1 Introduction

Faced with the massive growth that the Civil Engineering area has experienced in recent decades, construction on weak soils has been unavoidable. A known weakness of the soil is the lack of tensile strength, which makes it necessary to reinforce it for adequate performance in certain uses such as ground improvement techniques for soil retaining walls, slopes, foundations, and pavements, which use geogrids and other geosynthetics to improve the system performance [1, 2]. Despite the high cost of geogrids, they have gained popularity for their wide range of applications in soft soils, but it is important to consider the type of material, geogrid type, and aperture size of the mesh, as many researchers showed that it plays an important role in the performance of mechanically stabilized pavements [3]. The reinforcement of foundation soils, using geogrids, has been a subject of research in the last three decades, and according to the studies carried out, the main control parameters that must be considered when executing tests on soils reinforced with geogrids are the depth, number, and dimension of the geogrid layers [4, 5]. However, one of the main parameters was the optimum depth of the first geogrid reinforcement, which ranged between 0.10 and 0.25 times the footing width to generate an increase in BCR [5]. Furthermore, Lal et al. [5] mentioned that the use of more than three layers of reinforcement was no significant improvement. The effective length of the geogrid to be placed to achieve maximum load capacity is in a range of 3–6 times the width of the footing, and the optimal number of layers for reinforcement is 3–4, as long as a uniform arrangement of them is used [6]. Previous studies have confirmed that, from an engineering point of view, the response of the geogrid-reinforced soil is directly influenced by the type of grains of the soil, the characteristics of the geogrid, and the type of surface load [7]. To understand the behavior of soils in engineering and construction, it is important to understand the concept of effort within a soil mass and the way in which it acts. Generally, the stress on a point within the soil mass is not the same in all directions. For this reason, it is imperative to study the state of the stresses on the ground and their relations acting in different directions, however, the focus of most research on bearing capacity and foundation settlements is based on the load applied vertically. [8, 9].

Bearing capacity is the maximum stress that a foundation-level soil can withstand without failing, so it is considered an important factor for the design of foundations and retaining walls; because if the soil cannot resist, all structures with shallow foundations above it will fail [10]. Faced with this, the implementation of geosynthetics, specifically the use of geogrids, allows the creation of a Geogrid-Soil interface capable of receiving, transmitting, and distributing loads uniformly, increasing the bearing capacity of the soil [2]. Due to the combination of geogrids with soil creates a new material with improved mechanical properties resulting in a higher tensile strength in the soil, which is inherently low [1].

The use of geogrids in pavements, as reinforcement for pavement layers, provides lateral restraint that supports the base and subbase material, this type of reinforcement can help reduce the thickness of the aggregate layer [11]. Moreover, the use of geogrids offers several benefits such as the improvement of surface deformation characteristics to arrest downward deflections, the increase of bearing capacity and the control of the rutting of pavements under traffic loads when it is used in soft subgrades, and the increment of the pavement’s service life arresting longitudinal cracks [11,12,13]

Furthermore, Badakhshan et al. [14] studied the effect of the use of geogrids in footings, in which the researchers demonstrated that the shape of the footing and number of reinforcement layers, directly influenced the bearing capacity soil, in which the bearing capacity ratio (BCR) increases in a circular footing respect to a square footing. Moreover, it was shown that the point of application of the load influences the BCR and the deformation of the soil, where the use of geogrids decreases the settlement and inclination of the footings when an eccentric load is applied. In addition, Badakhshan et al. [15], studied the load-settlement response on a square base, the researchers showed that the depth of the geogrid reinforcement layer and the presence of a reinforced aggregate layer on top of the sand directly influenced the BCR value, in which the presence of an aggregate layer increases the BCR, while for a sand deposit, the effective depth was equal to 0.45 times the width of the base.

Consequently, the available conventional methods such as cement or lime stabilization or fiber reinforcement consume a lot of time and money, which is why several studies have demonstrated that its use is an economic, ecological, and easy-to-construct alternative to traditional techniques. [12, 16,17,18,19,20]. Furthermore, its installation on-site is simple, even in unfavorable weather conditions [6].

