Exact Second-Order Stiffness Matrix of Axial-Loaded Beam–Columns on Winkler Foundation

This paper establishes the exact second-order stiffness matrix of axial-loaded beam–columns on Winkler foundation. This Winkler foundation model has been widely used in engineering applications. The equilibrium analysis of an axial-loaded beam–column on Winkler foundation is conducted, and the element flexural deformations and forces are solved exactly from the governing differential equation. Two dimensionless factors for the axial force and foundation spring stiffness are defined, respectively. The exact second-order element stiffness matrix of the axial-loaded beam–column on Winkler foundation is then formulated, showing the relationship between the element-end deformations (translation and rotation angle) and the corresponding forces (shear force and bending moment). Such an exact second-order stiffness matrix may be used for the buckling and second-order solutions with allowance for the use of one element per member for the exact solution. The classical buckling problem of axial-loaded beam–columns on Winkler foundation with typical element-end boundary conditions is then analyzed by the developed matrix stiffness method. By comparing with classical buckling criterion expressions of axial-loaded beam–columns on Winkler foundation with various boundary conditions, it is demonstrated that the derived exact second-order stiffness matrix can be used to obtain the exact buckling solutions systematically.