A DISCRETE MODEL FOR THE ANALYSIS OF A THESIS IN CIVIL

A DISCRETE MODEL FOR THE ANALYSIS OF

SHORT PIER FOUNDATIONS IN CLAYS

by

FARIBORZ ALIKHANLOU B.S. in C.E.

A THESIS

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

CIVIL ENGINEERING

Approved

\y August 1981

Ac kn o w le ' l g nie n t s

The a u t h o r w i s h e s t o e x p r e s s h i s s i n c e r e a p p r e c i a t i o n

t o h i s a d v i s o r . Dr. C.V,G, V a l l a b h a n , for h i s v a l u a b l e

a d v i c e , e n c o u r a g e m e n t and g u i d a n c e t h r o u g h o u t t h i s

r e s e a r c h . S p e c i a l t h a n k s a r e a l s o e x p r e s s e d t o Dr. ^ a r r e n

K Wray, and Dr. Kishor C. ."iehta for their helpful

suggestions and constructive criticism.

In addition, the author would like to express his

deepest gratitude to his wife and parents for their

patience, understanding and encouragement during the period

or his research.

11

Table of Contents

Page No.

Acknowledgments - ii

List of Tables , v

List of Figures , vi

Chapter 1. Introduction 1

1-1. The Problem 1

1-2. Methods used i n P r e v i o u s R e s e a r c h

of t h e Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

C h a p t e r 2 . The P roposed D i s c r e t e Model f o r

R i g i d P i e r s , 8

2 - 1 . R i g i d P i e r - 8

2 - 2 . S o i l S p r i n g s . . . . . . . — . . . 10

2 - 3 - Development of t h e System E q u a t i o n s

f o r t h e Model , . . . . . . . ^ , 12

2 - 4 . P r o c e d u r e f o r D e t e r m i n i n g D e f l e c t i o n s . . . . 19

C h a p t e r 3 . F o r m u l a t i o n of S p r i n g c o n s t a n t s . . . . . . . . . . 22

3 - 1 . The L a t e r a l Sp r ing C o n s t a n t s . . . . . . . . . . . . . 22

3 - 2 - The Bottom V e r t i c a l Spr ing C o n s t a n t - 30

3 - 3 . The Bottom Moment S p r i n g C o n s t a n t . . . . . . . . 34

3 - 4 . The Bottom F r i c t i o n Spr ing C o n s t a n t 38

3 - 5 . The Skin F r i c t i o n S p r i n g C o n s t a n t s 41

C h a p t e r 4 . B e h a v i o r of t h e Model 44

4 - 1 . I n f l u e n c e of S p r i n g s 44

• • • 1 1 1

4 - 2 . Comparison of ?lodel Resu l t s with

F i e ld Tes t Data ^6

Chapter 5, Conc lus ions and Reconmendations - . 7 1

Peferf^nces 75

1 V

Lis t of Tables

Page No.

4-1 Soil Properties for Piers # 1 and 2 48

4-2 Soil Properties for Piers # 3,4,5,and 6 ,. .... 49

4-3 Soil properties for pier # 1 53



4-6 Soil properties for Ontario test 69

List of Figures

Page No.

1-1 Single Lateral Spring :iodel .- 6

2-1 The Discrete Multi Spring Model .- 9

2-2 Forces acting on the pier 11

2-3 Rotation of a rigid pier 13

2-4 Movements of a rigid pier 16

3-1 Illustration of Lateral Load Transfer 24

3-2a Family of p-y curves 26

3-2b Illustration of Secant Modulus 26

3-3 Parametric Study (J = 2.83, n = 1/2) ..... .... 31

3-4 Parametric Study (C. = 2.0, n = 1/2) 32

3-5 Parametric Study (C-i = 2.0, J = 2.83) 33

3-6 Variations of Moment with Angle of Rotation ..... 36

3-7 Relation between the frictional force and

sliding of the pier , 39

3-8 Side Friction on the Pier 42

4-1 Illustration of influence of springs 45

4-2 Shear strength of soil vs depth used

for piers # 1 and 2 50

4-3 Shear strength of soil vs depth used

for piers # 3,4,5, and 6 51

4-4 Load-Deflection Diagram for Pier # 1 54

4-5 Load-Deflection Diagram for Pier # 2 5b

vi

4-6 L o a d - D e f l e c t i o n Diagram f o r P i e r = 3 - ^^

4-7 L o a d - D e f l e c t i o n Diagram f o r P i e r * 4 61

4 -3 L o a d - D e f l e c t i o n Diagram f o r P i e r # 5 64

4 -9 L o a d - D e f l e c t i o n Diagram f o r P i e r # 6 66

4 -10 S o i l p r o p e r t i e s f o r t h e O n t a r i o Hydro

R e s e a r c h t e s t , 60

4-11 Load-Deflection Diagram for the

O n t a r i o Hydro R e s e a r c h t e s t 70

V I 1

chapter 1

Introduction

1-1. The Problem

Rigid piers have been used as foundations fcr

bui ldings , bridges , highway interchanges , and a number cf

other applications when there e x i s t s strong s o i l s trata at

r e l a t i v e l y shallow depths. Recently they have increas ingly

been used as foundations for power transmission poles .

Current design techniques for pier foundations fcr

transmission poles generally overcompensate for foundation

movements which r e s u l t s in larger piers than necessary

in f la t ing construction c o s t s . Designs may be overly

conservative by a factor that ranges from three to f ive

[ 1 0 ] .

The design of pier foundations for transmission poles

i s very different from the design of foundations fcr

s tructures , such as bridges and buildings ,because the

design of the former i s predominantly controlled by a

moment at the top of the pier and the design of the l a t t e r

i s primarily control led by vert ica l and horizontal forces .

Transmission pole foundations must be able to r e s i s t

environmental loads such as wind, i c e , service load cf

tension in cables , and dead load of weight of mater ia ls .

The design of transmission pole foundations i s control led

1

more by environmental and service loads rather than dead

loads. This is in contrast to the foundations of buildings

and bridges that must support very large dead loads as well

as live loads.

Additionally, design techniques for pier foundations

for transmission poles subject to high-moments are not as

advanced and as well understood as are techniques fcr

designing foundations subject to vertical compressive

loads. Unlike buildings and other structures that are

designed for the particular soil conditions at each site,

transmission lines extend several miles over a variety cf

soil types and have to be designed for many different soil

conditions. This emphasizes the need for a simple , fast,

and economical method of analysis and design of pier

foundations for transmission poles. A discrete model has

been developed and presented herein for the analysis cf

pier foundations of transmission poles in clayey soils.

This model is discussed in chapter 2 of this study.

When a transmission pole is subjected to lateral loads

at the top (wind or tension of cables), it transfers the

loads to its pier foundation as a bending moment, a

vertical load and a lateral load. The moment is usually

large due to the height of the pole, which typically ranges

from 75 to 120 feet. The above loads, together with the

weight of the pole and cables, act on the top of the pier

foundation. The pier , in turn, transfers the loads to the

surrounding s o i l . As a r e s u l t , the pier foundaticn

undergoes a trans lat ion and a rotation in the s o i l .

Since the l a t e r a l def lect ion of the pier foundaticn

i s very important from the design point of view, i t i s

des irable to predict these def lect ions from working to the

maximum loading conditions in a way that i s acceptable to

the design engineers . This needs to be done with

reasonable accuracy so that the foundation i s not grossly

overdesigned. Several methods have been previously

proposed for predicting the def lect ions of a l a t e r a l l y

loaded pier. In th i s study a new approach to the problem is

used. The model that i s developed for r igid piers r e l a t e s

the s o i l res i s tance to the movements of a l a t e r a l l y loaded

pier more ra t iona l ly and accurately than ex is t ing models.

Before discussing t h i s new model, a brief d iscuss ion cf

previous research on t h i s subject i s presented.

1-2- Methods Used in Previous Research of The Problem

Methods used for predicting latera l de f l ec t ions of a

l a t e r a l l y loaded pier can be c l a s s i f i e d into three

di f ferent categories :

i - The Ultimate Strength Method

This method as proposed by Broms [ 5 ] was used for the

ana lys i s of p i l e s in homogeneous s o i l s ( i . e . , no

s t r a t i f i c a t i o n ) . In t h i s method the s o i l i s assumed to

reach an u l t imate capac i ty of 95 B at depths below 1-1/2 I,

where S i s the shear strength of s o i l and B i s the p ier

diameter. S i a i l a r methods have a l so been used by Hansen

[ 8 ] , and a number of other i n v e s t i g a t o r s -

i i - The e l a s t i c method of a n a l y s i s .

