Predicting the Tensile Properties of Carbon FRCM Using a LASSO Model

Abstract

The use of Fibre Reinforced Cementitious Matrix (FRCM) for structural retrofitting requires prior assessment of the composite’s mechanical properties, particularly its tensile stress–strain response. This paper presents a LASSO regression model applied to 107 uniaxial tensile tests on Carbon FRCM in order to investigate the impact of both the material and testing parameters on FRCM performance. A highly effective LASSO regression model was trained using k-fold validation, resulting in concise and comprehensible models. Within the testing parameters, both the gripping system and load–speed ratio significantly affected the performance. A substantial impact on ultimate values was found for the load–speed ratio. From the material-related parameters, the most influential was the textile coating in terms of strength and the existence of bilinear or trilinear behaviour. It was also concluded that the combination of textile and matrix properties influenced the stress–strain response at all stages, with high-performance mortars resulting in better textile-to-matrix interaction.

1. Introduction

1.1. FRCM for Structural Retrofitting

Fibre Reinforced Cementitious Matrix (FRCM) is an alternative to Fibre Reinforced Polymers (FRP); it is broadly employed for strengthening and retrofitting structures thanks to its high strength-to-weight ratio [1,2,3]. The FRCM systems have several advantages with respect to FRP, such as vapour permeability, the ability to be applied onto wet surfaces, a higher compatibility with masonry structures, a higher temperature resistance, and even fire resistance; these perform similarly to concrete and masonry structures when exposed to fire [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. FRCM are composite materials consisting of a textile or fabric embedded in a mortar matrix. The textile is usually conformed with an orthogonal grid, which can be made of organic or inorganic fibres including carbon fibres, alkaline-resistant (AR) glass, basalt, aramid, or synthetic polymer reinforcements such as Polyphenylene Benzobisoxazole (PBO). Mortars can be cementitious (more frequently) or lime-based, and can include several additions (e.g., polymers, fibres, or fly ash) [10,18,19,20,21,22,23,24,25,26,27,28,29].

The lack of knowledge on this topic, together with the diversity of the performance of FRCM systems, makes it necessary to assess the mechanical properties of these materials through tensile tests before actual applications are performed (as encouraged by ACI434 (2011) [30] and CNR-DT 215/2018 Guide [31]). The mechanical properties can differ significantly among FRCMs depending on both the textile and matrix characteristics. The bond between the grid and the mortar has a strong influence on the overall behaviour of the composite, being responsible of the load transfer mechanism and potentially resulting in premature failure. FRCMs may present a premature slippage of the fibres within the matrix, resulting in a loss of the composite behaviour and thus limiting the overall performance of the reinforcements. If the matrix is unable to fully penetrate inside the yarn, fibre rovings are left partially unimpregnated, which results in different behaviours of the inner filaments with respect to the outer ones [32,33,34]. Therefore, the inner filaments may have a slippage (telescopic behaviour) which can be prevented by means of coatings or impregnation systems that are able to penetrate inside the rovings (Figure 1). At the same time, such coatings may serve to improve the adhesion to the matrix and to provide additional protection to the fibres [32,35].

FRCM systems may show, under uniaxial tensile loading, an idealized stress–strain curve, which can be bilinear or trilinear, as shown in Figure 2. Each branch of the curve corresponds to a cracking state. First, the elastic phase occurs (stage A), in which the load is supported by an uncracked matrix. Once the tensile strength of the mortar is reached, the first crack appears and thence the slope of the stress–strain curve decreases. The matrix is responsible for transferring the load to the embedded fibres, representing the beginning of the second stage (stage B). The third phase (stage C), which can either be present or not, starts when the matrix is fully cracked, and the cracks only increase their width; at this stage, the textile mainly sustains the load [3,19,36,37]. The observance of stage C mainly depends on the slippage between fibres and matrix, particularly the composites with a low-strength matrix [3,27,37,38]. According to the literature, the gripping system, as well as the mortar properties (tensile strength, and the presence of fibres or other additives) and textile coating, influence the presence (or not) of the third branch [39,40,41].

1.2. Direct Tensile Testing of FRCM

Direct tensile tests of FRCM are necessary before designing a retrofitting solution for an existing structure, as the mechanical properties of composites cannot be estimated with merely the characteristics of the matrix and the textile. Tensile tests provide the following data: stress, strain, and stiffness in each stage, which enables the identification of the stress–strain curve as either trilinear or bilinear. As stated above, the observance of bilinear behaviour can be related to the slippage of fibres within the matrix. To date, there exists no standardised tensile test set-up, and the CNR-DT 215/2018 Guide [31] provides no recommendations for testing arrangement. On the other hand, ACI434 (2011) [30] includes several recommendations regarding the following: (i) the velocity of the increasing load (v = 0.2 mm/min monotonically); (ii) the minimum width related to the spacing grid; and (iii) the use of metallic tabs in the gripping area to avoid damage.

The variability of test parameters and mechanical performance of FRCM systems in tension are the main motivations for this study. Previous results by the authors, presenting significant differences within the literature, suggest the relevant role of several aspects including the coating of the textile and test variables such as the speed rate of loading or the gripping system. Thence, the proposed modelling approach employs a database collected from direct tensile tests conducted on carbon FRCM, which is presented in more detail in Section 2.1.

