Theoretical study on the confine-stiffening effect and fractal cracking of square concrete filled steel tubes in tension loads

Abstract

Tension stress in steel-concrete composite is widely observed in engineering design. Based on an experimental program on tension performance of three square concrete-filled tubes (SCFT), the tension theory of SCFT is proposed using a mechanics-based approach. The tension stiffening effect, the confining strengthening effect and the confining stiffening effect, observed in tests of SCFTs are included in the developed tension theory model. Subsequently, simplified constitutive models of steel and concrete are proposed for the axial tension performance of SCFT. Based on the MSC.MARC software, a special fiber beam-column element is proposed to include the confining effect of SCFTs under tension and verified. The proposed analytical theory, effective formulas, and equivalent constitutive laws are extensively verified against three available tests reported in the literature on both global level (e.g., load-displacement curves) and strain level. The experimental verification proves the accuracy of the proposed theory and formulations in simulating the performance of SCFT members under tension with the capability to accurately predict the tensile strength and stiffness enhancements and realistically simulate the fractal cracking phenomenon.

Graphical abstract

Keywords

square concrete filled tubes / confine-stiffening / confine-strengthening / fractal cracking / fracture

1 Introduction

Steel-concrete composite structures integrate the advantages of the constituent materials. Structures utilizing square concrete-filled steel tube (SCFT) members have been reported in numerous engineering structures and have been widely investigated in the world through experimental, theoretical and numerical approaches [1–5]. A series of SCFT tension tests [6] have provided explicit evidence that the tensile strength and stiffness of SCFTs are notably increased compared to that of members with the same steel tubes sections. SCFT members are potentially loaded by tensile forces in bridge structures [6–8] as shown in Fig. 1(a) and high-rise buildings subjected to overturning moment collapse during a severe earthquake (Fig. 1(b)) [9]. In these cases, inaccurate prediction of the failure mode and inaccurate prediction of the nonlinear mechanical performance may occur when the tensile stiffness enhancement of the SCFT members is not addressed adequately. To improve these predictions, an understanding of the mechanism of the tensile stiffness and strength enhancement of SCFTs is needed rather than approximating these values with the stiffness and strength of the hollow tube.

Significant research has been devoted to finite element modeling of SCFT members. Among them, a fiber beam-column element using uniaxial constitutive laws to capture material nonlinear behavior is utilized by many researchers because it achieves a good balance between accuracy and efficiency. Several uniaxial constitutive laws have been proposed for modeling SCFT members [10–13]. The majority of these models neglect the influence of the in-filled concrete on the tensile response. Hajjar and Gourley [12] reported that including the tensile stress-strain relationship of concrete can predict the sectional strength of SCFT members with higher accuracy, and hence they used the tensile law of plain concrete recommended by Vecchio and Collins [14]. However, the interaction between the concrete and steel in SCFT members under tension has not yet been taken into account. A few experimental studies have been conducted specifically on the tension response of SCFT members [6]. Moreover, the majority of leading standards [15–17] adopt the assumption that the tension performance of SCFTs is the same as the steel section while neglecting the contribution of concrete to the stiffness and capacity of the SCFTs. However, existing test studies conducted by Zhou et al. [6] (as shown in Fig. 2(a)) all confirm that the concrete cores enhance the tensile capacity (Fig. 2(b)) of the hollow tubes by 3% to 5%, depending on different steel content ratios. More importantly, the tensile stiffness enhancement (Fig. 2(b)) of SCFTs over hollow tubes is also confirmed, with values ranging from 27% to 37% (28.5% on average), through comparison with the test results of all three groups of SCFT tensile tests. A similar experimental study and theoretical study have been completed for the circular concrete-filled tubes (CCFTs) [18,19]. However, the theory and simplified design method for SCFT in tension loads have not been established. In this work, we derive a theoretical formulation to explicitly explain and mimic the tensile response of SCFT members.

Recently, significant research has been developed for elaborate simulation of crack behavior of concrete material, including approaches based on the eXtended Finite Element Method (XFEM) [20], the element-free Garlerkin method (EFG) [21], the Strong Discontinuity embedded Approach (SDA) such as cracking particles and cracking elements [22–27], explicit Phase Field Method [28], and Nonlocal Operator Method [29,30]. These methods have been successfully adopted to predict the elaborate evolution of cracking and other damage to concrete material under complicated loading history and show good agreement with test results. This study is focused on deriving the theoretical solution of SCFT member from an engineering point of view and also for simplified constitutive model based on continuum approach. Besides, the “fractal cracking” behavior of the concrete in SCFT members under tension is observed and explained for the tests in this work, as briefly shown in Fig. 2. Hundreds of cracks, with a crack spacing ranging from 10 to 20 mm, were observed in the concrete core of each SCFT specimen after the test by removing the steel tube, significantly different from behaviour observed in traditional reinforced concrete (RC) members tested under uniaxial loads with large crack spacing above 50 mm. These large numbers of cracks were found to have distinctly different crack widths. Hence, in this work, we classify these cracks into several levels according to their different crack widths and introduce the concept of fractal cracking to describe this newly observed cracking phenomenon. The concept of fractal cracking is adopted since the concrete inside SCFT members under tension are subdivided by cracks into sub-SCFTs, and those sub-SCFTs generate new smaller cracks recursively during monotonic axial tensile loading, as shown in Fig. 2.

We develop a three-dimensional (3D) theory to describe tensile performance of SCFT members following Hooke’s Law, the fracture mechanics of concrete, the von Mises yield criterion of steel, and Coulomb’s law of friction for the steel-concrete interface. Effective formulas focused on design needs are then derived based on the proposed theory to quantify the strength and stiffness enhancements of SCFTs under tension. Moreover, equivalent uniaxial constitutive laws are developed for FEA, implicitly accounting for the salient behavior of tensile SCFT members, such as the confining-strengthening effect via modifying the yield strength of steel, the confining-stiffening effect via modifying the modulus of steel, and the tension-stiffening effect via calculating the residual stress of concrete. Following the equivalent uniaxial constitutive models of concrete and steel, the fractal cracking law is proposed through the steel and concrete interface via friction and confining to quantitative model the fractal cracking phenomenon. Utilizing the developed equivalent uniaxial constitutive model and fractal cracking law, a fiber beam-column finite element is established. Finally, the reliability and accuracy of the proposed theory are validated by test data in both global curves and at microscopic level. Work on the confining-strengthening ratio α_strength, confining-stiffening ratio α_stiffness, transverse strain over longitudinal strain ratio R, and fractal cracking law are highlighted.

2 Theoretical model for SCFT under tension

2.1 Definition of parameters

Xu et al. [19] reported theoretical model for tension behavior of CCFT. The objective of this section is to derive the theoretical solution of SCFT under uniaxial tension load and to propose the confine-ellipse model for SCFT members based on a previous test program. Figure 3 shows the stress and strain distribution of tensile SCFT members. As shown in Fig. 3, the average longitudinal strain of steel tube over the length equal to half of the crack spacing is denoted as {\overline{\epsilon }}_{\text{l}}. Based on strain compatibility, the average strain of concrete is formulated as follows:

(1){\overline{\epsilon }}_{\text{l}}=\frac{2}{L_{\text{m}}}{\int }_0^{\frac{L_{\text{m}}}{2}}{\epsilon }_{\text{s,l}}(x)\text{d}x,

(2){\overline{\epsilon }}_{\text{l}}=\frac{2}{L_{\text{m}}}\left({\int }_0^{\frac{L_{\text{m}}}{2}}{\epsilon }_{\text{c,l}}(x)\text{d}x+\frac{\omega }{2}\right),

where L_m denotes the average crack spacing, ω denotes crack width, ε_c,l and ε_s,l denote the longitudinal strain of concrete and steel tube, respectively.

Similar to the definition of average transverse strain {\overline{\epsilon }}_{\text{l}}, the average steel hoop strain is defined as {\overline{\epsilon }}_{\text{t}} and the average concrete radial strain is defined as {\overline{\epsilon }}_{\text{r}}.

(3){\overline{\epsilon }}_{\text{t}}=\frac{2}{L_{\text{m}}}{\int }_0^{\frac{L_{\text{m}}}{2}}{\epsilon }_{\text{s,t}}(x)\text{d}x,

(4){\overline{\epsilon }}_{\text{r}}=\frac{2}{L_{\text{m}}}{\int }_0^{\frac{L_{\text{m}}}{2}}{\epsilon }_{\text{c,r}}(x)\text{d}x,

where ε_s,t denotes hoop strain of steel tube, ε_c,r denotes concrete.

The average steel longitudinal stress {\overline{\sigma }}_{\text{s,l}}, the average steel hoop stress {\overline{\sigma }}_{\text{s,t}}, the average longitudinal stress of concrete {\overline{\sigma }}_{\text{c,l}} , and the average radial stress of concrete {\overline{\sigma }}_{\text{c,r}} are the average results in half crack spacing [19].

(5){\overline{\sigma }}_{\text{s,l}}=\frac{2}{L_{\text{m}}}{\int }_0^{\frac{L_{\text{m}}}{2}}{\sigma }_{\text{s,l}}(x)\text{d}x,

(6){\overline{\sigma }}_{\text{s,t}}=\frac{2}{L_{\text{m}}}{\int }_0^{\frac{L_{\text{m}}}{2}}{\sigma }_{\text{s,t}}(x)\text{d}x,

(7){\overline{\sigma }}_{\text{c,l}}=\frac{2}{L_{\text{m}}}{\int }_0^{\frac{L_{\text{m}}}{2}}{\sigma }_{\text{c,l}}(x)\text{d}x,

(8){\overline{\sigma }}_{\text{c,r}}=\frac{2}{L_{\text{m}}}{\int }_0^{\frac{L_{\text{m}}}{2}}{\sigma }_{\text{c,r}}(x)\text{d}x,

where σ_s,l and σ_s,t denote longitudinal stress and hoop stress of steel tube, respectively; σ_c,l and σ_c,r denote longitudinal stress and radial stress of concrete, respectively.

