Active and passive arching stresses outside a deep trapdoor

RESEARCH PAPER

Active and passive arching stresses outside a deep trapdoor

Received: 12 July 2019 / Accepted: 10 April 2020

 Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract

The classic trapdoor configuration has been useful to examine the changes in stresses expected on buried structures. However, the primary focus of previous studies has been on investigating the loads on the surface of the trapdoor, while stresses outside the trapdoor boundaries have generally been overlooked. This paper presents and discusses results of three-dimensional laboratory model tests conducted to investigate changes in the vertical soil pressure measured at various locations within a granular soil mass surrounding a deep rectangular trapdoor acting in both active and passive modes. The study aimed at investigating stress changes within the portion of the soil mass beyond the boundaries of the trapdoor. Redistributions of soil pressure were found to occur in a large zone of the soil outside the trapdoor under both active and passive conditions. Results indicate that active conditions induced the development of an unloading region in the soil, which includes the collapsing mass above the trapdoor and a portion of the soil surrounding the trapdoor. A stable transfer region could be identified in farther portions of the backfill. In passive conditions, the development of a load-transfer region above the trapdoor and an unloading region extending to farther zones in the backfill was also identified. The soil relative density, soil confinement and trapdoor shape were found to affect soil pressure distributions outside the trapdoor limits.

Keywords Sand Soil arching  Soil stress  Trapdoor tests  Underground structure List of symbols

B Width of the trapdoor or buried structure (m) Dr Soil relative density (%)

Es Young’s modulus of the soil (kPa) H Height of soil above the trapdoor (m) He Vertical distance from the trapdoor (m) L Length of the trapdoor or buried structure (m) q Applied surface pressure (kPa)

x Horizontal distance from the center of the model (m)

d Trapdoor vertical displacement (m) w Dilatancy angle of the soil ()

e1 Axial strain change

ev Volumetric strain change

ms Poisson’s ratio of the soil

r03 Effective confining pressure (kPa)

rv Vertical pressure in the soil (kPa)

rvo Vertical pressure prior to yielding of the buried

structure (kPa)

/cr Critical state friction angle of the soil ()

/0 Internal friction angle of the soil () /0_p Peak friction angle of the soil ()

1 Introduction

The arching phenomenon in soils results from the inter-action between a buried structure and the surrounding soil mass. Arching can be defined as the load transfer that occurs between the soil and the structure through mobi-lization of soil shear. Active (or positive) arching takes place if the structure yields more than the adjacent soil mass, and involves a decrease in the stresses acting on the structure in relation to the initial stresses before yielding. & Yuri D. J. Costa

ydjcosta@ct.ufrn.br

Jorge G. Zornberg zornberg@mail.utexas.edu

1 _{Department of Civil Engineering, Federal University of Rio}

Grande do Norte, Av. Sen. Salgado Filho, 3000, Natal, RN 59072-970, Brazil

2 _{Department of Civil, Architectural and Environmental}

Engineering, The University of Texas at Austin, E. Dean Keeton St., Stop C1792, Austin, TX 78712-1174, USA https://doi.org/10.1007/s11440-020-00969-x(0123456789().,-volV)(0123456789().,-volV)

The opposite condition, known as passive (or negative) arching, occurs when the buried structure yields less than the surrounding mass, resulting in an increase in the stresses acting on the structure relative to the initial stresses.

Terzaghi [31] described arching as one of the most universal phenomena of soil mechanics, in both the field and laboratory. Understanding arching is essential for the design of many geotechnical structures, such as buried pipelines, tanks, tunnels, anchors, trench excavations, piled embankments and earth dams, to name a few. An appro-priate analysis of soil–structure systems becomes very difficult to carry out without understanding of arching [1]. The development of arching in soils is governed by a number of parameters that are related to the geometric and rheological characteristics of the underground structure, rheological characteristics of the soil, depth of embedment below the ground surface and presence of external loads. Addressing the influence of each governing parameter is essential to a comprehensive understanding of the phenomenon.

Arching has been studied for many decades by mea-suring changes in the load acting on the surface of a hor-izontal rigid trapdoor apparatus in active or passive conditions. Experimental and numerical investigations involving measurements of the vertical load acting on a trapdoor with a length much larger than its width (B), to represent plane strain (2D) conditions, were carried out by Chevalier et al. [4], Evans [9], Iglesia [13], Koutsabeloulis and Griffiths [17], Tanaka and Sakai [29], Terzaghi [30], Vardoulakis et al. [33], Wang et al. [35] and Xu et al. [37]. Changes in the vertical load acting on a circular trapdoor to represent axisymmetric conditions were investigated by Dewoolkar et al. [6], McNulty [20] and Santichaianant [25].

While the existing research focuses almost exclusively on examining changes in the vertical load acting within the limits of the trapdoor, a limited number of studies have investigated load variations within the portion of the soil mass beyond the boundaries of the trapdoor. McNulty [20] measured vertical soil pressures at some locations outside a circular trapdoor in active arching. A reduction in the soil pressure was detected to a distance around 0.5 B from the edges of the trapdoor. However, trapdoor displacements were halted prematurely in the experiments, which pre-vented the full development of arching in the soil. Fur-thermore, vertical pressures in the outer soil mass were not recorded in models tested in passive mode. Han et al. [12] examined the progressive development of soil arching with trapdoor settlements in two-dimensional sand models. Soil pressures were recorded at the surface of an active trapdoor and at two stationary portions right beneath the edge of the trapdoor. However, additional comments on the behavior

of soil pressures developed off trapdoor limits were not made. Rui et al. [24] carried out laboratory model tests on a two-dimensional multi-trapdoor apparatus to study the transfer of loads in piled embankments. Vertical pressures in the soil were collected in some positions at the floor of the model, both inside and outside active trapdoors. Tests with passive trapdoors were not reported. Eskisar et al. [8] conducted a three-dimensional analysis of arching in reinforced and unreinforced piled embankments for dif-ferent fill materials and pile spacings using a multi-trap-door device. However, vertical load measurements were limited to the head of the pile (i.e., outside the yielding area).

