Flexibility coefficients and interaction factors for pile group analysis

BAHAA EL SHARNOUBY AND MILOS NOVAK Faculty of Engineering Science, The University of Western Ontario, London, Ont., Canada N6A 5B9

Received November 4, 1985 Accepted May 2, 1986

Flexibility coefficients of single piles and interaction factors established for groups of two piles are presented to facilitate analysis of arbitrary pile groups exposed to static horizontal loads. Such an analysis may yield pile group flexibility, stiffness, deflection, and distribution of loads on individual piles. The data given are complete in that they include horizontal translation, rotation in the vertical plane, and cross effects between the two, making it possible to establish complete stiffness and flexibility matrices of pile groups provided with either rigid caps or arbitrarily flexible caps. Homogeneous, parabolic, and linear (Gibson's) soil profiles are considered and the piles may have a free length sticking above the ground surface. The methods of group evaluation based on superposition of interaction factors are reviewed and compared and numerical examples are given.

Key words: piles, pile groups, lateral loads, flexibility, stiffness, load distribution.

Les coefficients de flexibilitk des pieux simples et les facteurs d'interaction Ctablis pour des groupes de deux pieux sont prCsentCs pour faciliter l'analyse de groupes arbitraires de pieux exposCs ?I des charges statiques horizontales. Une telle analyse peut foumir la flexibiliti, la rigidit&, la deflection et la distribution des charges pour un groupe de pieux. Les donnCes foumies sont completes en ce qu'elles comprennent la translation horizontale, la rotation dans le plan vertical et les effets de I'un sur l'autre, rendant possible la definition complete des matrices de rigidit6 et de flexibilitC de groupes de pieux pourvus soit de casques rigides ou arbitrairement flexibles. Des profils de sol homogenes, paraboliques et IinCaires (de Gibson) sont considCrCs, et les pieux peuvent avoir une longueur libre au-dessus de la surface du sol. Les methodes d'evaluation de groupe basees sur la superposition des facteurs d'interaction sont passCs en revue et comparCes, et des exemples nurneriques sont donnCs.

Mots clks : pieux, groupes de pieux, charges IatCrales, flexibilitC, rigidit&, distribution de charge.

Can. Geotech. J . 23,441-450 (1986) [Traduit par la revue]

Introduction It has been recognized for quite some time that piles in a

group interact with each other, increasing group settlement, redistributing loads on individual piles, modifying group stiffness, and under dynamic conditions, changing its damping.

Static analysis of these group effects was pioneered by Poulos (Poulos 1968, 1979; Poulos and Davis 1980), employing Mindlin's (1936) solution for the displacement field in an elastic half-space, and by Banerjee and Butterfield (Butterfield and Banerjee 1971; Banerjee and Driscoll 1976; Banerjee 1978; Butterfield and Ghosh 1980), using the boundary element method. A direct solution suitable for static and low-frequency response of large groups was formulated by El Sharnouby and Novak (1985). In the latter paper a detailed review is provided of the more general dynamic studies of pile groups, from which the static case follows as a special case for frequency approach- ing zero. More recent studies of this type are due to Sen et al. (1985) and Nogami and Paulson (1985).

Analysis of pile groups can be conducted in two ways: (a) accurately (in the theoretical sense) using a computer-based direct analysis of the whole group and (b) approximately using superposition of interaction factors. The direct analysis is preferable because it is accurate within the validity of the assumptions made and provides more information. The advan- tages of the interaction factor approach are that it is simple and the analysis can be conducted by means of a small computer programme or even longhand calculation; the only conditions are that a complete set of sufficiently accurate interaction factors be available and that the conditions of the current case do not preclude their use.

In a recent study, the authors compared their direct solution with the interaction factor approach for static and low-frequency loading (El Sharnouby and Novak 1985). It appeared that the interaction factor approach works very well for horizontal loading of all piles and vertical loading of friction piles, even for

very large groups; only for vertical loading of shorter, end- bearing piles does the accuracy of the interaction factor approach deteriorate, exaggerating the interaction effects. Thus, in many practical situations the interaction factor approach may provide sufficient accuracy.

The concept of interaction factors was introduced by Poulos (Poulos 1979; Poulos and Davis 1980), who, over the years, presented a wide range of charts for their evaluation and together with Randolph (Randolph and Poulos 1982) suggested empirical formulae for their calculation. Butterfield and Douglas (198 1) also published design curves facilitating pile group analysis but their data are limited to certain pile-soil parameters, pile arrangements, rigid caps, and small groups.

It is the purpose of this paper to remove some of the above limitations and to present flexibility coefficients and interaction factors that would facilitate the analysis of pile groups, allowing for either rigid or flexible caps, a broad range of pile-soil parameters and soil profiles, and pile separation from the soil. The paper complements the data previously presented by the authors (El Sharnouby and Novak 1985; Novak and El Sharnouby 1985). The study is limited to static horizontal loading of pile groups because there is already a sufficient number of charts available for vertical pile loading (e.g. Poulos and Davis 1980; Randolph and Poulos 1982; Novak and El Sharnouby 1983).

