Experimental and Explicit FE Studies on Flexural Behavior of Superposed Slabs

Abstract

This study explores the use of recycled brick powder (P_RB), derived from waste bricks, and calcined recycled slurry powder (P_CRS), sourced from waste cement blocks, as partial replacements for cement and fly ash in concrete. These materials can be utilized to produce concrete with favorable engineering properties. Five concrete mixtures with varying P_RB/P_CRS proportions were prepared. Uniaxial monotonic compression tests were conducted to generate stress-strain curves for each mixture. Corresponding physical superposed slabs were fabricated, and finite element (FE) models were developed for each slab. Both physical testing and explicit FE simulations were performed to evaluate the flexural performance of the slabs. The results demonstrated that the flexural performance of the P_RB/P_CRS recycled micro-powder concrete slabs was comparable to that of conventional concrete slabs. Notably, the slab incorporating a 1:1 mixture of P_RB and P_CRS instead of fly ash exhibited the highest yield and ultimate bearing capacities, reaching 99.3% and 98.4% of those of the conventional concrete slab, respectively. The FE simulations accurately predicted the flexural performance, with maximum deviations of 8.9% for the yield load and 6.5% for the ultimate load. Additionally, the simulation-based energy time-history curve provides valuable insights into the progression of slab cracking. This study contributes to the advancement of research on the engineering and mechanical performance of concrete members incorporated with P_RB/P_CRS.

1. Introduction

Waste bricks, waste cement blocks, and waste glass are some common types of construction waste, which can be crushed and milled to make recycled micro-powder (RMP); it is a hot topic in the industry to study the mechanical properties of recycled concrete made by replacing part of the cement with such RMP [1]. It is evident that using recycled brick powder (P_RB) prepared from waste bricks or calcined recycled slurry powder (P_CRS) prepared from waste cement blocks to partially replace cement or fly ash in concrete production, not only reduces carbon emissions in the building materials industry but also enhances the industrial value of these two types of construction waste.

Tests reported by [2,3] revealed that both P_RB and P_CRS contain chemical components with alkaline-stimulating effects. In other words, both materials can partially replace cement as cementitious materials. The chemical composition of P_CRS is relatively similar to that of cement; however, appropriate carbonation can enhance its reactivity [4,5,6]. Compared with ordinary cement, P_RB contains constituents that exhibit pozzolanic activity [2,7,8,9,10]. However, due to their higher SiO₂ and Al₂O₃ contents and lower CaO content, P_RB-based mortars require longer curing periods to ensure adequate strength development [9,10]. When cement is partially replaced by P_RB or P_CRS at rates below 20%, a weak positive correlation exists between the substitution rate and mortar strength, and the durability properties are enhanced [7,8,9,10,11,12,13]. Furthermore, studies [4,5,8,11] have shown that mortars prepared by blending P_RB and P_CRS or by mixing P_RB/P_CRS with other mineral admixtures as partial cement substitutes effectively improve mechanical properties. Additional studies [4,5,7,8,11] revealed that P_RB/P_CRS concrete typically exhibits lower early-age strength compared with conventional concrete, but achieves desirable strength at later ages. Based on the aforementioned experimental findings, it is assumed that members cast using concrete supplemented with P_RB/P_CRS should exhibit satisfactory engineering capabilities.

Concrete superposed slabs are a prevalent type of flexural member in prefabricated concrete structures. Due to their construction advantages, such as reduced seasonal restrictions and high mechanization, prefabricated concrete structures have become a popular topic of research and engineering applications [14,15,16,17]. Concrete superposed slabs are often used in prefabricated concrete structures. Many studies have been conducted on the flexural properties of concrete slabs (including concrete superposed slabs) prepared using recycled aggregates to replace part of the aggregate or recycled cement to replace part of the cement [18,19,20,21]. As a non-seismic-resistant component, a slab is considered a relatively minor structural element compared to framed columns and beams. It is evident that flexural performance is the key engineering performance for a slab. However, there have been no reports on the fabrication and flexural performance analysis of concrete superposed slabs cast with concrete produced using cement partially added with P_RB, P_CRS, or a mixture of P_RB and P_CRS, hereinafter referred to as P_RB/P_CRS RMP concrete superposed slabs.

It is evident that conducting a parametric analysis is imperative in order to comprehensively understand the flexural behavior of P_RB/P_CRS RMP concrete superposed slabs. Four objective facts must be acknowledged before the parametric analysis. The first fact is that the finite element (FE) method is recognized as a universal and cost-effective method for the parametric analysis of structural static properties. The second fact is that to ensure the correctness and accuracy of FE static analysis results, the selection of constitutive models must be reasonable (Requirement I). There are two points of consensus regarding Requirement I. The first consensus is that the P_RB/P_CRS recycled micro-powder concrete discussed in this study, along with various non-ordinary concretes discussed in previous studies, can be classified as concrete-like materials based on their comparable mechanical properties [22,23,24,25,26,27,28]. The second consensus is that material behaviors such as strain hardening, strain softening, plastic deformation, and unilateral effects, which are common in concrete-like materials, can be well formulated using the damage plasticity constitutive model (CDP) [29]. Therefore, the CDP is often used in the FE analysis of concrete structures [25,26,27,28,30,31,32,33]. The third fact is that traditional implicit iterative methods often fail to achieve a convergent solution when solving an FE static model containing at least one strain-softening zone characteristic with cracking and/or crushing [34,35]. One method for approximately solving this problem is to use an explicit algorithm that does not perform convergence testing and exhibits strong robustness [35]. In addition, the explicit solver has two key characteristics that reduce its computational cost in comparison [35]. First, the explicit analysis does not require the stiffness matrix K, which means that the computational cost associated with inverting K is eliminated. Second, the explicit solver performs only vector operations, which are inherently less computationally expensive than the matrix operations required for implicit analysis. Previous studies on the finite element (FE) static analysis of concrete structures have explicitly adopted the aforementioned explicit solution method [23,25,26,27,30,31,33]. The fourth fact is that to ensure the validity of a structural FE model established for parameter analysis, the employed FE model must be verified by physical experimental results, and prior studies have addressed this type of validation work [23,24,27]. In light of the aforementioned four facts, as well as the additional fact that there are no documented reports on the FE static model of P_RB/P_CRS RMP concrete superposed slabs, it is necessary to conduct research on an explicit FE simulation model used to simulate the flexural behaviors of P_RB/P_CRS RMP concrete superposed slabs.

Based on the aforementioned discussions, it can be concluded that experiments and explicit FE simulations on the flexural performance of P_RB/P_CRS RMP concrete superposed slabs are necessary. This study carried out these two investigations.

The remainder of this paper is structured as follows. An overview of the physical experiments is provided in Section 2, encompassing details such as concrete mixture proportions, specimen design, measurement scheme, loading strategy, and so forth. Section 3 details the FE simulation experiment, covering the creation and discretization of the numerical slab, the setup of the contact and boundary conditions, and the formulation of the constitutive relationships. Section 4 presents the results, analysis, and discussion, specifically including the energy-history curves observed during the simulation, cracking and crushing of slabs, stress-increasing process of longitudinal stressed rebars, load-displacement curves, and the values representing the flexural performance. Finally, the conclusions are presented in Section 5.

2. Overview of Physical Test

2.1. Concrete Mixture Proportion and Concrete Stress Strain Curve

In this study, five types of concrete materials, labeled as A_N, A_RB, A_CRS, A_RBSI, and A_RBSII, were prepared according to the mixture proportions listed in

Table 1. Concrete mixture proportion/(kg/m³).

Material ID	Cement	Fly Ash	PRB	PCRS	Natural Sand	Coarse Aggregate	Water	Water Reducer
AN	336	59	0	0	807	948	170	4.54
ARB	269	59	67	0	807	948	170	4.54
ACRS	269	59	0	67	807	948	170	4.54
ARBSI	269	59	33.5	33.5	807	948	170	4.54
ARBSII	328	0	33.5	33.5	807	948	170	4.54

Note: The water reducer is a liquid polycarboxylic-based admixture with a water reducing rate of 20%.

, and a superimposed slab was cast using each type of concrete. The subscript strings in the material marks indicate the admixture characteristics, as follows: N denotes ordinary concrete without recycled powder; RB denotes concrete with partial replacement of cement by P_RB; CRS denotes concrete with partial replacement of cement by P_CRS; RBSI denotes concrete where P_RB and P_CRS are mixed in a 1:1 mass ratio to partially replace the cement; and RBSII denotes concrete where P_RB and P_CRS are mixed in a 1:1 mass ratio to completely replace fly ash.

For the five types of concrete listed in Table 1, a total of 20 material specimens with dimensions of 100 mm × 200 mm (diameter × height) were prepared, following the rule of fabricating four specimens per concrete type. Based on these specimens, the curves of total stress ${\sigma }_{\mathrm{c}}$ to total strain ${\epsilon }_{\mathrm{c}}$ for these materials were firstly derived through physical tests, and subsequently, the corresponding ${\sigma }_{\mathrm{c}}\text{-}{\epsilon }_{\mathrm{c}}$ fitting curves were obtained. The specific sub-divisions are delineated below.

