Long-term creep behavior of expansive agent core concrete in full-scale concrete-filled steel tube from the world’s largest span arch bridge study

Abstract

The Tian’e Longtan Bridge, currently under construction and boasting the world’s largest span arch at 600 m, employs a concrete-filled steel tube (CFST) as its primary structural component, forming the stiffness skeleton upon which an outer reinforced concrete arch ring is constructed. As the internal defects of CFST, once encased by outer concrete, cannot be remedied, it becomes imperative to prevent long-term debonding of the core concrete. In recent years, expansion agents have been extensively utilized in CFST arch bridge engineering to compensate for early autogenous and thermal shrinkage. However, a comprehensive comprehension of the creep behavior of core concrete expansion under extended steel tube confinement remains elusive. To address this concern, the radial expansion process of the core concrete can be segmented into two stages: debonding and restriction. We derive a long-term deformation model for the radial expansion of core concrete during the restriction stage, based on elastic mechanics and linear creep mechanics. We represent the expansion process of these two stages uniformly using piecewise functions. Subsequently, in conjunction with the ongoing construction of the Tian’e Longtan Bridge, we measure the core concrete’s behavior in full-size CFST specimens (Φ0.92 m × 12 m × 0.01 m) both with and without expansion agents for up to 114 d. This validates the practicality of the creep model and enables us to determine its relevant parameters. Our results reveal that a debonding gap of 0.142 m occurred before the initial setting of the core concrete. The core concrete underwent a radial expansion of 290.1 × 10⁻⁶, with 158.3 × 10⁻⁶ being used to address early debonding, and the remaining 131.8 × 10⁻⁶ generating self-stress on the steel tube. The creep model indicates that radial creep of the core concrete persisted for approximately six months under the hoop limitation of the steel tube, resulting in a residual expansion deformation of 26.4 × 10⁻⁶ and a residual self-stress of 0.119 MPa. Additionally, axial deformation results of CFST without expansion agents demonstrated a decreasing constraining force of the steel tube on the core concrete from the outer to the inner sections, attributable to local core concrete yield. Conversely, the inclusion of an expansion agent altered the stress state of the core concrete, maintaining consistent constraining forces within the same section. As a result, we derive an axial long-term expansion model for the core concrete under steel tube restriction. Finally, the introduction of laboratory specimen deformations enhances the practicality of our model, with results demonstrating strong alignment between measured data and the model. The experimental findings and theoretical models developed provide critical support for quantifying the expansion behavior of CaO and MgO-based compound expansive agents in the vault of CFST arch bridges.

Graphical abstract

Keywords

CFST / autogenous shrinkage / restriction / expansion agent / principle of creep superposition

1 Introduction

Concrete-filled steel tubes (CFST) have garnered favor within the engineering community due to their distinctive load-bearing capabilities [1], particularly in the domain of arch bridges [2–4]. In recent years, China has witnessed the construction of CFST arch bridges with spans exceeding 300 m, as depicted in Tab.1. Capitalizing on the superb compressive strength and streamlined construction attributes of CFST, the development of 300 m span CFST arch bridges commenced as early as 2000. By 2005, a significant milestone was reached with a span of 460 m, and this achievement was further extended to 530 m within the following eight years. As arch bridge spans continue to expand, so too does the complexity of their construction, placing heightened demands on engineering expertise. Over the subsequent seven years, during the construction of the Pingnan Third Bridge, the span of CFST arch bridges escalated to 575 m. Presently, with the ongoing construction of the Tian’e Longtan Bridge, the anticipated span of CFST arch bridges is poised to breach the 600 m threshold [4]. The rugged terrain of western China necessitates CFST arch bridges with elevated rigidity and outstanding durability. However, catering to the demands of these challenging environments remains a formidable task, especially for 600-m-class span bridges. Consequently, the advancement of CFST arch bridge technology necessitates a commensurate evolution in academic research [5].

The development of CFST arch bridge spans is challenged by a significant hazard known as CFST debonding, which severely compromises the bearing capacity of the structure and poses potential safety risks [6–10]. This debonding issue tends to accumulate in the arch crown due to the horizontal orientation of the arch crown and the inability to vibrate the core concrete of closed CFST during construction [11]. Additionally, during the plastic stage, air bubbles trapped within the core concrete rise and accumulate at the top, exacerbating the debonding of the arch crown due to both core concrete bleeding and their presence [7]. In the subsequent hardening and service stage of CFST, the sharp shrinkage of high-strength core concrete [12] and the disparity in Poisson’s ratio between the two materials [8] may also contribute to debonding.

To mitigate CFST debonding, scholars have proposed various solutions, including the incorporation of expansion agents [13–22], internal curing materials [16], and fibers [14,16,19,20,23] to facilitate and compensate for shrinkage, respectively. A CFST extrusion test conducted by Lu et al. [14] under varying expansion agent content revealed that self-stress presence led to an average 42.7% increase in bond strength. According to Chang et al.’s study [15], self-stress can enhance bond strength by 1.2 to 3.3 times. Extensive research has also focused on exploring the ultimate bearing capacity of CFST. Shen et al. [16] studied the synergistic effect of internal curing materials and expansion agents to prepare core concrete with high residual expansion and exceptional compressive strength, suitable for CFST applications. Huang et al.’s research [17] demonstrated that incorporating an expansion agent significantly enhanced the axial compressive properties of CFST, leading to yield hardening. Ho et al. [18] showed that adding a 20% expansion agent enhanced the elastic modulus, yield strength, and ultimate strength of CFST. Li et al. [19] investigated the axial compressive properties of calcium sulfoaluminate cement and ordinary Portland cement CFST, finding that the presence of self-stress significantly restrained axial shortening while enhancing bearing capacity by 9.8% to 27.6%. Xu et al. [24] studied the axial compressive deformation of CFST specimens and observed that varying expansion agent content (0%–10%) resulted in self-stress-induced co-deformation of early core concrete under lateral confinement, leading to significant load sharing by the core concrete. These studies have primarily focused on investigating the mechanical characteristics of self-stressed CFST subjected to expansion agents, making the correlation between self-induced stress and the adhesive strength/load-bearing capacity of CFST clearer, and recognizing the advantageous effects of expansion agents on CFST.

However, limited research currently exists on the volume deformation of core concrete in CFST, and the investigation into the expansion behavior of expansion agents within the steel tube is still in its early stages. Li et al. [25] incorporated CaO and MgO compound expansion agents into vertically positioned CFSTs and investigated their radial and axial free and constrained deformations of the steel tube. The findings indicate that despite the reduction in expansion value due to the steel tube constraint, the calcium-magnesium composite expansion agent still offers sufficient expansion capacity to maintain the integrity of CFST. In horizontal sections with a high potential for debonding in CFST arch bridges, the predominant form of debonding is mostly spherical crown, which differs from vertically placed specimens where circumferential debonding occurs more frequently. Therefore, the current research on vertical specimen expansion development is insufficient to serve as a reference for debonding treatment of the arch crown. Investigating the contraction compensation and debonding relief effects of expansion agents in horizontal CFST structures is imperative. The expansion agent can only induce significant overall expansion in the concrete after it has attained sufficient strength and elastic modulus [26,27]. Prior to this, the core concrete has already detached from the steel tube in the upper section due to plastic shrinkage or bleeding. Before it can generate self-stress on the steel tube, the void needs to be filled with core concrete. The current calculation of CFST compensation shrinkage fails to account for this factor, resulting in the expansion agent often meeting laboratory requirements but exhibiting poor performance in actual applications. Furthermore, the utilization of expansion agents to mitigate core concrete shrinkage has been acknowledged in a significant number of CFST arch bridge projects [5,28,29]. During the prolonged construction period of CFST arch bridges, after the completion of concrete pouring in the tube, it is also imperative to install the hanger and bridge slab before opening it to traffic, a process that typically takes approximately six months to one year. The stability of the expansion development of core concrete under long-term constraints and the maintenance of self-stress are crucial considerations in practical projects. While the creep behavior of CFST under axial load has been extensively investigated in numerous studies [30–36], there is currently no available literature on the long-term creep development of self-stress in the core concrete when subjected to steel tube confinement. Quantitatively characterizing the long-term expansion behavior of the expansion agent in CFST arch bridges under the constraint of steel tubes warrants further investigation to determine whether self-stress will gradually subside or remain stable for an extended period.

