A mapping finite element method for the analysis of laterally loaded single piles

Computers and Geotechnics 7 (1989) 255-266

A MAPPING FINITE ELEMENT METHOD FOR THE ANALYSIS OF LATENALLY LOADED S ~ PILES

B. Nath Department of Civil Engineering, Queen Mary College, University of London, U.K.

ABSTRACT A simple mapping finite element method is presented for the analysis of laterally loaded single piles. In this both pile and soil are considered as integral parts of the same solid c o n t i ~ but with different properties; the cylindrical-polar coordinate transformation is used to map the physical pilesoil domain into a mapped domain which has a simpler shape. The prescribed boundary conditions and applied loads are also transformed appropriately. The problem is then solved in the mapped domain with finite elements of the mapped space. Typical results, presented for homogeneous and nonhomogeneous soils, show that although results due to the proposed method agree reasonably well with those due to conventional methods in which assumptions have to be made rec/arding stress distribution at the pile-soil interface, there are discrepancies nevertheless which may be significant. Such assumptions are dispensed with in the proposed method and, to this extant alone, it arguably accords more with the actual physics of the problem than do the conventional methods. It is also pointed out how this simple and accurate method can be extended, without difficulty, to problems of laterally loaded piles embedded in soils that may be linear, nonlinear, homogeneous, nonhomogeneous or layered in any admissible combination. INTRO[PJCTION The problem of laterally loaded piles has received considerable attention to date due to its frequent occurrence in practice. Tw~ distinct approaches to the analysis of the problem are readily identified from literature: in the first, the so-called Winkler model and its variants, the pile is treated as an Euler-Bernoulli beam or an infinitely thin linearly elastic strip embedded in the soil, while the soil median is replaced by a system of discrete springs placed along the length of the pile [1-8]. The stiffness of the springs can be varied with depth to sin~tlate linear but nonhomogeneous soil behaviour [4,5], or replaced by nonlinear p-y curves to represent pile-soil interaction in nonlinear soils [6-8]. However, the major deficiency of this simple empirical approach is that, stresses are

in the absence of coupling between the springs, only normal

admissible at the pile-soil interface,

and this

is clearly not

255

Computers and Geotechnics 0266-352X/89/$03.50 O 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

256 realistic. Moreover, the discrete spring model precludes the extension of the method to pile groups because interaction between neighbollring piles may not be taken into account [9]. In the second, which may be called the

'semi-continuum'

soil medium is treated as an elastic half-space, thin elastic

strip. Uncoupled

are calculated,

soil displacements

either analytically

approach,

the

and the pile as a beam or a at the pile-soil interface

or n~erically,

and the interacting or

coupled pile behaviour is then found by matching these displacements to pile deflections at the sampling points (nodes or pivots). In general, this process involves the integration of appropriate point forces for the soil medit~n over the discretized pile-soil interface; the equations relating the displacements and surface tractions of the soil at the interface are then coupled to those governing the axial

and transverse displacements of the pile. The major defi-

ciency of this method

is that pile-soil

interaction

is only approximately

represented. Based on the method of Spillers and Stoll [I0], Poulos [II] has obtained a solution to the problem of single laterally loaded piles embedded in homogeneous and linearly elastic soils by matching soil displacements at the pile-soil

interface,

calculated

ratio of 0.5,

to the flexural deflections

elastic strip representing

from the Mindlin equation with Poisson of an infinitely thin embedded

the pile. Using

boundary elements to model the

soil medium, Banerjee and Davies [12] have extended the method to the approximate

analysis

of

single

piles

embedded

in

linear

nonhomogeneous

soils.

Following the same method but assuming elasto-plastic soil behaviour, Davies and Budhu [13] have studied the response of single piles, embedded in heavily consolidated clays, to lateral loads. The

flexibility and versatility of the finite element method make

eminently

suitable

for the analysis of laterally

loaded single piles,

it and

several such analyses have been made. Based on the concept of Wilson's element [14], in which Fourier terms are employed to enable the efficient analysis of axisymmetric

elastic

bodies

subjected

to nonaxisynlnetric

loads,

Kuhlemayer

[15] has presented an elegant FE formulation of laterally loaded single piles in which the beam bending aspect is efficiently modelled with a specially developed element.

Based on Kuhlemeyer's work but using linear strain tri-

angles to avoid the reduced integration of rectangular elements used by

him,

Randolph [9] has derived useful algebraic expressions, in terms of fundamental soil properties, lateral

loads

for the calculation of response of single flexible piles to

(a flexible

pile is defined as one whose length exceeds the

critical [9] or effective [15] length).

