A Novel Simplified Analysis Model to Predict Behaviors of Single Piles Subjected to Reverse Faulting

Abstract

Pile foundations are vulnerable to fault deformations. However, both the physical and numerical modeling of pile foundations under fault deformations are complicated and time-consuming. A simplified model is required for design and engineering practices. This study proposed a novel simplified analysis model to predict the behaviors of single piles subjected to reverse faulting. A two-dimensional beam–spring model is applied. The calculations of the stiffnesses of soil springs, skin friction, ultimate soil resistances, and Young’s modulus of sand are presented and discussed. The numerical results show a good agreement with the results of previous centrifuge tests. The parametric studies using the novel model show that ultimate horizontal soil resistance, skin friction, Young’s modulus of soil, pile stiffness, and sand density exhibit apparent effects on the responses of a single pile. The ultimate soil resistance controls the maximum inner forces, while Young’s modulus affects the increment of inner forces. The bending moment increases with pile stiffness initially and then remains relatively stable. Larger sand density leads to larger inner forces of the pile, owing to greater ultimate soil resistance and stiffness of the soil spring.

1. Introduction

In a previous earthquake, faut-induced damage occurred to structures, such as houses [1,2], bridges [3,4,5,6], tunnels [7,8,9,10,11], pipelines [12,13,14], roads [15], and foundations [16]. The pile foundation was found to be more vulnerable to fault deformation than the shallow foundation. In the 1999 Chi-Chi earthquake, the pile foundations suffered great tilting or sheared off directly by faulting [16]. These instances motivated researchers to explore the responses of pile foundations through testing [17]. Cai and Ng [18] conducted two centrifuge tests for a single pile and a pile group subjected to normal faulting. They found that the single pile was displaced, inducing an additional bending moment along the pile. Li et al. [19] performed sandbox tests for pile groups subjected to thrust faults. They found that the piles in the shear zone could separate. Therefore, the pile foundation should be built far from the shear zone. Yao and Takemura [20] conducted six groups of centrifuge tests for single piles in reverse faulting. Cai et al. [21] summarized two types of loading transfer mechanisms for pile groups subjected to normal faulting using centrifuge tests and numerical studies. Additionally, a series of numerical simulations was applied to investigate the behaviors of piles subjected to faulting [22,23,24,25], based on three-dimensional numerical models. As for the anti-measurement, Rasoulia and Fatahib [26] proposed a novel cushioned piled raft foundation to protect buildings from normal faults. Nowkandeh and Ashtiani [27,28] put forward a novel connection between the raft and the piles and a cushioned helical-piled raft system to mitigate hazards in normal faults. Although a series of tests and numerical studies have been conducted for behaviors of piles subjected to faulting, both the physical tests and three-dimensional numerical modeling are complex and time-consuming. It is difficult to handle a large number of models in a short time using physical tests and three-dimensional numerical modeling. Therefore, a simple model is required to assess the pile damage.

In this study, a novel simplified analysis model is proposed to predict the behaviors of single piles subjected to reverse faulting. The novel model is based on a two-dimensional spring model. The input parameters of the novel model are presented and discussed, such as the stiffnesses of the soil springs, skin friction, ultimate soil resistances, and Young’s modulus of sand. The objectives of this study include the following: (1) to validate the reasonability of the novel model using previous centrifuge tests; and (2) to discuss the effects of some influencing factors on the pile behaviors in the reverse fault.

2. The Simplified Analysis Model

2.1. General Description of the Novel Model

This study proposed a novel two-dimensional beam–spring model for a single pile subjected to reverse faulting, as illustrated in Figure 1. In the model, the single pile is modeled as a beam, while the surrounding soil is treated as springs. The fault displacements are input through the end of the soil springs, including horizontal displacement Δ_h and vertical displacement Δ_v (Figure 1). Therefore, the interactions between the soil and the pile during faulting are modeled by the deformation of the soil springs. The horizontal springs simulate the lateral resistance from the surrounding soil during faulting. They are set at one side of the pile, which can exhibit both tension and compression. The vertical springs model the skin friction resistance between the pile and surrounding soil. Tip reaction springs for vertical end bearings are not considered as the prototype pile is a floating pile. The pile is meshed to N units with a unit length of ΔL. Therefore, the number of both horizontal and vertical springs is N + 1. The assumptions of the novel model are as follows: (1) the pile is a single pile with an unrestraint pile head; (2) the constitutive model of the pile and the soil springs are bilinear.

