Theoretical and Experimental Study on the Stress State of Joints in Two-Way Composite Slabs

Abstract

To investigate the stress state at the joints of two-way reinforced concrete composite slabs, this study conducted static load tests on four two-way concrete composite slabs. The primary focus was on analyzing the effects of lap reinforcement length and cross-sectional height at the joints on the load-bearing capacity, deformation behavior, and crack development of the slabs. The experimental results revealed that increasing the lap reinforcement length and cross-sectional height at the joints significantly enhanced the overall integrity and flexural capacity of the composite slabs, with load-bearing capacity increasing by up to 92.3% and deflection decreasing by as much as 40.2%. Additionally, a finite element model was used to simulate the mechanical behavior of the composite slabs, and the results were compared with experimental data, showing an error margin of within 10%. Based on the validated finite element model, the study further explored key factors influencing the stress performance at the joints of two-way concrete composite slabs and their impact patterns. Finally, the paper proposes a simplified formula for calculating the load-bearing capacity of composite slabs, which enables rapid estimation of slab performance, providing theoretical support and practical guidance for structural engineering and construction practices.

1. Introduction

As a crucial element in the industrialization of construction, prefabricated concrete structures have steadily developed with the support of national and local government policies as well as market promotion [1,2,3,4,5]. Among these structures, composite slabs have been widely adopted due to their ease of construction and high degree of industrialization [6,7,8]. A composite slab is divided into two parts along its thickness: the lower part is a precast concrete base, while the upper part consists of cast-in-place concrete. Due to structural deficiencies, ensuring that the concrete on both sides of the composite interface works together and bears loads in a coordinated manner is critical for the design of composite slabs [9,10]. Therefore, studying the stress transfer capacity and mechanism at the joints of prefabricated composite slabs is essential.

In recent years, to enhance the stress performance at the joints of composite slabs, both domestic and international scholars have optimized the joint design of composite slabs [11,12,13,14,15], conducting experimental studies and theoretical analyses. Extensive research has also been conducted on the flexural behavior of various composite materials, including studies on cold-formed thin-walled steel, fiber-reinforced polymer sheets, and reinforced concrete slabs [16,17,18,19,20,21]. Stehle et al. [22] performed experimental studies on composite slabs with closely spaced steel truss reinforcements, showing that the steel trusses not only increased the stiffness of the precast base but also aided in the anchoring and force transfer of additional reinforcement in the slab. Zhang et al. [23] investigated the tensile and shear properties at the joints of composite slabs, revealing that reinforcement or sleeves could increase the ultimate load-bearing capacity by approximately 20%. Lou [24] proposed a T-joint prefabricated composite slab and conducted bending tests. The results indicated that the load-bearing capacity and cracking load of the novel composite slab with a trapezoidal T-joint concrete layer were comparable to those of cast-in-place slabs, meeting engineering requirements. Han et al. [25] conducted bending tests on prestressed composite slabs, demonstrating that the cracking load and stiffness were significantly dependent on the reinforcement ratio and prestress level. Liu et al. [26] carried out bending tests on eight precast one-way slabs, exploring the effects of slab thickness, truss spacing, and overlap length of reinforcement on the mechanical properties of one-way slabs. The results showed that the steel trusses helped prevent brittle failure between the precast reinforced concrete slab and the joint, and both truss spacing and overlap length affected the load-bearing capacity of semi-precast reinforced concrete one-way slabs. Abokifa et al. [27] used ultra-high-performance concrete in localized areas to effectively connect the concrete slab and steel beams, although this method imposed higher demands on structural components.

In addition, the bonding performance between the precast and cast-in-place layers of composite concrete slabs is a critical factor in determining their overall performance. Good bonding ensures effective shear transfer at the interface, preventing delamination and maintaining the overall stiffness and load-bearing capacity of the slab. In recent years, extensive research has been conducted on the characteristics of the bonding interface and methods to enhance it. For example, Mohamed et al. [28] demonstrated that the bond strength between the precast and cast-in-place layers significantly affects the bending behavior of semi-precast reinforced concrete slabs. The experimental results showed that different surface treatment methods had a considerable impact on the shear strength of the interface, with the shear bond strength exceeding 1.0 MPa, ensuring the integrity of the slab system under bending loads. Jiang et al. [29] investigated the shear-friction behavior between high-strength precast concrete beams and lightweight cast-in-place concrete slabs, finding that the shear strength at the interface was significantly influenced by the reinforcement ratio and surface treatment. The study revealed that lightweight concrete slabs exhibited better shear transfer strength than normal-weight concrete slabs, and the application of interface reinforcement substantially improved shear capacity and residual resistance. Adawi et al. [30] evaluated the bonding performance between machine-cast hollow-core slabs and cast-in-place concrete toppings through pull-out, push-off, and full-scale tests. The results indicated that even with machine-cast surfaces, the tensile and shear strengths exceeded the requirements of North American design standards, demonstrating sufficient composite action. Wang et al. [31] proposed a novel ultra-high-performance concrete (UHPC) shear key wet joint designed to connect precast components and enhance their interconnection. The research demonstrates that the UHPC shear key significantly improves the ultimate shear strength and ductility of the interface, particularly under longitudinal loads. In contrast to traditional methods that extend reinforcement at the joints, the combination of UHPC shear keys with short straight bars enables more effective load and stress transfer without significant slippage or failure. This approach offers a reliable solution for the effective connection of composite precast slabs.

