Arching behavior and pressure distribution characteristics in calcareous sand strata: Insights from trapdoor tests

Abstract

Calcareous sand exhibits distinct physical and mechanical properties, including irregular particle shapes, high porosity, and crushability, leading to arching behaviors that differ from siliceous sand. This study investigates the development of soil arching and pressure redistribution and ground deformation in calcareous sand under trapdoor excavation. A series of model tests examined the effects of particle gradation, void ratio, and burial depth ratio. Comparative tests were also performed between calcareous and siliceous sands under wet conditions, and between dry and wet calcareous sand. Results indicate that soil arching evolves through four stages: initiation, formation, expansion and adjustment, and residual state. Compared to siliceous sand, calcareous sand shows stronger arching and lower surface settlement, attributed to higher interparticle friction and stronger particle interlocking. Moisture enhances arch stability, while dry calcareous sand exhibits ongoing loosening and greater settlement. Well-graded sands and lower void ratios promote arch formation and reduce relaxed earth pressure, whereas a smaller burial depth ratio limits arch development. To quantify these behaviors, three normalized indices are proposed to evaluate unloading efficiency, surface deformation and arch mobilization, enabling unified comparisons. These findings provide a systematic understanding of deformation mechanisms and offer practical insights for geotechnical design in calcareous sand strata.

Graphical abstract

Keywords

calcareous sand / soil arching effect / trapdoor test / particle image velocimetry / relaxed earth pressure

1 Introduction

Subsurface excavation activities such as tunneling often induce stress redistribution in the surrounding ground, typically manifested as soil arching. This phenomenon was first demonstrated by Terzaghi [1,2] through trapdoor experiments and has since been recognized as a key mechanism controlling the variation of vertical earth pressure during excavation-induced deformation [3–7].

Calcareous sand, widely encountered in coral reef and island regions, is mainly composed of biogenic debris from marine organisms. Owing to its complex genesis, it exhibits distinctive physical and mechanical features, including high porosity, irregular and fragile particles, and strong interparticle interlocking [8–11]. These characteristics lead to apparent cohesion and internal friction angles that are generally higher than those of conventional siliceous sand [12–15], and can significantly affect its deformation behavior and stress-transfer mechanisms under loading. In addition to its high angularity and strong interlocking, calcareous sand is crushable, and particle breakage has been shown to markedly influence its dilatancy, phase transformation, and shear strength [16–18]. Its compressibility and lateral earth-pressure response may also differ substantially from those of siliceous sands owing to crushing susceptibility under confined loading [19–21]. However, most of this evidence comes from oedometer or triaxial tests conducted under relatively high confinement, and how crushing manifests under low-stress, excavation-induced unloading conditions relevant to soil arching remains unclear. Despite the increasing use of calcareous sand as a foundation material in coastal and island engineering, the soil arching behavior of calcareous sand has received little attention [22]. Most existing studies on soil arching have been performed on siliceous sands, and direct extrapolation of their findings to calcareous sand may lead to inaccurate assessment or unsafe design.

Trapdoor tests are a fundamental experimental approach for investigating the formation and evolution of soil arching. Previous work has examined the influence of burial depth ratio, trapdoor sequence and device scale on failure modes and pressure redistribution [23–25]. With the aid of advanced techniques such as centrifuge modelling, transparent soil and particle image velocimetry (PIV), the progressive development of curved, triangular and prismatic arching structures and the associated load reduction have been clarified [26–29]. Other studies have explored additional influencing factors: Liang et al. [30] investigated the evolution of slip surfaces under gradual relative displacement; Rui et al. [31,32] and Gao et al. [27] highlighted the role of fill density in controlling minimum and ultimate arching ratios; and Xu et al. [33] incorporated seepage effects into two-dimensional trapdoor tests, finding that seepage flow increases vertical stress and particle displacement but has limited influence on the development of slip surfaces; Xu et al. [34] used custom devices to perform trapdoor tests under self-weight, static and cyclic loading, showing that both static and cyclic loads can weaken a stable arch, with the degree of weakening increasing with load amplitude and frequency and decreasing with a larger load application area. In addition, a recent analytical study on loosening earth pressure above a shallow trapdoor in unsaturated soil, considering different groundwater levels and matric suction distributions, has further extended Terzaghi-type arching theory to unsaturated conditions by deriving solutions for effective loosening earth pressure under non-uniform stress states [35]. However, these investigations have predominantly focused on granular materials with relatively uniform particle shapes and moderate interlocking. For calcareous sand, whose angular, crushable grains and strong interlocking may substantially alter stress redistribution, deformation patterns and arching evolution, systematic experimental evidence under different moisture contents, relative densities and burial depths is still scarce.

To address this gap, this study conducts a series of trapdoor model tests focusing on the arching behavior of calcareous sand. The experimental program includes comparisons between calcareous and siliceous sands under wet conditions, as well as between dry and wet calcareous sand. Furthermore, the effects of key influencing factors, including burial depth, relative density and particle gradation, on the deformation characteristics and stress evolution in calcareous sand are systematically investigated. Based on PIV-derived displacement fields and earth-pressure measurements, the evolution of soil arching is characterized in detail, and three normalized indices are proposed to quantify unloading efficiency, surface settlement and arch mobilization, providing a unified framework for comparing different materials and test conditions. The findings provide a clearer understanding of the soil arching mechanisms specific to calcareous sand and offer practical insights for engineering applications, particularly in tunnel excavation and foundation design in marine or island environments.

2 Trapdoor model tests in calcareous sand formations

2.1 Materials

As shown in Fig. 1, the calcareous sand used in the test exhibits diverse particle shapes, distinct edges and corners, and uneven surface textures with a rich pore structure. Since the calcareous sands are mainly composed of CaCO₃, the particles are mostly white in color. The specific gravity of the calcareous sand particles is Gs = 2.744.

Dried calcareous sand was sieved using a standard mechanical sieve shaker to produce three types of gradations with different coefficients of uniformity (C_u): Soil No. 1 (C_u = 2, homogeneous and poorly graded), Soil No. 2 (C_u ≈ 8, well-graded), and Soil No. 3 (C_u ≈ 16, gap-graded). The corresponding particle-size distribution curves are shown in Fig. 2(a). The characteristic particle sizes (D₁₀,D₃₀,D₆₀) and gradation parameters (C_u,C_c) for all sands are summarized in Table 1.