Method of analysis The method used in the analysis is described in full detail

elsewhere (El Sharnouby and Novak 1985) and therefore only its basic features and assumptions are summarized here for completeness.

The basic idea of the approach is to view the whole pile or group with the soil as one composite continuum whose conditions of equilibrium are specified for a number of discrete points (nodes). The conditions of equilibrium are expressed in

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.

CAN. GEOTECH. J. VOL. 23. 1986

PlLE NODE

X POINT LOAD

0 REFERENCE POINT

TYPICAL PlLE ELEMENT

( b ) POINT- LOAD SYSTEM ON PlLE

FIG. 1. Layout of pile elements and equivalent point loads for analysis of horizontal response of piles.

terms of the stiffness method in which the structural stiffness of the piles is combined with the stiffness of the soil medium. This idea is common to many of the approaches already in use; the principal differences are in the ways in which soil stiffness constants are established.

The piles are assumed to be vertical and of constant circular cross section. Each pile is divided into a number of elements of varying length (Fig. 1). A total of 12 elements was found to yield sufficient accuracy, with the uppermost element length being equal to one-fourth of the average element length and the length of the other elements increasing with depth. This variable element length yields greater accuracy than do equal elements. The nodal points are located on the axis of the pile, with the first and last nodes located at the top and bottom of the pile respectively.

The element stiffness matrix for pile bending is 4 X 4 and has the standard form [k] relating the translation u and rotation + to end forces P and moments M . Because external moments act only at the top nodes, the pile stiffness matrices are condensed by eliminating all nodal rotations except those of the butt.

The stiffness of the soil is derived using the Mindlin (1936) solution for the displacement field generated in the interior of the elastic half-space by a horizontal static point load. To simplify the analysis, the vertical displacements and the horizontal displacements normal to the applied loads are neglected because these displacements are much smaller than the principal horizontal displacements.

The nodal soil flexibility coefficients are generated by the application of discrete point loads located on the vertical axis of the pile. Each element is acted upon by two equal point loads at a distance of Li/4 from the tributary soil element ends (Fig. 1). This load system has been chosen because it yields displace- ments at the reference points almost equal to those produced by uniform horizontal stresses.

The displacements at the surface of the pile at a distance equal to the radius r vary. However, it was found that the average displacement around the pile is equal to the displacement at a surface point at 45O from the x-axis of the pile (the direction of

piles, the flexibility coefficient is calculated at and related to the pile node, i.

With the pile and soil stiffness established and superimposed, the analysis of single pile and pile groups readily follows for any case of interest. Symmetry of a group can be exploited to reduce the computing time. The Mindlin solution is strictly valid only for a homogeneous medium, but it can be used as an approximation even for nonhomogeneous soil if the representa- tive modulus is taken as the average value of the modulae at the two stations i , j being considered when evaluating the flexibility coefficient FV. This approximation was adopted for nonhomo- geneous soil profiles.

The whole analysis was efficiently computerized and flexibil- ity coefficients and interaction factors calculated for three profiles of Young's or shear modulus: homogeneous, parabolic, and linear (Gibson soil). The parabolic profile follows the equation of the quadratic parabola and reflects the well-known observation that soil shear modulus is proportional to the square root of confining pressure. In addition, a free length of pile sticking out of the ground is considered, which makes it possible to account for either such a free length or, approximately, for pile separation from soil often anticipated under cyclic loading.

Horizontal pile flexibilities and horizontal interaction factors were calculated and are presented in the form of charts and simple analytical expressions. The presentation of the charts and formulae is simplified by the observation that for piles whose length L 10 diameters, the horizontal interaction factors and flexibilities are largely independent of pile length. The charts are most useful for the design of single piles or small pile groups while the formulae can facilitate the analysis of larger groups using a computer.

The piles are assumed to be vertical and of constant circular cross section; for other piles an equivalent circular cross section can be established.

Flexibility coefficients of single piles If a pile head is loaded by a horiz~ntal force P and moment in

the vertical plane M the horizontal displacement of the head u and its rotation 0 can be expressed in terms of the flexibility coefficients J, as

The flexibility coefficients J l p , JuM, fop, and feM are displace- ments and rotations caused by units loading by P = 1 and M = 1 respectively, as indicated in Fig. 2, and J d M = fop.

