Compressive behavior of recycled aggregate concrete with different stirrup confinements subjected to axial loading after freeze–thaw cycles

Abstract

Recycled aggregate concrete (RAC) undergoes irreversible freeze–thaw damage, with more complex and stochastic crack propagation than natural aggregate (NA). Consequently, steel stirrups are commonly used to enhance its durability and mechanical performance. 36 specimens with four recycled aggregate (RA) replacement ratios (0%, 30%, 70%, and 100%) and three stirrup types (no stirrup restraint (PCC), rectangular stirrup restraint (RSC), spiral stirrup restraint (SSC)) were therefore tested for uniaxial compression after 100 freeze–thaw cycles. Afterward, the stress–strain behavior of specimens was analyzed. The results demonstrate that RA exhibits superior resistance to freeze–thaw cycles compared to NA under similar strength, the peak stress increases with higher RA replacement ratios. SSC exhibits a stronger strengthening effect than RSC due to the uniform confinement of spiral stirrups. Lateral strain increases with axial strain, following a cubic nonlinear relationship. Subsequently, by employing the cohesive elastic ring model and the modified elastic beam theory, the lateral effective stress of RSC and SSC can be accurately predicted, with a standard deviation of 0.113 and an average absolute error of 0.186. A novel compressive damage constitutive model for RAC, incorporating lateral stress effects, shows strong agreement with test stress–strain curves, with R² > 0.91. Finally, a comparison of bearing capacity calculations shows the proposed method aligns best with experimental results.

Graphical abstract

Keywords

RAC / stress–strain curves / peak stress / lateral effective stress / damage constitutive model

1 Introduction

As urbanization accelerates and the construction industry continues to grow, the demolition of aging concrete structures has become increasingly prevalent, resulting in substantial environmental stress due to the generation of construction and demolition waste [1]. Processing demolition waste into recycled aggregate (RA) through crushing and screening provides a sustainable solution that helps mitigate environmental impact and reduce the over-exploitation of natural stone resources [2,3]. To achieve green buildings, the reuse of waste aggregates is a key strategy [4]. However, the widespread adoption of recycled aggregate concrete (RAC) remains constrained due to its inferior mechanical and durability properties compared to nature aggregate concrete (NAC). These limitations arise from factors such as high water absorption [5], increased porosity [6], lower apparent density [7], and the presence of residual cement mortar on the RA surface [8,9]. Because of the intricate interface and internal imperfections found within RAC, its deformation capacity is less than that of NAC [10,11]. In addition, recent studies indicate that treatments such as acid leaching, mechanical extrusion, and quenching can enhance the performance of RAC. The impact of RA on the functionality of structural elements appears to be less significant than initially anticipated [12]. Research on structural components made from RAC, including beams [13], slabs [14], columns [15], shear walls [16], and beam–column connections [17], has demonstrated its suitability for applications extending beyond road construction and non-load-bearing elements, encompassing both horizontal and vertical load-bearing structures. These findings underscore the considerable market potential of RAC in the future.

Freeze–thaw damage is a gradual process characterized by the continuous buildup of deterioration linked to fatigue. The phase transformation strain of free water within the pores, similar to cyclic loading, results in increased porosity and the formation of micro-cracks during repeated freeze–thaw cycles. Ultimately, this progression leads to the cracking and spalling of concrete [18]. In frigid areas, reinforced concrete structures are susceptible to freeze–thaw damage, presenting a significant challenge to durability [19,20]. Due to the characteristics of RA, RAC structures face more complex freeze–thaw issues, which restricts their widespread application [21]. Relevant studies [22,23] indicate that freeze–thaw cycles can cause surface scaling or erosion of the structure, potentially resulting in the mortar detachment. Moreover, the physical properties of RA deteriorate with prolonged exposure to freeze–thaw conditions [24].

Although fiber-reinforced polymer (FRP) materials offer excellent corrosion resistance and durability in aggressive environments, they exhibit brittle failure modes and limited axial load-bearing capacity under compressive forces [25]. Hybrid systems combining FRP and steel have been proposed to leverage the benefits of both materials [26–28]. However, the complex construction process, uncertain long-term performance, and higher cost of hybrid systems limit their practical adoption in many applications [29,30]. In contrast, conventional steel stirrups provide well-established confinement efficiency, ductility, and load redistribution capacity, especially under dynamic or severe mechanical loading [31]. Therefore, in this study, steel stirrups were selected to investigate their effectiveness in improving RAC performance after freeze–thaw deterioration. In recent times, steel reinforcement has emerged as the primary load-bearing material in engineering structures. However, vertical elements necessitate enhanced seismic design to effectively endure horizontal forces. Additionally, the majority of structures are designed with sufficient lateral reinforcement, such as spiral, rhomboid, or rectangular stirrups, to reduce the risk of shear failure or sudden collapse of the core concrete [32,33]. A significant amount of research has investigated how lateral reinforcement influences the confinement of concrete. For instance, Assa et al. [34] proposed a standard related to peak strength and developed a segmented model designed to predict the stress–strain behaviors of confined specimens. In another investigation, Ren et al. [35] studied peak strength to stirrup spacing and different strain rates, resulting in the development of an innovative damage model. Sheikh and Uzumeri [36] introduced a coefficient that evaluates the effectiveness of confinement, taking into account the impact of reinforcement on concrete, as well as the configuration of stirrups and the shape of the specimens’ cross-section. Refs. [37,38] investigated confined concrete behavior, with Rong and Shi [37] employing the energy balance method to propose a passive strain model, boundary points, and a confinement coefficient, while Saatcioglu and Razvi [38] developed a model based on equivalent stirrup pressure, emphasizing the strong relationship between strength, ductility, and stirrup configuration.

In conclusion, while the use of RAC structures is on the rise, their application in cold regions is hindered by challenges such as large pores and interface defects. Incorporating stirrup reinforcement improves RAC’s resistance to freeze–thaw damage, with various stirrup configurations significantly affecting its mechanical properties and behavior. Furthermore, existing predictive models, which are primarily based on experimental data, demonstrate good performance but are constrained by factors such as cross-sectional structure, stirrup type, and variations in lateral confinement. Further investigation into the effects of changes in lateral confinement is warranted. Therefore, this study aims to investigate the combined effects of RA content and stirrup configuration on the mechanical behavior of RAC after freeze–thaw cycles. Specifically, 36 RAC short columns with varying RA replacement ratios (0%, 30%, 70%, and 100%) and confinement types (no stirrups, rectangular stirrups, spiral stirrups) were tested under uniaxial compression after 100 freeze–thaw cycles. The effects on peak stress, peak strain, and initial stiffness were analyzed. Following this, various stirrup configurations were analyzed to compute lateral effective stresses by applying the cohesive elastic ring model alongside the modified elastic beam theory. A constitutive model for RAC compressive damage, which takes into consideration the influence of lateral constraints, was developed utilizing principles from damage mechanics theory. Finally, experimental results were compared with predicted values, and the bearing capacity was calculated for verification. We believe this study will serve as a valuable reference for the design of RAC structures. This research fills a critical gap by clarifying how lateral confinement modifies the freeze–thaw resistance of RAC, providing a valuable reference for the design and application of RAC in cold regions.

2 Raw materials and test design

2.1 Materials and mix proportion

The cement applied in this research was P·O. 42.5 from Conch brand designation (C). The fine aggregate was comprised of natural river sand (S) featuring a fineness modulus of 2.98, along with natural tap water (W). The natural aggregate (NA) showcased a particle size distribution ranging from 5 to 31.5 mm, indicating a well-graded composition. RA was produced through the mechanical crushing of waste pavement concrete, followed by pre-cleaning and manual screening. As per GB/T 25177-2010 [39], RA is classified as class II. Details regarding the material properties of both aggregates can be found in Table 1 and Fig. 1. Before concrete pouring, the bars were tested according to GB/T 228.1-2010 China [40], yielding strength values of 407.6 MPa for 6 mm hoop bar, and 450 MPa for 8 mm longitudinal steel bars, respectively. The compressive strength of RAC, targeting a C40 grade, was evaluated in accordance with the specification standard [41]. Four mix ratios were applied to achieve a comparable compressive strength with varying RA content, as seen in Table 2. In this table, AW indicates additional water. The values 0, 30, 70, and 100 represent the respective replacement ratios of RA.

