# Unidirectional Geosynthetic Reinforcement Design for Bridging Localized Sinkholes in Transport Embankments

Authors: Peinan Li; Feng Su
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The durability of foundations with localized sinkholes can be improved by geosynthetic-reinforced soil. Current design methods for geosynthetic-reinforced structures are based on the assumptions that the vertical load is distributed uniformly on the geosynthetic above the sinkhole and it corresponds to the geostatic stress in the anchorage area. In this study, a new analytical method is proposed to consider the ‘secondary arching effect’ and the vertical load distribution in the anchorage area due to the ‘arching effect.’ The influence of vertical load acting on the geosynthetic is analyzed based on three different cases. Results indicate that the increment of vertical load on the geosynthetic in the anchorage area has little effect on the maximum tensile force and surface settlement. Compared to a uniform load distribution on the geosynthetic above the sinkhole, the assumption of an inverse triangular load distribution can reduce the maximum tensile force and surface settlement. A new design method is proposed to determine the minimum geosynthetic stiffness to meet design standards. The obtained results verify the method as an applicable tool to meet the serviceability limit state and the ultimate limit state criteria.


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## Body

## 1. Introduction

With the development of infrastructure in China, a massive number of highways and railways are built in karst areas. However, some underground cavities are neither readily detected nor do they appear after the construction of the roads, especially in the case of small cavities. Sinkholes usually result from the collapse of underground cavities. Transport routes are damaged when the underground cavities collapse suddenly. Therefore, it is necessary to reinforce roads to prevent localized sinkholes. Due to the cost-effectiveness and installation convenience, reinforcing embankments with geosynthetics is an attractive technique [1, 2, 3]. Reinforced soil can improve the service performance of buildings or infrastructure, consequently improving the toughness of urban structures.Experimental studies [4, 5, 6, 7, 8] have indicated that geosynthetics, if properly selected, protect embankments from small cavities, and unidirectional geosynthetics with machine direction oriented along the direction of traffic are the most appropriate. Based on simple assumptions, geosynthetic shape, surface settlement, and tension within the geosynthetic can be calculated theoretically and design methods are proposed (BS8006 2010) [9, 10, 11].The BS8006 (2010) design method is based on the tensioned membrane theory only. Combining tensioned membrane theory with arching theory, Giroud et al. [12] proposed a design method to span voids for soil layer-geosynthetic systems. Blivet et al. [13] adopted the ‘RAFAEL’ method to take the expansion of soil into account further, obtaining the relationship between surface settlement and geosynthetic deflection. These methods apply the tensioned theory to describe the membrane effect of the geosynthetic located above the cavity (implying the fixity of geosynthetic at the edge of the cavity). Villard and Briancon [7] considered the geosynthetic behaviors in the anchorage area and at the edge of the cavity.A key issue is a load imposed on the geosynthetic. If the embankment is high enough, the ‘arching effect’ will occur within the embankment when the underground cavity underneath the embankment collapses. The vertical stressq acting on the portion of geosynthetic located above the sinkhole decreases and becomes smaller than the sum of the vertical geostatic stress γH and the surcharge qs applied on the embankment surface q<γH+qs. The vertical stress qa acting on the portion of geosynthetic in the anchorage area increases and exceeds γH+qs. Most of the design methods considered the arching effect but none of them investigated the influence of the vertical load distribution on geosynthetic in the anchorage area owing to the arching effect. These design methods assume the uniform distribution of q; however, experiments reveal that q is not uniform [14, 15, 16, 17] for the nonuniform vertical displacement of the geosynthetic above the sinkhole.To improve the current design methods, a new approach is developed. An inverse triangular distribution forq and a Gaussian distribution for qa are applied based on experimental results [18, 19, 20, 21]. The influence of different distribution forms of q as well as qa on the mobilized tension and surface settlement is analyzed using different methods. Finally, analytical formulations and design methods for long and circular voids are proposed.


## 2. Arching Effect

### 2.1. Vertical Load on the Geosynthetic in the Collapse Area

The stress transfer, owing to relative movement within the soil, is commonly defined as the arching effect [22]. Embankment fills and underlying geosynthetic layers are generally located on firm soils. The vertical stress then equates to the geostatic stress of the fills and surcharge. If sinkholes below the geosynthetic collapse (after the construction of the geosynthetic-reinforced embankment), as illustrated in Figure 1, the geosynthetic deflects and the relative movement within the soil occurs between collapse and anchorage areas subjected to a vertical load. A portion of the vertical stress on the geosynthetic in the collapse area transfers to the anchorage area. Thus, the vertical stress on the geosynthetic in the collapse area decreases. Based on the assumptions that the vertical load distributes uniformly and slip planes are vertical, the vertical load on the geosynthetic in the collapse area can be calculated using the Terzaghi method (1943).Figure 1
Vertical loads on the geosynthetic for a long void.For a long void of widthB,(1)q=Bγ−2c/B2Ktanφu1−e−Ktanφu2H/B+qse−2KtanφuH/B.For a circular void of diameterD,(2)q=Dγ−4c/D4Ktanφu1−e−Ktanφu4H/D+qse−4KtanφuH/D.The load on the geosynthetic in the collapse area can be calculated conservatively for cohesive and cohesionless soils (c=0), using (1) and (2) [21]. Experimental data indicates that the value of K is larger than 1 [19, 22, 23]. Some design methods indicate that K is replaced by the active Earth pressure coefficient Ka [13, 24, 25].(3)Ka=tan245∘−φu2.Since the deflection of the geosynthetic is not uniform, the vertical stress on geosynthetic above the sinkhole redistributes in response to relative displacement between fills above the sinkhole. The portions of the geosynthetic close to the edge of the sinkhole deflect less and bear more vertical loads. Sloan [14] defined this phenomenon as a ‘secondary arching effect’. For the collapse area, an inverse triangular distribution is adopted to describe the stress redistribution on the geosynthetic [26].The ‘secondary arching effect’ changes the distribution of vertical stress; however, the magnitude of the average load and the total vertical forceQ on the geosynthetic in the collapse area do not undergo any changes. Thus, Q in the cases of cohesive and cohesionless soils can be approximately calculated by multiplying the vertical load by the loading area as given as follows.Long void:(4)Q=qB=B2γ−2c/B2Katanφu1−e−Katanφu2H/B+Bqse−2KatanφuH/B.Circular void:(5)Q=qπD24=πD3γ−4c/D16Katanφu1−e−Katanφu4H/D+πD24qse−4KatanφuH/D.


