Chapter 3 pile foundation

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A METHOD FOR THE ANALYSIS OF PILE SUPPORTED FOUNDATIONSCONSIDERING NONLINEAR SOIL BEHAVIORby

Frazier Parker Jr.William R. Cox

Research Report Number 117-1

Development of Method of Analysis of DeepFoundations Supporting Bridge BentsResearch Project 3-5-68-117

conducted for

The Texas Highway Department

in cooperation with theU. S Department of TransportationFederal Highway AdministrationBureau of Public Roads

by the

CENTER FOR HIGHWAY RESEARCHTHE UNIVERSITY OF TEXAS T AUSTIN

AUSTIN TEXAS1 June 1969


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he op1n1ons findings and conclusionsexpressed in this publication are thoseof the authors and not necessari ly thoseof the Bureau of Public Roads.


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PREFACE

This study presents a procedure which was developed for analysis ofpi le supported foundations.

In this study special emphasis is placed on pi le supported bridgebents. Two bridge bents which were designed and constructed by the TexasHighway Department have been analyzed.

The computer program included in this report is a modification of aprogram developed at The University of Texas at Austin by Lymon C Reese andHudson Matlock. The program is writ ten in FORTRAN IV. I t was developed forthe CDC 6600 system but i t is also operational on the IBM 360 system.

The assistance and advice of Messrs. H D Butler Warren Grasso andFred Herber of the Texas Highway Department and Mr Bob Stanford of the Bureauof Public Roads is great ly appreciated.

June 1969

i i i

Frazier Parker JrWilliam R Cox


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BSTR CT

This report contains a review of exis t ing methods of analysis offoundations supported on pi le groups consisting of ver t ical and bat ter pi lesThe method of analysis developed at The University of Texas a t Austin refer redto here as the UT method is modified to take into account the interact ioneffect of axial and l a tera l loading and also to consider some special boundarycondit ions associated with bridge bents .

The study also compares the UT method with other methods of analysisavai lable bringing out i t s features and advantages. The assumptions andl imitat ions involved in the U method are indicated.

generalized computer program has been wri t ten to aid in the solut ionof the problem. With the aid of this computer program i t is possible to takeinto account the nonlinear behavior of the so i l with respect to applied load.Documentation of the program is provided in the form of a l i s t of the notationused a l i s t ing of the program including subroutines and forms necessary forinput of data. Two example problems are solved using the computer program.

complete l i s t ing of input and output data for the example problems is pro-vided.

v


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PREFACE.

ABSTRACTLIST OF TABLESLIST OF FIGURESNOMENCLATURECHAPTER I INTRODUCTION

TABLE OF CONTENTS

CHAPTER II METHODS OF ANALYSIS OF BATTER PILE FOUNDATIONSGENERAL CONSIDERATION OF PROBLEM

Culmann s MethodVetter s Method.Hrennikoff s Method

COMPARISON OF METHODS WITH UT METHOD

Two Dimensional ConfigurationRigidity of the FoundationConnection of Piles to the FoundationPi le-Soi l Interact ion . . .Load Movement Relationships

CONCLUSIONSCHAPTER I I I THEORETICAL DEVELOPMENT

PURPOSE

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469

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13131415161717

COORDINATE SYSTEMS ND SIGN CONVENTIONS 17RELATIONS BETWEEN FOUNDATION MOVEMENTS ND PILE HEAD MOVEMENTS 21RELATIONS BETWEEN FOUNDATION FORCES ND PILE REACTIONS . . . . 23

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PILE HEAD MOVEMENT ND PILE REACTIONEQUILIBRIUM EQUATIONS

CHAPTER IV BEHAVIOR OF INDIVIDUAL PILES.AXIAL BEHAVIOR

Dynamic FormulasStat ic FormulasFull Scale Loading TestConclusions

LATERAL BEHAVIORFinite Difference Solution for Laterally Loaded PilesLateral Soil Pi le In teract ionSoil r i ter iaConclusions

CHAPTER V. COMPUTATIONAL PROCEDUREOUTLINE OF PROCEDURE FOR BENT 1

CHAPTER VI. EXAMPLE PROBLEMSGENERAL CHARACTERISTICS OF EXAMPLE PROBLEMSCOPANO B Y CAUSEWAYHOUSTON SHIP CHANNEL

CHAPTER VII. SUMM RY ND CONCLUSIONSAPPENDIX A. GUIDE FOR DATA INPUTAPPENDIX B. FLOW CHART FOR BENT 1APPENDIX C GLOSSARY OF NOTATION FOR BENT 1APPENDIX D. LISTING OF DECK FOR BENT 1

Page24263131333334343435464956575761616168757999

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PPENDIX E CODED INPUT FOR EX MPLE PROBLEMSPPENDIX F OUTPUT FOR EX MPLE PROBLEMS

REFERENCES

ix

age157163183


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LIST OF TABLES

I PILE LOCATION INFORMATION COP NO B YII PILE LO DS ND MOVEMENTS COP NO B Y . .III PILE LOCATION INFORMATION SHIP CH NNELIV. PILE LO DS ND MOVEMENTS SHIP CH NNEL

x

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LIST OF FIGURES

1. Graphical representation of Cu1mann s method2. Pile simulation for Vetter s method3. Dummy pile representation of Vetter s method4. Pile constants for Hrennikoff s method5, Foundation constants for Hrenniko ff s method.6. Geometry of foundation .7. Sign convention for foundation forces and movements8. Forces and moment on pi le head9. Pile head movements - x-y coordinate system

10. Movements of pile head - structural coordinate system11. Spring representation of pi le12. Hypothetical spring load-deflection curves13. Forces on the pi les and foundation14. Axial load-settlement curve15. Generalized beam column element16. Fini te difference representation of pi le17. Typical p-y curve18. Variat ion of soi l properties with depth19. Construction of p-y curve20. Stress-s t rain curve21. Approximate log-log plot of s t ress s t ra in curve22. Block diagram for i te ra t ive solution23. Copano Bay Causeway bent . . . . . . .

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111118182202225252732364047485050545862


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24. Foundation representat ion Copano Bay.25. Load deflection curve Copano Bay26. Soil Properties for generat ion of p y curves Copano Bay27. Houston Ship Channel bent28. Foundation representat ion Ship Channel29. Estimated axial load deformation curve Ship Channel30. Soil propert ies for generation of p y curves Ship Channel

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SymbolaAb

cC

E

Es

Fvh

I

J xJ yJm

KoM

p

Typical Unitsin

in

lb

lbin

41n

lb / in

lb / in

in lb/ in

in lbin lb

lb

NOMENCL TURE

Defini t ionHorizontal distance to pi le topRecursive coeff iceintVertical distance to pi le topRecursive coeff ic ientCohesionRecursive coeff ic ientModulus of elas t ic i ty Pile)Soil modulusHorizontal load on pi le

Vertical load on pi leIncrement lengthPile moment of iner t iaAxial secant modulusLateral secant modulusMoment secant modulusCoefficient of active ear th pressureCoefficient of passive earth pressureMomentMoment on pile topSoil reactionVertical load on foundation

x


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xvi

Symbol Typica1 Units DefinitionP lb Horizontal load on foundation

R l b- i n2 Pile s t i ffness E1)V lb Shearw in Pi le diameter or pi le widthxt in Axial movement of pi le topX in Distance from so i l surface

in Lateral pi le def lect ionQ rad Rotation of founda t ion

lb in3 Soil uni t weigh tll in Horizontal foundation movementw in Vertical foundation movement in in Strain3 rad Pile bat te rJ Ib/in Stressall lb in2 Deviator s t res s 0 1 - 3 ):b degrees Angle of internal f r ic t ion


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CH PTER IINTRODUCTION

The purpose of this study is to review and expand upon exis t ingmethods for analyzing foundations which are supported on pile groups consis t ing of vert ical and bat ter piles The expansions of the exis t ing methods areaimed a t solutions for problems of bridge bents supported on pil ing. t isbelieved that the resul t ing method will apply equally well to other types ofpi l ing supported foundations i f the cap connecting the piles is r igid inre la t ion to the f lex ib i l i ty of the pile .

When a grouping of ver t ical piles is subjected to horizontal loadingthe s t if fness of the piles may resu l t in a portion of the horizontal loadbeing t ransferred to the lower so i l s t ra ta larger portion of the horizontalload will be t ransferred directly to the upper so i l layers as the piles bendla te ra l ly f the upper so i l layers are weak and highly compressible thel a tera l def lect ion which occurs may be excessive.

By using batter piles in a pi le grouping the portion of the horizontalload t ransferred to the upper so i l layers is reduced since the component ofthe horizontal force paralle l to the axis of the bat ter pi le i s t ransferred tothe lower s t ra ta through axial loading. This t ransfer of horizontal loadinto axial load in bat ter pi les will usually reduce the def lect ion of the pilegroup since piles are s t i f f e r under axial loading than under bending typeloading and the lower so i l s t r a t a are usually s t i f fe r than the upper soils t ra ta

t is desirable to know the forces on each pile and the load def lect ionbehavior of each pi le in order to make a more complete appraisal of the adequacy of a pile-supported foundation. When only ver t ical piles are used and


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2

the only load applied is a ver t ical load through the centroid of the pilegroup the ver t ical load is dis t r ibuted equally to the individual pi les andonly the axial behavior of the pi les need be considered. However i f hori-zontal loads are also applied and i f batter piles are included in the pilegroup then the problem becomes more complex.

