Introduction

Performance of reinforced soil–walls is proven to be more effective in comparison to conventional retaining walls, such as gravity type, semi gravity type and cantilever walls in the seismic conditions [1, 2]. Apart from this, there are number of advantages associated with the use of this reinforced soil–walls in place of conventional retaining walls. These reinforced soil–walls are flexible, lightweight and economical. Thus, the use of reinforced soil structures has been increasing markedly over the recent years [3]. Numbers of studies, mostly based on limit equilibrium approach, were reported for the design of such reinforced soil–wall system under static conditions.

Bathurst and Cai [4] presented a design methodology for the segmental type reinforced soil–walls under seismic forces using pseudo-static approach. Ling et al. [5] developed a seismic design methodology for the geosynthetic reinforced soil structures based on pseudo-static limit equilibrium method of analysis by taking into account the effect of wall inclination angle. Later, the effects of vertical seismic acceleration on the total geosynthetic tensile reinforcement force (Ttot) needed for maintaining the stability of a geosynthetic reinforced soil structure was studied by Ling and Leshchinsky [6]. Similarly, Shahgholi et al. [7] developed a new methodology based on the horizontal slice method (HSM) for obtaining the value of Ttot. The simplified formulation of HSM requires satisfying two static equilibrium conditions arising from (i) the vertical equilibrium of each slice, and (ii) the horizontal equilibrium of the whole wedge. In addition to equilibrium conditions in the simplified formulation of HSM, Nouri et al. [8] have considered the moment equilibrium conditions as well, in order to compute the value of Ttot required for the reinforced-soil slopes and walls under seismic conditions together with the assumption of log-spiral failure surface. Furthermore, this research work was also extended by considering the pseudo-dynamic earthquake forces [9,10,11,12,13]. Recently, Chandaluri et al. [14] have obtained the values of Ttot for a reinforced soil–wall with general c–ϕ soil backfill adopting a simplified formulation of HSM without considering the effect of vertical acceleration and wall roughness.

From the review of the literature, it is evident that only a limited study has been reported for determining the total geosynthetic reinforcement tensile force required for ensuring the stability of a reinforced soil wall with general c–ϕ soil considering the influence of seismic loading, surcharge pressure and wall roughness. Thus, the aim of the present study is to propose a simplified method for determining the total geosynthetic reinforcement tensile force with general c–ϕ soil as backfill behind a reinforced soil wall in the presence of pseudo-static seismic forces considering the influence of surcharge pressure and wall roughness. In the proposed analysis, the equilibrium of potential failure mass bounded by a nonlinear slip surface, similar to that observed in the earlier experimental investigation, has been analyzed by using the simplified assumptions in the HSM. The effect of surcharge pressure acting on the top surface of backfill and the roughness of wall has also been included in the present analysis. In order to show the effectiveness of the proposed analytical approach, the magnitudes of Ttot computed in the present study were compared with (i) the pseudo-dynamic method of analysis by Shekarian et al. [12] and (ii) MSEW program by Leshchinsky [15].

Details of Analytical Formulation

Required Geosynthetic Tensile Force for Wall Stability

A vertical geosynthetic reinforced soil wall of height H carrying a uniform ground surcharge pressure q on the top horizontal surface of c–ϕ soil backfill is depicted in Fig. 1. The angle of internal friction of the soil wall interface is δ. The horizontally backfilled soil is assumed to be homogeneous and it possesses the unit cohesion c, the angle of internal friction ϕ and the unit weight γ, respectively. The geosynthetic reinforced soil wall is subjected to pseudo-static earthquake accelerations with horizontal and vertical components of magnitudes khg and kvg, respectively; where, g is the acceleration due to gravity and kh and kv are the pseudo-static earthquake acceleration coefficients in the horizontal and vertical directions respectively. It has been considered that the positive direction of horizontal and vertical inertia forces resulting from pseudo-static accelerations is shown in Fig. 1b. The geosynthetic reinforced soil wall is reinforced with m number of geosynthetic reinforcements placed parallel to the ground surface. The required tensile force by any horizontal reinforcement j in the geosynthetic reinforced soil wall is tj. Hence, the total tensile force Ttot needed by the geosynthetic reinforcement is simply equal to the addition of forces required by the individual reinforcements (t1, t2, t3, …tm−1, tm) anchored at various positions along the height of the reinforced geosynthetic wall, that is,

$${t_1}+{t_2}+{t_3}+ \ldots +{t_{m - 1}}+{t_m}=\sum\limits_{{j=1}}^{m} {{t_j}} ={T_{{\text{tot}}}}$$
(1)
Fig. 1
figure 1

