Machine learning-based prediction of ultimate load in pultruded glass fibre column under axial compression

ABSTRACT

This experimental study investigates the axial compression effect of rectangular pultruded glass fiber-reinforced polymer (P-GFRP) tubular column sections, examining the impact of width-to-thickness ratio (B/t), aspect ratio (H/B), and column height on their structural performance. A total of 27 GFRP columns were subjected to axial compression tests to evaluate their ultimate load and initial stiffness. The columns exhibited a uniform failure pattern, characterized by crushing, mid-section fractures, and longitudinal splitting at the corners. The results revealed a negative correlation between the ultimate load and the aspect ratio, as well as the width-to-thickness ratio. This study utilized advanced machine learning algorithms, namely Response Surface Methodology (RSM) and Artificial Neural Network (ANN), to develop predictive models for the ultimate load of GFRP columns. The RSM model achieved an R² value of 0.8347, demonstrating good accuracy in predicting ultimate load. The ANN model outperformed the RSM model, with an R criterion exceeding 0.68807 across training, testing, and validation phases, showing a stronger correlation between experimental and predicted outcomes. This research establishes a framework for forecasting the mechanical properties of column sections.

Keywords:
Axial compression; Pultruded GFRP Columns; Ultimate load; ANN; RSM

1. INTRODUCTION

The pultrusion method of fiber composite materials exhibits a unique combination of exceptional mechanical properties, including high strength-to-weight ratios, corrosion resistance, and versatility, making them an attractive option in the construction industry. The pultrusion process combines glass fibers with a polymer matrix, resulting in lightweight and durable profiles. These composites offer significant advantages over traditional materials, making them an attractive alternative for industries like construction, aerospace, robotics, etc. The growing use of GFRP composites reflects ongoing advancements in composite technology, enabling innovative solutions and expanding applications [¹]. The material’s crushing and stability influence the collapse processes, while local fiber buckling and resin fracture often lead to global failure. These components are generally manufactured by a pultrusion method and possess slender walls. Despite their numerous advantages, GFRP materials display significant deformability perpendicular to fiber alignment and tend to fail in a brittle manner, limiting their application and warranting further investigation [²]. Glass Fiber Reinforced Polymer (GFRP) composites are susceptible to various failure mechanisms that can jeopardize their structural integrity and long-term performance. Nevertheless, they offer a unique combination of benefits, leveraging the strengths of glass fibers and polymer matrices to achieve exceptional lightweight and high-strength properties. The pultrusion process enables the manufacture of GFRP composites with outstanding mechanical characteristics, including resistance to corrosion, environmental degradation, and enhanced durability. This makes them an attractive option for construction and infrastructure applications. Rigorous characterization of both the constituent materials and the final products is essential to ensure consistent quality and performance. Furthermore, GFRP’s adaptability, durability, and reliability have established it as a pivotal material in contemporary engineering, driving innovation and advancements in various fields [³]. It provides an economical substitute for the fabrication of high-quality prismatic components that mimic steel and possess excellent longitudinal properties. The subsequent section provides an in-depth examination of the pultrusion manufacturing process, including the various fiber configurations and characterization methods utilized. Additionally, it examines pultruded fiber reinforced polymer (FRP) as a sustainable material, emphasizing its advantages in infrastructure development, building construction, and related disciplines. This discussion sheds light on the prospects of pultruded FRP in revolutionizing traditional construction materials and methods [⁴]. To ascertain the ideal wear rate of the smaller characteristic, the composite section—which consisted of hand layup, VARTM, and RTM—was validated using Taguchi and ANNOVA [⁵].

A comprehensive understanding of their behaviour requires a combined experimental and computational approach. Mechanical testing provides valuable insights into their structural performance under various conditions, while computational modelling predicts their response and failure mechanisms. The synergy between these approaches reveals crucial details about fiber/matrix interactions, interphase characteristics, and overall performance [⁶]. Recent breakthroughs in interphase engineering have significantly enhanced the mechanical properties and durability of GFRP composites, expanding their potential applications. Research on pultruded Glass Fiber fibre-reinforced polymer (GFRP) materials has yielded valuable insights into their fracture behaviour. Studies have investigated the tensile and compressive properties of pultruded GFRP composites with varying fiber orientations, revealing a dependence on fiber alignment and loading direction. Additionally, the interlaminar fracture toughness of pultruded GFRP laminates has been evaluated using double-cantilever beam (DCB) testing, identifying interfacial debonding as the primary failure mechanism [⁷]. Glass and carbon fibers are the most prevalent in North America, with epoxy or vinyl ester serving as the predominant matrices. Aramid fibers and polyester resins are also used, while GFRP’s affordability makes them a popular choice. Conversely, glass fibers possess a low elastic modulus and are prone to degradation in alkaline environments [⁸]. Carbon FRPs exhibit a modulus of elasticity comparable to that of steel, although they are significantly highly costly. Aramid FRPs have limitations, including creep susceptibility and moisture absorption. In a comparison of matrix types, vinyl ester showed better fracture toughness and energy absorption than epoxy [⁹, ¹⁰].

