J. Cent. South Univ. (2020) 27: 2094-2112

DOI: https://doi.org/10.1007/s11771-020-4433-8

Evaluation of soil arching effect due to partially mobilized shear stress in piled and geosynthetic-reinforced embankment

LV Wei-hua(吕伟华)^{1, 2}, WU Tao(武涛)¹, GU Fan(顾凡)^{3, 4}, GAO Lei(高磊)²

1. School of Civil Engineering, Nanjing Forestry University, Nanjing 210037, China;

2. Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering,Hohai University, Nanjing 210098, China;

3. National Engineering Laboratory of Highway Maintenance Technology, Changsha University of Science & Technology, Changsha 410114, China;

4. National Center for Asphalt Technology, Auburn University, Alabama 36830, USA

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: In piled and geosynthetic-reinforced (PGR) embankment, the arching behavior determines the overburden load on piles and subsoils. Placement of geosynthetic is effective in reducing the relative displacement between pile and subsoil. When the mobilized shear stress is less than the shear strength, partially developed arching will occur. Consequently, existing analytical methods, adopting the ultimate shear strength failure criterion, need to be improved. This study developed a simplified 2D analytical method, which is based on the developing arching effect, to evaluate the load redistribution of the PGR embankment. Then, the influences of embankment height and internal friction angle, subsoil depth, ratio of pile cap width to pile clear spacing (RPC) and geosynthetic tensile stiffness on the critical height ratio, stress concentration ratio, soil arching ratio, geosynthetic tension and axial strain were investigated. This study suggests that a RPC of 1:1.0 and a one-way of single-layer geosynthetic tensile stiffness of 2000 kN/m should be considered as the sensitivity thresholds for the PGR embankment.

Key words: pile; geosynthetic; arching effect; mobilized shear stress; parametric analysis

Cite this article as: LV Wei-hua, WU Tao, GU Fan, GAO Lei. Evaluation of soil arching effect due to partially mobilized shear stress in piled and geosynthetic-reinforced embankment [J]. Journal of Central South University, 2020, 27(7): 2094-2112. DOI: https://doi.org/10.1007/s11771-020-4433-8.

1 Introduction

The piled and geosynthetic-reinforced (PGR) technology is usually utilized to prevent local/global instability of road infrastructures [1], overcoming the excessive post-construction settlements [2], increasing the foundation bearing capacity [3], enhancing the subsoil shear strength [4] and reducing the compressibility of foundation soils [5, 6], etc. In a PGR embankment, the fill layer between piles (or caps) has a subsidence trend due to the presence of interfacial compressible soil [7]. Fortunately, the subsidence would be partially resisted by the shear resistance from the soil medium on top of the pile heads (or caps) [8]. The friction decreases the load on the installed geosynthetic layer but increases the load transferred to adjacent drilled piles [9]. In a design framework, the stress on the pile (cap) or foundation soil should be accurately estimated [10, 11], indicating that the pile size and selection of geosynthetic material should be determined accordingly on proper evaluation of arching effect and geomembrane effect.

Developing from the traditional pipeline theory proposed in Ref. [12], it is accepted that the frictional arching behavior plays a significant influence on load redistribution regarding the adjustment of shear stress [13], which is due to the yield of the trap door. Subsequently, some soil arching models (see Refs. [14-22]) based on fixed arch profiles have been proposed, and these methods have tried to yield solutions for arching behavior evaluation and load distribution estimation. In addition, more considerable studies on arching mechanism have been performed qualitatively and quantitatively, with the assistance of numerical simulation technology or statistics from in-situ projects (see Refs. [23-32]). In summary, the initial research mainly focused on soil arching effect itself, and the subsequent research began to consider the geosynthetic geomembrane effect. However, these methods basically belong to a category of two-step analytical approach. First, the load on geosynthetic or interfacial subsoil is estimated, and then the geosynthetic tensile strain is calculated accordingly. However, almost all the aforementioned methods were based on a limit state condition that the stress state reaches the extreme condition, either at the foot or crown of an idealized arch porotype.

Comparatively, the analysis model deducing the stress state due to relative settlement between pile and subsoil seems to be more applicable in theory [33-36]. These methods indicate that the strength of arching behavior is dependent on the relative displacement and is influenced by the interaction between pile, soil and geosynthetic. Obviously, when the relative settlement between pile and soil is small, the arching behavior in embankment is weak, or else the arching effect is strong [37]. Magnitude of displacement between adjoining parts in the fill material exactly determines the shear stress level within the yielding slip surface due to localized subsidence [38, 39]. In frictional arching models, the positive arching action occurs when the subsoil is compressed, and subsequently decreases stresses on the subsoil but increases overburden load on the pile (cap) [40]. In an in-situ project, with a fixed layout of pile installation and the common filling material, the degree of soil arching probably essentially depends on the ratio of modulus of foundation soil to that of concrete pile. The geosynthetic reinforcement may only plays an auxiliary role in the later overburden redistribution.

