Influence of freeze–thaw damage gradient on stress–strain relationship of stressed concrete

Abstract

Freeze–thaw damage gradients lead to non-uniform degradation of concrete mechanical properties at different depths. A study was conducted on the stress–strain relationship of stressed concrete with focus on the freeze–thaw damage gradient. The effects of relative freeze–thaw depths, number of freeze–thaw cycles (FTCs) and stress ratios on the stress–strain curves of concrete were analyzed. The test results demonstrated that freeze–thaw damage was more severe in the surface layers of concrete than in the deeper layers. The relative peak stress and strain of concrete degraded bilinearly with increasing depth. The stress–strain relationship of stressed concrete under FTC was established, and it was found to agree with the experimental results.

Graphical abstract

Keywords

freeze–thaw damage / stress ratios / stress–strain relationship / damage variable

1 Introduction

Diaphragm walls have long been recognized as a versatile and dependable solution for addressing the complex challenges presented by deep excavations, foundation systems, and underground structures. Their capacity to support substantial depths, accommodate various soil types, and minimize the impact on surrounding environments has made them a cornerstone of geotechnical engineering. Diaphragm walls are not only known for their traditional applications but have also evolved into innovative forms, such as the lattice-shaped diaphragm wall (LSDW), which promises unique advantages for geotechnical and foundation engineering [1].

LSDW, an innovative diaphragm foundation structure, consists of an interconnected network of continuous underground wall structures forming a rectangular framework with an integrated top platform, as depicted in Fig.1(a). The construction of LSDW involves trench excavation, where each wall section is rigidly interconnected, as demonstrated in Fig.1(b). This construction method culminates in the formation of a closed lattice-shaped foundation, all while preserving an undisturbed soil core within the walls. While the external appearance of this foundation may bear a resemblance to caisson foundations, the trench excavation method employed in LSDW construction enhances the compaction between the walls and the surrounding soil. Consequently, this approach yields a heightened capacity to bear both vertical and horizontal loads. Furthermore, the presence of an undisturbed soil core within the foundation further bolsters its load-bearing capabilities. Field tests in comparable geological conditions with equivalent concrete usage, unequivocally demonstrated that LSDW outperformed caisson foundations significantly in terms of horizontal load-bearing capacity [2,3]. The LSDW, as a new breed of diaphragm wall foundation, exhibits a unique set of characteristics that set it apart.

Fig.1 Construction form and method of LSDW: (a) lattice-shaped wall; (b) construction method.

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1) Versatility and adaptability, suitable for construction near existing structures while minimizing environmental disruptions.

2) Cost-efficiency in construction, outperforming caisson foundations in terms of time, material use, and overall expenses.

3) Excavation optimization, substantially reducing the volume of excavation and concrete requirements, while enhancing flood resistance.

4) Superior load-bearing characteristics, offering higher stiffness and greater resistance to horizontal loads and seismic events than pile foundations.

The practical adoption of the LSDW foundation in bridge engineering, though firmly established in Japan, is progressively gaining traction worldwide. Its success in surmounting challenges such as extensive spans, deep foundations, soft soils, and demanding construction conditions has proven pivotal in achieving economic benefits and executing projects with remarkable efficiency. A notable case in point is the Aomori Bridge, where a LSDW has been utilized for the main towers, offering stability and load-bearing capacity that surpass conventional alternatives [4].

Field tests are an effective means of studying the bearing capacity of a foundation. Chen et al. [5] and Meng et al. [6] carried out a static load test on the vertical bearing capacity of LSDW foundation and found that the vertical bearing capacity of LSDW foundation is mainly provided by wall−soil lateral resistance, and the test found that LSDW foundation is more suitable for bridge foundation in loess area. Complementing these findings, Song et al. [7] obtained key data of LSDW foundation, such as wall bending moment, shear force, displacement, etc., by testing in the field under different horizontal loading conditions, and explored the change rule and bearing characteristics of LSDW foundation.

Moreover, Dai et al. [8] conducted model tests on three different sizes of single-chamber and four-chamber diaphragm wall foundations and concluded that the damage of single-chamber shaft diaphragm wall was characterized by overall inclined damage, and the four-chamber shaft diaphragm wall was characterized by rigidity damage at the time of damage. Wu et al. [9] investigated the vertical load-bearing characteristics of single-chamber LSDWs, double-chamber LSDWs, and four-chamber LSDWs in indoor modeling tests, and compared them with group pile foundations, according to the test, it proved the conclusion that the bearing capacity and settlement of the LSDW foundation are superior to that of the group pile foundation, and at the same time, with the increase of the grid compartments, the ultimate bearing capacity of the LSDW foundation increases continuously. Li et al. [10] conducted a study on the seismic performance of LSDW foundations in liquefied sites and the resistance to seismic liquefaction, and according to the test results, the excellent performance of LSDW foundations in resisting liquefaction was proved, while the seismic performance was also more excellent.

