Experimental Investigation on Deformation Characteristics of Strutted U-Shape Sheet Pile Flexible Retaining Structures in Excavations Using 3D Printing

Abstract

U-shape steel sheet piles are widely used in deep and large excavation engineering due to their excellent soil-retaining and water-stopping performances. To achieve deformation control in excavations, an experimental investigation on the deformation characteristics of a strutted U-shape sheet pile flexible retaining structure was conducted. Single-layer and double-layer strutted retaining structure excavation indoor model tests in sand, where U-shape sheet piles were formed by 3D printing, were successfully accomplished. Deformation was monitored in real time during the test. The results show that the lateral displacement mode of the retaining structure transformed with the change in excavation from the “cantilever” to the “bulge” and finally developed into an intricate “recurve bow”. The average maximum lateral displacement was 0.756% of the excavation depth. With a maximum settlement of 0.375% of the excavation depth on average, the distribution of ground settlement behind walls changed from “exponential” to “triangular” before ultimately transforming into “trough” or “trapezoid” mode. The maximum settlement to maximum lateral displacement ratio (s_max/δ_max) on average was 0.54; the maximum settlement deformation rate was always less than the maximum lateral displacement deformation rate.

1. Introduction

With the current trend of deeper and larger excavations [1], the deformation issues that come with excavating foundation pits are becoming particularly prominent. In addition, U-shape steel sheet piles are widely used as high-quality retaining materials in deep and large excavations due to their superior performance, and the deformation characteristics of the flexible retaining structures composed of them are more complex and diverse. The deformation of excavations is related to engineering safety and surrounding stability, which makes it a severe challenge for retaining excavations to gradually change from traditional strength control to deformation control. Therefore, it is necessary to investigate the deformation of flexible retaining structures, which could provide a theoretical basis for deformation control in the design and construction of excavation retaining engineering and aid in optimizing the design and ensuring the excavations’ safety.

At present, some research on the deformation characteristics of flexible retaining structures has been published both domestically and internationally. Clough [2] summarized a large amount of engineering experience and believes that deep foundation pit excavation causes three modes of deformation in strutted (anchored) retaining structures: cantilever deformation, deep inward deformation, and cumulative deformation, as specified in Figure 1a–c. Wu [3] distinguished the deformation of diaphragm walls into four modes based on monitoring data of retaining structures in the Taipei area: standard mode, rotating mode, multi-tortuous mode, and cantilever mode. The multi-tortuous mode is a subdivision of deep inward deformation, as shown in Figure 1d. Gong [4] studied numerous excavations in China and concluded that the deformation behavior of retaining structures could be divided into four modes: cantilever, convex, composite, and kick-in deformation modes. The kick-in mode is a type of convex deformation, as illustrated in Figure 1e.

Clough [2] found that the distribution of excavation ground settlement after walls in sand and stiff to very hard clays presents a triangular pattern, with an impact range of 2 Z and 3 Z (Z represents the excavation depth). Moreover, the distribution of excavation ground settlement in soft to medium clays follows a trapezoidal pattern, with the maximum settlement occurring within a horizontal distance of 0–0.75 Z and an impact range of 2 Z. Ou [5] analyzed the measured data from 10 excavations in Taipei’s soft clay area and concluded that the characteristics of ground surface settlement could be divided into spandrel and concave profiles. Furthermore, Hsieh [6] improved Ou’s prediction method and proposed a primary and secondary influence zone.

Long [7] investigated the deformation of excavations using data from deep excavation engineering. Xu, Wang, and Wang [8,9,10,11] analyzed the deformation behavior of retaining structures and ground surface settlement in numerous deep and large excavations in Shanghai’s soft clay area and summarized the relationship between deformation behavior and factors such as excavation depth and retaining system stiffness. Liang et al. [12] proposed an excavation method that could control the deformation of retaining structures and ensure construction safety through centrifugal testing. In addition, Wei and Liang [13,14] used indoor excavation model tests to determine the deformation characteristics of cantilever and strutted pipe pile retaining structures and then compared and analyzed the results. Pan Hong [15] and Du Chuang [16] used finite element analysis on steel sheet pile cofferdams to investigate the distribution of internal force and the deformation of flexible retaining structures of steel sheet piles, as well as their variation patterns. Cui [17] used the FLAC3D finite difference method (FDM) to simulate four excavation schemes and study the deformation of the main retaining structure.

In summary, most studies have focused on traditional retaining structure deformation characteristics. However, there is a lack of in-depth understanding and experimental evidence on the deformation characteristics, distribution patterns, and evolution mechanisms of the strutted flexible retaining structure of U-shape sheet piles. Therefore, we proposed to conduct indoor excavation model tests to further investigate the deformation characteristics of strutted U-shape sheet pile flexible retaining structures. However, the key to model testing was the production of U-shape sheet pile flexible retaining structures, which needed to achieve a consistent appearance, effective interlocking structures on both sides, and demonstrate material flexibility. Three-dimensional printing is a revolutionary manufacturing technology with efficient and environmentally friendly manufacturing processes. It can achieve complex structural designs, such as concave and convex surfaces and lock buckle structures. It can also replace printing materials as needed, from plastic to metal. Therefore, the U-shape sheet piles were produced using 3D printing technology, which could accurately realize the geometric shape, interlocking structure, and flexible features.

