A Review of the Methods Calculating the Horizontal Displacement for Modular Reinforced Soil Retaining Walls

: Most of the damage to reinforced retaining walls is caused by excessive deformation; however, there is no calculation method for deformation under static and dynamic loads in the design codes of reinforced soil retaining walls. In this paper, by collecting the measured displacement data from four actual projects, four indoor prototype tests and two indoor model tests under a total of 10 static load conditions, and comparing the calculation results with seven theoretical methods, the results show that the FHWA method is more applicable to the permanent displacement prediction of indoor prototype tests and that the CTI method is more applicable to the permanent displacement prediction of actual projects and indoor model tests. Two yield acceleration calculation methods and four permanent displacement calculation formulas were selected to calculate the displacement response of two reinforced soil test models under seismic loads and compared with the measured values, and the results showed that the Ausilio yield acceleration solution method was better. When the input peak acceleration ranges from 0.1 to 0.6 g, the Richards and Elms upper limit method is used, and when the input peak acceleration is 0.6–1.0 g, the Newmark upper limit method can predict the permanent displacement of the retaining wall more accurately.


Introduction
A modular reinforced soil retaining wall is a flexible retaining structure composed of a modular panel, reinforcement, and backfill. Because of their simple structures, strong adaptability, and many other advantages, modular reinforced soil retaining walls are widely used [1][2][3][4][5]. However, problems related to the loss of structural function due to excessive deformation of the retaining wall have arisen in construction and use [6][7][8]. Robert M. Koerner et al. [6] investigated 320 damaged retaining walls, of which 99 were damaged due to excessive deformation ( Figure 1). The reinforced soil retaining wall at the bridgehead approach of Xinzhuang Interchange at the turnout of Ningzhen Highway (312 National Highway)-Qixiashan Section in China [7]. In the late construction period, the southern panel of the eastern section of the bridge is obviously convex. Subsequently, due to many days of rain, the deformation of the whole wall gradually intensified, the road surface seriously subsided, and the wall was seriously tilted and bulging, which forced half of the traffic to be interrupted, and there was the possibility of collapse at any time. Hoe I. Ling et al. [8] investigated the failure of reinforced soil retaining walls in Highway 129 of  At present, the steps for the design of modular reinforced soil retaining walls a follows (based on the [9,10] railway subgrade retaining structure design specific 2006): (1) determine the engineering conditions. (2) Determine the engine parameters. (3) Determine the wall height, wall panel and reinforcement parameter Check the external stability. (5) When the internal stability is checked, there deformation checking process or steps. Because the panel, reinforcement, backfill foundation conditions will lead to and affect the deformation, it is very important to a reliable deformation calculation method. Many scholars have performed rel research on deformation calculations.
For the method for calculating the deformation of reinforced soil retaining under static forces, Wang et al. [11] calculated the equivalent reinforced soil retaining structure for horizontal isotropic vertical elastic beams, according to Rayleigh method, and the calculation method of horizontal displacement of reinforced retaining wall under static force was deduced, while the deformation was the sum o bending deformation and shear deformation.
Mahsa Khosrojerdi et al. [12] introduced six calculation methods for the horiz deformation of reinforced soil retaining walls and reinforced abutments (FHWA me GeoService method, CTI method, JeWell-Miigan method, Wu method, and A method). Among them, the FHWA method is an analysis formula based on the regre analysis of actual engineering and numerical simulation results, and the maxi horizontal deformation can be calculated only by the reinforcement length (L) Figure 1. Excessive deformation of the retaining wall [6].
Appl. Sci. 2021, 11, x FOR PEER REVIEW Hoe I. Ling et al. [8] investigated the failure of reinforced soil retaining walls in High 129 of Dagong City (Figure 2) after the Chi-Chi earthquake in Taiwan in 1999 analyzed the reasons for failure. They believed that failure was caused by the exce displacement of the modular panel during the earthquake. The main reason for the fa of reinforced soil is that the influence of deformation on the structure is not consider the design process.  At present, the steps for the design of modular reinforced soil retaining walls a follows (based on the [9,10] railway subgrade retaining structure design specifica 2006): (1) determine the engineering conditions. (2) Determine the engine parameters. (3) Determine the wall height, wall panel and reinforcement parameter Check the external stability. (5) When the internal stability is checked, there deformation checking process or steps. Because the panel, reinforcement, backfill foundation conditions will lead to and affect the deformation, it is very important to a reliable deformation calculation method. Many scholars have performed rele research on deformation calculations.
For the method for calculating the deformation of reinforced soil retaining under static forces, Wang et al. [11] calculated the equivalent reinforced soil retaining structure for horizontal isotropic vertical elastic beams, according to Rayleigh method, and the calculation method of horizontal displacement of reinforced retaining wall under static force was deduced, while the deformation was the sum o bending deformation and shear deformation.
