Load-Deflection response of transversely isotropic piles under lateral loads

*Correspondence to: J. Han, Tensar Earth Technologies, Inc., 5775B Glenridge Drive, Suite 450, Atlanta, GA 30328,U.S.A.

Contract/grant sponsor: U.S. National Science FoundationContract/grant number: CMS 9457549

Received 29 August 1998Copyright ( 2000 John Wiley & Sons, Ltd. Revised 30 August 1999

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS

Int. J. Numer. Anal. Meth. Geomech., 2000; 24:509}529

Load}De#ection response of transverselyisotropic piles under lateral loads

J. Han1 and J. D. Frost21 Tensar Earth Technologies, Inc., 5775B Glenridge Drive, Suite 450, Atlanta, GA 30328, U.S.A.

2 School of Civil and Environmental Engineering, the Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.

SUMMARY

In general, pile materials are assumed to be isotropic during the analysis of the load}de#ection response ofpiles under lateral loads. However, commonly used materials such as reinforced concrete and timber as wellas potentially promising new pile materials such as "ber reinforced polymers are typically transverselyisotropic materials. Experimental studies have shown that transversely isotropic materials have a high ratioof section longitudinal modulus to the section in-plane shear modulus (E

zz/G

xz) compared to the value for

isotropic materials. The high modulus ratio leads to a more signi"cant shear deformation e!ect in beambending. To account for the shear deformation e!ect, the Timoshenko Beam Theory has been adopted inderiving the solutions for the load}de#ection response of transversely isotropic piles under lateral loadsinstead of the Classical (Euler}Bernoulli) Beam Theory. The load}de#ection responses depend on the sheare!ect coe$cient, the lateral soil resistance, the embedment ratio, and the boundary conditions. Thede#ection of the pile, if the shear deformation e!ect is considered, is always larger than if it is neglected.Copyright ( 2000 John Wiley & Sons, Ltd.

KEY WORDS: pile; transversely isotropic material; lateral load; load}de#ection response; Timoshenkobeam; shear deformation

1. INTRODUCTION

In practice, piles are not only used to support vertical loads but in many cases are designed toresist lateral loads or combined lateral and vertical loads. Laterally loaded piles are typicallydesigned based on allowable lateral de#ections rather than ultimate lateral load capacities.1 It iswell recognized that load}de#ection responses of laterally loaded piles depend on pile geometryand material properties, soil conditions, boundary conditions, and loading. Subgrade reactionapproaches including p-y method (e.g. References 2}7) and elastic continuum theory (e.g. Refer-ences 8 and 9) are two major categories of procedures used to analyse the load}de#ectionresponses of laterally loaded piles. Between these two methods, the subgrade reaction approach isconsidered to be versatile and practical for routine analyses.10

NAG*079*Ravi*VVC*BG

Figure 1. Transversely isotropic materials (a) Reinforced concrete or polymer, (b) Timber.

Most of the factors known to a!ect the load}de#ection response of laterally loaded piles havebeen extensively investigated in the past few decades with the exception of the in#uence of pilematerial properties. It is always assumed that the pile material is isotropic and has an elasticmodulus, E, and a Poisson's ratio, l. This assumption is only valid for piles with isotropicmaterials such as steel. Inclusion of steel bars in reinforced concrete, the existence of grains intimber, and the presence of "bres in polymers result in these materials being transverselyisotropic. A transversely isotropic materials is de"ned as a material that has the same propertiesin the x and y directions but di!erent properties in the z direction as shown in Figure 1. Thetransversely isotropic material in a two-dimensional state has four independent elastic constantssuch as E

xx, E

zz, G

xz, and l

xz, in which E

xxis the transverse (x}x or y}y axis) elastic modulus; E

zzis

the longitudinal (z-z axis) elastic modulus; Gxz

is the in-plane (x}z plane) shear modulus; and lxz

isPoisson's ratio in the x}z plane.11

The development and utilization of Fiber Reinforced Polymer (FRP) composites for civilengineering applications has been increasing rapidly in the past few years as a result of extensive

