Static and low-frequency response of pile groups

BAHAA EL SHARNOUBY'AND MILOS NOVAK Faculty of Engineering Science, The University of Western Ontario, Lorzdon, Onr., Cn~zuda N6A 5B9

Received July 17, 1984 Accepted October 24, 1984

A simple, efficient method is presented for the analysis of large pile groups subjected to vertical loads, horizontal loads, and moments acting either statically or dynamically with low frequencies. The soil profile and the arrangement of the piles may be arbitrary but the solution is particularly efficient for groups with double symmetry or axisymmetry. The group is analyzed directly, avoiding the use of interaction factors, but the interaction factors are also presented. The piles can be of any type, i.e., floating, end-bearing, socketed, or with a pedestal.

Une mCthode simple et efficace est prCsentCe, qui permet l'analyse des grands groupes de pieux soumis ii des charges verticales et horizontales et ii des moments statiques ou dynamiques a basses frequences. Le profil de sol et I'arrangement des pieux peuvent Ctre quelconques mais la solution est particulikrement efficace pour les groupes ii symCtrie double ou axiale. Le groupe est analysC directement, sans passer par I'utilisation de facteurs d'interaction mais ceux-ci sont Cgalement prCsentCs. Ces pieux peuvent Ctre de n'importe quel type, soit flottants, ii risistance en pointe, encastrCs ou avec piedestal.

[Traduit par la revue] Can. Geotech. J. 22, 79-94 (1985)

Introduction Piles in a group interact with one another because they are

connected via the soil and thus the displacement of one pile contributes to the displacements of the others. This pile-soil- pile interaction increases the settlement of the group, redistri- butes the loads on individual piles, and modifies group stiffness as well as 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 displace- ment 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.

Dynamic studies of pile groups are more recent, and were prompted by the design of nuclear power plants, machine foundations, offshore towers, and other large facilities. Wolf and Trbojevic (Wolf and von Arx 1978; Wolf et al. 198 1 ; Trbojevic 1981) made use of the finite element technique while Aubry and Chapel (1981) and Wolf and Darbre (1983) used the boundary element discretization scheme. Approximate analytical solutions were formulated by Gyoten et al. (1980), Penzien et al. (1964), Liu and Penzien (1980), Nogami (1979, I980), Sheta and Novak (1982), and Thiel (1982). A beam on Winkler foundation was assumed by Kagawa (1983). A very efficient scheme based on the displacement field (Green's functions) due to ring loads was developed by Waas and Hartmann (1981, 1984); a similar approach was employed by Kaynia and Kausel(1980) and Tyson and Kausel(1983).

While the dynamic studies brought about considerable pro- gress in analytical capability and understanding of pile group behavior, they are all rather complex and rely on the use of large and often proprietary computer programs.

Static analysis is simpler and was made readily applicable thanks to the concept of interaction factors introduced and made available by Poulos (Poulos 1979; Poulos and Davis 1980). This concept has been quite popular and is very useful, particularly for smaller groups; for large groups, its applicability may suffer from three drawbacks: (1) the evaluation of pile interaction becomes tedious and the use of a computer desirable, (2) additional calculations may be needed to determine the loading of individual piles because with rigid caps all piles are not

'Present address: Alexandria University, Alexandria, Egypt.

loaded equally, and (3) the pile interaction effects may be overestimated, particularly for end-bearing piles under vertical loads. The latter may occur because the interaction factors to be superimposed are calculated for any two piles in the group ignoring the stiffening effect of the other piles.

The aim of this paper is to formulate a method that would eliminate the drawbacks of the interaction factor approach but remain simple, and that would be computationally very efficient, thus facilitating a fast, inexpensive computer analysis of very large pile groups. The method presented is based on Mindlin's displacement field in the elastic half-space. It allows for an arbitrary soil profile and evaluates group stiffness, forces on individual piles, and material damping, making it suitable for static and low-frequency loading. Vertical, horizontal, and rocking responses are considered. The theory is linear, as are most of the other approaches referred to above, but an approximate account of nonlinearity can be made by adjusting the soil stiffness and material damping to the level of strain expected and by incorporating a weakened zone around the pile as shown by Novak and Sheta (1980, 1982). The pile cap is assumed not to be in active contact with the soil.

Method of analysis The basic idea of the approach presented is to view the whole

pile 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 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.

Pile stiffness The piles are assumed to be vertical and of constant circular

cross section. A cross section which is not circular may be replaced by an equivalent circular cross section whose axial and bending stiffnesses are equal to those of the actual pile and whose diameter is representative of the shape and dimensions of the pile profile. The tip may be floating or fixed, or may rest on a pedestal.

For the analysis, each pile is divided into a number of elements (Fig. 1). Nodal points, for which pile displacements are to be specified, are located on the axis of the pile. The first

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FIG. 1. Model of pile for vertical response and location of point loads on pile surface and pile base.

node is placed at the top of the pile and the last node is located at the bottom of the lowest element to accommodate the base reaction.

The structural stiffness matrix of the ith element has the standard form, e.g. for the vertical direction,

in which E, = Young's modulus of the pile, A; = cross- sectional area of the pile, and 1; = element length. If the volume of the pile is included in the evaluation of the soil stiffness, it would be more accurate, but not practically significant, to replace E, by the difference (E, - E,) where E, = Young's modulus of the soil adjacent to the element. The overall (global) structural stiffness matrix of the whole pile group [K,] is obtained by superimposing the individual element stiffness.

Soil stiffness The stiffness of soil is described by means of the Mindlin

solution for the displacement field generated by a static point load applied in the interior of the elastic half-space. These displacements are given by eqs. 9 and 18 in Mindlin (1936). Mindlin's solution also provides the basis for Poulos's pile analysis (Poulos 1968, 1979; Poulos and Davis 1980). Poulos's procedure involves extensive analytical and numerical integra- tions along and around the pile associated with the determination of discrete nodal flexibility coefficients from continuously

ELEMENT I ELEMENT j

( 0 )

,- Average Displacement

Unit Point Load

0 Soil Reference Poinf

( b )

FIG. 2. Influence lines for soil displacement at reference nodes 0 due to point load acting on pile surface.

distributed shears. This rather rigorous but time-consuming procedure makes a direct solution of the whole group im- practical.

To avoid this difficulty and to cut down considerably on computing time, the approach employed in this paper generates the nodal soil flexibility coefficients by the application of discrete point loads applied and located such that the resultant flexibility coefficients are almost the same as those obtained from continuously distributed shears. For the vertical direction this is achieved in the following manner.

