Nonlinear Stress Analysis of Flexible Pile Composite Foundation by Energy Method

To calculate the stress of cement-soil pile (flexible pile) in composite foundation accurately and overcome the defective preposition that the pile and the soil are elastic, the nonlinear analysis and calculation of pile-soil stress by adopting the energy method are performed. First, on the basis of the universally adopted load transfer method, it is assumed that the friction and the relative displacement of the pile shaft conform with the elastic-plastic relationship. *en, the modified Cam-clay model is introduced to calculate the compressive amount of soil around the pile. Further, the stress and strain characteristics curve of the flexible pile is used to calculate the compression of the pile, deducing the energy equation of the pile and soil around the pile, establishing the total potential energy expression of the flexible pile composite foundation, and the load sharing value and the corresponding stress value of the pile-soil are obtained based on the minimum potential energy principle. Finally, this method is verified by indoor laboratory model test and the reported case. *e results show that the calculated solutions are in agreement with the measurements; therefore, the calculation method is reasonable.


Introduction
Cement-soil piles [1] or cement-soil pile composite foundations [2] have been used in ground treatment.In the design of composite foundation, pile-soil stress ratio constitutes an important parameter for the design of composite foundation.In the early stage, the pile-soil stress ratio of the cement-soil pile composite foundation stemmed mainly from field load test data [3,4].Baumann and Bauer [5] put forward an empirical formula taking deformation modulus, displacement ratio, and side pressure coefficient into consideration through the analysis of the deformation of gravel pile and soil around the pile as well as the soil pressure.Simplified methods were adopted to analyse the stress of the piles proposed by Poulos and Davis [6].Due to the deformation of mixing piles composite foundation and the distinctive distribution of soil pressure, Duan [7] and Kitazume and Terashi [8] found that the pile-soil stress ratio n is mainly related to the ratio of pile-soil modulus.Song et al. [9] held the idea that the stress ratio was proportional to the 1/2 of pile-soil modulus ratio, and the comprehensive empirical parameter α is introduced to reflect the nature of soil and load, and then the formula for calculating the stress ratio of mixing pile was proposed.Ishikura et al. [10,11] by using the average skin friction deduced the average stress distribution ratio after consolidation.
e above formula was a fitting formula based on measured data.Afterwards, many scholars set up equations on the basis of the theory of pile-soil interaction.Alamgir et al. [12] proposed a onedimensional displacement model that did not consider radial displacement and deduced an analytic calculation formula of the pile shaft stress, pile skin friction, and settlement of the reinforcement area.Based on the above study, Liu and Zhang [13] obtained an analytical solution with the two-dimensional displacement mode considering the radial displacement.However, these studies failed to consider the relative displacement between the pile and soil.Lv et al. [14] and Yu et al. [15] assumed that the pile-soil interface had relative displacement, and the settlement difference between the soil and pile at the same depth, modifying Alamgir displacement model, and deducing the analytic solution of the working characteristics of the composite foundation under the flexible foundation.In 2011, Yu et al. [16] improved on the assumption of the linear distribution of the side friction of a pile along the depth, using the nonlinear distribution of skin friction of the pile along the depth.e paper [5][6][7][8][9][10][11][12][13][14][15][16] performed a study of the pile and soil, assuming them to be an elastic material, which was different from the nonlinearity of materials in practice.erefore, Qin et al. [17] adopted the load-settlement hyperbolic model to analyse the pile-soil stress ratio.Huang and Huang [18] and Zheng and Huang [19] used the bilinear load transfer function and hyperbolic function to represent the pile and soil load-settlement curve, respectively, and deduced an analytical formula of the pile-soil stress ratio.In addition, Low et al. [20], Hu [21], and Zhou et al. [22] deduced a formula for pile-soil stress of composite foundations by analysis of the stress that the effect of soil arching exerts on the pile top.Zhao et al. [23], Yang et al. [24], and Zhao et al. [25] performed a comprehensive analysis of the effect of a reinforced cushion on the pile-soil stress ratio and derived the calculation of the pile-soil stress ratio and settlement of the pile-net composite foundation.Currently, the main problem in the study of pile-soil stress calculations is that the pile or soil is assumed to be an elastomer [5][6][7][8][9][10][11][12][13][14][15][16], which is obviously not realistic.In this paper, it is assumed that the soil satisfies the modified Cam-clay model, the constitutive behaviour of the cement-soil pile is introduced to describe the nonlinearity of the pile, and an equation is established to perform the calculation of pile-soil stress based on the fact that skin friction of the pile shaft and the relative displacement conforms with elastic-plastic relations.
e energy method based on the principle of energy conservation can be used to analyse the displacement, deformation, and internal forces of deformable solids.Shen et al. [26] used the minimum potential energy principle to analyse the interaction between the pile group and the pile cap.Zheng et al. [27] deduced the analytical solution of the composite modulus of the composite foundation by applying the parametric variational principle.Liu et al. [28] established the energy balance equation of the pile shaft and obtained the settlement of the foundation pile by adopting the pile skin resistance and end resistance model which explains the energy transfer process between the pile shaft and soil.Yang [29] and Zhao et al. [30] used the energy method to solve for the foundation pile axial force and displacement to obtain a calculation method for the settlement of elevated caps pile foundations.At present, the energy method is confined to the pile foundation calculation that does not take the load sharing of the soil into consideration.It does not work in the calculation of composite foundation as well.
erefore, this paper attempts to establish a new energy method analysis model for the stress calculation of composite foundation based on energy conservation law.e paper deduced the formula of the energy equation of the reinforced pile shaft and the soil based on the proper simplification of the model, and the total potential energy expression of the flexible pile composite foundation is established.It also adopts the minimum potential energy principle to obtain the load sharing ratio of the pile and soil, and the stress of the pile and the soil are calculated, thus providing a new calculation method for the stress calculation of the composite foundation.