Despite the lack of agreement on testing parameters, the most widely supported systems are clevis and clamping, as illustrated in Figure 3. Both support systems have similar prevalence; 43% of the specimens from the database were tested with clevis, and the remaining 53% with clamping. The speed of cross-head displacement varies between 0.2 mm/min and 1 mm/min. Some FRCM were tested non-monotonically, meaning that the velocity increased after cracking [2,37,42]. Regarding the size parameters of the specimens, the tested widths of the database range from 40 mm to 125 mm; the thickness ranges from 6 mm to 30 mm; and the length varies between 260 mm and 650 mm.

Several methods of data acquisition were employed in the database; some systems were used simultaneously [13,27,36,38,43,44]. Digital Image Correlation (DIC) was used in 26% of the tests [13,27,43,44,45], generally when the support system is clamping. DIC allows better observation of cracking patterns and, therefore, of the performance of the interface between fibres and matrix at all load stages.

1.3. Research Significance

The present research aims at evaluating the influence of different parameters affecting the performance of FRCM in a state of tension. The model has three main objectives: (i) to assess the mechanical properties and stress–strain curve main points as a function of the testing and material parameters; (ii) to evaluate the influence of test variables on the overall performance in tensile tests, to provide guidance for test standardization; and (iii) provide further insight into the material parameters (matrix and textile properties) on the mechanical properties of the composite material.

Points two and three are regarded as essential given the wide ranges of results and the lack of unified knowledge on FRCM mechanical characterization, which prevents its widespread use.

In this research, a LASSO (Least Absolute Shrinkage and Selection Operator) regression was adopted as presented in Section 2.3. Model introduction. LASSO regression, a variant of linear regression, is designed to tackle the issue of multicollinearity and overfitting in predictive modelling. It achieves this through the introduction of an λ regularization term that encourages sparsity, effectively shrinking less-influential predictor variables to zero. This dimensionality reduction technique is particularly beneficial when dealing with high-dimensional datasets, where feature selection is essential for model interpretability and efficiency [46]. This regression approach was selected in this study for its dual capability in variable selection and regularization, effectively addressing multicollinearity among numerous predictors. This feature is particularly valuable for creating a parsimonious model that identifies key variables impacting FRCM performance, thereby enhancing interpretability. Ridge and Principal Component Analysis (PCA) regressions, though valid alternatives, were not pursued as they do not facilitate the same degree of model simplicity and clarity in variable influence, respectively, which are critical for understanding complex FRCM behaviours [47,48]. On the other hand, the limited dataset of 107 observations constrained our ability to apply more complex machine learning models, which typically require larger sample sizes to perform effectively [49].

Therefore, in the context of our study, where numerous parameters may influence FRCM performance, LASSO provides a robust analytical framework for identifying the most influential factors. This approach not only enhances the interpretability of the model but also mitigates issues related to multicollinearity, offering a more accurate and parsimonious representation of the complex relationships within the data. Consequently, the use of LASSO models in this research offers methodological rigour and precision, ensuring that the insights gained contribute meaningfully to the broader understanding of FRCM behaviour under tensile loading. The steps followed in this research are detailed in Figure 4.

2. Data Collection and Analysis

2.1. Database

A database of 107 FRCM systems under uniaxial tensile load was collected from previously published works since 2015 [2,3,13,15,23,27,32,36,37,38,42,43,44,45,50,51,52,53,54,55,56] to investigate the parameters affecting test results. A protocol was designed for the creation of the database, in which the selection of papers was motivated by the inclusion of experimental research, within the Web of Science corpus of papers dealing with Carbon FRCM materials, by the inclusion of “carbon” either in the abstract or in the title. Only specimens that have results for strength, strain, and the Young’s modulus at each stage collected numerically are included in the database. This means that the research was restricted to Carbon Fibre Reinforced Cementitious Matrix (CFRCM), since carbon is the most widely used fibre type for concrete retrofitting. Therefore, the results of the model may not be applicable to other fibres as the potential differences in the fibre–matrix bond cannot be regarded merely through the studied parameters affecting the performance (the tensile strength of the textile and coating).

The database includes all the information related to the properties of the matrix and textile, the specimens’ characteristics, monitorization and testing, and results, given as follows:

• Matrix properties: compressive strength, tensile strength, and stiffness.
• Textile properties: grid spacing in the direction of loading and perpendicular to loading direction (also referred to as warp density and weft density), coating (if any), density, tensile strength, stiffness, and ultimate strength.
• Specimen characteristics: length, width, thickness, number of tensile longitudinal threads, textile cross-section, or volumetric percentage of reinforcement.
• Monitorization and testing: type of monitorization and data acquisition, test type, and test parameters (test speed).
• Results: identification of the modulus of elasticity and tensile strength for each branch, the number of identifiable branches in the stress–strain behaviour under tensile stresses.

In order to be able to perform a LASSO model, the variables without statistical significance were removed. The independent variables collected in the database and used in the model, as well as their corresponding coding, can be seen in

Table 1. Independent variables from the database.