The soil cylinder attached to the pile is divided into tributary soil elements (Fig. 1). Displacement influence lines are estab- lished for the reference points, 0, lying on a vertical line on the surface of the pile by first applying a unit point load at the surface of the cylinder at different depths z (Fig. 2a) and then at different positions 0 in the radical direction (Fig. 2b). From these influence lines, the average nodal displacement at point 0, 6, is established for each soil element. Then point load positions that yield a nodal displacement equal to the average displace- ment 6 are sought. This investigation suggests that the average displacement can be very nearly obtained from two equal point loads which act on the cylinder surface at a distance ofLi/4 from the soil element ends and which are related to the reference node 0 by the angles 0 = 80" or a = 40" (Fig. 2). Thus the horizontal distance between the reference nodes, 0 , and the points of load application is d sin 40°, where d = pile diameter.

The average soil displacement established in this way is taken as the soil flexibility coefficient related to pile node i. Thus a soil flexibility coefficient 5, is equal to the average of the vertical displacements at point 0 on the element i due to two equal unit loads acting on element j. This definition holds for the elements of one pile as well as for the elements of different piles.

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EL SHARNOUBY AND NOVAK 81

A similar simplification is adopted for the pile base. A vertical unit load distributed uniformly over the base produces nonuniform displacements, with the value at the base centre being Eb. Almost the same displacement Sb can be generated at the centre of the base by the unit load concentrated as a point load at a distance from the base centre equal to rb = db/n in which db = base diameter. To account approximately for the rigidity of the base, the displacement Sb is corrected by a factor of n / 4 to yield the expected base flexibility, fbb = Sb. The factor n / 4 is the ratio of the displacement of a rigid circular base resting on the surface of the elastic half-space to the displace- ment of the centre of a uniformly loaded circle, with the total load being equal in both cases. The ratio n / 4 is exact for the surface of the half-space. (A more accurate position of the equivalent point load could also be established.) A unit point load applied at the centre of the base is used to generate all the flexibility coefficients fib associated with the displacements of the other pile elements.

Using the flexibility coefficients fi,. andLb the soil flexibility matrix [F , ] is assembled. Its inversion yields the soil stiffness matrix [K,] = [F,]-'. (In actual computing, this inversion is circumvented by means of Gaussian elimination.)

The flexibility coefficients obtained using the described system of point loads are almost the same as those obtained by the analytical and numerical integrations of the displacement function due to vertical shear stresses distributed uniformlv around the periphery of the pile elements or over the base. The distinct advantage of this equivalent point load system is the considerable reduction in computing time and effort without which the direct analysis of large pile groups would not be practical.

In addition to computational efficiency, the approach pre- I sented offers a few other advantages: soil nonhomogeneity is

/ approximately accounted for by calculating the flexibility coefficients f, with a Young's modulus equal to the average of

I

the moduli pertinent to the stations i and j. (This approximate I measure was found to give sufficient accuracy by Poulos and

Davis (1980), who used finite element results for comparison, and is further supported by comparisons described later herein.) The pile tip is relaxed to a degree depending on the stiffness (Young's modulus) of the underlying stratum and of the layer surrounding the tip, and thus a continuous transition from floating piles to end-bearing piles or to socketed piles is provided. Finally, layering under the tip can be accounted for by either considering the average Young's modulus or by extend- ing the piles as soil piles into the underlying stratum.

The stiffness matrix of the soil-pile system, [ K ] , is obtained by superposition of the soil and pile stiffness matrices, i.e.,

Vertical response of pile groups The horizontal displacements associated with the vertical

displacements are small and can be neglected. Then the equilibrium condition can be expressed for all pile nodes in the standard way as

[31 { P ) = [ K l {u)

in which {P) = the vector of vertical forces acting on the nodes and {u) = the vector of nodal displacements.

Computational efficiency depends on the number of elements per pile. It was found that 10 elements per pile yields sufficient accuracy for vertical displacements.

FIG. 3. Node numbering for group of four piles.

Vertical stiffness and settlement of pile groups For an efficient analysis of pile groups, the nodal numbering

can be arranged by layers starting from the top as indicated in Fig. 3.

To generate group stiffness for the case of a rigid cap the displacements of all heads must be equal to unity. For example, for the group of four piles, the vector of displacements becomes

[4] {u) = [l 1 1 1 ug u,j . . . u,IT

where n is the number of degrees of freedom. The forces acting on the pile heads become the stiffness constants of individual piles, ki. Then, the force vector takes on the form

[5] { P ) = [ k , k 2 k 3 k 4 0 0 . . . 0 ] T

Substituting [4] and [5] into [ 3 ] , the unknown displacements are obtained. With these displacements, the stiffness constants of individual piles, ki, follow from [ 3 ] . The total stiffness of the pile group is

[61 KG = C ki

This group stiffness can be used to analyze the response to external loads. The elastic settlement of the group due to total load Q is uG = Q/KG and the loads on individual piles are qi = ki-uG. These loads are not equal; the peripheral piles carry larger loads than the other piles.

With an infinitely Jlexible cap and all the piles carrying the same load, Po, the load vector becomes

[7] { P ) = Po[l 1 1 1 0 0 ... 0lT

and the displacements follow from [3] as {u ) = [K] - ' {P) .

Exploiting symmetry Even with the simplifications introduced above, considerable

computational difficulties are encountered when analyzing large pile groups because the total number of degrees of freedom, and consequently the size of the matrices involved, becomes formidable or even prohibitive. For example, a group of 100 piles divided into elements may lead to a 1000 by 1000

