Русский
English
The article presents a solution to the problem of finding the optimal ratio of the height of the cross-section to the width for a rectangular and box-shaped section in the case of oblique bending and eccentric compression. Optimization is performed according to the strength criterion, and for the case of oblique bending of a rectangular beam, a solution was also obtained from the condition of a minimum full deflection. For a rectangular section, the solution is made analytically, and for a box section, numerically using the MATLAB environment and the Optimization Toolbox package. As a numerical method of nonlinear optimization, the interior point method is used. To simplify the solution, the box section is assumed to be thin-walled, i.e. it is assumed that the wall thickness is significantly less than the height and width of the cross section. An estimate of the error of such an assumption is also performed. It has been established that in the case of oblique bending of a rectangular beam, when optimizing according to the strength criterion, the optimal ratio of the cross-sectional height to width is equal to the cotangent of the angle between the force plane and the vertical axis, and when optimizing according to the rigidity criterion, it is the square root of the cotangent of this angle. In the case of eccentric compression of a rectangular beam with eccentricities in two planes, the optimal ratio of the height of the cross section to the width is equal to the ratio of the eccentricity along the vertical and horizontal axes. For a box-shaped section, graphs of the change in optimal parameters depending on the angle between the force plane and the vertical axis in the case of oblique bending, as well as depending on the ratio of eccentricities along the axes in the case of eccentric compression, are plotted.
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[2] Shariat M. et al. Computational Lagrangian Multiplier Method by using for optimization and sensitivity analysis of rectangular reinforced concrete beams. Steel Compos. Struct. 2018. 29 (2). P. 243 – 256.
[3] Villar-García J. R. et al. Cost optimisation of glued laminated timber roof structures using genetic algorithms. Biosystems Engineering. 2019. 187. P. 258 – 277.
[4] Qiao H., Li H. The discussion on optimization models of pure bending beam. International Journal of Advanced Structural Engineering. 2013. 5. P. 1 – 10.
[5] Mukherjee P., Punera D., Mishra M. Coupled flexural torsional analysis and buckling optimization of variable stiffness thin-walled composite beams. Mechanics of Advanced Materials and Structures. 2022. Vol. 29. 19. P. 2795 – 2815.
[6] Sohouli A., Yildiz M., Suleman A. Efficient strategies for reliability-based design optimization of variable stiffness composite structures. Structural and Multidisciplinary Optimization. 2018. 57. P. 689 – 704.
[7] Punera D., Mukherjee P. Recent developments in manufacturing, mechanics, and design optimization of variable stiffness composites. Journal of Reinforced Plastics and Composites. 2022. 41 (23-24). P. 917 – 945.
[8] Hao P. et al. An integrated framework of exact modeling, isogeometric analysis and optimization for variable-stiffness composite panels. Computer Methods in Applied Mechanics and Engineering. 2018. 339. P. 205 – 238.
[9] Zhu G. et al. Design optimisation of composite bumper beam with variable cross-sections for automotive vehicle. International journal of crashworthiness. 2017. 22 (4). P. 365 – 376.
[10] Duan L. et al. Parametric modeling and multiobjective crashworthiness design optimization of a new front longitudinal beam. Structural and Multidisciplinary Optimization. 2019. 59. P. 1789 – 1812.
[11] Kasperska R.J., Magnucki K., Ostwald M. Bicriteria optimization of cold-formed thin-walled beams with monosymmetrical open cross sections under pure bending. Thin-Walled Structures. 2007. 45 (6). P. 563 – 572.
[12] Manevich A.I., Raksha S.V. Two-criteria optimization of H-section bars–beams under bending and compression. Thin-Walled Structures. 2007. 45 (10-11). P. 898 – 901.
[13] Hämäläinen O.P., Björk T. Optimization of the cross-section of a beam subjected to bending load. Design, Fabrication and Economy of Metal Structures: International Conference Proceedings 2013, Miskolc, Hungary, April 24-26, 2013. Springer Berlin Heidelberg, 2013. P. 17 – 22.