The e l a s t i c method i s based on the concept of the

l i n e a r l a t e r a l subgrade r e a c t i o n theory [ 5 ] or on l i n e a r

e l a s t i c c o n s i d e r a t i o n s . In t h i s method , i t i s assumed

tha t the p i e r i s embedded in an e l a s t i c continuum

having a constant modulus of e l a s t i c i t y , E, o r , t h a t the

modulus of e l a s t i c i t y E increases l i n e a r l y with depth.

Examples of such analyses based on purely e l a s t i c

c o n s i d e r a t i o n s were presented by Douglas and Davis [ 6 ] acd

extended by Poulos [ 1 5 , 1 6 ] for s ing le p i l e s .

Purely e l a s t i c s o l u t i o n s do not o f f e r much

advantage over the l i n e a r subgrade reac t ion theory

because both are based on l i n e a r r e l a t i o n s h i p s . Both

methods require reasonably accurate va lues for e i t h e r the

modulus of e l a s t i c i t y or the c o e f f i c i e n t of subgrade

r e a c t i o n of the s o i l . In addi t ion they require i t e r a t i o n if

the s o i l i s s t r a t i f i e d . F i n a l l y , the e l a s t i c methods do

not take i n t o account the non l inear i ty of the s o i l response

and l ead to conservat ive p r e d i c t i o n s i n variance with

actua l behavior as demonstrated by f u l l - s c a l e t e s t

r e s u l t s [ 1 , 1 7 ] .

i i i - The E l a s t i c - P l a s t i c / N o n l i n e a r method of a n a l y s i s .

A more s o p h i s t i c a t e d approach i s the nonlinear method

based or the concept of the l a t e r a l load vs d e f l e c t i o n (p-

y) curves ( f i r s t proposed by McClelland and Focht [ 1 4 ] ) fcr

the a n a l y s i s of p i l e s . This concept was advanced ty

Matlock [ 1 2 ] , using the pr inc ip l e of beam on e l a s t i c

foundation (Figure 1-1) and the f i n i t e d i f f erence methcd

for the a n a l y s i s of l a t e r a l l y leaded p i l e s in s o f t c l a y s .

I t was l a t e r extended by Reese [ 1 9 ] for s t i f f c l a y s .

The concept y i e l d s nonlinear p r e d i c t i o n s that approximate

the ac tua l behavior of p i l e s under l a t e r a l loading

c o n d i t i o n s . The fourth order d i f f e r e n t i a l equation of a

beam on e l a s t i c foundation as used by Reese [ 20] in h i s

a n a l y s i s of the problem i s given below :

EI d!iL + p df^ . p = 0 (1-1)

where dx"* ^ dx^

p = a x i a l load on the pier X

p = soil reaction per unit length

y = lateral deflection of pier

EI = flexural rigidity.

Lateral Resistance

77?3 v77=55C

9 I

p .

•> ^

\ I I

. • • •

• «

I

< V

«»,

• V

'>

—AMAA—^

; - ^ V V W - ^

Figure 1-1. Single Lateral Spring Model

To apply the finite difference method to the above

problem, the length of the pier must be divided into a

number of small segments interconnected at points or nodes.

It is also necessary to consider the stiffness of the pier.

The model yields a set of linear simultaneous eguatiots

with unknown nodal deflections. The number of equations

depends upon the number of nodes that are used. The

nonlinear nature of the problem leads to an iterative

solution procedure. Therefore, the computer time required

can become relatively larg.

In the above procedure the resisting forces and moment

of the soil at the bottom of the pier and the skin fricticn

between the pier and the surrounding soil are neglected.

It will be shown in this research that, depending on the

soil properties and the dimensions of the pier, the

resisting forces and moment at the bottom, as well as the

skin friction on the sides, can significantly influence the

results.

Chapter 2

The Proposed Discrete Model For Rigid Piers

A new model for short piers with their support

conditions as shown in Figure 2-1 is presented in this

study. In this proposed model the pier is assumed to te

rigid. The validity of this assumption for transmission

pole foundations is discussed in the following section .

The horizontal soil springs are assumed to have their

individual soil resistances at each depth. In contrast, the

shear strength of soil is not averaged as done by Reese

[20] or the soil ultimate resistance is not modified as

done by Isaael and jrlym in their models [9]. Since it is

assumed that the pier is rigid the concept of beam on

elastic foundation as used in most of the previous analyses

is not required.

2-1- Rigid Pier

In this research, analysis of rigid piers with cr

without bells are considered. Piers can be assumed rigid if

the following condition is satisfied [23],

I/E < 2

8

Bottom Friction

Bottom Vertical Resistance

Lateral Resistance

Side Fricti on

Bottom Moment

Figure 2-1. The Discrete Multi Spring Model

where R k

I = l ength of the p i e r .

10

and K = horizontal modulus of subgrade reaction of the

soil.

For piers used for transmission pole foundations, this

condition is usually true because those piers have large

moment of inertia of the cross section and short lengths.

In addition, the angle of rotation of the pier is

assumed to be small. This can be easily verified through a

simple example. Assume that a rigid pier with a total

length of 15 feet has 2 inches of deflection at the tcp

(from the practical point of view this deflection is more

than allowable). Further assuming that the inflection

point is at 12 feet below the ground surface, the angle cf

rotation will be less than 1°. The relations tana = a and

Sin a = a are approximately true for a < 2 °. Therefore,

the above assumption is justified for short piers.

2-2. Soil Springs

The model presented in this report uses horizontal,

vertical and rotational springs to account for lateral

resistance , skin friction, and moment resistance of soil

(Figure 2-2).

n

Figure 2-2. Forces acting on the pier,

12

Spec i f ic springs, their function and the ir constants

are termed as fol lows : a set of horizontal springs , with

spring constant K , representing the l a t e r a l re s i s tance cf

s o i l . Figure 2-1, , a rotat ional spring, with a spring

constant K. , representing the r e s i s t i n g moment of the

s o i l at the bottom of the pier, a horizontal spring, with a

spring constant K. , representing the fr ic t ion at the

bottom of the p ier , a ver t i ca l spring, with a spring

constant K r representing the vert ica l res istance of s o i l bu

at the bottom of the pier, and a set of vertical springs

representing the skin friction between the pier and the

surrounding soil, with spring constant KJ for the right

side of the pier, and spring constant K for the left side

of the pier.

2-3. Development of The System Equations For The Model

To develop the system equations, the geometrical

relationships between the nodal displacements and rigid

body motions of the pier must be established. Assumirg

that the pier experiences a pure rotation. Figure 2-3, the

change in the coordinates of a point on the pier is

determined as follows :

13

V

ul

Figure 2-3. Rotation of a r i g i d p ier .

14

r = R t + X J , 2 . , ^ ,

cir = a X r (2-lb)

"^^ ' ^^ (2-lc)

Where R = radius of the pier

r = p o s i t i o n vector of any node on the pier

a = angle of r o t a t i o n of the p i e r

(c lockwise d i rec t ion)

i , j , k = the c a r t e s i a n unit vectors , a s shown in

Figure 2-3

S u b s t i t u t i n g Equations (2-1a) and (2-1c) i n t o Equation

(2- lb) r e s u l t s in :

dr = - R a j " + y a t (2-2)

using Y = L/2 - X in Equation (2-2) for transformation cf

coordinates

dr = - R a J + (4" - ^ ) ^ " (2-3)

Therefore,

u = R a (2-4) V = (-L. . X) a

where u and v are the vertical and horizontal movements cf

any node along the pier when it has rotated an angle a . If

15

there i s a r i g i d body t r a n s l a t i o n (UQ , VQ) of the c e n t r o i d

of the pier in a d d i t i o n to the pure r o t a t i o n , Figure 2 - 4 ,

the displacements along l i n e AB are as f o l l o w s :

u = Uo + R a (2 -5)

V = Vo + (-— - X) a

S i m i l a r l y , l i n e CD w i l l have the fol lowing displacements

u = Uo - R a (2 -6 )

V = Vo + {-J- - X) a

For a center point at the top of the p i e r the displacements

are :

t = "° L (2-7)

V^ = Vo + - o - ot

and, for a center point at the bottom of the p i er t t e

d i sp lacements are :

U. = Uo

^ (2-8) Vb = Vo - - ^ a

16

B — diamater of the pier

D — deplh of the pier

"o.Vo.a - * Rigid body motion of the pier

Figure 2-4. Movements of a r ig id pier

17

The above equat ions show that i f the r i g i d body

motions of the cen tro id (UQ, VQ) and the angle of r o t a t i o n

a f are known , then the movements of every node on the

p i e r can be determined-

In order to develop the system equat ions the Hinimum

Total P o t e n t i a l Energy Theorem i s used [ 1 0 ] . To do t h i s ,

equat ions are developed that w i l l g i v e the v a r i a b l e s

(Uo,Vor and a) for any loading V. , H., and M. ac t ing a t the

top of the p i e r .