2.2 Specimen design

The RA replacement ratio and stirrup configuration were the key parameters in this experiment. The RA replacement ratios (\gamma) were set at 0%, 30%, 70%, and 100%. The stirrup configurations included no stirrup restraint (PCC), rectangular stirrup restraint (RSC), and spiral stirrup restraint (SSC), resulting in 36 specimens. The PCC and RSC specimens were 150 mm × 150 mm × 450 mm, while the SSC specimens were \varnothing150 mm × 450 mm. RSC configuration featured four 8 mm longitudinal steel bars and 6 mm stirrups spaced at intervals of 50 mm, with 25 mm spacing at both ends. SSC specimens had a stirrup spacing of 50 mm, with a core-to-cross-sectional area ratio of 0.68. The thickness of the protective layer for the stirrup-restrained specimens was set at 10 mm, as detailed in Fig. 2 and Table 3. In Table 3, d_{\mathrm{c}\mathrm{o}\mathrm{r}} is the core concrete’s diameter. d_1 and d_2 represent the diameters of the stirrup and longitudinal ribs, respectively. {\rho }_{\mathrm{v}} refers to the stirrup volume ratio. Specimen names follow the format: stirrup constraint type-RC number. A total of 100 freeze–thaw cycles were conducted on the specimens. The specimens were cast according to the specified mix ratio, demolded, and cured under standard conditions for 28 d. After the curing period, the specimens were submerged in water for 4 d, ensuring the water level remained at least 20 mm above the specimens. Following this soaking period, the short columns were cleaned and placed in the freeze–thaw apparatus (Fig. 3). The rapid freezing method was employed in the freeze–thaw test [42], maintaining the specimen temperature at 15 °C. During each cycle, the temperature was lowered from 15 to 6 °C and subsequently raising it back to 15 °C over a period of 3.5 h.

2.3 Test procedure

The axial compression test was conducted using the YAW-10000J electro-hydraulic servo testing machine, as illustrated in Fig. 4. A load sensor was placed between the specimen and the loading plate to accurately measure the axial load. Strain measurements prior to cracking were recorded using strain gauges symmetrically positioned on both sides of the specimen. Additionally, two displacement gauges, positioned between the upper and lower loading plates, monitored strain after cracking. To assess lateral displacement, four displacement meters were strategically positioned at the center of each side. The loading system initially adopted a force-controlled approach, increasing the load to 20 kN at 5 kN/min. This was subsequently followed by a transition to a displacement-controlled loading phase, with a rate of 0.2 mm/min, until failure was observed. The displacement-controlled loading scheme was employed beyond the peak load to prevent sudden brittle failure, maintain experimental stability, and facilitate precise observation of post-peak behavior and crack propagation. This approach is particularly crucial when testing RAC columns, which are susceptible to abrupt strength degradation after reaching peak stress due to their heterogeneous and defect-prone internal structure.

3 Test results and analysis

3.1 Failure patterns

Figure 5 illustrates the failure patterns observed in the specimens. The concrete’s uniaxial failure involves the entire process of internal crack formation, from initial appearance to complete penetration. The failure process of RSC was analyzed: initially, the surface remained intact, with no visible failure signs. As the load increased, vertical oblique cracks appeared at the ends, widening and expanding diagonally as stress approached the peak. After reaching the peak stress, cracks propagated, penetrating the specimen at a 1/4 angle, causing failure. In the SSC specimen, cracks formed at the ends, evolving into small, dense, short fractures. As the load increased, these cracks expanded and penetrated, resulting in a lantern-like damage pattern. For the PCC specimen, cracks were initiated at the corners. Due to the absence of external confinement, diagonal cracks propagated, leading to sudden brittle failure both before and after peak load. Oblique cracks formed in the final failure stage. Observations indicated that the failure modes were not significantly influenced by the type of stirrup confinement or the replacement rate.

3.2 Stress–strain curve

Figure 6 presents the complete stress–strain curves for all specimens. Due to brittle failure, PCC was unable to capture the entire descending section of the curve. The green dotted curve represents the average curve. The axial force that was measured relates to the overall force produced by the core concrete under confinement, the protective layer, and the longitudinal reinforcement. It is assumed that all forces undergo the same axial strain when calculating the effective stress of the confined core concrete. Specifically, the effective stress of the confined core concrete (\sigma) is calculated:

(1)\sigma =\frac{F_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-F_{\mathrm{l}}-F_{\mathrm{h}}}{A_{\mathrm{c}\mathrm{o}}},

where F_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}} = the total load (i.e., measured load value); F_{\mathrm{l}} = the axial forces from the longitudinal reinforcement, F_{\mathrm{h}} = the load of the concrete protective layer, A_{\mathrm{c}\mathrm{o}} = the area of effective compression zone.

Based on the test results, Fig. 7 illustrates the typical uniaxial compressive stress–strain curve, emphasizing the key stages.

Elastic stage (OA): During the initial loading stage, the applied load remains low, resulting in elastic deformation. The stress–strain curve exhibits a nearly linear upward trend with a steep slope, while lateral expansion and stirrup strain growth are minimal.

Plastic stage (AB). As the load reaches between 40% and 100% of the maximum stress, the strain exceeds the concrete’s ultimate elastic strain, indicating the onset of plastic deformation.

Rapid decline stage (BC). When the stress surpasses the maximum compressive strength of the confined concrete, a swift reduction in stress takes place, along with the yielding of both the longitudinal reinforcements and the stirrups.

Slow decline stage (CE). As the stress reduces to about 50% of its maximum, the stress–strain curve begins to flatten. By the DE section, the stress either stabilizes or sees a minor decline. Ultimately, the specimen fails due to the deformation of the longitudinal reinforcement and the subsequent failure of the core concrete.

The peak stress and strain of the constrained RAC after 100 freeze–thaw cycles were determined through testing. The measured data are presented in Table 4, where f_{\mathrm{c}\mathrm{c}} represents the average peak stress, {\epsilon }_{\mathrm{c}\mathrm{c}} denotes the average peak strain, {\sigma }_{3\epsilon p} indicates the average lateral stress, {\epsilon }_{\mathrm{l}\mathrm{c}} corresponds to the maximum lateral strain, and f_{\mathrm{c}} refers to the uniaxial compressive strength of the prism.

3.3 Characteristic parameter analysis

3.3.1 Peak stress

Figure 8 illustrates the peak stress (f_co) of RAC short columns at various aggregate substitution ratios under different stirrup constraints. The data presented represent the average peak stress of three specimens per group, as shown in Fig. 6. Figure 8(a) demonstrates that with a rise in the amount of RA, the peak stress of the constrained recycled concrete also increases. This rise can be linked to the necessity for more cementitious material to adequately bond the increased volume of RAs, while the compressive strength of the specimen cubes remains relatively stable. Consequently, the increased cement content enhances axial compressive strength. Furthermore, due to its higher internal porosity and lower initial strength, RA demonstrates greater adaptability to freeze–thaw cycles compared to NA due to the presence of old mortar. However, after multiple freeze–thaw cycles, pores in RAs may fill or compact, stabilizing the internal structure and enhancing performance. In contrast, although NAs exhibit greater initial strength, they may develop microcracks or particle shedding after 100 freeze–thaw cycles, which can diminish their performance. In addition, the elevated water absorption capacity of RA likely reduced the amount of free water within the concrete matrix, subsequently lowering internal frost-induced pressure. This observation suggests that increasing the RA substitution rate improves frost resistance in recycled concrete.

Figure 8(b) illustrates the relationship between the normalization coefficient (f_{\mathrm{c}\mathrm{o}}/f_{\mathrm{c}\mathrm{c}}) under different stirrup configurations, where f_{\mathrm{c}\mathrm{c}} denotes the peak stress of PCC. For various substitution ratios (0%, 30%, 70%, and 100%), the normalization coefficients for the RSC series were recorded as 1.53, 1.44, 1.38, and 1.40, respectively, while those for the SSC series were 1.69, 1.60, 1.51, and 1.47. Notably, after 100 freeze–thaw cycles, specimens with a 0% substitution ratio exhibited the largest peak stress increase compared to PCC, whereas those with 100% substitution had the smallest increase. The exhibited performance of PCC columns is due to the lack of lateral restraint. As demonstrated in Fig. 9, damage from frost heave takes place when the force generated by frost heave surpasses the tensile strength of concrete, resulting in the formation of holes or cracks. However, stirrups are effective in preventing cracks in the core area. The axial compressive strength of SSC surpasses that of RSC due to the continuous spiral configuration of the spiral stirrup, which provides uniform lateral confining pressure and ensures uniform triaxial compression throughout the concrete section.

