The behavior of a rectangular closed diaphragm wall when used as a bridge foundation

Abstract

The rectangular closed diaphragm (RCD) wall is a new type of bridge foundation. Compared to barrette foundation, measuring the performance of RCD walls is relatively complicated because of their incorporation of a soil core. Using the FLAC3D software, this paper investigates the deformation properties, soil resistance and skin friction of a laterally loaded RCD wall as well as the settlement, axial force and load-sharing ratio of a vertically loaded RCD wall. Special attention is given to the analysis of factors that influence the performance of the soil core. It was found that under lateral loading, the RCD wall behaves as an end-bearing friction wall during the entire loading process. The relative displacement between the wall body and the soil core primarily occurs below the rotation point, and the horizontal displacement of the soil core is greater than that of the wall body. Under vertical loading, the degree of inner skin friction around the bottom of the soil core and the proportion of the loading supported by the soil core increase with increased cross-section size. The wall depth is directly proportional to the loading supported by the outer skin friction and the tip resistance of the wall body and is inversely proportional to the loading borne by the soil core.

Keywords

diaphragm wall / soil core / bridge foundation / FLAC3D / bearing behavior

Introduction

Diaphragm walls are underground walls constructed with reinforced concrete, concrete or other predominantly cement-based materials. Diaphragm walls can have a structural, water-retaining or protective function and are suitable for both temporary and permanent applications [1]. Often called concrete diaphragm walls [2-4], slurry walls [5-7] or cutoff walls [8-11], diaphragm walls are used throughout the world in various engineering fields.

The rectangular closed diaphragm (RCD) wall, shown in Fig. 1, is a new type of bridge foundation composed of a cap and diaphragm walls. The diaphragm walls under the cap are connected with rigid joints that form a rectangular frame in horizontal section [12]. As a result of their high construction efficiency, small cost, low noise, strong rigidity and good compactness, RCD walls are especially suitable for large-span bridge foundations [13–15]. Because of the incorporation of a soil core, the load transfer mechanism and bearing behavior of an RCD wall become complicated.

Diaphragm walls were originated in Europe, and closed diaphragm walls were devised in Japan. Although Verfel [16] and Nichol and Wilson [17] have introduced the application of barrettes in the construction of bridge foundations, the application of closed diaphragm walls used directly as bridge foundations has not yet been reported in Europe or the United States. In 1979, a rigid closed type of diaphragm foundation was used in an elevated bridge at Shinkansen, Japan, which was the first application of the diaphragm wall technique in a bridge foundation [18]. Subsequently, closed diaphragm walls have undergone substantial development in Japan [19–21].

Recently, various filed and indoor tests, theoretical calculations and numerical analyses of the bearing behavior of diaphragm walls have been performed, and there have been significant advancements, especially with respect to diaphragm wall bearing elements (i.e., barrettes) [22-28]. Nevertheless, there are several differences between RCD walls and barrettes, including their structural conformation and bearing behavior. Most precious field tests have addressed the bearing capacity but have not further studied or discussed other bearing properties and mechanisms [19,22]; for example, the research on wall-soil interactions and the physics governing the variation in soil resistance are far from complete [10,25,27,28].

This paper describes the bearing mechanisms of RCD walls under lateral and vertical loading determined using the FLAC3D software based on a total stress approach. The deformation properties, variations in the distribution of the moment and shear with wall depth, soil resistance and the skin friction of an RCD wall are systematically studied. Special attention is given to the performance of soil cores with variable cross-sectional dimensions and different wall depths. The conclusions will be useful in the subsequent study of RCD walls as bridge foundations.

Numerical modeling

Basic assumptions

The analysis of the bearing behavior of RCD walls is based on the following assumptions:

1) The wall concrete is assumed to be linear-elastic, and the soil is treated as an elasto-plastic material based on Mohr-Coulomb criteria.

2) The friction coefficient of the wall-soil interface element remains constant.

3) The self-weight stress field is considered and treated as the initial stress of the loading process.

Model dimension and material parameters

A square RCD wall model of dimensions D = 20.0 m, L = 8.0 m, l = 6.4 m, t = 0.8 m, T = 2.0 m, d = 1.0 m is simulated as the representative model in this paper. The wall concrete grade is C25, and the lateral soil and the bottom soil are of the same type of the typical loess and are homogeneous.

