# Mechanical Analysis of Ground Sleeve
## 5 Mechanical Analysis of Grouted Sleeve Lapping Connector
### Force State at the Middle Section  
From the force equilibrium in the middle section of the grouted sleeve lapping connector (as shown in Figure 17), the following is found.  
\[P-P_{a}-P_{a}^{\prime}=P_{\text{st}}+P_{\text{ct}}=P_{\text{re}} \tag{8}\]  
Figure 16: Bond pullout test with bond stress distributions (reproduced from Reference [44]).  
Figure 14: Mechanical equilibrium of the inserted bar in the grouted sleeve lapping connector.  
Figure 15: Ratios of \(F_{b}^{AB}\) and \(F_{b}^{BC}\) to specific loads in the grouted sleeve lapping connector.  
where \(P\) denotes the load and \(P_{a}\) and \(P_{a}^{\prime}\) denotes the tensile forces of the reinforcement bars, which can be calculated by using the strains measured by SG5 and SG6. \(P_{st}\) denotes the tensile force of the sleeve, which is also the resultant shear stress acting on the sleeve wall. \(P_{ct}\) denotes the resultant tensile stress of the grout and \(P_{\rm re}\) denotes the resultant of \(P_{\rm st}\) and \(P_{\rm ct}\). Substituting the relevant data into Equation (8) yields the value of \(P_{\rm re}\).  
Figure 18 shows the relationship between \(P\) and \(P_{\rm re}\) for specimens with different lap lengths. For the specimens of the 150, 200, 250, and 300 series, the developments of the curves were similar and could be divided into three stages. In the first stage, i.e., section AB, \(P_{\rm re}\) was tensile and increased with the load. \(P_{\rm re}\) reached its maximum (\(P_{\rm re}^{\rm m}\)) when the load reached a certain value (\(P_{1}\)). In the second stage, i.e., section BC, \(P_{\rm re}\) was still tensile, but it decreased as the load increased. At the moment that the load value reached \(P_{2}\), the value of \(P_{\rm re}\) decreased to 0. In the last stage, i.e., section CD, \(P_{\rm re}\) was compressive and increased with the load.  
As we can see from Table 4, \(P_{\rm re}^{\rm m}\), \(P_{1}\), and \(P_{2}\) increased with the lap length. The increment in the lap length engaged more bar ribs in interlocking with the grout keys from the loading end to the middle section, such that a larger part of the load was borne by the bond between the bar and grout. Thus, for the same load, the longer was the lap length, the smaller was \(P_{a}+P_{a}^{\prime}\), and the larger was \(P_{\rm re}\). Therefore, the maximum tensile value of \(P_{\rm re}\) (i.e., \(P_{\rm re}^{\rm m}\)) and the corresponding load (\(P_{1}\)) increased with the lap length.  
Figure 17: Force equilibrium at the middle of the grouted sleeve lapping connector.  
\begin{table}
\begin{tabular}{l r r r r r r r} \hline \hline
**Lap Length** & \multicolumn{3}{c}{**150 mm**} & \multicolumn{3}{c}{**200 mm**} & \multicolumn{3}{c}{**250 mm**} & \multicolumn{1}{r}{**300 mm**} \\ \hline Specim & 150-1 & 150-3 & 150-4 & 200-1 & 200-2 & 250-1 & 250-2 & 300-1 \\ \(P_{\rm re}^  
According to Equation (6), \(P_{a}+P_{a}^{\prime}\) was equal to \(P\) when \(P_{\rm re}\) was 0. The load transfer from the loading end to the middle section was slower when the lap length was longer. Therefore, it took more time for the value of \(P_{a}+P_{a}^{\prime}\) to increase to the value of \(P\) as the loading rate was fixed. This showed that the duration of section AC where \(P_{\rm re}\) was tensile was longer. Accordingly, the value of \(P_{2}\) was larger for the specimens with longer lap lengths.