510 J. HAN AND J. D. FROST

NAG=079=Ravi=VVC=BG

Copyright ( 2000 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 2000; 24:509}529

Table I. Structural properties of some FRP composite beams

Material Beam shape Ezz

(GPa) Gxz

(GPa) <f

Ezz/G

xzReference

E-glass/vinylester I-beam 22)5 1)25 0)42 18 12E-glass/vinylester Box 30)5 3)9 0)47 8 13Carbon/epoxy Rectangular 31)65 2)17 0)39 15 14E-glass/polyester I-beam 22)13 1)20 0)4}0)5 18 15E-glass/vinylester I-beam 22)82 1)23 0)4}0)5 19 15E-glass/poylester WF-beam 22)75 0)75 0)4}0)5 30 15E-glass/vinylester WF-beam 21)86 0)88 0)4}0)5 25)6 15

Note: (1) z is oriented along the longitudinal axis of member: (2) Ezz

for steel is 210 GPa: (3) <f

is the "ber volumefraction.

research that has shown that FRP materials have desirable properties including being non-corrosive, non-conductive, lightweight, having a high speci"c strength and sti!ness, exhibitinggood thermal stability, and having a high speci"c damping capacity. In particular, FRP com-posites are potentially more suitable for harsh environments (e.g. contaminated or marineenvironments) than conventional construction materials (steel, concrete, and wood) due to theirchemical and corrosion resistant properties. Results summarized in Table I show that FRPcomposites typically have an elastic modulus which is an order of magnitude less than steel in thelongitudinal direction. In addition, FRP composites have a high ratio of section longitudinalelastic modulus to the section in-plane shear modulus (E

zz/G

xz), in the range of 8 to 30 as opposed

to the value of 2)6 for isotropic materials (assuming a Poisson's ratio of 0)3). The term &section'was used by Bank16 to distinguish the modulus properties determined by the loaded beamde#ection method from those determined by the standard test method using coupon samples cutfrom beams. The data provided in Reference 17 indicated that "ve hardwoods (Balsa, Af.mahogany, Birch, Ash, and Beech) have the E

zz/G

xzratio ranging from 13 to 31.

2. SHEAR DEFORMATION AND TIMOSHENKO BEAM THEORY

As shown in Figure 2, the bending of a beam subjected to transverse loads can be considered theresult of bending moments and shear forces acting on its cross-sections.18 The transversedeformation at any location along the beam consists of a bending deformation and a sheardeformation created by their corresponding moments and forces. The theory that considers thetransverse shear deformation in analysing beams is generally termed as Timoshenko BeamTheory. If shear deformation alone is considered, it is assumed that linear (or planar) elementssuch as ab which are perpendicular to the centreline of the beam in the undeformed state moveonly in the vertical direction and also remain vertical during deformation (Figure 3). The totalslope dw/dz of the centreline resulting from shear deformation and bending deformation can thenbe expressed as the sum of two parts:

dw/dz"h"(z)#h

4(z) (1)

where w is the de#ection of the centreline, h"(z) is the rotation of line elements with respect to the

centreline due to bending only, and h4(z) is the rotation of elements along the centreline due to

shearing only.

TRANSVERSELY ISOTROPIC PILES UNDER LATERAL LOADS 511

NAG=079=Ravi=VVC=BG


Figure 2. Deformation based on Timoshenko beam theory18

Figure 3. Shear deformation component of Timoshenko beam theory19

For isotropic materials, the shear deformation is relatively small compared with the bendingdeformation and can be ignored without causing any signi"cant error under most circumstances.The simpli"ed theory is generally referred as the Classical Beam Theory or Euler}Bernoulli BeamTheory.