An assessment of previous studies on the trapdoor problem indicates that although good progress has been made to understand the arching phenomenon and its main causes of influence since the pioneering laboratory exper-iments conducted by Terzaghi [30], very little attention has been paid to the stress redistributions that occur beyond the trapdoor boundaries. As the ground construction is growing rapidly, the significance of evaluating the effect a buried structure has on the surrounding soil mass due to potential interactions with other structures becomes quite evident.

Studies that consider the three-dimensional nature of the trapdoor problem are also in need, since most investiga-tions on stress changes outside the trapdoor limits have used either plane strain (2D) or axisymmetric models. Though much easier to work with and useful for identify-ing the evolution of failure mechanisms, these two condi-tions may not reproduce realistically some important geotechnical problems as a three-dimensional analysis. Pipelines and other underground structures undergoing differential movements due to inadequate backfill com-paction, mining-related excavations, soil heave, karstic soils, etc., are examples in which three-dimensional anal-yses are more suitable. In this respect, well-instrumented three-dimensional experimental studies may generate reli-able quantitative information that can be used for a better interpretation of the response of buried structures.

Furthermore, there is a disparity between the data available on active versus passive arching modes among trapdoor modeling investigations involving soil pressure measurements outside the trapdoor limits. The volume of data generated thus far indicates more attention has been paid to the active rather than the passive mode. However, this minimizes the importance of passive arching in pro-viding further insight into the performance of many geotechnical engineering structures, such as onshore and offshore anchors, propped excavations and buried pipelines.

To address the aforementioned deficiencies, this paper aims at presenting and discussing the results of three-di-mensional laboratory model tests constructed to investigate

the changes in vertical soil stress at various locations within a granular soil mass surrounding a deep rigid door acting in both active and passive modes. Deep trap-door installations are characterized by heights of soil cover above the trapdoor (H) over 2 B [5, 17, 20, 25]. The experimental setup included reduced-scale models with rectangular and square horizontal trapdoors embedded in dense and loose sand backfills. Emphasis was placed on examining the changes in vertical soil pressure in zones of the sand fill outside the footprint of the trapdoor. A dis-cussion on arching evolution mechanisms is provided. The findings presented in this paper can be used as the basis for further improvements on the theoretical analysis of the interaction between underground structures.

2 Synthesis of arching results from previous

investigations with different buried

structures

The arching phenomenon has been typically quantified in terms of the arching ratio, which can be defined as the vertical stress over an underground structure after arching has taken place, normalized in relation to the vertical stress acting prior to yielding of the structure. As previously mentioned, arching has been studied by measuring changes in the vertical soil pressure acting on the surface of a horizontal structure. The main features of soil arching, observed from previous investigations, are outlined below. Figure1synthesizes the minimum (active) arching ratio and the maximum (passive) arching ratio as a function of the corresponding normalized displacement (d=B). The data were compiled from results of previous experimental studies involving vertically displaced rigid structures of different shapes (e.g., trapdoors, anchors and pipes). The data correspond to measurements made within the limits of the structure. Normalized displacements for the passive arching mode are considered negative. In spite of the sig-nificant scatter, a line is shown to highlight the observed trend. The information available includes results of labo-ratory models constructed with H/B ratios corresponding to shallow and deep conditions and granular backfills com-pacted with relative densities (Dr) ranging from 70% to 100% (medium dense to dense sands). Tests were con-ducted either under normal gravity (1g) conditions, or using a centrifuge, or by applying an external distributed pressure.

Aspects relevant to the arching problem identified in previous studies are illustrated in Fig.1. The scattered data exhibit a trend along the trend line. While comparatively small displacements are sufficient to mobilize active con-ditions, the development of passive conditions required comparatively larger movements. In the active condition,

displacements typically below 5% B are required to reach a minimum stress value, while displacements as large as 15% B or more are needed in the passive condition. Moreover, while the minimum active load usually drops below half the initial load, to nearly zero in most cases, the maximum passive load can reach as much as ten times the initial load, and sometimes higher. A thorough analysis of the results in Fig.1 also reveals that both passive arching and active arching are influenced by the depth of burial. An increasing soil cover ratio (i.e., increasing soil confinement) over the buried structure results in increasing maximum passive arching ratio and a decreasing minimum active arching ratio.

Despite the contributions of those studies, soil pressure changes outside the limits of an underground structure remain unclear, and many relevant aspects of the problem need further elucidation. Therefore, geotechnical engi-neering has a major need for reliable data on the redistri-bution of soil pressures in the vicinity of buried structures in order to improve the design and the performance of those structures.

3 Trapdoor model tests

3.1 Test apparatus

A schematic of the main features of the apparatus used in this study is shown in Fig.2. The tests were conducted in a rigid metal container with internal dimensions of 560 mm in height, 560 mm in depth and 1400 mm in length. The sides and base of the container were manufactured of thick steel plates and profiles. The internal walls of the container were coated with two layers of 0.075-mm-thick polyester film to minimize the effects of side friction.

A trapdoor system was mounted to the bottom of the container to induce both active and passive conditions on the backfill mass as it displaced downward or upward, respectively. This system consisted of three stainless steel prisms having plane dimensions of 100 mm 9 100 mm and a depth of 120 mm. The ascent or descent of the prisms was controlled by a threaded axis driven by gears con-nected to a hand crank. Depending on the arrangement of the gears, the system allowed conducting tests with either a square (100 mm 9 100 mm) trapdoor or a rectangular (100 mm 9 300 mm) trapdoor. Two linear variable dis-placement transducers (LVDT) were positioned below the test apparatus to monitor the vertical displacements of the prisms.

The base of the test container was instrumented with 15 small self-temperature-compensated miniature pressure sensors with a sensing surface measuring 23 mm in diameter. The ratio of the sensing surface diameter to the

soil average grain size was 100, which exceeds the mini-mum recommended value of 10 [28]. The thickness of the sensors is 5.7 mm, and the resolution of measurement is 1% at 200 kPa. Two types of sensors were used: interface (I) sensors and inclusion (M) sensors. The interface (I) sensors were placed into slots flush with the container floor. Units I1, I2 and I3 were positioned on top of the prisms and measured vertical total pressures in the soil mass within the limits of the trapdoor. Units I4 to I9 were used to record vertical pressures in the soil beyond the trapdoor boundaries.

