Study on Dynamic Response Characteristics of Stepped Reinforced Retaining Wall

Abstract

In order to further explore the influence of reinforcement materials laying position on the dynamic characteristics of reinforced retaining walls, based on FLAC3D finite difference methods for solving nonlinear problems, established with the same size of the retaining wall in practical engineering, the reinforcement material’s relative position in the retaining wall for the normalized processing, under seismic load, and the panel under the conditions of different reinforcement arrangement were analyzed, and the horizontal displacement of slope, the vertical and horizontal earth pressures behind the wall, and the distribution of potential sliding surface were calculated. The relationship between the maximum horizontal displacement of the panel and the laying position of the reinforcement was fitted by MATLAB. The results show that the horizontal displacement of the panel is about 40% smaller when the upper layer of the reinforcement is arranged than that in the lower layer for step 1, and the horizontal displacement of the reinforcement in step 2 is about 30% lower than that in other conditions when the reinforcement is arranged at the top of the slope. The reinforcement arranged in the lower layer of step 1 and the upper layer of step 2 can minimize the wall top displacement. In step 1, the vertical earth pressure and horizontal earth pressure are 19% and 5% smaller than those in other conditions when the reinforcement is arranged near the middle and lower layer of the step. In step 2, the difference between vertical and horizontal earth pressure is not obvious, and the difference between the two conditions is controlled within 5%. At the same time, soil liquefaction and uplift occur under the action of earthquake. The position of sliding crack surface has no obvious regularity with the position of reinforcement, but the position of reinforcement at the step classification is obviously better than other conditions. The fitting formula can describe the relationship between the panel displacement and the position of the reinforcement well. The conclusions can provide a point of reference for practical engineering.

1. Introduction

As a flexible structure, geosynthetics reinforced retaining walls can better absorb seismic loads and adapt to foundation deformation and have better seismic performance, mainly used to strengthen and improve the overall performance of foundations, roads and slopes [1,2,3,4,5,6]. Experimental studies and numerical simulations have been widely used to study the mechanical behavior and deformation characteristics of reinforced retaining walls. The static characteristics of reinforced retaining walls can be studied by indoor model tests. Although the dynamic characteristics of retaining walls can be simulated by shaking table tests [7,8,9,10], this experiment costs a lot of manpower and material resources, is economically unreasonable, and the correctness of the obtained data is difficult to verify. In recent years, numerical methods have been widely used to study the dynamic characteristics of reinforced retaining walls.

Quite a number of studies have shown that the properties of reinforcement materials have a great influence on the reinforcement effect of retaining walls [11,12,13]. Wu et al. [14] used finite element software to simulate and analyze the dynamic characteristics of reinforced earth retaining wall under seismic load and showed that the horizontal displacement of reinforced earth retaining walls could be effectively reduced by using reinforcement material with large length, and the horizontal displacement of the whole reinforced earth retaining wall panels could be effectively limited by reducing the spacing of reinforcement. Sitar [15] used numerical research to study the influence of dynamic load on reinforced soil walls, indicating that the stiffened stiffness, stiffened spacing and the type of filler in the stiffened area have an impact on the stiffened retaining wall. Konthesingha [16] considered two different wall aspect ratios and six different batten reinforcement schemes, and the study showed that the maximum load of the wall strengthened by the vertical and horizontal reinforcement schemes was improved, and the maximum bearing capacity was increased by 9%.

Researchers [17,18,19] have analyzed the influence of load factors on the dynamic characteristics of retaining walls. Latha [20] discussed the shaking table test of the retaining wall of soil workshop under the condition of ground shaking with different accelerations and frequencies. The results showed that soil displacement and amplification increased with the increase of vibration frequency, and the effect was more obvious at lower acceleration amplitude. According to the study of Sun et al. [21], the bearing capacity of the geoclear reinforced embankment under strip load is about 2.5 times that of pure sand, which improves the ultimate bearing capacity of the embankment and reduces the settlement of the embankment, and the lateral displacement of the slope surface of the geocell reinforced embankment is reduced by 75% compared with that of pure sand embankment. Xiao et al. [22] studied the influence of step width D on the reinforced earth retaining wall under the action of driving load. The maximum horizontal displacement of the panel appeared at the height of the upper wall about 0.85H, and the load amplitude and frequency had a more obvious influence on the strain of the upper retaining wall reinforcement.