Despite the promise inherent in the LSDW concept, comprehending its lateral bearing behavior is paramount to evaluating its feasibility and performance across diverse geotechnical conditions. Meanwhile, due to the special structure of the foundation of LSDW, the bearing characteristics of the inner soil core under horizontal load are very different from those of the outer wall, and at the same time, due to the small size of the model test, plus the very limited space inside the soil core, which poses a considerable challenge for this test to accurately obtain the test data of the inner and outer walls.

This research endeavors to bridge this knowledge gap through an extensive experimental study, presenting findings that contribute to the expanding body of knowledge on LSDWs. A cornerstone of this research is the novel incorporation of a double-layer wall configuration, coupled with a newly proposed testing principle and corresponding computational formulas. This innovative approach has enabled the capture of various components of internal and external wall forces within LSDWs for the first time, culminating in the acquisition of foundational p–y curves. The methodologies and findings detailed in this paper pave the way for guiding field and laboratory tests of LSDWs, as well as informing the design calculations for foundations subjected to horizontal loads.

2 Experimental design and overview

2.1 Similarity constants and design of experimental models

For this test, the main research is carried out for the LSDW foundation; for this reason, the single-chamber LSDW foundation and the double-chamber LSDW foundation are selected as the experimental research object. Since this test is an indoor scaled model test, the geometric similarity constant is determined as 1:30, and according to the similarity theory, the similarity constants of the wall, soil medium, and other related physical quantities can be determined, as shown in Tab.1.

Tab.1 Similarity constants

Item	Similarity ratio
Elastic modulus	$C_{\mathrm{E}}=C_{\mathrm{l}}=30$
Linear displacement	$C_{\text{δ}}=C_{\mathrm{l}}=30$
Linear dimension	$C_{\mathrm{l}}=30$
Strain	$C_{\text{ε}}=1$
Stress	$C_{\text{σ}}=C_{\mathrm{l}}=30$
Inner friction angle	$C_{\text{φ}}=1$
Posson’s ratio	$C_{\mathrm{v}}=1$

For the LSDW, there is an earth core inside the foundation, which is very different from the traditional embedded foundation, which is a great challenge for the study of soil resistance and friction force in the inner and outer walls, in order to solve this problem, this test divides the LSDW into double walls, and at the same time, due to the setup of the double walls, it puts forward a test on the test modeling accuracy and feasibility of the test model, etc., which is the best way to solve this problem, and it is a good way to solve this problem, and it is a good way to solve this problem. Therefore, the test model was fabricated in sections and assembled with strong adhesive and high-strength bolts. Fig.2 shows the dimensions and installation schematics of the single-chamber and double-chamber LSDW foundations used for this test.

Fig.2 LSDW models and geometric dimensions: (a) single-chamber LSDW; (b) double-chamber LSDW.

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In actual engineering construction, reinforced concrete is generally used as the construction material for LSDW foundations, but the indoor scaled-down model test cannot directly utilize the prototype construction material. After considering the similarity coefficients and the successful experiences of similar model tests by Wu et al. [9], Li et al. [10], Zhang et al. [11], etc., the present test adopts Plexiglas as the model material for the test of LSDW foundations. Therefore, considering the practical requirements of this experiment and drawing from successful cases in similar model experiments, organic glass was selected as the material, and its elastic modulus was determined through indoor testing, resulting in an elastic modulus of 2.85 × 10³ MPa. To accurately capture stress conditions within and outside the model wall, a dual-layer wall structure was employed. Following strength calculations and bending stiffness conversions, the final thickness of a single wall layer was determined to be 12 mm.

The test model uses plexiglass as the production material, in order to ensure the accuracy of the model, the plexiglass panels used in the test are all customized processing by the manufacturer. The professional technicians of the manufacturer assembled the test model on site after the necessary measuring equipment was installed, and the maximum error of the installation did not exceed 2 mm. in order to make the model foundation and the prototype foundation have the same load transfer characteristics, we refer to the existing research results, and do the sealing treatment on the bottom of the wall of the model foundation.

In addition, because the surface of the Plexiglas is very smooth, direct contact with the soil cannot well simulate the friction between the foundation and the soil contact surface in the actual situation; therefore, after the Plexiglas plate has completed the installation of the necessary measuring equipment, the use of AB glue on the exterior and interior walls for the adhesive sand treatment, as shown in Fig.3, adhesive sand used in the sand particle size and density of the adhesive sand with reference to the Wu et al. [1] on the wall−soil interface of the straight shear test and the research on the wall−soil interface straight shear test by Wu et al. Ultimately, with the above references, it was determined that this test would use standard sand of 0.6 to 2.0 mm at a density of 50 to 70 grain/cm² for the plexiglass surfaces to adhere to the sand to maximize the simulation of wall−soil contact in the actual project.