To address these gaps in research, this paper derives a creep model for the expansion development of core concrete under constrained conditions based on principles of elastic mechanics and linear creep mechanics. Subsequently, it conducts long-term monitoring of 114 d to observe the deformation of full-size CFST core concrete with horizontally placed undoped and doped expansion agents, revealing the chronological development of concrete expansion within the tube and validating the creep model. In this research process, the paper aims to achieve four primary research objectives: 1) explore the debonding gap caused by plastic shrinkage or air entrapment in the plastic stage of core concrete; 2) study the long-term creep behavior of concrete radial expansion constrained by a steel tube and establish a creep model; 3) investigate the expansion behavior and long-term development of expansion agents in the axial direction in addition to the radial deformation of core concrete; 4) examine the disparity and correlation between the development of the expansion process and the ultimate expansion of the core concrete in both laboratory and steel tube settings. Subsequently, incorporating deformation data from laboratory specimens into the model is proposed to enhance its applicability range.

2 Mechanical mechanism of radial expansion of core concrete

The mechanical mechanism behind the radial expansion of the core concrete in CFST structures is a crucial aspect of this study. Often, due to the influence of early plastic shrinkage, significant debonding occurs inside CFST during pouring, as illustrated in Fig.1. The expansion development of the core concrete can be classified into two distinct stages: the debonding stage and the subsequent restriction stage, based on the extent to which the steel tube limits its impact on the concrete.

During the debonding stage, the primary role of the expansion agent is to compensate for debonding gaps. In this phase, the steel tube has minimal influence on deformation development. However, in the latter stage, the core concrete gradually fills the debonding gap, and the expansion deformation is subject to effective constraints imposed by the steel tube. This encompasses both instantaneous (elastic) and long-term restrictions (creep).

The entire expansion process of the core concrete revolves around four critical variables. These variables encompass the total deformation ε_r,tot(t) of the stress-free concrete, the radial deformation ε_r,meas(t) measured by the sensor within the core concrete, and two values employed in mechanical calculations: ε_r,exp(t), representing the expansion effect of core concrete on the steel tube under restricted conditions, and ε_rc(t), denoting the radial strain of core concrete at the restriction stage (where subscript r indicates the radial direction). The ideal development curve for these four variables is depicted in Fig.2.

During the debonding stage, owing to the minimal degree of restriction, there is little difference between ε_r,tot(t) and ε_r,meas(t), while both ε_r,exp(t) and ε_rc(t) are essentially zero. However, it’s important to note that limitations imposed by steel tubes cannot be disregarded during the restriction stages, resulting in a gradual decrease in ε_r,meas(t) in comparison to ε_r,tot(t). Additionally, ε_rc(t) is also smaller than ε_r,exp(t).

To quantitatively describe the deformation process of core concrete, a mathematical model has been established to simulate the entire expansion process. This model is based on the following assumptions.

1) The restriction effect of the steel tube on the deformation of core concrete is disregarded during the debonding stage, implying that ε_r,meas(t) = ε_r,tot(t).

2) Since the restriction effect of the steel tube on the core concrete during the debonding stage is not considered, both ε_r,exp(t) and ε_rc(t) are assumed to be zero during this stage. In other words, the steel tube is not subjected to circumferential expansion force during the debonding stage.

3) The time point that marks the division between the two stages is denoted as τ₁. At this point, the debonding gap measures 2aε_r,meas(τ₁). Furthermore, the relationship between the core concrete strain ε_rc(t) and the sensor-measured concrete deformation ε_r,meas(t) in both stages satisfies Eq. (1). Meanwhile, the relationship between the expansion ε_r,exp(t) generated by core concrete on the steel tube and the total deformation ε_r,tot(t) under freedom conditions satisfies Eq. (2).

(1)

(2)

4) It is assumed that the total expansion displacement of the core concrete always surpasses the debonding gap, leading to the generation of expansion force on the steel tube. In simpler terms, there exists a value t such that t ≥ τ₁.

2.1 Radial expansion stress of concrete-filled steel tube based on elastic mechanics

Once the core concrete makes full contact with the steel tube and undergoes stress, it enters the restriction stage. During this stage, the stress in both the core concrete and steel tube can be calculated using principles of mechanics. We assume an expansion deformation ε_r,exp within the core concrete and a close connection between the inner wall of the steel tube and the concrete:

(3){\epsilon }_{\text{r,exp}}a+(u_{\text{rc}}^{\text{e}}{)}_{\mathrm{r}=\mathrm{a}}=(u_{\text{rs}}^{\text{e}}{)}_{\mathrm{r}=\mathrm{a}},

where the expansion deformation generated by the core concrete is denoted as ε_r,exp, and a represents the inner radius of the steel tube, which corresponds to the radius of the core concrete. The displacement of the concrete is represented as u_{\text{rc}}^{\text{e}}, while u_{\text{rs}}^{\text{e}} represents the displacement of the steel tube. In polar coordinates, Eq. (4) can be used to express both displacements.

(4)\{\begin{array}{c}{u}_{\text{rc}}^{\text{e}}=\frac{1+{\mu }_{\text{c}}}{E_{\text{c}}}\left[2\left(1-2{\mu }_{\text{c}}\right)C_{\text{c}}r-\frac{A_{\text{c}}}{r}\right]+I_{\text{c}}\cos \theta +K_{\text{c}}\sin \theta ,\\ u_{\text{rs}}^{\text{e}}=\frac{1+{\mu }_{\text{s}}}{E_{\text{s}}}\left[2\left(1-2{\mu }_{\text{s}}\right)C_{\text{s}}r-\frac{A_{\text{s}}}{r}\right]+I_{\text{s}}\cos \theta +K_{\text{s}}\sin \theta ,\end{array}

where the Poisson ratios of concrete and the steel tube are denoted as μ_c and μ_s, respectively. The variables r and θ represent the radius and angle in polar coordinates. E_c and E_s refer to the elastic moduli of concrete and the steel tube, respectively. A_c, A_s, C_c, C_s, I_c, I_s, K_c, and K_s are constants.

The steel tube is treated as a thick-walled cylinder with an inner radius ‘a’ and an outer radius ‘b’, while the concrete is considered a thick-walled cylinder with an outer radius ‘a’ and an inner radius of zero. Due to the central symmetric nature of CFST, I_c, I_s, K_c, and K_s are all equal to zero. Assuming that q represents the pressure between the steel tube and core concrete as depicted in Fig.1, the parameters A_c, A_s, C_c, and C_s in Eq. (4) can be determined as follows:

(5)\{\begin{array}{c}{A}_{\text{c}}=0,\\ 2C_{\text{c}}=-q,\end{array}

(6)\{\begin{array}{c}{A}_{\text{s}}=-\frac{a^2{b}^{2}q}{b^2-a^2},\\ 2C_{\text{s}}=\frac{a^{2}q}{b^2-a^2}.\end{array}

The combination of Eq. (3) to Eq. (6) yields:

(7)\begin{array}{rl}& {\epsilon }_{\text{r,exp}}\\ =& q\left[\frac{\left(1+{\mu }_{\text{c}}\right)\left(1-2{\mu }_{\text{c}}\right)}{E_{\text{c}}}+\frac{\left(1+{\mu }_{\text{s}}\right)\left(1-2{\mu }_{\text{s}}\right)a^2+\left(1+{\mu }_{\text{s}}\right)b^2}{E_{\text{s}}\left(b^2-a^2\right)}\right].\end{array}

From this, the total radial strain {\epsilon }_{\text{rc}}^{\text{e}} + ε_r,exp of core concrete can be obtained:

(8){\epsilon }_{\text{rc}}^{\text{e}}+{\epsilon }_{\text{r,exp}}=\frac{1-{{\mu }_{\text{c}}}^2}{E_{\text{c}}}\left({\sigma }_{\text{rc}}^{\text{e}}-\frac{{\mu }_{\text{c}}}{1-{\mu }_{\text{c}}}{\sigma }_{\text{θc}}^{\text{e}}\right)+{\epsilon }_{\text{r,exp}}=\left(1-\frac{\left(1+{\mu }_{\text{c}}\right)\left(1-2{\mu }_{\text{c}}\right)}{E_{\text{c}}\left[{\displaystyle \frac{\left(1+{\mu }_{\text{c}}\right)\left(1-2{\mu }_{\text{c}}\right)}{E_{\text{c}}}}+{\displaystyle \frac{\left(1+{\mu }_{\text{s}}\right)\left(1-2{\mu }_{\text{s}}\right)a^2+\left(1+{\mu }_{\text{s}}\right)b^2}{E_{\text{s}}\left(b^2-a^2\right)}}\right]}\right){\epsilon }_{\text{r,exp}}.