(1)

${\sigma }_{\mathrm{c}}\text{-}{\epsilon }_{\mathrm{c}}$ testing curves. Four ${\sigma }_{\mathrm{c}}\text{-}{\epsilon }_{\mathrm{c}}$ test curves were obtained for each type of material by uniaxial monotonic testing (refer to Figure 1), and the curves obtained after averaging are the ones labelled with the word “Testing” in Figure 2. The peak stress ${\sigma }_{\mathrm{c}\mathrm{p}}$ and the corresponding strain ${\epsilon }_{\mathrm{c}\mathrm{p}}$ from each curve are tabulated in

Table 2. Testing and fitting results of ${\mathit{\sigma }}_{\mathbf{c}}\text{-}{\mathit{\epsilon }}_{\mathbf{c}}$ curves.

Material ID	${\mathit{\sigma }}_{\mathbf{c}\mathbf{p}}$/MPa	${\mathit{\epsilon }}_{\mathbf{c}\mathbf{p}}/\mathit{\mu }\mathit{\epsilon }$	${\mathit{\alpha }}_{\mathbf{c}}$	${\mathit{\beta }}_{\mathbf{c}}$	${\mathit{E}}_0$/MPa	${\mathit{\alpha }}_{\mathbf{c}\mathbf{p}}^{\mathit{i}}$
AN	23.1	2150	2.587	1.239	29,181	--
ARB	21.4	1970	2.303	0.259	24,877	7.4%
ACRS	16.0	2160	2.809	0.320	21,879	30.7%
ARBSI	20.6	2013	2.142	3.139	23,060	10.8%
ARBSII	21.5	1892	2.154	0.906	25,766	6.9%

Note: The formula for the reduction coefficient ${\alpha }_{\mathrm{c}\mathrm{p}}^i$ is expressed in Equation (1), where the superscript “$i$” is the wildcard denoting the mixing characteristics of the RMP concrete.

. Based on the ${\sigma }_{\mathrm{c}\mathrm{p}}^i$ (the superscript “i” is the wildcard denoting the mixing characteristic of the tested concrete), the compressive strength reduction index,

$${\alpha }_{\mathrm{c}\mathrm{p}}^i={\displaystyle \frac{{\sigma }_{\mathrm{c}\mathrm{p}}^{\mathrm{N}}-{\sigma }_{\mathrm{c}\mathrm{p}}^i}{{\sigma }_{\mathrm{c}\mathrm{p}}^{\mathrm{N}}}},$$

(1)

was obtained for each type of concrete with the exception of ordinary concrete, as detailed in Table 2. The compressive strength reduction indexes of the four types of RMP concrete, along with the characteristics of the geometric shapes of the five ${\sigma }_{\mathrm{c}}\text{-}{\epsilon }_{\mathrm{c}}$ testing curves clearly demonstrate that the uniaxial compressive performance of each RMP concrete studied closely aligns with that of ordinary concrete.

(2)

${\sigma }_{\mathrm{c}}\text{-}{\epsilon }_{\mathrm{c}}$ fitting curves. The ${\sigma }_{\mathrm{c}}\text{-}{\epsilon }_{\mathrm{c}}$ testing curves were fitted using this stress-strain equation

$$\left\{\begin{array}{c}{\sigma }_{\mathrm{c}}=\left[{\alpha }_{\mathrm{c}}{\epsilon }_{\mathrm{c}}/{\epsilon }_{\mathrm{c}\mathrm{p}}+\left(3-2{\alpha }_{\mathrm{c}}\right){({\epsilon }_{\mathrm{c}}/{\epsilon }_{\mathrm{c}\mathrm{p}})}^2+\left({\alpha }_{\mathrm{c}}-2\right){({\epsilon }_{\mathrm{c}}/{\epsilon }_{\mathrm{c}\mathrm{p}})}^3\right]{\sigma }_{\mathrm{c}\mathrm{p}} \left({\epsilon }_{\mathrm{c}}\le {\epsilon }_{\mathrm{c}\mathrm{p}}\right),\\ {\sigma }_{\mathrm{c}}={\displaystyle \frac{({\epsilon }_{\mathrm{c}}/{\epsilon }_{\mathrm{c}\mathrm{p}}){\sigma }_{\mathrm{c}\mathrm{p}}}{{\beta }_{\mathrm{c}}{\left(({\epsilon }_{\mathrm{c}}/{\epsilon }_{\mathrm{c}\mathrm{p}})-1\right)}^2+{\epsilon }_{\mathrm{c}}/{\epsilon }_{\mathrm{c}\mathrm{p}}}} \left({\epsilon }_{\mathrm{c}}>{\epsilon }_{\mathrm{c}\mathrm{p}}\right),\end{array}\right.$$

(2)

which is proposed by Guo [36]. In this equation, ${\alpha }_{\mathrm{c}} \mathrm{and} {\beta }_{\mathrm{c}}$ are the material coefficients to be fitted. Based on this fitting equation and the ${\sigma }_{\mathrm{c}}\text{-}{\epsilon }_{\mathrm{c}}$ testing curve for each type of concrete, the values of ${\alpha }_{\mathrm{c}}$ and ${\beta }_{\mathrm{c}}$ for each type were determined, as shown in Table 2, and the resulting fitted curves are labeled as “Fitting” in Figure 2. Each stress-strain relationship fitting function yields a coefficient of determination ($R^2$), with a minimum value of 0.9908. This result demonstrates a highly accurate correlation between the models utilized in this study and the experimental data, thereby validating the reliability and robustness of the fitting functions.

2.2. Design of the Slab Specimens

A total of five RMP concrete superposed slabs were designed and fabricated, as detailed below.

(1)

Geometric dimensions. All slabs had the same dimensions of 2000 mm × 600 mm × 130 mm (length × width × thickness), with the prefabricated bottom layer being 60 mm thick and the cast-in-place top layer being 70 mm thick, as shown in Figure 3 and Figure 4.

(2)

The RMP admixture configuration. Each slab was cast using a specific concrete mixed with different types/specific gravities of RMP, as detailed in Table 1. The key material mix proportions for each type of concrete are presented in Table 1.

(3)

The rebar configuration. Rebars were placed only during the fabrication of the prefabricated bottom layer, and their positions are detailed in Figure 3, and their level is the same as that of HRB400. The rebars can be categorized based on their functions into three types: longitudinal stressed rebars on the bottom of the precast layer (referred to as “l-rebar(s)” and indicated by blue lines in Figure 3), transverse unstressed rebars (indicated by green lines in Figure 3), and triangular rebar trusses each of which is composed of two bottom chord rebars, some web rebars, and a top chord rebar. The diameters and the mechanical properties of the abovementioned rebars are shown in

Table 3. Properties of the used rebars.

$\mathbf{Diameter} {\mathit{d}}_{\mathbf{s}}$/mm	$\mathbf{Yield} \mathbf{Strength} {\mathit{f}}_{\mathbf{s}\mathbf{y}}$/MPa	$\mathbf{Elasticity} \mathbf{Modulus} {\mathit{E}}_{\mathbf{s}}$/GPa
6	480	206
8	514	200
10	540	200

.

(4)

The load-response law. The load-response law of all slabs is the same, with each slab being supported by two simple supports located on its bottom face, as depicted in Figure 4. Consequently, it is evident that the load-response law can be formulated using the same method applicable to simply supported members. Two symmetrically arranged and equally concentrated loads were applied on each studied slab, with their positions and other relevant details illustrated in Figure 4.

The aforementioned designs not only ensured the failure of slabs due to bending but also facilitated an analysis of the impact of the types and specific gravities of RMP admixtures on the flexural performance of slabs to a certain extent, thus partially achieving the research purpose.

Figure 3. The reinforced configuration of the precast layer (in mm).

Figure 3. The reinforced configuration of the precast layer (in mm).

Figure 4. Loading configuration.

Figure 4. Loading configuration.

2.3. Testing Equipment, Measuring-Point Arrangement, and Loading Program

The main tasks before loading a specimen and after its fabricating include mounting the specimen into the testing equipment, setting up the measuring scheme, and determining the loading program. These specific details utilized in this study are as follows.

(1)

Testing equipment. Each studied slab was fixed on the supports within the reaction frame, as shown in Figure 5a. Two-point loading was applied with the help of a distribution steel beam, and the concentrated load was generated by a jack fixed to the top beam of the reaction frame.

(2)

Layout of the measuring points. During the test, data on the applied load and displacements in the thick direction were gathered. The load was measured using a force sensor positioned between the distribution beam and jack, as shown in Figure 5a. The collected displacements were measured using five dial gauges, with their positions shown in Figure 5b.

