Using slippage theory to analyze shear behavior of loop-formed fiber reinforced soil composites

Abstract

Since composite structures with soft-material matrix do not have adequate pullout resistance with flat-type reinforcements such as fibers, there are increasing cases where reinforcements with passive resistance are used in conjunction. So, loop-formed polyethylene fibers were used to reinforce soil against shear loading. Afterward, shear behavior of both fiber and loop-formed fiber-reinforced soil composite samples was modeled by using “force-equilibrium method” and “slippage theory” of short fiber composites. The proposed model indicated that both fiber parameters and ambient conditions determine shear strength of a fiber-soil composite. In the next step, a set of laboratory direct shear tests was performed on different samples including neat soil, loop-formed fiber and fiber-reinforced treatments. Thus, it was found that the performance of polyethylene looped fibers in shear strength improvement of soil composite is more than that of the ordinary polyethylene fibers. In addition, a novel apparatus based on fiber pullout test was designed to determine the interfacial shear stress between fiber and soil. Finally, an artificial neural network technique and least square method were used to calculate “fiber reinforcing amplitude” and “slippage ratio”, as input parameters required for the model. Consequently, both the proposed models and established artificial neural network adequately predicted shear behavior of loop-formed fiber and fiber reinforced soil composite.

Introduction

As a soft material, soil is relatively strong against compressive external forces but very weak against tensile stresses [1]. Fibers can be useful in this way due to mechanical reinforcement and economical benefits [2,3]. Therefore, randomly distributed fiber-reinforced soils have recently attracted increasing attention in geotechnical engineering for the second time as mimicry of the past [4]. Mainly, fiber reinforced soil behaves as a composite material in which fibers of relatively high tensile strength are embedded in a matrix of soil. Shear stresses in the soil mobilize tensile resistance in the fibers, which in turn imparts greater strength to the soil [5–7].

Short fiber reinforced soil composites have been conventionally designed using composite approaches to characterize the contribution of fibers to stability. In these cases, the mixture is considered as a homogenous composite material [8]. A comprehensive literature review shows that composite models have been proposed by several investigators including mechanistic models [9 and 10], a statistical model [11] and an energy-based limit analysis model [12].

Models based on a volumetric homogenization technique but limited to the description of non-linear elastic behavior have been also presented by Ding and Hargrove (2006) [13] for monotonic loading and by Li and Ding (2002) [14] in cyclic loading conditions. A complete constitutive law for soils reinforced with continuous filament (Texsol) has been presented by Villard et al (1990) [15] and Prisco and Nova (1993) [16] employing the superposition of sand and fiber effects. The model proposed by Villard et al (1990) is the only one that recognizes the importance of fiber orientation as a parameter governing the effectiveness of fiber inclusion. Recently, a two-dimensional distinct element method (DEM) has been developed for the micromechanical analysis of mixtures of granular materials and flexible fibers [17,18]. Numerical analysis with finite difference code has been performed by Babu et al (2008) [19]. Abtahi et al (2008) [7] extended the shear lag theory proposed by Cox (1952) [20] to explain the role of fiber length and fiber diameter in short fiber soil composites. Thus, it was found that by increasing the fiber length and deceasing fiber diameter, the compressive strength of the soil will improve [20,21]. In another work, Diambra et al (2010) [22] presented a model based on the rule of mixtures of composite materials at conventional triaxial soil tests. The model considers that fibers behave linear elastically and the soil, when unreinforced, obeys the simple linear elastic perfectly plastic Mohr-Coulomb model [22]. As well, using artificial neural network (ANN) to predict the role of fiber parameters on shear strength of short fiber soil composites has been successfully reported by Abtahi et al (2010) [23,24].

However, in every respect, ordinary short fibers reinforcing soil can be damaged during the compaction of soil [25]. Moreover, since soft materials with insufficient friction do not have adequate pullout resistance with flat-type reinforcements, there are increasing cases where reinforcements with passive resistance are used in conjunction [26,27]. Accordingly, there is a need to develop a system that can maintain good pullout resistance properties, while minimizing the damage to reinforcements caused by soil compaction [28]. Based on these motivations, the idea of using loop-formed fibers capable of minimizing damage to reinforcement has been developed for soft-matrix composites in this study. For this purpose, polyethylene (PE) loop-formed fibers were used as reinforcing elements due to economical advantages, feasibility of using PE waste strands and excellent alkaline resistance [29,30].

On the other hand, soils mixed with randomly distributed fibers can be used as patches in the localized repair of failed slopes, soil protection and/or wall stabilization. Therefore, shear strength of fiber reinforced soil would be considered as an important concept [31]. So, this paper firstly introduces a novel method of reinforcement of composite materials, i.e. loop-formed fibers. After that, shear behavior of fiber and looped fiber reinforced soil composite will be investigated by using both theoretical models, based on force-equilibrium method and slippage theory of short fiber composites, and a set of direct shear tests. In this way, a novel apparatus based on fiber pullout test will be used to determine the interfacial shear stress between fiber and soil and to complete the required input parameters of the proposed model.

