FINITE-ELEMENT ANALYSIS OFANCHORED BULKHEAD BEHAVIOR

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Authors Sogge, Robert Lund, 1941-

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75-4945

SOGGE, Robert Lund, 1944-FINITE ELEMENT ANALYSIS OF ANCHORED BULKHEAD BEHAVIOR.

The University of Arizona, Ph.D., 1974 Engineering, civil

Xerox University Microfilms , Ann Arbor, Michigan 48106

© COPYRIGHTED

BY

ROBERT LUND SOGGE

1974

i i i

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.

FINITE ELEMENT ANALYSIS OF ANCHORED

BULKHEAD BEHAVIOR

by

Robert Lund Sogge

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY WITH A MAJOR IN CIVIL ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

19 7 4

THE UNIVERSITY OF ARIZONA

GRADUATE COLLEGE

I hereby recommend that this dissertation prepared under my

direction by Robert Lund Sogge .

entitled Finite Element Analysis of Anchored Bulkhead Behavior

be accepted as fulfilling the dissertation requirement of the

degree of Doctor of Philosophy

Dissertation Director Date

After inspection of the final copy of the dissertation, the

following members of the Final Examination Committee concur in

its approval and recommend its acceptance:-'*

J2 &

zf,

This approval and acceptance is contingent on the candidate's

adequate performance and defense of this dissertation at the

final oral examination. The inclusion of this sheet bound into

the library copy of the dissertation is evidence of satisfactory

performance at the final examination.

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to bor­rowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or re­production of this manuscript in whole or in part may be granted by the copyright holder.

SIGNED:

ACKNOWLEDG MENTS

I express my sincere appreciation to Prof. Hassan A. Sultan

for providing me with the interest for research, suggesting this stimu­

lating topic, and encouraging and guiding me through all aspects of

this study.

Special appreciation is also due Prof. Ralph M. Richard for

his stimulating courses, which provided many of the tools necessary

for this research, and whose professional advice was always available

to help engineer solutions to the problems that arose.

I wish to thank Michael Johnson for his critical and education­

al discussions during the progress of the work. Thanks is extended to

Prof. Rudolf A. Jimenez for reviewing the study. Also, acknowledgment

is given to the Department of Civil Engineering and Engineering Mechan­

ics at The University of Arizona and to its head, Quentin M. Mees, for

providing an inspiring research environment.

I am very grateful to Susan Sogge for the partnership I shared

with her in academia and for providing me with insights into myself from

which this work removed me.

iv

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS viii

LIST OF TABLES xii

ABSTRACT xiii

1. INTRODUCTION 1

Definition of an Anchored Bulkhead System 2 Object and Scope of Research 4

2. REVIEW OF EXPERIENCE WITH ANCHORED BULKHEADS 7

Description of General Design Process 7 Design Methods 9

Classical Methods 9 Danish Method 10 Equivalent Beam Method 10 Brinch Hansen's Method 11 Model Studies and Design Methods Proceeding

from Them 11 Tschebotarioff's Model Tests and Design Method. ... 12 Rowe's Model Tests and Design Method 14

Pressure Tests 14 Flexibility Tests 17

Subgrade Reaction Methods 23 Field Tests 26

Anchor-wall Design 29 Limitations of Available Design Methods 30

3. THEORY OF THE FINITE ELEMENT METHOD 34

The General Stress Analysis Problem 34 Finite Element Method 35

Finite Element Formulation 36 Convergence 40 Nonlinearities 42

Applications 44 Soil-Structure Interaction 45

v

vi

TABLE OF CONTENTS—Continued

Page

4. MODEL FOR SOIL BEHAVIOR 48

Review of Previous Soil Behavior Models 48 Nonlinear Soil Model 51

Tangent Modulus 51 Tangent Poisson's Ratio 53 Unloading and Reloading Modulus 55 No Tension Characteristics 55 Nonlinear Interface Soil Model 56

5. FINITE ELEMENT MODEL FOR ANCHORED BULKHEADS 58

Soil Model 58 Beam Model 59 Tie-rod Model 60 Anchor-wall Model 62 Interface Model 62 Computer Program Capabilities 65

System Degrees of Freedom 65 Nodal Equilibrium Check 66 Water-table Elevation 67 Initial Stre s se s 67 Driving of Sheet Pile 68 Initial Sheet-pile Displacements 69 Backfilling 70 Dredging 70 Surcharge Loads 73 Simulating Tie-rod Force 73 Modification for Linear Material Properties 74

6. VERIFICATION OF THE FINITE ELEMENT ANCHORED BULKHEAD MODEL 75

Burlington Beach Wharf 75 Results of Full-scale Test Observations 77 Finite Element Model Analysis 79

Finite Element Mesh Idealization of the Continuum. . . 81 Data Preparation 85

Comparison of Behavior of Model to Burlington Wharf Bulkhead 85

Moments 86 Deflections 89 Tie-rod Force 93 Model Results during Construction 94 Anchorage Stiffness 99

vii

TABLE OF CONTENTS—Continued

Page

7. INVESTIGATION OF BULKHEAD SYSTEM PARAMETERS 105

Parameters Representing System Behavior 105 Soil Stiffness Similitude Expression 107 System Stiffness Similitude Expression 108 Displacement Similitude Expression 110

Finite Element Model Study of Interaction Concepts 110 Structural Stiffness Ill Soil Stiffness 115 System Stiffness 122 Interaction Between Soil and Structure 124 Influence of Dredge Level Depth 129 Displacement Relations 136 Influence of Pois son's Ratio 140 Influence of Material Density 142 Anchor System Stiffness 142

Tie-rod Force—Anchor-wall Displacement Relations 145

Influence on Sheet-pile Moment 153

8. SIMULATION OF ROWE'S MODEL TESTS 157

Simulation of Pressure Test 157 Full-scale Simulation 160 Imposed Sheet-pile Displacements at the Tie-rod Level. . . 162 Effect of Construction Sequence 166

Backfill-Dredging Sequence 166 Tie-rod Release Sequence 172

Simulation of Flexibility Test 175

9. CONCLUSIONS AND RECOMMENDATIONS 179

Conclusions 179 Recommendations for Further Research 181

APPENDIX A: NOMENCLATURE 183

APPENDIX B: PROGRAM SSI DOCUMENTATION 187

APPENDIX C: LISTING OF PROGRAM SSI 201

REFERENCES 240

LIST OF ILLUSTRATIONS

Figure Page

1-1. Anchored Bulkhead 3

2-1. Method of Conducting Pressure Test 16

2-2. Pressure and Moment Distributions from Rowe's Pressure Tests 17

2-3. Bending Moment Versus Pile Flexibility from Rowe's Flexibility Test 19

2-4. Bending Moment Distribution from Rowe's Flexibility Test 19

2-5 Two Main Causes of Decrease of Bending Moment with Increase in Pile Flexibility 20

2-6. Cross Section, Burlington Beach Wharf, Test Location 2 . . . 29

2-7. Anchor Location 31

3-1. Iterative Procedure 43

3-2. Incremental Procedure 43

5-1. Shear Transfer Condition 61

5-2. Interface Element Deformation Modes 64

5-3 . Analytic Simulation of Excavation 72

6-1. Tests Results, Burlington Beach Wharf, Test Location 2 ... 78

6-2. Finite Representation of Infinite Body 83

6-3 . Extent of Finite Element Grid for Burlington Wharf Bulkhead . 84

6-4. Maximum Moment in Sheet Pile Versus Soil Modulus Number, Burlington Wharf Models 86

6-5. Moment and Deflection Distribution, Burlington Wharf Models 88

viii

ix

LIST OF ILLUSTRATIONS—Continued

Figure Page

6-6. Maximum Differential Sheet-pile Deflection Versus Soil Modulus Number, Burlington Wharf Models 89

6-7. Sheet-pile Displacement, I = 120 in.4, K = 190, Burlington Wharf Model 91

6-8. Sheet-pile Displacement, I = 289 in.4, K = 120, Burlington Wharf Model 92

6-9. Sheet-pile Pressure Distribution during Construction, Burlington Wharf Model, 1= 120 in.4, K= 190 95

6-10. Sheet-pile Moment Distribution during Construction, Burlington Wharf Model, 1= 120 in.4, K= 190 100

6-11. Tie-rod Force Versus Anchor-wall Displacement, Burlington Wharf Model 102

6-12. Secant Anchor-wall Soil Stiffness Versus Tie-rod Force, Burlington Wharf Model 102

7-1. T Versus Log p, Finite Element Model 114

7-2. T Versus Log s, Finite Element Model 116

7-3. T Versus Log s, Finite Element Model, Logp = -3.32 . 117

7-4. Sheet-pile Moment and Soil Pressures, Finite Element Model, H = 3.5 ft, K = 45 119

7-5. Sheet-pile Moment and Soil Pressures, Finite Element Model, H = 40 ft, K = 79 120

7-6. Sheet-pile Moment and Soil Pressures, Finite Element Model, H = 40 ft, K = 140 121

7-7. T—Log S Relation, Finite Element Model 123

7-8. Sheet-pile Moment and Soil Pressures for Two Different Sheet-pile Flexibilities, Finite Element Model 125

7-9. Sheet-pile Moment and Soil Pressures for Two Different Soil Modulus Numbers, Finite Element Model 126

7-10. Sheet-pile Moment during Construction 130

X

LIST OF ILLUSTRATIONS—Continued

Figure Page

7-11. Sheet-pile Pressure Distribution during Construction 131

7-12. r Versus Sheet-pile Flexibility for Different Dredge Level Depths 135

7-13 . T Versus Soil Modulus Number for Different Dredge Level Depths 135

7-14. Sheet-pile Tip Displacement Versus Soil Modulus Number . . 137

7-15. Sheet-pile Tip Displacement Versus Sheet-pile Height .... 137

7-16. Sheet-pile Tip Displacement Versus Sheet-pile Flexibility . . 138

7-17. Sheet-pile Moment and Soil Pressures, Finite Element Model, M= 0.30 141

7-18. Sheet-pile Moment and Soil Pressures, Finite Element Model, Y= 135 pcf 143

7-19. Tie-rod Force Versus Anchor-wall Displacement for Various K 146

7-20. Tie-rod Force Versus Anchor-wall Displacement for Various Log p 147

7-21. Tie-rod Force Versus Soil Stiffness 148

7-22. Tie-rod Force Versus Sheet-pile Height 148

7-23. Tie-rod Force Versus Sheet-pile Flexibility 149

7-24. Anchor-wall Displacement Versus Soil Modulus Number. . . . 150

7-25. Anchor-wall Displacement Versus Sheet-pile Height 150

7-26. Anchor-wall Displacement Versus Sheet-pile Flexibility ... 151

7-27. Secant Anchor Stiffness Versus Soil Modulus Number 152

7-28. T Versus Log s for Naturally Occurring Tie-rod Release .... 155

8-1. Sheet-pile Moment, Soil Pressures, and Deflection for Large Imposed Tie-rod Release, Finite Element Model. . . 163

8-2. Tie-rod Force Versus Imposed Tie-rod Release 165

xi

LIST OF ILLUSTRATIONS—Continued

Figure Page

8-3. Sheet-pile Moment, Soil Pressures, and Deflections for Backfill-Dredge Construction Sequence, Finite Element Model 168

8-4. Sheet-pile Moment, Soil Pressures, and Deflections for Dredge-Backfill Construction Sequence, Finite Element Model 169

8-5. Sheet-pile Tie-rod Level Displacements for Two Construction Sequences, Finite Element Model 170

8-6. Sheet-pile Moment, Soil Pressures, and Deflections for Backfill-Dredge Construction Sequence with Naturally Occurring Tie-rod Release, Finite Element Model 173

8-7. Sheet-pile Moment, Soil Pressures, and Deflections for Backfill-Dredge Construction Sequence with Imposed Tie-rod Release, Finite Element Model 174

LIST OF TABLES

Table Page

6-1. Maximum Bending Moments Induced by Dredging (Including Surcharge) 79

6-2. Tie-rod Stresses, Burlington Beach Wharf, Test Location 2 80

6-3. Material Properties for Finite Element Model of Burlington Beach Wharf 82

7-1. Material Properties for Parameter Study, Finite Element Model 112

7-2. Finite Element Analysis Results, No Tie-rod Release 113

8-1. Displacement of Sheet-pile at the Tie-rod Level for Two Types of Construction Sequence 167

xii

ABSTRACT

The literature containing the available knowledge on anchored

bulkheads shows the great emphasis in present design methods on the

assumption of a stress or deformation pattern. This study approaches

the problem by using a finite element computer program model developed

to analyze the soil-structure interaction inherent in the entire continuum

of a highly redundant bulkhead system.

The sheet pile is modeled using beam elements, and the soil

and anchor wall are modeled using triangular elements. The discontin­

uous displacements and frictional nature of the soil-structure boundaries

are simulated using interface elements. A bar connecting the sheet pile

and anchor wall is used to model the tie rod. The soil and interface

elements can model soil behavior that is nonlinear and is confining-

pressure dependent.

The programmed finite element model is verified by a compari­

son with the field results obtained from monitoring the Burlington Beach

Wharf bulkhead, Ontario, during construction. The construction se­

quence, including the initial stress state, driving of the sheet pile,

and horizontal displacement that occurs during driving, is modeled.

Agreement of moment and displacement patterns and magnitudes are

obtained using reasonable values for parameters to describe the soil

present.

A theory of bulkhead behavior formulated based on the subgrade

reaction equation relating pressure to displacements is evaluated using

xiii

xiv

the finite element model. The influence of the parameters are described

by analytical expressions. Sheet-pile behavior as characterized by

moment is related to the structural and soil stiffness components of the

system. Soil stiffness is expressed by a ratio relating the soil modulus

number to height and is further refined by defining it in terms of dredge

level depth.

An analysis of system stiffness shows that small-scale model

results are unconservative for full-scale moment predictions. Active-

pressure distribution on the sheet pile decreases as the stiffness of the

soil increases relative to the structural stiffness. Rotational and trans-

lational deflections are related to the system stiffness ratio. Other

parameters studied are Poisson's ratio, material density, and the anchor-

system stiffness.

A finite element simulation of small-scale pressure and flex­

ibility tests at model and full-scale proportions show the importance of

scale considerations. The effect of imposed tie-rod releases and con­

struction sequence is examined for a full-scale bulkhead. The finding,

using the finite element model, of a decrease in the moment with tie-

rod release is the significant difference between the two models.

CHAPTER 1

INTRODUCTION

The behavior of anchored sheet-pile bulkhead retaining struc­

tures represents a complex problem of soil-structure interaction. Much

of the complexity arises from the high redundancy of the system. The

earth pressures are dependent on the deformation patterns of the sheet

pile, and these patterns are in turn related to the sheet-pile flexibility

and soil stiffness. In order to scale the problem to a size he can handle,

the engineer introduces various simplifying assumptions in modeling the

bulkhead. These assumptions result in a solution which often must be

modified due to the lack of conformity between the real bulkhead system

and the mathematical model idealization of it.

Design methods are a result of an assumed bulkhead behavior

and alone do not provide knowledge of its actual behavior. The validity

of the assumed behavior can be ascertained by comparisons to model

studies and field tests. Classical design methods have been based on

stress or deformation assumptions at failure or on the concept of a hinge

at a known location in the bulkhead. The added knowledge gained from

model tests has been incorporated into the design process in some

methods. Also, field test results have precipitated modifications in the

theories of bulkhead behavior. Presently, numerical procedures that are

not limited by highly indeterminate systems as were previous methods of

analysis have been developed for digital computer application. These

1

2

methods have dealt with a beam on elastic foundation approach using a

finite element discretization of the beam. Due to their somewhat cum­

bersome applicability, they are used more often in research aimed at

refining the classical methods of design.

Definition of an Anchored Bulkhead System

A general anchored bulkhead system consists of a sheet pile

connected by a tie rod to an anchor some distance behind the wall.

Wales mounted parallel to the length of driven piles are used to transfer

the load from the sheet pile to the individual tie rods. The sheet pile

consists of timber sections or steel Z- or U-shaped sections interlock­

ing along their lengths to form a continuous flexible wall. The anchor

wall is embedded a small depth in the ground and consists of either a

continuous sheet-pile wall, a rigid continuous concrete wall, a pile cap

with battered timber piles, or discontinuous concrete deadmen. The con­

nections between the anchor wall and sheet piles are usually made by

steel tie rods, but often a rigid platform structure is used. The latter

system would allow no yield of the sheet-pile anchor wall. An illustra­

tion of a bulkhead system is present in Fig. 1-1.

The function of this type cf structure is to retain earth. The

restraining forces on the wall are the tie-rod force at the top and the

lateral pressure on the embedded portion of the wall. This earth-

retaining system is constructed by driving the wall of sheet piles and

either dredging out the material in front of the wall or backfilling be­

hind the wall, depending on the original ground surface elevations. The

construction method will not be a design variable, provided the anchor

Fig. 1-1. Anchored Bulkhead.—Courtesy of United States Steel Corporation

4

system yields sufficiently, usually an amount 1/1000 of the sheet-pile

wall height (Rowe, 1952). The term "yield," as applied to the tie rod,

refers to deformation of the tie rod and is not meant to apply specifically

to the plastic portion of the deformation. This provision is usually sat­

isfied by backfilled-constructed bulkheads but is harder to meet with

dredged-constructed systems.

Object and Scope of Research

The object of this research is to develop a finite element com­

puter program model of an anchored bulkhead system and to use it to

analyze the soil-structure behavior of the system. This method will al­

low modeling of the components of the system—sheet pile, anchor rod,

anchor wall, and soil—such that the simultaneous solution of the equa­

tions of equilibrium, force-deformation, and compatibility for a network

of discrete elements of the components yields a displacement and stress

solution for the continuum made up of the various components. The

method is not limited by highly redundant systems as were previous

methods of analysis. Thus, no stress or deformation approximation is

necessary, and the solution is not dependent on the environment of the

proper pattern of deformation as is true with the usual limit design ap­

proach .

The method will be developed to consider the following param­

eters in order that it accurately represents the significant characteris­

tics of actual prototype structures: nonlinear material properties in which

strength is dependent on confining pressure, incorporating an elastic un­

loading modulus and a non-tension state provision; surcharge loads;

5

the yield characteristics of an anchor system, consisting of a steel tie

rod and a continuous anchor wall; the frictional nature of the sheet pile-

soil interface; the method of construction, dredge or backfill; pile

driving; and water-table elevation. It will be appHed to bulkhead sys­

tems of various heights, tie-rod depths, dredge-level depths, sheet-

pile flexibilities, and geometric configurations.

The continuum will be modeled using bar elements for the steel

tie rod, beam elements for the steel sheet pile, triangular constant-

strain elements for the loose and medium dense sand soil and continuous

concrete anchor wall, and joint interface elements for the sheet pile-

soil interface and anchor wall-soil interface. A two-dimensional plane

strain idealization will be used for the continuum of unit thickness paral­

lel to the line of sheet piles.

The accuracy of the finite element model will be evaluated by

using a monitored field bulkhead. The validity will be assessed by com­

paring displacement and bending moment patterns for the sheet pile. The

comparison will provide some knowledge of the input material property

parameters considered from the point of view of small-scale laboratory

test versus large-mass behavior.

Once confidence in the programmed model has been obtained,

it will be used to study various aspects of the soil-structure system as

well as new combinations of parameters not previously considered by

model tests. Some of the factors to be analyzed that affect the interre­

lationship between the soil and the bulkhead are: dimensional similtude

of the sheet-pile height, equivalent stiffness of the tie-rod anchor wall

system, ratio of tie-rod level deformations to the sheet-pile height at

6

various stages of dredging, and lateral earth pressure distribution.

Evaluation of scale effect will be aided by using the results gained from

previously constructed small-scale anchored bulkhead models. A char­

acterization of the soil-structure interaction in terms of stiffness will

be made. The influence of each of the contributing components, the

soil and the sheet pile, will be analyzed separately.

The ultimate goal of this sophisticated model study is not to

eliminate the need for simpler methods but to gain new insight in order

that the simpler methods might be used more effectively. The program

used in this study will be presented and documented in order that it can

be used directly in design.

CHAPTER 2

REVIEW OF EXPERIENCE WITH

ANCHORED BULKHEADS

Theories of anchored bulkhead behavior are embodied in the

methods used to design them. The limitations imposed for analysis by

each design method cause the system to behave in a prescribed manner.

The more complex the model is, the more features of actual bulkhead

performance it can incorporate. The engineering literature contains

many design methods for analyzing the stability of anchored bulkhead

systems. Model studies and field tests have been performed to verify

and modify these theories. This chapter will examine the most impor­

tant contributions in these areas.

Description of General Design Process

The direction which any design procedure assumes is dependent

on the complexity of the system. If the system is determinate, enough

assumptions have been made to solve directly for the unknown variables.

In a highly indeterminate structure, such as a bulkhead system, it is

necessary to make some assumptions to gain enough equations to solve

for the unknowns. A set of assumptions might be the design configura­

tion, reducing the design procedure to an analysis of it. For a general

anchored bulkhead system, this latter approach still leaves the designer

with too many complexities, so rather than choose the design configura­

tion as an assumption, the approach taken is to delineate and then

7

8

analyze the collapse or failure condition and apply safety factors to pre­

vent such an occurrence. This approach is known as limit design.

One general class of failures is an instability of the entire

sheet pile and soil support system along a slip surface which may pass

in front of or behind the anchor wall. This mode will not be considered

in the discussion of design methods, even though its evaluation is an

essential step in the design process. Another class of failure modes

concerns the sheet pile itself. Structural yielding of the pile can occur

with the location dependent on the deformation characteristics of the

tie rod and the restraint provided by the embedment in the supporting

soil. A third class of failures deals with wall rotation, which occurs

when there is inadequate support from the soil below the depth of em­

bedment or when the anchor system fails or yields enough to allow rota­

tion about the bottom of the sheet pile. A combination of these latter

two types of failures in which both mechanisms occur simultaneously

is possible.

In order to simplify the limiting equilibrium mechanics portion

of the solution, the pattern of deformations accompanying failure is

envisioned. This leads to the assumption of an active and passive

lateral stress distribution. With this assumption, the problem is made

determinate, and it is possible to calculate directly the depth of sheet-

pile penetration, the maximum bending moment in the pile, and the

anchorage force without considering the deformation of the system. The

proper section of sheet piling and size of the anchor system can then be

selected.

9

Design Methods

Classical Methods

Prior to the turn of the century, anchored bulkheads were de­

signed without the benefits of analytical techniques. The first classical

design methods were based on active and passive pressure distributions

obtained by using Coulomb's sliding wedge theory and Rankine's theory

of earth pressure. These theories assume, perhaps tenuously, that the

amount of wall movement is adequate to ensure the development of full

active pressure on the inside face and full passive pressure on the out­

side face, but they do not consider the magnitude of this movement.

The classical methods are of two types, depending on the rela­

tive stiffness of the soil compared to that of the pile. In the "free earth

support" method, no reversal of bending moment below the dredge level

occurs since the wall is rigid compared to the supporting soil. This

yields a simple support condition at the tie rod and at the location of

passive resistance along the embedded portion of the sheet pile. This

condition occurs if the depth of embedment is small, causing only enough

soil resistance to maintain equilibrium. The "fixed earth support" method

assumes that the wall is flexible compared to the resistance offered by

the soil and that the pile is driven deep enough to cause fixity below the

dredge level.

These two approaches result in two limiting types of failure

mechanisms, a sheet-pile rotation about the tie-rod level or about the

dredge level. The free earth support assumption results in rotation about

the tie rod due to inadequate passive soil resistance in front of the sup­

port. The fixity assumed in the fixed earth support method would cause

failure by sheet pile yielding as the pile rotates at the dredge level.

Both cases assume a stable anchor system. As these theories involve

limiting assumptions, both are conservative. In the free earth support

method, large moments in the sheet pile occur due to the small embed­

ment and rigid pile assumed. In obtaining the rigidity of soil support as­

sumed by the fixed earth support method, large embedment depths result.

Danish Method

It was noted in Denmark that the application of the free earth

support method to steel sheet-pile walls resulted in larger sections than

were needed for timber sheet-pile walls designed empirically and still

properly functioning. When the stresses in the timber were checked us­

ing classical methods, the design was found to be unsafe. The Coulomb

active earth pressure distribution was then modified, causing reduced

bending moments and thus allowing smaller sections. The pressure dis­

tribution proposed is based on the concept of soil arching between the

points of support at the tie-rod and dredge level. The Danish regulations

(Tschebotarioff, 1951; Hansen, 1953), although resulting in structural

economy, see limited application today because of questions as to their

theoretical basis. Their use in practice is substantiated though by the

many successful sheet-pile bulkheads designed by this method.

Equivalent Beam Method

Due to the time-consuming nature of the fixed earth support

calculations, a simpler method was proposed by Blum (Tschebotarioff,

1951), called the "equivalent beam" method. This method developed a

relation to locate the point of contraflexure, which exists somewhere

below the dredge level. This point, a point of zero moment or a hinge,

makes possible the consideration of the two parts of the bulkhead

separately.

Brinch Hansen's Method

An equilibrium method proposed by Brinch Hansen (1953) deter­

mines the earth pressure from the statical equilibrium of a wedge. This

calculation is performed for five possible failure modes to check for the

most critical. As this method neglects wall deformations and considers

the wall to be either rigid or completely flexible, only limiting cases of

wall stiffness are considered. A disadvantage to the design engineer is

that the method entails lengthy iterative calculations.

Model Studies and Design Methods Proceeding from Them

Design methods based on large- and small-scale model testing

constitute a large portion of the body of knowledge from which design

proceeds. Some of the earliest work on anchored sheet-pile structures

is that of Stroyer (1935) and Browzin (1949), but their work will not be

discussed due to its limited scope compared to more recent tests.

Tschebotarioff (1949) conducted large-scale model tests at Princeton

University for the Bureau of Yards and Docks of the United States Navy.

These tests were followed by Rowe's (1952) small-scale tests on walls

ranging in height from 20 to 42 inches. Both Tschebotarioff and Rowe

developed design procedures based on their tests. Terzaghi (1954) made

design recommendations based on Rowe's tests. All these proposed

methods represented revisions of the classical methods and thus afford

the ease of design calculation inherent in them.

Tschebotarioff's Model Tests and Design Method

Tschebotarioff (1949) in his 1:5- and l:10-scale model tests

using SR-4 strain gages to measure bending moments noted that the

point of zero moment usually occurred at or near the dredge level. He

felt that the fixed earth support method agreed best with his findings.

Since his tests were conducted with relatively flexible piles, the rela­

tive soil support was quite stiff. Also, the soil below the dredge level

was vibrated into a dense state which explains why the agreement with

the fixed earth support condition. His design procedure, based on in­

formation gained in the model tests, is a modification of the equivalent

beam method and is known as the simplified equivalent beam method

(Tschebotarioff, 1951). It assumes a hinge at the dredge level. Further

evaluation of the tests resulted in the hinge being moved to the first

compact layer below the dredge level (Tschebotarioff, 1958). This de­

sign method offers the advantages of the fixed earth support method in

that economical sections result along with a very adequate embedment

ensuring a failure mode of yielding in the pile if the anchorage is safe.

His recommendation for the depth of embedment was 0.43 times the

height of the sheet pile above the dredge level. This depth will provide

a factor of safety of about two.

Tschebotarioff investigated the effect of the method of con­

struction on earth pressures by testing a few models in which dredging

had been the last construction operation. These tests were in addition

to. some in which backfilling was the last operation. Where dredging was

the last operation and the anchor support fixed, definite evidence of ver­

tical arching was obtained with larger pressures occurring at the anchor

and dredge levels. For the usual test where backfilling was the last

operation, no evidence of arching was found. The yielding of the anchor

rod during backfill resulted in the breakdown of the unstable shallow

vertical arch, and the existing pressure reverted to the active pressure

case for the small anchor movements. Since small anchor movements

will always occur in field situations, Tschebotarioff did not believe

that arching was a practical consideration. Thus moment reductions

could not be attributed to the phenomenon of arching, contradicting the

basis for reduction of the pressure distribution in the Danish method.

Tschebotarioff s third series of tests on backfilled bulkheads

showed that during the three sequential stages of vibrating the sand in

front of the bulkhead (below the dredge level), applying surcharge, and

releasing the tie rod an amount of approximately 1/80 the height of the

wall, the maximum positive bending moment decreased while the maxi­

mum negative bending moment below the dredge level increased. The

wall height for these l:10-scale tests ranged from 5 to 8 feet. In some

cases, the maximum negative bending moment became larger than the

maximum positive bending moment. The pressures acting on the pile

were reduced to the active case. For these large anchor level displace­

ments, the tensile strength of the sand above the water table became a

factor. It should be noted, however, that the deflection ratios were very

large compared to those encountered in field situations.

14

The tests showed that the center of passive pressure occurred

near the dredge level. Furthermore, the passive pressure very close to

the dredge level was 3 to 4 times the maximum Rankine values not con­

sidering wall friction. This result corresponds to a previous analysis

on a sheet pile that has sunk into the soil for the case of equal angles

of internal friction and wall friction.

Rowe's Model Tests and Design Method

Rowe (1952) conducted two types of model tests, denoted as

pressure tests and flexibility tests. The pressure tests were conducted

to ascertain the pressure distribution existing on a sheet pile undergo­

ing movement. This pressure had been a subject of much controversy due

to the questionable relation between arching and wall movement. The

flexibility tests were carried out in order to study the influence of pile

flexibility on the factors governing design. Tie-rod yield was incorpo­

rated so that no difference would exist due to the type of construction.

Pressure Tests. According to Rowe (1952, p. 32), the method

of conducting the pressure tests was as follows:

The pile was hung in a vertical position from supports at the side, and the zero readings of all the gauges were taken. The bin was filled with sand on each side of the pile, the sand level being raised evenly by the aid of guide-lines painted on both sides of the pile at 3-inch intervals. When the tie-rod level was reached, the tie-rods were placed in position and left loose. The filling was completed, the pile was released from its supports, and all gauge readings taken. The tie-rods were set to be just tight.

The sand on the outside of the pile was dredged away in stages by removing the steel plates at the front of the bin. All gauge readings were recorded at values of the dredged depth given by <? = 0.5, 0.6, 0.7, and 0.8 &£is the ratio of the height above the dredge level to the total height of the sheet pile]. The dredging was either continued until com­plete passive slip occurred or it was stopped at an earlier

stage and the effect of the addition of surcharge or tie-rod yield studied. When surcharge was added, the effect of the tie-rod yield was studied at the end of the test.

A diagrammatic representation of the procedure is given in Fig. 2-1.

Some important results of Rowe's (1952, p. 33-36) pressure

tests were:

1. When the bin had just been filled, the distribution of ac­tive pressure was that given by the Coulomb theory with no wall friction.

2. On dredging, with no anchor yield, the pressures in­creased at the tie-rod level and decreased at the centre. The total active load given by the area under the pressure diagram decreased, and at values of <X= 0.6-0.8 became equal to the Coulomb value assuming full wall-friction, that is taking 5 = 2/3^ GS is the angle of wall friction and f6 is the angle of internal friction]. The increase in pres­sure at the tie-rod level and decrease at the centre con­tinued until passive slip occurred, provided that no yield of the anchor was allowed. At a safe value of the dredge level, the relief of pressure at the centre due to arching was considerably less than that assumed by the Danish Society of Engineers.

3. Outward yield of the tie-rods caused a breakdown of the arching and the pressure distribution became triangular, remaining equal in total area to the Coulomb value. The yield necessary to relieve completely the concentration at the tie-rod level varied with the amount of surcharge and the tie-rod depth, but a maximum value equal to H/1000 was sufficient to destroy the arching in every case. This amount of yield is generally less than that which might be considered to occur in the field using elastic tie-rods anchored to plates subject to settlement . . . (H is the sheet-pile height].

4. The passive pressures at the dredge line increased with dredging, from the active pressure value at = 0 to the Coulomb passive value (S=0) when ct was approximately equal to 0.7. Only part of the complete triangular pressure-distribution was mobilized at this stage and the wall movements were very small. With farther dredging, the movements at the toe increased in size, developing wall-friction on the passive side. Ultimate failure oc­curred when the Coulomb-v-lue with full wall-friction was fully mobilized. . . .

5. During slip a considerable shear force acted between the toe of the pile and the subsoil.

16

(I) lift empty Pode* fcwft| vtrttul-Zt»o rckiA|i UWM

()) em bit. Tie>ro4t fiit«nctf

(Sj &« fiii«4 (*•<«} IM |*

77d fcr

Dal £»yj«i id mi ^oiil^on All like*

(4J SaH 4rt4i*4. tiki* M

vi?wf«of «t01 oi.o/.oe.i^ot

ZZZZ7, 55_J5EE5>a_- (S)Swr«h..|..«.4n >v/.v— ^ •' \ vi'wciof 4 • 04100*/|,*.'»* *. ''ill

(4} Ti«*«4i n'tuH (a nijei

• • • ^

Fig. 2-1. Method of Conducting Pressure Test.—From Rowe (1952, p. 32)

6. Before yield, the maximum bending moment agreed closely with that calculated from Stroyer's empirical formula. . . .

7. Upon yield, the bending moment increased to that given by the Coulomb theory. . . .

An example typical of the pressure and moment distributions in these

tests is shown in Fig. 2-2.

Rowe's Result 1 is extremely significant from a comparison

viewpoint to other models (perhaps even to mathematical models). The

area in each side of the pile was backfilled with soil to the same height

at which point the pile was released from its supports. Due to the non­

uniform geometry of the test bin (Fig. 2-1), a slight movement occurred at

this stage, resulting in a Coulomb active pressure distribution. At this

point, the tie rod was secured. This initial pressure state was different

than the earth pressure at rest value that would be present if the soil

were uniform in extent on both sides of the pile.

17

fUnkiMitor

;liO| We<;i Uwjf 'I • J

TKMtaurirf.

0*t«n»«4 - «• wKtor yfcltf.

OVttrvtf - M

ml-Off I»M«!*•««. h Ni At vifcifii/ hvwWr, W| p • • 312 v-

HQntHti il .INCHES *i* fOOt •(NOINC KOMINT CMACMH PattlwM ll.fl«IQUAM INCN

MUIUM OiACftAM

Fig. 2-2. Pressure and Moment Distributions from Rowe's Pressure Tests.—From Rowe (1952, p. 34)

Flexibility Tests. The method of conducting the flexibility

tests was as follows (Rowe, 1952, p. 38, 40-41):

The bin was filled evenly on both sides of the model until three-quarters of the model was embedded in the sand, when the initial strain-gauge readings were taken. The tie-rods were set, the filling completed at the rear, and the soil dredged on the outside to the first value & = 0.6. The strain gauge which was likely to be recording the maximum bending moment was then switched in, and the galvanometer was al­lowed time to steady. The anchorage was then released. While vertical sand-arching was being destroyed, the bend­ing moment increased. However, too much yield eventually caused a decrease, owing to cantilever action from the dredge level. Accordingly, the anchorage was released carefully until the galvanometer indicated maximum deflexion. The increment of yield required after each stage of dredging was approximately H/2,400, giving a total yield of H/800 at the end of the test.

The sides were next vibrated and the galvanometer read­ing increased to a further amount varying from 0 to 10 per

18

cent according to the type of test. All gauge readings were then recorded and the dredging was continued.

In the flexibility tests, similarity of model and prototype sheet-

pile flexibilities was ensured by making the flexibility numbers (p) equal

for both, or p = H^/EI, where H = height and EI is the stiffness composed

of Young's modulus E and the moment of inertia I. Another ratio (Tr),

which was equal for both the model and prototype, isT= M/H^ where M

is the moment in the sheet pile.

The flexibility tests using walls having log p values ranging

from -2.07 (flexible) to -3.74 (rigid) showed that decreasing the sheet-

pile stiffness decreased the moment in the pile. The results led to

Rowe's design procedure, which was presented in the form of charts re­

lating the moment reduction allowed, as a percentage of the free earth

support moment, versus the flexibility of the pile. The free earth support

method assumes a rigid wall, so there would be no reduction allowed for

a rigid wall. All moments were referenced to free earth support values,

since a passive failure occurred with all wall flexibilities. Full Coulomb

active and passive pressure distributions existed concurrently. In gen­

eral, for practical flexibilities, reduction factors ranged from 40 to 60

percent of the free earth support values (Fig. 2-3). The bending moment

distributions for this same case are shown in Fig. 2-4.

An increase in flexibility caused a rise in the center of the pas­

sive pressure distribution. This rise is a function of the deformation

pattern. The more flexible the pile, the more relative deformation be­

tween the toe and the dredge level, and thus an upward shift in the pas­

sive pressure distribution results (Fig. 2-5d). This rise in effect reduces

19

S»f« (ii(«

Fig. 2-3. Bending Moment Versus Pile Flexibility from Rowe's Flexibility Test.—From Rowe (1952, p. 42)

ft

»» il

SAND, LOOSE: £-0, q-0

Fig. 2-4. Bending Moment Distribution from Rowe's Flexibility Test.—From Rowe (1952, p. 48)

insert OOOS4

loose son otNU soa

WAIL TUVHN* * WALL UlAN'NG ASOUT THf TOC ' ABOUT Tn£ TO? MEASURED TTFCS Of PRESSURE DISTRIBUTION ON STlfF WALL ROTATING INTO THE SOIL

(c)

Fixing moment tfuc to pats<«c p."<stvr« }i«<e (he «Je it *cry tmafl antf »«ri»b(« vrilH t*>« c*c yi«M

ckanck in rA;m russuic onTAnungui with ikcmasc *n mi »if xwutv »/.