Relevant personnel also studied and analyzed the dynamic characteristics of the mixed retaining wall formed by placing a thin sand layer in the retaining wall [23], and the results showed that there was an optimal value for the thickness of the thin sand layer of the reinforced soil retaining wall. When the thickness of the thin sand layer increased from 3 cm to 10 cm, the horizontal displacement, acceleration amplification coefficient and maximum internal force of reinforcement all showed a decreasing trend first and then an increasing one. Song et al. [24] used numerical analysis to study the deformation characteristics of the flexible retaining wall of geocell under different aspect ratios, different slopes and soil filling surface acting loads. Ganbaatar [25] studied the in-plane shear strength and deformation capacity of wood-reinforced retaining walls under cyclic horizontal loads and static loads.

To sum up, the existing research on reinforced retaining walls is still limited to three aspects: (1) the properties of reinforcement materials, including length, modulus and distance between reinforcement materials; (2) load conditions, including static load and dynamic load; dynamic research less so; (3) reinforcement materials are basically geogrid, rarely involved in geogrid; (4) retaining wall types, most studies are only limited to single-stage retaining walls, not involving stepped retaining walls (secondary or multistage). Therefore, in view of the shortcomings of the existing research, this paper makes use of the geotechnical engineering finite element software FLAC3D, uses the built-in Geogrid unit to form the geocell unit in three-dimensional arrangement, takes the geocell as the reinforcing material, and researches the dynamic response of the stepped reinforced retaining wall under the action of earthquake load. The retaining wall deformation pattern, earth pressure and slip surface distribution are paid special attention. The research results have certain referential significance for the dynamic response of stepped reinforced retaining wall under seismic load.

2. Finite Element Model

2.1. Constitutive Model Selection of Soil and Panel

There are many kinds of FLAC3D built-in models, including elastic model, elastic-plastic model, strain softening model, joint model, etc. The Mohr–Coulomb constitutive model is the most common geotechnical constitutive model; it describes the shear sliding surface of the relationship between shear strength and normal stress on the surface. Taking soil reflection as the basic characteristic of repose of friction strength is relatively reasonable. Based on the Mohr–Coulomb yield criterion, the biggest Mohr’s circle soil material is damaged when there is a tangent between soil mass stress state and strength envelope, as shown in Figure 1. This model has the following two assumptions:

(1)

The principal stress has no effect on the strength of soil.

(2)

The strength envelope of soil is a straight line, that is, the internal friction angle is constant and has nothing to do with the confining pressure.

These assumptions generally do not cause large errors, but when the plane strain state and stress level are large, the yield plane has derivative discontinuous corner points on the plane, which is not convenient in numerical calculation, and the convergence speed is slow. In order to avoid the above problems, the strain-softening Mohr–Coulomb criterion model in FLAC3D has a smooth track on the π plane [26], which can describe the mechanical behavior of soil after damage under seismic load, as shown in Figure 2. For concrete slab, the linear elastic model can better describe the stress–strain behavior of concrete material under the low-stress condition.

2.2. Modeling

2.2.1. Geometric Model

The size of the built geometric model appears in Reference [27]. Considering the different types of retaining walls and in order to maximize the simulation of the actual situation of retaining walls, the final size of the model is determined as follows: the height of the retaining wall is 10 m, the height of the lower steps is 5 m, the height of the upper steps is 5 m, and the width is 4 m. The calculation depth of the foundation is 10 m, and the length is 30 m. The model is divided into three parts: foundation, wall and panel. The geometric dimensions and mesh division of the model are shown in Figure 3.

2.2.2. Material Parameters

(1)

Soil and face plate

The retaining wall foundation is soft soil, and the wall is backfilled soil. The material parameters of the foundation and wall are listed in

Table 1. Mechanical parameters of foundation and wall.

Type	Bulk/MPa	Density/(kg/m3)	Friction/(°)	Cohesion/kPa	Shear/MPa
Retaining wall	16.8	1720	28	25,000	7.2
Foundation	20.9	1520	23	35,000	9.8

.

The mechanical parameters of concrete panels are shown in

Table 2. Mechanical parameters of the panel.

Elastic/MPa	Porosity	Density/(kg/m3)	Thickness/mm
30 × 103	0.2	2400	0.5

. The shell element is used to simulate the panel, and the related parameters of C30 reinforced concrete are used.

(2)

Geocell

Considering that there is no built-in geocell model in Flac3D, Geogrid units provided with the software can be used to form geocell units through staggered arrangement, as shown in Figure 4. The friction characteristics between reinforcement and soil can be simulated by the spring-slider system, so as to reasonably solve the problem that there is no structural unit of geocell in FLAC3D. The specific model of geocell is shown in Figure 3, and the panel and geocell are rigidly connected. The mechanical parameters of geocell are listed in

Table 3. Mechanical parameters of geocell.

Elastic/MPa	Porosity	Density/(kg/m3)	Coupling Cohesion /kPa	Coupling Friction /(°)	Thickness/mm
5.5 × 102	0.3	1000	12,000	27	3

.