2.2. Stiffnesses and Ultimate Resistances of Horizontal Soil Springs

The horizontal soil springs follow a bilinear constitutive model. Poulos [29] and Pender [30] provided an expression of the bilinear constitutive model for horizontal soil springs to idealize the soil response as follows:

$$\left\{\begin{array}{c}{P}_h=k_h{\Delta }_h\\ {|P}_h|\le P_{hu}\end{array}\right.,$$

(1)

where P_h, P_hu, k_h, and Δ_h are horizontal soil resistance, ultimate horizontal soil resistance, stiffness of horizontal soil spring, and horizontal displacement, respectively.

The stiffness of a horizontal spring has a linear relationship with the coefficient of subgrade reaction, k_s:

$$k_h=k_s{D\Delta }_L,$$

(2)

where D is the diameter of the pile; Δ_L is the unit length of the pile.

According to Equations (1) and (2), the behavior of the horizontal springs is controlled by the coefficient of subgrade reaction, k_s, and the ultimate horizontal soil resistance, P_hu. According to previous studies, the coefficient of subgrade reaction, k_s, can be estimated from Young’s modulus of soil, E_s, in cohesionless soil, as listed in

Table 1. Coefficients of subgrade reaction in previous studies.

No.	Author(s)	ks
1	Terzaghi [31]	0.74Es/D
2	Broms [32]	(0.48–0.90)Es/D
3	Muskhelishvili [33]	2Es/D
4	Matlock [34]	1.8Es/D
5	Poulos [29]	0.82Es/D
6	Rani and Prashant [35]	0.424Es1.136/D

. As displayed in Table 1, the coefficient of subgrade reaction ranges from approximately 0.48 to 2 times E_s/D. Therefore, k_s can be evaluated by

$$k_s=C_{ks}{E}_s/D,$$

(3)

where C_ks is a constant ranging from 0.48 to 2.

For a passive pile in cohesionless soils, Poulos (1995) [29] suggested that the horizontal ultimate resistance could be expressed as

$$P_{hu}=C_{ph}{K}_P{\sigma }_v^{\prime } D{\Delta }_z,$$

(4)

where C_ph is a constant ranging from 3 to 5; K_P is the Rankine passive earth pressure coefficient; K_P = tan²(45° + φ/2), in which φ is the internal frictional angle; σ_v′ is the effective vertical stress; γ_s is the effective unit weight of soil.

Equation (4) was initially proposed by Broms [32] for laterally loaded piles, where C_ph was suggested to be 3. However, the value of constant C_ph is disputed for taking a value of 3 for passive piles. Some researchers found that the real soil resistance for passive piles was much smaller than 3K_P σ_v′ when the sand was flowing around the pile, which overestimated the evaluated bending moment of the pile by 2 to 3 times [36]. Therefore, the value of 3 for C_ph may also overestimate the bending moment of piles subjected to reverse faulting.

Another uncertainty is that the expression of ultimate horizontal soil resistance may not follow a linear distribution along depth in reverse fault conditions. The faulting leads to a more complicated soil pressure distribution at different locations. Therefore, the linear distribution of ultimate horizontal soil resistance may not satisfy such a complicated soil pressure distribution. The soil on the footwall experiences less compression, which may lead to overestimation of the soil resistance and bending moment of piles on the footwall. To reduce the overestimation, another assumption of ultimate horizontal soil resistance is put forward, as shown in Figure 2. In Assumption 2, the ultimate horizontal soil resistance above the rupture also follows a linear distribution, while it becomes a constant distribution below the rupture. For Assumption 2, this could be expressed as follows:

$$P_{hu}=\left\{\begin{array}{c}{C}_{ph}{K}_P{\sigma }_v^{\prime } D{\Delta }_{z }(above the rupture)\\ C_{ph2}{P}_{hu-r}(below the rupture)\end{array}\right.,$$

(5)

where C_ph2 is a constant no larger than 1; P_hu−r is the ultimate horizontal soil resistance at the rupture surface.

