A framework for analysis of piles with rectangular cross section

A Framework for Analysis of Piles with Rectangular Cross Section

Dipanjan Basu1, C Eng., M.ASCE, Rodrigo Salgado2, P.E., F.ASCE, and Monica Prezzi3, A.M.ASCE

1Assistant Professor, Department of Civil and Environmental Engineering, University of Waterloo, 200 University Avenue West, Waterloo, N2L 3G1, Canada; [email protected] 2Professor, School of Civil Engineering, Purdue University, 550 Stadium Mall Drive, West Lafayette, IN 47907; [email protected] 3Professor, School of Civil Engineering, Purdue University, 550 Stadium Mall Drive, West Lafayette, IN 47907; [email protected] ABSTRACT: An analysis framework is presented for piles with rectangular cross section. Existing analysis and design methods are mostly applicable for piles with circular cross section. For piles with cross section other than a circle, an equivalent circle is often assumed and the methods for circular piles are applied. The newly developed framework explicitly takes into account the rectangular cross section and produces the response of piles under axial and lateral loads. A rational soil displacement field surrounding the rectangular cross section of the pile is assumed, and the total potential energy of the loaded pile-soil system is considered. The potential energy is minimized using calculus of variations to obtain the differential equations governing the pile and soil displacements. Closed-form solutions are obtained for pile settlement and axial force in the case of axially loaded piles. For laterally loaded piles, closed-form solutions are obtained for lateral deflection, slope of the deflected curve, pile bending moment and shear force. The input parameters needed for the analysis are the pile geometry, applied load and the elastic constants of the soil and pile. The new analysis framework produces results in seconds and has the accuracy of equivalent three-dimensional finite element analysis. INTRODUCTION

Existing analysis and design methods for piles are mostly developed for circular cross sections (Dennis and Olson 1983a, b). Although piles with circular cross sections (e.g., drilled shafts, pipe piles, and CFA piles) are common, there are a large number of piles used in practice that have rectangular cross sections (e.g., precast concrete piles and barrettes). It is a common practice to convert the rectangular cross section into equivalent circle for the purpose of analysis and design. However, such conversions are approximate and more accurate results can be obtained if the actual rectangular cross section of the pile can be retained in the analysis. In this paper, a framework for analysis of axially loaded and laterally loaded piles with rectangular cross sections is presented. The analysis is developed following the variational principles of mechanics for an assumed soil displacement field surrounding the rectangular cross section. Closed-form solutions are obtained for pile settlement and axial force in the case of axially loaded piles. For laterally loaded piles, closed-form solutions are obtained for lateral deflection, slope of the deflected curve, pile bending moment and shear force.

Settlement analysis of axially loaded piles can be broadly classified into load-transfer methods and continuum-based methods. The load-transfer methods (Seed and Reese 1957, Coyle and Reese 1966, Randolph and Wroth 1978, Armaleh and Desai 1987, Guo and Randolph 1997,

558

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Mylonakis 2001) essentially model the soil as a series of Winkler springs with spring constants defined in a variety of ways (e.g., t-z curves). Approximate analytical or numerical solutions of pile settlement have been obtained using this approach. In the continuum approach, the soil is represented as a continuum, and numerical techniques such as the boundary integral method, the finite layer method or the finite element method are used to obtain solutions (Poulos and Davis 1968, Butterfield and Banerjee 1971, Randolph and Wroth 1978, Rajapakse 1990).

Analysis of laterally loaded piles can also be classified into load-transfer methods and continuum-based methods. The load-transfer methods typically include the p-y method and other similar approaches in which the lateral soil resistance is represented by Winkler springs, and approximate analytical or numerical solutions of pile response are obtained (Reese and Matlock 1956, Broms 1964, Matlock 1970, Reese et al. 1974, 1975, O’Neill et al. 1990). The continuum-based methods typically include the boundary element method, the finite difference method or the finite element method (Poulos 1971a, b; Banerjee and Davies 1978, Randolph 1981, Verruijt and Kooijman 1989, Trochanis et al. 1991, Ng and Zhang 2001).

