Experimental and Numerical Study on Flexural Behavior of Concrete Beams Using Notches and Repair Materials

Abstract

In a concrete beam, cracking is generated on the tension side under the effect of flexure, shear, and torsional loadings. Accordingly, these weak concrete members require repair and/or strengthening to increase or restore their internal load capacity. In the current experimental and numerical investigations on concrete beams, the impact of using notches with different width to depth ratios on the ultimate flexural load under a three-point test was considered. Further, the flexural behavior performance of a notched concrete beam repaired using the three repair materials—cement mortar, bacterial mortar, and adhesive—was also examined. Consequently, a comparative study was implemented between the experimental and numerical results. A concrete damage plasticity (CDP) model was used for the finite element numerical analysis of the beams. The differences in numerical and experimental measured results ranged from 0.65 to 22.20% for the ultimate load carrying capacity. As the notch size increased, the ultimate load carrying capacity of the beam reduced. Additionally, a linear regression model was used to predict the ultimate load values at a notch width interval of 5 mm up to a maximum notch width of 100 mm. It was observed that the ultimate load capacity for a repaired beam increased as compared to the notched beam for all three repair materials under consideration. And the maximum ultimate load increased in the case of notched beams repaired using adhesive. Furthermore, in comparison to the cement mortar, the performance of the bacterial mortar in terms of the ultimate load was more. The bacterial mortar was found to be more sustainable and more durable as a repair material for concrete structures.

1. Introduction

Nowadays, the sustainability of present-day civil infrastructure is being given more attention and significance. When considering the financial and environmental aspects of resource conservation into account, and the reduction in the construction industry’s overall carbon footprint, repairing and/or strengthening is a more sustainable option than simply destructing and rebuilding the civil infrastructure [1]. Moreover, maintenance should be prioritized over construction. Also, in cases of historical monuments, it is not possible to demolish and replace them; thus, repairing and/or strengthening are sustainable choices to preserve them. Concrete is widely employed in the construction of the majority of civil engineering structures because of its superior shielding ability, fire rating, long service life under normal and accidental conditions, ease of construction, and relative affordability. Despite these prominent characteristics, the majority of concrete structures may become damaged when subjected to the addition of structures, overloading, accidental loads, errors in design or during the erection process, environmental conditions, etc. Consequently, these weak concrete members require repair and/or strengthening to increase or restore their internal load capacity to resist additional external loads [2,3]. After the repair or strengthening work is successfully completed, the structural member’s safety, serviceability, and durability performance can be enhanced.

A concrete beam is a horizontal member in civil engineering structures subjected to external vertical loading. Due to such loading beams, deflections occur and flexural, shear, and torsional stresses are developed. Over the past two to three decades, researchers have been endeavoring to develop numerous repair and/or strengthening techniques for concrete beams, as shown in Figure 1. When selecting any repair and/or strengthening technique, there are a number of factors to take into account, such as repair time, localized changes in the stiffness of the member, strength, ductility, cost, minimizing disturbance to occupants during repair, preserving the aesthetic appearance of structure, preserving or enhancing durability, and work safety [4,5].

Table 1. List of repair and strengthening materials.

Sr No.	Types of Material	Specification	Applications
A. Concrete Repair
1	Bonding primer	Epoxy resin	Used as bonding agent between old and new concrete structures
2	Crack repair	Low-viscosity injection resin	Moisture incentive for sealing cracks > 5 mm
3	Site batch mortars	Synthetic rubber emulsion	Used for good adhesion and water resistance
4	Smoothing mortars	Epoxy-modified cement, thixotropic, fine-textured mortar	Used for levelling and finishing of concrete, mortar, or stone surfaces
5	Structural injection material	Low-viscosity injection resin	Moisture incentive for sealing cracks > 5 mm
B. Structural Strengthening
1	Prefabricated CFRP plates	Pultruded carbon fiber-reinforced polymer (CFRP) laminates	Strengthening concrete, timber, masonry, steel, and fiber-reinforced polymer structures
2	FRP fabrics	Uni-directional woven carbon fiber fabric	Strengthening concrete, timber, masonry, steel, and fiber-reinforced polymer structures
3	Structural adhesives	Epoxy impregnation resin	Structural strengthening application
C. Repairing Mortar
1	Cementitious mortars	Polymer-modified	Repairing mortar
2	Epoxy mortar	Epoxy resin	For surface sealing and patch repair/filling mortar
3	Additives for mortars	Synthetic rubber emulsion	For waterproofing and repair

shows the various repair and strengthening materials used for concrete structures. Fiber-reinforced polymer (FRP), epoxy resin, epoxy-modified cement, and synthetic rubber emulsion are a few of the materials that have been utilized. These materials are applied in sheets, laminates, sprays, and rods.

