Optimized Design of Piled Embankment Using a Multi-Effect Coupling Model on a Coastal Highway

Abstract

This study presents a multi-effect coupling model to optimize the design of a geosynthetic-reinforced pile-supported embankment (GRPSE) considering the coupling effects of soil arching, membranes, and pile–soil interaction on a coastal highway. The developed model could optimize the design of the GRPSE to fulfill the design and construction requirements at a relatively low project cost. This was achieved by adjusting the critical factors that govern the settlement of GRPSEs, such as pile spacing, tensile stiffness of geosynthetic reinforcement (GR), arrangement of piles, pile cap size, and cushion thickness. The model predictions were validated by a series of field tests using a range of geotechnical sensors. The results show that model predictions agreed with experimental measurements reasonably well. In addition, the results indicate that in comparison to a square arrangement of piles, a triangle net arrangement can decrease the differential settlement of pile soil. Furthermore, this study demonstrates that a change in the GR’s tensile stiffness has little impact on the settlement of GRPSEs. This study can help to improve the stability of roadbeds of coastal highways.

1. Introduction

As a soft soil layer is often encountered in the embankment construction of a coastal highway, soil treatment becomes necessary [1]. A geosynthetic-reinforced pile-supported embankment (GRPSE) is a soft foundation treatment technology that evolved from a composite foundation [2]. Compared with the traditional composite foundation, it has obvious advantages in reducing post-construction settlement, shortening the construction period, and simplifying the operation process [3]. It is difficult to comprehensively consider the soil arching effect, membrane effect, and pile–soil interaction in existing design codes. Therefore, in the design and construction process, measures such as reducing the pile spacing are usually adopted to control the settlement stability, but then the project cost is too high. This limits the development of GRPESs to some extent.

Research on GRPSEs in recent years has mainly focused on bearing mechanisms, settlement stability, and design standards. Studying the bearing mechanism of GRPSEs could contribute to a better understanding of embankment deformation and stability. Zhang et al. established a soil arch model to investigate the bearing mechanism of GRPSEs, which could accurately predict the working efficiency of piles and the development of geosynthetic reinforcement (GR) tension [3]. By studying three-layer GR, Ye et al. and Zhang et al. found that the bottom GR layer has a more significant influence on load transfer than other layers [4,5]. Briançon and Simon analyzed the load transfer of the bottom GR layer of an embankment through experiments, and proved that the geogrid has a positive impact on the stability of the embankment [6,7]. The field testing results of Lu et al. (2019) showed that the eccentricity of a two-dimensional planar soil arch affects the load distribution of pile caps and filling [8]. Zhou et al. proposed a coupled macroscopic and mesoscopic creep model for marine soft soils to predict the long-term deformation behavior of soils [9]. Lin et al. used a combination of DEM (a mechanical simulation method) and laboratory tests to study the correlation of structural characteristics between undisturbed soil and remodeled soil, providing a reference for the application of GRPSEs in structural soil [10]. Rui et al. believed that with the increase in embankment height, the deformation mode of the GRPSE would change from a concentric elliptical arch mode to an equal settlement mode [11].

The settlement of the embankment affects the stability of GRPSEs. The study by Zhang et al. (2012) on settlement stability based on the theory of elastic beam foundation showed that the maximum and uneven settlement of GRPSEs could be reduced by increasing the rigidity of piles, reducing the distance between piles, and increasing the elastic modulus of the beam [12], while the study by Wang et al. revealed that the maximum settlement of a GRPSE depends on the stiffness of geogrid layers [13]. Chen et al. studied the stability of a GRPSE under overload and consolidation, and proved that under overload, some movable arches weakened, and the excess pore pressure dissipated significantly during consolidation, resulting in a significant increase in differential settlement [14].

To improve the design of GRPSEs, Zhang et al. suggested that the shape of the pile cap has a great influence on the maximum strain of GR [15]. Liu et al. established a theoretical model for reinforcing river bank slopes with geocell anti-sliding piles, which provided a reference for the optimal design of GRPSEs [16]. Pham et al. found that when the shear stress moves along the upper and lower sides of the soil–GR interface, the soil provides a lot of support and reduces the tension of GR [17]. Ariyarathne and Liyanapathirana studied the load transfer mechanism of a GRPSE through the finite element method, compared different design methods, and reported that the content of the lateral deformation of the embankment could be added to the design method [18]. Pham proposed a design method that considers the subsoil effect [19]. Compared with the previous design method, which only considered the arch or membrane effect, this method is more comprehensive and easier to calculate. However, current GRPSE designs do not consider all critical factors, such as the arching effect, membrane effect, construction process, pile and soil interaction, etc., in an integrated manner. Therefore, the purpose of this study was to develop a comprehensive design method with the aim of achieving optimal GRPSE design outcomes.

In this study, we coupled the 3D arching effect and the influence of membrane and pile–soil interaction effects to form a multi-effect coupled embankment settlement calculation model, which provides an evaluation standard for preliminary design. The objective cost function with structural parameters as variables was established. The influence of structural parameters on pile–soil differential settlement was obtained through numerical calculation. On this basis, a design optimization method for a pile-supported reinforced embankment was formed. Combined with a comparison of engineering examples, it was proved that this method can comprehensively consider the 3D arching, membrane, and pile–soil interaction effects, reduce the engineering settlement and engineering cost, and provide a new method for further improving the design of GRPSEs.