### 2.2. Vertical Load on the Geosynthetic in the Anchorage Area

Adachi et al. [18], Jia [19], and Gao [20] measured the vertical stress qa on the fixed support through two-dimensional trapdoor experiments. The vertical stress on the fixed support increased when the trapdoor lowered. The Earth pressure increasing rate qa−q0/q0 was used to represent the increment of qa. Experimental data demonstrates that the Earth pressure increasing rate shows close correlations to the standardized distance from the pressure gauge to the centerline of the descending trapdoor by the width of the trapdoor (x/B) and the value of H/B. This experimental data is summarized in Figure 2. Figure 2 show that the value of qa−q0/q0 decreases rapidly with the increment of x/B. The decrease of qa−q0/q0 is more apparent with a smaller value of H/B. The relationship between qa−q0/q0, x/B, and H/B can be approximately described by a Gaussian distribution function from Figure 2 as(6)qa−q0q0=a112πσexp−x/B−0.522σ2.Figure 2
Comparison of the approximated and experimental results of the increments in Earth pressure outside the trapdoor. (a) Adichi (2003), (b) Jia (2007), and (c) Gao (2009).
(a)(b)(c)The vertical load on the fixed support can be calculated by(7)qa=a12πσexp−x/B−0.522σ2+q0,where σ is a function of H/B and the properties of the fills, a1=a/q0, where a is a coefficient determined by(8)∫B/2+∞qa−q0dx=B2q0−q.Thus,(9)a=q0−q.Substituting (9) into the (7), the following expression is obtained.(10)qa=q0−q12πσexp−x/B−0.522σ2+q0.Both the experimental results and the estimated results are illustrated in Figure2. The smaller the value of H/B is, the better agreement between the approximate results and the experimental data will be. Furthermore, there is a great difference between experimental and estimated results if H/B >4. Therefore, for long voids and H/B  = 1–4, the vertical load on geosynthetic in the anchorage area can be estimated by the proposed equation with the proper value of σ.


## 2.1. Vertical Load on the Geosynthetic in the Collapse Area

The stress transfer, owing to relative movement within the soil, is commonly defined as the arching effect [22]. Embankment fills and underlying geosynthetic layers are generally located on firm soils. The vertical stress then equates to the geostatic stress of the fills and surcharge. If sinkholes below the geosynthetic collapse (after the construction of the geosynthetic-reinforced embankment), as illustrated in Figure 1, the geosynthetic deflects and the relative movement within the soil occurs between collapse and anchorage areas subjected to a vertical load. A portion of the vertical stress on the geosynthetic in the collapse area transfers to the anchorage area. Thus, the vertical stress on the geosynthetic in the collapse area decreases. Based on the assumptions that the vertical load distributes uniformly and slip planes are vertical, the vertical load on the geosynthetic in the collapse area can be calculated using the Terzaghi method (1943).Figure 1
Vertical loads on the geosynthetic for a long void.For a long void of widthB,(1)q=Bγ−2c/B2Ktanφu1−e−Ktanφu2H/B+qse−2KtanφuH/B.For a circular void of diameterD,(2)q=Dγ−4c/D4Ktanφu1−e−Ktanφu4H/D+qse−4KtanφuH/D.The load on the geosynthetic in the collapse area can be calculated conservatively for cohesive and cohesionless soils (c=0), using (1) and (2) [21]. Experimental data indicates that the value of K is larger than 1 [19, 22, 23]. Some design methods indicate that K is replaced by the active Earth pressure coefficient Ka [13, 24, 25].(3)Ka=tan245∘−φu2.Since the deflection of the geosynthetic is not uniform, the vertical stress on geosynthetic above the sinkhole redistributes in response to relative displacement between fills above the sinkhole. The portions of the geosynthetic close to the edge of the sinkhole deflect less and bear more vertical loads. Sloan [14] defined this phenomenon as a ‘secondary arching effect’. For the collapse area, an inverse triangular distribution is adopted to describe the stress redistribution on the geosynthetic [26].The ‘secondary arching effect’ changes the distribution of vertical stress; however, the magnitude of the average load and the total vertical forceQ on the geosynthetic in the collapse area do not undergo any changes. Thus, Q in the cases of cohesive and cohesionless soils can be approximately calculated by multiplying the vertical load by the loading area as given as follows.Long void:(4)Q=qB=B2γ−2c/B2Katanφu1−e−Katanφu2H/B+Bqse−2KatanφuH/B.Circular void:(5)Q=qπD24=πD3γ−4c/D16Katanφu1−e−Katanφu4H/D+πD24qse−4KatanφuH/D.


