## 5 Limit State Design Method PSRPC Calculation of Column Bias Load Capacity
### Calculation of Load-Carrying Capacity  
In this study, bending damage occurred in specimens II-PSRPC-1 to 6, with the section flange in the tension zone breaking before the concrete in the compression zone, and the section web did not reach its yield strength. Due to the uncertainty of the location of the neutral axis, the section may or may not reach its yield strength during loading, so the following two scenarios are derived based on the location of the neutral axis, as shown in Figure 14.  
(1) As shown in Figure 14a, when \(1.2a_{a}^{\prime}\leq x\leq x_{b}\), it is known from the assumption of a plane section that the compression flange of steel reaches the yield strength. Since the section is symmetrically reinforced, \(f_{a}=f_{a}^{\prime}\), \(a_{a}=a_{a}^{\prime}\), \(a_{s}=a_{s}^{\prime}\), According to the equilibrium condition of force \(\sum x=0\), we can obtain:  
\[ \begin{array}{l}N_{cu}=f_{c,out}bh_{2}+2f_{c,out}b_{2}(x-h_{2})+f_{c,in}b_{1} (x-h_{2})\\ \\ +A_{af}^{\prime}f_{yf}^{\prime}+\bigg{(}x-a_{a}^{\prime}-\frac{t_{f}^{\prime} }{2}-d\bigg{)}t_{w}^{\prime}f_{aw}^{\prime}-A_{af}f_{af}\\ \\ -\Big{(}h-x-a_{a}-\frac{t_{f}}{2}-d\Big{)}t_{w}f_{aw}\end{array}  \tag{14}\]  
Figure 13: Comparison of the calculation results and test results. \[x=\frac{N_{cu}-f_{c,out}bh_{2}+2f_{c,out}b_{2}h_{2}+f_{c,in}b_{1}h_{2}+t^{\prime}_ {w}f^{\prime}_{yw}h}{2f_{c,out}b_{2}+f_{c,in}b_{1}+2t^{\prime}_{w}f^{\prime}_{yw}} \tag{15}\]  
According to the stress-strain distribution diagram in Figure 14a, the moment of the neutral axis is obtained from the moment balance equation:  
\[ \begin{array}{l}N_{cu}\Big{(}e_{0}-\frac{h}{2}+x\Big{)}=f_{c,out}bh_{2}\Big{(} x-\frac{h_{2}}{2}\Big{)}\\ \qquad+\frac{1}{2}f_{c,out}b_{2}(x-h_{2})^{2}+\frac{1}{2}f_{c,out}b_{1}(x-h_{2} )^{2}+A^{\prime}_{af}f^{\prime}_{yf}(x-a^{\prime}_{a})\\ \qquad+t^{\prime}_{w}f_{yw}\left[\frac{\left(x-a^{\prime}_{a}-\frac{t^{\prime} _{a}}{2}\right)^{2}}{2}-\frac{d^{2}}{6}\right]+A_{af}f_{yf}(h-x-a_{a})\\ \qquad+t^{\prime}_{w}f_{yw}\left[\left(h-x-a_{a}-\frac{t_{f}}{2}\right)^{2}- \frac{\rho}{6}\right]\\ \qquad\qquad\qquad\qquad x_{b}=\frac{0.8}{1+\frac{f_{a}}{2k_{cs}}}\big{(}h-a^ {\prime}_{a}\big{)}\end{array}  \tag{16}\]  
(2) As shown in Figure 14a, when \(a^{\prime}_{a}\leq x\leq 1.2a^{\prime}_{a}\), it is known from the assumption of a plane section that the compression flange of steel reaches the yield strength. Since the section is symmetrically reinforced, \(\sum x=0\). According to the equilibrium condition of force x = 0, we can obtain:  
\[ \begin{array}{l}N_{cu}=f_{c,out}bh_{2}+2f_{c,out}b_{2}(x-h_{2})+f_{c,in}b_{1 }(x-h_{2})\\ \qquad-\Big{(}h_{1}-2t_{f}\Big{)}t_{w}f_{yw}-A_{af}f_{yf}\end{array}  \tag{18}\]  
\[ \begin{array}{l}x=\frac{N_{cu}-f_{c,out}bh_{2}+2f_{c,out}b_{2}h_{2}+f_{c,in }b_{1}h_{2}+\Big{(}h_{1}-2t_{f}\Big{)}t^{\prime}_{w}f_{yw}+A_{af}f_{af}}{2f_{c,out}b_{2}+f_{c,in}b_{1}}\end{array}  \tag{19}\]  
From the moment balance, the moment of the neutral axis is obtained:  
\[ \begin{array}{l}N_{cu}\Big{(}e_{0}-\frac{h}{2}+x\Big{)}=f_{c,out}bh_{2} \Big{(}x-\frac{h_{2}}{2}\Big{)}+f_{c,out}b_{2}(x-h_{2})^{2}+\\ \frac{1}{2}f_{c,in}b_{1}(x-h_{2})^{2}+A_{af}f_{af}(h-a_{a}-x)+\\ \Big{(}h_{1}-2t_{f}\Big{)}t^{\prime}_{w}f_{yw}\left(\frac{h}{2}-x\right)\end{array}  \tag{20}\]  
in the formula:  
\(x\)**--**height of the concrete compression zone;  
\(f_{c,in}\), \(f_{c,out}\)--**axial compressive strength of RPC in effective and unconstrained zones of stirrups;  
\(f^{\prime}_{af}\), \(A^{\prime}_{af}\)--**yield strength and section area of the steel flange in the compression zone;  
\(f_{af}\), \(A_{af}\)--**yield strength and section area of steel flange in tensile zone;  
\(t_{f}\), \(t_{w}\)--**steel flange thickness and web thickness;  
\(f^{\prime}_{awf}\), \(A^{\prime}_{aw}\)--**yield strength and section area of steel web in compression zone;  
\(f_{aw}\), \(A_{aw}\)--**yield strength and section area of steel web in tension zone;  
\(d\)--**height of unyielding steel.