Local and Global Instability of the Slender Geometrically Nonlinear SystemWith Non-Prismatic Element Subjected to the Euler’s Load

A theoretical considerations and numerical calculations concerning the issue of the stability of the geometrically nonlinear system with non-prismatic element are presented in this work. The analysed columns were subjected to the Euler’s load. On the basis of the minimum potential energy principle as well as the small parameter method, the differential equations of displacements were formulated and its solutions were obtained. The assumption that the approximation of the non-prismatic rod satisfies the condition of constant total volume resulting from the value of the coefficient of flexural stiffness distribution has been made. The results of the carried out numerical simulations refer to the local and global stability loss. It has been proved that taking into consideration in the geometrically nonlinear system appropriate shaped rod of variable cross-section causes an increase in the value of bifurcation load and in a consequence an „exit” from the area of the local instability (loss of rectilinear form of static equilibrium).