# Study on Stability of Elastic Compression Bending Bar in Viscoelastic Medium
## 2 Methods
### Deflection Function for an Elastic Compression-Bending Bar within an Elastic Medium
#### 2.1.3 Mechanical Model of an Elastic Compression-Bending Bar in an Elastic Medium  
Figure 5 depicts the mechanical model of an elastic compression-bending bar in an elastic medium. A constant load with a width of \(L_{p}\) and magnitude of \(p\) acts on the surface of the elastic medium at a distance of \(d_{p}\) from one side of the bar to simulate subgrade filling on one side of the pile foundation; \(q(x)\) represents the real passive load on the bar body caused by pile loading and soil self-weight; axial force \(P\) acts on the end of the bar, inducing a small displacement \(\Delta x\) in the vertical direction at the end of the bar. The bottom of the bar is fixed and the top is free; the length of bar is \(L\), and \(k(x)\) represents the ground reaction coefficient at different depths; \(b_{1}\) denotes the calculated width of the elastic bar. Drawing upon the relevant theories of elasticity and soil mechanics, it can be shown that, within an elastic medium, the horizontal additional stress \(\sigma_{x}\) at a given point in the medium induced by a constant load \(p\) of a certain width can be determined based on the Boussinesq solution and Flamant solution [42; 43]. Subsequently, the real passive load \(q(x)\)  
Figure 5: Model of elastic compression bending bar in elastic medium. on the bar body can be obtained demonstrated in Equation (8) which is detailed introduced in Appendix A.  
\[q(x)=\frac{b_{1}p}{\pi}\left(\frac{(L-x)^{2}\text{arctan}\frac{L_{p}}{L-x}}{\left( L-x\right)^{2}+\left(L_{p}+d_{p}\right)d_{p}}-\frac{\left(L_{p}+d_{p}\right)(L-x)}{ \left(L-x\right)^{2}+\left(L_{p}+d_{p}\right)^{2}}+\frac{(L-x)d_{p}}{\left(L-x \right)^{2}+d_{p}{}^{2}}\right) \tag{8}\]  
where: \(q(x)\) is passive load on elastic bar caused by constant load and soil self-weight  
\(b_{1}\) is calculated width of the elastic bar  
\(L\) is length of bar  
\(L_{p}\) is width of constant load  
\(p\) is magnitude of constant load  
\(d_{p}\) is distance from the bar to constant load  
The objective of this study is to investigate the displacement response of piles under the influence of ground load in the adjacent area. A crucial step in this analysis is to ascertain the additional pile load induced by the ground load. To this end, ABAQUS is employed as finite element model (FEM) solution software, pile material and geometric parameters are applied with roadbed filling area layout parameters, etc. that are specified at the outset of this section. Detailed parameters are shown in Table 1 and entire model are shown as Figure 6, which has a grey solid part for soil and a constant force adding in the yellow part. As is shown in Figure 7, the numerical solution is in good agreement with the analytical solution. Therefore, the passive load of the pile body can be calculated by Equation (8). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Magnitude** & **Unit** \\ \hline Width of bar/\(b_{1}\) & 1 & m \\ Length of bar/\(L\) & 50 & m \\ Elastic Modulus of bar/\(E\) & \(3\times 10^{4}\) & MPa \\ Density of bar/\(\rho\) & 2385 & kg/m\({}^{3}\) \\ Poisson’s ratio/\(\lambda\) & 0.2 & / \\ Distance from the bar to constant load/\(d_{p}\) & 1 & m \\ Magnitude of constant force/\(q\) & 50 & kPa \\ Length of constant force/\(L_{p}\) & 20 & m \\ Reaction force coefficient/\(k\) & 4500 & kN/m\({}^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters in ABAQUS model. Figure 6: FEM model of single bar with elastic medium. 2.1.4.