.250 0.050

2.50 0.01 0.000 10. 2.

4

300. 0.4 0.7500 21. 0.720 0.250 0.050

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5

300. 0.4 0.7500 21. 0. 720 0.250 0.050

1.50 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0. 720 0.250 0.050

1.50 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0.020

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8

300. 0.4 0.7500 60. 0. 000 0.124 0.020

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43 4 1 1 0.

345 337 338 346

44 4 1 2 8 0.

10 2 3 11

86 4 1 2 0.

146

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2.50

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1.50

1.50

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1.90

0.73

0.63

0.53

0.42

0.42

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1.58

346 338

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129

347 339

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339 347

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172

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216 4

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301

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362 353

303 4

371 362

304 4

372 371

305 4

373 372

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1 7

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306

374

307

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373

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375 374

308 4

376 375

309 4

377 376

310 4

378 377

311 4

379 378

312 4

380 379

313 4

381 380

314 4

382 381

315 4

383 382

316 4

384 383

317 4

385 384

318 4

386 385

319 4

387 386

320 4

388 387

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363 354 353 362

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391 390 371 372

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338 4 1 8 1.

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339 4 1 8 1.

389 389 386 387

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364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

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353 4 1 8 2.

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354 4 1 8 2.

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365 356 355 364

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358 4 1 8 1 3.

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367 4 1 8 3.

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370 4 1 8 4.

366 357 356 365

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379 4 1 8 4.

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380 4 1 8 4.

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381 4 1 8 4.

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367 358 357 366

383 4 1 8 5.

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384 4 1 8 1 5.

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148

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363 359 358 367

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11 81 89

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16 121 129

17 129 137

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345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

538 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

34

1

625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

388 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

154

459 40.000 85.000

460 25.000 90.000

8

5

0. 640. -7.5 0. 0. 0.

0. 0. 0. -11.5 0. 125 0.0626 21

0. 640. -11.5 -7.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -15.5 -11.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21

2

0. 640. -19.5 -15.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21,

2

0. 640. -23.5 -19.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21.

2

0. 640. -27.5 -23.51 0. 0.

0. 0. 0. -11.5 0. 120 0.0576 21.

2

0. 640. -35.0 -27.51 0. 0.

0. 0. 0. -11.5 0. 130 0.0676 21.

3

0. 295. 0. 90. 0. 0.

0.1 0.50

2

1

409 4 10 8 13 4 2 16

1

300. 0.4 0.7500 21. .720 0.230 0.039

10.0 0.01 0.000 10. 2

2

300. 0.4 0.7500 21. 0..720 0.250 0.050

3.00 0.01 0.000 10. 2.

3

300. 0.4 0.7500 21. 0..720 0.250 0.050

2.50 0.01 0.000 10. 2.

4

300. 0.4 0.7500 21. 0.,720 0.250 0.050

2.00 0.01 0.000 10. 2.

5

300. 0.4 0.7500 21. 0. 720 0.250 0.050

1.50 0.01 0.000 10. 2.

6

300. 0.4 0.7500 21. 0. 720 0.250 0.050

1.50 0.01 0.000 10. 2.

7

300. 0.4 0.6630 21. 0. 720 0.124 0.020

1.02 0.01 0.000 10. 2.

8

300. 0.4 0.7500 60. 0. 000 0.124 0.020

8.0

1

0.01

4 1

0.125

1 8

0.

0.

2.

9 1 2 10

43 4 1 1 0.

345 337 338 346

44 4 1 2 8 0.

10 2 3 11

86 4 1 2 0.

155

3.00

2.5

2.00

1.5

1.50

1.02

1.90

0.73

0.63

0.53

0.42

0.42

0.30

1.58

346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

. 387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

156

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 384 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

418 417 401 402

354 4 1 8 2.

405 418 402 403

355 4 1 8 2.

405 405 403 404

356 4 1 8 3.

365 356 355 364

357 4 1 8 3.

420 365 364 406

358 4 1 8 1 3.

421 420 406 407

367 4 1 8 3.

430 429 415 416

368 4 1 8 3.

419 430 416 417

369 4 1 8 3.

419 419 417 418

370 4 1 8 4.

366 357 356 365

371 4 1 8 4.