Pultruded GFRP composites’ mechanical properties depend on microstructure, fiber orientation, and matrix properties. Fracture behaviour is crucial for reliability and safety, and loading conditions impact it. Cyclic loading causes fatigue failure, while static loading leads to brittle fracture. Research showed loading frequency, and stress levels affect fatigue life. Previous studies found fracture behaviour is highly orthotropic, varying with load direction [¹¹]. Pultruded GFRP composites typically exhibit two fracture modes: interlaminar (occurring between layers) and intralaminar (within a single layer). Research has demonstrated that internal flaws can cause material properties to vary spatially, potentially leading to premature failure in areas with lower fiber content [¹²]. Notably, the maximum strength of closed sections exceeded that of open sections by approximately 280%. Therefore, it is recommended to consider closed sections for temporary bridges, as the misuse of open-section composites has led to local damage and adverse effects. Micromechanical models, rooted in material mechanics, treat composites as a continuum and predict fracture behaviour through stress-strain relationships. Fracture mechanics models view composites as a collection of discrete elements and predict fracture behaviour using energy release rates [¹³].

Response Surface Methodology (RSM) is a powerful experimental design technique for modelling and analyzing complex issues where multiple variables affect a response of interest. This methodology has been widely used for optimizing experimental processes. RSM excels in detecting potential interactions between experimental variables, determining the optimal response when faced with conflicting priorities, and generating functional data frameworks with minimal effort [¹⁴]. RSM has proven effective in predicting the ultimate load of PFRP box profiles, providing reliable results. However, while RSM has been used to predict durability and various connection processes reliably, its application in evaluating the impact of adhesives and glass fibers on the ultimate load of PFRP profiles remains unexplored [¹⁵, ¹⁶]. Artificial Neural Networks (ANN) have emerged as a powerful computational tool for predicting and modelling complex systems, including material properties in construction. ANN operates by mimicking the structure of biological neural networks, with interconnected layers of neurons that process data and learn patterns through training. Unlike traditional statistical methods like Response Surface Methodology (RSM), ANN offers greater flexibility and accuracy in capturing nonlinear relationships between input variables and output responses, making it highly effective for material performance prediction. It has been demonstrated to be a potent modelling tool for complicated and nonlinear situations, with significant applications in learning and function approximation [¹⁷] Training ceases when the progression of generalization stagnates. The testing data do not influence training, therefore ensuring an unbiased performance. Following multiple experiments, the optimal output of the final load was achieved using a basic neural network architecture. This research investigates the use of statistical methods to optimize the ultimate load of glass fiber-reinforced PFRP box sections bonded with epoxy adhesives [¹⁸].

2. EXPERIMENTAL CHARACTERIZATION

2.1. Specimen details

A depiction of pultruded GFRP tubular columns is shown in Figure 1. The purpose of this test is to evaluate the impact of the pultruded GFRP rectangular tubular stub column’s axial compression performance. In total, twenty-seven test specimens were created. E-glass fibers and epoxy resin are combined to form the GFRP tubular column specimen, which is produced by the pultrusion process. The E-glass fiber composed 60% by volume, combined with epoxy resin known for its high strength and superior bonding properties. The column specimen has a various rectangular cross-section, lengths of 114, 203, and 310 mm, and a 6, 8 and 10 mm wall thickness. According to GB/T 1448–2005 were recommended the methods of testing compressive strength and ensures standardized loading rates, specimen preparation [¹⁹, ²⁰]. The details of pultruded GFRP columns have been shown in Table 1 below.

Figure 1
Schematic diagram and photos PFRP tubular column.

Table 1
Details of pultruded GFRP columns.