The property of the wall-soil interface is a key factor influencing the bearing behavior of the friction wall. Research by Potyondy [29] and Acer et al. [30] has shown that the relationship δ/φ = 0.6~0.7 (δ is the interface friction angle and φ is the internal friction angle) is suitable for cohesive soil. Therefore, in this paper, the c and δ of the wall-soil interface are 2/3 of the cohesion and internal friction angle of the soil body, respectively. All parameters used in the analysis of the RCD wall are shown in Table 1.

Constitutive model and boundary conditions

In the numerical analysis, a linear-elastic model is applied to the wall body, and a Mohr-Coulomb model is employed for the surrounding soil. In general, the selection of a model boundary depends on trial calculations. Those areas with insignificant impacts or no effects on the bearing behavior of the RCD wall can be considered as the boundary. Accordingly, the RCD model boundary is delineated by 4L in the horizontal direction and 2.5D in the vertical direction. The soil body is divided into two layers (a lateral soil layer and a bottom soil layer). In addition, hinges at the bottom and chain poles at the lateral sides are used to constrain the conditions of the calculation area.

To determine the initial stress field, the unit density of the RCD wall is initially considered to be identical to that of the soil body. The initial displacement field is then returned to zero, and the excess unit density of the RCD wall is overlain on the initial stress field in the form of stress generated by the second loading round. Thus, the initial stress field of the RCD wall and foundation can be accurately simulated. The initial stress is defined by

(1){\sigma }_z={\gamma }_z,{\sigma }_{\mathrm{h}}=K_{\mathrm{0}}{\gamma }_z,

where σ_z is the initial vertical stress, σ_h is the initial horizontal stress, γ is the specific weight of soil and K₀ is the horizontal pressure coefficient ranging from 0.95 to sinϕ [31].

The typical 3D FD mesh is shown in Fig. 2. The mesh consists of 23,020 zones with a total of 25,177 grid points. The outer boundary of the mesh is fixed against displacement.

Loading scheme

Figures 3(a) and (c) illustrate the lateral and vertical loading scheme, respectively, of the RCD wall. To facilitate the analysis, the loadings are applied in the middle of the back wall and the midpoint of the cap respectively because the cross section of the typical RCD wall is square. The Q is 57.60 MN, and the H₀ is varying from 12 to 60 MN. In addition, each level of loading is applied evenly on the wall top. For the sake of the efficiency of the analysis, specialized points are identified as shown in Figs. 3(b) and (d).

Analysis of a laterally loaded RCD wall

Deformation properties

Wall deformation

The relative horizontal displacement ΔS_h indicates the differential property of the horizontal deformation of each laterally loaded wall. Figs. 4(a) and (b) separately show the variation in the displacement ΔS_h with depth between points O₁ and O₂ (see the point locations in Fig. 3(b)), and between O₁ and O₃ or O₄. The ΔS_h-D curve appears to be nonlinear and ΔS_h increased with greater horizontal loading on the wall top. Overall, because of the large integral stiffness, high strength of the foundation and the small deformation difference of ΔS_h, the horizontal deformation of each wall is thought to be synchronous.

Figure 4(c) illustrates the S_h-D curve of the horizontal displacement of the wall body with varying depth (positive horizontal displacement values indicate that the displacement direction and the loading direction are uniform and negative values suggest the opposite; the same conditions are applicable below). It can be inferred that the laterally loaded RCD wall rotates around a certain point centered above the wall toe, and the failure pattern appears to be an overall incline failure. The horizontal displacement increases with additional loading, shows a linear reduction with depth and reaches zero at approximately 0.875 D of the wall body. From 0.875 D to the wall toe, the displacement direction of the wall body is inversely related to the loading direction, and the displacement of the wall body increases with depth and loading on the wall top.

The S_v-H₀ curves for each wall are superimposed in Fig. 4(d) (a positive vertical displacement value indicates uplift and a negative value indicates downward settlement; the same conditions apply below). Fig. 4(d) shows the following results: upward vertical displacement occurs at the back wall leading to uplift, and the extent of uplift increases gradually with increased loading, which enhances the uplift; downward vertical displacement and compression occur in the front wall, and the displacement shows a slight decreasing tendency as the loading develops; for the lateral wall, the vertical displacement gradually changes from compressive settlement to upward migration and increases with greater loading, which suggests that the rate of change of the uplift becomes greater.

Lateral soil deformation

3.1.2.1‚Deformation characteristic of the soil core

The lateral displacement property of the soil core is similar to that of the RCD wall, as shown in Figs. 4(c) and 5(a). Based on Fig. 5(a), it can be observed that the zero point of horizontal displacement is located at approximately 0.875D of the wall body. In addition, the lateral displacement of the soil core shows a linear decrease above the zero point, whereas the displacement increases in the opposite direction with additional loading below the zero point.