After the completion of the embankment project, it is unlikely that there will be considerable relative settlement between piles and subsoil, especially when the in-between soils are not very weak. Therefore, there should be a lot of incomplete arching behaviors in the situ. To consider this situation, this study proposed a simplified method to investigate the developing arching effect in a PGR platform. The key question is how to quantitatively determine the development of shear stress which is determined by the level of relative settlement between piles and subsoil. Based on a coupled iterative calculation process, this method is able to estimate the occurrence degree of soil arching and predict the geosynthetic tension. Then, the proposed method was validated according to several current methods, and a parametrical analysis was also conducted, as well as the discussion.

2 Theoretical derivation

2.1 Modelling basis

In Figure 1, an analytical prototype of arching was illustrated in which the physical model of the PGR embankment has been refined. In such a platform, the interactions among the embankment fill, the pile (or cap), the subsoil and the geosynthetic can be briefly illustrated in Figure 2. Due to the compression of interfacial subsoil, the above filled mass in-between piles tends to subside, but is partially resisted by the friction (shear stress) from adjacent motionless embankment fill. This friction decreases the pressure on the geosynthetic, and thus increases proportional overburden load to the pile. For arching behavior, the arch shape tends to follow the trajectory of the principal stress chains [33, 36], indicating that there is an influenced stress range boundary, which is corresponding to a critical height or an active arch affected area. The shear stress state of the fill within the slip plane, which may undergo a change from elastic to plastic state, has a direct relationship to the relative displacement between piles (caps) and subsoil. Simultaneously, settlement firstly comes up adjacent to the pile cap, and the induced shear stress mobilizes along a log spherical plane from the arch foot to the crown, accompanied with a decrease in magnitude from the maximum to zero.

Figure 1 Sketch of PGR embankment (modified from Ref. [21])

Figure 2 2D load transfer mechanism of PGR embankment

To simplify the analysis, the embankment fill material is assumed to be uniform and isotropic, and the lateral displacement is ignored. Piles are installed in square grids, the disengagement between fill material and geosynthetic layer is not allowed. The geosynthetic herein used for analysis is a single-layer uniaxial plastic geogrid. The main stress direction of the geogrid is installed perpendicularly to the route direction. For simplicity, the geomembrane effect and the reaction of the interfacial subsoil are all assumed to be uniformly distributed in the vertical direction. Moreover, the stress and displacement at the elevation of the geosynthetic layer satisfies the condition of coordination, indicating that relevant equality relations between upper and down can be established directly according to equilibrium conditions in the follow derivation.

2.2 Partially-developed frictional arching effect

In construction, the embankment is filled by layers, the differential settlement develops from the contact surface between the cap and the interfacial subsoil. A slip surface initiates from the elevation of the cap edge to a height of the embankment. Within the slip surface, the shear stress mobilizes and is determined by the magnitude of the relative displacement. In other words, the stress state point will move from one position to another in the Mohr circle (see Figure 3). Thereafter, the arching behavior is entering into a partially developing state, indicating that the shear strength of the fill material cannot be fully mobilized. According to the classic mechanical principle, there exists a relation:

                (1)

where σ_H is the horizontal stress; σ_V is the vertical stress; σ_min is the minor principal stress; σ_maj is the major principal stress; f is the angle of σ_maj to the abscissa axis.

Figure 3 Shear stress state in Mohr circle

Based on the vertical rough-wall theory, HANDY [33] proposed that the partially mobilized shear stress (at point A in Figure 3) can be estimated by utilizing an inclination angle (β). The relation is between the difference of the horizontal stress and the minor principal stress with a mobilized shear stress (τ_s), and yields the following relation:

               (2)

As previously assumed, the soil stress at the arch foot enters the active state first. Then, an earth pressure coefficient, which is the ratio of the horizontal stress to the vertical stress, can be expressed as:

          (3)

where K_a is the active lateral earth pressure coefficient,

According to Eqs. (1) and (2), we get:

                   (4)

Then, the shear stress at point A can be reformulated as:

      (5)

where is the coefficient of the shear stress and the vertical stress, which contains the information of influence of localized sinking of subsoil on the stress state of embankment fill. For convenience, the stress state as the foot represents the state of the whole slip plane, and the subsequent derivation can be simply spread out.

