Mechanical Characteristics of Deep Excavation Support Structure with Asymmetric Load on Ground Surface

Abstract

The purpose of this paper is to capture the mechanical response of the support structure of deep excavation subject asymmetric load. A two-dimensional (2D) numerical analysis model was established by taking a pipe gallery deep excavation subject to asymmetric load as an example. The numerical analysis results were in good agreement with the measured data, thus verified the validity of the numerical model. On this basis, the stress and displacement of support structure caused by the change in foundation asymmetric load were studied. According to the numerical results, horizontal displacement of the diaphragm wall (DW) was dominant, and the maximum horizontal displacement of the DW was 7.54 mm when the deep excavation was completed. With the increase in asymmetric load, the left wall displacement continued to increase, while the displacement of the right DW continued to decrease, and the maximum horizontal wall displacement occurred near the excavation face. The DW was the main bending component, and the maximum wall bending moment when the deep excavation was completed was 173.5 kN·m. The maximum wall bending moment increased with the increase in asymmetric load, and the maximum wall bending moment on the left of the deep excavation was greater than that on the right. The inner support sustained the main component of axial force, with the axial force peaking at 1051.8 kN when the deep excavation was completed. The axial force of the inner support increased with increasing the asymmetric load, and the axial force of the second inner support was obviously greater than that of the first inner support. This research has a positive effect on the design and optimization of deep excavation support structure subject to asymmetric load on ground surface.

1. Introduction

If the structure’s geometry, support, component stiffness, and section size are all axisymmetric with respect to a certain coordinate, the structure is called a symmetric structure [1]. Symmetrical load means that the load on both sides of the symmetry axis is equal in size, and the application point and action line of the load after folding are overlapped, and the direction is the same [2]. Symmetrical structures are widely used in engineering, such as bridges, buildings, planes and so on. In addition, symmetrical structures can also be used to optimize design by changing the symmetry of structures [3]. In addition, when the symmetric structure under the action of symmetric loads, it has many characteristics, which include symmetry of reaction and displacement, symmetry of internal forces, symmetry of geometric shapes, reduction of unknown quantities, simplification of calculation, and improvement of calculation accuracy [4].

The deep excavation subject to asymmetric load refers to a deep excavation under the asymmetric force state due to the large distinction of loads on two sides of the deep excavation. The problem of asymmetric load caused by different reasons is very common [5,6,7], and the deformation behavior of the deep excavation under asymmetric load is different from that of the ordinary deep excavation. It is mainly manifested in unbalanced pile load, different digging depth, different supporting conditions, and different soil quality nearby. If it is designed according to conventional methods, there may be security risks [8,9,10]. The measured results of relevant deep excavation show that asymmetric load will have adverse effects on deep excavations, and may even cause deep excavation damage in severe cases [11,12,13].

Research related to deep excavation has had more attention paid to it by scholars, and many advancements have been achieved [14,15,16]. Wang et al. [17] studied the influence of precipitation on deep excavation by field test. Besides, Ren et al. [18] studied the characteristics of large deep excavation, including lateral displacement, column displacement, and stress. On the basis of deformation control, Wang et al. [19,20,21,22] studied the problem of deep excavation with different methods. Ye et al. [23] created a loading model of tunnel segment under lateral unloading. In addition, Guan et al. [24,25,26] studied the construction scheme and supporting technology of deep excavation.

To sum up, the related problems of deep foundation pit have attracted many scholars to carry out research, and the exploration and development of related problems are still the current hot spot. However, in current similar studies, most scholars focus on symmetric structures. The construction load is symmetrical, and there is a lack of research on the influencing factors of the mechanical characteristics change of pipe corridor deep excavation engineering under asymmetric loads. In addition, since the integrated corridor projects are mostly located in densely populated areas of cities, the construction process will inevitably have an impact on neighboring buildings [27,28,29,30]. In this paper, New eXperience of Geo-Technical analysis System (MIDAS/GTS NX) is used to numerically simulate the construction process of a tunnel deep excavation. The constitutive model is modified molar Coulomb. The deformation of deep foundation pit under asymmetric load is studied. The data of the two methods are close and the rules are similar. On this basis, the influence of load variation on the stress and deformation of the supporting structure is studied. The conclusion can provide reference for the research of related problems.