In previous research [19], the theoretical study was completed for CCFTs. In this study, an analogy between the SCFT section and an equivalent CCFT section is used, and the section hoop strain of the SCFT section is averaged as follows.

First, hoop strain of steel tube is formulated as:

(9){\epsilon }_{\text{s,t}}(x)=\frac{1}{A_{\text{s}}}\underset{A_{\text{s}}}{\iint }{\epsilon }_{\text{s,t}}(x,y,\mathit{\text{z}})\text{d}{A}_{\text{s}},

where ε_s,t (x,y,z) denotes the hoop strain of steel tube at section x. ε_s,t (x) denotes the sectional average result of steel hoop strain, and A_s denotes the steel tube section area.

Second, the radial strain of concrete is formulated as:

(10){\epsilon }_{\text{c,r}}(x)=\frac{1}{A_{\text{c}}}\underset{A_{\text{c}}}{\iint }{\epsilon }_{\text{c,}\mathit{\text{z}}}(x,y,\mathit{\text{z}})\text{d}{A}_{\text{c}}=\frac{1}{A_{\text{c}}}\underset{A_{\text{c}}}{\iint }{\epsilon }_{\text{c,}y}(x,y,\mathit{\text{z}})\text{d}{A}_{\text{c}},

where ε_c,y(x,y,z) and ε_c,z(x,y,z) denotes the concrete strain component in the y-direction and z-direction, respectively. ε_c,r(x) denotes the average radial strain of concrete at section x, A_c denotes the concrete section area.

Third, the average steel hoop stress is formulated as:

(11){\sigma }_{\text{s,t}}(x)=\frac{1}{A_{\text{s}}}\underset{A_{\text{s}}}{\iint }{\sigma }_{\text{s,t}}(x,y,\mathit{\text{z}})\text{d}{A}_{\text{s}},

where σ_s,t(x,y,z) denotes the steel hoop stress and σ_s,t (x) denotes the average steel hoop stress at section x.

Fourth, the average radial stress of infilled concrete is formulated as

(12){\sigma }_{\text{c,r}}(x)=\frac{1}{A_{\text{c}}}\underset{A_{\text{c}}}{\iint }{\sigma }_{\text{c,}\mathit{\text{z}}}(x,y,\mathit{\text{z}})\text{d}{A}_{\text{c}}=\frac{1}{A_{\text{c}}}\underset{A_{\text{c}}}{\iint }{\sigma }_{\text{c,}y}(x,y,\mathit{\text{z}})\text{d}{A}_{\text{c}},

where σ_c,y(x,y,z) and σ_c,z(x,y,z) denote the stress component of concrete in SCFT in the y-direction and z- direction, respectively, σ_c,r(x) denotes the average radial stress of infilled concrete in SCFT.

The average crack spacing L_m is obtained by bond-slip law. First, for the same level of crack (the cracks occurring at the same load), the fractal influence is not included, and the bond-slip law is adopted to determine the minimum possible crack spacing L_min:

(13)4D{L}_{min}\cdot \overline{\tau }=D^2\cdot f,

(14)L_{min}=\frac{f_{\text{t}}}{4\overline{\tau }}\cdot D,

where D is the section length of SCFT, \overline{\tau } stands for the average interface friction stress, f_t denotes the tensile strength of concrete.

As recommended by Hognestad [31], the maximum possible crack spacing is twice the minimum crack spacing, and the average crack spacing L_m is formulated as follows:

(15)L_{max}=2L_{min}=\frac{f_{\text{t}}}{2\overline{\tau }}\cdot D,

(16)L_{\mathrm{m}}=\frac{L_{max}+L_{min}}{2}=1.5L_{min}=\frac{3f_{\text{t}}}{8\overline{\tau }}\cdot D.

2.2 Basic equations

Based on Hooke’s Law, the strain compatibility, and the section force equilibrium, the governing equations for the SCFT under uniaxial tension are formulated. First, Hooke’s Law before cracking is formulated as:

(17){\epsilon }_{\text{s,l}}(x)=\frac{{\sigma }_{\text{s,l}}(x)}{E_{\text{s}}}-{\nu }_{\text{s}}\frac{{\sigma }_{\text{s,t}}(x)}{E_{\text{s}}},

(18){\epsilon }_{\text{s,t}}(x)=\frac{{\sigma }_{\text{s,t}}(x)}{E_{\text{s}}}-{\nu }_{\text{s}}\frac{{\sigma }_{\text{s,l}}(x)}{E_{\text{s}}},

(19){\epsilon }_{\text{c,l}}(x)=\frac{{\sigma }_{\text{c,l}}(x)}{E_{\text{c}}}-2{\nu }_{\text{c}}\frac{{\sigma }_{\text{c,r}}(x)}{E_{\text{c}}},

(20){\epsilon }_{\text{c,r}}(x)=(1-{\nu }_{\text{c}})\frac{{\sigma }_{\text{c,r}}(x)}{E_{\text{c}}}-{\nu }_{\text{c}}\frac{{\sigma }_{\text{c,l}}(x)}{E_{\text{c}}},

where ε_c,l(x) denotes the longitudinal elastic tensile strain of concrete.

The strain compatibility is formulated as:

(21){\epsilon }_{\text{c,r}}(x)={\epsilon }_{\text{s,t}}(x).

The section equilibrium function is formulated as:

(22)-{\sigma }_{\text{c,r}}(x)\cdot D=2\chi t\cdot {\sigma }_{\text{s,t}}(x),

where χ is an analogy factor based on previous research on CCFT to reflect the section confinement effect. For CCFT, χ equals 1.0 [19]. For SCFT, however, this factor χ needs to be introduced because it may be too hard to derive this factor theoretically. Based on data regression in the following discussion, selecting χ to be 1.9 will make the theoretical result of steel transverse strain to longitudinal strain ratio (or Poisson ratio) to best fit with test results.

Based on previous equations, the average stress-strain relationship can be formulated as:

(23){\overline{\epsilon }}_{\text{l}}=\frac{{\overline{\sigma }}_{\text{s,l}}}{E_{\text{s}}}-{\nu }_{\text{s}}\frac{{\overline{\sigma }}_{\text{s,t}}}{E_{\text{s}}},

(24){\overline{\epsilon }}_{\text{t}}=\frac{{\overline{\sigma }}_{\text{s,t}}}{E_{\text{s}}}-{\nu }_{\text{s}}\frac{{\overline{\sigma }}_{\text{s,l}}}{E_{\text{s}}},

(25){\overline{\epsilon }}_{\text{l}}-\frac{\omega }{L_{\text{m}}}=\frac{{\overline{\sigma }}_{\text{c,l}}}{E_{\text{c}}}-2{\nu }_{\text{c}}\frac{{\overline{\sigma }}_{\text{c,r}}}{E_{\text{c}}},

(26){\overline{\epsilon }}_{\text{r}}=(1-{\nu }_{\text{c}})\frac{{\overline{\sigma }}_{\text{c,r}}}{E_{\text{r}}}-{\nu }_{\text{c}}\frac{{\overline{\sigma }}_{\text{c,l}}}{E_{\text{r}}},

(27){\overline{\epsilon }}_{\text{r}}={\overline{\epsilon }}_{\text{t}}.

The section equilibrium formula is also obtained as:

(28)-{\overline{\sigma }}_{\text{c,r}}\cdot D=2\chi t\cdot {\overline{\sigma }}_{\text{s,t}}.

In addition, the axial tension force P is formulated as:

(29)P=4Dt\cdot {\sigma }_{\text{s,l}}(x)+D^2\cdot {\sigma }_{\text{c,l}}(x)=4Dt\cdot {\overline{\sigma }}_{\text{s,l}}+D^2\cdot {\overline{\sigma }}_{\text{c,l}}.

2.3 Initial cracking

When the average longitudinal tensile stress of concrete {\overline{\sigma }}_{\text{c,l}} reaches f_t, concrete cracking occurs with very small or negligible crack width ( \omega =0). By adding this condition into the previous equations Eqs. (23)–(29), the average stress and strain results at initial cracking load are obtained as follows:

(30){\overline{\epsilon }}_{\text{l}}={\displaystyle \frac{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}^2)}{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}{\nu }_{\text{s}})}}\cdot \frac{f_{\text{t}}}{E_{\text{c}}},

(31){\overline{\epsilon }}_{\text{r}}={\overline{\epsilon }}_{\text{t}}=-{\displaystyle \frac{{\nu }_{\text{c}}(1-{{\nu }_{\text{s}}}^2)+{\displaystyle \frac{2\chi nt}{D}}{\nu }_{\text{s}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}^2)}{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}{\nu }_{\text{s}})}}\cdot \frac{f_{\text{t}}}{E_{\text{c}}},

(32){\overline{\sigma }}_{\text{s,l}}=\frac{1-{\nu }_{\text{c}}{\nu }_{\text{s}}+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}^2)}{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}{\nu }_{\text{s}})}\cdot n{f}_{\text{t}},

(33){\overline{\sigma }}_{\text{s,t}}=\frac{({\nu }_{\text{s}}-{\nu }_{\text{c}})}{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}{\nu }_{\text{s}})}\cdot n{f}_{\text{t}},

(34){\overline{\sigma }}_{\text{c,r}}=-\frac{{\displaystyle \frac{2\chi nt}{D}}({\nu }_{\text{s}}-{\nu }_{\text{c}})}{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}{\nu }_{\text{s}})}\cdot f_{\text{t}},

where n denotes the elastic modulus of steel divided by elastic modulus of concrete.

