## 3 Simulating the Behavior of the Concrete-Encased Steel Columns
The Mander model predicts that hoops will yield at the maximum strain of confined concrete, but other experimental research work showed that high-strength hoops might not yield when the confinement effect is minimal [24]. As a result, modifications are required to determine the real stress in hoops. Therefore, it is imperative to ascertain the real stress status of the hoops, and the iterative process was used in this instance [25]. Figure 4: Definition of confinement of CES columns, reprinted with permission from [23; 2023], _Journal of Building Engineering_. It should be noted that the parameters that can impact the resistance of the confined concrete are the distribution of the longitudinal reinforcing bar, the volumetric ratio of the lateral reinforcement, and the spacing of the stirrups [16]. A reduction in the stirrup spacing enhances the post-peak behavior of the axial load-deformation behavior. Furthermore, in order to investigate the confinement impact for highly and partially confined concrete, the confinement factors (\(K_{p}\) and \(K_{h}\)) have been suggested [22]. The concrete strength \(f_{cc}^{\prime}\) for partially and highly confined concrete is described as presented in Equations (1) and (2), respectively, while \(f_{co}^{\prime}\) is the compressive strength of the unconfined concrete. \[f_{cc}^{\prime}=K_{p}.f_{co}^{\prime} \tag{1}\]  
\[f_{cc}^{\prime}=K_{h}.f_{co}^{\prime} \tag{2}\]  
The relations of the tie spacing versus the confinement factor \(K_{p}\) for partially confined concrete for columns are shown in Figure 6[22]. The effectiveness of confinement by lateral reinforcement can be shown in Figure 6. It can be concluded that reducing stirrup spacing increases the confinement factor \(K_{p}\). The form of the structural steel section, which applies confining stress to the core concrete, affects the confinement factor \(K_{h}\) for highly confined concrete. The steel profile form versus confinement factor \(K_{h}\) relations and their impact on the efficiency of confinement are displayed in Figure 7. The structural steel, especially the cross-shaped section, increases the confining effect. The explanation for the change is that in contrast to I- or H-shaped steel sections, which confine the concrete in just one direction, cross-shaped steel sections can produce confining pressure in both directions, resulting in a greater value of the confinement factor \(K_{h}\)[22]. Figure 5: Stress–strain relationship of materials in CES section, reprinted with permission from [18], 2015, _Journal of Constructional Steel Research_. The effect of concrete confinement improves the ductility of CES columns but has little impact on the axial capacity since it activates after cover spalling. In addition, compared to shear-dominant CES columns, the flexural dominant CES columns benefit more from the confinement effect provided by stirrups [23]. Another research work has developed an analytical model to predict the axial compressive behavior of CES columns [19]. The confinement factors \(K_{p}\) and \(K_{h}\) for PC and HC, respectively, are described as follows:  
\[k_{p}=-1.254+2.254\sqrt{1+\frac{7.94f_{le,p}^{t}}{f_{co}^{t}}}-2\frac{f_{le,p}^ {t}}{f_{co}^{t}} \tag{3}\]  
Figure 6: Confinement factor (\(k_{p}\)) for partially confined concrete, reprinted with permission from [22], 2006, _Journal of Constructional Steel Research_. Figure 7: Confinement factor (\(K_{h}\)) for highly confined concrete, reprinted with permission from [22], 2006, _Journal of Constructional Steel Research_. Figure 4:0  
As presented in Equation (3), \(f^{\prime}_{le,p}\) is the effective confining pressure from stirrups. It can be obtained as follows:  
Figure 4:1  
where \(f_{rh}\) is the real stress in the stirrups, and \(\xi\) is the confinement effectiveness factor for the partially confined concrete.