In Figure 4, a chip element within the active arching zone is selected for analyzing, and a partial differential governing equation can be obtained as:

                         (6)

where b is the clear pile spacing; γ is the unit weight of embankment. Then, the solution of Eq. (6) can be formulated as:

                   (7)

Figure 4 Force balance of a soil slide element of dz

The constant C₀ here can be calculated by the formulaaccording to the vertical stress state, we obtain:

                (8)

Then, the vertical stress is solved to be:

 (9)

where z is the depth from the top of stress arch to the cap elevation; and h is the soil arch height. Within a single element scale of a pile in 2D condition, the stress acted on the cap can be expressed as:

                (10)

where a is the pile cap width; q₀ is the uniform surcharge on the embankment top. Then, the stress concentration ratio can be expressed as:

                (11)

In the aforementioned derivation, the shear stress along the slip surface decreases from the maximum at the foot of the arch to zero at the crown elevation. Obviously, deflection of the soil element at the footpoint will naturally be affected by the resistance from the deflected geosyntehtic and the compressed subsoil.

2.3 Geosynthetic geomembrane effect

Yielding of the embankment fill between piles (caps) subsides the geosynthetic, i.e. the geosynthetic layer deflects with the subsidence of soft/compressible soil. Figure 5(a) illustrates the mechanical behavior of a single-layer geosynthetic. According to the modelling basis, the vertical earth pressure within two adjacent piles (caps) and the inbetween subsoil can be assumed to be uniformly distributed (see Figure 5(b)). For simplification, the uniform stress (i.e. the equivalent geomembrane effect) acting on the geosynthetic is q, and the geosynthetic tensile stiffness is J_g. In the deflected geosynthetic, an assumed point M, with a horizontal distance x to the midpoint, is selected for analyzing the geomembrane mechanical behavior. In derivation, the tension at point M is assumed to be T_M, with an oriented angle φ_M to the horizontal direction. Due to the force balance of a microelement segment dx, the following relations can be deduced:

,         (12)

If we take where T_con is the horizontal component force of the geosynthetic, then we get in which

Figure 5 Geomembrane effect analysis:

Then, the geosynthetic deflection can be formulated as:

                    (13)

At midpoint, the maximum deflection is produced to be δ_g:

                        (14)

Defining the following relation:

                        (15)

At the edge of the basal subsidence, the oriented angle of the defected geosynthetic can be assumed to be φ_c, then an equivalent relationship can be established to be:

                      (16)

Equation (16) shows that the geosyntehtic deflection and the relative displacement between pile and soil are connected, satisfying the continuity condition along the geosynthetic layer. By modifying the methods recommended by Refs. [41, 42], the following integrated equation, based on the elongation of the tensile strain within the deflected geosynthetic, can be obtained:

 (17)

As long as the trajectory of the deflected geosynthetic is determined, the equivalent geomembrane effect can be estimated. However, the whole calculation procedure cannot be achieved independently, the reason is that the determination of the load acting on the reinforced layer needs to consider the coupling action of arching and geomembrane, and also includes the reaction from the foundation soil.

2.4 Iterative calculation method

Combining Eqs. (9) and (17), the vertical stress and the deflection of geosynthetic are still uncertain. However, if the embankment fill is high enough, the affected region due to arching must be limited to a certain extent, which can be termed as the soil arch active zone, i.e. the soil arch height. Consequently, Refs. [34, 43] utilized a virtual structure to represent the action character of arch, with the arch foot standing on a pile cap and the arch rib bearing the overburden load. Moreover, the active zone represented by an arch height can be determined using a balancing method of force and moment of a virtual arch structure. This method to determine the influenced area due to arching is also adopted in this study. However, this study takes into account the influences of geomembrane effect and reaction of subsoil. These two effects have been coupled in the assessment of partially developed frictional arching effect.

As illustrated in Figure 6, a half 2D arch model is selected for mechanical analysis. Some basic parameters can be regarded as the known condition, such as the embankment height (H), the soil arch height (h), the uniform stress on the embankment top (q₀). Ideally, a virtual horizontal force (F_d) is assumed to act on crown of the half arch to simulate the action from adjacent fill mass. Similarly, the non-uniform horizontal pressures and are assumed to act at different positions within the active arch zone between two adjacent piles (caps). The compressible subsoil (with a depth H_d) exerts a

uniform stress σ_sto the subsiding fill, where E_q is the weighted average modulus of foundation soil. Then, the following equations can be obtained:

                     (18)

                       (19)

Figure 6 Geostatic balance of 2D soil arch cross section (Modified from Refs. [34, 43])

Combining all the above equations with integration, Eq. (19) becomes:

         (20)

The ultimate implicit expression can be obtained as:

  (21)

In Eq. (20) or (21), the calculation formula for each symbol is as follows:

             (22)

           (23)

                         (24)

 (25)

            (26)

                      (27)

In light of Eq. (21), the critical arch height can be calculated out, and after two or three steps of iteration computation, a set of numerical solutions can be obtained. Once the soil arch height is determined, the geosynthetic deflection and tension, and the vertical stresses on the pile cap and the interfacial subsoil are able to be obtained synchronously. In general, a flow chat of the calculation procedure of the presented method is shown in Figure 7. Essentially, this proposed method is a one-step calculation approach when compared to the traditional two-step method. Comparatively, the two-step method calculates the soil arching first, and then determines the tensile force or resistance from the compressible subsoil, which ignores the interaction between the geomembrane effect and the arching effect.

Figure 7 Calculation procedure of presented method

3 Validation and comparison with several current analytical methods

Sponsoring proper comparison with measured data is necessary, but there will also be some contingency. Consequently, this study tried to do some verification and comparison with some existing methods to reveal the advantages and disadvantages of this present method. Occasionally, those referenced methods here were proposed in Refs. [14, 16, 17, 19-21]. These methods are all classic methods for arching assessment and are often used to verify and evaluate new artworks. The detailed derivations, features or defects of these methods can be referred to the review and analysis in Ref. [44], so their introduction and analysis procedure will not be repeated here. Comparing with these aforementioned methods, this study aims to make a comprehensive evaluation in terms of several arching indexes, such as the critical height ratio (the ratio of the soil arch height to the cap clear spacing), the stress concentration ratio (ratio of stress on pile cap to on subsoil), the soil arching ratio (the ratio of stress on subsoil to geostatic earth pressure due to fill depth), the geosynthetic tension and the tensile strain, etc. These indexes can reflect the effect of arching behavior and geomembrane action well, and it is also convenient to quantify the evaluation and design reference. Definitions and expressions of these indicators can be referred to in the relevant literatures. For validation and comparison, three different types of embankments are selected for computational analysis. Occasionally, the conditions are those that would probably have encountered in real projects but herein are idealized to be cases A, B and C. Details of embankment and foundation soil, as well as the specific information of pile net design are shown in Table 1.

Table 1 Embankment geometries and fill properties

The calculation results of this presented method together with the aforementioned methods for the three cases are shown in Tables 2-4. In theory, Refs. [16, 19, 20, 21] all adopted a same critical height ratio of 1.4 to define the arch influencing scale, but utilized different stress and membrane tension computational functions to obtain the redistribution of overburden load. In general, KEMPFERT et al [17] overestimated the arching degree and yields conservative results on stress concentration ratios for cases A, B and C. The geosynthetic tensions are overestimated in Refs. [14, 20], because they neglected the reaction from the compressible soil to the geosynthetic. On the contrary, LOW et al [16] underestimated the tension of reinforcement for all the three cases. However, the method in Ref. [21] and this study obtain comparable results for cases A and B, while there is no obvious trend to follow for case C.

It seems that this method underestimates the vertical load acting on the pile, but overestimates the geosynthetic geomembrane effect. Therefore, the lowest stress concentration ratio is herein obtained, indicating that an insufficient arching effect probably develops and is more reasonable and satisfactory for the non-soft soil foundation condition. For case A, the present method obtains a similar critical height ratio with Refs. [14, 17], and prefers intermediate values to these current design methods for cases B and C. Obviously, this study obtains the highest soil arching ratios against other methods, indicating that the soil arch takes effect to a lower extent due to the partially mobilized shear stress (instead of being assumed to play its full role). This mechanism can also be supported by the presence of the minimum stresses on caps and the maximum stresses on interfacial subsoils between piles.

Table 2 Results comparison for case A with geosynthetic

Table 3 Results comparison for case B with geo synthetic

Table 4 Results comparison for case C with geo synthetic

Inevitably, there are always differences in the results between the listed methods, and there are even some contradictions between them. Through comparison, it can be concluded that this study is applicable for evaluating the arching effect and geosynthetic tension in the PGR embankment for the three cases, although there may exist unavoidable differences in some evaluation indexes. Moreover, the computational results are inclined to locate in a compromise assessment side, which is neither as high as the methods proposed in Refs. [17, 20] to estimate the pile bearing capacity, nor does it fully emphasize the geomembrane effect as method in Ref. [14]. Comparing with the current approaches, the proposed method would yield a better performance for analyzing the engineering condition of foundation soil with certain bearing capacity, and is a rational analyzing method based on the mobilized shear stress forming a developing soil arching effect.