As mentioned earlier, transversely isotropic materials have a high ratio of section longitudinalelastic modulus to the section in-plane shear modulus. The shear deformation e!ect becomesmore important. Work performed by dos Reis and Goldman20 indicated that the maximumde#ection at the centre of an FRP composite tube obtained by the Classical Beam Theory wasabout 30 per cent less than that using a "nite element approach, which included a transverse shear


NAG=079=Ravi=VVC=BG


deformation. Therefore, the shear deformation e!ect should be considered when transverselyisotropic beams or piles during bending are dealt with.

To date, almost all solutions for load}de#ection responses of piles have been developed usingthe Classical Beam Theory. It is obvious that the existing solutions may not be appropriate fortransversely isotropic piles. In order to improve the solutions, the Timoshenko Beam Theory isadopted herein in the development of solutions for load}de#ection of laterally loaded trans-versely isotropic piles. Earlier studies,21~23 which computed the static critical (buckling) loadsand natural frequencies of Timoshenko beams on the surface of Winkler or Pasternak founda-tions (subgrade models) are valuable for this study. However, these earlier studies21~23

assumed that Winkler foundations or Pasternak foundations have a constant sti!ness(modulus) of subgrade reaction along the beams but Pasternak foundations have additionalshear interactions. These models are not always valid for piles embedded in soil. For example,Terzaghi et al.24 indicated that sti! overconsolidated clays have an approximately constantsti!ness of subgrade reaction with depth, while sands have a linearly increasing sti!nessof subgrade reaction. The solutions developed in this paper are based on the assumption ofa Timoshenko beam (pile) supported by a series of springs with a linearly increasing sti!ness ofsubgrade reaction with depth.

3. THEORETICAL SOLUTIONS

3.1. Assumptions

An illustrated in Figure 4, a laterally loaded pile is treated as a Timoshenko beam embedded inan elastic foundation with a linearly increasing coe$cient of subgrade reaction with depth. Inorder to study the load}de#ection relationship, the vertically applied loads are assumed to be lessthan the buckling load for the pile}soil system. Additional assumptions are made as follows inorder to simplify the problem:

(1) Only static loads are applied at the top of the pile. The vertically applied load is assumed toact along the centreline of the pile. The laterally applied load is assumed to act in the x}zplane and no torsional moment exists.

(2) The surrounding soil responds as a Winkler medium, which has a series of springs withlinear relationships between soil reaction and lateral displacement. The pile is formed bya transversely isotropic and elastic material, and the unidirectional reinforements areoriented along the z-axis.

(3) The elastic modulus of the pile under compression is equal to the elastic modulus undertension.

(4) The cross-section of the pile is constant along the length of the pile and symmetric about thex}z and y}z planes (Figure 4), and the shape of the cross-section does not change duringpile bending.

(5) The pile bends toward the x direction (Figure 4), and the lateral reaction forces from the soildue to pile bending are symmetric about the x}z plane.

(6) The in#uence of the friction along the pile on the lateral de#ection is expected to be smalland is neglected in this study.

(7) No local buckling occurs along the pile shaft.


NAG=079=Ravi=VVC=BG


Figure 4. Timoshenko Beam embedded in elastic foundation

3.2. General formulation

The general expressions for the total energy and the Euler}Lagrange equations of the pile}soilsystem for a transversely isotropic pile under a vertical load were previously derived by theauthors.25 In order to help the readers to better understand the formulation of the solutions forthis study, a brief review of the previous study is included.

The total energy % for the pile embedded in the elastic foundation is given as

%";#< (2)

where; is the strain energy stored in the pile and< is the work exerted by external forces appliedon the pile.


