Interaction factor methods

A widely used method of analysing the pile group settlement is based on the concept of interaction factors (Φ) defined as follows :

Φ = additional settlement caused by an adjacent pile under load

settlement of pile under its own load [7.2]

This is an extension of the elastic continuum method for analysis of settlement of single piles where the interaction effects in a pile group are assessed by superposition. Basic solutions for the group settlement ratio (Rgs) for incompressible friction or end-bearing pile groups are summarised by Poulos & Davis (1980). Correction factors can then be applied for base enlargement, depth to incompressible stratum, non-homogeneous soil, effect of pile slip, interaction between piles of different sizes, pile compressibility and rigidity of the bearing stratum. The relationship between group settlement ratio, Rgs and the number of piles derived by Fleming et al (1992) for two simple cases is shown in Figures 7.8(a) & (b). The solutions given are for key piles in uniformly loaded pile groups and also for pile groups loaded through a rigid pile cap. It can be seen that interaction effects are less pronounced in a soil with increasing stiffness with depth than in a homogeneous soil.

An alternative and simplified form of the interaction factor method was proposed by Randolph & Wroth (1979). Equations have been derived for shaft and base interaction factors for equally loaded rigid piles, which are summarised in Figure 7.9. For compressible piles installed in homogenous or non-homogenous soils, the base and shaft settlements are not equal. The pile head settlement should be adjusted according to the approach by Randolph & Wroth (1979).

Poulos (1988b) has modified the interaction factor method to incorporate the effects of strain-dependency of soil stiffness. The modified analysis shows that the presence of stiffer soils between piles results in a smaller group settlement ratio and a more uniform load distribution than that predicted based on the assumption of a linear elastic, laterally homogeneous soil.

The reinforcing effect of the piles on the soil mass is disregarded in the formulation of interaction factors. This assumption becomes less realistic for sizeable groups of piles with a large pile stiffness factor, K. This effect can be modelled by using a diffraction factor (Mylonakis & Gazetas, 1998) that will lead to a reduction of the interaction factor. Randolph (2003) expanded the solution to include pile groups with piles in different diameters.

The assumption of linear elasticity for soil behaviour is known to over-estimate interaction effects in a pile group. Jardine et al (1986) demonstrated the importance of non-linearity in pile group settlement and load distribution with the use of finite element analyses.

Mandolini & Viggiani (1997) incorporated the non-linear response of a single pile into the formulation of interaction factors. The method allows for modelling of piles with variable sectional area and in horizontally layered elastic soils. The procedures use boundary element method to calibrate soil model against load-settlement behaviour of a single pile.

This is then used to determine the interaction factor for pairs of piles at different spacing. It also establishes a limiting pile spacing, beyond which the effect of interaction is insignificant.

Legend :

Figure 7.8 – Typical Variation of Group Settlement Ratio and Group Lateral Deflection Ratio with Number of Piles (Fleming et al, 1992)

1 3 5 7 9 11

Group Lateral Deflection Ratio, Rh

20 Group Lateral Deflection Ratio, Rh

20

Group Settlement Ratio, Rgs sp

1 3 5 7 9 11

For axial loading on rigid piles with similar loading, the interaction between the pile shafts and the pile bases can

Total pile head settlement can be computed by assuming compatibility of pile base and shaft stiffness : Pt = δ_t (Pb

δ_b + Ps

δ_l )

Interaction factor from adjacent piles can be computed by rearranging the above equation and expressed as : δ_t = (1 + α') P_t

GLro where α' is the interaction factor Legend : δt = settlement at pile head due to load at pile head, Pt

δ_b = settlement at pile base due to load at pile base, Pb

δ_l = settlement due to shaft resistance in response to load along pile shaft, Ps

rm = maximum radius of influence of pile under axial loading, empirically this is expressed in term of the order of pile length, rm = 2.5 ρ L (1 - ν_s)

ν_s = Poisson's ratio of soil

Figure 7.9 – Group Interaction Factor for the Deflection of Pile Shaft and Pile Base under Axial Loading (Randolph & Wroth, 1979 and Fleming et al, 1992)

[ 2

Profile of soil shear modulus, G

Ø

Fraser & Lai (1982) reported comparisons between the predicted and monitored settlement of a group of driven piles founded in granitic saprolites. The prediction was based on the elastic continuum method, which was found to over-estimate the group settlement by up to about 100% at working load even though the prediction for single piles compares favourably with results of static loading tests. Similar findings were reported by Leung (1988). This may be related to the densification effect associated with the installation of driven piles or the over-estimation in the calculated interaction effect by assuming a linear elastic soil.

