Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3511310.22363/2658-4670-2023-31-2-174-188Research ArticleBuckling in inelastic regime of a uniform console with symmetrical cross section: computer modeling using Maple 18ChistyakovViktor V.<p>Candidate of Sciences in Physics and Mathematics, Senior Researcher of Laboratory of Physics of Rare Earth Semiconductors</p>v.chistyakov@mail.ioffe.ruhttps://orcid.org/0000-0003-4574-0857SolovievSergey M.<p>Candidate of Sciences in Physics and Mathematics, Leading Researcher (Head of Laboratory) of Laboratory of Physics of Rare Earth Semiconductors</p>serge.soloviev@mail.ioffe.ruhttps://orcid.org/0000-0002-9019-7382Physical-Technical Institute named after A.F. Ioffe of RAS3006202331217418829062023Copyright © 2023, Chistyakov V.V., Soloviev S.M.2023<p style="text-align: justify;">The method of numerical integration of Euler problem of buckling of a homogeneous console with symmetrical cross section in regime of plastic deformation using Maple18 is presented. The ordinary differential equation for a transversal coordinate <span class="math inline">\(y\)</span> was deduced which takes into consideration higher geometrical momenta of cross section area. As an argument in the equation a dimensionless console slope <span class="math inline">\(p=tg \theta\)</span> is used which is linked in mutually unique manner with all other linear displacements. Real strain-stress diagram of metals (steel, titan) and PTFE polymers were modelled via the Maple nonlinear regression with cubic polynomial to provide a conditional yield point (<span class="math inline">\(t\)</span>,<span class="math inline">\(\sigma_f\)</span>). The console parameters (free length <span class="math inline">\(l_0\)</span>, <span class="math inline">\(m\)</span>, cross section area <span class="math inline">\(S\)</span> and minimal gyration moment <span class="math inline">\(J_x\)</span>) were chosen so that a critical buckling forces <span class="math inline">\(F_\text{cr}\)</span> corresponded to the stresses <span class="math inline">\(\sigma\)</span> close to the yield strength <span class="math inline">\(\sigma_f\)</span>. To find the key dependence of the final slope <span class="math inline">\(p_f\)</span> vs load <span class="math inline">\(F\)</span> needed for the shape determination the equality for restored console length was applied. The dependences <span class="math inline">\(p_f(F)\)</span> and shapes <span class="math inline">\(y(z)\)</span>, <span class="math inline">\(z\)</span> being a longitudinal coordinate, were determined within these three approaches: plastic regime with cubic strain-stress diagram, tangent modulus <span class="math inline">\(E_\text{tang}\)</span> approximations and Hook’s law. It was found that critical buckling load <span class="math inline">\(F_\text{cr}\)</span> in plastic range nearly two times less of that for an ideal Hook’s law. A quasi-identity of calculated console shapes was found for the same final slope <span class="math inline">\(p_f\)</span> within the three approaches especially for the metals.</p>Euler problemplane cross-sections hypothesisbucklingconsoleplastic deformationstrain-stress diagramconditional yield pointcritical buckling loadMaple programmingnonlinear estimationAl/PTFEsteelпроблема Эйлерагипотеза плоских сеченийвыгибаниеконсольпластические деформациидиаграмма сжатияусловный предел текучестикритическая выгибающая силапрограммирование на Mapleнелинейная оценкатефлон Al/PTFEсталь[T. H. G. Megson, “Columns,” in Aircraft Structures for Engineering Students, 6th. Elsevier Ltd., 2022, pp. 253-324.][F. R. Shanley, “Inealstic Column Theory,” Journal of Aeronautical Sciences, vol. 14, no. 5, pp. 261-280, 1947.][A. Afroz and T. Fukui, “Numerical Analysis II: Branch Switching,” in Bifurcation and Buckling Structures, 1st. CRC Press, 2021, p. 12.][N. Shuang, J. R. Kim, and F. F. 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