## 2.2. Vertical Load on the Geosynthetic in the Anchorage Area

Adachi et al. [18], Jia [19], and Gao [20] measured the vertical stress qa on the fixed support through two-dimensional trapdoor experiments. The vertical stress on the fixed support increased when the trapdoor lowered. The Earth pressure increasing rate qa−q0/q0 was used to represent the increment of qa. Experimental data demonstrates that the Earth pressure increasing rate shows close correlations to the standardized distance from the pressure gauge to the centerline of the descending trapdoor by the width of the trapdoor (x/B) and the value of H/B. This experimental data is summarized in Figure 2. Figure 2 show that the value of qa−q0/q0 decreases rapidly with the increment of x/B. The decrease of qa−q0/q0 is more apparent with a smaller value of H/B. The relationship between qa−q0/q0, x/B, and H/B can be approximately described by a Gaussian distribution function from Figure 2 as(6)qa−q0q0=a112πσexp−x/B−0.522σ2.Figure 2
Comparison of the approximated and experimental results of the increments in Earth pressure outside the trapdoor. (a) Adichi (2003), (b) Jia (2007), and (c) Gao (2009).
(a)(b)(c)The vertical load on the fixed support can be calculated by(7)qa=a12πσexp−x/B−0.522σ2+q0,where σ is a function of H/B and the properties of the fills, a1=a/q0, where a is a coefficient determined by(8)∫B/2+∞qa−q0dx=B2q0−q.Thus,(9)a=q0−q.Substituting (9) into the (7), the following expression is obtained.(10)qa=q0−q12πσexp−x/B−0.522σ2+q0.Both the experimental results and the estimated results are illustrated in Figure2. The smaller the value of H/B is, the better agreement between the approximate results and the experimental data will be. Furthermore, there is a great difference between experimental and estimated results if H/B >4. Therefore, for long voids and H/B  = 1–4, the vertical load on geosynthetic in the anchorage area can be estimated by the proposed equation with the proper value of σ.


## 3. Analysis of Reinforced Embankment Bridging a Long Void

### 3.1. Geosynthetic Behavior in the Collapse Area

The behavior of the geosynthetic is assumed to be linear elastic:T=Jε. Half of the void is taken into consideration. The frictional forces between the fills and the geosynthetic in the collapse area are neglected.Considering an infinitesimal elementdx of the geosynthetic in the collapse area (Figure 3), the equilibrium equations of vertical and horizontal forces give(11)qx=dTVdx,(12)dTH=0,Figure 3
Schematic diagram of forces within the geosynthetic in the collapse area.For an infinitesimal elementdx,(13)TV=−THdzdx.Thus,(14)dTVdx=−dTHdxdzdx−THd2zdx2.Combining equations (11), (12), and (14), the following expression is obtained.(15)qxTH=−d2zdx2.Inverse triangular load distribution on the geosynthetic in the collapse area is given.(16)qx=4QB2x.Combining (15) and (16), the following expression is obtained.(17)4QTHB2x=−d2zdx2.Solving (17) subjected to the boundary conditions that z=0 for x=B/2 and z′=0 for x=0, one can obtain(18)z=−2Q3THB2x3+QB12TH.At the edge of the cavity (point A), the tensile force within the geosynthetic reaches the maximum valueT1.(19)T1=TH2+Q24.Substitutingx=0 into (18), the geosynthetic deflection w can be written as(20)w=QB12TH.Using (18), the increase in geosynthetic length ΔL on the half-width of the cavity can be calculated as follows:(21)ΔL=∫ds−B2=∫0B/21+z′2dx−B2=−B2+112B4+Q2TH2+4M−Q24TH2.where M−t is a hypergeometric function in the Wolfram Mathematica package, which can be expanded by the Maclurin series as(22)M−t=1−t10+t224−5t3208+35t42176−3t5256+231t625600−429t759392+195t832768+Ot9.The first three Maclurin series terms ofM−t are precise enough under the situation that Q/2TH<1 for w/B<1/6. Equation (21) can be rewritten as(23)ΔL=−B2+112B4+Q2TH2+4−Q210TH2+Q496TH4.The geosynthetic tension deformationχ on the half-width of the cavity can be calculated as follows:(24)χ=∫εds=∫0B/2ε1+z′2dx=THJB2+BQ240TH2.Combining (23) and (24), the displacement of point A UA can be written as(25)UA=ΔL−χ=−B2+112B4+Q2TH2−Q210TH2+Q496TH4−THJB2+BQ240TH2.


### 3.2. Surface Settlement

In Figure4, the subsidence will rise rapidly to the surface of the embankment when the geosynthetic deflects. The soil above the void expands in volume thus indicating dilation. Thus, surface displacement is less than geosynthetic displacement in the vertical direction. Assuming that the shape of surface settlements and the geosynthetic sag are the same (cubical parabola) which is just a first approximation, combined with (20), the relation among surface settlement ws, deflection of geosynthetics w, soil expansion coefficient Ce, and height of the fill H can be obtained [6, 24, 27]:(26)ws=w1−18THQHBCe−1.Figure 4
Schematic diagram of the surface settlement.


