# Study on Vertical Load-Carrying Capacity of Post-Grouting Bored Piles
## 2 Pile Load-Transfer Analysis  
The pile is generally elastic in its work. According to the theory of elasticity, the pile displacement \(w\) under vertical load is related to the pile side frictional resistance \(q_{s}\) as follows  
\[B\frac{\partial^{2}w}{\partial z^{2}}=Uq_{s} \tag{1}\]  
where \(B\) is the pile compressive stiffness and \(U\) is the perimeter of pile section. Since in practical engineering, pile generally traverses multiple layers of soil, the solution for the pile side frictional resistance needs to consider the influence of the nature of different soil layers on the pile side frictional resistance in addition to the factor of pile displacement. [PERSON] et al. [8] proposed the load-transfer method to express the relationship between pile lateral resistance and pile displacement as  
\[q_{s}=q_{s}(w) \tag{2}\]  
This relationship is known as the load-transfer function and can be solved for using the pile balance condition and deformation coordination conditions. Due to the different properties and transfer functions of the soil layers on the side of the pile, the specific calculations are analyzed by dividing the pile into several segments (as in Figure 1a). For segment \(i\) pile unit (as in Figure 1b) the top surface settlement is \(w_{i-1}\) and the bottom surface settlement is \(w_{i}\). Figure 1: Pile-soil interaction model:(**a**) Pile force diagram; (**b**) Pile unit force diagram. Based on the load-transfer function, the side frictional resistance \(q_{si}\) can be obtained for the segment \(i\) pile unit, and the total side frictional resistance for the segment \(i\) pile unit can be approximated as  
\[Q_{si}=q_{si}u_{i}l_{i} \tag{3}\]  
where \(u_{i}\) is the perimeter of the section of the segment \(i\) pile unit and \(l_{i}\) is the length of the segment \(i\) pile unit. Let the top surface axial force and bottom surface axial force of the segment \(i\) pile unit be \(P_{i-1}\), \(P_{i}\), respectively, then according to the balance condition of the pile body, we get  
\[Q_{si}=P_{i-1}-P_{i} \tag{4}\]  
Approximate the average axial force of the segment \(i\) pile unit as  
\[N_{i}=\frac{P_{i-1}+P_{i}}{2} \tag{5}\]  
Considering the pile body as an elastomer, the compression of the segment \(i\) pile unit is  
\[\Delta_{i}=\frac{N_{i}}{E_{ci}A_{ci}}l_{i} \tag{6}\]  
where \(E_{ci}\) is the elastic modulus of the segment \(i\) bored pile and \(A_{ci}\) is the cross-sectional area of the segment \(i\) bored pile. According to the deformation compatibility condition, then  
\[\Delta_{i}=w_{i-1}-w_{i} \tag{7}\]  
Using the pile balance and deformation compatibility conditions, the pile axial force, the pile side frictional resistance, and the pile settlement at the unit division interface of each pile segment can be calculated. For multilayer soils, the specific calculation uses the iterative method, presupposing a smaller pile tip settlement \(w_{b}\). Based on the load-transfer function, the relationship between the pile tip resistance \(q_{p}\) and the pile tip settlement \(w_{b}\) is  
\[q_{p}=q_{p}(w_{b}) \tag{8}\]  
The pile tip resistance \(q_{p}\) is calculated from Equation (8), and assuming a uniform distribution of pile tip resistance, the total pile tip resistance \(Q_{b}\) is  
\[Q_{b}=q_{p}A_{b} \tag{9}\]  
where \(A_{b}\) is the pile tip cross-sectional area. According to the balance condition and deformation compatibility condition of the pile body, the bottom settlement \(w_{n}\) and bottom axial force \(P_{n}\) of the segment \(n\) pile unit are  
\[q_{s}=q_{s}(w) \tag{2}\]0  
\[q_{s}=q_{s}(w) \tag{2}\]1  
The values of the top surface settlement \(w_{n-1}\) and top surface axial force \(P_{n-1}\) of the segment \(n\) pile unit, which satisfy both the pile balance condition and the deformation compatibility condition, are then calculated by iteration. Similarly, calculating from the bottom up can get the top surface settlement \(w_{n-2}\), \(w_{n-3}\),..., \(w_{0}\) of each pile unit and the top axial force \(P_{n-2}\), \(P_{n-3}\),..., \(P_{0}\) of each pile unit.