3.3.2 Residual stress

Figure 10 illustrates the relationship between residual stress (f_reo) and stirrup configurations (RSC and SSC) under varying RA substitution ratios. A comparison of the stirrup specimens reveals that SSC exhibits significantly higher residual strength than RSC, highlighting a clear correlation between residual stress and stirrup configuration. The failure surface begins to form at peak stress, as depicted in the BC stage of Fig. 7. As cracks develop, lateral reinforcement displaces, generating reverse pressure that increases cohesion and friction along the failure surface, converting interface force into residual stress (see CE stage of Fig. 7). The emergence of the failure surface and the crushing of aggregate cause only slight changes in residual stress. At this stage, frictional stress dominates the axial load, complicating the axial force-bearing mechanism. With similar compressive strength, freeze–thaw damage reaches a balance in specimens with varying substitution ratios, thereby reducing the influence of RA substitution on residual stress.

3.3.3 Axial and lateral strain

Figure 11 illustrates the peak strain of concrete short columns under different substitution ratios and stirrup configurations. As seen in Fig. 11(a), with the presence of stirrup constraints, the peak strain of RSC and SSC is greater than that of PCC. Figure 11(b) illustrates the normalization coefficient ({\epsilon }_{\mathrm{c}\mathrm{o}}/{\epsilon }_{\mathrm{c}\mathrm{c}}) under different stirrup configurations, where {\epsilon }_{\mathrm{c}\mathrm{o}} represents the peak strain of the stirrup-constrained specimens, {\epsilon }_{\mathrm{c}\mathrm{c}} denotes the peak strain of PCC. With various substitution ratios of 0%, 30%, 70%, and 100%, the peak strain of RSC was observed to rise by 37%, 65%, 36%, and 52%, respectively, in comparison to PCC. Meanwhile, the peak strain for SSC showed increases of 103%, 103%, 88%, and 113%, respectively, exceeding twice the peak strain recorded for PCC. Furthermore, the change of peak strain with substitution rate (≥ 30%) is not obvious between different stirrup constraints after freeze–thaw. This phenomenon can be attributed to the substantial effect on the microcracks and pores in the concrete, leading to a uniform deterioration trend regardless of the RA substitution rate. As the cube compressive strength remains stable, this observation indicates that the freeze–thaw cycles ‘homogenize’ the elastic modulus and ductility, thereby diminishing the influence of substitution ratios on peak strain.

Figure 12 presents the average dimensionless {\epsilon }_1/{\epsilon }_{1\mathrm{c}}–\epsilon /{\epsilon }_{\mathrm{c}\mathrm{c}} curve (\mu \left(\epsilon \right)) for different specimens. Where {\epsilon }_1 represents the lateral strain, {\epsilon }_{1\mathrm{c}} denotes the peak lateral strain, and \epsilon corresponds to the axial strain. As the axial strain increases, the lateral strain is observed to increase accordingly. The relationship between axial-lateral strain is described by the nonlinear regression equation shown in Fig. 12, where y={\epsilon }_1/{\epsilon }_{1\mathrm{c}}, and x=\epsilon /{\epsilon }_{\mathrm{c}\mathrm{c}}. As illustrated in the figure, the proposed cubic function equation effectively predicts the {\epsilon }_1/{\epsilon }_{1\mathrm{c}}–\epsilon /{\epsilon }_{\mathrm{c}\mathrm{c}} curves.

3.3.4 Initial stiffness

Figure 13 illustrates the initial stiffness of PPC, RSC, and SSC specimens under varying substitution ratios. The initial stiffness is calculated as the tangent modulus at 40% of the peak stress, corresponding to the slope of the OA section in the elastic stage (Fig. 7). The results indicate that SSC exhibits the highest initial stiffness, ranging from 27.7 to 28.6 GPa, followed by RSC with values between 26.6 and 27.8 GPa, and PPC ranging from 22.3 to 26.1 GPa across the different substitution rates. For each stirrup configuration, initial stiffness first increases and then decreases with the RA replacement rate, peaking at 30%. At lower replacement rates (0%–70%), RA fills internal pores, improving pore distribution and reducing crack propagation, thereby enhancing the elastic modulus. At higher replacement rates (70%–100%), poor bonding between old mortar and aggregate increases microcracks and pores, reducing stiffness, particularly after freeze–thaw cycles.

4 Determination of the lateral confinement model

4.1 External concrete constraints

The findings from the tests suggest that using lateral stirrups markedly enhances the axial strength and bearing capacity of confined RAC. Consequently, it is crucial to create a strength model that takes into account the effects of the protective layer, the crack confinement zone, and the effective confinement zone, as recommended by earlier research [43]. Acknowledging the lateral pressure exerted by the concrete protective layer, meanwhile, previous research [44] introduced a model based on an equivalent cylinder, as demonstrated in Fig. 14. In this model, the equivalent diameter corresponds to the diameter of the concrete core in each specimen, with the constrained specimens exhibiting elastic behavior. In the OA stage of Fig. 7, the protective concrete layer that surrounds the core mainly serves to contain it. According to the cohesive elastic ring model proposed by Ref. [45], which is depicted in Eqs. (2) and (3), the confining pressure ({\sigma }_{\mathrm{r}\mathrm{c}}) provided by this protective layer of concrete can be calculated as follows

(2){\sigma }_{\mathrm{r}\mathrm{c}}=\frac{{\phi }_1\cdot (r_{\mathrm{b}}^2-r_{\mathrm{a}}^2)\cdot {\sigma }_{\mathrm{r}}}{\left(1-{\displaystyle \frac{r_{\mathrm{b}}^2}{r^2}}\right)\cdot r_{\mathrm{a}}^2},

(3){\sigma }_{\mathrm{r}}=\frac{{\mu }_{\mathrm{r}}\cdot {\phi }_2\cdot E}{r\cdot \left[\left(1-{\displaystyle \frac{r_{\mathrm{b}}^2}{r^2}}\right){\displaystyle \frac{2r^2}{r^2-r_{\mathrm{b}}^2}}-1\right]\cdot (1+{\upsilon }_{\mathrm{c}})}.

In this context, r denotes the distance from the specimen’s center, while r_b indicates the equivalent column radius, r_{\mathrm{b}}=75\mathrm{m}\mathrm{m}. r_{\mathrm{a}} represents the radius of the core concrete. {\sigma }_{\mathrm{r}} stands for the radial stress, and the reduction factor {\phi }_1 is applied to account for the fragile nature and low strength of the recycled concrete protective layer, with {\phi }_1 = 0.45. {\mu }_{\mathrm{r}} signifies the radial deformation, recorded by the lateral displacement meter, and {\upsilon }_{\mathrm{c}} reflects the Poisson’s ratio of RAC, established at 0.21 for this study. Drawing from the test findings, a reduction coefficient, denoted as {\phi }_2 and valued at 0.8, was recommended to illustrate the reduction in the elastic modulus of RAC. The initial stiffness ratio of PCC compared to the hoop-constrained specimen was found to be 0.76 for RSC and 0.81 for SSC.

4.2 Confinement of rectangular stirrup

After the OA stage of Fig. 7, as the axial force increases, the concrete protective layer begins to peel off, and the stirrup undergoes deformation. Prior to yielding, the stirrup exerts significant lateral pressure, which contributes to the axial strength, although the pressure distribution is uneven. Therefore, accurate determination of stirrup constraints is essential. To explore the relationship between stirrups and the core concrete, Campione [46] utilized beam theory based on an elastic foundation along with symmetry principles to represent this interaction. Figure 15 showcases the interaction force, which includes the component transmitted by the diffusion spring:

(4)j=\frac{2E_{\mathrm{c}}}{L(1-{\upsilon }_{\mathrm{c}})},

where j is the equivalent spring stiffness, E_c is the elastic modulus of RAC, and L is the stirrup size. The differential equation governing an equivalent beam resting on an elastic foundation can be formulated with the functional analysis as follows [47]:

(5)\frac{{\mathrm{d}}^{4}w}{\mathrm{d}{x}^4}+\frac{k\left(\delta w\right)}{E_{\mathrm{s}}{I}_{\mathrm{s}}}=0,

where w represents the deflection of the equivalent beam, \delta signifies the transverse deformation of the core concrete, E_s and I_s are the elastic modulus and moment of inertia of the transverse reinforcement, respectively, with I_{\mathrm{s}}=\pi d^4/64. The value of d is the same as that in Table 3. To facilitate the analysis, a simplified parameter \eta, \eta =\sqrt[4]{{\displaystyle \frac{j\cdot d}{4\cdot E_{\mathrm{s}}{I}_{\mathrm{s}}}}}, is introduced into Eq. (5), yielding:

(6)w=\delta +m\cdot \cos h\eta x\cdot \cos \eta x+n\cdot \sin \mathrm{h}\eta x\cdot \sin \eta x,

where \delta is the lateral deformation of the core concrete, \delta ={\epsilon }_{1}L/2. {\epsilon }_1 represents the lateral strain of the concrete, which is measured using a strain gauge, and m and n are parameters that can be determined based on the boundary conditions, as follows:

(7)\omega \left(x\right)={\delta }_{\mathrm{s}},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}x=L/2,

(8)\mathrm{d}\omega \left(x\right)/\mathrm{d}x=0,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}x=L/2.

