J. Cent. South Univ. (2021) 28: 2527-2543

DOI: https://doi.org/10.1007/s11771-021-4784-9

Experimental study on bearing capacity of shell strip footings near geotextile-reinforced earth slopes

Seyed ALI GHAFFARI, Elham SATTARI, Amir HAMIDI, Gholamhosein TAVAKOLI MEHRJARDI,Abtin FARSHI HOMAYOUN ROOZ

Department of Civil Engineering, School of Engineering, Kharazmi University, Tehran, Iran

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: From a financial point of view, urbanization frequently enforces the clients to construct superstructures near the slopes, giving rise to increasing the risk of building instability. By conducting a series of small-scale plate load tests, this work aims to investigate the effects of installing geotextile reinforcement layers in sandy slope and reducing the apex angle of triangular shell strip footings. The results show considerable effect of using geotextile-reinforced layers and decreasing the apex angle on the ultimate bearing capacity of shell foundations. With increasing foundation distance from the slope, the adverse effect of the slope is reduced. However, as the distance decreases, the effect of reinforcement and apex angle is increased. For practical applications, empirical equations are also presented for determining the ultimate bearing capacity of the footings and scale effect as well. Finally, 3D numerical simulations are executed and compared with the experimental results.

Key words: slope; shell strip footing; geotextile-reinforced; apex angle; ultimate bearing capacity; numerical simulation

Cite this article as: Seyed ALI GHAFFARI, Elham SATTARI, Amir HAMIDI, Gholamhosein TAVAKOLI MEHRJARDI, Abtin FARSHI HOMAYOUN ROOZ. Experimental study on bearing capacity of shell strip footings near geotextile-reinforced earth slopes [J]. Journal of Central South University, 2021, 28(8): 2527-2543. DOI: https://doi.org/ 10.1007/s11771-021-4784-9.

1 Introduction

The crucial role of foundations on the stability of structures is now more perceivable than ever. Therefore, efficient and economical foundations have been the scope of numerous studies. On the other hand, bearing capacity of foundations near slopes has been a significant challenge [1-5]. Among different foundations, shell footings have considerable benefits. In effect, shells are thin-walled structures that attain stability and bearing capacity by using their specific shape. This feature enables them to generate maximum structural efficiency with minimum materials [6-9]. From a geotechnical view of point, the particular performance of shell footings has been directly related to different shapes consisting of triangular, conical, and pyramidal shell strip footings [10-12].

Based on the experimental investigations carried out to date about the triangular shell strip footings located on the sand [13, 14], the bearing capacity and settlement characteristics of the shell footings with a reduced apex angle can be noticeably improved compared to the conventional flat one. The bearing capacity of triangular shell strip footings on unreinforced sand has been studied by using experimental models. Moreover, the effects of soil compaction and geotextile reinforcement beneath the foundation on the maximum bearing capacity of shell footing have also been surveyed [15, 16]. The results confirmed the increase of bearing capacity of shell footings with a decrease in apex angle. Also, the reduction of bearing capacity has been observed due to the increase in the burial depth of geotextile layers. Furthermore, the investigations indicated that the failure wedge of shell foundations in reinforced backfill conditions is formed deeper than that of conventional flat ones [17, 18].

In the case of land limitation specifically in urban areas or construction near river banks, constructors are forced to conduct the operations in the vicinity of slopes. Thus, comprehensive studies are necessary to reach safe conditions as well as economical aspects. The advantages of shell footings on the extension of failure wedge into the soil depth impress the researchers to investigate the response of this foundation system adjacent to the slopes. A number of researchers have investigated the performance of conventional flat footings located next to the unreinforced and reinforced slopes [19-24]. The results have demonstrated that the ultimate bearing capacity of flat footings and their settlement can be remarkably improved using reinforcement layers.

The overriding purpose of the present study is to provide practical results about the performance of shell footings adjacent to the unreinforced and geotextile-reinforced slopes. In this regard, a series of small-scale plate load tests have been performed to fully investigate the behavior of triangular shell strip footings next to the sandy slopes. To find out the efficiency, similar tests have also been conducted for the shell footings located on the flat ground, namely the ground without slope having reinforced or unreinforced conditions. Then, major parameters such as bearing capacity, edge distance, and apex angle of shell footings are studied, and subsequently, the empirical equations are suggested for design engineers. The scale effect is also investigated, and finally, numerical simulations are executed to compare with the experimental ones.

2 Experimental investigation

2.1 Experimental program

For a comprehensive experimental program, a total of 40 small-scale plate load tests have been conducted to accurately investigate various factors relevant to shell footings adjacent to two different conditions of unreinforced (UR) and geotextile-reinforced (R) slopes. As fully summarized in Table 1, the variables are four apex angles for shell footing and four edge distances. The edge distance is a distance between slope crest and footing centerline. The experiments have been executed for shell footings located on sloped and flat ground.

Table 1 An experimental program for shell footings in different conditions

2.2 Testing box

Figure 1 shows the schematic design of the tests. The testing box was constructed with steel frames.

Figure 1 Experimental model:(1-Hydraulic jack; 2-Load ring; 3-Diall gauge; 4-Footing; 5-Fiberglass wall; 6-Box floor; 7-Beam; 8-Columns)

The internal dimensions are 1200 mm×700 mm in plane, and 700 mm in height. As shown in Figure 1(a), the test box has three sidewalls of 20 mm-thick made of fiberglass for monitoring the failure surfaces during the test which were held by two steel columns. To ensure rigidity of the tank, a steel plate was also utilized on the floor.

