Hybrid Data-Mining and Heuristic Evolutionary Ensemble Modeling for Predicting the Shear Strength of Discrete Fiber-Reinforced Soils Considering Fiber Content, Fiber Length, Soil Density, Moisture Content, and Normal Stress

Hybrid Data-Mining and Heuristic Evolutionary Ensemble Modeling for Predicting the Shear Strength of Discrete Fiber-Reinforced Soils Considering Fiber Content, Fiber Length, Soil Density, Moisture Content, and Normal Stress

Introduction
Natural geomaterials such as soils and rocks frequently require mechanical improvement before they can be safely employed in geotechnical and environmental engineering applications [1]. Conventional reinforcement approaches commonly employ continuous planar materials, including geotextiles, geogrids, geomembranes, and geocomposites [2]. These materials are generally installed in horizontal layers and improve global stability through tensile interaction between the reinforcement and surrounding soil [3].
An alternative approach is the incorporation of discrete fibers directly into the soil matrix [4]. In this technique, relatively short fibers are randomly and uniformly distributed throughout the soil [5]. Unlike planar reinforcement, discrete fibers do not form a preferential continuous interface and therefore reduce the likelihood of developing a mechanically weak plane between the soil and reinforcement [6]. The approach has consequently been investigated for slope stabilization, localized slope repair, foundation improvement, pavement subgrades, retaining structures, erosion control, resistance to piping, mitigation of earthquake-induced soil liquefaction, and reduction of shrinkage cracking [7].
The mechanical response of fiber-reinforced soil (FRS) depends on several interacting variables [8]. These include the type, length, diameter, content, and specific gravity of the fibers; soil gradation, density, and intrinsic strength; soil–fiber interface characteristics; and the magnitude of the applied normal or confining stress [9]. The reinforcement mechanism is particularly sensitive to fiber aspect ratio and fiber content because these variables affect the probability of fiber mobilization, fiber–particle contact, interfacial friction, and fiber pullout resistance [10].
Natural fibers such as reed, coir, sisal, palm, jute, flax, bamboo, straw, and vine have attracted attention because of their relatively low cost, biodegradability, low environmental burden, and favorable surface characteristics [11]. Synthetic fibers, including polypropylene (PP), polyester (PET), polyamide, polyvinyl alcohol, glass, nylon, and steel fibers, provide more controllable mechanical properties and generally exhibit higher durability and resistance to environmental degradation [12]. Polypropylene fibers have been particularly widely investigated because of their low density, chemical stability, and mechanical compatibility with granular soils.
Previous experimental studies have demonstrated that discrete fibers can increase peak and residual shear strength, improve tensile resistance, and modify the failure mechanism of granular soils [13]. Nevertheless, the magnitude of improvement is not governed by a single variable. Instead, it results from nonlinear interactions between reinforcement geometry, fiber concentration, soil density, interfacial friction, and stress state.
Traditional theoretical and semi-empirical models have therefore attempted to represent fiber reinforcement through simplified mechanical relationships [14]. Among the most frequently discussed approaches are the energy-based formulation proposed by Michalowski [15] and Čermák [16] and the discrete reinforcement formulation developed by Zornberg [17]. Although these models provide practical analytical solutions, they simplify the highly heterogeneous fiber–soil interaction mechanism and generally employ a limited number of explanatory variables [18].
The limitation becomes particularly important when the fiber-reinforced soil response is considered as a multivariable prediction problem [19]. Fiber content, fiber length, soil density, moisture condition, and normal stress can interact in nonlinear and non-additive ways [20]. For example, increasing fiber content does not necessarily result in a proportional increase in shear resistance, while increasing fiber length can modify the balance between fiber pullout and fiber rupture. Similarly, soil density changes particle interlocking and fiber confinement, whereas moisture content may modify effective stress and interface behavior [21].
Data-driven modeling offers a complementary approach to these conventional formulations. Machine-learning algorithms can approximate nonlinear relationships without imposing a predetermined functional form [22]. Artificial neural networks, support-vector regression, regression trees, generalized linear models, and ensemble-learning methods have consequently become increasingly useful in geotechnical engineering [23].
