UJCW/ZA-Chairman of ^ the^ Committe (^AiM e

LATERALLY LOADED RIGID PIERS IN SAND AND SANDY SOILS

by

TONG-JONG TSENG, B.S. in Eng.

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

Chairman of the Committee UJCW/ZA- ^ ^ (ÂiM

Accepted

Dean of t!ve Graauate School

August,1984

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to Dr.

C.V.G. Vallabhan for his kind guidance and encouragement

throughout the progress of this research and also during the

period of study. I also wish to thank Dr. W. K. Wray, and

Dr. H. S. Norville for their valuable suggestions.

I would like to express my gratitude to the Department

of Civil Engineering for supporting me during my graduate

studies. I would also like to thank Miss April Stigers and

Mr. K. N. Gunalan for helping me revise the draft.

I am deeply indebted to my parents for their support

and encouragement.

11

ABSTRACT

In this research a model has been described to analyze

and design rigid piers in sand and sandy soils. This model

utilizes several soil spring constants to analyze the

resistance-displacement relationship of the pier in sand and

sandy soils. The soil spring constants used are lateral

spring constants, bottom vertical spring constant, bottom

friction spring constant, bottom moment spring constant, and

friction spring constants on the periphery of the pier.

The most important aspect of obtaining the resistance-

displacement relationship between the pier and the soils is

to calculate the ultimate resistance capacity of each layer

of soil, and the slope of the initial part of the

resistance-displacement curve. These have been done

efficiently and the results are promising that are described

and discussed here. The minimum potential energy theoreom is

used to develop the system equations.

Ill

CONTENTS

ACKNOWLEDGEMENTS ii

ABSTRACT iii

LIST OF TABLES vi

LIST OF FIGURES vii

LIST OF SYMBOLS ix

CHAPTER

I. INTRODUCTION 1

The Problem 1 The Object of The Research 2

Previous Research of The Problem 3

II. THE MODEL FOR RIGID PIERS WITH SOIL SPRINGS 9

The Concept of Rigid Pier 9 Soil Springs 12 The System Equations For The Model 14 Procedure for Determining Deflections 18

III. SPRING CONSTANTS OF THE MODEL 21 Introduction 21 The Lateral Spring Constants 21 The Bottom Vertical Spring Constant 34 The Bottom Moment Spring Constant 36 The Bottom Friction Spring Constant 40 The Vertical Skin Friction Spring Constants 42 Evaluation Of Angle Of Internal Friction And

Modulus Of Elasticity For The Sand 44

IV. MODEL BEHAVIOR COMPARED WITH EXPERIMENTAL RESULTS 47

Introduction 47 Sample Pier Problem 48 EPRI Tests On Sandy Soils 51 Tests At Hager City 56

IV

V. CONCLUSIONS AND RECOMMENDATIONS 69

Conclusions 69 Recommendations For Future Studies 71

LIST OF REFERENCES 72

LIST OF TABLES

1. Values Of Moduli For Sandy Soil As Suggested By Terzaghi 29

2. Values Of Moduli For Sandy Soils As Suggested By

Reese 30

3. Soil Properties for EPRI Test No. 3 49



6. Soil Properties for Hager City Test No. 1 57






VI

LIST OF FIGURES

1. Idealized Ultimate Capacity Method by Broms 4

2. Single Lateral Spring Model 6

3. Forces Acting on The Pier 11

4. The Proposed Discrete Model 13

5. Movements of The Pier 15

6. Figure of Pier Segment 23

7. Pressure Distribution Before Loading 24

8. Pressure Distribution After Loading 24

9. Family of p-y Curves 25

10. Illustration of Secant Modulus 26

11. Effect of L/D on k 30

12. Lateral Bearing Capacity Factor for Granular Soil 33

13. Variations Of M And Rotation Angle 39

14. Relation Between The Frictional Force And Sliding Of

The Pier 41

15. Moment/Deflection Comparison With EPRI Test No. 3 50



18. Moment/Deflection Comparison With Hager City Test No. 1 63



Vll




Vlll

LIST OF SYMBOLS

A = cross-section area of pier

B = least lateral dimension of footing

D = diameter of the pier

E = modulus of elasticity of the pier

Eg = modulus of elasticity of soil

^si ~ "modulus of the initial portion of p-y curve

^m=v = maximum force

^max ~ " ^ i"iuni frictional force on the left side

^max ~ " ^ i" " frictional force on the right side

f- = factor of diameter

f^ = factor of angle of internal friction of the soil

I = moment of inertia of the pier

I = shape factor, equal to 6.0

I = an influence coefficient

J = nondimensional coefficient

K = horizontal modulus of subgrade reaction

k = soil modulus value suggested by Reese

K, = soil spring constant for resisting moment

at the bottom

K, = soil spring constant for vertical soil reaction

at the bottom

K, = soil spring constant for horizontal soil resistance

at the bottom

ix

K^. = stiffness matrix

K = Rankine passive earth pressure coefficient

K = soil spring constant for shear

K . = vertical soil spring constant on the left side

for the i-th segment

K . = vertical soil spring constant on the right side

for the i-th segment

K . = horizontal soil spring constant for the i-th segment

L = length of the pier

M„-,„ = maximum moment resistance at the bottom max

M. = applied moment at the top

N = standard penetration number

N = Terzaghi*s bearing capacity factor

N = Terzaghi's bearing capacity factor

P = resistance force at deflection y

p = soil reaction per unit length

P = ultimate resistance of sandy soil u

Q ,. = ultimate bearing capacity of the pier at the bottom ^L JL w

R = radius of the pier

S = section modulus of pier

s = settlement

S = shear strength of the soil u ^

U = strain energy due to the displacements

Uj = vertical displacement at the bottom of the pier

X

u. = maximum displacement on the right side of the pier

u. = maximum displacement on the left side of the pier

^max ~ P^sscribed maximum displacement for shear

u = vertical displacement at the center of the pier

u. = vertical displacement at the top of the pier

V = potential energy due to the external load

Vj = lateral deflection at the bottom of the pier

V. = horizontal displacement of the i-th soil spring

V = prescribed maximum displacement

V = horizontal displacement at the center of the pier

Vrp = total vertical load at the bottom of the pier

V = applied vertical load at the top of the pier

X. = middle depth of each layer of soil

y = lateral deflection of the pier

a = rotation of the pier

$ = potential energy of the system

it - angle of internal friction of soil

y = Poisson's ratio

Y = effective unit weight of soil

% = length of segment of the pier

XI

CHAPTER I

INTRODUCTION

The Problem

Piers are used frequently as foundations for structures

such as multistory buildings, bridges, steel transmission

poles, etc.. The analysis of pier foundations for

transmission poles is quite different from the analysis of

pier foundations for other structures, which primarily

support very large dead loads as well as live loads. The

analysis of the behavior of the pier foundations of steel

transmission poles is a complex soil-structure interaction

problem. The complexity of the problem of pier analysis

results from variable soil conditions, the large moments,

and large vertical and horizontal forces to which the piers

are subjected. Many methods are available for pier analysis.

Some of them are very simple and conservative and others are

very complicated. The need exists for a relatively simple

and economical method to analyze piers.

The results produced by a recent model of a pier

foundation presented by Davidson of GAI Consultants, Inc.