The inclusion (M) sensors recorded the vertical pressure within the soil mass and were placed in the center of the trapdoor above position I1. One additional inclusion sensor (M6) was placed at a height of 0.3 B above position I4. The use of sensor M6 provided additional insight into the load-transfer mechanisms outside the trapdoor limits because of its higher elevation in relation to the other external sensors. Model preparation involved different arrangements and numbers of inclusion sensors. The characteristics of the models are described next.

3.2 Backfill properties, model preparation

and testing program

The models were prepared using a clean, poorly graded quartz silica sand with rounded to subrounded particles. The sand classifies as SP according to the Unified Soil Classification System (USCS) and has an average particle size of 0.23 mm, a coefficient of uniformity of 2.7 and a coefficient of curvature of 1.09. The maximum void ratio equals 0.87, which corresponds to a minimum dry unit weight of 14.2 kN/m3. The minimum void ratio equals 0.5, which corresponds to a maximum dry unit weight of 17.7 kN/m3. The soil specific gravity is 2.65 and its moisture content was less than 1% during testing. Shear strength and volume change properties of the sand, obtained from conventional triaxial compression tests, are presented in Table1. These tests were carried out on two series of three specimens with different confining pressures (r3), prepared

with target relative densities (Dr) of 50% and 100%. To achieve a homogeneous soil density, models were prepared by pluviating air-dried sand into the test con-tainer. All models had a soil layer with 560 mm thickness, which corresponds to an H/B ratio of 5.6. The pluviation equipment was calibrated to yield backfills with target relative densities (Dr) of 50% and 100%. Pluviation was

Fig. 1 Minimum active arching and maximum passive arching versus corresponding normalized displacement from laboratory experiments with several underground structures

halted several times during filling to place the pressure sensors at pre-established locations within the sand mass. After the test container was filled, vacuum was used to remove excess pluviated sand.

Pressure was applied to the soil mass via a PVC bladder positioned on top of the model. A constant applied pressure was maintained in the test container using a lid as reaction to the bladder. Pressure was applied in increments of 10 kPa until the target pressure level (q) of 50 or 100 kPa was reached, after which point, pressure was maintained constant while the trapdoor displacements occurred. A data acquisition system recorded measurements from the stress and displacement transducers at predetermined displace-ments of the trapdoor. The scope of the testing program conducted in this investigation is summarized in Table2. Active arching models are designated as ‘‘AA,’’ and pas-sive arching models are designated as ‘‘PA’’ in Table2. The different arrangements of inclusion (M) sensors used in each model above position I1 are detailed in Table 3. Fig. 2 Schematic of trapdoor test apparatus: a plan view showing arrangement of interface pressure sensors on container floor; b transverse cross-section view; c longitudinal cross-section view (dimensions in mm)

Table 1 Summary of conventional triaxial test results with backfill sand

Series Dr(%) r03(kPa) /0p () /cr() w () ( dev=de1)max

50 38.0 … 11.7 0.508 1 50 100 36.7 33.4 8.8 0.363 200 34.0 31.0 6.8 0.271 50 39.9 34.6 15.5 0.731 2 100 100 39.2 33.1 13.4 0.604 200 38.2 31.8 8.9 0.368 r0

3, effective confining pressure; / 0

p, peak friction angle; /cr, critical

state friction angle; w, dilatancy angle; [( dev=de1)max] = maximum

4 Results of active arching tests

4.1 Soil pressure redistributions

above the trapdoor

The results obtained from testing one of the models in active arching (Model AA3) are presented to illustrate the typical information gathered for the different models tested as part of this study. Figure3 shows the arching ratio as a function of the normalized trapdoor displacement (d=B) in a vertical profile above the center of the rectangular trap-door. The soil arching ratio was calculated as the ratio between the vertical soil pressure measured after arching has taken place and the vertical soil pressure measured prior to displacing the trapdoor (rv=rvo). Vertical soil

pressures were obtained using interface sensor I1 and several inclusion (M) sensors placed above it. The vertical distance from the floor of the model (He) is normalized with the trapdoor width (B). The changes in soil pressure at the surface of the trapdoor (He/B = 0) are characterized by an abrupt initial reduction under very low displacements followed by a clear stabilization of the pressure for higher displacements. This behavior agrees with the typical trend reported in previous studies [12,15,20,25,29]. A recovery

of part of the load at large displacements was not observed in the results. The curve changes the general shape for increasing distance He- above the trapdoor, from a hyperbole-type shape into a reasonably linear trend for increasing normalized displacements. This indicates that the stress relief prompted by the arching effect gradually decreases with increasing vertical distance from the trap-door. Above He/B of 1.4, the magnitude of trapdoor dis-placements did not result in a constant residual soil pressure value.

Figure4a shows the evolution of the arching ratio along a profile directly above the center of the trapdoor as the normalized displacement increased from 0.25 to 20%. The load relief is substantial for comparatively small settle-ments. A zone of intense relief developed for conditions characterized by a large trapdoor settlement, extending up to a distance (He) of approximately 1.5 B above the trap-door. The plane along which the settlement of the soil mass Table 2 Characteristics of the trapdoor models

Model designation Relative density, Dr(%) External pressure, q (kPa) Trapdoor shape Arching mode

AA1 100 100 Rectangularb Active

AA2a _{100 100 Rectangular Active}

AA3 50 100 Rectangular Active

AA4 100 50 Rectangular Active

AA5 100 100 Squarec Active

PA1 100 100 Rectangular Passive

PA2 100 100 Square Passive

a_{Repeat test}

b_{100 9 300 mm trapdoor} c_{100 9 100 mm trapdoor}

Table 3 Number and elevation (in mm) of inclusion M sensors above position I1 in each model

