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.