2. Project Overview

This research is based on the integrated pipe gallery deep excavation engineering of an open cut section. Figure 1 shows the profile of the deep excavation. The stacking site of relevant materials and tools in the construction process is on the left of the deep excavation, and various construction materials such as steel bars were stacked at D = 4 m away from the pit. This deep excavation is a typical problem of an asymmetric load around the pit. Through the calculation and estimation of the relevant engineers of the construction unit, we believe that in the process of deep excavation construction, there are mainly two related construction loads. The first load appears in the construction stage of the DW. Specifically, it is a symmetrical load with a size of 15 kPa and an action range of 12 m around the deep excavation. The second load occurs after all excavation of the deep excavation is completed. Specifically, it is asymmetric load. It’s on the left side of the pit. Other relevant cases are: q = 15 kPa, B = 8 m, and D = 4 m. In addition, the shape is similar to a typical narrow strip. The depth, height, and thickness of the embedded soil of the DW are 8 m, 16 m, and 600 mm respectively. The deep excavation has two internal supports from top to bottom. Among them, the first inner support material is reinforced concrete. The shape of the section is square. The sides of the square are 600 mm. In addition, the second inner support material is steel. The cross-section shape is a round tube. The thickness and diameter of the round tube are 12 mm and 609 mm respectively. The upper support spacing is 6 m and the other is 5 m. The distance between the two inner supports is 2 m. Among them, the upper supports are located 1 m below the surface. In addition, the DW and reinforced concrete internal support were made of C30 concrete. The associated weights, Poisson’s ratio, and elastic modulus are 29 kN·m⁻³, 0.2, and 2.8 × 10⁷ kN·m⁻², respectively. The weight, Poisson’s ratio, and elastic modulus of steel pipe support are 77 kN·m⁻³, 0.2, and 2.0 × 10⁸ kN·m⁻², respectively. It should be noted that the object of this study is symmetrical structure. The loads in this study are mainly asymmetric loads.

The deep excavation was excavated four times, each excavation had the same depth of 2 m. The simplified soil layer has three layers. For details, see the schematic diagram of the deep excavation section. The excavation range is mainly pebble layer. It should be noted that the effects of groundwater were not considered. The soil mechanical parameters are shown in Table 1.

3. Model Building and Numerical Calculation

3.1. Basic Assumptions

There are three basic assumptions in this study: (1) the soil in each layer is distributed continuously and evenly; (2) The DW and the support are elastomer; (3) the same material in this study is an ideal material with uniform material and the same performance in each direction.

3.2. Calculation Model

In this study, MIDAS GTS NX (version 2022) software was used to establish the model, and relevant numerical simulation analysis and calculation were carried out. The purpose of this study is the influence of asymmetric load on the mechanical characteristics of deep foundation pit supporting structure. The length is about 100 times the width. The shape is narrow and long. Therefore, except for the corner area of the deep excavation, other parts can be analyzed with reference to the plane strain problem. Consider the actual engineering situation and combine the relevant research experience. The whole two-dimensional model is established in this study. Width x = 80, height y = 40 m.

The model coordinate system of this study is shown in Figure 2. Where, the x axis is consistent with the width of the deep excavation, and the y axis is consistent with the vertical upward direction. Specifically, the boundary on both sides of the model is only allowed to shift in the vertical direction. In addition, the top is free. Vertical and horizontal displacements can occur. Furthermore, the bottom is a fixed constraint. Considering that soil deformation has the characteristic of small strain, the modified molar Coulomb is adopted in this study. The elastic constitutive relationship between the envelope structure and internal support was adopted. Moreover, in the model, the soil mass was a 2D plane element considering plane strain. 1D beam elements are used for both the inner support and DW. The overall two-dimensional model grid division is shown in Figure 2.

