GRAPHS FOR THE DESIGN OF LATERALLY LOADED PILES IN CLAY

A. ALEM 1 and M. GHERBI2

ABSTRACT In the design of piles, we are frequently bound to study both the pile head deflection and the maximum

bending moment. By using a simplified analysis method to predict the response of laterally loaded piles in clay, graphical nondimensional relationships, load-lateral deflection at ground level, load-rotation at ground level and load-maximal bending moment in pile are proposed. Predicted responses by this method are compared with experimental results and are found to be in good agreement.

INTRODUCTION

Over the years, several methods have been proposed to predict the response of laterally loaded piles. The most frequently used is the well-known P-y analysis, which has been included into several codes of practice (Fascicule 62 (M.E.L.T.) 1993, P.H.R.I. 1980, A.P.I. 1993). In this approach, the soil around a pile is simulated by a series of independent non-linear springs, each spring representing the behaviour of a layer of soil of height unit. This procedure requires to input a series of P-y curves. To find load-deformation characteristics of piles needs generally to use a numerical method except in some cases where analytical solutions may be found.

This paper presents an example of analytical solution to predict the response of laterally loaded piles in clay. The principal results are presented in the form of graphical nondimensional relationships that permit to value easily pile head deflection and maximal bending moment in the pile.

LATERAL SOIL REACTION

The relationship between pressure P and deflection y at any point along a pile subject to a lateral load is non-linear. The simplest approach to account for this nonlinearity is to consider the elastoplastic model used in the present method and in which below the ultimate pressure Pu leading to failure, the soil reaction is P (z) = Esy (z). Es is the modulus of lateral subgrade reaction of the soil and y is the lateral displacement of the pile.

This model needs to define the ultimate lateral pressure Pu and the modulus of lateral subgrade reaction Es.

In the case of cohesive soil, it is generally assumed that the ultimate lateral pressure increases normally from 2CuBp to 3CuBp at ground level up to 8CuBp to 12CuBp at a depth of about 2Bp to 3Bp and beyond which it remains constant (Matlock, 1970; Reese and al., 1975). Cu is the undrained shear strength of the soil and Bp the pile width. According to Broms (1964a) a simple modified distribution of Pu is suggested (as shown in figure 1). The soil in the top zone up to a depth zd equal to 1.5Bp is neglected. Below this depth ultimate lateral pressure Pu is equal to 9CuBp.

A number of empirical correlations for Es with other soil properties are available. For cohesive soils Es can be expressed as a function of the undrained shear strength Cu. According to Banerjee and Davies (1978), secant value of Es in short-term loading can be estimated at 100Cu to 180Cu. In some investigations of laterally loaded piles in clay, it has been found that the tangent value of Es estimated at 250Cu to 400Cu yields more satisfactory correlations with experimental tests. In practice, a possible way of determining the modulus of lateral subgrade reaction is by using the results of pressuremeter tests (Fascicule 62 (M.E.L.T.) 1993). Es is related to the pressuremeter modulus and a factor dependent on the soil type.

It is important to note that modulus of lateral subgrade reaction is generally assumed to be constant with depth only in the case of stiff overconsolidated clay. For most normally consolidated clay, modulus of 1 Abdellah ALEM & 2 Mouloud GHERBI, Laboratoire de Mécanique, Université du Havre, 25 rue Philippe Lebon, BP 540, 76058 Le Havre , France. Tél : 02 32 74 49 57 ; Email : Abdellah.Alem@iut.univ-lehavre.fr

lateral subgrade reaction is assumed to increase linearly with depth,E (z) zs = ηη . To find solutions in this case needs to use numerical methods. This inconvenience is overcome by assuming an average value of Es taken in a significant part of the pile. After further investigations, the following approach to estimate the average value of Es is proposed.