A number of methods have been proposed for analyzing the generalproblem of ver t ica l and horizontal loading on a pile group which consists of

ver t ica l and bat ter piles . All of these methods involve approximations andassumptions but four methods have been selected which have a degree ofra t iona l i ty in their approach. Three of these four methods are outl inedbrief ly and the l imita t ions assumptions and approximations involved in thesethree methods are noted and compared with the fourth method which was devel-oped a t The University of Texas a t Austin by Lymon C Reese and HudsonM 1 k8 14 15at oc The method developed by Reese and Matlock referred to inthis report as the UT method was intended for use in analyzing off-shoredr i l l ing platforms which are supported on ver t ica l and batter pi les but themethod has been applied successful ly to other types of pile supported s t ruc-tures. The UT method has several def in i te advantages over other methods.These advantages will be discussed.

In this report the UT method will be presented with cer ta in modifica-t ions and additions as formulated by the author. The basic procedures involvedare not changed from those developed by Reese and Matlock but some al terat ionshave been made for the solut ion of individual l a tera l ly loaded piles . A pro-cedure is also presented for introducing the so i l propert ies into a calcula-t ion of the l a tera l in teract ion of the pile with the so i l . The modificationsto the UT method were incorporated into a computer program and two exampleproblems are solved by using the program.


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CH PTER I IMETHODS OF N LYSIS OF B TTER PILE FOUND TIONS

GENER L CONSIDER TION OF PROBLEMA procedure is available for design of pile supported foundations in

which a l l the piles are ver t ical and in which the applied loads may be resolvedinto a ver t ical force through the centroid of the pi le group. The procedureinvolves two steps. Firs t the allowable be aring capacit ies of the individualpiles are obtained by applying an appropriate safety factor to the ultimatecapacit ies of the piles as determined ei ther from load t es t s from drivingcharac ter i s t ics or from other theoret ical procedures. Second, the to ta lapplied load is divided by the number of piles in the foundation to obtainthe load on each pile . I f this load does not exceed the allowable bearingcapacit ies of the individual piles then the design is considered adequate.Terzaghi and Peck24 also recommended that the design be checked by computingthe allowable bearing capacity of the pile group against breaking into theground as a unit .

The above procedure for ver t ica l piles and ver t ica l loads gives noindication of the deflections which occur for intermediate loads, but onlythe allowable load which may be sustained with a safety factor againstexcessive settlement of the foundation. The procedure must also be consideredas an approximation since i t i s fe l t that al l piles do not carry the same load.The load which is carr ied by a pile is inf luenced by the spacing of adjacentpiles but the exact relationship of this influence is not known This in -fluence is frequently estimated by empirical rules of thumb or approximations.

I f the pi le group includes bat ter as well as ver t ical piles and i fthe group is subjected to horizontal and ver t ica l loading, the analysis be-comes more complicated. In a rigorous analysis the horizontal and moment

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resis tance offered by the piles must be considered, as well as the axialres is tance. b 19 d h do ertson 1scusses some 0 t e assumptions an approximationsfrequently employed to handle horizontal and moment resistance. Robertsonpoints out that some of these assumptions may misrepresent a bat te r pilestructure and that the methods of analysis which employ these assumptions mayhave l imited usefulness due to the inaccuracy of the approximations involved.

There is some degree of approximation in al l methods which have beenproposed for the analysis of foundations supported by bat ter pi les . Briefdiscussions of the methods proposed by Carl Culmann as reported by Terzaghi

24 25 7and Peck C . P. Vetter and Alexander Hrennikoff will be presented inthe following sections. The discussions will include l i s t s of the l imi tat ionsand approximations involved in each method. These methods are considered tobe representat ive of the available methods for analysis of foundations support-ed on bat ter piles .

Culmann s Method24According to Terzaghi and Peck t h e method proposed by Carl Culmann

is based on the resolution of the applied force into three components. Thesecomponents act in directions para l le l to the axes and through the centroid ofthree pile groups which support the foundation. A pi le group is defined asal l piles driven in a par t icu lar direct ion, and Culmann s method requires thatthe foundation be supported by three pi le groups. The basic procedure isshown graphically in Fig. 1. Definitions are as follows:

R Force applied to foundationComponent of force R acting on and paralle lto pi le groups 1, 2 and 3 respectively.


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\\\\\\\\\\

roup 3

P

----, I1 II\ I

\ I\ I\ I\ I\\ I

roup roup I

Fig. 1. Graphical representation of Culmann s method.

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The method is subject to the following l imitat ions:1. Solution is l imited to two dimensional configurations.2. The foundation must be supported by three nonparallel

groups of piles .3. No load-displacement relat ionships are considered for the

foundation or the pi les .

The assumptions and approximations involved are as follows:1. The pi les develop only-axia l forces.2 The foundation is s t a t i ca l ly determinate.

Vetter s Method25The method presented by C P Vetter is s imilar to the methods

developed ear l ier by Swedish engineers. Vetter mentions a number of ear l ie rworks in the acknowledgments to his paper.

This method ut i l izes the concept of an e las t ic center (center of

rotat ion) about which the foundation rotates . Forces through the elas t iccenter cause only t ranslat ion without rotat ion while a moment about thee las t ic center wil l cause a rotat ion without t ranslat ion. This t ranslat ionand rotat ion of the foundation wil l cause movement of the pi le heads. Themethod proposed by Vetter consis ts of locating the e las t ic center of thefoundation, and determining the forces required to produce small e las t ic de-formations in the pi les . The applied loads are resolved into a force throughthe e las t ic center , and a moment about the e las t ic center. By adjust ing theapplied forces in re la t ion to the forces required to produce elas t ic deforma-t ions in the pi les the forces on the pi les due to the applied load may befound.


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Axial, l a tera l , and rotat ional resistances of the piles are consi-dered. The forces developed will correspond to an axial def lect ion, a la te ra ldeflection and a rotat ion of the pile head. The l a tera l pi le res is tanceoffered by the pi le is simulated by assuming the pi le fixed a t some depthh as shown in Fig. 2. The pi le may be considered as pinned or fixed to

the structure , depending on the rotat ional res is tance offered by the pile .The effect of l a tera l and rotat ional res is tance is simulated by intro

ducing imaginary dummy piles perpendicular to the real pi les and consideringthe real piles as columns, pinned to the footing and pinned at some depth inthe soil . The dummy piles are also considered as pinned columns.

y introducing dummy piles the la te ra l load-deformation characteri s t i c s are simulated by the axial behavior of the dummy piles . The locationand length of the dummy pi les wil l depend on the manner in which the pi Ieis connected to the s t ructure and the location of the point of f ixi ty . Thecross-sectional area of the dummy pile is expressed in terms of the crosssectional area and s t if fness of the real pile . I f the pile shown in Fig. 2is considered fixed to the structure , the dummy pi le representat ion isshown in Fig. 3.

With the representat ion shown in Fig. 3, the res is tance of the pileis simulated by axial forces in the pin-connected columns. The magnitude ofthe axial forces in the columns are determined by the force and moment throughand about the elas t ic center , and by the location of the pi le head. From theforce in the pinned column representing the axial behavior of the real pile ,the axial pi le movement may be predicted. However, no method is available forpredicting the l a tera l pi le movement or the foundation movement.

Vet te r s method is subject to the following l imi tat ions :1. Solution is l imited to two-dimensional configurations.


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onnection May e Pinned or Rigid

Fig. 2. Pile simulation for Vetter s method.

Fig. 3. ummy pil representat ion for Vetter s method.


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2. o method is suggested for determining the point of f ixi ty3. Load-deformation behavior is limited to axial character is t ics

of pinned columns.4. o prediction of foundation movement is possible.

The assumptions and approximations involved are as follows:1. The foundation is r igid so that the pi le tops maintain the

same re la t ive posit ions.2. Pile deformations are e las t ica l ly proportional to the applied

loads.3. The pile which is loaded la te ra l ly along i t s ent i re length

may be simulated by a canti lever system.4. The behavior of a real pile may be simulated by pin-connected

columns.

Hrennikoff s Method

9

The method presented by Alexander Hrennikoff7

in 195 containedseveral important advances in technique. Probably the most important was theconcept of a relat ionship between pi le resistance and pile movements. Important relat ionships between movements and footing geometry were also developed.

The procedure consists of obtaining expressions for the forces and mo-ments exerted on the structure by the piles resul t ing from a unit horizontalt ranslat ion, a unit ver t ica l t ranslat ion, and a unit rotat ion of the s t ructure.These forces and moments are summed in three equations of equil ibrium, which aresolved simultaneously for the movements of the foundation. Movements of thestructure are related to the movement of the pi le heads through the geometryof the s t ructure. The movements of the pile heads are related to the forceson the pile heads through a se t of pile constants. I f these constants are


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known and the pile-head movements are known the pile forces and moments maybe found.

Hrennikoff defines the pile constants as the forces with which thepile acts on the foundation when the pile head is given a unit displacement.There are three sets of constants, corresponding to three di f fe rent kinds ofdisplacements. The five pi le constants (n, t 6 m6 t a mJ are shown inFig. 4 with the corresponding displacements t 6t a).5y the Betti theorem t ma 6 leaving only four pi le constants .The pile constant n is evaluated using an approximate formula. The con-s tants and m are evaluated by considering the pile as a beam onaan e las t ic foundation of in f in i te length, loaded a t the free end. The e las t icmodulus of the so i l is evaluated using approximate formulas developed by theauthor.