The p o t e n t i a l energy of the system $ i s def ined as

the sum of the s t r a i n energy 0 , and the p o t e n t i a l energy

of the ex terna l loads V , as f o l l o w s :

$ = U + V (2-9)

4 2K^V^ + i Z K ; ; ( U ^ ) ^ + i ZK^(U^)^ +

Vûo - H^(vo + ^ a ) - Mâ

=]- ZK^ Vo + ( ^ - X ) a ^ + \ EK[^(UO + Ra)^ +

\ E K J ( U O - R a ) ^ + \ Kj^û§ + \ K^^(vo - ^ a ) ^ +

\ S m ^ ' - Vtô - H^(vo ^ ^ a ) - Mâ (2 -10)

18

Minimizing $ with respect to the variables UQ, VQ, and a yields the

following :

j ^ - 2:K;J(UO + Ra) + ZK^(uo - Ra) + K ^ ^ U Q - V^ =0

av

9a

•= ZK^ Vo + ( ^ - X)a + K^^(vo- j a) - H^ =0 (2-11)

•= ZK^ Vo +{j - X)a (t . X) + ZK|^(UO + Ra)R -

S K ) (uo - Ra)R - K. (vo - T a)T +

•^Km^ - H.y - M. =0 bm t2 t

Writing these equations in a matrix form :

K. +ZK bu u I

IZK +K. V bv

Symmetric

ZK"^ . R - ZK|J| . R

^ (F- ) -hA ZK^(j - X)^ + ZK[| .R2 +

ZK .R2 + K. X + Kt, u bv 4 bm

r \

< Vo > = <

a V J

V.

H

K\ I '\

(2-12)

19

I t can be concluded from the above analys is that due to the

assumption of a r ig id pier , only three quant i t ies are

required to represent the r ig id body motion of the p ier .

Therefore, the computation time of th i s procedure is

considerably reduced.

As mentioned before, the s o i l spring constants used in

the above s t i f f n e s s matrix vary nonlinearly with the

displacements. The procedures for determining the spring

constants are discussed in chapter 3. In th i s study an

incremental procedure i s used for obtaining the load-

def l ec t ion response of the pier. This procedure is

discussed below.

2 -4 . Procedure for Determining Deflections

The steps involved in solving the nonlinear force-

def lect ion charac ter i s t i c s of a l a t e r a l l y loaded pier are

as fol lows :

Step 1 : The length of the pier i s divided into n

equal segments (generally 10 to 15). Also the load vectcr

i s developed in increments.

Step 2 : A small i n i t i a l value i s assumed for the

angle a equal to 0.0001 radians and the i n i t i a l values cf

UQ and Vo are set to be equal to zero.

20

Step 3 : Using UQ , VQ , and a in equat ions ( 2 - 5 ) , the

v e r t i c a l and hor i zonta l displacements of every node along

the p i e r are c a l c u l a t e d .

u . = Uo + R a

î = Vo + ( -^ - X) a

Step 4 : Using the s o i l propert i e s ( S , £50. Y ) , the

p-y curves (equations) are developed for each node at i t s

p a r t i c u l a r depth using the procedure in chapter 3 , s e c t i o n

3 - 1 . This s t e p i s executed only once.

Step 5 : The value of v. from s t e p 3 i s s u b s t i t u t e d

i n t o the s tep U equat ions t o obtain the corresponding

f o r c e , and the spr ing constant ( K ) of each s p r i n g .

These values of spring constants w i l l be l a t e r used to

determine the s t i f f n e s s matrix.

Step 6 : The spring c o n s t a n t s , K , found in s t e p 5

t o g e t h e r with the spring cons tant s K^ , K^ , K ^ , and \

determined from the procedure described in chapter 3 cf

t h i s report are s u b s t i t u t e d i n t o Equation ( 2 - 1 2 ) . This

system of equat ions i s then so lved to f ind the new va lues

of Un # Vn # and a -

21

Step 7 : The new values of UQ, VQ , and ^ are

s u b s t i t u t e d in to Equations (2-5) to f ind the new v a l u e s cf

u and v . The d i f f e r e n c e . De l ta , between the new and the

old va lues of v for a point at the top of the p ier i s

determined as f o l l o w s :

Delta = | ( v - V , . ) / v T , I ' * new old ' ^ old '

I f Delta i s l e s s than a prescr ibed to l erance then cont inue

with the next s t e p ; otherwise return to s t ep 3 and

i t e r a t e . The value of the to lerance i s chosen to be 10

in t h i s re search , because t h i s wi l l a l low the system cf

s p r i n g s t o become s u f f i c i e n t l y c l o s e to the s t a t i c

equi l ibr ium from a p r a c t i c a l point of view.

Step 8 : The load vector i s increased by another

increment and the whole process i s repeated. This i s

repeated u n t i l the maximum load i s reached.

Chapter 3

Formulation of Spring Constants

As described in previous chapters, when a tranmissicn

pole i s subjected to la tera l loads (wind or tension cf

cables) i t transfers these loads to i t s pier foundation as

a l a t e r a l load, a ver t i ca l load, and a bending moment.

Therefore, the pier trans lates and rotates in the s o i l as

shown in Figure 2-4. This motion of the pier causes

d i f ferent react ions in the s o i l (Figure 2-2) , including:

l a t e r a l re s i s tance , ver t i ca l res is tanceat at the bo t to i ,

moment at the bottom, f r i c t i on at the bottom, and f r i c t i c n

along the s ides of the pier . The p i e r - s o i l interact ion

c h a r a c t e r i s t i c s due t o these reactions are modeled with

equivalant s o i l spring constants . The development of these

spring constants are discussed below.

3 -1 - The Lateral Spring Constants

The s o i l response to the movements of a l a t e r a l l y

loaded pier i s characterized as a set of discrete springs

s imi lar to the Winkler e l a s t i c foundation concept (1867).

Bat these springs can have nonlinear load-deformaticn

responses , and the response at a point i s independent cf

p ier def lec t ion elsewhere. Obviously t h i s equivalant spring

constant assumption i s not s t r i c t l y valid for s o i l

22

23

continua, but the errors involved in i t s use are assumed to

be small.

A discussion of the physical meaning of the l a t e r a l

spring mechanisms i s given here. Consider the behavior cf

a pier i n s t a l l e d in ground as shown in Figure 3 - la . A thin

s l i c e through the pier and surrounding s o i l i s shown at a

depth X below the ground surface. Before any l a t era l load

i s applied to the p ier , the pressure dis tr ibut ion on the

pier wi l l be s imilar to that shown in Figure 3 - lb . For t h i s

condit ion the resultant force on the pier , obtained ty

in tegrat ing the pressure around the segment, wi l l be zero.

If a l a t e r a l load i s applied to the p i e r , i t wi l l de f l ec t

to the new pos i t ion and the pressure d is tr ibut ion wi l l

become as shown in Figure 3-1c. Integration of the

pressure around the segment for this condition y i e l d s a

resul tant force p per unit length along the pier . This

force p acts in the opposite direct ion of the def l ec t ion y.

(The l e t t e r y i s used here for def lect ion for convenience

of the reader. For la tera l deflection analys i s , the symbcl

V i s used to d i f f erent ia te from y which i s generally used

for representing the coordinate ax i s ) .

I f the above process i s repeated for a range of

d e f l e c t i o n s ( i . e . , d i f ferent y ' s ) , the success ive

in tegrat ions y ie ld the p-values corresponding to the

d i f ferent y-values , allowing the p-y curve to be developed

24

^ ^

\

V/////.

'/m//^ Ground Surface

A

" \

Xi

View A-A

b- Pressure Distribution

Before Loading

a- Pier Segment

c- Pressure Distribution after Loading

After Reese and Cox (1969)

Figure 3-1. Illustration of Lateral Load Transfer

25

f o r t h e d e p t h X. By a p p l y i n g t h e above p r o c e d u r e t o o t h e r

d e p t h s , a f a m i l y of p-y c u r v e s can be d e v e l o p e d a s in

F i g u r e 3-2a . C u r v e s of F i g u r e 3 -2a r e p r e s e n t t h e

c h a r a c t e r i s t i c s of p-y c u r v e s a t v a r i o u s d e p t h s . The

a b i l i t y t o p r e d i c t t h e b e h a v i o r of p i e r s s u b j e c t e d to

f o r c e s and moments a t t h e t o p i s d i r e c t l y d e p e n d e n t on t h e

a b i l i t y t o d e r i v e t h e p-y c u r v e s g i v i n g t h e s o i l r e s p o n s e

w i th t o l e r a b l e a c c u r a c y .