The {\sigma }_{\text{rc}}^{\text{e}} denoted as radial stress of core concrete, and {\sigma }_{\text{θc}}^{\text{e}} denoted as the circumferential stress of core concrete. By determining the parameter k_r that satisfies Eq. (9), the expression for the radial strain of concrete can be obtained using Eq. (10).

(9)k_{\text{r}}=1-\frac{\left(1+{\mu }_{\text{c}}\right)\left(1-2{\mu }_{\text{c}}\right)}{E_{\text{c}}\left[{\displaystyle \frac{\left(1+{\mu }_c\right)\left(1-2{\mu }_{\text{c}}\right)}{E_{\text{c}}}}+{\displaystyle \frac{\left(1+{\mu }_{\text{s}}\right)\left(1-2{\mu }_{\text{s}}\right)a^2+\left(1+{\mu }_{\text{s}}\right)b^2}{E_{\text{s}}\left(b^2-a^2\right)}}\right]},

(10){\epsilon }_{\text{rc}}^{\text{e}}+{\epsilon }_{\text{r,exp}}=k_{\text{r}}{\epsilon }_{\text{r,exp}}.

When r = b, the circumferential strain {\epsilon }_{\text{θs}}^{\text{e}} of the steel tube is obtained:

(11){\epsilon }_{\text{θs}}^{\text{e}}=\frac{1-{{\mu }_{\text{s}}}^2}{E_{\text{s}}}{\sigma }_{\text{θs}}^{\text{e}}=\frac{2\left(1-{{\mu }_{\text{s}}}^2\right)}{E_{\text{s}}\left({\displaystyle \frac{b^2}{a^2}}-1\right)\left[{\displaystyle \frac{\left(1+{\mu }_{\text{c}}\right)\left(1-2{\mu }_{\text{c}}\right)}{E_{\text{c}}}}+{\displaystyle \frac{\left(1+{\mu }_{\text{s}}\right)\left(1-2{\mu }_{\text{s}}\right)a^2+\left(1+{\mu }_{\text{s}}\right)b^2}{E_{\text{s}}\left(b^2-a^2\right)}}\right]}{\epsilon }_{\text{r,exp}},

where {\sigma }_{\text{θs}}^{\text{e}} is the circumferential stress of the steel tube. Given the parameter k_s so that it satisfies Eq. (12), the circumferential strain of the steel tube can be expressed as Eq. (13).

(12)k_{\text{s}}=\frac{2\left(1-{{\mu }_{\text{s}}}^2\right)}{E_{\text{s}}\left({\displaystyle \frac{b^2}{a^2}}-1\right)\left[{\displaystyle \frac{\left(1+{\mu }_{\text{c}}\right)\left(1-2{\mu }_{\text{c}}\right)}{E_{\text{c}}}}+{\displaystyle \frac{\left(1+{\mu }_{\text{s}}\right)\left(1-2{\mu }_{\text{s}}\right)a^2+\left(1+{\mu }_{\text{s}}\right)b^2}{E_{\text{s}}\left(b^2-a^2\right)}}\right]},

(13){\epsilon }_{\text{θs}}^{\text{e}}=k_{\text{s}}{\epsilon }_{\text{r,exp}}.

Thus, by considering the specific radial expansion ε_r,exp, we can determine the respective strains of both concrete and steel tube in their elastic state. In case ε_r,exp varies over time (i.e., ε_r,exp(t)), the previously obtained strains and stresses will also undergo temporal changes.

2.2 Radial expansion stress of concrete-filled steel tube based on linear creep mechanics

Over an extended service period, the confined stress within the steel tube induces creep in the core concrete, resulting in the gradual attenuation of both radial expansion deformation and stress. Due to the imperfections in creep mechanisms, various long-term prediction models have been developed, including exponential, logarithmic, power functions, hyperbolic functions, and combinations thereof [37]. Considering the advantageous mathematical properties of the exponential function in stress calculation, this paper adopts the creep form based on the exponential function. Therefore, for the restriction stage, the core concrete’s creep function is defined as follows [38–40]:

(14)C(t,\tau )=\left(C_0+\frac{A}{\tau }\right)\left(1-{\text{e}}^{-\gamma \left(t-\tau \right)}\right).

In this equation, C₀ + A/τ represents the continuous aging of the core concrete with time, and 1 − exp(−γ(t − τ)) represents the continuous decay of creep with time under load. Here, τ represents the initial time of creep, and A, C₀, γ are constants. t represents the age of the core concrete. Since the core concrete is exposed to sealed conditions, only its basic creep is considered. Additionally, given that the restriction force of the steel tube is significantly less than 0.4 times the compressive strength of the core concrete, it is reasonable to assume that the stress and creep exhibit a linear relationship. Therefore, Boltzmann’s superposition principle [41–43] is applicable to the creep deformation under various stress histories. The radial deformation of the core concrete, {\epsilon }_{\text{rc}}^{\text{c}}(t) considering creep, can be calculated using Eq. (15).

(15)\begin{array}{rl}{\epsilon }_{\text{rc}}^{\text{c}}(t)=& \frac{1-{{\mu }_{\text{c}}}^2}{E_{\text{c}}(t)}\left[{\sigma }_{\text{rc}}(t)-\frac{{\mu }_{\text{c}}}{1-{\mu }_{\text{c}}}{\sigma }_{\text{θ c}}(t)\right]+(1-{{\mu }_{\text{c}}}^2)\\ & \cdot {\int }_{{\tau }_1}^t\left[{\sigma }_{\text{rc}}(\tau )-\frac{{\mu }_{\text{c}}}{1-{\mu }_{\text{c}}}{\sigma }_{\text{θ c}}(\tau )\right]\frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\left(\frac{1}{E_{\text{c}0}}+C(t,\tau )\right)\mathrm{d}\tau .\end{array}

The first part of left-hand side in the aforementioned equation represents the elastic term, while the second part represents the creep term. As u_rc(t) = r{\epsilon }_{\text{rc}}^{\text{c}}(t), Eq. (3), when taking creep into account, should be expressed as follows:

(16)\begin{array}{rl}& {\epsilon }_{\text{r,exp}}+\frac{1+{\mu }_{\text{c}}}{E_{\text{c}}(t)}\left[-(1-2{\mu }_{\text{c}})q(t)\right]\\ & +(1-{{\mu }_{\text{c}}}^2){\int }_{{\tau }_1}^t\left[-(1-2{\mu }_{\text{c}})q(\tau )\right]\frac{\mathrm{\partial }}{\mathrm{\partial }\tau }C(t,\tau )\mathrm{d}\tau \\ =& \frac{1+{\mu }_{\text{s}}}{E_{\text{s}}}\left[(1-2{\mu }_{\text{s}})\frac{a^{2}q(t)}{b^2-a^2}+\frac{b^{2}q(t)}{b^2-a^2}\right].\end{array}

Equation (17) can be obtained by a simple transposition operation from Eq. (16).

(17)\frac{{\epsilon }_{\text{r,exp}}}{\left[{\displaystyle \frac{(1+{\mu }_{\text{c}})(1-2{\mu }_{\text{c}})}{E_{\text{c}}(t)}}+{\displaystyle \frac{(1+{\mu }_{\text{s}})(1-2{\mu }_{\text{s}})a^2+(1+{\mu }_{\text{s}})b^2}{E_{\text{s}}(b^2-a^2)}}\right]}=q(t)-{\displaystyle \frac{(1-{{\mu }_{\text{c}}}^2){\int }_{{\tau }_1}^t\left[(1-2{\mu }_{\text{c}})q(\tau )\right]{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }}C(t,\tau )\mathrm{d}\tau }{\left[{\displaystyle \frac{(1+{\mu }_{\text{c}})(1-2{\mu }_{\text{c}})}{E_{\text{c}}(t)}}+{\displaystyle \frac{(1+{\mu }_{\text{s}})(1-2{\mu }_{\text{s}})a^2+(1+{\mu }_{\text{s}})b^2}{E_{\text{s}}(b^2-a^2)}}\right]}}.