Shear modeling of fiber and loop-formed fiber reinforced soil composite

General description

The shear resistance force of loop-formed fibers has two major components including: (a) the friction force between soft-matrix and vertical elements of looped fiber (F₁) and (b) shear resistance force of soft-matrix within the loop (F₂). Therefore, the total improvement in shear strength of composite (ΔS_t) can be written in accordance with [28]:

$\Delta {\text{S}}_{\text{t}}=\Delta {\text{S}}_1+\Delta {\text{S}}_2$

(1)

where ΔS₁ and ΔS₂ represent the roles of vertical elements of looped fiber and shear resistance of soft-matrix within the loop, respectively. The authors suggest the term “ΔS₂” to be entitled “loop effect”, while it is clear that “ΔS₁” is related to the fiber reinforcing effect, i.e. reinforcement by vertical elements of looped fiber. So, “ΔS₁” will be introduced in this paper by the “fiber effect” term. Therefore, it is clear that the main benefit of using loop-formed fibers over the fibers in soil reinforcement is related to the “loop effect”. So, these two effects will be theoretically investigated in the following sections.

Modeling of “fiber effect”

Soil samples reinforced with short fibers may be analyzed as if it were a composite material in which fibers of relatively high tensile strength are embedded in a matrix of lower tensile strength. Force is mainly carried from point to point within the soil matrix by forces tangential to the fibers producing differing tensions along their length. These tangential forces may be carried by friction between fibers and surrounding matrix. So, in this work, the Waldron's method [32] is extended to model the role of fibers in shear improvement of composite soil.

For simplification, it is firstly assumed that reinforcing fibers have a general vertical orientation normal to the shearing surface of the soil matrix. Figure 1 shows the schematic of the proposed model. The following assumptions should be made in the model: (1) composite shearing occurs in a horizontal zone of thickness D which is penetrated by vertical fibers and/or vertical elements of looped fibers; (2) D does not change during shear; (3) fibers are flexible with circular cross-section, of uniform diameter d_f and are linearly elastic with the modulus of E_f; (4) the looped fibers have squared shape, i.e. $\overline{\text{EE}\text{'}}$  =  $\overline{\text{AE}}$ (see Figure 1).

Figure 1.

The schematic of the proposed model.

According to the short fiber composite theory proposed by Cox (1952) [20], the soft matrix (soil) inserts tension on the fibers and/or looped fibers through tangential stresses τ which has a maximum value at non-slippage zone (τ_max). When the upper mass of matrix in Figure 1 has been displaced a distance Δx, the fiber segments $\overline{\text{BC}}$ and $\overline{\text{B}\text{'}\text{C}\text{'}}$ will be extended to new lengths $\overline{\text{BF}}$ and $\overline{\text{B}\text{'}\text{F}\text{'}}$ (see Figure 1: after shearing section). Consequently, the average tensile stress σ_av within the fiber element will depend on the fiber elastic modulus E_f and the strain (ΔD/D) induced in the stressed length of each vertical fiber element given by:

${\sigma }_{\text{av}}=(\Delta \text{D}/\text{D})\times {\text{E}}_{\text{f}}\text{D}=\overline{\text{BC}}=\overline{\text{B}\text{'}\text{C}\text{'}}$

(2)

Consequently, fiber displacement ΔD within the shearing zone D can be obtained through:

$\Delta \text{D}=\text{D}\times (\text{sec}\beta -1)$

(3)

According to force equilibrium existing on a fiber element that has been shown in Figure 2, the following equation can be written [32]:

$\text{d}\sigma /\text{dD}=4\tau /{\text{d}}_{\text{f}}$

(4)

Figure 2.

Force equilibrium on a fiber element during the shearing of the composite.

Figure 3 shows shear and tensile stress distributions within a fiber by considering the “slippage phenomenon”. According to the “slippage theory” in short fiber composites, it is supposed that during extension, there is slippage near both ends of fiber and a central region along the fiber length gripped by the matrix which is called non-slippage region. If λ is defined as the fiber slippage ratio, which is the ratio of fiber length of the slipping portion to the whole fiber length D, the slippage from each end of the fiber will occur over a length of λD/2 [33].

Figure 3.

Tensile and shear stress distribution within a fiber during the shearing of the composite.

In other words, λ is the value of fiber cooperation and assistance in the bearing of induced tensile stresses during the shearing of the matrix. If M is the point on $\overline{\text{BC}}$ (see Figure 3) where non-slippage region starts and tensile stress rises to the maximum value (σ_max) then integration gives the tensile stress σ_max at point M:

${\int }_{\sigma =0}^{\sigma ={\sigma }_{\text{max}}}\text{d}\sigma ={\int }_{\text{B}}^{\text{M}}(4\tau /{\text{d}}_{\text{f}})\text{dD}$

(5)

Thus σ_max will be:

${\sigma }_{\text{max}}=(4\tau /{\text{d}}_{\text{f}})\times \overline{\text{MB}}$

(6)

Since $\overline{\text{MB}}$  = (λ/2) ×  $\overline{\text{BC}}$ , therefore:

${\sigma }_{\text{max}}=(4\tau /{\text{d}}_{\text{f}})\times (\lambda /2)\times \overline{\text{BC}}=(4\tau /{\text{d}}_{\text{f}})\times (\lambda /2)\times \text{D}$

(7)

So, the length of the fiber under tensile stress is:

$\text{D}={\sigma }_{\text{max}}{\text{d}}_{\text{f}}/(2\lambda \tau )$

(8)

From Figure 3, it is clear that the average tensile stress σ_av can be calculated in accordance with:

${\sigma }_{\text{av}}=(2\times [{\sigma }_{\text{max}}\times (\lambda \times \text{D}/2)/2]+(\text{D}-\lambda \times \text{D})\times {\sigma }_{\text{max}})/\text{D}$

(9)

Consequently, σ_av will be:

${\sigma }_{\text{av}}={\sigma }_{\text{max}}\times (1-\lambda /2)$

(10)

So, combination of equations (3) and (8) and substituting in equation (2) gives:

${\sigma }_{\text{av}}=(\Delta \text{D}/\text{D}){\text{E}}_{\text{f}}=2\lambda \tau {\text{E}}_{\text{f}}\text{D}(\text{sec}\beta -1)/({\sigma }_{\text{max}}{\text{d}}_{\text{f}})$

(11)

Therefore, considering equations (10) and (11) will be:

${\sigma }_{\text{max}}\times (1-\lambda /2)=2\lambda \tau {\text{E}}_{\text{f}}\text{D}(\text{sec}\beta -1)/({\sigma }_{\text{max}}{\text{d}}_{\text{f}})$

(12)

Thus, σ_max can be obtained through:

$\sigma \text{max}=(2\lambda \tau \text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5(1-\lambda /2)-0.5$

(13)

So, the maximum induced tensile force within the single fiber element can be divided in tangential force (F_t):

$\text{F}\text{t}=\text{A}\text{f}\times \sigma \text{max}\times \text{sin}\beta =(\pi d_{\text{f}}^2/4)(1-\lambda /2)-0.5(2\lambda \tau \text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5\text{sin}\beta$

(14)

and normal force (F_n) to the horizontal sheared plane:

$\text{F}\text{n}=\text{A}\text{f}\times \sigma \text{max}\times \text{cos}\beta =({\pi \text{d}}_{\text{f}}^2/4)(1-\lambda /2)-0.5(2\lambda \tau \text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5\text{cos}\beta$

(15)

where A_f is the fiber cross-sectional area. Consequently, for a given shear displacement, the total tangential force of a single fiber and/or a fiber element resisting against shear (F_sft) will be:

$\text{F}\text{sft}=\text{F}\text{t}+\text{F}\text{n}\times \text{tan}\phi =({\pi \text{d}}_{\text{f}}^2/4)(1-\lambda /2)-0.5(2\lambda \tau \text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\text{sin}\beta +\text{cos}\beta \times \text{tan}\phi )$

(16)

In which φ is the soil friction angle according to the Mohr-Coulomb equation. Currently, assume that there are N_f looped fibers within the shearing zone, i.e. there are 2N_f fiber vertical elements within the shearing zone. Therefore, the total tangential force of reinforcing looped fibers resisting against shear (F_ft) will be:

$\text{F}\text{ft}=2\times \text{N}\text{f}\times \text{F}\text{sft}=2\times \text{N}\text{f}\times (\pi {\text{d}}_{\text{f}}^2/4)(1-\lambda /2)-0.5(2\lambda \tau \text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\text{sin}\beta +\text{cos}\beta \times \text{tan}\phi )$

(17)

So, the total shear strength improvement of looped fiber reinforced soil composite ΔS₁ can be calculated by:

$\Delta \text{S}1=\text{F}\text{ft}/\text{A}\text{s}$

(18)

where A_s is the total sheared area. Consequently, combination of equations (17) and (18) gives:

$\Delta \text{S}1=2\times \text{N}\text{f}\times ({\pi \text{d}}_{\text{f}}^2/4)(1-\lambda /2)-0.5(2\lambda \tau \text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\text{sin}\beta +\text{cos}\beta \times \text{tan}\phi )/\text{A}\text{s}$

(19)

It is clear that the interfacial shear stress between fiber and soil τ can be obtained by:

$\tau =\mu \times \sigma \text{v}$

(20)

Therefore, equation (19) changes to:

$\Delta \text{S}1=2\times \text{N}\text{f}\times (\pi {\text{d}}_{\text{f}}^2/4)(1-\lambda /2)-0.5(2\lambda \mu \sigma \text{v}\text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\text{sin}\beta +\text{cos}\beta \times \text{tan}\phi )/\text{A}\text{s}$

(21)

In which µ is the coefficient of friction between fiber and soil, while σ_v represents vertical stress inserted on the soil composite. The parameter µ can be obtained by a novel pullout test technique that will be discussed in the following.

Modeling of “loop effect”

The “loop effect” is the main benefit and the key advantage of using loop-formed fibers over ordinary fibers in the reinforcement process. It means that the existence of soft-matrix, i.e. soil, within the loop leads to produce an additional resisting shear stress during shear loading of composite. The produced shear stress performs between “soft matrix surrounded by loop” and soft matrix mass tangential over the loop. Consequently, shear strength improvement of composite derived by the “loop effect” ΔS₂ can be calculated by division of effective shear force “ ${\text{N}}_{\text{f}}\times ({\sigma }_{\text{v}}\times \text{tan}\phi )\times (\text{D}\times {\text{L}}_{\text{f}})\times \text{cos}(\pi /2-\beta )$ ” within the loop to the total shear area of the composite A_s [28]:

$\Delta {\text{S}}_2={\text{N}}_{\text{f}}\times ({\sigma }_{\text{v}}\times \text{tan}\phi )\times (\text{D}\times {\text{L}}_{\text{f}})\times \text{cos}(\pi /2-\beta )/{\text{A}}_{\text{s}}$

(22)

where L_f is the mesh length of the looped fiber, i.e. $\overline{\text{EE}\text{'}}$ and/or $\overline{\text{AE}}$ in Figure 1.

