# Evaluation of Eccentric Compressive Behavior Between Active and Passive Preloading in CFST
## 3 Definition of the Specimens
### Application of Preloads  
All four preloaded specimens were slightly deformed before the application of the service load, although the direction of the applied preload and the object to which the preload was applied differed, the initial strain had the same magnitude. For the selection of the preload value, the preload applied to the steel tube concrete should be within the yield range. In addition, this study is dedicated to discussing the performance of each preload specimen for a small value of preload, according to reference [43], the applied preload should not exceed 0.5 \(f_{y}\). Therefore, the prestress value is assumed to be 0.2 \(\varepsilon_{y}\). For CFST-SP, the deformation was a hoop for steel. For CFST-CP, the deformation was a three-way expansion for concrete. For CFST-CC, the deformation was longitudinal for concrete, and for CFST-SC, the deformation was longitudinal for the steel tube. Although the direction of the applied prestress and the object of the prestressing force were different, the initial strain had the same magnitude, and because the standards of various countries limit the prestressing stress of steel pipes to 0.5-0.6 fy, it was assumed to be 0.2 \(\varepsilon_{y}\). For CFST-SP, the deformation is the circumferential deformation of the steel bar. For CFST-CP, the deformation is the triaxial expansion of the concrete. For CFST-CC, the deformation is the longitudinal deformation of the concrete, and for CFST-SC, the deformation is the longitudinal deformation of the steel pipe. As far as CFST-SP was concerned, when the hoop tensile strain of the steel tube was 0.2 \(\varepsilon_{y}\) (Figure 9a), the steel-concrete contact pressure \(p_{0}\) could be determined with the following formulas:  
Figure 9: CFST-SP with active preloading: (**a**) Model; (**b**) Steel; (**c**) Concrete; (**d**) Cross-section of steel tube; (**e**) Cross-section of concrete core. (a) Assuming no vertical slip between concrete and steel tube, the longitudinal strains of the concrete (\(\varepsilon_{cz}\)) and the steel tube (\(\varepsilon_{sz}\)) were equal; so,  
\[\varepsilon_{cz}=\varepsilon_{sz}\, \tag{5}\]  
(b) According to equilibrium theory, the vertical internal forces in the cross-section of the steel tube and the concrete were equal in magnitude and opposite in direction. In the expressions below, \(A_{c}\) and \(A_{s}\) are the cross-sectional areas of the steel tube and the concrete, respectively, while \(\sigma_{cz}\) and \(\sigma_{sz}\) are the normal stresses of the cross-sections in the steel tube and the concrete, respectively. \[\sigma_{cz}A_{c}=\sigma_{sz}A_{s}; \tag{6}\]  
\[\frac{\sigma_{cz}}{\sigma_{sz}}=\frac{A_{s}}{A_{c}}. \tag{7}\]  
(c) According to the generalized Hooke's Theorem, the expression for steel, is shown in Figure 9b, as follows:  
\[\varepsilon_{sz}= \frac{1}{E_{s}}(\sigma_{sz}-\mu_{s}\sigma_{sh}); \tag{8}\]  
\[\varepsilon_{sh}= \frac{1}{E_{s}}(\sigma_{sh}-\mu_{s}\sigma_{sz}). \tag{9}\]  
The expression for concrete, is shown in Figure 9c, is as follows:  
\[\varepsilon_{cz}=\frac{1}{E_{c}}((-\sigma_{cz})-\mu_{c}(-p_{0}-p_{0}))=\frac {1}{E_{c}}(2p_{0}\mu_{c}-\sigma_{cz}), \tag{10}\]  
In the expression, \(\varepsilon_{sh}=0.2\)\(\varepsilon_{y}\), and \(D\) is the diameter of the specimen. \(E_{c}\) and \(E_{s}\) are the elastic modulus of concrete and steel, respectively, \(\mu_{c}\) and \(\mu_{s}\) are the elastic Poisson's ratios of the concrete and steel, respectively, and \(\sigma_{sh}\) is the hoop stress of the steel tube.