Fig.3 Double-layer wall structure and assembly for diaphragm wall models.

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2.2 Soil property and loading scheme

This test is to simulate the basic condition of soft ground in actual engineering, so sandy soil is used as the filling soil of this test, and the detailed mechanical parameters of sandy soil are shown in Tab.2. The particle size distribution curve is shown in Fig.4. To prepare the soil for the experiment, a layered filling method was employed. The so-called layered filling method is to control the compactness of each layer of soil filling by dividing the thickness of the soil into several portions of the same thickness and the same weight. At the same time, after the completion of each layer of soil filling, use the ring knife to take samples at the edge of the soil of the outer wall and the edge of the inner wall for testing to confirm whether the density of the filled soil is the same as that of the experimental design as determined by indoor geotechnical tests, and to control the filling of soil to meet the requirements of the test by using the above methods. The above method is used to control the fill soil to meet the requirements of the test.

Tab.2 Soil performance parameters

Item	Value
Natural density ρ (g/cm3)	1.795
Saturated density ρs (g/cm3)	2.05
Natural water content (%)	8.44
Cohesion c (kPa)	0.24
Internal friction angle φ (° )	31.37
Compression modulus Es0.1–0.2 (MPa)	3.77
Natural sand relative density Dr	1.08
Saturated sand relative density Dr	0.44
Uniformity coefficient Cu	8
Curvature coefficient Cc	0.5

Fig.4 Particle size distribution curve.

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To better simulate the filling of the soil core and the soil outside the wall during the actual construction of the foundation of a LSDW, and after referring to the successful experience of similar tests in the past, the test soil will be filled by the layered filling method in this test. In the filling process in advance, at the bottom of the layered filling of a certain thickness of soil, and then the test model will be placed; at this time, the test model does not exert any external force, only with the bottom of the sandy soil in contact, before placing the test model in the model slot, the test model should be checked for horizontal and vertical errors using a checking instrument to ensure that there is no error before placing the test model in the model slot and to check again whether the placement of the model is accurate after the completion of the prevention. After making sure that the test model is placed correctly, we fill the soil body to the designed depth of the test.

It’s important to note that one significant difference between the model and the prototype lies in how the model’s experimental process and results are affected by their boundary conditions. Since the external structures of caisson foundations and LSDWs share certain similarities, we can draw from the selection of the range influenced by caisson foundation model tests. Based on field measurements conducted by Zhu et al. [12] and Che et al. [13], the influence range of a caisson foundation under horizontal load in its loading direction is generally 2.5 times the foundation diameter; accordingly, in order to eliminate the influence of the boundary conditions on the test, this test will determine the horizontal influence range as 3 times the bearing platform size. To realize the horizontal loading effect in this test, design a set of loading devices, including a loading bracket, pulley system, basket, and so on, as shown in Fig.5. In the test process, the weights are placed in the hanging basket, and the horizontal load applied by the weights is amplified through the pulley system and finally applied to the LSDW foundation model to achieve the horizontal loading test effect.

Fig.5 Loading scheme.

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2.3 Double wall setup and parametric determination methods

Due to the special structure of the LSDW foundation, the existence of soil core inside the walls causes the acquisition of test data to become very difficult in order to satisfy the demand that the change rule of the strain value measured on the inner and outer sides of the model wall under the action of a horizontal load of the LSDW foundation in this test coincides with the actual working condition. Therefore, this paper will set up the monitoring wall as a double-layer wall, as shown in Fig.6.

Fig.6 Double-layer configuration of LSDW.

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Specifically, by pasting the strain gauges in the main parts of the inner and outer walls and the central cavity of the double wall (Fig.2), we can accurately obtain the strain data of the test model in each position and then use this as the basis for calculating key data such as bending moments, wall displacements, and wall-side soil resistances to analyze the load-bearing characteristics of the LSDW.

Fig.6 shows the schematic diagram of the installation of the double wall and strain gauges, through which the required test data can be accurately obtained in the test so that through the above method, combined with Eq. (1), the bending moments along the wall body of the inner and outer wall can be calculated M

(1)$M=EI\frac{{\epsilon }_{\mathrm{s}}-{\epsilon }_{\mathrm{c}}}{h_0},$

where E is the modulus of elasticity of the material used in the test model, I is the overall moment of inertia of the cross-section of the test model, ε_s and ε_c are the tensile and compressive strain values of a test model wall in the same cross-section, and h₀ is the spacing between the tensile and compressive strain monitoring points in a cross-section.