Total shear improvement

As it was previously mentioned, the total shear improvement of loop-formed fiber reinforced soil composite ΔS_t is summation of “the fiber effect” and “the loop effect”. Therefore, according to equation (1) and merging equations (21) with (22), ΔS_t can be obtained by:

$\Delta \text{S}\text{t}=(\text{N}\text{f}/\text{A}\text{s})[2\times (\pi {\text{d}}_{\text{f}}^2/4)(1-\lambda /2)-0.5(2\lambda \mu \sigma \text{v}\text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\text{sin}\beta +\text{cos}\beta \times \text{tan}\phi )+\sigma \text{v}\times \text{tan}\phi \times \text{D}\times \text{L}\text{f}\times \text{sin}\beta ]$

(23)

It is clear that in the case of fiber reinforced soil composite, equation (21) should be only used to analyze shear strength envelopment ΔS_t.

Materials and methods

Sample preparation

The soil, as a soft-matrix material, used in the reinforcement tests was a silty sand kind prepared from Segzi Desert, Isfahan, Iran. Table 1 presents some physical and mechanical properties of the soil derived from test results.

Table 1.

Some physical properties of soil used in this study.

Property	Value
Specific gravity, Gs (gr/cm3)	2.680
Liquid limit, LL (%)	29.00
Plastic limit, PL (%)	10.62
Plasticity index, PI (%)	18.37
Maximum dry unit weight, γdmax (kN/m3)	18.05
Optimum moisture content, OMC (%)	13.82
California bearing ratio, CBR (%)	1.62
c (kPa)	19.07
φ (deg)	35.56

Polyethylene (PE) reinforcing elements were used to evaluate the proposed model, and as well to compare experimental results with the model outputs. The reinforcements consisted of both ordinary and looped PE fibers to evaluate separately the “fiber effect” and the “loop effect” and to validate the developed model. Therefore, four types of PE welded grids were prepared from Iran Tour Company, Tehran, Iran. After that, the grid samples were cut to produce discrete grids and/or looped fibers from each treatment. The reinforcements were selected to obtain a range of elastic modulus (E_f= 74 MPa and 230 MPa), fiber diameter (d_f= 1 mm and 2 mm), loop dimension (L_f= 7 and 14 mm) and grid fraction (N_f= 8, 16 and 32). In addition, staple fibers were made from one kind of grid samples (E_f= 230 MPa, L_f= 7 mm and d_f= 1 mm) to produce randomly distributed fiber reinforced soil samples at three fiber mass fractions of 0.5%, 1% and 2%. These fiber dosages equal discrete grid numbers (N_f= 8, 16 and 32). Table 2 summarizes the properties of the reinforcements (both fibers and looped fibers) at each treatment.

Table 2.

Description of different polyethylene (PE) treatments used in test program.

No.	Code No.	Mass fraction (%)	Type of reinforcement	Mesh size (mm)	Fiber diameter (mm)	Number of reinforcements	Elastic modulus (MPa)
1	PEG/7/8	0.5	Looped Fiber	7.00	1.0389	8	230
2	PEG/7/16	1	Looped Fiber	7.00	1.0389	16	230
3	PEG/7/32	2	Looped Fiber	7.00	1.0389	32	230
4	PEG/14/10/2	1	Looped Fiber	14.00	1.8663	10	74
5	PEG/14/10/1	1	Looped Fiber	14.00	1.0389	10	230
6	PEF/7/0.5%	0.5	Fiber	7.00 a	1.0389	32	230
7	PEF/7/1%	1	Fiber	7.00 a	1.0389	64	230
8	PEF/7/2%	2	Fiber	7.00 a	1.0389	128	230

a

Fiber length.

Determination of fiber diameter

As it can be seen from equation (23), fiber diameter d_f is an important input parameter to predict shear improvement of fiber and/or looped fiber reinforced soil composite. Therefore, microscopic method was used to measure d_f accurately. In this way, a digital microscope capable of connecting to PC with the zoom of 40 was selected for all samples. Thus, the values of d_f for different treatments were obtained that has been shown in Figure 4(a–c) and registered in Table 2.

Figure 4.

Fiber diameter d_f for different treatments: (a) No. 4 (PEG/14/10/2); (b) No. 1-3 and No. 5-9; (c) No. 10 (PEG/6/30/1).

Fiber pullout test

Coefficient of friction between PE fiber and soil µ was measured using a novel pullout test method. In this way, the Instron Tensile Tester (Zwick universal testing machine 1446 60, 1994, Germany) was modified to provide fiber pullout test through the soil matrix. The schematic of the apparatus is shown in Figure 5. The fiber embedded length L_embf was chosen as 10 mm, meanwhile the vertical stress σ_v on all samples was 25 kPa. The pullout test was performed with the speed of 20 mm/min. This speed was determined in accordance with the fiber embedded length and based on the fact that the total time of the test should be 30 to 60 sec.

Figure 5.

