Pile Side Resistance in Sands for the Unloading Effect and Modulus Degradation

C. F. Zhao^a, Y. Wu^a, C. Zhao^a,b,*, Q. Z. Zhang^a, F. M. Liu^a, F. Liu^a

a. Department of Geotechnical Engineering, Tongji University, (Shanghai, China)

b School of Engineering, Tibet University, Lhasa, (Tibet, China)

*zhaocheng@tongji.edu.cn

1. INTRODUCTIONTOP

Piles are generally used to transfer loads from the superstructure to a competent soil or rock layer (1). The use of pile side resistance analyses is becoming more important compared to solely using bearing capacity analyses. Extensive effort has been expended to develop theoretical methods to analyze pile side resistance behavior. These methods fall into four categories: (i) the load transfer method (1–3), which uses the pile side load-transfer function to describe the relationship between pile side resistance and pile-soil interface displacement; (ii) the shear displacement method (4–9), which assumes that the vertical displacement of the soil at any point around the pile is only related to the shear stress at this point and consider the vertical displacement of the soil induced by the shaft shear stress as a logarithmic relationship of the radial distance away from the pile side; (iii) the elastic theory method (10–12), which employs the Mindlin solution under the concentrated load in the elastic half-space to calculate the displacement of the soil. The equilibrium equation is established by the coordination conditions between the displacement of the pile body and soil to obtain the displacement and side resistance of the pile; (iv) the numerical analysis method (13–18), which includes the finite element method (17), boundary element method (13), discrete element method (16) and infinite layer method (14), and generally needs high computational requirements that caused them are not commonly used in practice.

The shear displacement method assumes that the vertical displacement of the soil at any point around the pile is only related to the shear stress at this point, the shear stress transfer causes the settlement of the surrounding soil, and thus the stress and deformation characteristics of the pile-soil system can be obtained (19). Although the conventional shear displacement method has some defects. For example, this method cannot reflect the pile side resistance softening phenomenon (20), but it is still widely used in the analysis of pile side load transfer mechanism due to its easy calculation process and theoretically reflecting the shear deformation properties of the soil around the pile. (7–8) derived the pile side resistance-displacement functions based on the shear displacement method. Guo et al. (21) modified the pile side resistance-displacement functions that derived from the shear displacement, considering the variations of soil stiffness and limiting pile side resistance with depth. Zhu at al. (22) considered the nonlinear elastic properties and modulus degradation characteristics of the soil into the pile side resistance-displacement functions. The stress-strain behavior of natural soils during shear is highly nonlinear, and the elastic modulus generally decreases with increase in shear strain (22).

On one hand, although above scholars considered many factors that affect the pile side resistance developed, the soil radial unloading for the non-displacement pile during boring stage almost has never been addressed. On other hand, the mechanical properties of soil around a non-displacement pile are very different compared with those of loading soil (23, 24). In particular, the radial unloading effect and modulus degradation characteristics of the soil are not considered comprehensively to a great degree in the pile side resistance-displacement model. As a result, it is difficult to quantify the radial unloading effect of the soil around the pile side on the pile side resistance developed through the field full scale pile tests. Furthermore, the static field full scale pile load tests are extensive and time-consuming although it is the most reliable means of assessing a single pile load transfer mechanism (25). However, the indoor model test has the advantages of parameter control, test repeatability, and lower cost (26). Hence, it can be achieved through the concrete-soil interface large direct shear tests to simulate the radial unloading during the boring stage and pile side-soil shear process (27).

In this paper, a series of sand-concrete interface loading and unloading direct shear tests are used to analyze the mechanical properties of the pile side-soil interface. First, the effect of the unloading process and interface roughness on the mechanical properties of the interface are discussed. Then, a pile side resistance-displacement model is established using the shear displacement method. The proposed τ–s function considers the effect of radial unloading during the non-displacement pile boring stage and modulus degradation of soil around the pile. The direct shear test data is used to validate the proposed model. At the end of this paper, the effect of radial unloading and sand-concrete interface roughness on the degradation of the equivalent shear modulus G_eq is analyzed using a single fitting parameter b.