2.3. Stiffnesses and Ultimate Resistances of Vertical Soil Springs

The vertical soil springs apply a bilinear constitutive model as well. The expressions are given by

$$\left\{\begin{array}{c}{P}_v=k_v{\Delta }_v\\ {|P}_v|\le P_{vu}\end{array}\right.,$$

(6)

where P_v, P_vu, k_v, and ${\Delta }_v$ are the soil resistance, ultimate soil resistance, stiffness, and displacement of the vertical spring, respectively. The stiffness of the soil in the vertical direction can be evaluated according to the work of Randolph and Wroth [37], as follows:

$$k_v={\displaystyle \frac{2\pi G_s{\Delta }_Z}{ln \left(\frac{r_m}{D}\right)}},$$

(7)

in which G_s is the shear modulus of the soil, G_s = E_s/(2(1 + ${\nu }_s$)), $r_m=2.5L(1-{\nu }_s)$, and ${\nu }_s$ is the Poisson’s ratio of soil. The ultimate vertical soil resistance is controlled by the sliding friction force on the pile shaft. According to this relationship (Pender [30]), the ultimate skin friction is

$$p_{vu}=C_{pv}{K}_p{\sigma }_v^{\prime }tan {\phi }_{sp}{C}_{peri}{\Delta }_L,$$

(8)

where $C_{pv}$ is the constant; ${\phi }_{sp}$ is the interfacial friction angle between the soil and the pile, the suggested values of which are shown in

Table 2. Suggested friction angle between soil and pile.

Soil Type	Steel Pile	Concrete Pile	Author(s)
Sand	20°	0.75φs	Tomlinson [38]
--	(0.5–0.7)φs	(0.8–1.0)φs	Kulhawy [39]
Dense sand	29.4°	--	O’Neill and Raines [40]
--	(0.7–0.9)φs	--	Yang et al. [41]

; $C_{peri}$ is the perimeter of the pile cross-section; $C_{peri}=\pi D$ for a circular pile and $C_{peri}=4a$ for a square pile; $a$ is the side length of square pile. The distribution of ultimate vertical soil resistance follows Assumption 2 in Figure 2. It could be expressed as follows:

$$P_{vu}=\left\{\begin{array}{c}{C}_{pv}{K}_p{\sigma }_v^{\prime }tan {\phi }_{sp}{C}_{peri}{\Delta }_L(above the rupture)\\ C_{pv2}{P}_{vu-r}(below the rupture)\end{array}\right.,$$

(9)

where $C_{pv2}$ is a constant no larger than 1, which is suggested to take the same value with $C_{ph2}$; $P_{vu-r}$ is the ultimate vertical skin friction at the rupture surface, calculated by Equation (8).

2.4. Young’s Modulus of Sand

Young’s modulus of soil is a key parameter for calculating the stiffness of horizontal springs. There are three Young’s modulus distribution models for the static condition (see Figure 3): con (constant soil modulus with depth), par (parabolic variation of soil modulus), and lin (linear variation of soil modulus with depth). For cohesionless soil, it is usually assumed that Young’s modulus varies linearly along depth. It is known that a higher stress level leads to a larger Young’s modulus. In reverse faulting, the soil above the rupture is highly compressed, while the soil below the rupture is only slightly compressed. A parabolic variation of soil modulus may be more suitable for faulting conditions. Therefore, the Young’s modulus applied in the analytical model could be expressed as follows:

$$E_s=\left\{\begin{array}{c}{N}_{h}Z (Z\le D)\\ E_{sD}\times \sqrt{Z}/D (Z>D)\end{array}\right.,$$

(10)

2.5. Analytical Procedures

The novel method supports a simple approach to evaluate the behaviors of piles subjected to reverse faulting by ANSYS. The analytical procedures are listed as follows:

(1)

Prepare the basic parameters of the pile (L, D, EI) and soil (H, φ, ψ, e, γ).