Mahsa Khosrojerdi et al. [12] introduced six calculation methods for the horiz deformation of reinforced soil retaining walls and reinforced abutments (FHWA me GeoService method, CTI method, JeWell-Miigan method, Wu method, and A method). Among them, the FHWA method is an analysis formula based on the regre analysis of actual engineering and numerical simulation results, and the maxi horizontal deformation can be calculated only by the reinforcement length (L) At present, the steps for the design of modular reinforced soil retaining walls are as follows (based on the [9,10] railway subgrade retaining structure design specification, 2006): (1) determine the engineering conditions. (2) Determine the engineering parameters. (3) Determine the wall height, wall panel and reinforcement parameters. (4) Check the external stability. (5) When the internal stability is checked, there is no deformation checking process or steps. Because the panel, reinforcement, backfill, and foundation conditions will lead to and affect the deformation, it is very important to find a reliable deformation calculation method. Many scholars have performed relevant research on deformation calculations.
For the method for calculating the deformation of reinforced soil retaining walls under static forces, Wang et al. [11] calculated the equivalent reinforced soil retaining wall structure for horizontal isotropic vertical elastic beams, according to Rayleigh-Ritz method, and the calculation method of horizontal displacement of reinforced soil retaining wall under static force was deduced, while the deformation was the sum of the bending deformation and shear deformation.
Mahsa Khosrojerdi et al. [12] introduced six calculation methods for the horizontal deformation of reinforced soil retaining walls and reinforced abutments (FHWA method, GeoService method, CTI method, JeWell-Miigan method, Wu method, and Adams method). Among them, the FHWA method is an analysis formula based on the regression analysis of actual engineering and numerical simulation results, and the maximum horizontal deformation can be calculated only by the reinforcement length (L) and retaining wall height (H). The GeoService method is based on the limit equilibrium theory. The horizontal deformation of each layer can be calculated only by knowing the limit strain or the maximum strain and length of each layer. Based on actual engineering and finite element analysis, the CTI method for calculating the structural deformation during the service period can be used to calculate the horizontal deformation of each layer if the ultimate strain or the maximum strain of each layer of the reinforcement and the height of the retaining wall are required. The Wu method is based on the lifting method of the Jewell and Milligan method, which can be used to calculate the horizontal deformation of each layer without considering the strain of the reinforcement and the influence of the panel stiffness. The Adams method assumes that the volume strain is 0, the synchronous deformation of the reinforcement and soil, and the horizontal strain is less than 1%, which can be used to calculate the horizontal deformation of the top of the structure.
Mahsa Khosrojerdi et al. [12] compared the six methods with the experimental data of 17 reinforced soil retaining walls and reinforced abutments to determine the practicability of the six calculation methods. Krystyna Kazimierowicz-Frankowska [13] reviewed five calculation methods for the deformation of reinforced soil retaining walls, and the advantages and disadvantages of these five calculation methods were explained. At the same time, an IBWPAN method that can evaluate strain-displacement was introduced. The IBW PAN method divides the displacement into the creep displacement of the free zone and the tensile displacement of the anchorage zone, and the horizontal displacement calculation is divided into three modes: (1) the mode where only the displacement generated by the free zone is calculated; (2) the mode without external loads, the displacement of the free zone and the anchorage zone; and (3) the mode with external loads, the displacement of the free zone and the anchorage zone. He [14] established a numerical model through FLAC 3D , summarized the horizontal deformation formula of reinforced soil retaining walls, and obtained the formula of each influencing factor and the horizontal deformation of retaining walls, as well as the horizontal deformation formula of reinforced soil retaining walls containing each influencing factor. Each calculation method is summarized in Table  S1.
For the deformation of reinforced soil retaining walls under seismic loads, Z. Cai and R.J. Bathurst [15] compared several calculation methods (Newmark upper bound method, Richards and Elms upper bound method, Whitman and Liao average fitting method, and Cai and Bathrust method) and showed that although different methods are calculated with different parameters, the permanent displacement values obtained by all methods are in a reasonable area. Budhu [16] used the sliding safety factor method to analyze the yield acceleration of a backpacked reinforced soil retaining wall. Younan et al. [17] simplified the retaining wall as a cantilever spring model and proposed a calculation method of the retaining wall deformation considering panel stiffness. Based on the calculation method of horizontal displacement of reinforced soil retaining walls under static action, Wang et al. [11] deduced the theoretical research formula for the horizontal seismic deformation of reinforced soil retaining walls under seismic action by using the quasi-static method. After analyzing the existing models (Ambraseys and Menu method, Jibson method, and Roberto method), Xu et al. [18] established a permanent displacement prediction model based on the critical acceleration ratio, Arias strength, and seismic residual strength by using a large number of strong motion records recorded during the Wenchuan earthquake. The energy method was used to convert the measured displacement of the slope into the permanent displacement of the slope without a supporting structure, and the validity of the model was verified by the measured data.