Schematic diagram of geosynthetic reinforced wall showing various internal and boundary forces (a) by considering slip surface bounded by region ABC as a rigid body; and b in a small elemental slice i of width zi

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Using the simplified formulation of HSM, the magnitude of Ttot can be explicitly computed with the help of two static equilibrium conditions formulated from the consideration of vertical equilibrium of individual slice and the horizontal equilibrium of the whole wedge. Therefore, if the reinforced soil wedge behind the wall is divided into n number of horizontal slices, then the total number of equations and unknowns becomes equal to 2n + 1.

On considering the vertical equilibrium of individual slices, the following expression can be obtained,

$${V_{\text{i}}} - {V_{{\text{i}}+1}}+(1+{k_{\text{v}}}){W_{\text{i}}} - {S_{\text{i}}}\sin {\theta _{\text{i}}} - {N_{\text{i}}}\cos {\theta _{\text{i}}}=0$$
(2)

Assuming the case of full mobilization of shear stress along failure surface of any slice i, the net shear force Si acting on the failure surface of slice i can be presented as follows,

$${S_{\text{i}}}=c\frac{{{z_{\text{i}}}}}{{\sin {\theta _{\text{i}}}}}+{N_{\text{i}}}\tan \phi$$
(3)

On substituting the Eq. (3) into Eq. (2) and with the further simplification, the net normal force Ni acting on the failure surface of slice i can be written in the form given below,

$${N_{\text{i}}}=\frac{{{V_{\text{i}}} - {V_{{\text{i}}+1}}+(1+{k_{\text{v}}}){W_{\text{i}}} - c{z_{\text{i}}}}}{{\cos {\theta _{\text{i}}}+\tan \phi \sin {\theta _{\text{i}}}}}$$
(4)

Here, Vi and Vi+1 are the interslice forces acting vertically on the top and bottom horizontal plane of any slice i whose width is zi (Fig. 1b). By integrating overburden pressures acting on the top and bottom horizontal plane of slice i, the magnitude of these forces can be easily computed [7, 16]. Wi refers to the weight of slice i. It is to be noted that the values of Ni computed from the Eq. (4) should be always greater than zero (i.e., non-negative value) or equal to zero if not. Now, by considering the sliding failure wedge mass as a whole rigid body and by the various forces acting on it, the expression for Ttot can be presented explicitly as,

$${T_{{\text{tot}}}}=\sum\limits_{{i=1}}^{n} {{N_{\text{i}}}} \sin {\theta _{\text{i}}}+\sum\limits_{{i=1}}^{n} {{k_{\text{h}}}{W_{\text{i}}}} +{k_{\text{h}}}ql - \sum\limits_{{i=1}}^{n} {{S_{\text{i}}}} \cos {\theta _{\text{i}}}$$
(5)

Substituting the Eqs. (3) and (4) into Eq. (5), the Ttot can be presented as,

$$\begin{aligned} {T_{{\text{tot}}}}= & \sum\limits_{{i=1}}^{n} {\left( {\frac{{{V_{\text{i}}} - {V_{{\text{i}}+1}}+(1+{k_{\text{v}}}){W_{\text{i}}} - c{z_{\text{i}}}}}{{\cos {\theta _{\text{i}}}+\tan \phi \sin {\theta _{\text{i}}}}}} \right)} \sin {\theta _{\text{i}}}+\sum\limits_{{i=1}}^{n} {{k_{\text{h}}}{W_{\text{i}}}} +{k_{\text{h}}}ql \\ - \sum\limits_{{i=1}}^{n} {\left[ {c\frac{{{z_{\text{i}}}}}{{\sin {\theta _{\text{i}}}}}+\left( {\frac{{{V_{\text{i}}} - {V_{{\text{i}}+1}}+(1+{k_{\text{v}}}){W_{\text{i}}} - c{z_{\text{i}}}}}{{\cos {\theta _{\text{i}}}+\tan \phi \sin {\theta _{\text{i}}}}}} \right)\tan \phi } \right]} \cos {\theta _{\text{i}}} \\ \end{aligned}$$
(6)

where l refers to the length of AC.