The load-transfer (spring-based) methods mentioned above are typically used in routine projects because of their ease of use and because of the speed with which results are obtained. However, these methods may not always produce accurate results because they are developed by calibrating numerical results with the results of a limited number of field tests performed mostly on circular cross section piles, and do not explicitly take into account the pile cross section shape. The continuum-based numerical methods, on the other hand, are theoretically more rigorous and can explicitly take into account the shape of the pile cross section. In spite of that, the use of continuum-based methods is rather limited, both in research and in practice, because they require specially trained personnel and are often time consuming. Recently, Basu et al. (2008) and Basu and Salgado (2008) developed continuum-based analysis of rectangular cross section piles subjected to axial and lateral loads, respectively, which are integrated in a unified framework and presented in this paper. The advantages of this analysis framework are that it is rigorous, explicitly captures the rectangular pile cross section, and produces results in seconds.

ANALYSIS FRAMEWORK Problem Definition

A pile with a rectangular cross section with dimensions 2a 2b and length Lp is considered embedded in a layered soil medium containing n layers (Figure 1). Each soil layer extends to infinity in all horizontal directions, and the bottom (nth) layer extends to infinity in the downward direction. The depth to the base of any intermediate layer i is Hi, which implies that the thickness of the ith layer is Hi Hi1 with H0 = 0. The top of the pile (head) is at the level of the ground surface, and the bottom (base) is embedded in the nth layer. The pile is subjected to a vertical axial force Fv at the head for the case of an axially loaded pile (Case 1) and to a horizontal transverse force Fh and moment Mo at the head for the case of a laterally loaded pile (Case 2). Fh and Mo are orthogonal as vectors (Figure 1); the vector Fh acts perpendicular to the face with a width of 2b, and the vector Mo acts perpendicular to the face with a width of 2a. A rectangular Cartesian coordinate system is chosen with its origin located at the center of the pile head such that the positive z-axis points downward and coincides with the pile axis and the vertical force Fv. The positive x-axis is aligned in the direction of the horizontal force vector Fh, and the moment vector Mo acts in the direction of the negative y-axis (Figure 1).

The soil medium is assumed to be elastic, isotropic and homogeneous within each layer, with Lame’s constants si and Gsi for the ith layer. There is no slippage or separation between the pile and the surrounding soil or between the soil layers. For the case of axially loaded pile (Case 1), the pile behaves as a column (rod) with an axial stiffness EpAp (Ep is the Young’s modulus of pile and Ap = 2a 2b is the area of pile cross section). For the case of laterally loaded pile (Case

FROM SOIL BEHAVIOR FUNDAMENTALS TO INNOVATIONS IN GEOTECHNICAL ENGINEERING 559


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2), the pile behaves as an Euler-Bernoulli beam with a flexural rigidity EpIp (Ip = 4ba3/3 is the second moment of inertia of the pile cross section for bending in the x-z plane).

Soil Displacement, Strain and Stress Fields

A soil displacement field, compatible with the rectangular geometry of the pile cross section, is assumed around the pile (Figure 2). A separable variable technique is used to describe the soil displacement so that the variations of the displacement in each of the coordinate directions are uncoupled. For the axially loaded pile (Case 1), it is assumed that the horizontal soil displacements ux and uy are negligible (i.e., ux = 0 and uy = 0) and that the vertical soil displacement uz is given by

( ) ( )v vz x yu u z x y (1a)

Figure 1. Axially and laterally loaded pile. Figure 2. Displacement and stresses in soil. where u(z) is a displacement function (with a dimension of length) varying with depth z, which describes the axial pile settlement; ( )v

x x is a dimensionless displacement function varying

along x; and ( )vy y is a dimensionless displacement function varying along y. These functions

are set equal to one at the pile-soil interface (so that the soil displacement uz becomes equal to the pile displacement u at the interface), which ensures displacement continuity (strain compatibility) between the pile and soil. The dimensionless functions are set equal to zero at infinite horizontal distance from the pile, which ensures that the soil displacements due to pile movement decrease with increase in horizontal distance from the pile and become negligible at sufficiently large horizontal distance.

For the laterally loaded pile (Case 2), it is assumed that the displacements in the z (vertical) and y directions are negligible (i.e., uz = 0 and uy = 0), and the horizontal displacement in the x direction is given by

y

x

Fh

2a

2b

Mo (Case 2)