NAG=079=Ravi=VVC=BG


From strain}displacement relations, the strains can be expressed by

exz"h

s(z) (3)

ezz"

du0(z)

dz!x

dh"(z)

dz"u

0,z!xh

",z(4)

exx"e

xy"e

yy"e

yz"0. (5)

where u0(z) is the deformation resulting from the compression of the beam (pile) under axial loads.

exx

, eyy

, and ezz

are the normal strains in the x, y, and z directions, respectively and exy

, exz

, and eyz

are the shear strains.The strain energy stored in the pile is

;"

1

2: :v

: pijeij

dv"1

2: :v

: (pxz

exz#p

zzezz

) dv (6)

where pzz

is the normal stress in the z direction, pxz

is the shear stress, and v is the volume of thepile.

In Timoshenko's beam equations the transverse shear strain is assumed as the ratio of theaverage shear stress on a cross-section to the product of the shear modulus and the shearcoe$cient k. The stress}strain relationship can be expressed in the following form:

pzz"E

zzezz

(7)

qxz"G

xzexz"kG

xzh4(z) (8)

where Ezz

and Gxz

are the longitudinal modulus and in-plane shear modulus of the pile.The dimensionless coe$cient k is introduced to account for the non-uniformly distributed

shear stress and shear strain over the cross-section.26 Cowper26 obtained shear coe$cients forisotropic beams, and Dharmarajan and McCutchen27 and Bank15 derived shear coe$cients fortransversely isotropic composite beams with di!erent cross-sections. The shear coe$cient, k, fortransversely isotropic beams is a function of four parameters as follows:

k"F(Ezz, G

xz, l

xz, #) (9)

where lxz

is Poisson's ratio in the x}z plane and # is a cross-section shape and geometry factor.Based on the studies conducted by Cowper26 and Bank15, the shear coe$cient k varies from 0)09to 0)90.

By substituting equations (3), (4), (7), and (8) into equation (6) and considering the fact that::A

x dx dy"0 due to the symmetry of the cross-section about the y}z plane, equation (6) yields

;"

1

2 GEzzAP

L

0

(u0,z

)2dz#Ezz

IyyP

L

0

(hb,z

)2dz#kGxz

APL

0

(w,z!h

")2dzH (10)


NAG=079=Ravi=VVC=BG


where A is the cross-sectional area of the pile, Iyy

is the moment of inertia of the cross-sectionabout the y-axis, and ¸ is the pile length. The work exerted by external forces is

<"PL

0C!PeN

zz#

1

2q (z) )w(z)!HwDdz (11)

where H is the lateral force applied at the top of the pile. e6zz

is the strain on the x}y plane, andincludes a small amount of strain resulting from the pile shortened during bending, that is

e6zz"

1

A::ACu0,z#xh

",z#

1

2w2,zDdxdy"u

0,z#

1

2w2,z

(12)

Based on the assumptions, the resistances q(z) from the elastic foundation are a function of thedisplacements w (z), and can be expressed as

q (z)"ksw"gw(h!z), z)h (13a)

q(z)"0, z*h (13b)

where ksis the coe$cient of subgrade reaction, which increases linearly with depth; g, the constant

of subgrade reaction; h, the embedment length of the pile into the ground. The ratio of theembedment length, h, to the pile length, ¸, is termed as the embedment ratio, e.

Substitution of equations (10)}(12) into equation (2) yields

%"

1

2 PL

0

[Ezz

Iyy

(h",z

)2#Gxz

kA(w,z!h

")2] dz#

1

2 PL

0

q (z)w dz

!

1

2 PL

0

P (w,z)2 dz!HwD

z/L(14)

The Euler}Lagrange equation for this case can be easily deduced from equation (14) by takingvariations as d%"0.

Ezz

Iyy

h",zz

#kGxz

A(w,z!h

")"0 (15)

where

hb,zz

"

d2h"

dz2, w

,z"

dw

dz.