Mechanical parameters of concrete panels are shown in

Table 4. Mechanical parameters of the panel.

Elastic/MPa	Porosity	Density/(kg/m3)	Thickness/mm
30 × 103	0.2	2300	0.5

. The shell element is used to simulate the panel, and the related parameters of C30 reinforced concrete are used.

(3)

Contact surface

In order to simulate the interactions between panels, between panels and soil fill, and between foundation and wall, contact surfaces are set at the three interfaces. Since the Geogrid unit is a structural plane unit, no contact surface is needed between the lattice and soil fill. The interface parameters of retaining wall and soil behind the wall and the interface between foundation and soil fill mainly include bond force, friction angle, normal stiffness and tangential stiffness. Chen [28] conducted simulation tests on a large number of engineering examples and concluded that the values of the bonding force and friction angle were about times of the adjacent soil layer on the contact surface, and the values of the normal stiffness and shear stiffness were calculated using the following formula:

$$k_n=k_s=10\max \left[\frac{\left(K+\frac{4}{3}G\right)}{\mathsf{\Delta }{z}_{\min }}\right]$$

(1)

where K is the volume modulus, G is the shear modulus, Δz_min is the minimum size of the normal connection area of the contact surface.

The specific parameters of the contact surface are shown in

Table 5. Contact surface parameters.

Contact Type	Shear Stiffness/$\mathbf{G}\mathbf{P}\mathbf{a}$	Normal Stiffness/$\mathbf{G}\mathbf{P}\mathbf{a}$	Cohesion/kPa	Internal Friction Angle/(°)
Foundation-wall	4.53	4.53	56	26
Panel-panel	7.35	7.35	1000	45
Panel-wall	5.25	5.25	53	31

.

2.3. Load Input

2.3.1. Seismic Load

In the analysis and study of dynamic load, the input of dynamic load is realized by reading files in TABLE format by FLAC. The Kobe earthquake in Japan is one of the most representative earthquakes because it caused great human and property damage. Since the frequency of seismic action is in the range of about 1~5 Hz, SeismoSignal seismic wave processing software was used for filtering operation to filter out the part of seismic wave with frequency greater than 5 Hz. The maximum acceleration was adjusted to 0.6 g, and the earthquake duration was 20 s. The maximum acceleration occurred at 8.5 s. The seismic acceleration time history is shown in Figure 5.

Since the foundation is soft soil and belongs to the flexible foundation, the dynamic load at the bottom cannot directly apply the velocity or acceleration, and the acceleration or velocity time history should be converted into the stress time history and applied to the bottom of the model. The conversion formula is as follows [28]:

$${\sigma }_n=a\left(\rho C_p\right){\upsilon }_n$$

(2)

$${\sigma }_s=a\left(\rho C_s\right){\upsilon }_s$$

(3)

where, σ_n and σ_s are normal and shear stresses imposed by seismic wave; $v_n$ and $v_s$ are normal stress and tangential stress applied on the static boundary at the bottom. a is the amplification coefficient related to the material parameters of the model, and its value is 1–2. ρ is the soil density; $C_p$ and $C_s$ are the P wave velocity and S wave velocity and can be expressed as [29]:

$$C_P=\sqrt{\frac{K+\frac{4}{3}G}{\rho }}$$

(4)

$$C_S=\sqrt{\frac{G}{\rho }}$$

(5)

where K is the mass modulus of soil, and G is the shear modulus of soil.

The velocity time history can be transformed into the corresponding stress time history by multiplying the velocity component by the corresponding coefficient. Dynamic load input is shown in Figure 6.

2.3.2. Boundary Condition

In terms of the selection of boundary conditions, static boundary conditions are used for static calculation. The model is fully constrained in three directions at the bottom, Y direction at the front and back and X direction at the left and right surfaces. In the dynamic analysis, the free field boundary is applied around the model, and the static boundary condition is applied at the bottom, that is, the two-dimensional plane grid and one-dimensional cylinder grid are generated on each side of the model, which are used to simulate the free field motion without ground structure, so as to simulate the same effect of infinite field. This is shown in Figure 7.