The pi le constants , number of piles and the geometry of the founda-t ion are combined to evaluate the foundation constants. The foundation con-s tants are defined as the resul tant forces with which l l pi les act on thefooting, when the footing is given a unit t ranslat ion in the posi t ive direct ionof one of the axes, or a unit rota t ion about the origin in a clockwise direc-t ion. The coordinate system and the foundation constants are shown in Fig. 5.The constants X Y M X Y M X Y and M are obtainedx x x y y y a a aby giving the foundation a displacement x 1, Y = 1 or a = 1 as mentionedpreviously.

y the Betti theorem Y = Xx y M Xx a and M = YY a leaving onlys ix constants to be evaluated. The equations of equilibrium for the footingare then


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Fig. 4. Pile constants for Hrennikoff s method.

, , : . ~

L . f . . . . - c : : ; . ~ . . . . . . . ~I::;::y

Fig. 5. Foundation constants for Hrennikoff s method.


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X 6 X 6 X X 0x x y yX 6 Y 6 Y Y 0Y x Y yX 6 Y 6 M M 0x y

where X Y and M are the forces and moment applied to the footingthrough and about the origin of the coordinate system. Once the structuremovements 6 , 6 and a are found the forces and moments exerted by thex ypiles may be found by working backwards. The movements of the pi le head mayalso be found.

Hrennikoff s method is subject to the following l imitations:1. Solution is l imited to two-dimensional configurations.2. All piles must behave alike with regard to the load-deformation

re la t ion.

The approximations and assumptions involved are as follows:1. Pile deformations are elas t ica l ly proportional to the applied

loads.2. The foundation is r ig id so that the pi le tops maintain the

same re la t ive posit ions.3. Foundation movements are small.4. The piles are in f in i te in length.

COMPARISON OF METHODS WITH DT METHODBefore beginning a detailed presentat ion of the DT method the basic

assumptions involved in the method will be presented and compared with assump-t ions in the three methods previously discussed. I t is fe l t that the advan-tages of the DT method will be apparent af ter this discussion.


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Two Dimensional Conf igura t ionThe methods o f Vet te r , Culmann, and Hrennikoff are l imi t ed to the

ana lys is of two dimensional problems, This does not l im i t the so lu t ion tofoundations with p i l es in only one plane, t does, however, l imi t the solu-t ion to problems which have a l l p i l es para l le l with , and symmetr ical withr e spec t to a v e r t i c a l plane o f symmetry, Simi l a r ly the r esu l t an t o f a l lexte rna l forces and moments must be loca ted in the plane o f symmetry,

The UT method i s a lso sub jec t to the l imi t a t ion o f two dimensionalanalys is , There are s t ruc tu res for which a three dimensional so lu t ion i sdes i rab le , However, for many prac t i ca l eng ineer ing problems a two dimensional

1 ff Th d . 1 1 . 1,21 '1 bl bna YS1S 1S su 1C1ent . ree 1menS10na so ut10ns are ava1 a e u twil l not be considered in t h i s s tudy,

Rig id i ty o f the Foundat ionCulmann's method, s ince it considers only equ i l ib r ium of the founda-

t ion , requi res no assumptions concerning the r i g id i t y of the foundat ion, ForV e t t e r s and Hrennikof f s methods, as well as the UT method, the p i l e cap i sassumed to be r ig id so t ha t the p i l e heads maintain the same r e l a t i v e pos i t ionsbefore and a f t e r movement.

Connect ion o f Pi les to the FoundationNo cons ide r a t ion i s given to the method o f connect ing the p i l es to the

foundation in Culmann's method s ince the ana lys is i s based on each p i l e groupexe r t ing a r esu l t an t fo rce p a r a l l e l to the p i l es in tha t group, For themethods of Vet te r and Hrennikoff the p i l e s may be f ixed o r pinned to thes t ruc tu re , For the UT method the p i l es may be f ixed, pinned or a t tached insuch a manner tha t the foundat ion exer ts some constant ro ta t iona l r e s t r a in t


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on the pile . That i s , the moment on the top of the pi le divided by theslope a t the top of the pi le will be a constant.

Pile-Soi l Interact ionFor Cu1mann s method no pi le-so i l interact ion is considered. Vet ter s

method simulates the axial interact ion by considering the pi le as a column.The l a tera l interact ion is simulated by considering the pi le as a beam witha fixed end.

The axial interact ion, for Hrennikoff s method, is characterized bya constant. This constant is obtained by considering the axial compressionfor the pi le as i f t were a free standing column. The l a tera l interact ionis characterized by a set of three constants obtained by considering thepile as a beam of in f in i te length on an elas t ic foundation.

For the UT method the axial pi le-so i l interact ion is obtained froma load-deformation curve. o specif ic pi le-so i l interact ion i s speci f ied,but the overal l axial behavior is speci f ied by the load-deformation curve.The l a tera l interact ion is speci f ied by a set of def lect ion-react ion curves.These curves, refer red to as p-y curves, establ ish the re la t ionship betweenthe deflect ion of the pi le and the react ion exerted by the so i l . These curvesare nonlinear as opposed to the l inear behavior for the methods of Vetter andHrennikoff. The procedure for obtaining p-y curves and the manner in whichthey are used in the analysis will be discussed l a ter , but the point to beemphasized here is that in the UT method the so i l -p i le interact ion is nonl inear as compared to the l inear behavior which is assumed for the othermethods of analysis .

Soils do not deflect l inear ly under load. This can be seen by notingthe nonlinear shape of the s t ress-s t ra in curves for soils as obtained from


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t r iax ia l tes t This would indicate that a consideration of a nonlinear inter-action will yield more rea l is t ic resu l t s

Load Movement RelationshipsSince Culmann s approach is based only on equations of equilibrium,

no predict ion of the movements resul t ing from the applied loads is possible.Similarly, Vetter s method provides no means for predict ing foundation move-ment.

With Hrennikoff s method the foundation movement is defined by a horizontal and ver t ical t ranslat ion and a rotat ion. These movements are relatedto the forces on the foundation by a set of foundation constants. The rela-tionship between applied load and foundation movement is l inear since theyare re la ted through a set o f constants . Similarly the force-deflect ion rela-t ionship between pile-head movement and applied force is l inear since theyare re la ted by the pi le constants.

For the DT method the movement of the foundation is defined by twot ranslat ions in the direct ion of the established coordinate system, and arotat ion about the or igin of the coordinate system. The loads on the foundat ion are resolved into two forces through the or igin of the coordinate systemand a moment about the or igin. The movements of the pi le heads are re la tedto the foundation movement by the geometry of the system. The forces on thepile heads are related to the pile-head movements by nonlinear factors . Allof these re la t ions are combined into three equations of equilibrium for thefoundation. From these equations the three movements of the structure areobtained. Since the relat ionships between pile-head def lect ion and pi lereaction are nonlinear , an i t era t ive process is necessary for es tabl ishingan equilibrium posit ion for the structure . Once the equil ibrium posit ion isfound, the def lect ion of the pi le head and reactions may be obtained.


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CONCLUSIONSThe UT method and Hrennikoff s method offer several major advantages

over the methods of ulmann and Vetter . The method of ulmann was the f i r s tmethod proposed and i t is l imited by i t s fa i lure to consider def lect ion ofthe foundation system. Vetter s method was the next method proposed and i tintroduces several improvements, but i t is s t i l l l imited by several assumpt ions.

The method of Hrennikoff and the U method are similar in theirapproach. However, the UT method introduces two major improvements. Probablythe most important of these is the use of nonlinear pi le so i l resistance relat ionships. The second major improvement of the U method is that t permitsthe rotat ional s t if fness of the structure or pile-head res t ra in t to be in cluded in the analysis .


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PURPOSE

CH PTER

THEORETIC L DEVELOPMENT

n the following sections the theory involved for the UT method wil lbe developed. n the f i r s t section the coordinate systems and sign convent ions for movements and forces wil l be established. n the second sect ionthe relat ionships between foundation movement and pile-head movements wil lbe developed. Relations between foundation forces and pi le reactions areestablished in sect ion three. n the fourth sect ion re la t ions between pi1ehead movement and pi le reaction wil l be developed. n the final section theequilibrium equations will be established.

COORDIN TE SYSTEMS ND SIGN CONVENTIONSTwo types of coordinate systems are established. Examples are i l lus -

t ra ted in Fig. 6. horizontal axis a and a ver t ica l axis b are establ i shed re la t ive to the foundation. Foundation movements, forces and dimensions are re la ted to these axes. The location of th is system is completelyarbitrary , but proper location wil l simplify calculat ions for most foundat ions.

For each pi le an x-y coordinate system is established. The xaxis is paralle l to the pi le and the y axis is perpendicular to the pile .Subscripts are used to indicate the part icular pile Pile def lect ion andforces are related to these systems.

The coordinates of the pile heads as re la ted to the a-b axes areshown in Fig. 6. n the example al l coordinates are posi t ive. The batter of

7


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18

b

01 +) ... .I0 21 1

b

Fig 6 Geometry of foundation

////

M11

P 1 PVI 1 /

Fig 7 Sign convention for foundationforces and movements

o

o


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19

the piles is posi t ive counter clockwise from the ver t ical and negative clockwise from the ver t ical as shown.

The external loads on the foundation are resolved into a ver t ica land horizontal component through the origin of the s tructural coordinatesystem and a moment about the or igin. The sign convention established isi l lus t ra ted in Fig. 7.

The external loads M PV and PH will cause the foundation tomove. I f the a-b coordinate system is considered to be r igidly attached tothe foundation, the movement of the foundation may be re la ted to the movementof the coordinate system. These movements (6V, 6H, and are shown inFig. 7 with posi t ive signs.

Due to the movement of the foundation, forces will be exerted on thefoundation by the piles The sign convention for these forces is i l lus t ra tedin Fig. S.

