Calculative Width of Pile Foundation on Slope Based on Particle Image Velocimetry (PIV)

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, Sichuan 610065, China College of Water Resource and Hydropower, Sichuan University, Chengdu, Sichuan 610065, China State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu, Sichuan 610059, China College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu, Sichuan 610059, China


Introduction
e weakening effect of slope on the horizontal bearing stability of pile foundation is a hotspot in engineering [1,2]. e change of calculative width with the slope has been considered as the major reason for this weakening effect. As an important factor affecting its horizontal bearing capacity, the calculative width is directly related to the scale of the soil exerted resistance, and it plays a key role in bearing capacity design.
Pile-soil interaction under lateral load has been widely studied but mainly on the horizontal ground [3,4]. Pile foundation shows different stress characteristics under slope, for example, the change of effective calculating width [5]. As an important parameter to measure the stress characteristics of the pile, effective calculative width plays a key role in horizontal bearing capacity [6]. e reason is that the skin friction along the pile will drive more soil to bear lateral load because the soil around the pile will diffuse at a certain angle, which has been called horizontal diffusion angle [7,8]. is phenomenon will make the actual width bearing load bigger than geometric width. Stress characteristics during pile-soil interaction can be researched by Static Lateral Loading Test [9] and PIV [10,11]. ere are many indexes that can be used to illustrate the change of lateral bearing capacity of the pile, like m-value [12,13] and p-y curve [14,15]. is paper mainly considers the skin friction along the single pile to carry out the horizontal static load model test of single pile. In order to study the effect of slope on calculative width of pile foundation, effective calculative width was obtained based on vectorial displacement field of soil by PIV system under the slopes of 0°, 15°, 30°, and 45°. Comparing indoor model test with numerical test and full-scale test, the reduction effect of the calculative width of pile foundation with the slope is quantified. Furthermore, m-value was used as a characterization parameter to illustrate the effect of the effective calculative width on the horizontal bearing capacity of the pile. e test scheme is shown in Table 1.

Indoor Physical Simulation
e physical simulation was carried out in the threedimensional mechanical model system test bed, the dimension of which is 4 m × 2 m × 1.5 m. e boundary effect considered in this test is 5-9 times the diameter of the pile [16], and the size of the three-dimensional model is 1.5 m × 1.1 m × 1.5 m. Plane sketch of the model is shown in Figure 1, profile under 15°slope is shown in Figure 2, and scene diagram of the indoor test is shown in Figure 3.

Design of Model.
According to the similar-ratio principle and the size of the laboratory test bench, the geometric similarity ratio of the model test is determined to be 10. e size and concrete grade of the prototype square pile are 1 m × 1 m × 10 m and C30, respectively. So the size and concrete grade of the model square pile are 0.1 m × 0.1 m × 1 m and C30, respectively, too. In order to conform to the similar conditions, 4 steel bars with 6 mm diameter are used as the main reinforcement of the model pile, and 2 mm fine wire is arranged as stirrups with equal spacing of 10 cm. e model soil was prepared based on the prototype soil according to the similarity ratio. Referring to the particle size distribution of the prototype soil, we replaced the particle of model soil according to the principle of equivalent substitution. e particles with a size of >40 mm were replaced by those with a size of 5-40 mm and the particles with a size of <5 mm were prepared according to the naturally graded curve. e physical and mechanical parameters of it are regulated to close to the undisturbed soil. In this test, the main materials of model soil are boulder and gravel, and fine-grained soil is cohesive soil. e physical and mechanical parameters of model soil are shown in Table 2. Model soil is shown in Figure 4.

Monitoring System.
e main monitoring elements are the displacement dial indicator and Particle Image Velocimetry (PIV) System. e horizontal displacement of pile top is collected by two dial gauges, one of which is placed on the top of pile and the other is placed 10 cm below the top of pile. e vectorial displacement field of soil around the pile is collected by a high-speed camera installed above the soil surface. e layout of the monitoring elements is shown in Figure 5.
In order to research the soil strain around the pile quantitatively, 13 gauge points were symmetrically arranged on both sides of the pile according to the diffusion angles (center, 0°, 15°, 30°, 38°, 45°, and 52°). ese points will move with the variation of displacement of soil. Gauge points are arranged in three rows and the distance between every row and the pile is 0 b, 0.75 b, and 1.5 b, respectively. e overhead view of the gauge points is shown in Figure 6. e 0 b row in front of the pile is shown in Figure 7 as an example. e hydraulic servo system was used to load and unload step by step and the process of loading and observation is shown as follows:    Conditions for termination are as follows: (1) the pile being damaged, (2) soil surface crack or uplift obviously in front of pile, and (3) the displacement of pile top exceeding 30 mm.

