LATERAL RESPONSE OF PILES UNDER EXTREME EVENTS

Kevin J. Bentley Graduate Program in Engineering Science

Department of Civil and Environmental Engineering

Submitted in partial fulfillment of the requirements for the degree of

Master's of Engineering Science

Faculty of Graduate Studies The University of Western Ontario

London, On tario April, 1999

O Kevin J. Bentley 1999

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ABSTRACT

Due to recent destructive earthquakes, the lateral response of piles and foundations has

been under intense investigation. The following study investigates the dynarnic lateral

response of piles under extreme events such as earthquakes, intense wind loading, or

iceberg impact. Both kinematic and inertial pile-soi1 interaction effects were studied in a

simple rational manner. The analysis was performed in the time domain to accurately

portray the nonlinear behaviour of the soil, separation and slippage at the pile-soi1

interface, and allow for direct transient dynamic loading. The first part of the study

investigated the kinematic interaction of piles using a three dimensional finite element

analysis. It was concluded that the pile head response closely resembled the fiee-field

response for seismic loading for the range of pile and soil parameters considered in this

study. The second part of the study focussed on developing a simple two dimensional

analysis to accurately model the inertial interaction of pile head loading. The proposed

model incorporates the static p-y curve approach and the plane strain assumption to

represent the soil reactions within the frarne of a Winkler model. The p-y curves are used

to relate pile deflection to nonlinear soil reaction. The wave propagation and energy

dissipation are also accounted for to generate "dynarnic p-y curves". A parametnc study

was performed using the developed analysis and the results were used to establish

dynarnic p-y curves. These dynarnic p-y curves are a fünction of the static p-y curve,

dimensionless frequency, and relative velocity of the soil nodes. Closed form solutions

were proposed for a range of soil types, that c m be used to model soil reactions for pile

vibration problems in readily available FEA and dynamic structural analysis packages.

KEYWORDS: dynamic, lateral, piles, p-y cuwes, kinematic, inertial

.-- I l l

Special thanks to Prof. M.H. El Naggar for his patience, understanding and

encouragement throughout the course of my studies.

1 would like to thank my fellow graduate students for their assistance and positive

spirit when the load was just too heavy for one person alone.

Sincere thanks to the staff at the Department of Civil and Environmentai

Engineering and Geotechnical Research Centre at the University of Western Ontario for

their help and guidance.

Lastly, the positive support fiom Kim, my parents and farnily, Mats, Curtis, and

Wong was greatly appreciated.

TABLE OF CONTENTS

L I S T OF FIGURES ..................................................................................... ix

CHAPTER 1 . INTRODUCTION ............................................................... 1

.................................................................. 1.2 FINITE ELEMENT ANALYSIS 3

1.3 WNKLER MODEL ...................................................................................... 4

1.3.1 Defining p-y curves .............................................................................. 5

1.4 OBJECTIVES ................................................................................................ 7

OUTLINE ................. OF STUDY ..........................*.......................... 7

CHAPTER 2 . LITERATURE REVIE W ................................................. 1 1

2.1 ~TRODUCTION ........................................................................................ 11

2.2 EMPIR.ICAL METHODS ........................................................................... 1 1

2.3 THEORETICAL METHODS ......................................*.....o................,. 14

........................ . CHAPTER 3 KIDEMATIC INTERACTION STUDY 20

3.1 INTRODUCTION ........................................................................................ 20

3.2 ASSUiMPTIONS AND RESTRICTIONS ..........,...,. . ............... ...... 21

............................................................ 3 3 3-0 FINITE ELEMENT MODEL 22

33.1 Mode1 Formulation ......................................................................... 22 3.3.2 Soi1 Properties ....................... ... .................O.......................... 2 3 3.3.3 Pile Properties .............O...... ... ....................................................... 24 33.4 Pile-Soi1 Interface ............................................................................. 25

3.4 BOUNDARY CONDITIONS ......... ...... .......................... 26

3.5.1 Initial Loading ............. ... ..... ..... .................................................... 28 3.5.2 Static Loading ................... ........ ..................................................... 28

.......................................................................... 3.53 Dynamic Loading 29

3.6 VERIFICATION OF FINITE ELEMENT MODEL ............................... 29

3.7 COMPUTATIONAL TIME AND METHOD ........................................... 33

3.8 NUMERICAL STUDY FOR KINEMATIC INTERACTION ............m... 33

CHAPTER 4 . INERTIAL INTERACTION S T m Y ............................. 53

4.2 MODEL DESCRIPTION .............. .. ..................................................... 55

Pile ........................ ............................. ..................................... 55 Soi1 Model: Hyperbolic Stress-Strain Relationship ......... .... ......... . 56 Inner field eiement ................... .. .......... .......... 56 Far field element ................... .... ......................O........................ 5 8 Soil-Pile interface ................................. .............................. 60 Soi1 Model: p-y Curve Approach ............. .. ................................ 60

........................................................... p-y curve generation for clay 62 .................... ............................... p-y curve generation for sand ... 63

Damping ................. ...... ................................................................ 6 5

....................................... 4 3 DEGRADATION FACTOR ............. 66

4.4 TIME-DOMAIN ANALYSIS AND EQUATIONS OF MOTION .......... 66

4.5 VERIlFICATION O F THE ANALYTICAL MODEL .............................. 68

4.5.1 Verification of Clay mode1 .................................. .................. 68

4.5.2 Verification of Sand mode1 ............................................................ 70

4.6 VALIDATION OF DYNAMIC MODEL WITH LATERAL

......................................................................................... STATNAMIC TESTS 72

4.6.1 Case Study . Jacksonville Test Site ................................................ 73

4.7 DYNAMIC P-Y CURVE GENERATION (PARAMETRIC STUDY) .m. 74

4.7.1 Description of Method .................................................................... 74 4.7.2 Results / Discussion ........................................................................ 75

4.8 DEVELOPMENT OF A SIMPLIFIED MODEL ..................................... 77

4.8.1 Complex Stiffness Mode1 .................................................................. 78 4.8.2 Obtaining Cornplex Stiffness Constants - Soft Clay Example ..... 80 4-83 Implementing dynamic p-y cuwes into ANSYS ................... ......... 82

CHAPTER 5 = SUMn/IARY AND CONCLUSION ............................m.. 108

5-2 CONCLUSIONS ................ ...,,.....o..............................................e....... 109

5.3 FUTURE CONSIDERATIONS ........ ............... ............... .... ....O............ 111

.......................................................................................... REFERENCES 113

APPENDIX II ............................................................................................ 121

vii

LIST OF TABLES

Table 3.1 . Verification of pile head response as cantilever beam ................ .......... 38

Table 4.1 a Description of parameters used for each numerical run .......................... 84

Table 4.2 . Dynamic p-y curve parameter constants for a range of soi1 types. (d = 0.25. L/d = 40. 0.015 <ao = o r f i < 0.225 ) ................................... 85

LIST OF FIGURES

................. . Figure 1.1 Typical applications of pile foundations (after Otani, 1990) 9

Figure 1.2 - (a) Kinernatic interaction analysis and (b) Inertial interaction analysis. Mass of structure in inertial interaction analysis shoivn as being lumped at the centre of the structure (after Kramer, 1996) .................................................................................. 9

Figure 1.3 - Graphical definition o fp and y : (a) Earth pressure distribution prior to lateral loading; (b) Earth pressure distribution after lateral loading .......................................................................................... 10

Figure 1.4 - Typical set ofp-y curves for a given soi1 profiie (after Poulos and Davis, 1980) .............................................................................................. 10

Figure 3.1 - Definition of the problern and terminology (actual acceleration, Us, acceleration due to kinernatic interaction only, U,, bedrock acceleration, U , and free-field acceleration, UR) ............... .................. 39

Figure 3.2 - (a) Detail of wedge shaped pile elements surrounded by soil elements (pian view), (b) Isometric view of complete soil and pile mesh mode1 .................... .........0.....................................9...........9... 40

Figure 3.3 - Finite element mesh (Mesh no.3) showing boundaw conditions: (a) plan view, (b) front cross section view with geostatic pressure distribution ............................................................................... 41

Figure 3.4 - (a) Block element used for soil and pile, (b) Surface contact element used between piie and soil to alIow for slippage and separation, (c) Transmitting Boundary element consisting of "spring 0'' and "dashpot (C)" to allow for radiating boundaries ................................ .......................................................... 4 2

Figure 3.5 - Two dimensional representation of floating and socketed pile in either homogeneous (used for verification) o r layered soil profile ....................................................................................................... 43

Figure 3.6 - Fourier amplitude spectrum for earthquake loading a t the bedrock level.. .......................................................................................... 44

Figure 3.7 - Response of single socketed pile for (a) elastic @) elastic-gapping (c) plastic-gapping ................................................................................... 45

Figure 3.8 - (a) Comparison of soil displacements along Iine of loading, @) Comparison of soil displacements normal to direction of Ioading ............................................................................~.....................t... 46

Figure 3.9 - One dimensional verification of finite element analysis (FEA using ANSYS) with SHAKE9I .................. .. ............................................ 47

Figure 3.10 - Response of underlying bedrock and free-field for homogeneous soi1 (using one-dimensional FEA) ................... ................................. 47

Figure 3.11 - EIastic free-field response for homogeneous soil (EFB) for one and three dimensional analysis ................... ................................... 47

Figure 3.12 - (a) Cornparison between calculated accelerations for elastic free-field (EFII) and plastic free-field (PFE) using the Drucker-Prager criteria for a homogeneous soil profile. (b) Fourier spectrum for the response at the bedrock level, elastic

................................................ soi1 free-field, and plastic soi1 free-field 48

Figure 3.13 - (a) Comparison between calculated accelerations for elastic free-field (EFH) and floating pile head (ESNFH) for a homogeneous elastic soil profile. @) Fourier spectrum for the response of the elastic soi1 free-field and floating pile head ................ 49

Figure 3.14 - (a) Comparison between calculated accelerations for elastic free-field (EFIQ and socketed pile head (ESNSfl) for a homogeneous soil profile. (b) Fourier spectrum for the response at the plastic soi1 free-field and socketed pile head .............. 50

Figure 3.15 - EFH and PFH response (Elastic and Plastic free-field, .................................................... ......... ES=20000kPa) ......... 51

Figure 3.16 - Pile head response for floating pile (EIastic, Elastic with Gapping) ..... ............................................................................................. 51

Figure 3.17 - Pile head response for floating pile (Elastic gapping, Plastic gapping). ................................... .......................................................... 52

Figure 3.18 - Pile head response for floating and socketed pile (Plastic Gapping). ................................................................................................. 52

Figure 4.1 - Element representation of proposed mode1 ............................................. 86

Figure 4.2 - Envelope of variations of horizontal stiffness and damping stiffness parameters between v = 0.25-0.4 (after Novak et al.,

Figure 4 3 - Cornparison behveen hyperbolic model and p-y model for the soi1 medium .......................................e......................................o.o....o.. 87

Figure 4.4 - Determination of stiffness (kpy) from an internally generated static p-y curve to produce a dynamic p-y curve (including damping) ..................,.............~........................0........0.............................. 87

Figure 4.5 - Definition of soil modulus variation for sand and clay profiles considered in the analysis .......... . ........ ........................~....o...........e.......... 88

Figure 4.7 - Pile head response under applied harmonic load equal to 0.10*PU (L/d=30, E&,(L)=1000, linear profile, P/J?,=O.l) ........... ............. 88

Figure 4.8 - Calculated dynarnic p-y cuwes for 1.5 metre depth using: a) hyperbolic model, b) p-y curve model (for prescribed displacement at pilehead equal to 0.05d) ................... ................... .. 89

Figure 4.9 - Calculated dynamic p-y curves for 3.0 metre depth using: a) hyperbolic model, b) p-y curve model (for prescribed displacement at pile head equal to 0.05d) ................... ... .............. 8 9

Figure 4.10 - Calculated dynamic soil reactions, 1.0 metre depth (for prescribed displacement at pile head equal to 0.075d, L/d=20, E~s(L)=lOOO) ...... .................................................~............................. 90

Figure 4.12 - Calculated dynamic p-y curves for 3.0 rnetre depth using: (a) h yper bolic model, (b) p-y curve model ................... L.............. ..... 0.......91

Figure 4.13 - Calculated dynamic p-y curves for 4.0 metre depth using: (a) hyperbolic model, (b) p-y curve model ................... .L......-..................... 9 1

Figure 4.14 - Soil profile and pile test set-up according to Brown et al. (1988) for measured static p-y curve data ....... . ........... .......o..a........................ . 92

Figure 4.15 - Static p-y curves for loose sand using computer generated p-y curve model and full-scale experimental results (Depth = 0.91rn) .....................~.o....e.............................o........................................... 93

Figure 4.16 - Soil profde and lateral Statnamic pile test set-up for case study at Camp Johnson, Jacksonville .....,..........................o...-e........................ 94

Figure 4.17 - Measured and Computed pile head displacement for Statnamic .......................... test using head load equal to: (a) 350 kN, @) 470 kN 95

. Figure 4.18 Description of soi1 and pile properties for CASE 1 and CASE II ........ 96

Figure 4.19 - Dynamic p-y curves and static p-y curve for numerical run C l ....... ...................... (depth = 1.5m) .., ....... ................................................. 97

Figure 4.20 - Dynamic p-y cuwes and static p - curve for numerical run C2 ....................... .............................................................. (depth = 1.5m) ., 98

Figure 4.21 - Dynamic p-y curves and static p-y curve for numerical run C3 ....................................................... .............................. (dep th = 1.5m) .. 99

Figure 4.22 - Dynamic p-y curves and static p-y curve for numerical run S5 ......... (depth = 1.5m) .................... .. .............................................. 100

Figure 4.23 - Dynamic p-y curves and static p-y curve for numerical run S6 ...................................................................................... (depth = M m ) 101

Figure 4.24 - Dynamic p-y curves and static p-y curve for numerical run S8 ...................................................................................... (depth = 1.5m) 102

Figure 4.25 - Dynamic p-y curves and static p-y curve for numerical run S9 ......................................... ................................. (depth = 1.5m) .... 103

Figure 4.26 - Dynamic p-y curves and static p-y curve for numerical run C l ................................................................ ..................... (depth = 1.0m) .. 104

Figure 4.27 - Dynamic p-y curves and static p j curve for numerical run CI ....................................................... ........................... (depth = 2.0m) ..... 105

. Figure 4.28 True stiffness parameter for numerical run C 1 (Soft Clay).. ........... 106

Figure 4.29 - Equivalent damping parameter for numerical run C l (Soft Clay) with dimensionless frequency.. ............... ..,.... .............-...... 1 06

Figure 4.30 - Calculated pile head response using 2-D analytical model cornpared with ANSYS using: (a) static p-y curves, @) dynamic

...................................................................... p-y curves -...107

Figure 4.31 - Catculated pile head response using 2-D analytical model compared with ANSYS using: (a) complex stiffness, (b) modified complex stiffness.. ........ ... .............................................. ..IO7

xii

CHAPTER 1

INTRODUCTION

1.1 GENEIRAL REMARKS

With increasing infrastructure and decreasing space, engineers are forced to build

larger, higher, and heavier structures such as massive skyscrapers, offshore platforms,

and two-story highways. These expensive and strategic structures result in much greater

risks and new problems for the design engineer. Recent destructive earthquakes in Kobe,

Japan and Northridge, U.S.A. are reminders of the importance of pile foundations and

their impact on the response of the supporling structures. Whether a structure fails from

ultimate limit state design or serviceability lirnit States design, pile foundations can ofien

be the governing factor for such failures. The cost of repairing deep foundation

problems and devising remedial action plans are extremely expensive, in terms of time

and cost.