Sketch drawing of the pullout test (not to scale).

According to the experimental design, pullout test for each treatment was repeated five times. Firstly, the maximum value of induced tensile force T₁ within the metal wire was obtained through the force-time diagram derived from PC. Afterward pullout force T₂ could be calculated by using [34]:

${\text{T}}_1={\text{T}}_2\times {\text{e}}^{\theta \mu *}$

(24)

where θ is the angle of yarn wrapped around the spool, while µ* is the coefficient of friction between yarn and spool that is previously determined 0.6998. In the third step, μ can be obtained through:

$\mu ={\text{T}}_1/({\text{e}}^{\theta \mu *}{\sigma }_{\text{V}}{\pi \text{d}}_{\text{f}}\times {\text{L}}_{\text{embf}})$

(25)

Thus, the mean value of μ was obtained 0.35.

Direct shear test

The shear tests were conducted using a standard, laboratory direct shear apparatus in accordance with ASTM D 3080. The specimens for the shear tests were shaped in a cylindrical mold with 20 mm height and 63 mm inner diameter by static compaction at the respective optimum moisture content (OMC) state of the soil. The OMC value was determined 13.82% through the standard Proctor compaction test designated by ASTM D 698.

The shear tests were strain controlled with the rate of 1 mm/min. The tests were also performed at three vertical normal stresses (σ_v) of 49, 98 and 196 kPa in order to determine the shear behavior of the soft-matrix composite at different compressive loads. According to the experimental design, three direct shear tests should be repeated for each treatment. Results of three different tests for the neat soil were presented in a chart with maximum measured shear stress on vertical axis and vertical normal stresses (σ_v) on the horizontal axis. A linear curve was fitted to the test result points. The intercept of this line with the vertical axis gives the cohesion C and its slope presents the peak friction angle φ. Consequently, the corresponding friction angle of neat soil φ and cohesion C measured in direct shear was 35° and 19 MPa, respectively.

According to the proposed model polyethylene fibers and grids were always placed in a regular pattern at approximately equal spacing from each other. As well, it was made an attempt that reinforcing elements were placed in a perpendicular orientation to the shear plane. Both shear stress and vertical deformation were recorded as a function of shear or horizontal displacement up to a total displacement of 9 mm. The experimental shear improvement of composite soil $\Delta \text{S}({\sigma }_{\text{v}})$ for each sample at each vertical stress σ_v can be calculated by:

$\Delta \text{S}({\sigma }_{\text{v}})={\text{S}}_{\text{comp}}({\sigma }_{\text{v}})-{\text{S}}_{\text{neat}}({\sigma }_{\text{v}})$

(26)

where S_comp( ${\sigma }_{\text{v}}$ ) and ${\text{S}}_{\text{neat}}({\sigma }_{\text{v}})$ are shear strength of composite soil and neat soil at the same vertical stress ${\sigma }_{\text{v}}$ , respectively, that can be straightly obtained through the shear test.

Establishment of an ANN

Architecture of ANN

Based on the proposed model, to analyze the performance of fiber and/or looped fiber reinforced soil composite, the first problem is calculation of D and λ as input parameters (see equation (23)). By considering Figure (1) and equations (2) to (23), each one can find that D is the length of the fiber which deforms during shear loading of the soil composite. Since the two parameters of D and λ refer to fiber performance during shearing of composite, a semi-empirical method should be used to analyze the performance of reinforcing elements and to compare experimental results with model outputs. For this purpose, an ANN was established on the experimental data obtained from direct shear test. So, it was tried to determine the best values of D and λ by using the predicted values of ΔS.

In this way, multilayer perceptron (MLP) was used as a common and powerful ANN that is capable of modeling complex functions with hidden layers. The number of hidden layers depends on the complexity of functions and input features. Frequently, using more than two hidden layers does not improve the performance of the system, and only increases calculations [35]. In this work, two hidden layers were used to train the network. Based on experiences, the number of hidden neurons should be lower than the number of input neurons. As well, experiments show if the number of hidden neurons takes nearly one-half of the number of input neurons, the ratio of performance to cost will be optimum. Therefore, the architecture of the designed ANN was an input layer that consisted of seven neurons (number of reinforcing elements N_f, fiber diameter d_f, Elastic Modulus of fiber E_f, coefficient of friction between fiber and matrix µ, vertical compressive stress σ_v, mesh length L_f and being “ordinary or looped fiber”); two hidden layers, each layer with four neurons and an output layer including one neuron ( $\Delta \text{S}$ ). Transfer functions of hidden neurons were log-sigmoid according to:

$\text{y}=1/(1+{\text{e}}^{-\text{x}})$

(27)

The transferred function of output layer was chosen linear. Figure 6 illustrates the architecture of the designed ANN.

Figure 6.

The architecture of the designed artificial neural network (ANN).

Training ANN

Training is a necessary part for classifying with ANN. MLP is a supervised ANN and needs target addition to input for the training. Standard algorithm for training MLP is fed forward back propagation. Levenberg-Marquardt is the most popular algorithm for training MLP ANN [35]. This algorithm is often the best method for MLP training, since the method is fast and numbers of training epochs are low, as well the performance of ANN after training with this method is closed to goal. Other network parameters for training phase were as follows: learning factor = 0.05, stop condition = maximum number of epochs reaches 1000 or the mean square error is less than 0.002. Initial bias and weights were randomly selected between −1 and 1.