(Case 1) Fv

2a

xMo

…

…

…

…

Hn‐1

Hn‐2

Hi

Hi‐1

H1 H2

Lp

Layer 1

Layer 2

Layer i

Layer n‐1

Layer n

z

Fh

Pile

Fv

Pile

xy

z

Displacement at a

Point in Soil

ux uz

uy

Stresses acting on a

soil element

x

yz

xy

xy

xz x

z

yz xz

FROM SOIL BEHAVIOR FUNDAMENTALS TO INNOVATIONS IN GEOTECHNICAL ENGINEERING560


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( ) ( )( ) ( ) ( )h hx x yu w z x y (1b)

where w(z) is a displacement function (with a dimension of length) varying with depth z, which describes the lateral pile deflection; and ( )h

x x and ( )hy y are dimensionless displacement

functions varying along x and y that ensures perfect contact between the pile and soil at the interface (displacement continuity) and describe how the displacements in the soil mass (due to pile deflection) decrease with an increase in horizontal distance from the pile ( ( )h

x x and

( )hy y are set equal to one at the pile-soil interface and equal to zero at an infinite horizontal

distance from the pile). The strain fields in the soil can be derived from the displacement fields (with the

compression positive sign convention) as

xx yy zz xy xz yz

( )( )( ) ( ) ( ) ( )1 1

0 0 02 2

vvyv v v vx

x y y x

ddduu u

dz dx dy

(2a)

for axially loaded piles and as

xx yy zz xy xz yz

( )( )

( ) ( ) ( ) ( )( )( ) 1 1 ( )( ) ( ) 0 0 ( ) ( ) ( ) ( ) 0

2 2

hhyh h h hx

y x x y

d yd x dw zw z y w z x x y

dx dx dz

(2b)

for laterally loaded piles, where pq (= , ,[ ] / 2p q q pu u ) is the strain tensor (the indices p and q can

be x, y or z and, q in the subscript means differentiation with respect to q). The corresponding stress fields can be obtained by using the elastic constitutive relationship, expressed in terms of indicial notation as

2pq s pq kk s pqG (3)

where pq is the stress tensor, pq is the Kronecker’s delta (summation is implied by repetition of index).

Potential Energy of Pile-Soil System and its Minimization

Using the above stress-strain fields, the potential energy of the pile-soil system can be quantified as

0

0

2

00

22 2 ( )2 2 2( ) ( ) 2 ( )

0

2( )22 ( )

0

1 1

2 2

1 12

2 2

p

p

L

p p pq pq v z

L vv v vx

p p s s x y s y

vyv

s y a z

duE A dz d F u

dz

ddu duE A dz G G u

dz dz dx

dG u d F u

dy

(4a)

for axially loaded piles, and as

0

0

22

2 000

222 ( )( )22 22 ( ) 2 ( )

20

1 1

2 2

1 12

2 2

p

p

L

p p pq pq h ozz

L hhyh hx

p p s s y s x

d w dwE I dz d F w M

dz dz

ddd wE I dz G w G w

dz dx dy



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2

2 2( ) ( )

00

h hs x y h oz

z

dw dwG d F w M

dz dz

(4b)

for laterally loaded piles. In the right hand side of equations (4a) and (4b), the first integral represents the internal potential energy of the pile, the second integral represents the internal potential energy of the soil, and the remaining term(s) represent(s) the external potential energy. In the second integral, the soil domain 0 extends from to + in the x and y directions and from 0 to + in the z direction, but excludes the volume Lp 2a 2b occupied by the pile.

The total potential energy is minimized to obtain the equilibrium configuration of the pile-soil system under the externally acting forces. This is achieved by setting the first variation of the total potential energy equal to zero (i.e., ) and manipulating the equation using calculus of variations, which produces an equation of the form:

0

( ) ( ) ( ) ( ) ( ) ( ) 0i j k

x x y y x x y yA v v B C d D v v E F

(5)

where the function v is to be replaced by u for axially loaded pile and by w for laterally loaded piles, x is ( )v

x or ( )hx and y is ( )v

y or ( )hy , 0 represents the domain and i represents a

boundary. In the above equations, A(), B(), C(), D(), E() and F() are differential operators. In order for the above equation to be valid, the individual terms must be equal to zero, i.e., A(v)v = 0, B(x)x = 0, C(y)y = 0, D(v)v = 0, E(x)x = 0 and F(y)y = 0. These individual terms can each be zero if the functions v, x and y are known a priori (in which case, their respective variations are zero, i.e., v = 0, x = 0 and y = 0) or if the differential operators are each equal to zero, leading to equations of the form: A(v) = 0, B(x) = 0, C(y) = 0, D(v) = 0, E(x) = 0 and F(y) = 0. Within the domain 0 and over part of the boundaries i (called the natural

boundaries), the functions v, x, and y are not known a priori and, therefore, the equations represented by A(v) = 0, B(x) = 0 and C(y) = 0 give the differential equations for the domain 0and the equations D(v) = 0, E(x) = 0 and F(y) = 0 give the natural boundary conditions for the natural boundaries. At the remaining portions of the boundaries i (called the essential boundaries), v, x, and y are known or prescribed and automatically satisfy equation (5). Since these differential equations and boundary conditions of the displacement functions v(z), x(x) and y(y) are obtained by satisfying the minimum potential energy, solutions of these equations give the equilibrium configuration of the pile-soil system. Differential Equations and Solutions of Soil Displacement Functions

