Realization of equipotential surfaces in structurally inhomogeneous rods in variational formulations of optimization problems

Introduction. A variational formulation of the problem of optimizing the geometric configuration of a layered heterogeneous rod under the condition of constant total cost of materials is considered. The integral criterion of minimum deformation energy is adopted as the optimality criterion when varying the geometric functions profiling the rod layers. Currently, this approach, applied in homogeneous systems, requires development and extension to complex heterogeneous environments, and the development of methods for application in building structures.

Materials and methods. Using the mathematical model of the Timoshenko rod, formulas are given for the main components of stress and rigidity characteristics of zero, first and second orders. An energy functional and a constraint on the total cost of materials were formulated. Euler equations were obtained for varying geometric functions.

Results. The optimization problems of a layered rod by varying the width and thickness of layers for symmetrical and arbitrary structures are solved. The cases of bending, tension, transverse shear and combined bending with tension were investigated. It has been analytically proven that in all the cases considered, surfaces with an equal level of specific deformation energy are formed in the system. It is shown that the isoperimetric variational formulation leads to the minimum cost of the construction’s materials.

Conclusions. A variational formulation with one constraint on the total cost of materials, necessary according to the meaning of the problem, provides a global minimum of the functional of the deformation energy and the cost of materials of the system and reflects, the so-called, reference project. Knowledge of such a project is valuable and useful from a practical point of view. In an optimal system, equipotential surfaces with identical specific deformation energy values are formed. Their shape and location are determined by the emerging efforts and the structure of the system. From the integral energy criterion follow practical criteria for equalizing the specific energy of deformation, as well as the main stress or deformation on the surfaces of areas received by varying of dimensions.

1. Perel'muter A.V. Synthesis problems in the theory of structures (brief historical review). Journal of Construction and Architecture. 2016; 2(55):70-106. EDN VSTVRX. (rus.).

2. Yuriev A., Panchenko L. Energy beginning in the theory of structure synthesis. Bulletin of Belgorod State Technological University named after V.G. Shukhov. 2023; 10:35-41. DOI: 10.34031/2071-7318-2023-8-10-35-41. EDN SGKZNZ. (rus.).

3. Yankovskii A.P. Refined Modeling of Flexural Deformation of Layered Plates with a Regular Structure Made from Nonlinear Hereditary Materials. Mechanics of Composite Materials. 2018; 53(6):705-724. DOI: 10.1007/s11029-018-9697-9. EDN XXUTAL.

4. Yankovskii A.P. Critical Analysis of the Equations of Statics in the Bending Theories of Composite Plates Obtained on the Basis of Variational Principles of Elasticity Theory 1. General Theories of High Order. Mechanics of Composite Materials. 2020; 56(3):271-290. DOI: 10.1007/s11029-020-09880-8. EDN ASGJMZ.

5. Troitsky V.A., Petukhov L.V. Optimization of the shape of elastic bodies. Moscow, Nauka, 1982; 432. (rus.).

6. Yuryev A.G. Variational statements of problems of structural synthesis in the statics of structures. Moscow, MISI, 1987; 94. (rus.).

7. Lyakhovich L.S., Perelmuter A.V. Some problems of building constructions optimal projecting. International Journal for Computational Civil and Structural Engineering. 2014; 10(2):14-23. EDN SXCOQB. (rus.).

8. Vasilkov G.V. Evolutionary tasks of structural mechanics. Synergetic paradigm. Rostov-on-Don, InfoService, 2003. EDN QNKGAT. (rus.).

9. Goldstein Yu.B., Solomeshch M.A. Variational problems of statics of optimal rod systems. Leningrad, Leningrad State University Publishing House, 1980; 208. (rus.).

10. Mishchenko A.V., Veshkin M.S. Application of the criterion of minimum deformation energy in problems of rational profiling of heterogeneous rods. News of higher educational institutions. Construction. 2025; 5(797):15-29. DOI: 10.32683/0536-1052-2025-797-5-15-29. EDN TCEUPL. (rus.).

11. Yuriev A.G., Nuzhniy S.N. Topology optimization of one-spaned one-storeyed frames. Fundamental Research. 2013; 10-4:742-746. EDN RCHQFH. (rus.).