Figure 5(b) shows that the vertical displacement of the soil core tends to occur in response to compressive settlement when the loading is relatively small. With increased loading on the wall top, the vertical displacement gradually changes to an upward direction. Below the bottom of the soil core, the vertical displacement is effected by loading on the wall top and decreases drastically and nonlinearly with depth.

The relative horizontal displacement between points O₁ and O (see Fig. 3(b)), in which positive values indicate soil compression and negative values indicate detachment, is illustrated in Fig. 5(c). It is clear that the relative displacement between the wall body and the soil core primarily occurs below the rotation at approximately 0.125D and appears to be negative, which indicates that the horizontal displacement of the soil core is greater than that of the wall body. In addition, above the rotation point, the relative displacement between the wall body and the soil core is positive and relatively small, which suggests that the distribution and value of the horizontal displacement of the soil core generally coincides with that of wall body.

3.1.2.2‚ Deformation characteristics of the soil outside the back wall

The point of zero horizontal displacement of the outside back wall is located at the rotation point of the wall body; i.e., the zero displacement point is below the ground surface at approximately 0.875D, as shown in Fig. 6(a). The horizontal displacement of the soil body above the zero point increases with greater loading on the wall top and shows a nonlinear change with depth. Below the zero point, the direction of the horizontal displacement is opposite to that of the loading direction.

The vertical displacement of the soil outside the back wall appears to be in the upward direction (Fig. 6(b)), and reaches a maximum value near the wall toe. When the loading on the wall top is relatively small, the maximum displacement value appears in the soil body above the wall toe over a specific depth range. From the top downward, the vertical displacement reduces nonlinearly and rapidly. With the growth of loading on the wall top, the vertical displacement increases, and the maximum extent of uplift mainly occurs at the wall toe.

The relative horizontal displacement between the soil outside the wall and the wall body (i.e., between the soil outside the back wall and point O₁ in Fig. 3(b)) reflects the extent of the wall-soil detachment. Figure. 6(c) indicates that with greater loading on the wall top, ΔS_h tends to show a nearly linear decrease and a detachment develops.

3.1.2.3‚Deformation characteristics of the soil outside the front wall

Along the wall depth, the lateral displacement of the soil outside the front wall resembles that of the wall body, as shown in Figs. 4(c) and 7(a). The vertical displacement under loading on the wall top indicates continuous settlement (Fig. 7(b)) and varies slightly across the wall body. Loading on the wall top has little influence on either the horizontal or vertical displacement of the soil body below the wall toe.

The relative horizontal displacement between the soil outside the wall and the wall body (i.e., the displacement between the soil outside the back wall and point O₂ in Fig. 3(b)) is plotted in Fig. 7(c). The results show that at approximately 2D/3 of the wall body, ΔS_h is zero, which means that the displacement of the wall and of the soil is the same under a critical contacting condition. Above the zero point of ΔS_h, the relative horizontal displacement ΔS_h is positive; i.e., the horizontal displacement of the wall body is greater than that of the soil and the front wall is under compression. Below the zero point of ΔS_h, the relative horizontal displacement ΔS_h is negative (i.e., the horizontal displacement of the wall body is less than that of soil), which indicates that a detachment occurs between the wall and the soil.

The horizontal displacement of the surface soil outside the front wall decreases drastically and nonlinearly with the increasing distance from the wall body, as shown in Figure. 8(a), and the maximum displacement occurs near the lateral wall. Figure 8(b) shows that the vertical displacement tends to show downward compression when the loading on the wall top is relatively small. With greater loading, plastic failure occurs in the lateral soil, and soil uplift forms along a certain depth of the surface soil outside the front wall. The maximum vertical displacement occurs at a distance of a/2 from the lateral wall. In addition, the displacement decreases sharply with the distance from the wall body.

The moment, shear and angular displacement of the wall body

Figure 9(a) shows that for a laterally loaded RCD wall, the moment of the wall body ascends with the increased loading; at the same time, the moment of the wall body first increases progressively, and then drops down as the embedded depth develops. At approximately 0.45D in depth below the ground, (i.e., near the central point of depth), the moment of the wall body reaches a maximum, and the position of maximum moment tends to move downward with increased loading on the wall top. The zero value of the moment does not appear in the overall wall body, which indicates that the foundation failure pattern is a general incline failure.