[5] do not agree well with the results of full-scale

experiments as reported by Davidson [5]. When the soil is

dense, the deflections of the piers predicted by Davidson

1

model yield lower values than those obtained experimentally.

When the soil is loose, the deflections predicted by the

model are much greater than those obtained experimentally

[7], Another model was presented by Vallabhan and Alikhanlou

[1,19] for the analysis of laterally loaded short circular

rigid piers, but their model was limited to clay and clayey

soils. More research is required to advance an adequate

model for the analysis of short piers with sand and sandy

soils.

The Object of The Research

The purpose of this research is to develop an alternate

analytical procedure to obtain the soil resistance-

displacement relationships, for laterally loaded rigid piers

in sand and sandy soils. The procedure presented in this

report is based on a two dimensional model employing the

minimum potential energy theorem with nonlinear soil

resistance-displacement properties for analysis. The soil

resistances include lateral resistance, vertical resistance,

frictional resistance and moment resistance at the bottom of

the pier and frictional resistance on the sides of the pier.

Previous Research of The Problem

Methods used to predict lateral deflections of a

laterally loaded pier can be classified into three different

categories.

The Ultimate Strength Method

Broms [3] utilizes the lateral earth pressure

distribution shown in Figure 1 to calculate the ultimate

capacity of sand and sandy soils. Similar methods have been

used by a number of other investigators. This method

neglects the effect of bottom reaction and skin friction.

Hence, this method generally leads to a conservative design

of the pier foundation. Also, this method does not yield

displacement data. However, this method is simple to apply

and has been accepted by many foundation engineers to design

small pier foundations.

The Elastic Method

In this model, the pier may be assumed to be either

rigid or to have linear stress-strain characteristics. The

soil is modeled by assuming either a linear relationship

between lateral deflection and lateral pressure (subgrade

modulus model), or a linear relationship between stress and

strain in the soil continuum (continuum model).

-TW^

k3 KprDXH

- D *-

RIGID BODY ROTATION

ASSUMED DISTRIBUTION OF SOIL PRESSURE

Figure 1: Idealized Ultimate Capacity Method by Broms

Purely elastic solutions offer mininal advantages over

the linear subgrade reaction theory, because they require

reasonably accurate values for either the modulus of

elasticity or the coefficient of subgrade reaction of the

soil. In addition the elastic solutions require iteration

or become complex if the soil is stratified. The elastic

solution does not account for nonlinearity of the soil

response and results in conservative predictions when

compared with actual behavior as demonstrated by full-scale

test results [5].

The Nonlinear Method

A more sophisticated approach is the nonlinear method

based on the concept of the lateral load versus deflection

commonly known as p-y curves for the analysis of piers.

These p-y curves were developed empirically by the use of

nondimensional coefficients and many sophisticated lateral

load tests [14]. The test procedure requires that the

deflection and slope at the groundline be measured for each

applied moment and lateral load; in addition, internal

strains along the pier must sometimes be measured.

Nondimensional solutions are generated using different

assumed variations of the moduli with depth until the

solutions agree with the measured values of deflection and

slope at the groundline. Agreement between the non-

dimensional solutions using measured deflections and slopes

approximately indicates the variation of soil moduli. The

soil moduli so obtained are used in conjunction with finite

element or finite difference methods to obtain deflections

as a function of depth. Thus, as both the soil moduli and

deflections are known along the length of the pier, the

values of resistances at desired depths can then be

computed. The procedure described above is repeated for

different values of applied loads and moments to generate a

family of p-y curves. This concept was advanced by Matlock

[10], using the principle of a beam on an elastic foundation

(Figure 2) and the finite difference method for the analysis

of laterally loaded piers. It was later extended by Reese

and others [16]. The concept yields nonlinear predictions

that approximate the actual behavior of piers under lateral

loading conditions. The fourth order differential equation

of a beam on elastic foundation, as used by Reese, is given

below [15]:

EI dV

dx r* T d = y

dx' - p = 0 (1)

y/Xv777>5J^^'^

Lateral Resistance

/

• —vvwv—k

• . *

—\V>AV-^

--VA/vVV—^

^ v V A ^ — ^

-ÂVvV—^

Figure 2: Single Lateral Spring Model

where




X = depth of the pier corresponding to y

V^ = total vertical load on the pier at x

p = soil resistance per unit length

To apply the finite difference method to the above

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

number of equal segments interconnected at points called

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

pier. For a given load, the model yields a set of linear

simultaneous equations with unknown nodal deflections. The

number of equations depends upon the number of nodes chosen.

The nonlinear nature of the problem leads to an iterative

solution procedure. Therefore, the computer time required

can become relatively long. In this procedure, the

resisting forces and the moment of the soil at the bottom of

the pier and the skin friction between the pier and the

surrounding soil are neglected. Depending on the soil

properties and the dimensions of the pier, the resisting

forces and the moment at the bottom, as v/ell as the skin

friction on the sides, can significantly influence the

behavior of the pier.

Another model is offered by Davidson of GAI

Consultants, Inc., in v/hich he utilizes a four-spring

subgrade modulus model [7]. These four springs are lateral

translational springs existing at each layer of soil,

vertical side shear springs at the perimeter of the pier, a

base shear translational spring at the bottom, and a base

moment spring at the bottom. Hov/ever, moment-deflection

curves obtained from field tests did not correlate very well

with those predicted by this model.

CHAPTER II

THE MODEL FOR RIGID PIERS WITH SOIL SPRINGS

The Concept of Rigid Pier

A simple discrete model for the analysis of short rigid

piers subjected to predominantly large moments was conceived

by Vallabhan and Alikhanlou [1,19]. They assumed that when

the length of the pier is less than three to four times its

diameter, the pier acts like a stiff or rigid beam. In other

words, the pier is sufficiently rigid to displace and rotate

under lateral loads without appreciable distortion from its

axis. The lateral displacement of the pier at any depth may

be described with reference to the displacements at its

center of gravity. This condition simplifies the displace

ment compatibility analysis and has generated a concept

of "rigid pier" solutions for the coraplex nonlinear soil-

pier interaction problem. A free-end pier can be assumed

rigid if the following condition is satisfied [20]:

- ^ ^ 2.0 (2)

,1-.

T

where


T = V(EI/K)


10


K = horizontal modulus of subgrade reaction

For piers used as foundations for steel transmission

poles, this condition is usually true because these piers

have a large moment of inertia of the cross section, a large

modulus of elasticity of reinforced concrete compared with

tiiat of soil, and relatively short lengths. Also, some

designers provide a bell at the bottom of the pier, which

increases the rigidity of the pier substantially.

In the design of the pier, the angle of rotation has to

be small. From a practical point of view, let us assume that

the allowable deflection is less than 4 inches for a pier of

total length of 15 ft. For this case, the center of rotation

would be at about 12 feet below the ground surface; thus,

the angle of rotation will be less than 0.03 radian. The

relations tan9 = 9, and sinS ^ 9 are true for 9 < 0.05

radian. Therefore, the above assumption that a is small is

justified for short piers.

Data obtained from full-scale tests on such pier

foundations indicate that they displace as rigid piers.

