Damage-constitutive model for seawater coral concrete using different stirrup confinements subjected to axial loading

Abstract

Recently, the application of detrital coral as an alternative to natural aggregates in marine structures has attracted increased attention. In this study, research on the compressive performance of coral aggregate concrete (CAC) confined using steel stirrups with anti-rust treatment was experimentally conducted. A total of 45 specimens were cast, including 9 specimens without stirrups and under different strength grades (C20, C30, and C40) and 36 specimens under different strength grades (C20, C30, and C40). Moreover, three stirrup levels (rectangular, diamond-shaped compound, and spiral stirrups) and different stirrup spacings (40, 50, 60, and 70 mm) were used. Subsequently, the stress−strain curves of specimens subjected to axial loading were measured. The effects of the stirrup spacing and stirrup configurations on the stress and strain were investigated, respectively, and the lateral effective stress of the different stirrups was calculated based on the cohesive-elastic ring model and modified elastic beam theory. Moreover, a damage-constitutive model of CAC considering the lateral stress was set up based on damage mechanics theory. The results indicated an increase in the stress and strain with a decrease in the stirrup spacing, and the adopted stirrup ratio had a better strengthening effect than the different concrete grades, and the variation in the deformation was restricted by the performance of coral coarse aggregate (CA). However, an increment in the lateral strain was observed with an increase in the axial strain. The lateral stress model showed a good agreement with the experimental data, and the proposed damage-constitutive model had a good correlation with the measured stress−strain curves.

Graphical abstract

Keywords

coral aggregate concrete / stress−strain curves / lateral effective stress / peak stress / axial−lateral curves / damage-constitutive model.

1 Introduction

Currently, due to the rapid growth in the demand for land, many marine structures have been intensified [1]. Concrete has been extensively used in a vast majority of marine structures. However, conveying the coarse aggregate (CA) and fine aggregates from the mainland for the construction of marine structures is costly and time-consuming. Therefore, it is important to replace the natural coarse and fine aggregate with local materials. Coral reef, a type of primitive local resource, is typically distributed around marine islands. The detrital corals are formed as a result of rain-swept and geological processes [2], and their application can be traced back to the World War II, with various studies indicating that detrital coral aggregates can substitute natural aggregates in marine structures [3–9]. Nonetheless, the use of detrital corals and seawater may introduce corrosive ions (such as Cl⁻ and {\mathrm{S}\mathrm{O}}_4^{2-}) into the concrete, which corrode the reinforcement steels and eventually cause the deterioration of marine structures. Additionally, the plastic hinge regions in marine concrete structures need to be confined to achieve sufficient ductility for resisting earthquakes. The aforementioned problems can be solved by using anti-corrosion fiber-reinforced polymer (FRP) materials (including FRP bars, tubes, and wraps) to obtain a lateral stress.

Numerous investigations have indicated that FRP materials have been applied in civil engineering for many years [10–13]. Most of the previous studies have focused on the application of different FRP materials in confined concrete [14,15], relationship between the axial and lateral strains [16], and confining stiffness and axial strength of concrete. The aforementioned documents have indicated that the confining stiffness is related to the lateral pressure, as compared with unconfined concrete, and the confinement of FRP materials can enhance the axial strength and ductility of concrete. Furthermore, due to the protection and anti-corrosion effect of FRP materials on marine structures, FRP-confined concrete is used in many marine structures. Some researchers have investigated marine concrete confined by FRP wraps, tubes, and composite materials. Moreover, sufficient axial strength models [17,18] and stress−strain models [19,20] have been conducted. Despite the anti-corrosiveness ability of FRP-confined marine concrete, the ultimate loading of FRP-confined concrete can cause brittle failure, and FRP materials have no ability of bearing axial forces [21]. Moreover, the sudden brittle failure of FRP confined coral aggregate concrete (CAC), which may cause the failure of structures. Thus, the use of ductile materials to overcome the brittle failure is important.

Compared with FRP materials, reinforcement steel bars have an evident yield point and satisfactory ability to resist sudden brittle failure. For many years, they have been the main load-bearing materials in engineering structures, but the vertical components of structures need to be refined during seismic design to resist horizontal loads. In addition, most structures are designed with sufficient lateral reinforcements in the form of circular hoops or rectangular stirrups to prevent the shear or sudden failure of core concrete [22–24]. However, the corrosion of reinforcements in marine structures cannot be ignored. To overcome the disadvantages of FRP-confined marine concrete, reinforcement steels with antirust treatment and high bearing capacity and corrosion resistance can be adopted. Therefore, their performance is quite important for marine structures.

Many studies have reported on the high capability of appropriate lateral steels in enhancing the strength and ductility of concrete, with some attempting to investigate the strength and deformation of steel-confined concrete. After comparing different stress−strain models of confined concrete determined by different researchers, Bousalem and Chikh [25] proposed an analytical model that included the above-mentioned features. The results showed an increase in the peak strain of the stress−strain curves increasing lateral stress, and the modified enhancement factor for strength could accurately predict the test results. Based on the study on the deformation behaviour of 24 confined specimens, Assa et al. [26] proposed the criterion of peak strength and finally established a fractional model for predicting the stress−strain relationship in confined specimens. Ren et al. [27] investigated the peak strength with different stirrup spacing and the constitutive model under different strain rates, respectively, based on the influence of strain rates. At last, a novel damage model was established. Nevertheless, the above-mentioned researches aimed to study the concrete cast with the same CA. To investigate the effects of different CAs, El-Dash [28] considered three types of aggregates for confined concrete, and the test results showed that the strength behaviour is related to the aggregate types, and aggregate type-induced variation in traditional parameters should be further investigated. Furthermore, considering the different stirrup shapes, Saatcioglu and Razvi [29] proposed a model to describe the strength and strain based on the equivalent pressure produced by stirrups. The results showed that the improvements in strengths and ductility were related to the configurations of stirrups.

However, the existing constitutive models of confined concrete have been developed, according to the test data. These models, in spite of showing accurate prediction in several applications, conform to the limitations in the field of oceanic concrete, sectional-configuration, stirrup types and the variation in lateral confinement. Moreover, the above-mentioned reinforced concrete neglects the corrosion protection of steel bars, and the variation in lateral restraint needs to be further studied, to establish a damage-constitutive model with the consideration of lateral stress. Therefore, a novel type of CAC confined within steels treated using antirust paint is proposed in this paper. To this end, the CAC specimens with different quantities and levels of lateral steels are tested. Subsequently, the peak and residual stress, axial-lateral strain, and feature relationships related to the damage-constitutive model are established. The proposed damage-constitutive model considering lateral stress is further verified based on the test results.

2 Material and methods

2.1 Materials

The P.O 42.5 Portland cement was used in this test. Seawater was from Beibu Gulf with the ion content shown in Tab.1. The coral CA was broken from detrital corals with the maximum diameter of 16 mm shown as Fig.1(a), while the fine aggregate consisted of coral fine sand and coral coarse sand shown as Fig.1(b) and Fig.1(c), respectively.