Pile foundation systems cm reach deeper, stronger soils and bedrock to provide

sufficient resistance as opposed to shallow foundations. Figure 1.1 shows typical pile

configurations for different structures. Although static loading (including Iive and dead

loads) is essential in designing pile foundations, it is the dynarnic loading that poses the

greatest challenge to the design engineer. Dynamic axial and lateral loading causes

additional forces on pile foundations. Although many studies have led to sufficient

understanding of the axial response of piles to dynamic loads, the predictions of lateral

response remains questionable. Lateral dynamic loads fiom wind, seismic activity and

offshore loading (waves) on a structure may be amplified or constrained by the type of

soil and/or pile foundation system used at a specific site. The process in which the

response of the soil influences the motion of the structure and the response of the

structure influences the motion of the soil is referred to as soil-stmcture interaction.

Soil-Structure Interaction

Ground motions that are not influenced by the presence of structures are referred

to as free-field motions. The response of a structure to earthquake Ioading is influenced

by the behaviour of the supporting soil. This influence stems from two factors. First, the

inability of the foundation to conforrn to the deformations of the fiee-field motion would

cause the motion of the base of the structure to deviate from the fkee-field motion

(kinematic interaction). Second, the dynamic response of the structure itself would

induce deformation of the supporting soil (inertial interaction).

In a seismic soil-structure interaction analysis, kinematic and inertial loading can

be examined directly or indirectly. The direct method analyzes the entire soil-

foundation-structure system and is ofien performed using finite element analysis which is

very complicated and requires an enormous amount of time and resources. The indirect

(multi-step) method considen the kinematic and inertial interaction as two separate

stages which can later be coupled together to obtain the actual response (Figure 1.2). The

principle of superposition is used in the indirect rnethod, thus the soIution is Iimited to

linear (equivalent linear) systems.

Soil-structure analyses are generally performed by one of two methods:

equivalent linear analysis or nonlinear analysis. Equivalent linear analyses are linear

analyses in which the soil stifiess and damping characteristics are adjusted until they are

compatible with the level of strain (effective level of strain) induced in the soil.

Nonlinear analyses consider the nonlinear inelastic stress-strain behaviour of soils by

integrating the equations of motion in small time steps. One, two or three-dimensional

analysis c m be used to solve the dynamic response of piles, increasing in complexity and

solution time respectively. Different models have been proposed based on theoretical

and empirical methods including finite element analyses and the Winkler models which

are discussed below.

The finite element method treats a continuum as an assemblage of discrete

elements (mesh) whose boundaries are defîned by nodal points, and assumes that the

response of the continuum can be described by the response of the nodal points. The

stress applied at any nodal point on the mesh is directly related to the sunounding nodal

stresses through the characteristics of adjoining elements. The stifhess, damping and

mass matrices of these elements are formulated and assembled to obtain the global

matrices. It is easy to account far nonlinear characteristics and discontinuities and the

FEM can be used directly or indirectly to solve for dynamic pile response-

1.3 U?NKLER MODEL

Winkler (1867) developed a model used to simulate soil-structure interaction.

The model is one-dimensional and is formed by dividing the pile-soi1 boundary into

horizontal divisions. The basic assumption of the model is that the deflection y(x,z) of

the soil medium at any point on the soil-structure boundary is directly proportional to the

stress p(x,z) applied at that point and independent of stresses applied at other locations.

The force-displacement relationship c m be directly related to the stress-snain

relationship and be expressed by p = ky, in which k is the stiffbess or modulus of

subgrade reaction. The stifhess can be derived theoretically (Mindlin's solution) or be

determined experimentally using p-y curves (discussed in next subsection). Nonlinearity

may be accounted for in the model. The simplicity and practical accuracy of the Winkler

model make it a widely accepted model in the geotechnical engineering community.

The implementation of Winkler models for the nonlinear anaIysis of the dynamic

response of pile foundations (modified dynarnic Winkler model) involves the use of

lumped pile and soil masses to induce inertial forces and includes damping effects using a

variety of rheological modek for the complex stiffness. The dynamic Winkler model c m

be divided into two types. One is the fiequency dornain model and the other is the tirne

domain model. Although the fiequency domain is computationally more efficient and

c m model many soil-structure systems accurately, permanent nonlinear effects and

gapping (separation) at the pile-soi1 interface cannot be included into the analysis

effectively. When nonlinearity with gapping and slippage effects is analyzed in a

rigorous manner, the model must be developed in the time dornain manhting a step-by-

step analysis.

1.3.1 Defining p-y cuwes

The concept of p-y curves is illustrated in Figure 1.3 showing a section through a

deep foundation. Figure 1.3(a) shows a typical stress distribution in the vicinity of the

pile after instal1ation. Figure 1.3(b) shows a typical stress distribution in the soil after it

has been loaded laterally inducing a deflection (y). Integration of the soil stresses yields

an unbalanced force, p, per unit of length. For the solution of a 1ateraIly loaded pile

subjected to ar,y static load, it is necessary to predict a set of p-y curves, such as those

shown in Fig. 1.4 to yield deflection, pile rotation, bending moment, and shear.

The p-y curve approach is readily implemented into the Winkler model because in

both methods the soil is represented by a set of discrete mechanisms with load-deflection

characteristics representing stifhess. The concept implies that the behaviour of the soil

at a particular depth is independent of the soil behaviour at al1 other depths, which in

reality is not true. However, experiments (after Matlock, 1970) seem to indicate that the

behaviour of soil at a certain point is dependent only on the deflection of the pile at that

point, and not on deflections of the pile above or below that point. The theory behind

empirical p-y curves is based on the assumption that the pile is linearly elastic and that an

equivalent line load can represent the soil reaction. The equilibrium of pile displacement

relative to soil reaction at a certain depth can be expressed by a single goveming

equation. For a single pile-soi1 system loaded laterally the solution involves solving a

fourth-order differential equation as given below,

where P, = axial load on pile, y = lateral deflection of the pile at depth x, p = soil reaction

per unit length, and EI is the flexural ngidity of the pile. The procedure for obtaining

expenmental p-y curves involves field testing of a deep foundation, instnimented with

strain gauges so that the bending moment can be measured along the length of the

foundation. From sets of experimental bending-moment c w e s , values of p and y at

points along the pile can be obtained by solving

The soil reaction per unit length @) can be nonlinear and dependent on parameters such

as depth and soil shear strength as shown fiom several experimental studies (Meyer &

Reese, 1979). Limited static p-y curves exist and recently cyclic p-y curves have been

generated dependent on the nurnber of load cycles applied to the pile. The effects of

dynamic loading on p-y curves have yet to be determined.

1.4 OBJECTIVES

Experimental studies involving dynamic pile foundation response are limited due

to the expense incurred in perfomiing such tests. Analytical methods denved fiom theory

and experiment must be comparable to actual measured dynamic response in order to

predict the behaviour of foundations and ultimately shmctures during transient dynamic

loading. The main objective of the following research is to develop a computationally

efficient mode1 to predict the response of piles in various soi1 profiles. The model will

take into account nonlinear soi1 behaviour, slippage and gapping at the pile-soi1 interface,

material and geometric damping. A modified dynamic Widder model is incorporated

into the analysis, which is forrriulated in the time domain using a step-by-step integration

method. The model is verified with similar computational models and is then used to

produce dynarnic p-y curves.

1.5 OUTLINE OF STUDY

Chapter 2 summarizes previous studies focussing on soil-structure interaction and

discusses relevant advancernents made in the field. Various approaches to the analysis of

dynamic pile response were studied. Based on this literature review, the most

econornical and reasonably accurate method was chosen to be used as a basis for the

remainder of the research. There are basically two components to soil-structure

interaction in deep foundations subjected to seismic loading, kinematic and inertial.

Therefore, kinematic loading (seismic loading) and inertial loading were investigated

separately in Chapters 3 and 4, respectively. Chapter 3 investigates the kinematic

interaction accounting for soil nonlinearity, and gapping and slippage at the pile-soi1

interface, incorporated into a three-dirnensional finite elernent model. A commercial

finite elernent package was used to directly rnodel the pile and soil under the influence of

a transient earthquake loading. The finite element model was verified against similar

computational models and exact solutions to determine the influence of kinematic

interaction on the foundation input motion. The results obtained fiom Chapter 3 were

used to confirm similar results in the literature review to develop the inertial loading

mode1 proposed in Chapter 4. The effects of soi1 damping, soil nonlinearity, gapping,

and slippage were incorporated into the 2-D dynamic Winkler model. Chapter 4 includes

the verification of the developed dynarnic Winkler model using semi-ernpirical p-y

curves to account for nonlinearity and different soil and pile profiles. A pararnetric study

was then performed to obtain dynarnic p-y curves for various soil and pile configurations.

Closed form solutions for the dynamic p-y curves were developed by curve fitting the

calculated data to a derived global equation. A simplified mode1 was then developed

using a complex stifhess method (real and imaginary parts) and implemented into a

structural analysis program. The conclusions are surnmarized in Chapter 5, and future

considerations for further analysis are mentioned.

U U U

(a) pile foundations for tank (b) multi-column foundations (c)

Figure 1.1 - Typical applications of pile foundations (Otani, 1990).

offshore platform

utr Fixed boundâry

Figure 1.2 - (a) LUnernatic interaction analysis and (b) Inertial interaction analysis. Mass of structure in inertial interaction analysis shown as being lumped at the centre of the structure (Kramer, 1996).

Figure 1.3 - Graphical definition ofp and y : (a) Earth pressure distribution prior to lateral loading; (b) Earth pressure distribution after lateral loading.

Figure 1.4 - Typical set of p-y curves for a given soi1 profile (Poulos and Davis, 1980).

CHAPTER 2

LITERATURE REVIE W

2.1 INTRODUCTION

The methods used to analyze the response of single piles and pile groups can be

categorized into two main approaches; empirical and theoretical. Ernpirical methods rely

on actual in-situ testing or physical laboratory experiments to backfigure the

characteristics of the pile-soi1 system such as p-y curves. Theoretical approaches use

analytical methods to characterize the pile-soi1 system and are based on derived solutions

including the finite element and boundary element solutions. The following section

provides a brief description of investigations focussing on dynamic pile-soi1 analysis

including relevant advancements made in the field.

2.2 EMPIRICAL METHODS

Reese and CO-workers (Reese and Welch, 1975; Reese et al., 1974; Reese et al..

1975) developed a number of criteria for developing single pile p-y curves in clay and

sand based on experimental studies. The critena were based on field tests performed on

0.3- 1.5 m diameter piles which were fitted with strain gauges to obtain moment data over

the length of the piles. The experiments were focussed primarily on flexible piles and

static loading. Matlock (1970) also performed expenmental tests on soft clay to denve p-

y curves for similar diameter single piles.

Bhushan and others (Bhushan et al., 1979; Bhushan and Haley, 1980; Bhushan et

al., 1981; Bhushan and Askari, 1984) suggested a rnodified cntena for generating p-y

curves based on an analysis of full-scale load tests and recommendations by Matlock

(1 970). Empirical equations were formed to predict p-y curves for clays and sands based

on laboratory tests, in situ testing, pile diameter, and ultimate soil strength.

Blaney and O'Neill (1983) performed full-scale dynamic load tests on

instrumented steel pipe piles capped by a rigid mass. The piles had a 0.273m outer

diameter and 9.27mm wall thickness. The piles were driven in a deposit of stiff to very

stiff overconsolidated clay. Loading fiequencies were in the range of seismic or low-

frequency machine loading. The results of the testing indicated the effect of gapping in

the top 6 pile diameters of the soil and a nonlinear zone in the vicinity of the pile.

Measured response was used for the cornparison with mathematical mode1 predictions

and small-scale tests to predict damping and soi1 degradation parameters.

Abendroth and Greimann (1990) performed eleven one-tenth scale tubular steel

pile tests (monotonic and cyclic loading) to establish the behaviour of pinned and fixed

head piles in sand. Both floating and end-bearing piles were fitted with strain gauges and

tested to determine soil resistance and displacement relationships. Modified Ramberg-

Osgood expressions were used to curve-fit measured strain data to obtain lateral p-y

curves at different depths. The characteristic soil parameters (p-y curves) were

implemented in a Winkler type 2-D h i t e element analysis which predicted the actual

measured results reasonably well.

Brown et al. (1987, 1988) tested an isolated single pile and a large-scale group of

piles subjected to two-way cyclic lateral loading. The tests reported in Brown et al.

(1987) were conducted on dnven piles in stiff, overconsolidated clay at a site in Houston,

Texas. Brown et al. (1988) performed tests in submerged fim to dense sand that was

placed and compacted around the piles. Al1 piles were instrumented and the measured

strain and load data were used to construct pile deflection and soil reaction relationships.

Polynornial curves were fitted to the bending mornent data in a manner similar to that

described by Matlock and Ripperger (1956), to obtain the p-y curves at different depths.

The piles had a diameter of 0.273 m and a wall thickness of 9.27 mm. The results

showed that the load-transfer curves @-y curves) for the single pile were approximately

equal to the resistance of the piles in the leading row of the pile group. However, the

trailing pile rows showed a pattern of decreasing resistance which was due to the effect of

"shadowing", or the reduction in resistance due to passive type wedge failure. Both

studies showed that the deflection of the piles in the group was significantly greater than

that of a single pile under a load equal to the average load per pile in the group.

Furthemore. the soil within the upper five to ten pile diameters clearly dominates lateral-

load response, and densification fiom two-way loading in sand resulted in less reduction

in soi1 resistance than in cIay. Only very low fiequency loading was performed,

fkequencies of 0.067Hz and 0.033 Hz, which is typical of wave loading on offshore

structures. Brown et al. (1988) introduced the concept of using a p-multiplier, f,, to

modiQ the single-pile p-y curve to obtain a grougpile p-y curve. However, methods of

predicting the variation cf f, with soil properties, pile stiffhess and spacing, depth, cyclic

loading, etc. have yet to be exarnined.

Anagnostopoulos (1983) and Prater (1979) describe a range of rnethods for

generating p-y c w e s for soft to stiff clays and some sands based on previous

investigations. Cyclic and monotonie p-y curves are investigated with the inclusion of a

gap formation in some clays. Soi1 degradation or "shakedown" is incorporated into some

p-y models which are dependent on the number of cycles of loading and strain amplitude

of the soil elements.

2.3 TmORETICAL METHODS

Flores-Bemones and Whitman (1982) analyzed a dynamic pile-soil-supported

mass interaction based on analytical solutions in the frequency dornain. A Winkler type

model was used to incorporate the derived stiffhess for end-bearing piles in linear elastic

homogeneous soils. The results showed that the pile followed the soil deflecticns for

kinernatic loading at the bedrock level for flexible piles. Also, the inertial loading of the

structure was introduced through the inertial loading of the rnass supported by the pile

muitiplied by the free-field accelerations (acceleration at pile top). This inertia loading

causes additional movements of the pile in both rotational and lateral modes. Gapping

and nonlinear effects were not incorporated into the model.