3.3. Boundary conditions

It is well recognized that the load}de#ection response of laterally loaded piles is highlydependent on the boundary conditions of the piles in the ground. Various boundary conditionsfor piles are encountered in practice. In this study, four boundary conditions as shown inFigure 5 were selected for study. Piles keyed into bedrock or hard soils may be considered to have"xed tip boundary conditions, while relatively long piles #oating in soft soils or seated on hardsoils may be assumed to have pinned boundary conditions. Similarly, piles keyed into pile


NAG=079=Ravi=VVC=BG


Figure 5. Di!erent boundary conditions for laterally loaded piles1,28

caps/foundations may be considered to have &"xed with sway' conditions. The contribution of tipboundary conditions may also be in#uenced by other factors, such as lateral soil con"nement.

3.4. Solutions for load}deyection responses

For the case with sway top and "xed tip boundary conditions (BC-2), the de#ection functionswhich satisfy the boundary conditions and the Euler}Lagrange solution are obtained as follows:

w"

N+

m/1

Bm A1!cos

mn¸

zB (16)

h""

N+

m/1

mmBmsin

mn¸

z (17)


NAG=079=Ravi=VVC=BG


where

mm"!

mnj2

m2n2#j2

1

¸

(18)

j"SG

xzkA¸2

Ezz

Iyy

(19)

in which j is termed as the shear e!ect coe$cient. Conceptually, the shear deformation of the pileis in"nite if the shear modulus/elastic modulus ratio equals to zero, i.e. j"0. On the other hand,the shear deformation equals to zero when the modulus ratio or the j value approaches in"nity.Therefore, the less the j value is, the more the shear deformation e!ect will be. There is no sheardeformation e!ect when the j value approaches in"nity. The solutions with no shear e!ect areequivalent to those using the Classical Beam Theory. The authors' previous study25 showed thatthe shear e!ect coe$cient j of a circular isotropic pipe pile ranged from 12 to 48 when thelength/radius ratio (¸/r

0) of the pile from 20 to 80. Considering the modulus ratio E

zz/G

xzfor

transversely isotropic piles ranging from 8 to 30, the j values were within the range 4}30. j"10 isselected as an example for most parametric studies in this paper. Extended parametric studiesbased on di!erent j values can be conducted using the formulae provided in Table II.

Using the de#ection functions in equations (16) and (17) and the following minimum potentialenergy principle

L%/LBg"0 (20)

Equation (14) leads to

1

2¸3g2n4E

zzIyy A

g2j2

g2n2#j2B Bg#g¸2 C

3

4e2!

15

8g2n2#

2

g2n2cos gne!

1

8g2n2cos 2gneDB

g

!

1

2¸g2n2PB

g#g¸2 +

mOg

Bm C

(m2#g2)

(m2!g2)2n2#

1

2e2!

1

m2n2!

1

g2n2#

1

m2n2cos mne

#

1

g2n2cos gne!

1

2(m!g)2n2cos(m!g)ne!

1

2(m#g)2n2cos(m#g)neD

"H[1!(!1)g]

g, m"1,2, N (21)

Normalizing the above equation using the dimensionless parameters:

P*"P¸2

n2Ezz

Iyy

, q*"g¸5

n2Ezz

Iyy

and H*"H¸2

n2Ezz

Iyy

equation (21) can be expressed in matrix form as

GB"H (22)


NAG=079=Ravi=VVC=BG


Tab

leII

.Sum

mar

yofboun

dar

yco

ndi

tions,

de#

ection

func

tions,

and

mat

rice

sG

and

H

Bou

ndar

yD

e#ec

tion

funct

ion

Ggg/G

gm/H

g(g

,m"

1,2

,N

,gO

m)

condi

tion

BC

-1w"

N + m/1

Bmsin

2m!

1

2¸nz

Ggg"

1 8(2

g!1)

2n2C

(2g!

1)2j2

(2g!

1)2n2

#4j

2 D#q*Ge2

#

2[co

s(2g

!1)

ne!

1]

(2g!

1)2n2

Hh b"

N + m~1

m mBmco

s2m

!1

2¸n

!

