Theoretical Research on the Shear Mechanical Properties and Size Effect of Recycled Concrete Beams Without Stirrups

Abstract

As a new type of building material with great potential, recycled concrete is playing a vital role in the context of the current construction industry’s pursuit of sustainable development. At present, the analysis method of recycled concrete structures is mainly based on the test results of small-scale specimens, but the reports relevant to the size effect of large components are not enough. Therefore, in this paper, the three-dimensional mesoscale numerical simulation is employed to conduct the static shear failure analysis of recycled concrete beams without web reinforcement. Based on existing experiments and verification of the rationality and accuracy of such numerical simulation, the influence of cross-sectional height, shear-span ratio, and the replacement rate of recycled aggregate on the shear mechanical properties and consequential size effect of recycled concrete beams are investigated. The research results reveal the dimension effect of nominal shear strength (NSS) and indicate that the shear strength of recycled concrete beams without stirrups shows a notable size-dependent effect, and the shear-bearing ability of recycled concrete beams reduces as the shear-span ratio and replacement rate of recycled aggregate rises. For every 20% increase in replacement rate, the shear-bearing capacity decreases by approximately 5%. The NSS shows a significant size effect, and it diminishes as beam height elevates. In addition, building on the material hierarchy of the Bažant size-effect law, a theoretical formula for the dimension effect on the shear strength of recycled concrete beams is proposed, considering the impact of shear-span ratio and replacement rate. The shear strength obtained from the supplied formula is subsequently compared with the standards of various countries, the results from existing calculation methods, and experiments. The accuracy and rationality of the supplied formula are verified. The research conclusion of this paper can provide a reference for engineering design.

1. Introduction

As a new type of construction material, recycled concrete can not only effectively cut down the consumption of natural resources but also alleviate the pollution of waste concrete to the environment, contributing crucially to the sustainable advancement of the construction industry. At present, the analysis methods for recycled concrete beams mainly lie on the experimental outcome of small-scale specimens. However, the research on the side effects of large-scale specimens is not sufficient, and the three-dimensional mesoscale numerical simulation is rarely applied in the study of recycled concrete beams. In addition, the shear behavior of recycled concrete beams is extremely complex, including many factors affecting the shear failure mode, and the mechanism is still not unified. Therefore, the shear failure of recycled concrete beams and their size effect have become critical concerns in the engineering field. For ordinary aggregate concrete beams, researchers have designed and performed numerous experiments to explore the mechanism of shear failure and the dimension effect of beams without stirrups. Iguro et al. [1] and Shioya et al. [2] observed that the shear-bearing ability of beams abated significantly as the cross-sectional sizes increased through the experiments of reinforced concrete (RC) beams under uniformly distributed loads. Tan et al. [3] conducted in-depth research on large-size RC beams, declaring that the cracking strength of diagonal cracks in RC beams is independent of the beam size, whereas the size effect of the shear-bearing ability of RC beams is significant. The experiment results obtained by Yu et al. [4], Che et al. [5], and Zararis et al. [6] demonstrated the presence of a significant size effect on the shear strength of RC beams without stirrups. Sherwood et al. [7] conducted thin plate shear experiments and found that the maximal aggregate size had a significant impact on the shear-bearing capacity. Sneed et al. [8] researched the shear performance of ordinary RC beams without stirrups via experiments, announcing that the shear strength is affected by both size effect and the variation in shear transference and modes at failure.

Some researchers have started to pay attention to the shear performance of recycled concrete beams over the years. For example, Lee et al. [9], Ignjatovic et al. [10], and Knaack et al. [11] studied the shear performance of recycled concrete beams by comparing natural aggregate concrete beams and recycled aggregate concrete beams with variable fine and coarse aggregate replacement rates. They concluded that the shear strength of beams decreases with the elevation of the recycled aggregate replacement rate. González-Fonteboa et al. [12] and Choi [13] conducted shear experiments on recycled concrete beams with variable aggregate replacement rates, shear-span ratios, and reinforcement ratios. The results showed that the ultimate bearing ability of recycled concrete beams diminishes with the improvement of recycled coarse aggregate replacement rate and shear-span ratio and raises with the elevation of reinforcement ratio. Zhang et al. [14] and Ni [15] conducted shear experiments on recycled concrete beams with variable recycled coarse aggregate replacement rates and shear-span ratios to investigate the stress and deformation performance of the inclined section of recycled concrete beams without stirrups. Chen et al. [16] studied the mechanical properties of recycled concrete beams and analyzed the impact of the recycled coarse aggregate replacement rate and the shear-span ratio on the ultimate bearing capacity. The measurement outcome showed that the failure process and mode of recycled concrete beams are close to those of ordinary concrete beams, and the ultimate bearing ability of recycled concrete beams progressively abates with the elevation of the recycled coarse aggregate replacement rate. The improvement of the shear-span ratio significantly influences the decrement of the bearing ability.