Kaynia and Kausel (1982) developed ngorous methods of solution to three-

dimensional dynamic boundary-value problems for piles and pile groups interacting with

soil. The method is described as a boundary-intepl-type formulation, which uses

Green's functions, defining the displacement fields due to unifom unit loads acting on an

elemental cylindrical surface and on a circular disk. The Green's functions are computed

by solving the wave equations through Fourier and Hankel transformations. The

solutions describe the dynamic soi1 ffexibility ma& that is cornbined with the pile

flexibility matrix. The method gives the cornplex valued irnpedance functions for

rotation and horizontal translation for a given soil profile which can be irnplemented into

standard structural programs.

Fan et al. (199 1) performed a numerical study on the kinernatic response of

vertical floating piles and pile groups based on the methods proposed by Kaynia and

Kausel (1982). The piles were subjected to vertically propagating harrnonic shear waves

(S-waves) in homogeneous, linear, and parabolic soil profiles. Fixed-head and pinned-

head pile conditions were investigated but the soil was assurned to be a linear hysteretic

continuum with constant radiation and material damping. The results consisted of

dirnensionless graphs relating the pile head defIections to the fiee-field deflections. The

results showed very little difference between the pile head and fiee-field deflections. The

study isolated soil profile, pile-head conditions, pile slendemess ratio, and pile rigidity as

determinhg factors in lateral response. Results showed that the pile head response could

slightly exceed the fkee-field displacements at certain fiequencies.

Trochanis et al. (1988) used a three-dimensional nonlinear study of piles to gain

some insight of the lateral performance of piles, which led to the development of a

simplified model. The commercial finite elernent program ABAQUS ( 1987) was used in

the study. The nonlinear behaviour of soil and gapping at the pile-soi1 interface was

examined for the axial and lateral response of piles due to rnonotonic and cyclic loading.

The Drucker-Prager inelastic model was used for the soil. The piles were made of

concrete and had a square cross-section. The results agreed well with elastic solutions

and experimental field tests. For lateral loads, pile-soi1 separation and, to a lesser extent,

plasticity of the soil were found to be the crucial factors affecting the horizontal response.

It was found that interaction between neighbouring piles was also influenced by nonlinear

behaviour, and ignoring these effects c m significantly overestimate the amount of

interaction between piles.

Wu and Finn (1996) used a quasi-three-dimensional method of analysis to

evaluate the response of single piles and a square group of four piles. The noniinear

dynamic analysis was conducted in the time domain and incorporated into the program

PILE-3D (Wu and Finn, 1996). Modified equivalent linear properties were used at 0.5s

time intervals to simulate nonlinear behaviour and darnping according to strain leveis

(similar to SNAXE program). Gapping was also included using a no tension cut-off for

the eight node block soil elements. Two-noded beam elements were used to simulate the

cylindrical piles. One directional lateral seismic loading was applied at the bedrock level

to the mode1 and compared with scaled model centnfbge tests carried out at the

California lnstitute of Technology (Gohl, 199 1). Both the finite element mode1 and the

centrifuge tests took into account both the nonlinear pile-soil-pile kinernatic interaction

and the superstructure-foundation inertial interaction. Radiation darnping was not

considered.

Makris and Gazetas (1992) developed a sirnplified three-step procedure for

estirnating the dynamic interaction between two vertical piles subjected to either inertial

or kinematic loading (vertically propagating shear waves). The method takes into

account pile-soiI-pile interaction in a homogeneous soi1 stratum through the use of a

dynamic Winkler model and dynamic interaction factors. The results show pile-to-pile

interaction effects are significant in the inertial loading (strong dependence on frequency)

but are insignificant for seismic (kinematic type) loading and could be neglected.

Novak and CO-workers (Novak and Sheta, 1982; Novak and Mitwally, 1990; El-

Marsafawi et al., 1992; and El Naggar and Novak, 1996) used plane strain solutions

(adopting a dynamic Winkler medium) to solve the Iateral response of piles and pile

groups. Novak and MitwaIly (1990) and El Marsafawi et al. (1992) used the

superposition approach, while Novak and Sheta (1982) used a direct analysis approach

(Mindlin-type solution) to calculate the impedance functions of single piles and pile

groups. These solutions assume linear elastic piles and linear visco-elastic soils, except

for the model by El Naggar and Novak (1996), which will be discussed at the end of the

section.

Nogami et al. (1992) developed a rational dynamic soil-pile interaction model

adopting Winkler's hypothesis with the consideration of nonlinearity in the vicinity of

single piles. The model uses frequency independent mas , sprïngs, and dashpots and uses

direct time-domain analysis to predict lateral response due to pile head loading or

prescnbed displacement at the pile head. Either the static behaviour of the pile-soi!

system or static p-y curves is used as initial pile-soi1 properties. With the consideration

of a gap formation at the pile-soi1 interface, the mode1 c m successfùlly predict the pile

response observed in dynamic load tests in the field. The study is limited to a single case

and no closed-form solutions are available.

El-Naggar and Novak ( 1995.1996) developed a computationally efficient time

domain mode1 for the lateral response analysis of single piles and pile groups. In their

model, the piles are assumed to be elastic, vertical, and embedded in a nonlinear

horizontally layered soil. Based on the Winkler hypothesis, the soil is divided into a

number of layers. In each layer the soil model consisrs of two regions; the first region is

an inner field to which nonlinearity is confined, and the second region is a linear visco-

elastic far field which accounts for wave propagation away from the pile. In the lateral

response model, the soil reactions at both sides of the pile are modeled separately to

account for the state of stress and discontinuity conditions (i.e. allowing for ga? forming

or closwe automatically) at both sides as the load direction changes. Pile-soil-pile

interaction is incorporated in the analysis through connecting each hvo piles in the group

with a viscoelastic spring whose constant is derived through subjecting the function

describing the fiequency dependent displacement field to an inverse Fourier transfom.

The force in this spring depends on the displacement of the source pile and is transmitted

to the pile on the other end of the spring. This force is to be accounted for in the group

analysis. This mode1 may be applied to analyze the response of the entire pile group,

accounting directly for the nonlinearity and the interaction between al1 piles

simultaneously. alternative!^, the superposition approach may be used to approxirnate

the group response. A limited parametric study was performed to calculate impedance

functions at the pile head.

The studies mentioned above represent the most sophisticated and detailed

analyses on pile-soi1 and pile-soil-pile foundation systems. The additional infornation

given later can be used as a reference to Further enhance the understanding of nonlinear

lateral dynamic analysis on piles.

CHAPTER 3

KINEMATIC INTERACTION STUDY

3.1 INTRODUCTION

The catastrophic darnage fkom recent earthquakes (e-g. Yugoslavia earthquake of

1998, Kobe earthquake of 1995_ North Ridge earthquake of 1994 and Cairo earthquake of

1992) has raised concems about the curent codes and approaches used for the design of

structures and foundations. In the past, free-field accelerations, velocities and

displacements have been used as input ground motions for the seismic design of

structures without considering the kinematic interaction of the foundation or the site

effects that have resulted from the introduction of piles or the soi1 stratigraphy.

Depending on the pile group configuration and soil profile, free-field response may

underestimate or overestimate actual in-situ conditions which as a result, wilI radically

change design criteria.

Earthquake induced loading can be separated into two basic loading conditions,

kinematic and inertial. The present chapter is concemed with the response of single piles

to kinematic loading over a range of soil and pile parameters which can be used to help

model the inertial interaction model presented in Chapter 4.

Fan et al. (1991) performed an extensive parametric study using an equivalent

linear approach to develop dimensionless graphs for pile head deflections verjus the free-

field response for various soil profiles subjected to vertically propagating harmonic

waves. Makris and Gazetas (1 992) applied fiee-field accelerations to a one-dimensional

Beam-on-Dynamic-Winkler-Foundation mode1 with frequency-dependent springs and

dashpots to analyze the response of single piles and pile groups. The results fiom both

studies concluded that interaction effects on kinematic loading are negligible but are

significant for pile head loading. These studies were limited to linear (equivalent linear)

analysis and one-dimensional harmonic loading.

A full three-dimensional transient nonlinear dynamic analysis was performed in

the current study to investigate the effects of kinernatic interaction on the input motion at

the foundation level. This analysis accounted for pile-soi1 gapping and slippage, soil

plasticity, and three-dimensional wave propagation. The finite element program (ANSYS)

was used in the analysis.

3.2 ASSUMPTIONS AhD RESTWCTIONS

Figure 3.1 depicts the problem addressed in this study. As s h o w in Fig.3.1, the

actual system consisted of a pile foundation supporting a bridge pier. The current codes

use the free-field motion as the input ground motion at the foundation level. The analysis

described herein attempted to evaluate the interaction of the pile-soi1 system and how it

alten the fiee- field motion and modifies the ground motion at the foundation level.

The dynarnic loading was applied to the rigid underlying bedrock (Fig.3.1) as

one-dimensional horizontal acceleration (X-direction in finite element model) and only

horizontal response was measured. Vertical accelerations were ignored because the

rnargins of safety against static vertical forces usually provided adequate resistance to

dynamic forces induced by vertical accelerations. Wu and Fim (1996), using a three-

dimensional elastic model, found that deformations in the vertical direction and normal to

the direction of shaking are negligible compared to the deformations in the direction of

horizontal s haking.

Although the finite element analysis used in this study includes important features

such as soi1 nonlinearity and gapping at the pile-soi1 interface, it does not account for

build-up of pore pressure due to cyclic loading. Thus, the potential of liquefaction is not

accounted for in the current analysis. The dilatational effect of clays and the compaction

of loose sands in the vicinity of piles are not accounted for. Furthemore, the inertial

interaction between the superstructure and the pile-foundation system is not considered.

The analysis is limited to the response of free-headed piles with no extemal forces

(D'Alembert forces) from the superstructure to better understand the kinematic

interaction effects in seismic events.

3.3 3-D FINITE ELEMENT MODEL

3.3-1 Mode1 Formulation

Full three-dimensional geomeû3c models were used to represent the pile-soi1

systems. Exploiting symmetry, only one half of the actual mode1 was built thus

significantly reducing computing effort. Figure 3.2(b) depicts the pile-soi1 system

considered in the analysis showing an isometric view of the considered half. Figs. 3.2

and 3.3 shows the finite element mesh (mesh No. 3) used in the analysis. The pile and

soil were rnodelled using eight-noded block elements. Each node had three translational

degrees of fkeedom, i.e. X, Y and Z coordinates, as shown in Fig.3.4(a). A three-

dimensional point-to-surface contact element was used at the pile-soi1 interface to allow

for sliding and separation in tension, but ensured compatibility in compression. The

contact element had five nodes with three degrees of freedom at each node, i.e.

translations in the X, Y and Z directions as shown in Fig.3.4(b). Transmiaing boundaries

were used to allow for wave propagation and to eliminate the "box effect" (Le. the

reflection of waves back into the mode1 at the boundaries) during dynamic loading. The

element used to mode1 the transmitting boundary consisted of a spring and a dashpot

arranged in parallel, illustrated in Fig.3.4(c).

3.3.2 Soil Properties

The soil was modeled using two approaches; a homogeneous elastic medium and

an elasto-plastic material using the Drucker-Prager failure cnteria, to evaluate the effect

of soil plasticity on the response. For cases involving plasticity, the angle of dilatency

was assumed to be equal to the angle of interna1 fiction (associated flow rule). There

was no strain hardening and therefore no progressive yielding was considered. Effective

parameters and drained conditions were assumed because excess pore pressures were not

considered in the analysis. The material damping ratio of the soil, C, was assumed to be

5% based on average cyclic shear strain laboratory tests (Kramer, 1996). The goveming

equations of the system are given by

where {fi}, {ti}, and {u} are the acceleration, velocity and displacement vecton,

respectively, and FI], [Cl and w] are the global mass, damping and stiffness matrices,

respectively. The darnping matrix, [CI = &], in which the damping coefficient, P

-- , where the predominant frequency of the loading (rads) is substituted for o, *O

(natural frequency). Material damping was assurned to be constant throughout the entire

seismic event although the damping ratio varies with the strain level.

3.3.3 Pile Properties

Cylindrical reinforced concrete piles with linear elastic properties were

considered in this study. The piles were rnodeled using eight-noded brick elernents. The

cylindncal geometry was approximately modeled using wedge shaped elements

(Fig.3.2a). No darnping was considered within the piles and relevant parameters are

Iisted in Fig.3.5.

3.3.4 Pile-Soi1 Interface

The modeling of the pile-soi1 interface is crucial because its effect on the response

of piles to lateral loading is significant (Trochanis et al., 1988). Two cases were

considered in the analysis. First, the pile and soil are perfectly bondedo in which case the

penmeter nodes of the piles coincide with the soil nodes (elastic with no separation).

Second, the pile and soil are connected by fnctional interface elements that are descnbed

below. The contact surface (pile) is said to be in contact with the target surface (soil)

when the pile node penetrates the soil surface. A very small tolerance was assumed to

prevent penetration and to achieve instant contact as pile nodes attempt to penetrate the

soil nodes (or vice versa). Coulomb fnction was ernployed between the pile and soi1

dong the entire pile length as well as the pile tip (for floating piles). The coefficient of

fnction relating shear stress to the nomial stress was chosen according to API

recornmendations (MI, 1991) and assumed to be 0.7 throughout the analysis. The

contact surface coordinates and forces were fully updated, for either large or small

deflections that occurred. The penalty function method was used to represent contact

with a normal contact stiffness (K,). The normal stiffness allowed the interface element

to deform elastically before slippage occurred and was chosen to be equal to the shear

modulus of the soil. Convergence was achieved and over-penetration was prevented

using K,=6800 Wh.

3.4 BOUNDARY CONDITIONS

Boundary conditions varied depending on the type of loading. For static loading,

the bottorn of the mesh (representing the top of the bedrock layer) was always fixed in al1

directions. Al1 symrnetry faces were fixed against displacernent normal to the syrnrnetry

plane, but were free to move on the swhce of the plane. The nodes along the top surface

of the mesh were free to move in al1 directions. The nodes along the sides of the mode1

were £kee to move vertically but were constrained in the horizontal direction by a Kelvin

element in order to represent a horizontally infinite soil medium during static and

dynamic analyses. The constants were calculated using the solution due to Novak and

Mitwally ( 1988), given by

where k, = total stifhess, G = soil shear modulus, ro = distance to finite element

boundary, Si and S2 = dimensionless parameters korn closed fom solutions, v =

Poisson's ratio, a, = dimensionless fiequency = rooNs, and o = circular fiequency of

loading and V, = shear wave velocity of the soil. The following assumptions were

adopted in the formulation of Eq.3.2:

(1) The medium is linear, homogeneous, and isotropic with hysteretic, fiequency

independent matenal damping;

(2) The cylinder (pile) is circular, massless, and infinitely long and is welded to the

medium;

(3) The displacements are small and uniform along the cylinder; and

(4) The vibration is harmonic and axisymmetric.

The real and imaginary parts of Eq.3.2 represent the stiffness and damping, respectively,

i.e.

GS, K=- GS2 and C = -

To determine the stifiess and damping of the Kelvin elements, the constants given by

Eq. 3.3 were multiplied by the area of the element face (normal to the direction of

loading) because they assume constant unit area of contact. For static loading, i.e. zero

fiequency. the damping term vanishes and the element reduces to a spring only.