The differential equations governing the functions ( )vx and ( )h

x have the same mathematical form and can be generically written as

22

20x x

xp

d

dx r

(6)

where x = ( )vx and ( )v

x x for axially loaded piles, x = ( )hx and ( )h

x x for laterally loaded

piles, and pr ab . The differential equations governing the functions ( )vy and ( )h

y also have

the same mathematical form as that of equation (6) and can be generically represented by replacing the variables x and x by y and y, respectively. The boundary conditions of x are: x = 0 at x = and x = 1 for a x a. The boundary conditions of y are also mathematically similar: y = 0 at y = and y = 1 for b x b. Therefore, the solutions of x and y have the same mathematical form, and are given by



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;

1;

;

xp

xp

x a

r

x

x a

r

e x a

a x a

e a x

(7a)

;

1;

;

yp

yp

y b

r

y

y b

r

e y b

b y b

e b y

(7b)

The dimensionless parameters ( )vx and ( )v

y for axially loaded piles are given by

1

2( )

( )2 2 2 1

( )11

1

2 22

i

i

vx H vn

p p nsi i sn n vz

iy y nH

r r tb G u dz b G u

k

1

1

2 ( )2

( )11 1

2 ( )2 2

1

2 2 28

i

i

i

i

H vnp i n

si si sn sn n vziy nH

Hv np y

si i sn nip H

r du kb G dz G u

dz t

LG u dz G u

r

( )

1( )1 2

vn

vzn

t

k

(8a)

1

2( )

( )2 2 2 1

( ) ( ) ( )11

1

2 22

i

i

vy H vn

p p nsi i sn nv v vz

ix x nH

r r ta G u dz a G u

k

1

1

2 ( )2

( ) ( )11 1

2 ( )2

1

2 2 28

i

i

i

i

H vnp i n

si si sn sn nv vzix nH

Hv np x

si iip H

r du ka G dz G u

dz t

LG u dz G

r

( )

2 1( )1 2

vn

sn n vzn

tu

k

(8b)

where p pL r . The dimensionless parameters ( )hx and ( )h

y for laterally loaded piles are given

by

1

2( )

( )2 2 2 1

( ) ( ) ( )11

1

2 2 2 22

i

i

hx H hn

p p nsi si i sn sn nh h hz

iy y nH

r r tb G w dz b G w

k

1 1

2 2 ( )2

( )1 1

(2

( )1

2

2

i i

i i

H Hhn np p yi

si si ihi iy pH H

hp n

sn n hzy

r Ldwb G dz G w dz

dz r

r kG w b

2 ( )) ( )1

( ) ( )18 2

h hp y n

h hn p n

L t

t r k

(9a)



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1

2( )

( )2 2 2 1

( ) ( ) ( )11

1

2 22

i

i

hy H hn

p p nsi i sn nh h hz

ix x nH

r r ta G w dz a G w

k

1 1

2 2 ( )2

( )1 1

2 ( )( )2

( ) ( )11

2 2

2 28

i i

i i

H Hhn np p xi

si si si ihi ix pH H

hhp p xn

sn n sn snh hzx n p

r Ldwa G dz G w dz

dz r

r Lka G w G

t r

( )2 1

( )1 2

hn

n hzn

tw

k

(9b)

In the above equations, the pile displacements u(z) and w(z) are expressed in terms of the normalized variables / pu u L , / pw w L and / pz z L with the subscript i associated with ũ and

w̃ describing the respective normalized functions within the ith layer; /i i pH H L ; and ( )vnk , ( )

1v

nt ,

( )hnk and ( )

1h

nt are normalized soil parameters defined later. Note that, for the purpose of the

analysis, the nth (bottom) layer is split into two parts, with the part below the pile denoted by the

subscript n+1; therefore, (i.e., 1)n p nH L H and 1 1(or )n nH H .