(3)

The loading program. The loading was performed using a load-displacement hybrid control loading scheme, which is divided into three sub-stages. ① During sub-stage I, the load was incrementally applied at about 10% of the cracking load $F_{\mathrm{c}\mathrm{r}}$ until a specimen showed a visible crack. According to Figure 4, the bending cracking load can be calculated using the following formula: $F_{\mathrm{c}\mathrm{r}}={2M}_{\mathrm{c}\mathrm{r}}/l_2$, where $M_{\mathrm{c}\mathrm{r}}$ is the cracking moment. In this study, the determination of the cracking moment was based on the widely accepted assumption that concrete undergoes cracking when the tensile stress at the bottom-edge of the tension zone reaches the axial tensile strength ${\sigma }_{\mathrm{t}\mathrm{p}}$, which can be determined using the plane section hypothesis regarding normal strain distribution. As proposed in the literature [37], the following formula,

$${\sigma }_{\mathrm{t}\mathrm{p}}=0.395\times {({\sigma }_{\mathrm{c}\mathrm{p}}/0.67)}^{0.55},$$

(3)

was adopted to estimate ${\sigma }_{\mathrm{t}\mathrm{p}}$ in this study, where ${\sigma }_{\mathrm{c}\mathrm{p}}$ is the axial compressive strength of the concrete. ② During sub-stage II, the commencement of which coincided with the appearance of a discernible crack, the load was increased incrementally by about 7% of the ultimate load $F_{\mathrm{u}}$ until the total load was approximately 0.7$F_{\mathrm{u}}$. In addition, the ultimate load was calculated using the formula $F_{\mathrm{u}}={2M}_{\mathrm{u}}/l_2$, where $M_{\mathrm{u}}$ is the ultimate moment. The method for calculating $M_{\mathrm{u}}$ can be found in a textbook on the fundamental principles of the concrete structure. The yield strength $f_{\mathrm{y}}$ and the sectional area $A_{\mathrm{s}}$ of l-rebars, both of which were used to calculate $F_{\mathrm{u}}$, are listed in Table 3, and the effective height of each slab $h_0=110 \mathrm{m}\mathrm{m}$. ③ During sub-stage III, whose starting point was the instant when $F\approx 0.7F_{\mathrm{u}}$, the loading style was switched to displacement control, and the load was increased incrementally, with each increment equaling about 30% of the mid-span displacement corresponding to $F\approx 0.7F_{\mathrm{u}}$, until the tested slab collapsed or the midspan deflection reached 1/50 of the calculated span length $l_0 (=1800 \mathrm{m}\mathrm{m}$). ④ The values of the two characteristic loads for all specimens can be calculated using the aforementioned methods, and as a result, yielding $F_{\mathrm{c}\mathrm{r}}\in [13.84, 19.99] \mathrm{k}\mathrm{N}$ and $F_{\mathrm{u}}\in [91.88, 93.24] \mathrm{k}\mathrm{N}$, respectively, and it is worth noting that the cracking load of the B_RBSII slab

$$F_{\mathrm{c}\mathrm{r},\mathrm{a}}^{\mathrm{R}\mathrm{B}\mathrm{S}\mathrm{I}\mathrm{I}}=18.60 \mathrm{k}\mathrm{N}.$$

(4)

The two values adopted to determine the consequent incremental loads for sub-stages I and II were estimated corresponding to the abovementioned two value ranges, as detailed in

Table 4. Controlled loads at various loading sub-stages.

Load Type	Sub-Stage I	Sub-Stage II	Sub-Stage III
Total load	15 kN (≈$F_{\mathrm{c}\mathrm{r}}$)	60 kN ($\approx$0.7$F_{\mathrm{u}}$)	10 mm
Incremental load	1.5 kN	6 kN	3 mm

, where the controlled displacements for sub-stage III are also provided.

3. Overview of Simulation Experiment

3.1. Creation of FE Model

In this study, the Abaqus code was utilized to develop the numerical model entities of the superposed slabs. The geometric dimensions and reinforcement configurations were determined according to the design details described in Section 2.2. A three-dimensional aerial view of the numerical slab is shown in Figure 6. What is worth mentioning is that, when creating a FE slab model, it was assumed that there was no slip between the bottom face of the cast-in-place layer and the top face of the precast layer; in other words, the two layers were considered as a continuum before embedding rebars.

3.2. Discrete, Contact, and Boundary Settings

In the process of discretizing numerical slabs and configuring contact and boundary conditions, the following five aspects were primarily involved.

(1)

Element type determination. Slabs and steel pads were discretized using an eight-node linear incongruous element, whose number is C3D8I in the Abaqus element library, while the rebars were discretized using a 3D two-node linear element, whose number is T3D2.

(2)

Element size determination. Discretization was carried out through the utilization of a pertinent Graphical User Interface (GUI) labeled as “Global Seeds”. The detailed settings for all types of entities (concrete-zones, rebars, and steel pads) were the same as follows: the parameter “approximate global size” was set to 25 mm, the parameter “curvature control” was set to 0.1, and the item named minimum size control used a relative size mode with a value of 0.1.

(3)

Determination of the bonding performance between rebars and concrete. It was assumed that there was no bond slip between the rebars and concrete. Therefore, the “Embedded constraint” option was adopted to embed rebars within the concrete entities.

(4)

Determination of the contacting mode between the steel pad and the concrete entities. It was assumed that each region near a contact interface between a steel pad and its corresponding slab entity could be considered a continuous medium; consequently, the “Tie” constraint mode was employed to merge the contact interfaces.

(5)

Boundary condition determination. To simulate the boundary conditions, only the rotational freedom around the transverse direction was assigned to the bottom boundary surface of a slab near the right support, and both the rotational freedom around the transverse direction and translational freedom along the longitudinal direction were assigned to the bottom boundary surface near the left support. Figure 5b shows the specific details.

3.3. Constitutive Relationships

Four constitutive relationships were formulated to ascertain the behaviors of the four types of entities concerning each slab simulation. The details are as follows:

(1)

The concrete constitutive relationship in the regions near the concentrated forces (Region I, shown in red in Figure 6). The elastic constitutive relationship employed for the steel pads mentioned in the “(4)” below was assigned to Region I to prevent calculation of failure due to excessive stresses surpassing the predefined value of the adopted concrete damage constitutive relationship for the main body of the simulation slab.

(2)

The concrete constitutive relationship in regions except Region I. This relationship was formulated using the concrete damage plasticity model (CDP) developed by J. Lubliner (1989) et al. and J. Lee (1998) et al., which has been collected into the Abaqus material library [38,39,40]. CDP is widely utilized in structural simulations. Recent applications include its use in simulating the shear performance of concrete T-beams [41] and in modeling the collapse behavior of industrial roof systems [42]. According to references [29,40], the established parameters for determining the material behavior are limited to the following: initial (undamaged) elastic modulus $E_0$ and Poisson’s ratio $\nu$; ratio $K_1$ of initial equibiaxial yield stress to initial uniaxial compressive yield stress, and the ratio $K_2$ of the second stress invariant on the tensile meridian to that on the compressive meridian; eccentricity parameter $\varphi$, dilation angle factor $\phi$, compressive stiffness recovery factor ${\omega }_{\mathrm{c}}$ and tensile stiffness recovery factor ${\omega }_{\mathrm{t}}$; viscosity parameter $\mu$; relation curve of uniaxial effective compressive cohesion stress ${\overline{\sigma }}_{\mathrm{c}}$ to equivalent compressive plastic strain ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}$(${\overline{\sigma }}_{\mathrm{c}}$-${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}$ curve); and the relation curve of uniaxial effective tensile cohesion stress ${\overline{\sigma }}_{\mathrm{t}}$ to equivalent tensile plastic strain ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}$ (${\overline{\sigma }}_{\mathrm{t}}$-${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}$ curve). In this study, the values of these constitutive parameters were established as follows:

• ① The value of $E_0$ was determined by calculating the right derivative of the ${\sigma }_{\mathrm{c}}\text{-}{\epsilon }_{\mathrm{c}}$ fitting curve at the point $\left({\sigma }_{\mathrm{c}},{\epsilon }_{\mathrm{c}}\right)=\left(0,0\right)$ and the resulting value of $E_0$ for each type of concrete is listed in Table 2. Additionally, it is noted that the values of $\nu$ for all types of concrete were uniformly established at 0.2.
• ② It was to take $K_1=1.16, K_2=0.67, \varphi =0.10, \phi =30°, {\omega }_{\mathrm{c}}=0$ and ${\omega }_{\mathrm{t}}=0$ according to the suggestions from the literature [29,40] for each type of concrete.
• ③ The value of $\mu$ was set to 0.005 for each type of concrete, as suggested in references [43,44]. Herein, it is worth noting that: firstly, in order to avoid potential convergence challenges arising from material softening, Abaqus offers the option of viscoplastic regularization of CDP constitutive equation when employing the implicit iterative algorithm to solve the FE equation set, and following this treatment, a viscosity coefficient is incorporated into the alternate stress-strain equation; secondly, the explicit algorithm was employed to approximate the solution of the FE equation set in this study, and unlike the implicit iterative algorithm, the explicit algorithm does not dictate convergence, leading us to believe that $\mu$ has no impact on simulation results. More comprehensive discussions on this topic are available in references [34,40].
• ④ When constructing ${\overline{\sigma }}_{\mathrm{c}}$-${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}$ curve: the equivalent compressive plastic strain

  $${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}={\gamma }_{\mathrm{c}}\left({\epsilon }_{\mathrm{c}}-{\sigma }_{\mathrm{c}}/E_0\right),$$