Furthermore, the CA and fine aggregate grading curves are shown in Fig.2(a) and Fig.2(b), respectively, which indicate that the CA sample has a gradation of 9.5–16.0 mm, and the fine aggregate sample has a fineness module of 3.0. The CA physical properties are shown in Tab.2. In addition, the diameters of the HPB300 lateral and longitudinal steel placed in the specimens are 6 and 8 mm, respectively, and the elastic module and strength are 2.05 × 10⁵ and 401 MPa, respectively.

2.2 Concrete mixture

Three mixtures (C20, C30, and C40) were designed according to the China Standard GB/T 25177-2010 [30], and additional water was added in terms of the water absorption of CA. The CAC mixtures and cubic compressive strength are shown in Tab.3, where C represents cement, R is the hydroxyl water reducer which consists of C₃H₄O₂, CH₂C(CH₃)CH₂SO₃Na and (NH₄)₂S₂O₈. SA denotes coral sand, W is seawater, AW is additional seawater, TW is the total of seawater and f_{\mathrm{c}\mathrm{u}} is cubic compressive strength of CAC.

2.3 Compressive specimens

Test specimens with different stirrups were cast for the axial compressive test with a size of 150 mm × 150 mm × 450 mm, and 3 stirrup levels and 3 concrete strength grades (C20, C30, and C40) were considered for specimens. Schematic representations of the sections are depicted in Fig.3, where JC represents the rectangular stirrup, LC is the diamond-shaped compound stirrups, YC is spiral stirrups and CC represents unconfined CAC. Moreover, regarding different stirrups, 4 stirrup spacing (40, 50, 60, and 70 mm) were designed, as shown in Fig.4, and the specific steel ratio are shown in Tab.4, where {\rho }_{\mathrm{l}} is the longitudinal steel ratio, {\rho }_{\mathrm{v}} represents the volumetric stirrup ratio. Therefore, 9 groups of unconfined specimens, and 36 groups of confined specimens were prepared. A two-part designation was proposed for the specimens, such as JC30-40, where JC, 30, and 40 represent the stirrup level, concrete strength grade, and stirrup spacing, respectively. As for unconfined specimens, such as CC40-1, CC represents the unconfined specimens, 40 is the concrete strength grade, and 1 is the specimen number. Before casting the specimens, the steels had been painted with galvanized anticorrosive paint and undergone 28 d of standard curing after casting.

2.4 Compressive test setup

After curing, all the specimens were loaded using the YE-1000F press machine shown as Fig.5(a). To avoid the abrupt spalling of concrete cover, the displacement–control was used to obtain the integral curves with the rate of 0.2 mm/min, 4 displacement meters were used to measure the axial and lateral deformation, and the strain gauges were pasted in the specimens and stirrups, shown as Fig.5(b). The concrete gauges were utilized to collect the axial and lateral strain data, while the steel gauges were used to obtain the deformation of stirrups with the increase in the axial force. All the specimens were placed to avoid the eccentricity, before applying axial loading.

3 Results and discussion

3.1 Failure pattern

Fig.6 shows the typical failure patterns of different specimens after axial loading. As shown in Fig.6(a), with the treatment of micro-cracks, a major longitudinal crack was observed on the lateral section, which indicates that the specimens without stirrups failed via splitting failure. In addition, the specimens in the presence of stirrups exhibited different failure patterns. When the axial force reached 70%–80% of the ultimate force, spalling of the concrete cover was observed, and finally the yielding of the stirrups and longitudinal steels, as shown in Fig.6(c)–Fig.6(d).

A typical non-dimensional curve, namely, {\sigma }^{\prime }−{\epsilon }^{\prime }, {\sigma }^{\prime } versus {\epsilon }^{\prime } relationship can be idealized in Fig.7(a), where the differences in the 4 loading stages are illustrated. The stage oa is the elastic stage, and the cracks began to emerge in stage AB. Moreover, macro-cracks appeared in the path of the ascending curve when the axial loading reached the peak loading (point b). Owing to the confining pressure produced by the stirrups, the axial loading gradually decreased from point b to c, as displayed in the descending stages (b−c−d) in Fig.7(a). Fig.7(b) depicts the relationship between the measured stirrup strain and the axial stress of the typical specimen YC20-40. From Fig.7(b), a slight deformation of the different stirrups is observed in the oa stage, and the stirrups yielded when the axial loading reached the peak value.

Additionally, the failure patterns of the specimens at different loading stages are illustrated in Fig.7(c). After stage ab, the spalling of concrete cover is found in stages bc and cd. Through the observation of the failure patterns, further considerations emerged. The poor strength and natural brittleness of CA resulted in a weak interlock force, which induced the restriction of the loading-bearing capacity of the specimens by the performance of CAC. Moreover, the brittleness of CAC caused the crushing of the damage plane; thus, the stirrups and longitudinal steels yielded significantly after the ultimate loading stage.

3.2 Axial stress−strain curves

Fig.8 depicts the measured stress−strain curves of different specimens, where the axial force tested is the sum of the forces produced by the confined core concrete, concrete cover, and longitudinal rebar. To calculate the effective stress of the confined core concrete \sigma, an assumption was established to ensure all the aforementioned forces are subjected to the same axial strain, and the axial forces produced by the longitudinal rebar (P_{\mathrm{h}}) and concrete cover (P^{\prime }) are subtracted from the total axial force (P), respectively. The calculation of \sigma was based on Eq. (1), where A_{\mathrm{c}\mathrm{o}} is the core area of the specimens. Moreover, the axial peak stress of the unconfined concrete in the ascending curves is defined by f_{\mathrm{c}\mathrm{c}}, which is obtained using Basset and Uzumeri’s [31] methods according to the test results of unconfined CAC.

(1)\sigma =\frac{P-P^{\prime }-P_{\mathrm{h}}}{A_{\mathrm{c}\mathrm{o}}}.

3.3 Analysis of the feature stress and strain

To systematically establish the damage-constitutive model, it is necessary to define the feature points on the stress−strain curves and confinement action by the different stirrups. Therefore, a detailed flowchart for the technical route, as shown in Fig.9, was added to reflect the specific steps used to define the above-mentioned relationship and to establish the model.

3.3.1 Peak stress

The values for the peak stress of the specimens with different series and incorporating different stirrup ratios ({\rho }_{\mathrm{v}}) are shown in Fig.10(a)–Fig.10(c). A normalized coefficient f_{\mathrm{c}\mathrm{o}}/f_{\mathrm{c}\mathrm{c}} was introduced, where f_{\mathrm{c}\mathrm{o}} denotes the peak stress of confined concrete, and f_{\mathrm{c}\mathrm{c}} represents the peak strength of unconfined specimens. Fig.10 reveals that all the specimens show an increase in f_{\mathrm{c}\mathrm{o}}/f_{\mathrm{c}\mathrm{c}} with the increase in {\rho }_{\mathrm{v}}. Under similar concrete strength conditions, LC specimens with diamond-shaped compound stirrups exhibit a more pronounced enhancement of f_{\mathrm{c}\mathrm{o}}/f_{\mathrm{c}\mathrm{c}} than the other series. Moreover, with the increase in the stirrup ratio from minimum ({\rho }_{\mathrm{v}min}) to the maximum ({\rho }_{\mathrm{v}max}), the JC, LC, and YC series showed an increase of 21.2%, 30.8%, and 16.2%, respectively. Relative to the unconfined CAC, the enhancement of the peak stresses can be attributed to the increase in the lateral steel areas. This demonstrates that the lateral reinforcement proposes sufficient confined stress, due to the fact that the equilibrium of the dilation. And the enhancement of the peak stress is related to the stirrup ratio. It can be concluded that increasing the quantities of steel results in the enlargement of the areas of lateral reinforcement, which in turn improves the confined force. Furthermore, the maximum increment f_{\mathrm{c}\mathrm{o}}/f_{\mathrm{c}\mathrm{c}} in C20 specimens is higher in contrast to C30 and C40 series. Compared with the other series, CA had a low strength, and the lateral dilation of the C20 series was more evident in the loading test. Therefore, the lateral steels are pushed by dilation of core concrete, next, the reaction confined pressure is applied to it in return. Therefore, the ductility of C20 series is enhanced, which increases the parameter f_{\mathrm{c}\mathrm{o}}/f_{\mathrm{c}\mathrm{c}} of C20 series.

