Buckling in inelastic regime of a uniform console with symmetrical cross section: computer modeling using Maple 18

The method of numerical integration of Euler problem of buckling of a homogeneous console with symmetrical cross section in regime of plastic deformation using Maple 18 is presented. The ordinary differential equation for a transversal coordinate (y) was deduced which takes into consideration higher geometrical momenta of cross section area. As an argument in the equation a dimensionless console slope (p=tg theta) 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 ((t),(sigma_f)). The console parameters (free length (l_0), (m), cross section area (S) and minimal gyration moment (J_x)) were chosen so that a critical buckling forces (F_text{cr}) corresponded to the stresses (sigma) close to the yield strength (sigma_f). To find the key dependence of the final slope (p_f) vs load (F) needed for the shape determination the equality for restored console length was applied. The dependences (p_f(F)) and shapes (y(z)), (z) being a longitudinal coordinate, were determined within these three approaches: plastic regime with cubic strain-stress diagram, tangent modulus (E_text{tang}) approximations and Hook’s law. It was found that critical buckling load (F_text{cr}) 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 (p_f) within the three approaches especially for the metals.