Stress Calculation of Composite Foundation
2.1.Pile-Soil Interaction Model.In the composite foundation, the cushion adjusts the pile-soil stress differently with the load, which eventually stabilises.To analyse the compression of the pile and soil in the reinforced area, the paper selects the centre pile in the composite foundation (as shown in Figure 1) as the object of analysis.Suppose that the analysis unit under the load is p, the central pile shares the load for the pile p p and that the soil shares the load p s .In Figure 1, s p0 is the displacement of the pile top.s b is the displacement of the pile end.s s0 is the displacement of the soil between the piles.τ i is the pile skin friction.e pile area is A p , and the calculated area of soil around the pile is A s .e pile diameter is d.erefore,

Calculation Model of Pile Side Friction.
e pile skin friction is usually related to the relative displacement between the pile and soil.It is generally considered that the pile skin friction and the relative displacement between the pile and soil accord with the elastic-plastic mathematics model [25].at is, when the relative displacement between the pile and soil reaches a certain critical value, pile skin friction will reach the maximum (as shown in Figure 2).e skin friction of each pile section along the depth is where s i is the relative displacement of the pile shaft i, s i � s pi − s si ; τ u is the ultimate friction resistance of the layer i of soil; s u is the minimum displacement of the pile-soil when the layer i achieves ultimate friction τ u ; and c s is the friction resistance transmission coefficient of the layer i, c s � (τ u /s u ).

Calculation Model of Pile Bottom Resistance.
As for the undamaged pile after the pile bottom pierced the underlying stratum, the relationship between the pile bottom resistance and displacement can be simulated by the double fold line model [31], as shown in Figure 3. e pile bottom resistance can be expressed as where s b is the displacement of the pile bottom; k 1 is the slope of the first straight line; s bu is the displacement of the first line corresponding to the limit displacement value, usually (0.5%∼2.5%) d, mm; k 2 is the slope of the second 2 Advances in Materials Science and Engineering line; and k 1 can be calculated according to the following formula [32]: where G b and υ b are the shear modulus and Poisson's ratio of pile bottom soil, respectively, and k 2 can be obtained by inverse analysis of the load-sedimentation curve in eld tests.It can be calculated by the following formula [33]: where ΔP t is the pile top load increment, ΔS t is the pile top settlement increase caused by the pile top load increase, and L is the pile length.e other symbols remain the same.