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82 CAN. GEOTECH. J. VOL. 22, 1985

flexibility matrix. Substantial savings in computing time can be type of symmetry. Double symmetry and axisymmetry are achieved if the group is symmetrically arranged or if a discussed. nonsymmetrical group is approximately represented by an equivalent symmetrical group. Then the number of unknown Doubly symmetric pile groups

displacements and system constants is reduced and efficient This type of symmetry is encountered when the group

solution schemes can be formulated to analyze very large pile features two vertical planes of symmetry. The formulation of an

groups by reducing them to much smaller systems without any efficient scheme for the analysis of such a group can be outlined

loss of accuracy. Symmetry also has been taken advantage of in using the example of a group of four piles (Fig. 3). Each pile is

other approaches, e.g. by Tyson and Kausel (1983) and Waas divided into an equal number of elements, with the total number

and Hartmann (1981), but the procedure employed here is of degrees of freedom being n. With the node numbering shown

somewhat different and is outlined below for completeness. The in Fig. 3, the equation of equilibrium, [3], can be written in full

degree of the reduction of the computing effort depends on the Seen in

P I

p2

P3

P4 - -

P~

- - 181

p8

- -

PI,

The matrices are partitioned according to the individual levels replaced by a circularly arranged equivalent group that has the of nodes. Because of double symmetry, the displacements and same number of piles, similar spacing between piles, and nearly forces on each level of nodes are the same, i.e., u l = 02 = u3 = the same plan area (Fig. 4c). v4, etc. and P I = P2 = P3 = P4, etc. Thus, the number of These types of symmetry were incorporated into an efficient unknown displacements to be calculated is reduced to one- computer program (El Sharnouby and Novak 19846). The fourth. For the solution, only one equation, i.e. the first one, of efficiency of the approach can be seen from the analysis of a each group of four has to be considered. Just as importantly, the group of 201 piles supporting a reactor building described in stiffness constants on each line of the stiffness submatrices more detail later herein; the computation took less than 40 s on a repeat although not in the same order. For example, for the first CDC CYBER 170/835 computer. (The computer program will two lines of the first submatrix, K1 = K22, K I 2 = K Z 1 , K I 3 = be made available by SACDA, The University of Western K24, and K14 = K23; hence the sum of the four constants is the Ontario, London, Ont.) same for each line of the submatrix.

Examples of vertical stiffness of pile groups Axisymmetric pile groups Even greater efficiency of analysis can be achieved when the The theory was used to the

group is axisymmetrically arranged in the sense that the piles are sing1e piles and pile groups. The were

outlined in concentrical rings, with each ring comprising the with published data and used to evaluate the validity of the

same number of piles (Fig. 4a). The spacing of the piles of one used

ring is constant and piles of individual rings lie on lines passing The interaction factors, introduced by Poulos, are defined for through the centre. For such a group, the displacements of all loaded piles as the

piles in one ring are the same and the number of degrees of Settlement of reference pile due to load on adjacent pile freedom to consider is reduced to that corresponding to one pile cx = per ring. Real groups are rarely arranged in such a truly Settlement of reference pile under its own load axis~mmetric fashion because groups have a For a larger group, these interaction factors are superimposed to different number of piles per ring (Fig. 46). It was found, yield the total settlement of the piles in the group. however, that even for such groups axisymmetry can be as- sumed, with only a marginal loss of accuracy. With somewhat Single piles less accuracy, square, rectangular, and irregular groups can be To verify the accuracy of the approximate solution based on

- K I I K12 K13 K14 1 K1.5 K I ~ K I ~ K I ~ 1 . . . . . . .KI ,~-

K21 K22 K23 K24 I K25 K 2 6 I 1 K2,l

I K31 K32 K33 K34 I K35..... I K ~ I K42 K43 k44 I K45... .. I I - - - - - - - - - - - - - I - - - - - - - - - - - - - 1 - - - - - - - -

K51 K52..... I I I I

I I I I

K8 1 I 1 - - - - - - - - - - - - - 1 - - - - - - - - - - - - - I - - - - - - - -

I I

I I

I I

- Kl, 1 1 Kt,- I I

U I

U 2

u3

u4

- -

U 5

u8

- -

U l ,

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JBY AND NOVAK 83

FIG. 4. Types of symmetric arrangements for group of 16 piles: (a) axisymmetric, (b) cyclic, and ( c ) doubly symmetric.

the point loads, the stiffness of single piles was calculated and compared with the results of Poulos and Davis (1980). Floating piles in homogeneous soil with a wide range of slenderness ratios L l d and stiffness ratios Kr were used. The stiffness ratio characterizes the relative stiffness of the pile and the soil, and for solid piles is defined as K, = Ep/Es , where E, and E, are the Young's moduli of the pile and the soil respectively. For L l d ranging from 10 to 50 and Kr from 100 to 10000, the two solutions do not differ by more than about 5%.

Pile groups Square groups Figure 5 shows the stiffness of a small 3 X 3 pile group

evaluated for different spacings by Poulos and Davis (1980), Kaynia and Kause1(1980), and the authors. The group stiffness KG is plotted in a dimensionless form as the

[9] Group efficiency ratio = KG/Nk

where KG = stiffness of the group, N = number of piles, and k = the stiffness of a single pile considered in isolation. (The ratio KG/Nk is the inverse of the settlement ratio.) For the floating piles used in Fig. 5, the agreement between the three approaches is excellent, and thus the interaction factor approach works very well. However, the accuracy of the latter approach may deteriorate when the piles rest on a stiffer bearing stratum. This can be seen from Fig. 6, showing the group efficiency ratios for square groups of 9, 25, and 100 piles supported by strata of different stiffnesses characterized by the ratio Eb/Es where Eb = Young's modulus of the bearing stratum. Figure 6 compares the results due to Poulos and Davis (1980) obtained using the interaction factor approach with the results of the direct analysis presented. (For the group of 100 piles, Poulos's results were extrapolated using his formula.) For floating piles (Eb/Es = 1) both approaches again give the same result; however, as Eb/Es increases, the two approaches diverge, with the direct analysis giving consistently higher group efficiency, i.e. higher group stiffness. The differences also increase with the number of piles but their magnitude depends on the parameters of the system. Figure 6 suggests that the superposition of interaction factors

Poulos 8 Davis

+ + + Koyn~o 8 Kousel

- - -- Present

FIG. 5 . Vertical stiffness of group of 9 floating piles vs. pile spacing (homogeneous soil, Ep/Es = 1000, Lld = 25, v = 0.5).

1 2 3 4 5 6 7 8 9 l 0 1 1

FLOATING - 'b/~, - END-BEARING

FIG. 6. Vertical stiffness of groups of 9, 25, and 100 piles vs. stiff- ness of underlying stratum; calculated using (a) interaction factors, (b) direct analysis (Lld = 25, Ep/E, = 1000, Sld = 5, v = 0.5, homogeneous soil).

may exaggerate pile-soil-pile interaction effects, particularly for large groups of end-bearing piles.

The direct solution also yields the distribution of forces acting on individual piles. For the groups of 25 and 100 piles, the forces acting on typical piles of each group are plotted in a normalized form in Fig. 7. The peripheral piles carry greater loads than the interior piles, with the comer piles carrying the most. (Similar observations were made by others.) The differ- ences in individual pile loads are greatest for floating piles (full lines) and diminish as Eb/Es increases.