It is interesting to note from the Eq. (6) that the number of slice n need not necessarily be equal to the number of geosynthetic reinforcement layers m. This indicates that the value of n can be chosen based on the required degree of accuracy in computing the magnitude Ttot. Similar to earth pressure coefficient, the magnitude of Ttot can be normalized by the force coefficient K, which is given by,

$$K=2\frac{{{T_{{\text{tot}}}}}}{{\gamma {H^2}}}$$
(7)

Critical Inclination of Failure Surface

The critical value of angles (θ1, θ2, θ3, …, θn−1, θn) defining the shape of failure surface can be obtained corresponding to the maximum magnitude of Ttot via any of the optimization techniques. However, in order to obtain a simplified analytical expression without deviating from the scope of the present study, the critical angle θn at base of the wall as identified by Shukla [17, 18] was employed. That is, the value of θn used was directly obtained from the below expression,

$$\tan {\theta _n}=\frac{{{a_1}\sin (\delta +\phi ) \pm \sqrt \alpha }}{{{b_1}\sin (\delta +\phi )+{m_2}\cos (\delta +\phi )}} \ge 0$$
(8)

in which

$$\alpha ={({a_1}\sin (\delta +\varphi ))^2}+{a_1}\cos (\delta +\varphi )({b_1}\sin (\delta +\varphi )+{m_2}\cos (\delta +\varphi )) \ge 0$$
(9a)
$$\psi ={\tan ^{ - 1}}\left( {\frac{{{k_{\text{h}}}}}{{1+{k_{\text{v}}}}}} \right)$$
(9b)
$${m_1}=\left( {\frac{{1+{k_v}}}{{\cos \psi }}} \right)\left( {1+\frac{{2q}}{{\gamma H}}} \right)$$
(9c)
$${m_2}=\frac{{2c}}{{\gamma H}}\cos \phi$$
(9d)
$${b_1}={m_1}\cos (\psi - \phi )$$
(9e)
$${a_1}={m_2} - {m_1}\sin (\psi - \phi )$$
(9f)

It should be mentioned that the effect of wall roughness has been explicitly considered in the formulation. This can be noticed from the base angle θn dependency on the magnitude of δ. Das [19] indicated that the critical failure surface obtained even with the log-spiral curve as a potential failure surface generally exits with an angle 90° with respect to horizontal ground. Further, in favor of the above statement, Sabermahani et al. [20] observed experimentally that the failure surface, during the bulging mode of deformation of a 1 m high reinforced-soil, intersects at an angle θ0 with the ground surface is closely equal to 90°. This assumption of the normal intersection of failure surface at the ground has also been adopted in a recent analytical study performed by Ahmad and Choudhury [21]. Therefore, in the present study, it has been assumed that the failure surface intersects normally at the ground surface (θ0 = 90°). Moreover, consistent with the experimental observations of Sabermahani et al. [20], the following relation is defined which can be used for obtaining the critical inclination θi of slip surface of any slice i,

$${\theta _{\text{i}}}={\theta _{\text{0}}}{\left[ {\frac{{{\theta _{\text{n}}}}}{{{\theta _0}}}} \right]^{({\text{i}} - 1)/({\text{n}} - 1)}}$$
(9)

Results and Comparisons

Variation of Force Coefficient (K)