For piles whose head cannot rotate (fixed-head piles), the horizontal force causes only horizontal translation and there is only one flexibility coefficient fuF equal to u / P . This coefficient can be determined by equating [I] to J I F P and setting the rotation equal to 0, which yields

J 0 M

Alternatively, f,, can be calculated directly from the governing equations of the pile.

loading). DimensionlessJexibility coefJicients The average established in this way for To facilitate the use and presentation of the flexibility coeffi-

point 0 is taken as the soil flexibility coefficient related to the cients, dimensionless flexibilities F , can be introduced as pile node i. Thus, a soil flexibility coefficient f i ; is eaual to the ;verage of the horizontal displacements at p o i n < ~ on'element i Flip = dEsfup FUF = dEsf i l~ due to two equal unit loads acting on element j. This definition [31 holds for the elements of one pile; for the elements of different F(,M = d ' E ~ , , M F ~ M = d 3 E s f w

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.

EL SHARNOUB IY AND NOVAK 443

FIG. 2. Generation of flexibility coefficients of single piles.

where d is pile diameter and E, is Young's modulus for the soil at the pile tip. (Similar expressions have been used by others).

Conversely, the true pile flexibility coefficients can be calculated using the dimensionless flexibility coefficients Fv as

The dimensionless flexibilities Fu are plotted versus the pile stiffness/soil stiffness characteristic ratio E,/E,, where E, = pile Young's modulus, in Figs. 3-5. The dimensionless flexibilities are given for the three soil profiles considered, i.e., homogeneous, parabolic, and linear. (Notice the two different scales in Figs. 4 and 5 .)

The charts were calculated for fully embedded piles with L l d = 25, E, = 10 MPa, variable E,, and soil Poisson's ratio v, = 0.5, but they can be used for other pile-soil data as well if L l d 3 10 for homogeneous soil and L l d 3 12.5 for the other two soil profiles. For the parabolic and linear (Gibson) soil profiles the nominal value of E, has to be specified for the depth of 25 pile diameters. The Poisson's ratio effect is not very strong particularly when the data are presented in terms of the E,/E, ratio. This generalization was verified by an extensive para- metric study.

From Fig. 3 it can be seen that FeM and FlrM change more dramatically with E,/E, than do F,,p and F,,,. The figure also shows that Fl,p is markedly larger than FldF as expected. The change in flexibilities with pile-soil stiffness ratio is more pronounced for linear and parabolic soil profiles (Figs. 4 and 5). The magnitudes of the flexibilities increase significantly with the changing of the soil profile from homogeneous to parabolic to linear. This indicates the importance of determining the soil profile correctly and the need to account for the reduction of the soil modulus towards ground surface associated with the reduction of confining pressure.

The dimensionless flexibilities vary with the ratio Ep/Es in a smooth fashion and therefore can be fitted by simple analytical expressions. A suitable expression for the dimensionless flexi- bility is

in which a and b are constant. These constants were established using an optimization subroutine and are given for the three soil profiles and L / d > 10 in Table 1. (Similar expressions were proposed by Kuhlemeyer (1976), Blaney et al. (1976), Krish- nan et al. (1983), Gazetas (1984), and others.)

E P / E ~

FIG. 3. Dimensionless pile flexibilities vs. pile-soil stiffness ratio for homogeneous soil profile.

FIG. 4. Dimensionless pile flexibilities vs. pile-soil stiffness ratio for parabolic soil profile.

E p / ~ S

FIG. 5. Dimensionless pile flexibilities vs. pile-soil stiffness ratio for linear soil profile.

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.

444 CAN. GEOTECH. 1. VOL. 23, 1986

TABLE 1. Constants a and b for the calculation of single-pile flexibility by [51

Soil Profile

Homogeneous Parabolic Linear

Flexibility a b a b a b

FUP 1.20 -0.160 6.95 -0.25 28.13 -0.34 F,F 0.76 -0.172 3.73 -0.27 12.22 -0.35 F,,,u 1.90 -0.425 6.78 -0.47 20.80 -0.547 FeM 8.80 -0.72 15.30 -0.73 26.70 -0.769

With the dimensionless flexibilities established from [ 5 ] or Figs. 3-5 the flexibility constantsfi,. follow readily from [4].

Effect of pile free length on pile jexibility It is well known that if the pile has a free length sticking out of

the ground its flexibility related to the butt increases. This effect is particularly marked for flexible free-head piles. In some cases, the free length is actually present; in others, a free length may be assumed even for a fully embedded pile in order to account in an approximate way for pile separation from the soil, which often occurs especially under cyclic loading. Such a separation or loosening of the piles was observed on offshore towers in the North Sea during storms. Continuous monitoring of the tower vibration has shown that during a heavy storm the fundamental natural frequencies of the towers distinctly de- crease but they slowly return to the original value as the soil around the piles recovers. ' Thus it is useful to examine the effect of pile free length (pile separation from soil) on pile flexibility.