Circular groups and equivalent cyclicly arranged groups Particularly great efficiency of direct solution can be achieved

for circular groups or for square or rectangular groups replaced by an equivalent axisymmetrical or cyclicly arranged circular group. For example, the 4 x 4 square group shown in Fig. 4 c can be approximately represented by the circular groups shown in Figs. 4a and 4b.

Table 1 shows, for a 4 X 4 group, the stiffness constants of individual piles and of the whole group, calculated for both

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FLOATING --- END-BEARING '

S .I.. . W . . . .

FLOATING

, 0.5

a END-

BEARING

6 5 4 5 6

b/E ( 0 ) GROUP OF 25 PILES

FIG. 7. Distribution of pile loads in groups of 25 and 100 piles vs. S/d = 5, homogeneous soil).

TABLE 1. Comparison of stiffness of 4 X 4 square group and its equivalent circular models (floating piles, L = 11.0 m, d = 0.3 m,

S = 1.5 m, v = 0.3, Ep = 210 000 MPa, Es = 11 700 kPa)

Stiffness of individual piles (kN/m) Stiffness of group

Pile 1 Pile 2 Pile 3 KG Group ki k2 k3 (MN/m)

Square 4 102.3 10 244.7 16 392.1 163.9 Cyclic (general) 4 151.6 12 177.4 12 194.3 162.9 Cyclic (axi-

symmetry) 4 157.3 12 188.5 12 188.5 162.9

TABLE 2. Static stiffness and damping of group of 102 test piles

Stiffness Damping K2 Method (kN/mm) (kN/(mm. s))

Vertical direction Writers' direct solution

Actual model 139.2 Equivalent cyclic model 142.9 28.3

Waas theory Actual model Equivalent cyclic model

Static interaction factors

Horizontal direction Experiment, measured in Y-direction

(short) 22.8

Writers' direct solution Y-direction 20.0 X-direction 19.1 Equivalent cyclic model 19.9 30.2

Waas theory Actual model Y-direction 23.1 Equivalent axisyrnmetric model 21.6 32.1 Actual model X-direction 21.9

Static interaction factors, Y-direction 11.9

FLOATING - EbIES -4 END-BEARING

( b ) GROUP OF 100 PILES

stiffness of underlying stratum (Lld = 25, Ep/Es = 1000, v = 0.5,

TABLE 3. Soil properties of Angra 2 site

Depth Soil Gs PS vs (m) ty pe (GN/m2) (kg/m3) (rnls) v

5.0- 9.5 Sand 0.435 1950 472 0.42 9.5-12.5 Clay 0.027 1680 127 0.49

12.5-16.5 Transition 0.944 2100 670 0.35 16.5-25.0 Residual 1 1.022 2050 706 0.36 25 .O-32.0 Residual 2 1.556 2050 871 0.34 32.0-39.0 Weathered

rock 3.018 2400 1121 0.43 39.0- Bedrock

a square arrangement (Fig. 4c) and an equivalent circular group (Fig. 4b). The circular group was analyzed twice, first accurately as a general group and secondly as an axisymmetrical group for which displacements of and forces on all piles of one ring are equal. The three solutions give almost the same group stiffness; the circular solutions also yield almost identical individual pile stiffnesses but these stiffnesses differ from those obtained for the comer pile of the square model by a significant amount. That may be expected but need not be very serious from a practical point of view because, in reality, the peak loads may be reduced due to nonlinear behavior of the soil.

A group of 102 test piles The authors conducted extensive field experiments with a

large group of 102 small test piles described in full detail by El Sharnouby and Novak (1984a). For this group, static stiffness was predicted using the Waas and Hartmann (1981) technique, the present direct solution, and Poulos's interaction factors (Table 2). The analysis carried out by Waas and Hartmann by means of their PILAX computer code and the authors' direct analysis yield very similar results; the interaction factor ap- proach gives a considerably lower stiffness than the first two methods.

Pile foundation of Angra 2 nuclear power plant The second unit of the nuclear power plant at Angra des Reis

in Brazil was one of the first facilities of this type to be founded on piles in a seismic region and its foundation prompted considerable research by Wolf and von Arx (1978), Waas and

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TABLE 4. Stiffness constants of Angra 2 nuclear power plant pile foundation (201 piles)

Method

Vertical Horizontal

Stiffness K I Group Stiffness K 1 Group (GN/m) efficiency (GN/m) efficiency

Present analysis 404.0 0.25 72.3 0.099 Wolf and von Arx (1978) 652.3 0.40 71.4 0.098

9 0 L I I I I I 2 ' 2 3 4 5 0 Number of Piles Per Group Side

Y z \ Wolf (F. E.M.)

Poulos

.- ._ -----

FIG. 8. Vertical group stiffness efficiency ratio for different number of end-bearing piles (Angra 2 site, S / d = 3). -5.00m

U .- L -, 0.2

Hartmann (1981, 1984), Trbojevic et al. (1981), and others. The finite element solution due to Wolf and von Arx (1978) is used here for comparison with the present approach. In this solution, the concrete piles are assumed to be drilled to the bedrock, which is regarded as rigid; pile diameter d = 1.3 m (4.27 ft) and Young's modulus E, = 30 000 M N / ~ ~ (4.35 x: lo6 psi). The soil profile and soil properties are given in Table 3.

For the Angra site, various square pile groups are analyzed. The groups are characterized by the number of piles per group side. The group efficiency ratios computed by means of the present direct method and using Poulos's interaction factors are compared with Wolf's finite element solution in Fig. 8. The present solution lies between the other two, with the interaction factor approach giving the lowest values.

The complete foundation of the reactor building comprising 201 piles is shown in Fig. 9. The vertical stiffness of the group evaluated using the present direct approach is compared with Wolf's finite element result in Table 4. The comparison indi- cates a difference that is marked but still reasonable considering the very large size of the group, the different assumptions made in the two approaches, the very irregular soil profile, and a very low pile-soil stiffness ratio (about 10). In his solution, Wolf lumped every four piles into one; this could contribute to the difference in stiffness.

-.-.-._, -

Rocking stiffness of pile groups When analyzing the response of pile-supported structures and

foundations to horizontal forces and overturning moments, the rocking stiffness of pile groups is needed. For pin-headed piles with thin caps, this stiffness derives solely from the vertical stiffness of individual piles, as indicated in Fig. 10, and thus can be established readily from [3].