2. LARGE DIRECT SHEAR TESTTOP

2.1. Test apparatusTOP

Although the large-scale direct shear apparatus has some inherent defects, it is frequently used in interface research due to its simplicity in principle and operation. The test apparatus used in this experiment is the large-scale multi-function direct shear test apparatus SJW-200, which is independently researched and developed by Tongji University, Shanghai, China. The test apparatus has a large size shear box with net size 600 mm × 400 mm × 200 mm (length × width × height) and wall thickness of 40 mm. The test apparatus in both the normal and tangential directions is equipped with an advanced server and control system; the range of the test displacement and applied load are enough to satisfy the requirements of this paper. Compared with a conventional direct shear device, this test apparatus has more accurate test results because it can effectively reduce the boundary effect (7). Figure 1 is the schematic diagram of the large-scale direct shear test apparatus.

2.2. Soil specimenTOP

The soil specimen was gray silt, which was obtained from a construction project in Shanghai City. The main properties of the soil are listed in Table 1; the grain-size distribution of the soil is shown in Figure 2.

2.3. Concrete plate specimenTOP

Because the bored pile side surface is not smooth in actual engineering, it is necessary to consider the influence of the physical form of the sand-concrete interface. The key is how to simulate and define the roughness of the concrete plate. The quantitative criteria for the interface roughness are as follows.

Dove et al. (28) proposed using the surface regular sawtooth to quantify the soil-concrete interface roughness, as shown in Figure 3. The roughness of the interface can be changed by changing the sawtooth angle, height and space position relationship.

Zhang et al. (29), (30) proposed using the peak-to-valley distance R to quantify the steel plate interface roughness, as shown in Figure 4. The interface roughness can be changed by changing the peak-to-valley distance.

Considering the advantages and disadvantages of the above interface roughness definition and the actual situation of the pile wall, the interface roughness model chosen in this paper is shown in Figure 5. Figure 5 shows that the shape of the rough concrete plate surface is a standard tooth, and its profile is trapezoidal. The tooth angle α = 45 ° remains constant, and the length of the bottom edge S₂ of the trapezoidal convex portion is equal to the bottom edge S₃ of the trapezoidal concave portion. The volume of the concave portion is always equal to the convex portion. By changing the tooth height h and keeping the other conditions unchanged, including tooth angle, space position relationship and so on, the roughness of the concrete plate interface can be adjusted. Hence, the interface roughness R can be expressed by the size of h.

Three variations of concrete plates are used in this experiment, namely, R = 0 mm, 10 mm and 20 mm according to the above definition of concrete interface roughness model. The concrete plate for testing was 600 mm long by 400 mm wide by 50 mm thick and is made of C25 concrete, and bidirectional bars with a diameter of φ8 @ 100 are added into it.

2.4. Test proceduresTOP

A total of 36 groups of sand-concrete interface direct shear tests with three different degrees of interface roughness and initial normal stress were used. The test soil is first consolidated under an initial normal stress σ_c for 1 hour and then unloaded to a specific applied normal stress σ_s to shear. Both the loading and unloading rates are 0.5 kPa/s. The three programs of loading and unloading for three different roughness of concrete-soil interface are: (i) σ_c = 300 kPa, and σ_s = 300, 250, 200, 150, 100, 50 kPa, respectively; (ii) σ_c = 200 kPa, and σ_s = 200, 150, 100, 50 kPa, respectively; (iii) σ_c = 100 kPa, and σ_s = 100, 50 kPa, respectively. The shear stage begins when the curve of the normal displacement to time is stable, which means the normal displacement is less than 0.001 mm within 1 minute under the applied normal stress. The shear rate is controlled at a constant rate of 2 mm/min. All data regarding the test are collected by a computerized data logging system; the results are monitored and saved using the computer software TEST.