The free length of different magnitudes e was included in the analysis and the results are plotted in Figs. 6-9. The curves plotted in these figures show the relative variation in pile flexibility with pile separation ratio e / d and stiffness ratio Ep/E, in terms of the dimensionless flexibility amplification coefficient CU. This coefficient CU is defined as the ratio between the flexibility of the pile with a separation e / d and the flexibility of an identical fully embedded pile, i.e.,

This dimensionless coefficient can be applied as a correction factor to the flexibility of a fully embedded pile, giving

[7] FU(e/d) = FU(e = O)CU

The subscripts used for CU are consistent with those used to identify the flexibility coefficients fi,. and Fij. A complete set of the ratios CU is given for the three soil profiles in Figs. 6-9. The effect of pile separation can be seen to be significant and more dramatic for the homogeneous soil profile than for the parabolic and linear soil profiles. This is understandable because the homogeneous soil provides much more lateral support to the uppermost part of the pile than do the other two soil profiles.

Interaction factors In the second part of the study, the method described was

applied to a group of two piles and used to analyze their interaction and to evaluate interaction factors for a number of situations.

The interaction factors are calculated considering two iden- tical piles equally loaded and are defined as

'Private communication with Dr. L. R. Wootton.

-- Soil Profile - - Homogeneous +

FIG. 6. Variation of flexibility coefficient C,,p with separation ratio e l d and stiffness ratio Ep/Es for different soil profiles.

FIG. 7. Variation of flexibility coefficient COP with separation ratio e l d and stiffness ratio Ep/Es for different soil profiles.

Deflection of reference pile due to load

01.. = on adjacent pile

" Deflection of reference pile under its own load

For larger groups, these interaction factors can be superimposed to yield the total behaviour of the group.

When evaluating the interaction factors, only the displace- ments occumng in the vertical plane parallel to the plane of load application are considered; the displacements perpendicular to that plane are neglected as insignificant, if they are present.

For the dimensionless ratios Ep/Es and S l d , where S is pile

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.

EL SHARNOUBY AND NOVAK 445

Soil Profile Homogeneous

--- Parabolic - - - - - - - Linear

FIG. 8. Variation of flexibility coefficient CeM with separation ratio e l d and stiffness ratio Ep/Es for different soil profiles.

I 0 I 2 3 4

e'd FIG. 9. Variation of flexibility coefficient CItF with separation ratio

e l d and stiffness ratio E p / E , for different soil profiles (fixed-head pile).

1 Soil Profile

- ---

spacing, the interaction factors established are plotted in Figs. 10-14. These charts are for L l d > 10, homogeneous soil, and the extreme values of the angle of incidence P equal to 0" and 90".

The interaction factors are given for free-head piles, i.e., those whose butt is allowed to rotate under the effect of horizontal loading, in Figs. 10-13. These figures actually

FIG. 10. Variation of interaction factor a,p with pile stiffness ratio and spacing for free-head piles.

FREE HEAD

-

- -

-

2 3 4 50.2 0.1 0

FIG. 1 1. Variation of interaction factor a g p with pile stiffness ratio and spacing for free-head piles.

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.

446 CAN. GEOTECH. J . VOL. 23, 1986

FREE HEAD

2 3 4 50.2 0.1 C

FIG. 12. Variation of interaction a , ~ with pile stiffness ratio and spacing for free-head piles.

W I- z o )

s/d d / ~

FIG. 13. Variation of interaction factor OM with pile stiffness ratio and spacing for free-head piles.

0.6

FIXED HEAD

-

-

04!- \ FIXED HEAD

FREE HEAD

LId > 10 - 8 = 90"

-

FIG. 14. Variation of interaction factor a , , ~ with pile stiffness ratio and spacing for fixed-head piles.

- -

2 3 4 50.2 0.1 C

represent a complete set of flexibility coefficients normalized by the deflections of the reference pile and pertinent to a group of two piles. Thus, butt translations as well as rotations are given for a wide range of parameters. Using these interaction factors a complete flexibility matrix for a larger pile group can be established approximately, as will be shown in more detail later herein.

It may be observed that a,, + aOp. This is so because Mindlin's equation does not produce a symmetric flexibility matrix. For practical use, however, aep can be taken equal to a,, in order to satisfy symmetry in the solution, since the difference in most cases is quite small.

In Fig. 14 interaction factors are presented for fixed-head piles. For such piles, the rotations of individual pile heads are assumed to be prevented by a rigid cap. This assumption reduces the number of interaction factors that have to be considered. Horizontal translation of a large cap whose rotation is neglected is then analyzed with particular ease.

Some of the interaction factors presented here are markedly smaller than those due to Poulos and Davis (1980), who used elements of equal length.

For intermediate values of p, the interaction factors may be calculated from the extreme values as suggested by Randolph and Poulos (1983) using the relation

[8] a@) = a(0") cos2 p + a(90") sin2 p This simple expression was found to yield sufficient accuracy.

All the above interaction factors were presented for homo- geneous soil. Hence it is important to examine how they vary

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.