When such a pier is subjected to lateral loads, the

surrounding soil resistance can be pictorially represented

as shown in Figure 3. Here, it is assumed that the forces

are in one plane, hence the force resultants per unit length

11

GROUND LEVEL

SOIL RESISTArJCE TO HORIZOfJTAL MOVEMENTS —

•.u'sA^m^w

VERTICAL FRICTIONAL RESISTANCE

SOIL AT

RESISTANCE BOTTOM

BASE SHEAR RESISTANCE

Figure 3: Forces Acting on The Pier (From Vallabhan and Alikhanlou [19])

12

of the pier are represented in the plane of the applied

forces. These force resultants are assumed to be activated

by the set of equivalent nonlinear soil springs discussed in

the following section.

Soil Springs

A set of equivalent soil springs in the plane of motion

are used to model the interaction of the soil v/ith the rigid

pier as shown in Figure 4. This set of springs contains five

different subsets of soil springs. As ' the lateral soil

behavior is of major concern, the most important subset of

soil springs consists of the horizontal springs, each

located at a discrete depth. Associated with these

horizontal springs are the vertical springs used to model

the skin friction of the soil. As equivalent soil springs,

their functions, and their constants, are defined as

follows:

1. A set of horizontal springs, with spring constant

K ., represent the lateral resistance of soil for the

i-th element.

2. A rotational spring, with a spring constant v. ,/

represents the resisting moment of the soil at the

bottom of the pier.

3. A horizontal spring with a spring constant j f

represents the friction at the bottom of the pier.

13

SIDE FRICTION ( kii )

Ht

BOTTOM FRICTION (K^y)

BOTTOM VERTICAL RESISTANCE

Vt

LATERAL RESISTArjCE

rAWvV-t^

^VvVvV—^

SIDE FRICTION ( k|,i)

BOTTOM MOMENT { K^^ )

Figure 4: The Proposed Discrete Model (Terms Inside the Parenthesis Are the Corresponding Spring Constant)

(From Vallabhan and Alikhanlou [19])

14

4. A vertical spring, with a spring constant K, , bu'

represents the vertical resistance of soil at the

bottom of the pier.

5. A set of vertical springs, with spring constants K^., ui'

and K ., represent the skin friction between the pier

and the surrounding soil on the right and left sides

of the pier.

The System Equations For The Model

To develop the system equations, the geometric

relations between the spring displacements and the rigid

body motions of the pier must be established. Assuming the

rigid body motions of the pier to be u , v and a, as shown •^ o o

in Figure 5, from simple geometrical considerations it can

be shown [1,18] that the displacements at the distance x.

measured from the top along line AB are: r u. = u + R a 1 o v ^ = v + ( 0 . 5 L - x . ) a (3) 1 o 1

Similarly, the displacements at the distances x. measured

from the top along line CD are shown as:

u. = u - R a 1 o

v- = V + (0.5L - X. ) a (4) 1 o 1

The superscripts 'r' and '1' denote the right and left sides

of the pier. The displacements at the center point on the

top are shown as:

15

-y-axis

X-axis

Figure 5: Movements of The Pier (From Vallabhan and Alikhanlou [19])

16

^t = o

v^ = v^ + 0.5L a (5)

and the displacements at the center point on the bottom are

shown as:

b o

Vj = v^ - 0.5L a (6)

The above equations show that if the rigid body motions

of the centroid of the pier (u , v and a) are known, then

the displacements of all the springs can be determined.

The Minimum Total Potential Energy Theorem [9] is used

to develop the system equations. The potential energy of

the system is defined as the sum of the strain energy and

the potential energy of the external loads, i.e.,

$ = U + V (7)

where

$ = potential energy of the system

U = strain energy due to the displacements of the springs

V = potential energy due to the external load

The function $ is a minimum for an equilibrium state.

Equations are developed that will give the variables (u^,

V , and a) for any applied loads acting at the top of the o

1 .. pier. Then substitution for v., u^, and u^, in terms of u^,

V , and a, and minimizing the function $ with respect to the

17

variables u^, v^, and a, gives a set of equations of the

form:

K 11

sym,

K

K

12

22

K

K

K

13

23

33

^ •"

"o

ô

a k

/

=

V

H,

M^ + 0.5L H^

(8)

where

K.. = K, „ + Z (K^. + K-'-. ) 11 bu ui ui

K^2 = 0

K -, = R Z (K^. - K- . ) 13 ui ui'

^22 = ^bv "" vi

^23 = - O - ^ V ^ ^ ^ ^ i <0-5L - x^)

^33 = '\m * 0-25Kbv ^'

+ I [ K . (0.5L - x.)=' + (K^. + K- .) R^ ] VI 1 Ul UI

V. = applied vertical load at the top of the pier

H. = applied horizontal force at the top of the pier

M. = applied moment at the top of the pier

Z = summation for all springs from 1 to n, along

the sides of the pier

This matrix is symmetric and positive definite;

hence, the solution is unique for all loading conditions.

It is seen that due to the assumption that the pier is

rigid, only three independent quantities are required to

18

represent the rigid body motion of the pier, thus, making

the analysis relatively simple. The above equations are

derived assuming linear, elastic behavior of the soil

springs. However, the soil springs behave nonlinearly with

displacements. The solution for a given loading,

incorporating the nonlinear behavior of the springs, is

obtained using an incremental iterative scheme, which is

discussed in the following section.

Procedure for Determining Deflections

The steps involved in solving the nonlinear force-

deflection characteristics of a laterally loaded pier for a

given set of equivalent soil spring constants are outlined

below:

Step 1 . The length of the pier is divided into n equal

segments (generally, 15 to 25 segments), and the total load

vector is developed in increments. A small initial value of

0.0001 radian is assumed for the angle a and the initial

values of u and v are set equal to zero, o o

Step 2. Using u , v , and a in Equation 3, the ^— ^ o o

vertical and horizontal displacements of every node along

the pier are calculated.

19

Step 3. Using the soil properties (shear strength,

angle of internal friction, cohesion, effective density, and

depth) the load-deformation characteristics corresponding to

the calculated u., v. displacements are developed for each

node i using the procedure discussed in chapter III. These

values of spring constants will then be used to determine

the initial stiffness matrix.

Step 4. These system equations are then solved for a

given set of load vectors to find the new values of u , v , o o'

and a.

Step 5. The new values of u , v , and a are o' o'

substituted into Equation 3 to find the new values of u. and

v.. As the problem is nonlinear, the convergence to the

correct displacement solution for a given set of loading is

monitored by comparing the new horizontal displacement at

the top of the pier to the old one. The relative difference

6 between the new and the old values of v for a point at the

top of the pier is defined as follows:

new old old

If 6 is less than a prescribed iteration tolerance of 0.001,

Steps 2 through 5 are repeated until the solution converges.

This value of the tolerance is chosen because it will allow

the system of springs to become sufficiently close to the

static equilibrium condition from a practical point of view.

20

The load vector is now increased by another increment

and Steps 2 through 5 are repeated until the total maximum

load is applied on the pier-soil system.

CHAPTER III

SPRING CONSTANTS OF THE MODEL

Introduction

Vallabhan and Alikhanlou [1,19] have discussed

techniques to evaluate soil spring constants for clay and

clayey soils for their rigid pier-soil interaction model.

Based on similar arguments, the author is discussing

techniques for the development of the various soil spring

characteristics of sand and sandy soils as a function of the

corresponding displacement parameters. It is assumed that

the soil properties are horizontally uniform, and the

magnitudes of these soil springs are pure functions of the

corresponding soil characteristics without considering any

interactions between soil layers and the effect of the

boundary especially at the top. The specific assumptions

used in each case of soil springs are discussed separately.