Because the experiment has the disadvantages of high cost, long time consumption, difficulty in implementing experimental conditions, and difficulty in covering all critical parameters, the numerical simulation as a crucial complementary research approach can expand the scope and extent of research on the performance of concrete beams. The shear performance and size effect of beams with and without stirrups were studied using numerical simulation methods by Jin et al. [17,18]. The results showed that the dimension effect is mainly manifested in the NSS of beams. When the shear-span ratio is small, the longitudinal reinforcement ratio has a greater influence on the shear strength. Caballero et al. [19] and Lopez et al. [20] performed numerical analyses on the tensile fracture behavior of concrete using the interface element approach. Jin et al. [21,22,23] adopted a micro-scale simulation method to investigate the shear failure performance and size effect of lightweight aggregate concrete beams without stirrups. From the microscopic level, concrete can be considered a triphasic composite, consisting of aggregates, cement matrix, and interfacial transition zone. Yu et al. [24] investigated the shear strength of RC beams with stirrups considering different cross-sectional sizes through numerical simulation. The results showed that the stirrups can reduce the dimension effect of the shear strength but cannot eliminate it, and the effectiveness of reduction in small-sized beams is more significant than that of large-sized beams. Vecchio et al. [25] proposed a modified compression field theory, which suggested that as the size increased, the crack width increased, resulting in a decrease in the aggregate bite force and the shear-bearing ability of the beam. Overall, the current investigation on recycled concrete beams without stirrups is not systematic, and the effects of shear-span ratio and recycled aggregate replacement rate on the dimension effect of recycled concrete beams are not yet clear.

This article focuses on the shear failure behavior and dimension effect of the shear strength of recycled concrete beams without stirrups, aiming to reveal the differences in shear performance between recycled and ordinary concrete beams. Given this, this article establishes a three-dimensional microscopic numerical model to analyze the shear failure of recycled concrete beams without stirrups. Building on the strong correlation with existing experimental outcomes, the influence mechanism of recycled aggregate substitution rate and shear-span ratio on the shear failure and dimension effect of shear strength is examined. Utilizing the Bažant size-effect law, a theoretical formula for the size effect of shear strength in recycled concrete beams without stirrups considering the impact of shear-span ratio and recycled aggregate replacement rate is proposed, which can provide a reference for engineering design.

2. Microscopic Numerical Analysis Model

2.1. Geometric Model

The primary reason for the dimension effect in concrete components is the intricate interaction between the concrete and reinforcement, as well as the heterogeneity and mechanical nonlinearity of concrete. In this article, at the microscopic level, the recycled coarse aggregate is considered a five-phased composite material, consisting of natural aggregate, an old interfacial transition zone (OTZ), old hardened mortar, interfacial transition zone (ITZ), and new hardened mortar. Accordingly, the microstructure of recycled concrete is more complex than that of ordinary concrete. Building on this, a microscopic numerical analysis model is presented based on the secondary development system of ABAQUS 2020 and Python 3.9 64-bit, where the aggregates are set as spatially randomly distributed spheres. The recycled coarse aggregates are set as high-strength elastomers, which are assumed to have no significant deformation under static loading. For the transition region between the mortar and aggregate, the mechanical properties are determined based on the stress–strain relationship suggested by Xiao [26]. The three-dimensional solid element C3D8R and three-dimensional truss element T3D2 are, respectively, used to simulate the concrete and reinforcement bar. The average grid size is set as 2 cm. To avoid stress concentration, steel plates are installed at the loading points and support, and the tie connection is used between the surface of the beam and the steel plate. A reference point is set at the center of each steel plate to apply displacement loads and pin supports, and the displacement-controlled loading method is adopted. The displacements are constrained in the Y and Z directions at the bottom of the left support, while the displacements are constrained in the X, Y, and Z directions at the bottom of the right support, as seen in Figure 1.

2.2. Constitutive Relationship of Microscopic Components

2.2.1. Constitutive Relationship of Concrete

The constitutive relation of ordinary concrete under compression refers to the recommended formula in the “Code for Design of Concrete Structures” (GB 50010-2010) [27], while the stress–strain curve of recycled concrete under compression adopts the stress–strain relationship recommended by Xiao [28]:

$$y=\left\{\begin{array}{l}{a}_{1}x+(3-2a_1)x^2+(a_1-2)x^3,0\le x<1\\ {\displaystyle \frac{x}{b_1{(x-1)}^2+x}},x\ge 1\end{array}\right.$$

(1)

$$a_1=2.2(0.748r^2-1.231r+0.975)$$

(2)

$$b_1=0.8(7.6483r+1.142){\rho }_c={\displaystyle \frac{f_{c,r}}{E_c{\epsilon }_{c,r}}}$$

(3)

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_c}}, y={\displaystyle \frac{\sigma }{f_c^{*}}}$$

(4)

where r, $f_c^{*}$, and ${\epsilon }_c$ represent the recycled aggregate replacement rate, the compressive strength of the recycled aggregate, and the peak strain of recycled concrete corresponding to uniaxial compressive strength, respectively. a₁ and b₁ are the parameters of the recycled coarse aggregate replacement rate, ρ_c represents the apparent density of recycled concrete, E_c represents the elastic modulus of recycled concrete, and f_c,r represents the prism compressive strength of recycled concrete.

For the rising segment of the stress–strain relation of recycled concrete under uniaxial tension, the constitutive relationship of recycled concrete proposed by Xiao [28] is adopted:

$$y=dx-(d-1)x^6,0\le x<1$$

(5)

$$y={\displaystyle \frac{x}{a_t{(x-1)}^{1.7}+x^{,}}},x\ge 1$$

(6)

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_t}}, y={\displaystyle \frac{\sigma }{f_t^{*}}}$$

(7)

where $f_t^{*}$ and ${\epsilon }_t$ represent the tensile strength of the recycled aggregate and the peak strain of recycled concrete associated with uniaxial tensile compressive strength. d is determined according to the following

Table 1. The value of parameter d.

Recycled Coarse Aggregate Replacement Percentage	0	30%	50%	70%	100%
d	1.19	1.21	1.23	1.24	1.26

.