2.3.3. Damping

The main source of resistance is friction within the material and possible sliding of the contact surface. In dynamic calculation, damping of the natural system under dynamic load needs to be reproduced in numerical simulation. Rayleigh damping was initially set up in the dynamic analysis of structures and elastos in order to reduce the amplitude of natural vibration of the system. The following formula is commonly used in calculation to show the relationship between damping matrix C, mass matrix M [29] and stiffness matrix K in the dynamic equation:

$$C=\alpha M+\beta K$$

(6)

where α represents the damping constant proportional to the mass, and β represents the damping constant proportional to the stiffness. Two parameters needed to be determined for Rayleigh damping are the minimum central frequency and the minimum critical damping ratio, which can be calculated by the following formula [28]:

$${\xi }_{\min }={\left(\alpha \cdot \beta \right)}^{1/2}$$

(7)

$${\omega }_{\min }={\left(\alpha /\beta \right)}^{1/2}$$

(8)

where ${\xi }_{\min }$ denotes the minimum critical damping ratio, and ${\omega }_{\min }$ denotes the minimum center frequency.

For geotechnical materials, the critical damping ratio is generally within the range of 2~5%, 0.05 in this paper. In simple models, the natural vibration frequency of the structure can be regarded as the minimum center frequency of Rayleigh damping. Under the above boundary conditions, no damping is set, gravity is applied, and 10,000 steps are solved to make the model oscillate. An oscillation period is found, and the natural vibration frequency of the model can be obtained by taking the reciprocal. The oscillation period of the model is 0.19 s, and the minimum center frequency of the system is 5.26 Hz.

3. Test Principle and Case Demonstration

3.1. Test Principle

In this study, the influence of the laying position of reinforcement material on the reinforcement effect of retaining wall is mainly considered. How to determine the position of reinforcement qualitatively becomes the focus of this paper. N layers of geocell arranged in each level of steps, and the spacing is ΔD. The N-layer cell of each layer is regarded as a whole, and the distance between the center height of the cell and the bottom surface is H₁, H₂, and the distance between the cell whole can be expressed as:

$$\mathsf{\Delta }H=H_2-H_1$$

(9)

Then, the relative position of reinforcement in the retaining wall can be determined as follows:

$$U\left(x,y\right)=f\left(H_1,H_2,\mathsf{\Delta }H\right)$$

(10)

In order to facilitate comparison, the distance $\mathsf{\Delta }H$ between the whole cell and the height H₁, H₂ of the step are normalized, and the relative position of the reinforcement in step 1 and step 2 can be expressed as:

$$U_I\left(x,y\right)\to \frac{\mathsf{\Delta }H}{H_1}$$

(11)

$$U_{II}\left(x,y\right)\to \frac{\mathsf{\Delta }H}{H_2}$$

(12)

When the distance ΔH between the whole cell is constant, H₂ can be determined by H₁; when H₁ (H₂) is larger, the reinforcement is closer to the upper layer of the retaining wall and vice versa. By adjusting two parameters ΔH, H₁ (H₂), the influence of reinforcement material position on the reinforcement effect of the reinforced retaining wall can be analyzed, as shown in Figure 8.

3.2. Model Verification

In the absence of indoor shaking table equipment, according to the principle of similar ratio [30,31], the numerical model was scaled down 1:5 to establish the indoor model. The step reinforced retaining wall model equipment includes model box, reaction frame, jack, data acquisition instrument, etc., as shown in Figure 9. The length, width and height of the model box are 200 cm × 100 cm × 120 cm, respectively. The overall frame is made of welded steel, and I-beam steel is welded on both sides to ensure the structural strength. Steel plates are welded on the back and bottom of the wall. Applying petroleum jelly evenly on the glass before the test can effectively reduce the friction on the side wall. The reaction frame is welded on the model box through two vertical channel steels, and the angle steel is welded at the joint of channel steels to meet the strength requirements.

The height of the two steps is 0.5 m; the wall material is backfilled soil, and the panel is made of C30 concrete. The physical and mechanical properties of the soil and the panel are consistent with the finite element model. The horizontal displacement of the panel is obtained by laser rangefinder.

Static load is applied through a jack at the top of the slope to obtain the variation trend of the horizontal displacement of the panel with the wall height, as shown in Figure 10. It can be seen that under a certain static load condition, the calculated lateral displacement of the wall panel is approximately consistent with the measured results, with an error of about 10%. The data show that the finite element numerical model used in this study can reliably estimate the deformation and mechanical behavior of the reinforced retaining wall.

4. Results and Discussion

As can be seen from the previous text, the variable $\mathsf{\Delta }H$ tends to indicate the distribution position of the reinforcement material in the whole retaining wall, while the variable $\mathsf{\Delta }H/H_1$ ($\mathsf{\Delta }H/H_2$) tends to explain the distribution position of the reinforcement material in a certain step. These two variables, respectively, reflect the position of reinforcement in the retaining wall from two aspects. The reinforcement effect is analyzed as follows.