Although there are many calculation methods under static load and seismic load, there is no recognized calculation method because of the accuracy of the data. Dunnicliff [19] judged the reliability of measured data by numerical calculation, and Pantelidis [20] compared solutions, design specifications and centrifuge tests to support the effectiveness of the proposed coefficients. In order to find a suitable deformation calculation method for modular reinforced earth retaining wall, this paper first describes the actual deformation of retaining wall under 10 static loads, and predicts the displacement by seven calculation methods. Then, through two indoor dynamic test models, the measured displacement values are compared with the calculated values under four calculation methods to verify the Appl. Sci. 2021, 11, 8681 4 of 15 accuracy of the calculation method. Finally, the calculation method that is most suitable for predicting displacement under static and dynamic loads is obtained. The research results can provide reference for the deformation design of modular reinforced earth retaining wall.

Study on the Deformation Mode
Many experts and scholars have performed relevant research on the deformation mode of modular reinforced soil retaining walls under static action (Figure 3), which is mainly divided into tilting types ( Figure 3a) and bulging types (Figure 3b).
Many experts and scholars have performed relevant research on the deformation mode of modular reinforced soil retaining walls under static action (Figure 3), which is mainly divided into tilting types ( Figure 3a) and bulging types (Figure 3b).
Bathurst et al. [21] introduced deformation monitoring of a 6.1 m-high modular reinforced soil retaining wall in Illinois, USA, after completion and after loading. The monitoring results are shown in Table 1.   N. Abu-Hejleh et al. [22] introduced the deformation response of reinforced soil retaining wall abutment structures completed near Denver, Colorado, in 1999 during the construction period. The maximum outward displacements of Section 200, Section 400, and Section 800 are 7, 9m and 10 mm, respectively. Among them, the maximum deformation positions of Section 200 and Section 800 are equal to 2/3 of the wall height, and the deformation mode is bulging. The maximum deformation position of Section 400 is at the top of the wall, and the deformation mode is an outward-dip type. This may be due to the difference in the deformation modes caused by different construction seasons and different construction processes.
Hatami et al. [23,24] and Bathurst et al. [25,26] introduced four 3.6 m high reinforced soil retaining walls constructed by the Royal Canadian Military Academy, and four Bathurst et al. [21] introduced deformation monitoring of a 6.1 m-high modular reinforced soil retaining wall in Illinois, USA, after completion and after loading. The monitoring results are shown in Table 1. N. Abu-Hejleh et al. [22] introduced the deformation response of reinforced soil retaining wall abutment structures completed near Denver, Colorado, in 1999 during the construction period. The maximum outward displacements of Section 200, Section 400, and Section 800 are 7, 9m and 10 mm, respectively. Among them, the maximum deformation positions of Section 200 and Section 800 are equal to 2/3 of the wall height, and the deformation mode is bulging. The maximum deformation position of Section 400 is at the top of the wall, and the deformation mode is an outward-dip type. This may be due to the difference in the deformation modes caused by different construction seasons and different construction processes.
Hatami et al. [23,24] and Bathurst et al. [25,26] introduced four 3.6 m high reinforced soil retaining walls constructed by the Royal Canadian Military Academy, and four models were different due to different spacings and stiffnesses of the reinforcement. In the observation after completion and after loading, the maximum deformation position is approximately 2.7 m high, and the deformation mode bulges.
Jennifer E. Nicks et al. [27] developed a model with dimension of 1.4 m (length) × 1.4 m (width) × 2.0 m (height) to study the deformation mode of reinforced soil columns.
The deformation modes under different loads, different ultimate tensile forces of the reinforcement, and different vertical spacings were obtained. The test results show that the maximum lateral deformation occurs at a height of one-third from the top regardless of whether the veneer is included in the test process, and the overall deformation trend is inclined.
Xiao et al. [28] made a retaining wall with dimensions of 1.0 m (length) × 0.54 m (height) to analyze the influence of the distance between the strip foundation and wall panel on the deformation characteristics of the reinforced retaining wall. The results show that when the foundation deviation distance D is less than 0.5 H, the top deformation of the retaining wall is the largest, and the overall deformation trend is inclined. When D is 0.6H and 0.8H, the deformation of the upper part of the retaining wall is the largest, and the overall trend changes to bulging.
In conclusion, due to different factors, such as the height of the retaining wall, the length of the reinforcement, the spacing of the reinforcement, the stiffness of the reinforcement, the construction season and the construction sequence, the modular reinforced soil retaining wall has different deformation modes and deformation amounts. Therefore, it is necessary to evaluate the applicability and accuracy of existing deformation calculation methods.

