## 5 Conclusions  
As a critical component of the duct system, the tie rod plays a vital role in ensuring the safety and reliability of the overall system. In this paper, thermal stress analysis of a specific engine duct system was conducted using the finite element method in an attempt to investigate the impact of temperature and pressure on the stress characteristics of the duct system. Based on these findings, a sleeve-type tie rod connection structure was proposed, and its influence on the stress characteristics of the duct system was thoroughly analyzed and validated. The following conclusions were finally obtained:  
Under atmospheric pressure conditions, as the temperature increased from 315.56 \({}^{\circ}\)C to 537.78 \({}^{\circ}\)C, the maximum stress of the duct system increased from 375.62 MPa to 771.25 MPa. When the temperature was maintained at 537.78 \({}^{\circ}\)C, the maximum stress of the duct system fluctuated between 754 MPa and 776.34 MPa as the pressure changed from 1 MPa to 2 MPa. The stress of the duct system varied significantly with temperature.  
Owing to the limited rotational freedom of the conventional spherical bearing at the tie rod end, a high level of restraint exists between the tie rod and the duct system, particularly at their contact position. In the case of the investigated air duct system, when the temperature reached 426.67 \({}^{\circ}\)C, the stress reached 590.61 MPa, which exceeded the yield limit of 565.1 MPa. However, the high temperature experienced in the engine air duct system often exceeds this limit, and the duct stress will be worse when high pressure is exerted.  
The problem of limited tie rod freedom is effectively addressed by the proposed sleeve-type tie rod as it allows for the release of axial displacement of the duct system at the tie rod position. Particularly during instances of significant deformation in high-temperature ducts, it can mitigate the additional force exerted by the tie rod on the duct system. Under the worst-case scenario of 537.78 \({}^{\circ}\)C and 2 MPa as an example, the sleeve-type tie rod proposed in this paper can reduce the stress at the tie rod connection from 757.61 MPa to less than 25 MPa, a reduction of more than 96%. The original tie rod tearing problem is solved, and the maximum stress of the whole duct system is reduced to 459.25 MPa, which is below the yield strength. Expanding the degrees of freedom while avoiding excessive constraints remains the best strategy for mitigating thermal stresses, and this principle is equally applicable in all cases.  
Conceptualization, X.L. and K.F.; methodology, X.L. and Y.W.; software, J.Y.; validation, R.W., K.Y. and R.Z.; formal analysis, Y.W.; investigation, K.Y.; resources, X.L.; data curation, K.F.; writing--original draft preparation, R.W.; writing--review and editing, J.Y.; visualization, R.Z.; supervision, X.L.; project administration, K.F.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the Research Fund of Key Laboratory of Aircraft Environment Control and Life Support, MIIT, Nanjing University of Aeronautics and Astronautics (Grant No. KLAECLS-E-202201).  
Not applicable.  
Data Availability Statement: The data used to support the findings of this study are available from the corresponding author upon request.  
The authors declare no conflict of interest.