SPECIMEN LABEL	SPECIMEN CROSS-SECTION	H (mm)	B (mm)	T (mm)	B/t	H/B
PR7-4-4-H1	75 × 40 × 4	114	40	4	10	2.85
PR7-4-4-H2	75 × 40 × 4	203	40	4	10	5.07
PR7-4-4-H3	75 × 40 × 4	310	40	4	10	7.75
PR7-4-6-H1	75 × 40 × 6	114	40	6	6.67	2.85
PR7-4-6-H2	75 × 40 × 6	203	40	6	6.67	5.07
PR7-4-6-H3	75 × 40 × 6	310	40	6	6.67	7.75
PR5-5-6-H1	50 × 50 × 6	114	50	6	8.33	2.28
PR5-5-6-H2	50 × 50 × 6	203	50	6	8.33	4.06
PR5-5-6-H3	50 × 50 × 6	310	50	6	8.33	6.2
PR10-5-6-H1	100 × 50 × 6	114	50	6	8.33	2.28
PR10-5-6-H2	100 × 50 × 6	203	50	6	8.33	4.06
PR10-5-6-H3	100 × 50 × 6	310	50	6	8.33	6.2
PR10-5-8-H1	100 × 50 × 8	114	50	8	6.25	2.28
PR10-5-8-H2	100 × 50 × 8	203	50	8	6.25	4.06
PR10-5-8-H3	100 × 50 × 8	310	50	8	6.25	6.2
PR10-5-10-H1	100 × 50 × 10	114	50	10	5	2.28
PR10-5-10-H2	100 × 50 × 10	203	50	10	5	4.06
PR10-5-10-H3	100 × 50 × 10	310	50	10	5	6.2
PR15-7-6-H1	150 × 75 × 6	114	75	6	12.5	1.52
PR15-7-6-H2	150 × 75 × 6	203	75	6	12.5	2.71
PR15-7-6-H3	150 × 75 × 6	310	75	6	12.5	4.13
PR15-7-8-H1	150 × 75 × 8	114	75	8	9.37	1.52
PR15-7-8-H2	150 × 75 × 8	203	75	8	9.37	2.71
PR15-7-8-H3	150 × 75 × 8	310	75	8	9.37	4.13
PR15-7-10-H1	150 × 75 × 10	114	75	10	7.5	1.52
PR15-7-10-H2	150 × 75 × 10	203	75	10	7.5	2.71
PR15-7-10-H3	150 × 75 × 10	310	75	10	7.5	4.13

2.2. Test process

One strain gauge is placed on the GFRP tubular column’s exterior at half of the column height, and another strain gauge is placed on the opposite side. The strain gauge was attached to a dual-channel strain indicator, and the strains were monitored in both longitudinal and transverse directions. The sensitivity coefficient of the BX120-3AA strain gauges was 2.081%, resolution of 120Ω resistance, <±0.1% tolerance. Accurate strain measurement is ensured by the sensitivity coefficient, which is essential for detecting slight changes in deformation during loading. A static strain acquisition system was used to measure the yield and ultimate displacement. Each specimen was positioned in the center of the 2000 kN hydraulic compression testing machine’s end plate before the test [²¹].

During the axial compression test, the load was automatically recorded by the testing apparatus. The loading mechanism and the strain gauge placement on the specimen are shown in Figure 2. The centroid of the column section aligned with the geometric center of the steel end plates, ensuring that the axial compression loading criterion was met without any eccentricity. One end plate was secured to a top fixture linked to the spiral arm via a hinge, while the other end plate was affixed to the hydraulics. The upper and lower fixtures established a pin-fixed condition for the column portion under compression. To minimize errors associated with rapid loading, a controlled load was applied. Initially, an initial load equivalent to 10% of the anticipated load-bearing capacity of the GFRP tubular column was used in incremental stages for each phase, ensuring a controlled and gradual loading process. Surface strain data were collected from the specimens 2-3 minutes after static loading. The axial load was subsequently applied continuously until it reached a level equivalent to 70% of the expected ultimate load. This measurement may be approximate and could encompass the closure of gaps between components of the loading apparatus. Nonetheless, as all columns were subjected to the identical configuration, their axial displacements could be qualitatively compared. The primary emphasis of the current investigation is the critical loading of the column, which has minimal association with the measurement of axial displacement. As the specimens approached their failure point, the increment of the load was reduced to 5% of the predicted ultimate load for each subsequent step, allowing for a more precise assessment of their behaviour under increasing load. When the column reached their ultimate load, they exhibited a sudden and pronounced load drop, which triggered the compression testing apparatus to automatically halted.

Figure 2
Specimen setup with strain gauges at heights (a) 114 mm, (b) 203 mm, (c) 310 mm.

2.3. Test results

This section provides an in-depth examination and discussion of the primary parameters for each specimen, encompassing failure modes, load-displacement curves, and load-strain curves. A detailed summary of the essential properties, including initial stiffness (K), ultimate load (Pu), yield displacement (Δy), ultimate displacement (Δu), and failure modes, is presented in a concise and organized manner in Table 2.

Table 2
Experimental test results.

