# Pull-Out Test and Mechanical Properties Analysis Based on Intelligent Bolt and Internet of Things
## 5 Theoretical Analysis  
Through field test analysis, under the influence of rod end tension, the frictional resistance of the super-long bolt is concentrated at the frontal one third of the bolt, while the plastic deformation in the remaining two thirds of the bolt is minimal. Therefore, based on the local common deformation theory, the force of the super-long bolt is theoretically analyzed in the elastic strain range. It is assumed that the stress in the micro-segment of the anchorage body is linearly related to the relative displacement between the anchorage body and the surrounding rock and soil mass. The mechanical calculation diagram is shown in Figure 11 below:  
The total displacement \(S\) produced by the anchor under rod-end tension consists of the elastic deformation \(S_{1}\) of the anchor solid and the relative slip \(S_{2}\) of the anchor solid and the soil:  
\[S=S_{1}+S_{2} \tag{1}\]  
Shear force per unit length of anchor solid:  
\[q=-\frac{dp}{dx} \tag{2}\]  
Relationship between anchor solid axial force \(p_{x}\) and elastic deformation \(S_{1}\):  
\[p_{x}=E_{a}A_{a}\varepsilon=\frac{1}{4}\pi D^{2}E_{a}\frac{dS_{1}}{dx} \tag{3}\]  
Figure 11: Mechanical distribution diagram.  
where \(A_{a}\) is the cross-sectional area of the anchor solid; \(D\) is the diameter of the anchor solid; and \(E_{a}\) is the equivalent modulus of elasticity of the anchor solid.  
Substituting Equation (3) into Equation (2) gives the following:  
\[q_{x}=-\frac{1}{4}\pi D^{2}E_{a}\frac{d^{2}S_{1}}{dx^{2}} \tag{4}\]  
Assuming a linear elastic relationship between the frictional resistance of the anchor solid and the geotechnical body and the relative slip between them, we obtain:  
\[q=kS_{1} \tag{5}\]  
where \(k\) is the integrated shear modulus.  
From Equation (1):  
\[S_{1}=S-S_{2} \tag{6}\]  
Substituting Equations (4)-(6) into Equation (2) and combining with the boundary condition: \(x=0,p_{x}=P_{0};x=L_{a},p_{x}=0\) solves for:  
\[S_{2}=\frac{\beta P_{0}}{k}\cdot\frac{\cosh[\beta(L_{a}-x)]}{\sinh(\beta L_{a})} \tag{7}\]  
where \(\beta=\sqrt{\frac{k}{E_{a}\Delta_{a}}}\), \(L_{a}\) is the anchoring length and \(P_{0}\) is the rod end tension.  
The axial and frictional forces of the anchor solid under rod end tension are:  
\[p_{x}=P_{0}\frac{\sinh[\beta(L_{a}-x)]}{\sinh(\beta L_{a})} \tag{8}\]  
\[\tau_{x}=\frac{\beta P_{0}}{\pi D}\cdot\frac{\cosh[\beta(L_{a}-x)]}{\sinh( \beta L_{a})} \tag{9}\]  
Substituting Equation (8) into Equation (3) and integrating, the elastic deformation of the anchor solid under the rod-end tension \(S_{1}\) can be obtained:  
\[S_{1}=\frac{\beta P_{0}}{k}\cdot\frac{\cosh(\beta L_{a})-\cosh[\beta(L_{a}-x) ]}{\sinh(\beta L_{a})} \tag{10}\]  
Make \(x=0\) in Equations (7) and (10) and bring in Equation (1) to determine the relationship between the displacement at the rod end and the tensile force at the rod end.  
\[S=\frac{\beta P_{0}}{k}\text{coth}(\beta L_{a}) \tag{11}\]  
According to the theoretical derivation, the calculation formulas of axial force and friction resistance of anchorage body are derived in the range of elastic strain. The comprehensive shear modulus k is calculated according to Equation (11) combined with field test research, and the calculation results are presented in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline
**NO.** & **Free Part/m** & **Anchorage Section/m** & **k/kPa** \\ \hline A-1-2 & 2 & 10 & 423.31 \\ A-1-9 & 3 & 12 & 315.75 \\ B-2-1 & 2 & 16 & 433.98 \\ B-2-9 & 2 & 20 & 566.59 \\ C-1-3 & 2 & 30 & 757.65 \\ \hline \end{tabular}
\end{table}
Table 3: Comprehensive shear modulus k calculation table.  
Through theoretical calculation, the comprehensive shear modulus is in the range of 300-800 kPa. The comprehensive shear modulus is influenced by soil properties, soil pressure, bolt length, and the shear characteristics of the interface between the anchorage body and the rock-soil layer. This information can serve as a reference for designing bolt supports in similar loess strata.