The variation in the shear curves for the wall body with depth is superimposed in Fig. 9(b). It can be observed that the shear is the greatest at the point of load application on the wall top and increases with the extent of loading. The shear shows a nonlinear decrease with the embedded depth and reaches a value of zero where the maximum moment occurs; below the zero value point, the shear distribution shows a parabolic change with depth. Near the wall toe, the shear decreases slightly as a result of the plastic state of the soil body and partial stress release surrounding the wall toe.

The angular displacement increases with greater loading, and the varying law of the θ-H₀ curve agrees with that of the S_h-D curve for the wall body; both values vary slightly, as illustrated in Figs. 10 and 4(c).

Lateral soil resistance

Two types of earth pressure, active and passive, are involved in loading process and occur in lateral soil resistance (negative resistance values represent active earth pressure and positive values indicate passive earth pressure; the same conditions apply below).

The lateral soil resistance of the outside wall shows two distinct distribution patterns bordering the rotation point of the wall body. For the soil outside the back wall, the lateral soil resistance above the rotation point appears to be active earth pressure, as shown in Fig. 11(a). As the loading increases, a detachment between the wall and soil develops, and the lateral soil resistance increases. From the beginning point of the wall-soil detachment down toward the rotation point, the value of the lateral resistance decreases to zero resulting from the block function of the wall body; below the rotation point, the soil outside the wall is compressed, and the lateral resistance, which increases with greater loading and embedded depth, appears to be passive earth pressure.

The distribution of lateral soil resistance outside the front wall is different from that of the lateral soil resistance outside the back wall (Fig. 11(c)). The soil outside the front wall is compacted above the rotation point of wall body, as shown in Fig. 7(c), and its lateral soil resistance, which increases along with greater loading and decreases with embedded depth, appears to be passive earth pressure. Below the rotation point, the wall and soil are in a state of detachment (Fig. 7(c)), and the lateral soil resistance shifts to active earth pressure.

In addition, the lateral soil resistance outside the front wall near the ground surface is greatest when the loading on the wall top is relatively small. When the loading is relatively large, uplift occurs in the soil outside the front wall along a critical depth below the ground surface, as shown in Fig. 8(b), and the lateral soil resistance is smaller than that in deeper location. In addition, the critical depth develops, and the maximum lateral soil resistance value shifts downward with increased loading.

The distribution of the lateral soil resistance of the soil core is relatively complicated compared to the infinite soil body of the outside wall because the soil core is equivalent to a one-dimensional soil column. Above the rotation point, the horizontal displacement of the soil core develops with approximately the same trend as that of the wall body, i.e., the relative wall-soil displacement is nearly zero. Below the rotation point, a relatively large displacement effected by the soil surrounding the wall toe occurs between the wall and the soil (Fig. 5(c)). Thus, the distribution and value of the lateral soil resistance of the soil core above and below the rotation point are significantly different, as shown in Figs. 11(b) and (d): above the rotation point, the soil resistance is so small that it can be ignored entirely, whereas below the rotation point, the lateral soil resistance of the inside back wall appears to be active earth pressure, as illustrated in Fig. 9(b), and the resistance of the inside front wall is passive earth pressure, as shown in Fig. 11(d).

Skin friction

The failure pattern of the laterally loaded RCD wall is a general incline failure: The vertical displacement of the back wall and the corresponding outside soil tends to show uplift, and the lift rate of the back wall is larger than that of the outside soil. These results lead to the commencement of vertical shear deformation between the wall and the soil and the progressive formation of downward outer skin friction, as shown in Fig. 12(a). The outer skin friction gradually increases with increased loading on the wall top. The outer skin friction near the detachment area between the back wall and the outside soil is zero. In addition, the detachment becomes larger with increased loading, which causes the effective depth of the skin friction to gradually diminish downward toward the bottom of the wall body.

Because the vertical displacements of the front wall and its outside soil tend to show settlement, and are influenced by the lateral soil resistance, the outer skin friction gradually develops in an upward direction as the wall depth and loading increase, as shown in Fig. 12(c). When the loading is relatively small, the outer skin friction around the cap is largest because the lateral soil resistance near the ground surface reaches its maximum value and the relative wall-soil displacement is active. Nevertheless, the decrease in the outer skin friction is small near the ground surface because of the weakening effect of the lateral soil resistance caused by surface soil uplift when the loading is fairly large.

