1. Introduction
As global environmental issues become increasingly prominent, the sustainable development of engineering construction activities is receiving growing attention [
1,
2,
3]. The excessive extraction of construction materials such as sand and gravel has led to a severe shortage of these resources. Simultaneously, production activities generate vast amounts of construction waste [
4,
5], occupying extensive land resources and imposing a significant burden on the environment [
6,
7,
8].
After smelting magnetite, the Panzhihua Iron and Steel Group produces large quantities of high-titanium heavy slag, an industrial solid waste [
9]. Current technology struggles to effectively refine this slag, resulting in its prolonged accumulation and occupation of land resources. Researchers have found that high-titanium heavy slag is a good alternative material for concrete aggregates, possessing significant recycling and reuse values as well as unique advantages [
10,
11,
12]. Sun J et al. [
13] determined the mix ratio of high-titanium heavy-slag concrete using orthogonal experiments, verifying the mechanical properties of lightweight composite slabs made from this concrete. Zhang T et al. [
14] found that high-titanium heavy slag can promote the hydration reaction of cement, reduce concrete porosity, and increase interfacial strength. Lin-Ze LI et al. [
15] explored the impact of high-titanium heavy-slag incorporation on the carbonation depth of concrete, proposing related functional relationships. Zhong [
16] confirmed the excellent frost resistance of high-titanium heavy-slag concrete. These studies have explored the properties of high-titanium heavy-slag concrete from various perspectives, partially revealing its material characteristics when used in different structures.
Prefabricated construction, which involves factory production, transportation to the site, and on-site assembly, is a new building model with advantages such as rapid construction, a high engineering quality, reduced labor consumption, and environmental sustainability [
17,
18,
19]. Composite concrete beams, commonly used in prefabricated buildings [
20,
21], offer high construction efficiency and a low environmental impact, making them widely utilized in practical engineering [
22]. Du [
23] studied the bending performance of glued laminated timber–concrete composite beams and proposed an optimization scheme for the load-bearing capacity of these beams. Ranjbar, Navid [
24] investigated the bending failure mode, ultimate load, and deflection characteristics of geopolymer concrete beams. Gohnert [
25] examined the horizontal shear performance of composite-beam interfaces, analyzing the relationship between concrete surface roughness and compressive strength. These studies have partially revealed the impact of different material properties on the mechanical performance of concrete composite beams.
The deflection and cracking of concrete beams are closely related to their safety performance, garnering significant attention from scholars [
26,
27]. Variations in material selection and structural composition can result in differences in the crack resistance of concrete beams, prompting extensive research into structural reinforcement and optimal material design. Khorasani [
28] studied the relationship between the reinforcement ratio and the development of deflection and cracking in concrete beams. Chaudhary [
29] developed a model to predict the inelastic mid-span deflection and elastic mid-span deflection of continuous composite beams. Zou [
30] discussed experimental results on the deflection and cracking of carbon-fiber-reinforced polymer (CFRP) prestressed concrete beams under sustained long-term loading. Andreaus [
31] proposed a new method for damage detection in multi-cracked beam structures based on a static deflection analysis. Gouda [
32] investigated the effects of glass-fiber-reinforced polymer (GFRP) on crack propagation, deflection behavior, and the variation in kb values under different crack widths in concrete. Pansare [
33] demonstrated the effectiveness of the static deflection method when detecting diagonal cracks in cantilever beams. Nie [
34] developed a time-dependent analysis model to predict the long-term deflection of cracked reinforced concrete beams under sustained loading. Mazaheripour [
35] studied the combination of a non-corrosive, low-modulus GFRP with prestressed steel strands, achieving a favorable balance in reinforcement effectiveness, ductility, durability, and economic cost. Through experimental investigation, theoretical analysis, and model prediction, the development of deflection and cracking in concrete beams has been partially revealed.