[14] Afzal W., Mufti R.A. Optimal cross-section of cross member for increased torsional and bending stiffness of ladder frame chassis. 2019 16th International Bhurban Conference on Applied Sciences and Technology (IBCAST). IEEE, 2019. P. 218 – 228.
[15] Ye J. et al. Development of optimum cold-formed steel sections for maximum energy dissipation in uniaxial bending. Engineering structures. 2018. 161. P. 55 – 67.
[16] Kim Y.Y., Kim T.S. Topology optimization of beam cross sections. International journal of solids and structures. 2000. 37 (3). P. 477 – 493
[17] Chepurnenko A.S. et al. Trihedral lattice towers with optimal cross-sectional shape. IOP Conference Series: Materials Science and Engineering. IOP Publishing, 2021. 1083 (1). Article 012012.
[18] Sabitov L.S., Badertdinov I.R., Chepurnenko A.S. Optimization of the shape of the cross section of the chords of trihedral lattice supports. Construction and architecture. 2019. 7 (4). P. 5 – 8.
[19] Marutyan A.S. Optimization of pentagonal profile pipes of a new modification. Construction mechanics and calculation of structures. 2016. 3. P. 25 – 35.
[20] Marutyan A.S. Bent-closed profiles and calculation of their optimal parameters. Structural Mechanics of Engineering Constructions and Buildings. 2019. 15 (1). P. 33 – 43.
[21] Magnucki K., Paczos P. Theoretical shape optimization of cold-formed thin-walled channel beams with drop flanges in pure bending. Journal of constructional steel research. 2009. Vol. 65 (8-9). P. 1731 – 1737.
[22] Novoselov O.G., Sabitov L.S., Sibgatullin K.E., Sibgatullin E.S., Klyuev A.S., Klyuev S.V., Shorstova E.S. Method for calculating the strength of massive structural elements in the general case of their stress-strain state (kinematic method). Construction Materials and Products. 2023. 6. (3). P. 5 – 17. https://doi.org/10.58224/2618-7183-2023-6-3-5-17
[23] Novoselov O.G., Sabitov L.S., Sibgatullin K.E., Sibgatullin E.S., Klyuev A.V., Klyuev S.V., Shorstova E.S. Method for calculating the strength of massive structural elements in the general case of their stress-strain state (parametric equations of the strength surface). Construction Materials and Products. 2023. 6 (2). P. 104 – 120. https://doi.org/10.58224/2618-7183-2023-6-2-104-120
[24] Shorstov R.A., Yaziev S.B., Chepurnenko A.S., Klyuev A.V. Flat bending shape stability of rectangular cross-section wooden beams when fastening the edge stretched from the bending moment. Construction Materials and Products. 2022. 5 (4). P. 5 – 18. https://doi.org/10.58224/2618-7183-2022-5-4-5-18
[25] Shein A.I. Optimal dimensions of a rectangular section of a bar with an oblique bend //Bulletin of the Eurasian Science. 2016. 8 (2-33). URL: https://naukovedenie.ru/PDF/116TVN216.pdf
[26] Moiseev G.D., Kolesnikov P.G. Minimization of the mass of structural elements of a box-shaped section at oblique bending. Forestry and chemical complexes-problems and solutions. 2016. P. 100 – 103.
[27] Moiseev G.D., Ivannikova E.A., Kolesnikov P.G., Box-section shape of minimum mass at oblique bending. Machines, Aggregates and Processes. Design, creation and modernization. 2018. P. 68 – 69.
[28] Byrd R. H., Hribar M. E., Nocedal J. An interior point algorithm for large-scale nonlinear programming. SIAM Journal on Optimization. 1999. 9 (4). P. 877 – 900.
[29] Ostroukh E. N. et al. On the issue of the effectiveness of methods and algorithms for solving optimization problems, taking into account the specifics of the objective function. Advanced Engineering Research. 2019. 19 (1). P. 81 – 85.
Chepurnenko A.S., Turina V.S., Akopyan V.F. Optimization of rectangular and box sections in oblique bending and eccentric compression. Construction Materials and Products. 2023. 6 (5). 2. https://doi.org/10.58224/2618-7183-2023-6-5-2