Beam p-version finite element for geometrically non-linear vibrations in space

A beam p-version, hierarchical finite element for geometrically non-linear vibrations in space is presented. Stresses are obtained by adding the effects of traction, torsion and bending. A simplified version of Green’s strain tensor and the generalised Hooke’s law are used. The mass and stiffness matrices are obtained by the principle of the virtual work, and the ensuing non-linear equations of motion are solved by Newmark’s method. Some examples of linear spacial motion and non-linear planar motion are tested, in order to verify the element derived, and it is demonstrated that this element requires a small number of degrees of freedom for accuracy. Non-linear vibrations in space are then studied.