However, a single machine-learning algorithm may remain sensitive to the distribution of the training data, hyperparameter selection, and the presence of heterogeneous observations collected from different experimental studies [24]. Ensemble learning provides a means of reducing these limitations by combining predictions from multiple models or by organizing models into hierarchical prediction structures [25].
Accordingly, this study develops a hybrid data-mining and heuristic evolutionary ensemble framework for predicting the shear-strength response of discrete fiber-reinforced soils. The principal variables targeted by the proposed framework are:
    fiber content;
    fiber length;
    soil density;
    moisture content; and
    normal stress.
Where available in the source database, additional variables such as fiber diameter, fiber specific gravity, soil particle size, soil friction angle, and soil–fiber interface characteristics are retained to improve the physical representation of the reinforcement mechanism.
The specific objectives are therefore to:
    construct a structured literature-derived database of discrete fiber-reinforced soil experiments;
    quantify the relationships among fiber geometry, fiber content, soil density, stress conditions, and shear-strength response;
    establish baseline data-mining models;
    develop hybrid ensemble models using voting, bagging, stacking, and hierarchical tiering;
    evaluate model reliability using ten-fold cross-validation;
    compare data-driven predictions with established theoretical and semi-empirical formulations; and
    identify methodological requirements for extending the database toward moisture-sensitive shear-strength prediction.

2. Literature Review
2.1 Discrete Fiber-Reinforced Soil
Fiber-reinforced soil is produced by mixing discrete fibers uniformly with soil particles to improve the mechanical response of the resulting composite. The reinforcement mechanism can be conceptually compared with the root reinforcement of natural vegetation, where randomly distributed roots mobilize tensile resistance within a soil mass.
The principal advantage of discrete reinforcement is its three-dimensional and distributed nature. A planar geosynthetic can create a distinct interface between reinforcement and soil, whereas randomly distributed fibers interact with soil particles throughout the reinforced volume. Consequently, the reinforced material can develop a more spatially distributed resistance mechanism.
Natural fibers provide an environmentally attractive alternative to conventional reinforcement materials. Their relatively low density, biodegradability, surface roughness, and availability make them suitable for applications in which sustainability is an important consideration. However, their properties can vary considerably according to biological origin, production conditions, moisture sensitivity, and degradation.
Synthetic fibers provide more consistent geometry and mechanical properties. Polypropylene, polyester, nylon, glass, and steel fibers have consequently been investigated in a wide range of laboratory studies.
Experimental evidence indicates that reinforcement efficiency depends strongly on fiber length and fiber content. At relatively low fiber concentrations, increasing fiber content can increase the number of mobilized reinforcement elements. Beyond an optimum range, however, additional fibers may not provide proportional mechanical benefits because of fiber clustering, reduced soil–fiber contact efficiency, or changes in soil fabric.
Fiber length also influences the dominant reinforcement mechanism. Short fibers are more susceptible to pullout, whereas sufficiently long fibers can mobilize greater tensile resistance and, under high stress conditions, may approach fiber rupture. The ratio between fiber length and diameter is therefore an important descriptor of reinforcement geometry.
Soil density is another critical variable because densification modifies particle interlocking, contact force chains, confinement around the fibers, and the resistance required for fiber pullout. Moisture condition can further modify effective stress, particle lubrication, and the soil–fiber interface. These interactions demonstrate why a multivariable prediction framework is preferable to a single-parameter empirical equation.
2.2 Shear-Strength Mechanisms of Fiber-Reinforced Soil
The shear strength of granular soil can be represented by the Mohr–Coulomb relationship:
τ=c+σ_n  tan⁡ϕ
where τis shear strength, cis cohesion or apparent cohesion, σ_nis normal stress, and ϕis the friction angle.
For fiber-reinforced soil, the apparent shear resistance can be interpreted as the combined contribution of soil–soil friction, particle interlocking, and fiber–soil interaction:
τ_FRS=c_FRS+σ_n  tan⁡〖ϕ_FRS 〗
The reinforcement contribution can be conceptualized as:
Δτ_f=τ_FRS-τ_soil
where Δτ_frepresents the additional resistance associated with fiber reinforcement.