Figure 12(e) shows that the position at which the greatest wall-soil shear deformation occurs is also where the outer skin friction is in full effect. Furthermore, when the loading is small, the outer skin friction of the lateral wall accumulates with increased loading, and the maximum value point moves gradually downward. Below the rotation point, the outer skin friction develops with an increase in wall depth and stays nearly constant with the depth at greater extents of loading. When the loading reaches 48 MN, the outer skin friction increases with the depth as the relative wall-soil displacement is high; however, with increased loading, the change in the outer skin friction is fairly small, which indicates that the outer skin friction is in a state of full exertion.

Figures 12 (b), (d) and (f) illustrate the variation in inner skin friction with depth. These diagrams show that the inner skin friction of the lateral wall mainly occurs below the rotation point and that there is little change as loading develops. Above the rotation point, it is difficult for inner skin friction to occur because the deformation of the wall body and soil core is essentially synchronous, and wall-soil shear deformation is typically not generated. Compared to the lateral wall, the inner skin friction of both the back and front walls develops reasonably well and increases with increased loading. In addition, at shallower depths, it is difficult to exert inner skin friction because the relative displacement between the soil core and the wall is relatively small or essentially zero.

The distribution of the inner skin friction is related to the overall incline failure of the RCD wall, which leads to the uplift of the back wall and compression of the front wall.

Analysis of vertically loaded RCD wall

Foundation settlement

In Figs. 13-15, variations in the soil settlement below the wall toe in the soil core and in the nearby soil outside the wall with depth are superimposed. The five curves in each figure represent situations with 20%, 40%, 60%, 80% and 100%. The relative displacement between the wall and soil core and between the wall and the soil adjacent to the outside wall are shown in Figs. 16 and 17, respectively.

The settlement of the wall body mainly consists of soil compression below the wall toe (Fig. 13); in addition, the settlement of the soil below the wall toe decreases steeply and nonlinearly with depth, which indicates that the load transferred from the wall top is relatively small and has a limited effect on the settlement of the soil below the wall toe.

As shown in Figs. 14 and 16, the settlements of the upper soil core and the wall body are in close agreement under each level of loading, i.e., the extent of compression of the upper soil core is relatively small because of the presence of the cap. As a result of substantial counterforce, the soil core bottom is compressed, leading to an upward displacement of the soil core relative to the wall body and a gradual reduction in the extent of the soil core settlement. For soil below the bottom of the soil core, the settlement decreases drastically and nonlinearly with depth, as shown in Fig. 14.

Figures 15 and 17 illustrate that when the loading on the wall top is relatively small, the settlement between the soil adjacent to the outside of the wall and the wall body is almost negligible; however, the differential settlement gradually increases, and a relatively large extent of soil-wall displacement occurs as loading develops. When the loading reaches its maximum value, the settlement of the bottom soil increases sharply (Fig. 15) because plastic failure occurs within the soil surrounding the wall toe. The settlement of the soil below the wall toe shows a nonlinear reduction, which suggests that the effect of loading on soil settlement is fairly small.

The axial force of the wall body

Figure 18 shows the axial force of the wall body with depth under different levels of loading.

It can be inferred that, on the wall top, loading is supported by the cap and wall body and that the axial force is relatively small. With increasing depth, the loading changes to be mainly supported by the wall body, and the axial force is greater at the joints between the cap and wall body. Therefore, the axial force decreases with depth and is affected by skin friction. When the loading is relatively small, the axial force curve appears as an approximately straight line, and the load transferred to the wall toe is small; the axial force curve gradually changes to be an arced line with greater curvature as loading develops, especially for those positions near the wall toe where the exertion of skin friction is relatively high.

Skin friction

In contrast to the infinite soil body outside the wall, the soil core is equivalent to a one-dimensional soil column. Therefore, shear deformation mainly occurs in the soil outside the wall, whereas compression is always found within the soil core. The varied distributions of the inner and outer skin friction with depth are given in Figs. 19 and 20, respectively, and the definition of the corner and side midpoints are shown in Fig. 3(d).

As shown in Fig. 19, the outer skin friction, excluding the friction on the wall top, increases linearly under small loads, and changes to a parabolic trend at the wall toe because the relative wall-soil displacement is considerably large (see Fig. 17). Because of the plastic deformation of the soil body surrounding the wall toe, the soil deviates from the outside wall and weakens the horizontal stress of the outside wall. Thus, the outer skin friction of the wall toe is minimized at this point in response to the stress softening effect. In addition, the maximum outer skin friction occurs near the wall top resulting from the stress concentration caused by soil compression.

A comparison of Figs. 19(a) and (b) indicates that the exertion of the outer skin friction is uneven, especially when the loading is small. The outer skin friction of the corner is exerted at a shallower depth and is greater than that of the side midpoint; as the loading increases, the increased outer skin friction of the corner is also greater than that of the side midpoint.