2. Materials and Methods

2.1. Similarity Ratio Design

This indoor model test used the engineering prototype from the Shengli Road excavation in Nanchang City, Jiangxi Province, China, as the background. The depth of the excavation reaches 11.7 m, which is a deep excavation. There are adjacent multistory buildings with a safety level of A, pipelines, etc. We selected the typical cross-section ZK63 of the prototype excavation as the object of the model test for analysis. According to the survey data, the soil layer information is shown in

Table 1. Soil layer parameters of ZK63 cross-section.

Number	Soil Layers	Layer Thickness h (m)	Volumetric Weight γ (kN/m3)	Cohesive Forces c (kPa)	Internal Friction Angle φ (°)	Scale Factor m (MN/m4)
1	Miscellaneous fill	3.1	16.5	8.0	10.0	1.8
2	Silty clay	1.8	18.8	22.7	10.1	3.3
3	Medium sand	0.2	20.3	0.0	32.0	17.3
4	Pebble	9.4	20.8	0.0	42.0	31.1
5	Strong-weathered rock	0.8	20.5	25.0	30.0	17.5
6	Medium-weathered rock	5.0	24.0	45.0	50.0	49.5

.

The excavation retaining system was designed as a “steel sheet pile + strut”, with the sheet pile being a hot-rolled U-sheet pile of type SKSP-IV, the cross-section size of which is shown in Figure 2. The pile length was designed to be 18.56 m, and the sheet pile was Q295 steel. The underground main structure was built by the inverse method, with the top (0.45 m), middle (0.35 m), and bottom (0.55 m) plates serving as the retaining structure’s horizontal internal struts.

The material characteristics and geometric dimensions of the selected model and prototype in the model test needed to conform to a certain similarity relationship. Considering the feasibility of the model test box and the layout of test components, the height of the model sheet pile was determined to be 1.8 m. Based on the similarity criterion [18], the basic physical quantities (geometric dimensions, material mechanical properties, etc.) of the model test were designed. The similarity relations and constants between the prototype and the model are listed in

Table 2. Similarity relations and constants.

Physical Property	Abbreviation	Unit	Dimension	Similarity Relation	Similarity Constant
Elastic modulus	E	Pa	[F][L]−2	SE	80.93
Length	L	m	[L]	SL	10.31
Area	A	m2	[L]2	SA = SL2	106.30
Section factor	W	m3	[L]3	SW = SL3	1095.91
Moment of inertia	I	m4	[L]4	SI = SL4	11,298.86
Moment of inertia per linear meter	I’	m4/m	[L]3	SI’ = SL3	1095.91
Line load	q	N/m	[F][L]−1	Sq = SESL	834.39
Bending moment	M	N·m	[F][L]	SM = SESL3	8602.54
Flexural rigidity	EI	N·m2	[F][L]2	SEI = SESL4	914,416.81
Stiffness	K	N/m	[F][L]−1	SK = SESL	834.39

by the three theorems of similarity. Furthermore, based on the similarity in volumetric weight, it could be determined that the elastic modulus of the model material was 1.5–2.5 GPa. Finally, the retaining structure primarily underwent bending deformation during excavation, and it should have theoretically met the similarity relationship of bending stiffness.

2.2. Test Device

2.2.1. Model Test Box

The model test box was the YT-3000 pseudo-static underground geotechnical engineering comprehensive model test system, independently developed by East China University of Technology in Nanchang, China, as shown in Figure 3. The effective length, width, and height inside the test box were 2.5 m × 2 m × 2 m.

2.2.2. U-Shape Sheet Pile Model

To accurately represent the pile–soil interaction of the retaining structure during excavation in the model test, the model pile had to exhibit cross-section characteristics and geometric similarities like the prototype while accurately displaying the details of the sheet pile’s interlocking structure. Therefore, the U-shape model sheet pile was innovatively formed using 3D printing SLA (stereo lithography appearance) technology, provided by Vistar 3D Printing Industry Co., Ltd. (Xiamen, China). The model pile was made of Vistar 6335 UV-curable resin, and its specific parameters are listed in

Table 3. Physical and mechanical parameters of 3D-printing material Vistar 6335 UV-curable resin.

Shore Hardness (HD)	Elongation at Break (%)	Flexural Modulus (MPa)	Flexural Strength (MPa)	Tensile Modulus (MPa)	Tensile Strength (MPa)	Density (g/cm3)
ASTM D2240-15R21 [20]	ASTM D638-22 [19]	ASTM D790-17 [21]	ASTM D790-17 [21]	ASTM D638-22 [19]	ASTM D638-22 [19]	—
85	10–13	1750–1940	92	2100–2200	65	1.125

. Among them, the elongation at break was assessed using the ASTM D638-22 [19] test method, and it was determined to be within the range of 10–13%, very close to that of steel (≥15%). This indicated that the material was pliable and capable of undergoing significant plastic deformation. The tensile modulus was 2100 MPa–2200 MPa, which agreed with the elastic modulus calculated using the similarity ratio. All parameters met the expectations of the experiment’s design.