Thanks to the double wall design, both interior and exterior wall bending moments along the wall can be calculated independently of each other with the formula:

(2)$M_{\mathrm{i}}=EI\frac{{\epsilon }_{\mathrm{s}\mathrm{i}}-{\epsilon }_{\mathrm{c}\mathrm{i}}}{h_1},$

(3)$M_{\mathrm{e}}=EI\frac{{\epsilon }_{\mathrm{s}\mathrm{e}}-{\epsilon }_{\mathrm{c}\mathrm{e}}}{h_2}.$

To eliminate data errors and ensure the accuracy of the test data, the strain data of the exterior and interior walls were obtained by averaging the data obtained from the two sets of strain gauges, ε_si = (ε_1i + ε_2i)/2 and ε_ci = (ε_3i + ε_4i)/2, where ε_si and ε_ci are the tensile and compressive strain values for the monitored cross sections of the exterior wall, respectively. Similarly, the strain data of the inner wall are averaged for two groups of strain gauges, ε_se = (ε_1e + ε_2e)/2 and ε_ce = (ε_3e + ε_4e)/2, where ε_se and ε_ce are the tensile and compressive strains of the inner wall sections, respectively. Where h₁ is the spacing from the strain gauges on the tensile side to the strain gauges on the compressive side in a section of the external wall of the test model, while h₂ is the distance from the strain gauges on the tensile side to the strain gauges on the compressive side in a section of the internal wall.

Since the LSDW foundation is a double wall and there is a soil core in the LSDW foundation, the moment of inertia I of the cross-section needs to be calculated using the parallel shift equation:

(4)$I=\frac{b_1{h}_1^3-b_2{h}_2^3-2b_3{h}_3^3-24b_2{h}_2{h}_4^2}{12}.$

To facilitate the fitting of the curve, the wall displacement y(x) can be assumed here as a polynomial function:

(5)$y(x)=c_1+c_{2}x+c_3{x}^2+c_4{x}^3+\cdots +c_j{x}^{j-1}+\cdots +c_m{x}^{m-1}.$

To ensure the accuracy of the test data, the least squares method can be utilized to eliminate the error between the actual data obtained from the test and the calculated data; the least squares method can be used as follows:

(6)$e=\sum _{i=1}^n{\left({\overline{y}}_i-y_i\right)}^2,$

where ${\overline{y}}_i$ is the measured wall displacement and $y_i$ is the theoretical calculated value. To minimize the value of e, then:

(7)$\mathrm{d}e=\frac{\mathrm{\partial }e}{\mathrm{\partial }{c}_1}\mathrm{d}{c}_1+\cdots +\frac{\mathrm{\partial }e}{\mathrm{\partial }{c}_k}\mathrm{d}{c}_k+\cdots +\frac{\mathrm{\partial }e}{\mathrm{\partial }{c}_m}\mathrm{d}{c}_m=0,$

where dc_k ≠ 0, then we can obtain:

(8)$\frac{\mathrm{\partial }e}{\mathrm{\partial }{c}_k}=\frac{\mathrm{\partial }e}{\mathrm{\partial }{y}_i}\frac{\mathrm{\partial }{y}_i}{\mathrm{\partial }{c}_k}=-2\sum _{i=1}^n\left(\overline{y_i}-y_i\right)x_i^{k-1}=0,$

(9)$\begin{array}{rl}\sum _{i=1}^n\overline{y}{x}_i^{k-1}=& \sum _{i=1}^n(c_1+c_{2}x+c_3{x}^2+c_4{x}^3+\cdots \\ & +c_j{x}^{j-1}+\cdots +c_m{x}^{m-1})x_i^{k-1}.\end{array}$

From the above, the following matrix exists

(10)$[T]=[S][C],$

where $[T]={\displaystyle \sum _{i=1}^n{\overline{y}}_i{x}_i^{k-1}}$, $[T]={\displaystyle \sum _{i=1}^n{\overline{y}}_i{x}_i^{k-1}}$, $[C]=[S{]}^{-1}[T]$.

From the above, the wall bending moment calculation formula is

(11)$M(x)=EI\cdot {\mathrm{d}}^{2}y(x)/\mathrm{d}{x}^2.$

The derivation of the perimeter soil reaction P(x) from the wall bending moment M(x) is similar to the previous discussion on the derivation of the front bending moment from the wall displacement. From the wall body shear at the mud face, c₂ = (dM/dx)_{x = 0} = P; from the horizontal soil resistance around the wall at the mud face, which is 0, c₃ = 0.5(d²M/dx²) = 0; and from the horizontal soil resistance around the wall at the base of the wall, which is 0, P(x) = (d²M/dx²)_{x = L} = 0. Therefore, the perimeter earth reaction force, P(x), is:

(12)$P(x)={\mathrm{d}}^{2}M/\mathrm{d}{x}^2.$

Through the above method, we can obtain the displacement of the wall body of LSDW foundation and the soil resistance of the inner and outer walls, and then we can obtain the p–y curve of LSDW foundation under the action of horizontal load.