3.3. Simulated Construction

The processes involved in this study are controlled by the “activation” and “passivation” commands of the software. The main construction procedures include: construction of DW, soil excavation, and setting of internal support, The principle of “first set up the enclosure structure, then carry out the excavation work” is followed. In addition, excavation and support are controlled by command passivation and activation in the software, respectively. The excavation process of numerical simulation was consistent with that of actual excavation. There were seven parts in the construction condition, which were as follows: In working condition 1, the initial ground stress analysis was carried out and the displacement was cleared. In working condition 2, there was the construction of the DW, and add a symmetrical construction load of 15 kPa. In working condition 3, excavation is carried out, and the depth is 2m. At the same time, add the first inner support to the specified location. In working condition 4, continue to excavate the deep excavation, and the excavation depth is also 2 m, and a second support was added. In working condition 5, the deep excavation was excavated to 6 m below the surface. In working condition 6, excavation was conducted to 8 m below the surface. In working condition 7, add an asymmetric load of 15 kPa at the specified position.

4. Comparison between Numerical Calculation and Field Measurement

The numerical calculation has the advantages of good visualization, simulation and convenient calculation. On-site monitoring can monitor and dynamically control the construction process. If these two different methods are combined effectively, this will help to improve the reference value of numerical calculation. Besides, reasonable predictions can be made on this basis. This will have a positive impact on the project [31,32,33]. Figure 3 shows the displacement cloud image of the DW. Obviously, with the increase of construction conditions, the displacement has a tendency to increase. Specifically, When the construction phase increases from 2 to 6, the deformation of DW is symmetrical., and the maximum displacement s are 1.64 mm, 2.23 mm, 2.85 mm, 3.98 mm, and 6.53 mm, respectively. It should be noted that in the calculation results of construction phase 7, the deformation calculation cloud image results of the DW show the characteristics of asymmetric graphs. The maximum displacement of the DW on the left is different from that on the right, which is 7.53 mm and 6.19 mm respectively. Obviously, the deformation results on the left side are larger relative to the other side. This may be due to the effects of asymmetric loads. The deformation of DW will produce axial pressure on the internal support. In order to achieve the new force balance, the axial pressure is transmitted through the internal support, which in turn creates pressure on the right DW. The pressure will hinder the displacement trend of the right DW towards the pit. This blocking effect exhibits an effect similar to “inhibition”. If the asymmetric load is too large, it will further cause the reverse deformation and displacement in the direction of the outside.

The reliability of the numerical model and the rationality of the model parameter values need to be tested. In this study, the calculation results at the end of the calculation of construction phase 7 were extracted, mainly the displacement value of the DW in the X direction was extracted, and the calculated results were compared with the corresponding monitoring value of the construction site. Figure 4 shows the comparison curve between relevant simulated values and field monitoring results. It is not difficult to find that the displacement curve in the numerical simulation results is relatively close to the actual displacement monitoring in the field. This shows that the model established according to the relevant data of the construction site has a certain feasibility. In addition, the values of the relevant calculation parameters of the numerical model are reasonable and reliable.

In addition, in terms of the maximum displacement of the left DW, the results of monitoring and simulation are about 7.3 mm and 7.5 mm, respectively. Analogously, the results on the right are 5.6 mm and 6.2 mm, respectively. The position of the maximum value was close to that of the excavation face. In addition, it was also found from Figure 4 that the field monitoring results were somewhat different from the numerical simulation results. The reason may be that the research process has been simplified. However, in the actual process, there will inevitably be many influencing factors near the deep excavation. Such as: the movement of construction machinery, long-term rainfall during construction. These factors have obvious contingency and uncertainty. Their presence can make a difference in outcomes. Therefore, project managers should strengthen management. On the one hand, try to avoid unnecessary construction machinery staying near the deep excavation. On the other hand, the construction material stacking area should be set up away from the deep excavation. In addition, attention should be paid to monitoring and protection during continuous rainfall.

