Research Article  Open Access
Mathematical Model and Analysis of Negative Skin Friction of Pile Group in Consolidating Soil
Abstract
In order to calculate negative skin friction (NSF) of pile group embedded in a consolidating soil, the dragload calculating formulas of single pile were established by considering Davis onedimensional nonlinear consolidation soils settlement and hyperbolic loadtransfer of pilesoil interface. Based on effective influence area theory, a simple semiempirical mathematical model of analysis for predicting the group effect of pile group under dragload was described. The accuracy and reliability of mathematical models built in this paper were verified by practical engineering comparative analysis. Case studies were studied, and the prediction values were found to be in good agreement with those of measured values. Then, the influences factors, such as, soil consolidation degree, the initial volume compressibility coefficient, and the stiffness of bearing soil, were analyzed and discussed. The results show that the mathematical models considering nonlinear soil consolidation and group effect can reflect the practical NSF of pile group effectively and accurately. The results of this paper can provide reference for practical pile group embedded in consolidating soil under NSF design and calculation.
1. Introduction
When vertical soil settlement is larger than that of pile, dragload (the additional compressive force) and downdrag (the excessive pile settlement) of pile caused by negative skin friction (NSF) occurred. It is one of common problems in the designing and construction of pile foundations in soft ground [1, 2]. In general, NSF on pile is caused by surcharge load or consolidating soil.
Previous studies results have shown that time effect were obviously presented on NSF on pile shaft caused by consolidating soil [3–9]. These studies were very helpful in understanding the NSF time effect behaviors of pile embedded in a consolidating soil, while few literatures focused on mathematical model analysis of NSF of pile embedded in a consolidating soil. NSF on pile shaft was increased with the increase of consolidation time and tended to be stabilized at last. Model tests and numerical simulation results have demonstrated that the dragload and downdrag on individual piles in a group was smaller than that on an isolated single pile because of group effect [10–13]. These studies were very helpful in understanding the behaviors of pile group under dragload. However, the literatures which focused on NSF group effect coefficient calculation method were scanty.
NSF theoretical research methods are mainly as follow: limit analysis method, elastic method, loadtransfer ( curves) method, shear displacement method, mixed method, and variation method [14–18]. While a few of those methods consider the time effect of dragload and downdrag, less attention is given to group effect.
It was evident from the literature as cited above that most of the available studies were limited to empirical methods or numerical methods, while a few were on mathematical models or theoretical methods. Hence, there was a need to develop mathematical models to predict the NSF with considering time effect and the group effect of pile group. Such an analysis was presented in this study. The NSF on single pile calculating mathematical model was established by considering Davis onedimensional nonlinear consolidation soils settlement and hyperbolic loadtransfer of pilesoil interface. One simple semiempirical mathematical model of analysis on predicting the group effect coefficient of pile group under NSF was described. In this group effect coefficient of mathematical model, the influence factors, such as, group arrange, pile space, soil type and its density, and pilesoil interface friction coefficient, were considered. The accuracy and reliability of mathematical models built in this paper were verified by practical engineering and model tests results comparative analysis. Then, the influences factors, such as, soil consolidation degree, the initial coefficient of volume compressibility, and the stiffness of end bearing soil, were analyzed and discussed.
2. Formula Built for Negative Skin Friction on Single Pile
2.1. Basic Assumptions
Soil was assumed as semiinfinite elastic body; foundation materials were assumed as homogeneous and isotropic materials. The radial deformation of pile and surrounding soils were ignored, and the consolidation processes of soils influenced by the driving of piles were not considered. Compressible homogeneous elastomers stressstrain relationship which obeys Hooke’s law model was chosen for piles. Davis onedimensional nonlinear consolidation theory model was chosen for the consolidation of soils. Hyperbolic loadtransfer model and linear transfer model chosen for pile sidesoils interface, and pile tipsoils interface mechanical transfer model, respectively.
Pile and its surrounding soil were chosen for research object in this paper, which was shown in Figure 1. The stress transferred from embankment to pile top was , and to the surface ground was , which means that pilesoil stress ratio equals . The pile and its surrounding soil were divided into unit in which node numbers were from 1 to . The elastic modulus and crosssectional area of pile were and , respectively.
2.2. Mechanical Model
(1) Control Equation of PileSoil System. The axial strain experienced by the pile can be related to the axial load by Hooke’s law; the following equation can be obtained: where is the axial force of pile at depth, is the settlement of pile at depth, is the elastic modulus of pile, and is the crosssectional area of pile.