  (5)

  where ${\gamma }_{\mathrm{c}}$ is the material coefficient; the effective compressive cohesion stress ${\overline{\sigma }}_{\mathrm{c}}=E_0\left({\epsilon }_{\mathrm{c}}-{\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}\right)$. The functional relationship between ${\epsilon }_{\mathrm{c}}$ and ${\sigma }_{\mathrm{c}}$ is shown in Equation (2) and Table 2. When doing quasi-static simulation, it is a common practice to adopt a simulation loading duration significantly shorter than the physical loading duration in order to reduce computational cost [45]. The duration utilized in this study is found in Section 3.4. During such an abbreviated simulation loading duration, there is naturally a great release of strain energy due to material strain softening. This released strain energy not only amplifies the inertia effect but also leads to unrealistic damage within the range of the inertia effect, ultimately resulting in unreasonable simulation results (Situation I). In order to mitigate the impact of Situation I as much as possible, in this study,

  $${\gamma }_{\mathrm{c}}=0.9$$

  (6)

  was taken to reduce the amount of strain energy released.
• ⑤ Considering the fact that the influence caused by the tensile behavior of concrete on the simulation results, such as bearing capacity and cracking, is tiny, ${\overline{\sigma }}_{\mathrm{t}}$-${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}$ curve, used in this study was determined based on the common values characterizing the properties of concrete under uniaxial monotonic loading, which are as follows: equivalent tensile plastic strain ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}={\gamma }_{\mathrm{t}}\left({\epsilon }_{\mathrm{t}}-{\sigma }_{\mathrm{t}}/E_0\right)$, where ${\gamma }_{\mathrm{t}}$ is a coefficient, and it was to take ${\gamma }_{\mathrm{t}}=0.9$ also to mitigate the impact of Situation I; uniaxial effective tensile cohesion stress ${\overline{\sigma }}_{\mathrm{t}}=E_0\left({\epsilon }_{\mathrm{t}}-{\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}\right)$. The functional relationship between ${\epsilon }_{\mathrm{t}}$ and ${\sigma }_{\mathrm{t}}$ is in the following form suggested by the literature [36]

  $$\left\{\begin{array}{c}{\sigma }_{\mathrm{t}}=\left(1.2{\epsilon }_{\mathrm{t}}/{\epsilon }_{\mathrm{t}\mathrm{p}}-0.2{\left({\epsilon }_{\mathrm{t}}/{\epsilon }_{\mathrm{t}\mathrm{p}}\right)}^6\right){\sigma }_{\mathrm{t}\mathrm{p}} \left({\epsilon }_{\mathrm{t}}\le {\epsilon }_{\mathrm{t}\mathrm{p}}\right),\\ {\sigma }_{\mathrm{t}}={\displaystyle \frac{{\epsilon }_{\mathrm{t}}/{\epsilon }_{\mathrm{t}\mathrm{p}}}{0.312{\sigma }_{\mathrm{t}\mathrm{p}}^2{\left({\epsilon }_{\mathrm{t}}/{\epsilon }_{\mathrm{t}\mathrm{p}}-1\right)}^{1.7}+{\epsilon }_{\mathrm{t}}/{\epsilon }_{\mathrm{t}\mathrm{p}}}}{\sigma }_{\mathrm{t}\mathrm{p}} \left({\epsilon }_{\mathrm{t}}>{\epsilon }_{\mathrm{t}\mathrm{p}}\right),\end{array}\right.$$

  (7)

  where ${\sigma }_{\mathrm{t}\mathrm{p}}$ is the peak tensile stress and ${\epsilon }_{\mathrm{t}\mathrm{p}}$ is the corresponding strain. In this study, the value of ${\sigma }_{\mathrm{t}\mathrm{p}}$ was determined using Equation (3), and ${\epsilon }_{\mathrm{t}\mathrm{p}}$ was calculated as $1.2{\sigma }_{\mathrm{t}\mathrm{p}}/E_0$ according to reference [36].

The specific constitutive relationship is derived after CDP is configured using the above material parameters, and this relationship allows for the determination of some properties and macroscopic characteristics. Firstly, in terms of material tension, this constitutive relationship enables the determination of the relationship between ${\epsilon }_{\mathrm{t}}$ and ${\sigma }_{\mathrm{t}}$ (${\epsilon }_{\mathrm{t}}$-${\sigma }_{\mathrm{t}}$ curve) and the relationship between ${\epsilon }_{\mathrm{t}}$ and tensile damage $d_{\mathrm{t}}$ (${\epsilon }_{\mathrm{t}}$-$d_{\mathrm{t}}$ curve). Here, the dependent variable $d_{\mathrm{t}}$ is solely a function of the independent variable ${\epsilon }_{\mathrm{t}}$ (refer to [29,40] for more information). Figure 7 illustrates the ${\epsilon }_{\mathrm{t}}$-${\sigma }_{\mathrm{t}}$ curve and ${\epsilon }_{\mathrm{t}}$-$d_{\mathrm{t}}$ curve for A_RBSⅡ concrete. It is evident from Figure 7 that for A_RBSⅡ concrete, when ${\epsilon }_{\mathrm{t}}$ > 1.3, indicating entry into the strain-softening stage, the material undergoes irreversible and spontaneous strain energy dissipation (Characteristic I), refer to the literature [46] for more information. From a physical perspective, it can be inferred that the moment at which concrete materials exhibit Characteristic I marks their cracking. In other words, the cracking strain

$${\epsilon }_{\mathrm{c}\mathrm{r}}^{\mathrm{R}\mathrm{B}\mathrm{S}\mathrm{I}\mathrm{I}}=1.3\times {10}^{-4}$$

(8)

for A_RBSII concrete and the corresponding cracking damage

$$d_{\mathrm{c}\mathrm{r}}^{\mathrm{R}\mathrm{B}\mathrm{S}\mathrm{I}\mathrm{I}}=0.03$$

(9)

Secondly, in terms of material compression, the relationship curve between ${\epsilon }_{\mathrm{c}}$ and compressive damage $d_{\mathrm{c}}$ (${\epsilon }_{\mathrm{c}}$-$d_{\mathrm{c}}$ curve) can be determined. By utilizing ${\epsilon }_{\mathrm{c}}$-$d_{\mathrm{c}}$ curve of A_RBSII concrete, and considering the widely acknowledged fact that ordinary concrete will crush when ${\epsilon }_{\mathrm{c}}\approx 0.003$, it can be inferred (omitting the analysis process here) that the crushing damage

$$d_{\mathrm{c}\mathrm{r}\mathrm{u}}^{\mathrm{R}\mathrm{B}\mathrm{S}\mathrm{I}\mathrm{I}}=0.25$$

(10)

when A_RBSII concrete crushes.

(3)

The constitutive relation of rebar. Before the slab fails, rebars within the slab typically exhibit either elastic or yield behavior, rather than strain-hardening behavior. Therefore, an ideal elastic-plastic model was selected to simulate the mechanical behaviors of rebars. The stress-strain curve of this model is represented as

$$\left\{\begin{array}{c}{\sigma }_l=E_{\mathrm{s}}{\epsilon }_l ({\epsilon }_l\le {\epsilon }_{\mathrm{s}\mathrm{y}}),\\ {\sigma }_l=f_{\mathrm{s}\mathrm{y}} {(\epsilon }_l>{\epsilon }_{\mathrm{s}\mathrm{y}}),\end{array}\right.$$

where ${\sigma }_l$ denotes the rebar stress, ${\epsilon }_l$ is the corresponding strain, ${\epsilon }_{\mathrm{s}\mathrm{y}}$ is the initial yield strain, $E_{\mathrm{s}}$ is the elastic modulus, and $f_{\mathrm{s}\mathrm{y}}$ is the yield stress. In this study, values for $E_{\mathrm{s}}$ and $f_{\mathrm{s}\mathrm{y}}$ for all types of rebars (depicted in Figure 3) are provided in Table 3; additionally, the Poison’s rates $v_{\mathrm{s}}$ of them were uniformly established at 0.3.

(4)

The constitutive relation of the steel pad. According to the local influence principle, the impact of the selection of the constitutive model for the steel pads on the static behavior of a loaded slab is tiny, thus allowing for an assumption that the steel pads remain in a linear elastic state under static loading. The constitutive relation equation for this model is

$${\mathit{\sigma }=\mathit{D}}_{\mathrm{e}}:\mathit{\epsilon },·$$

where $\mathit{\sigma }$ is the Cauchy stress tensor, $\mathit{\epsilon }$ denotes the Cauchy strain tensor, and ${\mathit{D}}_{\mathbf{e}}$ is the elastic stiffness tensor, which is dependent on both the elastic modulus $E_{\mathrm{b}}$ and Poisson’s ratio $v_{\mathrm{b}}$. In this study, $E_{\mathrm{b}}$ was set to 200 GPa and $v_{\mathrm{b}}$ to 0.3.