(2)

Ensure the number of meshed units N for the pile, and then calculate the stiffnesses of the horizontal and vertical soil springs and their ultimate resistance according to Equations (1), (5), (6), and (8).

(3)

Confirm the cross-point location and soil displacements. The location of the cross points and displacements of the surrounding soil could be obtained from the tests or designed values.

(4)

Build a beam–spring model by FEM, and calculate the inner forces and displacements of the pile and soil springs.

It should be pointed out that the vertical displacement of the pile and the increase in ground depth on the hanging wall should be considered in Steps 2 and 3.

3. Validation of the Novel Model

Yao and Takemura [20] conducted six groups of centrifuge tests to explore the responses of single piles subjected to faulting, using different fault types, pile locations, and pile stiffness. Three of the tests were conducted under the reverse fault condition. This study applies the square piles of test case RP80 to validate the novel model. The test setups are illustrated in Figure 4, where the prototype scale is presented. The sand with a depth of 11 m was subjected to reverse faulting. The dip angle was α = 60°, and the maximum bedrock offset (δb) was δb_max = 1.5 m. Two square piles (SP2 and SP3) were installed at X = 3.75 m and 8.75 m, respectively. The dimensions of the cross-section of the piles were 0.5 m × 0.5 m. The piles had a length of 11 m, where the lower 10 m was buried in the sand. Piles SP2 and SP3 were crossed by the rupture at about Z = −5 m and −2 m, respectively.

3.1. Validation by the Square Pile at X = 3.75 m in Case RP80

Table 3. Parameters of pile and sand used in the validation of pile at X = 3.75 m.

Sand	Dr. (%)	e	emax	emin	γ (kN/m3)
	80	0.682	0.973	0.609	15.4
	φ (°)	ψ (°)	νs	Nh (MPa)	φsp
	40	20	0.17	5	0.5φs
	Cks	Cph	Cph2	Cpv	Cpv2
	1	1	0.8	0.5	0.8
Pile	Ep (Pa)	L (m)	a (m)	N	ΔL (m)
	3.14 × 109	10	0.5	100	0.1

lists the parameters of the pile and sand used in the validation. The pile meshes as 100 units, corresponding to 101 horizontal and vertical springs, respectively. Considering that the frictional angle decreases with shear strain, a 40° frictional angle is applied, which is smaller than the maximum value. C_ph is valued as 1 to fit the maximum bending moment after rupture outcrops, which is smaller than that used by Broms [32] and Poulos [29]. C_pv is 0.5, meaning that only half of the superficial area of the pile contributes to the effective skin friction. Assumption 2, rather than linear distribution, is applied for the distribution of ultimate horizontal resistance and ultimate skin friction. Both C_ph2 and C_pv2 are assigned 0.8. In the validation, the displacements of the soil springs are obtained from the tests.

Figure 5 and Figure 6 compare pile behaviors obtained from the test and calculated from the novel model. In this study, the bending moment is defined as positive when the footwall side of the pile is under tension. The axial force is negative when the pile is under compression. Figure 5a shows that both the evaluated and tested bending moments display S-shaped distributions along the depth, where the extreme values occur at about Z = −5 m and −8 m. It should be pointed out that the input fault displacement of soil may inlcude errors, as it was measured at the front window of the fault simulator rather than at the location of the pile. Therefore, small differences occur in Figure 5a between the tested and estimated bending moments. Figure 5c presents the variations in the extreme values using the bedrock offset. Both the evaluated and tested values in Figure 5c increase initially, and then become relatively stable. The estimated bending moment at Z= −8 m is slightly overestimated from the bedrock offset δb = 0.5 m to 1.1 m. The axial forces distribute as a lying V-shape both for the calculated and tested values (Figure 5b). The extreme values occur at about Z = −6 m, where the rupture crosses the pile. The evaluated axial forces are a little larger than the measured values, experiencing a slightly larger compression. The value at Z = −6 m shows that the calculated axial forces are almost mimicked by the tests (Figure 5d).