With the increase in the lateral deformation, the lateral pressure produced by the stirrups also increases. According to the models proposed by Cusson and Paultre [32], the linear relationship between f_{\mathrm{c}\mathrm{o}}/f_{\mathrm{c}\mathrm{c}} and {\rho }_{\mathrm{v}} is expressed using Eq. (2), where {\phi }_1 is a correction factor calculated using f_{\mathrm{c}\mathrm{t}1}/f_{\mathrm{c}\mathrm{t}2}, where f_{\mathrm{c}\mathrm{t}1} and f_{\mathrm{c}\mathrm{t}2} are the numerical tube pressure of CA and light weight aggregate, respectively, and f_{\mathrm{y}\mathrm{h}} is the yielding strength of the lateral reinforcement. According to the test results and proposed value in [18], f_{\mathrm{c}\mathrm{t}1} and f_{\mathrm{c}\mathrm{t}2} are 3.1 and 6.2 MPa respectively, whereas the parameter a is equal to 5.11, 7.54, and 7.18 for the JC, LC and YC series, respectively. A comparison between the predicted peak stress f_{\mathrm{c}\mathrm{o}\mathrm{p}} and measured peak stress f_{\mathrm{c}\mathrm{o}\mathrm{t}} is shown as Fig.10(d).

(2)\frac{f_{\mathrm{c}\mathrm{o}}}{f_{\mathrm{c}\mathrm{c}}}=1+a{\phi }_1{\rho }_{\mathrm{v}}\sqrt{\frac{f_{\mathrm{y}\mathrm{h}}}{f_{\mathrm{c}\mathrm{c}}}}.

3.3.2 Residual stress

To comprehensively analyze the relationship between the residual stress and stirrup ratio, the relationship between the normalized coefficient f_{\mathrm{r}\mathrm{e}\mathrm{o}}/f_{\mathrm{r}\mathrm{e}\mathrm{c}} and stirrup ratio ({\rho }_{\mathrm{v}}) of specimens with different series are depicted in Fig.11(a)–Fig.11(c) respectively, where f_{\mathrm{r}\mathrm{e}\mathrm{o}} denotes the residual stress of specimens incorporating stirrups, and f_{\mathrm{r}\mathrm{e}\mathrm{c}} is the residual stress of unconfined specimens. As revealed by the comparison results of specimens with different stirrups, the residual strength shows an increasing trend with an increase in the lateral reinforcement area, while the diamond-shaped compound and spiral stirrups show a notable improvement in the residual stress with an increase in {\rho }_{\mathrm{v}}. Hence, it can be concluded that the residual stress is related to the stirrup ratio. When axial loading reaches the peak stress, the failure plane begins to form at stage bc in Fig.7(a).

Owing to the development of cracks, the lateral reinforcement is pushed, and reaction pressures are applied, the cohesion, and the friction force on the damaged plane are enhanced, which transforms the interfacial force into a residual stress. In addition, as shown in Fig.11, no relationship is evident between residual stress and strength grade because of the emergence of the failure plane and crushing of CA. At this stage, the friction stress is what determines the axial loading, while the damage at the failure plane makes it more complicated to bear axial force. Therefore, residual stress is slightly impacted by strength grade.

Moreover, based on the equation proposed by Afifi et al. [33], the relationship between the lateral stress f_{\mathrm{l}} and the stirrup ratio {\rho }_{\mathrm{v}} can be expressed using the equation f_{\mathrm{l}}=b\cdot {\rho }_{\mathrm{v}}\cdot f_{\mathrm{y}\mathrm{h}}, where f_{\mathrm{y}\mathrm{h}} signifies the yield strength of stirrups, b exhibits an undefined parameter. According to the test results of the unconfined specimens, the relationship between f_{\mathrm{r}\mathrm{e}\mathrm{c}} and f_{\mathrm{c}\mathrm{c}} is shown in Eq. (3). Further analysis points out that in the proposed model, the b values of JC, LC and YC series are 0.66, 1.39 and 2.97, respectively, as shown in Eq. (4), where f_{\mathrm{r}\mathrm{e}\mathrm{o}} is the residual stress of core concrete. A slight deviation is found between the predicted residual stress f_{\mathrm{r}\mathrm{e}\mathrm{o}\mathrm{p}} and the measured residual stress f_{\mathrm{r}\mathrm{e}\mathrm{o}\mathrm{t}}, which is shown in Fig.11(d).

(3)f_{\mathrm{r}\mathrm{e}\mathrm{c}}=0.126f_{\mathrm{c}\mathrm{c}}\text{+ 1}\text{.175},

(4)\frac{f_{\mathrm{r}\mathrm{e}\mathrm{o}}}{f_{\mathrm{r}\mathrm{e}\mathrm{c}}}=1+b{\rho }_{\mathrm{v}}\frac{f_{\mathrm{y}\mathrm{h}}}{f_{\mathrm{r}\mathrm{e}\mathrm{c}}}.

3.3.3 Axial and lateral strain

To describe the impact of lateral reinforcement on the axial deformation, the parameter {\epsilon }_{\mathrm{c}\mathrm{o}}/{\epsilon }_{\mathrm{c}\mathrm{c}} is utilised to assess lateral pressure-caused enhancement of the axial deformation behavior, where {\epsilon }_{\mathrm{c}\mathrm{o}} and {\epsilon }_{\mathrm{c}\mathrm{c}} are the peak axial strain of confined core concrete and unconfined concrete, respectively. As can be seen from Fig.12(a)–Fig.12(c), with the exception of a few points, and based on the observation of the tendency of peak strain, the normalized parameter {\epsilon }_{\mathrm{c}\mathrm{o}}/{\epsilon }_{\mathrm{c}\mathrm{c}} of the confined CAC increases slightly with the variation in the volumetric ratio of lateral reinforcement, and the axial strain increment of JC, LC and YC series are 2.5%, 4.4%, and 1.4%, respectively. The results show that the enhancement of ductility, which is related to the increase in {\rho }_{\mathrm{v}}, is responsible for the increment of axial strain. On the contrary, a few points perform an opposite trend, presenting that {\epsilon }_{\mathrm{c}\mathrm{o}}/{\epsilon }_{\mathrm{c}\mathrm{c}} decreases with the increase in {\rho }_{\mathrm{v}}. This phenomenon is concluded that the lower strength of CA makes it easier for macro-cracks to emerge and pass through the CA, which effects the interlock and friction mechanism on the failure plane. Therefore, the variation in deformation is restricted by the complicated failure plane.