For a smooth wall (δ = 0°) with angle of internal friction of backfill ϕ = 30°, the variation of force coefficient K with kh for different values of kv has been presented in Fig. 2a–d. Four different cases has been considered herein to examine the effect of backfill cohesion and ground surcharge pressure on the magnitudes of K, which are represented in the normalized fashion as (a) c/γH = 0 and q/γH = 0; (b) c/γH = 0.15 and q/γH = 0; (c) c/γH = 0 and q/γH = 0.5; and (d) c/γH = 0.15 and q/γH = 0.5. It can be seen from the Fig. 2a that the values of required tensile force in the geosynthetic reinforcements increase continuously with an increase in horizontal earthquake acceleration coefficient kh. The required tensile forces in the geosynthetic reinforcements are also increasing significantly depending on the magnitude of the vertical earthquake acceleration coefficient kv. However, in the presence of backfill cohesion, a considerable reduction in the total geosynthetic reinforcement tensile force is found as shown in Fig. 2b. On the other hand, Fig. 2c shows the presence of surcharge pressure on the soil backfill has an unfavorable effect of increasing the required magnitude of tensile forces in the geosynthetic reinforcements, which in turn increases the required quantity of reinforcing material. For the case of cohesive frictional soil, Fig. 2d represents the variation of K with kh for different values of kv. Depending on the component of cohesion towards the soil strength, the effect of surcharge pressure placed on the backfill may even diminish. Henceforth, when the soil backfill possesses cohesion, the overall stability of geosynthetic reinforced soil walls increases.

Fig. 2
figure 2

Variation of force coefficient K with kh for different values of kv with ϕ = 30°: a c/γH = 0 and q/γH = 0; b c/γH = 0.15 and q/γH = 0; c c/γH = 0 and q/γH = 0.5; and d c/γH = 0.15 and q/γH = 0.5

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For a soil backfill with ϕ = 30°, the variation of force coefficient K with δ/ϕ for different values of kv with kh = 0.25 has been presented in Fig. 3 for the two cases (a) c/γH = 0 and q/γH = 0.2; and (b) c/γH = 0.15 and q/γH = 0.2. The values of required tensile forces in the geosynthetic reinforcements are found to increases with an increase in the magnitude of δ. This is true for both frictional and cohesive frictional soils (Fig. 3). All the plots corresponded to ϕ = 30°; however, the magnitude in terms of K for cases other than presented can be easily obtained using the proposed approach. To examine the influence of n on the solution, Table 1 shows the magnitudes of K obtained with different values of n ranging from 5 to 80 for a particular case where ϕ = 30°, c/γH = 0.15 and q/γH = 0.5. The influence of value n on the solution becomes almost negligible for values of n > 20. Hence, all the results presented were obtained by considering n = 20. In fact, the selection of n = 20 gives good accuracy almost in all cases examined for estimating the force coefficient K.

Fig. 3
figure 3

Variation of force coefficient K with δ/ϕ for different values of kv with ϕ = 30° and kh = 0.25: a c/γH = 0 and q/γH = 0.2; b c/γH = 0.15 and q/γH = 0.2

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Table 1 The magnitude K for different values of n with ϕ = 30°, c/γH = 0.15 and q/γH = 0.5
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Failure Pattern at Collapse

The variation of the shape of rupture surface at critical collapses corresponding to a smooth wall (δ = 0) with ϕ = 30° for different values of kh and kv was obtained as shown in Fig. 4a–d. The effects of q/γH and δ on the failure pattern are not presented; however, specific remarks are made at the end of this section. It can be observed from the Fig. 4a that the soil mass bounded by the rupture surface in case of purely frictional soil, gradually increases with an increase in kh. That is, the active participation of soil mass towards failure would become larger in order to equilibrate the external seismic loads which depend on the magnitude of seismic acceleration. On the contrary, for a given ϕ and kh, decreased failure zone was observed as shown in Fig. 4b owing to the additional shear resistance acting on the failure plane due to the presence of cohesion. The changes of rupture surface shown in Fig. 4c for kh = 0.3 and q/γH = 0 observed to be quite less with an increase in kv. However, the change in the magnitude of K with kv is significant depending on the magnitude of kv. Similar to the previous observation in presence of cohesion, for a given kv, the active participation of soil mass bounded by rupture surface towards failure presented in Fig. 4d for the case of c/γH = 0.15 and kh = 0.3, has been found to be decreased considerably than that of the frictional soil.

