# Study on Stability of Elastic Compression Bending Bar in Viscoelastic Medium
## 3 Results and Discussion
### Time Dependence of the Effect of Axial Force on the Lateral Deflection of the Bar Top  
When an elastic bar is situated within a viscoelastic medium, its deflection over time can be determined using the deflection function Equation (10) of the longitudinal and transverse bending elastic bar. In this section, we investigate the compressive stability of the longitudinal and transverse bending elastic bar within the viscoelastic medium using the value of the bar end displacement \(w(L)\) under long-term lateral pile load as a key parameter. Taking a lateral pile load strength of 50 kPa and a bar length of 50 m as examples, and with a background of a typical bridge design life of 100 years in engineering, the total calculation time was set to 100 years. The change in transverse deflection, \(w(L)\), of the bar top was investigated when different magnitudes of axial force, \(P\), were applied to the bar  
When \(P\) is small, only its linear growth stage can be observed within the calculation period; when \(P\) increases to a certain level, \(w(L)\) will exhibit nonlinear acceleration growth within the calculation period but \(w(L)\rightarrow+\infty\) has not yet occurred; when \(P\) is large, \(w(L)\rightarrow+\infty\) can be observed within the calculation period. By analogy with the definition of critical force \(P_{cr}\) that causes simple compression bars to become unstable, we define the time that causes bars to become unstable as critical instability time \(t_{cr}\). The above results show that if \(w(L)\rightarrow+\infty\) is used to define bar instability, different axial forces \(P\) correspond to corresponding critical instability times \(t_{cr}\); alternatively, specifications can be referred to and \(w(L)=10\) cm can be used as the basis for determining critical instability time \(t_{cr}\). In order to clarify the characteristics of bar displacement throughout the process and in conjunction with [PERSON] body's creep law, it is necessary to study the deformation morphology of long-term loaded bars. Figure 9b shows the \(w(L)\)-\(t\) curve of bars subjected to axial force and horizontal additional load in their initial stage. Analysis of this figure reveals that instantaneous deformation occurs in bars at the moment of load action and that larger axial forces result in larger instantaneous deformation amounts. As time progresses, \(w(L)\) experiences rate of increase with time first increases and then decreases before finally entering a stable uniform growth stage. Larger axial forces result in larger growth amplitudes at each time point during nonlinear growth stage of \(w(L)\) and also result in larger rates during its uniform growth stage. On the graph it shows that when axial force is large, \(w(L)\)-\(t\) curve envelopes \(w(L)\)-\(t\) curve when axial force is small from top to bottom. In this section, considered a scenario as where the bar-side pile loading intensity is 50 kPa and the bar length is 50 m. The total calculation time is set to 100 years. We investigate the temporal evolution of lateral deflection \(w(L)\) at the top of the bar under varying magnitudes of axial force \(P\) as depicted in Figure 9.  
Figure 9a,b illustrate that the bar undergoes instantaneous deformation upon the application of load. As time progresses, the rate of increase of \(w(L)\) with respect to time initially increases before decreasing, ultimately reaching a stable and uniform growth phase. This displacement process closely resembles the three-stage creep characteristics of the [PERSON] model. Over the course of the 100-year calculation period, when P is small, \(w(L)\) exhibits an approximately linear growth trend during the long-term load application period following the initial stage. When \(P\) is larger, such as 5000 kN or 7500 kN, \(w(L)\) undergoes a linear growth phase in the early stage and exhibits a nonlinear acceleration trend with time in the later stage. When \(P\) continues to increase to 10,000 kN, \(w(L)\) does not reach 100 years  
Figure 9: The time history of displacement variation when \(L=50\).