4 Parametric analysis and discussion

Before analyzing, a baseline case should be introduced. The height of the baseline case embankment is H=7.2 m; the gravity of the embankment fill is γ=20.0 kN/m³; the internal friction angle is φ=30°; the elastic modulus is E_s=15.0 MPa. As an equivalent, a uniform load q₀=12 kPa can be used to simulate the dead and live surcharge of the traffic load action [23]. The piles are installed in square grid with a space S=2.0 m, with the pile clear spacing b=1.0 m, and the square cap width a=1.0 m. Without loss of generality, the elastic modulus of the compressible subsoil is equalized to be E_q=4.0 Mpa, and the depth is H_d=10.0 m. The ultimate tensile force of the installed single layer uniaxial plastic geogrid is T_g=120 kN/m; and the tensile stiffness is J_g=1000 kN/m; the thickness is t_g=1.0 mm, the limit tensile strain is ε_g=12%; the interaction friction of geosynthetic and embankment fill is φ_up=φ_low=25°. Using the proposed method, the convergent so-called soil arch critical height can be obtained through three to four steps of iteration, and then the calculation results are collected in Table 5.

Table 5 Calculation results of baseline case

Based on the baseline case embankment, the geometric sizes and material properties are set to vary with different design values, such as the embankment height, the internal friction angle, the compressible subsoil depth and the geosynthetic tensile stiffness. Occasionally, all the design parameters listed in Table 6 involved in this parametric analysis are those would probably encounter in real projects (see Refs. [24, 26, 27, 29, 45]). In parametric analysis, the unit weight of embankment soil remains at 20 kN/m³, and the cap width is kept as a constant unit length of 1.0 m, but with varied ratios of cap width to pile clear spacing. Then, critical soil arch height ratio, stress concentration ratio, degree of soil arching effect, geosynthetic tension and strain are investigated. At last, some design charts of parametric sensitivity are obtained, and the response analysis of parametric influences are discussed, as well as some suggestions.

Table 6 Parameter variations used for numerical analyses

4.1 Influence of embankment fill height

The embankment height directly determines the overburden load distribution or sharing proportion between pile and subsoil. Figure 8 shows the influence of fill height on critical height ratio with different ratio of pile cap width to pile clear spacing (RPC). Obviously, the critical height ratio increases with increasing embankment height and increasing RPC. Whether reinforced or not, such growth will have a limit of no more than 1.5 for all cases, which is slightly larger than the traditional assumption of 1.4. Moreover, the reinforced case obtains lower critical height ratios against the unreinforced case, with a reduction range of 2.4% to 8.5% due to the geomembrane effect. Indeed, the existence of reinforcement has weakened the influence area of soil arching.

Inclusion of a geosynthetic layer helps to transfer part of the overburden load from compressible subsoil to adjacent piles (caps). In Figure 9, the stress concentration ratio is directly dependent on the embankment height, whether reinforced or not. It is obvious that a higher stress concentration ratio is obtained with a higher RPC and with a higher embankment. If the stress on a cap is converted to the cross section of the pile shaft, the actual stress concentration ratios are between 5 and 20. However, with increasing embankment height, the growth rate of stress concentration ratio of the reinforced embankment is superior to the unreinforced case and it also applies to different RPCs. The change law can also be seen in the numerical simulation results conducted by HAN et al [23] and the field monitored results carried out by CHEN et al [18]. Because of the geosynthetic reinforcement, the stress concentration ratio increases from 10.1% to 36.0% for these embankments.

With increasing embankment fill height, the load acting on pile (cap) and compressible subsoil becomes more and more susceptible to soil arching effect. So, the soil arching ratio, defined as the ratio of actual vertical stress to geostatic earth pressure, is adopted to evaluate the arching degree. In Figure 10, when the embankment height increases, all soil arching ratios will develop to a limit value range, which is 0.45 to 0.52 for unreinforced cases and 0.31 to 0.47 for reinforced cases. With the help of reinforcement, the maximum increase of soil arching ratio is even up to 30% for all the current cases. However, higher soil arching ratio represents lower degree of arching completion, indicating that the geosynthetic reduces the shear stress within the arched embankment, but does not affect the efficiency of load borne by piles. Through inspecting many field studies, the measured soil arching ratio always stays between 0.4 and 0.5 with the ultimate embankment height of 6 m or higher, which is consistent with many researches (see Refs. [24, 26, 27, 29, 31]). This also indirectly proves the rationality of the proposed method.