To facilitate comparison, a well-graded siliceous sand sample (Soil No. 4) was prepared using the same processing method. Due to the particle size limitation of standard siliceous sand (less than 2 mm), it was not possible to replicate the full gradation of calcareous sand. Instead, the gradation of Soil No. 4 was adjusted to match the coefficient of uniformity (C_u) and coefficient of curvature (C_c) of the well-graded calcareous sample (Soil No. 2). The particle size distribution curve for the siliceous sand is shown in Fig. 2(b), and its characteristic sizes and gradation parameters are also included in Table 1. The limiting void ratios were determined according to standard methods: for calcareous sand, e_max = 1.626 and e_min = 0.695; for siliceous sand, e_max = 0.697 and e_min = 0.362. The relative density (D_r) for each specimen was calculated based on Dr = (e_max − e)/(e_max − e_min) × 100%. To address possible scale effects, the characteristic particle sizes are summarized in Table 1, including D₅₀. Based on the sieve curves, D₅₀ = 2.18 mm (Soil No. 1), 1.98 mm (Soil No. 2), 1.57 mm (Soil No. 3), and 0.99 mm (Soil No. 4). With the trapdoor width B = 200 mm, the ratios B/D₅₀ range from ~92 to 201, indicating that the model characteristic dimension is sufficiently larger than the particle size. The maximum particle size is d_max ≈ 5 mm for the calcareous sands and d_max ≈ 2 mm for the siliceous sand, giving B/d_max ≈ 40–100. In addition, the trapdoor was located 400 mm from the sidewall (≥ 2B), and the adopted cover depths (H = 100–400 mm) also yield sufficiently large H/D₅₀ values, suggesting that particle-size and boundary-related scale effects are limited in the present model tests.

Although the particle size distributions of the two sands were matched to minimize gradation effects, their particle morphologies differ markedly, particularly at the local texture scale. To quantify these differences, particle shape descriptors including three-dimensional sphericity, roundness, and roughness were adopted based on recent high-precision X-CT reconstruction studies on comparable calcareous and siliceous sands (e.g., Liang et al. [36]). Quantitative analysis reveals that while the overall form (sphericity) of the two sands is relatively high, significant differences exist in their angularity and surface texture. Specifically, the calcareous sand exhibits a mean sphericity of 0.892 and a mean roundness of 0.625. In contrast, the siliceous sand shows higher mean values of 0.921 for sphericity and 0.728 for roundness. Notably, the calcareous sand is approximately 14% lower in roundness and 6% higher in roughness (1.081 vs. 1.021) compared to the siliceous sand. These indices confirm that the morphological differences are dominated by local angularity and surface roughness rather than general sphericity. This rougher, more angular morphology of the calcareous grains significantly enhances inter-particle interlocking and friction, which serves as the primary mechanism for the observed differences in soil arching behavior.

To evaluate the shear strength characteristics of both calcareous and siliceous sands, a series of direct shear tests were conducted under different working conditions. For the standard calcareous sand (e = 1.2, C_u = 8), the cohesion (c) and internal friction angle (φ) were 63.4 kPa and 45.6°, respectively, which are significantly higher than those of the siliceous sand (c = 20.7 kPa, φ = 33.9°) under the same conditions. For the other calcareous sand samples, the friction angle φ remained relatively stable (ranging from 42.6° to 45.9°), while the cohesion c varied notably with density and gradation, ranging from 43.2 kPa (for the loose state e = 1.5) to 97.3 kPa (for the dense state e = 0.9)

2.2 Experimental apparatus

The test setup utilized in this experiment is depicted in Fig. 3. The setup comprises two main sections: the upper part being the model box with dimensions of 1000 mm in length, 500 mm in height, and 400 mm in width. The front side of the model box is constructed from a thick transparent acrylic board, while the other three sides are composed of welded stainless steel plates. The base of the model box features pre-drilled holes with a diameter of 9 mm to accommodate and secure earth pressure cells. A trapdoor measuring 200 mm in size is centrally positioned 400 mm away from the sidewall. Below the trapdoor, a displacement meter is installed to measure its movement, and the trapdoor is linked to a motor for control over its motion.

2.3 Testing procedures

In this study, earth pressures, ground surface settlements and internal displacement fields were measured by earth pressure cells, displacement meters and a PIV system. The earth pressure cells (type BWM28-0.02; capacity 20 kPa, diameter 28 mm, thickness 6.5 mm, accuracy 0.5%) and displacement meters (type YWC-100; range 0–100 mm) were used together with a PIV system consisting of a digital single-lens reflex camera and image-processing software. Digital images taken during each test were processed by PhotoInfor [37] to obtain particle displacements, and the displacement fields were post-processed and plotted using PostViewer [38]. The captured images have a resolution of 1554 × 618 pixels and were recorded at 1 fps. In PhotoInfor, the correlation was performed within a test-specific rectangular Region of Interest (ROI) selected to fully cover the active deformation zone above the trapdoor. Because the deformation pattern and its spatial extent vary among test conditions, the ROI size and position were adjusted slightly from test to test; however, the PIV processing parameters were kept identical for all tests, with a subset size of 21 × 21 pixels and a step size of 10 pixels. A representative ROI size is 560 × 240 pixels, which yields a 57 × 25 (1425) measurement grid. The sensor layout is shown in Figs. 3(b)–3(d): Cells Nos. 1–12 are mounted on the trapdoor, cells Nos. 13–16 at the base of the model box beside the trapdoor, and cells Nos. 17–19 within the soil mass at mid-depth (within 100 mm of the centerline). Two displacement meters, Nos. 21 (front) and 22 (back), are installed at the ground surface above the trapdoor to record central surface settlement.

Before filling, the inner walls of the model box, particularly the front acrylic panel, were thoroughly cleaned and coated with a thin layer of petroleum jelly to minimize boundary friction and ensure plane strain conditions. All sensors at the box base were installed, calibrated and zeroed. A thin bedding layer of fine sand (0.315–0.63 mm) was spread around the pressure cells to reduce local stress concentration. To control the target density and improve specimen fabric consistency, all tests adopted a layer-wise static compaction (tamping) procedure, rather than air pluviation. This method was selected to minimize particle segregation, which can be pronounced in well-graded and gap-graded sands during pluviation. The sand was placed in 50-mm-thick layers. For each layer, the mass of sand was predetermined from the target void ratio and the layer volume and then carefully controlled during filling. For the dry series, oven-dried sand was weighed, poured, levelled and statically compacted to the target layer height. For the wet series (moist tamping), dry sand was premixed with water to a target gravimetric water content of 15% prior to compaction. After each layer was compacted, the surface was lightly scarified to enhance interlayer bonding. The procedure was repeated until the target burial depth was reached. After each layer, earth pressure readings were checked to confirm uniform density. Layers were added until the target burial depth was reached, and the resulting earth pressures were recorded as the self-weight stress p₀. After installing the remaining sensors and the camera, all instruments were checked and re-zeroed if necessary. The trapdoor was then driven downward at a constant rate of 0.1 mm/s to a maximum displacement of 50 mm, while earth pressure, surface settlement and PIV images were recorded throughout.

2.4 Experimental program

To investigate the deformation behavior and soil arching of calcareous sand during trapdoor-induced excavation, three groups of laboratory model tests were carried out: 1) comparative tests between calcareous and siliceous sands under wet conditions; 2) tests on dry and wet calcareous sand; and 3) parametric tests on calcareous sand under dry conditions.

