The assessment of the lateral deflection of a pile group is a difficult problem. The response of a pile group involves both the lateral load-deformation and axial load- deformation characteristics as a result of the tendency of the group to rotate when loaded laterally. Only when the rotation of the pile cap is prevented would the piles deflect purely horizontally.
7.5.2.2 Methodologies for analysis
There are proposals in the literature for empirical reduction factors for the coefficient of subgrade reaction, nh (Table 7.2) to allow for group effects in the calculation of deflection, shear force, bending moment, etc. using the subgrade reaction method. Although these simplifying approximations do not have a rational theoretical basis in representing the highly interactive nature of the problem, in practice they are generally adequate for routine design problems and form a reasonable basis for assessing whether more refined analysis is warranted.
An alternative approach, which may be used for routine problems, is the elastic continuum method based on the concept of interaction factors as for the calculation of pile group settlement. Elastic solutions for a pile group subject to horizontal loading are summarised by Poulos & Davis (1980).
Table 7.2 – Reduction Factor for Coefficient of Subgrade Reaction for a Laterally Loaded Pile Group (CGS, 1992)
Pile spacing/ Pile Diameter Reduction Factor, Rn, for nh
3 0.25 4 0.40 6 0.70 8 1.00 Notes : (1) Pile spacing normal to the direction of loading has no influence,
provided that the spacing is greater than 2.5 pile diameter.
(2) Subgrade reaction is to be reduced in the direction of loading.
As a general guideline, it may be assumed that piles can sustain horizontal loads of up to 10% of the allowable vertical load without special analysis (CGS, 1992) unless the soils within the upper 10% of the critical length of the piles (see Sections 6.13.3.2 & 6.13.3.3 for discussion on critical length) are very weak and compressible.
Based on the assumptions of a linear elastic soil, Randolph (1981b) derived expressions for the interaction factors for free-head and fixed-head piles loaded laterally (Figure 7.11). It can be deduced from this formulation that the interaction of piles normal to the applied load is only about half of that for piles along the direction of the load. The ratio of the average flexibility of a pile group to that of a single pile for lateral deflection under the condition of zero rotation at ground level can also be calculated. This ratio, defined as the group lateral deflection ratio (Rh), is analogous to the group settlement ratio (Rgs). As an illustration, results for typical pile group configurations are shown in Figure 7.8 which illustrates that the degree of interaction under lateral loading is generally less pronounced compared to that for vertical loading. This approach by Randolph (1981b) is simple to use and is considered adequate for routine problems where the group geometry is relatively straight forward.
An alternative is to carry out an elasto-plastic load transfer analysis using the subgrade reaction method with an equivalent pile representing the pile group. In this approach, the group effect can be allowed for approximately by reducing the soil resistance at a given deflection or increasing the deflection at a given soil pressure (Figure 7.12). In practice, the actual behaviour will be complex as the effective H-δh curve for individual piles may be different and dependent on their relative positions in the pile group. Considerable judgement is required in arriving at the appropriate model for the analysis for a given problem.
7.5.2.3 Effect of pile cap
Where there is a pile cap, the applied horizontal loads will be shared between the cap and the pile as a function of the relative stiffness. The unit displacement of the pile cap can be determined following the solution given by Poulos & Davis (1974), whereas the unit displacement of the piles may be determined using the methods given in Sections 6.13.3 and 7.5.2.2. From compatibility considerations, the total displacement of the system at pile head level can be calculated and the load split between the cap and the piles determined. Care should be taken to make allowance for possible yielding of the soil where the strength is fully mobilised, after which any additional loading will have to be transferred to other parts of the system.
Definition of Departure Angle, αs
If the stiffness of a single pile under a given form of loading is KL, then a horizontal load H will give rise to a deformation δh given by :
δh = H KL
If two identical piles are each subjected to a load H, then each pile will deform by an amount δh given by : δh = (1+ α') H
KL
For fixed-head piles α' = 0.6 ρc' ⎝⎛
⎠⎞
Ep
Gc 1/7
ro(1 + cos2αs)
sp
At close spacing, the above expression over-estimates the amount of interaction. When the calculated value of α' exceeds 0.33, the value should be replaced by the expression 1- 2
27α' For free-head piles
α' = 0.4 ρc' ⎝⎛
⎠⎞
Ep
Gc 1/7
ro
(1 + cos2αs) sp
Legend :
α' = interaction factor for deflection of piles
αs = angle of departure that the pile makes with the direction of loading ρc' = degree of homogeneity = G0.25Lc
Gc
G = shear modulus of soil G* = G (1 + 0.75 νs)
G0.25Lc = value of G* at depth of 0.25Lc
Gc = average value of G* over Lc
Lc = critical pile length for lateral loading = 2 ro⎝⎛
⎠⎞
Epe
Gc 2/7
νs = Poisson's ratio of soil
sp = spacing between piles ro = radius of pile Ep = Young's modulus of pile Ip = moment of intertia of pile
Epe = equivalent Young's modulus of pile = 4EpIp
πro4
Figure 7.11 – Interaction of Laterally Loaded Piles Based on Elastic Continuum Method (Randolph, 1981a and Randolph, 1990)
sp
Pile A αs Pile B
H
Kim et al (1977) observed from full-scale tests on a group of vertical piles that the effect of contact between a ground-bearing cap and the soil is to reduce the group deflection by a factor of about two at working conditions. However, it was reported by O'Neill (1983) that the effect of cap contact is found to be negligible where the majority of the piles are raked.