Figure 8 Influence of embankment height on critical soil arch height ratio without geosyntehtic (a) and with geosyntehtic (b)

Figure 9 Influence of embankment height on stress concentration ratio without geosyntehtic (a) and with geosyntehtic (b)

Figure 11 shows how the embankment height influences the geosyntehtic tension and tensile strain with different RPCs. The higher the embankment is, the larger the tension and tensile strain of the reinforcement are obtained. Altogether, with decreasing pile cap width, the increasing rate of geosyntehtic tension and tensile strain becomes larger. The same condition also applies to the gap variation of tension or tensile strain influenced by increasing embankment height. Coincidentally, the conclusion is consistent with the study of Ref. [19], and the latter even gains a larger and exaggerated gap.

4.2 Influence of internal friction angle

In the backfilling engineering, there exists a universe mechanism of arching behavior. The Terzaghi theory has been widely used or modified for load evaluation within the soil and structures. According to Ref. [21], the arching model of this study belongs to the typical frictional arching model. Therefore, the development of Terzaghi theory reveals that a product of the coefficient of earth pressure governs the stress state in the soil medium. SINGH et al [46] argued that there is no need to investigate the backfill friction angle laboriously, because the earth pressure or frictional coefficient only changes slightly versus the change of frictional angle. However, whether this is applicable for the partially developed soil arching model of a PGR embankment needs further study.

Figure 10 Influence of embankment height on degree of soil arching effect without geosyntehtic (a) and with geosyntehtic (b)

Figure 11 Influence of embankment height on tension (a) and axial strain (b)

The variation of the critical height ratio versus the internal friction angle is illustrated in Figure 12. Obviously, the critical height ratio increases with increasing internal friction angle, accompanying with average increments of 30% for the unreinforced embankment and 45% for the reinforced case. Because of the installed geosynthetic, the critical height ratio becomes smaller. Although the influence is not significant, it cannot be neglected. As shown in Figure 13, the same condition is also applicable to the stress concentration ratio. Whereas, the geosynthetic enlarges the gap between stress concentration ratios for all the presented cases. With the help of geosynthetic, the stress concentration ratio increases substantially, especially for the RPCs of a:b=1:0.8 and 1:1.0.

On the contrary, the soil arching ratio, the geosynthetic tension and tensile strain decrease sharply with increasing internal friction angle in this study. Figure 14 shows that the geosynthetic layer reduces the gap of soil arching ratios at different RPCs. As shown in Figure 15, whether for tension or tensile strain, when the angle of internal friction increases from 15° to 35°, the maximum decrease reaches nearly 40%. However, the difference of geosyntehtic tension and tensile strain is not significant and is not sensitive to the RPC. In a word, the aforementioned analysis shows that the influence of the angle of internal friction on arching behavior cannot be neglected, which is against the conclusion drew by SINGH et al [46]. Therefore, engineers should pay more attention to direct effort towards the selection of the backfill material.

4.3 Influence of compressible subsoil depth

The subsoil depth herein can be considered to be the effective thickness of the compressible foundation soil within the pile length. In other words, the subsoil depth determines the magnitude of compression of the interstitial foundation soil, i.e. the deflection of the basal geosynthetic reinforcement. Although, the critical height ratio increases with increasing compressible subsoil depth (see Figure 16), the growth also has a threshold for all RPCs. The geosynthetic significantly reduces the increase trend when compared to the normal piled embankment. Occasionally, a higher RPC obtains a larger critical height ratio, indicating that more overburden load is transferred to the interfacial subsoil. Thus, insufficient arching degree takes place in case of a smaller RPC happens.

Figure 12 Influence of internal friction angle on critical soil arch height ratio without geosyntehtic (a) and with geosyntehtic (b)

Figure 13 Influence of internal friction angle on stress concentration ratio without geosyntehtic (a) and with geosyntehtic (b)

Figure 14 Influence of internal friction angle on degree of soil arching effect without geosyntehtic (a) and with geosyntehtic (b)

Figure 15 Influence of internal friction angle on tension (a) and axial strain (b)

Different from the findings of Ref. [19], the stress concentration ratios calculated by the present method increase with increasing subsoil depth with or without the geosynthetic layer (see Figure 17). The geosynthetic reinforcement significantly increases the increasing rate and enlarges the gap of stress concentration ratios for all presented RPCs. The larger the RPC, the higher the stress concentration ratio. On the contrary, increasing subsoil depth decreases the soil arching ratio, but the effect is weakened due to the appearance of geosynthetic reinforcement (see Figure 18). When the subsoil depth is small, the influence of geomembrane effect on soil arching is small, indicating that the compression of subsoil or the deflection of geosynthetic is at a minimal magnitude when compared to the maximum depth case. The ultimate state of the arching effect is achieved when the subsoil depth is 20 m, with soil arching ratios of 0.24 and 0.48 for the unreinforced cases of a: b=1:0.8 and 1:2.0, as well as 0.44 and 0.56 for the reinforced cases of a: b=1:0.8 and 1:2.0, respectively.