2.4.1 Wet-state comparison: Calcareous sand vs. siliceous sand

In the first group, the arching behavior of calcareous sand was compared with that of siliceous sand under identical wet conditions. The burial depth ratio (soil depth/trapdoor width) was fixed at 1.0 (soil depth 200 mm), and the relative density was 0.457. Calcareous sand Soil No. 2 (C_u = 8, C_c = 2) was used as a representative well-graded sand, while siliceous sand Soil No. 4 was prepared with matching gradation parameters. The wet condition was defined by the gravimetric water content (w). Based on the standard oven-drying method (representative samples dried to constant mass), the measured average water contents were 16.5% for the calcareous sand and 14.8% for the siliceous sand, indicating a comparable moist (unsaturated) state. This group isolates the influence of mineral composition and particle morphology on arching under the same boundary and initial conditions.

2.4.2 Moisture condition comparison: Dry vs. wet calcareous sand

The second group focused on the effect of moisture. Using Soil No. 2 with the same burial depth (200 mm) and relative density, tests were performed under dry and wet states to clarify how moisture affects deformation, load transfer and arch formation in calcareous sand.

2.4.3 Parametric analysis under dry conditions (calcareous sand only)

The third group consisted of parametric tests on dry calcareous sand. The baseline case had a burial depth ratio of 1.0 (200 mm soil thickness), void ratio e = 1.2 and well-graded Soil No. 2 (C_u = 8). Around this baseline, three key parameters were varied independently: Burial depth ratio: 0.5, 1.0 and 2.0 (soil thicknesses 100, 200, and 400 mm), to examine the effect of overburden and arch height on earth pressure redistribution. Particle gradation: uniform (Soil No. 1, C_u = 2), well-graded (Soil No. 2, C_u = 8) and gap-graded (Soil No. 3, C_u = 16), to assess how particle size distribution influences interlocking and arch formation. Void ratio: 0.9, 1.2 and 1.5. These values correspond to relative densities (D_r) of approximately 78% (dense), 46% (medium-dense), and 14% (loose), respectively.

Test IDs follow the format N_nC_ce_v, where n is the burial depth ratio, c represents the coefficient of uniformity (C_u), and v denotes the void ratio (e). For example, the baseline case is N₁C₈e_1.2. In total, seven cases were designed to cover the above parameter combinations.

3 Results and discussion

3.1 Comparative analysis on calcareous and silica sands under wet conditions

3.1.1 Soil deformation and failure mechanisms

To clarify the link between arch evolution and trapdoor displacement, soil responses at representative trapdoor descents of 5, 10, 20, 30, 40, and 50 mm are shown in Fig. 4 (PIV displacement fields) for both wet calcareous and wet siliceous sands. The corresponding earth-pressure response is discussed in Subsubsection 3.1.2. The four-stage arching evolution is identified using operational criteria combining 1) kinematic features in the PIV-derived displacement fields (i.e., the emergence and evolution of a clearly recognizable active-zone boundary), and 2) the characteristic response of the averaged vertical earth-pressure ratio on the trapdoor, {\overline{k}}_{\text{v}}\left(\text{Δ}\right). Here, {\overline{k}}_{\text{v}}\left(\text{Δ}\right) is defined as the mean value of the pressures measured by the trapdoor cells (Nos. 5–8), normalised by the initial self-weight pressure p₀ at the same depth.

1) Wet calcareous sand

The deformation evolution of calcareous sand is classified into four distinct stages based on the PIV-derived displacement fields and the averaged vertical earth-pressure ratio {\overline{k}}_{\text{v}}\left(\text{Δ}\right). Stage I––Initiation (Δ ≤ 10 mm): soil movements are initially limited. As {\overline{k}}_{\text{v}}\left(\text{Δ}\right) starts to deviate from the self-weight state, differential displacement initiates at the arch foot. When the trapdoor descent reaches about 10 mm, an incipient triangular loosening zone appears with a boundary inclined at approximately 40°–45°. Stage II––Formation (Δ ≤ 20 mm): a distinct collapse-arch profile becomes clearly recognizable as displacements concentrate along a curved band. Mechanically, this kinematic breakthrough coincides with {\overline{k}}_{\text{v}}\left(\text{Δ}\right) exhibiting a rapid reduction towards its minimum value. The arch height reaches about 70–80 mm, with deformation confined below the upper pressure cells. Stage III––Expansion and adjustment (20–40 mm): the geometry of the failure zone remains largely stable, changing only slightly as the loosened block settles as a whole. Mechanically, {\overline{k}}_{\text{v}}\left(\text{Δ}\right) varies gradually or fluctuates around a quasi-stable level, while downward displacement dominates within the loosening zone. Stage IV––Residual state (40–50 mm): the arch-controlled response is fully established. Kinematically, further descent mainly increases settlement within the existing loosening zone without extending the failure surface boundaries. Correspondingly, {\overline{k}}_{\text{v}}\left(\text{Δ}\right) converges to a final stable residual value. The final inclination at the arch foot remains about 40°–45°.

2) Wet siliceous sand

Siliceous sand follows the same four-stage framework, but the failure-zone geometry and deformation localization differ markedly. Stage I––Initiation (Δ ≈ 10 mm): limited deformation is observed initially, and the first clear failure surface develops from the arch foot; the emerging failure plane is much steeper (≈60°–65°) than that in calcareous sand, accompanied by the initial reduction of {\overline{k}}_{\text{v}}\left(\text{Δ}\right). Stage II––Formation (Δ ≈ 20 mm): a tall triangular loosening zone forms quickly and displacements become strongly concentrated within this zone; {\overline{k}}_{\text{v}}\left(\text{Δ}\right) decreases rapidly towards its minimum. The loosened region reaches a height of about 140–150 mm, nearly the full soil depth, and the arch is narrower and steeper than that in calcareous sand. Stage III––Expansion and adjustment (20–40 mm): while {\overline{k}}_{\text{v}}\left(\text{Δ}\right) evolves more gradually, deformation remains concentrated within the triangular zone; local collapse near the left arch shoulder may occur, leading to pronounced asymmetry in the displacement field. Stage IV––Residual state (40–50 mm): further descent mainly increases settlement within the already formed loosening zone and does not change the overall triangular geometry; {\overline{k}}_{\text{v}}\left(\text{Δ}\right) approaches its final stable level.

Overall, wet calcareous sand forms a lower, more curved (arc-shaped) arch with gentler failure angles and a relatively confined loosening zone, whereas wet siliceous sand develops a higher, steeper, triangular arch with larger settlement and stronger asymmetry. These differences arise from their contrasting particle characteristics: the irregular, crushable calcareous grains promote interlocking and higher shear resistance, favoring a stable, low-rise arch; the more regular siliceous particles provide lower frictional resistance, resulting in a taller arch with steeper failure planes and more extensive deformation.