2.4 After cracking and before steel tube yielding

After cracking initiates in SCFT, the interface between concrete and steel exhibits bond-slip and makes the stress and strain components of concrete and steel non-uniformly distributed in the longitudinal direction. Therefore, we cannot directly solve the average strain and average stress components from Eqs. (23)–(29). The equilibrium equation for the interface between concrete and steel is added as shown in Eqs. (35)–(37). In this study, Eq. (36) adopts the Mohr−Coulomb Friction Law,τ(x) denotes the interface frictional force in the longitudinal direction, σ_c,r(x) denotes the radial compressive stress of concrete, k denotes the friction coefficient, τ₀ denotes the initial bond stress.

(35)4Dt\cdot \frac{\text{d}{\sigma }_{\text{s,l}}(x)}{\text{d}x}=-D^2\cdot \frac{\text{d}{\sigma }_{\text{c,l}}(x)}{\text{d}x},

(36)\tau (x)=-k{\sigma }_{\text{c,r}}(x)+{\tau }_0,

(37)\tau (x)=t\cdot \frac{\text{d}{\sigma }_{\text{s,l}}(x)}{\text{d}x}.

Based on Eqs. (17)–(22) and Eqs. (35)–(37), the post-cracking stress and strain results are obtained as Eqs. (38)–(45):

(38){\epsilon }_{\text{s,l}}(x)=\frac{1}{E_{\text{s}}}\cdot \left[\frac{(1-\gamma {\nu }_{\text{s}})\alpha }{\gamma }\cdot \exp \left(\frac{2\chi k\gamma }{D}x\right)+\left(\frac{D{\nu }_{\text{s}}{\tau }_0}{2\chi kt}+C\right)\right],

(39)\begin{array}{rl}{\epsilon }_{\text{c,l}}(x)=& \frac{1}{E_{\text{c}}}\cdot [25-\frac{4\alpha t(1-\chi \gamma {\nu }_{\text{c}})}{D\gamma }\cdot \exp \left(\frac{2\chi k\gamma }{D}x\right)\\ & +\frac{1+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}^2)}{k{\nu }_{\text{c}}{\displaystyle \frac{2\chi nt}{D}}}\cdot {\tau }_0+{\displaystyle \frac{{\nu }_{\text{s}}}{n{\nu }_{\text{c}}}}\cdot C],\end{array}

(40)\begin{array}{rl}{\epsilon }_{\text{c,r}}(x)=& {\epsilon }_{\text{s,t}}(x)=\frac{1}{E_{\text{s}}}\cdot [\frac{(\gamma -{\nu }_{\text{s}})\alpha }{\gamma }\cdot \exp \left(\frac{2\chi k\gamma }{D}x\right)\\ & -\left(\frac{D{\tau }_0}{2\chi kt}+{\nu }_{\text{s}}C\right)],\end{array}

(41){\sigma }_{\text{s,l}}(x)=\frac{\alpha }{\gamma }\cdot \exp \left(\frac{2\chi k\gamma }{D}x\right)+C,

(42){\sigma }_{\text{s,t}}(x)=\alpha \cdot \exp \left(\frac{2\chi k\gamma }{D}x\right)-\frac{D{\tau }_0}{2\chi kt},

(43)\begin{array}{rl}{\sigma }_{\text{c,l}}(x)=& -\frac{4\alpha t}{D\gamma }\cdot \exp \left(\frac{2\chi k\gamma }{D}x\right)\\ & +\frac{1+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}})}{k{\nu }_{\text{c}}{\displaystyle \frac{2\chi nt}{D}}}\cdot {\tau }_0+{\displaystyle \frac{{\nu }_{\text{s}}}{n{\nu }_{\text{c}}}}\cdot C,\end{array}

(44){\sigma }_{\text{c,r}}(x)=-\frac{2\chi \alpha t}{D}\cdot \exp \left(\frac{2\chi k\gamma }{D}x\right)+\frac{{\tau }_0}{k},

(45)\tau (x)=\frac{2\chi \alpha kt}{D}\cdot \exp \left(\frac{2\chi k\gamma }{D}x\right).

In Eqs. (38)–(45), α and C are two coefficients introduced for solving ordinary differential equations, γ denotes the ratio between transverse strain increment and longitudinal stress increments of steel tube as formulated in Eq. (46):

(46)\gamma =\frac{{\nu }_{\text{s}}+{\displaystyle \frac{4nt}{D}}{\nu }_{\text{c}}}{1+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}})},

(47)\frac{\text{d}{\sigma }_{\text{s,t}}(x)}{\text{d}x}=\gamma \cdot \frac{\text{d}{\sigma }_{\text{s,l}}(x)}{\text{d}x}.

To solve for the coefficients α and C in Eqs. (38)–(45), the additional boundary condition is as shown in Eq. (48):

(48)\{\begin{array}{rl}& {\sigma }_{\text{c,l}}\left(x=\frac{L_{\text{m}}}{2}\right)={\sigma }_{\text{t}}(\omega )\\ & {\overline{\epsilon }}_{\text{l}}-\frac{\omega }{L_{\text{m}}}=\frac{{\overline{\sigma }}_{\text{c,l}}}{E_{\text{c}}}-2{\nu }_{\text{c}}\frac{{\overline{\sigma }}_{\text{c,r}}}{E_{\text{c}}}\end{array},

where ω is the current crack width, {\sigma }_{\text{t}}(\omega ) denotes the tensile stress of plain concrete when the crack width equals ω.

Substituting Eqs. (38)–(45) into Eq.(48), the coefficients of α and C are formulated as:

(49)\alpha =\frac{{\displaystyle \frac{E_{\text{c}}\omega }{L_{\text{m}}}}+{\displaystyle \frac{{\nu }_{\text{s}}-{\nu }_{\text{c}}}{{\nu }_{\text{s}}}}\cdot {\sigma }_{\text{t}}(\omega )+{\displaystyle \frac{1-{\nu }_{\text{s}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}{\nu }_{\text{s}})}{k{\nu }_{\text{s}}\cdot {\displaystyle \frac{2\chi nt}{D}}}}\cdot {\tau }_0}{{\displaystyle \frac{1-\gamma {\nu }_{\text{s}}+{\displaystyle \frac{4nt}{D}}(1-\chi \gamma {\nu }_{\text{c}})}{\chi nk{L}_{\text{m}}{\gamma }^2/\phantom{\chi nk{L}_{\text{m}}{\gamma }^{2}D}D}}\cdot \left(\exp \left({\displaystyle \frac{\chi k\gamma L_{\text{m}}}{D}}\right)-1\right)-{\displaystyle \frac{4t({\nu }_{\text{s}}-{\nu }_{\text{c}})}{{\nu }_{\text{s}}D\gamma }}\cdot \exp \left({\displaystyle \frac{\chi k\gamma L_{\text{m}}}{D}}\right)},

(50)C=\frac{n{\nu }_{\text{c}}}{{\nu }_{\text{s}}}\cdot \left[\frac{4t\alpha }{D\gamma }\cdot \exp \left({\displaystyle \frac{\chi k\gamma L_{\text{m}}}{D}}\right)-{\displaystyle \frac{1+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}})}{k{\nu }_{\text{c}}\cdot {\displaystyle \frac{2\chi nt}{D}}}}\cdot {\tau }_0+{\sigma }_{\text{t}}(\omega )\right].

The axial tension stress of plain concrete {\sigma }_{\text{t}}(\omega ) adopts the “β-ellipse” model, which was proposed and validated by a large number of previous tests by Xu et al. [32]. The formula for the post-crack concrete stress and crack width is formulated as Eq. (51):

(51)

By substituting Eqs. (1)–(12) into Eqs. (38)–(45), the results for average stress and average strain are formulated as Eqs. (52)–(58):

(52){\overline{\epsilon }}_{\text{l}}=\frac{\alpha D(1-\gamma {\nu }_{\text{s}})}{\chi k{\gamma }^2{L}_{\text{m}}{E}_{\text{s}}}\cdot \left(\exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-1\right)+\frac{1}{E_{\text{s}}}\cdot \left(\frac{D{\nu }_{\text{s}}{\tau }_0}{2\chi kt}+C\right),

(53){\overline{\epsilon }}_{\text{r}}={\overline{\epsilon }}_{\text{t}}=\frac{\alpha D(\gamma -{\nu }_{\text{s}})}{\chi k{\gamma }^2{L}_{\text{m}}{E}_{\text{s}}}\cdot \left(\exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-1\right)-\frac{1}{E_{\text{s}}}\cdot \left(\frac{D{\tau }_0}{2\chi kt}+{\nu }_{\text{s}}C\right),

(54){\overline{\sigma }}_{\text{s,l}}=\frac{\alpha D}{\chi k{\gamma }^2{L}_{\text{m}}}\cdot \left(\exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-1\right)+C,

(55){\overline{\sigma }}_{\text{s,t}}=\frac{\alpha D}{\chi k\gamma L_{\text{m}}}\cdot \left(\exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-1\right)-\frac{D{\tau }_0}{2\chi kt},

(56)\begin{array}{rl}{\overline{\sigma }}_{\text{c,l}}=& \frac{4\alpha t}{D\gamma }\cdot \exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-\frac{4\alpha t}{\chi k{\gamma }^2{L}_{\text{m}}}\\ & \cdot \left(\exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-1\right)+{\sigma }_{\text{t}}(\omega ),\end{array}

(57){\overline{\sigma }}_{\text{c,r}}=-\frac{2\alpha t}{k\gamma L_{\text{m}}}\cdot \left(\exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-1\right)+\frac{{\tau }_0}{k},

(58)\overline{\tau }=\frac{2\alpha t}{\gamma L_{\text{m}}}\cdot \left(\exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-1\right).

In Eq. (58), \overline{\tau } denotes the average bond stress between steel tube and concrete.