Let k₀(t) satisfy Eq. (18), and simplify this creep issue to a stress relaxation problem by considering the concrete elastic modulus as a fixed value E_c. Then, Eq. (17) can be converted to Eq. (19), and k₀(t) can also be simplified to k₀.

(18)k_0(t)=\frac{(1-2{\mu }_{\text{c}})(1-{{\mu }_{\text{c}}}^2)}{\left[{\displaystyle \frac{(1+{\mu }_{\text{c}})(1-2{\mu }_{\text{c}})}{E_{\text{c}}(t)}}+{\displaystyle \frac{(1+{\mu }_{\text{s}})(1-2{\mu }_{\text{s}})a^2+(1+{\mu }_{\text{s}})b^2}{E_{\text{s}}(b^2-a^2)}}\right]},

(19)q^{\text{e}}=q(t)-k_0{\int }_{{\tau }_1}^{t}q(\tau )\frac{\mathrm{\partial }}{\mathrm{\partial }\tau }C(t,\tau )\mathrm{d}\tau .

Equation (19) represents a prototypical stress relaxation problem, which can be resolved as Eq. (20).

(20)q(t)=q^{\text{e}}\left[1-\gamma k_0\left(C_0+\frac{A}{{\tau }_1}\right){\tau }_{{}_1}^{\gamma A{k}_0}{\mathrm{e}}^{\gamma (1+k_0{C}_0){\tau }_1}{\int }_{{\tau }_1}^t\frac{{\mathrm{e}}^{-\gamma (1+k_0{C}_0)\tau }}{{\tau }^{\gamma A{k}_0}}\mathrm{d}\tau \right].

By determining the relaxation coefficient K(t) that satisfies Eq. (21).

(21)K(t)=1-\gamma k_0\left(C_0+\frac{A}{{\tau }_1}\right){\tau }_{{}_1}^{\gamma A{k}_0}{\mathrm{e}}^{\gamma (1+k_0{C}_0){\tau }_1}{\int }_{{\tau }_1}^t\frac{{\mathrm{e}}^{-\gamma (1+k_0{C}_0)\tau }}{{\tau }^{\gamma A{k}_0}}\mathrm{d}\tau .

And the radial deformation of the steel tube ε_rs(t) can be obtained.

(22){\epsilon }_{\text{rs}}(t)=\frac{1-{{\mu }_{\text{s}}}^2}{E_{\text{s}}}\left[{\sigma }_{\text{rs}}(t)-\frac{{\mu }_{\text{s}}}{1-{\mu }_{\text{s}}}{\sigma }_{\text{θ s}}(t)\right]={\epsilon }_{\text{rs}}^{\text{e}}K(t).

The total radial deformation of concrete, ε_rc(t), is the sum of ε^c_rc(t) and ε_r,exp. Directly applying Eq. (20) to solve for the radial deformation of core concrete in Eq. (15) involves complex operations. However, since the total radial deformation of both concrete and steel tube are equal at r = a, it becomes possible to express ε_rc(t) in a similar form as Eq. (22), leading to the derivation of ε_rc(t) as Eq. (23).

(23){\left({\epsilon }_{\text{rc}}(t)\right)}_{\text{r = a}}={\left({\epsilon }_{\text{rc}}^{\text{e}}\right)}_{\text{r = a}}K(t)=k_{\text{r}}{\epsilon }_{\text{r,exp}}K(t).

The long-term creep model of core concrete subjected to confinement from steel tubes has been derived using the equations mentioned above.

3 Raw materials and test methods

The Tian’e Longtan Bridge employs a CFST structure with an inner diameter of 900 m as its stiffness skeleton, onto which the encased concrete is subsequently cast. Core concrete pouring for the CFST was successfully completed on August 10th, 2022, and the construction of the encased concrete was finished on April 30th, 2023. The Tian’e Longtan Bridge was opened to traffic on February 1st, 2024. Before the encased concrete reached sufficient strength, the CFST stiffness skeleton bore the construction load of the arch bridge. However, once the CFST arch ribs were enveloped by encased concrete, any internal defects in the CFST became irreparable. Therefore, our team had to ensure the compactness of the CFST during pouring and guarantee the long-term expansion of the core concrete without debonding. To investigate the long-term creep and expansion behavior of the expansion agent in the CFST before the construction of the Tian’e Longtan Bridge, we designed a full-scale CFST test based on the dimensions of the bridge’s arch rib. This test primarily involved measuring the long-term deformation of both the core concrete and the steel tube. Additionally, to ensure the validity of the test, a section of CFST was cut after 114 d to assess the uniformity of its core concrete. The entire testing process is illustrated in Fig.3.

3.1 Material property

The full-scale experimental design consists of two full-scale CFST specimens, each with a diameter of 0.92 m, a length of 12 m, and a thickness of 0.01 m, as depicted in Fig.3. The core concrete mix proportion is presented in Tab.2. CFST-A serves as the control specimen for investigating the autogenous shrinkage of core concrete, while CFST-B incorporates a 9% expansion agent to examine the shrinkage compensation. The core concrete possesses a strength grade of 70 MPa, and a polycarboxylate superplasticizer is employed to enhance its flowability, ensuring compliance with self-compaction requirements.

The properties of the raw materials used in the full-scale CFST specimens are detailed as follows:

1) P.O 52.5 cement was used;

2) manufactured sand was selected as the fine aggregate, with a fineness modulus of 2.9 and an apparent density of 2.676 g/cm³;

3) graded diabase crushed stone was used as the coarse aggregate, with an apparent density of 2.959 g/cm³ and a crush index of 9.8%;

4) the expansion behavior is influenced by factors such as the elastic modulus of the steel tube, based on the aforementioned deduction. Considering economic considerations and given that the elastic modulus of different types of steel is nearly identical, Q235 steel was chosen as the steel tube material for testing purposes [44].

The chemical composition of the cement, silica fume, fly ash, and expansion agent are provided in Tab.3. The expansion agent utilized in the experiment is a compound of calcium oxide and lightly burned magnesium oxide. Since the core concrete is enclosed after pouring and experiences high internal temperature rise during its early stages, an expansion agent with a large water demand and high-temperature instability, such as calcium aluminate sulfate [27,45,46], was not chosen. Previous research also supports this approach [25].

3.2 Test methods

3.2.1 Deformation measurement of core concrete

In this study, vibrating-wire strain gauges were employed to capture the spatial distribution of core concrete deformation. These strain gauges had a cylindrical shape, measuring 100 mm in length and 25 mm in diameter. A total of 10 strain gauges were strategically placed in the middle cross-section of the CFST for measurement purposes. Among these, seven strain gauges were arranged in the axial direction and labeled as S1 to S7. The remaining three strain gauges were positioned in the radial direction and designated as S8 to S10. The arrangement of these sensors is depicted in Fig.4.

The data directly measured by the vibrating-wire strain gauges are the resonant frequency f, and the obtained resonant frequency f can be converted into the deformation results by Eq. (24).

(24)\mathrm{\Delta }\epsilon =k_{\text{v}}\mathrm{\Delta }{f}^2+(a_1-a_2)\mathrm{\Delta }T,

where ∆ε denotes the calculated deformation change (×10⁻⁶); k_v represents a constant, calibrated by the manufacturer; ∆f is the change in resonant frequency; a₁ denotes the temperature correction coefficient, which is (13.5 × 10⁻⁶) °C⁻¹ according to the calibration from manufacturer; a₂ represents the thermal expansion coefficient of concrete, which is generally (1 × 10⁻⁶) °C⁻¹, ∆T is the change in temperature of the measured points (°C).

3.2.2 Circumferential deformation measurements of steel tube

To measure the circumferential deformation of the CFST-B specimen’s steel surface, resistance strain gauges were affixed at four annular positions, as illustrated in Fig.4. After installing the strain gauges, a protective layer of silica gel was applied to minimize potential damage. These four strain gauges were complemented with temperature compensation strain gauges, strategically positioned around each gauge. This accounted for the early heat release during concrete hydration and subsequent fluctuations in ambient temperature that could affect the accuracy of deformation measurements. The temperature-compensated strain gauges were mounted on the surface of small steel sheets (1 mm × 10 mm × 20 mm), composed of the same material as the steel tubes. The steel sheet was placed adjacent to its corresponding strain gauge and fully encapsulated with silica gel so that only temperature would affect it. The final deformation results of the steel tube were obtained by subtracting the measurement data of the strain gauge from its respective temperature compensation.