Figure 5 integrates the ground surface settlement profiles obtained under various testing conditions to facilitate a comprehensive comparison. As shown in Fig. 5(a), the surface settlement of siliceous sand remained relatively small under different trapdoor displacements (5–50 mm), not exceeding 1.5 mm. This limited settlement is attributed to the formation of a complete and stable collapse arch. However, compared to calcareous sand, siliceous sand exhibited slightly larger settlement, suggesting a broader loosening zone and weaker self-supporting capacity. This trend is consistent with the direct shear results, which show substantially lower shear-strength parameters for the wet siliceous sand than for the wet calcareous sand (φ = 33.9° vs. 44.9°; apparent cohesion c = 20.7 kPa vs. 55.6 kPa). The smoother and more rounded siliceous grains therefore provide weaker interlocking and a smaller wet-state bonding contribution, leading to slightly larger settlement. The measured gravimetric water contents were comparable (16.5% for calcareous sand vs. 14.8% for siliceous sand), indicating that the settlement difference is primarily associated with intrinsic material characteristics rather than moisture discrepancy. Additionally, a sudden drop in sensor No. 22’s reading was observed near 27 mm displacement, likely due to the detachment of a soil block above the sensor.

3.1.2 Variation of relaxed earth pressure

1) Earth pressure ratio on the trapdoor

Figures 6(a) and 6(b) show the lateral distribution of earth pressure ratio from sensors Nos. 5–8 during trapdoor descent, illustrating arch evolution in wet calcareous and siliceous sands. Consistent with the deformation patterns, the arching process progresses through four stages: Initiation, Formation, Expansion and adjustment, and Residual state. However, the two sands respond differently. In calcareous sand, the vertical stress on the trapdoor drops rapidly to nearly zero during the Initiation and Formation stages, indicating stronger unloading efficiency. Furthermore, its Expansion and adjustment stage lasts longer than that of siliceous sand. Siliceous sand develops a stable collapse arch at a trapdoor descent of about 10 mm, whereas calcareous sand only stabilizes at around 25 mm, reflecting its stronger interparticle interlocking and higher shear resistance. At large displacements, the final relaxed earth pressure on the trapdoor is lower in calcareous sand, indicating more effective stress transfer away from the trapdoor.

2) Earth pressure ratio adjacent to the trapdoor

Figures 6(c) and 6(d) show the evolution of earth pressure ratio at the box base adjacent to the trapdoor for wet calcareous and wet siliceous sands, respectively. The evolution of earth pressure ratio at the box base highlights the different deformation behaviors of the two sands. In siliceous sand, arching develops and stabilizes quickly, pressure fluctuations near the trapdoor are small, and the influence zone is narrow: sensors Nos. 14 and 16 are barely affected, indicating that disturbance is largely confined within about ±250 mm of the central axis. In calcareous sand, pressure fluctuates more during trapdoor descent due to the progressive upward extension of the arch and local sliding. Slight changes at sensors Nos. 14 and 16 suggest a broader influence zone approaching ±250 mm, consistent with the larger particle size, irregular shape and stronger interlocking of calcareous sand. Overall, siliceous sand maintains higher earth pressure near the trapdoor, which, combined with the deformation observations, can be attributed to its taller loosening zone and the resulting higher arch-foot stress after redistribution.

3) Lateral distribution of earth pressure ratio

Figure 7 shows the lateral distribution of earth pressure ratio at the model base for different trapdoor displacements. In siliceous sand, a stable arch forms at an early stage and the pressure distribution stabilizes rapidly. In calcareous sand, the distribution continues to evolve and only becomes relatively stable at a trapdoor descent of about 30 mm, consistent with the extended Expansion and adjustment stage observed in the deformation field. Because calcareous sand develops a more effective arch with a smaller loosening height, the relaxed earth pressure at the base is lower than in siliceous sand. As shown in Fig. 7(c), both the minimum pressure ratio p_min and the final pressure ratio p_fin at 50 mm are markedly smaller for calcareous sand, indicating more efficient arching, reduced vertical pressure in the crown region and a more pronounced arching effect than in siliceous sand.

3.2 Comparison of calcareous sand under dry and wet conditions

3.2.1 Soil deformation and failure mechanisms

Figure 8 shows the PIV-derived displacement fields of dry calcareous sand at representative trapdoor descents. Overall, the failure surface evolves in a manner broadly similar to the wet condition, but the deformation magnitude and the geometry of the loosening zone differ markedly. Following the same operational criteria used in Subsubsection 3.1.1 (i.e., kinematic features in the displacement fields, supported by the averaged vertical earth-pressure ratio {\overline{k}}_{\text{v}}\left(\text{Δ}\right), the deformation process of dry calcareous sand can be subdivided into four stages.

Stage I––Initiation (0–10 mm): at Δ = 5 mm, soil movements are very small and particle tracking is close to the correlation noise level. When Δ reaches about 10 mm, settlement initiates at the base and a distinct failure surface becomes identifiable. The loosening zone attains a height of approximately 40 mm and a top width of about 150 mm. Minor left–right differences may appear at this early stage due to the very small displacement magnitude and inevitable initial fabric heterogeneity; slight reflections from the acrylic viewing panel may further reduce local image contrast. With increasing Δ, the displacement pattern becomes clearer and largely symmetric, and the left/right earth-pressure measurements remain broadly comparable, indicating that wall-related artifacts are limited and do not affect the main trends.

Stage II––Formation (10–20 mm): at Δ ≈ 20 mm, the loosening zone propagates rapidly upward towards the ground surface. The disturbed region reaches a height of approximately 180 mm, while the top width reduces to about 100 mm as the failure planes converge. Displacements concentrate within a vertically elongated band directly above the trapdoor, accompanied by a rapid reduction of {\overline{k}}_{\text{v}}\left(\text{Δ}\right).

Stage III––Expansion and adjustment (20–40 mm): as Δ increases from 20 to 40 mm, the failure surface reaches and intersects the ground surface, leading to noticeable surface settlement. The loosening zone expands laterally and evolves from a trapezoidal to a near-rectangular configuration. The top width increases to about 120 mm at 30 mm and about 150 mm at 40 mm, while {\overline{k}}_{\text{v}}\left(\text{Δ}\right) evolves more gradually towards a quasi-stable level.

Stage IV––Residual state (40–50 mm): with further trapdoor movement to 50 mm, the relaxed region becomes almost rectangular, with a top width of approximately 200 mm. Additional deformation mainly manifests as increased settlement within the already formed loosening zone, whereas the outer boundary shows limited further expansion; correspondingly, {\overline{k}}_{\text{v}}\left(\text{Δ}\right) approaches and remains near its final level.

Compared with the wet condition, dry calcareous sand exhibits larger deformation and a much more extensive loosening zone. In the wet state, moisture increases apparent cohesion and self-supporting capacity, promoting a relatively stable collapse arch and limiting overall displacement. Under dry conditions, arching is weaker: the failure surface propagates more quickly to the ground surface, the loosening zone is larger at each displacement stage, and both surface settlement and the top width of the relaxed region are greater. The loosening-zone height and the inclination of the failure surface relative to the horizontal are consistently larger in the dry tests, highlighting the key role of moisture in controlling soil arching and the stability of calcareous sand strata.