It is evident that scholars both domestically and internationally have proposed various reinforcement methods for the joints of composite concrete slabs. Based on this, to further optimize the joint performance of composite slabs, this study investigates the stress performance of two-way composite slabs with lattice reinforcement at the joints. The stress state at the joints is examined from two perspectives. Through experimental research on four composite slabs, combined with validated finite element models, the influencing factors and variation patterns of the stress performance of two-way composite slabs are simulated. Additionally, using concrete damage theory, a more in-depth analysis of the stress performance is conducted, deriving a coefficient that reflects the influence of the interface on the anchorage capacity of reinforcement at the joints. A simplified method for calculating the load-bearing capacity is also proposed. In summary, this study not only identifies the key factors and mechanisms affecting joint stress but also provides a scientific theoretical basis for the design and construction of composite slabs, offering new insights and references for future research on optimizing joint stress performance.

2. Materials and Methods

2.1. Purpose of the Experiment

In this experiment, the design of the joint reinforcement diameter and spacing was determined based on the longitudinal reinforcement ratio. Static load tests were conducted on four two-way reinforced concrete composite slab specimens to investigate the effects of varying the lap reinforcement length and cross-sectional height at the mid-span joint on load-bearing capacity, deformation behavior, crack development patterns, normal section stress characteristics, and the stress transfer performance of the joint reinforcement. Understanding these factors is crucial for optimizing joint performance in precast systems.

2.2. Experimental Design and Production

The experiment involved the design and fabrication of four two-way reinforced concrete composite slabs, designated DB-1, DB-2, DB-3, and DB-4; their geometric dimensions and configurations are shown in Figure 1. Each slab had the same geometric dimensions: a length of 2400 mm, a width of 880 mm, and a height of 130 mm.

Table 1. Specimen parameters.

Plate Number	Length of Lap Bar (mm)	Location of Lap Bars	Average Number of Lapped Bars	Whether There Is an Extension into the Lattice Reinforcement
DB-1	500	Secondary casting surface	4B12	clogged
DB-2	500	Prefabricated layer surface	4B12	clogged
DB-3	350	Secondary casting surface	6B12	clogged
DB-4	900	Secondary casting surface	4B12	be

summarizes the key dimensions of the four specimens. The primary difference between the slabs was the overlap length of the reinforcement at the joint, which was as follows: DB-1: 500 mm, DB-2: 500 mm, DB-3: 350 mm, and DB-4: 900 mm. The placement of the overlapping reinforcement is shown in Figure 2. In DB-1, the overlapping reinforcement was placed on the upper surface of the composite interface, while in DB-2, the reinforcement was embedded in a groove at the lower surface of the interface with a groove depth of 15 mm. The diameter of the primary reinforcement was 12 mm, with a spacing of 150 mm, and the diameter of the structural reinforcement was 6 mm, spaced at 200 mm. The concrete cover thickness was 15 mm.

2.3. Material Properties Test

The material test results are presented in

Table 2. Mechanical Properties of Concrete.

Concrete	Measured Compressive Strength (MPa)			Average Measured Compressive Strength (MPa)
	Specimen 1	Specimen 2	Specimen 3
Precast bottom slab	32.1	33.5	33.1	32.9
Cast-in-place slab	33.5	32.2	33	32.9

. The specimens include two batches of concrete: one for the precast layer and the other for the cast-in-place layer, both using C30-grade concrete. To ensure the reliability of the concrete’s performance, three cube specimens measuring 150 mm × 150 mm × 150 mm were reserved during the casting process. These specimens were cured in a chamber at a temperature of (20 ± 2) °C and a relative humidity of 95% for 28 days. After curing, the concrete compressive strength was measured using a universal testing machine, as shown in Table 2. It should be noted that the compressive strength of the concrete could be influenced by curing conditions, including temperature, humidity, and time, so these conditions were strictly controlled during the experiment to minimize variability.

The reinforcement used in the specimens consisted of two types of rebar with diameters of 6 mm and 12 mm, both made of HRB335 ribbed steel. For each rebar size, three 300 mm-long samples were subjected to tensile testing, and the average values obtained from the tests were used as the yield strength and ultimate elongation. The rebar strength values are provided in

Table 3. Material Properties of Reinforcing Steel.

Diameter/mm	Average Yield Strength/MPa	Ultimate Elongation/%
6	348	19.5
8	358	18.7

. Additionally, variations in the quality of the rebar may arise due to differences in manufacturing processes; therefore, production standards and quality control were also considered when selecting the rebar to ensure the reliability of the experimental results.

2.4. Loading and Measurement Program

The loading apparatus and the arrangement of displacement measurement points are illustrated in Figure 3a,b. The specimens were subjected to concentrated loading at one-third points using a jack-distribution beam system for monotonic loading. A pressure sensor, with an accuracy of ±0.03%, was employed to control the loading value. Electronic displacement gauges were installed at mid-span and support locations to measure the deflection of the composite slab and support deformation, with a linear accuracy of ±0.05%. Prior to the experiment, the pressure sensors were calibrated using a pressure calibrator, and the displacement gauges were standardized with a ruler to ensure the accuracy of the experimental data. The experimental data were automatically collected by a computer and monitored in real time based on the load–displacement curve.

As shown in Figure 3c, multiple strain gauges were installed at the mid-span of the lap reinforcement at the joint to measure the strain of the reinforcement and investigate the strain transfer mechanism at the joint. The primary reason for installing multiple strain gauges is that a single strain gauge may be affected by installation errors or other external factors, leading to potential damage and data collection issues. By using multiple strain gauges, random errors can be minimized, and overall measurement accuracy can be improved.