The d e v e l o p m e n t of P-Y c u r v e s f o r c o h e s i v e s o i l s . As

d e s c r i b e d a b o v e , t h e s o i l r e s p o n s e t o l a t e r a l l o a d s i s

g i v e n by a f a m i l y of c u r v e s g i v i n g s o i l r e s i s t a n c e a s a

f u n c t i o n of p i e r d e f l e c t i o n . The s o i l modul i a r e s e c a n t s to

t h e p - y c u r v e s and can vary i n any a r b i t r a r y manner wi th

d e p t h (and w i t h d e f l e c t i o n of p i e r a s w e l l a s w i t h t h e s o i l

p r o p e r t i e s ) . T h i s i s shown in F i g u r e 3-2b-

A number of i n v e s t i g a t o r s (McCle l land and Foch t [ 1 3 ^ ,

Reese [ 1 8 ] r Mat lock [ 1 2 ] , G i l l and Denars [ 7 ] and R e e s e and

Welch [ 1 9 ] ) have s u g g e s t e d p r o c e d u r e s f o r o b t a i n i n g t h e

r e l a t i o n s h i p be tween t h e s o i l r e a c t i o n p , and p i e r

d e f l e c t i o n y , a t v a r i o u s d e p t h s . The g e n e r a l p r o c e d u r e

f o r o b t a i n i n g a s e t of p-y c u r v e s a t v a r i o u s d e p t h s a l o n g a

p i e r i n c l a y s a s p r o p o s e d by Matlock [ 1 2 ] and R e e s e a rd

Welch [ 1 9 ] i s : T,

26

-»-y

X = X1

Figure 3-2a. Family of p-y curves

X = X2

X = X3

X = X4

Figure 3-2b. Illustration of Secant Modulus

27

1. Determine the variat ions (values) of undrained shear

strength determined from t r i a x i a l compression t e s t s

(CO), e f f e c t i v e unit weight of the s o i l , and £50

the s train corresponding to one-half the maximum

principal s t r e s s di f ference, i . e . ( QI-GS ) max/2,

with depth. In the absence of t r i a x i a l compression

t e s t r e s u l t s , use unconfined compression t e s t

r e s u l t s .

2. Osing the £50 values , compute def lect ion Vso at

one-half the ultimate s o i l reaction by

yso = Ci B £50 ( 3 - 1 )

where

yso = def lec t ion at one-half the ultimate s o i l

react ion;

C, = a constant re lat ing pier def lect ion to the

labratory s tra in ;

B = pier diameter .

3. For a given depth , X , compute the ultimate s o i l

res is tance per unit length of p i e r , P , by

P, = ( 3 + ^ + J - f - ) S^ B 9 S B (3-2)

28

where

S|j = average undrained shear strength of s o i l

from ground surface to depth X;

J = a constant which controls the depth at which

Pu reaches 9cB for cohesive s o i l s ;

B = pier diameter;

Y = average e f f e c t i v e unit weight of s o i l

within the depth X.

U. Compute points describing the p-y curve at depth X

by

^ = 0.5 ( JL )n (3-3)

where

p = s o i l resistance per unit length of p i er ;

y = def lect ion corresponding to p;

n = a constant re lat ing s o i l resistance to pier

de f l ec t ion . 1/n

Note that p = p at y = 2 .yso (3-4)

5. Compute the secant modulus of the s o i l for any value

of y

K = - £ - = î! , (3-5) ' ' 2 ySo y'-'

29

Note that i n the computation of the spring

constant K , Eg, (3-5) g ive s i n f i n i t e modulus fcr

y=0 : t h e r e f o r e , in order to prevent the c r e a t i o n cf

such very large spring c o n s t a n t s at small

d e f l e c t i o n s , an a d d i t i o n a l c r i t e r i a i s used here.

For d e f l e c t i o n s l e s s than a d e f l e c t i o n of 0 .5 yso

which corresponds to a s t r a i n equal to 0 .5 £50 in the

s o i l , the spring constant K i s held cons tant at

that p a r t i c u l a r depth. Subs t i tu t ing i n equat icn

(3-5) for y , the i n i t i a l s o i l modulus can te

determined as fo l l ows :

P l/T - Û

\ - 'TZ^ 7?Pl (3-6) ^ • Yso

Values of the cons tant s C-j , j , and n are based en

empir i ca l c o n s i d e r a t i o n s and t a l l y i n g with a n a l y s i s of load

t e s t r e s u l t s . The proposed va lues of C-j for c l a y s range

from 0.5 (McClelland and Focht [13]) t o 2.5 (Matlock [ 1 2 ]

and Reese and Welch [ 1 9 ] ) . The genera l ly recommended value

of C, =2.0 [ 1 2 , 1 9 ] i s based on Sicempton's e l a s t i c - a n a l y s i s

[ 2 1 ] of a uniformly-loaded area with a l eng th - to -w id th

r a t i o of 10.

For the constant J , va lues in the range of 0.5

(Matlock [12] ) to 2 .83 (Reese [18]) have been sugges ted .

30

The t h e o r e t i c a l e v a l u a t i o n s [ 1 8 ] , load t e s t data [ 1 4 ] , and

b e a r i n g c a p a c i t y t h e o r y [ 2 1 ] i n d i c a t e t h a t the l i m i t i n g

v a l u e of u l t i m a t e l a t e r a l s o i l r e a c t i o n should be o b t a i n e d

a t depths o f about two t o four t imes the p i e r d iameter

c o r r e s p o n d i n g to v a l u e s of parameter J of about 1.5 t o 2 . 8 -

The recommended v a l u e s of exponent n range from 1/3

(Matlock [ 1 2 ] ) t o 1/a (Reese and Welch [ 1 9 ] ) . However, use

of a c t u a l l a b o r a t o r y s t r e s s - s t r a i n c u r v e s , c o r r e s p o n d i n g to

a p p r o x i m a t e l y a v a l u e of n=1/2 , has a l s o been s u g g e s t e d

[ 1 0 ] t o d e f i n e l o a d - d e f l e c t i o n behavior of p i e r s .

A p a r a m e t r i c s tudy was made of the e f f e c t cf

v a r i a t i o n s of C, , J , and n on the computed g r o u n d l i n e

d e f l e c t i o n of a t y p i c a l p i e r ( p i e r # 3 , s e e chapter 4 ) . The

r e s u l t s o f t h i s a n a l y s i s are shown i n F i g u r e ' s ( 3 - 3 to

3 - 5 ) . A s i m i l a r s t u d y was repor ted by Bhushan, Haley , and

Fong [ 2 ] . From t h e t h e o r e t i c a l c o n s i d e r a t i o n s and the

r e s u l t s of t h e above parametr ic s tudy , v a l u e s of C, = 2 . 0 ,

J = 2 . 8 3 , and n=0 .5 were s e l e c t e d f o r computing p-y

c u r v e s .

3-2. The Bottom Vertical Spring Constant

The vertical reaction at the bottom is replaced by a

linear spring placed vertically at the bottom of the pier,

as shown in Fig.2-1.

31

500

400

to Q .

- a

o

ns u

4->

300

200

100

0

-

/ 1

1 1

1 1 / 1 / 1 / /

/ / /

/ /

/ /

/

/ / / /

/ / /

•

y

y y

y y

^y ^

y

**• y

,y

<^

^ ^ -—

1.0 I

,J*^

'7'

^

"^ ' '

-2 .0

0 1 .0 2 .0 3 .0 4 . 0 5 .0

G r o u n d l i n e D e f l e c t i o n ( i n . )

Figure 3-3. Parametric Study (J=2.83, n= l /2 ) .

32

500

400

- 300

-a o

200

03

100

0

.

/ ^ >

II

r

/ /

f /y / y

• ^

y y

y

y y

y y

^ ^ ^

y

y

^ ^ ^ ^

^

- - • ^

^ ^ ^ • * •

^ * ' f

2.ij3

""^T (

1.5

J=0.5

0 1 .0 2 . 0 3 .0 4 .0 5.0


Figure 3-4. Parametric Study (C^=2.0, n= l /2 ) .

33

500

400

to

- 300

• a

o

^ 200 (T3 S-<U

4->

100

0

•

II

yy' < * - •

^ - . - ' • '

^ » ^

•*.