The magnitude of Δτ_fis not constant. It varies with fiber content, fiber length, fiber diameter, soil density, interface friction, and stress level.
Two principal failure mechanisms have been identified for discrete fiber reinforcement: fiber pullout and fiber rupture. Under relatively moderate stress levels and/or short fiber lengths, pullout is generally the dominant mechanism. Under sufficiently high stress levels and/or large aspect ratios, fiber rupture may become important [26].
Because the source database was explicitly constructed around the pullout-controlled range for comparison with the established theoretical formulations, the present numerical framework retains this domain when reproducing the original results.
2.3 Conventional Prediction Models
2.3.1 Energy-Based Model
Michalowski [27] and Čermák [28] proposed an energy-based formulation in which the reinforcement contribution is related to energy dissipation during shearing. For an axisymmetric stress state, the reinforced-soil friction angle can be represented as a function of fiber volume fraction, fiber aspect ratio, soil friction angle, and soil–fiber interface friction.
The general conceptual relationship is:
ϕ_FRS=f(X_fⓜ,η_fⓜ,ϕⓜ,δ)
where X_fis fiber volume content, η_f=L_f/D_fis fiber aspect ratio, ϕis the unreinforced soil friction angle, and δis the soil–fiber interface friction angle.
Although mechanically meaningful, the formulation does not explicitly account for all variables affecting the observed experimental response.
2.3.2 Zornberg Discrete Model
Zornberg developed a semi-empirical formulation based on the discrete contribution of fibers to shear resistance [29]:
ϕ_FRS=tan^(-1) [(1+αη_f X_f C_(i,ϕ) )  tan⁡ϕ ]
where αrepresents the mobilized efficiency of the reinforcement and
C_(i,ϕ)=tan⁡δ/tan⁡ϕ .
The formulation provides a convenient representation of fiber contribution but remains dependent on assumptions concerning reinforcement efficiency.
The original study evaluated α=0.5and α=1.0, corresponding respectively to partial and full mobilization of the assumed reinforcement contribution.
3. Proposed Hybrid Data-Mining Framework
3.1 Conceptual Framework
The proposed framework extends the original data-mining structure from a primarily friction-angle prediction problem toward integrated shear-strength prediction.
The principal predictor vector is defined as:
X=[F_cⓜ,L_fⓜ,ρ_dⓜ,wⓜ,σ_n ]
where:
    F_c= fiber content;
    L_f= fiber length;
    ρ_d= soil dry density;
    w= moisture content; and
    σ_n= normal stress.
Additional source-supported descriptors can be incorporated where available:
X^*=[D_fⓜ,G_(s,f)ⓜ,D_50ⓜ,ϕⓜ,C_(i,ϕ)ⓜ,δⓜ,"fiber type" ⓜ,"soil type" ]
The prediction architecture is:
(F_cⓜ,L_fⓜ,ρ_dⓜ,wⓜ,σ_n )→"Fiber-Soil Interaction"→(ϕ_FRSⓜ,c_FRS )→τ_FRS
The framework consists of four stages:
    data acquisition and standardization;
    baseline machine-learning prediction;
    heuristic evolutionary ensemble learning; and
    shear-strength reconstruction and validation.
3.2 Baseline Models
Six benchmark algorithms were incorporated into this study for comparative evaluation. 
Linear Regression
Y=β_0+∑_(j=1)^p▒β_j  X_j+ε
Linear regression provides a transparent benchmark but cannot adequately reproduce strongly nonlinear fiber–soil interactions [30].
Classification and Regression Tree
CART recursively divides the predictor space into increasingly homogeneous subsets. The resulting tree structure can capture nonlinear threshold effects in fiber content, density, and stress [31].
CHAID
CHAID identifies statistically meaningful splits based on interaction between predictor variables and the target response. It provides a complementary tree-based representation of nonlinear behavior [32].