3 Experimental results and analysis

3.1 Q–s curves

Because the overall stiffness of LSDW is very large, the horizontal displacement at each point of the top of the wall is basically the same, so it can be taken as the horizontal displacement at any point of the top of the wall (in this paper, the center of the top of the wall is taken as the center of the top of the wall) in place of the displacement characteristics of the whole top of the wall. Through the top of the wall (the monitoring point is located in the mud surface at 0.125 m) horizontally uniformly deployed two linear variable displacement transducer (LVDTs) (Fig.6); accordingly, it is possible to monitor in real-time the displacement of the top of the wall in the horizontal direction under all levels of horizontal loads. Take its average and record the corresponding load displacement value, and finally, the load–displacement (Q–s) curve of the single-chamber and two-chamber LSDW can be drawn, as shown in Fig.7.

Fig.7 Q–s curves.

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It can be seen that the displacement of the LSDWs increases with the lateral load. The increase in displacement is smaller at lower loads (around 1 kN) and becomes more significant at higher loads. However, overall, the Q–s curves for both types of LSDW foundations show a gradual trend without any sharp inflection points. The LSDW foundations exhibit greater overall rigidity compared to traditional pile foundations, which show a steep decline in their Q–s curves. The Q–s curves of LSDW foundations demonstrate a continuous, gradual change without obvious inflection points, reflecting their evolution and failure mode.

Therefore, the standard for judging the horizontal bearing capacity of LSDW foundations should be based on the allowable horizontal displacement of the structure built on the foundation. To facilitate a comparative analysis of the impact of the number of chambers on the horizontal bearing capacity of the LSDW foundation, the load corresponding to 10 mm displacement is used as a reference point. At this displacement, the bearing capacity of the single-chamber LSDW is about 1.9 kN, while for the two-chamber LSDW, it reaches 3.2 kN, approximately 1.68 times that of the single-chamber LSDW. This indicates that the increase in bearing capacity from single to two-chamber LSDW is not proportional, likely due to the group wall effect [14].

3.2 Displacement analysis of wall body

Using the strain gauges installed on the wall body, and based on Eq. (6), the wall displacement of LSDWs at different depths under typical load levels can be calculated, as shown in Fig.8 and Fig.9.

Fig.8 Displacement distribution of single-chamber foundation.

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Fig.9 Displacement distribution of double-chamber foundation.

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In Fig.8 and Fig.9, a positive horizontal displacement indicates that the displacement direction is the same as the loading direction, while a negative value indicates that the displacement direction is opposite to the loading direction. It can be inferred that, due to the significant stiffness of the LSDW foundation, under the action of horizontal loads, the displacement approximately follows a diagonal straight line along the wall. When subjected to horizontal loads, the LSDWs undergo rotation from a point above the wall end, exhibiting an overall tilting deformation and failure characteristic. The horizontal displacement of the wall increases with the increasing load, and it gradually decreases linearly with increasing depth into the soil. Both types of LSDW foundations experience zero displacements at a depth of approximately 610 mm below the mud surface (approximately 0.87D, where D is the foundation depth).

Within the range from 0.87D to the wall end, the wall generates displacements opposite to the loading direction, which increase with the load. In general, LSDW foundations under horizontal loads exhibit an overall tilting deformation and failure trend in the direction of loading, and they rotate around a specific turning point along the wall. For LSDW foundations, the location of the turning point for the overall foundation deformation is not influenced by the load level or the number of chambers. The turning point location (i.e., the neutral point of displacement) remains at a depth of approximately 0.87 times the foundation depth.

3.3 Angular displacement analysis

By utilizing strain gauges installed on the wall and based on Eq. (5), the angular displacement θ of the LSDWs at various depth positions under typical load levels (i.e., θ–D curve) can be calculated, as shown in Fig.10 and Fig.11.

Fig.10 θ–D curve of single-chamber LSDW.

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Fig.11 θ–D curve of double-chamber LSDW.

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It can be observed that under the horizontal load on the wall, due to the relatively high overall stiffness of LSDW foundation, it behaves like a rigid body, resulting in minimal bending of the foundation wall and only a small angular displacement at the foundation level, typically on the order of 0.001 radians. Additionally, with increasing wall depth, the change in angular displacement of the wall is extremely small. Notably, there is a relatively noticeable variation in wall angular displacement in the middle and upper portions of the single-chamber foundation; for the double-chamber foundation, with its large wall stiffness, the angular displacement of the wall under horizontal loading changes very little and is almost negligible. This further confirms the earlier mentioned characteristics of LSDWs having significant overall stiffness and exhibiting a tilting deformation and failure pattern around a specific pivot point.