2. Materials and Methods

2.1. Calculate the Settlement of GRPSE Based on Multi-Effect Coupling

(1)

Calculate the settlement of GRPSE based on the 3D arching effect

The soil arch formed by an embankment can be assumed to be hemispherical, and can be divided into a spherical soil arch (secondary arch) and four 2D-plane soil arches (primary arches). As shown in Figure 1, the height of the primary arch is f₁, the height of the secondary arch is f₂, the height of the 3D soil arch is f₀, and $f_0=f_1+f_2$. To simplify the 3D soil arch surface, the equation can be expressed as:

$$z=\frac{q_c}{\gamma }(ch\sqrt{\frac{\gamma }{H}}x+ch\sqrt{\frac{\gamma }{H}}y-2)$$

(1)

where $\gamma$ is the weight of embankment fill (kN/m³), $H$ is the horizontal force at the primary arch foot (kN/m³), and $q_c$ is the fill load at the whole arch crown (kN/m³). Then, the vertical load $V$ of the secondary arch skewback and horizontal load $H$ of the secondary arch skewback can be obtained [20]:

$$V=\frac{q_{c}s[{(T+\sqrt{T^2-1})}^2-1]}{4(T+\sqrt{T^2-1})\ln (T+\sqrt{T^2-1})}$$

(2)

$$H=\frac{\gamma s^2}{4{\ln }^2(T+\sqrt{T^2-1})}$$

(3)

where $T=\frac{f_2\gamma }{q_c}+1$; $s$ is the span of the secondary arch, and $f_2$ is the height of the secondary arch. Because the skewback of the secondary arch falls on the primary arch, the vertical load of the primary arch is the vertical force $V$ of the skewback of the secondary arch, and the horizontal load of the primary arch is the horizontal force $H$ of the skewback of the secondary arch.

As the thickness of the primary arch is small compared with the overall arch height but relatively large compared with the primary arch height, it is considered in the stress analysis of the primary arch. Supposing t is the main arch thickness, s is the pile spacing, a is the width of the pile cap, ${\sigma }_r$ is the radial earth pressure of the soil element, and ${\sigma }_{\theta }$ is the tangential earth pressure, taking the vault soil element of the primary arch as the target object for stress analysis, the radial balance equation of the vault soil element can be expressed by:

$$\frac{d{\sigma }_r}{dr}+\frac{2({\sigma }_r-{\sigma }_{\theta })}{r}=-\gamma$$

(4)

According to Mohr–Coulomb strength theory, the soil element under the equilibrium limit condition can be obtained:

$${\sigma }_r=K_p{\sigma }_{\theta }$$

(5)

where $K_p$ is Rankine’s passive earth pressure coefficient, and $K_p=\frac{1+\sin \phi }{1-\sin \phi }$. Combining Equations (4) and (5), we obtain:

$${\sigma }_r=C_1{r}^{2(K_p-1)}-\frac{\gamma r}{3-2K_p}$$

(6)

where $C_1$ is the integral constant.

According to the stress analysis of the secondary arch, when the boundary condition $r=f_1$ ($f_1$ is the height of the secondary arch), as shown in Figure 2, the vertical force of the primary arch is equal to the vertical load $V$ on the skewback of the secondary arch; that is:

$${{\sigma }_r|}_{r=f_1}=V$$

(7)

The solution of V can refer to Equation (2). The integral constant C₁ can be obtained by combining Equation (4) with Equation (3):

$$C_1=(V+\frac{\gamma f_1}{3-2K_p}){(\frac{1}{f_1})}^{2(K_p-1)}$$

(8)

Combining Equations (6) and (8), the radial earth pressure of the soil element can be obtained as:

$${\sigma }_r=(V+\frac{\gamma f_1}{3-2K_p}){(\frac{r}{f_1})}^{2(K_p-1)}-\frac{\gamma r}{3-2K_p}$$

(9)

As shown in Figure 3, when $r=f_1-t$, the soil stress at the lower part of the primary arch vault can be calculated by:

$${\sigma }_{st}={{\sigma }_r|}_{r=f_1-t}=(V+\frac{\gamma f_1}{3-2K_p}){(\frac{f_1-t}{f_1})}^{2(K_p-1)}-\frac{\gamma (f_1-t)}{3-2K_p}$$

(10)

The soil stress in the upper part of GR can be obtained by Equation (11):

$${\sigma }_s={\sigma }_{st}+\gamma (f_1-t)=\left(V+\frac{\gamma f_1}{3-2K_p}\right){\left(\frac{f_1-t}{f_1}\right)}^{2\left(K_p-1\right)}-\frac{\gamma \left(f_1-t\right)}{3-2K_p}+\gamma \left(f_1-t\right)$$

(11)

Since this formula assumes that the GR and the pile cap are at the same elevation, it is more suitable for the GR layout than the traditional GR layout shown in Figure 4.

By analyzing the soil element of the primary arch skewback (Figure 2), the radial balance equation can be obtained:

$$\frac{d{\sigma }_r}{dr}+\frac{{\sigma }_r-{\sigma }_{\theta }}{r}=0$$

(12)

From Equations (5) and (12), we obtain:

$${\sigma }_r=C_2{r}^{K_p-1}$$

(13)

where C₂ is the integral constant term.

As shown in Figure 3, taking $r=\frac{s-a}{2}$, ${\sigma }_r={\sigma }_i=K_p{\sigma }_s$ as the boundary condition, where ${\sigma }_i$ is the lateral stress of the fill to the arch foot of the soil arch, using Equation (13), we obtain:

$$C_2=K_p{\sigma }_s{(\frac{2}{s-a})}^{K_p-1}$$

(14)

Substituting Equation (14) into Equation (13), we obtain:

$${\sigma }_r=K_p{\sigma }_s{(\frac{2r}{s-a})}^{K_p-1}$$

(15)

The vertical stress of the pile cap can be obtained by substituting Equation (15) into Equation (5):

$${\sigma }_{\theta }=K_p{\sigma }_r=K_p^2{\sigma }_s{(\frac{2r}{s-a})}^{K_p-1}$$

(16)

The total load of the pile cap can be calculated by integrating the vertical force of the primary arch skewback in the whole range of the pile cap. As shown in Figure 3, the integral interval is $\left(s-a\right)/2$ to $s/2$, and Equation (17) is obtained:

$$p_u=4{\displaystyle {\int }_{\frac{s-a}{2}}^{\frac{s}{2}}2(\frac{s}{2}}-r){\sigma }_{\theta }dr=\frac{2s^2{K}_p{\sigma }_s}{K_p+1}[{(1-\delta )}^{1-K_p}-(1-\delta )(1+\delta K_p)]$$

(17)

where $\delta =a/s$. According to Equations (11), (16), and (17), the soil stress at the upper part of the GR and the top of pile cap and the total load of the pile cap can be calculated. It is assumed that the thickness of the soil arch is equal to the width of the skewback; that is:

$$t=\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$$

(18)

According to the balance equation of the overall vertical load of the embankment, we obtain:

$$\gamma h{s}^2=p_u+{\sigma }_s(s^2-a^2)$$

(19)

where s is the distance between piles, a is the width of the pile cap, p_u is the total load on the pile cap, ${\sigma }_s$ is the soil stress on the upper part of the reinforced cushion, h is the distance from the pile cap to the top of the embankment, and $\gamma$ is the bulk density of soil.