2.5 Yielding

The steel tube yields first at the location of cracking. According to von-Mises yield criterion, the governing equation for yield capacity of steel tube is formulated in Eq. (59):

(59)\begin{array}{rl}& \frac{\sqrt{2}}{2}\cdot \sqrt{{\left[{\sigma }_{\text{s,l}}\left(\frac{L_{\text{m}}}{2}\right)-{\sigma }_{\text{s,t}}\left(\frac{L_{\text{m}}}{2}\right)\right]}^2+{\left[{\sigma }_{\text{s,l}}\left(\frac{L_{\text{m}}}{2}\right)\right]}^2+{\left[{\sigma }_{\text{s,t}}\left(\frac{L_{\text{m}}}{2}\right)\right]}^2}\\ & =f_{\text{y}},\end{array}

where {\sigma }_{\text{s,l}}\left({\displaystyle \frac{L_{\text{m}}}{2}}\right) and {\sigma }_{\text{s,t}}\left({\displaystyle \frac{L_{\text{m}}}{2}}\right) denote the steel longitudinal stress and steel hoop stress, respectively, f_y is the yield strength of steel material.

At the yield capacity, it is assumed that the concrete tensile stress softens to 0 at the crack section ( {\sigma }_{\text{c,l}}\left({\displaystyle \frac{L_{\text{m}}}{2}}\right) equals 0). Substituting this condition into Eqs. (17)–(22), the relationship between steel hoop stress and steel longitudinal stress of is formulated as Eq. (60):

(60){\sigma }_{\text{s,t}}\left(\frac{L_{\text{m}}}{2}\right)=\lambda \cdot {\sigma }_{\text{s,l}}\left(\frac{L_{\text{m}}}{2}\right),

(61)\lambda =\frac{{\nu }_{\text{s}}}{1+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}})}.

Substituting Eq. (60) into Eq. (59), the yield capacity of SCFT in tensile load is formulated as follows.

Up to this point, the whole-process governing equations for the tension behavior of SCFT has now been developed in this paper.

(62){\sigma }_{\text{s,l}}\left(\frac{L_{\text{m}}}{2}\right)=\frac{1}{\sqrt{1-\lambda +{\lambda }^2}}\cdot f_{\text{y}}.

3 Simplified formulas for design

Based on the analytical theory proposed above, effective formulas for calculating the capacity and stiffness enhancements of SCFT members under tension are derived for practical design purposes.

3.1 Strength enhancement ratio α_strength

The strength enhancement of SCFT members under tension is due to the confine-strengthening effect. The so-called confine-strengthening effect means the concrete confinement due to the shrinkage of the steel tube under longitudinal tension. This confinement causes the steel in biaxial tensile stress state and hence higher longitudinal steel stress exists at the crack {\sigma }_{\text{s,l}}\left({\displaystyle \frac{L_{\text{m}}}{2}}\right) compared to the uniaxial yield strength of steel f_y as previously given in Eq. (62). The tensile strength of SCFT members P_SCFT is governed by the longitudinal steel stress at the crack {\sigma }_{\text{s,l}}\left({\displaystyle \frac{L_{\text{m}}}{2}}\right) as shown in Eq. (63). The strength enhancement ratio α_strength is formulated by subdividing the tensile strength of SCFT by that of hollow steel tube as:

(63)P_{\text{SCFT}}=\frac{1}{\sqrt{1-\lambda +{\lambda }^2}}\cdot f_{\text{y}}\cdot 4Dt,

(64)P_{\text{Hollow}}=f_{\text{y}}\cdot 4Dt,

(65){\alpha }_{\text{strength}}=\frac{P_{\text{SCFT}}}{P_{\text{Hollow}}}=\frac{1}{\sqrt{1-\lambda +{\lambda }^2}}.

3.2 Stiffness enhancement ratio α_stiffness

The stiffness enhancement of SCFT members under tension is attributed to a combination of two stiffening effects, i.e., the tension-stiffening effect and confine-stiffening effect. The tension-stiffening effect is similar to the tension stiffening effect of a tensile RC member [33]. The confine-stiffening effect results in the longitudinal steel strain being smaller when the longitudinal steel stress is the same for the steel of SCFTs compare to the hollow tubes. This is due to the bi-axial tensile stress state of the steel tube of SCFTs. In a similar way as for the confine-strengthening effect, this biaxial tension stress state is also attributed to the difference of Poisson ratio between steel tube and infilled concrete.

To derive the stiffness enhancement ratio, the stiffness is defined as the secant stiffness at 50% of the strength [34]. Therefore, the axial tension stiffness of a SCFT member K_SCFT can be obtained as

(66)K_{\text{SCFT}}=\frac{0.5P_{\text{SCFT}}}{{\overline{\epsilon }}_{\text{l}}(0.5P_{\text{SCFT}})}.

At this point, corresponding to half of the strength, it is assumed that the crack width is fully open, i.e., \omega ⩾{\omega }_{\mathrm{u}}, and hence the longitudinal concrete stress at the crack is zero, i.e., {\sigma }_{\text{c,l}}\left({\displaystyle \frac{L_{\text{m}}}{2}}\right)=0. Therefore the longitudinal steel stress at the crack {\sigma }_{\text{s,l}}\left({\displaystyle \frac{L_{\text{m}}}{2}}\right) is {\displaystyle \frac{1}{2\sqrt{1-\lambda +{\lambda }^2}}}\cdot f_{\text{y}}. Substituting these two boundary conditions into Eqs. (41) and (43) provides the values of the coefficients α and C:

(67)\begin{array}{rl}\alpha (0.5P_{\text{SCFT}})=& \left(\frac{\lambda }{2\sqrt{1-\lambda +{\lambda }^2}}\cdot f_{\text{y}}+\frac{D{\tau }_0}{2\chi kt}\right)\\ & \cdot \exp \left(-\frac{\chi k\gamma L_{\text{m}}}{D}\right),\end{array}

(68)C(0.5P_{\text{SCFT}})=\frac{1}{2\sqrt{1-\lambda +{\lambda }^2}}\cdot f_{\text{y}}-\frac{\alpha }{\gamma }\cdot \exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right).

Substituting Eqs. (67) and (68) into Eq. (52) solves {\overline{\epsilon }}_{\text{l}}(0.5P_{\text{y,SCFT}}). Then K_SCFT can be solved as:

(69)K_{\text{SCFT}}={\displaystyle \frac{4Dt{E}_s}{\left(1-{\displaystyle \frac{\lambda }{\gamma }}\right)+{\displaystyle \frac{D(1-\gamma {\nu }_{\text{s}})}{\chi k{\gamma }^2{L}_{\text{m}}}}\cdot \left(\lambda +{\displaystyle \frac{D{\tau }_0}{\chi kt{f}_{\text{y}}\sqrt{1-\lambda +{\lambda }^2}}}\right)\cdot \left(1-\exp \left({\displaystyle \frac{-\chi k\gamma L_{\text{m}}}{D}}\right)\right)+\left({\nu }_{\text{s}}-{\displaystyle \frac{1}{\gamma }}\right)\cdot {\displaystyle \frac{D{\tau }_0\sqrt{1-\lambda +{\lambda }^2}}{\chi kt{f}_{\text{y}}}}}}.

Then, noting that the axial tensile stiffness of a hollow tube K_Hollow is formulated in Eq. (70), the stiffness enhancement ratio α_stiffness is derived as:

(70)K_{\text{Hollow}}=E_{\text{s}}\cdot 4Dt,

(71){\alpha }_{\text{stiffness}}={\displaystyle \frac{K_{\text{SCFT}}}{K_{\text{Hollow}}}}={\displaystyle \frac{1}{\left(1-{\displaystyle \frac{\lambda }{\gamma }}\right)+{\displaystyle \frac{D(1-\gamma {\nu }_{\text{s}})}{\chi k{\gamma }^2{L}_{\mathrm{m}}}}\cdot \left(\lambda +{\displaystyle \frac{D{\tau }_0}{\chi kt{f}_{\text{y}}\sqrt{1-\lambda +{\lambda }^2}}}\right)\cdot \left(1-\exp ({\displaystyle \frac{-\chi k\gamma L_{\text{m}}}{D}})\right)+\left({\nu }_{\text{s}}-{\displaystyle \frac{1}{\gamma }}\right)\cdot {\displaystyle \frac{D{\tau }_0\sqrt{1-\lambda +{\lambda }^2}}{\chi kt{f}_{\text{y}}}}}}.

In Eq. (71), the value of {\displaystyle \frac{D{\tau }_0}{kt{f}_{\text{y}}}} is close to 0, so Eq. (71) can be simplified for the design formula of stiffness enhancement ratio:

(72)\begin{array}{rl}{\alpha }_{\text{stiffness}}& ={\displaystyle \frac{K_{\text{SCFT}}}{K_{\text{Hollow}}}}\\ & ={\displaystyle \frac{1}{\left(1-{\displaystyle \frac{\lambda }{\gamma }}\right)+{\displaystyle \frac{D\lambda (1-\gamma {\nu }_{\text{s}})}{\chi k{\gamma }^2{L}_{\text{m}}}}\cdot \left(1-\exp \left({\displaystyle \frac{-\chi k\gamma L_{\text{m}}}{D}}\right)\right)}}.\end{array}

Equation (72) may be still complicated for engineering practice because the triaxial stress state is elaborately considered in derivation. To meet the requirement of engineering design, and ensure the accuracy and efficiency of the theoretical model, a simplified design formula for the effective stiffness is developed. The assumption is that the average longitudinal concrete stress in Eq. (56) can be re-formulated as:

(73){\overline{\sigma }}_{\text{c,l}}=\varsigma \cdot f_{\text{t}},

where \varsigma is average coefficient to consider the complicated non-uniform longitudinal stress distribution of concrete, f_t is the concrete tensile strength. Subsequently, this assumption is used to derive the simplified design formula for effective stiffness.

Based on Eqs. (23)–(28) and Eq. (73), the following solution is obtained:

(74){\overline{\sigma }}_{\text{s,t}}=\lambda {\overline{\sigma }}_{\text{s,l}}-\frac{n{\nu }_{\text{c}}}{{\nu }_{\text{s}}}\lambda {\overline{\sigma }}_{\text{c,l}},

where λ is the ratio between transverse stress and longitudinal stress and is calculated using Eq. (61).