By substituting the above known conditions into Eq. (6), parameters m and n can be gained according to Eqs. (9) and (10), respectively, where {\delta }_{\mathrm{s}} is the elongation of a single stirrup.

(9)m=-2({\delta }_{\mathrm{s}}-\delta )\cdot \frac{\sin {\displaystyle \frac{\eta L}{2}}\cos \mathrm{h}{\displaystyle \frac{\eta L}{2}}+\cos {\displaystyle \frac{\eta L}{2}}\sin \mathrm{h}{\displaystyle \frac{\eta L}{2}}}{\sin \mathrm{h}(\eta L)+\sin (\eta L)},

(10)n=-2({\delta }_{\mathrm{s}}-\delta )\cdot \frac{\cos {\displaystyle \frac{\eta L}{2}}\sin \mathrm{h}{\displaystyle \frac{\eta L}{2}}-\sin {\displaystyle \frac{\eta L}{2}}\cos \mathrm{h}{\displaystyle \frac{\eta L}{2}}}{\sin (\eta L)+\sin \mathrm{h}(\eta L)}.

Based on the results obtained from Ref. [48], the lateral pressure caused by the interaction between stirrup and core concrete deformation is not uniform. Assuming that the stirrup contributes the lateral pressure, it can be expressed as follows:

(11)q\left(x\right)=j\cdot \left[\delta -\omega \left(x\right)\right].

To further explore the effective lateral pressure \overline{q} that the stirrup exerts, an equivalent uniform stress is introduced in Eq. (12). Utilizing the principle of mechanical equilibrium, the lateral pressure F related to a single lateral stirrup is articulated in Eq. (13). The deformation of the longitudinal steel bars leads to a discontinuity in lateral pressure between neighboring stirrups. To address this issue, a reduction factor j_{\mathrm{s}} is incorporated, with s signifying the distance separating the two stirrups and L indicating the section’s side length.

(12)\overline{q}=\frac{{\displaystyle {\int }_{-L/2}^{L/2}j\cdot (\delta -\omega )\mathrm{d}x}}{L},

(13)F=0.5L\cdot j_{\mathrm{s}}\cdot s\cdot \overline{q}.

Ref. [49] suggests that j_{\mathrm{s}} is related to the deflection of longitudinal reinforcement and the lateral expansion of concrete. In Fig. 16, {\delta }_1 represents the transverse deformation of the concrete core, {\delta }_1={\upsilon }_{\mathrm{c}}\epsilon L. The microgeometric relationship shown from Fig. 16, it can be observed that the microsegment of the steel bar is denoted as \mathrm{\Delta }L. Using the first-order Taylor series expansion, the microdeformation angle \varphi and \mathrm{\Delta }L can be derived as {\upsilon }_{\mathrm{c}}\epsilon \mathrm{\Delta }l/\mathrm{\Delta }l=\tan \varphi =\sin \varphi. The micro deflection of the longitudinal reinforcement is assumed to follow the form {\delta }_3=0.5s, and j_{\mathrm{s}} is obtained from Eq. (14):

(14)j_{\mathrm{s}}=\frac{{\delta }_1-{\delta }_3}{{\delta }_1}=\frac{{\upsilon }_{\mathrm{c}}\cdot \epsilon \cdot L-0.5\cdot s\cdot {\upsilon }_{\mathrm{c}}\cdot \epsilon }{{\upsilon }_{\mathrm{c}}\cdot \epsilon \cdot L}.

The relationship between F and the reinforcement elongation {\delta }_{\mathrm{s}} can be expressed as

(15)F=\frac{2E_{\mathrm{s}}\cdot A_{\mathrm{s}\mathrm{t}}\cdot s\cdot {\delta }_{\mathrm{s}}}{L}.

A_{\mathrm{s}\mathrm{t}} represents the cross-sectional area of the stirrup. Eq. (16) can be derived by substituting the Eqs. (5)–(12) and Eqs. (14) and (15) into the Eq. (13).

(16)F=\frac{j\cdot \left[(\tilde{m}-\tilde{n})\cdot C+(\tilde{m}+\tilde{n})\cdot D\right]\cdot j_{\mathrm{s}}}{\eta \cdot \left[1-(\tilde{m}-\tilde{n})\cdot (C+D)\cdot j_{\mathrm{s}}/A_{\mathrm{s}\mathrm{t}}{E}_{\mathrm{s}}\right]}\cdot u\left(\epsilon \right)\cdot L,

where

(17)\tilde{m}=\frac{-m}{2({\delta }_{\mathrm{s}}-\delta )},

(18)\tilde{n}=\frac{n}{2({\delta }_{\mathrm{s}}-\delta )},

(19)C=\cosh \left(\frac{\eta L}{2}\right)\sin \left(\frac{\eta L}{2}\right),

(20)D=\cos \left(\frac{\eta L}{2}\right)\sinh \left(\frac{\eta L}{2}\right).

The axial-lateral strain relationship, referred to as u\left(\epsilon \right), is outlined in Fig. 12. According to Eq. (16), it is evident that the force F increases as the axial strain rises prior to reaching the yield threshold, F=f_{\mathrm{y}}\cdot A_{\mathrm{s}\mathrm{t}}, (f_{\mathrm{y}} denoting the stress in the stirrup). Furthermore, Fig. 17(a) presents the lateral stress model for RSC. A constrained pressure model [48], is introduced, as detailed in Eq. (21)

(21)f_{\mathrm{l}\mathrm{e}\mathrm{l}}=2\frac{j_{\mathrm{e}\mathrm{l}}\cdot f_{\mathrm{y}\mathrm{h}}\cdot A_{\mathrm{s}\mathrm{t}}}{L\cdot s},

where f_{\mathrm{l}\mathrm{e}\mathrm{l}} is the peak lateral stress, f_{\mathrm{y}\mathrm{h}} is the yield strength of the steel bar. The constraint effective coefficient j_{\mathrm{e}\mathrm{l}} in Eq. (21) is expressed as

(22)j_{\mathrm{e}\mathrm{l}}=2\frac{\left[1-{\displaystyle \sum _{i=1}^n\frac{{c_{\mathrm{i}}}^2}{a_1{L}^2}}\right]{\left[1-0.5s\tan ({\theta }_1/L)\right]}^2}{1-{\rho }_1},

where {\rho }_1 denotes the volume ratio of the longitudinal rebar, c_i refers to the distance separating two neighboring longitudinal rebars, {\theta }_1 represents the angle of the parabola, and a_1 signifies the adjusted parameter. This study suggests values of {\theta }_1=\pi /6 and a_1=12 based on the evaluation of the test results. In Eq. (21), F = f_{\mathrm{y}\mathrm{h}}\cdot A_{\mathrm{s}\mathrm{t}}. The effective lateral stress of the rectangular stirrup, expressing its elastic–plastic characteristics, is described by Eq. (23)

(23)f_{\mathrm{l}\mathrm{e}1}=j_{\mathrm{e}\mathrm{l}}\cdot u\left(\epsilon \right)\cdot L\frac{2j\cdot \left[(\tilde{m}-\tilde{n})\cdot C+(\tilde{m}+\tilde{n})\cdot D\right]\cdot j_{\mathrm{s}}}{\eta \left[1-(\tilde{m}-\tilde{n})\cdot (C+D)\cdot j_{\mathrm{s}}/A_{\mathrm{s}\mathrm{t}}{E}_{\mathrm{s}}\right]\cdot L\cdot s}.

