# Study on Stability of Elastic Compression Bending Bar in Viscoelastic Medium
## 4 Conclusions  
This study constructs a mechanical model of an elastic bar constrained by elastic and viscoelastic medium, fixed at the base and free at the top under Winkler foundation hypothesis, The Rayleigh-Ritz method is employed to derive the deflection function of the bar when subjected to both axial force and local horizontal distributed load. The primary conclusions are as follows:  
1. A mechanical model of an elastic slender compression-bending bar, constrained by elastic-viscoelastic medium wrapping and fixed at the base while free at the top, was established. Utilizing the energy method criterion and the Rayleigh-Ritz method, an approximate solution for the deflection function of the bar body was derived when subjected to axial force \(P\) at the top and horizontal additional load caused by pile loading at the side.
2. Employing the elastic-viscoelastic correspondence principle, the approximate solution for the deflection function of the bar body in an elastic medium was transformed into an approximate solution for the deflection function of an elastic compression-bending bar in a viscoelastic medium. Using a Kelvin body as an example for the side medium of the bar, a deflection function under a second-order triangular series was derived. MATLAB is applied in this study to compute the corresponding theoretical solutions under the burgers model.
3. The influence of the magnitude of axial force on the change of \(w(L)\) over time under equal bar length and equal pile load strength is studied. The calculation results show that when there is constant load on the side of the bar, axial force acts on the top of the bar, and the larger the axial force, the faster the rate of change of \(w(L)\) with time, and the earlier it develops towards instability. The influence of pile load strength on the change of \(w(L)\) over time under equal bar length condition is studied. The results show that the change of bar length has little effect on the stability of elastic bar in viscoelastic medium, and the research should focus on the influence of axial force and pile load strength on the surface of side medium on bar stability.  
This research presents the derivation of the deflection equation of an elastic rod in a viscoelastic medium and discusses the potential implications for the durability of a long rod under this assumption. However, the working condition assumed in this study has not been validated or compared with the actual engineering scenarios. In future research, more realistic engineering situations can be examined by incorporating boundary conditions, and contrasted with the actual engineering cases, to demonstrate the applicability and accuracy of this method in practical work.  
Methodology, X.Z.; Software, J.H.; Investigation, S.C.; Resources, S.C.; Writing--original draft, J.H.; Writing--review & editing, X.Z. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
The authors declare no conflict of interest.