The review of previous experimental and numerical studies carried out on concrete beams are discussed herein. A numerical investigation of rectangular and T section concrete beams for bending moment capacity was carried out by [5]. An experimental investigation of prestressed concrete beams for flexural strength was performed by [6,7]; the results suggested that, in plain concrete, brittle failure occurred due to a flexural crack. In higher-strength concrete, beam brittle failure is caused by shear force and the development of a diagonal crack. The different types of openings reduce the stiffness of reinforced cement concrete (RCC) beams [8]. Dong et al. (2002) [9] suggested that the load carrying capacity and stiffness of pre-cracked RC beams can be increased by providing carbon fiber-reinforced polymer (CFRP) at the tension side of the beam; the failure of the concrete cover and the debonding between the CFRP fabric and the concrete should also be investigated. Rahim et al. (2020) investigated the experimental behavior of deep beams with web openings under a four-point load [10]. It was found that, as the size of the web opening increased, the shear strength of the concrete deep beam decreased. As discussed in Figure 1, various techniques can be used, like jacketing [11,12] and FRP [13,14].

Researchers generally investigate the performance of any repair and strengthening technique, as discussed above, by imitating damages, defects and cracks in the concrete beam through embedding a notch approximately in the middle of the beam. The notch is embedded in U, V, or key-hole shapes [15,16]. The notch formation simulates the cracks or damages and then repair techniques are studied using experimental and numerical approaches.

Table 2. Review of effects of notches on concrete beams.

Authors	Type of Material	Specimen Size	Loading	Properties of Material	Remarks
[17]	Polymer mortars	250 mm × 60 mm × 30 mm notches 2 mm thick at different positions, i.e., 0, 24 mm, 48 mm, and 72 mm, towards the support.	Three-point load	-	Cracked mouth-opening displacement, cracked tip-opening displacement, and values of energy release rate.
[18]	Concrete and CFRP plate	100 mm × 100 mm × 500 mm beams with notches at the center with depths of 10 mm, 20 mm, and 30 mm, wrapped in CFRP plates with lengths of 100 mm, 200 mm, and 350 mm. Thickness of CFRP at 1 mm and adhesive at 0.5 mm.	Three-point load	fck = 41.4 MPa EC = 32.89 GPa ECFRP = 150 GPa Eadhesive = 4.3 GPa MIConcrete = 8.3 × 106 mm4	With and without CFRP-laminated plates, visually analysis the brittle failure, shear failures, and delaminate failure.
[19]	Foamed concrete	Foamed concrete beam 840 mm × 100 mm × 100 mm with a notch (5 mm thickness; 42 mm height) at the center.	Three-point load	EFoamed Concrete = 1000 GPa µFoamed Concrete = 0.2, ε = 0.2, φ = 15°	Extended finite element method (XFEM) model of foamed concrete.
[20]	Foamed concrete	Foamed concrete beam 750 mm × 150 mm × 150 mm with a 30 mm long V-notch at the center.	Cube compressive strength and three-point load test	$\mathsf{\rho }$concrete =2400 kg/m3, $\mathsf{\rho }$foamed concrete = 1400–1600 kg/m3	Compressive strength of cube and fracture energy of foam concrete is lower.
[21]	RCC beam, CFRP, and adhesive	RCC beam sized 3000 mm × 200 mm × 300 mm with a notch at the center of beam and 4 mm thick CFRP plates applied using a 2 mm thick layer of adhesive.	Three-point load	EC = 30 GPa ECFRP = 140 GPa EAdhisieve = 3 GPa µc = 0.18 µCFRP = 0.28 µAdhesive = 0.35	CFRP plates bond with the surface of the beam and significantly enhance the stiffness and ultimate load.
[22]	PCC beam	1640 mm × 200 mm × 400 mm in size with a notch at the center (a/d = 0.5).	Three-point load	EC = 30,570 MPa fck = 21.9 MPa ftk = 2.4 MPa	Acoustic emission (AE) and digital image correlation (DIC) techniques.
[23]	PCC beam with a steel fiber hook	A 600 mm × 150 mm × 150 mm plain concrete beam with different notch depths of 13 mm, 25 mm, and 50 mm at the center, and the a/d ratios are 0.08, 0.16 and 0.33.	Three-point load	Steel fiber of 50 mm length, 1 mm diameter, and 1130 MPa tensile strength. The water/cement ratio is 0.55, and the cement/sand/aggregate ratio is 1:2.93:2.31.	a/d ratio notch is increased by 0.08, 0.16, and 0.33 as the load carrying capacity decreases. Due to presence of high volumes of steel fiber, the fracture energy increases with increases in the a/d ratio.
[1]	RCC beam with a rectangular section and a T section.	1600 mm × 300 mm × 150 mm CFRP ropes were used for retrofitting and shear strengthening.	Three-point load	Compressive and tensile strengths of concrete were found to be 28.0 MPa and 2.70 MPa. The area of the CFRP rope was 28 mm2 and the modulus of elasticity was 240 GPa.	CFRP ropes improve the structural performance of deep RC beams by converting brittle shear failure into ductile flexural failure.