The rocking stiffness of pile groups is generated by prescrib- ing a unit rotation, $ = 1, to the cap. This rotation produces vertical dispjacements of pile heads equal to ui = xi$ = xi where xi = the horizontal distance of the pile from the axis of rotation (Fig. 10). Substituting these displacements into [3] and

FIG. 9. Reactor building of Angra 2 power plant supported by 201 piles (1 m = 3.28 ft).

solving for the loads { P ) , the forces acting on pile heads, being the only nonzero nodal loads, are obtained as

[lo] { P ) = [ P I P2 .. . PN 0 0 .. . OIT

in which N = the number of piles. The rocking stiffness of the group, K+, follows as the total moment of the forces {P) , i.e.,

[ I I I K* = pix; = X lpillxil

Low-frequency response The static analysis of pile groups presented in this paper can

be extended to describe dynamic response at low frequencies. This extension can be formulated because the studies of dynamic behavior of elastic strata suggest that, for frequencies lower than the fundamental natural frequency of a stratum, no

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FIG. 10. Generation of rocking stiffness of pile group.

progressive wave is generated in the absence of soil material damping and only a weak progressive wave occurs in the presence of material damping (Nogami and Novak 1976). Consequently, the low-frequency dynamic stiffness of the pile group can be assumed to be approximately equal to the static stiffness, and damping can be assumed to stem only from material damping (hysteresis) of the soil and piles. The upper-bound frequency for the applicability of these assump- tions is the fundamental natural frequency of the soil layer. For a homogeneous layer with soil shear wave velocity Vs and depth h, this fundamental natural frequency in the vertical direction is

and in the horizontal direction is

In [12a], v = Poisson's ratio. In many practical applications, particularly those involving large structures such as offshore towers, nuclear reactor buildings, or skyscrapers, and those exposed to dynamic loading due to wind; eaihquakes, or sea waves, the dominant frequency of the response is very low and often lower than those given by [12]. Then the pile group stiffness and damping can be evaluated using the simplification suggested and the much more complicated true dynamic analysis may not be necessary. If, for example, V, = 150 m/s, h = 30m, and v = 0.33, then frequency oh = 7.9 s-I (1.3Hz) and w, = 15.6 s-' (2.5 Hz). The fundamental frequency of many large structures is often lower than these values. For nonhomogeneous soil profiles, the layer natural frequencies can be evaluated using the technique due to Dobry et al. (1976).

Material damping is usually assumed to be of the frequency- independent, hysteretic type and is defined either by material damping ratio P or in terms of the loss angle 6 such that

[13] tan 6 = G'/G = 2P

in which G' = the complex complement of the real shear modulus G. Then the complex shear modulus is

in which i = m. For materials with hysteretic damping, all elements of the soil and pile stiffness matrices in [2] become complex, i.e.,

in which the subscript "s" indicates the magnitudes pertinent to soil and "p" pertains to piles.

For low frequencies of o and particularly for o + 0, the inertia of the system is not very important. Consequently, the mass of the system can be approximately represented by the diagonal mass matrix [MI listing the lumped masses of pile elements augmented by some covibrating mass of the soil. (Such representation of mass is quite common in the often used lumped mass approaches, e.g. Matlock et al. (1978).) For harmonic nodal forces {P(t)) = {P) exp (iwt) in which {P) = the vector of complex force amplitudes, w = circular frequency of excitation, and t = time, the equation of equilibrium of the soil- pile system becomes

[I61 [[a - 02[M11 (0 ) = {P)

where { u ) = the vector of complex amplitudes. This equation replaces [3] and can be solved for displacements or forces using the same schemes as those employed for the static response. The main difference is that the computer storage requirements are doubled because all matrices have imaginary parts.

Dynamic group stiffness in the vertical direction Dynamic group stiffness, which is needed in the prediction

of dynamic response of pile-supported facilities, is obtained by prescribing unit displacement amplitudes to pile heads in [16]. Dynamic stiffness (impedance function) is obtained in complex form as

[17] KG = K1 + iK2

in which K, = the real stiffness and K2 = the imaginary part of stiffness, which represents damping due to energy dissipation in the soil and piles. For low frequencies and hysteretic material damping, both K, and K2 are constant. Alternatively, the damp- ing can be expressed in terms of the equivalent viscous damping constant

Single piles For single piles and a wide range of ratios Ep/Es, the results

calculated by [16] were compared with those obtained by Novak and El Sharnouby (1983) using a completely different method. For both stiffness and damping, the results of the two methods do not differ by more than 10%. Examples of K I , K2, and the ratio K2/K1 are given in Tables 5 and 6. For floating piles, the ratio K2/Kl is approximately equal to tan 6, = 2Ps of the soil. For end-bearing piles, the ratio K2/K1 can be smaller, depend- ing on the damping of the pile and varies with P,, Pp, and Ep/Gs as shown in Table 6.

Pile groups In Table 5, the real and imaginary parts of complex stiffness

calculated using [16] are given for a group of 16 piles, both end-bearing and floating; single pile data are also shown for comparison. Table 5 and a detailed examination of other cases indicate that, for floating piles, the ratio K2/K, is close to the soil material damping characterized by tan 6, = GSf/Gs = 2P, (0.05 for the examples shown in Table 5). 'Thus, for low frequencies, the damping of single floating piles and groups of floating piles can readily be estimated on the basis of soil material damping alone and evaluated by means of the simple relation

[I91 K2 = tan 6, K,

where K, is dynamic stiffness, which can be taken as equal to static stiffness. For end-bearing piles the damping may be less

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TABLE 5. Impedance functions for single piles and square group of 16 piles (Lld = 36.7, S ld = 5, Ep/E, = 18000, 2P, = 0 . 0 5 ; ~ = 0.5, L = ll.Om, E, =

11.7 MPa)

Single pile 16 pile group

Type of pile 2Pp KI K2 K ~ / K I KI KZ KzIKL

Floating 0.01 56008 2357 0.042 163063 7815 0.048 0.05 55 995 2800 0.05 163052 8153 0.05

TABLE 6. Effect of pile rigidity and material damping on imaginary part of single pile stiffness (Lld = 25, end-bearing piles)

Imaginary part KZ/real part KI for 2P, =

than that given by [19] or close to pile material damping depending on the parameters of the system.