4.1. Effect of $\mathsf{\Delta }H$ on Reinforcement Effect of Retaining Wall

4.1.1. Horizontal Displacement of Panel

The initial positions of the whole of the two cells start from the step; the position of the whole of the cell 1 moves down, and the position of the whole of the cell 2 moves up. Figure 10 shows the influence of different $\mathsf{\Delta }H$ values on the maximum horizontal displacement of the panel.

It can be seen from Figure 11 that the overall horizontal displacements of step 1 and step 2 decreased with the increase of wall height. The influence of $\mathsf{\Delta }H$ on the horizontal displacement of the panel is mainly reflected in three aspects: (1) When $\mathsf{\Delta }H$ is small, the overall horizontal displacement of step 1 is the largest, the maximum and minimum horizontal displacements are 5.5 cm and 2.6 cm. The reason is that the two room whole near the reinforced area overlap, the soil reinforcement scope is limited, with the increase of $\mathsf{\Delta }H$, geotechnical reinforcement scope of lattice room and wall back horizontal earth pressure parts shall be borne by the reinforcement material; the effect on the front panel of earth pressure decreases, and horizontal displacement decreases. Under the condition of $\mathsf{\Delta }H$ = 6, the maximum and minimum horizontal displacements are 4.2 cm and −0.7 cm, and the average horizontal displacements of the panel are reduced by about 40% compared with those under the condition of $\mathsf{\Delta }H$ = 3. (2) When $\mathsf{\Delta }H$ = 4, 6 and 7, the panel of step 1 has a displacement to the back of the wall because the soil in the middle of the wall is liquefied due to the earthquake, the bearing capacity of the soil becomes weak, and the panel slips backward. This is highly consistent with the shaking table test results in Reference [7]. As the position of reinforcement moves down, the liquefaction area increases, as shown in the red box. (3) The horizontal displacement of step 2 is similar when $\mathsf{\Delta }H$ = 3–6. On the one hand, the seismic energy is dissipated due to the higher retaining wall, and on the other hand, the upward propagation seismic wave energy is greatly reduced due to the enhanced filtering effect of soil in the reinforced area. When $\mathsf{\Delta }H$ = 7, the panel displacement of step 2 is the smallest, and compared with the previous four working conditions, the displacement decreases are 37.1%, 34.2%, 33.5% and 25.1%, respectively.

4.1.2. Wall Top Displacement Time History

Under seismic load, while focusing on the horizontal displacement of the reinforced retaining wall panel, the horizontal pattern of the top of the retaining wall is also a problem worth considering because the horizontal displacement of the top of the wall significantly affects the safety of the superstructure. The creep effect of soil and reinforcement occurs in the retaining wall under dynamic load. The correlation shows that the obvious creep effect of soil and reinforcement leads to a great increase in the total horizontal displacement of the wall after an earthquake [32]. The center point at 1 m distance behind the panel of step 1 and step 2 is selected, and the point distribution is shown in Figure 12 to analyze its horizontal displacement mode under seismic load.

Figure 13 shows the time history of horizontal displacement of the wall tops of step 1 and step 2. Since the acceleration time history is small in the range of 0−7 s, the displacement of step 1 and step 2 in this stage is almost 0. From 8.5 s, the horizontal displacement begins to change greatly, and the horizontal displacement of the wall top reaches the maximum at about 18.8 s. The influence of parameters on wall top displacement is mainly reflected in three aspects: (1) When $\mathsf{\Delta }H$ = 4 and 5, the displacement time history curves of step 1 and step 2 are close to coinciding, indicating that the displacement control effects of the two reinforcement laying conditions on the slope top are similar. (2) The displacement amplitude of the time–history curve of step 1 at $\mathsf{\Delta }H$ = 3 was the largest, and the average amplitude was about 34% higher than that at $\mathsf{\Delta }H$ = 4 and 5. The final displacement is the largest in all working conditions, which is about 2.3 cm. When $\mathsf{\Delta }H$ = 7, the amplitude and displacement of the displacement time–history curve are both the smallest, indicating that the safety of the slope top is the best when the reinforcement is arranged on the upper layer of step 1, which is consistent with the relevant conclusions in literature [27]. The negative value of the displacement at the top of the slope is due to the creep effect of the soil, which causes the soil to slip towards the wall. (3) For step 2, when $\mathsf{\Delta }H$ = 4 and 5, the final displacement is the largest, reaching 1.3 cm. The displacement time history curves of step 2 are similar under the conditions of = 6 and 7, but the final displacement of step 2 is smaller when $\mathsf{\Delta }H$ = 6, about 0.25 cm. The reason is that the layout of geocell in the lower layer improves the compactness of soil and effectively limits the upward propagation of seismic load.