In the first step, there was 81 input data to implement the designed ANN algorithm in Matlab 6.1 software (9 treatments, 3 vertical stresses and 3 repeats). Training set was built from 70 samples which were normalized before the training phase. After the training phase, the network was evaluated by validation of the rest of samples using MSE as an indicator index. According to the diagram of MSE vs. number of epochs that has been shown in Figure 7, the best validation performance is 0.00136 at 8'th epoch.

Figure 7.

Performance of training of artificial neural network (ANN).

Results and discussions

Stage analysis of the proposed model

As it was mentioned previously, the first problem is calculation of D and λ as input parameters in equation (23). D is the length of the fiber and/or looped fiber which deforms during shear loading of soil composite. This deformation induces a tensile stress within the fiber that can resist against shearing of soil composite. Thus, shear improvement ΔS can be achieved by the attendance of fibers through the soil composite. In other words, D is the thickness of the sheared zone in which fibers and/or looped fibers corporate perfectly with the soil matrix throughout shearing. Therefore, let the authors to entitle D as “the reinforcing amplitude of fiber” within the soil composite. However, it is difficult to determine the reinforcing amplitude of fiber D as an input parameter in equation (23). It is clear that D is a function of total fiber length and/or loop mesh L_f. The same comment is applicable for slippage ratio λ, i.e. the more the value of λ leads to the more fiber disobedience during the insertion of tensile stresses. Fundamentally, λ determines the induced tensile stress distribution derived from shear loading of soil composite on fiber segment D. Altogether, the two parameters of D and λ refer to fiber performance during shear loading of soil composite. Consequently, a semi-empirical method should be used to obtain ΔS through the proposed model, to evaluate the fiber and/or looped-fiber performance (determination D and λ) and to compare experimental results with model outputs.

Therefore, it was decided to establish an ANN system on data obtained from direct shear tests performed on treatments No. 1 to 9. It means that the parametric characters of the treatments of No. 1 to 9 were firstly considered as input data for ANN; meanwhile ΔS was designed as output parameter. After that the training process was finished, the predicted values of ΔS were used to determine the best values of D and λ. In this way, least square error (LSE) method was generalized to find the superlative values of D and λ with the best fitness that will be discussed in the following. In the final, two PE looped fibers as two new treatments (No. 10 and 11) were prepared with certain specifications. The shear strength improvement of these treatments was mutually obtained by experimental work (direct shear test) and by the proposed model (according to obtained D and λ). Figure 8 demonstrates the schematic of the stage analysis of the research work.

Figure 8.

The schematic of the stage analysis of the research work.

Using LSE method to find D and λ

ANN-predicted values of ΔS for the treatments of No. 1-3 can be employed to find D and λ by using LSE method. Therefore, in the case of looped fiber reinforced soil composite, equation (23) can be rewritten on the basis of independent variables of N_f and ${\sigma }_{\text{v}}$ in accordance with:

$\Delta {\text{S}}_{\text{t}}=\text{A}{\text{N}}_{\text{f}}{\sigma }_{\text{v}}^{0.5}+\text{B}{\text{N}}_{\text{f}}{\sigma }_{\text{v}}$

(28)

where

$\text{A}=({\pi \text{d}}_{\text{f}}^2)(1-\lambda /2)-0.5(2\lambda \mu \text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5(\text{sin}\beta +\text{cos}\beta \times \text{tan}\phi )/(2\text{A}\text{s})$

(29)

and

$\text{B}=\text{tan}\phi \times \text{D}\times \text{L}\text{f}\text{sin}\beta /\text{A}\text{s}$

(30)

As if D is considered as a function of fiber length and/or looped mesh L_f, i.e. D = L_f/a, equation (30) changes to:

$\text{B}=\text{tan}\phi \times {\text{L}}_{\text{f}}^2\times \text{sin}\beta /(\text{a}\times \text{A}\text{s})$

(31)

where “a” is the ratio of fiber reinforcing magnitude. Assuming N_f= x, ${\sigma }_{\text{v}}^{0.5}$  = z and ΔS_t= y, equation (28) changes to:

$\text{y}=\text{Axz}+\text{Bxz}2$

(32)

By using LSE method, the authors have proved that coefficients of A and B can be obtained through dependent and independent variables of Y_i, X_i and Z_i for i = 1 to n, respectively (see Appendix A):

$\text{A}=\frac{\overline{\text{XYZ}2}\times \overline{\text{X}2\text{Z}3}-\overline{\text{XYZ}}\times \overline{\text{X}2\text{Z}4}}{(\overline{\text{X}2\text{Z}3})2-\overline{\text{X}2\text{Z}2}\times \overline{\text{X}2\text{Z}4}}$

(33)

$\text{B}=\frac{\overline{\text{XYZ}}-\stackrel{\wedge }{\text{A}}\times \overline{\text{X}2\text{Z}2}}{\overline{\text{X}2\text{Z}3}}$

(34)

with being $\overline{\text{XYZ}2}$  = (1/n) $\sum _{\text{i}=1}^{\text{n}}\text{X}\text{i}\text{Y}\text{i}{\text{Z}}_{\text{i}}^2$ and so on. Consequently, coefficients of A and B can be calculated by applying 27 data (three samples (No. 1–3), three vertical stresses and three repeats) in equations (33) and (34). According to equation (31), the ratio of fiber reinforcing magnitude “a” will be:

$\text{a}=(\text{tan}\phi \times {\text{L}}_{\text{f}}^2\times \text{sin}\beta )/(\text{B}\times \text{A}\text{s})$

(35)

Thus, “a” was obtained as “2.92”. By understanding “A” and putting D in equation (29), fiber slippage ratio λ can be derived:

$(2\text{A}\text{s}\text{A})/[{\pi \text{d}}_{\text{f}}^2(2\mu \text{E}\text{f}\text{D}/\text{d}\text{f})0.5(\text{sec}\beta -1)0.5(\text{sin}\beta +\text{cos}\beta \times \text{tan}\phi )]=(\lambda /(1-\lambda /2))0.5$

(36)

From equation (36), λ was calculated “0.52”.

Experimental and theoretical analysis of results

The analytical calculations in the past section demonstrated that one-third of the fiber and/or looped fiber deforms approximately during shear loading of soil composite (a = 2.92). This deformation induces a tensile stress within the fiber that can resist against shearing of soil composite. As well, the derived value of λ indicates that about half of the deformed segment of fiber length D bears completely the maximum induced tensile stress σ_max (λ = 0.52). It is emphasized that the derived values of λ and “a” are only applicable in the case of the used soil and PE fibers. However, the proposed model provides an analytical method to investigate the performance of fiber-soil composite and to predict the role of soil and fiber parameters on soil shear improvement ΔS.

Table 3 illustrates shear strength improvement (ΔS) for treatments No. 1 to 9 derived by both direct shear test results and theoretical model outputs (based on λ = “0.52” and a = “2.92”). In general, both fiber and looped-fiber reinforcements increase ultimate shear strength of fiber-soil composite. Regardless of applied vertical stress, shear envelopment of soil composite will be increased as the fiber mass fraction and/or the number of reinforcing elements N_f enhances (compare neat soil, PEF/7/0.5%, PEF/7/1% and PEF/7/2%, PEG/7/8, PEG/7/16 and PEG/7/32 samples). It is clear that according to the proposed model, the presence of more fibers oriented across the shear plane creates more reinforcing influence. Equations (21) and (23) state that as the fiber and/or looped fiber content enhances the shear envelopment ΔS will be increased linearly. Another outcome is that by raising the vertical stress σ_v from 50 kPa to 200 kPa, the shear improvement ΔS is increased which shows a complete consistency with the proposed model.

Table 3.

Shear strength improvement (ΔS): comparison between experimental results and model outputs (λ = 0.52, a = 2.92).

Code No.	Experimental Results (Pa)	Model outputs (Pa)
	σv = 49000 Pa		e%
Neat soil	0	0.00	0.00
PEG/7/8	4397	5164.97	17.47
PEG/7/16	9423	10329.95	9.627
PEG/7/32	19788	20659.89	4.41
PEG/14/10/2	15077	15945.30	5.80
PEG/14/10/1	12250	11067.66	9.65
PEF/7/0.5%	3769	4565.64	21.14
PEF/7/1%	7853	9131.27	16.28
PEF/7/2%	15705	18262.55	16.29
	σv = 98000 Pa		e%
Neat soil	0	0.00	0.00
PEG/7/8	8795	7655.46	12.96
PEG/7/16	16019	15310.92	4.42
PEG/7/32	31410	30621.84	2.51
PEG/14/10/2	22930	24305.47	6.00
PEG/14/10/1	15077	17407.46	15.46
PEF/7/0.5%	5654	6456.79	14.20
PEF/7/1%	11308	12913.57	14.20
PEF/7/2%	24500	25827.14	5.42
	σv = 196000 Pa	e%
Neat soil	0	0.00	0.00
PEG/7/8	12878	11528.62	10.49
PEG/7/16	26699	23057.24	13.64
PEG/7/32	43032	46114.48	7.16
PEG/14/10/2	39263	37883.96	3.51
PEG/14/10/1	27327	28128.69	2.93
PEF/7/0.5%	9109	9131.27	0.24
PEF/7/1%	19161	18262.55	4.69
PEF/7/2%	32353	36525.10	12.90

On the other hand, it is sufficient to compare pair samples of (PEF/7/0.5%, PEG/7/8), (PEF/7/1%, PEG/7/16) and (PEF/7/2%, PEG/7/32). So, it can be found that apart from applied vertical stress, the value of ΔS in the case of looped fiber reinforced samples is more than that of fiber reinforced treatments. This result is due to the “loop effect” that was discussed previously. A looped fiber as a reinforcing element includes two vertical components. These vertical elements perform like two vertical fibers in a soft-matrix composite. As well, looped fibers have looped-form configurations which create some additional shear resistance. Therefore, at the same orientation, fiber content and fiber length, loop-formed fibers reinforce the soil (as a soft-matrix material) superior than ordinary fibers.