4.3 Confinement of spiral stirrups

The mechanical function of the rectangular stirrup differs from that of the spiral stirrup; hence, it is essential to modify the effective constraint coefficient and lateral stress to consider the influence of the spiral stirrup. Figure 17(b) depicts the effective constraint model related to the spiral stirrup. According to Ref. [50], the variation in spiral spacing can be analyzed through A_{\mathrm{e}}, where {\alpha }_1 denotes the horizontal angle. The radius of the section, labeled as r\left({\alpha }_1\right), diminishes as s increases, as demonstrated in Eq. (24), where {\alpha }_2 is a parameter adjusted based on the constrained RAC results, with {\alpha }_2 set to 0.8. The effective constraint parameters for the spiral stirrup are obtained from Eq. (25).

(24)r\left({\alpha }_1\right)=r_2-a_2\frac{\varpi \left({\alpha }_1\right)}{4}=r_2-a_2\frac{s{\alpha }_1}{4\pi },

(25)\begin{array}{rl}{A}_{\mathrm{e}}=& 2\cdot \frac{1}{2}{\int }_0^{\pi }\left(r_{\mathrm{s}}-a_2\frac{s}{4\pi }{\alpha }_1\right)\mathrm{d}{\alpha }_1\\ =& r_{\mathrm{s}}^2\pi -\frac{1}{4\pi }{a}_2\cdot r_{\mathrm{s}}\cdot s\cdot {\pi }^2+a_2\cdot \frac{{\pi }^2{s}^2}{48}.\end{array}

The spiral stirrup model is simplified by Ref. [48] and expressed by

(26)j_{\mathrm{e}2}=\frac{A_{\mathrm{e}}}{A_{\mathrm{c}\mathrm{o}}}={\displaystyle \frac{\pi r_{\mathrm{s}}^2-{\displaystyle \frac{1}{4\pi }}{a}_2\cdot r_2\cdot s\cdot {\pi }^2+a_2\cdot {\displaystyle \frac{s^2\pi }{48}}}{\pi r_{\mathrm{s}}^2\cdot \left(1-{\rho }_1\right)}},

where A_{\mathrm{c}\mathrm{o}} represents the primary concrete region, excluding the impact of the vertical rod. Initially, N is used to calculate the effective confinement pressure, as demonstrated in Eq. (27), with r_{\mathrm{s}} = L/2. The connection between the effective confinement pressure f_{\mathrm{l}\mathrm{e}2} and \epsilon can be established based on the model illustrated in Fig. 17(b). Let {\delta }_{\mathrm{l}\mathrm{e}} denote the deflection along the spiral stirrup, while {\delta }_{\mathrm{c}} signifies the expansion of the core concrete. As per Ref. [51], the following relationship can be derived

(27){\delta }_{\mathrm{s}}=\frac{N\cdot r_{\mathrm{s}}}{E_{\mathrm{s}}\cdot A_{\mathrm{s}}\cdot \cos {\alpha }_2},

(28){\delta }_{\mathrm{l}\mathrm{e}}=\frac{f_{\mathrm{l}\mathrm{e}2}\cdot r_{\mathrm{s}}\cdot (1-{\upsilon }_{\mathrm{c}})}{E_{\mathrm{c}}},

(29){\delta }_{\mathrm{c}}={\upsilon }_{\mathrm{c}}\cdot r_{\mathrm{s}}\cdot \epsilon ,{\alpha }_2=\frac{\arctan (2s)}{L},

(30){\delta }_{\mathrm{s}}={\delta }_{\mathrm{c}}-{\delta }_{\mathrm{l}\mathrm{e}}.

In this study, the above relationship is substituted into Eq. (31) and ultimately expressed by Eq. (32). The parameters in the equation are consistent with those previously defined.

(31)f_{\mathrm{l}\mathrm{e}2}=k_{\mathrm{e}3}\cdot \frac{N}{s\cdot r_{\mathrm{s}}}\cdot \cos {\alpha }_2,

(32)f_{\mathrm{l}\mathrm{e}2}=k_{\mathrm{e}2}\cdot \frac{{\upsilon }_{\mathrm{c}}\cdot \epsilon }{s\cdot r_{\mathrm{s}}}\cdot \frac{1}{{\displaystyle \frac{1}{\cos {\alpha }_2}}\cdot {\displaystyle \frac{4}{E_{\mathrm{s}}\cdot \pi \cdot d^2}}+{\displaystyle \frac{\left(1-{\upsilon }_{\mathrm{c}}\right)}{E_{\mathrm{c}}\cdot s\cdot r_{\mathrm{s}}}}}.

According to Eqs. (26) and (32), the lateral pressure generated by the stirrup is associated with axial strain. In the case of the confined concrete specimens, we can consider the confining pressure ({\sigma }_{\mathrm{r}\mathrm{c}}) from the protective layer concrete, which signifies the lateral stress (designated as f_{\mathrm{l}\mathrm{e}1} for RSC or f_{\mathrm{l}\mathrm{e}2} for SSC) exerted by the stirrup during the OA phase illustrated in Fig. 7, to be {\sigma }_3. The typical dimensionless relationship between {\sigma }_3/{\sigma }_{3\mathrm{p}} and \epsilon /{\epsilon }_{\mathrm{c}0} is illustrated in Fig. 18, where {\sigma }_{3\mathrm{p}} represents the peak lateral stress, \epsilon and {\epsilon }_{\mathrm{c}0} correspond to the axial strain and peak strain of the confined core concrete, and x = \epsilon /{\epsilon }_{\mathrm{c}0}, and y = {\sigma }_3/{\sigma }_{3\mathrm{p}}. In Fig. 18, the increasing rate of {\sigma }_3/{\sigma }_{3\mathrm{p}} increases with the increasing \epsilon /{\epsilon }_{\mathrm{c}0}. Additionally, Both the RSC curves (Fig. 18 (a)) and SSC curves (Fig. 18 (b)) that have been fitted, indicating that the suggested model successfully forecasts the experimental curve. To evaluate the precision of the newly introduced lateral stress model, three evaluation metrics–average ratio (AV), standard deviation (SD), and average absolute error (AAE) [52] were utilized to compare the theoretical and experimental outcomes, with calculations depicted in Fig. 19. Alongside the experimental findings from this research and those referenced in Ref. [29], Fig. 19 illustrates a comparison of the predicted lateral stresses against the measured values. Here, {\sigma }_{3\mathrm{p}\mathrm{p}} represents the predicted lateral stress for RSC or SSC, and {\sigma }_{3\mathrm{p}\mathrm{t}} denotes the experimental results. Calculations based on the formulas in Fig. 19 yield an SD of 0.1129 and an AAE of 0.1864. These results demonstrate a strong alignment between the predicted and experimental lateral stresses, thereby validating the accuracy of the model. The AV is 0.9649, suggesting that the predicted values are generally lower than the experimental results, with a certain safety margin retained.

5 Compressive damage constitutive model

This paper introduces a computational model designed to analyze the bearing capacity of RAC specimens exposed to freeze–thaw cycles. Building on the model established in concrete damage mechanics and referenced in literature [53], the axial damage constitutive framework is further applied to a triaxial compression condition. In Eq. (33), D denotes the damage index, and all remaining parameters align with those outlined earlier.

(33){\sigma }_{\mathrm{l}}=E\cdot \epsilon \cdot (1-D)+{\upsilon }_{\mathrm{c}}\cdot ({\sigma }_2+{\sigma }_3).

D equals 0 in the OA stage in Fig. 7. From the experimental form in this paper, we can get {\sigma }_2={\sigma }_3. The microdamage distribution of concrete follows a Weibull distribution [54]

(34)D=1-\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left(\frac{\hat{M}}{{\hat{M}}_0}\right)}^t\right].

Combining Eq. (34) with Eq. (33), the RAC damage constitutive model considering lateral stress is expressed as Eq. (35)

(35){\sigma }_{\mathrm{l}}=E\cdot \epsilon \cdot \mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left(\frac{\hat{M}}{\hat{M}0}\right)}^t\right]+{\upsilon }_{\mathrm{c}}\cdot ({\sigma }_2+{\sigma }_3),

where \hat{M} is the damage parameter of the microunit; t and \hat{M}0 are the shape parameters of the damage evolution curve, respectively, which are calculated as follows

(36)\hat{M}=E\cdot \epsilon \cdot \varphi \left(\sigma \right)/({\sigma }_1-2{\upsilon }_{\mathrm{c}}{\sigma }_3),

(37)\varphi \left(\sigma \right)=\frac{{\varsigma }_{\mathrm{c}}}{f_{\mathrm{c}}}+a_1\left(\frac{\xi }{f_{\mathrm{c}}}\right)+a_2{\left(\frac{\xi }{f_{\mathrm{c}}}\right)}^2,

(38){\varsigma }_{\mathrm{c}}=\frac{1}{\sqrt{15}}\sqrt{({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_1-{\sigma }_3{)}^2+({\sigma }_2-{\sigma }_3{)}^2},

(39)\xi =\frac{1}{3}({\sigma }_1+{\sigma }_2+{\sigma }_3).