For the descending segment of the stress–strain relationship of recycled concrete, the tensile constitutive relationship of ordinary concrete recommended in “Code for Design of Concrete Structures” (GB 50010-2010) [27] is adopted. In modeling, the research results of Liu [29] and Yue [30] are referenced. The material properties of recycled coarse aggregate and old and new mortars in the finite element analysis model are obtained by averaging the experiment results, and the material properties of the transition zone between OTZ and ITZ are obtained by reducing the mortar matrix. A comparative analysis is accomplished to evaluate the impact of the properties of recycled coarse aggregate, old and new mortars, OTZ, and ITZ on the physical and mechanical features of materials. For example, when the elastic modulus of new mortar increases (decreases) by 20%, the elastic modulus of recycled concrete increases (decreases) by about 10%. When the strength of new mortar increases (decreases) by 20%, the strength of recycled concrete increases (decreases) by 10%; when the elastic modulus of old mortar raises (decreases) by 20%, the elastic modulus of recycled concrete raises (decreases) by 5%; when the unit strength of old mortar increases (decreases) by 20%, the compressive strength of recycled concrete increases (decreases) by about 8%, and the tensile strength raises (decreases) by 10%. Therefore, the mechanical property parameters of recycled coarse aggregate, old and new mortars, OTZ, and ITZ in the finite element models in such investigation refer to the research results of references, and according to the comparison between the analysis and experiment results, the selection of the above parameters is reasonable and effective. The corresponding five-phased material parameters are collected in

Table 2. Relevant parameters of the constitutive model.

Composition	Elastic Modulus E/GPa	Compressive Strength σc/MPa	Compressional Strain εc	Ultimate Strength σ/MPa	Ultimate Strain ε	Tensile Strength σt/MPa	Tensile Strain εt	Poisson’s Ratio ν
Aggregates	80.00	80.00	-	-	-	10.00	-	0.16
Old mortar	19.27	18.17	3.77 × 10−3	6.06	0.010	1.66	1.0 × 10−4	0.22
New mortar	23.68	26.78	2.80 × 10−3	8.93	0.008	1.91	1.0 × 10−4	0.22
OTZ	19.27	10.20	3.77 × 10−3	3.40	0.010	1.02	1.0 × 10−4	0.20
ITZ	23.68	12.20	2.80 × 10−3	4.06	0.008	1.22	1.0 × 10−4	0.20

. The stress–strain curve of the transition zone between the old and new concrete interface is shown in Figure 2.

2.2.2. Constitutive Relationship of Reinforcement Bar

The elastic-hardening double-line model, also known as the ideal elastoplastic model, is used as the constitutive relationship model for reinforcement bars. The yield strength is determined through experiment and the elastic modulus is set based on conventional values. The Poisson’s ratio is adopted as 0.3.

$${\sigma }_s=\left\{\begin{array}{l}{E}_s{\epsilon }_s,0\le {\epsilon }_s\le {\epsilon }_y\\ f_y,{\epsilon }_y\le {\epsilon }_s\le 0.01\end{array}\right.$$

(8)

where E_s, ε_y, and f_y refer to the elastic modulus, yield strain, and yield strength of reinforcement bars, respectively.

2.3. Model Verification

In this section, a specimen with a cross-section of 120 mm × 180 mm based on the shear failure experiments accomplished by Zhao et al. [31] is employed to evaluate the rationality and applicability of the supplied microscopic numerical analysis model. The longitudinal bar ratio, the ratio of shear span to the height of the cross-section, and the recycled aggregate replacement rate of the specimen are 1.4%, 1.7, and 100%, respectively. An HRB400 steel bar with a diameter of 14 mm is chosen as the longitudinal reinforcement, the mixing ratio is listed in

Table 3. The mixing ratio set of recycled concrete.

Replacement Rate r/%	Water–Cement Ratio	Cement/(kg/m3)	Water/(kg/m3)	Sand/(kg/m3)	Recycled Coarse Aggregate/(kg/m3)
100	1:0.34	623.3	211.8	491.4	1073.5

, and other relevant parameters are collected in

Table 4. Material parameters of microscopic components of concrete and reinforcement bar.

Composition	Aggregates	Old Mortar	New Mortar	OTZ	ITZ	Longitudinal Reinforcement
Compressive strength σc/MPa	80	18.17	26.78	10.20	12.20	-
Tensile strength σt/MPa	10	1.66	1.91	1.02	1.22	528
Elastic modulus E/GPa	80	19.27	23.68	19.27	23.68	189
Poisson’s ratio ν	0.16	0.22	0.22	0.2	0.2	0.3
Yield stress fy/MPa	-	-	-	-	-	459

. Based on the parameters shown in Table 4, this article conducted three-dimensional microscopic numerical analyses of the shear failure of recycled concrete beams with variable grid dimensions. Figure 3 presents a comparison of the failure modes between experimental and corresponding simulation results. It is evident that the failure mode derived from the microscopic numerical analysis model closely aligns with the experimental result presented in reference [31], and both modes are shear failures. To clarify the impact of grid size on simulation outcomes, microscopic numerical models with grid sizes of 1 cm, 2 cm, 3 cm, and 4 cm are established, and the comparison of the load–displacement relationship curves is depicted in Figure 4. It can be observed that the load–displacement curves from the numerical analyses with the four different grid sizes are consistent with experimental curves. Compared with the experiment, the errors range from 11.07% to 15.87%, and the maximum error occurs at a displacement of 1.8 mm. When the grid size is 2 cm, the maximum error compared to the experimental value is 11.07%. Compared with other grid sizes, the error is the smallest when the grid size is 2 cm. In addition, a smaller grid size can generally accelerate the convergence speed, because the smaller grid size can accurately capture the details of physical phenomena, and make the calculation results approach to the stable value faster. The smaller grid size can satisfy the stricter convergence criteria and provide higher calculation accuracy. Taking into account both calculation efficiency and accuracy, a grid size is taken as 2 cm in the subsequent analyses.