Substituting Eq. (74) into Eq.(23):

(75){\overline{\epsilon }}_{\text{l}}=\frac{{\overline{\sigma }}_{\text{s,l}}}{E_{\text{s}}}-{\nu }_{\text{s}}\lambda \frac{{\overline{\sigma }}_{\text{s,l}}}{E_{\text{s}}}+{\nu }_{\text{c}}\lambda \frac{{\overline{\sigma }}_{\text{c,l}}}{E_{\text{c}}}.

Substituting Eq. (73) into Eq. (75):

(76){\overline{\epsilon }}_{\text{l}}=(1-{\nu }_{\text{s}}\lambda )\frac{{\overline{\sigma }}_{\text{s,l}}}{E_{\text{s}}}+{\nu }_{\text{c}}\lambda \varsigma \frac{f_{\text{t}}}{E_{\text{c}}}.

In this study, the effective stiffness of specimen is defined as the secant stiffness corresponding to half of ultimate capacity, the average longitudinal stress of steel tube can be obtained from Eq. (29):

(77){\overline{\sigma }}_{\text{s,l}}=0.5{\alpha }_{\text{strength}}{f}_{\text{y}}-\frac{D}{4t}\varsigma f_{\text{t}}.

Substituting Eq. (77) into Eq. (76), the average longitudinal strain at half of ultimate capacity is derived as:

(78){\overline{\epsilon }}_{\text{l}}=(1-{\nu }_{\text{s}}\lambda )\frac{0.5{\alpha }_{\text{strength}}{f}_{\text{y}}-{\displaystyle \frac{D}{4t}}\varsigma f_{\text{t}}}{E_{\text{s}}}+{\nu }_{\text{c}}\lambda \varsigma \frac{f_{\text{t}}}{E_{\text{c}}}.

From Eqs. (66), (70), and (78), the formula for stiffness enhancement ratio α_stiffness is formulated as:

(79)\begin{array}{rl}& {\alpha }_{\text{stiffness}}=\\ & \frac{1}{\left(1-{\nu }_{\text{s}}\lambda \right)\left(1-{\displaystyle \frac{\varsigma D{f}_{\text{t}}}{2t{f}_{\text{y}}}}\cdot \sqrt{1-\lambda +{\lambda }^2}\right)+2n{\nu }_{\text{c}}\lambda {\displaystyle \frac{\varsigma f_{\text{t}}}{f_{\text{y}}}}\sqrt{1-\lambda +{\lambda }^2}}.\end{array}

Note that the parameter 2n{\nu }_{\text{c}}\lambda \varsigma \sqrt{1-\lambda +{\lambda }^2}\cdot {\displaystyle \frac{f_{\text{t}}}{f_{\text{y}}}} is relatively small, therefore, the Eq. (79) can be simplified as:

(80){\alpha }_{\text{stiffness}}=\frac{1}{\left(1-{\nu }_{\text{s}}\lambda \right)}\cdot \frac{1}{\left(1-{\displaystyle \frac{\varsigma D{f}_{\text{t}}}{2t{f}_{\text{y}}}}\cdot \sqrt{1-\lambda +{\lambda }^2}\right)},

where the factor \varsigma is average coefficient to consider the complicated non-uniform longitudinal stress distribution of concrete and is recommended to be set as 0.5 based on the comparison between test data and theoretical model.

4 Equivalent constitutive laws for FEA

The equivalent constitutive laws include the uniaxial constitutive models of constituent steel and concrete and the fractal cracking law. The equivalent constitutive laws of concrete and steel are derived based on the proposed analytical theory using the average strain-stress formulation to include the tension-stiffening effect, confine-strengthening effect, and confine-stiffening effect. The fractal cracking law is developed to quantitatively explain and model the fractal cracking phenomenon. The formulations of both the equivalent uniaxial constitutive laws and the fractal cracking law are geared for FE. Utilizing these formulations, a specific fiber beam-column element is established based on the prior work on conventional fiber beam-column elements [35,36] in MSC.Marc to realistically simulate the mechanical behavior and fractal cracking phenomenon of SCFT members under tension.

4.1 Equivalent constitutive law of concrete

4.1.1 Cracking stress and strain

Two characteristic points and three distinct regions are determined for the proposed equivalent uniaxial constitutive law of concrete as shown in Fig. 4. The initial cracking point is denoted as ({\epsilon }_{\text{t}}^{\ast },f_{\text{t}}^{\ast }) and the fully cracking point is denoted as ({\epsilon }_{\text{resi}}^{\ast },f_{\text{resi}}^{\ast }), where {\epsilon }_{\text{t}}^{\ast } and f_{\text{t}}^{\ast } are the cracking strain and cracking stress; {\epsilon }_{\text{resi}}^{\ast } and f_{\text{resi}}^{\ast } are the residual strain and residual stress. The three distinct regions bounded by these two characteristic points are successively the elastic region, the descending region, and the platform region. Detailed derivation and discussion of the formulation of this proposed uniaxial concrete law are presented in the following sub-sections.

Assuming the concrete core cracks when its longitudinal tensile stress reaches its tensile strength f_t, i.e., the cracking stress f_{\text{t}}^{\ast } is equal to f_t, the cracking strain is expressed as Eq. (81) according to Eqs. (30)–(34).

(81){\epsilon }_{\text{t}}^{\ast }=\frac{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{{\nu }_{\text{c}}}^2)}{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}{\nu }_{\text{s}})}\cdot \frac{f_{\text{t}}}{E_{\text{c}}},

where n denotes the ratio of steel Young’s modulus divided by concrete elastic modulus.

Equation (81) indicates that the ratio of the concrete cracking strain of tensile SCFTs {\epsilon }_{\text{t}}^{\ast } over the cracking strain of plain concrete {\displaystyle \frac{f_{\text{t}}}{E_{\text{c}}}} is {\displaystyle \frac{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2nt}{D}}(1-{\nu }_{\text{c}}-2{{\nu }_{\text{c}}}^2)}{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}{\nu }_{\text{s}})}}, and is exclusively dependent on the value of {\displaystyle \frac{2nt}{D}} for SCFT members of various geometrical and material parameters. Noting that {\displaystyle \frac{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2nt}{D}}(1-{\nu }_{\text{c}}-2{{\nu }_{\text{c}}}^2)}{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}{\nu }_{\text{s}})}}\approx 1 for a wide range of the value of {\displaystyle \frac{2nt}{D}} as shown in Fig. 5, the cracking strain is approximately the same as that of plain concrete as follows:

(82){\epsilon }_{\text{t}}^{\ast }=\frac{f_{\text{t}}}{E_{\text{c}}}.

4.1.2 Residual stress and strain

The residual strain {\epsilon }_{\text{resi}}^{\ast } and stress f_{\text{resi}}^{\ast } are determined as the strain and stress corresponding to the time when the primary concrete crack is fully open, i.e., ω = ω_u. Thus, the residual stress f_{\text{resi}}^{\ast } and strain {\epsilon }_{\text{resi}}^{\ast } can be derived according to Eqs. (49), (50), (52), and (56). The positive residual tensile stress f_{\text{resi}}^{\ast } of concrete of SCFT members under tension is a result of the tension-stiffening effect. The tension-stiffening effect causes the concrete between cracks still has the capability of carrying tensile stress and hence the average longitudinal concrete stress should be positive residually after the complete cracking of the primary crack.

(83)f_{\text{resi}}^{\ast }=\frac{4\alpha t}{D\gamma }\cdot \exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-\frac{4\alpha t}{\chi k{\gamma }^2{L}_{\text{m}}}\cdot \left(\exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-1\right),

(84){\epsilon }_{\text{resi}}^{\ast }=\frac{\alpha D(1-\gamma {\nu }_{\text{s}})}{\chi k{\gamma }^2{L}_{\text{m}}{E}_{\text{s}}}\cdot \left(\exp \left(\frac{\chi k\gamma L_{\text{m}}}{D}\right)-1\right)+\frac{1}{E_{\text{s}}}\cdot \left(\frac{D{\nu }_{\text{s}}{\tau }_0}{2\chi kt}+C\right),

where the coefficients α and C are determined based on Eqs. (49) and (50), respectively. The crack width ω needs to be set to the ultimate crack width ω_u.

The descending branch (see Fig. 4) is assumed to be a quarter-ellipse curve, taking the recommendation of Xu et al. [32] expressed as per Eq. (85). The quarter-ellipse curve has shown to be good in mimicking the convex downward characteristic of the tensile load-average strain curve of plain concrete, and particularly convenient in calculating the fracture energy via using the area of a rectangle minus one-quarter of an ellipse.

(85)\begin{array}{rl}& \frac{{({\overline{\sigma }}_{\text{c,l}}-f_{\text{t}}^{\ast })}^2}{{(f_{\text{t}}^{\ast }-f_{\text{resi}}^{\ast })}^2}+\frac{{({\overline{\epsilon }}_{\text{l}}-{\epsilon }_{\text{resi}}^{\ast })}^2}{{({\epsilon }_{\text{resi}}^{\ast }-{\epsilon }_{\text{t}}^{\ast })}^2}=1\text{;}\\ & {\epsilon }_{\text{t}}^{\ast }⩽{\overline{\epsilon }}_{\text{l}}⩽{\epsilon }_{\text{resi}}^{\ast },\text{and}{f}_{\text{resi}}^{\ast }⩽{\overline{\sigma }}_{\text{c,l}}⩽f_{\text{t}}^{\ast }.\end{array}

4.2 Equivalent constitutive law of steel

A bilinear curve adopted by Wang et al. [37] in numerically modeling various steel-concrete composite structural members including SCFTs is utilized for the equivalent steel uniaxial constitutive law to mimic the hardening effect of steel, as shown by the grey line in Fig. 6. Moreover, an initial plastic strain is included to account for the plastic strain induced by the cold-forming process of the steel tube during fabrication as recommended by Refs. [38–40], resulting in the black line in Fig. 6.