Energy Balance Equation of Composite Foundation System
3.1.Basic Assumptions.After the soil is subjected to load in the reinforced area, the soil particles are compressed Advances in Materials Science and Engineering and slipped to the equilibrium state, without considering the shear deformation of soil around the pile; ultimately, the soil surface is flat.Regardless of the interaction between the units, the compression of the analysis unit can be considered the compression of the composite foundation.
When analysing a typical unit, the following is supposed: (1) Without considering the impact that lateral deformation of the pile has on the settlement, the pile skin friction is exerted before that of the pile bottom under the external load.(2) e soil around the pile is isotropic homogeneous material, adopting a modified Cam-clay model.e soil surface is flat initially and after loading.(3) e selected analysis unit is the central part of the foundation, and the soil is subjected to uniaxial compression consolidation without lateral displacement.

Pile Shaft Energy Equation.
e total potential energy increment of the pile shaft in the foundation can be divided into two parts: the pile deformation energy W u and the potential energy increment W gp caused by the settlement, namely, In (7), σ z is the pile stress, ε z is the pile strain, m z is the pile mass at z, and g is the gravitational acceleration.In general, the stress in the design is half of the maximum stress, so it represents the rising section in the stress and strain curve, and the pile stress can be expressed as follows: where q u is the unconfined limit strength and ε u is the strain value corresponding to the ultimate strength.
External force potential consists of work by the load in the pile top W p and work by the pile bottom resistance W b : where p b is pile bottom resistance.

Energy Equation of Soil.
e soil around the pile and underlying stratum is calculated using the modified Camclay model.In 1968, Roscoe and Burland [34] further amended the Cam-clay model, believing that the yield surface trajectory of the Cam-clay model should be elliptical, with the following energy equation: dW e � pdε e p + qdε e q , (12) where p is the mean effective stress, q is the deviator stress, dε e p is the elastic volumetric strain increment, and dε e q is the elastic deviator strain increment.Assuming that all shear deformation is irreversible, then where e is the void ratio of the soil, κ is the gradient of unloading or swelling line of structured clay, dε p p is the plastic volumetric strain increment, and dε p p is the plastic deviator strain increment.e incremental stress-strain of the modified Cam-clay model is where λ is the gradient of isotropic compression line of destructured clay.e shape of the yam of the modified Cam-clay model is elliptical and its yield criterion is where M is the gradient of critical state line in the q-p plane.
Since the selected element does not consider lateral deformation, the potential energy increment W gs caused by soil subsidence is where m s is the soil quality at z. e external force potential is worked by W s mainly for the soil surface:

Calculation Process
e total potential energy Π of the composite foundation under the external load p is equal to the deformation energy U of both the pile and soil and the external potential energy V, that is, Π � U + V.According to the principle of minimum potential energy, for δΠ � 0, the true displacement makes the total potential energy of the system take an extreme value in all allowable displacements u, v, and w that satisfy the known geometric boundary conditions.In the actual calculation, the pile top load p p and the load of soil between the pile p s are unknown.To search the level of the load, the 4 Advances in Materials Science and Engineering minimum potential energy principle is used to calculate.at is, under the external load p p,k and p s,k (k 1, 2, 3, ... , n pos ), the total potential energy for Π k is calculated by the iteration to nd the total potential energy minimum in the possible n pos , which is calculated by the real pile-soil load.
e calculation steps are as follows: (1) Load is divided into n gra numbers to load, and then j is loaded (j 1, 2, 3, ... , n gra ), the external load at the pile top is jp p,k /n gra , and the load of the soil surface is calculated as jp s,k /n gra according to (1).( 2) e piles and soil around the piles are divided into n str layers, and the thickness and unit weight of the pile-soil are calculated.(3) e layer i (i 1, 2, 3, ... , n str ) is selected for pile calculation, z (il/n str ), the unit body i is loaded externally by jp p,k /n gra (σ 0 ); the corresponding pile stress is, ) e deformation energy and settlement potential of the pile are calculated according to ( 7) and ( 8), respectively.(5) e soil around the pile calculation, unit body i, is subjected to the external load increment jp s,k /n gra (σ s,0 ), soil stress is σ s,i σ s,i−1 + m s g, and then p and q are calculated; ( 16) is used to determine whether it yields.If not, ( 14) is used to calculate the energy; otherwise, ( 11)-( 13) are adopted.Its strain should be accumulated in the calculation of the soil.(6) e j-level load and soil around the pile potential can be calculated according to (17); the total potential energy of soil is its cumulative value.(7) Repeat steps (1) to ( 6), and j is selected from 1 to n gra .(8) e total potential energy of the pile and soil system Π, Π W u + W gp − W p − W b + ΣdE + ΣW gs − W s ; (9) Repeat steps (1) to (8), and k is selected from 1 to n pro .(10) Within a reasonable range, when Π k is the minimum, then p p,k and p s,k stand for the actual load value of the pile and soil, and the corresponding stress is that of the reinforcement area.