Fig.12 depicts the variation of wall top angular displacement θ with load Q for LSDW foundations (i.e., θ–Q curves). It can be observed that the wall top angular displacement increases gradually with increasing load. As can be seen from the figure, whether at the beginning or at the end of loading, when the double-chamber LSDW is subjected to the same horizontal load as the single-chamber LSDW, the horizontal displacement of the top of the wall is significantly smaller than that of the single-chamber LSDW. Whether it is a single- or a double-chamber LSDW, the θ–Q curve follows a similar trend to the Q–s curve, showing a gradual change.

Fig.12 θ–Q curves.

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3.4 Bending moment analysis

For foundations with soil cores, such as tube pile foundations, the bending moment is often measured by attaching strain gauges on the inside and outside surfaces of the foundation body to calculate the strain and subsequently derive the bending moment. However, this method has significant limitations in studying the overall mechanical behavior and soil−structure interaction mechanisms of LSDW. It does not effectively capture the load response and deformation mechanisms of LSDW under horizontal loads, particularly the challenging study of the interaction between the soil core and the wall. Therefore, in this experiment, a double-layer wall structure was used for the LSDW test model (Fig.6), with the specific testing principles detailed in Subsection 2.3. As a result, we obtained, for the first time in indoor experiments, the bending moment curves for the outer and inner walls of single- and double-chamber LSDW models under typical load levels, as shown in Fig.13 and Fig.14.

Fig.13 Bending moment analysis of outer wall: (a) single-chamber; (b) double-chamber.

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Fig.14 Bending moment analysis of inner wall: (a) single-chamber; (b) double-chamber.

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From Fig.13, it can be observed that under horizontal static loading, the bending moment distribution along the wall of a single-chamber LSDW increases with the increasing load. It exhibits a trend of initially increasing and then decreasing with depth, and the bending moment does not return to zero at the wall end. The bending moment along the wall shows nonlinear variation and reaches its maximum at a depth of around 200 mm below the mud surface (i.e., 0.29D). With increasing load, there is a tendency for the maximum moment location to move deeper below the mud surface.

As can be seen from the above figure, whether it is single-chamber foundation or double-chamber foundation, the overall trend of bending moment distribution under the action of horizontal load is relatively similar, and it shows the trend of increasing first and then decreasing. The bending moment curves along the wall body of both types show similar trends, but there are still some differences. Due to the greater overall stiffness of the wall body of the double-chamber LSDW foundation, the change in the position of the maximum value of the bending moment with the increase of load is not very obvious compared with that of the single-chamber type, and the reduction of the bending moment at the end of the wall is also significantly weakened due to the greater stiffness and the strong soil resistance of the wall side.

Fig.14 represents the bending moment distribution curve for the inner wall of the LSDW foundations. It can be observed that, compared to the bending moment distribution of the outer wall, the bending moment distribution of the inner wall is in the opposite direction. The maximum bending moment also occurs at a depth of around 200 mm below the mud surface (i.e., 0.29D), but with increasing load, there is a trend for the maximum moment location to move upward, opposite to the trend observed for the outer wall bending moment, as shown in Fig.15. The bending moment distribution along the inner wall of LSDW increases initially with depth and then decreases. Below the mud surface, at a depth of approximately 0.9D, the bending moment changes from negative to positive.

Fig.15 Earth pressure analysis of outer wall: (a) single-chamber; (b) double-chamber.

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From the above figure, it can be seen that the maximum value of the bending moment of either the interior wall or the exterior wall is located at 0.29D below the mud surface. However, in contrast to the outer wall bending moment, the inner wall bending moment distribution has a zero point, and its location is almost consistent with the position where the wall undergoes rotation, namely, below the mud surface (0.9D). This may be due to the soil core being enclosed by the inner wall, effectively forming a one-dimensional soil column. During the rotation process, the soil core and the wall maintain almost the same pace of movement, which partly differs from the deformation and stress pattern of the surrounding soil mass.

3.5 Earth pressure analysis

Depicted in Fig.15 are the soil resistance curves on the side of the exterior wall for the two foundation forms explored in this test, distributed along the depth, under horizontal loading. For the single-chamber LSDW, the distribution begins with positive earth pressure values at the top of the wall, suggesting that the top experiences passive pressure as the soil pushes against the wall, creating a state of compression. This pressure gradually transitions to negative values toward the bottom, indicative of active earth pressure where the soil moves away from the wall. The pivot point, where the pressure shifts from positive to negative, suggests a rotation point around which the wall is likely deflecting under the applied load.

In the double-chamber configuration, the pressure profile along the external wall is slightly different. The earth pressure distribution along the external wall shows a greater range of both positive and negative pressures compared to the single-chamber variant. This is indicative of the double-chamber LSDW’s enhanced horizontal load-bearing capacity, as reflected in the higher magnitudes of applied loads. While the pressure at the top remains positive, the transition to negative pressures at the bottom is more abrupt, and the range of pressures is broader. This change in the pressure profile indicates a more complex soil−structure interaction, where the additional chamber alters the distribution of lateral forces along the wall.