EL SHARNOUBY AND NOVAK 447

6 L / d t I0

j3 = 00"

P 4

\

a-

PARABOLIC DISTRIBUTION \ \\, \,

I UNIFORM DISTRIBUTION HEAD

C - - - - LINEAR DISTRIBUTION -\

L / d 2 10

,3 - 0.0'

P

a- UNIFORM DISTRIBUTION

b.-.-- PARABOLIC DISTRIBUTION / - --- LINEAR DISTRIBUTION \\ I

FIG. 15. Effect of Young's modulus distribution on horizontal interaction factor.

with the soil profile. This is shown for both fixed-head and free-head piles in Fig. 15 in which the effect of Young's modulus variation with depth on the interaction factors is plotted. For small spacing, the interaction factors tend to be slightly smaller for linearly or parabolically increasing modulus than for the constant modulus; for greater spacing ratios, the difference diminishes or even changes its sign for free-head piles. For practical purposes, the effect of soil modulus variation with depth on the interaction factors is generally small and the values of CY given for homogeneous soil profiles in Figs. 10-14 may be used for other soil profiles as well. The flexibility of piles and groups depends, however, on the soil profile, as evidenced by Figs. 3-5.

To facilitate the evaluation of the interaction factors and their inclusion into computer programmes users may write, the curves plotted in Figs. 10-14 were fitted by simple analytical expressions reflecting the dependence of interaction factors on the pile-soil stiffness ratio Ep/Es and the spacing ratio d/S . Hence, the interaction coefficients CYV can be calculated using the relation

In [9], i stands for the displacement and j for the associated force. The constants A , B, and C were evaluated using an optimization subroutine and are given for the extreme values of p equal to 0" and 90" in Table 2. For intermediate values of P, [8] can again be used. Table 2 gives the values of a,,,,,, and aep that are different. However, for the sake of symmetry, and considering all the approximations made, it is recommended to

TABLE 2. The constants A , B, and C for the calculation of interaction factors by [9]

a ij P (deg) A B C

0 ~ , L F 90

0 ~ I , P 90

0 ati~ 90

0 a%p 90

%JM = ~ O P 0 (average values) 90

0 "OM 90

-- - Parabolic

FIG. 16. Variation in the interaction reduction factor R, with separation ratio eld and stiffness ratio E,/Es for different soil profiles and pinned-head piles.

use the values of constants A , B, and C that are the average of a , , ~ and a@,, which are given in the second last group of Table 2.

Effect of pile separation on interaction factors The free pile length affects not only the flexibility of

individual piles but also to a considerable degree the interaction between the piles. The general trend is for the free length to reduce pile interaction. This reduction can be expressed in terms of the dimensionless interaction reduction factor

which is the interaction factor for the separation e /d divided by the interaction factor for the fully embedded pile. Figures 16 and 17 show the values of R versus dimensionless separation e /d for a range of stiffness ratios Ep/E, and the three typical soil profiles, namely the homogeneous, the parabolic, and the linear soil profiles. The interaction reduction factor Rp plotted in Fig. 16 was calculated for the horizontal interaction factor of

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.

448 CAN. GEOTECH. 1. VOL. 23, 1986

- Soil Profile Hwnoseneous

i ------- Linear

o b I I I

I 2 3 4

e'd

FIG. 17. Variation in the interaction reduction factor RF with separation ratio e l d and stiffness ratio E,/E, for different soil profiles and fixed-head piles.

free-head piles sup; however, a detailed parametric study indicated that with an error not exceeding 5%, this reduction factor Rp can also be used for the other pinned-head pile interaction factors a,,,,,, and aeM. Figure 17 shows the reduction factor for fixed-head piles, RF, the only one needed for this type of pile.

The values shown in Figs. 16 and 17 were calculated for S l d = 5 and p = 00; however, with an error of up to about +lo%, these figures can be used for other parameters as well. Thus, using the reduction factors Rp and RF, the interaction factors available for fully embedded piles can readily be corrected for the effect of pile free length or separation. This correction counteracts the effect of free length on single-pile flexibility to some extent.

Analysis of pile groups The flexibility coefficients and interaction factors presented

above can be used to evaluate the flexibility and stiffness of the whole group of piles, bypassing the dependence on a large computer programme for direct analysis. This can be done in a number of ways depending on the type of pile cap and the accuracy required.

General formulation of group flexibility and stzffness When the pile cap is flexible or even missing, the pile heads

can move vertically and horizontally and also rotate in the vertical plane. For vertical piles, the vertical flexibility of individual piles can be taken as independent of the other motions and the flexibility matrix associated with horizontal forces Pi and moments in the vertical plane Mi assembled as follows.