1 8(2

g!1)

2n2

P*

Ggm"

!

q* 2Gco

s(m!

g)ne

(m!

g)2n2

!

cos(

m#

g!1)

ne

(m#

g!1)

2n2

!

16(2

g!1)

(2m!

1)

[(2m

!1)

2!(2

g!1)

2]2n2H

Hg"

!H

*¸(!

1)g

BC

-2w"

N + m/1

BmA1!

cos

mn¸

z BG

gg"

1 2g2

n2A

g2j2

g2n2

#j2B#

q*C3 4

e2!

15

8g2

n2#

2co

sgne

g2n2

!

cos2g

ne

8g2n2D

h ""N + m/1

m mBmsin

mn¸

z!

1 2g2

n&P*.

Ggm"

q*C

m2#

g2

(m2!

g2)2

n2#

1 2e2!

1

m2n2

!

1

g2n2

#

1

m2n2

cosm

ne

#

1

g2n2

cosgne!

1

2(m

!g)

2n2

cos(

m!

g)ne

!

1

2(m

#g)

2n2

cos(

m#

g)ne D

Hg"

H*¸

[1!

(!1)

g]


NAG=079=Ravi=VVC=BG


Tab

leII

.C

ontinu

ed

Bou

ndar

yD

e#ec

tion

funct

ion

Ggg/G

gm/H

g(g

,m"

1,2

,N

,gO

m)

condi

tion

BC

-3w"

N + m/1

BmA1!

cos

2m!

1

2¸nzB

Ggg"

1 8(2

g!1)

2n2C

(2g!

1)2j2

(2g!

1)2n2

#4j

2 D#q*G3 4

e2#

8

(2g!

1)2n2

cosAg!

1 2 Bne

h b"N + m/1

m mBmsin

2m!

1

2¸nz

!

15

2(2

g!1)

2n2

!

1

2(2

g!1)

2n2

cos(

2g!

1)neH!

1 8(2

g!1)

2n2

P*

Ggm"

q*G4[

(2m!

1)2#

(2g!

1)2]

[(2m

!1)

2!(2

g!1)

2]2n2

!

4

(2g!

1)2n2

#

4

(2m!

1)2n2

cos Am

!

1 2 Bne

#

1 2e2!

4

(2m!

1)2n2

#

4

(2g!

1)2n2

cosAg!

1 2 Bne

!

1

2(m

!g)

2n2

cos(

m!

g)ne

!

1

2(m

#g!

1)2n2

cos(

m#

g!1)

neH

Hg"

H*¸

BC

-4w"

B0z

¸

#

N + m/1

Bmsin

mn¸

zG

00"

q*A1 12

e4B!

P*

h ""!

B0

¸

#

N + m/1

m mBm

cos

mn¸

zG

0m"

q*C!

e

m2n2

sinm

ne!

2

m3n

3(c

osm

ne!

1)D

Gg0"

1 2gn

q*[!

gne

sin

gne!

2(c

osgn

e!1)

]!1 2

g2n2

P*

Ggg"

1 2

g4n2

j2g2

n2#

j2#

1

8g2n2

q*[2

g2n2

e2#

cos2g

ne!

1]!

1 2g2

n2P*

Ggm"

1 2q*C!

1

(m!

g)2n

2[c

os(m

!g)

ne!

1]#

1

(m#

g)2n2

[cos(

m#

g)ne

!1]D

H0"

H*¸

and

Hg"

0


NAG=079=Ravi=VVC=BG


The elements for matrix G are given by

Ggg"

1

2g2n2 A

g2j2

g2n2#j2B#q* C3

4e2!

15

8g2n2#

2

g2n2cos gne!

1

8g2n2cos 2gneD (23a)

!

1

2g2n2P*

Ggm"q* C

m2#g2

(m2!g2)2n2#

1

2e2!

1

m2n2!

1

g2n2#

1

m2n2cos mne

#

1

g2n2cos gne!

