A Simplified Nonlinear Method for a Laterally Loaded Pile in Sloping Ground

A simplified nonlinear method was proposed to evaluate lateral behavior of a pile located in or nearby a slope, based on the traditional p-ymethod. +is method was validated with field test results of a steel pipe pile in clay and model tests of piles in sand slopes. +e comparison indicated that the calculated horizontal displacement and bending moment of piles agree well with experimental results. +en, parametric studies were performed, and it shows that horizontal displacement, rotation, bending moment, and shear force increase along with increasing slope angles; the depth of maximummoment locates at about 1.6D below ground surface for horizontal ground, while this value turns to be about 3.6D and 5.6D for sloping ground of 30° and 60°, respectively. +e study clearly shows that slope angle has a significant effect on the deflection and lateral capacity of piles.


Introduction
e urban infrastructure development in China increases the possibility to construct piles in or nearby slopes, to support bridges, high-rise buildings, transmission towers, off-shore structures, retaining walls, etc. [1][2][3][4][5].e lateral bearing behavior of these piles is extremely complicated due to sloping ground [6][7][8].Compared to conventional piles, they may undergo severe reduction in horizontal bearing behavior.Although conventional laterally loaded piles have been studied by many researchers [9][10][11][12][13][14][15], limited literature can be found for piles in sloping ground [16][17][18][19][20][21][22][23].In the past, the lateral behavior of piles could be evaluated with assumed earth pressure distribution, which can also be determined from field or model tests.is method usually assumes linearly increasing subgrade reaction modulus and cannot account for the influence of sloping ground.us, further studies have to be carried out to fill this gap.
is paper presented a simplified p-y method of laterally loaded piles located in or nearby slopes.A field test result of a steel pipe pile in clay and small-scale pile tests in sand slopes were employed to verify the proposed method and to assess the influence of slope angles on the maximum deflection and required length of a pile.

The p-y Analysis for Sloping Ground
For a laterally loaded pile, the p-y method is a simple and practical way to account for the relationship between soil reaction p and pile deflection y along pile shaft.e p-y curve can be measured in field, by loading cells installed on pilesoil interface, or stress meters installed on steel cages as proposed by McClelland and Focht [24].

e p-y Curve of Clay Slopes.
According to field test results, a p-y curve of Houston soft clay was proposed by Matlock, as shown in Figure 1, and has been adopted by the American Petroleum Institute (API).It can be expressed as follows [9,10,25]: where p is the soil reaction; y is the pile de ection; p ult is the ultimate soil resistance; p/p ult is the ratio of soil resistance; y/y 50 is the ratio of pile de ection; and y 50 is the pile deection when soil resistance reaches 50% of its ultimate value, and it can be evaluated by the following equation: where ε 50 is the strain when the soil resistance reaches a half of its ultimate value and d is the pile diameter.When a laterally loaded pile locates in a clay slope of a slope angle θ, the ultimate soil resistance in front of a pile can be computed by the following equation [24,26]: where c u is the average undrained shear strength; c is the average unit weight of soil; z is the depth from ground surface to a studied point on the pile; θ is the slope angle; and z r can be computed by the following equation:

e p-y Curve of Sand Slopes.
A hyperbolic model was proposed to best-t normalized p-y curves for laterally loaded piles in sand ground, as shown in Figure 2. is p-y curve is featured with an initial sti ness k in and the relevant equation is as follows [27][28][29][30][31][32]: where k in is the initial sti ness, which depends on the soil sti ness, the pile sti ness, and the pile diameter.e initial sti ness, k in , can be assumed to increase linearly with depth in sands as [33][34][35]: where n h is the coe cient of horizontal subgrade reaction, which is related to the internal friction angle, the relative density, etc. n h can be determined according to internal angle and relative density of sands [26].e ultimate soil resistance in front of a pile located in sand slope can be described by the following equation [24,26]: where p p ult 0.5p ult p/p ult = 0.5(y/y 50 ) (1/3)   y/y 50 where K 0 is the coe cient of static earth pressure; ϕ is the internal friction angle; β 45 °+ (ϕ/2); and α is the angle of the wedge.Bowman [36] suggested that α ϕ/3 ∼ ϕ/2 for loose sand and ϕ/2 ∼ ϕ for dense sand; K a is the coe cient of active earth pressure.

Basic Equations.
Assuming that the slope is stable and ignoring friction on the pile, a simpli ed method for laterally loaded piles in sloping ground can be established as shown in Figure 3. is yields to the di erential equation as follows [20,26]: where EI is the exural sti ness of a pile and p i (y, z) is the soil resistance of sloping ground.