3. Analysis of Shear-Span Ratio on the Shear Damage and Size Effect

To quantitatively study the impact of the shear-span ratio on the shear failure and shear-bearing capacity of recycled concrete beams without web, variable shear-span ratios and sections are considered in the shear failure experiment, as detailed in

Table 5. Parameters of recycled concrete beams without web.

Specimens	Section Size b × h (mm)	Virtual Height h0/mm	Shear Span Ratio λ	Beam Length l/mm	Longitudinal Reinforcement Ratio ρ/%	Recycled Aggregate Replacement r/%
S-group	100 × 200	175	1.2~3.0	2000	1.2	20
M-group	150 × 300	275	1.2~3.0	3000	1.2	20
L-group	200 × 400	370	1.2~3.0	4000	1.2	20
U-group	250 × 500	470	1.2~3.0	5000	1.2	20
V-group	300 × 600	565	1.2~3.0	6000	1.2	20

. It is significant to denote that in this section, the numerical analyses are conducted based on a fixed recycled aggregate replacement rate of 20%.

3.1. Shear Damage Mode

Figure 5 shows the shear damage modes of recycled concrete beams with a replacement rate of 20% under different shear-span ratio parameters and cross-sectional sizes. Evidently, the failure types of the beams with shear-span ratios of 1.2 and 2 are shear-compression failures. In this failure mode, bending cracks first appear in the mid-span and near the loading point, and diagonal cracks begin to emerge in the shear span zone when the load increases to a certain level. As the load boosts to half of the failure load, the development of the bending cracks in the mid-span nearly stops. As the load increases, the diagonal cracks in the shear span zone extend along the height of the beam in a short distance and then extend diagonally. One of the diagonal cracks develops into a main diagonal crack that runs through the entire section. When the load approaches the ultimate load, the main diagonal crack quickly progresses, and eventually destroys the concrete in the shear-compression zone, resulting in a brittle failure and loss of bearing capacity. The generation and extension process of diagonal cracks in recycled concrete beams without stirrups is similar to that of ordinary concrete beams without stirrups. The beam damage usually occurs in the region subjected to the combined effects of shear and moment, with the failure surface located in the shear span zone beam near the supports.

The failure type of the beam with shear-span ratios of 3 is a diagonal-tension failure. In this failure mode, bending cracks initially appear at the mid-span and the bottom of the loading points, and as the load increases, the height of the cracks increases. Bending cracks also appear in the shear span zone, and the heights of the cracks are lower than the longitudinal reinforcement. As the load increases, these bending cracks turn into diagonal cracks and extend towards the loading points. Finally, one diagonal crack near the support suddenly increases and expands, resulting in a diagonal-tension failure. This is because, in the concrete beam, the bending moment and shear force produced by the load act together. As the load increases, the shear and normal stresses of each point in the beam change constantly, giving rise to the principal tensile and compressive stresses. As far as the principal tensile stress surpasses the tensile strength of concrete, cracking will occur. Since the direction of the principal tensile stress is at a certain angle with the longitudinal axis of the beam, the crack presents an oblique direction, forming inclined flexural-shear cracks. Stress concentration occurs near the support of the beam and below the loading point. The stress distribution in these parts is uneven, and the local stress value is high. When the load increases, the concrete in the stress concentration parts is easier to reach its strength limit, leading to cracks appearing and expanding rapidly.

The entire failure process of the beam is quick and sudden, and the increase in the failure load is merely slightly higher than the load at which the diagonal cracks appear. The failure of recycled concrete beams is more sudden than that of ordinary concrete beams. This is probably because the porosity of recycled coarse aggregates is higher and there are many microcracks, which are prone to be pulled at the critical diagonal cracks.

Compared with small-sized beams, large-sized beams have fewer bending cracks, and the starting points of diagonal cracks are farther from the loading points when the beams fail. This is because the stiffness of large-sized beams is greater, and the concrete beside the loading point is less prone to cracking under shearing.

3.2. Load–Displacement Curves

According to the different shear-span ratios, the shear failure modes of the inclined section of a beam can be classified into oblique-compression, shear-compression, and diagonal-tension failure. In addition, it also affects the shear-bearing ability of the beam. Such an article mainly focuses on the impact of the shear-span ratio on the size effect of beam shear strength. For this purpose, the failure behavior of recycled concrete beams with variable shear-span ratios of 1.2, 2, and 3 was investigated.

The load–deflection of beams vs. variable shear-span ratios can be observed in Figure 6. Evidently, for the recycled concrete beam with a height of 200 mm, when the parameters λ are 1.2, 2, and 3, the bearing capacities are, respectively, 42.1 kN, 27.4 kN, and 21.3 kN. Compared to the beam with a parameter λ of 1.2, the shear-bearing capacities are reduced by 34.9% and 49.4% when the shear-span ratios are adopted as 2 and 3, respectively, indicating that the parameter λ has a great impact on the ultimate strength of recycled concrete beams without stirrups. Under the same conditions, the rate of increase in the displacement of recycled concrete beams diminishes with the enhancement of the shear-span ratio, and the shear-bearing ability decreases with the increase in the shear-span ratio. This indicates that the ultimate deformation of recycled concrete beams with variable shear-span ratios is different. The reason is that the shear-span ratio of a beam presents the ratio of the bending moment to the shear force, which reflects the stress state and proportion in the shear span failure zone, and the stiffness of the beam with a small shear-span ratio is higher than that of the beam with a large shear-span ratio, resulting in the slope of the load–displacement curve decreases as the shear-span ratio rises. In addition, as the dimension of the component is enlarged, the shear-bearing ability of the component is enhanced, indicating that the dimension effect becomes more pronounced as the size of the component increases.

