analysis of laterally loaded pile

7/29/2019 Analysis of Laterally Loaded Pile

1/16

Analysis of Laterally Loaded Pile in

Layered Soils

Rongqing Li

PhD candidate, State Key Lab. of Coastal and Offshore Eng.,

Dalian Univ. of Technol., Dalian, China

e-mail: [email protected]

Jinxin Gong

Professor, State Key Lab. of Coastal and Offshore Eng.,

Dalian Univ. of Technol., Dalian, China

e-mail: [email protected]

ABSTRACT

An analytical method is developed to predict the responses for single pile subjected to

lateral load in layered soils. The method uses fundamental basis of structural mechanics to

obtain the governing Equations of the soil and pile systems. Both free head and fixed head

piles are considered in this method. The pile deflection, bending moment and soil reaction

can be calculated using this method. This method is easy to understand for engineers and is

simple enough to be adapted for computer use. An example is included to demonstrate its

use. Deflections and bending moments calculated using this method are found to be in good

agreement with those obtained from finite element method, thus verifying the reliability of

the proposed method. This method can be used to predict the response of laterally loaded

pile in preliminary design and then help engineers to make informed engineering decisions.

KEYWORDS: laterally loaded pile; deflection; moment; layered soils

INTRODUCTION

Piles have been widely used for supporting axial and lateral loads for a variety of civil

engineering structures such as high rise buildings, transmission lines, bridge piers and port

structures. In many cases, lateral loads govern the design of piles. Two aspects of interest which


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should be considered in the design of laterally loaded pile are the maximum deflection and the

bending moment in the pile.

Several methods have been published for predicting the response of single piles under

lateral loading (Hetenyi, 1946; Broms, 1965; Desai, 1974; Sun, 1994; and Hsiung, 2006). One

of these methods is called subgrade reaction method which considers the pile as a flexible

beam on the elastic foundation and replaces soil as a series of elastic, closely spaced but

independent springs. This method has the advantage of being relatively simple, and layered

foundation (Davisson and Gill, 1963; Dai, 2007) can be considered. The responses of pile can

be calculated by solving differential Equations of deflection curve or using finite difference

method or finite element method (FEM) (Poulos, 1971a; Poulos and Davis, 1980).

Compared with those solutions above, the procedure developed in this paper uses

fundamental basis of structural mechanics to obtain the governing Equations of the soil and

pile systems. The method is easy to understand for engineers and is simple enough to be

adapted for computer use. Both free head and fixed head piles are considered in the method.

The pile deflection, bending moment and soil reaction can be calculated using this method.

The proposed method has been validated by comparison of the results with those calculated

using FEM.

FREE HEAD PILE

Modeling for the pile-soil system

The interaction model for the pile-soil system is shown in Figure 1. The pile is assumed to

be a line of length L with constant flexibility EI, and to be fully embedded into soil.

Symbols0

H and0

represent the horizontal load and moment applied at top of the pile

respectively; and 1 nk k

denote the stiffness of the springs. The deflections and moments in

the pile can be obtained from finite element analysis. In this analysis, an alternative procedure

for analyzing the response of laterally loaded pile is presented based on the model shown in

Figure 2. Substitute the springs for soil reactions represented by1 n

p p as shown in Figure

2. Two virtual supports represented by B and D are set at the top and the tip of the pile,

respectively. The soil reaction and pile deflection at a pointi

x below the ground surface are

denoted by ip and iy , respectively.


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Based on field measurements of instrumented piles, different p y curves have been

developed for different soils and pile types by a number of investigators (Briaud, 1997; Gabr,

1994; and Ashour, 2000). Ye and Shi (2000) presented a nonlinear p y curve based on

lateral pile-load test for 39 piles in China. In addition, the linear p y curve was widely

used in engineering practice (Dai, 2007). Most of those p y curves in engineering practice

can be expressed in the general form:

1 2sgn( )( )n np y a mx y= + (1)

where p is the pressure at a point (kN/m2); x is the depth below ground surface (m);

y is pile deflection (m) ;1 2

, , ,a m n n are model factors, which can be determined by

different p y curve;