For dynarnic loading, o was taken as the predominant frequency of the

earthquake load and was determined fiom a discrete Fourier aansform of the time history

of the input motion. Figure 3.6 shows the Fourier amplitude (c.) versus fiequency (a,)

content for the strong motion record used in the study. It is evident that a narrow

s p e c t m exists at a dominant frequency of approximately 2 Hz. Time dependent

displacements were applied to the straatum base to simulate seismic loading. Al1 other

boundary conditions remained unchanged and are graphically portrayed in Fig.3.3.

3.5 LOADING CONDITIONS

3.5.1 Initial Loading

The state of stress in the pile-soi1 system in actual in-situ conditions was

replicated as an initial loading condition pnor to any additional dynamic or static external

load. That is, geostatic stresses were modeled by applying a global gravitational

acceleration, g, to replicate vertically increasing stress with depth. A linearly increasing

pressure with depth was applied to the periphery of the soil block to replicate horizontal

stresses as shown in Fig.3.3(b). A coefficient of lateral earth pressure, &=0.65, typical

of many geological conditions, was used. Due to the difference in density and stiffness

for the pile and soil, the soil tended to settle more than the pile in the vertical direction

resulting in premature slippage at the pile-soi1 interface. To eliminate this false

representation of initial conditions, the difference between the relative displacement

between the soil and the pile were accounted for by adding a corresponding body load to

the pile. The resulting mesh represented in-situ conditions, especially for ddled caissons

(Trochanis et al, 1988).

3.5.2 Static Loading

Al1 static loads were applied as distributed load along the perimeter of the pile

head which was level with the ground surface. Only one half of the total load was

applied to the pile in the finite element analysis due to the symrnetric geometry of a full

circuIar pile.

3.5.3 Dynamic Loading

Strong motion records fiom the Lorna Prieta earthquake in Califomia (Mr=7. 1) in

1989 were used in the finite element study. The accelerogram and displacement data

used were from the Yerba Buena Island rock outcrop station in the Santa Cniz Mountain

(NCEER, 1998). The measured displacements were applied to the top of the rigid

bedrock layer at 0.02sec intervals. Considenng that the maximum acceleration of the

measured one dimensional motion was 0.03g, the data were multiplied by a factor of

seven to simulate a PHA (peak horizontal acceleration) of approxirnately 0.2g for the

bedrock input motion. It is important to note that the acceleration data were for bedrock

motions and not fiee-field motions, which can either increase or decrease in terms of

PHA due to the site effects. Motions of 20sec duration were modeled to capture al1 the

important features of the earthquake. The predominant frequency was approximately 2

Hz, which is typical of destructive earthquakes (Kramer, 1996).

3.6 VERIFICATION OF FINITE ELEMENT MODEL

The verification process followed incremental steps to ensure that pile, soil, and

b o u n d q conditions were separately accounted for to minimize error accumulation. The

size of the mesh was mainly dependent on the loading conditions (static or dynamic) and

geometry of the piles. The mesh was refined near the pile to account for the severe stress

gradients and plasticity encountered in the soil, with a gradua1 transition to a coarser

mesh away fiom the pile in the horizontal X and Y directions. The vertical Z-direction

subdivisions were kept constant to allow for an even distribution of vertically propagating

SH-waves. The maximum element size, E,, was less than one-fifi to one-eighth the

shortest wavelength (A) to ensure accuracy m e r , 1996), i.e.

where h = V J f , V, is the soil shear wave velocity and f is the excitation fiequency in Hz.

n i e minimum V, was 80 m/s and the dynarnic loading had a cut-off fiequency equal tu

20Hz, thus a maximum element length of 0.5m was adopted. The proposed element

division was verified by results from a sensitivity study focussing on vertical pile shafi

discretization. El-Sharnouby and Novak (1985) found that using 12 to 20 elements gave

accurate results with a minimum of computational effort, hence this range was adopted in

this study.

The pile mesh was first verified by considering the pile as a fixed cantilever in air

(no soil). Lateral deflections due to a static load for three different pile mesh sizes were

compared to one-dimensional beam flexure theory (Table 3.1). As can be noted from

Table 3.1, the maximum difference was 8%. The results were very close and the small

differences, however, could be explained as bearn theory is not exact (ignoring shear

deformations) and the finite model was not a perfect cylinder. Due to the Iirnitations of

the maximum number of elernents (6000) and nodes (1 1000) available, 180 elements

were used to model the pile (accuracy within 8% of theoretical solutions). When soil and

boundary elements were added, the total nurnber of elements was close to the limitations.

The soil was then added to the model assurning a homogeneous soil straturn (Fig.

3.5). The elastic responses of socketed and floating single piles in homogeneous soi1

stratum were compared to the results from two different analyses. First, the results f?om

Poulos and Davis (1980) using Mindlin's equations and enforcing pile-soi1 cornpatibility.

Second, the results presented by Trochanis et al. (1988) using a 3-D fmïte element

analysis, however, their pile had a square cross-section but same flexural rigidity. Three

different soil meshes were built with increasing refinement to determine an acceptable

level of accuracy while maintaining a computational efficient model. Mesh No. 1

consisted of 1080 elements, mesh No.2 consisted of 2640 elements, and mesh No.3 had

3280 elements. Other meshes with a total number of elements equal to 6000 were also

tested but were not used due to unreasonable cornputer processing time.

The results for the linear elastic response are shown in Fig.3.7(a). The mesh that

yielded the closest match (Mesh No. 3, depicted in Figs.3.2,3.3) was used in the analysis.

The deflections obtained in this study were slightly greater than those from Poulos and

Davis (1980). However, they pointed out that their solution rnight underestimate the

response of long piles in sofi soils. Figures 3.7(b) and 3.7(c) show pile head deflections

considering separation at the pile-soi1 interface and soil plasticiv, respectively. It cm be

seen that good agreement with the results from Trochanis et al. (1988) exists. The

differences in the plastic soi1 case may be attributed to using a different model for soi1

plasticity (modified Drucker-Prager model). Figures 3.8(a) and 3.8(b) show the elastic

soi1 surface displacements away fiom the pile compared to results from elastic theory

(Poulos & Davis, (1980); other FEA (Trochanis et al., (2988)). It can be seen from

Figs.3.8(a) and (b) that the results obtained using mesh No.3 agree well with both

solutions, especially close to the pile. The pressure distribution in the soi1 agreed equally

well.

The final step in the verification process was accomplished by solving the ground

response to an earthquake signal using the finite element model and comparing the elastic

ftee-field response to that obtained using the prosam SHAKE91 (Idnss and Sun, 1992).

Considering that SHAKE9 1 is a 1-D analysis, constraints were applied to the finite model

to allow only displacements in the direction of shaking (one degree of freedom per node)

to replicate 1-D results. The results from the finite element analysis and SHAKEBI are

plotted in Fig.3.9 for elastic response using exactly the same parameters. A constant

shear modulus and material darnping ratio were used in both the SHAKE91 and the FEA

models. It can be seen from Fig.3.9 that the agreement is quite good along the entire time

period considered. The maximum free-field accelerations for the FEA and SHAKE91

were both amplified to approximately 0.6g fiorn 0.2g (bedrock input motion) and are

cornpared with bedrock accelerations in Fig. 3.10. The same FEA rnodel was modified to

allow for three dimensional response and the free-field response is plotted against the one

dimensional results in Fig.3.11. The maximum fiee-field acceleration obtained from 3-D

analysis was onIy 0.35g (Fig. 3.1 1). The amplification of the accelerations calculated

fiom the 3-D analysis are closer to those observed during actual seismic events. Hence, it

was concluded that the 3-D analysis resulted in realistic acceleration magnitudes.

Therefore, al1 M e r models assumed full 3-D capability and are discussed herein.

3.7 COMPUTATIONAL TIME AND METHOD

The ANSYS finite element program can solve the dynamic response of structural

systems in either the fkequency or time domain. Because nonlinearity and gapping were

deemed to be important in seismic response, and hence were introduced into the model,

the time domain was chosen for the analysis. The response was calculated at intervals of

approximately 0.02s. The Newmark integration method (Bathe, 1982) was used with

a=1/2 and 6=1/4 to obtain an unconditionally stable scheme. The Modified Newton-

Raphson iteration technique (Bathe, 1982) was used and convergence criteria were force

and displacement dependent. Static solution processing time averaged between five and

forty-five minutes, whereas the dynamic solutions were extremely time demanding. For

a 20-second earthquake with 0.02 sec time intervals, the nonlinear solution with gapping

lasted approximately 10 days on a Pentiurn 233 MHz persona1 cornputer. Five Pentium

cornputers (three 233 MHz and bvo 266 MHz with 64 and 128 Mb RAM respectively)

were used simultaneously at the University of Western Ontario to optimize time

efficiency.

3.7 NUMERICAL STUDY FOR KINEMATIC INTERACTION

The kinematic ef5ects of piles in a homogeneous soi1 medium were evaluated by

comparing acceleration time histones and Fourier Spectra of the pile head and the fke-

field. The sarne dynamic loading was applied in al1 cases (Le. Loma Prieta data) to the

underlying bedrock for a hornogeneous soil profile. Results from seven different pile-soi1

configurations were obtained and are referred to in Figs.3.12-3.18. The following

notation is used throughout the graphs and literature to identie each numerical nin case

(see Fig.3.5 for soi1 and pile parameters):

EFH - refers to the fiee-field response using linear isotropic visco-elastic constitutive

relations (Elastic, Free-field, Homogeneous).

PFH - refers to the fkee-field response using a perfect elastic plastic soil model,

Drucker-Prager criteria (Plastic, Free-field, Homogeneous).

ESNFH - refers to the floating single pile head response using a linear isotropic visco-

elastic soil with no separation at the pile-soi1 interface, LID = 15, Efi, = 1000 (Elastic

Single pile, No separation, Floating, Hornogeneous)

ESNSH - refers to the socketed single pile head response using a linear isotropic visco-

elastic soil with no separation at the pile-soi1 interface, L/D = 20, E@, = 1000 (Elastic,

Single pile, No separation, Socketed, Homogeneous).

ESSFH - same as ESNFH case, but allows for separation at the pile-soi1 interface

(Elastic, Single pile, Separation, Floating, Homogeneous).

PSSFa - refers to the floating single pile head response using a perfect plastic elastic

soil model with separation allowed at the pile-soi1 interface, L/D = 15, E+, = 1000, c, =

34 Wa, = 16.5" (Plastic, Single pile, Separation, Floating, Homogeneous).

PSSSH - refea to the socketed single pile head response using a perfect plastic elastic

soil model with separation allowed at the pile-soi1 interface, L/D = 20, E&, = 1000, c, =

34 kPa, y = 16.5' (Plastic, Single pile, Separation, Socketed, Homogeneous).

3.8.1 Results

Figure 3.l?(a) compares the free-field response for the elastic and plastic soil

cases respectfully. The difference between the two cases is not evident over the 20

second duration, but a more detailed evaluation is presented in Fig.3.15 for the 2 to 10

second interval. The acceleration response is slightly arnplified using a plastic soil model

which cm be attributed to the lirniting ultimate effective stress and thus limiting shear

strength. Figure 3.12(b) compares the Fourier Spectra for elastic and plastic soi1 profiles

against the input bedrock spectnim using a cut-off fiequency of 20 Hz. it is evident fiom

the figure (Fig.3.12(b)) that there is an amplification of the fourier amplitudes for the

free-field response compared to the bedrock. There is notable increase in amplitude for

the plastic soil model over the elastic soi1 model, suggzsting the reduction in soil stifiess

reduces the natural frequency of the homogeneous layer. The increase in acceleration

and amplitude rnay be attributed to the fact that the fmt natural frequency of the elastic

homogenous layer is slightly greater than 2 Hz, whereas the natural frequency of the

plastic soil layer is slightly decreased to approach the predominant fkequency at the free-

field (approximately 1.5 Hz). For higher frequencies (Fig.3.12b), the srna11 amplitude

peaks seen at the bedrock level diminish as the seismic waves propogate throughout the

soil until they reach the fiee-field. Both the elastic and plastic free-field amplitudes

dirninish at frequencies higher than 10 Hz, above which is usually considered to induce

Iittle response in most structures.

Similar results for the floating and socketed pile head response are plotted in

Figs.3.13 and 3.14. Figures 3.13(a) and (b) represent the corresponding acceleration and

founer spectrum for ESNFH compared to EFH. Both diagrarns are almost identical,

except the founer amplitudes are slightly greater for the floating pile (especially for

frequency above 5 Hz). Figure 3.14 shows the response of the ESNSH socketed pile

case. Again, the overall acceleration of the pile head is similar to that of the elastic free-

field. The fourier amplitudes of the socketed pile (no separation) show both a decrease

and an increase in magnitude over the elastic fiee-field depending on the frequency

range. At the predominant frequency amplitude (2 Hz), ESNSH seems to slightly

decrease compared to EFH, and at frequencies above and below the predominant

frequency the amplitudes are increased. The increased stifmess of the system by

introducing the socketed pile may be responsible for the increased amplitude at higher

fkequency ranges compared to the fkee-field.

Figure 3.16 introduces the effects of separation between the pile and soil for the

floating pile case. Only the 2-10 second time interval is shown to provide a more

detailed analysis. The overall response is very similar for both cases shown in Fig.3.16.

The floating pile with gapping seems to eliminate small fluctuations of acceleration seen

when no gapping was allowed.

Figure 3.17 introduces the effects of the soi1 plasticity in addition to separation for

the floating pile. PSSFH is compared with the elastic model (ESSFH) and the results are

very similar. The random scatter shown by introducing plasticity may be attributed to the

solution procedure used in the finite element program. For convergence reasons, smaller

timesteps had to be used for the plastic soil model which led to numerical instabilities.

Figure 3.18 compares the floating and socketed pile head response including both

separation and soil nonlinearity. The floating pile showed slightly higher peaks over the

socketed pile, but the response remained alrnost identical.

Table 3.1 - Verificatioa of pile head response as cantilever beam

MODEL Bearn-Flexure Theory

Pile mesh No.1

Pile mesh No.2

Pile mesh No.3

NOTES:

'A error O O

5.2 5.0

6.1 6.6

7.0 8.1

Beam Rexure theory 6 3El

All mcdels assume L = 7.5m. D=0.5m. ~,=2'l0' kPa. 1 = it*(~/2)'/4 Pile mesh No.? - 15 vertical divisions, 54 elernents/division equaliing a total of 810 elements Pile mesh No.2 - 30 vertical divisions, 12 elements/division equalling a total of 360 elements Pile mesh No.3 - 15 vertical divisions, 12 elements/division equalling a total of 180 elements

Figure accompanies Table 3.1

P = applied static lateral load (kN) O = pile diameter (m) L = pile length (m) 8 = defiection of pile head (rn)

SUPERSTRUCTURE

c- LrhPERLYWG BEDROCK

ACTUAL SYSTEM SYSTEiM ANALYZED

Figure 3.1 - Definition of the problem and terminology (actual acceleration, Us, acceleration due to Lÿnernatic interaction only, U,, bedrock acceleration, U,, and free-field acceleration, Un)

Figure 3.2 - (a) Detail of wedge shaped pile elements surrounded by soi1 elements (plan view), (b) Isometric view of compiete soi1 and pile mesh model.

Figure 3.3 - Finite element mesh (Mesh No.3) showing boundary conditions: (a) plan view, (b) front cross section view with geostatic pressure distribution.