Considering the aforementioned phenomenon, the relationship between {\epsilon }_{\mathrm{c}\mathrm{o}}/{\epsilon }_{\mathrm{c}\mathrm{c}} and {\rho }_{\mathrm{v}} can be expressed using Eq. (5), which was suggested by Baduge et al. [34], where f_{\mathrm{y}\mathrm{h}} is the yield strength of stirrups, and c is the representation of an undefined parameter associated with the stirrup configurations. According to the regression analysis, the fitting values of the JC, LC and YC series are 0.022, 0.049, and 0.094, respectively. Considering the interaction between core concrete and stirrups, the relationships between transverse and axial strain are of great importance to evaluate the transverse dilation of core concrete, which determines the reactive lateral stress produced by stirrups. As seen in Fig.12(d), the average non-dimensional lateral-axial curves u\left(\epsilon \right) of different specimens are plotted, where {\epsilon }_{\mathrm{l}} is the lateral strain, {\epsilon }_{\mathrm{l}\mathrm{c}} represents peak lateral strain. With the increase in axial strain, the increment of lateral stress is observed. Based on the nonlinear regression, the equations for axial-lateral strain calculation are shown in Fig.12(d), where x=\epsilon /{\epsilon }_{\mathrm{c}\mathrm{c}}, y={\epsilon }_{\mathrm{l}}/{\epsilon }_{\mathrm{l}\mathrm{c}}.

(5)\frac{{\epsilon }_{\mathrm{c}\mathrm{o}}}{{\epsilon }_{\mathrm{c}\mathrm{c}}}=1+c\frac{{\rho }_{\mathrm{v}}}{{\epsilon }_{\mathrm{c}\mathrm{c}}}\left(\frac{f_{\mathrm{y}\mathrm{h}}}{300}\right).

3.4 Determination of lateral confinement model

3.4.1 Confinement of outer concrete

The aforementioned test results reveal that the bearing capacity of the confined CAC is improved by the lateral stirrups, and the axial strength and constitutive model are effective in terms of enhancing the performance of stirrups-confined CAC. Therefore, it is essential to establish a strength model, which is determined as the sum of the different strength contributions. According to the strength model proposed by Campione et al. [35], the cover concrete, cracked confined zone and effective confined zone, are taken into consideration. In addition, considering the lateral stress contributed by concrete cover, Légeron et al. [36] suggested an equivalent circular column model as shown in Fig.13, where L₁ is the outer diameter of specimens (equals to 150 mm).

The equivalent diameter of the model was the same as that of the concrete core of different specimens, and while the confined specimens present elastic behaviour, the concrete core is predominantly restrained by cover concrete in stage oa in Fig.7(a). Based on the cohesive-elastic ring model proposed by Timoshenko et al. [37], as shown in Eqs. (6) and (7), the cover concrete-contributed restrained pressure p_{\mathrm{r}\mathrm{c}} could be obtained, where r is the distance from the centre of specimens, R_{\mathrm{b}} represents the radius of equivalent column (equals to 75 mm), and R_{\mathrm{a}} exhibits the radius of core concrete (equals to 59 mm), {\sigma }_{\mathrm{r}} is the radial stress, {\sigma }_{\theta } is the hoop stress, {\phi }_1 denotes the same parameter as that in Eq. (2), u_{\mathrm{r}} reflects the radial deformation measured by the lateral displacement meter, and {\upsilon }_{\mathrm{c}} is Poisson’s ratio of coral concrete. Considering the CAC cover is brittle and low in strength, a reduction factor {\phi }_2 (equals to 0.25) is proposed for the purpose of describing reduction of CAC elastic modules, according to the test results.

(6){\phi }_1{\sigma }_{\mathrm{r}}=\frac{p_{\mathrm{r}\mathrm{c}}{R_{\mathrm{a}}}^2}{{R_{\mathrm{b}}}^2-{R_{\mathrm{a}}}^2}\left(1-\frac{{R_{\mathrm{b}}}^2}{r^2}\right),

(7)u_{\mathrm{r}}=\frac{1+{\upsilon }_{\mathrm{c}}}{{\phi }_{2}E}r{\sigma }_{\mathrm{r}}\left[\left(1-{\upsilon }_{\mathrm{c}}\right)\frac{2r^2}{r^2-{R_{\mathrm{b}}}^2}-1\right].

3.4.2 Confinement of rectangular and diamond-shaped compound stirrups

After the oa stage, with the increase in axial force, the concrete cover spalls, and the deformation of stirrups are found. Prior to stirrups yielded, the lateral pressure should not be neglected and the increase in axial strength is mainly contributed by stirrups, while the distribution of lateral pressure is uneven. Therefore, defining the confinement of stirrups is critical. To investigate the interaction between stirrups and core concrete, based on the elastic foundation beam theory and symmetry principle, Campione [38] has proposed a model indicating that the interaction force between stirrups and core concrete could be described in Fig.14(a). The model further takes into account that the interaction force supported by the diffused springs as shown in Fig.14(b), and the stiffness of equivalent springs k is equal to 2\cdot E_{\mathrm{c}}/\left(L\cdot \left(1-{\upsilon }_{\mathrm{c}}\right)\right), where E_c is the elastic module of CAC, L represents the dimension of stirrups, and {\upsilon }_{\mathrm{c}} is the same parameter as that in Eq. (7) .

As derived from Campione [35], based on the functional analysis, a differential equation of equivalent beam on elastic foundation can be expressed as Eq. (8), where \omega denotes the deflection along the equivalent beam, \delta represents the lateral deformation of core concrete, E_s reflects the elastic module of lateral steels and the moment of inertia of stirrups could be obtained by I_{\text{s}}\text{=}\text{π}{d}^4/\phantom{\pi d^{4}64}64, where d is the diameter of stirrups (equals to 6 mm).

(8)E_{\mathrm{s}}{I}_{\text{s}}\frac{{\mathrm{d}}^4\omega }{\mathrm{d}{x}^4}+k\left(\delta -\omega \right)=0.

According to the test results of confined CAC, a simplified parameter \beta expressed as \beta =\sqrt[4]{{\displaystyle \frac{k\cdot d}{4\cdot E_{\mathrm{s}}\cdot I_{\mathrm{s}}}}}, is introduced so as to obtain the solution of Eq. (8), where the modified parameter \beta is proposed by \beta =1.5\cdot d\cdot \sqrt[4]{{\displaystyle \frac{E_{\mathrm{c}}\cdot d}{4\cdot E_{\mathrm{s}}\cdot L\cdot \left(1-{\upsilon }_{\mathrm{c}}\right)}}}, and the solution of Eq. (8) is finally described as the following equation:

(9)\omega =\delta +A\text{cosh}\beta x\cdot \text{cos}\beta x+B\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\beta x\cdot \text{sin}\beta x.