UNOiNC noMiKT KtovciiON out 10 cnd thrust

MomtM fcduU4A • / i l k

Nat to * bi;'4f.:ic d p.k «K«r« & * 0, f « 0

Fig. 2-5. Two Main Causes of Decrease of Bending Moment with Increase in Pile Flexibility.—From Rowe (1952, p. 46)

21

the span of the pile, and since bending moment is proportional to the

cube of the span distance, a small change in span results in the same

effect as increasing the fixity below dredge level. Thus another way of

decreasing the moment is to increase the depth of embedment or the soil

stiffness. In Rowe's (1952) study, the soil stiffness was characterized

by density, being either loose or dense.

The soil stiffness was much more significant than the depth of

embedment in causing fixity, the latter variable being essentially of no

importance. A change in the pressure distribution with soil density is

caused by the different sheet-pile deformation patterns which arise de­

pending on the stiffness of the supporting soil (Fig. 2-5a).Both soil stiff­

ness and pile flexibility were of approximately equal significance in

reducing moments because in both the passive pressure resultant is

raised with increase in density or flexibility.

Since lowering the tie rod will decrease the span, reduction in

moment will occur. This reduction is not as large as for fixity below the

dredge level, since the effect of soil pressure or fixity condition above

the tie level enters in. The effect of the soil above the anchor level on

the moment reduction is related to the pile deformation pattern. As the

pile deforms, that portion above the anchor level is pushed into the soil,

assuming small tie-rod deformation. Thus, the soil displays fixity or

stiffness. Another way of characterizing the occurrence of moment reduc­

tion is to specify the fixity above the tie-rod level, which increases

with the length of pile above the tie-rod level.

A secondary effect that caused additional moment reduction with

flexible piles was the resisting force set up along the tie rod when it is

dragged into the fill by a settling wall. Other tests showed that defor­

mation in the tie rod would cause moment reductions of up to 10 percent.

This reduction is allowed for anchor deformations greater than H/1000,

since in Rowe's (1952) flexibility tests he found that the maximum mo­

ment in the pile increased with movements smaller than this amount. In

the field, deformations considerably greater than H/1000 can be ex­

pected .

In general, moment reduction occurs when fixity below the

dredge level or above the tie-rod level is present. The fixity refers to

the soil support condition. Below the dredge level the fixity increases

with increasing pile flexibility.

Tie-rod loads are reduced by increasing pile flexibility which

is equivalent to an increase in fixity below the dredge level. An in­

crease in the tie-rod load occurred for fixity above the tie-rod level.

In general, the load increase was independent of tie-rod yield for dense

soils.

An optimum design consisted of a dredge level at a depth of

0.73H and the tie rod located at 0.20H. Rowe's depth of embedment,

0.27H, compares closely to the 0.30H recommended by Tschebotarioff

(1949) in his fixed earth support oriented procedures. In Rowe's method,

factors of safety are incorporated into the depth of embedment either

directly or by solving for the depth using "factorized," or reduced soil

parameters with failure conditions. The sheet pile and anchor rod are

given a safety factor by designing them for the load or moment they will

be subject to at failure as well as providing a material factor of safety.

Terzaghi's (1954) recommendations based on Rowe's model tests

suggest the use of additional embedment, values for wall friction, and

the use of a factor of safety of 2 to 3 on passive failure without a toe

shear force.

Subgrade Reaction Methods

Another approach to the design of anchored bulkheads that

varies in concept from classical design methods is known as the "sub-

grade reaction method." In general, it idealizes the bulkhead system,

sheet pile, soil and anchor, as a beam supported by a series of springs.

The governing equation is

EI~dx4" = Pa " Pp = f(x,y)

where (pa - Pp) represents the resultant pressure of the active and pas­

sive distribution at any point that is dependent on the deflection y and

the location along the beam x.

Rowe (1955a) determined that the pressure coefficient pp is

linearly dependent on confining pressure represented by depth x as well

as the soil stiffness modulus m, which is related to the constant of

subgrade reaction £. A nonlinear relation exists for deflections y and

model scale D, and the nonlinear exponent is n. The equation given by

Rowe took the form:

Values of Jl have been established by Terzaghi (1955), but due to their

limited accuracy, he recommended that they be used for stress, not de­

flection calculations.

24

In 1951, Blum (cited in Rowe, 1955a) solved the governing

equation by using a second-order deflection approximation and a linear

relation for pp. Rowe's (1955a) approach considered a linear relation for

all terms in the expression for pp and expanded the deflection into a 30th-

order equation. On substitution in the governing differential equation and

using boundary conditions, he obtained an expression for deflection.

From this deflection expression it was evident that the second-order

equation Blum used was insufficient for piles of medium flexibility.

The above approaches are extremely tedious and are not suited

to normal design practice. Therefore, a moment-flexibility curve was

presented by Rowe (1955c) for use in design. Richart (1957) handled part

of the numerical complexities by using Newmark's numerical method on

separate sections above and below the dredge line. He also made the

assumption that zero deflection occurred at the anchor level. Richart's

analysis yields results similar to Rowe's (1955a, 1955b) analyses for

pinned or encastr£ walls as well as an equivalent beam analysis.

A more sophistaicated idealization involving less assumptions

was done by Rauhut (1966). The bulkhead system was represented as

interconnected beam elements attached to springs at their endpoints.

The iterative procedure used consisted of assuming a load pattern on the

structure, computing the soil stiffness based on the modulus of subgrade

reaction corresponding to the load level and solving the beam on elastic

foundation equation for deflections using finite elements. If this deflec­

tion pattern is not close to that corresponding to the original load pat­

tern, assume new loads and iterate again. A nonlinear soil and wall

friction are incorporated into the method, which requires a computer for

25

solution. The main problem in the approach, which is characteristic of

all subgrade reaction methods, is in characterizing the soil stiffness

modulus. The approach is realistic in that actual test results relating

the nonlinear subgrade reaction of the soil to plate deflections are used.

An extension of some of Rauhut's work is found in Haliburton's (1968)

study in which he increases the generality of the analysis. An advantage

of the computer approach used by these researchers is that it eliminates

the computational difficulties of finding the moments (stresses) in the

sheet pile. A favorable aspect of the coefficient of subgrade approach

was stated by Terzaghi (1955, p. 306): "The errors in the evaluation of

the stresses in the mat due to an error of + 50% in the evaluation of ks

are negligible" [ks = coefficient of subgrade reaction].

A "distribution" method (Turabi and Balla, 1968) considers the

effect of wall deformations on the earth pressure distribution by putting

spring supports at five locations above and five locations below the

dredge level. The deflections are computed by using standard indeter­

minate beam theory. A linear variation with depth of a linear soil stiff­

ness was incorporated into both the theory and results. A flexural

rigidity number equal to logpm represents the soil stiffness at the toe

of the sheet pile.

The main limitation of the subgrade reaction method is in the

use of a coefficient of subgrade reaction for calculating deflection.

Terzaghi (1955) cautions against the use of these methods for the pur­

pose of estimating displacements. Accuracy of deflections is essential

since deflections are a significant output in the analysis on which pres­

sures and sheet-pile stresses are based.

Field Tests

Measurements on bulkhead systems in the field have been con­

ducted to ascertain their actual behavior. These measurements allow

the behavior predicted by various design methods to be evaluated. The

examination of field tests cannot be limited to cohesionless materials,

as was done for the review of design methods, due to the varied distri­

bution of soils encountered. A description of the behavior of clays must

include such concepts as consolidation, effective stresses, and creep.

Positive volume changes yielding negative pore-water pressure can

arise with overconsolidated clays. These factors pose added complex­

ities in analyzing data obtained from bulkheads with a clay supporting

soil. Results that relate to these factors of clay behavior are not dis­

cussed.

Lea (1953) reported on a bulkhead for which movements, tie-

rod loads, and varying ore surcharge loads had been monitored for 10

years. He noted that the tie-rod load may be increased significantly by

the large value of friction existing between it and the soil and by heavy

surcharge loads.

Using a Wiegman slope indicator, Tschebotarioff and Ward

(1957) measured the slope from which the moment could be calculated

on five bulkheads of varying types of construction, anchoring, depth of

embedment, surcharge, soil types, etc. They found the moments to be

between one third and two thirds of free earth support values, agreeing

in general with Rowe's model tests. Also, the maximum bulkhead mo­

ments predicted by Tschebotarioff' s (1951) simplified equivalent beam

method were close to the maximum moments measured.

27

Tschebotarioff (1958) measured moments below the dredge level

using a slope indicator run down inside box piles. The results caused

him to modify his hinge at dredge-level assumption to a hinge at the

first compact layer below dredge level.

Thompson and Matich (1961) made slope indicator readings on

bulkheads in Hamilton Harbor and Burlington Channel in Hamilton,

Ontario, and in Seven Islands and Sorel in Quebec. These piles were of

the U type for which the interlocks are located at the neutral axis of the

wall where the shear force is greatest. Thus, slip may occur, depending

on the friction of the interlocks, causing a decrease in the section

modulus for the wall section. The moments were compared to theoreti­

cal calculations based on Rowe's method, using limiting values of sec­

tion moduli for full interlock friction and no interlock friction.

The earth pressure measurements obtained by Mead (1963) were

similar to those obtained by Duke (1953) and Tschebotarioff (1949).

Further substantiation of their results was given by the pressure mea­

surements Beverly (1963) obtained by double differentiation of the

moment curve derived from strain gage measurements. For this bulkhead,

which was excessively driven below the dredge level, full passive pres­

sure was mobilized near the dredge level with small passive pressures

occurring near the toe on the back side of the sheeting.

Hakman and Buser (1962) tried three approaches to measure

stresses in the anchored bulkheads during construction at Toledo, Ohio.

The strain gages malfunctioned, and the surveying was of limited ac­

curacy, leaving only the slope indicator measurements from which the

moments and deflections could be established. All moments calculated

were less than those obtained using Tschebotarioff's equivalent beam

method. Hakman and Buser also noted that large differential water *

levels occurred even though weep holes were provided to drain the porous

backfill material.

A more extensive study was undertaken from 1960 through 1964

on the bulkheads located at Burlington Beach wharf in Hamilton Harbor

and the Ship Channel Extension in Toronto Harbor (Matich, Henderson,

and Oates, 1964). The moments and deflections were measured using a

slope indicator operating inside a grooved plastic guide casing which

had been built into the structure during construction. Tie-rod measure­

ments were made with strain gages. Since the bulkheads were monitored

during construction, it was possible to measure the deflections and

moments induced by driving and dredging. The readings showed that

large deflections of up to 20 inches from the vertical resulted from the

driving operation causing moments of a magnitude great enough to affect

any comparison to theoretical moments. Moments obtained for the Ship

Channel Extension were very close to those calculated by Rowe's method,

although the tie-rod force was 50 percent greater than the calculated

value.

The bulkhead at Burlington Beach wharf, location number 2

(Matich et al., 1964) consists of a 50-foot U section pile driven into

sand, anchored to a continuous concrete anchor wall constructed in

backfilled material (Fig. 2-6). The last stage of construction was the

dredging of material from in front of the pile. The observed moments

based on full interlock friction were equal to those calculated by the

equivalent beam method. For the case of no interlock friction the

29

ANCHOR WALL

230-1

SURCHARGE *600 LSS /SO. FT.

riH.'Vyu.V

SIEJIQD

VERY DENSE AND. WlIHJjRAyEL. DREDGE DEPTH'

lilSTEEL SHEET PILING ^LARSSEN «A OEEP ARCH • • .:

Fig. 2-6. Cross Section, Burlington Beach Wharf, Test Location 2.—From Matich et al. (1964, p. 175)

moments were 27 percent less than those calculated using Rowe's method.

The tie-rod force after dredging conformed best to that force calculated

using the equivalent beam method, agreeing to within 3 percent.

The agreement between calculated and observed data does not

appear to be conclusive in establishing the superiority of a particular

design method or in supporting any of the proposed design models. The

problem appears to lie not with the design methods as much as with the

state of knowledge concerning field testing. Also, the effects of field

conditions, such as soil variability, anchor structures, and variation in

anchor-rod forces along the sheet-pile length, are uncertain.

Anchor-wall Design

Various types of anchoring systems are used to restrain tie-rod

movement. In general, they can be divided into two types of construc­

tion: (1) wood, concrete, or H piles and (2) "deadman" structures vary­

ing from concrete blocks, walls, or footing structures to sheet-pile

anchor walls. Continuous forms of the latter type of construction will

be considered in this discussion.

30

If such deadman structures are long in a horizontal direction

parallel to the sheet pile or if they are continuous, it is possible to com­

pute the tie-rod restraining force (T) offered by them as the difference in

the resultant forces of the passive (Pp) and active (Pa) pressure distribu­

tions acting on the deadman divided by a factor of safety (FS):

T = (Pp - Pa)/FS.

Ideally, the tie rod should be located at the point where the resultant

earth pressure acts on the anchor. This location will cause only trans­

lation with no rotation and allows full development of the anchor

capacity.

The location of the anchor can best be discussed by consider­

ing the three zones shown behind the bulkhead in Fig. 2-7. If the anchor

is located in the active wedge represented by abc, no resistance will

be offered. The location of the anchor in the zone to the right of ad

will result in no transfer of load to the wall and full anchor resistance.

In zone acd, the amount of resistance offered and the magnitude of the

force transferred to the sheet-pile bulkhead is dependent on the overlap

of the passive failure wedge of the anchor wall and the active failure

wedge of the bulkhead.

More complex methods of design are available which include

empirical results, various types of failure arcs, or coefficients of sub-

grade reaction. They are not presented here since they are deemed be­

yond the basic consideration necessary to design anchor walls.

Limitations of Available Design Methods

The limitations of the classical design methods and their modi­

fications lie in their assumptions. The hypothesis that failure conditions*

Anchor wall

^ = angle of internal friction for soil

Estimated point of zero moment in sheet pile

Fig. 2-7. Anchor Location.—From Terzaghi (1943)

32

divided by some factor of safety yield working conditions is untrue for a

nonlinear soil system in which deflections are dependent on soil pres­

sures. These methods cannot ascertain the pressure conditions under

working load due to their neglect of the stress-strain properties of the

soil.

Subgrade reaction methods depend on load-deformation charac­

teristics for the springs, which are not readily available. Also, a knowl­

edge of the various idiosyncracies inherent in each computer program, its

input data and performance, is required.

In general, from an economic viewpoint, there is little differ­

ence between the design methods except for the classical methods (free-

earth support and fixed-earth support). The classical solutions, being

limiting procedures, are conservative. The classical free-earth support

method is conservative with respect to pile section, and the classical

fixed-earth support method is conservative with respect to depth of

embedment. In the design process, there are more important considera­

tions than the design method. Failures have occurred more often due to

improper identification of the soils present in the field or characteriza­

tion of the soil-strength parameters than are caused by choosing the

wrong design method (Terzaghi, 1954). Therefore, a more fruitful area

of research in design methods may be in avoiding the tedious calcula­

tions or in improving the characteristics of the soil-strength input

parameters. An acceptable computer approach appears to offer tangible

possibilities.

The finite element idealization of the entire bulkhead system

will circumvent the problems of limiting assumptions and tedious

calculations. It will use a more fundamental soil characteristic, the

stress-strain curve, rather than the modulus of subgrade reaction or

other parameters that only represent the soil at failure. The solution,

which relies on the availability of a computer, provides the deflections

and soil stresses throughout the system. A feasibility study of the use

of a finite element approach with the sheet-pile problem was undertaken

by Bjerrum, Clausen, and Duncan (1972). They found that the method

showed promise in studying the factors influencing the behavior of sheet-

pile walls.

Various factors, the consideration of which is not inherent in

any specific design method, have not been discussed. These include

corrosion, scour, overall stability, various surcharge loading distribu­

tions, and differential water pressures. Also, the presentation of the

design methods and the model test results have been restricted to in­

clude only those portions that apply to cohesionless materials.

CHAPTER 3

THEORY OF THE FINITE ELEMENT METHOD

The theory of the finite element method has been refined to a

sophisticated level during the 17-year period since its introduction.

The various areas of specialization, such as element development,

equation solution, and convergence, have received intensive research.

Some of the theoretical developments of those portions of the finite ele­

ment method that are utilized in the idealization of the bulkhead system

will be discussed in this chapter.

The General Stress Analysis Problem

In solving any indeterminate problem in stress analysis, it is

necessary to satisfy simultaneously equilibrium of forces, force-

deformation relations, and geometric compatibility. In the discrete

analyzation, these equations are considered at each node point where

the corners of the elements join.

Determinations of stress distribution and associated stability

state of soil-structure masses have always been tedious problems. In

certain situations, loading and system configurations may be of such a

nature that if elastic properties are assumed, a solution from elasticity

may apply, for example, when the Boussinesq solution is used. If the

problem is approached through the equations of elastic theory, mathe­

matical techniques, such as finite difference methods and numerical

integration of the partial differential equations that arise are often

34

employed. In this approach, the exact equations of the actual physical

system are solved by approximate mathematical procedures. No solution

has been attempted for an anchored bulkhead using the theory of elas­

ticity due to the complex nature of the boundary conditions.

The elasticity requirements may be satisfied by using either

the displacement method or the force method of structural analysis. The

former considers the nodal displacements as unknowns, the latter con­

siders the internal element forces. As the displacement method is easier

to use for a general approach to all problems, it is almost exclusively

used.

Finite Element Method

The finite element method assumes that the continuum can be

considered as an assemblage of a finite number of discrete elements

interconnected at node points. Each of the individual elements retains

all of the material properties of the original system. The approximation

involved is the substitution of a modified structure for the actual con­

tinuum. No approximation is necessary in the analysis of this substitute

system (Zienkiewicz, 1971).

The power of the finite element method lies in its applicability

to many disciplines, as well as its ability to handle discontinuities. In

structural system analysis, the method can readily handle nonlinear,

jointed, nonhomogeneous, anisotropic, viscoelastic or creeping materi­

als in a discontinuous or complex geometrical configuration with mixed

boundary conditions. The method provides solution results over the en­

tire domain at all intermediate loading conditions up to the limiting

failure state. The procedure used applies to many boundary value prob­

lems and is easily programmed.

Limitations in the development of the finite element method

arise with certain problems. Analysis of discontinuous media formed by

cracking or separated joints along with strain-softening materials poses

problems in application of the method. Also difficulties can arise in

zones of high stress concentrations. One severe criticism of the finite

element approach as compared to a limit design approach is that the

elastic material properties that are required in the former method are

more difficult to measure than the strength properties needed for the lat­

ter method. Other limitations concern the generation of errorless input

data from a feasible grid and the proper interpretation of the results. A

requirement is access to a large computer with adequate core memory.

Finite Element Formulation

The general theory of the finite element method has been pre­

sented by Zienkiewicz (1971) and Przemieniecki (1968). Regardless of

the element types used, the finite element displacement method of anal­

ysis has the following basic steps (R. W. Clough, 1965):

1. Evaluation of the stiffness properties of the individual struc­

tural elements in terms of a local (element) coordinate system.

2. Transformation of the element stiffness matrix from the local

coordinate system to a global coordinate system representing

the complete structural assemblage.

3. Superposition of the individual element stiffnesses contributing

to each nodal point to obtain the total nodal stiffness matrix

ClQ for the system.

37

4. Applying boundary conditions and solving the system equilibrium

equations = [iQ which express the relationship be­

tween the applied nodal forces [P'j and the resulting nodal dis­

placement .

5. Evaluation of the element deformations from the computed nodal

displacements by using kinematic relationships and, finally,

determination of element forces from the element deformations

by using the element stiffness matrices.

The first step involves the determination of element stiffnesses.

An equilibrium array f A^ relates the external (global) forces |p$ applied

at the nodes to the internal (local) forces (stresses) |>F]. The relation is

[PX = JjOjF j. (3-1)

The element force-deformation, - £e$, (stress-strain, \6l )

properties are expressed by the constitutive relations [S] as follows:

iFi = IsJfef . (3-2)

A compatibility array, [B] , relates the local internal displacements [ej

(strains,) to the global external displacements at the nodes by

the expression,

*e! = [B] lul. (3-3)

Array [B] is the transpose of [A] divided by the element's volume. The

element stiffness array can be formed by combining the equilibrium,

force-deformation, and compatibility arrays, yielding,

= [ASB] i u i , o r ( 3 - 4 )

{?} = LXf iuj . (3-5)

The derivation of the stiffness properties of triangular,

constant-strain membrane elements with three nodes (TRIM3 elements)

requires the assumption of a displacement pattern along the element

boundaries. For a TRIM3 element, a displacement function that insures

displacement compatibility with adjacent elements of the same type is

assumed. This assumption causes equilibrium between elements to be

violated, although general equilibrium of nodal force resultants is pro­

vided. In the limit, as the mesh size is reduced, equilibrium is satis­

fied at element boundaries.

An alternative approach to assuming displacement patterns is

assuming stress or force patterns. The resulting flexibility array can be

inverted to yield the stiffness array.

Idealizations that involve only bars and beams have at times

not been considered to be finite element problems, since exact solutions

are possible. Such solutions result from the use of a displacement func­

tion that accounts for all possible displacement configurations.

The element stiffness matrix can be transformed from the local

coordinate system to a global coordinate system by use of the following

transformation relations,

where P isglobal force and U' is global displacement. The transformation

matrix contains the direction cosines between the local and global orien­

tation. The transformed element stiffness array in the global coordinate

system is

{P'5 = M [p]

( U j = P T T l U ' j

(3-6)

(3-7)

{ ? ' ] = D k ^ T ( u ' } , o r ( 3 - 8 )

{?'} = WtU'l. (3-9)

The system stiffness array [K] can be formed by adding all the

element stiffness contributions at each node and storing the result in

the global location, corresponding to that node. Depending on how the

system's nodes are numbered, the coefficients in the [K] matrix will oc­

cupy a certain bandwidth around the main diagonal. This bandwidth is

a function of the greatest node separation between any one element's

nodes. Since the stiffness matrix is symmetric, it is necessary to store

only that portion above or below the diagonal. The combination of the

banding technique and the symmetric property surmounts a tremendous

computer problem, that of available core memory.

The applied nodal forces must be consistent with the distributed

loads and body forces. For a triangular element, a third of the body

force would be applied at each node and for a distributed force, a half

at aach node. In order to have an independent set of equations, the [K]

matrix should consist of only those rows and columns for which a support

is not specified. Rather than remove those elements in the array, the

equations can be decoupled by putting unity on the diagonal and zeros in

the rest of the row or column corresponding to the coordinate that is sup­

ported. Support displacements can be read in through the load vector by

adding unity to the diagonal coefficient of the [Kj matrix at the supported

degree of freedom location. This number, as well as the displacement in

the load vector, are multiplied by a very large number. The end result is

decoupling of that equation (Zienkiewicz, 1971).

40

The equations are now ready for solution. For any given load­

ing at the nodes, the equations can be solved for the nodal displace­

ments. Since there are as many equations as there are total degrees of

freedom of the structure, a computer is necessary for even the simplest

structure. A very efficient method for solving these equations for the

nodal displacements is Gauss elimination with back substitution. This

method is preferable to matrix inversion for the cases where the equa­

tions are banded as less core memory is required, where the stiffness

matrix is dependent on the previous loading and must be regenerated

each time and, where only one loading condition is being analyzed.

Once the displacement pattern is known. Eqs. 3-7, 3-3, and

3-2 can be used to solve for the local displacements and element strains

and stresses, respectively. A check on these latter results as well as

the displacements is available, using the calculated stresses in Eqs.

3-1 and 3-6 to compute the applied nodal forces. These should be equal

to the originally applied forces.

Another theoretical development of the finite element method of

analysis is available by using the calculus of variations (Desai and

Abel, 1972; Zienkiewicz, 1971). The minimization of the potential ener­

gy integral with respect to each of the non-zero nodal loads leads direc-

ly to the stiffness array. This procedure is analogous to that of Ritz.

Convergence

Convergence to the correct solution is insured, if

1. Compatibility is satisfied.

41

2. The element can undergo rigid body displacements without

strain.

3. The element can represent constant strain states.

For a triangular constant strain element, the linear displacement func­

tion that is used satisfies compatibility everywhere; therefore, dis- •

placements will converge from below the actual value. Due to the en­

forced linearity along the boundaries, the structure is stiffer than actual.

This increased stiffness yields smaller than actual displacements. The

stresses in the TRIM3 elements will not be affected by a uniform change

in stiffness among all elements but rather by a discontinuous change in

stiffness between TRIM3, beam, and bar elements. Since the correct

displacement function is assumed for a bar or a beam element, the cor­

rect displacements and stresses will only be limited by the accuracy for

the stiffnesses of adjoining TRIM3 and interface elements.

Accuracy can be increased by using displacement functions of

higher order. These functions may or may not be accompanied by more

nodes on the element. If the displacement function chosen is noncon­

forming for which compatibility is satisfied only at the nodes and not

necessarily along the boundaries, convergence will occur from either

above or below.

The accuracy of a finite element solution is also dependent on

the mesh size. As the mesh becomes progressively smaller, convergence

occurs, since in the limit, displacements between elements are correctly

satisfied. Deflection patterns will be more accurate than the distribution

of strain or stress values because the latter quantities are represented by

a lower order polynomial, which is the derivative of the deflection

42

polynomial. Fine meshes should be used in regions where the stress

gradients are high and coarser meshes at other locations.

Nonlinearities

The two types of nonlinearities, material and geometrical, are

handled by two methods of solution, iterative and incremental. Geomet­

rical nonlinearities occur when the deformations are large compared to

the size of the system or individual element strains are large. In such

cases, it becomes necessary to analyze the deformed configuration rather

than to assume that the configuration of the original system correctly

simulates the deformed system. In general, this latter assumption is

valid for anchored bulkhead systems.

In considering nonlinear material problems by the direct itera­

tive approach, the following steps are followed:

1. Apply the full load to the system.

2. Assemble the stiffness matrix using element stiffnesses based

on some initial tangent modulus.

3. Compute the changes in system displacements and element

strains.

4. Using the element stress-strain curve find the stress (load)

compatible with the computed strains (displacements).

5. Determine the out-of-equilibrium nodal forces.

6. Repeat this procedure applying the loads computed in step 5,

and using a new stiffness in step 2 based on the present load

level.

These steps are diagrammatically portrayed in Fig. 3-1. Convergence is

Deformation

Fig. 3-1. Iterative Procedure

Deformation

Fig. 3-2. Incremental Procedure

44

not assured for bilinear or ideally plastic materials, since two different

loading states may be in equilibrium. The iterative procedure is able to

handle strain softening but poses problems when initial stresses are

present.

In the incremental procedure the system is loaded in steps. At

the end of each step a new modulus is calculated based on the existing

stress state. The cycle is repeated until the full load is on the struc­

ture, as shown in Fig. 3-2. Accurate modeling of the load deformation

curve is dependent on step size. The method is particularly applicable

to systems for which construction steps, such as backfilling or dredging,

must be analyzed incrementally. It is for this reason that the incremen­

tal approach will be used in this study.

The incremental method can readily handle geometrical non-

linearities by introducing the new geometric configuration after each

increment. It also has the advantage of readily treating initial stresses

and also providing an analysis for each increment of load applied, up

to the full load. It is not presently possible to analyze systems that

display a general strain-softening behavior using an applied incremental

load analysis. It is necessary to increment deflections in such a situa­

tion. A mixed procedure can be used that combines the advantages of

both the incremental and iterative procedures (Desai and Abel, 1972).

Applications

Due to the generality of the finite element method of analysis,

it has been applied to a wide range of engineering problems. Stresses

and deformations in embankments have been examined by Clough and

Woodward (1967), Kulhawy, Duncan, and Seed (1969), Duncan and Dun-

lop (1969), Chang and Duncan (1970), Nobari and Duncan (1972), and

Kulhawy and Duncan (1972). In each analysis, nonlinear material prop­

erties were considered. The modeling was such that the construction

sequence of either cut or fill was considered.

Christian and Boehmer (1970) analyzed consolidation using

finite elements. The inclusion of drained and undrained earth dam be­

havior was done by Wroth and Simpson (1972). An extension to three

dimensions was performed by Lefebvre, Duncan, and Wilson (1973).

Dynamic and earthquake studies, using the finite element method,

studied the influence of the base of an embankment or dam and the extent

of the boundaries necessary to model the system. These studies were

conducted by Clough and Chopra (1966), Idriss and Seed (1967), Finn

(1967), and Idriss (1968).

Soil-Structure Interaction

The design of anchored sheet-pile structures is one portion of a

larger class of problems denoted as soil-structure interaction problems.

These problems concern those systems in which the deformations of the

structure are large enough to cause substantial redistribution of the

pressure at the interface (Peck, 1972). This definition can be extended

to refer to systems composed of two different stiffnesses. "Structure"

refers to the stiffer system, which is sometimes steel, concrete, or

rock but may be any material forming a portion of the system that dis­

plays a different stiffness than the soil. Thus, investigations of the

interaction between a rigid base and a soft embankment on top can be

considered to be in the general class of interaction problems. Also, an

46

extension to rigid structures that displace bodily is often made so as to

include rigid retaining walls and rigid raft footings. The general design

procedure for all types of problems within this class follows a limit de­

sign approach whereby some failure state is analyzed and the result may

be modified by experimental data.

The finite element method by its universality solves all types

of problems within the larger class of soil-structure interaction problems

in a very general manner. Even the restricted usage of finite elements in

Haliburton's (1968) bulkhead analysis allowed him to place the beam

element idealization horizontally on the soil surface in order to model

footing behavior.

A brief review of recent finite element soil-structure interaction

studies will be undertaken. A more complete survey of finite element ap­

plications in earth-structure interaction has been presented by G. W..

Clough (1972).

Footings resting on soil present an interaction problem (Huang,

1968; Girijavallabhan and Reese, 1968). The effect of various footing

flexibilities can easily be analyzed (Radhakrishnan and Reese, 1969;

D'Appolonia and Lambe, 1970; and Desai and Reese, 1970). The analy­

sis of pavements resting on layered soils presents a situation in which

there is interaction between the more rigid pavement and the relatively

softer soil (Duncan, Monismith, and Wilson, 1968). The interaction be­

tween rigid retaining walls and soil was studied by Morgenstern and

Eisenstein (1970). Interaction in three dimensions on buried cylinders

was studied by Ruser and Dawkins (1972).

47

A breakthrough in the modeling of soil-structure interaction

problems came with the development of a one-dimensional element to

represent the interface between the components of the system (Goodman,

Taylor, and Brekke, 1968). This element was employed by Duncan and

Clough (1971) and Clough and Duncan (1971) on ship-channel locks and

rigid retaining walls.

Tied-back excavations were investigated in a parameter study

by Wong (1971) and in a study on soldier-pile walls and slurry-trench

walls by Clough, Weber, and Lamont (1972). This latter study showed

that the maximum deflection for a soldier-pile wall was 2 inches com­

pared to 1.5 inches for a ten times stiffer slurry-trench wall. Both

walls were restrained by prestressed tie backs.

There is a definite lack of application of the finite element

method to the dynamic interaction of structures and soil. The models

most frequently employed represent the soil by an equivalent spring-

da'shpot system of generally no more than three degrees of freedom, a

Winkler foundation, an eleastic half-space, or a lumped parameter sys­

tem. A review of dynamic finite element applications by Seed and

Lysmer (1972) discusses some aspects of dynamic soil-structure inter­

action .

CHAPTER 4

MODEL FOR SOIL BEHAVIOR

A stress-strain plot for cohesionless soil exhibits nonlinearity.

The initial slope of the curve, known as the initial tangent modulus, is

dependent on the confining pressure. Failure of the material occurs at

the maximum point on the stress-strain curve and can be represented by

a failure theory. This point varies for different confining pressures.

The graphic stress-strain relations for soils show very little tensile

stress capacity. During unloading and reloading, rebound occurs along

a path dissimilar from the loading curve. The need for an explicit rela­

tionship describing the above behavior is evident. Also, the behavior

of the soil at an interface with a wall is similar. This chapter will

describe the development of the theory used to represent soil behavior.

Review of Previous Soil Behavior Models

Seed and Idriss (1967) represented the elastic and shear modu­

lus values of soil materials as a function of the 1/3 power of the con­

fining pressure. Clough and Woodward (1967) described plastic materials

by attributing all volume change to shear deformation with normal defor­

mations being negligible. The constitutive equations are composed of a

constant bulk modulus, E/2(1 + ji) (1 - 2p)), which is dependent on the

initial value of Poisson's ratio, j j, less than 0.5, and a distortion or

shear modulus, E/2(1+ j j), which can incorporate values of Poisson's

ratio greater or equal to 0.5. Such values account for incompressible

48

and dilatant soils for which u approaches or exceeds 0.5 and causes

the bulk modulus to be infinite or negative.

Clough and Woodward (1967) also conducted an investigation of

the variation of stress and strain quantities with p. Their results strong­

ly suggest that u must be considered as a variable in any soil behavior

model for a proper stress and strain determination.

Nonlinear studies by Huang (1968) showed that nonlinearity of

soils has a relatively small effect on stresses but a large effect on dis­

placements. The elastic modulus increased with increase in the confin­

ing stress invariant (6\ +62 + 6*3) and decreased as the deviatoric

stress (6i - 6"i) increased. The latter effect was more influential with

clays, and the former with sands. Later on, Huang (1969) described the

variation of the modulus for sand as a function of the first stress invari­

ant and for clay as a function of the second stress invariant. In this last

study, it was determined that of 0.45 gave very accurate results in

representing an incompressible material.

Girijavallabhan and Reese (1968) used octahedral normal and

shear stresses to describe the hydrostatic and deviatoric components of

a Von Mises yield theory. A similar proposal was made by Newmark

(1960) in which the general soil stress-strain relationships and failure

hypothesis were expressed in terms of octahedral stresses and strains.

A drawback in Girijavallabhan and Reese's (1968) approach was the ex­

clusive use of empirical results to describe the octahedral stress versus

strain relations rather than the use of an explicit functional form.

Domaschuk and Wade (1969) considered the volumetric and

deviatoric components of strength. The volumetric component they

50

described by a function bilinearly increasing with normal stress and the

deviatoric component by a hyperbolic function of the mean normal stress,

relative density, and deviatoric stress. Graphic representations were

used for the variations with the mean normal stress of the initial shear

modulus and the failure deviatoric stress.

The dilatant property exhibited by dense or overconsolidated

soils can be considered by a variational theorem proposed by Herrmann

(1965) and later modified by Christian (1968) and Hwang, Ho, and Wil­

son (1969). A less complex representation of a contractive or dilative

elastic work-hardening soil was given by Smith and Kay (1971). A more

general representation defines the constitutive equations in terms of the

bulk and deformation moduli rather than Young's modulus and Poisson's

ratio. A perfectly plastic mode occurs after failure in which the material

cannot take additional shear stress but can sustain additional bulk con­

fining stress. Dilation is accounted for by permitting values of Poisson's

ratio greater than 0.5 in the deformation modulus.

The combination of a bulk and shear modulus approach, along

with a hyperbolic strain-softening relation proposed by Richard (1973),

yields a very general nonlinear stress-strain formulation. Such a theory

is needed since materials that dilate during shear usually display strain-

softening behavior. This approach makes it possible to handle incremen­

tally, situations where the elements connecting into a node cause the node

to display a strain-softening behavior. Loads can be incremented up to the

point where the general system load-deformation curve displays a strain-

softening behavior and catastrophic failure occurs.

51

Nonlinear Soil Model

A theory that embodies by an analytical equation all of the

stress-strain curve properties noted previously except dilatency has

been proposed by Duncan and Chang (1970). This theory will be recapit­

ulated here because it is used in the finite element formulation.

Tangent Modulus

A hyperbolic equation is used to describe the stress-strain re­

lation:

*1 " *3 " ~a"+~Er <4"1)

where <5"i ana (5*3 are the major and minor principal stresses, respective­

ly, £ is axial strain, and a and b are experimentally determined con­

stants. The value of a corresponds to the inverse of the initial tangent

modulus, Ej[, and b corresponds to the inverse of the asymptotic or ul­

timate stress difference, (<5"j - <5"3)uit. With a hyperbolic relation, it is

found that the soil's failure compressive strength, (61 - <5"3)f, is a frac­

tion, Rf, of the ultimate or asymptotic stress difference. Incorporating

this ratio into Eq. 4-1 yields

«1 - «3 = j ^ f gj . (4-2)

EI (<5"I - <Y3)F

The variation of the initial tangent modulus with confining pres­

sure, as expressed by the minor principal stress 63 is given in the fol­

lowing relation:

El = KPatm(iwr) (4"3)

where patm is atmospheric pressure expressed in the same units as Ej

and 63, K is a pure number called the modulus number, and n is a real

number describing the variation of Ei with <53. These latter two param­

eters can be determined experimentally.

The Mohr-Coulomb strength criterion is used to represent the

failure state. This relation expresses a linear increase in compressive

strength with increase in confining pressure. The Mohr-Coulomb crite­

rion predicts a general three-dimensional failure state by a two-

dimensional failure law whose parameters are determined from a

three-dimensional test. It assumes the stress in the third dimension

does not affect the strength.

If failure occurs with no change in 6 3 , the criterion defining

failure strength in terms of confining pressure can be written as follows:

(61 - 6-3)f ~ 2ccos<S+ 2*3 SIM (4_4) 1 - sinp

where c and f6 are the cohesion and angle of internal friction for the ma­

terial, respectively.