Figure 6a shows that both the calculated displacements at the pile top generally agree with those measured by LDT scanning from the test. However, the calculated pile top rotation is larger than that measured from the test (Figure 6b). This may be due to the overestimation of ultimate horizontal soil resistance below the rupture. In general, the novel model yields reasonable results, agreeing with those observed from the test.

3.2. Validation by the Square Pile at X = 8.75 m in Case RP80

Table 4. Parameters of pile and sand used in the validation of pile at X = 8.75 m.

Sand	Dr (%)	e	emax	emin	γ (kN/m3)
	80	0.682	0.973	0.609	15.4
	φ (°)	ψ (°)	νs	Nh (MPa)	φsp
	40	20	0.17	5	0.5φs
	Cks	Cph	Cph2	Cpv	Cpv2
	1	2.7	0.8	1.35	0.8
Pile	Ep (Pa)	L (m)	a (m)	N	ΔL (m)
	3.14 × 109	10	0.5	100	0.1

presents the parameters of the pile at X = 8.75 m with sand applied in the validation. All parameters are the same as those applied for the pile at X = 3.75 m, except C_ph and C_pv. Their values are 2.7 and 1.35, which are 2.7 times those for the pile at X = 3.75 m. The value of C_ph is close to that used by Broms [32] and Poulos [29].

Figure 7 displays the comparison of the evaluated and observed inner forces of the pile. The bending moment presents a D-shaped distribution along the depth (Figure 7a). The estimated bending moments are in better agreement with the tested values above Z = −4 m. The calculated value at Z = −3 m shows a good agreement with the observed measures (Figure 7c). The distributions of axial force are similar for the estimated and measured data, forming a D-shape along the depth (Figure 7b). The axial forces calculated by the novel model are slightly larger than the tested values (Figure 7b). The values at Z = −4 m are generally consistent with each other, especially when δb is larger than 0.5 m.

The comparison of the evaluated and observed displacements and rotation of the pile top are presented in Figure 8. Figure 8a shows that the calculated displacements generally reproduce the values and trends of the tested values. The estimated pile top rotations follow the trend of measured data, with slightly larger values (Figure 8b).

Generally, the predictions by the novel model are reasonable, which shows a good agreement with the tested data. The novel model can be applied to the single piles subjected to reverse faulting.

4. Parametric Studies

In this section, the effects of some parameters on the responses of single piles in a reverse fault are discussed, such as ultimate horizontal soil resistance, skin friction, Young’s modulus of soil, pile stiffness, and sand density. The parametric studies are based on the tests conducted by Yao and Takemura [20] to evaluate the reasonability of the parameters. All parameters are the same as those in the validations, unless otherwise stated. The ranges of variables in the parametric studies are listed in

Table 5. Ranges of variables in the parametric studies.

Discussed Parameters	Variable	Range of Variable
Ultimate horizontal soil resistance	Cph	0.5–50
skin friction	Cpv	0.25–2.5
Young’s modulus of soil	Nh	1.5–15
Pile stiffness	EI	9.6–1 × 106 MN·m2
Sand density	Dr	30–95%

.

4.1. Ultimate Horizontal Soil Resistance and Skin Friction

The ultimate horizontal soil resistance and skin friction are significant parameters that control the horizontal and vertical responses of the single pile, respectively. In this section, different ultimate horizontal soil resistances and skin frictions are applied for the square piles at X = 3.75 m and 8.75 m in Case RP80, performed by Yao and Takemura [20]. The C_ph ranges from 0.5 to 5.0 for the pile at X = 3.75 m, while the C_pv is 0.25 to 2.5. For the pile at X = 8.75 m, the ranges of C_ph and C_pv are 1.0–5.0 and 0.5–2.5, respectively.