As shown in Eq. (9), the lateral deformation of core concrete \delta can be replaced by {\epsilon }_{\mathrm{l}}\cdot L/\phantom{L2}2, and {\epsilon }_{\mathrm{l}} is the lateral strain defined by the model proposed in Fig.12(d), A and B are the undefined parameters, respectively. To determine the A and B, the boundary conditions are shown as follows: 1) \omega {\left(x\right)}_{x=\frac{L}{2}}={\delta }_{\mathrm{s}}, 2) {\mathrm{d}\omega \left(x\right)/\phantom{\mathrm{d}\omega \left(x\right)\mathrm{d}x}\mathrm{d}x}_{x=\frac{L}{2}}= 0. By imposing the aforementioned boundary conditions into Eq. (9), parameters A and B could be calculated as Eqs. (10) and (11), respectively, where {\delta }_{\mathrm{s}} is the elongation of single stirrups.

(10)\begin{array}{rl}A=& -2\left({\delta }_{\mathrm{s}}-\delta \right)\\ & \cdot \frac{\sinh \left({\displaystyle \frac{\beta L}{2}}\right)\cdot \cos \left({\displaystyle \frac{\beta L}{2}}\right)+\cosh \left({\displaystyle \frac{\beta L}{2}}\right)\cdot \sin \left({\displaystyle \frac{\beta L}{2}}\right)}{\sinh \left(\beta L\right)+\sin \left(\beta L\right)},\end{array}

(11)\begin{array}{rl}B=& 2\left({\delta }_{\mathrm{s}}-\delta \right)\\ & \cdot \frac{\sinh \left({\displaystyle \frac{\beta L}{2}}\right)\cdot \cos \left({\displaystyle \frac{\beta L}{2}}\right)-\cosh \left({\displaystyle \frac{\beta L}{2}}\right)\cdot \sin \left({\displaystyle \frac{\beta L}{2}}\right)}{\sinh \left(\beta L\right)+\sin \left(\beta L\right)}.\end{array}

According to the conclusion by Mander et al. [39], the confinement pressure which induced by the deformation coordination of stirrups and concrete core is uneven, Campione et al. [35] assumed that the stirrups-contributed confinement stress conformed to the expression: q(x)=k\cdot (\delta -\omega \left(x\right)). Furthermore, to investigate the effective lateral pressure caused by stirrups, the equivalent uniform stress \overline{q} is introduced and defined as Eq. (12). Based on the principle of mechanical equilibrium, the confinement force of a single lateral stirrup bar F could be expressed as Eq. (13). Because of the deformation of the longitudinal steel bar, it is observed that the lateral pressure between two stirrups is discontinuous, therefore, a reduction factor k_{\mathrm{s}} is introduced, while s is the distance between two stirrups, and L represents the side length of the cross-section.

(12)\overline{q}=\frac{1}{L}\underset{-L/2}{\overset{L/2}{\int }}k\cdot \left(\delta -\omega \right)\mathrm{d}x,

(13)\text{2}\cdot F=L\cdot s\cdot \overline{q}\cdot k_{\mathrm{s}}.

To determine the parameter k_{\mathrm{s}}, Braga et al. [40] indicated that k_s is related to the longitudinal bar deflection and lateral dilation of CAC. As shown in Fig.15, where {\delta }_1 represents the lateral deformation of concrete core concrete (equals to {\upsilon }_{\mathrm{c}}\epsilon L), and {\upsilon }_{\mathrm{c}} is Poisson’s ratio of CAC.

From which, by using the micro-geometric relation in Fig.15, it could be assumed that the micro-segment of longitudinal bar is \mathrm{\Delta }l, therefore, the relationship between the micro-deformation angle \theta and \mathrm{\Delta }l is produced by {\upsilon }_{\mathrm{c}}\epsilon \mathrm{\Delta }l/\phantom{{\upsilon }_c\epsilon \mathrm{\Delta }l\mathrm{\Delta }l}\mathrm{\Delta }l\text{=}\tan \theta \text{=}\sin \theta according to the first-order Taylor series expansion. As outlined in Ref. [40], the longitudinal steel micro-deflection is assumed to be the form of {\delta }_3\text{=}{\displaystyle \frac{s}{2}}\cdot \sin \theta, and the expression of the parameter k_{\mathrm{s}} is obtained using Eq. (14).

(14)k_{\mathrm{s}}=\frac{{\delta }_1-{\delta }_3}{{\delta }_1}=\frac{\epsilon {\upsilon }_{\mathrm{c}}L-{\displaystyle \frac{s}{2}}{\upsilon }_{\mathrm{c}}\epsilon }{\epsilon {\upsilon }_{\mathrm{c}}L}.

The tensile force F in the stirrups is determined after substituting Eqs. (9)–(12) and Eq. (14) into Eq. (13), respectively. While F is related to {\delta }_{\mathrm{s}} expressed by F=2\cdot E_{\mathrm{s}}\cdot A_{\mathrm{s}\mathrm{t}}\cdot {\delta }_{\mathrm{s}}/\phantom{2\cdot E_{\mathrm{s}}\cdot A_{\mathrm{s}\mathrm{t}}\cdot {\delta }_{\mathrm{s}}L}L, where A_{\mathrm{s}\mathrm{t}} is the sectional area of stirrups. After utilizing Eq. (13) through analyzing the aforementioned relation including Eqs. (8)–(12) and Eq. (14), F is expressed by Eq. (15), where parameter A′ and B′ are calculated by A/[-2\left({\delta }_{\mathrm{s}}-\delta \right)] and B/2\left({\delta }_{\mathrm{s}}-\delta \right), parameters C and D are equal to \cosh \left({\displaystyle \frac{\beta L}{2}}\right)\cdot \sin \left({\displaystyle \frac{\beta L}{2}}\right) and \cos \left({\displaystyle \frac{\beta L}{2}}\right)\cdot \sinh \left({\displaystyle \frac{\beta L}{2}}\right), respectively. In addition, u\left(\epsilon \right) signifies the axial-lateral strain relationship defined by the proposed model in Fig.12(d).

(15)F=\frac{k/\beta \cdot \left[\left(A^{\prime }-B^{\prime }\right)\cdot C+\left(A^{\prime }+B^{\prime }\right)\cdot D\right]\cdot k_{\mathrm{s}}}{\left[1-\left(A^{\prime }-B^{\prime }\right)\cdot \left(C+D\right)\cdot k_{\mathrm{s}}/E_{\mathrm{s}}\cdot A_{\mathrm{s}\mathrm{t}}\right]}\cdot u\left(\epsilon \right)\cdot L.

From Eq. (15), F increases with the increase in the axial strain \epsilon before reaching the yield value, which is expressed as \text{F}=f_{\mathrm{y}}\cdot A_{\mathrm{s}\mathrm{t}}, where f_{\mathrm{y}} is the stress in stirrups, and A_{\mathrm{s}\mathrm{t}} is the area of lateral reinforcement. In addition, Fig.16(a) and Fig.16(b) depict the lateral stress model of the JC and LC specimens, respectively. Mander et al. [39] proposed a confinement pressure model and indicated that the ultimate lateral stress f_{\mathrm{l}\mathrm{e}1} is proportional to the yield strength of reinforcement f_{\mathrm{y}\mathrm{h}}, as shown in Eq. (16).

(16)f_{\mathrm{l}\mathrm{e}1}=k_{\mathrm{e}1}\frac{2\cdot f_{\mathrm{y}\mathrm{h}}\cdot A_{\mathrm{s}\mathrm{t}}}{s\cdot L}.