shows the review of some of studies where the effects of notches on concrete beams were studied. Recently, carried out an experimental investigation on notched concrete beams under three-point loading for constant width and varying depths of the beam section. The results were obtained as size effect, fracture energy, failure modes, peak loads, and load–displacement analysis.

In recent decades, bacterial mortar and concrete have also been used as sustainable materials for repairing microcracks and increasing the self-healing property of concrete. As it is organic material, its properties do not change with time, and thus it can be used for healing deep, internal microcracks. Cracks of more than 0.8 mm can be easily healed [24]. Bacteria that are rich in calcium are added to mortar or concrete during mixing. These bacteria precipitate calcium carbonate in the event that cracks emerge in the concrete, thus sealing the cracks and repairing them. Bacterial mortar or concrete has higher strength and durability than normal concrete. Kumari et al. (2017) [25] used Bacilus Cohnii (BC), which can biologically increase durability and mechanical strength without producing any harmful gases. The encapsulation of the bacteria spores and nutrients by expanded perlite helps to heal the cracks, when 20% of the weight of the fine aggregates were replaced by expanded perlite [26]. Sierra et al. (2014) [26] used bio-based repaired mortar for the concrete beam. The flexural and compressive strength and dry shrinkage were measured. The performance of the bio-based repair mortar was studied through restrained shrinkage and pull-out tests [25]. Valdez Aguilar et al. (2021) [23] presented a detailed review on the various types of bacteria used for healing concrete and discussed the different concrete properties affected by the addition of bacteria. Vijay and Kunamineni et al. (2019) [27] used cultured bacillus subtilis in spore powder form and culture form in mortar concrete. From their experimental works, it was concluded that the compressive strength increases and also that evidence of aggregates in pores, which are responsible for the high concentration of Ca+ ion calcium silicate hydrate. In their study, Abo-El-Enein et al. (2019) [28] used microbes to precipitate the calcite by filling microcracks in order to increase compressive strength and decrease water absorption. The permeability of cracked concrete in wet–dry cycle conditions decreases because crack healing by calcium carbonate is precipitated by microbes [29].

Table 3. Review based on bacterial mortar and concrete.