A single model pile measured 38.737 mm in length, 16.463 mm in width, 1.8 m in height, and 2 mm in thickness, with interlocking structures on both sides. Figure 4 depicts the cross-section dimensions, and

Table 4. U-shape sheet pile cross-section geometric characteristics.

Area S (mm2)	Centroid Moment of Inertia Ic (mm4)	Section Factor W (mm3)	Maximum Centroid Distance lmax (mm)
149.188	7228.761	632.548	11.428

lists the specific geometric characteristics of the cross-section. In the table, area (S) is the cross-section area of the model sheet pile. The centroid moment of inertia (I_c) is the second moment calculated by the cross-section area based on the neutral axis passing through the centroid point, and it is closely related to the flexural rigidity of the sheet piles. The maximum centroid distance (l_max) is the maximum vertical length of a point in the cross-section from the neutral axis. The section factor (W) is the ratio of the centroid moment of inertia to the maximum centroid distance (I_c/l_max), which was used to verify the maximum stress on the surface of the sheet pile.

To ensure the correctness of the sheet pile design, it was also necessary to verify the physical and mechanical properties of the materials. The materials’ flexural stiffness EI was verified through a simply supported beam test, as described in Figure 5. We applied a concentrated load to the mid-span in four stages, increasing the load by 5 N at each stage. The dial indicator measured the deflection at the quarter point, which increased by 0.1 mm per stage. According to the theoretical equation of material mechanics (Equation (1)), the model’s elastic modulus E was 2.17 GPa.

$$EI=\frac{0.014323F{l}^3}{y}$$

(1)

In this formula, EI is the flexural stiffness (N·m²); F is the mid-span load (N); l is the span length of a simply supported beam, (l = 0.28 m); and y is the average deflection at the quarter point (mm).

2.2.3. Strut System

We converted the equivalent stiffness of the prototype strut into the equivalent stiffness of the rod-shape strut through the similarity relation in Table 2, K_m = 40.983 MN/m. We selected FR-4 epoxy resin fiberglass pipe as the main structure of the strut and calculated the cross-section area according to the strut stiffness formula (Equation (2)).

$$K=\frac{2\alpha EA}{Ls}$$

(2)

In this formula, K is the strut stiffness (N/m); α is the relaxation coefficient, taken as 1; E is the elastic modulus (GPa); A is the cross-section area (m²); L is the span (m); and s is the strut spacing (m).

The elastic modulus of the epoxy resin fiberglass pipe ranged from approximately 20 GPa to 30 GPa. The excavation width was 1 m, resulting in a span of L = 1 m. Only one strut was specified for each layer of the excavation. The width of the test model sheet pile wall was 0.66335 m, resulting in a spacing of s = 0.663 m. The cross-section area of the strut fell within the range of 452 mm² to 679 mm², so it was designed with an inner diameter of 20 mm, an outer diameter of 34 mm, and a length of 40 cm. To ensure the correctness of the strut design, it was also necessary to verify the physical and mechanical properties of the strut. Similarly, the elastic modulus of the epoxy fiberglass pipe could be measured as 20 GPa through a simply supported beam test, as per Equation (1). Figure 6 shows the strut system, which consisted primarily of antislip supports at both ends, the main structure of epoxy resin fiberglass pipe in the middle, and the aluminum threaded rod connection in the center.

2.3. Test Soil

The test soil was filled with building sand from Nanchang. To clarify the characteristics of the sand used in this test, particle analysis was conducted in strict accordance with geotechnical testing standards [22]. We drew the sand particle grading curve shown in Figure 7. The sand particles were fairly uniform, and the mass of particles larger than 0.25 mm exceeded 50% of the total weight. The particle size distribution of the test sand was between 0.25 mm and 0.5 mm on average. According to the sand classification criteria, the fineness modulus (M_x) of the test sand was 2.43, indicating that it belonged to the category of medium sand. In addition, it was necessary to conduct unconsolidated–undrained (UU) triaxial shear tests, water content tests, and consolidation tests to obtain detailed sand parameters, as shown in

Table 5. Physical and mechanical parameters of test sand.

ρ (g/cm3)	Mx	Cu	Cc	w (%)	c (kPa)	φ (°)	Es (MPa)
1.70	2.43	3.38	1.32	4.20	6.88	32.85	20.00

. Among them, the nonuniformity coefficient (Cu) was 3.38, and the curvature coefficient (Cc) was 1.32, indicating that the sand particles were relatively uniform. The moisture content (w) was 4.2%, the cohesion (c) was 6.88 kPa, the internal friction angle (φ) was 32.855°, and the compressive modulus (E_s) was 20 MPa.

2.4. Test Monitoring

This model test focused on the deformation of the retaining structure during excavation. Therefore, the main monitored physical quantities included the vertical strain of the pile, the lateral strain of the strut, the lateral displacement of the pile top, and the ground surface settlement after the pile.