Denote the vector of forces acting on the heads of n piles in a group

{P) = [PIP2 ... M1M2 ...IT

Then, the flexibility matrix lfG] of a group of identical piles has diagonal elements equal to the flexibility coefficients of indi- vidual piles, i.e., f j$, f 13 .. . f ig , f& . . . f&; the off- diagonal elements can be expressed using the interaction factors as a9fUp, and a$'J)fep = a(uuM with i, j = 1, 2, ..., n.

This procedure would be accurate if all the piles of the group were present when evaluating the numerical values off and a. Because an isolated pile was considered when calculating f and two piles when calculating a , the procedure is approximate. Nevertheless, the accuracy of the approximate approach is quite satisfactory, particularly for the horizontal response, as shown by El Sharnouby and Novak (1985).

The inversion of the group flexibility matrix yields the group stiffness matrix [KG] = LfG] - I . This matrix can be complemented by the vertical stiffness submatrix. Then the pile group stiffness

- -

matrix can be combined with the stiffness matrix of the superstructure, or flexible cap, modelled by finite elements and the analysis of the response to any external loads can proceed. For rigid caps, the pertinent displacement conditions can be imposed readily.

This procedure is quite general, but may be tedious. In some cases simpler approaches may suffice.

Simpl$ed evaluation of group stiffness A very simple approximate formula for group stiffness can be

obtained by evaluating the flexibility of a typical representative pile (the reference pile) in the group using the interaction factors without imposing the condition of a rigid cap. Then the horizontal stiffness of a group of fixed-head piles, K,,, can be evaluated as (Novak 1977)

in which k$ = l/f$:,k is the horizontal stiffness of a single pile, i.e. for n identical piles x/& = nkliF, and 4% are the

i

interaction factors between the reference pile (number 1) and all the other piles, with o$f,? = 1 being used for the reference pile. The reference pile should not be at the extremity or right in the centre of the group. Even so, the results of [ l I] vary some- what with the position of the reference pile. Despite this, the very simple [ l l ] yields quite satisfactory results. For pinned- head piles a , , ~ is replaced by and kllF by klip = l/&. For vertical stiffness, [I I.] also applies if the proper constants are substituted.

For rigid caps a more rigorous formular can be derived by imposing identical displacements on all pile heads and using, again, the interaction factors to describe group flexibility. This procedure gives the group horizontal stiffness as (Novak 1980)

[12] K u = k , F C 2 ~ i j i j

in which eij are the elements of the matrix

The matrix [au] lists the interaction factors between any two piles in the group, the diagonal elements orii are all equal to unity and the matrix is symmetrical with dimensions n X n. Fixed-head or pinned-head values are substituted for k,, and aij. The double sum actually indicates summation of all elements of the matrix [E]. With the pertinent constants substituted, [12] applies for the vertical displacement as well.

While [12] is more rigorous than [1.L], the differences between the results of the two are not great. (For four symmetrically arranged piles, the results are identical; this is so because the displacements of such piles are identical even when the cap is not rigid.) The assembly of the matrix [aij] is, however, tedious and matrix inversion is required. Hence, for large groups, a

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.

EL SHARNOUB

direct computer analysis is preferable if a suitable code is available.

Finally, for small groups whose caps are either rigid or com- pletely flexible, group stiffness and flexibility can be obtained by writing the conditions of equilibrium at pile heads using the superposition of the interaction factors and applying the boundary conditions. This procedure is described in detail in the example that follows.

Example A group of five identical piles, shown in Fig. 18, has a rigid

cap with fixed heads and is embedded in a homogeneous deposit. Establish the horizontal stiffness of the group in the direction X using the various approaches available, compare the results with those obtained assuming both parabolic and linear variation of soil shear modulus with depth, and determine the distribution of the total horizontal load P on individual piles. Given: d = 0.3m, L = 7.5m, E, = 30000MPa, E, = 30 MPa, v, = 0.5.

Properties of a single pile-L/d = 7.510.3 = 25, E,/E, = 30 000/30 = 1 000. The dimensionless flexibility can either be read from Fig. 3 as FtrF = 0.23 or calculated using [5] and Table 1. The latter approach gives

FUF = 0.76 X ~ o o o - ~ . ~ ~ ~ = 0.23

Flexibility follows from [4] as fUF = FuF/dEs = 2.55 X

m/kN for the homogeneous soil profile. For the parabolic soil profile the same procedure yields ffl = 6.33 x m/kN; for the linear soil profile fuF = 12.11 X lop5 m/kN.

Interaction factors-For fully embedded piles, interaction factors can be taken as independent of the soil profile. For the separations involved and for P = 0" and 90°, the interaction factors read from Fig. 14 are listed in Table 3. Alternatively, the interaction factors can be established using [9] and Table 2. The values thus obtained are also given in Table 3. The dif- ferences between the values obtained from the chart and those calculated are insignificant. Using the values of a , , ~ established for p = 0' and 90°, the interaction factors for = 45" are calculated by [8]. These values are also listed in Table 3.