Authors	Materials	Name of Bacteria	Preparation of Specimen and Curing	Test Method	Result and Discussion
[30]	Cement, sand, aggregate, bacteria liquid, cyclic-enriched ureolytic powder (CERUP), and activated compact denitrifying core (ACDC) granules.	B. Cohni	Specimens cured in water for 28 days.	Compressive strength, water absorption and recovery of water tightness.	Compressive strength increased by 25%, water absorption decreased, and due to the aerobic oxidation of organic carbon O2 consumption by bacteria, the rebar corrosion was reduced.
[31]	Cement, sand, aggregate, with a ratio of 1:2.5 of water–bacterial liquid in a clay ball.	Genus. Bacillus	Specimens cured in water and a wet/dry cycle for 28 and 56 days.	Crack water permeability. Recovery of water tightness. Oxygen consumption measurements and ESEM analysis.	Crack permeability was lower than in normal concrete.
[27]	Portland slag cement and fine sand at a ratio of 1:6; bacterial solution w/c 0.55.	B. Cohni	Specimen cured in water for 28 days.	Standard consistency, setting time, cement soundness, compressive strength, sorptivity, drying shrinkage, microstructure, morphology, field emission scanning electron microscope (FESEM), and X-ray diffraction (XRD) techniques.	For 28 days, compressive strength increased by 49.8% and sorptivity decreased.
[28]	Portland cement, sand fly ash, silica fume, calcium lactate, calcium acetate, and encapsulated material.	S. Pastteurii and others ureolytic bacteria.	Specimen cured for 28 days.	Setting time of concrete, compressive strength, permeability, chloride ion permeability, and microstructure calcite.	Compressive strength increased and permeability decreased.
[32]	Cement, sand, bacterial liquid, encapsulated material.	Ureolytic bacteria	Specimen cured in buffer solution for 7 and 28 days.	Scanning electron microscope, compressive strength, permeability.	Permeability decrease.
[33]	Portland fly ash cement and sand at a ratio of 1:3; bacteria perlite, sodium silicate, water, calcium acetate, and yeast extract; w/c 0.5.	B. Pseudoformu	Specimens were cured in a controlled environment, then after demolding, the specimen was cured in water at 20 °C for 28 days. Cracked sample cured in a moist and humid environment for 165 days.	Surface water absorption and visualization of crack healing.	Water absorption decreases.
[29]	OPC, sand, and w/c 0.46 buffer solution.	B. Pasturi	Specimen cured in a humidity chamber with a relative humidity of 100% for 24 h. After demolding, the bacterial specimens were cured in buffer solution for 28 days.	Compressive strength and water absorption.	For 28 days, compressive strength increased by 33% and water absorption.
	Portland cement, sand, water, and encapsulated material.	Ureolytic bacteria	Specimens cured for 28, 60, 90, 365, and 730 days.	Compressive strength, flexural strength, and water absorption.	Compressive strength and flexural strength increased, construction cost increased, and maintenance cost decreased.
[34]	Portland cement, sand, water, and encapsulated material.	Bacillus Cereus	Specimen cured for 180 days.	Compressive and split tensile strength, ultrasonic pulse velocity and water absorption capacity.	Compressive and split tensile strengths were improved and reduced water absorption capacity.

lists the types of bacteria, preparation curing specimens, test methods and major remarks based on the previous researchers on bacterial mortar and concrete. In case of bacterial mortar or concrete the bacteria used for the bio-precipitation process is an important stage.

Literature on based on recent advancements in non-destructive techniques have also been published: Adhikary et al. (2012) [35] three-dimensional finite element models were used in a parametric investigation, and their validity was checked against experimental data. The impacts of longitudinal and transverse reinforcement ratios, as well as the shear range-to-effective depth ratio, were investigated. Finally, empirical formulas were provided to predict the maximal resistance of RC beams at different loading rates as a function of the dynamic increase factor (DIF). Furthermore, IS 456 [36] incorporated non-destructive evaluation methods to continuously check concrete beams for deterioration or corrosion without endangering the structure. Bao et al. (2001) [37] advanced experimental methodologies that have been used to investigate the mechanical properties of concrete beams under various loading scenarios. These techniques include a variety of methods, such as measuring strain and stress distributions longitudinally along the beam with strain gauges and load cells. Roux et al. (2006) [38] adopted digital image correlation (DIC) for crack propagation analysis and surface deformation capture. Kim et al. (2003) [39] studied the matrix microcracks and fracture events of concrete using acoustic emission (AE) monitoring. By monitoring wave velocities and attenuation, ultrasonic testing (UT) is an important technique for assessing the integrity and homogeneity of concrete beams. Moreover, Hafezolghorani, Milad, et al (2017) [40] incorporated fiber optic sensors facilitate the tracking of variations in temperature and moisture penetration in concrete beams. Yehia et al. (2007) [41] proposed that without producing any structural damage, non-destructive evaluation (NDE) techniques are essential for monitoring the performance and condition of concrete beams. The authors of [18,42] used ground-penetrating radar (GPR) to find corrosion, cavities, and cracks in concrete buildings. Villain et al. (2012) [43] created a 2D data fusion model using GPR and developed an eco-friendly impact approach to forecast the service life of concrete structures in maritime environments. Bogas et al. (2013) [44] used ultrasonic pulse velocity (UPV) testing measures the propagation of ultrasonic waves and provided a dependable way to evaluate the homogeneity and quality of concrete. Tran et al. (2017) [45] noted that the use of thermal imaging to detect thermal abnormalities potentially containing moisture intrusion, delamination, or other flaws in concrete beams has shown promise. Murad et al. (2021) [46] suggested acoustic emission techniques for the assessment of concrete dams. These non-destructive evaluations collectively contribute to a comprehensive understanding of the conditions and performances of concrete beams, facilitating timely maintenance and ensuring long-term structural reliability.