For JC specimens, the confinement effectiveness coefficient k_{\mathrm{e}1} in Eq. (16) is expressed by Eq. (17), where {\rho }_{\mathrm{l}} is the volumetric ratio of longitudinal bar, c_i is the distance of two adjacent longitudinal steel, {\theta }_1 is the parabola angle, and a_1 is a modified parameter. According to the analysis of test results, considering the brittleness of CA, the values of {\theta }_1=\text{π}/6 and a_1\text{=}12 are proposed in this paper.

(17)k_{\text{e1}}=\frac{\left[1-\sum _{i=1}^n{\displaystyle \frac{{(c_i)}^2}{a_1{L}^2}}\right]{\left(1-0.5s\tan {\theta }_1/L\right)}^2}{1-{\rho }_{\mathrm{l}}}.

Based on the force equilibrium and the model depicted in Fig.16(b), the effective lateral pressure of LC series f_{\mathrm{l}\mathrm{e}2} is calculated by Eq. (18), where k_{\mathrm{e}2} is the LC specimens confinement effectiveness coefficient that is expressed by Eq. (17), c_i is equal to 53 mm.

(18)f_{\mathrm{l}\mathrm{e}2}=k_{\mathrm{e}2}\frac{2\cdot f_{\mathrm{y}\mathrm{h}}\cdot A_{\mathrm{s}\mathrm{t}}+{\displaystyle \frac{2\sqrt{2}}{2}}\cdot f_{\mathrm{y}\mathrm{h}}\cdot A_{\mathrm{s}\mathrm{t}}}{L\cdot s}.

As shown in Eq. (19), a parameter \lambda is introduced to simplify the confinement pressure with diamond-shaped compound section to that of the rectangular section, according to the lateral stress equilibrium of the stirrups.

(19)\lambda =\frac{k_{\mathrm{e}2}}{k_{\mathrm{e}1}}\left(1+\frac{\sqrt{2}}{2}\right).

Most importantly, by substituting f_{\mathrm{y}\mathrm{h}}\cdot A_{\mathrm{s}\mathrm{t}} with F in Eq. (16), before expressing the elastic-plastic performance of stirrups, the effective lateral stress of rectangular stirrups f_{\mathrm{l}\mathrm{e}1} is expressed by Eq. (20). As for diamond-shaped compound stirrups, f_{\mathrm{l}\mathrm{e}2} can be obtained by replacing k_{\mathrm{e}1} with \lambda k_{\mathrm{e}1}.

(20)f_{\mathrm{l}\mathrm{e}1}=k_{\mathrm{e}1}\frac{2k/\beta \cdot \left[\left(A^{\prime }-B^{\prime }\right)\cdot C+\left(A^{\prime }+B^{\prime }\right)\cdot D\right]\cdot k_{\mathrm{s}}}{s\cdot L\cdot \left[1-\left(A^{\prime }-B^{\prime }\right)\cdot \left(C+D\right)\cdot k_{\mathrm{s}}/E_{\mathrm{s}}\cdot A_{\mathrm{s}\mathrm{t}}\right]}\cdot u\left(\epsilon \right)\cdot L.

3.4.3 Confinement of spiral stirrups

Owing to the difference in stress mechanism between rectangular and spiral stirrups, modifying the confinement effectiveness coefficient and the lateral stress is crucial for considering the impacts caused by spiral stirrups. Fig.17(a) depicts the effective confinement model of spiral stirrups, and the mechanical model of stirrup steels is shown in Fig.17(b). West et al. [41] indicated that the variation in spiral space could be observed by {\text{A}}_{\mathrm{e}}, where {\alpha }_1 is the angle of the horizontal plane. The radius of the cross-section r\left({\alpha }_1\right) decreases with the increase in s, as shown in Eq. (21), where a_2 represents the modified parameter (equals to 0.8), based on the results of the confined CAC. Moreover, the confinement effective parameter of the spiral stirrups k_{\mathrm{e}3} is based on the concrete core area A_{\mathrm{e}}, as shown in Eq. (22).

(21)r\left({\alpha }_1\right)=r_{\mathrm{s}}-a_2\frac{\varpi \left({\alpha }_1\right)}{4}=r_{\mathrm{s}}-a_2\frac{s}{4\text{π}}{\alpha }_1,

(22)\begin{array}{rl}{A}_{\mathrm{e}}=& 2\cdot \frac{1}{2}\underset{0}{\overset{\text{π}}{\int }}{\left(r_{\mathrm{s}}-a_2\frac{s}{4\text{π}}{\alpha }_1\right)}^2\text{d}{\alpha }_1,\\ =& r_s^2\text{π}-\frac{1}{4\text{π}}{a}_2\cdot r_{\mathrm{s}}\cdot s\cdot {\text{π}}^2+a_2\cdot \frac{s^2{\text{π}}^2}{48}.\end{array}

The model of spiral stirrups is simplified by Mander et al. [39], in which k_{\mathrm{e}3} is expressed by Eq. (23), where A_{\mathrm{c}\mathrm{o}} is the core concrete without considering the vertical bar effect.

(23)k_{\mathrm{e}3}=\frac{A_{\mathrm{e}}}{A_{\mathrm{c}\mathrm{o}}}=\frac{r_{\mathrm{s}}^2\text{π}-{\displaystyle \frac{1}{4\text{π}}}{a}_2\cdot r_{\mathrm{s}}\cdot s\cdot {\text{π}}^2+a_2\cdot {\displaystyle \frac{s^2\text{π}}{48}}}{\text{π}{r}_{\mathrm{s}}^2\cdot \left(1-{\rho }_{\mathrm{l}}\right)}.

The effective confinement pressure {\delta }_{\mathrm{c}} is initially calculated by N as shown in Eq. (24), where r_{\mathrm{s}} is equal to L/2, and the correlation among the effective confinement pressure f_{\mathrm{l}\mathrm{e}3} and \epsilon could be determined according to the models depicted in Fig.17(a) and Fig.17(b), respectively. Assuming that the deflection along the spiral stirrups is {\delta }_{\mathrm{l}\mathrm{e}}, and the dilation of core concrete is {\delta }_{\mathrm{c}}, thus the effective deformation between lateral reinforcement and concrete can be expressed as {\delta }_{\mathrm{s}}={\delta }_{\mathrm{c}}-{\delta }_{\mathrm{l}\mathrm{e}} [42], where {\delta }_{\mathrm{s}}={\displaystyle \frac{r_{\mathrm{s}}\cdot N}{E_{\mathrm{s}}\cdot A_{\mathrm{s}}\cdot \cos {\alpha }_2}}, {\delta }_{\mathrm{l}\mathrm{e}}={\displaystyle \frac{f_{\mathrm{l}\mathrm{e}3}\cdot r_{\mathrm{s}}\cdot \left(1-{\upsilon }_{\mathrm{c}}\right)}{E_{\mathrm{c}}}}, {\delta }_{\mathrm{c}}={\upsilon }_{\mathrm{c}}\cdot r_{\mathrm{s}}\cdot \epsilon, {\alpha }_2 is determined by {\alpha }_2=\text{arctan}2s/\phantom{2sL}L. This paper substitutes the aforementioned relationships into Eq. (24), f_{\mathrm{l}\mathrm{e}3} is finally expressed by Eq. (25), where {\upsilon }_{\mathrm{c}} is Poisson’s ratio of CAC (equals to 0.19), d stands for the diameter of stirrups, E_{\mathrm{s}} and E_{\mathrm{c}} are the elastic module of steels and CAC, respectively.