In an incremental analysis, it is necessary to have a value for

the tangent modulus. Differentiating the relation between (61 - 63) and

(leads to the following expression for the tangent modulus:

Et = + ^ Rf 1 2' *4~5* 1 E i | _ E i ( f f l - c r 3 ) f j

It is logical to eliminate strain in this expression due to its arbitrarily

chosen reference state. Rewriting Eq. 4-2 as strain in terms of stress

yields

£ = SI -g3 [", . Rf(<?l -<?3>] "2 (4_6)

El L «ri-<S3)fJ • W

Incorporating the expressions for Ej, - d*3)f, and € into

Eq. 4-5 yields the following expression for tangent modulus, which can

be easily applied in an incremental analysis

f - l~i RfU - slnjflfrri ( <?3 \n (a 7v Et L 2c cos (6 - 2<y3 sin/J KPatm^Patmy * (4 7)

Tangent Poisson's Ratio

The other parameter that is necessary to describe isotropic

materials by the generalized Hooke's law is Poisson's ratio, ji. A de­

velopment similar to that for the modulus of elasticity of Duncan and

Chang (1970) was presented by Kulhawy, Duncan, and Seed (1969).

The tangent Poisson's ratio for a uniaxially loaded cylinder,

is defined by the following equation:

in which £r is radial strain and Ca is axial strain. The nonlinear rela­

tion between axial and radial strain can be represented by a hyperbolic

equation of the form:

in which f and d are empirically determined parameters. The parameter

f corresponds to the tangent Poisson's ratio at zero strain, denoted as

the initial Poisson's ratio, pj. Parameter d corresponds to the slope of

the transformed hyperbolic strain plot.

The value of is known to decrease with increasing confining

pressure. This variation may be represented by ar. equation of the form

in which G is the value of jii at a confining pressure of one atmosphere

d£r

" d f a (4-8)

(4-10)

54

and F is an experimentally determined parameter representing the rate of

decrease of;ui with increasing confining pressure, 63.

The mathematical manipulation of differentiating Eq. 4-9 with

respect to 6r and substituting for ( a ln terms of £r in the result yields

= a -£fad)^ • <4-u>

Substituting the expression for jii given in Eq. 4-10 for f in Eq. 4-11

gives

G-Flog(-^-)

(1 - fad)2

The axial strain can be rewritten in terms of stresses and stress-strain

parameters, as in Eq. 4-6 of the tangent modulus derivation. Substi­

tuting Eqs. 4-10 and 4-11 into this equation produces the following ex­

pression for fa:

6*1 " g3 = ( 3 \n ( Rf (gl ~ g3) 0- " sin/flX- (4"13)

Patm^ patmj ^ 2c cos 16 + 2 <5"3 sin/6 j

The introduction of Eq. 4-13 into Eq. 4-12 provides the following ex­

pression for the tangent Poisson's ratio,

* Mt = (6l " 6'3)d

1 ~ i Rf (6 1 - 6 3) (! - slnjrf)

atm(r,_i_l ( 2ccos 16+ 2<5"3sin/rf

2 (4-14)

In using this expression, it is necessary to be cognizant of the

fact that pt cannot be allowed to exceed or equal 0.5 if the constitutive

equations are described by the two parameters, Young's modulus and

Poisson's ratio.

Unloading and Reloading Modulus

The stress-strain characteristic of soil is inherently nonlinear

during primary loading with rebound occurring inelastically with respect

to the primary loading curve. If the major principal stress difference,

(<5"i - <5"3), either in unloading or reloading, is less than any previously

higher stress difference, the stress-strain curve will be linearly elastic.

The inelastic or nonconservative behavior during unloading or reloading,

as opposed to primary loading, is accounted for by utilizing one relation­

ship for each type.

The modulus for both unloading and reloading will be essential­

ly equal and dependent not on the stress difference but only upon the

confining pressure, <5*3. Thus the unloading-reloading tangent modulus

can be described by one modulus value, Kur, yielding

Eur = KurPatmf ^ ^ (4-15) \Patm /

whare the exponent n is the same value as for primary loading. The

value of Kur will be greater than for primary loading.

No Tension Characteristics

A granular soil material cannot sustain a tensile loading state.

Such a state exists if the all-around confining pressure, average princi­

pal stress, is tensile. When initial stresses are present, it is not pos­

sible to monitor strains in order to determine if a tensile stress state

exists, since the displacements or strains in the initial stress state are

taken as zero.

56

For a tensile situation, the modulus is set equal to a very small

value, 10~6 times atmospheric pressure, and all stresses are zeroed.

The modulus should not be set equal to zero or a region of tensile ele­

ments could produce a node with zero stiffness. This node would be

decoupled during formulation of the stiffness matrix, resulting in zero

displacement for the node. It is important that the small value be chosen

with consideration toward the significant figures carried by the computer

in order that accuracy is not lost in the simultaneous equation solution.

Unless the increments of load are very small, one load interval

could produce a tensile state in an element from which the element would

never recover. In order to promote recovery from such a condition, the

modulus used for computing the element stresses from the nodal point

displacements is given a magnitude of approximately 10 times atmos­

pheric pressure. Also, the zeroing of tensile stresses will help in this

regard and will better simulate the actual stress state in the field.

Thus, if a displacement state produced by a load increment

yields a compressive stress state, this state will be used rather than

the stress state, perhaps tensile, which is compatible with the total

displacement of all load increments up to that point. This procedure

will affect any nodal equilibrium check that depends on element forces.

Nonlinear Interface Soil Model

The nonlinear stress-dependent interface behavior in the shear­

ing mode may be represented by an equation completely analogous to the

one developed for a soil continuum. It can be developed from the non­

linear soil strength equation by rewriting the Mohr-Coulomb failure law

as Tf = (5"n tan/rfw where /z$w is the angle of wall friction, if is the shear

stress on the failure plane, and <5"n is the normal stress on the failure

plane. The following adaptation was presented by Clough and Duncan

(1971).

The interface shear stress, T, is related to the interface rela­

tive shear displacement, As, by the following hyperbolic equation,

t = (4-16) a + bAs

where a and b are empirical coefficients. The initial shear stiffness,

ESi, is proportional to confining pressure by the relation,

^si = Kiltw (—r—) (4-17) \ Patm I

where Kj is the interface modulus number. The resulting equation for

the tangent shear stiffness, Est, is

e« - (i <4-i8)

The terms patm and Rf correspond to similar values in the previously

noted derivation and lfw is the density of water.

CHAPTER 5

FINITE ELEMENT MODEL FOR ANCHORED BULKHEADS

An anchored bulkhead system is composed of many structural

types. These are modeled using four basic finite elements: bar, beam,

TRIM3, and interface elements. The element properties are incorporated

into a finite element formulation which is implemented by a computer

program (Appendices B and C). Soil and interface behavioral models de­

veloped by others have been included in the model. The programmed

representation has been given the capabilities to consider various fea­

tures which are unique to the anchored bulkhead problem.

Soil Model

Constant strain triangles are used to model the soil material.

The nonlinear strength properties are represented by the method pro­

posed by Duncan and Chang (1970). Inclusion of variation of Poisson's

ratio with strain is handled using the procedure of Kulhawy et al. (1969).

The large extension of the system parallel to the line of sheet

piles leads to no deformation in that direction, or a plane strain state.

This situation can be modeled by using either a plane stress constitu­

tive equation formulation with plane strain parameters or by using a

plane strain constitutive equation formulation with plane stress param­

eters. The parameters are related by the following expressions:

58

59

ej

^"plane stress

r _ ^lane stress , Lplane strain ~ 2 » anc* (5-1)

* " W nlario ctrocc

•Hplane stress /p. ^ /•plane strain r~; . (5-2)

1 ^plane stress

A conventional triaxial test simulates the plane stress condi­

tions under which the parameters E and are defined. Thus, the use of

triaxial parameter test data together with an approximate conversion fac­

tor related to confining pressure that was suggested by Radhakrishnan

and Reese (1969) is used. Since confinement is higher in a plane strain

situation than for a triaxial case with the same minor principal stress,

it is reasonable to express the confining pressure in the plane strain

case as the average of the intermediate, 62, and minor, 63, principal

stresses.

One limiting type of soil is considered in this study, a sand in

a loose and medium dense state. The dilatant condition and strain-

softening that occur with a dense sand are not considered, since for

such a material constitutive relations used in the Duncan and Chang

(1970) method have no meaning.

Beam Model

The steel sheet pile is idealized as a series of interconnected

beam elements whose material behavior is approximated by a linear

stress-strain relation. The flexural rigidity of the pile wall is repre­

sented by that for a beam, EI, rather than that of a plate, EI/(1 - p?).

This representation is reasonable, since the arched pile cross section

will allow movement along the length of the sheet pile and will not offer

the constraint and thus the increased rigidity of a plate. The beam will

be a one-foot wide section of the entire wall displaying apportioned

properties.

The interlock friction between adjacent sheet piles will be

handled by considering two limiting cases. First, the friction between

them is assumed to be zero, yielding no shear transfer (Fig. 5-la).

This assumption is standard American practice and amounts to assuming

the neutral axis to exist at the centroid of each section rather than at

the centroid of the entire wall section. The second assumption is that

adequate friction is developed between the interlocks to allow full shear

transfer (Fig. 5-lb). This less conservative European practice more

closely approximates field behavior (Brewer and Fang, 1969) and will

yield a neutral axis at the centroid of the entire wall section.

The boundary displacements of adjacent INFACE and beam ele­

ments are not compatible. Nonconvergence occurs, since the displace­

ment pattern for interface and beam elements is described by first- and

third-order displacement functions. In the limit, convergence results as

the mesh size is made finer.

Tie-rod Model

The steel tie rod, which extends from the sheet pile to the

anchor wall, is modeled by using one bar element. By its connection at

only the sheet pile and anchor wall, the simulation is for a fully incased

tie rod that does not bear on the soil. Its material characteristics are

represented by a linear relation. Its size is apportioned for the one-foot

U or Arch Section Pile

NA

-n^v

Bending Stress

(a) United States Practice (no shear transfer)

U or Arch Section Pile

(b) European Practice (shear transfer)

Shear Stress

Bending Shear Stress Stress

Z section pile

Bending Shear Stress Stress

(c) No shear transfer since no shear at extreme outer edges

Fig. 5-1. Shear Transfer Condition

section of pile being analyzed. The force then applied by the tie rod to

the anchor wall and to the sheet pile is an equivalent force per unit of

their length. A three-dimensional analysis considering wall stiffness

and tie-rod spacing will not be considered.

Anchor-wall Model

The continuous rigid wall used to anchor the tie rod is simu­

lated by three triangular constant-stress plane strain TRIM3 elements.

A linear stiffness corresponding to reinforced concrete is prescribed.

Interface Model

A special joint interface element developed by Goodman et al.

(1968) is used where finite relative displacements or slip could occur.

For a bulkhead system, such a location is usually at the interface be­

tween two different materials. The presence of shear stiffness elimi­

nates the need for the assumption of a perfectly rough or smooth interface

and allows the presence of shear displacement discontinuities in what is

otherwise a continuous displacement field.

The stiffness properties of a joint interface element were de­

rived by Goodman et al. using an energy approach. Another approach

considers the one-dimensional interface element as a special case of a

two-dimensional linear strain, rectangular element. The stiffness of the

one-dimensional model can be obtained from either a plane stress or

plane strain characterization of the rectangular element stiffness by mul­

tiplying the terms in the stiffness relation by the short length and then

setting the short-length term along with Poisson's ratio equal to zero.

63

The resulting element stiffness matrix is

0 Es 0 -Es 0 -2Es 0

2En 0 En 0 "En 0 -2En

2ES 0 -2ES 0 -Es 0

2En 0 "2En 0 "En

2ES 0 Es 0

Symmetrical 2En 0

2ES

En

0

2En_

where L is the element length and t is the thickness. The stiffnesses

in the tangential and normal directions, Es and En, respectively, have

replaced the shear and Young's modulus. A nonlinear relation previously

developed in Chapter 4 is used to represent this shear stiffness. The

possible deformation modes for this element are shown in Figure 5-2. In

order to insure that there is no significant overlapping of adjacent ele­

ments, the normal stiffness, En, is set equal to a large number 10®

times atmospheric pressure.

The concept of elastic unloading and reloading is applied to the

interface shear modulus. Here, load reversal or a change between a

positive and negative shear condition can occur. Such a state exists

during the sheet-pile driving and the backfill or dredge sequence. Load­

ing into a shear state which is opposite in sign to the previous loading

state is considered to be a primary loading.

This element is used on the sides of the sheet pile, on the sides

and bottom of the continuous anchor wall, and occasionally along the

64

•6

Fig. 5-2. Interface Element Deformation Modes.—From Clough and Duncan (1971, p. 1661)

65

bottom interface of the entire finite element idealization. The friction

force, which is considered at the tip in conventional methods, is not

really an applied force, due to slip. Instead, it is a force equivalent

to the effect of the forces on the soil continuum below the pile tip eleva­

tion on the sheet pile. In the finite element formulation, this latter situ­

ation is represented.

Computer Program Capabilities

System Degrees of Freedom

The total number of degrees of freedom of the system, and thus

the total number of simultaneous equations that must be solved, is de­

pendent on the degrees of freedom that any node in the finite element

idealization possesses. The nodal degree of freedom is a function of

the type of elements that connect to that node point. Bar, TRIM3, and

interface elements all have two degrees of freedom at a node, while

beam elements have three. For the case where most nodes possess

three degrees of freedom, it is easier to analyze the whole system as

having three nodal displacement degrees of freedom at every node and

then to introduce a support in the rotational direction at all two degree

of freedom nodes.

For the sheet pile—soil system, two degree of freedom nodes

predominate, and the above procedure would be wasteful of computer

storage space. Therefore, a modification of the usual procedure is made

in which a vector of the cumulative system degrees of freedom for any

node was used in order that only those degrees of freedom present would

be considered.

66

Nodal Equilibrium Check

A check on the equation solution for displacements and on the

stress and strain computations is performed by considering nodal force

equilibrium. The nodal forces that are equivalent to the internal element

forces must balance the applied load.

When initial stresses are present, the nodal forces caused by

them are subtracted from the total equivalent nodal forces. Another pos­

sibility would have been to compute the nodal forces equivalent to the

change in stress from the initial state. Where displacements are im­

posed by reactions, the reaction force will be calculated. These reac­

tions are not the total reaction value if initial stresses have been input

rather than calculated. A no-tension provision on element stresses will

violate nodal equilibrium at those nodes belonging to tensile elements.

A nodal equilibrium check with interface elements needs more

interpretation. Since the element stress is location dependent, the dis­

tribution of equivalent nodal forces will depend on the location at which

the stress is specified. The true equilibrium nodal force distribution is

dependent on an integration of the linear displacement interpolation for­

mula over the element volume. Its distribution could be calculated di­

rectly from the incremental nodal point displacements, using a tangent

stiffness. A secant stiffness with total displacements cannot be ustSd,

because of the confining pressure dependency.

A disadvantage of a check that is dependent on nodal displace­

ments is that the stress and strain computations are not checked. Also,

this method of checking cannot be combined with a check on other ele­

ments which compute nodal loads from internal force states, since the

67

former method is not compatible with the zeroing of the initial displace­

ment state. For this reason, the interface elements were checked by

computing nodal forces based on internal forces.

If the nodal forces are calculated from the stresses at the cen-

troid of the element, they will not equal zero at any individual node of

the element. Overall equilibrium of all nodes of the element will be

maintained and can be checked by summing the nodal forces at all nodes,

which should equal zero.

Water-table Elevation

The uniform presence of water below some elevation in the sys­

tem can be easily handled by using effective buoyant weights and effec­

tive strength parameters for all soils below that level. A uniform water

level on both sides of the bulkhead produces only uniform pressures on

both sides and thus no moment by itself.

Where differing water levels exist on opposite sides of the bulk­

head, seepage pressures exist in the soil as well as a hydrostatic pres­

sure differential. This pressure differential can be replaced by

equivalent nodal forces acting on the sheet pile. The case of differing

water-level elevations has not been considered in this study.

Initial Stresses

The initial stress state of the soil mass before driving the pile,

dredging, or backfilling can be considered as being hydrostatic if the top

surface of the mass is horizontal, the material possesses a horizontal

homogeneity, and the mass is large in lateral extent. For the general

case of zones of different material properties in masses of irregular

configurations or for the previous situation with many horizontal layers

of different densities, it becomes easier to compute the stresses in the

earth mass directly from a body force loading. The stress distribution in

the mass is dependent on the distribution of soil strength, and the latter

distribution is dependent on the confining pressure. Since the way mate­

rials formed the existing masses is rarely known, except in a geologic

sense, it is impossible to simulate the construction of the mass.

Therefore, for the case where initial stresses are not known and

cannot be directly input, they will be approximated by a one increment

loading of all body forces. These forces are dependent on the element

densities. The distribution of the resulting stresses are dependent on

the material stiffnesses. The displacement state at the end of this cal­

culation is taken as the reference state or zero displacement state. Dur­

ing the application of the initial stress state, the sheet pile will not be

present.

Driving of Sheet Pile

In order to create the stress state that exists after driving of

the sheet pile, it is necessary to simulate the pile driving. Driving is

achieved by imposing a vertical displacement on the tip of the sheet

pile. This displacement cannot be imposed in one large step, due to the

nonlinear nature of the soil properties. An overshooting of the ultimate

strength of the soil surrounding the pile would result if only the initial

tangent modulus value were used. Therefore, the displacement is ap­

plied in several increments of the total value. This total value of dis­

placement is not the actual total displacement incurred by the pile while

69

being driven, but it is a vaiue which results in an ultimate or failure

strain being applied to the interface elements. A value of 1 percent of

the average interface element length is sufficient to impose a failure

strain but is not excessive to cause the continuum to deform in a manner

yielding an unrealistic stress pattern. After driving, all displacements

and stresses are zeroed.

In situations where large load reversal occurs suddenly, it is

possible to get only a small decrease in load and a strain of opposite

sense to the load direction. This occurs since a loading modulus rather

than an unloading modulus is used for an unloading condition. The prob­

lem can be eliminated by calculating the stresses before and after a load

application and reanalyzing the load increment, using the proper modulus

depending on whether the element undergoes primary loading, unloading,

or reloading (Chang and Duncan, 1970).

The condition of sudden stress reversal occurs in interface

elements during the transition from sheet-pile driving to backfilling.

Rather than analyzing such load increments twice, as suggested pre­

viously, another procedure is followed. A check on the shear strain of

interface elements is made and if the strain is of opposite sense to the

shear stress, the shear stress is given a value having the correct sense

which is compatible with the strain state. This correction is not applied

to TRIM3 elements, since the sense of the normal stresses and normal

strains can be reversed and still be compatible.

Initial Sheet-pile Displacements

Initial horizontal sheet-pile displacements and the ensuing

bending moments occur in actual field driving of the piles. These

70

displacements from a vertical configuration are input as support dis­

placements at the beam nodes. The bending moment and stresses in the

beam cannot be computed from the horizontal displacements alone. The

rotations at the nodes must be known as well. Therefore, it is necessary

to solve simultaneous equations consisting of a stiffness matrix for the

beam elements only with a loading of imposed horizontal sheet-pile dis­

placements in order to get the complete displacement pattern. Soil ele­

ments need not be considered since they supply no rotational stiffness,

their vertical stiffness is small compared to that of the beam, and their

horizontal stiffness is in the direction of imposed displacements.

The imposed horizontal sheet-pile displacement state affects

the stresses in the soil or interface elements, which cannot be zeroized.

The displacements at this point can be either zeroized or not, depend­

ing on whether the change in displacement from this initial situation is

desired.

Backfilling

A backfill sequence is simulated by activating those elements

to be added and then loading all nodes common to those elements with

the body forces of the added elements.

Dredging

To simulate dredging of soil, it is necessary to remove those

elements in the area of the mesh being dredged. Two variations of ac­

complishing dredging have been practiced in the past. One method is to

apply upward nodal forces at the nodes along the bottom of the dredged

layer which are equivalent to the body forces of the soil in the layer to

be removed. The disadvantage of this procedure is that it does not in­

sure a zero stress state at the top of the remaining layer.

To improve on this deficiency, a second method has been pro­

posed by Duncan and Dunlop (1969). By this method, upward nodal

forces equivalent to the stresses at the bottom of the dredged layer are

applied such that the net result is zero stress at the top portion of the

remaining elements (Fig. 5-3). In practice, nodal forces equivalent to

the stresses in those elements to be removed are used. Those nodal

forces for which the node is at the top of the remaining elements are

applied in an upward direction. A nodal load, equivalent to the body

force distributed to the layers below the removed layer, acts at this

interface. It causes no stress in the layer above but must be added to

the upward-applied nodal forces so as not to get distributed to the layer

below.

It should be noted that the nodal forces due to the stresses in

some of the upper elements being removed are not applied directly to

them. Their effect enters into the stresses in the layers below and into

the equivalent nodal forces applied below.

In computing the nodal forces equivalent to a stress state, it

is possible to use an array of constants, unless the stresses are loca­

tion dependent, as is true with interface elements. Since the calcula­

tion of nodal forces from displacements would require a matrix of stored

nodal forces for each interface element, an equivalent nodal force based

on the stress at the centroid of interface elements is used. A provision

has been incorporated into the program which allows the application of

the load in small increments within a dredge or backfill layer.

72

luijlllllo^

(0)

n i l L U » » P f A(T

(b)

1w

w«c cr • cr t i f f -0

(c )

Fig. 5-3. Analytic Simulation of Excavation.—From Duncan and Dunlop (1969, p. 476)

Surcharge Loads

A surcharge loading is applied by loading the nodes with a

statically equivalent loading. Since the soil properties are nonlinear,

it will not be possible to superimpose stress states and the surcharge

loading will be considered in its proper sequence with the other loads.

All loads can be applied in increments of any size.

Simulating Tie-rod Force

Rather than have an anchor wall and tie rod, it may be desir­

able to replace them by a tie-rod stiffness. This simulation could be

done if it has been determined that the anchor wall's zone of influence

does not intersect that of the sheet pile. Such a determination can be

made by comparing the results of analysis performed, using the two dif­

ferent approaches.

Another reason for only having a tie-rod stiffness is to mini­

mize the bandwidth. The bar element that extends from the sheet pile to

the anchor wall accounts for the greatest nodal point difference for most

nodal point numbering schemes. Great difference occurs because it

spans many triangular soil elements but is not connected to any of their

nodes.

Proper simulation of the tie rod—anchor system entails adding a

tie rod-anchor system stiffness equivalent to the diagonal of the stiff­

ness matrix corresponding to tie-rod direction degree of freedom. Adding

an equivalent tie-rod stiffness to the diagonal terms of the stiffness ar­

ray corresponding to the tie-rod direction degree of freedom at the sheet

pile and at the anchor wall is not proper. Such a scheme would allow

74

independent displacements at those locations and would not consider the

necessary off-diagonal coupling terms.

The specification of a tie-rod stiffness does not interfere with

tie-rod release sequences. The latter are conducted by specifying a

horizontal displacement for the sheet-pile node at the tie-rod level.

Modification for Linear Material Properties

An analysis based on linear soil properties can be performed by

setting Rf = 0 and inserting the linear value of the elastic modulus for

TRIM3 elements. Confining pressure dependency without nonlinear

strength reduction can be imposed by setting Rf equal to a very small

number and specifying the confining pressure relation by a value of n.

The modulus number on unloading, Kur, can be set equal to that for pri­

mary loading, K. The interface element properties can be made linear in

a similar manner. In both cases, the no-tension provision is excluded

in a linear analysis.

Poisson's ratio is taken to be a constant when a linear value of

the elastic modulus is employed which is or is not confining pressure de­

pendent. The constant value is input through the parameter G. Param­

eters F and d are ignored. For a constant value of }x, regardless of the

strength variation, both F and d should be zero. The value of is input

through the parameter G.

CHAPTER 6

VERIFICATION OF THE FINITE ELEMENT ANCHORED BULKHEAD MODEL

The finite element anchored bulkhead model is verified by a

comparison to the data from a full-scale bulkhead test program. This

check is necessary in order to determine whether the component parts

of the system model function reasonably well together. The intention

of the verification is not to detect numerical inaccuracies, as the nodal

equilibrium check performed by the computer program does this.

Burlington Beach Wharf

The literature contains many accounts of monitored bulkheads.

For this study it is necessary to choose a system in which cohesionless

material without clay lenses was present, for which accurate measure­

ments were obtained, and for which the anchor system was a continuous

type. The first restriction is imposed due to the difficulty of modeling

clay material, as noted in Chapter 2. The last restriction is enforced in

order that the size of the anchor system would not have to be scaled to

represent an equivalent anchor for the one-foot width of bulkhead that

is analyzed.

The full-scale bulkhead chosen is the Burlington Beach Wharf

in Hamilton Harbour, Ontario, Canada. A full documentation of the bulk %

head and the test program is given in the three reports by Matich,

75

Henderson, and Oates (1964), Henderson and Matich (1962), and Thomp­

son and Matich (1961).

The wharf is 1000 feet long and 600 feet wide. Test location 2

along this wharf has been chosen for the comparison bulkhead, since a

crushed rock dike along with a large amount of hydraulic fill sand was

not needed at this location. The design cross section is given in Fig.

2-6. The material consists of naturally dense sand and gravel having a

density of 110 pcf and an effective angle of friction (6 of 35°. A hydrau­

lic sand fill with a density of 130 pcf and a /6 of 40° is placed in front

of the anchor wall up to the sheet pile. The low water-table level in the

harbor is at elevation 244. No differential water head between each side

of the sheet pile is assumed.

The bulkhead is constructed from Larssen 4A deep-arch sheet

piling. Since the interlocks for this section are at the center of the pile

cross section, it is necessary to consider the cases of no-interlock and

full-interlock friction (Fig. 5-1). For no-interlock friction, a lineal foot

section of the sheet pile has a moment of inertia, I, of 120 in.4, a sec­

tion modulus SM of 22.3 in.3# and 46.4 in.3 for the interlock and arch

side, respectively. With full-interlock friction, I is 289 in.4 and SM is

40.9 in.3, both quantities being per lineal foot.

The 52-foot-high sheet pile is tied to a 9-foot-high continuous

concrete anchor wall by 60-foot-long, 2.5-inch diameter tie rods spaced

at 7.9 feet. The angle of wall friction at the sheet pile—soil interface is

assumed to be 2/3 jeJ. An assumption for the wall friction angle is not

needed for the computer program analysis as the wall interface shear

load versus deformation characteristics are used instead.

77

In order to accommodate instrumentation, a prefabricated hollow

full-length, box-shaped steel pile with the same section modulus as ad­

jacent piles was driven at the test location. Readings were made using a \

Wilson slope indicator in a tube inserted in the box pile. Measurements

on three tie rods adjacent to the sheet-pile test location were made us­

ing a Whittemore mechanical strain gauge.

The history of construction and instrumentation begins with the

steel sheet pile being driven with the aid of water jetting. Initial slope

indicator readings were taken at this point. Backfilling and dredging

were carried out in that order, with slope measurements being made at

the end of this stage. At test location 2, a 600-psf surcharge loading

was applied and final readings were taken.

Results ofFull-scale Test Observations

A particularly important aspect which this testing program eval­

uated is the deflected shape and induced bending moments due to driv­

ing. These results are shown in Fig. 6-1. In this figure, the driven

deflected shape is referenced with respect to the top of the sheet pile

and the post-dredging shape has been referenced to the computed elon­

gation at the tie-rod level. Driving produces horizontal deflections of

approximately 1 inch, which are enough to cause moments which are

three to four times the magnitude of those due to dredging and surcharge

application. This realization is especially noteworthy if any sort of

valid comparison is to be made to a design method which assumes a ver­

tically driven pile. At other locations, maximum horizontal driven de­

flections of up to 20 inches from the vertical were measured.

ORIVEN SHAPE meMCt :J<>

t u H i.5 j

•ENDING MOMENT INDUCED 6Y CHIVING •ftCH'lftS/'T. «»».<• o •» o»#o*

- — ̂,

-t~: iJ—! ~~i 1 I

OEFIECTION DUE TO OREOOINO 1NCHC1

UNO If

< 1 1 1 . I V«r(M«-v—IA«D

(MIM rt'Mtl •* 15* 0

X ««•»* •

eCNOlNO MOMENTS INDUCED ORE9CINO

IZ-

l^tO» 9

Mil (»••••«< »•••« t.i: MOf otliritl »r»« >« filli p TKO. fMvaiittl ««••! •>••4 (i'>» i<«»l

Fig. 6-1. Tests Results, Burlington Beach Wharf, Test Location 2.—From. Matich et al. (1964, p. 175)

VJ 00

79

The maximum bending moments induced by backfilling, dredg­

ing, and surcharge application in the field observation are given in

Table 6-1 along with comparison values calculated by various design

methods. A similar comparison for results of the tie-rod measurements

is presented by Table 6-2. The results of the full-scale field observa­

tions are within the values computed using the three design procedures.

Table 6-1. Maximum Bending Moments Induced by Dredging (Including Surcharge)

(Test location 2, Burlington Beach Wharf.)

Maximum Moments (in. -lb/lineal foot)

No-interlock Friction Full--interlock Friction

Full-scale observation9 0.270 x 106 0.625 x 106

Equivalent beam method a 0.615 0.615

Free earth support3 1.200 1.200

Rowe's method3 0.367 0.486

Finite element model 0.272 0.619

a. Comparative data from Matich et al. (1964, Table II, p. 176).

Finite Element Model Analysis

The finite element analysis is conducted using the computer

program listed in Appendix C with the following construction sequence:

1. Compute initial stress state from a one-increment loading.

2. Drive pile—three increments.

3. Impose horizontal bulkhead displacements due to driving.

80

Table 6-2. Tie-rod Stresses, Burlington Beach Wharf, Test Location 2

Full-scale observation3

Three readings after dredg ing and surcharge appli­cation

Average

Equivalent beam method3

Free-earth support3

Rowe's method3

Finite element model

after backfilling

after dredging layer 1

after dredging layer 2

after dredging layer 3

after dredging layer 4

after surcharge

Tie-rod Stress, psi

1= 120 in.^ 1 = 289 in .4

6 ,230 12 ,190

9 .860

9 ,430

10 ,200

14 ,100

12 ,700

3 ,560 3 ,480

5 ,300 5 ,450

6 ,350 6 ,690

7 ,020 8 ,230

7 ,630 8 ,930

16 ,640 20 ,140

a. Comparative data from Matich et al. (1964, Table 1, p. 171).

81

4. Backfill behind pile—1 layer of 1 increment each.

5. Dredge in front of pile—4 layers of 1 increment each.

6. Apply surcharge load in 2 increments.

The parameters used to describe the material behavior in the

analysis are presented in Table 6-3. Variations in the values of K and Kur

have been made in order to investigate their influence on factors such as

moment and deflection.

Finite Element Mesh Idealization of the Continuum

The discretization of the continuum has been performed from the

viewpoint of ensuring that the mesh is fine enough to represent the

changes in stress along the sheet-pile length and that it is of sufficient

extent to model the infinite expanse of the continuum. The imposed

boundary conditions represent those present in a system of infinite ex­

tent using rollers on the vertical side boundaries and pin connections

with interface elements along the horizontal base boundary. The impor­

tance of pinning the lower boundary rather than placing it on rollers has

been shown by Morgenstern and Eisenstein (1970).

The adequacy of the mesh idealization has been examined by

two methods. First, reference has been made to the experience gained

by other investigators of similar problems. Mesh patterns used by Dun­

can and Dunlop (1969) for cut slopes and Morgenstern and Eisenstein

(1970) for retaining walls assumed the boundaries shown in Fig. 6-2.

Starting with this idealization, representations of more limited

extent were tried. Changes in the sheet-pile moment and deflection pat­

terns with changes in boundary extent were observed. Similarly, the

Table 6-3. Material Properties for Finite Element Model of Burlington Beach Wharf

(Where two values are given, top value is for case of 1= 120 in.4, the bottom value for 1 = 289 in.4.)

,h ^ c w <0 m T: • • T? T* ® 2 s ® ^ . S - 5 3 ^ S £ .2 £ - * - S - 3

WH W HH < CO "O. O OS S»S UJ G CM O "O

1 Bar 29,000 0.62

2 Beam 29,000 120 289

12.1 22.3 40.9

3 TRIM3 0.068 40 0 0.75 190 120

228 144

0.5 0.18 0.42

4 INFACE .068 27 .90 60,000 70,000 1.0 .42

5 TRIM3 3,900 .150 .00 .30

6 INFACE .068 35 .90 70,000 81,000 1.0 .42

7 INFACE .068 35 .75 8,000 4,400

9,600 5,300

.5 .42

8 TRIM3 .111 35 0 .80 285 180

342 216

.5 .23 .42

a. Material location: 3, generally, where dense sand and gravel (buoyant); 4, steel sheet pile-soil interface; 5, concrete anchor wall; 6, concrete anchor wall-soil interface; 7, horizontal base boundary interface; 8, dense sand fill in front of anchor wall.

83

gradation of the grid size was varied from coarse to fine by halving the

mesh size. Changes in the area of concern, that region around the sheet

pile, were noted.

The resulting mesh pattern and its dimensions are shown in

Fig. 6-3. The nodal point coordinate data for the mesh pattern is chosen

so as to model the Burlington Wharf bulkhead by providing an anchor-wall

and backfill layer height of 0.17H and four dredge layers starting from

0.17H which are located at 0.34H, 0.45H, 0.57H, and 0.69H. The con­

figuration departs from that used by other researchers (Fig. 6-2) for em­

bankment analysis. The side on which dredging occurs, the outside of

the pile, extends out from the sheet pile a distance of 4H rather than

the 6H shown in Fig. 6-2. This shorter distance for the sheet-pile sys­

tem is reasonable since the anchor on the sheet pile transfers much of

the load to the region behind the sheet pile. Due to this transfer, the

side on which backfilling occurs, the inside, extends a distance 4H be­

hind the sheet pile rather than the distance 3H of Fig. 6-2.

• 2H

1

1 •

2H

1 H

-n— 3H

«tj -

t t

Fig. 6-2. Finite Representation of Infinite Body

00 CO

LO to

CNJ

u

^[4- pinned 4H = 208 ft 4H = 208 ft

Fig. 6-3. Extent of Finite Element Grid for Burlington Wharf Bulkhead

oo >u

Data Preparation

Once the grid has been drawn, the data preparation consists of

numbering the node points and elements. Other necessary input data are

specified in the computer program documentation given in Appendix B of

this study. The coordinate points of each node point are specified, and

the node points for each element along with the element type and its

material type are prescribed. At this stage, the maximum node point

difference existing between the nodes of any element is not necessarily

a minimum. The minimization of this quantity is essential if the band­

width of the simultaneous equations is to be as small as possible.

In order to perform this bandwidth minimization in as efficient

a manner as possible, an iterative scheme similar to that proposed by

Grooms (1972) is used. The scheme has been modified to eliminate the

need for storing two large connectivity arrays. This savings is accom­

plished from calculating the maximum node separation directly by com­

paring the original or revised nodal point difference for each element.

A new data deck of node point coordinates and element node points is

punched out.

Comparison of Behavior of Model to Burlington Wharf Bulkhead

The comparison of the finite element model and the full-scale

monitored structure is made on the basis of deflections, moments, and

tie-rod forces, as these were the quantities measured in the field study.

The most accurate quantity for comparative purposes is the moment. The

deflections involve an integration of the slope readings and must be ref­

erenced to some known deflection.

Moments

The most influential of the soil stiffness parameters is the soil

stiffness modulus number. As it is varied, sheet-pile moments and de­

flections change due to the change in the soil strength. In this study,

the modulus number for the predominant sand and gravel material is

varied with the intention of matching the maximum moments and differen­

tial deflections between the maximum deflection near the center of the

sheet pile and the deflection at the bottom of the sheet pile.

The variation in the maximum moment, which always occurs be­

tween the anchor and dredge level, with modulus number is given in

Fig. 6-4 for a sheet pile with I = 289 in.4, or assuming full-interlock

friction. This figure shows that the soil can be idealized as being stiff

for all modulus numbers greater than approximately 500. Throughout the

range of the soil stiffnesses, the moment patterns exhibited were typical

1000

"max k-in. I = 289 in

500

1000 800 600 200 400

Fig. 6-4. Maximum Moment in Sheet Pile Versus Soil Modulus Number, Burlington Wharf Models

of a relatively flexible pile with two points of inflection occurring in the

region located a few feet above the dredge level to the bottom tip of the

pile. The flexible behavior is ensured as the log of the flexibility num-

ber,p, is equal to -3.06 and -2.68 for I = 289 in.4 and I = 120 in.4,

respectively, for the full-interlock and no-interlock friction conditions.

The pattern of the bending moment distribution over the length

of the pile is similar for the model (Fig. 6-5) and the actual structure

(Fig. 6-1). Modulus numbers of 120 and 190 were required for the full-

interlock friction and no-interlock friction cases, respectively, in order

to match moment magnitudes. An evaluation of these moduli values in

terms of the density and f6 angles for the material provides an assessment

of the finite element model and the degree of interlock friction present.

Typical stress-strain parameters for dense sand and gravels

are given by Kulhawy et al. (1969). They suggest an average modulus

value of 300 for a well-graded sand. A range on this value based on all

the data presented is on the order of+100. Therefore, of the two moduli

values used, 120 and 190, the latter value appears the more reasonable,

as it is in the range for the dense sand and gravel material.

In addition, it is pointed out that, since much of the area on

both sides of the sheet pile is in a state of unloading, the unloading

modulus number is used, which is 20 percent higher than the unloading

modulus number referred to as the modulus number in the previous dis­

cussions. Furthermore, the TRIM3 soil elements used allow linear dis­

placements between the nodes and are thus slightly stiffer than if the

true displacement pattern were allowed. This results in a more flexible

I = 120 in K = 190 ,

I = 289 in. K = 120 /

120 in 190 K = K = 120

mmntmmrsBnmto&rizsz vnstaimmmmk

500 8 in M k-in

Fig. 6-5. Moment and Deflection Distribution, Burlington Wharf Models

00 00

89

pile by comparison, requiring a lower modulus number in order to achieve

the desired moment compatibility.