The sign conventions i l lus t ra ted by Fig. Sa are consistent with thosepreviously established for the structure . The conventions i l lus t ra ted byFig. Sb are consistent with those established in the solut ion of l a tera l lyloaded pi1esS The differences should be careful ly noted. The inconsistenciesare taken care of when the re la t ions between foundation forces and pi le forcesare developed.

The sign conventions for movements of the pi le head are consistentwith the x-y coordinate system. A movement in the posi t ive x direct ion,which const i tutes an axial compression, is considered as a posi t ive movement.A movement in the posi t ive y direct ion is considered as a posi t ive move-ment. A rotat ion of the pi le head wil l cause a change in the slope at the topof the pile The sign convention for slope is consistent with the usual


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2

p in

P in

a Forces and moment structuresign convention b. Forces and moment pi lsign conventionFig. 8 Forces and moment on pi l head.

Fig. 9. Pile head movementsx-y coordinate system.


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manner in which slope is defined. The movements of the pi le head are i l lus-t ra ted in Fig. 9.

RELATIONS BETWEEN FOUNDATION MOVEMENTS ND PILE HEAD MOVEMENTS

21

When the structure moves the pile heads move Two assumptions aremade in order to re la te structure movement to pile-head movement The f i r s tassumption is that the foundation is r igid so that the pile heads maintainthe same re la t ive positions before and after'movement. The second assumptionis that the foundation movements are small. Because of this assumption theapproximation

t R::: tan t 1)

is val id.In Fig. lOa diagrams are given of the l ineal movements a t the pi le

head of a given pile in terms of the s t ruc tura l movements. The movement ofthe structure is defined by the sh i f t of the a-b axes to the posit ion indicated by the a b axes. The pi le head movement is from point Q to pointQ/. The tota l movement of the pi le head is resolved into a component paralle lto the a axis Llli + bet) and a component para l le l to the b axisIN + aet).

Figure lOb i l lus t ra tes the resolution of the horizontal and ver t icalcomponents of movement into components paralle l and perpendicular to the direct ion of the pi le These movements are designated as xt and Yt. ConsideringFig. lOb the axial component of pi le head movement may be writ ten as

l i l i + bet) sin e + IN + aet) cos e (2)and the corresponding la te ra l movement as


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b

fb

JJJIJ

I

a

Pile ead

b

- \VPileIII

O ~ - r - - - ~ - - ~ ~ - - - - - - - - - - - - - - - - - - - - ~ - - - - 4 1 - - - - ~0.:..__________ :H ___ I/

a. Lineal movements of pi l head.

b. Resolution of movement into components.

---- '0'JIIIIII

Fig. 10. Movements of pil head s tructura lcoordinate system.


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23

6H ba cos e - 6V aa s in e 3)In addition to the l ineal displacements of the pile head, the change

in slope of a tangent to the elas t ic curve will be considered. The change inthe slope wil l depend on the manner in which the pile is attached to thefoundation. I f the pile is fixed to the s t ructure , then the change in slopewil l be equal to the rota t ion of the foundation. For the restrained case thechange in slope will depend on the moment applied to the pi le top. For apinned connection the slope will depend on the deflected shape of the pile .

RELATIONS BETWEEN FOUNDATION FORCES AND PILE REACTIONSThe forces acting on the foundation and pi le are i l lus t ra ted along

with sign convention, in Fig. 8. I t has been noted that inconsistencies inthe sign conventions are present. These will be taken care of in the re la -t ions between the forces.

Considering Fig. 8 the relationship between moments on the s t ructureand moment on the pile may be writ ten as

Ms -M t 4)

The re la t ions between forces are obtained by resolving the forces on the pi leinto components in the horizontal and ver t ical directions. With the sign con-ventions considered, the components are summed as follows:

Fv Pt s in e - Px cos e-P s in e - P cos ex t

5)

6)


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24

PILE HEAD MOVEMENT ND PILE REACTIONIn the preceding sections the movement of the pi le head and the forces

acting on the pi le head have been defined. In th is section relations betweenpile reaction and movement will be developed.

For computational purposes the pi le shown in Fig. l la may be simulatedby the set of springs as shown in Fig. l lb . The springs will produce a forceparal lel to the pi le axis, Px and a force acting perpendicular to the pileaxis, Pt The rotational spring will produce a moment about the pi le top,

The forces produced by the springs will depend on the deflection ofthe springs. Since the springs are nonlinear the movement and reaction arenot related by a single constant. t is assumed that curves can be obtainedwhich show spring reaction as a function of deflection. In Fig. 12 a hypo-thet ical set of load-deflection curves are drawn for a set of springs. fthe curves are single valued then the spring reactions may be calculated fora part icular deflection by

p J x (7)x x t

Pt = J Yt (8)yM J Yt (9)t m

where J J y and J are the secant modulus values as i l lus t ra ted inx mFig. 12.

t should be noted that the moment produced by the rotational springis proportional to the lateral deflection, rather than the rotation. For arota t ional spring th is procedure is inconsistent with usual concepts. This


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25

a. Pile and foundation. b. Springs and foundation.Fig. 11. Spring representat ion of pile .

P J P Y

Y

J ,= -M, Y

YFig. 12. Hypothetical spring load-deflection curves.


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26

concept is used because t provides a convenient means for deriving andsolving the equilibrium equation for the s t ructure.

The curves shown in Fig. 2 do not adequately explain the behavior ofa pile . t is not necessary that the exact nature of the curves be known.The representation shown is only for the formulation of the equil ibriumequations. The procedure for calculat ing values for J x J y and J wil lmbe discussed in the following chapters . However for the formulation of theequilibrium equations Eqs. 7 8 and 9 are suf f ic ien t since they wil l beapplicable no matter what kind of relat ionship exis ts between the loads andthe displacements.

EQUILIBRIUM EQUATIONSThe relat ions between forces and movements for the structure and the

pi le have been developed in the preceding sections. In this sect ion theserelat ions wi l l be combined to form three equations of equil ibrium for thestructure . The form of the equations i s such that an i te ra t ive type solutionmay be used. This is necessary since the system is nonlinear.

Consider a foundation supported by n piles . The coordinate systemand the i t h pi le are shown in Fig. 13. The external loads applied to thefoundation are resolved in to the forces and moment through and about the or iginof the coordinates as shown in Fig. 13. The forces and moment exerted by eachpi le are shown as .V1 and M .S1 in Fig. 13. The three equations areobtained by summing forces in the horizontal and ver t ica l direct ions and bysumming moments about the or igin of the a-b coordinate system. Performingthese operations the equil ibrium equations may be wri t ten as


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7

Fig 13 Forces on the piles nd foundation


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28

nL: F . Pv 0i=l 10)

11)

nL: M . a. F . b. Fh . ) M = 0 i=l 12)

Substi tuting Eqs. 4, 5, and 6 into Eqs. 10, 11, and 12 and rearranging

nPv L P . cos e. - Pt i sin e. )i=l 13)

nP = L P . cos e Pxi sin e.H . 1 14)

n[Mti= L a. P . cos e. - P t i sin e.i=l

b. P t cos e. Pxi sin ei8 15)Subst i tu t ing Eqs. 7, 8, and 9 into Eqs. 13, 14, and 15 the equilibrium

equations may be written as

nL (J . x . cos e.i=l x ~ t ~n

P = L (J . Yt . cos e ; J Xi x t ; sin e.)i=l

16)

17)


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30

+ b ~ J .COS 2 e.+J .s in2 e.)+2(J . - J . ) Sine.cose.)a .b . ] O }. (21)1 y 1 1 X1 1 X1 y 1 1 1 1 1Equations 19, 20, and 21 const i tute the complete set of equilibrium

equations for a foundation. The loads on the foundation, the distance to thepi le tops, and the batter of the piles are known quant i t ies . I f the springmodulus values are known the three equations may be solved simultaneouslyfor eN MI and 0 . But, since the system is nonlinear, Jm J x and Jwill not be constants. Because of this an i t era t ive solut ion is required.Chapter IV will present methods for handling the behavior of the individualpiles . Chapter V will give a brief summary of the i t era t ive procedure usedin the computer program for solving the equilibrium equations.


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CHAPTER IVBEHAVIOR OF INDIVIDUAL PILES

In the preceding chapter equilibrium equations were developed for apile supported foundation. These equilibrium equations contain secant modulusvalues obtained from the nonlinear load-deformation curves for individualpi les This chapter deals with the methods used for obtaining the secantmodulus values for the individual piles

The modulus J is obtained from the axial behavior of the pi lexModulus values J and J are obtained from the la te ra l behavior of them ypi le

AXIAL BEHAVIORIn order that a value for J be calcula ted an axial load-deflectionx

re la t ion i s necessary. The procedure employed involves finding a load-sett1e-ment curve for the pi le A typical load-sett lement curve is shown in Fig. 14.The curve shown consists of two branches corresponding to bearing and pulloutof the pi le

I f a load-sett lement curve is available a value of secant modulus maybe obtained for any value of axial def lect ion by applying Eq 7. This is asimple procedure for obtaining J x af te r the correct value for axial def1ec-t ion i s found. The problem which arises is to find a load-sett lement curvewhich wil l accurately describe the axia l behavior of a pile Earl ier methodsof analysis did not require that an exact load-sett lement curve be found. Acomputed ultimate axial load or an ultimate load obtained from a ful l scaleload tes t was usually considered adequate for design purposes. For the pro-posed method a relat ionship between load and def lect ion is necessary. The

3


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3

Fig. 14. Axial load settlement curve.