The Lateral Spring Constants

For the behavior of laterally loaded rigid piers, the

lateral spring constants are the most important ones. Here

the soil response to the movements of a laterally loaded

pier is characterized by a set of discrete springs similar

21

22

to the Winkler elastic foundation concept (1867). But these

springs can have nonlinear load-deformation responses and,

as stated above, the response at a point is independent of

pier deflection elsewhere. This equivalent spring constant

assumption is not strictly valid for soil continua, but the

overall error involved in its use is assumed to be small

from practical considerations.

A discussion of the physical meaning of the lateral

spring mechanisms is given here. Consider the behavior of a

pier installed in the ground as shown in Figure 6. A thin

segment through the pier and surrounding soil is shown at a

depth X. below the ground surface. Prior to application of

any lateral load to the pier, the pressure distribution on

the pier will be similar to that shown in Figure 7. For this

condition the resultant force on the pier, obtained by

integrating the pressure around the segment, will be zero.

If a lateral load is applied to the pier, it will deflect to

the new position and the pressure distribution will change

to that shown in Figure 8. Integration of the pressure

around the segment for this condition yields a resultant

force p and acts in the opposite direction of the deflection

y.

If the process discussed above is repeated for a range

of deflections by successive integration, different p-values

23

^ ^ ^

i

I.

I'

V/////.

- I h

Ground Surface

' / / ^ / / ^

Xi

] ^ ^ -

Figure 6: Figure of Pier Segment (After Reese and Cox, 1969)

24

View A-A

Figure 7: Pressure Distribution Before Loading (After Reese and Cox, 1969)

Figure 8: Pressure Distribution After Loading (After Reese and Cox, 1969)

25

are obtained corresponding to the different y-values. Thus,

the p-y curve is developed for the depth x.. Applying the

above procedure to other depths, a family of p-y curves is

developed as shown in Figure 9. This concept of p-y curves

was originated by Matlock [10] and Reese [15]. The ability

to predict the behavior of piers subjected to forces and

moments at the top is primarily dependent on the ability to

derive the p-y curves expressing the soil response within

a resonable tolerance.

fx Figure 9: Family of p-y Curves

• > - y

X = Xl

X = X2

X = X3

X = X4

26

The p-y Curves For Sandy Soils

As described above, the soil response to lateral loads

is given by a family of curves showing soil resistance as a

function of pier deflection. The soil moduli vary with

deflection of pier and the soil properties as shown in

Figure 10.

Deflection

Figure 10: Illustration of Secant Modulus

As may be seen in Figure 10, the initial portion of the

curve is a straight line indicating the linear elastic

behavior of soil for relative deflection resulting from

small lateral loads. For large deflection, the soil

27

resistance attains a limiting value, defined as the ultimate

soil resistance, at which there is no increase in value with

any further deflection. In most of the practical problems,

deflections fall within the nonlinear portion of the p-y

curves far beyond the straight line portion. Therefore, the

soil modulus is a nonlinear function of deflection and

depth. It is almost impossible to predict an entire p-y

curve from a purely theoretical basis, so empirical methods

were developed. For prediction of the p-y curve in this

report, the initial and final portions of the curves are

used which are discussed below.

The initial portion of the p-y curve has been discussed

by Parker and Reese (1972), based on the work of

Terzaghi(1955). Terzaghi used the following equation as the

basis for his recommendations:

y = q D I / E (10) • ^ y s

where

q = unit pressure


I = an influence coefficient y E = modulus of elasticity of soil s

This equation is derived from a theory of elasticity

solution for a line pressure q acting on an elastic layer of

thickness 3D. From the elastic solution, we obtain a value

28

of 1.35 for the influence factor. Thus, the above equation

can be rewriten in this form:

P = y Eg / 1.35 (11)

By substituting I and noting that p is equal to q D

the modulus of the initial portion of p-y curve can be

computed from the value of modulus of elasticity of the

sand. The modulus of elasticity of sand and sandy soils is

found from stress-strain curves which are usually plotted

from compression tests in which the confining pressure is

equal to the in situ overburden pressure. If no laboratory

data is available, the modulus of elasticity of soil can be

approximated with an expression suggested by Terzaghi. As

can be expected, the modulus of elasticity increases

linearly with depth for homogeneous sands with a constant

effective unit weight, i.e.;

E = J Y X (12)

where

J = nondimensional coefficient

y = effective unit weight of soil

X = depth below ground surface

The expression E = k x has been used by Reese and

Matlock (1956) in solutions of the laterally loaded pier

problems where the value of k is equal to J Y / 1.35. The

values proposed by Terzaghi for a pier with a width of 1 ft

are reproduced in Table 1.

29

TABLE 1

Values Of Moduli For Sandy Soil As Suggested By Terzaghi (after Reese and Cox, 1969)

Relative Density of sand Loose Medium Dense

Range of values of J

Adopted values of J

Dry or moist k (ton/ft^)

(Ibs/inM

Submerged k (ton/ft^)

(Ibs/inM

100-200

200

7

8.1

4

4.6

300-1000

600

21

24.3

14

16.2

1000-2000

1500

56

64.8

34

39.4

The angle of internal friction is assumed to be 30° for

loose sand, 35° for medium dense sand, and 40° for dense

sand [6]. From Table 1, it is seen that the value of the

modulus of elasticity for dry or moist sand is different

from that for submerged sand. These values of moduli of

elasticity of soil as suggested by Reese[16] are given in

Table 2.

For a linearly elastic soil, the modulus of elasticity

is independent of pier diameter. It is, therefore, assumed

that the values remain constants in the expression for k or

J. However, field experiments indicate that the L/D ratio

has some influence on the k values. Figure 11 indicates the

30

TABLE 2

Values Of Moduli For Sandy Soils As Suggested By Reese

sand soil type

internal angle used by Monahan

used by Davidson value of modulus

dry or moist (kip/ft^)

(Ib/inM

submerged (kip/ft^)

(Ib/inM

loose

31 °

30°

43.2

25

34.6

20

medium

36°

35°

155.6

90

103.7

60

dense

41°

40°

388.9

225

216.0

125

I 1 r i l l

10 T r—1—I I I I I

100 I I

Coefficient of Subgrode Reaction, k (pci)

1 , I I i I

1000

Figure 11: Effect of L/D on k (After Monahan, 1977)

31

approximate relationship between L/D ratio for a rigid

foundation and an assumed subgrade modulus distribution

incresing linearly with depth as is commonly assumed for

cohesionless soil [3]. The maximum L/D ratio for a rigid

foundation in cohesionless soil is in the approximate range

of 3 to 6. The foundations tested had L/D ratios ranging

from 1 to 2 and are, therefore, rigid foundations [12]. So,

the modulus of the initial portion of p-y curve are defined

using the following equation:

E . = k X. D / f- (13)

where

k = soil modulus value suggested by Reese

X. = middle depth of each layer of soil


f- is a factor of diameter equal to

0.6, if diameter is equal to 10 ft


Analysis Of Ultimate Lateral Resistance

Several researchers have discussed the ultimate

resistance of sand and sandy soils for laterally loaded

piers. Broms [5] has proposed the ultimate lateral capacity

of the soil as:

32

û = ^-O ^p Y x^ (14)

Hansen [6] has derived the lateral bearing capacity at a

given depth along a pier in cohensionless soil using the

following equation:

Hansen, using theoretical considerations, obtained values of

N , which are dependent on the ratio of depth to diameter

and the angle of internal friction. These results are given

in Figure 12. The value of N , is varied between 8 and 10

for loose sand and between 12 and 15 for medium-dense sand.