432 366 365 420

372 4 1 8 1 4.

433 432 420 421

379 4 1 8 4.

440 439 427 428

380 4 1 8 4.

431 440 428 429

381 4 1 8 4.

431 431 429 430

382 4 1 8 5.

367 358 357 366

383 4 1 8 5.

442 367 366 432

384 4 1 8 1 5.

443 442 432 433

389 4 1 8 5.

448 447 437 438

390 4 1 8 5.

441 448 438 439

391 4 1 8 5.

157

441 441 439 440

392 4 1 8 6.

368 359 358 367

393 4 1 8 6.

450 368 367 442

394 4 1 8 1 6.

451 450 442 443

397 4 1 8 6.

454 453 445 446

398 4 1 8 6.

449 454 446 447

399 4 1 8 6.

449 449 447 448

400 4 1 8 7.

369 360 359 368

401 4 1 8 7.

456 369 368 450

402 4 1 8 7.

457 456 450 451

403 4 1 8 7.

458 457 451 452

404 4 1 8 7.

455 458 452 453

405 4 1 8 7.

455 455 453 454

406 4 1 8 8.

370 361 360 369

407 4 1 8 8.

460 370 369 456

408 4 1 8 8.

459 460 456 457

409 4 1 8 8.

459 459 457 458

1 20 2 2 1 8

1

0.

1 1 9

067 083 .16

2 9 17

3 17 25

4 25 33

5 33 41

6 41 49

7 49 57

8 57 65

9 65 73

10 73 81

11 81 89

12 89 97

13 97 105

14 105 113

15 113 121

16 121 129

17 129 137

18 137 145

19 145 153

20 153 161

158

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Computer Program InputFigure A. 19.1 Case 18 - Contours of Horizontal DisplacementFigure A. 19. 2 Case 18 - Contours of Vertical DisplacementFigure A. 19. 3 Case 18 - Contours of Principal Stress RatioFigure A. 19.4 Case 18 - Contours of Local Factors of Safety


460 1 1 18 1

1 15 .01

0.0 .5 9.0

1 1 0.000 0.000

2 1 0.000 -7.500

7 1 0.000 -27.500

8 1 1 0.000 -35.000

9 12.500 0.000

17 25.000 0.000

337 625.000 0.000

345 1 640.000 0.000

10 12.500 -7.500

15 12.500 -27.500

16 1 12.500 -35.000

18 25.000 -7.500

338 625.000 -7.500

346 1 640.000 -7.500

351 1 640.000 -27.500

352 1 1 640.000 -35.000

19 25.000 -11.500

339 625.000 -11.500

20 25.000 -15.500

340 625.000 -15.500

21 25.000 -19.500

341 625.000 -19.500

22 25.000 -23.500

342 625.000 -23.500

23 25.000 -27.500

343 625.000 -27.500

24 1 25.000 -35.000

344 1 625.000 -35.000

353 1 0.000 10.000

361 1 0.000 90.000

362 12.500 10.000

370 12.500 90.000

371 25.000 10.000

387 265.000 10.000

38S 280.000 5.000

389 250.000 15.000

390 25.000 20.000

404 235.000 20.000

405 220.000 25.000

406 25.000 30.000

418 205.000 30.000

419 190.000 35.000

420 25.000 40.000

430 175.000 40.000

431 160.000 45.000

432 25.000 50.000

440 145.000 50.000

441 130.000 55.000

442 25.000 60.000

448 115.000 60.000

449 100.000 65.000

450 25.000 70.000

454 85.000 70.000

455 70.000 75.000

456 25.000 80.000

458 55.000 80.000

163

459

460

8

1

0.

0.

1

0.

0.

2

0.

0.

2

0.

0.

2

0.

0.

2

0.

0.

2

0.

0.

3

0.

0.1

2

1

300.

10.0

2

300.

3.00

3

300.

2.50

4

300.

2.00

5

300.

1.50

6

300.

1.50

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300.

1.02

640.

0.

640.

0.

640.

0.

640.

0.

640.

0.

640.

0.

640.

0.

295.

0.50

409 4

0.4

0.01

0.4

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0.4

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0.4

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0.4

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0.4

0.01

0.4

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40.000

25.000

-7.5

0.

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0.

