Stability Of Variable Flexural Stiffness Euler Columns Placed On Elastic Foundation

Axial compression Euler column is widely applied in the engineering areas.Most of the engineering accidents caused by its buckling instability often bring serious economic losses and casualties,thus,the research on the stability of the axial compression Euler column is of great practical significance.With the development of material science,the inhomogeneous Euler column has attracted more and more interests than the traditional Euler column owing to its superior mechanical properties.However,the bending stiffness of the inhomogeneous Euler column is not necessarily constant,which makes it difficult to solve its critical buckling load.Meanwhile,the working environment of the Euler columns may be very complicated in practice,expecially,the transverse direction of columns often subjected to the elastic constraints such as the rails on the subgrade.All these problems make it complicate to study the stability of the axial compression Euler column.In the case of the constant volume(weight)and the constant length,it is an important research problem for designing the cross-section of the Euler column so that it not only can fully exert the mechanical properties of the material but also maximize the stability of the Euler columns.In this paper,we first deduces the differential equations of the Euler column placed on an elastic foundation with stiffness coefficient varying along the axial direction,considering its bending stiffness also varying along the axial direction.Then,substituting the three kinds of boundary conditions(clamped-clamped,clamped-pinned,clamped-guided)into the differential equations,we obtain the characteristic equation containing a dimensionless critical buckling load.An approximate method is used in this paper,to solve the dimensionless critical buckling load of the Euler column with considering the three boundary conditions mentioned above.The numerical results are in agreement with the previous results,which proving the applicability of the approximate method to this problem.Then,some cases of the dimensionless critical buckling load for the Euler columns subjected to the axial compression are given.And the effect of the gradient coefficient variation on the dimensionless critical buckling load is discussed for the case of bending stiffness and the stiffness coefficient of elastic foundation both varying in different forms.Finally,we discuss the optimization design of the Euler column with the homogeneous circular cross-section and under the C-G boundary conditions.The optimized design scheme which can effectively improve the anti-destabilization ability of the Euler column is presented for the cases of no elastic foundation and several constant elastic foundations.