As shown in Fig. 5(b), surface settlement is negligible in wet sand due to the well-developed arching effect, which effectively transfers loads and limits vertical deformation at the ground surface. In contrast, under dry conditions the loosening zone reaches the surface, resulting in significant settlement. An anomalous reading was recorded by sensor No. 21 when the trapdoor descent was about 42 mm. Subsequent inspection revealed that this sensor, located near the model boundary, rotated and became stuck due to differential settlement, so it no longer followed the surrounding soil movement.

3.2.2 Variation of relaxed earth pressure

1) Earth pressure ratio on the trapdoor

As shown in Figs. 9(a) and 9(b), the evolution of the arching effect in dry calcareous sand is broadly similar to that in wet sand, but the post-arching behavior differs. In wet sand, moisture enhances interparticle attraction and apparent cohesion, allowing a relatively stable collapse arch to form and causing the earth pressure on the trapdoor to stabilize once the arch is fully developed. In dry sand, however, the absence of moisture-induced bonding means that even after the initial arch formation, the loosening zone continues to extend upward to the ground surface during the Expansion and adjustment stage. As a result, the relaxed earth pressure on the trapdoor does not stabilize but increases with further trapdoor descent, eventually exceeding that in the wet condition.

2) Earth pressure ratio adjacent to the trapdoor

Figures 9(c) and 9(d) show the variation of earth pressure ratio at the box base adjacent to the trapdoor edges. The overall trend is similar in dry and wet sand, indicating that the direct influence of arching on regions beyond the trapdoor is limited, but relaxed pressures in dry sand are slightly higher due to the larger loosening zone without moisture-related bonding. At a trapdoor displacement of about 16 mm, readings from sensors Nos. 13 and 15 begin to decrease, implying that the collapse arch has expanded roughly 25 mm beyond the trapdoor edge and started to affect the stability of the overlying soil there. Asymmetry in arch formation produces more pronounced changes in the right-side sensors (Nos. 15 and 16), indicating uneven load transfer during arch development.

3) Lateral distribution of earth pressure ratio

Figures 10(a) and 10(b) show the lateral distribution of earth pressure ratio at the model base for different trapdoor displacements. Under both dry and wet conditions, the earth pressure ratio peaks near the center and decreases toward the sides, but its magnitude and evolution differ between the two cases.

In wet sand, moisture strengthens the self-supporting capacity of the soil skeleton and promotes a pronounced arching effect, so the relaxed earth pressure stays low and becomes almost stable once the trapdoor displacement exceeds about 30 mm. In dry sand, the soil keeps loosening and settling as the trapdoor descends, and the earth pressure continues to rise without a clear stabilization stage.

In these trapdoor tests, deformation is governed by progressive loosening rather than intense shearing or crushing, so capillary suction induced by moisture is more influential than frictional softening. Stronger interparticle suction in wet sand leads to more effective arching and lower vertical pressures than in dry sand.

Figure 10(c) compares the minimum (p_min) and final (p_fin) earth pressure ratios in the x-direction for dry and wet calcareous sand. Both p_min and p_fin are consistently higher in the dry condition, confirming that wet sand more readily forms a stable arch with lower relaxed pressures. The difference is small when the pressure ratio first reaches its minimum, indicating that the arch is still evolving, but at a trapdoor displacement of 50 mm the pressure ratio in dry sand becomes much higher because the absence of interparticle suction makes the arch harder to maintain and allows pressure to keep accumulating.

3.3 Effects of different parameters on calcareous sand

3.3.1 Effects of buried depth ratios

Figure 11 shows the evolution of ground deformation at different trapdoor descents for N_0.5C₈e_1.2 and N₂C₈e_1.2. In all tests, a similar pattern is observed: as the trapdoor descends, the failure surface initiates at the arch foot, propagates upward towards the ground surface and gradually widens, so that the soil arch evolves from an initially narrow shape to a broader loosening zone.

For N_0.5C₈e_1.2 (H/B = 0.5), the failure surface quickly reaches the ground surface, doing so at a trapdoor descent of about 10 mm. At this stage, the soil arch attains its maximum surface width and the failure planes are steep, but the overall arch height is limited by the shallow overburden. For N₁C₈e_1.2 (H/B = 1.0), the failure surface also propagates rapidly upwards and approaches the ground surface at a descent of about 20 mm. With further movement, the arch evolves from a triangular shape into a trapezoidal form and eventually approaches a rectangular loosening zone. For N₂C₈e_1.2 (H/B = 2.0), the failure surface must travel a much longer distance before reaching the ground surface, which occurs only at a descent of about 40 mm. In this deeper case, the arch remains relatively slender: the final loosening zone does not develop into a fully rectangular shape and exhibits the smallest surface width and the smallest inclination of the failure planes among the three cases.

These results indicate that the burial depth ratio significantly influences the evolution of the soil arch. Smaller H/B leads to rapid breakthrough to the ground surface and a low, wide loosening zone, whereas larger H/B results in a longer period of internal arch development and a higher but narrower arch profile.

Figure 5(c) shows the evolution of ground surface settlement with trapdoor descent for different burial depth ratios. For N_0.5C₈e_1.2, the loosening zone quickly reaches the surface and settlement increases almost linearly with trapdoor displacement. For N_0.5C₈e_1.2, a soil arch develops as the trapdoor starts to move, leading to slower initial settlement; when the descent approaches 20 mm and the failure surface reaches the surface, settlement then increases more rapidly and again shows an approximately linear trend. For N_0.5C₈e_1.2, arch development is more pronounced: surface settlement is negligible while the failure surface remains inside the soil, begins gradually at a descent of about 20 mm, and accelerates once the trapdoor displacement reaches 40 mm and the loosening zone breaks through to the surface. For the same trapdoor displacement, settlement decreases with increasing burial depth, reflecting the stabilizing effect of a thicker overburden.

1) Earth pressure ratio on the trapdoor

As shown in Figs. 12(a)–12(c), the Expansion and adjustment stage is shortest for N_0.5C₈e_1.2, giving generally higher earth pressure ratios on the trapdoor than in N₁C₈e_1.2 and N₂C₈e_1.2. For N₂C₈e_1.2, the arching process is more fully developed, with a higher failure surface and a stable arch, leading to the lowest overall pressure ratios. Irregular fluctuations recorded by pressure cell No. 6 are attributed to local compaction during filling and did not reoccur. With increasing burial depth ratio, the earth pressure outside the relaxed zone changes from a decreasing to an increasing trend, indicating more pronounced stress redistribution at greater depth.

2) Earth pressure ratio adjacent to the trapdoor

Figures 12(d)–12(f) depict the variation of earth pressure ratio at the base of the model box adjacent to the trapdoor edges. For N_0.5C₈e_1.2, the loosening zone extends approximately 25 mm beyond the trapdoor at a displacement of about 10 mm, causing a reduction in pressure at sensors Nos. 13 and 15. For N₁C₈e_1.2, the pressures at these sensors initially increase due to arching and then decrease when the loosening zone reaches a similar 25 mm extent at a trapdoor descent of around 20 mm. For N₂C₈e_1.2, the earth pressure at sensors Nos. 13 and 15 rises rapidly during the initial arch formation stage and then tends to stabilize as the arch expands and redistributes the stresses.