2.4.1. Strain Monitoring

In the test, a BX120-3AA foil strain gauge with a resistance of 119.9 ± 0.1 Ω and a sensitivity of 2.08 ± 1% was used, manufactured by Kechengchenxing Electronics, Zhejiang, China. to It effectively moni-tored the strain distribution and changes in the pile. It was arranged horizontally and symmetrically on the central axis of both sides, A and B, of the sheet pile wall, from the top to the bottom of the pile, with a strict 10 cm interval. The specific arrangement is shown in Figure 8a. The A side of the sheet pile wall was the side facing the filling soil behind the wall, and the B side was the side facing the excavation in front of the wall. In addition, in order to monitor the change in strut strain, strain gauges were symmetrically arranged at the center of the outer surface of the epoxy resin fiberglass pipe. There were two layers of strut systems, A and B, and four strain gauges were arranged for each layer of strut, as shown in Figure 8b.

2.4.2. Lateral Displacement of Pile Top and Ground Surface Settlement Monitoring

The lateral displacement at the pile’s top was measured with an ejector rod displacement meter, model ZY-DT100, produced by Jincheng Testing Instrument Factory in Changzhou, China, which was installed at the pile’s top. The ground surface settlement behind the wall was monitored using a digital display dial indicator at five measuring points. The horizontal distances from the measuring points to the wall top were 25, 300, 600, 1000, and 1500 mm, respectively. The on-site layout is shown in Figure 9.

2.5. Test Method

The interlocking connections of the sheet pile wall are shown in Figure 10a. The sheet pile wall was made up of 17 single piles through a mechanical interlocking structure to form a whole, with a height of 1.8 m and a width of 0.66 m. The bending neutral axis of the sheet pile wall passed through each interlocking structure, and its moment of inertia per linear meter (I’) was 4.43 × 10⁻⁷ m⁴/m. Vaseline and polyethylene film had to be used on both sides of the wall to prevent slipping, and sponge sealing strips had to be used to prevent sand leakage. In addition, Figure 10b shows the layout of the sheet pile wall, which was 1 m away from the inner wall of the model test box to reserve working space. The sheet pile wall was reinforced with a four-layer device called a fixture, which fixed the sheet pile wall like a clamp. The function of this device was to ensure that the sheet pile wall was always vertical during the sand-filling process. Finally, sand had to be layered every 30 cm and allowed to stand for 24 h. As each layer of sand was filled, the fixture was gradually removed without affecting the testing process. After the soil stabilized, excavation tests were carried out.

The model test was divided into two groups: single-layer strutted and double-layer strutted retaining structures, as shown in Figure 11. The total depth of the excavations was 1.11 m, and they were carried out in three layers, with each layer having a depth of 0.37 m. The first layer of struts was set at 0.32 m, and the second layer was set at 0.69 m. After each excavation and setting of strut stages, the structure was allowed to stand for one hour to ensure the overall stability of the excavation.

3. Results

3.1. Processing Method for Strain Data of the Retaining Structure

The curvature distribution of the pile body was used to calculate the lateral displacement of the retaining structure. The distribution of the pile body’s curvature (ω) along the depth (z) of the pile body was as follows, according to the elastic beam theory:

$$\omega \left(z\right)=\frac{{\epsilon }_A-{\epsilon }_B}{D}$$

(3)

In this equation, ε_A is the strain value of side A of the sheet pile; ε_B is the strain value on side B; and D is the thickness of the sheet pile wall (m).

The lateral displacement calculation method introduced the finite integral method (FIM) [23]. The principle is based on the known values of the derivative function (δ’), which are used to perform numerical analysis within the defined domain [0, h] to obtain the corresponding function values δ. We conducted a numerical analysis of the curvature distribution of the U-shape sheet piles based on this principle. The U-shape sheet pile’s height range [0, h] was divided into even segments each with a length of c. As shown in Figure 12, the design divided the sheet piles into 18 segments along the axial direction, yielding c = 1.8 m/18 = 0.1 m.

This article used the first-order derivative function value (δ′) of displacement at nodes within the interval of depth z ∈ [ic, (i+2) c] for the U-shape sheet pile to perform quadratic parabolic interpolation, where i = (0, 2, 4, 6, 8, 10, 12, 14, 16), and after integration, we obtained:

$$\left\{\begin{array}{c}{\delta }_{i+1}=\frac{c}{12}\left(5{\delta }_i^{\prime }+8{\delta }_{i+1}^{\prime }-{\delta }_{i+2}^{\prime }\right)+{\delta }_i\\ {\delta }_{i+2}=\frac{c}{3}\left({\delta }_i^{\prime }+4{\delta }_{i+1}^{\prime }+{\delta }_{i+2}^{\prime }\right)+{\delta }_i\hfill \end{array}\right.$$

(4)

Based on Equation (4), the relationship between the lateral displacement value {δ} and the first-order derivative function value {δ′} of displacement could be obtained as follows:

$$\left\{\delta \right\}=\frac{c}{12}\left[N\right]\left\{{\delta }^{\prime }\right\}+{\delta }_0\left\{I\right\}$$

(5)

In this equation, {I} is the unit column vector of 19 elements; [N] is a square matrix of size 19; and δ₀ is the lateral displacement of the pile top (m).