3.2.3 Deformation measurements of laboratory concrete

In addition to the full-size CFST specimen, two groups of prism specimens measuring 515 mm × 100 mm × 100 mm were prepared in the laboratory at a temperature of (20 ± 1) °C to quantify autogenous shrinkage and expansion deformation. Each group consisted of three specimens, and the average test results were recorded. To minimize constraints on the concrete specimens from the mold, oil was applied inside the test mold, and two layers of plastic film were used. Immediately after pouring the concrete, a target was inserted and then covered with plastic film to maintain a sealed condition, as illustrated in Fig.5. The laser testing system was activated 6 h after the initial setting time of the concrete, and the instrument automatically recorded data every 30 min. After 24 h of pouring, the recording was temporarily halted, and six specimens were demolded to eliminate any influence of mold constraints on specimen deformation. Subsequently, data collection resumed for continuous observation.

3.2.4 Mechanical properties and working performance of core concrete

Cube specimens with dimensions of 150 mm × 150 mm × 150 mm and prismatic specimens measuring 100 mm × 100 mm × 300 mm were used to determine the compressive strength and elastic modulus of the concrete, respectively. The measured results are presented in Tab.4.

The uniformity of the concrete within the tubes was confirmed by cutting CFST-A and CFST-B specimens at the measured cross-sections after 6 months of pouring. The obtained cross-sections are illustrated in Fig.6. The cutting position of CFST-A precisely matches the sensor layout location, revealing both the half-cut vibrating-wire strain gauges and a portion of the support. The aggregate ratio was quantitatively analyzed using picture recognition technology in Fig.7, with ‘h’ representing the section height ranging from 0 to 900 mm. The aggregate ratio within the cross-section typically falls within the range of 30% to 40%, corresponding to the specified volume fraction in the mix proportion. Overall, both the material selection process and pumping process during testing have proven to be reliable, ensuring a uniform distribution of cement paste and aggregate. Building upon this foundation, the analysis of spatial shrinkage and expansion in the core concrete also demonstrates its effectiveness.

4 Results and discussion

4.1 Radial deformation of core concrete

4.1.1 Radial deformation results and creep model of core concrete

The test results of radial autogenous shrinkage and expansion deformation of core concrete are presented in Fig.8. It is evident that the autogenous shrinkage and expansion deformation of CFST core concrete exhibit contrasting trends. In the case of CFST-A, the autogenous shrinkage of the lowermost part of the core concrete is intensified by gravity, while the uppermost part undergoes free contraction resulting in minimal shrinkage. Conversely, for the expansion of CFST-B, the effect of gravity results in significantly smaller radial expansion in the lower part of the core concrete compared to the upper and middle parts. This indicates that gravity exerts a restraining effect on concrete expansion.

Referring to previous studies conducted by other researchers [47–49], it can be observed that autogenous shrinkage follows an exponential function form as depicted in Eq. (25).

(25){\epsilon }_{\text{G}}(t)={\epsilon }_{\text{G}}\left(1-e^{-ct}\right),

where ε_G(t) denotes the shrinkage value of core concrete at time t, ε_G denotes the ultimate shrinkage of core concrete, and c is a constant.

Under unconstrained conditions, the total radial deformation ε_r,tot(t) after adding expansion agent to concrete under unconstrained condition, can be expressed as the summation of autogenous shrinkage ε_G(t) and expansion ε_E(t):

(26){\epsilon }_{\text{r,tot}}(t)={\epsilon }_{\text{G}}(t)+{\epsilon }_{\text{E}}(t).

Equation (2) indicates that when t is less than τ₁, the radial deformation of core concrete, ε_r,exp(t) = 0. However, when t ≥ τ₁, ε_r,exp(t) = ε_tot(t) − ε_tot(τ₁). By referring to Eq. (23), we can derive the expression for the long-term creep-induced radial deformation of core concrete as presented in Eq. (27).

(27)

In fact, Eq. (27) is expressed as ε_rc(t) calculated based on elastic and creep mechanics, whereas the deformation measured by vibrating-wire strain gauges (i.e., the expression of deformation measured in Fig.8) differs from this expression. In the debonding stage, the measured radial deformation ε_r,meas(t) of the core concrete should be equal to the total radial deformation. While in the restriction stage, the measured deformation was expected to be constrained by the steel tube and exhibited creep behavior, as demonstrated in Eq. (28).

(28)

The expansion development of core concrete can be also considered as an exponential function [50,51]. Therefore, Eq. (28) can be expressed as:

(29)

where ε_r,G and ε_r,E represent the final values of radial shrinkage and expansion of core concrete, respectively, and c₁ and c₂ are constants.

To characterize the deformation of the entire section, the average value of B8–B10 is used to represent the radial deformation of the entire section, denoted as B_r,avg. Simultaneously, the mean value of the data obtained from the strain gauge after temperature compensation is employed to determine the circumferential strain ε_θs of the steel tube, as illustrated in Fig.9. The circumferential strain of the steel tube goes through three distinct processes: initial flattening, rapid increase, and gradual decrease. Initially, the circumferential strain of the steel tube remains close to 0, but subsequently experiences expansion due to the concrete pressure on its inner surface. Finally, the strain of the steel tube gradually decreases due to the creep of the concrete. The dividing point between the first two processes coincides with the transition from the debonding stage to the restriction stage. In this experiment, the partition time point of the two stages, τ₁, is found to be 0.839 d, with its corresponding B_r,avg(τ₁) at 158.3 × 10⁻⁶. This indicates that the initial expansion of concrete fills up the debonding gap of 0.142 mm generated during the plastic stage of concrete. Even for full-size specimens, this debonding gap remains exceedingly small and poses challenges for measurement using conventional detection methods. The findings presented in this paper can serve as valuable references for engineering applications.

By subtracting B_r,avg from B_r,avg (τ₁), the radial strain ε_rc of core concrete in the restriction stage is compared with the circumferential strain ε_θs of the steel tube. The development trend of the two is found to be essentially identical. In fact, in the calculation results of the mechanical mechanism part of the paper, it is observed that the circumferential deformation ε_θs of the outer surface of the steel tube is proportional to the radial deformation ε_rc of the concrete, where ε_θs = 0.948ε_r,expK(t), ε_rc = 0.901ε_r,expK(t) (the calculation parameters are shown in Tab.5, wherein the Poisson ratio and elastic modulus of concrete represent measured values, while those of steel are obtained from the manufacturer). However, the measured circumferential deformation of the steel tube is slightly smaller than the radial deformation of the concrete, which is inconsistent with the calculated results. As the mechanical calculation treats the CFST as a plane strain problem, disregarding axial strain, this may lead to inconsistencies between measured and calculated results. After a measurement period of 114 d, it can be observed that the expansion strain of the steel tube remains at 40 × 10⁻⁶. This indicates that the addition of an expansion agent allows for the long-term maintenance of expansion microstress, ensuring the sustained synergistic effect between the steel tube and concrete.

Based on this, τ₁ is determined to be 0.839 d. Then, Eq. (29) can be employed to fit their respective measured values ε_r,meas, and their average values ε_r,avg of B8–B10 in the restriction stage, respectively. First, Eq. (25) is used to fit the core concrete autogenous shrinkage, and parameters ε_r,G and c₁ of each point are determined (refer to Tab.6 for details). To ensure convenience in fitting, a uniform value of 0.8 is adopted due to the small difference in c₁ values between each individual shrinkage. Then, by substituting autogenous shrinkage parameters and τ₁ into Eq. (29), the respective parameters ε_r,E and c₂ of the expansion function and the parameters of the relaxation function K(t) can be obtained. The parameters for fitting are listed in Tab.6, while the fitting results for each data point and their average result are illustrated in Fig.10 and Fig.9, respectively. The determination coefficients for each fitting range from 0.78 to 0.96, indicating that the model proposed in this paper exhibits superior capability in accurately capturing the early radial expansion and long-term creep of core concrete constrained by steel tubes.

Based on the deformation model presented in this paper, the radial strain of core concrete ε_cr (i.e. B_r,avg − B_r,avg(τ₁)), the circumferential strain of the steel tube ε_sθ, and the interfacial stress q between concrete and the steel tube are calculated. The calculated age t of the concrete ranges from 0.839 to 360 d, equivalent to one year, and the results are illustrated in Fig.11. The creep predominantly progresses over a period of approximately six months under the specified test conditions outlined in this paper, subsequently reaching a state of stability. In the scenario where the total deformation ε_tot = ε_G + ε_E equals 290.1 × 10⁻⁶, 158.3 × 10⁻⁶ is utilized to fill the debonding gap resulting from plastic shrinkage, and the subsequent expansion deformation of 131.8 × 10⁻⁶ is observed to persist at a level of 26.4 × 10⁻⁶ after one-year, indicating a remaining expansion of only 20%.