ID	Name of Variable	Type of Variable
fcmat	Compressive strength of matrix	Scale
ftf	Tensile strength of the textile	Scale
Ef	Stiffness of the textile	Scale
SL	Spacing of the grid in the direction of loading	Scale
ST	Spacing of the grid in the direction perpendicular to loading	Scale
C	Coating (if any) of the fibres	Nominal
v	Test speed of increasing load	Scale
SS	Support system; this can be clevis or clamping	Nominal
t	Thickness of the specimen	Scale
b	Width of the specimen	Scale
l	Length of the specimen	Scale

. The response variables are detailed in

Table 2. Response variables.

ID	Symbol	Name of Variable	Type of Variable
O1	σTA	Tensile strength of stage A	Scale
O2	ETA	Young’s modulus of elasticity of stage A	Scale
O3	σTB	Tensile strength of stage B	Scale
O4	εTB	Strain of stage B	Scale
O5	ETB	Young’s modulus of elasticity of stage B	Scale
O6	σu	Tensile strength at peak	Scale
O7	εu	Ultimate strain	Scale
O8	Eu	Young’s modulus of elasticity of stage C	Scale

.

Table 3. Descriptive_input.

	fcmat	ftf	Ef	SL	ST	v	t	b	l
count	105.00	91.00	92.00	99.00	99.00	104.00	106.00	103.00	103.00
mean	40.45	2737.22	198.70	14.22	14.98	0.46	13.54	70.72	461.54
std	22.49	1269.22	69.78	5.69	6.63	0.28	7.23	27.39	91.97
CV	55.60%	46.37	35.12%	40.01%	44.26%	60.87%	53.40%	38.73%	19.93%
min	6.50	441.00	1.80	8.50	7.00	0.15	6.00	40.00	260.00
25%	17.00	1863.00	196.40	10.00	10.00	0.25	10.00	50.00	410.00
50%	39.56	2125.00	223.00	10.00	10.00	0.30	10.00	60.00	450.00
75%	70.20	4300.00	240.00	20.00	20.00	0.50	15.00	85.00	525.00
max	79.40	4900.00	263.00	30.00	30.00	1.00	30.00	125.00	650.00
nan	1.00	15.00	14.00	7.00	7.00	2.00	0.00	3.00	3.00
mode	16.40	4300.00	240.00	10.00	10.00	0.50	10.00	50.00	410.00

and

Table 4. Descriptive_output.

	O1	O2	O3	O4	O5	O6	O7	O8	O9
count	98.00	97.00	102.00	106.00	104.00	82.00	106.00	104.00	83.00
mean	659.50	0.04	2620.13	1108.17	0.66	109.80	1400.46	0.80	155.20
std	511.10	0.05	2667.23	804.18	0.50	109.16	1202.14	0.52	170.14
CV	77.50%	125.00%	101.80%	72.57%	75.76%	99.42%	85.84%	65.00%	109.63%
min	86.10	0.01	18.00	196.00	0.01	1.00	196.00	0.01	8.60
25%	204.75	0.02	512.50	524.22	0.24	46.63	617.78	0.43	51.00
50%	569.65	0.03	1540.93	1031.50	0.62	76.00	1201.50	0.80	105.90
75%	888.28	0.05	3727.25	1358.75	0.96	135.00	1498.83	1.12	190.00
max	2495.00	0.48	11,707.00	5808.00	2.25	680.00	6159.00	2.25	799.00
nan	8.00	9.00	4.00	0.00	2.00	24.00	0.00	2.00	23.00

illustrate a comprehensive descriptive analysis of the raw database presented in Figure 5 and Figure 6.

The input and output parameters were graphically collected in the correlation matrix as presented in Figure 7, from which it is possible to observe several trends and connections between the variables.

As can be observed from Figure 7, there is a strong correlation between related output variables, i.e., between tensile stresses for stages A, B, and C, and for strains for stages B and C. This is caused by the possibility of the non-existence of branch C in some specimens, but it also indicates that there is a major influence of the first stage upon the subsequent behaviour of the composite material. This aspect could be appreciated in [45], in which the idealized stress–strain curves of FRCM systems from several tests from the bibliography were collected in a graph, allowing us to observe that the higher stresses in the pre-cracked state were related to the higher ultimate stresses. Finally, Figure 7 suggests a weak-to-moderate correlation between sample size parameters and output variables. Specifically, parameter b shows a moderate negative correlation with O1, O3, and O5, and parameter l with O5 and O8, indicating a potential, but limited, influence of specimen configuration on test results—a relationship that has been noted in other studies on fibre-reinforced cementitious materials [57].