Deflections

A plot of the differential deflection, 6, between the point of

maximum deflection near the center of the sheet pile and the deflection

at the bottom of the sheet pile versus the modulus number is given in

Fig. 6-6 for a sheet pile of I = 289 in.4. At a modulus number of 110,

this deflection corresponds to the 1.5-inch deflection of the actual bulk­

head (Fig. 6-1). For the other case, a sheet pile without interlock

friction, a soil modulus number of 190 produced the field differential

deflection of 1.5 inches (Fig. 6-5). For both cases of interlock friction,

the modulus number that causes the moments to be equal to the measured

field values results in the maximum differential deflection equaling the

field value.

S in.

I = 289 in

200 400 600 800 1000

Fig. 6-6. Maximum Differential Sheet-pile Deflection Versus Soil Modulus Number, Burlington Wharf Models

90

Thus the validity of the finite element model is ensured in that

in using soil property values that reasonably describe the materials pres­

ent in the field, it yields results comparable to field measurements. An­

other conclusion supported by the modulus numbers obtained is that the

interlock friction in the Burlington Wharf bulkhead is best represented

by a no-interlock friction assumption.

A comparison of translational sheet-pile displacements for the

model and the field bulkhead reveals, for each, the relative stiffness of

the anchor system compared to the constraining soil. The anchor wall

location is theoretically far enough away from the sheet pile that no

transfer of load to the wall should occur. The deflections for the finite

element models of different stiffness are presented in Figs. 6-7 and 6-8.

The values at the tie rod include the anchor deflections as well as the

tie-rod elongation. The field measurements are referenced to the tie-rod

elongation. The tie-rod level displacements, after surcharge application

but excluding initial displacements due to driving, in the finite element

model analysis are 2.6 and 3.9 in. for the no-interlock and full-interlock

bulkheads, respectively. These values are significantly larger than the

0.4-in. elastic deformation of the tie rod.

The displacement patterns for both the no-interlock and full-

interlock friction models (Figs. 6-7 and 6-8) show that during all dredg­

ing stages the extreme bottom of the sheet pile displaced more relative

to the extreme top. During the surcharge stage, the opposite was true.

The backfill operation produced a uniform displacement at the top and

bottom. The net result of all operations is an approximately equal rela­

tive displacement of the extreme top and bottom of the sheet pile. In

Displacement, in.

Fig. 6-7. Sheet-pile Displacement, 1= 120 in.4, K= 190, Burlington Wharf Model

Backfill top 17% of H; dredge in four layers tod = 0.7H; apply surcharge.

<o

Displacement, in.

Fig. 6-8. Sheet-pile Displacement, 1 = 289 in.4

Backfill top 17% of H; dredge in four layers to cl

K = 120, Burlington Wharf Model

0.7H; apply surcharge.

to [S3

93

Fig. 6-5, the model deflections are referenced to the dredge level dis­

placement, as this value was zero for the monitored bulkhead. The de­

flected shapes of the measured and model bulkheads are similar. The

monitored field bulkhead produced deflections after surcharge which were

0.3 in. greater at the extreme top than at the bottom of the sheet pile

(Fig. 6-1). The relation of the magnitude and pattern of deflections to

the soil stiffness and sheet-pile flexibility will be fully developed in

Chapter 7.

Tie-rod Force

A comparison of tie-rod stresses is provided in Table 6-2. The

field tie-rod stress of 9.43 ksi after surcharge applications is derived

from the average of three readings, 6.23, 12.19, and 9.86 ksi. The

model stresses of 16.6 and 20.1 ksi are considerably higher. The

model tie-rod stresses of 7.63 and 8.93 ksi after dredging indicate that

much less of the 600-psf surcharge load was transferred to the tie rod

in the field than in the model. This result can be explained by the

somewhat larger outward rotation of the field sheet pile than of the

model. Thus, less lateral stress is exerted on the sheet pile, causing

a stress relaxation in the field tie rod. This decrease in the tie-rod

force suggests that the anchor system for the model is more rigid than

for the field bulkhead system. No discussion of the relation of tie-rod

force to soil and sheet-pile stiffness will be done here as a complete

parametric relation will be developed in the next chapter.

Model Results during Construction

Some results that cannot be compared due to the lack of mea­

sured data include the moments and deflections after each stage of con­

struction, together with the pressure distributions. The value of these

quantities for the Burlington Wharf bulkhead model having I = 120 in.4

will be discussed. A more thorough understanding of the deflections is

obtained by a simultaneous consideration of the pressure distribution

along the pile. The pressure distributions during construction for the

case of I = 120 in.4 are presented in Fig. 6-9. This distribution is in­

fluenced greatly by the stresses incurred from the horizontal sheet-pile

displacements that arise due to driving the sheet pile.

The pressure distribution curves show a general decrease in the

active pressure distribution and a localized increased in the passive

pressure just below the ground surface as the dredging proceeds. During

the latter dredging stages, the passive pressure increases above the

maximum Coulomb value using = 2/3*5. This trend was noted by both

Rowe (1952) and Tschebotarioff (1949). The shift in the pressure distri­

bution has been related to pile flexibility by Rowe, as noted previously

in Fig. 2-5. By this reasoning, the passive pressure distribution is ex­

plained by reference to the rotational movement of the sheet pile at the

dredge level due to the bending deformation occurring between the tie-

rod and dredge levels. This movement, together with the rotational

movement of the entire sheet pile, causes the large passive pressure

near the dredge level with decreasing pressure below.

The average translation of the bulkhead is equivalent to the

tie-rod level movement. The movement during backfilling, the four

2.21 kips

TZZnTTTTTTTTTTTm 0 . 0 ft 77777777777777777777 <07777777777777777777775

Initial Conditions Due to Driving B"ackfill

20 10 20

Pressure, psi

Fig. 6-9. Sheet-pile Pressure Distribution during Construction, Burlington Wharf Model, I = 120 in.4, K = 190

3.29 kips 3.94 kips

Dredge Layer 1 Dredge Layer 2

8.6 ft TT7TT7THTTT77T

14.3 ft

30 20 10 0 10 20 30 40 30 20 10 0 10 20 30 40

Pressure, psi

Fig. 6-9. Sheet-pile Pressure Distribution—Continued

to o>

4.36 kips- 4.73 kips

Dredge Layer 3 Dredge Layer 4

TTTTTTVTTTrnTm 20.07 ft

27.0 ft i i i i f i m i i m i i i n m n t t t

Pressure, psi

Fig. 6-9. Sheet-pile Pressure Distribution—Continued

(O •vj

10.32 kips

Surcharge

27.0 ft rrrr

Pressure, psi

Fig. 6-9. Sheet-pile Pressure Distribution—Continued

99

dredge stages, and surcharge application is given in Fig. 6-7. Total

movement is about 3H/800 and differential movement between the top

and bottom during each stage is approximately H/2000. An active pres­

sure distribution (Fig. 6-9) is encountered only in the region of maxi­

mum deflection of the sheet pile.

The moments during the construction stages for the sheet pile

having no-interlock friction are presented in Fig. 6-10. The largest in­

crease in bending moment occurs during the first dredging layer that

lowers the dredge level 33 percent of the distance to its final level.

The next largest increase occurs during surcharge application. The flex­

ible nature of a sheet pile having I = 120 in.4 is indicated by the con-

traflexure and equal positive and negative bending moment values below

the dredge level as well as by the magnitude of the moment.

Anchorage Stiffness

The stiffness of the anchor system connected to the sheet pile

comprises two components, the tie-rod stiffness and the stiffness of the

soil support on the comparatively rigid anchor wall. As these stiffnesses

are in series, their combined effect can be calculated as

KA = KTRKAWS/ (ktr+ KAWS)

where the subscript TR denotes tie rod and the subscript AWS denotes

anchor-wall soil. The tie-rod stiffness is constant, being numerically

equal to AE/L, where A is the area of the tie rod and L is its length.

For Burlington Wharf, the tie-rod stiffness is 300 k/ft.

The stiffness of the anchor-wall soil support is nonlinear due

to the nonlinear nature of the soil. This behavior is shown in the

anchor force-displacement plot of Fig. 6-11. The secant stiffness of

Backfill Dredge Layer 1 Dredge Layer 2 Dimunnin

300 200 100 100

Moment, k-in.

100 200 100 100 300 200 100

Fig. 6-10. Sheet-pile Moment Distribution during Construction, Burlington Wharf Model, _I = 120 in.4, K= 190

1 1 *4^ 1 1 1 ~±md 1 1 1 JU \ 1 1

Dredge ^ Dredge Q Surcharge Layer 3 \ Layer 4

\ > ., ,<

1 1 1 N i i i 1 ^ i l 300 200 100 0 100 300 200 100 0 100 300 200 100 0 100 200

Moment, k-in.

Fig. 6-10. Sheet-pile Moment Distribution—Continued

12

I = 120 in.4

K = 190 JSTR = 300 kips/ft

Surcharge

Backfill

. 1 0 . 2 0 Displacement, ft

Fig. 6-11. Tie-rod Force Versus Anchor-wall Displacement, Burlington Wharf Model

Surcharae

Backfill

I = 120 in.4

K = 190 Ktr = 300 kips/ft

Fig. 6-12. Secant Anchor-wall Soil Stiffness Versus Tie-rod Force, Burlington Wharf Model

103

The soil support system is dependent on the ratio of total load to total

anchor-wall deformation. A plot of the secant stiffness versus tie-rod

force, T, is presented in Fig. 6-12. Since the average stiffness of the

anchor-wall soil support of 25 k/ft is much less than the tie-rod stiff­

ness, the series combination of the stiffnesses, 23 k/ft, is close to the

value for the soil support stiffness.

Secant stiffness can be related to the coefficient of subgrade

reaction, k, as given by the equation presented by Terzaghi (1955):

k =/z/D, where ji equals the constant of subgrade reaction for z/D= 1,

z is depth below the ground surface, and D is depth of the bottom of the

wall below ground surface.

Due to the rotation of the anchor wall, only the top 4 feet of it

is in compression on the sheet-pile side. An average value for k will

be at z/d = 1/2. This unit stiffness of the wall per foot of wall for a

contact area of 4 square feet is 25 k, or 6.25 kef. An £ value of 12.5

kef or 6.25 tcf results. This value is reasonable for a sand of medium

density, since Terzaghi (1955) gave an average £ value of 8 tcf for a

medium dense sand. This magnitude suggests that the anchor system is

somewhat more flexible than anticipated for a dense backfill.

The behavior of the soil support around the anchor changes

during the surcharge application stage. The anchor-wall soil stiffness

increases to approximately 50 k/ft, yielding a constant of subgrade

reaction of 12.5 tcf. A large increase in stiffness occurs during the

surcharge application, since the soil behaves as in a confined com­

pression test. A slightly larger value for,/ would be obtained if J were

104

assumed to vary as Vz. The appropriateness of this relation is con­

sidered in the next chapter.

In conclusion, it can be stated that the finite element model

provides results that compare to field test results within the accuracy

of the measuring instruments. Also, insight into the behavior of the real

structure is provided.

CHAPTER 7

INVESTIGATION OF BULKHEAD SYSTEM PARAMETERS

A study of bulkhead systems should involve the determination of

the parameters that influence the bulkhead performance. Previously, the

sheet-pile flexibility, p, which incorporates H, E, and I of the sheet

pile, has been taken as the major factor along with the system configu­

ration in determining bulkhead behavior. This study ascertains the effect

of the soil stiffness on the sheet-pile performance.

Parameters Representing System Behavior

A dimensional analysis of the parameters involved in a two-

dimensional statically loaded structure should include the applied sur­

charge load, q; soil modulus, m; depth to any point on the structure,

z; height of the structure, H; displacement of the structure, y; soil

density, V; and sheet-pile bending stiffness, EI. The resulting nondi-

mensional expression is

_Z_ = f/iilnL J 3_\ . (7-1) H \ EI ' H TH)

The term H^m/EI represents the stiffness of the structural system includ­

ing the influence of the soil support stiffness on that quantity.

A similar term results from solving the differential equation for

the deflection curve of a beam supported on an elastic foundation.

Hetenyi (1946) classified the stiffness of a beam interacting with soil

by the dimensionless quantity, H(k/4EI)*/4, where k represents the

105

106

elasticity of the surrounding medium and is denoted as the coefficient of

subgrade reaction. The coefficient of subgrade reaction is influenced by

the size of the system, as will be seen later. Curvature and thus mo­

ments are dependent on this stiffness quantity.

The characterization of the stiffness of a soil support system

has been achieved by quantifying the stiffness of soil on an elemental

level and then using scale factors to transform the representation of the

system to a large-scale level. One such characterization has been done

by Terzaghi (1955) in his work on coefficients of subgrade reaction. For

a sheet pile embedded a depth D, an expression for the coefficient of

horizontal subgrade reaction, k, is

where I is a constant of subgrade reaction and z is the distance below

the ground surface. The above equation assumes that the subgrade reac­

tion increases in simple proportion to depth. Rowe's (1955a) equation,

presented previously, related the pressure p to the deflection y, as

follows:

where n is a constant. Model scale is represented by D, and z repre­

sents a linear increase in shear strength with confining pressure. The

soil stiffness modulus, analogous to /, is represented by m. This value

is known to increase with large increases in z and y. The quantity y/D

includes the effect of pressure bulb size and length of slip path on the

scale factor. The exponent n, representing the change in pressure with

(7-2)

p = mz (-X-j " (7-3)

107

scale size, if taken as unity, yields an expression equivalent to Ter-

zaghi's given previously.

Both Rowe and Terzaghi have used the same two parameters to

relate pressure to deflection. These are soil stiffness and distance re­

lated to height of the structure. Both investigators use values of It that

are at the depth H and take z/H amounts of /at lesser depths. This is

the same as considering a //H value increasing with depth z. In es­

sence, the I value is constant and it is only divided by H since H is

the scale factor for the deflection y.

The nonlinear nature of the soil produces a decrease with de­

flection, which is not included in their theories. Also, the soil stiffness

modulus is known to increase in general as a function of the square root

of the confining pressure or depth below the ground surface.

Soil Stiffness Similitude Expression

The work of the previous researchers that characterized the soil

stiffness in terms of the coefficient of subgrade reaction and a term in­

volving the scale of the system will be extended to yield expressions for

the soil support stiffness in terms of K and H. The relation between the

soil modulus number, K, and the constant of subgrade reaction,^/, must

be determined. K times a constant and a factor related to the depth ex­

presses the stress-strain properties of a soil element. The constant,^,

is independent of the dimensions of the system and is one component

term in the expression for the stiffness relation between pressure and

displacement in a soil continuum. Therefore, / is an intrinsic property

of a soil element and is dependent only on soil density just as K is.

108

The Young's modulus, E, for soil varies as the 0.4 to 1.0

power of the confining pressure, with an average value being 0.5 (Lambe

and Whitman, 1969; Kulhawy et al., 1969). Since confining pressure is

directly related to depth of overburden, E varies as the 0.5 power of

depth. In the expression used for the tangent soil modulus, E^, it is

directly dependent on K. The constant,, may be directly represented

by K, as discussed previously. Furthermore, m, the soil stiffness modu­

lus, is analagous to /. Thus Eq. 7-3 may be represented by

p = J? \/z y, (7-4) D

if a square-root increase in strength with confining pressure is incor­

porated and n is equal to unity.

Substituting K for ./and considering the quantities z and D,

which are dependent on the scale, to be represented by height H yields

the following soil support stiffness, denoted as s, for the system rather

than element level,

s = . (7-5) V/ H

Its value decreases as H increases, causing longer piles having the

same^o value to behave in a relatively more rigid manner.

System Stiffness Similitude Expression

The behavior of a structural system can be characterized by a

quantity defining its stiffness or flexibility. This quantity could be com­

posed of the stiffnesses of the component parts of the system, the struc­

ture and the support. For a sheet-pile system, both the soil (support)

stiffness, s = K/v/h" and the sheet-pile (structural) flexibility,

109

p = H4/EI, form the system stiffness, S, which determines the sheet-

pile behavior.

A form for the system stiffness can be obtained from Hetenyi's

(1946) work with beams on elastic foundations. In his study, the elas­

ticity of the soil, as represented by k, was multiplied by the flexibility

of the sheet pile, H^/4EI, to yield a term representative of the flexibility

of the system, kH^/4EI. This relation arises due to the inverse nature of

the influence of the structural and soil stiffnesses on the system. Large

soil stiffnesses promote a flexible behavior of the structure.

Thus, the magnitude of the sheet-pile moment is indicated by

the ratio of the structural stiffness to the soil stiffness. The structural

stiffness is the inverse of the sheet-pile flexibility, p. In symbols,

Moc t oc (7-6) s

or inverting and writing as a functional relation

Y = f(s-yo) (7-7)

r = f(S). (7-9)

In these relations, f represents some functional relation of the given

parameters that is not specified but which will be represented graphical­

ly.

It should be noted that it is not the magnitude of either the

sheet-pile flexibility or the soil stiffness alone that influences the mo­

ment value, but the relative value of one to the other. Sheet-pile

110

systems that have the same S value should have identical slopes along

the pile regardless of scale and thus should display equal M/H^ ratios.

Displacement Similitude Expression

The relation between displacements for sheet piles of different

scales can be generated from the pressure-displacement relationship

given by Eq. 7-4

p = ~ ^ - y . ( 7 - 4 )

The pressure can be written as an earth pressure coefficient, Ks, times

the vertical stress at a depth z, yielding

Ks/z = ^ ViL y . (7-10)

Since Ks, y, and J. are independent of scale, the following expression is

obtained for z = D = H.

K -T c = constant = Ks (7-11) H1 • o H

where K has been substituted for l/y. For soils having the same soil

support stiffness, s = K/\/H , the deflection similitude relation reduces

to y/H.

Finite Element Model Study of Interaction Concepts

An investigation of sheet piles constructed in accordance with

Rowe's (1952) pressure test procedure is performed in order to ascertain

the influence of the structural stiffness, p, and soil support stiffness,

s = K/\/H , on the sheet-pile behavior as well as the evaluation of the

expression for s. The sheet-pile behavior is quantified in terms of the

I l l

maximum bending moment value and the displacement. Unlike Rowe's

pressure tests, having heights of approximately 3 to 4 feet and loose

and dense soils under low confining pressures, sheet piles ranging in

height from 3.5 to 60 feet are used with soils havingKvalues from 10 to

800. Other constant input parameters used are presented in Table 7-1.

The pressure test simulation consists of a backfill sequence

followed by four dredge increments to a depth of 0.7, all with the

tie-rod level position held fixed. At the end of all dredging stages the

tie rod is released. In order to study the influence of the soil support

stiffness, the effect of tie-rod yield has been eliminate by using

values for bending moment obtained before the tie-rod release stage.

The results of the computer analyses for the case of no tie-rod yield are

presented in Table 7-2. The finite element grid used for these analyses

is the same as in Fig. 6-2 except that the tie rod is located at the top of

the sheet pile, the backfill layer and anchor-wall height is 0.25H, and

the dredge levels are at 0.4H, 0.5H, 0.6H, and 0.7H. The difference

in maximum moment between using three and four layers is 6.5 percent.

This change is small enough that little would be gained in using five

layers to dredge between the same levels.

Structural Stiffness

The specific case considered here assumes^ equal to zero.

This configuration eliminates any influence of flexure above the tie-rod

level. The first relation plotted, Fig. 7-1, shows the effect of the sheet

pile or structural flexibility, p, on the moment in the pile. The moment

has been divided by in order that two sheet piles having the same

Table 7-1. Material Properties for Parameter Study, Finite Element Model

Mat

eria

l T

yp

e a

Ele

men

t T

yp

e u

w"

c •H *

CO C

<

00 c •M

£ CO

4-1 o M

£ M-H

ca u. 3

G U, O T3

1 Bar 29,000

2 Beam 29,000 V V V

3 TRIM3 0.090 30 0.8 50 V o

cn

o

• to

0.40 4.5

4 INFACE .090 20 .9 60,000 70,000 1.0 .40

5 TRIM3 3,900 .150 .0 • CO

o

6 INFACE .090 30 .9 70,000 81,000 1.0 .40

7 INFACE .090 30 .8 20,000 24,000 .5 .40

a. Material location: 3, generally where soil exists; 4, steel sheet pile-soil interface; 5, concrete anchor wall, 6, concrete anchor-wall-soil interface; 7, horizontal base-boundary inter­face.

Table 7-2. Finite Element Analysis Results, No Tie-rod Release

(Based on input parameters given in Table 7-1.)

H I log p K log K/VH log S M, k-in. T/ in. -Ib/ft3 ytip* ft

3.5 0.0108 -3.32 10 0.73 -2.59 0.817 19.1 0.0897 45 1.38 -1.94 .405 9.45 .0199

300 2.21 -1.11 .155 3.62 .0031

20 11.53 -3.32 50 1.05 -2.27 86.6 10.82 .1859 300 1.83 -1.49 40.6 5.06 .0315 800 2.25 -1.07 23.9 2.98 .0118

120 -4.34 50 1.05 -3.29 267 33.4 .1892 300 1.83 -2.51 126 15.75 .0311

30 287 -4.01 50 0.96 -3.05 719 26.6 .3250 150 1.44 -2.57 428 15.85 .1066 300 1.74 -2.27 303 11.22 .0532 460 1.86 -2.15 271 10.04 .03 97

40 184 -3.32 50 0.90 -2.42 783 12.23 .4623 76 1.08 -2.24 617 9.64 .3077 79 1.10 -2.22 602 9.41 .2881 79 a 1.10a - 2 . 2 2 a 624 a 9.75 a .24453 79^ 1.10 b -2.22° 765 b 11.95 b .3338b

81 1.11 -2.21 597 9.33 .2811 100 1.20 -2.12 540 8.44 .2271 140 1.35 -1.97 455 7.11 .1611 300 1.68 -1.64 311 4.86 .0758 800 2.10 -1.22 206 3.22 .0288

60 14.64 -1.52 300 1.59 +0.07 81 0.38 .1262 800 2.01 + 0.49 29 0.12 .0480

a. For p = 0.3. b. For K = 135 pcf.

I I L -4 -3

log p

Fig. 7-1. T Versus Log p, Finite Element Model

115

structural flexibility but different heights can be represented as having

the same behavior. Equal M/H^ for the sheet piles insures slope cor­

respondence between sheet piles of different lengths. The T versus

log p curve shown in Fig. 7-1 is similar in form to those obtained by

Rowe(1952).

For some very rigid sheet piles, the rvalues obtained are

greater than the Coulomb free earth support values. This difference oc­

curs due to the restricted movement at the top of the pile which allows

a pressure distribution considerably different from the Coulomb distri­

bution to exist.

Soil Stiffness

The plot of r versus log s is presented in Fig. 7-2. For an in­

creasing stiffness of the soil support, the soil receives more of the load

and the sheet pile less, resulting in a decreased moment in the sheet

pile. For the same soil stiffness, a more rigid pile receives more

moment.

There is a variation within the data having the same log p

value. This shows up in Fig. 7-3, where the data for log p = -3.32 are

expanded on the vertical scale. The T value for 40.0-foot sheet piles is

30 percent less than for 3.5-foot sheet piles at the same soil stiffness.

In order to make these curves identical, a soil support stiffness of

K/Hn where n is less than 0.5 should be used.

The difference in the T versus log s relation for sheet piles hav­

ing the same structural stiffness but different heights can be eliminated

by choosing the moment similitude value as M/Hn where n is slightly

log/3 = -4.34

30 logp = -4.00

20

logp = -3.32

10

logp = -1.52

0 1 3 2 log s = JL_

v/H

Fig. 7-2. T Versus Log s, Finite Element Model

T = M H3

H = 3.5 ft.

H = 20 ft

H = 40 ft

log s = log -J=r-M

Fig. 7-3. T Versus Log s, Finite Element Model, Log p = -3.32

118

less than 3. The exponent 3 holds for beams with linearly increasing

distributed loads. As can be seen from the pressure distributions (Figs.

7-4 and 7-5), a linearly increasing distribution is only approximated,

thus indicating a value of n of other than 3.

The correct expression forT and s can be obtained by modify­

ing the exponents on H in the expressions such that each relation is the

same for two sheet piles of different heights that display identical mo­

ment patterns. Such a procedure is followed for two sheet piles having

heights of 3.5 and 40 feet. The bending moment patterns are shown in

Figs. 7-4 and 7-5, respectively.

The two sheet piles of Figs. 7-4 and 7-5 have equal maximum T

values but different moment patterns. The soil support stiffness of the

shorter pile is greater than for the longer pile producing a more flexible

pattern in the shorter pile below the dredge level even though the maxi­

mum T values near the top of the pile are identical. Increasing the mod­

ulus number of the 40-foot pile to 140 produces a relatively more flexible

behavior in the pile (Fig. 7-6) and a moment pattern more similar to that

of the shorter pile. The log s value is then 1.35 for the long pile as

compared to 1,38 for the 3.5-foot pile. Equating K/Hn for each yields a

value for n of 0.46. Working backward from the expression K/H^«46

the expression/\/z/D yields,

K = KH°-5" = KHO-5 ^ -A/I~ , (7_12)

jj0.46 jj fj0.96 i)0.96

indicating that the soil stiffness modulus increases as the 0.54 power of

depth or that the exponent on the scale factor D should be less than unity

if a square-root increase in soil stiffness with depth has been specified.

In this study the latter concept holds.

1X9

fa-yj tie-rod release of no releas. tie-rod releise of H/500,

no release

log p = -3.32 H = 3.5 ft

log s = 1,38

0.500 0 1 . 0 0 1.0 Moment, k-in. Pressure, psi

Fig. 7-4. Sheet-pile Moment and Soil Pressures, Finite Element Model, H = 3.5 ft, K = 45

no tie-rod release

tie-rod release no tie-rod^* release/v'tie-rod _ release of

/X H /500

log p = -3.32 H = 40 ft K = 79

log s = 1.10 p = 0.40

500 500 20 20 Moment, k-in. Pressure, psi

Fig. 7-5. Sheet-pile Moment and Soil Pressures, Finite ^Element Model,H = 40 ft, K = 7S

H = 40 ft K = 140

log s = 1.35 log p = - 3.32

no tie-rod release

500 500 Moment, k-in. Pressure, psi

Fig. 7-6. Sheet-pile Moment and Soil Pressures, Finite Element Model, H = 40 ft, K = 140

122

Equation of the M/Hn expressions for moments of 0.405 and

455 k-in. for the 3.5- and 40-foot piles of Figs. 7-4 and 7-6 results in

a T expression of M/H2.88. This is close to the relation M/H^ for a

uniformly increasing distributed load. The corrections for the expres­

sions for r and s will produce the correct shifts in the curves of Fig.

7-3 to make them form one curve.

System Stiffness

The validity of the characterization of a combined stiffness for

the sheet-pile support system as the one variable that influences the

moment in the pile is investigated, using finite element model studies.

A plot of T versus logS is presented in Fig. 7-7. This plot is obtained,

using data (Table 7-2) spanning the complete range of heights, 3.5 to

60 feet, structural stiffnesses, logp = -4.0 to logp= -1.5, and soil

stiffnesses as represented by K = 10 to K = 800. A consistent charac­

terization of the sheet-pile response in terms of the one variable

o - H4 JK_ b VH

is evident throughout this entire range.

The expressions Tof M/H^ and for s of K/VH are used through­

out this study even though the more accurate expressions of M/h2«88

and K/h0'46 will produce a t versus log S relationship with less devia­

tion of data points than given in Fig. 7-7.

General demarcations between rigid, intermediate, and flexible

sheet-pile behavior can be made fror?. the T versus log S plot in Fig. 7-7.

For values of log S less than -2.5, the pile will behave in a rigid man­

ner, and for values of log S greater than -1.0, in a flexible manner.

M "H3"

-2 -1

log S = log ps

7-7. T —Log S Relation, Finite Element Model

124

Intermediate values yield intermediate behavior. This relation can be

written as

log S < -2.5 Rigid

-2.5 — log S — -1.0 Intermediate

log S > -1.0 Flexible

using practical units of feet for H, in.^ for I, and lb/in.2 for E. This

classification is analogous to Hetenyi's (1946) classification of beams

according to stiffness. It differs in that all sheet piles do not fall into

the flexible range for usual values of soil and sheet-pile stiffness.

Interaction Between Soil and Structure

The influence of sheet-pile flexibility on the soil pressure dis­

tribution is seen in Fig. 7-8 where the moments and soil pressures at

the end of a naturally occurring construction sequence for two 40-foot

sheet piles having equal K values are given. The expression, naturally

occurring construction sequence, denotes that the tie rod is secured to

an anchor wall and allowed to displace during the backfilling and dredg­

ing stages of construction. A comparison of the pressure distributions

for two sheet piles having the same flexibility number but different soil

support stiffnesses provides the relation between the soil system stiff­

ness, s, and the soil pressure distribution. The data for this situation

are given in Fig. 7-9 for a 40-foot pile having log/D = -3.32 and two

different soil stiffness values.

The pressure intensity on the active side is less for the more

flexible pile since the soil behaves in a relatively stronger manner.

Vertical arching between the tie-rod and dredge levels is present in both

T = 2.74 kips logp = -1.5 T= 5.92 kips logp = -4.0

log p = -4.0 log p = -1.5 log p = -4.0 1,5

-4.0

-1.5"

log f

log p H = 40 ft K = 200

log s = 1.5 KTR = 300 k/ft

500 Displacement, ft Moment, k-in. Pressure, psi

Fig. 7-8. Sheet-pile Moment and Soil Pressures for Two Different Sheet-pile Flexibilities, Finite Element Model

T=2.86 kips K = 600 T= 5.81 kips K = 79

K = 600 log s = 2.0

K = 79 K = 600 600 K= 79

H = 40 ft log p = -3.32

KTR = 300 k/ft

500 20 20 Moment, K-in. Pressure, psi Displacement, ft

Fig. 7-9. Sheet-pile Moment and Soil Pressures for Two Different Soil Modulus Numbers, Finite Element Model

127

pressure distributions. In general, arching is expected to be greater

with stronger soils and where differential displacement along the pile is

large but translational movement is small. The results of Figs. 7-8 and

7-9 show that it is more dependent on flexing deformation than on soil

strength or translational deflection. This conclusion is justified by

greater arching pressures existing even though the translation is large

and the soil is weak (Fig. 7-9) and arching pressure existing even when

the soil is relatively weak or strong compared to the pile (Fig. 7-8).

Differential displacement is dependent on the integral of M/EI

between two points. A rigid moment pattern can produce large differen­

tial displacement over a larger length of the pile, since the moment is

large and does not change sign. A flexible moment pattern can produce

large local differential displacements, since the change in moment is

great. In between these cases of large and small S, smaller differential

deflections result yielding less arching. The arching, which is present,

is not greatly dependent on whether the flexing deformation occurs local­

ly or generally, but in the local case the arching pressure may occur

more toward the tie rod, as in Fig. 7-8. For this case, log S is zero

compared to the other two cases where more general flexing and arching

are present and log S is -2.5 and -2.2. For the case where arching is

small, log S is -1.3 (Fig. 7-9).

Another viewpoint on the concept of interaction can be had by

comparing the pressure intensity for two sheet piles of equal flexibil­

ities, p, but different soil modulus numbers (Fig. 7-9). The pile with

the less stiff soil behaves in a more rigid manner. Despite the fact that

a pile's flexibility in relation to the soil support increases as the soil

128

becomes stiffer, more differential displacement of the center of the

pile with respect to the ends occurs with the weaker soil. This beha­

vior occurs since the moment sustained by the pile is greater due to

the increased loads on it. The result of the flexing is increased arching

pressure for the weaker soil. Also, in a similar manner to differingp

values, greater pressures occur on the active side of the pile for a

weaker soil.

The passive pressure distribution is dependent on the relative

flexibility of the sheet pile in relation to the soil. A more flexible sheet

pile rotates more at the dredge level concentrating more of its passive

resistance closer to the. dredge level, as in Fig. 2-5. Similarly, as the

soil becomes stiffer, less angular rotation of the pile tip with respect to

the dredge level occurs but more curvature at the dredge level results,

since the pile is relatively more flexible. This transfers the resultant of

the passive pressure distribution upwards.

The influence of flexibility on the passive pressure resultant is

not as evident in Fig. 7-8, since only three beam elements exist below

the final dredge level. The more uniform passive pressure distribution

below the dredge level shown for the more flexible pile is in accordance

with the larger number of inflection points occurring below the dredge

level for such a case.

It should be noted that the pressure distribution differences are

small compared to the differences in moment patterns that occur with

changing soil stiffnesses. This similarity in pressures occurs, since

the stresses obtained are required for similar stability considerations for

129

the sheet piles. The deflections are not solely dependent on the stabil­

ity of the sheet pile and can vary with soil stiffness.

Influence of Dredge Level Depth

The soil support stiffness is dependent on the extent of the soil

support over the length of the pile. This extent is proportional to

(1 - ct)H, the depth below the dredge level. This section will show the

relation between the soil stiffness and #and the influence of &. on the

sheet-pile moment.

As dredging proceeds, less of the sheet pile is supported, re­

sulting in a lower soil support stiffness, which, in turn, causes the pile

to behave in a more rigid manner, gaining moment. The increase in mo­

ment that occurs during construction is shown in Fig. 7-10 for a 40-foot

sheet pile attached to an anchor wall 10 feet tall located a distance

2.5H behind the sheet pile by an anchor having a stiffness of 300

kips/ft.

An explanation of the increase in sheet-pile moment during the

construction process is obtained by an analysis of the distributions of

pressure on the sheet pile during the construction stages (Fig. 7-11).

The rise in the passive pressure resultant as a sheet pile behaves in a

more flexible manner is shown in the passive pressure distributions dur­

ing dredging. Even though the pile becomes relatively more rigid during

dredging compared to the soil, its deflection pattern displays a more

flexible nature and thus the passive pressure resultant rises.

The horizontal pressure distribution on the pass_/e side is

greater than the Coulomb value, using a wall friction of angle of 2/3/6.

i

< D a = 0 . 2 — ® Q H = 40 ft

^Yv\. P = - 3 . 3 2 Vi\ \ \ K = 4 0 0 VO\ \ KTR = 300 k/ft

— © a= 0 . 4 —

V^o\

© a= 0 . 5

— V. %

• < 2 = 0 . 6 —

B>

o a= o . 7

, , i 1

300 200 100 0 Moment, k-in.

100

Fig. 7-10. Sheet-pile Moment during Construction

Backfill Initial Conditions

cC= 0.25 4 = 0.25 •7777777777777 7777777777777777777

H = 40 ft log p = -3.32

K = 400 Ktr = 300 k/ft

Pressure, psi

Fig. 7-11. Sheet-pile Pressure Distribution during Construction

I

777777777777 CL= 0.4

T 1

H = 40 ft log p = -3.32

K = 400 KTR = 300 k/ft

Dredge Layer 2

i—r

> > i / i n i i i m n / f i i m i * a= 0.5

Kr

20 10 10 20 Pressure, psi

Fig. 7-11. Sheet-pile Pressure Distribution—Continued

Dredge Layer 3 Dredge Layer 4 H = 40 ft log o = -3.32

K = 400 = 300 k/

20 Pressure, psi

Fig. 7-11. Sheet-pile Pressure Distribution—Continued

co GO

134

This indicates that the maximum obliquity of wall friction is developed.

The corresponding average ratio of shear to normal stress for the inter­

face elements of 0.73 verifies this result. The wall friction angle of 36°

that results is slightly greater than 16 due to the approximation of the

nonlinear relation by a series of linear steps.

ibility and soil modulus number for different dredge level depths is pre­

sented in Figs. 7-12 and 7-13. These plots show that more drastic

reductions occur in the early dredging stages. A comparison of the two

figures shows that changes in structural stiffness cause greater changes

inTthan those due to changes in soil stiffness. The data presented in

Fig. 7-13 show the relative influence of^on the soil stiffness. As cL

changes from 0.5 to 0. 7 for K= 300, the change in Ci is about half the

amount caused by increasing K from 300 to 450. Thus, the soil stiff­

ness, s, as quantified by K/Vff plays a greater role in determining

sheet-pile moments than does the depth of embedment (1 -ct)H.

plotted versus K(1 -tf)n where n very closely approximates 0.5 for large

OL values. Incorporating this result into the soil stiffness expression

yields the relation,

The sheet-pile moment is proportional to the ratio of the structural stiff­

ness to the soil support siffness. Expressing the system stiffness as

the inverse of this ratio yields

The variation of the sheet-pile moment with sheet-pile flex-

The three curves presented in Fig. 7-13 will coincide if T is

(7-13)

(7-14)

135

12

OL = 0.70

H = 40 ft K = 200

Ktr = 300 k/ft 8

a = 0.25

4

-4 logyo

Fig. 7-12. TVersus Sheet-pile Flexibility for Different Dredge Level Depths

12

H = 40 ft log p = -3.32 Ktr = 300 k/ft 8 <X = 0.70

d= 0.25 4

150 300 450 600 K

Fig. 7-13. T Versus Soil Modulus Number for Different Dredge Level Depths

136

Displacement Relations

A comparison of translational deflections at the tip of the sheet

pile after dredging with the tie-rod deformation fixed is made in order to

evaluate the proposed similitude expression Ky/H^»5. Values of H rang­

ing from 3.5 to 60 feet, for K from 10 to 800, and for log p from -1.5 to

-4.0 are used. As can be seen from the data for ytip in Table 7-2 and

Fig. 7-14, the deflection is linearly related to K. The translational de­

flection varies as H**30 within a narrow range of + 0.03 (Fig. 7-15).