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33

axial behavior of piles is usually determined by one of three methods. Theseare as follows:

1. Dynamic formulas2. Stat ic formulas3. Full scale loading tes t .

Dynamic FormulasDynamic formulas such as that of Hiley as described by Chellis 2 give

only a maximum pile capacity with no regard to corresponding movements. I thas been demonstrated that the dynamic formulas give very errat ic resul ts with

3 13poor correla t ion between calculated and measured values of pi le capacity ,The various formulas have l imited usefulness for the method considered becauseof the lack of load-settlement data.

Sta t ic FormulasThe s t a t i c formulas re la te the load carrying capacity of the pi le to

the so i l propert ies . The usual procedure is to calculate a t ip load using11 12some bearing capacity formula such as that suggested by Meyerhof a n dsome shaft load which is t ransfer red to the so i l through skin f r ic t ion alongthe pile . Accurate predict ion of skin fr ic t ion i s d i f f icu l t but suggestedvalues are available 2. The bearing capacity and shaf t load are added to ob-ta in the to ta l p i le capacity. This method is also limited by the lack ofload-settlement data. I f the dynamic and s ta t ic formulas are to be of anyvalue to the analysis under consideration some method must be found to re la teload to deflection.

6The method proposed by Reese seems to offer a great deal of promisefor predict ing load-settlement curves from so i l data. 4Coyle has comparedmeasured values with values calculated using this method for s tee l f r ic t ion


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34

piles in clay. The correlat ion obtained was quite good. However the use-fulness of this method is l imited by the lack of correlat ion for a range ofpi le and soil types.

Full Scale Loading TestThe use of loading tes ts is the most re l iable method present ly avai l -

able for predic t ing load-sett lement curves. A pullout tes t and a bearing tes twil l give the desired load-deflection relat ion

ConclusionsOf the methods discussed the loading tes t gives resul ts which best

represent the axial behavior of a pi le 16The method suggested by Reesewill give re l iab le resul ts provided the load t ransfer can be accurately pre-dicted. The s t a t i c and dynamic formulas have l imited usefulness because ofthe lack of load-def lec t ion information. A load-deflection curve may be ob-tained by assuming some relat ion between load and def lect ion based on thecalculated ultimate load. The accuracy of this procedure wil l depend on theaccuracy of the assumption and i t wil l probably give only a rough estimate.

L TER L BEH VIORFor the calculat ion of the modulus value J a relat ionship betweeny

the shear at the top of the p i le and the la te ra l def lect ion of the pi le topmust be known. For the calculat ion of J some relat ionship between momentma t the top of the pi le and top def lect ion must be known. In the precedingsect ion on the calculat ion of J x a load-deflection curve was used. Thisis possible since i t is assumed tha t the axial behavior of the pi le is un -affected by any l a tera l effects That is to say that the axial load on thepi le is dependent only on the axial def lect ion of the pi le A similar


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assumption concerning la te ra l behavior is not t rue. Simple single-valuedcurves for Pt vs. and M vs. as shown in Fig. 12 do not exis tfor a pi le which is attached to a foundation.

35

Since a single-valued load-deflect ion relat ionship cannot be found, adifferent approach must be taken for calculat ing J m and J x The approachtaken involves the solut ion for the deflected shape of the pi le using f in i tedifference equations. Once the deflected shape is known the shear and momentscan be calculated and modulus values may then be calculated using Eqs. 8 and 9.The in teract ion is nonlinear so that an i t era t ive process must be employed tofind the correct modulus values. The i te ra t ive procedure wil l be explainedin deta i l in Chapter V For the following discussion assume that the i te ra t iveprocedure is complete and that correct boundary condit ions are applied to thepi le With this in mind the f in i te difference solut ion for the la te ra l lyloaded pi le will be discussed and the calculat ion of the modulus value ex-plained. The so i l cr i ter ia used to determine the la te ra l interact ion wil lalso be explained.

Fini te Difference Solution for Lateral ly Loaded PilesThe f in i te difference approach to the solut ion of la te ra l ly loaded

6piles was f i r s t suggested by Gleser This idea was further extended by9 17Reese and Matlock The method presented here is for the special case of

a la te ra l ly loaded pi le and is similar to the method presented in Refs. 9and 17, the differences being in the applicat ion of boundary conditions andthe addition of the effects of axial load on the la te ra l deflect ion.

The dif fe ren t ia l equations are derived by considering an element ofthe pi le as shown in Fig. 15. The sign of al l forces, deflect ions, and slopesshown are posit ive. I t should also be noted that the axial load is constantover the length of the pi le For piles th is assumption is not consistent


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36

b

y

V

p

V

x

X

Fig 15 Generalized beam column element


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7

with observed behavior, since i t is known that some of the applied axial loadis t ransferred to the so i l by skin f r ic t ion along the shaft . The validity ofthis assumption is based on the fact that the errors introduced wil l be in-signif icant . Considering the problem from a physical s tandpoint i t is knownthat for most cases the skin f r ic t ion increases with depth. This, plus thefact that any l a tera l movement will cause a decrease in skin f r ic t ion, leadsto the conclusion that the axial load removed by the skin f r ic t ion in theupper portion of the pile is small. Since the maximum moment occurs in thetop portion of the pi le , and since i t is the def lect ion of the pi le top whichi s of in te res t , the assumption of constant axial load will not signif icant lyaffect the resul ts of in te res t .

The reason for having the assumption of axial load constant onthe top of the pile is one of convenience. The addition of a variable axialload could have been handled analyt ical ly but the effor t required for obtain-ing a solution would not be warranted because of uncer taint ies involved inobtaining the nature of the var iat ion.

Referring to Fig. 15 the equilibrium equations for the element may bewrit ten as

and

where

dM V Pdx x dx

dVdx = P

M = Bending moment

E Ys

o

x Distance along pi le

22)

23)


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38

V ShearP Axial load constant)x

y Lateral deflectionp = Soil react ion per unit length

E Soil modulus.sy combining Eqs. and 3 and differentia t ing, the following equation is

obtained:

d vY PX dx 2 o

The equation for shear is writ ten as

v d Pdx x dx

and the equation for moment is writ ten as

M

whereE Modulus of elas t ic i ty of the pileI Moment of iner t ia of pi le sect ionR EI f lexural r ig id i ty ) .

24)

25)

26)

Equations 24, 25, and 6 may be writ ten in f in i te difference formusing the centra l -dif ference approximations. The equations will be writ tenfor a general point referred to as s ta t ion i . Stat ion numbering increasesfrom top to bottom of piles . The equations obtained for s ta t ion i are asfollows:


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where

+ R 1 - 2P h2 + E .h 4 + y. 1(-2R. - 2R. 1 + P h2 1 X S1 1 1 1 X

+ y. 2 (R. 1) = 0 (27)

- R 1) + y. 1(2R. 1 - P h2 ) + y. 2(-R. l )J (28)1 1 1 X 1 1

h = Increment length.

2y.1 (29)

The f in i te difference equations are used to obtain the deflected

39

shape of the pile . Once the deflected shape is obtained any other informationabout the pi le may be obtained by the appl icat ion of the appropriate equations.

The pile is divided into n increments of length h as shownin Fig. 16. In addit ion, two f ic t i t ious increments are added to the top andbottom of the pile . The four f ict i t ious s ta t ions are added for formulatingthe set of equations but they will not appear in the f inal set of equations.The coordinate system and numbering system used i s i l lus t ra ted in Fig. 16.

The procedure used is to write Eqs. 27, 28, and 29 about s ta t ionn+3. This results in 3 equations involving 5 unknown deflect ions (Yn+5Yn+4 Yn+3 Yn+2 Yn+1)' Two boundary conditions, Vn+3 = 0 andMn+3 = 0, are applied a t s ta t ion n+3. The deflect ions for the f ict i t iouss ta t ions n+4 and n+5 are eliminated from the three equations and the


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40

r---I II II II I2 ~ ~I II II II II Ir ---4

5

6n

n

I II II II II In 4 ~ ~ ~I II In 5

I II II__ . . . . Ix

Fig 16 Fini te difference representat ion of pile .


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41

deflect ion for s t t ion n 3 is found in terms of the deflect ion a t stationsn 2 and n 3. The equation obtained may be wri t ten as:

30)

where

31)

and

32)

Equation 27 is written for s t t ion n 2. This equation is combined withEqs. 28 and 29 for s t t ion n+3, and Eq. 30 to determine the deflect ion fors t t ion n 2. The deflection Yn 2 is found in terms of the deflect ion ofs ta t ions n 1 and n. The equation obtained is as follows:

A Y - B Yn 2 n 1 n 2 n 33)

where

34)

and

35)


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42

The def l ec t ion for s t a t i o n n+l may be found in a s imi la r manner. Froms t a t i o n n+l to the top o f the p i l e the expressions for the def l ec t ion havethe same form. The general form o f the equa t ion i s as follows:

where

and

A

B

36)

2R i _ l + Ri 2-2Bi + l + Ri + l Ai + 2 Bi + l - 2B i + l - Pxh 2 1-B i + l c.

R 1C

37)

38)

39)

With the general express ion the def l ec t ion of each s t a t i o n may beexpressed as a func t ion of the def l ec t ion of the two s t a t i o n s immediatelyabove it I f the def lec t ions for s t a t ions 3, 4, and 5 are wr i t t en a s e t o fthree equa t ions involving f ive unknown def lec t ions wi l l be obta ined I f two

boundary condi t ions are introduced the def lec t ions for the f i c t i t i ous s t a t i o n smay be e l imina ted and the equat ions solved for the def l ec t ions Once the de-f l ec t ions for s ta t ions 3 and 4 are found the def lec t ions for the remainder ofthe p i l e may be obta ined by back subs t i tu t ion in to the equat ions obta ined forthe def l ec t ion of a s t a t i o n in terms of the def l ec t ion of the two s t a t i o n sd i r e c t l y above it


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43

The expressions obtained for Y3 and Y4 will depend on the boundaryconditions applied to the top of the pile . Three sets of boundary conditionsare used resul t ing in three sets of equations.