(a) - 8 noded Block element, DOF: X, Y, Z

SOlL SURFACE ,' PILE NOOE -----'

(b) - 5 noded Contact element. DOF: X, Y, Z

(c) - 2 noded Boundary element, DOF: X, Y, Z

Figure 3.1 - (a) Block element used for soil and pile, (b) Surface Contact element used between pile and soil to aliow for slippage and separation, (c) Transmitting Boundary element consisting of "spnng(K)" and "dashpot(C)" to allow for radiating boundaries.

PILE CONFIGURATIONS SOIL PROFILES

FLOATING PILE SOCKETED PILE HOMOGENEOUS LAYERED Lf = 7.5 m L , = l O m Es= Soit Young's Modulus

E,=Pile Young's Modulus = 2'1 0 'k~a D = 0.5m y, = Soil unit weight = 1.5 k ~ l r n ~ 6 = horizontal deflection y, = Pile unit weight = 2.3 k ~ l r n ~ P = horizontal load v,= Soil Poissons Ratio = 0.45 Cu = 34 kPa, 4 = Y = 16.5" (Plasticity) v,= Pile Poissons Ratio = 0.30

Figure 3.5 - Two dimensional representation of floating and socketed pile in either homogeneous (used for verifkation) or layered soi1 profile.

5 10 15 20 25

Frequency (a, 1 2 x = Hz)

Figure 3.6 - Fourier amplitude spectrum for earthquake Ioading at the bedrock level.

Horizontal deflection (cm)

+ present study UD=20 (Mesh#3)

+ 1988 FEA studv

O 0.5 1 1-5 Horizontal deflection (cm)

-&- present study UO=20 (Mesh#3)

+ 1988 FEA study UD=20

O 0.5 1 1.5 Horizontal deflection (cm)

Figure 3.7 - Response of single socketed pile for: (a) elastic, @) elastic- gapping, (c) plastic-gapping.

O Poulos & Davis

0 1988 F E . Study

+ present study (Mesh#3)

A

)r Y

0.1 - 0.0 4 m m

lm w

A

Figure 3.8 - (a) Comparison of soi1 displacements along line of loading, s(x), relative to pile head displacement, s(O), (b) Cornparison of soi1 displacements normal to direction of Ioading, w(y), relative to pile head displacement, w(0).

* . - - - - 1 -D FREE-FIELD (FEA)

- FREE-FIELD (SHAKE91)

O 5 15 20 10 Tirne (s)

Figure 3.9 - One dimensional verification of finite element analysis (FEA using ANSYS) with SHAKE91.

0.7 - a - - . - 1-0 FREE-FIELD

0.5 - BEDROCK (1-0 8 3-0)

G 0.3 I i C 0 0.1 C

2 $ -0.1 U

$ -0.3

-0.5

-0.7 O 5 'O Time (s) 15 20

Figure 3.10 - Response of underlying bedrock and free-field for hornogeneous soil (using one-dimensional FE-4)

- - - - * 1-D FREE-FIELD (1-0 EFH)

3-D FREE-FIELD (3-0 EFH)

O 5 10 Time (s) 15 20

Figure 3.11 - Elastic free-field response for hornogeneous soil (EFH) for one and three dimensional anaiysis.

O 2 4 6 8 10 12 14 16 18 20 Time (s)

O 2 4 6 8 10 12 14 16 18 20 Frequency (Hz)

Figure 3.12 - (a) Comparison between calculated accelerations for elastic frea field (EFH) and plastic free-field (PFH) using the Drucker-Prager criteria for a homogeneous soii protile. @) Fourier spectrum for the response at the bedrock level, elastic soi1 free-field, and plastic soi1 free-field.

8 10 12 Time (s)

ESNFH

8 10 12 Frequency (Hz)

Figure 3.13 - (a) Cornparison between calculated accelerations for elastic free- field (EFH) and floating pile head (ESNFH) for a homogeneous elastic soil profile. (b) Fourier spectrum for the response of the elastic soil free-field and floating pile head.

Time (s)

O 2 4 6 8 10 12 14 16 18 20 Frequency (Hz)

Figure 3.14 - (a) Comparison between calculated accelerations for elastic free- field (EFH) and socketed pile head (ESNSH) for a homogeneous soil profile. (b) Fourier spectrum for the response at the plastic soil free-field and socketed pile head.

2 3 4 5 6 7 8 9 10 Time (s)

Figure 3.15 - EFH and PFH response (Elastic and Plastic free-field, E,=20000

- - - - - - ESNFH

ESSFH

2 3 4 5 6 7 8 9 10 Time (s)

Figure 3.16 - Pile head response for Boating pile (Elastic, Eiastic with Gapping).

I I I

PSSFH

2 3 4 5 6 7 8 9 1 O Time (s)

Figure 3.17 - Pile head response for floating pile (Elastic gapping, Plastic gapping).

PSSSH

2 3 4 5 6 7 8 9 10

Time (s)

Figure 3.18 - Pile head response for floating and socketed pile (Plastic Gapping)

CHAPTER 4

INERTUL INTERACTION STUDY

4.1 INTRODUCTION

Pile foundations are often subjected to lateral loading due to forces on the

supporting structure, i.e. inertial interaction. In Chapter 3, the effects of the kinematic

interaction and the response of piles to loading from excitation at the bedrock level were

examined. This chapter focuses on inertial Ioading or excitation at the pile head. The

horizontal loads at the pile head can be the goveming design constraint for single piles

and pile grououps supporthg different types of structures in many situations. Such

conditions may include environmental loading (wind, water, and earthquakes) and

machine loading on structures such as buildings, bridges, and offshore structures.

Extensive research efforts have been directed to the lateral response of piles

recently due to the significant influence of the foundation flexibility on the performance

of structures in extreme loading events. Offshore oil platforms could cost miIIions of

dollars due to overly conservative designs to address the effects of wave and wind action,

as well as iceberg impact. On the other band, unsafe design could result in catastrophic

human and economic losses. Lateral movement of the supporting foundations also

influences the response of buildings and bridges to earthquake loading or ship impacts.

Traditionally, large factors of safety have been used resulting in over-conservative design

and cost ineffectiveness. Most building and bridge codes use factored static loads to

account for the dynarnic effects of pile foundations. Although very low frequency

vibrations may be accurately rnodeled using factored loads, the introduction of

nonlineanty, damping, and pile-soi1 interaction during transient loading may significantly

aIter the response.

Novak et al. (1978) developed a fiequency dependent soil model, however, it

assumes strictly linear or equivalent linear soil properties. Gazetas and Dobry (1984)

introduced a sirnplified linear method to predict fixed-head pile response accounting for

both geometric and radiation damping and using avaiiable static stiffhess (denved from

finite element or any other accepted method). This method is not suitable for the seismic

response analysis because of the linearity assumptions. In general, there is much

controversy over advanced iinear solutions (frequency domain) as they do not account for

permanent deformation or gapping at the pile soil interface.

Nogami et aL(1992) have developed a time domain analysis method for single

piles and pile groups by integrating plane strain solutions with a nonlinear zone around

each pile using p-y curves. El Naggar and Novak (1995,1996) also developed a

computationally efficient model for evaluating the lateral response of piles and pile

groups based on Winkler hypothesis accounting for nonlinearity using a hyperbolic

stress-strain relationship, and slippage and gapping at the pile-soi1 interface. The model

also accounts for the propagation of waves away from the pile and energy dissipation

through both material and geometnc darnping.

The p-y curves (unit load transfer curves) approach is a widely accepted method

for predicting pile response under static loads because of its simplicity and practical

accuracy. In the present study, the model proposed by El Naggar and Novak (1996) is

modified to utilize existing or developed cyclic or static p-y curves to represent the

nonlinear behaviour of the soil adjacent to the pile. The model uses unit load transfer

curves in the time domain to model nonlinearity, and incorporates both rnaterial and

radiation darnping to generate dynamic p-y cuves.

4.2 MODEL DESCRIPTION

4.2.1 Pile

Piles are assurned to be vertical, flexible wiîh circular cross-section. Non-

cylindrical piles are represented by cylindrical piles with equivalent radius to

accommodate any piedpile configurations. The pile, and the surrounding soilt is

subdivided into n segments, with pile nodes corresponding to soil nodes at the same

elevation. The standard bending stifhess rnatrix of beam elements models the siructural

stifiess matrix for each pile element. The pile global stiffhess matrix is then assembted

h m the eiement stifiess matrices and is condensed to give horizontal translations at

every layer and the rotational degree of fieedom at the pile head.

4.22 Soil Model: Hyperbolic Stress-Strain Relationship

The soil is divided into n layers with different soil properties assigned to each

layer according to the soil profile considered. Within each layer, the soi1 medium is

divided into two annulus regions as shown in Figure 4.1. The first region is an inner zone

adjacent to the pile and accounts for the soil nonlinearity. The second region is the outer

zone, which allows for the wave propagation away from the pile and provides for the

radiation damping in the soil medium. The soil reactions and the pile-soi1 interface

conditions are modeled separately on both sides of the pile to account for slippage,

gapping and state of stress as the ioad direction changes.

4-2.3 Inner field element

The inner field is modeled with a nonlinear spring to represent the stiffness and a

dashpot to simulate material (hysteretic) darnping. The stifkess is calculated assuming;

plane strain conditions, the inner field is a homogeneous isotropic viscoelastic medium,

the pile is rigid and circular, there is no separation at the soil-pile interface. and

displacements are small. Novak and Sheta (1982) obtained the stifiess under these

conditions as,

where ro is the pile radius, ri is the outer radius of the imer zone, and v is the Poisson's

ratio of the soil stratum. G, is the rnodified shear modulus of the soil and is

approxirnated, according to the strain level, by a hyperbolic Iaw as

G- is the maximum shear modulus (small strain modulus) of the soil according to lab or

field tests and q =PPu is the ratio of the horizontal soil reaction in the soil spring, P, to

the ultirnate resistance of the soil element, Pu. The ultimate resistance of the soil element

is calculated using standard relations given by the API (Arnerican Petroleum Institute,

199 1). For clay. the ultimate resistance is given as a force per unit length of soil by

where Pu is the minimum of the resistances calculated by Eqs.4.3 and 4.4, c, is the

undrained shear strength, d is the diarneter of the pile, y is the effective unit weight of the

soil, and J is an empirical coefficient dependent on the shear strength. A value of J=0.5

was used for soft clays (after Matlock, 1970) and J= lS for stiff clays (after Bhushan et

al., 1979).

The corresponding criteria for the ultimate Iateral resistance of sands at shallow

depths Pu! or at large depth Pu2 are (API, 199 1)

K,X tan#sinp + (d + x tanptana) + K, x tan,b(tan#sin,8 - tana) - ~ . d tan@ - 4 cosa tan(,& 4

In the previous equations, A is an empirical adjustment factor dependent on the depth

from the soi1 surface, K, is the earth pressure coefficient at rest, 4 is the effective friction

angle of the sand, P = 4/2 + 45O, a = 4/2, and K, is the Rankine minimum active earth

pressure coefficient defined as Ka = tan2(45~-$/2).

4.2.4 Far fieid element

The outer field is modeled with a linear spring in paralle1 with a dashpot to

represent the linear stiffness and darnping (mainly radiation damping). The outer zone

allows for the propagation of waves to infinity. Novak et al. (1978) developed explicit

solutions for the soi1 reaciions, complex stifkess ( K ) , of a unit length of a cylinder

embedded in a linear visco-elastic medium given by

where a, = or,,N, is the dimensionless frequency, o = the frequency of loading, V, =

shear wave velocity of the soi1 layer, and D = the matenal damping constant of the soi1

layer. Figure 4.2 shows the generaI variations of Sul and Sd with Poisson's ratio and

material damping. Rewriting Eq. 4.7, the complex stiffness, K, c m be represented by a

spring coefficient, k ~ , and a damping coefficient, CL, to represent the complex stiffness,

K, as

Frorn Figure 4.2, it can be noted that for the dimensionless frequency range between 0.05

and 1.5, Sui maintains a constant value and Su2 increases linearly with h. The majority of

dynamic loading on foundations falls within this fiequency range including destructive

earthquake loading. Therefore, for the purpose of the time dornain analysis, the spring

and dashpot constants, Sui and Su2, respectively, are considered frequency independent

and depend only on Poisson's ratio and are given as

4.2.5 Soii-Pile interface

The soil-pile interface is modeled separately on each side of the pile allowing for

gapping and slippage to occur on each side independently. The soil and pile nodes in

each layer are connected using a no-tension spring. That is, the pile and soil will remain

connected and will have equal displacement for compressive saesses. The spring is

discomected if tensile stress is detected in the soi1 spring to allow a gap to develop. This

separation or gapping results in permanent displacement of the soil node dependent on

the magnitude of the load. The development of such gaps is often observed in

experiment, during offshore loading, and afier earthquake excitation in clays. These gaps

eventually fil1 in again over time until the next episode of lateral dynarnic loading. The

pile-soi1 interface for sands does not allow for the gap formation, but instead the sand

caves in resulting in the virtual back filling of sand particles around the pile during

repeated dynarnic loading. When the pile is unloaded the sand on the tension side of the

pile follows the pile with zero stiffkess instead of rernaining permanently displaced as in

the clay model. In the unloading phase, the stiffness of the inner field spring is assumed

to be linear in both clay and sand rnodels.

4.2.6 Soi1 ModeI: p-y Cuwe Approach

The soil reaction to transient loading comprises stifmess and damping. The

stiffhess is established using the p-y curve approach and the damping is established from

analytical solutions that account for wave propagation. A similar approach was

suggested by Nogami et al (1 992) using p-y curves.

Based on model tests, p-y curves relate pile deflections to the corresponding soil

reaction at any depth (element) below the ground surface. The p-y curve represents the

total soil reaction to the pile motion, Le. the i ~ e r and outer zones' reactions combined.

The total stiffhess, Kpy, derived from the p-y curve is equivalent to the m e stiffness (real

part of the complex stifhess) of the soi1 medium. Thus, relating to the hyperbolic law

model, the combined inner zone stiffhess ( k ~ ~ ) and outer zone stiffhess (kL) c m be

replaced by a unified equivalent stiffhess zone (kpy) as shown in Figure 4.3. Hence, to

ensure that the bue stiffhess is the same for the two soil models, the flexibility of the two

models is equated, i.e.

The stiffhess of the nonlinear strength is then calculated as

The constant of the linear elastic spring, k ~ , is established from the plane strain solution,

i.e. Eq. 4.7. The static soi1 stifiess, kpy represents the relationship between the static soil

reaction, p, and the pile deflection, y, for a given p-y c -me at a specific load level. The p-

y curves are established using empîrical equations (Matlock, 1970; Reese and Welch,

1975; Reese et al, 1975; and more) or c w e fit to measured strain data using an accepted

method such as the modified Ramberg-Osgood model @esai and Wu, 1976). In the

present study, intemally generated static p-y curves are established based on cornrnonly

used empirical correlations for a range of soil types.

4.2.7 p-y curve generation for clay

The general procedure for cornputing p-y curves in clays both above and below

the groundwater table and corresponding parameters are recommended by Matlock

(1970) and Bhushan et al. (1979), respectively. The p--v relationship was based on the

following equation

where p = soil resistance, y = deflection corresponding to p, Pu= uItimate soil resistance

fiom Eqs. 4.3 and 4.4, n = a constant relating soi1 resistance to pieripile deflection, and

= corrected deflection at one-half the ultimate soil reaction determined fkom lab tests.