,'^'

\ )

rY=l/4

1/3

0 1 .0 2 .0 3.0 4 .0 5.0


Figure 3-5. Parametric Study (C,=2.0, 0=2.83)

34

the value of the spring constant i s obtained from the

f o l l o w i n g equation for the se t t l ement of a c i r c u l a r f o o t i r g

which i s based on the theory of e l a s t i c i t y [ 3 ]

S = q B ( L I - H ! ) I^ (3.7)

where

S = s e t t l e m e n t , f t

q = i n t e n s i t y of contact pressure, psf

B = l e a s t l a t e r a l dimension of f o o t i n g , f t

y= Poisson's ra t io

E= modulus of e l a s t i c i t y of s o i l , p s f

I = influence factor, equal to 0.88 [31

Therefore, the spring constant of an equivalent spring to

replace t h i s s o i l reaction i s

'''" '- B (1 - ,n I„ '^-^'

3 - 3 . The Bottom Moment Spring Constant

When a l a t e r a l force or a moment i s applied on the top

of a r ig id p i er , the pier tends to ro ta te , and a r e s i s t i r g

moment i s developed at the bottom of the pier, as shown in

Figure 2-2. This react ion of the s o i l at the bottom is

replaced by the rotat ional spring shown in Figure 2 -1 . The

35

spr ing constant i s determined from the fo l lowing equat icn

which was developed by Lee £3 ] for the rotat ion of r i g i d

f o o t i n g s subjected t o a moment M ;

Tan a = J ^ -Lu^ I B3 r m T3 — : % <3-9)

where

a =rotation of footing (radians)

I =shape factor, equal to 6.0 [3]

Note that for small a , tan a = a

Hence

Equation (3-10) represents a linear relationship

between the angle of rotation a and the moment at the

bottom, n. With K. being constant, at larger values of a bm

some u n r e a l i s t i c v a l u e s of moments would r e s u l t . In order

to prevent t h i s , i t i s assumed that the M- a diagram i s as

shown in Figure 3-6 where M represents the maximum ^ max

moment that the soil can develop (resist) under the pier.

This moment is controlled by the soil conditions as well as

the pier geometry. The actual maximum moment capacity cf

the soil requires further testing and research.

36

M

Actual

a

bm initial spring constant

Figure 3-6. Variations of Moment with Angle of Rotation .

37

In the absence of data on this subject it is assumed

that the maximum allowable stress between the bottom cf

pier and the soil is a function of the shear strength cf

soil S and the vertical stress due to the total vertical

load at the bottom of the pier, P/A , thus:

^ = - -f (3-11)

M max

or "max = S(S^^-£-) ,3.,2,

where

S = sec t ion modulus of the pier cross sect ion

equal to -^—

P = t o t a l v e r t i c a l force at the bottom of the

pier which includes the weight of the pier

L = t o t a l length of the pier

Equation (3-12) s impl i f i e s to the following

M = 4 - ^ ( s + - ^ ) (3-13) max 32 ' u A

The M obtained above i s very conservative because of the max "*

38

s i m p l i f i c a t i o n t h a t i s made t o the M-° diagram (Figure

2 - 9 ) . By knowing the maximum moment t h a t t h e s o i l can

t o l e r a t e and the i n i t i a l v a l u e of \m ' ^^^ M-a

r e l a t i o n s h i p i s deve loped- Then by knowing t h e v a l u e cf

t h e r o t a t i o n a f o r each l oad ing c o n d i t i o n , the

c o r r e s p o n d i n g va lue of K, can be determined . To accompl i sh bm

this, an initial value of K. is determined first and then

the moment M is calculated as ^ = 'bm ° • ^^^' ^^ ^^^

new calculated moment is more than \g^x ' ^ ® secant spring

constant is calculated by the following equation :

M K = ^^^ (3-14) bm a

3-4- The Bottom Frict ion Spring Constant

The behavior due to the fr ic t ion at the bottom is

represented by a horizontal spring at the bottom (Figure

2 - 1 ) . In the absence of experimental data, the

re la t ionsh ip between the f r i c t i o n a l force and the l a t e r a l

movement of the bottom of pier i s assumed as shown in

Figure 3-7. The f r i c t i o n a l force w i l l reach a constant

maximum value after a prescribed displacement. I t i s

further assumed that the value of the prescribed

dislplacement i s about 0.01 f t . This assumption i s based on

experiments conducted by other inves t igators [ 2 2 ] .

39

u i-o

C

o •r-

o

Maximum Frictional Force

Displacement

Figure 3-7. Relation between the frictional force

and sliding of the pier.

The maximum frictional stress at the bottom is given

by:

a = Sj + -^ Tan 4) (3-15)

where S = shear s trength of s o i l

P = load a t the bottom of the p ier

A = area at the bottom (be l l )

L = length of the pier

Therefore , the maximum f r i c t i o n a l force w i l l be

F = (S + 4 - Tan (J) ) A (3-16) rmax ^ u A

Hence, i n i t i a l l y ,

>/i ^rmax. (3-17) • bv " 0.01

where i t i s assumed that the F ^ ^ i s reached when t t e

d i f f e r e n t i a l displacement between the p ier and the s o i l i s

0.01 f t .

I f the displacement at the bottom of the p ier i s found

to be more than 0.01 f t . then the new spring constant w i l l

be a secant modulus determined from the f o l l o w i n g

r e l a t i o n :

F K = rmax (3-18) bv V

41

where v i s the l a t e r a l d e f l e c t i o n of the bottom of the

p i e r .

3 - 5 . The Skin F r i c t i c n Spring Constants

For s i m p l i c i t y , i t i s assumed that the force cf

f r i c t i o n has a l i n e a r var iat ion with the v e r t i c a l

d e f l e c t i o n up to a maximum of F (Figure 3-7) , and that max ^ ^ ' '

it remains constant at that level.

Using the above assumptions, the relationship between

the frictional force and the vertical displacement can te

found. Assuming the pier has rotated an angle a , every

point on the pier will undergo a vertical displacement with

a maximum downward displacement u. on the right hand side

and a maximum upward displacement u- on the left hand side

as shown in Figure 3-8a- Figure 3-8b shows the plot cf

these displacements around the pier which is a helix. The

frictional surfaces for the right and left sides of the

pier are determined by approximating the area covered by a

strip of width W after the pier has rotated, as follows,

(see Figure 3-8) :

i i " 2 TT R w ( .H ,. ) (3-19)

u.

^ " " u. i u. ' t3-20)

42

Ui 4^77>>.^ 1 -^^^.iV^ luc

a- Rotation of a unit strip

u. A.. = 2 , R w (--"i--")

u. A. ^ 2 TT R w ( 1 - — )

u. + u. ' 1 J

b- Area covered by movement of unit strip

Figure 3-8. Side Friction on the Pier.

43

T h e r e f o r e , t h e maximum f r i c t i o n a l f o r c e on a s t r i p of width

W w i l l be

or

and

F = A. . . S. max 1 or J u

(3-21)

F K. = -TTTTr Ôr ^ 0-01 f t - (3-22) u inrr

F K = "^^^ fo r u 0.01 f t . (3-23) ^ ^ i or j

Since not a l l points on the s tr ip have a motion equal

to u. o r u . and the movements of dif ferent points vary frcra

0 to u. or u.# and the center of gravity of the f r i c t i o n a l

force i s not at the boundary, a coe f f i c i ent equal to 1/2 i s

introduced to account for these variat ions .

Chapter 4

Behavior Of The Model

TO assess the behavior of the a n a l y t i c a l model, t*io

types of study a re conducted, (1) s tudy of inf luence cf

va r ious s o i l sp r ings on load -de f l ec t ion response foe a

t y p i c a l p i e r , and (2) comparison of the a n a l y t i c a l l y

obta ined l o a d - d e f l e c t i o n responses with f i e ld t e s t data fcr

seven d i f f e r e n t p i e r s . These s t u d i e s a re presented below.

4 - 1 . Inf luence Of Soi l Springs

As discussed prev ious ly in chapter 2 (Ref. Figure

2 - 1 ) , the a n a l y t i c a l model incorpora tes s o i l sp r ings for

s i d e f r i c t i o n , bottom f r i c t i o n , bottom v e r t i c a l r e s i s t a n c e ,

and bottom moment i n addi t ion to the l a t e r a l r e s i s t a n c e

s p r i n g s . To assess the inf luence of these var ious s o i l

s p r i n g s , a p i e r of dimensions 4 f t . diameter and 15 f t .

long for which f i e ld t e s t data are a v a i l a b l e i s analyzed to

ob ta in l o a d - d e f l e c t i o n response. s o i l p r o p e r t i e s ard

r e l a t e d p i e r data are given in Example 1 in the next

s e c t i o n . The l oad -de f l ec t i on response for t h i s p i e r i s

f i r s t determined using only the l a t e r a l s o i l sp r ings (curve

1, Figure 4 - 1 ) . Then other spr ings are added one a t a t i n e

to t h i s t y p i c a l p i e r model and the corresponding load-

d e f l e c t i o n responses a re determined.

44

LA'.er^ I - f . : •. l^-CP

500

400

to

- 300

-a

o

200 03 S-

100

0

45

Curve 1 : Lateral springs only

2: Lateral+moment springs

3: Lateral+moment+bot.

friction springs

4: All springs

5: Field test data.