Differential Equations and Solutions of Pile Displacement

The differential equation governing the normalized axial pile displacement is given by:

2

( ) ( )2

1 2 0v vii i i

d ut k u

dz

(10)

with the following boundary conditions:

( ) 111 2 v

v

dut F

dz

(at the pile head, i.e., at 0z z ) (11)

1i iu u (at the interface between adjacent layers, i.e., at z = Hi or iz H ) (12a)

( ) ( ) 111 2 1 2v vi i

i i

du dut t

dz dz

(at the layer interfaces, i.e., at z = Hi or iz H ) (12b)

( ) ( ) ( )11 2 2 0v v vn

n n n n

dut k t u

dz

(at the pile base, i.e., at z = Lp or 1z ) (13)

The differential equation governing the normalized lateral pile displacement is given by: 4 2

( ) ( )4 2

2 0h hi ii i i

d w d wt k w

dz dz

(14)

The corresponding boundary conditions are: 3

( )1 113

2 hh

d w dwt F

dz dz

(15a)

and

1 0dw

dz

(applicable if a cap is present at the pile head) (15b)

or 2

12

0o

d wM

dz

(applicable for a free-head pile) (15c)

at the pile head (i.e., at 0z z );

1i iw w (16a)



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d.

1i idw dw

dz dz

(16b)

3 3( ) ( )1 1

13 32 2h hi i i i

i i

d w dw d w dwt t

dz dz dz dz

(16c)

2 21

2 2i id w d w

dz dz

(16d)

at the interface between two adjacent layers (i.e., at or i iz H z H ); and 3

( ) ( ) ( )13

2 2 0h h hn nn n n n

d w dwt k t w

dz dz

(17a)

2

20nd w

dz

(17b)

at the pile base (i.e., at or 1pz L z ).

The normalized applied forces and moment in the above equations are defined as ( )v v p pF F E A , 2 ( )h h p p pF F L E I and ( )o o p p pM M L E I . The normalized soil parameters

are defined as

2

( ) ( ) ( ) ( )

( )

( ) ( )

21 1 2 2; 1, 2, ...,

2

212 2 ; 1

2

si si p

v v v vp p x y x p y pv

i

p psi si

v vp p x y

G r b ai n

E A r rt

r rGa b i n

E A

(18a)

2 ( )( )( )

( ) ( )2 2

vvsi p p p yv x

i v vp p p y x p

G L r rk b a

E A r r

(18b)

2 2

( ) ( ) ( ) ( )

( )

2

( ) ( )

1 2 2; = 1, 2, .....

2

2 2 ; = 12

p si p

h h h hp p x y x p y ph

i

si p p p

h hp p x y

r G L b ai n

E I r rt

G L r ra b i n

E I

(19a)

4 ( )( )

( )( ) ( )

2 2 2hh

p p y ph xi si si sih h

p p p y p x

L r rk G b G a

E I r r

(19b)

The governing differential equation for axially loaded pile resembles that of a column (or rod) supported by an elastic foundation (i.e., elastic soil mass). The soil parameter ( )vt accounts for the resistance of the soil due to vertical compression, while the parameter ( )vk represents the resistance of soil against shearing in the vertical direction. The boundary condition at the head ensures that the external vertical force is balanced by the resistive axial force in the pile. The boundary conditions at the interfaces of the adjacent layers ensure the continuity of displacements and axial forces. The boundary condition at the pile base ensures that the axial force in the pile infinitesimally above the base is equal to that infinitesimally below it.

The governing differential equation for laterally loaded pile resembles that of an Euler-Bernoulli beam supported by an elastic foundation (i.e., elastic soil mass). The parameter ( )hk is analogous to the Winkler spring constant and determines the compressive resistance of soil against pile (beam) movement. The parameter ( )ht determines the soil resistance due to shearing in the horizontal direction. The boundary conditions at the pile head ensures that the applied



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horizontal force is equal to the shear force of the pile (which comprises of the shear resistance of the pile section and the horizontal shear resistance of soil) and that either the slope is zero (which is a reasonable assumption if a pile cap is present) or the applied moment is equal to the pile bending moment (if the head is free to rotate). The boundary conditions at the interface of any two layers imply that the pile deflection, slope, bending moment and shear force in the adjacent layers have the same magnitude at the interface. The boundary conditions at the pile base states that the pile bending moment is zero and that the shear force infinitesimally above the pile base is equal to that infinitesimally below it.