12. Ebrahimi M., Fakoor M. Design-oriented fracture criteria for orthotropic composites based on minimum strain energy density theory. Theoretical and Applied Fracture Mechanics. 2025; 140:105182. DOI: 10.1016/j.tafmec.2025.105182. EDN EHNYRE.

13. Beck R., da Silva Ja.A.P., da Silva L.Fm., Tita V., de Medeiros R. Assessing critical fracture energy in mode I for bonded composite joints: A numerical–experimental approach with uncertainty analysis. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications. 2024. DOI: 10.1177/14644207241229601. EDN QPBVWG.

14. Lyakhovich L.S. Special properties of optimal systems and the main directions of their implementation in the methods of calculating structures. Tomsk, TSUACE Publishing House, 2009; 371. EDN QNOOHF. (rus.).

15. Wenzel C., Vidal P., D'Ottavio M., Polit O. Coupling of heterogeneous kinematics and Finite Element approximations applied to composite beam structures. Composite Structures. 2014; 116:177-192. DOI: 10.1016/j.compstruct.2014.04.022

16. Biscani F., Giunta G., Belouettar S., Carrera E., Hu H. Variable kinematic beam elements coupled via Arlequin method. Composite Structures. 2011; 93(2):697-708. DOI: 10.1016/j.compstruct.2010.08.009. EDN OEMSBZ.

17. Vidal P., Polit O. A family of sinus finite elements for the analysis of rectangular laminated beams. Composite Structures. 2008; 84(1):56-72. DOI: 10.1016/j.compstruct.2007.06.009. EDN KUKZVN.

18. Liu S., Soldatos K.P. On the prediction improvement of transverse stress distributions in cross-ply laminated beams: advanced versus conventional beam modelling. International Journal of Mechanical Sciences. 2002; 44(2):287-304. DOI: 10.1016/S0020-7403(01)00098-4

19. Matsunaga H. Interlaminar stress analysis of laminated composite beams according to global higher-order deformation theories. Composite Structures. 2002; 55(1):105-114. DOI: 10.1016/S0263-8223(01)00134-9

20. Tahani M. Analysis of laminated composite beams using layerwise displacement theories. Composite Structures. 2007; 79(4):535-547. DOI: 10.1016/j.compstruct.2006.02.019. EDN KULDMX.

21. Gorynin A.G., Gorynin G.L., Golushko S.K. Mathematical Modeling of Three-dimensional Stress-strain State of Homogeneous and Composite Cylindrical Axisymmetric Shells. Journal of Siberian Federal University. Mathematics and Physics. 2024; 17(1):27-37. EDN IRZWVG.

22. Hu H., Belouettar S., Potier-Ferry M., Daya E.M., Makradi A. Multi-scale nonlinear modelling of sandwich structures using the Arlequin method. Composite Structures. 2010; 92(2):515-522. DOI: 10.1016/j.compstruct.2009.08.051

23. Atashipour S.R., Challamel N., Girhammar U.A., Folkow P.D. Flexible N-layer composite beam/column elements with interlayer partial interaction imperfection — A novel approach to structural stability and dynamic analyses. Composite Structures. 2025; 367:119219. DOI: 10.1016/j.compstruct.2025.119219. EDN HBOLFG.

24. Mishchenko A.V. Methods of calculation and rational design of structurally heterogeneous rods. Novosibirsk, NGASU (Sibstrin), 2025; 146. EDN AWFVQL. (rus.).

25. Mishchenko A.V. Stressstate of structurally not uniform rods made from different module materials under thermo-force influences. Structural Mechanics of Engineering Constructions and Buildings. 2016; 4:43-52. EDN WCEXDF. (rus.).

26. Wasiutynski Z. On the congruency of the forming according to the minimum potential energy with that according to the equal strength. Bull. Acad. Pol. Sci. Tech. 1960; 8(6):259-268.

27. Mischenko A.V., Kucherenko I.V. On the conformity of general physical and technical-economic criteria of optimization of mechanical systems. Innovative technologies for strengthening soil massifs on transport communications in complex geological conditions : Proceedings of the All-Russian scientific and practical conference. 2025; 183-191. EDN ZDLFSL. (rus.).

28. Nemirovsky Yu.V. Equal-strength layered elastic arches and beams. News of higher educational institutions. Construction. 1996; 8:20-25. EDN XPVISD. (rus.).