Introduction

Thousands of years ago, many attempts were performed aiming to improving the soft clay bearing capacity, by adding agricultural fibers as a reinforcement material. In reinforced soil structures, single or multiple layers of a reinforcing material and squeezed soils are located underneath the foundation element to get an intensively improved performance characteristics material. The reinforced earth structure systems are widely used because of the relatively low cost, the easiness of construction, and the convenience of the utilized materials. The reinforced soil techniques are characterized by the raised tension forces and the shear resistance built up by the mobilized frictional resistance at the fill-reinforcement boundaries and passive resistance along the reinforcement transverse members. For a geosynthetic reinforcing material, the major engineering characteristics are tensile strength, tensile modulus, and interface shear strength, as they are resisting the tensile stresses conveyed from the fill material under deformations (Pinto 2002).

The insufficient bearing capacities of the shallow footings due to the improper soil formations cause major defects in foundations and super-structure due to shear strength and settlement considerations (Chao 2006). A solution for this problem could be by using the soil reinforcement systems to enhance the engineering properties of soil (Wu 2003; Chao 2008). Researches in soil reinforcement systems initiated at 1980s, using scale models. As a result of these researches, the reinforced soil with geosynthetics was approved as a good tool to overcome the bearing capacity and settlement problems associated with low quality soil formations (Holtz et al. 1997).

A state-of-the-art article concerning the geosynthetic-reinforced slopes was carried out by Shukla and Sivakugan, 2011, in which analytical, numerical, and experimental models, case studies, and detailed design processes were reported. Nevertheless, more investigations were conducted for more understanding of the advantages of using planar reinforcement inclusions in soil improvement as tabulated in Table 1.

Table 1 List of researches investigated the planer reinforcement inclusions as a soil improving technique
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Tests on geosynthetic-reinforced clay showed a raise in limit loads (Sakti and Das 1987). Adams and Collin 1997 confirmed the benefit of geosynthetic reinforcement for soil using a large-scale laboratory model tests on square footings. A primary estimation of limit loads on soils improved by geosynthetics as foundation strata was tried in the area of unpaved roads (Giroud and Noiray 1981).

A numerical investigation related to the optimal burial depth of the reinforcement within sand formations was done by Aria et al. 2017. The study concluded that the optimum burial depth depends significantly on the angle of internal friction (ϕ) of sand. For any value of (ϕ), there are two optimal burial depths: the first is constant depth equals 0.2 times the foundation width (B), which is mostly possessing the maximum improvement in bearing capacity, and the second is optimal burial depth which ranges between 0.38B and 0.5B as (ϕ) increases from 25 to 30°.

Investigations for the geocell reinforcement were carried out by Hegde and Sitharam 2015 through the implementation of a 3D numerical modeling for geocell and geogrid using finite difference package FALC3D. The model was validated with experimental studies, and then was used in the evaluation of geocell properties on the performance of the reinforced foundation beds.

A hypothetical mechanical models were carried out by Shukla and Chandra, 1994a, b, in which they presented a generalized model to assess the settlement characteristics of the reinforced granular fill overlaying soft soil formation. They concluded that:

  • The compressibility of the granular fill has an appreciable influence on the settlement response of the system, as long as the granular fill stiffness is less 50 times than that of the soft formation.

  • The granular fill compressibility increase results in more settlement at the edge of the loaded area.

  • At higher load intensities; the reinforced granular fill-soft soil system behaves as much stiffer system.

The improvement of shallow foundations performance using reinforced soil techniques was studied and it was revealed that the improved soil with geogrid insertion is one of the most economic modifications for insufficient soil conditions. The enhancement gained by using geosynthetics is highly related to its usage form.

This research is presenting outcomes of the proposed numerical model and the previously studied experimental model tests in addition to analysis for a rigid strip footing founded on reinforced dune sand bed. The current investigation is conducted to suggest the soil reinforcement system using geogrids as an efficient alternate to solve the technical problems associated with shallow foundations rested on weak soils.

Finite element modeling

Two-dimensional finite element code (PLAXIS) was employed for more knowledge about the behavior of the reinforced soil with geogrid beneath a rigid shallow foundations. The Mohr-Coulomb constitutive model is used for fill material and elastic tensile model for geogrid reinforcement to envisage the bearing capacities and settlements characteristics. The assigned materials for all of the model individuals, namely, fill geogrid, strip footing, and interface elements, were defined with the proposed boundary conditions and loading system as described herein.