Equation [16] was also used to establish the low-frequency impedance of the group of 102 test piles. The stiffness Kl obtained is almost identical with the values established by the purely static solution and given in Table 2, in which the damping K2 is also shown together with the value computed for w GO by Waas and Hartmann. The analysis was conducted with p, = 0.1 and p, = 0.01. For both stiffness and damping, the results of the two different approaches agree remarkably well. An example of the variations of K l , K2, and c with frequency is shown for a group of 16 piles in Fig. 11. Groups of floating and end-bearing piles, both embedded in a stratum of limited depth, are compared. (The natural frequency of the soil layer is w, = 14.16 s-'.) It can be seen from Fig. 1 1 and Table 5 that damping constants K2 and c are smaller for floating piles than for end-bearing piles in this case. This is so because geometric (radiation) dimping is absent for o < w,. At higher frequencies, for which geometric damping is the principal source of energy dissipation, floating piles usually provide more damping than end-bearing piles (Novak 1977). For higher values of soil stiffness, the differences between floating piles and end-bearing piles diminish.

I \

I 001 0 1 10 10 0

FREQUENCY w i i ' i

FIG. 11. Vertical stiffness and damping constants of group of 16 piles vs. frequency (Sld = 5, L = 11.0 m, d = 0.3 m, E, = 210 G N / ~ ~ , E, = 11 700 kPa; 1 m = 3.28 ft).

* PlLE NODE

X POINT LOAD

0 REFERENCE POINT

TYPICAL PlLE ELEMENT

(b) POINT LOAD SYSTEM ON PlLE

FIG. 12. Layout of pile elements and equivalent point loads for analysis of horizontal response.

length of the other elements increasing with depth. The nodal Horizontal response of pile groups points are located on the axis of the pile, with the first and last

The response of pile groups to horizontal loads can be nodes located at the top and bottom of the pile respectively. analyzed in the same fashion as the response to vertical loads. The element stiffness matrix for pile bending is 4 X 4 and has For this case, the piles are divided into a number of elements of the standard form [k] relating the translation u and rotation IJJ to varying length (Fig. 12). A total of 12 elements was found to end forces P and moments M. Because external moments act yield sufficient accuracy, with the uppermost element length only at the top nodes, the pile stiffness matrices are condensed being equal to one-fourth of the average element length and the by eliminating all nodal rotations except those of the butt.

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Soil stiffness The stiffness of 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. 12). 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 45" from the x-axis of the pile (the direction of loading). (Parmelee and Ludtke (1975) also found that the average displacement around a horizontal buried pipe is equal to the displacement at 135" below the vertical axis.)

The average soil displacement established in this way for points 0 is taken as the soil flexibility coefficient related to the pile node i. Thus, a soil flexibility coefficient& is equal to the average of the horizontal displacements at point 0 on element i due to two equal unit loads acting on element j. This definition holds for the elements of one pile; for the elements of different piles the flexibility coefficient is calculated at and related to the pile node, i.

Analysis of horizontal response With the pile and soil stiffnesses established and super-

imposed as in [2], the analysis of single piles and pile groups follows from [3] for any case of interest. Symmetry of a group can be exploited in a way analogous to that discussed in relation to Figs. 3 and 4 and [8]. For low-frequency dynamic response, soil and pile material damping is accounted for in terms of complex shear modulus of soil ([15]) and complex Young's modulus of piles Ep* = (1 + i2Pp)Ep, where Ep is real Young's modulus and the upper-bound frequency is given by [12b]. The horizontal analysis is described in more detail in El Sharnouby (1984).

Single piles under horizontal load To verify the accuracy of the present formulation and to study

the behavior of single piles, flexibilities of free- and fixed-head piles were evaluated and are compared with Poulos and Davis' (1980) results in Figs. 13 and 14. Piles in homogeneous soil were analyzed for L / d = 25 and a wide range of stiffness ratios K,. The stiffness ratio characterizes the relative stiffness of the pile and the soil and is usually defined as K, = E,Ip/EsL4. Also plotted in Figs. 13 and 14 are results for piles projecting a free length e as well as some values of pile stiffness from the dynamic plane strain solution (Novak and El Sharnouby 1983).

For free-head piles, the agreement between Poulos's solution and the present method is quite good whereas for fixed-head piles, the flexibility evaluated by Poulos is somewhat over- predicted when compared with the results of the present formulation. It can also be seen that the results based on the dynamic solution agree reasonably well with those by the present method. It is noticed that the free length of the pile e has a significant effect on pile flexibility; flexibility increases with e, particularly for flexible free-head piles. The curves in Figs. 13 and 14, which were calculated for L l d = 25, have been made

k I/ FREE HEAD

FIG. 13. Horizontal flexibility and stiffness of free-head single piles with free length.

I I I I 1 1 1po 10- 10-A lo-3 10-

K, = ~ ~ l ~ / ~ ~ ( 2 5 d ) ~

FIG. 14. Horizontal flexibility and stiffness of fixed-head single piles with free length.

applicable for all values of L /d > 25 by replacing L~ in the stiffness ratio K, by the term (256)'. (A study of the dimen- sionless parameters will be outlined later herein.) The effect of the free length is indicative of the effect of pile separation from soil.

The evaluation of material damping effect for a wide range of parameters showed that for low frequencies, not exceeding those given by [12b], the damping of the soil-pile system, K2, can be estimated for homogeneous soils as

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K, = 1': L / d = 25 FREE HEAD

--- POULOS

- PRESENT

0 7 1 FIXED HEAD K, = ~ / d = 25

- - - \ POULOS

\\Q* --- PRESENT

I I I

S /d d /S S /d d / S

FIG. 15. Horizontal interaction factors vs. pile spacing ratio by the present formulation and Poulos for K, = K, = EP~, /EsL4

I - - - POULOS I ---- PRESENT

( a )

1 1 1 1

1 2 3 4 5 0 . 2 0.15 0.1 0.05 C

FIG. 16. Horizontal interaction factors vs. pile spacing ratio by the present formulation and Poulos for K, = K, = E ~ I ~ / E , L ~ .

0.7

i - P

L / d 2 2 5

z 0 4 -

( 0 ) FREE HEAD

0 7 - I d f ,

0 ji) L / d 2 2 5 4 S/d = 3

'"06 -

( b ) FIXED HEAD

I I I - 1

0 I lo-= I O - ~ lo-3 I O - ~ LO-'

K~ = Epip/ E5 ( 2 5 d )4

- FIXED HEAD

FIG. 17. Effect of pile free length on horizontal interaction factors (S l d = 3, homogeneous soil).

~ / d z 2 5 , K, =

in which K , = static stiffness and P, = soil material damping ratio. For the parabolic variation of the soil shear modulus, a coefficient of about 0.75 appears more adequate than 0.85.