SPECIMEN LABEL	B/t	H/B	Pu, (KN)	K, (kN/mm)	Δu, (mm)	Δy, (mm)	FAILURE MODE
PR7-4-4-H1	10	2.85	87.61	50.90	1.92	1.68	Mode one
PR7-4-4-H2	10	5.07	63.47	15.86	4.24	3.87	Mode one
PR7-4-4-H3	10	7.75	54.28	9.12	5.98	5.72	Mode two
PR7-4-6-H1	6.67	2.85	91.57	55.92	1.94	1.6	Mode one
PR7-4-6-H2	6.67	5.07	82.61	23.17	4.05	3.44	Mode one
PR7-4-6-H3	6.67	7.75	76.57	12.22	6.42	6.03	Mode two
PR5-5-6-H1	8.33	2.28	212.34	136.00	1.91	1.54	Mode one
PR5-5-6-H2	8.33	4.06	208.84	64.97	3.45	3.17	Mode one
PR5-5-6-H3	8.33	6.2	198.53	41.98	5.15	4.67	Mode two
PR10-5-6-H1	8.33	2.28	222.24	228.90	1.32	0.96	Mode one
PR10-5-6-H2	8.33	4.06	256.37	90.99	3.22	2.79	Mode one
PR10-5-6-H3	8.33	6.2	208.81	42.45	5.04	4.86	Mode two
PR10-5-8-H1	6.25	2.28	235.17	249.32	1.24	0.93	Mode one
PR10-5-8-H2	6.25	4.06	198.24	67.45	3.31	2.89	Mode one
PR10-5-8-H3	6.25	6.2	190.63	37.02	5.43	5.06	Mode two
PR10-5-10-H1	5	2.28	240.86	212.11	1.34	1.12	Mode one
PR10-5-10-H2	5	4.06	202.49	137.61	2.94	1.45	Mode one
PR10-5-10-H3	5	6.2	211.38	52.76	4.79	3.95	Mode two
PR15-7-6-H1	12.5	1.52	205.42	192.82	1.32	1.05	Mode one
PR15-7-6-H2	12.5	2.71	190.53	193.37	1.24	0.97	Mode one
PR15-7-6-H3	12.5	4.13	187.64	59.85	3.46	3.08	Mode two
PR15-7-8-H1	9.37	1.52	211.21	223.55	1.21	0.93	Mode one
PR15-7-8-H2	9.37	2.71	201.38	110.65	2.05	1.79	Mode one
PR15-7-8-H3	9.37	4.13	196.75	64.91	3.53	2.98	Mode two
PR15-7-10-H1	7.5	1.52	219.24	242.39	1.03	0.89	Mode one
PR15-7-10-H2	7.5	2.71	208.47	169.39	1.57	1.21	Mode one
PR15-7-10-H3	7.5	4.13	199.29	62.35	3.48	3.14	Mode two

2.3.1. Failure modes

The GFRP tubular column specimen shows signs of crushing mode of failures. As the load progressively rises, the specimens’ failure process can be divided into four phases. Stage 1: During the initial loading phase, the material exhibits no discernible deformation. However, as compression damage evolves at the interface between the test specimen and the end plates, signs of corner damage become progressively more pronounced. Stage 2: The specimen starts to exhibit mid-span buckling, accompanied by subtle sounds indicative of fiber rupture. Stage 3: Longitudinal ripping events appear as the buckling deformation of the stub column in the middle portion gets stronger. Stage 4: The longitudinal ripping phenomena become more noticeable, and finally, the GFRP tubular columns fail with a loud tearing sound. The specimens’ failure processes are comparable in stages (1) and (2). On the other hand, in stage (3), The longitudinal rupture of the rectangular tube demonstrates two separate causes of failure. The first mode of failure involves minimal longitudinal rupture at the corners of the column, with the majority of the damage concentrated in the mid-height region.

Figure 3(a) illustrates the GFRP tube fractures in one-third of the column from the top edge as the load increases. This is primarily because the pultruded GFRP tube fibers are oriented in an axial orientation, giving more strength in an axial direction. Nonetheless, the GFRP tube’s strength is comparatively low perpendicular to the fiber arrangement in the transverse direction [²²]. The specimen fractures in the middle because of the column’s bigger cross-section and lesser transverse strength. The second mode of failure: As seen in Figure 3(b), the longitudinal rip is mostly focused on the corner and buckling at mid-section, with less obvious damage in half of the column height.

Figure 3
(a,b,c) Failure mechanisms of Pultruded GFRP tubular columns.

