Group effect in laterally loaded piles
When a pile group, with n piles, joined by a rigid cap, is subjected to a lateral force applied at cap level, it is observed that:
i) the pile head lateral displacement of all piles is equal to the cap displacement;
ii) the lateral displacement of the group, ug, is greater than the displacement of a single isolated pile, with the same boundary conditions at the head, when subjected to a lateral load corresponding to a uniform load distribution;
iii) the distribution of the load by the piles is not uniform and depends on the location of each pile.
The above findings are the result of the development of pile-soil-pile interaction effects, since the displacement of a given pile contributes to the displacement of the remainder. When a pile group is submitted to a lateral load, each pile pushes the adjacent soil, creating a localized shear zone. With the increase of the load, the shear zone of each individual pile enlarges in order to mobilize the required soil reaction, and overlapping of effects occurs, as depicted in Figure 7.31. The overlapping effect between piles in different rows is called the shadowing effect and it is responsible for a lower mobilization of soil reaction in the piles located in the intermediate rows. Since the overlapping effect can also occur due to the proximity of piles of the same row (the edge effect), the piles located in the corners usually make a larger contribution to the balance of the global lateral load.
From the representation depicted in Figure 7.31, it is possible to conclude that the reaction offered by the piles located in the peripheral corners is only slightly lower than the reaction given by an isolated pile under the same conditions, whereas the piles located at the group interior provide a significantly lower reaction due to shadowing and edge effects. Since all piles suffer the same head displacement, and the reaction offered by the soil is dependent on pile location, piles in the external rows are subjected to greater shear forces than are piles located in the inner rows.
Experimental evidence of this interaction problem was presented by Brown et al. (1988) Rollins et al. (1998), among others. It should be highlighted that, in pile groups with a large number of piles, this effect can be very significant, and its omission can lead to serious underestimation of mobilized internal forces on the peripheral piles.
As in other geotechnical problems, the complexity of the phenomena under analysis can be understood in detail only through an approach based on a continuous formulation. However, it is also well known that such approaches, taking into account the 3-D character
Figure 7.31 Laterally loaded pile group: shadowing and edge effects (Walsh, 2005).
of the problem and the soil non-linearity, are very demanding from the computational point of view, discouraging its use as a regular technique for design support. Therefore, simplified methods have been proposed, such as the method of influence factors, proposed by Poulos (1971), and the P-multipliers method proposed by Brown and Bollman (1993).
7.7.3. I Method of the influence factors
The application of the method of influence factors is limited to linear-elastic analysis, since it is based on superposition of effects. The method can be faced as an extension of the Winkler model for the analysis of pile groups, introducing influence coefficients in order to take into account the pile-soil-pile interaction effects. The influence factor is a multiplier of the pile deflection due to the presence of other piles, subjected to similar loading conditions, in its vicinity. Thus, assuming K, as the horizontal stiffness of an isolated pile, then the displacement of the pile head i belonging to a group with n equal piles is given by:
where u, is the horizontal displacement of the head of pile, is the influence factor between pile i and pile /, and is the lateral load transmitted to pile /.
On the other hand, taking into account the expressions given in Table 7.10, it is easy to find that, for a pile with fixed rotation head, the horizontal stiffness is equal to:
Randolph (1981), through the calibration of the results predicted by the present method with those obtained by means of continuous three-dimensional analysis, proposed a set of analytical expressions for the calculation of the influence factors. Given the generality of practical situations, the most interesting is the expression for influence coefficients related to the horizontal displacement of the pile head with fixed rotation and lateral loading. In this context, and considering the ground to be an isotropic and homogeneous elastic medium, the author proposes the following expression for the evaluation of the influence factors:
where щ is the angle formed by the load direction and the hypothetical line that links pile i to pile /, and is a dimensionless coefficient given by:
where Ep is the pile Young’s modulus, Gs and v are the soil shear modulus and Poisson ratio, respectively, В is the pile diameter, and, finally, s/( is the center-to-center distance between piles i and /.
The distribution of horizontal loads is obtained from the prescription of displacement compatibility and load equilibrium at the pile heads. From a mathematical point of view, the load attributed to each pile is obtained by solving the following system of equations:
Notice that the pile head displacement is given by Equation 7.85.
After assessing the load distribution by the different piles, the estimation of the pile response (in terms of displacements, internal forces, etc.) is a simple step. Actually, after solving the system of Equations 7.89, the head displacement (w„ which is the same as ug due to the displacement compatibility imposed by the pile cap) and the applied load (H;) are known variables. Thus, the pile response can be obtained directly by application of expressions presented in Table 7.10, but considering a hypothetical Winkler coefficient, which depends on pile location. This hypothetical coefficient, is obtained taking into account the following relationship:
Therefore, after some mathematical manipulation:
A final comment must be presented about the suitability of the method. This method can be considered to be a balance between simplicity and accuracy. However, there are some drawbacks that the reader must be aware of when applying it:
i) it is assumed that the soil is elastic, homogeneous, and linear;
ii) the comparison of experimental results with the results provided by the method shows that there is usually an overestimation of the loads in the peripheral piles;
iii) there is no distinction of load for piles located in symmetrical positions relative to the group center of gravity, i.e., the load attributed to front piles is the same as that for rear piles, the adherence of which to experimental evidence is dubious.
22.214.171.124 P-multipliers method
One way to evaluate the group response to applied lateral load is to modify the P-у curves to reflect the pile-soil-pile interaction effects. Brown and Bollman (1993) propose a procedure whereby the P-у curves for an isolated pile are modified as functions of the pile location. An illustration of this concept is depicted in Figure 7.32. Application of the method should follow these steps:
i) select a P-у curve compatible with the soil and pile characteristics for a single pile;
ii) apply a P-multiplier, Pm to the selected P-у curve, generating a revised P-у curve as shown in Figure 7.32. For s = 3B, the recommended P„, values are 0.8 for the lead row,
Figure 7.32 Illustration of P-multiplier concept for lateral group analysis (Hannigan et al., 2016).
0.4 for rhe second row, and 0.3 for the third and subsequent rows. For s = 5B, the recommended P„, values are 1.0 for the lead row, 0.85 for the second row, and 0.7 for the third and subsequent rows. For intermediate pile spacing, linear interpolation may be used;
iii) develop a numerical model (using the finite element method, for instance), where the piles are simulated by beam elements linked by a rigid cap and assuming the modified P-у curves, as functions of the pile location in relation to the load direction (see Figure 7.32).
Usually, all the piles of the group have the same structural configuration, so the pile design is based on the shear forces and bending moments mobilized in the critical row, i.e., the front row, as represented in Figure 7.32.