## 4 Calculation Method for Vertical Bearing Capacity of SDRN File
### Calculation Method for Core Pile Penetration Failure Mode  
Considering the similar joints on pile shaft and the same fail mode, the calculation method of cast-in-situ piles with expanded branches and plates can be referred to. The calculation formula for the characteristic value of vertical bearing capacity of the cast-in-situ pile is as follows [14]:  
\[Q_{u}=u_{p}\sum q_{sik}l_{i}+\sum\eta q_{pkj}A_{pj}+\eta q_{pk}A_{p}, \tag{4}\]  
where \(Q_{u}\) is the standard value of vertical ultimate bearing capacity of the cast-in-situ pile, \(u_{p}\) is the perimeter of the main pile, \(q_{ski}\) is the standard value of ultimate pile shaft resistance in the \(i\)-th soil layer, \(l_{i}\) is the effective thickness of the \(i\)-th layer of soil or rock around pile excluding the length of branches and plates. \(q_{pkj}\) is the resistance of soil beneath the \(j\)-th plate, \(A_{pj}\) is the \(j\)-th plate horizontal sectional area excluding the main pile sectional area, \(q_{pk}\) is the tip resistance of the pile, and \(A_{p}\) is the pile tip horizontal sectional area. The second expression in the formula is to calculate the bearing capacity of soil beneath the \(j\)-th plate. The coefficient \(\eta\) is related to the position and radium of the plates. According to the analysis of Lu [15], the compaction coefficient of the squeezed branch and plate, \(\beta_{j}\) = 1.1\(\sim\)1.3, should also be taken into consideration. The axial force on the upper and lower surface of plates is not the same due to the load borne by the soil beneath the plates directly. In contrast, the nodular joints do not bear the axial force directly [8], so there is no need to derive an expression to calculate the bearing capacity of the joints separately.  
Based on the construction mechanism, the standard value of the upper pile shaft resistance \(q_{sik2}\) should be greater than the resistance between cemented soil and soil. The upper PRHC or PHC pipe pile shaft resistance is calculated by the method of precast pile.  
The lower nodular pile shaft resistance is calculated based on the diameter of the joints. Due to the joints, the axial force declining rate of the nodular piles with depth is faster than that of the PRHC or PHC piles. The compaction coefficient \(\eta_{2}\) of nodular joints is related to the length of joints and interval between joints [3]. Compared with cast-in-situ pile with expanded branches and plates, \(\eta_{2}\) should be smaller than \(\beta_{j}\). Therefore, it is suggested that \(\eta_{2}\) = 1.0\(\sim\)1.1, which is close to the compaction coefficient \(\eta_{s}\) = 1.05\(\sim\)1.10 proposed by [PERSON] [16],  
\[Q_{sk2}=u_{ns}\sum q_{sik2}l_{i}+u_{nx}\sum\eta_{2}q_{sik2}l_{i}, \tag{5}\]  
where \(Q_{sk2}\) is the standard value of pile shaft resistance in core pile penetration failure mode, \(u_{ns}\) is the perimeter of the upper PRHC or PHC pile; \(q_{sik2}\) is the standard value of ultimate core pile shaft resistance in the \(i\)-th soil layer, the precast pile shaft resistance is taken; \(u_{nx}\) is the perimeter of nodular joint, \(\eta_{2}\) is the compaction coefficient considering nodular joints, \(\eta_{2}\) = 1.0\(\sim\)1.1, \(l_{i}\) is the thickness of the \(i\)-th layer of soil or rock around pile, excluding the length of the expanded section and 2\(d\) upward, and \(d\) is the diameter of the cemented soil pile not expanded.