Ultimate Lateral Capacity of Rigid Pile in c–φ Soil

To date no analytical solution of the pile ultimate lateral capacity for the general c–φ soil has been obtained. In the present study, a new dimensionless embedded ratio was proposed and the analytical solutions of ultimate lateral capacity and rotation center of rigid pile in c–φ soils were obtained. The results showed that both the dimensionless ultimate lateral capacity and dimensionless rotation center were the univariate functions of the embedded ratio. Also, the ultimate lateral capacity in the c–φ soil was the combination of the ultimate lateral capacity (fc) in the clay, and the ultimate lateral capacity (fφ) in the sand. Therefore, the Broms chart for clay, solution for clay (φ=0) put forward by Poulos and Davis, solution for sand (c=0) obtained by Petrasovits and Awad, and Kondner’s ultimate bending moment were all proven to be the special cases of the general solution in the present study. A comparison of the field and laboratory tests in 93 cases showed that the average ratios of the theoretical values to the experimental value ranged from 0.85 to 1.15. Also, the theoretical values displayed a good agreement with the test values.


Introduction
Piles are widely used in bridges, high-rise buildings, wharf infrastructures, power transmission towers, offshore platforms, wind power infrastructures, and other civil engineering projects. The ultimate lateral capacities of these piles are important contents of engineering designs and safety assessments. To date no analytical solution of the ultimate lateral capacity for the general c-φ soil has been obtained, and only under the condition of c = 0 or φ = 0, the analytical solution is available. Although the calculation formula was not obtained, Broms (1964a) put forward the well known calculation charts. Poulos and Davis (1980) further presented the calculation formulas for the ultimate lateral capacity and rotation center in clay under the condition of φ = 0. More relevant research on clay can be found in the study of Reese and Welch (1975) and other researchers. On sand, Petrasovits and Award (1972) put forward a calculation formula of the ultimate lateral capacity of sand under the condition of p u =kz. However, a calculation method for the rotation center was not obtained. For the general c-φ soil, Hansen (1961) proposed an evaluation method that requires repeated iterations, with a more complex process and more time spent on it.
In this study, a new dimensionless embedded ratio was proposed. And the strict solutions of the dimensionless ultimate lateral capacity and rotation center for three kinds of soils (c=0, φ=0, and c-φ soil) were obtained. The results showed that both the dimensionless ultimate lateral capacity and the dimensionless rotation center were the univariate functions of the new embedded ratio, and that the ultimate lateral capacity in c-φ soils is the combination of the ultimate lateral capacity (f c ) in clay, and the ultimate lateral capacity (f φ ) in sand. The ultimate bending moment for c-φ soil was obtained also. In the comparison of the published field and laboratory test results of 93 cases, the average ratios of the theoretical values to the experimental values ranged from 0.85 to 1.15.

Ultimate lateral resistance of soils
For rigid piles the estimation of the ultimate lateral load capacity requires knowledge of the magnitude and distribution of the limit lateral soil resistance for a given soil condition. The ultimate lateral resistance represents the limit load capacity applied by the surrounding soil per unit length. For clay, there have been several methods to estimate the ultimate lateral resistance, such as Broms (1964a), Matlock (1970), Hansen (1961), Shen (1961), Hays et al. (1974), Reese and Welch (1975), Ito and Matsui (1975), Randolph and Houlsby (1984), Murff and Hamilton (1993), Chen and Poulos (1997), Martin and Randolph (2006), Georgiadis et al. (2013), and Zhang et al. (2017). Generally the ultimate lateral resistance can be written as follows: where, N p is the lateral bearing factor; c u is the undrained strength of clay; and D is the pile diameter. The limiting value of N p for the Matlock, Randolph, Reese, and Hays theories is 9, while Hansen (1961) defined the limit as 8.14.
The Randolph and Houlsby's (1984) study shows that the bearing factor is related to the friction characteristics of the pile surface. The values vary from 9.14 to 11.94 for different friction conditions. The commonly adopted values of the ultimate lateral pressure for clay are in the range of N p = 9-12. For cohesionless soils, the methods for estimating the ultimate lateral resistance were given by several researchers (Broms, 1964b;Hansen, 1961;Shen, 1961;Petrasovits and Award, 1972;Reese et al., 1974;Meyerhof and Sastry, 1985;Prasad and Chari, 1999;Chari and Meyerhof, 1983;Barton, 1982;Fleming et al., 1992). Although these methods produce significantly different values, the ultimate lateral resistance for cohesionless soils can be simplified as follows in general, where, γ is the unit weight of soil, z is the depth from the ground surface, and K is the ultimate lateral resistance coefficient for sand. For Broms theory, K=3K p ; Hansen theory K=K p ; Barton ; for Reese theory tan φ-K a , where K p is passive earth pressure coefficient, and K a is active earth pressure coefficient.
For c-φ soils, according to the recommendations of Hansen (1961), Shen (1992), Evans and Duncan (1982), and Reese and van Impe (2001), the ultimate lateral resistance of c-φ soil can be expressed by the following equation: where, q u =N p c u D and k=γKD. Obviously Eq. (3) is applicable for clay (φ=0), sand (c=0) and c-φ soils.
3 Ultimate lateral capacity of pile 3.1 Ultimate lateral capacity of a pile in clay For clay (φ = 0), the limit state of a pile is shown in Fig.  1a. In accordance with the force and moment equilibrium conditions, the following two equations can be obtained: where H u and z c are the unknown variables, while the remaining are the known variables. By the definition of f c = H u /(q u L 0 ), the dimensionless can be conducted. The analytical solution of the dimensionless ultimate lateral capacity of a pile in clay can be obtained as follows: where f c =H u /(q u L 0 ) is the dimensionless ultimate lateral capacity of a pile in clay; H u is the ultimate lateral capacity; r is the embedded ratio, r = L/L 0 ; q u is the ultimate pressure in Eq.
(1); and z c is the dimensionless rotation center of the pile. The distance from the rotation center to the mud surface was Z c , Z c = z c L = z c rL 0 , as shown in Fig. 1a. Fig. 2a shows the variations of the dimensionless ultimate lateral capacity, and the dimensionless rotation center with the embedded ratio. Eq. (5) and Fig. 2a show that both the dimensionless ultimate lateral capacity and the dimensionless rotation center were the univariate functions of the embedded ratio, and that the reasonable dimensionless format was able to simplify the expressions of the dimensionless ultimate lateral capacity, and that of the rotation center. Comparing with the traditional bivariate (L/D, e/D) function, the expression in this study was dramatically simplified.

