1. Introduction

A concrete-filled steel tube (CFST) has higher strength and ductility than traditional reinforced concrete, owing to the confinement effect therein. It has become widely adopted in bridges, e.g. Hejiang Bosiden Bridge and Guangzhou Yajisha Bridge, and tall buildings, e.g. Shenzhen KK100. However, the effectiveness of confinement in CFST is dependent on the section shape and loading type. In theory, a circular CFST column under axial loading has uniform confinement over the entire concrete section before any local buckling occurs, whereas a non-circular (rectangular, elliptical or polygonal) CFST column under any loading or a circular CFST under eccentric loading has non-uniform and generally less effective confinement.

Many empirical axial strength and stress-strain models of confined concrete have been developed by incorporating the effects of confinement, whether uniform or non-uniform, in terms of empirical factors dependent on the geometric and material parameters without explicit consideration of the actual distribution of confining stresses [1-7]. In a recent study by Yu et al. [8, 9] on concrete confined by fibre-reinforced polymer (FRP), the finite element (FE) method was employed to analyse the confining stresses. A pivotal step was taken to simulate the interaction between the laterally expanding concrete and the confining FRP by making solution-dependent adjustments to the dilation angle of the plastic flow potential of the concrete.

To avoid such complicated adjustments, the authors have developed a novel FE model, which allows directly computation of the inelastic components of the lateral strains, from which the confining stresses can be evaluated [10, 11]. This paper presents the key features of the FE model and some numerical results obtained for square CFST.

2. Concrete Modelling

This model simulates the constitutive behaviour of concrete under confinement using the lateral stain-axial strain relation developed by Dong et al. [12], the triaxial failure surface developed by Menétrey and Willam [13] and the axial stress-strain relation of confined concrete developed by Attard and Setunge [14]. Detailed mathematical formulations of the three models are presented in Table 1 for reference.

According to Dong et al. [12], the lateral strains of concrete in the two in-plane directions each comprises of two components, an elastic component and an inelastic component. Based on this postulation, the in-plane principal lateral strains ε₁ and ε₂ in each concrete element can be expressed as and in which and are the elastic components, and and are the inelastic components. With the inelastic components taken as initial strains, the constitutive equation of concrete at element level may be expressed as:

(1a)

(1b)

where E_c and ν_c are the Young’s modulus and Poisson’s ratio of the concrete. The inelastic components in Eq. (1a) are dependent on the axial strain in the longitudinal direction ε₃, the lateral confining stresses σ₁ and σ₂, and the concrete cylinder strength f_c, as given by:

(2a)

(2b)

The expressions of and can be found in Table 1.

Table 1. Adopted concrete models

Triangular three-noded (T3) elements are used. Hence, the axial strain ε₃ at the centroid of the T3 element is used in Eqs. 1 and 2. The stiffness matrix equation of the concrete elements in the global coordinate system is derived as:

(3)

in which ∆ is the area of the T3 element; B is the strain-displacement matrix of the T3 element; A is the strain transformation matrix converting the global lateral strain vector {ε_x ε_y γ_xy}^T to the local principal strain vector {ε₁ ε₂ γ₁₂}^T; u is the nodal displacement vector {u₁ υ₁ u₂ υ₂ u₃ υ₃}^T for the T3 element; is the local inelastic strain vector is the axial strain vector {ε₃ ε₃ 0}^T in both local and global coordinate systems; and C is the constitutive matrix of the concrete.

The triaxial failure surface developed by Menétrey and Willam [13] is given by:

(4)

where ξ is the hydrostatic length; ρ is deviatoric length; θ is the Lode angle; m is the friction parameter; e is the out-of-roundness parameter; and c is the cohesion parameter. The uniaxial tensile strength f_t is assumed as -0.1f_c. When Eq. (4) is only describing the failure surface, c should be equal to 1. The value of e can be derived by putting σ₁ = 0 and σ₂ = σ₂ = 1.5∙f_c^0.925 into Eq. (4):

(5)

as per the suggestion by Papanikolaou and Kappos [15] that the biaxial-to-uniaxial compressive strength ratio of concrete should be given by 1.5∙f_c^-0.075. As far as the failure surface is concerned, the compressive strength of confined concrete f_cc is equivalent to σ₃ in Eq. (4) and can be calculated from the lateral confining stresses σ₁ and σ₂ at each iteration step.

