A new static–dynamic equivalence beam bending approach for the stability of a vibrating beam

A new static–dynamic equivalence approach for beam bending is developed to determine the critical stability load of a vibrating beam resting on an elastic foundation. It is validated that the critical stability load corresponding to an arbitrary natural frequency parameter can be obtained via both the vibration and the static analysis approaches. By introducing a new elastic foundation stiffness parameter, the natural frequency parameter can be substituted, thus allowing a new formulation to solve the critical stability that corresponds to the zero or negative elastic foundation constant. In the vibration analysis, the governing equation of motion is solved by a harmonic response assumption. In this case, the critical stability loads can be determined by solving the determinant of the dynamic system under different boundary conditions. For zero or negative elastic foundation stiffness, a beam acted by a specific point load on such an elastic foundation can yield an infinite deflection. Under such circumstances, each real natural frequency has an equivalent relationship with the corresponding negative elastic stiffness. Consequently, using this equivalent system, directly solving of the transcendental equation in the dynamic frequency analysis can be avoided. Some numerical examples are presented and it is demonstrated that highly accurate numerical critical stability solutions can be derived by this equivalent static bending approach. The results are validated by comparing with the classical beam buckling solutions. In conclusion, a new theoretical model with analytical solution procedure is put forward and it yields highly convergent numerical solutions that compare well with classical analytical solutions.