3.4. Loading Strategy and Solving Algorithm Configuration

It was necessary to determine the loading strategy and the solving algorithm before the simulation. The displacement control loading mode was adopted, with a constant loading rate. The total loading time ($t_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$) for each simulation was 12 s, during which the displacement at the loading point increased linearly from the initial value of 0 mm to 10 mm over the time span of 0.0 s–3.5 s, and from 11 mm to 30 mm over the time span of 3.5 s–12.0 s. The static responses of the slab models were solved using the explicit solver, which was configured through the explicit solver GUI as follows: the “consider geometric nonlinearity” option was checked; the “automatic/global/improved” options were selected for determining the time increment, with scaling factor set to 1.0; linear volume viscosity parameter set to 0.06, and the secondary volume viscosity parameter set to 1.2.

4. Results and Discussion

Several types of results were obtained from the physical and simulation tests. Figure 8 shows the time history curves of the kinetic energy, internal potential energy, and energy ratio (ratio of kinetic energy to internal potential energy) obtained by simulating the B_RBSII slab. The crack evolution of the B_RBSII slab during the physical test is shown in Figure 9. The simulation is capable of generating contour plots that depict the loading responses. Figure 10 shows the $d_{\mathrm{t}}$ and ${\sigma }_l$ contour plots for the B_RBSII slab, and Figure 11 shows the $d_{\mathrm{c}}$ contour plots. Here, it is worth noting that the threshold for deciding whether to average the response values at a FE node had been set to 0.75 before plotting. The relation curves of load $F$ to displacement $\Delta$ ($F$-$\Delta$) obtained through the physical tests and simulations are depicted in Figure 12, with key points summarized in

Table 5. The testing values and simulation values of several typical forces and their comparisons.

Slab ID	${\mathit{F}}_{\mathbf{c}\mathbf{r},\mathbf{e}}$/kN	${\mathit{F}}_{\mathbf{c}\mathbf{r},\mathbf{s}}$/kN	${\mathit{\delta }}_{\mathbf{c}\mathbf{r}}$/%	${\mathit{F}}_{\mathbf{y},\mathbf{e}}$/kN	${\mathit{F}}_{\mathbf{y},\mathbf{s}}$/kN	${\mathit{\delta }}_{\mathbf{y}}$/%	${\mathit{F}}_{\mathbf{u},\mathbf{e}}$/kN	${\mathit{F}}_{\mathbf{u},\mathbf{s}}$/kN	${\mathit{\delta }}_{\mathbf{u}}$/%
BN	15.5	20.4	31.6	87.8	93.5	6.5	101.1	94.5	6.5
BRB	13.5	18.4	36.3	85.5	91.7	7.3	92.7	95.4	2.9
BCRS	10.7	18.3	71.0	85.1	91.7	7.8	90.5	92.3	2.0
BRBSI	13.6	19.9	46.3	86.3	93.6	8.5	96.9	94.8	2.2
BRBSII	14.5	19.1	31.7	87.2	95.0	8.9	99.5	95.3	4.2

Note: The calculation formula for the deviation ${\delta }_{☐}$ is shown in Equation (11), where the subscript ☐ refers to cr, y, or u, corresponding to cracking, yielding, or ultimate load, respectively.

, and especially the relative variations in yield load and ultimate load for each slab obtained from physical tests are shown in Figure 13. These results will be further described, analyzed, and discussed in detail below.

4.1. Time History Curve of Energy

A type of dynamic analysis algorithm is employed by the explicit solver built into the Abaqus code to solve quasi-static problems [35]. Naturally, the time history curves of various kinds of mechanical energies can be produced as byproducts when using the explicit solver. The simulation results demonstrated that the geometric characteristics of the energy time history curves given by different numerical slabs are essentially identical. Therefore, this paper only presents the curves given by the B_RBSII slab. The time history curves of kinetic energy $E_{\mathrm{k}}$, the internal potential energy $E_{\mathrm{i}\mathrm{p}}$ and the energy ratio

$${\gamma }_{\mathrm{E}}\triangleq {\displaystyle \frac{E_{\mathrm{k}}}{E_{\mathrm{i}\mathrm{p}}}},$$

abbreviated as $E_{\mathrm{k}}\text{-}t$ curve, $E_{\mathrm{i}\mathrm{p}}\text{-}t$ curve and ${\gamma }_{\mathrm{E}}\text{-}t$ curve, respectively, are shown in Figure 8. According to this figure, the following two comments can be made.

(1)

The loading program adopted for the B_RBSII slab and the cracking failure characteristics of the B_RBSII slab can be reflected through the geometric characteristics of the $E_{\mathrm{i}\mathrm{p}}\text{-}t$ and $E_{\mathrm{k}}\text{-}t$ curves. ① According to the $E_{\mathrm{i}\mathrm{p}}\text{-}t$ curve, it is evident that $E_{\mathrm{i}\mathrm{p}}$ increases monotonically (Phenomenon A₁) as a whole, with a higher rate in the time domain $\left[\begin{array}{cc}3.88& 7.24\end{array}\right] \mathrm{s}$ compared to the time domains $\left[\begin{array}{cc}0.00& 3.88\end{array}\right) \mathrm{s}$ and $\left(\begin{array}{cc}7.24& 12.00\end{array}\right] \mathrm{s}$ (Phenomenon A₂). Similarly, analysis of the $E_{\mathrm{k}}\text{-}t$ curve indicates a significant increase in noise intensity (Phenomenon B₁) during $\left[\begin{array}{cc}0.00& 3.88\end{array}\right] \mathrm{s}$, followed by a notable decrease in both amplitude and intensity of the noise (Phenomenon B₂) within $\left[\begin{array}{cc}3.88& 7.24\end{array}\right] \mathrm{s}$ (Excluding the short time interval after the yielding of all l-rebars at $t=5.48 \mathrm{s}$, as detailed in Section 4.2.2), and ultimately culminating in a marked surge in both amplitude and intensity of the noise during $\left(\begin{array}{cc}7.24& 12.00\end{array}\right] \mathrm{s}$ (Phenomenon B₃). It is clear that Phenomenon A₁ confirms the loading program described in Section 3.4. ② B_RBSII slab obviously underwent three mechanical processes. The first process denoted as Procedure C₁ can be described as follows: the tensile zone on the bottom face of the studied slab (B-face), particularly in the pure bending section (see Figure 9b), would continuously experience cracking (see Section 4.2 below) due to the pronounced tensile brittleness of concrete, and as soon as a tensile crack emerged, the internal potential energy $E_{\mathrm{i}\mathrm{p}}$ originally stored near this emerged crack dissipated irreversibly, accompanied by a natural conversion of $E_{\mathrm{i}\mathrm{p}}$ to $E_{\mathrm{k}}$. The second process, denoted as Process C₂, can be described as follows: the emerging frequency of fresh tensile cracks between two adjacent cracks had decreased gradually (see Section 4.2.2 below) due to the deformation localization effect at the cracks. The other process denoted as Process C₃ can be described as follows: when crushing occurred on the top face of the studied slab (T-face), there was irreversible dissipation of $E_{\mathrm{i}\mathrm{p}}$, and it is evident that the strain energy released from crushing exceeded that released from tension cracking by a significant margin. ③ The evolution characteristics of damages formulated by Procedures C₁, C₂ and C₃ can explain Phenomena A₂, B₁, B₂ and B₃. Specifically, as follows: firstly, Phenomenon A₂ can be attributed to the fact that a greater amount of internal potential energy is dissipated through the frequent germination of tension fractures during the early loading period or compression fractures during the late period compared to the intermediate period; secondly, the emergence of a great numerous tension cracks in the initial phase causes a substantial and concentrated release of kinetic energy, naturally leading to the occurrence of Phenomenon B₁; thirdly, Phenomenon B₂ occurring during the intermediate phase attributes to both the reduced initiation of fresh cracks, leading to limited and intermittent release of kinetic energy, and the accelerated extending-rate of existing cracks in the thick direction within a short time interval following the yielding of all l-rebars, resulting in large and closely-spaced release of kinetic energy; lastly, the initiation of some fresh crushing areas caused by compression in the later stage, accompanied by the significant and concentrated release of kinetic energy, gives rise to Phenomenon B₃.

(2)

The value-range characteristic of ${\gamma }_{\mathrm{E}}$ indicates that the B_RBSII slab remained in a quasi-static state for the majority of the loading time. Analysis of the ${\gamma }_{\mathrm{E}}\text{-}t$ curve reveals that

$${\gamma }_{\mathrm{E}}<0.01$$

during the time interval $\left[\begin{array}{cc}0.0& \delta \end{array}\right] \mathrm{s}$, where $\delta$ is a slightly positive number greater than zero. And according to the quasi-static standard specified in reference [35], where ${\gamma }_{\mathrm{E}}$ ≤ 0.10, it can be concluded that the B_RBSII slab was in a quasi-static state when $t\in \left(\begin{array}{cc}\delta & 12\end{array}\right] \mathrm{s}$. Consequently, it can be inferred that other results, presented in the subsequent sections, are statistically valid despite being produced by the explicit solver.