3. Experimental Analysis

3.1. Experimental Phenomena

After the completion of preloading, as formal loading commenced and the applied load increased, the deflection of the specimens progressively grew, and the number and width of cracks increased accordingly. In all four tested slabs, initial cracking appeared at the joint in the mid-span and gradually extended to the composite interface. With increasing load, the cracks propagated along the interface toward the slab ends. Notably, slab B-3 exhibited rapid fracture failure when the load reached 7 kN, as the cracks continued to spread along the interface. The fracture occurred along the ends of the connecting reinforcement, causing the reinforcement to disengage from the slab. This failure was attributed to the short length of the connecting reinforcement (300 mm), which did not extend into the lattice reinforcement, leading to poor overall integrity of the slab.

In comparison, slabs B-1 and B-2 demonstrated that changing the effective section height altered the crack path, resulting in improved crack development. For slab B-4, where the connecting reinforcement extended into the lattice reinforcement, the slab’s integrity was significantly enhanced, effectively controlling crack propagation. The crack development patterns on the sides of the slabs are shown in Figure 4.

As shown in Figure 5, cracks began to appear at the mid-span of specimens DB-1, DB-2, and DB-4 when the load reached 2 kN, while cracks in specimen DB-3 only appeared when the load reached 4 kN. This delay is attributed to the use of six reinforcing bars for lap joints at the mid-span of DB-3, compared to four bars in the other three specimens. However, when the load reached 5 kN, the cracks in DB-3 rapidly propagated, and the crack width increased sharply as the lap reinforcement yielded. At 7 kN, DB-3 failed and ceased to carry load. This brittle failure was caused by the reduced length of the connecting reinforcement within the tear zone along the composite interface, which could no longer effectively transfer stress.

For specimens DB-1 and DB-4, when the load reached 12 kN and 11 kN, respectively, the crack width at the mid-span rapidly increased, indicating that the lap reinforcement began to slip or yield. Ultimately, DB-1 failed at 13 kN and DB-4 at 12 kN, as the crack width at the mid-span joint exceeded 2.0 mm. Conversely, specimen DB-2 exhibited increased load-bearing capacity because the lap reinforcement was placed at the surface of the precast layer, increasing the effective cross-sectional height. Cracks in DB-2 rapidly developed at 24 kN, and it failed at 25 kN when the crack width at the mid-span exceeded 2.0 mm.

These results demonstrate that two-way concrete composite slabs are prone to tearing failure along the composite interface at the joints, with significant crack development during failure. Increasing the length and cross-sectional height of the joint reinforcement effectively mitigates tearing failure along the interface.

3.2. Load–Mid-Span Deflection Curve

The load–mid-span deflection curves for each specimen are shown in Figure 6. All four specimens exhibited small deflections during the initial loading phase, with a linear growth trend. After cracking, the deflection growth rate significantly increased, and the slope of the curve gradually decreased, indicating nonlinear characteristics. Specimen DB-3, due to its shorter lap reinforcement length, experienced brittle fracture at a load of 7 kN, resulting in early failure and withdrawal from the experiment. The remaining specimens showed a noticeable increase in deflection when the load reached 8 kN.

At higher load levels, the failure modes of the specimens varied: DB-1 failed at a load of 13 kN, while DB-4 failed at 21 kN. Specimen DB-2 demonstrated the highest load-bearing capacity, ultimately failing at 25 kN.

Table 4. Comparison of deflection of experimental plates.

Specimen Number	DB-1	DB-2	DB-3	DB-4
Specimen deflection value (mm)	0.510	0.205	devastation	1.075
Relative value	1	0.5	devastation	2.107

presents a comparison of the deflection of each test specimen under a 10 kN load. Using the deflection value of specimen DB-1 as a baseline (1.0), it can be observed that the effective height increase in specimen DB-2 resulted in a 40.2% reduction in deflection, significantly enhancing the load-bearing capacity and structural stability of the composite slab.

Additionally, the influence of lap reinforcement length is quite pronounced. Comparing DB-1 with DB-3 reveals that when the lap reinforcement length is reduced from 500 mm to 350 mm, the load-bearing capacity of the composite slab decreases significantly, with DB-3 already failing under a 10 kN load. In contrast, comparing the deflection values of DB-1 and DB-4 indicates that when the lap reinforcement length is increased to 900 mm, the ability to control deflection reaches saturation, and further increases in length do not yield significant improvements.

From this analysis, it is evident that increasing the cross-sectional height and optimizing the lap reinforcement length can effectively control the deflection of the composite slabs. However, once the reinforcement length exceeds a certain threshold, the effectiveness of deflection control diminishes.

3.3. Carrying Capacity Analysis

From the experimental analysis, it can be concluded that increasing the effective cross-sectional height and lap reinforcement length can enhance the flexural capacity of composite slabs.

Table 5. Comparison of specimen load capacity.

Specimen Number	DB-1	DB-2	DB-3	DB-4
Does it extend into the lattice reinforcement	clogged	clogged	clogged	be
Crack status	break apart	Crack width 2.0 mm	break apart	Crack width 2.0 mm
Breaking load (kN)	13	25	7	12
Relative value	1	1.923	0.538	0.923
Increase or decrease in Load capacity	0	+0.923	−0.462	−0.077

provides a comparison of the load-bearing capacities of the tested slabs.

From the table, it can be observed that different lap reinforcement lengths and cross-sectional heights significantly impacted the load-bearing capacity and failure modes of the specimens.