This result indicates a smaller exponent than unity for the scale factor

D or for the depth z in the expression

p = Ksrz = i^y (7-15)

since Ks is a constant. The smaller value suggested for D is in agree­

ment with the smaller value indicated for use in the soil stiffness term.

The variation of deflection with sheet-pile stiffness for piles with the

same K and H is 1 percent for a change in logp of 1.02 (Table 7-2;

Fig. 7-16). The stiffer pile has slightly less deflection. Thus, the re­

lation between sheet-pile deflections for models of different scale is

adequately represented by the equation

yoc-Mil^, (7-16)

which is independent of the sheet-pile stiffness.

As shown by Fig. 7-9, differential deflection between the cen­

ter of the pile and a chord through its endpoints increases as the soil

becomes weaker. Even though a pile behaves more rigidly with a softer

soil, in the sense of receiving more moment, this rigidity concept cannot

137

5

H = 40 ft log p= -3.32

.4

versus 1/K

.3

. 2

1 versus K

400 K 600 0.010 1/K 0.015

800 0 . 0 2 0 0.005

Fig. 7-14. Sheet-pile Tip Displacement Versus Soil Modulus Number

.15

log -3.32 300

.10

a

.05

Fig. 7-15. Sheet-pile Tip Displacement Versus Sheet-pile Height

138

. 0 6

.04

a •H 4->

.02

1 1 1

—

—

0

H = 20 ft

1 1

K = 300

1 -4 -3 -2

log p -l

Fig. 7-16. Sheet-pile Tip Displacement Versus Sheet-pile Flexibility

139

be applied to the deflection pattern. This result has been mentioned

previously in the section on interaction between soil and structure.

The specification of the tie-rod level displacement necessary

to yield a fully active pressure condition necessitates the relation of

pressure to deflection. An active pressure distribution is encountered

in the region of the bulkhead between the tie-rod level and a point a

few feet above the dredge level (Fig. 7-11). In this area, larger deflec­

tions and increased rotations exist at the top than in the embedded por­

tion of the bulkhead (Fig. 7-9). The pressure distribution indicates

that the active failure wedge is occurring only in the zone from just

above the dredge level to the surface, regardless of the total movement

of the sheet pile. Differential, translational, and rotational movements

within the soil mass on both sides of the sheet pile cause local failure

zones, which are not propagated to the surface in the form of failure

wedges. No slip occurs at the pile tip, as this entire region is part of

the continuum according to the idealization of the bulkhead system.

The movement necessary to develop a general slip surface and

prevent any arching from occurring has been specified by Terzaghi (1934)

for a completely rigid wall as some fraction of the wall height. The tie-

rod level deformation is dependent on the current dredge level at any

dredging stage. An expression for the tie-rod level deformation required

to produce fully active soil pressures should involve the distance from

the top of the sheet pile to the current dredge level, <X H, rather than the

total sheet-pile height, H. From Fig. 7-9 it is evident that for sheet-

pile walls less movement is required to develop a fully active pressure

for a stronger soil than for a weaker soil. The more rigid the wall, the

140

greater the movement that is necessary (Fig. 7-8). Thus, the expres­

sion of the movement required to develop an active earth pressure con­

dition solely as some fraction of the sheet-pile height is not adequate.

Influence of Poisson's Ratio

The influence of the value of Poisson's ratio was investigated

by using a finite element model of a 40-foot sheet pile having identical

properties to the 40-foot sheet-pile model used to generate the results

of Fig. 7-5. The only difference is that Poisson's ratio is 0.30 instead

of 0.40. The moments and pressures are shown in Fig. 7-17. The maxi­

mum moment before tie-rod release is 3.5 percent more with the smaller

}i of 0.30.

This result can be explained as follows. The constitutive

strength relations for soil can be written in terms of the bulk modulus

and deviatoric or shear modulus. As p increases, the shear modulus

E/2 (1 + p) decreases. The opposite is true for the bulk modulus as

represented by E/2(l + ju)(l -2 j j) . The increase occurring in the bulk

modulus is greater than the decrease occurring in the shear modulus.

Thus, the strength of the soil, though smaller in shear, is greater in

the normal stress direction for a larger p value. It is this latter strength

that controls the behavior of the system to a greater extent, as can be

seen from a comparison of Figs. 7-17 and 7-5. The soil with the greater

j j value causes the pile to display a more flexible moment pattern with

a sligthly smaller maximum moment.

The pressure distribution on the inside of the sheet pile for the

25 percent smaller j j is less by up to 30 percent, as would be expected.

141

log p = -3.32 H = 40 ft K = 79 p = 0.30 _

500 0 0 20 Moment, k-in. Pressure, psi

Fig. 7-17. Sheet-pile Moment and Soil Pressures, Finite Element Model, p = 0.30

142

The decrease, as can be seen from a comparison of Figs. 7-17 and 7-5

is in the region near the tie-rod level. Less vertical arching pressure is

present at the support levels. Near the dredge level, the pressure dif­

ference is less for the cases of the two j j values, although just below

the dredge level on the passive side, the pressure increased 20 percent

with the smaller p. This increase is due to the more rigid confined con­

ditions that occur with the less lateral expansion in the p= 0.30 case.

Influence of Material Density

The results of an analysis of a bulkhead system having identi­

cal properties to the system shown in Fig. 7-5, except that the density

of the soil is 135 pcf instead of 90 pcf, are shown in Fig. 7-18. The

angle of internal friction, 30°, and the soil modulus number, 79, used

in the previous analysis are retained. An increase in strength for the

soil having the greater density occurs due to the increase in confining

pressure. Even though the soil is stronger and the pile therefore rela­

tively more flexible, a 20 percent greater moment exists with the

denser soil having a 50 percent greater loading and a 22 percent greater

soil stiffness. Modifying the soil stiffness component of the system

stiffness to include density yields s = The sheet-pile moment

is directly dependent on the loading regardless of the stiffness, and the

results presented show that large decreases in moment can be expected

when the bulkhead is below water, reducing the soil density and thus

loading to the buoyant value.

Anchor System Stiffness

This study, so far, has characterized the stiffness of the sup­

porting soil system and has eliminated the influence of the anchor

log p = -3.32 H = 40 ft. K = 79 Y - 135 pcf

500 0 20 0 20 Moment, k-in. Pressure, psi

Fig. 7-18. Sheet-pile Moment and Soil Pressures/ Finite Element Model, Y = 135 pcf

144

system support by comparing quantities obtained through keeping the

tie-rod level fixed. This section will develop the relations for the anchor

system stiffness, consisting of the anchor wall, tie rod, and anchor sup­

porting soil, by analyzing the factors of tie-rod level displacements,

soil modulus number, system scale as represented by H, and the sheet-

pile stiffness. The influence of the tie-rod force and the related anchor

system stiffness on the sheet-pile behavior, as quantified by T, will be

presented in a general manner.

The stiffness of the anchor system consists of a series combina­

tion of its component parts. Since E = 5 x 10^ ksf, the effect of the

anchor-wall stiffness, can be eliminated because it is rigid compared to

the other components. The stiffness of the tie rod is AE/L, where A is

the area and L is the length of the tie rod. The stiffness of the anchor

soil support is dependent on the modulus number, K, and some factor of

H, representing the scale of the system as developed previously for a

soil mass.

In a naturally occurring construction sequence, the tie-rod

force and moment increase rather than decrease as dredging and thus

sheet-pile movement occur. Therefore, rather than imposing displace­

ments, the anchor stiffness is studied by allowing the displacements to

occur naturally, thus including the influence of the anchor wall. This

latter approach is performed in one of two ways. Either the sheet pile

is attached to the anchor wall by a tie rod or an anchor wall stiffness

is added to the sheet pile at the tie-rod level and the area of the tie rod

is set equal to zero. In this chapter, an anchor wall is always attached

to the sheet pile if the displacements are naturally occurring.

145

Tie-rod Force—Anchor-wall Displacement Relations. The stiff­

ness relation between tie-rod forces and anchor-wall displacements at

the end of construction can be evaluated by ascertaining the parameters

that influence each. The parametrical expression for force can then be

divided by that for displacement to yield the stiffness. In the finite

element analysis of the parameters, the anchor-wall height is 25 per­

cent of the sheet-pile height and the anchor is located a distance 2.5H

behind the pile. The tie-rod area is apportioned to yield a K^r of

300 k/ft.

The data representing the horizontal component of the tie-rod

force versus anchor displacement relation as a function of the soil stiff­

ness and sheet-pile flexibility are graphically presented in Figs. 7-19

and 7-20 for a 40-foot sheet pile. All data are from computer runs simu­

lating the displacements that naturally occur in the field during construc­

tion. The close correspondence between the T^-y curves and the

nonlinear stress-strain behavior of the soil is apparent.

Numerically, the end of construction T varies as K"® *35 (Fig.

7-21). A stiffer soil produces less pressure on the pile, resulting in

less tie-rod force (Fig. 7-9). The exponent on K is different from unity

due to the interaction between the sheet pile and anchor force. Intro­

ducing a variation in the height of the sheet-pile model allows the ex­

pression of T in terms of H, the height of the sheet pile that represents

the system size. The end of construction tie-rod force varies as

(Fig. 7-22). Generally, this variation is assumed to be proportional to

H2. The influence of sheet-pile flexibility on the tie-rod force, within

practical ranges, is small compared to the effect of sheet-pile height

K = 79

5

K = 200

4

Dredge 1

K = 600 3

log p = -3.32

H = 40 ft Kfp = 300 k/ft

Backfill

2

1

533 400 267 1000 800 320

.150 .175 .075 yt ft .100 .125 .025 .050

Fig. 7-19. Tie-rod Force Versus Anchor-wall Displacement for Various K

6

H = 40 ft K = 200

TR = 300 k/ft 4 log p = -4 .Q,

og p = -3.32

2

Dredge 1 Backfill

. 0 1 02 .03 Anchor-wall Displacement, ft

.04 .05 .06 .07

Fig. 7-20. Tie-rod Force Versus Ancho^wall Displacement for Various Log p

•Cfc

148

6

4

H = 40 ft log p = -3.32

Ktr = 300 k/ft

2

300 150 K 450 600 1.38 1.68 1.85 1.98

log VM

Fig. 7-21. Tie-rod Force Versus Soil Stiffness

6

4

2

log p = -3.32 K = 2 0 0 '

KTR = 300 k/ft

10 20 40 H, ft

Fig. 7-22. Tie-rod Force Versus Sheet-pile Height

149

6

4 01 a iH

H = 40 ft K = 200

= 300 k/ft 2

3 - 2 -5 •4 -1 log p

Fig. 7-23. Tie-rod Force Versus Sheet-pile Flexibility

(Fig. 7-22). Larger tie-rod forces are obtained with stiffer sheet piles,

since the pressure acting on a stiffer pile is greater due to the soil be­

ing relatively weaker (Fig. 7-8). The tie-rod force for the flexible pile

is smaller even though the arching pressure is more concentrated at the

tie-rod level (Fig. 7-8).

The similarity between the anchor-wall displacements at the

end of construction with a naturally occurring tie-rod release and the

sheet-pile tip displacements during a fixed tie-rod level construction

sequence is shown by a comparison of Figs. 7-24, 7-25, 7-26 with

Figs. 7-14, 7-15, and 7-16. The same relation, yocH* with

little effect of sheet-pile flexibility exists for both cases. This result

is to be expected since the anchor wall is out of the zone of influence

150

.16 1 1 1 1

.15 H = 40 ft

log p = -3.32 yS Ktr = 300 k/ft

.10 — V versus 1/K —

.05 — / versus K

1 1 1 I

—

150 300 K 450 600 .003 .006 1/K .009 .012

Fig. 7-24. Anchor-wall Displacement Versus Soil Modulus Number

.075

log p = -3.32 K = 200

Ktr = 300 k/ft .050

.025

10 20 30 40 H, ft

Fig. 7-25. Anchor-wall Displacement Versus Sheet-pile Height

151

• lb 1 1 1

H = 40 ft K = 2 0 0

.10= — K t r = 3 0 0 k/ft —

. 0 5

I I 1 -4 -3 -2 -1

log p

Fig. 7-26. Anchors-wall Displacement Versus Sheet-pile Flexibility

of the sheet pile and the displacements for the pile and anchor wall are

practically equal. Less displacement occurs with a stiffer soil since

such a soil exerts less pressure on the active side of the sheet pile by

supporting itself more (Fig. 7-24). Stiffer sheet piles, which produce

larger tie-rod forces yield approximately the same anchor-wall dis­

placements (Fig. 7-26). Since

Tot H^SO/K0-35 (7-17) and

yoc H1,30/K1,00, (7-18)

the anchor-wall soil stiffness from a force viewpoint is

KAWS = T/y * K0.65H0.60. (7_ig)

The secant stiffness from a force viewpoint at the end of con­

struction is plotted in Fig. 7-27 for the anchor system consisting of the

•AWS

100

Secant Anchor Stiffness

k/ft 50 log p = -3.32

KTR = 300 k/ft

100 200 300 400 K

Fig. 7-27. Secant Anchor Stiffness Versus Soil Modulus Number

153

anchor-wall soil and tie-rod stiffnesses in series. For a low value of K,

the tie rod is stiff compared to the soil, permitting the system stiffness

to be adequately represented by the anchor soil stiffness.

The stiffness of the anchor soil support represents the relation

between the pressure p and displacement y. If the entire anchor wall is

under a compressive soil pressure, the average pressure is proportional

to T/H and the stiffness of the anchor-wall support from a stress view­

point, KawS, at the end of construction is

KAWS K°'65/H0'40. (7-20)

This quantity is comparable to K//H used as the soil support stiffness

previously, but the former quantity includes the interactive effect of the

sheet pile on the anchor loads by the nonlinear exponent on K.

Influence on Sheet-pile Moment. An assessment of the effect

of the anchor stiffness on the sheet-pile moment can be made by evalu­

ating the tie-rod level movement. Small s values produce a greater

lateral pressure on the wall and thus more movement. Even if the lateral

forces were the same, more movement would result, since KawS *s ^ess

for small s values. Thus, it is necessary to separate the influence of

the force and stiffness on the resulting movement.

As log s increases from 1.1 to 2.0, the tie-rod force decreases

by half from 5.4 to 2.7 kips (Fig. 7-21). The value of KawS =

1^0.65^0.60 increases by a factor of 3.6 from 160 to 580 for the same

change in log s. The tie-rod level movement decreases by a factor of

7.7 from H/256 to H/1970 (Fig. 7-25). This last factor is of course the

change in tie-rod force divided by the change in KawS.

154

The change in moment with tie-rod level movement is given in

Fig. 7-28 for two sheet piles having identicalp and H values. The top

curve of this figure represents the change in moment without any effect

of anchor stiffness, since the tie-rod level displacement is held fixed.

The bottom curve represents the moment value after a naturally occurring

tie-rod release. The difference in the ordinates of the two curves repre­

sents the change in moment due to tie-rod release. This value is essen­

tially a constant amount, ranging from 1.6 to 1.1, as log s changes from

1 . 1 t o 2 . 0 .

Since the deviation from the fixed tie-rod, no K^WS effect con­

dition is constant even though large changes in tie-rod level movement

occur, it can be concluded that the amount of the moment change is not

sensitive to the magnitude of the tie-rod level movement or to changes

in K. The changes in moment due to changes in soil stiffness, K/v/H are

greater than any changes induced by movement and any of its components

T or KawS* The effect of anchor-wall height, which is one variable com­

ponent of the anchor stiffness, would also have an unmeasurable effect

on the sheet-pile moment.

Rowe (1952) justified the larger values of moment with tie-rod

release, than for the fixed tie-rod case, by the breakdown of arching.

Thus, he finds moments increasing as the stiffness of the anchor system

decreases. In the current study, moment decreases with tie-rod release

are observed. Less stiff anchors, which allow more tie-rod deflection,

provide reductions in the moment values calculated using fixed tie-rod

level conditions. This is opposite to the expected reduction due to a

stiff soil surrounding the pile. However, the former reductions are so

15

CO a £ ii

K

10

tie-rod movement

1 2 log s = log JL_

/h

Fig. 7-28. TVersus Log s for Naturally Occurring Tie-rod Release

156

small as to be insignificant compared to changes inT induced by soil

stiffness variations, as seen previously.

Some factors that have not been considered in assessing the

anchorage stiffness and its influence on sheet-pile moments are the

location of the anchor wall in relation to the sheet pile, the ratio of the

anchor wall to sheet-pile height, and the point of application of the

stiffness, i.e., the tie-rod level, £H. In regard to this last variable,

Rowe (1952) presented curves showing the decrease inT with increasing

y 3 . T y p i c a l l y , a 3 0 p e r c e n t r e d u c t i o n i n m o m e n t s o c c u r r e d w i t h 4 = 0 . 7

f o r £ = 0 . 2 i n s t e a d o f 0 . 0 .

CHAPTER 8

SIMULATION OF ROWE'S MODEL TESTS

Rowe's (1952) extensive series of model bulkhead tests pro­

vided an insight into the effect of the configuration parameters a and ,3,

the flexibility of the sheet pile, and the influence of dredge and fill

sequences during construction. A major drawback of this test series

is that the effect of scale was not considered. Also the soil stiffness

contribution was only generally considered by the specification of small

variations in moment reduction for loose and dense soils. The stiffness

of the soil was assumed to be independent of depth. The importance of

scale will be seen in the comparison between Rowe's pressure tests and

the finite element simulation of them, using sheet piles of different

lengths.

Simulation of Pressure Test

A slight difference in the method of conducting the pressure

test from that used by Rowe and presented in Chapter 2 is used with the

finite element model. Rather than starting with the soil height at the ex­

treme top of the sheet pile on both sides and then dredging down one

layer, the soil is backfilled on the inside on the top 25 percent of the

sheet-pile height. The difference in soil behavior during loading and

unloading results in two different modulus numbers being used which

simulate loading and unloading for the backfilling and dredging cases,

respectively. The difference in sheet-pile behavior is very small,

157

158

since the backfilling step that is substituted for a dredging step to yield

the same configuration occurs at the same region of the sheet pile. It

will be shown later that the difference between backfill and dredge se­

quences is greatest when the construction sequences occur at different

regions of the sheet pile. This modification makes the initial portion of

the test similar to the flexibility test performed by Rowe, except for the

tie-rod release portion.

Rowe's pressure test for a 3,5-foot-high sheet pile provides

pressure and moment values at the end of the final dredging stage. These

results are presented in Fig. 2-2. The corresponding results for the

finite element model sheet pile are presented in Fig. 7-4, where the soil

modulus number, K, is varied until the maximum bending moment in the

sheet piie before tie-rod release corresponds to that of Rowe's model.

This correspondence is achieved by using a K value of 45. The other in­

put parameters used in the simulation are presented in Table 7-1.

Rowe's value for the constrained modulus is 333 psi. Convert­

ing this to an analogous triaxial compression modulus, using a Poisson's

ratio of 0.4 yields 155 psi. In order to obtain this value for the initial

modulus with 3.5 feet of overburden pressure, a modulus number, K, of

48 is necessary. This value is very close to the K value of 45 needed to

obtain comparable moment results.

The slope of the bending moment curve for the finite element

model displays a more flexible pattern than Rowe's test results. This

comparison indicates that the square root increase of the soil modulus

with confining pressure is too large. In general, a larger value of the

exponent on confining pressure than 0.5 applies for loose sands. Since

159

the density in this case is loose, it is unlikely that a smaller value than

0.5 should be used.

An explanation for the flexible sheet-pile behavior for the finite

element model lies in the manner in which Rowe conducted his tests. As

was noted in the review of his method, due to the nonuniform conditions

on each side of the sheet pile, a Coulomb pressure distribution occurred

initially. In the finite element model, the initial pressure is equal to the

rest condition. The soil modulus is stiffer in this condition than in the

limiting passive or active cases, since it is on the linear portion rather

than on the nonlinear portion of the stress-strain curve. Thus, the sheet-

pile would be relatively more flexible.

The significant difference in the bending moment diagram occurs

after imposed tie-rod release. In Rowe's model (Fig. 2-2), the maximum

moment increased 20 percent during a tie-rod release of approximately

H/1000. In the 3.5-foot-high finite element model (Fig. 7-4), the mo­

ment decreases 7 percent during a larger tie-rod release of H/500. For

all the smaller increments up to this amount, smaller decreases in

moment occur.

In Rowe's tests, it is reasonable to expect a moment decrease

for any tie-rod level displacements greater than H/1000, since this

value is sufficient to destroy the arching to which the moment reduction

is attributed. Such a magnitude is greater than the deflection that he

finds is necessary to cause a decrease in bending moment due to canti­

lever action being approached.

The pressure distribution in each case is very similar initially

before tie-rod yield. Arching between the dredge level and tie-rod level

is present. A tie-rod release of H/500 on the finite element model de­

creases the intensity of the pressure distribution by 16 percent in the

vicinity of the tie rod only. A more radical decrease was found by

Rowe to occur generally for the same tie-rod yield. The essentially

constant pressure below the center of the sheet pile accounts for the

small change in moment during tie-rod release with the finite element

model.

Full-scale Simulation

In order to obtain information on the effect of scale on Rowe's

tests, a finite element pressure test simulation of a 40-foot sheet pile

is performed. The results are presented in Fig. 7-5. The maximum T

value for the actual si2;e sheet pile is identical to the one for the 3.5-

foot sheet pile when a soil modulus number of 79 is used. The soil

stiffness, log s = 1.10, is less than the value of 1.38 used for the

3.5-foot sheet pile. Since the structural stiffness, p, is the same for

each pile, different system stiffnesses, S, result. This difference is

due to the enforced correspondence of M/H^ using an s expression of

K/'(/if. A redefinition of the comparison quantity, J-, and the soil support

stiffness, s, utilizing different exponents on the H term would yield the

same magnitude T for the same system stiffness, S.

The difference in soil stiffnesses is apparent in the moment

patterns. The slightly stiffer soil surrounding the 3.5-foot sheet pile

produces a more flexible sheet-pile pattern. If the same soil modulus

number, K, is used with two sheet piles of different heights, the longer

sheet pile will have the softer soil support stiffness, s = k//H. This

161

soil support is less for the long pile due to the scale of the system,

even though the soil stress-strain modulus at the bottom of that pile is

greater. This result will cause theT value to be greater for the taller

sheet pile. For the case of a 3.5-foot and 40-foot sheet pile with the

same logp = -3.32 and K value of 300, T for the longer pile is 34 per­

cent greater than for the short pile. Thus, Rowe's models of the pres­

sure tests predict moments for full-size sheet piles that are on the

unsafe side.

An analysis of two bulkheads having heights of 3.5 and 40 feet

with linear soil stiffnesses of 500 ksf and linear interface properties

shows the importance of scale even with linear strength relations. In the

strength relation, no increase in soil stiffness with depth or confining

pressure is incorporated. The moments in the sheet piles are related to

the 3.73 power of the ratios of the height, and the pile tip deflections

before tie-rod release to the 2.00 power of the height ratio for the two

soil support stiffnesses. The soil support stiffness is greater for the

shorter pile, producing less moment in it than the value related to H^,

which is associated with systems of comparable system stiffnesses.

Thus, a soil stiffness of the form K/Hn where n is close to 1.0 is sug­

gested, which decreases with increasing height. This is in agreement

with previous relations, which are modified to remove the depth z in­

fluence on confining pressure. From this height relation, it is seen that

the predictions of the moments in a full-scale sheet pile from the mo­

ments in a smaller sheet pile using an relation is unconservative.

162

Imposed Sheet-pile Displacements at the Tie-rod Level

An investigation of imposed tie-rod release after dredging is

completed on a full-scale bulkhead is conducted in order to ascertain

the deflections needed to break down arching and to understand better

the events occurring during the breakdown. In these tests, where tie-

rod level displacement is imposed, the area of the tie rod is set equal

to zero in order that there is no anchor-wall influence on the sheet pile.

The influence of imposed tie-rod release on sheet piles of

different heights is the same, as can be seen from the identical changes

in the pressure distributions for the 3.5-foot and 40-foot sheet piles un­

dergoing a release of H/500 after construction (Figs. 7-4 and 7-5).

Therefore, any release needed to decrease the pressure is dependent on

a linear power of scale for systems having approximately the same s.

As tie-rod release occurs, the soil support below the dredge

level becomes relatively stiffer. This change shows up in the relatively

more flexible sheet-pile moment pattern throughout the pile and especial­

ly below the dredge level. The moment diagrams for a series of release

increments up to H/50 are presented in Fig. 8-1. The maximum moment

location, which was previously near the tie-rod level, shifts to a loca­

tion toward the center of the pile. Its magnitude decreases even with

respect to the previous value of the moment at this new location of the

maximum moment.

The tie-rod release influences the moment in the sheet pile

more from the deformation occurring generally throughout the pile than

from the change in pressure, which is concentrated at the tie-rod level.

H/50

H/IOO

H/50 H = 40 ft K = 79

log s = 1.10 log p = -3.32

H/500J H/50 H/10

/50

0 .05 .10 Deflection, ft

500 500 Moment, k-in. Pressure, psi

Fig. 8-1. Sheet-pile Moment, Soil Pressures,and Deflection for Large Imposed Tie-rod Release, Finite Element Model

164

During tie-rod movement the arching pressure decreases only near the

tie-rod level and pressure is transferred to the region just above the

dredge level. A release of H/50 is needed to break down all evidence

of arching near the tie-rod level (Fig. 8-1). This ratio is, of course,

for the specified log s of 1.10.

Information about the changes in the tie-rod load during imposed

sheet-pile anchor level displacements is obtained by a series of tests in

which the force necessary to hold the sheet pile under imposed sheet-

pile displacements is obtained. An example of the tie-rod force variation

with imposed tie-rod release is given in Fig. 8-2 for a sheet-pile flex­

ibility, logp = -3.32, and varying soil modulus numbers. For low

values of K, a linear decrease in the tie-rod force occurs as the pile is

released at the tie-rod level. This result is consistent with previous

pressure distribution patterns, which show a decrease in intensity with

outward sheet-pile movement. No increase in load due to the breakdown

of arching is evident, implying that anchor systems that are flexible and

allow more deformation should experience load reductions.

The variation of the tie-rod force with K is shown in Fig. 8-2

for a sheet-pile height of 40 feet. The tie-rod force, T, increases by

240 percent as K changes from 300 to 50, since the weaker soil exerts

more pressure on the sheet pile. A nonlinear decrease occurs if K is

large, since the imposed wall displacement is relatively large for this

stiffness and the force, therefore, decreases rapidly. These changes

are comparable to those occurring during construction, since a 40-foot

sheet pile would undergo a tie-rod movement of less than H/350 if the

soil modulus number is greater than 200.

H = 40

K = 50

logP = -3.32

H/2000 H/1000 Tie-rod Release

H/500

Fig. 8-2. Tie-rod Force Versus Imposed Tie-rod Release CT> cn

166

Thus, the pressure intensity near the top of the sheet pile on

the active side decreases with tie-rod movement (Fig. 8-1). This de­

crease in pressure intensity is manifested in a smaller tie-rod force.

Both factors lead to a moment decrease for an imposed tie-rod displace­

ment. Therefore, it is evident that the condition of imposing tie-rod

displacements cannot be compared to the release that occurs naturally

during the dredging stages of construction and results in moment and

tie-rod force increases.

Effect of Construction Sequence

The construction sequence has an influence on sheet-pile mo­

ments due to the different sheet-pile deformations that occur. Both

Rowe (1952) and Tschebotarioff (1949) have observed that a backfill-

dredge sequence will result in greater pressure distributions due to less

tie-rod movement than will a dredge-backfill sequence. A backfill stage

relieves the pressure buildup behind the sheet-pile by permitting defor­

mation. An investigation of the difference in moments caused by tie-rod

release occurring naturally during construction or imposed after construc­

tion will relate model conditions to field conditions.

Backfill-Dredging Sequence

This effect is investigated with the finite element model by

simulating the two different construction methods with a 40-foot sheet

pile having a tie-rod anchor stiffness of 35 k/ft. Backfilling takes

place behind the top 25 percent of the sheet-pile height and dredging

occurs between the region of 0.25 H and 0.70H, as measured from the

top of the sheet pile. The moments, pressure distributions, and

167

deflections at the end of construction are shown in Fig. 8-3 for the situ­

ation where dredging is the last operation and in Fig. 8-4 for the case

where backfilling is the last operation.

The pressure distribution on the inside of the sheet pile is

slightly less for the case where dredging is the last operation. The

smaller pressure yields a maximum moment, which is 9 percent less

than the bulkhead system constructed with backfilling being the last

operation. As would be expected, more deflection at the tie-rod level

occurs during the backfill stage in the dredge-backfill method of con­

struction and during the dredge stage for the backfill-dredge method of

construction (Table 8-1 and Fig. 8-5). The reason is that in each of the

cases cited, less of the sheet pile is supported.

Table 8-1. Displacement of Sheet-pile at the Tie-rod Level for Two Types of Construction Sequence

Displacement, ft

Backfill-Dredge Method Dredge-Backfill Method

Backfill

Dredge 0.0275 0.0164 0.0187 0.0143

0.0707

0.0769

Dredge

Backfill

0.0177 0.0082 0 . 0 1 0 0 0.0052

0.0411

0.1046

Total 0.1476 Total 0.1457

H = 40 ft K = 79

log p = -3.32 Ka= 35 k/ft

500 Moment, k-in. Pressure, psi Deflection, ft

Fig. 8-3. Sheet-pile Moment, Soil Pressures, and Deflections for Backfill-Dredge Construction Sequence, Finite Element Model

H = 40 ft K = 79

log p = -3.32 KA= 35 k/ft

Moment, k-in. Pressure, psi Deflection, ft

Fig. 8-4. Sheet-pile Moment, Soil Pressures, and Deflections for Dredge-Backfill Construction Sequence, Finite Element Model

o Dredging last operation 6

• Backfilling last operation

4

H = 40 ft K = 79

KawS = 35 (applied) 2

H/400 H/267 H/800

.15 . 1 0 .05 Deflection, ft

Fig. 8-5. Sheet-pile Tie-rod Level Displacements for Two Construction Sequences, Finite Element Model

171

The effect of the backfill layer is felt to a greater degree when

it is the last operation and dredging has exposed one side of the pile.

Where backfilling is the first operation, the soil level is the same on

both sides of the sheet pile and backfilling has less influence on the

pressure on the inside of the pile. As dredging occurs, the pressure

caused by the backfill layer acting as a surcharge is relieved by a trans­

lation of the bottom as well as the center of the sheet pile. The transla­

tion occurs near the bottom since that is the center of dredge activity.

Where backfilling is the last operation, the center portion of

the pile is closer to the load application and translates more than the

bottom, causing a pressure and moment buildup. The relatively more

stiff soil condition is evidenced in Fig. 8-4, where a more flexible

bending moment pattern exists in the sheet pile below the dredge level.

Larger tie-rod level deformations occur during this particular construc­

tion sequence, since more of the pile is exposed and is therefore less

supported during the stage in which activity is centered near the tie-

rod level.

The larger tie-rod movement that occurs during the backfilling

operation results in a larger tie-rod force, since an anchor system stiff­

ness of 35 kips/ft is provided (Fig. 8-5). In the naturally occurring re­

leases during construction, pressure distributions are nearly identical

(Figs. 8-3 and 8-4) but the increased tie-rod force during the backfill

stage of the dredge-backfill sequence provides a moment increase.

The results obtained show that the difference in the moments,

soil pressures, and displacements as a function of the method of con­

struction is small and is related to the relative displacements of the

172

center and bottom of the sheet pile rather than to the magnitude of total

deflection. During the dredging stage of each construction sequence, a

tie-rod level movement of H/200 takes place. This movement is more

than that assumed by Rowe to occur during dredging, and it will account

for the pressure and moment decrease occurring in that stage. In each

of the dredge-backfill or backfill-dredge sequences, the tie-rod level

deformation due to dredging was not significantly less than that due to

backfilling.

Tie-rod Release Sequence

An associated aspect of construction sequence is the relation

between the sequence of occurrence of tie-rod movement and the result­

ing sheet-pile moments. This relation is investigated by comparing the

moments, pressures, and deflections for a bulkhead constructed by

backfilling followed by dredging with tie-rod movement occurring natur­

ally after each stage to those of a bulkhead constructed by backfilling

followed by dredging but with all tie-rod movement imposed after the

final dredging step. In both cases, the total tie-rod movements are

made equal.

The sheet-pile moments, pressures, and deflections for the

two cases are presented in Figs. 8-6 and 8-7. The patterns and magni­

tudes are almost identical. The increase in the moment in the sheet

pile during construction composed of backfilling the top 25 percent of

H and four dredging stages between A - 0.25, 0.4, 0.5, and 0.7 is

shown in Fig. 8-6. As dredging occurs, the soil support weakens,

causing the sheet pile to behave in a relatively more rigid manner

156 H = 40 ft K = 79

log s = 1.10 log p = -3.32 K t r = 300 k/ft

Ct = 0.25

500 500 Moment, k-in. Pressure, psi Deflection, ft

Fig. 8-6. Sheet-pile Moment, Soil Pressures, and Deflections for Backfill-Dredge Construction Sequence with Naturally Occurring Tie-rod Release, Finite Element Model

156

Before release After

~ releas H = 40 ft K = 79

log s = 1.10 logp - -3.32

500 Deflection, ft Moment, k-in. Pressure, psi

Fig. 8-7. Sheet-pile Moment, Soil Pressures, and Deflections for BackfillrDredge Construction Sequence with Imposed Tie-rod Release, Finite Element Model

175

accepting more moment. The maximum moment that occurs near the

Gt = 0.4 level increased by 15, 16, and 12 percent as otincreased from

0.4 to 0.7 in increments of 0.1.

The sheet-pile moments for the case where release occurs si­

multaneously with dredging are 1 percent more than for the case where

release occurs in five steps after final dredging is completed. This in­

dicates that a tie-rod movement of H/257 is approximately equivalent to

the same movement distributed throughout construction in increments of

H/677, H/1172, H/2022, H/1575, andH/2292.

This information can be applied to Rowe's tests. In his flexure

tests, a tie-rod release of H/2400 was imposed at the end of each

dredging stage to yield a total tie-rod level deformation of H/800 at the

end of the test. Since the movement H/800 is small in comparison to the

H/257 release occurring during actual construction of a 40-foot bulkhead,

it would indicate that the flexure test could be adequately modeled by a

pressure test having a tie-rod release of H/800 imposed at the end of

the final dredging stage.

A larger difference in the moments for the various sequences of

dredge and backfill could occur if the tie rod were not located at the top

of the sheet pile, k/3 greater than zero provides a lever arm by which a

change in pressure above the tie-rod level due to breakdown of arching

can exert its influence on moment.

Simulation of Flexibility Test

Rowe's flexibility test series were run in order to ascertain the

influence of tie-rod movement during dredging. The tie-rod release was

176

performed at the end of each dredging level,CL= 0.6 to <X= 0.8. In each

case, he observed a moment increase, which he attributed to the break­

down of vertical arching. At a release of H/2400 for each layer, he found

that the moment began to decrease as a cantilever situation was ap­

proached. At this point, dredging on the next layer was begun. As the

tie-rod release was stopped just at the point where the moment ceased

to increase, all of Rowe's flexibility test results should provide greater

moments than those obtained from pressure tests at stages before tie-rod

release.

A flexibility test series was simulated using a finite element

model of a 40-foot sheet pile. After each dredge layer from cL = 0.6 to

d = 0.8, a series of three small tie-rod releases of magnitude H/7200,

yielding a total release of H/2400 at the end of each dredge layer and

H/800 total deformation at the end of all dredge layers, is imposed. A

nfoment decrease rather than the increase observed by Rowe is obtained

after every increment of tie-rod release.

The moment decrease is consistent with the findings of the

pressure test. It occurs since for any magnitude of tie-rod release only

that region of soil surrounding the tie rod is subject to a decrease in

pressure. There is no general influence on the soil pressures over the

entire length of the sheet pile. Thus, pressures near the tie rod ap­

proach the active case, while soil pressures on the center portion of the

sheet pile remain virtually constant.

The decrease in bending moment with anchor movement is less

if the tie rod is located below the top of the sheet pile. This lesser

177

change in moment occurs due to the decrease in pressure above the

anchor level, causing a decrease in the pile's rotational deflection.

scale size is performed by using a pressure test with tie-rod release

occurring at the end of the final dredge stage. This approach can be

taken since the sequence of tie-rod release has been shown to be un­

important. For the case of H = 3.5 feet, <2 = 0.7, /3= 0.0, log p = -3.32,

and a loose sand with total tie-rod yield equal to H/800, Rowe's flex­

ibility test results yield a T equal to 11.0. This value is 22 percent

greater than the 9.05 value achieved for the finite element pressure test

configuration with a tie-rod release of H/1000 at the end of the final

dredging stage. As with the pressure tests, the finite element model

gives lower moments than Rowe's flexibility test results due to the dif­

ferent changes in moment with tie-rod release that each model displays.

lease for both his pressure and flexibility tests, while the finite element

models yield moment decreases with tie-rod release, the latter method

will have values of Tthat are less than Rowe's at any construction stage

where tie-rod release is incorporated.

tion due to sheet-pile flexibility are more conservative than those sug­

gested by the finite element model, since the latter analysis did not

find the breakdown of arching with anchor level displacement to increase

the sheet-pile moment. Also, field tie-rod level displacements are

greater than H/2400, the point at which Rowe says moment decrease

A simulation of Rowe's flexibility tests at the original 3.5-foot

Since Rowe (1952) observed moment increases with tie-rod re-

This comparison indicates that Rowe's values for moment reduc-

178

begins, thus substantiating the use of the smaller moment reduction

values obtained in the finite element model study.