Various interpretations of full-scale and model tests have

led to the deduction that the ratio of ultimate lateral

capacity and overburden pressure may range from two to four

times the coefficient of passive earth pressure K . Based

on the above expressions, Vallabhan suggested an equation of

this type:

P = f, f^ K Y X. (16) u 1 2 p ' 1

where

K = Rankine passive earth pressure coefficient

= tan^ (45° + 0.5 «J)

(h - angle of internal friction of soil, in degrees

f. is a factor of diameter equal to



33

N qh

2 3 4 5 67 8910 20 30 40 60 80

O 8 •H 4J <n 10 u

X 12

14

16

IB

20

_ :N . ' . A - V ^ . - ^ ^

v-^-X _.-V-\-A

X-lA _... L_n —- t-t _„. ]__.t

i . I LTL E T J..T

_.-. -1_T T IE IT

r :::::::: 3 V - — _ , I t^

t L ± l_... _. 1 1„. -J _L... . . ^ 4 L 1 r

1 i I__L. _. I__L. .. 4-t- -j__t -1_1_..

^ • 25" 30® 35*» 400

Figure 12: Lateral Bearing Capacity Factor for Granular Soil (After Brinch Hansen, 1961)

f is a factor of angle of internal friction of the soil

equal to

3.0 if ^ is equal to 30°

6.0 if is equal to 45°

Y = density of the soil

From the ultimate resistance

straight line portion of p-y curve, the resistance force for

P and the initial u

34

any deflection y can be calculated using the following

proposed equation:

P = P^ ( 1.0 - e^ ) (17)

where

P = resistance force at deflection y

P^ = ultimate resistance of sandy soil

E . y SI ^

G = -

^ u ^

^si " "^°^^l^s of the initial portion of p-y curve



More realistic p-y curves will be developed as research

in this area progresses.

The Bottom Vertical Spring Constant

The bearing capacity of piers on, or into, a hard

strata or soft rock is very difficult to determine, and

usually can only be estimated by approximate formulae. In

major foundation projects, this estimated value must be

verified by a full-scale loading test. If the pier tip is

embedded in a granular soil to a depth, h, the ultimate

vertical bearing capacity can be estimated using the

following equation proposed by Terzaghi [18]:

ûlt = " ' (Y L N + 0.6 Y R N ) (18)

35

where

Q^lt ~ ultimate bearing capacity of the pier at the bottom



Y = effective unit weight of soil


= e^""^^^^^ tan^ (45° + 0.5 ?S)

i> - angle of internal friction of the supporting soil


= 2.0(N + 1 .0)tan?S

The vertical reaction at the bottom is replaced by a

linear spring placed vertically at the bottom of the pier,

as shown in Figure 4. The value of the initial spring

constant is obtained from the following equation for the

settlement of a footing which is based on the theory of

elasticity [2].

s = q B (1.0 - y^ ) I„ / E^ (19)

where

s = settlement

q = intensity of contact pressure


U = Poisson's ratio

E = modulus of elasticity of soil s

I = influence factor, equal to 0.88 [2] W

36

Therefore, the initial spring constant of an equivalent

spring to replace this soil reaction is:

^s ^ \u = (20)

B ( 1.0 - y ) I w

where

^bu ~ ^°^^ spring constant for vertical soil reaction

A = cross-section area of pier at the bottom

In many practical cases, the modulus of elasticity of

the bottom soil is not usually available. So two empirical

equations are suggested and employed in this investigation

to evaluate the modulus of elasticity. Great caution is

necessary in the use of these empirical equations. The

details of these empirical equations are given at the end of

this chapter.

The Bottom Moment Spring Constant

The resisting moment developed at the bottom of the

pier due to the rotation of the pier is modeled by the

rotational spring shown in Figure 5. A good way to obtain

this spring characteristic is to perform experiments under

controlled condition. Due to lack of availability of data

for "moment versus rotation" of circular piers on soils, the

following procedure is recommended and used herein. The

37

spring constant is determined from the following equation

which is given in Ref [2] for the rotation of rigid

footings on semi-infinite elastic half space subjected to a

moment:

M ( 1 .0 - P M I tana = ^^ (21)

B^ E s

where

a = angle of rotation of the pier, radian

y = Poisson's ratio

1^ = shape factor, equal to 6.0 [2]


When a is small, tana = a, hence

(22)

where

K, = soil spring constant for resisting moment

at the bottom

This equation represents a linear relationship between

the angle of rotation and the moment at the bottom. To

obtain the ultimate moment resistance, a simple equation

using elementary mechanics has been developed. It is

assumed that the maximum allowable stress between the bottom

V

bm

B^ E s

( 1.0 - y= ) l „

38

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

of soil, s^, and the vertical stress at the bottom of the

pier, given by:

^T q = S^ + - X - (23)

M max

or

»„=.„ = S (S„ + -t- ) (24) *max " '''u • A

where

S = section modulus of pier, equal to fr B^ / 32.0

B = diameter of pier

V_ = total vertical load at the bottom of the pier

A = cross-section area

This equation can be rewritten as:

M = - ^ m ^ ( S + - ^ ) (25) max 32 u A

By knowing the maximum moment, M , , which the soil can •* ^ max

tolerate, and the initial value of K, , the M-a relationship

is developed as shown in Figure 13. The maximum moment

obtained from Equation (25) is very conservative because of

the simplification employed in Figure 13. For this reason.

39

the value of the maximum moment so obtained is multipled by

a factor of 1.5 to achieve better results. For any value of

the angle of rotation, a, the corresponding value of Kj ^ is

determined as a secant modulus if K, a > M , bm max'

the initial modulus is used.

otherwise

Actual

M

a

K. = initial spring constant

Figure 13: Variations Of M And Rotation Angle

40

The Bottom Friction Spring Constant

The behavior due to the soil friction at the bottom of

the pier is represented by a horizontal spring at the bottom

as shown in Figure 5. In the absence of data, the relation

ship between the frictional force and the movement at

the bottom of the pier is assumed as shown in Figure 14.

The frictional force will reach a maximum value after a

prescribed maximum displacement. It is further assumed that

the value of the prescribed maximum displacement is 0.01 ft.

This assumption is based on experiments conducted by other

investigators [17].