4.2. Cracking and Collapse of the Slab (Phenomena I) and Stress-Increasing Process of Longitudinal Rebars (Process II)

4.2.1. Phenomena I Given by Physical Test

The physical test results indicated that the crack distribution pattern and the failure mode of the five tested slabs are basically identical. Therefore, this paper exclusively presents these two kinds of results given by the B_RBSII slab. The development of cracks on the side face of the studied slab (S-face) is depicted in Figure 9a, while the distribution of cracks on the B-face and the crushing pattern of concrete on the T-face at failure are shown in Figure 9b and Figure 9c, respectively. Detailed descriptions and corresponding discussions are as follows.

(1)

The crack development illustrated by Figure 9a indicates the following three reasonable experimental facts. ① The initial crack originated on the B-face when $F_{\mathrm{e}}=14.5 \mathrm{k}\mathrm{N}$, which closely approximates the estimated value obtained from the corresponding analysis method introduced in Section 2.3 and shown by expression (4). ② Fresh cracks progressively developed from the center of the slab towards both support ends, with pre-existing cracks extending in a thick direction as the applied load increased. ③ In the purely flexural section, the spacing between adjacent cracks exhibited statistically similarity, whereas in the flexural-shear section, there was an increase in crack spacing accompanied by a decrease in crack length.

(2)

The crack distribution on the B-face when the slab is crushed is illustrated by Figure 9b. It is evident that the distribution characteristic of the cracks is extremely similar to that of the cracks on the S-face as a whole. It is worth noting that on the B-face, a tiny number of longitudinal cracks had emerged, which had been driven out by the radial components of the forces caused by the extrusion between concrete and rebars.

(3)

The concrete crushing on T-face is depicted by Figure 9c.

(4)

In addition, there are two more points worth noting. Firstly, the absence of slippage between the precast layer and the cast-in-place layer throughout the entire loading process provides empirical support for treating the slab as a unified entity during simulation. Secondly, the yielding of all l-rebars could be deduced from the appearance of the second inflection point in the load-displacement curve (see Figure 12), corresponding to this special yielding load $F_{\mathrm{y},\mathrm{e}}=88.9 \mathrm{k}\mathrm{N}$.

As previously mentioned, it is evident that the crack distribution pattern and the failure mode of these tested superposed slabs align with those of typical flexural members with appropriate reinforcement, which are consistent with established common knowledge.

4.2.2. Phenomena I and Process II Given by FE Simulation

Before analyzing and discussing the typical simulation results, it is useful to briefly describe the objective mapping relationship between $d_{\mathrm{t}}$ and $w_{\mathrm{c}\mathrm{r}}$. As stated in point (2) in Section 3.2, a material point is in a critical state of crack initiation when its $d_{\mathrm{t}}=d_{\mathrm{c}\mathrm{r}}$ being similar to the one shown in Equation (9), and it is in the cracked state when $d_{\mathrm{t}}$ > $d_{\mathrm{c}\mathrm{r}}$, and obviously, the greater the difference given by the formula ($d_{\mathrm{t}}-d_{\mathrm{c}\mathrm{r}}$), the greater the $w_{\mathrm{c}\mathrm{r}}$ that has appeared. Based on the relationship of $d_{\mathrm{t}}$-${\sigma }_{\mathrm{t}}$ implicitly given in Figure 7, it is known that the relationship between $w_{\mathrm{c}\mathrm{r}}$ and ($d_{\mathrm{t}}-d_{\mathrm{c}\mathrm{r}}$) has been proposed and confirmed by a study [46]. And this relationship has been applied directly or indirectly in FE simulation of many concrete structures, as demonstrated in a study [47]. Therefore, the tensile damage value can be used to determine whether a point in a concrete entity is cracked and the width of a crack.

The simulation results indicate that the $d_{\mathrm{t}}$ contour plots of the five slabs and the ${\sigma }_l$ contour plots of l-rebars of the five slabs are similar, respectively (see Figure 10, for example), and the $d_{\mathrm{c}}$ contour plots of the top faces of the five plates are also similar (see Figure 11, for example). Therefore, this paper takes the B_RBSII slab as an example to elaborate on the evolution processes of the above-motioned three responses, where the $d_{\mathrm{t}}$ evolution law can faithfully represent the crack evolution law. The $d_{\mathrm{t}}$ contour plots as well as the $d_{\mathrm{c}}$ contour plots largely confirm the test phenomena described in Section 4.2.1. The evolution law of the stress of the l-rebars, as given by simulation, is consistent with objective common sense. Detailed descriptions and corresponding discussions are as follows.

(1) When the displacement at the min-span of the slab, ${\Delta }_{\mathrm{s}}$, equals to $0.7 \mathrm{m}\mathrm{m}$, meanwhile the applied force $F_{\mathrm{s}}=19.1 \mathrm{k}\mathrm{N}$ (see Figure 12) and the simulation time $t=0.24 \mathrm{s}$, there were several simulation phenomena with causal relationships as described below: two cracks simultaneously appeared on the bottom and side faces for the first time, and the emergence of them not only cause sudden increases in stress in l-rebars (see Figure 10(c2)) which were across the two abovementioned cracks and a distinct decrease of stiffness (see Figure 12), but also a sudden increase in kinetic energy $E_{\mathrm{k}}$ also for the first time (see Figure 8).

(2) When ${\Delta }_{\mathrm{s}}$ increased monotonically in $\left(0.7,16.0\right] \mathrm{m}\mathrm{m}$ in which $F_{\mathrm{s}}$ accordingly increased monotonically from the cracking load $F_{\mathrm{s},\mathrm{c}\mathrm{r}}$ equaling $19.1 \mathrm{k}\mathrm{N}$ to the yielding load $F_{\mathrm{s},\mathrm{y}}$ equaling $95.0 \mathrm{k}\mathrm{N}$ and $t\in \left(0.24,5.48\right] \mathrm{s}$, four noteworthy simulation phenomena were observed. The detailed descriptions and the accompanied discussions are as follows.

Firstly, the evolution pattern of the cracks in the thick direction closely corresponds to the observed patterns in the physical experiments, specifically referring to “Figure 10(c1,d1,e1)” and “Figure 9a” for comparison.

Secondly, the evolution law of the number and density of transverse cracks on the B-face of the numerical slab is consistent with empirical observations and resembles the evolution law of the thickness-direction cracks on the S-face of the physical slab. Most transverse cracks had already appeared before ${\Delta }_{\mathrm{s}}=11.0 \mathrm{m}\mathrm{m}$ when $F_{\mathrm{s}}=79.0 \mathrm{k}\mathrm{N}$ and $t=3.88 \mathrm{s}$, according to comparative study of Figure 10(d2,e2). When ${\Delta }_{\mathrm{s}}$ increased within (11.0, $16.0$] mm, fresh cracks rarely generated (comparative study of Figure 10(d2,e2)), but existing cracks continued to extend in the thick direction and widen longitudinally (comparative study of Figure 10(d1,e1)). These two simulated phenomena confirm two empirical observations followed below. The first one is that concrete cracking, which can be considered as strain softening, causes concrete deformation to exhibit a localization effect. That is to say, the strain values of material points in close proximity to either of two adjacent cracks exhibit a monotonically increasing trend, whereas the strain values of these points located between two adjacent cracks but at a distance from any crack remain relatively stable, making it difficult for fresh cracks to emerge between them (comparative study of Figure 10(d2,e2)). The second empirical observation is that the emergence of fresh cracks and extension of existing ones represent different orders of magnitude of strain energy release. More precisely speaking, the frequent emergence of new cracks with larger strain energy release should result in more noticeable fluctuations on the $E_{\mathrm{k}}\text{-}t$ curve, whereas the thickness-directional extension of existing cracks with smaller strain energy release (except for the short time interval after the yielding of all l-rebars at $t=5.48 \mathrm{s}$), should result in subtler fluctuations on the $E_{\mathrm{k}}\text{-}t$ curve (Referring to Figure 8 for evidence on this logical reasoning).

Thirdly, the evolutionary patterns of longitudinal cracks on B-face and S-face, wherein there are numerous cracks on B-face (Figure 10(d2)) and some cracks on S-face (Figure 10(d1)), manifest clear distinctions from those observed in the physical tests, wherein a few longitudinal cracks are evident on B-face (Figure 9b) but none are observed on S-face (Figure 9a). The authors contend that the disparity between the simulation and experimental observations is to be expected, and the underlying mechanism accounting for this variance is as follows: the elevated loading rate employed in the simulation (elaborated in Section 3.4) results in an excessive release of strain energy at the instant of concrete’s cracking (Mechanical process I), and this kind of uncontrolled energy release induces a heightened impact force on the concrete near the crack, which induces the development of multi-directional cracks within the affected zone (Mechanical phenomenon I). The causal relationship between Mechanical phenomenon I and Mechanical process I was substantiated by checking the simulation results.