3. TEST RESULTS ANALYSIS TOP

3.1. Effect of loading and unloading on the sand-concrete interface mechanical propertiesTOP

Figure 6 shows the interface shear stress-shear displacement curves for different initial and applied normal stresses to study the effect of loading and unloading on the interface mechanical properties. For the sake of simplicity, the legend is expressed as initial normal stress σ_c-applied normal stress σ_s. For example, the legend showing 300–100 means the interface is consolidated at the initial normal stress of 300 kPa, and sheared at the applied normal stress of 100 kPa. Figure 6 also shows that the interface peak shear stress τ_f increases with the initial normal stress under the unloading conditions when the interface roughness R = 0 mm and 10 mm, but both τ_f are less than the loading conditions. For R = 20 mm, the interface peak shear stress increases with the initial normal stress. For the rough soil-concrete interface, the interface residual shear stress tends to be a constant regardless of the loading and unloading conditions and increases with the interface roughness. However, the interface residual shear stress tends to be a constant only for the unloading condition in the smooth soil-concrete interface.

The above can be explained by these points. Higher initial normal stress contributes to a higher density of the interface, which results in soil particles around the interface needing a higher shear stress to make them move along the roughness interface and each other. Therefore, interface peak shear stress increases with the initial normal stress at high roughness. If the interface roughness is low, the interface peak shear stress is mainly influenced by the water content of the interface soils rather than the density. Higher initial normal stress lowers the water content of the interface soils, and if the water content is below the optimal water content, the interface peak shear stress under unloading may be less than the loading condition.

The shapes of the curves for loading and unloading are both logarithmic, and the strain softening phenomena of the curves are also consistent between loading and unloading. These phenomena show that the effect of loading and unloading on the sand-concrete interface mechanical properties change the value of interface peak shear stress, but there is no fundamental change in the failure mode of the interface.

3.2. Effect of unloading degree on the sand-concrete interface mechanical propertiesTOP

Figure 7 shows the shear stress-shear displacement curves for the initial normal stress at 300 kPa unloaded with different applied normal stress to shear. The unloading degree is defined as the ratio of the increment between the initial and applied normal stress to the initial normal stress. Figure 7 also shows that the unloading degree has a significant effect on the sand-concrete interface mechanical properties. The peak shear stress, the shear displacement corresponding to the peak shear stress and the interface initial shear modulus G₀ all decrease with the unloading degree for the same interface roughness R.

3.3. Effect of interface roughness on the sand-concrete interface mechanical properties TOP

Figure 8 shows the fitting curves between peak shear stress and applied normal stress for different interface roughness and initial normal stress. Table 2 shows the interface equivalent frictional angle φ calculated from the fitting curves. Figure 8 and Table 2 also show that the interface equivalent friction angle φ increases positively with roughness for the same initial normal stress; the rate of increase is positive with the initial normal stress. The interface equivalent friction angle φ is mainly affected by the interface roughness and the initial normal stress.

3.4. Effect of loading and unloading on the sand-concrete interface shear dilatancy and shrinkageTOP

Figure 9 shows the variation curves of maximum interface, shear dilatancy and shrinkage amount for different conditions. In this paper, the sign of normal displacement corresponding to the interface shear dilatancy is negative, and the sign of normal displacement corresponding to the interface shrinkage is positive. Figure 9 also shows that the maximum amount of interface shear dilatancy decreases with the applied normal stress σ_s, while the maximum amount of interface shear shrinkage increases with the applied normal stress σ_s as the σ_s increases to a value for the same interface roughness and initial normal stress. The maximum amount of interface shear dilatancy increases with the initial normal stress, and the maximum amount of interface shear shrinkage decreases with the initial normal stress for the same interface roughness and applied normal stress.

4. MODEL DESCRIPTIONTOP

The proposed pile side resistance-displacement model curve, as shown in Figure 10, consists of two parts: (i) a nonlinear pre-failure portion and (ii) a perfectly plastic after-failure portion. Figure 10 shows that the pile side resistance increases nonlinearly with the pile-soil interface displacement increased at first. When the pile-soil interface displacement reaches the value s_u, the pile side resistance achieves its peak value τ_f. The pile side resistance will keep the fixed value τ_f with the pile-soil interface displacement increased. The softening or hardening behavior of the soil, as illustrated in Figure 10, is not considered in this paper.