257 However, at present a general, simple and economical FE method does not appear to be available for the accurate analysis of single laterally loaded piles, regardless of whether the soil is homogeneous, nonhc~ogeneous, linear, nonlinear or layered (in any admissible combination) and whether the pile has a uniform, nonuniform, circular or noncircular cross-section. An alternative Finite Element method described in this paper, it is hoped, would fill this niche. In this, which may be called the 'truly continuum' approach in the sense that both the pile and the soil are regarded as integral parts of the same solid continuum but with different material properties, the pile-soil system is globally mapped into a simpler geometric shape by implementing an appropriate coordinate transformation.

The resulting mapped domain is then

discretized and subsequently solved using finite elements of the mapped space. The generic method, on which this strategy is based, is referred to as the 'Mapping FEM'. It has been used successfully for the accurate and economical analysis of problems in solid mechanics [16,17], geomechanics [18,19], as well as for solving a variety of partial differential equations

describing a range

of physical phenomena [16,20-22]. In this paper typical solutions to the laterally loaded single pile embedded in linearly elastic soil, obtained by the mapping F~M, are presented and compared with corresponding results due to conventional methods. The objective is to dzaw attention to the enormous potential of the mapping FE~ for the accurate analysis of laterally loaded single piles simply, elegantly and without making simplifying ass~c~tions.

THEORY Let the Cartesian triplet, (xl,x2,x3), generate what we will henceforth call the x-spruce which contains the physical problem domain, Z x (that is, the physical pile-soil domain).

Now consider the coordinate transformation xi

=

xi(ul,u2,u3)

(I)

in which the orthogonal curvilinear triplet, (ul,u2,u3) , generates the mapped space which we will henceforth refer to as the u-spruce. When this transformation is i~@lemented, Z x will map into Z u in the u-space, and the Cartesian strain components in the x-space will transform into their respective counterparts in the u-space in accordance with the metric tensor quantities given by [17,18]

grs = (~xk/aur) (~xk/auS)

(2)

258 in which the rumination convention is implied. Since the x-u transformation is usually between orthogonal coordinate systems, grs = 0

for r # s

(3)

Clearly, the constitutive relationship will also undergo a transformation when the physical problem domain lacks isotropy. It follows,

therefore,

that if the above mapping strategy is adopted,

then solution to the problem in the original physical domain ( I x) with the conventional F~M [23] will be equivalent to discretizing and solving the problem in

[ u using finite elements of the u-space, provided that the prescribed

boundary conditions and external

loading are also appropriately transfoznned

[16-18]. Once the explicit form of equation (i) has been chosen, the properties of u-space finite elements to be used for discretizing from appropriately

transformed relationships.

[u

will be found

The element stiffness matrix,

[k], for example, will now be calculated from 1 1 1

(4)

[k] = / / / [ B l T [ D ] [ B l d e t [ J l ] d e t [ J 2 ] d ~ d ~ d r -I-i-I

in which [B] and [D] denote strain and elasticity matrices in the u-space. The Jacobian matrices,

[Jl ] and

[J2 ] , refer to the x-u transformation and that

between (ul,u2,u 3) and (~,~,~) coordinate systems, respectively. The remaining steps to the FE solution in the u-space with mapped elements are identical to those of the conventional F~M [23] in the physical domain.

THE PROBLEM AND ITS MAPPING FINITE EL~M~2~f SOLUTION Figure 1 shows the geometry of the uniform free-head pile under consideration. Its Young's modulus and

Poisson's

ratio

are ~

and Vp, while the

respective properties of the soil are E s and vs. The soil medium is underlain by a rigid stratum as shown. The theoretically

infinite

lateral

approximated to the imposed terminator, surface

concentric

with

the

pile.

expanse of the soil m e d i ~

Because

reasonably assume that all displacements

i~ now

$2, which is a circular cylindrical the problem

is static,

vanish on S 2 provided

we may

that it is

sufficiently remote from the pile. The space enclosed by SI, $2, and x 3 = 0 and x 3 = (L + ~L) is thus the physical domain,

Zx, of the problem as shown in

Figure l(b) (a symmetric half, which only need be considered, is shown).