In summary, utilizing high-titanium slag as an aggregate in concrete can alleviate the global shortage of sand and gravel resources while effectively addressing the issue of industrial solid waste occupying land resources. Although scholars have studied the deformation performance of components made from high-titanium heavy slag, composite beams with different materials, and flexural concrete members, the use of high-titanium heavy-slag concrete in composite beams remains limited. The following issues urgently require clarification. 1. The relationship between the fracture characteristics and deflection development of flexural composite beams made from high-titanium heavy-slag concrete and variables such as the reinforcement ratio, prefabricated height of composite beams, and material structure composition remains unclear. 2. There are currently no established standards or regulations regarding the cracking and deformation of flexural members made from high-titanium heavy-slag concrete. To better utilize and promote high-titanium heavy slag in composite-beam construction, it is essential to conduct a comparative analysis of the factors causing crack development and deformation deflection in composite beams. Clarifying the relationship between these variables and the deformation characteristics of composite beams will help to improve the relevant theoretical framework.
Based on this, our study tracks the crack development and deflection progression of seven concrete composite beams under flexural loading using the reinforcement ratio, prefabricated height, and composite type as variables. For comparison, the study establishes a correlation between the material properties and deformation performance, comparing the results with those obtained from standardized empirical formulas. Finally, relevant coefficients are introduced through linear fitting to modify significantly divergent empirical formulas. This research provides a comprehensive exploration of the flexural performance of high-titanium heavy-slag concrete composite beams, offering a theoretical foundation and technical support for their application in composite-beam construction.
3. Experimental Design
To investigate the damage process of high-titanium slag concrete composite beams under bending loads, hydraulic jacks were used for two-point concentrated loading through a distribution beam. Sensors monitored the load values in real-time. Fine sand was used to level the supports as well as the simply supported points of the distribution beam before loading.
Before the experiment, the components were preloaded to check the functionality of the loading equipment and other instruments. After preloading, the formal loading commenced. Until the specimen cracked, loading was conducted at a rate of 10 kN/min. Upon reaching 50 kN, the rate was reduced to 5 kN/min. Once the specimen cracked, the loading rate was restored to 10 kN/min. When the load reached 90% of the bending capacity, the rate was again reduced to 5 kN/min.
We observed the deflection development and crack propagation process of composite beams under different variables to explore the factors affecting beam deformation. We compared the experimental values with the theoretical values calculated by empirical formulas provided in the standards, and adjusted the formulas with significant deviations. Finally, we compared the modified theoretical formulas with the experimental values. The research methodology is illustrated in
Figure 4.
4. Results and Discussion
To investigate the relationship between different variables and the performance of composite concrete beams, the beams were grouped based on various variables. Beams L1, L2, and L3 had different reinforcement diameters to explore the impact of the reinforcement ratio on the composite-beam performance. Beams L2, L4, L5, and L6 were used to study the effect of the composite height on the beam performance. Beams L4, L6, and L7 were used to examine the influence of material structure combinations on the beam performance.
4.1. Test Results and Analysis of Deflection Development in Concrete Beams
This experiment investigated the effect of different variables on the deflection development of high-titanium heavy-slag concrete composite beams. The relationship between various variables and the deflection development of the concrete composite beams is shown in
Figure 5.
As shown in
Figure 5a, when other variables were constant, the deflection of the concrete composite beams with different reinforcement ratios significantly varied. Specifically, at the same load, the concrete composite beam with a reinforcement ratio of 0.84 exhibited smaller deflection, while the beam with a reinforcement ratio of 0.54 showed greater deflection. This indicated that increasing the reinforcement ratio within a certain range could reduce the deflection of high-titanium heavy-slag concrete composite beams under loading. This was because the tensile strength of the steel reinforcement far exceeded that of the concrete and an increased reinforcement ratio enlarged the contact area between the steel and concrete, significantly enhancing the concrete, thereby effectively reducing the deformation of the concrete beam under loading. Additionally, variations in the prefabricated height also affected the deflection of the concrete composite beams, warranting a further in-depth analysis. Compared with monolithic casting, the composite beams exhibited a smaller deflection under the same load, with the high-titanium heavy-slag concrete composite beams showing the least deflection. Although high-titanium heavy slag has a relatively high density, the incorporation of fly-ash ceramsite reduces the composite beam’s weight. Both materials possess excellent properties that effectively decrease the deflection of flexural members. The mechanisms behind the improved deformation performance of the high-titanium heavy-slag concrete composite beams require a further detailed analysis and an explanation.