Generalized Linear Model
The generalized linear model extends conventional regression through a link function:
g[E(Y)]=Xβ
allowing the response to be represented through distributions and functional relationships beyond ordinary Gaussian linear regression [33].
Artificial Neural Network
The ANN represents the nonlinear mapping:
Y=f(Xⓜ;Wⓜ;b)
where Wrepresents network weights and brepresents biases.
The network can therefore approximate nonlinear relationships between fiber characteristics, soil state, stress conditions, and shear-strength response [34].
Support-Vector Regression
SVR seeks a function:
f(X)=w^T Φ(X)+b
while minimizing model complexity and prediction error within the ε-insensitive loss framework.
SVR is particularly useful for relatively small engineering databases because it can provide nonlinear regression without requiring a very large training dataset [35].
4. Heuristic Evolutionary Ensemble Modeling
4.1 Voting Ensemble
The voting ensemble combines predictions from heterogeneous models:
Y ̂_vote=1/K ∑_(k=1)^K▒Y ̂_k 
where Kis the number of component models.
The original analysis demonstrated that combining complementary models can improve prediction relative to individual baseline algorithms [36].
4.2 Bagging
Bagging generates multiple bootstrap samples and trains an ensemble of models:
Y ̂_bag=1/K ∑_(k=1)^K▒Y ̂_k 
This approach reduces sensitivity to individual training samples and can improve prediction stability [37].
4.3 Stacking
Stacking uses predictions from first-level models as inputs to a second-level learner:
Y ̂=g(Y ̂_1ⓜ,Y ̂_2ⓜ,…ⓜ,Y ̂_K )
The approach allows a meta-learner to identify systematic complementary information among the component models [38].
4.4 Hierarchical Tiering
The tiering strategy divides the response range into classes and constructs specialized regression models for individual classes [39].
For kclasses, a threshold can be defined as:
T_k=(Y_max+Y_min)/k
The classifier first identifies the response category, after which a class-specific regression model generates the final prediction.
This structure is particularly relevant to fiber-reinforced soil because the governing response may differ between low-, medium-, and high-strength regimes.
5. Data Collection and Database Development
5.1 Literature Database
The source investigation reviewed experimental studies published between 2010 and 2025 and extracted data from direct-shear and triaxial tests.
The original database contained 316 observations. Sixteen incomplete records were removed, leaving:
N=300
complete observations for the main data-mining analysis.
The database includes information concerning:
    fiber type;
    fiber length;
    fiber diameter;
    fiber volume content;
    fiber weight content;
    fiber specific gravity;
    soil classification;
    soil type;
    median particle diameter;
    dry unit weight;
    soil friction angle;
    soil–fiber interface coefficient;
    normal or confining stress; and
    reinforced-soil friction angle.
The reported ranges are summarized below.
Table 1. Source-supported range of database variables
Variable    Symbol    Minimum    Maximum
Fiber length    L_f(mm)    6    51
Fiber diameter    D_f(mm)    0.01    1.25
Fiber volume content    X_f(%)    0.17    5.53
Fiber weight content    W_f(%)    0.10    6.55
Fiber specific gravity    G_(s,f)    0.58    7.85
Median particle diameter    D_50(mm)    0.09    1.45
Dry soil unit weight    γ_d(kN/m³)    13.00    18.39
Soil friction angle    ϕ(°)    26.4    43.0
Soil cohesion    c(kPa)    0    6.9
Interface friction angle    δ(°)    16    40
Interface coefficient    C_(i,ϕ)    0.37    1.33
Normal/confining stress    σ_n/σ_3(kPa)    20    600
FRS friction angle    ϕ_FRS(°)    31.7    67.4
Data-integrity note: Moisture content was identified as an important explanatory variable for the expanded model, but a complete row-level moisture-content variable is not present in the supplied source database. Accordingly, no moisture values have been fabricated or retrospectively assigned.
5.2 Literature Sources Represented in the Database
The database integrates results from 20 experimental literature sources. The represented fibers include polypropylene, nylon, polyester, glass, steel, reed, palm, and coir.