Model

Inclusion sensor AA1 and AA2 AA3 AA4 AA5 PA1 PA2

M1 30 70 30 30 30 30

M2 – 140 140 140 140 140

M3 – 220 260 200 200 200

M4 – 290 – 300 300 300

M5 – 370 – – 390 –

Fig. 3 Typical changes in arching ratio above a rectangular trapdoor in active mode (Model AA3)

above the trapdoor equals that of the adjacent mass outside the trapdoor is known as the plane of equal settlement— PES [2, 26]. If the soil layer is thick enough, a PES is expected to occur above the mobilized zone, resulting in a condition referred to as ‘‘incomplete projection.’’ Other-wise, if a PES does not develop within the soil mass, the condition is termed ‘‘complete projection.’’ Ideally, the ultimate arching ratio equals one above the PES since shearing in the soil cannot develop beyond this elevation. The arching ratio profile shown in Fig.4a can be used as a preliminary estimate of the height of the PES. Since soil pressures measured in the models were within a distance of 3.7 B from the trapdoor, the curves needed to be extrapo-lated by extending a straight line from the last two points to define the location of the PES. The elevation of the PES can be defined at the intersection of the straight line and the vertical line at rv=rvo = 1. According to this simplified

approach, a PES developed at an elevation of about 4 B at very small settlements and then raised to about 5 B at larger settlements. Indeed, as the trapdoor settlements evolved,

the disturbance of the backfill was found to extend to a larger zone, forcing the PES to move gradually upward.

Figure4b compares the arching ratio profiles of the different active arching tests (Models AA1 to AA5) con-ducted in this study using different relative densities, external applied pressures and trapdoor shapes. The data presented in the figure correspond to the maximum induced normalized displacement (d=B), which was 10% for Test AA1 and 20% for all other tests. A zone of intense pressure relief was identified over the trapdoor in all models, regardless the differences in backfill density, soil confine-ment level and trapdoor shape. The zone of intense relief extended to a height of approximately 1.5 B above the trapdoor of Model AA3, which was built with loose backfill sand. Results from Models AA4 and AA5 indicate that this zone reached elevations of 1.5 B in the dense soil and suggest that the PES extended beyond a height of 4 B and perhaps exceeded the backfill height (i.e., a complete projection condition). A PES at an elevation of roughly four times the width of the trapdoor was reported by [6].

Stress redistribution in active arching has been postu-lated as essentially unaffected by backfill density and trapdoor shape [6, 17, 33]. Both variables had an insignificant effect on the arching ratio within a height of about 1.5–2 B above the trapdoor in the models of the present study. The confinement level, on the other hand, somewhat influenced the changes in soil pressure at the surface of the trapdoor (He/B = 0), where the arching ratio was slightly higher than in the other models and became negligible at larger elevations.

4.2 Soil pressure redistributions outside

the trapdoor

The experimental results showed that the soil pressure redistribution was not limited to the footprint of the trap-door, but extended to a comparatively large area beyond the limits of the trapdoor. The load originally carried by the trapdoor before yielding was transferred to the soil near the structure and was then transmitted to farther regions of the backfill with increasing magnitude of settlements. Fig-ure5a presents the arching ratio as a function of the nor-malized displacement at the central transverse cross section of Model AA3 (plane xz). The results shown in the fig-ure are for soil pressfig-ures recorded at the floor of the model (He/B = 0) and at elevations Heof 0.3 B above position I4. The portion of soil adjacent to the trapdoor, at a distance of 0.2 B from its boundaries (position I4), experienced an initial increase in vertical soil pressures under small trap-door displacements, followed by a sharp decrease with increasing trapdoor displacements. The portion of soil at 0.8 B (position I5) from the trapdoor followed a similar path, but with a higher peak and comparatively softer Fig. 4 Arching ratio variations: a above a rectangular active trapdoor

at different induced displacement levels (Model AA3); b above active trapdoors tested under different conditions

decay in vertical pressures. A similar trend was observed at an elevation (He) of 0.3 B above I4 (position M6), but with lower arching ratios from d=B = 2.5%. Changes in the arching ratio in the central longitudinal section of the model (plane yz) are shown in Fig.5b. Although the arching behavior close to the trapdoor boundary (0.2 B) was similar to that of the central transverse cross section, a different trend was identified in farther areas along this direction. Arching ratio values above one were recorded from a distance of 0.7 B for all induced displacements of the trapdoor. The data presented in Figs.5a, b indicate the development of an unloading region that included not only the soil above the trapdoor, but also a portion of the mass beyond its footprint. Conversely, a load-transfer region formed in portions of the soil at a greater distance.

The spatial distributions of measured vertical soil pres-sures in the transverse and longitudinal central sections of Model AA3 are shown in Fig.6 for selected normalized

trapdoor displacements. Only one side of the model, where data were collected, is presented in the figure. The data displayed in Fig.6provide evidence that the unloading and load-transfer regions moved within the backfill. As the induced settlements evolved, the unloading region pro-gressively spread to the remoter portions of the backfill, reducing the size of the load-transfer region.

A comparison between the experimental results and predictions made using the elastic solution reported by Finn [10] for the trapdoor problem is also shown in Fig.6a. Finn’s solution represents the trapdoor as a rigid horizontal discontinuity, contained in an elastic, semi-infinite, homogeneous and isotropic medium, to which prescribed displacements are applied. Predictions were made for the selected normalized displacements of 0.1%, 1% and 10%, using Young’s modulus (Es) of 35.2 MPa and a Poisson’s ratio (ms) of 0.34 for the soil. According to Finn [10], the

arching ratio is given by the following equation: rv rvo ¼ 1 þ d Es 2p 1ð  msÞ 1 xþB 2  1 xB 2   ð1Þ where x is the distance from the center of the model (de-scriptions of the other variables can be found in the list of symbols).

Poor predictions close to the edges of the trapdoor were expected with Finn’s solution, since it represents a dis-continuous function at x = B/2, with infinite limits approaching from both sides. Reasonable estimates were only obtained far from the edge of the trapdoor and at very small settlements, while unrealistically high compressive and tensile stresses were predicted near the edge of the trapdoor. Tensile stresses were obtained inside the trapdoor at large settlements and were omitted from Fig. 6a. The dramatic divergency of the predicted results at large dis-placements occurred due to the elastic nature of Finn’s proposition.