2 Research objectives

The large-scale laboratory tests, such as Saride and Baadiga [21], are useful to reproduce field stress conditions for soil [22]. Consequently, we use different arrangements of geogrids to determine the resistance to the uniaxial load of the soil.

Hence, the density and compaction mechanism of the soil were significant parameters for the analysis prior to developing the test. Furthermore, the use of different arrangements of geogrids is important to determine the best arrangement based on the resistance to the uniaxial load of the soil.

With the results obtained the effects of each geogrid distribution and type of geogrid were evaluated with respect to the applied uniaxial load, determining the optimum arrangement and type of geogrid according to the higher soil resistance.

The article consists of the following six sections: Sect. 3 presents the materials used in the investigation, classification, mechanical parameters, laboratory tests, and the characteristics of the geogrids used. Section 4 presents the test setup, laboratory test models, description of the equipment used, reinforcement distribution, test procedure, and axial load application. Section 5 presents the results obtained from the verification of density, maximum stress, and bearing capacity ratio. These results are discussed in Sect. 6, where an analysis of the type of failure is presented, as well as an incremental analysis of the bearing capacity based on numerical, theoretical, and physical models. In addition to the justification of the best geogrid arrangement used. Finally, the conclusions are detailed in Sect. 7.

3 Materials

3.1 Granular soil

The soil used in this research was obtained from San Antonio, Pichincha province’s quarry on the northern side of Ecuador. The San Antonio quarry is in the coordinates 0° 00′ 25.99″ N; 78° 28′ 20.62″ W. The quarry has more than 10 million cubic meters of material, and the gravel has an abrasion resistance of 34% [23]. This granular material is sold by the name of fine sand, and its properties are presented in Table 1, while the particle size distribution is presented in Fig. 1.

Table 1 Summary of soil properties
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Fig. 1
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Grain size distribution curve of the soil

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3.2 Direct shear test

The Consolidated Drained direct shear test to obtain the friction angle and the cohesion value was carried out on a sample of the soil from the mold after the Axial Load Test in a controlled relative humidity (55%) and temperature (19 °C) environment, with the objective that the specimen maintains its moisture content during the test, which was based on the ASTM D3080 [27]. A summary of properties that were obtained from the Consolidated drained shear test was presented in Table 2, while the direct shear test results, for unsaturated conditions, are presented in Fig. 2.

Table 2 Summary of soil mechanical properties of the consolidated drained direct shear test
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Fig. 2
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Direct shear test results, unsaturated condition

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Under the same test criteria, the soil was analyzed in a saturated consolidated-drained condition, originally the soil sample was obtained from the mold after the Axial Load Test and tested under the same environmental conditions as the unsaturated condition. The test was run to obtain the cohesion parameters and internal friction angle of particles to be used in the calculation of ultimate load capacity with the use of numerical modeling as shown in Sect. 5. A summary of the soil mechanical properties for a saturated consolidated-drained condition was presented in Table 3, while the direct shear test results, for unsaturated conditions, are presented in Fig. 3.

Table 3 Summary of soil mechanical properties of saturated consolidated-drained condition
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Fig. 3
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Direct shear test results, saturated condition

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3.3 Geogrid reinforcement

Two types of polypropylene geogrids were used as soil reinforcement, as shown in Fig. 4. The specifications of B × 1100 Biaxial geogrid were presented in Table 4, and the specifications of T × 5 Multiaxial geogrid were presented in Table 5.

Fig. 4
figure 4

a Biaxial geogrid, b multiaxial geogrid

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Table 4 Technical characteristics of B × 100 geogrid, [28]
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Table 5 Technical characteristics of Tx5 geogrid, [31]
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4 BCR determination in the laboratory

4.1 Test setup/laboratory model tests

To compact the soil, two types of plates were used (Fig. 5), a wooden plate that was placed on the surface of the soil layer to compact it and reach a density close to the maximum dry density, and a metal plate that simulates a square foundation. The wooden board consists of the union of 6 Triplex boards bolted together. The dimension of the wooden plate is 48 cm × 48 cm. On the other hand, the metal plate has dimensions of 7 cm × 7 cm × 2 cm.