3.3. Analysis of Size Effect of Shear-Span Ratio on the Nominal Shear Strength

This article uses the NSS τ_u to investigate the correlation between the shear-bearing capacity and the cross-sectional dimension of recycled concrete beams without stirrups. The NSS is described as follows:

$${\tau }_u=P_u/(b{h}_0)$$

(9)

where P_u is the ultimate load, b, and h₀ are specified as the section width and effective section height.

Figure 7 shows the NSS of recycled concrete beams without stirrups, considering variable shear-span ratios and the dimension of the cross-section. Figure 8 presents the error analysis results of the NSS of beams with varying shear-span ratios. Significantly, the NSS of the beam with identical size gradually reduces with the increase in parameter λ. As for the beam with a height of 200 mm, compared to beams with parameter λ of 1.2, the NSS of beams is reduced by 34.5% and 50% when the shear-span ratios are 2 and 3, respectively, indicating that the parameter λ has a remarkable impact on the NSS of the recycled concrete beam. Moreover, under the same parameter λ, the NSS of beams with variable sizes reduces with the enhancement of the cross-sectional height. When parameter λ is 1.2, the nominal shear-bearing capacity of the beam with a height of 600 mm is reduced by approximately 39% compared to the beam with a height of 200 mm, showing an obvious size effect. In Figure 8, the length of the error bar reflects the variability or uncertainty of the data. A longer error bar means that the data fluctuate greatly, while a shorter error bar means that the data are concentrated. For the above three different parameters of λ, the variation trend of the NSS of recycled concrete beams with variable dimensions is generally parallel. This demonstrates that the impact of parameter λ on the dimension effect of shear-bearing capacity is relatively small, and the quantitative relationship of the dimension effect of the NSS of recycled concrete beams without stirrups is consistent with variable shear-span ratios.

4. Influence of Replacement Rate on the Shear Failure and Size Effect

To investigate the impact of replacement rate on the shear failure and dimension effect of beams, based on the influence of different parameters λ on the shear failure, five different replacement rates of 0%, 20%, 50%, 80%, and 100% are considered in this section to study the shear behavior of recycled concrete beams without stirrups. The geometric parameters and corresponding symbols of the beams are in

Table 6. Geometric parameters of beams.

Specimens	Section Size b × h (mm)	Virtual Height h0/mm	Shear Span Ratio λ	Beam Length l/mm	Longitudinal Reinforcement Ratio ρ/%	Recycled Aggregate Replacement r/%
S-group	100 × 200	175	1.2~3.0	2000	1.2	0–100
M-group	150 × 300	275	1.2~3.0	3000	1.2	0–100
L-group	200 × 400	370	1.2~3.0	4000	1.2	0–100
U-group	250 × 500	470	1.2~3.0	5000	1.2	0–100
V-group	300 × 600	565	1.2~3.0	6000	1.2	0–100

.

4.1. Shear Failure Mode

Figure 9 depicts the shear failure modes of recycled concrete beams with variable replacement rates and cross-sectional sizes when the parameter λ is 1.2. Obviously, when the recycled concrete beams fail, the shear diagonal cracks are obvious, and the cracks extend diagonally from the loading points to the supports owing to the small shear-span ratio. Due to the high stiffness of large-sized beams, the concrete adjacent to the point of loading is not easy to crack, so the cracking point is farther away from the loading point. Furthermore, the distribution of cracks in recycled concrete beams with different replacement rates is very similar, and most of the cracks are not penetrating. Moreover, the higher the replacement rate of recycled aggregates, the greater the width of diagonal cracks in recycled concrete beams.

4.2. Load–Displacement Curve

Figure 10 exhibits the load–displacement curves of the recycled concrete beams with different replacement rates. Noticeably, the replacement rate possesses an obvious impact on the shear-bearing ability. For the beam with a height of 200 mm and a shear-span ratio of 1.2, when the replacement rates are 0%, 20%, 40%, 60%, 80%, and 100%, the shear-bearing capacities are 44.1 KN, 42.05 KN, 40.03 KN, 37.91 KN, and 35.83 KN, respectively. Compared to the beam with a parameter r of 0%, the shear-bearing capacities are reduced by 4.8%, 9.2%, 14%, and 18.9%, respectively, indicating that the shear-bearing ability of recycled concrete beam abates as the replacement rate rises. This is because the mechanism of interlocking recycled aggregate concrete in the oblique cracks is different from that of ordinary concrete. For ordinary concrete, the strength of the natural coarse aggregate is much greater than that of cement stone, and the aggregate may extend from the surface of diagonal cracks, forming the interlocking force. Due to the many microcracks and high porosity, the recycled aggregate concrete is easily snapped at the diagonal cracks, and the interlocking force of aggregate is smaller than that of ordinary concrete. Therefore, the shear ability of recycled concrete beams is slightly lower than that of ordinary concrete beams, and as the replacement rate increases, the shear-bearing ability of recycled concrete beams reduces.