Using the correct calculation method, the number of freeze–thaw cycles (100) is substituted into the values from Ref. [55], yielding a₁ = 0.9219 and a₂ = 0.1743. By substituting Eqs. (37) and (38) into Eq. (39), the \epsilon into peak strain {\epsilon }_{\mathrm{c}\mathrm{o}}, and also substituting into the axial stress {\sigma }_{1\mathrm{P}} and transverse peak value {\sigma }_{3\mathrm{p}} of Eqs. (38) and (39), the damage parameter {\hat{M}}_{\mathrm{p}} at the maximum point can be gained

(40){\hat{M}}_{\mathrm{p}}=\frac{E\cdot \epsilon \cdot \left({\displaystyle \frac{{\varsigma }_{\mathrm{c}}}{f_{\mathrm{c}}}}+a_1\left({\displaystyle \frac{\xi }{f_{\mathrm{c}}}}\right)+a_2{\left({\displaystyle \frac{\xi }{f_{\mathrm{c}}}}\right)}^2\right)}{{\sigma }_{1\mathrm{p}}-{\sigma }_{3\mathrm{p}}-{\sigma }_3\cdot (2{\upsilon }_{\mathrm{c}}-1)}.

Based on boundary conditions:

(41){\sigma }_1|(\epsilon =0)=0,\mathrm{d}{\sigma }_1/\mathrm{d}\epsilon |{}_{(\epsilon =0)}=0.

Bringing the above formula into Eq. (35), then the m, {\hat{M}}_0 and {\hat{M}}_{\mathrm{p}} are

(42)t={\left[\ln (\frac{{\sigma }_{1\mathrm{p}}-{\sigma }_{3\mathrm{p}}-{\sigma }_3\cdot (2{\upsilon }_{\mathrm{c}}-1)+E{\epsilon }_{\mathrm{c}\mathrm{o}}}{E{\epsilon }_{\mathrm{c}\mathrm{o}}})\right]}^{-1},

(43){\left(\frac{{\hat{M}}_0}{{\hat{M}}_{\mathrm{p}}}\right)}^t=-\frac{1}{\left[\ln ({\displaystyle \frac{{\sigma }_{1\mathrm{p}}-{\sigma }_{3\mathrm{p}}-{\sigma }_3\cdot (2{\upsilon }_{\mathrm{c}}-1)+E{\epsilon }_{\mathrm{c}\mathrm{o}}}{E{\epsilon }_{\mathrm{c}\mathrm{o}}}})\right]}.

Equations (38) and (39) are substituted into Eq. (37). Based on Eq. (37), the relationship between {\sigma }_1 and {\sigma }_3 can be expressed by Eq. (44). Finally, Eqs. (42) and (43) are linked to {\sigma }_3, being expressed through the lateral stress-axial strain relationship discussed earlier. Here, f_c represents the strength of the RAC.

(44)\frac{{\sigma }_1}{f_{\mathrm{c}}}=\left(3.643\cdot \sqrt{1+5.877\frac{{\sigma }_3}{f_{\mathrm{c}}}}-2.849\right)-\frac{2{\sigma }_3}{f_{\mathrm{c}}}.

In the descending section, considering the residual confinement provided by the stirrup, Baduge et al. [56] introduced a model for the descending section, illustrated in Eq. (45). The other parameters are consistent with those previously mentioned.

(45){\sigma }_1=\left(f_{\mathrm{c}\mathrm{o}}-f_{\mathrm{r}\mathrm{e}\mathrm{o}}\right)\exp \left[-\lambda (\epsilon -{\epsilon }_{\mathrm{c}\mathrm{o}})\right]+f_{\mathrm{r}\mathrm{e}\mathrm{o}}.

The parameter \lambda represents the unconstrained value, which is calculated based on experimental results and sourced from Ref. [53]. Using the descending section model presented in Eq. (45), the damage constitutive model for confined recycled concrete, accounting for lateral stress, is derived in Eq. (46). As shown in Fig. 20, the correlation coefficients for both the experimental and predicted values exceed 0.91, indicating a high level of accuracy in the model put forth.

(46){\sigma }_1=\{\begin{array}{rl}& E\cdot \epsilon +2\cdot {\upsilon }_{\mathrm{c}}\cdot {\sigma }_3,\\ & \epsilon \le \epsilon |({\sigma }_1\le 0.4f_{\mathrm{c}\mathrm{o}}),\\ & E\cdot \epsilon \cdot \exp \left[-{\left({\displaystyle \frac{\hat{M}}{{\hat{M}}_0}}\right)}^m\right]+2\cdot {\upsilon }_{\mathrm{c}}\cdot {\sigma }_3,\\ & \epsilon |({\sigma }_1\le 0.4f_{\mathrm{c}\mathrm{o}})\le \epsilon \le \epsilon |({\sigma }_1\le f_{\mathrm{c}\mathrm{o}}),\\ & \left(f_{\mathrm{c}\mathrm{o}}-f_{\mathrm{r}\mathrm{e}\mathrm{o}}\right)\exp \left[-\lambda (\epsilon -{\epsilon }_{\mathrm{c}\mathrm{o}})\right]+f_{\mathrm{r}\mathrm{e}\mathrm{o}},\\ & \epsilon \ge \epsilon |({\sigma }_1\ge f_{\mathrm{c}\mathrm{o}}).\end{array}

6 The bearing capacity prediction method

The short columns of reinforced concrete analyzed are all specimens. According to the previous analysis, when lateral deformation becomes considerable due to loading, the lateral confinement significantly restrains the specimen. Yet, at this stage, the concrete protective layer fails to maintain its confinement effect. As a result, the longitudinal bar experiences yielding, the core concrete attains its maximum strength (f_mL), and the theoretical bearing capacity can be expressed as follows

(47)N_C=f_{\mathrm{c}\mathrm{c}}\cdot A_{\mathrm{c}\mathrm{o}\mathrm{r}}+f_{\mathrm{y}}\cdot A_{\mathrm{s}},

where f_mL refers to the peak stress of reinforced concrete, with the maximum value from Eq. (46) being utilized. The yield strength of longitudinal reinforcement is denoted as f_y, while this paper employs the measured yield strength. The term A_cor signifies the concrete core’s area, and A_s represents the cross-sectional area of longitudinal reinforcement.

Various nations utilize distinct calculation techniques to assess the compressive strength of structural elements. The frequently adopted approaches include the following.

The Chinese standard GB50010-2010 [41] specifies that the load-bearing capacity (N_CN) of a rectangular cross-section is

(48)N_{\mathrm{C}\mathrm{N}}=0.9\cdot \phi \cdot (f_{\mathrm{c}}\cdot A+f_{\mathrm{y}}\cdot A_{\mathrm{s}}),

where φ accounts for the stability coefficient of length and slenderness, taking 1.0. f_c denotes the axial compressive strength of RAC. A is the cross-sectional area of the specimen.

The stirrup spacing of SSC (s) in this paper 50 mm, and when 40 mm ≤ s ≤ 80 mm, it is

(49)N_{\mathrm{C}\mathrm{N}}=0.9(f_{\mathrm{c}}\cdot A_{\mathrm{c}\mathrm{o}\mathrm{r}}+f_{\mathrm{y}}\cdot A_{\mathrm{s}}+2\alpha \cdot f_{\mathrm{y}\mathrm{v}}\cdot A_{\mathrm{s}\mathrm{s}\mathrm{o}}),

where f_yt represents the stirrup’s tensile strength. A_{\mathrm{s}\mathrm{s}\mathrm{o}} refers to the equivalent cross-sectional area of the stirrup, and A_{\mathrm{s}\mathrm{s}\mathrm{o}}=\pi d_{\mathrm{c}\mathrm{o}\mathrm{r}}{A}_{\mathrm{s}\mathrm{t}/\mathrm{s}}. \alpha is the reduction coefficient for the restraint effect of the stirrup, which is set to 1.0. The remaining parameters are consistent with those described above.