Ultimate lateral capacity of a pile in sand
According to the existing research, in regards to sand (c=0), the distribution of the ultimate pressure can be written as p u = kz. The force and moment equilibrium equations Fig. 1. Ultimate lateral load and ultimate pressure of pile in limit state: (a) in clay (Broms, 1964;Poulos and Davis, 1980); (b) in sand (Petrasovits and Award, 1972); (c) in c-φ soil; (d) ultimate bending moment under pure-bending conditions. of the pile (Fig. 1b) are given in Eq. (6), And the following solutions are obtained: where, f φ is the dimensionless ultimate lateral capacity of the pile in sand, f φ = H u /(kL 0 2 ); r is the embedded ratio, r = L/L 0 ; and z φ is the dimensionless rotation center of the pile. Eqs. (7) and (8) indicate that both the dimensionless ultimate lateral capacity and dimensionless rotation center of the pile in sand both are the univariate functions of the embedded ratio r , as shown in Fig. 2b.
For the purpose of simplification and practice, the following equations can be used to replace Eqs. (7) and (8) in engineering designs. Eq. (9) is also plotted in Fig. 2b in dashed line, which will satisfy the precision requirements of engineering design in general.
3.3 Ultimate lateral capacity of a pile in c-φ soil The analysis model of the laterally loaded pile in c-φ soil is displayed in Fig. 1c. The distribution of the pile-soil ultimate pressure along the depths can be expressed as p u = q u +kz. The limit equilibrium equations of the pile are established as follows: where H u represents the ultimate lateral capacity of the pile; L 0 is the pile length; L is the embedment length of the pile, L= L 0 r; r represents the embedded ratio, r=L/L 0 ; e is the loading eccentricity, e = L 0 (1-r); and Z 0 = z 0 L 0 r is the distance from the surface to the rotation center. After nondimensionalization, the solutions of Eq. (10) can be written as follows: when y < 0 f 1 =H u /(q u L 0 ) is the dimensionless ultimate lateral capacity of the pile in c-φ soil; f c =H u /(q u L 0 ) is the dimensionless ultimate lateral capacity of the pile in clay (see Eq. (5)); f φ = H u /(kL 0 2 ) is the dimensionless ultimate lateral capacity of the pile in sand (see Eq. (7)); r is the embedded ratio, r=L/L 0 ; and z 0 is the dimensionless rotation center of the pile.
Since kq=kL 0 /q u in Eqs. (11)-(13) is the parameter depending on γ, c and φ of the soil, after the soil is specified, kL 0 /q u can also be determined. This indicates that the dimensionless ultimate lateral capacity of the pile in c-φ soil is also the univariate function of the embedded ratio.
In Eqs. (11)-(13), when q u =0, a singularity problem will appear. Therefore, in this case, another dimensionless ultimate lateral capacity is defined as f 2 = H u /(kL 0 2 ). Using ZHANG Wei-min China Ocean Eng., 2018, Vol. 32, No. 1, P. 41-50 43 this new definition, based on Eq. (10), the following solutions are obtained: The two groups (Eqs. (11)-(13) and Eqs. (14)- (16)) of ultimate lateral capacity expressions can be used in the analysis of pile engineering. In general, for the clay with a smaller φ, Eqs. (11) and (12) are recommended, while for the sand with a smaller c, Eqs. (14) and (15) are suggested. Both Eq. (11) and Eq. (14) indicate that the dimensionless ultimate lateral capacity in the c-φ soil is a combination of f c in clay, and f φ in sand. However, on the other hand, both f c and f φ are particular solutions of f 1 and f 2 , respectively. Fig. 3 shows that the variations of the ultimate lateral capacity with the embedded ratio are under the two different dimensionless formats, respectively. Fig. 3a is more suitable for the clay with a smaller φ, while Fig. 3b is more suitable for the sand with a smaller c. Fig. 4 shows the relationship of the rotation center of the pile to the embedded ratio.
For the purpose of simplification and practice, Eq. (12) can be replaced by the following equation for the calculation of the rotation center in engineering designs, being plotted in dashed line, as shown in Fig. 4.