4.2.1 Modified yield stress

The equivalent uniaxial constitutive model of steel in tensile SCFT members is different from that of the steel tubes because of the biaxial tension stress (i.e., the confine-strengthening effect). Another difference is the non-uniform distribution of the longitudinal steel stress along the length of L_m due to the tension-stiffening effect. These two different results in equivalent yield stress f_y^* are expressed in Eq. (86) by substituting Eq. (83) to Eq. (29). The first term of Eq. (86) (i.e., {\displaystyle \frac{1}{\sqrt{1-\lambda +{\lambda }^2}}}\cdot f_{\text{y}}) is a reflection of the confine-strengthening effect, whereas the second term of Eq. (86) (i.e., -{\displaystyle \frac{D}{4t}}{f}_{\text{resi}}^{\ast }) reflects the tension-stiffening effect.

(86)f_{\text{y}}^{\ast }=\frac{1}{\sqrt{1-\lambda +{\lambda }^2}}\cdot f_{\text{y}}-\frac{D}{4t}{f}_{\text{resi}}^{\ast }.

4.2.2 Modified modulus

The equivalent uniaxial constitutive model of steel in tensile SCFT members is different from the curve of the reference steel tubes by having a steeper elastic region also because of the bi-axial tensile stress state due to the confine-stiffening effect. According to Eqs. (30)–(37), the modified yield strain {\epsilon }_{\text{y}}^{\ast } corresponding to the modified yield stress f_{\text{y}}^{\ast } is expressed as Eq. (87).

(87){\epsilon }_{\text{y}}^{\ast }=\frac{f_{\text{y}}^{\ast }}{E_{\text{s}}}-{\nu }_{\text{s}}\lambda \frac{f_{\text{y}}^{\ast }}{E_{\text{s}}}+{\nu }_{\text{c}}\lambda \frac{f_{\text{resi}}^{\ast }}{E_{\text{c}}}.

Based on Eqs. (86) and (87), the modified elastic modulus E_{\text{s}}^{\ast } can be derived as Eq. (88).

(88)E_{\text{s}}^{\ast }=\frac{f_{\text{y}}^{\ast }}{{\epsilon }_{\text{y}}^{\ast }}=\frac{1}{1-\lambda {\nu }_{\text{s}}+n\lambda {\nu }_{\text{c}}{\displaystyle \frac{f_{\text{resi}}^{\ast }}{f_{\text{y}}^{\ast }}}}\cdot E_{\text{s}}.

Noting that {\displaystyle \frac{f_{\text{resi}}^{\ast }}{f_{\text{y}}^{\ast }}} is small, Eq. (88) can be simplified as:

(89)E_{\text{s}}^{\ast }=\frac{1}{1-\lambda {\nu }_{\text{s}}}\cdot E_{\text{s}}.

The ratio {\displaystyle \frac{1}{1-\lambda {\nu }_{\text{s}}}} in Eq. (89) distinctly indicates the influence of the confine-stiffening effect which is directly brought by the steel tube due to its bi-axial tensile state. The confine-stiffening effect can be clearly distinguished from the tension-stiffening effect by comparing Eq. (89) with Eq. (86) because the tension-stiffening effect is a result of the concrete core possessing the capability of carrying tensile stresses between cracks. The hardening modulus of hollow steel tube E_h is determined as 0.6% E_s based on the best fit of the hollow tube test result. The use of this ratio of hardening modulus over elastic modulus is extended and applied for the steel of tensile SCFT members in estimating the relationship between its modified elastic modulus E_{\text{s}}^{\ast } and modified hardening modulus E_{\text{h}}^{\ast } (i.e., E_{\text{h}}^{\ast } = 0.6% E_{\text{s}}^{\ast }).

4.2.3 Initial plastic strain

During the fabrication of the cold-formed steel tubes, residual stresses are generated. Due to these residual stresses, the tubes have been shown to have no obvioius yield plateau [38]. As recommended by Refs. [38–40], an initial plastic strain {\epsilon }_{\text{p,initial}}^{\ast } is included to consider the influence of the cold-forming process. {\epsilon }_{\text{p,initial}}^{\ast }=0.0006 is adopted as per Denavit and Hajjar [38]. Then, an unloading point on the backbone curve ({\epsilon }_{\text{history}}^{\ast },{\sigma }_{\text{history}}^{\ast }) is assumed to exist in the history, where {\epsilon }_{\text{history}}^{\ast }={\epsilon }_{\text{y}}^{\ast }+{\epsilon }_{\text{p,initial}}^{\ast } and {\sigma }_{\text{history}}^{\ast }=f_{\text{y}}^{\ast }+E_{\text{h}}^{\ast }\cdot {\epsilon }_{\text{p,initial}}^{\ast }. Thus, the tensile loading of the tubes is considered as reloading from point (0,0) to that historic unloading point. This reloading curve, as given in Eq. (90), is adopted as per Legeron et al. [41]:

(90){\overline{\sigma }}_{\text{s,l}}=E_{\text{s}}^{\ast }\cdot {\overline{\epsilon }}_{\text{l}}-{\left(\frac{{\overline{\epsilon }}_{\text{l}}}{{\epsilon }_{\text{history}}^{\ast }}\right)}^p\cdot \left(E_{\text{s}}^{\ast }\cdot {\epsilon }_{\text{history}}^{\ast }-{\sigma }_{\text{history}}^{\ast }\right),

where the coefficient p={\displaystyle \frac{(E_{\text{s}}^{\ast }-E_{\text{h}}^{\ast })\cdot {\epsilon }_{\text{history}}^{\ast }}{E_{\text{s}}^{\ast }\cdot {\epsilon }_{\text{history}}^{\ast }-{\sigma }_{\text{history}}^{\ast }}}, E_{\mathrm{h}}^{\ast } denotes the hardening modulus.

4.3 Transverse strain to longitudinal strain ratio R

The transverse strain to longitudinal strain ratio R is determined as the ratio of the steel transverse strain over steel longitudinal strain as expressed in Eq. (91). This ratio R is important evidence of the combination of the confine-strengthening effect and confine-stiffening effect.

(91)R=\frac{{\overline{\epsilon }}_{\text{t}}}{{\overline{\epsilon }}_{\text{l}}}.

The initial value of the ratio R, documented as R₀, can be derived theoretically. Before concrete cracks, substituting Eqs. (30) and (31) into Eq. (91), the initial transverse stain to longitudinal strain ratio R₀ is derived as Eq. (92).

(92)R_0=\frac{{\nu }_{\text{c}}(1-{{\nu }_{\text{s}}}^2)+{\displaystyle \frac{2\chi nt}{D}}{\nu }_{\text{s}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}^2)}{1-{{\nu }_{\text{s}}}^2+{\displaystyle \frac{2\chi nt}{D}}(1-{\nu }_{\text{c}}-2{\nu }_{\text{c}}^2)},

where the Poisson’s ratio of steel ν_s is 0.3 and that of concrete is 0.2, respectively. It is indicated that the value of R₀ is strictly between 0.2 to 0.3 as illustrated in Fig. 7.

After the initial cracking (i.e., {\overline{\epsilon }}_{\text{l}}⩾{\epsilon }_{\text{t}}^{\ast }), an exponential function of the average longitudinal strain {\overline{\epsilon }}_{\text{l}}, as expressed in Eq. (93), is proposed for calculating the ratio R through an extensive calibration against the experimental R− {\overline{\epsilon }}_{\text{l}} curves measured by Zhou et al. [6] as shown in Fig. 8. The data regression results showselcting M as 400 best fits the test results.

(93)

4.4 Fractal cracking law

In the prior work [6], a unique and intriguing phenomenon named fractal cracking was experimentally observed for SCFT members under tension. This phenomenon was called fractal cracking [42], because during the tensile loading the SCFTs under tension were divided by cracks into sub-SCFTs and those sub-SCFTs formed smaller cracks and so on, self-similarly and recursively. As shown in Fig. 9, despite the two primary cracks, more secondary cracks existed between the two primary cracks and there were even more tiny small non-through cracks forming between every two secondary cracks. These cracks were so-called fractal cracks. In the present study, a fractal cracking law is proposed in the following sub-sections to quantitatively explain and model the fractal cracking phenomenon and fractal cracks.

4.4.1 Average crack spacing

In fractal cracking, the cracks are classified into different levels, from 0, 1 to i, according to their order of formation, from the primarily formed crack, secondly formed crack, to the thirdly formed crack, and so forth. Reasonably speaking, the cracks formed earlier open up larger than the cracks formed later. Thus, the cracks from 0-level to ith-level can be distinguished according to their order of width, from large to small, as shown in Figs. 9 and 10 where L_{\text{m}}^{(i)} is the average crack spacing of the ith-level cracks.

Because once the (i−1)th-level cracks form, they will not disappear or move places, and hence theith-level cracks can only form between every two existed (i−1)th-level cracks in an integer number. Thus, the relationship between the average crack spacings of theith-level cracks L_{\text{m}}^{(i)} and (i−1)th-level cracks L_{\text{m}}^{(i)} can be expressed as Eq. (94).

(94)L_{\text{m}}^{(i)}=\frac{1}{N}\cdot L_{\text{m}}^{(i-1)}\iff L_{\text{m}}^{(i-1)}=N\cdot L_{\text{m}}^{(i)},

where N is the fractal coefficient, which is an integer larger than one.

In this study, a tracking-back approach is proposed to determine the value of N as in the following steps.

a) First it is assumed that the cracking of the core concrete has reached the ith-level and the average crack spacing is L_{\text{m}}^{(i)}.

b) Then, the loading procedure using a reverse perspective is trackbacked. As we track back the loading procedure, the normal pressure on the interface decreases, and hence the bonding and friction stress \overline{\tau } decreases, and hence the mathematically calculated crack spacing increases continuously as shown in Fig. 11, where L⁽ⁱ⁻¹⁾ is the possible value of crack spacing of the (i−1)th-level cracks. Thus, the question becomes when theith-level cracks vanish and the average crack spacing jumps back from L_{\text{m}}^{(i)} to L_{\text{m}}^{(i-1)}. As shown in Fig. 11, when tracking back the loading procedure to search for the (i−1)th-level cracks, three distinct cases are found.