The maximum interface normal stress q exerted on the steel tube is 0.512 MPa, while the corresponding residual normal stress after one-year amounts to 0.119 MPa, equivalent to a remaining stress of 23%. Therefore, the results indicate that the core concrete of CFST can effectively compensate for shrinkage and induce micro-stresses by incorporating a compound expansion agent consisting of calcium and magnesium oxide. Moreover, even taking into consideration the creep behavior, residual micro-stresses and micro-expansion are retained.

4.1.2 Radial expansion development results and expansion model of core concrete

Based on the fitted parameters ε_r,E and c₂, the expansion development models for each measuring point can be derived. Since the measured results contained both creep and autogenous shrinkage, the expansion function ε_r,E (t) can be inversely calculated from Eq. (29), as shown in Eq. (30). After inputting the data from points A8–A10 and B8–B10 into Eq. (30) for computation, the radial expansion of the core concrete at each location can be determined. The expansion results and model curves are illustrated in Fig.12(a), demonstrating a favorable agreement with the exponential model.

(30)

Furthermore, the final expansion value of concrete at different heights exhibits a significant linear relationship with height h, as shown in Fig.12(b). This indicates the feasibility of utilizing the average value of B8–B10 to represent the mean radial deformation of the section, as mentioned earlier. Based on this, a development model for radial expansion at each height within the core concrete can be established.

(31)\{\begin{array}{c}{\epsilon }_{\text{r,E}}(t)={\epsilon }_{\text{r,E}}\left(1-e^{-c_{\text{2}}t}\right),\\ {\epsilon }_{\text{r,E}}=k_1\left(\frac{h-h_{\text{0}}}{2R}\right)+{\epsilon }_{\text{r,E0}},\end{array}

where k₁ is the influence coefficient of gravity, that is, the slope of the straight line in Fig.12(b), which was measured to be 425.68; ε_r,E0 represents the radial expansion of the core concrete at the height of h₀; c₂ is the expansion development coefficient, which is between 0.793 and 0.914 in the test.

4.2 Axial deformation of concrete-filled steel tube core concrete

4.2.1 Measurement results of axial deformation of core concrete and the finite element analysis

The axial deformation of core concrete is presented in Fig.13, with the initial time selected as 6 h after the completion of pouring. The majority of autogenous shrinkage in concrete without an expansion agent was observed within the initial 6 d, followed by a stabilization of deformation. The axial autogenous shrinkage of core concrete is constrained by the steel tube, exhibiting maximum shrinkage at the midpoint (A3), with decreasing shrinkage closer to the inner surface of the steel tube. Furthermore, no significant creep was observed within the measurement period of 114 d. However, the core concrete deformation of CFST-B exhibits distinct patterns. The deformations measured at points B1–B4 are highly consistent, with only a slight elevation observed at the midpoint (B3) compared to the other three points (B1, B2, and B4). Moreover, the deformation at the bottom point (B5) is approximately 80 × 10⁻⁶ lower compared to other regions. To further analyze the stress of both the steel tube and core concrete under autogenous shrinkage and expansion, a finite element model was established, as depicted in Fig.14(a).

The steel tube has an elastic modulus of 2 × 10⁵ MPa and a Poisson’s ratio of 0.27, while the core concrete has an elastic modulus of 3.7 × 10⁴ MPa and a Poisson ratio of 0.14. Due to the relatively low tensile strength of core concrete, it is prone to plastic tensile yielding and even cracking under severe shrinkage. To characterize the softening behavior of core concrete, the Menetrey–Willam model [52] is adopted, in which the yield tensile strength is 3 MPa, the plastic tensile limit strain is 1 × 10⁻³, and the residual stress is equal to one-tenth of the tensile strength, i.e., 0.3 MPa. The core concrete was subjected to a shrinkage of 400 × 10⁻⁶ (corresponding to the maximum shrinkage measured at midpoint A3) and an expansion of 300 × 10⁻⁶ (corresponding to the maximum expansion measured at midpoint B3), respectively, in order to obtain the distribution of axial stress in the section, as depicted in Fig.14(b) and Fig.14(c). When shrinkage occurs, the concrete experiences axial tensile stress as a result of the steel tube’s restraint, leading to certain areas of the concrete exceeding their tensile strength and entering into a plastic softening stage. In contrast, the concrete undergoes axial compressive stress during expansion. Its significantly higher compressive strength compared to tensile strength prevents it from reaching the plastic stage. After selecting a representative section for analysis, it is observed that the axial tensile stress of the core concrete in CFST-A exhibits a phenomenon characterized by higher values at the edges and lower values toward the middle, and the stress gradually decreases from 2.8 to 1.9 MPa. The compressive stress of the core concrete in CFST-B is uniformly distributed throughout the section, measuring approximately −3 MPa (with a negative sign indicating compressive stress). Only a small area near the bottom bearing experiences a slightly higher compressive stress of −5.3 MPa.

In fact, this stress attenuation phenomenon in CFST-A is indeed analogous to the edge interference problem in elastic mechanics, a common occurrence in wall engineering [53]. The constraint effect of the ground foundation on the wall increases as the distance between them decreases, resulting in a greater shrinkage constraint for the wall closer to the foundation and vice versa for those farther away from it. The diminishing impact of this constraint is associated with the H/L ratio, which represents the height-to-length ratio of the wall. A higher H/L value corresponds to a more pronounced decline in constraint effect.

Similarly, CFST can be considered as the concrete core constrained by a steel tube. In CFST, H/L is equivalent to R/L, which in this test is 0.0375 for the full-size CFST specimen, an extremely small value. As a result, the restraining effect in the test of the steel tube on the core concrete is nearly identical within the same section, resulting in consistent axial stress. For instance, finite element models of CFST specimens with R/L ratios of 0.5 and 1 were established, along with the application of a 300 × 10⁻⁶ expansion. The axial stress distribution results are shown in Fig.13(d) and Fig.13(e). As the ratio of R/L increases, a significant decrease in stress distribution across the cross-section is observed from the periphery toward the center. When R/L = 0.5, the stress decreases from −2.6 to −1.5 MPa, whereas for R/L = 1, the stress reduces from −2.4 to −0.9 MPa, indicating that a higher value of R/L leads to a more pronounced reduction in stress.

Although both specimens CFST-A and CFST-B have an R/L ratio of 0.0375, the core concrete in CFST-A undergoes shrinkage, leading to localized plastic softening and even crack formation. Consequently, the tensile stress load in this region is relieved, causing fragmentation of the core concrete into discrete fragments. As a result, the R/L generally increases to approximately 1, as depicted in Fig.14(b). In this case, the stress decline effect of the core concrete cannot be ignored, resulting in a reduced confinement from the inner surface of the steel tube to the center of the concrete and a gradual increase in its shrinkage. This is also the essential reason why the deformation distribution of the core concrete was inconsistent when absence or presence of an expansion agent. In other words, the inclusion of an expansion agent altered the stress state of the core concrete, thereby influencing its distribution of deformation. Additionally, this is also the underlying reason why no significant creep was observed in CFST-A. The segmentation of the core concrete into smaller units effectively mitigates the stress exerted by the steel tube, thereby being insufficient to induce noticeable creep behavior. The non-uniformity and non-continuity of the concrete itself may also aggravate the stress attenuation by hindering the effective transfer of shear stress from the steel tube to the concrete center.