Several researchers have reported the extension of the failure zone up to about 2 to 2.5 times of the footing width away from the footing center [25-27]. Hence, this consideration was regarded in the test box dimensions, and thereby, the box width was considered equal to 1200 mm. Indeed, the ratio of the box width to the footing width was considered equal to 12, which surely guarantees zero lateral deflection of the sides during loading as well as no interference of the failure surface with the box.

2.3 Testing set-up

As illustrated in Figure 1(b), the embedment depth of all footings was considered equal to zero. The 400 mm height of the embankment was also constructed in two layers of 150 mm and one layer of 100 mm. According to ASTM standards [28, 29], the maximum and minimum unit weights of sand were determined. The required weight of soil was specified based on the box volume. The unit weight of the poured sand was determined considering a relative density of 70%. To ensure reaching the desired relative density, each soil layer was compacted to the required height of the box. The compaction of each layer was carried out using a vibratory compactor until reaching the expected relative density. Shear strength components of the soil, i.e., cohesion and friction angle, were attained using consolidated drained triaxial tests under different confining pressures. The moisture content of the backfill was considered equal to 4.5%, and it was highly tried to keep it constant which helped the stability of slope during construction and early loading. Finally, the embankment was trimmed to reach an angle of 45° sloped backfill. This is a representative slope angle at which the embankment remains stable under its weight.

The effect of depth ratio (the effective depth of an embankment to the foundation width) on the footing settlement is of particular importance. FATTAH et al [30] studied the depth ratio from 0.5 to 4. The settlement ratio remained at its lowest value and remained nearly constant when the depth ratio was equal to 2. They also investigated the width ratio (the effective width of an embankment to the foundation width) and concluded that the lowest settlement ratio occurs when the width ratio equals 0.75. Hence, as shown in Figure 1(b), two layers of geotextile reinforcement were employed in the present tests. The upper and lower geotextile layers will be called the first and second layers, respectively. According to FATTAH et al [30] and other previous studies such as by ALTALHEA et al [31], the first layer embedment depth (u) was considered equal to 0.5B=50 mm, and the distance between reinforcement layers (h) was chosen equal to 0.7B=70 mm based on the study by ALAMSHAHI et al [20]. Enough reinforcement length (6B=600 mm) in fine sands, proposed by TAVAKOLI et al [21], was considered regardless of the footing distance from the slope’s crest.

In practical works, shell foundations are constructed using cast-in-place or precast methods. In the cast-in-place method, the soil is cut to fit the core beneath the shell foundation. Then, the subgrade is grouted to reach a smooth surface, and next, shell footing made of reinforced concrete is constructed. For the precast method, there are two construction strategies. The precast shell can be transferred from the factory and placed on the cutting edge at the same core shape; moreover, the gap between footing and soil are filled with pressure-controlled grout. In the second method, the precast shell is placed on the flat subgrade, and the gap between it and soil is filled with sand. The sand is poured using the holes placed on the precast foundation. The compaction of the sand core can be performed using a small rotary vibrator.

In the present study, to prepare the soil core under the shell model, the space under the shell was filled with sand considering the required unit weight according to HANNA et al [13] and AZZAM et al [16]. In fact, the bottom of shell footings was first placed upward. After that, the sand filling process of shell models was carried out by placing a thin steel plate on the bottom of the shell models before placing it on its location. Then, the steel plate underneath the shell footing was slowly and horizontally pulled out.

2.4 Testing procedure

A total number of 40 tests were conducted in different conditions, including flat/sloped backfill, unreinforced/reinforced backfill, several shell footing apex angles, and various edge distances. Figure 2 displays the loading set-up of the foundation. The monotonic loading system comprises a hand-operated hydraulic jack and pre-calibrated load ring, which was mounted on the footing at a certain distance from the slope edge. The hydraulic jack located between the loading shaft and footing applies loading through a pre-calibrated load ring having a capacity of 5000 kg. The accuracy is ±0.01% of the full range. The loading was applied on a small plate welded at the footing centerline without any fixed connection. To measure any possible rocking or tilting of the footings, settlements were monitored using two dial gauges located on opposite edges of the loading shell, which had an accuracy of 0.01% of the full range (60 mm). The average value of recorded settlements of footing was reported at each loading step. The load was applied with increments of 0.4 kN and was maintained until the footing settlement reached a constant value. However, the monotonic loading was incrementally continued in all tests as long as reaching the peak settlement of 0.25B=25 mm.

Figure 2 Illustration of foundation under loading

2.5 Materials

2.5.1 Soil

A uniform and clean quartz beach sand is utilized for the tests, which is locally available in the shores of Babolsar city, Iran. In Figure 3, the particle size distribution of the Babolsar sand is presented. The physical property of the soil is also summarized in Table 2. Based on the Unified Soil Classification System [32], the soil is classified as SP.

Shear strength components of the soil, including cohesion and friction angle, were obtained from consolidated-drained (CD) triaxial tests under different confining pressures (50, 100, and 150 kPa).

Figure 3 Particle distribution curve for Babolsar sand

Table 2 Properties of Babolsar sand

2.5.2 Geotextile

Table 3 presents the mechanical properties of the geotextile used as slope reinforcement. The geotextiles, which are expanded over the slope backfills, were made of high strength woven polyester manufactured from high tenacity and high molecular weight multifilament polyester yarns.

Table 3 Mechanical properties of geotextile

2.5.3 Model footings

Four models for the footings were employed in the tests, including three types of triangular shell footings and one type of simple footing. Figure 4 depicts all four footings. To simulate the plane strain conditions, the dimensions of footings were considered equal to 100 and 640 mm for the width and length, respectively. The apex angles of 60°, 90°, 120° (shell footing), and 180°(simple footing) were also constructed to investigate the response of the foundation system. These values can be representative of a practical range for construction purposes [13]. All the other dimensions of footings’ geometry were kept the same. As displayed in Figure 4(a), the top segment thickness in the footing with an apex angle of 60° was increased to 57 mm to avoid bending or breakage of the model under loading without a significant effect on the results. In Figures 4(b)-(d), the height of the footings is identical (50 mm). The aspect ratio of the metallic footings is an important factor that controls the buckling of foundations during axial loading. The height and width of most footings were considered the same to provide a similar aspect ratio (h/B=50/100=0.5).