Integrating (1) with limits to , the following can be obtained: where and are the axial force of upper point of unit , , respectively; and are the settlement of upper point of unit , , respectively and is the thickness of unit pile. The symbols schematic are shown in Figure 2.
Equation (2) can be rewritten into a matrix form: where is the column vector of axial force of pile unit node, is the column vector of settlement of pile unit node, is the column vector of constant, and is the coefficient matrix.
(2) Pile ShaftSoil LoadTransfer Model. The basic differential equations governing the deformation behavior of a compressible elastic pile under surcharge can be established by considering a small horizontal section as shown in Figure 1 as follows: where is the side friction of pile at depth and is the radius of pile.
Integrating (4), (5) can be obtained: where is the side friction of unit and and are the depth of unit , , respectively.
Equation (5) can be rewritten into a matrix form: where is the column vector of side friction of pile unit node, is the column vector of constant, and is the coefficient matrix.
Hyperbolic relationship between the friction of pile shaft and the relative displacement of pile and soil was chosen which is shown in where is the average relative displacement of interface unit , is the initial shear stiffness coefficient of interface unit , which is determined by , is the shear modulus of soil, is the thickness of pile unit, is the influence radius of pile, in which , is the ratio between shear modulus at the pile mid depth and shear modulus at pile tip, is the pile length, is Poisson’ ratio, is the limit shear stress of interface unit , which is determined by , is the lateral pressure coefficient of soil, is the vertical effective stress of surrounding soil, is the friction angle of pilesoil interface, and is the coefficient of negative skin friction.
Equation (7) can be rewritten into a matrix form: where is the average relative displacement of pile and soil, in which , and is the coefficient matrix of pile sidesoil loadtransfer. Consider
Substituting (8) into (6), the following equation can be obtained:
(3) PileBearing Soil Stratum LoadTransfer Model. Linear relationship between the tip resistance of pile and the relative displacement of pile and soil was chosen which is shown in where is the tip resistance of pile, is the relative displacement of pilesoil at pile tip, where , is the coefficient function of pile tip loadtransfer, where , is the compression modulus of soil, and coefficient .
Combine (3), (10), and (11), then iterative solution. The axial force of pile , side friction of pile , and displacement of pile unit can be obtained.
(4) Consolidation Settlement of Surrounding Soil. Based on Davis onedimensional nonlinear consolidation theory, excess pore water pressure of soil at depth in time, can be written as follows: where is the total stress, is the initial effective stress, and and are the effective stress and the pore water pressure in time, respectively. where , is the natural number, is the time factor, where , is the soil consolidation factor, and is the thickness of drainage path.
The soil settlement with time and depth variation under uniform load can be expressed as follows: where is the initial volume compressibility, where , and is the soil compression factor. is the vertical strain of soil, is the void ratio of soil, and is the initial void ratio of soil.
3. Formula Built for Negative Skin Friction on Piles Group
3.1. Pile Group Effect Coefficient
For the purpose of explanation first assume that a given vertical pile group was arranged at a spacing in a rectangular arrangement as shown in Figure 3(a). Taking notice of a typical location of pile arrangement, it was possible to define the unit typical location of pile effective influence area as that shown on Figure 3(b). Then, consideration of the balance of forces in the vertical direction leads to (15) for the soil region whose horizontal section was single repeated in Figure 3(b) as follows: where is the submerged unit weight of soil, is the vertical effective stress, is the vertical coordinate, is the ultimate shaft friction, is the pile diameter, and is the influence area of soil. Note that the parameters and are expressed as follows: where is the influenced radius and is the pile diameter, for corner pile , , , , , , and , edge pile , , , , , , and , interior pile , , , , , , and . Consider
The influence areas of typical location of piles were shown in Table 1; the overlap times of each influence area part were shown in Figure 3(c).

The result of interior pile was
The result of edge pile was
The result of corner pile was where is the horizontal effective stress, is the angle of friction mobilized between pile and soil, and is the ratio of the horizontal to the vertical effective stress.
Substitution of (21) into (15) yields where . The boundary condition assumed here was where is the intensity of surface loading. Substitution of (23) into (22) yields
It should be noted here that putting the parameter to be zero in (24) restores the expression for the single pile; that is,
Next, in order to express the degree of reducing dragload by installing piles in group, the group efficiency parameter was introduced in the form where is the distance from the ground surface to the neutral plane of the given pile. Substitution of (24) and (25) into (26) allows the group efficiency parameter to be expressed as where .
