## 5 Conclusions  
In the present study, it is shown that TPO/MA is an efficient method in solving plane strain problems too whether the constitutive equation is linear or not. Until now, TPO/MA was applied to truss systems [5; 6; 7; 8], tensegrity structures [9], cable structures [10; 11], and plane stress problems [12; 13] for a wide range of examples. By including plane-strain problems in the application record, it is shown in this paper that the application area of this technique is not limited to previous ones. It is hoped that in the future, the method TPO/MA will be shown to apply to many other types of structural analysis problems. Currently, there are quite a big number of metaheuristic algorithms. Although all of them can be implemented in TPO/MA, it is seen that there may be differences in inaccuracies and time consumed, probably depending on problems. In this paper, 10 metaheuristic algorithms were compared with each other on an exemplary problem that was a 4 member pipe with a linear stress-strain relationship. It is seen that among the algorithms tried, five were more efficient than the others, and AHS among them was the most successful if the CPU time is taken into account. Three different nonlinear behaviors are reflected with stress-strain relationships that are different from the linear material. The nonlinear stress-strain relations in case 2 are very close to those of the linear case since it differs from the linear problem by only with the shifting of the value of the elasticity modulus according to the values of strains. For example, displacements of nonlinear case 2 in structure 1 were double those in the linear case due to the tendency of the nonlinear part with the half value of the elasticity modulus of the linear situation. Differently, displacements in nonlinear case 2 of structure 2 tend to have the same value of the elasticity modulus of the linear case. Both displacements and total potential energy of nonlinear case 1 and case 3 are very different from those of linear case since these cases involve cubic powers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{v}\) & **TPOMA (Linear)** & **TPOMA (Nonlinear 1)** & **TPOMA (Nonlinear 2)** & **TPOMA (Nonlinear 3)** \\ \hline
0.15 & \(-\)0.031403525 & \(-\)654458.4484 & \(-\)0.031403525 & \(-\)39.59712014 \\
0.16 & \(-\)0.031457332 & \(-\)771097.8575 & \(-\)0.031457332 & \(-\)39.81729795 \\
0.17 & \(-\)0.031501216 & \(-\)900090.841 & \(-\)0.031501216 & \(-\)40.03945641 \\
0.18 & \(-\)0.031534625 & \(-\)1044225.23 & \(-\)0.031534625 & \(-\)40.26378793 \\
0.19 & \(-\)0.031556952 & \(-\)1205801.743 & \(-\)0.031556952 & \(-\)40.49051621 \\
0.2 & \(-\)0.031567536 & \(-\)1385982.025 & \(-\)0.031567536 & \(-\)40.71990187 \\
0.21 & \(-\)0.031565648 & \(-\)1586562.796 & \(-\)0.031556548 & \(-\)40.95224928 \\
0.22 & \(-\)0.031550489 & \(-\)1812681.81 & \(-\)0.031550489 & \(-\)41.18791476 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The total energy values for the system with 14 members by changing Poisson’s ratio (AHS). The method TPO/MA can automatically deal with nonlinear stress-strain relationship formulations by only changing the coded equation without the need for additional user interventions like in FEM. In addition, TPO/MA can handle the total potential energy process quickly. For example, the process for the linear case of structure 1 via AHS can reach the optimum solution after 0.4 s of computing time. As indicated above, the nonlinear analyses were done by employing AHS after seeing its success in applications on the linear case. Indeed, it was among the five algorithms that reached the minimum energy level in all cycles, and the one necessitating minimum CPU time. For nonlinear cases, the processing time is between 1 and 2 s according to the number of design variables. This study has shown that TPO/MA is a real and powerful alternative to other methods in solving statical analysis problems especially when there are nonlinearities in the problem.