Group of two piles The present method has been employed to verify the accuracy

of the widely used interaction factor approach. The interaction factors are plotted in Figs. 15 and 16 together with those produced by Poulos for free- and fixed-head piles, respectively, and for two values of the stiffness ratio K,. The interaction factors obtaind by Poulos are up to 20% greater than those from the present formulation for flexible piles, but the differences

close up for stiffer piles. For flexible piles, results similar to those of Poulos were obtained by the present method when elements of equal length were used.

In Fig. 17, the interaction factors are plotted versus the stiffness ratio for a range of free pile length to diameter ratio e l d . The dramatic effect of the free length of the pile on reducing the interaction between piles can be seen. This observation suggests that a similar reduction of interaction may occur as a result of pile-soil separation when gaps open at the uppermost parts of the piles, as often happens under cyclic loading. More details on this effect are given in Novak and El Sharnouby (1985).

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90 CAN. GEOTECH. J. VOL. 22, 1985

I .o KG / Nk (EFFICIENCY)

e /d

FIG. 18. Effect of pile free length on horizontal group stiffness and efficiency of 3 X 3 group (Lld 2 25, S/d = 3, K, = E,I,/E,L~).

K, =

x x X KAUSEL

- P O ~ L O S & 0.1 FIXED-HEAD GROUP i - - - - W PRESENT 1

FIG. 19. Horizontal stiffness of group of 9 piles vs. pile spacing (homogeneous soil, L/d = 25, v = 0.5).

Larger groups under horizontal loading The free pile length also reduces pile interaction and stiffness

of larger groups as can be seen from Fig. 18, in which group efficiency, KG/Nk, and the group stiffness ratio, KG(e)/ KG(e=O), are plotted versus e /d for a group of nine piles. The effect of pile free length is more pronounced in flexible piles where the group efficiency increases with e /d while group stiffness decreases considerably.

Figure 19 shows the stiffness of a 3 X 3 pile group evaluated for different spacings by Kaynia and Kausel(1980), Poulos and Davis (1980), and by the present direct method. The group stiffness, KG, is plotted in a dimensionless form as the group efficiency ratio, KG/Nk. The agreement between the three approaches is very good except for the difference of up to 15% between the results obtained by Poulos and the present analysis for more flexible piles. (This may be due to the use of equal pile elements by Poulos.) Thus, the interaction factor approach works very well for smaller groups exposed to horizontal loads.

The present direct solution also yields the distribution of forces acting on individual piles. For a group of 16 piles, the forces acting on typical piles of the group are plotted in a normalized form together with those published by Poulos in Fig. 20. The peripheral piles carry greater loads than the interior piles, with corner piles carrying the most. However, the difference between the loads carried by different piles is markedly smaller in the present method's results than that predicted by the interaction approach. It seems that the accuracy of the interaction factor approach deteriorates in the case of load distribution in a group of piles.

18 \ \

'4, ---- POULOS

/ - e * I * * 0 e . e

0 I I I I I

2 4 6 8 10 I2 S/d

FIG. 20. Horizontal load distribution on 4 x 4 fixed-head square pile group (K, = homogeneous soil).

FIG. 21. Horizontal load distribution on 16 fixed-head piles arranged in cyclic pile group (K, = homogeneous soil).

In circular groups a simpler pattern emerges, as can be seen from Fig. 21. A circular group of 16 piles was analyzed as a general group and the load distribution plotted versus different pile spacings. The difference between a pile load and an average pile load in the same ring does not exceed 3% whereas the difference between the maximum and the minimum pile loads in one ring is about 6% for these particular pile-soil data and geometry. From the practical point of view, the load variation in one ring is insignificant and, thus, the presumption of equal pile loads for each ring in the analysis may be a reasonable assumption to adopt, particularly with regard to the great efficiency gained.

This group of 16 cyclicly arranged piles is equivalent to the 4 X 4 square group for which the load distribution is plotted in Fig. 20; this is so because, for comparable spacings, the two groups feature almost the same total area and moment of inertia. The stiffness constants of the square group and of the equivalent circular group were evaluated, for a spacing ratio S/d = 5 and L/d = 33, as 55 833 and 56 496 KN/m, respectively. The group stiffness constants differ by only I%, but the difference in the load distribution pattern between the two groups is quite noticeable.

For the 102 test piles used in the field experiments described in El Sharnouby and Novak (1984a), the horizontal stiffness and damping evaluated using the present direct method are very close to the results of the Waas and Hartmann analysis; Poulos

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EL SHARNOUBY AND NOVAK 91

FREE HEAD

0.6

0 1 I I I I I 0 IOd 2Od 3 0 d 4 0 d 5 0 d

PILE LENGTH L

T r FIXED HEAD I

0 1 I 8 I I 0 IOd 2Od 3 0 d 4 0 d 5 0 d 6 0 d

PILE LENGTH L

FIG. 22. Variation of horizontal interaction factors with pile length for S / d = 5.

FIG. 23. Variation of horizontal interaction factor with pile stiffness and spacing for fixed-head piles.

interaction factors give much lower stiffness (Table 2). The Kaynia and Kausel (1980), the interaction factor approach agreement of the predicted stiffness with the experimental value works quite well for horizontal loading of small and moderately is also very good. large groups. For this reason, a few more observations relevant

For the Angra 2 nuclear power plant pile foundation the to this approach are described below. horizontal stiffness obtained bv means of the Dresent method compares very well with the value calculated by Wolf and von Arx (1978) using the finite element method, as can be seen from Table 4. It may be noticed that for this large pile group the efficiency ratio is very low, indicating strong interaction effects.

Many other observations can be generated for the horizontal response of pile groups but most of the phenomena observed resemble those discussed when dealing with the vertical response. The material damping of the group can be evaluated approximately as

[21] K2 = 0.85 tan 6,KI

for homogeneous soil and as

[22] K2=0 .75 tanS ,K1

Dimensionless parameters The deformations of laterally loaded piles are confined to the

upper part of the pile and the overall length of the pile does not significantly affect the response of the pile. Hence the length of the pile is rarely a relevant parameter when presenting inter- action factors for laterally loaded piles. Yet the traditional form of presentation of influence and interaction factors for laterally loaded piles is unnecessarily complicated for the majority of piles encountered in practice because the pile length is intro- duced into the stiffn'ess ratio K, = (EI~ , /E ,L~~ With this stiffness ratio, a set of charts is needed for each value of pile length.