4.3. Analysis of Size Effect of Replacement Rate on the Nominal Shear Strength

Figure 11 shows the NSS of recycled concrete beams with variable replacement rates and beam heights with three shear-span ratios, and Figure 12 displays the error analytic results of the NSS of beams with variable replacement rates and shear-span ratios. Notably, the dimension effect of the recycled concrete beams is very significant. When the shear-span ratio is the same, the replacement rate has a distinguished impact on the NSS of recycled concrete beams, and the NSS abates by increasing the replacement rate. When the shear-span ratio is 1.2 and the beam height is 200 mm, the NSS of recycled concrete beams with replacement rates of 0%, 20%, 40%, 60%, 80%, and 100% are 4.43 MPa, 4.21 MPa, 4.01 MPa, 3.75 MPa, and 3.55 MPa, respectively, and the NSS of the recycled concrete beam with a replacement rate of 100% is about 20% lower than that of the beam with a replacement rate of 0%. In addition, as the shear-span ratio is identical, the variation trend of the NSS in recycled concrete beams with variable replacement rates is nearly parallel, indicating that the replacement rate affects the NSS but has a small influence on its quantitative relationship.

5. Size-Effect Law of Shear Strength

5.1. Comparison of the Simulated Results and Bažant Size-Effect Law

Based on the fracture mechanic theory of material, Bažant [32] supplied a theoretical formula for the dimension effect in quasi-brittle materials:

$${\tau }_u=V_0/\sqrt{1+D/D_0}$$

(10)

where V₀ and D₀ refer to the related empirical coefficients of concrete, and D is specified as the cross-sectional height in beams.

$$Y=AX+C$$

(11)

where $Y={\left(1/{\tau }_u\right)}^2$; $X=D$; $C=1/\left(V_0^2\right)$; $A=C/D_0$.

By conducting regression analysis on the NSS, a double logarithmic curve relating the NSS to the cross-sectional size of the recycled concrete beam without stirrups can be obtained, as exhibited in Figure 13. The Bažant size-effect law can be represented by the logarithmic curve, where the diagonal dashed line with a slope of −1/2 illustrates the linear elastic fracture mechanics (LEFM) theory for those fully brittle matters, and the horizontal dashed line demonstrates the strength criterion for those plastic matters, excluding the dimension effect. The correlation coefficient between simulated outcomes and the Bažant size-effect law is around 0.92, indicating that the aforementioned law effectively describes the quantitative relationship of the dimension effect of NSS of recycled concrete beams.

It is clear that for different shear-span ratios and replacement rates, V₀ and D₀ obtained from the regression analysis are different. Essentially, the material hierarchy of the Bažant size-effect law cannot directly reflect the quantitative impact of replacement rate and shear-span ratio on the dimension effect of the shear strength in recycled concrete beams.

5.2. Theoretical Formula for the Dimension Effect of Nominal Shear Strength

Stemmed from the simulation results in Section 2, the NSS of recycled concrete beams abates as the shear-span ratio rises. Therefore, the effect of the shear-span ratio on the shear strength of recycled concrete beams should be considered, where the symbol β is used to represent the effect coefficient.

According to the simulation results in Section 3, the replacement rate has a distinguished impact on the NSS of recycled concrete beams, and the NSS decreases as the replacement rate rises. Therefore, the impact of replacement rate on the shear strength of recycled concrete beams should be considered, where the effect coefficient is represented by γ.

Related to the above conclusion and Bažant size-effect law, a theoretical formula for the dimension impact of shear strength of recycled concrete components is supplied, where the quantitative influences of both the replacement rate and the shear-span ratio on the NSS of recycled concrete beams are considered:

$${\tau }_c=V_0/\sqrt{1+D/D_0}\gamma \beta$$

(12)

Evidently, it reflects the quantitative impact of the replacement rate and component size on the NSS of recycled concrete beams.

5.2.1. Determination of the Effect Coefficient of Shear-Span Ratio β

The computation formula for the shear-bearing ability V_cs of reinforcement concrete beams according to Chinese standards is as follows:

$$V_{cs}={\alpha }_{cv}{f}_{t}b{h}_0+f_{yv}{\displaystyle \frac{A_{sv}}{s}}{h}_0$$

(13)

where α_cv is the shear-bearing ability coefficient of concrete, which is taken as 0.7, and $1.75/(\lambda +1)$ for general bending components and beams under concentrated load, respectively. λ refers to the shear-span ratio.

Based on the form of the formula in Chinese standard, the simulation results are fitted and shown in Figure 14. The formula of the effect coefficient of the shear-span ratio on the NSS of recycled concrete beams is obtained as follows:

$$\beta =2/(\lambda +0.16)$$

(14)

5.2.2. Determination of the Effect Coefficient of Replacement Rate γ

For the effect coefficient of the replacement rate on NSS, the following formula can be used:

$$\gamma =1-\alpha r$$

(15)

where α is the reduction coefficient of ultimate bearing capacity, and r is the replacement rate of recycled aggregate.

The effect coefficients of the replacement rate are fitted, and the fitting result is shown in Figure 15. The formula of the effect coefficient of the replacement rate is obtained as follows:

$$\gamma =1-0.21r$$

(16)

5.2.3. Fitting Formula for the Dimension Effect of NSS

Figure 16 shows the fitting results of the NSS dimension effect from the formula. In the figure, the slanted line with a slope of −1/2 describes the LEFM theory for entirely brittle matters, while the horizontal line illustrates the strength theory of plastic matters (strength criterion), assuming no consideration of the dimension effect. It is well realized that the correlation coefficient between simulated outcomes and the size-effect law proposed in this paper is 0.95, which is closer to 1 than the correlation coefficient of the aforementioned law, indicating that the dimension-effect law proposed in the current paper can better quantitatively describe the dimension effect of the NSS in recycled aggregate concrete beams than the Bažant size-effect law.