Ultimate bending moment of pile in c-φ soil under pure-bending conditions
The effects of a bending moment are independent of its applied position. Therefore, under the condition of pure bending, only the embedment length of the pile under the surface should be taken into consideration. In accordance with Fig. 1d, the limit equilibrium equations of the pile under the condition of pure bending are shown as follows: where M u is the ultimate bending moment, L is the embedment length of the pile, and z 0 =Z 0 /L represents the dimensionless rotation center. The solutions of Eq. (18) can be written as: When c =0, a singularity may occur in Eq. (19). Under the condition of φ ≠0, the other sets of dimensionless variables of M u /(kL 3 ) and q u /(kL) are used, and another set of solutions can be obtained.
The variations of the dimensionless ultimate bending moment, and the dimensionless rotation center (Eqs. (19) and (20)) under the two different dimensionless formats are shown in Figs. 5a and 5b, respectively.
For the clay, when φ = 0 and kq= 0, the following formulas can be obtained from Eq. (19), For the sand, when c = 0 and qk = 0, the following formulas can be obtained from Eq. (20), The above results show that, due to the characteristics of the bending moment, the ultimate bending moment of the pile can be simplified further under the condition of pure bending, and both the dimensionless ultimate bending moment and rotation center under pure-bending conditions are equal to a constant, respectively.

Comparison with the field and laboratory tests
Some methods have been proposed over the years to interpret the ultimate load of lateral tests, and sometime the ultimate loads obtained by those different methods may have several times of difference (Chen and Lee, 2010). The ultimate capacity of the pile can be divided into two categories (Reese and van Impe, 2001): the serviceability limit capacity, and the ultimate limit capacity. The serviceability limit capacity involves considering the allowable deforma-tion of the structure. The ultimate limit capacity involves not considering the working requirement of the structure, and giving consideration instead to the theoretical maximum capacity. The hyperbolic function method (Kondner et al., 1964;Coyle and Bierschwale, 1983;Manoliu et al., 1985;Mayne and Kulhawy, 1991) can eliminate not only the influence of artificial factors to a certain extent, but also the concept of the hyperbolic asymptote consistent with the definition of the limit load in plastic mechanics. In the present study, the following hyperbolic function was used to analyze the field and laboratory tests, and to compare these with the theoretical values.
where H is the test lateral load; and δ is the pile displacement at the mud surface. According to the characteristics of the hyperbola, K i is the initial tangent slope of the curve, and H u is the ultimate load. The comparisons between the computed ultimate capacities and the test results of 93 cases are shown in Fig. 6 and in Table 1-Table 4. The comparisons show that the average ratios between the theoretical solution and the test results range from 0.85 to 1.15.