The general solution of equation (10) for the axially loaded pile is given by: ( ) ( )

( ) ( )1 2( ) ( )

( ) sinh cosh1 2 1 2

v vi ii i

i v vi i

k ku z C z C z

t t

(20)

The integration constants ( )1

iC and ( )2

iC for any layer i can be determined using the relevant boundary conditions.

The general solution of equation (14) for the laterally loaded pile is given by: ( ) ( ) ( ) ( )1 1 2 2 3 3 4 4( ) i i i i

iw z C C C C (21)

where, ( )1

iC , ( )2

iC , ( )3

iC and ( )4

iC are integration constants of the ith layer; 1 = sinh( )cos( )i iz z ,

2 = cosh( )cos( )i iz z , 3 = cosh( )sin( )i iz z and 4 = sinh( )sin( )i iz z (i = [½{ ( )hik +

( )hit }] and i = [½{ ( )h

ik ( )hit }]) if ( ) ( )h h

i ik t , and 1 = sinh( )i z , 2 = cosh( )i z , 3 =

sinh( )i z and 4 = sinh( )i z (i = [ ( )hit + {( ( )h

it )2 ( )hik }] and i = [ ( )h

it {( ( )hit )2 ( )h

ik }])

if ( ) ( )h hi ik t . The integration constants can be determined using the boundary conditions.

Adjustment of Soil Modulus

Modifications of the Lame’s constants of the elastic soil are necessary to counterbalance the excessively stiff pile response obtained by solving the above equations because the assumption of zero horizontal displacements in the case of axially loaded piles and that of zero displacements in the y and z directions in the case of laterally loaded piles artificially restrain the pile movement. It was found by comparing the results of this analysis with those of equivalent finite element analysis that setting si = 0 (which is equivalent to setting the soil Poisson’s ratio si = 0) removed the artificial stiffness in the system and that the effect of soil Poisson’s ratio can be indirectly

taken into account by replacing the shear modulus Gsi by * 20.6 1 1.25si si siG G for the case of

axially loaded piles and by * 0.7 1 0.75si si siG G for the case of laterally loaded piles. These

adjustments were made before the final results were obtained.

Solution Algorithm The dimensionless parameters x and y are required to be known in order to estimate the

parameters ki and ti without which pile displacements cannot be determined. The parameters x and y, in turn, depend on the pile displacement. Hence, an iterative solution scheme is required. In the first iteration, initial guesses on x and y are made and, using these guessed values, ki and ti are determined and pile displacements are calculated. Using the calculated pile displacements, new values of x and y are calculated and compared with the assumed initial values of x and y, respectively. If the differences are greater than a tolerance limit, iterations are continued with the calculated values of x and y taken as the new guesses. Successive iterations are continued until the values of both x and y obtained from two consecutive iterations fall below the tolerance limit. A value of one is chosen as the initial guess for both x and y, and the tolerance limit



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prescribed on both x and y between the ith and (i+1)th iteration is | 1/i

x y /i

x y | < 0.001 (this

tolerance limit was determined by a systematic convergence study). In most problems, solutions are obtained in three iterations.

RESULTS

The use of the analysis framework is demonstrated with the use of examples. In the first

example, a 15-m-long axially loaded precast pile with a square cross section of 0.5 m 0.5 m is considered. The pile is embedded in a four-layer soil deposit with H1 = 2 m, H2 = 5 m and H3 = 10 m; Es1 = 10 MPa, Es2 = 15 MPa, Es3 = 30 MPa and Es4 = 100 MPa; s1 = 0.4, s2 = 0.35, s3 = 0.3 and s4 = 0.15 (Esi and si are the Young’s modulus and Poisson’s ratio of the ith soil layer, respectively; 2(1 )si si siG E , [(1 )(1 2 )]si si si si siE ). The pile is subjected to an

axial force Fv = 3000 kN at the head. Figure 3(a) shows the pile deflection as a function of depth, as obtained from our analysis and equivalent three-dimensional (3D) linear finite element (FE) analysis performed using Abaqus. It is evident that the match between the results of the present analysis and that of the FE analysis is good the difference in the pile settlement at the head is about 0.5%.