1 0

sgn( ) 0 0

1 0

y

y y

y

>

= =


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1( )n

i i ik a mx bd = + ( 1, ,i n= ) (3)

then the soil reaction can be rewritten as:

2sgn( ) ni i i ip y k y= ( 1, ,i n= ) (4)

Deformation compatibility and static equilibrium

By using the method of superposition, the total loading system on the pile can be

subdivided into loading conditions that produce deflections which are already known. In

Figure 2, the pile deflection at a point is made up of three parts: (1) a deflectioni caused

by the movement of the supportsB andD, (2) a deflection ,i p caused by the soil reactions,

and (3) a deflection0,i M

caused by the moment applied at pointB. The superposition of all

three deflections gives the total deflection:\

0, ,i i i p i M y = ( 2 , 1i n= ) (5)

The first part of the deflectioni

can be expressed as follows:

( ) 11i i i nr y r y = + ( 2 , 1i n= ) (6)

where1

andn

represent movements of supports B and D , respectively; iirl

= .

Deflections due to soil reaction jp can be calculated based on mechanics principle (Gere,

2004):

( )

2 2 2

,3

2 2 3

( )( ) ,

6

( )( ) ,

6 ( )

j

j j i

i j

i pj j

i j j i i

j

p l x xl x l x i j

EIl

p l x lx l l x x x i j

EIl l x

= + >

( 2 , 1i n= ; 2 , 1j n= ) (7)


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Using the method of superposition once again, one can get the deflections due to soil

reactions:

1

, ,

2

n

i p i j j

j

c p

=

= ( 2 , 1i n= ) (8)

where

( )

2 2 2

,3

2 2 3

( )( ) ,

6

( )( ) ,

6 ( )

j i

i j

i jj

i j j i i

j

l x xl x l x i j

EIlc

l x lx x l l x x x i j

EIl l x

= + >

( 2 , 1i n= ; 2 , 1j n= )

(9)

The deflection due to0

can be calculated:

0

2 20

,

( )( )

6

i

i M i

M l xl l x

EIl

=

( 2 , 1i n= ) (10)

Substituting Equations (6) and (8) into Equation 5, one can get:

(11)

Substituting Equation (4) into Equation (11) yields:

(12)

When solving the Equation, the deflections i will be transferred to the left-hand sides,

so that the Equation appears in the form:

( ) 20

1

1 , ,

2

1 sgn( )n

n

i i j i j j j i n i M

j

r y y y c k y r y

=

+ + = ( 2 , 1i n= ) (13)


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Solving the Equations of static equilibrium 0D

M = and 0BM = , respectively,

the support reactionsB

R andD

R can be expressed as follows:

210

0

1

sgn( ) ( )nni i i i

B

i

y k y l xR H

l l

=

= +

(14)

20

2

1sgn( )

nn

D i i i i

i

MR y k y x

l l ==

(15)

Boundary conditions

Since the supports B and D are not exist actually, reactionsB

R andD

R should be

equal to zero. Thus, after rearranging Equations (14) and (15), one can get the following two

Equations of boundary conditions:

2

1

0 0

1

sgn( ) ( )

n

ni i i i

i

y k y l x M H l

=

= + (16)

2

0

2

sgn( )n

n

i i i i

i

y k y x M=

= (17)

Determination of moments and deflections of the pile

As previous discussion, the soil reaction in Equation (2) is expressed in a general form.

The linear p y curve and the nonlinear p y curvedeveloped by Ye and Shi (2000)

were used in the following analysis.

For linear p y curve, the factors in Equation (2) are a =0, 1n =1 and 2n =1,

Equation (2) reduces to the following:

i i ip k y= (18)


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wherei i i i

k m x bd = ;i

m is constant of horizontal subgrade reaction.

Combining Equations (13), (16) and (17), one can get:

Solving Equations 19 for iy , and then substituting i into Equation 18, soil reactions are

found. Finally, considering a free body diagram and using Equations 20 and 21, the shears and

bending moments in the pile can be calculated:

for moment: ( ) ( )n

j j

j i

x p x x=

= (20)

for shear: ( )n

j

j i

Q x p=

= (21)

where i is the segment number of which depth is just greater than x.