In summary, to reduce the deflection of concrete beams, it is advisable to appropriately increase the reinforcement ratio and use the material structure combination of high-titanium heavy-slag composite beams.
4.2. Comparison and Correction of Empirical Formulas for Mid-Span Deflection
4.2.1. Comparison of Empirical Formulas for Mid-Span Deflection
To further explore the relationship between deflection development in composite beams and various variables, experimental values were compared with theoretical values. The theoretical range for the calculation of beam deflection applies from the cracking of the beam component until the yielding of the longitudinal reinforcement. According to relevant standards [
40], the deflection calculation formulas for flexural members are as follows:
where
is the bending moment borne by the beam’s span,
is the length of the beam’s span,
is the short-term stiffness of the flexural member,
is the coefficient of the non-uniform strain of reinforcement,
is the ratio of the elastic modulus of the steel reinforcement to the elastic modulus of the concrete,
is the longitudinal reinforcement ratio of the tensile steel reinforcement,
is the ratio of the tensile flange section area to the effective section area of the web,
is the modulus of elasticity of the reinforcing steel,
is the cross-sectional area of the tensioned reinforcing steel, and
is the effective depth of the beam’s cross-section.
Based on empirical formulas and experimental results, the comparison chart of the short-term deflection experimental values and the standard theoretical values is shown in
Figure 6. Panels (a) to (g) represent the deflection development of concrete beams L1 to L7, respectively.
As shown in
Figure 6, the experimental mid-span deflection values for the L6 high-titanium heavy-slag monolithic beam closely matched the standard theoretical values. However, for the other composite beams, there were significant discrepancies between the experimental and theoretical deflection values. Under the same load, the experimental deflection values of the composite beams were greater than the theoretical values, and this difference increased with the load. This suggests that the actual deflection of the composite beams under loading may have exceed the theoretically designed values, potentially negating the safety margin considered during the design [
41].
4.2.2. Correction and Re-Comparison of Empirical Formulas for Mid-Span Deflection
To improve the accuracy of the deflection calculation formulas for composite beams and improve the precision of the theoretical system, the theoretical and experimental deflection values of the composite beams were linearly regressed based on the prefabricated height. A correction coefficient
φ was introduced to the linear regression equation as follows [
40]:
where
represents the prefabricated height of the composite beam in meters. The definitions of the other parameters were the same as those listed in
Section 4.2.1.
Using the modified theoretical formulas, the deflection of the concrete beams was calculated and the experimental values were compared with the modified theoretical values. The results are shown in
Figure 7.
Figure 7 illustrates that by introducing a modification factor
φ for the height of the prefabricated part of the composite beam, the modified theoretical short-term deflection values of the concrete composite beam aligned more closely with the experimental values, resulting in a smaller deviation in the load–deflection curve. To further quantify the error between the modified theoretical values and the standard theoretical values, a calculation formula for the deflection error was introduced, as shown below [
36].
where
is the calculation error,
is the deflection of the concrete beam obtained after experimental testing, and
is the deflection of the concrete beam derived from the formula. The error comparison between the modified theoretical deflection values and the standard theoretical values of the concrete composite beam according to the calculations is shown in
Figure 8.
Figure 8 shows that before modification, the deviation between the standard theoretical values and the modified theoretical values of the concrete composite beam was significant, with L4 having a deviation of 10.9%, L5 a deviation of 29.2%, and the remaining composite beams around 20%. After introducing the modification factor, the deviation between the theoretical calculation values and the experimental values of the composite beam was within 5%, demonstrating that the modification of the theoretical formula significantly improved the accuracy.