Other previous studies related to the effects of microstructures in influencing crack propagation paths and ultimately impacting material behavior under loads are described as follows. Xie et al. (200) [47] suggested that microstructure analysis plays a critical role in identifying the fracture properties of cementitious specimens. Li et al. (1986) and Wahab et al. (2012) [48,49] used the technique for examining the fine details of fracture surfaces and demonstrated that the fundamental principles driving crack propagation are microstructure-related [50,51]. Furthermore, Devi at al. (2022) [50] carried out the three-dimensional reconstruction of microstructures by sophisticated imaging modalities like focused ion beam-scanning electron microscopy (FIB-SEM) and X-ray microtomography (XMT), which shed light on the spatial distribution of phases and interfaces inside the material. Landis and Keane et al. (2010) [51] demonstrated that fracture properties, such as crack morphology, crack route, and the existence of microcracks and flaws on the fracture surface of cement specimens, can be discovered through microstructure analysis. Kotula et al. (2014) [52] studied the behavior of the material under loads and its ability to withstand the propagation of cracks by establishing a correlation between these observations and mechanical characteristics including tensile strength, fracture toughness, and energy dissipation.

Furthermore, Lange et al. [53] analyzed microstructural aids in identifying important microstructural characteristics, such as aggregate presence, porosity distribution, and interfacial bonding between phases, that affect fracture behavior. Launey and Ritchie (2009) [54] observed that knowledge obtained from a study on fracture surface microstructure has important ramifications for material design and performance optimization. Through comprehending the ways in which microstructural characteristics impact fracture patterns, scientists can devise methods to improve the mechanical characteristics and service lives of cementitious materials.

Contribution of Work

From this literature review it can be seen that no comparison had been conducted on the cement, bacterial mortar, and commonly used adhesive materials. In the present experimental and numerical investigations, the effects of ultimate flexural loads on concrete beams with different width-to-depth ratios of the notches under a three-point loading test were studied. Further, the flexural performance of notched concrete beams repaired with the three repair materials (cement mortar, bacterial mortar, and adhesive) was also investigated. Consequently, a comparative study was carried out between the experimental and numerical results in order to study the performance of these repair materials for the concrete structures. Also, the application of bacterial mortar as a repair material was studied. A concrete damage plasticity (CDP) model was used for finite element numerical analysis of beams. Additionally, a linear regression model was used to predict the ultimate load values at a notch width interval of 5 mm up to a maximum notch width of 100 mm.

2. Experimental Study

Concrete beam specimens were fabricated with a specific composition for each constituent material for M40 grade, as per IS 10262:2019 [55]. The strength of pozzolana Portland cement (PPC) with a standard consistency 31.5% and a specific gravity is 3.14, and Narmada River sand acts as a fine aggregate that remains in confining zone II of IS 383:2016 [55], and maximum size of 20 mm coarse aggregates were used. The SIKA plasticizer was used as an admixture. The mix proportion cement: fine aggregates: coarse aggregates adopted was 1:1.52:2.72 with a water-cement ratio 0.40 and admixture 1.1% by weight of cement. A cement content of 445 kg/m³ and workability with a 0.85 compaction factor were used. The fresh concrete mixed workability as per the designed mix proportions was first investigated, a further total 21 specimens of concrete beams sized 100 mm × 100 mm × 500 mm were cast. The solid concrete beam specimens were cast as shown in Figure 2a. The beam specimens were removed from their molds 24 h after casting and then cured in water at room temperature for 28 days. Further, the required size of the rectangular-shaped notch was marked on the middle span of the concrete beam on the tension side (bottom face) after 28 days curing the hardened concrete beam, as shown in Figure 2b. The notch was cut with the help of grinder with a circular blade. The notch was cut from one side, and then we moved slowly cut-by-cut to the other side along the width of the beam specimen. Precautions were taken to prevent the incorrect cutting of a notch and to cut the desired size of notch without damaging the specimens. The notch was cut three hours before testing the specimens. The notch was considered suitable to simulate failure or crack or defect in a beam that were necessary to repair. The different width-to-depth ratios 0.33, 1.66, and 3.33 of the notches were considered in the present investigation. Rectangular notches with widths of 10 mm, 50 mm, and 100 mm and constant depths of 30 mm at the center of the tension side of the beam were considered.