The maximum frictional stress at the bottom is given

by:

^T q = S + —r^ tanai (26) ^ u A

v/here

S = shear strength of the soil

V^ = total vertical load at the bottom of the pier

A = area at the bottom

^ = angle of internal friction of soil

Therefore, the maximum frictional force, â^' '^^^^ ^^

41

u o

to c o

u •r-

Maximum Frictional Force

Displacement

Figure 14: Relation Between The Frictional Force And Sliding Of The Pier

Initially, the value of the bottom horizontal spring

constant is

K, = F / V bv max max (28)

where

K, = soil spring constant for horizontal soil resistance

v = prescribed maximum displacement

If the displacement at the bottom of the pier is found

to be more than the prescribed maximum displacement, then

42

the new spring constant will be given by the secant modulus

determined from the following relation:

where

V, = lateral deflection at the bottom of the pier

The Vertical Skin Friction Spring

Constants

These constants correspond to the vertical springs on

both sides of the pier. When the pier rotates, the vertical

displacements on the right and left sides are different and

can be of opposite signs. If they are of opposite signs, as

in the case of the transmission pier foundations, the

frictional surfaces for the right and left sides of the pier

are determined by approximating the area covered by a segment of length of the pier as:

A. = 2 TT R £ u. / ( u^ + u. )

A. = 2 TT R jl u. / ( u^ + u. ) (30)

where


i = length of segment of the pier

u. = maximum displacement on the right side of the pier

u. = maximum displacement on the left side of the pier

43

Therefore, the maximum frictional force on the segment of

length will be

F max

where

= A^ (or A^) S^ (31)

S^ = shear strength of the soil

or the initial spring constant for shear is given by:

K . = F / u for u < u (32)

Ul max ' max max ^^^'

where

K . = initial spring constant for shear

u = prescribed maximum displacement for shear

and the general spring constant for shear is given by:

^„ = m . / ,- (o^ Ô f°^ ^ > ^m^^ (33) u max 1 J max

where

K = soil spring constant for shear

Since not all points on the segment have the motion

equal to u. or u., and the movements of different points

vary from 0.0 to u. or u., and the center of gravity of the

frictional force is not at the boundary; a coefficient,

equal to 0.5, is introduced to account for these variations.

44

Evaluation Of Angle Of Internal Friction And Modulus Of Elasticity

For The Sand

For some equations in this chapter, we need the values

of the angle of internal friction and modulus of elasticity

of sand. Two empirical equations are suggested: one equation

for obtaining the angle of internal friction and the other

for obtaining the modulus of elasticity of the soil. For

sand and sandy soils, the in situ strength characteristics

are usually determined in practice by using the standard

penetration test. The number of blows per foot of

penetration of the split spoon sampler is taken as an

indication of the relative density of sand. Peck, Hanson,

and Thornburn [13] suggest that the angle of internal

friction is related to a standard penetration number by the

following equation:

4> = 0.3N + 27** (34)

where

$ - angle of internal friction of soil

N = standard penetration number

while Meyerhof [11] recommends the equation as:

^ = 25° + 0.15D (35)

where

D = Relative density of sand r

45

Based on the data of the modulus of elasticity versus

the angle of internal friction given by Terzaghi [2], a

multi-linear interpolation formula is recommended herein:

i = 0.5N + 27.5** for N < 10

^ = 0.25N + 30.0° for 10 < N < 30

(^ = 0.15N + 33.0° for N > 30 (36)

The standard penetration number for sand is also

influenced by the depth at which it is measured. Peck,

Hanson, and Thornburn [13] recommended a correction factor:

C^ = 0.77 log^Q(40.0 / Y) (37)

where

C = correction factor between 2.0 and 0.4 n

Y = effective pressure in ksf

Equation (36), along with the correction factor C , is

used in this research.

Another equation is required to evaluate the modulus

of elasticity of sand from the angle of internal friction.

Using a regression analysis on the soil data [5], the writer

has developed an empirical equation relating angle of

internal friction and modulus of elasticity as given below:

E = 5 ' - 300 <^ + 4500 ksf for > 35°

= 3.3 ^ for flJ < 35° (38)

v/here

E = modulus of elasticity of soil

^ = angle of internal friction of soil

46

This equation is used in the analysis. It is

recommended that it be used only when no other value of

modulus of elasticity of sand is available. More research

in this area is necessary, after which appropriate equations

can be established.

CHAPTER IV

MODEL BEHAVIOR COMPARED V7ITH EXPERIMENTAL RESULTS

Introduction

During the last few years, several investigators have

performed detailed full-scale experiments on short rigid

drilled piers in different kinds of soils. Two known sets

of test results are from GAI Consultants, Inc. for EPRI at

14 different test sites spread in various locations around

the United States [5]. Six test sites are in Hager City

[12]. The first set of tests for EPRI is very comprehensive,

while the second set at Hager City is not. Hence, the

analysis of the piers are given in two parts. The first

part of this chapter discusses the results using the EPRI

tests, while the second part discusses the results of the

tests at Hager City. Before presenting the comparison of

the results with the proposed model, a sample pier problem

is selected and analyzed for load deflection characteristics

using the lateral soil springs and adding the other

springs in succession. This way, the relative merit of

the use of the different springs representing the soil

behavior can be seen.

47

48

Sample Pier Problem

This problem is shown to illustrate the influence of

the various soil springs in the model. The pier test data

is from the EPRI set identified as EPRI test No. 3 [5].

The pier dimensions and soil properties at the site are

presented in Table 3.

The pier is divided into 21 segments and the soil

properties for the various segments are linearly

interpolated from the data. The field experimental test

results are shown in Figure 15, along with the results

obtained by the PADLL model created by Davidson of the GAI

Consultants Inc. [5]. In the proposed model, only the

lateral horizontal springs are provided and the resulting

solution (curve 1) is very close to the solution obtained by

PADLL. Then the bottom moment spring is added to the system

in the proposed model, and the results are shown by curve 2.

As this pier has a large L/D ratio (i.e., small pier

diameter), the contribution from the bottom moment

resistance is not significant. However, in cases where the

pier diameter is large, or if the pier has a bell at the

bottom, and the soil strength at the bottom is high, the

contribution from the bottom moment resistance can be

substantially high. As a third step, the bottom shear spring

is added and the result is shown by curve 3. Substantial

49

TABLE 3

Soil Properties for EPRI Test No. 3 (After Davidson, 1982)

The length above soil is 1 ft The length embedded in soil is 21 ft Maximum Applied Bending Moment is 2600 ft-kip Maximum Lateral Load is 32.5 kip

Depth (ft)

-1.0--2.0-

-4.0-

-6.5-

-9.5-

11.0-

15.0-

18.0-

21 .0-

Generalized Description

Topsoil

Very Loose Silty Sand

Loose Silty Sand

Loose Silty Sand Some Gravel

Loose Sand Trace Silt

Med. Dense Sand Trace Silt

V. Loose-Loose Silty Sand Trace Clay

Diameter (ft)

5.0

5.0

5.0

5.0

5.0

5.0

5.0

5.0

Unit Weight (pcf)

110

110

110

110

110

62

62

62

62

Modulus of Elasticity (ksi)

0.21

0.21

0.45

0.45

1.63

0.45

1 .10

0.60

1 .20

Internal I Friction Angle (")

28

29

29

29

30

30

30

28

28

improvement in the overall behavior of the soil-pier system

can be seen. This means the bottom shear spring constant of

this model is high. In the next step, all the springs are

50

I

4000H

3500 -

3000-

z 2500 . Z UJ

2 o LJ

O

o o

2000^

1500 '.

1000-

y 5 0 0 -Q. Q. <

TEST DATA CURVE 4

" CURVE 3

-CURVE 2 CURVE I

NONLINEAR PREDICTION BY FOUR-SPRING MODEL (PADLL)

U r> M I I I M I I I I I I I I I I I M I M I I I I I I I

0 1 2 3 I * T T T T I I I I I I I I I I I I I I I I r I I I r I I I I

2 3 4 5

DEFLECTION AT TOP OF PIER (IN.)

