## 8 Conclusions  
The relationships between the CLS of pipelines and the relevant influencing parameters was studied herein, and a comparison was made between three different loading conditions; uniform axial compression, uniform bending, and combined axial compression and bending. The distinction in the nonlinear strain-hardening of YPT and RHT pipes has been observed in previous studies [62; 63; 64; 65; 66] to significantly impact on the deformational capacity of pipes, but marginally affect the limit stress and limit moment. An extensive parametric study was, therefore, conducted to derive the functional relationships between nondimensionalized parameters representing the relevant factors and the deformational performance of pipes made of API X80 grade line pipe steel, while separately considering the material behavior under YPT and RHT classifications.  
The shape constants of a new material model were used to fully describe the shape of the nonlinear portion of the stress-strain curve of the material, and were included as dimensionless parameters in developing semi-empirical models for predicting the CLS. Individual variable functions were developed to mathematically represent the trends of the CLS with respect to the parameters that constitute the semi-empirical models. Advanced nonlinear multiple regression was performedusing Wolfram Mathematica to obtain the nonlinear regression coefficients for each of the derived semi-empirical equations, and effort was made to achieve the highest possible \(R^{2}\) value for each nonlinear expression by accounting for the complex interrelationships between the different parameters. The nonlinear regression analysis yielded high \(R^{2}\) values (>0.95), indicating an excellent goodness-of-fit between the model prediction and the FEA-derived results of the CLS.  
The semi-empirical models developed in this paper are considered as handy tools for evaluating the deformational capacity of both unpressurized and pressurized API X80 line pipes. The results clearly demonstrate that the CLS values for combined axial compression + bending typically exist in a spectrum between the CLS for uniform bending and the CLS for uniform axial compression. It is, however, the prerogative of the designer to determine the level of axial compression to consider, especially in the case of displacement-controlled loading conditions, where the axial stress is not readily derivable. For cases such as this, it may be necessary to conservatively adopt the models for uniform axial compression as a lower bound measure of the CLS.  
This research paper is a product of assiduous collaboration between the authors: conceptualization, O.N. and S.A.; methodology, O.N. and S.A.; validation, O.N., J.J.R.C., and S.A.; formal analysis, O.N.; investigation, O.N.; resources, M.M., J.J.R.C., and S.A.; writing--original draft preparation, O.N.; writing--review and editing, O.N. and S.A.; supervision, S.A.; project administration, M.M.; funding acquisition, M.M. and S.A.  
This research and the APC for publication were funded by TransCanada Pipelines Ltd.  
The authors wish to express their appreciation to TransCanada Pipelines Ltd. for providing the financial support and technical input necessary for conducting this research study.  
The authors declare no conflict of interest.