(1)

The lap reinforcement lengths for specimens DB-1 and DB-2 were both 500 mm; however, DB-2 exhibited a crack width of 2.0 mm when it failed under a load of 25 kN, representing a 92.3% increase in load-bearing capacity compared to DB-1. This indicates that even with the same reinforcement length, the cross-sectional height of the composite slab has a significant impact on load-bearing capacity, and increasing the height can substantially enhance the reinforcement’s strength.

(2)

Specimen DB-3 had a shorter lap reinforcement length (350 mm), resulting in a failure load of only 7 kN, a reduction of 46.2% compared to DB-1. This specimen failed earlier, indicating that the insufficient length of the lap reinforcement weakened the overall integrity of the slab, preventing the reinforcement from fully exerting its strength. Thus, lap reinforcement length has a notable effect on the early failure behavior of the specimen.

(3)

Specimen DB-4, with a lap reinforcement length of 900 mm, showed only a 7.7% increase in load-bearing capacity compared to DB-1, ultimately failing under a load of 12 kN with a crack width of 2.0 mm. This suggests that once the lap reinforcement length reaches a certain threshold, further increases in length have a limited impact on load-bearing capacity, as evidenced by the small increase in capacity.

To assess the significance of the observed differences, the standard deviation of the failure loads for the specimens was calculated, yielding a result of 6.63 kN. This indicates a certain degree of dispersion in the failure load data. The failure load of DB-2 was significantly higher than that of the other specimens, exhibiting substantial variability, which suggests that changes in its cross-sectional height play a critical role in enhancing load-bearing capacity. In contrast, although the lap reinforcement length in DB-4 was significantly increased, the effect on load-bearing capacity was limited. This finding aligns with the standard deviation results, indicating that the relationship between increased lap reinforcement length and enhanced load-bearing capacity is not linear.

In summary, DB-2 demonstrated the best load-bearing capacity, showing a 92.3% increase compared to the baseline specimen DB-1. Conversely, DB-3 exhibited a 46.2% reduction in load-bearing capacity due to its shorter lap reinforcement. While DB-4 had a significant increase in lap reinforcement length, its load-bearing capacity only improved by 7.7%. This analysis indicates that a lap reinforcement length that is too short will significantly reduce structural load-bearing capacity, while excessive length provides limited benefits for subsequent capacity enhancement.

3.4. Strain Analysis of Reinforcement in Mid-Span Splices

Based on the data analysis presented in Figure 7, significant differences were observed in the strain of the lap reinforcement bars for specimens DB-1, DB-2, DB-3, and DB-4, primarily influenced by the length of the reinforcement and the cross-sectional height. First, specimen DB-2, with a lap reinforcement length of 500 mm, exhibited well-developed strains under loading, exceeding 2300 με, indicating strong load-bearing capacity. This enhancement is attributed to the notched design of DB-2, which increased the effective height of the reinforcement, thereby improving the bonding force between the composite layer concrete and the precast bottom slab.

In contrast, specimen DB-3 had a shorter lap reinforcement length of only 350 mm, resulting in significantly lower strain levels that did not reach the yield strain, suggesting a minimal contribution to load-bearing capacity. The limited reinforcement length in DB-3 led to poor overall integrity, causing separation at the interface between the old and new concrete, which subsequently resulted in rapid strain development in the lap reinforcement and a yield load considerably lower than that of the other specimens.

Additionally, although specimen DB-4 featured a lap reinforcement length of 900 mm and demonstrated enhanced load-bearing capacity, the rate of strain development was relatively slow, indicating that the strength of the reinforcement was not fully utilized. Finally, the strain in the reinforcement of specimen DB-1 began to increase significantly when the load reached 13 kN, suggesting that yielding of the reinforcement had commenced and indicating strong load-bearing capacity.

In conclusion, optimizing the length and cross-sectional height of the reinforcement is crucial for enhancing the overall load-bearing capacity of the composite slabs.

4. Numerical Simulation

Finite element models of the bidirectional concrete composite slab specimens were developed using ABAQUS software to analyze their bending performance. The results from the finite element analysis were compared with experimental data to verify the accuracy of the finite element models. This verification provides a foundation for further in-depth analysis of the load-bearing performance of bidirectional concrete composite slabs.

4.1. Material Intrinsic Relationship

The concrete is modeled using a plastic damage model, with the stress–strain relationship for tension and compression based on the stress–strain curves recommended in the “Code for Design of Concrete Structures” (GB 50010-2010) [32]. The Poisson’s ratio for concrete is set to 0.2, the dilation angle is 38°, the eccentricity of the subsequent yield surface’s plastic potential function is 0.1, the ratio of biaxial to uniaxial initial yield strength is 1.16, the shape factor is 0.66667, and the viscosity coefficient of the plastic damage model is set to 0.0005. The steel is modeled using a bilinear reinforcement model, with a Poisson’s ratio of 0.3, a hardening modulus of 0.01 times the elastic modulus, and the strength of the reinforcement is based on the measured values from material tests.

To enhance the reliability of the analysis, a brief sensitivity study was conducted to examine the impact of variations in key material parameters on the finite element results. By altering the Poisson’s ratio, dilation angle, and viscosity coefficient of the concrete, we assessed their influence on the stress distribution and deformation in the simulation. The results showed that when the Poisson’s ratio fluctuated between 0.18 and 0.22, the displacement variation in the calculations remained below 3%. However, changes in the dilation angle (±2°) had a more pronounced effect on the failure mode and crack distribution, especially at higher load levels. As for the viscosity coefficient, increasing or decreasing it by 0.0002 had a noticeable impact on the concrete’s failure point, leading to approximately a 5% variation in the failure load.

