Cap IX, “Dynamic Stability of Beams Using a Higher Order Theory”

SEBASTIÁN P. MACHADO; CORTÍNEZ VÍCTOR HUGO

Structural Materials and Engineering

Nova Science Publishers, Inc.

Lugar: New York, Estados Unidos de America.; Año: 2009; p. 295 - 327

The dynamic stability of thin walled beams subjected to different sets of boundary conditions is investigated in this presentation. The analysis is based on a seven-degree-of-freedom shear deformable beam theory. The theory is formulated in the context of large displacements and rotations, through the adoption of a displacement field considering moderate bending rotations and large twist. This geometrically non linear formulation is used for analyzing regions of dynamic instability of simply supported, cantilever and fixed-end beams subjected to axial and transverse periodic loads. The influence of shear deformation and inertial effects (corresponding to loading plane) on the unstable regions is analyzed, for mono- and bi-symmetric cross-section beams. This last influence is generally neglected in most of the dynamic instability studies. Such an assumption is valid to a certain extent when the exciting frequency is small in comparison with the free frequency of the loading plane. This is the case frequently assumed for analyzing the dynamic stability of bars subjected to axial excitation. However for transverse excitation, the frequency at which a parametric resonance occurs can be the same order as the natural frequency of the loading plane vibrations. Ritz variational method is used to reduce the governing equation, the independent displacements vector is expressed as a linear combination of given x-function vectors and unknown t-function coefficients. Regions of dynamic instability of simple and combination resonances are determined by applying Hsu’s procedures to the Mathieu equation. The static load parameter and the dynamic amplitude of the excitation load are scaled with the buckling load. This critical value is obtained taking into account the effect of prebuckling deflections. The excitation frequency is also scaled with the lowest frequency value of parametric resonance.