Comparison of the Measured Data and Calculation Methods
To analyze the applicability of the deformation calculation methods, this paper uses the measured values of 10 modular reinforced soil retaining walls to calculate the results of the FHWA (F) method, GeoService (G) method, CTI (C) method, Wu (1,2) method, Adams (A) method, and Wang (W) method, and analyzes the practicability of seven calculation methods. Table S2 summarizes the parameters of the 10 structures. Models 1, 2, 3, and 4 are practical engineering, models 5, 6, 7, and 8 are indoor prototype tests (1:1 indoor model test is carried out according to the test prototype), and models 9 and 10 are indoor model tests (according to the test prototype, the indoor model test designed by similarity ratio). Table S3  In this paper, the η value (1-calculated value/measured value) is defined. When η is 0, the predicted value is the same as the measured deformation. When the η value is greater than 0, the prediction method is not conservative, and the prediction method with an η value less than 0 represents the conservative prediction method. In 10 cases η the values are shown in Table 2 and Figure 5. The displacement value of case 9 under FHWA method in Figure 5 is −466.71, which is excessively conservative, so it is not shown in Figure 5.       As can be seen from Figure 7, the calculation results under seven calculation method ηn value analysis shows that, in 10 cases, the relative difference between the CTI metho and the GeoService method is the smallest, because most ηn value is less than 0, so th existing calculation methods are conservative in predicting deformation, and the FHW method is the most conservative. In cases 1-4 of the practical engineering cases, th GeoService method and CTI method show good practicability. Although the two method need accurate reinforcement strain in calculation, the CTI method has less difference, s the CTI method is more accurate in displacement prediction in practical engineering. I The seven calculation methods are note applicable to all cases, and there may not be parameters in the formula in the case. The L/H values of case2-4under FHWA method are 1.778-2.667, 1.355-2.033, 1.355-2.033, respectively, which are not between 0.3 and 1.175, which do not meet the conditions of formula, so the permanent displacement value cannot be obtained. In case 1 of the Wu (1) method, the reinforcement stiffness (K) parameter is missing. Case 9 cannot get a displacement because L/H = 0.5 is less than 0.7. In cases 1-4 and 10 under Wu (2) method, due to the lack of the friction angle (δ) between the module bricks and the friction angle between the brick and soil (β), in case 9, since L/h is 0.5 less than 0.7, the displacement value cannot be obtained. Due to the lack of data in case 1 and case 10 under the Adams method, the displacement value cannot be obtained. Under the W method, case 1 lacks the geogrid elastic modulus (E r ) geogrid Poisson's ratio (v r ), cases 2-4 lack the geogrid elastic modulus (E r ) and the fill elastic modulus (E s ), and case 9 and case 10 lack the fill elastic modulus (E s ), so the calculated value cannot be obtained.
As can be seen from Figure 7, the calculation results under seven calculation methods, η n value analysis shows that, in 10 cases, the relative difference between the CTI method and the GeoService method is the smallest, because most η n value is less than 0, so the existing calculation methods are conservative in predicting deformation, and the FHWA method is the most conservative. In cases 1-4 of the practical engineering cases, the GeoService method and CTI method show good practicability. Although the two methods need accurate reinforcement strain in calculation, the CTI method has less difference, so the CTI method is more accurate in displacement prediction in practical engineering. In cases 5-8 of the indoor prototype test, the relative difference between the Adams method and the Wang (W) method is the largest, so both methods are not suitable for the prediction of indoor prototype test displacement, while the CTI method, GeoService method, and Wu (1) method show good practicability. In case 7, the GeoService method is more conservative than the Wu (1) method; therefore, the Wu (1) method is more appropriate. In cases 9-10, the relative differences obtained by GeoService method, CTI method, and Wu (1) method are similar. Therefore, the GeoService method and CTI method have shown good performance in predicting indoor model tests. Comparing the calculated/measured values of the GeoService method and the CTI method, the ratios of the two calculation methods are 1.5, 3.96, 0.55, and 2.048, respectively, in cases 9 and 10. Because the difference between the calculated value and the measured value of CTI method is smaller, the CTI method is selected as the displacement prediction method of the indoor model test. The GeoService method and CTI method can get the predicted value when predicting 10 cases, so they are more practical than the other five methods. method have shown good performance in predicting indoor model tests. Comparing the calculated/measured values of the GeoService method and the CTI method, the ratios of the two calculation methods are 1.5, 3.96, 0.55, and 2.048, respectively, in cases 9 and 10. Because the difference between the calculated value and the measured value of CTI method is smaller, the CTI method is selected as the displacement prediction method of the indoor model test. The GeoService method and CTI method can get the predicted value when predicting 10 cases, so they are more practical than the other five methods.

Discussion
The model fabrication process in the model test and indoor prototype test is rigorous and precise, so the deformation is small. The construction quality of actual projects is often difficult to guarantee due to factors such as the construction period and cost. Robert M. Koerner et al. [6] considered that improper filler, poor compaction effect, unreasonable design, and poor drainage were the main reasons for the failure of 320 reinforced soil retaining walls. At the same time, because the modulus of the foundation soil, the stiffness of the reinforcement, the vertical spacing of the reinforcement, the length of the reinforcement, the nature of the backfill soil, the change in the wall height, the additional load, the depth of the panel foundation and other factors will affect the deformation of the reinforced soil retaining wall, it can be speculated that an accurate deformation calculation method does not exist.
According to the suggestion of the η value, the FHWA method is used to predict the deformation value before the construction of a modular reinforced soil retaining wall. Under the premise of known reinforcement deformation, the Wu (1) method and CTI method are used to estimate the deformation of modular reinforced soil retaining walls in normal use.