Figure 9 and Figure 10 display the evaluated inner forces of single piles with different ultimate soil resistances and skin frictions. As shown in Figure 9a, the bending moment grows with the increase in ultimate soil resistance. The bending moment is limited by the soil stiffness as well, as the values for C_ks = 1 become stable after C_ph reaches 2.5. The axial force follows a similar trend in which a larger skin friction leads to a larger axial force (Figure 9b,d). The best-fitting parameters for the test data are C_ph = 1.0 and C_pv = 0.5. For the response of the pile at X = 8.75 m in Figure 10, both the bending moment and the axial force display the same trend as those in Figure 9. The parameters C_ph = 3.0 and C_pv = 1.35 show the best fit for the experiment. The above results suggest that ultimate soil resistance controls the maximum inner forces. Additionally, it seems that the soil resistances under revere faulting are different at the two locations. Therefore, further discussions are required for the two parameters.

Broms [32] applied 3 K_pσ_v′ for ultimate horizontal soil resistance in his analysis, which was larger than the passive lateral resistance. Because it corresponds to an infinitely long wall due to lateral stress distribution in the soil, it works well under smaller soil displacement conditions for both active and passive piles. However, the soil displacement is large under reverse faulting, and the sand may flow around the pile. From the results of Yao and Takemura [20], the pile at X = 3.75 m exhibited significantly different horizontal displacements with the soil on the footwall, where the sand flowed around the pile. The one at X = 8.75 m was almost static on the footwall, with very small displacements. Similar phenomena were also observed in previous studies. Viggiani [42] summarized the ultimate horizontal soil resistance values from previous works and applied an ultimate horizontal soil resistance value in the stable soil in his calculation for piles subjected to landslide, which was 2 times that applied for the flowing soil. In Leung et al. [43]’s back analysis for piles subjected to excavation-induced soil movement, they found that 3 K_pσ_v′ worked well for small soil displacement conditions; however, the evaluated bending moment of the pile was about 2.75 times that of test data when soil displacement was very large flowing around the pile. Smethurst and Powrie [36] also conducted back analysis for piles used to stabilize a railway embankment. They found that horizontal resistance in the flowing sand was about half the limiting pile–soil pressure calculated by 3 K_pσ_v′. The above findings indicate that the ultimate soil resistances are about 1/3~1/2 of 3 K_pσ_v′ for passive piles experiencing flowing sand. This agrees with the ultimate soil resistance applied for the pile at X = 3.75 m in this study, which is K_pσ_v′.

Another major concern for the ultimate soil resistance is the distribution with depth. To further investigate the effects of ultimate soil resistance distribution on pile behaviors, some comparisons of Assumptions 1 and 2 are conducted, based on the novel model. Figure 11 shows the evaluated inner forces of piles with different assumptions of ultimate soil resistance distributions. As presented in Figure 11a, Assumption 1 enlarges the bending moment and axial force of the pile at X = 3.75 m, especially at the lower part of the pile, while Assumption 2 agrees well with the test results. However, the different assumptions have no apparent influence on the inner forces of the pile at X = 8.75 m. This suggests that the pile behaves like an active pile when the rupture crosses the upper part of the pile. This can be calculated using the analytical method for active piles with a linear ultimate soil resistance distribution. However, when the rupture crosses the middle part of the pile, it can no longer be calculated by the analytical method for the active pile. Both the sand flow and a non-linear ultimate soil resistance distribution should be considered.

4.2. Young’s Modulus of Soil

Young’s modulus of soil is a factor controlling the stiffness of the soil spring. In the novel model, Young’s modulus of soil is expressed as N_h times the depth. To investigate the effect of Young’s modulus on pile behavior, parametric studies are conducted, where N_h ranges from 1.5 to 15.

Figure 12 and Figure 13 present the evaluated inner forces of piles with different Young’s modulus at δb = 1.2 m. As shown in Figure 12, both the bending moment and axial force of the pile at X = 3.75 m increase with Young’s modulus and then become relatively stable. This indicates that Young’s modulus is a key factor for soil stiffness, affecting the increment of inner forces. However, the inner forces in Figure 12c,d become stable with the limitation of the ultimate soil resistance. In the stable part, the inner forces are controlled by the ultimate soil resistance. For the pile at X = 8.75 m (Figure 13a,c), the bending moment shares the same trend. However, the axial force keeps increasing with Young’s modulus (Figure 13b,d). This may be because the lower part of the pile mainly experiences static friction. Therefore, increasing the stiffness of the vertical springs would increase the axial force. In general, a larger Young’s modulus leads to larger inner forces before soil resistance reaches the ultimate values.