1

2(m!g)2n2cos(m!g)ne!

1

2(m#g)2n2cos(m#g)neD (23b)

The elements for matrix H are

Hg"H*¸ [1!(!1)g] (24)

The matrix B can be solved from equation (22) by using equations (23) and (24) and then thede#ections of the pile can be computed according to equation (16). Similar solutions are providedin Table II for other boundary conditions.

4. PARAMETRIC STUDY OF INFLUENTIAL FACTORS

According to Matlock and Reese,4,5 the normalized lateral de#ection is de"ned as

w*"w )5J(E

zzIyy)2g3

H(25)

where w* is the normalized lateral de#ection and other terms are as previously de"ned.As noted earlier, the load}de#ection responses of laterally loaded piles are dependent on

a variety of factors. The in#uences of these factors on the load}de#ection responses are discussedin the following parametric studies for the solutions developed in this paper.

4.1. Normalized lateral force H*

Figure 6 shows that the normalized lateral force H* has no in#uence on the normalized lateralde#ection w*. As expected from the assumption of a linear pile}soil system, the lateral de#ectionw has a linear relationship with the lateral force H based on equation (25). P*"0 indicates thatno vertical loads are considered.

4.2. Shear ewect coezcient j

The results in Figure 7 clearly show that the shear e!ect coe$cient j plays an important role inthe normalized lateral de#ection w*, especially when j is relatively small. The normalized lateralde#ection w* decreases with an increase of j. For this speci"c situation, the in#uence of j may beneglected when j is larger than about 20. Based on the results previously presented by the


NAG=079=Ravi=VVC=BG


Figure 6. In#uence of normalized lateral force H* on normalized lateral de#ection w* at pile head

authors,25 a circular pipe pile with a modulus ratio Ezz/G

xz"20 and a shear e!ect coe$cient

j"20 would have a length to radius ratio ¸/r0

of the pile equivalent to 80. In other words, theshear deformation e!ect on the lateral deformation of the pile can be neglected for a pile with the¸/r

0ratio more than 80.

4.3. Normalized soil resistance q*

Based on Broms's de"nition,29 a pile with Z.!9

)2 is considered as a short rigid pile and onewith Z

.!9*4 as a long #exible pile, in which

Z.!9

"¸/¹ (26)

¹"(EIyy/g)1@5 (27)


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Figure 7. In#uence of shear e!ect coe$cient j on normalized lateral de#ection w*

where E is the elastic modulus of an isotropic pile and ¸ is the pile length. The relationshipbetween Z

.!9and q* can be established mathematically as

q*"n2Z5.!9

(28)

The range for q* values in Figure 6 of 0 to 400 corresponds the Z.!9

values of 0 to 5)2 (i.e. fromrigid to #exible piles). q*"200, which is adopted in most of the parametric studies, correspondsto a Z

.!9value of 4)6, which would result in the pile being classi"ed as a long #exible pile. As

shown in Figure 8, the normalized lateral de#ection w* varies signi"cantly with the normalizedsoil resistance q* at a low q*. It is apparent that piles with boundary conditions BC-1 and BC-4have de#ection modes changing from one of rotation of rigid piles to one of bending of #exiblepiles as the q* value increases. However, the e!ect of q* on w* becomes less important as q*increases.


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Figure 8. In#uence of normalized soil resistance q* on normalized lateral de#ection w*

4.4. Normalized vertical load, P*

As shown in Figure 9, the in#uence of the normalized vertical load P* on the normalized lateralde#ection becomes more signi"cant as P* increases. As P* approaches the normalized criticalload P*

#3, the normalized lateral de#ection w* increases exponentially. The P*

#3values were

determined using the solutions previously presented by the authors.25 The lateral de#ection of thepile at 50% of the critical load is almost double that for the same pile without the vertical load forfree pile head cases and 67 per cent larger than that without the vertical load for "xed pile headcases.