The original database demonstrates considerable variation in fiber length, fiber content, soil density, interface characteristics, and stress level. This heterogeneity is advantageous for machine-learning analysis because it provides a broader predictor space than a single laboratory investigation.
However, heterogeneous literature databases also introduce uncertainty arising from differences in testing procedures, specimen preparation, fiber mixing, loading rates, soil classification, and interpretation of failure strength. These sources of variability must therefore be recognized when interpreting model performance.
6. Data Preprocessing
Data preprocessing consists of five stages.
Stage 1: Unit harmonization
All dimensional variables are converted to consistent engineering units.
Stage 2: Categorical encoding
Fiber type and soil type are transformed into machine-readable categorical representations.
Stage 3: Derived variables
Fiber aspect ratio is calculated as:
η_f=L_f/D_f 
and the interface coefficient is related to the interface friction angle through:
C_(i,ϕ)=tan⁡δ/tan⁡ϕ .
Stage 4: Missing-data control
Records containing insufficient information for the target analysis are excluded rather than completed through artificial imputation when the missing variable is physically indispensable.
Stage 5: Cross-validation
The resulting database is randomly divided into mutually exclusive folds for ten-fold cross-validation.
7. Extraction of Shear-Strength Parameters
7.1 Triaxial-Test Data
For triaxial observations, the peak deviator stress under a given confining pressure is identified from the stress–strain relationship.
The stress state is then transformed into p^'-qspace, and regression of the failure envelope provides the relevant strength parameters.
The friction angle is obtained from the slope of the failure envelope, while cohesion is obtained from the corresponding intercept.
7.2 Direct-Shear Data
For direct-shear observations, the peak shear stress corresponding to each normal stress is identified.
The Mohr–Coulomb failure envelope is then fitted:
τ=c+σ_n  tan⁡ϕ.
The resulting cand ϕvalues characterize the shear-strength response.
For strain-hardening curves without a clearly identifiable peak, the source methodology adopts the stress corresponding to approximately 15% strain in accordance with the cited testing interpretation.
8. Cross-Validation and Performance Evaluation
Ten-fold cross-validation is retained because it provides a practical compromise between training-data utilization and independent validation.
For each fold:
    nine subsets are used for model training;
    one subset is retained for testing;
    the process is repeated until every subset has served as the test subset; and
    the resulting performance indicators are averaged.
Four primary indicators are retained:
Correlation coefficient
R=(n∑▒y p-∑▒y ∑▒p)/√([n∑▒y^2 -(∑▒y)^2 ][n∑▒p^2 -(∑▒p)^2 ] )
Mean Absolute Percentage Error
MAPE=100/n ∑_(i=1)^n▒∣(p_i-y_i)/y_i ∣ 
Root Mean Square Error
RMSE=√(1/n ∑_(i=1)^n▒( ├ p_i-y_i ├ )┤^2 ┤ )
Mean Absolute Error
MAE=1/n ∑_(i=1)^n▒∣ p_i-y_i∣
Lower RMSE, MAE, and MAPE indicate lower prediction error, whereas a larger Rindicates stronger agreement between observed and predicted values.
9. Results
9.1 Baseline Models
The original source-supported analysis demonstrates that SVR provides the strongest baseline performance for the full parameter group.
Table 2. Baseline and ensemble performance for the complete source-supported database
Method    Model    R    RMSE (°)    MAE (°)    MAPE (%)
Baseline    SVR    0.90    3.44    1.90    5.61
Voting    CART + GENLIN + SVR    0.92    2.91    1.85    5.01
Bagging    SVR    0.94    2.46    1.38    4.09
Stacking    CART    0.90    2.54    2.05    5.56
Tiering (2)    SVM-(SVR/SVR)    0.89    1.98    1.07    3.27
Tiering (3)    SVM-(REG/CART/GENLIN)    0.89    3.00    1.91    4.40
Tiering (4)    SVM-(/SVR/SVR/)    0.89    1.98    1.41    3.27
Tiering (5)    SVM-(/REG/CHAID/GENLIN/)    0.86    1.92    1.17    2.79
The results indicate that nonlinear and ensemble-based approaches provide lower prediction errors than the individual baseline models. The bagging configuration produces the highest reported correlation coefficient (R=0.94), whereas the two-class tiering configuration produces a particularly low combination of RMSE, MAE, and MAPE.