4.3. Test Results and Analysis of Concrete-Beam Crack Development
The experiment investigated the effects of different variables on the deflection development of high-titanium heavy-slag concrete composite beams. Based on the load value
obtained from the experiment, the moment
experienced by the specimen could be calculated as follows [
40]:
where
l represents the distance between the loading point at the top of the beam and the support point at the bottom. In this study,
= 0.6 m.
The relationship between various variables and crack development in the concrete composite beams is illustrated in
Figure 9.
Through experimentation, it was observed that the crack development in seven different beams was similar. The initial cracks all originated near the point of the pure bending moment, accompanied by a slight noise. As the load increased, the crack width widened and new cracks continued to appear both in the bending–shear zones and pure bending zones, propagating towards the loading point.
Figure 9a shows that under the same load, an increase in the reinforcement ratio appropriately reduced the maximum crack width in the beam, indicating that higher reinforcement ratios could moderately suppress crack development. Additionally, when the reinforcement ratios were 0.54, 0.68, and 0.84, the cracking moments
were 25.5 kN·m, 27.2 kN·m, and 28.5 kN·m, respectively. This demonstrated that increasing the reinforcement ratio could somewhat inhibit crack development. This was because the reinforcement could replace the tensile role of the concrete, resulting in smaller deformations of the concrete beam under the same load and reducing the likelihood of cracking.
Figure 9b shows that appropriately increasing the prefabricated height resulted in an increase in the crack width, with the cracking moments slightly varying for different prefabricated heights. Furthermore,
Figure 9c reveals that the crack widths under the same load differed among the three composite beams. Overall, the crack width of the high-titanium heavy-slag composite beam was smaller, while the crack width of the ordinary concrete composite beam was larger. This demonstrated that the high-titanium heavy slag and fly-ash ceramsite exhibited excellent properties in the flexural members, with both materials enhancing the composite beam’s crack-resistance performance.
The experiment investigated the effects of different variables on the flexural cracks in high-titanium heavy-slag concrete composite beams. Our overall analysis suggests that it is advisable to appropriately increase the reinforcement ratio and utilize a material structure combination incorporating high-titanium heavy slag to enhance the flexural crack resistance of concrete beams.
4.4. Comparison and Modification of Empirical Formulas for Fracture Characteristics
To further explore the practical applicability of the fracture theory for composite beams, the standard theoretical values of the fracture characteristics were compared with the experimental values. The theoretical formulas were then modified accordingly, with significant deviations.
4.4.1. Comparison of Theoretical and Experimental Values of the Cracking Moment
According to the standards, the cracking moment
(when the concrete in the tensile zone cracks) can be calculated using the following formula [
40]:
where
is the standard value of the axial compressive strength of concrete;
is the plasticity influence coefficient of a specimen’s section resistance moment, which is 1.55 for a rectangular section;
is the elastic section modulus of the converted section’s tensile edge, measured in mm
3; and
is the standard value of the concrete’s tensile strength.
is calculated using the following formulas [
40]:
where
is the width of the specimen’s cross-section, in millimeters;
is the height of the specimen section, in mm;
is the distance from the centroid of the converted section to the compression edge;
is the moment of inertia of the converted section about the centroidal axis;
is the elastic modulus of the reinforcement; and
is the elastic modulus of the concrete. The definitions of the other parameters were the same as those listed in
Section 4.2.1.
Based on the empirical formulas and experimental results, the experimental cracking-moment values
and the theoretical values
for the concrete beams were obtained. The cracking-moment error rate
was defined and calculated as follows [
36]:
The comparison between the experimental cracking-moment values and the standard theoretical values for the seven composite beams after data processing and analysis is shown in
Figure 10.