In general, laying the reinforcement near the lower layer is conducive to reducing the slope top displacement of step 1, while laying the reinforcement in the upper area is more beneficial to step 2. For the retaining wall as a whole, the reinforcement should be laid at the bottom and the top, respectively.

4.1.3. Earth Pressure Distribution

The influence of different $\mathsf{\Delta }H$ on the horizontal and vertical earth pressure behind the panel is shown in Figure 14.

It can be seen from Figure 14a that the vertical earth pressure has a similar distribution law under different $\mathsf{\Delta }H$: the vertical earth pressure of the two steps has an approximately ‘S’ shaped distribution. The maximum vertical earth pressure of step 1 appears at 0.2H, and the maximum vertical earth pressure of step 2 appears at 0.5 H. The positive earth pressure of the upper wall in the range of 6~8 m may be caused by the soil uplift there under the action of earthquake. The vertical earth pressure is different under different conditions: (1) In step 1, when the reinforcement material condition is $\mathsf{\Delta }H$ = 3, the average vertical earth pressure behind the panel is the smallest, but the absolute value of vertical earth pressure at the bottom reaches 180 kPa, which is much higher than other working conditions (50–80 kPa). When $\mathsf{\Delta }H$ = 4−7, the vertical earth pressure of each measuring point is similar. The reason is that the reinforcement is close to the upper layer, the compactness of the upper layer increases, and the stress diffusion angle is larger when the vertical earth pressure propagates, while the lower earth pressure is smaller. The vertical earth pressure is about 19% lower than that under other conditions when $\mathsf{\Delta }H$ = 3. (2) The vertical earth pressure distribution is similar at different condition for step 2, the average earth pressure difference under different conditions is controlled within 5%, because there is energy dissipation when the seismic wave propagates upward. In addition, due to the nonlinear characteristics of soil, the filtering effect of soil is strengthened, and the reinforcement effect of reinforcement material is not obvious.

It can be seen from Figure 11b that the variation trend of horizontal earth pressure is consistent at different $\mathsf{\Delta }H$. The maximum value of the horizontal earth pressure of step 1 is at 0.15 H, and the maximum value of step 2 is at the bottom of the upper wall. The horizontal earth pressure of step 1 and step 2 is the smallest when $\mathsf{\Delta }H$ = 3 because the panel displacement is large at this time. Step 1 height is 3~5 m, and bench height is 6~8 m within the scope of the lateral earth pressure because the earthquake is under the action of soil liquefaction and panel one-way passive earth pressure. Step 1 and step 2 are the sizes of the active earth pressure control in the 2~8 kPa and 4~15 kPa range, and it can be seen step 2 of soil liquefaction is more severe.

In general, under dynamic load, the laying of geocell in the middle and upper layers is very beneficial to reduce the overall earth pressure of the retaining wall, but the control of local earth pressure in other laying conditions is better.

4.1.4. Slip Plane

Figure 15 shows the cloud images of the overall horizontal displacement of the retaining wall under different $\mathsf{\Delta }H$ conditions.

As can be seen from the figure above, the location of the potential slip surface is different for different $\mathsf{\Delta }H$: (1) When $\mathsf{\Delta }H$ = 3, there is no obvious slip surface of the retaining wall, indicating that the reinforcement laid at the retaining wall classification is conducive to the control of the overall deformation of the retaining wall. (2) When parameter $\mathsf{\Delta }H$ changes from 4 to 5, the slip develops towards the interior of the wall, and the shape and position of the slip surface of step 1 are similar to the fracture surface of the “0.3 h” method [33]. The shape of the slip fracture surface of step 2 is divided into two sections. The fracture surface of the first section is approximately at an angle of 30° with the horizontal direction and extends to about 5 m behind the panel, while the fracture surface of the second section is approximately at an angle of 15° with the horizontal direction. It is worth noting that when $\mathsf{\Delta }H$ = 6, the slip surface only exists in the interior of step 1, while there is no obvious slip surface in the interior of step 2, which may be related to soil liquefaction phenomenon during the earthquake. (3) When $\mathsf{\Delta }H$ = 7, the distribution trend of the slip surface is similar to that when $\mathsf{\Delta }H$ = 5. In conclusion, the reinforcement laid at the step classification can effectively limit the overall horizontal displacement of the step retaining wall, and the sliding area of the retaining wall is also minimal.

4.2. Influence of $\mathsf{\Delta }H/H_1$ on Reinforcement Effect of Retaining Wall

4.2.1. Horizontal Displacement of Panel

Under the condition of $\mathsf{\Delta }H$ = 5, the influence of reinforcement effect is studied. Figure 16 shows the variation trend of the horizontal displacement of the panel.