2.2. Data Preprocessing

2.2.1. Data Imputation

Data imputation is a critical step in ensuring the integrity and reliability of our dataset for analysis. In Table 3, missing values (NaN) in various input variables are identified as follows: ‘fcmat’ with 1 missing value, ‘ftf’ with 15 missing values, ‘Ef’ with 14 missing values, ‘SL’ with 7 missing values, ‘ST’ with 7 missing values, ‘v’ with 2 missing values, ‘t’ with no missing data, ‘b’ with 3 missing values, and ‘l’ with 3 missing values. Notably, the output data also contained missing values, as evident in Table 4; however, we made the deliberate decision not to impute these output values. The imputation method employed here leverages the mode, the most frequently occurring value within each respective input variable, whose values for each case are indicated in Table 3. This approach ensures that our imputed data retain the inherent characteristics of the original dataset while allowing us to proceed with robust analyses, even in the presence of incomplete information. This mode-based imputation approach aligns with established practices in statistical analysis, as it preserves data distribution and maintains dataset integrity for reliable regression modelling [58,59].

2.2.2. Outliers’ Treatment

In the context of this research, Cook’s distance analysis is applied to identify possible outliers by assessing the influence of individual data points on regression models involving the considered eight distinct output variables. This method gauges the impact of each observation on the regression coefficients by systematically fitting the model both with and without that observation, subsequently quantifying the magnitude of parameter changes. Specifically, Cook’s distance is computed by measuring the multivariate distance between model parameters estimated using the entire dataset and parameters obtained when a particular data point is excluded. This distance is then scaled by the residual sum of squares and the number of predictors. Points with Cook’s distance values exceeding a predefined threshold are identified as influential outliers, potentially skewing model results. This procedure forms a critical component of the research methodology, aiding in the detection and management of influential data points across multiple regression analyses. This technique has been reported as a very effective method in improving regression performance, as per several pieces of research [60,61,62]. Figure 8 presents Cook’s distance analysis for the O2 response, while the number of observations removed in each case is presented in

Table 5. Number of observations per response before and after removing outliers.

Response	Original Number of Observations	Observations to Be Removed	Final Number of Observations for Training LASSO Models
O1	98	6	92
O2	102	4	98
O3	106	5	101
O4	104	3	101
O5	82	6	76
O6	106	5	101
O7	104	3	101
O8	83	2	81

.

2.3. Model Introduction

This paper explores the use of LASSO (Least Absolute Shrinkage and Selection Operator) regression, a powerful statistical method, to accurately adjust regression models for predicting the performance of FRCM materials under direct tensile test conditions. To the authors’ knowledge, this study represents the first known application of LASSO regression for predicting the tensile FRCM performance; however, previously published works have presented successful applications in predicting the behaviour of other complex fibre-reinforced cementitious composites under direct tensile or flexural loadings [63,64].

In the context of polynomial regression, the selection of parameters β₀, β₁, …, β_p that effectively minimize the sum of squares of the residuals (RSS) is determined by Equation (1) [46,64] as follows:

$$RSS=\sum _{i=1}^n{\left(y_i-{\beta }_0-\sum _{j=1}^p{\beta }_j{x}_{ij}\right)}^2$$

(1)

In contrast, the coefficients derived from the LASSO regression are determined by minimizing the expression depicted in Equation (2) [46,64] as follows:

$$\sum _{i=1}^n{\left(y_i-{\beta }_0-\sum _{j=1}^p{\beta }_j{x}_{ij}\right)}^2+\mathsf{\lambda }\sum _{j=1}^p\left|{\beta }_j\right|$$

(2)

The penalty parameter, denoted as λ, is a non-negative value. When the value of λ exceeds a certain threshold, it is possible for certain coefficients to be precisely reduced to zero. This property of the method serves as a variable selector, allowing for the creation of simpler models and facilitating the interpretation of the resulting model. The selection of an optimal value for the parameter λ holds significant importance in this statistical modelling, as the efficiency and effectiveness of the approach are contingent upon this choice. Consequently, it becomes imperative to employ a systematic approach or methodology to determine the appropriate value for the penalty parameter. One viable approach is to employ k-fold cross-validation. During the training process of the model utilizing k-fold cross-validation, a range of λ values is selected (i.e., from λ = 0.00001 to λ = 100, which encompasses 10,000,001 interactions for each k partition); subsequently, the validation root-mean-square error (RMSE) is computed for each λ value. In the study presented herein, a k value of 10 was selected. The optimal value of λ, which minimizes the error, is determined through a selection process. Subsequently, the model is recalibrated using all available observations and the chosen value of λ [46,64,65]. The selected λ value for each response model is presented in

Table 6. Number of observations per response before and after removing outliers.