4.2. Finite Element Model

The finite element model for the bidirectional concrete composite slabs was developed using the following specifications:

Concrete: both the precast bottom slab and the composite layer concrete were modeled using 8-node reduced integration elements (C3D8R).

Reinforcement Bars: the reinforcement in the precast bottom slab, lap reinforcement bars, and truss reinforcement bars were modeled using 2-node 3D truss elements (T3D2).

Steel Tubes: the steel tubes in the truss were modeled using 4-node reduced integration shell elements (S4R).

To determine the optimal mesh size, a mesh sensitivity analysis was performed. The final mesh size for the concrete was set to 20 mm, while the optimal mesh size for the reinforcement bars was determined to be 10 mm. Reducing the concrete mesh size from 20 mm to 15 mm resulted in a less than 2% change in results, but increased computation time by approximately 30%. Similarly, reducing the mesh size for the reinforcement from 10 mm to 8 mm yielded negligible accuracy improvements, while increasing computation time by about 20%. Therefore, the selected mesh sizes provided a good balance between accuracy and computational efficiency.

In the model, no relative slip or cracking occurred between the precast bottom slab and the composite layer at the lap joints, so a TIE constraint was applied at the interface between the new and old concrete layers. This TIE constraint specifically restricted the relative degrees of freedom in the X and Y translational directions, while allowing the Z-direction translation and Y-axis rotation to remain unconstrained. Contact conditions were defined at both ends of the lap joints, with frictionless tangential contact and hard contact in the normal direction. The reinforcement bars in the bottom slab and lap joints were embedded into the concrete using the embedded command.

Constraints were applied 80 mm from both the left and right ends of the composite slab. At the left end, the bottom nodes were constrained in the Y and Z translational directions, as well as the X and Z rotational degrees of freedom. At the right end, the bottom nodes were constrained in the X, Y, and Z translational directions, as well as the X and Z rotational degrees of freedom.

Displacement-controlled loading was applied at reference points RP-1 and RP-2, located at one-third of the length of the component. The finite element model of the specimen is shown in Figure 8.

Computational Resources: The simulation was performed using a six-core processor with 16 GB of RAM. Under these conditions, the computation time for each model was approximately 3 h. Readers wishing to replicate this analysis should consider using similar computational resources and mesh configurations to ensure a balance between computation time and result accuracy.

4.3. Finite Element Model Validation

Taking specimen DB-1 as an example, as shown in Figure 9, a comparison was made between the crack distribution observed in the experiment and the tensile damage distribution obtained from the finite element analysis. The red areas in the figure indicate regions with severe tensile damage, representing the development of cracks. In the central region of the composite slab (particularly in the tensile zone), damage is concentrated and widely distributed, indicating that this area likely experienced the highest tensile stress, which caused cracks to initiate and propagate from these locations. This behavior is consistent with the stress characteristics of the slab under pure bending, where the location of the maximum bending moment often coincides with the concentration of tensile stress and the formation of cracks.

Based on Figure 9, it can be seen that the locations of tensile damage and crack distribution in the concrete specimen from the finite element analysis (FEA) generally align with the experimental results. However, despite the overall good agreement, there are minor local discrepancies between the FEA and experimental results. These differences may stem from the simplifications made during the modeling process, particularly in handling the nonlinear properties of materials and boundary conditions.

Specifically, the cracking and damage behavior of the concrete was simulated using a plastic damage model. While this model is effective for representing the overall damage, it may have limitations in capturing the precise path and dynamic characteristics of crack propagation. Furthermore, the actual loading conditions, support structures, and the complexity of contact surfaces in the experiment could lead to localized stress concentrations and cracking in certain regions of the specimens, which were not fully captured in the finite element model.

As shown in Figure 10, the mid-span moment–deflection curves for the two-way composite slab specimens obtained from the FEA match well with the experimental curves, though slight deviations are observed at specific load levels. These deviations are primarily due to the initial crack development and stress concentration regions not being fully represented in the finite element model. The ultimate load values presented in

Table 6. Comparison of specimen test bearing capacity and numerical simulation results.

Laminated Plate Number	Ultimate Load of Test Plate (KN)	Numerical Simulation of Plate Ultimate Load (KN)	Error (%)
DB-1	13	12.3	5.4
DB-2	25	23.6	5.6
DB-3	7	6.2	11.4
DB-4	21	20.5	7.1

show a maximum discrepancy of approximately 10% between the FEA and experimental results. This difference may be attributed to the homogenization of material parameters and the omission of interface shear slip effects in the model.

In conclusion, the finite element analysis shows good overall consistency with the experimental results in terms of crack distribution, moment–deflection curves, and bending capacity. However, the simplifications made in the model—particularly regarding material nonlinearity, boundary conditions, and local stress concentrations—are the main reasons for the observed discrepancies. Future research can focus on refining material constitutive models and optimizing boundary condition definitions to reduce the error between the simulation and experimental results.

5. Simplified Carrying Capacity Calculation Theory

To reveal the stress characteristics at the joint of composite slabs and to investigate the performance of joints without truss reinforcement, this study explores the stress state at the joint from two perspectives: (1) By analyzing the anchorage stress characteristics of reinforcement at the joint, a formula for the ratio of tensile bearing capacity between reinforcements with and without composite surfaces has been derived. This provides insight into the influence of the composite surface on the overall anchorage stress. (2) The critical bending moment for composite slabs with joints is derived from the bending state of the composite surface at the joint.