Fig. 5
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a Wooden plate, b metal plate

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The axial load application system is shown in Fig. 6. After an iterative process, the number of layers to obtain uniform compaction was determined to be 7, where the first 4 layers are 7 cm thickness each and the remaining 3 layers are 2.1 cm thickness each. The geogrid was placed between the last three layers as shown in Fig. 6c. After all the layers were compacted, the axial load was applied to the metal plate.

Fig. 6
figure 6

a Diagram of the test models, b cut view of the diagram of the test models, c example of the distribution of soil layers and geogrids

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The system presented in Fig. 6, represents the experimental model that was used in this research, it was composed of a steel box, a universal testing machine, soil layers, and geogrids. The steel box had a square cross-section of 48 cm width and a height of 48 cm (Fig. 6a). It was used to restrain lateral and horizontal deformation, setting a free vertical deformation during the test. In addition, it was used to compact each of 7 soil layers by compression through a Universal Testing Machine, reaching a soil height of 34.3 cm inside the box (Fig. 6b). Furthermore, the geogrids were placed between the last 3 layers separated by 2.1 cm from each other (Fig. 6c).

4.2 Deluxe super “L”: universal testing machine of 2000 kN

The Universal testing machine presented in Fig. 7, is commonly used for tension or compression testing of materials and has a capacity of 2000 kN. It consists of four circular metal supports separated by a distance (A) of 762 mm from each other, with two thick heads with a length of 267 mm (C) located along the supports. In the intermediate head, the metal load pin is placed allowing the coupling of different plates. It should be noted that the machine is used to perform the soil compaction process and load axial test. For the compaction process, the main head was in contact with the wooden plate to perform the compaction of the soil, and for the axial test, the main head was in contact with the metal plate which simulates a square foundation. Thus, the series of loads applied that allow it to reach the desired soil density are indicated in the subsequent paragraphs.

Fig. 7
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Deluxe super “L”: universal testing machine

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4.3 Reinforcement distribution

Considering that the depth of the reinforcement zone beneath a square foundation is only effective up to around twice its width and that adding reinforcing layers beyond that depth will not result in a meaningful increase in the foundation's bearing capacity [6], the maximum depth of the implemented geogrids layers was set to 6.3 cm (6.3 cm < 2B = 14 cm).

Based on previous studies developed by [6, 32] who conducted a series of laboratory model footing tests on sand reinforced with different types of geosynthetics and multiple numbers of layers “N”, they determined that the optimum number of layers for higher capacity is 3. Therefore, three layers of geogrid were used in all types of reinforcement, with a vertical spacing of 0.3B (2.1 cm), where B is the width of the metal plate under consideration. The first distribution, Fig. 8a, indicates the adoption of the conventional geogrid distribution model, where the layers have the same dimension and are labeled as the Uniform geogrid distribution. The width of the reinforcement was equal to 3B in both directions, a value that is within the allowed range (3B–6B) to achieve maximum load capacity [32].

Fig. 8
figure 8

a Uniform distribution of the geogrid reinforcement, b trapezoidal distribution of the geogrids reinforcement, c inverse trapezoidal distribution of the geogrids reinforcement

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The second distribution, Fig. 8b, was a Trapezoidal one, where the length of the geogrid layers increases with depth. The first geogrid layer had a length of 4B, the second layer a length of 3B, and the final layer with a length of 1.4B [6].

The third configuration, Fig. 8c, consisted of an Inverted Trapezoidal Distribution, which is an inverse arrangement to the second distribution. The first layer of geogrid has a length of 1.4B, the second layer a length of 3B, and finally the third layer a length of 4B.

