Determination of Required Reinforcement Force in Geosynthetic Reinforced Soil Walls Under Seismic Loadings
Abstract
This paper presents a simplified analytical approach for computing the required reinforcement tensile force of the geosynthetic reinforced soil walls subjected to earthquake loads using the method of horizontal slices. In the analysis, the earthquake load is taken into account by following the pseudostatic approach. The proposed formulation considers the equilibrium of potential failure mass bounded by a nonlinear slip surface similar to that observed in the earlier experimental investigation. The effects of pseudostatic seismic acceleration in horizontal and vertical directions, as well as the friction angle along the soil–wall interface, have been considered in the analysis. It has been found that the computed values of total tensile force required by the geosynthetic reinforcements from the present study compare favorably well with the methods reported in the literature. Furthermore, the present approach is straightforward and can be easily implemented through a simple spreadsheet application.
Keywords
Geosynthetic Tensile force Backfill Limit equilibrium Reinforced wall EarthquakeIntroduction
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 pseudostatic approach. Ling et al. [5] developed a seismic design methodology for the geosynthetic reinforced soil structures based on pseudostatic 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 (T_{tot}) 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 T_{tot}. 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 T_{tot} required for the reinforcedsoil slopes and walls under seismic conditions together with the assumption of logspiral failure surface. Furthermore, this research work was also extended by considering the pseudodynamic earthquake forces [9, 10, 11, 12, 13]. Recently, Chandaluri et al. [14] have obtained the values of T_{tot} 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 pseudostatic 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 T_{tot} computed in the present study were compared with (i) the pseudodynamic 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
Using the simplified formulation of HSM, the magnitude of T_{tot} 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.
Critical Inclination of Failure Surface
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 k_{h} for different values of k_{v} 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 k_{h}. The required tensile forces in the geosynthetic reinforcements are also increasing significantly depending on the magnitude of the vertical earthquake acceleration coefficient k_{v}. 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 k_{h} for different values of k_{v}. 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.
For a soil backfill with ϕ = 30°, the variation of force coefficient K with δ/ϕ for different values of k_{v} with k_{h} = 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.
The magnitude K for different values of n with ϕ = 30°, c/γH = 0.15 and q/γH = 0.5
n  k_{v} = 0.0 k_{h}  k_{v} = 0.5 k_{h}  k_{v} = 1.0 k_{h}  

k_{h} = 0.1  k_{h} = 0.3  k_{h} = 0.5  k_{h} = 0.1  k_{h} = 0.3  k_{h} = 0.5  k_{h} = 0.1  k_{h} = 0.3  k_{h} = 0.5  
5  0.320  0.571  0.958  0.329  0.609  0.984  0.344  0.647  1.098 
10  0.325  0.575  0.967  0.335  0.610  1.007  0.345  0.657  1.105 
20  0.324  0.577  0.970  0.334  0.612  1.014  0.346  0.659  1.106 
40  0.323  0.578  0.973  0.334  0.613  1.016  0.346  0.661  1.106 
80  0.323  0.578  0.974  0.334  0.613  1.018  0.346  0.661  1.106 
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 k_{h} and k_{v} 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 k_{h}. 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 k_{h}, 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 k_{h} = 0.3 and q/γH = 0 observed to be quite less with an increase in k_{v}. However, the change in the magnitude of K with k_{v} is significant depending on the magnitude of k_{v}. Similar to the previous observation in presence of cohesion, for a given k_{v}, 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 k_{h} = 0.3, has been found to be decreased considerably than that of the frictional soil.
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 θ_{n}. These two modes of failures can be visualized in Fig. 4a for curves corresponding to k_{h} = 0 (shallow failure mode) and k_{h} = 0.5 (steep failure mode). Under static condition (k_{h} = k_{v} = 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 k_{h} and k_{v} 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 pseudodynamic 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 pseudostatic 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/m^{3}, 10°, 7 m and 10 kN/m^{2}, respectively. Table 2 presents the comparison of T_{tot} 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 T_{tot} 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 T_{tot} are almost similar to those computed from MSEW program developed by Leshchinsky [15] based on pseudostatic approach. On the other hand, solutions obtained with the assumption of linear failure surface from the pseudodynamic approach by Shekarian et al. [12] is considerably lower than that of the present study and Leshchinsky [15].
k _{h}  ϕ = 25°  ϕ = 30°  ϕ = 35°  

(1)  (2)  (3)  (1)  (2)  (3)  (1)  (2)  (3)  
0  181.21  179.02  179.81  147.00  146.16  148.85  118.22  116.71  122.00 
0.05  196.66  192.20  204.20  160.73  156.71  170.45  130.39  126.14  141.14 
0.15  232.89  218.11  247.75  192.41  180.74  209.3  158.29  148.82  175.32 
0.25  279.31  249.22  284.34  231.48  208.40  241.42  191.94  171.56  204.25 
k _{h}  ϕ = 25°  ϕ = 30°  ϕ = 35°  

(1)  (2)  (3)  (1)  (2)  (3)  (1)  (2)  (3)  
0  202.88  210.02  204.44  164.39  170.14  169.22  132.09  136.98  138.69 
0.05  220.24  221.43  229.89  179.79  181.80  191.92  145.73  147.06  158.56 
0.15  260.98  246.72  273.44  215.35  204.78  230.29  176.96  168.16  192.75 
0.25  313.20  277.24  310.04  259.22  231.51  262.68  214.71  192.03  221.47 
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 (k_{h} and k_{v}), 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 T_{tot} for maintaining the stability of the reinforced soil wall increases with an increase in the magnitudes of both horizontal and vertical seismic acceleration coefficients (k_{h} and k_{v}).

For a given surcharge and seismic loadings, the magnitude of T_{tot} 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 T_{tot}; whereas, the presence of cohesion in the backfill reduces the magnitude of T_{tot} 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 T_{tot}.
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