In summary, the results of the two different methods show that the deformation rules are close. This shows that the model design is reasonable, and also verifies that the selection of numerical simulation parameters has a certain feasibility. Therefore, based on the established model, the correlation analysis of the change of asymmetric load q on the stress and deformation of the support structure can be further carried out.

5. Influence of Asymmetric Load Change on Mechanical Characteristics of Deep Excavation Support Structure

Only the size of the asymmetric load q is changed, and D = 4 m, H = 8 m, and B = 8 m were kept unchanged. The model is used to calculate the deformation of deep excavation after the completion of excavation, when the asymmetric load q is 15 kPa, 30 kPa, 45 kPa, 60 kPa, 75 kPa, and 90 kPa, respectively. The main contents are: horizontal displacement, axial force of support structure, and bending moment.

5.1. Influence of Asymmetric Load on Horizontal Displacement of Support Structure

Figure 5 shows the displacement cloud image in the X direction. It is observed that the asymmetric load will cause the horizontal displacement of the support structure. When q changes, the displacement cloud image also changes. When q is different, the displacement of support structure in X direction also changes constantly. In addition, with the increasing of q, the displacement of the left support structure increases. On the contrary, the right side is getting smaller. The maximum value on both sides occurs near the excavation face. The components with displacement are mainly DW. Although the internal support will also produce some horizontal displacement due to asymmetric load. But it’s not big. In order to highlight the research focus, the following is a detailed analysis of the effects of different q on DW horizontal displacement.

In order to better study the influence of q on the displacement of the support structure in the X direction more intuitively, the results of numerical calculation are extracted. According to the calculation results, the software is used to create the X direction displacement comparison curve of q support structure at different times. Figure 6 shows the comparison curve of displacement of DW in X direction when q is different. Obviously, when the asymmetric load on the left support structure gradually increases from 0 to 90 kPa, the X direction displacement of the support keeps increasing. The absolute value changes greatly. The right supporting structure has a displacement in the X direction away from the deep excavation. Probably because the pressure generated by q needs to be transmitted through the internal support. When q is too large, the support structure even tends to move in reverse [34].

The table can more vividly show the change law of the X-direction displacement of the DW under the action of q. The relevant results of numerical calculation are extracted and sorted out. Table 2 shows the comparison table of maximum displacement in X direction of DW. It is observed that, when the asymmetric load gradually increases from 0 to 90 kPa, the maximum displacement of the left DW are 6.53 mm, 7.54 mm, 8.58 mm, 9.62 mm, 10.62 mm, 11.59 mm, and 12.55 mm, respectively. On the right side are 6.53 mm, 6.19 mm, 5.82 mm, 5.40 mm, 4.94 mm, 4.45 mm, and 3.95 mm, respectively.

The fitting curve can better reveal the general variation law of the relation between the displacement in X direction and q of the DW. The relevant calculation results are fitted. Figure 7 shows the fitting curve. Obviously, although when q changes, the displacement of the DW in the direction of X is also different. But there is a certain law of change between the two, showing a linear relationship. Among them, the relation between the maximum displacement of the left DW and the change of q is shown in Formula (1). The change relationship on the right side is shown in Formula (2). After calculation, the correlation coefficient R² between X and q can be obtained. It is not difficult to see that the value of R² is very close to 1 on both sides. This reflects that it is more appropriate to use linear relationship for fitting.

X = 0.0671q + 6.5547(1)

X = 0.0289q − 6.6244(2)

5.2. The Influence of Asymmetric Load Variation on Bending Moment

Figure 8 shows bending moment cloud image of support structure under asymmetric load variation. Obviously, under the action of asymmetric load, both the DW and the inner support will produce bending moments. The bending moment changes with the change in asymmetric load. Moreover, when the conditions are the same, the construction with greater bending moment is the underground diaphragm. We can find that the position where the maximum value of the left DW appears is between the second support and the excavation face. And the right side is the joint between the DW and the second support. In addition, the DW is the main bending component, and the internal support will also produce a bending moment internal force. However, compared to the DW, the bending moment is relatively small. The following is a detailed analysis of the influence of different asymmetric loads on the bending moment of the DW.

