Journal of Engineering Geology

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Distribution of static active earth pressure on an inclined retaining wall, with frictional or cohesive-frictional backfill, has been studied in the present research. Based on the limit equilibrium concept, and by implementing the horizontal slices method (HSM), two formulations have been proposed for determination of critical failure wedge. Results obtained from these formulas and results of the suggested equations by other researchers have been compared. Findings of current study show that horizontal slices method is capable of predicting the stress distribution and angle of failure wedge for inclined walls with high degree of accuracy. In addition, this method is applicable for various conditions of soil and wall and is able to consider the slope of backfill, friction between soil and wall, cohesion of soil and the effect of surcharge, simultaneously. Application of achieved formulation from horizontal slices method reveals that active earth pressure on inclined walls is nonlinear for both frictional and cohesive-frictional soils and the center of mass point of the resultant force would be located in an elevation less than one third of the height of wall.

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The shape of slip surface of the wedge creating lateral thrust on rigid retaining walls plays an important role in the magnitude, distribution, and height of point of application of lateral thrust.  Considering the shape of slip surface as linear, circular, logarithmic spiral, or a combination of them has been used in the literature. In the Coulomb lateral earth pressure method, a linear distribution of soil pressure on retaining walls is tentatively assumed and thus the point of application of total thrust is placed at one third of the wall height from the wall bottom. However, some experimental studies have revealed non-linear distribution of lateral earth pressures and that the point of application of resultant thrust is placed upper than one third of the wall height. In the present study, a plasticity equation is used to determine the reaction of the stable soil on cohesionless backfill supported by a retaining wall using an empirical equation derived from experiments performed in the field by others. A new analytical solution for determining the total resultant thrust on the wall is introduced and the distribution of pressures and the point of application of total thrust are computed. The results have been compared with some analytical methods, experimental data, and also with available data reported from field, demonstrating the accuracy and capability of the developed method. The results show that the distribution of the active lateral earth pressure is nonlinear and the point of application of total thrust is located about 0.42H from the wall bottom (H=wall height). In addition, the application point of total thrust is nonlinear function of soil-soil, wall-backfill soil friction angels and the height of the wall

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As usage of reinforced soil structures is highly increased in seismic active zones, the analysis of dynamic behavior of these structures begins to be of great significance.  The present paper is an attempt to study the seismic behavior of reinforced soil retaining walls with polymeric strips. The consequences of the most principal parameters counting the length of reinforcement, reinforcement arrangements (zigzag vs. parallel), maximum base input acceleration and wave frequency on the wall displacement have been investigated for sensitivity analyses. The main drawback of numerical methods in dynamic analysis is being very time consuming. Therefore, determination of equivalent coefficients is a suitable, easy and beneficent approach to converge   results of   pseudo-static and dynamic methods. In this case, a relatively accurate design is achieved by using pseudo-static method that takes less time. To this end, an earthquake equivalent horizontal acceleration coefficient is proposed by considering horizontal displacement of the wall as the basis for comparison