Variable	Type	O1	O2	O3	O4	O5	O6	O7	O8
λ	Parameter	0.17185	8.74607	43.71260	0.00021	1.53808	37.71957	0.00011	2.76502
Intercept	Lineal	−1080.5492	4220.71564	1529.57966	−1.951815	129.69499	1851.83024	−1.84532	246.51386
t	Lineal	-	-	-	0.05244	-	-	0.07136	-
fcmat	Lineal	49.97333	16.02402	-	-	-	-	−0.01103	-
v	Lineal	−533.68788	-	-	0.71211	-	-	-	-
b	Lineal	-	-	−10.36494	0.02960	-	-	0.03176	-
l	Lineal	0.11113	-	-	0.00133	-	-	0.00254	−0.10579
SL	Lineal	126.02625	-	-	0.08360	-	-	0.10760	-
ST	Lineal	1.02557	-	-	0.04541	-	-	0.08398	-
Ef	Lineal	-	-	-	−0.00196	-	-	-	-
ftf	Lineal	−0.03275	-	-	-	-	-	−7.37 × 10−05	-
SS	Lineal	1884.8536	1026.35973	-	0.85759	-	-	-	-
C	Lineal	2416.64108	-	-	0.72912	-	-	-	-
t_fcmat	Interaction	−0.85624	−0.29003	-	−0.00147	-	−0.49378	−0.00063	−0.07319
t_v	Interaction	-	-	-	0.19599	-	-	0.08785	−1.63038
t_b	Interaction	0.07833	-	-	-	−0.00477	-	0.00017	-
t_l	Interaction	0.09640	-	-	7.30 × 10−05	−0.00199	-	0.00011	-
t_SL	Interaction	0.00065	2.80652	-	−0.00821	-	-	−0.00643	-
t_ST	Interaction	-	-	-	-	-	-	-	−0.08217
t_Ef	Interaction	-	-	-	-	-	-	−8.80 × 10−05	-
t_ftf	Interaction	-	-	-	7.84 × 10−06	-	-	-	-
t_SS	Interaction	−15.79442	-	-	0.020627	2.25390	-	0.03677	3.02028
t_C	Interaction	−50.04543	-	−9.47848	0.04389	-	−44.15476	−0.00621	−9.07314
fcmat_v	Interaction	17.48055	-	-	−0.01036	-	-	−0.00569	-
fcmat_b	Interaction	−0.12322	-	-	−0.00061	-	-	−0.00065	-
fcmat_l	Interaction	0.01957	-	-	0.00010	-	-	3.98 × 10−05	-
fcmat_SL	Interaction	−0.40605	8.55617	-	0.00031	-	-	0.00028	−0.02669
fcmat_ST	Interaction	-	-	0.01973	5.75 × 10−05	-	0.31957	0.00056	-
fcmat_Ef	Interaction	0.02112	-	-	−4.25 × 10−05	−0.00447	-	−4.21 × 10−05	-
fcmat_ftf	Interaction	−0.00882	−0.02369	-	-	-	-	-	-
fcmat_SS	Interaction	−17.01157	37.02676	1.36691	−0.00802	-	-	0.00827	-
fcmat_C	Interaction	−1.05882	−31.33745	-	−0.01118	−1.18493	-	0.00534	-
v_b	Interaction	−13.05169	−45.76146	-	0.009964	−0.52841	−6.47701	0.01270	-
v_l	Interaction	0.85194	-	-	−0.00455	-	-	−0.00265	−0.15473
v_SL	Interaction	−54.60457	-	-	0.14743	-	-	0.07710	-
v_ST	Interaction	-	−109.39538	17.19090	0.01577	0.43927	-	0.07210	-
v_Ef	Interaction	5.58488	11.12098	1.18158	−0.01199	-	3.15255	−0.00536	0.43078
v_ftf	Interaction	-	1.55710	-	−9.95 × 10−05	-	-	0.00018	0.02196
v_SS	Interaction	−3.30525	−3619.3552	-	−1.17770	-	-	−2.15643	-
v_C	Interaction	−772.23127	3869.89198	298.91922	0.35410	126.69180	4074.7616	0.03091	342.81744
b_l	Interaction	0.00372	-	-	−2.84 × 10−05	−0.00028	-	−3.51 × 10−05	-
b_SL	Interaction	-	0.97261	−0.00242	0.00104	-	-	0.00150	-
b_ST	Interaction	-	-	-	0.00016	−0.01343	-	0.00032	-
b_Ef	Interaction	0.02377	0.05445	-	-	-	-	−3.33 × 10−05	-
b_ftf	Interaction	0.00046	-	-	3.88 × 10−06	-	-	8.71 × 10−06	-
b_SS	Interaction	18.07249	14.11763	-	0.00846	-	-	7.24 × 10−03	-
b_C	Interaction	−2.86340	−27.10386	-	−0.01165	−0.11839	−15.46691	-	−1.96501
l_SL	Interaction	−0.02455	-	-	−0.00033	-	-	−0.00042	-
l_ST	Interaction	−0.03044	-	-	-	-	-	-	-
l_Ef	Interaction	-	-	-	1.95 × 10−05	-	-	3.41 × 10−06	-
l_ftf	Interaction	−0.00062	−0.00120	-	−5.27 × 10−07	1.85 × 10−05	-	1.40 × 10−07	-
l_SS	Interaction	−5.90392	−4.09624	-	−0.00020	-	-	0.00014	-
l_C	Interaction	-	2.30317	-	1.54 × 10−05	-	-	−0.00037	-
SL_ST	Interaction	−1.99106	−7.15118	-	-	-	-	−0.00234	-
SL_Ef	Interaction	-	-	-	−0.00013	-	-	−2.81 × 10−05	-
SL_ftf	Interaction	0.02314	0.05316	-	−4.27 × 10−06	-	-	−1.00 × 10−05	-
SL_SS	Interaction	26.72357	-	-	−0.08911	-	-	−0.07794	-
SL_C	Interaction	−35.66590	-	-	0.00036	0.52578	-	0.01819	-
ST_Ef	Interaction	−0.12982	−0.58577	-	−8.36 × 10−05	-	-	−5.75 × 10−05	-
ST_ftf	Interaction		-	-	−1.15 × 10−05	-	-	−9.69 × 10−06	-
ST_SS	Interaction	2.44207	-	-	-	-	-	-	-
ST_C	Interaction	1.41889	83.40750	-	0.00910	-	-	0.02932	-
Ef_ftf	Interaction	-	-	-	-	3.33 × 10−05	-	-	-
Ef_SS	Interaction	0.99707	-	-	0.00222	−0.10975	−2.17723	0.00088	-
Ef_C	Interaction	2.16414	-	-	-	0.09757	1.22845	−0.00192	0.05136
ftf_SS	Interaction	0.09861	−0.01918	-	−5.96 × 10−05	-	-	0.00013	−0.37311
ftf_C	Interaction	−0.09481	0.05063	-	−0.00012	-	-	1.14 × 10−05	0.06120
SS_C	Interaction	−1942.8705	−2102.3335	-	−0.15304	-	-	−0.52274	−7.93388
t2	Squared	-	-	-	−0.00091	-	-	−0.00105	-
fcmat2	Squared	−0.16403	−0.29107	-	0.00026	-	-	0.00033	-
v2	Squared	14.35983	-	-	0.19151	-	-	−0.61315	-
b2	Squared	−0.02899	-	-	−0.00019	-	−0.02203	−0.00023	-
l2	Squared	0.00283	−0.00087	-	-	-	-	2.04 × 10−06	-
SL2	Squared	−2.86847	-	-	0.00059	-	−0.97159	-	-
ST2	Squared	1.06919	-	-	0.00085	-	-	−0.00076	-
Ef2	Squared	−0.01157	−0.04597	-	-	-	-	6.78 × 10−06	-
ftf2	Squared	3.13 × 10−05	-	-	5.14 × 10−08	-	-	−4.31 × 10−08	-
SS2	Squared	-	-	-	-	-	-	-	-
C2	Squared	-	-	-	-	-	-	-	-