It can be seen from Figure 16 that the horizontal displacement of retaining wall panel decreases with the increase of height. The influence of parameter $\mathsf{\Delta }H/H_1$ on the horizontal displacement of the panel is reflected in the following two aspects: (1) For step 1, except for $\mathsf{\Delta }H/H_1$ = 5/3.5, the panel displacement curves under the other four working conditions are close to coincide. The overall horizontal displacement of step 1 is the largest when $\mathsf{\Delta }H/H_1$ = 5/3.5, and the average horizontal displacement is about 1.1 cm higher than that under other working conditions. It shows that under certain conditions, the reinforcement effect in the middle and upper area of the retaining wall is better than that in other positions, which is consistent with the previous conclusion. (2) When $\mathsf{\Delta }H/H_1$ = 5/1.5, the overall displacement of step 2 reaches the maximum, and the average displacement of the panel reaches 6.2 cm, which is much larger than other working conditions. The minimum overall displacement of step 2 occurs when $\mathsf{\Delta }H/H_1$ = 5/3.5, and the average displacement of the panel is about 2.4 cm, with a difference of about 61.3%. It can be seen that the laying position of reinforcement has a significant influence on the horizontal displacement of the retaining wall, and the laying of geogrid close to the upper layer is beneficial to improve the safety performance of the retaining wall.

4.2.2. Earth Pressure Distribution

It can be seen from Figure 17a that the distribution of vertical earth pressure is consistent under different parameters. The influence on the vertical earth pressure is mainly reflected in two aspects: (1) when the parameter $\mathsf{\Delta }H/H_1$ = 5/3.5, the overall vertical earth pressure of the retaining wall is the minimum, and when the parameter $\mathsf{\Delta }H/H_1$ = 5/2.0, the overall vertical earth pressure is the maximum, indicating that the change trend of the earth pressure does not change in a single direction with the movement of the reinforcement but shows a nonlinear trend. In the height ranges of 1–3 m and 6–8 m, the difference between the maximum and minimum vertical earth pressure is about 45%, and the difference between the earth pressure in other areas is only about 5%.

It can be seen from Figure 17b that the horizontal earth pressure of step 1 decreases with the height of the wall, and the overall horizontal earth pressure is the minimum when parameter $\mathsf{\Delta }H/H_1$ = 5/2.5 and the maximum when parameter $\mathsf{\Delta }H/H_1$ = 5/3.0, indicating that the reinforcement laid in the middle area of the wall is conducive to reducing the horizontal earth pressure behind the panel, while the horizontal earth pressure in the opposite direction appears at the top of step 1. This is due to a certain degree of deflection of the panel along the toe of the wall under seismic load caused by soil liquefaction. The horizontal earth pressure of step 2 decreases first and then increases. When parameter $\mathsf{\Delta }H/H_1$ = 5/3.5, the overall horizontal earth pressure is the maximum, and when parameter $\mathsf{\Delta }H/H_1$ = 5/1.5, the overall horizontal earth pressure is the minimum. The average horizontal earth pressure of $\mathsf{\Delta }H/H_1$ = 5/3.5 is about 10kPa smaller than that of  = 5/1.5.

It can be seen that under certain conditions of parameter $\mathsf{\Delta }H$, the geocell laid in the upper area of the retaining wall can effectively control the earth pressure behind the panel and improve the overall security of the retaining wall. This is highly consistent with the previous conclusion.

4.3. Influence of $\mathsf{\Delta }H$ and $\mathsf{\Delta }H/H_1$ on Maximum Horizontal Displacement of Retaining Wall

The maximum horizontal displacements of upper and lower walls are extracted under different conditions, as shown in

Table 6. Maximum horizontal displacement of panel (Unit: cm).

$\mathsf{\Delta }\mathit{H}$	$\mathsf{\Delta }\mathit{H}/{\mathit{H}}_1$	${\mathit{X}}_{1\mathbf{max}}$	${\mathit{X}}_{2\mathbf{max}}$
3	3/3.5	5.5	3.2
4	4/2.5	4.1	3.0
	4/3.0	4.2	3.1
	4/3.5	4.2	3.2
5	5/1.5	3.8	2.8
	5/2.0	3.9	2.9
	5/2.5	4.2	3.2
	5/3.0	4.3	2.9
	5/3.5	4.3	3.0
6	6/1.5	3.8	2.9
	6/2.0	4.2	3.1
	6/2.5	4.3	3.1
7	7/1.5	3.0	1.2

. $X_{1\max }$ represents the maximum horizontal displacement of step 1, and $X_{2\max }$ represents the maximum horizontal displacement of step 2.