CHAPTER 9

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

The following conclusions are made based on the results of this

study for a sheet pile with its tie rod located at the top of the pile:

1. The finite element method provides a means of analyzing com­

plex soil-structure systems, as exemplified by anchored sheet

piles.

2. The moments, displacements, and tie-rod forces obtained from

a monitored sheet pile in the field compare to those calculated

using a continuum idealized by using finite elements for the

sheet pile, tie rod, soil, and sheet pile—soil interface.

3. A mathematical theory is predicted to ascertain the behavior of

anchored bulkheads. The evaluation of this theory using a

finite element model supports its performance.

4. The stiffness of the soil support s surrounding a sheet pile can

be characterized by the expression s = K//H\ This term in­

cludes the important effect of system scale.

5. The moment in a sheet pile is dependent on the ratio of the soil

stiffness to the structural stiffness or the system stiffness ratio

S = (K//H) (H4/EI). Larger structural stiffnesses and smaller

soil stiffnesses cause the sheet pile to behave in a relatively

more rigid manner in that it receives more moment and tie-rod

179

180

force. The soil support stiffness is somewhat less important

than the structural stiffness.

A small-scale test prediction of field sheet-pile behavior over­

estimates the soil stiffness and results in unsafe moments.

Stiffer soils, those with large K//tT, produce less pressure on

the active side of the sheet-pile wall and a smaller, more flex­

ible moment pattern in the pile. Piles that are flexible relative

to the soil cause less pressure to be exerted on the active side

and also display less moment.

The sheet-pile movement at the tie-rod level during construction

that is necessary to develop a fully active zone is dependent

on the soil stiffness, with less movement being required for a

stronger soil.

The arching pressure acting on the active side of the sheet pile

near the tie-rod level is more dependent on the amount of dif­

ferential deflection along the pile than on soil strength or trans­

lation. Either local or general flexing occurring for low and

high log S values results in greater arching pressures.

Incorporating dredge level depth into the expression for soil

support stiffness yields the approximate expression s =

K/(l -tf)//H.

Sheet-pile translational displacements are proportional to

h1.30/k but are independent of sheet-pile flexibility. Flexing

or differential displacements are greater when the soil is weak­

er even though the pile behaves in a relatively more rigid man­

ner with respect to receiving more moment.

181

12. Decreases in Poisson's ratio slightly increase the moment in a

sheet pile.

13. Increases in material density significantly increase lateral pres­

sures and sheet-pile moments.

14. Decreases in moment and tie-rod force occur as the lateral

pressure decreases with imposed tie-rod level movement.

15. Tie-rod force at the end of construction varies as

and increases as the sheet pile becomes stiffer. Anchor-wall

displacements are equal to sheet-pile displacements, and

anchor-wall soil stiffness from a force viewpoint is therefore

proportional to Since this stiffness is in the range

of 25 to 75 k/ft for a 40-foot sheet pile, it is little influenced

by the tie-rod stiffness, which is in series combination. An­

chor system stiffness exerts much less influence on moment in

the pile than does the soil stiffness surrounding the sheet pile.

16. The effect of altering the construction sequence, backfilling

and dredging, on sheet-pile moments and displacements is

small.

17. The maximum sheet-pile moment at the end of construction for

the case of tie-rod movement occurring naturally during dredg­

ing is approximately equal to that for the case where the move­

ment is imposed in five stages after dredging is complete.

Recommendations for Further Research

Investigate the stress state in the soil, using plots of stress

trajectories, noting whether the area is subject to loading or unloading.

182

Ascertain the arching pattern, both horizontal and vertical, existing in

the active zone adjacent to the sheet pile.

Analyze a bulkhead system at failure and quantify the design

conditions of the system in terms of a factor of safety.

Study the influence on sheet-pile behavior of zones of various

sizes and materials surrounding the sheet pile.

Examine the effect of locating the anchor wall within the zone

of influence. Also investigate the optimum size of the anchor wall.

Characterize the influence of the tie-rod location parameter^

on the system stiffness and sheet-pile moment and deflection response.

Develop a relation between loading of the bulkhead system as

characterized by density and the response as characterized by sheet-

pile moment.

APPENDIX A

NOMENCLATURE

[A] element equilibrium array

A area

a inverse of initial tangent modulus

[B] element compatibility array

b inverse of asymptotic deviatoric stress value

c cohesion

D model height or scale distance

d rate of change of initial tangent Poisson's ratio with strain

E Young's modulus of elasticity

EI stiffness

Ei initial tangent Young's modulus

En normal stiffness for interface element

Es shear stiffness for interface element

Esj initial shear stiffness for interface element

Est tangent shear stiffness for interface element

Et tangent Young's modulus

Eur Young's modulus for unloading, reloading

{e] element, internal, local displacements

[F} element, internal, local forces

F rate of change of initial tangent Poisson's ratio with confining pressure

f initial tangent Poisson's ratio

183

184

G initial tangent Poisson's ratio at one atmosphere confining pressure

H sheet-pile height

I moment of inertia

[k J system stiffness array in global coordinate system

K soil modulus number

Ka stiffness of anchor system

Kaw stiffness of anchor wall

KAWS stiffness of anchor-wall soil support

Ka coefficient of active earth pressure

Kj interface modulus number

K0 coefficient of earth pressure at rest

Kp coefficient of passive earth pressure

Ks coefficient of earth pressure

Ktr stiffness of tie rod

Kur soil modulus number for unloading, reloading

[k ̂ system stiffness array in local coordinate system

k coefficient of subgrade reaction

L length

A constant of subgrade reaction

M moment

m soil stiffness modulus

n real number

[Pi system, external nodal forces in local coordinate system

fP'} system, external nodal forces in global coordinate system

Pa resultant force of active pressure distribution

185

Pp resultant force of passive pressure distribution

p pressure

pa active pressure

Pp passive pressure

Patm atmospheric pressure

q surcharge pressure

Rf failure ratio

5 element stiffness array

[s] system stiffness ratio, s/(l/p) or sp

SM section modulus

s soil support stiffness (K/i/H)

T tie-rod force, subscript h denotes horizontal component

t thickness

system, external nodal displacements in local coordinate system

system, external nodal displacements in global coordinate system

x distance quantity

y displacement

z distance

CL ratio of distance above dredge level to total height of sheet pile

ft ratio of distance above tie-rod level to total height of sheet pile

Y density

yw density of water

As relative shear displacement

6 deflection

186

€ strain; subscripts 1,2,3 denote principal strains or direction

P Poisson's ratio

Mi initial Poisson's ratio

tangent Poisson's ratio

W coordinate transformation array

P flexibility number, equals H^/EI, ft4/lb-in.2

G stress; subscripts 1,2,3 denote principal stresses or direction

r moment-height ratio, M/H3# in.-lb/ft^

Tf shear stress on failure plane

Ti interface shear stress

* angle of internal friction

angle of wall friction

APPENDIX B

PROGRAM SSI DOCUMENTATION

187

188

1. Program Identification

1.1 Program Title: Soil-Structure Interaction.

1.2 Program Code: SSI.

1.3 Writer: Robert L. Sogge.

1.4 Organization: Funding for the development of program SSI was pro­vided by the Department of Civil Engineering and Engineering Me­chanics, University of Arizona, Tucson, Arizona 8S721.

1.5 Date: First documentation, March 1974.

1.6 Updates, Versions: None. Version No.: 0.

1.7 Source Language: FORTRAN IV.

1.8 Availability: Listing provided in Appendix C from which card decks can be produced.

1.9 Abstract: Program computes the displacement, moments, and stresses in a soil-structure system using a nonlinear, plane strain finite element idealization.

2. Documentation

2.1 Narrative Description: Program SSI, "Soil-Structure Interaction," computes the displacements, moments, and stresses in a finite element system comprising soil and structural elements. Stress states initially due to gravity loads and stress states due to cut, fill, and surcharge conditions can be computed. The program pro­vides for complete analysis of an anchored bulkhead system by simulating initial horizontal sheet-pile displacements, the driving of the sheet pile, and the anchor wall.

Bar, beam, TRIM3, and interface elements are provided to model the structure, soil, and discontinuous behavior of the two at the interface. The TRIM3 elements model a nonlinear soil continuum, using equations proposed by Duncan and Chang (1970) and Kulhawv et al. (1969). A no tension state and elastic unload­ing are provided in the program.

2 . 2 Method of Solution: The program sets up and solves the equations of equilibrium, force-deformations, and geometric compatibility for

189

the continuum idealized as an assemblage of discrete elements. The theory has been presented in Chapter 3 of this study. The equations are solved using a banded symmetrical equation solver. Special fea­tures are explained in the text of this study.

2.3 Program Capabilities: As dimensioned, the program can accommo­date 196 elements, 127 nodes, having a total number of degrees of freedom of 261 with a bandwidth of 54. Also, 15 materials, 20 sur­charge loads, 55 support directions, 60 change elements (cut and fill), and 11 nodal points along any cut can be specified. This size will necessitate a storage capacity of 73,000 (octal).

The input units must be compatible with the units for the atmos­pheric pressure and density of water. Output units will be the same as input units.

2.4 Data Input: The data are input from punched cards according to the format shown.

Data Block 1 1 card

Column Format Variable Description

1-5 15 NNODES Number of nodes

6-10 15 NELEMS Number of elements

11-15 15 NMATLS Number of materials

16-20 15 NSUPTS Number of supports

Data Block 2 1 card

Column Format Variable Description

0

h

1 h F10.0 HB Height of bulkhead

Data Block 3 NNODES cards

Column Format Variable Description

1-5 15 NPT Node point number

6-15 F10.0 X(NPT) X coordinate of node NPT

16-25 F10.0 Y (NPT) Y coordinate of node NPT

Positive coordinate direction in to the right and down.

190

Data Block 4 NELEMS cards

Column Format Variable Description

1-5 15 J Element number

6-10 15 IP (J) P node of element J

11-15 15 10 (J) Q node of element J

16-20 15 IR(J) R node of element J

21-25 15 IS (J) S node of element J

26-30 15 NTYPE(J) Element type of element J

31-35 15

1 = bar 2 = beam 3 = TRIM3 4 = interface

MATYPEfl) Material type of element J

Elements must be numbered clockwise. Interface elements must have P,Q nodes and RfS nodes on the two long sides.

Data Block 5 1 card

Column Format Variable

1-10

11-20

21-30

31-40

41-50

51-60

110

F10.0

F10.0

F10.0

F10.0

F10.0

J

EMOD(J)

XXI (J)

AREA (J)

Description

Material type number

Linear elastic modulus of mate­rial J

Moment of inertia of material J (beam elements)

Cross-sectional area of material J (bar or beam elements)

SECMOD(J) Section modulus of material J (beam elements)

DENSTY(J) Density of material J (TRIM3 and INFACE elements)

191

Data Block 6 1 card

Column Format Variable

1-5 15 J

6-10 F5 .0 PHI (J)

11-15 F5 .0 COH(J)

16-20 F5 .0 RF(J)

21-25 F5 .0 FK(J)

26-30 F5 .0 FN (J)

31-35 F5 .0 FF (J)

36-40 F5 .0 GI(J)

41-45 F5 .0 DD(J)

46-50 F5 .0 FKUR(J)

Data Block 7 1 card

Column Format Variable

1-10 F10.0 PATM

11-20 F10.0 DWATER

Description

Material type number

Angle of internal friction $ of material J; angle of wall fric­tion if interface element

Cohesion c of material J

Failure ratio Rf of material J

Modulus number K of material J

Exponent n for stress-dependent modulus of material J

Rate of change of initial tangent Pois son ratio with confining pressure F of material J

Initial tangent Pois son's ratio at one atmospheric pressure G of material J

Rate of change of initial tangent Poisson's ratio with strain d of material J

Unloading, reloading modulus number Kur of material J

Description

Atmospheric pressure

Density of water

192

Data Block 8 1 card

Column Format Variable

1-10 110 ICSD

11-20 F10.0 STIFF

Description

Coordinate direction of added stiffness

Stiffness to be added in direction ICSD

Data Block 9 1 card

Column Format Variable

1-80 1615 NS(I)

Description

Coordinate direction number of support points

Data blocks 10 through 16 can be chosen by the user to model the desired construction sequence. Data block 17 terminates the program when used after any of these data blocks.

Key to Data Blocks 10 through 17

Data Block 10: INDEX = 0--initial stress state

Data Block 11: INDEX = -2 —drive sheet pile

Data Block 12: INDEX = -1 —initial horizontal displacement of sheet pile

Data Block 13: INDEX = 1--backfill

Data Block 14: INDEX = 2--dredge

Data Block 15: INDEX = 3--surcharge (and/or support dis­placements)

Data Block 16: INDEX = 4--tie-rod release

Data Block 17: INDEX = 10 —ends program

193

Data Block 10 INDEX=0, Initial stress state

Column Format Variable

Card 1

Card 2

Card 3

1-5

1-5

6-80

1-5

15

15

15F5.0

15

INDEX

NINT

PLC (I)

NCE Cards 4 through 3 + (NCE/16)

1-80 1615 NEC (I)

Description

0 = initial stress state

Number of intervals in which load or displacement is analyzed

Cumulative percentage of total load or displacement to be applied at end of Ith interval

Number of changing elements

Element numbers of changing elements, i.e., elements not to be included in analysis

Data Block 11 INDEX=-2, Drive sheet pile

Card 1

Card 2

Column Format Variable

INDEX

NINT

1-5

1-5

15

15

6-80 15F5.0 PLC (1)

Card 3 1-5 15 NCE

Cards 4 through 3 + (NCE/16)

1-80 1615 NEC(I)

Card 4 + (NCE/16) 1-5 15 NTB

6-10 15 NBB

Card 5 + (NCE/16)

1-10 F10.0 BDVI

Description

-2 = drive sheet pile

Number of intervals in which load or displacement is analyzed

Cumulative percentage of total load or displacement to be applied at end of Ith interval

Number of changing elements

Element numbers of changing elements

Node at top of bulkhead

Node at bottom of bulkhead

Vertical bulkhead displacement

194

Data Block 12 INDEX=-1, Initial horizontal displacement of sheet pile

Column Format Variable

Card 1

Card 2

Card 3

Card 4

1-5

1-5

6-80

1-5

15 INDEX

15 NINT

15F5.0 PLC (I)

15 NDISP

Description

-1 = initial horizontal displace­ment of sheet pile

Number of intervals in which load or displacement is analyzed

Cumulative percentage of total load or displacement to be applied at end of Ith interval

Number of initial horizontal sheet-pile displacements due to driving

1-10 110 NODE Sheet-pile node number 11-20 F10.0 AMOUNT Initial horizontal displacement of

node due to driving

Input data for variables of Card 4 NDISP times.

Data Block 13 INDEX=1, Backfill

Column Format Variable

Card 1

Card 2

Card 3

Card 4

1-5

1-5

6-80

1-5

1-5

15 INDEX

15 NINT

15F5.0 PLC 0)

15

15

NINC

NCE Cards 5 through 4 + (NCE/16)

1-80 1615 NEC (I)

Card 5 + (NCE/16)

1-5 15

Description

LEEN

1 = backfill

Number of intervals in which load or displacement is analyzed

Cumulative percentage of total load or displacement to be applied at end of Ith interval

Number of layers into which sequence is broken

Number of changing elements

Element numbers of changing elements

Ending element number of backfill

Input data for variable LEEN NINC times.

195

Data Block 14 INDEX=2, Dredge

Column Format Variable

Card 1

Description

Card 2

Card 3

Card 4

1-5

1-5

6-80

1-5

1-5

15

15

15F5.0

15

15

INDEX

NINT

PLC (I)

NINC

NCE Cards 5 through 4 + (NCE/16)

1-80 1615 NEC 0)

2 = dredge

Number of intervals in which load or displacement is analyzed

Cumulative percentage of total load or displacement to be applied at end of Ith interval

Number of layers into which sequence is broken

Number of changing elements

Element numbers of changing elements

Ending element number of backfill

Number of nodal points along cut

Card 5 + (NCE/16) 1-5 15 LEEN

Card 6 + (NCE/16) 1-5 15 NNPCUT

Cards 7 + (NCE/16) through 6 + (NCE/16) + (NNPCUT/16) 1-80 1615 NPCUT(I) Node number of nodal points

along cut

Input data for LEEN, NNPCUT, and NPCUTfl) NINC times.

Data Block 15 INDEX=3, Surcharge (and/or support displacements)

Column Format Card 1

Card 2

Card 3

Card 4

1-5

1-5

6-80

1-5

1-10

15

15

15F5.0

15

110

Variable

INDEX

NINT

PLC 0)

NLOADS

LP (I)

11-20 F10.0 PT(I)

Description

3 = surcharge (and/or support displacements)

Number of intervals in which load or displacement is analyzed

Cumulative percentage of total load or displacement to be applied at end of Ith interval

Number of surcharge loads

Coordinate direction number of load I

Load I magnitude

Input data for LP(I) and PT(I) NLOADS times.

196

Support displacements can be imposed by specifying the coor­dinate direction of the displacement as a support and inputting the mag­nitude of the displacement as a load in the direction of displacement.

Data Block 16 INDEX=4, Tie-rod release

Card 1

Card 2

Card 3

Column Format Variable

1-5 15 INDEX

1-5 15 NINT

6-80 15F5.0 PLC (I)

1-10 F10.0 TDI

Description

4 = tie-rod release

Number of intervals in which load or displacement is analyzed

Cumulative percentage of total load or displacement to be applied at end of Ith interval

Imposed tie-rod level displace­ment

Data Block 17 INDEX = 10, Ends program 1 card

Column Format Variable Description

1-5 15 INDEX 10 = ends program

2 . 5 P r o g r a m O p t i o n s : T h e p r o g r a m t e r m i n a t e s u p o n t h e r e c e i p t o f a value of 10 for IND2X. No debugging options have been provided.

j 2 . 6 P r i n t e d O u t p u t : N o p r o v i s i o n i s m a d e f o r r e d u c i n g p r i n t e d o u t p u t

except by modifying program.

2 . 7 O t h e r O u t p u t s : N o n e

2 . 8 F l o w C h a r t : P r e s e n t e d i n F i g . B - l .

2 . 9 S a m p l e R u n : D u e t o l e n g t h o f p r o g r a m , n o s a m p l e r u n h a s b e e n p r o ­vided. A check can be made by using the grid presented in Fig. 6-3 and the input data presented in Table 6-3 of this study to arrive at the sheet-pile moments in Fig. 6-1.

3. System Documentation

3 . 1 C o m p u t e r E q u i p m e n t : P r o g r a m S S I w a s d e v e l o p e d o n a C D C 6 4 0 0 computer.

3 . 2 P e r i p h e r a l E q u i p m e n t : C a r d r e a d e r , l i n e p r i n t e r .

3 . 3 S o u r c e P r o g r a m : T h e s o u r c e l i s t i n g o f p r o g r a m S S I i s p r e s e n t e d i n Appendix C.

3 . 4 V a r i a b l e s a n d S u b r o u t i n e s : P r o g r a m S S I c o n s i s t s o f t h e f o l l o w i n g parts. The variables in it, other than the input variables are not defined due to the length of the program. All output variables are generally defined in the following program description.

197

«• < t

SSI

r BAR — BEAM - TRIM3 -i L- INFACE -J

NPASS = 1

NPASS

1 STORE

I EQSOL

STRBAR STRESB STREST

\-¥ STRESI

NPASS = 3

Fig. B-l. Sequential Flow Chart

All calling is done from the main program, SSI, thus all sub­routines return back to the main program before proceeding to the next subroutine.

198

Main Program BLKBW4

Reads and writes data. Modifies coordinate numbering. Com­putes the reduced coordinate degrees of freedom. Initializes arrays. Deactivates or activates element, depending on whether dredge or backfill sequence, and monitors layer information on the sequence. Populates the load vector. Calls all subroutines. Subtracts nodal loads equivalent to stresses in removed layer. Imposes initial hor­izontal bulkhead displacements resulting from driving. Adds stiff­ness in specified coordinate direction. Eliminates body force when support or support displacements. Simulates vertical driving of sheet pile. Computes fraction of total loads to be applied during load interval. Outputs applied nodal loads. Applies boundary con­ditions. Decouples deactivated nodes. Outputs nodal displace­ments. Performs nodal equilibrium check and outputs nodal forces.

Subroutine BAR

Computes a bar element stiffness matrix and the force-d i s p l a c e m e n t a r r a y .

Subroutine BEAM

Computes a beam element stiffness matrix and the force-displacement array.

Subroutine TRIM3

Computes a TRIM3 element stiffness matrix, the stress-displacement array, and the compatibility array. Loads one-third of the body forces in the vertical direction at each of the element's nodes, if an initial stress state or backfill sequence is specified.

Subroutine INFACE

Computes an INFACE element stiffness matrix, the stress-displacement array at the centroid of the element, and the compat­ibility array.

Subroutine STORE

Stores the components of the element stiffness arrays in the proper location in the system stiffness array. This latter array is stored as a narrow bandwidth rectangle of width NBW and length NODOF and contains only elements from the symmetrical top half of the system stiffness array.

Subroutine EQSOL

Solves the NODOF simultaneous equations of bandwidth NBW by using banded Gauss elimination, yielding the nodal displace­ments .

Subroutine STRBAR

Determines and outputs the axial force, stress, and strain in a bar element. Computes the nodal loads for the equilibrium check on the solution.

Subroutine STRESB

Determines and outputs the moments, axial force, shear and bending stresses at the ends of a beam element. Computes the nodal loads for the equilibrium check on the solution.

Subroutine STREST

Determines and outputs the stresses and strains in the global orientation and the principal stresses. Computes the nodal loads for the equilibrium check on the solution. Calculates the nodal loads necessary for a dredge or cut sequence.

Subroutine STRESI

Determines and outputs the tangential and normal stresses and strains in the local element orientation. Computes the nodal loads for the equilibrium check on the solution. Calculates the nodal loads necessary for a dredge or cut sequence.

3 . 5 D a t a F i l e s : N o f i l e s a r e c r e a t e d o r r e a d b y t h i s p r o g r a m .

3 . 6 S t o r a g e R e q u i r e m e n t s : T h e p r o g r a m r e q u i r e s 7 3 , 0 0 0 ( o c t a l ) s t o r a g e as presently dimensioned. Of this amount 44,000 (octal) is blank common storage.

3 . 7 M a i n t e n a n c e a n d U p d a t e s : N o n e t o d a t e . P r o v i d e d b y a u t h o r , a s needed.

200

4 . 1 O p e r a t o r I n s t r u c t i o n s : P r o g r a m S S I i s r e a d f r o m c a r d s p u n c h e d i n IBM 026 language or from a compiled binary deck. The latter form is preferred due to the compilation time of 24 decimal seconds.

4 . 2 O p e r a t i n g M e s s a g e s : N o r m a l s y s t e m m e s s a g e s o n l y .

4 . 3 C o n t r o l C a r d s : S t a n d a r d C D C c o n t r o l c a r d s .

4 . 4 E r r o r R e c o v e r y : P r o g r a m m u s t b e r e s t a r t e d o n e r r o r .

4 . 5 R u n T i m e : T h e r u n t i m e d e p e n d s o n t h e n u m b e r o f I N D E X s e q u e n c e s , load intervals, etc. For a typical case of one initial horizontal sheet-pile displacement sequence, initial stress state, three pile driving intervals, one backfill and four dredge layers of one load interval each and two surcharge intervals, the central processing execution time is 130 seconds on a CDC 6400, excluding compila­tion time. The data for the above problem consists of 181 elements, 121 nodes, having 253 total number of degrees of freedom, and a b a n d w i d t h o f 4 6 .

APPENDIX C

LISTING OF PROGRAM SSI

201

PRGGRAM SSI SSI 1 i(INPUT,OUTPUT,TAPE5=INPUT,TAPE6=OUTPUT) SSI 2 COMMON NUM,NT,MT,NPASS,IPNF,IQNF,IRNF,ISNF,NIT,NINT, SSI 3

2XP,XQ,XR,YP,YQ,YR,S(8,8) ,B(3,6),D(8,8),ESB(8,3 ),DEF(8), SSI k 3A,H3,INDEX,PATM,QWATER SSI 5 COMMON EMOD(15>,XXI(15).AREA(15),SECMOO(15).DENSTY(15)« SSI 6

2PRATI 0(15) ,PHI (15) , COH (15) , RF (15 ) , FK(15) ,FN(15), SSI 7 3FF (15 ) ,G1<15 ) , DO (15 ) ,FKUR(15 ) • SSI 8 4SS<261.54),P(261),PTEMP(261),DISP(261),X1127),Y(127),NNF(127), SSI 9 5NTYPS(196),MATYPEf196),IP(196),IQ(196),IR(196)*IS(196)* SSI 10 6STr?ESS(3,19o) , NVAE (196 ) , STRMAX(196) , SSI 11 7LP (20),PT(20),NS(55),PCKI(261) SSI 12 DIMENSION PLC(15),PTOT (261),NVSE(196),NEC(60),NPCUT(11) SSI 13

100 FORMAT (1615) SSI 1<»

101 FORMAT (110, 7F10.0) SSI 15 102 FORMAT ( I5.15F5.0) SSI 16 10 3 FORMAT ( 8F10 .0 ) SSI 17 10U FORMAT (I5,2F10.0) SSI 18

1 CONTINUE SSI 19 READ (5,100) NNOD£S,NELEMS»NMATLS,NSUPTS SSI 20 IF (NNOOES.EQ. 0) GO TO 9999 SSI 21 WRITE (6,25) SSI 22

25 FORMAT (1H1,3X,*N0 NODES NO ELEMS NO MATLS NO SUPTS*) .SSI 23 WRITE (6,27) N NODES ,NELEMS,NMATLS, NSUPTS SSI 2<»

27 FOR"AT (*• (>*X , I 5) ) SSI 25 READ (5,103) HB SSI 26 WRITE (6,203) HB SSI 27

208 FORMAT (//4X,*HEIGHT OF BULKHEAD =*,F10.3) SSI 28 WRITE (6,29) SSI 29

29 FORMAT (//*X,*NODE POINT X-COORDINATE Y-COORDINATE*) SSI 30 DO 5 1 = 1,NNODES SSI 31 READ (5,10^) NPT,X(NPT),Y(NPT) SSI 32 X(NPT)=X(NPT) SSI 33 Y ( NPT ) = Y ( NPT) SSI 3k

5 WRITE (6,33) NPT,X(NPT),Y(NPT) SSI 35 33 FORMAT (5X, 15, 6X, F10.2, 5X, F10.2) SSI 36

WRITE (6,35) SSI 37 35 FORMAT (//*»X,*ELE NO P NOOE Q NODE R NODE S NODE N TYPE MATLSSI 38

2 TYPE*) SSI 39 00 7 I=1,NELEMS SSI to REAO (5,100) J,IP(J),IQ(J),IR(J)* IS(J),NTYPE(J)» MATYPE CJ) SSI kl

7 WRITE 16,39) J,IP(J),IQ(J),IR(J),IS(J),NTYPE(J),MATYPE(J) SSI <•2 39 FORMAT (<»X , 15, <« (3X , 15) ,2X , 15, tX, 15) SSI t»3

WRITE (6,t«*0) SSI kt* **«t0 FORMAT (/MX,*MATL TYP*,2X,*ELASTIC MOO*, «*X , *MOM INRTA*, 8X»*AREA* ,SSI kS

1 UX,*SECT M0D*,5X,*DENSITY*) SSI t* 6 DO 8 I=1,NMATLS SSI U7 READ (5,101) J,EMOD(J),XXI(Jl,AREA(J),SECMOO(J),DENSTY(J) SSI t*a

8 WRITE(6, <^2) J,£MOD(J),XXI(J) , AREA (J) , SECMOD( J), DEN STY (J) SSI 1*9 kkZ FORMAT ('•X^S^X.ElO.ij^X^lO.^.SX.Eg.S^X^g.a.SX.Eg, 3) SSI 50

WRITE (6,^6) SSI 51 ^6 FO-IMAT (//7X, *KT* , <*X , *PHI *, <*X, *COH* ,5X,*RF* ,5X , *FK*,5X ,*FN* ,5 X, SSI 52

1*FF*,5X,*G1*,5X,*D0*,3X,*FKUR*) SSI 53 00 10 1=1,NMATLS SSI 5 REAO (5,102) J,PHI(J),COH(J),RF(J),FK(J),FN(Jl,FF(J),Gi<J),00(J) SSI 55 2 ,FKUR(J) SSI 56

10 WRITE (6,^8) J,PHI (J) , COH (J ) , RF (J),FK(J) ,FN(J),FF(J) ,G1(J),DD(J) SSI 57

2 ,FKUR(J) SSI 58

8 FORMAT (t»X,I5,3X,F'».l,2X,F5.3,2X,F5.3,lX,F6.0,3X,F«»«2*2X,F5.3,2X, SSI 59

1F5.3,3X,F^.2,1X,F6.0) SSI 60

REAO (5,103) PATM,DWA TER SSI 61

WRITE (6,1*50) PATM,OWATER SSI 62

450 FORMAT (//^X,*ATMOSPHERIC PRESSURE =*,F15.5,2X,*WATER DENSITY = *• SSI 63

2 F15.5) SSI 6t»

READ (5,101) ICSD,STIFF SSI 65

WRITE (6,33) ICSO,STIFF SSI 66

38 FORMAT (//<»X,*COORO STIFF DIRECTION =*,15,2X,*STIFFNESS =»,F10.3) SSI 67

WRITE (6,^5) SSI 68

i»5 FORMAT (//*»X , *COORDINATE NO. OF POINTS SUPPORTED*) SSI 69

READ (5,100) (NS(I),1=1,NSUPTS) SSI 70

WRITE(6,^7) (NS(I), 1=1, NSUPTS) SSI 71

1*7 FORMAT (16 (IX«!<»)) SSI 72

•TOTAL NUMBER OF OEGREES OF FREEOOM SSI 73

NOOOF= 0 SSI 7 «•

MN0FPN=2 SSI 75

DO 126 NK=1, NNOOES SSI 76

00 122 NE=1,NELEMS SSI 77 NT=N T YPE <NE) SSI 78 NNF(NN)=N0D0F SSI 79

GO TO (122,12^,122,122,12^), NT SSI 80

I Z k IF (NN.NE.IP(NE).ANO.NN.NE.IQCNE)) GO TO 122 SSI 81 MNDFPN=3 SSI 82

NOOJF=NOOOF+3 SSI 83

GO TO 126 SSI 8<*

122 CONTINUE SSI 85

N0T0F=N0D0F*2 SSI 86

126 CONTINUE SSI 87

NODE SEPARATION ANO BANOWIOTH SSI 88

NOW= 1 SSI 89 N00SFP=1 SSI 90

DO 131 1=1,NELEMS SSI 91

NT=NTYPE(I) SSI 92 IPP=IP (I) SSI 93 IQQ=IQ(I) SSI 91* NG=MAX 0 (IPP,IGQI SSI 95 NL=MINO (IPP,ICQ) SSI 96

IF (NT.LT. 3.0R.NT.ECU5) GO TO 130 SSI 97

IRR=IR(I) SSI 98 NG-1AX0 (NG,IRR) SSI 99

NL=HINO (NL,IRR) SSI 100

IF (NT . NE• *•) GO TO 130 SSI 101 ISS=IS(I» SSI 102 NG=MAX 0 (NG,ISS) SSI 103

NL=MINO (NL,ISS) SSI 10U

139 NNFG=NOOOF SSI 105 IF (NG.NE.NNODES) NNFG=NNF(NG+1) SSI 106

ITN3W=NNFG-NNF(NL) SSI 107 ITNS=NG-NL SSI 108

IF (ITK8W.GT.NeW) NBW=ITNBW SSI 109 IF (ITNS.GT.NOOSEP) NODSEP=ITNS SSI 110

131 CONTINUE SSI 111 WRITE <6,132) N00CF,N8W,N00SEP,MNDFPN SSI 112

132 FORMAT (1H1,3X,*N0 DEG OF FRFQ =*,15,3X,*BANDWI0TH =•, I*»,3X .•NODE SSI 113 2SEP = * , 13,3X,*MAX NODE OOF =*,121 SSI lit*

WRITE (6,133) SSI 115

133 FORMAT (//i»X,»VALUES OF NNF (I) *) SSI 116 WRITE (6,i»7) (NNF(I) , 1= 1, NNOCES) SSI 117

C***»*CONSECUTIVE COORDINATE NUMBERING MODIFICATION SSI 118 00 1^0 K=1,NSUPTS SSI 119 N=(NS(K)-1)/MNQFPN+1 SSI 120

lttQ NS(K)= NNF(N)+NS(K)-N*MNDFPN+MNDFPN SSI 121 N=(ICSO-l)/MNDFPN+1 SSI 122 ICSD=NNF(N)+ICSD-N*MNOFPN+MNDFPN SSI 123

•INITIALIZE VALUES SSI 12^ UO 55 1=1,NODCF SSI 125 PTEMP(I) = 0 .0 SSI 126 PCKI(I)=0.0 SSI 127 PTOT(I)=0.0 SSI 128

55 OISP(I)=0.0 SSI 129 00 56 1=1,NELEMS SSI 130 NVSTCI>=0 SSI 131 STRMAX(I) = 0•0 SSI 132 DO "6 K=1,3 SSI 133

56 STRESS(K,I)=0.0 SSI 13U NST=NSU PTS SSI 135

801 CONTINUE SSI 136

READ (5,100) INDEX SSI 137 WRITE (6,160) INDEX SSI 138

160 FORMAT (1H1,oX,*INDEX =*,I3> SSI 139 IF (INOEX.EQ.1C) GO TO 9999 SSI 1<»0 READ (5,102) NINT,(PLC(I),I=1,NINT> SSI 1<*1

WRITE (6,53) SSI 1U2 53 FORrlAT ( / /<»X,*N0 LOAO INTVLS FRACTIONS OF TOTAL LOAD*) SSI 1U3

WRITE (6,5^) NINT,IPLC(I),I=1,NINTI SSI !<•<•

SW FORMAT (7X,I5,6X,15C2X,F5.3)) IF (INDEX.LT.l) GO TO 803 IF (INDEX.GE.3) GO TO 8*t0 REAO (5,100) NINC WRITE (6,<*8) NINC

it8 FOR 1AT i//UXt*NO OF INCREMENTS (LAYERS) = *,Hf> C*****CONSIDERING DEACTIVATED ELEMENTS

803 CONTINUE RE AO (5,100) NCE WRITE (6,80^) NCE

80*» FORMAT (/i*Xt*NO OF CHANGING ELEMENTS =*,15) READ (5,100) (NEC(I),1=1,NCE) WRITE (6,806)

806 FORMAT (/*»X ,* ELEMENT NOS OF ELEMENTS CHANGING*) WRITE (6,<*7) (NEC(I),1=1,NCE) IF (INDEX.EQ.2) GO TO 812 DO 81<* 1 = 1,NCE J=NEC (I)

81<» NVSE(J)=1 812 CONTINUE

NLAY=0 L EEN = 0

C*****ANALYZE FOR NUMBER OF INCREMENTAL LAYERS***** 800 NLAY=NLAY*1

IF (INDEX.NE.l.AND.INDEX*NE»2) GO TO 8^0 WRITE (6,862) NLAY

862 FORMAT C*X,*LAYER NUMBER =»,I3)

C*****NEW LAYER INFORMATION LBEN=LtEN+1 READ (5,100) LEEN WRITE (6,818) LEEN

818 FORMAT (//4X,*LIFT ENDING ELEMENT (OF NEC(I)) NO =*,I5> IF (INDEX.NE.2) GO TO 83b READ (5,100) NNPCUT WRITE (6,820) NNPCUT SSI 179 g

820 FORMAT t/«*X,*NO OF NODAL POINTS ALONG CUT =*,15) SSI 180 CT>

SSI lk5 SSI 1^6 SSI 11*7 SSI 11*8 SSI 1<*9 SSI 150 SSI 151 SSI 152 SSI 153 SSI 15<* ssr 155 SSI 156 SSI 157 SSI 158 SSI 159 SSI 160 SSI 161 SSI 162 SSI 163 SSI 16<* SSI 165 SSI 166 SSI 167 SSI 168 SSI 169 SSI 170 SSI 171 SSI 172 SSI 173 SSI 17 <• SSI 175 SSI 176 SSI 177 SSI 178 SSI 179 SSI 180

READ (5,100) (NPCUT(I),I=1,NNPCUT> WRITE (6,322)

