# Nonlinear Vibration and Post-Buckling Behaviors of Metal and FGM Pipes Transporting Heavy Crude Oil
## 7 Conclusions  
This research explores the post-buckling and nonlinear vibration behavior of functionally graded material (FGM) pipes transporting heavy crude oil, modeled using Timoshenko beam theory. Two distinct FGM configurations are examined, with their material properties considered to be temperature dependent. The nonlinear governing equations (NGEs) are derived by applying [PERSON]'s principle, Timoshenko beam theory, and [PERSON]'s nonlinear strain-displacement relations. These equations are then discretized via [PERSON]'s method. This study highlights the influence of both material properties and geometric parameters on the system's nonlinear response. The main findings can be outlined as follows:  
* By analyzing pipes made from functionally graded materials (FGMs), this study offers valuable insights into how material gradation affects structural behavior. Results reveal that the natural frequency (NF) of Type I pipes is higher than that of Type II.
* As the volume fraction index (\(N\)) increases, the NF and critical flow velocity (CFV) decrease for Type I pipes, whereas they increase for Type II pipes. This demonstrates that, beyond their inherent resistance to corrosion and fatigue, FGM pipes offer a unique advantage. Their vibration characteristics can be tailored by adjusting the gradient index \(N\). This tunability allows engineers to control the NF within a desired operational range. Additionally, the flow resistance--reflected by the critical flow velocity--can be effectively optimized by selecting the suitable FGM type and modifying the value of \(N\).
* As \(N\) increases, the critical flow velocity decreases for Type I FGM pipes and increases for Type II.  
* Increasing the length results in a decrease in the CFV. Additionally, an increase in the outer radius leads to a higher CFV, while an increase in the inner radius causes it to decrease.  
The originality of this study lies in its unified treatment of nonlinear vibration and postbuckling behaviors in FGM pipes transporting heavy crude oil, which is a domain where the combined effects of temperature-dependent properties, realistic fluid characteristics, and structural gradation have been largely overlooked. By comparing two distinct FGM configurations under identical thermal and flow conditions and integrating these into a temperature-sensitive nonlinear model, this work offers new perspectives on how to tailor structural performance in thermomechanical environments.  
The findings of this study have practical implications in engineering scenarios where systems are subject to complex loading and vibration, such as fluid-conveying pipes operating under high pressure and temperature, aerospace panels exposed to turbulent aerodynamic forces, and advanced structural elements in robotic or mechatronic systems. By tuning geometric and material parameters, engineers can tailor the dynamic behaviors of such systems to mitigate resonance, enhance stability, and optimize performance. Future work could extend this framework to include active vibration control or real-time monitoring strategies for critical infrastructure.  
Conceptualization, K.F. and F.T.; Methodology, K.F.; Software, K.F. and A.P.P.; Validation, K.F., F.T. and A.P.P.; Formal analysis, K.F.; Investigation, K.F. and A.P.P.; Resources, K.F.; Data curation, K.F. and F.T.; Writing--original draft, K.F.; Visualization, K.F. and F.T.; Supervision, F.T.; Project administration, F.T. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.