4.5. Embedment ratio, e

Figure 10 shows that the normalized lateral de#ection w* decreases with an increase of theembedment ratio e since the pile with a higher embedment ratio e has a longer portion of the pile


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Figure 9. In#uence of normalized vertical load P* on normalized lateral de#ection w*

in the ground and can gain more support from the surrounding soil. The piles with BC-1 andBC-3 have the same w* at e"0 (the whole pile is in the air) because they have the same boundaryconditions with the exception that the locations of the top condition and the tip condition areexchanged.

4.6. Boundary conditions

As shown in Figures 6}10, the pile response is related to the pile top condition. BC-1 and BC-2have a "xed top condition while BC-3 and BC-4 have a free top condition. Since a "xed topcondition has more constraint than a free top condition, the pile under BC-1 or BC-2 has a lowernormalized lateral de#ection w* than the pile under BC-3 or BC-4. The lateral de#ection of thepile depends not only on the pile top condition but also on the pile tip condition. The e!ect of thepile tip condition becomes less important when the shear e!ect coe$cient or/and the lateral soilresistance or/and the embedment ratio increase.


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Figure 10. In#uence of embedment ratio e on normalized lateral de#ection w*

5. THEORETICAL COMPARISON FOR TWO SPECIAL CASES

Since the solutions for a Timoshenko beam embedded in an elastic foundation subject to lateralforces are not available in the literature, the comparison can only be made based on availableinformation for a pile assumed as a classical beam. The results obtained by Matlock and Reese4,5are adopted herein as examples for this comparison. The modi"cation of the solutions obtained inthis study to those for a classical beam can be accomplished by assuming the shear e!ectcoe$cient j equal to in"nity. The comparison of Figures 11 and 12 shows that they are in goodagreement, where Z is the depth coe$cient de"ned as

Z"z/¹"z (Ezz

Iyy

/g)~1@5 (29)

6. CONCLUSIONS

By considering the shear deformation e!ect using Timoshenko Beam Theory, solutions forload}de#ection responses of transversely isotropic piles under lateral loads have been presented


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Figure 11. Normalized lateral de#ection for a free-headed pile in cohensionless soil without shear e!ect

Figure 12. Normalized lateral de#ection for a "xed-headed pile in cohensionless soil without shear e!ect


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in this paper. For four pile models with various boundary conditions, comprehensive parametricstudies were conducted to investigate the e!ect of the lateral force, the vertical load, the sheare!ect coe$cient, the lateral soil resistance, and the embedment ratio on the lateral de#ection.Since linear elastic behaviour of the soil and the pile material was considered, the lateralde#ection had a linear relationship with the lateral force. The in#uence of the vertical load on thelateral de#ection became more signi"cant as the vertical load increased. When the vertical loadcontinued to increase and approached the critical load, the lateral de#ection increased exponenti-ally. The lateral de#ection of the pile at 50 per cent of the critical load was almost double thatwithout the vertical load for free pile head cases and 67 per cent larger than that without thevertical load for "xed pile head cases. The shear e!ect coe$cient, especially when relatively small,played an important role in in#uencing the lateral de#ection. As the shear e!ect coe$cientincreased, the lateral pile de#ection decreased. An increase in the lateral soil resistance from smallto large values resulted in a change in the de#ection mode from rigid pile rotation to #exible pilebending. Both top and tip "xity conditions controlled the de#ection of the pile for cases withrelatively small lateral soil resistance. For cases with large lateral soil resistance, however, thelateral de#ection mainly depended on the pile top "xity condition. The e!ect of the pile tip "xitycondition became less important when the shear e!ect coe$cient, or the lateral soil resistance, orthe embedment ratio increased.

ACKNOWLEDGEMENTS

The research presented in this paper was supported by the U.S. National Science FoundationGrantdCMS 9457549. This support is gratefully acknowledged.

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Load-Deflection response of transversely isotropic piles under lateral loads

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