2.3.2. Load vs displacement curves

The load-displacement curves exhibit three distinct stages: elastic, elastoplastic, and decreasing, which share similar shapes across various specimens. Figure 4 illustrates the load-displacement curves for pultruded GFRP tubular columns, showcasing these characteristic stages. Nine variations of width-to-thickness ratios and twenty-seven variations of pultruded GFRP tubular column sections were presented and compared against the ultimate load. The specimens PR7-4-4-H1 to PR7-4-6-H3 exhibit an average ultimate load of approximately 78 kN and a yield load of approximately 61 kN. A brittle failure mode was seen in all specimen types during loading. Figure 4(a) to (h) indicates that all specimens attained their maximum ultimate load due to the brittle characteristics of fiber materials. Specimens PR5-5-6-H1 to PR15-7-10-H3 provide superior outcomes, with an average ultimate load of approximately 210 kN and a yield load of around 185 kN, determined by the width-to-thickness ratios [²³]. The B/t ratios have a crucial influence on the strength performance of pultruded GFRP materials. The increase in the B/t ratio demonstrates enhanced load-carrying ability up to a certain optimum level; beyond this level, the ultimate load of the column member progressively diminishes. The test findings indicate that the optimal B/t ratio for the pultruded GFRP column member is PR10-5-6-H2, with an ultimate load of 256.37 kN. Similarly, an increase in the height of the specimen influences the ultimate load of the column. The specimens exhibit enhanced deformation ability when the B/t ratio increases, as seen by the elastic stage marginally decreasing and the descending stage lagging.

Figure 4
Load vs displacement plots for pultruded GFRP tubular columns B/t = (a) 5, (b) 6.25, (c) 6.67, (d) 7.5, (e) 8.33, (f) 9.37, (g) 10 and (h) 12.5.

The correlation between the aspect ratio and the ultimate load is described in 5(a). Figure 5(b) illustrates the correlation between the B/t ratio and the ultimate load. Likewise, there exists an inverse correlation between the B/t ratio and ultimate load [¹³]. As the B/t ratio increases, the ultimate load diminishes. A reduced cross-sectional area under compression arises from an increased B/t ratio, which corresponds to a decrease in the tube’s thickness. The ultimate load decreased by 2%, 46.24%, 49.74%, 15.36%, 48.35%, 51.28%, 12.05%, 9.46%, and 13.23% when the B/t ratio was augmented from 5, 6.25, 6.67, 7.5, 8.33, 8.33, 9.37, 10, to 12.5. Table 2 illustrates the correlation between the B/t ratio and initial stiffness. The B/t ratio and initial stiffness exhibit no significant correlation, and an increase in the B/t ratio does not substantially affect the initial stiffness. As the B/t rises, the specimen’s initial stiffness diminishes, as seen. From 5 to 12.5, the B/t ratio increases, but the initial stiffness falls by 4.23%, 35.68%, 41.24%, 12.42%, 39.29%, 42.36%, 8.92%, 5.46% and 7.57%. The primary reason behind this is that the energy needed to cause damage increases with the GFRP column’s width and thickness. The yield displacement and final displacement for each specimen are shown in Table 2.

Figure 5
The impact of test parameters on the ultimate load (a) aspect ratio and (b) B/t ratio.

3. MACHINE LEARNING METHODS

3.1. RSM

RSM is a powerful statistical approach in machine learning that uncovers the complex relationships between multiple input variables and an output response. By constructing a mathematical model, RSM reveals the optimal conditions for achieving desired outcomes, enabling the fine-tuning of processes. This technique excels at analysing variable interactions and identifying the ideal combination of inputs to optimize performance, whether maximizing or minimizing outcomes. RSM’s efficiency and effectiveness have made it a popular choice in machine learning for parameter optimization, streamlining the process while reducing the need for extensive experimentation and fostering data-driven decision-making. The approach involves three primary steps: designing experiments to collect data, developing and validating a numerical model, and optimizing mixture constituents’ proportions to achieve desired responses. A higher F-value indicates a more significant effect of the corresponding parameter, while the P-value represents the significance or insignificance of the model’s output. RSM is valuable for identifying optimal conditions, providing insight into variable interactions, and creating predictive models that guide input factor tuning to improve performance, particularly in machine learning applications where computational resources are a constraint [¹⁸]. A probability value less than 0.05 indicates that the model or criteria are statistically significant. Historical data has been analysed using Design Expert software to develop a Response Surface Methodology (RSM) model [²⁴].

3.2. ANN

The mechanical properties of Pultrude GFRP stub column data have been modelled utilizing an Artificial Neural Network (ANN) with MATLAB ‘nntool’ version R2020. The feed-forward artificial neural network employed the multilayer Levenberg-Marquardt algorithm and the perceptron model method for learning propagation to process the experimental data and perform the task [¹⁶, ¹⁷]. The output layer comprises neurons that generate the final network output based on the inputs from the preceding layers. In a feedforward network, data flows unidirectionally from input to output. As shown in Figure 6, the input, hidden, and output layers are interconnected through a straightforward linkage. The utilized ANN consists of three layers: an input layer (incorporating aspect ratio and B/t ratio data of GFRP stub columns), a hidden layer (indicating ultimate load), and an output layer [¹⁷].

Figure 6
ANN model functional diagram.