5.1. Anchorage Stress State at the Joint

When deformed steel bars are subjected to stress, the protruding ribs generate diagonal bearing pressure on the surrounding concrete. The horizontal component of this pressure induces axial tension and shear forces in the concrete surrounding the rebar, while the radial component creates radial tension in the concrete. The axial tension and shear forces cause internal diagonal conical cracks, and the radial tension results in internal radial cracks within the concrete. Determining the splitting stress values for rebar anchorage typically involves two approaches: a semi-theoretical, semi-empirical method or statistical analysis of experimental data using regression analysis to derive empirical formulas.

The semi-theoretical method simplifies the concrete surrounding the rebar into a thick-walled tube. Based on the bearing pressure exerted by the rebar ribs on the concrete, elastic or plastic theory is applied to derive an approximate calculation formula. In this model, the following assumptions are made:

(1)

Uniform Tensile Stress Distribution: It is assumed that, when the concrete cover reaches its critical splitting state, the tensile stress on the splitting surface is uniformly distributed and reaches its axial tensile strength, denoted as $f_t$. This assumption is reasonable because it simplifies the stress transfer process at the rebar-concrete interface, ignoring the possibility of localized stress concentration in the concrete.

(2)

Angle of Rib Bearing Pressure: It is assumed that the angle between the rib bearing pressure and the rebar axis is 45°. This assumption is based on a simplified analysis of the combined effects of tensile and shear forces, reflecting a balance between these forces under stress conditions.

These assumptions are essential for reducing the complexity of modeling the stress behavior at the rebar–concrete interface and allow the development of practical calculation methods. However, they also introduce limitations, as they may overlook certain stress concentration phenomena or non-uniform stress distributions that could occur in real structural conditions.

The following can be obtained based on equilibrium conditions:

$$p_r\times d=2c\times f_t$$

(1)

where $p_r$ represents the radial component of the inclined bearing pressure, $d$ is the diameter of the reinforcement, and $c$ is the thickness of the concrete cover.

$${\tau }_{cr}\approx p_r={\displaystyle \frac{2c}{d}}{f}_t$$

(2)

where ${\tau }_{cr}$ denotes the splitting stress.

According to Tepfers [33], the calculation formula for the splitting stress of reinforcement anchorage is:

$${\displaystyle \frac{{\tau }_{cr}}{f_t}}=0.3+0.6{\displaystyle \frac{c}{d}}$$

(3)

According to Wang et al. [34], the calculation formula for the splitting stress of reinforcement anchorage is:

$${\displaystyle \frac{{\tau }_{cr}}{f_t}}=0.5+{\displaystyle \frac{c}{d}}$$

(4)

According to Xu et al. [35], the calculation formula for the splitting stress of reinforcement anchorage, derived from regression analysis of experimental data, is:

$${\displaystyle \frac{{\tau }_{cr}}{f_t}}=1.6+0.7{\displaystyle \frac{c}{d}}$$

(5)

The average ultimate bond strength between reinforcement and concrete, typically determined through regression analysis of experimental data, is given by the formula proposed by Wang et al. [28]:

$${\displaystyle \frac{c}{d}}\le 2.5 {\displaystyle \frac{{\tau }_u}{f_t}}=\left(1.325+1.6{\displaystyle \frac{d}{l}}\right){\displaystyle \frac{c}{d}}$$

(6)

where ${\tau }_u$ represents the ultimate bond strength of the reinforcement, and $l$ is the bond length of the reinforcement.

$$2.5<{\displaystyle \frac{c}{d}}<5 {\displaystyle \frac{{\tau }_u}{f_t}}=\left(5.5{\displaystyle \frac{c}{d}}-9.76\right)\left({\displaystyle \frac{d}{l}}-0.4\right)+1.965{\displaystyle \frac{c}{d}}$$

(7)

The formula proposed by Xu et al. [35] is:

$${\displaystyle \frac{{\tau }_u}{f_t}}=1.6+0.7{\displaystyle \frac{c}{d}}+20{\rho }_{sv}$$

(8)

where ${\rho }_{sv}={\displaystyle \frac{A_{sv}}{c\cdot s_{sv}}}={\displaystyle \frac{\pi }{4c}}{\displaystyle \frac{d_{sv}^2}{s_{sv}}}$ is the area ratio of the stirrups, and $d_{sv}$ and $s_{sv}$ are the diameter and spacing of the stirrups, respectively.

The above formula for the ultimate bond strength is derived based on experiments with relatively short embedment lengths (${\displaystyle \raisebox{1ex}{$l$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}=2~20$). For longer embedment lengths and when calculating the minimum anchorage (or lap) length of the reinforcement, the results differ. An empirical formula applicable for ${\displaystyle \raisebox{1ex}{$l$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}\le 80$ has been provided by Orangun et al. [36] (units have been converted to match the reference format):

$${\tau }_u=\left(1+{\displaystyle \frac{2.5c}{d}}+{\displaystyle \frac{41.6d}{l}}\right){\displaystyle \frac{\sqrt{f_c}}{10}}$$

(9)

where $f_c$ is the concrete’s axial compressive strength.

As shown in Figure 11, when there is a restraining hoop, the action of the restraining hoop on the bond is:

$${\tau }_{utr}={\displaystyle \frac{A_{sv}{f}_y}{3.45d_{sv}{s}_{sv}}}\sqrt{f_c}$$

(10)

where $f_y$ is the yield strength of the stirrups.

Figure 11. Schematic diagram of restraining hoops.

Figure 11. Schematic diagram of restraining hoops.

When the stirrup reinforcement ratio reaches a certain value, the ultimate bond strength will no longer increase. In the study, this value is specified as ${\displaystyle \frac{A_{sv}{f}_y}{500d_{sv}{s}_{sv}}}\le 3$. This is primarily due to the confining effect of the stirrups, which prevents the component from experiencing splitting failure.