A chain of two-dimensional finite element models for fine sand-geogrid-strip footing system was implemented to simulate the obtained experimental model tests, to define the deformations pattern in the sandy fill body, and to validate the numerical model. The used 2D finite element code supports mesh construction using triangular elements of six or 15 nodes for each element of the simulated fill material.

Material modeling

Soil constitutive law

The recognized Mohr-Coulomb constitutive law was used as a first-order (linear) simulation of actual soil performance. This elasto-plastic, stress-strain relation involves five elementary input parameters: the dry and wet fill unit weight, Young’s modulus [Eref], Poisson ratio [vnu], cohesion [Cref], friction angle [ɸ], and the dilatancy angle [Ψ]. The values of these parameters were determined through the laboratory testing program for the used fill and are presented in Table 2.

Table 2 Soil parameters used in FEM
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Geogrid modeling

Geogrid is an elastic flexible element with only normal stiffness that can only withstand tensile forces. The geogrid was modeled using elastic tensile constitutive model as that available in Plaxis code Version 8.2, with the parameters determined using the universal axial tension machine as shown in Table 3, where (EA) represents axial/normal stiffness and (Np) represents the geogrid ultimate tensile strength.

Table 3 Geogrid parameters used in FEM
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Strip footing modeling

Six strip footings (A), (B), (C), (D), (H), and (L) were modeled with the dimensions shown in Table 4.

Table 4 Strip footings—steel model and R.C. real full scale dimensions
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The steel strip footings (A), (B), and (C) were modeled as a plate with the steel parameters shown in Table 5; however, the reinforced concrete strip footings (D), (H), and (L) were modeled as a reinforced concrete element with proper interface properties. All footings input parameters were presented in Table 6.

Table 5 Model steel plate footing parameters used in the FEM
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Table 6 Real concrete footing parameters used in the FEM
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Interface elements

The roughness of the interface elements (contact surfaces between fill material and geogrid reinforcement) were represented by selecting a proper value for the reduction factor of the shear strength at the interface (Rinner). This factor relates the interface element shear characteristics as a percentage of the fill shear strength parameters (friction angle and cohesion). In this model, the interface element between the soil and the geogrid had the typical value of Rinner = 0.85. The interface element between the soil and the steel plate was chosen as rigid interface (Rinner = 1), and between the concrete footing and the soil was chosen with the typical value of Rinner = 0.6. The values of the Rinner were chosen based on the previous experience with the interaction between soils and geogrids which develop bearing and friction stresses.

Boundary conditions and vertical load on the strip footing

The demonstrated boundary conditions were constructed such that the boundaries of both model sides were free vertically and constrained horizontally, while the bottom horizontal boundary was fully fixed to simulate the fill particle movements in all directions as in the physical model as shown in Fig. 1. The load on the strip footing was simulated by employing an incremental pre-described load (load control method) that was associated with a multi-step loading procedure up to failure.

Fig. 1
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Finite element model—geometry, vertical load, and boundary conditions

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Mesh generation and initial conditions

Once the model geometry was defined, the material properties were assigned to all clusters (different zones in the generated mesh elements including certain number of finite elements) and the structural objects were defined. The geometry inputs have to be divided into elements as shown in un-deformed mesh in Fig. 2 to perform finite element calculations. A refined mesh (more elements with smaller dimensions) was implemented to reduce the effect of mesh dependency for cases involving variations in the reinforcement layers: number, length, and location relative to the footing. Then, the initial stress condition was conducted by applying the gravity force due to soil own weight with the existence of geogrid reinforcements. The initial conditions included the initial groundwater circumstances, the initial geometry alignment, and the initial effective stress state.

Fig. 2
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Finite element model—un-deformed mesh

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Finite element modeling matrix

A chain of two-dimensional numerical models were implemented and analyzed for achieving the research aim. Figures 3 and 4 show the FEM matrices for the single- and multi-layered reinforcement cases, which is the same matrix carried out for the previous conducted experimental testing program. In the multi-layered case, the FEM was made on footing type (B) (refer to Table 5) for h/B = 0.75. Figure 5 shows the geogrid layout and terms definition, where (N) is the geogrid inclusions number, (L) is the geogrid extension length relative to the footing width, (u) is the embedment of the first geogrid layer measured from bottom of the strip footing, (h) is the vertical spacing between consecutive layers of the geogrid, and (B) is the footing width.