Equation 4.13 is discussed in more detail with relevant parameters in Appendix 1. The

stiffness constant, /r,,, of any soi1 element at time step t+At represents the slope of the

tangent to the p-y curve at the specific Ioad level as shown in Fig. 4.4. This dope is

established fiom the soil deflections at t h e steps t and t-At and the corresponding soil

reactions calculated fiom Eq.4.13, Le.

Therefore, Eq.4.9 and Eq.4.14 can be substituted into Eq.4.12 to obtain the nonlinear

stifiess representing the inner field element in the analysis. Thus, the linear and

nonlinear qualities of the unit load transfer curves have been logically incorporated into

the outer and inner zones, respectively. It should be noted that the maximum shear

modulus for any soil layer is calculated in this mode1 by (Hardin and Black, 1968)

3230(2.97 - e)' 0 5

G,. = w / m ' l + e

where e = void ratio and a. = the mean principal effective stress at the soil layer.

4.2.8 p-y curve generation for sand

Several different methods have been used to obtain p-y curves for sandy soils

from experiment. Abendroth and Greimam (1990) perfonned eleven scaled pile tests

and used a modified Ramberg-Osgood mode1 to approximate the nonlinear soil resistance

and displacement behaviour for loose and dense sand. The most commonly used criteria

for development of p-y c w e s for sand were proposed by Reese et al. (1974) but tend to

give very conservative results. Bhushan et al. (198 1) and Bhushan and Askari (1984)

used a different procedure based on full-scale load test results to obtain nonlinear p-y

curves for saturated and unsaturated sand. A step-by-step procedure for developing p-y

curves in sands (suggested by Bhushan and Haley, 1980; Bhushan et al., 198 1) was used

to estimate the static unit load transfer curves for different sands below and above the

water table. The procedure used to generate p-y curves for sand is different than the

procedure suggested for clays. The secant modulus approach is used to approximate soil

reactions at specified lateral displacements. The soil resistance can be calculated using

the following equation,

where (k) is a constant relating the secant rnodulus of soil to depth (ES=h) , (x) is the

depth at which the p-y curve is being generated, (y) is the lateral deflection, (Fl) is a

density factor, and (F2) is a groundwater (saturated or unsaturated) factor. The equation

and corresponding parameters are discussed in more detail in Appendix II. The constant

(k) assumes a linear Es profile with depth that is typicaI for many sands. The main

factors affecting k are the relative density of the sand (Ioose or dense) and secant

rnodulus reduction with increasing lateral displacement. Table II. 1 in Appendix LI shows

values of k with corresponding y/d (deflection normalized with pile diameter) proposed

by Bhushan et a1 (198 1). The values in Table II. 1 are applicable for sands with a relative

density of 85%. Therefore, Meyer and Reese (1979) suggested a correction factor (FI)

shown in Table 11.2 to obtain the k values for sands with a range of relative densities.

The relative density (Dr) of sand can be detennined using lab or in-situ techniques.

Appendix II also shows two numerical methods used to determine Dr (also referred to as

the density index, ID) from correlation with either the angle of intemal fiction (Figure

II. 1) or Standard Penetration Blow Count value (N-value). PolynomiaI functions were fit

to the values shown in Tables 11.1 and 11.2 and incorporated into the analytical model.

The groundwater factor (F2) varied between 0.5-1 .O for saturated and unsaturated sand,

respectively .

The stifhess b,, was calculated using Eq.4.14 based on calculated soil reactions

from the corresponding pile displacements for two consecutive time steps (using

Eq.4.16). The secant modulus decreases with increasing displacement and thus the

nonlinearity of the sand can be modeled accurately if small time-steps are used.

4.2.9 Damping

The dampinj (irnaginary part of the complex stifhess) is incorporated into both

the p-y approach and hyperbolic model to allow for energy dissipation throughout the

soil. The nonlinearity in the vicinity of the pile, however, drastically reduces the

geometric darnping in the inner field. Therefore, both material and geometric (radiation)

darnping are modeled in the outer field. A dashpot is comected in parallel to the far-field

spring, and its constant is derived from Eq. 4.10. The addition of the darnping resistance

to static resistance represented by the static unit load m s f e r (the p-y curve) tends to

increase the total resistance as shown in Figure 4.4. This effect is investigated M e r in

this chapter.

4.3 DEGRADATION FACTOR

Transient loading, especially cyclic loading, may result in buildup of pore water

pressures and/or change of the soi1 structure that causes the shear strain amplitudes of the

soil to increase with increasing number of cycles (Idriss et al., 1978). Idriss et al. (1978)

reported that the shear stress amplitude decreased with increasing number of cycles for

harmonically loaded clay and s a m t e d sand specimens under strain-controlled undrained

conditions. These studies suggest that repeated cyclic loading results in the degradation

of the soil stifhess. For cohesive soils, the value of the shear modulus after N cycles,

&, can be related to its value in the first cycle, G,,, by

where the degradation index, d; is given by S=N't and t is the degradation parameter

defined by Idriss et al. (1978). This is incorporated into the proposed 2-D p-y curve

mode1 by updating the nonlinear stiffhess kXr by an appropriate factor each loading cycle.

4.4 TIME-DOMAIN Ah'ALYSIS AND EQUATIONS OF MOTION

To include al1 aspects of nonlinearity and examine the transient response logically

and realistically, the tirne domain analysis was chosen. The governing dynarnic equation

of motion is given by

where [Ml, [Cl and [KI are the global mass, damping and stiffhess matrices, respectively,

and {ii}, {Ù}, {u} and F m are acceleration, velocity, displacement, and extemal load

vectors, respectively. Referrîng to Fig4.1, the mass of the imer field is lurnped at two

nodes. One half (ml) is lumped at the node adjacent to the pile (node l), and the other

half (mz) is at the node adjacent to the outer field (node2). The equations of motion at

each element for the inner field are

where z i ~ and 142 are displacements of nodes 1 and 2, and Fi is the force in the nonlinear

spring including the confining pressure, and F2 is the soi1 resistance at node 2.

The equation of motion for the outer field is written as

Assuming compatibility and equilibrium at the interface between the inner and outer

zones leads to the following equation, which is valid for both sides of the pile

Am, + Be, + kNL kNL + Am, + Bc, + kL

where F ~ " ~ and FZ'" are the sums of inertia forces and soil reactions at nodes 1 and 2,

respectively. The values A and B are constants of numerical integration for inertia and

damping.

The linear acceleration assumption was used and the Newmark f! method (Bathe,

1982) was implemented for direct time integration of the equations of motion. The

modified Newton-Raphson iteration scheme was used to solve the nonlinear equilibrium

equations.

4.5 VERIFICATION OF THE ANALYTICAL MODEL

4.5.1 Verification of clay model

Different soi1 profiles were considered in the analys is. Figure 1 g.5 shows i

typical pile-soi1 system and the soil profiles considered including linear and parabolic soil

profiles. The p-y model was first verified against the hyperbolic model (El Naggar and

Novak, 1996). Figures 4.6 and 4.7 compare the dynamic soil reaction and pile head

response for both the hyperbolic and p-y curve models for a single reinforced concrete

pile in soft clay. A 0.5m diameter, 15m long pile was used with an elastic modulus (E,)

equal to 35GPa. A parabolic soil profile was used and the ratio E@s=lOOO at the bottom

depth &) of the pile. Fig.4.6 that shows the calculated dynamic p-y cuves for a

prescribed displacement of 0.003d at the pile head with a fkequency of 2 Hz. It can be

noted fiom Figure 4.6 that the soi1 reactions obtained from the two models are very

similar, and approach stability afier 5 cycles. The pattern shown in Figure 4.6 is also

similar to that obtained by Nogami et al. (1992) showing an increasing gap and stability

afier approximately 5 cycles. Figure 4.7 shows the displacement-time history of the pile

head installed in the sarne soil profile. The load was applied at the pile head and was

equal to approximately 10% of the ultimate lateral loading capacity of the pile. The

hyperbolic and p-y models show very similar responses at the pile head and both stabilize

after approximately 5 cycles.

The dynamic soil reactions are, in general, larger than the static reactions because

of the contribution from darnping. Employing the same definition used for static p-y

curves, ddmamic p-y c w e s can be established to relate pile deflections to the

corresponding dynamic soil reaction at any depth below the ground surface. The

proposed dynamic p-y curves are fiequency dependent. These dynamic p-y curves can be

used in other static analyses that are based on the p-y curve approach to account

approximately for the dynamic effects on the soil reactions to transient loading.

Figures 4.8 and 4.9 show dynamic p-y curves established at two different clay

depths for a prescnbed displacernent at the pile head equal to O-OSd, for a fiequency

range from 0-10 Hz. A parabolic soi1 profile was assumed to exist along the 12.5m

length of a 0.5m diameter concrete pile. The elastic modulus of the concrete was 35 GPa

and the ratio E@,= 1000 at the 12Sm depth. Both the p-y curve and hyperbolic models

were used and the comparison was good, especially for lower fiequencies, as c m be

noted from Figures 4.8 and 4.9. It c m also be observed from Figures 4.8 and 4.9 that the

soil reaction increased as the fkequency increased. This increase was more evident in the

results obtained fiorn the p-y c u v e model

4.5.2 Verification of Sand model

The p-y curve and hyperbolic models were used to analyze the response of piles

installed in sand. The sand was assumed to be unsaturated and a linear soil modulus

profile was adopted. The same pile used in the clay example was used, except the total

ernbedded length was 10m instead of 15m. Figure 4.10 shows the calculated dynamic

soil reactions at Im depth for a piescribed displacement equai to 0.075d at the pile head

with a frequency of 2 Hz. As can be noted from Figure 4.10, the two models feature very

similar dynamic soil reactions. It should be noted that the soil reactions at both sides of

the pile are traced independently. The upper part of the curve in Figure 4.10 represents

the reactions for the soil element adjacent to the right face of the pile, when it is loaded

nghtward. The lower part represents the reactions of the soil element adjacent to the lefi

face of the pile, as it is loaded leftward. Both elements offer zero resistance to the pile

movement when tensile stresses are detected in the nonlinear soi1 spnng during unloading

of the soil elernent on either side. However, the soil nodes remain attached to the pile

node at the sarne level, allowing the sand to "cave in" and fil1 the gap. Observations tkom

field and laboratory pile testing confirmed that, unlike clays, sands usually do not

experience gapping dunng harmonic loading. Thus both analyses model the physical

behaviour of the soi1 realistically and logically.

The pile head displacement-time histories obtained fkom the p-y curve and

hyperbolic models for a pile installed in a sand with linearly varying elastic modulus due

to an applied harmonic load are shown in Figure 4.11. It can be noted fiom Figure 4.11

that good agreement exists between the results fkom the p-y curve model and hyperbolic

model.

Figures 4.12 and 4.13 show dynamic p-y curves established at two different

depths for a prescnbed displacement at the pile head for a steel pile driven in sand for a

frequency range from 0-10 Hz. The results from both the p-y curve and hyperbolic

models displayed the same trend, as can be noted fiom Figures 4.12 and 4.13.

An additional verifkation of the p-y method was performed to determine the

model's ability to predict the generation of static p-y curves. Brown et al. (1988) tested

an isolared single pile to two-way cyclic lateral loading at the pile head in subrnerged

dense sand cverlying stiff clay. The test consisted of a 0.2731-11 outside diameter steel pile

with a wall thickness of 9.27mm. Figure 4.14 shows the test set-up and site conditions

fiom penetration tests. The relative density of the sand was approximately 50% with a

uniform gradation and an angle of intemal fiction (measured f?om direct shear tests) of

38.5". The maximum displacement obîained at the pile head during the expenment was

used as the maximum prescribed displacement in the computational model. The

measured angle of interna1 friction, SPT value, and density of each soil layer was input

into the program to calculate stahc p-y curves. The calculated p-y curves at a depth of

0.9h are presented in Figure 4.15, compared with the measured p-y curve at the

equivalent depth. The experimental measured data used polynomial curves fitted to the

bending moment data in a rnanner similar to that described by Matlock and Ripperger

(1956).

It is obvious fiorn Fig.4.15 that the cornputed static p-y curves underpredicted the

soil resistance significantly using the measured soil properties. Brown et al. (1988) using

a similar analytical model observed the same trend. The reason for the observed increase

of the soi1 resistance in the experiment could be attributed to significant densification

o c c u ~ n g in the sand due to two-way cyclic loading. The soil parameters were then

modified (backfigured) to obtain the best-fit p-y curve shown in Figure 4.15. The

cornparison between the rneasured and best fit calculated static p-y curves revealed that

there is good agreement between the initial and ultimate soil stiffness evaluated from

both curves.

4.6 VALIDATION OF DMYAMIC MODEL WITH LATERAL STATNAMIC TESTS

In order to veriQ that the p-y cuve mode1 can accurately predict dynarnic

response, it was employed to analyze a lateral Statnamic load test and the computed

response was compared with measured values.

3.6.1 Case Study - JacksonvilIe Test Site

Statnamic lateral loading tests were perfonned by M. Janes and P. Bermingharn,

both of Berminghammer Foundation Equipment, a Division of Baltidaniel Inc., Hamilton,

Ontario. The test site was located north of the New River at the Kiwi maneuvers area of

Camp Johnson in Jacksonville, North Carolina. The soil profile is shown in Figure 4.16,

consisting of a medium dense sand extending to the water table, underlain by a very

weak, gray silty clay. A layer of gray sand is encountered at a depth of 7m and is

underlain by a calcified sand strata. The single pile tested at the site consisted of

reinforced concrete with an outer steel casing having an outer diameter of 0.6 lm (13m.m

wall thickness). Statnamic testing was conducted at the site two weeks after lateral static

testing was performed. More details of the full study and procedure can be found in the

paper by El Naggar (1998). The computed lateral response of the pile head at the site is

compared with the measured response in Figd. 17 for two separate tests with peak load

amplitudes of 350 and 470 W. The agreement between the measured and computed

values was excellent, especially for the first load test. The initial displacement was

slightly adjusted for the cornputer generated model to accommodate initial gapping from

previous tests that had developed during the experiment. The static p-y c w e for the top

soil layer was reduced significantly in order to model the loss of resistance due to

permanent gap developed near the surface.

4.7 DYNAMIC P-Y CURVE GENERATION (PARAMETRIC STUDY)

4.7.1 Description of Method

In this study, dynamic p-y curves were generated over a fiequency range of 0- 10

Hz (2 Hz intervals) for different classifications of sand and clay based on standard lab

and field measurements (SPT-value, relative density, c., etc.). Al1 results were obtained

after one or two cycles (no degradation) using a fiee head single pile with a prescribed

displacement at the pile top.

A prescribed displacement equal to 0.01d was applied at the pile head, which

allowed for the development of plastic deformation in the soil along the top quarter of the

pile length. Steel pipe piles were considered in the analysis. The pile properties are

given in Figure 4.18. n i e numerical mns were divided into two separate cases involving

clays (CASE 1) and sandy soils (CASE II). The sand was assumed to have a linear soil

profile while the clay was assumed to have a parabolic profile, to match the soil profile

employed to derive the static p-y curves used in the analysis. The shear wave velocity

profiles are shown in Figure 4.18.