Compared to the outer skin friction, the inner skin friction is much smaller because the relative displacement between the wall and soil core appears to be elastic. The exertion of the inner skin friction of the upper soil core is small because the displacement of the upper soil core closely agrees with that of the wall body under the cap (see Fig. 16). However, an upward compressive displacement of the soil body occurs below the soil counter force at the bottom of the soil core, which causes the inner skin friction of the lower section of the soil core to increase exponentially, as shown in Fig. 20. The scope of exertion of the inner friction is restricted within the range of 1/4 of the wall depth adjacent to the wall toe; at the wall toe, the inner skin friction decreases sharply in response to the stress-softening effect resulting from the occurrence of plastic deformation within the soil body. A comparison of Figs. 20(a) and (b) shows that the distribution of the inner skin friction is also uneven. The exertion of the inner skin friction near the corner is greater than the exertion at the side midpoint, particularly for the upper soil core.

Load-sharing properties

In Fig. 21, the inner and outer skin friction is derived from the sum of the product of the shear and unit area; the counterforce on the top of the soil core is the sum of the product of the vertical stress and unit area; and the tip resistance of the wall body is derived from the value of the loading on the wall top minus the skin friction and the counterforce on the top of the soil core.

The load-sharing process can be divided into two stages: The load sharing is stable in the first stage and begins to change in the second stage. In the initial stage when the loading ratio is less than 0.27, all of the curves in Fig. 21 are essentially linear, which indicates that the skin friction, tip resistance of the wall body and counterforce on the top of the soil core increase in the same proportion (i.e., the load-sharing ratio remains constant and the loading is undertaken by all bearing components without change). In the first stage, the loading is mainly supported by the outer skin friction, and the distribution of the load-sharing ratio is as follows: 78.88% by outer skin friction, 15.19% by tip resistance, 5.67% by inner skin friction and 0.26% by the counterforce on the top of the soil core. When the loading ratio exceeds 0.27, the equilibrium state is broken, leading to the start of the second stage. Although the outer skin friction continues to increase with depth, its load-sharing ratio gradually decreases, whereas the load-sharing ratio of the tip resistance of the wall body continues to increase. The remaining increased loading is supported by a combination of the outer skin friction, the tip resistance of the wall body and the inner skin friction.

Thus, the exertion of skin friction and tip resistance do not occur at the same loading step; only after the outer skin friction and tip resistance of the wall body occur in sequence, can the inner skin friction be increased. During the entire loading process, the load-sharing proportion for the counterforce on the top of the soil core is consistently small and can be ignored.

Analysis of factors influencing the bearing behavior of the soil core

The performance of soil cores with different cross-sectional dimension

In this section, five RCD models with D = 20 m, t = 0.8 m and l/t = 4, 6, 8, 10, 12 (i.e., l = 3.2, 4.8, 6.4, 8.0 and 9.6 m) are simulated and analyzed.

Figures 22(a), 23(a) and 24(a) show the settlement contour of the RCD wall with different soil core cross-sectional dimensions. It can be inferred that the settlement distributions for the wall top and the wall toe are largely uniform in different circumstances. Under the ultimate load (the corresponding load at which the settlement of the wall top is 60 mm [32]), the amounts of compression of the soil derived from the differential settlement of the top and bottom of the soil core were 7.387, 8.429 and 10.820 mm for each model, respectively, which indicates that the amount of compression increases with the enlargement of the dimension. These results illustrate that the compressive depth of the soil core increases as the cross-section dimension increases, which occurs primarily because the influential area of the confining effect of the inside wall is reduced and the amount of free space available for compression becomes greater.

Figures 22(b), 23(b) and 24(b) illustrate the vertical stress contours of the RCD wall with different cross-sectional dimensions. These figures show that with an increase in the cross-sectional dimension, the contour of σ_z surrounding the bottom of the soil core becomes increasingly moderate, i.e., σ_z shows relatively little change in the vertical direction but shows a large change in the horizontal direction. When l/t is 4, 8 and 12, the final loads on the wall top are 23.91, 33.73 and 42.44 MN, respectively; the load-sharing ratios on the top of the soil core are 0.074%, 0.26% and 0.68%, respectively, and the load-sharing ratios of the skin friction are 2.10%, 5.02% and 7.81%, respectively. These results show that the loading on the wall top and the load-sharing ratio on the top of the soil core increase correspondingly as the dimension of the soil core increases, which stimulates an increase in the amount of compression in the soil core. As a result of a strong exertion of inner skin friction, the load-sharing ratio of the skin friction for the soil core also increases.