, along with the model definition.

Despite the advantages of variable selection and regularization offered by this statistical procedure, a notable limitation is its potential challenge with highly correlated predictors. This can lead to biased coefficient estimates and diminished predictive accuracy [64]. However, as illustrated in Figure 7, although certain output variables exhibit strong correlations, the absolute correlation value among predictors remains below 0.7 in all instances, except for the relationship between ST and SL. This correlation is expected, as it pertains to grid spacing, but due to their mechanical implications in terms of responses, both variables must be considered in the model.

3. Results and Discussion

3.1. Model Definition

Table 6 puts forward the models’ definition, including the λ value for each one.

3.2. Model Performance Evaluation

The performance evaluation of the LASSO models is a pivotal aspect of our study, providing a comprehensive understanding of their predictive capabilities. A multifaceted approach to assess model accuracy is presented. Firstly, regression plots for all eight models, visualizing the relationships between the actual and predicted values, are put forward in Figure 9. These plots not only offer valuable insights into the goodness of fit but also reveal the predictive trends and variations within our dataset. In addition to graphical analysis, we calculate statistical metrics to quantitatively assess model performance. The coefficient of determination (R²) provides a measure of how well the models explain the variance in the data. Moreover, we compute the Normalized Mean Bias Error (NMBE) and Mean Absolute Error (MAE) to assess bias and overall accuracy. Finally, we calculate the p-value for each model, allowing us to determine the significance of their predictive power.

Table 7. Models’ significance and performance evaluation.

Response	R2	NMBE (%)	MAE	p-Value
O1	0.921	0.000%	85.244	2.22 × 10−16
O2	0.726	0.000%	910.625	1.21 × 10−15
O3	0.4297	0.000%	361.313	9.06 × 10−03
O4	0.796	0.000%	0.136	1.62 × 10−02
O5	0.637	0.000%	32.435	1.91 × 10−04
O6	0.830	0.000%	396.861	3.82 × 10−06
O7	0.829	0.000%	0.118	4.40 × 10−02
O8	0.884	0.000%	41.955	1.77 × 10−07

provides the afore mentioned information about LASSO models’ performance. The combination of graphical and statistical evaluation ensures a comprehensive appraisal of the LASSO models, enabling us to draw robust conclusions about their suitability for our specific research context.

3.3. Impact of Material and Test Parameters on Performance

The impact of material-related and test-related parameters on the behaviour of the composite is analysed using the equation for tensile strength at peak (O6) obtained with the LASSO model. The stress variation is compared by changing the input parameters one by one, aiming at isolating the ideal effect each parameter could have on the output.