4.4 Test procedure

To obtain a homogeneous surface for the wood plate during the compaction process, it was necessary to eliminate the part of soil retained in sieve No. 4, because this part caused an irregular surface to distribute the load. Therefore, the predominant part of the soil, which is less than 0.475 mm and represented 99.43% of the material, was used for this study. Second, the moisture content of the soil has been increased to 14.32% because the soil is compacted on the wet side of the compaction curve to avoid dust and facilitate compaction. To obtain the moisture content of the soil the ASTM D2216 standard test methods for laboratory determination of water (Moisture) content of soil and rock by mass was used [33].

The necessary material for each layer was calculated and separated according to the required density of the soil and the volume of each layer, using Eq. (1). A summary of the results of the wet mass used for each layer was presented in Table 6.

Table 6 Wet mass for each layer
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Equation to determine the mass of material for each layer.

$$d = \frac{m}{V}$$
(1)

A liquid release agent is applied to the mold walls before inserting the soil in the metal box; in this case, a glycerin-based releasing agent was employed to avoid adhesion between the soil and the mold and possible contamination of the soil. The material is then placed in a metal box, which is then placed in the Universal Machine Fig. 6a.

4.4.1 Replication of soil layers

4.4.1.1 Layer 1st–4th

Once the material is ready for the first layer, it is placed in the mold and its surface is flattened, so the material is well distributed. In addition, a bubble level is used to verify that the surface of the layers is not tilted. Subsequently, the wooden plate is placed on top of the soil layer, which is to be compacted, applying a load that gradually increases from 300 to 900 kN. This type of load was defined during the iterative process to define the number of layers to obtain a uniform compaction. Finally, the compacted layer is flattened again, and its inclination is verified. This procedure is the same for the 2nd, 3rd, and 4th layers.

4.4.1.2 Layer 5–7th

According to the distribution of the geogrid used, the reinforcement is positioned before placing the layer of soil, and the geogrid must be attached to the ground with small metal clips. Once the geogrid is placed, the material is deposited in the mold, and it is flushed. Similarly, the wooden plate is placed on the top of the soil layer to be compacted, and then was applied a gradual load from 900 to 1000 kN, which was defined in the same way as mentioned for the 1st–4th layers. Finally, the layer tilt level is verified, and the procedure is repeated for the 6–7th layers.

4.5 Axial load test

A vertical load that increases over time is applied until the soil-geogrid system reaches failure. The load is transmitted to the soil specimen using a metal plate with dimensions of 7 cm × 7 cm × 2 cm. Load data and vertical plate displacement are registered for later analysis.

5 Test results

5.1 Sample density

A measurement of the density of the soil is performed after the load test to ensure that the soil tested reached the densities obtained in the Modified Proctor test. The determination of the density for each soil sample was performed under the procedure of ASTM D7263, the results are shown in Table 7 [34].

Table 7 Post test density verification
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5.2 Axial load test

Figure 9 presents the variation of the displacement with the application of the stress load under the uniaxial load criterion, in all the proposed geogrid distributions. The displacement refers to the vertical distance that the metal plate penetrates the soil by the application of the load.

Fig. 9
figure 9

Stress (kPa) versus displacement (mm)

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The displacement reached over 70 mm (Fig. 9). However, 3 additional analyses were developed to determine the increase in bearing capacity. The first analysis considers the peak of stress for each analysis, which took place between 5 and 10 mm, as seen in Fig. 10a. The second analysis considers the s/B ratio equal to 10%, in which the BCR was determined based on the stress corresponding to 7 mm of displacement for each analysis (Fig. 10a). The third analysis considers the s/B ratio equal to 20%, in which the BCR was determined based on the stress corresponding to 14 mm of displacement for each analysis (Fig. 10b). The last two analyses were developed due to an observable slight fall in a value of s/B equal to 15% (Fig. 11), which is also observed in the study presented by Ahmad et al. [35]. This behavior may occur due to a non-100% uniform compaction of the layers and a possible variation between each layer.