Figure 4 Geometrical configuration of model footings (Unit: mm)

The model footings were fabricated of high-quality aluminum alloy (Type 6061) using the computer numerical control (CNC) method. Aluminum shell footings have been employed in different experimental studies [11]. This is because aluminum is a light deformable metal, which can be simply trimmed to construct small-scale shell foundations with various angles and shapes. To attain a uniform structure without nodes and hinges, each model footing was made by shaving an ingot of alloy.

3 Results and discussion

3.1 Bearing capacity in unreinforced slope condition

Figure 5 shows the applied load-settlement ratio (s/B) curves for shell footings (i=60°, 90° and 120°) and simple footing (i=180°) located near the unreinforced slopes and at different edge distances. The rate of change in bearing capacity is in the range of 18%-56% for unreinforced slope conditions, depending on the edge distance. The highest bearing capacity of all footings is for the flat ground condition (without slope). The bearing capacity of footings adjacent to the sloped backfill is significantly decreased in comparison with the flat backfill condition, because of removing an effective part of the resistant zone from the induced failure surface. The failure mode of the footings located on the flat ground is the local shear; however, a general shear mode of failure is also observed for the foundations near the sloped backfill regardless of the edge distance. In all edge distances, a decrease in the apex angle is led to an increase in the bearing capacity of foundations. Depending on the apex angle, the bearing capacity of shell foundations in flat ground conditions is in the range of 8%-18% more than the simple foundation. This can be attributed to the fact that the shell footings extend the failure zone into the depth of the foundation to provide more resisting forces against failure, which is in agreement with previous experimental studies [8, 14, 33].

To evaluate the reduction rate of bearing capacity in sloped backfill condition compared to the flat one, the parameter of ultimate bearing capacity decrease D_ubc, is defined as below.

                         (1)

Figure 5 Settlement variations of foundation located near unreinforced slope and flat backfills:

where Q_u,f and Q_u,_s are the ultimate bearing capacity of footings in flat and sloped backfills, respectively. As can be seen in Table 4, D_ubc is continuously increased with decreasing the edge distance. Moreover, by comparing D_ubc of shell footings with different apex angles, it can be concluded that the shell footing with the apex angle of 60° reflects the minimum amount of bearing capacity reduction. Obviously, this kind of geometry is suggested to be used in practical works due to the highest benefits in terms of stability by providing a deeper failure wedge. It is also inferable that if footings locate at the edge distances larger than 4B, the ultimate bearing capacity of foundation in sloped backfill will become closer to that of the flat condition. Thus, this finding agrees well with the recommendation of 3.5B as the minimum safe edge distance of footing by TAVAKOLI et al [21]. In general, there is no consensus over the exact safe edge distance in the literature. A few studies concluded that the safe edge distance does not affect the bearing capacity beyond the distance of 2B and 3B. Some other studies found this value to vary up to 5B and 6B [34-36]. Hence,the safe edge distance of 4B in the current study is in correspondence with earlier studies.

Table 4 Ultimate bearing capacity decrease D_ubc of different shell footings, edge distances, and reinforcement conditions

3.2 Bearing capacity in geotextile-reinforced slope condition

Figure 6 illustrates the load-settlement ratio curves in geotextile-reinforced soil for different apex angles. Based on the obtained results, when the shell footing apex angle becomes more acute, the settlement related to the ultimate bearing load is smoothly increased, approaching that of the flat condition. Furthermore, the increased rate of ultimate bearing capacity for shell footing with the apex angle of 60° is 43% and 17% compared to the simple footing (i=180°) in sloped backfill (with an edge distance of 1B) and flat ground, respectively.

Figure 6 Settlement variations of foundation located near reinforced slope and flat backfills:

Figure 7 presents the variations of ultimate bearing capacity for different edge distances and shell footing apex angles. The rate of change in bearing capacity is in the range of 17%-43% for geotextile-reinforced sloped backfills, depending on the edge distance. This can be explained through the fact that the reinforcement layers cause horizontal extension of the stress zone which prevents the stress distribution into the soil depth. As a result, the overall mutual effects led to the debilitation of shell foundation efficiency. Additionally, values of D_ubc in the geotextile-reinforced state are smaller than those in the unreinforced condition, as listed in Table 4,and thus, reinforcement layers can improve the critical conditions of a foundation located near a sloped backfill.

Figure 7 Relationship between edge distance and maximum applied load for simple and shell footings on unreinforced and reinforced slope backfill

It should be pointed out that the soil movements and cracks were visually monitored during the tests to examine the failure mechanism of shell foundations located near the geotextile-reinforced slopes. Furthermore, the soil layer above the geotextiles was removed at the end of each test to investigate the deformations of the reinforcement in terms of the length and depth of the affected area. Based on the visual inspection of reinforced tests, two following conclusions can be drawn:

1) The slope failure surface is expanded to the second layer, continuing parallel with the second layer until intersecting the slope face. This finding is in agreement with the results of TAVAKOLI et al [21], which concluded crossing the failure surface around the second reinforced layer. Also, the cracks with an approximate width of 10 mm were observed in the backfill as well as the end of geotextiles layers.

