Analysis on Laterally Loaded Group Piles by Plaxis 3D Foundation

It has been known that a group pile lateral capacity is smaller than the sum of each pile capacity composing the group. A reduction factor, also known as efficiency factor, is required to determine the effective lateral capacity of group piles. To the authors' knowledge, in most geotechnical text books, only the spacing of piles is considered in evaluating the pile group lateral capacity. No consideration on the effects of soil stiffness modulus and the total number of piles forming the group is taken into account. This research tried to investigate the influence of those factors by using geotechnical 3D finite element software, namely Plaxis 3D foundation. It is found that the bigger the number of piles in a group, the lower the efficiency factor is; the higher the soil stiffness modulus, the greater the efficiency is.


INTRODUCTION
Apart from axial loads, lateral loads induced by wind, earthquakes, berthing of ships, vehicle acceleration and braking forces on the bridges, etc, have to be taken into account in designing a foundation system. Therefore, determination of the lateral capacity of pile foundation is one of the utmost important in foundation engineering.
It is well understood that a group pile lateral capacity is smaller than the sum of each pile capacity composing the group. A multiplier is required to determine the effective lateral capacity of group piles. The multiplication factor is known as group efficiency factor or reduction factor. In the available geotechnical text books, the only factor considered in determining the reduction factor is the center-to-center spacing of the piles.
To the authors' knowledge, no in-depth study has been carried out to investigate the influence of the total number of piles forming the group and the effects of soil stiffness on the said efficiency factor. To answer whether there is any influence of these factors on the carrying capacity of the laterally loaded group pile -under the supervision of the second author -the first author carried out numerous analyses by using Plaxis 3D Foundation software version 2.2.
Due to time constraints, the extend of the study is limited to the following scope: (1) only 1 m diameter bored pile of 40 meter length is considered; (2) the center-to-center spacing of piles considered are 4D, 5D and 6D, where D is the diameter of the pile; (3) in evaluating the effect of pile spacing, the numbers of piles taken into account in a group are 4, 9, 16, 25, 36, 49 and 64 piles; (4) the value of soil stiffness ranges from 3000 kN/m 2 to 25000 kN/m 2 ; (5) in evaluating the effect of soil stiffness modulus, the numbers of piles considered in a group are 2, 4, 6, 9, 12, 25, 36, 49 and 64; (6) no existence of axial load; (7) pile head deflection is limited to 6 mm; (8) no pile cap effect is considered; (9) only fixed head capacity of piles is considered. As the present Plaxis version cannot determine the single pile fixed head capacity; the pile single pile capacity of pile is determined from finite difference method.

Literature Review
There are numerous methods available in determining the lateral capacity of a single pile, e.g. Reese Matlock method, Chang method, finite difference method, etc. From all the available methods, only finite difference method can directly evaluate the pile lateral carrying capacity of layered soils without going through the averaging of the horizontal sub grade reaction coefficient. Therefore, this method is adopted in determining the lateral capacity of a single pile.

Finite Difference Method
Winkler (1867) (1) and (2) The solution to the above differential equation can be obtained either analytically or numerically. An analytical solution is easy to obtain when the value of k h is constant throughout the pile. When the value of k h varies with depth, a numerical solution by finite difference method is employed (Palmer and Thompson, 1948;Gleser, 1953).
In this method, the basic differential equation (3) is written in the form of finite difference as follows: Equation (4)   A total simultaneous equation n+5 is needed to calculate the n+5 displacement which is unknown at the point (-2, -1, n +2 and n +3). Equation (6) can be employed from point 2 to point n in order to provide (n-1) equations. Further equations can be obtained from boundary conditions at the pile head. At the pile head there are two conditions to consider, i.e. free head and fixed head conditions. Free head pile Shear force: Another way to solve the above is to ignore the shear force equation at the pile end (tip) and the pile head (top), i.e., equations (8) or (12) and (16). Therefore, ignore the two displacement variables at the point -2 and n +3. In this case only n +3 equations have to be solved. This procedure gives similar results to the previous procedure.

Group Pile Analysis
To find the lateral capacity of the group pile, Prakash in 1962 (Poulos, 1980) proposed to reduce the value of coefficient of horizontal sub grade reaction (k h ), as shown in Figure 2. The group pile can then be performed using the finite difference method described earlier by entering reduced k h values.

Modeling in Plaxis 3D
Plaxis 3D Foundation is a three-dimensional Plaxis program, developed for the analysis of three dimensional foundation and geotechnical problems. It is part of the Plaxis suite finite element software used worldwide for geotechnical engineering design. The software allows the complex finite element model to be solved quickly. The various available output facilities can be used to display the detail computational results. In this study the effect of pile cap is not considered. To eliminate friction between the soil and the pile cap, a dummy soil of 10 cm thick beneath the pile cap is introduced in the modeling (Figure 3). The dummy soil has the characteristics of water, so as to eliminate the friction.

RESULTS AND DISCUSSION
The results of the study are presented in Table 2 and Figure 5 and 6 below.

CONCLUSION
Based on the analysis using Plaxis 3D Foundation program, it is found out that the spacing and the number of piles, as well as the soil stiffness do have significant effects to the lateral capacity of group pile. The conclusions are summarized below: (1) the greater the spacing between piles in a group pile, the greater the efficiency factors are. This is due to the fact that the reaction area of the soil behind each pile is larger. Therefore, the interaction region among the piles (i.e. the overlapping reaction areas) becomes smaller. Hence, the lateral capacity of the group pile becomes greater; (2) the total number of piles in a group has significant influence on the efficiency factor of the group pile. The greater the number of piles in a group pile, the lower the efficiency factor is; giving the lowest efficiency factor to around 0.20; (3) the stiffness modulus of the soil also affects the efficiency factor of the group pile. The efficiency factor increases a long with the higher stiffness modulus of the soil.
The above is the interim results of the study. To make the study complete, the following shall be investigated further: (1) pile spacing of 2, 3, 7 and 8 times pile diameter; (2) various pile lengths and diameter; (3) pile cap effects; (4) comparative study by using different geotechnical finite element program, such as the GeoStudio or other.