Fig. 3
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FEM matrix for single-layer reinforced soil bed

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Fig. 4
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Previous experimental testing program matrix for multi-reinforced soil bed

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Fig. 5
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Geogrid reinforcement layout

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Validation of finite element model

To validate the proposed numerical model, the geometry and materials of the geogrid-soil-strip footing system were made to be completely similar to the experimental model and included steel strip footing, geogrid, and sandy fill. The results are presented in Figs. 6, 7, and 8 in terms of stress under the footing in (kN/m2) vs settlement in (mm) for unreinforced, single-reinforced, and multi-reinforced soil beds. All results are presented for finite element model (FEM) versus previously studied experimental model for comparison and validation.

Fig. 6
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Relationship between bearing stress and vertical displacement for footing type (B) on a unreinforced soil bed and b single reinforced soil bed

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Fig. 7
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Effect of first geogrid embedment (u/B) on bearing stress–footing vertical displacement relationship for footing type (B) on single-reinforced soil bed for N = 1, L/B = 7: a u/B = 0.25 and b u/B = 0.5

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Fig. 8
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Effect of geogrid extension (L/B) on bearing stress–footing vertical displacement relationship for footing type (B) on single-reinforced soil bed for N = 1 and u/B = 0.25: a L/B = 5 and b L/B = 7

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Unreinforced soil bed

Stress–settlement relationship of the rigid strip footing resting on unreinforced sand dunes was produced for both experimental and numerical models to be reference tests. Figure 6 a illustrates the comparison between FEM and the experimental model results for footing (B) on unreinforced soil bed.

Single-reinforced soil bed for footing type (B)

To examine the influence of the single-reinforcement burial depth (u/B) and the extension of the reinforcement relative to the footing width (L/B), these two parameters were studied using the proposed numerical model and were compared with the results of the previously constructed experimental model for the validation purpose as shown in Figure 6b.

Effect of u/B for footing (B)

The embedment ratio (u/b) was investigated as a variable with all of the other parameters influencing the soil-geogrid-strip footing system were kept fixed. Figure 7 a and b illustrate the comparison between the proposed FEM and the previous experimental model results for footing (B) with N = 1, L/B = 7.5 and u/B = 0.25 and 0.5 respectively.

Effect of L/B with footing (B)

As previously known that the bearing capacity ratio (BCR) rises as the reinforcement inclusion length spreads further than the failure pattern under the footing, the reinforcement length is the major parameter governing the (BCR). Figure 8 a and b illustrate the comparison between the proposed FEM and the previously obtained experimental model test results of bearing stress versus strip footing settlement for footing (B) with N = 1, u/B = 0.25, and L/B = 5 and 7 respectively.

As described in the above Figs. 6, 7, and 8 in the FEM validation process, the test results of the single-layer reinforced soil bed and the comparison indicated that there is a considerable matching for the experimental model and FEM results which is accepted.

Multi-reinforced soil bed

Figure 9 a, b, and c illustrate the comparison between the proposed FEM and the previously obtained experimental results for footing (B) with L/B = 7.5, u/B = 0.25, and h1/B = h2/B = h3/B = 0.75 for N = 2, 3, and 4 respectively.

Fig. 9
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Effect of number of geogrid layers (N) on bearing stress–footing vertical displacement relationship for footing type (B) on multi-reinforced soil bed for u/B = 0.25 and L/B = 7: a N = 2, h1/B = 0.75, b N = 3, h1/B = h2/B = 0.75, and c N = 4, h1/B = h2/B = h3/B = 0.75

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It can be noticed by relating the finite element model and the previous physical model results for the cases of unreinforced, single-reinforced, and multi-reinforced soil beds that there is a considerable matching for the FEM and the experimental model results. Therefore, the FEM is validated and ready for the parametric study.