Example 1.
To study the accuracy of the calculation, the model test of the soil-cement pile composite foundation with a lateral constraint was conducted in a trough with dimensions of 6000 mm × 3000 mm × 4000 mm (length × width × height) in the laboratory, and the geometric similarity ratio of the piles was 1 : 6.After the lling, the top of the lling soil was covered with thin lm to retain moisture, which formed the pile after deposit for 28 days.e cementsoil mixture that was made for the cement-soil pile was mixed clay with cement.e same clay was used in the soilcement mixture and lling soil, for which the air drying, crushing, screening, and measuring of the moisture content of the soil are similar to those of the former, which was only sieved with a sieve of 1 mm mesh to accelerate the interaction with cement and avoid a great di erence of cement-soil mixture.Ordinary Portland cement was used, of which the strength grade was 32.5 according to China General Cement Code (GB175-2007), and the dosage was α w 10%.e moisture of the mixture achieved the optimal 19.2%.e cement-soil pile was made by ramming the cement-soil mixture into a borehole in layers, with a compaction factor of 0.9.Chinese Fir was used in the pile with the constraint parameters shown in Figure 4. e central three piles of each group of piles served as test piles, and the pile spacing was three times the pile diameter.Pile bottom stress was tested by the earth pressure box, and pile shaft stress tests adopted a PVC pipe paste strain gauge; its speci c process is shown in [35].
e load test was carried out after 29 days of forming the pile.
e load plate size (steel plate) for loading was 707 mm × 707 mm × 20 mm. e test adopted the fast maintenance load method.Each load is 8 kPa and the total load is 56 kPa in the test, as shown in Figures 4 and 5.
e soil was selected to carry out the experiment.e calculation parameters of the soil were obtained, as shown in Table 1.
In this paper, the Maple software is used to compute the above calculation program, which divided the pile and soil around the pile into 120 elements.e load is divided into 100 loadings.e 0.1 m gravel cushion without restraint di uses the upper load.According to the Technical Code for Ground Treatment of Buildings of China (JGJ-2012), the load at the bottom of cushion is calculated to be 46.37 kPa.
e pile-soil stress under the last load was calculated.It is calculated that when the pile top load reaches 0.837 kN, the Advances in Materials Science and Engineering total potential energy of the system reaches its minimum (Figure 6), and its stress is 106.6 kPa.e calculation results of the pile-soil stress are shown in Table 2.
As seen from Table 2, the calculated value is slightly larger than the measured value, which occurred because the cushion di used the upper load in the test and the load on the centre pile was less than the upper load.As seen from Figure 7, the calculated pile shaft stress shows a similar curve to that of the measured value, and the maximum stress of the pile lies at 0.3 times the pile length from the pile top.e calculated values are well close to the experimental values above the maximum stress of the pile, but the deviations exist down the maximum stress of the pile.It is analysed that the main reason lies in the compaction on the soil around the pile during pile forming, which increased the side friction of the soil and reduced the pile stress.e maximum stress is 1.56 times the pile stress of the pile top, which is close to the measured value of 1.47.e results show that the calculation method proposed in this paper is reliable.