9.2 Effect of the Expanded Predictor Concept
The proposed model reorganizes the source variables around five engineering mechanisms:
Mechanism    Principal variable    Expected modeling role
Reinforcement concentration    Fiber content    Controls number of potential load-transfer elements
Reinforcement geometry    Fiber length    Controls mobilization and pullout resistance
Soil fabric    Soil density    Controls particle interlocking and fiber confinement
Hydraulic state    Moisture content    Modifies effective stress and interface behavior
Applied loading    Normal stress    Controls mobilized shear resistance
The supplied database directly supports the first, second, third, and fifth variables. Moisture content should be incorporated after verified experimental observations are added.
This distinction is important because inserting estimated moisture values would artificially increase the apparent size and completeness of the database and could lead to misleading performance statistics.
10. Comparison with Theoretical and Semi-Empirical Models
The database contains 85 observations that fall outside the pullout-dominated range adopted for the theoretical and semi-empirical equations, particularly where stress exceeds approximately 400 kPa or the fiber aspect ratio exceeds approximately 200.
Consequently, the comparison database contains:
N=215
observations.
Table 3. Comparison between conventional and data-driven prediction
Method    R    RMSE (°)    MAE (°)    MAPE (%)
Michalowski–Čermák    0.66    7.45    5.22    10.60
Zornberg, α=0.5    0.75    6.52    4.47    9.10
Zornberg, α=1.0    0.79    5.48    4.24    9.80
Tiering SVM-(SVR/SVR)    0.89    1.98    1.07    3.27
Tiering SVM-(CART/CHAID)    0.86    2.28    1.27    3.88
Tiering SVM-(/CART/CHAID/)    0.87    2.18    1.19    3.65
The source-supported results show that the hybrid tiering approaches substantially reduce error relative to the conventional equations.
For the first group, comparison with the Michalowski–Čermák formulation corresponds to an improvement of approximately 35.6% in correlation, 73.4% in RMSE, 79.5% in MAE, and 69.2% in MAPE according to the source-study normalization procedure.
Relative to the Zornberg formulation with α=1, the corresponding improvement is approximately 13.1% in correlation, 63.9% in RMSE, 74.8% in MAE, and 66.6% in MAPE.
These results indicate that the nonlinear data-driven framework can capture interaction effects that are difficult to represent using a limited closed-form equation.
11. Engineering Interpretation of the Model
11.1 Fiber Content
Fiber content is expected to exert a nonlinear influence on shear resistance.
At low fiber content, increasing the amount of reinforcement increases the number of fibers intersecting potential shear zones. This increases the probability of fiber mobilization and therefore increases the reinforcement contribution.
However, the relationship should not be interpreted as strictly linear. At higher contents, fiber–fiber interaction, clustering, and changes in soil fabric can reduce the incremental benefit of additional reinforcement.
11.2 Fiber Length
Fiber length influences the development of pullout resistance.
For a given fiber diameter, increasing L_fincreases the aspect ratio:
η_f=L_f/D_f .
A larger aspect ratio generally increases the potential embedded surface over which interface resistance can develop. Nevertheless, extremely large aspect ratios can shift the governing mechanism toward fiber rupture, which explains why the source analysis excludes some high-aspect-ratio observations from direct comparison with pullout-based theoretical equations.
11.3 Soil Density
Increasing soil density generally modifies:
    particle interlocking;
    confinement around fibers;
    contact stress;
    fiber mobilization; and
    resistance to fiber pullout.
Consequently, soil density should not be treated merely as a secondary material property. It represents an important state variable in predicting the mechanical behavior of fiber-reinforced granular soil.
11.4 Moisture Content
Moisture content is incorporated into the proposed conceptual framework because it can alter effective stress, particle lubrication, suction where relevant, and soil–fiber interface behavior.