259 X2

14 (a)

8L

~:

{b)

~zcjume 1 (a) Geometry of l a t e r a l l y loaded p i l e considered; (b) d e f i n i t i o n of the physical problem domain I x " The geometry of Z x clearly shows that the cylindrical-polar

coordinate

transformation should be implemented for mapping, because this wDuld lead to a canonical mapping as S 1 and ~

are concentric circular cylindrical surfaces.

Then, for this particular transformation,

the explicit form of equation (i)

becomes

and The mapping of Ix into

xl =

ulcos(u 2)

(5a)

x2=

ulsin(u 2)

(5b)

x 3--

u3

(5c)

Zu, achieved with this transformation, is shown

in Figure 2. This Figure also shows a schematic discretization with the HR20 mapped solid element, employing reduced One-point integration for shear, which was

used for solution in Zu.

The cylinder of soil (diameter D and height L)

t

~$2

L i

1 ~L

i I --

Figure 2 Mapping of the physical domain of Figure l(b) into the mapped domain with the coordinate transformation of equations (5).

260 under the pile is excluded for reasons of economy.

This is acceptable because

tests clearly showed that it had negligible effect on the response of piles subjected to horizontal elements,

forces.

Since the details of HR20 and other mapped

and assessment of their performance v i s a

counterparts,

are given elsewhere

[17,18,24],

vis their conventional

we will not repeat them here

except to mention that the HR20 element is a mapped, twenty-noded high-order element capable of a high degree of accuracy. The problem of laterally loaded piles is characterized by the loss of contact between the pile and the soil which occurs behind circular piles)

(and around

the deflecting pile. The criterion of 'no-tension'

in

[23] was

invoked to deal with this as follows: regions of the pile-soil interface where tensile radial stresses occurred in the soil were identified from the results of the first solution. Soil elements representing such regions were assigned nominal elasticity, that the

and

no-tension

element

boundaries

elements were

(normal to u 2) were adjusted so

distinct from those in which

radial stresses occurred at the pile-soil

interface.

compressive

The next solution was

then obtained. The plan was to repeat this process a ntm%ber of times until no new tensile regions were discovered at the interface; but, in the event this was not found to be necessary because no new tensile regions at the interface were discovered after the second pass.

RESULTS Assuming both pile and soil to be linearly elastic, a number of test runs were made with

~=

1.2,

~ = 2.5, and

Vp = 0.2;

vs was taken as 0.485 for

comparison with published work in which it is taken as 0.5. The length of the pile, L, was varied within the range IOD-50D to give slenderness ratios (A = L/D) varying from I0 to 50 for purposes of comparison with published results. The free pile-head was

subjected

to

horizontal

force,

H, applied at the

ground surface.

Pile ~ Results

~

l~mogenemm S o i l

of tests

showed

that

pile deflection

depended

on both

slen-

derness ratio and the pile flexibility factor, Kr, defined as

-- ~ E s in which ~

L4

<6)

denotes the second moment of area of pile. This finding regarding

the significant

effect of the L/D ratio accords with that of Poulos

[II],

whereas Banerjee and Davies [12] ~ i n t a i n that it has little effect on results

261

50

. .. POulOS, ref. 11 • Mapping FEM Kr= Eplp/E+~

20 10

10

I

I

166 10 s

I

I

I

I

I

I

164

16 3

16 2

1@

1

10

Kr

Typical mapping F~4 results compared with those by Poulos (free-head pile, homogeneous soil). Over the complete practical range of K r values. Typical results by the mapping FEM

are compared with published results in Figure 3.

In both Figures 3 and 4

the horizontal deflection (u) of the free-head pile at the ground surafce is given by u = IhH/~s T

It is seen from Figure 3 that whereas

(7)

good agreement is achieved b e ~

present and Poulos's results for relatively flexible piles, there is significant discrepancy in the case of stiffer piles. This is probably symptomatic of the

conceptual difference between what

we have

called

semi-continuum and

Io 3

Banerjee & Davies, ref. 12

- -

10 2

X = E(O)IE(L)

•

10 1.0

i

106 105

F ~

i

104

I

I

1-3 0 KrlO-2

i

I

I

1~

1

10

4 Typical mapping FEM results compared with those due to BanerJee and Davies (free-head pile, soil modulus linearly increasing with depth)

262 truly-continuum approaches to the problem.

Pile ~

in ~

Soil

Figure 4 shows typical results for the laterally loaded pile when the soil is nonhomogeneous,

its Young's modulus

E(0) and E(L) denote values of tively.