Let’s consider linear theory and a free-head pile of embedded length D, bending rigidity EI and subjected to a horizontal load Ho at an eccentricity L above the ground surface. By equating solutions giving horizontal displacement of pile at ground line for the simple cases of constant and linearly increasing lateral subgrade reaction modulus with depth, the following implicit equation giving average value of Es is obtained.

Figure 2 presents the graphical dimensionless solution of this equation.

41s

2s

4EI

E

4EI

EL

1

+

=

+

5/35/25/25/3s

)(EI)(

0.80L

)(EI)(

1.2E

ηη (1)

Figure : 1 Pile movement and lateral soil pressure distribution

MATHEMATICAL FORMULATION

When the pile is subjected to a horizontal load, it will move until the counteracting pressure is distributed

in such a way that the equilibrium conditions are satisfied. The resulting distribution of soil reaction against the pile may be decomposed into two zones. A plastic zone which extends to a depth zp function of the horizontal force level and its point of action, and an elastic zone below this depth.

The governing equation for the pile deflection at any depth z is:

EId y

dz

4

4 + P (z) = 0 (2)

Equations that give the load P, the shear V, the bending moment M, the rotation ω and the lateral deflection y in each zone are as follow:

In the plastic zone ( )z zp≤

pu4

4

B9Cdz

ydEIP(z) =−= (3)

V (z) = EI 1u3

3

Cz Pdz

yd +−= (4)

21

2

u2

2

Cz C2

z P

dz

ydEI=M(z) ++−= (5)

elastic zoneP(z)=Esy(z)

1.5Bp

Ho

plastic zoneP(z) = Pu = 9CuBp

zp

0

(b) Assumed lateral soil reaction distribution

z

assumed ground surface

D

0 .00 5.00 10.00 15.00 20 .00

1 .15

1.20

1.25

1 .30

L β1

β/β1

Figure : 1 Curve giving average value of lateral soil modulus

( )β η10 2= EI.

; ( )β = E 4EIs0 25.

; L = e+1.5Bp

(a) Pile subjected to lateral load

+++−= 32

2

1

3

u Cz C2

z C

6

z P

EI

1

dz

dy=(z)ω (6)

+++−= 43

2

2

3

1

4

u C+z C2

zC

6

z C

24

zP

EI

1y(z) (7)

In elastic zone (z zp≥ ), soil reaction on pile is given by:

P (z) = Es (z)y(z) (8)

Assuming a long pile and a soil modulus independent from depth, we find the deflection:

( ) ( )( )zcosCzsinCey(z) 6 5z- βββ += (9)

( )β = E 4EIs0 25. is the so-called characteristic length of the pile.

The expressions that give the rotation ω , the bending moment M, the shear V and the load P are found by successive derivations.

The seven unknowns, six Ci constants and zp are obtained by imposing for moment and shear equilibrium at z = 0 and z = zp, and for displacement and slope compatibility at z = zp. The last boundary condition is obtained from the fact that at depth zp, the rate of shear variation is equal to the ultimate soil resistance/unit length.

Complete solution depends on the load level and needs to be identified into stages as follow. For pile subject to a lateral force Ho and moment Mo at the ground surface, we have:

For )1L(2

PH

uo

+≤

ββ, where oo HML = , load level is such that soil is elastic. Thus zp = 0.

Solutions are obtained from elastic theory. For deflection and moment we have the following:

( ) ( )

++−=

−

zcosMH

zsinMEI2

ey(z) o

o

2

z

ββ

ββ

β

o (10)

( ) ( )

++= − zsinM

HzcosMeM(z) o

oz ββ

ββo (11)

The position and value of maximal bending moment are:

+

=ββ L21

1arctg

1zo ( ) ( )

++= −

ooo

oz

max zcosMH

zsinMeM ββ

ββo (12, 13)

For )1L(2

PH

uo

+>

ββ , zp > 0

From boundary conditions at z = 0, we find:

C1 = Ho , C2 = Mo (14)

and from boundary conditions at z = zp,

u

o

2u

2o

u

o

P

M2

P

H

P

H1zp +++−=

β (15)