4.2.2 Axial stain and expansion model of core concrete

As analyzed above, the R/L ratio of the tested CFST was sufficiently small, indicating that there is minimal change in axial constraint within the same section after the addition of the expansion agent. Therefore, the axial strain model of the core concrete assumes that the strain is uniform across the entire section. Additionally, for simplification purposes, it is assumed that the axial displacement of both the core concrete and steel tube is equal, as follows:

(32)\left({\epsilon }_{\text{a,exp}}+\frac{{\sigma }_{\text{ac}}^{\text{e}}}{E_{\text{c}}}\right)L=\frac{{\sigma }_{\text{as}}^{\text{e}}}{E_{\text{s}}}L,

where ε_a,exp represents the axial expansion generated by the core concrete, {\sigma }_{\mathrm{a}\mathrm{c}}^{\mathrm{e}} represents the elasticity axial stress of the concrete, {\sigma }_{\mathrm{a}\mathrm{s}}^{\mathrm{e}} represents the elasticity axial stress of the steel tube. The equilibrium equation is expressed as:

(33){\sigma }_{\text{ac}}^{\text{e}}{A}_{\text{ac}}+{\sigma }_{\text{as}}^{\text{e}}{A}_{\text{as}}=0,

where A_ac and A_as represent the cross-section area of the core concrete and steel tube, respectively. By combining Eqs. (32) and (33), the total axial strain of core concrete {\epsilon }_{\mathrm{a}\mathrm{c}}^{\mathrm{e}}+{\epsilon }_{\mathrm{a},\mathrm{e}\mathrm{x}\mathrm{p}} is:

(34){\epsilon }_{\text{ac}}^{\text{e}}+{\epsilon }_{\text{a,exp}}=\frac{{\sigma }_{\text{ac}}^{\text{e}}}{E_{\text{c}}}+{\epsilon }_{\text{a,exp}}=\left(-{\displaystyle \frac{1}{1+{\displaystyle \frac{a^2{E}_{\text{c}}}{(b^2-a^2)E_{\text{s}}}}}}+1\right){\epsilon }_{\text{a,exp}}.

Given the parameter k_a such that it satisfies Eq. (35), then Eq. (34) can be converted to Eq. (36).

(35)k_{\text{a}}=1-\frac{1}{{\displaystyle \frac{a^2{E}_{\text{c}}}{(b^2-a^2)E_{\text{s}}}}+1},

(36){\epsilon }_{\text{ac}}^{\text{e}}=k_{\text{a}}{\epsilon }_{\text{a,exp}}.

Taking into account the creep, then the problem is similar to the radial creep of core concrete, where Eq. (32) transforms to:

(37)\left({\epsilon }_{\text{a,exp}}+\frac{{\sigma }_{\text{ac}}^{\text{c}}(t)}{E_{\text{c}}(t)}\right)L=\frac{{\sigma }_{\text{as}}(t)}{E_{\text{s}}}L.

The solution is:

(38){\sigma }_{\text{ac}}^{\text{e}}={\sigma }_{\text{ac}}^{\text{c}}(t)+{\displaystyle \frac{{\int }_{{\tau }_1}^t{\sigma }_{\text{ac}}^{\text{c}}(\tau ){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }}C(t,\tau )\mathrm{d}\tau }{{\displaystyle \frac{1}{E_{\text{c}}(t)}}+{\displaystyle \frac{a^2}{(b^2-a^2)E_{\text{s}}}}}}.

Let k'₀(t) satisfy Eq. (39), and transform the problem into a stress relaxation scenario, that is, where E_c(t) is considered as a constant E_c. Consequently, Eq. (40) can be derived.

(39)k_0^{\prime }(t)=\frac{1}{{\displaystyle \frac{1}{E_{\text{c}}(t)}}+{\displaystyle \frac{a^2}{(b^2-a^2)E_{\text{s}}}}},

(40)\begin{array}{rl}& {\sigma }_{\text{ac}}^{\text{c}}(t)\\ =& {\sigma }_{\text{ac}}^{\text{e}}\left[1-\gamma {k^{\prime }}_0\left(C_0+\frac{A}{{\tau }_1}\right){\tau }_{{}_1}^{\gamma A{k^{\prime }}_0}{e}^{\gamma (1+{k^{\prime }}_0{C}_0){\tau }_1}{\int }_{{\tau }_1}^t\frac{e^{-\gamma (1+{k^{\prime }}_0{C}_0)\tau }}{{\tau }^{\gamma A{k^{\prime }}_0}}\mathrm{d}\tau \right].\end{array}

Similarly, the axial strain of the core concrete can also be expressed as:

(41){\epsilon }_{\text{ac}}(t)={\epsilon }_{\text{ac}}^{\text{e}}(t)K^{\prime }(t)=k_{\text{a}}\left({\epsilon }_{\text{a,G}}(t)+{\epsilon }_{\text{a,E}}(t)\right)K^{\prime }(t),

where ε_ac(t) represents the total axial strain of the core concrete at time t, and ε_a,G(t) and ε_a,E(t) are the free autogenous shrinkage and expansion of the core concrete at time t, respectively. Since the central part of the core concrete has the smallest shrinkage restriction and can best reflect the real free autogenous shrinkage, ε_a,G(t) are taken as the measurement result of A3, and the ultimate shrinkage ε_a,G is 403.3 × 10⁻⁶ and c₁ is 0.8. As τ₁ cannot be 0 in the proposed creep model, it is assumed that the creep begins when the concrete finally sets, with τ₁ = 0.26 (time zero is the initial setting time). The value of ε_ac(t) is obtained by averaging the results from points B1 to B4. Equation (41) is employed to fit all the parameters, and the resulting fitting curve is illustrated in Fig.15. The specific fitting parameters can be found in Tab.6.

The expansion development model can be obtained based on the fitting parameters ε_a,E and c₂. Since the measured data contains creep and autogenous shrinkage, the expansion function ε_a,E(t) is extracted from Eq. (41), as shown in Eq. (42). To calculate the average axial expansion development of core concrete, the ε_a,E(t) obtained from collected results is substituted into Eq. (42), as illustrated in Fig.15. It is indicated that the expansion results are in good agreement with the proposed expansion development model.

(42){\epsilon }_{\text{a,E}}(t)=\frac{{\epsilon }_{\text{ac}}(t)}{k_{\text{a}}K(t)}-{\epsilon }_{\text{a,G}}(t).

4.3 The incorporation of laboratory specimen deformation into the creep model

4.3.1 Comparison of shrinkage and expansion results between laboratory and core concrete

The laboratory-sealed specimens subjected to constant temperature conditions are depicted in Fig.16, illustrating the results of shrinkage and expansion. Shrinkage and total deformation are the measured parameters, and the expansion results reflect the difference between them. In the absence of an expansion agent, the concrete displayed substantial shrinkage behavior, which persisted for 28 d. However, the introduction of an expansion agent not only compensated for the shrinkage but also induced some degree of expansion.

Establishing a relationship between the laboratory deformation behavior of core concrete and its real-world performance in CFST is challenging due to practical measurement difficulties. To address this, ε_r,G(t) and ε_a,G(t) are used to represent the shrinkage of core concrete under free-stress conditions. ε_a,G(t) is derived from the minimal steel tube restriction experienced at A3, while ε_r,G(t) is represented by the average shrinkage of A8 to A10. These values are compared to laboratory shrinkage data, as shown in Fig.17. The laboratory shrinkage is fitted using Eq. (25), resulting in ε_Glab of −462.6 and c₂ of 0.43. Notably, the laboratory shrinkage rate is slower, but its ultimate shrinkage at 28 d exceeds that of the core concrete in CFST. This is attributed to the elevated temperature within the CFST core, which accelerates the shrinkage rate while reducing its final magnitude, a phenomenon consistent with prior research [48,54].

Similarly, radial mean expansion ε_r,E(t) (obtained from B_r,avg) and axial average expansion ε_a,E(t) (obtained from ε_ac(t)) are compared with laboratory expansion data in Fig.17. The laboratory expansion is fitted using Eq. (25), resulting in ε_Elab of 682.1 and c₂ of 0.39. The final expansion ε_Elab and the development rate c₂ in the laboratory are both lower than those observed in the core concrete. This indicates a slower expansion process and lower final expansion value when subjected to room temperature confinement. This discrepancy can be primarily attributed to the temperature sensitivity of magnesium oxide, which undergoes a more rapid reaction and achieves a higher final expansion when exposed to the elevated curing temperatures within CFST [55,56]. Additionally, gravity effects result in a final radial mean expansion ε_r,E that is 117 × 10⁻⁶ smaller than the axial mean expansion ε_a,E. However, when compared to radial expansion at point S8, which experiences less influence from gravity, it becomes evident that both internal radial and axial expansion magnitudes are similar, measuring approximately 870 × 10⁻⁶.

Based on this, the radial and axial average shrinkage (expansion) of the core concrete can be obtained from the shrinkage (expansion) in laboratory. Equations (43) and (44) are obtained by replacing the shrinkage and expansion of core concrete in Eq. (25) with ε_Glab and ε_Elab measured in the laboratory, respectively.