However, as can be seen from Fig. 22, the horizontal

for parabolic soil profiles just as with single piles. The limit to interaction factors are practically independent of pile length

the applicability of these formulae is again [12b]. Equations beyond 10-15 diameters, depending on the pile stiffness.

[21] and [22] hold for horizontal response of both floating and Consequently, the stiffness ratio K, may be replaced by the ratio

end-bearing piles. KD = EpIp/Esd4. Furthermore, for piles of solid circular cross section the ratio 1,/d4 is constant, and thus the Young's

Interaction factor approach modulus ratio Ep/E, act'ually is the main dimensionless param- As was shown above and also observed by others, e.g. eter; for more general cross sections, E,/Es may be replaced by

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FIG. 24. Variation of horizontal interaction factor with pile stiffness and spacing for free-head piles.

The second dominant dimensionless parameter is, of course, the p = o .oO

spacing ratio, S / d . P 0.4 - - I

u I

Interaction factors / I

n I

For the dimensionless ratios E J E , and S l d , the horizontal o I

interaction factors are plotted in Figs. 23 and 24 calculated using 2 0.3 -

the present direct solution. These charts are for L l d 10, 0- z homogeneous soil, and the extreme values of P of 0" and 90". o

The values shown are somewhat smaller than those presented by 5 0.2 - , - - - - DISTRIBUTION Poulos. For intermediate values of P, the interaction factors 5 may be calculated from the extreme values as suggested by Randolph and Poulos (1982), i.e., 0.1 -

[24] a(P) = a(O") cos2 p + a(90°) sin2 p Figure 25 shows the effect of Young's modulus variation with o depth on the interaction factors. For small spacing, the inter- 2 3 4 510.2 0.15 o I 0.05 o action factors tend to be slightly smaller for linearly or para- s / d d / S

bolically increasing modulus than for the constant modulus; for o 5

greater spacing ratios the difference diminishes or even changes i / d 2 1 0

its sign for free-head piles. For practical purposes, the effect of p = 0.0" modulus variation with depth on the interaction factors is 0.4 -

u generally small and the values of a for constant modulus given n in Figs. 23 and 24 may be used for varying modulus soils as

I well. The stiffness of piles and groups depends, however, on the 0.3 - I

soil profile. Similar observations are made by Randolph (198 l), who pro- 6

poses simple, easy to use empirical expressions for the evalua- 5 0.2 -

tion of interaction factors. However, some of his data markedly g differ from those obtained by means of the present method as 2

0- shown in Figs. 23-25. Most notably, Randolph suggests that 0.1 - UNIFORM DISTRIBUTION

the interaction factor for p = 90' is half that for P = 0°, that b PARABOLIC DISTRIBUTION

the interaction factor for soil stiffness proportional to depth is - --- LINEAR DISTRIBUTION about half that for homogeneous soil, and that the interaction

0 factors are inversely proportional to pile spacing. Empirical ex- 2 3 4 510.2 '0.15 0.1 0.05 o pressions for the interaction factors calculated by the present s / d d l 5

method are given in Novak and El Sharnouby (1985) to facilitate FIG. 25. Effect of Young's modulus distribution on horizontal inter- their evaluation. action factor.

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EL SHARNOUB :Y AND NOVAK 93

Concluding remarks Nonlinearity of pile behavior is important for large displace-

ments but cannot be incorporated into the pile-soil-pile interaction analysis without great difficulties. An approximate account of it can be made by adjusting soil properties according to the level of strain, allowing for pile separation, and considering a weakened zone around the piles (Novak and Sheta 1980). For small displacements the linear theory may be adequate (Novak and El Sharnouby 1984).

Experimental investigation of pile group behavior is very desirable, particularly under dynamic loading, to clarify some of the more peculiar features predicted by the theories and to complement the reported small-scale experiments (El Sharnouby and Novak 1984a; Novak and Grigg 1976) with full- scale data-.

Summary and conclusions A direct, computer-based analysis of pile groups is formu-

lated. The approach uses equivalent point loads to describe soil flexibility and exploits symmetry of pile arrangement. Static and low-frequency response in the vertical, rocking, and horizontal directions is considered. Group stiffness, damping, and settlement as well as distribution of the pile loads are analyzed. The approach is used in a comprehensive parametric study, tested against experimental data and other direct ap- proaches, and employed to examine the interaction factors. The following conclusions emerge:

1. The direct analysis presented is simple and computation- ally very efficient even for very large groups; its results agree well with field test data and those of other direct methods.

2. The often used interaction factor appoach gives satisfac- tory results for vertical response of floating piles and horizontal response of smaller groups but may considerably overestimate the settlement and underestimate the vertical stiffness of groups of end-bearing piles. The accuracy of the interaction factor approach deteriorates when predicting load distribution in pile groups.

3. For low frequencies, the vertical damping of floating piles and the horizontal damping of both floating and end-bearing piles can be estimated as a fraction of static stiffness. The vertical damping of end-bearing piles depends on the pile-soil stiffness ratio and material damping of both the pile and soil.

4. The damping may be smaller for floating piles than for end-bearing piles when the frequencies are low and geometric damping is negligible.

5 . Horizontal interaction factors are practically independent of pile length and the soil profile and are provided, in the form of a single set of curves, as a function of stiffness ratio E,/E, and pile spacing S ld .

6. Further research should include nonlinearity and field experiments.

Acknowledgements This study was supported by a grant in aid of research from

the Natural Sciences and Engineering Research Council of Canada.

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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 Engin- eers, London, England, pp. 109-1 18.

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List of symbols cross-sectional area of pile constant of equivalent viscous damping pile diameter pile base diameter Young's modulus of soil under pile base Young's modulus of pile Young's modulus of soil complex complement of shear modulus complex shear modulus pile shear modulus soil shear modulus depth of soil layer real and imaginary parts of complex stiffness, respectively element of stiffness matrix of soil-pile system stiffness constant of pile group pile-soil stiffness ratio rocking stiffness of pile groups stiffness constant of single isolated pile stiffness constant of ith pile in group pile length length of soil tributary element length of pile element number of piles in group load o n pile node vertical load on each pile in group with flexible c a p total vertical load on pile group load on ith pile of group with rigid cap distance between piles horizontal displacement shear wave velocity of soil = vertical displacement vertical displacement of pile group interaction factor material damping ratio of piles material damping ratio of soil loss angle of piles loss angle of soil soil Poisson's ratio mass density of soil rotation in vertical plane (rocking) circular frequency fundamental natural frequency of soil layer in vertical direction fundamental natural frequency of soil layer in horizontal direction

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