### 3.3. Geosynthetic Behavior in the Anchorage Area

The friction laws at the soil/geosynthetic interfaces are supposed to be the Coulomb friction law in Figure5. In Figure 6, the tensile force within the geosynthetic decreases from T1 to T2 at the edge of the cavity based on the limit equilibrium of segment of geosynthetic sheet acting by friction on a circular arc (7) and (27).(27)T2=T1e−βk0αtanφl,where k0=1 for UA≥U0, k0=UA/U0 for UA<U0; and tanβ=z′B/2=Q/2TH.Figure 5
Coulomb friction law.Figure 6
Edge of the void.In Figure7, the geosynthetic AC in the anchorage area is divided into n elements evenly, and the length of each element equals Δx. A horizontal stress is caused by the friction between the geotextile and the upper layer; however, the influence is not very high, so this horizontal stress can be neglected. Thus, the shear stress on the upper and lower interfaces at any element of the geosynthetic is expressed as(28)τu=KUqaαtanφu,τl=KUqaαtanφl,where KU  = 1 for Uak≥U0 or KU=Uak/U0 for Uak<U0.Figure 7
Schematic diagram of the forces within the geosynthetic in the anchorage area.The equilibrium of the elementk is expressed as(29)Tak+1=Tak−KUqaαtanφu+tanφlΔx,(30)Uak+1=Uak−Tak+1JΔx.The equilibrium of the point A (k=0) is given as(31)T2=Ta0,UA=Ua0.At the end of the geosynthetic (point C), the tensile force within the geosynthetic equals 0; that is,(32)Tak|k⟶n=Tac=0.Combining equations (29) and (30) with the boundary conditions that Tac=0, Ta0=T2 and UA=Ua0, the system can be solved by an iterative calculation with the following equations.(33)Tak+1=Tak−KUqaαtanφu+tanφlΔx,Uak+1=Uak−Tak+1JΔx.


### 3.4. Determination ofT1 and ws Based on an Example

To be more specific, an example is taken herein to elaborate the calculation procedure of the maximum tensile force and surface settlement. The relevant parameters used in the calculation example are given in Table1.Table 1
Parameters in the calculation example.
ParameterValueCavity width (B)2 (m)Fill soil thickness (H)4 (m)Internal friction angle of the fill soil (φl)35 (°)Internal friction angle of the foundation soil(φu)35 (°)Unit weight of fill soil (γ)20 (kN/m3)Displacement of the geosynthetic at the point a (U0)10 (mm)Frictional coefficient (α)0.9Expansion coefficient of the fill soil (Ce)1.04Anchorage length (Lac)3 (m)Vertical overload applied on the fill soil surface (qs)0From equations (1), (3), and (4), we obtain q=56 kPa and Q=112 kN. Let q0=280 kPa; the value of q/q0 is 0.2. Based on Figure 2(b), σ2 is assumed to be 0.4. From equations (10), (19), and (25), we can obtain(34)qa=15exp−x/2−0.520.8+280,T1=TH2+3136,UA=−13+164+1122TH2−112210TH2+112496TH4−TH20001+627TH2.LetΔx  = 0.001 m. From (33) with Ta0=T2 and UA=Ua0, the relationship between Tac and TH is obtained in Figure 8. When Tac=0, we obtain TH= 74.7 kN/m and T1=93 kN/m.Figure 8
Relationship betweenTac and TH.Combining (20) and (26), we obtain ws  = 37 mm.


### 3.5. Influence of Vertical Loads Applied to the Geosynthetic

To investigate the influence of the form of vertical load distribution and the increment of vertical load in the anchorage area on the maximum tensile force and surface settlement, three calculation models are adopted for comparison.Model 1: an inverse triangular load distribution on the geosynthetic in the collapse area and a Gaussian load distribution on the anchorage area (Figure9(a)).Model 2: an inverse triangular load distribution on the geosynthetic in the collapse area and a uniform load distribution on the anchorage area (Figure9(b)).Model 3: a uniform load distribution on the geosynthetic in the collapse area and anchorage area (Figure9(c)).Figure 9
Schematic diagram of the vertical loads on geosynthetics for three calculation models: (a) model 1, (b) model 2, and (c) model 3.
(a)(b)(c)Figures10, 11, 12, 13, 14, 15, and 16 illustrate the variation tendency of the maximum tensile force and surface settlement against various parameters. From the holistic perspective, the same variation tendency of the maximum tensile force and surface settlement can be observed from these three models in Figures 10–16. It can be seen that the surface settlement calculated by model 1 is the lowest and the maximum tensile force calculated by model 2 is the smallest. Apart from that, compared with a uniform load distribution on the geosynthetic in the collapse area in model 3, an inverse triangular load distribution used in model 1 and model 2 induces much smaller maximum tensile force and surface settlement, indicating that a uniform load distribution in the collapse area will overestimate the tensile force of geosynthetic and surface settlement of the embankment. Meanwhile, the vertical stress at the edge of the void in model 1 is larger than that in model 2 because of different load distributions in the anchorage area used in these two models. Generally, large vertical stress at the edge of the void will lead to less elongation of the geotextile. In consequence, a smaller surface settlement in model 1 should be observed. Unexpectedly, calculation results show that little discrepancy exists, which means the increment of vertical stress in the anchorage area due to the ‘arching effect’ has little effect on both tensile force and surface settlement. Hence, model 2 is considered the best calculation model for its accuracy and simplicity.Figure 10
Influence of the width of the void on the maximum tensile force and surface settlement.Figure 11
Influence of the height of the embankment on the maximum tensile force and surface settlement.Figure 12
Influence of the tensile stiffness of the geosynthetic on the maximum tensile force and surface settlement.Figure 13
Influence of the internal friction angle on the maximum tensile force and surface settlement.Figure 14
Influence ofU0 on the maximum tensile force and surface settlement.Figure 15
Influence ofLac on the maximum tensile force and surface settlement.Figure 16
Influence of the expansion coefficient on the maximum tensile force and surface settlement.From the perspective of different parameters, all parameters are insensitive to the load distribution on the geosynthetic in the anchorage area but susceptible to the load distribution on the geosynthetic in the collapse area, which demonstrates the importance of the form of load distribution in the collapse area in practical engineering. As expected, Figure10 shows that the size of cavity B has the most significant effect on the maximum tensile force and surface settlement. Therefore, it requires that the size of the underground cavity should be known as precisely as possible. It can be seen from Figures 11–14 that the maximum tensile force and the maximum surface settlement have the opposite trends as an increase in the height of embankment H, the tensile stiffness of the geosynthetic J, internal friction angle φu, and relative displacement U0, respectively. Furthermore, φu and U0 have little effect on the tensile force and surface settlement, compared to H and J. Figure 15 indicates that the anchorage length Lac has no effect on the maximum tensile force and surface settlement when Lac exceeds 1 m. Hence, the influence of the anchorage length can be neglected for that the anchorage length is long enough in practice. In spite of difficult determination to the expansion coefficient α in practice, the results indicate that α only affects the surface settlement. Consequently, the maximum tensile force remains constant and an increase in α involves a decrease in the maximum surface settlement, as shown in Figure 16.