2) The deformations of the influenced area over the geotextiles demonstrate that the penetration of shell footings into the sloped backfill is led to the wider stress distribution comparing to the simple footing. In other words, shell footings result in a larger failure zone, and thereby, they lead to an increase in the bearing capacity.

3.3 Effect of geotextile reinforcement

The bearing capacity ratio, i.e., R_ubc, is defined according to Eq. (2) as the ratio of ultimate bearing capacity of footings on the reinforced slope, i.e., Q_u,Re, to that of the unreinforced slope condition, i.e., Q_u,Ur. In fact, R_ubc indicates how much the shell footing performance is improved by the presence of reinforcement.

                             (2)

Figure 8 shows the reinforcement effect on the response of shell footings using the introduced R_ubc. According to the figure, installing reinforcement layers under the shell footings in the sloped backfill conditions can highly improve the bearing capacity ratio of foundations in the range of 2.4-3.0, depending on the test conditions. As a matter of fact, the geotextiles can successfully mobilize the anchorage potential using the pull-out resistance behind the sliding zone. The resistance results in stress reduction at depth and fortification of the influenced area.

Evidently, the maximum bearing capacity ratio belongs to the footings with the minimum edge distance not only because of the small value of bearing capacity in shorter edge distance for unreinforced slope condition but also because of the greater effective length of geotextile behind the sliding zone, which leads to the more mobilization of pull-out resistance. Generally, the effect of apex angle on the bearing capacity decreases when the footing gets away from the slope. In other words, increasing the apex angle decreases the failure depth; so that, the failure surface will not intersect the slope face in the greater edge distances. Hence, the increment rate of bearing capacity due to the increasing apex angle is higher for X/B=1.0 compared to the other values. Moreover, the results proved that the reduction of apex angle tends to decrease the reinforcement efficiency. Unlike the flat ground condition that the effect of apex angle on the bearing capacity is negligible, it is significantly effective in the sloped backfill conditions.

Figure 8 Bearing capacity ratio of foundations with respect to edge distance and apex angle

Furthermore, R_ubc of shell footings in sloped backfill is approached to that of the flat one when the edge distance becomes greater than 4B. This criterion for the effect of sloped backfill on the bearing capacity of the foundation was previously observed in the unreinforced slope condition. In some studies, a bearing capacity ratio of 0.9-1.0 has been reported at distances more than 4B for the flat ground having a friction angle of 35°, which implies consistency with the results of current study [35].

3.4 Effect of edge distance (X)

To investigate the bearing capacity variations influenced by edge distance, settlement factor (F_δ) is introduced in Eq. (3) as a non-dimensional parameter.

                               (3)

where δ_u is the settlement associated with the ultimate bearing capacity (m), γ is the unit weight of the soil (kN/m³), A is the area of the footing in horizontal projection (m²), and Q_u is the ultimate bearing capacity (kN). Since the unit weight of the backfill is kept constant in all tests, a lower value of the settlement factor indicates greater bearing capacity, which reveals the better performance of the foundation.

Figure 9 displays the variations of settlement factor for different apex angles of footings in both reinforced and unreinforced conditions. By getting the footings closer to the slope, without considering the apex angle and reinforcement condition, the settlement increases while the bearing capacity decreases, and thus, it leads to the greater settlement factor. In edge distances more than 4B, implying a safe distance for shell footings near the sloped backfill, the settlement factor varies with a small rate. Also, the rate of increase in weight of moving soil is greater for foundations near to the sloped edge, which results in the reduction of the bearing capacity and increase of the settlement factor.

Figure 9 Variations of settlement factor versus apex angle for all footings near reinforced and unreinforced backfill conditions

3.5 Effect of shell footing apex angle

In practical works, the influence of apex angle will depend noticeably on shell length, shell rigidity, and applied load. In this study, shell footing characteristics, including length, width, material property as well as the applied loading conditions are the same in all tests. Furthermore, the shell footing stiffness in the present study is almost the same. The stiffness is slightly higher only for the shell footing with the apex angle of 60° because of the extra height considered to avoid possible bending. Therefore, the effect of different shell footing apex angles has been assessed under comparable conditions.

The shell efficiency factor (SE, E_s) is defined in Eq. (4) to quantify how much the shell footing apex angle, representative of shell footing geometry, affects the bearing capacity of foundations in sloped and flat backfill conditions.

                          (4)

where Q_u,sf and Q_u,ff are the ultimate bearing capacity of shell and simple footings, respectively. The shell efficiency factor of all shell footings for various edge distances in reinforced and unreinforced slope conditions is shown in Figure 10. The results show that the shell footing with the apex angle of 60° is the most efficient foundation system because the failure wedge of shell foundations is formed deeper than the simple foundations one, i.e., near the slope crest. The deeper failure wedge has led to the formation of larger shear resistance zones.

Figure 10 Shell efficiency factors of all shell footings located in unreinforced and reinforced slope conditions

In Figure 11, the failure wedge for different shell foundations is presented in the unreinforced slope conditions. The visual inspection shows that the wedge angle of the failure surface is increased by decreasing the apex angle. When the apex angle changes from the 180° (flat one) to 120°, 90° and 60°, the triangular wedge angle increases from 65° to 71°, 74°, and 77°, respectively. This indicates a deeper failure mode caused by shell foundations compared to the flat ones. The results are in agreement with previous studies [16], which have pointed out the deeper formation of failure wedge surface of shell footings than that of simple footing.

Figure 11 Failure wedges induced by different shell foundations in unreinforced slope condition:

As shown in Figure 12, the wedge failure angle (α) plays the main role in the efficiency of shell foundations, which can be computed using Eq. (5) suggested by ABDEL-RAHMAN [37].