3) Lateral distribution of earth pressure ratio

Figure 13 shows that, despite differences in burial depth, the lateral pattern of earth pressure ratio at the base of the model remains broadly similar for all three tests. For N_0.5C₈e_1.2, the initially narrow loosening zone causes pressure concentration near the center, followed by gradual redistribution as the arch expands. For N₁C₈e_1.2, the loosening zone largely remains within the projection of the trapdoor, leading to relatively stable and elevated pressures on both sides of the relaxed region.

Figure 13(d) compares the minimum and final (stabilized) earth pressure ratios for different burial depth ratios. For N_0.5C₈e_1.2, the loosening zone quickly extends to the surface, indicating a weak arching effect and resulting in higher pressure ratios. In contrast, N₁C₈e_1.2 and N₂C₈e_1.2 both develop well-formed arches, with similar pressure ratios at the peak arching stage. As the burial depth ratio increases, the final earth pressure ratio decreases, reflecting an enhanced arching effect at greater depth. A local anomaly at x = 25 mm for N₂C₈e_1.2 is attributed to sensor behavior discussed previously.

3.3.2 Effects of particle gradation

Figure 14 presents the displacement fields for Soil Nos. 1 and 3, respectively. Comparison with Fig. 8 (Soil No. 2) shows that particle gradation affects the failure mode of calcareous sand to a noticeable extent. For the poorly graded Soil No. 1 (N₁C₂e_1.2) and the gap-graded Soil No. 3 (N₁C₁₆e_1.2), the failure surface reaches the ground surface at a trapdoor descent of about 20 mm, becomes nearly vertical at 40 mm and subsequently expands outward. The loosening zone thus develops into a tall, wide rectangular block.

In contrast, for the well-graded Soil No. 2 (N₁C₈e_1.2), the failure surface reaches the ground surface and begins to expand at a descent of 20 mm, but only becomes approximately vertical at 50 mm and shows little outward expansion thereafter. At the same trapdoor displacement, the arch height, top width and failure-plane inclination are greater for N₁C₂e_1.2 and N₁C₁₆e_1.2 than for N₁C₈e_1.2, and the trapdoor descent required for the loosening zone to approach a rectangular shape also differs among the three soils. Overall, the ground response is most stable for well-graded calcareous sand, less stable for uniformly graded sand and least stable for gap-graded sand, highlighting the beneficial role of continuous gradation and enhanced interparticle interlocking for arch stability.

Figure 5(d) shows ground surface settlement versus trapdoor displacement for different gradations. Particle gradation has little influence on settlement: deformation is minimal while the loosened zone remains below the surface, and once the arch reaches the surface, settlement increases rapidly and almost linearly with trapdoor descent. Consistent with Figs. 8 and 14, N₁C₈e_1.2 gives the smallest settlements, but differences among the three soils are modest.

1) Earth pressure ratio on the trapdoor

Figures 15(a)–15(c) show the variation of earth pressure ratio recorded by pressure cells Nos. 5–8 under different particle gradation conditions during trapdoor descent. For the uniformly graded case (N₁C₂e_1.2), the pressures at cells Nos. 6 and 7 increase steadily, while those at Nos. 5 and 8 tend to stabilize, indicating asymmetric development of soil arching.

In the gap-graded condition (N₁C₁₆e_1.2), the arching behavior is generally similar to the well-graded case (N₁C₈e_1.2), except at 20 mm trapdoor displacement: in N₁C₈e_1.2, pressure continues to rise slowly due to ongoing Expansion and adjustment stage, while in N₁C₁₆e_1.2, the pressure stabilizes early, indicating a transition toward a rectangular arch form. Among the three gradations, N₁C₈e_1.2 shows the lowest overall earth pressure, suggesting it promotes the most effective arching. In contrast, N₁C₁₆e_1.2’s discontinuous gradation leads to poor particle interlocking and weaker arch development compared to both well-graded and uniformly graded sands.

2) Earth pressure ratio adjacent to the trapdoor

Figures 15(d)–15(f) show that the general variation pattern of earth pressure on both sides of the trapdoor is similar across different gradations. However, minor differences are observed in the displacement at which pressure begins to drop, indicating variations in the onset of surface settlement and transition to the trapezoidal arch phase. These differences reflect the influence of particle gradation on the progression and timing of soil arching.

3) Lateral distribution of earth pressure ratio

Figure 16 shows the lateral distribution of earth pressure ratio at different trapdoor displacements. The lateral distribution of earth pressure on the trapdoor shows minimal variance when the soil arch is active under different particle gradation conditions. In N₁C₂e_1.2, the earth pressure at the center of the trapdoor consistently increases, while the pressure on both sides tends to stabilize as the trapdoor descends by 20 mm. Conversely, in N₁C₁₆e_1.2, as the trapdoor descends by 20 mm, the soil arch transitions gradually into a rectangular arch, leading to overall stabilization of earth pressure.

Figure 16(d) compares the minimum and final earth pressure ratios along the x-direction for all three gradations. Initially, N₁C₂e_1.2 and N₁C₁₆e_1.2 show similar minimum values, but the latter exhibits a slightly higher value, indicating weaker interparticle interlocking and a less effective initial arching process. At the final stage, both N₁C₂e_1.2 and N₁C₁₆e_1.2 exhibit higher stabilized pressures than N₁C₈e_1.2, confirming that poor gradation significantly impairs the development and stability of the soil arch.

3.3.3 Effects of void ratios

Comparison of Figs. 8 and 17 highlights the substantial influence of void ratio on the failure mode of calcareous sand. For N₁C₈e_0.9 (dense state), the failure surface only approaches the ground surface after a trapdoor descent of about 30 mm and remains inclined, without becoming vertical, even at a 50 mm descent. For N₁C₈e_1.2 (medium-dense state), the failure surface reaches and begins to expand at the ground surface at a descent of 20 mm and becomes approximately vertical at 50 mm. For N₁C₈e_1.5 (loose state), the failure surface reaches the ground surface as early as a 10 mm descent and becomes nearly vertical at 30 mm. The loose structure associated with a high void ratio leads to a reduced internal friction angle and allows the soil arch to expand outward continuously as the trapdoor descends. This outward expansion is essentially absent for N₁C₈e_0.9, which has the smallest void ratio and the densest packing. At the same trapdoor displacement, the arch height, top width and failure-plane inclination are smallest for N₁C₈e_0.9 and largest for N₁C₈e_1.5. These trends indicate that a smaller void ratio corresponds to a more pronounced and stable soil arching effect, whereas a larger void ratio promotes earlier breakthrough of the failure surface to the ground surface and the development of a larger loosening zone.