$$[N]=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& \\ 5& 8& -1& 0& 0& \\ 4& 16& 4& 0& 0& \cdots \\ 4& 16& 9& 8& -1& \\ 4& 16& 8& 16& 4& \\ & & ⋮& & & \end{array}\right]$$

(6)

Similarly, according to Equation (5), the relationship between the first-order derivative value {δ′} and the second-order derivative value of lateral displacement {δ^”} could be obtained as follows:

$$\left\{{\delta }^{\prime }\right\}=\frac{c}{12}\left[N\right]\left\{{\delta }^{\prime \prime }\right\}+{\delta }_0^{\prime }\left\{I\right\}$$

(7)

By solving Equations (5) and (7) simultaneously, we established a relationship between the lateral displacement value {δ} and the second-order derivative value of lateral displacement {δ″}. According to the theory of small deflection, the curvature was treated as (ω = −δ″). The FIM could be further rewritten as the relationship between curvature and lateral displacement:

$$\left\{\delta \right\}=-\frac{c^2}{144}\left[D\right]\left\{\omega \right\}+{\delta }_0^{\prime }\left\{z\right\}+{\delta }_0\left\{I\right\}$$

(8)

In this equation, [D] = [N][N], where [D] is the matrix of lateral displacement; {ω} is the column vector of pile curvature; and δ′₀ is the pile top rotation angle (°).

$$[D]=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& \\ 36& 48& -12& 0& 0& \\ 96& 192& 0& 0& 0& \cdots \\ 144& 384& 84& 48& -12& \\ 192& 576& 192& 192& 0& \\ & & ⋮& & & \end{array}\right]$$

(9)

The pile top rotation angle (δ′₀) could be obtained by solving the system of equations using two displacement boundary conditions. Given the known lateral displacements at the pile top (δ₀) and at a specific depth h (δ_h), then:

$${\delta }_0^{\prime }=\frac{{\delta }_h-{\delta }_0+\frac{c^2}{144}\left\{d\right\}\left\{\omega \right\}}{h}$$

(10)

In this formula, {d} is the element in the h-th row of the matrix [D], and {ω} is the curvature of the pile.

The above Formulas (3)–(10) represent the entire reasoning process of the FIM. Firstly, curvature ω in Equation (3) was the key to calculating lateral displacement. Secondly, based on the calculation principle (4), we obtained the lateral displacement value δ through Formula (5), the displacement first derivative function value δ′ using Formula (7), and the matrix N (6). Solving Equations (5) and (7) simultaneously, we obtained the FIM Formula (8), which was based on the curvature ω of numerical analysis to determine the lateral displacement δ. Finally, matrix D (9) was defined as the lateral displacement matrix. The pile top rotation angle δ′ (Equation (10)) required two displacement conditions.

To verify the effectiveness of the FIM (8), it was compared with the results of the simply supported beam test, as shown in Figure 13. Under the action of a mid-span load, the model sheet pile generated deflection, and the sheet pile presented a parabolic distribution with a total of three loadings. At the first loading, the load was 0.38 N, and the maximum deflection at the mid-span was −2 mm. During the second loading, the load was 0.64 N, and the maximum deflection was −3.42 mm. During the third loading, the load was 1.25 N, and the maximum deflection was −6.7 mm. From the figure, it can be seen that the calculated values of the FIM were basically consistent with the measured values, with an average relative error of 1.76%. This method could accurately reflect the retaining structure’s deformation distribution overall. As a result, the FIM was used for lateral displacement numerical analysis.

3.2. Lateral Displacement Distribution and Analysis of the Retaining Structure

Figure 14 shows the lateral displacement distribution curves of the single-layer and double-layer strutted retaining structures at various stages of the excavation. As shown in the figure, during the excavation of the first layer (Z = 0.37 m), there was significant lateral displacement in the middle and upper parts of the wall due to sand unloading in front of the retaining structure and displacement towards the excavation side under soil pressure behind the wall. The maximum displacement was at the top of the pile, with single-layer supports and double-layer struts measuring −4.64 mm and −4.99 mm, respectively. The lateral displacement of the retaining structure began to significantly decrease below the excavation surface, and the closer it was to the pile bottom, the closer it approached zero. The curve shape shows that the retaining structure’s lateral displacement resembled the deformation characteristics of a cantilever beam under uniformly distributed loads, presenting a “cantilever” type displacement mode.

When excavating the second layer (Z = 0.74 m), compared to excavating the first layer, the lateral displacement of the pile top was significantly reduced, but the maximum lateral displacement of the retaining structure remained at the top, at −4.02 mm. Due to sand lateral unloading, the middle of the retaining structure protruded towards the excavation side, indicating a preliminary trend of a “bulge”-type displacement mode. The above phenomenon suggested that the strut played a role, with just 0.017 mm of shrinkage at the strut. At this point, the strut axial forces in the two test groups were −0.63 kN and −0.64 kN, respectively. It was noticed that the displacement of the local area near the retaining structure’s strut did not alter considerably. The retaining structure exhibited the typical recurve deformation of a flexible retaining structure at the strut and a certain point below the excavation bottom surface.