822 FORMAT (/4X,*NCDE NO OF NOOAL POINTS WRITE (6,47) (NPCUT(I),I=1,NNPCUT) 00 828 1=1,NELEMS

828 NVAE(I)=0 DO 830 I=L3EN,LEEN J=NrC(11 NVAE(J)=1

830 NVSE(J)=1 GO TO 340

834 00 838 1=1,NELEMS 838 NV AE (I ) =0

DO 336 I=LBEN,LtEN J = N P C ( I ) NVAE(J)=1

836 NVSE(J)=0 840 CONTINUE

C»***•POPULATE THE LCAJ VECTOR, P(I) DO 60 I = 1, NOOOF

60 P(I> = 0.0 IF (IN0EX.NE.3) GO TO 63 READ (5,100) NLOADS WRITE (6,52) NLOADS

52 FORMAT (//4X,*NO OF LOADS =*,151 WRITE (6,49)

49 FORMAT 1//4X,*C00R0INATE NO. OF LOAD DO 11 I = 1, NLOADS RE AO(5,101) LP(I)* PT(I) .

11 WRITE(6,51) LP(I), PT(I) 51 FORMAT (12X,I5,13X,F10.3)

DO 142 K=l,NLOADS ,N=<LP(K)-1)/MNDFPN+1

142 LP(K)=NNF(N)+LF(K)-N*MNOFPN+MNOFPN DO 62 1=1,NLOACS J = LP(I)

SSI 181 SSI 182 SSI 183 SSI 184 SSI 185 SSI 186 SSI 187 SSI 188 SSI 189 SSI 190 SSI 191 SSI 192 SSI 193 SSI 194 SSI 195 SSI 196 SSI 197 SSI 198 SSI 199 SSI 200 SSI 201 SSI 202 SSI 20 3 SSI 204 SSI 205 SSI 206 SSI 207 SSI 208 SSI 209 SSI 210 SSI 2ll SSI 212 SSI 213 SSI 214 SSI 215 SSI 216

62 P(J) = PT(I) SSI 217

63 CONTINUE SSI 218 C**»**C0KPUTE NODAL LOADS EQUIVALENT TO STRESSES IN REMOVED LAYER SSI 219

IF (INDEX.NE.21 GO TO 882 SSI 220 DO 867 1=1 ,NODCF SSI 221

86 7 SS(1,1>=Q.0 SSI 222 NPASS=3 SSI 223 DO 868 NUM = itNELEMS SSI 224 IF (NVAECNUH).NE.l) GO TO 868 SSI 225 NT = NTYPE (NUM) SSI 226 GO TO (868,868,865*866) NT SSI 227

865 CALL TRIM3 SSI 228 CALL STRFST SSI 229

GO TO 863 SSI 230

866 CALL INFACE SSI 231

CALL STRESI SSI 232

863 CONTINUE SSI 233 C»»**»SUGTRACT NODAL LOADS EQUIVALENT TO STRESSES IN REMOVED LAYER SSI 234

DO 872 1=1,NNPCUT SSI 235

J=NPCU T(I) SSI 236 K=NNF(J)+1 SSI 237 P(K)= SS (K , 1) SSI 238

872 P(K+1>= SS(K+1,1) SSI 239

882 CONTINUE SSI 240 NIT = 0 SSI 241

€•••*•ANALYZE FOR NUMBER OF ITERATIONS SSI 242

65 NIT=NIT«-1 SSI 243

WRITE (6,864) NIT SSI 244

864 FORMAT (4X,*ITERATI0N NO =* • 13) SSI 245 DO 64 I = 1, NCDOF SSI 246

DO 64 J=1,NBW SSI 247

64 SS(I,J) = 0.0 SSI 248 NPASS = 1 SSI 249

DO 66 NUM=1»NELEMS SSI 250

IF (NVSE(NUM).KE.O) GO TO 66 SSI 251 MT=NTYPE(NUM) SSI 252

GO TO (91,92,92,9^1 NT SSI 253

91 CALL BAR SSI 25 k GO TO 96 SSI 255

92 CALL BEAM SSI 256 GO TO 96 SSI 257

93 CALL TRIM3 SSI 258 GO TO 96 SSI 259

9k CALL INFACE SSI 260 96 CALL STORE SSI 261 66 CONTINUE SSI 262

C*****ADD STIFFNESS IN SPECIFIED COORDINATE DIRECTION SSI 263 IF (IN3EX.LT.1) GO TO 69 SSI ZbU SS(ICSO,l)=SS(ICSD,l) • STIFF SSI 265

69 CONTINUE SSI 266 C*#*»*ELIMINATING BOOY FORCE WHEN SUPPORT OR SUPPORT DISP SSI 267

IF (INOEX.LT.O.OR.INOEX.GT.3) GO TO 71 SSI 268 IF (INDEX.NE.3) GC TO 67 SSI 269 DO 68 1 = 1, NSUPTS SSI 270 NSI=NS(I) SSI 271 P(NSI>=0.0 SSI 272 DO 68 J = l, NLOA CS SSI 273 LPJ=LP(J) SSI 2 7i*

68 IF (NSI.EQ.LPJ) P(LPJ)=PT(J) SSI 275 GO TO 71 SSI 276

67 DO 70 1=1,NSUPTS SSI 277 NSI=NS(I) SSI 278

70 P(NSI)=0.0 SSI 279 71 CONTINUE SSI 280

IF (NIT.NE.1) GO TO 281 SSI 281 C»,*,»IMP0S£0 VERTICAL BULKHEAD DISPLACEMENT SSI 282

IF (INDEX.NE.-2) GO TO 920 SSI 283 READ (5,100) NTB,N8B SSI 28<* WRITE (6,922) NT3,NBB SSI 285

922 FORMAT (//*»X,*NQD£ AT TOP OF BULKHEAD =»,I5,2X,*NODE AT BOTTOM OF SSI 286

2 BULKHEAD =*,15) SSI 287

RFAO (5.103) BOVI SSI 288

WRITE (6*923) eDVI SSI 289

923 FORMAT (//*»X,* IMPOSED VERT BULKHEAD DISP =*,F10.6) SSI 290 J=NNF(KTB)+2 SSI 291 P(J) =(1DVI SSI 292 NSUPTS=MSUPTS+3 SSI 293 NS(NSUPTS-2)=J SSI 29«*

NS (NSJPTS-1)=J-1 SSI 295

NS(NSUPTS) =NNF(NBB)«-1 SSI 296 923 CONTINUE SSI 297

C»»***IMP0SE IMXIAL HOrtZ BULKHEAD DISPLACEMENTS SSI 298

IF (INOEX.NE.-l) GO TO 167 SSI 299 161 REAO (5,100) NOISP SSI 300

WRITE (6,162) NOISP SSI 301

162 FOR -1AT (// *• X , * N 0 OF INDUCED 1 HORZ BULKHEAD DISPLACEMENTS =*,15) SSI 302 NSUPTS=NST+NOISP+l SSI 303 DO 166 I=1,NDISP SSI 30<*

REAO (5,101) NODE,AMOUNT SSI 305 N= NNF ( N'OCE ) +1 SSI 306 NS(NST+I)=N SSI 307

166 P(N)=AMOUN T SSI 308 NS(NSUPTS)=N*1 SSI 309

167 CONTINUE SSI 310 C*****IMPOSED TIEROD DISPLACEMENTS SSI 311

IF (INDEX.NE.«t) GO TO 9U0 SSI 312 READ (5,103) TDI SSI 313 WRITE(6,932) TDI SSI 31*»

932 FORMAT (//**X,*IMPOSED TIEROD DISPLACEMENT =*,F10.6> SSI 315 P( ICSD) =TD I SSI 316

9**0 CONTINUE SSI 317 C»»***COM^UTE FRACTION OF LOAD FOR THIS ITERATION SSI 318

DO 280 1=1,NODOF SSI 319

280 PTEMP(I)=P(I)*FLC(1) SSI 320 GO TO 281 SSI 321

281 DO 282 1=1,NOOCF SSI 322

282 PTEMP(I)=P(I)*(PLC(NIT)-PLC C NIT-1)) SSI 323 283 CONTINUE SSI 32^

DO 600 I=l,NODCF SSI 325 600 PTOT(I)=FTOT(Il+PTEMPCI) SSI 326 602 CONTINUE SSI 327

WRITE (6,7«f) SSI 328 7t* FORMAT (1H0 ,<fX ,*N00E*,12X,*H0RZ LOAO * »12X»*VERT LOAD** 15X»*MOMENT*)SSI 329

DO 75 I = l» NNODES SSI 333 IF (I.EQ.NNODES) GO TO 72 SSI 331 NDFPN=NNF(I+1)-NNF(I) SSI 332 GO TO 73 SSI 333

72 NDFPN=NODOF-NNF(I) SSI 33<t 73 J=NNF{I) + 1 SSI 335

IF (N0FPN.EQ.2) GO TO 77 SSI 336 WRITE <6,12) I,PTEHP(J),PTEMPCJ*1) , PTEMP(J+2) SSI 337 GO TO 75 SSI 338

77 WRITE (6*82) I»PTEMP(J)»PT£MP(J+i) SSI 3 39 75 CONTINUE SSI 3<t0

C*****flPPLY 30UNDARY CONDITIONS - SUPPORT DISPLACEMENTS SSI 3«*1 DO 78 I=l» NSUPTS SSI 3U2 J=NS(II SSI 3k3 SSIJ,l) = (SS(J,l)+1.0)*i.0E'»0 SSI 3 kk

79 PTEMP(J)=PTEMP(J)*SS(JtH SSI 3k5 C J M M M f * D E C 0 ( J p L I N G DEACTIVATED nodes SSI 3<*6

DO 8^2 1=1•NODOF SSI 31*7 IF (A9S(SS(I,1)),GT.1.0E-10) GO TO 8t»2 SSI 3i*8 NR=I SSI 31*9 DO 8 t*i* K=2.N8W SSI 350 SS(I»K)=0.0 SSI 351 IF (NR.LE.l) GO TO 8^ SSI 352 NR=NR-1 SSI 353 SS(NR,K) =0.0 SSI 35*»

8<»<» CONTINUE SSI 355 SS(Iti)=1.0 SSI 356 PTEMP(I)=0•0 SSI 357

8 U?. CONTINUE SSI 358 WRITE (6,862) NLAY SSI 359 WRITE (6,86«») NIT SSI 360

CALL EQSOL (SS,PTEMP,NODOF, NBW) SSI 361 DO 286 1=1 *NODCF SSI 36 2

286 DISP(I)=DISP(I)•PTEMP(I) SSI 363

287 WRITE 16,80) SSI 361*

80 FORMAT ( 1H0 , <4X ,*NOOE*, 6X, *HORZ DEFLECTION** 6X,*VERT DEFLECTION*, SSI 365 113X,*R0TATI0N») SSI 366 OO ««• I = 1» NNODES SSI 367

IF (1.EQ.NNODES) GO TO 83 SSI 368 NOFPN-NNF(I + l)-NNF (I) SSI 369

GO TO 85 SSI 370

83 NDFPN=NCHOF-NNF CI) SSI 371 85 J=NNF{I)+1 SSI 372

IF (NDFPN.EQ.2) GO TO 81 SSI 373 WRITE (6*8 2) I,OISP(J>,OISP(J4i>,OISP(J+2) SSI 37 k

82 FORMAT UX tI5 , 3( 2X,F19. 8) ) SSI 375 GO TO 8<» SSI 376

81 WRITE (6,82) I*DISP(J),DISP(J+l) SSI 377 8<* CONTINUE SSI 378

173 WRITE (6,87) SSI 379

87 FORMAT (///,UX ,*ELE NO*,5X,*AVG-X*,5X,*AVG-Y*»2X,*STRESS-X*12X» SSI 380 2*STRESS-Y*,IX,*STRESS-XY*,2X,*STRESS-Z*, 2X, *STRAIN-X*,2X,*STRAIN-YSSI 381 3*,1X,*STRAIN-XY*,3X,*SIGMA-I*,2X,*SIGMA-II* ,1X,*MAX -SHEAR*) SSI 382

86 CONTINUE SSI 38 3 NPASS=2 SSI 38«»

OO 605 1=1,NODOF SSI 385 605 SS(I,1)=0.0 SSI 386

OO 120 NUM=1,NELEMS SSI 387

IF (NVSE(NUM).NE.O) GO TO 120 SSI 388

NT =NTY PE(NUM) SSI 389 GO TO (111,112,113,11^) NT SSI 390

111 CALL OAR SSI 391

CALL STR3AR SSI 392 GO TO 120 SSI 393

112 CALL BEAM SSI 39<f

CALL STRESB SSI 395 GO TO 120 SSI 396

113 CALL TRIM3 SSI 397 CALL STREST SSI 398 GO TO 120 SSI 399

ll^ CALL INFACE SSI <•00

CALL STRESI SSI <*01

120 CONTINUE SSI <•02

C***»*FERFORK NODAL EClUlLIBRIUM CHECK* P = A * F SSI <•03

DO 609 1 = 1»NODOF SSI <•0^

609 SS(I,1)=SS(I,l)-PTOT(I)-PCKI(I) SSI <•05

WRITE (6*620) • SSI <•06

620 F0R;1AT C/MX,*NODAL EQUILIBRIUM CHECK*) SSI <•07

WRITE (6,7i») SSI <•08

DO 625 1-1»NNOOES SSI <•09

IF (I.EO.NNODES) GO TO 622 SSI MO NOFPN=NNF(I+1)-NNF(I) SSI <•11

GO TO 623 SSI <•12

622 NDFPN=NODOF-NNF(I) SSI <•13

€23 JsNNFlI)•! SSI <•1^

IF (NDFPN.EQ.2) GO TO 627 SSI <•15

WRITE (6,82) I,SS(J,1>,SS(J+1, 1)»SS(J+2, 1J SSI <•16

GO TO 625 SSI <•17

627 WRITE (6,82) I,SS(J,1),SS(J+1, 1) SSI <•18

625 CONTINUE SSI *•19 IF (NIT.NE.NINT) GO TO 65 SSI <•20

IF (INDEX.GT.0) GO TO 630 SSI <•21

IF (INDEX.EQ.-l) NSUPTS=NST SSI <•22 IF (INDEX.EQ.-21 NSUPTS=NSUPTS-3 SSI <•23

DO 338 I=1,NELEMS SSI <•2^

. 808 NVSE(I)=0 SSI <*25

DO 57 1=1,NODOF SSI <•26

57 DISP(I)=0.0 SSI <•27

630 CONTINUE SSI <•28 IF (INOEX.NE.1.ANO.INOEX.NE.2) GO TO 801 SSI <•29

IF (NLAY.NE.NINC) GO TO 800 SSI <•30

GO TO 801 SSI <•31

9999 CONTINUE SSI <•32

214

ro ro .»

t-4 Vi (/>

O z Ui

SUBROUTINE BAR SSI t*3k COMMON NUM,NT,*T,NPASS, IPNF,IQNF,IRNF, ISNF,NIT, NINT, SSI 435

2XP,XQ,XR,YP,YQ,YR,S(8,8),8(3, 6),0 (8,8) ,ESB(8,6) ,DEF (8), SSI 436 3A,HT, INDEX,PAT!",DWATER SSI 4 37 COMMON EMUDt15),XXI(15) ,AREA(15),SECM0D(15)* DENSTY(15), SSI 4 38

2PRATI0(15),PHIC15),C0H(15)» RF(15),FK(15),FN(15) * SSI <•39 3FF(15),G1(15)»D0(15) ,FKUR(15) f SSI 440 4SS (261, 5*») » P (261) , PT EMP(261), DISP(261) ,X(127),Y(127),NNF(127>, SSI 441 5NTYPE(196) , MAT YPE(196), IP(196),13(196) ,IR(196), IS(196) , SSI 442 6STRESS(3,196),NVAE(196) ,STRMAX(196), SSI 44 3 7LP(2 0),PT(20),NS(55) ,PCKI(261) SSI 444 GENERATE OAR STIFFNESS ARRAY, S,IN ELEKENT GLOBAL COORDINATES SSI 445 IPP=IP(NUM) SSI 446 IQQ=IQfNUM) SSI 44 7 IPNF=NNF(IPP) SSI 44 8 IQNF=NNF (ICQ) SSI 449 XL=X ( IFP)-X(IGC) SSI 4 50 YL=Y(IPP)-Y(IQG) SSI 451 EL=SORT(XL*XL+YL*YL) SSI 452 MT=MATYPE(NUM) SSI *•53 A=AREA(MT) SSI 454 E=EMOD(MT) SSI 455 H=A*E/(EL*EL*EL) SSI 456 IF (NPASS.NE.l) GO TO 16 SSI 457 S(l,1)=H*XL*XL SSI 458 S(1,2)=H*XL*YL SSI 4 59 S(1, 3)=-S(1,11 SSI <*6 0 S (1, <•) = -S (1, 2) SSI 461 S(2,2)=H*YL*YL SSI 462 S(2,3)=S(1,4) SSI 463 S(2,4)=-S(2,2) SSI 464 S(3,3)=S(1,1) SSI 465 S (3,<t)=S (1,2) SSI 466 S(«*,<f)=S(2,2) SSI <•67

DO U J=1»3 SSI 468 JP1=J+1 SSI 469

00 14 K=JP1,4 14 S<K,J)=S<J,K>

RETURN 16 H=H*EL

ES9{1»1)=H*XL ESQ(1» 2)=H*YL ES3d,3)=-ESBd,l> ESdd,4)=-ESBd,2> B(l.l)=XL/EL B(1,2)=YL/EL B(l,3)=-B(l,l) P<1,4)=-3<1,2> RETURN ENO

SSI 470 SSI 471 SSI 472 SSI 473 SSI 474 SSI 1*75 SSI 476 SSI 477 SSI 478 SSI 479 SSI 4B0 SSI 481 SSI 482 SSI 483

S U B R O U T I N E B E A V SSI 484 C O M M O N N U M , N T , K T , N P A S S , I P N F , I Q N T , I R N F , I S N F . N I T , N I N T , SSI **85

2 X P , X Q , X R , Y t > , Y Q , Y R , S ( d , 8 ) » B ( 3 , 6 ) , 0 ( 8 , 6 ) , E S 9 ( 3 , 9 ) , D E F ( 8 ) t SSI 486 3 A . H 3 , I N D E X , P A T V ! , O W A T E R SSI 487

C O M M O N E M 0 0 ( 1 5 ) , X X I ( 1 5 ) , A R E A ( 1 5 It SECMOO(15)t DENSTYC15)» SSI 488 2 P R A T I 0 ( 1 5 ) » P h i ( 1 5 ) » C O H ( 1 5 ) , R F ( 1 5 ) » F K ( 1 5 ) , F N ( 1 5 ) * SSI 489 3 F F ( 1 5 ) » G 1 ( 1 5 ) , C D ( 1 5 ) » F K U R ( 1 5 ) , SSI 490 4 S S ( 2 6 1 » 5 4 ) , P ( 2 6 1 ) , P T E M P ( 2 6 1 ) » D I S P ( 2 6 1 ) • X ( 1 2 7 ) « Y ( 1 2 7 ) » N N F C 1 2 7 ) t SSI 491 5 N T Y ° E ( 1 9 6 ) » M A T Y P E ( 1 9 6 ) t I P ( 1 9 6 ) , I Q ( 1 9 6 ) , I R ( 1 9 6 I * I S ( 1 9 6 ) « SSI 4 9 2 6 S T R H S i > ( ? » 1 9 6 ) , K V A E ( 1 9 6 ) , S T R M A X ( 1 9 6 ) , SSI 493 7 L P ( 2 0 > » P T ( 2 0 ) » N S ( 5 5 ) , P C K I ( 2 6 1 ) SSI 494

C***GEMERATc BEAM STIFFNESS ARRAY IN E L E M E N T G L O B A L C O O R D I N A T E S . S SSI 495 I P ° = I P M U M ) SSI 496 I Q Q = I Q ( N U M ) SSI 4 9 7 I P N F = N N F ( I P P ) SSI 4 9 8 I Q N F = N \ ' F ( I O Q 1 SSI 499 X L = X ( I P P ) - X ( I Q Q ) SSI 5 0 0 Y L = Y ( I P P ) - Y ( I Q G ) SSI 501 E L = S Q R T ( X L * X L + Y L * Y L ) SSI 5 0 2 M T = M A T Y P E ( N ' U M I SSI 5 0 3 A = A R E A ( M T ) SSI 5 0 4 E = E M O D ( M T > SSI 5 0 5 X X I F = X X I ( M T ) S S I 5 0 6 G = A * E / ( E L * E L * E L ) SSI 5 0 7 H = 1 2 . 0 * E * X X I F / E L * * 5 SSI 5 0 8 G G = 6 . Q * E * X X I F / ( E L * E L * E L > SSI 5 0 9 I F ( N P A S S . N E . l ) G O T O 1 6 SSI 510 S ( l t 1 ) = G * X L * X L + H * Y L * Y L SSI 511 S ( I t 2 ) = n * X L * Y L - H * X L * Y L S S I 512 S ( 1 , 3 ) = G S * Y L SSI 513 S ( l , 4 ) = - S ( l * l ) SSI 514 S ( l , 5 ) = - S ( l , 2 ) SSI 515 S ( 1 , 6 ) = S ( 1 , 3 ) SSI 516 S ( 2 , 2 ) = G * Y L * Y L + H » X L * X L SSI 517 S ( 2 , 3 ) = - G G * X L SSI 518 S ( 2 , 4 ) = - S I 1 , 2 ) SSI 519

S ( 2 , 5 ) = - S ( 2 , 2 ) S ( 2 « 6 ) = S C 2 , 3 ) S ( 3 , 3 > = 4 . 0 » E » X X I F / E L S ( 3 , 4 ) = - S ( l , 3 ) S ( 3 » 5 ) = - S ( 2 * 3 ) S ( 3 » 6 ) = 0 . 5 * S ( 3 » 3 ) S ( 4 , 4 ) = S ( 1 , 1 ) S ( 4 , 5 ) = S ( 1 * 2 ) S ( 4 » b ) = - S ( 1 » 3 ) S ( 5 , 5 ) = S ( 2 » 2 ) S ( 5 , 6 ) = - S ( 2 , 3 1 S ( 6 , 6 ) = S ( 3 » 3 ) 0 0 1 4 J = 1 » 5 JP1=J+1 00 14 «=JP1,6

1 4 S ( K » J ) = S ( J , K ) return

C # * * » * G E N E R A f E E L E M E N T E S S A R R A Y 1 6 H = H » £ l

G=G*EL E S B ( 1 * 1 ) = G " X L E S S ( 1 » 2 ) = G * Y L E S 9 ( 1 , 3 ) = Q . O E S B ( 1 , 4 ) = - E S B ( 1 , 1 ) E S 3 ( 1 , 5 ) = - E S B ( 1 , 2 ) E S 3 ( l , 6 ) = C . O f c S 3 l 2 » i ) = - H » Y L E S 3 ( 2 , 2 ) = H * X L E S b ( 2 , 3 ) = - G G * E L E S 3 ( 2 , 4 ) = - E S 8 < 2 , 1 ) E S 3 ( 2 » 5 ) = - E S 3 ( 2 , 2 ) E S 3 ( 2 , 6 ) = E S B ( 2 , 3 ) E S 3 ( 3 , 1 ) = G G * Y L E S B ( 3 , 2 ) = - G G * X L E S 3 ( 3 , 3 ) = 4 . 0 * E * X X I F / E L E S B ( 3 , 4 ) = - E S B ( 3 , 1 )

SSI 5 2 0 SSI 521 SSI 522 SSI 5 2 3 SSI 5 2 4 SSI 525 SSI 526 SSI 5 2 7 SSI 5 2 8 SSI 529 SSI 5 3 0 S S I 531 SSI 532 S S I 533 SSI 5 3 4 SSI 535 SSI 536 SSI 537 SSI 538 SSI 539 SSI 5 4 0 SSI 541 S S I 5 4 2 SSI 543 SSI 544 SSI 545 SSI 546 SSI 5 4 7 SSI 548 SSI 549 SSI 550 SSI 551 SSI 552 SSI 553 SSI 554 SSI 555

E S B C 3 , 5 > = - E S B < 3 , 2 ) E S B < 3 , 6 ) = Q . 5 * E S B ( 3 , 3 > 0 0 1 7 I = « t , 5 D O 1 7 J = 1 1 6

1 7 E S B ( I , J ) = - E S B ( I - 3 » J J F S B { 6 , l ) = E S n ( 3 , l l E S B ( 6 , 2 > = E S 3 ( 3 , 2 1 E S 3 ( 6 » 3 ) = E S B ( 3 , 6 ) E S 3 { 6 , 4 ) = E S 3 ( 3 , U E S J ( 6 , 5 ) = ? S 3 ( 3 t 5 > E S B ( 6 , 6 ) = E S B ( 3 , 3 ) D O ? 0 1 = 1 , 3 D O 2 0 J = l » 3

2 0 B ( I , J ) = O . 0 B ( 1 » 1 ) = X L / E L B ( 1 » 2 ) = Y L / E L B ( 2 , 1 ) = - B C 1 » 2 ) B ( 2 , 2 ) = B ( l , l i B < 3 , 3 ) = 1 . 0 R E T U R N E N D

S S I 5 5 6 S S I 5 5 7 S S I 5 5 8 S S I 5 5 9 S S I 5 6 0 S S I 5 6 1 S S I 5 6 2 S S I 5 6 3 S S I 5 6 < * S S I 5 6 5 S S I 5 6 6 S S I 5 6 7 S S I 5 6 8 S S I 5 6 9 S S I 5 7 0 S S I 5 7 1 S S I 5 7 2 S S I 5 7 3 S S I 5 7 t * S S I 5 7 5 S S I 5 7 6

S U B R O U T I N E T R I P 3 S S I 5 7 7 C O M M O N N U M » N T t f c T , N P A S S , I P N F , I Q N F « I R N F , I S N F , N I T , N I N T , S S I 5 7 8

2 X P , X Q , X R , Y P , Y Q , Y R , S ( 8 , 8 ) , B ( 3 , 6 ) , D ( 8 , 8 ) , E S B ( 8 , 8 ) « D E F ( 8 ) , S S I 5 7 9 3 A , H 3 , I N D E X , P A T M , D W A T E R S S I 5 9 0

C O M M O N E M 0 D ( 1 5 ) , X X I ( 1 5 ) , A R E A ( 1 5 ) , S E C M O O ( 1 5 ) , O E N S T Y ( 1 5 ) , S S I 5 8 1 2 P R A T I 0 ( 1 5 ) , P H I ( 1 5 ) , C 0 H ( 1 5 ) » R F ( 1 5 ) , F K ( 1 5 ) , F N ( 1 5 ) S S I 5 8 2 3 F F ( 1 5 ) , G 1 { 1 5 ) , C D ( 1 5 ) , F K U R ( 1 5 ) » S S I 5 8 3 < • S S ( 2 6 1 , 5 < » ) » P ( 2 6 1 ) * P T E M P ( 2 6 1 ) , 0 I S P ( 2 6 1 ) , X ( 1 2 7 ) , Y C 1 2 7 > , N N F ( 1 2 7 ) , S S I 5 8 < • 5 N T Y P E ( 1 9 6 ) , M A T Y P E ( 1 9 6 ) , I P ( 1 9 6 ) , I Q ( 1 9 6 ) , I R ( 1 9 6 ) , I S ( 1 9 6 ) « S S I 5 8 5 6 S T R I S S ( 3 , 1 9 6 ) , N V A E ( 1 9 6 ) , S T R M A X ( 1 9 6 ) , S S I 5 8 6 7 L P ( 2 0 ) , P T ( 2 0 ) , N S C 5 5 ) , P C K I ( 2 6 1 ) S S I 5 8 7

C O M P U T E S T I F F N E S S P R O P E R T I E S F O R E A C H T R I M 3 E L E M E N T S S I 5 8 8 I P P = I P (Nl!M) S S I 5 8 9 I U C = I Q ( N U M ) S S I 5 9 0 I R R = I R ( N U M ) S S I 5 9 1 I P N F = N N F . ( I P P ) S S I 5 9 2 I Q N P = N N F ( I Q Q ) S S I 5 9 3 I R N F = N N F ( I R R ) S S I 5 9 * » X P = X ( I P P ) S S I 5 9 5 X Q = X ( I C Q ) S S I 5 9 6 X R = X ( I R R ) S S I 5 9 7 Y P = Y ( I P P ) S S I 5 9 8 Y U = Y ( I Q Q ) S S I 5 9 9 Y R = Y ( I R R ) S S I 6 0 9 X R Q = X R - X O S S I 6 0 1 X R P = X R - X P S S I 6 0 2 X Q P = X Q - X P S S I 6 0 3 Y R Q = Y R - Y Q S S I 6 0 k Y R P = Y R - Y P S S I 6 0 5 Y Q P = Y Q - Y P S S I 6 0 6 A 2 = X R Q * Y Q P - X Q P * Y R Q S S I 6 0 7 A = A 3 S ( A 2 ) / 2 . 0 S S I 6 0 S D O 6 1 = 1 , 3 S S I 6 0 9 D O 6 J = 1 , 6 S S I 6 1 0

f > B ( I , J ) = 0 . 0 S S I 6 1 1 B ( 1 , 1 ) = Y R Q / A 2 S S I 6 1 2

b (i* 3) = -yrp/a2 ssi 613 3(1,51 = yqf/a2 ssi 614 b(2t 2) = -xrq/a2 ? ssi 615 b(2»4) = xrp/a2 ssi 616 b (21 6) = -x0p/a2 ssi 617 b(3,l) — 8(2,2) ssi 618 b(3,2) a b(1* 1) ssi 619 b(3,3) = b c2,«• > ssi 620 b(3,4) = 3(1,31 ssi 621 b(3,5) = b (2, 6 ) ssi 622 b ( 3 • 6) = b (1,5) ssi 623 mt = '1atypf(num) ssi 624 if (npass.eq.3) go to 800 ssi 625 gp=g1(mt) ssi 626 rfp=rf(mt) ssi 627 if (rfp.eq.0.0) go T O 69 ssi 628 h1=0.5*(stress(1,num)+stress(2, num) ) ssi 629 h= .5* (sqrtt ( (stress (i»num)- stress (2* num))**2+4.0* S T R E S S ( 3 , N U M ) * * 2 ) )ssi 630 sigma1=h1-h ssi 631 sig^a2 = hh-h ssi 632 avgy=(yp + ych-yr)/3.0 ssi 633

cs=.5msigma2+g1(mt)•(sigma2+sigma1)*1.11) ssi 634 if (inoex.eq.o) cs=-densty(mt)*gp/(1»0-gp)* H B * Q .10ssi 635 if (index.eq.l.and.nvae(nuh) .eq .1)cs=-densty(mt) * G P / t l . - G P ) * H B * •10ssi 636

(^••••provision for to tension in material ssi 637 if (cs.ge.-1.0e-10) go to 52 ssi 638 fnp=fn(mt) ssi 639 if (-sigma1+sigma2.lt.0.9999*strmax(num)) go to 56 ssi 640 fkp=fk(mt) ssi 641 e0=f.<p*patm* (-cs/patm) **fnp ssi 642 fp=ff(mt) ssi 643 vo=gp-fp*alogio(-cs/patm) ssi 644 if (sigma1-cs.gt.-0.01) go T O 60 ssi 645 phip=phi(mt)» 0.017.453 ssi 646 c1=1.-rfp*(1.-sin(phip)) M - S X G P A 1 + C S I / C 2 . * ( - C S ) » S I N ( P H I P ) ssi 647 2 +2.*coh(mt)*cos(phip)) ssi 648

I F ( C l . L T . 0 . 1 ) C l = 0 . 0 1 SSI 6 < * 9 E T = E 0 * C 1 * C 1 S S I 6 5 0 D P = O n < M T ) S S I 6 5 1 V T = V 0 / ( 1 . 0 - D P * A B S ( S I G M A 1 - C S ) / ( E 0 * C 1 ) ) * * 2 S S I 6 5 2 E = E T S S I 6 5 3 V = V T S S I b S i * I F ( V . G T . 0 . ^ 7 ) V = 0 . < » 7 S S I 6 5 5 I F ( N P A S S . N E . i ) G O T O 5 9 S S I 6 5 6 W R I T E ( 6 , 5 3 ) N U M , E T , V T S S I 6 5 7

5 3 F O R M A T C * X , * E L E N O = • ,I«f,2 X , * E T = * t £ 1 0 • 3 « 2 X » * V T = ** F6« 3 ) S S I 6 5 8 G O T O 5 9 S S I 6 5 9

6 0 E = E 0 S S I 6 6 0 V = V 0 S S I 6 6 1 I F ( V . G T . O V = Q . ^ 7 S S I 6 6 2 I F ( N P A S S . N E . I ) G O T O 5 9 * S S I 6 6 3 W R I T E ( 6 , 6 2 ) N U M , E 0 , V 0 S S I 6 6 < *

6 2 F O R M A T U X , * E L E N O = * , H » « 2 X , » E 0 = * , E 1 0 . 3 , 2X, *vo =»,F6. 3 ) S S I 6 6 5 G O T O 5 9 S S I 6 6 6

6 9 E = F M O D ( M T ) S S I 6 6 7 G O T O 5 7 S S I 6 6 8

5 2 I F ( N P A S S . N E . i ) G O T O S S I 6 6 9 E = P A T M * 1 . 0 E - 6 S S I 6 7 0 G O T O 5 8 S S I 6 7 1

5 6 E = F < U R ( M T ) * P A T M * ( - C S / P A T M ) * * F N P S S I 6 7 2 5 8 V = G P S S I 6 7 3

I F ( N P A S S . N E . I ) G O T O 5 9 S S I 6 7 1 * W R I T E 1 6 , 6 6 ) N U M , E , V , S T R M A X ( N U M ) S S I 6 7 5

6 6 F O R M A T U X , * E L E N O = * , I ^ * 2 X , * E = » , E 1 0 • 3, 2X» *v =<SF6. 3 , 6 X , S S I 6 7 6 2 * M A X S T R E S S = * , F 1 0 . 3 ) S S I 6 7 7

G O T O 5 9 S S I 6 7 8 *•0 E = P A T M * 1 0 . 0 S S I 6 7 9 5 7 V = G P S S I 6 8 0 5 9 G = E / ( 2 . 0 * ( 1 . 0 + V ) ) S S I 6 8 1

B K = E / ( 2 . 0 * ( i . O + V ) * ( 1 . 0 - 2 . 0 * V ) ) S S I 6 8 2 D O i * 1 = 1 , 3 SSI 6 8 3 0 0 « • J = l , 3 SSI 6 8 < »

I * D ( I * J ) = 0 • 0 D ( 1 , 1 ) = B K + G 0 ( I t 2 ) = 3 K - G 0 ( 2 * 1 ) = D ( 1 , 2 ) D ( 2 , 2 ) = D ( 1 , 1 ) D ( 3 , 3 ) = G I F ( N P A S S . E Q . 2 ) R E T U R N D O 8 3 1 = 1 , 3 0 0 8 3 < = 1 , 6 H= 0 . 0

D O 8 2 J = l » 3 8 2 H = H + 0 ( I » J ) * B ( J , K ) 8 3 E S B ( I , K ) = H

D O 8 5 L = 1 , 6 D O 8 5 K = L » 6 H = Q . 0 D O f i t * 1 = 1 , 3

6 k H = H + B ( I , L ) * E S 8 < I , K ) 8 5 S C L , K ) = H * A

D O 2 J = 1 , 5 J P 1 = J + 1 D O 2 K = J P 1 , 6

2 S ( K » J ) = S ( J , K ) I F C N I T . N E . l ) R E T U R N I F ( I N O E X . E Q . O ) G O T O 8 0 1 I F ( I N O E X . N E . l . O R . N v y A E ( N U M ) . N E . 1 J

8 0 1 P L O A 0 = A * D E N S T Y ( M T ) / 3 • 0 P ( I P N F + 2 ) = P ( I P N F + 2 > + P L O A D P < I Q N F + 2 ) = P ( I G f * F + 2 ) + F L 0 A D P ( I R N F + 2 ) = P ( I R t v F + 2 ) + P L 0 A D R E T U R N

8 0 0 P L O A D = A * D E N S T Y ( M T ) / 3 . 0 S S ( I P N F + 2 , 1 ) = S S ( I P N F + 2 . 1 ) - P L 0 A 0 S S { I Q N F + 2 , 1 ) = S S ( I Q N F + 2 , 1 ) - P L O A O S S ( I R N F + 2 , 1 > = S S ( I R N F + 2 , 1 J - P L 0 A D R E T U R N

RETURN

S S I 6 8 5 S S I 6 8 6 S S I 6 8 7 S S I 6 8 8 S S I 6 8 9 S S I 6 9 0 S S I 6 9 1 S S I 6 9 2 S S I 6 9 3 S S I 6 9 < * S S I 6 9 5 S S I 6 9 6 S S I 6 9 7 S S I 6 9 8 S S I 6 9 9 S S I 7 0 0 S S I 7 0 1 S S I 7 0 2 S S I 7 0 3 S S I 7 0 < » S S I 7 0 5 S S I 7 0 6 S S I 7 0 7 S S I 7 0 8 S S I 7 0 9 S S I 7 1 0 S S I 7 1 1 S S I 7 1 2 S S I 7 1 3 S S I 7 1 < « S S I 7 1 5 S S I 7 1 6 S S T 7 1 7 S S I 7 1 8 S S I 7 1 9 S S I 7 2 0

E N D S S I 7 2 1

to to •Ck

r S U B R O U T I N E I N F A C E S S I 7 2 2 C O M M O N N U M , N T , I " T , N P A S S , I P N F , I C l N F , I R N F , I S N F » N I T , N I N T , S S I 7 2 3

2 X P , X Q , X R , Y P , Y Q , Y R , S ( 8 , 8 ) , 9 ( 3 , 6 ) , 0 ( 8 , 8 ) , E S B ( 8 , 8 ) , O E F ( 8 ) . S S I 7 2 4 3 A , H 3 , I N O i i X , P A T f ' , O W A T E R . S S I 7 2 5