The bearing capacity specified in the European Code [57]

(50)N_{\mathrm{E}\mathrm{U}}=\kappa \cdot \varpi \cdot f_{\mathrm{c}\mathrm{d}}\cdot \left(A-A_{\mathrm{s}}\right)+f_{\mathrm{y}}\cdot A_{\mathrm{s}},

(51)\kappa =\{\begin{array}{lr}0.8,& f_{\mathrm{c}\mathrm{k}}\le 50\mathrm{M}\mathrm{P}\mathrm{a},\\ 1.0-{\displaystyle \frac{\left(f_{\mathrm{c}\mathrm{k}}-50\right)}{400}},& 50\mathrm{M}\mathrm{P}\mathrm{a}\le f_{\mathrm{c}\mathrm{k}}\le 90\mathrm{M}\mathrm{P}\mathrm{a},\end{array}

(52)\varpi =\{\begin{array}{lr}1.0,& f_{\mathrm{c}\mathrm{k}}\le 50\mathrm{M}\mathrm{P}\mathrm{a},\\ 1.0-{\displaystyle \frac{\left(f_{\mathrm{c}\mathrm{k}}-50\right)}{200}},& 50\mathrm{M}\mathrm{P}\mathrm{a}\le f_{\mathrm{c}\mathrm{k}}\le 90\mathrm{M}\mathrm{P}\mathrm{a}.\end{array}

According to Ref. [27], f_cd=,0.667f_ck, f_ck = 0.8f_cu,k [57], f_cu,k is the axial compressive strength of the cylinder specimen. \kappa and \varpi are effective height coefficients. {\alpha }_{\mathrm{c}\mathrm{c}} denotes long-term load effect, a value of 0.85 is recommended by the specification. {\gamma }_{\mathrm{c}} represents the material coefficient, with a value of 1.5 for concrete.

According to the provisions of ACI 318-19 [58], the maximum axial bearing capacity:

(53)N_{\mathrm{A}\mathrm{C}\mathrm{I}}=\alpha \cdot \Phi \cdot \left(0.85\cdot f_c^{\prime }\cdot \left(A-A_{\mathrm{s}}\right)+f_{\mathrm{y}}\cdot A_{\mathrm{s}\mathrm{t}}\right).

In the formula, \alpha is taken as 0.85 for RSC and 0.81 for SSC. The strength reduction coefficient, Ф, takes on a value of 0.75 for SSC and 0.65 for other elements. 0.85{\gamma }_{\mathrm{c}} represents the compressive strength of PCC. In this study, the measured concrete strength, f_c0 = 0.85{\gamma }_c.

The calculation results of the above methods are shown in Figs. 21 and 22. The calculation results indicate that, compared to other standard calculation methods, the peak stress predicted by the damage constitutive model proposed in this study is closest to the experimental value, 0.925 times the N_test value. The Chinese standard yields the second-best result, with a ratio of 0.746 to the test value, closely aligning with the test values. In contrast, the predicted values from the US and European codes are more conservative. The US code predicts between 63% and 74% of the actual bearing capacity (with a mean of 0.683N_test), while the European code predicts between 59% and 70% (with a mean of 0.642N_test).

7 Conclusions

This study investigates the axial compressive properties of 36 reinforced RAC columns subjected to 100 freeze–thaw cycles. The key conclusions are drawn.

1) Under equivalent strength conditions, the replacement ratio of RA has no significant impact on the damage degree of RSC, SSC, and PCC. In comparison to NA, RA exhibits superior frost resistance in freeze–thaw environments. Additionally, although peak stress rises with a higher RA substitution ratio, there is no clear correlation between residual stress, peak strain, and the substitution ratio.

2) Due to the presence of lateral constraints, both rectangular and spiral stirrups provide enhanced confinement to the concrete core compared to PCC. The spiral stirrup, characterized by its continuous and uniform confinement of the concrete, yields greater strength, deformation, and initial stiffness than RSC. Furthermore, the axial strain increases in conjunction with lateral strain, and this relationship can be accurately modeled using a cubic nonlinear function.

3) The lateral stress increases effectively with the rise in axial strain. To compute the lateral stress during the elastic phase, an enhanced cohesive elastic ring model is utilized. Furthermore, a novel elastic beam model is proposed, which incorporates the effective constraint parameters prior to stirrup yielding. The SD between the calculated and measured results is 0.113, and the AAE is 0.1864, indicating that the model accurately predicts the variation in lateral stress.

4) Drawing upon the Weibull damage distribution theory and the failure criteria associated with RAC, a novel damage constitutive model has been developed to estimate the uniaxial compression behavior of confined RAC. The correlation coefficient exceeds 0.91, indicating a strong agreement between the predicted outcomes and the observed stress–strain curves.

5) Based on the peak stress predicted by the damage constitutive model developed in this study, we derive a theoretical formula for calculating the bearing capacity of RAC. The results indicate a strong correlation between the theoretical predictions and the experimental data, with the predicted values averaging 92.5% of the experimental results.

References

[1] Salesa A , Esteban L , Lopez-Julian P L , Perez-Benedicto J A , Acero-Oliete A , Pons-Ruiz A . Evaluation of characteristics and building applications of multi-recycled concrete aggregates from precast concrete rejects. Materials, 2022, 15(16): 5714

[2] Saberi S , Mohamed A , Eltwati A S . Mechanical and physical properties of recycled concrete aggregates for road base materials. Journal of Physics: Conference Series, 2021, 1973: 012236

[3] Xiao J Z , Wang C H , Ding T , Akbarnezhad A . A recycled aggregate concrete high-rise building: Structural performance and embodied carbon footprint. Journal of Cleaner Production, 2018, 199: 868–881

[4] Yang W R , Huang Y W , Li C W , Tang Z , Quan W , Xiong X , He J , Wu W . Damage prediction and long-term cost performance analysis of glass fiber recycled concrete under freeze–thaw cycles. Case Studies in Construction Materials, 2024, 21: e03795

[5] Guo H , Shi C J , Guan X M , Zhu J , Ding Y , Ling T C , Zhang H , Wang Y . Durability of recycled aggregate concrete––A review. Cement and Concrete Composites, 2018, 89: 251–259

[6] Guo S Q , Ding Y H , Xu P , Wu J , Bao J . Stress–strain relationship of steel fiber reinforced fully recycled coarse/fine aggregate concrete under cyclic loading. Case Studies in Construction Materials, 2025, 22: e04139

[7] Wang Y G , Xie M , Zhang J . Mechanical properties and damage model of modified recycled concrete under freeze–thaw cycles. Journal of Building Engineering, 2023, 78: 107680

[8] Omar T , Sherif Y , Akmal A , Mohamed E . Sustainable concrete production: The potential of utilizing recycled waste materials. Journal of Building Engineering, 2024, 98: 111467

[9] Olorunsogo F , Padayachee N . Performance of recycled aggregate concrete monitored by durability indexes. Cement and Concrete Research, 2002, 32(2): 179–185

[10] Pereiro-Barceló J , Lenz E , Torres B , Estevan L . Mechanical properties of recycled aggregate concrete reinforced with conventional and recycled steel fibers and exposed to high temperatures. Construction & Building Materials, 2024, 452: 138976

[11] Chen Y L , Wang Q , Ye P H , Zhang W . Experimental and damage modeling research on direct shear behavior of recycled concrete with pebble/gravel coarse aggregate. Journal of Building Engineering, 2024, 98: 111218

[12] Gao S , Li Q Y , Luo J L . Fractal characteristic of recycled aggregate and its influence on physical property of recycled aggregate concrete. Reviews on Advanced Materials Science, 2021, 60(1): 663–677

[13] Arezoumandi M , Smith A , Volz J A , Khayat K H . An experimental study on flexural strength of reinforced concrete beams with 100% recycled concrete aggregate. Engineering Structures, 2015, 88: 154–162

[14] Mohamad N , Khalifa H , Abdul Samad A A , Mendis P , Goh W I . Structural performance of recycled aggregate in CSP slab subjected to flexure load. Construction & Building Materials, 2016, 115: 669–680

[15] Saribas I , Goksu C , Binbir E , Ilki A . Seismic performance of full-scale RC columns containing high proportion recycled aggregate. Bulletin of Earthquake Engineering, 2019, 17(11): 6009–6037