For the second example, a laterally loaded rectangular barrette, 40 m long, 2.8 m 0.8 m in cross section with Ep = 25 106 kPa is considered embedded in a four-layer soil profile with H1 = 1.5 m, H2 = 3.5 m, and H3 = 8.5 m; Es1 = 20 MPa, Es2 = 25 MPa, Es3 = 40 MPa and Es4 = 80 MPa; and s1 = 0.35, s2 = 0.3, s3 = 0.25 and s4 = 0.2. A horizontal force Fh = 3000 kN acts at the pile head along the longer dimension of the pile cross section (i.e., 2a = 2.8 m and 2b = 0.8 m). Figure 3(b) shows the pile deflection profiles, as obtained from the present analysis and 3D linear FE analysis performed using Abaqus. The deflection profiles match well; the difference in the head deflection is equal to 5%.

(a) (b) Figure 3. (a) Settlement of a 15-m long precast pile and (b) lateral deflection of a 40-m long barrette.

Several other examples have been solved and compared with equivalent 3D FE analysis, and

the ability of the analysis framework to predict accurate pile response have been established the details are available in Basu et al. (2008) and Basu and Salgado (2008). Moreover,

0 2 4 6 8 10

Axial Pile Deflection (mm)

15

10

5

0

De

pth

, z

(m)

FE Analysis

Analytical

-4 0 4 8 12 16 20

Lateral Pile Deflection (mm)

40

30

20

10

0

De

pth

, z

(m)

FE AnalysisAnalytical



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comparisons with the response of piles with circular cross section obtained using a similar analytical framework developed by the authors (Basu et al. 2009, Seo et al. 2009, Basu and Salgado 2007) have shown that conversions of rectangular cross section into equivalent circles may produce reasonably accurate results if the equivalent circle has the same area of cross section as that of the actual rectangular cross section in the case of axially loaded pile and if the equivalent circle has the same second moment of inertia as that of the actual rectangular cross section in the case of laterally loaded pile. However, such conversions into equivalent circular cross sections may not produce accurate results when soil nonlinearity is considered because, as demonstrated in the analysis, pile geometry has a role in determining the soil strain and stress fields, which are different for circular and rectangular cross sections.

It is also important to note that the inputs required for the analysis pile geometry and Young’s modulus, soil layering and elastic constants, and magnitude of applied forces can be given in the form of a simple text file and no elaborate numerical mesh generation is required. Moreover, because closed-form solutions are used, the results are obtained within a few seconds.

CONCLUSIONS

An analysis framework for axially and laterally loaded piles with rectangular cross section

embedded in multi-layered elastic media is developed. The analysis produces the pile settlement and axial resistance force for axially loaded piles, and lateral deflection, slope of the deflected curve, bending moment and shear force for laterally loaded piles. The inputs required are the pile cross sectional dimensions and length, thicknesses of the soil layers, Young’s modulus of the pile material, Young’s modulus and Poisson’s ratio (or any pair of elastic constants) of the soils in the various layers, and the magnitudes of the applied forces and moment.

The governing differential equations for the pile deflections are obtained using the principle of minimum potential energy. Closed-form solutions are obtained. The solution depends on two parameters, x and y, that describe the rate at which the deflections in the soil medium decrease with an increase in the distance from the pile. These parameters are not known a priori and must be determined iteratively. The method produces pile responses in seconds.

The shape of the pile cross section has a bearing on the pile response. The analysis presented in the paper can be used to make realistic predictions of the response of axially and laterally loaded rectangular piles.

ACKNOWLEDGEMENTS This material presented in this paper is based upon work supported by the National Science Foundation under Grant No. 0556347. The authors are thankful for this support. REFERENCES

Armaleh, S. and Desai, C. S. (1987). “Load-deformation response of axially loaded piles.” J.

Geotech. Engng., 113(12), 1483-1500. Banerjee P. K., and Davies T. G., (1978). “The behaviour of axially and laterally loaded single

piles embedded in nonhomogeneous soils.”, Geotechnique, 28(3), 309-326. Basu, D., and Salgado, R. (2007). “Elastic analysis of laterally-loaded pile in multilayered soil.”

Geomech. Geoeng.: An Int. J., 2(3), 183-196. Basu, D., and Salgado, R. (2008). “Analysis of laterally loaded piles with rectangular cross

sections embedded in layered soil media.” Int. J. Numer. Anal. Meth. Geomech., 32(7), 721-744.

Basu, D., Prezzi, M., Salgado, R. and Chakraborty, T. (2008). “Settlement analysis of piles with rectangular cross sections in multi-layered soil.” Computers and Geotechnics, 35, 563-575.