Fig. 10
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Stress (kPa) versus displacement (mm)

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Fig. 11
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S/B (%) versus stress (kPa)

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The results obtained from the maximum stress, BCR, and BCR increment are shown in Tables 8, 9 and 10. The settlement ratio (s/B, %) is defined as the ratio of the footing settlement (s) to the footing width (B). A comparison of the s/B ratio against the applied stress is shown in Fig. 11.

Table 8 Maximum experimental stresses and bearing capacity ratio (BCR) for the first peak of each analysis
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Table 9 Maximum experimental stresses and bearing capacity ratio (BCR) for s/B = 10% of each analysis
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Table 10 Maximum experimental stresses and bearing capacity ratio (BCR) for a displacement of s/B = 20% of each analysis
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6 Axial load test

6.1 Numerical modeling

The development of the 2D model was realized based on the research presented by M. Abu-Farsakh et al. [36] and Zheng et al. [37], which indicates that 3D simulations are slightly smaller than the 2D simulations, however, both generally agreed with the field measurements [36, 37]. Furthermore, the simulation results of Shen et al. [38] indicate that 2D plane strain conditions are more conservative than 3D conditions due to lateral facing displacements from 2D analysis being permitted for opposite sides of the mini piers but for all sides in 3D. Nevertheless, it is recommended that 3D modeling is performed in further research to obtain more precise results for these types of geogrid arrangements. The experimental and theoretical results were compared with results obtained in a 2D model of the test realized in the Optum G2 program. Optum G2 is a finite element program for the analysis of the strength and deformation of geotechnical problems. The program was chosen due to the variety of calculation methods, which allow to find the upper and lower limits of the real answer, or the approximate answer itself, using a Limit Analysis. The Limit Analysis allows a quick analysis of the load capacity of geostructures (OptumG2, 2014).

The Optum G2 program allows the modeling of different structural elements such as reinforcements or geomaterials (OptumG2, 2014). A simulation between two different materials can be performed and the interaction of the soil with a reinforcement material analyzed. Specifically in our research, a limit analysis was used through the application of the Gaussian 15 Nodes Theory as shown in Fig. 12. The soil modeling properties that were used, are presented in Table 11, while the results and the ultimate bearing capacity (BCR) increase calculated for the numerical modeling results are presented in Table 12.

Fig. 12
figure 12

Optum G2, numerical model in different conditions, a generalized 2D model, b mesh and deformation of the soil without reinforcement, c mesh and deformation of the soil with uniform distribution reinforcement of geogrids, d mesh and deformation of the soil with trapezoidal distribution reinforcement of geogrids, e mesh and deformation of the soil with inverse trapezoidal distribution reinforcement of geogrids

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Table 11 Summary of soil modeling properties used in the numerical model
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Table 12 Maximum load results, numerical modeling stress, and results of ratio of the numerical modeling load increase
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7 Discussion

7.1 Type of soil failure

Using the fault mode diagram (Fig. 13), according to the “index of compactness for foundations on sand” proposed by Vesić [39], retrieved from Abul-Haija et al. [40], a relative density of 0.88 and a \(\frac{Df}{B}\) ratio that starts from zero (the sample does not have an initial depth from the level + 0.00), the type of fault that is generated below the foundation is a “Local Shear Failure”.

Fig. 13
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Failure Modes of a Foundation. Taken from: Abul-Haija et al. [40] p. 18

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7.2 Increment of the bearing capacity

For the experimental tests, Fig. 14 shows the trapezoidal distribution showed a greater increase in the ultimate bearing capacity of the soil when compared to both types of geogrids: biaxial and multiaxial geogrid. Additionally, the biaxial geogrid showed a 3% increase over the multiaxial geogrid when a Trapezoidal arrangement is used. When a uniform and reverse trapezoidal distribution pattern was adopted, the multiaxial geogrid had a larger increase in the soil's load capacity.

