## 2 Numerical Study
### Material Modelling  
To consider the inelastic behavior of materials under cyclic loading, a nonlinear combined isotropic/kinematic hardening material model was adopted in this paper for finite element analysis. This model was developed based on work by [PERSON] and [PERSON] [19], and can be used to represent cyclic hardening and the kinematic shift of the yield surface in a multi-dimensional stress space, as shown in Figure 4. The non-linear kinematic hardening element retains the same size and shape of the yield surface, but changes with additional stress cycles. The kinematic hardening comprised two terms: a linear kinematic phase followed by the Ziegler hardening law, and a back stress interval that defines the non-linearity. The mathematical expression of non-linear kinematic hardening is as follows:  
\[\dot{\alpha}_{k}=C_{k}\frac{1}{\sigma^{0}}(\sigma-\alpha)\dot{\frac{\cdot}{ \epsilon}}p-\gamma_{k}\alpha_{k}\dot{\frac{\cdot}{\epsilon}}p \tag{1}\]  
where \(\sigma^{0}\) is the preliminary yield stress, \(\dot{\frac{\cdot}{\epsilon}}p\) denotes the corresponding plastics strain level, and \(C_{k}\) and \(\gamma_{k}\) are material constants. \(C_{k}\) signifies the first kinematic hardening moduli and \(\gamma_{k}\) governs the kinematic hardening moduli diminution rate with respect to the accelerating plastic disfigurement. The total back-stress (\(\alpha\)) is calculated using the given formula:  
\[\alpha=\sum_{k=1}^{N}\alpha_{k} \tag{2}\]  
where \(N\) represents the total number of back-stresses. The isotropic hardening element, which tends to maintain an identical shape of the yield surface and allows the size to expand with increasing stress, can be illustrated as a direct function of the corresponding plastic strain, \(\epsilon^{p}\), which jointly reflects the softening and hardening of the material. This can be computed as follows:  
\[\sigma^{0}=\sigma^{0}(\overline{\epsilon}^{p})=\sigma\big{|}_{0}+Q_{\infty} \Big{(}1-e^{-b\overline{\epsilon}^{p}}\Big{)} \tag{3}\]  
where \(Q_{\infty}\) and \(b\) are material parameters, and \(\sigma\!\mid\!_{0}\) signifies the yield stress at zero strain. \(Q_{\infty}\) represents the extreme variation in yield surface size, whereas \(b\) indicates the rate of variation for the yield surface during the development of plastic straining.  
The nonlinear parameters of hot-rolled carbon steel required for simulation of the aforementioned model were taken from [20], as summarized in Table 1. These were applied to all numerical models throughout the study irrespective of brace size, geometry, and displacement levels of loading. All tubular structural members contain initial geometric imperfections, which mainly originate during the manufacturing process and have a significant influence on the endurance of the members. In this numerical study, both  
Figure 4: (**a**) Rule of isotropic hardening; (**b**) rule of kinematic hardening; (**c**) joint hardening.  
properties were applied as in the experimental test specimens. The comparison between FE predicted hysteresis axial force-displacement responses and test results are displayed in Figure 7. These results demonstrate that the material modeling approach adopted in the current FE study can precisely replicate the key features of experimental hysteresis behavior of the steel braces. Therefore, the FE modeling approach adopted in the current study is able to simulate the hysteretic behavior well for a large number of square and circular braces under cyclic loading. The dissimilarities between the experimental and FE results shown in Figure 7 could be due to the initial elastic stiffness through the elastic unloading, which resulted in a steeper drop in the unloading curves than those of the test specimens.