Figures 25(a)–(c) demonstrate the load-sharing ratio of the outer skin friction (f_o/Q_i), of the tip resistance of the wall body (R_t/Q_i) and of the soil core (Q_fc/Q_i), respectively, with variation in the wall top loading. In Fig. 25(a), f_o/Q_i decreases with greater loading on the wall top, and the increased rate of change in the outer skin friction gradually decreases as the loading develops; f_o/Q_i diminishes progressively for an l/t of 4 to 10, but shows a slight increase when l/t is 12, which indicates that the RCD wall acts as an end-bearing friction wall throughout the loading process. Figure. 25(b) illustrates that the variation in the load-sharing ratio of the tip resistance of the wall body is the exact inverse of the outer skin friction.

Figure 25(c) illustrates that the Q_fc/Q_i appears to be a catenary curve that changes according to the enlargement of the cross-sectional dimension. When Q_i/Q is less than approximately 0.6, Q_fc/Q_i decrease with an increase in Q_i/Q, i.e., the increasing rate of change in Q_tc/Q_i gradually diminishes with the increase of loading on the wall top. When Q_i/Q is greater than 0.6, Q_fc/Q_i increases with the higher Q_i/Q values, i.e., the rate of change in Q_fc/Q_i gradually increases with greater loading on the wall top. When l/t is 4, 6, 8, 10 and 12, the corresponding ultimate load-sharing ratio (when the settlement of the wall top is 60 mm) is 2.17%, 3.73%, 5.28%, 6.34% and 8.49%, respectively. These results show that the exertion of the inner skin friction around the bottom of the soil core and the loading supported by the soil core increase with the enlargement of the cross-sectional dimension.

Soil cores with different wall depths

The settlement and stress contours of three RCD wall models (t = 0.8 m, D = 15 m with 8 m × 8 m cross section; t = 0.8 m, D = 25 m with 8 m × 8 m cross section; t = 0.8 m, D = 40 m with 8 m × 8 m cross section) are plotted in Figs. 26–28.

A comparison of Figs. 26(a), 27(a) and 28(a) illustrate that the settlement of the wall top and the wall toe are nearly the same at different wall depths. When under the ultimate load, the amounts of compression of the soil core are 9.708, 8.625 and 7.525 mm, respectively, which indicates that the extent of soil compression decreases with depth. This occurs mainly because it is difficult to fully exert the outer skin friction, and the amount of load transferred to the wall toe decreases gradually with wall depth. The differential settlement of the central bottom of the soil core and wall toe tends to decrease with a greater wall depth because the extent of the wall compression increases and the settlement of the wall toe diminishes with the depth.

Figures 26(b), 27(b) and 28(b) illustrate the vertical stress contours of the RCD wall at different wall depths. The results show that when D is 15 m, 25 m and 40 m, the corresponding ultimate load on the wall top is 30.65, 34.48 and 38.26 MN, respectively, and the load-sharing ratio on the top of the soil core is 0.32%, 0.25% and 0.25%, respectively, and the skin friction is 7.41%, 4.55% and 3.68%, respectively. Thus, the ultimate load increases correspondingly and the load-sharing ratio on the top of the soil core tends to decrease as the wall depth increases. This change occurs because the full exertion of the outer skin friction beneath the deeper wall requires greater wall-soil relative displacement, which makes it difficult to quickly transfer the load from wall top to the toe; thus, the extent of soil core compression decreases, and there is little exertion of the inner skin friction, leading to a reduction in the skin friction of the soil core.

Figure 29 shows the load-sharing ratios of the outer skin friction, tip resistance of the wall and soil core, respectively, with varied loading on the wall top. In Fig. 29(a), f_o/Q_i decreases with greater loading on the wall top, i.e., the increase in the rate of change in the outer skin friction diminishes as the loading develops; f_o/Q_i increases gradually, and the f_o/Q_i - Q_i/Q curves tend to be smooth when D is 15, 20, 25 and 30 m. However, f_o/Q_i decreases slightly when D reaches 40 m. When Q_i/Q<0.33, the relative displacement between the soil and the outside wall causes the exertion of outer skin friction, and f_o/Q_i diminishes from 87.99% to approximately 73.66%. When Q_i/Q>0.33, the outer skin friction changes into a stabilized developing phase, and f_o/Q_i remains constant at approximately 73% with an increase in Q_i/Q.