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yrig

ht A

SCE

. For

per

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l use

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Basu, D., Salgado, R. and Prezzi, M. (2009). “A continuum-based model for analysis of laterally loaded piles in layered soils.” Géotechnique, 59(2), 127-140.

Broms, B. B. (1964). “Lateral resistance of piles in cohesionless soils.” J. Soil Mech. Fdn. Div., 90(SM3), 123-156.

Butterfield, R. and Banerjee, P. K. (1971). “The elastic analysis of compressible piles and pile groups.” Géotechnique, 21(1), 43-60.

Coyle, H. M. and Reese, L. C. (1966). “Load transfer for axially loaded piles in clay.” J. Soil Mech. Fdn Div., 92(SM2), 1-26.

Dennis, N. D. and Olson, R. E. (1983a) “Axial capacity of steel pipe piles in clay.” Proc. Conf. Geotech. Practice Offshore Eng., New York, 370-388.

Dennis, N. D. and Olson, R. E. (1983b) “Axial capacity of steel pipe piles in sand.” Proc. Conf. Geotech. Practice Offshore Eng., New York, 389-402.

Guo, W. D. and Randolph. M. F. (1997). “Vertically loaded piles in non-homogeneous media.” Int. J. Num. Analyt. Meth. Geomech., 21(8), 507-532.

Matlock, H. (1970). “Correlations for design of laterally loaded piles in soft clay.” Proc. 2nd Offshore Tech. Conf., Houston, Texas, 1, 577-594.

Mylonakis, G. (2001). “Winkler modulus for axially loaded piles.” Géotechnique, 51(5), 455-461. Ng, C. W. W. and Zhang, L. M. (2001). “Three-dimensional analysis of performance of laterally

loaded sleeved piles in sloping ground.” J. Geotech. Geoenv. Engng., 127(6), 499-509. O’Neill, M. W., Reese, L. C., and Cox, W. R. (1990). “Soil behavior for piles under lateral

loading.” Proc. 22nd Offshore Tech. Conf., Houston, Texas, 3, 279-287. Poulos, H. G. (1971a). “Behavior of laterally loaded piles: I – single piles.” J. Soil Mech. Fdn

Div., 97(SM5), 711-731. Poulos, H. G. (1971b). “Behavior of laterally loaded piles: III – socketed piles.” J. Soil Mech.

Fdn Div., 98(SM4), 341-360. Poulos, H. G. and Davis, E. H. (1968). “The settlement behavior of single axially loaded

incompressible piles and piers.” Géotechnique, 18(3), 351-371. Rajapakse, R. K. N. D. (1990). “Response of an axially loaded elastic pile in a Gibson soil.”

Géotechnique, 40(2), 237-249. Randolph, M. F. and Wroth, C. P. (1978). “Analysis of deformation of vertically loaded piles.” J.

Geotech. Div., 104(GT12), 1465–1488. Randolph, M. F. (1981). “The response of flexible piles to lateral loading.” Geotechnique, 31(2),

247-259. Reese, L. C. and Matlock, H. (1956). “Non-dimensional solutions for laterally loaded piles with

soil modulus assumed proportional to depth.” Proc. 8th Texas Conf. on Soil Mech. Fdn. Engng., Austin, Texas, 1-41.

Reese, L.C., Cox, W. R. and Koop, F. D. (1974). “Analysis of laterally loaded piles in sand.” Proc. 6th Offshore Tech. Conf., Houston, Texas, 2, 473-483.

Reese, L.C., Cox, W. R. and Koop, F. D. (1975). “Field testing and analysis of laterally loaded piles in stiff clay.” Proc. 7th Offshore Tech. Conf., Houston, Texas, 2, 671-690.

Seed, H. B. and Reese, L. C. (1957). “The action of soft clay along friction piles.” Transactions ASCE, 122.

Seo, H., Basu, D., Prezzi, M. and Salgado, R. (2009). “Load-settlement response of rectangular and circular piles in multilayered soil.” J. Geotech. Geoenviron. Eng., 135(3), 420-430.

Trochanis, A. M., Bielak, J. and Christiano, P. (1991). “Three-dimensional nonlinear study of piles.” J. Geotech. Engng., 117(3), 429-447.

Verruijt, A. and Kooijman, A. P. (1989). “Laterally loaded piles in a layered elastic medium.” Geotechnique, 39(1), 39-46.



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A framework for analysis of piles with rectangular cross section

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