Fig. 14
figure 14

Load capacity increase (BCR%)-experimental results based on the peak on each stress–strain curve

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As shown in Fig. 15 for a s/B ratio equal to 10%, the arrangement that presented an increase in bearing capacity using biaxial geogrids was the trapezoidal distribution presenting an increase of 30%. While for multiaxial geogrids, the analysis presents two options for the best arrangement. The first is the Uniform Distribution which presented an increase of 32% and the Trapezoidal Distribution with an increase of 31%. Hence, as the BCR values are close between the two arrangements of multiaxial geogrids, the trapezoidal distribution is considered the best as in biaxial geogrids.

Fig. 15
figure 15

Load capacity increase (BCR%)-experimental results based on a s/B = 10%

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For an s/B ratio equal to 20%, Fig. 16 presents that the best arrangement for biaxial and multiaxial was the trapezoidal distribution with an increase of 56% and 81% respectively. These increases are bigger than the values present in Fig. 15 with a difference of 26% for biaxial geogrids, while for the multiaxial geogrids, the difference is 50%.

Fig. 16
figure 16

Load capacity increase (BCR%)-experimental results based on a s/B = 20%

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As for the numerical modeling results, presented in Fig. 17, it can be said that the increase in bearing capacity of the numerical models for the trapezoidal and inverted trapezoidal distribution are close, while according to the results of the numerical model, the best arrangement is the uniform distribution.

Fig. 17
figure 17

Load capacity increase (BCR%)-numerical Modeling Results

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7.3 Justification of the best arrangement

The shape of the failure shown in Fig. 18a.1 was obtained from Terzaghi’s model (1943), using the friction angle “\(\varnothing\)” and the width “B” of the foundation, which was presented in Fig. 18a.2.

Fig. 18
figure 18

a.1 Local failure for ground cutting; measured in cm, a.2 failure of bearing capacity in soil under a foundation. Terzaghi's model [41], from Jabbar et al. [42], b uniform distribution of geogrids, c trapezoidal distribution of geogrids, d inverted trapezoidal distribution of geogrids

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Once the soil-geogrid failure has been identified under the application of a 7 × 7 cm area load, it is observed that the implementation of the geogrid reinforcement using a Trapezoidal Distribution presents a major improvement in the load capacity of the soil used. This is because geogrids are better distributed based on the type of fault produced, which is a “General Failure by Shear”.

The theoretical results were calculated by Sharma Method [43] and Binquet and Lee Method [44], the equations to determine the increment of bearing capacity are shown below:

Sharma method: Equation to determine the bearing capacity by the Sharma Method

$$\Delta T = \mathop \sum \limits_{i = 1}^{N} \frac{{4*T_{i} *\left[ {u + \left( {i - 1} \right)*h} \right]}}{{B^{2} }}$$
(2)

where N: number of reinforcement layers. Ti: tensile strength, kN/m. B: footing width, m. u: the distance between the footing and the first reinforcement layer, m.

Binquet and Lee Method: Equation to determine the bearing capacity by the Binquet and Lee Method.

$$\Delta T = \left( {\frac{N*T}{{J*B - I*\Delta h}}} \right)$$
(3)

where N: number of reinforcement layers. T: tensile strength, kN/m. B: footing width, m. u: the distance between geogrids, m. J, B: a dimensionless factor obtained from the following graphic (Fig. 19), using the relationship z/B.

Fig. 19
figure 19

Dimensionless factors of I, M, J, obtained from Binquet et al. [44]

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Based on the results produced by the theoretical procedures, it is observed that the Sharma Method [43] is more conservative than the Binquet and Lee Method [44], with 13.5% less ultimate capacity in the results. In addition, a similarity was found between the results obtained by the Sharma Method, the numerical modeling, and the experimental results.

8 Conclusions

Performing a numerical model offers the possibility of knowing an approximate value of the load increase, which can be used for the pre-design of a structure. Additionally, a simulation of the behavior of the test samples was performed in the numerical model, so the displacement of the soil with the geogrid’s reinforcement can be observed and analyzed.