During the excavation of the third layer (Z = 1.11 m), the lateral displacement of the retaining structure altered significantly, with notable variances between the two groups of tests. Firstly, the lateral displacement of the pile top of the single-layer strutted retaining structure was no longer the maximum, at −3.57 mm. The middle part of the retaining structure completely protruded towards the excavation side, showing a “bulge-style” displacement mode, with the maximum lateral displacement of −5.24 mm occurring at a sheet pile depth of 0.7 m. Secondly, the maximum lateral displacement of the double-layer strutted retaining structure remained at the top, at −4.24 mm. The lateral displacement of the sheet pile’s middle part protruded at the corresponding excavation site. Additionally, the strut contributed further, with an axial force of −1.39 kN for a single-layer strut, which was roughly 2.2 times that of the excavation of the second layer. The first strut of the double-layer strut group had an axial force of −0.52 kN, which was 18.60% less than the excavation of the second layer, and the second strut had an axial force of −1.565 kN, which was roughly three times that of the first strut. This showed that the second strut had a significant impact.

The significant recurve deformation at the retaining structure’s strut suggested that the strut had a powerful limiting influence on the retaining structure’s lateral displacement. It was also a characteristic of flexible retaining structure deformation, manifesting as the strut pushing the wall back, greatly limiting the lateral displacement in the middle part of the wall. Similarly, the retaining structure below the excavation bottom-surface sand had obvious recurve deformation. As a whole, the retaining structure evolved into a “recurve bow”-type complex displacement mode with protruding in the middle and recurving on both sides, as shown in Figure 15.

The relationship between the maximum lateral displacement of the single-layer and double-layer strutted retaining structures and excavation depth is shown in Figure 16. The measured maximum lateral displacement was between 0.3% Z and 3.5% Z, which was in line with the theoretical range of the deformation laws of deep excavations protected by steel sheet pile retaining structures proposed by Mana [24]. U-shape steel sheet piles are flexible structures with an average maximum displacement of 1.5% Z. In these two groups of tests, the average lateral displacement of the retaining structure was 0.756% Z. The overall trend of nondimensionalized maximum lateral displacement decreased with an increasing excavation depth, reflecting the constraint effect of the strut on the retaining structure deformation.

3.3. Ground Surface Settlement Distribution and Analysis behind the Wall

Figure 17 depicts the ground surface settlement distribution curve behind the wall. As the excavation depth increased, so did the settlement, and the closer it was to the wall, the greater the settlement. When the excavation depth was shallow (the excavation of the first layer Z = 0.37 m), the maximum settlement value was measured at the first measuring point 25 mm away from the retaining structure. The results for the two groups of tests were −2.05 mm and −1.97 mm, respectively. The settlement distribution was similar to the exponential function curve distribution mode. When the excavation depth was relatively deep (the excavation of the second layer Z = 0.74 m), the maximum settlement remained at the first measurement point, with settlement values of −2.31 mm and −2.38 mm, respectively. The second measuring point had the most variation in settlement, with measured settlements of −1.97 mm and −1.86 mm, respectively. There was also a trend of “concave” surface settlement, which resembled a triangular distribution mode, similar to the retaining structure protruding into the excavation side. When the excavation depth was high (the excavation of the third layer Z = 1.11 m), the maximum settlement value was moved to the second measuring point position, which was −2.78 mm and −2.68 mm, respectively. There were differences in settlement distribution, with the settlement of the single-layer strutted test exhibiting a “trough” distribution mode and the settlement of the double-layer strutted test exhibiting a “trapezoid” distribution mode. The above evolution phenomenon of the ground surface settlement distribution mode reflected the control effect of struts on the fill behind the wall, dividing its settlement into primary and secondary influence zones.

Figure 18 shows the relationship between the maximum ground surface settlement and the excavation depth of the single-layer and double-layer strutted retaining structures. The maximum ground surface settlement value gradually increased at each excavation depth, ranging from 1% Z to 0.1% Z, which met Peck’s statistical description of the maximum ground surface settlement of a deep-excavation steel sheet pile flexible retaining structure (s_max < 1% Z) [25]. The average maximum ground surface settlement of these tests was 0.375% Z, which was larger than the value of 0.15% Z calculated by Clough for solid piles, steel sheet piles, diaphragm walls, and bored pile retaining structures [2]. The right axis represents the nondimensionalized maximum ground settlement. When the first layer of the excavation was excavated, the ratio of the two groups of the strutted retaining structures increased sharply to a maximum of 0.0055 and 0.0053, respectively. Subsequently, under the action of the strut, the ratio decreased continuously with an increasing excavation depth.