When using thepy curvedeveloped by Ye and Shi (2000), factors in Equation 2 are a

=0,1

n =2/3 and2

n =1/3, then Equation 2 becomes:

1/ 3sgn( )i i i ip y k y= (22)

where2/ 3

i Ni i ik k x bd = ; Nik is coefficient of horizontal subgrade reaction.

Combining Equations 13, 16 and 17, one can get nonlinear Equations ofi. Using the

proposed method for solving nonlinear Equations (Poulos and Davis, 1980), one can get pile

deflectionsyi . The soil reaction and the internal forces in the pile can be calculated using the

previous procedure.


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FIXED HEAD PILE

Model for fixed head pile

For fixed head pile, the rotation at top of the pile is restrained. After setting two virtual

supports at top and bottom of the pile, respectively, the fixed head pile becomes a statically

indeterminate structure to the first degree carrying loadsi

p and subjected to horizontal force,

as shown in Figure 3 (a). To obtain the governing Equation, the reaction moment at supportB

is selected as the redundant and then the rotational restraint at B must be released; the

resulting released structure is a simple beam, as shown in Figure 3 (b). The reaction moment

atB consists of the moment due to the movement of supportsB andD , and the moment due

to soil reactions, denoted byB

and Bp respectively in Figure 3 (b).

Similar to free head pile, the total pile deflection can be given using the method of

superposition:

, , , pi i i p i M i M y

= + + ( 1 ,i n= ) (23)

where the former two terms in Equation (23) are the same as those for free head pile in

Equations (6) and (8), respectively; the latter two terms, ,, pi M i M are deflections caused by

Band Bp , respectively, which are discussed as follows.

a) b)

Figure 3: Model for fixed ile head

MB

MBp

H0

B

D1 D

B1

x

y

i,p-i,M0

i

yi

RB

RDyn

y1

H0

pi

l

p1

pn

B

D

x

y

l

pi

p1


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Pile deflection due toB

As shown in Figure 4, the value of23 /EI l of the fixed-moment will occur due to unit

movement at support B. Thus, the fixed-moment MB caused by the relative displacement

betweenB andD, (y1yn), is expressed by:

1

2

3 ( )nB

EI y yM

l

=

(24)

Figure 4: Moment at top of the pile due to support movement

After getting the momentB

, the pile deflection due toB

can be calculated by

considering pile as a simple beam with applied moment at end of it:

2 2

,

( )( )

6BB i

i M i

M l xl l x

EIl

= (25)

Substituting Equation (24) into (25), Equation (25) can be rewritten as:

1, 2

3 ( )B

i ni M

EIs y y

l

= (26)

where 2 2( ) ( )6

ii il xs l l x

EIl = .


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Pile deflection due to Bp

The moment Bp can be characterized as the function of soil reaction by analyzing the

structure as shown in Figure 5 (a). By using the flexibility method, one obtains the following

Equation of compatibility:

i11 B,pM

+ 1p

=0 (27)

where11

is the deflection at point B due toB,piM =1; 1p is the deflection at location

B due to ip .

The bending moment diagrams for the released structure corresponding toB,pi

M =1 and

ip are shown in Figure 5(b) and Figure 5 (c), respectively. The deflections 11 and 1p can

be found by using the unit-load method (Gere, 2004):

pi

11d 3

MMl/ EI

EI = =

1p

( )(2 )d

6

i i i ip l - x l - x xMM

xEI EIl

= =

Substituting 11 and 1p into Equation (27), one obtains:

(a) (b) (c)

Figure 5: Moment at pile head due to soil reaction:

(a) released structure; (b) moment curve due top

(c) moment curve due to

B

D

MB,Pi

pi

l

xi

B

D

pi

B

D

MB pi =1

1

pixi(l-xi)/l

x

y


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i

1p

B,p 2

11

( )(2 )

2

i i i ip l - x l - x x

Ml

= =

(28)

Using simple superposition, the moment due to all soil reactions can be obtained:

(29)

The pile deflection due to Bp at a point can be obtained based on mechanics principle:

2 2

,

( )( )

6pBp i

i M i

M l xl l x

EIl

=

(30)

Substituting Equation (29) into (30), Equation (30) can be expressed by:

(31)

Deformation compatibility and static equilibrium

Substituting Equations (6), (8), (26) and (31) into Equation (23) yields:

11 , 2

1 1

3 ( )(1 )

n ni n

i i i n i j j j j i

j j

EIs y yy r y r y c p p e s

l= =

= + + +

(32)

Transferringi

to the left-hand sides, and substituting Equation (4) into Equation (32),

the deformation compatibility Equation for head-fixed pile appears in the form:

2

1 ,2 2

1

3 3(1 ) sgn( )( ) ( ) 0

nni i

i i j i j j i j j i n

j

EIs EIsr y y y c e s k y r y

l l=

+ + + = (33)


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In Figure 3 (b), considering the conditions of static equilibrium 0D

M = and

0B

M = , respectively, the support reactions BR and DR can be expressed as follows:

2

1

0 1

1

1 3sgn( ) ( ) ( )

nn

B i i i i i n3i

EIR H y k l x e y y y

l l

=

= + (34)

2

1

1

1

1 3( ) ( )

nn

D i i i i n3i

EIR k e x y y y

l l

=

= + (35)

Boundary conditions

Similar to free head pile,B

R andD

R should be equal to zero. Thus, after rearranging

Equations 34 and 35 yields:

2

1

1 02 2

1

3 3sgn( ) ( )

nn

i i i i i n

i

EI EIy y k l x e y y H l

l l

=

+ + = (36)

2

12 22

3 3sgn( ) ( ) 0

nn

i i i i i n

i

EI EIy y k e x y y

l l=+ = (37)

Combining Equations 33, 36 and 37, and then solving the equations fori

y , the pile

deflection are obtained. The internal force can be further determined by procedure similar to

that for free head pile.

For the case where pile extends through ground to air or water, the procedure is similar to

that fully embedded in soil except that the modulus of horizontal subgrade reaction associated

with those segments above ground surface should be taken as zero.

ILLUSTRATIVE EXAMPLE

A typical pile of square cross section 650 mm 650 mm in a four-layer soil profile is

loaded as shown in Figure 6. The pile length is 18 m, and the Youngs modulus for thematerial is 3.25 104 MPa. The soil profile from the top is clay, silty clay, silty sand, and


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coarse sand. The soil parameters are given in Table1. Free and fixed head piles are

considered. The analysis is carried out using the p y curve developed by Ye and Shi

(2000).

Table1: Soil Parameters

Soil type

Top elevation of

layer

(m)

Bottom elevation

of layer

(m)

m

(kN/m4)

Nk

(kN/m3)

Clay 0.0 -9.0 4000 400

Silty clay -9.0 -10.9 6000 600

Silty sand -10.9 -13.5 8000 800

Coarse sand -13.5 -16.0 10000 1000

Using the method proposed in this paper, the calculated results are shown in Figure 7. For

free head pile, the deflection at the top of the pile is 0.9 cm, and the maximum bending

moment in the pile is approximately 250 kN m . For fixed head pile, the deflection is 0.12 cm,

and the maximum bending moment is 122.3 kN m

. For verification, the response of the pilewas analyzed by finite element method (FEM) with software ANSYS (Moaveni, 2003). The

calculated results are shown in Figure 8. It can be seen that the deflections and bending

moments calculated using the proposed method and FEM are in good agreement.