Figure 19 shows that the effect of subsoil depth on geosynthetic tension and tensile strain is significant. The geosynthetic tension and tensile strain increase sharply with increasing compressible subsoil depth, i.e. larger relative settlement occurs between pile and subsoil. With a higher RPC, the tension or tensile strain reaches the critical value faster while the corresponding subsoil depth is smaller. The general rule of change analyzed in this study is also similar to that obtained by Ref. [19]. It can also be noted that the geosynthetic tension and tensile strain increase with increasing RPC when the subsoil depth is less than 9 m, and the opposite result is obtained when the subsoil depth is larger than 13 m.

Figure 16 Influence of compressible subsoil depth on stress concentration ratio without geosyntehtic (a) and with geosyntehtic (b)

Figure 17 Influence of compressible subsoil depth on soil arching ratio without geosyntehtic (a) and with geosyntehtic (b)

Figure 18 Influence of compressible subsoil depth on tension (a) and axial strain (b)

4.4 Influence of geosynthetic tensile stiffness

To date, few studies can be referenced for evaluating the effect of reinforcement on arching action. One important reason is that the most current evaluation methods are developed based on the two-step method in Refs. [17, 19, 20, 21]. Independently, stresses on the pile head or the interfacial soil is firstly calculated by arching, and then the geomembrane tension is proportionally calculated out according to a given overburden load. Although there is a consensus that inclusion of geosynthetic will weaken the soil arching effect [23, 35, 47], the quantitative relationship is still uncertain and a definite mathematical solution is difficult to obtain. Therefore, this study refined the isolated two-step method by combining the effects of arching and geomembrane, and derived an iterative method for calculating the soil arch height instead of being presumed.

The influence of geosynthetic tensile stiffness on critical height ratio with different RPCs is shown in Figure 20. Obviously, the critical height ratio decreases with increase of geosynthetic tensile stiffness, but the effect becomes weaker when the stiffness exceeds 2000 kN/m in this case study. Moreover, with a smaller RPC than a:b=1:1.0, the critical height ratio has a low sensitivity to the tensile stiffness, and the same condition applies to the stress concentration ratio, too. As illustrated in Figure 21, although the stress concentration ratio increases with increasing geosynthetic strength, the degree of influence for different RPC is different.

Figure 19 Influence of geosynthetic tensile stiffness on critical soil arch height ratio

Figure 20 Influence of geosynthetic tensile stiffness on stress concentration ratio

Overall, a tensile stiffness 2000 kN/m of the geosyntehtic and a RPC a:b=1:1.0 are the sensitive design parameters for the PGR embankment in this study.

The inhibitory effect on arching behavior due to geomembrane can also be reflected by the soil arching ratio (see Figure 22). Obviously, the soil arching ratio increases with the increase of geosynthetic tensile stiffness. The higher the RPC, the smaller the soil arching ratio, and is more sensitive to the geosynthetic strength. When the RPC is at a:b=1:0.8, the soil arching ratio is increased by 0.3 at most due to the geomembrane effect. Moreover, even 80% of the increment has been accomplished with the geosynthetic tensile stiffness no more than 2000 kN/m. Smaller soil arching ratio means that the arching effect is developed into a higher degree, and larger soil arching ratio takes the lower arching effect. As shown in Figure 22, the stronger the geosynthetic, the larger the tension, and the smaller the tensile strain. This is because a stiffer geosynthetic promotes a slower mobilization of the tensile strength, and therefore contributes to a lower tensile strain in the geosynthetic. In other words, selection of the geosynthetic should also consider the magnitude of the tensile stiffness and the induced tensile strain.

Figure 21 Influence of geosynthetic tensile stiffness on soil arching ratio

Figure 22 Influence of geosynthetic tensile stiffness on tension (a) and axial strain (b)

5 Conclusions

1) An arching evaluating method, based on partially mobilized shear stress, was presented for the PGR embankment, as well as the verification and parameter sensitivity analysis. In derivation, the developing soil arching effect was utilized to investigate the coupling effects of arching and geomembrane by an iterative calculation process.

2) Parametric study indicates that the stress concentration ratio always increases with the increasing parameters, while the soil arching ratio decreases except for the increasing geosynthetic tensile stiffness. Although inclusion of geosynthetic has a significant weakness on soil arching, it improves the bearing efficiency of the pile.