With the values given in Table 3, the complete interaction factor matrix can be written as

1 0.19 0.31 0.31 0.20 1 1 0.31 0.20 0.31 2

[I41 [aijI = 1 0.31 0.31 3 Sym. 1 0.19 4

1 5 ! Group stiffness

Calculation by [I 11-Taking the pile No. 1 as the reference pile, the sum of a ' s appearing in [ l 11 becomes

With this value and kllF = 1 If,,,, the group stiffness calculated by means of [ l 11 is

For the parabolic and linear soil profiles respectively the values of 39.30 and 20.54MN/m are obtained.

Calculation by [Id]-The more rigorous [12] requires the

IY AND NOVAK

4Y

FIG. 18. Pile group used in example.

TABLE 3. Interaction factors for example

%F

P Pile S l d (deg) Fig. 14 Eq. [9]

inversion of the full interaction matrix [aij], eq. [14], and the summation of all elements of the matrix [E] = [aij]- l. This sum- mation yields Z Zeij = 2.448 and the group stiffness by [12]

i j

attains a value of 96.00MNlm. This value is only slightly different from the result of [l 11, i.e. 97.55 MN/m.

Calculation using equilibrium conditions-The interaction coefficients can also be used to write the conditions of equili- brium of the pile group and to evaluate load distribution on piles as well as group stiffness.

Because of symmetry, the displacements of piles No. 1 ,2 ,4 , and 5 are identical and these piles carry identical loads. Using the interaction factors, the displacement of pile No. 1 due to so-far-unknown loads on individual piles, Pi, is

ul = f U ~ [ ( l + a 1 2 + a 1 4 + a15)Pl + ai3P31

For the displacement of the third pile, u3, it is similarly

U 3 = f,rF[(a31 + a 3 2 + a34 + a35)Pl f 1 P3I

Substituting for a ' s from Table 3 or [14], these equations become [15a] ul = fuF(1.7Pl + 0.31P3)

[15b] ~3 =f,,F(1.24Pl + P3)

Also,

[15c] P = 4P, + P3

For a rigid cap, ul = u3, and the solution of [15] yields

P1 = 0.214P, P3 = 0.144P, U; = o.408fUFP

The loads are not distributed uniformly, with the middle pile

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.

450 CAN. GEOTECH. J. VOL. 23, 1986

TABLE 4. Results of direct analysis of exalnple pile group

Soil profile

Homogeneous Parabolic Linear

Single pile flexibility f , 4 ~ ( m / m x loe5) 2.55 6.36 12.00

Pile group stiffness Ku (MN/m) 95.81 39.72 22.02

Individual pile stiffness kl = kZ = k4 = k5 (MN/m) 20.8 1 8.53 4.68 k3 (MN/m) 12.53 5.60 3.29

carrying less than the others, and the deflection of the group is significantly greater than the deflection of an isolated pile loaded by P/5. The latter loading would result in the pile deflection ui = 0.205,F'P.

For stiffness of the group with a rigid cap K,, u l = u3 = 1 and P = K,, and [15] gives K,, = 96.12 MN/m for the homo- geneous soil profile. This value is quite close to those obtained by the other approaches and is almost identical with the value of 96.00 MN/m obtained from [12]. This is so because the deriva- tion of [12] also satisfies the equilibrium conditions and is basically the same as the last procedure.

Direct analysis-Finally, the problem was solved using the direct computerized solution due to El Shamouby and Novak (1984b, 1985) in which no use is made of interaction factors. The direct analysis was efficiently programmed and took less than 6 s on the CYBER 351170 computer. For the three soil profiles, the results are listed in Table 4. The values shown are in excellent agreement with those obtained using the interaction factor approach. Similarly, good agreement can be expected for other pile systems under horizontal loads.

Concluding remarks The approach proposed is based on the assumption of pile

and soil linearity, which is quite justified for small displace- ments. For these conditions experiments with model pile groups gave very encouraging results (El Sharnouby and Novak 1984a; Novak and El Sharnouby 1984; Novak and Grigg 1976). The pile data given can also be used for low-frequency dynamic loading with group damping evaluated as proposed by El Sharnouby and Novak (1985) and Novak and El Shamouby (1985).

For large displacements and associated nonlinearity no rigorous solution is available as yet. An approximate account of it can be made by adjusting the soil properties to the level of strain, allowing for pile separation as suggested in this paper and by considering a weakened zone around the piles as pro- posed by Novak and Sheta (1980). A Winker-type medium also allows for an approximate account of nonlinearity as shown by Nogami and Paulson (1985).

Acknowledgement The study was supported by a grant in aid of research from

the Ministry of Energy, Mines and Resources, Canada, which is gratefully acknowledged.

BANJERJEE, P. K. 1978. Analysis of axially and laterally loaded pile groups. In Developments in soil mechanics. Edited by C. R. Scott. Applied Science Publishers, London, England, pp. 3 17-346.