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Introduction
Retaining walls are geotechnical structures built to resist the driving and resistant lateral pressure. In terms of serviceability life, these walls are divided into two groups including short-term structures (temporary), such as urban excavation project, and long-term (permanent) structures, such as Mechanically Stabilized Earth Walls (MSE Walls). Retaining walls are implemented by two main methods including Top-down and Bottom-up. Among the reinforcements applied in the Bottom-up walls, one can name geocells, geogrids, metal strips, and plate anchors. On the other hand, the common reinforcements applied in the Top-down walls are grouted soil nails and anchors and helical (screw) soil nails and anchors.
Plate anchors are burial mechanical reinforcements that have one or multiple bearing plates with a bar or cable to transfer the load to an area with stable soil. Among different types of plate anchor applied in onshore and offshore projects, one can name simple horizontal, inclined, and vertical plate anchors, deadman anchors, multi-plate anchors, cross-plate anchors, expanding pole key anchors, helical anchors, drag embedment anchors, vertically loaded anchors (VLAs), suction-embedded plate anchors (SEPLAs), dynamically-embedded plate anchors (DEPLAs) like Omni-max and torpedo anchors, and duckbill, manta ray and stingray anchors.
The present research reports the results from physical modeling of plate anchor retaining walls under static loading. The evaluation parameters in this work include the geometry, dimension, and reinforcement configuration of plate anchors on wall stability. PIV technique was employed to observe critical slip surface. It is worth mentioning that PIV is an image processing technique firstly used in the field of fluid mechanics to observe the flow path of gas and fluid particles. This method was used in geotechnical modeling by White et al. (2003) and few reports are already available about its application to observe wedge failure of mechanically stabilized retaining walls.
Material and methods
To carry out tests at a laboratory scale, a dimensionality reduction ratio of 1/10 was applied. Thus, all dimensions of the designed retaining wall were divided by 10. As a result, a retaining wall with a height and length of 3000 mm was reduced to a wall with 300×300 mm² dimensions. To build a retaining wall, a chamber was designed with a length, width, and depth of 1000 mm, 300 mm, and 600 mm, respectively.
The soil used in all tests was the sandy soil supplied from Sufian (in Eastern Azerbaijan, Iran). According to the Unified Soil Classification System (USCS), the soil is classified as poorly graded sand with letter symbol ‘SP’.
To create a perfect planar strain condition and prevent any friction between the footing and the lateral sides of the test box, the footing length was selected 1 mm smaller than the 300 mm width of the test chamber. Therefore, the length, width, and thickness of footing were selected as 299, 70, and 30 mm, respectively.
The length and diameter of applied tie rods were respectively 300 mm and 4 mm, which are the smaller scales of 3000 mm length and 40 mm diameter tie rod. The two sides of the tie rods were threaded to plate anchors and wall facing. Four polished square and circular anchor plates with two different areas were used. The area of small and medium circulars are respectively equivalent to the area of small and medium square plates.
Because no post-tensioning occurs in these plate anchors, the horizontal and vertical distances were both selected as 1500 mm. By applying a dimensionality reduction coefficient of 1/10, a 150 mm center-to-center distance was obtained for reinforcements in the wall. Accordingly, three applied reinforcement configurations including 5-anchor, diamond, and square configurations were used.
To construct permanent retaining wall facing, prefabricated or precast concrete blocks with a thickness of 300 mm were used. Wood (2003) conducted a dimensional analysis and introduced four types of material with different thicknesses for a 300 mm concrete facing in laboratory modeling. Accordingly, a 0.9 mm thick aluminum plate was used in the experiments performed in the present work.
Results and discussion
With an increase in dimensions of anchor plates, an increase in bearing capacity of footing and a decrease in horizontal displacement of the wall are noticed. By comparing the 24 mm footing settlement in three configurations, with changing dimension of the plates from C1 to C2 and S1 to S2 respectively, 63% increases are observed in bearing capacity of the wall.
An increase in anchor plate dimensions results in a significant decrease in wall displacement. Therefore, changing the plates from C1 to C2, S1 to S2 leads to 24% and 28% declination in wall displacement.
By changing reinforcement configuration from square to diamond, diamond to 5-anchore, and square to 5-anchor, respectively, 27%, 31%, and 67.5% increases in bearing capacity for small plates, 9.2%, 27%, and 38% for medium plates are achieved using a comparison of the final loading steps in experiments. An analogy of percentages shows that a decrease in the effect of changing the reinforcement configurations on the bearing capacity of the wall with an increase in plate anchors dimensions is reached. 
Conclusion
In the present research, a set of laboratory experiments were carried out to evaluate the stability of mechanical retaining walls reinforced with plate anchors with different geometries (square and circular), sizes (small and medium), and configurations (diamond, square, and 5-anchor). The main results of the present work can be outlined as follows:
• The maximum bearing capacity is for the 5-anchor configuration since it has one more reinforcement. After 5-anchor configuration, the diamond configuration results in a higher bearing capacity compared to the square configuration.
• Circular anchor plates compared to square anchor plates provide a higher wall stability and in the most of the experiments lead to higher bearing and lower displacement in the wall.
• Wall displacement in a diamond configuration with one less reinforcement shows a little difference with 5-anchor configuration. The maximum wall displacement occurs in a square configuration and more wall swelling is observed in the wall middle height due to inefficient anchors configuration in the wall.
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One of the effective parameters in the dynamic behavior of reinforced soil walls is the fundamental vibration frequency. In this paper, analytical expressions for the first three natural frequencies of a geosynthetic reinforced soil wall are obtained in the 3D domain, using plate vibration theory and the energy method. The interaction between reinforced soil and the wall is also considered by modeling the soil and the reinforcement as axial springs. The in-depth transverse vibration mode-shapes, which were impossible to analyze via 2D modeling, are also analyzed by employing plate vibration theory. Different behaviors of soil and reinforcements in tension and compression are also considered for the first time in a 3D analytical investigation to achieve a more realistic result. The effect of different parameters on the natural frequencies of geosynthetic reinforced soil walls are investigated, including the soil to reinforcement stiffness ratio, reinforcement to wall stiffness ratio, reinforcement length, backfill width and length to height ratio of the wall, using the proposed analytical expressions. Finally, the results obtained from the analytical expressions proposed are compared with results from the finite element software Abaqus and other researchers’ results, showing that the proposed method has high accuracy. The proposed method will be a beginning of the 3D analytical modeling of reinforced soil walls.