For the f i r s t case the following boundary condit ions are applied:

40)

41)

where Mt and Pt are the moment and la te ra l load applied to the top of thepi le The applicat ion of these boundary conditions resul ts in the followingexpressions for Y3 and Y4:

where

Yo =f [R 2 Ao B. - 4 B.) + Ro 2 - 2 B.) + 2Pxh2 BJ+ Do Gj/ f [ Ro 2 B. - 2) + R. 4 B. - 2 Ao B.)- 2 Pxh

2B J + G2 [R3 4 - 2 ~ + R4 2 A4 As

- 2 B5 - 4 A + 2) + Pxh (- 2 + 2 A.) + s o h ~ } 42)Y4 Y3 A - Baa G) 11.:G2 43)

M h2D tR3 44)

D3 2P h3t 45)

G1 = 2 - 46)


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44

The second set of boundary conditions applied are as follows:

Vs = P tYi - Yah

47)

41)

48)

These boundary conditions resul t in the following expressions for a and Y4:

where

Ya f 1 B,) D, [2 R, 2 B, - ... B,) 2 Ra B, - 1)

- Pxh2 B ]} t R, [ . . . - B, - B, B,Y4 = Y

D4 = 28 ht

4 Ra 1 - B4 )- 1 + Eh}3

The third set of boundary conditions applied are as follows:

49)

50)

51)

41)

52)

These boundary conditions resul t in the following expressions for Y and


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where

Y. = D [1 B, + D5 1 + B,)] / {2 D,; 2 .. + 2.. B2 R3 A4 + R4 As - R4 B4 Bs 2 R4

+ B4 Bs 2 + 3 B4 + 1) + 2P h2 ~ B4 1x+ Po,. D - D5 - B, D5 + E h [1 B, + D 1 + B,8

53)

54)

55)

When the f i r s t set of boundary conditions i s used the calculation ofJ and J involves only the applicat ion of Eqs. 8 and 9. The moment andy m

45

l ter l load applied are divided by the calculated deflection of the pi le top.When the second and th i rd sets of boundary conditions are used the

moment applied must be calculated. This is obtained by applying the followingequation:

56)

Since the l t er l load is known the modulus values may be obtained.


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46

Lateral Soil-Pi le Interact ionn the preceding section the effect of the so i l on the pi le was shown

as a distr ibuted reaction p. The so i l react ion p was defined as:

p E Ys 23)

where E is the so i l modulus and y is the la te ra l def lect ion. The so i lsreaction res is t s the def lect ion of the pi le . For the der ivat ion of the f in i tedifference equat ions i t was assumed that the so i l modulus values were knownSince the so i l p i l e in teract ion is usually nonlinear an i te ra t ive procedureis required to find the correct values of Es The following discussion dealswith the development of the relat ionship between la teral pile movement andso i l reaction. n the f inal section of this chapter the so i l cr i t e r ia usedwil l be discussed.

A typical relat ion between p and y is shown in Fig. 17. The so i lmodulus is defined by Eq 23. From Fig. 17 i t is seen that the so i l modulusis the slope of the secant drawn from the origin to any point along the curve.Since p i s defined as the distr ibuted so i l reaction with units of force peruni t of length along the pile the so i l modulus E wil l have uni ts of forcesper unit length squared. Since the p-y curve for most so i l s i s nonlinear ,an i te ra t ive procedure wil l usually be required to find the correct so i lmodulus, and the corresponding deflec ted shape.

The p-y curves will depend on the so i l propert ies. For most casesthe proper t ies of the so i l in a prof i le is not constant with depth. The usualcase being that the strength of the so i l increases with depth. A typicalvariat ion of shear strength of so i l with depth i s shown in Fig. l8a. Sincethe s t rength of the so i l will affect the p-y curves obtained, a varia t ionsimilar to that i l lus t ra ted in Fig. l8b might be expected. t should be


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c

.c r E, p y

y in

Fig. 17. Typical p-y curve.

47


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48

t::

eo

hear trength

a. Variat ion of shear s t rengthwith depth.

p

y

p

y

b Variation of p ycurves with depth.

Fig. 18. Variation of so l propert ies with depth.


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49

pointed out that the shear strength is not the only parameter which willaffect the p-y curve although t does have considerable influence. Thepurpose of the varia t ion shown in Fig. l8b is only to i l lus t r a te the var iabi -l i ty of the p-y re la t ion.

For use in the equations for def lect ion a value of so i l modulus isrequired for each stat ion. I f a p-y curve is available at a s ta t ion andthe def lect ion is known then a value for so i l modulus can be obtained. I fa p-y curve is not available for a part icular s ta t ion then a so i l modulusvalue is obtained by l inear interpolat ion between p-y curves above andbelow the part icular s ta t ion. The E values obtained are used in the solust ion for the deflections. The i t era t ive process is continued unt i l closureis obtained for the def lect ions .

Soil Criter iaThe so i l cr i ter ia presented here for obtaining p-y curves is derived

from theoretical and empirical considerations. I t is limited by the fact thatcr i ter ia is available only for clay and sand. No cr i t e r ia is available forso i l which has cohesion and also some angle of internal f r ic t ion. I t musta lso be used with r ese rva t ion s ince su f f ic ien t cor r e l a t ion with measured valuesis not available. Work of this nature has been done but i t is s t i l l confi-dent ial information. Once this information becomes available to the engineer-ing profession i t will be possible to obtain more rea l is t ic p-y curvesthan are obtainable from the theory presented.

I t is assumed that the p-y curves can be divided into two segments.The portion designated as O A and the portion designated as A B in Fig. 19.The segment O A represents the early part of the curve and the segment A Brepresents the ultimate part of the curve. Because of this division theconstruction of p-y curves may be carr ied out in two steps. Fi r s t the


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50

'

,''A B

PUll - - ~ - - - - - - - - . . , ~ - - - - - - - - - - -Ultimate Port of Curve

arly Port of Curve

o ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~o

Fig. 19. Construction of p y curve.

o ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~o

Fig. 20. Stress s tra in curve.


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51

ultimate soil resistance is calculated and then the shape of the early parto f the curve is obtained. The horizontal l ine representing the ultimate so i lres is tance and the early part of the curve are then joined to form a continuouscurve. In the following sect ions the procedure wil l be explained for clay andthen sand.

For clay two methods are employed to obtain p-y curves. I f s tresss t ra in data are available the method proposed by Bramlette McClelland and John

1A Focht, Jr . is used, with one modification. For th is method s t ress s t ra incurves s imilar to the one shown in Fig. 2 are required. The curve is obtainedfrom a t r iaxial tes t in which the confining pressure 03 is as close aspossible to the confining pressure on the soil in the f ie ld. McClelland andFocht recommend that the p-y curve be obtained by using the following relat ions:

p 57)

and

y 12 w 58)

wherew Pile diameter or width

= Strain. 20A W Skempton has suggested the following relationship for calcu-la t ing deflect ions of footings:

y 2 w 59)


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52

I t i s fe l t that the best value to use for deflection would be one between thevalues calculated using Eqs. 58 and 59. The equation suggested i s :

y W 60)

Using Eqs. 57 and 60 and the s t ress-s t ra in curve a corresponding p-y curvemay be obtained.

I t is assumed that the t es t is run unt i l fai lure is obtained. Thatis the maximum value for obtained wil l represent the ult imate valuewhich may be carr ied by the soi l . Because of th i s the value for p calcu-lated using the ult imate value of is considered to be the ult imate soi lres is tance.

I f no s t ress -s t ra in curves are available, but the shear strength andunit weight are known p-y curves can be obtained. Two expressions areavailable for calculating the ult imate soi l res is tance for clay. These equa-

8t ions were suggested by Reese and are as follows:

and

where

y w X 2 c w 2 83 c X

11 cw

Y = Unit weightw Pile diameter or widthX Depth from soi l surfacec = q /2 = Cohesion.u

61)

62)

The smaller of the two values obtained from Eqs. 61 and 62 is used. Equation61 wil l usually control near the surface since i t is based on the occurrence


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54

elCIo

2

lo

Fig. 21. Approximate log log plot ofs t ress s t ra in curve.


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where

= ywX

Ko

45 0 - /2) [ tan 8 45 0 +

tan tan 45 0 /2

o = Coefficient of earth pressure at res ty Unit weight of sandx = Depthw = Pile diameter or width = Angle of internal f r ic t ion

= 45 0 /2a /2 to /3 (loose sand)

(dense sand),

(64)

Equation 63 is for wedge shaped fai lure and 64 is for flow around fai lure,

55

The early par t of the curves are obtained by applying theory developed

b 1 h,22

y Kar Terzag 1 This resul ts in a l inear variat ion between p and y,with the slope given by Eq. 65,

s A y X1. 35S = Slope of early part of curveA = Constant depending on re la t ive densi ty of sand.

(65)

Suggested values for A are 200 for loose sand, 600 for sand withmedium densi ty, and 1500 for dense sand. The unit weight used is the effect iveuni t weight,

I f the slope of the early par t o f the curve is known, the p-y curvecan be constructed by connecting a s tra ight l ine through the origin, with aslope defined by Eq. 65, to the horizontal l ine defined by the ultimate so i l


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6

r es i s t ance This re su l t s in a p y curve which consis ts o f two s t r a i gh tl ines hen one cons iders the behavior of a sand t wi l l be noted i t s be-havior i s not l inea r Because of th i s the p y curve obta ined should beconsidered as an approximat ion.