Example 2.
To further verify the reliability of the calculation method, the case of a eld test is calculated.A clay with thickness of approximately 16.0 m was developed in the shallow foundation of a highway test section.e cementsoil mixing pile with φ0.5 m × 8.0 m and φ0.5 m × 17.0 m is designed to carry out reinforcement treatment for the soft foundation.
e mixing pile spacing was 1.4 m and was arranged as a plum blossom.e nal embankment height was 4.9 m, which is equivalent to the embankment load of 96.0 kPa.In load testing of the test section, a circular load plate with a diameter of 2.73 m was used to carry out the load test of the three-pile composite foundation.Six vibrating earth pressure boxes (three on the pile top and three in the pile) were arranged to observe the stress distribution between the centre pile and soil around the pile in the composite foundation and the relationship with the load level.At the end of the test, the pile-soil stress ratio was 2.61.According to [36] and experience, the calculation parameters were selected as shown in Table 3.
In this paper, the Maple software is used to compute the above calculation program, which divides the pile and soil around the pile into 80 elements, and the load is divided into 100 loadings.
e pile-soil stress under the last load is calculated.It is calculated that when the pile top load is 40.7 kN, the total potential energy of the system reaches its minimum, and its stress is 207.3 kPa, as shown in Figure 8.
e calculation results of the pile-soil stress are shown in Figure 8.As shown in from Table 4, the calculated value is close to the measured value.e two cases show that the calculated results in the paper are in agreement with the measured values and therefore the calculation method is reliable.

Discussion.
It is generally believed that the parameters which exert great in uence on the results are as follows: M, λ, and κ in the correction of the parameters for the modi ed Cam-clay model. is paper analyses the sensitivity of the three parameters.
e results of the study are shown in Figure 9.
e horizontal line represents the increase and decrease of parameters, while the vertical line indicates the stress ratio of pile-soil.
As can be seen from Figure 9, the stress ratio rst decreases and then increases with the increase of M. When M decreases by 20%, the stress ratio increases from 2.69 to 4.30.When M increases by 20%, the stress ratio increases from 2.69 to 3.60.When M decreases, which means that the soil is to deteriorate under the lower stress, the shear strength of the soil decreases; correspondingly, the pile undergoes great load, and therefore the stress ratio of the pilesoil increases.When M increases, the increase of q will improve the compressibility of the soil, so that the di erential settlement of pile-soil will increase, the load will be transferred to the pile around, which undergoes greater load.Generally speaking, when M increases or decreases to the same extent, the decrease of soil bearing capacity is more than the load transfer, so the stress ratio of pile-soil when M decreases by 20% is greater than that when M increases by 20%.
e stress ratio of pile-soil shows approximate linear increase with the increase of λ. e parameter λ indicating compressibility means that the soil has more porosity ratio and compression when λ increases.
e di erential settlement of the pile-soil increases with the increase of λ, and the load will be transferred to the pile around; hence, the stress ratio of the pile-soil increases.Zhang [37] shows that the horizontal displacement increases when λ increases, while the horizontal displacement is proportionate to the vertical displacement.It is safe to say that the settlement increases with the increase of λ, which is similar to the results in this paper.
e stress ratio of pile-soil shows approximate linear decrease with the increase of κ. e main reason is that when κ increases, the soil resilience will increase, signifying small compression of the soil and the slight settlement di erence of the pile-soil; therefore, the soil shares more load, and the stress ratio of the pile-soil tends to be smaller.When κ increased from 0.8 to 1.2 κ, the stress ratio of pile-soil changed from 3.28 to 2.03.

Conclusions
(1) e modi ed Cam-clay model is used to establish the equation of the deformation energy and potential energy of soil around the pile in the composite foundation.On the basis of the nonlinear stressstrain relationship of the cement-soil pile, the expression of deformation energy and potential energy of the pile is obtained.Advances in Materials Science and Engineering (2) According to the principle of minimum potential energy, the total energy equation of the composite foundation system is established and programmed.e verification of two cases shows that the calculated values are in good agreement with the measured values; therefore, that the calculation method is reliable.
(3) e parameters of the modified Cam-clay model exert great influence on the calculation results.erefore, the accuracy of the parameters determines the reliability of the stress calculation.It is suggested that the representative core-drilling holes be tested and the number of tests should be increased.

Figure 5 :
Figure 5: Picture of the pile group with load.

Table 1 :Figure 6 :
Figure 6: Energy with load on pile top.

Figure 7 :
Figure 7: Pile shaft stress along with depth.

Figure 8 :
Figure 8: Energy with load on pile top.

Table 2 :
Comparison between calculated and measured stress.

Table 3 :
Physical mechanical index of soils.

Table 4 :
Comparison between calculated and measured stress ratio.