### 3.6. Analytical Formulation

Based on the analysis above, the vertical stress on geosynthetic in the anchorage area can be simplified with the initial uniform loadq0 instead of a Gaussian load distribution.Thus, the analysis of geosynthetic in the anchorage can be simplified with Briancon’s method [23].(35)ifUA≤U0:UA=T2Jr,ifUA>U0:UA=U0+T22−T022Jτ0.Withτ0=αq0tanφu+tanφl, r2=τ0/JU0, and T0=JrU0, T2=TH2+Q2/4/eβk0αtanφl.Combining (25) and (35), the solution of the system is(36)ifUA≤U0:T2Jr=−+112B4+Q2TH2−Q210TH2+Q496TH4−THJB2+BQ240TH2,ifUA>U0:U0+T22−T022Jτ0=−B6+112B4+Q2TH2−Q210TH2+Q496TH4−THJB2+BQ240TH2.The solution forTH from (36) is obtained by an iterative calculation. Determination together with equations (19), (20), and (26) makes it possible to calculate the maximum tensile force within the geosynthetic and surface settlement.


## 3.1. Geosynthetic Behavior in the Collapse Area

The behavior of the geosynthetic is assumed to be linear elastic:T=Jε. Half of the void is taken into consideration. The frictional forces between the fills and the geosynthetic in the collapse area are neglected.Considering an infinitesimal elementdx of the geosynthetic in the collapse area (Figure 3), the equilibrium equations of vertical and horizontal forces give(11)qx=dTVdx,(12)dTH=0,Figure 3
Schematic diagram of forces within the geosynthetic in the collapse area.For an infinitesimal elementdx,(13)TV=−THdzdx.Thus,(14)dTVdx=−dTHdxdzdx−THd2zdx2.Combining equations (11), (12), and (14), the following expression is obtained.(15)qxTH=−d2zdx2.Inverse triangular load distribution on the geosynthetic in the collapse area is given.(16)qx=4QB2x.Combining (15) and (16), the following expression is obtained.(17)4QTHB2x=−d2zdx2.Solving (17) subjected to the boundary conditions that z=0 for x=B/2 and z′=0 for x=0, one can obtain(18)z=−2Q3THB2x3+QB12TH.At the edge of the cavity (point A), the tensile force within the geosynthetic reaches the maximum valueT1.(19)T1=TH2+Q24.Substitutingx=0 into (18), the geosynthetic deflection w can be written as(20)w=QB12TH.Using (18), the increase in geosynthetic length ΔL on the half-width of the cavity can be calculated as follows:(21)ΔL=∫ds−B2=∫0B/21+z′2dx−B2=−B2+112B4+Q2TH2+4M−Q24TH2.where M−t is a hypergeometric function in the Wolfram Mathematica package, which can be expanded by the Maclurin series as(22)M−t=1−t10+t224−5t3208+35t42176−3t5256+231t625600−429t759392+195t832768+Ot9.The first three Maclurin series terms ofM−t are precise enough under the situation that Q/2TH<1 for w/B<1/6. Equation (21) can be rewritten as(23)ΔL=−B2+112B4+Q2TH2+4−Q210TH2+Q496TH4.The geosynthetic tension deformationχ on the half-width of the cavity can be calculated as follows:(24)χ=∫εds=∫0B/2ε1+z′2dx=THJB2+BQ240TH2.Combining (23) and (24), the displacement of point A UA can be written as(25)UA=ΔL−χ=−B2+112B4+Q2TH2−Q210TH2+Q496TH4−THJB2+BQ240TH2.


## 3.2. Surface Settlement

In Figure4, the subsidence will rise rapidly to the surface of the embankment when the geosynthetic deflects. The soil above the void expands in volume thus indicating dilation. Thus, surface displacement is less than geosynthetic displacement in the vertical direction. Assuming that the shape of surface settlements and the geosynthetic sag are the same (cubical parabola) which is just a first approximation, combined with (20), the relation among surface settlement ws, deflection of geosynthetics w, soil expansion coefficient Ce, and height of the fill H can be obtained [6, 24, 27]:(26)ws=w1−18THQHBCe−1.Figure 4
Schematic diagram of the surface settlement.