Further, in order to study the effects of the three repairing materials, i.e., cement mortar, bacterial mortar, and adhesive (Sikadur-31), the notches were filled with repairing materials and a 10 mm thick layer of material was placed on the tension side, as shown in Figure 2c. The effect of the repairing material on the flexural strengthening resistance of concrete beam was studied. In the case of cement mortar, a cement-to-sand ratio of 1:3 and a water-to-cement ratio of 0.50 was used and cured for 28 days.

In case of bacterial mortar, Bacillus cohnii (MTTCC3616) was procured from IMTECH, CSIR Labs Chandigarh, India. All revival cells were incubated in calcium precipitate culture for 72 h using a shake flask incubator at 37 °C and 150 rpm, as shown in Figure 3. After incubation time, the culture was mixed with cement and sand at a 1:3 proportion, and the culture-to-cement ratio was 0.5. Figure 3 depicts the step-by-step procedure for preparing the bacterial mortar and its use as a repair material. The bacterial cement mortar specimens were prepared similarly to the above method. The specimens were cured in water with a 0.1 Mol calcium lactate powder solution in order to provide the external calcium resource for the bacteria used in the mortar. The third repair material, thixotropic epoxy adhesive, was used, and after its application, it was left to dry for at least 15 days at room temperature. In order to eliminate the uncertainty of the concrete material, three concrete beams were cast for each solid beam, notched beam, and beam with repair material.

The casting and testing procedure of the beam indicated 4 (a-g). The concrete beam specimens were mounted and simply supported on a servo-hydraulic testing machine, and single point load was applied at the middle of the length (three-point flexural test) of the concrete beam specimen, as shown in Figure 4e. The reaction was carried out by the two roller-supports near the ends of the concrete beam specimens. The strength tests on the concrete beam specimens were conducted as per the Indian Standard—IS 456:1959 [36]. The values for the ultimate flexural load for each specimen were recorded, and flexural behavior was studied, as shown in Figure 4f.

3. Numerical Study

Further, in the present work, a numerical study was also carried out. The numerical study is a more efficient and economical tool as compared to the experimental study for structural analysis and engineering research. Thus, numerical studies can capture complex damage behavior accurately. The finite element modeling of beams was carried out using the commercial software ABAQUS (Abaqus 6.21-1, Simulia, Dassault Systemes, Paris, France) [56]. Concrete was modeled using the smeared crack concrete model and the concrete damage plasticity (CDP) model. The purpose of the model was to calculate beam performance during experimental testing. In the present study, the concrete damage plasticity (CDP) model [47,48,49,50] was used. This model is based on the proposed models by Lee and Fenves (2018) [57] and Lubliner et al. (1989) [58], which can simulate the tensile cracking and compressive crushing of concrete materials while considering the isotropic elastic damage and plastic behaviors of the materials. The CDP model is used for applications where concrete is subjected to arbitrary loading circumstances, such as cyclic loading, and is predicated on the idea of scalar (isotropic) damage. The model accounts for the plastic straining-induced deterioration of elastic stiffness in both tension and compression. The effects of stiffness recovery during cyclic loading are also taken into account. The material properties, such as elastic modulus, Poisson’s ratio, and damage parameters must be specified [45].

3.1. Validation of the Numerical Model

The present numerical model was validated using the published results in the literature.

Table 4. Verification of the numerical results.

References	Material Properties	Dimensions	Maximum Failure Load (N)		% Difference
			Published Results	Present Study Results
[52]	E= 32.89 GPa and fck = 41.2 MPa	500 mm × 100 mm × 100 mm, notch 10 mm × 10 mm × 100 mm.	6933.3	6606	4.72
[53]	fck = 38.24 MPa	L = 1200 mm, d = 200 and b = 100 mm.	11,200	10,770	3.83
	fck = 54 MPa, ftk = 3.16	L = 1400 mm, d = 230 mm and b = 140 mm.	16,000	15,963	0.23

shows the comparison of the three-point flexural test maximum failure load with published results and represents a numerical study. The percentages of difference are negligible and good agreement is achieved between both set of results.