By substituting Equations (11) and (17) into Equation (19), we can determine the primary arch height. Because the primary arch height is equal to the secondary arch height, the overall arch height can be obtained. In addition, the soil stress ${\delta }_s$ between the piles on top of the GR can also be calculated and introduced into the analysis of the membrane and pile–soil interaction effects in an integrated manner. The soil arch model proposed in this paper mainly analyzes the stress state when the soil arch is completely formed. According to BS8006, when the soil arch is completely formed and H/(s − a) > 1.4, this calculation model is applicable [21].

(2)

Calculate the settlement of GRPSE based on the membrane effect of GR.

Van Eekelen analyzed the membrane effect of stiffeners and obtained the calculation formula for the tensile force of stiffeners [22]. Since the bearing capacity of soil between piles is not considered in this formula, based on Van Eekelen‘s study, the stress (${\sigma }_s^{\prime }$) caused by the soil at the bottom of the reinforcement body is introduced in this section, and then the bearing capacity of the soil between piles is considered.

Figure 5a shows a GR force diagram, where ${\sigma }_s$ is the soil stress in the upper part of the reinforced body, ${\sigma }_s^{\prime }$ is the soil stress in the lower part of the GR, and $\Delta s$ is the settlement of the GR. The micro-element stress is shown in Figure 5b, where T_H is the component of tensile force T along the y axis, and T_V is the component of tensile force T along the x axis. The static equilibrium leads to:

$${\displaystyle \sum X=0} T_H-(T_H+d{T}_H)=0$$

(20)

$${\displaystyle \sum y=0} -T_V+{\sigma }_{s}dx-{\sigma }_s^{\prime }dx+T_V+d{T}_V=0$$

(21)

Solving Equations (20) and (21) leads to:

$$d{T}_H=0$$

(22)

$${\sigma }_s-{\sigma }_s^{\prime }=-\frac{d{T}_V}{dx}$$

(23)

From Figure 4b, we obtain:

$$\tan \alpha =\frac{T_V}{T_H}=\frac{dy}{dx}$$

(24)

$$T_V=T_H\frac{dy}{dx}$$

(25)

The second derivative of Equations (24) and (25) leads to:

$$\frac{d{T}_V}{dx}=\frac{d{T}_H}{dx}\frac{dy}{dx}+T_H\frac{d^{2}y}{d{x}^2}$$

(26)

Using Equations (22) and (24), we obtain:

$$\frac{d^{2}y}{d{x}^2}=-\frac{{\sigma }_s-{\sigma }_s^{\prime }}{T_H}$$

(27)

The combination of Equations (23) and (27) leads to:

$$T_V=-\left({\sigma }_s-{\sigma }_s^{\prime }\right)x+C_3$$

(28)

$$T_H\times y=-\frac{1}{2}\left({\sigma }_s-{\sigma }_s^{\prime }\right)x^2+C_{3}x+C_4$$

(29)

where C₃ and C₄ are integral constants.

By substituting the end points (0,0) and (s − a,0) of GR into Equation (29), we obtain:

$$\begin{array}{c}{C}_3=\frac{1}{2}\left({\sigma }_s-{\sigma }_s^{\prime }\right)\left(s-a\right),C_4=0\\ T_V=-({\sigma }_s-{\sigma }_s^{\prime })x+\frac{1}{2}({\sigma }_s-{\sigma }_s^{\prime })(s-a)\end{array}$$

(30)

$$T_H\times y=-\frac{1}{2}({\sigma }_s-{\sigma }_s^{\prime })x^2+\frac{1}{2}({\sigma }_s-{\sigma }_s^{\prime })(s-a)x$$

(31)

When x = 0, Equation (30) becomes:

$$T_V=\frac{1}{2}({\sigma }_s-{\sigma }_s^{\prime })(s-a)$$

(32)

When x = s − a, Equation (30) becomes:

$$T_V=-\frac{1}{2}({\sigma }_s-{\sigma }_s^{\prime })(s-a)$$

(33)

When $x=\frac{s-a}{2}$, the GR settlement can be obtained using Equation (31):

$$\Delta s=\frac{({\sigma }_s-{\sigma }_s^{\prime }){(s-a)}^2}{8T_H}$$

(34)

$$T_H=\frac{({\sigma }_s-{\sigma }_s^{\prime }){(s-a)}^2}{8\Delta s}$$

(35)

Bouma obtained the deformation formula of GR length. The deformation of GR ($\Delta l$) can be expressed as [23]:

$$\Delta l=\frac{8\Delta s^2}{3(s-a)}$$

(36)

The strain of the GR can be described as:

$$\epsilon =\frac{\Delta l}{s-a}=\frac{8\Delta s^2}{3{(s-a)}^2}$$

(37)

The combination of Equations (34), (35) and (37) leads to:

$$T_H=\frac{\left({\sigma }_s-{\sigma }_s^{\prime }\right)\left(s-a\right)}{\sqrt{24\epsilon }}$$

(38)

From the GR force analysis diagram shown in Figure 4b, it can be seen that the tensile force of the GR can be obtained from its components, and is expressed as:

$${T|}_{\mathrm{x}=0}=\sqrt{T_V^2+T_H^2}=\frac{({\sigma }_s-{\sigma }_s^{\prime })(s-a)}{2}\sqrt{1+\frac{1}{6\epsilon }}$$

(39)

Because $T=J\epsilon$, where J is the elastic modulus of the GR, we obtain:

$$\left({\sigma }_s-{\sigma }_s^{\prime }\right)=\frac{64J\Delta s^3}{3{\left(s-a\right)}^3}\sqrt{\frac{1}{16\Delta s^2+{\left(s-a\right)}^2}}$$

(40)

Substituting the value of ${\sigma }_s$ obtained in the previous section into Equation (40) and combining the result of the pile–soil interaction effect in the following section, we can obtain the value of $\Delta s$.