3) Some conclusions of the parametric study are in good agreement with the existing studies. To the partially developed arching, the internal friction angle has a non-negligible influence on all the calculated indexes, which extends the research work conducted by SINGH et al [46] for the piled and geosynthetic reinforced embankment.

4) If only the one-way force in a single layer reinforcement and the two-dimensional arching state analysis are considered, it is preliminarily suggested that a geosynthetic tensile stiffness of 2000 kN/m and a RPC of 1:1.0 are the sensitivity thresholds for the piled and geosynthetic-reinforced embankment. Further measurements and data verification are also needed in the future study.

Nomenclature

a

Pile cap width, m

b

Pile clear spacing, m

C₀

Constant dependent on the critical soil arch height

E_q

Elastic modulus of the compressible subsoil, MPa

E_s

Elastic modulus of the embankment fill, MPa

F_d

Concentrated force on top of the arch, kN

h

Soil arch height, m

H

Embankment fill height, m

H_d

Subsoil depth, m

J_g

Geosynthetic tensile stiffness, kN/m

K_a

Active earth pressure coefficient

K

Earth pressure coefficient

n

Stress concentration ratio

q

Uniform load acted on the deflected geosynthetic, kPa

q₀

Uniform static traffic load acted on the embankment surface, kPa

S

Pile spacing, m

t_g

Average geosynthetic thickness, mm

T₀

Geosynthetic horizontal component force, kN/m

T_g

Geosynthetic tensile force, kN/m

T_M

Geosynthetic tensile force at point M, kN/m

β

Friction angle due to mobilized shear stress, (°)

γ

Unit weight of the embankment fill, kN/m³

δ_g

Deflection of the geosynthetic at the mid, mm

ε_g

Tensile strain of the geosynthetic, %

λ

Ratio of deflection to clear subsided width

ρ

Soil arching ratio

σ_H

Vertical stress, kPa

σ_V

Horizontal stress, kPa

σ_V0

Vertical stress at the elevation of the soil arch height, kPa

σ_maj

Major principal stress, kPa

σ_min

Minor principal stress, kPa

Vertical stress on pile head (or cap), kPa

Horizontal pressure acting on the centerline of the pile cap, kPa

Horizontal pressure acting on the centerline between two piles, kPa

σ_s

Resistance of the compressible subsoil, kPa

τ_s

Shear stress in the embankment fill, kPa

φ

Internal friction angle of embankment fill, (°)

φ_sub

Internal friction angle of subsoil, (°)

φ_c

Deflection angle of force to the horizontal direction at position C, (°)

φ_up

Friction angle of upper contact interface between geosynthetic and soil, (°)

φ_low

Friction angle of lower contact interface between geosynthetic and soil, (°)

ψ

Coefficient of shear stress and vertical stress

f

Angle of σ_maj related to the horizontal direction, (°)

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(Edited by ZHENG Yu-tong)

中文导读

基于剪应力发展型桩承加筋路堤的渐进土拱效应评价

摘要：在桩承式加筋路堤中，土拱效应影响着桩与地基土之间的荷载分担。土工加筋对减少桩土差异沉降作用显著，故路堤中剪应力未能达到临界水平，此时桩承路堤中发生的是部分发展的渐进土拱效应，而大部分现有研究方法由于采用了极限土拱效应原理进行路堤荷载分担计算而无法适用。因此，本文提出了一种基于剪应力发展型土拱效应的桩承加筋路堤荷载效应评价模型并对其合理性进行了验证，系统分析了路堤高度与内摩擦角、地基土厚度、桩帽宽与净距比、土工加筋抗拉刚度等设计参数对拱跨比、桩土应力比、土拱度、加筋拉应力和拉应变的影响敏感性。结果表明，桩帽宽与净距比为1:1.0且抗拉刚度为2000 kN/m的单层单向加筋是桩承式加筋路堤的设计敏感阈值。

关键词：桩；土工加筋；土拱效应；发展剪应力；参数分析

Foundation item: Project(51508279) supported by the National Natural Science Foundation of China; Project(KFJ170104) supported by the Open Fund of National Engineering Laboratory of Highway Maintenance Technology of Changsha University of Science & Technology, China; Project(BK20150885) supported by the Jiangsu Provincial Natural Science Fund,China; Project(2019003) supported by the Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering of Hohai University, China

Received date: 2020-03-26; Accepted date: 2020-05-20

Corresponding author: LV Wei-hua, PhD, Associate Professor; Tel: +86-13776673646; E-mail: whlnjfu@njfu.edu.cn; ORCID: 0000- 0003-3041-2041