Boundary Conditions.
e boundary condition at pile top can be free, hinged, and partially or fully xed, while that at pile toe can be xed or hinged [20,26].In this paper, a bending moment M 0 and a shear force Q 0 are considered as external load on the pile head, and the pile toe is xed, which yields to the following boundary conditions: Free pile top: Fixed pile toe:

Finite Di erence Solution.
Subdividing the pile into N sections, the length of each section is h l/N, as shown in Figure 4.According to the principle of the central difference method, two virtual nodes are added at the pile head and toe, respectively.us, there are N + 1 nodes on the pile shaft (node number: from 0 to N), 2 virtual nodes at the top (node number: −2 and −1), and another 2 virtual nodes at the toe (node number: N + 1 and N + 2).Let the horizontal displacement at each node be y i (where i 0 ∼ n), then (9) can be rewritten as e slope, φ i , the moment, M i , and the shear force, Q i , along the pile shaft can be obtained by using the di erence method: e boundary conditions of the pile, namely, ( 10) and (11), can also be rewritten as follows: Advances in Civil Engineering en, we can establish the basic equation of the pile as where F { } is the matrix of load at pile nodes and [K] is the horizontal stiffness matrix.
ese two matrices are as follows: where en, ( 16) can be rewritten as e horizontal displacement, y i , slope, φ i , bending moment, M i , and shear, Q i , at each pile node can be solved by (19) by using an iteration process as shown in Figure 5.
e iteration criterion is indicated as ε in the process.

Field Test in Clay.
e field test result of a steel pipe pile in clay by Matlock [9] was used to verify the proposed method.Parameters adopted in this analysis are as follows: the diameter, d � 0.324 m; the wall thickness of the pipe, t � 12.7 mm; the bending rigidity, EI � 31.28MN•m 2 ; the length, l � 12.81 m; the average unit weight of soils, 4 Advances in Civil Engineering c 18 kN/m 3 ; the undrained strength of soils, c u 39.1 kPa; ε 50 0.012; and the slope angle, θ 0 °. e comparison between the calculated and measured pile de ection and the bending moment is plotted in Figure 6.
It is clear in Figure 6 that the calculated horizontal displacement and bending moment agree well with the measured and API method.

Model Tests of Piles in Sand Slopes.
e objective of the model tests is to verify the proposed method.e model piles were made by the PPR (polypropylene random) pipe, which is of 63 mm in outside diameter, 58 mm in inner diameter, and 1680 MPa in elastic modulus.e total length of piles is 1200 mm, and the embedded depth is 900 mm, as shown in Figure 7. e model slope was lled by sand using   e laboratory test carried out and the details of the model test preparation have been presented in the reference [37].e predicted horizontal displacement and bending moment of piles are compared to the measured in Figure 8 and Table 1.
We can learn from Figure 8 and Table 1 that the pile head de ection and bending moment predicted by the proposed method agree well with the measured in the model tests, and the discrepancy is 2.7% for bending moment and 15.9% for pile head de ection.
As we can learn from Figure 8, the depth of the maximum moment increases from about 10 cm (1.6 D, D is the pile diameter) below ground level in even ground (θ 0 °) to 22.5 cm (3.6 D) and 36 cm (5.6 D) below ground level in sloping ground of 30 °and 60 °in slope angle, respectively.
e results also show that the pile head de ection on slope surface rises from 3.3 mm in even ground (θ 0 °) to 7.5 mm (127%) and 15.3 mm (364%) in sloping ground of 30 °and 60 °in angle, respectively; the rotation at the top of the pile rises from −1.15 °in even ground (θ 0 °) to −1.84 in sloping ground, which are 34% and 37% increase, respectively; the maximum shear force increases from −78 N in even ground to −83 N (θ 30 °) and −93 N (θ 60 °) in sloping ground.

Conclusion
A simpli ed p-y method of piles located in slopes was proposed and solved using di erence method in this paper.
e proposed method was veri ed by the eld tests of a steel pipe pile in clay and the model tests of piles in sand slopes.
e main in uence factor, namely, the slope angle, was discussed by parametric study.e results indicate that the horizontal displacement, rotation, bending moment, and shear force increase with increasing slope angle; the depth of maximum moment is about 1.6 D below ground level for even ground and about 3.6 D and 5.6 D for sloping ground of 30 °and 60 °, respectively.It is suggested that steep slope should be avoided when designing a laterally pile in sloping ground.Advances in Civil Engineering

Figure 4 :Figure 3 :
Figure 4: De ection and di erential points of the pile.

Figure 6 :Figure 7 :Figure 8
Figure 6: e comparison of the horizontal displacement (a) and bending moment (b).