Relation between Displacement and Load of Pile Top.
e displacement-load curves of pile top under 4 slopes are drawn in Figure 8.
Displacement of pile top increases with the growth of load nonlinearly under different slopes and the growth of slope under the same load, and the required load for the plastic failure of the soil decreases with the increase of slope. In other words, the bigger the slope is, the weaker the resistance of the soil around the pile to the horizontal load on the pile is and the easier the pile is to lose stability.
Horizontal load-displacement gradient curves of pile top are drawn according to the load-displacement of pile top curves, as shown in Figure 9.
For the load-displacement gradient curve under 0°slope, the first obvious inflection point appeared at 5.11 kN. In other words, the critical load under 0°slope is 5.11 kN. e second obvious inflection point appeared at 7.12 kN. In other words, the ultimate load under 0°slope is 7.   Advances in Civil Engineering

Vectorial Displacement Field of Soil.
Particle Image Velocimetry (PIV) system is mainly composed of a highspeed camera and a computer. Its basic principle is as follows: PIV divides the collected image into many grid windows. en, the computer selects the window matching between the two frames according to the cross-correlation algorithm to match the displacement and direction of the center point of the window, representing the average displacement and direction of the particles in the window [17]. e PIV used in this test is shown in Figure 10.
High-speed photos of soil around piles with 4 slopes under different loads are collected by the PIV system and the vectorial displacement field of soil is analyzed by the PIV process Flow Field processing system. Vectorial displacement field of soil with 15°is shown in Figure 11 as an example.
Under the slopes at 0°, 15°, 30°, and 45°, the displacement vectors paralleling the load appeared on the soil in front of the pile firstly when the load was small and appeared along the broadside of the pile at a certain angle with the load subsequently. With the load increases, the displacement vectors in front and the side of piles will increase further. With the beginning of the plastic deformation of soil, the soil fissures began to appear in the soil around the pile, and the displacement vectors of each area increase further along the original direction. When the soil is completely destroyed, the displacement vectors will not change anymore. At this time, the displacement vectors diffuse and attenuate in an approximately annular shape around the front of the pile [18]. For example, the slope of 45°is shown in Figure 12.

Effective Calculative Width.
In China, the current calculative width of pile is mostly calculated according to the Technical Code for Building Pile Foundations (JGJ94-2008), as shown in Table 3.
Pile-soil interaction is a complex spatial problem. Because of the existence of skin friction along with the pile, more soils around the pile form a whole and exert resistance together to bear horizontal load. is leads to the fact that the width bearing the horizontal load is larger than the actual  width of pile. is phenomenon is called Horizontal Diffusion Principle and is shown in Figure 13 [19]. In order to illustrate the effective calculating width, Horizontal Diffusion Principle of the single pile is simplified as Figure 14 based on the introduction of diffusion angle-ϕ.
Under horizontal load (F), the skin friction (f ) on both sides of pile drives more soil to participate in bearing horizontal load due to the interaction of forces. erefore, the calculative width bearing the horizontal load of pile foundation is composed of two parts: (1) the actual width of the pile b and (2) the width in which skin friction drives more soil to participate in bearing load-2 × htgϕ.
For the convenience, to calculate, the effective calculative width of the pile is converted to the width in plane view. So the formula for effective calculative width proposed in this paper is shown as follows: where b is the pile width perpendicular to the direction of the force (m), h is the pile height (m), ϕ is the horizontal diffusion angle (°), k φ is the conversion coefficient of foundation shape (the value of round pile is 0.9 and the value of square pile is 1), and k is interaction coefficient between piles. (When is the pile spacing paralleling to the direction of external force (m), h 1 is the embedment depth of pile below ground (m), and b 1 is the coefficient determined by the number n 1 of piles in a row parallel to the direction of external forces adopted by Table 4). In order to quantify the horizontal diffusion angle and eliminate the errors caused by the position of camera, the displacement value of each row of central point on the same slope was set to be 1, and other displacement values were nondimensionalized according to the same proportion. In this way, the displacement distribution curve  On the same slope, the distribution curve of soil displacement under the same row is similar bearing the different loads. e relationship curve between the soil displacement and the layout angle of gauge points is like normal distribution and the central displacement is the largest. With the increase of distance for gauge points in the same row to pile, the displacement of soil decreases nonlinearly. In addition, under the same slope, the concentrative trend of the curve is the 0 b row >0.75 b row >1.50 b row. e reason for this phenomenon is that the farther away from the pile, the weaker the tendency of shear failure caused by soil movement driven by skin friction of pile.   In order to analyze the relationship between soil displacement and horizontal diffusion angle quantitatively, the distribution curves of soil displacement ratio by the position of gauge points were integrated based on the Interpolation Method. en, the horizontal diffusion angle was calculated by integral value. Finally, effective calculative width was calculated by combining formula (1). e results are shown in Table 5.