Calculation Method of the Yield Acceleration
Under seismic loading, the displacement calculation of reinforced soil retaining walls is generally based on the Newmark sliding block method. The Newmark sliding method theory was originally used to estimate the permanent displacement of embankment slope

Discussion
The model fabrication process in the model test and indoor prototype test is rigorous and precise, so the deformation is small. The construction quality of actual projects is often difficult to guarantee due to factors such as the construction period and cost. Robert M. Koerner et al. [6] considered that improper filler, poor compaction effect, unreasonable design, and poor drainage were the main reasons for the failure of 320 reinforced soil retaining walls. At the same time, because the modulus of the foundation soil, the stiffness of the reinforcement, the vertical spacing of the reinforcement, the length of the reinforcement, the nature of the backfill soil, the change in the wall height, the additional load, the depth of the panel foundation and other factors will affect the deformation of the reinforced soil retaining wall, it can be speculated that an accurate deformation calculation method does not exist.
According to the suggestion of the η value, the FHWA method is used to predict the deformation value before the construction of a modular reinforced soil retaining wall. Under the premise of known reinforcement deformation, the Wu (1) method and CTI method are used to estimate the deformation of modular reinforced soil retaining walls in normal use.

Calculation Method of the Yield Acceleration
Under seismic loading, the displacement calculation of reinforced soil retaining walls is generally based on the Newmark sliding block method. The Newmark sliding method theory was originally used to estimate the permanent displacement of embankment slope caused by earthquake. When the ground acceleration exceeds the critical acceleration of the soil, the block will move. Other calculation methods are modified and improved on the basis of the Newmark method. For example, Whitman and Liao carried out regression analysis on Newmark displacement data and put forward formulas for estimating permanent displacement.
The test data selected Li et al.'s [29] single-step, two-tiered modular reinforced soil retaining wall model scale test, and the test parameters are shown in Table 3. The model size of the single-step modular reinforced soil retaining wall is 2.0 × 1.5 × 1.8 m (length × width × height), which is divided into 12 layers, and the height of each layer is 0.15 m. To monitor the displacement response of the retaining wall under input acceleration, a rod displacement meter is arranged at the middle position of each layer module, for a total of 12 displacement meters, and an accelerometer is arranged from the bottom to the top of each pair of layers in the reinforced area and non-reinforced area, for a total of 12 locations, to obtain the acceleration response. The size of the two-tiered modular reinforced soil retaining wall model and the layout of the accelerometer and the top rod displacement meter are the same as those of the single-step modular reinforced soil retaining wall model. The design diagram of the two models is shown in Figure 8. 12 displacement meters, and an accelerometer is arranged from the bottom to the top of each pair of layers in the reinforced area and non-reinforced area, for a total of 12 locations, to obtain the acceleration response. The size of the two-tiered modular reinforced soil retaining wall model and the layout of the accelerometer and the top rod displacement meter are the same as those of the single-step modular reinforced soil retaining wall model. The design diagram of the two models is shown in Figure 8. The yield acceleration method proposed by Newmark gives the acceleration value when the safety factor is 1. When the yield acceleration value is greater than the input acceleration value, displacement accumulation will occur. By summarizing and analyzing the results of previous studies and large shaking table tests, Muni obtained the yield acceleration formula through the safety factor of anti-slip earthquakes. E. Ausilio applied the limit analysis method to obtain the yield acceleration method under seismic loads. Table 4 introduces four calculation methods for the yield acceleration. The calculated values of the yield acceleration of reinforced retaining walls with two different forms of panels under the Muni method and E. Ausilio method are shown in Table S4.
(a) Single-step modular reinforced retaining wall. (b) Two-tiered modular reinforced retaining wall.   The yield acceleration method proposed by Newmark gives the acceleration value when the safety factor is 1. When the yield acceleration value is greater than the input acceleration value, displacement accumulation will occur. By summarizing and analyzing the results of previous studies and large shaking table tests, Muni obtained the yield acceleration formula through the safety factor of anti-slip earthquakes. E. Ausilio applied the limit analysis method to obtain the yield acceleration method under seismic loads. Table 4 introduces four calculation methods for the yield acceleration. The calculated values of the yield acceleration of reinforced retaining walls with two different forms of panels under the Muni method and E. Ausilio method are shown in Table S4.

Calculation Method of the Permanent Displacement
The empirical formula method is one of the main contents of the residual displacement estimation method of reinforced soil retaining walls after earthquakes.  [15], and Newmark method [33]. The five methods are based on the critical acceleration coefficient kc, peak acceleration km and propagation velocity Vm of seismic waves. The Vm values of the two shaking table tests are shown in Table 5.