                    (5)

where φ is the friction angle of soil, and R_s is the shell ratio, as representative of the footing configuration in the vertical direction (R_s=(π+2θ)/4, θ=90-0.5i). Considering the measured wedge angles and using the equilibrium equations [37], Eq. (5) has been modified for strip shell foundations and better rewritten in the form of Eq. (6) as:

               (6)

Figure 12 Definition of wedge failure angle (a) for shell and flat footings

3.6 Empirical relationships for flat ground condition

The empirical relationships are derived from the test results. The aim is to determine the ultimate bearing capacity of triangular shell strip footings located on the flat and sloped backfills. Using the normalization of the relation between the ultimate bearing capacity of shell footings, i.e., Q_u,_shell, and the simple footings, i.e., Q_u,_simple, Eq. (7) is definable for the unreinforced and reinforced sloped backfills. In the case of flat backfills (β=180°), the edge distance is meaningless. For this reason, the equation should be modified for the flat ground condition, which is led to Eqs. (8) and (9) for estimation of the bearing capacity of shell footings located on the unreinforced and reinforced flat backfills, respectively. In Table 5, the proposed empirical and experimental results are compared for eight tests, i.e., four reinforced and four unreinforced conditions.

Table 5 Values of defined parameters for reinforced and unreinforced backfills in flat ground condition

                 (7)

R²=0.95 (8)

R²=0.94 (9)

where X/H is the ratio of edge distance to backfill height, i/β is the ratio of shell apex angle to backfill slope angle, and p_shell/p_simple is the applied stress p=W_f/A due to the footing weight W_f to shell area in horizontal projection A. Due to the small and thin cross-section of shell footing and uncertainties in stress distribution compared to the simple footings, the bearing capacity of the shell foundation can also be expressed based on forces rather than stresses [7].

Figure 13(a) shows the comparison of empirical relation results with the current experimental one in flat ground condition. Also, the comparison between the empirical relation results with experimental results of HANNA et al [14] and SHALIGRAM [15] are carried out in Figure 13(b). According to the Figures, there is a satisfactory consistency in the obtained empirical values. The reason for close results can be attributed to the normalization of vertical axis, Q_u,_shell by the associated simple footings value, Q_u,_simple.

3.7 Empirical relationships for sloped condition

Based on the defined parameters for reinforced and unreinforced sloped backfills (Table 6) and using MATLAB software, the ultimate bearing capacity of shell footing in unreinforced and reinforced sloped backfills (β=45°) can be estimated through Eqs. (10) and (11), respectively. As shown in Table 6, the proposed empirical and experimental results were compared using 32 test results, i.e., 16 reinforced and 16 unreinforced conditions.

Figure 13 Comparison between the ultimate bearing capacities obtained from experimental results and empirical relations in flat ground condition:

R²=0.83                            (10)

R²=0.87                            (11)

3.8 Effect of scale

To generalize any small-scale results to other cases, paying particular attention to the effect of scale is vital. A dimensional analysis using an extrapolation towards the prototype case was performed by TAVAKOLI et al [21] for the major physical parameters to survey the response of geogrid-reinforced slopes. The resulted equation for the bearing capacity of shell foundations located on the geotextile-reinforced slopes is Eq. (12), which can be represented in a non-dimensional form of Eq. (13).

Table 6 Defined parameters for reinforced and unreinforced backfills in sloped condition

                      (12)

               (13)

where D_r is the relative density of soil, v is the Poisson ratio, E_soil is the elasticity modulus of soil and E_geo is the elastic modulus of geotextile reinforcement. For the prototype shell footing with width (B_p) which is n times greater than that of the experimental model (B_m), Eq. (14) can be written as:

                                (14)

For instance, when a prototype shell footing has a width of 1.2 m, n will be 12. Regarding the equality of 15 non-dimensional variables in Eq. (13) for both prototype and model, the values of X, u, h, L, H and D₅₀ should be considered 12 times of the model variables. Moreover, assuming the same unit weights for the soil used at model and prototype, values of c, E_soil, E_geo and p_shell should be taken into consideration as 12 times of model parameters as well. In this regard, the following equation can be obtained for bearing capacities of the prototype (Q_u/A)_p and model (Q_u/A)_m.

       (15)

It is necessary to point out that the proposed equations are based on the superposition law and are only valid at the elastic deformation range of the soil and geotextile. To evaluate the scaling law for this system with higher accuracy, nonlinear behavior should be investigated. To do so, the stress values for both the geotextile and soil should be determined based on the consistency of deformations, and subsequently, the relevant equations can be derived.

3.9 Numerical simulation

This study also employed a finite difference method program, FLAC3D (ITASCA), to perform the numerical simulations. The three-dimensional numerical simulations were executed for triangular shell foundations with two different apex angles of 60° and 120°. A Mohr-Coulomb elastic-plastic failure criterion was considered for the sandy backfill, and an elastic behavior was assigned to the foundation. The geotextile properties presented in Table 3 were used for the simulations, and the backfill and foundation properties employed in the numerical models were summarized in Table 7. The primary boundary conditions, fixation of x-direction for lateral boundaries as well as x- and y- directions for the bottom boundary, were applied for the boundaries of the model.