4.1. Theory function of pile side resistance-displacementTOP

The vertical displacement of the soil at any point around the pile is only related to the shear stress at this point, according to the shear displacement method assumptions, and can be expressed as [1] (19):

where s(z) is the vertical displacement of the soil at a point where the depth is z; τ_s is the shear stress of the soil at that point.

Randolph et al. (7) proposed an approximate analytical solution for the pile side resistance-displacement based on the shear displacement method and expressed the solution as [2]:

where r₀ and r_m are the radius of the pile and the limiting radius where the shear stress becomes negligible, respectively; τ₀ and G(τ_s, r) are the shear stress of the pile side and the shear modulus of the soil mass, respectively.

Soil Modulus DegradationTOP

The stress-strain behavior of most geomaterials is highly nonlinear at all phases of loading (22). There is a significant reduction in stiffness with increasing strain level. The larger value of soil shear strain, the lower value of soil shear modulus during the shearing process. According to the assumptions of the shear displacement method, the shear strain of the soil around the pile is decreased with the radial distance from the pile increased. Hence, the degradation model of the soil shear modulus G(τ_s, r) along the pile radial direction can be assumed in Figure 11. The equation of the soil shear modulus G(τ_s, r) is expressed as [3]:

where r₁ represents the turning point where the soil shear modulus G(τ_s, r) begins to degenerate and the value of r₁ ≤ r_m; M represents the soil shear modulus degradation coefficient.

By introducing Eq. [3] into Eq. [2] and integrating them, Eq. [2] becomes [4]:

The concept of equivalent shear modulus G_eq is adopted to facilitate the solution of Eq. [4] where the shear modulus varying in Eq. [4] is equivalent to a constant shear modulus G_eq expressed as [5] and [6]:

The shear modulus G (τ_s, r) (or G_eq) is related to the soil shear stress τ_s around the pile (or pile side resistance τ₀) and soil initial shear modulus G_s0. Fahey et al. (31) proposed a modified hyperbolic model to express the degradation of soil sear modulus G (τ_s, r) expressed as [7]

where the coefficient a and b are the degree and rate, respectively, of the shear modulus degradation. τ_s and τ_max are the soil shear stress and maximum soil shear stress around the pile. For the equivalent shear modulus G_eq, it is necessary to replace the τ_s and τ_max of the pile side resistance τ₀ and peak pile side resistance τ_f in Eq. [7], expressed as [8]

By introducing Eq. (8) into Eq. (5) and integrating them, Eq. [5] becomes [9]

Evaluation of Peak Pile Side Resistance τ_fTOP

There are three methods for determining the peak pile side resistance τ_f : (i) the α method belonging to the total stress method (32); (ii) the β method belonging to the effective stress method (33); and (iii) the λ method belonging to the mixing method (34). For the object of this paper, the β method is modified to determine the peak pile side resistance τ_f considering the unloading effect.

The β method can be used to calculate the pile side resistance of both cohesive soil and non-cohesive soil. It was proposed by Chandler et al. (33), and the calculated formula is expressed as [10]

where σ’_v is the average vertical effective stress of the pile side-soil layer; K₀ is the static lateral pressure coefficient; δ is the pile-soil interface friction angle.

For normal consolidated soil, K₀=1-sinφ′, where φ’ is the effective internal friction angle of the soil δ=φ′, and β = (1- sinφ')∙tanφ'. For over-consolidated soil, the effect of over-consolidation OCR should be considered. Mayne et al. (35) proposed a modified formula for calculating the value of β for over-consolidated soil [11]

The over-consolidation ratio OCR is calculated as [12]

where σ_c and σ_s are the initial consolidation normal stress and applied normal stress in the process of shearing, respectively. The unloading ratio ξ is introduced when considering the unloading effect and is expressed as [13]

Introducing Eq. [11], Eq. [12] and Eq. [13] into Eq. [10] becomes [14]

Eq. [22] is the peak pile side resistance calculation function that considering the unloading effect.