By programming in MATLAB, the relationship between $\mathsf{\Delta }H$ and $\mathsf{\Delta }H/H_1$ and the maximum horizontal displacement of the step is fitted. The relationship between the maximum horizontal displacement of the panel and $\mathsf{\Delta }H$ and $\mathsf{\Delta }H/H_1$ can be fitted by the Gaussian polynomial:

$$\{\begin{array}{l}{X}_1{}_{\max }=9.84{\left(\mathsf{\Delta }H\right)}^{-0.5942}\left(R^2=0.8066\right)\\ X_2{}_{\max }=1.12e^{15}{\left(\mathsf{\Delta }H\right)}^{18.04}+3.18\left(R^2=0.997\right)\end{array}$$

(13)

$$\{\begin{array}{l}{X}_1{}_{\max }=-0.099{\left(\frac{\mathsf{\Delta }H}{H_1}\right)}^4+1.8{\left(\frac{\mathsf{\Delta }H}{H_1}\right)}^3-12{\left(\frac{\mathsf{\Delta }H}{H_1}\right)}^2+36\left(\frac{\mathsf{\Delta }H}{H_1}\right)-35\left(R^2=0.9244\right)\\ X_2{}_{\max }=0.0014{\left(\frac{\mathsf{\Delta }H}{H_1}\right)}^4-0.23{\left(\frac{\mathsf{\Delta }H}{H_1}\right)}^3+3.2{\left(\frac{\mathsf{\Delta }H}{H_1}\right)}^2-16\left(\frac{\mathsf{\Delta }H}{H_1}\right)+30\left(R^2=0.9566\right)\end{array}$$

(14)

Figure 18 shows the fitting effect of Equations (13) and (14) on the original data. It can be seen that the above formula can well describe the relationship between the maximum horizontal displacement of the panel and $\mathsf{\Delta }H$ and $\mathsf{\Delta }H/H_1$. Through the fitting curve, the maximum horizontal displacement of step 1 and step 2 panel under different reinforcement laying position can be predicted in a certain range, which has certain guiding significance for the actual project construction.

5. Conclusions

In this paper, the nonlinear finite difference method is adopted to establish the calculation model of stepped reinforced retaining walls, and the correctness of the model is verified by the indoor static model test. The effect of reinforcement laying position on deformation and mechanical properties of retaining walls under seismic load is further studied. The results show that:

(1)

The position of reinforcement material has significant influence on the horizontal displacement pattern and earth pressure distribution of stepped reinforced retaining wall. Reinforcement laying in the upper layer can effectively reduce the horizontal displacement of the step 1 panel and laying in the upper layer can minimize the horizontal displacement of the step 2 panel. Under the condition of certain $\mathsf{\Delta }H$, the horizontal displacement of the panel in step 1 is 15% higher than the other four conditions when the reinforcement is arranged on the upper layer, and the horizontal displacement of the panel in step 2 is 61.3% lower when the reinforcement is arranged on the top of the wall than at the bottom of the wall. Secondly, the reinforcement laid in the upper layer is the most favorable to reduce the displacement of the top of the step 1, and the control effect of the horizontal displacement of the top of the step 2 is the best when laid on the lower layer. In addition, when the reinforcement is arranged at the bottom of step 1 and step 2, the vertical and horizontal earth pressures of the retaining wall are small, and when $\mathsf{\Delta }H$ = 3, the vertical earth pressure and horizontal earth pressure are about 19% lower than the other four conditions. At a certain time, the vertical earth pressure and horizontal earth pressure under the condition of parameter $\mathsf{\Delta }H/H_1$ = 5/3.5 decrease by about 16% on average compared with other conditions. The negative value of vertical earth pressure is caused by soil uplift inside the wall, while the negative value of horizontal earth pressure is caused by soil liquefaction.

(2)

The location of the fracture surface of the retaining wall changes with the change of parameters. When $\mathsf{\Delta }H$ = 3, no obvious slip surface appears, while when $\mathsf{\Delta }H$ = 6, the fracture surface only exists in the step 1 area. The position change of the sliding surface has no obvious regularity with respect to the parameters, but the reinforcement layout at the step classification is obviously due to other conditions.

(3)

Through the formula fitting of the maximum horizontal displacement of step 1 and step 2 panels, it is shown that there is a certain quantitative relationship between the maximum horizontal displacement of step 1 panel and parameters $\mathsf{\Delta }H$ and $\mathsf{\Delta }H/H_1$, which has certain reference value for relevant practical projects.

(4)

In this paper, the consideration of the variable of reinforcement location is not comprehensive enough, and it does not involve all possible locations of reinforcement laying. In the future, correlation functions may be introduced to describe the position of reinforcement materials in retaining walls, but this study is still of considerable significance for the construction of reinforced retaining walls.