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

(25)f_{\mathrm{l}\mathrm{e}3}=k_{\mathrm{e}3}\cdot \frac{{\upsilon }_{\mathrm{c}}\cdot \epsilon }{{\displaystyle \frac{1}{\cos {\alpha }_2}}\cdot {\displaystyle \frac{4}{E_{\mathrm{s}}\cdot \text{π}\cdot d^2}}+{\displaystyle \frac{(1-{\upsilon }_{\mathrm{c}})}{s\cdot r_{\mathrm{s}}\cdot E_{\mathrm{c}}}}}\cdot {\displaystyle \frac{1}{r_{\mathrm{s}}\cdot s}}.

3.5 Lateral stress−axial strain relationship

Based on the proposed confinement pressure model of JC, LC and YC series, the lateral stress produced by stirrups is related to the axial strain \epsilon. As for the core concrete, it is assumed that the lateral stress supplied by cover concrete or stirrups is {\sigma }_3, as shown in Eqs. (6), (7) and Fig.7(a). {\sigma }_3 can be represented by p_{\mathrm{r}\mathrm{c}} in stage oa, with the further increase in axial loading, which is equal to f_{\mathrm{l}\mathrm{e}1}, f_{\mathrm{l}\mathrm{e}2} and f_{\mathrm{l}\mathrm{e}3} for JC, LC and YC series, respectively, at inelastic stage in Fig.7(a). Moreover, the variation in {\sigma }_3 is shown in Fig.18. The typical non-dimensional relationship between {\sigma }_3/{\sigma }_{3\mathrm{p}} and \epsilon /{\epsilon }_{\mathrm{c}\mathrm{o}} is plotted in Fig.18(a), where {\sigma }_{3\mathrm{p}} is the peak lateral stress, and {\epsilon }_{\mathrm{c}\mathrm{o}} is the same parameter as that in Eq. (5), and x and y represent \epsilon /{\epsilon }_{\mathrm{c}\mathrm{o}} and {\sigma }_3/{\sigma }_{3\mathrm{p}}, respectively. As revealed in Fig.18(a), the growth rate of {\sigma }_3/{\sigma }_{3\mathrm{p}} increases with the increase in \epsilon /{\epsilon }_{\mathrm{c}\mathrm{o}}, and the fitting curves are shown in Fig.18(a). This indicates that the test curves can be predicted well by the proposed model. For further assessing the accuracy of the newly proposed models of lateral stress, the predicted results along with the measured results based on Mander’s [39] methods are plotted in Fig.18(b), where {\sigma }_{3\mathrm{p}\mathrm{p}} is the predicted stress and {\sigma }_{3\mathrm{p}\mathrm{t}} is the measured stress. From all the above, the plotted points indicate that the measured results are consistent with the predicted results.

3.6 Compressive damage-constitutive model

For investigating the compressive response of the confined CAC, a compressive damage-constitutive model considering the confinement stress is required. According to the concrete damage mechanics (CDM) and the model proposed by Cui et al. [43], the axial damage-constitutive model can be expanded to the triaxial compressive state. As shown in Eq. (26), where E is defined as the secant axial strain at the 0.4 time of the peak stress, {\upsilon }_{\mathrm{c}} is the Poisson’s ratio, whereas D is the damage parameter. For the confined concrete, the lateral stress in the lateral directions is similar ({\sigma }_2 = {\sigma }_3) [41], and the relationships between {\sigma }_3 and \epsilon aforementioned can be arranged to express the variation in {\sigma }_3 in Eq. (26).

(26){\sigma }_1=E\cdot \left(1-D\right)\cdot \epsilon \text{+}{\upsilon }_c\left({\sigma }_2\text{+}{\sigma }_3\right).

According to the results from Cui et al. [43], it could be noted that the distribution of micro-damage in concrete conforms to Weibull distribution expressed by Eq. (27), where F^{\prime } is the damage parameter of micro-element, D is equal to 0 at the elastic path in Fig.6(a), m and F_0^{\prime } represent the shape parameter of the damage evolution curves, respectively.

(27)D=1-\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left(\frac{F^{\prime }}{F_0^{\prime }}\right)}^m\right].

By substituting Eq. (27) into Eq. (26), the damage-constitutive model of CAC with the consideration of lateral stress is expressed by Eq. (28).

(28){\sigma }_1=E\cdot \exp \left[-{\left(\frac{F^{\prime }}{F_0^{\prime }}\right)}^m\right]\cdot \epsilon +{\upsilon }_{\mathrm{c}}\left({\sigma }_2+{\sigma }_3\right).

The expression of parameter F^{\prime } is obtained by F^{\prime }=E\epsilon f\left(\sigma \right)/\phantom{E\epsilon f\left(\sigma \right)\left({\sigma }_1-2{\upsilon }_c{\sigma }_3\right)}\left({\sigma }_1-2{\upsilon }_{\mathrm{c}}{\sigma }_3\right) [43], where f\left(\sigma \right) can be defined by the failure criterion of CAC supported by Zhou et al. [44] as shown in Eq. (29), where \beta =1.084, {\alpha }_1=0.0952, {\alpha }_2=0.9422, {\alpha }_3=0.1605, {\rho }_{\mathrm{c}} and \xi are the coordinates proposed by William and Warnke [45], which are expressed by Eqs. (30) and (31), respectively.

(29)f\left(\sigma \right)=\frac{{\rho }_{\mathrm{c}}}{f_{\mathrm{c}}}+{\alpha }_2\beta \left(\frac{\xi }{f_{\mathrm{c}}}\right)+{\alpha }_3\beta {\left(\frac{\xi }{f_{\mathrm{c}}}\right)}^2={\alpha }_1\beta ,

(30){\rho }_{\mathrm{c}}=\frac{1}{\sqrt{3}}\sqrt{{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2},

(31)\xi =\frac{1}{\sqrt{3}}\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right).

Setting Eq. (29) into the parameter F^{\prime }, and substituting \epsilon by the peak strain {\epsilon }_{\mathrm{c}\mathrm{o}}, the damage parameter at the peak point F_{\mathrm{p}}^{\prime } could be obtained by Eq. (32).

(32)F_{\mathrm{p}}^{\prime }=\frac{E{\epsilon }_{\mathrm{c}\mathrm{o}}\left[{\displaystyle \frac{{\rho }_{\mathrm{c}}}{f_{\mathrm{c}}}}+{\alpha }_2\beta \left({\displaystyle \frac{\xi }{f_{\mathrm{c}}}}\right)+{\alpha }_3\beta {\left({\displaystyle \frac{\xi }{f_{\mathrm{c}}}}\right)}^2\right]}{{\sigma }_{1\mathrm{p}}-{\sigma }_{3\mathrm{p}}-\left(2{\upsilon }_{\mathrm{c}}-1\right){\sigma }_3}.