Comparison between the Experimental and Calculated Values
By analyzing the η n values of two different panel forms of modular reinforced soil retaining walls under the action of WL waves and EL waves, the practicability of four calculation methods for the displacement prediction of modular reinforced soil retaining walls is judged. The meaning of η n is shown in Table 6. The η n obtained by the four calculation methods in Figures 9 and 10 show that with the increase of input ground motion, the relative difference under each calculation method increases gradually, indicating that the deviation between the calculated value and the measured value is getting larger and larger, and the difference between each method is increasing. When the η n value was at the initial 0.1 g, each η n value was the closest. With the increase of acceleration, the gap between each η 1 , η 2 and η 3 , η 4 values showed an increasing trend, and the size and trend of η 1 and η 2 were almost consistent from beginning to end. When the similarity ratio is 1:4, the WL wave is input. When the peak acceleration is less than 0.6 g, the η n values are all greater than 0, and the calculated values are less than the measured values. When the peak acceleration reaches 1.0 g, the η n values are all less than 0, and the calculated values are greater than the measured values. The calculated values are conservative. When the peak acceleration is 0.1-0.6 g, the η1 values are smaller than other η n values, which can be predicted by the Richards and Elms upper bound method. When the peak acceleration is 0.6-1.0 g, the η 1 < η 2 < η 4 < η 3 ; that is, the Richards and Elms upper bound method < Cai and Bathurst average upper bound method < Newmark upper bound method < Whitman and Liao average fitting method. Correspondingly, EL wave is input at the similarity ratio of 1:2. When the input peak acceleration is 0.1-0.6 g, η n values are greater than 0, η 1 values and η 2 values are close to 0, and when the input peak acceleration is 0.8-1.0 g, η n values are all less than 0, η 3 and η 4 are much larger than η 1 and η 2 . At this time, η 3 minimum distance from 0 line, and the Whitman and Liao average fitting method is more suitable for predicting the displacement value. Comparing Figures 8 and 9, it can be seen that the η n value under the Ausilio calculation method is generally closer to the 0 value line than that under the Muni method, so the Ausilio method is more suitable. acceleration is 0.1-0.6 g, ηn values are greater than 0, η1 values and η2 values are close 0, and when the input peak acceleration is 0.8-1.0 g, ηn values are all less than 0, η3 and are much larger than η1 and η2. At this time, η3 minimum distance from 0 line, and Whitman and Liao average fitting method is more suitable for predicting displacement value. Comparing Figures 8 and 9, it can be seen that the ηn value under Ausilio calculation method is generally closer to the 0 value line than that under the Mu method, so the Ausilio method is more suitable.  Figures 11 and 12 show that the value of ηn decreases with the increase of input pe acceleration, and the change amplitude is larger and larger. Under the condition similarity ratio 1: 4 and similarity ratio 1: 2, the change trend of ηn value is consistent. W the increase of peak acceleration, the trend line of η1 value and η2 value is gradually aw from η3 and η4. At 1.0 g, η1 and η2 are farthest from η3 and η4. In the case of similarity ra 1: 4, when the input peak acceleration is 0.1-0.6 g, the η1 value is less than 0 under 0.2 g Figure 11, and the other values are greater than 0, which shows that each calculat method is conservative in predicting the displacement value under small earthquak acceleration is 0.1-0.6 g, ηn values are greater than 0, η1 values and η2 values are close 0, and when the input peak acceleration is 0.8-1.0 g, ηn values are all less than 0, η3 and are much larger than η1 and η2. At this time, η3 minimum distance from 0 line, and Whitman and Liao average fitting method is more suitable for predicting displacement value. Comparing Figures 8 and 9, it can be seen that the ηn value under Ausilio calculation method is generally closer to the 0 value line than that under the M method, so the Ausilio method is more suitable.  Figures 11 and 12 show that the value of ηn decreases with the increase of input pe acceleration, and the change amplitude is larger and larger. Under the condition similarity ratio 1: 4 and similarity ratio 1: 2, the change trend of ηn value is consistent. W the increase of peak acceleration, the trend line of η1 value and η2 value is gradually aw from η3 and η4. At 1.0 g, η1 and η2 are farthest from η3 and η4. In the case of similarity ra 1: 4, when the input peak acceleration is 0.1-0.6 g, the η1 value is less than 0 under 0.2 g Figure 11, and the other values are greater than 0, which shows that each calculat method is conservative in predicting the displacement value under small earthquak Figure 10. Ausilio [31] method under the WL wave with the η n value. Figures 11 and 12 show that the value of η n decreases with the increase of input peak acceleration, and the change amplitude is larger and larger. Under the condition of similarity ratio 1:4 and similarity ratio 1:2, the change trend of η n value is consistent. With the increase of peak acceleration, the trend line of η 1 value and η 2 value is gradually away from η 3 and η 4 . At 1.0 g, η 1 and η 2 are farthest from η 3 and η 4 . In the case of similarity ratio 1:4, when the input peak acceleration is 0.1-0.6 g, the η 1 value is less than 0 under 0.2 g in Figure 11, and the other values are greater than 0, which shows that each calculation method is conservative in predicting the displacement value under small earthquakes, and the η 1 value is relatively more reasonable, so Richards and Elms upper bound method is more reasonable. When the input peak acceleration is 0.8-1.0 g, the η 1 and η 2 values are gradually away from the zero line, so Richards and Elms upper bound method and Cai and Bathurst average upper bound method are not applicable at this time. Relatively speaking, η 3 and η 4 values show good applicability, η 4 value is more conservative, so the Whitman and Liao average fitting method is more reasonable. When the similarity ratio is 1:2, the input peak acceleration is 0.1-1.0 g, and the ηn value gradually decreases. When the similarity ratio is 0.1-0.4 g, the η n values are all greater than 0, and the η 1 value is smaller. Therefore, the Richards and Elms upper bound method is suitable for this stage. At 0.6-1.0 g, the η 4 value is more conservative than the η 3 value, and the Whitman and Liao average fitting method is more suitable.