Use triangle of hatched sections ( and ) which were shown in Figure 3(a) to calculate the effective influence area. The simplified style of was obtained. Its precision results satisfied the demands of practical engineering. Consider where equal to 0.77, 1.77, and 2.70 and equal to 1, 2, and 3, respectively. is the spacing of pile.
Pile groups were arranged in a hexagonal arrangement which were shown in Figure 4. From the same reasoning as stated above it can be concluded that (27) applies to the pile group of hexagonal pattern as well, provided that the parameter was modified as follows.
And for the need of practical engineering, the area of the hatched section was calculated by the triangle instead of curve side triangle. Then, where is equal to 7/3, 3, 5/3, 4, and 11/3 from to 5, respectively. .
Before closing this subsection it may be appropriate to supplement that (27) reduced to (30a), when the surface loading term (term) dominates, whereas (27) reduced to (30b), when the bodyforce term (term) dominates. Considerfor , .
3.2. Pile Group Calculation
Based on the above analyze, the dragload and downdrag of pile group can be obtained as follows:where is the dragload of pile group, is the dragload of single pile; is the downdrag of piles group, is the downdrag of single pile, is the number of piles.
4. Theoretical Model Verification
4.1. Single Pile Calculation Method
In order to verify the accuracy and reliability of the suggested mathematical model, one of field tests investigating on negative skin friction of pile on soft ground was comparatively analyzed [24]. Drilled shaft with 43 m long and 1.0 m diameter, was loaded by 5.05 m embankment fill height that was constructed in the site. Physical and mechanical parameters of soils on site are shown in Table 2. The bulk density of filled soils equals 21.5 kN/m^{3}, the stiffness modulus of pile () equals 25 GPa, the coefficient of soil consolidation () equals m^{2}/s, the initial volume compressibility factor () equals m^{2}/kN, base coefficient of pile tip () equals 23.1 MPa/m, the coefficient of negative skin friction () equals 0.3, and Poisson’s ratio () of soil equals 0.3.

The relative displacement of pilesoil contact surface has nonlinear relationship with friction of pile shaft; hence, secant shear stiffness coefficient () was used in loadtransfer function shown in where is the initial value of shear stiffness coefficient , is the mean value of relative displacement, and absolute value was chosen in iterative calculation process.
Take the end of embankment filling (4.5 months after embankment filling) for an example; the friction of pile shaft and axial force of pile which is obtained by field test and calculated method are shown in Figures 5 and 6, respectively. The stress ratio of pilesoil on the end of embankment filling equals about 4.5. The pile top load and surcharge load of measured equal 487.9 kPa and 108.6 kPa, respectively. Figure 5 shows that the friction of pile shaft calculated by proposed method in this paper is similar to that obtained on field test. Above neutral plane, the friction of pile shaft nearly increased on linear; below neutral plane, the friction of pile shaft increased on nonlinear. The measured value near the bottom of the pile is improved greatly; it may be because of the enhancement effect on the pile tip. Figure 6 shows that the axial force of pile calculated by mathematical model is similar to that obtained by field test. The maximum value of axial force of pile is occurred on about 14 m deep. In other words, the neutral plane of calculated by mathematical model equals 14.4 m; the neutral plane of measured value by field test is between 11.5 to 14.7 m.
4.2. Piles Group Calculation Method
In order to verify the accuracy and reliability of pile group calculated method, series field tests and model tests results on dragload comparative analysis were taken out. The details of the pile arrangements were shown in Figure 7, and the principal items regarding each pile test were showed in Table 3.
 
The pile code for the relative position of the piles within the group shown in Figure 7. **Hexagonal arrangement. 
(a)
(b)
(1) Example Case Studied by Kong et al. One piles group with pile spacing equals ( equal to 0.041 m) was presented by Kong et al. [19] (for the position of piles, refer to Figure 7(a)). The model piles were driven to 0.7 m clay, approximately 0.2 m above the silt layer. For the surrounding clay layer, the soil modulus equals 2.01 MPa, the unit weight of soil () equals 17.4 kN/m^{3}, and the effective cohesion () and effective friction angle () equal 7.0 kPa and 19.1°, respectively. For the stratum silt layer, the soil modulus equals 2. 31 MPa, the unit weight of soil () equals 13.9 kN/m^{3}, the effective cohesion () and effective friction angle () equals 16.2 kPa and 33.9°, respectively. The surface load equals 5.1 kPa. The pilesoil interface friction coefficient () was found to be 0.59 from the direct shear test. The results of measured and predicted values are shown in Figure 8.