Figure 5(e) shows the ground surface settlement versus trapdoor displacement for calcareous sand with different porosities. At N₁C₈e_0.9, the soil is relatively dense, so surface settlement is negligible at small displacements; once the trapdoor has descended about 30 mm, the loosened zone reaches the surface and settlement increases rapidly. At N₁C₈e_1.5, the soil is very loose, the relaxed zone reaches the surface early, and settlement grows almost linearly with trapdoor displacement, with N₁C₈e_1.2 showing intermediate behavior. Settlements recorded by meter 21 are generally smaller than those from meter 22, indicating longitudinally non-uniform settlement, as the meter closer to the boundary is more constrained by boundary effects and friction.

1) Earth pressure ratio on the trapdoor

Figures 18(a)–18(c) show the variation of earth pressure ratio on the trapdoor for different void ratios. When the void ratio is 0.9 (N₁C₈e_0.9), the dense packing leads to strong particle interlocking and a pronounced arching effect. As a result, the overall earth pressure on the trapdoor is lower and the arch expansion phase is prolonged, as the compacted soil gradually loosens during trapdoor movement. In contrast, for a void ratio of 1.5 (N₁C₈e_1.5), the soil is loose, interparticle contacts are weaker and the arching capacity is reduced. This leads to higher earth pressures and a shorter expansion phase. When the trapdoor displacement reaches about 15 mm, the loosening zone extends to the surface and the arch evolves from a triangular to a trapezoidal shape, causing the rate of pressure increase to slow down. The response at e = 1.2 is intermediate between these two cases.

2) Earth pressure ratio adjacent to the trapdoor

Figures 18(d)–18(f) present the earth pressure response at the base of the model box. For the dense case (e = 0.9), the pressures at sensors Nos. 13 and 15 first increase as the arch forms and then decrease after a trapdoor descent of about 25 mm, when the loosening zone reaches the ground surface. The trend for e = 1.2 is similar, but the pressure decrease begins earlier, at a displacement of around 20 mm. For the loose case (e = 1.5), the pressures at sensors Nos. 13 and 15 increase immediately as arching starts, then gradually decrease between 10 and 20 mm of displacement and stabilize once the arch has transformed into a rectangular loosening zone beyond 20 mm.

3) Lateral distribution of earth pressure ratio

As shown in Fig. 19, the lateral distribution pattern of earth pressure ratio at the model base is broadly similar for all void ratios; the main differences lie in the magnitude of the pressures and the displacement required to reach a stabilized state. Figure 19(d) shows that both the minimum and final stabilized earth pressure ratios increase with void ratio. Lower void ratios, corresponding to denser soil, promote stronger particle interlocking and more effective arching, which reduces the earth pressure on the trapdoor and in the surrounding ground.

It should be noted that the present trapdoor experiments were conducted at relatively low stress levels: the vertical earth pressures measured on the trapdoor generally remained below 20 kPa. This range is substantially lower than the stress levels at which measurable particle breakage is commonly reported for calcareous sands in oedometer and triaxial tests (typically hundreds of kPa to MPa) [19,21]. In addition, the active trapdoor mode induces a predominantly unloading/arching stress path, which further limits the likelihood of pervasive crushing. Therefore, the observed differences in arching behavior with void ratio are interpreted primarily in terms of changes in density, particle rearrangement and interlocking, rather than pervasive grain breakage. However, local micro-crushing at highly stressed contacts cannot be completely excluded, especially in the dense case with e = 0.9, and may contribute to the gradual reduction of dilatancy during arch evolution. A more detailed quantification of particle breakage (e.g., by comparing pre- and post-test grain-size distributions or micro-CT imaging) is left for future work.

3.4 Normalized indices for quantifying soil arching

3.4.1 Definitions and physical meanings

To enable a systematic comparison of the different test conditions and to compress the complex soil response into a few dimensionless measures, three normalized indices are introduced to quantify soil arching in the trapdoor tests. These indices describe, respectively, the efficiency of stress redistribution, the severity of ground-surface deformation, and the mobilization process of arching of arch mobilization.

First, a stress-based arching index, denoted by ξ, is defined from the vertical earth pressure ratio on the trapdoor. For each test, the vertical earth pressure measured by cells Nos. 5–8 on the trapdoor is normalized by the initial self-weight pressure p₀ at the same depth, giving the earth pressure ratio as defined in Eq. (1):

(1)k_{\text{v}}(\text{Δ})=\frac{p(\text{Δ})}{p_0},

where Δ is the vertical displacement of the trapdoor. The spatially averaged ratio on the trapdoor is then obtained according to Eq. (2):

(2){\overline{k}}_{\text{v}}(\text{Δ})=\frac{k_5(\text{Δ})+k_6(\text{Δ})+k_7(\text{Δ})+k_8(\text{Δ})}{4}.

The corresponding arching index is subsequently defined by Eq. (3):

(3)\xi \left(\text{Δ}\right)=1-k_{\text{v}}\left(\text{Δ}\right).

Physically, ξ represents the fraction of the initial overburden (self-weight) load that is shed from the trapdoor and transferred to the adjacent stationary soil due to arching. A larger value of ξ indicates a greater reduction of vertical stress on the trapdoor relative to the self-weight state, and thus a stronger arching effect. In the following, the minimum and final values of ξ during the test are used to characterize the maximum and residual arching intensity.

Second, a deformation-based index, I_s, is introduced to quantify the normalized surface settlement above the trapdoor. The vertical settlements recorded by displacement meters Nos. 21 and 22, located at the front and back of the ground surface above the trapdoor center, are averaged to obtain a representative surface settlement S_max at the end of the test (Δ = 50 mm). This value is normalized by the trapdoor width B = 200 mm, as defined in Eq. (4):

(4)I_{\text{s}}=\frac{s_{max}}{B}.

From a practical perspective, I_s serves as a proxy for the potential damage to overlying infrastructure. A smaller I_s corresponds to smaller ground-surface deformation for a given amount of unloading, and thus reflects a more favorable deformation performance of the soil arch.

Third, an evolution-type index, I_f, is proposed to describe the trapdoor displacement required to mobilize a stable arch. For each test, the average earth pressure ratio {\overline{k}}_{\text{v}}\left(\text{Δ}\right) is again considered. The value at the end of the test, {\overline{k}}_{\text{v,final}} at Δ = 50 mm, is taken as the stabilized pressure ratio. Moving backwards along the loading history, the arch stabilization displacement Δ_f is defined as the first trapdoor displacement at which {\overline{k}}_{\text{v}}\left(\text{Δ}\right) enters and then remains within a ±5% band around {\overline{k}}_{\text{v,final}}. The corresponding normalized index I_f is given by Eq. (5).

(5)I_{\text{f}}=\frac{{\text{Δ}}_{\text{f}}}{B}.