Equation (3) for calculating the approximate tensile strength at peak is given as follows:

σ_u ≈ 1851.8302 − 0.4938·t·fcmat + 0.3196·fcmat·ST − 6.4770·v·b + 3.1525·v·Ef − 2.172·Ef·SS − 44.1548 t·C + 4074.7616·v·C − 15.4669·b·C + 1.2284·Ef·C − 0.0220·b² − 09716·SL²

(3)

The independent variables length of specimen (l) and tensile strength of the textile (ftf) do not appear in the equation; therefore, according to the proposed model, they have no influence on the tensile strength at peak of the FRCM. With regard to this, it is worth mentioning that as a consequence of the model being proposed for carbon composites, the tensile strength of the textile has a wide range of values; however, the mode is 4300 MPa, close to the highest value, indicating that the results from the model may vary if non-carbon fibres are included, providing the database with a more significant weight of low values.

First, the influence of the fibre coating on tensile strength at peak is evaluated. This is a material-related parameter that may have some influence, as previously reported, on the bilinear or trilinear behaviour of the composite. Test parameters are set to the minimum values of the database (t = 6 mm, v = 0.2 mm/min and b = 40 mm), and the tensile strength at peak obtained for each support system and the extreme properties of material as a function of the presence or absence of coating are compared.

Table 8. Values of tensile strength at peak depending on the coating of the fibres.

		O6 (σu)
Material Properties	SS	Uncoated	Coated
fcmat = 6.5 MPa; SL = 30 mm; ST = 30 mm; Ef = 66 GPa	Clevis	831 MPa	844 MPa
fcmat = 6.5 MPa; SL = 30 mm; ST = 30 mm; Ef = 66 GPa	Clamping	975 MPa	987 MPa
fcmat = 79.4 MPa; SL = 8.5 mm; ST = 7 mm; Ef = 263 GPa	Clevis	1230 MPa	1485 MPa
fcmat = 79.4 MPa; SL = 8.5 mm; ST = 7 mm; Ef = 263 GPa	Clamping	1803 MPa	2057 MPa

shows the results obtained for each combination.

It can be observed from Table 8 that the O6 values are higher when fibres are coated, regardless of the combination of parameters. The influence of coating has a greater impact when the number of fibres is higher, and the properties of the matrix and fibres are better. Predictions from Table 8 are consistent with the observed results for O6 (See Figure 10) for coated and uncoated carbon fibres, in which the use of a clamping system achieves higher ultimate strengths; the results are remarkable for some of the data corresponding to coated textiles.

To assess the effect of the support system (SS), one of the main test parameters considered in this study, the test setup from previous analysis is retained (t = 6 mm, v = 0.2 mm/min and b = 40 mm), and the results obtained with the extreme values as a function of the mesh coating are compared in

Table 9. Values of tensile strength at peak depending on support system.

		O6 (σu)
Material Properties	Coated	Clevis	Clamping
fcmat = 6.5 MPa; SL = 30 mm; ST = 30 mm; Ef = 66 GPa	Uncoated	831 MPa	975 MPa
fcmat = 6.5 MPa; SL = 30 mm; ST = 30 mm; Ef = 66 GPa	Coated	844 MPa	987 MPa
fcmat = 79.4 MPa; SL = 8.5 mm; ST = 7 mm; Ef = 263 GPa	Uncoated	1230 MPa	1803 MPa
fcmat = 79.4 MPa; SL = 8.5 mm; ST = 7 mm; Ef = 263 GPa	Coated	1485 MPa	2057 MPa

.

The values obtained with clamping as a support system are always higher than those with clevis. This effect is more pronounced with better material properties, particularly if the fibres are uncoated, meaning that slippage is more common. The predictions are in good agreement with the data for clamping and clevis systems, as can be observed from Figure 11.

The test speed of the increasing load is evaluated by comparing the result obtained with the maximum and minimum values from the database (v = 0.2 mmm/min and v = 1 mm/min). These comparisons are made for specimens with a width of 40 mm (b) and a thickness of 6 mm (t), along with all the combinations of support systems, coatings, and extreme properties of the materials. The results are collected in

Table 10. Values of tensile strength at peak depending on the speed test of the load increased.

		O6 (σu)
Material Properties	Coated-SS	v = 0.2 mm/min	v = 1 mm/min
fcmat = 6.5 MPa; SL = 30 mm; ST = 30 mm; Ef = 66 GPa	Uncoated Clevis	831 MPa	791 MPa
fcmat = 6.5 MPa; SL = 30 mm; ST = 30 mm; Ef = 66 GPa	Uncoated Clamping	975 MPa	934 MPa
fcmat = 6.5 MPa; SL = 30 mm; ST = 30 mm; Ef = 66 GPa	Coated Clevis	844 MPa	4063 MPa
fcmat = 6.5 MPa; SL = 30 mm; ST = 30 mm; Ef = 66 GPa	Coated Clamping	987 MPa	4206 MPa
fcmat = 79.4 MPa; SL = 8.5 mm; ST = 7 mm; Ef = 263 GPa	Uncoated Clevis	1230 MPa	1686 MPa
fcmat = 79.4 MPa; SL = 8.5 mm; ST = 7 mm; Ef = 263 GPa	Uncoated Clamping	1803 MPa	2259 MPa
fcmat = 79.4 MPa; SL = 8.5 mm; ST = 7 mm; Ef = 263 GPa	Coated Clevis	1485 MPa	5200 MPa
fcmat = 79.4 MPa; SL = 8.5 mm; ST = 7 mm; Ef = 263 GPa	Coated Clamping	2057 MPa	5773 MPa

.