(2) Example Case Studied by Little. Two piles groups (one friction pile type and one endbearing pile type) with pile spacing equals ( equals 0.406 m) field test on dragloads were presented by Little [20]. (for the position of piles, refer to Figure 7(a)). The friction piles were driven to 20.4 m, approximately 1 m above the gravel layer. The endbearing piles were driven onto the gravel layer; the piles length were 20.8 m. The soil modulus of 3.5 MPa was backcalculated for the clay layer from the measured ground settlement (180 mm). The interface friction coefficient of 0.35 was used. Assume that the submerged unit weight of soil () equals 9 kN/m^{3}, the value of equals 0.3, the effective cohesion () equals 3.0 kN/m^{2}, and effective friction angle () equals 25°. An embankment loading of 40 kPa was applied on the top of the clay after driving the piles. The results of measured and predicted values are shown in Figure 8.
(3) Example Case Studied by Combarieu and Jeong. One piles group with pile spacing field test on dragloads was presented by Combarieu [21] and Jeong [25]. The model pile length () equals 20 m and pile diameter () equals 0.5 m. The soil properties were as follows: the submerged unit weight of soil () equals 18 kN/m^{3}, the value of equals 0.3, the effective cohesion () equals 30.0 kN/m^{2}, and the surface load () equals 200 kPa. Assuming a value of 0.60.7 for normally consolidated clay and a value of 0.3 as given by Combarieu, a rough estimate for interface friction coefficient () equales 0.4 and 0.5 was obtained. The stiffness modulus () of clay was taken to be 5 and 10 MPa, respectively. The results of measured and predicted values are shown in Figure 8.
(4) Example Case Studied by Shibata et al. Model tests for a single pile and piles in group were carried out by Shibata et al. [22]. The model steel pipe piles with an outer diameter () of 0.06 m, a length () of 0.7 m, and wall thickness () of 1.2 mm were carried out. The pile group consisted of piles with a spacing () of (for pile positions, refer to Figure 7(a)). Assume that the modulus of the steel pipe pile () equals kPa. The piles were assumed to be endbearing. The clay model was formed from kaolin slurry under selfweight consolidation, which had the following index properties: equals 52%, equals 39%, clay fraction equals 39%, and specific gravity equals 2.61. The effective unit weight of the clay slurry () equals 5.3 kN/m^{3}. A surface loading of 20 kPa was applied on the top of the clay surface to produce soil settlement after completion of the selfweight consolidation. The soil modulus () of 150 kPa was assumed from the consideration of measured soil settlement (65 mm). The measured results showed that the dragload of single pile reached to 410 N after consolidation; the corner pile, edge pile, and central pile of pile group reached to 358 N, 350 N, and 294 N, respectively. The group effects results of measured and predicted values are shown in Figure 8.
(5) Example Case Studied by Okabe. Field test on dragload in a pile group, resulting from a combination of draining and surcharge loading was reported by Okabe [23]. The pile group consisted of 38 piles with pile spacing, the diameter of pile () was 0.71 m, and embedded lengths () were 40 m (for the arrangement of pile group see Figure 7(b)). There were 14 external piles that can be free to move; it takes most of the NSF; the other 24 internal endbearing piles are connected to a rigid pile cap. After installation of the piles, the embankment as a surcharge was placed on a very soft clay layer with SPT values of 0–10 and unconfined compressive strength of 0–110 kN/m^{2} from 0 to 40 m depth. Below the soft clay was a medium sandy gravel layer with values of about 50. Ground settlement was reported both by the fill and dewatering from the underlying aquifer for urban water supply. The properties of the soft clay were assumed. The submerged unit weight of soil () equals 18 kN/m^{3}, the value of equals 0.3, the soft clay Young’s modulus () equals 5 MPa, the effective cohesion () equals 5.0 kN/m^{2}, and effective friction angle () equals 25°. The soil settlement reached 10 cm at ground surface after 300 days at a 5.6 m surcharge placement, and the surface load () equals 250 kPa. The results of measured and predicted values are shown in Figure 8.
Computations of group efficiency parameter value were made by using (27) and (28) for the pile group of rectangular arrangement and (27) and (29) for the pile group of hexagonal arrangement. Comparative pile group efficiency parameters obtained by using the present method and field test results reported by various investigators are shown in Figure 8. It is evident from Figure 8 that there is a good agreement between the measured and predicted values of the group efficiency parameter.