The dynarnic p-y Cumes presented in Figures 4.8,4.9, 4.12, and 4.13 showed that

a typical family of curves exist related to depth, much like the static p-y curve

relationships. Thus, dynarnic p-y curves could be established at any depth and be

representative of the soil resistance at this specific depth. In this study, they were

obtained at a depth equal to 1.5m that was f o n d to illustrate the characteristics of the

dynamic p-y curves. Table 4.1 surnrnarizes the characteristics of each numerical nin and

relevant pile and soil parameters.

4.7.2 Results / Discussion

Figures 4.19 through 4.27 show the computer generated dynamic p-y curves (solid

lines) for the numerical run cases along with fitted curves (dashed lines), which will be

discussed later. The results fiom the computational mode1 show a general trend of

increase in the soil resistance with increase in Frequency. The staticp-y c w e s calculated

from Eqs.4.13- 4.17 (neglecting darnping) are also plotted in the figures, for each

numerical run.

The dynamic p-y curves shown in Figs. 4.19 through 4.27 seem to have three

distinct stages or regions. The initial stage (at srnall displacements) shows an increase in

the soi1 resistance (compared with the static p-y curve), that corresponds to increasing

velocity of the pile to a maximum. This increase in the soil resistance is larger for higher

frequencies. In the second stage, the dynamic p-y curves have almost the same dope as

the static p-y cuve at equal displacement. This stage occurs when velocity is fairly

constant and consequently the darnping contribution is also constant. The third stage of

the dynamic p-y curve is characterized by a slope approaching zero as plastic

deformations start to occur (similar to the static p-y curve at the sarne displacement).

niere is also a tendency for the dynamic curves to converge at higher resistance levels

approaching Pu.

The overall relationship between the dynarnic soil resistance and loading

fiequency for each numerical run was established in the form of a global equation. The

equation was developed from regression analysis relating the static p-y curve,

dimensionless frequency, and apparent velocity (oy) so that

where Pd = dynamic p-y curve at depth x (N/m), P, = static soi1 reaction (obtained fkom

static p-y curve) at depth x (Nlm), a, = dimensionless frequency (or, / V,), w = frequency

of Ioading (radk), d = pile diarneter (m), y = lateral pile deflection at depth x when soi1

and pile are in contact during loading (m), Pu = the ultimate lateral resistance of the soil at

depth x determined fiom Eqs.4.3-4.6, and u , P , K, and n are constants determined from

curve fitting Eq.4.23 to the computed dynamic p-y curves ( show as solid lines) in

Figs.4.19-4.27. A summary of the best fit values for the constants are provided in Table

4.2. The constant a is taken equal to unity to ensure that Pd = Ps, for O = O. For Large

ftequencies or displacements, the maximum dynamic soil resistance is lirnited to the

ultimate lateral resistance of the soil (Pu) as determined fiom experiment.

Figures 4.19 through 4.27 show dynamic p-y curves (as dashed lines) established

using Eq.4.23 and the best-fit constants. The approximate dyiiamic p-y curves

established from Eq.4.23 seem to represent soWmedium clays (Figs.4.19,4.20,4.26,4.27)

and loose/medium dense sands (Figs.4.22,4.23) reasonably well. However, the accuracy

is less for stiffer soils (higher V, values). The precision of the fiaed curves also increases

with eequency (a 2 4 Hz) where the dynarnic effects are important. The low accuracy at

lower frequency (a,,< 0.02) may be attributed to the application of the plane strain

assumption in the dynarnic analysis. This assumption is suitable for higher frequencies as

the dynarnic stifiess of the outer field mode1 vanishes for < 0.02 due to the

assumption of plane strain.

Numerical run C 1 (Fig-4. L 9) was also used to obtain dynamic p-y cunTes at depths

of l.Om and 2.0m to justi@ Eq.4.23 at more than one depth in the soil profile. Figs. 4.26

and 4.27 show that the sarne constants used f?om Table 4.2 in Eq.4.23 accurately predict

the dynamic p-y curves at depths above and below 1 Sm.

4.8 DEVELOPMENT OF A SIMPLIFIED MODEL

For many structural dynamic programs, soil-structure interaction is modeled using

static p-y curves to represent the soil reactions along the pile length. However, the use of

static p-y curves for dynamic analysis does not include the effects of velocity-dependent

darnping forces. The dynamic p-y curves established using Eq. 4.23 and the parameters

given in Table 4.2 allow for the generation of different dynamic p-y curves based on the

frequency of loading and soil profile. Substituting dynamic p-y curves in place of

traditional static p-y curves for analysis should result in better estimates of the response

of structures to dynarnic loading.

Alrematively, the dynamic soil reactions can be represented using a simple spring

and dashpot model. This rnodel can still capture the important charactenstics of the

nonlinear dynarnic soil reactions. A simplified dynamic model that can be easily

implemented into any general finite element program is proposed in the following

section.

4.8.1 Complex Stiffness Mode1

As discussed previously, Eq.4.23 c m be used directly to represent the dynarnic

relationship between the soi1 reaction and corresponding pile displacement. The total

dynamic soil reaction at any depth is represented by a nonlinear sprùig whose stiffness is

frequency dependent.

A more conventional and widely accepted method of calculating dynamic

stiffness is through the development of the complex stiffhess. The complex stiffhess has

a real part Kt and an imaginary part K2 , Le.

The real part represents the true stifkess and can be defined directly from static p-y curve

The imaginary part of the complex stifiess, Kz, descnbes the out-of-phase cornponent

and represents the darnping due to the energy dissipation in the soi1 element. Because

this damping cornponent generally grows with frequency, resembling viscous damping, it

c m also be defined in terms of the constant of equivalent viscous damping (the dashpot

constant) given by

Then the complex stiffhess can be rewritten as

and the p-y curve relation can be described either as

in which both k and c are real and represent the spring and dashpot constants,

respectivety, and y = dyldt is velocity. Using Eq.4.23, the complex stiffness of the

dynarnic p-y curve can be written in the form of Eq-4.24 as

The stifhess and damping constants are then calculated as

The complex stiffness can be generated at any depth along the pile using the static p-y

curves and Eqs.4.3 1 and 4.32.

4.8.2 Obtaining Complex Stiffness Constants - Soft Clay Example

The complex stiffhess constants were calculated for numerical run Cl (See Table

4.2) using the method descnbed in the previous section. The values of the mie stiffhess,

k, were obtained for the range of displacements experienced by the pile for the frequency

range spanning 0-10 Hz. The stifkess parameter (Si), was obtained by nomalizing k

with G,,, and is plotted in Figure 4.28.

nie constant of equivalent damping, c, was obtained by averaging the value fiom Eq.4.32

for the range of velocities expenenced by the pile for each fiequency of loading. The

equivalent damping parameter (S&, was obtained for each loading case and is ploaed

against the corresponding frequency in Figure 4.29.

The real and imaginary parts of the total complex stiffness can be rew-itten in a rnanner

similar to that proposed by Novak et al. (1978) for the outer field model;

where both Si and Sz are functions of dimensionless fiequency, Poisson's ratio, and pile

diameter.

Fig.4.28 shows the true stiffness calculated £kom the static p-y curve and it can be

noted this stiffness is identical at al1 loading fkequencies considered. There is a definite

trend of decreasing stifiess with increased displacement due to the soil nonlinearity.

The constant of equivalent darnping shown in Fig.4.29 shows a slightly decreasing

pattern with frequency that can be attributed to separation at the pile-soi1 interface. The

values fiom Figs.4.28 and 4.29 c m be directly input into a finite element program as

spring and dashpot constants to obtain the approximate dynamic stifkess of a soil profile

similar to numerical nin C 1.

4.8.3 Implementing dyoamic p-y curves into ANSYS

A pile and soil system sirnilar to numerical run C 1 was modeled using ANSYS to

venQ the applicability and accuracy of the dynamic p-y curve model in a standard

structural analysis pro_garn. A dynamic load with a peak amplitude of 100 kN was

applied at a frequency of 6 Hz to the head of the same steel pipe pile used in numerical

run C 1. The soil stiffhess was modelled using three procedures as follows; 1) static p-y

curve, 2) dynamic p-y curve using Eq.4.23, and 3) complex stifhess method using

equivalent damping constant. The pile head response for each numerical run was

obtained and compared to the results fiom the two-dimensional p-y curve model.

The pile was modelled using two noded bearn elements and was descretized into

10 elernents, increasing in length with depth. At each pile node, a spnng or spring and

dashpot was attached to both sides of the pile to represent the appropnate loading

condition at the pile soil interface. The pile and soil remained comected and had equal

displacement for compressive stresses. The spnng and dashpot disconnect if tensile

stress is detected in the soil and allow a gap to develop.

The soil was first modelled using nonlinear springs with force displacement

relationships calculated directly from static p-y c w e s . The soil stifiess was then

modelled using the approximate dynarnic p-y curve relationship caiculated for numencal

run C l using Table 1.2. The last numerical run considered a spring and dashpot in

parallel representing the static p-y curve stiffness (Fig.4.28) and constant damping

parameter found in Fig.4.29 respectively.

The pile head response for each numerical run is s h o w in Figs.4.30 and 4.31,

along with the calculated response from the 2-D analytical p-y model. Figure 4.30(a)

shows that the static p-y c w e cornputed larger displacements with increasing amplitudes

with the nurnber of cycles. The sofi soi1 allowed the pile head to "drift" to one side, but

the mode1 eventually stabilized with a longer time penod. Figure 4.30(b) shows that the

response computed using the dynarnic p-y curve mode1 was in a good agreement with the

response computed using the 2-D analytical model. The results using the complex

stiffness model are presented in Fig.4.31(a) which show a decrease in displacement

amplitude. The overdamped response can be attnbuted to using an average damping

constant. Figure 4.31(b) shows the response of the 2-D model compared to a modified

complex stifhess approach in which the same static p-y curve was used to mode1 soil

stiffness, but the average dmping constant was reduced over 50%. The results in

Fig.4.3 1 (b) show good agreement using the modified complex stiffhess approach.

Table 4.1 - Description of parameters used for each numerical run.

CASE I (clays)

CI

C2

C3

"Note: al[ values represent those calculated at a depth of 1.5m.

v S O U TYPE

I I 1 I l I i I I I

SOFT CLAY

MEDIUM CLAY

STIFF CLAY

D (m)

C u P a )

32

34

34

38

38

D, (%)

35

50

50

90

90

CASE n (sands)

S4

SS

S6

S8

S9

(50

80

SOIL TYPE

LOOSE SAND (saturated)

MEDIUM SAND (saturated)

MEDIUMSAND (samted)

DENSE SAND (saturated)

DENSE SAND (unsaturated)

Lm

v

0.3

0.3

0.45

0.45

Efls

HO0 1 0.45

D (m)

0.25

0.25

0.25

0-25

Grn, &Pa)

0.25

LID

40

40

Vs (mfs)

40

40

3800 20

40

40

0.3

0.3

0.3

40

E ~ ,

6300

3800

0.50

0.25

0.25

10000

4500

2.0e7

1600

100

m a

@Pa)

1.2e7

2.0e7

8.3e7 1 200

6.6e6

L6e7

150

220

V, (mh)

70

100

1580

70

150

4.7e7

1 1 .

790 9.7e7

Table 4.2 - Dynamic p-y curve parameter constants for a range of soi1 types (d = 0.25, L/d =40, 0.015 Ca, = orJV, c0.225 ).

SOIL TYPE ( DESCRIPTION i I

a ( p ! K 1 a,, c 0.025 1 a, > 0.025 1

SOFT CLAY

n

I l l 1

80 Cu< 50 kPa 1 1 1 -180 1 -200 Vs < 125 m/s 1

l

0.18

STIFF CLAY

I i

1 Cu > 100 kPa Vc >175 m/s

-825 -2900

0.1

0.15

1

1

I 1 DENSE SAND 50 <Dr< 85 % (unsaturated) 1 125 4 V, c 175 m/s

I

1960

6000 DENSE SAND (saturated)

l

D,> 85 % V, > 175 m/s

100

960 1 -20

0.19

1876 -100

...........................................a. o.......... o....... ............**...*......................... -...a...-.. l t

pile elernent (i) ___---.-----.-__.*._.

furJeld inrzer field

Figure 4.1 - Element representation of proposed model.

DIiMENSIONLESS FREQUENCY (G)

Figure 4.2 - Envelope of variations of horizontal stiffness and damping stiffness parameters between v=0.25-0.40 (after Novak et al., 1978).

HYPERBOLIC STRESS-STRAIN MODEL P-Y CURVE MODEL

Figure 4.3 - Cornparison between hyperbolic model and p-y model for the soi1 medium.

C - DYNAMIC P-Y Z ce--

# #

CURVE C - 4

C

STATIC P-Y CURVE

Y

Figure 4.4 - Determination of stiffness (kpy) from an internally generated static p-y curve to produce a dynamicp-y curve (including damping).

PILESOIL SYSTEM LINEAR Es (Pa)

O 2E98 (€4 6EdB

Figure 4.5 - Definition of Soil Modulus Variation for Various Soil Profiles Considered in the Analysis.

l5OOOO

100000 . . A

E 50000 -

C O - - O v m

! - -50000 - hyperbolic relationship O V)

-1 O0000 ------ p-y curve mode1

-1 50000 -0.002 -0.0015 -0.001 -0.0005 O 0.0005 0.001 0.0015 0.002

Pile deflection (m)

Figure 4.6 - Calculated dynamic soi1 reactions at 1.0 metre depth. (prescribed displacement at pile head = 0.001Sm,L/d=30,Ep/Es(L)=l000)

0.00008 *

ô 0.00006 hyperbolic relationship ------- p-y curve mode1 - 2, - = 0.00004 al g 0.00002 m - P U)

. . 1 m 1 . m m - 0 - u O g -0.00002 - t

2 -0.00004 - " V V V ~ ~ V \ ' 'u V V V v' V* n -0.00006 -

O 1 2 3 4 5 6 7 8 9 Tirne (s)

Figure 4.7 - Pile head response under applied harmonic load equal to 0.10 of the uttirnate load (L/d=30, E&(L)=1000, linear profile, P/P,=O.lO).

Parabolic Halfspace E&,(L)=1000 L/d=25 d=0.5m pdp,=1.25

O 0.002 0.004 0.006 0 0.002 0.004 0.006 pile displacement (yld) pile displacernent (yld)

(a) (b) Figure 4.8 - Calculated dynamic p-y curves for 1.5 metre depth using: (a) hyperbolic model, (b) p-y curve model (for prescribed displacement at pile head equal to 0.05d).

O 0.0002 0.0004 0.0006 0.0008 0.001 O 0.0002 0.0004 0.0006 0.0008 0.001

pile displacement (yld) pile displacement (yld)

(4 (b)

Figure 4.9 - Calculated dynamic p-y curves for 3.0 metre depth using: (a) hyperbolic model, ( b) p-y curve model (for prescribed displacement at pile head equal to 0.05d).

-0.025 -0.02 -0.015 -0.01 -0.005 O 0.005 0.01 0.015 0.02 0.025 Pile deflection (y/d)

Figure 4.10 - Calculated dynamic soi1 reactions, 1.0 metre depth (prescribed displacement at pile head equal to 0.075d, Lid= 20, EJE,(L) = 1000 ).