As shown in Fig. 20, it was found that the best distribution of geogrids was the Trapezoidal Geogrid Distribution which provides an increase in the load capacity of the soil by 36% when the biaxial geogrid is used and 33% when the multiaxial geogrid is used. Furthermore, the Trapezoidal geogrid distribution is more economical than a Uniform Distribution as less material is used, and the geogrids are better distributed based on the type of fault produced, which is a “General shear failure”.

Fig. 20
figure 20

Summary of bearing capacity ratios (BCR) for the physical models, numerical models, and theoretical equations based on the peak

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In addition, Fig. 20 shows that the numerical model underestimated the BCR by 8%, obtaining 28% in the trapezoidal model versus the trapezoidal biaxial physical model with 36% BCR. In the case of theoretical equations, the Binquet and Lee equation, 1975, overestimated the BCR by 13%, reaching 49%, while the Sharma equation, 2009, is the closest to the physical model, with 29% BCR. Additionally, if an analysis is carried out from an economic and environmental point of view, using the trapezoidal arrangement saves 7% of material compared to the use of a uniform arrangement.

Based on the analysis which considers different s/B ratios of 10% and 20%. Figure 21 presents that the physical analysis for an s/B ratio of 10% is close to numerical models which had an average increase of 30%, with a maximum difference of 21% and a minimum difference of 1% for all types of analyses. However, it is important to show that all physical models for this s/B ratio were close to the bearing capacity increase calculated with the Theoretical equation of Sharma, [43]. Moreover, the trapezoidal distribution was considered the best arrangement, because it presented the highest values in physical models for the two types of geogrids. Furthermore, this distribution for biaxial geogrids presented a BCR that was closer to the BCR obtained from the numerical model for the same type of geogrid (biaxial) and to the BCR obtained from Sharma's equation.

Fig. 21
figure 21

Summary of bearing capacity ratios (BCR) for the physical models, numerical models, and theoretical equations, based on s/B ratios of 10% and 20%

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Furthermore, Fig. 21 shows the physical models considering an s/B ratio of 20% present the highest values for multiaxial geogrids in comparison to the others. Meanwhile, the physical models for biaxial geogrids were slightly close to the bearing capacity increment calculated with the Theoretical equation of Binquet and Lee [44]. Nevertheless, the BCR results considering this s/B ratio were higher than the results obtained from numerical models with a maximum difference of 53% and a minimum of 7%, which is a deviation of the physical models that can be caused by the small dimensions of the footing and that must be investigated in future research. On the other hand, Fig. 21 showed the maximum values of BCR in physical models correspond to the Trapezoidal arrangement of s/B equal to 20%. Obtaining a difference of 1% for biaxial geogrids and 25% for multiaxial geogrids with respect to s/B equal to 10%.

In comparison with similar research performed by [45], where the inverse trapezoidal layout was found to yield the greatest BCR values, attributed to the fact that putting lengthy layers of reinforcement at upper levels near the footing base may lead to higher bearing capacity improvement; and the least BCR values were found for trapezoidal layout, considered as an indication that placing small size layers in upper levels of sand bed may yield low reinforcement efficiency [45], we found that the trapezoidal arrangement, for both biaxial and multiaxial geogrids, leads to a higher BCR. Considering that the tests were carried out under existing standards and procedures, and the test procedures were controlled, this variation could probably be explained by considering that a trapezoidal arrangement covers a larger fault surface, as can be seen in Fig. 18c, thus giving a greater increase in load capacity, which still needs to be studied in future research.

In addition, in this research, it was found that for a vertical displacement of 10 mm of the foundation (metal plate), a considerable increase in the resistance is observed provided by the multiaxial over the biaxial geogrid. However, it should be considered that there is a scale effect since the geogrids used were not scaled to the test’s samples, especially considering that the width of the foundation is 70 mm, and the aperture size of the biaxial and multiaxial geogrid are 33 and 40 mm respectively, and which should be investigated in future research.