Fourthly, the evolutionary pattern of the stress of l-rebars corresponds to the previously described configuration of rebars and the evolutionary pattern of cracks along the widthwise direction on the B-face. When ${\Delta }_{\mathrm{s}}=0.7 \mathrm{m}\mathrm{m}$ (corresponding to $t=0.24 \mathrm{s}$), the initial cracks were observed on the B-face, as illustrated in Figure 10c. Concurrently, a significant increase in stress occurred within some l-rebars, which acted as a barrier to the crack propagation. Additionally, as shown in Figure 3, Furthermore, these l-rebars with the larger diameter (10 mm) exhibited reduced stress compared to their smaller diameter counterparts in the truss bottom chords. These two findings concordantly reflect the objective reality. Subsequently, as ${\Delta }_{\mathrm{s}}$ escalated within $\left(0.7,16.0\right] \mathrm{m}\mathrm{m}$, the number of rebar-sections resisting the expansion of cracks was increasing, and within a given section, ${\sigma }_l$ exhibited a monotonic ascendance until it attained the yield strength specified in Table 3 (this simulation result also fits the objective reality obviously), as depicted in Figure 10c–e. Notably, when ${\Delta }_{\mathrm{s}}=12.1 \mathrm{m}\mathrm{m}$, corresponding to $F_{\mathrm{s}}=84.5 \mathrm{k}\mathrm{N}$ and $t=4.24 \mathrm{s}$, it was recorded for the first time the born of a yielding section in a truss bottom chord. Subsequently, at ${\Delta }_{\mathrm{s}}=16.0 \mathrm{m}\mathrm{m}$ ($F_{\mathrm{s}}=95.0 \mathrm{k}\mathrm{N}$, $t=5.48 \mathrm{s}$), it was also observed for the first time the existence of yielded sections in all l-rebars. Refer to Figure 10(e2) for further details.

(3) The compressed zone (CP zone) on the top of the purely-bended section of the B_RBSII slab underwent three distinct periods when ${\Delta }_{\mathrm{s}}$ increased in the interval $\left(16.0,36.0\right] \mathrm{m}\mathrm{m}$, during which $t\in \left(5.48,12.0\right] \mathrm{s}$ and $F_{\mathrm{s}}$ monotonically increased from the yielding load $F_{\mathrm{s},\mathrm{y}}=95.0 \mathrm{k}\mathrm{N}$ to the ultimate load $F_{\mathrm{s},\mathrm{u}}=95.3 \mathrm{k}\mathrm{N}$ and then decreased to $F_{\mathrm{s},36}=44.0 \mathrm{k}\mathrm{N}$ corresponding to the ultimate displacement equaling 36 mm. During the first period, CP zone evolved from a state of no noticeable compressive damage to a state of having noticeable but scattered compressive damage sub-zones, and an instance of the latter condition, corresponding to ${\Delta }_{\mathrm{s}}=22.0 \mathrm{m}\mathrm{m}$, $t=7.24 \mathrm{s}$ and $F_{\mathrm{s}}=94.5 \mathrm{k}\mathrm{N}$, is depicted by Figure 11b. The second period began following the end of the first period, during which the CP zone evolved into a state characterized by localized interconnection of some aforementioned scattered subzones. This state is exemplified by Figure 11b, with ${\Delta }_{\mathrm{s}}=22.0 \mathrm{m}\mathrm{m}$, $t=7.24 \mathrm{s}$ and $F_{\mathrm{s}}=94.5 \mathrm{k}\mathrm{N}$ as an example, and was experimentally confirmed by Figure 9c. Finally, the CP zone developed into the third period, characterized by the complete interconnectedness of all scattered compressive damage zones, as depicted in Figure 11d, occurring at ${\Delta }_{\mathrm{s}}=33.7 \mathrm{m}\mathrm{m}$, $t=10.56 \mathsf{s}$ and $F_{\mathrm{s}}=48.0 \mathrm{k}\mathrm{N}$.

4.3. Load Displacement Curve and Characteristic Load Values

The load-displacement curves ($F$-$\Delta$ curves) of each slab obtained from testing and simulation are shown in Figure 12. These curves can reveal the characteristics of the studied slabs, including stiffness, strength, and the degree of variation in the results obtained from testing and simulation. The three points of detailed description are listed as follows.

Firstly, the ultimate bearing capacity $F_{\mathrm{u},\mathrm{e}}$ obtained from the test closely approximates the ultimate bearing capacity $F_{\mathrm{u},\mathrm{s}}$ obtained from the simulation for each slab. In this study, $F_{\mathrm{u},\mathrm{e}}$ or $F_{\mathrm{u},\mathrm{s}}$ for a certain slab is the maximum load had applied prior to the mid-span displacement reaching 1/50 of the calculated span length $l_0$, and $l_0$ = 1800 mm according to Figure 5b. $F_{\mathrm{u},\mathrm{e}}$ for a slab is denoted by the mark “●” at the corresponding position of its $F$-$\Delta$ curve, while $F_{\mathrm{u},\mathrm{s}}$ is marked with the mark “◆”. All specific values of $F_{\mathrm{u},\mathrm{e}}$ and $F_{\mathrm{u},\mathrm{s}}$ are presented in Table 5. From the perspective of deviation

$${\delta }_{☐}={\displaystyle \frac{\left|F_{☐,\mathrm{e}}-F_{☐,\mathrm{s}}\right|}{F_{☐,\mathrm{e}}}},$$

(11)

(where the ☐ denotes cr, y, or u, corresponding to cracking, yielding, or ultimate load, and here ☐ is u), $F_{\mathrm{y},\mathrm{e}}\approx F_{\mathrm{y},\mathrm{s}}$ for a certain slab (refer to Table 5).

Secondly, the stiffness change characteristics exhibited by the $F$-$\Delta$ curve of each slab generally adhere to the typical stiffness change characteristics of a reinforced concrete bending member. Based on these characteristics, it is possible to infer the cracking load $F_{\mathrm{c}\mathrm{r}}$ and yielding load $F_{\mathrm{y}}$.

According to Figure 12, not only $F_{\mathrm{c}\mathrm{r}}$ of a physical or numerical slab can be determined by observing the initial but slight decrease in stiffness attributed to the tensile cracking (as indicated in “Section 4.2.1→(1) and Section 4.2.2→(1)”), but also $F_{\mathrm{y}}$ can be formulated by observing the subsequent but evident decrease in stiffness caused by the tensile yielding of all l-rebars (as shown in “Section 4.2.1→(4) and Section 4.2.2→(2)”). The deduced $F_{☐,\mathrm{e}}$, $F_{☐,\mathrm{s}}$ and the deviation degree ${\delta }_{☐}$ (Note: The subscript ☐ in the sentence denotes cr, y, or u, corresponding to cracking, yielding, or ultimate load.) between them are presented in Table 5.

According to ${\delta }_{\mathrm{y}}$, it can be inferred that $F_{\mathrm{y},\mathrm{e}}\approx F_{\mathrm{y},\mathrm{s}}$ with a similar feature of the aforementioned $F_{\mathrm{u}}$, however, it can be obtained that $F_{\mathrm{c}\mathrm{r},\mathrm{e}}\not\approx F_{\mathrm{c}\mathrm{r},\mathrm{s}}$ judged by ${\delta }_{\mathrm{c}\mathrm{r}}$. The authors believe that the reason for these differences is that in this study, ${\sigma }_{\mathrm{c}\mathrm{p}}$, which can determine $F_{\mathrm{u},\mathrm{s}}$ and $F_{\mathrm{y},\mathrm{s}}$, was obtained through experimental measurement, while ${\sigma }_{\mathrm{t}\mathrm{p}}$, which governs $F_{\mathrm{c}\mathrm{r},\mathrm{s}}$, was estimated through empirical Equation (3). Therefore, it can be inferred that the phenomenon that ${\delta }_{\mathrm{c}\mathrm{r}}>{\delta }_{\mathrm{y}}\approx {\delta }_{\mathrm{u}}$ holds certain validity. Although in general engineering practice, emphasis is placed on $F_{\mathrm{y}}$ and $F_{\mathrm{u}}$ for a bending member rather than $F_{\mathrm{c}\mathrm{r}}$, it is advisable for future research endeavors to experimentally determine ${\sigma }_{\mathrm{t}\mathrm{p}}$ to enhance calculation accuracy or uncover new phenomena warranting further investigation.

Thirdly, the $F$-$\Delta$ curves of a certain slab obtained from testing and simulation demonstrate favorable agreement prior to reaching the yield point $({\Delta }_{\mathrm{y}}, F_{\mathrm{y}})$, while the agreement after this point may not be ideal. The authors posited two underlying factors contributing to this phenomenon. One factor is that, as the applied force approaches $F_{\mathrm{y}}$, some of the concrete in the compression zone enters the strain-softening stage. This phenomenon, the existence of strain-softening, raises the mesh-dependence of FE analysis [48]. This is to say, different results would be gained when employing various FE mesh division strategies for a component with strain-softening subzones under an applied force near $F_{\mathrm{y}}$ [48]. Another contributing factor is the adoption of a hypothetical yet reasonable value for the material coefficient ${\gamma }_{\mathrm{c}}$ (Equation (5)) in determining ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}$ of kinds of concrete mixed in this study, despite utilizing a fitted ${\sigma }_{\mathrm{c}}\text{-}{\epsilon }_{\mathrm{c}}$ curve (see Figure 2) corresponding to the testing result as a fundamental variable for constructing the corresponding CDP constitutive relationship for each numerical slab. After the yield of a member, the plastic strain of concrete in the compression zone exhibited a more significant increase, thus leading to a deviation of the $F$-$\Delta$ curve gained by simulation from the one gained by physical test to some extent due to an inherent assumption in the material coefficient.

