A modified stress-strain model accounting for the local buckling of thin-walled stub columns under axial compression
Introduction
The steel members with non-compact sections which are commonly used in modern steel frames or in concrete-filled steel tube (CFST) columns may buckle locally before reaching the ultimate load, and thus leading to a reduction in their load-carrying capacity. This local buckling effect is dependent on initial imperfections and residual stresses and can be explicitly captured when two or three dimensional (2D or 3D) elements are used [1], [2], [3]. For beam-column ‘line’ elements, the local buckling effect was implicitly taken into consideration using either design equations from Specifications (e.g. Kim et al. [4] employed the design equations from AISC-LRFD Specification) or modified stress–strain curves [5], [6]. Compared with the use of design equations, the modification of stress–strain curves is more attractive since it is simple and convenient to programme and more importantly it can simulate the behaviour in the post-local buckling range. This approach was therefore adopted in this study. It is worth noting that the modified stress–strain curve proposed by Chan et al. [5] was determined from rigorous numerical analyses, whilst Skallerud and Amdahl [6] constructed the stress–strain curve based on a combination of experimental data and a simple plastic theory. In this paper, the modified stress–strain curve was constructed using the energy method.
The fibre beam-column element can be developed using either displacement-based or force-based approaches. The displacement-based formulation which is commonly used in a standard finite element programme is based on the assumptions of a linear distribution of curvatures and a constant axial strain along the element, and consequently, requires many elements per member in the modelling. The force-based formulation, on the other hand, is based on the assumptions of linear bending moments and a constant axial force which strictly satisfy the equilibrium of the element in the absence of element loads. As a result, only one element per member is required in the modelling. The implementation of the force-based formulation in a standard finite element programme requires an iteration during the element state determination (i.e. the determination of the element resisting forces) to satisfy the element equilibrium and compatibility [7]. However, this iteration can be eliminated by using the procedure proposed by Neuenhofer and Filippou [8] where both unbalanced section forces and residual element displacements were used in computing element resisting forces, and thus further expanding the benefits of the force-based model. Ayoub [9] modified the force-based element to include the partial interaction between the steel and concrete in composite steel-concrete beams. Jeffers and Sotelino [10] employed the force-based element to simulate the response of steel frames in fire. Neuenhofer and Filippou [11] included geometrically nonlinear effect in the force-based element using a curvature-based displacement interpolation (CBDI) procedure. De Souza [12] later modified the CBDI procedure to account for both geometric and material nonlinearities.
Although the force-based element has been developed and well discussed in the above-mentioned studies, no literature has been reported for the force-based model incorporating the effect of local buckling of thin-walled steel plates. This objective of this paper is therefore to include the local buckling effect in the force-based element. The co-rotational formulation and CBDI procedure were used to capture the geometric nonlinearities due to the large and small P-Delta effects, respectively. The spread of plasticity over the cross section and along the member length is captured by monitoring the uniaxial stress–strain relationship of the fibres on the cross sections located at the integration points along the member length. To account for the local buckling effect, the stress–strain relationship was modified based on the energy method. The proposed model accounted for residual stresses but omitted the effect of initial local imperfections due to the use of Euler-Bernoulli beam theory. The effects of shear deformation and lateral-torsional buckling are neglected in this study. The accuracy of the present element is verified by comparing the obtained predictions with experimental results and those generated by ABAQUS using shell and solid elements.
Section snippets
Geometrically nonlinear force-based element
The displacements of a beam-column element can be decomposed into two parts as the rigid displacements and natural deformations. Since the rigid body modes do not introduce any straining on the element, the element stiffness matrix is derived only from the natural modes. The natural forces and deformations of a beam-column element AB at the section and element levels are shown in Fig. 1. The section force vector consists of one axial force N(x) and two bending moments Mz(x) and My(x) with
Modified stress–strain relationships accounting for local buckling of steel plates
Local buckling of the component plates in thin-walled sections under compression was taken into account using the stress–strain curves as shown in Fig. 3. These curves are dependent on the width-to-thickness ratio b/t. For compact section with lower value of b/t, the whole section will reach the yield stress fy and then buckle inelastically at a strain greater than the buckling strain εcr. The inelastic post-local buckling response is determined based on the consideration of the strain energy
Numerical examples
The present formulation was implemented in a computer programme for the advanced analysis of framed structures [27], [28]. An incremental-iterative scheme based on the generalized displacement control method [29] was used to trace the equilibrium path due to its numerical stability. The internal iteration during the element state determination is eliminated by using the procedure proposed by Neuenhofer and Filippou [8]. This procedure was modified herein to account for the local buckling as
Conclusions
A fibre force-based beam-column element accounting for local buckling effects has been developed for modelling thin-walled and CFST stub columns under axial compression. A new stress–strain equation accounting for elastic and inelastic local buckling of steel plates was proposed based on the energy method. As shown in the numerical examples, the present element is capable of reasonably predicting the ultimate strength as well as the elastic and inelastic local buckling behaviour of thin-walled
Nomenclature
- Ac
area of concrete
- Ai
area of i fibre
- As
area of steel plate
- b
force interpolation matrix
- b
width of section
- be
effective width of section
- d
section deformation vector
- D
section force vector
- E
Young's modulus of steel
- Ei
tangent modulus of i fibre
- Ec
elastic modulus of concrete
- f
section flexibility matrix
- F
element flexibility matrix
- fc
compressive strength of unconfined concrete
- f0
compressive strength of confined concrete
- fre
residual stress of confined concrete
- fr
confining pressure on concrete
- fy
yield stress of steel
- k
Acknowledgements
This work was supported by the Australian Research Council (ARC) under its Discovery Early Career Researcher Award scheme (Project No: DECRA140100747). The financial support is gratefully acknowledged.
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2021, StructuresCitation Excerpt :In the process of modifying the uniaxial constitutive model of steel, the original yield stress fy is replaced by the buckling stress σcr of the stiffened plate, which is determined by Japanese code. Thai et al. [32] deduced the ACSSC of plates based on the energy method, and introduced the curves into the flexible beam-column FE model. He then used the model to analyze the ultimate capacity of high-strength steel box section members with no stiffeners, I-beam members and square tube concrete composite section members, and the proposed model is verified through the ABAQUS shell FE model.