0 1 .0 2.0 3.0 4.0 5.0

Ground l ine D e f l e c t i o n ( i n . )

Figure 4 -1 . I l lus t ra t ion of influence of springs

46

In t h i s particular example the e f f e c t of the bottom

r e s i s t i n g moment (curve 2) , Figure 4-1 , on reducicg

de f l ec t i ons i s not very s igni f icant due to the small

diameter of the b e l l . This feature can become more

e f f e c t i v e for a pier with a larger b e l l , or i f the s o i l

strength at the bottom i s much higher than at the top. The

addit ion of the bottom f r i c t i o n spring (curve 3), Figure

4 - 1 , and also the s ide f r i c t i on springs (curve 4 ) , Figure

4 - 1 , seem to have s ign i f i cant e f f ec t s on reducing the pier

d e f l e c t i o n s . As a resu l t of a l l the features that have been

added, the f i n a l r e s u l t s are reasonably c lose to the f i e ld

t e s t data up to a load of about 250 kips (Ref. Figure 4-1) .

4 -2 . Comparison Of Bodel Results With Field Test Data

Bhushan, Haley, and Fong [ 2 ] have reported f i e l d t e s t

data on twelve pier foundations. Of these , s ix piers were

on sloping ground or they were r e l a t i v e l y f l ex ib le p iers .

The remaining s ix piers could be c l a s s i f i e d as r ig id piers

(L/R varies from 1.5 to 1 .9) . These s i x piers are used fcr

the comparison of model r e s u l t s with f i e l d t e s t data. An

addit ional pier foundation was chosen from a f i e l d t e s t

reported by Ismael and Klym [ 9 ] . This could a l so te

regarded as a r ig id pier (L/R=1.3).

47

The actual properties of the soil as presented ty

Bhushan, Haley, and Fong [2] are reproduced in Tables 4-1

and 4-2 for convenience. Figure 4-2 and 4-3 (data fcr

ploting these Figures are obtained from Tables 4-1 and 4-2)

are also presented here as a convenient way fcr

interpolating the shear strength of soil for the depths at

which the soil data are not given in Tables 4-1 and 4-2.

In the absence of the value of cohesion and angle cf

internal friction of the soil, the Standard Penetration

Test results are used to predict the average strength

parameters. In order to relate the SPT blow counts N, to

the shear strength of soil an approximate equation c = N/8

ksf is used [ 4 ].

Results of analysis using the model are compared with

field test data for following seven piers. In each case

pier dimensions and soil properties are given, and lateral

load-deflection results are compared.

48

Table 4 - 1 . S o i l P r o p e r t i e s for P i e r s * 1 and 2

Depth

f t .

2

4

6

9

11

14

SPT^

N

19

34

—

-

35

44

20

20

0.'

ksf

-

0 . 7 2

1 . 4 4

2 . 8 8

—

0

-

0

cu s

uu

ksf

—

10.8

11.2

12.3

>

8.8

-

3.6

a c

ksf

3,8 17°

a £^50

%

—

0 . 7

0 . 9

1 .0

-

1 . 7

—

0 . 8

used

S u

ksf

2 . 4

4 . 0

4 . 4

4 . 4

2 . 5

1 .8

used

%

0 . 9 4

0 . 9 4

0 . 9 4

0 . 9 4 1

0 . 9 4

0 . 9 4

a. From t e s t s by Bhushan, Haley, and Fong [ 2 ]

^9

Table 4 - 2 . S o i l P r o p e r t i e s for P i e r s # 3 , 4 , 5 , and 6

D e p t h

f t .

2

4

5

6

9

11

14

SPT^

N

19

30

-

—

32

22

43

24

31 r 1 _

—

—

a 03

k s f

—

0 . 7 2

1 .44

2 . 8 8

0

-

0

-

0

0 . 7 2

1 .44

2 . 8 8

CO 5^

00

k s f

-

6 . 6

9 . 6 J>

14 .6

4 . 6

-

1 4 . 0

-

3 .9 \

7 . 7

1 1 . 8 >

c^

k s f

0.9

1.0

* ^

1

41°

35°

a £ 5 0

%

—

0 -6

0 . 8

0 . 8

0 . 7 6

-

—

-

—

0 .6

0 . 7

1 . 1

used

S

k s f

2 . 4

1.3

2 . 3

2 . 8

7 . 0

3 . 0

2 . 2

u s e d

£ 5 0

^

0 . 7 2

0 . 7 2

0 . 7 2

0 . 7 2

0 . 7 2

0 . 7 2

0 . 7 2

a . From t e s t s by Bhushan, Haley, and Fong [ 2 ]

50

2.0

S, (ksf)

4.0

^ 6

+

+

4-> Q. CU O

+

+

11

14

+

+

Figure 4-2. Shear strength of soil vs depth used for piers # 1 and 2.

51

4

5

6

Q.

O

11

14 I

2.0

\ (ksf)

4.0

+

+ -h

+

+

•f

Figure 4-3. Shear strength of soil vs depth used for piers # 3,4,5, and 6 .

52

Example 1, F i g u r e 4-4 , P i e r # 1. This p i e r was t e s t e d ty

Bhushan , Ha ley , and Fong [ 2 ] . The p i e r dimensions and the

s o i l p r o p e r t i e s are as f o l l o w s :

P i e r d i m e n s i o n s : Length = 15 f t .

Diameter = 4 f t .

Has a 5 .5 f t . d iameter b e l l at t h e

bottom 2 f t .

S o i l p r o p e r t i e s are a s shown i n Table 4 -1 and Figure 4 - 2 .

Tab le 4-3 g i v e s t h e a c t u a l v a l u e s of s o i l p r o p e r t i e s used

i n t h e computer program.

In t h i s example , t h e r e s u l t s of the a n a l y s i s are

r e a s o n a b l y c l o s e to t h e a c t u a l t e s t data up t o a l oad cf

about 250 k i p s . The model r e s u l t s d e v i a t e c o n s e r v a t i v e l y

from t h e t e s t data a t h igher l o a d s .

Table 4-3. Soil properties for pier # 1

53

D e p t h

0 . 8

2 . 3

3 . 8

5 . 3

6 . 8

8 . 3

9 . 8

1 1 . 3

1 2 . 8

1 4 . 3

1 5 . 0

<ft) S^ (ks f )

2 . 0

2 . 6

3 . 8

4 . 3

4 . 4

4 . 4

3 . 6

2 . 4

2 . 0

1.8

1.8

^ 5 0

0 . 0 0 9 4

0 . 0094

0 . 0094

0. 0094

0 . 0094

0. 0094

0. 0094

0. 0094

0 . 0094

0 . 0 0 9 4

0 . 0094

54

500

400

to

- 30() .1^

O

200 <0

TOO

0

•

i

h fi 11

II ii

It ji ii

II

II ji II

If

i 1 11

11

/ /

/ /

/ X / y / /

/ f

•

0

y 0

•

Mo

^ . ' ^

^

del — F i p l ^ â.e-¥ ^ a ^ r 1 C 1 VJ 1<C

0 1 .0 2.0 3.0 4.0 5.0

Groundline Deflection (in.)

Figure 4-4. Load-Deflection Diagram for Pier # 1

55

Example 2, Figure 4-5 , P i e r # 2- This pier was t e s t e d ty

Bhushan , Haley, and Fong [ 2 ] . The s o i l propert ies are the

same as for p ier # 1 , pier dimensions are as fo l l ows :

Pier dimensions : Length = 15 f t .

Diameter = 4 ft ,

In t h i s example, the predicted l a t e r a l d e f l e c t i o n s cf

the p i e r remain c l o s e to the ac tua l va lues up to a load cf

about 250 k. The r e s u l t of the a n a l y s i s d e v i a t e s

c o n s e r v a t i v e l y from the actual t e s t data at higher l o a d s .

'>r ::-,_ Ai--:^ry? .•

55

;; uu

400

- 300

• o f8 O

200

tera

l

1

100

0

-

/ / /

/ / / /

/ / r

11 If 11

/ / / /

/ / / / / /

7 1

y

/ ^

> y'

y

1

/ y

y y

Mo

Fi 1

y y

del --'

eld te'^''' ' ^^ 1 1 <

1,0 2.0 3.0 4.0 5.0



r i

57

Example 3, Figure 4-6 , P i er # 3. This p ier was t e s t e d ty

Bhushan , Haley, and Fong [ 2 ] . The pier dimensions and the

s o i l p r o p e r t i e s are as fo l l ows :

P ier dimensions : Length = 12.5 f t .

Diameter = 4 f t .

Has a 5.5 f t . diameter b e l l at the

bottom 2 f t .

S o i l p r o p e r t i e s are a s shown in Table 4-2 and Figure 4 -3 .

Table 4-4 g i v e s the ac tua l values of s o i l proper t i e s used

in the computer program.

In t h i s example the r e s u l t of the a n a l y s i s i s c l o s e to

the a c t u a l t e s t data up to an ul t imate load of about 200 k.