Fig.16, to be interpreted in a corresponding manner, displays the earth pressure distribution along the inner walls of LSDWs for single and double-chamber designs. Unlike the external walls, the internal walls face the confined soil core, resulting in a different pattern of pressure distribution. i.e., the distribution begins with negative values at the top of the wall and gradually transitions to positive values toward the bottom. In addition, the inner wall shows a more evenly distributed profile of earth pressure at the top wall range compared to the outer wall, and this is particularly evident in the greater variation and magnitude of earth pressures. This indicates that the internal wall, interfacing with the confined soil inside the chambers, shows a more uniform pressure profile, indicative of the moderating influence of the internal soil core on the wall’s lateral deflections.

Fig.16 Earth pressure analysis of inner wall: (a) single-chamber; (b) double-chamber.

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3.6 p–y curves analysis

The development of a robust method for fitting the bending moment curves of LSDW is an ongoing challenge in geotechnical engineering. In the current study, we explored the suitability of various polynomial fitting methods, referencing approaches commonly used for caisson and pile foundations. After comparative analysis of the cubic spline interpolation and the fifth-order polynomial fitting methods, we observed substantial fluctuations in the calculated earth pressures from the second derivative of the cubic spline interpolation. Similarly, the fifth-order polynomial method introduced unrealistic inflections in the computed earth pressure curves.

Our experimental results indicated that higher-degree polynomials, specifically the sixth and seventh-order, provided a superior fit for the bending moment curves of the LSDW. Furthermore, these higher-degree polynomials yielded earth pressure distributions that were reasonable upon second differentiation [15,16]. Therefore, we adopted a sixth-order polynomial to fit the bending moment curves in this study. The earth pressure along the wall was obtained by differentiating the bending moment twice, and the wall displacement was determined by integrating the bending moment curve twice, as deduced in Eqs. (11) and (12).

Across all tests, the LSDW foundation’s p–y response curves demonstrated slower responses with increasing depth, as shown in Fig.17 and Fig.18. When comparing the outer and inner wall p–y curves, the latter exhibited a faster response at the same displacement levels. This is attributed to the more complex distribution of lateral earth resistance within the soil core. Compared to the semi-infinite soil body surrounding the wall, the soil core behaves akin to a one-dimensional soil column, resulting in significantly greater maximum earth pressures on the outer wall. As depth increases, the inner wall’s p–y curve shows a more pronounced decreasing trend in earth pressure, gradually stabilizing at the displacement’s end. Conversely, the outer wall’s p–y curves exhibit an opposite trend with increasing depth, where the earth pressure and growth trend are more pronounced, and the curves approach stability more slowly than the inner wall’s.

Fig.17 p–y curves of outer wall: (a) single-chamber; (b) double-chamber.

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Fig.18 p–y curves of inner wall: (a) single-chamber; (b) double-chamber.

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When examining double-chamber LSDWs, the horizontal loads applied were greater than those for single-chamber foundations, and the surface area of the double-chamber LSDW wall was also larger, hence resulting in greater maximum earth pressures for the double-chamber LSDW foundation. Compared to single-chamber configurations, the double-chamber inner wall p–y curves responded faster, although the overall trend was similar, with an apparent increase in earth pressure growth with depth, responding slower. For the outer walls, the double-chamber p–y curves with decreasing depth showed a gradual increase in earth pressure growth compared to the single-chamber configurations.

Currently, the p–y curve method is extensively used for the analysis of forces and displacements in foundations subjected to horizontal loads [17–20]. Among various p–y curve methods, the hyperbolic p–y curve method is particularly favored for its ability to accurately represent the developmental trends observed in measured p–y curves. Its general expression is:

(13)$p=\frac{y}{{\displaystyle \frac{1}{k}}+{\displaystyle \frac{y}{p_{\mathrm{u}}}}},$

where p_u represents the ultimate soil resistance, and k is the initial subgrade reaction coefficient. For sandy soils, these coefficients can be obtained using the following [21]:

(14)$p_{\mathrm{u}}=K_{\mathrm{p}}^2\gamma \mathit{\text{z}}B,$

(15)$k={\eta }_{\mathrm{i}}\mathit{\text{z}},$

where η_i is the initial subgrade reaction coefficient; K_p is the coefficient of active earth pressure; γ is the effective unit weight of saturated sand; z is the depth, and B is the effective width of the foundation in the direction of the load.