Furthermore, based on the stress−strain model of concrete, the following are the obtained boundary conditions: (1) \epsilon =0, {\sigma }_1=0 and (2) \epsilon ={\epsilon }_{\mathrm{c}\mathrm{o}}, \mathrm{\partial }{\sigma }_1/\phantom{\mathrm{\partial }{\sigma }_1\mathrm{\partial }\epsilon }\mathrm{\partial }\epsilon =0, in which the boundaries (1) and (2) are substituted into Eq. (28), and the relationship between m, F_0^{\prime } and F_{\mathrm{p}}^{\prime } could be expressed as Eqs. (33) to (34), where {\sigma }_{1\mathrm{p}} and {\sigma }_{3\mathrm{p}} represent the axial and lateral peak stress, respectively, in Eq. (34).

(33)m=\frac{1}{\ln {\displaystyle \frac{{\sigma }_{1\mathrm{p}}-{\sigma }_{3\mathrm{p}}-\left(2{\upsilon }_{\mathrm{c}}-1\right){\sigma }_3+E{\epsilon }_{\mathrm{c}\mathrm{o}}}{E{\epsilon }_{\mathrm{c}\mathrm{o}}}}},

(34){\left(F_0^{\prime }/\phantom{F_0^{\prime }{F}_{\mathrm{p}}^{\prime }}{F}_{\mathrm{p}}^{\prime }\right)}^m={\left(-\ln \frac{{\sigma }_{1\mathrm{p}}-{\sigma }_{3\mathrm{p}}-\left(2{\upsilon }_{\mathrm{c}}-1\right){\sigma }_3+E{\epsilon }_{\mathrm{c}\mathrm{o}}}{E{\epsilon }_{\mathrm{c}\mathrm{o}}}\right)}^{-1}.

By substituting Eqs. (30) and (31) into Eq. (29), the relationship between {\sigma }_1 and {\sigma }_3 is expressed by Eq. (35) according to Eq. (29), and finally Eqs. (33) and (34) are related to {\sigma }_3, which is expressed by the lateral stress−axial strain relationships mentioned above, where f_{\mathrm{c}\mathrm{c}} is the strength of CAC.

(35)\frac{{\sigma }_1}{f_{\mathrm{c}\mathrm{c}}}=\left(4.11\cdot \sqrt{1+4.33\frac{{\sigma }_3}{f_{\mathrm{c}\mathrm{c}}}}-3.40\right)-\frac{2{\sigma }_3}{f_{\mathrm{c}\mathrm{c}}}.

As for the descending branch, considering the residual confinement of stirrups, Baduge et al. [34] established a descending branch model as shown in Eq. (36), where k^{\prime } is an unconfined parameter calculated according to the test results, other parameters are consistent with Sections 3.3.1 and 3.3.2.

(36){\sigma }_1=\left(f_{\mathrm{c}\mathrm{o}}-f_{\mathrm{r}\mathrm{e}\mathrm{o}}\right)\cdot {\mathrm{e}}^{-k^{\prime }\left(\epsilon -{\epsilon }_{\mathrm{c}\mathrm{o}}\right)}+f_{\mathrm{r}\mathrm{e}\mathrm{o}}.

Based on the descending branch model as shown in Eq. (36), the damage-constitutive model of the confined CAC considering lateral stress is shown in Eq. (37). The proposed model is verified by the tested results as shown in Fig.19.

(37){\sigma }_1=\{\begin{array}{rl}& E\epsilon +{\upsilon }_{\mathrm{c}}\left({\sigma }_2+{\sigma }_3\right),{\sigma }_1⩽0.5f_{\mathrm{c}\mathrm{o}},\\ & E\exp \left[-{\left(\frac{F^{\prime }}{F_0^{\prime }}\right)}^m\right]\epsilon +\upsilon \left({\sigma }_2+{\sigma }_3\right),\\ & 0.5f_{\mathrm{c}\mathrm{o}}⩽{\sigma }_1⩽f_{\mathrm{c}\mathrm{o}},\epsilon ⩽{\epsilon }_{\mathrm{c}\mathrm{o}},\\ & \left(f_{\mathrm{c}\mathrm{o}}-f_{\mathrm{r}\mathrm{e}\mathrm{o}}\right){\mathrm{e}}^{-k^{\prime }\left(\epsilon -{\epsilon }_{\mathrm{c}\mathrm{o}}\right)}+f_{\mathrm{r}\mathrm{e}\mathrm{o}},\epsilon ⩾{\epsilon }_{\mathrm{c}\mathrm{o}},\end{array}

For comparing the damage-constitutive parameters of specimens with different stirrups distance s. Two non-dimensional parameters m/m_1 and F_0^{\prime }/F_{01}^{\prime } are introduced, respectively, where m_1 and F_{01}^{\prime } represent the ascending parameters of the specimens with the stirrups distance s = 40 mm, respectively. The variations between the aforementioned parameters and the coefficient of stirrups distance s/L₁ are shown in Fig.20(a)–Fig.20(b), where L₁ represents the outer diameter of the specimens (equals to 150 mm). As depicted in Fig.20, the increase in parameters can be observed with the increase in stirrup distance. Moreover, the increase in m/m_1 indicates that the increase in stirrup ratio decreases the brittleness of confined core concrete, while the brittleness of CA at the initial damage of coral concrete affects the enhancement of the strength of core concrete. Moreover, the parameter F_0^{\prime }/F_{01}^{\prime } represents that the shape of ascending curves is affected by s. In addition, the non-dimensional parameter k^{\prime }/k_1^{\prime } versus s/L₁ of the descending curves are plotted in Fig.20(c), where k_1^{\prime } is the parameter depending on the results of the specimens incorporating a stirrup distance of 40 mm in diameter. k^{\prime }/k_1^{\prime } increases with the increase in s/L₁, indicating that stirrups distance affects the shape of the descending curves. Taken together, it can be concluded that the friction force of failure plane enhanced by lateral reinforcement, which affects the exponential decay of the descending branch in Eq. (37).

4 Conclusions

In this study, forty-five coral concrete specimens were cast to research their axial compressive properties, and the following conclusions were drawn.

(1) The volumetric stirrup ratio increases the peak stress, residual stress, and peak strain for confined coral concrete. No evident relationship was found between the strength grade and peak strain. Under similar strength conditions, an evident enhancement in the above-mentioned strength and deformation were observed in specimens with a higher stirrup ratio. In addition, when compared with the rectangular stirrups, the diamond-shaped compound and spiral stirrups had a higher confinement of the concrete core.

(2) The further increment in the residual stress and peak strain were restricted by the performance of the coral CA, whereas the proposed strength and deformation models produced satisfactory test results. Owing to the increase in the lateral stress, there was a cubic function relationship between the lateral and axial strains, which were well predicted by the proposed models.

(3) The effective lateral stress increases with an increase in the axial strain, and the lateral stress at the elastic stage can be precisely calculated using a modified cohesive-elastic ring model. Before the yielding of the stirrup steels, the new elastic beam model could predict the variation in the lateral stress under the consideration of the effective confinement parameter.

(4) Based on the Weibull damage distribution theory and failure criterion of coral concrete, a novel damage-constitutive model was established to predict the stress−strain relationship in confined coral concrete subjected to axial loading, and the results were in good agreement with those of the measured stress−strain curves.

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