(3)

Calculate the settlement of GRPSE based on multi-effect coupling

Based on the analysis of micro-elements of pile and soil, the soil pressure reduction between piles is obtained [24] (Qiang 2009):

$$S_s=S_{s1}+S_{s2}=\frac{{\sigma }_s^{″}}{E_S\beta }(e^{\beta (l-2l_0)}-2e^{-\beta l_0}+1)$$

(41)

where $S_{s1}$ is the compression above the equal settlement section, $S_{s2}$ is the compression below the equal settlement section, $S_s$ is the soil compression between piles, ${\sigma }_s^{″}$ is the earth pressure between piles at the pile top plane, which is the initial stress in the soil, E_S is the elastic modulus of soil between piles, $\beta =\frac{\pi d{K}_0\tan \phi }{A_s}$, $K_0$ is the coefficient of static lateral pressure, $\phi$ is the friction angle of soil, $A_s=s^2-\frac{\pi d^2}{4}$ is the area of soil between piles, $l_0$ is the position of the neutral point of the pile, and $l$ is the length of the pile. Stress analysis of pile micro-elements leads to:

$$d{\sigma }_p(z)=\pm \eta {\sigma }_s(z)dz$$

(42)

where ${\sigma }_p(z)$ is the stress of the pile and ${\sigma }_s\left(z\right)$ is the soil stress between piles. $\eta =\frac{\pi d{K}_0\tan \phi }{A_p},A_p=\frac{\pi d^2}{4}$.

The stress expression of the pile can be obtained from Equation (42):

$${\sigma }_{p1}(z)=-\frac{1-m}{m}{\sigma }_{s1}(z)+C_2$$

(43)

where $m=\frac{A_p}{s^2}$, and A_p is the cross-sectional area of the pile. The boundary condition ${{\sigma }_{p1}(z)|}_{z=0}=p_u^{\prime }$ can be obtained when z = 0, and $p_u^{\prime }$ is the pressure at the bottom of the pile cap. The vertical stress expression of the pile body can be obtained by:

$${\sigma }_{p1}(z)=p_u^{\prime }+\frac{(1-m){\sigma }_s^{″}}{m}(1-e^{-\beta z})$$

(44)

Similarly, the stress of the pile can be obtained by:

$${\sigma }_{p2}(z)=-\frac{1-m}{m}{\sigma }_{s2}(z)+C_3$$

(45)

The continuous boundary condition ${{\sigma }_{p2}(z)|}_{z=l_0}={{\sigma }_{p1}(z)|}_{z=l_0}$ can be obtained by letting z = l₀. Using Equation (45), the vertical stress expression of the pile body can be obtained by:

$${\sigma }_{p2}(z)=p_u^{\prime }+\frac{(1-m){\sigma }_s^{″}}{m}(1-e^{\beta (z-2l_0)})$$

(46)

The compression of the piles above the equal settlement section can be calculated using Equation (44); that is:

$$S_{p1}={\displaystyle {\int }_0^{l_0}\frac{{\sigma }_{p1}(z)}{E_p}}dz=\frac{p_u^{\prime }{l}_0}{E_p}+\frac{(1-m){\sigma }_s^{″}{l}_0}{m{E}_p}-\frac{1-m}{m}\frac{E_s}{E_p}{S}_{s1}$$

(47)

where $E_p$ is the elastic modulus of the pile and m is the replacement ratio of the pile–soil area, which is the cross-sectional area of a pile divided by the area of foundation treatment shared by a pile. Compression of the piles below the equal settlement section can be calculated using Equation (46):

$$S_{p2}={\displaystyle {\int }_0^{l_0}\frac{{\sigma }_{p2}(z)}{E_p}}dz=\frac{p_u^{\prime }(l-l_0)}{E_p}+\frac{(1-m){\sigma }_s^{″}(l-l_0)}{m{E}_p}-\frac{1-m}{m}\frac{E_s}{E_p}{S}_{s2}$$

(48)

Using Equations (45) and (46), the compression of the overall pile can be obtained:

$$S_p=S_{p1}+S_{p2}=\frac{p_u^{\prime }l}{E_p}+\frac{(1-m){\sigma }_s^{″}l}{m{E}_p}-\frac{1-m}{m}\frac{E_s}{E_p}{S}_s$$

(49)

The amount of soil compression between piles is equal to the sum of pile compression, soil deformation when the pile just enters the soil, and soil deformation when the pile completely enters the soil, expressed as $S_s$ = $S_p$ + $\Delta a$ + $\Delta b$, while the differential settlement of pile soil is similar to the deformation amount of the pile top. Thus, the equation of differential settlement between piles and soil can be obtained:

$$\Delta \mathrm{s}=\Delta a=S_s-S_p-\Delta b$$

(50)

Using Winkler’s model, deformation of the pile bottom can be obtained [16]:

$${\Delta b=\frac{1}{k}\left({\sigma }_{p2}\left(z\right)-{\sigma }_{s2}\left(z\right)\right)|}_{z=l}$$

$$=\frac{1}{mk}[\gamma h-{\sigma }_s^{″}{e}^{\beta (l-2l_0)}]$$

(51)

where k is the coefficient of the subgrade soil between the piles of the GRPSE.

Substituting Equations (41), (49), and (51) into Equation (50), the differential settlement of pile–soil can be obtained:

$$\begin{array}{c}\Delta s={\sigma }_s^{″}[\frac{1}{E_s\beta }(1+\frac{1-m}{m}\frac{E_s}{E_p})(1+e^{\beta (l-2l_0)}-2e^{-\beta l_0})\\ +\frac{1}{mk}{e}^{e^{\beta (l-2l_0)}}]-(p_u^{\prime }+\frac{1-m}{m}{\sigma }_s^{″})(\frac{1}{E_p}+\frac{1}{k})\end{array}$$

(52)

Equation (52) can be simplified as:

$${\sigma }_s^{″}=\frac{\Delta s+p_u^{\prime }A}{B-\frac{1-m}{m}A}$$

(53)

where $A=\left(\frac{1}{E_p}+\frac{1}{k}\right)$ and $B=\left[\frac{1}{E_s\beta }\left(1+\frac{1-m}{m}\frac{E_s}{E_p}\right)\left(1+e^{\beta \left(l-2l_0\right)}-2e^{-\beta l_0}\right)+\frac{1}{mk}{e}^{\beta \left(l-2l_0\right)}\right]$.

