## 4 Conclusions  
This paper proposed a beam FE model to investigate the bending mechanism for a typical single-layered strand subjected to multi-axial loads. A penalty stiffness algorithm incorporating a Coulomb friction model is implemented into the FE model to ensure the precise estimation for interwire friction without any assumptions. It is suitable for strands with more complex geometry and longer length because the present model remarkably reduces the number of beam elements.  
The influence of strand length on deflection, seldom mentioned in previous research, is discussed here. It indicated that the influence was diminished with the increase of the length and almost disappeared as the strand length exceeds six times pitch length.  
Subsequently, a general deflection expression was extended from a classic Euler beam theory to predict the lateral deformation for frictional strands subjected to multi-axis loads based on the small deformation assumption. This expression fitted well with the simulation results. The resistance of the friction on the bending deformation was clearly characterized. The deflection consists of two terms, one is related to the bending stiffness of individual constituent wire and another term corresponds to the resistance of friction. Moreover, this expression was proved to be quietly accurate since it precisely predicted the sag deflections in accordance with the experimental measurements for a simple strand under free bending. The relative error of the proposed model with respect to experimental measurements is less than 10%.  
Furthermore, additional axial tension induced by the interwire friction was apparently presented and the uneven force distribution caused the out-plan deformation of the spiral strand under the in-plan loading. An expression analogous to the parametric equation of Archimedes helix was proposed and utilized to estimate the additional tension for the individual wire in the outer layer of the strand. The coefficient for this expression was determined by means of RSM based on the simulation results. Compared with the FE results, this present method can well predict the tension for individual wires for frictional spiral strands under free bending. It provided an innovative insight for further analytical study to alter the assumptions on the slippage criterion.  
Figure 21: (**a**) The comparison of fitted results and simulation results of the component of force \(r_{i}\) along the longitudinal axis for the 12-pitch strand in condition where \(\mu=0.1\), \(T=5\) kN and \(P=500\) N and (**b**) for the 24-pitch-long strand in condition where \(\mu=0.2\), \(T=5\) kN and \(P=500\) N; (**c**) The comparison of fitted results and simulation results of the axial tension force of each outer layer wire along the longitudinal axis for the 12-pitch-long strand in condition where \(\mu=0.1\), \(T=5\) kN and \(P=500\) N and (**d**) for the 24-pitch-long strand in condition where \(\mu=0.2\), \(T=5\) kN and \(P=500\) N.  
**Author Contributions:** Conceptualization, B.Z. and X.Z.; methodology, B.Z.; software, B.Z.; validation, B.Z., X.Z. and H.Z.; formal analysis, B.Z.; investigation, B.Z.; resources, Y.H.; data curation, X.Z.; writing--original draft preparation, B.Z.; writing--review and editing, B.Z.; visualization, H.Z.; supervision, Y.H.; project administration, Y.H.; funding acquisition, Y.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Program on Key Basic Research Project (Grant No. 2014 CB049401), and Open Fund of the State Key Laboratory of Mechanical Transmission (Grant No. SKLMT-MSKKT-202008).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.