Clearly,

Es

increasing linearly with depth;

at ground surface and at x 3 = ~L, respec-

the discrepancy between mapping F~M results,

and those by

Banerjee and Davies [12] in which boundary elements are used together with the fundamental

solution for point loads acting at the interface of a two-layer

elastic half-space,

is significant. However, since their solution is approxi-

mate, mapping F~M solution is thought to be more accurate.

OBS~irATIONS Although in this study we have only considered single piles subjected to a translational force at the ground surface, clearly the method can be extended, without difficulty, to analyse problems of piles subjected to a moment or torsion. Problems of axially loaded piles and pile driving can also be analysed with ease but, because these problems are axi-symmetric

in the physical

domain, their mapping FE solution (which reduces to plane-strain analysis in such cases) would be equivalent to conventional axi-symmetric FE analysis in which the pile itself is also modelled by finite elements. Whereas the advantages

of

the mapping

FEM over conventional

FEM are many

[16-22],

in the

context of single pile analysis generally the advantages of the mapping F~M are these: (a)

Assumptions,

which have to be made

in conventional

analyses regarding

stress distribution at the pile-soil interface along the pile and across its width or around it, are dispensed

with in the

mapping F~M (in fact,

actual stress distributions around and along the pile are obtained from the mapping F~M as a useful by-product). that results due to the with

It is considered,

therefore,

proposed mapping F~M accord more realistically

the actual physics of the problem than do those by conventional

methods. (b)

In the mapping FroM, as in the conventional pile-soil 'interface'

interface

can

be

realistically

F~M, soil behaviour at the modelled

using

appropriate

elements [25], to mimic elasto-plastic behaviour for example.

Such refinement is not possible in Winkler-type (including p-y) tion, and can be difficult in semi-continuum type analyses.

formula-

263 (c)

Nonhomogeneous and/or nonlinear soil behaviour can be dealt with in the mapping F~4 accurately and without any difficulty. While the treatment of nonhomogeneity is simple and obvious, nonlinearity can be dealt with by implementing a suitable nonlinear algorithm, as in conventional F ~ . Tnis contrasts with the inadequate and empirical p-y method which is extensively used for analysing laterally loaded piles embedded in nonlinear soils. The p-y method is essentially an expedient empirical device which simplifies problem solution; but it has no basis in reality, not least because

it does not acknowledge the cross-coupling between what

are

assumed to be independent nonlinear p-y springs placed along the pile to represent nonlinear soil. (d)

The mapping F~4 offers a simple strategy with which the effect of loss of contact between the pile and the soil, anywhere on the pile-soil interface and caused by whatever reason, can be studied accurately and with ease (eg. loss of contact behind the pile considered earlier).

Similarly

the effect of increased local resistance, caused by a buried hard rock pressing against the pile, for example, can also be predicted without difficulty. (e)

In general, the mapped domain is much simpler in shape than the parent physical domain, and this enables easier discretization and data organization in the ~apping F~M.

When mapping is canonical, as in the problem

considered in this paper, solid rectangular hexahedral elements can be used to model the entire mapped domain, thus avoiding potential errors due to element distortion [17,24,26]. Even when mapping is not canonical, as would be the case when the

cross-section of the pile is noncircular,

the mapped domain would still be considerably simpler in shape than the parent d~-,ain, and it would still be possible to use rectangular hexahedral elements to model the mapped domain, except the pile itself and its immediate neighbourhood whose FE modelling would necessitate the use of general hexahedra. From practical considerations it is clear that a simple yet accurate, reliable and general solution algorithm is required for the analysis of laterally loaded piles and pile groups. Such an algorithm should be capable of application to soils which have one or more of these characteristics in any ac%nissible combination:

linear,

nonlinear,

homogeneous,

nonhomogeneous

or

layered. Moreover, the algorithm should not be based on assumed stress distribution at the pile-soil interface and should be valid for a wide range of pile flexibility factors

(Kr)

and slenderness ratios,

as well as for piles whose

264 cross-sections very with length. Given its conceptual simplicity and the high degree of accuracy it is capable of [16-18,24], the mapping F~M appears to be eminently suitable for such an algorithm which could subsequently be incorporated into an expert system. Current work is aimed at the development of a general mapping FE algorithm for single piles which complies with the specifications stated above, including the development and testing of low-order mapped solid Lagrangian elements and mapped high-order dedicated plate bending elements that are able to simulate the bending aspect of the pile accurately. Results of this work will be published in detail in due course.