Case 1) when the maximum crack spacing of the (i−1)th-level cracks L_{\text{max}}^{(i-1)} reaches 2\cdot L_{\text{m}}^{(i)}, the minimum crack spacing of the (i−1)th-level cracks L_{\text{min}}^{(i-1)} reaches 1\cdot L_{\text{m}}^{(i)}, and hence the possible range of the crack spacing of the (i−1)th-level cracks is L^{(i-1)}\in \left[1\cdot L_{\text{m}}^{(i)},2\cdot L_{\text{m}}^{(i)}\right], whereby the possible value of N is 2.

Case 2) when L_{\text{min}}^{(i-1)} reaches between 1\cdot L_{\text{m}}^{(i)} and 2\cdot L_{\text{m}}^{(i)}, L_{\text{max}}^{(i-1)} reaches between 2\cdot L_{\text{m}}^{(i)} and 4\cdot L_{\text{m}}^{(i)}, and hence the possible range of the crack spacing of the (i−1)th-level cracks is L^{(i-1)}\in \left(\begin{array}{cc}1\cdot L_{\text{m}}^{(i)},& 4\cdot L_{\text{m}}^{(i)}\end{array}\right), whereby the possible values of N are 2 and 3.

Case 3) when L_{\text{min}}^{(i-1)} reaches 2\cdot L_{\text{m}}^{(i)}, L_{\text{max}}^{(i-1)} reaches between 4\cdot L_{\text{m}}^{(i)}, and hence the possible range of the crack spacing of the (i−1)th-level cracks is L^{(i-1)}\in \left[2\cdot L_{\text{m}}^{(i)},4\cdot L_{\text{m}}^{(i)}\right], whereby the possible values of N are 2, 3, and 4. Moreover, in Case 3, because the minimum value of the range of the crack spacing of the (i−1)th-level cracks has already reached 2\cdot L_{\text{m}}^{(i)}, it means that at least one crack between two (i−1)th-level cracks has vanished, and hence the cracking state has jumped back fromith-level to (i−1)th-level.

Thus, it is concluded that the three cases discussed above cover all the possible cracking patterns and the possible values of N are 2, 3, and 4, as illustrated in Fig. 12.

4.4.2 Fractal crack width

Neglecting fractal cracking, the crack width in Eq. (17) is formulated by the following equation:

(95)\mathrm{\Delta }{\omega }^{\ast }=L_{\text{m}}^{(0)}\cdot \left(\mathrm{\Delta }{\overline{\epsilon }}_{\mathrm{l}}-\frac{\mathrm{\Delta }{\overline{\sigma }}_{\text{c,l}}}{E_{\text{c}}}+2{\nu }_{\text{c}}\frac{\mathrm{\Delta }{\overline{\sigma }}_{\text{c,r}}}{E_{\text{c}}}\right),

where \mathrm{\Delta }{\omega }^{\ast } is the nominal crack width increment neglecting fractal cracking; \mathrm{\Delta }{\overline{\epsilon }}_{\text{l}}, \mathrm{\Delta }{\overline{\sigma }}_{\text{c,l}}, and \mathrm{\Delta }{\overline{\sigma }}_{\text{c,r}} are each increment of the average longitudinal strain, average longitudinal concrete stress, and average radial concrete stress.

Considering the fractal cracking, it is assumed that the crack width of every existing crack increases equally. In other words, it means that every existing crack shares the current nominal crack width increment equally. Thus, the formula for calculating the crack width of fractal cracks is expressed as follows:

(96)

where \mathrm{\Delta }{\omega }^{(0)},\mathrm{\Delta }{\omega }^{(1)},\mathrm{\Delta }{\omega }^{(2)},\dots ,\mathrm{\Delta }{\omega }^{(i)},\mathrm{\Delta }{\omega }^{(i+1)},\dots represents the crack with increments of the primary cracks, secondary cracks, tertiary cracks, \dots, (i−1)th-level cracks,ith-level cracks \dots

5 Experimental validation

The proposed analytical theory, effective formulas for design, and equivalent constitutive laws for FEA of SCFT members under tension are verified against three available sets of SCFT tensile tests reported in Ref. [6]. Extensive test results are compared, including the force-average longitudinal strain curves, tensile strength, effective stiffness from a general perspective, and most importantly the strain results of steel and the fractal cracks of the core concrete from a micro perspective.

5.1 Average force-axial strain curves

As shown in Fig. 13, the proposed analytical theory and equivalent constitutive laws are validated by comparing the ANA (analytical) results and FEA results with the EXP (experimental) results of the force-average longitudinal strain curves within the average longitudinal strain ranging from 0 to 10000 με. The ANA results represent the analytical results using the proposed analytical theory following the theoretical model in Section 2. The FEA results represent the finite element analysis results of the developed fiber beam-column elements. The EXP results represent the experimental results. It is shown that both the ANA and FEA curves match well with the EXP curves, whereby the FEA curves give a better simulation of the transition region to hardening for both hollow tubes and SCFTs than the ANA curves because the developed equivalent constitutive law of steel accounts for the influence of the initial plastic strain.

Moreover, full curves of the tests conducted by Zhou et al. [6] are calculated and compared, as shown in Fig. 14, to validate the accuracy of the equivalent constitutive laws in modeling the hardening region of the tension force-average longitudinal strain curves. A good agreement between the finite element results and test results is achieved for both the hollow tubes and SCFTs.

5.2 Strength and stiffness

Table 1 verifies the accuracy of the proposed analytical theory, effective formulas, and equivalent constitutive laws in predicting the SCFT tensile stiffness and capacity against the experimental results. The EXP and FEA in Table 1 each represent the experimental results and predicted results using the test and the fiber beam-column element developed based on the equivalent constitutive laws. For the tensile strength, the developed FEA element has high accuracy. The mean error of strength enhancement ratio α_strength of SCFT is 5.2%. For the tensile stiffness, the developed FEA element predict the stiffness enhancement ratio α_stiffness with a relative error of –14.2% (due to slight underestimation of the test stiffness). In general, the accuracy of the developed fiber beam-column element is well validated.

5.3 Strains of steel tube

From a micro perspective, three clusters of the transverse strain to longitudinal strain ratio R-tensile force curve of the three SCFT specimens tested by Zhou et al. [6] are used for an intensive verification of the proposed analytical theory and equivalent constitutive laws, as given in Fig. 15. Both the ANA and FEA results correcdtly predict the descending tendency of the EXP curves. It is indicated that the ANA curves distinctly present two derivative-discontinuous points with strong mechanical meanings, i.e., the initial cracking point and steel yielding point as highlighted in Fig. 15. But the ANA curves descend more steeply than the EXP curves, which may be caused by the neglect of the initial plastic strain of the steel tube due to the cold-form process during the fabrication. Whereas the FEA curves, which are based on the R− {\overline{\epsilon }}_{\text{l}} function in Section 4.3. calibrated by the experimental R− {\overline{\epsilon }}_{\text{l}} curves, match well with the EXP curves, presenting only one obvious derivative-discontinuous point, i.e., the yielding point.

5.4 Fractal cracks

The proposed fractal cracking law in Section 4.3.2, included as a sub-law in the equivalent constitutive laws, is verified against the experimentally recorded results of N of two SCFT tensile specimens reported by Zhou et al. [6], as shown in Fig. 16. The experimentally recorded results of N of T-200-6 and T-200-3 each contain 44 and 47 recorded values of N and they are all within the theoretical range of N (i.e., N = 2, 3, or 4) predicted in Section 4.3.2.

Additionally, the proposed fractal cracking law is further verified against the experimental photo of the fractal cracks using the developed fiber beam-column element, as presented in Fig. 17. The experiment photo of the fractal cracks shows the crack distribution of specimen T-200-6 between primary crack No. 5 and 6 as in Fig. 17(a). The core concrete section is discretized into 4 × 12 fibers in this study as illustrated in Fig. 17(b). The value of the fractal coefficient N is set to be its middle value 3 for this case. The crack width and crack spacing of each fiber are read measured to draw the crack distribution pictures in Figs. 17(c) –17(e). Figures 17(c)–17(e) indicate that the proposed fractal law enables the developed fiber beam-column element to have the capability of giving a realistic simulation of the intriguing and complex fractal cracking phenomenon recently observed in the test [6].

Moreover, the crack width of the fractal cracks of specimen T-200-6 is calculated by the proposed fractal law, using the developed fiber beam-column element. This result of the fractal crack width is compared with the result of the non-fractal crack width calculated by the conventional fiber beam-column element [35,36] in Fig. 18. Figure 18(a) indicates that the calculated non-fractal crack width is almost 5 to 10 times larger than those fractal crack widths, reaching an high value of nearly 25, 50, and 12mm for specimen T-200-6, T-200-3, and T-100-3 [6]. In addition, Fig. 18(b) demonstrates that the calculated non-fractacrack width-load curves are steeper than those fractal crack width-load curves before and after the yielding point. This difference between the non-fractal and fractal crack width is because the non-fractal theory neglects the fractal cracking phenomenon and significantly underestimates the number of cracks, and hence significantly overestimates the crack width since the crack width increment of all cracks is blindly assumed to be due to primary cracks. Besides, the significant overestimation of the crack width using non-fractal theory proves the theoretical value of the proposed fractal cracking law through a negative example.

6 Conclusions

This study reports a theoretical model of SCFT members in tension. The effects of the strength enhancement ratio α_strength, stiffness enhancement ratio α_stiffness, transverse strain to longitudinal strain ratio R, and the fractal law are highlighted. The main research work in the present study is summarized as follows.