It is assumed that the stress transfer between the upper and lower soil layers of the pile cap meets the following conditions:

$${\sigma }_s^{\prime }(s^2-a^2)={\sigma }_s^{″}(s^2-\frac{\pi d^2}{4})$$

(54)

$$p_u\frac{a^2}{4}=p_u^{\prime }\frac{\pi d^2}{4}$$

(55)

where a is the pressure on top of the pile cap. Equation (53) can be transformed into:

$${\sigma }_s^{\prime }=\frac{\Delta s+p_u\frac{a^2}{\pi d^2}A}{B-\frac{1-m}{m}A}•\frac{s^2-\frac{\pi d^2}{4}}{s^2-a^2}$$

(56)

By combining Equations (40) and (56), we can obtain the soil stress between piles and differential settlement of pile soil under the reinforced cushion. According to the overall equilibrium condition of the embankment vertical load within the scope of a single-pile-equivalent treatment, the top stress of the pile cap can be calculated by Equation (57):

$$\gamma h{s}^2={\sigma }_p^{\prime }{a}^2+{\sigma }_s^{\prime }(s^2-a^2)$$

(57)

The pile–soil stress ratio n can be expressed as the ratio of soil stress on the pile cap to soil stress between piles; that is:

$$n=\frac{{\sigma }_p^{\prime }}{{\sigma }_s^{\prime }}$$

(58)

The load sharing ratio E of piles can be described as the proportion of piles bearing the total load of the embankment; that is:

$$E=\frac{{\sigma }_p^{\prime }{a}^2}{\gamma h{s}^2}$$

(59)

In summary, the GRPSE settlement calculation based on multi-effect coupling can be described in detail as follows:

(a)

Based on the 3D soil arch effect of the GRPSE, the thickness of the soil arch can be obtained based on Equation (18). Solving Equations (11), (17), and (19) obtains the height $f_0$ of the soil arch and the soil stress ${\sigma }_s$ between the piles on top of the reinforced cushion.

(b)

To consider the membrane effect and pile–soil interaction effect of the GRPSE, ${\sigma }_s$ is substituted into Equations (40) and (56), and the differential settlement $\Delta s$ of pile–soil and soil stress ${\sigma }_s^{\prime }$ between piles under the reinforced cushion can be obtained.

(c)

Based on the force equilibrium condition of the embankment, Equation (57) can be used to solve the stress value ${\sigma }_p^{\prime }$ of the pile cap under the reinforced cushion.

(d)

Based on the soil arching, membrane, and pile–soil interaction effects, the obtained pile–soil differential settlement $\Delta s$, pile cap stress ${\sigma }_p^{\prime }$ below the reinforced cushion, and soil stress ${\sigma }_s^{\prime }$ between piles can be substituted into Equations (38), (58), and (59) to obtain the tensile force T of the GR, pile–soil stress ratio n, and load sharing ratio E of the pile.

2.2. Design of Optimized GRPSE Based on Multi-Effect Coupling

(1)

Design method for a GRPSE based on the multi-effect coupling settlement calculation model

According to the theoretical settlement calculation model proposed in Section 2.1, design standards are provided for the GRPSE. As shown Figure 6, the design procedure for the GRPSE, considering multi-effect coupling, can be described as follows:

Select pile type and construction materials based on the engineering requirements.

Determine the pile design (e.g., pile diameter, spacing, length, cap size, and layout). The pile design can refer to the research by Zhou Jing et al. and the Technical Code for Ground Treatment of Buildings JGJ 79-2012 [25,26].

Calculate pile bearing capacity. The pile top load can be calculated using Equation (17). To maintain the stability of the GRPSE, the load on the pile top should be less than the ultimate bearing capacity of the pile; that is, $P_u\le P_{uk}/{\gamma }_p$, where $P_{uk}=u{\displaystyle \sum q_{sik}{l}_i+\psi P_{sk}{A}_p}$, and ${\gamma }_p$ is the partial coefficient of the pile’s bearing capacity.

Determine the reinforced cushion design (e.g., strength, number of layers, and construction method). The geogrid used in the design optimization method introduced in this paper is a bidirectional geogrid. When checking GR strength, it can be solved based on the multi-effect coupling settlement calculation model.

Calculate differential settlement and analyze the post-construction stability of the GRPSE. The differential settlement can be calculated by the multi-effect coupling settlement calculation model, and the allowable settlement value should be met.

(2)

GRPSE design optimization based on multi-effect

Design optimization

To save on construction costs by reducing the number of piles, increasing the pile spacing and the size of the pile cap becomes necessary. On the other hand, to compensate for the reduced stiffness of the GRPSE due to the reduced number of piles, it is necessary to increase the size of the pile caps or use multiple layers of GR. However, this enhancement could increase the construction cost. The optimization design mainly finds the factors that control the differential settlement of pile soil through numerical simulations and field tests. At the same time, the cost influence coefficient is introduced. In the end, an optimized method is obtained that can not only ensure the safety of the engineering, but also save on cost.

Define the objective function

The objective cost function of the GRPSE can be developed based on the optimization parameters, such as pile spacing, pile cap size, GR stiffness, and cushion thickness.

$$C=C_p+C_c+C_g+C_l$$

(60)

where $C_p$ is the total cost of the pile body, $C_c$ is the total cost of the pile cap, $C_g$ is the total cost of the GR material, and $C_l$ is the total cost of the cushion. The constraints of the optimization process can be described as follows:

Settlement constraints:

$$\Delta S\le \left[\Delta S\right]$$

(61)

Equal settlement section constraints:

$$h_{es}<h$$

(62)

After the optimization design is completed, the pile spacing, pile cap size, GR stiffness, and cushion thickness can be determined. Finally, the overall bearing capacity of the project should be checked to determine its safety.