BANERJEE, P. K., and DRISCOLL, R. M. C. 1976. Three dimensional analysis of raked pile groups. Proceedings-the Institution of Civil Engineers, 61, Part 2, pp. 653-671.

BLANEY, G. W., KAUSEL, E., and ROESSET, J. M. 1976. Dynamic stiffness of piles. 2nd International Conference on Numerical Methods in Geomechanics, ASCE, New York, pp. 1001-1009.

BUTTERFIELD, R., and BANERJEE, P. K. 1971. The elastic analysis of compressible piles and pile groups. GCotechnique, 21, pp. 43-60.

BUTTERFIELD, R., and DOUGLAS, R. A. 1981. Flexibility coefficients for the design of piles and pile groups. Construction Industry Research and Information Association, 6 Storey's Gate, London, England, Technical Notes 108.

BUTTERFIELD, R., and GHOSH, N. 1980. A linear elastic interpretation of model tests on single piles and groups of piles in clay. In Numerical methods in offshore piling. Institution of Civil Engineers, London, England, pp. 109-1 18.

EL SHARNOUBY, B. , and NOVAK, M. 1984a. Dynamic experiments with a group of piles. ASCE Journal of Geotechnical Engineering, 110(6), pp. 719-737.

19846. "PGROUY-A computer program for static and low frequency analysis of pile groups. Faculty of Engineering Science, The University of Western Ontario, London, Ont.

1985. Static and low-frequency response of pile groups. Canadian Geotechnical Journal, 22, pp. 79-94.

GAZETAS, G. 1984. Seismic response of end-bearing single piles. Soil Dynamics and Earthquake Engineering, 3(2), pp. 82-93.

KRISHNAN, R., GAZETAS, G., and VELEZ, A. 1983. Static and dynamic lateral deflexion of piles in non-homogeneous soil stratum. GCotechnique, 33, pp. 307-325.

KUHLEMEYER, R. L. 1976. Static and dynamic laterally loaded piles. Department of Civil Engineering, The University of Calgary, Calgary, Alta., Research Report No. CE76-9, p. 48.

MINDLIN, R. D. 1936. Force at a point in the interior of a semi-infinite solid. Physics (New York), 7, pp. 195-202.

NOGAMI, T., and PAULSON, S. 1985. Transfer matrix approach for nonlinear pile group response analysis. International Journal of Numerical and Analytical Methods in Geomechanics, 9 , pp. 299-316.

NOVAK, M. 1977. Vertical vibration of floating piles. ASCE Journal of the Engineering Mechanics Division, 103(EM1), pp. 153- 168.

1980. Soil-pile interaction under dynamic loads. In Numerical methods in offshore piling, Institution of Civil Engineers, London, England, pp. 59-68.

NOVAK, M., and EL SHARNOUBY, B. 1983. Stiffness constants of single piles. ASCE Journal of Geotechnical Engineering, 109(7), pp. 961-974.

1984. Evaluation of dynamic experiments on a pile group. ASCE Journal of Geotechnical Engineering, 110(6), pp. 738-756.

1985. Pile groups under static and dynamic loading. Proceed- ings, 1 lth International Conference on Soil Mechanics and Founda- tion Engineering, San Francisco, Vol. 3, pp. 1449-1454.

NOVAK, M., and GRIGG, R. F. 1976. Dynamic experiments with small pile foundations. Canadian Geotechnical Journal, 13, pp. 372-385.

NOVAK, M., and SHETA, M. 1980. Approximate approach to contact effects of piles. Proceedings of Session on Dynamic Response of Pile Foundations: Analytical Aspects, American Society of Civil Engineers National Convention, Florida, pp. 53-79.

P o u ~ o s , H. G. 1968. Analysis of the settlement of pile groups. GCotechnique, 18, pp. 449-471.

1979. Group factors for pile-deflection estimation. ASCE Journal of the Geotechnical Engineering Division, 105(GT12), pp. 1489- 1509.

P o u ~ o s , H. G., and DAVIS, E. H. 1980. Pile foundation analysis and design. John Wiley & Sons, New York, Chichester, Brisbane, Toronto, 397 p.

RANDOLPH, M. F., and P o u ~ o s , H. G. 1982. Estimating the flexibility of offshore pile groups. Proceedings of the 2nd International Conference on Numerical Methods in Offshore Piling, University of Texas, Austin, TX, pp. 313-328.

SEN, R., DAVIES, T. G., and BANERJEE, P. K. 1985. Dynamic analysis of piles and pile groups in homogeneous soils. Journal of Earthquake Engineering and Structural Dynamics, 13, pp. 53-65.

Can

. Geo

tech

. J. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y Sa

nta

Bar

bara

(U

CSB

) on

10/

07/1

3Fo

r pe

rson

al u

se o

nly.