speaking, η3 and η4 values show good applicability, η4 value is more conservative, so Whitman and Liao average fitting method is more reasonable. When the similarity ra is 1:2, the input peak acceleration is 0.1-1.0 g, and the ηn value gradually decreases. Wh the similarity ratio is 0.1-0.4 g, the ηn values are all greater than 0, and the η1 value smaller. Therefore, the Richards and Elms upper bound method is suitable for this sta At 0.6-1.0 g, the η4 value is more conservative than the η3 value, and the Whitman a Liao average fitting method is more suitable.  Table   Figures 13 and 14 show that the change trend of the two-tiered modular reinforc earth retaining wall under earthquake is the same as that of the single-step modu reinforced earth retaining wall. When the input peak acceleration is small, the value of is greater than 0. With the increase of the peak acceleration, the value of ηn decrea gradually. When the input peak acceleration is 0.2-0.4 g, Richards and Elms upper bou method can better predict the displacement value. When the similarity ratio is 1:10, Cai and Bathurst average upper bound method is more suitable for the input of WL wa When EL wave is input, Whitman and Liao average fitting method and Newmark up bound method are both suitable. Since η4 is more conservative, the Whitman and L Figure 11. Under the yield acceleration obtained by the Muni method [16] under the El wave and η n value.
gradually away from the zero line, so Richards and Elms upper bound method and C and Bathurst average upper bound method are not applicable at this time. Relativ speaking, η3 and η4 values show good applicability, η4 value is more conservative, so Whitman and Liao average fitting method is more reasonable. When the similarity ra is 1:2, the input peak acceleration is 0.1-1.0 g, and the ηn value gradually decreases. Wh the similarity ratio is 0.1-0.4 g, the ηn values are all greater than 0, and the η1 value smaller. Therefore, the Richards and Elms upper bound method is suitable for this sta At 0.6-1.0 g, the η4 value is more conservative than the η3 value, and the Whitman a Liao average fitting method is more suitable.  Table   Figures 13 and 14 show that the change trend of the two-tiered modular reinforc earth retaining wall under earthquake is the same as that of the single-step modu reinforced earth retaining wall. When the input peak acceleration is small, the value of is greater than 0. With the increase of the peak acceleration, the value of ηn decrea gradually. When the input peak acceleration is 0.2-0.4 g, Richards and Elms upper bou method can better predict the displacement value. When the similarity ratio is 1:10, Cai and Bathurst average upper bound method is more suitable for the input of WL wa When EL wave is input, Whitman and Liao average fitting method and Newmark up bound method are both suitable. Since η4 is more conservative, the Whitman and L  Table   Figures 13 and 14 show that the change trend of the two-tiered modular reinforced earth retaining wall under earthquake is the same as that of the single-step modular reinforced earth retaining wall. When the input peak acceleration is small, the value of ηn is greater than 0. With the increase of the peak acceleration, the value of η n decreases gradually. When the input peak acceleration is 0.2-0.4 g, Richards and Elms upper bound method can better predict the displacement value. When the similarity ratio is 1:10, the Cai and Bathurst average upper bound method is more suitable for the input of WL wave. When EL wave is input, Whitman and Liao average fitting method and Newmark upper bound method are both suitable. Since η 4 is more conservative, the Whitman and Liao average fitting method is selected. The ηn values of yield acceleration obtained by the Ausilio method were 0.76372, −0.14144, −9.89728 and −56.87885 under the EL wave at 0.2-1.2 g, respectively. The ηn values of yield acceleration obtained by Muni method were 0.7029, −0.43424, −12.69613, and −10.83064. At this time, the calculated value obtained by the Ausilio method is closer to the measured value. Thus, the Ausilio method is more accurate than the Muni method.
average fitting method is selected. The ηn values of yield acceleration obtained by Ausilio method were 0.76372, −0.14144, −9.89728 and −56.87885 under the EL wave at 0 1.2 g, respectively. The ηn values of yield acceleration obtained by Muni method w 0.7029, −0.43424, −12.69613, and −10.83064. At this time, the calculated value obtained the Ausilio method is closer to the measured value. Thus, the Ausilio method is m accurate than the Muni method.

Discussion
According to the Newmark sliding block method theory, the reason for accumulation of permanent displacement after an earthquake is that when the input p acceleration km is greater than the critical acceleration kc, there is no displacement wh kc is less than km. In the actual process, when kc is less than km, there is still a grad decrease in the internal soil; that is, there will be a small displacement. Therefore, althou the yield accelerations calculated by the Muni method and E. Ausilio method are grea than the peak acceleration of the partial input, various calculation methods can be used predict the generation of the displacement.