Table 7 Properties used in numerical model

The 3D numerical model of foundation with an apex angle of 60° is depicted in Figure 14. For a better simulation of the stress generation, smaller elements were utilized for the soil beneath the foundation. The dimensions of sandy backfill are 1200 mm×670 mm×400 mm, and the total 3D elements are 6072. Also, two geotextile layers were modeled, as presented in Figure 15. To achieve the in-situ stresses of the model, analyses under self-weight were initially performed for each layer (lower backfill layer, first geotextile layer, middle backfill layer, second geotextile layer, and upper backfill layer), and subsequently, the displacements were set to zero while the stresses were kept at their initial values. Then, loading was applied to the foundation. The numerical simulation results in a failure wedge beneath the footing in the soil. The failure wedge can be displayed using shear strain increment contours. Figure 16 displays the failure wedge of shell footings having apex angles of 60° and 120° in unreinforced and reinforced backfill conditions. The increase in depth of the failure wedge is caused by decreasing the apex angle in which the results of previous experimental studies such as by HANNA et al [14] corroborates the conclusion. Furthermore, Figure 17 shows the comparison of the numerical simulation results of foundation with an apex angle of 60° with the experimental one, and for the foundation with an apex angle of 120°, the numerical results are compared in Figure 18. The results indicate that for a fixed footing width, an increase in the apex angle of shell footings in the unreinforced conditions leads to the reduction of bearing capacity. The reason can be attributed to the fact that any increase in the apex angle will make the footing much closer to the flat foundation conditions, which has less bearing capacity than the shell footings. Moreover, the employment of geotextile reinforcement significantly increases the bearing capacity of shell footings. The increase in the bearing capacity by decreasing the apex angle is also inferable in the reinforced conditions.

Figure 14 FLAC3D numerical mesh for shell foundation with an apex angle of 60°

Figure 15 Two geotextile layers in numerical model:

Figure 16 Failure wedge induced by numerical simulation for different shell footings located on reinforced/unreinforced slope condition:

Figure 17 Comparison between numerical and experimental results of load-settlement curves for shell foundation with an apex angle of 60° in different sloped conditions:

Figure 18 Comparison between numerical and experimental results of load-settlement curves for shell foundation with an apex angle of 120° in different sloped conditions:

4 Conclusions

In this paper, the experiments were conducted on three different shell footings and one simple footing, namely flat footing, located near unreinforced and geotextile-reinforced slopes. Based on the laboratory investigations and interpretation of the data as well as numerical modeling, the following conclusions can be drawn.

1) Locating the foundations adjacent to an earth slope will dramatically affect the bearing capacity, compared to the flat ground. In unreinforced flat ground condition (without slope), the ultimate bearing capacity is improved 8% to 18% by employing shell footings (with a reduced apex angle of 120°, 90°, and 60°) instead of simple footing (with an apex angle of 180°).

2) To improve the ultimate bearing capacity in an unreinforced slope condition, utilization of shell footings rather than a simple footing is concluded as an effective solution; so that, the improvement of the bearing capacity is in the range of 18%-56%, depending on the edge distance. Better improvement of bearing capacity belongs to a shell footing with a smaller apex angle. Moreover, by making the edge distance farther than 4B, the adverse effect of slope condition is eliminated, and the results are approaching the flat ground condition.

3) In geotextile-reinforced slope conditions, reducing the apex angle of shell foundations from 180° to 60° resulted in a continuous increase in the bearing capacity in the range of 17%-43%, depending on the edge distance. Comparisons between shell footings and simple footing in both unreinforced and reinforced backfills demonstrate the higher bearing capacity of shell footings due to the formation of a larger failure zone.

4) Application of the geotextile as soil reinforcement under the shell footings can remarkably improve the bearing capacity. In sloped backfill condition, the bearing capacity ratio of shell foundation (reinforced to the unreinforced conditions) is calculated in the range of 2.4-3.0. Therefore, geotextiles as the reinforcing material could successfully mobilize the anchorage potential, provided by pull-out resistance behind the sliding zone that leads to more stabilization of the affected area.

5) By visual inspection of the failure wedge surfaces, it is inferred that shell footings tend to resist penetration, and therefore, they extend the failure zone into the depth of soil. In the unreinforced slope condition, changing the apex angle from 180° (flat one) to 60° increases the triangular wedge angle from 65° to 77°.

6) For practical purposes, several equations were proposed for the bearing capacity of shell footings in the unreinforced and reinforced flat ground as well as sloped conditions. The accuracy of the proposed empirical equations was evaluated by comparing the results with the present experiments and some other previous experimental studies. The comparison shows an acceptable agreement.

7) The numerical simulation was successfully conducted for shell footings with two different apex angles (60° and 120°) in the unreinforced and reinforced conditions. Then, the resulted failure wedges were graphically compared using shear strain increments. The comparison indicates a remarkable improvement of bearing capacity through decreasing the apex angle of shell footing as well as the application of geotextile reinforcement layers.

Contributors

Seyed ALI GHAFFARI carried out the literature review, laboratory experiments, data gathering, and wrote the first draft of the manuscript. Elham SATTARI executed the numerical simulations and verified them with the experimental results. Amir HAMIDI designed and supervised the whole process of the investigation. Gholamhosein TAVAKOLI MEHRJARDI supervised the numerical simulations. Abtin FARSHI HOMAYOUN ROOZ edited and modified the first draft, and also took responsibility as a corresponding author. All authors have read and agreed to the published version of the manuscript.

Conflict of interest

The authors declare that there is no conflict of interest in this work.