5.3. Comparison with Existing Methods for the Shear-Bearing Capacity of Beams

5.3.1. Comparison of the Simulated Results with Those Obtained from Existing Calculation Methods

Table 7. The existing calculation method of recycled concrete beams without stirrups.

Order Number	The Existing Calculation Method	The Main Calculation Expressions and Key Parameters
1	Chinese code 6.3 Calculation of bearing capacity of the inclined section (GB 50010-2010) [27]	$V_c={\displaystyle \frac{1.75}{\lambda +1}}{f}_{t}b{h}_0$ λ: shear-span ratio, when λ ≤ 1.5, λ = 1.5, when λ ≥ 3, λ = 3; ft: design value of axial tensile strength of concrete; b: rectangular section width; h0: effective depth of section. ${\tau }_c={\displaystyle \frac{V_c}{b{h}_0}}$ τc: NSS of beams; Vc: the calculated value of shear-bearing capacity; b: section width of the beam; h0: the effective height of the beam section.
2	American standards ACI 318 Punching Shear Design ACI 318-11 [35]	$V_c=0.166\gamma \sqrt{f_c^{\prime }}{b}_w{h}_0$ fc′: compressive strength of concrete cylinder; ordinary concrete γ = 1; bw: calculate the width of the cross-section; h0: effective depth of section. ${\tau }_c={\displaystyle \frac{V_c}{b{h}_0}}$ τc: NSS of beams; Vc: the calculated value of shear-bearing capacity; b: section width of the beam; h0: the effective height of the beam section.
3	Canadian code CSAA23.3 Punching Shear Design CSAA23.3-04 [36]	$V_c=\gamma \beta \sqrt{f_c^{\prime }}{b}_w{d}_v$ $\beta ={\displaystyle \frac{0.4}{1+1500{\epsilon }_x}}.{\displaystyle \frac{1300}{1000+\frac{35S_Z}{15+a_g}}}$ εx: longitudinal strain at the midpoint of the section height; Sz: crack spacing parameters, value with dv; dv: calculate the effective shear width of the section, taken 0.9 h0; ordinary concrete γ = 1; ag: maximum aggregate size; bw: calculate the width of the cross-section. ${\tau }_c={\displaystyle \frac{V_c}{b{h}_0}}$ τc: NSS of beams; Vc: the calculated value of shear-bearing capacity; b: section width of the beam; h0: the effective height of beam section.
4	European standard Punching shear strength of flat slabs—a critical review of Eurocode 2 and fib Model Code 2010 design provisions [33]	$V_c={\left[C_{1}k(100{\rho }_s{f}_c^{\prime })\right]}^{1/3}{b}_w{h}_0$ $k=1+\sqrt{{\displaystyle \frac{200}{d}}}$ fc′: compressive strength of concrete cylinder; ρs: longitudinal reinforcement ratio; d: Beam depth; bw: calculate the width of the cross-section; h0: the effective height of beam section; C1: correlation coefficient of concrete aggregate, conventional aggregate C1 = 0.1. ${\tau }_c={\displaystyle \frac{V_c}{b{h}_0}}$ τc: NSS of beams; Vc: the calculated value of shear-bearing capacity; b: section width of the beam; h0: the effective height of beam section.
5	Size-effect law of Jin [34]	${\tau }_c=V_0/\sqrt{1+D/D_0}\gamma \phi$ $\gamma =2/(\lambda +1)$ $\phi ={(0.95\rho )}^{0.3}$ τc: NSS of beams; D: section height of beam; V0 and D0: empirical coefficient in Bažant size-effect law; γ: influence coefficient of shear-span ratio on nominal shear-bearing ability; β: influence coefficient of longitudinal reinforcement ratio on NSS; λ: shear-span ratio; ρ: longitudinal reinforcement ratio of beam.

lists five different calculation methods for the shear strength of concrete beams without stirrups. According to Table 5 and Table 6, the simulated values of shear-bearing ability and the calculated values in the standards are compared. As depicted in Figure 17, the computation results of shear-bearing ability for concrete beams vary greatly among the different standards of countries, but the simulated values of shear-bearing capacity are all above the design values in the standards of four countries, indicating that the standard of each country is inclined to conservative. Figure 18 exhibits the variation in the NSS with size, as obtained from standards and simulation. Significantly, the design shear strength calculated by the standards is below the simulation value, and the difference slowly reduces as the component size increases. In addition, the European standard (EC 2010) [33] is the most conservative, and the correction method studied by Jin et al. [34] is closest to the simulation.

5.3.2. Comparison Between the Proposed Formula and Experimental Results

In the presented theoretical formula for the dimension effect of NSS in recycled concrete beams in this article, the values of V₀ and D₀ need to be determined. This article takes a recycled concrete beam with a shear-span ratio of 1.2 and a recycled aggregate replacement rate of 20% as a reference. According to the regression analysis, V₀ = 6.37 and D₀ = 489.

Based on this, more experimental results are shown in

Table 8. Experimental data on the shear capacity of concrete beams with varying section sizes.