In the single-step and two-tiered reinforced soil retaining wall models, the ηn val obtained under the yield acceleration values obtained by the E. Ausilio method and Muni method are compared. The ηn value under the E. Ausilio method is smaller th that under the Muni method, and the measured value is closer to the calculated va Therefore, the E. Ausilio method is more suitable for the displacement prediction modular reinforced soil retaining walls. The E. Ausilio method is more suitable beca Figure 13. Muni method [16] for yield acceleration at a value of η n . average fitting method is selected. The ηn values of yield acceleration obtained by Ausilio method were 0.76372, −0.14144, −9.89728 and −56.87885 under the EL wave at 0 1.2 g, respectively. The ηn values of yield acceleration obtained by Muni method w 0.7029, −0.43424, −12.69613, and −10.83064. At this time, the calculated value obtained the Ausilio method is closer to the measured value. Thus, the Ausilio method is m accurate than the Muni method.

Discussion
According to the Newmark sliding block method theory, the reason for accumulation of permanent displacement after an earthquake is that when the input p acceleration km is greater than the critical acceleration kc, there is no displacement wh kc is less than km. In the actual process, when kc is less than km, there is still a grad decrease in the internal soil; that is, there will be a small displacement. Therefore, althou the yield accelerations calculated by the Muni method and E. Ausilio method are grea than the peak acceleration of the partial input, various calculation methods can be used predict the generation of the displacement.
In the single-step and two-tiered reinforced soil retaining wall models, the ηn val obtained under the yield acceleration values obtained by the E. Ausilio method and Muni method are compared. The ηn value under the E. Ausilio method is smaller th that under the Muni method, and the measured value is closer to the calculated val Therefore, the E. Ausilio method is more suitable for the displacement prediction modular reinforced soil retaining walls. The E. Ausilio method is more suitable beca

Discussion
According to the Newmark sliding block method theory, the reason for the accumulation of permanent displacement after an earthquake is that when the input peak acceleration k m is greater than the critical acceleration k c , there is no displacement when k c is less than k m . In the actual process, when k c is less than k m , there is still a gradual decrease in the internal soil; that is, there will be a small displacement. Therefore, although the yield accelerations calculated by the Muni method and E. Ausilio method are greater than the peak acceleration of the partial input, various calculation methods can be used to predict the generation of the displacement.
In the single-step and two-tiered reinforced soil retaining wall models, the η n values obtained under the yield acceleration values obtained by the E. Ausilio method and the Muni method are compared. The η n value under the E. Ausilio method is smaller than that under the Muni method, and the measured value is closer to the calculated value. Therefore, the E. Ausilio method is more suitable for the displacement prediction of modular reinforced soil retaining walls. The E. Ausilio method is more suitable because the yield acceleration obtained by this method is slightly larger than Muni method, so the displacement value is more accurate.

Conclusions
In this paper, the experimental and calculated values are compared, and the calculation methods suitable for predicting the behavior of modular reinforced soil retaining walls under static and dynamic loads are obtained, and suggestions are provided for future experimental analysis and practical engineering.
(1) In the prediction of retaining wall calculation method under static action (P), since the FHWA method only requires few parameters (wall height and reinforcement length), the deformation value can be roughly estimated before the construction of modular reinforced retaining wall; the FHWA method is the most conservative method and the Wu (1) method is the least conservative method. Under the premise of knowing the deformation of reinforcement, the GeoService method is an accurate method to predict the lateral deformation. It is more practical to select the CTI method without knowing the strain of reinforcement. Therefore, the CTI method is recommended to estimate the normal deformation of modular reinforced earth retaining wall.
(2) By comparing the η n values of single-step and two-tiered modular reinforced earth retaining walls, it can be seen that the η n values are quite different when the peak acceleration is less than or greater than 0.6 g, which also leads to the need to use different calculation methods to predict the results. When the input peak acceleration is 0.1-0.6 g, the actual displacement value can be calculated by the Richards and Elms upper bound method through numerical calculation. When the input peak acceleration is 0.6-1.0 g, the Whitman and Liao average fitting method can truly reflect the permanent displacement of the retaining wall.
(3) Since the measured values of permanent displacement of modular reinforced earth retaining wall under static and dynamic actions are relatively large, the panel is prone to damage. Therefore, engineers should evaluate the displacement in the early and after the actual construction, and take the horizontal displacement as one of the indicators to evaluate the safety of the project.
Supplementary Materials: The following are available online at https://www.mdpi.com/article/ 10.3390/app11188681/s1, Table S1: Calculation methods of the deformation of RSRW under static loading, Table S2: Reinforced soil retaining wall parameters of 10 case histories, Table S3: Measured and predicted maximum lateral deformations of GRS walls, Table S4: Introduction of the yield acceleration, Table S5: Calculation method of the horizontal displacement for the retaining wall under earthquake action. Reference [34] refer to the supplementary material.  Data Availability Statement: This study did not report any data.