Figure 29(b) shows that the variation in the load-sharing ratio of the tip resistance of the wall body is nearly the opposite to that of the outer skin friction; R_t/Q_i increases with increasing Q_i/Q and decreases when D is 15, 20, 25 or 30 m. As the wall depth increases, the length of the flat segment of the R_t/Qi – Q_i/Q curve increases. When the wall depth reaches 40 m, R_t/Q_i shows an abnormal change; when Q_i/Q<0.33, R_t/Q_i increases from the initial 8.50% to 21.50% and then remains at approximately 23.18% regardless of changes in Q_i/Q.

Figure 29(c) shows that when the depth is relatively shallow (e.g., 15 or 20 m), the Q_fc/Q_i changes as a catenary curve with increased Q_i/Q, i.e., the change in Q_fc/Q_i shows a decreasing and increasing trend with increased Q_i/Q. As the depth increases, the slope of the Q_fc/Q_i – Q_i/Q curve appears to be small, which suggests that the rate of change in Q_fc/Q_i does not vary with increased loading on the wall top. When D is 15, 20, 25, 30 and 40 m, the load-sharing ratio of the soil core is 7.73%, 5.28%, 4.80%, 4.62% and 3.93%, respectively. This shows that with a greater wall depth, a greater proportion of the loading is supported by the outer skin friction and tip resistance of the wall body, and a smaller fraction of the loading is borne by the soil core. When D exceeds a certain range, the load-bearing contribution of the soil core can be ignored.

Conclusions

The behaviors of an RCD wall and corresponding soil core were studied using the finite difference program of FLAC3D. Based on the findings of this study, the following conclusions can be drawn:

1) The laterally loaded RCD wall rotates around a specific point above the wall toe. The horizontal displacement increases with increased loading, which shows that a linear reduction occurs with depth and reaches zero at approximately 0.875D of the wall body. The relative displacement between the wall body and the soil core primarily occurs below the rotation point at approximately 0.125D, and the horizontal displacement of the soil core is greater than that of the wall body. Above the rotation point, the distribution and horizontal displacement values for the soil core are basically in line with that of the wall body.

2) The moment of the wall body first increases progressively to reach a maximum at approximately 0.45D of depth below ground, and then decreases as the embedded depth increases. The shear of the wall body is the greatest at the point of the load on wall top and reaches its zero value where the maximum moment occurs. In the soil outside the back wall, the lateral soil resistance above the rotation point appears to be active earth pressure and changes to passive pressure below the rotation point.

3) The outer skin friction at the upper part of the back wall is zero because of the detachment that occurs between the wall and the outside soil. The position at which the greatest shear deformation occurs between the wall and the soil is also where the outer skin friction is exerted; at shallower depths, the inner skin friction is difficult to exert because the relative displacement between the soil core and the wall is relatively small or zero. The settlement of the upper soil core and wall body are closely aligned under each loading level; when the loading on the wall top is relatively small, the difference in the settlement between the soil adjacent to the outside wall and the wall body is almost negligible.

4) As the depth increases, the loading changes to be borne by the wall body, and the axial force is greatest at the joints between the cap and the wall body. The shear deformation mainly occurs in the soil outside the wall, whereas compression always occurs within the soil core; not only are the distribution trends and generation mechanisms of the inner and outer skin frictions different, but the frictions are also dissimilar.

5) The exertion of skin friction and tip resistance does not occur simultaneously: Only after the outer skin friction and tip resistance of the wall body occur in sequence can the inner skin friction be initiated. During the entire loading process, the load-sharing ratio of the counterforce on the top of the soil core is relatively small and can be ignored.

6) The amount of compression and the compressive depth of the soil core increase as the cross-sectional dimension increases. As the soil core dimension changes, the loading on the wall top and the load-sharing ratio on the top of the soil core increase correspondingly, which leads to an increase in the degree of soil core compression. As the exertion of the inner skin friction becomes stronger, the load-sharing ratio of the skin friction for the soil core also increases. The RCD wall behaves as an end-bearing friction wall during the entire loading process. The exertion of inner skin friction around the bottom of the soil core and the proportion of loading supported by the soil core increases with the enlargement of the cross-sectional dimension.

7) The differential settlement of the central bottom of the soil core and the wall toe tends to decrease with a greater wall depth because the degree of wall body becomes greater and thus the settlement of the wall toe diminishes. As the wall depth increases, a greater proportion of the loading is supported by the outer skin friction, the tip resistance of the wall body increases, and a smaller proportion of the loading is borne by the soil core. When D exceeds a certain limit, the bearing contribution of the soil core can be ignored.

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