Data Sources	Number	b (mm)	fc (MPa)	h (mm)	l (mm)	λ	r (%)	ρs (%)	Vu (KN)	τu (MPa)
Literature [37]	4	120	53.4	120~300	740~1550	1.7	100	1.4	86.3~177.7	4.72~6.54
Literature [38]	8	200	31.4~52.7	200~1600	1440~7140	2	0	1.74~1.85	394~1674	3.76~11.6
Literature [14]	13	151~154	28.7~38.7	300~309	1700~2100	1~3	0~100	2.49	114~530	2.78~4.44
Literature [4]	7	250~600	39.2~42.8	500~1206	2600~6400	2.6	0	0.67~1.19	299-1260	0.94~1.2
Literature [39]	12	200	40~50.2	300	2700	2~3.3	0~100	1.21~2.42	84~289.8	1.4~3.21

, which are compared with the design shear strengths of existing calculation methods and the proposed formula for the dimension effect of NSS, as shown in Figure 19 and

Table 9. The modified formula and the existing calculation method compared with the nominal shear strength of the experimental value.

Data Sources	GB 50010-2010 [27]		ACI 318-11 [35]		CSAA23.3-04 [36]		EC 2010 [33]		Jin’s Correction Method [34]		The Correction Method in This Paper
	τu Calculated Value	Error (%)	τu Calculated value	Error (%)	τu Calculated Value	Error (%)	τu Calculated Value	Error (%)	τu Calculated Value	Error (%)	τu Calculated Value	Error (%)
Literature [37]	2.11~2.15	54~67	1.19~1.2	74~81	1.41~1.44	69~78	0.69~0.7	85~89	3.17~3.23	32~51	4.25~4.33	9~35
Literature [38]	1.01~1.02	73~84	0.57~0.58	84~91	0.67~0.68	81~86	0.33~0.34	91~94	1.51~1.52	60~76	2.01~2.02	46~68
Literature [14]	1.18~1.41	55~68	0.66~0.79	75~82	0.79~0.94	70~79	0.39~0.47	85~89	1.76~2.11	33~52	2.37~2.38	10~35
Literature [4]	0.7~0.71	25~41	0.39~0.4	58~67	0.47~0.48	50~61	0.23~0.24	75~81	0.76~0.78	17~35	0.8~0.81	12~34
Literature [39]	0.68~0.96	51~70	0.37~0.53	72~83	0.46~0.64	69~80	0.22~0.32	84~90	1.02~1.44	27~55	1.37~2.02	9~40

. Obviously, the European Code (EC2010) is the most conservative, with an error ranging from 75% to 94% compared to experimental results. The computed values from the proposed formula of NSS closely match the experimental outcome, with an error ranging from 9% to 40%. This reveals that the supplied theoretical formula can better accurately predict the shear strength of beams in practical engineering. In addition, from the relevant theoretical analysis, when the section size of the beam changes within a certain range, the theoretical formula for size effect can better reflect the variation law in the NSS of beams. However, when the size of the beam is too large or too small, and beyond the range of experiment data on which the formula is based, the accuracy of the formula may be reduced. For example, special attention should be paid to the thin plate component with a height of less than 100 mm and the large-sized beam with a height of more than 1 m. When the shear-span ratio is located between 1.0 and 3.0, the shear-failure beam modes are predominantly diagonal-compression and shear-compression failure, and the theoretical formula of size effect generally performs well. When the shear-span ratio is beyond 3.0, the failure mode of the beam may change to diagonal tension failure, the law of size effect may be different, and the applicability of the formula requires further verification.

6. Conclusions

This article establishes a shear failure analysis model for recycled concrete beams without stirrups, based on a three-dimensional microscopic numerical analysis method, and studies the dimension effect of the shear-span ratio and replacement rate of recycled concrete beams. Based on the Bažant size-effect law, a theoretical formula is supplied that accurately reflects the influence of the shear-span ratio and replacement rate on shear strength. The main conclusions of the study are drawn as follows:

• The three-dimensional microscopic analysis model can capture the heterogeneity of concrete and describe the crack development and failure process in recycled concrete beams, showing good agreement with experimental results. The NSS of recycled concrete beams possesses a distinguished dimension effect. The NSS of the beam with a height of 600 mm is reduced by nearly 40% compared to the beam with a height of 200 mm as the shear-span ratio reaches 1.2.
• The NSS of recycled concrete beams abates as the shear-span ratio rises. Specifically, the NSS of a beam with a shear-span ratio of 3 is nearly 50% lower than that of a beam with a shear-span ratio of 1.2. As the shear-span ratio increases, the failure mode of the beam transitions from shear compression to diagonal tension. Compared with small-sized beams, large-sized beams have fewer bending cracks at failure, and the initiation point of the penetrating diagonal crack is farther away from the point of loading. This is due to the greater stiffness of large-sized beams, which reduces the likelihood of concrete near the loading point cracking under shear stresses.
• The replacement rate of recycled aggregate significantly affects the shear-bearing ability of recycled concrete beams without stirrups. As the replacement rate of recycled aggregate increases, the NSS of recycled concrete beams decreases. The shear-bearing capacity of beams with a 100% replacement rate is reduced by 18.9% compared to that with a replacement rate of 0%. In addition, the higher the replacement rate of recycled aggregate, the wider the diagonal cracks in recycled concrete beams.
• The design shear strengths of formulas of various countries’ standards and proposed formulas for the NSS are all below the experimental results. European Code (EC2010) is the most conservative, with a minimum error of 75% compared to experimental results. The calculated results of the proposed formula closely match the experimental results, with a minimum error of 9%, which demonstrates the closest agreement with experiment values. Notably, the formula proposed in this study has certain limitations, for instance, it is only applicable to static loading and rectangular cross-sections. Future work will expand to dynamic loading conditions, non-rectangular cross-sections, or other recyclable material types, to narrow the gap between research and practical application.