Figure7 provides a comparison of the spatial distribu-tion of the arching ratio in all active arching models at the same relative displacement (d=B) of 10%. Soil pressure relief was found to be similar in the different experimental models, in both the transverse and longitudinal directions. More perceivable differences were noted at an elevation of 0.2 B from the trapdoor, which is close to the location where soil pressures switch from relief to increase. Due to differences in shape, the longitudinal arching ratio distri-bution of the square trapdoor (Model AA5) diverged from that of the other models, which were tested with a rect-angular trapdoor. However, due to the width common to both shapes, very similar arching ratio distributions were obtained in the transverse section of the models.

Insight into the effect of backfill density on arching was gained by assessing the results of models AA1 and AA3, Fig. 5 Soil pressure changes outside the active trapdoor of Model

AA3: a transverse central cross section; b longitudinal central cross section

tested with relative densities (Dr) of 100% and 50%, respectively. Differences in the arching ratio changes inside the trapdoor were insignificant for both models (as shown in Fig.7), which is in agreement with the results reported by Dewoolkar et al. [6] for backfills compacted with relative densities of 40% and 80%. Nevertheless, backfill density was found to affect soil pressure changes beyond the limits of the trapdoor. Figure8a, b shows the development of soil pressures on the transverse and lon-gitudinal central cross sections of the model, respectively. While a more significant relief in soil pressure occurred for the case of the loose backfill, the dense backfill experi-enced a comparatively larger increase in soil pressure. The

lower shear strength developed in the loose material was found to lead to a reduced ability of the soil to transfer stresses from the regions close to the trapdoor to regions that are more distant. The effect of relative density was greater in the longitudinal direction of the model (Fig. 8b). Soil confinement level has been investigated by varying the soil cover height above the trapdoor and/or application of external distributed loads of different magnitudes. Lar-ger soil confinement levels have been found to contribute to greater reductions in the load recorded at the surface of active trapdoors [6, 13,17, 20]. In the present study, the influence of soil confinement was investigated by com-paring the results of Models AA1 and AA4, tested with Fig. 6 Spatial distribution of vertical soil pressures on the base of Model AA3 and evolution with increasing normalized displacements: atransverse central cross section; b longitudinal central cross section

Fig. 7 Spatial distribution of vertical soil pressures on the base of all active arching models at d=B = 10%: a transverse central cross section; blongitudinal central cross section

external pressures (q) of 100 kPa and 50 kPa, respectively. These results revealed a trend similar to that reported in previous studies regarding vertical load variations on the trapdoor surface (Fig.7). Minimum arching ratios at the center of the trapdoor (position I1) were 0.07 for q = 50 kPa and 0.03 for q = 100 kPa, which, in terms of absolute loading, represent virtually the same weight of soil above the trapdoor. Iglesia et al. [16] also found the same minimum absolute load over the trapdoor for all models run with varying H/B ratios.

Figure9 compares the behavior of the arching ratio outside the trapdoor limits. Similar results were obtained at both confinement levels in the transverse direction of the model (Fig.9a) and at a distance of 0.2 B in the longitu-dinal direction (Fig.9b). However, the arching ratio developed very differently at a distance of 0.7 B in the longitudinal direction (Fig.9b). The higher confinement level triggered a greater increase in vertical soil pressures,

and the arching ratio remained above unity during the entire trapdoor translation. Conversely, the pressure in the soil under the lower confinement gradually decreased after reaching a peak and eventually became smaller than the initial soil pressure at large trapdoor displacements (arch-ing ratio below unity).

Differences in the performance of a rectangular versus a square trapdoor were evaluated in the present study by comparing the results of Models AA1 and AA5. Differ-ences in arching ratio variations on the surface of both trapdoors (position I1) were insignificant. Very similar arching ratios, obtained at the maximum displacement of the trapdoors, are shown in Fig. 7. These findings are consistent with results reported in the literature, which indicate that the load measured on trapdoors in active arching is essentially independent of the trapdoor shape [17,33].

Fig. 8 Influence of backfill relative density on development of soil pressures outside active trapdoors: a transverse central cross section; blongitudinal central cross section

Fig. 9 Influence of soil confinement on development of soil pressures outside active trapdoors: a transverse central cross section; b longi-tudinal central cross section

Figure10 compares the development of the arching ratio outside both square and rectangular trapdoors, in the transverse central cross section of Models AA1 and AA5. Differences were negligible at a distance of 0.2 B from the edge of the trapdoor, but were perceptible at 0.7 B, where the decrease in arching ratio was more substantial with the rectangular trapdoor. The differences between the results of both shapes increased with increasing trapdoor settlements. No significant changes were found in the arching ratio in the longitudinal direction of the models.

5 Results of passive arching tests

5.1 Soil pressure redistributions

above the trapdoor

Figure11a shows the arching ratio (rv=rvo) as a function

of relative displacements (d=B) in several locations above the center of the trapdoor (position I1) of Model PA1. Negative normalized displacement values indicate an upward movement of the trapdoor. Despite the compara-tively small induced relative displacement, the effect of trapdoor movement on the redistribution of pressures in the soil mass was significant. Specifically, an arching ratio of about 4 was reached at the elevation of the trapdoor for the maximum relative displacement of - 0.63%. Gradually smaller arching ratios were obtained with increasing ver-tical distance (He) from the trapdoor. Figure11b shows the impact of the trapdoor shape on the arching ratio for increasing vertical distance above the centers of the rect-angular trapdoor of Model PA1 and square trapdoor of Model PA2. The arching ratio values shown in the fig-ure correspond to the maximum induced upward

displacement. Very similar arching ratio profiles were found for both models. The results suggest that the PES was reached at an elevation (He) of 3 to 4 B for the case of the maximum induced upward displacement.

5.2 Soil pressure redistributions outside

the trapdoor

The redistribution of vertical soil pressures in the trans-verse and the longitudinal central cross sections of Model PA1 is shown in Fig.12a, b, respectively. The passive movement of the trapdoor induced changes in soil pres-sures over a comparatively large volume of backfill. While a distinct relief in the arching ratio occurred around the trapdoor, an increase in the arching ratio was not observed at any other location within the backfill. The relief in the arching ratio increased with increasing trapdoor Fig. 10 Influence of trapdoor shape on development of soil pressures

outside the limits of trapdoors in active arching

Fig. 11 Experimental results under passive conditions: a arching ratio as a function of trapdoor displacement at different heights above the center of trapdoor; b variation in arching ratio above the centers of passive rectangular and square trapdoors

displacements and decreased with increasing distance from the trapdoor. Notably, arching ratio changes no longer occurred at a distance of 2.7 B from the trapdoor (position I8), which indicates that the zone of influence of the trapdoor was at a horizontal distance somewhere between 0.7 B and 2.7 B from the edge of the trapdoor.

