## 4 Conclusions  
In addition to the advantages presented by the upper bound theorem by means of triangle rigid zones (TRZ) under modular approach, such as easy implementation, low computational cost, or quick calculation response, this work shows an extension of its possibilities of application. The initial analysis was limited to parts produced by forging with dies of plane-parallel plates. The modular approach allowed, in a simple way, to be used in parts with symmetrical profiles. Finally, this paper extends its application to nonsymmetrical profiles.  
Considering a nonsymmetrical profile presents the difficulty of defining the vertical plane from which the material flows in opposite directions, which does not happen in symmetrical parts. In this case, the plane of symmetry is the position from which the yield occurs. Determining the new plane of symmetry requires a redistribution and a reconfiguration of modules to be adapted to the new profile. This ingenious application allows to generate the yield plane and, according to the results obtained, a lower value of the maximum limit calculated, adjusting the final result to the real value of the load required to reach the plastic deformation of the material.  
Another conclusion drawn from this study is the fact that the use of very distorted modules (very high relation between height and width) does not mean a significant negative influence, because its low module width provides a very reduced weight in the overall calculation of the nondimensional relation \(P/2k\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**h1** & **Total Conf 1 ws** & **Total Conf 2 ws** & **Total Profile ws** \\ \hline
6 & 2.053 & 2.259 & 2.158 \\
5.8 & 2.014 & 2.199 & 2.109 \\
5.6 & 1.975 & 2.141 & 2.060 \\
5.4 & 1.936 & 2.083 & 2.012 \\
5.2 & 1.898 & 2.027 & 1.964 \\
5 & 1.860 & 1.971 & 1.917 \\
4.8 & 1.823 & 1.916 & 1.871 \\
4.6 & 1.786 & 1.863 & 1.826 \\
4.4 & 1.751 & 1.812 & 1.782 \\
4.2 & 1.717 & 1.762 & 1.741 \\
4 & 1.684 & 1.715 & 1.700 \\ \hline \hline \end{tabular}
\end{table}
Table 6: \(P/2k\) values: total configuration 1 ws, configuration 2 ws, and complete profile.  
Figure 13: (a) \(P/2k\) evolution: configuration 1 ws; (b) \(P/2k\) evolution: configuration 2 ws.  
The possibilities of application through this triangular rigid zones (TRZ) model remain intact, both in parameters contemplated here, reflected in the theoretical development (shear and Coulomb friction), and in others not presented in this work (temperature or strain hardening), that provide solutions through the application of an extremely low computational cost method of immediate response, strongly competitive with numerical methods.  
Conceptualization and investigation: M.J.M., F.M., and M.J.C.; methodology: M.J.M., F.M., and M.J.C.; formal analysis and writing--original draft preparation: F.M.; writing--review and editing: M.J.M.; visualization and supervision: F.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors want to thank the University of Malaga-Andalucia Tech, International Campus of Excellence, for its financial support of this paper.  
The authors declare no conflict of interest.