A. K. H. Kwan, Y. Ouyang
Department of Civil Engineering, The University of Hong Kong, Hong Kong, China
Correspondence to: A. K. H. Kwan, Department of Civil Engineering, The University of Hong Kong, Hong Kong, China.
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Abstract
Most theoretical models of nonuniform concrete confinement use empirical equations and thus are applicable only to limited ranges of geometric and material parameters. In light of such shortcomings, a novel finite element (FE) method is proposed to explicitly compute the nonuniform lateral confining stresses in the concrete section. To determine the lateral stresses, the lateral strains are divided into two components: an elastic component and an inelastic component, and a 2dimensional FE analysis of the concrete section with the inelastic laterals strains taken as initial strains is carried out. Since the lateral stresses and inelastic lateral strains are interrelated, an iterative process of evaluating the lateral stresses from the inelastic lateral strains by FE analysis and then evaluating the inelastic lateral strains from the lateral stresses by a lateral strain model is employed until convergent results are obtained. This FE method is herein applied to square concretefilled steel tube columns.
Keywords:
Concretefilled steel tube (CFST), Confinement effect, Finite element analysis
Cite this paper: A. K. H. Kwan, Y. Ouyang, A Novel Finite Element Method for Analysing the Confinement Effect in ConcreteFilled Steel Tubes, Journal of Civil Engineering Research, Vol. 7 No. 2, 2017, pp. 5762. doi: 10.5923/j.jce.20170702.04.
1. Introduction
A concretefilled 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 noncircular (rectangular, elliptical or polygonal) CFST column under any loading or a circular CFST under eccentric loading has nonuniform and generally less effective confinement.Many empirical axial strength and stressstrain models of confined concrete have been developed by incorporating the effects of confinement, whether uniform or nonuniform, in terms of empirical factors dependent on the geometric and material parameters without explicit consideration of the actual distribution of confining stresses [17]. In a recent study by Yu et al. [8, 9] on concrete confined by fibrereinforced 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 solutiondependent 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 stainaxial strain relation developed by Dong et al. [12], the triaxial failure surface developed by Menétrey and Willam [13] and the axial stressstrain 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 inplane directions each comprises of two components, an elastic component and an inelastic component. Based on this postulation, the inplane principal lateral strains ε_{1} and ε_{2} 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 ε_{3}, the lateral confining stresses σ_{1} and σ_{2}, 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 threenoded (T3) elements are used. Hence, the axial strain ε_{3} 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 straindisplacement 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 {ε_{1} ε_{2} γ_{12}}^{T}; u is the nodal displacement vector {u_{1} υ_{1} u_{2} υ_{2} u_{3} υ_{3}}^{T} for the T3 element; is the local inelastic strain vector is the axial strain vector {ε_{3} ε_{3} 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 outofroundness 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 σ_{1} = 0 and σ_{2} = σ_{2} = 1.5∙f_{c}^{0.925} into Eq. (4):  (5) 
as per the suggestion by Papanikolaou and Kappos [15] that the biaxialtouniaxial 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_{c}_{c} is equivalent to σ_{3} in Eq. (4) and can be calculated from the lateral confining stresses σ_{1} and σ_{2} at each iteration step. The relation between the axial strain ε_{3} and the axial stress σ_{3} (different from σ_{3} 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 a_{1}, a_{2}, a_{3} and a_{4} are coefficients governing the shape of the stressstrain 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. σ_{1} = σ_{2} = 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 σ_{1} and σ_{2} are not equal to each other, f_{r} is assumed to be the minimum of σ_{1} and σ_{2}, 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 ε_{3} 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 strainaxial strain relation. Likewise, the plastic strain vectors and of steel can be determined by vonMises yield criterion and the associated plastic flow. Subsequently, the global stiffness matrix equation can be assembled as:  (7) 
where F_{p} and F_{3} 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 σ_{1} and σ_{2} 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 ε_{3}, f_{cc}, f_{r} = min{σ_{1}, σ_{2}}, and the use of Attard and Setunge axial stressstrain relation, the axial stress σ_{3} of each concrete element can be evaluated. Meanwhile, the axial stress σ_{3} of each steel element is determined also by vonMises yield criterion and the associated plastic flow. The force P can be calculated by integrating σ_{3} 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 CR4A41, CR4C41 and CR4D41 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 CR4A41 and 41.1 MPa for the other two. In Fig. 2, the peak loads of CR4A41, CR4C41 and CR4D41 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 ε_{3 }= 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. Loadstrain 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 ε_{3} = 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 ε_{3} = 4.0% 
5. Conclusions
This paper has presented a novel FE method that utilizes Dong et al.’s lateral strainaxial strain model, Menétrey and Willam’s triaxial failure surface and Attard and Setunge’s axial stressstrain 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 loadstrain 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.
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