The loop samples of PEG/14/10/2 and PEG/14/10/1 are only different in modulus of elasticity and fiber diameter (see Table 2). The modulus of elasticity of the later sample is about three times greater than that of the former, while the former treatment has approximately twice fiber diameter compared to the later. As it can be seen, loop sample PEG/14/10/2 gives more shear improvement compared to PEG/14/10/1 in all tested vertical stresses. This outcome was as well predicted by the proposed model and shows that the effect of fiber diameter in shear improvement is more important than modulus of elasticity of fiber. This is because the failure mechanism of such a soft-matrix composite is governed by the pullout of the fibers. The more the fiber diameter, the more induced tension force is obtained. Here, the authors emphasize that as the fiber diameter decreases, the stress transfer from matrix to fiber will be improved due to less slippage, but, the induced tension force resisting against inserted shear stress is small.

Altogether, the proposed model adequately predicts shear behavior of soft-matrix composites reinforced with fibers and/or loop-formed fibers. According to the obtained results, the percentage of error for the model is estimated to be 9.64% (12.58% in the case of σ_v = 50 kPa, 9.40% in the case of σ_v = 100 kPa and 6.94% in the case of σ_v = 200 kPa).

Evaluation of the proposed method by new treatments

New treatments of No. 10 and 11 were prepared to evaluate the proposed method (see Figure 8). Treatment of No. 10 consisted of PE looped fibers of sample No. 2, but the surface of the fibers was impregnated by a rapid curing epoxy resin. It was expected that the interface friction of fiber skin and soil particles would be modified by this procedure. In other words, the fiber parameters of sample No. 10 differed from sample No. 2 only in coefficient of friction µ. The pullout test estimated µ to be 0.67 in the case of sample No. 10. The fiber parameters of treatment No. 11 are shown in Table 4.

Table 4.

The fiber parameters of treatment No. 11.

No.	Code No.	Mass fraction (%)	Type of reinforcement	Mesh size (mm)	Fiber diameter (mm)	Number of reinforcements	Elastic modulus (MPa)
11	PEG/5/30	1	Looped Fiber	5.00	1.0413 a	30	230

a

See Figure 4(c).

Accordingly, Table 5 illustrates the experimental results and model outputs (based on λ = “0.52” and a = “2.92”) of shear improvements in the case of these two treatments. As it can be seen, the proposed method predicts ΔS accurately with the error estimation of 8.91%.

Table 5

shear strength improvement (ΔS): comparison between experimental results and model outputs in the case of treatments of No. 10 and 11 (λ = 0.52, a = 2.92).

Code No.	Experimental results (Pa)	Model outputs (Pa)
	σv = 49000 Pa		e%
Neat soil	0	0.00	0.00
PEG/7/16 a	15076	13832.50	8.25
PEG/5/30	21987	17502.33	20.40
	σv = 98000 Pa		e%
Neat soil	0	0.00	0.00
PEG/7/16 a	23243	20264.27	12.82
PEG/5/30	24814	25719.30	3.65
	σv = 196000 Pa		e%
Neat soil	0	0.00	0.00
PEG/7/16 a	27955	30062.34	7.54
PEG/5/30	38006	38307.12	0.79

a

Modified surface.

In the final, an important outcome can be achieved by comparing treatments of No. 10 and 11. The two samples are different in fiber frictional property (0.67 vs. 0.35) and number of reinforcing elements (16 vs. 30). Both of experimental results and model outputs illustrate that ΔS is greater in the case of the later sample compared to the former one (see Table 5). This concept shows that the importance of number of reinforcing elements in soil shear strength improvement is more with regard to fiber frictional property.

Conclusion

The main aim of this paper was to introduce loop-formed fibers as a novel method to reinforce composite materials. Thus, an analytical model was developed to investigate the performance of both fiber and looped fiber soil composites. The presented model demonstrated that shear resistance of loop-formed fibers has two major components including “fiber effect” and “loop effect”. The fiber effect was modeled by using “force-equilibrium method” and “fiber slippage theory” derived from mechanics of short fiber composite. The model is based on the fact that shear stresses in the soil composite mobilize tensile resistance within the fibers, which in turn imparts greater strength to the soil. The “loop effect” means that the existence of soft-matrix, i.e. soil, within the loop leads to produce an additional resisting shear stress for the duration of shear of composite. The produced shear stress performs between “soft matrix surrounded by loop” and soft matrix mass tangential over the loop.

The proposed model indicated that number of reinforcing elements N_f, fiber diameter d_f, Elastic Modulus of fiber E_f, coefficient of friction between fiber and soil matrix µ, vertical compressive stress ${\sigma }_{\text{v}}$ , reinforcing amplitude of fiber D and fiber slippage ratio λ determine shear strength of a fiber reinforced soil composite. Since the two parameters of D and λ refer to fiber performance during shear of soil composite, a semi-empirical method was used to obtain ΔS through the proposed model. Therefore, an ANN was established on data obtained from direct shear tests performed on some treatments. In this way, LSE method was generalized to find the superlative values of D and λ. It was found that one-third of the fiber and/or looped fiber deforms during shear of soil composite, while about half of the deformed segment of fiber bears completely the maximum induced tensile stress.

In addition, coefficient of friction between fiber and soil was measured by a novel pullout test method. Consequently, the Instron Tensile Tester was modified to provide fiber pullout test through the soil matrix. Accordingly, the proposed method predicts ΔS accurately with the error estimation of about 9%.

Funding

The authors gratefully acknowledge the generous support of the New Developments in Civil Engineering R&D Group at the Isfahan University of Technology (IUT) and the funding provided by the Department of Textile Engineering, IUT.