The nondimensionalized ground surface settlement profiles behind the wall are shown in Figure 19. Firstly, the settlement profile of s/Z is shown in Figure 19a. Peck summarized the typical normalized surface settlement distribution profiles of a steel sheet pile retaining structure [25]; defined three types of surface settlement areas; and divided them into zones I, II, and III by three dotted lines. Obviously, the ground surface settlement was not significant, and most of the data points were scattered in zone I, corresponding to the excavation area in the sand and soft to hard clay soil layers. At the same time, this also indicated that the strut could effectively control the sand after the U-shape sheet pile retaining structure. In this article, the exponential function (11) expresses the profile of the ground settlement of the strutted retaining structure. In the settlement profile, the influence zone of the ground settlement behind the wall was within 2.5 Z.

$$\frac{s}{Z}=0.6337e^{-1.1502\frac{d}{Z}}-0.03356$$

(11)

Secondly, Figure 19b is the settlement profile of s/s_max. For comparative analysis, the surface settlement profiles summarized by Clough [2] and Hsieh [6] are also shown in the figure. The settlement profile of this case was in the primary influence zone at 0 ≤ d ≤ 1.5 Z and presented a trapezoidal distribution pattern, where 0 ≤ d ≤ 0.27 Z had the maximum settlement value, which was more consistent with the regular curve proposed by Clough. In the range of 1.5 Z ≤ d ≤ 4.05 Z, the ground settlement was in the secondary influence zone, presenting a triangular distribution pattern. The trend of change was similar to that described by Hsieh, but the range was smaller.

3.4. Normalized Maximum Ground Surface Settlement and Maximum Lateral Displacement Relationship

There was a close relationship between the ground settlement behind the wall and the lateral displacement of the retaining structure, which varied with the excavation depth. We adopted a normalization treatment for the maximum settlement and maximum lateral displacement, as shown in Figure 20. The vertical axis represents the nondimensionalized maximum ground settlement (s_max/Z), and the horizontal axis represents the dimensionless maximum lateral displacement (δ_max/Z). As illustrated in the figure, the majority of the data points fell within the range (s_max/δ_max = 0.51–1), which was consistent with the work of Ou [5] and Hsieh [6], who concluded that the maximum lateral displacement was 50–75% of the maximum ground surface settlement. This test obtained the average value (s_max/δ_max = 0.54), and the ratio increased gradually with the excavation depth. This indicated that the maximum settlement behind the wall was about half of the maximum lateral displacement. The settlement deformation rate gradually increased with the depth of the excavation, but it was always lower than the deformation rate of lateral displacement.

3.5. Bending Moment Distribution and Analysis of Retaining Structure

The bending moment distribution of the pile body was calculated from the curvature distribution:

$$M\left(z\right)=EI\omega \left(z\right)$$

(12)

In this formula, M (z) is the bending moment of the pile body (N ∙ m); E is the elastic modulus (Pa); I denotes the cross-sectional moment of inertia (m⁴); and ω (z) is the curvature of the pile body.

Figure 21 displays the distribution of bending moments on the pile body of the retaining structure at various excavation depths. A positive number was designated for the bending moment that bent toward the soil side and a negative value for the opposite direction. The bending trend and degree of the retaining structure could be represented by the bending moment. The bending moment had an extreme value at various excavation stages and continuously increased with the depth of the excavation. The zero point of the bending moment generally coincided with the depth at the zero point of the lateral earth pressure, located below the excavation bottom surface, and continuously moved downward as the excavation depth deepened. Due to the constraint effect of the strut and the embedding effect of unexcavated sand, the bending moment distribution presented a multi-segment bending curve.

4. Discussion

In this article, to explore the complex deformation issues of strutted flexible retaining structures in deep and large excavations, we adopted an indoor model test investigation. We analyzed some deformation phenomena of the excavations and found that the lateral displacement of the flexible retaining structures of the excavations in sand varied with the excavation depth, which conformed to the descriptions of the deformation modes of retaining structures by scholars such as Clough [2], Wu [3], and Gong [4]. However, the flexible retaining structures eventually evolved into a complex displacement mode of the “recurve bow” type, which was characterized by recurving on both sides and protruding in the middle. The above deformation differed from the conventional mode for three main reasons. Firstly, the flexural rigidity of the retaining structure itself was not large, belonging to the flexible category, and it was prone to deformation under external forces. Secondly, the strut’s stiffness was high enough to limit the retaining structure’s further displacement, resulting in recurve deformation. Finally, there was interaction between the structure and the soil, which was the least intuitive but most complex reason. This meant that during excavation, the structure and the soil were in dynamic equilibrium, and the uneven deformation generated a vertical soil arch. The earth pressure behind the wall redistributed to the arch feet on both sides [26], so its impact on the lateral deformation of the retaining structure was reflected in the reduction in the protruding deformation trend in the middle of the retaining structure, and there was a trend of deformation towards the excavation side at the strut and a certain position below the excavation bottom. This showed that there was more significant structure–soil interaction in the strutted flexible retaining structure.

Furthermore, the test results showed that the lateral deformation of the single-layer strutted retaining structure was greater than that of the double-layer strutted structure, and its “recurve bow”-type displacement mode was more obvious. There are two explanations for the emergence of such a unique “recurve bow” displacement mode. The first factor is the number of struts. The more struts there are, the more load they bear behind the wall; additionally, the lower the recurve deformation of the retaining structure, the lower the total displacement. As a result, it could be boldly predicted that the overall lateral displacement of the multi-layer strutted retaining structure would be small, and the displacement mode would be of the “wave” type. The second factor is the strut’s location. Closer to the pile top, the smaller the lateral displacement at the top of the retaining structure, the smaller the recurve deformation, presenting a “bow”-type displacement mode. Closer to the middle, the greater the lateral displacement at the top of the retaining structure, the greater the recurve deformation, and the more obvious the “recurve bow”- type displacement mode.