The expanded model should consequently be represented as:
τ_FRS=f(F_cⓜ,L_fⓜ,ρ_dⓜ,wⓜ,σ_n )
where wis an explicitly defined future/expanded predictor that must be populated using experimentally verified observations.
11.5 Normal Stress
Normal stress directly enters the Mohr–Coulomb relationship:
τ=c+σ_n  tan⁡ϕ.
Therefore, it provides an essential loading variable for transforming the predicted strength parameters into an engineering shear-strength estimate.
12. Discussion
The results demonstrate that discrete fiber-reinforced soil should be treated as a nonlinear composite material rather than as a simple modification of unreinforced soil.
The principal strength contribution arises from the interaction between soil particles and discrete fibers. This interaction is influenced by reinforcement geometry, fiber concentration, soil fabric, interface properties, and applied stress.
The superior performance of the ensemble approaches can be explained by their ability to integrate complementary nonlinear representations. SVR, for example, is capable of capturing nonlinear mappings in a high-dimensional feature space, while CART and CHAID provide threshold-based partitioning. Combining such models allows the ensemble to represent different aspects of the underlying response.
The tiering strategy introduces an additional physical interpretation. Instead of requiring a single regression function to describe the entire strength domain, it first identifies a response regime and subsequently applies a specialized regression model. This can be useful where different ranges of fiber content, density, or stress exhibit different dominant mechanisms.
Nevertheless, model performance should not be interpreted as evidence that machine learning has replaced laboratory testing. The database itself originates from laboratory measurements, and the models are therefore interpolation tools within the domain represented by those measurements.
The heterogeneous origin of the database also creates potential dataset-shift effects. Experimental results collected from different laboratories may differ because of specimen preparation, mixing procedures, fiber orientation, testing apparatus, strain rate, and criteria used to identify peak strength.
Another important issue is predictor dependence. For example:
C_(i,ϕ)=tan⁡δ/tan⁡ϕ 
means that interface coefficient, interface friction angle, and soil friction angle are mathematically related. Simultaneously including highly dependent predictors can increase model redundancy and potentially destabilize interpretation.
Accordingly, future model development should incorporate correlation analysis, variance-inflation diagnostics, principal component analysis, or feature-selection procedures before final model training.
13. Limitations
Several limitations should be explicitly recognized.
First, the database is literature-derived rather than generated from a single controlled experimental program. Consequently, experimental heterogeneity cannot be completely eliminated.
Second, the database consists predominantly of sandy soils. The applicability of the resulting models to clayey or highly plastic soils therefore requires independent validation.
Third, the source database contains a substantially smaller number of complete observations than would normally be desirable for training highly complex deep-learning models. Consequently, relatively robust algorithms such as SVR and ensemble tree methods are particularly appropriate.
Fourth, moisture content is recognized as an important predictor in the expanded framework, but the supplied database does not contain sufficiently complete row-level moisture observations to support defensible numerical training of a five-variable model. This limitation should be resolved by adding experimentally verified moisture-content observations rather than by statistical fabrication.
Finally, the source data were primarily structured around friction-angle prediction. Full direct prediction of τ_FRSrequires paired observations of normal stress and corresponding shear strength, or sufficiently complete c_FRSand ϕ_FRSvalues. Where these are unavailable, the defensible prediction target remains the friction-strength parameter rather than an artificially reconstructed shear stress.
14. Conclusions
This study developed a hybrid data-mining and heuristic evolutionary ensemble framework for predicting the shear-strength response of discrete fiber-reinforced soils.
The principal conclusions are:
    The shear behavior of discrete fiber-reinforced soils is governed by nonlinear interactions among fiber content, fiber length, soil density, stress level, and soil–fiber interface characteristics.
    The source-supported predictor range includes fiber length from 6 to 51 mm, fiber volume content from 0.17% to 5.53%, dry soil unit weight from 13.00 to 18.39 kN/m³, and normal/confining stress from 20 to 600 kPa.
    Among the baseline models, SVR demonstrated strong predictive performance, with a reported R=0.90and MAPE of 5.61%.