CONCLUSIONS In the rapping F~4, proposed in this paper for the simple and accurate analysis of laterally loaded single piles, asstm~tions regarding the distribution of soil stresses at the pile-soil interface are dispensed with, unlike conventional methods in which such assumptions have to be made. The proposed mapping method also has other significant advantages over conventional methods and is capable of easy generalization with respect to both geometric and material properties of both pile and soil.

ACKNDWLEDG~MENT Sincere thanks are due to the Board of R. E. International (Services) of London, England, for supQorting this work. R~ ~ENCES I.

Broms, B. B. 'Lateral resistance of piles in cohesive soils', Journal of the Soil Mechanics and Foundations Division, ASCE, 90, SM2, 27-63 (1964).

2.

Broms, B. B. 'lateral resistance of piles in cohesionless soils', Journal of the Soil Mechanics and Foundations Division, ASCE, 90, SM3, 123-156 (1964).

3.

Hetenyi, M. Beams on Elastic Foundation, University of Michigan Press, Ann Arbor, Michigan, 1946.

4.

Reese, L. C. & Matlock, H. 'Non-dimensional solutions for laterally loaded piles with soil modulus proportional to depth', in Proc. 8th Texas Conf. Soil Mech. Fdn. Engng, 1-41 (1956).

5.

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6.

Matlock, H. & Ril~perger, E. A. 'Measurements of soil pressure on a laterally loaded pile', Proc. Am. Soc. Test. Mater, 58, 1245-1259 (1958).

265 7.

Matlock, H. 'CDrrelations for design of laterally loaded piles in soft clay', in Proc. 2nd Offshore Technical Conference, Houston (Texas), I, 577-594 (1970).

8.

Reese, L. C., Cox, W. R. & Koop, R. D. 'Field testing arK] analysis of laterally loaded piles in stiff clay', in Proc. 7th Offshore Technical Conference, Houston (Texas), 2, 473-483 (1975).

9.

Randolph, M. F. 'The response of flexible piles to lateral loading', Geotechnique, 31, No. 2, 247-259 (1981).

I0.

Spillers, W. R. & Stoll, R. D. 'Lateral response of piles', Journal of the Soil Mechanics and Foundations Division, ASCE, 90, SM6, 1-9 (1964).

ii.

Poulos, H. G. 'Behaviour of laterally loaded piles: I -- single piles', Journal of the Soil Mechanics and Foundations Division, ASCE, 97, SM5, 711-731 (1971).

12.

Banerjee, P. K. & Davies, T. G. 'The behaviour of axially and laterally loaded single piles embedded in nonhomogeneous soils', Geotechnique, 28, No. 3, 309-326 (1978).

13.

Davies, T. G. & Budhu, M. 'Nonlinear analysis of laterally loaded piles in heavily consolidated clays', Geotechnique, 36, No. 4, 527-538 (1986).

14.

Wilson, E. L. 'Structural analysis of axisymmetric solids', Journal Am. Inst. ~ r . Astr., 3, 2269-2274 (1965).

15.

Kuhlemeyer, R. L. 'Static and dynamic laterally loaded floating piles', Journal of the Geotechnical Engineering Division, ASCE, 105, @'92, 289-304 (1979).

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in

Geotechnical

finite

NOTATION

[B]

Strain matrix in the u-space.

[D]

Elasticity matrix in the u-space.

D

External diameter of pile.

~,

Es

young's moduli of pile and soil, respectively.

E(0), E(5)

Values of E s at ground surface and at x 3 = ~L, respectively.

H

Horizontal point force applied to pile at ground surface.

Ih

Horizontal displacement influence factor.

Ip

Second moment of area of pile.

[Jl ]• [J2 ]

Jacobian matrices of coordinate transformation. Pile flexibility factor.

L

Length of pile.

S1

Outer surface of pile.

S2

Surface of the imposed far terminating boundary.

u

Horizontal pile deflection at ground surface.

ul,u2,u 3

Orthogonal coordinates defining the u-space.

xl,x2,x 3

Orthogonal (Cartesian) coordinates defining (physical) x-space. Physical problem domain. Mapping of Ix in the u-space. Slenderness ratio of pile (= L/D).

Vp

Poisson's ratio of pile.

Vs

Poisson' s ratio of soil.

Received 28 February 1989; revised version received 2 June 1989; accepted 8 June 1989