4.3. Pile Stiffness

The piles used in the tests of Yao and Takemura [20] are much softer than the actual piles. Figure 14 presents the effects of pile stiffness on pile behaviors. The modeled pile stiffness ranges from 9.6 to 1 × 10⁶ MN·m². Additionally, the test data for both the circular piles (CP, EI = 9.6 MN·m²) and square piles (SP, EI = 16.4 MN·m²), from Yao and Takemura [20], are displayed.

Figure 14a,b show that the evaluated bending moments of circular and square piles at X = 8.75 m and 13.75 m agree well with the test results. Both bending moment axial forces increase with pile stiffness initially (Figure 14c). However, a larger pile stiffness makes it difficult to induce additional inner forces. The critical point for the bending moment is EI = 100 MN·m², which for axial force is EI = 1000 MN·m².

4.4. Sand Density

Sand density is a combination of several effect factors, such as void ratio, unit weight, frictional angle, and Young’s modulus. In this section, the relative density of sand is considered to be from 30% to 95%, where the test results of Dr = 60% and 80%, obtained by Yao and Takemura [20] are presented as well. The input parameters for the analysis are summarized in

Table 6. In put parameters for sand with different densities.

Dr	30	40	50	60	70	80	90	95
e	0.86	0.83	0.79	0.75	0.72	0.68	0.65	0.63
γs (kN/m3)	13.92	14.20	14.49	14.79	15.10	15.43	15.77	15.95
φs (°)	31.42	32.82	34.64	36.45	38.43	40	42.14	43.13
Nh (MPa)	1	1.2	1.5	2.5	3.5	5	10	15

. It should be noted that the calculations ignore the effect of sand density on rupture trace and soil displacement.

Figure 15 and Figure 16 show the evaluated inner forces of piles with different sand densities. As presented in Figure 15, both the bending moment and axial force of the pile at X = 3.75 m increase with sand density. For the pile at X = 8.75 m in Figure 16, the inner forces increase with sand density as well. The increase in inner force is mainly owing to an increment of φ_s and N_h, which enlarge the ultimate soil resistance and the stiffness of the soil springs.

5. Conclusions

This study proposed a novel simplified analysis model to predict behaviors of single piles subjected to reverse faulting. The novel model was based on a two-dimensional beam–spring model. The calculation methods of the input parameters and the analytical procedures of the novel model are presented. Parametric studies are performed using novel models, such as the ultimate horizontal resistance, skin friction, Young’s modulus of soil, pile stiffness, and sand density. The following conclusions are drawn:

(1)

The novel model can simulate reasonable results of the inner forces and displacements of single piles. It can be applied to predict behaviors of single piles subjected to reverse faulting.

(2)

For passive piles experiencing significant sand flow, an ultimate horizontal soil resistance of K_pσ_v’ rather than 3K_pσ_v’ is in agreement with the test results. A linear ultimate soil resistance will overestimate the inner forces of the lower part of the pile.

(3)

The responses of single piles are affected by the ultimate horizontal soil resistance, skin friction, Young’s modulus of soil, pile stiffness, and sand density. The ultimate soil resistance controls the maximum inner forces, while Young’s modulus is a key factor for soil stiffness, affecting the increment of inner forces. The bending moment increases with pile stiffness initially, and the increment of pile stiffness rarely induces additional bending moment. Larger sand density leads to larger inner forces of the pile, owing to greater ultimate soil resistances and stiffness of the soil springs.

The novel model is a simplified model, which is easy to operate and is also time-saving. It could be used in design and engineering practices. The application conditions of the novel model are the single piles in sand subjected to reverse faulting. The novel model in this study cannot applied to the group piles, as a three-dimensional model is required. However, the parameters of the soil springs should be modified along the depth according to the geometries when the ground consists of layered sand.