The sensitivity of four structural variables of a GRPSE to engineering cost and settlement differs. In order to determine the influence of each variable on the total cost of the embankment, when each variable changes separately, the ratio of the change in cost to the change in settlement is defined as the cost influence coefficient of the optimization variable:

$${\beta }_n=\frac{\Delta C}{\Delta K}$$

(63)

where $\Delta C$ is the change in cost when the variable changes and $\Delta K$ is the change in differential settlement when the variable changes. The larger the cost influence coefficient, the more sensitive the variable is to cost; if the cost influence coefficient is smaller, the variable is more sensitive to settlement. Therefore, in the design optimization, we can first determine the variables with a low cost sensitivity and adjust those with a high cost sensitivity according to the influence of each variable on differential settlement in order to optimize project cost.

2.3. Field Experiment and Numerical Analysis

(1)

Project overview

The developed design method was implemented to study the GRPSE used in a section of a coastal highway (A2, K47 + 100 – K59 + 201) in South China. The length of the test section was 160 m. According to the geological survey data, this section is a soft foundation. The characteristics and mechanical properties of the soil are shown in

Table 1. Characteristics of soil (section A2 of coastal highway).

Geotechnical Type	Area Number of Soil Layer	Length (m)	Thickness (m)	Overburden Layer	Substratum
Silty clay	K58 + 942 – K59 + 201	259.6	15.0–16.3	Plain fill thickness: 3.70–3.80 (m)	Silty clay
Strongly weathered granite and silty sand	K58 + 942 – K59 + 201	259.6	4.5–4.8	Plain fill, silt, muddy silty clay, and silty clay 20.7–22.6 (m)	Fine sand, gravelly sand

and

Table 2. Mechanical properties of soil used in this study.

Soil Layer	Thickness (m)	Saturated Bulk Density (kN/m3)	Compression Modulus (Mpa)	Poisson’s Ratio	Cohesion (kPa)	Internal Friction Angle (°)
Embankment soil	3.5	18	18	0.2	15	20
Muddy silty clay	11	16.6	9.65	0.3	13	16.2
Silt	9	15.4	6.12	0.35	10	7
Medium sand and silt	10	22.5	20	0.15	39	22

, respectively.

The height of the filling was 4 m, the slope ratio of the left side slope was 1:1.5, the width of the freeway pavement was 12 m, and the width of embankment ground was 18 m. The geogrid and geocell, with a thickness of 0.5 m, were used as GR. Prestressed pipe piles (0.3 m in diameter and 21 m in length) were used in the GRPSE. The thickness of the pile cap was 0.4 m. A rigid connection was adopted between the pile and pile cap. The material properties of the GRPSE used in this study are shown in Table 2 and

Table 3. Mechanical properties of piles, pile caps, and GR.

Material	Type and Shape	Diameter/Thickness (m)	Unit Weight (kN/m3)	Elastic Modulus (Gpa)	Compression Modulus (Mpa)	Poisson’s Ratio	Tensile Stiffness (kN/m)
Pile	Cylindrical	0.3	24	50	/	0.1	/
Pile cap	Square	0.4	24	24	/	0.1	/
GR	Geogrid/geocell	0.5	19	/	20	0.2	1000

. Figure 7 shows the field layout of reinforcement. The laying of GR was divided into traditional and new construction methods. The new construction method is to set a fixed joint composed of several steel bars at the top of the pile cap, and then set a geogrid on top of the pile cap, and pour concrete to form the fixed end. This method is composed of a fixed joint and fixed end to form a fixed connection system to connect the geogrid and pile top. The new construction method fixes the geogrid and the pile top together. The geogrid can restrain the lateral displacement of the embankment slope and deep subgrade with restraining piles, which improves the utilization efficiency of reinforcement materials. This method completely breaks the limitation of the interface friction between the geogrid and the soil, and can use a geogrid with a higher strength in the GRPSE to give full play to its membrane effect.

(2)

Field test setup

As shown in Figure 8 and Figure 9, the settlement and bearing capacity of the GRPSE were monitored on site. The measurement results verify the theoretical results. The monitoring system consisted of the following components, and the details of the monitoring points are shown in

Table 4. Monitoring design parameters for each monitoring point.

No.	Location Number of Monitoring Area	Length (m)	Filling Height (m)	Pile and Pile Cap(m)			GR
				Pile Spacing	Pile Cap	Clear Spacing	Type	Layers
1	K58 + 940 – K58+970	30	4.0	3.0	1.0	2.0	Geogrid	3
2	K58 + 970 – K59 + 100	30	4.0	3.0	1.2	1.8	Geogrid	2
3	K59 + 000 – K59 + 030	30	4.0	3.0	1.2	1.6	Geogrid	2
4	K59 + 030 – K59 + 060	30	3.0	3.0	1.6	1.4	Geogrid	1
5	K59 + 060 – K59 + 080	20	3.0	3.0	1.6	1.4	Geocell	1
6	K59 + 080 – K59 + 100	20	3.0	3.0	1.6	1.4	Geocell	1

.

The influence of the pile spacing and reinforced cushion on embankment settlement was measured using a field test. The test content and instrument embedding method are as follows:

Load tester: Bury between the top of the pile and the cap to measure the load on the pile during the filling process. In the process of embedding, the pile head of the pipe pile needs to be leveled and drilled at the pile head with an impact drill.

Soil settlement gauge: Put the water level gauge into the water tank, bury the water tank 1 m outside the slope toe, and pour concrete to fix the water tank in place. Connect a water pipe between the two water tanks and pour clean water into the water tank so that the water surface is flush with the top of the tank.

Flexible strain gauge: Fix the gauge on the grid of GR to measure the GR tension.

Earth pressure sensor: Bury in the ground of GR to measure the load on soil between piles. Bury the earth pressure sensor in the designed position, wrap it with 30 cm of fine sand and compact it. Lead the lead wire out of the slope toe, connect the instrument, and set the instrument to zero.

Inclinometer tube: Bury at the toe of the embankment; this measures the horizontal displacement of the subgrade. Drill holes at the design position, put the inclinometer tube into the hole, and inject clean water. When the first tube is 40–50 cm above the ground, connect the second tube and inject clean water into the tube at the same time, and so on. Fill the hole with soil and compact it. The tube exposed to the ground should not be too long; generally, 50 cm is appropriate, and the excess part should be sawed off.