Nomenclature

c

Cohesion intercept

φ

Internal friction angle

D₁₀

Effective grain size

D₅₀

Mean grain size

e_max

Maximum void ratio

e_min

Cohesion intercept

C_c

Curvature coefficient

C_u

Uniformity coefficient

G_s

Curvature coefficient

w

Specific gravity

B

Water content

B_m

Width of model shell footing

B_p

Width of prototype shell footing

i

Apex angle of footing

W_f

Weight of footing

E_soil

Elasticity modulus of soil

E_geo

Elasticity modulus of geotextile

H

Backfill height

β

Backfill slope angle

X

Edge distance to footing centerline

α

Wedge failure angle

θ

Angle between shell’s edge and ground

u

Embedded depth of the first reinforcement layer

L

Length of reinforcement

h

Distance between reinforcement layers

A

Area of the footing in horizontal projection

n

Size ratio of prototype and model footing

p_shell

Stress induced by weight of shell footing

p_simple

Stress induced by weight of simple footing

E_s

Shell efficiency

R_s

Shell ratio

F_δ

Settlement factor

δ_u

Settlement at ultimate bearing capacity

Q_u

Ultimate bearing capacity

Q_u,_shell

Ultimate bearing capacity of shell footing

Q_u,simple

Ultimate bearing capacity of simple footing

D_ubc

Bearing capacity decrease in unreinforced condition

R_ubc

Bearing capacity ratio

Q_u,f

Bearing capacity in flat ground

Q_u,_s

Bearing capacity in sloped ground

Q_u,_ff

Bearing capacity of simple footing

Q_u,_sf

Bearing capacity of shell footing

Q_u,_Re

Bearing capacity in reinforced slope condition

Q_u,_Ur

Bearing capacity in unreinforced slope condition

References

[1] CHEN C F, DONG W Z, TANG Y Z. Seismic ultimate bearing capacity of strip footings on slope [J]. Journal of Central South University of Technology, 2007, 14: 730-736. DOI: http://doi.org/10.1007/s11771-007-0139-4.

[2] YANG X L, WANG Z B, ZOU J F, LI L. Bearing capacity of foundation on slope determined by energy dissipation method and model experiments [J]. Journal of Central South University of Technology, 2007, 14: 125-128. DOI: http://doi.org/10.1007/s11771-007-0025-0.

[3] NAEINI S A, KHADEM R B, MAHMOODI E. Bearing capacity and settlement of strip footing on geosynthetic reinforced clayey slopes [J]. Journal of Central South University, 2012, 19: 1116-1124. DOI: http://doi.org/ 10.1007/s11771-012-1117-z.

[4] LU L, WANG Z J, ARAI K. Numerical and experimental analyses for bearing capacity of rigid strip footing subjected to eccentric load [J]. Journal of Central South University, 2014, 21: 3983-3992. DOI: http://doi.org/10.1007/s11771-014-2386-5.

[5] ATTARZADEH A, GHANBARI A, HAMIDI A. A study of bearing capacity of shallow foundations next to sand slope with experimental models [J]. Journal of Engineering Geology, Kharazmi University, 2015, 9(1): 2695-2710. (in Persian)

[6] KURIAN N P, JAYAKRISHNA DEVAKI V M. Analytical studies on the geotechnical performance of shell foundations [J]. Canadian Geotechnical Journal, 2005, 42(2): 562-573. DOI: http://doi.org/10.1139/t04-110.

[7] KURIAN N P. Shell foundations: Geometry, analysis, design and construction [M]. Harrow, UK: Alpha Science International, 2006.

[8] YAMAMOTO K, LYAMIN A V, ABBO A J, SLOAN S W, HIRA M. Bearing capacity and failure mechanism of different types of foundations on sand [J]. Soils and Foundations, 2009, 49(2): 305-314. DOI: http://doi.org/10.3208/sandf.49.305.

[9] FERNANDO N, SENDANAYAKE E, SENDANAYAKE D, de SILVA N. The experimental investigation of failure mechanism and bearing capacity of different types of shallow foundations [R]. Sri Lanka: Department of Civil Engineering, University of Moratuwa, 2011.

[10] KURIAN N P, MOHAN C S. Contact pressures under shell foundations [C]// Proceedings of 10th International Conference on Soil Mechanics and Foundation Engineering. Stockholm, Sweden, 1981: 165-168.

[11] ESMAILI D, HATAF N. Experimental and numerical investigation of ultimate load capacity of shell foundation on reinforced and unreinforced sand [J]. Iranian Journal of Science and Technology, 2008, 32(5): 491-500. DOI: http://doi.org/10.22099/ijstc.2008.726.

[12] THILAKAN S, NAIK N. Geotechnical behaviour of shell foundations [C]// Proceedings of 50th Indian Geotechnical Conference. Pune, India, 2015: 1-9.

[13] HANNA A, ABDEL-RAHMAN M. Ultimate bearing capacity of triangular shell strip footings on sand [J]. Journal of Geotechnical Engineering, 1990, 116(12): 1851-1863. DOI: http://doi.org/10.1061/(ASCE)0733-9410(1990)116:12 (1851).

[14] HANNA A, ABDEL-RAHMAN M. Experimental investigation of shell foundations on dry sand [J]. Canadian Geotechnical Journal, 1998, 35(5): 847-857. DOI: http://doi.org/10.1139/t98-049.

[15] SHALIGRAM P S. Behavior of triangular shell strip footing on georeinforced layered sand [J]. International Journal of Advanced Engineering Technology, 2011, 2(2): 192-196.

[16] AZZAM W R, NASR A M. Bearing capacity of shell strip footing on reinforced sand [J]. Journal of Advanced Research, 2015, 6(5): 727-737. DOI: http://doi.org/10.1016/j.jare.2014.　04.003.

[17] FATTAH M Y, WARYOSH W A, AL-HAMDANI M A E. Investigation on the behavior of conical shell foundations composed of reactive powder concrete embedded on sandy soil [J]. Advances in Structural Engineering, 2015, 18(11): 1859-1873. DOI: http://doi.org/10.1260/1369-4332.18.11.　1859.

[18] FATTAH M Y, WARYOSH W A, AL-HAMDANI M A E. Experimental and theoretical study on bearing capacity of conical shell foundations composed of reactive powder concrete [J]. Acta Geodynamica et Geomaterialia, 2015, 12(4): 1–16. DOI: http://doi.org/10.13168/AGG.2015.0037.