## 3.3. Geosynthetic Behavior in the Anchorage Area

The friction laws at the soil/geosynthetic interfaces are supposed to be the Coulomb friction law in Figure5. In Figure 6, the tensile force within the geosynthetic decreases from T1 to T2 at the edge of the cavity based on the limit equilibrium of segment of geosynthetic sheet acting by friction on a circular arc (7) and (27).(27)T2=T1e−βk0αtanφl,where k0=1 for UA≥U0, k0=UA/U0 for UA<U0; and tanβ=z′B/2=Q/2TH.Figure 5
Coulomb friction law.Figure 6
Edge of the void.In Figure7, the geosynthetic AC in the anchorage area is divided into n elements evenly, and the length of each element equals Δx. A horizontal stress is caused by the friction between the geotextile and the upper layer; however, the influence is not very high, so this horizontal stress can be neglected. Thus, the shear stress on the upper and lower interfaces at any element of the geosynthetic is expressed as(28)τu=KUqaαtanφu,τl=KUqaαtanφl,where KU  = 1 for Uak≥U0 or KU=Uak/U0 for Uak<U0.Figure 7
Schematic diagram of the forces within the geosynthetic in the anchorage area.The equilibrium of the elementk is expressed as(29)Tak+1=Tak−KUqaαtanφu+tanφlΔx,(30)Uak+1=Uak−Tak+1JΔx.The equilibrium of the point A (k=0) is given as(31)T2=Ta0,UA=Ua0.At the end of the geosynthetic (point C), the tensile force within the geosynthetic equals 0; that is,(32)Tak|k⟶n=Tac=0.Combining equations (29) and (30) with the boundary conditions that Tac=0, Ta0=T2 and UA=Ua0, the system can be solved by an iterative calculation with the following equations.(33)Tak+1=Tak−KUqaαtanφu+tanφlΔx,Uak+1=Uak−Tak+1JΔx.


## 3.4. Determination ofT1 and ws Based on an Example

To be more specific, an example is taken herein to elaborate the calculation procedure of the maximum tensile force and surface settlement. The relevant parameters used in the calculation example are given in Table1.Table 1
Parameters in the calculation example.
ParameterValueCavity width (B)2 (m)Fill soil thickness (H)4 (m)Internal friction angle of the fill soil (φl)35 (°)Internal friction angle of the foundation soil(φu)35 (°)Unit weight of fill soil (γ)20 (kN/m3)Displacement of the geosynthetic at the point a (U0)10 (mm)Frictional coefficient (α)0.9Expansion coefficient of the fill soil (Ce)1.04Anchorage length (Lac)3 (m)Vertical overload applied on the fill soil surface (qs)0From equations (1), (3), and (4), we obtain q=56 kPa and Q=112 kN. Let q0=280 kPa; the value of q/q0 is 0.2. Based on Figure 2(b), σ2 is assumed to be 0.4. From equations (10), (19), and (25), we can obtain(34)qa=15exp−x/2−0.520.8+280,T1=TH2+3136,UA=−13+164+1122TH2−112210TH2+112496TH4−TH20001+627TH2.LetΔx  = 0.001 m. From (33) with Ta0=T2 and UA=Ua0, the relationship between Tac and TH is obtained in Figure 8. When Tac=0, we obtain TH= 74.7 kN/m and T1=93 kN/m.Figure 8
Relationship betweenTac and TH.Combining (20) and (26), we obtain ws  = 37 mm.


## 3.5. Influence of Vertical Loads Applied to the Geosynthetic

To investigate the influence of the form of vertical load distribution and the increment of vertical load in the anchorage area on the maximum tensile force and surface settlement, three calculation models are adopted for comparison.Model 1: an inverse triangular load distribution on the geosynthetic in the collapse area and a Gaussian load distribution on the anchorage area (Figure9(a)).Model 2: an inverse triangular load distribution on the geosynthetic in the collapse area and a uniform load distribution on the anchorage area (Figure9(b)).Model 3: a uniform load distribution on the geosynthetic in the collapse area and anchorage area (Figure9(c)).Figure 9
Schematic diagram of the vertical loads on geosynthetics for three calculation models: (a) model 1, (b) model 2, and (c) model 3.
(a)(b)(c)Figures10, 11, 12, 13, 14, 15, and 16 illustrate the variation tendency of the maximum tensile force and surface settlement against various parameters. From the holistic perspective, the same variation tendency of the maximum tensile force and surface settlement can be observed from these three models in Figures 10–16. It can be seen that the surface settlement calculated by model 1 is the lowest and the maximum tensile force calculated by model 2 is the smallest. Apart from that, compared with a uniform load distribution on the geosynthetic in the collapse area in model 3, an inverse triangular load distribution used in model 1 and model 2 induces much smaller maximum tensile force and surface settlement, indicating that a uniform load distribution in the collapse area will overestimate the tensile force of geosynthetic and surface settlement of the embankment. Meanwhile, the vertical stress at the edge of the void in model 1 is larger than that in model 2 because of different load distributions in the anchorage area used in these two models. Generally, large vertical stress at the edge of the void will lead to less elongation of the geotextile. In consequence, a smaller surface settlement in model 1 should be observed. Unexpectedly, calculation results show that little discrepancy exists, which means the increment of vertical stress in the anchorage area due to the ‘arching effect’ has little effect on both tensile force and surface settlement. Hence, model 2 is considered the best calculation model for its accuracy and simplicity.Figure 10
Influence of the width of the void on the maximum tensile force and surface settlement.Figure 11
Influence of the height of the embankment on the maximum tensile force and surface settlement.Figure 12
Influence of the tensile stiffness of the geosynthetic on the maximum tensile force and surface settlement.Figure 13
Influence of the internal friction angle on the maximum tensile force and surface settlement.Figure 14
Influence ofU0 on the maximum tensile force and surface settlement.Figure 15
Influence ofLac on the maximum tensile force and surface settlement.Figure 16
Influence of the expansion coefficient on the maximum tensile force and surface settlement.From the perspective of different parameters, all parameters are insensitive to the load distribution on the geosynthetic in the anchorage area but susceptible to the load distribution on the geosynthetic in the collapse area, which demonstrates the importance of the form of load distribution in the collapse area in practical engineering. As expected, Figure10 shows that the size of cavity B has the most significant effect on the maximum tensile force and surface settlement. Therefore, it requires that the size of the underground cavity should be known as precisely as possible. It can be seen from Figures 11–14 that the maximum tensile force and the maximum surface settlement have the opposite trends as an increase in the height of embankment H, the tensile stiffness of the geosynthetic J, internal friction angle φu, and relative displacement U0, respectively. Furthermore, φu and U0 have little effect on the tensile force and surface settlement, compared to H and J. Figure 15 indicates that the anchorage length Lac has no effect on the maximum tensile force and surface settlement when Lac exceeds 1 m. Hence, the influence of the anchorage length can be neglected for that the anchorage length is long enough in practice. In spite of difficult determination to the expansion coefficient α in practice, the results indicate that α only affects the surface settlement. Consequently, the maximum tensile force remains constant and an increase in α involves a decrease in the maximum surface settlement, as shown in Figure 16.