(3)

Numerical analysis

The numerical geometric model was established by the Plaxis 3D Foundation. The filling height of the embankment was 4 m, the slope ratio of the left side slope was 1:1.5, and the pavement width of the auxiliary road was 12 m. The height of the GR was 0.15 m, and the thickness of the cushion was 0.5 m. The diameter of the pile was 0.3 m, and the pile length was 21 m. The pile body passed through the upper soil (muddy silty clay) for 11 m, the middle weak area (silt) for 9 m, and the bottom underlying layer (medium sand + silt) for 10 m.

The numerical model adopted the Mohr–Coulomb constitutive model. The pile element was the embedded element, and the GR element was simulated by the floor element.

The construction process was as follows: (a) initial state; (b) pile construction; (c) reinforced cushion filling; (d) embankment filling; and (e) calculation of post-construction settlement. The embankment filling adopted the layered filling method, with each layer 0.5 m high, and the embankment filling was divided into eight layers. The model parameters are shown in Table 2 and Table 3. The geometric model is shown in Figure 10.

3. Results and Discussion

3.1. Cost Influence Coefficient

There were 60 piles in the studied section, with a square pile cap (the steel plate fixed at the top of the pile was used to bear the partial load, 1 m × 1 m). The thickness of the gravel cushion was 0.5 m, and the stiffness of the geogrid was about 1000 kN/m. Based on the highway engineering budget quota in China [27], the unit price of a pipe pile, geogrid, geocell, gravel cushion, and pile cap was 130 RMB/m, 8 RMB/m², 10 RMB/m², 160 RMB/m³, and 310 RMB/m³, respectively. The optimization process involved increasing the pile spacing to reduce the project cost and adjusting the stiffness of GR to reduce differential settlement. The initial parameter values and optimization schemes 1 and 2 are shown in

Table 5. Design details of GRPSE system used in this study.

Cases	Pile Spacing (m)	Number of Piles	Pile Cap Size (m)	Cushion Thickness (m)	GR Tensile Stiffness (kN/m)
Control	3	60	1 × 1	0.5	1000
Scheme 1	2.5	84	0.5 × 0.5	0.4	500
Scheme 2	3.5	45	1.5 × 1.5	0.6	1500

, and the variable cost influence coefficients of schemes 1 and 2 are shown in

Table 6. Cost influence coefficient for each variable in scheme 1.

Variable	Initial Cost (CNY)	Cost of Comparison Scheme (CNY)	Change in Cost (CNY)	Initial Displacement (mm)	Displacement of Comparison Scheme (mm)	Change in Displacement (mm)	Cost Influence Coefficient
Pile spacing (m)	218,760	284,280	65,520	29.28	19.215	−10.065	0.628
Pile cap size (m)	218,760	213,924	−4836	29.28	48.555	19.275	0.024
Cushion thickness (m)	218,760	210,120	−8640	29.28	32	2.72	0.307
GR tensile stiffness (kN/m)	218,760	217,680	−1080	29.28	31.83	2.55	0.041

and

Table 7. Cost influence coefficient for each variable in scheme 2.

Variable	Initial Cost (CNY)	Cost of Comparison Scheme (CNY)	Change in Cost (CNY)	Initial Displacement (mm)	Displacement of Comparison Scheme (mm)	Change in Displacement (mm)	Cost Influence Coefficient
Pile spacing (m)	218,760	177,810	−40,950	29.28	52.5	23.22	0.300
Pile cap size (m)	218,760	223,875	5115	29.28	13.08	−16.2	0.050
Cushion thickness (m)	218,760	227,400	8640	29.28	27	−2.28	0.601
GR tensile stiffness (kN/m)	218,760	219,840	1080	29.28	26.805	−2.475	0.069

, respectively. The results show that the cost influence coefficients of pile spacing and cushion thickness were relatively large, and the pile cap size and GR had a great influence on settlement.

3.2. Optimization Based on GRPSE Settlement

(1)

Optimization of pile cap size

In this section, numerical simulation is used to analyze the influence of variables on settlement, so as to achieve optimization. Refer to Section 2.3 for engineering situations and parameters. The established 3D geometric model is shown in Figure 9.

It can be seen from Figure 11 that the differential settlement of pile soil reached its steady state when the pile cap size increased to 1.5 m × 1.5 m. However, a too-large pile cap could increased the risk of bending failure and resulted in a higher project cost. The optimization outcomes show that the differential settlement could be reduced by 55% without increasing the project cost too much when the pile cap size was in a range from 1 m × 1 m to 1.5 m.

(2)

Optimization of GR

Figure 4 shows the details of embankment structures using the traditional GRPSE construction method (GR laid on top of the pile caps) and the new construction method (GR laid on the same level of the top surface of the pile caps). It can be seen from Figure 12 that the increased GR stiffness had little influence on reducing differential settlement. While increasing the number of GR layers could reduce settlement, it could increase the cost influence coefficient. Figure 13 shows that the pile soil produced by the new construction method had smaller differential settlement than that of the two-layer GR when the stiffness of the GR was between 800 and 1500 kN/m. The comparison of results from Figure 13 and Figure 14 shows that the influence on differential settlement was relatively small when the tensile stiffness of the GR was >1500 kN/m and the pile cap size was >1.5 m. This indicates that the required reinforced cushion stiffness could be achieved under an optimized combination of cushion thickness and GR tensile stiffness.

(3)

Optimization of cushion thickness

The numerical simulation results in Figure 14 show that the differential settlement of pile soil changed greatly when the cushion thickness changed from 0.3 to 0.5 m. When the cushion thickness changed from 0.6 to 0.8 m, the pile–soil differential settlement change was small, due to the cushion cap effect. The greater the cushion thickness, the greater the cushion cap effect [28]. The thickness of the cushion layer had a larger cost influence coefficient. In order to reduce the cost, reducing the thickness of the cushion layer while keeping the differential settlement increase small can be considered. After adjusting the cushion thickness from 0.5 to 0.4 m, the differential settlement increased by 11.2%. However, according to the strength calculation of the reinforced cushion mentioned in Section 2.2, the overall stiffness of the reinforced cushion was calculated to meet the design requirements.