[19] EL SAWWAF M A. Behavior of strip footing on geogrid-reinforced sand over a soft clay slope [J]. Geotextiles and Geomembranes, 2007, 25(1): 50-60. DOI: http://doi.org/ 10.1016/j.geotexmem.2006.06.001.

[20] ALAMSHAHI S, HATAF N. Bearing capacity of strip footings on sand slopes reinforced with geogrid and grid-anchor [J]. Geotextiles and Geomembranes, 2009, 27(3): 217-226. DOI: http://doi.org/10.1016/j.geotexmem.2008.11. 011.

[21] TAVAKOLI M G, GHANBARI A, MEHDIZADEH H. Experimental study on the behaviour of geogrid-reinforced slopes with respect to aggregate size [J]. Geotextiles and Geomembranes, 2016, 44(6): 862-871. DOI: http://doi.org/ 10.1016/j.geotexmem.2016.06.006.

[22] VAHIDIPOUR A, GHANBARI A, HAMIDI A. Experimental study of dynamic compaction adjacent to a slope [J]. Proceedings of the Institution of Civil Engineers- Ground Improvement, 2016, 169(2): 79-89. DOI: http://doi.org/10.1680/grim.14.00007.

[23] JAWAD F W, FATTAH M Y. Finite element analysis of the bearing capacity of footings nearby slopes [J]. International Review of Civil Engineering, 2019, 10(1): 1-7. DOI: http://doi.org/10.15866/irece.v10i1.16365.

[24] FATTAH M Y, AL-HADDAD S A, HAMEEDI M. Load carrying capacity of rectangular foundation on geogrid reinforced sloped sandy soil [C]// IOP Conference Series Materials Science and Engineering, Fourth International Conference on Buildings, Construction and Environmental Engineering – BCEE4. Istanbul, Turkey, 2019: 1-9.

[25] SELIG E T, MC KEE K E. Static and dynamic behaviour of small footings [J]. Journal of the Soil Mechanics and Foundation Division, 1961, 87(6): 29-50.

[26] CHUMMER A V. Bearing capacity theory from experimental results [J]. Journal of the Soil Mechanics and Foundations Division, 1972, 98(12): 1311-1324.

[27] DASH S K, BORA M C. Improved performance of soft clay foundations using stone columns and geocell-sand mattress [J]. Geotextiles and Geomembranes, 2013, 41: 26-35. DOI: http://doi.org/10.1016/j.geotexmem.2013.09.001.

[28] ASTM D 4253. Standard test methods for maximum index density and unit weight of soils using a vibratory table [S].

[29] ASTM D 4254. Standard test methods for minimum index density and unit weight of soils and calculation of relative density [S].

[30] FATTAH M Y, AL-BAGHDADI W, OMAR M, SHANABLEH A. Analysis of strip footings resting on reinforced granular trench by the finite element method [J]. International Journal of Geotechnical Engineering, 2010, 4(4): 471-482. DOI: http://doi.org/10.3328/IJGE.2010.04.04.471-482.

[31] ALTALHEA E B, TAHA M R, ABDRABBO F M. Bearing capacity of strip footing on sand slopes reinforced with geotextiles and soil nails [J]. Jurnal Teknologi, 2013, 65(2): 1-11. DOI: http://doi.org/10.11113/jt.v65.2184.

[32] ASTM D 2487. Standard practice for classification of soils for engineering purposes (unified soil classification system) [S].

[33] COLMENARES J E, KANG S R, SHIN Y J, SHIN J H. Ultimate bearing capacity of conical shell foundation [J]. Structural Engineering and Mechanics, 2014, 52(3): 507-523. DOI: http://doi.org/ 10.12989/sem.2014.52.3.507.

[34] HUANG C C, KANG W W. The effects of a setback on the bearing capacity of a surface footing near a slope [J]. Journal of GeoEngineering, 2008, 3(1): 25-32. DOI: http://doi.org/ 10.6310/JOG.2008.3(1).3.

[35] SHUKLA R P, JAKKA, R S. Critical setback distance for a footing resting on slopes [J]. Acta Geotechnica Slovenica, 2017, 14(2): 19-31.

[36] SHUKLA R P, JAKKA R S. Critical setback distance for a footing resting on slopes under seismic loading [J]. Geomechanics and Engineering, 2018, 15(6): 1193-1205. DOI: http://doi.org/10.12989/gae.2018.15.6.1193.

[37] ABDEL-RAHMAN M. Geotechnical behavior of shell foundations [D]. Concordia University, 1996.

(Edited by FANG Jing-hua)

中文导读

土工织物加筋土坡附近带状壳体的基础承载力研究

摘要：从经济角度来看，城市化经常迫使居民在斜坡附近建造上层建筑，但是这增加了建筑物失稳的风险。本文通过一系列小规模的板载试验，在沙质边坡中安装土工织物来加固层，并研究减小三角壳条形基础的顶角的影响。结果表明，使用土工织物增强层并减小顶角对壳基础的极限承载力具有显著效果。随着距斜坡地基距离的延长，斜坡的不利影响减小。但是，随着距离的缩短，增强效果和顶角增大。对于实际应用，本文提出了经验方程式用于确定基础的极限承载力和比例效应。最后，进行3D数值模拟并与实验结果进行比较。

关键词：边坡；壳体条形基础；土工织物增强；顶角；极限承载力；数值模拟

Received date: 2020-04-16; Accepted date: 2020-11-22

Corresponding author: Abtin FARSHI HOMAYOUN ROOZ, Master, Geotechnical Engineer; Tel: +98-9126509425; E-mail: std_abtinfhr@alumni.khu.ac.ir; ORCID: https://orcid.org/0000-0002-3176-3188