(43)\{\begin{array}{c}{\epsilon }_{\text{r,G}}(t)={\alpha }_{\text{rG}}{\epsilon }_{\text{Glab}}\left(1-e^{-{\beta }_{\text{rG}}{c}_{1}t}\right),\\ {\epsilon }_{\text{a,G}}(t)={\alpha }_{\text{aG}}{\epsilon }_{\text{Glab}}\left(1-e^{-{\beta }_{\text{aG}}{c}_{1}t}\right),\end{array}

(44)\{\begin{array}{c}{\epsilon }_{\text{r,E}}(t)={\alpha }_{\text{rE}}{\epsilon }_{\text{Elab}}\left(1-e^{-{\beta }_{\text{rE}}{c}_{\text{2}}t}\right),\\ {\epsilon }_{\text{a,E}}(t)={\alpha }_{\text{aE}}{\epsilon }_{\text{Elab}}\left(1-e^{-{\beta }_{\text{aE}}{c}_{\text{2}}t}\right),\end{array}

where α_rG (α_aG) and β_rG (β_aG) are temperature coefficients, indicating the impact of temperature on the radial (axial) ultimate shrinkage and its development process. In this test, the values for the radial shrinkage are 1.08 and 2.2, while for the axial shrinkage they are 1.26 and 1.85. Similarly, α_rE (α_aE) and β_rE (β_aE) indicate the impact of temperature on ultimate radial (axial) expansion and its development process, respectively. In the radial aspect, 1.08 and 2.2 are taken in this test, and in the axial aspect, 1.26 and 1.85 are taken.

4.3.2 Improvement of the model based on deformation in laboratory

Furthermore, incorporating the laboratory shrinkage and expansion deformation into Eqs. (27) and (41), respectively, yields an improved model for core concrete strain, as demonstrated in Eqs. (45) and (46), correspondingly.

(45)

(46){\epsilon }_{\text{ac}}(t)=k_{\text{a}}\left({\alpha }_{\text{aG}}{\epsilon }_{\text{Glab}}\left(1-e^{-{\beta }_{\text{aG}}{c}_{1}t}\right)+{\alpha }_{\text{aE}}{\epsilon }_{\text{Elab}}\left(1-e^{-{\beta }_{\text{aE}}{c}_{\text{2}}t}\right)\right)K^{\prime }(t).

Similarly, by bringing the shrinkage and expansion in the laboratory into Eq. (20), an improved model of the normal stress at steel-concrete interface can be obtained, expressed as Eq. (47).

(47)q=\frac{\left[{\alpha }_{\text{rG}}{\epsilon }_{\text{Glab}}\left(1-e^{-{\beta }_{\text{rG}}{c}_{1}t}\right)+{\alpha }_{\text{rE}}{\epsilon }_{\text{Elab}}\left(1-e^{-{\beta }_{\text{rE}}{c}_{\text{2}}t}\right)-{\epsilon }_{\text{r,tot}}({\tau }_1)\right]K(t)}{\frac{\left(1+{\mu }_{\text{c}}\right)\left(1-2{\mu }_{\text{c}}\right)}{E_{\text{c}}}+\frac{\left(1+{\mu }_{\text{s}}\right)\left(1-2{\mu }_{\text{s}}\right)a^2+\left(1+{\mu }_{\text{s}}\right)b^2}{E_{\text{s}}\left(b^2-a^2\right)}}.

The improved model is compared to the test results, which included the radial strain ε_rc(t) (mean value of B8–B10), the axial strain ε_ac(t) (mean value of B1–B4), and the normal stress at steel-concrete interface (obtained by the circumferential strain ε_θs(t) of steel tube). These comparisons are illustrated in Fig.18. It can be observed that the measured data generally fall within a ±10% error range around the model curve, indicating a strong agreement between the proposed long-term creep model and experimental results.

Furthermore, it’s worth noting that a critical parameter in the radial creep model is the expansion ε_r,tot(τ₁) used to fill the debonding gap, which was determined to be 158.3 × 10⁻⁶ in the full-scale test. However, it’s important to highlight that quantitatively measuring the early plastic debonding gap within CFST remains a significant challenge, requiring further technical and theoretical research. The results obtained from this experiment can serve as a valuable reference for advancing the research and development of CFST debonding mechanisms. Overall, the full-size test underscores the utility of the proposed creep models, which can accurately predict the long-term radial and axial deformation of core concrete, as well as the normal self-stress at the interface between the steel tube and concrete, based on laboratory deformations of concrete. This capability enables precise quantitative design and construction of CFST structures with controlled micro-expansion or self-stress.

5 Conclusions

In this study, we conducted theoretical and experimental research on the restricted radial and axial expansion behavior of core concrete doped with an expansion agent when CFST is placed horizontally. We proposed a long-term creep model for core concrete and a normal stress model between the steel tube and core concrete based on elastic mechanics and linear creep mechanics, specifically in the debonding stage and the restriction stage. Two full-size CFST specimens were designed and tested to validate our proposed models comprehensively. Our main research findings are as follows.

1) Early plastic shrinkage of the core concrete leads to debonding at the top, necessitating compensation of the debonding gap before the expansion agent applies pressure to the steel tube. This led us to categorize CFST core concrete deformation into the debonding stage and the restriction stage. In our full-scale test, we determined that the transition between these stages occurred at 0.839 d, during which the core concrete experienced an average radial deformation of 158.3 × 10⁻⁶ in the debonding stage, filling a debonding gap of 0.142 mm. Notably, quantitatively detecting these small debonding gaps generated during the plasticity period remains challenging using existing detection methods. Therefore, our findings offer valuable insights for further research on CFST debonding and compensation.

2) The radial expansion of the core concrete during the restriction stage results from the combined effects of shrinkage, expansion, and creep. We derived a long-term creep model for the core concrete based on the stress conditions of CFST. Validation through full-scale tests demonstrated that this model accurately represents the long-term deformation behavior of core concrete in the restriction stage. The radial creep of the core concrete was predicted for a period of 360 d, revealing that out of a total deformation value of 290.1 × 10⁻⁶, 158.3 × 10⁻⁶ is utilized to fill the debonding gap during the debonding stage. The remaining 131.8 × 10⁻⁶ expansion strain maintained a final value of 26.4 × 10⁻⁶ after one year, representing a 20% residual, with a corresponding residual normal stress of 0.119 MPa. These findings emphasize that even with concrete creep, expansion agents can maintain their ability to generate micro-stresses on the steel tube to some extent without fully diminishing.

3) We observed variations in the radial deformation of core concrete across its cross-section. The lower part of the core concrete, subjected to gravitational forces, experienced greater radial shrinkage compared to the upper part. Conversely, the lower concrete, influenced by gravity, exhibited smaller radial expansion than the upper concrete, with this expansion linearly correlated with height h. Building on these observations, we established time-varying models for the radial expansion of core concrete at different heights.

4) Severe shrinkage can cause the core concrete to fracture into distinct blocks due to tensile softening or damage. This results in an increase in the diameter-to-length ratio (R/L) and a rapid decline in binding force between the steel tube and core concrete along the radial direction. Consequently, greater shrinkage occurs closer to the core concrete’s center. The inclusion of an expansion agent can improve the stress state of CFST, preserving its R/L ratio. This maintains relatively constant bond strength between the steel tube and core concrete, resulting in minimal variation in axial expansion within the cross-section. We represented the axial shrinkage and expansion of core concrete using the measured shrinkage at the middle point and the mean expansion measured in full-scale CFST, and we established a long-term creep axial expansion model for core concrete.

5) The core concrete experienced a higher temperature history in its early stages, leading to faster shrinkage and expansion development compared to laboratory specimens. This resulted in ultimately smaller shrinkage and larger expansion of the core concrete, respectively. We incorporated the shrinkage and expansion results obtained from laboratory tests into the radial and axial long-term creep models, as well as the normal stress model at the steel-concrete interface proposed in this paper, enhancing their practical applicability. Our results demonstrated that the measured data generally fell within a ±10% error limit of the improved model, indicating a good agreement.

While our proposed long-term creep model exhibited excellent agreement with measured expansion, further enhancing the theoretical mechanism of the model by considering elastic modulus as a variable remains a challenge. Additionally, reliable methods for verifying whether the debonding gap measures 0.142 mm, as stated, require further advancements in detection technology.

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