5. Parametric Analysis
Based on the pile and soil parameters shown in Section 4.1, series influence factors, such as, soil consolidation degree, initial volume compressibility coefficient of surrounding soil, soil stiffness of pile base, and pilesoil stress ratio, were analyzed and discussed.
5.1. Influence of Soil Consolidation Degree
The relationships of skin friction versus pile depth influenced by soil consolidation degree were shown in Figure 9. It shows that the neutral plane position is a little higher in the beginning of consolidation time than that in the end of consolidation time; the neutral plane position increased with consolidation time increasing. In other words, the areas of negative skin friction gradually expanded with consolidation time. In this analysis condition, the neutral plane position increased from 16.2 m to 22.1 m, with increase ratio nearly equals 36.4%. It also shows that the neutral plane position becomes a stable value after a long period of consolidation time; neutral plane position has an obvious time effect.
5.2. Influence of Initial Volume Compressibility Coefficient of Surrounding Soil
For the same soil consolidation degree, the skin friction and neutral plane position influenced by different initial volume compressibility coefficient () of surrounding soil are shown in Figure 10. It shows that the negative skin friction increased with initial volume compressibility coefficient increasing neutral plane position increased with initial volume compressibility coefficient increasing. In this analysis condition, the neutral plane position increased from 14.2 m to 16.6 m, with increase ratio of nearly 16.9%. In other words, the negative skin friction is obviously influenced by the initial volume compressibility coefficient.
5.3. Influence of Bearing Soil Stiffness
The negative skin friction and neutral plane position influenced by different soil stiffness of pile base () are shown in Figure 11. It shows that the neutral plane position increased with soil stiffness of pile base increasing. In this analysis condition, the neutral plane position increased from 13.8 m to 18.2 m, with increase ratio nearly equals 31.9%. In other words, soil stiffness of pile base is one of key influence factors to neutral plane position.
5.4. Influence of PileSoil Stress Ratio
In the same soil consolidation degree, the maximum axial force of pile, maximum dragload, tip resistance, and neutral plane position influenced by different pilesoil stress ratio were shown in Table 4. It shows that maximum axial force of pile increased with pilesoil stress ratio increasing, the maximum dragload decreased with pilesoil stress ratio increasing, and the tip resistance increased with pilesoil stress ratio increasing. It also shows that neutral plane position increased with pilesoil stress ratio increasing. When pilesoil stress ratio increased, more loadtransfers to pile top, which leads to the settlement of pile. This new pile settlement leads to the relative displacement of pilesoil. Hence, the neutral plane position increased and maximum dragload decreased.

6. Conclusions and Discussions
In this paper, one NSF of single pile calculating formula and one group effect coefficient calculating formula were described. Based on case studies and influence factors analysis, the following conclusions can be drawn.
(1) In the simple mathematical models on negative skin friction of single pile calculation method and group efficiency parameter calculation method built in this paper, physical meaning is clear, calculated process is simple and operable, and the parameters in the formula are easy to obtain, especially for the simplified effective influence area which is according to engineering demand. In the calculation method, nonlinear soil consolidation settlement can be considered in negative skin friction of single pile. Pile group arrangement, embedment length to diameter ratio of piles, pile spacing, soil properties such as angle of internal friction, density, and pilesoil interface friction coefficient can be considered in group efficiency parameter of pile group. Case studies show that the calculated results and measured data from previous observations are in excellent agreement, which indicates that the presented approach is feasible for applications in engineering practice.
(2) Negative skin friction and neutral plane position exists a significant time effect its values increased with consolidation time increasing, and eventually they tends toward stable values. Neutral plane position is obviously influenced by soil consolidation degree and soil stiffness of pile base. In this paper’s analysis condition, the neutral plane position increased nearly 36.4% when 2.5 times soil consolidation degree increased; the neutral plane position increased nearly 31.9% when 5 times soil stiffness of pile base increased.
(3) Increasing pilesoil stress ratio leads to a reduction in pilesoil relative movement, which decreases the negative skin friction on pile shaft.
Disclosure
This work is an original one conducted at Hohai University. The work described has not been submitted elsewhere for publication, in whole or in part.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Authors’ Contribution
All the authors have contributed to and approved this paper.
Acknowledgments
The authors wish to thank the National Science Foundation of China (no. 51278170, U1134207), the Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1125) and the 111 Project (no. B13024) for financial support.
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Copyright
Copyright © 2013 Gangqiang Kong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.