4.4. Yield Load and Ultimate Load

Yield load $F_{\mathrm{y}}$ and ultimate load $F_{\mathrm{u}}$ are two more commonly used indexes when evaluating the engineering performance of a slab compared with cracking load.

The relative variation of $F_{\mathrm{y},\mathrm{e}}$ and $F_{\mathrm{u},\mathrm{e}}$ for the five slabs is illustrated in Figure 13, from which two key points can be inferred.

Firstly, $F_{\mathrm{y},\mathrm{e}}$ and $F_{\mathrm{u},\mathrm{e}}$ for a certain slab prepared with RMP concrete is found to be smaller than $F_{\mathrm{y},\mathrm{e}}^{\mathrm{N}}$ and $F_{\mathrm{u},\mathrm{e}}^{\mathrm{N}}$ for the slab made with ordinary concrete, respectively (here the superscript denotes the type of concrete used for preparing the corresponding slab). However, on one hand, compared to $F_{\mathrm{y},\mathrm{e}}^{\mathrm{N}}$ and $F_{\mathrm{u},\mathrm{e}}^{\mathrm{N}}$, the holding percent in $F_{\mathrm{y},\mathrm{e}}$ and $F_{\mathrm{u},\mathrm{e}}$ for each slab is significantly high, with the minimum holding percent in $F_{\mathrm{y},\mathrm{e}}$ being 96.9% and in $F_{\mathrm{u},\mathrm{e}}$ being 89.5%. On the other hand, the decrease in $F_{\mathrm{y},\mathrm{e}}$ and $F_{\mathrm{u},\mathrm{e}}$ for a specific slab is essentially identical.

Secondly, it is possible to infer, to some extent, the beneficial significant engineering implications of the recycled micro-powder mixture type based on the variation law of the reduction range of $F_{\mathrm{y},\mathrm{e}}$ and $F_{\mathrm{u},\mathrm{e}}$. It can be observed from Figure 13 that $F_{☐,\mathrm{e}}^{\mathrm{N}}>F_{☐,\mathrm{e}}^{\mathrm{R}\mathrm{B}\mathrm{S}\mathrm{I}\mathrm{I}}>F_{☐,\mathrm{e}}^{\mathrm{R}\mathrm{B}\mathrm{S}\mathrm{I}}>F_{☐,\mathrm{e}}^{\mathrm{R}\mathrm{B}}>F_{☐,\mathrm{e}}^{\mathrm{C}\mathrm{R}\mathrm{S}}$, where the subscript ☐ refers to y or u. Considering this observation in conjunction with Section 2.1, it is evident that the kind of slab achieving superior engineering performance is the one cast using concrete prepared by completely replacing fly ash with powder produced by mixing P_RB and P_CRS at a ratio of 1:1, while the type of slab achieving inferior one is the slab cast using concrete prepared by partially substituting cement with P_CRS.

Thirdly, considering the preservation amplitude of yield load and ultimate load, the bending performance of P_RB/P_CRS recycled micro-powder concrete composite board studied in this paper is higher than that of recycled concrete composite board reported in literature [18].

First, based on the three points discussed above, the differences between P_RB/P_CRS slabs and conventional concrete slabs in key engineering performance metrics—specifically yield load ($F_{\mathrm{y}}$) and ultimate load ($F_{\mathrm{u}}$)—are marginal (Section 4.3, Table 5). Second, as demonstrated in Figure 12 (Section 4.3), the stiffness of P_RB/P_CRS slabs exhibits minimal deviation from that of conventional concrete slabs. These findings collectively indicate that the P_RB/P_CRS materials developed in this study meet the requisite engineering performance standards. This further suggests that novel P_RB/P_CRS materials, formulated through adjustments to admixture proportions, may also achieve comparable compliance. Consequently, future investigations should prioritize: The sensitivity of P_RB/P_CRS material properties to admixture dosage and composition; The flexural performance of P_RB/P_CRS structural elements (e.g., beams and large-scale slabs); The fire resistance and fatigue behavior of P_RB/P_CRS slabs and beams.

5. Conclusions

In order to investigate the flexural properties of PRB/PCRS superposed slab, the flexural properties of four P_RB/P_CRS slabs and one common concrete slab (BN slab) were tested and simulated. Consequently, the conclusions from this study are as follows.

(1)

All P_RB/P_CRS physical slabs have flexural properties that meet the engineering requirements, and they are generally stronger than the traditional recycled concrete superposed slab, of which the B_RBSⅡ plate has the best performance. The failure mode of each P_RB/P_CRS slab corresponds to that exhibited by a classic balanced-reinforced flexural member, and the superposed interface remains intact without any tearing. The evolution laws of side cracks, the distribution characteristics of bottom cracks, and the load $F$-displacement $\Delta$ curves provided by the experiments and simulations conducted on P_RB/P_CRS slabs and B_N slabs are similar, respectively. Compared to the yield load $F_{\mathrm{y},\mathrm{e}}^{\mathrm{N}}$ or the utmost load $F_{\mathrm{u},\mathrm{e}}^{\mathrm{N}}$ of the B_N slab, the yield load $F_{\mathrm{y},\mathrm{e}}^{i\ne \mathrm{N}}$ or the utmost load $F_{\mathrm{u},\mathrm{e}}^{i\ne \mathrm{N}}$ of P_RB/P_CRS slabs show a slight decrease within $\left[0.7\%, 3.1\%\right]$ or $\left[1.6\%, 10.5\%\right]$. Notably, the slab with 1:1 mixture of P_RB and P_CRS demonstrated the highest yield and ultimate bearing capacities, showing minimal decreases 0.7% and 1.6%, compared to the conventional concrete slab.

(2)

The superior performance of each P_RB/P_CRS physical slab can be attributed to the excellent compressive performance of its corresponding P_RB/P_CRS concrete. The compressive stress ${\sigma }_{\mathrm{c}}$-compressive strain ${\epsilon }_{\mathrm{c}}$ test curves of A_N concrete and four kinds of P_RB/P_CRS concrete are similar. Notably, compared to the peak stress ${\sigma }_{\mathrm{c}\mathrm{p}}^{\mathrm{N}}$ of A_N concrete, the peak stresses ${\sigma }_{\mathrm{c}\mathrm{p}}^i$ ($i\ne \mathrm{N}$, CRS) of the three types of P_RB/P_CRS concrete exhibits modest reductions ranging from 6.9% to 10.8%, with the exception being a 30.7% decrease in ${\sigma }_{\mathrm{c}\mathrm{p}}^{\mathrm{C}\mathrm{R}\mathrm{S}}$ of A_CRS concrete.

(3)

The proposed FE model for P_RB/P_CRS slabs provides both reasonable simulation solutions and ease of creation. The simulation results, consistent with the experiment ones or common sense, include the crack evolution history, $F$-$\Delta$ curve, and stress evolution history of stressed longitudinal rebars. Specifically, compared to $F_{\mathrm{y},\mathrm{e}}^{i\ne \mathrm{N}}$ or $F_{\mathrm{u},\mathrm{e}}^{i\ne \mathrm{N}}$, $F_{\mathrm{y},\mathrm{s}}^{i\ne \mathrm{N}}$ or $F_{\mathrm{u},\mathrm{s}}^{i\ne \mathrm{N}}$ given by simulations demonstrate a tiny degree of deviation within $\left[6.5\%, 8.9\%\right]$ or $\left[2.0\%, 6.5\%\right]$. The geometric features of the kinetic energy history curve obtained from simulation of a numerical slab provide validation for the crack propagation process in the corresponding physical slab.

Despite the positive research results, further studies are necessary. These include, but are not limited to: (1) Investigating the fire resistance and thermo-mechanical behavior of P_RB/P_CRS recycled micro-powder concrete superposed slabs, particularly under elevated temperatures, to evaluate their structural integrity and failure mechanisms in fire scenarios [49]; (2) parametric analysis on the flexural behavior of a P_RB/P_CRS slab, for example, using “P_RB and P_CRS mixing ratio (Ratio I) and P_RB/P_CRS replacing ratio (Ratio II)” as independent variables; (3) formulating the ${\sigma }_{\mathrm{c}}$-${\epsilon }_{\mathrm{c}}$ curve function for P_RB/P_CRS concrete with Ratios I and II; (4) examining the variation law of constitutive parameters in the CDP model, excluding the ${\sigma }_{\mathrm{c}}$-${\epsilon }_{\mathrm{c}}$ curve, such as studying the tensile stress ${\sigma }_{\mathrm{t}}$-tensile strain ${\epsilon }_{\mathrm{t}}$ curve (${\sigma }_{\mathrm{t}}$-${\epsilon }_{\mathrm{t}}$ curve) to improve the simulation accuracy of the cracking load $F_{\mathrm{c}\mathrm{r}}$ of a slab, with which the larger deviation between $F_{\mathrm{c}\mathrm{r},\mathrm{e}}^{i\ne \mathrm{N}}$ and $F_{\mathrm{c}\mathrm{r},\mathrm{s}}^{i\ne \mathrm{N}}$ caused by Method I could be reduced.