C O M M O N E M 0 D ( 1 5 ) , X X I ( 1 5 ) , A R E A ( 1 5 ) , S E C M O O ( 1 5 ) , D E N S T Y C 1 5 ) * S S I 7 2 6 2 P R A T I 0 ( 1 5 ) , P H I ( 1 5 ) , C O H ( 1 5 ) , R F ( 1 5 ) • F K C 1 5 ) » F N ( 1 5 ) » S S I 7 2 7 3 F F ( 1 5 I , G 1 ( 1 5 ) , C D ( 1 5 ) , F K U R ( 1 5 ) , S S I 7 2 8 U S S ( 2 6 1 , 5 t « ) , P ( 2 6 1 ) , P T E M P ( 2 6 1 ) , 0 I S P ( 2 6 1 ) , X ( 1 2 7 ) , Y ( 1 2 7 ) » N N F ( 1 2 7 ) t S S I 7 2 9 5 N T Y P E ( 1 9 6 ) , M A T Y P E ( 1 9 6 ) , I P ( 1 9 6 ) , I Q ( 1 9 6 ) » I R ( 1 9 6 ) , I S ( 1 9 6 ) , S S I 7 3 0 6 S 7 " G ? T S S ( 3 , 1 9 6 ) , N V A E ( 1 9 6 ) , S T R M A X ( 1 9 6 ) , S S I 7 3 1 7 L P ( 2 3 ) , P T ( 2 0 ) , N S ( 5 5 ) , P C K I ( 2 6 1 ) S S I 7 3 2

I P F = I P ( N U M ) S S I 7 3 3 I Q Q = I Q ( N U M ) S S I 7 3 4 I R R = I R ( N U M ) S S I 7 3 5 I S S = I S ( N U M ) S S I 7 3 6 I P N F = N N F ( I P P ) S S I 7 3 7 I Q h F = N N F ( I Q Q ) S S I 7 3 8 I R N F = N N F ( I R R ) S S I 7 3 9 I S N F = N N ' F ( I S S ) S S I 7 4 0 X P = X ( I P P ) S S I 7 4 1 X O = X ( I Q Q ) S S I 7 4 2 Y P = Y ( I P P ) S S I 7 4 3 Y Q = Y ( I Q Q ) S S I 7 4 4 X L = X a - X P S S I 7 1 * 5 Y L = Y P - Y Q S S I 7 4 6 E L = S Q R T ( X L * X L + Y L » Y L ) S S I 7 4 7 B ( l , l ) = X L / E L S S I 7 t * 6 8 ( 1 , 2 ) = - Y L / E L S S I 7 k 9 B ( 2 , 1 ) = B ( 1 , 2 ) S S I 7 5 0 B ( 2 , 2 ) = - 3 ( 1 , 1 ) S S I 7 5 1 M T = M A T Y P E ( N U M ) S S I 7 5 2 I F ( N P A S S . E Q . 3 ) R E T U R N S S I 7 5 3 G P = G 1 ( M T ) S S I 7 5 4 D E L T A = P H I ( M T ) » 0 . 0 1 7 4 5 3 S S I 7 5 5 R F P = R F ( M T ) S S I 7 5 6 I F ( R F P . E Q . 0 . 0 ) G O T O 4 1 S S I 7 5 7

S S H E A R = S T R E S S ( 1 * N U M ) S S I 7 5 8 S N O R M = S T R E S S ( 2 , N U M ) S S I 7 5 9 I F ( I N D E X . E C . 0 > G O T O 7 S S I 7 6 0 I F ( I N D E X . N E . l . O R . N V A E ( N U M ) . N E . l ) G O T O 6 S S I 7 6 1

7 S N O R M = - O E N S T Y ( M T ) * i , P / ( 1 . . 0 - G P ) * H B * 0 . 1 0 S S I 7 6 2 S S H t A R = - S N O R M * R F F * T A N ( D E L T A ) • O . 3 3 S S I 7 6 3

6 C O N T I N U E S S I 7bk • P R O V I S I O N F O R h O T E N S I O N I N M A T E R I A L S S I 7 6 5

I F ( S N O R M . G E . - 1 . 0 E - 1 0 ) G O T O 1 2 S S I 7 6 6 F N P = F N ( M T ) S S I 7 6 7 I F ( A B S ( S S H E A R ) . L T . 0 . 9 9 9 9 * A B S ( S T R M A X ( N U M ) 1 ) G O T O 1 6 S S I 7 6 8 F K P = F K ( M T ) S S I 7 6 9 f c O = F < P * D W A T E R * ( A B S ( S N O R M ) / P A T M ) * * F N P S S I 7 7 0 C l = l . C - R F P * A B S ( S S H E A R / S N O R M ) / T A N C O E L T A ) S S I 7 7 1 I F ( C I . L T . 0 . 1 ) C l = 0 . 9 1 s s t 7 7 2 E S = E 0 * C 1 * C 1 S S I 7 7 3 G O T O ik S S I 77k

1 2 I F ( N P A S S . N E . l ) G O T O S S I 77 5 E S = D H A T E R * 1 . 0 E - 6 S S I 7 7 6 E N = D W A T E R * 1 . 0 £ - 6 S S I 7 7 7 G O T O 1 8 S S I 7 7 8

1 8 E S = F K U 3 ( M T ) * D W A T E R * ( A B S ( S N O R M ) / P A T M ) ^ * F N P S S I 7 7 9 Ik F N = D W A T E R * 1 . 0 E 6 S S I 7 8 0

I F ( N P A S S . N E . l ) G O T O 1 9 S S I 7 8 1 1 8 W R I T E ( 6 , 1 3 ) N L M , E S » E N » S T R M A X ( N U M ) S S I 7 8 2 1 3 F O R M A T ( * + X , * E L E N O = * , I 4 , 2 X , * E S = * , E 1 0 . 3 , 2 X , * E N = » , E 1 0 . 3 , 2 X , S S I 7 8 3

2 * M A X S T R E S S = * , F 1 0 . 3 ) S S I 7 8 * » G O T O 1 9 S S I 7 8 5

k 0 E S = D K A T F R * 1 0 . 0 S S I 7 8 6 E N = 3 W A T E R * 1 0 Q • 0 S S I 7 8 7 G O T O 1 9 S S I 7 8 8

< • 1 E S = E M O D ( M T ) S S I 7 8 9 E N = D W A T E R * 1 . 0 E 6 S S I 7 9 0

1 9 C O N T I N U E S S I 7 9 1 D O 2 0 1 = 1 , 8 S S I 7 9 2 O O 2 0 J = I , « S S I 7 9 3

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S U 9 R 0 U T I N E S T O R E S S I 8 3 2 C O M M O N N U M , N T , ! " . T , N P A S S , I P N F , I Q N F , I R N F , I S N F , N I T , N I N T , S S I 8 3 3

2 X P , X a , X R , Y P , Y Q , Y R , S ( 8 , 8 ) , B ( 3 , 6 ) , 0 ( 8 , 8 ) , £ S B ( 8 , 8 ) , D £ F ( 8 ) , S S I 8 3 4 3 A , H 3 , I N 0 E X , P A T f , O W A T E R S S I 8 3 5

C O M M O N F M 0 0 ( 1 5 > , X X I ( 1 5 ) , A R E A ( 1 5 ) , S F C M 0 D ( 1 5 ) , O E N S T Y ( 1 5 ) • S S I 8 3 6 2 P R A T 1 0 ( 1 5 ) , P H I ( 1 5 ) , C 0 H ( 1 5 ) , R F ( 1 5 ) , F K ( 1 5 ) , F N ( 1 5 ) » S S I 8 3 7 3 F F ( 1 5 ) , G 1 ( 1 5 ) , C 0 ( 1 5 ) , F K U R ( 1 5 ) , S S I 8 3 8 < * S S ( 2 6 1 , 5 * # > , P ( 2 6 1 ) , P T E M P ( 2 6 1 ) , D I S P ( 2 6 1 ) , X ( 1 2 7 ) , Y ( 1 2 7 ) , N N F 1 1 2 7 ) , S S I 8 3 9 5 N T Y P E ( 1 9 6 ) , M A T Y P E ( 1 9 6 ) , I P ( 1 9 6 ) , I Q ( 1 9 6 ) , I R ( 1 9 6 ) , I S ( 1 9 6 ) , S S I 8 4 0 6 S T R £ S S ( 3 , 1 9 6 ) , N V A E ( 1 9 6 ) • S T R M A X ( 1 9 6 ) . S S I 8 4 1 7 L P ( 2 0 ) , P T ( 2 0 ) , N S < 5 5 ) , P C K I ( 2 6 1 ) S S I 8 4 2

D I M E N S I O N M ( 8 ) S S I 8 4 3 O F S Y M M E T R I C A L S T I F F N E S S A R R A Y F O R E L E M E N T S S I 8 4 4

S S I 8 4 5 G O T O ( 1 , 2 , 3 , ) N T S S I 8 4 6

S S I 8 4 7 1 D O 5 1 = 1 , 2 S S I 8 4 8

M ( I ) = I P N F + I S S I 8 4 9 3 M ( 1 + 2 ) = I Q N F + I S S I 8 5 0

L = t * S S I 8 5 1 G O T O 1 5 S S I 8 5 2

S S I 8 5 3 2 D O 1 0 1 = 1 , 3 S S I 8 5 4

M ( I ) = I P N F + I S S I R 5 5 1 J M ( J + 3 ) = I Q N F * I S S I 8 5 6

L = 6 S S I 8 5 7 G O T O 1 5 S S I 8 5 8

C * » * » * T R H 3 E L E M E N T S S I 8 5 9 3 D O 1 1 1 = 1 , 2 S S I 8 6 0

M ( I ) = I P N F + I S S I 8 6 1 M ( 1 + 2 ) = I Q N F + I S S I 8 6 2

1 1 M ( I + £ » ) = I R N F + I S S I 8 6 3 L - 6 S S I 8 6 4 G O T O 1 5 S S I 8 6 5

S S I 8 6 6 k 0 0 1 3 1 = 1 , 2 S S I 8 6 7

HCI) = I P N F + I M ( 1 + 2 ) = I Q N F + I M ( I * t * ) = I R N F + I

1 3 M ( 1 + 6 ) = I S N F + I L = 8

1 5 D O 2 0 1 = 1 * L I I = M ( I ) D O 2 0 J - l « L I J = - 1 ( J ) I F ( I I . G T . I J ) G O T O 2 0 K K = I J - I I + l S S I I I , K K ) = S S ( I I , K K ) + S C I t J )

2 0 C O N T I N U E R E T U R N E N D

S S I 8 6 8 S S I 8 6 9 S S I 8 7 0 S S I 8 7 1 S S I 8 7 2 S S I 8 7 3 S S I 8 7 k S S I 8 7 5 S S I 8 7 6 S S I 8 7 7 S S I 8 7 8 S S I 8 7 9 S S I 8 8 0 S S I 8 8 1 S S I 8 8 2

S U B R O U T I N E E Q S O L ( A » B » N » N B W J C » » » * B A N D E O G A U S S E L I M I N A T I O N O N U P P E R H A L F

D I M E N S I O N A ( 2 6 1 t 5 t f ) » Q C 2 6 1 ) D O 1 1 = 1 , N D = A ( I » 1 ) I F C A 3 S ( D ) . L T . l . O E - l t f ) G O T O 7 11=1*1 N E = I • • N H W - l D O 3 L = I I » N E I F ( L . G T . N ) G O T O 1 N C 1 = L - I * 1 C = A ( I » N C I I / O I F ( C . E O . O . O ) G O T O 3 N R E = L * N 3 W - 1 D O K = L , N R E I F ( K . G T . N ) G O T O 8 N C < + = K - I + 1 I F I N C U . G T . N B W ) G O T O 8 N C 3 = K - L « - 1 A t L , N C 3 ) = A ( L , N C 3 ) - A ( I , N C M * C

k C O N T I M U E 8 B ( L ) = 3 ( L ) - B ( I ) * C 3 C O N T I N U E 1 C O N T I N U E

C * * * * B A C < S U d S T I T U T I C N B ( N ) = 3 ( N ) / A ( N , i ) D O 5 1 = 2 , N J = N + 1 - I K = I - 1 D O 6 L = 1 , K L L = K - L • 1 M = N 4 - i - L L N C = M - J + 1 I F ( N C . G T . N B W ) G O T O 5

6 B ( J ) = B ( J ) - A ( J » N C ) * B ( M ) 5 B ( J ) = U ( J ) / A ( J , 1 I

S S I 8 8 3 S Y M M E T R I C A L M A T R I X S S I B Q k

S S I 8 8 5 S S I 8 8 6 S S I 8 8 7 S S I 8 8 8 S S I 8 8 9 S S I 8 9 0 S S I 8 9 1 S S I 8 9 2 S S I 8 9 3 S S I 8 9 * » S S I 8 9 5 S S I 8 9 6 S S I 8 9 7 S S I 8 9 8 S S I 8 9 9 S S I 9 0 0 S S I 9 0 1 S S I 9 0 2 S S I 9 0 3 S S I 9 0 U S S I 9 0 5 S S I 9 0 6 S S I 9 0 7 S S I 9 0 8 S S I 9 0 9 S S I 9 1 0 S S I 9 1 1 S S I 9 1 2 S S I 9 1 3 S S I 9 1 t f S S I 9 1 5 S S I 9 1 6 S S I 9 1 7 S S I 9 1 8

R E T U R N S S I 9 1 9 7 W R I T E ( 6 , 2 0 ) I S S I 9 2 0

2 0 F O R M A T ( < * X , » S T I F F N E S S C O E F F I C I E N T » . I 5 , » O N D I A G O N A L E Q U A L S Z E R O » ) S S I 9 2 1 S T O P S S I 9 2 2 E N D S S I 9 2 3

S U B R O U T I N E S T R 8 A R S S I 9 2 4 C O M M O N N U M , N T , M T , N P A S S , I P N F , I Q N F , I R N F , I S N F , N I T , N I N T , S S I 9 2 5

2 X P , X 1 , X R , Y P , Y G , Y R , S ( 8 , 8 ) , 3 ( 3 , 6 ) , D ( 8 » a > , E S B ( 8 , 8 ) , 0 E F ( 8 > • S S I 9 2 6 3 A , H t i , I N D E X , P A T M , D W A T E R S S I 9 2 7

C O M . I O N E K C O ( 1 5 ) , X X I ( 1 5 ) , A R E A ( 1 5 ) , S E C M 0 D ( 1 5 ) • D E N S T Y C 1 5 ) » S S I 9 2 8 2 P R A T I 0 ( 1 5 ) » P H I ( 1 5 ) , C 0 H ( 1 5 ) , R F ( 1 5 ) , F K ( 1 5 1 , F N ( 1 5 ) * s s r 9 2 9 3 F F ( 1 5 ) , G l ( 1 5 ) , C D ( 1 5 ) , F K U R ( 1 5 > , S S I 9 3 0 4 S S ( 2 6 1 , 5 4 ) , P ( 2 6 1 ) , P T E M P ( 2 6 1 ) , 0 I S P ( 2 6 1 ) , X ( 1 2 7 ) * Y ( 1 2 7 ) , N N F ( 1 2 7 ) « S S I 9 3 1 5 N T Y ? E ( 1 9 6 ) , M A T Y P E ( 1 9 6 ) , I P ( 1 9 6 ) , I d ( 1 9 6 ) , I R ( 1 9 6 ) , I S ( 1 9 6 ) t S S I 9 3 2 6 S T R E S S ( 3 , 1 9 b ) , N V A E ( 1 9 6 ) , S T R M A X ( 1 9 6 ) » S S I 9 3 3 7 L P ( 2 0 ) , P T ( 2 0 ) , N S ( 5 5 ) , P C K I ( 2 6 1 ) S S I 9 3 4

C * * * * * E L t £ M E N T F O R C E S I N L O C A L C O O R D I N A T E S Y S T E M S S I 9 3 5 0 0 3 1 = 1 , 2 S S I 9 3 6 O E F ( I ) = D I S P ( I P N F + I ) S S I 9 3 7

3 O E F ( I + ? ) = O I S P ( I Q N F + I ) S S I 9 3 8 F = 0 . 0 S S I 9 3 9 0 0 4 J = l , 4 S S I 9 4 0

4 F = F + t S B ( l , J ) * O E F ( J ) S S I 9 4 1 I F ( A . L T . 1 . 0 E - 1 Q ) G O T O 6 S S I 9 4 2 S T R S = F / A S S I 9 4 3 S T R N = S T R S / E M O 0 ( M T ) S S I 9 4 4 G O T O 8 S S I 9 4 5

6 S T R S = 0 . 0 S S I 9 4 6 S T R N = 0 . 0 S S I 9 4 7

8 C O N T I N U E S S I 9 4 8 W R I T E ( 6 , 1 0 ) S S I 9 4 9

1 0 F O R M A T ( 4 X , * E L E M N O * , 7 X , * A X I A L F O R C E * » 7 X , * S T R E S S * , 7 X , • S T R A I N * ) S S I 9 5 0 W R I T E ( 6 , 1 2 ) N U M , F « S T R S , S T R N S S I 9 5 1

1 2 F O R M A T ( 5 X , I 4 , 5 X , F 1 5 . 3 , 1 X , F 1 2 . 5 , 3 X , F I O . 8 ) S S I 9 5 2 0 0 6 5 0 1 = 1 , 4 S S I 9 5 3

6 5 0 D E F ( I ) = B ( 1 , I ) * F S S I 9 5 4 0 0 6 5 6 1 = 1 , 2 S S I 9 5 5 S S ( I P N F + I , 1 ) = S S ( I P N F + I , 1 ) + 0 E F ( I ) S S I 9 5 6

6 5 6 S S ( I Q N F + I , 1 ) = S S ( I Q N F « - I , 1 I + 0 E F C H - 2 ) S S I 9 5 7 R E T U R N S S I 9 5 8 F N D S S I 9 5 9

S U B R O U T I N E S T R E S S C O M M O N N U i l , N T , M T , N P A S S , I P N F , I Q N F , I R N F , I S N F , N I T , M I N T ,

2 X P , X Q , X R , Y P , Y Q , Y R , S ( 8 , 8 ) , B ( 3 , 6 ) , D ( 8 , 8 ) , E S B ( 8 , 8 ) , D E F ( 6 ) , 3 A , H f ? , l N 0 E X , P A T M , D W A T E R

C O M M O N E M O O ( 1 5 ) , X X I ( 1 5 ) , A R E A ( 1 5 ) , S E C M O D ( 1 5 ) , 0 E N S T Y ( 1 5 ) , 2 P R A T 1 0 ( 1 5 ) , P H I ( 1 5 ) , C O H ( 1 5 ) , R F ( 1 5 ) , F K ( 1 5 ) , F N ( 1 5 ) , 3 F F ( 1 5 ) , G 1 ( 1 5 ) , 0 0 ( 1 5 ) , F K U R ( 1 5 ) , i * S S ( 2 S 1 , 5 * » ) » P ( 2 6 1 ) , P T E M P ( 2 6 1 ) , O I S P ( 2 f t 1 ) , X ( 1 2 7 ) , Y ( 1 2 7 ) , N N F ( 1 2 7 ) , 5 N T Y P E ( 1 9 6 ) , M A T Y P £ ( 1 9 6 ) , I P ( 1 9 6 ) , I Q ( 1 9 6 ) , I R ( 1 9 6 ) , I S 1 1 9 6 ) , 6 S T R £ S S ( 3 , 1 9 6 ) , N V A E ( 1 9 6 ) , S T R M A X ( 1 9 6 ) , 7 L P ( 2 0 ) , P T ( 2 0 ) , N S ( 5 5 ) , P C K I ( 2 6 1 )

D I M E N S I O N F ( 6 ) C O M P U T E A N Q W R I T E E L E M E N T F O R C E S C O M P U T E S Y S T E M O E F C R M A T I O N S A T E L E M E N T G L O B A L C O O R D I N A T E S

2 D O 3 1 = 1 , 3 D E F ( I ) = O I S P ( I P I v F + I )

3 D c F ( I + 3 ) = D I S P ( I Q N F + I ) D O 5 1 = 1 , 6 H = 0 . 0 D O J = 1 , 6

t * H = H + - £ S 2 ( I , J ) * D E F I J ) 5 F ( I ) = H

S M = S E C M O D ( M T ) S I G M A P = F ( 3 ) / S M S I G M A Q = F ( 6 ) / S M F 1 = F ( 1 ) F 2 = F ( 2 ) F 5 = F ( 5 ) F 3 = F ( 3 ) F 6 = F ( 6 ) W R I T E ( 6 , 1 0 )

1 0 F O R M A T ( , * E L E N O * , 1 0 X , * M C M E N T S - P , Q E N D S * , 1 0 X , * A X I A L F O R C I 2 1 0 X , * S H E A R S - P , Q E N D S * , 1 0 X , * B E N D I N G S T R E S S E S - P , Q E N O S * )

W P I T E ( 6 , 1 2 ) N U M , F 3 , F 6 , F 1 , F 2 , F 5 , S I G M A P , S I G M A Q 1 2 F O R M A T U X , I 5 , < » X , F i 2 . ' t , 2 X , F i 2 . < * , 6 X , F 1 2 . ' » , * » X , F 1 2 . < * , 2 X , F 1 2 . t » , ' » X , S S I 9 9 * * £

2 F 1 2 « 3 , 2 X , F 1 2 « 3 ) S S I 9 9 5

S S I 9 6 0 S S I 9 6 1 S S I 9 6 2 S S I 9 6 3 S S I 9 6 % S S I 9 6 5 S S I 9 6 6 S S I 9 6 7 S S I 9 6 8 S S I 9 6 9 S S I 9 7 0 S S I 9 7 1 S S I 9 7 2 S S I 9 7 3 S S I 9 7 k S S I 9 7 5 S S I 9 7 6 S S I 9 7 7 S S I 9 7 8 S S I 9 7 9 S S I 9 8 0 S S I 9 8 1 S S I 9 8 2 S S I 9 8 3 S S I 9 8 < * S S I 9 8 5 S S I 9 8 6 S S I 9 8 7 S S I 9 8 8 S S I 9 8 9 S S I 9 9 0

* S S I 9 9 1 S S I 9 9 2 S S I 9 9 3 S S I 9 9 1 * S S I 9 9 5

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l + I ) J 3 0 + ( T 4 I + J N D I ) S S = ( T 4 I + J H C I ) S S 9 S 9 ( I ) J 3 0 + ( T 4 I + J N d I ) S S = ( T 4 I + J N d I ) S S

£ 4 T = I 9 5 9 O U C £ + D i * C l 4 r ) G + ( £ + I > d 3 G = ( £ + I > J 3 0 0 S 9

(Dd*(I4r)B+(I) 330 = <1)330 P4T=r t?S9 00 0'0=C£. + I) J3U

0 * 0 = ( I ) 3 3 0 £4T=I 099 00 9T

S U B R O U T I N E S T R E S T C O M M O N N L M , N T , M T , N P A S S , I P N F , I Q N F , I R N F , I S N F , N I T , N I N T ,

2 X P , X Q , X R , Y P , Y Q , Y R , S ( 8 , 8 ) » B ( 3 , 6 ) * D ( 8 , 8 ) , E S B ( 8 , 8 ) , 0 E F ( 8 ) , 3 A , H < 3 , I N D E X , P A T f . D W A T E R

C O M M O N E M O D C 1 5 ) , X X I ( 1 5 ) , A R E A ( 1 5 ) , S E C H O O C 1 5 ) . D E M S T Y C 1 5 » , 2 P R A T I 0 C 1 « 5 > , P H I ( 1 5 ) , C 0 H ( 1 5 ) , R F C 1 5 ) , F K ( 1 5 ) . F N ( 1 5 ) • 3 F F I 1 5 ) , G i ( 1 5 ) , 0 D ( 1 5 ) , F K U R ( 1 5 ) , ^ S S ( 2 6 l , 5 < * ) , P ( 2 6 1 ) , P T E H P ( 2 6 1 ) , D I S P ( 2 6 1 ) , X ( 1 2 7 ) , Y ( 1 2 7 ) , N N F ( 1 2 7 > , 5 N T Y " E ( 1 9 6 ) » M A T Y P E ( 1 9 6 ) , I P ( 1 9 6 ) , I Q ( 1 9 6 ) , I R ( 1 9 6 ) , I S < 1 9 6 1 t 6 S T R E S S ( 3 , 1 9 6 ) , N V A £ ( 1 9 6 ) , S T R M A X ( 1 9 6 ) • 7 L P ( 2 0 > , P f ( 2 0 > , N S ( 5 5 ) , P C K I ( 2 6 1 )

D I M E N S I O N S T R A I N ( 3 ) I F ( N P A S S . E Q . 3 ) G O T O 6 ^ 8 D O 5 5 1 = 1 , 2 D t F ( I ) = P T E M P ( I P N F * I ) D E F ( 1 * 2 ) = P T E M P ( I Q N F + I )

5 5 D E F ( I « - U = P T E M P ( I R N F + I ) C O M P U T E S T R E S S E S I N T E R N A L T O C A L C E L M T S T R N S I N H C R Z ( X ) ,

D O 7 0 1 = 1 , 3 H = 0 . 0 D O 6 9 J = l » 6

6 9 H = H « - 3 ( I , J ) * 0 E F ( J ) 7 0 S T R A I N ( I ) = H

C A L C E L M T S T R E S S I N H O R Z ( X ) , 7 2 D O 7 U 1 = 1 , 3

H = 0 • 0 D O 7 3 J = l , 3 H = H + 0 ( I , J ) * S T R A I N ( J ) S T R E S S ( I , N ' l i M ) = S T R E S S ( I t N U M ) + H D O f t O 1 = 1 , 2 D E F ( I ) = O I S P ( I P N F + I ) 0 E F ( I + 2 ) = D I S P ( I Q N F + I ) D E F ( I * < » ) = O I S P ( I R N F + I ) 0 0 6 2 1 = 1 , 3 H = 0 . 0

I N T E R V A L U S I N G T A N G E N T P R O P E R T I E S V E R T ( Y ) , X Y , A N D Z D I R E C T I O N S

t f E R T ( Y ) , X Y , A N D Z D I R E C T I O N S

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to co o>

D O 6 4 J = i , 6 6 4 H = H + B ( I , J ) * D E F ( J ) 6 2 S T R A I N C I ) = H

C » * * » * M A J O R A N D M I N O R P R I N C I P A L S T R E S S E S H l = 0 . 5 * ( S T R E S S < 1 , N U M ) + S T R E S S ( 2 , N U M ) ) H = . 5 * ( S Q R T ( ( S T R E S S d , N U M ) - S T R E S S C 2 , N U M ) ) * * 2 + 4 . 0 » S T R E S S C 3 , N U M ) » * 2 ) S I G M A 1 = H 1 - H S I G M A 2 = H H - H T A U M A X = H

C » » * » » M O T E N S I O N P R O V I S I O N I F ( R F ( M T ) . E Q . G . O ) G O T O 1 4 C S = 0 . 5 * ( S I G M A 2 + G 1 ( M T ) * ( S I G M A 2 + S I G M A 1 ) * 1 . 1 1 ) I F ( C S . L T . U . O ) G O T O 1 4 0 0 1 6 1 = 1 , 3

1 6 S T R E S S ( I , N U M ) = 0 . 0 W R I T E < 6 , 1 8 ) N U M

11 1 4

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F O R M A T ( 4 X , * E L E M E N T N O * , 1 4 , * I S I N T E N S I O N * ) C O N T I N U E S T R Z = G K M T ) * ( S T R E S S ( 1 , N U M ) + S T R E S S ( 2 , N U M ) ) I F ( 2 . 0 * T A ! J M A X . G T . S T R M A X ( N U M I ) S T R M A X ( N U M I = 2 . 0 * T A U M A X A V G X = ( X P + X Q + X R » / 3 . 0 A V G Y = ( Y P + Y Q « - Y R » / 3 . 0 W R I T F ( 6 , 2 0 ) N U M , A V G X , A V G Y , ( S T R E S S C I , N U M ) , I = i » 3 ) t S T R Z ,

2 ( S T R A I N ( I ) , 1 = 1 , 3 ) , S I G M A 1 , S I G M A 2 , T A U M A X F O R M A T ( 4 X , I 5 , 2 X , F 9 . 2 , 1 X , F 9 . 2 , 4 t 1 X , F 9 . 2 ) , 3 ( 1 X , E 9 « 2 ) » 3 ( 1 X , F 9 . 2 ) ) D O 6 5 0 J = 1 , 6 D E F ( J ) = 0 . 0 D O 6 5 0 1 = 1 , 3 D E F ( J ) = D E F ( J ) + E ( I , J ) * S T R E S S ( I , N U M ) * A D O 6 5 6 1 = 1 , 2 S S ( I P N F + I , 1 ) = S S ( I P N F + I , 1 ) + O E F ( I ) S S ( I Q N F + I , 1 ) = S S ( i a N F + I , 1 ) » O E F ( 1 + 2 ) S S ( I R N F f 1 , 1 ) = S S ( I R N F + I , 1 ) + 0 ^ ( 1 + 4 ) R E T U R N E N O

5 5 1 1 0 4 3 5 5 1 1 0 4 4 5 5 1 1 0 4 5 5 5 1 1 0 4 6 5 5 1 1 0 4 7

) S S I 1 0 4 3 S S I 1 0 4 9 S S I 1 0 5 3 S S I 1 Q 5 1 5 5 1 1 0 5 2 5 5 1 1 0 5 3 5 5 1 1 0 5 4 5 5 1 1 0 5 5 5 5 1 1 0 5 6 5 5 1 1 0 5 7 5 5 1 1 0 5 8 5 5 1 1 0 5 9 5 5 1 1 0 6 0 5 5 1 1 0 6 1 5 5 1 1 0 6 2 5 5 1 1 0 6 3 5 5 1 1 0 6 4 5 5 1 1 0 6 5 5 5 1 1 0 6 6 5 5 1 1 0 6 7 5 5 1 1 0 6 8 5 5 1 1 0 6 9 5 5 1 1 0 7 0 5 5 1 1 0 7 1 5 5 1 1 0 7 2 5 5 1 1 0 7 3 S S I 1 0 7 4 5 5 1 1 0 7 5 5 5 1 1 0 7 6 5 5 1 1 0 7 7 (O

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S U B R O U T I N E S T R E S I S S I 1 0 7 8 C O M M O N N U M , N T , M T , N P A S S , I P N F , I Q N F , I R N F , I S N F , N I T , N I N T , S S I 1 0 7 9

2 X P , X Q , X R , Y P , Y Q , Y R , S ( 3 , 8 ) , 9 ( 3 , 6 ) , D ( 8 , 8 ) , E S B ( 8 , 8 ) , D E F ( 8 ) t S S I 1 0 8 0 3 A , H b , I N D E X , P A T K , O W A T E R S S I 1 0 8 1

C O M M O N E M 0 D ( 1 5 ) » X X I ( 1 5 ) » A R E A ( 1 5 ) » S E C M 0 D ( 1 5 ) » O E N S T Y I 1 5 ) » S S I 1 0 8 2 2 P R A T I O ( 1 5 ) , P H I ( 1 5 ) , C O H ( 1 5 ) , R F ( 1 5 ) , F K ( 1 5 ) , F N ( 1 5 ) , S S I 1 0 1 3 3 F F ( 1 5 ) , G 1 ( 1 5 ) , 0 L ) M 5 ) , F K U R ( 1 5 ) , S S I 1 0 8 < * * » S S ( 2 5 1 , 5 < * ) , P ( 2 6 1 ) , P T £ M P ( 2 6 1 ) , O I S P ( 2 6 1 ) , X ( 1 2 7 ) , Y ( 1 2 7 ) • N N F ( 1 2 7 J « S S I 1 0 8 5 5 N T Y P E ( 1 9 6 ) , M A T Y P E ( 1 9 6 ) , I P ( 1 9 6 ) , I Q ( 1 9 6 ) , I R ( 1 9 6 ) , I S ( 1 9 6 ) , S S I 1 0 8 6 6 S T R < E S S ( 3 , 1 9 6 ) , N V A E ( l c . 6 ) , S T R M A X ( 1 9 6 ) , S S I 1 0 8 7 7 L P ( 2 0 ) , P T ( 2 0 ) , N S ( 5 5 ) , P G K I ( 2 6 1 ) S S I 1 0 8 8

D I M E N S I O N S T R A I N ( 2 ) S S I 1 0 8 9 X L = X Q - X P S S I 1 0 9 0 Y L = Y P - Y Q S S I 1 0 9 1 F L - S Q R T ( X L » X L + Y L * Y L ) S S I 1 0 9 2 I F ( M P A S S . E Q . 3 ) G O T O b k S S S I 1 0 9 3 0 0 2 2 1 = 1 , 2 S S I 1 0 9 ^ H = 0 . 0 S S I 1 0 9 5 0 0 2 U J = l , 2 S S I 1 0 9 6

Z i * H = H + ( B ( I , J ) * P T E M P ( I P N F + J ) + 8 ( I , J ) * P T E M P ( I Q N F + J ) S S I 1 0 9 7 1 - B ( I , J ) * P T E M P ( I R N F + J ) - B ( I , J ) * P T E M P ( I S N F + J ) ) * 0 . 5 S S I 1 0 9 8

2 2 S T R E S S ( I , N U M ) = S T R E S S ( I , N U M > + ( 0 ( I , I ) / 2 . 0 ) * H S S I 1 0 9 9 D O 1 « • 1 = 1 , 2 S S I 1 1 0 0 H = 0 . Q S S I 1 1 0 1 D O 1 5 J = l , 2 S S I 1 1 0 2

1 6 H = H + ( D ( I , J ) * D I S P ( I P N F + J ) • 8 ( I , J ) * D I S P ( I Q N F + J ) S S I 1 1 0 3 1 - d ( I , J ) * D I S P ( I R N F + J ) - 3 ( I , J ) * D I S P ( I S N F + J ) ) * 0 . 5 S S I l l O ^

i e » S T R A I N ( I ) = H S S I 1 1 9 5 C » * * * * S H F A R S T R E S S R E V E R S A L F R O V I S I O N S S I 1 1 0 6

I F ( S T R A I N ( l ) . L T . 0 . 0 . A N D . S T R E S S ( 1 , N U M ) . G T . 0 . 0 ) S T R E S S ( 1 , N U M ) = S S I 1 1 0 7 1 ( 0 ( 1 , 1 ) / 2 . 0 ) * S T R A I N ( 1 ) S S I 1 1 0 8

I F ( S T R A I N ( l ) . G T . 0 . 0 . A N D . S T R £ S S ( l * N U M ) a L T « 0 « 0 ) S T R E S S ( 1 , N U M ) a S S I 1 1 0 9 1 ( 0 ( 1 , 1 ) / ? . 0 ) * S T R A I N ( 1 ) S S I 1 1 1 0

C » * * * * N O T E N S I O N P R O V I S I O N S S I 1 1 1 1 I F ( R F ( M T ) . E Q . 0 . 0 ) G O T O 3 6 S S I 1 1 1 2 I F ( S T R E S S ( 2 , N L M ) . L T . 0 . 0 ) G O T O 3 6 S S I 1 1 1 3

to OJ 00

S T R E S S ( 1 , N U M ) = 0 « 3 S S I 1 1 1 4 S T R E S S ( 2 , N U M ) = 0 « 0 S S I 1 1 1 5 W R I T E ( 6 * 1 8 ) N U M S S I 1 1 1 6

1 8 F O R M A T ( * » X , * E L E M E N T N O * , I < » , * I S I N T E N S I O N * ) S S I 1 1 1 7 3 6 C O N T I N U E S S I 1 1 1 8

A V G X = ( X P + X Q ) / 2 . 0 S S I 1 1 1 9 A V G Y = ( Y P * Y G ) / 2 . 0 S S I 1 1 2 0 I F ( A O S ( S T R E S S ( 1 , N U M ) ) . G T . A 8 S ( S T R M A X ( N U M ) ) • O R . A B S ( S T R M A X ( N U M ) - S S I 1 1 2 1

2 S T ^ E S S ( 1 , N U M ) ) . G T . A f c ) S ( S T R M A X ( N U M ) ) ) S T R M A X ( N U M ) ^ S T R E S S ( 1 » N U M ) S S I 1 1 2 2 S T R 1 = S T R E S S ( 1 , N U M ) S S I 1 1 2 3 S T R ? = S T R E S S ( 2 , N U M ) S S I 1 1 2 « » W R I T E ( 6 , 3 0 ) N U M , A V G X , A V G Y , S T R 1 , S T R 2 t ( S T R A I N ( I ) , 1 = 1 , 2 ) S S I 1 1 2 5

3 0 F O R M A T ( , 1 5 , 2 X , F 9 . 2 , I X , F 9 . 2 , 2 ( I X , F 9 . 2 ) , 2 0 X , 2 ( I X , E 9 . 2 ) ) S S I 1 1 2 6 6 < » 8 D E F ( 1 ) = ( 3 ( 1 * 1 ) * S T R E S S ( 1 , N U M ) + 3 ( 2 , 1 ) * S T R E S S ( 2 , N U M ) ) * E L / 2 » 0 S S I 1 1 2 7

O E F ( 2 ) = ( 0 ( 1 , 2 ) * S T R E S S ( 1 , N U M ) + B ( 2 , 2 ) * S T R E S S ( 2 , N U M ) ) * E L / 2 . 0 S S I 1 1 2 8 D O 6 7 0 1 = 1 , 2 S S I 1 1 2 9 S S ( I P N F * I , 1 ) = S S ( I P N F + I , 1 ) + D E F ( I ) S S I 1 1 3 0 S S I i a W F « - I , l ) s S S ( I Q N F * I , l ) + n E F ( I ) S S I 1 1 3 1 S S ( I R N F + I , l ) = S S ( I R N F + I , l ) - O E F ( I ) S S I 1 1 3 2

6 7 0 S S ( I S N F + I , l ) s S S ( I S N F « I , l ) - O E F ( I ) S S I 1 1 3 3 R E T U R N S S I 1 1 3 ^ £ N C S S I 1 1 3 5

to co (O

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