Discussion and comparison with the existing calculation methods
6.1 Comparison with the results of Broms (1964a) Figs. 7a and 7b show the limit analysis model and the design calculation chart presented by Broms (1964a) for clay, respectively. Broms assumed that the surrounding soil within a 1.5D depth below the mud surface had already been damaged before the limit state was reached, and therefore that part of the soil had no effect on the results. Broms did not give a theoretical analytic solution. However, the design chart shown in Fig. 7b was given in the form of bivariate implicit function chart. By comparing Fig. 7a with Fig. 1a, it can be found that Broms's assumption ignored the   ZHANG Wei-min China Ocean Eng., 2018, Vol. 32, No. 1, P. 41-50 effect of the pile surrounding soil within a 1.5D depth below the mud surface, which can actually be seen as the embedment length reduced by 1.5D. Therefore, the embedded ratio of Broms's model can be written as r=(L-1.5D)/(L+e). By substituting the embedded ratio (L-1.5D)/(L+e) instead of r, and 9c u D instead of q u directly into Eq. (5), the explicit calculation formula for Broms's chart can be easily obtained as follows: where P ult is the ultimate lateral load after Broms; L is the embedment length of the pile; e is the load eccentricity; and D is the diameter of the pile. The curves of Eq. (24) are plotted in Fig. 7b, which are identical to those of Broms. Figs. 7b and 7c display the comparison between the results presented by Broms and the solution obtained in this study. Fig. 7c is drawn according to Eq. (5), in which the coordinates are after Broms's bivariate coordinate system (L/D, e/D). The univariate curve in Fig. 2a is obviously far simpler than the bivariate chart put forward by Broms (Fig.  7b), and the working mechanism of the pile is more clear also.
6.2 Comparison with the solution of Poulos and Davis (1980) When p u is a constant, the formula of the ultimate lateral capacity (Eq. (25)) have been obtained by Poulos and Davis (1980). In fact, the results presented by Poulos and Davis can also be conveniently derived from the solution of this study. For example, by substituting L/(L+e) instead of r into Eq. (5), the formula of Poulos and Davis,Eq. (25) can be obtained.
where, p u in this study is equal to p u D in the study presented by Poulos and Davis due to the different definitions. Similarly, according to r=L/(L+e), Eq. (25) can be briefly returned to Eq. (5). It was indicated that the Poulos and Davis's equation is the same as the solution of this study. Therefore, the Poulos and Davis's equation will be a particular solution of the c-φ solution of this study when φ = 0 and p u = const. (see Fig. 3a). Petrasovits and Award (1972) The calculation Eq. (26) of the ultimate capacity in sand with c=0 were presented by Petrasovits and Award (1972). The formula can also be conveniently derived from the solution presented in this study. According to its definition, when k = γεD, r = L/L 0 , and f φ = H u /(kL 0 2 ) were substituted into the solution Eq. (7), Eq. (26a) can be obtained. It has been proven that the solution presented by Petrasovits and Awad (1972) is the same as that of Eq. (7) in this study, and therefore, the solution of Petrasovits and Awad (1972) is another particular solution of the c-φ solution when c = 0 also. (see Fig. 3b). In addition, Petrasovits and Awad (1972) did not come to the solution of the center of rotation (z φ ), merely presented an implicit formula (26b), so their solutions were incomplete. While that in Eq. (8) of this study was an explicit expression of the solution.

Comparison with the results of
6.4 Comparison with results presented by Kondner et al. (1964) Kondner et al. (1964 obtained a conclusion through the dimensional analysis and model test that under the condition of pure bending, and the dimensionless ultimate bending moment M u /(γCL 3 ) in sand was found equal to a constant. From Eq. (22) it can be seen that Kondner et al.'s conclusion was consistent with that of this study. For the dense sand, when γ =17.14 kN/m 3 and φ = 37, a test dimensionless number was obtained by Kondner et al. as follows: where C is the perimeter of the pile. By substituting k=K(φ)γD instead of k into Eq. (22), the solution Eq. (22) can be converted as follows: where K(φ) is the ultimate pressure coefficient. By comparing Eq. (27)    Q u = test ultimate load determined by the hyperbolic function; b H u = computed lateral capacity, the combination of Broms's formula and Reese's, p u =9c u D+(K p 3 +K 0 K p 2 tanφ-K a ) γDz, was used.  Reese's (1974) formula, K(φ) =(K p 3 +K 0 K p 2 tanφ-K a ) is used, the ultimate pressure coefficient, K(φ) = K(37) =69.7, and Kondner et al.'s dimensionless number will be obtained as follows: The computed result in Eq. (29) is reasonably close to that of Kondner et al.'s test in Eq. (27).

Conclusions
In the present study, the strict solution of the dimensionless ultimate lateral capacity of the pile in c-φ soil was obtained. The results showed that the ultimate lateral capacity in c-φ soil was the combination of the ultimate lateral capacity (f c ) in clay, and the ultimate lateral capacity (f φ ) in sand. The solutions of Poulos and Davis (φ = 0), Petrasovits and Awad (c = 0), and pure bending results of Kondner et al. were special cases of the solution.
The new proposed dimensionless embedded ratio is rather effective. It transforms the complex bivariate implicit charts into a univariate explicit function, simplifies the analysis process and shows a better understanding on the working mechanism of pile. The results show that the dimensionless ultimate lateral capacity of the pile depends on the embedded ratio only.
By comparing the field and laboratory tests of 93 cases, the average ratios of the theoretical values to the experimental were found to range from 0.85 to 1.15, and the computed values displayed good agreement with the testing results.
Not only can the hyperbolic function method eliminate the influences of artificial factors to a certain extent, but also the hyperbolic asymptote concept is close to the definition of the limit load in plastic mechanics.