Furthermore

( ) ( )( ) po

2p

o

3p

upp3 z M2

zH

6

zPzBzAEIC −−++−= β (16)

( ) p3

2p

o

3p

o

4p

up4 z C2

zM

6

zH

24

zPzEIAC −−−+= (17)

( ) ( ) ( ) ( )( )pppp5 zcoszBzsinzAC βββ −= pze (18)

( ) ( ) ( ) ( )( )pppp6 zsinzBzcoszAC βββ += pze (19)

where:

( )s

up

E

PzA = ( )

++−= opo

2p

u2

p MzH2

zP

EI.2

1zB

β (20, 21)

The two values C3 and C4 divided by the bending rigidity EI of pile give respectively rotation and horizontal deflection of pile at ground surface.

The position and value of maximum bending moment are:

For )1L(2

PH

uo

+≥

ββ and

2u

o

u

o 12L

P

H

P

H

β≤

+ , maximum bending moment occurs in elastic zone.

( ) ( )[ ] ( ) ( ) ( )[ ] ( )( ) ( )[ ] ( ) ( ) ( )[ ] ( )

−++−++

=pppppp

ppppppo

zsinzBzAzcoszBzA

zcoszBzAzsinzBzAarctg

1z

ββββ

β (21)

( ) ( )( )o6o5

z2max zcosCzsinCEIe2 = M βββ β +−− (22)

For )1L(2

PH

uo

+≥

ββ and

2u

o

u

o 12L

P

H

P

H

β≥

+ , maximum bending moment occurs in plastic zone.

u

oo

P

Hz =

+= L

2P

HHM

u

oomax (23, 24)

For pile subjected to a load Ho at level L above the ground surface, lateral deflection at the point of applied load is given by the following equation.

3EI

LH + L (0)y(0)y(L)

3oω−= (25)

in which y(0) and ω(0) are respectively lateral deflection and rotation of pile at ground surface.

RESULTS AND DISCUSSION In order to check the suggested formulation, some of the available published data obtained from tests are

used. Tests presented in this paper were issued in (Poulos 1980). The details of pile materials and soil are given in table 1. The modulus of lateral subgrade reaction is valued by using relation from pressuremeter test results with α = 2/3 and EmBp/Pu=9 (C.F.E.M. 1985). α is a rheological coefficient of soil and Em is the pressuremeter modulus.

Figures 3 show typical experimental and calculated load-deflection curves. The predicted responses are reasonably in good agreement with the experimental responses.

Test No pile diameter

(mm) pile length (mm)

pile bending rigidity (kNm)

shear strength of soil

Cu (kPa)

lateral subgrade reaction modulus

Es (kPa) 1 10 3000 622000 14.4 4360 2 3.8 5250 31600 14.4 4360 3 7.6 5250 246000 14.4 4360 4 10 5250 320000 14.4 4360

Table 1 : Details of pile materials and soil properties (Poulos 1980)

Figure 3 : Load-deflection curves

------Measured , ---o--- predicted (Poulos 1980)

GRAPHICAL DIMENSIONLESS SOLUTIONS

In the design of piles, we are frequently bound to study both the pile head deflection and the maximum

bending moment. By using the above solution, it is possible to define respectively the nondimensional lateral deflection and rotation at ground surface,(y E ) (Cu B )o s p and(ωE ) (Cu B )s p , and maximal bending

moment (M (Cu B )max pβ2 ) in the pile, which can be expressed as function of the nondimensional applied load

(H (Cu B )o pβ) and the dimensionless length βL, as shown in figures 4. Pile head deflection and maximal-bending moment can easily be valued from these graphical relationships. CONCLUSIONS

A simple analytical formulation has been proposed to predict the response of laterally loaded piles in

clay. Calculated solutions by this method are compared with experimental results and are found to be in good agreement. Nondimensional curves relationships to value easily pile head deflection and maximal bending moment in pile are presented. Similar nondimensional curves are obtained for piles embedded in sand and will be published later on.