The spatial distributions of the arching ratio in the transverse and longitudinal central sections of passive arching Models PA1 and PA2 are compared in Fig.13a, b, respectively. The data correspond to the maximum induced upward displacement. As previously mentioned, important changes in stresses took place in the soil mass after the elevation of the trapdoor. The movement of the trapdoor promoted load transfer from the outer soil mass to the portion of soil above the trapdoor. Specifically, significant stress gradients occurred near the edges of the trapdoor due to the significant shearing that occurs within the soil at this location. As illustrated in Fig.13b by the results of Model PA1 (rectangular case), the vertical pressure over the

trapdoor but near its edge considerably exceeded the ver-tical pressure recorded at the other two internal locations. Beyond the footprint of the trapdoor, the vertical pressure decreased to almost zero close to the trapdoor edge. The induced movement of the structure modified the stress field in the longitudinal direction to maximum distances of 2.7 B and 1.7 B from the borders of the rectangular and square trapdoors, respectively (Fig.13b).

Experimental evidence shows that the arching ratio on the surface of buried structures decreases as the length-to-width ratio, L/B, increases [3,11,22,23]. This effect was found to be particularly intensified by the burial depth [17,21,23]. A similar trend was observed in the results of the present study, which revealed a slightly larger arching ratio for the square trapdoor as compared to that for the rectangular trapdoor (Figs.11b and13a, b).

The influence of structure shape on the development of the arching ratio outside the limits of the trapdoor of Models PA1 and PA2 can be observed in Fig.14. The arching ratio appeared to be essentially unaffected by the trapdoor shape at very small displacements (below approximately 0.15% B). Beyond this displacement level, the rectangular trapdoor induced a significant reduction in the arching ratio than the square trapdoor, a trend that was observed in both directions of the model. Furthermore, the shape appeared to have a greater influence on soil pressure changes at positions farther away from the edge of the trapdoor (0.7–0.8 B). Despite the asymmetrical shape, the rectangular and square trapdoors yielded remarkably sim-ilar results in both transverse and longitudinal directions.

6 Discussion on arching evolution

mechanisms

6.1 Load-transfer mechanisms under active

arching

Typical failure mechanisms identified from image analysis of shallow installations include the development of one or more inclined failure surfaces originating at the corners of the trapdoor and propagating toward the center of the model with continued displacements [6,9,27,29,33]. The soil above the trapdoor was found to remain essentially rigid. The path followed by the failure surfaces is governed by soil density and confinement level, which are variables that also govern soil dilatancy. A decreasing soil density and increasing confinement were found to lead to failure surfaces with comparatively higher inclination in relation to the horizontal. Since soil density decreases with increasing trapdoor displacements, each new failure sur-face evolves with a smaller inclination angle to the vertical than the previous one. The condition associated with a final Fig. 12 Changes in vertical soil pressures outside the passive trapdoor

of Model PA1: a transverse central cross section; b longitudinal central cross section

trapdoor movement was found to be represented by an approximately vertical failure surface.

Costa et al. [5] observed significant differences in the failure mechanisms induced by the active movement of a rectangular trapdoor under deep conditions. A series of reduced-scale model tests with a soil cover ratio of 4.5 was conducted under normal gravity as well as under an increased gravitational field using a centrifuge facility. Failure mechanisms in the transverse direction of the models were characterized by the development of multiple failure surfaces propagating first inside and then outside the footprint of the trapdoor. Visible external failure surfaces evolved and eventually reached lateral extensions ranging from 1.4 to over twice the trapdoor width (B). As with internal failure surfaces, the path followed by external

failure surfaces depends on soil density and confinement level. Lower soil densities and higher confinement levels result in external failure surfaces with a higher inclination in relation to the horizontal. Moreover, other important features observed in the tests, which deviate from the typical mechanisms described for shallow conditions, were: (i) the soil directly above the trapdoor showed sig-nificant dilation in the direction of the trapdoor movement and (ii) both internal and external failure surfaces changed inclination during trapdoor movements.

Further experimental evidence of external failure sur-faces can be found in results of small-scale tests, with H/ B ratios ranging from 2 to 5.3, reported by Ladanyi and Hoyaux [18] and Santichaianant [25]. By using upper and lower bound computational limit analyses, Wang et al. [35] reported failure mechanisms shifting from an internal failure to a critical yield surface that developed externally when H/B was greater than two.

The development of external failure surfaces is dictated not only by the soil cover thickness, but also to a combi-nation of other factors. Soil shear strength was found to play an important role in the development of external failure surfaces. Wang et al. [35] reported failure mecha-nisms with internal and external failure surfaces for back-fills with low internal friction angles (/0). Backfills with comparatively higher friction angles produced patterns involving only the development of internal failure surfaces. Sidewall friction in the transparent panel of trapdoor model tests using image analysis can affect the development of failure patterns. Friction can restrict the settlement of sand soil due to interaction with the transparent surface of the test container, which may result in underdeveloped failure surfaces. External failure surfaces in particular can be more Fig. 13 Spatial distribution of vertical soil pressures on the base of Models PA1 and PA2: a transverse central cross section; b longitudinal central cross section

Fig. 14 Changes in soil pressures outside rectangular and square trapdoors in passive mode

affected throughout the process than internal failure sur-faces because comparatively larger settlements are needed to reach effects that are noticeable. Santichaianant [25] compared the section of a model in contact with a Plexiglas wall to another section of the same model distant from the Plexiglas wall and found that friction prevented visualiza-tion of a fourth external failure surface that emerged after three previous internal failure surfaces. Distinguishable external failure surfaces may also be more difficult to develop in multiple trapdoor installations due to the interference of neighboring trapdoors. Rui et al. [24] per-formed tests with setups including soil cover ratios (H/B) ranging from 0.7 to 8 and normalized trapdoor displace-ments (d=B) below 40%. However, patterns involving external failure surfaces were not visually detected.

