THEORETICAL SOLUTION OF PILING COMPACTION AND THE INFLUENCE OF PILE-SOIL-BOUNDARY CURVE HYPOTHESIS

Research is ongoing to find theoretical solution to three-dimensional piling compaction. Considering the spacial-axis-symmetric characteristics, the boundary surface of pile-soil interaction is expressed by polynomials of different orders. First, the curve family parameter is introduced to construct the displacement and integral function. Then, the solution of pile-soil interaction is derived by combining the constitutive relation model of Duncan-Chang and the variational theory. Results of engineering computing show that the theoretical solution converges to the classical CEM and the limit equilibrium theory well at the corresponding computing area. Moreover, the effects of polynomial of different orders on the calculation results are not obvious. The conclusion in this paper can be used for reference in the derivation and application for other interaction of structure and soil problems.


INTRODUCTION
For cavity expansion, a source-sink [1,2] method is proposed to solve the displacement field caused by pile-soil interaction. Displacement method [3,4]is also applied to solve the problem of cavity expansion in semi-infinite space. With the truth that the ground must be free or zero stress surface for semi-infinite space, the stress superposition [5] is adopted to correct the source-sink method. The above studies should have the following limitations: (1) Assumption of linear elasticity constitutive model of nonlinear material. (2) The superposition principle is used to solve the nonlinear displacement field.
In the above equation, i F is the volume force, 1 S is the boundary of the known force, and i P is the known force on the boundary;   ij A  is the potential energy density. The density of the pile-soil interaction can be given as Equation (2)

SETTING DISPLACEMENT FUNCTION
The interaction between pile and the soil around it in semi-infinite space can be regarded as a spatial axisymmetric problem. The mechanical components are independent of the coordinates  and functions of the coordinates   , rz. The equation of boundary curve is given as Equation (3) and In order to set the displacement function , r uw to satisfy the known displacement boundary condition, the curve family parameter can cover any spatial point in the calculation range.

GEOMETRIC AND CONSTITUTIVE EQUATIONS
Since the parameters 0 z are introduced into the displacement function, the Straindisplacement relationship are given as Equation (7) 00 00 00 00 , rr rz Duncan-Chang model is adopted [8,9], as shown in Equation (8) and (9). The relationship curve between stress and primary strain of the constitutive model is shown in Figure 3.
Modulus of elasticity is given as where 1  is the first main strain [10], determined by Equation (11).
The volume deformation modulus of Duncan-Chang model is expressed as Equation (12) Km is the experimental constant.

POTENTIAL ENERGY DENSITY
The increment of stress components [11] can be expressed as Equation (13)  (7), 0  is Poisson ratio. According to Equation (13), the total stress can be expressed as where  is the buoyant unit weight, z  is the consolidation stress. Therefore, the potential energy density can be expressed as: Since the curve parameter 0 z is introduced, the potential energy density can be written as Equation (14), which is a function of where 0 ,, z r z meet the constraint conditions of

FUNCTIONAL CONSTRUCTION AND COEFFICIENTS DETERMINATION
According to energy density of Equation (14), Equation (1) can be written as Equation (15)  where: (14).
The coefficient , AB can be solved by the extreme calculation of Equation (16), and the theoretical solution of displacement and stress field can be obtained then.

PROJECT CASES
Equation (3) is set as the polynomial function as shown in Equation (17)   Family curves can be written as Equation (19), as shown in Figure 1, (20) are given for calculating the stress in the plastic and elastic zone around the pile by the cavity expansion method (CEM) [14,15].  Equation (21) is given for calculating the stress in the r direction based on the limit equilibrium theory [17]. Figure 5 shows a comparative analysis of the lateral pressure results based on the limit equilibrium theory and the theoretical solution derived in this paper based on Equation (19).  condition, it is only suitable for solving the plane strain problem. For finite-length pile threedimensional expansion, the region below or around the pile tip is obviously inconsistent with the assumption of the plane strain. Compared with the CEM results, the theoretical solution of this paper is more reasonable with the rapidly decreasing of radial pressure around or below the pile tip. Figure 5 shows that the results of theoretical solution are basically matching with the results of limit equilibrium theory at plastic line located in the range of 0m Compared with the limit equilibrium results, the theoretical solution of this paper is more reasonable, with the rapidly decreasing of radial pressure around or below the pile tip. In addition, because the limit equilibrium results are directly related to the volume-weight of the soil, the curve of passive earth pressure shown in Figure 5 has inflection points at the interface of the soil layer. Figure 5 also shows that the results of theoretical solution are basically matching with the results of numerical simulation results at

Example 2
Family curves can be written as Equation (22)  To sum up, with the increase of undetermined coefficients from 0 to 4, the additional stress generated by pile compaction rapidly converges to the reasonable value and polynomial with lower order can also be used to simulate the initial hole wall boundary to reduce the computational workload.   The test site is located in a section of Taiwan high-speed railway project [18]. The precast concrete pile length is 34m, and diameter is 80 cm. The distribution of the test pile and measuring pipes are shown in Figure 8. Burial depth of survey tubes are 40m. The soil layers in the site are inter-bedded sandy soil and cohesive soil, and the critical values of sandy and clay are taken as parameters for calculating. Parameters of soil layer [9,18]