-15.5

0.

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0.

-23.5

0.

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85.000

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164

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-23.51

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90.

10 8 13

0.7500

0.000

0.7500

0.000

0.7500

0.000

0.7500

0.000

0.7500

0.000

0.7500

0.000

0.6630

0.000

21.

10.

21.

10.

21.

10.

21.

10.

21.

10.

21.

10.

21.

10.

0.

0.125

0.

0.120

0.

0.120

0.

0.120

0.

0.120

0.

0.120

0.

0.130

16

0.720

2.

0.720

2.

0.720

2.

0.720

2.

0.720

2.

0.720

2.

0.720

2.

0.

0.0626 21. 10.0

0.

0.0576 21. 3.00

0.

0.0576 21.* 2.5

0.

0.0576 21. 2.00

0.

0.0576 21. 1.50

0.

0.0576 21. 1.50

0.

0.0676 21. 1.02

0.

0.230 0.039 1.90

0.250 0.050 0.73

0.250 0.050 0.63

0.250 0.050 0.53

0.250 0.050 0.42

0.250 0.050 0.42

0.124 0.020 0.30

300

3.0

1

9

43

0.4

0.01

4 1

1 2 10

4 1 1

345 337 338 346

44 4 1 2

10 2 3 11

86 4 1 2

0.7500

0.125

1 8

60.

0.

0.

0.

0.

0.

0.000

2.

0.124 0.020 0.73

346 338 339 347

87 4 1 3 8 0.

11 3 4 12

129 4 1 3 0.

347 339 340 348

130 4 1 4 8 0.

12 4 5 13

172 4 1 4 0.

348 340 341 349

173 4 1 5 8 0.

13 5 6 14

215 4 1 5 0.

349 341 342 350

216 4 1 6 8 0.

14 6 7 15

258 4 1 6 0.

350 342 343 351

259 4 1 7 8 0.

15 7 8 16

301 4 1 7 0.

351 343 344 352

302 4 1 8 0.01

362 353 1 9

303 4 1 8 0.01

371 362 9 17

304 4 1 8 0.01

372 371 17 25

305 4 1 8 0.01

373 372 25 33

306 4 1 8 0.01

374 373 33 41

307 4 1 8 0.01

375 374 41 49

308 4 1 8 0.01

376 375 49 57

309 4 1 8 0.01

377 376 57 65

310 4 1 8 0.01

378 377 65 73

311 4 1 8 0.01

379 378 73 81

312 4 1 8 0.01

380 379 81 89

313 4 1 8 0.01

381 380 89 97

314 4 1 8 0.01

382 381 97 105

315 4 1 8 0.01

383 382 105 113

316 4 1 8 0.01

384 383 113 121

317 4 1 8 0.01

385 384 121 129

318 4 1 8 0.01

386 385 129 137

319 4 1 8 0.01

387 386 137 145

320 4 1 8 0.01

388 387 145 153

321 4 1 8 0.01

165

388 388 153 161

322 4 1 8 1.

363 354 353 362

323 4 1 8 1.

390 363 362 371

324 4 1 8 1 1.

391 390 371 372

337 4 1 8 1.

404 403 384 385

338 4 1 8 1.

389 404 385 386

339 4 1 8 1.

389 389 386 387

340 4 1 8 2.

364 355 354 363

341 4 1 8 2.

406 364 363 390

342 4 1 8 1 2.

407 406 390 391

353 4 1 8 2.

418 417 401 402

354 4 1 8 2.

405 418 402 403

355 4 1 8 2.

405 405 403 404

356 4 1 8 3.

365 356 355 364

357 4 1 8 3.

420 365 364 406

358 4 1 8 1 3.

421 420 406 407

367 4 1 8 3.

430 429 415 416

368 4 1 8 3.

419 430 416 417

369 4 1 8 3.

419 419 417 418

370 4 1 8 4.

366 357 356 365

371 4 1 8 4.

432 366 365 420

372 4 1 8 1 4.

433 432 420 421

379 4 1 8 4.

440 439 427 428

380 4 1 8 4.

431 440 428 429

381 4 1 8 4.

431 431 429 430

382 4 1 8 5.

367 358 357 366

383 4 1 8 5.

442 367 366 432

384 4 1 8 1 5.