According to Xu et al. [35], the relationship between the axial tensile strength and the standard cube compressive strength of concrete is given by the following equation:

$$f_t=0.26f_{cu}^{2/3}$$

(11)

where $f_{cu}$ is the standard cube compressive strength of concrete.

The relationship between the design value of the concrete axial compressive strength and the standard cube compressive strength of concrete, as proposed by Guo et al. [37], is given by:

$$f_c=0.8f_{cu}$$

(12)

Then:

$$f_{\mathrm{c}}=6f_t^{3/2}$$

(13)

$${\tau }_u=\left(1+{\displaystyle \frac{2.5c}{d}}+{\displaystyle \frac{41.6d}{l}}\right){\displaystyle \frac{f_t^{3/4}}{4}}$$

(14)

$${\tau }_{utr}={\displaystyle \frac{A_{sv}{f}_y}{1.4d_{sv}{s}_{sv}}}{f}_t^{3/4}$$

(15)

For composite slabs, as shown in Figure 12, additional reinforcement is anchored in the new and old concrete at the interface. Due to the lower tensile strength of the interface, splitting occurs when the bond stress between the reinforcement and concrete reaches its average value, leading to rapid failure of the component. Let $\alpha$ be the reduction factor for the tensile strength of the interface; then:

$${\displaystyle \frac{{{\tau }^{\prime }}_u}{{\tau }_u}}={\displaystyle \frac{\left(1+\frac{2.5c}{d}+\frac{41.6d}{l}\right)\frac{{f^{\prime }}_t^{3/4}}{4}}{\left(1+\frac{2.5c}{d}+\frac{41.6d}{l}\right)\frac{f_t^{3/4}}{4}}}={\alpha }^{3/4}$$

(16)

It is evident that the ratio of the tensile load-carrying capacity $T^{\prime }$ of reinforcement anchored with the interface to that $T$ of reinforcement anchored without the interface is given by:

$${\displaystyle \frac{T^{\prime }}{T}}={\alpha }^{3/4}$$

(17)

For typical reinforced concrete slabs, in the case of unrestrained reinforcement pull-out failure, the tensile load-carrying capacity is ${\tau }_u=\left(3~6\right)f_t^{3/4}$. For composite slabs, however, the presence of the interface leads to a reduced load-carrying capacity of ${{\tau }^{\prime }}_u=$${\alpha }^{3/4}\left(3~6\right)f_t^{3/4}$$=\left(2.1~4.2\right)f_t^{3/4}$, with a reduction magnitude of $\left(0.9~1.8\right)f_t^{3/4}$.

Figure 12. Schematic section of laminated floor slabs at splices.

Figure 12. Schematic section of laminated floor slabs at splices.

From the perspective of the concrete splitting tensile strength at the reinforcement anchorage, the stress state at the composite slab joints was investigated, leading to the derivation of the ratio between the tensile load-carrying capacity $T^{\prime }$ of reinforcement with the interface and $T$, the tensile load-carrying capacity of reinforcement without the interface. Figure 13 illustrates the relationship between the bond length and the ultimate bond strength at the interface. It can be observed that the average ultimate bond strength decreases rapidly between 0 and 10 mm, and then levels off with only a slight reduction thereafter. Therefore, the bond length of the reinforcement does not need to be excessively long to achieve effective bond performance at the interface; a sufficient length is adequate to ensure the required bond strength.

5.2. Bending State at the Joint

The bending state of the flexural member is shown in Figure 14. According to the principles of material mechanics, when neglecting the effects of shear forces on deformation:

$${\displaystyle \frac{1}{\rho (x)}}={\displaystyle \frac{M(x)}{E{I}_z}}$$

(18)

Figure 14. Bending state of a flexural member.

Figure 14. Bending state of a flexural member.

According to mathematical principles:

$${\displaystyle \frac{1}{\rho }}=\pm {\displaystyle \frac{\frac{d^{2}y}{d{x}^2}}{\sqrt{{[1+{(\frac{dy}{dx})}^2]}^3}}}$$

(19)

Ignoring higher-order small terms, the equation simplifies to:

$${\displaystyle \frac{1}{\rho }}=\pm {\displaystyle \frac{d^{2}y}{d{x}^2}}$$

(20)

$$\pm {\displaystyle \frac{d^{2}y}{d{x}^2}}={\displaystyle \frac{M(x)}{E{I}_z}}$$

(21)

Based on the sign convention of the bending moment, it can be determined that the sign of the bending moment is consistent with the sign of the second derivative of the deflection curve. Therefore, the approximate differential equation for the deflection curve is given by:

$${\displaystyle \frac{d^{2}y}{d{x}^2}}={\displaystyle \frac{M(x)}{E{I}_z}}\Rightarrow E{I}_z{\displaystyle \frac{d^{2}y}{d{x}^2}}=M(x)$$

(22)

Integrating once yields the equation for the rotation angle:

$$E{I}_z{\displaystyle \frac{dy}{dx}}=E{I}_z\theta ={\displaystyle \int M(x)dx+C}$$

(23)

Integrating once more yields the deflection equation:

$$E{I}_{z}y={\displaystyle \iint M(x)dxdx+C}x+D$$

(24)

The constants $C$ and $D$ are determined based on the displacement boundary conditions and the smooth continuity requirements of the structure.