The relation between the axial strain ε₃ and the axial stress σ₃ (different from σ₃ in Eq. (4)) of each concrete element may be determined by Attard and Setunge’s model [14]. The mathematical expression of this model is given by:

(6)

where ε_cc is axial strain at peak stress corresponding to f_cc, and a₁, a₂, a₃ and a₄ are coefficients governing the shape of the stress-strain curve. It should be stressed that Attard and Setunge’s original mathematical expressions for f_cc is only applicable to the cases in which the confining stresses along the two principal axes have the same magnitude, i.e. σ₁ = σ₂ = f_r, and is replaced by Menétrey and Willam’s triaxial failure surface since the latter is more suitable for anisotropic cases. f_r is also used to determine other parameters in Attard and Setunge’s model. When σ₁ and σ₂ are not equal to each other, f_r is assumed to be the minimum of σ₁ and σ₂, as a compromised approach.

3. Nonlinear FE Analysis of Square CFST

The analysis procedure for axially loaded square CFST columns is illustrated in Fig. 1.

Figure 1. Procedures for the FE analysis

The axial strain ε₃ of each concrete is the prescribed input, the inelastic lateral strain vector of concrete in Eq. (3) can be determined by Dong et al.’s [12] lateral strain-axial strain relation. Likewise, the plastic strain vectors and of steel can be determined by von-Mises yield criterion and the associated plastic flow. Subsequently, the global stiffness matrix equation can be assembled as:

(7)

where F_p and F₃ are load vectors related to the residual strains (inelastic lateral strains of concrete and plastic strains of steel) and axial strains in the concrete elements and steel elements. In Eq. (7), the residual strain vector on the right hand side is dependent on the nodal displacement vector on the left hand side. An iteration process is adopted to calculate the approximate solutions to Eq. (7) in each loading step. More specifically, a nodal displacement vector u_i can be calculated using the current values of axial strains and confining stresses in Step i:

(8)

Then the new nodal displacement vector can be used to produce a new stress vector which is used to compute the confining stresses for the i+1^th iteration:

(9)

where r is the relaxation factor. Normally the value of r is set between 0.3 and 0.7 to maintain the convergence rate during the iteration process.

After the principal lateral stresses σ₁ and σ₂ of each concrete element are converging to steady values, i.e. their approximate solutions are found, they can be used to evaluate f_cc via Menétrey and Willam’s triaxial failure surface. With the input of ε₃, f_cc, f_r = min{σ₁, σ₂}, and the use of Attard and Setunge axial stress-strain relation, the axial stress σ₃ of each concrete element can be evaluated. Meanwhile, the axial stress σ₃ of each steel element is determined also by von-Mises yield criterion and the associated plastic flow. The force P can be calculated by integrating σ₃ over the whole CFST section.

If there is flexural behaviour involved other than axial compression, an extra level of iteration involving member analysis should also be added to the procedure.

4. Applications

The FE model is verified against the experimental results of 3 axially loaded square CFST columns from Sakino et al.’s publication [16]. The sectional edge length for the columns CR4-A-4-1, CR4-C-4-1 and CR4-D-4-1 are 148 mm, 215 mm and 323 mm respectively; the steel tubes for all three have yield strength of 262 MPa and thickness of 4.38 mm; the cylinder strengths of concrete are 40.5 MPa for CR4-A-4-1 and 41.1 MPa for the other two. In Fig. 2, the peak loads of CR4-A-4-1, CR4-C-4-1 and CR4-D-4-1 predicted by the FE model are 0.94, 1.02 and 0.98 times their respective experimental results; as far as the residual strength at ε₃ = 4.0% is concerned, those multiples will become 0.94, 0.90 and 0.86. Overall, the predictions by the FE model agree quite well with the test results.

Figure 2. Load-strain curves

The corner effect is also studied with the help of the FE model. Fig. 3 shows the simulated distributions of axial stress for a series of axially loaded square CFST columns with edge length of 200 mm, tube thickness of 4 mm, S355 steel, Grade 80 concrete and corner radii of 10, 30 and 50 mm. It is discovered that the confinement effect is better at the corners and centre of the section. As the corner radius increases, the effectively confined concrete areas at ε₃ = 4.0% will increase, hence the residual strength of the column will be maintained at a higher level.

Figure 3. Concrete axial stress contour at ε3 = 4.0%

5. Conclusions

This paper has presented a novel FE method that utilizes Dong et al.’s lateral strain-axial strain model, Menétrey and Willam’s triaxial failure surface and Attard and Setunge’s axial stress-strain model under confined condition to analyse the passive confinement effect induced by the lateral expansion of concrete within CFST. The use of initial strains in the formulation of the global stiffness matrix equation is the key to compute passive confining stresses in the FE analysis. Owing to this new tool, the load-strain relation and the axial stress contour of CFST can be simulated, thus enabling further exploration on various phenomena in CFST, such as the corner effect.