# Analysis on Response of a Single Pile Subjected to Tension Load Considering Excavation Effects
## 3 Load-Transfer Model Considering Excavation Effects
### Load Transfer Relationship of Pile-Soil Interaction  
The laboratory and field test results show that the nonlinear relationship between the pile-soil shear stress and the relative displacement at the pile-soil interface can be expressed as a hyperbolic or exponential function. [PERSON] et al. [29] carried out load tests on instrumented tension piles to investigate the reliability of the hyperbolic function to model the load transfer between pile shaft and surrounding soil. The results showed a close agreement between test results and predicted values estimated by the hyperbolic function in different soils. Thus, the hyperbolic function is adopted to model the behavior of the pile-soil interface, as shown in Figure 3. The hyperbolic function can be expressed as:  
\[\tau_{sp1}=\frac{\omega_{sp1}}{f+g\omega_{sp1}} \tag{11}\]  
where \(\tau_{sp}\) and \(\omega_{sp}\) are the shear stress at pile-soil interface and the corresponding pile-soil relative displacement, respectively; The \(f\) and \(g\) are model parameters. The parameter \(f\) represents the inverse of the initial tangent stiffness of the load-transfer curve, and \(g\) is the inverse of the asymptote of the load-transfer curve. The parameters \(f\) and \(g\) can be determined using Equations (12) and (13):  
\[f=\frac{r_{0}}{G_{\mathrm{s}}}\ln\left(\frac{r_{m}}{r_{0}}\right) \tag{12}\]  
\[g=\frac{R_{sf}}{\tau_{su}} \tag{13}\]  
where \(r_{0}\) is the pile radius; \(G_{\mathrm{s}}\) is the shear modulus of surrounding soil; \(r_{m}\) is the limiting radius where the shear strain induced by the pile can be negligible; \(R_{sf}\) is the failure ratio of shaft resistance, \(R_{sf}=\tau_{su}/\tau_{sf}\); \(\tau_{sf}\) is the pile shaft resistance at which the pile-soil relative displacement is infinity; \(\tau_{su}\) is the ultimate pile shaft resistance. The ultimate pile shaft  
Figure 2: Schematic of basal heave calculation. resistance \(\tau_{su}\) is determined by the effective stress method. As suggested by [PERSON] et al. [30], the ultimate pile shaft resistance \(\tau_{su}\) can be calculated as:  
\[\tau_{su}=K_{0}\bigg{(}\frac{K}{K_{0}}\bigg{)}\sigma^{\prime}_{v}\tan\bigg{(} \varphi^{\prime}\bigg{(}\frac{\delta}{\varphi^{\prime}}\bigg{)}\bigg{)} \tag{14}\]  
where \(K_{0}\) is the lateral earth pressure coefficient at rest, which is determined by \(K_{0}=(1-\sin\varphi^{\prime})OCR^{\sin\varphi^{\prime}}\), \(OCR\) is the overconsolidation ratio; \(\varphi^{\prime}\) is the constant-volume (or critical-state) effective friction angle of surrounding soil; \(\delta\) is the friction angle of pile-soil interface; \(K\) is the lateral earth pressure coefficient; \(\sigma^{\prime}_{v}\) is the effective overburden pressure. The ultimate pile shaft resistance \(\tau_{su}\) is determined by effective stress analysis. Hence, both the cohesive and cohesionless soil can be modeled. The friction angle \(\delta\) depends on the properties of the pile shaft and surrounding soil, and can be determined by shear box tests and empirical correlations. The later earth pressure coefficient \(K\) depends on several factors including soil stress state, pile geometry, pile installation method. The later earth pressure coefficient \(K\) is related to the lateral earth pressure coefficient at rest \(K_{0}\). The empirical correlations to determine later earth pressure coefficient \(K\) and friction angle of pile-soil interface \(\delta\) are summarized in Table 1.  
The overconsolidation ratio \(OCR\) is defined as the ratio of preconsolidation stress to effective overburden stress. Excavation will cause mechanical unloading for soils below the excavation surface. For the soil near the excavation surface, the overconsolidation ratio seems to be infinite due to the infinitesimal effective overburden stress. It is assumed that the lateral earth pressure coefficient \(K_{0}\) of overconsolidated soil cannot be greater than the passive earth pressure coefficient \(K_{p}\).