## 3.6. Analytical Formulation

Based on the analysis above, the vertical stress on geosynthetic in the anchorage area can be simplified with the initial uniform loadq0 instead of a Gaussian load distribution.Thus, the analysis of geosynthetic in the anchorage can be simplified with Briancon’s method [23].(35)ifUA≤U0:UA=T2Jr,ifUA>U0:UA=U0+T22−T022Jτ0.Withτ0=αq0tanφu+tanφl, r2=τ0/JU0, and T0=JrU0, T2=TH2+Q2/4/eβk0αtanφl.Combining (25) and (35), the solution of the system is(36)ifUA≤U0:T2Jr=−+112B4+Q2TH2−Q210TH2+Q496TH4−THJB2+BQ240TH2,ifUA>U0:U0+T22−T022Jτ0=−B6+112B4+Q2TH2−Q210TH2+Q496TH4−THJB2+BQ240TH2.The solution forTH from (36) is obtained by an iterative calculation. Determination together with equations (19), (20), and (26) makes it possible to calculate the maximum tensile force within the geosynthetic and surface settlement.


## 4. Analysis of Reinforced Embankment Bridging a Circular Void

As for general reinforced materials, such as woven geotextiles and geogrids, they usually have different tensile characteristics in the two principal directions. It is necessary to consider the anisotropy of geosynthetics when they are used for circular voids. However, Gourc and Villard [5] illustrated that unidirectional geosynthetics placed with machine direction along the direction of traffic are the most appropriate types. Hence, it is essential to analyze the cases that unidirectional reinforcement is used to bridge circular voids. A conservative approach can be applied to analyze these cases. It is assumed that a long void with a width, B, equal to the diameter, D, of the circular void was used to replace the circular void for mechanical analyses and the transverse tensile strength of the geosynthetic was neglected [11]. The analysis procedure for the circular void will be the same as the long void. Figure 17 presents the schematic diagram of the vertical load distribution on the geosynthetic layer spanning a circular void.Figure 17
Schematic diagram of the vertical loads on geosynthetic spanning a circular void.


### 4.1. Geosynthetic Behavior in the Collapse Area

The vertical load can be written as(37)qx=12QπD3x.Combining (15) and (37), the expression obtained is(38)z=−2QπTHD3x3+Q4πTH.From (38), we obtain(39)T1=TH2+3Q2πD2.(40)w=Q4πTH.Ifw/D<1/6, the displacement of point A UA can be written as(41)UA=−D6+112D4+9λ2−34λ2+2732λ4−THJD2+9D40λ2,where λ=Q/πDTH.


### 4.2. Surface Settlement

Assuming that the shape of surface settlement and the geosynthetic sag are both cubical parabola of revolution, combined with (40), the relation among surface settlement ws, deflection of geosynthetic w, soil expansion coefficient Ce, and height of the fill H can be obtained.(42)ws=w1−20πTH3QHCe−1.


### 4.3. Geosynthetic Behavior in the Anchorage Area

The comparison between two-dimensional (2D) and three-dimensional (3D) trapdoor experiments demonstrates that the vertical stress on the fixed support in 3D trapdoor experiments increases by a much smaller amount than that in 2D [17]. Thereafter, consideration of the increments of the vertical stress in anchorage areas for circular voids is unnecessary. The relationship between UA and T2 can be determined by equation (35).


### 4.4. Analytical Formulation

Combining (35) and (41), the solution of the system is(43)ifUA≤U0:T2Jr=−D6+112D4+9λ2−34λ2+2732λ4−THJD2+9D40λ2,ifUA>U0:U0+T22−T022Jτ0=−D6+112D4+9λ2−34λ2+2732λ4−THJD2+9D40λ2.The solution forTH from (43) is obtained by an iterative calculation. The maximum tensile force T1 and the surface settlement ws can be determined by (39) and (42).