An increase in the speed test of the increasing load generally results in an improvement in the composite ultimate strength. In the concurrence of a low strength matrix, small fibre content with low stiffness, and uncoated fibres, an increase in speed will cause a slight decrease in ultimate strength. The strength at peak increases significantly (more than doubles) when the mesh is coated, irrespective of the other material properties with the highest speed ratio.

The influence of the compressive strength of the matrix on the strength at peak of the composite is evaluated by maintaining the test setup (t = 6 mm, b = 40 mm, and v = 0.2 mm/min) and textile properties (SL = 30 mm, ST = 30 mm, and Ef = 66 GPa) and comparing how the tensile strength at peak changes according to the coating and support system.

Table 11. Values of tensile strength at peak depending on the compressive strength of the matrix.

		O6 (σu)
Coated	SS	fcmat = 6.5 MPa	fcmat = 79.4 MPa
Uncoated	Clevis	831 MPa	1314 MPa
Uncoated	Clamping	975 MPa	1458 MPa
Coated	Clevis	844 MPa	1327 MPa
Coated	Clamping	987 MPa	1470 MPa

shows the values obtained.

The strength at peak of the composite increases with the strength of the matrix if the composite has a low content of low-stiffness fibres. The variation is virtually zero when the textile has better properties since the mesh is responsible for bearing the loads. A similar trend can be observed for pairs of predicted values when fcmat is changed. In this sense, apart from the beneficial effects of coating and clamping, the highest value of fcmat results in strength at peak values for each combination of support system and coating/uncoating ranging from 1.49 to 1.58, corresponding to the lowest value of matrix compressive strength.

The model predicts a maximum value for strength at peak corresponding to the clamping support system, with a load–speed ratio of 1 mm/min (maximum from the database), which, with proper maximisation of the remaining parameters, provides a prediction of 5773 MPa. This is the theoretical maximum attainable value for the database. This supports the notion that the test speed has a significant impact upon the overall performance of FRCM in tensions, thus suggesting the need to create testing protocols that set both the support system and the speed. The remaining parameters of greater impact are the strength of the matrix, related to the contribution in the pre-cracked and cracked state, and the textile coating, which is related to the adherence of the textile matrix and to the potential to develop a third phase. Also, as previously noted in [3], the gripping method should be included when defining a procedure for material characterization.

4. Conclusions and Future Works

A LASSO model was elaborated based on a database comprising 107 carbon FRCM systems subjected to uniaxial tests. The development of the model allowed for the identification of some material-based and test-based parameters with a high impact upon the stress –strain performance of the FRCM. The following conclusions are drawn:

• The models’ significance and performance allow for the conclusion that LASSO regression has proven to be highly effective, accurately predicting the properties of FRCM under direct tensile loadings while providing streamlined and interpretable models through optimal variable selection and mitigation of multicollinearity.
• With respect to the test parameters, both the gripping system and load–speed ratio have a significant influence on the results. The influence is greater for ultimate values (ultimate load and, if present, stiffness in the third branch of the stress–strain curve). The observed high impact of the load–speed ratio should be properly addressed when designing an experimental set-up. It is proposed that a relatively low speed be employed for conservative performance values until further evidence on this parameter exists. For the support system, the beneficial influence of clamping is not considered unconservative, as actual FRCM reinforcements are not subjected to equivalent stress concentrations as those caused by the clevis system, which leads to relatively poor performance.
• Of the material-related parameters, it is concluded that the combination of textile and matrix properties has a significant impact upon the stress–strain curve at all stages.
• The parameter of highest impact is the coating of the textile, as it affects not only the ultimate load and strain but also the existence of bilinear or trilinear behaviour, thus indicating a better interaction between fibres and matrix when the textile is coated. The influence of coating is greater upon the ultimate tensile load when combined with high matrix strengths.
• Similarly, the relevance of the mechanical properties of the matrix on the cracking development and ultimate load was observed, consistent with the enhanced load-carrying capacity of the matrix during the evolution of cracking.

Moreover, the application of LASSO regression in this study not only showcased its ability to effectively address the challenges inherent in predicting FRCM properties, but also demonstrated its potential as a valuable tool for advancing the state of the art in engineering research on construction materials. The combination of accurate predictions, interpretability, and resilience to multicollinearity positions LASSO as a robust analytical approach for unravelling the intricate dynamics of FRCM under direct tensile loadings.

For future works, the results from the LASSO model will be compared with other methodological approaches to ensure comprehensive analysis. The identification of the key parameters of the FRCM performance is a departure point for the regularization of test set-up and for the study of the optimal combination of material properties for the performance of the composite material. Particularly, subsequent investigations should explore coating systems with optimized mortars, which appears to be a promising avenue for optimization. Additionally, standardizing the procedure for FRCM material characterization is imperative. Future efforts will involve testing the same material using a clamping system and fixed specimen size while varying the load–speed ratio. These endeavours will provide valuable insights into this parameter and contribute to further improving model accuracy and predictive capabilities.