In general, the interaction factor method based on linear elastic assumptions should, in principle, give a conservative estimate of the magnitude of the pile group settlement. This is because the interaction effects are likely to be less than assumed.

7.5.1.6 Numerical methods

A number of approaches based on numerical methods have been suggested for a detailed assessment of pile group interaction effects. They usually provide a useful insight into the mechanism of behaviour. The designers should be aware of the capability and limitations of the available methods where their use is considered justifiable for complex problems. Examples of where numerical methods can be applied more readily in practice include design charts based on these methods for simple cases, which may be relevant for the design problem in hand. Some such design charts are discussed in the following, together with the common numerical methods that have been developed for foundation analysis.

A more general solution to the interaction problem was developed by Butterfield &

Bannerjee (1971a) using the boundary element method. Results generally compare favourably with those derived using the interaction factor method (Hooper, 1979). An alternative approach is to replace the pile group by a block of reinforced soil in a finite element analysis (Hooper & Wood, 1977).

Butterfield & Douglas (1981) summarised the results of boundary element analyses in a collection of design charts. The results are related to a stiffness efficiency factor (Rg), which is defined as the ratio of the overall stiffness of a pile group to the sum of individual pile stiffness. This factor is equal to the inverse of the group settlement ratio (i.e. Rg = 1/Rgs).

Fleming et al (1992) noted that the stiffness efficiency factor is approximately proportional to the number of piles, np, plotted on a logarithmic scale, i.e. Rg = np-a. Typical design charts for calculating the value of the exponent a are given in Figure 7.10. For practical problems, the value of a usually lies in the range of 0.4 to 0.6. It is recommended that this simplified approach may be used for pile groups with simple geometry, i.e. regular arrangement of piles in a uniform soil.

Other numerical methods include the infinite layer method for layered soils (Cheung et al, 1988) and the formulation proposed by Chow (1989) for cross-anisotropic soils. Chow (1987) also put forward an iterative method based on a hybrid formulation which combines the load transfer method (Section 6.13.2.2) and elastic continuum approach (Section 6.13.2.3) for single piles using Mindlin's solution to allow for group interaction effects.

0 20 40 60 80 100 0.60

0.58 0.56 0.54 0.52

0.50

0.0 0.2 0.4 0.6 0.8 1.0

1.10 1.00 0.90 0.80 0.70

2 4 6 8 10 12

2.0 2.4 2.8 3.2 3.6 4.0

Poisson's Ratio and Homogeneity Factor, ρ

Spacing Ratio, sp/D

Log10 (Stiffness ratio, Ep

GL ) Slenderness Ratio, L/D

Efficiency Exponent, a Exponent Correction Factors

(a) Base Value

(b) Correction Factors Legend :

Ep = Young's modulus of pile Rg = stiffness efficiency factor a = exponent for stiffness efficiency factor L = length of pile

D = pile diameter ν_p = Poisson's ratio of pile

sp = pile spacing GL = shear modulus of soil at pile base

np = number of piles in a group ρ = rate of variation of shear modulus of soil with depth (homogeneity factor)

Note :

(1) Rg = np –a where the efficiency exponent, a, is obtained by multiplying the base value from (a) and the correction factors selected from (b).

Figure 7.10 – Calculation of Stiffness Efficiency Factor for a Pile Group Loaded Vertically (Fleming et al, 1992)

Stiffness ratio, Ep/GL

Homogeneity, ρ

Spacing ratio, sp/D Poisson's ratio, νp

Results of numerical analyses of the settlement of a pile group that are socketed into a bearing stratum of finite stiffness are presented by Chow et al (1990) in the form of design charts.

Computer programs based on the 'beam (or slab) on spring foundation' model may be used where springs are used to model the piles and the soil (Sayer & Leung, 1987; Stubbings

& Ma, 1988). This approach can reasonably be used for approximate foundation-structure interaction analysis. For a more detailed and rational assessment of the foundation-structure interaction and pile-soil-pile interaction, iterations will be necessary to obtain the correct non-uniform distribution of spring stiffness across the foundation to obtain compatible overall settlement profile and load distribution between the piles.

There is a relatively wide range of approaches developed for detailed studies of interaction effects on the settlement of a pile group. Different formulations are used and it is difficult to have a direct comparison of the various methods. The applicability and limitations of the methods for a particular design problem should be carefully considered and the chosen numerical method should preferably be calibrated against relevant case histories or back analysis of instrumented behaviour. In cases where a relatively unfamiliar or sophisticated method is used, it would be advisable to check the results are of a similar magnitude using an independent method.

7.5.2 Lateral Loading on Vertical Pile Groups