Evaluation of the influence area of radial limiting radius r_mTOP

Randolph et al. (23) proposed the following equation as an estimation method to calculate the r_m [15]

where C is the empirical coefficient, and its value is 2.5 for a pile in the semi-infinite space and 2.0 for the rigid layer below the pile end 2.5 times the pile length; L is the depth of the pile in the soil; ρ is the ratio of soil shear modulus at mid-depth to that at the pile tip, and ρ = 1.0 for uniform soil; v is the Poisson ratio, and its value generally takes v = 0.2 ~ 0.4 for Shanghai sand.

From Eq. [15], it is relatively simple to determine the r_m, but this equation does not consider the impact of the load level of the pile and the change of r_m with the pile buried depth. Hence, this paper uses the following method to determine the value of r_m.

The downward displacement of the soil at r = r_m is 0, according to the shear displacement method assumptions. Thus, differentiating Eq. [2] on both sides is expressed as [16]

Eq. [16] shows that the hold condition is r_m→∞. In the actual engineering, it is impossible to meet the conditions of r_m→∞. Therefore, there exists a treatment measure as follows [17]

where η is a finite small value. Solving r_m from Eq. (17) can be expressed as [18]

Eq. [18] shows that the r_m is related to the pile side resistance τ₀ (reflecting the pile load level), the soil initial shear modulus G_s0 (reflecting the soil properties) and the pile radius r₀ (reflecting the pile shape).

By introducing Eq. [14] and Eq. [18] into Eq. [9], the pile side resistance-displacement model considering the unloading effect and soil modulus degradation is obtained as follows [19]:

4.2. Determination of the values of parameters a, b, η, δ and G_s0TOP

Evaluation of Parameters a, b and ηTOP

The range of the coefficient a is from 0 to 1.0, and (36) suggests that the results of most cases can be well fitted when a = 0.98. Mayne et al. (37) suggests that the range of the coefficient b is from 0.2 to 0.4 for most types of soil. In this paper, the range of empirical coefficient b is extended to b = 0.02 ~ 0.4.

The value of parameter η can be obtained by experiment. Randolph et al. (7) suggests that the range of value for η is 1 × 10^–7 ~ 1 × 10^–5, its value can take 1 × 10^–6.

Evaluation of Parameters δTOP

The parameter δ is the pile-soil interface friction angle, and its value varies with the type of pile and soil properties and generally takes as Table 3 recommended.

Evaluation of Initial Soil Shear Modulus G_s0TOP

The initial shear modulus of the soil G_s0 is generally calculated by the E-v model in actual engineering [20]

where E and v are the elastic modulus and Poisson’s ratio of the soil, respectively. For Shanghai soil, E = 3.5E_s1-2 according to Wang et al. (39). v is the Poisson ratio, and its value generally is v = 0.2 ~ 0.4 for Shanghai sand.

4.3. Algorithm for analysis of the pile side resistance-displacement modelTOP

The algorithm for the analysis of the pile side resistance-displacement model can be summarized as follows:

5. MODEL VALIDATION AND ANALYSISTOP

The test results obtained by the large-scale direct shear test are used to validate the pile side resistance-displacement model. The parameter r₀ in Eq. [19] cannot be reflected in the direct shear test. The pile-soil interface shear strength τ_f can be derived directly from the test results and do not need to be calculated by Eq. [14]. Hence, Eq. [19] can be transformed as follows [21]

where τ₀ and τ_f are the current shear stress and peak shear stress in the direct shear test, respectively. Using the parameter τ to replace τ₀ in Eq. [21], s is the shear displacement.