In addition, it was also discovered from the experimental investigation that the tendency of the ground surface settlement behind the wall to alter with the excavation depth was comparable with the findings of investigators such as Peck [25], Clough [2], and Hsieh [6]. The distribution of the ground surface settlement ultimately evolved into “trough” or “trapezoid” mode, which may have been due to three reasons. Firstly, it was closely related to the lateral deformation of the retaining structure. The lateral protrusion deformation of the single-layer strut was more severe, resulting in more significant surface subsidence, thus presenting a “trough” type. Secondly, there was an interaction between the structure and the soil. Coordinated deformation in space between the structure and the soil occurred, and the sand behind the wall could have experienced significant plastic flow deformation, resulting in settling [27].

More importantly, in this study of the relationship between normalized maximum ground settlement and the maximum lateral displacement of retaining structures in excavations, it was also found that the mutual relationship between the excavations’ deformation largely conformed to the summaries of Ou [5] and Hsieh [6]. However, after excavating the first layer (Z = 0.37 m), the ground settlement rate was slightly less than half of the lateral displacement. This was likely due to the shallow excavation depth and the limited plastic flow range of sand, resulting in minimal settlement. It is also possible that the contact friction between the structure and the earth resulted in reduced settling.

Compared with a previous investigation of excavation deformation also using 3D printing [13,14], the experimental conditions were different. Firstly, U-shape sheet piles were used in this article, while pipe piles were used in the above investigation, and the flexural rigidity of the retaining structure was 13.4 times that of this test. Secondly, the stiffness of the strut was 4.5 times that of the above investigation [14]. The strut location was also different, and the first strut was set from the top in the above investigation. The research results indicated that during the excavation process, the retaining structures in the two investigations had similar deformation patterns, and the ground surface settlement patterns were roughly the same. However, the lateral displacement of the retaining structure in this article was greater, and there were significant differences in the final deformation mode. This was because the retaining structure in this investigation was generally more flexible, with a large amount of deformation, and because of the influence of the location of the strut.

The experimental investigation yielded numerous discoveries, but it also had limitations. On the one hand, the experimental results were limited to sand conditions and cannot be used to generalize the deformation characteristics of all retaining structures. On the other hand, no in-depth investigation of the interaction between the structure and the soil was conducted to investigate the specific influence of the interaction on the deformation of the excavations. However, these limitations could be improved. For example, by considering conducting experiments in various soil environments or using numerical model methods to study the excavations under complex soil conditions. Additionally, the interaction between structure and soil should be studied together with lateral earth pressure.

In summary, the deformation characteristics of strutted flexible retaining structures are complex and related to the struts, soil, and structure–soil interactions. Therefore, future investigations should concentrate on strut variations, multiple soil choices, and the relationship between earth pressure and displacement.

5. Conclusions

In this article, according to the engineering prototype, a model was designed based on similarity theory, and the retaining structure was innovatively manufactured by 3D printing technology. Two groups of indoor model excavation tests of the strutted retaining structures were carried out in sandy soil. The deformation characteristics and laws of the U-shape sheet pile strutted flexible retaining structures were compared and analyzed, and the following conclusions could be drawn:

• The finite integral method could effectively analyze strain data to obtain the lateral displacement distribution, with an average relative error of 1.76%, providing an effective method for calculating the lateral displacement of retaining structures.
• With the increase in excavation depth, the displacement mode of the U-shape sheet pile strutted flexible retaining structure changed from the “cantilever” type to the “bulge” type and finally evolved into a complex displacement mode of the “recurve bow” type with both sides recurving and the middle protruding. The retaining structure’s maximum lateral displacement spanned from 0.3% Z to 3.5% Z, with an average value of 0.756% Z, which was within the deformation range of a typical flexible retaining structure. This showed that the strut could successfully confine the wall’s lateral displacement and achieve deformation control.
• The distribution of the ground surface settlement of sand behind the wall increased with the excavation depth, from an “exponential” to a “triangular” mode, and ultimately evolved into a “trough” or “trapezoid” distribution mode. The maximum settlement at each excavation depth ranged from 0.1% Z to 1% Z, with an average value of 0.375% Z, indicating that the strutted retaining structure could effectively control the sandy filling. The nondimensionalized surface settlement points of the two groups of tests were scattered in zone I, which conformed to the settled law of sandy soil.
• The normalized deformation relationship curve of the excavation reflected that the maximum lateral displacement and the maximum ground surface settlement increased with the increase in the excavation depth, and the ratio of maximum settlement to maximum lateral displacement (s_max/δ_max) was between 0.5 and 1, with an average value of 0.54. This indicated that the maximum settlement behind the wall was about half of the maximum lateral displacement of the retaining structure, and the settlement deformation rate was always lower than the lateral displacement deformation rate.
• The distribution of the bending moment varied with the excavation depth. As the excavation depth increased, the maximum bending moment increased, and the zero point of the reverse bending moved downward, presenting a multi-segment bending curve distribution.