Figure15shows a conceptual arching evolution model that captures the arching-induced changes in vertical stresses observed from the multiple sources of soil pressure measurements gathered as part of this testing program. The conceptual model also builds on the failure patterns iden-tified in the experiments conducted by Costa et al. [5], as well as other experimental data generated by other researchers. The conceptual model identifies internal and external failure surfaces as part of an arching-unloaded region in contact with a stable arching-loaded region in the soil, as schematically shown in Fig.15a. As presented in the figure, the most external failure surface, line CG, divides the unloading region into a collapsing subregion ABCGF and a metastable subregion CDG. The collapsing subregion is where internal and external failure surfaces develop. In this simplified representation, the boundary between the unloading and load-transfer regions is assumed to correspond to the vertical line CG, which intersects the edge of the external failure surface CG. Points 1 and 2, from DG toward the trapdoor, experience relief in vertical stress, and because the external failure surface propagates with an inclination, the stress relief is greater at 1 than at 2.

Point 3, at a vertical distance He, undergoes more stress relief than point 2 because it is closer to the external failure surface. The arching ratio is expected to decrease with increasing Hewithin the metastable subregion, unlike what happens within the collapsing subregion, where the arching ratio increases with increasing He. Point 4, away from DG, undergoes an increase in the vertical stress, since it is located within the arching-loaded region. The PES in the backfill represents an upper limit for the arching-loaded and arching-unloaded regions. The portion of the backfill above the PES remains essentially unaffected by the movement of the underlying trapdoor. The mechanics of the problem involves a gradual expansion of the unloading region toward the outer soil mass as the trapdoor moves down, pushing the load-transfer region farther into the backfill. That is, as the external failure surface propagates during expansion, line DG moves away from the trapdoor and the PES moves upward. The vertical stress at point d gradually decreases with the approaching line DG.

Some aspects of the arching behavior identified from the results of the active arching models can be interpreted using the idealized load-transfer mechanism presented in Fig.15a. The increase in the arching ratio at small dis-placements near the trapdoor, as shown in Fig.5, can be attributed to the gradual approximation of the boundary DG to that portion of the soil. The arching ratio drops abruptly in the portions of soil within the arching-unloaded region, as exhibited by the curves of positions I4, I5, I6 and M6 in Fig.5. Areas of the backfill that remained within the arching-loaded region throughout the movement of the trapdoor experience an increase in vertical stress, as indi-cated by the soil pressure measurements from positions I7 to I9 (Fig.5b). Particularly, a slight decrease in the soil pressure at large trapdoor movements was detected by sensor I7, while the most remote sensors I8 and I9 revealed a constant increase in the soil pressure. The different trend followed by I7 is due to the gradual approximation of the

arching-unloaded region to the outer soil mass, which reached that distance after comparatively large trapdoor displacements. The greater pressure relief at M6 as com-pared to I4 can be explained by recognizing that M6 is comparatively closer to the zone where failure surfaces develop within the backfill (i.e., the collapsing subregion). Additional evidence of a comparatively greater relief in vertical soil pressure with increasing He within the metastable subregion was observed from the results of all models tested under active arching conditions.

The influence of backfill relative density was evaluated considering the data presented in Fig.8. Since soils with lower relative density have a reduced ability to dilate, external failure surfaces are expected to follow paths with a higher inclination to the horizontal, resulting in a com-paratively wider unloading region. This feature justifies the more significant decrease in vertical soil pressure in the vicinity of a trapdoor embedded in the loose backfill (D r-= 50%) (positions I4, I5 and I6 in Figs.8a, b). On the other hand, the reduced shear strength of the loose backfill affects the load transfer from the arching-unloaded region to the arching-loaded region, as illustrated by the trends shown in Fig.8b for the case of position I7.

A comparison of the evolution of the arching ratio in models subjected to different confinement levels is pre-sented in Fig.9. Increased soil confinement was found to lead to steeper external failure surfaces because the soil shows less dilatancy under this condition, resulting in a wider unloading region [5]. Considering a wider unloading region in the model subjected to higher confinement (q = 100 kPa) explains the slightly higher relief in vertical soil pressure observed in the region surrounding the trap-door positions I4, I5 and I6 in Fig.9a, b. Additionally, the higher shear strength of the soil subjected to the higher confinement level facilitates the load transfer from the arching-unloaded region to the arching-loaded region, as the changes in vertical soil pressure at position I7 of the models illustrate (Fig.9b).

Figure10 shows that the rectangular trapdoor experi-enced a more significant load relief than the square trap-door. This trend was clear at position I5, located at a distance of 0.8 B from the structure. A narrower unloading region is expected to have developed due to the movement of the square trapdoor, resulting in less stress relief in the soil mass surrounding the structure.

6.2 Load-transfer mechanisms under passive

arching

Failure patterns resulting from the upward vertical trans-lation of a horizontal rigid trapdoor in a noncohesive material involve the development of one or more failure surfaces that initiate at the corners of the trapdoor structure

and evolve with an inclination toward the outer soil mass [18, 29, 34, 35]. Since soil density decreases with increasing trapdoor displacements, each new failure sur-face develops more inclined to the vertical than the pre-vious one. The condition corresponding to a trapdoor with comparatively large movements has been reported to be represented by an approximately vertical failure surface. However, Wang et al. [35] reported the formation of minor internal failure surfaces under large trapdoor displacements in cases involving a soil cover of considerable thickness.