1) An analytical theory is proposed to explicitly explain and model the mechanical behavior of SCFT members under tension following the fracture mechanics of concrete, von Mises yield criterion of steel, and Coulomb’s Law of Friction for the steel-concrete interface. In the analytical theory, basic mechanical parameters and equations are determined and derived, respectively, serving as the fundamental work of this paper.

2) Effective formulas of the strength enhancement ratio and stiffness enhancement ratio are derived using theory to address the tensile strength and effective stiffness enhancements of SCFT members under tension for design purposes, respectively. According to the derived effective formulas, the mechanism of the enhanced capacity and stiffness are revealed, being the confine-strengthening effect, confine-stiffening effect, and tension-stiffening effect.

3) Equivalent constitutive laws, including the equivalent constitutive models of concrete and steel, the transverse strain to longitudinal strain ratio R−average longitudinal strain {\overline{\epsilon }}_{\text{l}} function, and the fractal law, are developed for FEA purpose. The confine-strengthening effect, confine-stiffening effect, tension-stiffening effect, and the concrete confinement to the transverse shrinkage of the steel tube are each implicitly modeled in the equivalent constitutive laws via modifying the yield stress of steel, modifying the elastic modulus of steel, utilizing the positive residual average longitudinal concrete stress, and using the proposed R− {\overline{\epsilon }}_{\text{l}} function. Fractal cracking is first quantitatively modeled by the fractal cracking law.

4) The proposed analytical theory, effective formulas, and equivalent constitutive laws are extensively verified against all three available tests reported in the literature on both general levels (e.g., load-displacement curves) and strain level. The experimental verification proves the capability of the proposed theory and formulations in modeling the behavior of SCFT members under tension, the most important being the capability to accurately predict the tensile strength and stiffness enhancements and realistically simulate the fractal cracking phenomenon.

References

[1] Nie J G, Wang J J, Gou S K, Zhu Y Y, Fan J S. Technological development and engineering applications of novel steel-concrete composite structures. Frontiers of Structural and Civil Engineering, 2019, 13( 1): 1– 14

[2] Shao X D, Deng L, Cao J H. Innovative steel-UHPC composite bridge girders for long-span bridges. Frontiers of Structural and Civil Engineering, 2019, 13( 4): 981– 989

[3] EN 1994-1-1: 2004. Eurocode 4: Design of Composite Steel and Concrete Structures. Part1-1: General Rules and Rules for Buildings. Brussels: European Committee for Standardization (CEN), 2004

[4] Leon R T, Gao Y. Resiliency of steel and composite structures. Frontiers of Structural and Civil Engineering, 2016, 10( 3): 239– 253

[5] Deng S W, Shao X D, Zhao X D, Wang Y, Wang Y. Precast steel–UHPC lightweight composite bridge foraccelerated bridge construction. Frontiers of Structural and Civl Engieering, 2021, 15( 2): 364– 377

[6] Zhou M, Fan J S, Tao M X, Nie J G. Experimental study on the tensile behaviour of square concrete-filled steel tubes. Journal of Constructional Steel Research, 2016, 121 : 202– 215

[7] Li W, Han L H, Chan T M. Tensile behaviour of concrete-filled double-skin steel tubular members. Journal of Constructional Steel Research, 2014, 99 : 35– 46

[8] Li W, Han L H, Chan T M. Numerical investigation on the performance of concrete-filled double-skin steel tubular members under tension. Thin-walled Structures, 2014, 79 : 108– 118

[9] Silva A, Jiang Y, Macedo L, Castro J M, Monteiro R, Silvestre N. Seismic performance of composite moment-resisting frames achieved with sustainable CFST members. Frontiers of Structural and Civil Engineering, 2016, 10( 3): 312– 332

[10] Shams M, Saadeghvaziri M A. Nonlinear response of concrete-filled steel tubular columns under axial loading. ACI Structural Journal, 1999, 96( 6): 1009– 1017

[11] Susantha K A S, Ge H, Usami T. Uniaxial stress-strain relationship of concrete confined by various shaped steel tubes. Engineering Structures, 2001, 23( 10): 1331– 1347

[12] Hajjar J F, Gourley B C. Representation of concrete-filled steel tube cross-section strength. Journal of Structural Engineering, 1996, 122( 11): 1327– 1336

[13] Inai E, Mukai A, Kai M, Tokinoya H, Fukumoto T, Mori K. Behavior of concrete-filled steel tube beam columns. Journal of Structural Engineering, 2004, 130( 2): 189– 202

[14] Vecchio F J, Collins M P. The modified compression field theory for reinforced concrete elements subjected to shear. ACI Structural Journal, 1986, 83( 2): 219– 231

[15] Architectural Institute of Japan (AIJ). Recommendations for design and construction of concrete filled steel tubular structures. Study on Concrete Properties Subjected Impact Loading, 2008, 12: 3– 10 (in Japanese)

[16] ANSI/AISC 360-05. Specification for Structural Steel Buildings. Chicago: American Institute of Steel Construction (AISC), 2005

[17] JGJ 138-2016. Code for Design of Composite Structures. Beijing: Ministry of Housing and Urban-Rural Development (MOHURD), 2016 (in Chinese)

[18] Zhou M, Xu L Y, Tao M X, Fan J S, Hajjar J F, Nie J G. Experimental study on confining-strengthening, confining-stiffening, and fractal cracking of circular concrete filled steel tubes under axial tension. Engineering Structures, 2017, 133 : 186– 199

[19] Xu L Y, Tao M X, Zhou M. Analytical model and design formulae of circular CFSTs under axial tension. Journal of Constructional Steel Research, 2017, 133 : 214– 230

[20] Rabczuk T, Zi G, Gerstenberger A, Wall W A. A new crack tip element for the phantom-node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 2008, 75( 5): 577– 599

[21] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61( 13): 2316– 2343

[22] Zhang Y M, Lackner R, Zeiml M, Mang H A. Strong discontinuity embedded approach with standard SOS formulation: Element formulation, energy-based crack-tracking strategy, and validations. Computer Methods in Applied Mechanics and Engineering, 2015, 287 : 335– 366

[23] Zhang Y M, Zhuang X Y. Cracking elements: A self-propagating strong discontinuity embedded Approach for quasi-brittle fracture. Finite Elements in Analysis and Design, 2018, 144 : 84– 100

[24] Zhang Y M, Zhuang X Y. Cracking elements method for dynamic brittle fracture. Theoretical and Applied Fracture Mechanics, 2019, 102 : 1– 9

[25] Zhang Y M, Mang H A. Global cracking elements: A novel tool for Garlerkin-based approaches simulating quasi-brittle fracture. International Journal for Numerical Methods in Engineering, 2020, 121( 11): 2462– 2480

[26] Zhang Y M, Gao Z R, Li Y Y, Zhuang X Y. On the crack opening and energy dissipation in a continuum based disconnected crack model. Finite Elements in Analysis and Design, 2020, 170 : 103333–

[27] Zhang Y M, Huang J G, Yuan Y, Mang H A. Cracking elements method with a dissipation-based arc-length approach. Finite Elements in Analysis and Design, 2021, 195 : 103573–

[28] Ren H L, Zhuang X Y, Anitescu C, Rabczuk T. An explicit phase field method for brittle dynamic fracture. Computers & Structures, 2019, 217 : 45– 56

[29] Rabczuk T, Ren H L, Zhuang X Y. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. CMC, 2019, 59( 1): 31– 55

[30] Ren H L, Zhuang X Y, Rabczuk T. A nonlocal operator method for solving partial differential equations. Computer Methods in Applied Mechanics and Engineering, 2020, 358 : 112621–

[31] Hognestad E. High strength bars as concrete reinforcement—Part 2: Control of cracking. Journal of the PCA Research and Development Laboratories, 1962, 4( 1): 46– 62

[32] Xu LY, Nie X, Zhou M, Tao MX. Whole-process crack width prediction of reinforced concrete structures considering bonding deterioration. Engineering Structures, 2017, 142: 240–254

[33] Belarbi A, Hsu T T C. Constitutive laws of concrete intension and reinforcing bars stiffened by concrete. ACI Structural Journal, 1994, 91( 4): 465– 474

[34] Nie J G, Tao M X, Cai C S, Li S J. Deformation analysis of prestressed continuous steel-concrete composite beams. Journal of Structural Engineering, 2009, 135( 11): 1377– 1389

[35] Tao M X, Nie J G. Fiber beam-column model considering slab spatial composite effect for nonlinear analysis of composite frame systems. Journal of Structural Engineering, 2015, 140( 1): 04013039–

[36] Tao M X, Nie J G. Element mesh, section discretization and material hysteretic laws for fiber beam-column elements of composite structural members. Materials and Structures, 2014, 48( 8): 2521– 2544

[37] Wang Y H, Nie J G, Cai C S. Numerical modeling on concrete structures and steel-concrete composite frames structures. Composite: Part B, 2013, 51( 1): 58– 67

[38] Denavit M D, Hajjar J F. Nonlinear seismic analysis of circular concrete-filled steel tube members and frames. NSEL Report No. NSEL-023. 2010

[39] Tort C, Hajjar J F. Reliability-based performance-based design of rectangular concrete-filled steel tube (SCFT) members and frames. Structural Engineering Report No. ST-07-1. 2007

[40] Hajjar J F, Molodan A, Schiller P H. A distributed plasticity model for cyclic analysis of concrete-filled steel tube beam-columns and composite frames. Engineering Journal (New York), 1998, 20( 4−6): 398– 412

[41] Legeron F, Paultre P, Mazars J. Damage mechanics modeling of nonlinear seismic behavior of concrete structures. Journal of Structural Engineering, 2005, 131( 6): 946– 955

[42] Zhou M. Study on basic theory and method of steel-concrete composite tension problem. Dissertation for the Doctoral Degree. Beijing: Tsinghua University, 2016 (in Chinese)

RIGHTS & PERMISSIONS

Higher Education Press 2021.