# Reliability Analysis of the Main Pier during the Construction Period of HLCR Bridges
## 2 Theoretical Foundation
### Resistance Probability Modeling  
The piers of rigid bridges are mainly thin-walled and of the column type. HLCR bridges usually use single or double, vertical, thin-walled piers, such as the Guizhou-Baishuichuan Bridge, Shaanxi-Shibaochuan River Bridge, Chongqing-Huangshi Bridge, etc. Compared to the single alternative, double thin-walled piers have large comprehensive flexural stiffness, great overall stability, and larger horizontal displacement is allowed. The research object of this paper is a double-limb, thin-walled hollow piers. According to China's regulations [31], when reinforced concrete members are designed for transient conditions, the positive cross-sectional compressive stress at the edge of the concrete in the compression zone should not exceed \(0.80f^{\prime}_{ck}\). Therefore, the resistance of the main pier under the strength failure condition can be written as [30]:  
\[R_{1}=k_{p1}[\sigma]=0.80k_{p1}f^{\prime}_{ck}=0.80k_{p1}k_{mh}f^{\prime}_{k} \tag{16}\]  
where \(f^{\prime}_{ck}\) is the standard value of concrete axial compressive strength corresponding to concrete cubic compressive strength \(f^{\prime}_{cuk}\) during the construction phase (kPa); \(k_{mh}\) is the parameter of material property uncertainty in structural members, and its mean value and coefficient of variation can be taken as in [30]; \(f^{\prime}_{k}\) is the standard value of the material properties of the specimen in the specification (kPa); \(k_{p1}\) is the uncertainty factor of the resistance calculation model.  
In the construction stage of the lower structure of the bridge, the pier mainly bears the effect of its own gravity and wind load. Then, during the construction of the upper beam structure, the load acting on the pier becomes complex. In addition to its own gravity, it must also bear the construction of the upper structure generated by the self-weight of the beam, hanging basket self-weight and other construction live load. At this time, the pier is a typical eccentric compression member. When the load applied to the structure exceeds its ultimate bearing capacity, the high pier structure will lose its original balance, causing an instability phenomenon.  
The critical force at the top of the combined pier is derived from the Rayleigh-Ritz method as [30]:  
\[\begin{array}{c}N_{cr}=k\frac{EI_{2}}{H^{2}}-0.3k_{G}qH\\ k=-\frac{8}{3}\sqrt{C_{1}(n-1)^{2}+C_{2}n^{2}+C_{3}n+(C_{4}+31)}+\frac{1}{3}[5 2+C_{5}(n-1)]\\ C_{1}=400B^{6}-900B^{5}+855B^{4}\\ C_{2}=-405B^{3}+81B^{2}\\ C_{3}=1025B^{3}-387B^{2}+72B\\ C_{4}=-620B^{3}+306B^{2}-72B\\ C_{5}=160B^{3}-180B^{2}+72B\\ B=H_{1}/H\\ B=H_{1}/H\\ n=I_{1}/I_{2}\end{array} \tag{17}\]  
where \(E\) is the modulus of elasticity of concrete (kPa); \(k_{G}\) is the statistical parameter of constant load indeterminacy; \(I_{1}\) is the bending stiffness of a single pier section (m\({}^{4}\)); \(I_{2}\) is the double-limb, thin-walled part of the pier flexural stiffness (m\({}^{4}\)); \(H_{1}\) is the free length of a single part of the pier (m); \(H\) is the pier height (m); \(B\) is the relative position parameter of the variable section of the bridge pier (the dividing point between the two-armed and one-armed piers); \(n\) is the ratio of cross-sectional moments of inertia of a single to a double, thin-walled pier; \(q\) is the bridge self-weight set degree (kN/m).  
Therefore, the resistance of the bridge pier instability can be expressed as [30]:  
\[R_{2}=k_{p2}N_{cr}=k_{p2}\bigg{(}k\frac{EI_{2}}{H^{2}}-0.3k_{G}qH\bigg{)} \tag{18}\]  
where \(k_{p2}\) is the uncertainty factor of the resistance calculation model.