Also, for the passive trapdoor condition, a conceptual arching evolution model has been refined considering the soil pressure changes collected from the experimental data in this study. Together with failure patterns described in previous investigations, the conceptual model provides insight into the evolution of the vertical stresses in the passive arching models of this study (Models PA1 and PA2). Consistent with the rationale adopted for the active mode, the most external failure surface of the passive mode can be considered a boundary between an arching-loaded region (above the trapdoor) and an arching-unloaded region beyond the footprint of the trapdoor limits (Fig.15b). Failure surfaces develop within the load-trans-fer region, and as the boundary failure surface propagates upward, points 1 and 2 in Fig.15b experience a reduction in vertical stresses. The stress decrease at point 1 is larger than that at point 2 because of the comparatively shorter distance to the boundary failure surface (the influence of a shearing zone in the soil increases with proximity). Point 3 at an elevation Heexperiences less stress relief than point 2 below since it is closer to the boundary failure surface. Within the arching-unloaded region, the arching ratio is expected to increase with increasing He, unlike the trend occurring within the arching-loaded region, where the arching ratio decreases with increasing He. The PES rep-resents an upper limit for the arching-unloaded and -loaded regions, with the backfill above the PES remaining unaf-fected by the movements of the underlying trapdoor. The upward translation of the trapdoor promotes a progressive widening of the arching-loaded region toward the outer soil mass.

Certain features identified in the results of Model PA1 (Fig.12a, b) can be explained via the conceptual repre-sentation presented in Fig. 15b. Specifically, the arching ratio of one reached at positions I8 and I9 indicates that the limit of the arching-unloaded region in the longitudinal direction reaches a maximum horizontal distance of 2.7 B from the trapdoor boundary. Also, consistent with the conceptual model, the soil pressure relief recorded at M6 was larger than that at I4 (for d=B beyond - 0.25%) because M6 was closer to the arching-loaded region.

A comparison between the results obtained in rectan-gular square trapdoors under passive arching models

(Models PA1 and PA2, respectively) is presented in Figs.13and14. The results shown in Fig.13indicate that the widening of the arching-loaded regions in the longitu-dinal direction for the square and rectangular trapdoors was 1.7 B and 2.7 B from the trapdoor edge, respectively. In Fig.14, the lower arching ratio obtained with the rectan-gular trapdoor suggests that the boundary failure surface propagated from the rectangular trapdoor at a compara-tively higher inclination to the horizontal, resulting in a comparatively wider arching-loaded region expanding toward the outer soil mass. The differences between the arching ratios obtained with the square and rectangular trapdoors increased with increasing normalized displace-ments and distance from the trapdoor. This may be due to the path followed by the boundary failure surface in both cases, which may coincide at the beginning of the trapdoor movements and separate from each other with increasing trapdoor displacements. However, experimental evidence from inspection of failure mechanisms involving passive trapdoors of different shapes is needed to better clarify these arching ratio trends.

7 Conclusions

An experimental investigation of soil arching development was performed via a series of three-dimensional reduced-scale trapdoor model tests. Focus was placed on evaluating vertical soil pressure changes outside the footprint of the trapdoor. The experimental program included models with rectangular and square trapdoors acting in both active and passive modes in a granular backfill soil. The models were constructed with a soil cover with a thickness of 5.6 times the width of the trapdoor (B), which corresponds to deep conditions. Vertical soil pressures were recorded at several locations within the central longitudinal and central trans-verse sections of the models. Conclusions from this investigation are as follows.

Vertical soil pressure redistributions within the sand mass extended to a comparatively large region beyond the trapdoor in active arching conditions. An assessment of the experimental results indicates that an arching-unloaded region and an arching-loaded region developed in the soil due to the trapdoor movement. The arching-unloaded region included the collapsing mass of soil above the trapdoor and a portion of the soil surrounding the trapdoor, while the arching-loaded region comprised a stable, more distant zone of the backfill. The data collected indicate that the arching-unloaded region reached distances above the trapdoor in excess of 4 B and extended to horizontal dis-tances from the edges of the structure of approximately 1 B. The arching-loaded region reached a horizontal distance of nearly 5 B. As the trapdoor moved down, the unloading

region gradually expanded to the outer soil mass, pushing the load-transfer region farther into the backfill. The soil relative density, confinement level and trapdoor shape were found to affect more significantly the changes in soil pressure within the arching-unloaded region outside the trapdoor and the arching-loaded region than the changes within the collapsing mass right above the trapdoor. While comparatively significant soil pressure relief occurred in the arching-unloaded region surrounding the trapdoor embedded in loose backfill soil, a comparatively larger increase in soil pressure was measured in the arching-loa-ded region of the dense backfill soil. An increase in soil confinement level was found to result in higher soil pres-sures being transferred to the arching-loaded region of the backfill. The outer arching-unloaded region experienced comparatively more pressure relief when using a rectan-gular trapdoor than with a square trapdoor of equal width and three times shorter.

When acting under passive conditions, the trapdoor modified the stress field of a comparatively large region of the backfill sand. Similar to the active mode, analysis of the experimental data suggests that the trapdoor upward movement resulted in the development of an arching-un-loaded region and an arching-arching-un-loaded region in the soil. However, the arching-loaded region included the soil mass above the trapdoor, while the arching-unloaded region involved the soil surrounding the trapdoor extending to farther zones in the backfill. The results revealed that the arching-loaded region reached elevations over the trapdoor greater than 3 B. The arching-unloaded region reached a horizontal distance of nearly 3 B from the edge of the trapdoor. As the trapdoor displaced upward, the arching-loaded region gradually expanded to the outer soil mass, pushing the arching-unloaded region farther into the backfill. Soil pressures in the arching-unloaded region experienced more relief with the rectangular trapdoor than with the square trapdoor.

Acknowledgements The authors dedicate this paper to Prof. Benedito S. Bueno, who actively participated in this research and, unfortu-nately, passed away on August 1, 2015. Prof. Bueno was an enthusiast researcher in geosynthetics and underground structures. He was greatly admired by his students and colleagues for his competence and character. His legacy includes the Laboratory of Geosynthetics of the University of Sa˜o Paulo at Sa˜o Carlos, Brazil, which he founded in 2001. We gratefully acknowledge his vital contributions to this research. The authors also express their gratitude to the Brazilian Research Agency Fapesp for the financial support provided for this research (Grant No. 00/09397-0), the Geotechnical Engineering Department of the University of Sa˜o Paulo at Sa˜o Carlos, Brazil, and the Civil Engineering Department of the University of Colorado at Boulder, USA, where the first part of this investigation was conducted.

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