[16] Zhang J W , Zhao Y R , Li X Y , Li Y , Dong H . Experimental study on seismic performance of recycled aggregate concrete shear wall with high-strength steel bars. Structures, 2021, 33: 1457–1472

[17] Letelier Gonzalez V A , Moriconi G . The influence of recycled concrete aggregates on the behavior of beam–column joints under cyclic loading. Engineering Structures, 2014, 60: 148–154

[18] Xu Y Q , Ye H S , Yuan Q , Shi C , Gao Y , Fu Q . The durability of concrete subject to mechanical load coupled with freeze–thaw cycles: A review. Archives of Civil and Mechanical Engineering, 2022, 22(1): 47

[19] Lin H W , Han Y F , Liang S M , Gong F , Han S , Shi C , Feng P . Effects of low temperatures and cryogenic freeze–thaw cycles on concrete mechanical properties: A literature review. Construction & Building Materials, 2022, 345: 128287

[20] Zahedi Z , Komar A , Sanchez L , Boyd A . Global assessment of concrete specimens subjected to freeze–thaw damage. Cement and Concrete Composites, 2022, 133: 104716

[21] Zhang P , Wittmann F , Vogel M , Müller H S , Zhao T . Influence of freeze–thaw cycles on capillary absorption and chloride penetration into concrete. Cement and Concrete Research, 2017, 100: 60–67

[22] Richardson A , Coventry K , Bacon J . Freeze/thaw durability of concrete with recycled demolition aggregate compared to virgin aggregate concrete. Journal of Cleaner Production, 2011, 19(2–3): 272–277

[23] Ma Z M , Zhu F Z , Ba G Z . Effects of freeze–thaw damage on the bond behavior of concrete and enhancing measures. Construction & Building Materials, 2019, 196: 375–385

[24] Wang Y R , Cao Y B , Zhang P , Ma Y , Zhao T , Wang H , Zhang Z . Water absorption and chloride diffusivity of concrete under the coupling effect of uniaxial compressive load and freeze–thaw cycles. Construction & Building Materials, 2019, 209: 566–576

[25] Yu T , Hu Y M , Teng J G . Cyclic lateral response of FRP-confined circular concrete-filled steel tubular columns. Journal of Constructional Steel Research, 2016, 124: 12–22

[26] Yang D , Lin S Q , Zhao Y G . Components interaction mechanism and strength of axially loaded FRP-confined concrete stub columns from energy path perspective. Structures, 2025, 71: 107981

[27] Tobbi H , Farghaly A S , Benmokrane B . Strength model for concrete columns reinforced with fiber-reinforced polymer bars and ties. ACI Structural Journal, 2014, 111(4): 789–798

[28] Yang H F , Li M H , Jiang J S , Zhang Y , Mei J , Deng Z . Mechanical properties of steel reinforced coral aggregate concrete column under uniaxial compression. Construction & Building Materials, 2023, 369: 130508

[29] Yang H F , Liu C L , Jiang J S . Damage constitutive model of stirrup-confined recycled aggregate concrete after freezing and thawing cycles. Construction & Building Materials, 2020, 253: 119100

[30] Zhang Y R , Wei Y , Miao K T , Li B . A novel seawater and sea sand concrete-filled FRP-carbon steel composite tube column: Cyclic axial compression behaviour and modeling. Engineering Structures, 2022, 252: 113531

[31] Qian K , Liang S L , Xiong X Y , Fu F , Fang Q . Quasi-static and dynamic behavior of precast concrete frames with high performance dry connections subjected to loss of a penultimate column scenario. Engineering Structures, 2020, 205: 110115

[32] Weng Y H , Qian K , Fu F , Fang Q . Numerical investigation on load redistribution capacity of flat slab substructures to resist progressive collapse. Journal of Building Engineering, 2020, 29: 101109

[33] Deng X F , Liang S L , Fu F , Qian K . Effects of high-strength concrete on progressive collapse resistance of reinforced concrete frame. Journal of Structural Engineering, 2020, 146(6): 04020078

[34] Assa B , Nishiyama M , Watanabe F . New approach for modeling confined concrete. II: Rectangular columns. Journal of Structural Engineering, 2001, 127(7): 751–757

[35] Ren X D , Liu K , Li J , Gao X L . Compressive behavior of stirrup-confined concrete under dynamic loading. Construction & Building Materials, 2017, 154: 10–22

[36] Sheikh S A , Uzumeri S M . Analytical model for concrete confinement in tied columns. Journal of the Structural Division, 1982, 108(12): 2703–2722

[37] Rong C , Shi Q X . Analysis constitutive models for actively and passively confined concrete. Composite Structures, 2021, 256: 113009

[38] Saatcioglu M , Razvi S . Strength and ductility of confined concrete. Journal of Structural Engineering, 1992, 118(6): 1590–1607

[39] GB/T25177-2010. Recycled Coarse Aggregate for Concrete. Beijing: General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China, 2011

[40] GB/T228.1-2010. Metallic materials-Tensile Testing-Part 1: Method of Test at Room Temperature. Beijing: General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China, 2011

[41] GB/T50107-2010. Standard for Test and Evaluation of Concrete Compression Strength. Beijing: Ministry of Housing and Urban-Rural Development of the People’s Republic of China, 2010

[42] GB/T50082-2024. Standard for Long-term Performance and Durability Testing Method of Ordinary Concrete. Beijing: Ministry of Housing and Urban-Rural Development of the People’s Republic of China, 2025

[43] Wang Y , Yang G F , Zhang J W , Cao C , Jin F . Mechanical properties of uniaxial compression of total tailings paste under lateral restraint effect. Construction & Building Materials, 2024, 412: 134702

[44] Légeron F , Paultre P . Uniaxial confinement model for normal- and high-strength concrete columns. Journal of Structural Engineering, 2003, 129(2): 241–252

[45] TimoshenkoSGoodierJAbramsonH. Theory of Elasticity. New York, NY: McGraw-Hill, 1970, 37

[46] Campione G . Analytical model for high-strength concrete columns with square cross-section. Structural Engineering and Mechanics, 2008, 28(3): 295–316

[47] Campione G , Cannella F , Minafò G . A simple model for the calculation of the axial load-carrying capacity of corroded RC columns. Materials and Structures, 2016, 49(5): 1935–1945

[48] Mander J , Priestley M , Park R . Theoretical stress−strain model for confined concrete. Journal of Structural Engineering, 1988, 114(8): 1804–1826

[49] Braga F , Gigliotti R , Laterza M . Analytical stress−strain relationship for concrete confined by steel stirrups and/or FRP jackets. Journal of Structural Engineering, 2006, 132(9): 1402–1416

[50] West J , Ibrahim A , Hindi R . Analytical compressive stress−strain model for high-strength concrete confined with cross-spirals. Engineering Structures, 2016, 113: 362–370

[51] Campione G , Fossetti M , Minafò G , Papia M . Influence of steel reinforcements on the behavior of compressed high strength R. C. circular columns. Engineering Structures, 2012, 34: 371–382

[52] Yang H F , Yang Q M , Luo J H , Jiang J , Mei J , Liu A . Shear strength and failure criterion of geopolymer coral aggregate concrete under compression-shear loading. Journal of Building Engineering, 2023, 76: 107241

[53] Jiang J S , Yang H F , Deng Z H , Zhang Y . Damage-constitutive model for seawater coral concrete using different stirrup confinements subjected to axial loading. Frontiers of Structural and Civil Engineering, 2023, 17(3): 429–447

[54] Cui T , He H X , Yan W M , Zhou D . Compression damage constitutive model of hybrid fiber reinforced concrete and its experimental verification. Construction & Building Materials, 2020, 264: 120026

[55] Sun C W , Zhu B W , Luo T , Liu K , Wei T , Yang S . Uniaxial compressive mechanical properties and stress–strain model for roller-compacted concrete with initial damage subjected to freeze–thaw cycles. Construction & Building Materials, 2024, 411: 134256

[56] Kristombu BadugeSMendisPNgoT. Stress–strain relationship for very-high strength concrete (> 100 MPa) confined by lateral reinforcement. Engineering Structures, 2018, 177: 795–808

[57] EN1992-1-1:2004. Eurocode 2: Design of Concrete Structures-Part 1-1: General Rules and Rules for Buildings. Brussels: European Committee for Standardization, 2004

[58] ACI318M–2011. Building Code Requirements for Structural Concrete and Commentary. Farmington Hills, MI: American Concrete Institute, 2011

RIGHTS & PERMISSIONS

Higher Education Press