In a composite slab subjected to pure bending, when the composite interface is not cracked, both the precast and cast-in-place concrete work together to resist the bending moment, as shown in Figure 15b. Due to the inherent weakness of the composite interface, it will reach its tensile strength before other parts of the slab, leading to rapid failure of the structure once a certain bending level is reached. To analyze the cracking capacity at the joint more clearly, this study focuses on the critical state just before failure, where the interface bonding strength reaches its maximum tensile strength, placing the structure in a pre-failure critical condition. This scenario is equivalent to the case illustrated in Figure 16. The actual deformation in segment AB is represented by the red dashed line, which corresponds to applying a certain force along A′B, resulting in a displacement of y_A. For simplification, the deformation curve under load is not assumed to be identical, and shear effects are neglected.

Based on the preceding analysis, let ${\mathit{M}}_0$ denote the moment at the critical state, and ${\mathit{I}}_1$ represent the moment of inertia of the composite slab section A′B. It can be easily derived that:

$$y_A={\displaystyle \frac{M(x)}{2E{I}_z}}{l}^2$$

(25)

$${\theta }_A={\displaystyle \frac{M_0}{E{I}_1}}l$$

(26)

$$y_A={\displaystyle \frac{M_0}{2E{I}_1}}{l}^2$$

(27)

For computational convenience, the forces can be equated to Figure 17, with a uniform load plus reverse triangular load on A′B:

$$y_A={\displaystyle \frac{b{f}_t}{8E{I}_1}}{l}^4-{\displaystyle \frac{b{f}_t}{30E{I}_1}}{l}^4={\displaystyle \frac{11b{f}_t}{120E{I}_1}}{l}^4$$

(28)

Figure 17. Equivalent force state of laminated surfaces at splices.

Figure 17. Equivalent force state of laminated surfaces at splices.

From Equations (25) and (28):

$${\displaystyle \frac{M_0}{2E{I}_1}}{l}^2={\displaystyle \frac{11b{f}_t}{120E{I}_1}}{l}^4$$

(29)

For simplification, it can be deduced that under critical conditions, the bending moment applied to the composite slab with a joint is given by:

$$M_0={\displaystyle \frac{11b{f}_t}{60}}{l}^2$$

(30)

Due to the absence of coarse aggregates and the presence of additional defects (microcracks) at the interface between the old and new concrete during the casting process, the tensile strength of the interface is reduced compared to the bulk concrete. If the reduction factor for the tensile strength of the interface is defined as $k$, the actual critical bending moment is given by:

$$M_0={\displaystyle \frac{11kb{f}_t}{60}}{l}^2$$

(31)

This study analyzes the state of composite slab joints prior to failure and examines the stress conditions at the joints in three distinct stages. The formula for the actual critical bending moment has been derived. Figure 18 illustrates the relationship between the overlap length of composite slabs and the critical deflection. It is evident from the figure that the initial increase in critical deflection is minimal with increasing overlap length. However, after the symmetric overlap length reaches 150 mm, the rate of increase in critical deflection accelerates. As the overlap length continues to increase, the critical deflection grows more rapidly. Experimental results indicate that the overlap lengths for specimens DB3, DB1, and DB4 are 350 mm, 500 mm, and 900 mm, respectively. The critical deflection observed in the experiments exhibits a trend similar to the calculated values, with experimental values being higher. The reason for the experimental values being higher than the calculated values is that the experimental data were measured after crack initiation, with cracking occurring later than predicted by the calculated values. Additionally, the assumption that tensile stress is uniformly distributed across the splitting surface, without accounting for potential stress concentration phenomena, is another factor contributing to the delayed crack development in the experimental results compared to the calculated values.

6. Conclusions

Through static load tests on four concrete composite slabs, examining the effects of varying lap reinforcement lengths at the joint and cross-sectional height on the load-bearing capacity of the composite slabs, combined with numerical simulation and theoretical research, the following conclusions can be drawn:

(1) By setting different lap lengths for the reinforcement at the joint and comparing the test results, it is evident that reasonably increasing the lap length contributes to enhancing the load-bearing capacity of the composite slabs. However, when the lap length exceeds a certain threshold, its impact on deflection resistance in the later stages becomes negligible.

(2) Placing lap reinforcement in the grooves of the precast slab, i.e., increasing the effective height of the cross-section where the joint reinforcement is located, delays the crack propagation rate and effectively controls the deflection of the composite slabs, resulting in an increase in flexural capacity of up to 92.3%. This demonstrates the significant influence of changing the cross-sectional height on improving crack resistance and flexural performance.

(3) The finite element model’s results align well with experimental data, confirming the accuracy of the finite element model in predicting the bending capacity and crack propagation patterns of two-way composite slabs under static loading conditions. The deviation between the simulation and test results is within 10%.

(4) By analyzing the anchoring stress characteristics of reinforcement at the joint and the bending state at the composite surface of the slabs, the derived mechanical characteristics and the formula for calculating the critical bending moment of the composite slab joint can quickly and effectively compute the ultimate load-bearing capacity. The critical deflection values obtained from the calculations match well with the experimental values, providing a reference for engineering design. However, a limitation arises from the assumption that tensile stress on the concrete splitting surface is uniformly distributed and reaches the axial tensile strength of the concrete. This idealized stress distribution may not fully materialize under actual conditions, especially in the presence of localized stress concentrations.

Additionally, while this study focuses on the flexural and crack resistance performance of composite slabs, future research should further explore other geometric configurations of composite slab designs and their influence on load-bearing capacity, as well as investigate the role of different reinforcement materials (such as high-strength steel or fiber-reinforced materials) in enhancing the joint’s load-bearing performance. This will facilitate broader applications of the findings in engineering practice.