O 0.5 1 1.5 2 2.5 3 3.5 4

Time (s)

Figure 4.11 - Pile head response to applied harrnonic load equal to 80% of the ultimate load (Lld =2O, Efl,(L)= 1000, linear profile, P/Pu=0.80).

Loose Sand - Linear Halfspace - E@,(L)=1000 L/d=25 p&=l.25

O 0.002 0.004 0.006

Pile displacement (yld)

(a)

O 0.002 0.004 0.006

Pile displacernent (yld)

(b) Figure 4.12 - Calculated dynamic p-y curves for 3.0 metre depth using: (a) hyperbolic model, @) p - curve mode1 (for prescribed displacement at pile head equal to 0.025d).

-0 Hz (static p-y curve) -2 Hz * - - - -

4 Hz 6 Hz

- 8 Hz -10 Hz

O 0.001 0.002 0.003 O 0.001 0.002 0.003

Pile displacernent (yld) Pile displacement (yld)

(a) Cb) Figure 4.13 - Calculated dynamic p-y curves for 4.0 metre depth usinp: (a) hyperbolic model, (b) p-y curve model (for prescribed displacement at pile head equal to 0.025d).

Medium fine clean sand + = 3 0 1 1 pom

Medium dense sand 1 1

Stiff red and gray clay SPT = 10

Very stiff tan and sandy clay SPT = 12

Figure 1.14 - Soi1 profile and pile test set-up according to Brown et al. (1988) for measured static p-y curve data.

C

0.1 Calculated Curve (Best Fit Soil Properües) - - - - - -Calculated Cuwe (Based on Direct Shear Data)

O w . . 1

O w

0.005 0.01 0.015 0.02 0.025 0.03

Pile displacernent (yld)

Figure 4.15 - Static p-y curves for loose sand using cornputer generated p-y curve mode1 and full-scale experimental results (Depth=O.glm).

STATNAMIC DEVICE

Tan brown fine sand SPT = I O

- - Gray silty clay

Cu = 20 kPa -

Gray green fine sand SPT=l6

Gray calcareous coarse sand SPT = 35

Gray fine sand

Figure 4.16 - Soi1 profile and lateral Statnamic piIe test set-up for case study at Camp Johnson, Jacksonville.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0-1

The (s)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (s)

Figure 4.17 - Measured and Computed pile head displacement for Statnarnic test ushg head load equal to: (a) 350 kN, (b) 470 kN.

General Properties for al1 tests

SOIL CASE t (Parabolic)

P L E (steel pipe)

Ep = 200 GPa p, = 7500 kglrn3 L = l O m

CASE II (Linear)

Figure 4.18 - Description of soi1 and pile properties for CASE 1 and CASE II.

Pile displacernent (yld)

Figure 4.19 - Dynamic p-y curves and static p-y curve for numerical run Cl (depth = 1.5 m).

Computed curves Closed f o m curves static D-v (Equation 4.23)

0.001 O 0.0020 0.0030

Pile displacement (yld)

Figure 4.20 - Dynamic p-y curves and static p-y curve for numerical run C2 (depth = 1.5 m).

Computed curves Closed f o m curves (Equation 4.23)

static p-y ....o.*. - 2Hz 2 Hz ....O .. - - 4Hz 4 Hz

0.0000 0.001 0 0.0020 0.0030 0.0040

Pile displacement (yld)

Figure 4.21 - Dynamic p-y curves and static p-y curve for numerical run C3 (depth = 1.5 m).

0.0000 0.0005 0.001 0 0.001 5 0.0020

Pile displacernent (yld)

Figure 4.22 - Dynamic p-y curves and static p-y curve for numerical nia S5 (depth = 1.5 m)

Computed curves Closed f o m curves static p-y (Equation 4.23)

0.0000 0.0004 0.0008 0.0012

Pile displacernent (yld)

Figure 4.23 - Dynamic p-y curves and static p-y curve for numerical run S6 (depth = 1.5 rn)

0.0000 0.0010 0.0020 0.0030 0.0040

Pile displacement (yld)

Figure 4.24 - Dynamic p-y curves and static p-y curve for numerical run SS (depth = 1.5 m).

O O

Computed curves Closed fonn curves (Equation 4.23) - 2 Hz ....O ... 2 Hz - 4 HZ ....a... 4Hz - 6 Hz ....o... 6Hz - 8 Hz

. . . .o... 8 Hz - 10 Hz ....o... 10 Hz

1 1 1

0.0000 0.0010 0.0020 0.0030 0.0040

Pile displacement (ytd)

Figure 4.25 - Dynamic p-y curves and static p-y curve for numerical run S9 (depth = 1.5 m).

Pile displacement (yld)

Figure 4.26 - Dynamic p-y curves and static p-y curve for numerical run Cl (depth = 1.0 m).

1 .O

0.9

0.8

0.7

0.6

0.5

0.4

O. 3

0.2

O.?

0.0

(Equation 4.23)

....o... 2Hz

.... * .. . 4 Hz

....o... 6Hz

... o... 8 Hz

0.0000 0.0010 0.0020 0.0030 O. O040 0.0050

Pile displacement (yld)

Figure 4.27 - Dynamic p-y curves and static p-y curve for numerical run Cl (depth = 2.0 m).

Figure 4.28 - True stiffness parameter for numerical run Cl (Soft Clay).

O 0.02 0.04 0.06 0.08 0.1 0.12

dimensionless frequency, a,

Figure 4.29 - Equivalent damping parameter for numerical run Cl (Soft Clay) with dimensionless frequency.

I - STATIC p-y (ANSYS) I O 0.2 0 -4 0.6 1 1.2 1-4 1 -6 1.8 2

Time (s)

. - 2-0 ANALYTICAL MODEL

] D Y N A M I C p-y (ANSYS) v O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

(b) Figure 4.30 - Calculated pile head response using 2-D analytical model cornpared with ANSYS using: (a) static p-y curves, (b) dynamic p-y curves.

I Y Y - 2-D ANALYTICAL MODEL

. C O M P L U D-v (ANSYS)

O 0.2 0.4 0.6 0.8 i 1.2 1.4 1.6 1.8 2 Time (s)

1 , - 2-D ANALMICAL MODEL 1

/ -MODIFIED COMPLU( p-y (ANSYS) ! + L

O 0.2 0 -4 0.6 0 -8 1 1.2 1.4 1 -6 1.8 2 Time (s)

(b) Figure 431 - Calculated pile head response using 2-D analytical model compared with ANSYS using: (a) complex stiffness, @) modified complex stiffness.

CHAPTER 5

Sl.JMMARY AND CONCLUSION

A 3-D finite element analysis was performed to investigate site effects and pile

kinematic interaction effects fiom seismic loading. The analysis considered floating and

socketed piles including nonlinear soil properties, slippage and gapping at the pile-soi1

interface, and dissipation of energy through damping. Based on the results fiom the

kinematic interaction study, it was concluded that the pile (floating and socketed) head

response closely resembled the fiee-field response for seismic loading.

The second part of the study focussed on developing a simple two dimensional

analysis to model the inertial interaction of pile head loading. The time domain was

chosen to efficiently model transient nonlinear response of the pile-soi1 system. Static p-

y curves were used to generate the nonlinear soil stifkess in the fiame of a Winkler

model. The piles, assumed to be vertical and circular, were modeled using standard beam

elements. An accurate and cornputationally efficient model was developed that included

soil nonlinearity, slippage and gapping at the pile-soi1 interface, and viscous and material

damping.

Dynarnic soil reactions (dynamic p-y curves) were generated for a range of soil

types and harmonic loading fiequencies applied at the pile head. Closed form solutions

were derived from regression analysis relating the static p-y c w e , dimensionless

frequency, and apparent velocity of the soil particles. A simple spring and dashpot model

was also proposed whose constants were established by splitting the dynamic p-y curves

into real (stiffiess) and imaginary (damping) components. This model could be used in

equivalent linear analysis for harmonic loading at the pile head. The dynamic p-y curves

were incorporated into a commercial fmite elernent program (ANSYS) and the pile head

response was verified against the prediction of the two dimensional analytical model.

5.2 CONCLUSIONS

The following conclusions were drawn fiom the present study:

1) The effect of a soil layer overlaying the bedrock was to amplifi the bedrock motion,

resulting in a higher free-field motion for the soil parameters used in the analysis:

2) The effect of allowing a three dirnensional model as opposed to a one dimensional

model, with seismic loading applied in one dimension, was to decrease the

acceleration amplitudes by a factor of two.

3) The effect of soil plasticity was to increase the Fourier amplitudes at the predominant

frequency but to slightly decrease the maximum acceleration amplitudes;

4) The elastic kinematic interaction of single piles @oth floating and socketed) bas

slightly amplified the bedrock motion when compared to the fiee-field response and

slightly decreased the Fourier amplitudes of a11 fiequencies considered (0-20 Hz).

5) Overall, the kinematic interaction response including soil nonlinearïv, slippage and

gapping at the pile soil interface, and darnping is equivalent to the fkee-field response.

However, the conclusions are limited to the pile and soi1 parameters, and earthquake

loading used in the analysis.

6) The soil reactions and pile head displacements using the two-dimensional analytical

model were in a good agreement with experimental static loading and lateral

statnamic load test results.

7) The results using thep-y model are very similar to those using the hyperbolic model.

8) The soil resistance to the pile motion increases with the frequency of the pile head

loading (inertial loading) for single piles.

9) Using a predefined static p-y curve, the modified dynarnic p-y curve can be

represented by

where Pd is the modified dynarnic soil resistance, P, is the static soil resistance at a

certain pile displacement, and a, B, K, and n are constants relating to various

parameters obtained fTom a regression analysis, oy is the apparent velocity, and d is

the diameter of the pile.

10) The pile head response under hannonic loading can be approximately modeled using

dynarnic p-y curve functions in most of the available structural analysis prograrns.

11) The complex stifiess method gives a slightly stiffer response at the pile head than

the two dimensional analytical method for the soil profile studied.

5.3 FUTURE CONSIDERATIONS

The results from the study assume perfect vertical piles and do not consider the

pile installation procedure. In the field, piles are rarely installed perfectly vertical, and

the installation procedure disturbs the soi1 surrounding the pile. The soil and pile

parameters can be adjusted accordingly. Pore water pressure build-up is important in

some soils, and should be accounted for in the analysis. There are no reliable

experimental dynarnic p-y curve data, and the nurnencal solutions presented in the study

need to be proven with full scale field tests.

The pile response to earthquake loading can be approximated by applying the

free-field accelerations at the pile head. To incorporate both kinernatic and inertial

interaction directly, the fiee-field accelerations obtained from wave propagation theory

(SHAKE) can be applied along length of the pile, connected to the outer field nodes.

In terms of dynamic p-y curve generahon, the next step would be to introduce two

or more piles and investigate the effects of pile-soil-pile interaction. The final results

could include a factor @-multiplier) multiplied by the single pile dynamic p-v curve to

give an equivalent resistance of pile groups to various loading frequencies applied at the

pile cap or head.

REFERENCES

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ANSYS (1996). "General Finite Element Analysis Program," Version 5.4, ANSYS, Inc., PA.

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Bhushan K., and Haley, S. C.. (1980)."Development of Cornputer Prograrn Using P-Y Data from Load Test Results for Lateral Load Design of Drilled Piers" a research report prepared for Woodward-Clyde Consultants Professional Development Cornmittee, San Francisco, California.

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Procedures for developing static p-y cuwes in clays

1. Step by Step Procedure (Bhushan et al., 1979; Bhushan and Haley, 1980)

a. From the results of laboratory tests, determine the variation of undrained shear

strength and effective unit weight of the soil with depth, and the value of cjo, the

strain corresponding to one-half the maximum principal stress difference, (oi -03),, /

2.

b. Compute the deflection at one-half the ultimate soil reaction by:

Y50 = C i D550

in which y50 = deflection at one-half the ultimate soil reaction:, Ci = a constant

relating pier / pile deflection to the laboratory strain: D = pier / pile diameter: and

= strain at (cri - ~ ~ ) m ~ ï 1 2 (Recommended values shown in Table 1.1).

c. For a given depth , x, compute the ultimate soil resistance per unit lenth of pied pile,

Pu, using Eqs.4.3-4.6.

d. Compute points descnbing the p-y curves at depth x by:

2. Selection of Parameters

Based on a review of available information , the following values of CI, J, and n were

adopted in the equations above :

Soft CIays (c c 50 kPa) CI = 2.5 (after Matlock, 1 970) J = 0.5

n = 0.33

Stiff Clays (c > 50 kPa) Ci = 2.5 (afier Bhushan et al., 1979) J = 1.5

n = 0.4

The relationship between y and D rnay not be linear as assumed in Eq. 1.1. The use of the

present procedures for piedpile diameters outside the range of 0.2 - 1.5m should be done

with caution.

Table 1.1 - Recommended values for 550 (after Meyer and Reese, 1979)

APPENDIX II

Proeedures for developing Static p-y curves for sands

(II. 1)

For sands, the secant modulus of soil reaction, Es, is defined as

Es = p/y

which is generally assumed to Vary linearly with depth such that

ES = Kx (11.2)

where K = constant relating the secant rnodulus of soil reaction to depth (kN/m3)

Combining Eqs. II. 1 and U.2_

p = f i y

where p = soil reaction / unit length (kN/m)

x = depth at which p-y curve is defined (m)

y = lateral deflection at depth x (m)

Major factors affecting K are relative density of the sand and lateral deflection. Tables

II. 1 and 11.2 give relationships for variation of K with y D (deflection, y normalized with

respect to the pier diameter, D) and a simplified function to obtain the K values of

relative densities other than 85 %.

1. Step bg Step Procedure (suggested by Bhushan and Haley, 1980; Bhushan et al.,

a. From SPT-values, CPT-values, or other data, estimate the relative density (Dr) of the

Sand deposit. For example, the following formula (suggested by Bazarra, 1967) was

used in the program based on Standard Peneatration Blow Count,

where N = standard penetration blowcount (blows/foot)

a', = effective overburden pressure (Pa)

b. From laboratory tests or other correlations, estimate the angle of interna1 fiction, and

soil unit weight (See Figure 11.1).

c. For p-y curve at any depth x and for a pier with diarneter D, cornpute a set of values

of lateral defection, y 1 , y2, y3,. . . xorresponding to y/D values in Table II. 1 by:

d. Compute corresponding values of soil resistance, p, by:

in which

y = yl,y2,y3,.. .. . .are values of y given by Eq.II.6.

K = K 1 ,K2,K3 ,. . . .. are corresponding values of K from Table II. 1

FI = relative densisty factor f o m Table H.2

F2 = groundwater factor - Use 1.0 No Groundwater

0.5 Below Groundwater

(II. 7)

Figure II.1- Relative density vs. Angle of Interna1 Friction (after Meyer and Reese, 1979).

20 40 60 80 1 O0

Relative Density (%)

Table II.1- Variation of K with deflection (Meyer and Reese, 1979).

0.0750 8.13*10'

Note: A polynomial curve was fit to the data so that: ,

where y = pier / pile deflection at groundline D = pier 1 pile diameter K = Constant relating the secant modulus of soi1 reaction to depth

(Es = Kx), ~ l r n ~

Table II.2 - Variation of K with relative density (after Meyer and Reese, 1979).

Note: A polynomial curve was fit to the data above to give;

Dt (%)

100

Density Factor (FI)

1.25