F i g u r e 15 : Moment /Def l ec t ion Comparison With EPRI Tes t No 3

51

added to the system and the result is represented by curve

4. The correlation between the experimental data and the

proposed model is remarkable. This correlation was possible

primarily because of the extensive soil test data available

for this test. Even though the comparison between the field

tests and the new model are very close, it should be noted

that the model represents the total behavior of the soil.

There are wide possibilities of errors due to sampling

procedures, local geology, and modeling of spring constants.

Since the model was previously proposed by Vallabhan for

clay and clayey soil [1,19], it will hereafter be identified

as the Vallabhan model.

EPRI Tests On Sandy Soils

Among the 14 pier foundation tests, only three could be

said to be piers on sandy soils. Others were on clayey soils

and combination of sand and clayey soils. The actual

properties of the soils are presented in Table 4 and 5. Each

pier is divided into 16 equal segments and the values of the

properties of the soil are linearly interpolated for every

segment from the soil data. The moment-deflection

characteristics of the pier-soil system using the proposed

model is compared to the field data and the results obtained

by the PADLL computer model. The results are presented in

52

Figure 16 and 17. The good correlation between experimental

data and model results can be seen in Figure 16 and 17. For

larger moments, the curves deviate away from the

experimental data on the conservative side. For the third

problem, the PADLL model results gave resistances which were

larger than the actual field results.

TABLE 4

Soil Properties for EPRI Test No. 8 (After Davidson, 1982)

The length above soil is 1 ft The length embedded in soil is 16.2 ft

Depth (ft)

. 1 "î...

"4.0

-6.0-

-7.5-

10.0-

1 *> r> 1 Z . 0

1 o r 13.5

16.2-


Loose Silt Trace Sand And Gravel

Loose Silty Sand

Med. Dense Sand

Dense Sand

Med. Dense Sand

Diameter (ft)

5.5

5.5

5.5

5.5

5.0

5.0

5.0

Unit Weight (pcf)

110

110

110

48

48

48

48

48


0.65

1 .08

1 .08

1.48

1.48

3.78

1.92

1.92

Internal Friction Angle (°)

32

38

36

36

36

45

36

36

53

TABLE 5

Soil Properties for EPRI Test No, (After Davidson, 1982)

The length above soil is 1 ft The length embedded in soil is 16 ft

10

Depth (ft)

. T R .

. . 7 0 . •• / . U

1 n n. -1 u. u- -

12.0-

16.0-


Loose Silty Sand And Gravel

Med. Dense Silty Sand And Gravel

Dense Cemented Silty Sand And

Gravel

Dense Silty Sand And Gravel

V. Dense Cemented Silty Sand And Gravel

Diameter (ft)

4.82

4.82

4.82

4.82

4.82

4.82

Unit Weight (pcf)

120

120

130

130

130

130

130


0.89

1.30

3.85

5.04

28.36

55.50

55.50

Internal Friction Angle (°)

30

32.5

43

38

38

45

45

54

TEST DATA 4000H

rr 3500 -H U.

O 2

Ui z -1 I

o o a: o

o UJ - J

3000-

2500-

2000-

1500-

1000-

500-^

NEW MODEL

0 1 2 3 4 5 DEFLECTION AT TOP OF PIER (IN.)

Figure 16: Moment/Deflection Comparison With EPRI Test NO. 8

55

u. I

H Z Ui 2 O

UJ

I

o o Q: o

o Ul a. CL <

4 0 0 0 ^

3500-

3000-

2500-

2000-

1500

1000

CM ^NONLINEAR PREDICTION BYy. FOUR SPRING MODEL (PADLL)

^TEST DATA

UJ 5 0 0

NEW MODEL

0 "n I M11II i| 11M111111'

0 I 2 T 1

5

DEFLECTION AT TOP OF PIER (IN.)

Figure 17: Moment/Deflection Comparison With EPRI Test No. 10

56

Tests At Hager rii-y

There are six full-scale field tests at Hager City.

The soil conditions at the Hager City test site consist of

terrace deposits of sand and gravel within the Mississippi

River Valley. Approximately four feet of a sandy clayey silt

exists on top of the sand and gravel. Field density tests

indicated the dry density in the sand ranged from 100 to 116

pcf with a moisture content of approximately 5%. The dry

density of the clayey silt averaged at 98 pcf with a

moisture content of approximately 17% [12].

The comparison between the test results and model

results are not as accurate as the previous ones. This data

does not contain soil unit weight, angle of internal

friction of soil, and modulus of elasticity of soil for each

layer. Only standard penetration blow counts are available

for each layer and the general description of soil density.

Using Equations (36) and (38), the values of the angle of

internal friction and modulus of elasticity of soil are

computed from the standard penetration blow counts. The

unit weight of the soil is assumed to be 110 pcf [12].

The soil properties at these sites are reported here in

Tables 6 through 11 . The results of analysis for the six

piers using the proposed model are compared with full-scale

test data in Figures 18-23, respectively. Each pier is

57

constants for each layer are linearly interpolated from the

soil data.

TABLE 6

Soil Properties for Hager City Test No. 1

The length embedded in soil is 6 ft

Depth (ft)

-3.5-

—4 "=;

0 . \J


Sandy clayey silt, dark brown to tan, moist, rather loose

Sand,slightly silty, a trace of gravel, fine to medium-grained, reddish brown, moist, medium-dense

Sand,no gravel,fine to medium-grained, reddish brown, moist, loose to medium-dense

Diameter (ft)

6.0

6.0

6.0

Unit Weight (pcf)

110

110

110

110

penetration number (BPF)

8

7

12

12

58

TABLE 7



Depth (ft)

-3.5-

4 t *i . ^

-6.5-

-8.5-

-9.0-


Sandy clayey silt, dark brown to tan, moist, rather loose

Sand,slightly silty, a trace of gravel, fine to medium-grained, reddish brown, moist, medium-dense

Sand, no gravel, fine to medium-grained, reddish brown, moist, loose to medium-dense

Diameter (ft)

6.0

6.0

6.0

6.0

6.0

6.0

Unit Weight (pcf)

110

110

110

110

110

110

110


8

7

12

12

7

8

8

59

TABLE 8



Depth (ft)

-2.0-

A f\ 4 . U

-4.5-

D . U'

0 . u -8.5-

10.0-

11 n 1 1 . U " ••

11.5-

12.0-


Sandy clayey silt, dark brown to tan

Sand,fine to medium-grained, reddish brown,moist,medium dense

Sand, a little gravel,medium to fine-grained, reddish brown, moist, medium dense

Sand, no gravel,fine-grained, brown, moist, medium dense

Sand, no gravel,fine to medium-grained, brown, moist, medium dense

Diameter (ft)

6.5

6.5

6.5

6.5

6.5

6.5

6.5

Unit Weight (pcf)

110

110

110

110

110

110

110


7

10

10

16

13

12

11

60

TABLE 9



Depth (ft)

-2.0-

-4.0-

-5.5-

O.J

-7.5-

10.0-


Silty sand, dark brown to 3.5ft then tan, a few lenses of clay at 5ft, moist, medium-dense

Sand, slightly silty, finegrained, reddish brown, moist, medium-dense

Sand, with some gravel, medium to fine-grained, brown, moist, rather dense

Diameter (ft)

11.0

11.0

11.0

11.0

11.0

11.0

11.0

Unit Weight (pcf)

110

110

110

110

110

110

110

110


23

26

20

23

23

29

29

37

61

TABLE 10



Depth (ft)

-2.0-

-4.0-

-5.5-

-7.5-

10.0-

15.0-


Silty sand, dark brown to 3.5ft then tan, a few lenses of clay at 5ft, moist, medium-dense

Sand, slightly silty, finegrained, reddish brown, moist, medixim-dense

Sand, with some gravel, medium to fine-grained, brown, moist, rather dense

Diameter (ft)

11.0

11.0

11.0

11.0

11.0

11.0

11.0

11.0

Unit Weight (pcf)

110

110

110

110

110

110

110

110

110


23

26

20

23

23

29

29

37

37

62

TABLE 11


The length embedded in soil is 20 ft(broken at 16 ft)

Depth (ft)

-1.5-. *> n. .