## 4.1. Geosynthetic Behavior in the Collapse Area

The vertical load can be written as(37)qx=12QπD3x.Combining (15) and (37), the expression obtained is(38)z=−2QπTHD3x3+Q4πTH.From (38), we obtain(39)T1=TH2+3Q2πD2.(40)w=Q4πTH.Ifw/D<1/6, the displacement of point A UA can be written as(41)UA=−D6+112D4+9λ2−34λ2+2732λ4−THJD2+9D40λ2,where λ=Q/πDTH.


## 4.2. Surface Settlement

Assuming that the shape of surface settlement and the geosynthetic sag are both cubical parabola of revolution, combined with (40), the relation among surface settlement ws, deflection of geosynthetic w, soil expansion coefficient Ce, and height of the fill H can be obtained.(42)ws=w1−20πTH3QHCe−1.


## 4.3. Geosynthetic Behavior in the Anchorage Area

The comparison between two-dimensional (2D) and three-dimensional (3D) trapdoor experiments demonstrates that the vertical stress on the fixed support in 3D trapdoor experiments increases by a much smaller amount than that in 2D [17]. Thereafter, consideration of the increments of the vertical stress in anchorage areas for circular voids is unnecessary. The relationship between UA and T2 can be determined by equation (35).


## 4.4. Analytical Formulation

Combining (35) and (41), the solution of the system is(43)ifUA≤U0:T2Jr=−D6+112D4+9λ2−34λ2+2732λ4−THJD2+9D40λ2,ifUA>U0:U0+T22−T022Jτ0=−D6+112D4+9λ2−34λ2+2732λ4−THJD2+9D40λ2.The solution forTH from (43) is obtained by an iterative calculation. The maximum tensile force T1 and the surface settlement ws can be determined by (39) and (42).


## 5. Design Chart

The reinforcement must satisfy the requirement of the ultimate state and the serviceability limit state of the geosynthetic-reinforced embankment. That is, the maximum tensile forceT1 is lower than the acceptable tensile force Tadm of the geosynthetic and the maximum surface settlement ws is less than the allowable surface settlement wc. Design charts are established from the derivations presented in Sections 3 and 4. These charts present the maximum tensile force and surface settlement for given parameters. Inadvertent changes in embankment height and properties of fill for designed transport routes occur in engineering practice. Therefore, the geosynthetic stiffness is determined to be the main factor in the consideration. Generally, there is a positive correlation between the tensile stiffness J and strength Tc of the geosynthetic.It is assumed thatTc=0.1J and q0=γH in the following examples. Charts for long and circular voids are plotted to find proper values of J complying with the surface settlement and tensile force criteria (ws<wc and T1<Tadm). These charts are prepared without any safety factors and strength reduction factors of geosynthetics due to creep, damage during construction, connections, environmental impacts, etc. (i.e., Tadm=Tc).Long void.B=2m, H=4m, φu=φl=35∘, γ=20 kN/m3, U0=0.01m, Ce=1.04, wc=0.03m, and α=0.9.In the example presented in Figure18, for criteria ws<0.03m and T1<Tc, we determine the minimum geosynthetic stiffness: J=2400 kN/m.Figure 18
Chart for a long void.Circular void.D=2m, H=4m, φu=φl=35∘, γ=20 kN/m3, U0=0.01m, Ce=1.04, wc=0.03m, and α=0.9.The same design procedure can be applied to a circular void. From Figure19, we determine the minimum geosynthetic stiffness: J=500 kN/m.Figure 19
Chart for a circular void.


## 6. Conclusions

This study proposes a new analytical method to consider the ‘secondary arching effect’ and the resultant distribution of vertical loads acting on the unidirectional geosynthetic in the anchorage area. It is argued that the distribution of vertical stress on the geosynthetic in the anchorage area for a long void andH/B=1∼4 could be approximately described by a Gaussian function. The existing analysis methods were modified by replacing a uniform vertical load distribution with an inverse triangular vertical load distribution in the collapse area. The differential method was adopted to analyze the geosynthetic in the anchorage area. Analytical equations and design charts were proposed based on the assumptions that an inverse triangular vertical load acts on the geosynthetic in the collapse area and the vertical uniform load on the geosynthetic in the anchorage area corresponding to the geostatic stress. The increment of vertical load in the anchorage area is neglected and the arching effect within cohesive soils is also investigated. The salient findings of this study are summarized as follows:(i)
The results of three calculation models indicated that the increment of vertical load in the anchorage area has little influence on the maximum tensile force and surface settlement if the anchorage length is long enough. For a situation with a short necessary anchorage length, this will be different when the minimum necessary anchorage length is several meters.(ii)
The vertical load distribution on geosynthetic in the collapse area has a significant effect on the maximum tensile force and surface settlement. A uniform vertical load distribution on the geosynthetic in the collapse area will overestimate the maximum tensile force and surface settlement, leading to economic waste. By contrast, the proposed inverse triangular distribution on the geosynthetic in the collapse area will cause rather small results which are also accepted. It is economical to adopt an inverse triangular distribution on the geosynthetic in the collapse area for engineering interests.(iii)
Due to uncertain changes in embankment height and properties of fill, a new design method is presented to determine the minimum geosynthetic stiffness to meet design standards. The minimum geosynthetic stiffness can be obtained by not allowing the maximum tensile force and surface settlement, calculated by specific parameters of geosynthetic reinforcement to exceed the allowable threshold.



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*Source: 9577348-2022-02-12.xml*