4. RESULTS AND DISCUSSION

Preliminary research indicates that glass fiber significantly influences the mechanical properties of Pultruded GFRP tubular stub column. Table 3 presents the outcomes of axial compression tests performed on PFRP box sections of several diameters to enhance the profiles’ cross-sectional design. The findings indicated that the cross-section of PFRP profiles with PR10-5-6-H2 exhibited superior ultimate load compared to the others. The ultimate load of PFRP box profiles demonstrates that including 0.45% (by volume) of glass fibers at the four corners of the box section maximizes the ultimate load. Soft computing methodologies, such as RSM and ANN, have been utilized to ascertain the predictive capability of a variable amalgamation that enhances ultimate load. [¹⁵]. RSM regression analysis was employed to construct expert software, while MATLAB software (version R-2020) was utilized to develop artificial neural network models.

Table 3
Axial compression test results.

SPECIMEN LABEL	SPECIMEN CROSS SECTION (mm)	H (mm)	B/t	H/B	ULTIMATE LOAD (kN)
PR7-4-4-H1	75 × 40 × 4	114	10	2.85	87.61
PR7-4-4-H2	75 × 40 × 4	203	10	5.07	63.47
PR7-4-4-H3	75 × 40 × 4	310	10	7.75	54.28
PR7-4-6-H1	75 × 40 × 6	114	6.67	2.85	91.57
PR7-4-6-H2	75 × 40 × 6	203	6.67	5.07	82.61
PR7-4-6-H3	75 × 40 × 6	310	6.67	7.75	76.57
PR5-5-6-H1	50 × 50 × 6	114	8.33	2.28	212.34
PR5-5-6-H2	50 × 50 × 6	203	8.33	4.06	208.84
PR5-5-6-H3	50 × 50 × 6	310	8.33	6.2	198.53
PR10-5-6-H1	100 × 50 × 6	114	8.33	2.28	222.24
PR10-5-6-H2	100 × 50 × 6	203	8.33	4.06	256.37
PR10-5-6-H3	100 × 50 × 6	310	8.33	6.2	208.81
PR10-5-8-H1	100 × 50 × 8	114	6.25	2.28	235.17
PR10-5-8-H2	100 × 50 × 8	203	6.25	4.06	198.24
PR10-5-8-H3	100 × 50 × 8	310	6.25	6.2	190.63
PR10-5-10-H1	100 × 50 × 10	114	5	2.28	240.86
PR10-5-10-H2	100 × 50 × 10	203	5	4.06	202.49
PR10-5-10-H3	100 × 50 × 10	310	5	6.2	211.38
PR15-7-6-H1	150 × 75 × 6	114	12.5	1.52	205.42
PR15-7-6-H2	150 × 75 × 6	203	12.5	2.71	190.53
PR15-7-6-H3	150 × 75 × 6	310	12.5	4.13	187.64
PR15-7-8-H1	150 × 75 × 8	114	9.37	1.52	211.21
PR15-7-8-H2	150 × 75 × 8	203	9.37	2.71	201.38
PR15-7-8-H3	150 × 75 × 8	310	9.37	4.13	196.75
PR15-7-10-H1	150 × 75 × 10	114	7.5	1.52	219.24
PR15-7-10-H2	150 × 75 × 10	203	7.5	2.71	208.47
PR15-7-10-H3	150 × 75 × 10	310	7.5	4.13	199.29

4.1. RSM analysis

The design considerations for the RSM technique are presented in Table 4. Twenty-seven distinct tests have been conducted to ascertain the design type using standard evaluation criteria. Regression model results of the ultimate load of a column and the different cross-sections and thicknesses of the PFRP profiles are shown in Table 5. In the analysis, we calculate the sum of squares (SS), degrees of freedom (df), mean squares (MS), F statistic, and p-value at the 5% significance level. The modified R² improves upon R² by reducing the chance of artificially increasing the initial value when more variables are added. This conclusion is based on the findings. Variables with a p-value exceeding 5% indicate that their occurrence is infrequent. An F-value of 0 substantiates the null hypothesis and indicates deficiencies in the predictions. The null hypothesis posits that there is no correlation between the two data sets [¹³]. The model exhibits statistical significance, as evidenced by an F-value of 9.54. The likelihood of obtaining such a significant F-value by random chance is merely one-hundredth of one percent. Significant model parameters demonstrate that p-values got below 0.05 [²⁵]. In this context, the pertinent model terms in the criteria include A, B, C, AB, AC, BC, A2, B2, and D2. The model can be improved through simplification if it contains multiple extraneous terms, omitting those vital for preserving hierarchy. Because of insignificant parameters, the p-values exceed 0.1000.

Table 4
Design of RSM.