However, they d e v i a t e conserva t ive ly from the actua l t e s t

data a t higher l o a d s .


58

D e p t h

0 . 6

1.9

3 . 1

4 . 4

5 . 6

6 . 9

8 . 1

9 . 4

1 0 . 6

1 1 . 9

1 2 . 5

( f t ) S u

2 . 0

2 . 4

1.9

1.8

2 . 6

4 . 0

6 . 0

6 . 0

3 .5

2 . 7

2 . 7

i^-s f ) £ 5 0

0 . 0 0 7 2

0. 0072

0 . 0 0 7 2

0 . 0 0 7 2

0 . 0 0 7 2

0 . 0 0 7 2

0 . 0 0 7 2

0. 0072

0 . 0 0 7 2

0 . 0 0 7 2

0 . 0 0 7 2

59

500

400

to

- 300

-a o

200 rt3 s .

(T3

100

0

0 1.0 2 . 0 3 .0 4 .0 5 .0



60

Example 4, Figure 4-7 , P i e r » 4. This pier was t e s t e d ty

Bhushan , Haley, and Fong [ 2 ] . The pier dimensions and the


Pier dimensions : Length = 12.5 ft.

Diameter = 4 ft.

S o i l Properties are the same as for pier # 3 .

In t h i s example the resul t of the analysis i s c l o s e to

the actual t e s t data up to an ultimate load of about 2C0

k ips , but i t deviates conservatively from the actual t e s t

data at higher loads .

61

500

400

to Q.

o

+-> (T3

300

200

100

0 L 0 1.0 2.0 3.0 4.0 5.0



62

Example 5, Figure 4-8 , Pier # 5. This pier was tested ty

Bhushan r Haley, and Fong [2]. The pier dimensions and the

soil properties are as follows :

Pier dimensions : Length = 15.5 ft

Diameter = 4 ft.

Has a 5.5 ft. diameter bell at the

bottom 2 ft.

Soil properties are as shown in Table 4-2 and Figure 4-3.

Table 4-5 gives the actual values of soil properties used

in the computer program-

In this example the result of the analysis is very

close to the actual test data up to the maximum load of

400 k.


€3

Depth

0.8

2.3

3.9

5.4

7.0

8.5

10. 1

11.6

13.2

14.7

15.5

(ft) S u

2.0

2.2

1.5

2.4

3.8

6.0

5.2

2.9

2.5

2.0

2.0

(ksf) ^50

0. 0072

0.0072

0.0072

0.0072

0. 0072

0. 0072

0. 0072

0.0072

0. 0072

0.0072

0. 0072

64

500

400

to

- 300

o

200

i.

«T3

100

0

•

/

/ /

/ /

/ /

^ //

/ / / /

/ / / /

^

/ / / /

'/

.,yy^

"^ 7 5 ^

- — — — -

1

Model

Field test data

— 1 _ _ J. I 1

0 1.0 2.0 3.0 4.0 5.0



65

Example 6, Figure 4-9 , P i er # 6. This pier was t e s t e d ty

Bhushan , Haley, and Fong [ 2 ] - The pier dimensions and the


Pier dimensions : Length = 15.5 f t .

Diameter = 4 f t .

soil Properties are the same as for pier • 5 .

In this example, the predicted deflections are very

close to the actual deflections up to a load of about 250

k. At higher loads the results of the analysis deviate

from the actual test data conservatively.

66

500

400

to

- 300

-o O

200 CO S-(U

«o

100

0

.

—'h 1 1 1

/ / / /

/ / / /

/

/ / /

//

7

/ X

/^

y y*

y

y y

•

^ ^^

M(

F-

•

)del -

le 1 u c CO c u a

1

•l-T C U

r 1

0 1.0 2.0 3.0 4.0 5.0



67

Example 7, Figure 4-11, the Ontario Hydro Research Tests

[9], the pier dimensions and the soil properties for this

test are as follows :

Pier dimensions : Length = 17 ft.

Diameter = 5 ft-

Has a 10 f t . diamter b e l l a t the

bottom 5 f t .

The a c t u a l s o i l propert i e s as published by Ismael and

Klym [ 9 ] are presented in Figure 4-10. Table 4-6 g i v e s the

a c t u a l v a l u e s of s o i l proper t i e s used in the computer

program.

In t h i s example, the e f f e c t of the bottom r e s i s t i n g

moment i s more than other examples which are s tud ied in

t h i s a n a l y s i s . The predicted d e f l e c t i o n s in t h i s c a s e are

very c l o s e to the ac tua l d e f l e c t i o n s up to the maximum load

of 160 k.

68

A X I A L . S T R A I N ( ( 5 0 )

0 . 0 1 0 . 0 2

S H E A R S T R E N G T H (P S F)

0 500 1000 1500 2000

I , « —

Q. U C

50 50

S K E M P T O N fi E L - A T I O N -SHI P *

-^ U N C O N F l N E D 'J S T R E N G T H

X Q U STI7 E N G T H

O CU S T R E N G T H

A P R E S S U R E M E T E R STR E N G T H

Figure 4-10. Soil properties for the Ontario Hydro Research test

[For the shear strength of soil use CU values)

Table 4-6. Soil properties for Ontario test.

69

Depth (ft)

0.8

2.5

4. 3

5.9

7.6

9.3

11.0

12-8

14.4

16. 1

17.

s

1.8

1.8

1.3

1.1

1.0

0.9

0.9

0.9

1.0

1.0

1.0

(ksf)

•

^50

0.01

0.01

0.01

0.01 1

0.012

0. 013

0.013

0. 013

0.013

0. 0 13

0.013

200

160

Q. ? 120

T3 ea O

^ 80

40

1.0 2.0 3.0 4 .0

Ground l ine D e f l e c t i o n ( i n . )

70

/ / / /

/ / / /

/ / / / / / / / / /

/ / / / i / / / 1

A

// //

//

fi / /

7 /

/ /

ML Hol _

F I Q I / H +QC-t- /â 1

1 •

w o o u a

— 1

¥~i . . _ .

•

5.0

Figure 4-11. Load-Deflection Diagram for the

Ontario Hydro Research test

Chapter 5

Conclusions and Recommendations

In t h i s s t u d y , a d i s c r e t e model for the a n a l y s i s cf

short p ier foundat ions i s developed. The fol lowing are the

b a s i c d i f f e r e n c e s between t h i s a n a l y s i s and previous

a n a l y s e s done by other i n v e s t i g a t o r s :

1. The p i e r i s assumed to be i n f i n i t e l y r i g i d .

2 . The concept of beam on e l a s t i c foundation i s not

used.

3 . Springs represent ing the bottom r e s i s t i n g moment,

bottom f r i c t i o n , bottom v e r t i c a l r e a c t i o n , and skin

f r i c t i o n are included in the model.

4 . The shear s t rength of s o i l ' i s not averaged over the

d i s tance from the ground surface to a depth X as was

done by Reese [ 2 0 ] .

5. The s o i l u l t i m a t e strength i s not modified as was

done by Ismael and Klym [ 9 ] .

71

72

The f o l l o w i n g c o n c l u s i o n s are der ived from t h i s s tudy :

1. The d i s c r e t e model p r e s e n t e d here has t h r e e d e g r e e s

of freedom, but i t r e p r e s e n t s t h e o v e r a l l behav ior

of t h e s o i l - s t r u c t u r e i n t e r a c t i o n .

2 . The l o a d - d i s p l a c e m e n t re sponse of the p i e r o b t a i n e d

us ing o n l y t h e l a t e r a l s p r i n g s n e g l e c t i n g t h e s k i n

f r i c t i o n and t h e bottom r e s i s t a n c e s , showed very

poor agreement with f i e l d t e s t d a t a .

3. The addit ion of the res i s t ing forces at the bottom

and on the s ides of the pier resulted in a more

r e a l i s t i c model, and the resu l t s of the analys i s

compared f a i r l y well with the f i e l d t e s t data fcr

most of the example problems.

4. The addition of b e l l to the pier increases the

e f f ec t of the bottom res is t ing moment and the bottom

f r i c t i o n a l force substant ia l ly , depending on the

s i z e of the be l l and the s o i l strength d i r e c t l y

below the b e l l .

5. In a few c a s e s , the predicted def lect ions of the

l a t e r a l l y loaded piers were not as c lose to the

•?3

ac tua l measured d e f l e c t i o n s as was des i r ed . This

may be because of pos s ib l e e r ro r in the measuremett

of the shear s t r eng th of so i l or the assumpt icrs

made in developing p-y curves and spring c o n s t a n t s .