Finite element model outputs

A sample of the finite element model outputs was presented for only one of the assessed configurations. These outputs included effective stresses within the tested soil under the footing, the soil elements that reaches the plastic state, horizontal and vertical displacements within soil mass and in geogrid reinforcement, and tension force distribution in geogrid. Figures 10, 11, 12, 13, and 14 show these outputs for the case of N = 1, u/B = 0.25, and L/B = 7.5 for footing (B).

Fig. 10
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Finite element model—deformed mesh

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Fig. 11
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Effective vertical stresses contours (Sigyy)

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Fig. 12
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Vertical displacements contours (Uy)

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Fig. 13
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Geogrid reinforcement total displacements (Ulot)

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Fig. 14
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Geogrid reinforcement axial forces

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Results, analysis, and discussion

The analysis was carried out for both single and multi-reinforced soil model test results and was presented through the bearing capacity ratio (BCR). This factor is expressed as the percentage of the footing ultimate pressure with reinforced foundation (q reinforced) to that for unreinforced bed (qun unreinforced).

Single reinforcement soil bed

Effect of first reinforcing layer depth (u/B)

The depth of the first reinforcing layer was changed in terms of u/B = 0.25, 0.5, and 0.75 for L/B = 7.5 for footings (A), (B), (C), (D), (H), and (L) (refer to Table 3) to get the optimum embedment depth. Figure 15 displays the variation of the bearing capacity ratio (BCR) of the soil versus the assessed parameter u/B. It was noticed that the (BCR) values for u/B = 0.25 and L/B = 7.5 were 1.67, 1.67, 1.63, 1.18, 1.10, and 1.07, respectively, with footings changed from (A) to (L) which indicate an improvement of 67 to 7% respectively. That means that as the width of the footing decreases, the improvement in terms of BCR increases with an optimum value at u/b = 0.25

Fig. 15
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Relationship of bearing capacity ratio BCR vs first geogrid embedment (u/B) for footings (A), (B), (C), (D), (H), and (L)

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Effect of reinforcing layers width (L/B)

The geogrid layer width relative to the strip footing width was changed in terms of L/B = 2.5, 5, 7.5, and 12 for u/B = 0.25 for footings (B), (D), (H), and (L) to get the optimum width for the maximum (BCR). Figure 16 illustrates the (BCR) of the soil versus the assessed parameter L/B.

Fig. 16
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Relationship of bearing capacity ratio BCR vs geogrid extension (L/B) for footings (B), (D), (H), and (L)

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It can be noticed that the effect of using L/B = 7.5 and 12 was almost the same; therefore, the optimum value of the parameter L/B = 7.5 for all tested footings. Also, It can be noticed that the values of (BCR) for L/B = 7.5 and u/B = 0.25 were 1.6, 1.13, 1.10, and 1.07 for footings (B), (D), (H), and (L), respectively. This means that as the width of the reinforcement increases, the improvement of the BCR increases up to L/B = 7.5 and as the width of the footing decreases the improvement of the BCR increases.

Multi-reinforcement soil bed

Effect of the spacing between the first and second geogrid (h1/B)

For the case of using two reinforcing layers with u/B = 0.25, L/B = 7.5, and h1/B = 0.75, the inspection of the results in Fig. 17 revealed that the values of (BCR) were 2.5, 2.12, 1.85, 1.32, 1.20, and 1.14 for footings (A), (B), (C), (D), (H), and (L), respectively. This indicates an inverse proportion between footing width (B) and the improvement in the BCR that ranges between 14 and 150% with the (h1/b) optimum value = 0.75 for all of the tested footings.

Fig. 17
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Relationship of bearing capacity ratio (BCR) vs first and second geogrid spacing (h1/B) for footings (A), (B), (C), (D), (H), and (L)

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Effect of the spacing between the second and third geogrid (h 2 /B)

Bearing capacity ratio (BCR)

In this part, the results were analyzed and discussed for the case of using three geogrid reinforcement layer, with fixing the spacing between the first and the second layers (h1/B) and varying the spacing between the second and the third layers (h2/B) to be assessed. From Fig. 18, it can be noticed that the values of (BCR) for using three reinforcing layers with u/B = 0.25, L/B = 7.5, and h1/B = 0.75, h2/B = 0.75 were 2.5, 2.27, 2.14, 1.36, 1.3, and 1.25 for footings (A), (B), (C), (D), (H), and (L), respectively. This indicates an inverse proportion between footing width (B) and the improvement in the BCR that ranges between 25 and 150%, with the (h2/b) optimum value = 0.75 for all of the tested footings for three reinforcement layers. The same (BCR) improvements were traced as shown in Fig. 19, for the case of using four reinforcement layers. So there was no gain from using the fourth layer.