FRAMEWORK DESIGN	QUADRATIC
Subtype	Randomized
Classification of design	Blank Spreadsheet
Classification of study	Response surface
Blocks	No Blocks
No. of. Runs	27

Table 5
Results obtained from ANOVA.

SOURCE	SUM OF SQUARES (SS)	df	MEAN SQUARE (MS)	F-VALUE	p-VALUE
Model	75557.57	9	8395.29	9.54	<0.0001	Significant
A-H	1290.19	1	1290.19	1.47	0.2426
B-B/t	191.52	1	191.52	0.2175	0.6468
C-H/B	4722.25	1	4722.25	5.36	0.0333
AB	661.81	1	661.81	0.7517	0.3980
AC	25787.23	1	25787.23	29.29	<0.0001
BC	31.07	1	31.07	0.0353	0.8532
A2	20430.29	1	20430.29	23.21	0.0002
B2	45.34	1	45.34	0.0515	0.8232
C2	20811.27	1	20811.27	23.64	0.0001
Residual	14966.45	17	880.38
Core Total	90524.01	26

As shown in Table 6, the Adequate Precision metric assesses the signal-to-noise ratio. It reveals that the Response Surface Methodology (RSM) regression analysis yields a robust correlation between the predictor variables and the response variable, with an R² value of 0.8347, indicating approximately 83.47% of the response variable’s variance. This paradigm facilitates extensive exploration of the design space. The presentation in relation to a conclusive dataset is as follows in Equation 1. This regression model facilitates accurate predictions of strength behaviour by accounting for the complex interdependencies between various parameters and provides an ultimate load equation [²⁶].

Table 6
Regression of RSM.

SD	MEAN	C.V. %	R2	ADJUSTED R2	PREDICTED R2	Adeq PRECISION
29.67	180.07	16.48	0.8347	0.7471	0.6007	10.5624

(1) $$\text{Ultimate load}\text{=}\text{+334}\text{.71241}+\text{1}\text{.18358H}-\text{11}\text{.86301B/t}-\text{99}\text{.81609H/B +0}\text{.071576*B/t + 1}\text{.47283H*H/B}-\text{0}\text{.767923B/t*H/B}-\text{0}{\text{.015331H}}^{\text{2}}-\text{0}{\text{.316973B/t}}^{\text{2}}-\text{32}{\text{.49079H/B}}^{\text{2}}$$

Figure 7(a, b) shows the residual versus run and deviation from reference plots. Figure 7(c) presents three-dimensional graphs constructed using RSM, which examine the relationships between the design parameters of PFRP profiles with varying cross-sectional shapes and heights. The incorporation of glass fiber at the corner of the box section significantly enhances the ultimate load. The predicted values exhibit a clustered pattern along the diagonal, indicating a strong correlation between the variables [²⁷]. The RSM model shows the ability for accurate predictions, displaying a relatively consistent distribution of data along the horizontal and vertical coordinates. Furthermore, the charts illustrate both the actual values and the projected values for each of the distinct regression models. In this context, the model demonstrates no tendencies toward overprediction or underprediction bias. The actual and projected response values are closely aligned, as indicated by their comparison. Figure 7(d, e) displays the enhanced and projected ultimate load values achieved through the Response Surface Methodology (RSM). The statistical analysis demonstrates a strong correlation between the actual and predicted response levels, indicating exceptional model precision [²⁸]. Furthermore, the residual plots display a random scatter pattern, confirming the adequacy of the selected model for predicting the strength and interactions of the materials and suggesting that the model errors are normally distributed.

Figure 7
(a, b) Residual/run and deviation plots; (c) surface response plots; (d, e) contour and predicted-actual response.

4.2. ANN analysis

This research employs the backpropagation training method, a widely recognized technique for addressing intricate engineering challenges. In this study, an ANN model is developed to predict the ultimate load of PFRP box profiles, utilizing data from 27 experimental datasets. The ANN analysis involves creating a network of interconnected neurons that process input data such as material proportions, mix designs, and environmental factors. Through training, the network adjusts its internal weights to minimize prediction errors, enabling the capture of complex interactions and dependencies that traditional methods may overlook [²⁹]. The optimal artificial neural network models for ultimate load are displayed in Table 7. Table 8 shows the actual and projected outcomes for an ultimate load. The anticipated outcomes from both models aligned with the data acquired from the tests. The precision of the test outcomes in evaluating the ultimate load has been emphasized [³⁰].

Table 7
Criteria of the ANN model.

CRITERIA	OPTIMUM VALUE CHOSEN
Adoption learning mechanism	LEARNGDM
Mechanism Training	TRAINLM
Classification of Network	Feed forward back propagation
Performance	MSE
Transfer function	TANSIG
Hidden layer	2 hidden layers with 10 and 1 neurons

Table 8
Lists of actual values and predicted values by RSM and ANN.