6. The s i z e of the s t i f f n e s s matrix for t h i s model i s

only 3x3, no matter how many nodes on the p i e r are

chosen. This i s in con t r a s t to the prev ious ly

repor ted ana lyses where the s i z e of the s t i f f n e s s

matrix depended on the number of nodes which were

used as in the case of the team on e l a s t i c

foundat ion. Consequently , consider ing the non

l i n e a r nature of t h i s problem , the computer t i n e

requ i red to so lve the above system of equat ions over

and over u n t i l convergence is achieved i s reduced

cons iderably because in each i t e r a t i o n only a system

of th ree equat ions needs to be solved for the

unknowns.

^Recommendations for future s t ud i e s :

1. Ihe model can be programed on a oiinicomputer very

e a s i l y .

2. As more experimental data become a v a i l a b l e , the

74

model should be checked and spring constants revised

for obtaining better comparisons. For example, the

p-y curves need to be improved in order to be able

to determine the initial spring constants more

accurately and to represent its behavior at loads

close to its ultimate capacity.

3. A better knowledge and understanding of the soil

reactions (the bottom resisting moment, bottom

friction, and skin friction) through a series cf

tests and studies would be helpful to improve the

model.

5. The ana lys i s should be extended for sands and sandy

s o i l s .

R e f e r e n c e :

1 . Adams, v7. I . , and R a d h a k r i s h na, H . S . , "The L a t e r a l C a p a c i t y of Deep Augered F o o t i n g s , " P r o c e e d i n g s of t h e 8 th I n t e r n a t i o n a l C o n f e r e n c e of S o i l Mechanics a rd F o u n d a t i o n Engineering:! , V o l . 2 . 1 , Moscow, U . S . S . E . , 1973 , p p . 1-8-

2 . Bnushan , K . , H a l e y , S . C . , and Fong, P . T . , " L a t e r a l Load T e s t s on D r i l l e d P i e r s i n s t i f f C l a y s , " Tournal o r t h e G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , Vo l . 105, NO. GTf, AUG., 1979 , P P . 9 6 9 - 9 6 5 .

3 . B o w l e s , J o s e p h E . , F o u n d a t i c n A n a l y s i s and D e s i g r , Second E d i t i o n , 1977, p p . 157-159 .

4 . Bowles , J o s e p h E . , F o u n d a t i o n A n a l y s i s and Des ign , Second E d i t i o n , 1977, p p . 8 2 - 8 7 .

5 . Broms, B. B. , " L a t e r a l R e s i s t a n c e of P i l e s in C o h e s i v e S o i l s , " jQurna 1 of t h e S o i l "1 ec ha n i c s ?.nd F o u n d a t i o n s D i v i s i o n , ASCE, Vol . 90 , No. SM2, P r o c . Paper 3825, M a r . , 19 6 4 , p p . 2 7 - 6 3 .

6» D o u g l a s , D . J . , and D a v i s , E . H . , "The Movement or Bur ied F o o t i n g s Due t o Moment and H o r i z o n t a l Load and t h e Movement o f Anchor P l a t e s , " G e o t e c h n i q u e , Vo l . 14, London, E n g l a n d , 1964, p p . 11 5-132.

7 . G i l l , H. L. ^ and Deraars, K. R. , " D i s p l a c e m e n t cf L a t e r a l l y Loaded S t r u c t u r e s i n N o n l i n e a r l y R e s p o n s i v e S o i l s , " T e c h n i c a l Report R670, Naval C i v i l E n g i n e e r i r g L a b o r a t o r y , P o r t Hueneume, C a l i f - , 1970 , pp. 1-59.

8 . Hansen , J . B . , "The U l t i m a t e R e s i s t a n c e of R ig id P i l e s A g a i n s t T r a n s v e r s F o r c e s . " Danish G e o t e c h n i c a l I n s t . , B u l l . 12 ( 1 9 6 1 ) .

9 . I s m a e l , N-F. , and ' lym, T. W., " B e h a v i o r of R i g i d P i e r s i n Layered C o h e s i v e S o i l s , " J o u r n a l of t h e G e o t e c h n i c a l E n g i n e e r i n g D i v i s i o n , ASCE, Vol . 104, No. GTS,Aug. , 1978 , p p . 1061-1074-

10. L a n d e r s , P h i l l i p , " R e s e a r c h on F o u n i a t i o n S y s t e m s , " J o u r n a l of t h e E l e c t r i c Power R e s e a r c h I n s t i t u t e , V o l . "4, No. 6 , J u l y / A u g u s t , 1979, pp. 3 3 - 3 4 .

75

75

11 .

12.

18.

L a n g h a a r , Henry L - , Energy Methods i n Appl ied Mechan ics . Viiley 5 S o n s , VJ62, pp - 1 - 3 3 .

M d t l o c k , H-, " C o r r e l a t i o n s for Design of L a t e r a l l y Loaded F i l e s in S o f t C l a y , " ^ r e p r i n t s , Second Annual O f f s h o r e Techno logy C o n f e r e n c e , ' l o u s t o n , T e x a s , V o l . No. OTC 1204, 1970 , p p . 1 - 5 7 7 - 5 8 8 .

I ,

13 . M c C l e l l a n d , B. , and F o c h t , J . A . , J r . , " S o i l Modulus f c r L a t e r a l l y Loaded P i l e s , " J o u r n a l of t h e S o i l ^ . echan ic s

20.

and Foundation Division, ASCE, Vol. 82, No. Paper 1081, Oct., 1956, pp.1-22.

S'fi'4 Proc-

14. McClelland, B., and Focht, J.A., Jr., discussion cf "Soil Modulus for Laterally Loaded Piles," ty McCelland, B. , and Focht, J.A-, Jr-, Transactions, ASCE. vol. 123, Paper No. 2954, 1950, p. 104:»-1086.

15. Poulos, H.G., "Behavior of Laterally Loaded Piles; I-single Piles," Journal of the Soil Mechanics and Foundations Division, ASCF, Vol. 97, Paper 8092, May, 1971, pp.71 1-731.

No. S M 5 , P r o c .

16» P o u l o s , H. G., "Behav io r of L a t e r a l l y Loaded P i l e s - I I I s o c k e t e d P i l e s , " J o u r n a l of t h e S o i l Mechanics ard F o u n d a t i o n s D i v i s i o n . ASCE, Vol. 9 8 , No. SM4, P r o c . P a p e r 8 8 3 7 , A p r . , 1972, pp . 341-360 .

17. R a d h a k r i s h n a , R . S . , "Highbury J e t t o London N o r t h e a s t TS , p r o p o s e d 230-KV 2 c c t S t e e l P o l e T r a n s m i s s i o n Line F u l l - s c a l e Founda t ion T e s t s , " O n t a r i o Hvdro R e s e a r c h D i v i s i o n P e £ o r t lo_. 71-4C9-K, T o r o n t o , Canada , 1971 ,

R e e s e , L . C - , d i s c u s s i o n of " S o i l Modulus f o r L a t e r a l l y Loaded P i l e s , " by B- McCle l l and , and J . A, F o c h t , J r . , T r a n s a c t i o n s . ASCE. Vol . 1 2 3 , Paper No. 2954 , 1 9 5 8 , pp . 1071-1074 .

19. R e e s e , L . C - , and Welch, R - C , " L a t e r a l Loading of 'êep " ~ "• "^ ^ ' - ' ^ e o t e c h n i c a l

GT7, P r o c . F o u n d a t i o n s i n s t i f f C l a y , " J o u r n a l of_ t h e E n g i n e e r i n g D i v i s i o n , H C E , Vol. 10 1 . No. p a p e r 11456, J u l y , 1975, pp

101 , 6 3 3 - 6 4 9 .

R e e s e , L. C. , Documenta t ion , " E n g i n e e r i r g Pi v i s i o n , ASCE, Vol-Paper 12862, A p r . , 1977, p p . 2 8 7 - 3 0 5 .

" L a t e r a l l y Loaded P i l e s : Program J o u r n a 1 of t h e G e o t e c h n i c a l

10 3 , No. G r 4 , P r c c .

77

21. Skeropton, A-l-J-, "The Bearing Capacity of Clays," Proceedings, Building Research Congress. Division 1, London,' England, 1951, pp. 180-188.

22. Stott, J. p.

23

H T e s t s on M a t e r i a l s f o r Use in S l i d i r g L a y e r s Under C o n c r e t e Road S l a b s , " c i v i l and P u b l i c Works R e v i e w , V o l . 5 6 , Nos . 1961 , p p . 1 2 9 7 , 1 2 9 9 , 1 3 0 1 , 1 4 6 6 , 1 6 0 3 , 1 6 0 5 .

E n g i n e e r inq 6 6 3 , 6 6 4 , 6 6 5 ,

Woodward , R i c h a r d J , , G a r d n e r , W i l l i a m S. , G r e e r , D a v i i M. , D r i l l e d P i e r F o u n d a t i o n s , M c G r a w - H i l l , I r . c . , 1972 , p p . 6 4 - 6 7 -

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