*4.2. Bearing Capacity Calculation Model*

According to the results of the tests and extended numerical analysis, the calculation method used for the axial compression bearing capacity of concrete members strengthened involving the enlarging section method found in [24], and Xu's study on the axial compression bearing capacity of self-stressed, concrete-filled steel tube columns [28], the calculation model for the axial compression bearing capacity of specimens strengthened by IPCSAM was established. In light of the structure and working characteristics of the components strengthened by the IPCSAM, the following assumptions were made on the bearing capacity calculation model of the IPCSAM reinforcement members:


concrete needed to be adjusted. The contribution of the LCSS on the bearing capacity of the specimens was neglected to simplify the calculation;


Based on assumption (1), the calculation formula for the axial compression bearing capacity of the strengthened specimens (*N*) was proposed as follows:

$$N = N\_0 + N\_\sf{s} + N\_\sf{p} \tag{1}$$

where *N*0, *Ns*, and *Np* are the bearing capacity of the column and the filled concrete, the bearing capacity improved by the confinement, and the bearing capacity improved by self-stress, respectively.

Based on assumption (2), the calculation formula for the bearing capacity of the unreinforced column and the filled concrete (*N*0) was proposed as follows:

$$N\_0 = 0.9 \,\mathrm{\rho} \left[ f\_{co} A\_{co} + f\_{yo}' A\_{so}' + \alpha (f\_{sc} A\_{sc} + f\_y' A\_s') \right] \tag{2}$$

where *ϕ* is the stability coefficient of members, taking the value according to code [26]; *fco* and *fsc* are the strength of the column concrete and filled concrete, respectively; *Aco* and *Asc* are the cross-sectional areas of the column and filled concrete, respectively; *fyo* and *fy* - are the strength values of steel bars in the column and filled concrete, respectively; *Ayo* and *Ay* are the cross-sectional areas of steel bars in column and filled concrete, respectively; *α* is the utilization coefficient of the filled concrete.

According to [24] and Huang's study [30], the value of α should be determined by the interface bonding between the column and filled concrete. Assuming that the influencing factors of the interface bonding are the roughness, concrete strength of the column, filled concrete strength, and the usage of an interface agent, the calculation formula of *α* is as follows:

$$
\alpha = A(0.15 + 0.025\Delta) \ln f\_{\mathcal{L}} \zeta\_{\ast} \tag{3}
$$

where *A* is the adjustment coefficient considering the preloading of concrete in the filled concrete, which is an undetermined constant; Δ is the roughness of the interface when Δ ≥ 2.5, taking Δ = 5 mm; *fsc* is the average strength of the column concrete and filled concrete; *ζ<sup>s</sup>* is the influence coefficient of the interfacial agent, which is an undetermined coefficient.

Based on assumption (3), the calculation formula for the bearing capacity improved by the confinement (*N*0) was proposed as follows:

$$N\_{\mathbb{S}} = \beta \theta N\_0 \tag{4}$$

$$\beta = B(\frac{f\_l}{f\_{co}})^{0.7} \tag{5}$$

$$\theta = \frac{A\_{\S^c} f\_{\S^t} + A\_{\S^y} f\_{\S^y} + A\_{sc} f\_{st} + A'\_s f'\_{y}}{A\_{co} f\_{co}} \tag{6}$$

where *θ* is the confinement coefficient when *θ* ≥ 0.25, taking *θ* = 0.25; *β* is the adjustment coefficient related to the concrete strength of the SSAWC, the LCSS, and the column; *B* is an undetermined constant; *fst*, *fgt*, and *fgy* are the tensile strengths of the SSAWC, the LCSS concrete, and steel bars reinforcement, respectively; *Agc* and *Agy* are the cross-sectional areas of the sleeve concrete and steel bars, respectively; the value of *fl* takes the minimum of *fsc* and *fgc*.

Based on assumption (4), the calculation formula for the bearing capacity improvement caused by self-stress (*N*0) was proposed as follows:

$$N\_p = \gamma N\_0 \tag{7}$$

$$
\gamma = C\eta^3 + D\eta^2 + E\eta \tag{8}
$$

where, *γ* is the improvement coefficient; *η* is the self-stress level, which is the ratio of the longitudinal self-stress value to the SSAWC strength; *C, D,* and *E* are coefficients of cubic equations to be solved.

To sum up, the calculation formula of axial the compression bearing capacity of the components strengthened by the IPCSAM is as follows:

$$N = 0.9\varvarphi \left[ f\_{\text{co}}A\_{\text{co}} + f\_{\text{yo}}'A\_{\text{so}}' + a(f\_{\text{sc}}A\_{\text{sc}} + f\_{y}'A\_{\text{s}}') \right] (1 + \beta\theta + \gamma) \tag{9}$$

#### *4.3. Determination of Parameters*


$$
\kappa = 0.75(0.15 + 0.025\Delta) \ln f\_c \zeta\_s \tag{10}
$$

The value of *ζ<sup>s</sup>* is 1 when no interface agent is used. When the filled concrete is SSAWC, the value of *ζ<sup>s</sup>* is 1.07.

(3) The *β* value: The values of B can be determined by regression fitting the data from Tables 7 and 8. Then, the relationship between *β* and *fl*, *fco* is as follows:

$$
\beta = 2.698 \left( \frac{f\_l}{f\_{co}} \right)^{0.7} \tag{11}
$$

(4) The *γ* value. The values of C, D, and E can be determined by regression fitting the data from Tables 7 and 8. Then, the relationship between *γ* and *η* is as follows:

$$
\gamma = 6.683\eta^3 - 58.275\eta^2 + 9.556\eta \tag{12}
$$

For the derivative of Formula (12), the self-stress level corresponding to the *γ* maximum value can be obtained; that is, the optimum self-stress level of the strengthened component. The optimum self-stress level of the specimen strengthened by the IPCSAM is 0.0875.
