Transverse free vibration of Euler-Bernoulli beam with pre-axial pressure resting on a variable Pasternak elastic foundation under arbitrary boundary conditions

Abstract

With consideration of pre-axial pressure and two-parameter elastic foundation (Pasternak), a new method was put forward for analysis of transverse free vibration of a finite-length Euler-Bernoulli beam resting on a variable Pasternak elastic foundation. Matrices and determinants corresponding to arbitrary boundary conditions were provided for engineers and researchers according to their own demand. In derivation process of new proposed method, Schwarz distribution was adopted for simplifying the partial derivative of Dirac function and compound trapezoidal integral formula was adopted for discretizing the shape function of beam. For the symmetrical boundary conditions, it was concluded that natural frequency of transverse free vibration obtained by FEM highly agreed with the new proposed method. In contrary, for the asymmetrical cases, the calculation results were different from each other. For solving the ordinary differential equation with nonlinear partial derivative terms of shape function, the key point of new proposed method was to establish stiffness equation set composed of obtained matrices, rather than a single equation on the basis of classical theory. This point should be treated as a great advantage. New proposed method can be generalized to solve more complicated problems, which were illustrated in conclusion and prospect.

Key words:
Euler-Bernoulli beam; Variation of parameters; Natural frequency; Compound trapezoidal integral formula; Schwarz distribution theory; Pasternak elastic foundation; stiffness equation

Graphical Abstract