Fig. 4
figure 4

Variation of collapse pattern at different values of kh and kv with ϕ = 30° and q/γH = 0: a c/γH = 0 and kv = 0.0; b c/γH = 0.15 and kv = 0.0; c c/γH = 0.0 and kh = 0.3; and d c/γH = 0.15 and kh = 0.3

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In case of partly rough walls (0 < δ < ϕ), the mode of failure changed from steep to shallow for greater values of δ. Steep and shallow mode of failures is characterized by the high and low value of base angle inclination θn. These two modes of failures can be visualized in Fig. 4a for curves corresponding to kh = 0 (shallow failure mode) and kh = 0.5 (steep failure mode). Under static condition (kh = kv = 0), no changes in the collapse pattern was noted with the variation in q/γH and c/γH. Further, the failure zone predicted in the study exactly coincides irrespective of the values of q/γH for a given kh and kv with c/γH = 0 which was observed previously by Shukla [17, 18].

Comparison with Published Results

In order to validate the proposed simplified approach, the results from the present study have been compared with the analytical method presented by Shekarian et al. [12] and MSEW program by Leshchinsky [15] for a wall with and without surcharge pressure on the backfill. Shekarian et al. [12] have developed analytical formulation in the framework of pseudo-dynamic approach and HSM for determining the total reinforcement tensile force required for a vertical geosynthetic reinforced soil wall by assuming the development of a simple linear rupture surface. In their formulation, both force and moment equilibrium conditions for the individual slices in the whole failure mass have been considered for finding the critical inclination of the slip surface and the total reinforcement tensile force. With the application of the vertical slices method and pseudo-static approach for seismic forces, Leshchinsky [15] have developed the MSEW program for estimating the values of total reinforcement tensile force required for a geosynthetic reinforced soil wall. For the comparison of the present solutions with the aforementioned studies, the unit weight of soil (γ), angle of friction between soil and wall (δ), height of wall (H) and surcharge pressure acting on the backfill (q) were selected to be 20 kN/m3, 10°, 7 m and 10 kN/m2, respectively. Table 2 presents the comparison of Ttot for the case of the wall without backfill surcharge pressure. For the case of the wall with backfill surcharge pressure, Table 3 compares the required tensile forces in the geosynthetic reinforcements Ttot from the present study with Shekarian et al. [12] and MSEW program by Leshchinsky [15]. The obtained results for the required tensile forces in the geosynthetic reinforcements Ttot are almost similar to those computed from MSEW program developed by Leshchinsky [15] based on pseudo-static approach. On the other hand, solutions obtained with the assumption of linear failure surface from the pseudo-dynamic approach by Shekarian et al. [12] is considerably lower than that of the present study and Leshchinsky [15].

Table 2 Comparison of Ttot from present method with analytical method by Shekarian et al. [12] and MSEW program by Leshchinsky [15] for a wall without surcharge
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Table 3 Comparison of Ttot from present method with analytical method by Shekarian et al. [12] and MSEW program by Leshchinsky [15] for a wall with surcharge
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Conclusions

By making use of the simplified assumptions in the HSM, the analytical expressions have been developed for obtaining the required reinforcement tensile force of the geosynthetic reinforced soil walls with general c–ϕ soil backfill under seismic loading based on a nonlinear slip surface, similar to that observed in the earlier experimental investigation. The variation of force coefficient K, which is proportional to total reinforcement tensile force, was examined with the changes in the magnitudes of the ground surcharge pressure (q), horizontal and vertical seismic acceleration coefficients (kh and kv), soil properties, and wall roughness, respectively. Based on the results obtained in this study, the following specific conclusions are obtained.

  • The required total reinforcement tensile force Ttot for maintaining the stability of the reinforced soil wall increases with an increase in the magnitudes of both horizontal and vertical seismic acceleration coefficients (kh and kv).

  • For a given surcharge and seismic loadings, the magnitude of Ttot varies significantly depending on the magnitudes of the angle of internal friction of the soil wall interface.

  • The presence of surcharge pressure on the soil backfill has an unfavorable effect of increasing the magnitude of Ttot; whereas, the presence of cohesion in the backfill reduces the magnitude of Ttot and enhances the overall stability of the geosynthetic reinforced soil wall.

  • The results obtained from the present simplified expression are in good agreement with those solutions reported in the literature.

  • The explicit form of the various analytical expressions derived in this study is one of the key advantages of the proposed approach which is of great benefit in practical use. Furthermore, the proposed approach could be easily implemented in spreadsheet application as it does not require any iteration to obtain the critical inclination angles defining the shape of the failure surface developed behind the wall at limiting condition and the magnitude of Ttot.