The table can more vividly show the change law of the bending moment of the DW under the action of q. The relevant results of numerical calculation are extracted and sorted out. Table 3 shows the maximum internal forces of bending moments. It is not difficult to see when the asymmetric load gradually increases from 15 kPa to 90 kPa, the maximum value of the left DW are 173.5 kN·m, 190.4 kN·m, 204.4 kN·m, 215.5 kN·m, 224.6 kN·m, and 233.4 kN·m, respectively. On the right are 120.3 kN·m, 132.9 kN·m, 145.2 kN·m, 157.5 kN·m, 170.3 kN·m, and 184 kN·m. It can be found that, with the increase in asymmetric load, the maximum bending moment of the DW is constantly increasing. And under the same load condition, the maximum bending moment value on the left side is greater. Therefore, on the one hand, it is necessary to strengthen the monitoring of the bending moment internal force of the DW which is closer to the asymmetric load. On the other hand, when the actual situation requires, relevant measures can be taken to protect. The relevant measures mainly include: increasing the thickness of underground continuous wall, increasing the embedment depth, and improving the strength grade of concrete.

The fitting curve can better reveal the general variation law of the relationship between the bending moment and q of the DW. The relevant calculation results are fitted, and a comparison graph of fitting curves is formed. Figure 9 shows comparison of fitting curves of the relationship between the bending moment value and the change of q. It is observed that, although the bending moment of the DW changes differently with the change of the asymmetric load. However, there is a certain law of change between them. Among them, the relevant fitting curve relationship on the left is shown in Formula (3). The change relationship on the right side is shown in Formula (4). After calculation, the correlation coefficient R² between M and q can be obtained. It is not difficult to see that the value of R² is very close to 1 on both sides. This demonstrates that the effect of linear relationship fitting is relatively good. In addition, the relationship between M and q in Formula (3) is an open downward parabola, while the relationship between M and q in Formula (4) is a linear relationship, which is different. This may be because the deep excavation was subjected to asymmetric load during excavation. This generates a different response of the two DWs in terms of bending moment. Therefore, it is necessary to design and monitor the internal force of relevant member to ensure the safety and stability of the deep excavation under asymmetric load.

M = −0.0048q² + 1.2879q + 155.63(3)

M = 0.8438q + 107.4(4)

5.3. The Influence of Asymmetric Load on Inner Support Axial Force

Figure 10 displays the cloud diagram of the support structure when q changes. We can discover that asymmetrical load will cause axial force in supporting structure. The axial force changes with the change in asymmetric load, and under the same conditions, the axial force generated by the internal support is greater. Besides, each inner support has the same axial force on any section. It should be noted that the axial force supported in the two channels is not all pressure. The first is the pull. Conversely, the second is stress. In addition, the internal support is the main component of axial force. Although the DW will also produce a certain axial force under the action of q. However, compared with the internal support, the axial force is small. Therefore, to strengthen the focus of this study, the following content will carry out a specific analysis of the influence of q on the internal support axial force.

In order to show the change in axial force of internal support under asymmetric load more intuitively, the numerical calculation results were extracted and sorted (see Table 4 for details). We can discover that when the asymmetric load gradually increases from 15 kPa to 90 kPa, the axial force of the first inner support are 344.1 kN, 382.2 kN, 410.5 kN, 431.8 kN, 447.5 kN, and 461.4 kN, respectively. The second are −1051.8 kN, −1103.5 kN, −1153.5 kN, −1203.4 kN, −1254.2 kN, and −1308.3 kN, respectively. It can be found that, with the increase in asymmetric load, the axial force is increasing. Moreover, under the same conditions, the absolute value of the supporting axial force in the second passage is greater than that in the other passage. Therefore, in the actual project, internal supports with greater axial force need more attention.