Fig. 18
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Relationship of bearing capacity ratio(BCR) vs second and third geogrid spacing (h2/B) for footings (A), (B), (C), (D), (H), and (L)

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Fig. 19
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Relationship of bearing capacity ratio( BCR) vs third and fourth geogrid spacing (h3/B) for footings (A), (B), (C), (D), (H), and (L)

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Settlement

Figure 20 a and b illustrate the settlement (mm) for footings (A), (B), (C), (D), (H), and (L), respectively, in order to define the optimum value for h2/B regarding their associated settlement values.

Fig. 20
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Relation of footing settlement (mm) vs. second and third geogrid spacing (h2/B) for footings a (A), (B), and (C), and b (D), (H), and (L)

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It can be concluded that h2/B = 0.75 had the least settlement; therefore, it was the optimum value.

Optimum number of geogrid layers

A package of finite element models was carried out to assess the effect of number of geogrids on the mobilized BCR and the associated settlements. Figure 21 a and b represent both of BCR and settlement (mm) vs number of geogrids (N) for footings (A), (B), and (C). The inspection of the results revealed that the BCR value rises with the increase of geogrid layers up to N = 3 but the settlement (mm) for N = 4 and 5 is less than the settlement for N = 3, with minor reduction in settlement for the fifth geogrid layer. The same results were observed for the finite element models of the footings (D), (H), and (L), as shown in Fig. 22a, b.

Fig. 21
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Effect of number of geogrid layers (N) for footings (A), (B), and (C) on a bearing capacity ratio (BCR) and b footing settlement

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Fig. 22
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Effect of number of geogrid layers (N) for footings (D), (H), and (L) on a bearing capacity ratio (BCR) and b footing settlement

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So, it can be concluded that for the settlement considerations, the optimum number of geogrid reinforcement layers under rigid strip footing overlaying a reinforced sand dunes is four layers.

Conclusions

A series of numerical model tests was carried out to assess the behavior of a rigid strip footing founded on unreinforced and reinforced sand dunes in terms of BCR and settlement. The research aimed to define the optimum parameters influencing the (footing-sand dune-geogrid) system, the embedment depth for the first and successive geogrid reinforcement layers (u/B and h/B), the extension of the reinforcement under the footing relative to its width (L/B), and the optimum number of geogrid reinforcement (N). Based on the numerical model test results, the following conclusions may be drawn:

  1. 1.

    As comparing the output of the proposed numerical model test results with those of the previously studied experimental model, an acceptable agreement was traced for the behavior pattern and the critical values of the geogrid parameters.

  2. 2.

    The numerical modeling output proved that the placement of the geogrid reinforcement layer/s at the proper place inside the sand dunes body caused a respectable enhancement in the mobilized bearing capacity with a decrease in settlements in the reinforced soil supporting the strip footing.

  3. 3.

    The ideal first geogrid burial expressed as (u/B) that mobilizes the maximum ultimate bearing capacity of the (dune sand-geogrid) composite was about 0.25 times the footing width, which shows an acceptable agreement with that of Shukla and Mohyeddin 2017 (0.2 B).

  4. 4.

    The optimum vertical spacing of each two successive layers of geogrids in terms of (h/B) that activates the maximum ultimate bearing capacity of (dune sand-geogrid) composite was about 0.75 times the footing width.

  5. 5.

    The most favorable geogrid length beneath the strip footing in terms of (L/B) that mobilizes the maximum ultimate bearing capacity of the (dune sand-geogrid) composite bed was about 7.5 times the footing width.

  6. 6.

    The optimum number of geogrid layers below strip footing is three layers for the bearing capacity considerations and four layers for the settlement considerations.

  7. 7.

    The study showed a direct proportional relation between the number of reinforcement layer and the BCR and an inverse proportional relation between the footing width and the mobilized BCR.