ConclusionsIn th i s chapter the behavior of a s ing le i so la t ed p i l e has been consi -

dered. The ax ia l and l a t e r a l behavior o f the p i l e was considered and themethods for calcula t ing the spr ing modulus values explained. The s o i l c r i t e r i aused for obtaining p y curves was a lso considered. Cer ta in l imi t a t ions o fthe procedures used were discussed . Further l imi t a t ions w i l l be considered inChapter VII.


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CHAPTER VCOMPUTATIONAL PROCEDURE

BENTI is a computer program writ ten to solve problems involving pilesupported foundations. I t is a modification of programs developed previouslyat The University of Texas a t Austin. I t consists of an i te ra t ive solutionfor the three equil ibrium equations developed in Chapter I I I using methodsdeveloped in Chapter IV to handle the nonlinear behavior of individual pi les

A general explanation of the computational scheme for the program willbe presented in this chapter. Example problems are considered in Chapter VI.Detailed guides for preparing input data are given in Appendix A. A completeflow chart is given in Appendix B. A l i s t of the notation used is given inAppendix C and a complete l i s t ing of the program is given in Appendix DLis of the coded input and output for the example problems are given inAppendices E and F.

OUTLINE OF PROCEDURE FOR BENTIThe general procedure used for solution of the equilibrium equations

is shown in Fig. 22. The purpose of the i te ra t ive procedure is to find thedeflected posit ion of the structure so that equilibrium and compatibili ty aresat i s f ied The procedure followed by the computer program is essentia l ly thatshown in Fig. 22. Rather than present a complete flow diagram for the programthe basic procedure employed will be described. I t will be noted that theprocedure described is essentia l ly that shown in Fig. 22.

To begin the solut ion input data for the problem are read in Thegeometry of the foundation and the axial behavior of the piles are described.The l a tera l behavior of the piles may be described by inputing p-y curvesor soi l propert ies may be input and p-y curves generated by SUBROUTINE MAKE

57


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58

Set tN ll and xequal to zero.

Set the deflectionof each pile topx . y . equal to1. 0;

Set in i t i a l boundaryconditions for usein la teral ly loadedpile solution.

Calculate FJX. usingx . and load set t lef lment curve or e.

Calculate FJY. andFJM. using la teral lyl o ~ e d pi le solutionwith appropda teboundary conditions.

Calculate 6V llHand x by simul taneoussolution of threeequilibrium equations.

Compare calculated6.V till and x valueswith previous values.

Closure obtained. Makefinal calculations.

Closure not obtained.Calculate new valuesfor deflection of pile1- - - - - - - - - - -1 tops. Set new boundaryconditions for later-ally loaded pile .solution.

Fig. Block diagram for i terat ive solution.


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59

cedure is s ta r ted . To s t a r t the procedure two assumptions are made. F i r s tthe foundation movements ~ V , ~ H , and a are se t equal to zero . Next thedef lec t ions of the p i l e heads and y are se t equal to one inch. Theseassumptions are made to get the i t e r a t i v e procedure s ta r ted . Once the procedurei s s t a r t ed it i s continued un t i l the equi l ibr ium pos i t ion fo r the s t ruc tu re i sfound.

The next s tep i s to se t the boundary condi t ions for the l a t e r a l l y load-ed p i l e so lu t ion SUBROUTINE COM62). For the i n i t i a l i t e r a t i o n one boundarycondi t ion i s tha t the l a t e r a l def lec t ion of the p i l e tops i s one inch . Thesecond boundary condi t ion w i l l depend on the manner in which the p i l e i s con-nected to the s t ruc tu re . The value of the second boundary condi t ion w i l l bese t equal to zero fo r the i n i t i a l i t e r a t i on . For pin connect ions the secondboundary condi t ion used is the moment a t the p i l e top. This means tha t i f thep i l e i s pinned to the s t ruc tu re the moment a t the p i l e top i s se t equal to zero.For f ixed connect ions t h i s se t s the s lope a t the p i l e top equal to zero andfor r es t r a ined connect ions the r e s t r a in t a t the top i s se t equal to zero .

With the i n i t i a l assumptions and the i n i t i a l boundary cond i t ionsvalues fo r the spr ing moduli are ca lcu la ted . FJX. i s ca lcu la ted from theax ia l load-def lect ion curve using the ax ia l def l ec t ion . To c a l c u l a t e FJY.and FJM. COM6 i s ente red with the i n i t i a l boundary cond i t ions . The def l ec ted shape the shear a t the top and the moment a t the top are ca lcu la tedand thus the spr ing modulus values ob ta ined .

With the spr ing moduli fo r each p i l e the equi l ibr ium equat ions aresolved for the foundation movement. One cycle is complete when the p i l e headmovements are c a l c u l a t e d using the components of the foundation movementobta ined. The so lu t ion obta ined i s checked by comparing the ca lcu la ted


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components of foundation movement with values from the previous i te ra t ionThe correct solu t ion is obtained when the movements agree to within theallowable tolerance. The allowable tolerance is set by the input variableTOL For 6V and 6H the i te ra t ion procedure is controlled by the inputvalue of TOL For control of TOL is multiplied internal ly by 0.001.I f closure is not obtained the procedure is repeated.

To s tar t the second cycle and each preceding cycle the boundarycondit ions for the la te ra l ly loaded pile routine are set One boundary condi-t ion is the shear at the top of the p i le This is found by multiplying FJYby the la te ra l deflection of the pi le tops as calculated from the foundationmovements. The second boundary condit ion wil l depend on the manner in whichthe pi le is connected to the foundation. For pinned connections the secondboundary is that the top moment is zero. For fixed connections the slope atthe top is se t equal to the rotat ion of the structure . And for restrainedconnections the second boundary condition is the res tra int provided by thestructure . The remainder of the procedure is the same as for the i n i t i a lassumption. This procedure is continued unt i l the correct foundation movementis obtained. When the correct movement is found a control is set and theforces and moment exerted by each pi le on the s t ructure are found. Thedeflected shape moment distr ibut ion and so i l reaction for each pi le are alsocalculated. Examples of the output information for program BENTI are presented

in Appendix F.


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CHAPTER VIEXAMPLE PROBLEMS

The two example problems presented in this chapter are associated withactual bents used by the Texas Highway Department for supporting bridges onthe Gulf Coast of Texas. The geometry of the bents propert ies of the pilesand soi l and the loads on the bents were obtained from highway department f i les .

GENERAL CHARACTERISTICS OF EXAMPLE PROBLEMSThe bents considered in the example problems are used in bridges

located on the Gulf Coast of Texas. There are two basic reasons why bents ofthis type were selected for analysis by the proposed method. The f i r s t reasonis that soi l conditions in this area are consis tent ly bad which makes pilesnecessary for bridge foundations. The second reason is that high l a te ra l loadsare common These are due primarily to wind and wave action. During hurr i -canes the la teral loads may be quite high. The use of long piles and highla teral loads makes the proposed method of analysis seem very a t t rac t ive forthese bents.

COPANO BAY CAUSEWAYThe f i r s t example considered wil l be one of the bents used in the

Copano Bay Causeway. The bridge i s located in Aransas County on State Highway35 between Port Lavaca and Rockport. The bridge is 920 f t in length and provides 50 f t ver t ica l clearance a t the center of the span. The roadway is supported by precast -prest ressed concrete girders. The bent caps columns andfootings are reinforced concrete. The bent heights vary from 20 to 50 f t . Thebent analyzed is shown in Fig. 23. The piles used are battered in 4 direct ions to

6


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Li--of l----- 31 f t -- - - -- t o o o I W - 133 in.T

...........

30 in.

........It:)C\J

2-ft Die

2-f t 6-in Die

L 1 0 1 - - - 13 f t ~6 int36 in. PrestressedConcr t Pile18 in. SQ93 ft Long

a. Bent elevat ion3

.:Q

18 in.

8 in.t

b. Top view o f footingA-A)

c. Side view o ffoot ing B-B)

Fig. 23. Copano Bay Causeway bent.


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6

r es i s t horizonta l forces perpendicular and paralle l to the roadway. Only thecase where the horizontal load is perpendicular to roadway will be considered.For th is case the two inter ior pi les in each footing, which are batteredpara l le l to the roadway, wil l be t reated as ver t ica l piles . The bottom t i ebeam is considered to provide suf f ic ien t r ig id i ty so that the assumption thatthe pi le heads remain in the same plane af ter movement is valid.

The geometry necessary for describing the foundation for the computersolut ion is shown in Fig. 24 and Table I The coordinate system and theresul t ing forces on the bent are also shown in th is f igure. The piles are18 in. square prestressed concrete piles . They have an effect ive flexuralr ig id i ty of 4.374 x 1 1 lb_in. 2 assuming a modulus of elas t ic i ty for concreteof 5 x 1 6 psi) and a length of 93 f t .

A pi le similar to the ones used in the bent was driven near the s i t eof the bent. A load tes t was performed on this pi le . The load sett lementcurve obtained and used in the computer solut ion is shown in Fig. 25.

The piles are driven through what is c lass i f ied as muck or very softclay, to bearing on a dense sand or firm sandy clay. The locat ion of thes t i f f e r s t ra ta is variable and so the length of piles and length of embedmentin the s t i f fe r s t ra ta will be variable . For th is analysis the piles areassumed to be 9 f t in length with an embedment length of 83 f t .