The fitting curve can better reveal the general variation law of the relationship between the axial force and q. The relevant calculation results are fitted, and a comparison graph of fitting curves is formed. Figure 11 presents the fitting curve of the relation between the axial force and the asymmetric load. Evidently, with the change of the asymmetric load, the axial force changes differently. It is not difficult to find that the relationship between F_N and q of the first inner support axial force is quadratic polynomial. However, there is a linear relationship between F_N and q for the other internal support axial force. Among them, the change relationship of the first support is shown in Formula (5). Another change relationship is shown in Formula (6). Through calculation, it can be concluded that the correlation coefficients R² of F_N and q are close to 1. This shows that the fitting effect is relatively good. It should be noted that the relationship between F_N and q in Formula (5) is an open downward parabola, while the relationship between F_N and q in Formula (6) is a linear relationship, which is different. This may be because the deep excavation was subjected to asymmetric load during excavation. It causes a different response in terms of the axial force of the support.

F_N = − 0.0136q² + 2.9592q + 303.98(5)

F_N = −3.399q − 1000.7(6)

6. Discussion

The research object of this paper is symmetrical structure, and the load of this study is mainly asymmetric load. Through this study, we can find that when there is an asymmetric load near the deep excavation, the supporting structure exhibits different deformation rules compared with the symmetric condition. If we do not pay enough attention to the adverse effects of asymmetric loads, it may bring unnecessary losses to the project and even increase the project risk. So, it is important to pay attention to the adverse effects of asymmetric loads in projects under similar conditions. If necessary, relevant measures can be taken to reduce the asymmetric load, so as to reduce the risk in the construction process. However, due to the regional nature of deep excavation, there are relatively many influencing factors. This time, only from a single aspect of relevant research. Despite some research results. Besides, many research results have reference significance for understanding the dynamic evolution process of deep excavation under asymmetric load and preventing deformation and failure. However, the content of this study is not rich enough. For example, the influencing factors considered in this study are relatively simple. The reality is that there are relatively many factors affecting deep excavation engineering. Some of these factors are uncertain. There may also be correlations between some factors. The situation is relatively complicated. In addition, in related projects, in addition to the bending moment and horizontal displacement of the supporting structure, there are also many mechanical-related indexes affected. For example: vertical displacement, bending moment, surrounding soil level, and vertical deformation. The emergence of these problems will also enrich the direction and content of the next stage of research. In addition, the research on the deformation characteristics of deep excavation under symmetric load is also an important research direction in the future.

7. Conclusions

In this paper, finite element method is used to study the mechanical characteristics of deep excavation supporting structure under asymmetric load. The main conclusions are as follows:

• (1). MIDAS/GTS NX was used to simulation the process of deep excavation. The deformation law of support structure caused by excavation under asymmetric load is studied. The rationality of the model is verified by comparing with the actual monitoring data.

• (2). The DW was the main component of horizontal displacement. The maximum displacement of the DW in X direction was 7.54mm. With the increasing of q, the displacement of the left DW shows a trend of increasing. On the contrary, the displacement on the right side shows a decreasing trend. The maximum displacement in the X direction occurs near the excavation face.

• (3). The DW was the main component of bending. The maximum bending moment when the deep excavation was completed was 173.5 kN·m. With the increasing of q, the maximum values of the left DW also tend to increase. Under the same load condition, the left DW is larger in terms of the maximum bending moment. The maximum values of both sides appear in different positions. The left side is between the second support and the excavation face. On the right side is the joint between the DW and the second support.

• (4). The internal support was the main component of axial force. The maximum axial force is 1051.8 kN, which occurs when the excavation of the deep excavation is completed. The first support is the pull, the other supports are the pressure. With the increasing of q, the axial force shows a trend of increasing. The axial force of the second inner support is obviously greater.

• (5). The next step will be to carry out more research on the influence of asymmetric influencing factors on deep excavation, such as changing the scope of asymmetric load, changing the position of asymmetric load, and adopting asymmetric support structure. Relevant research will further enrich the relevant achievements of deep excavation under asymmetric conditions.