RUN ORDER	ACTUAL VALUE	PREDICTED VALUE BY RSM	PREDICTED VALUE BY ANN
1	87.61	109.92	91.00
2	63.47	73.83	163.50
3	54.28	53.73	253.94
4	91.57	147.13	94.90
5	82.61	95.51	82.15
6	76.57	56.76	79.18
7	212.34	189.3	216.16
8	208.84	190.49	254.89
9	198.53	215.6	209.65
10	222.24	189.3	216.16
11	256.37	190.49	254.89
12	208.81	215.6	209.65
13	235.17	210.26	234.71
14	198.24	201.04	198.79
15	190.63	213.64	191.37
16	240.86	221.53	280.20
17	202.49	206.06	201.94
18	211.38	211.14	213.72
19	205.42	188.41	198.49
20	190.53	202.11	166.61
21	187.64	187.4	191.26
22	211.21	225.35	201.16
23	201.38	221.98	201.50
24	196.75	186.7	158.73
25	219.24	244.46	247.06
26	208.47	230.88	207.35
27	199.29	183.32	201.33

This research features an artificial neural network architecture with one concealed layer. Figure 8(a) illustrates the layers of the artificial neural network architecture employed in this study to evaluate the ultimate load [³¹]. Figure 8(b) illustrates the regression plots for the suggested artificial neural network model. The data demonstrates that all R values employed in the model’s development, testing, and validation phases surpass 0.69 and approach one, indicating complete accuracy. A plot of Mean Squared Error (MSE) versus epoch, shown in Figure 8(c), was used to determine the optimal ANN model for ultimate load prediction in the training, testing, and validation networks. The epochs represent iterations where all data configurations are presented to the neural network, with a greater number of epochs indicating increased computational time for training. Each epoch processes the entire dataset, and more epochs lead to increased training time and computational costs but also better learning and accuracy [³²]. Figure 8(c) shows that training converged after nine epochs for ultimate load prediction, with optimal validation occurring at an MSE of 6. The error patterns for both the test and validation sets exhibit similar characteristics, with no significant overfitting observed. The highest reliability was achieved at the minimum MSE over an extended duration of 6 epochs for the final load.

Figure 8
(a) ANN model in MATLAB, (b) regression plots for ANN, (c) validation performance for PFRP profiles.

5. CONCLUSION

• The primary failure mode observed in rectangular pultruded GFRP tubular stub column specimens is characterized by crushing failure, where the material collapses in a localized area due to excessive compressive stress. This indicates two separate failure modes: the lowest height specimens (H1 and H2) demonstrate mid-section fractures, whereas specimen (H3) exhibits longitudinal ripping at the corners, attributed to differences in aspect ratio and B/t ratio.

• Ultimate load inversely corresponds with the aspect ratio and the B/t ratio. The yield and final displacement progressively increase with a higher aspect ratio. The relationship among the ultimate load, initial stiffness, aspect ratio, and B/t ratio is complex.

• The test findings indicate that the optimal width-to-thickness ratio for the pultruded GFRP column member is PR10-5-6-H2, with an ultimate load of 256.37 kN. Similarly, an increase in the height of the specimen influences the ultimate load of the column.

• RSM investigations indicated that the R2 value of the ultimate load is 0.8347. The models forecast the ultimate load with an accuracy of 83.47% based on this data.

• The ANN analysis yielded a regression coefficient of 0.6887 for the final load, indicating a strong correlation between the predicted and actual values. Furthermore, the ANN model demonstrated excellent validity, with a correlation value approaching unity (1), thus confirming its high accuracy.

• This study compared the RSM and ANN predictions for the ultimate load of GFRP composites. The results showed that both models had potential, but ANN predictions were more accurate and had a stronger correlation with actual values. In contrast, RSM predictions had some deviations from actual values.

• The experimental findings measuring ultimate load have been meticulously examined utilizing ANN and RSM methodologies, facilitating precise effect estimation.

• The highest ultimate load for the specimen PR10-5-6-H2 has been determined, representing the optimal aspect ratio and B/t ratio of the pultruded GFRP rectangular stub column, validated by machine learning methodologies, specifically RSM and ANN.

Furthermore, the software can improve the compositions predominantly utilized in civil engineering endeavours. In applications of engineering, established models can provide a framework to save excessive time devoted to various tests. In the future, academics may explore diverse models, including support vector machines decision tree techniques are applied to various civil engineering applications, improving predictive accuracy and decision-making.

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Publication Dates

• Publication in this collection
  27 Jan 2025
• Date of issue
  2025

History

• Received
  14 Nov 2024
• Accepted
  12 Dec 2024