...A n

"O . o

-8.0-

-9.0-

10.5-

13.0-

16.0-

17.5-

19.0-

20.0-


Silty sand to sandy silt, dark brown, moist, rather loose

Sandy silt, tan to reddish brown, moist, loose

Sand, slightly silty, finegrained, moist, loose

Sand, mostly medium to finegrained, a little gravel, reddish brown, moist, rather dense

Diameter (ft)

11.0

11.0

11.0

11.0

11.0

11.0

11.0

11.0

11.0

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Figure 18: Moment/Deflect ion Comparison With Hager City Test No. 1

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Figure 19: Moment/Deflection Comparison With Hager City Test No. 2

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68

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CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

A discrete soil spring model is developed here for the

analysis of short rigid piers subjected to large lateral

loads and overturning moments. A stiffness matrix of order

three for the rigid body motion of the pier is used. Springs

representing the bottom resisting moment, bottom friction,

bottom vertical reaction, and side skin friction are used in

addition to the lateral soil springs. The following

conclusions have been reached from this study:

1. The curves of the load versus ground line deflection,

of all full-scale field tests in sandy soils are

nonlinear even at low stress levels.

2. The size of the stiffness matrix for this model is

only 3x3, in contrast to the previously reported

analyses where the size of the stiffness matrix

depended on the number of nodes which were used. The

procedure used here is very efficient for use with

the digital computer.

3. From the test data, we find that the shallow pier

foundations are less efficient when the piers are

69

70

subjected to primarily lateral forces. For better

efficiency, the ratio of depth to shaft diameter

should be 3.0 or more.

4. This model has only three degrees of freedom, but

represents the overall behavior of the soil-pier

interaction.

5. The load-displacement response of the pier obtained

by using only the lateral springs and neglecting the

skin friction and bottom resistances, showed

comparatively poor agreement with field test data.

6. The addition of the resisting forces at the bottom

and on the sides of the pier resulted in a more

realistic model. The results of the analysis of the

model agreed largely with the field test data for

most of the example problems.

7. The addition of a bell to the bottom of the pier

increased the effect of the bottom resisting moment

and the bottom frictional force substantially,

depending on the size of the bell and the soil

strength directly below the bell.

8. In a few cases, the predicted deflections of the

laterally loaded piers were not as close to the

loaded piers were not as close to the actual measured

deflections. This may be due to the error in the

71

assumptions made in developing the p-y curves and

other spring constants and possibly due to the errors

in the prediction of the angle of internal friction

from the standard penetration test results on the

soil.

Recommendations For Future Studies

Based on this research, the writer would like to

recommend the following future studies be made:

1 . As more experimental data become available, the model

should be revised and spring constants should be

revised. For example, the p-y curves need to be

improved in order to determine the initial spring

constants more accurately and to represent behavior

at loads close to the ultimate capacity.

2. A better knowledge and understanding of the soil

reactions of the bottom resisting moment, bottom

friction, and skin friction, through a series of

tests and studies would be helpful to improve the

model.

3. The analysis should be extended for all kinds of

soil.

LIST OF REFERENCES

1 . Alikhanlou, F. A Discrete Model For The Analysis of Short Pier Foundations in Clay, M.S. Thesis, Texas Tech University, August 1981.

2. Bowles, J. E., Foundation Analysis and Design, 2nd ed., McGraw Hill Book Co., New York, 1977, pp. 157-159.

3. Broms, B. B.,"Lateral Resistance of Piles in Cohesionless Soils," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 90, No. SM3, May, 1964, pp. 123-156.

4. Broms, B. B., "Lateral Resistance of Piles in Cohesive Soil," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 90, No. SM2, Proc. Paper 3825, Mar., 1964, pp. 27-63.

5. Davidson, H. L. Laterally Loaded Drilled Pier Research, Volume 2: Research Documentation. EPRI EL-2197, Volume 2, Project 1280-1, Final Report January 1982, pp. 4-73, 4-84, 4-87.

6. Davidson, H. L. Laterally Loaded Drilled Pier Research, Volume 2: Research Documentation. EPRI EL-2197, Volume 2, Project 1280-1, Final Report January 1982, pp. 2-2.



9. Langhaar, H. L., Energy Method in Applied Mechanics, John Wiley and Sons, Inc., New York, N. Y., 1962, pp. 1-33.

10. Matlock, H., "Correlations for Design of Laterally Loaded Piles in Soft Clay," Preprintsf Second Annual Offshore Technology Conference, Houston, Texas, Vol. I, No. OTC 1204, 1970, pp. 1-577-588.

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73

11. Meyerhof, G. G., Penetration Tests and Bearing Capacity of Cohesionless Soils. ASCE, Vol. 82, No. SMI, pp. 1-19.

12. Monahan, D. R. and Fiss, R. A. Evaluation Of Full-Scale Test Results Of Transmission Pole Foundation, GAI Consultants, Inc. 1982 Project 76-527.

13. Peck, R. B., Hanson, W. E. and Thornburn, T. H. Foundation Engineering, 2nd Edition, John Wiley & Sons, 1973, pp. 114, 312.

14. Reese, L. C , Study Guide For The Analysis of Piles Under Lateral Load, Geotechnical Engineering Center, U. of Texas at Austin, May 1980.

15. Reese, L. C., "Laterally Loaded Piles: Program Documentation," Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT4, Proc. Paper 12862, Apr., 1977, pp. 287-305.

16. Reese, L. C , Cox, W. R. and Koop, F. D., "Analysis of Laterally Loaded Piles in Sand," Sixth Annual Offshore Technology Conference, Houston, Texas, 1974.

17. Stott, J. P., "Tests on Materials for Use in Sliding Layers Under Concrete Road Slabs," Civil Engineering and Public Works Review, Vol. 56, 1961, pp. 1297, 1299, 1301 , 1466, 1603, 1605.

18. Teng, W. C. Foundation Design, Prentice-Hall, Inc. Englewood Cliffs, N. J., 1962, pp. 212, 213.

19. Vallabhan, C.V.G. and Alikhanlou, F., Short Rigid Piers in Clays, Journal of the Geotechnical Engineering Division, ASCE, Vol. 108, No. GTIO, Proc. Paper 17410, October, 1982, pp. 1255-1272.

20. Woodward, R. J. Jr., Gardner, W. S. and Greer, D. M. Drilled Pier Foundations, McGraw-Hill Book Company, pp. 64, 65.

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