Author Contributions

Conceptualization, P.Z. and P.G.; methodology, P.G.; software, Y.S.; validation, Y.S., Z.W., and P.Z.; formal analysis, P.Z. and Z.W.; investigation, Y.S.; resources, P.G. and Z.W.; data curation, P.Z.; writing—original draft preparation, P.Z.; writing—review and editing, P.G. and Z.W.; visualization, Y.S.; supervision, P.G.; project administration, Z.W. and P.Z.; funding acquisition, P.G. and Z.W. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Deep excavation profile. d was the thickness of soil, D was distance from the edge of asymmetric load to the edge of deep excavation, B was the width of the asymmetric load, and q was the size of the asymmetric load.

Figure 2. Two-dimensional grid rendering of numerical model. q is the asymmetric load; purple, yellow, and green areas represent, respectively, artificial fill, cobble, and sandy claystone layers.

Figure 3. Displacement cloud image in the X direction: (a) construction phase 2; (b) construction phase 3; (c) construction phase 4; (d) construction phase 5; (e) construction phase 6; (f) construction phase 7.

Figure 4. Comparison graph between simulated and monitored values: (a) left side; (b) right side.

Figure 5. Displacement cloud image in the X direction:(a) q = 15 kPa; (b) q = 30 kPa; (c) q = 45 kPa; (d) q = 60 kPa; (e) q = 75 kPa; (f) q = 90 kPa.

Figure 6. Comparison curve of displacement in X direction when q changes: (a) left side; (b) right side.

Figure 7. The fitting curve of the relationship between the maximum displacement in X direction and q: (a) left side; (b) right side.

Figure 8. Moment cloud image of support structure with different asymmetric loads: (a) q = 15 kPa; (b) q = 30 kPa; (c) q = 45 kPa; (d) q = 60 kPa; (e) q = 75 kPa; (f) q = 90 kPa.

Figure 9. Comparison of fitting curves of the relationship between the maximum bending moment value and the change of q: (a) left side; (b) right side.

Figure 10. Axial force cloud diagram of support structure with different asymmetric load: (a) q = 15 kPa; (b) q = 30 kPa; (c) q = 45 kPa; (d) q = 60 kPa; (e) q = 75 kPa; (f) q = 90 kPa.

Figure 11. The fitting curve of the relation between the axial force and the asymmetric load: (a) first inner support; (b) second inner support.

Table 1

Soil layer mechanical parameter.

Soil Layer	E50ref (kN·m−2)	Eoedref (kN·m−2)	Eurref (kN·m−2)	v	γ (kN·m−3)	c (kN·m−2)	φ (°)
Artificial fill	4100	4100	12,300	0.38	18.2	18	22
Cobble	11,000	11,000	33,000	0.33	20.2	2	43
Sandy Claystone	8100	8100	24,300	0.3	21.3	36	33

where, E50ref represents the shear stiffness of the triaxial test, Eoedref represents the tangential stiffness of the loading test, Eurref represents the unloading elastic modulus, v represents Poisson’s ratio, γ represents the bulk density, c represents the cohesion force, and φ represents the friction angle during shear failure.

Table 2

Comparison table of maximum displacement in X direction.

Asymmetric Load	q = 0	q = 15 kPa	q = 30 kPa	q = 45 kPa	q = 60 kPa	q = 75 kPa	q = 90 kPa
Left side (mm)	6.53	7.54	8.58	9.62	10.62	11.59	12.55
Right side (mm)	−6.53	−6.19	−5.82	−5.40	−4.94	−4.45	−3.95

Table 3

Table of maximum bending moments.

Asymmetric Load	q = 15 kPa	q = 30 kPa	q = 45 kPa	q = 60 kPa	q = 75 kPa	q = 90 kPa
Left side (mm)	173.5	190.4	204.4	215.5	224.6	233.4
Right side (mm)	120.3	132.9	145.2	157.5	170.3	184

Table 4

Internal support axial force.

Asymmetric Load	q = 15 kPa	q = 30 kPa	q = 45 kPa	q = 60 kPa	q = 75 kPa	q = 90 kPa
First inner support (kN)	344.1	382.2	410.5	431.8	447.5	461.4
Second inner support (kN)	−1051.8	−1103.5	−1153.5	−1203.4	−1254.2	−1308.3

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