For generat ion of p-y curves the so i l is t rea ted as a clay. That i sthe so i l i s treated as a f r ic t ion less material with the shear strength composedent i re ly of cohesion. Some th in sand layers are encountered but their effect isconsidered insignif icant . The t ip of the pile may also be buried to severalfeet in a sand or sandy clay, but the effect on the l a tera l behavior will beins ignif icant and wil l be ignored.


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64

500

400II 300Jt;

0l :Io...J 200

100

b

PH 36 4 kips

Fig 24. Foundation representation Copano Bay.

o ~ ~ ~0.1 0 2 0.3 0 4 0 5 0 6 0 7l ial Deflection t in

Fig L5 Load deflect ion curve Copano Bay.

a


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65

TABLE I . PILE LOCATION INFORMATION - COPANO B Y

Pile a No Piles BatterLocation Coordinate Coordinate atin. in. Location radians)

1 -126 0 1 -0.2442 - 90 0 2 0.03 + 90 0 2 0.04 +126 0 1 +0.244

TABLE I I PILE LOADS ND MOVEMENT - COPANO B Y

Pile Axial Load Lateral Load Moment Axial LateralLocation per Pile per Pile per Pile Movement Movementkips) kips) in. -kips) in. in.

1 78.7 1.7 -253.3 0.0397 0.11342 133.4 1.5 -218.9 0.0689 0.10043 156.5 1.5 -218.8 0.0843 0.10044 193.6 1.1 -155.2 0.1091 0.0763


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66

After considering boring logs from the vicini ty of the bent and af te ra review of t r i ax ia l data a var ia t ion of cohesion with depth was assumed andused for predic t ing la te ra l p i le - so i l interact ion. This assumed dis t r ibut ionof cohesion along the pi le length i s shown in Fig. 26. The depth given isthe distance from the so i l surface. The top of the pi les are located a t thewater surface which is 10 f t above the so i l surface. The scour l ine is assumedto be 5 f t below the so i l surface. The sa tura ted uni t weight of the so i l i staken as 92 pcf and the consistency is sof t .

A solut ion was obtained for this problem by using the program BENTIwhich was described in Chapter V The movement of the bent is described bythe following movements of the or ig in of the a-b coordinate system.

tV 7.664 x 10- 2 in .tH = 1.004 x 10 1 in.

= 8.536 x 10- 5 radiansThe loads t ransferred to each pi le and the movements of each pi le top are tabulated in Table I I . The forces and movements a t the pi le tops are re la ted tothe x-y coordinate system set up for each pi le .

The def lec t ion of the a-b coordinate system defines the equil ibr iumposi t ion for the s t ructure. When the foundation is in this posi t ion the pilesexert on the foundation the given forces and moments which sa t i s fy the threeequil ibr ium equations. A complete l i s t ing of the coded input and output arepresented in Appendices E and F.

I f the movement of the s t ructure and the loads carr ied by each pi leare considered i t would appear that the design is conservative. This isprobably t rue but i t should be pointed out that factors such as set t lementcaused by consolidat ion and cyclic loading have not been considered.


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Depth, fo

10

20

30

40

50

60

T

eo

h .... .,. ,...,> -- Assumed Scourline

Very Soft Silty Sandy Clayey Muck> sot : 9 pcf

cAv; = 3.8 psi

Very Soft Sandy ClayYsot : 9 pcfC Vil = 15 psi

Fig. 26. Soil proper t ies for generat ion o p y curves.

67


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8

HOUSTON SHIP CHANNELThe second example considered wi l l be one of the bents used in a

br idge across the Houston Ship Channel. The br idge i s located in Harr i s Countyon In t e r s t a t e Highway 610. Deta i l s of the bent analyzed a re shown in Fig. 27.The bent is re inforced concre te and i s suppor ted by 142 18 i n . square pre cas t -p res t re ssed concre te p i l e s . The pi le s in t h i s example are ba t t e redpa r a l l e l to the roadway to r e s i s t hor izon ta l loads from the super s t ruc tu re .I t i s assumed tha t the 7 f t th ick p i l e cap provides s u f f i c i e n t r i g i d i t y sotha t the assumption of plane movement i s va l id .

The geometry necessa ry for desc r ib ing the foundation for the computerso lu t ion i s shown in Fig . 28 and Table I I I . The coordina te system and theloads on the s t ruc tu re are a l so shown in the f igure . The p i l e s have an e f fec t i ve f l exu ra l r i g i d i t y o f 4.374 x 10 10 l b - i n . 2 assuming a modulus o f e la s t i c i t y of concre te of 5 x 106 ps i and a length of 44 f t .

No axia l load-def lec t ion curves obta ined from load t e s t s are ava i l ab le

for the pi le s used in the bent . Because of th i s it was necessary to es t imatethe ax ia l behavior o f the p i l e s . The u l t ima te bear ing capac i ty of the p i l e swas es t imated as 650 kips in compression and 600 kips in tens ion . The u l t i mate def lec t ion i s est imated as 0.5 i n . The load -def lec t ion re l a t i onsh ip i sassumed to be l i nea r re su l t i ng in a curve as shown in Fig . 29.

The proper t i e s of the so i l used for pred ic t ing the l a t e r a l p i l e - s o i li n t e r a c t i o n were obta ined from highway department bor ings . The proper t i e sused for genera t ion of p-y curves are shown in Fig . 30. It should be pointedout tha t the pr o f i l e shown i s a s imp l i f i ca t ion of the ac tua l pro f i l e . The top13 f t of s o i l , def ined as very dense sandy silt wi l l be t r ea ted as a sandwhen p-y curves are generated. That i s , it wi l l be t r ea t ed as a cohes ionl e ss mate r i a l . The bottom 31 f t , def ined as very s t i f f s i l t y c l ay , wi l l be


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I -

t

f

4f t 4 ft 4f t 4ft8 ft

46 a t 3ft 6 in = 161 f t ~ . ~ I0 0 0 0 D C 0-- r , r1 ..- ..--0 10 1 , 1 , , , 1 , 1 III 01 01 1 1 1 , , , 1 , , , 10 q 0 I , , , , , , _____ D Ie 0L. L________ L. ,cr-------i [ ' --------1 r-------j i - - - - - - - i [' -----11-10 Pq 0

: 0 : : : l : : : I : r C:... ... _ J L_ J L _ L 10 0 D 0 0 D 0

Fig. 27. Houston Ship Channel bent .

f

2 2

3f t 6in.5 ft5f t5f t5f t5fti 3ft 6in.


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70

b

in-kips

PH : 1 126 kips Pv: 27 600 kips

2 2 21

Fig 28 Foundation representat ion Ship Channel

a


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Cc0

u0

Load kip.100

O ~ ~ ~ ~ ~ ~ ~70000 300 400 500 600

0.5

1.0

1 5

Fig. 29. Estimated axial load deformation curve Ship Channel.

Depth f t

1

20

30

40

Very Dense Sandy SiltYSat : 115 pcff J : 5

Very Stif f Silty ClayYSat : 92 pcfc 14 psi

Note: Water Table AboveSection Shown

Fig. 30. Soil propert ies for p y curves Ship Channel.

71


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TABLE III PILE LOCATION INFORMATION - SHIP CHANNEL

Pile a b No. Piles BatterLocation Coordinate Coordinate tin. in. Location radians)

1 -150 0 24 -0.1662 - 9 0 23 -0.0833 - 30 0 24 -0.0424 30 0 24 0.0425 9 0 23 0.0836 150 0 24 0.166

TABLE IV. PILE LOADS ND MOVEMENTS - SHIP CHANNEL

Pile Axial Load Lateral Load Moment Axial LateralLocation per Pile per Pile per Pile Movement Movementkips) kips) in. -kips) in. in.

1 106.3 3.3 -46.0 0.0818 0.04742 143.6 2.5 0.4 0.1104 0.04253 178.3 2.0 32.8 0.1372 0.03904 214.5 0.3 122.1 0.1650 0.02635 248.3 0.2 83.8 0.1910 0.01746 281. 5 0.0 -15.2 0.2165 -0.0026


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t r ea ted as a c lay. That i s t w i l l be t r e a t e d as a f r i c t i o n l e s s mater ia l .Depths given are measured from the top of the p i l e . From the g iven s o i lp r o p e r t i e s p-y curves are genera ted . These a r e shown in Appendix F.

A so lu t ion was obtained for the Ship Channel problem by us ing theprogram BENT1 The movement of the bent when loaded i s descr ibed by thefo l lowing movements o f the or ig in of the a-b coord ina te system.

6V 1.512 X 10- 1 in .H 3.321 x 10 - 2 in .a 4.183 x 10-

4radians .

73

The loads t r ans fe r red to each p i l e and the movements o f each p i l e top are t ab ula ted in Table IV. The forces and movements a t the p i l e tops are r e la ted tothe x-y coordinate systems se t up for each p i l e . A complete s e t of codedinpu t i s given in Appendix E. The outpu t i s shown in Appendix F.

The small def lec t ions and loads obtained for the p i l es would tend toi nd ica t e tha t the design i s conserva t ive . This i s probably t r u e and i s tobe expected. But t should be pointed out t h a t a number of f ac to r s such asconso l ida t ion and cyc l i c loading have not been cons idered . I t must a l so bepOinted out tha t the load def lec t ion curve used i s only a rough approximation.The value used fo r ul t ima te load i s probab ly f a i r l y r e l i a b l e but the def lect ion a t which the load s tops i nc r ea s ing i s only an educated guess . Because oft h i s a l i nea r v a r i a t i o n of load with movement was considered to provide s u f f i c ien t ref inement . The e f f e c t o f th i s w i l l be r e f l ec t ed in the loads and def l ec t ions obtained f

Documents

Chapter 3 pile foundation