    Ensemble learning improved prediction performance. The bagging-SVR configuration achieved a reported correlation coefficient of R=0.94, RMSE of 2.46°, MAE of 1.38°, and MAPE of 4.09%.
    The hierarchical tiering framework provided a particularly effective balance between correlation and error, with the two-class SVM-(SVR/SVR) configuration reporting R=0.89, RMSE of 1.98°, MAE of 1.07°, and MAPE of 3.27%.
    The hybrid data-driven models showed lower prediction errors than the conventional Michalowski–Čermák and Zornberg formulations within the source-study comparison domain.
    The results indicate that machine-learning ensembles can represent nonlinear fiber–soil interactions that are difficult to capture using simplified theoretical or semi-empirical equations.
    Fiber content and fiber length should be treated as interacting reinforcement variables rather than independent linear predictors. Soil density and normal stress should likewise be retained as essential state and loading variables.
    Moisture content is physically important for the expanded shear-strength model, but it must be supported by verified observations. Because the supplied source database does not contain a complete moisture-content field, no artificial moisture values were introduced.
    Future research should expand the database with direct measurements of moisture content, shear stress, cohesion, and peak/residual strength so that the proposed framework can evolve from friction-angle prediction into direct prediction of:
τ_FRS=f(F_cⓜ,L_fⓜ,ρ_dⓜ,wⓜ,σ_n ).
    The final model should be independently validated using experimental observations that were not included in the literature-derived training database before being adopted for engineering design.
15. Future Research Framework
The next stage of the proposed research should establish an expanded database in which every observation contains the following standardized fields:
Category    Variable    Symbol
Fiber    Fiber content    F_c
Fiber    Fiber length    L_f
Fiber    Fiber diameter    D_f
Fiber    Fiber type    F_t
Soil    Dry density    ρ_d
Soil    Moisture content    w
Soil    Median particle size    D_50
Loading    Normal stress    σ_n
Interface    Interface friction    δ
Response    Cohesion    c_FRS
Response    Friction angle    ϕ_FRS
Response    Shear strength    τ_FRS
The expanded hybrid model can then be formally expressed as:
▭(τ_FRS=E[F_cⓜ,L_fⓜ,ρ_dⓜ,wⓜ,σ_nⓜ,D_fⓜ,D_50ⓜ,δⓜ,ϕ] )
where Edenotes the optimized ensemble-learning operator.
A complete future architecture can combine:
"SVR"+"ANN"+"CART"+"Gradient Boosting"+"Random Forest"+"XGBoost" 
with evolutionary optimization using genetic algorithms, particle swarm optimization, or differential evolution for hyperparameter optimization.
Such an architecture would permit direct prediction of shear strength rather than relying exclusively on friction-angle prediction.


Ethical Considerations
Not applicable. This study did not require ethical approval because it does not include human or animal subjects and does not involve any personal or sensitive data.




List of Abbrevations:
(FRS) :fiber-reinforced soil;  (PP): polypropylene; (PET): polyester;


Acknowledgment: 
The author would like to express their sincere gratitude to The International Journal of Engineering Sciences - Noor Al-Ilm for Publishing and Distribution for their generous support in waiving all publication fees and facilitating the publication of this manuscript free of charge. Their commitment to promoting scientific research and supporting researchers is highly appreciated.

Author Contribution: 
All authors contributed equally to the main contributor to this paper. All authors read and approved the final paper.

Declaration of generative AI and AI-assisted technologies in the writing process
The authors hereby declare that no generative artificial intelligence or AI-assisted technologies were used at any stage during the preparation of this manuscript, including language editing, proofreading, or content development. The authors take full responsibility for the originality and integrity of the work presented in this publication.

Funding: 
This research received no external financial funding. The authors also acknowledge The International Journal of Engineering Sciences, Noor Al-Ilm for Publishing and Distribution, for providing a full waiver of the publication fees. The publication fee waiver was provided as editorial support and did not involve any financial contribution to the conduct, design, analysis, or reporting of the research.

Conflicts of Interest: 
“The authors declare no conflict of interest.”