A brief sensitivity check was performed by recalculating I_f with tolerance bands ranging from ±2% to ±10%. While the absolute value of I_f decreases as the tolerance widens, the comparative trends are stable for a reasonable range (about ±3%–±8%). Therefore, ±5% is adopted as a practical compromise to avoid both premature stabilization (too wide) and excessive sensitivity to small residual fluctuations (too narrow). A smaller I_f means that the earth pressure on the trapdoor approaches its stable, arch-controlled level after a relatively small trapdoor movement, whereas a larger I_f indicates that a larger displacement is required to fully mobilize and stabilize the arching mechanism. When interpreted together with ξ and I_s, I_f provides additional insight into whether the soil rapidly evolves towards a favorable arching state or quickly collapses into an extensive loosening zone.

3.4.2 Result analysis

The three indices were evaluated for the key test series, including the comparison between wet calcareous and wet siliceous sands, the influence of burial depth ratio, particle gradation and void ratio, and the contrast between dry and wet calcareous sand. The corresponding results are summarized in Table 2.

For the wet-state comparison, wet calcareous sand exhibits the most favorable combination of the three indices. Its stress-based index ξ is consistently larger than that of wet siliceous sand, indicating a stronger reduction of vertical pressure on the trapdoor. Meanwhile, its normalized settlement index I_s is almost an order of magnitude smaller (I_s ≈ 0.06%) than that of wet siliceous sand (I_s ≈ 0.57%), confirming that the ground surface above the trapdoor remains nearly undeformed. The stabilization index of wet calcareous sand (I_f ≈ 12.98%) is also smaller than that of wet siliceous sand (I_f ≈ 23.19%), meaning that a stable, arch-controlled stress state is reached at a much smaller trapdoor displacement. In contrast, wet siliceous sand requires a larger displacement to stabilize and still retains a higher residual pressure ratio and significantly larger surface settlement. Overall, wet calcareous sand develops a more efficient soil arch, mobilizing stronger unloading on the trapdoor while keeping ground deformation very limited.

The parametric tests on calcareous sand reveal systematic trends in the indices. As the burial depth ratio decreases, the normalized settlement index I_s increases markedly. The values of I_s are about 8.8%, 21.1%, and 24.4% for N = 2.0, N = 1.0, and N = 0.5, respectively, indicating that shallow cover is associated with larger surface settlements and weaker confinement of the loosening zone. The shallow-buried case also exhibits the smallest stabilization index (I_f ≈ 6.55%), reflecting that the earth pressure on the trapdoor rapidly reaches its final level as the loosened zone quickly propagates to the ground surface and the system evolves into a global settlement block. For N = 1.0 and N = 2.0, I_f is larger (about 18.71% and 23.22%, respectively), while the residual pressure ratios are lower and the settlements smaller, suggesting a more gradual mobilization of arching and the formation of a more effective and stable arch under greater cover depth.

Regarding particle gradation, the well-graded calcareous sand (C_u = 8) shows a relatively low settlement index Is and an intermediate stabilization index I_f, consistent with its confined loosening zone and stable failure pattern revealed by PIV. The uniformly graded sand (C_u = 2) and the gap-graded sand (C_u = 16) both yield larger I_s values, implying larger surface settlements. The gap-graded sand, in particular, has a smaller I_f, indicating that its arching response and loosening zone develop more rapidly, in line with the more pronounced outward expansion of the failure zone.

The influence of void ratio is also clearly reflected in the indices. The dense calcareous sand with e = 0.9 has the smallest settlement index (I_s ≈ 18.33%) and a stabilization index similar to that of the medium-dense case (e = 1.2), highlighting the beneficial role of higher density in enhancing arching and restricting surface deformation. The loose sand with e = 1.5 exhibits slightly larger I_S and I_f, indicating larger settlements and a more gradual mobilization of the arch as the loosening zone extends more easily upward and outward.

Finally, comparison between dry and wet calcareous sand shows that drying significantly degrades the arching performance. The dry sand has a much larger settlement index (I_s ≈ 21.12%) and a larger stabilization index (I_f ≈ 18.71%) than the wet sand, meaning that a larger displacement is needed to reach a quasi-stable arching state and that this state is associated with considerably larger surface settlements. This corroborates the earlier observations that moisture and capillary suction enhance interparticle bonding, promoting the formation of a low-rise, stable arch and more efficient stress redistribution.

Overall, the combined use of the three normalized indices ξ, I_s and I_f provides a compact and physically transparent framework for quantifying soil arching in calcareous sand. Within this framework, wet calcareous sand consistently shows stronger arching, smaller deformation and a more favorable mobilization process than wet siliceous sand and less favorable calcareous sand conditions, which has important implications for the design of underground excavations and foundations in calcareous sand deposits.

4 Conclusions

This study investigated the soil arching behavior in calcareous sand using a series of trapdoor model tests. The research focused on comparative analyses of wet calcareous sand versus wet siliceous sand, dry versus wet calcareous sand, and explored the effects of burial depth ratio, particle gradation, and void ratio on arch formation and pressure redistribution. The main conclusions are as follows.

1) Under identical moisture and compaction conditions, calcareous sand demonstrates a stronger arching effect than siliceous sand. It forms a lower and more stable collapse arch with reduced surface settlement and lower relaxed earth pressure. This behavior is attributed to the angular particles, higher internal friction angle, and stronger interparticle interlocking in calcareous sand.

2) Moisture has a significant influence on the deformation characteristics of calcareous sand. In wet sand, capillary suction generates apparent cohesion, which enhances interparticle forces and facilitates the rapid formation of a stable soil arch. In contrast, dry sand experiences extended arch expansion, continuous loosening to the surface, and greater pressure and settlement, indicating weaker structural stability in the absence of moisture.

3) A smaller burial depth ratio inhibits arch development, leading to larger loosening zones, higher earth pressure, and increased surface deformation. Particle gradation affects the magnitude of pressure rather than the stages of arching. Well-graded sands enhance particle contact and arch stability, resulting in lower pressure. A lower void ratio (denser packing) improves particle interlocking, intensifies the arching effect, and reduces the final relaxed earth pressure.

4) Across all conditions, the soil arching effect in calcareous sand evolves through four distinct stages: Initiation, Formation, Expansion and adjustment, and Residual state. These stages correspond to observable transitions in pressure response and deformation patterns.

5) The three normalized indices proposed in this study-ξ for unloading efficiency, I_s for normalized surface settlement and I_f for stabilization displacement-provide a compact and physically transparent framework for quantifying soil arching in calcareous sand. Applied to the present tests, they consistently highlight the superior arching performance of wet calcareous sand and clarify the quantitative roles of burial depth, gradation and density. Their combined use allows different materials and loading conditions to be compared on a unified basis, identifying regimes that achieve strong unloading at acceptable deformation levels with a favorable mobilization process.

6) These findings support improved geotechnical design in calcareous sand formations (e.g., island foundations, coastal infrastructure and subsurface excavations). The frictional and interlocking characteristics of calcareous sand should be mobilized through well-compacted, well-graded configurations, with careful control of burial depth and moisture, which are crucial for pressure redistribution and arch stability.

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The Author(s). This article is published with open access at link.springer.com and journal.hep.com.cn