3.2. Present Numerical Study

Further, the concrete beam was modeled using an eight-node linear brick C3D8R element with six faces and reduced integration with an hourglass control, as shown in Figure 5. The boundary condition was designated as a simply supported beam with two hinged supports at a distance of 50 mm from each end of the beam and was modeled. The central point load was applied at a reference point on the middle top of the beam. Based on the convergence study, a 20 mm sized mesh was utilized, and a fine mesh of 10 mm was used near the notch section. All material properties in the CDP model for the M40-grade concrete were adapted from [14,59]. The mechanical properties for the M40-grade concrete are given in

Table 5. Mechanical properties for M40-grade concrete.

Material	fck (MPa)	ftk (MPa)	Modulus of Elasticity (MPa)	Density (kg/m3)	Poisson’s Ratio	Dilation Angle	Viscosity
Concrete	40	4.6	32,890	2400	0.2	31°	0.00001

, and

Table 6. Mechanical properties of repairing materials.

Repairing Material	Density	Compressive Strength (N/mm2)	Modulus of Elasticity (MPa)	Poisson’s Ratio
Cement mortar	2200 (kg/m3)	36.22 in 28 days	14,108.08 [60]	0.2
Bacterial mortar	2200 (kg/m3)	63.43 in 28 days	30,387.43 [60]	0.2
Adhesive	1.8 (kg/lit)	65 in 15 days	11,000 [61]	0.25

shows the mechanical properties of the repairing materials. In total, 21 FE concrete beam models were simulated and compared with the experimental results for solid, notched, and repaired beams.

4. Results and Discussion

In the current experimental and numerical investigations, concrete beams with different width-to-depth ratios of the notches on its ultimate flexural load were studied. Further, the flexural performance of repaired notched concrete beams using the three repair materials, namely cement mortar, bacterial mortar, and adhesive, was also investigated. The experiment’s validity and accuracy are demonstrated by the strong agreement between the numerical and the experimental observations.

4.1. Effect of Notching

The effect of the different width-to-depth ratios of 0.33, 1.66, and 3.33 of the notches were considered in the present investigation. The rectangular notches had widths of 10 mm, 50 mm, and 100 mm with a constant depth of 30 mm and were located at the center of the tension side of the beams. Figure 6 shows the ultimate flexural load versus the displacement behavior of the solid beams and the notched beams with different notch sizes. According to the experimental study, in the case of the solid beam, the ultimate load capacity was 12.33 kN, and according to the numerical study, the load obtained was 10.09 kN. Thus, the percentage difference in both the experimental and numerical studies is 0.65–22.20%. Figure 7 shows the comparison of the ultimate loads of the experimental study and the numerical study. Thus, the CDP model can accurately simulate the flexural behavior of the concrete beam.

The notched beam with a notch size of 10 × 30 mm had a 0.6% reduction in volume as compared to the solid beam, and the percentage reduction in load carrying capacity was observed to be 41.20%; this was similar to the case of the notched beam with a notch size of 50 × 30 mm, which saw 3% reduction in volume as compared to the solid beam, and the percentage reduction in load carrying capacity was observed to be 50.04%. Further, the notched beam with a notch size 100 × 30 mm saw a 6.0% reduction in volume as compared to the solid beam, and the percentage reduction in load carrying capacity was observed to be 67.56%. It can be observed that as the notch size increased the ultimate load carrying capacity of beam was reduced. Thus, the defects and cracks reduce the load performance capacity of the beam. Thus, in the case of the notched beam, the ligament area can be enhanced by decreasing the width-to-depth ratio of the notches, which increases the resistance to flexural damage and enhances the flexural capacity of the concrete beam.