(4)

Optimization of pile spacing

The combined effects of pile spacing and pile cap size on differential settlement are shown in Figure 15. It shows that increasing the pile spacing from 3 to 3.5 and 4 m resulted in an increased differential settlement by 30% and 48%, respectively. The relationship between the number of piles and pile spacing is shown in

Table 8. Number of required piles under square and triangle net arrangements for different pile spacings.

Pile Spacing	2	2.5	3	3.5	4	4.5
Number of piles in triangle net arrangement	135	84	60	45	40	28
Number of piles in square arrangement	153	98	66	45	41	32

. When the pile spacing was increased from 3.5 to 4 m, the number of piles used was only reduced by five. Therefore, the adjustment of pile spacing should not be too large, as too-large pile spacing cannot reduce the cost but will produce a large differential settlement. When the pile spacing was increased from 3.5 m to 4 m, the differential settlement produced was relatively large, and the number of piles used did not change much, so 3.5 was taken as the optimal pile spacing.

(5)

Optimization of pile arrangement

Compared to a square arrangement, a triangle net arrangement in Figure 16 generally led to more piles and a higher project cost. However, the optimization analysis results indicate that with pile spacing of 3.5 m, both triangle net and square arrangements require 45 piles to fulfill the design requirements. In addition, compared to a square arrangement of piles, the differential settlement of pile soil caused by a triangle net arrangement can be decreased by around 17%.

The influence of the pile arrangement (triangle net vs. square) on the required pile number is shown in Figure 17 and Figure 18 and Table 8.

(6)

Comparison of theoretical predictions with monitoring results

The monitoring location was a coastal highway project in South China. Figure 19 and Figure 20 show that the settlement trend of the 3D numerical simulation results and the actual monitoring results are similar to the filling time, indicating the correctness of the numerical simulation model. When the tensile stiffness of the GR was equal, the maximum settlement value of the soil between piles increased gradually with the increase in time; when the pile spacing increased from 2.6 m to 3 m, the maximum settlement value of soil between piles increased gradually at the same time. When the pile spacing was constant, the greater the tensile stiffness of the GR was, the smaller the maximum differential settlement of the soil between the piles was. The results in Figure 21 show that the numerical predictions agreed with the monitoring data reasonably well. The multi-effect coupling model was used to calculate the GRPSE, and the bearing capacity of a single pile was checked. According to the bearing capacity calculation method introduced in Section 2.2, the bearing capacity met the design requirements. Comparing the theoretical prediction in Figure 21 with the multi-effect coupling and numerical simulation prediction, the theoretical value was generally smaller than the numerical value and closer to the measured data. The error of the theoretical model in calculating the maximum settlement was about 3.3%. The error of the maximum settlement calculated by the numerical model was about 5.6%. In addition, this shows that the theoretical predicted maximum settlement was generally higher than that shown by the monitoring data, and the theoretical value was less than the allowable settlement value (20 cm) [29]. This indicates that the developed theoretical model could produce conservative/safe results. Therefore, the correctness and advantages of the multi-effect coupling model are proved.

(7)

Analysis of optimization results

The optimization results are shown in

Table 9. Design details of GRPSE before and after optimization.

	Pile Spacing (m)	Pile Cap Size (m)	GR Tensile Stiffness (kN/m)	Layers/Construction Method	Cushion Thickness (m)	Pile Arrangement	Cost (CNY)
Before optimization	3	1	1000	1-layer geogrid	0.5	Square	218,760
After optimization	3.5	1.5	1000	New method geocell	0.4	triangle net	163,089

. When considering the allowable requirements of bearing capacity [30], the total project cost could be reduced by 25%, while the differential settlement of pile soil could be decreased by 6%. Although increasing the pile spacing increased the differential settlement, the settlement could be compensated for by increasing the overall GR stiffness. This demonstrates that the developed model is capable of reducing the project cost while ensuring that all design requirements are met. The results demonstrate that the optimized GRPSE design can be achieved by optimizing the critical parameters, such as pile spacing, pile cap size, GR tensile stiffness, and cushion thickness.

4. Conclusions

This paper comprehensively considered the 3D arching, membrane, and pile–soil interaction effects, and combined the three together to establish a GRPSE settlement calculation model based on multi-effect coupling and used it as a control standard for design optimization. On this basis, the objective cost function was established and combined with 3D numerical simulation in order to propose the design optimization method for a GRPSE. This method does not only consider the arch, membrane, and pile–soil interaction effects and accurately reflect the actual settlement of the project, but also reduces the embankment settlement and project cost of the coastal highway through scheme adjustment. The results show that the theoretical predictions agreed with the monitoring data reasonably well. Due to the limitation of monitoring data, this theoretical model only considers the complete soil arching effect. The comparative study of a complete soil arch and an incomplete soil arch could be carried out by collecting engineering monitoring data of different embankment filling heights. The following are some of the major findings:

The embankment settlement calculated by the multi-effect coupling model is more accurate and safer. After optimizing the design of the project, the cost was reduced by 25%, and the differential settlement of soil between piles was reduced by 6%.

The pile cap size and GR tensile stiffness have a great influence on the settlement of the GRPSE. When the tensile stiffness of GR is less than 1500 kN/m and the size of the pile cap is less than 1.5 m, the differential settlement of the pile soil will be more affected.

The pile spacing and cushion thickness have a great influence on the cost coefficient, while GR has little influence, and can effectively reduce the pile–soil differential settlement.

Increasing the number of GR layers can reduce the differential settlement of pile soil more than increasing its tensile stiffness.

The thickness of the cushion layer has a larger cost influence coefficient. In order to reduce the cost, reducing the thickness of the cushion layer while keeping the differential settlement increase small can be considered. The optimal cushion thickness is 0.4 m.

The layer of geocell implemented is smaller than the geogrid.

With pile spacing of 3.5 m, the differential settlement is small. In addition, compared to a square arrangement of piles, the differential settlement of pile soil caused by a triangle net arrangement could be decreased by around 17%.

When the pile spacing increases from 2.6 m to 3 m, the maximum settlement value of soil between piles increases gradually at the same time. When the pile spacing is constant, the greater the tensile stiffness of the GR is, the smaller the maximum differential settlement of the soil between the piles is.