0 .00 4 .00 8 .0 0 1 2.0 0 1 6.00G round line deflec tion (m m )

0 .00

4 .00

8 .0 0

12 .0 0

Late

ral l

oad

(kN

)

T est 1

0.00 4.00 8.00 12.00 16.00G round line deflection (m m )

0.00

2.00

4.00

Lat

eral

load

(kN

)

Test 2

0.0 0 4 .0 0 8 .00 12 .0 0 16 .00G round line de flec tion (m m )

0 .00

4 .0 0

8 .00

12 .00

Late

ral l

oad

(kN

)

T est 3

0.0 0 4 .0 0 8 .00 12 .0 0 16 .00G round line de flec tion (m m )

0 .00

4 .0 0

8 .00

12 .00

Late

ral l

oad

(kN

)

T es t 4

Figure 4.a : Ground line deflection versus applied load Figure 4.b : Ground line rotation versus

applied load

Figure 4.c : Maximal bending moment versus applied load

REFERENCES

American Petroleum Institute (A.P.I) RP 2A-LRFD - Section G : Foundation Design, A.P.I, pp. 64-77. Banerjee, P.K. and Davies, T.G. (1978). ‘ The behaviour of axially and laterally loaded single piles

embedded in nonhomogeneous soils’, Geotechnique 28, No. 3, pp. 309-326. Broms, B.B. (1964a). ‘ Lateral resistance of piles in cohesive soils ’, Journal of the Soil Mechanics Division,

ASCE, Vol. 90, No. SM3, May, pp. 27-63. Canadian Foundation Engineering Manual (C.F.E.M.), 2nd Edition (1985)., Canadian geotechnical Society. Fascicule 62 - Titre V (1993). ‘Règles techniques de conception et de calcul des fondations et des ouvrages

de génie civil’, Ministère de l’Equipement, du Logement et des Transports (M.E.L.T.), France. Matlock, H. (1970). ’ Correlations for design of laterally loaded piles in soft clay ’, Proceedings of the

Offshore Technology Conference, Houston, Texas, OTC 1204, pp. 544-593. Poulos, H. G., and Davies, E. H. (1980). ‘Pile foundation analysis and design’, John Wiley and Sons, New

York, NY. Port and Harbour Research Institute (P.H.R.I.) (1980). ‘Technical standard for port and harbour facilities in

Japan’, Japan Ministry of Transport, 371p. Reese, L.C. and Welch, R.C. (1975). ‘Lateral loading of deep foundations in stiff clay ’, Journal of the

Geotechnical Engineering Division, ASCE, Vol.101, No. GT7, July, pp. 633-649.

0.00 20 .00 40.00 60.00 80.00

0.00

2.00

4.00

6.00

8.00

10.00

N o nd im e ns io na l g rou nd line d e flec tion yo

Non

dim

ens

iona

l lat

eral

load

Ho

β=0 β=0.2

β=0.5

β=1

β=1.5

β=2.5

β=4

β=10

β=6

L L

L

L

L

L

L

L

L

-100 .00 -80 .00 -60 .00 -40 .00 -20.00 0 .00

0.00

2.00

4.00

6.00

8.00

10 .00

N ondim ensional ground line ro tation w o

Non

dim

ensi

onal

late

ral l

oad

Ho

β=1.5

β=1

β=0

β=0.2β=0.5

β=2.5

β=10

β=6

L LL

L

L

L

L

L

β=4L

0.00 4 .00 8.00 12.00 16 .00

0.00

2.00

4.00

6.00

8.00

10.00

N ond im ens ional m ax im a l bend ing m om en t M m ax

No

ndim

ensi

ona

l lat

era

l lo

ad H

o

β=1.5

β=1β=0

β=0.2 β=0.5

β=2.5

β=10

β=6

LL L

L

L

β=4L

L

L

L