5.1. Comparison curves between the model calculation and test values with different σ_cTOP

The curves of s/r₀-τ/τ_f are plotted and compared with the τ-s curves obtained by the large-scale direct shear test. The τ/τ_f -s/r₀ curves are plotted on the main coordinate axis, and the τ-s curves are plotted on the sub-coordinate axis. The main coordinate axis and sub-coordinate axis in the figure have strict correspondence. The curves of both model calculation and test values are the pre-failure portion; after-failure portion is not considered for it is assumed perfectly plastic. The range of empirical coefficient b is extended to b = 0.02 ~ 0.4.

Figures 12 a, b and c are the comparison curves for the interface roughness R = 0 mm and σ_c = 100 kPa, 200 kPa and 300 kPa, unloaded to σ_s = 50 kPa. Figure 12 shows that the test values fit well with the model calculation, and the value of empirical coefficient b gradually changed from 0.05 to 0.3 as the σ_c increased. This phenomenon shows that the value of empirical coefficient b increases positively with the initial normal stress σ_c in the same applied normal stress and interface roughness.

5.2. Comparison curves between the model calculation and test values under different σ_sTOP

Figures 13 a, b and c are the comparison curves for the interface roughness R = 0 mm and σ_c = 300 kPa unloaded to σ_s = 300 kPa, 200 kPa and 100 kPa, respectively. Figure 13 show that the test values in the loading conditions fit the best with the model calculation values when the empirical coefficient a = 0.98 and b = 0.05. The test curves that fit with the model calculations gradually shift up with unloading degree increase; the value of empirical coefficient b is gradually changed from 0.05 to 0.2 with the unloading degree increased. This phenomenon shows that the value of empirical coefficient b increases with the unloading degree increased.

5.3. Comparison curves between the model calculation and test values under different RTOP

Figures 14 a and b, and Figures 15 a and b are both the comparison curves of the interface roughness R = 10 mm and R = 20 mm, respectively. Figure 14 (a) ~ (b) show that the test values fit best with the calculated values at b = 0.05~0.1 for R = 10 mm and b = 0.02 for R = 20 mm, respectively. Figures 15 a and b show that b = 0.1~0.2 and b = 0.05~0.1 for R = 10 mm and R = 20 mm, respectively. Combined with Figures 13 b and c, Figure 14 and Figure 15 show that the value of empirical coefficient b decreases with interface roughness.

5.4. Effect of unloading and roughness on the equivalent shear modulus degradationTOP

Figure 16 shows the degradation curves of the equivalent shear modulus G_eq, which is calculated from the modified hyperbolic function given by Eq. (8) where a = 0.98 and b is 0.02 to 1. Figure 16 shows that the degradation rate of G_eq decreases with the value of b during the early shearing stage (about τ < 0.05τ_f) and increases with the value of b after the early shearing stage.

The above analysis (section 5.1 to 5.3) shows that the value of the empirical coefficient b increases with the unloading degree but decreases with interface roughness. Unloading will slow down the degradation of G_eq during the early shearing stage; the degradation rate of G_eq decreases with the unloading degree. After the early shearing stage, unloading accelerates the degradation of G_eq; however, the effect of interface roughness on the degradation of G_eq is opposite to the unloading degree. The increase of interface roughness will accelerate the degradation of G_eq during the early shearing stage and decelerate the degradation of G_eq after the early shearing stage.

6. CONCLUSIONSTOP

In this study, large scale direct shear tests are conducted to analyze the mechanical properties of the pile side-soil interface. The effects of the unloading process and interface roughness on the mechanical properties of the interface are discussed. A pile side resistance-displacement model is developed using the shear displacement method. The proposed function considers both the radial unloading effect and modulus degradation of soil around the pile. The main conclusions are drawn as follows:

ACKNOWLEDGEMENTSTOP

This work was supported by the National Key Research and Development Plan [grant number 2017YFC0806000], the National Natural Science Foundation of China under Grant [No. 41672265, No. 41572262, No. 41502275]; and Shanghai Rising-Star Program under Grant No. 17QC1400600. The authors are deeply grateful for this support. The anonymous reviewers’ comments have improved the quality of this paper and are also greatly acknowledged.

REFERENCESTOP