Numerical Simulation
In order to supplement the laboratory model test, numerical simulations were carried out to calculate taking the full-scale test as prototypes. In this way, distribution of Earth pressure at 0.2 m below the horizontal surface of the soil was obtained and horizontal diffusion angle based on this Earth pressure was analyzed. Combining with formula (1), effective calculative width was calculated.

Establishment of Model.
A Finite-Difference Method-(FDM-) FLAC3D program was used to simulate the effect of slope on the pile foundation under different slopes. Taking a steep slope in Sima Village, Xuecheng Town, Lixian County, Sichuan Province, China, as the prototype, and a threedimensional numerical model was established.
e numerical model takes into account the working conditions of 0°, 15°, 30°, and 45°slope under the same soil and pile parameters. Tetrahedron was used to mesh in order to improve the running speed and reduce the boundary effect [16]. e dimension of the numerical model is 200 m × 100 m × 80 m. According to the full-scale test, the size of the model pile is 1 m × 1 m × 10 m and the length of the pile body exposed is 1 m. e generalized numerical model for 45°slope is shown in Figure 19.

Model Parameters.
e parameters of numerical model include cohesion, internal friction angle, Poisson's ratio, shear modulus, and bulk modulus. e normal and shear stiffness of the pile-soil interface was set to be 10 times of the equivalent stiffness of the surrounding soil, and all the parameters are shown in Table 6.

Loading of the Model.
150 kN is used as the increment to load step by step in the horizontal direction. When the maximum unbalanced force is unchanged, that is, when the corresponding entity test soil is in completely plastic destruction, the loading is stopped.

Result Analysis.
e diffusion angle is obtained by analyzing the stress spread of the soil around the model piles in FLAC 3D . e effective calculative width of different slopes obtained by combining formula (1) is shown in Table 7.
Tables 5 and 7 are used to plot the relationship between calculative width and slope angle, as shown in Figure 20. e results of numerical and physical simulation show good synergy. Under the level ground, calculative width based on Horizontal Diffusion Principle is greater than that based on code but it decreases substantially linearly with the increase of slope angle. Calculative width decreases by about 0.33 m with the increase of slope by 15°in this test. e decrease of calculative width means the scale of soil that resists horizontal load is reduced and slope has obvious weakening effects on this scale.

Terrain Landform.
e full-scale test is located on a high and steep slope on the left bank of Sima Village and its geographic coordinates are east longitude 103°12′36″ and north latitude 31°36′36″. e elevation of the slope is between 1700-1900 m. Pile position of the site is shown in Figure 21.

Stratigraphic and Lithologic.
e overlying layer is dominated by quaternary loose slope sediment (Q4col + dl) with a thickness of 3-15 m and the thickness of the test pile is More soil driven by friction  Advances in Civil Engineering more than 10 m which is more than the length of the pile. is layer is gravel soil, which is slightly medium in density, the particle size is generally 3-8 cm, and the maximum particle size can reach 50 cm filling a small amount of clay and gravel between the stones. For example, please refer to 30°slope, as shown in Figure 22. A series of laboratory physical and mechanical tests were carried out on undisturbed soil. e test results are shown in Table 8.

Data Collection and Loading.
A total of three dial indicators were installed on the left and right sides and the center of the pile top to measure the displacement of the pile top, and another dial indicator was installed about 40 cm below the pile top dial indicator to measure the displacement of the mud surface of the pile. Two inclinometers were symmetrically tied to the steel cage on the neutral plane of the pile to measure the displacement after each increase of load. e lateral load adopts a one-way slow-speed maintenance method, and each grade of loading is 1/10-1/15 of the estimated ultimate load. e standard for pile stability is that the displacement of the pile top does not exceed 0.1 mm within each hour. After loading, take the reading of the dial indicators of pile top every 5, 10, 15, and 30 minutes.

m-Value
Based on m-method of pile, calculative width based on Horizontal Diffusion Angle and that based on Technical where H cr is the critical load of single pile (kN), X cr is the displacement corresponding to the critical load (m), v x is the displacement coefficient of pile top which is determined by code, b 0 is the calculative width of pile (m), and EI is the flexural rigidity of pile (kN·m 2 ).

m-Value Based on Horizontal Diffusion Principle.
Calculative width-b 0 of formula (2) was substituted by effective calculative width-b h obtained from this test; m-values of 4 slopes were calculated as shown in Table 9.          Table 10. e synergy of m-value between model test and full-scale test under different slopes had been verified already [20]. is paper compared the m-values based on Horizontal Diffusion Principle and code as shown in Figure 23.   Pile Foundation (JGJ94-2008) in China, and the reduction effect of slope on the effective calculative width is not considered in the code.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.

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