Figure 10 shows that the experimental cracking-moment values for the seven composite beams ranged from 25.5 kN·m to 28.5 kN·m, while the standard theoretical values ranged from 26.28 kN·m to 28.05 kN·m. The beam with the largest error, L1, had an error rate of only 4.67%, demonstrating that the standard calculation formula for the cracking moment of concrete beams was also applicable to high-titanium heavy-slag concrete composite beams.
4.4.2. Comparison of Empirical Formulas for the Maximum Crack Width
To further compare the development of the crack width in the concrete composite beams and discuss the applicability of the crack-width formulas provided by the standards to composite beams, both were analyzed. In accordance with the standards, the maximum crack width
for concrete beams under bending was calculated as follows [
40]:
where
represents the characteristic coefficient for the member under load, with a value of 1.9 for beams under bending;
denotes the coefficient of strain non-uniformity for the longitudinal tensile reinforcement in cracks;
represents the stress in the longitudinal tensile reinforcement calculated according to a quasi-permanent combination;
represents the distance from the outer edge of the longitudinal reinforcement to the bottom edge;
and
represent the effective cross-sectional area of the tensile concrete zone and the cross-sectional area of the tensile reinforcement;
represents the equivalent diameter of the tensile reinforcement;
represents the reinforcement ratio of the tensile steel bars; and
, and
represent the nominal diameter, number, and relative bond characteristic coefficient of the tensile reinforcement, with a value of 1.0 for the ribbed reinforcement. The definitions of the other parameters were the same as those listed in
Section 4.2.1.
The data for the maximum crack width of the concrete beams based on the calculations and experiments are summarized in
Figure 11.
From
Figure 11, it was observed that beam L6, which was a monolithic beam, showed close agreement between the standard theoretical values and the experimental values for the maximum crack width, validating the accuracy of the experiment. However, for the other composite beams—except for L4, where the standard theoretical value was relatively close to the experimental value—significant errors were noted. Generally, the theoretical values were higher than the actual values and the errors increased with an increase in the bending moment. Our analysis suggested that due to structural differences, the load-carrying mechanisms of the composite beams differed from the monolithic beams, necessitating some adjustments to the empirical formulas provided by the standards to better suit composite beams.
4.4.3. Correction and Re-Comparison of Empirical Formulas for the Maximum Crack Width
To improve the accuracy of the calculations of the maximum crack width in composite beams, a linear regression was performed between the theoretical values and experimental results, introducing a correction factor
. The linear regression equations were [
40]:
where
represents the prefabricated height of the composite beam, in meters. The definitions of other parameters were the same as those listed in
Section 4.2.1.
The results of the corrected maximum crack widths compared with the experimental results are shown in
Figure 12.
Figure 12 illustrates that by introducing the correction factor
, and based on the prefabricated height of the composite beam, the theoretical values of the maximum crack width for six concrete composite beams aligned more closely with the experimental values. Similar to previous analyses, the error rate
was defined to further quantify the deviation of the corrected theoretical values and was calculated as follows [
36]:
where
represents the calculation error,
denotes the maximum crack width of the concrete beam obtained from the experimental tests, and
represents the maximum crack width of the concrete beam calculated using the formula.
The error comparison between the corrected theoretical values and the standard theoretical values for the maximum crack width of the concrete composite beams according to the calculations is shown in
Figure 13.
Figure 13 shows that before the modification, the standard theoretical formula for the maximum crack width for concrete beams had significant deviations, with the largest error for beam L5 at 31.6% and the smallest for beam L4 at 12.6%. Our analysis suggested that the differing heights of the prefabricated sections (250 mm for L5 and 150 mm for L4) contributed to the discrepancies between the theoretical and actual values.
After introducing the modified formulas, the theoretical calculations for all seven composite beams aligned more closely with the experimental values, with errors reduced to within 10%. This indicated that the modified formulas were effective, significantly improving the accuracy of the theoretical formulas for the maximum crack width of flexural composite beams, thereby enhancing the safety and reliability of composite-beam design.