This paper attempts to analyze the p–y curves of the outer walls of LSDWs using the American Petroleum Institute (API) method and the hyperbolic p–y curve method. Fig.19 and Fig.20 present the measured p–y curves for single and double-chamber LSDWs, alongside the calculated results from the API standard and the hyperbolic method. Comparisons reveal that for LSDW foundations, the p–y curves calculated using the API standard exhibit a significantly higher stiffness at the initial stages of wall deformation, leading to notably smaller calculated horizontal displacements at the wall top. This discrepancy could pose safety concerns in engineering design. In contrast, the p–y curves derived from the hyperbolic method closely match the experimentally measured p–y curves at various depths, indicating that the hyperbolic method provides a better fit for the experimentally obtained p–y curves of LSDW foundations compared to the API method (refer to Tab.3 and Tab.4 for standard deviation values). Therefore, using the p–y curve method for analyzing the horizontal behavior of LSDW foundations is feasible, and the hyperbolic p–y curve method offers higher accuracy in calculations.

Tab.3 Standard deviation of fitted curves for single-chamber LSDW

Depth (mm)	Loading level
	1	2	3	4	5	6	7
0	17.5281	20.6474	23.8454	27.5515	26.0873	22.3301	17.7906
45	21.2171	31.4857	57.0488	58.7918	49.9364	41.0099	21.4055
90	25.5288	50.4426	63.4890	57.7204	57.0783	54.8756	34.0495
135	42.4511	38.3420	51.1908	52.1908	61.9346	60.7458	29.1561
180	20.1072	30.6487	35.4827	31.6581	52.2648	64.4427	18.2228
225	3.2652	31.2092	32.6572	67.6244	93.1528	78.4559	40.9744

Tab.4 Standard deviation of fitted curves for double-chamber LSDW

Depth (mm)	Loading level
	1	2	3	4	5	6	7	8
0	7.0794	9.9426	11.3952	29.1704	37.3760	31.1812	30.3165	24.7891
45	7.20351	37.1666	57.8779	50.4880	64.2755	87.3306	66.3622	55.4472
90	5.2315	9.5806	18.7175	52.2120	66.8491	97.3975	117.776	113.872
135	4.2246	6.3044	23.0769	29.4978	35.0645	122.749	186.143	180.498
180	2.0158	8.5567	15.4916	6.0323	59.9049	80.4946	226.198	202.319
225	5.5596	9.18542	4.56173	27.1229	24.3787	106.359	203.821	181.629

Fig.19 Comparison of p–y curves for single-chamber LSDW.

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Fig.20 Comparison of p–y curves for double-chamber LSDW.

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4 Conclusions

In this study, an in-depth investigation into the lateral load-bearing behavior of LSDWs has led to several key findings that enhance our understanding of their structural characteristics and performance. These findings include.

1) The Q–s curves of LSDW foundations demonstrate a continuous, gradual change without obvious inflection points, reflecting their evolution and failure mode. Therefore, the standard for judging the horizontal bearing capacity of LSDW foundations should be based on the allowable horizontal displacement of the structure built on the foundation.

2) The bearing capacity for the double-chamber LSDWs was found to be approximately 1.68 times that of the single-chamber structure, pointing to a complex interplay between chamber number and structural capacity that extends beyond a linear relationship and incorporates the group wall effect.

3) Displacements recorded in the LSDW models under various load levels illustrated a direct correlation with the applied lateral loads, decreasing in a linear fashion with depth. This behavior underscores a consistent tilting deformation pattern, with a distinct rotation point identified at approximately 0.87D from the mud surface, regardless of the chamber configuration.

4) Angular displacements were minimal, suggesting that the LSDWs behaved predominantly as rigid bodies when subjected to horizontal loads. The double-chamber models, in particular, showed reduced angular displacements, reflecting their increased structural rigidity.

5) Nonlinear variations in the bending moment along the wall depth highlighted the intricate load distribution within the LSDWs. Both LSDW configurations reached a peak bending moment at a depth of 0.29D, with indications of the peak shifting downward as load levels escalated.

6) The p–y response curves of LSDW foundations indicated that inner walls respond faster to displacement than outer walls, with the latter experiencing higher maximum earth pressures due to the soil core’s complex lateral resistance, which decreases more markedly with depth. In contrast, double-chamber LSDWs bear higher horizontal loads and exhibit greater earth pressures than single-chamber configurations, with both showing increased pressure growth with depth, albeit more slowly for the double-chamber variant.

7) The hyperbolic p–y curve method demonstrates superior accuracy over the API standard in aligning with experimental p–y curves of LSDWs, ensuring more reliable analysis of their horizontal behavior. This method’s heightened precision makes it a preferable choice for calculating and assessing the lateral responses of LSDW foundations.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant Nos. 52178163 and 51808437), Xi’an Science and Technology Plan (No. 22SFSF0005), the Key R&D Program of Shaanxi Province (No. 2022SF-403), and the China Scholarship Council (No. 201908610062).

Conflict of Interest

The authors declare that they have no conflict of interest.

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