The relationship between ultimate load carrying capacity and concrete beams was uncovered based on the proposed experimental and numerical results. Almusallam et al. (1997), Murad et al. (2021), and Zhong et al. (2021) also predicated the load capacity of the concrete beams [62,63]. Further, the experimental and numerical results were used for applying a prediction model linear regression (LR) for precise notch width for a 30 mm constant depth. To apply the LR model, first, the existing experimental and numerical ultimate load values for notch widths (10 mm, 50 mm, and 100 mm) were used to validate the LR regression model. In the second stage, the LR model was used to predict the ultimate load values at a notch width interval of 5 mm up to maximum notch width of 100 mm, as shown in Figure 8. The LR mathematical model is defined as follows:

Where p₁ and p₂ are linear regression coefficients. The predicated coefficients of the proposed linear regression model are presented in

Table 7. Parametric values of the prediction model.

Parameters	Experimental	Numerical
Coefficients	p1 = −0.0364, p2 = 7.745	p1 = −0.0496, p2 = 8.259
R2	0.984	0.973
RMSE	0.288	0.522

. It can be seen that the correlation coefficient R² with experimental and numerical results shows a difference of nearly 1.12%. The root mean square errors (RMSE) for experimental and numerical results are 0.288 and 0.522, respectively.

4.2. Effect of Repairing Materials

The flexural load carrying capacity of notched beams repaired with the three repair materials, i.e., cement mortar, bacterial mortar, and adhesive, was also investigated. In this case, the notch beam with a notch size of 10 × 30 mm was considered and repaired with cement mortar, bacterial mortar, and adhesive. After the repair, the ultimate flexural load of the beam was determined experimentally and also numerically. Figure 9 illustrates the ultimate load–displacement curve for the repaired concrete beam with all three repair materials under consideration. It can be observed that the ultimate load capacity for the repaired beam was increased as compared to the notched beam.

In the case of cement mortar, the ultimate load was observed to be 7.60 kN as compared to the notched beam with an ultimate load of 7.25 kN. The maximum ultimate load in case of notched beam that was repaired using adhesive was increased to 12.24 kN. Figure 10 shows the comparison of the ultimate load in repaired beams for different materials and also a comparison between the experimental and numerical results. It can be seen that the CDP model can explicitly simulate the repairing material. The numerical result depends on the precision of the input data of the material. Figure 11 shows the numerical simulation results for the repaired beam. The adhesive repair material initially provides more load capacity, but it is costly, and the performance also decreases with respect to time and variation in the materials’ properties due to environmental conditions. The procedure for using the adhesive material is easy as compared to the cement and the bacterial mortars. The ultimate load for the bacterial mortar is 7.3% more than that for the cement mortar. Thus, as compared to the cement mortar, the performance of the bacterial mortar was better in terms of the ultimate load. The bacterial mortar was more sustainable and more durable as a repair material for concrete structures.

5. Conclusions

In the current experimental and numerical investigations, the effect of the ultimate flexural load on concrete beams with different width-to-depth notch ratios under three-point loading tests was studied. Further, the flexural performance of repaired notched concrete beams using the three repair materials, namely cement mortar, bacterial mortar, and adhesive, was also investigated. Consequently, a comparative study of the experimental and numerical results was carried out. The following conclusions can be summarized as follows:

• There is an excellent correlation between the experimental and finite element numerical results at every loading stage up to the point of failure to predicate the flexural behavior of the concrete beam. Additionally, the FE numerical platform can overcome the drawbacks of experimental testing.
• The differences in numerical and experimental results ranged from 0.65% to 22.20% for the ultimate load carrying capacity.
• As the notch size increases, the ultimate load carrying capacity of the beam is reduced. The notch volume ranged from 0.6 to 6% with a percentage reduction in load carrying capacity in the range of 41.20–67.56%.
• A linear regression (LR) model was proposed to predict the ultimate load values at a notch width interval of 5 mm up to maximum notch width of 100 mm. It can be observed that the ultimate load capacity of the repaired beams was increased when compared to the notched beams with all three repairing materials under consideration.
• It can be seen that the correlation coefficient with experimental and numerical results shows a difference of nearly 1.12%. The root mean square errors (RMSE) for the experimental and numerical results are 0.288 and 0.522, respectively.
• The maximum ultimate load was increased in case of the notched beam repaired using adhesive. Additionally, the FE numerical platform can be used to simulate the repair materials and to study their performance. As compared to the cement mortar, the performance of the bacterial mortar was better in terms of the ultimate load. The bacterial mortar is more sustainable and more durable as a repair material for concrete structures.
• The selection of any repair and/or strengthening material depends on repair time, localized changes in the stiffness of the member, strength, ductility, durability, cost, and work safety.