Figure 6: Laterally loaded pile in a four-layer

2.0m

-9.0m

-10.9m

-13.5m

-16.0m

0.0m

Clay

Silty Clay

Silty Sand

Coarse Sand

100kN.m

50kN


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CONCLUSIONS

A new analytical method for calculating the response of laterally loaded pile in layered

soil is proposed based on mechanics principle. Both free head and fixed head piles are

considered in this method. The analysis for a laterally loaded pile embedded in a four-layer

soil profile is carried out. The results show that the calculated deflections and bending

moments using the proposed method agree well with those obtained from the FEM with

Figure 7: Response of pile subjected to laterally load: (a) For free head pile; (b) For fixed head pile

(b)

-16

-12

-8

-4

0

-150 50

Moment (kN.m)

Elevation(m)

-16

-12

-8

-4

0

-0.2 -0.1 0

Deflection (cm)

Elevation(m)

-16

-12

-8

-4

0

-4 -2 0 2 4 6

Soil Reaction (kN)

Elevation(m)

-16

-14

-12

-10

-8

-6

-4

-2

0

2

-50 50 150 250

Moment (kN.m)

Elevation(m)

-16

-14

-12

-10

-8

-6

-4

-2

0

2

-1 0

Deflection (cm)

Elevation(m)

-16

-12

-8

-4

0

-8 -4 0 4 8

Soil Reaction (k N)

Elevation(m)

(a)

(a) (b) (c) (d)

Figure 8: Deflection and moment calculated using FEM:

(a) Deflection for free head pile (m); (b) Moment for free head pile (kN.m);

(c) Deflection for fixed head pile (m); (d) Moment for fixed head pile (kN.m)


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software ANSYS. This method has the advantage over other procedures in that it is easy to

understand and is adaptable to simple computer programs by engineers. In addition, the

proposed method is especially useful when it is only need to approximately predict theresponse of laterally loaded pile in preliminary design.

REFERENCES

1. Ashour, M., and G. Norris (2000) Modeling lateral soilpile response based onsoilpile interaction, Journal of Geotechnical and Geoenvironmental Engineering,

ASCE, 126(5), pp 420428.

2. Briaud, J.L. (1997) Sallop: simple approach for lateral loads on piles, Journal ofGeotechnical and Geoenvironmental Engineering, ASCE, 123(10), pp 958964.

3. Broms, B.B. (1965) Design of laterally loaded piles, Journal of the Soil Mechanicsand Foundations Division, ASCE, 91(3), pp 79-99.

4. Dai, Zihang, Linjing CHEN (2007) Two numerical solutions of laterally loaded pilesinstalled in multi-layered soils by m method, Chinese Journal of Geotechnical

Engineering, 29(5), pp 690-696 (in Chinese).

5. Davisson, M.T., and H.L.Gill (1963) Laterally loaded piles in a layered soil system,Journal of the Soil Mechanics and Foundations Engineering, ASCE, 89(3), pp 63-94.

6. Desai, C.S. (1974) Numerical design-analysis for piles in sands, Journal of theGeotechnical Engineering Division, ASCE, 100(6), pp 613-635.

7. Gabr, M.A., T. Lunne, and J.J. Powell (1994) Py analysis of laterally loaded pilesin clay using DMT, Journal of Geotechnical Engineering, ASCE, 120(5), pp

816837.

8. Gere, James M. (2004) Mechanics of materials, Brooks/Cole, United States.9. Hetenyi, M. (1946) Beams on elastic foundations, University of Michigan Press,

Ann Arbor, Mich., United States.

10.Hsiung, Y.M. (2006) Analytical Solution for Piles Supporting Combined LateralLoads, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 132, No.

10, pp 1315-1324.

11.Moaveni, Saeed (2003) Finite element analysis: theory and application withANSYS, Pearson Education, United States.

12.Poulos, H. G. (1971a) Behavior of laterally loaded piles. I: single piles, Journal ofthe Soil Mechanics and Foundations Division, ASCE, 97(5), pp 711-731.

13.Poulos, H. G., and E. H. Davis (1980) Pile foundation analysis and design, JohnWiley & Sons, Inc, United States.


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Vol. 13, Bund. J 16

14.Sun, K. (1994) Laterally Loaded Piles in Elastic Media, Journal of GeotechnicalEngineering, ASCE, 120(8), pp 1324-1344.

15.Ye, W. L. and B. L. Shi (2000) A practical non-linear calculation method of pileslateral bearing capacity NL method, Rock and Soil Mechanics, 21(2), pp 97-101(in

Chinese).

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analysis of laterally loaded pile

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