443 442 432 433

389 4 1 8 5.

448 447 437 438

390 4 1 8 5.

441 448 438 439

391 4 1 8 5.

166

441 441 439 440

392 4 1 8 6.

368 359 358 367

393 4 1 8 6.

450 368 367 442

394 4 1 8 1 6.

451 450 442 443

397 4 1 8 6.

454 453 445 446

398 4 1 8 6.

449 454 446 447

399 4 1 8 6.

449 449 447 448

400 4 1 8 7.

369 360 359 368

401 4 1 8 7.

456 369 368 450

402 4 1 8 7.

457 456 450 451

403 4 1 8 7.

458 457 451 452

404 4 1 8 7.

455 458 452 453

405 4 1 8 7.

455 455 453 454

406 4 1 8 8.

370 361 360 369

407 4 1 8 8.

460 370 369 456

408 4 1 8 8.

459 460 456 457

409 4 1 8 8.

45V 459 457 458

167

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APPENDIX B


FOR A COPY OF THE INPUT DATA FILES,

CONTACT SCOTT LUDLOW AT PURDUE UNIVERSITY

APPENDIX C

List of References

172

LIST OF REFERENCES

Geotechnical Investigation, (1986), Project No. RS-5124(2),"Indiana State Route 1 Realignment near Cedar Grove inFranklin County, Indiana," prepared by The H.C. NuttingCompany of Cincinnati, Ohio, June 27, 1986.

Holtz, R.D., and Kovacs, W.D. (1981), An Introduction toGeotechnical Engineering . Prentice-Hall, Inc. , New Jersey, 73 3

pp.

Huang, T.K. , and Chen, W.F. (1990), "Simple Procedure forDetermining Cap-Plasticity-Model Parameters," Journal ofGeotechnical Engineering , ASCE, Vol. 116, No. 3, pp. 492-513.

Huang, T.K., and Chen, W.F. (1991), "Embankment Widening andGrade Raising on Soft Foundation Soils: Computer ProgramImplementation," Report No. CE-STR-91-3 . School of CivilEngineering, Purdue University, West Lafayette, Indiana,January, 2 44 pp.

Huang, T.K. , Chen, W.F., and Chameau, J.L. (1990), "TheApplication of Cap-Plasticity-Model to Embankment Problems,"Report No. CE-STR-90-21 , School of Civil Engineering, PurdueUniversity, West Lafayette, Indiana, 37 pp.

Humphrey, D.N. (1986), "Design of Reinforced Embankments,"Report No. JHRP-86-16 , Joint Highway Research Project, Schoolof Civil Engineering, Purdue University, West Lafayette,Indiana, October, 423 pp.

Kulhawy, F.H. , and Mayne, P.W. (1990), "Manual on EstimatingSoil Properties for Foundation Design," Research Project 1493-6, Prepared for Electric Power Research Institute, Prepared byCornell University, Geotechnical Engineering Group, Ithaca,New York, August.

Ladd, C.C., Foote, R. , Ishihara, K. , Schlosser, F. , andPoulos, H.G., (1977), "Stress-Deformation and StrengthCharacteristics," State-of-the-Art Report, Proceedings of theNinth International Conference on Soil Mechanics andFoundation Engineering , Tokyo, Vol. 2, pp. 421-494.


Ludlow, S.J., Chen, W.F., Bourdeau, P.L., and Lovell, C.W.

,

(1991) , "Embankment Widening and Grade Raising on SoftFoundation Soils: Example 1 - Indiana State Route 55 overTurkey Creek in Lake County, Indiana," Report No. FHWA/IN/JHRP

173

- 91-18 , Joint Highway Research Project, School of CivilEngineering, Purdue University, West Lafayette, Indiana,December, 56 pp.

Ludlow, S.J., Chen, W.F., Bourdeau, P.L., and Lovell, C.W.

,

(1992) , "A Sensitivity Analysis of the Parameters for a CapPlasticity Model," Report No. FHWA/JHRP/IN - 92-1 . JointHighway Research Project, School of Civil Engineering, PurdueUniversity, West Lafayette, Indiana, January, 23 pp.

Documents

Embankment Widening and Grade Raising on Soft Foundation Soils, Phase 2